Microwave Studies on Fese Superconductors

Microwave Studies on FeSe Superconductors

Meng Li

B.Sc., Shandong University, 2014

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

The Faculty of Graduate and Postdoctoral Studies

(Physics)

THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2016 c Meng Li 2016 Abstract

In this thesis, the microwave electrodynamic properties of stoichiometric FeSe are measured by a cavity perturbation technique based on a 940 MHz loop-gap resonator. Measurements surprisingly show, for the ﬁrst time, that cˆ-axis conductivity in stoichiometric FeSe superconductors exhibits a broad peak in its temperature-dependence, a phenomenon which was only observed for in-plane electrodynamics in the cuprates. This result implies that the charge transfer between FeSe-planes is enhanced by the development of long transport quasiparticle lifetimes below Tc.

ii Preface

The work presented here is conducted in the Superconductivity lab at Uni- versity of British Columbia, Vancouver campus. The apparatus used for the cavity perturbation measurements described in this thesis were previously designed and constructed by Walter Hardy and Saeid Kamal (loop-gap resonator). The pure FeSe samples were grown by Shun Chi during the course of his Postdoctoral program. I was involved in later growing the Co-doped (0.3%) samples. For chapter 4, the work is original and unpublished. All the data collection and data analyses, presented in chapter 4, were carried out by me. Pinder Dosanjh played a major role in maintaining experimental equipment and assisting in data collection.

iii Table of Contents

Abstract ...... ii

Preface ...... iii

Table of Contents ...... iv

List of Tables ...... vi

List of Figures ...... vii

Acknowledgements ...... xi

1 Introduction ...... 1 1.1 Conventional superconductivity ...... 1 1.2 Unconventional superconductivity ...... 2 1.2.1 Cuprate superconductors ...... 3 1.2.2 Heavy-fermion superconductors ...... 6 1.2.3 Organic superconductor ...... 6 1.2.4 Iron-based superconductors ...... 7

2 Microwave electrodynamics of superconductors ...... 13 2.1 Penetration depth ...... 13 2.2 Generalized two-ﬂuid model ...... 13 2.3 Microwave surface impedance ...... 15 2.3.1 General overview ...... 15 2.3.2 Surface impedance of a superconductor ...... 16

3 Experimental techniques: microwave spectroscopy .... 18 3.1 Cavity perturbation technique ...... 18 3.2 Loop-gap resonator ...... 21 3.3 Swept-frequency cavity transmission measurement ...... 23

iv Table of Contents

4 In- and out-of-plane microwave electrodynamics of FeSe . 25 4.1 Sample preparation ...... 25 4.2 Cleave method ...... 26 4.3 Finite size consideration ...... 27 4.4 Experimental results and analysis ...... 30 4.4.1 Penetration depth ...... 30 4.4.2 Surface resistance ...... 30 4.4.3 Microwave conductivity ...... 30 4.4.4 Superﬂuid density ...... 33

5 Discussion and comparison with YBa2Cu3O6+δ ...... 35

Bibliography ...... 45

v List of Tables

1.1 The allowed singlet pairing states for a square CuO2 lattice. Table adapted from Refs [1,2]...... 5 1.2 The transition temperature of iron-based superconductors, cited from [3]...... 8

5.1 Two-gap ﬁt parameters for ab-plane andc ˆ axis...... 36 5.2 Two-band extended s-wave model ﬁt parameters [63]. . . . . 38

vi List of Figures

1.1 Cooper pair formation via electron-phonon interaction. Pic- ture taken from [4]...... 2 1.2 The superconducting gap in k-space (momentum-space) for (a) isotropic (s-wave) and (b) dx2−y2 (d-wave) superconductors. The cylindrical Fermi surfaces are shown as bold circles. The hatched region denotes the filled electrons states. For a conventional s-wave superconductor, the energy gap 2∆ has the same sign in all directions. However, for a d-wave superconductor, the sign and magnitude of the gap is a function of direction in the kx and ky-plane. Figure taken from Ref [2,5].3 1.3 The schematic phase diagram of the high Tc cuprate superconductors as a function of temperature and hole-doping. Figure taken from [6]...... 4 1.4 The superfluid density (inverse square of the superconducting penetration depth) as a function of temperature [7]. The linear T dependence at low temperature is a consequence of the linear density of states N(E) (inset) of a dx2−y2 superconductor with a cylindrical Fermi surface...... 6 1.5 Crystallographic structure of the iron-based superconductors. Iron pnictides have FeAs layers, identical to the FeSe layered structure, which are the central ingredients for the superconductivity. Figure taken from [8]...... 8 1.6 The electronic phase diagrams for two different iron-based superconductors as a function of chemical substitution. . . . . 10 1.7 (a) FeAs lattice indicating As above and below the Fe plane. Dashed green and solid blue squares indicate 1- and 2-Fe unit cells, respectively. (b) Schematic 2D Fermi surface in the 1-Fe BZ whose boundaries are indicated by a green dashed square. (c) Fermi sheets in the folded BZ whose boundaries are now shown by a solid blue square. Figure taken from ref. [9]. . . . 11

vii List of Figures

1.8 FeSe lattice structure: simplest tetragonal structure. Figure taken from [10]...... 12

2.1 Schematic view of σ1(ω), σ2(ω, T ) for a superconductor [11]. 16

3.1 Schematic diagram of cavity perturbation technique. Within an empty cavity, the resonant frequency is determined by the dimensions of the cavity. When a superconducting sample is introduced into the cavity, superﬂuid screening currents cause the sample to screen external magnetic ﬁeld and thereby shift the cavity resonant frequency by changing the cavity volume. Figure provided courtesy of Richard Harris [12]...... 19 3.2 Digital photograph of the loop-gap resonator, with a side view schematic diagram of the resonator assembly. Dimensions displayed are in millimetres. Figure provided by courtesy Jake Bobowski [2] based on original material by Saeid Kamal [13]...... 22 3.3 Top: Schematic view of the loop-gap assembly with a sample loaded. There are coupling loops on either side of the resonator. Bottom: Equivalent circuit of loop-gap resonator. Figure provided by courtesy Jake Bobowski [2]...... 23 3.4 The electronics setup for the swept-frequency cavity transmission measurement. Figure provided by Jordan Baglo [6]. . 24

4.1 A photograph of a pure FeSe sample measured in experiments. 26 4.2 Schematic view of sample measurement geometry. Figure provided by Jordan Baglo [6]...... 27 4.3 Cleaved samples to measure c-axis penetration depth. S3 sample is cleaved from S2 which is cleaved from S1. They have the same broad area, with diﬀerent thickness 166 µm, 87 µm, 32 µm for S1, S2 and S3 respectively. Bottom graphs show their respective cross section...... 28 4.4 a) The penetration depth measured at 1 GHz of a thin crystal before (S1) and after cleaving it into thinner ones (S2 and S3). Inset shows scaled penetration depth and the scaled constants for S1, S2 and S3 are 1.00, 1.40 and 2.15 respectively. b) The penetration depth ∆λ(T ) for S1, S2 and S3, is plotted over the full temperature range measured...... 29

viii List of Figures

4.5 a) The temperature dependence of the surface resistance for S1, S2 and S3 below Tc. Inset shows scaled surface resistance and the scaling constants for S1, S2 and S3 are 1.00, 1.26 and 2.10 respectively. b) The surface resistance Rs(T ) plotted in a log-linear scale over the full temperature range measured. . 31 4.6 The real part of the microwave conductivity σ1ab(T ) ( ab- plane, red triangles)) and σ1c(T ) (ˆc-axis, blue squares) were extracted from measurements of ∆λ and ∆Rs. Surprisingly, σ1c(T ) also has a broad peak at low temperature...... 32 4.7 The superﬂuid fraction λ2(0)/λ2(T ) for ab-plane (black squares) andc ˆ-axis (λc0 = 5µm, red circles; λc0 = 3µm, blue triangles) versus reduced temperature T/Tc. In both the ab-plane and cˆ-axis direction, the temperature dependence of the super- ﬂuid fraction varies almost linearly with temperature below Tc...... 33

5.1 Equation 5.1 was used to fit the ab-plane andc ˆ-axis superfluid density. The green dots denote the experimental data and the blue curve represents the two-gap model fit. a) Two- gap model fit for ab-plane superfluid density: x = 17%, ∆S(0) = 0.23 meV , ∆L(0) = 1.06 meV . b) Two-gap model fit forc ˆ-axis superfluid density: x = 26%, ∆S(0) = 0.20 meV , ∆L(0) = 0.94 meV ...... 37 5.2 a) A two-band extended s-wave model is fitted to the superfluid density, and the inset shows a polar plot of two gaps (∆1 and ∆2) at zero temperature, for various values of the DOS parameter [63]. b) The Temperature dependence of the rms gap amplitudes on two bands, and the overall gap minimum. 38 5.3 The superfluid fraction in all principal crystallographic directions of YBa2Cu3O6.95 [5]. Thec ˆ-axis superfluid density is qualitatively different from the behaviour in either direction in ab-plane...... 39 5.4 Surface resistance along the three crystal directions of YBa2Cu3O6.99. Thec ˆ-axis surface resistance is different from that of the ab- plane [14]...... 41 5.5 Extracted microwave conductivity σ1c for two YBa2Cu3O6.99 samples along thec ˆ-direction [14]...... 42

ix List of Figures

5.6 The ab-plane microwave conductivity of a high purity YBa2Cu3O6.99 crystal exhibits a broad, frequency dependent peak caused by the development of long-lived quasiparticles in the superconducting state [11]...... 43 5.7 The anisotropy of the microwave conductivity of YBa2Cu3O6.95 is illustrated by plottingc ˆ axis conductivity (open squares) measured at 18 GHz [15] along with the ab-plane conductivity (ﬁlled squares) taken at 1 GHz [15]...... 44

x Acknowledgements

First of all, I would like to express my sincere gratitude to my research super- visor Doug Bonn, for his patience, excellent guidance, immense knowledge, and oﬀering me the opportunity to be a graduate student in his group. Without Doug’s guidance, this thesis would not have been completed or written. He has never failed to inspire me with his deep insight into the physics problems. A very special thanks goes out to Pinder Dosanjh. Without Pinder’s guidance, I won’t be able to run the cavity-perturbation and bolometric spectroscopy apparatus. Pinder always provided me with direction, technical support whenever I ran into a trouble spot in my research. More im- portantly, Pinder taught me how to be critical thinking and prudent before and while proceeding any task. His patience, kindness and broad technical knowledge truly made a diﬀerence in the course of this thesis. I would like to thank David Broun for the wonderful collaboration at Simon Fraser University. The joy, enthusiasm and inspiration he has for his research was contagious and motivational for me. I would also like to thank Shun Chi, not only for the excellent crystals he grew, but also for being such a good friend who is always willing to help and give his best suggestions. I’m also so thankful to my fellow students James Day, Damien Quentin, Robert Delaney and Gelareh Farahi. Lastly, I must express my gratitude to my parents and brother for pro- viding me with unfailing support and continuous encouragement throughout my years of study.

xi Chapter 1

Introduction

Superconductivity, discovered in 1911 by Onnes, [16] is one of the few macro- scopic quantum phenomena in nature, and is characterized by the complete absence of DC electrical resistance and expulsion of magnetic ﬁelds under particular temperature and magnetic regimes. The development of BCS theory, the discovery of high temperature cuprate superconductors, and iron-based superconductors have marked several milestones in the history of superconductivity. Tremendous eﬀorts have been devoted to discover new superconductors and unveil the secret of superconducting mechanisms. However, there are still a lot of fundamental questions about superconductivity remaining unanswered and under investigation.

1.1 Conventional superconductivity

According to the pairing mechanism, superconductors usually can be divided into two branches: conventional and unconventional superconductors. The superconductivity occurring in conventional superconductors can be well explained by BCS theory proposed by Bardeen, Cooper, and Schrieffer in 1957 [17], for which they received the Nobel Prize in 1972. The BCS theory builds on the assumption that superconductivity emerges when the attractive Cooper pair interaction dominates over the Coulomb repulsion. A conventional Cooper pair is a weak electron-electron bound pair possessing zero total spin, zero angular momentum (s-wave) and zero total momentum, and the attraction is mediated by the electron-phonon interaction. More specifically, the electrons are bound together by their interaction with the vibrations of the lattice (phonon): one electron in a pair trav- eling through the crystal lattice will distort the lattice by attracting the nuclei towards it, creating a temporary region of excess positive charge in the vicinity. This region in turn attracts another electron at some distance - the positively charged nuclei thus mediate an attraction between the neg- atively charged electrons and overcome the Coulomb repulsion. Figure 1.1 demonstrates this simplified process. Cooper pairs behave more like bosons, and can condense into the lowest energy level.

1 1.2. Unconventional superconductivity

Figure 1.1: Cooper pair formation via electron-phonon interaction. Picture taken from [4].

Most elemental superconductors, such as Nb, Pb and Sn, are conventional. The highest Tc among the family of conventional superconductors is 39 K found in magnesium diboride (MgB2)[18].

1.2 Unconventional superconductivity

Unconventional superconductors refer to all the superconductors that can- not be understood in the context of BCS theory driven by electron-phonon interaction, for example the high-Tc and heavy fermion superconductors. A fundamental difference between conventional and unconventional superconductors is the symmetry of the superconducting gap function, ∆(k), which is defined as the energy difference between the ground state energy of the superconductor and the energy of the lowest quasiparticle excitation. For conventional superconductors, the zero total angular momentum of electron pairs (cooper pairs) is caused by isotropic attractive forces between two electrons in all spatial directions, which produces an isotropic superconducting gap over the entire Fermi surface as shown in Figure 1.2 (a). In contrast, for the unconventional superconducting state of the cuprates, the electron pairing state has finite angular momentum associated with the electron correlations caused by the large Coulomb repulsion between electrons on a Cu site in the CuO2 planes. The pairing symmetry for high temperature cuprates is antisymmetric spin singlet state [1], of which four possible distinct singlet pairing states are listed in Table 5.2, based on the group-theoretic calculations for a square CuO2 plane [19].

2 1.2. Unconventional superconductivity

Figure 1.2: The superconducting gap in k-space (momentum-space) for (a) isotropic (s-wave) and (b) dx2−y2 (d-wave) superconductors. The cylindrical Fermi surfaces are shown as bold circles. The hatched region denotes the ﬁlled electrons states. For a conventional s-wave superconductor, the energy gap 2∆ has the same sign in all directions. However, for a d-wave superconductor, the sign and magnitude of the gap is a function of direction in the kx and ky-plane. Figure taken from Ref [2,5].

1.2.1 Cuprate superconductors In 1986, Bednorz and M¨ullerdiscovered superconductivity in a ceramic oxide La2−xBaxCuO4+δ with a record-breaking transition temperature Tc near 29 K [20], soon followed by the discovery of high temperature superconductivity in YBa2Cu3O6+x with Tc = 93 K, above the boiling point of liquid nitrogen [21], and later reached the highest value of 138 K in Tl-doped HgBa2Ca2Cu3O8+δ [22]. Many other cuprate superconductors also were discovered and intensively studied, such as Tl2Ba2CuO6 and Bi2Sr2CaCu2O8, and they form the large family of high temperature superconductors. All the materials in this family share a common feature: weakly-coupled stacks of two-dimensional copper oxide planes. It is widely believed that the superconductivity occurs within these CuO2 planes. A lot of cuprate superconductors have very high critical temperatures which can’t be explained by the BCS theory. A new theory is needed to explain the unexpected high Tc superconductivity. Although these cuprate superconductors are the most widely studied family of superconductors during the past three decades, many of the fundamental questions, such as the pairing mechanism, are still challenging condensed matter physicists. The generic phase diagram for hole-doped cuprate materials is presented in Figure 1.3. The diﬀerent phases can be accessed by changing the tem-

3 1.2. Unconventional superconductivity

Figure 1.3: The schematic phase diagram of the high Tc cuprate superconductors as a function of temperature and hole-doping. Figure taken from [6].

4 1.2. Unconventional superconductivity

Group- Order Wave function theoretic parameter Nodes name notation basis function accidental line + s (s-wave) A1g const. nodes possible

− 2 2 s (g − wave) A2g xy(x − y ) line 2 2 line dx2−y2 (d-wave) B1g x − y

dxy B2g xy line

Table 1.1: The allowed singlet pairing states for a square CuO2 lattice. Table adapted from Refs [1,2]. perature T and the charge carrier doping p. Below the N´eeltemperature TN , the undoped (p = 0) material is an antiferromagnetic Mott-insulator. As the hole doping increases, this antiferromagnetic phase vanishes rapidly and superconductivity emerges with critical temperature Tc roughly varying parabolically with hole doping. An empirical parabolic relationship between the superconducting transition temperature Tc and hole doping level p has been formulated [23, 24] as follows:

T 1 − c = 82.6(p − 0.16)2 (1.1) Tc,max

The maximum Tc,max is obtained at a doping level of p = 0.16 (optimal doping); lower and higher doping levels are known as underdoped and overdoped regimes, respectively. Another important property of cuprate superconductors is the symmetry of the pairing states. The NMR measurements of the Knight shift showed that spin susceptibility declines rapidly below Tc, indicating that the superconducting states are spin-singlet [25, 26]. In 1993, Hardy et al. [7] observed the linear temperature dependence of London penetration depth in YBa2Cu3O6.95 (Figure 1.4), a proof for line nodes in ∆(k). However, these measurements could not distinguish between the three allowed symmetry basis function shown in Table 5.2, all of which have, or could have, line nodes. Subsequently, the phase-sensitive scanning SQUID measurements yielded convincing evidence for a predominant d-wave pairing symmetry in a wide variety of cuprate systems [1]. The corresponding superconducting gap in d-wave superconductors is shown in Figure 1.2 (b).

5 1.2. Unconventional superconductivity

Figure 1.4: The superﬂuid density (inverse square of the superconducting penetration depth) as a function of temperature [7]. The linear T dependence at low temperature is a consequence of the linear density of states N(E) (inset) of a dx2−y2 superconductor with a cylindrical Fermi surface.

1.2.2 Heavy-fermion superconductors Heavy-fermion superconductors are another class of unconventional superconductors, containing rare earth or actinide elements and possessing enor- mous eﬀective mass of charge carriers. The unexpected discovery of superconductivity in CeCu2Si2 [27] opened up the door to the heavy fermion non-BCS superconductivity. In heavy-fermion systems, it is believed that the pairing of quasiparticles is not mediated by an electron-phonon interaction, but by the spin ﬂuctuations of a nearby antiferromagnetic phase [28, 29]. The pairing symmetry is a key piece of information to understand how this works. Recently, a nodal d-wave character of superconducting pairing state has been demonstrated to exist in CeCoIn5 via high-resolution STM techniques [30].

1.2.3 Organic superconductor An organic superconductor refers to a synthetic organic compound exhibit- ing superconductivity below Tc. The ﬁrst organic superconductor (TMTSF)2PF6 was synthesized by Klaus Bechgaard in 1979, with Tc = 1.1 K at an external

6 1.2. Unconventional superconductivity pressure of 6.5 kbar [31]. This discovery inspired the creation of a large family of related organic compounds, known as Bechgaard salts, which exhibit a variety of unique properties. The structure of Bechgaard salts is very different from those of metallic superconductors; a lot of effort has been poured into investigating the superconducting mechanism. NMR Knight shift measurements [32] and specific heat measurements [33] on organic superconductor (TMTSF)2CIO4 provided strong evidence of spin-singlet nodal superconductivity. Nuclear spin relaxation rate measurements showed the superconducting pairing is mediated by interstack antiferromagnetic spin fluctuations [34].

1.2.4 Iron-based superconductors

In 2008, the discovery of superconductivity in LaFeAsO1−xFx with a transition temperature Tc ≈ 26 K [35] generated tremendous interest in the superconductivity community, not only because of its unusual properties (such as the coexistence of superconductivity and magnetism), but also pro- viding an exciting opportunity to finally resolve the mechanism of high Tc superconductivity. To date, the newly discovered iron-based superconductors (FeSCs) have roughly five classes of structures, with examples being FeTe (the 11 compounds), LiFeAs (the 111 compounds), LaFeAsO (the 1111 compounds), BaFe2As2 (the 122 compounds), and Sr2VO3FeAs (the 21311 compounds). These compounds all share a common layered structure, based on a planar layer of iron atoms joined by tetrahedrally oriented chalcogen (S, Se, Te) or pnictogen (P, As) anions arranged in a stacked sequence separated by alkali, alkaline-earth or rare-earth and oxygen/fluorine ‘blocking layers’. The superconducting transition temperature for some iron-based superconductors is summarized in Table 1.2.

7 1.2. Unconventional superconductivity

Figure 1.5: Crystallographic structure of the iron-based superconductors. Iron pnictides have FeAs layers, identical to the FeSe layered structure, which are the central ingredients for the superconductivity. Figure taken from [8].

Tc (K) Non- Oxypnictide Tc (K) oxypnictide SmFeAsO∼0.85 55 Sr0.5Sm0.5FeAsF 56 GdFeAsO0.85 53.5 Ba0.6K0.4Fe0.2As0.2 38 NdFeAsO0.89F0.11 52 FeSe < 27 ErFeAsO1−y 45 Ca0.6Na0.4Fe2As2 26 La0.5Y0.5FeAsO0.6 43.1 NaFeAs 9-25 SmFeAs0.9F0.1 43 CaFe0.9cO0.1AsF 22 CeFeAs0.84F0.16 41 (b) Non-oxypnictide materials. LaO0.9F0.2FeAs 28.5

LaO0.89F0.11FeAs 26

BaFe1.8Co0.2As2 25.3 (a) Oxypnicide materials.

Table 1.2: The transition temperature of iron-based superconductors, cited from [3]. 8 1.2. Unconventional superconductivity

Similar to high-Tc cuprates, the properties of FeSCs change dramati- cally with chemical modification. Most of FeSC parent compounds exhibit an antiferromagnetic spin-density wave (SDW) ground state, and superconductivity is induced upon charge-carrier doping or by applying an external pressure. In these materials, the superconductivity occurs within FeAs, FeP, FeSe or FeTe crystallographic planes, analogous to the CuO2 planes in cuprates. A compilation of experimental phase diagrams is shown in Figure 1.6a for the Ba-based 122 system [8], widely thought to share the main traits of all FeSCs. In BaFe2As2, it has been shown that the systematic substitution of either the alkaline-earth (Ba), transition-metal (Fe) or pnictogen (As) atom with a different element almost universally shares the same phase diagram presented in Figure 1.6a. The superconducting phase (SC) is approximately centred at the critical doping where the antiferromagnetic (AFM) order vanishes. The coexistence of AFM and SC phases in a small doping range is believed to be an intrinsic property of the typical FeSc phase diagram, whereas, in fluorine-doped 1111-systems such as LaFeAsO1−xFx, the superconducting phase is completely separated from AFM, as shown in Figure 1.6b[36]. All the Fe-based compounds have similar electronic band structure consisting of hole pockets centred at Γ (0, 0) and electron pockets centred at M(±π, ±π) or (±π, 0) in the Brillouin zone (BZ). Figure 1.7 shows the square FeAs lattice and the corresponding Fermi surface (folded and un- folded) for a stoichiometric parent compound. The electron Fermi surface pocket is strongly connected to the hole Fermi sheets by a reciprocal lattice vector, known as Fermi surface nesting. This strong nested structure leads to the SDW instabilities [40] and suggests the possibility of s±-wave pairing symmetry mediated by antiferromagnetic spin fluctuations with superconducting gaps having opposite sign between the electron and hole pockets. Although most experimental results favor s±-wave for the majority of materials, a universal consensus regarding the pairing symmetry of the iron-based superconductors has not been reached. A particularly interesting compound among the family of Fe-based superconductors is FeSe which has the simplest structure, comprised of edge- sharing FeSe4-tetrahedra stacked layer by layer, as shown in Figure 1.8. Furthermore, FeSe compounds have very low transition temperature Tc ∼ 8 K, but its Tc can be enhanced to 37 K by applying high pressure [41]. FeSe compounds have advantageous features for measuring and elucidating its superconducting properties: (1) A vapour transport technique allows us to grow clean stoichiometric FeSe crystals with low disorder; (2) FeSe has the

9 1.2. Unconventional superconductivity

(a) The phase diagram of the BaFe2As2 system, shown for K [37], Co [38] and P[39] substitutions. The dotted line represents tetragonal (T) to orthorhombic (O) structural phase transition obsserved for Co doped samples.

(b) The electronic phase diagram of the LaFeAsO1−xFx system obtained by µSR measurements [36].

Figure 1.6: The electronic phase diagrams for two diﬀerent iron-based superconductors as a function of chemical substitution. 10 1.2. Unconventional superconductivity

Figure 1.7: (a) FeAs lattice indicating As above and below the Fe plane. Dashed green and solid blue squares indicate 1- and 2-Fe unit cells, respectively. (b) Schematic 2D Fermi surface in the 1-Fe BZ whose boundaries are indicated by a green dashed square. (c) Fermi sheets in the folded BZ whose boundaries are now shown by a solid blue square. Figure taken from ref. [9]. simplest Fermi surface consisting of a hole pocket and an electron pocket; (3) The sample is superconducting without chemical substitution, which causes disorder as well as tuning the doping. Therefore, this provides a good opportunity to better understand the superconducting mechanism of this family of unconventional superconductors.

11 1.2. Unconventional superconductivity

Figure 1.8: FeSe lattice structure: simplest tetragonal structure. Figure taken from [10].

12 Chapter 2

Microwave electrodynamics of superconductors

2.1 Penetration depth

A superconductor is characterized by the complete absence of DC resistivity and by the Meissner eﬀect, at temperatures below Tc. These properties were ﬁrst formulated in a phenomenological picture by the London brothers [42]. They proposed that the electromagnetic response of superconducting carriers with number density ns is governed by the London equation:

∂J~ n e2 s = s E~ (2.1) ∂t m∗

∗ where J~s, e and m are respectively the supercurrent density, charge and eﬀective mass of the superconducting electrons. Combining Equation 2.1 with Maxwell’s equations, we can get a frequency independent skin depth, the London penetration depth

m∗ 1/2 λL = 2 (2.2) µ0nse

λL for a typical superconductor is less than a micron, implying that the magnetic ﬁeld is almost completely excluded from a bulk sample.

2.2 Generalized two-ﬂuid model

The generalized two-fluid model builds on the London model, by including the effects of the normal fluid, and serves as a phenomenological descrip- tion of the high frequency electrodynamics of superconductors.1 The model

1Strictly speaking, the two ﬂuid model is most easily justiﬁed in the clean limit where the quasiparticle spectrum has its spectral weight well below the superconducting gap frequency.

13 2.2. Generalized two-fluid model postulates that the electromagnetic response of a superconductor can be divided into two components: a normal fluid (associated with quasiparticles excited out of the condensate) and a dissipationless superfluid (associated with Cooper pairs in the condensate), with electron densities nn and ns respectively. In an electromagnetic field, normal electrons are assumed to behave like electrons in the normal state and therefore dissipate energy by scattering. The total electron density n is temperature independent and equal to the sum of the normal fluid density nn and the superfluid density ns at each temperature:

n = nn(T ) + ns(T ) (2.3)

At T > Tc, all the electrons are in the normal state n = nn(T ). When the temperature decreases below Tc, quasiparticles start to condense into the superconducting state, and ns begins to build up. Ultimately, in the clean limit where the energy scale of the superconducting gap is larger than the quasiparticle scattering rate, all of the electrons are supposed to condense into the superconducting state at T = 0 such that n = ns(T = 0). The total conductivity is the superposition of the normal-fluid and superfluid conductivityσ ˜(ω, T ) =σ ñ(ω, T ) +σ ˜s(ω, T ). At low temperature below Tc, the microwave properties are dominated by the superfluid. For now we only consider the superfluid conductivity, which can be introduced with a Drude model:

2 2 2 nse τs nse ωτs σ˜s(ω) = σ1s − iσ2s = ∗ 2 2 − i ∗ 2 2 (2.4) ms 1 + ω τs ms 1 + ω τs

∗ where n/ms is the ratio of superconducting electron density over effective mass and τ is the current relaxation time of the electrons. Since the superfluid is dissipationless, τs will go to infinity.

n e2 τ n e2 ωτ 2 σ˜ (ω)| = lim s s − i lim s s (2.5) s τs→∞ ∗ 2 2 ∗ 2 2 τs→∞ ms 1 + ω τs τs→∞ ms 1 + ω τs So, the imaginary conductivity component becomes:

2 nse σ2s(ω) = ∗ (2.6) msω

14 2.3. Microwave surface impedance

2 As for σ1s, it can been obtained by the Kramers-Kronig formula, which yields Z ∞ 0 2 ωσ1s(ω ) 0 σ2s(ω) = − 02 2 dω (2.7) π 0 ω − ω Combining with the above equation, we get the solution 2 πnse σ1s(ω) = ∗ δ(ω) (2.8) ms which is a delta function. Then, the total conductivity for a superconductor can be written as 2 2 πns(T )e ns(T )e σ˜(ω, T ) = ∗ δ(ω) − i ∗ +σ ˜n(ω, T ) (2.9) ms msω A schematic view of real and imaginary component of complex conductivity varying with the temperature and frequency has been shown in 2.1, taken from the reference [11].

2.3 Microwave surface impedance

2.3.1 General overview The complex surface impedance of a metal or superconductor is a measure of the absorption and screening characteristics of the material in the presence of an electromagnetic field. This quantity is directly accessible through microwave experiments and is defined to be the ratio of the tangential electric and magnetic field at the surface of the sample (e.g. Ex/Hy) in an electromagnetic field. In the case of local electrodynamics ( where the penetration depth is much larger than the coherence length, λ ξ ), the microwave surface impedance Zs is related to microwave conductivity σ via s iµ0w Z˜s(w, T ) ≡ Rs(w, T ) + iXs(w, T ) = (2.10) σ1(w, T ) − iσ2(w, T ) where Rs(w, T ) is the surface resistance and Xs(w, T ) is the surface reactance. The surface resistance Rs(w, T ) is directly related to the power ab- sorbed by the sample in the electromagnetic field and the surface reactance

2 The Kramers-Kronig formula relates the real and imaginary parts ofσ ˜ = σ1 − 0 0 2 R ∞ ω σ2(ω ) 0 iσ2 in terms of of the paired expression: σ1(ω) = + π P 0 ω02−ω2 dω , σ2(ω) = 0 0 2 R ∞ ω σ1(ω ) 0 + π P 0 ω02−ω2 dω , where P denotes principal value integration.

15 2.3. Microwave surface impedance

Figure 2.1: Schematic view of σ1(ω), σ2(ω, T ) for a superconductor [11].

Xs(w, T ) is a measure of the screening of magnetic ﬁeld from the interior of the sample. Equation (2.10) shows the direct link between the microwave conductivity and measurable surface impedance.

2.3.2 Surface impedance of a superconductor

For a superconductor, within the superconducting state well below Tc, normal electrons condense into the superconducting state, and one expects the imaginary part of the conductivity to be dominated by the superconducting condensate, with σ2 σ1. The surface impedance can be approximately

16 2.3. Microwave surface impedance expressed as: 1 R (ω, T ) ' µ2ω2λ3 (T )σ (ω) (2.11) s 2 0 L 1n

Xs(ω, T ) ' µ0ωλL(T ) (2.12)

These approximations are simple but quite accurate for T bellow Tc when the measurement frequency is smaller than the quasiparticle scattering rate 1 (ω < τ ). As we can see from (2.11) and (2.12), Xs is primarily related to the 3 penetration depth λL, while Rs is proportional to σ1n - but with a large λL dependence on the penetration depth. We can clearly observe that measurements of surface reactance alone provide the penetration depth directly, and the conductivity σ1n can be extracted from the measured surface resistance if the penetration depth is known. The measurements of surface impedance of bulk samples can be accom- plished by the cavity perturbation method or broadband bolometric spectroscopy. The experiments presented in this thesis focus on the cavity perturbation measurements.

17 Chapter 3

Experimental techniques: microwave spectroscopy

3.1 Cavity perturbation technique

The cavity perturbation technique can be used to measure both the change in the magnetic penetration depth and the surface resistance of a superconducting sample. Basically, the sample is inserted into the centre of a resonant cavity, and the small perturbation caused by the sample will provide information about the penetration depth and surface resistance. A schematic 3 4 diagram of a high-Q cylindrical resonator operating in the TE011 mode is shown in Figure 3.1. In this mode, an electric field node and a magnetic field antinode5 will exist at the centre of the cavity. The red lines inside the cavity of Figure 3.1 represent the microwave magnetic fields which are fairly uniform in the middle of cavity. The microwaves generated by a synthesized sweeper are coupled into (TX), and out of (RX), the cavity through small holes drilled in the side walls of the cavity. When temperature is varied, changes in the resonance frequency f of the cavity as well as the resonance width wB can be related to the change of magnetic penetration depth and surface resistance. The resonance frequency f is determined by the cavity dimensions, and the width of the resonance wB is related to cavity quality factor Q: peak energy stored f Q = 2π = (3.1) energy dissipated per cycle wB

3This is only used for illustration. All experiments done in this thesis were carried out on a loop-gap resonator. 4Most real resonant systems have multiple (often an infinite spectrum of) resonant modes and the framework to be presented can be adapted to this general case . However, in the cavity perturbation measurements presented in this thesis, only the single lowest- frequency resonant mode was considered, and higher modes are well separated and quite justifiably neglected. 5In the loop-gap resonator, the electric field only exists in the gap and the magnetic field only exists within the half-loop cavity.

18 3.1. Cavity perturbation technique

Without any sample in the cavity, the energy dissipated per cycle primarily comes from the dissipation in the cavity walls where the currents ﬂow back and forth. In order to minimize the dissipation and optimize the Q of the resonator, the inner surfaces of the copper cavity are coated with a P b0.95Sn0.05 superconducting alloy with a transition temperature Tc ≈ 7K.

Figure 3.1: Schematic diagram of cavity perturbation technique. Within an empty cavity, the resonant frequency is determined by the dimensions of the cavity. When a superconducting sample is introduced into the cavity, superﬂuid screening currents cause the sample to screen external magnetic ﬁeld and thereby shift the cavity resonant frequency by changing the cavity volume. Figure provided courtesy of Richard Harris [12].

Now consider a thin superconducting platelet sample that is introduced into a region of the cavity resonator where the magnetic field is fairly uniform. The screening effects cause the sample to shield microwave magnetic field from its interior within a penetration depth λ. The screening of magnetic field from the sample decreases the effective volume of the cavity, which then shifts the resonance to a higher frequency f + δf. Moreover, the dissipation in the sample results in the reduction of the stored energy within the resonator, causing a broadening of the resonance. A temperature-dependent surface impedance means that a change of sample temperature will result in a shift of resonance frequency and change

19 3.1. Cavity perturbation technique in cavity Q. A complex frequency notation is introduced to formulate the cavity perturbation problem [43]: ω ω˜ = ω + i 0 (3.2) 0 0 2Q where ω0 = 2πf0 is the resonant angular frequency. The relative shifts of the complex frequency due to the temperature change are related to the changes in surface impedance via :

∆w ˜0 ∆f0 i 1 ΓAs ΓAs = + ∆( ) = i (∆Rs + i∆Xs) = i ∆Z˜s (3.3) w0 f0 2 Q f0 f0 where Γ is the resonator constant and As is the surface area of sample. A detailed derivation of the cavity perturbation equation 3.3 can be found in [6]. A key assumption of the technique is that the perturbation by the sample is weak enough that the field distribution does not change significantly elsewhere in the cavity(i.e. no non-perturbative effect). This must remain reliable when we consider the perturbations from temperature changes of the sample. For a thin platelet superconductor (c a, b) in a low-demagnetization geometry with magnetic field H ⊥ cˆ, the change in the resonance frequency and Q of the cavity upon insertion of the sample are given by [44, 45]: δf V tanh(˜κc/2) 0 = s 1 − < (3.4a) f0 2Vc κc/˜ 2 1 V tanh((˜κc/2) δ( ) = s = (3.4b) Q Vc κc/˜ 2 where V is the sample volume, V is the effective volume of the resonator, √ s c κ˜ = iωµ0σ is the complex propagation constant of the electromagnetic field inside the sample, and < and = represent the real and imaginary parts of the quantities in curly brackets. For a normal metal with skin depth δ,κ ˜ = (1 + i)/δ and for a superconductor with penetration depth λ well below Tc,κ ˜ ≈ 1/λ. In the thick limit c 2λ for a superconductor at low temperature, the hyperbolic tanh term approaches 1 and Equation 3.4a is simplified to: δf V 2λ 0 ' s 1 − (3.5) f0 2Vc c In our experiments, the sample temperature can be set independently of the cavity temperature by a sapphire hot finger as shown in Figure 3.1. There- fore this allows tracking changes in δf due to changes in λ as the temperature is changed. The resonant frequency shift ∆f from a base (reference)

20 3.2. Loop-gap resonator

temperature T0 is deﬁned as:

f0As ∆(δf0) ≡ δf(T ) − δf(T0) ' − ∆λ(T ) (3.6) 2Vc where ∆λ(T ) ≡ λ(T ) − λ(T0). The above analysis, however, neglects the thermal expansion effects and the c-axis penetration depth λc. The sample dimensions change slightly with the change of the sample’s temperature, which causes the change of sample thickness c, and sample volume Vs. However, the thermal expansion is a small effect for temperatures below 70 K [5]. In this thesis, all experiments are carried out on thin samples at temperatures below 20 K and therefore the effects of thermal expansion are ignored. The procedure to separate thec ˆ-axis penetration depth’s contribution will be discussed in detail in Chapter 4.

3.2 Loop-gap resonator

Microwave surface impedance was measured through cavity perturbation of a superconducting loop-gap resonator (shown in Figure 3.2 and 3.3) which operates at a resonant frequency ∼ 940 MHz. The cavity is comprised of a copper half-loop bolted tightly on one side to a copper support block, with a sapphire plate held firmly in the gap between the other side of the loop and the block. In order to get a high-Q factor in this design, the superconducting joint (shown in Figure 3.3) between the loop-gap and the support base should be lossless. To achieve this, screws are used to tighten the loop-gap and the support base which both have been previously electroplated with P b0.95Sn0.05 (Tc ≈ 7 K ). The joint is superconducting while the apparatus is operating at base temperature (∼1.2 K). The quality factor (Q-factor) has been found to slowly decrease with air exposure over time, due to degradation of the plating and superconducting joints, from initial value (> 106) to a current value of 3 × 105 at 1.2 K which is still adequate for the penetration depth measurements, but somewhat low for accurate surface resistance measurements. In such an experiment, the sample is mounted on a thin sapphire plate and held fixed in the centre of the resonator, with the magnetic field applied parallel to the ab-plane of the sample. This orientation produces currents primarily running across the broad face of the slab in the ab-plane, with a small contribution from currents running down the side of the slab in the cˆ-axis direction.

21 3.2. Loop-gap resonator

Figure 3.2: Digital photograph of the loop-gap resonator, with a side view schematic diagram of the resonator assembly. Dimensions displayed are in millimetres. Figure provided by courtesy Jake Bobowski [2] based on original material by Saeid Kamal [13].

22 3.3. Swept-frequency cavity transmission measurement

Figure 3.3: Top: Schematic view of the loop-gap assembly with a sample loaded. There are coupling loops on either side of the resonator. Bottom: Equivalent circuit of loop-gap resonator. Figure provided by courtesy Jake Bobowski [2].

3.3 Swept-frequency cavity transmission measurement

This implementation of the microwave cavity perturbation method involves the measurement of transmission of the microwave cavity as a function of frequency ω. Since the measurements are not sensitive to phase, only the magnitude of total time-averaged power reaching a “square-law” diode detector is measured [6]:

P0 Pout(ω) = (3.7) 1 + 4Q2( ω−ω0 )2 ω0 where Q, ω0 and P0 are quality factor, resonant frequency, and maximum power, respectively. The hardware for the swept-frequency transmission measurement is shown in Figure 3.4. An HP 83620A microwave synthesized sweeper (with a frequency range of 10 MHz to 20 GHz) is connected to the input coupling loop

23 3.3. Swept-frequency cavity transmission measurement

Figure 3.4: The electronics setup for the swept-frequency cavity transmission measurement. Figure provided by Jordan Baglo [6]. of the loop-gap resonator via flexible coaxial cable; the output loop is coupled to an HP 423A crystal detector whose near-DC output is proportional to the incident microwave power. The output low-frequency signal from the detector is typically amplified by a factor of 103 through an amplifier and then connected to an input channel of a Tektronix TDS 520B digital oscilloscope. The experiment control and data acquisition is performed by a com- puter running the graphical programming environment LabVIEW (National Instruments). A LabVIEW “Virtual Instrument” (VI) is employed to auto- matically control the sample stage temperature, synthesized sweeper, digital oscilloscope, and perform the Lorentzian fits to obtain f0 and Q. Given a set of desired measurement temperatures and tolerances, this VI will handle the entire process of collecting and fitting resonance curves at the programmed temperatures, including tracking appropriate centre frequency, span, and amplitude of the synthesizer sweeps as they change with temperature. The output data file contains thermometry data, together with the Lorentizian fit parameters, for each temperature setpoint [6].

24 Chapter 4

In- and out-of-plane microwave electrodynamics of FeSe

Since the discovery of superconductivity in LaFeAsO1−xFx with a transition temperature Tc ≈ 26 K [35], great effort has been devoted to finding several other superconducting Fe-based materials [10, 46, 47] and investigating their pairing mechanisms and symmetry of the superconducting order parameter. In the family of Fe-based superconductors, the FeSe compound possesses several advantageous features for measuring and interpreting its physical properties. First, vapour transport technique yields clean stoichiometric FeSe crystals with low disorder. Furthermore, FeSe superconductor has the simplest crystal structure and also the simplest Fermi surface, comprised of a hole pocket and an electron pocket. Finally, it is superconducting without the added disorder of chemical substitution. Layered superconductors such as the iron-based superconductors and the cuprates display anisotropic transport properties and therefore need to be studied in all principal crystallographic directions. In the high-Tc cuprates (such as YBa2Cu3O7−δ), it has been established experimentally [14] and theoretically [48] that the interplane (ˆc-axis) transport properties are quite different from those observed in the ab-plane and thatc ˆ-axis transport can be interpreted in terms of a complex hopping process. In contrast, we will show here that for FeSe superconductors the transport properties in thec ˆ-axis direction are qualitatively similar to those in the ab-plane. In both directions, quasiparticles develop long transport lifetimes below Tc.

4.1 Sample preparation

Single crystals of β-FeSe were grown by the vapor transport technique ﬁrst reported by Chareev et al. [49] and Bohmer et al. [50]. The iron and

25 4.2. Cleave method

Figure 4.1: A photograph of a pure FeSe sample measured in experiments. selenium powder was ﬁrst mixed in the ratio Fe : Se = 1.1 : 1 and placed into a quartz ampoule. AlCl3 and KCl with the ratio AlCl3 : KCl = 0.6 : 0.4, as the vapor transport media, were sealed together with iron and selenium under vacuum. The sealed ampoule was then placed in a two-zone furnace with the high temperature end 686 K and the low temperature end 554 K. Millimeter-size FeSe single crystals were obtained after 25 days growth in this temperature gradient. The critical temperature of the crystals is Tc=8.8 K and the transition width is ∆T = 0.5 K. A photograph of a measured FeSe sample is shown in Figure 4.1. The sample was mounted on a thin sapphire plate.

4.2 Cleave method

Microwave surface impedance was measured through cavity perturbation of a superconducting loop-gap resonator operating at ∼ 1 GHz. In such an experiment, the sample is mounted on a thin sapphire plate and held ﬁxed in the centre of the resonator, with the magnetic ﬁeld applied parallel to the ab-plane of the sample. Figure 4.2 displays a schematic view of the sample measurement geometry. This orientation produces currents primarily running across the broad face of the slab in the ab-plane, with a small fraction of currents running down the side of the slab in thec ˆ-axis direction.

26 4.3. Finite size consideration

H || b

c b a

λa

λc c a

Figure 4.2: Schematic view of sample measurement geometry. Figure provided by Jordan Baglo [6].

The current contribution due toc ˆ-axis is then decreased by cleaving the slab into a set of thinner ones, while the ab-plane’s contribution almost stays the same because the cleaving process only slightly changes the dimensions of the ab-plane. Then, cleaved crystals are remeasured with H~ lying in the same orientation and with negligible change of the sample’s position in the cavity. This technique is particularly reliable because it has no signiﬁcant change in demagnetizing factors, and no change in the distribution of ab- plane currents. A simple cartoon shown in Fig.4.3 illustrates the cleaving sequence.

4.3 Finite size consideration

In highly anisotropic superconductors such as FeSe, ﬁnite-size eﬀects need to be taken into account in interpreting microwave data. In the limit of local electrodynamics, the complex screening length δ˜ is related to other electrodynamic variables in the following way:

Zs(T ) = Rs(T ) + iXs(T ) = iωµ0δ˜ (4.1)

1 σ(T ) = σ1(T ) − iσ2(T ) = (4.2) ˜2 iωµ0δ

27 4.4. Experimental results and analysis

Figure 4.3: Cleaved samples to measure c-axis penetration depth. S3 sample is cleaved from S2 which is cleaved from S1. They have the same broad area, with diﬀerent thickness 166 µm, 87 µm, 32 µm for S1, S2 and S3 respectively. Bottom graphs show their respective cross section.

In the superconducting state, the complex screening length is approximately equal to penetration depth δ˜ ≈ λ. With the magnetic ﬁeld lying in the ˆb direction of a thin crystal which has width a ina ˆ-axis direction and thickness c inc ˆ-axis direction,a ˆ-axis andc ˆ-axis penetration depth can be extracted by numerically solving the following equations [51]: ˜ ac 4 X 1 tanh αn tanh βn δeff = 2 2 + (4.3) a + c π n αn βn odd n<0

1  !2 2 c nπδ˜c αn = 1 +  (4.4) 2δ˜a a

1  !2 2 a nπδ˜a βn = 1 +  (4.5) 2δ˜c c ˜ ˜ ˜ where δeff , δa and δc correspond to the eﬀective,a ˆ-axis andc ˆ-axis complex screening length.

28 4.4. Experimental results and analysis

Figure 4.4: a) The penetration depth measured at 1 GHz of a thin crystal before (S1) and after cleaving it into thinner ones (S2 and S3). Inset shows scaled penetration depth and the scaled constants for S1, S2 and S3 are 1.00, 1.40 and 2.15 respectively. b) The penetration depth ∆λ(T ) for S1, S2 and S3, is plotted over the full temperature range measured.

29 4.4. Experimental results and analysis

4.4 Experimental results and analysis

4.4.1 Penetration depth Figure 4.4a shows the temperature dependence of the measured penetration depth ∆λ(T ) = λ(T ) − λ(1.2 K) for three pure FeSe crystals: S1, S2 and S3, with thickness t1= 166 µm, t2= 87 µm and t3= 32 µm respectively. Both thin slab S2 and S3 were cleaved from S1 and all of them have the same ab-plane dimensions: 1.040 mm×1.306 mm. As seen in Fig.4.4a, the considerable change in ∆λ after cleaving S1 into thinner samples (S2, S3) is attributed to the decrease inc ˆ-axis contribution and this indicates the penetration depth in thec ˆ direction is quite large. The inset displays the scaled penetration depth, obtained from the measured ∆λ for S1, S2 and S3 multiplied by 1.00, 1.40 and 2.15 respectively. These three curves agree ex- tremely well over the whole temperature range (T/Tc < 0.7), which implies that the penetration depth within the ab-plane and along thec ˆ-axis have nearly identical functional form but with diﬀering magnitude. The penetration depth across the full temperature range measured is shown in Figure 4.4b.

4.4.2 Surface resistance

Figure 4.5a shows the measured values of surface resistance ∆Rs(T ) = Rs(T ) − Rs(1.2 K) in the superconducting state for S1, S2 and S3. Similar to the in-plane surface resistance observed in YBa2Cu3O7−δ superconductors, the broad peak seen in all three samples is caused by a very large peak in the temperature dependence of microwave conductivity σ1(T ). This increase in conductivity below Tc has been attributed to a rapid increase in quasiparticle lifetime in the superconducting state [52]. The inset shows the scaled surface resistances, which were obtained in the manner described above but with diﬀerent scaled numbers: 1.00, 1.26 and 2.10 corresponding to S1, S2 and S3. The scaled ∆Rs agree well at low temperature, and from the comparison between the measured and scaled ∆Rs, we can still conclude that the temperature dependence of the surface resistance in thec ˆ-axis direction is quite similar to that in ab-plane. So, for both directions, a large conductivity peak would be expected to develop below Tc.

4.4.3 Microwave conductivity In order to directly compare the electrical transport properties in the ab- plane with those along thec ˆ-axis, the conductivity σ1ab(T ) and σ1c(T ) shown

30 4.4. Experimental results and analysis

Figure 4.5: a) The temperature dependence of the surface resistance for S1, S2 and S3 below Tc. Inset shows scaled surface resistance and the scaling constants for S1, S2 and S3 are 1.00, 1.26 and 2.10 respectively. b) The surface resistance Rs(T ) plotted in a log-linear scale over the full temperature range measured.

31 4.4. Experimental results and analysis

0.3 ab

c -1

30 -1 )

) 0.2 m m (

( c

0.1

0 0.0

0 2 4 6 8 10

T(K)

Figure 4.6: The real part of the microwave conductivity σ1ab(T ) ( ab- plane, red triangles)) and σ1c(T ) (ˆc-axis, blue squares) were extracted from measurements of ∆λ and ∆Rs. Surprisingly, σ1c(T ) also has a broad peak at low temperature. in Figure 4.6 were extracted from the data sets in Fig. 4.4a and Fig. 4.5a.A remarkable feature of this ﬁgure is that thec ˆ-axis conductivity also exhibits a large peak at low temperature (near 0.4Tc), which is only observed in the planar direction of the cuprate superconductors: YBa2Cu3O7−δ [14, 53], Bi2Sr2CaCu2O8+δ [54, 55], HgBa2Ca2Cu3O8+δ [53], La2−xSrxCuO4 [56, 57], Tl2Ba2CaCu2O8 and Tl2Ba2CuO6 [58]. This feature strongly suggests that, in the superconducting state, the inelastic scattering of quasiparticles is suppressed for charge transport in thec ˆ-direction, as well as within the ab- plane [59, 60] and that the charge transfer is enhanced by the development of long transport lifetime below Tc. However, thec ˆ-axis conductivity overall is two orders of magnitude smaller than that in the ab-plane.

32 4.4. Experimental results and analysis

1.0

ab-plane

c-axis( (0)= 5 m)

0.8

c-axis( (0)= 3 m)

0.6 (T) 2

0.4 (0)/ 2

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0

T/T

Figure 4.7: The superﬂuid fraction λ2(0)/λ2(T ) for ab-plane (black squares) andc ˆ-axis (λc0 = 5µm, red circles; λc0 = 3µm, blue triangles) versus reduced temperature T/Tc. In both the ab-plane andc ˆ-axis direction, the temperature dependence of the superﬂuid fraction varies almost linearly with temperature below Tc.

4.4.4 Superfluid density In figure 4.7, the superfluid fraction λ2(0)/λ2(T ) for ab-plane andc ˆ-axis were extracted from the measured penetration depth combined with mag- netization measurements of λab(0) = 445 nm [61], and the inferred value of λc(0) = 3000 nm obtained from Homes’ law [62] based on the measured −2 −1 conductivity σ1c(0) = 3 × 10 (µΩ m) . A possible limit is set by the red dotted line where λc(0) = 5000 nm. Both ab-plane andc ˆ-axis superfluid density exhibit a linear temperature dependence over a fairly wide temperature range (T/Tc < 0.8). However, further measurements carried out on the same sample at much lower temperature [63] observed that the superfluid density saturates at very low temperature, which indicates the presence of a non-zero gap minimum. This finding of a small gap is consistent with a recent thermal conductivity study on similarly grown UBC samples [64],

33 4.4. Experimental results and analysis and penetration depth measurements [65] on other similar FeSe crystals. Other measurements, however, observed the existence of gap nodes in FeSe [66, 66, 67]. One possible reason for this disagreement may come from whether those measurements can diﬀerentiate between a gap node and a very small gap. The quality of samples, to some extent, can also aﬀect the measurement results.

34 Chapter 5

Discussion and comparison with YBa2Cu3O6+δ

In this thesis, the 1 GHz loop-gap resonator was employed to study the electrodynamics of FeSe crystals. The measurements show that the microwave properties of the superconducting state in our FeSe crystals, such as penetration depth and surface resistance, have similar behaviors for both in-plane andc ˆ-axis direction. In this chapter I will compare these results to those found for YBa2Cu3O6+δ, where there is a large difference between in-plane and out-of-plane electrodynamics. In Figure 4.7, the superfluid density displays an approximately linear temperature dependence at low temperature, due to many thermal excita- tions (quasiparticles) out of the groundstate, like YBa2Cu3O6+δ [5]. How- ever, much lower temperature data reveal strong evidence for the existence of a very small minimum gap [63], rather than the nodal gap in the d-wave pairing state of YBa2Cu3O6+δ [7]. Therefore, a small gap is needed to model the exponentially saturating superfluid density at very low temperature [63] and a large one to interpret the superfluid density behaviour all the way up to Tc. This situation can be modelled with a multi-gap model, which is a reasonable approach since, FeSe has a complex Fermi surface comprised of an electron and hole pocket, and therefore mulitband models are needed to interpret the superfluid density data. Among multiband models, a two-gap model [68] is the simplest next step, and was successfully applied to metallic MgB2, which has two separate superconducting gaps. In this model, for two s-wave isotropic gaps, the low-temperature superfluid density λ2(0)/λ2(T ) can be expressed as [69]:

2 s s λ (0) 2π∆S(0) −∆S(0) 2π∆L(0) −∆L(0) 2 ≈ 1 − x exp[ ] − (1 − x) exp[ ] λ (T ) kBT kBT kBT kBT (5.1) 2 2 where x = λ (0)/λS(0) is the fractional contribution of the small gap ∆S(0) to the total superﬂuid density, which is the sum of the contribution from the two gaps:

35 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

1 1 1 2 = 2 + 2 (5.2) λ (0) λS(0) λL(0) Figure 5.1 illustrates the comparison of Equation 5.1 with the superfluid fraction from Figure 4.7. From this fit, we can obtain all of the parameters shown in Table 5.1. As we can see, for both ab-plane andc ˆ-axis, the gap ∆S(0) is fairly small compared with the gap ∆L(0): a factor of 5 smaller than the large gap. In reference [63], David Broun uses a more complicated model (two-band extended s-wave model), which has two gaps and one of those two gaps is anisotropic, to fit the experimentally determined superfluid density. The fit is shown in Figure 5.2 and the fitting parameters are summarized in Table 5.2. In such a model, a power law fit at low temperature is possible, but yields a power law close to T 3, rather than T 2 that arises from a superconducting gap with nodes in the presence of disorder. The small gap in our fit is in good agreement with the minimum of the small gap (∆min = 0.198 meV ) found by fitting the 202 MHz data to the two gap anisotropic model. The large gap does not agree very well, but that is because it’s probably some average of the maximum of the small gap and the large gap. To further support a multiband model with a non-zero gap minimum, the two-band character without nodes is also observed in the thermal conductivity study [64] and penetration depth measurements [65] on high-quality FeSe crystals, all indicating that the gap magnitude on one pocket of Fermi surface is an order of magnitude smaller than that on the other pocket [64]. A number of STM measurements show evidence for the presence of nodes at the Fermi surface by detecting a V-shaped tunneling spectra [66, 67, 70]. However, the STM measurements may not be quite low enough in temperature to cleanly resolve the minimum gap. Furthermore, other measurements of heat capacity [71, 72], lower critical field [73], and µSR [74] studies are consistent with non-zero gap minimum.

x ∆S(0) (meV ) ∆L(0) (meV ) ab-plane 17% 0.23 ± 0.035 1.06 ± 0.050 cˆ axis 26% 0.20 ± 0.030 0.94 ± 0.047

Table 5.1: Two-gap ﬁt parameters for ab-plane andc ˆ axis.

The two-gap model generates quantitatively similar ﬁt parameters for the ab-plane andc ˆ-axis, which is expected since the temperature dependence

36 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

1 Exp. data Two-gap model

0.9

(T) 0.8 2 ab λ (0)/

2 ab 0.7 λ

0.6

0.5 0 1 2 3 4 5 Temperature (K)

1 Exp. data Two-gap model 0.9

0.8 (T) 2 c

λ 0.7 (0)/ 2 c λ 0.6

0.5

0.4 0 1 2 3 4 5 Temperature (K)

Figure 5.1: Equation 5.1 was used to fit the ab-plane andc ˆ-axis superfluid density. The green dots denote the experimental data and the blue curve represents the two-gap model fit. a) Two-gap model fit for ab-plane superfluid density: x = 17%, ∆S(0) = 0.23 meV , ∆L(0) = 1.06 meV . b) Two- gap model fit forc ˆ-axis superfluid density: x = 26%, ∆S(0) = 0.20 meV , ∆L(0) = 0.94 meV . 37 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

Figure 5.2: a) A two-band extended s-wave model is ﬁtted to the superﬂuid density, and the inset shows a polar plot of two gaps (∆1 and ∆2) at zero temperature, for various values of the DOS parameter [63]. b) The Temper- ature dependence of the rms gap amplitudes on two bands, and the overall gap minimum.

∆1(0) (meV ) ∆2(0) (meV ) ∆min(0) (meV ) 1.68 ± 0.12 0.52 ± 0.04 0.198 ± 0.03

Table 5.2: Two-band extended s-wave model fit parameters [63]. ofc ˆ-axis superfluid density is similar to that of ab-plane as seen in Figure 4.7. In contrast, in YBa2Cu3O6+δ, thec ˆ-axis superfluid density shows a temperature dependence between T 2 and T 3 [14], discernibly different from the linear-T dependence in botha ˆ and ˆb directions, as shown in Figure 5.3. According to Xiang and Wheatley’s calculations [15, 48],c ˆ-axis superfluid density would have a large theoretical T 5 dependence which stems from two sources: one T term comes from the d-wave linear density of state (DOS); 4 4 2 the other T term is due to the (cos kx − cos ky) factor in t⊥(kk)[48]. In contrast, the in-plane superfluid density varies linearly with temperature due to the nodes in YBa2Cu3O6+δ. Xiang et al. [48] pointed out this strikingly anisotropic behaviour in the superfluid response of cuprates is caused by the interplay between the d-wave superconducting order parameter symmetry and the underlaying Cu 3d orbital based electronic structure. The interlayer

38 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

Figure 5.3: The superfluid fraction in all principal crystallographic directions of YBa2Cu3O6.95 [5]. Thec ˆ-axis superfluid density is qualitatively different from the behaviour in either direction in ab-plane.

39 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

hopping integral t⊥ is a function of in-plane momentum kk, and in a perfect 2 tetragonal system, t⊥ is proportional to (cos kx − cos ky) according to local density approximation (LDA) band structure calculations. In the diagonal directions (kx = ±ky) of a 2D Brillouin zone, the d-wave gap nodes ∆kk = ∆(cos kx − cos ky) perfectly coincide with the zero’s of t⊥, and thec ˆ-axis superﬂuid density takes a form of T 5. However, FeSe likely doesn’t have such a complex t⊥ hopping problem. Theoretically, the FeSe bandstructure is more 3 dimensional. Furthermore, as we can see from Figure 4.6, the c-axis resistivity is metallic at low temperature [75]. Experiments [76, 77] have observed strongly warped Fermi surface, which also suggests a much more 3-dimensional process in thec ˆ direction. A similar story is apparent when contrasting the in-plane and out-of- plane conductivity for these two diﬀerent superconductors. Both the in- plane and out-of-plane microwave properties of YBa2Cu3O6+δ superconductors were extensively studied over the last three decades. In 1998, Hosseini at al. [14] cleaved a thin plate of YBa2Cu3O6.99 crystal, measured the surface resistance Rs in all three crystal directions, as shown in Figure 5.4, and extracted the microwave conductivity σ1(T ) in the c direction shown in Figure 5.5. Both Rsa(T ) and Rsb(T ) show a broad peak at low temperature, which are respectively caused by a large peak in σ1a(T ) of 22.7 GHz in Fig- ure 5.6 and σ1b(T ) (not shown). The broad peak seen in the conductivity was attributed to a competition between two temperature dependent quantities: the number density of thermally excited quasiparticles nn(T ) that declines with temperature, and their transport scattering time τ(T ) that increases with decreasing temperature below Tc [14, 52]. The rapid increase in transport time has been interpreted as a collapse of the inelastic scattering processes that are responsible for the large normal state resistivity of the high temperature superconductors.

40 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

Figure 5.4: Surface resistance along the three crystal directions of YBa2Cu3O6.99. Thec ˆ-axis surface resistance is diﬀerent from that of the ab-plane [14].

However, Rsc(T ) as well as σ1c(T ) are qualitatively diﬀerent from those observed in either of the planar directions. As shown in Figure 5.5, σ1c(T ) drops rapidly below Tc, with no sign of the peak seen in the ab-plane. A weak rise appears at low temperature, but is thought to be an artefact which was caused by the process of cleaving. To circumvent this problem, Hosseini at al. [15] polished a relatively thick crystal into a thin blade, measured the surface resistance and penetration depth of that blade, and successfully extracted σ1c(T ) as illustrated in Figure 5.7. The absence of a peak in thesec ˆ-axis data indicates that charge transport between CuO2 planes is not inﬂuenced by the development of large scattering lifetime below Tc.

41 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

Figure 5.5: Extracted microwave conductivity σ1c for two YBa2Cu3O6.99 samples along thec ˆ-direction [14].

As we discussed in the superfluid density comparison, thec ˆ-axis quasiparticle will mainly come from momenta away from the zone diagonal nodes, due to the dependence of the hopping on the in-plane momentum. On the other hand, photoemission experiments [78] have directly shown the quasiparticle lifetime τ is unusually small everywhere on the Fermi surface except close to the zone diagonals. Therefore, the long lifetime quasiparticles have momenta mainly in the nodal direction in the CuO2 planes and enhance the in-plane conductivity, but not thec ˆ-axis conductivity. Similar to YBa2Cu3O6+δ, FeSe also has many quasiparticles below Tc and they are long-lived in the superconducting state, as shown by the peak in σ1(ω, T ) at low temperature. As with superfluid density, the big difference between YBa2Cu3O6+δ and FeSe is that FeSe exhibits this phenomenon in all directions. It seems experimentally that quasiparticles at the gap minimum in FeSe have long lifetime and they also can move freely in thec ˆ direction. This is consistent with the absence of strongly anisotropic behaviour in the superfluid density of FeSe.

42 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

Figure 5.6: The ab-plane microwave conductivity of a high purity YBa2Cu3O6.99 crystal exhibits a broad, frequency dependent peak caused by the development of long-lived quasiparticles in the superconducting state [11].

43 Chapter 5. Discussion and comparison with YBa2Cu3O6+δ

Figure 5.7: The anisotropy of the microwave conductivity of YBa2Cu3O6.95 is illustrated by plottingc ˆ axis conductivity (open squares) measured at 18 GHz [15] along with the ab-plane conductivity (ﬁlled squares) taken at 1 GHz [15].

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