Charged Domain Walls in Ferroelectric Single Crystals

CHARGED DOMAIN WALLS IN FERROELECTRIC SINGLE CRYSTALS

Dissertation zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.)

vorgelegt von Thomas Kämpfe geboren am 12.01.1988 in Dresden

Institut für Angewandte Physik Fakultät Mathematik und Naturwissenschaften Technische Universität Dresden

Dresden, November 2016 Eingereicht am 07.11.2016

1. Gutachter: Prof. Dr. Lukas M. Eng 2. Gutachter: Prof. Dr. Marty Gregg Rastlos vorwärts mußt Du streben, nie ermüdet stille stehn, willst Du die Vollendung sehn; mußt ins Breite Dich entfalten, soll sich Deine Welt gestalten; in die Tiefe mußt Du steigen, soll sich Dir das Wesen zeigen. Nur Beharrung führt zum Ziel, nur die Fülle führt zur Klarheit, und im Abgrund wohnt die Wahrheit.

F. Schiller

ABSTRACT

Charged domain walls (CDWs) in proper ferroelectrics are a novel route towards the creation of advancing functional electronics. At CDWs the spontaneous polarization obeying the ferroelectric order alters abruptly within inter-atomic distances. Upon screening, the resulting charge accumulation may result in the manifestation of novel fascinating electrical properties. Here, we will focus on electrical conduction. A major advantage of these ferroelectric DWs is the ability to control its motion upon electrical fields. Hence, electrical conduction can be manipulated, which can enrich the possibilities of current electronic devices e.g. in the field of reconfigurability, fast random access memories or any kind of adaptive electronic circuitry. In this dissertation thesis, I want to shed more light onto this new type of interfacial electronic conduction on inclined DWs mainly in lithium niobate/LiNbO3 (LNO). The expectation was: the stronger the DW inclination towards the polar axis of the ferroelectric order and, hence, the larger the bound polarization charge, the larger the conductivity to be displayed. The DW conductance and the correlation with polarization charge was investigated with a multitude of experimental methods as scanning probe microscopy, linear and nonlinear optical microscopy as well as electron microscopy. We were able to observe a clear correlation of the local DW inclination angle with the DW conductivity by comparing the three-dimensional DW data and the local DW conductance. We investigated the conduction mechanisms on CDWs by temperature-dependent two- terminal current-voltage sweeps and were able to deduce the transport to be given by small electron polaron hopping, which are formed after injection into the CDWs. The thermal activated transport is in very good agreement with time-resolved polaron luminescence spectroscopy. The applicability of this effect for non-volatile memories was investigated in metal-ferroelectric-metal stacks with CMOS compatible single-crystalline films. These films showed unprecedented endurance, retention, precise set voltage, and small leakage currents as expected for single crystalline material. The conductance was tuned and switched according to DW switching time and voltage. The formation of CDWs has proven to be extremely stable over at least two months. The conductivity was further investigated via microwave impedance microscopy, which revealed a DW conductivity of about 100 to 1000 S/m at microwave frequencies of about 1 GHz.

v ZUSAMMENFASSUNG

Geladene Domänenwände (DW) in reinen Ferroelektrika stellen eine neue Möglichkeit zur Erzeugung zukünftiger, funktionalisierter Elektroniken dar. An geladenen DW ändert sich die Polarisation sehr abrupt - innerhalb nur weniger Atomabstände. Sofern die dadurch hervorgerufene Ladungsträgeranreicherung elektrisch abgeschirmt werden kann, könnte dies zu faszinierenden elektrischen Eigenschaften führen. Wir möchten uns hierbei jedoch auf die elektrische Leitfähigkeit beschränken. Ein großer Vorteil für die Anwendung leit- fähiger DW ist deren kontrollierte Bewegung unter Einwirkung elektrischer Felder. Dies ermöglicht die Manipulation das Ladungstransports, welches zum Beispiel im Bereich der Rekonfigurierbarkeit, schneller Speicherbauelemente und jeder Art von adaptiven elektro- nischen Schaltungen Anwendung finden kann. In dieser Dissertationsschrift möchte ich diesen neuen Typus grenzflächiger elektroni- schen Ladungstransports an geladenen DW hauptsächlich am Beispiel von Lithiumniobat/- LiNbO3 (LNO) untersuchen. Die Annahme lautete hierbei: umso stärker die DW zur ferroelektrischen Achse geneigt ist, also desto stärker die gebundene Polarisationsladung und folglich die elektrische DW-Leitfähigkeit. Die elektrische DW-Leitfähigkeit und die Korrelation mit der Polarisationsladung wurde mit verschiedenen experimenteller Methoden wie Rasterkraftmikroskopie, linearer und nichtlinearer optischer Mikroskopie als auch Elektronenmikroskopie untersucht. Es konnte eine klare Korrelation durch Vergleich der dreidimensionalen DW-Aufzeichnungsdaten mit der lokalen Leitfähigkeit gezeigt werden. Wir haben weiterhin den Leitfähigkeitsmechanismus an geladenen DW mittels tempera- turabhängiger Strom-Spannungskennlinien untersucht und konnten hierbei einen Hopping- Transport kleiner Elektronenpolaronen nachweisen, welche nach Elektroneninjektion in die geladene DW generiert werden. Der thermisch aktivierte Ladungsträgertransport ist in guter Übereinstimmung mit zeitaufgelöster Polaron-Lumineszenzspektroskopie. Die Anwendbarkeit dieses Effektes für nicht-volatile Speicherbauelemente wurde an Metall- Ferroelektrika-Metall Schichtstrukturen mit CMOS-kompatiblen einkristalliner Filmen untersucht. Die Filme zeigen bisher nichtgesehene Durchhalte- und Speichervermögen, ge- nau definierte Schaltspannung sowie sehr geringe Leckageströme wie dies für einkristalli- ne Materialsysteme erwartet wird. Die Leitfähigkeit konnte mittels entsprechender Wahl der elektrischen Schaltzeiten und -spannungen zielgerichtet manipuliert und geschalten werden. Es konnte darüber hinaus gezeigt werden, dass die hergestellten geladenen DW über eine Zeitspanne von mindestens zwei Monaten stabil sind und hierbei leitfähig blei- ben. Die Leitfähigkeit der DW wurde weiterhin mittels Mikrowellenimpedanzmikroskopie untersucht. Dabei konnten DW-Leitfähigkeiten von 100 bis 1000 S/m für Mikrowellenfre- quenzen von etwa 1 GHz ermittelt werden.

vi CONTENTS

1 introduction 1

I theoretical basics 5 2 fundamentals 7 2.1 Ferroelectricity ...... 7 2.1.1 Spontaneous polarization ...... 8 2.1.2 Domains and domain walls ...... 9 2.1.3 Charged domain walls ...... 13 2.1.4 Conductive domain walls ...... 16 2.2 Visualization of ferroelectric domains and domain walls ...... 21 2.2.1 Light microscopy ...... 22 2.2.2 Second-harmonic generation microscopy ...... 22 2.2.3 Cherenkov second-harmonic generation microscopy ...... 25 2.2.4 Optical coherence tomography ...... 28 2.2.5 Piezo-response force microscopy ...... 30 2.2.6 Ferroelectric lithography ...... 31 2.2.7 Further methods ...... 34 2.3 Lithium niobate and tantalate ...... 37 2.3.1 General Properties ...... 37 2.3.2 Stoichiometry ...... 38 2.3.3 Optical properties ...... 40 2.3.4 Intrinsic and extrinsic defects ...... 43 2.3.5 Polarons ...... 47 2.3.6 Ionic conductivity ...... 51 3 methods 53 3.1 Sample Preparation ...... 53 3.1.1 Poling stage ...... 53 3.1.2 Thermal treatment ...... 56 3.1.3 Ion slicing of LNO crystals ...... 57 3.2 Atomic force microscopy ...... 59 3.2.1 Non-contact and contact mode AFM microscopy ...... 59 3.2.2 Piezo-response force microscopy (PFM) ...... 60 3.2.3 Conductive atomic force microscopy (cAFM) ...... 62 3.2.4 Scanning microwave impedance microscopy (sMIM) ...... 63 3.2.5 AFM probes ...... 66 3.3 Laser scanning microscope ...... 67 3.4 Time-resolved luminescence spectroscopy ...... 71 3.5 Energy-resolved photoelectron emission spectromicroscopy ...... 72

vii viii contents

II experiments 75 4 results 77 4.1 Three-dimensional proﬁling of domain walls ...... 78 4.1.1 Randomly poled LNO and LTO domains ...... 78 4.1.2 Periodically Poled Lithium Niobate ...... 81 4.1.3 AFM-written Domains ...... 83 4.1.4 Thermally treated LNO ...... 84 4.1.5 Laser-written domains ...... 86 4.2 Polarization charge textures ...... 90 4.2.1 Random domains in Mg:LNO and Mg:LTO ...... 90 4.2.2 Thermally-treated LNO ...... 92 4.3 Quasi-phase matching SHG ...... 92 4.4 Photoelectron microspectroscopy ...... 97 4.5 Activated polaron transport ...... 101 4.6 High voltage treated LNO ...... 113 4.7 Conductive domain walls in exfoliated thin-ﬁlm LNO ...... 115 4.7.1 Conductance maps ...... 116 4.7.2 Resistive switching by conductive domain walls ...... 120 4.8 Microwave impedance microscopy ...... 134 4.8.1 Finite-element method simulation ...... 134 4.8.2 Scanning microwave impedance microscopy ...... 136 5 conclusion & outlook 143

III epilogue 147 a appendix 149 a.1 Laser ablation dynamics on LNO surfaces ...... 149 a.2 XPS across a conductive DW in LNO ...... 150 a.3 XRD of thin-film exfoliated LNO ...... 151 a.4 Domain writing in exfoliated thin-film LNO ...... 152 a.5 Retention in conductance at DWs in thin-film exfoliated LNO ...... 155 a.6 sMIM on DWs in thin-film exfoliated LNO ...... 157 a.7 Domain inversion evolution under a tip by phase-field modeling ...... 159 a.8 Current transients in exfoliated LNO ...... 161 a.9 Surface acoustic wave excitation damping at DWs ...... 162 a.10 Influence of UV illumination on domains in Mg:LNO ...... 162

acronyms 165 symbols 169 list of figures 172 list of tables 176 bibliography 177 publications 225 erklärung 233 1

INTRODUCTION

There is nothing more difﬁcult to take in hand, more perilous to conduct, or more uncertain in its success, than to take the lead in the introduction of a new order of things. — Niccolò Macchiavelli

Modern ubiquitous electronics is so far primarily deploying the physics occurring when defects are incorporated into an otherwise perfectly ordered crystal structure. This results in p- and n-type semiconductor areas, which are basically owing all their special properties from the charge transfer of these implanted defects. These defects are, hence, commonly known as acceptor and donor sites determining the major charge carrier to be either positive (p-type) or negative (n-type). The advances in material growth, nanofabrication and microscopy enabled the study of interfaces and boundaries, that often possess properties quite different from the residual bulk material. Hence, rapidly further ways to control charge accumulation using e.g. heterostructures were found. An important example is the LaAlO3/SrTiO3 (LAO/STO) interface [1]. This in the recent years heavily investigated interface has been shown to not only exhibit electrical conductance, but as well other emerging properties from superconductivity, ferromagnetism, anisotropic magnetoresistance through to giant photoconductivity [2–4]. The interface was reported to be metal-like electrically conductive under the proper conditions. The angular dependence of Shubnikov-de Haas oscillations indicated their conductivity to be two-dimensional, resulting in the reference of a two-dimensional electron gas (2DEG)[5]. However, two-dimensional does not necessarily mean that the conductive channel has zero thickness, but rather the electron motion is conﬁned in two directions. Hence, LAO/STO interfaces are a very famous example of lower-dimensional condensa- tion of states. The enormous interest in the existence of such lower-dimensional phases of matter resulted in this years physics Nobel prize to be awarded to D. J. Thoules, F. D. M. Haldane, and J. M. Kosterlitz for their "theoretical discoveries of topological phase transitions and topological phases of matter", explaining the origin of the quantum Hall effect and introducing the concept of topologically-invariant quantum numbers and the breaking of the time-reversal symmetry [6]. Topological invariants are strongly geometrically protected, which boosted great interest for ultrafast electronics close to the atomic level. Fur- ther examples of lower-dimensional states of electrons are the Dirac cone in graphene [7],

1 introduction

the presence of multiple valleys in the Brillouin zone in transition metal dichalcogenide monolayers, which gave rise to valleytronics [8, 9], the helical Dirac fermions in bismuth telluride/Bi2Te3 [10] as well as the existence of Weyl fermions [11] in the up-to-now only Weyl semimetal tantalum arsenide/TaAs [12]. All these topological and low-dimensional electronics are investigated with the hope to further control and generate new forms of electronics and spintronics. The emergence of the aforementioned lateral electronic properties on LAO/STO interfaces was under debate for the last decade and three models were given to explain the phenomenon. Very recently these properties were proven to be a result of the formation of a charged domain wall (CDW) due to local imprinted ferroelectricity, i.e. inverse ion displacement, at the interface of the intrinsically non-ferroelectric materials [13]. A com- 1 parable constellation was found in formerly antiferroelectric materials , which showed a 2 ferroelectric order at the anti-phase boundary (APB) [14]. So far, all these emerging electronic effects were bound to structural interfaces, either material interfaces or APBs in antiferroelectrics. Hence, these are not movable upon an external electric field. Even in improper ferroelectrics, the CDW is pinned to the crystallographic structure. But such flexible interfaces are possible and open up a wide scope of malleability. Thus, the investigation of CDWs in proper ferroelectrics extents the efforts to understand lower-dimensional electronic properties, which might give rise to a broad scope of new functionalities [15] as well as the aforementioned emerging phenomenons. At CDWs, the spontaneous polarization changes abruptly within inter-atomic distances. This is quite similar to the previously discussed heterostructures. The abrupt change in polarization charge results in the necessity of charge carrier compensation. This potentially very strong charge accumulation is expected to trigger new fascinating electrical properties. Yet, in massive contrast to all the previously described lower-dimensional or topologically-protected conductors, there is a major advantage of these ferroelectric DWs. These DWs can be moved or destroyed entirely upon an external electric field and, hence, it is possible to tune the electrical properties. This can greatly enrich the possibilities of current electronic devices e.g. in the field of reconfigurability, fast random access memories (RAM) or any kind of adaptive and reconfigurable electronic circuitry. Several ferroelectric materials have already been shown to possess conductive DWs as 16 17 18 19 e.g. BiFeO3 [ ], PbZr0.2Ti0.8O3 [ ], BaTiO3 [ ], and Pr(Sr0.1Ca0.9)2Mn2O7 [ ]. But also the piezo-electric, mechanical, and dielectric properties are expected to be altered by the presence of CDWs [20–22]. In this thesis, we are investigating lithium niobate/LiNbO3 (LNO) and LTO. These two very similar materials are unique as to exhibit not only a uniaxial proper ferroelectricity, but also the fabrication of single-crystals with a single ferroelectric domain was rendered possible. Moreover, LNO is CMOS compatible [23] and already produced in large quantities in the framework of a Czrochalski process especially for GHz acousto-optic filters [24], which makes the application into conventional production techniques very promising.

1 Antiferroelectric materials follow an alternating ion displacement order and, hence, develop no macroscopic polarization. 2 At APBs the order is inverted, which is very similar to the case at ferroelectric DWs.

2 introduction

Evolving from the ﬁrst report by Schröder et al. on photo-induced DW conductivity in single-crystalline LNO wafer pieces, I started to further elucidate the conductance and polarization charge textures in LNO DWs, with the idea in mind: provided it became possible to tune the bound and screening polarization charge in LNO, then, as a consequence the conductive properties, but also the suggested change in other physical parameters e.g. piezo-electrical response would be switchable and tunable. This would especially result in DW conductivity without the need to illuminate the crystal. In the next chapters, I will discuss the main physical properties of ferroelectrics, the existence of ferroelectric DWs, material properties of LNO, and methods to visualize ferroelectric DWs in LNO.

Part I

THEORETICALBASICS

FUNDAMENTALS

Science is beautiful when it makes simple explanations of phenomena or connections between different observations. — Stephen Hawking

In this chapter, we will present a speciﬁc material property, ferroelectricity. The theoretical framework will be brieﬂy introduced, which includes typical characteristics of ferroelectric materials. The formation of ferroelectric domains and their typical shape are discussed. Ferroelectric domain walls (DW) are introduced and divided into neutral (NDWs) and charged domain walls (CDWs). Afterwards, several methods to visualize DWs are presented, especially an optical microscopy technique called Cherenkov second-harmonic generation (CSHG) microscopy, which is the most important technique being used in this thesis to three-dimensionally visualize such DWs. The materials lithium niobate (LNO) and lithium tantalate (LTO) are introduced and their main material parameters are given.

2.1 ferroelectricity

Ferroelectricity was discovered in 1920 by Valasek, who observed hysteretic loops of the in- 26 duced dielectric polarization upon applied electric field for Rochelle salt/ C4H4KNaO6[ – 28], very similar to the hysteretic behavior of the magnetization upon an external magnetic field in ferromagnetism, known at the time [29]. For almost 15 years, ferroelectricity was recognized as a very specific property of Rochelle salt, until Busch and Scherrer discov- 1935 ered ferroelectricity in potassium dihydrogen phosphate/KH2PO4 (KDP) in . During World War II, the anomalous dielectric properties of barium titanate/BaTiO3 (BTO) were discovered in ceramic specimens independently by Wainer and Solomon in the USA, by Ogawa in Japan, and by Wul and Goldman in the Soviet Union [30–33]. Since then, many ferroelectrics have been discovered and the research activity has rapidly increased. The most common ferroelectric single crystals nowadays are LNO, BTO 34 68 69 71 [ – ], strontium barium niobate/SrxBa1 xNb2O6 (SBN) [ – ], and lithium tantalate/- − LiTaO3 (LTO). With modern evaporation techniques, typically pulsed laser deposition (PLD) [72], atomic layer deposition (ALD) [73], or RF magnetron sputtering [74], ferroelectric thin films have been fabricated. The most conventional ferroelectric thin films are 75 76 bismuth ferrite/BiFeO3 (BFO)[ , ] and lead zirconate titanate/ Pb(ZrxTi1 x)O3 (PZT) −

7 fundamentals

[17, 77–79]. Apart from these oxide materials trigylcine sulfate/(NH CH COOH) H O 2 2 · 2 (TGS)[80, 81], antimony sulphoiodide/SbSI [82, 83] as well as the polymer polyvinylidene 2 2 2 84 92 fluoride/-(C H F )n-(PVDF)[ – ] are of importance. We can characterize ferroelectricity in the terms of crystallography [see Fig. 2.1]. There are 32 known point groups of crystals, of which 11 show centrosymmetric and 21 non- centrosymmetric behavior. Among the non-centrosymmetric point groups 20 of these show piezoelectricity. Hereby, the crystal will generate a polarization ~P upon external strain σ. 10 of the latter point groups give rise to pyroelectricity, which is defined by a temperature-dependent remanent polarization. If this remanent polarization can be inverted upon the application of an external electric field ~E the crystal behaves ferroelectric. Ferroelectric materials retain their polarization ~P even under absence of an external field, which is why it is as well defined by the existence of a spontaneous polarization ~PS. In general, the polarization is induced by a lattice distortion, resulting in an ion displacement. Thus, separated charged centers are created, forming a unit cell electric dipole ~p = q ~l/V · with the separated charge q, the displacement~l and unit cell volume V.

32 point groups (PG)

11 centrosymmetric PG 21 non-centrosymmetric PG

1 non-piezoelectric PG 20 piezoelectric PG

10 pyroelectric PG 10 pyroelectric PG

non-ferroelectric ferroelectric

Figure 2.1: Crystallographic analysis of the point groups (PG) with different properties like cen- trosymmetricity, piezoelectricity, pyroelectricity and ferroelectricity, inspired by [93].

2.1.1 Spontaneous polarization

Ferroelectricity can be treated as a phase transition with an intrinsic loss in symmetry. The conventional theory to describe such processes was developed by Landau. Hereby, the crystal state is described by a set of parameters such as temperature T, entropy S, electrical ﬁeld ~E, polarization ~P, stress σ or strain ε in thermodynamic equilibrium. The free energy of the crystal system can be expressed as a power series of the polarization ~P [94]

1 1 1 F(~P, T, ~E) = ~E ~P + a~P2 + b~P4 + c~P6 + ... (2.1) − · 2 4 6

8 2.1 ferroelectricity

with a = γ(T T ), T the Curie temperature of the ferroelectric phase transition, ~P − C C S the spontaneous polarization and a, b, c, . . . the so-called ferroelectric coefficients. F is depicted in Fig. 2.2(a) for a uniaxial ferroelectric at different temperatures. For temperatures 1 below TC the free-energy being expressed up to forth order , results in two local minima, resulting in a spontaneous and remanent non-zero polarization. Above TC, one observes the minimal free-energy for a zero polarization, therefore TC resembles the transition temperature between the ferroelectric and paraelectric phases. A prototype material is BTO, which has six free energy minima at room temperature. The associated ferroelectric polarization states are either parallel or perpendicular to the crystallographic axis. Upon the application of an external electric field exceeding the coercive field ~EC, the polarization can flip. For the discussed idealized and especially uniaxial ferroelectric, the hysteresis loop of ~E vs. ~P is given in Fig. 2.2(b). In the case of b > 0, there is a second-order phase transition at T = TC. The temperature-dependence of the spontaneous polarization ~PS can be expressed in this case by q ~P = ~P0 a/b(T T) for T < T , (2.2) S C − C 2 2 ~ ~ which is illustrated in Fig. . (c). The electrical susceptibility χ = ∂P/∂E ~P0 shows a 2 2 | singularity at TC as can be seen in Fig. . (d). For real ferroelectrics, one often identifies an asymmetry of the hysteresis loop. The coercive field EC deviates for backward and forward poling, which is why one introduces an internal field ~Eint = ~EC, f + ~EC,b/2. The physical reason for this internal field can reach from internal charge carriers to incomplete poling processes [95]. In the case of LNO the real hysteresis loop is depicted in Fig. 2.2(e) as reproduced from [96]. One can observe a significant asymmetry in the material. It was further shown by Gopalan and Gupta [96], that the internal field might reduce upon annealing above 100 ◦C.

2.1.2 Domains and domain walls

So far, the polarization in a ferroelectric material was treated as entirely uniform, which is far from the real case. In a realistic ferroelectric system, domains – regions of common polarization orientation – are created. There are many reasons for the existence of domains, including non-uniform strain, microscopic defects, and the thermal and electrical history of the sample. But even in an ideal crystal, domains are to be expected for energetic reasons associated with electrostatics under absence of screening, which is further discussed in Sec. 2.1.3. Single-domain states are commonly unfavorable due to the necessary depolarization energy at the crystal surface which are caused by the electrostatic ﬁeld. The resulting electrostatic energy can be minimized by the formation of different domains. These domains are separated by the so-called DWs. The formation of DWs yet requires additional energy, hence, a certain domain size will correspond to the energy minimum. An arbitrary antiparallel or ﬂux-closure domain pattern is formed to minimize the free-energy of the system. These considerations hold for ferromagnetic systems as

1 One typically designates such potentials as Mexican hat potentials.

9 fundamentals

T > TC T TC a) F ⇠ b) P

P E

T < TC

P c c) d) P0

T T TC TC 2 Ps[µCcm ] e) 50

EC,b EC, f

1 -10-20 10 20 30 E[kVmm ] Eint

-50

Figure 2.2: Landau theory of uniaxial ferroelectrics; a) Free energy F as a function of polarization P for different temperatures T below, at or above TC, retrieving two definite local minima for T < TC for an uniaxial ferroelectric, b) schematic of the hysteresis loop upon application of an external electric field E and depicting the free-energy and polarization configurations for the different external electric field configurations for an ideal ferroelectric, c) Temperature dependence of the spontaneous polarization PS for b > 0, resulting in a second-order phase transition at T = TC, d) associated susceptibility χ = ∂P/∂E P0 , e) experimental hysteretic loop of a single crystal of LNO reprinted 96 | from [ ], an internal field Eint is observed, therefore the introduction of backward and forward coercive fields EC,b and EC, f , respectively, to switch the polarization is mandatory.

10 2.1 ferroelectricity

a c+ c+ c a c+ c+ a c+

Figure 2.3: Schematic of domains and domain walls (DWs) present in the triaxial ferroelectric BTO in a [001]-cut at room temperature. Three polarization axes are allowed. One can classify two different domain types concerning the projection to the surface normal: a domains with in-plane polarization and c domains with out-of-plane polarization. De- + pending on the polarization orientation at the surface c (outwards) and c− (inwards) domains are introduced. Adapted from [97].

well. In a model introduced by Kittel [98] the domain width w for a regular antiparallel ferromagnetic stripe domain pattern was shown to depend on the material thickness d in the way that w2/d equals a constant c. Hereby, he assumed periodic boundary conditions as well as the absence of screening at the surfaces. Later, this model was transferred to the case of ferroelectrics by Mitsui and Furuichi [99]. It yields for the ferroelectric domain stripe width w: q w = 0.54 √ε ε γd/ ~P (2.3) a b | S| with γ the free-energy domain energy density, d the crystal or film thickness. The constant c in Kittel’s model was later refined to be Gδ with G a geometry parameter and δ the DW thickness [100, 101]. This relationship is typically referred to as the classical Landau- Lifshitz-Kittel scaling law. Such analytical models give valuable insight into the stability of domains in various cases, especially for ferroelectric films. However, they depend on the assumption of a one-dimensional configuration, hence, assuming domains to be rectilinear with straight edges. Yet, this is far from the truth in three-dimensional and non-infinite crystals. Additionally, these models cannot describe kinetically limited systems or are dominated by finite aperiodic boundary conditions [38]. In Fig. 2.3 we depict a typical domain configuration in thermodynamic equilibrium for [001] BTO in tetragonal phase. 2 As BTO has six spontaneous ferroelectric polarization manifestations different wall types separating these can be expected. Due to strain reduction a typical a c chain can be − observed in these [001] crystals as shown, where a and c domains are coined due to there surface polarization component with a domains with primarily in-plane and c domains 3 with primarily out-of-plane component . Additionally, we introduce the notation of c+

2 BTO possesses within the tetragonal phase at room temperature three axes of spontaneous polarization, which are the result of its crystallographic point group m3m.¯ Therefore, three axes of tetragonal deformation exist, 6 resulting in local minima of F for T < TC. 3 We have to state "primarily" in the case of BTO as the unit cells of a and c domains are practically turned by 90°, which results in additional strain costs, being reduced with the formation of a corrugated topography with an angle of 0.6° at the DWs separating a and c DWs [102].

11 fundamentals

head-to-head (h2h) DW a) + - c- ~PS tail-to-tail a ~nDW ~n (t2t) DW DW surface 3 inclined domain 2 through- DW 1 domain - b) c ~nDW a

Figure 2.4: a) Types of DWs in a uniaxial crystal. Within these we can both observe through 1 and surface domains 3 as well as inclined DWs 2 with the local inclination angle α. Hereby, both h2h as well as t2t DWs are possible. b) Interpretation of DW inclination: local alternation of CDW and NDW segments, inspired by the observation of these staircases in PZT [103, 104]. Such staircase behavior can be assumed due to mechanical compatibility.

and c− domains with the polarization in out-of-plane orientation to be outwards and inwards, respectively. Vertically the a c domain configuration shows an angle of 45° − to the polar axis along z, mainly due to reduction of the electrostatic costs, as discussed further. In the case of an uniaxial crystal the domain pattern is much simpler, see Fig. 2.4. From the same thermodynamical expectations only so-called through-domains [marked as 1 ] are preferred. These are domains, which show a zero inclination angle to the polar axis. However, the main focus of this dissertation is the investigation of deviations from this simple picture. We will show experimental investigations where we observe both so- called inclined DWs [marked as 2 ] as well as surface domains [marked as 3 ]. Inclined DWs possess a DW normal vector ~n with ~n ~e to be non zero. As we will show in DW DW · z Sec. 4.1.1, the inclination angle α can be constant throughout the whole crystal and the complete DW, but as well, there are deviations from this simple image. Surface domains do not protrude through the whole crystal within the poling procedure after nucleation on one surface, therefore these DWs show very high local inclination angles, as indicated by a stronger red color in the depicted case 3 . One can differentiate the inclined DWs between h2h and t2t DWs. In the case of h2h DWs the polarizations present in adjacent domains are oppositely pointing towards each other (α > 0), in t2t DWs these point away from each other (α < 0). This has influences in a lot of DW-specific properties as discussed further, e.g. both bear either a positive or negative polarization charge at the DW, which lead to the generation/attraction of screening charges.

12 2.1 ferroelectricity

2.1.3 Charged domain walls

We now introduce screening, which is an energetically favored reaction of ferroelectric crystals to reduce the depolarization ﬁeld associated with the bound charge density σ, calculated by

σ = ~n ~P , (2.4) · S with the surface normal vector ~n. Bound surface charges q are present as depicted in Fig. + 2.5(a) depending on the surface. For c domains these are positive, for c− domains meanwhile negative, and for a domains these are zero. As unfavorable due to the depolarizing electric ﬁeld

inﬁnite charged plane ~E = q/ε0 ε ~E = σ/2e0 e ~n (2.5) ∇ · r −→ r with q the bound surface charge, σ the bound surface charge density, and ~n the surface normal. Bound surface charges are compensated by screening surface charges σs, yet it is possible the screening to be unable to dynamically adapt rapid change in the spontaneous polarization e.g. in the case of an alternation of the ambient temperature condition, as given in eq. (2.2). Therefore, we can divide different screening efﬁciency regimes: σs/σ = 0 not screened σs/σ < 1 partially screened σs/σ = 1 completely screened σs/σ > 1 overcompensated Both internal and external screening mechanisms are discussed in literature, e.g. in the case of LNO basically external screening by adsorbate molecules is expected [105–107], whereas for PZT and BTO primarily internal screening is under discussion, due to bound ionic charges, defect dipoles and free charge carriers [108–111]. The very same concept holds for DWs, which can bear a bound polarization charge. DWs are charged if there is a change in the adjacent spontaneous polarization vectors across the DW:

σ = ∆~P ~n. (2.6) DW S · 4 Various cases of charged 180° and 90° DWs are given in Figs. 2.5(c)–(f). Hereby, a in this schematic non-compensated polarization charge is bound at the DW, resulting in either h2h [Fig. 2.5(c)–(d)] or t2t [Fig. 2.5(e)–(f)] DWs. It is generally conceived, that for pure ferroelectrics the larger the DW polarization charge, the less stable the CDWs are due 5 to electrostatic energy costs . Hence, the formation of these is unexpected, meanwhile,

4 Hereby, the angle describes the relation of the polarization axes of the adjacent domains to each other at the DW. Thus, at a 180° DWs the polarization axis is similar, whereas for 90° DWs these are perpendicular to one another. 5 95 The electric depolarizing field E often exceeds the coercive field EC [ ]. This can be further supported by a thermodynamic argument. The Curie temperature TC of the ferroelectric order would be shifted by C/εr, 35 which is significantly larger than typical values for TC [ ].

13 fundamentals

a) ------+ + + + + + + + b) - - - - - + + + + + - + - + + + ~ ~ - - PS PS + + - ~PS - ~PS - + - + + + + + + + + + ------+ + + + + - - - - - c) d) ------+ + - + + - + + - - - - + + - - + + - + + - ~ ~ - ~P + + ~P - - PS + + PS - - S S - + + - - + + - h2h DW + + + + + + + + + + e) f) + + + + + + + + + + + - - + + + + - - + - - + + + - - + + - - + ~ - - ~ + ~P - - ~P + + PS - - PS + S S + - - + + - - + t2t DW ------

Figure 2.5: a),b) neutral 180° and 90° DWs, c),d) head-to-head (h2h) 180° and 90° DWs, e),f) tail-to- tail (t2t) 180° and 90° DWs. The uncompensated surface and DW charges are given by + and for positive and negative bound charges, respectively. Further very common − cases are 71° and 109° DWs in rhombohedral crystals. Similar charged and neutral DWs can be constructed.

very dependent on the ability of the material to deliver sufﬁcient internal screening to 6 compensate these. Moreover, proper ferroelectrics will in general behave as an improper ferroelastic, which in turn results in a further impediment to the formation of CDWs, as the system will tend towards reduced strain [95]. However, recently, several research groups have shown the existence of CDWs. The CDW in 2.5(c) resembles a typical case investigated in this study, which is further discussed in Sec. 4.1.4. 90° CDWs as given in the Figs. 2.5(d) and (f) were observed in [110] BTO under so-called frustrated poling conditions [18]. 180° t2t and h2h CDWs have been 7 observed in the improper multiferroics erbium manganite/ErMnO3 (EMO), holmium manganite/HoMnO3 (HMO), and yttrium manganite/YMnO3 (YMO)[113–119]. A further important case studied here are inclined DWs as already introduced in Sec. 2.1.2 and illustrated in Fig. 2.4. Hereby, microscopically a staircase can be assumed with alternat-

6 Proper ferroelectrics are materials, whose primary order parameter is the ferroelectric distortion. In improper ferroelectrics a structural phase transition causes the ferroelectricity. 7 Multiferroic materials possess at least two ferroic – remanent and spontaneous – phase transitions, that can be described by an order parameter e.g. ferroelectricity, ferromagnetism, ferroelasticity or ferrotorrodicity. Whether these are connected and therefore result in a steering e.g. magnetic order by electric ﬁelds is of general debate. A popular example is the demonstration in the case of strontium manganite/SrMnO3(SMO!) [112].

14 2.1 ferroelectricity

ing CDW and NDW segments. The average polarization charge at the DW σDW is then calculated by

σ = 2 ~P sin α (2.7) DW · | S| CDWs were proposed to be thicker than NDWs and for 180° DWs the thickness w is given by Yudin et al. and analytically calculated upon degenerate screening, i.e. very large screening charge carrier accumulation [120]

1/5 4 6 ! A degenerate screening 9π h¯ 2 8 wCDW→|← = wCDW→|← = ( . ) a/2 β ~P γ −→ q5m3 a/2 3 ~P | | | S| | | | S| with A a temperature-independent material parameter, 0 β 1/2, 0 γ 1 de- ≤ ≤ ≤ ≤ pending on the screening regime. Hereby, wCDW→|← = wCDW←|→, which can be altered upon an asymmetry of major charge carrier concentration. 8 Experimentally, the width of DWs in PZT was investigated by HRTEM resolving the local ion displacement across the DW. Sections showed a thickness of 0.6 nm for NDWs, ∼ yet h2h CDWs gave a thickness of 4 nm [103]. For the same Ti content of x = 0.8 in ∼ lead zirconate titanate/Pb(ZrxTi1 x)O3 (PZT) a DW thickness of 7 nm was calculated, − ∼ reasonably comparable to the given experimental result. Another property to be specific for CDWs is an increase in piezoelectric properties as calculated by Sluka et al. [20]. Hereby, the in-built depolarizing field in the case of an imperfectly compensated CDWs is expected to lead to an increase by a factor 2 of 9 the relevant integrated piezoelectric coefficient d33 for domain widths of 2 µm at room temperature. The formation energy of CDWs is much larger than for NDWs as already discussed. It was analytically derived in the non-linear screening regime by Gureev et al. [121]

σDWEg 2 W W , W→|← = W←|→ 10 W↑|↓ (2.9) CDW ≈ q NDW CDW CDW ∼ NDW

where WNDW↑|↓ = wNDW↑|↓ Wferro with q the elementary charge, Eg the bandgap, wNDW↑|↓ the NDW thickness and Wferro the ferroelectric order energy. Again there is no difference for 2 2 h h and t t DWs WCDW→|← = WCDW←|→, yet, this can be modiﬁed in materials with asymmetric major charge carrier concentration, e.g. upon moderate doping. The necessary charge carrier density to screen the DW is given by [122]

σCDW PS 2 10 ρCDW = , ρCDW→|← = ρCDW←|→ = . ( . ) 2qwCDW − qwCDW→|← CDWs in proper ferroelectrics require almost complete compensation of their polarization charge. Typical ferroelectrics are insulating materials, yet, there are several sources of

8 High-resolution transmission electron microscopy (HRTEM) was carried out by Jia et al. with a resolution of 80 pm, the ion displacement of Ti and Zr were given to be 40 pm and 20 pm, therefor the evaluation had ∼ ∼ to employ the super-resolution approach as will be introduced and discussed in Sec. 4.1.1. 9 Piezoelectricity is explained in detail in Sec. 2.2.5

15 fundamentals compensating charges. Typical ferroelectrics have wide, but ﬁnite, bandgaps (e.g. BTO: 3 eV, LNO: 4 eV). The thermally activated intrinsic free carrier concentration in ∼ ∼ such wide-bandgap materials is basically zero. Yet, charge carriers of any origin (photoexcitation, defect ionisation, injection) can occupy the available states in the conduction band (CB) and valence band (VB) if it is energetically favorable, e.g especially when they can compensate a bound charge. Additionally, movable charged defects like oxygen or lithium vacancies can be assumed.Hence, the total charge density ρ is a sum of different types of charge [122]:

2 11 ρ = ρpolarization + ρfree + ρmobile dopants + ρimmobile dopants + ... ( . ) where ρpolarization is the volume density of the polarization charge, ρfree the free charge density of electrons and holes, ρmobile dopants the charge density of mobile dopants and defects, and ρimobile dopants the charge density of immobile dopants and defects, etc. An analogical expression can be written for surface charge densities. The mobile charge carriers ρfree and ρmobile dopants are attracted and therefore collected by the polarization charge ρpolarization which locally reduces the total charge density ρ and, thus, the depolarizing electric ﬁeld.

2.1.4 Conductive domain walls

For the last 5 years, domain wall conductivity (DWC) in ferroic materials has witnessed a strong rise mostly triggered by the possibility of integrating charged domain walls (CDWs) into prospective nanoelectronic applications when used as ultra-confined contacts for electronic transport [123]. The early works on conductive CDWs in BFO [124, 125] were soon complemented by PZT [17] investigations as well. Today, also bulk insulating ferroic single crystals of mm thickness exhibit such CDWs as recently documented both experimentally [25, 113, 118] and theoretically [18, 126–128]. Such a metallic conductivity of ferroelectric CDWs due to a surface charge carrier density of ρ 1021 cm 3 at the DW was already CDW ∼ − predicted by Vul et al. [126] in 1972. Very important is the fact, that such a conductivity would be insensitive on the bulk conductivity, which could result in giant resistance variation in the material [121]. Especially interesting is the investigation whether the current is governed by one of the basic conduction types (CoTs) as given in the listing 2.1. In the further paragraphs, several materials and material classes will be discussed, having shown promising results concerning domain wall conductivity. ferroelastic dw tungsten oxide wo : 3 x The first proof for DW-specific conduc- 134 135− tance was delivered by Aird and Salje [ , ] for insulating tungsten oxides (WO3 x). − In these materials ferroelastic twin walls were observed to show superconductivity upon reduction of oxygen and insertion of sodium. The critical temperature for superconductivity was observed to be 3 K with a critical field of 15 T. Despite the possibility to ap- ∼ ply such confined superconductive areas for specific Josephson contacts [136] no further backup reports have been published for this property since the year 2000. Simulations

16 2.1 ferroelectricity

Listing 2.1: Standard conduction types (CoTs) and their linearizations to I V curves, which could − be present in domain wall conductivity (DWC), with I the current, V the voltage, T the temperature, and n a ﬁt parameter.

CT- 1 limitation due to interface effects 2 1 Fowler-Nordheim (F-H) tunneling [ln(I/V ) ∝ V− ][129] Schottky thermionic emission (STE) [ln (I/T2) ∝ V1/2] Richardson-Schottky-Simmons (RSS) emission [ln(I/V) ∝ V1/2][130, 131] CT- 2 limitation due to bulk conductance properties Poole-Frenkel (P-F) emission [ln (I/V) ∝ V1/2][130] Mott variable-range hopping [ln (I/V) ∝ T 1][132] − − CT- 3 limitation by electrode-injection and mobility of the charge carriers as in space- charge-limited current (SCLC)[I ∝ Vn][133].

propose critical temperatures of up to 120 K at such ferroelastic twin walls upon hydrogen insertion. 137 2010 DWC has been demonstrated in WO3 x by Kim et al. [ ] in reporting a twin − wall conductance of 75 nA at 2 V in comparison to about 2 nA in the bulk material. Yet, besides electrical conduction, ferroelastic DWs are expected to be speciﬁcally good ionic conductors, as e.g. in the case of calcium titanate (CaTiO3)[138], which is expected to be the reason for the aforementioned changes in the critical temperature in host material and twin wall after ion insertion.

bismuth ferrite (bfo) The most abundantly investigated ferroelectric material to date concerning domain wall conductivity (DWC) is bulk insulating and multiferroic bismuth ferrite/BiFeO3 (BFO). Since its inception by Seidel et al. [124] in 2009 showing metallic DWC at room temperature, many works have addressed the conductive properties of ferroelectric DWs in BFO [123, 139, 140]. A large advantage was the now easy experimental availability for this nanoscopical property by scanning probe microscopy 10 techniques such as conductive atomic force microscopy (cAFM) , which rose the interest of several other research groups. In bismuth ferrite/BiFeO3 (BFO) three equilibrium DW 11 types are possible : 71°, 109°, and 180°. Seidel et al. observed, that 71° DWs did not show DWC. However, at elevated temperatures also 71° DWs were shown to exhibit DWC, yet diode-like, e.g. at 155 ◦C a ratio of bulk/DWC of 10 was observed at 4 V for 40 to 70 nm thick films on SrRuO3 by Farokhipoor and Noheda [125]. The controlled writing of h2h and t2t DWs was shown by means of scanning probe microscopy [141]. Moreover, DWC was observed both in La-doped and Ca-doped films [16, 142]. Local written domains resulting in arbitrary DW angles confirmed h2h CDW to result in an increase in the lo-

10 This technique is introduced in Sec. 3.2.3 11 BFO shows 8 remanent polarization orientations with rhombohedral distortions of the lattice connected to an ion displacement at room temperature.

17 fundamentals

cal conductivity, whereas t2t DWs showed no increase compared to the insulating bulk. Vortexes, a prominent type of topological defects, gave increased conductivity [143]. Yang et al. [144] showed BFO DW to be photovoltaic and demonstrated the photovolt- age to be proportionally dependent on the number of contacted DWs. Stolichnov et al. [145] observed persistent conduction for La-doped BFO at lines previously being spots of conductive DWs, which supported speculation the DWC to be defect-mediated as well as polarization controlled by Schottky injection as in ferroelectric Schottky diodes [146, 147]. Moreover, it was shown that the bulk conductivity in Ca-doped BFO can as well by controlled by the polarization state, which is contradictory to the aforementioned DW-speciﬁc conductivity[148]. At the orthorombic-rhombohedral phase boundary enhanced conductivity was observed [149].

lead zirconate titanate (pzt) In 2011 DWC was first observed in lead zirconate titanate/Pb(ZrxTi1 x)O3 (PZT) films with a thickness of 70 nm on SrRuO3 fabricated by RF − magnetron sputtering [17]. These films show 180° DWs, therefore the polarization charge state of these DWs can typically not be investigated by piezo-response force microscope 12 (PFM) . However, Guyonnet et al. observed significant lateral shear strain at DWs upon the application of an electric field by in-plane PFM. The shearing stress of both adjacent domains was observed to be at least remanent over three months. This observation can be connected either with a change in the local in-plane piezo-electric coefficient at the DW or inclined DWs [78, 79]. Guyonnet et al. observed DWC and investigated it according to CoTs- 1 - 3 given in list 4.1. Reasonable linearizations of the I V sweeps were observed for all CoTs. Hence, − no discrimination between the different CoTs was possible. Mainly, this is a consequence of the small voltage window accessible from 0.5 to 1.625 V, as already at a voltage − − of 2.25 V significant poling processes were invoked. Guyonnet et al. claim SCSL to − be unlikely, due to the large parameter n necessary to be able to fit the experimental results and rather assume oxygen vacancies acting as traps to be the relevant mechanism to comprehend the conduction. A support for this hypothesis was found by Gaponenko et al. [77] in the observation of an oxygen vacancy density dependent DWC in these films. With control of the oxygen vacancy distribution inside the ferroelectric film via cycling between vacuum thermal annealing and ambient conditions a reversible control in the DWC was established. Stolichnov et al. [150] found the DW conductivity σDW to behave metallic-like with a still present current at cryogenic temperatures i.e. 4 K, with an almost linear temperature increase [σDW ∝ T], therefore scattering on acoustic phonons 5 e.g. as described by Bloch-Grünberg [σDW ∝ T ] appear to be negligible. The nonlinear I V sweeps are explained by Fowler-Nordheim-Tunneling, which is facilitated at the DW. − Another possible explanation of the experienced currents at cryogenic temperatures are temperature-independent hot carrier injection under trap assistance, which are a typical effect in low-temperature MOSFET structures at electrical fields in the range of MV/cm. Recently, the formation of CDWs was found to be connected with pinning to defects and crystallographic shear planes [151].

12 This technique is explained in detail in Sec. 2.2.5

18 2.1 ferroelectricity

lithium niobate (lno) Reported only one year after the discovery of DWC in PZT, photo-induced DWC was observed in wafers of z-cut Mg:LNO single crystals. Upon illumination with light of the wavelength λ of about 310 nm, corresponding to the absorption edge or band gap, DWC was observed [25, 152]. The photoexcitation-assisted transport was observed to give current when illuminated both from above or below the sample by the application of transparent electrodes. The Mg-doping concentration correlates positive with the observable current, yet at a concentration of above 5mol.% a strong increase was present. Schröder et al. were able to connect that behavior to an inclination of the DWs by deducing the domain areas with PFM on both crystal faces and assuming a constant inclination angle α within the crystal as discussed in Sec. 2.1.2 and can be seen in Tab. 2.1, which is in accordance with the theoretical forecast for inclined DWs by Eliseev et al. [128] in LNO. Shur et al. [153] reported an abnormal conductance in LNO after poling at temperatures of 120 to 250 ◦C, which was connected with CDW, yet without local resolution measurements. However, large area CDWs have already been reported in LNO [154]. As LNO is a uniaxial ferroelectric with 180° DWs, these are both mechanically compatible 13 and non-ferroelastic. At the phase boundary between proton exchanged and as-grown Mg:LNO phases photocurrents under UV illumination were observed in a wedge shaped crystal with varying thicknesses [155]. The DW character in LNO is under debate. Lee et al. calculated by DFT that DWs in LNO not only show standard Ising-like character, but similarly Bloch-Neel-Ising mixing, which can have an influence on stability of CDWs [156]. The frequency dependent alternating current (AC) upon a constant bias field (DC) was evaluated by nanoimpedance microscopy (NIM) and local AC conductivity was extractable at frequencies up to 5 kHz [157]. With macroscopic contacts, an assemble of DWs 14 was contacted and investigated by standard impedance spectroscopy. The Bode plot for a frequency range from 30 Hz to 30 kHz was successfully fitted by applying a constant 15 phase element .

barium titanate (bto) The first proof for DWC in BTO was delivered by Sluka 18 2 2 110 et al. [ ], who created areas of alternating h h and t t DWs in [ ]c BTO by means of frustrated poling of 200 µm thick crystals. Yet, besides these DW sections similar to Figs. 2.5(c) and (e) also energetically favorable zig-zag CDW structures are created, reducing the electrostatic costs [35, 161]. Therefore, these alternating structures are not expected to be a very stable domain configuration. Sluka et al. were able to show, that only h2h DWs show significantly increased conductivity when compared with the bulk. The bulk-h2h-ratio was shown to reach 108 to 1010 upon the application of a voltage as high as 150 V between the circular contact patches with a diameter of 200 µm assuming a DW thickness of 10 to 100 nm. The large conductance was demonstrated to be long-term stable over at least

13 Mechanical compatibility is given if the spontaneous strain in both domains is equal. 14 Within this plot one draws magnitude and phase of the impedance Z over the frequency f of the applied alternating voltage [158]. 15 A constant phase element is an equivalent electrical circuit, which is typically applied to model the behavior of an imperfect capacitor with leakage current [159]. For 50 nm thick LTO ﬁlms a thermally activated DWC was observed [160].

19 fundamentals

Table 2.1: Inclination angles α for Mg-doped LNO samples as measured with PFM for different Mg doping concentrations c, annealing conditions and thicknesses d. The photo-assisted current I was evaluated at a voltage V of 10 V and an UV intensity I =12.5 µW/mm2 at wavelength λ = 310 nm. Moreover, the thickness-normalized current I/d is given. Data reproduced from [25, 152]. Mg content Annealing Inclination Thickness Current Conductivity 1 1 c [mol%] @ 200 ◦C α [°] d [µm] IDW [pA] σ [Ω− cm− ] 0 0.006 300 <0.3 <1.6 10 6 × − 1 0.006 500 <0.2 <2.6 10 6 × − 5 0.016 300 <0.2 <2.6 10 6 × − 2 0.024 500 0.2 5.2 10 6 × − 3 0.050 500 0.4 1.6 10 5 × − 5 0.074 300 2.1 1.6 10 4 × − 5 0.225 300 13.6 1.1 10 3 × −

105 s. t2t DWs show similar conductivity as the bulk BTO material. The h2h DWC showed metallic-like thermal dependence, i.e. showed a dependence of I ∝ (1 + a(T T0 )) 1 − − with a the temperature coefficient and T0 the reference temperature. Yet, this dependence was confirmed only over a very small temperature range as 10 to 50 ◦C. The CDW at h2h was confirmed to drop to bulk values crossing the phase boundaries to orthorhombic and paraelectric phases at 6 ◦C and 102 ◦C, respectively. The asymmetry in conductance of both CDW was not sufficiently explained up to now, Sluka et al. speculate about the local suppression of free-carrier concentration resulting in the formation of an quasi-2D hole gas by oxygen vacancy accumulation. Interestingly, the aforementioned different conductance properties are only to be stabilized after a time of 200 s, which is connected ∼ with the formation of a wedge domain at one side of the crystal prohibiting the electrical conductance. improper ferroelectrics: manganites An exotic family of DWs was found in the improper ferroelectrics as erbium manganite/ErMnO3 (EMO), yttrium manganite/ YMnO3 (YMO), and holmium manganite/HoMnO3 (HMO). These materials structurally lock ferroelectric DWs with a wide range of orientations including stable charged configurations and vortex-antivortex network patterns [41, 116, 117, 162]. t2t DWs in EMO were observed to result in a conductivity increase by a factor of about 2 as compared to the semiconducting bulk whereas h2h DWs resulted in a conductivity decrease by a factor of almost 2 [113]. The resulting I V curves can be described by interface-limited RSS. − However, unlike in ordinary ferroelectrics, ~PS is a secondary order parameter induced by a structural phase transition in these improper ferroelectrics, i.e. the mechanisms to drive the polarization are substantially different than in ordinary ferroelectrics [163]. Thus, properties of CDWs in proper and improper ferroelectrics cannot be linked easily, in general. E.g. manganites, unlike ordinary ferroelectrics, cause the unusual topological locking

20 2.2 visualization of ferroelectric domains and domain walls

of the polar and antiphase boundaries and do not require compensation of CDWs [164]. Due to the large internal conductivity, photoelectron emission microscopy (PEEM) has 16 first been demonstrated in EMO to resolve t2t DWs in these crystals as image resolution and interpretation typically suffers from charge-up, which is a general impediment in spatial and energy-resolved PEEM [114, 115]. Wu et al. [118] retrieved influences in the conduction in HMO at DWs, i.e. t2t and h2h DWs lead to an increase in conductivity of about 50 % and 5 % compared to the bulk. However, even NDWs resulted in an increase in conductivity of about 25. Another material of this class – terbium manganite/TbMnO3 (TMO) – was discussed in terms of DWC, however, no conclusive picture can be drawn about the conduction in different domains and at the DW for TMO thin films [165].

further materials Recently, new material classes have been brought into discussion for localized conduction such as calcium-strontium titanate, lead-strontium titanate, mangan-cobalt tungstanate, and bilayer manganites. The improper ferroelectric calcium- strontium titanate/(Ca,Sr)3Ti2O7 (CSTO) was reported to exhibit strong DWC at both switchable in-plane ferroelectric DWs and orthorhombic ferroelastic DWs i.e. twin walls [166]. In mangan-cobalt tungstanate Mn0.95Co0.05WO4 the formation of ferroelectric CDWs can be controlled by an external magnetic field [167], i.e. the ferroelectric polarization was observed to be rotatable by second-harmonic generation microscopy up to 90° upon application of a magnetic field of 6 T, resulting in a tuning range from NDWs to h2h CDWs. In lead-strontium titanate the controlled and creation of CDWs in a thin film was demonstrated [168]. The half hole-doped bilayer manganite Pr(Sr0.1Ca0.9)2Mn2O7 (PSCMO) displays local 3+ 4+ conductance for charge order (Mn /Mn ) DWs below the Verwey temperature TV of 370 K as present in the scanning microwave impedance microscope (sMIM) images con- ∼ 19 ducted for different temperatures below and above TV [ ]. DWs of all-in-all-out magnetic order in the pyrochlore neodymium iridate/Nd2Ir2O7 showed to be conductive below the Néel temperature of the metal-insulator transition[169]. This is explained by an extinction in magnetic-order mode-separation across the Ising-like DW in this specific material, as it is conductive without any magnetic order. At ferroelastic DWs of bismuth telluride/Bi2Te3 a free-electron charge was found [170].

2.2 visualization of ferroelectric domains and domain walls

For the investigation of ferroelectric materials it is valuable to be able to extract the ferroelectric domain conﬁguration. Tenths of methods applied to reveal the domain pattern in these ferroelectric materials have been proposed over the last decades [171]. We want to introduce here methods and techniques, which have been applied in this study or are of signiﬁcance to understand the current literature on ferroelectric domains and DWs.

16 The methodology of photoelectron emission microscopy (PEEM) is explained in detail in Sec. 3.5. PEEM investigations conducted on h2h DWs in insulating LNO are presented in Sec. 4.4.

21 fundamentals

2.2.1 Light microscopy

A large portion of ferroelectric materials are transparent, which generally appears to not facilitate any optical microscopy technique. Yet, to investigate these materials, typically ferroelectric single crystals, by means of optical microscopy or especially the materials investigated in this thesis – LNO and LTO – two main methods can be applied, that reveal the ferroelectric domain pattern: defocused light microscopy (DLM) cross-polarized light microscopy (PLM) In defocused light microscopy (DLM), one utilizes the fact, that in freshly poled LNO and 2 1 2 LTO internal fields Eint are present as already discussed in Sec. . . , which have influence 1 3 in the refractive index of the material via the electro-optical effect ∆n = /2n r33Eint with r33 the electro-optical coefficient, resulting in phase alterations for wavefronts passing through both virgin and switched domains. Hence, at the back crystal face the domain structure is imprinted in the local phase of the optical wavefront and can thereby be extracted through phase contrast microscopy. However, upon defocusing these phase patterns can be made visible in the image even in ordinary transmission light microscopy 172 [ ], which was generally applied in this study, as Eint does not significantly change over months at room temperature and therefore gives reliable domain visualization. Cross-polarized light microscopy (PLM) can be applied due to stress birefringence close to the DWs. Shortly after the poling process, rearrangements within the crystal lead to slowly relaxing mechanical stress. In order to make them visible one can utilize polarization microscopy with crossed polarizers, as LNO is a birefringent material. A typical example for cross-polarized light microscopy can be seen in Fig. 2.6(a). After severe changes in the DW pattern by alteration of the surface screening state, significantly higher resolution images can be obtained as depicted in Fig. 2.6(b). To conclude, light microscopy is a facile technique for domain pattern visualization, however, the general disadvantage for both discussed light microscopy techniques are their poor lateral resolution when compared with the further discussed techniques as well as their vertically integrated image, which cannot resolve changes throughout the crystal.

2.2.2 Second-harmonic generation microscopy

Second harmonic generation (SHG) microscopy is a sensitive tool to investigate material properties obeying a symmetry-breaking, including ferroelectricity, ferromagnetism, and antiferromagnetism [117, 173–175] as well as ordered structures in biological tissues [176, 177]. SHG is a second-order nonlinear optical effect to occur under illumination with intense light, typically pulsed laser beams. Therefore it is no surprise that the inception of nonlinear optics happened in 1961 by the discovery of SHG by Franken et al. [178], just one year after the invention of the laser by Maiman. Nonlinear processes give, in general, responses of the material to not depend linearly on the strength of excitation.

22 2.2 visualization of ferroelectric domains and domain walls

a) b)

100 µm 50 µm

Figure 2.6: Examples of PLM on ferroelectric domains in LNO; a) laser-written domains, which are further discussed in 3.1.1, b) a high-voltage treated hexagonal domain as discussed in 4.6

In conventional optics the induced optical polarization ~P(~r, t) depends linearly on the applied electrical ﬁeld ~E(~r, t). Often this can be given in the following way

~P(~r, t) = e0 χ(1)~E(~r, t) (2.12) with the susceptibility tensor χ(1). Upon high intensity illumination this needs to be generalized introducing higher-order terms to describe also anharmonic oscillations. The polarization in the medium with neglected retardation is then given as a power series of the following form [179]

2 3 ~P(~r, t) = e0 χ(1)~E(ω)(~r, t) + χ(2) ~E(ω)(~r, t) + χ(3) ~E(ω)(~r, t) + ...

= ~P(ω)(~r, t) + ~P(2ω)(~r, t) + ~P(3ω)(~r, t) + ... = ~P(ω)(~r, t) + ~PNL(~r, t) with the deﬁnition ~P(nω)(~r, t) = e0 χ(n)(Eω(~r, t))n. χ(n) are the nth-order susceptibility ten- sors, which describe the complete nonlinear optical response of the material. ~P(nω)(~r, t) are the nth-order polarizations as part of the nonlinear optical polarization ~PNL(~r, t). The nonlinear response to an incident monochromatic plane wave with frequency ω is described by

(ω) 1 (ω) iωt ~E (~r, t) = ~E (~r)e− + c.c (2.13) 2 with c.c. denoting the complex conjugated preceding term. This deﬁnition can be substituted in eq. (2.12) and reveals the complete polarization within the medium

1 h (1) (ω) iωt i ~P(~r, t) =e0 χ (~E (~r)e− + c.c.) + 2 (2) (ω) (ω) 1 (2) (ω) 2 2iωt e0 χ ~E (~r)(~E )∗(~r) + (χ (~E (~r)) e− + c.c.) + ... .(2.14) 2 The ﬁrst component represents the linear optical response of the material to the applied ﬁeld. The second term describes the second-order polarization, which is the quadratic

23 fundamentals

nonlinear response of the material. Terms appear that oscillate with different frequencies 0, 2ω, 3ω and so forth. This implies that radiation with double, triple and so forth frequency of the incident frequency should be visible in the coherently scattered field. Hence, under the presence of a non-zero second-order susceptibility a dipole can be excited oscillating with twice the frequency of the driving external field. This corresponds to a coherent emission of photons with the double frequency and energy, i.e. two photons are absorbed and one photon is emitted coherently with the energy of both. The second-order susceptibility tensor can be defined element wise in the following form [180]

(2) 2 15 Pi(2ω) = ∑ χijk Ej(ω)Ek(ω) ( . ) jk

The description in frequency domain is more convenient, as in the previously used description in time domain retardation effects are neglected. The electric ﬁelds in eq. (2.15) are permutable, which gives a ﬁrst symmetry equation for the second-order susceptibility

(2) (2) 2 16 χijk = χikj . ( . )

(2) (2)0 (2) Upon arbitrary symmetry operations T, χ transforms to χlmn = TliTmjTnkχikj and in (2)0 (2) case T to be a point group symmetry operation χlmn = χijk . By applying the inversion operation T = δ , we ﬁnd that the nth order susceptibility obeys ij − ij χ(n) = ( 1)(n+1)χ(n). (2.17) − Even-order susceptibility tensor elements vanish for inversion symmetry. Thus, the most efﬁcient second-harmonic generation sources are ferroelectrics as their crystal structure 2 15 is non-centrosymmetric manifested by ~PS. Eq. ( . ) can be rewritten more explicitly by 1 17 introducing another tensor dijk = 2 χijk and contracted notation

 (ω)2  Ex  2       (ω)  (2ω)  Ey  Px d11 d12 d13 d14 d15 d16  (ω)2  (2ω)  (2ω)     E  ~P =  P  = 2  d d d d d d   z  .(2.18)  y   21 22 23 24 25 26   (ω) (ω)  (2ω)  2Ey Ez  Pz d31 d32 d33 d34 d35 d36    (ω) (ω)   2Ex Ez  (ω) (ω) 2Ex Ey

The number of independent elements dlk is determined by the point group of the crystal. For different domains, e.g. in the case of BTO, differences in SHG emission of a and c domains are present. Materials of the point group 3m as LNO and LTO possess just three independent elements d22 = d21 = d16; d33; d31 = d32 = d24 = d15 and all other elements are zero. 17 Contracted notation (dlk := dijk) with 1 := 11, 2 := 2, 3 := 33, 4 := 23, 32, 5 := 31, 13, and 6 := 12, 21

24 2.2 visualization of ferroelectric domains and domain walls

The spatial evolution of E(2ω) in forward direction, when assuming negligible loss and slowly varying envelope approximation, is then obviously given by [181]

( ) ∂E 2ω 2 ∝ d E(ω) eiδkz, (2.19) ∂z eff with an effective tensor element d and phase mismatch δkz = (k 2k0 )z. In a disper- eff 2 − sive medium the refractive indices at fundamental and second-harmonic wavelength, n0 and n , respectively deviate and, thus, their wavevectors i.e. δ~k =~k0 2~k = 0 and funda- 2 − 2 6 mental and second-harmonic fronts of constant phases propagate with different velocities. If uncorrected, the phase mismatch of fundamental and second-harmonic wavefront will increase while traveling through the crystal. Solving eq. (2.19) under the assumption, that E(ω) E(2ω), we obtain Z z (2ω) (ω)2 iδkz (ω)2 iδkz/2 E (z) ∝ d E e 0 dz0 ∝ E z sinc (δkz/2π)e . (2.20) eff 0

Hence, the intensity alternates following I(2ω)(z) ∝ z2 sinc2 (δkz/2π). Up to the coherence length lc = π/δk the second harmonic will be built-up, followed by a decrease in intensity through destructive interference. When introducing an alternating phase pattern with a period length lc for the excited second-harmonic dipoles, e.g. by inversion of the crystals anharmonicity, one can achieve quasi-phase matching as ﬁrst suggested by Armstrong et al. [182] in 1962. Quasi-phase matching second-harmonic generation (qpmSHG) shows a quadratic increase over the travel [183], as in a complete phase-matching (δk = 0) gives I(2ω)(z) ∝ z2, however in maximum reduced by the factor 2/π. A periodic alteration in anharmonicity can be accomplished in ferroelectrics via periodic domain patterns as e.g. in periodically poled lithium niobate (PPLN)[184]. To conclude, if a laser is tightly focused into a ferroelectric single-domain crystal, only a weak SHG emission will be present, if no phase matching is present e.g. by birefringence.

2.2.3 Cherenkov second-harmonic generation microscopy

The obstacle of inhibited far-field SHG emission upon tightly focused excitation can be overcome upon alteration of χ(2) within the beam waist. Both surfaces and DWs show a step-like alteration of χ(2) over only a single to several unit cells, which introduces a broad spectrum of lattice vectors G~ . Their availability can be calculated by A Z ~ Ω ∞ 1 i ~ (2) ~ 2πik~r0 → 2 21 (G) = ~r0 [χ (~r0)](k) = e− χ(~r0)d~r0 = δ(kx) δ(ky)δ(kz),( . ) A F Ω 2 − πkx assuming the illumination waist Ω infinite in comparison to the step-edge width and the DW normal vector ~n ~e . However, taking into account that the illuminated area is DW k x finite, the distribution of lattice vectors G~ will be less strict especially along the y and z axes and (G~ ) should be derived from an integral over the local tight focus to be more A

25 fundamentals

accurately. Yet, with eq. (2.21) we can state, that a DW can always sufﬁciently fulﬁll the wavevector conservation:

! δ~k =~k0 2~k + G~ = 0. (2.22) − 2 DW 2 7 A schematic can be seen in Fig. . (a). Two distinct lattice vectors, denoted G~ DW, can fulﬁll the wavevector conservation [marked by 1 ]. This case is denoted Cherenkov second 18 harmonic generation (CSHG) or longitudinal phase matching . The beams are emitted under the Cherenkov angle θ derived from

cos θ = 2k0 /k2 = n0 /n2. (2.23)

In birefringent materials one has to consider the refractive indices of ordinary no and extraordinary ne axis, therefore in general there are two Cherenkov angles for type I: e + e e, o + o e and one for type-II: o + e o [186]. As all our measurements were → → → conducted with~k along ~c only type I CSHG o + o e has to be taken into account. 0 → In general, a series of lattice vectors can lead to non-collinear SHG [marked by 2 ], only (2.22) has to be fulfilled, which is depicted by a circle arc. Non-collinear SHG can only be present for finite illumination under e.g. tight focus condition, as (G~ ) can be assumed to A be low. For the discussed cases 1 and 2 , the DW normal vector ~nDW 1 , 2 was expected to be perpendicular to the propagation axis. If the normal vector is parallel, then phase matching is similarly always fulfilled [marked by 3 ]. However, as a much smaller lattice vector needs to be delivered and (G~ ) ∝ 1/k , the SHG emission efficiency is larger as in A x 1 . If the domain structure is periodic and the illuminated area covers a significant number of domains, so-called Raman-Nath second-harmonic emission [187] can be observed. (G~ )(m : integer) with the lattice vectors of the periodic structure can be extremely A m large for the lattice vectors of the periodic structure G~ m and determines the nonlinear scattering efficiency. It is therefore not required to have complete phase-matching as can be seen [marked by 4 ]. The residual lattice vector is expected to be delivered by e.g. surface, defects or inhomogeneities in the material. However, if the lattice vector delivers the matched lattice vector, we can observe nonlinear Bragg scattering, which is strongly enhanced [marked by 5 ]. Similarly, this can be achieved for illumination upon an angle by superimposing Cherenkov and Raman-Nath phase matching [188]. So far, we have discussed the extremal cases of DWs perpendicular or parallel to the wavefront propagation. If the DW is inclined, we observe an asymmetric angular splitting of two still CSHG beams [marked by 6 , 7 ]. One CSHG beam will be emitted under a smaller angle 6 , whereas the other is emitted under a larger angle 7 , which require a smaller and larger k , respectively with ~n ~e . Therefore, the intensity of both will x0 DW k x0 deviate. 18 The name Cherenkov second harmonic generation (CSHG) is inherited from Cherenkov radiation, known from particle physics, which is emitted upon ionization by a particle exceeding the speed of light in an ultra-thin gas medium [185]. The induced radiation is preferentially emitted under the so-called Cherenkov angle. In CSHG the fundamental wavefront travels faster than the induced second-harmonic wavefront, hence, significant far-field emission is similarly observed only under the Cherenkov angle.

26 2.2 visualization of ferroelectric domains and domain walls

a) b) c) 1 5 ~P ~k ~k ~ S 6 2 ~ 2 G2 4 GDW 2 inclined ~ GDW DW 1 , 2 G~ 1 DW a ~ ~ ~ 2k0 3 2k0 2k0 DW3

periodic ~k2 domains 7

~ d) ~ 2k0 k2

G~ DW c(2) c(2) c(2) c(2) (2) c(2) c

e) log I [a.u.] 0 1 90 3 collinear Cherenkov

a SHG SHG qc angle [°] inclination 6 7 0 90 0 90 internal emission angle [°] Figure 2.7: a) Schematic of the non-collinear SHG emission process promoted by a DW specific ~ lattice vector GDW, Cherenkov second harmonic generation (CSHG) 1 , general non- collinear 2 , and collinear 3 SHG emission with ~k0 and ~k2 the fundamental and second-harmonic wavevectors, respectively for DWs parallel [DW 1 , 2 ] or perpendicu- ~ 0 lar [DW 3 ] to k , b) Schematic of Raman-Nath SHG condition 4 and nonlinear Bragg (2) condition 5 for a periodic χ pattern with its lattice vectors G~ m = mG~ 1,(m: integer) and entire pattern illumination, c) Schematic of CSHG condition for inclined DWs with inclination angle α resulting in small 6 and large 7 angle nonlinear scattering, d) Schematic for tightly focused pulsed laser beam CSHG excitation in LNO. e) Inclination angle dependence of the internal type I SHG emission efficiency in LNO with ordinary excitation (o + o e) compared to the collinear emission due to qpmSHG. CSHG is → dominant, whereas general non-collinear SHG is strongly prohibited. Forward external emission for 7 is only expected up to the critical angle of total internal reflection θc.

27 fundamentals

In Fig. 2.7(d) the real case of pulsed laser excitation at the DW is depicted with the red fundamental beam and the two blue second-harmonic beams. A calculation for the emission efficiency in LNO at a wavelength of λ = 990 nm under the assumption of a tight focus for different inclination angles α is given in Fig. 2.7(e) and plotted over the emission angle. We can observe, that the efficiency discrepancy of 6 and 7 increases with the inclination angle α. The Cherenkov beam 6 reaches the collinear case at an inclination angle 90°, which coincides with 3 . In forward direction the beam 7 will only be present for emission angles below the critical angle θC = arctan (n2/n1) of the upper crystal face. The given critical angle assumes a crystal to ambient air interface. Often the aforementioned effects are discussed for two-dimensional nonlinear photonic structures [189–193], waveguides [181, 194–204] or random-domain structures [205–207], as sources for high efficiency laser systems. The hereby given considerations concerning Cherenkov and Raman-Nath phase matching can similarly be extended to other nonlinear optical process such as third harmonic generation, sum frequency generation or difference frequency generation [203, 208–212]. CSHG microscopy finds apart from ferroelectrics application in biological systems [213] or semiconductors [214]. Fibers can be used as high efficiency source of CSHG [215] and CSHG can be used for femtosecond- pulse characterization [216]. With three-dimensional stacking as further discussed in 3.3 three-dimensional visualization of ferroelectric DWs can be realized [217–220].

2.2.4 Optical coherence tomography

Ferroelectric DWs are expected to both possess different refractive indices than the bulk material as well as to be a boundary in refractive index present in different domains [221, 222]. Domains are expected to have different refractive indices mainly due to the already 2 1 in . discussed internal field Eint. The refractive index is thereby mainly altered due to 3 the linear electrooptical effect ∆n = 1/2 n r33Eint. At the DW stress-mediated changes in the crystal structure and persistent electrostatic fields are expected to lead to changes in 4 3 the refractive index in the order of 10− to 10− . Both contributions will lead to a minute reflectance R at the DW, which can be described with the associated Fresnel equations. The reflectance for a multidomain sample is governed by a set of these equations,

2 2i 4 2 24 Ii = I0 T1 T − R (for i > 1, i : integer) ( . ) 4n 4n(n + ∆n) T = , T = (2.25) 1 (n + 1)2 (2n + ∆n)2 ∆n 2 n 12 R = , I = I0 − (2.26) 2n + ∆n 1 n + 1 with Ii the intensity of the reﬂectance on the ith DW, n the refractive index of the bulk ferroelectric, ∆n the change in refractive index at a DW, T the transmittance trough the DW, T1 the in- and outcoupling transmittance and I1 the incoupled intensity. The minute reﬂectance intensities Ii associated to ∆n can be recorded by use of optical coherence tomography (OCT)[223], a technique which is based on low-coherence interferometry.

28 2.2 visualization of ferroelectric domains and domain walls

Both time-domain (TD) and frequency domain (FD) realizations are known, however FD- OCT is nowadays preferred both due to a higher speed imaging and signal-to-noise ratio. In FD-OCT the scattered light from the sample interferes with light from the reference arm. Fourier-transforming the fringes in the acquired spectrum, it is possible to deliver spatially encoded maps of the reﬂective power along an axial line. Scanning the incident laser by galvo mirrors, a three-dimensional visualization can be realized. The ﬁrst demonstration of DW visualization applying OCT was reported by Pei et al. [224] in undoped PPLN. Similar examples for both undoped and 5mol% Mg-doped PPLN crystals with period lengths of 3.5 µm and 10 µm, respectively are given in Fig. 2.8.

~Einc 5 ~ kinc

500 µm 500 µm 50 µm OCT I 50 µm OCT I

Figure 2.8: FD OCT images recorded for PPLN samples with an incident s-polarized picosecond supercontinuum laser beam, which was incident under an angle of 5°. The crystal surface was polished at the narrow face being previously cut under an angle of 5° to the x-axis. Left: 5mol% Mg-doped PPLN (periode: 3.5 µm), Right: undoped PPLN (periode: 10 µm)

The samples were cut and polished under an angle of 5° to the crystal x-axis to reduce the strong back-reﬂection at the air-crystal surface with the laser beam incident on the narrow face. To satisfy a wavefront propagation parallel to the y-axis within the crystal the laser is illuminated under an angle of 5° in s-polarization condition. In the ∼ two-dimensional FD-OCT map, one can clearly identify the DWs due to the signiﬁcant increase in the scattered light. From the deduced scattering strength, it is possible to calculate the change in refractive index at the DW. Therefore, it is necessary to both get a high axial resolution image, but also to compensate for normal dispersion within the medium. High axial resolution is commonly achieved by spectrally broad laser systems,

29 fundamentals e.g. superluminescent diodes, ultrafast lasers or supercontinuum lasers. In the given example in Fig. 2.8, a low-coherence pico-second supercontinuum laser was used with an emission wavelength range of 541 to 927 nm, which results in an axial resolution of 0.65 µm in a medium of n = 1 (air), thus, an axial resolution of 0.28 µm inside the LNO crystal. Images can be realized with a depth of several mm’s inside the crystal. With the numerical aperture of the objective used, a lateral resolution of 3 µm was obtained. The in- ﬂuence in dispersion, which results in a broadening of the imaged DWs particularly well visible in the undoped PPLN sample, was only compensated with a linear approximation of the dispersion of LNO. However, this linear approximation is far from the truth. As the dispersion curve is strongly nonlinear, it requires for a step-wise post processing for slabs at different depths inside the crystal to sufﬁciently obtain a dispersion-corrected image.

2.2.5 Piezo-response force microscopy

Piezo-response force microscopy (PFM) is the most frequently applied technique since its inception by Güthner and Dransfeld [225] to obtain the ferroelectric domain pattern and, thus, the local position of ferroelectric DWs. Hereby, one stirs a conductive tip into contact with a piezoelectric surface i.e. a ferroelectric surface and applies a voltage between probe tip and grounded back electrode in order to excite a deformation of the sample through the converse piezoelectric effect, as given by     σ11 d11 d21 d31            σ22   d12 d22 d32      Ex  σ33   d13 d23 d33      = 2    E  . (2.27)      y   (σ32 + σ23)/2   d14 d24 d34      Ez  (σ + σ )/2   d d d   31 13   15 25 35  (σ21 + σ12)/2 d16 d26 d36 with the σii the tensile and compressive strain components and σij i=j the shear strain | 6 components. For materials of point group 3m, due to symmetry considerations 2d22 = 2d = d ; d ; d = d ; d = d and all other elements are zero. Under the − 21 − 16 33 31 32 24 15 assumption that the electrical field can be considered to have mainly z-component, we obtain σ11 = σ22 = 2d31Ez, and σ33 = 2d33Ez, thus an out-of-plane (oop) and in-plane (ip) strain. With a laser being focused on the reflective side of the cantilever the resulting movement of the probe can be detected via the laser deflection on a screen. For an oop deformation [see Fig. 2.9(a)] the laser spot will be deflected vertically, subse- quently for an ip deformation [see Fig. 2.9(b)] a horizontal deflection will be present. The laser deflection can be recorded automatically via four-quarter photodiodes. With their associated photoresponses A to D, the current center position of the laser spot can be precisely deduced as follows: r = [(A + C) (B + D)] /(A + B + C + D) and ip − r = [(A + B) (C + D)] /(A + B + C + D) . By recording these deflections upon scan- oop − ning it is possible to trace the ferroelectric domain pattern. However, the piezoelectric

30 2.2 visualization of ferroelectric domains and domain walls

displacement of the tip would not be recordable with practically attainable sensitivity in 19 this case . Hence, in current PFM setups an AC voltage is applied and an oscillating deﬂection will be recorded. Small oscillation amplitudes can be registered with higher sensitivity than constant deformations by the use of a lock-in ampliﬁer. A further advantage is the occurrence of mechanical resonances increasing the oscillation amplitude up to 4 10 at the resonance frequency fres. Rendering an image by scanning the tip over the sample, domain patterns with a size of up to 100 100 µm2 and lateral resolutions down to ∼ × 5 nm can be visualized.

a) b)

Figure 2.9: Schematic of piezo-response force microscope (PFM). a) in out-of-plane (oop) and b) in-plane (ip) mode for the special case of a material of point group 3m as LNO. The cantilever deﬂection delivers a measure to determine the local deformation induced by an electrical ﬁeld applied between tip and back contact.

2.2.6 Ferroelectric lithography

Ferroelectric domains and DWs have shown to be photochemically reactive [226, 227, 227– 234], which is commonly attributed to electron-hole pair generation within the absorption depth of incident super-bandgap light. Electrostatic ﬁelds arising from the spontaneous polarization lead to a separation of these charge carriers and drive them towards different locations at the surface, where they promote reduction or oxidation reactions. The separation ﬁeld distribution drastically depends on the screening behavior of the ferroelectric material. As already discussed in 2.1.3 within BTO and PZT internal screening is dominant mainly due to oxygen vacancies. The band bending at the polar surface of c+ domains drives photo-excited electrons to the surface where they can reduce positively charged metal ions or attract molecules with an internal dipole moment. The reduction of positively charged metal ions leads to metal cluster formation at the surface, which is the seed for photochemical growth of metal (i.e. Au, Pt, Ag) particles [226, 235, 236]. For

19 Example: With the relevant piezoelectric coefﬁcient of 85.6 pm/V for BTO and an applied voltage of 1 V over the ferroelectric, this would only result in a deformation of 85.6 pm, which is too minute even for high precision deﬂection acquisition.

31 fundamentals materials as e.g. LNO external screening is dominant, which results in electrostatic fields + to peak at the DW separating c and c− domains and imply large electrophoretic forces acting on e.g. metal ions above the DW and, thus, confine the photochemical reduction at these. In conclusion, the forces acting on charged particles (electrophoretic force ~Eep) and uncharged particles (dielectrophoretic force ~Edep), can be described by ε ε 3 p m ~ 2 2 28 ~Fep = qp ~E ~Fdep = 2πr ε0 εm − (~E ) ( . ) · ε p + 2εm ∇ with qp the particle charge, r the particle radius, εm and ε p the dielectric coefficient of the solvent medium and the particle, respectively as well as the electrostatic field ~E.A conventional setup for ferroelectric lithography (FL) is given in Fig. 2.10(a). Hereby, the ferroelectric crystal, e.g. LNO, is mounted on a support and slided into the dilute metal ion or molecule solution within an UV transparent cuvette and illuminated with an UV lamp, e.g. a mercury arc lamp. In Fig. 2.10(b) the temporal evolution upon UV illumination is presented. In the case of LNO molecules or metal ions are attracted by dielectrophoretic forces ~Fdep towards the DW 1 , either get reduced or physisorbed at the surface and create a nucleus 2 , which locally grows to finally form a nanowire 3 . As an example to demonstrate the capabilities of FL in directed deposition of organic molecules at the DW [see Figs. 2.10(c)-(f)] the tenside molecules sodium cholate/C24H39NaO5 2 10 [see Figs. . (c),(d)] and dodecyltrimethylammonium bromide/ CH3(CH2)11N(CH3)3Br (DTAB) [see Figs. 2.10(e),(f)] have been deposited under illumination of a high-pressure mercury arc lamp with an intensity I(λ < 310 nm) = 1.6 mW/cm2 [228]. We observe various deposition characteristics at different spots on the crystal, which can be characterized in the following way: A DW decoration B domain decoration C undecorated domain area close to DW D film formation on crystal The associated spots are marked accordingly within the given AFM images. In both given cases a DW decoration by nucleus formation can be observed A . However, also undesired domain decoration is present for sodium cholate B , yet typically with an area close to the DW being undecorated C . This is an effect, which has already been observed for the deposition of rhodamine 6G molecules [228]. As artifact for molecule deposition of DTAB unwanted film growth can be present for unsuitably high molecule concentrations D , yet with subsequent rinsing this undesired deposition can be reduced. Therefore, we can conclude, that FL is an interesting tool to fabricate organo-metallic nanowires over a large distance with nanoscopic control, but nevertheless can also be utilized to replicate the underlying ferroelectric domain pattern. In the case of Pt nanowire formation this gives better optical resolution in dark field microscopy.

32 2.2 visualization of ferroelectric domains and domain walls

metal salt/molecules a) in solution

LiNbO3 UV transparent cuvette Hg lamp

b) 3 molecules/ ~ ~ 2 Fdep 1 Fep ~E metal - ~p + c-

P~ t c) d) A A

B B C

C DW + c c

5 µm I [a.u.] 2 µm I [a.u.] 0nm0 40nm1 0nm0 40nm1 e) f) A A

D DW

+ c c

5 µm I [a.u.] I [a.u.] 0 1 500 nm 0 1 0nm 20nm 0nm 10nm Figure 2.10: a) Schematic of the setup for ferroelectric lithography applied b) Schematic of nu- cleated growth of molecules on ferroelectric DWs of LNO upon UV illumination. c),d) Ferroelectric lithography with 2wt% sodium cholate with an illumination time of t = 25 min and I(λ < 310 nm) = 1.6 mW/cm2. e),f) Ferroelectric lithography of 1wt% DTAB with an illumination time of t = 25 min and I(λ < 310 nm) = 1.6 mW/cm2.

33 fundamentals

2.2.7 Further methods domain-specific etching Etching is a surface-destructive visualization method, based on the finding, that surfaces of different ferroelectric domains show different etch rates upon hydrofluoric (HF), phosphoric, nitric, and hydrochloric acid. First being intro- + duced for c and c− domains in BTO in 1955 [237]. A decade later, this technique was applied by Nassau et al. on LNO by means of a mixture of hydrofluoric and nitric acid (1 : 2 vol. %) at 110 ◦C, which is still the most common and efficient reported condition, yet depending on different dopants and the stoichiometry [171, 239]. Etching is fast, relatively simple, and reveals a resolution < 0.1 µm. Yet, attainable lateral resolutions are limited by sidewall etching, where non-polar crystal faces are also etched-off, which becomes important with increasing etching times [240, 241]. However, etching has two main drawbacks: (i) it is a destructive method, i.e. at least the top 50 nm of the crystal are affected and (ii) it does not permit in situ imaging of domain-reversal processes. nonlinear talbot imaging The Talbot effect is a near-field diffraction phenomenon in which self-imaging of a grating or other periodic structure replicates at certain imaging planes. The nonlinear Talbot effect, i.e., the formation of second-harmonic self-imaging instead of the fundamental has first been documented in periodically poled LTO crystals [242] and further theoretically described [243]. Further measurements have been carried out to use this effect in quasi-phase matching and Cherenkov-type configuration [244, 245]. However, both spatial and vertical resolution are not competitive to the further measurements. defect luminescence microscopy LNO has been doped with luminescent lan- thanides as erbium. Primarily intended for the use as active materials in lasers due to their very strong radiative relaxation, it later turned out these actinide ions in sufficiently low concentration to be act as local probes [246]. It is expected the local crystal field to has an influence on the strength of the present luminescence, both in intensity and energy shift. Measurements under cryogenic conditions revealed, that by combined excitation- emission spectroscopy different maximum spots can be observed, which were associated with different defect centers. As-grown and poled LNO shows differences in these patterns, i.e. poled LNO has a smaller amount of defects, which results in a weaker luminescence. These properties can be correlated to local electric fields in the different domains, i.e. the intrinsic field. [247]. The contrast between as-grown and poled domains vanished under thermal annealing at 200 ◦C, which is in good agreement to the results for internal fields. However, at DWs persistent shifts remain. This would result in an electrical field of 5 kV/mm across the DW. This value is derived from the contrast in samples prior to thermal treatment and the well-known internal field in these samples [248]. The apparent contrast in luminescence have further been used for the inspection of PPLN and other LNO configurations, especially to visualize defects [249–252]. As it is always necessary to use lanthanide doped LNO it is severely difficult to understand the influence of

34 2.2 visualization of ferroelectric domains and domain walls other dopants, as codoping is increasingly difficult and can result in a much worse crystal quality. multiphoton luminescence microscopy We invented another luminescence microscopy technique to visualize domains upon multiphoton excitation in LNO and LTO. It relies on a similar finding as in the case of defect luminescence microscopy, that poled domains emit significantly less multiphoton luminescence as virgin domains. A typical domain image visualization as well as the intensity and contrast dependence upon excitation wavelength is given in Fig. 2.11. Similarly, temporally and spectrally resolved luminescence measurements revealed different behavior in virgin and singly inverted domains. A drawback for the application as a 3D visualization is the weak contrast of e.g. only about 3 % in the case of Mg:CLN. Further details are discussed by Reichenbach et al. [253, 254]. Hereby, mainly a three-photon absorption-driven was observed. a) b) c) 4 1

0.8 3

0.6 [%] 2 [a.u.] C I 0.4 1 0.2

0 0 760 780 800 820 840 25!m I [a.u.] 760 780 800 820 840 [nm] 0.960 1 [nm]

Figure 2.11: a) Multiphoton luminescence scan recorded on the domain introduced in Fig. 2.6, b, c) Intensity and contrast dependence upon excitation wavelength. nearfield microscopy Optical nearfield microscopy is a scanning-probe technique, which can evaluate changes in the permittivity at a specific operating wavelength with an AFM tip being as close to evaluate the scattering properties of the material. Hereby, mainly two techniques exist: aperture-type and scattering-type scanning nearfield microscopy (SNOM), which are physically only different in the way the incident and scattered electromagnetic wave propagate. Both techniques have been used so far to observe domains and domain walls. Kim and Gopalan applied aperture-type SNOM and observed long- reaching changes in the refractive index around a domain wall [221]. Apart from that, changes in the transmission due to the electro-optic effect were studied and revealed even polarization orientation discretization [66]. Scanning-type SNOM revealed the changes in the permittivity for wavelengths close to the phononic resonance 10 µm. Most likely the ∼ changes in permittivity are connected to the present internal field in these materials. Like all scanning-probe techniques the results are surface-bound and typically have a vertical resolution in the range of <30 nm. However, this is not true for electrooptical aperture-

35 fundamentals

type SNOM, however in this case the detected signal is integrated over the sample with a nanoscopic source.

thz microscopy In hydrogen-bonded supramolecular ferroelectric 1:1 salt of 5,5’- dimethyl-2,2’-bipyridine and deuterated iodanilic acid domain speciﬁc Terahertz radiation was observed upon ultrafast excitation [37]. Hereby, the emitted THz radiation both depends on the polarization and propagation orientation. With this a pseudo three- dimensional imaging can be realized with up to four layers including the two surfaces. However, the reported lateral resolution with this technique is very limited and on the order of tens of µm.

raman imaging Raman spectroscopy is used to observe vibrational, rotational, and other low-frequency modes in a system. Hereby, light being inelastically scattered due to the existence of these modes is collected, with the energy loss or gain resembling the mode energy. Therefore, confocal Raman imaging or µ-Raman is especially suitable to pinpoint towards structural changes. Between ferroelectric domains and at ferroelectric DWs of LNO both changes in the Raman scattering efﬁciency as well as the Raman shift have been observed [255–257]. Especially, Raman shifts being present at DWs conclude their to be either structural changes or accumulation of defects, which was investigated especially for different Mg doping concentrations [258]. Hereby, an apparent change in the sign of the Raman shift have been observed for concentrations above or equal 5 mol%.

transmission electron microscopy Transmission electron microscopy (TEM) is a very powerful tool to investigate ferroelectric domain on the atomic scale. Samples being thinned down to less than 100 nm by either milling or focused ion-beam are bombarded with a monochromic electron beam. Both bright and dark field microscopy can be applied to differentiate domains. By applying scanning TEM (STEM) the atomic displacement of atoms can be revealed with an accuracy down to 10 pm. By accurate post-processing of the recorded images it is possible to extract the different domains by these ionic displacements. Typically, such measurements are conducted with high-angular annular dark field (HAADF) microscopy, as it gives a clear ion contrast. Coupling with a local probe, can further increase the knowledge about domain formation under electrical fields. How- ever, up until very recently, no STEM images of the crystallographic properties at DWs in LNO and LTO were reported. Very recently, ADF TEM has been reported for LNO and a 20 bending of niobium ions in reference to the oxygen octahedras observed [259] . Similarly, Gonnissen et al. observed kinks and wall-meanders at macroscopically NDWs in PPLN with [1100¯ ] HAADF-STEM [260]. The DW was shown to have a very variable width from 200 pm to 5000 pm with a variation on the unit cell level of about 35 over about 100 unit cells. A niobium ion displacement of 12 pm was observed, which is consistent with X-ray based crystallographic values [261].

20 In general, oxygen is not visible in TEM investigations as the intensity scales as I Z1.6 1.9 with Z the ∼ − average atomic number [Z(Nb) = 41, Z(O) = 8, Z(Li) = 3].

36 2.3 lithium niobate and tantalate

2.3 lithium niobate and tantalate

a) b) 2.31 A˚

0.258 A˚

0.690 A˚ y

z x O (lower) Nb +PS vacancy O (upper) Li

Figure 2.12: a) Crystal structure in LNO with the ion displacement b) Crystal structure viewed along the crystal axis c, a possible ferroelectric domain is colored lightly red. Repro- duced from [262].

Lithium niobate/LiNbO3 (LNO) and lithium tantalate/LiTaO3 (LTO) are intensively applied optical materials well known for their special acoustooptical, electrooptical, photorefractive, birefringent, non-linear optical, photochromic, and ferroelectric properties [263]. Most common applications incorporate LNO into optical waveguides [264], Mach- Zehnder interferometers [265], nonlinear optical upconverters [266, 267], downconvert- ers/THz generators [268–271], pyroelectric infrared detectors [272, 273] and gas sensors [274], as well as surface and bulk acoustic wave modulators [275–278]. The application of the almost "fatigue-free" LNO in ferroelectric ﬁeld effect transistors (FeFET)[279] or non-volatile random access memories (RAM)

2.3.1 General Properties

For simplicity, we only explain the physical properties of only LNO in detail. This is mainly due to the fact, that for most properties only minute differences between these two materials are present. If this is not the case, we will further describe these differences. LNO possesses a very high Curie temperature T 1255 K, even above its melting temper- C ≈ atures. The crystal consists of oxygen ion sheets separated by 2.31 Å in a distorted hexagonal close packed conﬁguration. The octahedral interstices inside this structure are ﬁlled with one-third each by lithium (Li+) ions (off-centered by 0.258 Å), niobium (Nb5+) ions

37 fundamentals

(off-centered by 0.690 Å) and structural vacancies [see complete unit cell in Fig. 2.12(a)].

Table 2.2: General ferroelectric, crystallic, and electrical properties of congruent (CLN) and stoichiometric lithium niobate (SLN). Property CLN SLN Ref. 247 280 Forward coercive ﬁeld EC, f [kV/mm] 20.7 4.1 [ , ] 247 280 Backward coercive ﬁeld EC,b [kV/mm] 16.9 3.3 [ , ] 2 − − 247 Spontaneous polarization PS [µC/cm ] 75 . . . 80 75 . . . 80 [ ] 263 Curie temperature TC [◦C] 1255 < 1200 [ ]

Melting point [◦C] 1140 1206 [263] Density ρ [g/cm3] 4.647 4.635 [263] Mohs hardness 5.0 5.0 [281] Thermal expansion coefﬁcient 4.1 ( c), 15.0 ( c) 93 6 1 k ⊥ [ ] αt [10− K− ] (T = 300 K) Lithium content [Li]/([Li]+[Nb]) 0.485 0.498 [247] Lattice constant a [nm] 0.5150 0.5147 [263] Lattice constant c [nm] 1.3864 1.3856 [263] Static dielectric coefﬁcients eT = 84 11 [263] T T εij e33 = 30 Bulk resistivity (DC) [Ω m] >5 1015 [282, 283] × Bulk resistivity (107 Hz) [Ω m] 3 104 [284] ∼ ×

LNO exhibits a three-fold rotation symmetry around the crystal axis c and possesses three glide mirror planes, thus, belongs to the point group 3m/C3V (international/Schön- ﬂiess notation) and the space group R3c. A view along c onto the crystal structure is given 21 in 2.12(b), well displaying the hexagonal unit cell . The lightly red colored area depicts a possible ferroelectric domain. The commonly found 120° angles in domain formation are crystallographically supported. The typical domain formation in LTO with 60° is similarly crystallographically supported. The difference in these domain formation manifestations is presumed to be the result of interfacial screening [293].

2.3.2 Stoichiometry

Single-crystalline LNO and LTO are commonly fabricated by Czrochalski process [294]. To satisfy a homogeneous composition growth, the crystalline solid should satisfy the same composition as the melt. However, the so-called congruent composition deviates from the stoichiometric material given by the structural formula. In fact, the congruently melting composition is Li deﬁcient with a composition [Li]/([Li]+[Nb])= 0.485 for

21 For both LNO and LTO similarly a rhombohedral unit cell can be constructed.

38 2.3 lithium niobate and tantalate

Table 2.3: General ferroelectric, crystallic, and electrical properties of congruent (CLT) and stoichiometric lithium tantalate (SLT). Property CLT SLT Ref. 285 287 Forward coercive ﬁeld EC, f [kV/mm] 21.2 1.7 [ – ] 285 287 Backward coercive ﬁeld EC,b [kV/mm] 12.6 1.5 [ – ] 2 − − 281 Spontaneous polarization PS [µC/cm ] 60 55 [ ] 281 Curie temperature TC [◦C] 605 695 [ ]

Melting point [◦C] 1650 [281] Density ρ [g/cm3] 7.456 7.458 [281] Mohs hardness 5.5 5.5 [281] Thermal expansion coefﬁcient 4.2 ( c), 12.0 ( c) 288 6 1 k ⊥ [ ] αt [10− K− ] (T = 300 K) Lithium content [Li]/([Li]+[Nb]) 0.485 0.498 [289, 290] Lattice constant a [nm] 0.5154 0.5151 [291] Lattice constant c [nm] 1.3864 1.3856 [263] Static dielectric coefﬁcients eT = 54 11 [292] T T εij e33 = 42 Bulk resistivity [Ω m] >5 1015 [283] ×

congruent lithium niobate (CLN). In Fig. 2.13 the phase diagram of Li2O:Nb2O5 in the high temperature regime is given. Ferroelectric crystals can be grown with a composition [Li]/([Li]+[Nb]) of 0.485 to 0.5. Only for temperatures very close to TC the ferroelectric phase widens to larger non-stoichiometry. The different structural, ferroelectric, and paraelectric phases are denoted. Perfect stoichiometry is achieved for [Li]/([Li]+[Nb])= 0.5. However, the practically attainable cases are either congruent or near-stoichiometric LNO. The general ferroelectric, crystallic, and electrical properties of both LNO and LTO are summarized in Tabs. 2.2 and 2.3, respectively for both congruent and (near)-stoichiometric material. Due to the immanent nonstoichiometry defects are incorporated, which have a strong inﬂuence in both the structural but also optical properties of LNO.

39 fundamentals

Li2O [mol %] 525456586062 50 48 46 44 42 40 1,300

1,250 congruent liquid [Li]/[Nb] = 48.5:51.5 1,200

1,150 TC

C] para- [ electric T 1,100 ferroelectric 1,050 LiNbO3 + Li3NbO4 LiNbO3 + LiNb3O8

1,000 stoichiometric [Li]/[Nb] = 50:50 LiNbO3 950 38 40 42 44 46 48 50 52 54 56 58 60

Nb2O5 [mol %] 2 13 Figure . : Phase diagram for Li2O:Nb2O5. The different structural, ferroelectric, and paraelectric phases are denoted. Perfect stoichiometry is achieved for [Li]/([Li]+[Nb])= 0.5. How- ever, the practically attainable cases are either congruent or near-stoichiometric LNO. Reproduced from [263].

2.3.3 Optical properties

absorption LNO has an ultra-wide transparency window over the visible and infrared part of the spectrum from 320 nm to 5 µm. The maximum transmission 70 % ∼ ∼ ∼ is solely limited by the reflection on both crystal faces. The only dip occurring within the transmission window at 2.87 µm is associated with the presence of OH-groups. In Fig. ∼ 2.14 transmission curves for a wavelength range 290 nm. . . 360 nm for differently doped crystals of CLN and SLN are shown. In the far infrared, a series of absorption peaks is observable associated with lattice vibrations up to wavelengths of about 60 µm. The electronic band structure of LNO around the Fermi level EF is determined mainly by the NbO6 octahedra since the upper level of the valence band is formed by the oxygen 2p states (electron configuration 1s22s22p4) and the lower level of the conduction band is formed by the niobium 4d states (electron configuration 5s14d4). The experimental band gap of LNO is around 3.7 to 3.9 eV [295]. However, so far it is not completely understood, whether this is defect-mediated, as DFT simulations of the absorption reveal significantly 22 larger values [296–299] , e.g. Schmidt et al. [297] assume a bandgap of 5.3 eV to be most likely. Riefer et al. found out, that spin-orbit coupling does not have a major influence by widening the bandgap by only 10 meV, whereas relativistic corrections increase it by

22 These results were obtained both in ground-wave approximation (GWA), generalized-gradient approximation (GGA) under the application of the Perdew-Burke-Enzerhof functional [300] as well in hybrid DFT under application of the Hyde-Scuseria-Enzerhof functionals [301].

40 2.3 lithium niobate and tantalate

a) b) 80 80

60 [%] 60 [%] T T

40 40 pure Hf 0.2% Sc 1.0% Mg 2.5% pure 20 20 Mg 5.0% transmission Zr 0.085% Zn 1.0% transmission Er Tm In 0.5% Mg 4.5% Nd 0 0 300 320 340 360 300 400 500 600 wavelength λ [nm] wavelength λ [nm]

Figure 2.14: Transmission through differently doped 500 µm thick samples of a) SLN b) CLN. Ap- parent are the shift of the cut-off wavelength for some dopants as Mg as well as the distinct atomic absorption lines visible for dopants as Er with a 4 f electronic shell.

150 meV and, hence, assume a bandgap value of about 5.4 eV at 300 K to be most likely [302]. Even though, spin-orbit coupling is not relevant, a Rashba-like spin texture can be found with a Rashba parameter of 0.06 eV Å and Rashba splitting in the quasi-parabolic band structure [302]. However, due to limited reported results on VUV reflectometry, the bandgap is often assumed to be 4 eV, which is severely smaller than the reported theoretical predictions. The to-date sole report on VUV reflectometry by Mamedov et al. [303] for photon energies ranging from 0 to 35 eV, however, supports the assumption of a larger bandgap of about 5.3 eV, with a peak in the reflection at 5.27 eV, but similarly at 3.90 eV. refractive index LNO exhibits birefringence with an optic axis c. As the ordinary k refractive index is larger than the extraordinary refractive index (ne < no) the crystal is said to be negatively uniaxial. The birefringence of LNO is comparably large (n n =∼ 0.1). o − e The refractive indices of LNO were thoroughly studied by Schlarb and Betzler [304], who carefully measured both no and ne of LNO in the composition range from 47 to 50 mol% Li2O over a wavelength range from 400 to 1200 nm, and a temperature range of 50 to 600 K. The authors then proposed a Sellmeier equation for refractive index as

2 A0,i(cLi, cMg) 2 ni = 2 2 AIR,iλ + AUV λ0,i(T)− λ− −  −  A0,i + (cthr cMg)ANbLi,i + cMg AMg,i for cMg < cthr A0,i(cLi, cMg) = − A0,i + cMg AMg,i for cMg > cthr λ (T) = λ + µ [ f (T) f (T )] 0,i 0,i 0,i − 0 261.6 f (T) = (T + 273)2 + 4.0238 105 coth 1 × T + 273 −

41 fundamentals

where cLi is the mol% of Li2O; λ is the wavelength in nm in vacuum; i = e or o, T the temperature in ◦C; and reference temperature T0 = 24.5 ◦C. The above general Sellmeier equation is only valid from the far infrared up to the UV region. Experimental values of the refractive index for various materials being investigated are given in Fig. 2.15. In LTO the birefringence is obviously much smaller.

a) b) ordinary ordinary 2.4 extraordinary 2.4 extraordinary n CLN n SLT 5% Mg:LNO CLT 2.3 2.3 8% Mg:LTO refractive index 2.2 refractive index 2.2

0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2 wavelength [µm] wavelength [µm] Figure 2.15: Refractive index of LNO. The blue and red rectangles signify the wavelength ranges of the fundamental and second-harmonic wavelength used in nonlinear-optical microscopy of the material. nonlinear optical properties Ferroelectrics like LNO possess large optical nonlinearities, as being already discussed in Sec. 2.2.2. It belongs to the point group 3m and, thus, only three coefficients of the second-order susceptibility tensor are non-zero: d33, d22, and d15. Hereby, d33 is by about one order of magnitude the largest coefficient. 2 2 As ISHG ∝ d I0 it typically gives the by far strongest SHG response. SHG, which is not amplified in an optical cavity, can solely be generated upon intense illumination, which is typically achieved by ultrashort pulses. However, under illumination of intense lasers the refractive index of the illuminated material will be altered according to [179]

n(I) = n0 + n I2 (2.29) 2 · with n2 the nonlinear refractive index. This effect is well known as the Kerr effect or quadratic electro-optic effect and arises from a third-order nonlinear process. Upon focused beam, this results in self-focusing, which can independently on this term either further focus or defocus. The shift in focal position ∆ f inside the material depends on the sign of n2, being either positive or negative as a result of third-order susceptibility. The focal length of the collateral Kerr lens fK is deduced from

2 w 2 30 fK = ( . ) 4an2tI0

42 2.3 lithium niobate and tantalate with w the beam width, I0 the peak-intensity at the focal center, t the medium thickness, and a the beam profile correction factor; for a Gaussian beam a = 1.723. The focal length shift ∆ f is calculated using the effective focal length feff of the lens system 1 1 1 d = + (2.31) feff fO fK − fO fK with fO the focal length of the objective, d the distance between the principal planes of objective and Kerr lens. Eq. (2.31) holds true under the assumption of no refraction between the nonlinear medium and the ambience. However, LNO possesses a rather large refractive index. Thus, refractive index matching is experimentally difficult to fulfill and has to be encountered in real case [305, 306]. This is further discussed in Sec. 3.3. Upon ultra-high intensity laser illumination ablative processes and bulk material damage can be invoked, which is particularly due to two- or multiphoton absorption processes, severely increasing the absorption in this widely transparent material [307]. For two-photon absorption and negligible linear absorption, the typical Beer’s law for absorption along z has to be transformed accordingly

αz two-photon absorption I0 I(z) = I0 e− I(z) = (2.32) −→ 1 + βzI0 with α the linear absorption coefﬁcient, β the two-photon absorption coefﬁcient and I0 = I(z = 0). A second important origin of augmented absorption upon illumination with ultrashort laser pulses in LNO is the two- or multiphoton absorption mediated formation of polaronic centers [308]. The types of polarons present in LNO will be further discussed in Sec. 2.3.5. To conclude, for assurance of non-destructive microscopy applying ultrashort lasers, the surface and bulk laser damage thresholds should not be exceeded. In general, the surface laser damage threshold is smaller than its counterpart in the bulk, due to eased plasma formation conditions. The optical properties are summarized in Tab. 2.4.

2.3.4 Intrinsic and extrinsic defects intrinsic lattice defects Despite LNO being grown as a single crystal the congruent melt LNO (CLN) exhibits a signiﬁcant level of Li deﬁciency and, hence, intrinsic point defects being formed to satisfy crystal charge compensation [318]. Possible defects to be formed are vacancies, such as of lithium VLi and niobium VNb, as well as antisite 319 321 defects such as NbLi and LiNb[ – ]. The existence of oxygen vacancies VO in CLN and SLN was brought up for discussion as well [322]. Analysis by means of X-ray diffraction [323], conductivity [320], and Raman spectroscopy [324, 325] as well as theoretical considerations using density functional theory (DFT) and ab-initio calculations have been performed to elucidate the type of occurring defects [326– 329]. 321 327 Niobium and oxygen vacancies VNb,VO are energetically unfavorable [ , ], mainly due to the long-range strain being introduced into the crystal. The occurrence of VNb and NbLi is unlikely due to their inconsistency with X-ray and neutron diffraction [323].

43 fundamentals

Table 2.4: Summary of linear and nonlinear optical properties of congruent and stoichiometric lithium niobate (CLN, SLN). Property CLN SLN Ref. Refractive index at λ = 990 nm n = 2.2372 n = 2.2370 o o [309, 310] for ordinary and extraordinary wave ne = 2.1599 ne = 2.1567 a = 4.9048 a = 4.91300 Sellmeier coefficients (ordinary) 1 1 2 a2 = 0.11775 a2 = 0.118717 311 no(λ[µm]) = [ ] 2 2 a3 = 0.21802 a3 = 0.045932 a1 + a2/(λ[µm] a3) a4λ[µm] − − a4 = 0.027153 a4 = 0.0278 T T Linear electrooptic r13 = 10.0 r13 = 10.4 coefficients (free) T [pm/V] T = 6.4 T = 6.8 rij r22 r22 [312–314] T T at λ = 632.8 nm r33 = 31.5 r33 = 38.3 T and low frequencies r51 = 32.6 1 0.2 (λ = 1064 nm) 315 Absorption coefficient α [m− ] ∼ [ ] 1500 (λ = 388 nm) ∼ UV absorption edge [nm] 316 308 [247] 1 (α > 2 mm− )

Second-order susceptibility d31 = 4.35 0.44 d31 = 4.64 0.66 − − 263 coefﬁcients dil [pm/V] d = 27.2 2.7 d = 41.7 7.8 [ ] 33 − 33 − at λ = 1.06 µm d = 2.10 0.21 d = 2.46 0.23 22 22 Two-photon absorption 0.9 for λ = 776 nm [316] coefﬁcient β [cm/GW] and τp = 240 fs Laser-induced bulk 60. . . 300 [315] 2 damage threshold [GW/m ] for τp = 10 . . . 30 ns Laser-induced surface 5. . . 111 [315] 2 damage threshold [GW/m ] for τp = 7 . . . 12 ns Nonlinear refractive index 8 1 (λ = 532 nm) [263] n [10 14 cm2/W] 9 1 (λ = 1064 nm) 2 − Group velocity dispersion [fs2/mm] 309.87 (λ = 990 nm) [317]

The most consistent description of the LNO crystal structure is composed of NbLi antisite defects, niobium vacancies VNb in an illmenite-type structure (. . . , Li, , Li, Nb, , Nb, Li, 26 3 , Li, . . . ) with concentrations of 4 % VLi (cVLi 8 10 m− ) and 2 1 % NbLi2(cNbLi 26 3 ≈ × ∼ ≈ 2 10 cm )[320, 326]. Other defects as Nb2O5 or more complex clusters are possible × − [330, 331]. Despite the given broad number of investigations on this material, the complete structure is still rather speculative, as LNO is a highly disordered material with different cluster formation and stacking orders [321, 331]. To reduce the intrinsic defect concentration, several techniques have been attempted to minimize the lithium deﬁciency, thus, to obtain near-stoichiometry: growth in lithium ex- cessive melts [332], by double crucible method [333, 334, 334], from a potassium rich melt [335, 336], and by vapor transport equilibration [337–339]. Upon doping with magnesium

44 2.3 lithium niobate and tantalate

(Mg), yttrium (Y), indium (In), scandium (Sc), or zink (Zn) the intrinsic niobium antisites NbLi defect concentration can be reduced to almost zero [340, 341, 341–344]. These intrinsic defects seem to play a significant role in the creation of an internal field. Upon electrical poling VLi complexes are not inverted, thus, a remaining polarization ~PD2 in the former direction is present, reducing the coercive field in this direction. However, these complexes can rearrange upon thermal annealing over 100 ◦C[345]. This influence is schematically depicted in Fig. 2.16.

Li Nb VLi NbLi +z +z z z a) b) c) d)

+P P P S S S +P P P D1 D1 D1 +PS +P +P P D2 D2 D2

F F F F +P +P P D2 D2 D2 +PS +P P P D1 D1 D1 +P P P S S S

P P dF P P dF+ dF+

Figure 2.16: Influence of defects present in LNO onto the hysteresis loop and the internal crystal field. a) Defect-free SLN, b) common defect configuration in CLN, containing both VLi and NbLi. Apart from spontaneous polarization by niobium and lithium displacement PS, additional contributions PD1, PD2 are present. These represent displacements of NbLi and VLi, respectively, c) Upon domain inversion both ~PS and ~PD1 are inverted, however, ~PD2 stays put, d) lithium vacancies can be thermally activated to invert. Free energy graphs reproduced from [345]. optical damage resistant impurities The optical damage resistance can be increased upon doping with the aforementioned ions Mg2+, In3+, Sc3+, Zn2+ above an element-specific concentration threshold, which is significantly larger for divalent ions as for trivalent ions [346, 347]. Similarly, this concentration is significantly larger for congruent as for (near)-stoichiometric crystals. It is generally conceived the impurity ions are incorporated on Li sites in the lattice, hence, resulting in a concentration reduction of NbLi antisites [348–352]. Above a certain concentration antisite defects nearly vanish, which results in a drastically increased bulk conductivity [353]. This explanation was

45 fundamentals

supported by optical absorption and ESR measurements, which indicate the presence of 4+ 4+ NbLi :NbNb bipolarons in reduced material below the optical damage resistance threshold 354 355 4+ 356 [ , ], while above this threshold free small NbNb polarons are found [ ]. Later analyses, however, reported the antisites to already vanish below the optical-damage resistance threshold (ODRT)[357, 358]. SLN has a smaller value of lithium vacancies and, hence, a smaller ODRT [359].

photosensitive impurities Transition metal ions as Fe, Cu, Mn, and Ti act as photosensitive impurities in LNO [263, 360, 361]. With the former three elements being known 23 24 to have both photochromic and photorefractive features , the inﬂuence on LNO concerning photochromism and photorefraction due to Ti ion indiffusion are so far not as clear, with a change in refractive index as the most dominant inﬂuence. Fe impurities appear to be the best-understood of these mainly due to their relevance for persistently and switchable photorefraction in holographic data storage devices [362]. Since this element has the highest residual concentration in nominally pure material of 2 ppm to 5 ppm [263], which makes it most likely to similarly affect the polaronic behavior of undoped LNO. Therefore, the properties of the very general case of Fe impurities in both nominally pure LNO and samples doped with optical-damage resistant ions are presented.

iron Fe impurities are shown to give rise to strong bulk photovoltaic and photorefractive effects [363]. Mössbauer spectroscopy revealed Fe to be incorporated in the two main valence states Fe2+ and Fe3+ [364]. In as-grown material the concentration of Fe2+ 3+ 364 impurities cFe2+ is much smaller than that of Fe impurities cFe3+ [ ]. However, upon reduction or oxidation these concentrations can be adjusted widely [365]. Both valence states are accompanied by optical absorptions in the visible and near-infrared spectral range. Fe2+ reveals optical absorption near 1.1 eV (5T 5 E d d transitions) and near 2 − − 2.6 eV (Fe2+-Nb5+ intervalence transfer). The latter energy both fortuitously is similar to 4+ 4+ 2+ those of the O− trapped hole absorption and of the NbLi :NbNb bipolarons. Fe shows sharp spin-forbidden d d transitions at 2.55 and 2.95 eV as well as O2 –Fe3+ charge − − transfer absorption starting at 3.1 eV, superimposed of the fundamental absorption. It is possible to determine the number density of Fe2+ from the absorption band at 2.6 eV in arbitrary LNO samples, given there is no additional absorption [364]. In congruent LNO both Fe2+ and Fe3+ are incorporated on Li sites [366]. However, there is a strong evidence for a change of this behavior in near-stoichiometric samples and crystals codoped with optical-damage resistant ions. While the incorporation of Fe2+ remains unchanged [367], Fe3+ partially occupies Nb sites in this case [368, 369], thereby losing its electron acceptor properties. It is assumed that the partial incorporation of Fe3+ on Nb site strongly enhances the optical-damage resistivity [263]. The contribution of Fe3+ to the thermally activated charge transport has been described in a polaronic way, assuming hopping tran-

23 Photochromism is deﬁned as the reversible conversion a compound can undergo between two forms due the photoabsorption. Hereby, the two states possess different absorption spectra. 24 Photorefraction is a nonlinear optical effect in that a material responds to light by persistently altering their refractive index.

46 2.3 lithium niobate and tantalate sitions from Fe2+ to Nb5+ [370]. The activation energy for this process has been reported to be between 0.9 eV and 1.3 eV. It is worth noting, that the formation of CDW can be screened by a transition of Fe2+/Fe3+. In this work, we will commonly refer on the anti- 2+/3+ site defects FeLi .

2.3.5 Polarons

Excited free charge carriers can undergo self-trapping. First works on self-trapped charge carriers were performed by L. D. Landau [371]. Self trapping was found to be caused by charge carriers distorting the surrounding lattice through Coulomb or other short-range interactions [372, 373]. Hereby, the displacement of the surrounding ions leads to a potential well in which the otherwise free carriers are localized. Since this type of electron-phonon coupling was ﬁrst considered in ionic (polar) materials, the quasiparticle composed of polaron. Nowadays this term is used in a wider scope, and one may associate polaron with the fact the surrounding lattice to be polarized by the carrier’s charge. Depending on the spatial range of the interaction one distinguishes between large and small polarons. In the former case the radius of the displacement pattern is considerably larger than one lattice constant. With increasing coupling strength the polaronic radius shrinks, eventually restricting the potential well to a single lattice site. This latter object is then called small polaron. It turns out that the physical properties do not change steadily in the transition regime between large and small polarons [374]. While, e.g. the movement of a large polaron may be described like that of a free electron with an increased effective mass [375, 376], small polarons merely move by hopping between nearest neighbor sites at least at temperatures above half the Debye temperature [373]. Similarly, the optical absorption spectra of large and small polarons also yield remarkable qualitative differences [375]. In this work, the term polaron will always mean small polaron, unless it is otherwise stated. general theory of small polarons A small polaron only occurs if the short- range interactions are sufﬁciently strong, i.e. for large electron-phonon coupling, which can be described with Holstein’s one-dimensional Molecular-Crystal Model (MCM) [372]. The following considerations are mainly derived from Imlau et al. [377], which is suggested for further detailed information. The polaron is localized at an ion site due to an appropriate relaxation. Therefore, instead of a number of neighbors in a real crystal we only assume two of them, forming a one-dimensional chain. The polaron at site p is then assumed to be stabilized by a displacement of the particles in the molecule qp. In the beginning an excess electron is placed at an originally undistorted site (q = 0). The electronic energy at q will change to E = Fq, with F denoting the short-range C − electron lattice coupling factor considering Coulomb and short-range interaction [378]. At the same time the elastic energy is enhanced due to the induced lattice distortion

47 fundamentals

1 2 Vel = /2 Kq . The total energy of the system is given as the sum of electronic and elastic energies [372, 373, 379]:

1 E = Kq2 Fq (2.33) tot 2 −

1 2 By minimizing this energy (qmin = F/K) one obtains the polaronic energy Ep = /2 F /K. If Ep exceed the electronic bandwidth 2J – the case of strong electron-phonon coupling – a polaron is formed. The electron can be localized at any site of the crystal lattice. As real crystals contain intrinsic and/or extrinsic point defects, whose formal charge may differ from that of the regular sites, the energy of a polaron at such a site is additionally decreased by W . Due to different ionic environments the values of the polaronic force − D F constant may slightly differ from those of the above addressed situation to F0. Eq. (2.33) transforms to

1 2 E = Kq F0q W (2.34) tot,D 2 − − D and the polaron energy becomes E = E W . Taking the induced lattice distor- min,D − P0 − D tion with them, trapped electrons can still meander inside the crystal lattice. The polaron transport is distinctly different at low and high temperatures. At low temperatures polarons are able to tunnel between different sites and the transport may be described in an electronic band picture with a large effective mass and a polaronic bandwidth 2J [375]. This bandwidth decreases exponentially with increasing temperature until ﬁnally a transition temperature Tt is reached, where the band picture breaks down. Above Tt the charge 375 transport proceeds via hopping transitions [ ]. The condition for Tt is given by

1 h¯ τ− J, (2.35) · band ≈

where τband is the mean lifetime of polaronic band states at T < Tt. Appel further found that in the regime of Tt the mean lifetimes for band and hopping transitions are equal 380 (τband = τhop)[ ]. The transition regime is rather sharp: it turns out that Tt is in the 1 range of half the Debye temperature Θ = h¯ ω /k , the lifetime being τ ω− . D LO B band ≈ LO For real crystals the transition temperature may be substantially less then ΘD/2 due to imperfections and disorder [376].

thermally activated transport In order to describe the thermally activated hopping transport of free polarons the situation of two equivalent sites, qp, with p = 1, 2 is examined. The total energy is written as

1 E = E + E = K(q2 + q2) F(q n + q n ). (2.36) tot 1 2 2 1 2 − 1 1 2 2 The minimum energy required for the transition in a hopping process is found to be

1 F2 1 E (q q ) = = E (2.37) min 1 → 2 4 K 2 FP

48 2.3 lithium niobate and tantalate

This thermal activation energy therefore equals half the polaron binding energy. It was shown by Austin and Mott, that in ionic crystals the activation energy may be less than 376 EFP/2 [ ]. This is very obvious, as the aforementioned theory assumes the very speciﬁc case of a potential dominated by Coulomb interaction, e.g. in the more realistic model of a Morse potential. adiabatic transport In the case of adiabatic transport the electron can coherently follow the motion of the ions. This means the time required for the electronic wave function to adjust itself to the position of the ions is smaller than the relaxation time of the lattice. The system then remains on the lower potential curve E( ) and the transition − probability is given by

ω E W(p p 1) = LO exp a (2.38) → 2π − kBT

The activation energy Ea for the site transition is substantially diminished: 1 E = E J (2.39) a 2 FP −

An adiabatic transitions occurs if the probability for a transition of the system from E( ) − to E(+) is small:

1 1 k TE /4 h¯ ω /2 J > B FP LO . (2.40) π π non-adiabatic transport If the polaronic bandwidth is much smaller than half the polaron binding energy, the electron cannot follow the ionic motion. Upon hopping the system can jump from a lower adiabatic E( ) to the higher potential E(+), thus ascending − in energy. Eventually, it descends again or ascends again. Summing up all the possibilities one obtains for the site transition probability in the classical limit (2kBT > h¯ ωLO):

1 J2 π /2 E W(p p 1) = exp a (2.41) → h¯ 4EakBT − kBT Note, that in this case the activation energy is given by

1 1 E = E J E . (2.42) a 2 FP − ≈ 2 FP Having obtained the transition probabilities W(p p 1) one ﬁnds the relation for the → mean lifetime of the localized polaron state:

N 1 τ− = W(p p0), (2.43) hop ∑ → p0=1

49 fundamentals where N is the number of nearest-neighbor ions; in one dimension N = 2. One may further determine the polaron mobility µdiff, which is related to the diffusivity D by the Einstein relation 2 44 µ = eD/kBT ( . ) where e is the elementary charge. Together with the hopping distance a – which may be identiﬁed as one lattice spacing – the diffusivity is written as

D = ga2W(p p 1). (2.45) → The parameter g depends on the dimensionality and crystal structure of the system under observation. For the one-dimensional MCM one has g = 1, several authors use a value 3 381 382 of g = /2 for real crystals, in particular for LNO [ , ]. The explicit form µdiff then reads 2 ad ea ωLO Ea 1 Ea µ = g exp ∝ T− exp (2.46) kBT 2π − kBT − kBT for adiabatic hopping, and

2 2 1/2 nad ea J π Ea 3/2 Ea µ = g exp ∝ T− exp (2.47) kBT h¯ 4EakBT − kBT − kBT if non-adiabatic hopping occurs. In the case of bound polarons the activation energy is given by Ea0 = Ea + WD. polarons in lithium niobate In LNO several sites and antisites are known to result in the formation of polarons [383–390]: 4+ the free small polaron NbNb; 4+ the bound small polaron NbLi ; 4+ 4+ the bipolaron NbLi :NbNb; and the free hole polaron O−. 2+ Additionally, polarons bound to impurity defects as e.g. FeLi have to be taken into account. As already discussed, iron is an impurity always present in conventional LNO material. Due to the large change in ion radius, it is expected such defects to form polaronic states [391]. Above the ODRT iron is expected to be present in the higher acceptor-state 3+ 4 FeLi , with an energy level above the NbNb+ energy level. It is further worthwhile to note, that bipolarons are only known to exist in reduced LNO [392]. The schematic orbitals of these polarons, being extracted from the typical valence of Nb, inside the LNO unit cell are depicted in Fig. 2.17. The polaron orbitals can be calculated by density ﬁeld theory (DFT). The physical shape of such orbitals has been discussed for strontium titanate/SrTiO3 (STO) [393, 394] and gadolinum titanate/GdTiO3 (GTO) [395]. As undoped CLN is known to contain NbLi free polarons will relax into bound polarons on the ps-timescale. As in Mg- doped CLN both VLi and NbLi are not present [396], different polarons at antisites as e.g.

50 2.3 lithium niobate and tantalate

MgLi would be possible, yet are energetically unfavorable [350], thus mainly free small polarons will contribute. Hence, in this work we will typically classify two cases: undoped CLN and heavily Mg-doped CLN. The energy levels of the electron polarons were investigated by time-resolved absorption spectroscopy [397–399], while up to now the energy levels of the hole polarons are unknown. Assuming a bandgap of 4 eV and energy levels of bound and free electron polaron of about 1 eV and 1.5 eV below the CB, one can deduce the energy of the hole polaron above VB from photoluminescence measurements as discussed further in Sec.4.5. The observed photoluminescence energy of 2.6 eV, thus would yield an energy level of the hole polaron of about 0.9 eV above VB. The energy levels of the up-to-now known polarons in LNO are depicted in Fig. 2.18.

Nb Li O vacancy

polaron orbital

4+ 4+ 4+ 4+ NbLi NbNb O –VLi NbLi – NbNb 2 17 4+ 4+ Figure . : Polarons present in LNO: bound electron polaron NbLi , free electron polaron NbNb, bound hole polaron O V , and bipolaron Nb4+:Nb4+. −− Li Li Nb

2.3.6 Ionic conductivity

The origin of the dark conductivity in as-grown undoped LNO at elevated temperature has been the subject of extensive investigation both upon DC [406–412] and AC [284, 413, 414]. First investigations revealed a proportional relationship between H+ concentration and conductivity of LNO crystals at elevated temperatures up to 1000 ◦C[415]. However, recent publications qualify the statement about the role of protons on the origin of the ionic conductivity. Yet, there is still a debate with some still supporting the explanation of protons to reveal electrical transport [416, 417], while others argue Li+ ions being accountable for it [418] due to both a smaller activation energy for SLN as compared with CLN as well as an absence of a signiﬁcant change in conductivity on the H+ concentration over a larger range [419]. Still, apart from the exact ion to result in the conductivity of LNO σ, it obeys an Arrhenius-type dependence, which illustrates its thermal activation

2 Ea Nq D Ea σ = σ0 exp = exp (2.48) − kBT kBT − kBT

51 fundamentals

2.6 eV 1.5 eV 1 eV 0.5 eV Fe3+ 4+ 2.5 eV NbNb

4 eV Nb4+ Li 2.6 eV =

g 4+ 4+ E Fe2+ Nb +Nb 3.1 eV Li Nb

0.9 eV

Figure 2.18: Sketch of conduction and valence bands and polaronic levels with colored arrows for UV interband excitation (violet), three-photon interband excitation (dark red), opti- 4+ cal absorption (black), NIR luminescence of bound polaron NbLi (red), luminescent 4+ recombination of free and hole polarons NbNb and hole polarons O− (green) and non- radiative relaxations (gray). The energy levels of iron impurities are added for sake of clarity. Values derived from: [400–405].

with Ea the activation energy, N the concentration of the charge carriers, and D the diffusion constant. The reported activation energies for CLN are in a range of 0.87 to 1.24 eV along different axes of the x and z crystallographic axes. [420]

52 3

METHODS

We see past time in a telescope and present time in a microscope. Hence the apparent enormities of the present. — Victor Hugo

In this chapter, we will present the sample preparation techniques, which includes both techniques to engineer domain patterns and speciﬁcally charged DWs as well as to create especially single-crystalline thin ﬁlms. We will introduce scanning probe microscopy techniques to visualize domains as well as to measure the local conductivity. Furthermore, we will present the technical conditions of the nonlinear-optical microscopy technique CSHG to three-dimensionally visualize the DWs and of an electro-microscopical method (PEEM) to retrieve the local surface potential.

3.1 sample preparation

In our measurements lithium niobate (LNO) and tantalate (LTO) samples with different dopants, doping concentrations, crystal cuts, stoichiometry, and producers were used. The list of materials summarizing these is given in Tab. 3.1.

3.1.1 Poling stage

We typically operated with crystals being purchased as single domain crystals. To generate a pattern of domains, and hence domain walls, it is a prerequisite to locally invert the ferroelectric order inside the material. As previously described in Sec. 2.1 the associated coercive ﬁelds in the investigated materials are relatively high. When operating with wafer-thick crystals it is necessary to apply a voltage in the order of kV to perform domain reversal. For this purpose a poling stage was used, which by proper insulating sealing avoids any discharges and breakthroughs from bottom to top contact around the crystal, otherwise being present in this voltage range. It has been described in more detail by Haußmann [228]. The poling stage requires the sample to have a size of at least 5 5 mm2. Hereby, the crystal is clamped by insulating O-rings into the plastic holder. × The voltage is applied through an opening in the upper plastic mount. Quartz windows were plugged in on both sides to both be able to visualize the domain growth process

53 methods

quartz window

holder V LNO/LTO sc A O-ring

HeCd laser UV beam

Figure 3.1: Poling device for random incomplete poling and laser-assisted poling. The electrodes necessary for field application are salt water liquid electrodes. optically and to illuminate the crystal with an UV laser. To be able to apply the required high voltage a voltage amplifier with a maximum voltage of 20 kV was used. random domain poling While ramping up the applied voltage the current is measured. At the coercive voltage current spikes are recorded. The domain growth is visualized through the quartz windows with polarization microscopy. The domain pattern growth supervised by PLM. The apparent sample-to-sample differences in stochastic nucleation are significant, therefore one rests upon the visual domain growth control. For specific measurements such as the multiphoton microscopy as well as time-resolved luminescence spectroscopy the complete are inside the O-ring salt-water electrodes are inverted one time. For a very homogenous random domain pattern multiple poling cycles were carried out until a homogeneous pattern was visible. Depending on requirements of the successive experiments small with either reverse or original polarization can be produced. laser-assisted poling Upon UV laser illumination the coercive field can be modulated locally. Therefore, local domain inversion can be realized upon an applied electric field well below the standard coercive field. This has been used for the creation of waveguides in various cases [252, 421–427]. The applied static electric fields were in the range of 20 to 50 % of the materials reported undisturbed coercive field. An image of the obtained domain inversion generated by surface second-harmonic generation microscopy is given in Fig. 3.2. The written lines of inverted polarization can be very well differentiated from the domains of as-grown polarization by the surface SHG reduction at the DW separating these areas. The stronger the incident UV laser power the wider the written line structure. Furthermore, for larger incident power crystallographically determined domain reversal can be observed. For the formation of these domain inversion lines the laser was moved with a constant velocity of 1 mm/s.

54 3.1 sample preparation

Table 3.1: Listing of all samples of LNO and LTO used in this work. # Sample Material Doping [mol%] Cut Source 1 CLN – z CrysTec GmbH 2 CLN Mg 1 z United Crystals 3 CLN Mg 2 z United Crystals 4 CLN Mg 3 z United Crystals 5 CLN Mg 5 z United Crystals 6 CLN Mg 5 z Yamaju Ceramics 7 CPPLN Mg 5 z EQ Photonics 8 CLN Mg 7 z Union Optic 9 CPPLN Mg 5 y EQ Photonics 10 CLT – z CrysTec GmbH 11 CLT Mg 7 z Yamaju Ceramics 12 CLT Mg 8 z Yamaju Ceramics 13 CLN – x E. Soergel1 16 SLN Hf 0.2 z L. Kovacs2 17 SLN Zr 0.5 z L. Kovacs 18 SLN Zn 1 z L. Kovacs 19 SLN Sc 0.5 z L. Kovacs 20 SLN In 1 z L. Kovacs 21 SLN Mg 1 z L. Kovacs 22 SLN Mg 2.5 z L. Kovacs 23 SLN Mg 4.5 z L. Kovacs 24 CLN – z NanoLN 25 CLN – z NanoLN 26 CLN Mg 5 z M. Schossig3

1 Institute of Physics, University of Bonn, Germany 2 Wigner Research Centre for Physics, Hungarian Academy of Sciences, Hungary 3 Institute of Solid State Electronics, TU Dresden, Germany

55 methods

y ~x

~x ~x

9.4 µW 54 µW 280 µW

50µm

Figure 3.2: Surface SHG image of UV laser written surface domains with different intensities. A weak CSHG contribution is present for domain walls some microns inside the crystal. The features of the laser written domains show crystallographic inﬂuence.

3.1.2 Thermal treatment

Besides the aforementioned sample preparation, influences by a specific thermal treatment were investigated. Hereby, the maximum temperature may or may not exceed the Curie temperature TC. While the treatment, the polarization inverts at only half the crystal along the polar axis and forms the aforementioned h2h DWs. A theory for this effect was given by Kugel and Rosenman [428]. The authors claimed this domain reversal to be due to a bias field as a result of a redistribution in the local oxygen vacancy concentration. It is possible to simulate the dynamics and the temporal evolution of the associated bias fields. Such a simulation for the bias field is given in Fig. 3.3. One can observe, that after 5 h the largest field gradient is central inside the 500 µm thick crystal, in good agreement to previous observation of this domain inversion in both CLN and CLT [429, 430]. The bias field easily exceeds the coercive field of only about 5 V/cm close to the Curie temperature TC. However, later researchers largely dismissed the occurrence of oxygen vacancies to be present in LNO, as already mentioned in Sec. 2.3.4 and no significant oxygen outdiffusion to happen and merely pinpoint towards lithium outdiffusing lithium [431]. Yet, the internal field to be created is very similar. However, in both cases the calculated bias field only exceeds the coercive field for temperatures very close to TC. Therefore, upon cooling the domain reversal would not be supported sufficiently to be stable if the electrostatic costs by the formation of the CDW were not bearable. However, the CDWs are very stable, especially upon heating. Probably, a bias field is still present across the DW, which would stabilize the DW. A similar effect, yet for 180° DWs, was already discussed by Dierolf and Sandmann in Er:LNO. Hereby, the electrical field was deduced from an apparent

56 3.1 sample preparation

600 h2h DW 10 min 400 ~ 30 min Ebias 60 min 200 [V / cm] 5h E 0

200

Electric field 400 ~PS ~PS 600 200 100 0 100 200 coordinate x [µm] 3 3 Figure . : Electrical field simulation for LNO being treated close to TC under the assumption of oxygen vacancy redistribution. As an effect of the in-built electrical field the ferroelectric order is inverted. The domains of inverted polarization grow towards the crystal center. Reproduced from [428].

3+ 4 4 Er luminescence energy Stark shift of the S3 I15 transition by 0.02 meV for both /2 → /2 as-poled and annealed samples (5 h, 250 ◦C) across the DW [248].

3.1.3 Ion slicing of LNO crystals

The previously discussed techniques are dealing with wafer-thick materials, hence resulting in very large voltages necessary to generate domain patterns, which in general is an impediment in the utilization of these. Yet, it was very difficult to fabricate thin-film material of LNO or LTO by evaporation techniques as ALD [432], PLD [433], MBE [434], laser ablation [435], sol-gel routes [436] resulting in thin-films with very different properties then known from single crystal wafers. Epitaxially grown LNO was investigated, yet being deposited on insulating substrates as e.g. sapphire and LTO [437–441], which can not be applied to gain insight into the electrical properties of the samples. However, a general difficulty to guarantee for single-crystalline properties for thinned LNO below the thickness limit of approximately 1 µm was overcome in the recent years by applying a technique primarily introduced to single crystalline silicon, called ion slicing or "smart cut", a schematic is given in Fig.3.4. Hereby, a high dose H+ or He+ ion implantation [marked as 2 ] is applied onto the crystal LNO A . At a well defined depth inside the crystal, well described by the Bethe- Bloch formula, the main ion momentum drop and thus ion accumulation occurs. After wafer bonding [marked by 3 ] to another LNO B wafer already coated with a metallic electrode as e.g. platinum or gold [marked by 1 ] as well as an insulating silicon oxide buffer layer one heats the crystal to a temperature as e.g. 400 ◦C [marked by 4 ], resulting in cleaving the topmost part of the ion-implanted wafer. After a chemical-mechanical polishing (CMP) process a smooth, single-crystalline LNO film is left with a thickness

57 methods

SiO2 Au/Pt + 1 2 He

LNO A LNO B I metal evaporation ion implantation

LNO B 3 4 z typical ion LNO A deposition ion slicing depth wafer bonding T 400 C ⇡ 5 6 600 nm ~ 100 nm PS 1 µm

chemical mechanical 500 µm polishing

Figure 3.4: Schematic of the fabrication steps to create single-crystalline thin-films of LNO. 1 Au/Pt evaporation on LNO A , 2 ion implantation of He+ into LNO B , 3 wafer-bonding of both LNO wafers A and B , 4 thermal annealing at temperatures about 400 ◦C, 5 chemical mechanical polishing (CMP) of the residual, 6 final stack of the thin-film sample. well below 1 µm. Fabrication of LNO films by this "Smart Cut" method was first reported by Levy et al. [442], still without a back electrode. In recent years, single-crystalline LNO films fabricated by ion slicing have attracted considerable interest [443, 444], as embedding of an ion-sliced film in low-index insulator substrate provides a high-index- contrast optical waveguide. As ion slicing is based on high-dose implantation into a bulk crystal by H+ or He+ ions and subsequent cleaving of the formed film from the crystal it is very important to control the vertical distribution of the implantation dose very precisely, otherwise resulting in an undesired deterioration of the film. To investigate whether changes due to strain as well as treatment related changes in the lattice constant occurred XRD measurements were carried out, which confirmed the lattice constants to only deviate by 0.001 Å from the literature wafer-size values of congruent LNO. A typical application for these LNO films based devices is optical frequency conversion, especially on ferroelectric domain patterns, based on the principle of quasi-phase matching (QPM). Radojevic et al. [445] prepared free-standing periodically poled LNO films by ion-slicing PPLN crystals. In this thesis, we utilized the domain formation in these ion-sliced LNO films with an additionally included electrically conductive metallic layer as platinum and gold. Domains were created by local poling as introduced in Sec. 2.2.5 with a biased AFM tip. The samples were generously provided by NanoLN Elec- tronics (Jinan, China). Further details of fabriaction as ion dose and kinetic energy of the implanted He2+ ions or thermal treatments after CMP are kept as company secret. Similar

58 3.2 atomic force microscopy results on the local poling of similar samples distributed by the very same company have been reported just recently by Gainutdinov et al. [446]. The resistive switching behavior after reduction of similarly fabricated exfoliated LNO has been observed after annealing in vacuum and Argon environment at 573 K [447].

3.2 atomic force microscopy

For microscopic resolution of a multitude of physical properties one can nowadays rely on a range of scanning probe or atomic force microscope (AFM) techniques. In this work, mainly two AFM systems have been in use, the Asylum Research Cypher S and the AIST- NT SmartSPM™1000. These made it possible to render the topography in tapping mode as well as contact mode AFM, to visualize the ferroelectric domains by piezo-response force microscope (PFM), the conductivity by conductive atomic force microscopy (cAFM), the alteration in microwave impedance properties by scanning microwave impedance microscope (sMIM) and to extract the local coercive voltage by switching spectroscopy PFM. In general, atomic force microscopy relies on the continuous scanning of a probe tip in- teracting with the sample. The atomic force microscope setups used in this work are of cantilever-type. Hereby, the tip position is obtained by the deflection of a laser beam focused onto the lever. The reflected beam is recorded in a four-quarter photodiode and the difference signal between both upper and lower photodiodes are amplified and is used for the analysis of either deflection or oscillation amplitude.

3.2.1 Non-contact and contact mode AFM microscopy

The most widespread AFM microscopy technique to render the topographical profile of the investigated sample is tapping mode. Hereby, the AFM cantilever is oscillating in its mechanical resonance, excited by the alternating deformation of a piezo plate. Upon approach towards the sample surface the attractive forces between tip and sample will alter the resonance frequency and thus decrease the oscillation amplitude. The proper appellation tapping mode is derived from the fact, that physical contact of sample and tip cannot be excluded, i.e. the tip to get into the repulsive regime. The alterations in the tip deflection amplitude are monitored and controlled to remain at the set point. For samples, that are sufficiently robuste i.e. not get peeled off easily contact-mode AFM can be applied. Here, the deflection is the value being controlled to remain at the set-point value. As this condition is sufficiently fulfilled for the materials being investigated here, the topography information is extracted from scans in contact-mode. An important parameter for AFM systems is the field of view allowed by the scanner. The Cypher S system is restricted to a maximum scan area of 30 30 µm2, whereas the SmartSPM™1000 delivers a scan area of × 100 100 µm2. Typical scan rates ranged from 0.1 Hz to 5 Hz with resolutions of 256 256 × × to 1024 1024. The vertical topography resolution being achieved was about 2 Å. ×

59 methods

3.2.2 Piezo-response force microscopy (PFM)

Piezoresponse force microscopy (PFM) delivers information on the piezoelectrical properties of the sample [448–452]. A schematic description is given in Fig. 3.5(a). As already discussed in 2.2.5, vertical and lateral deflection are acquired under the application of an AC voltage by lock-in amplifiers. The AC voltage, as delivered by a function generator, is applied on the AFM cantilever. The resulting deflection is detected with the four-quadrant photodiode and the differential signals, associated with a vertical and lateral oscillation due to the converse piezoelectric effect of the sample, are pre-amplified. The lock-in amplifiers compare these deflection signals with the reference signal and deliver both amplitude and phase of the deflection oscillation. In the system we applied, two built-in lock-in amplifiers as well as frequency generators are used to achieve both in-plane and out-of-plane PFM. The PFM operation is done close to the mechanical resonance frequency of the tip- sample system fres.

3.2.2.1 Advanced PFM modes Several additions have been made to PFM that substantially increase the ﬂexibility of the technique to probe nanoscale features [453, 454]. A typical issue in PFM is the instability of the resonance frequency while scanning, e.g. by gradually blunting the tip apex. Such a change has a drastic change both in amplitude and phase information, which can not be corrected afterwards. Therefore, in our measurements a technique was used to compensate for it, the so-called DART™PFM mode. Another issue is the acquisition of local ferroelectric hystereses of the ferroelectric system. dart™pfm Recalling that in PFM an AC bias of a certain frequency causes a deformation of the sample material at that same frequency the system can be considered as a driven harmonic oscillator. As such there exists a resonance as a function of driving frequency. This effect has been exploited in PFM to provide an enhancement in the signal, thus allowing for a higher signal-to-noise or similar signal-to-noise at lower driving bias amplitude. Typically this contact resonance is in the kilo- to megahertz range and typically about 3 to 5 times larger than the free-air cantilever resonance. However, a major drawback is the contact resonance to depend not only on the dynamic response of the cantilever but also on the elastic modulus of the sample material the tip being in contact with and thus may be altered during scanning. One method of bypassing the inherent disadvantages of contact resonance PFM is to change the driving frequency in order to shadow or track the changes in the frequency of the contact resonance [455]. This method, being made commercially available by Asylum Research, is called Dual AC Resonance 3 5 1 1 Tracking (DART™) [see Fig. . (b)] uses two limit frequencies f1 , f2 on either side of the contact resonance peak and so can sense changes in the peak position, when the probe is scanning over the sample to another spot, resulting in a different oscillation amplitude. It is then possible to compensate for these alteration by altering the AC bias driving fre-

60 3.2 atomic force microscopy a) B Â A A four-quadrant photo diode

Â B laser diode function generator D Â C C foop Â Â D oop ⇠

Aoop, joop differential ip Lockin A ampliﬁer

+ fip (A + B) (C + D) out-of-plane (oop) deﬂection ⇠ Aip, jip + Lockin B (A + C) (B + D) in-plane (ip) deﬂection

b) laser diode fc f1 Â ⇠ Ref.

1 2 A1, j1 Lockin A Recorder fc fc f2 Controller cantilever deﬂection ⇠ Ref. 1 2 fc fc A2, j2 A 1 2 A2 Lockin B A1 2 1 A A1 2 f1 = fc D f ! 1 2 A1/A2 = const. f2 = fc + D f

1 2 1 2 f f1 f1 f2 f2 c) VDC 1 j 2 j 3 poling repoling off p/2 VC, f p VC,b

VC,b VC, f VDC on t 0 p/2 t f tb t t f tb t Figure 3.5: a) Schematic of vector PFM with parallel in-plane and out-of-plane deflection detection with two separate lock-in amplifiers detecting the signal. Operation is separated in frequency to reduce mechanical crosstalk b) Schematic of oop DART™PFM with a controlled center frequency. It can be operated either in vertical or in lateral mode. The samples is excited with two AC frequencies close to each other. The resulting beat is analysed by two lock-in amplifiers. A control loop is used to satisfy the ratio of both oscillation amplitudes to remain constant over the scan by altering the center frequency. c) 1 Switching spectroscopy (ss) PFM to obtain local hysteresis curves. The PFM phases are analyzed in the off-case (VDC = 0), 2 polarization over the measurement cycle, 3 extracted ssPFM hysteresis loop.

61 methods quency correspondingly in order to maintain the signal boost that results from the contact 2 2 resonance and change to f1 , f2 . switching spectroscopy pfm In this technique the area underneath the PFM tip is switched with simultaneous acquisition of a hysteresis loop that can be analyzed to obtain information about the sample’s ferroelectric properties[456, 457]. A series of hysteresis loops can be acquired across the sample surface in order to map the ferroelectric homogeneity of the sample. The voltage cycle applied to the tip is drawn in Fig. 3.5(c). The resulting piezo-response, i.e. amplitude and phase of the deﬂection, are recorded in both on- and off-state. From the temporal evolution of both parameters in off-state a hysteresis loop can be extracted. Moreover, in this way a spatial distribution of switching properties such as coercive voltage, remanent polarization, imprint and work of switching amongst others can be displayed in which each pixel displays the desired data from the hysteresis loop acquired at that point. This allows spatial comparison switching properties with sample topography. In the case of the ferroelectric single crystal thin ﬁlm addressed in 3.1.3 the homogeneity can be cross-checked.

3.2.3 Conductive atomic force microscopy (cAFM)

In conductive atomic force microscopy (cAFM) the current conducted through the tip is measured. To achieve very high current sensitivity an I-V-converter located close to the cantilever chip is used. Their current noise floor was 1 pA and 200 fA for the Cypher S and the SmartSPM™1000, respectively. Apart from scanning the tip across the materials surface it is similarly possible to record local current-voltage-curves, which was especially helpful at the CDW. The internal voltage output served in both cases up to 10 V. The Cypher S was additionally equipped with a high-voltage-amplifier, which could deliver voltages up to 150 V. A main difficulty for cAFM is a very good contact condition as well as a small sheet resistance. To fulfill these prerequisites pure-platinum wire tips were applied. The macroscopic I V measurements were taken with W tip contacts inside a − Lakeshore PS-100 four-terminal cryogenic probe station and a Keithley 2400 sourcemeter.

current ampliﬁer

+ - A

Figure 3.6: Schematic of the conductive-type atomic force microscopy (cAFM).

62 3.2 atomic force microscopy

3.2.4 Scanning microwave impedance microscopy (sMIM)

Scanning microwave impedance microscopy (sMIM) is one realization of scanning nearfield microscopy, first being described by Synge [458, 459], which has found widespread use in the recent years, especially for the investigation of 2D materials such as MoS2, WSe2 or GaN, and materials with modulated conductance as strained manganate films or even coal [460–471]. Hereby, the interaction strength of a microwave locally impinging into the sample at the tip-sample area is monitored by the reflected microwave. As the impingement below the tip into the material is very limited, e.g. only about 100 nm for silicon with a conductance of 1 S/m and the spatial distribution to be determined primarily by the tip radius, e.g. 50 nm the spatial resolution is far better than expected from Abbe’s diffrac- 1 tion limit . E.g. assuming a resolution of 60 nm at an operating frequency of 50 MHz, which corresponds to a wavelength of 6 m in air a figure of merit of the resolving power of λ/108 can be reached. Since, the tip-sample interaction takes place at the length-scale smaller than the wavelength, one can drop the spatial dependent term and represent the interaction using an equivalency circuit as depicted in Fig. 3.7(a). In the reflection mode, microwave microscopy effectively measures the reflection coefficient S11, which is defined as

ZL Z0 3 1 S11 = − ( . ) ZL + Z0

with Z0 being the characteristic impedance of the transmission line, typically 50 Ω and ZL being the load impedance, i.e. the tip-sample impedance. ZL can be related to the complex permittivity, effectively resembling the dielectric constant ε and conductivity σ of the sample. These quantities are important for the study of condensed matter systems, such as insulators, semiconductors, and metals and the nanoscale. With the silicon chip or cantilever mount to bear a certain impedance the frequency has to be tuned to find a impedance matching condition, which is done by the insertion of a matching network very close to the tip. Ideally the frequency dependence can be simply drawn as circle in a Smith chart and close to an impedance matching condition S11 follows a curve given in Fig. 3.7(b). sMIM measures the detune of this free-standing case in close contact with the 3 7 sample. In Fig. . (c) a real case of the measured absolute value of S11 for a frequency sweep is given. It may seem obvious the highest signal-to-noise ratio to be achievable at this frequency fres, as any change in S11 due to tip-sample interaction results in larger relative change, however, as the signal background is anyway cancelled by attenuators and phase shifters the influence is not as strong as long as a dip of approx. 20 dB are − reached. At the resonance the highest field can be achieved at the tip apex for an antinodal boundary condition, which is important to obtain a large sample-related backscattering. For a number of frequencies the impedance matching can be sufficiently fulfilled. In order to

1 According to Ernst Abbe the resolution maximally achievable with microscopy by means of electromagnetic waves is given to be d = λ/2NA with NA the numerical aperture of the microscopy system. Techniques which undergo this limit and show a resolution of better than λ/2 are called superresolution techniques.

63 methods

a) matching coupler attenuator network bias tee coaxial cable micro- I wave Â j ⇠ source I phase-shifter mixer ref. cancellation R (S ) (S ) C f < 300 kHz⌘< 11 = 11 ⌘ lock-in A lock-in B Z D tip DC R Ref. ⇠ AdR/dV, jdR/dV AdC/dV, jdC/dV

0 b) (S11)[dB] c) d) = = < 10 background

f [dB] 20 nearﬁeld

res 11 f (S11)[dB] sMIM 0 < S component -40 -20 0 20 40 30 fres

40 2.8 3 3.2 0 0.5 1 1.5 2 f [GHz] d [µm] e) f) g) 1 1 15 = 10 x z [nm] sMIM sMIM [a.u.] reference 5 value 0

I10[a.u.] µm I10[a.u.] µm 0 2 4 6 8 10 0 1 0 sMIM 1 sMIM x [µm] = = h) i) 1 metallic tip ! 0.8 # air [a.u.] 0.6 150 nm ~ x E i | ⇡ z | h 0.4 material with 0.2 300 nm s = 1S/mand #r = 4 ⇡ 0 0.5 1 1.5 2 I2 µm[a.u.]0.2 1 0 ~E 1 x or z [µm] h| |i Figure 3.7: a) Schematic of the scanning-type microwave impedance microscopy (sMIM) with the local capacitance and resistance measurement under the tip b) Smith chart of frequency dependence of the impedance matching c) transmission line absorbance S11 d) retract curves of the imaginary and real part components of S11, showing clearly nearfield behavior e) raw sMIM data of reference sample consisting of Al dots on silicon oxide f) line corrected data to e) g) comparative section of sMIM and topography data h) FEM simulation of the time-averaged electric field in the tip-sample region driven at a frequency of f = 3 GHz i) sections of the time-averaged electric field in lateral and vertical direction with marked HWHM.

64 3.2 atomic force microscopy change the matching frequency it is possible to tweak the shunt capacitor in the matching network, being checked with a vector network analyzer. The microwave sources, being set to this resonance frequency fres, serve hereby both as the excitation being coupled into the coax cable, as a reference for cancellation of the backscattered background as well as a reference of the mixing to acquire imaginary and real part of the back-scattered microwave. As the phase shifter in the mixer can only reach a change in phase of π/2 four different contrast representations are possible. Therefore, a reference sample like Al dots on silicon oxide (SiO2) as well as p- or n-doped Si samples are necessary to define the sign of change in the measurement. Imaginary and real part of S11 are associated with the capacitive (C) and resistive (R) properties of the sample [463]. A metallic or capacitive sample reflects the microwave with a phase change and therefore gives a signal in the imaginary channel. The two channels are thus similarly labeled sMIM-C and sMIM-R. The signal itself is proportional to the admittance of the sample. The admittance of the sample was simulated three-dimensionally with the finite-element-method solver COMSOL Multiphysics v.5.0 equipped with the AC/DC module for different conductivities of the domain wall, domain wall thicknesses, excitation frequencies fres and different tip geometries being in use. Moreover, the profile of sMIM signals scanning across a CDW as well as retract curves [experimental results shown in Fig. 3.7(d)] were simulated. Apart from the fundamental sMIM-technique, it is possible to add a bias tee, which makes it possible to additionally apply a DC or low-frequency AC bias voltage between the tip and sample. A change under AC bias can be monitored using two lock-in amplifiers to acquire both the alternating capacitive and resistive part of the signal, i.e. dC/dV and dR/dV, which is possible as the AC frequency is orders of magnitude smaller than the lower bandpass filter limit of the channel input. The phase information of both can typically answer the question of the majority charge carrier, whereas the amplitude delivers information on the charge carrier density [472, 473]. For the measurements mainly two types of AFM tips were used: 1) shielded tips [see SEM image in Fig. 3.8 2 ][474–476], 2) pure-platinum etched wire tips [see SEM image in Fig. 3.8 3 ][477]. In the measurements several matching networks were used to span a large frequency range: A 20 to 50 MHz B 0.8 to 1.2 GHz C 2.7 to 3.3 GHz D 3 to 20 GHz with a λ/2 resonator as matching network with tip-dependent fixed impedance matching frequencies Hereby, the case C was realized by the use of the commercial system as distributed by PrimeNano. In this case the microwave output power was set to 100 µW ( 10 dBm), − resulting in an AC amplitude of about 300 mV [478]. In the other cases A , B and D self-built sMIM electronics was plugged to a Park Systems XE-70 AFM in the group of Prof. Shen at the Department of Physics, Stanford University with both varying excitation powers and especially AC amplitudes on the tip as being strongly depending on the matching conditions. To compensate large-range variations in the signal as visible in

65 methods

Fig. 3.7(e), which are the result of cantilever-sample interaction, line averaging methods are used. To compensate for these both matching-network-dependent but similarly tip- dependent variations, the reference Al dot sample as shown in Fig. 3.7(f) was used and sMIM-C was calibrated as schematically shown in Fig. 3.7(g). The main reason for the small frequency window in case C is the very large load attributed to the cantilevers in use as distributed by PrimeNano. For etched-platinum- wire tips bear a smaller load and thus the accessible frequency range increases. Besides the inspection of local charge carriers being mobile enough to follow high- frequency excitation, one reason to use microwave frequencies is the distinctly large contrast in response between metals and insulators at microwave frequencies, compared to the optical frequency regime. This gets obvious, when comparing the dispersion relation of a dielectric insulator to a good conductor [479]. For dielectric materials or insulators, this is given by

2 2 3 2 k0 = ω ε0(ε0 + iε00)µ ( . ) with ε00 the loss of the dielectric, which is small compared to the real part of the dielectric constant ε : ε + iε ε . E.g., for silicon dioxide SiO this number is 4. For conductors it 0 0 00 ≈ 0 2 holds σ k2 = ω2ε ε + i µ (3.3) 0 0 ωε E.g. for aluminum at 1 GHz, σ/ωε = 6.3 107. Therefore, the imaginary part dominates 0 × and is large compared to 4 for SiO2. Thus, large contrasts between aluminum and SiO2 are expected in microwave microscopy. Yet, for the optical frequency, f > 1014 Hz, the contrast reduces by at least 5 orders of magnitude [480, 481]. Another advantage of sMIM is the ability to look through transparent material [482] as in the case of super-lenses in infrared nearﬁeld microscopy [483]. Another realization of microwave inspection, scanning nonlinear dielectric microscopy (SNDM), as introduced by Cho cannot differentiate between resistive and capacitive changes in the material [484, 485], yet, alterations in the dielectric permettivity in different domains can be visualized by monitoring the demodulated change in the resonance frequency upon a slow bias AC voltage.

3.2.5 AFM probes

In this work, different types of AFM probes served to characterize the samples, mainly specific for a certain AFM mode. For tapping-mode AFM short and stiff cantilevers were applied with a free oscillation resonances around 300 kHz. The cantilevers were equipped with an additional high-reflectance coating to increase the deflection signal. For PFM less stiff cantilevers were applied, to both sufficiently guarantee for tip-sample contact, but also to achieve sufficient cantilever deflection. To operate PFM an additional prerequisite is the conductivity of the probe, therefore we either used platinum/iridium (Pt/Ir) coated silicon cantilevers or platinum-wire etched tips. In general, with Pt/Ir silicon probes a higher spatial resolution can be obtained. For cAFM the platinum etched-wire tips

66 3.3 laser scanning microscope resulted in superior current detection for several samples. Therefore, we operated cAFM solely with these probes. sMIM was operated both by platinum etched-wire tips and shielded tips, both being commercially available. The list of AFM probes being used is given in Tab. 3.2.

Table 3.2: AFM probes used in this work. Tip Height Cantilever Spring Const. Resonant Manufacturer Name Mode [µm] Length [µm] [N/m] Freq. [kHz] BudgetSensors Tap300Al-G 17 125 40 300 KPFM Bruker SCM PIT 8 200 1 75 PFM Bruker FMV 10 250 1 75 PFM RMN 25Pt300B 80 300 18 20 cAFM RMN 25Pt400B 80 400 8 10 cAFM PrimeNano 150 5 150 8 75 sMIM PrimeNano 300 5 300 1 19 sMIM

1 2 compressed 3 cantilever

Pt/Ir Pt TiW

Figure 3.8: A selection of some representative AFM probes used in this work imaged by SEM. 1 Bruker SCM-PIT (silicon cantilever with a 10 nm platinum/iridium (Pt/Ir) coating, tip radius: 20 nm), 2 Rocky Mountain Nanotechnology (RMN) 25Pt400B (etched platinum wire with a ﬂat cantilever by compression, tip radius: 20 nm), and 3 PrimeNano 300 (fully-shielded cantilever with a gold (Au) shield and a titanium/tungsten (TiW) tip, tip radius: 50 nm). Scale upper row: 10 µm, lower row: 2 µm.

3.3 laser scanning microscope

Confocal laser scanning microscopy (CSLM) is a technique to obtain high spatial resolution optical three-dimensional visualization. The key feature of confocal microscopy is its

67 methods ability to acquire in-focus images from selected depths, which is mainly enabled due to projection onto a back-focal aperture, only to pass light which stems from the very focal volume. Point-by-point reconstruction of the acquired back-reflected light intensity allows for three-dimensional visualization of topologically complex objects for non-opaque specimens, as e.g. LNO. For interior imaging, the quality of the image is greatly enhanced over simple transmission or reflection microscopy as the image information from multiple depths in the specimen is not superimposed. A conventional reflection-type microscope observes as far into the specimen as the light can penetrate, while a confocal microscope only observes images one depth level at a time. In effect, CLSM achieves a controlled and highly limited depth of focus, given by the point-spread-function (PSF).

2 6 3 1 8 5 1 Femtosecond laser 2 Electro-optical modulator 3 Galvo-mirror pair 4 Scan optics 6 5 Microscope objective 6 Multiphoton blocking ﬁlter 7 Transmission photomultiplier (T-PMT) 8 Reﬂection photomultiplier (NDD) 7

Figure 3.9: Schematic of the Leica SP5 MP laser scanning setup being in use. The excitation laser is a SpectraPhysics MaiTai laser 1 , whose power is altered by an electro-optical modulator 2 . The focal position inside the sample is modulated and scanned by a galvanometric mirror pair 3 . The beam is weakly focused by scan lenses 4 , and tightly focused into the sample with interchangeable microscope objectives given in Tab. 3.3, 5 . The excitation laser light is blocked in transmission and reflection direction by Semrock multiphoton filters 6 . The SHG is finally detected by photomultier tubes in forward 7 and backward 8 direction, called transmission photomultiplier (T-PMT) and non-descanned detector (NDD).

In 1978, Thomas and Christoph Cremer created a laser scanning process with three- dimensional detection of objects, scanning a surface point-by-point using a focused laser beam. In this work, we use a particular advancement, even though no back-focal aperture

68 3.3 laser scanning microscope

a) b)

z y x

Figure 3.10: a) Representative CSHG microscopical scan inside a LNO crystal with randomly distributed domains, b) sketch of the stacking process with a typical vertical stacking separation of about 1 µm. is plugged into the beam path, still the emitted and detected light stems only out of a very limited focal region. This technique is called multiphoton microscopy. All main components for multiphoton microscopy are similar to those applied in conventional laser scanning microscopy, apart from an ultrafast laser invoking the nonlinear optical processes as e.g. SHG. As nonlinear-optical processes will only be initiated in the very focal region, the SHG detectable has to stem from its close vicinity. Therefore, the attainable resolution can be similar to confocal laser scanning microscopy and mainly depends on the nonlinear- optical wavelength, e.g. the second-harmonic. For the experiments, a commercial multiphoton laser scanning microscope (SP5 MP, Le- ica) at the Light Imaging Facility of the Center for Regenerative Therapies Dresden (CRTD) was used, being equipped with a tuneable Ti:Sapphire laser (Mai Tai BB, Spectra Physics) at a frame rate of about 1 Hz (typical Galvo mirror pair line scanning rate of 600 Hz). The laser provides 100-fs-pulses at < 13 nJ pulse energy with 80 MHz repetition rate; the center wavelength can be tuned from 780 to 990 nm. The laser beam is focused through an objective. The microscope objectives used in this work are given in Tab. 3.1, which mainly differ in magnification, but also in the free working distance and the immersion liquid being applied. Hence, different lateral and axial resolutions can be realized. The laser focus is stirred by a galvanometric mirror pair. The emitted SHG signal is collected both in forward direction using a condenser lens (0.55 NA) as well as in backward direction by photomultipliers. The fundamental light beam is effectively blocked off the detector using a fluorescence filter (SemrockF75-680). We record images of the second-harmonic radiation at predefined focal depth, from the surface and up to few hundred nside the crystal. A representative scan for the case of LNO is given in Fig. 3.10(a). The images are stacked to gain three-dimensional domain or rather domain wall visualization follows the sketch given in Fig. 3.10(b). Initially, a piezo scanner system was used for the experiments, which resulted in a too slow image capture necessary for 3D visualization, yet, facilitated

69 methods

Table 3.3: Microscope objectives used in this work by magniﬁcation, numerical aperture (NA), immersion medium, free working distance (FWD) as well as lateral and axial resolution attainable under best condition (RESxy, RESz, respctively)

Type Magniﬁc- NA Immersion FWD RESxy [µm] RESz [nm] ation [nm] at λ = 990nm at λ = 990nm CFI APO TIRF 100 1.49 oil 130 110 200 × HCX PL APO CS 100 1.4 oil 90 139 236 × HCX PL APO blue 63 1.4 oil 100 139 236 × HCX PL APO lambda 63 1.2 water 300 139 236 × HCX APO U-V-I 40 0.75 dry 280 260 649 × HCX IRAPO L 25 0.95 water 2400 205 550 × HCX PL APO CS 10 0.4 dry 2200 488 2630 ×

the proof of CSHG emission, visible by the bright CSHG beams. This setup was described in [486]. CSHG generally proved to be non-invasive in all our investigations on CDWs as long as the pulse energy was kept below 2 nJ at 80 MHz repetition rate. However, for doses exceeding 6 nJ at wavelengths below 930 nm, three-photon-absorption processes may in- voke laser ablation when focusing the laser spot at the crystal surface, leading to severe deteriorations, examples of such deteriorations as well as bubble formation on the surface are given in [487]. The microscopical investigations inside LNO and LTO crystals have to be operated in non-index matching condition, due to their high indexes of refraction, as given in Sec. 2.3. Non-index matching results in a decrease in SHG for larger focal depths due to focus broadening, both axially and radially. These artifacts are compensated with local contrast normalization, in general applied for illumination speciﬁc image inhomogeneities. For the image acquisition and postprocessing we use the free imaging software compilation Fiji v. 2 1.0 of ImageJ v.1.49n [488, 489] . The associated radial resolution reduction does not affect the further super-resolution analysis, yet a decrease in axial resolution limits the maximal working distance, depending on the numerical aperture of the applied microscope objective. Moreover, the non-index matching condition also results in a focal shift within the material [306], which can be treated by

1 tan sin− (NA/n ) NA 0 n 1 → 2 3 4 f1 = 1 f0 f0 ( . ) tan sin− (NA/n2) ≈ n1 Thus, all reconstructed images have been corrected according to eq. (3.4). The aforementioned model is yet very fundamental. More accurate treatments of the focus deterioration e.g. using Fermat’s principle as delivered by Hell et al. [305], especially the inﬂuence of anisotropic point-spread-functions for large focal depths, are not investigated. Further- more, inﬂuences due to birefringence, which can result in the formation of two separate

2 For digital image stabilization of the tree-dimensional image stack the compilation was augmented by the template matching plugins [490].

70 3.4 time-resolved luminescence spectroscopy

a) b)

1 1

25 µm I 25 µm I

8 c) 1 backward 6 forward [a.u.] I 4

intensity 2

0 0 2 4 6 8 10 x [µm] Figure 3.11: Comparison of a) forward propagated and b) backward scattered CSHG detection in a 7 % Mg:LTO sample with random domains by T-PMT and NDD at the Leica SP5 MP setup, respectively, c) representative cross section, which reveals, the backward scattered CSHG to show a smaller background and a higher resolution of DW features. foci, are neglected [491]. After local contrast normalization Abbe limited Gaussian blurred DW reproduction is realized. Starting with the locally normalized three-dimensional stack a contour surface can be created. The projection onto the polar axis of the local normal vector of this by triangulation spanned DW surface reveals the DW surface polarization charge according to eq. (2.6). For this purpose we use the gradient surface and contour coloration of the image visualization software ParaView v4.3.1 [492].

3.4 time-resolved luminescence spectroscopy

All time-resolved luminescence spectroscopy (TRLS) experiments were carried out at the Institute of Resource Ecology of the Helmholtz Zentrum Dresden Rossendorf (HZDR). The sample was illuminated with an UV excitation pulses as delivered by a cascaded frequency-doubled Nd:YAG laser system (Continuum Minilite II, Continuum Electro Op- tics) with an operating wavelength of 266 nm, pulse energies of up to 5 mJ, pulses width of

71 methods

4 ns and a repetition rate of 10 Hz. The trigger input of the detection system was coupled directly to the trigger output of the laser. The luminescence signal was collected with an optical quartz ﬁber and directed into a spectrograph (iHR 550, HORIBA Jobin Yvon) by means of an optical ﬁber cable, and then detected by an ICCD camera system with 1024 illuminated pixels and built-in delay generator (HORIBA Jobin Yvon). The sample was mounted within a liquid nitrogen cryostat and cooled down to 100 K in dry nitrogen gas at atmospheric pressure. A schematic of the setup is given in Fig. 3.12. LN crystal

N2 vapor inlet ns-UV laser cryostate at T [100 . . . 200 K] excitation spectrograph trigger quartz ﬁber

gate control ICCD probe trigger_ [t, t + Dt] I(l, [t, t + Dt]) I(l, t) ⇡ Figure 3.12: Schematic of the time-resolved luminescence spectroscopy setup.

The whole functions and setting of the laser spectrometer were controlled by the computer software LabSpec5 (HORIBA Jobin Yvon). The TRLS spectra were recorded in a wavelength range from 250 to 650 nm averaging over 50 laser shots for each single spectrum. The gate time was adaptively set with a minimum value of 2 ns. The delay was varied from 1 ns to 20 ms.

3.5 energy-resolved photoelectron emission spectromicroscopy

To extract the local surface potentials as well as the energy levels and local chemistry, energy resolved photoelectron emission spectroscopy was executed at the Leibniz Insitute for Solid State and Materials Research Dresden (IFW) by both Andreas Koitzsch and Anna- Sophie Pawlik. For this purpose, a Lab-based ARPES system was utilized (NanoESCA, Omicron NanoTechnology) [see schematic in Fig. 3.13]. Thereby, it was possible to separate the emitted photoelectrons upon either X-ray (Al Kα line with E = 1486.295 eV) or UV illumination (UV laser at E = 6 eV, Hg arc lamp with E = 5.2 eV) according to their kinetic energy and project them onto the imaging screen without severe aberration errors using projection lenses and energy ﬁlters [493]. In the microscopical investigations, the secondary electrons were the observation tar- gets, as non-scattered electron, i.e. primary electron, emission is almost negligible [494]. To be more precise, electrons loose almost their entire energy by numerous inelastic scat-

72 3.5 energy-resolved photoelectron emission spectromicroscopy

Energy Filter CCD

MCP Screen Excitation

XPS

Sample CCD

Emission Projection lenses

Figure 3.13: Schematic of the Lab-based ARPES system (NanoESCA, Omicron Technology), which was used both for photo-electron emission spectromicroscopy as well as X-ray photoelectron spectroscopy. tering events, being very likely in a dielectric material. Hence, one can observe a very strong emission only for very small kinetic energies. Knowing the photon energy of illumination, the minimum kinetic energy of emitted electrons reveals the work function of the material. In insulating materials, this similarly incorporates influences of an electric charge-up. As electrons are insufficiently reposed in insulators, their emission results in a positive charge-up of the surface. This can finally result in a net-negative energy by a pulling ,i.e. negative, bias field. For conductivity modulated materials this technique makes it possible to contactless analyze the surface potential and thus reveal an image of the conductivity distribution inside the material. This feature is particularly interesting, in the case of CDW. Charge-up modulation as in the case of embedded CDWs in an insulating bulk domain can result in a reduction of the electron microscopy by bending of the emitted electrons [494]. The materials charge-up can be reduced by the creation of very small windows [495]. In LNO a small advantage in this case was observed limited by the small surface photoconductivity. By localizing an aperture with a size as small as 1 µm local XPS measurements are possible to investigate chemical differences between the DW and domain regions, which could be a result of ionic screening of CDW.

Part II

EXPERIMENTS

RESULTS

It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong. — Richard Feynman

In this chapter, we will present the outcome of the conducted investigations on DWs. We will start with the presentation of three-dimensional visualization results for differently Mg-doped LNO and the comparison with PFM measurements. We will present embedded domains, which are a result of multiply inverted crystals. We will show defects in periodically poled crystals, which give locally inclined DWs. Furthermore, typical DW configurations in LNO by AFM-writing, thermal treatment and laser-writing are presented. We will show typical polarization charge textures for various cases and introduce a faster SHG visualization by quasi-phase matching. The local surface potential is extracted from photo-emitted secondary electron microscopy and compared with SHG visualizations. A comparison of local inclination and surface potential reveals a good match. UV-excited polaron transport is investigated by the radiative recombination of electron and hole polarons and the activation energy of the thermally activated polaron hopping is extracted. The local conductivity in high-voltage treated LNO was compared and found to be connected with the local inclination angle by CSHG close to the surface. The local conductivity in thin-film LNO was investigated, revealing a rectifying behavior. Conductive DWs are tested for applicability in resistive-switching memories. Very abrupt current switching over about 5 orders of magnitude is observed. Upon domain reversal the stack can be switched back into a highly resistive state. Cyclic tests reveal a high endurance and retention. The conduction type is analyzed and a space-charge limitation is found. Temperature-dependent measurements reveal a thermally activated conductance with activation energies similar to theoretical values for bound polaron hopping. The macroscopic current were emulated by a local probe, memristivity was observed. The DW impedance in the GHz-regime was investigated by microwave impedance microscopy. Do- main wall conductivities of about 100 to 1000 S/m for different geometries are extracted from a comparison with simulated profiles across the DW.

77 results

4.1 three-dimensional profiling of domain walls

As we have discussed in Sec. 2.1.3 the shape of ferroelectric DWs and its associated polarization charge texture alters a large range of properties as e.g. electric, piezoelectric, ﬂexoelectric, and acoustooptical as compared to the bulk. In this section, we will present typical DW shape examples, that were created and can be present in either doped or undoped LNO and LTO single crystals and, thus, brieﬂy showcase the possibilities CSHG microscopy offers to obtain a 3D image of such shapes.

4.1.1 Randomly poled LNO and LTO domains

We commenced our systematic investigation with thick 0 to 5 % Mg:CLN single crystals. A random domain configuration is achieved through incomplete poling as already explained in Sec. 3.1.1. A typical domain pattern of such 180° DWs is shown in Fig. 3.10(a) being rendered by the help of CSHG. In an earlier work by Schröder et al. an averaged inclination angle α for these samples was deduced by a comparison of the effective surface area of through domains at the c+ and c−-faces, respectively, using PFM [see Tab. 2.1], which is certainly correct when assuming the freshly grown domain to be of a symmetrical truncated hexagonal pyramidal shape [25]. However, the effective and local DW might deviate from this ideal behavior in both its straightness and symmetry, having a strong impact on both local conductive, piezoelectrical, pyroelectrical or flexoelectrical properties. To elucidate the typical DW embodiment inside the bulk, it is pivotal to trace DWs three-dimensionally. The reported values for α by Schröder et al. ranged from 0 to 0.225° requesting a lateral resolution of approximately 50 nm at a focal depth of 100 µm. In general, this is out of scope for ordinary optical techniques, as e.g. confocal microscopy. However, with the recent efforts in pushing the lateral resolution to even smaller distances by the usage of super-resolution techniques, especially for fluorescence-based methods as e.g. stochastic optical reconstruction microscopy (STORM) [496], we already know that the displayed optical resolution of the microscopical setup not to be of a major concern, yet rather the ability to track the center position of the ideally Gaussian image at the detection plane of nanoscopically small emitting objects. Hence, the only prerequisite is the affirmation the detected light to stem from individual and nanoscopic sources only, e.g. single fluorescent dye molecules or quantum dots. Since ferroelectric DWs especially in LNO expand to definitely less than 10 nm in width [25] we are confident that the application of the super-resolution approach shall be equally possible on these CDWs. Hence, we experimentally tracked the center of a domain wall by CSHG and recorded the minute changes in the mean position while scanning the laser focus through the bulk of that particular crystal. In the case of mainly hexagonally shaped CLN domains, we can simplify the extraction of DW inclination by assessing the domain width at different depths within the crystal, assuming that the DW to stay hexagonal in projection throughout the crystal. Analyzing tenths of DWs, a proportional increase or reduction in domain width was always observed. With the very small changes in the domain width across the crystal, the increase

78 4.1 three-dimensional profiling of domain walls

a) Dl b) rear l surface 1 5% Mg:CLN [ µ m] 0.8 5% Mg:CLN annealed l 0% Mg:CLN 0.6 0.25 0.4

0.2 d 0.08 0 front <0.02 surface domain width increase 0.2 0 20 40 60 80 100 120 focal depth d [µm]

Figure 4.1: a) Three-dimensional visualization by CSHG of a ferroelectric DW inside a 5 % Mg:CLN single crystal. The domain is mapped across the whole crystal with a thickness of 200 µm. The width of the domain l along its crystallographic axes is extracted. b) The change in width of such domains ∆l is extracted across the crystal for various cases. In all these cases an almost linear domain width increase with depth inside the crystal was observed. The associated inclination angles α are attached. Heavily Mg-doped CLN shows the largest inclination of about 0.25°, however which is undoubtedly not the thermodynamic equilibrium as it can be reduced upon thermal annealing at 200 ◦C for 2 h.

Table 4.1: Comparison of the deduced DWs inclination angles α inside CLN single crystals with different Mg doping concentrations. Mg doping Inclination [25] Inclination concentration [mol %] via PFM [◦] via CSHG [◦] 0 0.006 <0.02 2 0.024 0.02 0.02 3 0.050 0.03 0.02 5 0.225 0.25 0.02 5 (annealed) 0.074 0.08 0.02

79 results or reduction of domain width was accounted equally on both DWs. We observe a very proportional domain wall width increase with depth inside the crystal. Consequently, the three-dimensional CSHG analysis of CDWs in Mg:CLN proofs the simple picture of straight but inclined DWs being valid [see a typical example in Fig. 4.1(a)], hence supporting the veracity of the former procedure to claim to be able to deduce α from subsequent PFM measurements on both upper and lower crystal surface. This finding is similarly in accordance with previous reports on scanning electron microscopy (SEM) xz images of HF-etched CLN [154]. Yet, as surface effects can alter the DW shape, a completely in-situ and non-destructive method as CSHG was necessary to proof this relation. Fig. 4.1(b) presents recorded domain width increases at different depths in differently prepared CLN. The deduced inclination angles α are in very good agreement with the previous PFM results [see Tab. 3.1]. Hence, for Mg:CLN we expect very little changes in properties distinctly different at CDWs as e.g. wall conductivity at different depths, indicating that e.g. the effective transport properties along such CDWs are solely con- trollable through the inclination angle α and the injection barriers to the top and bottom electrodes, respectively. Varying α even allows us to explore the optical limitations encountered when applying the 3D CSHG method. The minimal inclination angle to be resolvable lies in the range of 0.02°/100 µm, taking into account the effective experimental focal depth and focal size, respectively. With CSHG it is feasible to investigate mm-thick crystals, balancing lateral resolution and focal impingement. Hence, for highest lateral resolution measurements, hence we chose thinner crystals with a maximum thickness of 500 µm. enclosed domains The existence of fully-enclosed ferroelectric domains of reversed polarization, whose DW is not connected with the crystal surface at all, and, hence, re- mained hidden to surface-sensitive techniques, has already been proclaimed to exist due to an unexpected increase in nonlinear response in LNO [497]. With CSHG microscopy, however, as penetrating through the whole crystal, it was possible to reveal these pre- liminary hidden domains in the bulk [see schematic in Fig. 4.2(a)]. In this work, such enclosures were observed to appear after several re-poling cycles and to always be located within the center of hexagonal domains, yet it should be assumed their existence not be restricted to this very special geometrical case. Their typical lateral size reaches approximately 1 µm, which is right at the resolution limit by CSHG. Hence, we illustrate such an enclosure domain in Fig. 4.2(b) in three cross-sectional views taken at different depths d of 50 µm, 55 µm, and 60 µm, respectively. The enclosure is well visible at the 55 µm penetra- tion while no signature appears in the two adjacent imaged focal planes. The arguments to confirm these spots to be local enclosed domains and not artifacts are their nonex- istence before poling as well as the existence of a very same signature in multiphoton luminescence [254]. Although manifesting a highly-charged microscopic enclosure embedded in a matrix of opposite polarization, such enclosed domains proved to be sufficiently time stable even after moderate thermal heating mediated through the laser focus, which is extremely important, as a randomized and dense creation of such enclosed microscopic domains

80 4.1 three-dimensional profiling of domain walls could enable the generation of more efﬁcient high power lasers and optical systems e.g. in frequency conversion on the basis of a pure and uniaxial ferroelectric crystal as LNO, due to its larger nonlinear coefﬁcient d33, lower scattering, and smaller absorption losses.

a) b) z = 50 µm

z = 55 µm h2h

~PS

~ z = 60 µm t2t PS

~PS

z y enclosed 2 µm domain x ICSHG Figure 4.2: a) Schematic of enclosed domains b) CSHG visualization of an enclosed domain buried inside the crystal of 5 %Mg:CLN after repeated re-poling.

4.1.2 Periodically Poled Lithium Niobate

Periodically poled LNO (PPLN) crystals are used in many ﬁelds of optical frequency conversion. The main quality factors hereby are poling period, crystal thickness, and period stability. Nowadays, the realization of periodic-poling of 1 mm-thick Mg:CLN crystals has been shown to be feasible, however with distinct drawbacks concerning treatable crystal thickness and period stability as compared to undoped CLN, yet with the advantage of a pronouncedly higher optical damage threshold. Both starting from the optical frequency conversion point of view as well as from the perspective of artiﬁcial and controlled creation of periodic CDW it is important to investigate these crystals. Moreover, the question how relevant poling defects are and whether these are a manifestation of the formation of CDW are further important questions to ad- dress. Therefore, we investigated 5 % Mg:PPLN crystals as purchased from EQ Photonics with a nominal period length of about 7 µm and strip lengths of about 1 mm and a crystal thickness of 500 µm. In Fig. 4.3(a) sequential CSHG xy scan images can be observed for different focal depths within the crystal (0 µm, 25 µm, and 50 µm). One can observe slight changes in the domain wall representation, however, these variations are mainly due to changes in the focal quality inside the both birefringent and non-index-matched material. In backward detection, such alterations are commonly less prominent. Yet, when investigating xz cropped scans of the three-dimensional stack, one can observe a distinctly asymmetric wall movement,

81 results

a) z = 0 µm z = 25 µm z = 50 µm

y 1

x 20 µm ICSHG b) 1 c)

2 3

z y

20 µm 50 µm x 25 µm x ICSHG ICSHG d)

ICSHG

4 25 µm z

y 20 µm x Figure 4.3: CSHG inspection of 5 % Mg:PPLN. a) Series of CSHG scans for different focal depths z inside the crystal, b) Comparison of xz CSHG scans at different strip positions y pre- senting asymmetric growth, c) fabrication deviance to expected thermodynamic equilibrium case with domains not entirely grown [marked by 3 ], d) 3D visualization of a representative PPLN segment with an observable change along z [marked by 4 ].

82 4.1 three-dimensional profiling of domain walls the DW is canted towards the crystallographic axes in the xy PLane [see Fig. 4.3(b) with 1 and 2 referring to the y-position of these xz scans given in Fig. 4.3(a)]. This asymmetric behavior is in good agreement with previous reports on speciﬁc DW corner "roundening" in CLN by Haußmann as well as present in the aforementioned ferroelectric lithography as presented in Sec. 2.2.6 [228]. Yet, here the effect of preferred "roundening" of so-called "A" corners is observed to stretch out tenths of microns along the strip domain.

y x

6 z 6

5 I 5 CSHG 6 6 z 25 µm 25 µm

y x 20 µm 20 µm Figure 4.4: CSHG inspection of a thinned 5 % Mg:PPLN wafer being polished to a thickness of 100 µm. The polishing process resulted in minute changes in the DW appearance. Both merging of domains can be observed [marked by 5 ] as well as the sharpening of the very round cap on one side to the other [marked by 6 ]. Both projective images show the same 3D arrangement from a lower and an upper perspective.

The 3D CSHG visualization is important to detect deviances from the expected thermodynamic equilibrium in domain formation [see a typical deviance marked by 3 in Fig. 4.3(c)]. These deviances are very strongly connected with CDW formation in LNO, thus if such deviances prove stable it is possible to stabilize CDW in LNO. Moreover, other deviances are found to be present as e.g variations of the domain structure at the caps of the domain strips [marked by 4 in Fig. 4.3(d)], merging of domains [marked by 5 in Fig. 4.4], and sharpening of the cap of the strip [marked by 6 in Fig. 4.4]. A 3D visualization of a representative PPLN segment is given in Fig. 4.3(d).

4.1.3 AFM-written Domains

The local inversion of the ferroelectric order can be realized by AFM techniques, i.e. PFM with a DC offset sufﬁciently large to exceed the coercive voltage as already discussed in Sec. 3.2. For a polished wafer (original thickness: 500 µm) down to 70 µm domains were written applying a voltage of 140 V between the conductive tip and gold back electrode with the tip being in contact with the crystal surface for 5 s.

83 results

1 2 3

5 µm ICSHG

1 z = 0 µm

2 z = 20 µm

3 z = 40 µm

ICSHG z x 10 µm y 5 µm

Figure 4.5: 1 – 3 Series of CSHG scans at different depths z over domains being written by a sharp AFM tip with a voltage of 70 V into undoped CLN, associated 3D representation of the CSHG stack showing the inclination of these DWs as well as merging spots.

The resulting domains were visualized by CSHG, a 3D image of this system is shown in Fig. 4.5. As very obvious a very large inclination of the domains can be realized, a cone- like domain is created. The typical inclination angle is about 1 to 2°, which is signiﬁcantly larger than under homogeneous poling condition. Inﬂuences by the reduced thickness might additionally occur.

4.1.4 Thermally treated LNO

The thermal treatment results in theory in CDWs with an inclination angle of 90°. How- ever, already the SEM images of slightly HF-etched crystals revealed a peculiar domain shape, the DW was wavy and appeared not straight [429], with the waviness growing over annealing time. Yet, the internal structure of such DWs was unknown by now. We

84 4.1 three-dimensional profiling of domain walls recorded the DW shape and observed, that the wavy structure is maintained as well in the third dimension. A 3D representation of such a DW segment is given in Fig. 4.6(a),(b) under two different perspectives the segment being rotated around the axis marked by A . As can be clearly seen, a large range of possible inclinations is present, the DW waviness reaches values as small as 3 µm rms.

a) b) A

~PS

turned

c) A d) ~ PS ~ ~PS PS ~PS t2t T T ~P h2h S ~PS ~PS

turned

Figure 4.6: a),b) 3D representation of the DW shape inside thermally treated crystals of 5 % Mg:CLN crystals, c), d) 3D visualization of a DW segment with an unexpected behavior, the formation of "tunnels" marked by T . Hereby, the polarization charge condition continuously alters from h2h to t2t along the DW. Both 3D representations are given twice under two perspectives, each rotated around axis A . Scales are 20 µm each.

Furthermore, CSHG is also capable of revealing topologically nontrivial structures such as the one illustrated in Fig. 4.6(c),(d), again under two perspectives turned around the axis marked A . In fact, this h2h DW exhibits an unexpected condition very similar to a "tunnel" or "teacup handle", marked by T . The DW, hereby, undergoes a continuous change in DW polarization charge from h2h to t2t. The existence of such both fully charged h2h and t2t DWs underlines the possibility to create DWs with any possible shapes inside this uniaxial material. The reason such structures can be created is not fully understood, yet inhomogeneous growth conditions of the domain wall perpendicular to the growth direction can result in these peculiar DW shapes, which are stable over time. The crystal was heated up to 37 ◦C with an additional incubator system while CSHG images were taken, however, no detectable change was registered. Hence, one can conclude, these DWs to be

85 results strongly pinned and very stable. Furthermore, it might be possible, that these DWs are connected with a structural order. The origin of such DWs is not very intuitive. However, only recently the DW motion was found to follow hydrodynamic laws, which implies that also hydrodynamic instabilities as whirls can be created [498, 499]. Similarly, such instabilities showed to be astonishingly stable upon external stimuli.

4.1.5 Laser-written domains

a) l = 325 nm

~PS ~x

~v ~ z PS y b) x x ~x +Vext d = 500 µm

Vext = 800 V b) PUV = 280 µW v = 1 mm/s surface

~PS

z ~P 5 µm x S

Figure 4.7: a) Schematic of laser writing of surface domains into LNO. Hereby, the laser is scanning with a speed of ~v over the surface, while a voltage Vext is applied, the domain formation shows self-assembling character along ~x. b) An xz CSHG visualization of laser-written surface domains, being written along the crystallographic vector ~y. The domains were written upon Vext = 400 V, v = 1 mm/s, and a UV illumination power of 280 µW.

It has been well known for some time, that surface domains can be created upon laser illumination [422, 427, 500–509]. Two main approaches have been used over the time: i) illumination just before poling and ii) illumination while poling. The ﬁrst method was typically carried out especially with deep-UV lasers, while the second method worked out similarly for higher wavelengths. In this work, we will present results of laser written domains inside CLN upon illumination of a HeCd laser by Kimmon with a wavelength

86 4.1 three-dimensional profiling of domain walls of 325 nm and a typical output power of 15 mW. In Fig. 4.7(a) the schematic of the poling inversion process is sketched. Such an experimental realization is given in Fig. 4.7(b). Is very obvious, that very largely inclined DWs are formed with a typical inclination angle of 3°. These DWs, yet, are of t2t polarization charge, thus it would not be expected from theory, that such CDWs behave conductive [128].

a) z = 0 µm b) z = 0 µm

y y 10 µm 10 µm x x

z = 30 µm z = 30 µm

z z y y 10 µm 10 µm 10 µm 10 µm x x x x

c) z = 0 µm d) z = 0 µm

y 10 µm 10 µm x

z = 30 µm z = 30 µm

y z z 10 µm 10 µm 10 µm 10 µm x x x

Figure 4.8: 3D visualization of laser-written surface domains by CSHG microscopy in 5 % Mg:LNO at v = 1 mm/s and V = 800 V for different laser intensities: a) I = 1.8 104 W/cm2, ext × b) I = 8 104 W/cm2, c) I = 4.6 105 W/cm2, d) I = 2.4 106 W/cm2. × × × The formation of such CDWs is expected to be very sensitive concerning illumination strength. Hence, it was very important to observe the surface domain formation upon illumination intensities. We took four different illumination intensities I [1.8 104 W/cm2, × 8 104 W/cm2, 4.6 105 W/cm2, 2.4 106 W/cm2] for the very similar external voltage × × × Vext = 800 V. Note, this external voltage Vext is about 5 times smaller then the coercive voltage of the ferroelectric order. The result of this CSHG measurement series can be observed in Fig. 4.8. One can clearly observe an increase in protrusion depth of the created

87 results surface domains and the number of nucleation centers ﬁnally resulting in a continuous line with illumination intensity. Yet, there is no drastic change in inclination angle of the created and grown surface domains. Thus, both the broadening and deepening should keep their speed over growing time. Only for very short surface domains a reduction in inclination can be clearly observed. Such a behavior is very expected in the standard model of Müller-Weinreich model and its correction by Liu et al. [510, 511]. For very high illumination intensities, a self-assembling process can be observed, resulting in very reproducible zig-zag domains, which are marked by 1 . Such h2h CDW could result in conductive DW arrays in a positively doped environment.

a) z = 0 µm b)

z = 5 µm

z = 20 µm

z = 40 µm

z = 100 µm

z = 250 µm

z z = 500 µm y x x 25 µm 25 µm 25 µm

Figure 4.9: 3D visualization of laser-written surface domains by CSHG microscopy in 7 mol% Mg:LNO at v = 1 mm/s and Vext = 1200 V. On the illuminated surface a very large inverted area can be observed, yet only 5 µm below this surface non-inverted domain regions can be observed. As this only happen under the illuminated line, this area shows poling inhibition, previously known from pre-poling illumination.

Apart from that, one can observe domains protruding massively deeper into the crystal than the surface domains, sometimes even forming through domains, surrounding the laser focus region. This effect can be explained with changes in the coercive voltage around the laser focus due to optically promoted lithium ion migration. A schematic of such a possible mechanism to result in this domain formation is given in Fig. 4.10.

88 4.1 three-dimensional profiling of domain walls

1 2 3 ~v ~v ~v

t2t t2t t2t h2h ~PS ~PS ~PS

+Vext +Vext +Vext A B C ~v ~v ~v

T c Li = Ec ~PS + ~PS ~PS

+Vext +Vext +Vext Figure 4.10: Sketch of the working principle in UV illumination assisted domain writing. For small intensities loosely surface domains are created with t2t DWs 1 , for higher intensities nonlocal further result in deeper protruding domain formations, due to formation of lithium ion concentration gradients 2 and which result in even higher illumination and slightly higher voltages being applied in poling inhibition conditions and the formation of h2h domains 3 . Nonlocal effects are explained by thermal gradient A induced lithium concentration changes B , resulting in a decrease in coercive ﬁeld around the focal region C .

Upon larger external voltages Vext being applied this results even in through domains for wafer-thick crystals [marked by 1 in Fig. 4.8]. Following these arguments, another effect that was observed for laser assisted writing in 7 mol% Mg:CLN can be explained in good accordance: the poling inhibition at the focal line and creation of h2h DWs at the written line. In Fig. 4.9(a) through domains are visualized by xy scans at different depths z inside this crystal [marked by 2 ]. Yet, at the laser line, only the surface domain formation can be observed, with a typical random shape of surface domains interconnecting, and, hence, forming a completely inverted area at the surface. Below, this area the through domains enclose the domain and form a h2h CDW, which can be especially seen in Fig. 4.9(b). Thus, using CSHG inspection we found a facile way to readily create CDWs with h2h polarization charge with a suitable protocol by laser-writing. As for even thinner samples, the inclination angles are expected to be even larger, and the formation of CDW at the opposite crystal surface than illuminated, designed circuit boards are possible. This approach can even be based on standard lithographical systems’ UV exposure.

89 results

4.2 polarization charge textures

Clearly, the display of the 3D shape of ferroelectric DWs is an important step towards the understanding of e.g. the conductance properties of the sample. However, it is still not trivial to extract quantitative and local information on the inclination angle from this CSHG data stack, especially for topologies not as simple as truncated prisms with a hexagon base. Therefore, a procedure was established which made it feasible to extract a single surface. Experimental CSHG data has several insufﬁciencies, e.g. local inhomogeneity, inhomogeneous background, scattering efﬁciency, depth dependence, destructive interference, and drift. These artifacts have to be corrected. To correct for local CSHG inhomogeneities local normalization was used to homogenize the CSHG intensity along the DW. The background is corrected by thresholding. Depth dependence is corrected by image normalization. To correct for drift through the recording of these 3D stacks, the template matching package was applied [490]. These efforts result in normalized data of a Gaussian blurred DW, which can be visualized in contour mode by ParaView [492]. To visualize the local DW inclination, the projection of the local normal vector of the triangulated contour surface onto the polar axis is created.

4.2.1 Random domains in Mg:LNO and Mg:LTO

~ ~PS PS s = 2PS sin a ~nh2h h2h a = 0.45

CDW n

~nt2t ~n t2t a = 0.45 NDW ~nn 5 µm 50 µm

Figure 4.11: 3D visualization of a representative DW observed in 5 % Mg:LNO. The DW is dyed according to its local polarization charge, red for h2h, blue for t2t CDW and green for NDW sections. The exemplary random polarized domain shows a very homogeneous h2h polarization texture over the entire DW with an average inclination angle of about 0.25°. random domains in mg:lno The ﬁrst system to be analyzed were random domain conﬁgurations in Mg:CLN. As was already pointed out, these domains can be assumed to follow the picture of truncated pyramids. Yet, a localized view is important to give a

90 4.2 polarization charge textures clearer picture about the existence of local defects, as such local inconsistencies can have a major inﬂuence to the overall. In Fig. 4.11 the representation of the local inclination on the DW is visualized as a false colored surface. As is visible the overall picture of a homogeneous polarization charge over the DW is still present, yet some small deviations are visible, especially at the surfaces. random domains in mg:lto The next sample to investigate was Mg:LTO with a different Mg doping concentrations. This important as especially doping concentrations close to the optical damage resistance threshold (ODRT) are interesting, which is in the case of Mg:LTO about 7 mol% Mg. In Mg:LNO doping close or above the ODRT revealed a signiﬁcant increase in DW inclination [512]. Surprisingly, in 7 mol% Mg:LTO we not only observe h2h - marked by B - but also t2t CDW formation - marked by C - as well as neutral DWs - marked by A [see Fig. 4.12]. As one can see, homogeneous DW inclination is present, however, also larger deviations than present in Mg:LNO.

A B C

t2t h2h 5 µm 50 µm s = 2PS sin a a = 0.45 a = 0.45 Figure 4.12: 3D visualization of both charged and uncharged DWs observed in 7 mol% Mg:LTO. These exemplary randomly polarized domains show both h2h and t2t polarization textures, mostly homogeneous over the DW.

For 8 % Mg:LTO we can observe an even larger range of inclinations, even resulting in enclosure domains at the optical resolution limit [marked by 1 in Fig. 4.13] as well as enclosure domains with a macroscopic size as large as 20 µm [marked by 2 in Fig. 4.13]. Hereby, the vertical dimension was corrected with the refractive index by [513].

91 results

a) z = 20 µm z = 40 µm

1 b) 2

y y ~PS x x 1 2 z = 60 µm z = 80 µm z

x 1 1 1 y y 2 x x 5 µm ICSHG[a.u.] 0 1 Figure 4.13: a) 3D visualization of domains in 8 % Mg:LTO. a) xy scans for different focal depths z from 20 to 80 µm are given. b) Schematic of the polarization charge textures on the relevant DWs.

4.2.2 Thermally-treated LNO

In thermally-treated LNO we already observed DW shapes being very different to those present in electrically poled samples. As already pointed out, there is a continuous change from h2h towards t2t DW sections inside the crystal. In Fig. 4.14, the extracted false color representation of the contour surface according to polarization charge is shown. As is visible in this region, an island structure is observable over an area of about 800 µm. ≈ Hence, such a deviation from the thermodynamic equilibrium, hence, not expected due to the electrostatic costs, is still found and stable in the medium.

4.3 quasi-phase matching shg

We now introduce here a 3D technique, quasi phase-matching second harmonic generation (qpmSHG) microscopy, which provides both a high spatial and temporal resolution. Undoubtedly, the most powerful technique applied for DW imaging so far was CSHG [217, 512], which has proven to measure DW inclination angles down to 0.02° only. This impressive angular resolution, however, is at the cost of data acquisition time, since CSHG requires both time-consuming image stacking and post processing procedures. As

92 4.3 quasi-phase matching shg

~PS

y h2h

x z

20 µm 50 µm t2t

Figure 4.14: 3D visualization of thermally-treated LNO. The DWs are rendered in false color according to their local DW inclination. DW areas with both entire h2h and t2t polarization charge are visible. a consequence, transient or dynamical structural changes such as DW motion cannot be monitored. In addition, the high numerical aperture (NA) of the microscope objective generally used in CSHG, provides on the one hand high spatial resolution while on the other hand limits the focal depth in the sample. Quasi-phase-matching-SHG (qpmSHG), as introduced here, allows for DW mapping at a mm depth into the high refractive index sample with a superb angular resolution, while additionally being able to monitor the DW dynamics. In our experiments reported here we investigate CDWs in thermally treated LNO, as these constitute DWs with both the largest radius of curvature and the largest bound polarization charge on DWs in LNO.

In the case of inclined DWs, both non-collinear and collinear SHG radiation is emitted [see Fig. 4.15(a)]. The collinear SHG depends on the local lattice vector determined by the interface distance, which includes both surfaces and DWs. The required reciprocal lattice vector G~ m of the second-order susceptibility modulation convolved with the electric ﬁeld of the optical excitation must fulﬁll the following condition:

4π 4π G~ = (n n ) = ∆n, (4.1) | m| λ 2ω − ω λ with G~ m the lattice vector, and nω, n2ω being the refractive indices at the fundamental and the second-harmonic wavelength. Depending on the path difference between two interfaces, either constructive or destructive interference may be apparent, resulting in

93 results

(2) (2)

G~DW

z 2k~ y 0 ~ x k2

G~m

b) I(x) c) = 800nm = 900nm ~k d) x oil oil LNO g 1 LNO 2(n + 1)⇤ g 2n⇤ 2 ↵ A 3 B DW P~ z z z ~ ~ P~ P P 15 µm 15 µm x x DW x I [a.u.] I [a.u.] 0 1 0 1 e) f) g) 0.2 3.5 5 Experiment Ref. A Theory A B 4 0.18 3 B

↵ 3 0.16 2.5 n tan ⇤ [ µ m] 2 0.14 L = 2.3 µm 2 1 0.12 1.5 0 0.1 0 0.2 0.4 0.6 0.8 1 800 850 900 950 800 850 900 950 1 1 g [µm ] [nm] [nm]

Figure 4.15: a) Schematic of collinear (i.e. quasi-phase-matching/qpm) and non-collinear (i.e. longitudinal phase matching) second harmonic generation (CSHG) emission at inclined DWs, b) schematic of the investigated DW, c) and d): SHG image recorded at a fundamental wavelength λ of 800 nm and 900 nm, respectively, showing both interference fringes by qpmSHG at e.g. 1 and 3 between surface and DWs, as well as the superposition of qpmSHG and CSHG at e.g. position 2 at the DW e) plot of interference 1 fringe period g− vs. tan(α) with α the DW inclination angle, f) comparison of refractive index difference ∆n between fundamental and second-harmonic wavelength of reference and by qpmSHG for two DW sections A and B ; g) characteristic DW separation length Λ for SHG interference.

94 4.3 quasi-phase matching shg

z = 0 µm z = 20 µm 2nd section

x x

st 1 section 20 µm 20 µm x 10 µm y y z z = 40 µm z = 60 µm y 20 µm

x x

20 µm 20 µm y y z 0 1 x 10 µm I [a.u.] y 20 µm Figure 4.16: Depth proﬁling with quasi-phase-matching second-harmonic generation (qpmSHG) at different focal depths z at λ = 900 nm with an axial separation characterization factor Λ = 2.3 µm and an acquisition time for each image of 20 ms. a sinusoidal modulation of the measured second-harmonic intensity I(x) ∝ sin (2πx/g). Hence, we may easily relate the local DW inclination to the interference fringe separation g through the coherence length lc of the second-harmonic emission as sketched in Fig. 4.15(b):

2π 1 λ 4 2 2Λ = 2lc = = = g tan(α). ( . ) G~ 2 ∆n | m| As a result, we extract the local inclination angle α from the fringe separation with in- terferometric precision. In Fig. 4.15(c) and Fig. 4.15(d) we show xz SHG scans of the LNO crystal, centering a h2h DW upon ordinary fundamental excitation at a wavelength of 800 nm and 900 nm. As a matter of fact, we see both the interference pattern from collinear SHG (i.e. qpmSHG, marked at positions 1 and 3 in Fig. 4.15(c)) and the local enhancement at the DW by non-collinear SHG (i.e. CSHG, marked by 2 in 4.15(c)) being superposed to each other in this figure. As indicated by the DW-specific SHG emission, the inspected DWs display sections that have totally different inclination angles [marked A , B in Fig. 4.15(d)]. We extract the inclination angle from this graph which is possible as the DW sections are quasi linear. Also, we compare inclination angles α obtained from theory describing the qpmSHG fringes, to the local inclination angles that we extract from 1 the CSHG. In Fig. 4.15(e) we show that the relation g− ∝ tan(α) is in fact fully reproduced and that it scales with the slope of 2Λ for a fixed wavelength, as stated in eq. (4.2). To further verify this behavior, we have used oblique sections of DWs in periodically poled LNO fabricated at an inclination angle of 6° to the optical axis. Hence, the distance modulation

95 results between interfaces (i.e. DWs) appears only close to the sample surfaces. As domains are distributed quasi periodically, the bulk contributes with a constant second-harmonic background. The refractive indices and thus ∆n depend on the fundamental wavelength. We may easily deduce this refractive index difference between fundamental and second-harmonic wavelength for two different DW sections with different inclinations. We compared the obtained difference at A and B with the index of refraction for ordinary beams through the well-known Sellmeier equation as:

r a n (λ[µm]) = a + 2 a (λ[µm])2 (4.3) 0 1 (λ[µm])2 a − 4 − 3 using the parameters a1 = 4.9048, a2 = 0.11775, a3 = 0.21802, and a4 = 0.027153 for LNO [514]. We recognize that the difference in refractive indices is correctly reproduced [see Fig. 4.15(f)]. In Fig. 4.15(g) we calculate Λ, which speciﬁes the axial separation in qpmSHG. For λ = 780 nm, Λ reaches values of about 1.5 µm. Therefore, the angular resolution is comparable to the best high-NA optical microscopy using indexed-matched materials. Since we work here without any index matching and over a large focal depth, qpmSHG proves to be superior even in this respect. Theoretically, Λ might even be improved for instance by lowering the fundamental wavelength. Fig. 4.16 illustrates such an example. We show here xy scans at different focal depths z. Every bright line represents an intersection of the DW with levels at different depths. The axial displacement between these sections is deﬁned by Λ(λ = 900 nm) = 2.3 µm. The depth of independent sections recorded in a single xy scan can be increased when decreasing the NA of illumination. Hence, knowing the applied fundamental wavelength, we may directly compute the local DW topology from these xy scans. As shown, there is no need to stack the images now, in order to obtain a 3D representation. The images given in Fig. 4.16 were recorded with an acquisition time of only 20 ms. Since qpmSHG is ways faster than any previously reported technique, we will be able to track also dynamical processes in ferroelectric materials, i.e. DW dynamics. Also, when requiring an increased lateral resolution, qpmSHG provides the possibility to combine two or more fundamental wavelengths in parallel; hence, two or more sets of correlated SHG fringes result which then increase the precision to delineate the DW topology. As LNO is a birefringent material it is mandatory to investigate the impact on qpmSHG for ordinary and extraordinary incident radiation as well [see Fig. 4.17(a)]. A comparison of the observed xy scans is displayed for extraordinary (ee) [see Fig. 4.17(b)] and ordinary (oo) [see Fig. 4.17(c)] excitation, respectively. For ordinary excitation, the interference fringes appear narrower, which is explained by the slightly smaller difference in refractive indices. For ordinary excitation ∆no(λ = 800 nm) = 0.183 and extraordinary excitation ∆ne(λ = 800 nm) = 0.156, which results in a change of Λ. Therefore, it is advantageous to operate qpmSHG under ordinary excitation resulting in a larger axial image resolution. In Fig. 4.17 the aforementioned complex domain formation apparent in thermally treated LNO is shown. Here, both embedded ’tunnel’ and ’island’ structures can be directly per- ceived in this qpmSHG image. The positions are marked by T and I , respectively. These

96 4.4 photoelectron microspectroscopy

a) b) ee c) oo E~0 E~0 ~ 2k0 ee oo I

T T T 1 T T 1 z x x 0 1 x x I [a.u.] y I 20µm I 20µm y y 0 0

4 17 [a.u.]

Figure . : a) Schematic of qpmSHG scans on complex topologies at a formerly head-to-headI DW under extraordinary (ee) and ordinary (oo) excitation, qpmSHG scans under b) extraordinary and c) ordinary excitation at λ = 800 nm. For ordinary incidence the interference fringe separation is smaller, resulting in a higher axial resolution. The complex topology contains both tunnel and island structures, which are marked by T and I , respectively. structures incorporate both h2h and t2t DW regions with continuously altering the polarization charge.

4.4 photoelectron microspectroscopy

We conducted energy resolved photo emission electron microscopy (PEEM) images on a h2h DW in the thermally-treated LNO crystal [see Fig. 4.18]. Depending on the sample bias voltage, determining the kinetic energy of electrons reaching the detector, the edges of the domains (panels (a), (b) are visible, the DW (panel (c)) or the overall intensity di- minishes (panel (d)). These observations can be understood as follows: the photoemission spectrum of an arbitrary material consists of nonscattered primary electrons reﬂecting the valence band and the core levels and secondary electrons, which undergo scattering events and form a diffuse background with a pronounced maximum at low kinetic energies, since scattered electrons tend to loose energy, the so-called secondary electron cutoff. For insulating samples the charge of the photoemitted electrons cannot be compensated and the sample charges up positively, which exerts an attractive force to all leaving photoelectrons and reduces their kinetic energy up to the point they cannot overcome the work function barrier of the sample anymore. In effect, the charging shifts the spectrum to lower kinetic energies. LNO is a transparent wide bandgap insulator and, hence, charges up very easily. We concentrate in Fig. 4.18 on the secondary electron cutoff since it is by far the most intense feature of the spectrum. It turns out that, without further precautions, the charging is large enough to suppress electron emission at all. In order to compensate the positive charging, we applied a negative bias voltage to the sample. As can be seen in Fig. 4.18(c), the region that appears for the lowest bias voltage, i.e. experiences the smallest charging, corresponds to the DW. This is an indication that the DW has a larger local conductance to

97 results

a) E E = 3.3 eV b) E E = 4.8 eV F F

~PS ~PS ~PS ~PS x x

z z

c) E E = 0.5 eV d) E E = 1.4 eV F F

~PS ~PS ~PS ~PS x 2 1 3 4 5 x 30 µm z z

e) F + DE[eV] f) 1 spot 1

[a.u.] 5.5

I 2 0.75 3 6.5 4 5 7.5 0.5 8.5 9.5 0.25 electron emission 10.5 4 3 2 1 0 1 2 3 4 5 g) E EF [eV] 30 µm IAl Ka 1 [a.u.] 0.44 [a.u.] 0.8 I 0.5 h) 0.56 2 0.6 0.62 [eV]

0.69 ) 1 max

0.4 0.75 I (

0.82 F 0.88 E 0 0.2 electron emission 0.94 E 1 0 1 4 2 0 2 4 6 1.5 2 2.5 3 IAl K [a.u.] E EF [eV] a Figure 4.18: Photoelectron microspectroscopy of thermally-treated LNO. a)-d) energy-selected PEEM images for bias voltages close to the secondary electron cutoff, e) local electron emission spectrum at the secondary electron cutoff, f) surface potential map (SPM) extracted by secondary electron cutoff energies, g) X-ray excitation intensity dependence, h) secondary electron cutoff dependence on X-ray excitation intensity. PEEM data acquired by Anna-Sophie Pawlik (IFW Dresden).

98 4.4 photoelectron microspectroscopy the ground than its surrounding. Fig. 4.18(e) shows the cutoff spectra for several positions across the DW. The spectrum exactly at the DW appears at the lowest energy, whereas the spectra taken from positions away from the DW are successively shifted to higher and higher energies. Therefore, in Fig. 4.18(a) and 4.18(b), the DW region becomes dark, because the energy position of the secondary energy cutoff is passed already, and the edges of the DW become bright. Figs. 4.18(a)–(d) are summarized in Fig. 4.18(f) where the energy shift is represented by a color scale. The energy shift is not an arbitrary, spectrometer specific quantity. It directly reflects the potential at the surface for a given charge state. It is therefore justified to term Fig. 4.18(f) a surface potential map (SPM). The surface potential, on the other hand, depends on the local conductance. We will show below how further information can be extracted from this map. Hence, Fig. 4.18 demonstrates that energy resolved PEEM is capable of monitoring differences in the local conductivity. This is achieved by a true contact free imaging procedure rather than by a scanning probe. The results in Fig. 4.18 are consistent with comparable measurements on EMO [114]. The secondary-electron cutoff shows a proportional shift on illumination strength as can be seen in Fig. 4.18(g) and 4.18(h). Even more complex situations than the single DW feature in Fig. 4.18 are observed. In Fig. 4.19(a) a second DW structure appears at the left hand side of the field of view. Fig. 4.19(c) presents an image of the same sample region recorded by qpmSHG. The positions and shapes of the DWs are nearly identical except for the fact that the DW 2 feature is actually a closed structure whose inner side is missing in the PEEM image. The explanation of this effect is sketched in Fig. 4.19(a). If the DW 1 and DW 2 are of h2h type the missing inner feature must be t2t. t2t DWs do not increase the local conductance, which is expected from theory, and are therefore invisible to PEEM. The according SPM is given in Fig. 4.19(b). The favorable comparison of two independent measurement techniques ultimately confirms that the PEEM image contrast is due to local conductance variations. So far, we have argued on a qualitative level. However, the SPMs in Figs. 4.18 and 4.19 contain also useful quantitative information. In Fig. 4.19(a) it is seen that the local charging varies not only perpendicular but also along the DW. This is explicitly shown in Fig. 4.19(b) for the two visible DWs. In Figs. 4.19(f) and 4.19(g) the energy shift ∆E, i.e. the surface potential, is shown along the DWs extracted from the SPM in Fig. 4.19(b). Hence, this constitutes a surface potential curve (SPC). A model to explain the variations of ∆E by the properties of the DW is developed. As the energy shift ∆E reflects the charging related potential at the sample surface and the charge density at a given position on the surface decreases the better the electrical contact to the ground and increases the higher the incoming photon flux, one therefore can assume the equilibrium state to follow the ansatz:

I(~r) ∆E(~r) (4.4) ∼ Se(~r)

99 results

a) E E = 3.2 eV b) F ~P S 8 ~PS h2h 12 DW2 ] eV 16 [ E

h2h D ~PS DW1 t2t DW 18

20 h2h h2h t2t 50 µm t2t 50 µm neutral neutral 1 2 3 c) 4 5 ~~P 6 PS 8 ~ 7 P~PS 8 S 10

9 [eV] 10 2

11 h2h DW2 E

12 + 12

13 15 spots 14 17 14 1 16 h2h t2t h2h 2 18 19 3 0 50 100 150 200 250 4 21 x [µm] h2h DW1 5 20 6 t2t DW 22 23 7 25 ~ 24 8 PS 26 27 3 1 9 28 10 11 30 29 12 31 spots 13 32 14 33 local surface potential 15 34 35 F + DE[eV] 16 36 17 18 37 50 µm 38 39 8 13 40 80 d) a e) 60 a ] ] b b [ 1 [ 2 3 60 j j 40 40

20 20 DW inclination DW inclination

0 0 0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 25 30 35 40 11 f) exp. 13 exp. spots g) spots ﬁt ﬁt 12 10 [eV] [eV] E 11 E D D 9 10

9 8 0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 25 30 35 40 spots spots Figure 4.19: Photoelectron microspectroscopy of thermally-treated LNO. a) PEEM at bias condition E E = 3.2 eV, b) SPM derived by energy-resolved PEEM stack, with the conduc- − F − tive h2h DWs forming a ridge in the SPM, inset: SPC across the ﬁeld of view along the dashed line, c) qpmSHG image of the spot, deriving the entire 3D information in a single 2D image, overlaid with the extracted ∆E of selected spots on the h2h DWs; d),e) derived inclination angles ϕ at the marked spots, with both vertical and horizontal projections α and β; f),g) comparison of experimental local surface potential on h2h DWs and ﬁt according to eq. (4.7). PEEM data acquired by Anna-Sophie Pawlik (IFW Dresden).

100 4.5 activated polaron transport

where I(~r) is the photon ﬂux arriving at the position~r and Se is the local conductance. For the latter holds:

S σ = eµ n (4.5) e ∼ e e e where e is the elementary charge, σe is the conductivity and µe is the mobility. ne describes the amount of bound polarization charge carriers, which can be calculated with the ferroelectric polarization PS and the total inclination angle of the DW relative to the surface normal ϕ by: 4 6 ne = 2PS sin ϕ ( . )

The total inclination angle ϕ of the DW at ~r can be evaluated by the qpmSHG measurement presented in Fig. 4.19(c). The extracted inclination angle projections - vertical and horizontal - α and β and the total inclination angle ϕ is shown in Figs. 4.19(d) and 4.19(e) for both DW segments. One can observe, that ϕ scales to ∆E indicating that it largely determines the potential at the surface. Note, the DW segments for which mainly the vertical projection of the inclination determines are marked by 1 3. − In the following, we take together the given equations and assume a Gaussian beam ~r ~r 2/a2 proﬁle I(~r) = I0e| − 0| , with ~r0 the center of the beam and a the beam diameter. We introduce empirical ﬁtting parameters A and ∆Eoffset which absorb all other parameters and get:

2 2 e ~r ~r0 /a ∆E(~r) = A | − | + ∆E (4.7) sin ϕ(~r) offset From the agreement of model and experiment in Figs. 4.19(f) and 4.19(g) we can i) conﬁrm that the energy shifts observed by PEEM depend on the local conductance; ii) conclude that the local charge density, and hence the conductance, at the DW is indeed determined by the total inclination angel over a wide range of angles. Conversely, it seems possible to derive the angle of the DW from PEEM measurements by using the above empirical parameters. iii) shows the feasibility of quantitative analysis of PEEM images despite the presence of a large extractor ﬁeld inherent to this technique.

4.5 activated polaron transport

In this section, we want to discuss the origin of the previously reported photo-induced DW conductance in LNO. As it was observed, the current to decay following a Kohlrausch- Williams-Watts (KWW) equation, it was proposed, that polaronic hopping transport to play the major role. With optical means we now want to investigate the relaxation of these excited charge carriers in LNO. As is known, upon ultra-short laser pulse excitation, higher order absorption processes are invoked in LNO [253, 254, 401]. Generally, excited charge-carriers should then distribute over the conduction and valence band (CB, VB) and ﬁnally relax into inter-gap states that concurrently may increase the absorption cross section. Moreover, in order to increase and engineer the power threshold beyond

101 results the natural ODRT in materials as LNO, it is necessary to quantify and understand all the occurring optical transition processes with respect to both their temporal and spectral behavior. For such an analysis, time-resolved absorption (TRA) spectroscopy most preferentially has been applied, stating for instance that the decay of inter-gap states similar to the aforementioned conductance measurements follows the famous KWW relation [402]. Photo-luminescence (PL) spectroscopy, on the contrary, has rarely been used although being an as-powerful method as compared to time-resolved absorption (TRA). From the luminescence photon energy, we may directly deduce the energy positions of the acceptor inter-gap states. PL measurements thus may provide valuable information on inter-gap states hidden to TRA, which is important to further understand the kinetics of excited charge carriers, which e.g. can exist at CDWs and determine conductance. It is well known, that inter-gap states can form an ensemble of emitters with distributed decay rates, e.g. in distorted materials. The effective number of involved excited states can then be quantified via Laplace transformation of the decay-rate distribution [515] and one obtains: h i n(t) = n exp (t/τ)β , (4.8) 0 − with n0 the number of excited charge carriers, τ the total decay time, and β the stretch parameter that may vary between 0 and 1. β qualitatively expresses the underlying distribution of decay rates: a small β value thus represents a broad rate distribution, while a β close to 1 defines a narrow and discrete distribution. Even though eq. (4.8) was derived from macroscopic phenomenological findings, its application to TRA data was shown to be very successful. The number of excited charge carriers in inter-gap states depends proportional on the absorption for sub-bandgap illumination. In TRA measurements stretched exponential decays were consistently reported for LNO [388, 397, 403, 516, 517].

T [K] 170 150

[a.u.] 130 I 110 Exc. Intensity

200 300 400 500 600 wavelength λ [nm] Figure 4.20: Corrected steady-state luminescence spectrum in 7 mol% Mg:LNO for different cryogenic temperatures T and excitation laser spectrum. See our paper [518].

As radiative and non-radiative decay channels may superpose at room temperature, it is favorable to perform PL experiments at cryogenic conditions as the non-radiative recombination processes will be inhibited completely, resulting in a dramatically increased PL [see Fig. 4.20]. Assuming negligible non-radiative decays and the measured PL to

102 4.5 activated polaron transport

stem solely from these inter-gap states, the luminescence must fulﬁll I(t) = ∂ n(t)/n , − t 0 revealing the response to decay in the following way:

β 1 h βi I(t) = I t − exp (t/τ) . (4.9) 0 −

β with I0 = β τ− , and τ being the generalized decay rate. The luminescence decay in Mg:LNO upon super-bandgap illumination with a spectral resolution, and recordings over several orders of magnitude in both time and intensity 1 is investigated. Stretched exponentials [see eq. (4.9)] are found to ﬁt our data as well, similar to TRA measurements. In addition, our spectral PL analysis allows associating these energies to relaxation channels and recombination processes from inter-gap states in LNO. Therefore, we can introduce time-resolved PL measurements as a general tool to further understand photorelaxation properties in LNO, which apart from analysis of the activation of charge carriers in LNO and LTO appears to be necessary to further enhance the ODRT. Both properties are linked in the way, that charge carriers being excited upon ultra-fast excitation shall have a sufﬁciently large mobility to dispose thermalized energy outside the illuminated volume. Hence, the higher the ODRT the higher the charge carrier mobility within these materials.

a) b) 109 109 [s] [s] t t 108 108

107 107

106 106 temporal delay temporal delay

400 500 600 400 500 600 wavelength λ [nm] wavelength λ [nm] I [a.u.] 0 0.2 0.4 0.6 0.8 1 Figure 4.21: Time-resolved luminescence decay in virgin 7 mol% Mg:LNO for small delay a) normalized decay b). See our paper [518].

Luminescent properties of non-doped LNO have previously been reported from electron beam, X-ray, photo- and thermo-stimulated studies [396, 519], proposing a band- scheme that contains several inter-gap states. Doping congruent LNO with the lanthanide Er3+ as a local probe resulted in the measurement of the crystal ﬁeld induced energy shifts luminescence emission energy [246, 248]. Also, time resolved luminescence measurements were conducted for highly Mg-doped LNO spanning a time frame between 200 ns to 1 µs [520], however, without providing any spectral differentiation. Comparing the lumines-

1 We refer here on stretched exponentials, even though, mathematically the ﬁt function is the derivative of a stretched exponential to underline to coincidence of TRA and TRPLS measurements.

103 results cence reports in LNO two main ﬂuorescence energies were observed: One around 2.6 eV in the visible range and another around 1.5 eV in the NIR range [390, 521]. Luminescence was not only observed upon super-bandgap excitation, but also upon ultrafast multiphoton excitation in Mg:LNO [254]. The reported luminescence showed an onset close to the ODRT that resulted even in a domain contrast; virgin samples, for instance, showed a stronger luminescence in their work as compared to singly-inverted domains. It was reported the luminescence contrast to decrease under thermal annealing, being thermally activated with an energy of about 1 eV [253]. The reported threshold temperatures were similar to previous reports [96], concluding that the luminescence must be driven by the speciﬁc defect concentration and formation.

a) b) 108 108 7 7 [s] 10 [s] 10 t t 106 106 105 105 104 104 103 103 temporal delay temporal delay 102 102 400 500 600 400 500 600 wavelength λ [nm] wavelength λ [nm]

log10 I [a.u.] -6 -5 -4 -3 -2 -1 0 c) d) 1 100 1 0.8 [a.u.] 10 [a.u.] I I 102 0.6 3 10 0.4 104 0.2

luminescence 5 10 luminescence 0 107 105 103 2 2.5 3 temporal delay t [s] energy E [eV]

Figure 4.22: Time-resolved luminescence decay in virgin 7 mol% Mg:LNO for longer delay a) TRPLS graph b) best ﬁt TRPLS c) TRPL at λ = 475 nm d) luminescence spectrum. See our paper [518]. virgin lno7 We start by discussing the results of the virgin, single domain LNO7. Fig. 4.20 displays the well-known steady-state (cw) luminescence spectra for different cryogenic temperatures, while Fig. 4.21(a) reports the time-resolved photo-luminescence spectroscopy (TRPLS) of the same sample recorded at T = 150 K over the spectral and temporal range from 400 to 650 nm and 10 9 to 3 10 6s, respectively. cw and TRPLS spectra are − × − pretty identical over the whole wavelength range as shown in Fig. 4.21(b) upon normalization. Exhibiting equal decay rates as is typical to a single radiative transition process,

104 4.5 activated polaron transport we can assume an achromatic luminescence decay of Gaussian distribution with a central energy and width of Emax = 2.62 0.05 eV and w = 0.30 0.03 eV, respectively. Fig. 4.22(a) illustrates the TRPLS but recorded for up to 16 ms delay times. Surprisingly, a second transition peak pops up as best seen for t > ms. Moreover, this peak follows a stretched exponential behavior but is clearly separated from the fast radiative decay (as reported above) [Fig 4.22(c)]. The normalized luminescence decay [Fig 4.22(c)] hence may best ﬁtted to the following superposition of two stretched exponential decays:

h i βi 1 βi 4 10 I(t) = ∑ Ii t − exp (t/τi) . ( . ) i=1,2 − Fitting this distribution with eq. (4.10) readily yields τ = (12 2) 10 6 s and τ = 1 × − 2 (1.1 0.5) 100 s, as well as β = 0.48 0.07 and β = 0.59 0.08. The ratio of radiative × 1 2 strength between the two transitions measures I /I = 1 : (3.5 0.5) 10 2. Comparing 1 2 × − the modeled results in Fig 4.22(b) ﬁnally rationalizes our assumption of two independent decay processes to co-exist in LNO7.

a) b) 108 108 [s] [s] t 107 t 107 106 106 105 105 104 104 temporal delay temporal delay 103 103 400 500 600 400 500 600 wavelength λ [nm] wavelength λ [nm]

log10 I [a.u.] -4 -3 -2 -1 0 c) d) 100 1 0.8 [a.u.] [a.u.] I 101 I 0.6

0.4 102 0.2 luminescence luminescence 3 10 0 10 8 10 6 104 2 2.5 3 temporal delay t [s] energy E [eV]

Figure 4.23: Time-resolved luminescence decay in virgin 3 mol% Mg:LNO for longer delay a),TRPLS graph, b) best ﬁt TRPLS c) TRPL at λ = 475 nm d) luminescence spectrum. See our paper [518]. virgin lno3 Virgin LNO3 shows a luminescence that is by two orders of magnitude weaker as compared to LNO7, as displayed in Fig. 4.23(a) at T = 150 K. The time delay

105 results was varied here from 3 10 8 to 3 ms. Applying eq. (4.10) yields decay times τ = × − 1 (2.0 0.5) 10 6 s and τ = (1.0 0.5) 10 2 s with β = 0.52 0.05 and β = 0.70 0.05. × − 2 × − 1 2 These values that are pretty identical to the LNO7 sample. Nevertheless, the ratio of both 7 transition intensities shows an inverted behavior as compared to LNO with I1/I2 = (2.3 0.3) 10 3 : 1. The central energy is Emax = 2.62 0.05 eV with a width w = × − 1 0.28 0.03 eV, which is very close to the ﬁndings in LNO7. For larger delay times t, a gradual blue-shift and peak-narrowing is observed. Tab. 4.2 ﬁnally summarizes these two transition processes in virgin LNO3 and LNO7 samples by the comparison of all the relevant data, i.e. decay times, stretched-exponential parameters, intensities, center energies, and peak widths for T = 150 K.

4 2 Table . : Characteristic lifetimes τ1,2, stretched-exponential parameters β1,2, the equivalent inten- max max sities I1,2 , the mean energy E and width w for virgin Mg:LNO at T = 150 K. max max cMg [mol%] i τi [s] βi Ii [a.u.] E [eV] w [eV]

1 (12 2) 10 6 0.48 0.05 1 7 × − 2.6 0.1 0.30 0.03 2 (1.1 0.2) 100 0.59 0.05 (3.5 0.5) 10 2 × × − 1 (2.0 0.5) 10 6 0.52 0.11 (2.3 0.3) 10 3 3 × − × − 2.6 0.1 0.28 0.03 2 (1.0 0.5) 10 2 0.70 0.05 1 × −

linearity To exclude nonlinear temporal luminescence decay processes in Mg:LNO, e.g. associated with inhomogeneous population densities, the TRPL graphs were recorded with different pump intensities. In Fig. 4.24(a) TRPL at λ = 475 nm is given. As shown in Fig. 4.24(b) the PL signal shows a linear dependence on incident power over almost two decades, which excludes non-linear processes to be accounted for in the present TRPL study. singly-inverted lno3 and lno7 We now investigate the impact on TRPL by singly- inverting the ferroelectric polarization in LNO3 and LNO7. As shown in Fig. 4.25 the luminescence decays faster in LNO samples with 1 inverted domain conﬁguration for both × high and low Mg concentrations as compared to virgin LNO [see Fig. 4.25(b),(c) for LNO7 and 4.25(e),(f) for LNO3]. Since β is determined by the fast decaying processes at t τ, we record TRPL for up to µs [see Fig. 4.25(a), (d)] and extract the stretched-exponential parameters β = 0.23 0.03 and β = 0.44 0.04 for LNO3 and LNO7, respectively. 1 1 These values are much smaller when compared to virgin LNO [see Tab. 4.3]. Moreover, the smaller β for singly-inverted LNO samples clearly hints towards a larger variation of decay rates after poling, resulting in a reduced integrated luminescence intensity. Note, however, that no alterations in the spectral shape between any of these samples was observed.

106 4.5 activated polaron transport

a) b) 104 100 t [s]

[a.u.] 7

I 10 103 106 101 102

Ipump [a.u.] 2 1 10 10 0.025 0.25 luminescence intensity 1 108 107 106 10 1 100 temporal delay t [s] pump intensity Ipump [a.u.]

Figure 4.24: a) TRPL for different pump intensities. b) Normalized luminescence strength for different excitation powers. A clear proportional dependence can be seen, nonlinear relaxation processes can therefore be largely excluded. See our paper [518]. temperature dependence As we have seen in Fig. 4.20 the steady-state PL shows strong intensity rise while cooling. We, therefore, can assume the transitions to be thermally activated. Such thermally-activated radiative recombination processes follow an Arrhenius-like behavior described as:

1 4 11 τ(T) = Z− exp [Ea/(kBT)] ( . )

with kB the Boltzmann constant and Z the frequency factor. We, thus, inspected LNO7 samples by TRPL over the temperature range from T = 100 K to 200 K. Representative TRPL graphs are given in Fig. 4.26 for T = 100 K and 200 K, respectively. The experimental TRPLS graphs are compared to the optimal fits taking into account the stretched exponential decay with two transitions as given by eq. (4.10). One can observe a strong decrease in the second transition for lower temperatures. The normalized luminescence at λ = 475 nm is given in Fig. 4.27(a) for the given temperature range. We observe a strong change in the overall signal. The luminescence intensity was fitted again assuming two stretched exponential decays. Plotting the fitted decay times τ1,2 by means of an Arrhenius plot as is done in Fig. 4.27(c), we observe a thermally activated process for temperatures above 150 K and 140 K, respectively. ( ) ( ) The extractable activation energies are E 1 = (140 10) meV and E 2 = (110 10) meV. a a For lower temperatures a much smaller dependence on temperature results, therefore we can not apply the interpretation of a thermally excited process, as such a drastic change in slope in an Arrhenius plot does suggest a change to an non-thermally activated process. The extracted activation energy is in good agreement with previously reported activation energies in the case of 6.5 % Mg:LNO, which is detailed given in Tab. 4.4. The stretch parameters β1,2 undergo a strong alteration over the investigated temperature range [Fig. 4 27 . (b)]. Both β1 and β2 increase towards smaller temperatures, which conclusively sug-

107 results

a) b) c) 100 10−8 [s] t

[a.u.] −

0 1 10 −7

/ I 10 I

10−2 virgin × 10−6 intensity 1 inverted temporal delay

10−9 10−8 10−7 10−6 400 500 600 400 500 600 temporal delay t [s] wavelength λ [nm] wavelength λ [nm] d) e) f) 100 10−8

[s] 0

t 10

[a.u.] −

0 1 10 −7

/ I 10 I 10−1

−2 10 virgin − 10 6 intensity × −2 1 inverted temporal delay 10 − − − − I [a.u.] 10 9 10 8 10 7 10 6 400 500 600 400 500 600 temporal delay t [s] wavelength λ [nm] wavelength λ [nm]

Figure 4.25: TRPLS for 7 mol% Mg:LNO (a-c) and 3 mol% Mg:LNO (d-f). Normalized luminescence (a,d), TRPL graph virgin (b,e) and singly inverted (c,f). See our paper [518]. a) b) 10 8 10 8 a) b) 78 78 [s] 10 [s] 10 t t 7 7 [s] 6 [s] 6 10 10 t t 56 56 10 10 45 45 10 10 34 34 10 10 temporal delay 23 temporal delay 23 10 10 temporal delay 2 400 500 600 temporal delay 2 400 500 600 10 10 400wavelength500 [nm] 600 400wavelength500 [nm] 600 c) d) 10 8 wavelength [nm] 10 8 wavelength [nm] c) d) 78 78 [s] 10 [s] 10 t t 7 7

[s] 6 [s] 6 10 10 t t 56 56 10 10 45 45 10 10 34 34 10 10 temporal delay 23 temporal delay 23 10 10 temporal delay 2 400 500 600 temporal delay 2 400 500 600 10 10 400wavelength500 [nm] 600 400wavelength500 [nm] 600

wavelengthlog10 I [a.u.] [nm] wavelength [nm] -5 -4 -3 -2 -1 0 log10 I [a.u.] Figure 4.26: Normalized TRPLS graphs for 7-5 mol%-4 -3 Mg: -2LNO-1 for0 different temperatures T = 200 K (a) and T = 100 K (c). The ﬁtted data is given in (b,d), respectively. See our paper [518].

108 4.5 activated polaron transport

Table 4.3: Luminescence decay parameters for different doping concentrations and preparations at T = 130 K.

Mg [mol%] preparation β1 virgin 0.40 0.05 3 1 poled 0.23 0.03 × virgin 0.52 0.07 7 1 poled 0.44 0.04 ×

gests, that for smaller temperatures hopping transport appears to be a less correct description for the transport and recombination process. The absolute strength of both transitions I1,2 drastically changes over the temperature range, for higher temperatures the slower transition is mainly determining the overall relaxation, whereas for lower temperatures the faster transition determines the relaxation [Fig. 4.27(c)].

4 4 Table . : Comparison of time-resolved luminescence and absorption at different wavelengths λabs in above-threshold Mg:LNO (1) Mg [mol%] λabs [nm] Ea [eV] Z [Hz] Ref. 488 0.16 0.05 (3.0 1.0) 108 [386] 6.5 × 405 0.14 0.05 (1.0 0.5) 108 [386] × 7 0.14 0.02 (3.0 0.5) 108 [518] ×

Our combined spectral and temporal analysis allows us to exclude recombination processes that might be invoked in LNO upon illumination. Mainly, this refers to radiative relaxations following a mono- or biexponential decay as well as luminescence energies reported with different photon energies. LNO is known to incorporate impurities even for a nominally non-doped, pure stoichiometry. In congruent LNO we most commonly ﬁnd Fe2+/3+ ions with a concentration of typically 5 ppm [318]. However, neither the impurity concentration nor the luminescence spectra reported for highly Fe-doped LNO can explain the PL data that we report here, since the center luminescence energy was observed at E 1.4 eV in Fe:LNO [521] ≈ and undergoes a monoexponential decay. 4+ 4+ Notably, we can exclude the presence of NbLi :NbNb bipolarons, which have only been 2+ observed in reduced LNO. Upon Mg-doping MgLi defects are formed, reducing the 4+ number of NbLi anti-sites, which in turn dramatically affect the reported optical dam- 4+ age resistance threshold (ODRT) as the concentration of light-induced NbLi bound small polarons reduces [351]. An increase in luminescence by two orders of magnitude was 520 2+ reported above ODRT [ ]. A possible explanation is the formation of MgNb defects at these conditions. It is very likely that the induced disorder results in the increase in PL.

109 results

a) 100

1 10 [a.u.]

I 2 10

3 10 100 K 120 K 4 10 140 K 160 K luminescence 180 K 200 K 5 10 6 5 4 3 2 10 10 10 10 10

temporal delay t [s] 1 1000/T [K ] 5 6 7 8 9 10 b) 1 c) 0.8 6 0.6 10 µ s ] [

0.4 2 1 ⌧ 0.2 2 105 0 200 180 160 140 120 100 1 100 10 T [K] 100 1 1 10 µ s ] 10 [ / I 2 1

I 2 10 ⌧ 3 10 1 4 10 200 180 160 140 120 100 200 180 160 140 120 100 T [K] T [K]

Figure 4.27: Temperature-dependent analysis of TRPL curves. a) TRPL curves at λ = 475 nm for a temperature range from 100 K to 200 K. Hereby, dots represent experimental data and lines the ﬁtting curves according to eq. (4.10). b) Temperature dependence of the 1 2 ratio of fast ( ) and slow ( ) transition luminescence intensities I2/I1 and temperature dependence of the stretch parameters βi for fast (i = 1) and slow (i = 2) transition. c) Arrhenius plot of the characteristic lifetimes τ1,2 of the fast and slow decay versus (1) (2) temperature. Theoretical ﬁt with E = (140 10) meV and E = (110 10) meV. See a a our paper [518].

110 4.5 activated polaron transport

a) b)

Nb4+ Nb I(, t) radiative 2 recombination O (lower) hopping 1 O (upper) Li 3 O Nb Nb y Li O vacancy z x

Figure 4.28: a) Schematic of radiative recombination of electron and hole polaron in LNO after hopping transport b) Different radiative recombination properties: 1 single-step hopping recombination and 2 several step recombination after several hopping steps 3 . See our paper [518].

Moreover, phonon-coupling was readily shown to depend on Mg-doping concentration, and shows a step-like increase at the ODRT [350]. The observed non-exponential relaxation dynamics are in good agreement with theoretical expectations for the localized carrier dynamics in oxide crystals, even without structural or energetic disorder. A thermally-activated diffusive hopping transport under trap saturation can result in stretched-exponential carrier concentrations [386, 522]. Notably, in this context the hole polaron was treated as traps with free electron polarons being allowed to distribute homogeneously over the lattice. In our work here, the recorded PL in Mg:LNO follows a stretched-exponential decay. Taking the emission energy of approximately 2.6 eV and the proposed energies of polaronic interband-gap energies in LNO [384] into account, we conclude that only a polaronic recombination of the free electron polaron 4+ NbNb and the hole polaron O− can fully explain our time-resolved luminescence spectra. A two-step recombination process, as was apparent in the presented TRPLS, has already been reported for reduced LNO [398] by means of room-temperature TRA. It was associated to recombination processes of free and bound electron polarons. However, in our case, we can exclude bound polarons to be related to the slow recombination process, since no red-shifted luminescence for longer time delays was observed. A two-step stretched exponential decay was theoretically discussed as a result of the site-correlation effect in random-walk Monte Carlo simulations [399]. In the case of site-correlation the walk starts directly in the vicinity of the trap. As we assume the luminescence to be a result of recombination of electron and hole polaron formed upon light-induced electron-hole pair relaxation, where one typically generates a large fraction of electron and hole polarons in direct vicinity, only a single to a few hopping events are required for recombination, resulting in the fast transition. Only a small fraction of generated electron and holes are further

111 results separated before relaxation, resulting in trap saturation and the stretched-exponential decay. As a larger number of steps for recombination is required, it gets less probable to recombine for lower temperatures. We observe this behavior as depicted in Fig. 4.27(c). For low temperatures the luminescence is mainly governed by the fast transition, only requiring a small number of hopping events. The main idea of the radiative recombination process observed in our luminescence measurements is given in Fig. 4.28. A one-site hopping recombination gives a fast relaxation, which is understood in site-correlation, whereas the slow relaxation can be attributed to trap saturation for few to many hopping events. The difference in luminescence strength and decay characteristics of LNO3 and LNO7 2+ can be attributed to the influence of MgNb defects resulting in local distortions. Above 2+ ODRT a significant amount of MgNb centers is generated with a larger defect potential 2+ 351 2+ energy as MgLi [ ]. Concurrently, the MgNb centers give rise to larger perturbations of the local crystal field. This would explain the reduction of the stretch parameters upon ferroelectric poling, e.g. as a result of increased distortion. This corresponds to previously observed domain-specific multiphoton luminescence [253] and suggests an incomplete poling process, only to be completed upon thermal annealing. For free small polarons in LNO, Ep was reported to measure approximately 0.54 eV [384]. Assuming an harmonic oscillator potential this results in an hopping transport activation energy Ea = Ep/2 of 0.27 eV, which is comparably larger than the observed thermal activation in this measurement. Nevertheless, the latter assumption constitutes a generalized case and, thereby, systematically overestimates the activation energy, when compared to more realistic potential curves. The activation energies for polaron recombination reported here are in similar good agreement with electrical conductance measurements in ferroelectric DWs in the investigated material [25]. Accordingly, for lower temperatures a decrease in the activation energy was observed [523]. For lower temperatures the transition times do show a much smaller increase upon cooling, which can not be explained by the concept of thermal activation. A possible explanation of this phenomenon could be an increase in internal conversion. Hence, for the first time measurements on the polaron luminescence in Mg-doped LNO with both high spectral and temporal resolution were conducted. A clear significance of two independent radiative relaxation components was observed. The recorded temporal luminescence decay is a consequence of the light-induced hopping transport that finally results in the radiative recombination of free electron and hole polarons. Hence, further understanding for induced distortion upon ferroelectric poling, resulting in a change in radiative relaxation dynamics, is obtained. Moreover, this investigation suggests, that time-resolved photo-luminescence spectroscopy (TRPLS) can be applied as a complementary method to time-resolved absorption (TRA). For doped LNO, TRPLS provides an indispensable method to quantify and op- timize the optical damage resistance as needed for instance in high-power applications especially for nonlinear optics as the observed luminescence at E = 2.6 eV is an optimal reference tool to investigate the optical damage resistance. But moreover, the extracted

112 4.6 high voltage treated lno

a) b) 100 100

10 1 10 1

2 [a.u.] 10 [a.u.] 2

I I 10 0.2% Hf:SLN 3 10 1% Mg:SLN 10 3 0.5% In:SLN 2.5% Mg:SLN 0.5% Sc:SLN intensity 4 intensity 10 4.5% Mg:SLN 0.085% Zr:SLN 10 4 5% Mg:CLN 1% Zn:SLN 5 10 5 10 10 6 10 5 10 4 10 3 10 2 10 6 10 5 10 4 10 3 10 2 time delay t [s] time delay t [s]

Figure 4.29: a) Time-dependent luminescence at Mg-doped CLN and SLN. b) Time-dependent luminescence at SLN doped with other dopants. See our paper [518]. activation of small polarons is important to understand both the photo-induced as well as charge carrier injection driven conductivity in LNO and LTO. The TRPLS measurements were extended to differently doped SLN single crystals. The results are given in Fig. 4.29. The TRPL graphs of SLN crystals with different Mg-doping concentrations are given in Fig. 4.29(a). As one can clearly see, for large doping concentrations, the second transition intensity decreases with doping concentration, while crystals being doped with a variety of dopants right at the ODRT show a very similar temporal decay, which can be seen in Fig. 4.29(b). Taking these informations together, this is a clear indication of a decrease in bound polaron formation upon super bandgap illumination for larger doping concentrations. The larger amount of free small polarons similarly increases the mobility of the majority of charge carriers in the system. As it is known from nonlinear absorption measurements, free polarons are formed only about 100 fs after electron-hole pair formation and the relaxation to bound polarons happens about 400 fs after excitation [308], hence, measurements of the radiative relaxation of these charge carriers give an important insight into the conductive properties of these crystals, mainly the mobility of the excited charge carriers, as the number density of created electron-hole pairs and thus formed small polarons are kept constant in these measurements. Hence, if in conductivity measurements an activation energy of about 140 meV to 200 meV is detected in CLN this would clearly pinpoint towards a condition in which the majority charge carriers driving the current are really free or bound small polarons.

4.6 high voltage treated lno

In this section we discuss the implications of CDW formation and an experimental procedure that allows enhancing DWC in Mg:LNO by 3 to 4 orders of magnitude as developed

113 results

a) ~v b) ~v 2

c) d) ~v ~v

1 20 µm I [a.u.] 0 1 0pA 200pA e) contacted DW f) g)

1 2 3

corners with higher inclination

50 µm s a 50 µm I [a.u.]t2t h2hI [a.u.]1.6 1.6 0 1 0 1

Figure 4.30: Comparison of local conductivity and DW inclination in high voltage treated LNO. a)-d) cAFM with a 10 V bias upon 4 different AFM probe directions [marked by ~v to elucidate the ’smear-out’ effect to be independent on the crystal axes or any other inﬂu- ence than the contact movement, the three corners are marked revealing signiﬁcantly larger currents e)-g) Polarization charge texture on the conductive DW viewed under three different angles, highlighting the three corners with stronger inclined DWs near the probe contacted surface.

114 4.7 conductive domain walls in exfoliated thin-film lno

2 by Christian Godau . Such a procedure enables to readily drive a current through wafer- thick LNO DWs even in the dark, i.e. without needing any external stimuli such as super bandgap illumination, which has been discussed previously [25]. This method is based on stressing the LNO DWs by the application of a moderate electric field of up to 1 104 Vcm 1 using planar chromium (Cr) electrodes. This field, × − significantly smaller than the coercive field, physically moves and bends the DWs (as directly visualized by CSHG microscopy) until conductive paths are formed up to the LNO/electrode interface, resulting in a release in electric field being applied on the electrodes. A major advantage of this procedure is, that it provides DWC being tunable to any desired value. As-tuned DWs remain stable for further investigations over weeks; while CSHG is used to map the DW inclination close to the sample surface, piezo-response force microscopy (PFM) and conductive atomic force microscopy (cAFM) are applied in order to allocate the domain/DW polarization and transport currents across domains and DWs, respectively. As a result, it was possible to directly correlate the DWC to the DW inclination angle α in these highly conductive DWs. The local conductance measurements on such high- voltage treated LNO crystals and the associated polarization charge texture by 3D CSHG microscopy are compared in Fig. 4.30. Is is clearly visible, that the larger the local DW inclination close to the surface the larger the detected current. A closer analysis reveals a proportional dependence of local conductivity and inclination angle. Hence, this experiment delivers already the second experimental evidence, that larger inclination, yet even far away from 90°, yields large conductivities. In the first experiments reported by Schröder et al. photo-excitation was still a prerequisite. The cAFM measurements typically reveal ’smeared-out’ currents around the DW. This effect only depends on the speed and direction of the probe movement, which is illustrated by the four motion directions of the probe. This effects was minimized by the use of Pt wire-etched tips.

4.7 conductive domain walls in exfoliated thin-film lno

Exfoliated CLN single crystalline thin-films as distributed by NanoLN are an excellent improvement to study large conduction at CDWs. As being prepared from single crystalline wafer material and almost strain-free attachment similar material properties are expected. Only, the He+ ion bombardment could potentially lead to altered properties. Despite, these strong similarities to wafer-thick LNO the ferroelectricity of these films is still to be proven. In Fig. 4.31 domains were written into the film. Directly after writing the film, a PFM amplitude contrast is visible. Yet, after some time, e.g. 15 h, the amplitude contrast vanishes and a clear DW contrast is visible as expected for ferroelectrics. The amplitude contrast can be attributed to surface-charging, as the ferroelectric can act as an electret.

2 The results of these experiments have already been submitted to a peer-reviewed journal.

115 results

20min 15h

5 µm I [a.u.]PFM j 5 µm I [a.u.]PFM A 0 1801 00 pm 8001 pm Figure 4.31: PFM analysis of written domain sections at the contact resonance by DART. a) PFM phase b) PFM amplitude. The PFM amplitude clearly undergoes a clear change in contrast towards the expected DW contrast.

The clear PFM phase contrast over 180° remains. Furthermore, the stability of these domains was checked to remain for at least as long as 3 months, which is contradiction to very recent reports [524], which observe a stability of only 25 h. In our investigations such little stability was never observed. Similarly much clearer switching spectroscopy PFM hystereses were observed. This underlines, that the defect density or electrode conﬁgura- tion plays a major role in a longterm application. The smallest domain to be written in the 600 nm thick ﬁlms was 15 nm at a bias voltage of 23 V.

4.7.1 Conductance maps

The written domains were investigated by cAFM concerning the emergence of DWC. And, indeed, this phenomenon was observable in these CLN films. In Fig. 4.32 the comparison of PFM and subsequent cAFM measurements are given, which reveal a prefect match of the derived DWs from PFM and the conductive areas in the film. Similarly, a ’smear-out’ is visible. The extraction of the local current detected for a range of external bias voltages is given in Fig. 4.32(d). One can observe a very strongly nonlinear behavior upon increase of the bias voltage. The complete set of cAFM scans is given in Fig. 4.33, however, these images are less intuitive for higher voltages as the ’smear-out’ gets distinctly stronger. Hence, the extracted results shall not be understood too quantitatively. To reveal better insight into the conduction properties of CDWs in LNO thin films local I V measurements were carried out. − These are given in Fig. 4.34. One can still see a very strong nonlinear behavior in these

116 4.7 conductive domain walls in exfoliated thin-film lno

~v ~v a) b)

A F D C E

1 µm I [a.u.] 1 µm I [a.u.] 00 pA 1601 pA 00 pA 1401 pA c) d)

103

102

[pA] 101 A I B C 100 D E F 10 1 0 2 4 6 8 1 µm I [a.u.] PFM j V [V] 0 ext 0 1801

Figure 4.32: DWC in CLN thin films. a), b) trace and retrace cAFM scans at a bias voltage of 3 V, c) domain configuration rendered by PFM, d) I V curves extracted from cAFM scan − stack. curves given in Fig. 4.34(a) [spots are marked in Fig. 4.34(d)]. Moreover, a memristive behavior is visible, i.e. forward and backward curves do not completely overlap. The temporal stability measurements show a small increase in current over time, consistent with the I V measurements. Influences by drift can be neglected. Spot 1 is taken as − reference and shows no increase in conductivity. An example is given in Fig. 4.34(b). How- ever, after e.g. 200 s a saturation is visible with an increase of about 50 % at an external bias voltage of 10 V. Local probe measurements on these CDW reveal a stable conductance still after 3 h with a minute increase over the measurement of 5 % between 1 h and 3 h. For such long measurement times, drift cannot be excluded. If the I V currents are − plotted in a lin-log plot deviations from the results in scanning-type cAFM measurements are visible, yet, a good agreement can be observed. In the next section on macroscopic measurements the current behavior will be further discussed.

117 results

1 V 2 V 3 V 4 V

5 V 6 V 7 V 8 V

~vslow Figure 4.33: cAFM scans for a set of external bias voltages Vext from 1 to 8 V. For very large bias voltage a very signiﬁcant ’smear-out’ is visible. This smear-out is both connected to fast axis movement as well as to the slow axis movement direction, which is labeled by the vector of vslow.

If a negative bias voltage is applied, even for voltages clearly smaller as the coercive voltage, e.g. 3 V, current spikes are observed in thecAFM scans, which are not repro- − ducible in trace and retrace scans and cover the whole beforehand inverted domain area. A subsequent PFM measurement reveals, the domain to be partially inverted. The PFM amplitude does not show clear dips at the DW, as expected, but gradual changes. Yet, at areas where there is still a local dip, the cAFM still reveals a current. Hence, the following picture is expected to be present: even though, the film is very thin, surface domains are created, which interrupt the contact path. Such a behavior was very recently shown to be present in PZT films by TEM analysis, and, hence, would not be untypical in ferroelectric films [151]. Further confirmation for the existence and stability of CDWs in ferroelectric thin films is obtained by phase-field simulations. Phase-field simulations of the ferroelectric domain configuration are widely known for BTO, BFO, and PZT [45, 526–530]. Here, we give the first investigations of domain kinetics in LNO films. A biased probe tip was simulated upon the ferroelectric LNO film with a thickness of 100 nm and a bias voltage of 50 V. Within these simulations the voltage is ramped up until the given bias voltage. A nuclei is formed [shown in Fig. 4.36(a) 1 ], which protrudes through the single crystalline film. Finally, the inverted domain reaches the rear surface and further grows sideways. The formed domain stabilizes, the equilibrium case is given in Fig. 4.36(a) 2 . Afterwards, the bias voltage is released. The domain relaxes, yet, a stable CDW is formed with an inclination angle of even 7°, which is given in 3 . With a larger gradient coefficient the CDW will collapse into a NDW to reduce stronger polarization gradients as a result of

118 4.7 conductive domain walls in exfoliated thin-film lno

a) b) 1.6 1 bulk 2 1.5 1 3 DWs 4 1.4 [nA] [nA] temporal I I 0.5 1.3 stability 1.2 @ pos. 4 0 1.1 0 2 4 6 8 10 0 200 400 600 V [V] t [s] c) d)

100 1 1 2 3 1 4 2 10 4 [nA] I 2 10 3

3 10 0 2 4 6 8 10 1 µm I [a.u.] V [V] 00 pA 1601 pA Figure 4.34: a) Local I V measurements on CDWs at four marked spots, b) temporal development − of the current at spot 4 at an external bias voltage of 10 V, c) logarithmic plot of the detected current, d) cAFM scan at an external voltage of 3 V with marks referring on the spots investigated with the AFM probe at rest. inclination. The physical reason for such effects can be e.g. pinning by local defects. The gradient energy coefficient of LNO and LTO are significantly smaller than for other well-known pure ferroelectrics. Whereas for LNO g = 4 10 11 Nm4C 2 and for LTO × − − g = 2.8 10 11 Nm4C 2 [525], the coefficient of BTO is between 5.1 10 10 Nm4C 2 [531] × − − × − − and 9 10 10 Nm4C 2 [532], for PZT between g = 1.1 10 10 Nm4C 2 [533, 534] and × − − × − − g = 4.5 10 10 Nm4C 2 [535], for BFO between g = 6 10 10 Nm4C 2 [536] and g = × − − × − − 8 10 10 Nm4C 2 [537, 538], and of lead titanate/PbTiO (PTO) g = 1.2 10 9Nm4C 2 × − − 3 × − − [539]. Whereas a small DWC was reported for 180° DWs in PZT, no report has shown local DWC at these in PTO and BTO films so far, besides a recent report on increased leakage in multidomain films of PTO [540]. This spread in gradient energy coefficient could explain the spread in DWC, however, it has to be stated, that the coefficients are generally not well known and influences of anisotropy can be influential. Yet, the smaller value g for LTO in comparison to LNO is in good agreement with the observed larger inclination angles.

119 results

a) - 5 V b) - 5 V c)

1 µm I [a.u.] 1 µm I [a.u.] 1 µm 0 1 0 1 IPFM[a.u.]A 0 pA 15 pA 0 pA 55 pA 0 1 ~v ~v d)

Figure 4.35: a),b) Trace and retrace cAFM scan images at an external bias voltage of 5 V, respec- − tively, c) subsequent PFM scan after cAFM scan with an external bias voltage of 5 V, − d) typical polarization representation across the ﬁlm after cAFM scan with a negative external bias voltage below the coercive voltage.

4.7.2 Resistive switching by conductive domain walls

As we have seen, it is possible to write and erase CDW in such CLN films. This could be 3 a new paradigm for resistive switching . Yet, to apply this property on resistive switching memories, it is necessary to proof such effects can similarly be reproduced using homogeneous electrodes. Hence, the question arises, is it possible to create CDW under a small test electrode? We applied Cr/Au test electrodes with a size of A 2000 µm2. The very ≈ first proof of concept measurement with a probe station is given in Fig. 4.37. As is visible, the absolute value of the current driven through the metal-ferroelectric-metal stack can be tuned over five orders of magnitude with switch on and off voltage of 50 V and 30 V and − a probe voltage of 10 V. To further analyze the properties of the conductance, I V sweeps were taken. These − are given in Fig. 4.38(a). We can observe a very strong increase over five order of magnitude in current at a very defined set voltage Vset = 21.05 V with an accuracy of ∆Vset/Vset = 3 10− . This value of Vset is not only reproduced for a single device, but also for 50 individual devices on a single wafer, which clearly underlines the very precise and reproducible behavior of single crystalline resistive switching devices. For larger voltages than Vset no significant hysteretic behavior can be seen in these sweeps. Yet, on a linear scale a small hysteretic behavior with an increase in current of less than 20 % is visible, which shows the DWC to show to be weakly memristive. Up to a a voltage of 3 V, a very symmetric − current-voltage relation can be observed. Yet, for larger negative biases the absolute value of the current saturates and is not stable anymore, but reduces with time. The cycles are

3 Resistive switching refers to the physical property of a dielectric to suddenly change its resistance under the action of a strong electric ﬁeld or current. The change of resistance is non-volatile and reversible [541].

120 4.7 conductive domain walls in exfoliated thin-film lno

50 V 50 V a) 1 2 3

x non-equilibrium equilibrium equilibrium

z t x y

Figure 4.36: Two- and three-dimensional phase field simulation of CDW formation under an AFM tip [a) for a 100 nm thick film upon 50 V and b) for a 20 nm thick film upon 20 V, respectively ], 1 nucleation formation upon bias voltage, 2 through domain upon bias voltage application, equilibrium condition, 3 equilibrium domain condition after retraction of biased probe tip. Simulation with µ-pro by Bo Wang (Long-Qing Chen group, Penn State University) for a 100 nm thick LNO crystal. The calculation was performed under the assumption of a small isotropic gradient energy term with a gradient coefficient g = 4 10 11 Nm4C 2 as derived from the typical DW thickness × − − of 2 nm [525].

121 results

6 10

7 10

8 10

[A] 9

| 10 I | 10 10

11 10

10 12 0 200 400 600 800 1,000 1,200 1,400 t [s]

Vread,off = 10 V Vswitch,on = 30 V V = 10 V V = 50 V read,on switch,off Figure 4.37: Proof of concept measurement on resistive switching in CLN thin ﬁlms. The resistance of the ﬁlm was switched on and off at 50 V and 30 V, and measured at 10 V. −

obtained with a cycle frequency of 1.5 mHz. This behavior is very similar to the previously observed back-switching upon current injection from the top electrode at small voltages as was shown in Fig. 4.35. Hence, it is assumed, that upon the application of a negative bias t2t DWs are formed or domain inversion is invoked, thus there to be no conductive channel anymore, which prohibit current flow similar to the case being schematically shown in Fig. 4.35(d). 4 Before switching on, one can observe no current larger than 10 pA up to a bias voltage of 200 V, which corresponds to an electric field of 3.4 MV/cm, which is comparably − large for dielectric layers. The resistance in this case is at least 20 TΩ. With even more sophisticated measurements resistances of at least 1 PΩ were observed. This similarly holds for positive read-out voltages. Hence, the high resistance state (HRS) is expected to have a resistance of at least 25 TΩ, the low resistance state of around 10 MΩ. Both values are good in the sense, there to be a small power loss for applications as non-volatile memories in the on-state. In Fig. 4.38(b) the half cycle I V are shown upon the boundary condition the current − having reduced to 10 8 A at 210 V. We can observe the reproducible set voltage. How- − − ever, below this specific voltage a slight increase in current can be observed. In general, this current, which deviates from the first cycle is the smaller the lower the current at V = 210 V. It saturates after several cycles. Several reasons are possible, e.g. deep − traps, which are incorporated into the film upon large current flow. Hence, for cycled

4 This value is the lower limit for current detection of the applied Keithley 2400, yet measurements with a Keithley 6517B revealed even smaller currents smaller than 200 fA at a voltage of 200 V, which corresponds − to a resistance of at least 1 PΩ.

122 4.7 conductive domain walls in exfoliated thin-film lno

a) b) 5 LRS 10 6 10

7 10 set Vset

[A] 8 [A]

| reset 10 I | I 9 10

10 Vread 11 HRS 10 10 200 150 100 50 0 50 0 5 10 15 20 V [V] V [V] c) d) 6 10 4

7 LRS 10 104 resistance Projected lifetime [A]

8 [ µ A] 8 I (80%): 10 s 10 window I 2

9 10 HRS 10 0 10 0 1 2 3 4 5 1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 10 Cycles t [s]

e) f) 10–3 101 0

0.4 LRS 10–2 [s] HRS -1 0.2 10 write t 10–1 probability [%] 0 0 I [a.u.] 11 10 9 8 7 6 5 100 10 10 10 10 10 10 10 20 30 40 50 60 I [A] Vwrite [V] [a.u.] Figure 4.38: Investigation of switching behavior, endurance, stability, and tunability of resistive I switching of the Pt/LNO/Cr/Au stack with a contact area of 2000 µm2. a) full I V − cycle ( f = 1.5 mHz) with a very defined set voltage V = 21.05 V (∆V /V 10 3), set set set ≈ − hence a comparably small electric field Eswitch,off = 0.3 MV/cm and strongly rectifying behavior without significant leakage upon an electric field of Eswitch,off = 3.4 MV/cm with a resistance of >20 TΩ, b) switch-on I V cycle with constant switch-off voltage − V = 210 V, c) endurance of high resistance and low-resistance state (HRS, switch,off − LRS, respectively) over at least 105 cycles with a resistance window of >104 and a read voltage of 10 V, d) time stability of low resistant state over 104 s, which yields an 80 % reliability over 108 s or 3 years, e) probability of the current in HRS and LRS at 10 V for 50 tested devices on the same single crystalline thin-film f) tunability of read- out current Iread,on under modulation of writing time twrite and writing voltage Vwrite. The read-out current Iread,on reduces for larger writing voltages Vwrite,on. Iread,on is the average value over 100 writing cycles each.

123 results

a) b) c) ] [ ] ] 1 V [V] V [V] [ [

' 10 d [nm] ' 5 ' 20 5 40 10 10 20 15 15 50 20 20 100 20 without bias 10 without bias inclination angle inclination angle 0 1

0 ﬁnal inclination angle 10 103 104 102 103 104 0.2 0.4 0.6 0.8 1 time steps [a.u.] time steps [a.u.] electric ﬁeld [V/nm]

d) e) 1 10 2 10 [ µ A] [ µ A] read read I I 2 10

3 10 20 30 40 50 60 70 20 40 60 80 100

Vwrite [V] Vwrite [V] Figure 4.39: Comparison of extracted CDW inclination angles by phase field simulation for different poling voltages and film thickness. Simulation with µ-pro by Bo Wang (Long-Qing Chen group, Penn State University) with similar conditions as discussed to Fig. 4.36. a), b) simulated time-dependent inclination angle upon a singular external bias voltage with a thickness of 50 nm and 20 nm, respectively c) equilibrium inclination angle upon external electric field, d), e) measured current Iread at a bias voltage of 5 V upon ordered and random writing with voltages Vwrite. measurements high voltage treatment was kept as small as possible. In Fig. 4.38(c) the endurance of such cycling is shown. The HRS is created on application of Vswitch, on = 21.1 V, hence only slightly above Vset. The system is released into the LRS upon application of V = 210 V. This results in a very enduring resistive switching device over at least switch, off − 105 cycles. The temporal stability of the HRS is given in Fig. 4.38(d). As is visible, the conductance is very stable over 104 s. From this data, assuming an exponential decrease and a minimum current of 80 %, a stability over 108 s can be predicted. Similar measurements with an AFM tip revealed a stability of the current on CDWs over at least 3 h, which is about the longest time to measure due to probe drift. In the phase-field models it was observed, that the stable CDW inclination angle can be tuned by the inversion voltage. In Fig. 4.39(a)-(c) these results can be seen for thinner films of 20 nm and 50 nm thickness. The simulation was carried out until no change in the inclination was visible anymore, after 2 104 steps the bias voltage was switched off × without any significant change in the inclination angle. We can derive, that the stable inclination angle depends exponentially on the electric field for inversion. Similarly, the writing time should have an influence on the inclination. The longer the writing voltage is applied, the smaller the inclination angle. Hence, the read-out current was investigated

124 4.7 conductive domain walls in exfoliated thin-film lno

upon a series of writing voltages and times. We can observe a very similar behavior, which is given in Fig. 4.38(e). The read-out current can be tuned by almost two orders of magnitude in this case. It can be severely increased by the application of much shorter writing pulses, as the dynamic is expected in a temporal range of about 100 ns. To compare the phase-field modeling the read-out current at 10 V upon structured and randomized writing is depicted in Figs. 4.39(d),(e), which similarly shows a decrease of about two orders of magnitude in current for a ratio of 5 in voltage. We need to compare these results in this way, as phase-field modeling is known to overestimate the coercive field, hence the calculated electric fields are significantly larger than in real terms. It is expected this to mainly be due to local nucleation. In Fig. 4.39(e), we can observe, that the read-out current for large writing voltages does not have a significant effect on the erase voltage as long as it is larger than the coercive voltage, yet for very small writing voltages we can see, that the current which we observed emerging at about 10 V are similarly present and are not erased completely.

Listing 4.1: Standard conduction types (CoTs) and their linearizations to d log (I)/dV curves with Vbias internal bias voltage, ND internal donor density, me mass of electrons, Φ the work function, εh the high frequency permittivity, ε0 the static permittivity under abrupt junction approximation [542].

q 3/2 Fowler-Nordheim tunneling (FN) d log I = 1 + 4π 2meεε0 Φ dV V+Vbias 3he 2eND (V+Vbias) 1 3 /4 d log I e NDe 3/4 Schottky thermionic emission (STE) dV = 4kT 2 2 2 (V + Vbias)− 8π ε0εhε h q i d log I 1 1 e3 1/2 Poole-Frenkel emission (PF) = + V− dV V kT 4πε0εh h i d log I 2 Space-charge-limited current (SCLC) dV = V

The CoT was investigated by I V measurements. As already discussed in Sec. 2.1.4, − it possible to linearize these to judge which CoT is present. In Fig. 4.41(a) the I V − measurements are plotted according to SCLC theory, which predicts I V2 in log-log ∼ plot. A fit is given, which shows a perfect fit over almost two voltage magnitudes. In Fig. 4.41(b) the I V cure is linearized according to STE. A good fit can be seen in a − medium voltage range, but both for small and large voltage significant deviations are visible. Similarly, in the case of PF linearization given in Fig. 4.41(c), we obtain a reasonable fit for medium voltages, but discrepancies for very low and high voltages. Yet, it would be not sufficient to formally exclude these two CoTs. In common FN absolutely no fit can be seen. In the LRS we obtain a flat line. Hence, three CoT still have to be considered. Thus, we used an interpretation as introduced by Maksymovych et al. in abrupt 5 junction approximation [542] to further distinguish. Even though, the main advantage of this method is to understand impedance upon a DC bias, it is still helpful to analyze the

5 An abrupt junction is an approximation of a pn-junction with an abrupt doping proﬁle.

125 results

a) b) SCLC plot STE plot 6 6 10 10 [A] [A] 8 8 I I 10 10 subsequent subsequent 1st stat. cycles stat. cycles 10 10 st 10 cycle 10 1 cycle

100 101 2 4 6 V [V] pV [V]

c) d) 6 10 PF plot 100 [V] 10 8 STE subsequent

[A / V] stat. cycles SCLC 10 [A]) / dV

/ V 10

I 1 10 st 1 cycle log ( I

12 d 10 2 4 6 100 101 pV [V] V [V]

Figure 4.40: CoT analysis on CDWs in LNO a) SCLC I V plot, b) STE I V plot, c) PF I V plot, − − − d) derivative plot as introduced by Maksymovych et al. [542]. The I V curves show − the first cycle and the stationary cycles after 20 cycles. The off-state was obtained by applying V = 210 V until the current reduced to 10 8 A. − − plain DC I V curves as the d log (I)/dV is independent on most parameters, especially − the contact area, it is, hence, more robust especially for temperature dependence and it is possible to differentiate between FN and PF. Hence, in Fig. 4.40(d) the I V curves − in a temperature range T between 300 K and 340 K are given. With the in List. 4.1 given 1 linearizations we reveal, that only SCLC and PF [both d log(I)/dV V− ] can describe 3/4 ∼ such a curve, whereas STE [d log(I)/dV V− ] cannot describe this behavior. Yet, in ∼ 1/2 the case of PF also a very strong component ∝ V− should be present, which can not be seen in the measurements. Moreover, even though PF is a typical model to describe the current in metal-insulator-metal stacks, however, in recent time the realization of PF- linearized earned broad skeptics. This is because, unphysically large hopping rates are necessary and the typical PF linearization of I V curves cannot be realized even upon − the standard model of distributed traps and thermionic emission [543]. Hence, for dielectric thin films, PF got a speculative description of current transport in thin films within the recent years. However, as already been shown especially in Sec. 4.5, small polaronic transport has been observed both optically and electronically. Small polaron transport in thin films was so-far typically treated upon SCLC [544], which is typically assumed to be the main contribution to room-temperature conductivity in LNO. The I V SCLC curve −

126 4.7 conductive domain walls in exfoliated thin-film lno

fit is almost perfect over two orders of magnitude and, thus, significantly better than those of PF and STE. For now, we solely discussed electronic transport. Yet, especially resistive switching is believed to be similarly related to ionic movement. Hence, it is necessary to further rule out scenarios of mixed electronic-ionic conduction. We can follow here the argumentation given by Maksymovych et al. for conductive transport in PZT [542]. We can start with the Nernst-Ernstein equation, which is used to analyze whether ionic conduction is a significant contribution and given by

∆ = τeDE/kT. (4.12) with τ the diffusion time and D the diffusion constant. Using a conservative value of D 10 18 m2/s, which is derived from experimental values of lithium transport in Li Si ∼ − x [545], the estimated ionic movement over 100 ms is about 0.8 nm at a field of 2 107 V/m ∼ × and room temperature T = 300 K, which is about three orders of magnitude smaller than the film thickness. Reported values for ionic transport in LNO (e.g. Li, H, D, Na, Mg) diffusion constants at elevated temperatures and interpolated to room temperature are significantly smaller [546]. Hence, we can exclude ionic current to have a major share. Further, switching processes by ionically driven Schottky barrier modulation as in BFO [547] is unreasonable, which is, moreover, not consistent with the abrupt increase in conductivity at the very definite set voltage. A further evidence, that SCLC is the main conduction type (CoT) to be present is the apparent temperature dependence. The exact Poole-Frenkel I V curve should follow √aV − I ∝ Ve /kT and hence a change in the slope should be expected to be present for different temperatures, which is, yet, not present as can be seen in Fig. 4.41. We can conclude, that SCLC seems to be the relevant CoT. The conduction in DWs, thus, follows the Mott- Gurney equation (Child’s law) [548]:

9εµ(T)V2 I(T) = A (4.13) eff 8d3 with Aeff the effective contact area, d the thickness of the dielectric ﬁlm, ε the static permittivity, and µ the mobility of the major charge carrier. Using the cAFM measurements, we can derive a mobility of about 5 3 10 2 cm2/Vs, which is in good agreement with × − previously reported macroscopic photo-induced current measurements [523]. As one can see, SCLC transport shows a drastic thickness dependence. The temperature dependence in a SCLC transport regime is solely determined by the temperature dependence of the mobility of the major charge carrier. We assume small polarons to contribute most in this process, which has already been suggested by many authors [25]. As previously discussed, at CDW a band bending is apparent. The extension and lateral dimensions are determined by the carrier concentration for compensation. In LNO iron impurities were discussed [523]. Iron levels are much deeper than in reported simulations of the band bending in CLN as about 1.5 eV instead of 0.1 eV. The typical impurity 17 3 concentration of Fe in the investigated samples was about 10 cm− .

127 results

reset 4 10

6 10

[A] T |

| I 300 K 10 8 366 K set 10 10

20 10 0 10 20 V [V]

Figure 4.41: Complete I V cycle for room temperature (T = 300 K) and an elevated temperature − (T = 366 K). Both the curve slope as well as the set-voltage remain constant. Due to the increased current at higher temperatures a similarly increased current in the switch-off cycle can be seen. The current increases by two orders of magnitude, which underlines a large thermal activation in this temperature range.

The main reason there is no purely electronic conduction in bulk LNO is the very short electron lifetime, which has been extracted from pump-probe optical spectroscopy, hence, it is very unlikely the current to be driven purely by electron band or hopping conduction. In this case, as we have discussed in Sec. 2.3.5, the mobility of adiabatic and non-adiabatic E polaronic transport behaves like Tme a/kT with m = 1/2 and m = 3/2, respectively. ∼ ad − nad − The temperature dependence of the conductance in the Au/Cr/LNO/Pt stack is given in Fig. 4.42. As we can see, there are three current regimes, with different activation energies and transition temperatures. In 1 for temperatures above 300 K an activation energy of 0.63 eV was obtained, in 2 between 300 K and about 100 K we obtain an activation energy of 0.18 eV. For temperatures below 100 K 3 we obtain an activation energy of only 0.03 eV. Very similar regimes have been proposed very recently for the lifetime of 4+ 549 bound polarons NbNb in lightly-doped Fe:LNO via Monte-Carlo simulations [ ]. Monte- Carlo simulations are a very useful way to describe the dynamics of polaronic diffusion processes in LNO, which already explained the origin of stretched exponential decays upon hopping transport as well as the site-correlation effect (SCE), which explains the very likely trapping of excited photo-charge carriers at the trap they were created [399, 550]. Mhaouech and Guilbert similarly observed the SCE, but now also incorporated into the simulation that a list of individual processes can happen similarly. The list of these hopping, trapping and conversion processes as well as there associated activation energies are given in Tab. 4.5. As these processes have seemingly different thermal activations, the actual lifetime of e.g. bound polarons will, thus, be different. Hereby, three regimes were observed:

128 4.7 conductive domain walls in exfoliated thin-film lno

4 5 4+ 4+ Table . : General conversion processes in LNO under assumption of the existence of NbLi , NbNb, 2+ and FeLi . The activation energies are calculated with the experimentally derived polaron energy, binding energies, as well as peak energy and width derived from TRA measurements [384].

Name Process Ea [eV] Bound polaron trapping Nb4+ + Fe3+ Nb5+ + Fe2+ 0.043 Li Li −→ Li Nb Bound polaron hopping Nb4+ + Nb5+ Nb5+ + Nb2+ 0.290 Li Li −→ Li Li Polaron conversion Nb4+ + Nb5+ Nb5+ + Nb4+ 0.635 Li Nb −→ Li Nb Free polaron hopping Nb4+ + Nb5+ Nb5+ + Nb4+ 0.272 Nb Nb −→ Nb Nb Free polaron bounding Nb4+ + Nb5+ Nb5+ + Nb4+ 0.070 Nb Li −→ Nb Li Free polaron trapping Nb4+ + Fe3+ Nb5+ + Fe2+ 0.003 Nb Li −→ Nb Li

1 conversion regime with polaron conversion Nb4+ + Nb5+ Nb5+ + Nb4+, E 0.65 eV Li Nb → Li Nb a ∼ 2 hopping regime with bound polaron hopping Nb4+ + Nb5+ Nb5+ + Nb4+ E 0.2 eV Li Li → Li Li a ∼ 3 trapping regime with bound polaron trapping Nb4+ + Fe3+ Nb5+ + Fe2+, E 0.03 eV. Li Li → Li Li a ∼ These values are in very good agreement with the previously revealed activation energy of the optically excited charge carriers by TRPL given in Sec. 4.5. Thus, we can expect these to correctly describe the hopping processes in our crystals upon illumination. Thus, the simulated bound polaron lifetime by Mhaouech and Guilbert is overlaid on the aforementioned temperature-dependent I V curves given in Fig. 4.42. − Thus, we can conclude, that the current through the exfoliated LNO thin-ﬁlm is mainly governed by an interplay of bound and free small polaron transport, depending on the given temperature. As iron impurities act as deep traps, they will inhibit any further 2+ 384 conduction. The hopping activation energy of FeLi is given to be 0.35 eV [ ], hence, only little larger than for bound and free polarons, but as the hopping rate w e r/a e r[Å] ∼ − ≈ − hopping transport is very unlikely for the given concentrations and even for lightly doped Fe:LNO not relevant [549]. A conversion to bound and small polarons is unlikely due to the high binding energy of 1.22 eV [384]. This is believed to be the reason for the long lifetime of written holograms in Fe:LNO [417]. However, in these experiments a relaxation was found with activation energies much smaller, which suggests there are other channels 2+ of relaxation. And thus, the deep traps of FeLi can relax e.g. by electron tunneling. Yet, at the CDW this would be energetically unfavorable due to the band-bending. Thus, we 2+ believe, that at the CDW a persistent FeLi concentration above the standard value of 15 % of the impurity concentration cFe as usual in LNO [364] will be present to compensate the polarization charge.

129 results

1 1000/T [K ] 2 4 6 8 10 12 14 conversion regime 3 1 3 c [cm ] V [V] 10 NbLi Nb4+ + Nb5+ Nb5+ + Nb4+ 0.63 eV 5 1019 10 Li Nb ! Li Nb ⇥ 20 = 5 2 10 7.5 Ea 0.635 eV 10 ⇥ 5 ! 2.5 2

i [a.u.] hopping regime 7 [A] 10 4+ 5+ 5+ 4+ I 0.18 eV NbLi + NbLi NbLi + NbLi bound ! 9 Ea = 0.29 eV 0.2 eV 10 1 0.033 eV 1 / h t ! 3 trapping regime 11 2 10 3 Nb4+ + Fe3+ Nb5+ + Fe2+ Li Li ! Li Li E = 0.043 eV 0.03 eV 500 300 200 150 120 100 85 75 a ! T [K] Figure 4.42: Temperature dependence of the current for thin-ﬁlm LNO. Measured current at different voltages from 360 K to 77 K. We can observe different regimes of thermal activation 1 above 300 K with Ea = 0.63 eV, at 2 between 300 K and 100 K with Ea = 0.18 eV and 3 below 100 K with a thermal activation of Ea = 0.03 eV. These are overlaid with the inverse bound polaron lifetime, as calculated by Monte Carlo (MC) simulations by Mhaouech and Guilbert [549].

Now, we want to further discuss whether it is possible to perfectly correlate and understand the conductance in such a metal-insulator-metal stack with a primarily single- domain and ferroelectric insulator medium LNO by the formation of conductive CDWs. Therefore, we conducted local I V measurements with higher voltages being applied. − We can observe very abrupt increases in the current ﬂow, similar to the case of a large con- 6 tact area, yet at an altered nominal voltage 37 V . We did this measurements in an ordered array with 10 10 points. In the I V curves apparent steps can be observed, which are × − very similar to these in the macroscopic I V curves, yet with a larger set voltage, than − in the macroscopic measurements. This can be due to the different conductance behavior of the system as the tip is easily not connected with the CDW. An evidence for this behavior is the I Vn behavior with n > 2, which indicates trap assisted processes, e.g. ∼ current through the domain area. Afterwards, we scan the area and can clearly observe DWC, which is furthermore very strongly correlated with the previous current maximum in the array, which is e.g. especially visible at spot 1. The tip was positioned on the CDW and a voltage of 30 V was applied for 1 min. A similar approach was done on the further red encircled spots in Fig. 4.42(c).A cAFM scan shows a drastic increase at these CDW, which underlines the memristive property of the device for voltages above the coercive voltage, which is already very apparent in the I curves in Fig. 4.42(b) [551]. The cAFM −

6 This local cAFM set voltage value was similarly observed by contacting the contact area with the platinum tip, hence, we conclude this to be due to an increased sheet resistance in the system as compared to the macroscopic probe station measurements.

130 4.7 conductive domain walls in exfoliated thin-film lno

a) Au

comparability Pt Pt b) 40 10 V 1 subsequent spot 1 cAFM spot 2 local set 2 scan spot 3 20 voltage [nA]

I AFM max. 3 IV current map 0 0 10 20 30 40 50 1 µm 0 nA 40 nA 1 µm 0 nA 20 nA I [a.u.] I [a.u.] V [V] 0 1 0 1

c) 10 V — 3 V

1 µm 0 nA 75 nA 1 µm 0 nA 1 µm PFMI [a.u.]j 150 pA 0 1 I [a.u.] I [a.u.] 0 1 0 1 Figure 4.43: a) Schematic of AFM-based and plate electrode-like local switching on the basis of local nucleation, b) local I V measurements with sharp increases in conductivity − and, hence, largely hysteretic behavior, map of maximum currents of I V sweeps, − a subsequent cAFM scan at 10 V reveals a comparable current pattern, which shows only DWC with a comparable modulation, c) at the red marks the tip was positioned and a voltage of 30 V was applied, we can see a strong increase in the current at scan voltage of 10 V accompanied with the smearing out, at 3 V the DWC can be seen − much clearer, yet is not stable, after this scan the pattern vanishes and no DWC is detectable anymore, a PFM image shows the written domain pattern.

131 results scan shown in the middle at a voltage of 3 V reveals a very clear DWC behavior without − smear-out. The according PFM measurement is given on the right side. It yet also shows, that upon the high-voltage treatment the domains have moved. To conclude, we have proven, that the abrupt increases in current are connected with the creation of CDW under the electrodes conductively connecting both plane electrodes as depicted in Fig. 4.43(a) on the right. Now we want to summarize these measurements and develop a model, which can conclusively describe these results as well as previous photo-induced DWC results. In our model, it is important to describe the inversion process, the polarization charge compensation as well as the explicit conductance of the DW. Upon local inversion, generally charge carrier injection is discussed, even though, it has been speculative for a long time. We assume, that we inject charge carriers at the DW, which are needed to compensate the inclined DW of the spike-domain-like nucleus as depicted in Fig. 4.44(a). These free charge carriers – electrons – however, will condensate into electron polaronic states inside the single crystal material, which has been observed in the bulk material e.g. by optical time-resolved spectroscopy to happen within 200 fs [308]. As the band-bending is not large enough to create a 2DEG we obtain a 2D polaron gas (2DPG). The bound polarons 4+ NbLi will very fastly be trapped by iron impurities. Upon pure-diffusion transport, thus without an external electric field, the average mean free path of the bound polaron until trapping with FeLi is tNb4+ = 100 Å and for the free polaron is tNb4+ = 80 Å. These h Li i h Nb i values were derived by interpolation from the data of Mhaouech and Guilbert [549] under 17 3 consideration of the smaller iron impurity concentration cFe = 10 cm− and a niobium 21 3 antisite defect concentration of cNbLi = 10 cm− . Upon drift condition, these values can be larger. Hence, the bound polaron has a larger mean free path. However, the hopping 7 1 rate of bound polarons is only w 4+ = 10 s− . Assuming the hopping rate to depend NbLi r[Å] mainly on the nearest neighbor distance like w e− , we can calculate, that the hopping ∼ 9 1 rate of free polarons is two orders of magnitude larger, hence w 4+ = 10 s− . Thus, the NbLi electric current with free polaron transport can be significantly larger. As the polaronic states are trapped and the trap concentration moderately low, after very short time a trap saturation will be apparent. As the conductance band level EC is significantly reduced at the DW with a lateral extension down to 10 [552] and a tail depending on the polarization charge up to 1 µm, the injection will happen only on a very confined area. As the traps are saturated the average hopping distances will pro- 2+ 2+ portionally increase with cFeLi . Moreover, by electron tunneling the FeLi can follow the CDW and further compensate the polarization charge. This is schematically depicted in Fig. 4.44(c). The change in c 2+ is connected to an increase in the refractive index n as FeLi sketched in Fig. 4.44(d). Such apparent changes in the refractive index at DWs up to 1 10 3 have been already observed e.g. by OCT as shown in Sec. 2.8 and in literature × − [247]. These index changes have been observed to be independent on external voltage as well as moderate heating. Yet, despite the relatively high activation energy of these traps at 200 ◦C these will be slightly activated and the DW will not be compensated anymore.

132 4.7 conductive domain walls in exfoliated thin-film lno

Hence, the inclination is reduced. This was ﬁrst observed by Schröder et al. with PFM [25] and conﬁrmed by CSHG [512].

a) injection e— b) 4+ NbNb 2+ FeLi

100 nm 0.15 cFe 100 nm c 2+ Fe EC c) d) 2+ 2+ FeLi FeLi

100 nm 0.15 cFe 100 nm cFe2+ n

4 44 Figure . : Schematic model to describe the formation and stabilization9 1 of CDW in nominally7 1 + 4+ 10 s 4+ 80 A˚ l 4+ 100 A˚ w 4 10 s wNb undopedlNb LNO with ironNb impurities a) nuclei formationNbNb with electron injection,Li bound h Nb i⇠ h Li i⇠ ⇠ ⇠ polaron formation and trapping, b) alteration of conductance band due to compen- averagesated diffusionCDW formation,of bound and c) Sideways free polaronsDW movement, hopping which rate is of accompanied bound and with valence transportuntil trapping of iron at impurities, FeLi d) alteration of the refractive index due to local 2+ free polarons uponincrease drift conditions in FeLi . larger distance

Upon illumination first experiments by Schröder revealed photo-induced DWC for as- 17 3 21 3 c 1 10 cm cNb 10 cm congruent LNO grown DWsFe in⇠ wafer-thick⇥ LNO upon LiUV⇠ illumination. Upon ultrashort pulse illumination in the visible regime between 400 nm and 800 nm we have observed photo-induced almost infinite lifetime of FeLi DWC. These results were discussed in the dissertation thesis by Schröder [152]. Yet, no electron-hole pair generation by multiphoton absorption was present as a linear dependence of the current was observed upon intensity. This can be easily understood in our 2+ model. Upon ultrafast optical excitation deep trap levels FeLi compensating the small as- grown polarization charge are photo-excited and form polaronic states, which can drive a current through the DW. Upon continuous illumination the intensity is not sufficient due to the very small concentration of these deep trap states and the fast relaxation of the bound and free polarons. Hence, this does not give a significant photo-induced current. Upon super-bandgap illumination an orders of magnitude larger concentration of polarons can be created, hence, a photo-induced DWC can be observed. The current decreases again for wavelengths far below the cutoff-wavelength, as the polaron gas is not created over the whole CDW path. This model can also describe, why smear-out is visible in the cAFM measurements. The injected electrons, which form electron polarons, can be dragged off the CDW by a sufficiently large electric voltage. Differences for different domains can be attributed to different mean-free path or lifetime in the different domains, which has similarly been observed in the multi-photon luminescence microscopy in Sec. 4.5. Upon cryogenic tem-

133 results peratures, no current switching was observed, which can be explained by the fact, that not sufﬁciently mobile charge carriers are available to stabilize the DW.

4.8 microwave impedance microscopy

So far, all measurements were performed upon DC conditions. Now, we want to further investigate the high-frequency performance of DWC in the GHz regime. Therefore, we use scanning microwave impedance microscopy (sMIM). The technical conditions were discussed in Sec. 3.2.4. The measurements were performed with various working frequencies from 50 MHz to 10 GHz. sMIM can be made in a quantitative way by using calibrated samples [553]. However, this is very time-consuming and scarcely possible as the tip status has to be proven to be perfectly the same before and after the measurement of the interesting sample. Additionally, these calibrations are only shown for simple systems and a singular DW is expected to behave very different than extended layers or doping profiles. Hence, most of the times it is necessary to compare the experimental results with the simulation of the admittance of the tip being in contact with the sample for several tip radii. Hence, to extract conductivity data we generated a finite-element model (FEM) and simulated the sMIM profile across the CDW.

4.8.1 Finite-element method simulation

The FEM simulations were carried out with COMSOL Multiphysics 5.0. The chosen geometry, is given in Fig. 4.45(a). A mesh generated with the tip at the DW is shown in Fig. 4.45(b). The size of the DW is determined by the condition to obtain a FEM mesh as well as a converging result. The structural parameters are given in Fig. 4.45(c). We use the infrared permittivity of LNO for these simulations. The sMIM curve for the tip on top of the DW is given in Fig. 4.45(d). This curve is qualitatively very general and typically observed with sMIM. We can observe, that at about 300 S/m a drastic change in the imaginary part ( ) of the = admittance of the microwave is present, which literally means the material starts to reflect the microwave above this conductivity. The loss channel, the real part ( ), does only < give a signal around this transition conductivity. For conductivities larger than this value the imaginary part of the admittance does settle, thus, does not give anymore quantitative information. However, it is possible to obtain a quantitative value by comparing the signal strengths of real and imaginary part until about 105 S/m. We further simulated the sMIM profiles across the CDW with different associated conductivities spanning from 0.1 S/m till 1000 S/m. In Fig. 4.49(a),(b) the sMIM real and imaginary part profiles across the DW are simulated for a tip radius of 50 nm, which is about the size in the case of the PrimeNano tips being in use. We can observe, that a conductive DW with a high conductivity can show very long reaching influence. The imaginary part barely shows any contrast below 100 S/m. Both, imaginary and real part, show two decays around the DW. A very rapid decrease of about the tip size and a very

134 4.8 microwave impedance microscopy

a) b)

air sMIM j tip

d b c

c f DW a LiNbO3

a a e d) 6 7 c) 10 10 7.6 · · 4 Parameter Value a [µm] 40 7.4 sMIM- 3 b [µm] 20 = 7.2 sMIM- < c [µm] 8 = < d [µm] 5 2 7 @ 3GHz sMIM e [µm] 4 sMIM f [nm] 10 6.8 1 j [] 90

#L#iLNONbO3 4.2 6.6 s [S/m] 6 107 0 tip 2 1 0 1 2 3 4 5 6 ⇥ 10 10 10 10 10 10 10 10 10 conductivity s [S/m]

Figure 4.45: FEM simulation of the microwave admittance at a DW with different levels of conductivity, a) geometry of device with tip, surrounding air, DW, and hosting ferroeelctric, b) generated mesh, c) parameters used for the simulation, d) simulated imaginary and real part of sMIM at a DW. slow almost linear decay over even more than the shown 2 µm. In Fig. 4.49(c),(d) the simulation was carried out for a tip with a radius of 200 nm. Besides the broadening around the DW due to the increased tip size, the signals increase moderately with tip size, but the relative increase is even sublinear. This is not the case for a two-dimensional sheet conductor, which gives a quadratic increase in signal upon tip radius. Hence, in the case of a one-dimensional conductor as DWs the tip radius is almost negligible in terms of signal strength, as a tip radius of 200 nm is already extremely large. Thus, it is not strictly necessary to calibrate the tip in the aforementioned way.

135 results

7 7 10 10 a) 2 · b) 4 · r = 50 nm DW r = 50 nm DW 1.5 s [S/m] 0.1 [a.u.] [a.u.] 1 2 1 = < 10 0.5 100 sMIM sMIM 1000 0 0 2 1 0 1 2 2 1 0 1 2 x [µm] x [µm] 10 7 10 7 · · c) 3 d) r = 200 nm DW 6 r = 200 nm DW s [S/m] 2 0.1 [a.u.] [a.u.] 4 1 = < 10 1 2 100 sMIM sMIM 1000 0 0 2 1 0 1 2 2 1 0 1 2 x [µm] x [µm]

Figure 4.46: Simulation of the sMIM proﬁles as apparent for CDWs with different levels of conductivity. The calculation was carried out a frequency of 3 GHz. a) imaginary part b) real part of sMIM signal. 50 nm large tip.

4.8.2 Scanning microwave impedance microscopy

The ﬁrst entity we investigated with sMIM was the CDW in thermally treated LNO with a microwave frequency of 3 GHz as distributed by the PrimeNano system. The topography of the structure is given in Fig. 4.47(a). One can clearly see, that a cusp-like proﬁle is apparent due to polishing. In Figs. 4.47(b),(c) the out-of-plane PFM or VPFM of the structure can be seen, which clearly shows a laterally rather straight CDW with a significant inclination towards the crystal polar axis. In Figs. 4.47(d),(e) the real part ( ) and < imaginary part ( ) of the sMIM measurement at a frequency of 3 GHz is given. Sections = across the CDW are given in Fig. 4.47(f). We can observe a clear peak in the real part and a dip in the imaginary part of the sMIM signal. For now, we want to skip the dip as well as the broad band of increased signal around the CDW in the real part. These features will be explained in the appendix by SAW excitation modulation in Sec. A.9. The imaginary part of the sMIM signal shows a distinct peak, which supports the high-frequency conductance of the DW. This increase in sMIM- is correlated with an increase in the dark conductivity. The measurements were carried =

136 4.8 microwave impedance microscopy

a) b) c)

~PS

x x

2 µm 2 µm I [a.u.] 2 µm I [a.u.] 0 1 0 1 0 12 nm VPFM j VPFM A d) e) f) sMIM = 2 µm ⇠ 150 nm sMIM ⇠ <

VPFM A x x VPFM j

0 2 4 6 2 µmI [a.u.] 2 µmI [a.u.] 0sMIM 1 0sMIM 1 x [µm] = < 10 7 · g) h) i)2 sMIM s [S/m] < theory 10 1.5 exp. 100 300 [a.u.] 1 = DW

sMIM 0.5

2 1 0 1 2 2 µm I [a.u.] 2 µm I [a.u.] sMIM0 ∂ /∂V1 A sMIM0 ∂ /∂1V A x [µm] = < Figure 4.47: sMIM measurement with a microwave excitation at f = 3 GHz and 300 mV on a CDW in thermally treated LNO. a) topography of the sample, b), c) out of plane PFM, d), e) real-part and imaginary part of sMIM measurement, f) sections across the CDW, g), h) amplitude of ∂ /∂V and ∂ /∂V upon an AC voltage of 4 V, i) comparison of = = experimental and simulated sMIM- proﬁles derives a conductivity of about 100 S/m. =

137 results out upon an external AC bias voltage of 4 V to additionally extract amplitude and phase of ∂ /∂V and ∂ /∂V. The amplitude scans are given in Figs. 4.47(g),(h). < = The sMIM- profile was compared with the simulated profiles for different conductivi- = ties. From the topography scans we can deduce the tip radius to be definitely smaller than 70 nm. Hence, we can compare the experimental results with the aforementioned simulated profiles with a tip radius of 50 nm. In this tip size range no change in the profiles was observed in the simulations. The comparison with these curves gives the best fit for a conductivity of about 100 S/m.

a) b) c) ~P 1 S h2h

t2t 2 h2h3

~PS

2 µm I [a.u.] 2 µm I [a.u.] 2 µm 0 0 12 nm 0LPFM j1 LPFM A1

d) e) f) sMIM = 1 1 h2h ~ ~ 1 PS PS x CDW 2 ~ ~ x 2 PS PS sMIM = t2t 2 120 nm ⇠

2 µmI [a.u.] 2 µm 0[a.u.] I 0.2 0.4 0.6 0.8 1 0sMIM 1 sMIM1 0 = < x [µm] Figure 4.48: sMIM measurement on CDW sections in thermally treated LNO with both h2h and t2t polarization charge, similar conditions as in Fig. 4.47 a) topography of the area, b), c) in-plane PFM, d), e) real-part and imaginary part of sMIM measurement, f) sections across the CDW with h2h and t2t polarization charge.

Furthermore, we need to discuss inﬂuences by the topography. sMIM is sensitive to topographical features as visible by some dirt particles on the sample. The difference in the global admittance coupling with the cantilever is altered at sharp topographical features. As the polished sample shows a cusp-like feature, the sMIM signal can be altered by this. However, we can not derive the observed inverse behavior over the whole sample range and additionally at steeper steps and corners, which is especially apparent in another

138 4.8 microwave impedance microscopy position at the CDW discussed below, no larger signal was identiﬁed. Hence, we conclude, that the signal is not a manifestation of the sample topography but its electronic properties.

a) b)

I [a.u.] 5 µm 5 µm 0 1 0 nm 1 nm PFM j

c) d)

I [a.u.] I [a.u.] 5 µm 0 1 5 µm 0 1 1.3 nA sMIM 0 nA =

Figure 4.49: sMIM measurement on DW in an exfoliated thin-ﬁlm LNO, similar conditions as in Fig. 4.47. a) topography of sample, b) out-of-plane PFM phase, c) cAFM scan at 10 V bias, d) imaginary part of sMIM measurement.

Now, we want to present the measurements at this particular further position, which includes an island formation, which was already discussed and investigated in Sec. 4.4. This area is, hence, particularly interesting as to show both h2h and t2t CDW sections. The sMIM measurements and in-plane PFM characterizations are given in Fig. 4.48(b)-(e). We can clearly see, that the sMIM- contrast is preserved at the h2h CDW, however, at = the t2t CDW a dip in sMIM- can be observed, which, however, is not visible along the = whole DW. A comparison of the proﬁle of sMIM- across h2h and t2t CDW is given in Fig. 4.48(f). = The signal at the t2t CDW does not seem very strong and one might doubt as there are similarly strong dips in the scan, however, all these dips can be correlated with dust particles, whereas this is not the case at the CDW. This result might be interpreted with a simulation result for very small conductivities below 0.1 S/m. Then, we can observe a dip in several simulations. However, the picture is not very conclusive and the result

139 results

a) b) c)

10 µm 10 µm I [a.u.] 10 µm I [a.u.] 0 nm 2.2 nm 0PFM A1 0PFM '1

d) e) f)

1 x 2 x

10 µm I [a.u.] 10 µm I [a.u.] 10 µm [a.u.] I 0 pA0 1 0sMIM 1 sMIM1 0 200 pA = <

g) h) i) @ pos. 1 300 nm @ pos. 2 ⇠ < = < = sMIM sMIM sMIM sMIM 1.5 µm ⇠

0 0.5 1 1.5 2 2.5 0 5 10 15 x [µm] x [µm]

10 µm I [a.u.] sMIM0 ∂ /∂1Vj = Figure 4.50: sMIM on a high-voltage treated LNO sample. a) topography b),c) amplitude and phase of the PFM, d) cAFM scan at a bias voltage of 10 V, e),f) imaginary and real part of sMIM, g) amplitude of ﬁrst harmonic of the imaginary part of sMIM at a frequency of 25 kHz and a voltage amplitude of 4 V, h),i) cross section of the imaginary and real part at the indicated lines of g) and h).

140 4.8 microwave impedance microscopy depends significantly on the mesh conditions and convergence limit. As the experimental evidence for this behavior is comparably weak we can not claim, that t2t CDWs show a significantly larger conductivity than the bulk in the high-frequency regime, which we already have shown upon DC conditions. In the real channel of sMIM we can observe a reduction inside the domain area. This result will be discussed in the context of SAW excitation. Furthermore, we investigated DWs in the exfoliated thin-film LNO. The topography, out-of-plane PFM and cAFM scans are shown in Fig. 4.49(a)-(c), which confirm conductive DWs. sMIM- is given in Fig. 4.49(d). As there are no topographical features present, = there are no interferences with topography expected. We derived further measurements on fractured domain configurations after application of 4 V AC bias revealing sMIM profiles with a sharp resolution of about 50 nm on the fractured small domains.

a) b) 1,400 (1 MHz) # ~P ~P 1,200 S S

1,000

800 dielectric permittivity 15 10 5 0 5 10 15 10 µm I [a.u.] sMIM0 ∂ /∂1Vj electric field [kV/cm] = Figure 4.51: Explanation of sMIM-∂ /∂V domain contrast. a) sMIM-∂ /∂V of the aforementioned = = domain, b) exemplary permittivity curve upon an external bias field at a frequency of 1 MHz of a PZT film, data extracted from [554].

The last sample to be discussed was high-voltage treated. The nuclei was formed upon UV laser illumination and grown. Due to the limitation in scan range of the Cypher system, a very small domain was grown with a size of about 30 µm. The conductivity of the DW was enhanced in a two-terminal device with Cr contacts up to a current value of 2 µA at about 400 V through the entire DW by Christian Godau. The topography and PFM measurements are given in Fig. 4.50(a)-(c). It can be seen that an internal domain is present, which was already observed in such domains by previous 3D CSHG analyses. Yet, for larger-grown domains with a diameter of about 100 µm only embedded internal domains could be observed. In Fig. 4.50(d) the cAFM measurement at 10 V are given. We can observe very clear dark conductivity confined at the outer DW. The inner DW does not show conductivity. From previous CSHG it is assumed, that this inner DW does not reach through and is of t2t type. A similar finding can be observed in sMIM- . The DW, = which appeared conductive upon DC conditions, similarly gives an increased imaginary part signal, whereas the inner DW does not. The real part of the signal shows a broader dip and wavy signal, which is attributed to changes in the SAW excitation efficiency. The

141 results sMIM ∂ /∂V amplitude channel at the first harmonic reveals a very clear domain contrast. = This can be explained, as the permittivity is a function of the external and internal field [555]. An exemplary butterfly curve of the permittivity of a PZT film is given in Fig. 4.51(b). Such curves are very common in pure ferroelectrics. Hence, the phase contrast is a result of the change in the dielectric response of the bulk permittivity in the two adjacent domains.

142 5

CONCLUSION&OUTLOOK

The future depends on what you do today. — Mahatma Gandhi

In this dissertation thesis the properties of charged domain walls (CDWs) in ferroelectrics, especially lithium niobate (LNO) and lithium tantalate (LTO) was investigated. Based on previous results, that inclined DWs might be the reason of conductive properties, we investigated different types of DWs and the polarization charge and conductivity and obtained a clear signiﬁcance, that the conductance can be tuned and switched according to the polarization charge sin ϕ. ∼ We started with the three-dimensional visualization of DWs in LNO and LTO. We were able to visualize the inclination angle of these and relate them with previous PFM- based measurements. We observed, that a temperature treatment can similarly alter the inclination angle. Upon visualization of multiple times ferroelectrically inverted samples we observed enclosed domains. In periodically poled LNO crystals we were able to observe general poling defects, which can result in inclined DWs. AFM written domains in thinned, but still almost wafer-thick samples revealed very clear inclined DW formation as well as surface domains created upon laser-writing. We calculated the three-dimensional polarization charge textures in the investigated samples. In thermally-treated LNO samples with almost maximal bound polarization charge, we showed intriguing DW formation with both head-to-head (h2h) and tail-to- tail (t2t) polarization charge. Topologically interesting geometries like tunnels and islands were observed. The polarization charge texture was investigated in z-cut Mg-doped LNO and LTO. We observed a very homogeneous polarization charge with an inclination angle of about 0.2° in Mg:LNO. In high-voltage treated Mg:LNO samples an increase in inclination up to about 4° was observed and correlated with cAFM measurements. We invented a faster method to three-dimensionally visualize DWs on the basis of qpmSHG and interaction of different domains and interfaces, which can result in a three- dimensional imaging over an area of 40 µm 40 µm 20 µm within 20 ms. × × We used PEEM as an imaging tool to retrieve local surface potential in thermally treated LNO by secondary electron emission upon both X-ray and UV illumination. We were able to correlate the extracted local surface potential with the local inclination angle as derived from three-dimensional SHG DW tracking.

143 conclusion & outlook

We investigated the UV light excitation and recombination of electron and hole polarons in LNO and LTO for different doping concentrations and dopants as well as undoped. We observed a thermally activated stretched exponential decay, which revealed an activation energy, which was already found in previous electrical characterizations as well as a reduction in thermal activation energy for very low temperatures. Doping around the optical damage resistance thresholds gave very similar decays, which shows, that the particular dopant is not important on the polaronic transport, unless the concentration is sufficient. For singly-inverted domains a faster decay was observed, which can explain the domain contrast previously observed in multi-photon luminescence. The usage of exfoliated thin-film LNO opened new possibilities in the study of conductive DWs, as the voltage to switch individual domains decreased from the kV-regime towards the V-regime. Scaling to even thinner films will further boost the current through CDWs according to the Mott-Gurney Law. Polishing the films to a thickness of e.g. 60 nm by CMP an increase by 103 is expected. Furthermore, the set voltage will be decreased to about 2 V, comparable to now-a-days resistive switching devices. We have shown, that the set voltage is very precisely given for a large set of devices and robust upon temperature and cycling. Further materials to be tested should include stoichiometric thin-films with a smaller defect concentration, ionic conduction and, hence, smaller leakage current as well as Mg-doped thin films, which not only can increase the inclination angle but also leads to a decrease in maximal bound polaron saturation concentration towards zero. Current, which is determined by free polarons will be much larger and switch-off, whereas still following a stretched exponential curve are expected to increase by at least two orders of magnitude as compared to a congruent thin-film with equal thickness. Furthermore, we can expect, that in the low temperature regime even smaller thermal activation will by present, from theory 3 meV. ∼ Hence, we can state, that CDWs are a very interesting new material property, which can find applications in nonvolatile memories as RRAM or FeRAM. Similarly, the application for transistor devices appears reasonable as a FeFET. By scanning microwave impedance microscope (sMIM) we have shown, that the DW conductivity can reach in a much higher frequency regime as previously shown by nanoimpedance microscopy. CDWs have significant conductivity even in the GHz-regime. Fur- thermore, it is possible to tune the plasmon frequency of CDWs, which would make it possible to alter the THz optical transmittance, which can result in switchable mirrors and other electrooptical devices in the THz regime. The temperature-dependent measurements revealed a similar behavior as already observed in the time-resolved luminescence measurements, which revealed that the activation energy reduces strongly. In the CLN film it reduces to about 30 meV. As this correlated with the trapping energy of bound polarons at iron impurities, it would be worthwhile to study it for highly Mg-doped LNO films, as we would expect an even small activation energy for very low temperatures. The trapping energy of free polarons at iron impurities is as small as 3 meV, hence the thermal activation would be almost negligible in this case, and, thus behave metallic-like.

144 conclusion & outlook

The creation of CDWs while the inversion process can result in a decrease in the voltage over the sample, which is the prerequisite in the emerging field of negative capacitance [556]. As the CDW inclination angle shows a very similar temporal change upon the application of an external field, and hence conductivity, as discussed in the voltage drop with time by displacement currents by Khan et al. [557]. These authors used PZT films, which are known to almost not show CDWs due to larger gradient energies, hence, it is possible, that for LNO thin films with different thickness a much broader frequency range for the negative capacitance response can be observed. As we have already observed the strong memristive behavior for electric fields above the coercive field, it is expected, that threshold switching as observed in niobium and hafnium oxide films [558, 559] are expected to be observed in LNO, too. Beneficial for this would be non-adiabatic polaron hopping, which, hence, would result in an even stronger positive thermal feedback. A necessity to observe this would be thinner films and larger voltages to be applied. Threshold switching is attractive for selector applications. We were able to show, that domains in exfoliated thin-film LNO can preserve the domain formation over months. Small domains can be realized in these thin films, which can be applied in optical waveguides. Especially interesting are backward mirrorless optical parametric oscillators (OPOs), which comprise much higher efficiency [560, 561]. Inher- ently such OPOs are also switchable. The integration of conductive and hence plasmonically active materials into LNO can result in an increase of the overall nonlinear conversion, as has already been shown in metallo-ferroelectric nanostructures [562]. Such processes can similarly be applied for larger wavelengths. Hence, CDWs are potentially promising for higher nonlinear optical conversion in the THz regime. As now TEM measurements get available for ferroelectric materials as LNO, it is very important to investigate CDWs with TEM methods, which can resolve electrical fields as DPC STEM to further understand the thickness of CDWs and screening lengths [563, 564]. Piezotronics is a technology to alter the conduction through semiconducting ferroelectrics by strain [565, 566]. As we have observed, CDWs to behave like confined semiconductors it is possible, that CDWs observe alteration upon externally applied mechanical stress. Perovskite solar cells observed a remarkable rise in the last years due to the strong increase in quantum efficiency. So far, the physical reason for the high efficiencies are not conclusively investigated. However, several groups proclaimed the photovoltaic effects to dependent on the very good charge carrier separation by nanoscale ferroelectric domains and the formation of CDWs [567–571]. Drawbacks of the used molecule films are the small persistence and their lead containment. Similar results can be observed in LNO, which is known to be photovoltaic [572, 573], but is both CMOS-compatible and persistent. The resistive switching in LNO thin films has been proven to be electroforming-free and abrupt. This is in stark contrast to the mainly investigated memristive films as TaOx or HfOx with filamentary-type resistive switching, which results in very drastic material change as e.g. by formation of dendritic-like structures over up to micron ranges [574] or can even damage the device due to the very large electroforming currents, but has been

145 conclusion & outlook previously observed upon oxygen vacancy transport mechanisms in YMO and BFO. Yet, the formation time is by two orders of magnitude faster than in these films. In LNO no material change is apparent and similarly no ionic transport. The switching voltage is very precisely given by the ferroelectric order. The singe-crystalline films have, hence, very low fatigue, a high retention and endurance. The films are CMOS compatible and can be integrated as all smart-cut processes into the regular fabrication line, yet with the necessity to wafer-bond bottom and top electrodes. Further potential in a reduction in the resistance of the LRS is given by the fact, that the current is found to be determined by the Mott-Guerney Law, which proposes an increase in DWC by three orders of magnitude upon a reduction in film thickness by 10, which would promote a more accessible HRS/LRS ratio close to the maximal value of 1011. Moreover, the use of Mg:LNO will facilitate the formation of CDWs with larger inclination angles and drastically smaller polaron activation energy. This will result in a drastic decrease in switch-off times. The current state is not destroyed upon read-out and non-volatile. CDWs could hence be especially helpful for the integration into crossbars for 3D RRAM, where especially abrupt resistive-switching is necessary to ensure switching only at the cross-section of word- and bit-line [575, 576]. Moreover, we were able to show, that the read-out current can be analogously tuned, which is especially necessary for neuromorphic and cognitive computing [577], but also for fuzzy logic and reconfigurable logic elements. Cognitive approaches have the potential to revolutionize big-data processing, internet-of-things, real-time computing, exact optimization, machine learning, signal detection and data encryption. It is well known, that the human brain does 2000 MFlops under consumption of 0.01 W or 1016 operations per second, which would equal 105 W with conventional supercomputers. Hence, cognitive computing which was already shown to greatly enhance the computing power can serve very well with the emulation of memrisitive devices. Universal memcomputing machines could replace the well-known von-Neumann-machines [578, 579] and are proven to even solve NP-complete problems in polynomial time and are Touring-complete [580]. A hardware realization by CMOS can serve for large progress in computation [581]. As LNO is transparent completely transparent electronic devices can be realized upon the application of transparent electrodes as e.g. aluminum-doped zinc oxide (AZO) and indium-tin oxide (ITO). Very thin support material could, hence, also result in bendable electronics. Due to the thickness-dependent scaling power electronic switches can be realized with similar or higher switching speeds than in insulated-gate bipolar transistors or integrated- gate commuted thyristors. Switching in the kV-regime with switching speeds of about can be expected. With flexoelectric coupling it is possible to mechanically write into the film. Hence, mechanical sensors can be realized with threshold stress [582, 583]. The transgression of this threshold can be read-out electronically.

Semper aliquid addiscendum est. k

146 Part III

EPILOGUE

APPENDIX a.1 laser ablation dynamics on lno surfaces

1 t = 1 s t = 2 s t = 3 s

t = 4 s t = 5 s t = 6 s

t = 9 s t = 12 s t = 15 s

Figure A.1: Time series of a surface SHG images on a randomly poled LNO crystal. A laser ablative process is initiated [marked by 1 ], branches along the slow scan direction are created after some time [marked by 2 ], further branches are created later on [marked by 3 ].

We want to brieﬂy give an overview of the observed laser ablative processes being observed upon illumination with a wavelength of 800 nm. Below a wavelength of 900 nm

149 appendix three-photon absorption processes are possible for the focus centering at the surface, which can result in a local heat-up. In Fig. A.1 a time series of such a SHG measurement is given. Following a circular destruction 1 , very easily the ablative area increase upon laser-scanning along the slow scan direction 2 . After some seconds further branches are created 3 . a.2 xps across a conductive dw in lno

To further investigate the chemical properties of the material close and at the DW XPS measurements were conducted on the thermally treated CLN. As it was pointed out before, such highly charged DWs require almost complete compensation, which cannot be delivered electronically in the material. Hence, already in the ﬁrst theoretical explanations ionic movement was suggested, which would result in valence changes. However, it is not trivial to perform XPS measurements on surfaces, in particular on insulating ones. Typically, surface layers of contamination can violate the result. Hence, it was tried to clean the sample upon Argon sputtering. Yet, the ion bombardment leads to a completely deteriorated PEEM image [see Fig. A.2], which suggests, that the surface is modiﬁed upon the sputtering process. The XPS measurement shows a reduction in charging, the primary electron peaks get more prominent.

before after ] [a u I

0 100 200 300 400 500 600

EB [eV]

Figure A.2: XPS measurement on the DW before and after argon sputtering. Data provided by Anna-Sophie Pawlik (IFW Dresden)

But, as the sample is damaged upon this procedure, all further measurements were carried on the pristine material without sputtering. In Fig. A.3 XPS measurements are shown at the DW and of the bulk. It is clearly visible, that the primary electron emission is only present at the conductive DWs. In the bulk, a large charge-up is present, which leads to highly-efﬁcient scattering. Hence, the peaks in XPS are vanishing.

150 A.3 xrd of thin-film exfoliated lno

O 1s bulk DW Nd 3d Nd 3p3/2 Nd 4d Nd 3p1/2

[a u ] O 2s Nd 3s

I C 1s Nd 4d Li 1s

0 100 200 300 400 500 600

EB [eV]

Figure A.3: Local XPS measurements with an aperture of 10 µm on the DW and inside the bulk. A clear promotion of primary electron emission at the conductive DW can be observed. The local assignment is conﬁrmed with XPS measurements with a larger aperture of 100 µm. Data provided by Anna-Sophie Pawlik (IFW Dresden). a.3 xrd of thin-film exfoliated lno

The ferroelectric thin films were characterized by X-ray diffraction (XRD) pattern. The samples were investigated in a θ 2θ scanning geometry upon Cu K , 1s2p 1s2, and K − α − β co-radiations. From the given diffraction pattern in Fig. A.4(a) one can derive, that the strong (00n) reflections of LNO are present. The lattice parameter of the thin-film LNO was determined by high-resolution x-ray diffraction (HRXRD) ω 2θ scan. The contribution to the spectrum can be expected to − both stem from the LNO thin-film as well as the LNO substrate, which possess an almost equal periodic lattice structure. The SiO2 amorphous layer should not give any contribution. The measurements were performed with Cu-Kα radiation source (λ = 1.540 56 Å). After a rocking curve to assess the crystal plane of the LNO substrate, the ω 2θ scan − was conducted at the (006) peak of the LNO substrate around 38.94° with a step of 0.0005° to find the position of the (006) peak of the LNO thin film. After using another rocking curve to assess the crystal plane of the LNO thin film, a similar ω 2θ scan was conducted − around the position of the (006) peak of the LNO thin film. The high-resolution XRD ω 2θ scan resolves the peak doublet of the (006) reflections − Kα with an FWHMs of 0.18° as is given in Fig. A.4(b). Precisely, at 38.806° and 38.890° for the substrate and film with FWHMs of 0.02° and 0.0392°, respectively, which confirms the high crystallinity of the film to match the bulk value without the phase impurity. Yet, a small change in lattice parameters is documented by these measurements, with a bulk and film value of 13.8661 Å and 13.8655 Å. Similar XRD measurements for exfoliated CLN thin-films on the Pt electrode as prepared by NanoLN are reported by Han et al. [584, 585].

151 appendix

104 a) 103 b) ·

α 5 K α K 4 bulk 2 (006) (0012) 102 thin-ﬁlm α

1s2p-1s 3 K α K 2 [a u ] [a u ] β (006) I I 2 K β (006) 1s2p-1s K 101 Pt (111) 1 (0012) (006) (0012) * * 0 100 20 30 40 50 60 70 80 90 100 38.6 38.8 39

2✓ [] 2✓ [] Figure A.4: Left: XRD of the ferroelectric LNO thin ﬁlm with a thickness of 600 nm on Pt/Cr/SiO2/LNO. Right: HRXRD of the same sample resolves a separation of the

(006)Kα reﬂections between bulk and thin ﬁlm. XRD data provided by Prof. Hui Hu (Shandong University, China) a.4 domain writing in exfoliated thin-film lno

150 surface domain Vset ]

formation [ 100 '

Phase 50

40 20 0 20 40 V [V] Figure A.5: PFM phase loop on an exfoliated LNO thin-ﬁlm. We can clearly see a ferroelectric inversion at Vset, whereas for negative voltages a transient behavior is visible, which can explain the fractured domain conﬁgurations as observed for very small negative voltages. This switching spectroscopy PFM indicates surface domain formation.

The switching spectroscopy PFM reveals a phase loop as given in Fig. A.5. We observe a ferroelectric inversion at Vset, which is very well connected with the set voltages observed in I V sweeps. In the negative voltage regime, we can clearly see a broad transition − region, which indicates the formation of surface domains. Domain writing into thin-ﬁlm LNO has been previously observed in wedges and very recently on similar thin-ﬁlms [446, 586]. We conducted an ordered tip-induced domain

152 A.4 domain writing in exfoliated thin-film lno

a) b) V [V] 70

5 µm 0 nm 1 nm -3 -2 -1 0 1 log t [s] c) d)

5 µm I [a.u.]PFM A 5 µm I [a.u.]PFM j 0 1 0 1

e) f) 0.8 V [V] 75 70 0.6 65 60 0.4 55 50 45 0.2 40 domain width [ µ m] 35 30 0 3 2 1 0 1 2 5 µm I [a.u.]PFM f 10 10 10 10 10 10 0 1 t [s]

Figure A.6: Dot-like domain writing into the LNO thin-ﬁlm. a) topography after writing, no alteration can be observed, b) schematic of the array of pulse time and voltage being applied in the array, c) PFM amplitude, d) PFM phase, e) DART PFM frequency, f) extracted domain widths for the given pulse time and voltage, a logarithmic domain growth can be observed.

153 appendix writing with different voltages and times. In Fig. A.6(a) the topography after writing can be seen, which shows a very ﬂat surface. The switching voltages and times are given in Fig. A.6(b). The DART PFM analysis is given in Figs. A.6(c)-(e). We can observe, that at every spot a domain was written. The extracted domain diameters are given in Fig. A.6(f). The domain grows logarithmically in time and linearly in voltage. The smallest written domain had a size of about 15 nm at a voltage of 23 V. Hence, the sideways growth velocity can be described according to Rodriguez et al. [586] by

α/E(r+r ) v(r) e− 0 (A.1) ∼ with r CtipV ε R + δ E(r) = a (A.2) 3/2 2πε0√ε0εa + 1 εc ((R + δ) + r2)

a) b)

5 µm 5 µm SKPMI [a.u.]j 0 nm 7 nm 1.20 V 2.3 1V

c) d)

5 µm EFMI [a.u.]j 5 µm EFMI [a.u.]A 0°0 8°1 0 1 Figure A.7: Surface potential extraction on written domains. a) topography after writing, b) Kelvin- probe force microscopy (KPFM) extraction of the surface potential, c),d) electrostatic force microscopy (EFM) amplitude and phase.

The surface potential after writing the domains was investigated by Kelvin probe force microscopy (KPFM) and electrostatic force microscopy (EFM). The experimental results are given in Fig. A.7. We can see, that a surface potential domain difference of about 1 V s

154 A.5 retention in conductance at dws in thin-film exfoliated lno present. This voltage is only about 5 % of the coercive voltage and, hence, only marginally interferes the cAFM measurements. One reason is the memrisitive property of the CDW. Another reason can be, that the DWs do not form through-domains, as this cannot be revealed by scanning probe microscopy techniques.

a) b)

5 µm I [a.u.]I 5 µm I [a.u.]log I 0 1 0 1 0 nA 5 nA

Figure A.8: cAFM measurement on written dot array of Fig. A.6, a) linear scale, b) logarithmic scale. Clearly, domains with comparable diameter but shorter voltage application give smaller current.

cAFM measurements on the dot array are shown in Fig. A.6 are given in Fig. A.8. We can observe, that domains with equal diameter show a different level of conduction. Domains, which have been written slower gave an increased current. a.5 retention in conductance at dws in thin-film exfoliated lno

The retention of domains and conduction at DWs was investigated. We checked the domain one month and two months after writing. We could not observe significant alteration of the domains. In Fig. A.9(a),(b) the initial domain shape is shown with PFM. The domain shape after two months (63 days) is given in Fig. A.9(c),(d). This result is in stark contrast to very recent reports, which find a weak retention of only about 25 h [524]. We can observe, that even very small features of this complex structure depicting a Christmas tree with sizes smaller than 100 nm at the star and below the lowest left branch are still preserved. The domain is only altered upon the application of 4 V in the lower − area. This resulted in a fracturing of the domain structure as has been discussed. The DWC was checked for this specific domain. The cAFM scans are shown in Fig. A.10 for bias voltages of 8 V and 10 V. The fractured domain area gives a reference for freshly poled domains. We can see, that the conductivity of DWs is still present and the absolute strength is at least 80 % of the initial conductivity and to the DWC at these reference areas. As previously The conductivity appears increased at corners of the domain configuration, this could be related to a larger inclination angle of the DW.

155 appendix

a) b)

2 µm 2 µm IPFM[a.u.]A I PFM[a.u.]j 0 1 0 1 c) d)

2 µm 2 µm IPFM[a.u.]A I PFM[a.u.]j 0 1 0 1

Figure A.9: Domain retention in a 600 nm thick thin-film exfoliated LNO. a),b) initial domain configuration by PFM, c),d) domain configuration after two months (63 days).

a) b)

2 µmI [a.u.] 2 µm I [a.u.] 0 0pA 1801 pA 0 0pA 9001 pA

Figure A.10: DWC on domains 2 months after poling a),b) cAFM scan at a bias voltage of 8 V and 10 V.

156 A.6 smim on dws in thin-film exfoliated lno a.6 smim on dws in thin-film exfoliated lno

a) VAC = 3V b) VAC = 3V c) VAC = 3V

1 1

2 2

VAC = 1V VAC = 1V VAC = 1V

I [a.u.] I [a.u.] I [a.u.] 1 µm 0 1 1 µm 0 1 1 µm 0 1 sMIM sMIM sMIM ∂ /∂V 10 7 = 10 7 < = · · d) 3 e) r = 50 nm 6 r = 50 nm s f = 1 GHz f = 1 GHz [S/m] 2 0.1 [a.u.] [a.u.] 4 1 = < 10 1 2 100 sMIM sMIM 1000 0 0 2 1 0 1 2 2 1 0 1 2 x [µm] x [µm] 10 2 10 2 · · f) g) sMIM- /sMIM- 8 sMIM- 6 1 6 2 < = <⇠ sMIM- = 4 4 s [S/m] 300 ⇠ @ 1 GHz 2 2 sMIM [a.u.] sMIM [a.u.]

0 0

0 1 2 3 4 5 0 1 2 3 4 5 x [µm] x [µm] Figure A.11: sMIM characterization of DWs in exfoliated thin-film LNO. a), b) imaginary and real part sMIM, c) demodulated imaginary part upon an AC modulation with 3 V and 1 V, both at 1 kHz, d), e) FEM simulated sMIM profiles for a film of d = 600 nm and a back electrode for different DW conductivities, f), g) sMIM cross sections at 1 and 2 , revealing a ratio in sMIM- / of about 8, which is indicating a DW conductivity = < of about 300 nm, which, yet, further depends on the tip diameter.

In Sec. 4.8 we have shown first sMIM measurements on DWs in thin-film exfoliated LNO, which were conducted with the PrimeNano system at f = 3 GHz. In this appendix section, we want to give further measurements at room temperature, which were conducted at f = 1 GHz at the system in the group of Prof. ZX Shen with an adaptive microwave source. The given measurements were performed on fractured domain configurations. In Fig. A.11(a),(b) the imaginary and real part of sMIM is given for different AC amplitudes of 1 V and 3 V at an AC frequency of 1 kHz. The at this frequency demodulated

157 appendix

a) b)

1 2 1 2 x x

I [a.u.] I [a.u.] 1 µm 0 1 1 µm sMIM 0sMIM 1 = <

c) 1 d) 1

500 nm I [a.u.] 500 nm I [a.u.] 0sMIM 1 0sMIM 1 = <

e) f) g)

1 µm 1 µm I [a.u.] 1 µm I [a.u.] 0 1 0 1 0 nm 5 nm PFM A PFM j

h) sMIM 2 < theory for exp. 1 r[nm] f = 1 GHz 50 sDW = 1000 S/m 100 200 0.5 sMIM exp. = sMIM [a.u.] r[nm] 50 0 100 200 0 0.2 0.4 0.6 0.8 x [µm] Figure A.12: sMIM measurements at f = 1 GHz on a fractured domain configuration in the exfoliated thin-film of LNO, a), b) imaginary and real part of the sMIM signal, c), d) zoom into very dense domain configuration at 1 , revealing a spatial resolution of about 50 nm, e), f), g) associated topography and PFM scans of the fractured domain configuration with new conductive tip, h) comparison of experimental and simulated DW sMIM profiles for a conductivity of 1000 S/m and different tip radii, revealing a 158 best fit for 100 nm. A.7 domain inversion evolution under a tip by phase-field modeling imaginary part, sMIM ∂ /∂V is given in Fig. A.11(c). We can observe, that the sMIM sig- = nal is stronger upon a higher AC bias voltage. The demodulated signal, however, shows a contrast reversal of the DW contrast. Apart from this, we can see a small domain contrast in the amplitude, which we already discussed in Sec. 4.8. The stronger DW signal upon larger AC voltage suggests increased conductivity by charge carrier injection at these voltages. The imaginary signal is about a factor of 8 larger than the real part, which can be correlated with a conductivity of about >300 S/m, as can be seen from the simulated sMIM profiles in A.11(d) and (e) for a tip radius of r = 100 nm. This value is also in good accordance with the step widths in the topography profiles. In Fig. A.12 we can show high spatial resolution measurements on the fractured domain configuration as well as correlate the experimental profiles with the simulated profiles for various tip radii and a DW conductance of 1000 S/m. In the Figs. A.12(a) and (b) the imaginary and real part of the sMIM signal is shown. The according characterization topography and PFM scans are shown in A.12(e)-(g), which show a very good agreement. A high resolution sMIM scan at 1 is given in Figs. A.12(c) and (d). We can clearly see individual DWs with a resolution of about 50 nm. The profiles as extracted along 2 were compared with simulated sMIM profiles for a conductivity of 1000 nm and three different tip radii between 50 and 200 nm. We can observe, that for a tip radius of 100 nm a very good agreement can be realized, both in terms of ratio of real and imaginary part as well as of the profile shape. a.7 domain inversion evolution under a tip by phase-field modeling

The temporal evolution of the ferroelectric domain structures during switching was modeled for the LNO thin film using the phase field approach by Bo Wang (group of Prof. Long-Qing Chen, Penn State University). We will give some brief description of the method here, which is based on reported descriptions of the group, especially Britson et al. [527, 528] The ferroelectric domains in the system are described using a continuous vector field representing the three principal components of polarization in the system. Evolution of this vector field with time t during ferroelectric switching was modeled by solving the time dependent Ginzburg-Landau equations.

∂P δF = L (A.3) ∂t − δPi which evolve the components of the polarization Pi towards the minimum of the free energy F of the thin film with respect to the distribution of polarization in the film. In eq. (A.3) L is a kinetic coefficient related to the domain wall mobility of the system. Ferro- electric domain structures were modeled in a free-standing thin film, hence no influences to mismatch between thin-film and substrate were incorporated into the calculation. The system was assumed to be much larger in lateral extent than in thickness and as a result periodic boundary conditions were used in the two in-plane directions (in case of

159 appendix three-dimensional calculation), while speciﬁc boundary conditions were used in the out- of-plane direction. 3 The total free energy in eq. (A. ) consists of four contributions, the bulk free energy fbulk, the gradient energy fgradient, the electrostatic interaction energy felectric, and the elastic interaction energy felastic integrated over the volume of the ﬁlm V: Z 4 F = fbulk + fgradient + felectric + felastic dV (A. )

The bulk free energy density describes the local energy of the stress free crystal using the sixth order Landau polynomial 2 2 2 2 2 2 5 fbulk = αiPi + βijPi Pj + γijkPi Pj Pk (A. ) with the phenomenological coefﬁcients αi, βij, and γijk. For sake of simplicity, the gradient energy was assumed to be isotropic with a coefﬁcient of g = 4 10 11 Nm4/C 2, hence 13 × − −

2 6 fgradient = gij(∂Pi) (A. ) Interactions between bound charges in the system lead to an electrostatic energy density in the system 1 f = E P ε κ E E (A.7) electric − i i − 2 0 ij i j where Ei is the ith component of the total electric field of the applied electric field and the depolarizing electric field, ε0 is the vacuum permittivity and kij are the background dielectric constant components. The PFM-like top electrode is simulated by a potential given by

ξ2 ϕ(x, y) = ϕ (A.8) 0 (x x )2 + (y y )2 + ξ2 − 0 − 0 with ξ = 10 nm. Electric fields around bound charges are determined by first solving the Poisson equation ε∂i(κii∂i ϕ) = ∂iPi for the electric potential ϕ and then using the relationship Eb = ∂ ϕ. For simplicity, fully compensating electrodes with negligible screening i − i lengths were assumed, which describes very well the case of a wafer-bonded film. The Poisson equation fixes the potential at both surfaces of the film. Elastic strain energy introduced by the spontaneous deformation associated with the ferroelectric transition was described using the expression

1 f = C ε ε0 ε ε0 (A.9) elastic 2 ijkl ij − ij kl − kl The important parameter, which tunes the inclination angle in this simulation is the gradient energy coefﬁcient, which is comparably small and deduced from the wall diameter of about 2 nm. All parameters of the simulation were extracted from the work of Scrymgeour et al. [525], given in Tab. A.1.

160 A.8 current transients in exfoliated lno

Table A.1: Parameters for phase-ﬁeld modeling of inversion under an AFM tip. Parameter Formula Value Unit

α 1/2ε 2.012 109 Nm2/C2 1 33 × α 3.608 109 Nm2/C2 2 × α 1/ε 1.345 109 Nm2/C2 3 11 × β 1/2C 12.25 109 N/m2 1 33 × β 1/4(C + C ) 6.4 109 N/m2 2 11 12 × β 1/4(C C ) 3.75 109 N/m2 3 11 − 12 × β C 7.5 109 N/m2 4 13 × β 1/2C 3 109 N/m2 5 44 × β C 0.9 109 N/m2 6 14 × γ 1/2(C + C )Q + 1/2C Q 0.216 109 Nm2/C2 1 11 12 31 13 33 × γ 1/2C Q + 1/2C Q 1.848 109 Nm2/C2 2 33 33 13 31 × γ 2C Q 1/2(C C )Q 0.33 109 Nm2/C2 3 14 44 − 11 − 12 42 − × γ C Q 3.9 109 Nm2/C2 4 44 44 × g 3.98 10 11 Nm4/C2 × − ε11 84.3 ε33 28.9 11 2 C11 1.9886 10 N/m × 11 2 C12 0.5467 10 N/m × 11 2 C13 0.6726 10 N/m × 11 2 C14 0.0783 10 N/m × 11 2 C44 0.5985 10 N/m × 4 2 Q31 0.003 m /C − 4 2 Q33 0.016 m /C 4 2 Q42 0.003 m /C − 4 2 Q44 0.0375 m /C a.8 current transients in exfoliated lno

The current in exfoliated LNO in LRS shows a transient behavior upon the application of a negative voltage, as can be seen in Fig. A.13. The current can be approximated with a stretched exponential decay as given by the Kohlrausch-Williams-Watts formula I(t) = I exp (t/τ)β . Such a decay is an indication towards hopping transport inside 0 − the film. The extracted stretch parameters and the temperature dependence of the decay time for the temperature-dependent decay curves are in good agreement with interpolated values extracted from MC simulations of the decay of bound polarons in lightly Fe:LNO. However, the decay time in the electrical transport measurements are two to three orders of magnitude larger. This can be explained by the fact, that the mean free path of bound polarons is about two orders of magnitude smaller than the film thickness. The large electric fields required to reduce the current value in the low-positive voltage regime is, 2+ hence, expected to be the result of a de-trap of FeLi deep traps.

161 appendix

0 a) 1 b) 10

1 0 0 10 / I / I I 0.5 I

300 K 340 K 300 K 340 K 2 10

2 1 0 1 2 3 2 1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 time t [s] time t [s] Figure A.13: Transient current upon the application of 210 V at different temperatures, a) linear − plot, b) logarithmic plot. The experimental current trace can be ﬁtted with a stretched exponential decay with a stretch parameter β of 0.6 and a decay time of τ = 4 s for T = 300 K and β = 0.3 and τ = 3 s for T = 340 K. These experimental values are in good agreement with MC simulations [549]. Solid lines represent experimental transients, dashed lines are ﬁtted curves.

The transient average decay times are comparably large, yet, upon very precise control the occupied deep trap density can be signiﬁcantly lowered, which was used for the endurance measurements. Furthermore, in Mg:LNO these transients are expected to be faster by about two orders of magnitude, due to the increase hopping rate of free polarons. Such a transient behavior, can only be observed in the case of inhibited injection, which would be the case in a single domain ﬁlm. a.9 surface acoustic wave excitation damping at dws

The apparent dip in sMIM- across the CDW in thermally treated LNO is found to be a < result of SAW excitation damping close to the DW. This feature was proven by frequency- dependent sMIM measurements spanning from 0.8 GHz to 3 GHz. Cross sections for selected frequencies can be seen in Fig. A.14(a). For a comparison we use the fundamental wave equation λ = v/ f . The SAW velocity of LNO are discussed by various authors [587, 588]. The point-like excitation of surface- acoustic waves is inhibited around the DW until a distance of λ/4. The acquired half- width-half-minimum (HWHM) of the sMIM- proﬁles, which is given in Fig. A.14(b), < can be ﬁtted with a velocity of 3.5 km/s, which is the expected value for y-cut CLN in z-direction. a.10 influence of uv illumination on domains in mg:lno

In the ferroelectric lithography measurements we observed, that DWs can move upon UV application. An exemplary case is given in Fig. A.15.A PPLN crystal with a period of 3.5 µm, hence about 1.7 µm large domain strips, shows a signiﬁcant decrease of the

162 A.10 influence of uv illumination on domains in mg:lno

10 2 · a) b) DW 1.2 experimental 1 f [GHz] l/4 with v = 3.5 km/s < 3 0.8 1.2 0.8 0.6 sMIM

HWHM [µm] 0.4

0.2

4 2 0 2 4 0 1 1.5 2 2.5 3 3.5 x [µm] f [GHz] Figure A.14: Real part of the measured admittance in sMIM across DWs in the GHz frequency regime. a) experimental cross sections for various frequencies spanning a half octave, b) extracted half-width half minima compared with the SAW excitation damping model. inverted domain after 25 min of UV illumination with a total irradiance of 12.3 /cm2 and 1.6 mW/cm2 for wavelengths below 310 nm. Such a DWs mobility highlights, that a strong internal ﬁeld is present in the LNO wafer with a speciﬁc preferred polarization orientation. Similar observations have been reported by Haußmann [228].

a) b) c)

1 µmI [a.u.] 1 µm 1 µm 0 1 IPFM[a.u.]A IPFM[a.u.]j 0 nm 40 nm 0 1 0 1 Figure A.15: PFM analysis of domain dynamics upon UV illumination over 25 min with a total irradiance of 12.3 mW/cm2 and 1.6 mW/cm2 for wavelengths below 310 nm a) topography of deposition result with 1 wt% DTAB b) PFM amplitude c) PFM phase image

163

ACRONYMS

3D three-dimensional

2D two-dimensional

2DEG two-dimensional electron gas

2DPG two-dimensional polaron gas

4PPL four-photon photoluminescence

APB anti-phase boundary

AC alternating current

ADF annular darkﬁeld

AFM atomic force microscope

AOBS acousto-optical beamsplitter

AOTF acousto-optical tunable ﬁlter

APD avalanche photodiode

BFO bismuth ferrite/BiFeO3

BTO barium titanate/BaTiO3 cAFM conductive atomic force microscopy

CB conduction band

CBM conduction band maximum

CCD charge-coupled device

CDW charged domain wall

CLN congruent lithium niobate

CLT congruent lithium tantalate

CMP chemical mechanical polishing

CoT conduction type

165 acronyms

CSHG Cherenkov second harmonic generation

CSTO calcium-strontium titanate/(Ca,Sr)3Ti2O7

CMOS complementary metal-oxide-semiconductor

DART dual AC resonance tracking

DC direct current

DFT density functional theory

DLM defocused light microscopy

DW domain wall

DWC domain wall conductivity

EMO erbium manganite/ErMnO3

ESR electron spin resonance

FeFET ferroelectric ﬁeld-effect transistor

FeRAM ferroelectric random memory

FEM ﬁnite element method

FN Fowler-Nordheim tunneling

FWHM full width half maximum h2h head-to-head

HAADF high angular annular dark ﬁeld

HeCd helium cadmium (laser)

HRS high resistance state

HRXRD high-resolution x-ray diffraction

HF hydroﬂuoric acid

HMO holmium manganite/HoMnO3

ICCD intensiﬁed charge-coupled device

KPFM Kelvin-probe force microscope

KTP potassium titanyl phosphate/KTiOPO4

LAO lanthanum aluminate/LaAlO3

166 acronyms

LRS low resistance state

LED light emission diode

LIA light-induced absorption

LNO lithium niobate/LiNbO3

LTO lithium tantalate/LiTaO3

MC Monte Carlo

NA numerical aperture

NIM Nano-impedance microscopy

NDD non-descanned detector

NDW neutral domain wall

OCT optical coherence tomography

OD optical density

ODRT optical damage resistance threshold

PEEM photoelectron emission microscopy

PF Poole-Frenkel emission

PFM piezo-response force microscope

PL photoluminescence

PLM polarization light microscopy

PMT photomultiplier tube

PPLN periodically poled lithium niobate

PTO lead titanate/PbTiO3 2 2 2 PVDF polyvinylidene ﬂuoride/-(C H F )n-

PZT lead zirconate titanate/Pb(ZrxTi1 x)O3 − qpmSHG quasi-phase matching second harmonic generation

RAM random-access memory

RRAM resistive random-access memory

SAW surface acoustic wave

167 acronyms

SBN strontium barium niobate/SrxBa1 xNb2O6 − sc single crystal

SCLC space-charge limited current

SEM scanning electron microscopy

SHG second harmonic generation

SLN stoichiometric lithium niobate

SLT stoichiometric lithium tantalate

SPC surface potential curve

SPM surface potential map

STE Schottky thermionic emission

STEM scanning transmission electron microscopy

STO strontium titanate/SrTiO3

STORM stochastic optical reconstruction microscopy sMIM scanning microwave impedance microscope t2t tail-to-tail

TEM transmission-electron microscopy

TGS trigylcine sulfate/(NH CH COOH) H O 2 2 · 2 TMO terbium manganite/TbMnO3

TRA time-resolved absorption

TRPLS time-resolved photoluminescence spectroscopy

TRPL time-resolved photoluminescence

UV ultraviolet

VB valence band

VBM valence band minimum

XRD X-ray diffraction

XPS X-ray photoelectron spectroscopy

YMO yttrium manganite/YMnO3

168 SYMBOLS

sDW surface charge density at domain wall ai Sellmeier coefﬁcient i, second order phenomenological Landau coefﬁcient a domain with in-plane polarization

A amplitude c concentration c+ domain with upward out-of-plane polarization c− domain with downward out-of-plane polarization

C contrast d thickness

∆n refractive index difference

E (photon) energy, electric ﬁeld

Ea activation energy

EC coercive ﬁeld

EC, f forward coercive ﬁeld

EC,b backward coercive ﬁeld

EF Fermi energy

Eint internal ﬁeld f free energy, frequency g SHG fringe separation gij gradient energy coefﬁcient

G~ m lattice vector of order m

I intensity, current lc coherence length

169 acronyms

kB Boltzmann constant

Λ interface distance parameter

λ wavelength

µ mobility

n refractive index

nω refractive index at fundamental wavelength

n2ω refractive index at second-harmonic wavelength

no refractive index at fundamental wavelength under ordinary excitation

ne refractive index at fundamental wavelength under extraordinary excitation

PS spontaneous polarization

~r position

r tip radius

tFWHM pulse width at full width half maximum

t time (delay)

ts switching time

T temperature

TC Curie temperature

v speed

V volume, voltage

VAC AC voltage

VC coercive voltage

w f oc focal width

x position along x axis

y position along y axis

z position along z axis

α domain wall inclination angle

β stretch parameter, horizontal inclination angle

170 acronyms

βij fourth order phenomenological Landau coefﬁcient

ε0 free space permittivity

γijk sixth order phenomenological Landau coefﬁcient

κij background dielectric constant parameters

ϕ phase, total inclination angle, electric potential

σ conductivity

τ decay time

τ averaged decay time h i

171 LISTOFFIGURES

Figure 2.1 Crystallographic analysis of the point groups (PG) ...... 8 Figure 2.2 Landau theory of uniaxial ferroelectrics ...... 10 Figure 2.3 Schematic of domains and domain walls (DWs) present in the triaxial ferroelectric BTO ...... 11 Figure 2.4 Types of DWs in a uniaxial crystal, through (1) and surface domains (3) as well as inclined DWs (2) with the local inclination angle α..... 12 Figure 2.5 Neutral and charged domain walls ...... 14 Figure 2.6 Cross-polarized light microscopy on LNO samples ...... 23 Figure 2.7 SHG conditions at domain walls ...... 27 Figure 2.8 FD OCT images recorded for PPLN samples ...... 29 Figure 2.9 Schematic of piezo-response force microscope (PFM), out-of-plane and b) in-plane contribution ...... 31 Figure 2.10 Ferroelectric lithography on periodically poledLNO...... 33 Figure 2.11 Multiphoton luminescence microscopy on LNO domains...... 35 Figure 2.12 Crystal structure in LNO...... 37 Figure 2.13 Phase diagram for Li2O:Nb2O5...... 40 Figure 2.14 Optical transmission of LNO samples...... 41 Figure 2.15 Refractive index of LNO...... 42 Figure 2.16 Influence of defects present in LNO onto the hysteresis loop and the internal crystal field...... 45 Figure 2.17 Polarons present in LNO...... 51 Figure 2.18 Sketch of conduction and valence bands and polaronic levels...... 52 Figure 3.1 Poling device for random incomplete poling...... 54 Figure 3.2 Surface SHG image of UV laser written surface domains...... 56 3 3 57 Figure . Electrical field simulation for LNO being treated close to TC...... Figure 3.4 Schematic of the fabrication steps to create single-crystalline thin-films of LNO...... 58 Figure 3.5 a) Schematic of vector PFM, DART™PFM and switching spectroscopy (ss) PFM...... 61 Figure 3.6 Schematic of the conductive-type atomic force microscopy...... 62 Figure 3.7 Schematic of the scanning-type microwave impedance microscopy. . . . 64 Figure 3.8 AFM probes used in this work...... 67 Figure 3.9 Schematic of the Leica SP5 MP laser scanning setup...... 68 Figure 3.10 CSHG microscopical scan inside a LNO crystal...... 69 Figure 3.11 Comparison of forward propagated and backward scattered CSHG detection...... 71 Figure 3.12 Schematic of the time-resolved luminescence spectroscopy setup. . . . 72

172 List of Figures

Figure 3.13 Schematic of the Lab-based ARPES system for photo-electron emission spectromicroscopy and X-ray photoelectron spectroscopy...... 73 Figure 4.1 Three-dimensional visualization by CSHG of a ferroelectric DW inside a 5 % Mg:CLN single crystal...... 79 Figure 4.2 CSHG visualization of an enclosed domain buried inside the crystal. . 81 Figure 4.3 CSHG inspection of 5 % Mg:PPLN...... 82 Figure 4.4 CSHG inspection of a thinned 5 % Mg:PPLN wafer being polished to a thickness of 100 µm...... 83 Figure 4.5 CSHG scans at different depths z over domains being written by a sharp AFM...... 84 Figure 4.6 3D representation of the DW shape inside thermally treated crystals of 5 % Mg:CLN crystals...... 85 Figure 4.7 Schematic of laser writing of surface domains into LNO...... 86 Figure 4.8 3D visualization of laser-written surface domains by CSHG microscopy in 5 % Mg:LNO...... 87 Figure 4.9 3D visualization of laser-written surface domains by CSHG microscopy in 7 mol% Mg:LNO...... 88 Figure 4.10 Sketch of the working principle in UV illumination assisted domain writing...... 89 Figure 4.11 3D visualization of bound polarization charge at a representative DW observed in 5 % Mg:LNO...... 90 Figure 4.12 3D visualization of both charged and uncharged DWs in 7 mol% Mg:LTO. 91 Figure 4.13 3D visualization of domains in 8 % Mg:LTO...... 92 Figure 4.14 3D visualization of thermally-treated LNO...... 93 Figure 4.15 Schematic of collinear and non-collinear SHG emission at inclined DWs. 94 Figure 4.16 Depth proﬁling with quasi-phase-matching second-harmonic generation (qpmSHG)...... 95 Figure 4.17 qpmSHG scans on complex topologies at the h2! DW under extraordinary (ee) and ordinary (oo) excitation...... 97 Figure 4.18 Photoelectron microspectroscopy of thermally-treated LNO, illumination intensity dependence...... 98 Figure 4.19 Photoelectron microspectroscopy of thermally-treated LNO, comparison with inclination angles extracted from qpmSHG...... 100 Figure 4.20 Steady-state luminescence spectrum in 7 mol% Mg:LNO for different cryogenic temperatures T...... 102 Figure 4.21 Time-resolved luminescence decay in virgin 7 mol% Mg:LNO...... 103 Figure 4.22 Time-resolved luminescence decay in virgin 7 mol% Mg:LNO...... 104 Figure 4.23 Time-resolved luminescence decay in virgin 3 mol% Mg:LNO...... 105 Figure 4.24 TRPL for different pump intensities...... 107 Figure 4.25 TRPLS for virgin and singly inverted domains in 7 mol% Mg:LNO and 3 mol% Mg:LNO...... 108

173 List of Figures

Figure 4.26 Normalized TRPLS graphs for 7 mol% Mg:LNO for different temperatures T = 200 K (a) and T = 100 K (c). The fitted data is given in (b,d), respectively. See our paper [518]...... 108 Figure 4.27 Temperature-dependent analysis of TRPL curves...... 110 Figure 4.28 Schematic of radiative recombination of electron and hole polaron in LNO after hopping transport...... 111 Figure 4.29 Time-dependent luminescence at differently doped CLN and SLN... 113 Figure 4.30 Comparison of local conductivity and DW inclination in high voltage treated LNO...... 114 Figure 4.31 PFM analysis of written domain sections at the contact resonance by DART...... 116 Figure 4.32 DWC in CLN thin films...... 117 Figure 4.33 cAFM scans for a set of external bias voltages Vext from 1 to 8 V. . . . . 118 Figure 4.34 Local I V measurements on CDWs at four marked spots...... 119 − Figure 4.35 Trace and retrace cAFM scan images at an external bias voltage of 5 V.120 − Figure 4.36 Two- and three-dimensional phase field simulation of CDW formation under an AFM tip...... 121 Figure 4.37 Proof of concept measurement on resistive switching in CLN thin films.122 Figure 4.38 Investigation of switching behavior, endurance, stability, and tunability of resistive switching...... 123 Figure 4.39 Comparison of extracted CDW inclination angles by phase field simulation with experimental read current...... 124 Figure 4.40 CoT analysis on CDWs in LNO...... 126 Figure 4.41 Complete I V cycle for room temperature (T = 300 K) and an ele- − vated temperature (T = 366 K)...... 128 Figure 4.42 Temperature dependence of the current for thin-film LNO...... 130 Figure 4.43 Schematic of AFM-based and plate electrode-like local switching on the basis of local nucleation...... 131 Figure 4.44 Schematic model to describe the formation and stabilization of CDW in nominally undoped LNO with iron impurities...... 133 Figure 4.45 FEM simulation of the microwave admittance at a DW with different levels of conductivity...... 135 Figure 4.46 Simulation of the sMIM profiles as apparent for CDWs with different levels of conductivity...... 136 Figure 4.47 sMIM measurement with a microwave excitation at f = 3 GHz and 300 mV on a CDW in thermally treated LNO...... 137 Figure 4.48 sMIM measurement on CDW sections in thermally treated LNO with both h2h and t2t polarization charge...... 138 Figure 4.49 sMIM measurement on DW in an exfoliated thin-film LNO...... 139 Figure 4.50 sMIM on a high-voltage treated LNO sample...... 140 Figure 4.51 Explanation of sMIM-∂ /∂V domain contrast...... 141 = Figure A.1 Laser ablation on LNO surfaces...... 149 Figure A.2 XPS measurement on the DW before and after argon sputtering. . . . . 150

174 List of Figures

Figure A.3 Local XPS measurements with on DW and inside the bulk...... 151 Figure A.4 XRD and HRXRD on LNO thin film with a thickness of 600 nm on Pt/Cr/SiO2/LNO...... 152 Figure A.5 PFM phase loop on an exfoliated LNO thin-film...... 152 Figure A.6 Dot-like domain writing into the LNO thin-film...... 153 Figure A.7 Surface potential extraction on written domains...... 154 Figure A.8 cAFM measurement on written dot array...... 155 Figure A.9 Domain retention in a 600 nm thick thin-film exfoliated LNO...... 156 Figure A.10 DWC on domains 2 months after poling...... 156 Figure A.11 sMIM characterization of DWs in exfoliated thin-film LNO...... 157 Figure A.12 sMIM measurements at f = 1 GHz on a fractured domain configuration in the exfoliated thin-film of LNO...... 158 Figure A.13 Transient current upon the application of 210 V at different temper- − atures...... 162 Figure A.14 Real part of the measured admittance in sMIM across DWs in the GHz frequency regime...... 163 Figure A.15 PFM analysis of domain dynamics upon UV illumination...... 163

175 LISTOFTABLES

Table 2.1 Inclination angles α for Mg-doped LNO samples as measured with PFM for different Mg doping concentrations c...... 20 Table 2.2 General ferroelectric, crystallic, and electrical properties of congruent (CLN) and stoichiometric lithium niobate (SLN)...... 38 Table 2.3 General ferroelectric, crystallic, and electrical properties of congruent (CLT) and stoichiometric lithium tantalate (SLT)...... 39 Table 2.4 Summary of linear and nonlinear optical properties of congruent and stoichiometric lithium niobate (CLN, SLN)...... 44 Table 3.1 Listing of all samples of LNO and LTO used in this work...... 55 Table 3.2 AFM probes used in this work...... 67 Table 3.3 Microscope objectives used in this work ...... 70 Table 4.1 Comparison of the deduced DWs inclination angles α inside CLN single crystals with different Mg doping concentrations...... 79 4 2 Table . Characteristic lifetimes τ1,2, stretched-exponential parameters β1,2, the max max equivalent intensities I1,2 , the mean energy E and width w for virgin Mg:LNO at T = 150 K...... 106 Table 4.3 Luminescence decay parameters for different doping concentrations and preparations at T = 130 K...... 109 Table 4.4 Comparison of time-resolved luminescence and absorption at differ- 109 ent wavelengths λabs in above-threshold Mg:LNO ...... Table 4.5 General conversion processes in LNO under assumption of the exis- 4+ 4+ 2+ 129 tence of NbLi , NbNb, and FeLi ...... Table A.1 Parameters for phase-ﬁeld modeling of inversion under an AFM tip. . 161

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224 PUBLICATIONS

Some ideas and ﬁgures have appeared previously in the following publications:

ARTICLES

[1] T. Kämpfe, P. Reichenbach, M. Schröder, A. Haußmann, L. M. Eng, T. Woike, and E. Soergel. Optical three-dimensional proﬁling of charged domain walls in ferroelectrics by Cherenkov second-harmonic generation. Physical Review B, 89:035314, 2014.

[2] P. Reichenbach, T. Kämpfe, A. Thiessen, A. Haußmann, T. Woike, and L. M. Eng. Multiphoton-induced luminescence contrast between antiparallel ferroelectric domains in Mg-doped LiNbO3. Journal of Applied Physics, 115(21): 2135069, 2014.

[3] P. Reichenbach, T. Kämpfe, A. Thiessen, A. Haußmann, T. Woike, and L. M. Eng. Multiphoton photoluminescence contrast in switched Mg:LiNbO3 and Mg:LiTaO3 single crystals. Applied Physics Letters, 105(1):122906, 2014.

[4] T. Kämpfe, A. Thiessen, A. Haußmann, T. Woike, and L. M. Eng. Real-time 3- dimensional proﬁling of ferroelectric domain walls. Applied Physics Letters., 107(15):- 152905, 2015.

[5] S. C. Kehr, R. McQuaid, L. Ortmann, T. Kämpfe, F. Kuschewski, D. Lang, J. Döring, J. M. Gregg, L. M. Eng. A Local Superlens. ACS Photonics, 3(1):20, 2016.

[6] Y. Zhang, T. Kämpfe, G. Bai, M. Mietschke, F. Yuan, M. Zopf, J. Hao, L. M. Eng, R. Hühne, J. Fompeyrine, F. Ding, and O. G. Schmid. Upconversion photoluminescence 3+ 3+ of epitaxial Yb /Er codoped ferroelectric Pb(Zr,Ti)O3 ﬁlms on silicon substrates. Thin Solid Films, 607, 32-35, 2016.

[7] T. Kämpfe, P. Reichenbach, A. Haußmann, A. Thiessen, T. Woike, R. Steudtner, and L. M. Eng. Time-resolved luminescence spectroscopy on LiNbO3. Physical Review B, 93:174116, 2016.

[8] W. Thiessen, A. Thiessen, F. Börrnert, T. Kämpfe, V. Fiehler, J. Kunstmann, A. U. B. Wolter, and L. M. Eng. Synthesis of ultrathin boride-nanosheets by means of exfoliation of layered boron Zintl-phases. accepted at Europ. J. Inorg. Chem., 2016.

[9] A. Haußmann, M. Schröder, A. Gemeinhardt, T. Kämpfe, and L. M. Eng. Bottom-up assembly of molecular nanostructures by means of ferroelectric lithography. accepted at Langmuir, 2016.

225 publications

[10] C. Godau, T. Kämpfe, A. Thiessen, A. Haußmann, and L. M. Eng. Enhancing the domain wall conductivity in lithium niobate single crystals. submitted to Nature Com- munications, 2016.

[11] A.-S. Pawlik, T. Kämpfe, A. Koitzsch, A. Haußmann, L. M. Eng, U. Treske, M. Knupfer, B. Büchner, E. Soergel, and L. M. Eng. Electronic reconstruction at conducting domain walls in LiNbO3 observed by photoemission electron microscopy. submitted to Nature Communications, 2016.

[12] T. Kämpfe, P. Reichenbach, A. Haußmann, T. Woike, E. Soergel, and L. M. Eng. 3D Polarization-charge textures in ferroelectric domain walls. in preparation, 2016.

[13] L. Wehmeier, T. Kämpfe, A. Haußmann, and L. M. Eng. Real-Time Observation of Ferroelectric Domain Wall Dynamics in Bulk Triglycine Sulfate at the Curie Temper- ature by Second Harmonic Generation Microscopy. in preparation.

[14] P. Reichenbach, T. Kämpfe, A. Haußmann, L. M. Eng, T. Woike, R. Steudtner, I. Hajdara, S. Szaller, L. Kovacs. Polaronic luminescence and its domain contrast in doped lithium niobate and lithium tantalate. in preparation, 2016

[15] T. Kämpfe, S. Johnston, E. Y. Ma, C. Godau, A. Haußmann, Z-X. Shen, B. Wang, and L. M. Eng. Giant resistive switching in exfoliated lithium niobate by domain wall conductivity. in preparation.

TALKS

[16] T. Kämpfe, M. Schröder, P. Reichenbach, A. Haußmann, T. Woike, and L. M. Eng. Three-dimensional ferroelectric domain mapping in lithium niobate. 77th Annual Meeting of the Deutsche Physikalische Gesellschaft, Regensburg, 10.03 – 15.03.2013.

[17] T. Kämpfe. Conductive Domain Wall Inspection by c-AFM and Cherenkov-SHG. 8th Dachshund Workshop, Bad Schandau, 13.11 – 15.11.2013.

[18] P. Reichenbach, T. Kämpfe, A. Thiessen, M. Schröder, A. Haußmann, T. Woike, and L. M. Eng. Domain contrast in photoluminescence at Mg doped LiNbO3 and LiTaO3 single crystals. 78th Annual Meeting of the Deutsche Physikalische Gesellschaft, Dresden, 29.03 – 03.04.2014.

[19] L. M. Eng, M. Schröder, T. Kämpfe, P. Reichenbach, A. Thiessen, T. Woike, E. Soergel, X. Chen, D. Bonnell, J. Poppe, and A. Haußmann. AC/DC DWC-AC & DC Domain Wall Conductivity in Charged Ferroic Domain Walls. 12th International Symposium on Ferroic Domains and Micro- to Nanoscopic Structures (ISFD-12), Nanjing, China, 02.11 – 05.11.2014.

[20] T. Kämpfe, P. Reichenbach, M. Schröder, A. Haußmann, T. Woike, and L. M. Eng. 3D-mapping of ferroelectric domain walls by Cherenkov second-harmonic generation. 79th Annual Meeting of the Deutsche Physikalische Gesell-schaft, Berlin, 13.03 – 15.03.2015.

226 publications

[21] P. Reichenbach, T. Kämpfe, A. Thiessen, M. Schröder, A. Haußmann, T. Woike, and L. M. Eng. Domain contrast in photoluminescence at Mg doped LiNbO3 and LiTaO3 single crystals. 79th Annual Meeting of the Deutsche Physikalische Gesellschaft, Berlin, 13.03.– 15.03.2015.

[22] T. Kämpfe, P. Reichenbach, A. Haußmann, T. Woike, E. Soergel, and L. M. Eng. Domain wall topologies and topological defects in ferroelectric lithium niobate. E- MRS Spring Meeting, Lille, France, 11.05 – 15.05.2015.

[23] L. M. Eng, T. Kämpfe, P. Reichenbach, M. Schröder, T. Woike, and A. Haußmann. Domain wall topologies and topological defects in ferroelectric lithium niobate. In- ternational Workshop on Topological Structures in Ferroic Materials (Topo2015), Sydney, Australia, 19 – 21.05.2015.

[24] L. M. Eng, A. Haußmann, T. Kämpfe, P. Reichenbach, M. Schröder, A. Thiessen, D. A. Bonnell, J. Poppe, E. Soergel, T. Woike. Optimizing Domain Wall Conductivity in Ferroelectric Domain Walls. 2015 Joint IEEE International Symposium on Applications of Ferroelectric (ISAF), International Symposium on Integrated Function- alities (ISIF), and Piezoresponse Force Microscopy Workshop (PFM), Singapore, 24.05 – 27.05.2015.

[25] T. Kämpfe, A. Thiessen, M. Schröder, A. Haußmann, T. Woike, and L. M. Eng. Visu- alization of ferroelectric domain walls in LiNbO3. International Workshop on Lithium Niobate, Tianjin, China, 13.11.– 15.11.2015.

[26] L. M. Eng, D. A. Bonnell, C. Godau, T. Kämpfe, J. Poppe, P. Reichenbach, M. Schröder, E. Soergel, A. Thiessen, T. Woike, and A. Haußmann. Conductive domain walls in single crystalline LiNbO3. International Workshop on Lithium Niobate, Tianjin, China, 13.11.– 15.11.2015.

[27] A-S Pawlik, A. Koitzsch, T. Kämpfe, A. Haußmann, M Knupfer, L. M. Eng, B Büch- ner. Imaging of conducting domain walls in lithium niobate with energy ﬁltered photoelectron microscopy. 80th Annual Meeting of the Deutsche Physikalische Gesellschaft, Regensburg, 06.03.– 11.03.2016.

[28] A. Haußmann, T. Kämpfe, C. Godau, P. Reichenbach, A. Gemeinhardt, A.-S. Pawlik, A. Koitzsch, L. Kirsten, E. Koch, and L. M. Eng. A closer look inside out - Do- main wall functionalities in LiNbO3 single crystals go 3D. 80th Annual Meeting of the Deutsche Physikalische Gesellschaft, Regensburg, 06.03.– 11.03.2016.

[29] C. Godau, T. Kämpfe, A. Thiessen, A. Haußmann, and L. M. Eng. Enhancing the domain wall conductivity in lithium niobate single crystals. 80th Annual Meeting of the Deutsche Physikalische Gesellschaft, Regensburg, 06.03.– 11.03.2016.

[30] S. Xiao, T. Kämpfe, Y. Jin, A. Haußmann, X. Lu, and L. M. Eng. Anisotropic Domain Wall Conductivity in LiNbO3 single crystals. 80th Annual Meeting of the Deutsche Physikalische Gesellschaft, Regensburg, 06.03.– 11.03.2016.

227 publications

[31] A. Haußmann, T. Kämpfe, C. Godau, A.-S. Pawlik, A. Koitzsch, L. Kirsten, E. Koch, and L. M. Eng. 2D and 3D investigations of novel domain wall functionalities in LiNbO3 single crystals. International Workshop of Topological Structures in Ferroic Mate- rials (Topo2016), Dresden, 16.08 – 19.08.2016

POSTER

[32] T. Kämpfe, P. Reichenbach, S. Grafström, and L. M. Eng. Photostable imaging by second-harmonic-generating nanoparticles. 75th Annual Meeting of the Deutsche Physikalische Gesellschaft, Dresden, 14.03 – 18.03.2011.

[33] P. Reichenbach, U Georgi, T. Kämpfe, S. Grafström, B. Voit, and L. M. Eng. Metallic Nanoparticles as Non-linear Optical Antennae in Photoactive Polymer Matrices. 75th Annual Meeting of the Deutsche Physikalische Gesellschaft, Dresden, 14.03 – 18.03.2011.

[34] T. Kämpfe, P. Reichenbach, S. Grafström, and L. M. Eng. Three-dimensional ferroelectric domain imaging of poled Mg:LiNbO3 single crystals by means of conjoint multiphoton photoluminescence and Cherenkov-type second harmonic generation (SHG) acquisition. IONS-NA 3 Stanford 2011, Palo Alto, USA, 13.10 – 15.10.2011.

[35] T. Kämpfe, P. Reichenbach, S. Grafström, and L. M. Eng. Ferroelectric-metal core- shell nanoparticles. 76th Annual Meeting of the Deutsche Physikalische Gesellschaft, Berlin, 26.03 – 30.03.2012.

[36] T. Kämpfe, P. Reichenbach, S. Grafström, and L. M. Eng. Ferroelectric-metal core- shell nanoparticles. 1st Dresden Nanoanalysis Symposium, Dresden, 26.04.2013

[37] T. Kämpfe, L. M. Eng. Ferroelectric CNTFETs. 3rd cfaed Research Festival, Dresden, 01.10.2014

[38] D. Lang, S. C. Kehr, T. Kämpfe, M. Schröder, A. Haußmann, and L. M. Eng. Near- ﬁeld optics of conductive ferroelectric domain walls. 78th Annual Meeting of the Deutsche Physikalische Gesellschaft, Dresden, 29.03 – 03.04.2014.

[39] D. Lang, S. C. Kehr, T. Kämpfe, M. Schröder, A. Haußmann, and L. M. Eng. Near- ﬁeld optics of conductive ferroelectric domain walls. Nanofair, Dresden, 01.07 – 03.07.2014.

[40] T. Kämpfe, L. M. Eng, J. Kabalcova, D. Zahn. Characterization of carbon nanotubes for CNTFETs. 5th cfaed Research Festival, Dresden, .10 – 24.10.2015

[41] A.-S. Pawlik, T. Kämpfe, A. Koitzsch, A. Haußmann, M. Knupfer, L. M. Eng, and B. Büchner. Photoelectron spectroscopy and microscopy at conductive domain walls in lithium niobate. International School of Oxide Electronics 2015, Cargèse, France, 12.10 – 24.10.2015

228 publications

[42] T. Kämpfe, A. Haußmann, S. Johnston, E. Y. Ma, Z.-X. Shen, H. Hu, and L. M. Eng. Giant resistive switching by conductive domain walls in exfoliated thin-ﬁlm lithium niobate. International Workshop of Topological Structures in Ferroic Materials 2016, Dresden, 16.08 – 19.08.2016

[43] L. Wehmeier, T. Kämpfe, A. Haußmann, and L. M. Eng, In-situ observation of ferroelectric domain wall dynamics close to the Curie temperature. International Workshop of Topological Structures in Ferroic Materials 2016, Dresden, 16.08 – 19.08.2016

[44] C. Godau, T. Kämpfe, A. Thiessen, A. Haußmann, and L. M. Eng. Enhancing domain wall conductivity in lithium niobate single crystals. International Workshop of Topological Structures in Ferroic Materials 2016, Dresden, 16.08 – 19.08.2016

[45] S. Xiao, T. Kämpfe, Y. Jin, A. Haußmann, X. Lu, and L. M. Eng. Anisotropic Domain Wall Conductivity in LiNbO3 single crystals. International Workshop of Topological Structures in Ferroic Materials 2016, Dresden, 16.08 – 19.08.2016

[46] T. Kämpfe, A. Haußmann, S. Johnston, E. Y. Ma, Z.-X. Shen, H. Hu, and L. M. Eng. Resistive switching of exfoliated thin-ﬁlm lithium niobate by conductive domain walls. MEMRIOX International Workshop 2016, Lohmen, 25.09 – 27.09.2016, Best Poster Award

SUPERVISEDTHESES

[47] J. Weßling. Leitfähige Domänenwände in dünnem Lithiumniobat. Bachelor thesis, Dresden, 03.02.16.

[48] L. Wehmeier. Second Harmonic Generation For Real-Time Tracking Of Ferroelectric Domain Walls Under External Stimuli. Master thesis, Dresden, 04.04.16.

BOOKSANDBOOKCHAPTERS

[49] P. Reichenbach, U. Georgi, U. Oertel, T. Kämpfe, B. Nitzsche, B. Voit, and L. M. Eng. Optical antennae for near-ﬁeld induced nonlinear photochemical reactions of photolabile azo-and amine groups In: Optically Induced Nanostructures: Biomedical and Technical Applications, 267–281, 2015.

[50] T. Kämpfe. Nanoscopically localized second-harmonic radiation sources for imaging and lithography. GRIN Verlag, 2011.

229

Art is I; science is we. Le germe n’est rien, le terrain est tout! — Claude Bernard

ACKNOWLEDGMENTS

First and foremost I have to acknowledge my supervisor Prof. Lukas M. Eng for the opportunity to work on this interesting topic, his inspiring ideas, and fruitful discussions. I like to thank him for his patience, guidance, encouragement, funding of the project, freedom to pursue it, the possibility to present the work in conferences, and his general support throughout these years.

I have to acknowledge Prof. Marty Gregg for his immediate and enthusiastic consent to review this dissertation thesis.

I sincerely thank Prof. Zhi-Xun Shen for the possibility to stay in his group at Stanford University to perform scanning microwave microscopy and temperature dependent transport measurements on the conductive DWs.

The most heartfelt thanks to Dr. Alexander Haußmann for the introduction to the topic of conductive domain walls in ferroelectrics, helpful assistance in almost any ways, aid from ferroelectric lithography, sample poling to scanning probe microscopy, advising my work over the years, proofreading this thesis and especially always having an open door for discussion,

Further I want to thank: Dr. Mathias Schröder and Dr. Andreas Thiessen for the introduction into photo- induced conductive domain walls in lithium niobate and the open support of the CSHG measurements with samples, Dr. Philipp Reichenbach for the introduction into second-harmonic generation and luminescence microscopy, Dr. Theo Woike for helpful discussions on polaronic transport and lifetimes as well as access to the diamond thread saw for sample preparation, Dr. Eric Yue Ma and Scott Johnston for their introduction to frequency-dependent scanning microwave microscopy, cryogenic transport measurements, discussion of surface-acoustic wave excitations, and for the nice time in the lab and the ofﬁce at Stanford, Dr. Yongliang Yang for the helpful support in FEM simulation of scanning microwave microscopy on domain walls,

231 publications

Christian Godau for the fabrication of high voltage treated lithium niobate, Anna-Sophie Pawlik and Dr. Andreas Koitzsch for willingness to perform energy- resolved PEEM and XPS measurements on conductive domain walls and there on- going patience to run measurements, Bo Wang for the phase-field model simulations of the poling process and the persistent inclination angle in lithium niobate, Jan Weßling for the macroscopic and microscopic measurements on conductive domain walls in thin-film exfoliated single crystalline lithium niobate, Markus Burkhardt for the possibility to use the light microscopy facility of CRTD for CSHG and luminescence microscopy, Shuyu Xiao for the preparation of tip-poled samples of thin-polished lithium niobate, Prof. Hui Hu for the provision of the investigated exfoliated thin-film samples, Dr. Elisabeth Soergel for the provision of thermally-treated lithium niobate samples, Dr. Robin Steudtner for the opportunity to investigate the time-dependent luminescence decay on his setup in HZDR, cfaed for funding of the PhD thesis, new insights into technology and the financial support of a research stay with an Inspire grant, Dr. Susan Derenko, Eric Jehnes, and Fabian Patrovsky for a more then pleasant office time, Peter Leumer and Kai Schmidt for immediate IT support, Johanna Katzschner, Tatyana Sereda, and Susann Störmer for support in bureau- cracy, the whole ferroix and SPM2 group for interesting meetings, lunch and coffee breaks. Finally, I need to thank my parents for their on-going support in all occasions throughout the last decades.

232 ERKLÄRUNG

Diese Dissertation wurde am Institut für Angewandte Physik der Fakultät Mathematik und Naturwissenschaften an der Technischen Universität Dresden unter wissenschaftlicher Betreuung von Prof. Dr. Lukas M. Eng angefertigt. Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Form einer anderen Pü- fungsbehörde vorgelegt. Ich versichere weiterhin, dass bislang keine Promotionsverfahren stattgefunden haben. Ich erkenne die Promotionsordnung in der Fassung vom 23.02.2011 der Fakultät Mathematik und Naturwissenschaften an der Technischen Universität Dres- den an.

Dresden, November 2016

Thomas Kämpfe