Atomistic Modeling of Cu Doping in the Lead-Free Ferroelectric Potassium Sodium Niobate

Atomistic modeling of Cu doping in the lead-free ferroelectric potassium sodium niobate

Sabine Korbel¨

Dissertation zur Erlangung des Doktorgrades der Fakultat¨ fur¨ Mathematik und Physik der Albert–Ludwigs–Universitat¨ Freiburg im Breisgau Dekan: Prof. Dr. Michael Ru˚ziˇ ckaˇ Erstgutachter: Prof. Dr. Christian Elsasser¨ Zweitgutachterin: Prof. Dr. Elizabeth von Hauff Tag der mundlichen¨ Prufung:¨ 12.12.2012 Atomistic modeling of Cu doping in the lead-free ferroelectric potassium sodium niobate

Dissertation zur Erlangung des Doktorgrades der Fakultat¨ fur¨ Mathematik und Physik der Albert–Ludwigs–Universitat¨ Freiburg im Breisgau

vorgelegt von Sabine Korbel¨ aus Munster¨

Freiburg 2012

Parts of this work have already been published:

• E. Erunal,¨ R.-A. Eichel, S. Korbel,¨ C. Elsasser,¨ J. Acker, H. Kungl, and M, J. Hoffmann. ”Defect structure of copper doped potassium niobate ceramics“. Functional Materials Letters, Vol. 3, pp. 19–24, January 2010. •S.K orbel,¨ P. Marton, and C. Elsasser.¨ “Formation of vacancies and copper substitutionals in potassium sodium niobate under various processing conditions“. Phys. Rev. B, Vol. 81, pp. 174115-1– 174115-11, May 2010. •S.K orbel¨ and C. Elsasser.¨ “Cu substitutionals and defect complexes in the lead-free ferroelectric KNN”. In: W. E. Nagel, M. Resch, and D. B. Kroner¨ (eds.), “High Performance Computing in Science and Engineering ’10”. Springer-Verlag Berlin Heidelberg, pp. 181–188, 2011. •S.K orbel¨ and C. Elsasser.¨ “Ab initio and atomistic study of ferroelectricity in copper-doped potassium niobate”. Phys. Rev. B, Vol. 84, pp. 014109-1–8, July 2011. • E. Erunal,¨ P. Jakes, S. Korbel,¨ J. Acker, H. Kungl, C. Elsasser,¨ M.J. Hoffmann, and R.-A. Eichel. “CuO-doped NaNbO3 antiferroelectrics: Impact of aliovalent doping and nonstoichiometry on the defect structure and formation of secondary phases”. Phys. Rev. B, Vol. 84, pp. 184113-1–11, November 2011.

• M. J. Hoffmann, H. Kungl, J. Acker, C. Elsasser,¨ S. Korbel,¨ P. Marton, R.-A. Eichel, E. Erunal,¨ and P. Jakes. “Inﬂuence of the A/B stoichiometry on defect structure, sintering, and microstructure in undoped and Cu-doped KNN”. In: S. Priya and S. Nahm (eds.), “Lead-Free Piezoelectrics”. Springer-Verlag New York, pp. 209–254, 2012. vi Contents

List of Figures ix

List of Tables xi

List of Acronyms xiii

List of Symbols xv

Summary xvii

1 Introduction 1

2 Computational Methods 7 2.1 Density-functional-theory calculations ...... 7 2.2 Atomistic simulations with shell-model potentials ...... 8 2.3 The Nudged-Elastic-Band Method (NEB method) ...... 8

3 Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN 9 3.1 Introduction ...... 10 3.2 Thermodynamic formalism ...... 10 3.3 Cu substitutionals and vacancies in KNN 50/50 ...... 14 3.4 The Virtual Crystal Approximation for KNN 50/50 ...... 26 3.5 Effect of the alkali stoichiometry on the substitution site of Cu ...... 27 3.6 Fe substitutionals in KNN 50/50 ...... 29 3.7 Discussion ...... 33

4 Ferroelectricity in KNO with isolated Cu substitutionals 37 4.1 Introduction ...... 38 4.2 Method ...... 38 4.3 Results ...... 40 4.4 Discussion ...... 49

5 Ab initio thermodynamics of defect complexes in KNO and KNN 51 5.1 Introduction ...... 52 5.2 Formation of defect complexes in Cu-doped KNO and KNN ...... 52 5.3 Atom diffusion in KNO ...... 58 5.4 Ferroelectricity in KNO with Cu-defect complexes ...... 61 5.5 Discussion ...... 68

6 Conclusion 71

Bibliography 73

Acknowledgement 81 viii Table of Contents

Curriculum Vitae 83 List of Figures

1.1 The perovskite structure ...... 2 1.2 Phase diagrams of PZT and KNN ...... 3 1.3 Ferroelectric “hardening“ and the “large-strain effect” ...... 5

3.1 Phase stability diagram of KNN ...... 13 3.2 40-atom and 80-atom supercells of KNN ...... 14 3.3 Defect formation energies of Cu in KNN as a function of the Fermi level ...... 17 3.4 Defect formation energies of Cu in KNN as a function of the chemical potentials . . . . . 18 3.5 Defect formation energies of vacancies in KNN as a function of the Fermi level ...... 19 3.6 Defect formation energies of vacancies in KNN as a function of the chemical potentials . . 21 3.7 Density of states of CuK-doped KNN ...... 23 3.8 Density of states of CuNb-doped KNN ...... 24 3.9 Density of states of KNN with vacancies ...... 25 3.10 VCA-DOS of KNN ...... 27 3.11 Effect of the alkali stoichiometry on the most stable site for Cu in KNN ...... 28 3.12 Defect formation energies of Fe substitutionals in KNN ...... 31 3.13 Phase stability diagram of Fe doped KNN ...... 32

4.1 Ferroelectric energy profile in undoped KNbO3 ...... 43 4.2 Ferroelectric energy profile in CuK-doped KNbO3 ...... 43 4.3 Ferroelectric instabilities in CuK-doped KNbO3 as a function of the Cu content ...... 44 4.4 Histogram of atomic displacements in CuK-doped KNbO3 ...... 45 4.5 Atomic displacements and ionic dipole moments in CuK-doped KNbO3 ...... 46 4.6 Ferroelectric energy profile in CuNb-doped KNbO3 ...... 47 4.7 Histogram of atomic displacements in CuNb-doped KNbO3 ...... 47 4.8 Atomic displacements and ionic dipole moments in CuNb-doped KNbO3 ...... 48

5.1 Defect complex conﬁgurations ...... 53 5.2 Formation energies of defect complexes in KNO ...... 54 5.3 Electronic DOS of KNO with defect complexes ...... 54 5.4 Binding energies of defect complexes in KNO ...... 55 5.5 Binding energies of defect complexes in KNN ...... 55 5.6 Binding energies of defect complexes in KNN with a bandgap correction ...... 56 5.7 Energy barriers for atom diffusion in KNO ...... 59 5.8 Defect complex conﬁgurations in ferroelectric KNO ...... 61 5.9 Energies of defect complexes in orthorhombic KNO ...... 63 5.10 Polarization around defect complexes in orthorhombic KNO ...... 64 5.11 NEB relaxation with the SMP ...... 65 5.12 Energy and polarization along transition paths in KNO:CuNb −VO ...... 66 5.13 Energy and polarization along transition paths in KNO:VO−CuNb −VO ...... 67 5.14 Energy and polarization along transition paths in KNO:VO−CuNb −VO ...... 68 x List of Figures List of Tables

3.1 Chemical potential of O as a function of pressure and temperature ...... 13 3.2 Lattice constants and formation energies of elements and compounds ...... 15 3.3 Atomic chemical potentials for special points in the phase diagram ...... 16 3.4 Displacements of atoms surrounding Cu substitutionals ...... 20 3.5 Displacements of atoms surrounding vacancies ...... 20 3.6 Differences of spin-up and spin-down electrons of Cu substitutionals ...... 22 3.7 Numbers of electrons and holes generated by Cu substitutionals ...... 22 3.8 Numbers of electrons and holes generated by vacancies ...... 26 3.9 VCA for KNN 50/50 ...... 26 3.10 Magnetic spin moments of Fe substitutionals on K and Nb sites in KNN ...... 30

4.1 Ferroelectric displacements in undoped KNbO3 ...... 40 4.2 Ferroelectric instabilities in undoped KNbO3 ...... 41 4.3 Shell-model potential parameters for CuK-doped KNbO3 ...... 41 4.4 Phonon frequencies in undoped KNbO3 ...... 42 4.5 Energies of ferroelectric and rotational modes in CuK-doped KNbO3 ...... 45

5.1 Defect formation energies, binding energies and concentrations in KNO ...... 56 5.2 Defect formation energies, binding energies and concentrations in KNN ...... 57 5.3 Defect formation energies, binding energies and concentrations in KNN after a bandgap correction ...... 57 5.4 Energy barriers and diffusion times of K and O vacancies in undoped and Cu-doped KNO 60 5.5 Shell-model potential parameters for CuNb-doped KNbO3 ...... 62 xii List of Tables List of Acronyms

CBM Conduction-Band Minimum DFT Density Functional Theory DOS Density Of States EPR Electron Paramagnetic Resonance fcc face-centered cubic GGA Generalized-Gradient Approximation GULP General Utility Lattice Program LDA Local Density Approximation

KNN Potassium sodium niobate, (K,Na)NbO3

KNO Potassium niobate, KNbO3 MBPP Mixed-Basis PseudoPotential (program) MD Molecular Dynamics MEP Minimum Energy Path MPB Morphotropic Phase Boundary NEB Nudged Elastic Band (method)

NNO Sodium niobate, NaNbO3

PTO Lead titanate, PbTiO3

PZT Lead zirconate titanate, Pb(Zr,Ti)O3 PE ParaElectric sc simple cubic SC SuperCell SMP Shell-Model Potential TST Transition state theory VBM Valence-Band Maximum VCA Virtual Crystal Approximation xiv List of Acronyms List of Symbols

A 12-fold coordinated cation (A) site in the perovskite ABO3 structure ...... 11 a lattice constant ...... 15 B 6-fold coordinated cation (B) site in the perovskite ABO3 structure ...... 10 B bulk modulus ...... 41 EB binding energy ...... 53 E f defect formation energy ...... 10 0 ∆E f formation energy ...... 15 EF Fermi energy ...... 10 EG band-gap energy ...... 22 G Gibbs free energy ...... 12 0 ∆Hf formation enthalpy ...... 11 kB Boltzmann’s constant ...... 52 P ionic dipole moment ...... 39 P polarization ...... 39 p, p0 pressure ...... 12 q defect charge ...... 10 S entropy ...... 53 S strain ...... 53 T temperature ...... 52 Z ionic charge ...... 40 ∆µ relative chemical potential (of atoms) ...... 11 µ chemical potential ...... 10 × × + 0 Kroger-Vink¨ notation for a neutral defect charge, e.g. CuK ≡(CuK+ ) ...... 16 • • 2+ + Kroger-Vink¨ notation for a positive defect charge, e.g. CuK ≡(CuK+ ) ...... 16 0 0 − Kroger-Vink¨ notation for a negative defect charge, e.g. VK ≡ (VK+ ) ...... 16 xvi List of Symbols Summary

Ferroelectrics are part of many piezoelectric applications, such as actuators and sensors. A commonly used ferroelectric material is lead zirconate titanate (PZT), but efforts are made to develop or improve lead- free alternatives. A possible way to optimize material properties is doping, for example with transition metals. One candidate for future lead-free ferroelectrics is potassium sodium niobate ((K,Na)NbO3, KNN). According to the literature, Cu doping in KNN improves the density of the sintered KNN ceramics. In this work, Cu and for comparison also Fe doping in KNN were investigated on the atomic and electronic level in order to obtain a better understanding by which microscopic mechanisms the dopants inﬂuence the macroscopic ferroelectric properties of KNN. In this work, two possible doping mechanisms were investigated. Some dopants can cause a polymorphic or a morphotropic phase transition (a phase transitions with temperature or composition), or they can shift them to a desired position. A phase boundary between two compositions is called a morphotropic phase boundary (MPB). Such an MPB exists, for example, in PZT and in Li-doped KNbO3. Another doping effect in ferroelectrics is ferroelectric “hardening”. In “hard” ferroelectrics the ferroelectric domain walls are less mobile than in “soft” ferroelectrics, which is sometimes desired because in this case the hysteresis is small and the electromechanical strain depends almost linearly on an applied voltage. The domain wall mobility is related to the strain response of the ferroelectric to an applied voltage. Defect complexes consisting of dopants and lattice vacancies possibly contribute to ferroelectric “hardening” by impeding domain wall motion. In this work, density-functional theory (DFT) in the local-density approximation (LDA) and molecular dynamics (MD) simulations with classical interatomic potentials were combined to search for MPB’s in Cu-doped KNN and to clarify if and how defect complexes consisting of Cu dopants and oxygen vacancies increase the energy needed for ferroelectric domain wall motion. We found a MPB in the CuNbO3-KNbO3 system at a few mol% Cu, similar to the MPB in Li-doped KNbO3, and a hardening effect of the defect complexes consisting of Cu substitutionals and oxygen vacancies. xviii Summary 1 Introduction

Motivation and background Motivated by the vision of an environment-friendly industry, in 2003 the European Union adopted the “Directive on the restriction of the use of certain hazardous substances in electrical and electronic equipment” (RoHS directive), which restricts the use of six elements and compounds, including lead [EU 03]. Piezoelectric devices containing lead are so far excluded from the RoHS regulation [EU 03] because the lead-free alternatives cannot yet compete in performance with the currently best material lead zirconate titanate (PZT). One possible substitute for PZT is the lead-free piezoelectric material potassium sodium niobate ((K,Na)NbO3, KNN) [Sait 04]. This is investigated in this work.

Piezoelectrics If a material responds to mechanical stress with a voltage, this phenomenon is called piezoelectricity. It results from changes in dipole moments in the crystal due to stress, e.g. when sublattices of positive and negative ions are shifted relative to each other. The occurrence of strain in response to an electric field is called the converse piezoelectric effect. Hence, piezoelectric materials can trans- form electrical into mechanical energy and vice versa. A well-known piezoelectric material is quartz [Curi 82]. There is a wide range of technological applications for piezoelectrics, such as displacement actuators [Damj 05], force generators, ultrasonic devices, microphones, underwater sound sources, ac- celerometers, and electric wave filters [Scot 89, Jaff 58]. Except for quartz, most of the piezoelectric materials used for piezoelectric applications are ferroelectrics [Damj 05]. Ferroelectrics belong to the family of piezoelectric materials, but not every piezoelectric is ferroelectric. Ferroelectrics were named in analogy with ferromagnets [Jaff 58]. Like ferromagnets exhibit a spontaneous magnetic polarization, ferroelectrics exhibit an electric one (ferroelectricity is discussed more extensively in the next paragraph). Some ferroelectrics, especially PZT, have much higher piezoelectric constants with respect to other, non-ferroelectric piezoelectrics [Damj 05, Jaff 58]. This is why they play such an important role among the piezoelectrics. In addition to the piezoelectric applications mentioned above, ferroelectrics are used in, e.g., ferroelectric capacitors, in particular ferroelectric DRAM’s (FRAM’s) [Scot 89, Wiki 11]. Among the industrially relevant ferroelectrics the ones with a perovskite structure are the most common ones [Damj 05]. The prominent piezoelectric PZT is one of them, and it is a strong ferroelectric [Jaff 58]. In order to find a suitable substitute for PZT, it is obvious to begin the search among compounds with the same or a similar crystal structure. KNN is such a compound. Like PZT, it is ferroelectric and crystallizes in the perovskite structure.

Ferroelectric perovskites Perovskite is the name of the mineral calcium titanate (CaTiO3). Several other oxides with the formula ABO3 (also some non-oxides) also crystallize in the perovskite structure, which is depicted in Fig. 1.1. Cations (A and B ions) occupy the cube corners (A sites) and the cube centers (B sites), oxygen anions sit on the centers of the cube faces. There are three oxygen anions per unit cell, each with a formal charge of −2. The formal charges of the cations add up to +6 so that the unit cell is altogether neutral. Typically the formal charge of the cation at the cube corner is +2 (e.g. in BaTiO3, SrTiO3, PbTiO3, PbZrO3), +1 (KNbO3, KTaO3, NaNbO3), or +3 (e.g. in LaAlO3). While some perovskites remain unpolar also at low temperatures (SrTiO3, KTaO3), others undergo one or several transitions to ferroelectric or antiferroelectric phases (BaTiO3, PbTiO3, PbZrO3, KNbO3, NaNbO3). 2 1. Introduction

A A O B B x O ↑  P ↓

↑

Figure 1.1: The cubic, paraelectric unit cell of an oxide with perovskite structure (left), and a tetragonally distorted unit cell in which the negative O ions and the positive B site ions are shifted in opposite directions, resulting in a spontaneous ferroelectric polarization P (right). The atomic displacements and unit cell distortion are those of tetragonal KNbO3 [Hewa 73b].

Above the Curie temperature, the unit cell of an oxide with perovskite structure is cubic and paraelectric. Below the Curie temperature of a ferroelectric perovskite, the positive and negative ions are displaced from their cubic positions into different directions, so that the unit cell is spontaneously electrically polarized. The unit cell is elongated along the direction of the ferroelectric polarization, which is usually either a h001i, h011i, or h111i direction. The magnitudes of both the ionic displacements and the elongation are typically of the order of one or few percent of the lattice constant, which is about 4 A.˚ Some perovskite oxides, e.g. NaNbO3, have an antiferroelectric ground state. Instead of or in addition to the ferroelectric or antiferroelectric displacements of the ions, the oxygen octahedra can be tilted [Glaz 72], e.g. in NaNbO3 and KxNa1−xNbO3 up to x ≈ 0.5 [Ahte 78] or in SrTiO3 [Glaz 72]. The origin of ferroelectricity is mainly ascribed to hybridization between electronic states of the B and the O atoms [Post 94, Migo 76, Cohe 92]. The cubic, paraelectric crystal can lower its energy by shifting the atoms off center and reducing the distance between B and O atoms. In some cases, like PbTiO3, also hybridization between the A and the O states contributes to ferroelectricity [Cohe 92].

Lead-free ferroelectrics Environment-friendly lead-free ferroelectrics are currently investigated in order to substitute lead zirconate titanate [Pb(Zi,Ti)O3, PZT]. There are two main candidates for possible lead-free substitutes for PZT: Bismuth-based perovskite oxides ((Bi,Na,K)TiO3 − BaTiO3, BNKT [Take 05]) and alkali niobates ((K,Na)NbO3, KNN [Sait 04]). Both BNKT and KNN crystallize in the perovskite structure like PZT. In a specially textured KNN sample with Li, Sb, and Ta additives [Sait 04], piezoelectric characteristics similar to those of PZT were measured. In this work, only KNN is investigated. The most commonly used ferroelectric, PZT, is a mixture of approximately 50% PbTiO3 and 50% PbZrO3. At this composition, PZT exhibits enhanced piezoelectric properties due to a morphotropic phase boundary [Zhan 07]. At a morphotropic phase boundary (MPB), a phase transition with composition takes place. Near a composition of K:Na=50:50 a morphotropic phase boundary exists also in KNN. Figure 1.2 shows the phase diagrams of PZT and KNN, including the MPB’s around PZT 50/50 and KNN 50/50 (K0.5Na0.5NbO3). Below the Curie temperature of PZT, which lies roughly between 600 and 700 K, PZT is ferroelectric. At the MPB in PZT with a Zr:Ti ratio of about 50:50 (PZT 50/50), a transition between a tetragonal phase (polarization in a h001i direction) and a rhombohedral phase (polarization in a h111i direction) occurs. KNN is ferroelectric below about 400° C. In this work the composition region between 0 and 50 mol% Na is investigated. In this composition range, the order of the ferroelectric phases is the same as in the end member KNbO3, whose cystal structure changes upon cooling from cubic and paraelectric to tetragonal and ferroelectric with a polarization along a h001i direction at about 435° C, to orthorhombic with a polarization along a h011i direction at about 225° C, and to rhombohedral with a polarization in a h111i direction at about −10° C (this is analoguous to BaTiO3) [Hewa 73a, Shir 54a]. KNN undergoes a transition between two different orthorhombic structures at at K:Na ratio of about 50:50 (KNN 50/50), one with and one without tilted oxygen octahedra [Ahte 76]. In a more recent study, a 3 coexisting monoclinic phase was found at this composition [Bake 09]. Near the MPB the two structures are approximately degenerated in energy, which facilitates structural transformations and therefore the reorientation of the ferroelectric polarization [Zhan 07]. Other than polymorphic phase transitions (phase transition with temperature), which also facilitate switching, phase transitions at MPB’s allow operation at various temperatures, if the MPB depends only weakly on the temperature. For both PZT and KNN this is the case. Possibly the MPB in KNN 50/50 leads to enhanced ferroelectric properties like the one in PZT 50/50, but according to [Zhan 07] the enhanced piezoelectric coefﬁcients measured in KNN at this composition are caused by the polymorphic phase transition (phase transition with temperature).

MPB FR

0 . 0 5

Figure 1.2: Phase diagrams of PZT (left, from [Nohe 00]) and KNN (right, from [Bake 09]). In the phase diagram of PZT, PC is the paraelectric, cubic phase, and FT,FM,FO, and FR are the cubic, tetragonal, monoclinic, orthorhombic, and rhombohedral ferroelectric phases. In the phase diagram of KNN the superscripts denote the sense of oxygen tilting (cf. section 4.3 for details) and the subscripts the cation displacements.

Tailoring ferroelectric properties by doping Analoguous to the magnetization of ferromagnets, the polarization in ferroelectrics as a function of the external electric field follows a hysteresis loop. Whereas in ferroelectric memory applications a strong hysteresis (a large remanent ferroelectric polarization and a large coercive field, hence a long-time stable ferroelectric memory effect) is desirable, one wants the displacement of a piezoelectric actuator to increase approximately linearly with the applied voltage. The amount of hysteresis can be tailored to a large extent by doping a standard ferroelectric material like PZT or KNN. Typically, in analogy to ferromagnets, ferroelectrics are devided into “hard” and “soft” ferroelectrics. “Soft” ferroelectrics exhibit a high compliance, permittivity and piezoelectric coefficients, accompanied by strong hysteresis and nonlinearity, “hard” materials have low values of the piezoelectric coefficients but a reduced hysteresis, nonlinearity, conductivity and frequency dispersion [Damj 05], and they have smaller losses [Dibb 95]. “Hard” and “soft” ferroelectrics can be obtained from the same base material, e.g. PZT, by acceptor doping (lower valent ions, e.g. Fe3+ on a Ti4+ site) or by donor doping (higher valent ions, e.g. Nb5+ on a Ti4+ site), respectively [Damj 05]. While the origin of “softening“ is still unclear [Damj 05], one possible ”hardening“ mechanism is the interaction of defect complexes with ferroelectric domain walls [Damj 05]. Ferroelectric domains are regions in which the ferroelectric polarization has the same orientation in each unit cell. Normally a single crystal or a grain of a polycrystal of a ferroelectric material consists of many domains with different orientations of the polarization [Dibb 95]. These differently oriented domains form in order to lower the electrostatic and elastic energy of the system [Damj 05]. The domains are separated by domain walls. In a tetragonal ferroelectric, the domain walls can separate domains with polarization directions that differ by 90° (90° domain walls) or by 180° (180° domain walls). In orthorhombic and rhombohedral phases, additional angles between the polarization directions of adjacent domains are possible. A newly fabricated ceramic is macroscopically isotropic because there are many domains with different polarization directions. Piezoelectricity is achieved by ”poling“, i.e. orienting the ferroelectric domains 4 1. Introduction along one direction in an electric field on the order of 1 kV/mm, usually at elevated temperatures (e.g. 150°C) [Dibb 95]. The remanent polarization after ”poling“ is the working point of the ceramic [Dibb 95]. For small electric fields, the strain in the poled ceramic is approximately proportional to the electric field [Dibb 95].

The motion of ferroelectric domain walls through the crystal or through a grain usually is the domi- nant switching mechanism in a ferroelectric, although in thin films or under irradiation with strong electric pulses [Qi 09] homogeneous switching (the polarization in a whole domain changes its direction instantaneously) may occur. According to [Damj 05], it is ”a reasonable assumption in most cases encountered in practice“ that ”domain-wall displacement is the main source of the dielectric, elastic and piezoelectric hystereses“. Figure 1.3(a) shows schematically the motion of a 90° domain wall inside a tetragonal perovskite in an electric field. One possible ”hardening“ mechanism is the pinning of domain walls by electric dipoles that originate from acceptor doping (cf. Fig. 1.3 (b)). An Fe3+ or Cu2+ substitutional on 4+ − 2+ a Ti site in PbTiO3, for example, may trap an oxygen vacancy and form a defect dipole FeTi −VO or 2− 2+ CuTi −VO [Erha 07b] which may pose an obstacle to domain wall motion [Mart 11]. In [Erha 07b], the relative energies of the nonequivalent orientations of the defect dipoles CuTi −VO and FeTi −VO parallel, antiparallel, and perpendicular to the surrounding bulk polarization in PbTiO3 are reported. According to [Erha 07b], the alignment of defect and bulk polarization is energetically favored by 1.21 eV (Cu doping) and 0.45 eV (Fe doping), respectively, compared to the antiparallel orientation. In [Mart 11], the energy barrier for reorienting the FeTi −VO defect complex from the parallel (low energy) to the antiparallel (high energy) configuration was determined to be approximately 1 eV, so that altogether about 1.5 eV have to be overcome to reorient the defect complex. The mechanism underlying ferroelectric hardening is believed to be the following [Damj 05]: In each ferroelectric domain atomic defect dipoles and the ferroelectric polarization tend to align. As a consequence it becomes hard to change the ferroelectric polarization direction of the domain by an applied electric field, and the motion of domain walls is impeded. Polarization switching can in general occur via the motion of non-180° or of 180° domain walls. However, in the following the focus lies on the motion of non-180° domain walls (e.g. 90°domain walls). For tetragonal KNbO3 (this work, not shown) and for PbTiO3 it has been found that non-180° domain walls (in this case 90°domain walls) have a lower formation energy than 180°domain walls [Meye 02] and should therefore be more abundant. In a ferroelectric domain the crystal structure is elongated along the polarization direction, i.e., a ferroelectric polarization along the (positive or negative) z direction is accompanied by an elongation along z. If a ferroelectric crystal was poled along the z direction and an electric field in the same direction (±z) is applied, the field-induced changes in ferroelectric polarization and in ferroelectric strain are also oriented along z. The ferroelectric strain obtained this way is approximately reversible, since the ferroelectric re- turns to its original poled state once the electric field is turned off. An especially large ferroelectric strain is obtained when the ferroelectric polarization is switched, e.g., from the z to the x direction by an electric field along x. In this case the strain along x and z is very large, but this large strain normally is obtained only once, since after the polarization has been switched into the new (x) direction, there is no mechanism that restores the original polarization direction (z) [Ren 04]. However, defect dipoles can provide such a restoring force that allows reversible switching by, e.g., 90°, by the following mechanism [Ren 04]: When the ferroelectric crystal or ceramic is produced at a temperature above the Curie temperature, the material is paraelectric, hence the defect dipoles are initially orientated randomly along all possible directions. After cooling down below the Curie temperature, ferroelectric domains form and give rise to preferential orientations of the defect dipoles in each domain. If the material is given enough time to equilibrate, the defect dipoles orient along the ferroelectric polarization in their respective domain. If an electric field is applied to the equilibrated material, some of the randomly oriented domains will have a polarization perpendicular to the electric field, such that their polarization is switched by 90°. However, since the new polarization direction is no longer aligned along the defect dipoles in the domain, the polarization will switch back once the electric field is turned off (cf. Fig. 1.3 (c)), since in typical applications the switching cycles are too short to allow the defect dipoles to reorient. This reversible large strain effect was first proposed and observed in [Ren 04] for acceptor doped BaTiO3 and PbTiO3. It was also observed in acceptor doped KNN [Feng 09], which is orthorhombic at room temperature with a polarization in a h011i direction. 5

time (a) P S DW (b)

(c)

Figure 1.3: Ferroelectric “hardening“ and the “large-strain effect” caused by defect dipoles. (a): In a ferroelectric crystal region without defect dipoles the spontaneous ferroelectric polarization P is switched by 90° by an applied electric field E. Because different polarization directions are equivalent, the domains remain in their new polarization state after the electric field approaches zero. The large strain S accompanying this switching process is irreversible. (b): A defect dipole with a dipole moment PD parallel to the surrounding spontaneous polarization impedes switching in a hard ferroelectric. The domain wall (DW) gets stuck at the defect. A small strain occurs. (c): The polarization around a defect dipole is switched temporarily by an electric field, but as the electric field approaches zero, the defect dipole restores the original polarization direction of its domain. This is the reversible “large-strain effect”.

Cu doping in KNN Piezoelectric ceramics are commonly produced via a carbonate and oxide processing solid state powder technology [Li 08, Herb 07]. Metal oxide and carbide powders, like KCO3 and Nb2O5, are mixed and solidified and densified by calcining and sintering. In industrial production the ceramics are preferentially sintered at atmospheric pressure [Mats 05]. Pure KNN ceramics fabricated in this way are less dense than desired [Mats 05]. This problem can be overcome using advanced sintering techniques, such as hot pressing or the field assisted sintering technique (FAST, also called spark plasma sintering) [Acke 10], but these techniques are unfeasible for industrial mass production [Acke 10]. A cheaper way to obtain denser KNN ceramics is liquid phase sintering using a sintering aid. Adding CuO as a sintering aid improves the densification of the ceramic KNN [Sait 04, Mats 04], possibly by forming a liquid phase, which can act as a vehicle for atom transport and therefore facilitate grain growth [Mats 05]. Combining CuO as a sintering aid with a small Nb excess (on the order of 1%), densities close to those obtained with hot-pressing were obtained [Mats 05]. Cu doping is therefore a possible approach to improve the quality of KNN ceramics in industrial production processes. Besides its positive influence on the grain and pore structure of the KNN ceramics, Cu may be incorporated into the KNN lattice, where it acts as a dopant and may affect the electrical and ferroelectric properties of KNN. But it is not as obvious which lattice site Cu occupies in (K,Na)NbO3 as it is for Ta dopants, for example. If one compares formal ionic charges and ionic radii of host and dopant atoms, in the case of Ta doping it is very likely that Ta substitutes on Nb sites. Ta is isovalent with Nb, and the ionic radii of Ta5+ and Nb5+ are equal (0.64 A)˚ [Shan 76]. In the case of Cu doping the substitution site is less obvious. The typical formal charges of Cu ions in oxides + 2+ + + + are Cu (as in Cu2O) and Cu (as in CuO). Cu is isovalent with the A site ions in KNN (Na and K ), but the ionic radius of Cu+ is smaller (Cu+: 0.77 A;˚ Na+: 1.02 A;˚ K+: 1.38 A˚ in 6-fold coordination). Cu2+ (0.87 A˚) has a similar ionic radius as Nb5+ (0.64 A)˚ [Shan 76], but its ionic charge is much smaller. Therefore it is hardly possible to determine the substitution site just by means of ionic radius and charge. As far as ionic charge and radius are concerned, A and B sites are the only plausible substitutions sites, but neither of them is ideally suited for Cu dopants. Cu might therefore substitute on A or B sites, or even on both. Depending on its substitution site, Cu can act either as a donor (Cu2+ on A sites) or as an acceptor (Cu2+ on B sites), or it may be incorporated as an isovalent dopant (Cu+ on A sites). As a donor Cu may lead to softening, as an acceptor to hardening of KNN ceramics. As an isovalent dopant it may alter the piezoelectric properties in a way similar to Li doping, which causes an additional MPB with enhanced 6 1. Introduction piezoelectric coefficients [Guo 04]. Therefore it is important to know the substitution site of Cu dopants in KNN in order to understand how Cu dopants modify the ferroelectric properties of KNN.

Goals and structure of this work This work is a theoretical atomic- and electronic-level study of transition-metal doping in the lead-free ferroelectric perovskite KNN. Cu and Fe were chosen as representative doping elements. In addition, Cu and Fe are EPR-active (EPR: electron paramagnetic resonance, also called electron spin resonance) so that the theoretical findings can be compared to EPR experiments. The focus lies on Cu doping, which has been more intensively investigated experimentally than iron doping because of its positive effect on the sintering behavior of KNN ceramics. This work is divided into two parts: In the first part isolated Cu and Fe substitutionals, in the second part defect complexes of Cu substitutionals and vacancies are investigated. After this introduction, a short general description of the methods applied here follows in chapter 2, “Computational Methods”. The first goal of this work is to determine whether Cu and Fe dopants substitute on A (alkali) or B (Nb) sites, or on both. The results are compiled and discussed in Chapter 3, “Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN”. The second goal is to find or refute a possible MPB in Cu-doped KNN. The results are given and discussed in Chapter 4, “Ferroelectricity in KNO with isolated Cu substitutionals”. The third goal is to gain a better understanding if and how defect complexes in Cu-doped KNN contribute to ferroelectric hardening. The results are compiled and discussed in Chapter 5, “Ab initio thermodynamics of defect complexes in KNO and KNN”.

Methods For the thermodynamics of point defects an accurate method is required which is transferable to different chemical environments. Especially if point defects are charged, the electronic structure should be taken into account. A suitable method for such cases is density-functional theory (DFT). DFT was developed by Hohenberg and Kohn [Hohe 64]. Its main statements are that the quantum-mechanical ground-state properties of a system of electrons, like valence electrons in a crystal, are a functional of the electron density, and that the total energy follows a variational principle with respect to the electron density: the ground-state density minimizes the total energy. Therefore DFT allows to obtain electronic ground-state properties for atoms, molecules and solids, such as energies and structures. In this work DFT is applied in order to obtain defect thermodynamics and electronic and structural properties of undoped and doped KNN. Density-functional theory calculations are regularly applied in materials modeling because of their accuracy, but they are computationally demanding compared to other methods and therefore limited to systems that contain on the order of a few hundred to thousand atoms, whereas classical atomistic simulation using interatomic potentials is much faster and therefore applicable to much larger systems. The simplest form of such a potential a two-body interaction between two atoms each. Then the potential energy of the crystal is obtained by superposing the interactions between all pairs of atoms. One example for the analytical form of the atom-atom interaction is the Buckingham potential. In this model the atoms interact via Coulomb forces between ions, van der Vaals forces, an short-range repulsive forces, which mimic the repulsion of overlapping electronic wave functions of neighboring atoms according to the Pauli exclusion principle. In this work, atomistic simulations with Buckingham potentials were combined with DFT calculations to obtain structural and energetical properties of undoped and doped KNO. By this combined approach it is possible to take advantage of both the accuracy of DFT and the speed of classical atomistic simulations. Details of the method and the computational approach are given in chapter 2, “Computational Methods“, and in the chapters 3 to 5. 2 Computational Methods

In this chapter it is described shortly how the density-functional theory and the classical atomistic simulations in this work were performed. For the theoretical background of the DFT and its numerical imple- mentation used here the reader is referred to Ref. [Meye 99], and for details of the program GULP, which was used for the classical atomistic simulations, to, e.g., Ref. [Gale 97], and references therein.

2.1 Density-functional-theory calculations

Density functional theory (DFT) in the local (spin-)density approximation (L(S)DA) was employed to determine defect formation energies of Na, K, Nb and O vacancies, substitutional Cu and Fe atoms, and Cu- defect complexes in KNbO3,K0.5Na0.5NbO3, and NaNbO3. The defect formation energies were calculated for atomistic supercell models containing 2 × 2 × 2 sc (40 atoms) or 2 × 2 × 3 sc (60 atoms) or 2 × 2 × 2 fcc (80 atoms) unit cells. Optimally smooth norm-conserving pseudopotentials as proposed by Vanderbilt [Vand 85], the Ceper- ley-Alder [Cepe 80] LDA exchange-correlation functional as parametrized by Perdew and Zunger [Perd 81], and Chadi-Cohen [Chad 73] k-point meshes that are equivalent to an 8 × 8 × 8 k-point mesh for the simple cubic 1 × 1 × 1 unit cell with Gaussian broadening [Fu 83] of 0.2 eV were applied. The calculations were performed with the mixed-basis pseudopotential (MBPP) method [Meye, Elsa 90, Ho 92, Meye 95, Lech 02], employing a basis of plane waves up to a maximum energy of 340 eV combined with atom- centered basis functions for alkali metal s + p semicore states, oxygen p valence states, Nb s + p semicore and d valence states, and Cu and Fe d valence states. The atomic positions were relaxed until all forces were smaller than 10 meV/A.˚ The differences in total energies between successive relaxation steps were typically of the order of 10−6 eV per formula unit. In the case of charged defects a compensating homogeneous background charge density was included in the calculations of electrostatic energies and potentials. The LDA bandgap of bulk KNN obtained with a band structure calculation is about 1.7 eV, whereas the experimental bandgap of KNbO3 is about 3.3 eV [Wies 74]. Because this serious underestimation of the bandgap by LDA can affect the gap states and thus the defect formation energies, in some cases the LDA bandgap was corrected by rigidly shifting the conduction band states upwards by the difference between the experimental and the LDA bandgap. This crude method was chosen because there is no preferential ﬁrst-principles method to correct for LDA bandgap errors, which is computationally efﬁcient enough to be applicable to rather large defect supercells. Two potentially applicable approaches are LDA+U (see, e.g., Refs. [Mose 07, Cart 08, Lech 04]) or SIC-LDA (see, e.g., Refs. [Korn 10] and [Voge 97]). But an application to the KNN system was postponed to a future work.

The Virtual Crystal Approximation for KNN 50/50 Modelling arbitrary compositions of KNN is hardly possible using the supercell approach because large supercells are required. E.g., the ratio K0.51Na0.49NbO3 requires a supercell consisting of 100 unit cells (51 KNbO3 and 49 NaNbO3 unit cells). The virtual crystal approximation (VCA) overcomes this difﬁculty by replacing the two atom types of a binary system by one virtual atom type which is created by mixing the properties (in this case the pseudopotentials that represent the atoms) in the required ratio [Nord 31]. Sev- 8 2. Computational Methods eral mixing procedures were employed for PZT by other authors [Rame 00, Bell 00]. In parts of this work pseudopotentials of K and Na were linearly mixed to model KNN. Further details are given in section 3.4.

2.2 Atomistic simulations with shell-model potentials

In the shell model [Dick 58], the atoms are represented as point-like ionic cores and spherical electronic shells, which enables atoms to be electrically polarized, in contrast to the so-called rigid-ion potentials, in which each atom or ion is represented by one particle only. The cores and shells of different atoms interact via the electrostatic Coulomb potential,

Coul qiq j Vi j (ri j) = , (2.1) 4πε0ri j where qi is the charge of particle i and ri j the distance between particle i and particle j. The shells additionally interact via a two-body Buckingham potential,

r − i j C Buck ρi j i j Vi j (ri j) = Ai je − 6 , (2.2) ri j where ri j is the distance between two shells. The ﬁrst term is the short-range repulsion, the second the Van-der-Waals interaction. The core and shell of an atom interact via a spring potential, 1 1 V spring = k r2 + k r4 , (2.3) i,cs 2 2 i,cs 24 4 i,cs where ri,cs is the distance between core and shell of atom i. The Buckingham interaction was truncated at a distance of 6.5 A,˚ the spring interaction at 1 A.˚

2.3 The Nudged-Elastic-Band Method (NEB method)

In order to obtain energy barriers for switching the spontaneous polarization, the ”nudged-elastic-band“ (NEB) method [Henk 00] was applied. The NEB method is a method to obtain the ”minimum energy path“ (MEP) between two structures (initial and final structure, e.g. two different polarization directions) that correspond to local minima on the potential energy landscape of the system which are separated by an energy barrier. The energy barrier (together with entropy differences) determines how probably or at which rate the system changes from one state to the other. In the NEB method a chain of structures are set up which interpolate the initial and final structure in some way, for example linearly. These structures are called images. The energy of the images is then minimized subject to the constraint that the images may not move into the direction of the initial or final structure. This constraint is implemented by subtracting from the forces that act on the particles the force component along the NEB path. In order to keep the images in approximately equal distances from each other, the images are connected by an artificial spring force. At the end of the NEB calculation the images lie along the MEP, and the energy barrier is the energy maximum along the MEP. In this work the NEB was used along with the DFT in Chapter 5.3 and with SMP calculations in Chapter 5.4. 3 Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN 10 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

The content of this chapter, with the exception of the sections 3.4, 3.5, and 3.6, has been published in [Korb 10] (Korbel,¨ Marton, Elsasser,¨ Phys. Rev. B, Vol. 81, pp. 174115-1–11, May 2010. Copyright 2010 by the American Physical Society).

In this chapter the most stable substitution sites of Cu and Fe dopants in KNN are determined for different thermodynamic conditions.

3.1 Introduction ([Korb 10], Section I)

In this chapter, the substitution site of Cu dopants in KNN is determined as a function of the processing conditions. In Ref. [Zuo 08] it is assumed that Cu atoms substitute Nb atoms (on B sites in the perovskite structure ABO3), and oxygen vacancies are formed as a charge compensation mechanism. In Ref. [Li 07] Cu is assumed to substitute both alkali and Nb atoms (on A or B sites, respectively). While Cu2+ on a Nb5+ site can act as an acceptor dopant and lead to ferroelectric hardening, it can act as a donor on a Na+ or K+ site and cause ferroelectric softening. In order to determine the substitution site of Cu dopants and the prevailing vacancy type as function of the chemical processing conditions, first-principles density functional theory (DFT) calculations were carried out to obtain the thermodynamic stability of defect configurations and charge states of vacancies and Cu and Fe substitutionals in KNN. The defect formation enthalpy, which determines the substitution site of the dopant, depends on the chemical potentials of the atomic reservoirs [Wall 04], which for gases (oxygen in the case of KNN) correspond to the temperature and the partial pressures of the elemental constituents during processing [Reut 01]. In the present work the influence of the processing conditions is investigated according to the thermodynamic first-principles formalism outlined in, e.g., Ref. [Erha 07a]. First-principles calculations have already been applied to other ferroelectric materials by other authors, e.g., copper and iron doping in lead titanate [Erha 07b], vacancies in alkali niobates [Shig 04, Shig 05], and iron impurities in potassium niobate [Post 98] have been investigated. No such studies of Cu doping of alkali niobates have been reported so far, and comparable published studies of vacancies in the KNN system have been limited to neutral vacancies.

3.2 Thermodynamic formalism ([Korb 10], Section II)

The thermodynamically most stable lattice site for a dopant is governed by the defect formation enthalpy, which for most cases of inorganic solid state systems can be approximated by the defect formation energy [Wall 04]. The defect formation energy E f for a defect X in a charge state q is given by [Wall 04]

f q q E [X ] = Etot[X ] − Etot[bulk] − ∑µini + q · (EF + EVBM + ∆V), i

q where Etot[X ] is the total energy of the supercell containing a defect X with charge q, and Etot[bulk] is the total energy of a perfect cell of the same size. µi is the chemical potential of atom species i, ni is the number of atoms of species i that is exchanged with a reservoir in the defect formation reaction (ni > 0 for species that are added to, ni < 0 for species that are removed from the host crystal), and EF is the Fermi energy relative to the energy of the valence band maximum, EVBM. At zero temperature the Fermi energy is identical with the chemical potential of an electron, and the total energy per atom of a species is identical with the chemical potential of the species. ∆V is a correction term which aligns the energy zero of the crystal with a defect to that of the perfect crystal [Wall 04]. For charged defects, −q is the number of excess electrons assigned to the defect. Assuming chemical equilibrium conditions, the chemical potentials of the constituents can vary in ranges that are given by requiring the perovskite phase to be stable and the fact that the chemical potentials cannot exceed those of the composing elements in their most stable phases (see Tab. 3.3 for the numerical values). 3.2 Thermodynamic formalism 11

In the following, relative chemical potentials ∆µi are used:

0 ∆µi = µi − µi , (3.1) 0 where µi is the chemical potential of species i in its most stable elemental phase. In equilibrium, the alkali niobate perovskite ANbO3 is stable if

0 ∆µA + ∆µNb + 3∆µO = ∆Hf (ANbO3), (3.2) 0 where ∆Hf (ANbO3) is the formation enthalpy of ANbO3 from its elemental constituents (metallic A, metallic Nb, and gaseous oxygen). The upper boundary of the chemical potentials is

∆µi ≤ 0, (3.3) since otherwise the elemental phase of component i would precipitate. The outer triangle in Fig. 3.1 represents the allowed range of the chemical potentials for the alkali (A) and Nb atoms according to Eqs. (3.2) and (3.3). The allowed region is further conﬁned by the competing oxide phases A2O, A2O2, NbO, NbO2, and Nb2O5:

0 2∆µA + ∆µO ≤ ∆Hf (A2O), (3.4) 0 2∆µA + 2∆µO ≤ ∆Hf (A2O2), 0 ∆µNb + ∆µO ≤ ∆Hf (NbO), 0 ∆µNb + 2∆µO ≤ ∆Hf (NbO2), 0 2∆µNb + 5∆µO ≤ ∆Hf (Nb2O5).

In the case of a mixed perovskite, where the A sites are occupied partly by K and partly by Na atoms, there is an additional degree of freedom for the two A species. However, because metallic Na and K crystallize in the same structure, and the formation energies for the two alkali metals and their oxides are of similar magnitude, in the calculation of the chemical potentials it was assumed that ∆µK = ∆µNa = ∆µA and 1 ∆H0(bccA) = ∆H0(bccK) + ∆H0(bccNa), (3.5) f 2 f f 1 ∆H0(A O) = ∆H0(K O) + ∆H0(Na O), f 2 2 f 2 f 2 1 ∆H0(A O ) = ∆H0(K O ) + ∆H0(Na O ). f 2 2 2 f 2 2 f 2 2 The allowed ranges of the chemical potentials that remain after taking into account Eqs. (3.4) are indicated by the gray area in Fig. 3.1. Different processing conditions correspond to different combinations of the chemical potentials. The points indicated by the numbers 1 to 8 in Fig. 3.1 cover all extremes of the possible conditions. These extremes correspond to the equality in two of the Eqs. (3.3) and/or (3.4). For each of the points 1 to 8, the ANbO3 perovskite and two reservoir materials are in thermodynamical equilibrium. For instance, at point 2, ANbO3,O2, and A2O2 are in equilibrium. Together with Eq. (3.2), which must hold for all points, three equations for each point determine the three chemical potentials of A, Nb, and O:

Point 1:

0 2∆µA + ∆µO = ∆Hf (A2O), 0 2∆µA + 2∆µO = ∆Hf (A2O2). 12 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

Point 2:

∆µO = 0, 0 2∆µA + 2∆µO = ∆Hf (A2O2).

Point 3:

∆µO = 0, 0 2∆µNb + 5∆µO = ∆Hf (Nb2O5).

Point 4: 0 2∆µNb + 5∆µO = ∆Hf (Nb2O5), 0 ∆µNb + 2∆µO = ∆Hf (NbO2).

Point 5: 0 ∆µNb + 2∆µO = ∆Hf (NbO2), 0 ∆µNb + ∆µO = ∆Hf (NbO).

Point 6:

∆µNb = 0, 0 ∆µNb + ∆µO = ∆Hf (NbO).

Point 7:

∆µNb = 0, ∆µA = 0.

Point 8:

∆µA = 0, 0 2∆µA + ∆µO = ∆Hf (A2O).

In the following, the chemical potentials µ were approximated by the total energies per atom E for the 0 0 crystalline materials and the formation enthalpies ∆Hf were approximated by the formation energies ∆E f . When calculating the defect formation energies E f , a temperature and pressure correction was applied for the oxygen gas as follows. The chemical potential of oxygen is related to the oxygen partial pressure and the temperature via the ideal gas equation [Reut 01]:

1 1 µ (T, p) = G (T, p) = (H (T, p ) − TS (T, p )) O 2 O2 2 O2 0 O2 0 1 p + kBT ln (3.6) 2 p0 1 p = µO(T, p0) + kBT ln , 2 p0 where p0 is a reference oxygen partial pressure (the standard atmospheric pressure, about 1 bar), T is the temperature, G is the Gibbs free energy per molecule, and H and S are enthalpy and entropy per oxygen molecule, which can be found in thermochemical tables and were taken from Ref. [Lide 08]. For a deriva- tion of Eq. (3.6) see e.g. Ref. [Reut 01]. The oxygen chemical potentials were calculated (cf. Table 3.1) that correspond to air at the standard atmospheric pressure (pO2 ≈ 0.2 bar) and to air at 1 µbar (pO2 ≈ 0.2 µbar), indicated by diagonal lines in Fig. 3.1, at about room temperature (300 K) and at 1300 K, the latter being in the range of typical sintering temperatures for KNN [Acke 10]. 3.2 Thermodynamic formalism 13

300 K 1300 K 0.2 bar 0.2 µbar 0.2 bar 0.2 µbar -0.295 -0.473 -1.576 -2.350

Table 3.1: Relative chemical potential of oxygen, ∆µO, in eV for oxygen partial pressures of 0.2 bar and 0.2 µbar, for the temperatures 300 K and 1300 K, respectively. ([Korb 10] Tab. I)

µA [eV] −14 −12 −10 −8 −6 −4 −2 6 0 7 0 0 0 0 0 .2 bar .2 .2 bar .2 µ µ bar bar 5 , , 4 , , 300 K 300 K 1300 K 1300 8 −2

K −4

−6 [eV] Nb

−8 µ

1 −10 3

2 −12

−14 Figure 3.1: Region of possible values for the relative chemical potentials as deﬁned in Eq. (3.1). The area inside the outer triangle is given by the Eqs. (3.2) and (3.3), the shaded area remains after taking into account the Eqs. (3.4). The diagonal lines indicate the chemical potentials of oxygen that correspond to oxygen partial pressures in air at the standard atmospheric pressure and in air at 10−6 bar, for the temperatures 300 K and 1300 K. ([Korb 10] Fig. 1, [Hoff 12] Fig. 8) 14 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

Figure 3.2: Sc 2 × 2 × 2 (40 atoms, dashed line) and fcc 2 × 2 × 2 (80 atoms, solid line) supercell of KNN with a rocksalt-like ordering of K and Na. a is the cubic lattice parameter, cf. Tab 3.2. Oxygen atoms are omitted for clarity. ([Korb 10] Fig. 2)

3.3 Cu substitutionals and vacancies in KNN 50/50 ([Korb 10], Sections III and IV)

Density functional theory (DFT) in the local density approximation (LDA) was employed to determine defect formation energies of Na, K, Nb and O vacancies and substitutional Cu atoms in K0.5Na0.5NbO3. The defect formation energies were calculated for atomistic supercell models containing 2 × 2 × 2 fcc unit cells (80 atoms). Some results obtained with 2 × 2 × 2 sc supercells (40 atoms) are also presented for comparison. Figure 3.2 shows the supercells used here. The cubic high-temperature structure of KNN 50:50 (the Curie temperature is close to 400° C) and an ordered solid solution with the K and Na sublattices forming a rocksalt-like structure were assumed throughout. The LDA lattice constant of perfect KNN was used for the defective supercells. The total energy of the oxygen molecule was calculated in a periodic cubic supercell with an edge length of about 30 A.˚ The binding energy of the oxygen molecule and the formation energies of the metals were obtained as the total-energy differences compared to single atoms in the same cubic supercell as the O2 molecule. For the single atoms and the oxygen molecule, spin-polarization was taken into account. Spin-polarization of the substitutional Cu impurities was taken into account when calculating the defect formation energies with the larger (80 atoms) supercells. In the calculations with the smaller supercells (40 atoms), spin-polarization was neglected. Test calculations including spin-polarization yielded energy differences of about 0.2 eV between the polarized and the unpolarized state, thus the spin-polarization of the Cu defects has only a small effect on the defect formation energies. The conduction band states were rigidly shifted upwards by the difference between the experimental and the LDA band gap.

Lattice parameters and cohesive energies [Korb 10], Section IV.A) In order to validate the computational settings of the mixed basis and the pseudopotentials, the equilibrium lattice parameters and the formation enthalpies (here approximated by the formation energies) with respect to the elements of potassium niobate (KNO), sodium niobate (NNO), KNN 50/50, and the elementary and binary reference materials were determined. The results are compared to DFT and to experimental data from the literature in Tab. 3.2. The lattice constants determined by LDA are systematically underestimated with respect to experimental data in this work as well as in the literature. The cited DFT results of other authors were obtained with 3.3 Cu substitutionals and vacancies in KNN 50/50 15

0 0 a ∆E f ∆Hf calc. in calc. by calc. in calc. by this work others Expt this work others Expt 1 2 KNbO3 3.943 3.954 4.00 −14.296 1 2 3 NaNbO3 3.904 3.914 3.87 −13.852 −13.600 4 K0.5Na0.5NbO3 3.924 3.94 −14.067 K (bcc) 4.984 5.095 5.2256 −0.963 −1.067 −0.947 Na (bcc) 4.049 4.055 4.2248 −1.309 −1.219 −1.11310 Nb (bcc) 3.231 3.2711 3.30312 −8.551 −8.1511 −7.5711 Cu (fcc) 3.538 3.5613 3.6114 −4.423 −4.727 −3.4910 15 16 K2O 6.176 6.44 −3.792 −3.764 17 16 K2O2 6.529 6.733 −5.120 −5.139 15 18 Na2O 5.407 5.55 −4.214 −4.299 19 16 Na2O2 6.146 6.22 −4.687 −5.319 NbO 4.148 4.21020 −4.666 −4.20621 22 16 NbO2 4.792 4.846 −8.555 −8.239 23 a,24 Z − Nb2O5 5.278 5.219 −18.124 −19.687 25 B − Nb2O5 12.600 12.73 −20.805 CuO 4.631 4.68426 −1.440 −1.6427 28 27 Cu2O 4.191 4.269 −1.372 −1.75 29 29 29 29 O2 1.220 1.22 1.21 −7.467 −7.54 −5.2

1 [King 94] 9 [Vosk 80] 17 [Brem 92] 24 [Pozd 02] 2 [Shir 54b] 10 [Kitt 96] 18 [OHar 72] 25 [Lave 64] 3 [Pozd 02] 11 [Garc 92] 19 [Tall 57] 26 [Asbr 70] 4 [Atti 05] 12 [Neub 31] 20 [Bowm 66] 27 [Weas 84] 5 [Fran 95] 13 [Hash 07] 21 [Scha 64] 28 [Kirf 90] 6 [Barr 56] 14 [Stra 69] 22 [Bolz 97] 29 [Pain 82] 7 [Bagn 89] 15 [Dove 91] 23 [Zibr 98] 8 [Abe 94] 16 [Chas 98] a Unknown structure modiﬁcation.

Table 3.2: Calculated and experimental values for the lattice constants a in A˚ the calculated formation 0 0 energies ∆E f , and the experimental formation enthalpies ∆Hf in eV per formula unit for KNN, its elemental and oxidic reservoirs. The structures of Nb2O5, CuO, and A2O2 were optimized with respect to volume and internal coordinates. In the case of O2, a denotes the dimer bond length. ([Korb 10] Tab. II) 16 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

Point ∆µA ∆µNb ∆µO Equilibrium between 1 −1.551 −9.815 −0.9001 A2O2, A2O, ANbO3 2 −2.452 −11.615 0.000 O2, A2O2, ANbO3 3 −3.664 −10.402 0.000 O2, Nb2O5, ANbO3 4 −1.817 −1.165 −3.695 Nb2O5, NbO2, ANbO3 5 −1.623 −0.777 −3.889 NbO2, NbO, ANbO3 6 −0.069 0.000 −4.666 NbO, Nb, ANbO3 7 0.000 0.000 −4.689 A, Nb, ANbO3 8 0.000 −2.058 −4.003 A, A2O, ANbO3 Table 3.3: Chemical potentials in eV for the points 1 to 8 in the phase stability diagram in Fig. 3.1. ([Korb 10] Tab. III)

the LDA as well. Our results for the lattice constants deviate from the experimental data by less than 3% except for metallic sodium and potassium (4% and 5% deviation, cf. Ref. [Fran 95]). Some of the formation energies of the metals and the oxides, like bcc Na, show larger deviations from experiment than the results from the literature, others are closer to experiment, like fcc copper or the oxygen molecule. Altogether, all these deviations from experiment are within the typical range for LDA results. The computational setup was therefore kept for the following investigations of point defects.

Defect formation energies of vacancies and substitutional copper atoms in KNN [Korb 10], Section IV.B)

In order to calculate the defect formation energies of vacancies and substitutional Cu atoms in KNN the chemical potentials for the points 1 to 8 in Fig. 3.1 were determined using Eq. (3.1). The chemical conditions at each point and the values of the chemical potentials are compiled in Tab. 3.3. The points 1 to 3 correspond to an oxygen-rich atmosphere, the points 4 to 8 to an oxygen-poor one. Since the choice of the chemical potential of Cu does not affect the relative stability of the substitutional defect sites, the Cu reservoir was always assumed to be metallic (∆µCu = 0). For the points 1 and 4 in Fig. 3.1, the defect formation energies of substitutional Cu atoms as a function of the Fermi energy are shown in Fig. 3.3. The Fermi energy is allowed to vary within the range of the experimental band gap (indicated by vertical lines) of 3.3 eV. The upper region of the VBM and the lower part of the CBM are included in the graphs, so that charge transition levels [Wall 04] near the band edges are visible. These additional energy regions were taken into account in case the positions of the charge transition levels are not yet completely converged with respect to the size of the supercell. The top of the band gap obtained with the LDA is also marked by a vertical line (CBM (LDA) in Figs. 3.3 and 3.5), but in the following the experimental band gap or CBM is used (CBM (Exp.) in Figs. 3.3 and 3.5). In an oxygen-rich environment (Fig. 3.3, top), Cu substitutionals have a lower formation energy on the Nb sites compared to the alkali sites for Fermi energies in the entire range of the band gap. This changes in a reducing atmosphere (Fig. 3.3, bottom). Here the substitution on an alkali site has a lower formation energy for Fermi energies in the largest part of the band gap, the Na site being 0.4 eV lower in energy than the K site for equal chemical potentials of K and Na (Eq. (3.5)). Using the uncorrected band gap obtained with the LDA substantially changes the defect formation energies (by up to about 7 eV for a Fermi energy near the CBM), but unless the Fermi energy is close to the CBM, the difference is only quantitative (the most stable substitution site stays the same). For most Fermi energies in the band gap, the most stable charge states for substitutional Cu atoms are q = 0 for Cu on an alkali site and q = −2, q = −3, or q = −4 for Cu on a Nb site depending on the Fermi level. These defect charges correspond to Cu+ on a K+ or Na+ site and to Cu3+, Cu2+, or Cu+ on a Nb5+ × 00 000 0000 site (in Kroger-Vink¨ notation (cf. List of Symbols ): CuA and CuNb, CuNb, or CuNb, respectively). Whether an alkali site or a Nb site is the energetically preferred lattice site for substitutional Cu atoms hence depends on the chemical potentials and the Fermi level. In Figure 3.4 the defect formation energies of Cu substitutionals are depicted as a function of the chemical conditions. In the oxygen-rich region (points 1 to 3), Cu substitutes for Nb for Fermi energies in the range of the entire band gap (except for point 3, where for a Fermi level at the VBM the two sites are energetically degenerated), in a reducing atmosphere 3.3 Cu substitutionals and vacancies in KNN 50/50 17

VBM BGC CBM

15 O2 −rich (point 1)

5 Cu +2 +1 K 0 (eV) CuNa 1 f 0 0 1 3 E 2 CuNb 4 −5

−10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 EF (eV) VBM BGC CBM

15 O2 −poor (point 4)

0 1 2 10 CuNb 3 4 5 Cu +2 +1 K 0 (eV) CuNa 1 f 0 E

−5

−10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 EF (eV)

Figure 3.3: Defect formation energies of substitutional Cu atoms on K, Na and Nb sites for oxygen-rich conditions (point 1, top) and oxygen-poor ones (point 4, bottom). Circles indicate the charge transition levels. Solid lines and ﬁlled circles: fcc 80-atoms supercells, dashed lines and open circles: sc 40-atoms supercells. ([Korb 10] Fig. 3) 18 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

Figure 3.4: Defect formation energies of Cu dopants on K, Na, and Nb sites in their most stable charge states for the points 1 to 8, for an electronic chemical potential at the VBM (bottom), the center of the band gap (BGC, center) and the CBM (top). Crosses and dotted lines: without any band-gap correction; circles and solid lines: after shifting the conduction-band states upwards by the difference between the experimental and the band gap obtained with the LDA. ([Korb 10] Fig. 4), [Hoff 12] Fig. 9 3.3 Cu substitutionals and vacancies in KNN 50/50 19

VBM BGC CBM

15 O2 −rich (point 1)

10 4 V 5 Nb +1 5

(eV) V +1 0 K f 0 VO 1 E +2 V +3 Na 2 −5 6 −10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 EF (eV) VBM BGC CBM

15 VNb 5 10

+1 5 V (eV) 0 V K f +1 2 0 Na 1 E V +2 +3 O −5 6

−10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 EF (eV)

Figure 3.5: The same as in Fig. 3.3 for K, Na, Nb, and O vacancies. ([Korb 10] Fig. 5) 20 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

CuK, q = 0 CuNa, q = 0 CuNb, q = −3 40 at. 80 at. 40 at. 80 at. 40 at. 80 at. K 0.0 0.0 0.0 −0.1 −2.1 −1.3 Na 0.0 0.0 0.0 0.0 −11.6 −9.4 Nb −0.3 −0.3 −0.1 −0.1 0.0 −1.5 O −0.6 −0.5 −0.1 −0.1 1.7 1.5 Table 3.4: Changes in distance to the defect sites as compared to the ideal structure of neighbor atoms surrounding the substitutional Cu atom in the sc 40-atoms and fcc 80-atoms 2 × 2 × 2 supercells of K0.5Na0.5NbO3 (in percent of the cubic lattice constant a). Positive values indicate displacements away from the defect. ([Korb 10] Tab. IV)

VK, q = −1 VNa, q = −1 VNb, q = −5 VO, q = +2 K 0.0 0.0 −3.1 1.6 Na 0.8 0.0 −18.6 2.8 Nb −0.9 −0.6 −1.3 4.2 O 0.8 1.5 3.1 −4.0 Table 3.5: The same as in Tab. 3.4, for K, Na, Nb, and O vacancies. ([Korb 10] Tab. V)

(points 4 to 8), the substitution on an alkali site is energetically more favorable, if the Fermi level lies in the lower half of the band gap or slightly above the center of the band gap. In Fig. 3.5, the defect formation energies of vacancies are depicted. In an oxygen-rich atmosphere (Fig. 3.5, top), oxygen and alkali vacancies prevail for a Fermi energy close to the VBM, whereas Nb vacancies have a lower formation energy for a Fermi energy close to the CBM. In a reducing atmosphere (point 4), oxygen vacancies have the lowest formation energy over the entire band gap (Fig. 3.5, bottom). − 2+ 5− The vacancies are most stable in their nominal charge states VA , VO , and VNb (in Kroger-Vink¨ notation; 0 •• 00000 VA, VO , and VNb , respectively). For equal chemical potentials of K and Na, Na vacancies have a 0.5 eV lower formation energy than K vacancies. In Fig. 3.6 the defect formation energies of K, Na, Nb and O vacancies in their most stable charge states are depicted as a function of the processing conditions for a Fermi energy at the VBM (top), the center of the band gap (BGC, center) and at the CBM (bottom), respectively (as in Fig. 3.4, but for vacancies). In agreement with results in Ref. [Shig 05], oxygen vacancies have the lowest formation energy in a reducing atmosphere (points 4 to 8), for a Fermi energy in the lower half of the band gap. In an oxygen-rich environment (e.g. in air at ambient pressure and temperature) and a Fermi level in the upper half of the band gap or slightly below the band-gap center, cation vacancies, especially VNb, have a lower formation energy than oxygen vacancies. If the uncorrected LDA band gap is used, the formation energy of VNb is increased by about 8 eV for a Fermi energy near the CBM, so that the alkali vacancies have a lower formation energy than VNb under oxidizing conditions. However, if the Fermi energy lies close to the band-gap center or lower, the effect of the band-gap correction is mainly quantitative. The defect formation energies of Cu substitutionals on alkali sites of of alkali and oxygen vacancies obtained with 40-atoms and 80-atoms supercells are close, apparently they are nearly converged with respect to the size of the supercell. (at most about 0.24 eV energy difference between the small and the large supercells), and thus a 40-atoms supercell is sufﬁcient to model these defects. When the Nb site is involved, as for substitutional Cu atoms on Nb sites and for Nb vacancies, the results obtained with the 40- atoms supercell are not converged (up to about 1.35 eV energy difference between 40-atoms and 80-atoms supercell).

Atomic displacements and electronic densities of states in KNN with Cu substitutionals or vacancies [Korb 10], Section IV.C) In Table 3.4 the displacements of the atoms of each species nearest to the Cu substitutional are listed in percent of the equilibrium lattice constant of KNN. For each defect conﬁguration the most stable charge state close to the band-gap center was chosen. For Cu on an alkali site, the nearest O and Nb atoms, and 3.3 Cu substitutionals and vacancies in KNN 50/50 21

Figure 3.6: The same as in Fig. 3.4 for K, Na, Nb, and O vacancies. ([Korb 10] Fig. 6) 22 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

CuK/Na CuNb q +1 0 −1 −2 −1 −2 −3 −4 ∆n↑↓ 1 0 0 0 3 2 1 0

Table 3.6: Differences in numbers of spin-up and spin-down electrons ∆n↑↓ of Cu substitutionals on alkali and Nb sites for different charge states q. ([Korb 10] Tab. VI)

CuK/Na CuNb q +1 0 −1 −2 −1 −2 −3 −4 nh 1 0 0 0 1 0 0 0 ne 0 0 1 2 0 0 0 0

Table 3.7: Number of holes in the valence band (nh) and of electrons in the conduction band (ne) for substitutional Cu defects in different charge states q. ([Korb 10] Tab. VII)

for CuNa also the nearest K atoms, move towards the defect. This can be explained by Cu having a smaller ionic radius than both alkali elements. On the K site this effect is stronger, corresponding to the larger ionic + + + 2+ radius of K as compared to Na. The ionic radii (K : 1.78 A˚, Na : 1.53 A˚, CuK/Na : 1.35 A˚, CuNb : + 0.87 A˚) were taken from Ref. [Shan 76], the one for twelve-fold coordinated CuK/Na was extrapolated from those for lower coordination numbers. For Cu on a Nb site, the nearest oxygen ions move away from the defect. This can be explained by the smaller electrostatic Coulomb attraction by the Cu2+ ion as compared to the Nb5+ ion. The nearest cations move towards the negatively charged defect. For Cu on an alkali site, the atomic displacements obtained in a 40-atoms supercell are already close to the results from an 80-atoms supercell, but for Cu on a Nb site, the displacements obtained with the two supercell sizes differ more strongly. Especially the displacements of the Nb atoms cannot be obtained in a 40-atoms supercell, because if a Cu atom occupies a Nb site, these are zero for symmetry reasons. Table 3.5 shows the displacements of the atoms of each species nearest to a K, Na, Nb, or O vacancy in the 80-atoms 2 × 2 × 2 fcc supercell in the charge state with the largest stability range in the band − 5+ gap obatined with the LDA. An alkali vacancy, VK/Na, causes the nearest Nb ions to move towards the vacancy, whereas the nearest O2− ion moves away from it, as would be expected due to Coulomb attraction + − and repulsion, respectively. In the case of a K vacancy, the nearest Na ions move away from the VKa defect as well, possibly caused by the movement of the oxygen ions and/or the niobium ions. 5− A Nb vacancy VNb attracts the nearest cations and repels the oxygen ions, whereas an oxygen vacancy, 2+ VO , repels the nearest cations and attracts the nearest oxygen ions, as would be expected because of Coulomb interaction. In the Figures 3.7 to 3.9 the electronic densities of states (DOS) for perfect KNN, for KNN with a Cu dopant on a K or Nb site, and for KNN with K, Na, Nb or O vacancies are depicted. The VBM of the perfect and the defective crystals are set to zero. The shaded areas beneath the curves mark occupied energy levels. The energy for which the DOS of the pure material changes from zero to a ﬁnite number was taken as the VBM, and the CBM was set to EVBM + EG, with EG being the experimental band gap. The defective DOS was aligned with the perfect one in such a way that perfect and defective valence band and perfect and defective conduction band roughly lie at the same energy. The differences in number of spin-up and spin-down electrons ∆n↑↓ for each defect and charge state are listed in Tab. 3.6. For Cu on the two alkali sites (Fig. 3.7), almost identical densities of states are obtained, so only the one for Cu on a K site is shown. In the charge state q = +1, stable for a Fermi energy close to the VBM, the Cu substitutional generates a hole in the VBM (cf. Tab. 3.7). There are no gap levels. The Cu atom is spin-polarized (cf. Tab. 3.6). In the charge state q = 0, which is stable for Fermi energies over the most part of the band gap, Cu on an alkali site causes gap states approximately 0.8 eV above the VBM, which consist of Cu eg and t2g states. These gap states are fully occupied by electrons, and the Cu impurity is unpolarized. If more electrons are added, they occupy the conduction band without modifying the defect levels. 3.3 Cu substitutionals and vacancies in KNN 50/50 23

VBM CBM 2000

Cu K Cu eg 1800

Cu t2g q =−2 total 1600

Cu eg 1400

Cu t2g q =−1 total 1200 units) .

Cu eg 1000

Cu t2g DOS (arb q = +0 total 800

Cu eg ×4 600

Cu t2g ×4 q = +1 total 400

pure −1200 0 1 2 3 4 E (eV)

Figure 3.7: Total and partial (Cu-d) electronic densities of states (DOS) for pure KNN and for KNN with substitutional Cu atoms on K sites in their stable charge states. The VBM is set to zero, the VBM and the CBM are indicated by vertical lines. For the two curves q = +1 the partial DOS are scaled by a factor 4 for better visibility. ([Korb 10] Fig. 7) 24 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

VBM CBM 2000

Cu Nb Cu eg ×4 1800

Cu t2g ×4 q =−4 total 1600

Cu eg ×4 1400

Cu t2g ×4 q =−3 total 1200 units) .

Cu eg ×4 1000

Cu t2g ×4 DOS (arb q =−2 total 800

Cu eg ×4 600

Cu t2g ×4 q =−1 total 400

pure −1200 0 1 2 3 4 E (eV)

Figure 3.8: The same as in Fig. 3.7 for Cu atoms on Nb sites. The Cu-d DOS is scaled by a factor of 4 for better visibility. ([Korb 10] Fig. 8) 3.3 Cu substitutionals and vacancies in KNN 50/50 25

VBM CBM 1800

VNb q =−6 1600 q =−5

q =−4 1400

VO q =−1 1200 q =0

units) q =1 . 1000 q =2

q =3 DOS (arb 800

VK q =−2 600 q =−1

q =0 400

pure −1200 0 1 2 3 4 E (eV)

Figure 3.9: The same as in Fig. 3.7 for K, Nb, and O vacancies. ([Korb 10] Fig. 9) 26 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

VK/Na VNb VO q 0 −1 −2 −4 −5 −6 +3 +2 +1 0 −1 nh 10010010000 ne 00100000123 Table 3.8: The same as in Tab. 3.7 for vacancies. ([Korb 10] Tab. VIII)

0 a ∆Hf KNbO3 (cub.) 3.943 −14.296 NaNbO3 (cub.) 3.904 −13.852 K0.5Na0.5NbO3 (cub., SC) 3.924 −14.067 K0.5Na0.5NbO3 (cub., VCA) 3.924 −13.765 K (bcc) 4.984 −0.963 Na (bcc) 4.049 −1.309 A (bcc) 4.376 −1.045 K2O 6.176 −3.792 Na2O 5.407 −4.214 A2O 5.912 −3.253 Table 3.9: Lattice constants a in A˚ and formation energies in eV/formula unit for KNO, NNO and KNN 50/50, and the alkali metals (A denotes a “mixed” alkali atom, K0.5Na0.5NbO3 in the VCA) and oxides obtained with the supercell (SC) approach and the virtual-crystal approximation (VCA).

For Cu on a Nb site (Fig. 3.8), for all charge states that are stable for a Fermi energy in the band gap there are gap states. The Cu atom is spin-polarized for all Fermi levels in the lower half of the band gap. In the charge state q = −1, which is the most stable charge state for a Fermi energy in a range of 0.3 eV above the VBM, the Cu substitutional generates a hole in the valence band. The Cu eg states are split by spin polarization. Two eg levels lie inside the valence band, the other two are unoccupied and lie about 0.6 eV above the VBM. In the charge state q = −2, the valence band is completely occupied by electrons, the gap states are unoccupied. For q = −3, which is stable near the center of the band gap, all the Cu eg states have shifted into the band gap. There are two levels about 0.7 eV above the VBM which are fully occupied, the other two lie about 1.3 eV above the VBM and are half-occupied. In the charge state q = −4, which is stable in the upper part of the band gap, one of the defect levels has moved upwards and now lies close to the band gap center. All defect levels are now fully occupied by electrons, and the Cu atom is unpolarized. In the charge states that are stable over the most part of the band gap, the K, Na (again omitted because very similar to K), Nb, and O vacancies generate neither holes in the valence band nor free electrons in the conduction band (cf. Fig. 3.9 and Tab. 3.8). Only for a Fermi energy close to the VBM each vacancy type generates a hole in the valence band. For charge states that are stable slightly above the CBM, all vacancy types lead to electrons in the conduction band.

3.4 The Virtual Crystal Approximation for KNN 50/50

In the following, in addition to using the supercell approach the VCA for KNN was applied to describe the mixed A site occupation in Cu-doped KNN. The VCA for KNN was validated for a K/Na ratio of 50/50 by comparing lattice constants, formation energies and electronic densities of states obtained with the VCA and the supercell approach. The numerical values for lattice constants and formation energies are compiled in Tab. 3.9. For KNN and the alkali metals, the VCA gives lattice parameters and formation energies close to the averaged values of the two end members KNO and NNO, as does the supercell approach. For the oxide A2O, the results obtained with the VCA and the supercell approach differ more strongly. Figure 3.10 shows the electronic densities of states of the two end members and the solid solution obtained with the VCA and the supercell approach. Over a large energy range, the DOS of the end members 3.5 Effect of the alkali stoichiometry on the substitution site of Cu 27

SC VCA NNO

KNN DOS [arb. u.]

KNO

−25 −20 −15 −10 −5 0 5 10 E [eV] Figure 3.10: Electronic densities of states for KNO, NNO, and KNN from a supercell (SC) and a virtual crystal calculation (VCA). The DOS obtained with the VCA and the supercell approach are very similar close to the valence and conduction band edges, in more distant energy regions (about 5 eV from the band edges) they differ more strongly.

are almost identical, and so are the DOS of the mixture obtained with the VCA and the supercell approach. Differences occur in the higher energy area of the conduction band, and at the positions of the Na 2p peak in NNO at about -22 eV and the K 3p peak at -11 eV in KNO. The KNN DOS obtained in a supercell looks like a superposition of the NNO and the KNO DOS and exhibits both peaks with half intensity. One would expect that the VCA DOS exhibits again one peak (that of the virtual alkali atom), at an intermediate position between the Na and the K peak. In fact, this is the case, though the peak lies not at the averaged position of the two peaks, but much closer to the K peak (at about -12 eV).

3.5 Effect of the alkali stoichiometry on the substitution site of Cu

In Fig. 3.11 phase stability diagrams of NNO, KNN, and KNO obtained with sc 2 × 2 × 2 (40 atoms) supercells are depicted. Since the LDA underestimates the band gaps of NNO, KNN, and NNO in a similar way, the defect formation energies obtained without any band-gap correction were used to compare the different stoichiometries. For KNN 50/50, both the rocksalt-like ordering of K and Na on the alkali sublattice and the VCA were applied. The perovskite phase is stable for chemical potentials in the grey areas, the shade of grey indicates if Nb (dark grey) or alkali sites (light grey) are the most stable sites for Cu substitutionals. In the medium grey areas in-between, the relative stabilities depend on the chemical potential for electrons (the Fermi energy). Here the curves of the defect formation energies on alkali and Nb sites as function of the Fermi energy cross inside the LDA band gap (e.g. in Fig. 3.3, bottom), so that the position of the Fermi level determines the most stable lattice site for Cu substitutionals. For all alkali stoichiometries, the CuNb defect is more stable than CuA under oxygen-rich conditions (near the points marked with “2” and “3”), in KNO CuA is more stable than in NNO (the light grey area is larger). The stability regions of CuNb and CuA in KNN lie between those of the end members NNO and KNO both in the supercell and the VCA calculation. When going from NNO to KNO the CuA substitution becomes more favorable (the region of chemical potentials for which Cu substitutes on alkali sites becomes larger), but the effect of the alkali species is rather small. 28 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

µ [eV] Na 7 −12 −10 −8 −6 −4 −2 0 0 0 0 0 0 .2 bar .2 .2 bar .2 µ µ 5 bar bar 8 , , 4 300 K , 1300 K , 300 K 1300 −2 CuNa K

−4

µO −6 [eV]

CuNb Nb µ

−8

−10 3 1

2 −12

7 µA [eV] −12 −10 −8 −6 −4 −2 0 0 0 0 0 0 .2 bar .2 .2 bar .2 µ µ bar bar 5 , , CuA 4 300 K , 1300 K , 300 K 1300 −2 K

8 −4

µO −6 CuNb [eV] Nb µ

−8

1 −10 3

2 −12

µ [eV] K 6 7 −14 −12 −10 −8 −6 −4 −2 0 0 0 0 0 0 .2 bar .2 .2 bar .2 µ µ bar bar 5 , , , , 4 300 K 300 K 1300 K 1300 −2 CuK K 8 −4

µO 11 −6 [eV] CuNb Nb

10 −8 µ 1 −10 3 9

2 −12

−14

Figure 3.11: Most stable lattice sites for Cu substitutionals in NNO (top, compare [Erun 11], Fig. 3 and [Korb 11b], Fig. 3), KNN 50/50 [center, rocksalt-like ordering of K and Na on the alkali sublattice (left) and VCA (right)] and KNO (bottom, compare [Erun 10], Fig. 4) and [Korb 11b], Fig. 3. 3.6 Fe substitutionals in KNN 50/50 29

3.6 Fe substitutionals in KNN 50/50

The defect formation energies of isolated Fe substitutionals in KNN 50/50 are depicted in Fig. 3.12 as a function of the Fermi level under oxygen-rich and oxygen-poor conditions (points 1 and 4 in the phase stability diagram in Fig. 3.1 and in Tab. 3.3). The defect formation energies were calculated for an fcc 2 × 2 × 2 (80 atoms) supercell, assuming a rocksalt-like ordering of Na and K on the alkali sublattice. The LDA band gap of about 1.7 eV was used. +1 The charge states which are most stable over the largest part of the LDA band gap are (FeK/Na) , −1 −2 • 0 00 (FeNb) , and (FeNb) , in Kroger-Vink¨ notation: FeK/Na, FeNb, and FeNb. Under oxygen-rich conditions, the Nb substitution has a lower formation energy as compared to the alkali substitution for Fermi levels in the entire LDA band gap, in a reducing atmosphere the alkali substitution has a lower formation energy, as for Cu. Figure 3.13 shows the preferred lattice site for Fe dopants in KNN for various chemical conditions obtained with an sc 2 × 2 × 2 (40 atoms) supercell (spin polarization included). In the area marked dark- gray, the Nb substitution has a lower formation energy for Fermi energies in the entire LDA band gap. In the medium-gray area, the curves for the defect formation energies for Fe on A and B sites as function of the Fermi level cross in the band gap, so that the most stable lattice site depends on the position of the Fermi level. In the light-gray area, the alkali substitution has a lower formation energy for all Fermi levels in the band gap. For conditions corresponding to air at atmospheric pressure and temperatures up to about 1300 K the Nb substitution has a lower formation energy. Compared to Cu, the range of chemical potentials for which the alkali site is stable is smaller, so that Fe is less likely to occupy an alkali site in KNN than Cu. In order to analyze the spin and charge density distribution around the impurity, charge and spin density were projected on atomic spheres with a radius of 1/4 of the perovskite lattice constant around the nuclei. Since these spheres do not cover the whole volume of the unit cell, the sum of the averaged numbers of electrons conﬁned to the spheres does not exactly equal the number of valence electrons. This radius is the largest possible one for which adjacent spheres do not overlap in the perfect cubic structure. As a consequence of the atomic diplacements caused by the Fe dopant, a small overlap can occur in the defective supercell. The results were obtained with an sc 2×2×2 supercell of KNN 50/50 with a rocksalt- like ordering of the alkali sublattices at the theoretical lattice constant of undoped KNN (a = 7.416 Bohr). Table 3.10 lists the numbers of valence electrons ne and the magnetic spin moments ms contained in the atomic spheres of the Fe dopant and its neighbor atoms. For Fe on both alkali and Nb sites, the magnetic spin moment is conﬁned to the Fe atom. For Fe on a K site and transitions between the charge states q = 0 → q = +1 and q = +1 → q = +2 the largest difference in the number of electrons is found at the Fe atom, indicating that the Fe atom changes its valence state, other than for the transition q = +2 → q = +3, for which the charge difference is distributed over all atoms. This suggests that the Fe atom is oxidized from Fe+ to Fe3+ but not further. Instead of a reduction to Fe4+, an electron is removed from the valence band. 30 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

0 +1 +2 +3 0 −1 −2 −3 (FeK) (FeK) (FeK) (FeK) (FeNb) (FeNb) (FeNb) (FeNb) ms 3.472 4.000 4.928 5.381 −2.886 2.016 1.003 0.445 ms (K) 0.000 0.000 0.000 0.000 −0.001 0.001 0.000 0.000 ms (Na) 0.000 0.000 0.000 0.000 −0.000 0.000 0.000 0.000 ms (Nb) −0.017 −0.001 −0.004 −0.007 0.007 −0.011 −0.014 −0.012 ms (O1) 0.013 0.021 0.068 0.095 −0.079 0.037 0.011 0.004 ms (Fe) 3.190 3.382 3.638 3.680 −2.047 1.637 0.933 0.465 ne (K) 6.535 6.530 6.524 6.519 6.537 6.542 6.547 6.552 ne (Na) 7.834 7.832 7.831 7.829 7.835 7.839 7.844 7.849 ne (Nb) 8.715 8.691 8.680 8.669 8.699 8.715 8.737 8.762 ne (O) 5.924 5.914 5.893 5.870 5.787 5.817 5.849 5.870 ne (Fe) 5.633 5.552 5.439 5.415 5.595 5.635 5.702 5.753 ∆n↑↓ 3 4 5 6 3 2 1 0

Table 3.10: Total and atom resolved magnetic spin moments ms (in units of Bohr magnetons µB) of Fe substitutionals on K and Nb sites in KNN, number of electrons ne = n↑ + n↓ contained in spheres around the atoms, and difference in the numbers of electrons in spin-up and spin-down states ∆n↑↓. 3.6 Fe substitutionals in KNN 50/50 31

VBM CBM

15 O2 −rich (point 1)

0 5 Fe +1 1 +2 K

(eV) +3 FeNa f

E 0 FeNb +1 0 1 2 −5 3 4

−10 −0.5 0.0 0.5 1.0 1.5 2.0 EF (eV)

VBM CBM

15 O2 −poor (point 4)

FeNb 3 5 +1 0 1 Fe +1 2 1 +2 K

(eV) 0 +3 Fe 4

f Na

E 0

−5

−10 −0.5 0.0 0.5 1.0 1.5 2.0 EF (eV)

Figure 3.12: The same as Fig. 3.3 but for Fe substitutionals, and without any band gap correction. The VBM and the CBM obtained with the LDA are marked by vertical lines. 32 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

Figure 3.13: The same as Fig. 3.11 for Fe in KNN 50/50 (rocksalt-like ordering assumed for the alkali sublattices). For chemical potentials corresponding to air at atmospheric pressure and temperatures up to about 1300 K, Fe substitutionals are most stable on the Nb site. 3.7 Discussion 33

3.7 Discussion ([Korb 10], Section V)

Pinning of the Fermi level by defects In Ref. [Zhan 02] Zhang relates the Fermi energies, at which p- type and n-type defects have zero formation energy, to the minimum and maximum possible Fermi energy (pinning energies). E.g., negative defect formation energies for VO in a reducing atmosphere and for Fermi energies in the lower half of the band gap indicate that O2 gazes out and the Fermi level increases until it reaches the range where the defect formation energies are positive. In this work negative formation energies were obtained for p-type vacancies near the VBM and for n-type ones near the CBM, so that the Fermi energy is pinned to a region around the center of the band gap (BGC). In the following discussion of defect formation energies, it is therefore assumed that the Fermi level lies close to the BGC. However, the boundaries of this stability region depend on the atomic chemical potentials, e.g. via the oxygen partial pressure.

Substitution site of Cu dopants Mainly the temperature and the oxygen partial pressure determine the substitution site of Cu dopants in KNN. In air at ambient pressure and temperature Cu is energetically driven to substitute on Nb sites, but for temperatures around 1300 K and reduced O2 partial pressures (µbar or lower), substituting on the alkali sites is more favorable. Substituting Nb is favorable for typical processing conditions (in air at temperatures up to about 1300 K), but the alkali substitution cannot be ruled out completely. K2O is volatile and gazes out to some extent, which can lead to K vacancies. The vacant K sites might then be occupied by Cu, which could explain why in Ref. [Li 07] evidence was found for Cu substituting on A sites for small Cu concentrations. Experimental studies using EPR spectroscopy [Eich 09] found evidence for Cu substituting for Nb. Since in their most stable charge states, the CuNb defects have unpaired electrons for most Fermi energies in the band gap, whereas the CuK/Na defects do not, the latter may not be detected by EPR spectroscopy. So far, only the cubic phase was considered. In the ferroelectric phases, the ferroelectric instability can have an effect on the stability of Cu dopants on alkali and Nb sites, which can differ on the two sites. For sufficiently low Cu concentrations the ferroelectric instability is stronger in CuA-doped KNbO3, but weaker in CuNb-doped KNbO3 (cf. chapter 4), which lowers the energy of CuK/Na slightly more than that of CuNb. The negative defect formation energies of Cu substitutionals for some conditions are caused by our choice of metallic Cu as reservoir (the absolute number of µCu is not needed here because only the relative stability of CuK/Na as compared to CuNb is of interest) and do not indicate instability of the host perovskite with respect to exchange of e.g. Nb for Cu. × 000 0000 • 00 The Cu impurities mainly occur in the charge states CuA and CuNb/CuNb, although CuA and CuNb are also stable over a certain energy range in the lower part of the band gap. The charge transition levels between q = −2 and q = −3 and between q = −3 and q = −4 of the CuNb defect are located slightly below the center of the LDA band gap and slightly above the LDA CBM (which is located about 1.7 eV above the VBM), respectively, analogous to the transitions between q = −1 and q = −2 and between q = −2 and q = −3 found for CuTi in PTO [Erha 07b]. In Ref. [Erha 07b], Erhart et al. report the differences in spin-up and spin-down electrons ∆n↑↓ for 0 00 CuTi (∆n↑↓ = 2) and CuTi (∆n↑↓ = 1) in PTO. For the corresponding charge states of Cu on Nb sites in 00 000 00 KNN (CuNb and CuNb), the same differences in spin-up and spin-down electrons, ∆n↑↓ = 2 for CuNb and 000 ∆n↑↓ = 1 for CuNb, were found here. However, for determining defect formation energies, the effect of the spin-polarization is rather small. An unpolarized calculation is already sufficient for obtaining the relative stability of the possible lattice sites and the approximate positions of the charge transition levels for Cu substitutionals in KNN. Cu on an alkali site is an uncharged impurity for most Fermi energies inside the band gap, and the atomic displacements are rather small in this case (0.6% of a at most). Therefore neither electrostatic nor elastic interactions are large or long-ranged, so that already rather small supercells containing 40 atoms apparently are sufficient to model these defects. The same holds for alkali and oxygen vacancies, even though the latter defects are charged. If Cu occupies a Nb site, the atomic displacements (up to 11.6% of a) and the defect charges are large, and a 40-atoms supercell does not yet yield well converged results. This is only partly caused by the symmetry of the sc 2×2×2 supercell, which prevents displacements of the Nb ions towards or away from 34 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN

the impurity in the case of CuNb. A band structure calculation (not shown) for the smaller supercell (40 atoms) shows that the Cu eg states still show a considerable dispersion, which is reduced in the 80-atoms supercell, in accordance with results from Ref. [Post 98] for Fe impurities in KNbO3. The displacements of the atoms nearest to a Cu substitutional or a vacancy can mostly be explained in terms of Coulomb interaction between ions and charged defects, i.e. negative defects attract the nearest cations and repel the nearest anions, and vice versa for positive defects. The atomic displacements near a vacancy are in qualitative and for the cations also in quantitative agreement with results reported in Ref. [Shig 05] for vacancies in KNbO3.

Vacancies In a reducing atmosphere, oxygen vacancies are the prevailing vacancy type, also alkali vacancies form easily. If the oxygen partial pressure is of the order of the standard atmospheric pressure, cation vacancies, especially VNb, become energetically more favourable.

Effect of the band-gap correction Assuming that the chemical potential of the electrons lies close to the band-gap center, the band-gap correction changes the defect formation energies by up to 2−3 eV in the case of the Cu substitutionals and by up to 4−5 eV in the case of vacancies. For an electronic chemical potential near the CBM the energy differences are even higher. These differences in defect formation energies lead to differences by orders of magnitude in the defect concentrations. In this work, for electronic chemical potentials near the band-gap center, the order of the formation energies of the different defects is hardly changed by the band-gap correction, apart from the cases where two defects are almost degenerated. In [Erha 08] the chemical potential of BaTiO3 with point defects was calculated self-consistently. There it turned out that in the presence of acceptor dopants (which in the case of Cu-doped KNN could be CuNb substitutionals) the electronic chemical potential does not rise much above the band-gap center, and it is reasonable to assume that the same is the case here. Nevertheless it is apparent that the defect formation energies of charged point defects are strongly affected by the band-gap correction, and in order to obtain meaningful (relative) defect formation energies it is important to know the precise position of the electronic (defect) states in the band gap.

Inﬂuence of Cu dopants and vacancies on the electronic properties Over the most part of the band 0 00000 •• gap, the vacancies are most stable in their formal charge states VK,Na, VNb , and VO , and thus do not generate free charge carriers. In the defect charge states of vacancies and Cu substitutionals that are stable over the most part of the band gap, the valence band is completely occupied, and the substitutional Cu atoms introduce deep defect levels. Thus, in this case it is unlikely that free charge carriers are generated and the electronic conductivity of the material should hardly be enhanced by either the Cu substitutionals or the vacancies. Similar to Ref. [Shen 08], a DFT-GGA study of CuK-doped KNbO3, we found Cu d-like states centered in the band gap for Cu on K/Na sites in KNN, but we differ with respect to the interpretation. Although these defect states lower the effective band gap, they cannot increase the carrier density much because they lie too deep in the band gap. Only for a Fermi energy in a range of about 0.3 eV above the VBM, charge transition levels were found for all types of defects considered here, Cu substitutionals and vacancies, and the defects generate holes in the valence band. Since there are empty or partly ﬁlled deep levels for Cu on Nb sites for a Fermi energy over the most part of the band gap, the CuNb defect likely acts as a trap for electrons. Trapping of charge carriers at impurities is assumed to play a role in electrical fatigue of ferroelectrics [Warr 97].

Inﬂuence of the alkali stoichiometry on the substitution site of Cu dopants The alkali species apparently has hardly any effect on the stability of the two sites in cubic KNN. This is not surprising, since K and Na have the same electronic conﬁguration, and their oxides have the same stoichiometry and similar formation energies.

Fe substitutionals in KNN Other than Cu, Fe in KNN is preferentially a B-site dopant. It may be possible to drive it to the A sites in a reducing atmosphere, but under typical conditions up to sintering temperatures, the B (Nb) site is the thermodynamically most stable one. Here the charge mismatch is smaller than on the A sites (assuming Fe3+), and the ionic radii of Fe (low-spin state: 0.69 A,˚ high-spin 3.7 Discussion 35

state: 0.79 A)˚ and Nb (0.78 A)˚ are closer than those of Cu (0.87 A)˚ and Nb [Shan 76]. For FeNb in KNN we obtained the same differences in ∆n↑↓ (numbers of electrons with positive and negative spin) as Erhart and Albe in Ref. [Erha 07b] for FeTi in PTO, which were also obtained using pseudopotentials (and DFT- LDA), whereas our results differ from those of Ref. [Post 98], which were obtained using LMTO-ASA together with the LDA. Experimentally a high-spin state has been found for Fe on the B site of another perovskite-type compound: in Sr2FeMoO6 (Ref. [Serr 07] and references therein), which is in accordance with [Post 98] but not with [Erha 07b] and this work. The possible origin of the discrepancy can be the different approach (LMTO-ASA vs. pseudopotential approach), or the larger unit-cell volume used in [Post 98], which may accommodate the larger ionic radius of Fe in the high-spin state more easily. But since a test calculation at a larger unit-cell volume yielded the same low-spin state of Fe (not shown), it appears more likely that the approach using pseudopotentials and the LDA is responsible. 36 3. Ab initio thermodynamics of isolated Cu and Fe substitutionals in KNN 4 Ferroelectricity in KNO with isolated Cu substitutionals 38 4. Ferroelectricity in KNO with isolated Cu substitutionals

In this chapter it is investigated how Cu substitutionals inﬂuence the ferroelectric properties of KNO, for example, if they create a morphotropic phase boundary.

4.1 Introduction ([Korb 11a], Section I)

A solid solution of two or more perovskite oxides with a morphotropic phase boundary (MPB), like PZT, can exhibit enhanced piezoelectric properties [Zhan 07]. The (K,Na,Li)NbO3 system is such a case. It has a MPB between a tetragonal phase and the orthorhombic room-temperature phase of (K,Na)NbO3 [Guo 04]. At temperatures below 435° C, KNbO3 is a ferroelectric perovskite, its ground state is rhombohedral with a ferroelectric polarization along h111i (see chapter 1, “Introduction”). LiNbO3 crystallizes in the so-called LiNbO3 structure [Abra 66], which is related to the perovskite structure [Mega 68] and has a ferroelectric instability (the ferroelectric state has a lower energy than the paraelectric one) as well. If it is forced into the perovskite structure in a solid solution with KNbO3 as the other component, it stabilizes the tetragonal perovskite phase [Guo 04, Bilc 06]. Because the structure of KNbO3 is not tetragonal for temperatures below 225 °C, the solid solution (Li,K)NbO3 has a MPB. If CuO is added to KNN as a sintering aid, it is expected that part of the Cu is incorporated into the perovskite matrix. In chapter 3 it was found that Cu can occupy both A or B sites in the perovskite ABO3 structure, depending on the processing conditions. For Cu-doped (K,Na)NbO3 and KNbO3 produced with conventional solid-state powder processing techniques, there is experimental evidence that Cu substitutes for Nb (on the B site) and forms defect complexes with oxygen vacancies for charge compensation [Lin 07, Erun 10]. Nevertheless, some isolated CuNb ions may be present in the material as well. There are reasons to expect that Cu+ and Li+ can substitute for each other: Cu+ and Li+ have almost the same ionic radii [Shan 76]. Cu probably substitutes for Li in LiNbO3 [Paul 95], and Li in turn substitutes for K or Na in (K,Na)NbO3 [Guo 04]. The structure of CuNbO3 differs both from the perovskite and from the LiNbO3 structure [Mari 84], but CuTaO3 is isostructural with LiNbO3 [Slei 70]. If the solid solution (K,Cu)NbO3 can be produced, it can be expected to have a MPB similar to that in (K,Li)NbO3. In this chapter, the effect of Cu on K and Nb sites on the ferroelectric instabilities in KNbO3 was studied, and KNbO3 doped with CuK was investigated for a MPB by comparing the energetical ordering of the tetragonal, orthorhombic, and rhombohedral perovskite structures as a function of the Cu content. Such an approach to ﬁnd MPB’s was applied, e.g., to solid solutions of AgNbO3 and PbTiO3, BaZrO3, and BaTiO3, respectively, in Ref. [Grin 04].

4.2 Method ([Korb 11a], Section II)

The energies of the different ferroelectric phases were determined using ab initio density-functional theory (DFT) in the local-density approximation (LDA) and atomistic simulation (molecular statics) with classical interatomic potentials. The ferroelectric instabilities of undoped KNbO3 were determined by minimizing the total energy of a unit cell with respect to the atomic positions and the cell parameters starting from structures with a cubic (Pm3m), tetragonal (P4mm), orthorhombic (Amm2), or rhombohedral symmetry, respectively. The rhombohedral structure optimization was initialized by displacing the ions in the unit cell along [111] with respect to each other. During the optimization of the rhombohedral structure no symmetry constraint was applied, but the structure maintained its (R3m) symmetry, which is the experimentally observed one [Hewa 73a]. The cell parameters were optimized as follows: In the cubic, paraelectric unit cell only the volume was allowed to change. In the tetragonal phase, the c/a ratio was optimized as well. The orthorhombic phase has a primitive monoclinic unit cell. Therefore three cell parameters (the cell volume, 4.2 Method 39 the monoclinic angle, and the c/a ratio) can be optimized. In this work, in order to reduce the computational effort, the monoclinic cell angle was fixed to 90° (its experimental value is 89.74°) [Hewa 73a], and only the cell volume and the c/a ratio (and the atomic positions in the unit cell) were optimized. For comparison, the same calculations were also performed with the cell parameters fixed to the cubic ones. In the DFT supercell calculations for Cu-doped KNbO3, the cell parameters were fixed to the cubic ones throughout, so that ferroelectric strain was neglected. Cu concentrations of 12.5 and 6.25 mol % were taken into account by replacing one K or Nb atom by Cu in a simple cubic (sc, 40 atoms) and a face-centered-cubic (fcc, 80 atoms) supercell, respectively. Volume effects due to Cu doping were considered by performing the calculations at two different volumes, the theoretical one of undoped KNbO3 and the volume that minimizes the total energy of the cubic, paraelectric Cu-doped supercell. In the case of CuK doping, the magnitudes of the ferroelectric instabilities are almost the same for the optimized volumes of pure and CuK-doped KNbO3 due to the small volume change caused by CuK (0.6% shrinkage at a doping level of 12.5 mol %). Since this volume change of the cubic supercell is rather small, in the following all calculations for CuK-doped KNbO3 were performed at the theoretical volume of undoped KNbO3. For CuNb doping, the volume effect is stronger (about 2% expansion in the 40-atom supercell) and was therefore included in the following calculations for CuNb-doped KNbO3. In the 80-atom supercell calculations spin polarization of the Cu ions was taken into account, whereas it was neglected in the 40-atom supercell calculations because test calculations (not shown) revealed that the magnitudes of the ferroelectric instabilities are almost the same with or without spin-polarization.

The energies of undoped and CuK-doped KNbO3 obtained with the DFT were used to adjust a classical interatomic shell-model potential (SMP) in order to determine the ferroelectric instabilities for lower CuK concentrations by atomistic simulation with larger supercells for which DFT calculations are unfeasibly expensive. A Buckingham potential from the literature was used as a starting point ([Sepl 05], see next section). The potential parameter set for the Cu-O interaction was obtained for CuK by adjusting it to the ferroelectric instabilities of Cu-doped KNbO3 obtained with DFT for 12.5 and 6.25 mol % Cu concentrations. The atomistic structure optimizations were performed using the general utility lattice program (GULP) [Gale 03]. In order to determine how the Cu substitutionals disturb the arrangement of dipole moments in the surrounding host crystal, the changes in ionic dipole moments of the individual unit cells were calculated in a supercell containing Cu substitutionals. While the absolute value of the electric dipole moment of a crystal is not accessible, the change in dipole moment with respect to a reference structure is well deﬁned and measurable [Rest 92]. The structure in which all atoms occupy the lattice sites of the cubic paraelectric (ideal) perovskite structure was taken as the reference structure. In order to obtain accurate values for changes in electric dipole moments, both ionic and electronic contributions must be taken into account, e.g. by employing Born effective charges or the Berry-phase approach [Berr 84]. The ﬁrst attempt to determine the ferroelectric polarization in KNbO3 using the Berry-phase approach was made by Resta et al. [Rest 93] for the tetragonal phase. Born effective charges of 0.82 (K), 9.13 (Nb), −6.58 (O1) and −1.68 (O2, O3) were obtained, where the O1-Nb bonds are those along the direction of the polarization, and the 2 O2-Nb and O3-Nb bonds are perpendicular to it. A polarization of 35 µC/cm was found (experiments: 2 2 30 µC/cm [Trie 56]; 37 µC/cm [Klee 84]). In KNbO3 and other perovskite oxides, the Born effective charges deviate strongly from the formal ionic ones. According to Posternak et al. [Post 94] the “giant values” of the Born effective charges in KNbO3 are caused by covalent bonding due to hybridization of O 2p and Nb 4d states. The Born effective charges have the same sign as the formal ones but a larger value, which means that the electronic polarization follows the ionic one. Applying Born effective charges obtained in Ref. [Wang 96] for the cubic phase of KNbO3 to undoped, ferroelectric KNbO3 enhances the polarization by an almost constant factor of 1.8−1.85 with respect to a calculation with formal charges. In Ref. [Wang 96] it is shown that the Born effective charges of KNbO3 depend markedly on the structure. Using the Born effective charge tensor for tetragonal KNbO3 from Ref. [Wang 96], the polarization in [001] direction in a cubic unit cell is reduced by 25% compared to the one obtained with Born effective charges for the cubic phase. In principle it is necessary to determine the Born effective charges for each (defect) structure separately, but qualitative effects can already be obtained with a simple approximation using formal ionic charges and neglecting the electronic contribution. This approximation was applied here and only changes in ionic dipole moments were calculated. The change in ionic dipole moment ∆P of an 40 4. Ferroelectricity in KNO with isolated Cu substitutionals

DFT-LDA Expt. Ref. [King 94] This work Ref. [Hewa 73a] K 0.0101 0.008 0.0112 Nb 0.000 0.000 0.0000 O1,2 0.019 0.021 0.0295 O3 0.020 0.021 0.0308 2 ∆E[111] (meV) −10.9 −12.5 −13.3 1 Obtained from Table VII in Ref. [King 94]. 2 Obtained from Eq. (20) and Table V in Ref. [King 94].

Table 4.1: z component (in units of the lattice constant) of the ferroelectric displacements u[111] along [111] and ferroelectric instability in the rhombohedral phase of KNbO3. ([Korb 11a], Tab. I)

individual unit cell is

∆P = e ∑wiZi ∆ri, (4.1) i where e is the absolute value of the electron charge, Zi is the nominal charge of ion i, wi is its weight (e.g., in the B-site centered unit cell in Fig. 1.1 the weight of the eight A atoms at the corners of the cubic perovskite unit cell is 1/8 because they are shared by eight unit cells) and ∆ri is its distance vector from the reference position, for which in each unit cell the position of the B atom was chosen. The charges used were K+, Nb5+,O2−, Cu+ on K sites, and Cu2+ on Nb sites. In the case of Cu, the charge states were used that according to Ref. [Korb 10] are most stable at the center of the bandgap.

4.3 Results ([Korb 11a], Section III)

Undoped KNbO3 ([Korb 11a], Section III.A)

In Table 4.1 atomic displacements and the ferroelectric instability of the rhombohedral phase of KNbO3 obtained in this work using DFT-LDA are compared to those obtained also theoretically in Ref. [King 94] and to experiment. Our DFT-LDA setup yields a smaller soft-mode amplitude (smaller ionic displacements) than experiment (by about 1/3 in the case of O), which was also found in Ref. [King 94], whereas the ferroelectric instability obtained with the LDA lies close to the experimental one.

Initially the SMP simulations were performed using the SMP of Ref. [Sepl 05]. This potential reproduces the experimental phase transition temperatures closely [Tint 97], but the 0 K total energy differences between ferroelectric structures are overestimated with respect to our DFT results (cf. Table 4.2). In order to correct for this overestimation of the ferroelectric instabilities, the potential parameters were modified as follows: First the Buckingham potential and the spring force constants were multiplied by a factor of 1.016, then the theoretical equilibrium lattice parameter was reduced by rescaling all distances with a factor of 1.0025. The SMP parameters thus obtained and used in this work are compiled in Table 4.3. In Ref. [Yu 95] phonon frequencies of KNbO3 were calculated using the LAPW method and a linear- response approach. Unstable TO phonon modes were found, indicating that ferroelectric instabilities exist. Table 4.4 contains the phonon frequencies at the Γ point from Ref. [Yu 95], as well as those obtained with the SMP obtained in this work and those measured for the cubic structure at 710 K in Ref. [Font 84]. The SMP obtained in this work gives phonon frequencies that lie about as close to the experimental ones as the first-principles results from Ref. [Yu 95]. The SMP obtained in this work reproduces the ferroelectric structure and the energy surface of pure KNbO3 as obtained with DFT and from measured heats of formation [Shir 54a] closely, in the calculations with optimzed unit cell parameters, whereas it underestimates the ferroelectric instabilities if the unit cell parameters are fixed to the cubic ones (cf. Fig. 4.1 and Table 4.2). An LDA study of BaTiO3, which is isostructural to KNbO3, by Fu and Cohen [Fu 00] revealed that no energy barriers have to be overcome in 4.3 Results 41

SMP DFT Expt.1 Ref. [Sepl 05] This work This work ∆E[001] (meV) −61.6 −9.8 −10.4 −8.2 ∆E[011] (meV) −73.3 −11.4 −11.6 −11.9 ∆E[111] (meV) −80.4 −12.9 −12.5 −13.3 c/a ([001]) 1.042 1.017 1.016 1.017 c/a ([011]) 1.029 1.010 1.010 1.017 α (°) 89.34 89.74 89.91 89.83 V (PE, A˚ 3) 63.57 63.86 61.72 64.04 B (GPa) 199 184 223 165 172 u[001] (K) 0.021 0.016 0.012 0.023 u[001] (Nb) 0.000 0.000 0.000 0.000 u[001] (O1,2) 0.061 0.040 0.034 0.042 u[001] (O3) 0.062 0.037 0.034 0.040 1 Refs. [Hewa 73b, Shir 54a, Cher 99]. Table 4.2: Energies of the ferroelectric phases relative to the paraelectric cubic phase (∆E), c/a ratio, rhombohedral cell angle α, cubic volume V, bulk modulus B, and atomic displacements u[001] of pure tetragonal KNbO3 with a polarization in the [001¯] direction obtained with the SMP from Ref. [Sepl 05], the SMP obtained in this work, DFT, and experiment. ([Korb 11a], Tab. II)

Atom pair A ρ C K−O 126870.4000 0.194514 0.0 Cu−O 731.4800 0.282190 0.0 Nb−O 1053.2161 0.389027 0.0 O−O 3657.8642 0.282693 200.1785

Atom qcore qshell k2 k4 K 1.237854 −0.418377 229.74443 0.0 Cu 4.944000 −3.777500 767.50000 1215.0378 Nb −2.984072 7.816735 255.95572 410.47927 O 1.122198 −3.006245 76.581476 1539.2972

Table 4.3: SMP parameters [cf. Eqs. (2.2) and (2.3)] for CuK-doped KNbO3 obtained and used in this work. ([Korb 11a], Tab. III) 42 4. Ferroelectricity in KNO with isolated Cu substitutionals

Ref. [Yu 95] this work Expt. TO modes 147i 149i 170 207 198 262 232 2801 477 489 521 LO modes 168 205 190 232 2791 405 360 419 743 835 826 1 Measured in the tetragonal structure at 585 K. −1 Table 4.4: Phonon frequencies in cm in KNbO3 from Ref. [Yu 95], this work, and experiment (Ref. [Font 84]). ([Korb 11a], Tab. IV)

order to rotate the ferroelectric polarization. According to both DFT and the SMP also in KNbO3, like in BaTiO3, there are no energy barriers between the different polarization directions.

Cu on K sites ([Korb 11a], Section III.B)

For tetragonal K0.5Cu0.5NbO3 the DFT result for the c/a ratio is 1.016, and therefore the same value as in pure KNbO3, whereas the SMP result is a c/a ratio of 1.044. This finding indicates that the SMP obtained in this work for the Cu-O interaction overestimates strain effects. Because the Cu-O interaction was only adjusted to DFT energies of a narrow target range of structures and Cu concentrations, it may not necessarily be widely transferable to arbitrary perovskite phases or high Cu concentrations. The ferroelectric energy profile of KNbO3 doped with 12.5 and 6.25 mol % CuK, respectively, obtained ab initio and with the SMP, is shown in Fig. 4.2. Only the atomic coordinates were optimized; the cell parameters were fixed to the cubic ones in both the DFT and the atomistic calculations. If CuK is added, the ferroelectric instabilities are enhanced, and for 12.5 mol % Cu the relative order of the ferroelectric instabilities is reversed, so that the ground state has no longer rhombohedral ([111]), but tetragonal ([001]) symmetry, resulting in a MPB, at least at low temperatures. The SMP was used to simulate lower CuK concentrations than those accessible with DFT in order to determine the MPB composition at which the ground-state symmetry changes. In Fig. 4.3 the ferroelectric instabilites of the tetragonal ([001]), the orthorhombic ([011]), and the rhombohedral ([111]) ferroelectric phases with respect to the paraelectric cubic phase as a function of the Cu content are depicted. In the SMP simulation, other than in Fig. 4.2, both atomic coordinates and cell parameters were optimzed. In the DFT calculations the cell parameters were again fixed to the cubic ones. For all three ferroelectric phases the instabilities vary almost linearly with composition. A linear fit was applied using the data for 0, 1.5625, 3.125, and 6.25 mol % Cu content. The SMP predicts a MPB (determined as the crossing point of the linear fits for the tetrahedral and the rhombohedral structure) at a Cu content of 2 mol % (DFT: 4 mol %; SMP with cell parameters fixed to the cubic ones: 3.6 mol %). The orthorhombic phase does not become the ground state for any composition in the range 0−12.5 mol % Cu, but around the MPB composition, the three phases are approximately degenerated in energy. In order to determine whether the ferroelectric instabilities are more stable than rotational ones, the energy of rotations of the oxygen octahedra around [001] (R001, corresponding to a0a0c+ and a0a0c− in Glazer’s system [Glaz 72]) and around [111] (R111, corresponding to a−a−a−) were calculated using DFT and the SMP for the compositions 12.5 and 50 mol % Cu content. These three tilt systems were 0 0 + 0 0 − observed, among others, in other perovskite compounds [a a c in NaNbO3 (T2 phase), a a c in SrTiO3 − − − (low temperature phase), and a a a in NaNbO3 (N phase)] [Glaz 72]. The superscripts denote whether the direction of rotation alternates along the corresponding axis (−) or stays the same (+), or if there is no rotation around that axis (0). The results are compiled in Table 4.5, together with those for the ferroelectric 4.3 Results 43

0 [001] [111] -2 [011]

-4 PE -6 (meV/f.u.) -8 PE

E-E -10 Expt. -12 DFT SMP -14 [111] PE [001] [011] [111] [001] PE [011]

Figure 4.1: Energy profile in meV per perovskite formula unit along linear transitions between the different polarization directions (see inset; PE: paraelectric cubic perovskite structure) in pure KNbO3. Black squares: experiment; green triangles: DFT; red circles: SMP. Open symbols: unit-cell parameters fixed to the cubic ones; filled symbols: optimized unit-cell parameters. The lines only serve as a guide to the eye. ([Korb 11a], Fig. 2)

Figure 4.2: The energy proﬁle obtained ab initio and with the SMP for cubic supercells of pure (top) and CuK- doped KNbO3. Green triangles: DFT; red circles: SMP. The curves for undoped KNbO3 are the same as in Fig. 4.1. ([Korb 11a], Fig. 3) 44 4. Ferroelectricity in KNO with isolated Cu substitutionals

0 DFT -20 -40 -60 (meV/f.u.)

PE -80

E-E -100 SMP -20 -40 -60 (meV/f.u.)

PE -80 [001] [011] E-E -100 [111] 0 2 4 6 8 10 12 Cu conc. (mol%)

Figure 4.3: [001], [011], and [111] instabilities in meV per perovskite formula unit relative to the cubic, paraelectric phase as a function of the Cu content in mol % obtained with DFT (top) and the SMP (bottom). ([Korb 11a], Fig. 4)

phases. Other than in Table 4.2, the cell parameters were ﬁxed to the cubic ones in the DFT calculations. In the SMP calculations, they were optimized, and also the monoclinic cell angle of the orthorhombic phase was allowed to deviate from 90°. The SMP yields a monoclinic angle of 89.65° (experiment [Hewa 73a]: 89.74°). According to both DFT and the SMP the rotational modes are unstable at 12.5 mol % CuK but stable at 50 mol % CuK. According to both DFT and SMP results the R111 mode is more stable than the R001 modes if the lattice parameters are ﬁxed to the cubic ones. The two R001 modes have similar energies in both DFT and SMP, but opposite energetical ordering is obtained with the two methods. The SMP generally overestimates the instabilities of the rotational modes in Cu0.5K0.5NbO3 by a factor of about 1.5 − 1.7. If the cell parameters are optimized, according to the SMP the R001 modes are slightly more stable than the R111 mode. In all cases, the rotational modes are less stable than the ferroelectric ones. In Fig. 4.4 histograms of the atomic displacements in undoped and CuK-doped KNbO3 in a fcc 4×4×4 supercell (320 atoms) are depicted. The displacements of the host atoms in the undoped and the doped supercell are similar, apart from a few K atoms, which are displaced into the opposite direction (with respect to Nb) as in the undoped crystal. The Cu atoms are much more strongly off-centered than the K atoms (by about 28% of the single unit cell lattice constant with respect to Nb). In Fig. 4.5 the atomic displacements and the change in ionic dipole moment of the individual unit cells obtained by using Eq. (4.1) are depicted. In one unit cell the atomic positions and displacements are drawn as well for a perfect reference cell (∆Pr denotes the change in ionic dipole moment of a perfect unit cell). The dipole moments of the individual unit cells are similar to those in the undoped crystal (shaded unit cell).

Cu on Nb sites ([Korb 11a], Section III.C)

In Fig. 4.6 the ferroelectric instabilities of undoped KNbO3 and of KNbO3 doped with 6.25 % and 12.5 % Cu on Nb sites obtained with DFT are depicted. At a doping level of 6.25 % the ferroelectric energy differences are reduced compared to the undoped crystal at both the unoptimized and the optimized volume. For 12.5 mol % CuNb the ferroelectric instabilities are reduced to almost zero, if the volume is ﬁxed to that of the undoped crystal, whereas if volume relaxation is taken into account, an the volume increases by 4.3 Results 45

Cu [001] [011] [111] conc. (%) DFT SMP DFT SMP DFT SMP f o f o f o 0.0 −6.2 −9.8 −9.8 −12.1 −12.2 −12.9 1.5625 −21.7 −21.5 −22.3 3.125 −34.0 −31.5 −30.8 6.25 −55.5 −56.8 −51.7 −52.5 −52.5 −51.6 12.5 −80.8 −96.1 −77.9 −91.9 −72.4 −83.5 50.0 −317.0 −491.3 −313.9 −448.8

Cu a0a0c+ a0a0c− a−a−a− conc. (%) DFT SMP SMP DFT SMP SMP DFT SMP SMP f f o f f o f f o 12.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 50.0 −30.3 −46.1 −104.2 −28.5 −49.7 −108.9 −35.5 −56.2 −97.0 Table 4.5: Energies of the [001], [011], [111], R001 (a0a0c+ and a0a0c−), and R111 (a−a−a−) instabilities with respect to the cubic paraelectric phase in meV per perovskite formula unit as a function of the Cu concentration in % obtained with DFT and the SMP. f : cell parameters ﬁxed to the cubic ones, o: optimized cell parameters. ([Korb 11a], Tab. V)

60 40 K 20

4 6.25 % Cu 3 CuK 2 undoped 1 60 40 Nb 20

150 O 100 50 # atoms -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05

u[001]

Figure 4.4: Atomic displacements u[001], in units of the lattice constant of a single KNbO3 unit cell, with respect to the paraelectric positions for all unit cells in a supercell polarized in the [001¯] direction for undoped KNbO3 and 6.25 mol % CuK doping. The averaged Nb displacement was set to zero. ([Korb 11a], Fig. 5) 46 4. Ferroelectricity in KNO with isolated Cu substitutionals

Figure 4.5: Atomic displacements in the [100] − [001] plane (thin gray and red arrows) and change in ionic dipole moment (bold orange arrows) of the unit cells in a supercell which contains 6.25 mol % CuK and is polarized in the [001¯] direction. The arrows that mark the displacements of K, Nb, and O (all except for those of Cu) are scaled by a factor of 10 for better visibility. Black circles, black arrows, and the bold orange arrow in the shaded cell on the left mark the atom positions, displacements and change in ionic dipole moment ∆Pr of a perfect reference cell, respectively. ([Korb 11a], Fig. 6)

about 2%, and the ferroelectric instabilities are larger than in the undoped crystal. The inset contains the ferroelectric instability in undoped KNbO3 as a function of the lattice constant. The vertical lines denote the equilibrium lattice constant of undoped KNbO3 and those of KNbO3 doped with 6.26 mol % and 12.5 mol % CuNb, respectively. In Fig. 4.7 histograms of the ferroelectric displacements in KNbO3 doped with 6.25 mol % CuNb, and in undoped KNbO3 are depicted. Although the displacements mainly have the same directions as in the undoped material, the numbers scatter strongly (more than for CuK-doping, cf. Fig. 4.4). Cu is hardly shifted off center at all, and the averaged displacements are smaller than in the undoped crystal. In Fig. 4.8 the atomic positions, displacements, and change in ionic dipole moment [c.f. Eq. (4.1)] are depicted for each unit cell in a supercell that contains 6.25 mol % CuNb and is polarized along the [001¯] direction (cf. Fig.4.5). The dipole moments of the individual unit cells are inﬂuenced strongly by the neighboring Cu substitutional, some dipole moments even point in opposite directions. 4.3 Results 47

Figure 4.6: The energy proﬁle as in Fig. 4.2, but for Cu on Nb sites, and only DFT results are shown. Filled symbols: optimized volume, open symbols: volume ﬁxed to the cubic volume of the undoped crystal. Inset: ferroelectric instability of undoped KNbO3 for atomic displacements in the [001] direction as a function of the lattice constant. Vertical lines denote the equilibrium lattice constant of undoped KNbO3 (solid line) and KNbO3 doped with 6.25 mol % Cu (dotted line), and 12.5 mol % Cu (dashed line). ([Korb 11a], Fig. 7)

60 K undoped 40 6.25 % Cu 20 60 40 Nb 20 4 3 CuNb 2 1 150 O 100 50 # atoms -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

u[001]

Figure 4.7: The same as Fig. 4.4 for CuNb doping. Note the change of scale.([Korb 11a], Fig. 8) 48 4. Ferroelectricity in KNO with isolated Cu substitutionals

) K lat Nb O Cu z (a ΔPr [001]

[100] |a | x (a ) Figure 4.8: The same as Fig. 4.5 for 6.25 mol % CuNb. The arrows that mark the displacements, including those of Cu, are scaled by a factor of 10. ([Korb 11a], Fig. 9) 4.4 Discussion 49

4.4 Discussion ([Korb 11a], Section IV) Cu on K sites ([Korb 11a], Section IV.A)

According to the SMP results, the ground state of (Cu,K)NbO3 changes from rhombohedral to tetragonal symmetry at about 2 mol% Cu. According to DFT results for cubic supercells, this change occurs at about 4 mol% Cu (cf. Fig. 4.3). Because the SMP overestimates the influence of CuK on the ferroelectric instability (cf. Fig. 4.2), it may underestimate the amount of Cu needed to change the ground state structure, or in other words, the SMP may underestimate the MPB composition. Therefore the 2 mol % obtained in this work using the SMP should be regarded as a lower barrier. An upper barrier is the DFT prediction for unit-cell parameters fixed to the cubic ones (4 mol %). At finite temperatures, the MPB may shift further because then also entropy plays a role. While at low temperatures the MPB lies between the rhombohedral ground state structure of KNO and the tetrahedral one of (K,Cu)NbO3, at room temperature it lies between the orthorhombic room temperature structure of KNO and the tetrahedral one of (K,Cu)NbO3, which may result in a positionof the different MPB. In an earlier atomistic study of the instabilities for the [001], [011], and [111] displacements of an isolated LiK dopant in KTaO3 by Exner et al. [Exne 94], the same energetical ordering ([111] → [011] → [001]) was found as in the present work for CuK. The tetragonal instability of the hypothetical composition K0.5Cu0.5NbO3 found with the DFT is 317 meV (SMP: 491 meV, cf. Table 4.5),which is a similar value as the 222 meV found in Ref. [Bilc 06] for K0.5Li0.5NbO3. The very large off-center displacements of CuK obtained in this work were also found by Bilc and Singh for LiK (from Fig. 2 in Ref. [Bilc 06] we estimate about 23% of a single perovskite lattice constant with respect to Nb). These findings support the similarity of the effects of Li and Cu doping on K sites in KNbO3. In spite of the large off-center displacements of CuK the dipole moments are similar to those of the bulk even in the cells that contain the Cu dopants. Analogous to the weak ferroelectric instability of ≈ 12 meV per formula unit found for the octahedral rotation around [001] and [111] in Ref. [Bilc 06] for Li0.5K0.5NbO3, in the present work an energy gain of ≈ 30 meV per formula unit was obtained for the rotation around [001] and ≈ 36 meV for the rotation around [111], in Cu0.5K0.5NbO3, which are both much smaller than the 317 meV of the ferroelectric displacements in the [001] direction. Due to competing incorporation of Cu on Nb sites [Korb 10] and second phase formation [Mats 05] the MPB may be difficult to obtain experimentally. Another drawback may be the semiconduction found for CuNbO3 [Slei 70], which might limit the ferroelectric performance also in the solid solution. On the other hand, in chapter 3 it was found that a reducing processing atmosphere should drive Cu to the K sites, and + that the most stable charge state is CuK for a large range of electronic chemical potentials. In this case no free charge carriers are generated.

Cu on Nb sites ([Korb 11a], Section IV.B) Cu on Nb sites causes two counteracting effects: an increase of the ferroelectric instability at (unrealistic) high doping levels (e.g., 12.5%), and a decrease of the ferroelectric energies at lower doping levels (e.g., 6.25%). In both 40-atom and 80-atom supercells, in all individual unit cells the change in ionic dipole moment is smaller than in the undoped bulk cells. A possible explanation for these ﬁndings is the following: The larger ionic radius of Cu2+ as compared to Nb5+ (0.73 vs 0.64 A)˚ ([Shan 76]) leads to an overall volume increase. Because the ferroelectric energy is very sensitive to volume changes, the ferroelectric energy of the expanded cell can increase. The inset in Fig. 4.6 shows the increase in ferroelectric energy gain as function of the lattice constant in an undoped KNbO3 unit cell. A cubic unit cell was imposed. Increasing the lattice constant of undoped KNbO3 by 2% (like for 12.5 mol % Cu doping) with respect to the equilibrium one increases the ferroelectric energy gain by a factor of 5.6. Increasing the lattice constant by 0.8%, like in the case of 6.25 mol % Cu doping, increases the ferroelectric instability by the much smaller factor of 2.3. The increase of the lattice constant with the Cu doping level is only weakly nonlinear, but the ferroelectric instability increases with the lattice constant in a strongly nonlinear way. On the other hand, the smaller change in ionic dipole moments of the individual cells in KNbO3 doped 50 4. Ferroelectricity in KNO with isolated Cu substitutionals

with CuNb reduces the ferroelectric energy gain. In the 40-atom cell, the volume effect overcompensates the reduced change in ionic dipole moments, so that the ferroelectric energy gain increases. In the 80-atom cell, the volume effect is less strong and cannot compensate the reduced change in ionic dipole moment, so that the ferroelectric energy gain is smaller than in the undoped bulk crystal. The isolated CuNb dopants strongly disturb the arrangement of dipole moments of the surrounding bulk region. In the 80-atom supercell containing one Cu atom, in all unit cells the atomic displacements are different from those in the pure crystal. This ﬁnding can be attributed to the rather shallow ferroelectric instability in KNbO3 as compared to, e.g., PbTiO3 (11 meV/f.u. in tetragonal KNbO3 as compared to 53 meV/f.u. in tetragonal PbTiO3 [Umen 06]), so that the atomic structure can be more strongly inﬂuenced by lattice defects than in PbTiO3. However, isolated CuNb substitutionals will be rather an exception because of the large charge difference between Nb and Cu, and the effects of defect complexes of Cu and oxygen vacancies are likely much stronger than those of a small concentration of isolated CuNb substitutionals and hence determine the overall ferroelectric behavior of the material. 5 Ab initio thermodynamics of defect complexes in KNO and KNN 52 5. Ab initio thermodynamics of defect complexes in KNO and KNN

Parts of this chapter have been published in [Erun 10], [Hoff 12], and [Korb 11b].

In this chapter it is investigated which defect complexes of Cu substitutionals and vacancies are stable, and if they can lead to ferroelectric hardening.

5.1 Introduction

As motivated in the introduction (chapter 1), defect complexes can contribute to ferroelectric hardening. They can form due to Coulomb attraction between charged defects, like the FeTi −VO defect in PTO. In Cu- doped KNN three defect complexes, (CuK/Na −VK/Na), (CuNb −VO), and (VO − CuNb −VO) (cf. Fig. 5.1) + + 5+ 2− 2+ + − 2+ are most plausible. Assuming formal ionic charges (K , Na , Nb ,O , Cu or Cu , VK/Na, and VO (V 0 and V •• in Kroger-Vink¨ notation), then (Cu2+ )3− (Cu 000 in Kroger-Vink¨ notation) can be partly K/Na O Nb5+ Nb •• 0000 charge-compensated by one or over-compensated by two VO . CuNb can be exactly charge-compensated •• 2+ + • 0 by two VO . (CuK/Na+ ) (CuK/Na in Kroger-Vink¨ notation) can be charge-compensated by a VK/Na. In the orthorhombic room temperature phase of KNN and KNO the ferroelectric polarization is oriented along one of the 12 equivalent h110i directions of the pseudocubic lattice. In a defect-free crystal region these 12 polarization directions are degenerated in energy. A defect complex can lift the degeneracy and define a preferential orientation of the ferroelectric polarization, a “soft” direction. Switching the polarization to a direction that is not the preferential one or one of the preferential ones (a “hard” direction) then costs energy. Therefore the polarization close to the defect complex becomes harder to switch by an applied electric field. If it switches at all, it is likely to return to its original, preferential direction when the electric field is turned off. How strongly a defect complex impedes polarization switching is determined by the energy differences between “soft” and “hard” polarization directions and the energy barriers for switching between these. In the following sections the defect complexes (CuK/Na −VK/Na), (CuNb −VO), and (VO −CuNb −VO) in KNO and KNN are investigated for their thermodynamic stability and their ability to impede polarization switching.

5.2 Formation of defect complexes in Cu-doped KNO and KNN

Parts of this section have been published in [Erun 10], [Hoff 12], and [Korb 11b].

In this section ab-initio thermodynamics are applied to obtain relative concentrations of the defect complexes CuK/Na −VK/Na, CuNb −VO, and VO − CuNb −VO.

Method In order to obtain equilibrium concentrations of defect complexes (cf. Fig. 5.1) and isolated defects, formation energies and binding energies for (CuA −VA) (A=K or A=Na), (CuNb −VO) and (VO − CuNb −VO) and the isolated defects in KNO and KNN were determined. The Fermi energy was varied within the bandgap and for each value of the Fermi energy the defect formation energies of the most stable charge states were used. The defect formation energies were calculated using a 2 × 2 × 3 supercell, a 4 × 4 × 3 Monkhorst-Pack k-point mesh, and a plane wave cutoff-energy of 340 eV. The LDA band gap was used (but for KNN a bandgap correction was also applied for comparison). Spin-polarization was neglected since in chapter 3 its effect on the formation energies was found to be very small. The equilibrium concentration c of a defect was calculated using the defect formation energies assuming a Boltzmann-like distribution,

E f − k T c = Ncon f e B , (5.1) f where Ncon f is the number of different possible conﬁgurations of the defect, E is the defect formation energy, kB is Boltzmann’s constant and T is the temperature. Ncon f is related to the conﬁgurational entropy 5.2 Formation of defect complexes in Cu-doped KNO and KNN 53

(a) (b) (c) (d)

Figure 5.1: Impurity-vacancy conﬁgurations: (a) (CuK −VK), (b) (CuNb −VO), (c) straight (VO −CuNb −VO), and (d) bent (VO − CuNb −VO).

Scon f as follows: −Scon f /kB Ncon f = e . Another possibility to calculate the concentration ratios of the defect complexes is to use the binding energies EB. The formation energies depend on the chemical conditions, whereas the binding energies do not. Using the formation energies only requires that thermal equilibrium is reached, using the binding energies additionally requires assumptions about the vacancy concentration (here it was assumed that enough vacancies are available (more than Cu substitutionals)). Both methods were applied here. The binding energy EB of each complex was determined as the difference between the defect formation energy of the defect complex and the sum of the defect formation energies of the isolated defects as follows:

f f f EB(CuNb −VO) = E (CuNb −VO) − (E (CuNb) + E (VO)). (5.2) The defect formation energies depend on the Fermi energy, hence the binding energies do so as well. If the binding energies are used, only relative concentrations can be obtained. One compares the concentration of a defect complex with that of two or more isolated defects, and from Eq. (5.1) it follows that

c Ncon f (1) 1 = e−(EB(1)−EB(2))/(kBT). (5.3) c2 Ncon f (2) For example,

[V − Cu −V ] N (V − Cu −V ) O Nb O con f O Nb O (EB(CuNb−VO)−EB(VO−CuNb−VO))/(kBT) = iso e (5.4) [CuNb −VO] Ncon f (VO ) · Ncon f (CuNb −VO) and analogous for the other defects. The numbers for Ncon f that were used here are listed in Table 5.1. f Ncon f (E ) is the Ncon f inserted in Eq. (5.1) and Ncon f (EB) is the Ncon f inserted in Eq. (5.3).

Results

Figure 5.2 contains the formation energies of the defect complexes (CuK −VK), (CuNb −VO), and (VO − CuNb −VO) at the points 1 and 4 in the phase stability diagram in Fig. 3.11. • The defect charge states with the largest stability range in the LDA bandgap are (CuK −VK) and (CuK − 0 × 0 •• • VK) , (CuNb −VO) and (CuNb −VO) , and (VO −CuNb −VO) and (VO −CuNb −VO) , although additional charge transition levels were found in the bandgap or close to the band edges for all these defects. Electronic densities of states (DOS) indicate electronic defect levels in the bandgap for all the defects considered here (cf. Fig. 5.3). In Fig. 5.4 binding energies EB of the defect complexes (CuK −VK), (CuNb −VO), and (VO −CuNb −VO) in KNbO3 calculated according to Eq. 5.2 are depicted as a function of the Fermi level. Over the entire LDA band gap, the binding energies of the three defect complexes are negative (between - 1.1 and -2.8 eV). This means that a crystal with defect complexes has a lower energy than one with isolated 54 5. Ab initio thermodynamics of defect complexes in KNO and KNN

6.0 8.0 (a) +1 -1 (b) 0 5.0 7.0 -1 4.0 6.0 +1 3.0 5.0 +2 2.0 +1 4.0 (eV) (eV) f f E 1.0 0 E 3.0 +1 -1 +2 -1 0.0 2.0 -1.0 1.0 -2.0 0.0 -0.5 0.0 0.5 1.0 1.5 2.0 -0.5 0.0 0.5 1.0 1.5 2.0

EF (eV) EF (eV)

Figure 5.2: Formation energies of the defect complexes (CuK −VK) (red squares), (CuNb −VO) (green circles), and (VO − CuNb −VO) (blue triangles) as function of the Fermi energy in KNO. (a) thermodynamic conditions of point 1 (oxygen-rich); (b) of point 4 (oxygen-poor conditions) in Fig. 3.11. The LDA valence band maximum (VBM) and conduction band minimum (CBM) are indicated by vertical lines. The numbers attached to the curves mark the charge states of the defects.

VBM CBM

V0 CuNb-2 O

V1+ CuNb-2 O

V2+ CuNb-2 O

V2- CuNb- O

V1- CuNb- O

V0 DOS (arb. units) CuNb- O

V2- CuK- K

V1- CuK- K

V1+ CuK- K

undoped -2 -1 0 1 2 3 E (eV)

Figure 5.3: Electronic DOS of KNO with the defect complexes (CuK −VK), (CuNb −VO), and (VO −CuNb −VO). The LDA valence band maximum (VBM) and conduction band minimum (CBM) are indicated by vertical dotted lines. The colored regions under the curves mark occupied levels. 5.2 Formation of defect complexes in Cu-doped KNO and KNN 55

VBM CBM 0.0

−0.5

−1.0 V V (CuNb− O) (CuK− K)

(eV) −1.5 B E −2.0

−2.5 V V ( O−CuNb− O)

−3.0 −0.5 0.0 0.5 1.0 1.5 2.0 EF (eV)

Figure 5.4: Binding energies of the defect complexes (CuK −VK), (CuNb −VO), and (VO − CuNb −VO) versus Fermi energy in KNbO3. The LDA valence band maximum (VBM) and conduction band minimum (CBM) are indicated by vertical lines. ([Korb 11b], Fig. 4; ([Erun 10], Fig. 5; [Hoff 12] Fig. 10)) defects for all Fermi levels in the LDA band gap at least at low temperatures. In Fig. 5.5 the same is depicted for KNN. The results were obtained with the VCA as described in sections 2.1 and 3.4. In KNN the complexes are less strongly bound than in KNbO3 (as an example, for a Fermi level at the VBM EB of (VO − CuNb −VO) is about −1.5 eV in KNbO3 and about −0.8 eV in KNN).

VBM CBM 0.0

−0.5

V (Cu −V ) −1.0 (CuNb− O) K K

(eV) −1.5 B E −2.0 V V ( O−CuNb− O) −2.5

−3.0 −0.5 0.0 0.5 1.0 1.5 2.0 EF (eV) Figure 5.5: The same as Fig. 5.4 for KNN 50/50.

In Fig. 5.6 the same as in Fig. 5.5 is depicted after a bandgap correction. The binding energies vary with the Fermi level in a similar range as without a bandgap correction, but the values at the bandgap center are lower (their absolute values are higher) for (CuNb −VO) and (VO − CuNb −VO), whereas it is higher (the absolute value is lower) for (CuK −VK). The defect formation energies at room temperature (300 K) and atmospheric pressure (0.2 bar oxygen partial pressure, point 9 in Fig. 3.11 (bottom)), and close to the sintering temperature (1300 K) at the same and at a reduced oxygen partial pressure (0.2 bar and 0.2 µbar, points 10 and 11 in Fig. 3.11 (bottom)), the binding energies, and the concentrations of the defects obtained using the formation energies and the binding energies are compiled in Table 5.1. Since with the binding energies only the concentration ratios of the defect complexes can be determined, the concentration of one defect was always ﬁxed. This was done such that the two defect complexes together 56 5. Ab initio thermodynamics of defect complexes in KNO and KNN

VBM CBM 0.0

−0.5

V (Cu −V ) −1.0 (CuNb− O) K K

(eV) −1.5 B E −2.0 (V −Cu −V ) −2.5 O Nb O

−3.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 EF (eV) Figure 5.6: The same as Fig. 5.5 with a bandgap correction.

CuK VK CuK −VK CuNb VO CuNb −VO VO − CuNb −VO f Epoint 9 2.14 1.21 2.16 -0.07 2.46 0.96 2.68 f Epoint 10 3.04 2.11 3.96 2.87 1.18 2.62 3.06 f Epoint 11 3.44 2.51 4.76 4.79 0.40 3.77 3.43 f Ncon f (E ) 1 1 6 1 3 6 3 f −40 −21 −40 −3 −41 −20 −49 c (Epoint 9) 1 · 10 5 · 10 4 · 10 2.5 · 10 2 · 10 8 · 10 5 · 10 f −6 −9 −8 −5 −5 −3 −5 c (Epoint 10) 9 · 10 7 · 10 1 · 10 4 · 10 8 · 10 2.42 · 10 2 · 10 f −4 −10 −8 −9 −2 −4 −3 c (Epoint 11) 5 · 10 2 · 10 2 · 10 3 · 10 8 · 10 2 · 10 2 · 10 EB 0 -1.19 0 -1.43 -2.16 Ncon f (EB) 1 1/cCu 6 1 3/cCu 6 3 1300 K −6 −3 −6 −3 −4 c (EB) 4 · 10 2.5 · 10 1 · 10 1.9 · 10 6 · 10

Table 5.1: Formation energies, binding energies, and concentrations of defects in KNO in equilibrium in air at 1 bar and 300 K (point 9 in Fig 3.11), at 1 bar and 1300 K (point 10), and at 1 µbar and 1300 K (point 11). The vacancy concentrations are those of intrinsic vacancies. When using the formation energies E f to calculate concentrations c, it is assumed that altogether 0.25 mol% Cu are incorporated into the perovskite lattice. When using the binding energies EB, it is assumed that the 0.25 mol% Cu are incorporated either completely on K or completely on Nb sites. The cells with the concentrations of the most abundant Cu defects are shaded in yellow. give the experimental Cu concentration of 0.25 mol% used in [Erun 10]:

[CuK] + [CuK −VK] = 0.25% and [CuNb] + [CuNb −VO] + [VO − CuNb −VO] = 0.25%. When using the formation energies it was assumed that 0.25 mol% Cu are altogether incorporated into the perovskite lattice:

[CuK] + [CuK −VK] + [CuNb] + [CuNb −VO] + [VO − CuNb −VO] = 0.25%.

Assuming error bars in the formation energies of 1 eV (cf. Table 3.2) and of 0.5 eV in the binding energies (cf. Figs. 5.5 and 5.6), the error bar in the concentrations is 17 orders of magnitude when using the formation energies and eight orders of magnitude when using the binding energies at 300 K (point 9). At 1300 K the error bar is four orders of magnitude using the formation energies and two orders of magnitude using the binding energies. Taking the error bars into account, according to the formation energies at 300 K the 5.2 Formation of defect complexes in Cu-doped KNO and KNN 57

CuK VK CuK −VK CuNb VO CuNb −VO VO − CuNb −VO f Epoint 9 2.06 0.98 1.64 -0.22 2.29 0.79 2.60 f Epoint 10 2.86 1.78 3.24 2.82 1.01 2.55 3.08 f Epoint 11 3.26 2.18 4.04 4.74 0.23 3.70 3.45 f Ncon f (E ) 1 1 6 1 3 6 3 f −41 −17 −34 −3 −38 −19 −50 c (Epoint 9) 1 · 10 3 · 10 8 · 10 2.5 · 10 1 · 10 1 · 10 3 · 10 f −5 −7 −6 −5 −4 −3 −5 c (Epoint 10 3 · 10 1 · 10 5 · 10 4 · 10 4 · 10 2.42 · 10 1 · 10 f −3 −9 −6 −9 −1 −4 −4 c (Epoint 11 2 · 10 4 · 10 9 · 10 3 · 10 4 · 10 2 · 10 8 · 10 EB -1.40 -1.27 -1.75 Ncon f (EB) 1 1/cCu 6 1 3/cCu 6 3 1300 K −7 −3 −6 −3 −5 c (EB) 6 · 10 2.5 · 10 6 · 10 2.42 · 10 7 · 10

Table 5.2: The same as in Table 5.1 for KNN.

CuK VK CuK −VK CuNb VO CuNb −VO VO − CuNb −VO f Epoint 9 2.06 0.19 1.18 -1.96 3.87 0.00 3.39 f Epoint 10 2.86 0.99 2.78 1.08 2.59 1.76 3.87 f Epoint 11 3.26 1.39 3.58 3.00 1.81 2.91 4.24 f Ncon f (E ) 1 1 6 1 3 6 3 f −71 −4 −55 −3 −65 −35 −93 c (Epoint 9) 7 · 10 7 · 10 2 · 10 2.5 · 10 3 · 10 1 · 10 8 · 10 f −10 −4 −9 −3 −10 −5 −13 c (Epoint 10 3 · 10 1 · 10 4 · 10 2.47 · 10 3 · 10 3 · 10 1 · 10 f −5 −6 −6 −4 −7 −3 −9 c (Epoint 11 2 · 10 4 · 10 6 · 10 2 · 10 3 · 10 2 · 10 8 · 10 EB -1.06 -1.90 -2.38 Ncon f (EB) 1 1/cCu 6 1 3/cCu 6 3 1300 K −5 −3 −8 −3 −5 c (EB) 1 · 10 2.49 · 10 2 · 10 2.43 · 10 7 · 10

Table 5.3: The same as in Table 5.2 after a bandgap correction.

most abundant Cu defects are CuNb and possibly CuNb −VO, whereas according to the binding energies VO − CuNb −VO is more abundant at 300 K than CuNb −VO and CuNb, and CuK −VK is more abundant than CuK. At 1300 K (point 10) according to the formation energies CuNb −VO is more abundant than CuK −VK. According to the binding energies at 1300 K again CuK −VK is more abundant than CuK and CuNb −VO and possibly also VO − CuNb −VO is more abundant than CuNb. The same as in Table 5.1 is compiled in Table 5.2 for KNN, and regarding the most abundant Cu defects in KNN the same is concluded as for KNO. The same as in Table 5.2 after a bandgap correction is compiled in Table 5.3. Using the formation energies the only abundant Cu defect at 300 K (point 9) is CuNb. At 1300 K (points 10 and 11) it is again CuNb and/or possibly CuNb −VO. Using the binding energies the same conclusions are drawn as for KNO and KNN without a band-gap correction. 58 5. Ab initio thermodynamics of defect complexes in KNO and KNN

5.3 Atom diffusion in KNO

In order to obtain an estimate how fast KNN with Cu dopants and vacancies reaches thermal equilibrium at room temperature, diffusion times for atoms in KNN were calculated using classical transition state theory. If the system reaches thermal equilibrium very fast, it is appropriate to assume an equilibrium defect distribution at room temperature.

If one wants to determine a defect distribution, one needs to know if the system is in thermodynamic equilibrium. Thermodynamic equilibrium can be assumed if the processes of interest, here the diffusion of atoms, happen sufﬁciently fast, for example within seconds or days. Otherwise the system stays in a metastable state, here for example the equilibrium state at typical sintering temperatures. In this work classical transition state theory (TST) was applied to estimate the timescale on which alkali and oxygen vacancies diffuse through KNO. In [Erha 07a] the energy barrier for jumps of the transition metal in BaTiO3 (Ti) was found to be as high as 9.84 eV. Since the energy barriers for jumps of Ba and O in BaTiO3 found in [Erha 07a] (6.00 and 0.89 eV, respectively) are in a similar range as those found here for K and O in KNbO3 (4.2 and 0.59 eV, respectively), it was assumed that also the energy barrier for jumps of the transition metal in KNbO3 is similar to that in BaTiO3, so that Nb is almost immobile at typical sintering temperatures and below. The diffusion time was calculated for the cubic phase of KNO. Using the Einstein relation [Eins 05], the diffusion constant D for a given species in cubic crystals is

hR2i D = , (5.5) 6t where hR2i is the averaged quadratic diffusion length after a time t. According to classical TST [Ball 05], the diffusion constant is

−E /kBT 2 D = NNNνae barrier λ f /6, (5.6) where NNN is the number of nearest neighbor sites of the diffusing species, νa is the so-called “attempt frequency”, at which the atom oscillates in its initial position from which it eventually jumps to neighboring empty lattice site, λ is the distance to that neighboring lattice site (the hopping distance), f is a correlation factor which accounts for deviations of the diffusion behavior from a random walk, and Ebarrier is the energy barrier the atom needs to overcome in order to jump from one lattice site to the nearest neighbor site. According to [Ball 05], the correlation factor f is approximately

1 − 1/N f = NN . (5.7) 1 + 1/NNN

Like in Ref. [Erha 07a], the attempt frequency νa was approximated by the lowest optical phonon frequency −1 of 190 cm obtained experimentally for KNO (cf. chapter 4), which gives νa = 5.7 THz. Combining the equations 5.5 and 5.6, the time an atom typically needs to diffuse a distance R is

R2eEbarrier/kBT t = 2 . (5.8) NNNνaλ f There are two limiting cases for the diffusion of the vacancies to the Cu substitutionals: One is the random walk (if the vacancies are only weakly attracted by the substitutional, or not at all), the other is the direct walk to the substitutional (if the substitutional strongly attracts the vacancies). In the case of a random walk, assuming a random distribution of the Cu substitutionals and of the initial K and O vacancy positions, the average number of hops a vacancy needs to encounter a Cu substitutional is given by

1 hNhopsi = , pCu where pCu is the probability that a given lattice site is a nearest neighbor of a Cu substitutional. The numbers for these parameters are listed in Table 5.4. The experimental Cu concentration of cCu = 0.25 mol% 5.3 Atom diffusion in KNO 59 used in [Erun 10] was assumed also here. From the average number of hops a vacancy needs to end up next to a Cu substitutional, the corresponding diffusion length R is q 2 R = hNhopsiλ f .

Inserting into Eq. 5.8 ﬁnally gives the diffusion time for the random walk:

eEbarrier/kBT trandom walk = . (5.9) pCuNNNνa In the case of a direct walk of the vacancies towards the Cu substitutionals the diffusion time is obtained directly from Eq. 5.8 using half of the typical distance between the substitutionals for the diffusion length R: s 1 a3 R = 3 ≈ 15 nm, 2 cCu so that the time for the direct walk is

a2eEbarrier/kBT tdirect walk = . 2/3 2 4cCu NNNνaλ f

The energy barriers Ebarrier for nearest-neighbor jumps of K and O atoms in KNO were calculated with DFT. A sc 2 × 2 × 2 supercell was used, and the lattice parameters were ﬁxed to those obtained with DFT for perfect cubic KNO. Plane waves with energies up to 25 Ry (340 eV) and a k-point mesh corresponding to a 8 × 8 × 8 k-point mesh for the single perovskite unit cell were used. A K or O vacancy was created in the supercell, and using the nudged-elastic band (NEB) method the minimum-energy path (MEP) for a jump of a nearest-neighbor atom into that vacancy was determined. The results for Ebarrier and the resulting diffusion times are listed in Table 5.4. Figure 5.7 shows the energy barriers for K and O vacancy migration. In order to estimate the time treorient a CuNb-VO defect dipole needs to reorient spontaneously, the energy

4.5 V 4.0 VK VO 3.5 O at CuNb 3.0 2.5 2.0 (eV)

E 1.5 1.0 0.5 0.0 -0.5 0 1 2 3 4 5 6 NEB image Figure 5.7: Energy barriers for atom diffusion in KNO. barrier for the jump of an oxygen vacancy from one nearest neighbor site of a CuNb substitutional to another was calculated. It amounts to 2.0 eV and is therefore much higher than that in the defect-free crystal regions. Assuming the same attempt frequency for an oxygen vacancy to overcome this energy barrier, the typical time for such a jump is −Ebarrier/kBT treorient ≈ 1/Njump = 1/ νaNsitese , (5.10) where Njump is the number of jumps per second and Nsites is the number of possible sites to jump to (4 in −6 12 this case). At 1300 K one obtains treorient ≈ 2 × 10 s, at 300 K treorient ≈ 6 × 10 a. 60 5. Ab initio thermodynamics of defect complexes in KNO and KNN

VK VO Ebarrier in eV 4.2 0.59 Ebarrier at CuNb in eV 2.0 pCu 6cCu 2c√Cu λ a a/ 2 NNN 6 8 f 0.714 0.778 58 trandom walk in s (300 K) 4 × 10 0.03 58 tdirect walk in s (300 K) 1×10 0.006 4 −10 trandom walk in s (1300 K) 3 × 10 8×10 3 −10 tdirect walk in s (1300 K) 9×10 1×10 20 treorient in s (300 K) 2 × 10 −6 treorient in s (1300 K) 2 × 10

Table 5.4: Energy barriers and diffusion times of K and O vacancies in undoped and Cu-doped KNO. 5.4 Ferroelectricity in KNO with Cu-defect complexes 61

5.4 Ferroelectricity in KNO with Cu-defect complexes “Soft” and “hard” polarization directions in KNO with defect complexes

In this section it is investigated if the defect complexes CuNb −VO and VO − CuNb −VO cause “hard” and “soft” directions for the ferroelectric polarization in KNO.

In this section it is investigated if and how the defect complexes CuNb −VO and (VO − CuNb −VO) impede the motion of ferroelectric domain walls. The (VO − CuNb −VO) defect may occur in a straight configuration, in which the VO occupy opposite sites of the CuNb substitutional, or in a bent one. The CuNb − VO defect and the two possible configurations of the (VO − CuNb − VO) defect are depicted in Fig. 5.8. The orange solid arrow that points to the orange dot labeled as 1 marks one possible orientation of the ferroelectric polarization in orthorhombic KNO (the room temperature phase). In the perfect crystal all these polarization directions (the h110i directions) are equivalent, but close to a defect complex this is no longer the case. The CuNb −VO defect is shown in Fig. 5.8(a). This defect complex has an electric dipole moment, indicated by the dashed arrow, and there are three inequivalent orientations of the ferroelectric polarization with respect to the defect dipole, labeled 1, 2, and 3. Orientations that are still equivalent by symmetry in spite of the defect complex are marked with the same number, e.g., configuration 1’ is equivalent to configuration 1. The straight (VO − CuNb − VO) defect (cf. Fig. 5.8(b)) does not have an electric dipole moment. There are 2 nonequivalent orientations of the ferroelectric polarization with respect to this defect complex (configurations 0 and 6 in Fig. 5.8(b)). Figure 5.8(c) shows the possible orientations of the bent (VO − CuNb −VO) defect. This defect has an electric dipole moment, and there are five nonequivalent orientations of the ferroelectric polarization with respect to this dipole (configurations 1−5).

Figure 5.8: Defect complex conﬁgurations of CuNb (black circle) and VO (small black square) in a unit cell of orthorhombic ferroelectric KNO with different orientations of the ferroelectric polarization P. (a) CuNb −VO; (b) straight VO−CuNb −VO; (c) bent VO−CuNb −VO. The dashed blue arrows mark the electric dipole moments of the defects PD, the orange dots mark the possible directions of the ferroelectric polarization P in the orthorhombic phase (they do not mark atoms). One polarization direction is indicated by a solid orange arrow.

Defect complexes can lead to ferroelectric hardening if they increase the energy needed for the migration of a domain wall. This can be the case either if different relative orientations of the defect complex and the surrounding spontaneous polarization differ in energy or if the energy barriers between different relative orientations are higher than in the defect-free case. In this work it was investigated if and to which extent these two mechanisms apply here. In a first step, the energies of the different configurations were calculated using DFT in order to determine if different relative orientations have different energies, and if yes, which relative orientations have the lowest energy. Each defect complex was placed in a 2 × 2 × 2 fcc (80-atoms) supercell. This corresponds to 6.25% of the Nb atoms being replaced by Cu. The lattice parameters were fixed to those obtained for undoped orthorhombic KNO (cf. chapter 4). An energy cutoff of 22 Rydberg (≈300 eV) and a Monkhorst-Pack 2 × 2 × 2 k-point mesh was chosen. In a second step, using 62 5. Ab initio thermodynamics of defect complexes in KNO and KNN

Atom pair A ρ C K−O 126870.4000 0.194514 0.0 Cu−O 712.8000 0.32698 Nb−O 1053.2161 0.389027 0.0 O−O 3657.8642 0.282693 200.1785

Atom qcore qshell k2 k4 K 1.237854 −0.418377 229.74443 0.0 Cu 0.000000 2.000000 999999.0 0.0 Nb −2.984072 7.816735 255.95572 410.47927 O 1.122198 −3.006245 76.581476 1539.2972

Table 5.5: SMP parameters [cf. Eqs. (2.2) and (2.3)] for CuNb-doped KNbO3 used in this chapter.

the results of the DFT calculations as a benchmark, the energies of the same structures were recalculated with the SMP in order to validate the SMP for a more extensive mapping of the potential energy landscape of these systems. In a third step, the validated SMP was used in order to calculate energy barriers for polarization switching as function of the type and the concentration of the defect complexes. In particular, the energy barriers for transitions between the different relative orientations of the defect complexes and the ferroelectric polarization were determined. These transitions correspond to homogeneous switching (instantaneously switching the polarization in the whole domain). Therefore the resulting energy barriers should be regarded as an upper limit of the energy barrier for the motion of a domain wall. In the SMP calculation, a Buckingham potential for the Cu−O interaction from [Baet 90] was combined with the SMP for KNO. The set of potential parameters is listed in Tab. 5.5. In the SMP calculations both orthorhombic and cubic lattice parameters were used as an experiment, and lower defect concentrations (3.125 and 1.5625%) were taken into account additionally. The resulting relative energies of the defect configurations in Fig. 5.8 obtained with DFT and the SMP are depicted in Fig. 5.9. With DFT different energies are obtained for the different orientations of the ferroelectric polarization with respect to the defect complexes, whereas in the defect-free case the different orientations of the ferroelectric polarization are equivalent and therefore have the same energy. The configurations where the relative orientation of the defect complex and the ferroelectric polarization is as perpendicular as possible are highest in energy (for CuNb −VO these are the configurations 1 to 1”’ in Fig. 5.8(a), for VO−CuNb −VO these are the configurations 0 to 0”’ in Fig. 5.8(b) and the configurations 1 to 1”’ in Fig. 5.8 (c), respectively). The orientation that is as parallel as possible has the lowest energy for both CuNb −VO and VO−CuNb −VO (for CuNb −VO these are the configurations 3 to 3”’ in Fig. 5.8(a), for VO−CuNb −VO these are the configurations 6 to 6(VII) in Fig. 5.8(b) and configuration 5 in Fig. 5.8 (c), respectively). The energy differences between the structures with lowest and second lowest energy obtained with DFT are 47 meV for the CuNb −VO defect (configurations 3 and 2 in Fig. 5.8(a)) and 24 meV for the VO−CuNb −VO defect (configurations 5 and 4 in Fig. 5.8(b)). Hence the 12 different equivalent orientations of the spontaneous polarization in orthorhombic KNO are no longer equivalent near these defect complexes. Some polarization directions remain almost degenerated in energy, but for all three defect complexes some polarization directions have a higher energy near the defect complex. In other words, the defects create locally “hard” directions for the ferroelectric polarization. Including spin-polarization in the DFT calculations has no significant influence on the relative energies of the structures. The SMP reproduces the DFT energies of the CuNb −VO (Fig. 5.9(d)) and the bent VO−CuNb −VO (Fig. 5.9(f)) defects both qualitatively and in most cases also quantitatively, if orthorhombic lattice parameters are used. It overestimates the energy of the straight VO−CuNb −VO (Fig. 5.9(e)) defect (configurations 0 and 6 in Fig. 5.8) by more than 1 eV relative to the bent one (not shown), but the relative energies of the configurations 0 and 6 are in better agreement with DFT (SMP: 0.23 eV; DFT: 0.14 eV). If the Cu concentration is reduced to 1.5625% ((Fig. 5.9(g)−(i)), the order of the energies is the same as for the higher concentration (6.25%), and also the numerical values are similar. Apparently the energy differences obtained for the Cu concentration of 6.25% are already approximately converged with respect to the defect 5.4 Ferroelectricity in KNO with Cu-defect complexes 63

0.3 (a) 0.3 (b) 0.3 (c)

0.2 0.2 0.2 (eV) (eV) (eV)

E 0.1 E 0.1 E 0.1

0.0 0.0 0.0

0.3 (d) 0.3 (e) 0.3 (f)

0.2 0.2 0.2 (eV) (eV) (eV)

E 0.1 E 0.1 E 0.1

0.0 0.0 0.0

0.3 (g) 0.3 (h) 0.3 (i)

0.2 0.2 0.2 (eV) (eV) (eV)

E 0.1 E 0.1 E 0.1

0.0 0.0 0.0

1 2 3 0 6 1 2 3 4 5 configuration configuration configuration

Figure 5.9: Relative energies of the configurations depicted in Fig. 5.8. (a), (d), (g): CuNb −VO; (b), (e), (h): straight VO−CuNb −VO; (c), (f), (i): bent VO−CuNb −VO. (a)-(c): DFT, 6.25 mole% Cu; (d)-(f): SMP, 6.25 mole% Cu; (g)-(i): SMP, 1.5625 mole% Cu. Small filled and large open circles in (a)-(c): spin-polarized and non-spin-polarized DFT calculations, respectively. Filled and open symbols in (d)-(i): SMP calculations using orthorhombic and cubic lattice parameters, respectively. The configurations are labeled as in Fig. 5.8. The lines only serve as a guide to the eye.

concentration. According to both the SMP and DFT the straight VO−CuNb −VO defect is higher in energy than the bent one, which means that both the CuNb −VO and the VO−CuNb −VO defect have an electric dipole moment in their ground state configuration. Using cubic lattice parameters, not only different energies are obtained, but also their order is different than that for orthorhombic lattice parameters. This finding will be discussed below together with the results for the ferroelectric polarization. The polarization of the supercell was calculated from the sum of the dipole moments of the individual unit cells with respect to the position of their central atom (Nb or CuNb, respectively). In the DFT calculations only the ionic polarization was calculated. Figure 5.10 shows the ferroelectric polarization obtained with DFT and the SMP. The ferroelectric polarization obtained with DFT (cf. Fig. 5.10(a)−(c)) and the SMP (cf. Fig. 5.10(d)−(i)) deviates from that in the defect-free case (cf. Fig. 5.10(j)−(l)) for all three defect types. The ferroelectric polarization obtained with DFT is qualitatively reproduced correctly by the SMP if orthorhombic lattice parameters are used (filled symbols and solid lines in Fig. 5.10(d)−(i)), with the exception of configuration 2 in Fig. 5.10(d), where DFT predicts a small negative Pz, whereas the SMP predicts a positive one, although its absolute value is reduced with respect to the defect-free case as well. Therefore the SMP gives the correct tendency but overestimates the effect of the defect complex on the ferroelectric polarization in this case. If cubic cell parameters are used (open symbols in Fig. 5.10(d)−(i)), in some cases (e.g. Pz in Fig. 5.10(d)) the ferroelectric polarization differes qualitatively from that obtained using orthorhombic cell parameters. In particular, in these cases the polarization direction imposed at the beginning of the calculation is unstable and relaxes into one which has a lower energy, which explains why the relative energies of these structures are not reproduced correctly (cf. Fig. 5.9). If the Cu concentration is reduced to 1.5625% (Fig. 5.10(g)−(i)), the influence of the defect complexes on the ferroelectric polarization is still pronounced, but in all cases the ferroelectric polarization is oriented approximately along the same direction as in the defect-free case. Since with DFT only the ionic polarization was calculated, its numerical value is smaller than that obtained with the SMP (cf. chapter 4). 64 5. Ab initio thermodynamics of defect complexes in KNO and KNN

(a) (b) (c) ) ) ) 2 20 2 20 2 20 10 10 10

C/cm 0 C/cm 0 C/cm 0 µ µ µ ( -10 ( -10 ( -10 P P P -20 -20 -20 (d) (e) (f) ) ) ) 2 20 2 20 2 20 10 10 10

C/cm 0 C/cm 0 C/cm 0 µ µ µ ( -10 ( -10 ( -10 P P P -20 -20 -20 (g) (h) (i) ) ) ) 2 20 2 20 2 20 10 10 10

C/cm 0 C/cm 0 C/cm 0 µ µ µ ( -10 ( -10 ( -10 P P P -20 -20 -20 (j) (k) (l) ) ) ) 2 20 2 20 2 20 10 10 10

C/cm 0 C/cm 0 C/cm 0 µ µ µ ( -10 ( -10 ( -10 P P P -20 -20 -20 1 2 3 0 6 1 2 3 4 5 configuration configuration configuration

Figure 5.10: The three cartesian components (red circles: Px; green squares: Py; blue triangles: Pz) of the ferroelectric polarization of the conﬁgurations depicted in Fig. 5.8. The labels (a)-(i) and the meaning of ﬁlled and open symbols are the same as in Fig. 5.9. The graphs (j)-(l) show the ferroelectric polarization obtained with the SMP in the defect-free case.

The SMP results using cubic cell parameters are not in sufficient agreement with the DFT results for orthorhombic latticed parameters. Since using orthorhombic cell parameters for the orthorhombic phase is more appropriate than using cubic ones anyway, in the following all SMP calculations were performed using orthorhombic cells. In this case the SMP reproduces the DFT energies and the ferroelectric polarization sufficiently well to be employed in the calculation of the energy barriers between the configurations studied above, which is the subject of the next section.

Switching between “soft” and “hard” polarization directions In this section upper limits of the energy barriers for switching between different “soft” directions are calculated.

To determine how the defect complexes influence the polarization switching and contribute to the macroscopic ”hardness“ or ”softness“ of the ferroelectric, in addition to the energy differences between the different orientations of the ferroelectric polarization with respect to the defect complexes also the energy barriers for switching between them are important. The SMP was used in the following to simulate such switching processes, since it apparently reproduces the potential energy landscape obtained with DFT to a large extent, and the corresponding DFT calculations would be computationally much more expensive. Switching the spontaneous polarization in a region which contains a defect complex was simulated using the ”nudged-elastic-band“ (NEB) method [Henk 00] as implemented in GULP. The minimum energy paths (MEP’s) for switching and thus the energy barriers were obtained by optimizing the atomic coordinates until the force norm was below 1.5 meV/A.˚ The force norm || f || is √ ∑ fi || f || = i , N where fi is the force acting on an individual particle coordinate and N is the number of degrees of freedom 5.4 Ferroelectricity in KNO with Cu-defect complexes 65 of the particles (3 per particle). The initial and final structure of each NEB path were optimized with respect to the particle coordinates, but not with respect to the lattice parameters, which were fixed to those obtained for undoped orthorhombic KNO. Therefore the initial and final structures correspond to local minima or saddle points on the energy landscape subject to the constraint of fixed lattice parameters. If the lattice parameters do not change along a NEB path, the initial and final structures remain local minima or saddle points under constraints, but this is no longer the case if the lattice parameters vary along a NEB path. For example, when the ferroelectric polarization is switched from the [011] to the [011¯] direction, the ferroelectric strain in the initial and final configuration is the same (along [011]), so that the lattice parameters do not change along this transition, whereas switching the polarization from the [011] to the [110] direction changes the ferroelectric strain from the [011] to the [110] direction as well, and hence the orthorhombic lattice parameters change between initial and final structure. The NEB calculations were performed as follows: The initial structures of the NEB images were obtained by linearly interpolating the initial and final structure of each NEB path. Also the lattice parameters, in the cases where they differ between initial and final structure, were interpolated linearly. During the NEB calculation the lattice parameters were fixed to their interpolated values, whereas the particle coordinates were optimized by the NEB algorithm (cf. Fig. 5.11 for a sketch).

Figure 5.11: Sketch of the NEB relaxation with the SMP. The blue boxes are the unit cells of the NEB images 0 to 4; the black sphere is an atom.

CuNb − VO defect Figure 5.12 shows the resulting ferroelectric polarization in the supercell (top) and the corresponding MEP’s (center and bottom) along a chain of configurations of the CuNb − VO defect (configurations 1 − 3 in Fig. 5.8(a)). The chain does not cover all possible transition paths, but with the included paths all configurations can be reached. E.g., the path 1 → 10 can be divided into the paths 1 → 2 and 2 → 10, the latter being equivalent to the path 2 → 1. The energy difference per Cu defect is the energy penalty for reorienting the polarization of a crystal region containing a Cu defect (because there is no energy penalty in the defect-free regions in this case). Assuming that the Cu defects are diluted, the total energy penalty could be obtained by multiplying this number with the number of Cu defects present in the whole crystal. 000 The ground state of the CuNb −VO defect is four-fold degenerated (states 3 − 3 in Fig. 5.8 (a)). In other words, there are four “soft” directions for the ferroelectric polarization. Switching the polarization from one ground state or “soft” direction to another (e.g. from 3 to 30) requires no energy (cf. Fig. 5.12). Then there are four ”slightly hard“ directions (points 2 − 2000 in Fig. 5.8 (a)). Switching from one of the ground states 3 − 3000 to one of these requires an energy of about 50 meV. Switching from one of the points 3 − 3000 to one of the points 1 − 1000 requires a larger amount of energy (about 150 meV). These are the four “hard” directions. In Fig. 5.12 (bottom) the energy per formula unit (i.e. per perovskite unit cell) along the transition paths is drawn. There are no energy barriers for switching between two ”nearest” configurations, for example between the configurations 1 and 2. An energy barrier of about 8 meV per formula unit has to be overcome to switch between two configurations that lie “further apart”, like the configurations 20 and 30. In the system with defects this energy barrier is strongly reduced, the more, the higher the defect concentration (cf. Fig 5.12 bottom), and is much smaller than the energy difference between the configurations. But whereas in the defect-free system the effective energy barrier of the path 20 → 30 vanishes because the 66 5. Ab initio thermodynamics of defect complexes in KNO and KNN

Figure 5.12: Energy and polarization along transition paths in KNO with a CuNb −VO defect. The configurations are labeled as in Fig. 5.8(a). The lines only serve as a guide to the eye. system can switch barrierless in two steps (20 → 10 → 30), for the system with defects this is not possible because the path 20 → 1 → 30 has a higher energy barrier than the path 20 → 30. Therefore in the system with defects an effective energy barrier (though small) for the path 20 → 30 remains. For 1.5625% Cu the energy difference between the configurations 20 and 30 is about 100 meV per Cu defect (the energy difference between different polarization directions of a supercell containing one Cu defect, cf Fig. 5.12 (center)) or about 2 meV per formula unit, for 6.25% Cu it is about 36 meV per Cu defect or again 2 meV per formula unit. This defect complex therefore creates “hard” directions for the ferroelectric polarization (it lifts the degeneracy of the energies of the different orientations of the ferroelectric polarization), but the energy barriers it introduces in addition are small in comparison. The energy differences per Cu defect (cf Fig. 5.12 (center)) are not all yet converged at the higher defect concentration of 6.25%, which corresponds to the strong deviation of the ferroelectric polarization from the one in the defect-free case (filled symbols in Fig. 5.12 (top)).

Straight VO−CuNb −VO defect Figure 5.13 shows the same as Fig. 5.12 for the straight VO−CuNb −VO defect (configurations 0 and 6 in Fig. 5.8(b)). The straight VO−CuNb −VO defect, which does not have an electric dipole moment, has a ground state that is eight-fold degenerated (points 6 − 6(VII) in Fig. 5.8 (b)), so there are eight soft directions. The spontaneous polarization can be switched barrierless between each of the configurations 6 to 6”’ and between each of the configurations 6(IV) to 6(VII) (here the polarization rotates around the axis through the defect complex; these paths are equivalent to the path 6(VII) → 6(VI), which is depicted in Fig. 5.13), whereas for switching between one of the configurations 6(VI) to 6(VII) and one of the configurations 6(IV) to 6(VII) (e.g., 6 → 6(VII)) a small energy barrier of about 4-8 meV per formula unit has to be overcome, depending on the Cu defect concentration. In the defect-free system this energy barrier is 8 meV, whereas it is reduced to 4 meV in the system with 6.25% Cu. Whereas in the defect-free system the effective barrier vanishes because the system can switch barrierless along the path 6 → 0 → 6(VII), the (concentration-dependent) 5.4 Ferroelectricity in KNO with Cu-defect complexes 67

Figure 5.13: The same as in Fig. 5.12 for the straight VO−CuNb −VO defect. The conﬁgurations are labeled as in Fig. 5.8(b). barrier remains in the defective system because the path 6 → 0 → 6(VII) also has an energy barrier of about 4-14 meV per formula unit for defect concentrations of 1.5625-6.25% Cu, respectively. Therefore in addition to lifting the degeneracy of the energies of the different directions of the ferroelectric polarization this defect complex also introduces energy barriers for switching that are effectively absent (can be avoided) in the defect-free system, but these are very small compared to the energy differences between “soft” and “hard” directions. Then there are four “hard” directions (point 0 to point 0000), switching along which requires rather much energy (about 270 meV).

Bent VO−CuNb −VO defect Figure 5.14 shows the same as Fig. 5.12 and Fig. 5.13 for the bent VO−CuNb − VO defect (conﬁgurations 1 − 5 in Fig. 5.8(c)). The bent VO−CuNb −VO defect is by about 0.2 eV lower in energy than the straight one and therefore more likely. It has only one ground state or soft direction (point 5 in Fig. 5.8 (c)). There are two “slightly hard” switching directions (points 4 and 40), switching to which requires an energy between 24 meV (not counting the energy barrier) and 63 meV (counting the energy barrier) for a Cu concentration of 6.25%. For lower Cu concentrations this energy difference rather increases. All other switching directions require at least about 140 meV (cf. Fig. 5.14) and are therefore “hard”. Like for the two other defect complexes, for the bent VO−CuNb −VO defect the energy barriers that are present in the defect-free system are lowered in the system with defects, the more, the higher the defect concentration. Since in the defect-free system any of the transition paths considered here that has an energy barrier can be replaced by a barrierless one (for example, path 1 → 2 can be replaced by the barrierless path 1 → 100 → 2), in the defective system this is not possible. This defect therefore effectively introduces energy barriers for polarization switching in addition to lifting the degeneracy of the different polarization directions. However, like in the other cases the energy barriers are small compared to the energy differences between “soft” and “hard” directions. Because this defect creates more “hard“ switching directions than the other two, it is likely to be the strongest obstacle to domain wall motion. 68 5. Ab initio thermodynamics of defect complexes in KNO and KNN

Figure 5.14: The same as in Figs. 5.12 and 5.13 for the bent VO−CuNb −VO defect. The conﬁgurations are labeled as in Fig. 5.8(c).

5.5 Discussion

Formation of defect complexes in Cu-doped KNO and KNN

In KNN and KNO, like in PbTiO3, defect complexes of dopants and vacancies are lower in energy than the isolated defects. Cu on an alkali site in KNN can trap an alkali vacancy, Cu on a Nb site can trap one or two oxygen vacancies to form (CuNb −VO) or (VO − CuNb −VO). The (VO − CuNb −VO) defect has a lower energy than the (CuNb −VO) defect and an isolated VO, but (CuNb −VO) and an isolated VO have a much higher conﬁgurational entropy because of the many possible sites for the isolated VO, so that at high temperatures the concentration of (CuNb −VO) may exceed that of (VO − CuNb −VO).

How to determine the defect concentrations There are two limiting cases for the concentrations of the defects: in the first case (low mobility of the defects and/or fast cooling rates) the equilibrium defect distributions at the sintering temperature persist after cooling down to room temperature. In the second case (high mobility of the defects and/or slow cooling rates) the defect distributions are in thermal equilibrium at room temperature. In thermal equilibrium the defect concentrations follow a Boltzmann-like distribution, which is a function of the defect formation energies and the temperature (Eq. (5.1)). If the defect distribution at the sintering temperature persists down to room temperature, the Boltzmann-like distribution at this high temperature should be used also for the distribution at room temperature. Another approach to find the most abundant defect configuration is to use the binding energies of the defect complexes. This approach is only justified if enough vacancies exist and if they are sufficiently mobile to diffuse to the Cu substitutionals.

CuNb−VO and VO − CuNb−VO defect The theoretical equilibrium concentrations of oxygen vacancies in KNO and KNN at 1300 K obtained without a band-gap correction are of a similar order of magnitude as the 2.5 mol% Cu doping, whereas at room temperature they are many orders of magnitude lower. The diffusion time obtained in section 5.3 for oxygen vacancies towards a Cu substitutional is small (less than a ns at 1300 K and still less than a second at 300 K), so the oxygen vacancies are indeed sufﬁciently mobile to diffuse to the CuNb substitutionals. Therefore the concentrations of the CuNb defect complexes should reach their equilibrium values already during cooling down to room temperature after sintering. Using the formation energies, in thermal equilibrium at 300 K and at atmospheric pressure the most abundant Cu 5.5 Discussion 69

defects are CuNb and CuNb −VO. It is justiﬁed to use the binding energies to obtain defect concentration ratios at the sintering temperature (high VO mobility, high VO concentration) but not at room temperature (still high VO mobility, but low VO concentration). When calculating the defect formation energies, only the straight VO − CuNb −VO defect complex in cubic KNbO3 (with the vacancies on opposite sides of the Cu atom) was considered, but according to this work the total energy of the bent VO −CuNb −VO in ferroelectric KNbO3 differs by about 0.2 eV only (cf. section 5.4), so that the binding energies can be expected to differ by a similar amount. This energy difference is small in comparison to the binding energies themselves, so that within the given accuracy the concentrations of the bent and the straight VO − CuNb −VO defects are similar (at least when compared to that of the other defects) and the discussion above holds for both conﬁgurations.

Comparison with experiment Experimentally for KNO two types of Cu defects were obtained with a concentration ratio of ≈ 1 : 3 by ﬁtting spin-Hamiltonian parameters to EPR spectra [Erun 10]. These two Cu defects were ascribed to CuNb − VO and VO − CuNb −VO, whereas according to this work the most abundant defects in thermal equilibrium at room temperature are CuNb and CuNb −VO. Therefore the experimental and the theoretical results are only compatible assuming that the equilibrium defect distribution at the sintering temperature at least partly persists after cooling the material down to room temperature.

CuK−VK defect The concentration of the (CuK −VK) defect is lower than those of the CuNb defects. Only for KNN it is comparable to those of the CuNb defects, but only if the uncorrected LDA band-gap is used. For K vacancies the diffusion time is about two hours at 1300 K (cf. section 5.3), at 300 K they are practically immobile. The much lower mobility of the K vacancies is another reason why the CuK −VK defect is less likely to form than the CuNb −VO and the VO−CuNb −VO defect. Therefore the (CuK −VK) defect was not investigated further. Nevertheless its formation should not be excluded.

Oxygen vacancy concentration and diffusion In [Erha 08] O and Ba vacancy concentrations in BaTiO3 were obtained as a function of the oxygen partial pressure and the temperature using DFT-LDA as well. At 1300 K and 1 bar oxygen partial pressure, from their Fig. 5 the VO concentration was estimated to [VO] ≈ −7 −6 −6 −5 10 ···10 mol%, in a reducing atmosphere at 1 µbar oxygen partial pressure [VO] ≈ 10 ···10 mol% −10 was estimated, whereas in the present work [VO] ≈ 10 mol% was obtained at 1300 K and 0.2 bar oxygen −7 partial pressure and [VO] ≈ 10 mol% at 1300 K and 1 µbar oxygen partial pressure. Taking into account that in this work the concentrations were determined only within an accuracy of a few orders of magnitude, the equilibrium oxygen vacancy concentration in KNbO3 obtained here for 1300 K is comparable to that in BaTiO3 obtained in [Erha 08]. Using DFT-LDA an oxygen vacancy migration energy of 0.89 eV was found in [Erha 07a] for BaTiO3 and of 0.87 eV in [Park 03] for PbTiO3. In [Sing 10] two activation energies (0.69–0.75 eV and 1.20-1.56 eV) were found in dielectric relaxation and electrical conductivity measurements of KNbO3. Both activation energies are apparently related to oxygen vacancies and may correspond to energy barriers for oxygen vacancy migration. The energy barrier for oxygen vacancy migration of 0.59 eV found in this work for KNbO3 is close to the smaller of the two experimental values and is smaller than those of PbTiO3 and BaTiO3.

“Soft” and “hard” polarization directions in KNO with defect complexes

The 12 h111i orientations of the spontaneous polarization in orthorhombic KNbO3 are no longer equivalent if there is a CuNb −VO or VO−CuNb −VO defect. Some polarization directions remain almost degenerated in energy, but for all three defect types studied (CuNb − VO and the straight and bent VO−CuNb − VO) close to the defect complex some orientations of the spontaneous polarization are higher in energy than others. Therefore all three defect types can impede the motion of ferroelectric domain walls. The bent VO−CuNb −VO complex has a lower energy (by 0.2 eV) than the straight one. Therefore both the CuNb −VO and the VO−CuNb −VO defects have electric dipole moments in their ground states, which can interact electrostatically with the spontaneous polarization and with external electric ﬁelds. 70 5. Ab initio thermodynamics of defect complexes in KNO and KNN

Compared to PbTiO3 the energy differences for different orientations of the defect dipoles are smaller. The energy difference between parallel and antiparallel orientation of the CuTi −VO defect and the spontaneous polarization is about 1.4 eV [Erha 07b] in tetragonal PbTiO3, in orthorhombic KNbO3 the energy difference between (partly) parallel and (partly) antiparallel orientation of (VO−)CuNb −VO is only about 0.2 eV. This difference is not surprising: In PbTiO3 the ferroelectric instability is much larger than in KNbO3 (53 meV/f.u. in tetragonal PbTiO3 [Umen 06] as compared to 11.4 meV/f.u. in orthorhombic KNbO3). Both in PbTiO3 [Erha 07b] and in KNN the defect dipoles align parallel to the spontaneous polarization. In the DFT and SMP calulations presented here no depolarization field is taken into account. The energy of a point-like dipole (in this case a defect dipole) in an electric field (in this case the depolarization field) is lowest if the dipole is aligned parallel to the electric field (in this case antiparallel to the spontaneous polarization). This energy contribution depends on the strength of the depolarization field, which depends on the boundary conditions (domain walls, grain boundaries, charged surfaces, electrodes,...) and was not taken into account here. That the defect dipoles align parallel to the spontaneous polarization must therefore be caused by a different interaction. This interaction could be an elastic one or a short-range electrostatic one, or a combination of both.

Switching between “soft” and “hard” polarization directions In order to enable ferroelectric ”hardening“ or the ”large-strain“ effect, the defect dipoles must create at least one ”hard“ direction for polarization switching (switching to a ”hard“ direction requires more energy than switching to a “soft” direction). This is the case for all three defect complexes studied here. But the three defect complexes differ with respect to the number of ”hard“ and ”soft“ directions. For the CuNb −VO defect there are four ”soft“, four ”slightly hard“, and four “hard” directions; for the straight VO−CuNb −VO defect there are eight “soft” and four “hard” directions; and for the bent VO−CuNb − VO defect there is one “soft”, two “slightly hard”, and nine “hard“ directions. Because this defect creates more “hard“ switching directions than the other two, it is likely to be the strongest obstacle to domain wall motion.

The energy needed to reorient the ferroelectric polarization around a defect complex depends on the initial and final orientations of the polarization with respect to the defect complex. For a single domain wall the pinning effect of the defect complexes therefore depends on the orientation of the applied electric field. If the single crystal or polycrystalline ceramic contains many ferroelectric domains with random orientations, a fraction of the domain walls is always oriented such that an applied electric field moves it along a “hard” direction. Therefore the “hardening“ or the “large-strain” effect macroscopically has an isotropic behavior. The energy differences between the different orientations of the ferroelectric polarization with respect to the defect complexes are independent of the switching mechanism. However, the energy barriers between the different orientations of the ferroelectric polarization with respect to the defect complexes were obtained for homogeneous switching and may differ from those for the motion of a domain wall. They should therefore be regarded as an upper limit to the energy barriers for domain wall motion. Since the energy barriers found for homogeneous switching are already rather small, the main contribution of the defect complexes to ferroelectric hardening will be the creation of “hard” directions. Whereas the energy barriers for switching the ferroelectric polarization for fixed defect configurations are well below one eV, switching the defect dipoles themselves apparently requires a much higher energy. In section 5.3 an energy barrier of 2 eV for reorienting the CuNb −VO defect dipole was found. At room temperature this process therefore practically does not happen spontaneously, but a driving force such as an electric field or additional thermal energy are needed. 6 Conclusion 72 6. Conclusion

In this work the microscopic effects of doping on the crystal properties of a ferroelectric were studied in detail. A typical lead-free ferroelectric, KNN, and a typical dopant, Cu, were chosen for this study. Quantum-mechanical and classical computational simulation methods were combined to enable both precise and fast modeling. In order to describe the thermodynamics of defect incorporation, a precise and transferable method like DFT is needed and was applied. Switching the ferroelectric polarization is asso- ciated with only slight changes in crystal structure and could therefore be simulated using computationally inexpensive but less transferable classical atomistic potentials. With this combined approach the doping process was studied from the materials processing level, where the dopants are incorporated, to the operat- ing level, where they contribute to the macroscopic piezoelectric response of the ferroelectric.

On the materials processing level the inﬂuence of the chemical synthesis conditions on the substitution site of Cu and Fe was determined. The most important factor that determines the substitution site of Cu and Fe in KNN is oxygen. The chemical potential of oxygen, which is controlled experimentally by the oxygen partial pressure and the temperature, determines whether Cu substitutes on alkali or Nb sites in KNN. In a reducing atmosphere, Cu substitutes alkali atoms, at atmospheric pressure at room temperature and in a more strongly oxidizing atmosphere Cu substitutes on Nb sites. Fe substitutes more likely on Nb sites than Cu, so that a more strongly reducing atmosphere is necessary to drive Fe to substitute on an alkali site than for Cu.

If Cu substitutes on an alkali site, it leads to a morphotropic phase boundary, similar to Li doping. Like in Li-doped KNN, the MPB in CuK-doped KNbO3 lies at a doping level of a few mol %. Cu substitutionals on Nb sites do not cause a MPB.

Cu substitutionals on Nb sites are a trap for oxygen vacancies. Depending on the concentration of oxygen vacancies and the temperature, CuNb can trap one or two oxygen vacancies. For three possible defect complexes in CuNb-doped KNbO3 it was investigated how the defects can contribute to ferroelectric “hardening” and/or the “reversible large strain” effect. For three defect complexes, CuNb −VO and two conﬁgurations, a straight and a bent one, of VO − CuNb − VO it was investigated whether they impede domain wall motion. All three defect complexes can impede polarization switching, e.g., by domain wall motion, and therefore they can lead to ferroelectric “hardening“ and/or to the “reversible large strain” effect. The bent VO − CuNb −VO complex is slightly lower in energy than the straight one and is expected to impede domain wall motion more strongly than the other two defects.

This work focused on Cu doping, but part of the results is representative and holds for other dopants as well. As an example, the morphotropic phase boundary in Li-doped KNN can be expected to have similar properties as the one in CuA-doped KNN which was found in this work. Any B-site dopant that is lower-valent than Nb could attract an oxygen vacancy, form a defect dipole and create “hard“ directions for switching the polarization similar to Cu. The exact shape of the potential energy landscape and the energy barriers, however, will differ for different dopants.

The finite temperature behavior of the ferroelectric with defects was not investigated directly. The atomistic potential for Cu-doped KNbO3 optimized in this work could be used to study the morphotropic phase boundary in (Cu,K)NbO3 with molecular dynamics or Monte-Carlo simulations at finite temperatures, and to simulate domain wall motion through Cu-doped KNbO3 also at finite temperatures. In this way the ferroelectric ”hardening” and/or the ”reversible large-strain effect” caused by the defect complexes could be observed directly. Bibliography

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First of all my thanks go to Professor Dr. Christian Elsasser,¨ for this position, this dissertation topic, ideas and goals, a DFT program to work with, very good lectures, motivation, optimism, constant support, pa- tience, and for giving me a lot of freedom and enough time to do a thorough job. All shortcomings of this dissertation are my own responsibility. Thanks to the project partners, Dr Jer´ omeˆ Acker, Professor Rudiger¨ Eichel, Professor Michael Hoff- mann, and Dr Hans Kungl from the Karlsruhe Institute of Technology (KIT), and Dr Ebru Erunal¨ from the University of Freiburg, for initializing the project and pushing it forward, for your ideas, the good atmosphere in all our project meetings, your support, and the joint publications. Thanks to my ofﬁce mates Martin Reese, Dr Wolfgang Korner,¨ Dr Janina Zimmermann, Dr Francesco Colonna, and Davide Di Stefano for your good company in and outside the ofﬁce, and to Dr Wolfgang Korner¨ for help with getting started with ab-initio thermodynamics, critical reading of Chapter 4 and good discussions about DFT and other topics. Many many thanks to the physical modelling group: Dr Jan- Michael Albina, Dr Nedko Drebov, Dr Adham Hashibon, Dr Pierre Hirel, Eva Marie Kalivoda, Dr Wolf- gang Korner,¨ Dr Pavel Marton, Dr Matous Mrovec, Martin Reese, and Dr Pim Schravendijk, for accepting me so friendly in the group, for always being ready to help and to discuss, the very good atmosphere, and for everything else! Thanks to Franz Doll, Jan Hulsberg,¨ and Dr Matous Mrovec for help with computers and software, and to Dr Monika Gall and Davide Di Stefano for helpful comments about the thesis (and your good company). Thanks also to all the colleagues of the GF5, GF1.6, the former GF4, and the IWM. I enjoyed very much the friendly atmosphere, the good working conditions, and the opportunity to try out different methods and to work both on academic and industrial projects. Financial funding by the German Research Foundation (DFG, grants EL 155/21-1,2) is gratefully ac- knowledged. The calculations for this work were performed on computers provided by the Fraunhofer IWM, the Fraunhofer Ernst Mach Institute (EMI), and the Steinbruch Centre for Computing (SCC) at the Karlsruhe Institute of Technology (KIT), Germany. The DFT calculations were performed with the MBPP code developed by Prof. Dr Bernd Meyer, the atomistic ones with the GULP code developed by Prof. Ju- lian Gale. Crystal structures were visualized with VESTA [Momm 08] and XCrySDen [Koka 99], the other graphics were made with gnuplot, the matplotlib library for python, gimp, and inkscape. This thesis was written on a Linux (openSUSE) desktop PC, the calculations were performed on Linux computer clusters. The layout of this dissertation is based on a LATEX template by Robert Dahlke and Sigmund Stintzing. 82 Curriculum Vitae

Sabine Korbel¨

30.04.1981 Born in Munster,¨ Germany 2000 – 2006 Physics studies at the Westfalische¨ Wilhelms-Universitat,¨ Munster,¨ Germany 2006 Diploma thesis “Pump-Probe-Spektren von BCS-Supraleitern“ (“pump-probe spectra of BCS-type superconductors”) at the Institut fur¨ Festkorpertheorie¨ (Institute of Condensed Matter Theory) 2006 – 2007 Gap year at the university hospital of Munster,¨ Germany 2007 – 2012 PhD student at the Fraunhofer Institute for Mechanics of Materials IWM, Freiburg, Germany