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Microwave studies on FeSe superconductors Li, Meng 2016

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Microwave Studies on FeSeSuperconductorsbyMeng LiB.Sc., Shandong University, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2016c© Meng Li 2016AbstractIn this thesis, the microwave electrodynamic properties of stoichiometricFeSe are measured by a cavity perturbation technique based on a 940 MHzloop-gap resonator. Measurements surprisingly show, for the first time, thatcˆ-axis conductivity in stoichiometric FeSe superconductors exhibits a broadpeak in its temperature-dependence, a phenomenon which was only observedfor in-plane electrodynamics in the cuprates. This result implies that thecharge transfer between FeSe-planes is enhanced by the development of longtransport quasiparticle lifetimes below Tc.iiPrefaceThe work presented here is conducted in the Superconductivity lab at Uni-versity of British Columbia, Vancouver campus. The apparatus used for thecavity perturbation measurements described in this thesis were previouslydesigned and constructed by Walter Hardy and Saeid Kamal (loop-gap res-onator). The pure FeSe samples were grown by Shun Chi during the courseof 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 I was involved in later growing the Co-doped(0.3%) samples.For chapter 4, the work is original and unpublished.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 col-lection and data analyses, presented in chapter 4, were carried out by me.Pinder Dosanjh played a major role in maintaining experimental equipmentand assisting in data collection.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Conventional superconductivity . . . . . . . . . . . . . . . . 11.2 Unconventional superconductivity . . . . . . . . . . . . . . . 21.2.1 Cuprate superconductors . . . . . . . . . . . . . . . . 31.2.2 Heavy-fermion superconductors . . . . . . . . . . . . 61.2.3 Organic superconductor . . . . . . . . . . . . . . . . . 61.2.4 Iron-based superconductors . . . . . . . . . . . . . . . 72 Microwave electrodynamics of superconductors . . . . . . 132.1 Penetration depth . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Generalized two-fluid model . . . . . . . . . . . . . . . . . . 132.3 Microwave surface impedance . . . . . . . . . . . . . . . . . . 152.3.1 General overview . . . . . . . . . . . . . . . . . . . . 152.3.2 Surface impedance of a superconductor . . . . . . . . 163 Experimental techniques: microwave spectroscopy . . . . 183.1 Cavity perturbation technique . . . . . . . . . . . . . . . . . 183.2 Loop-gap resonator . . . . . . . . . . . . . . . . . . . . . . . 213.3 Swept-frequency cavity transmission measurement . . . . . . 23ivTable of Contents4 In- and out-of-plane microwave electrodynamics of FeSe . 254.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . 254.2 Cleave method . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Finite size consideration . . . . . . . . . . . . . . . . . . . . . 274.4 Experimental results and analysis . . . . . . . . . . . . . . . 304.4.1 Penetration depth . . . . . . . . . . . . . . . . . . . . 304.4.2 Surface resistance . . . . . . . . . . . . . . . . . . . . 304.4.3 Microwave conductivity . . . . . . . . . . . . . . . . . 304.4.4 Superfluid density . . . . . . . . . . . . . . . . . . . . 335 Discussion and comparison with YBa2Cu3O6+δ . . . . . . . 35Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45vList of Tables1.1 The allowed singlet pairing states for a square CuO2 lattice.Table adapted from Refs [1, 2]. . . . . . . . . . . . . . . . . . 51.2 The transition temperature of iron-based superconductors,cited from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.1 Two-gap fit parameters for ab-plane and cˆ axis. . . . . . . . . 365.2 Two-band extended s-wave model fit parameters [63]. . . . . 38viList of Figures1.1 Cooper pair formation via electron-phonon interaction. Pic-ture taken from [4]. . . . . . . . . . . . . . . . . . . . . . . . 21.2 The superconducting gap in k -space (momentum-space) for(a) isotropic (s-wave) and (b) dx2−y2 (d-wave) superconduc-tors. The cylindrical Fermi surfaces are shown as bold circles.The hatched region denotes the filled electrons states. For aconventional s-wave superconductor, the energy gap 2∆ hasthe same sign in all directions. However, for a d-wave super-conductor, the sign and magnitude of the gap is a function ofdirection in the kx and ky-plane. Figure taken from Ref [2, 5]. 31.3 The schematic phase diagram of the high Tc cuprate supercon-ductors as a function of temperature and hole-doping. Figuretaken from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 The superfluid density (inverse square of the superconduct-ing penetration depth) as a function of temperature [7]. Thelinear T dependence at low temperature is a consequence ofthe linear density of states N(E) (inset) of a dx2−y2 super-conductor with a cylindrical Fermi surface. . . . . . . . . . . 61.5 Crystallographic structure of the iron-based superconductors.Iron pnictides have FeAs layers, identical to the FeSe layeredstructure, which are the central ingredients for the supercon-ductivity. Figure taken from [8]. . . . . . . . . . . . . . . . . 81.6 The electronic phase diagrams for two different iron-basedsuperconductors as a function of chemical substitution. . . . . 101.7 (a) FeAs lattice indicating As above and below the Fe plane.Dashed green and solid blue squares indicate 1- and 2-Fe unitcells, respectively. (b) Schematic 2D Fermi surface in the 1-FeBZ whose boundaries are indicated by a green dashed square.(c) Fermi sheets in the folded BZ whose boundaries are nowshown by a solid blue square. Figure taken from ref. [9]. . . . 11viiList of Figures1.8 FeSe lattice structure: simplest tetragonal structure. Figuretaken from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Schematic view of σ1(ω), σ2(ω, T ) for a superconductor [11]. 163.1 Schematic diagram of cavity perturbation technique. Withinan empty cavity, the resonant frequency is determined by thedimensions of the cavity. When a superconducting sample isintroduced into the cavity, superfluid screening currents causethe sample to screen external magnetic field and thereby shiftthe cavity resonant frequency by changing the cavity volume.Figure provided courtesy of Richard Harris [12]. . . . . . . . 193.2 Digital photograph of the loop-gap resonator, with a side viewschematic diagram of the resonator assembly. Dimensionsdisplayed are in millimetres. Figure provided by courtesyJake Bobowski [2] based on original material by Saeid Kamal[13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Top: Schematic view of the loop-gap assembly with a sam-ple loaded. There are coupling loops on either side of theresonator. Bottom: Equivalent circuit of loop-gap resonator.Figure provided by courtesy Jake Bobowski [2]. . . . . . . . . 233.4 The electronics setup for the swept-frequency cavity trans-mission measurement. Figure provided by Jordan Baglo [6]. . 244.1 A photograph of a pure FeSe sample measured in experiments. 264.2 Schematic view of sample measurement geometry. Figureprovided by Jordan Baglo [6]. . . . . . . . . . . . . . . . . . . 274.3 Cleaved samples to measure c-axis penetration depth. S3sample is cleaved from S2 which is cleaved from S1. Theyhave the same broad area, with different thickness 166 µm,87 µm, 32 µm for S1, S2 and S3 respectively. Bottom graphsshow their respective cross section. . . . . . . . . . . . . . . . 284.4 a) The penetration depth measured at 1 GHz of a thin crystalbefore (S1) and after cleaving it into thinner ones (S2 and S3).Inset shows scaled penetration depth and the scaled constantsfor S1, S2 and S3 are 1.00, 1.40 and 2.15 respectively. b) Thepenetration depth ∆λ(T ) for S1, S2 and S3, is plotted overthe full temperature range measured. . . . . . . . . . . . . . . 29viiiList of Figures4.5 a) The temperature dependence of the surface resistance forS1, S2 and S3 below Tc. Inset shows scaled surface resistanceand the scaling constants for S1, S2 and S3 are 1.00, 1.26 and2.10 respectively. b) The surface resistance Rs(T ) plotted ina log-linear scale over the full temperature range measured. . 314.6 The real part of the microwave conductivity σ1ab(T ) ( ab-plane, red triangles)) and σ1c(T ) (cˆ-axis, blue squares) wereextracted from measurements of ∆λ and ∆Rs. Surprisingly,σ1c(T ) also has a broad peak at low temperature. . . . . . . 324.7 The superfluid fraction λ2(0)/λ2(T ) for ab-plane (black squares)and cˆ-axis (λc0 = 5µm, red circles; λc0 = 3µm, blue triangles)versus reduced temperature T/Tc. In both the ab-plane andcˆ-axis direction, the temperature dependence of the super-fluid fraction varies almost linearly with temperature belowTc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1 Equation 5.1 was used to fit the ab-plane and cˆ-axis super-fluid density. The green dots denote the experimental dataand 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 modelfit for cˆ-axis superfluid density: x = 26%, ∆S(0) = 0.20 meV ,∆L(0) = 0.94 meV . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 a) A two-band extended s-wave model is fitted to the super-fluid density, and the inset shows a polar plot of two gaps (∆1and ∆2) at zero temperature, for various values of the DOSparameter [63]. b) The Temperature dependence of the rmsgap amplitudes on two bands, and the overall gap minimum. 385.3 The superfluid fraction in all principal crystallographic direc-tions of YBa2Cu3O6.95 [5]. The cˆ-axis superfluid density isqualitatively different from the behaviour in either directionin ab-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4 Surface resistance along the three crystal directions of YBa2Cu3O6.99.The cˆ-axis surface resistance is different from that of the ab-plane [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.5 Extracted microwave conductivity σ1c for two YBa2Cu3O6.99samples along the cˆ-direction [14]. . . . . . . . . . . . . . . . 42ixList of Figures5.6 The ab-plane microwave conductivity of a high purity YBa2Cu3O6.99crystal exhibits a broad, frequency dependent peak caused bythe development of long-lived quasiparticles in the supercon-ducting state [11]. . . . . . . . . . . . . . . . . . . . . . . . . 435.7 The anisotropy of the microwave conductivity of YBa2Cu3O6.95is illustrated by plotting cˆ axis conductivity (open squares)measured at 18 GHz [15] along with the ab-plane conductivity(filled squares) taken at 1 GHz [15]. . . . . . . . . . . . . . . 44xAcknowledgementsFirst 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 orwritten. He has never failed to inspire me with his deep insight into thephysics problems.A very special thanks goes out to Pinder Dosanjh. Without Pinder’sguidance, I won’t be able to run the cavity-perturbation and bolometricspectroscopy apparatus. Pinder always provided me with direction, techni-cal support whenever I ran into a trouble spot in my research. More im-portantly, Pinder taught me how to be critical thinking and prudent beforeand while proceeding any task. His patience, kindness and broad technicalknowledge truly made a difference in the course of this thesis.I would like to thank David Broun for the wonderful collaboration atSimon Fraser University. The joy, enthusiasm and inspiration he has for hisresearch was contagious and motivational for me.I would also like to thank Shun Chi, not only for the excellent crystalshe grew, but also for being such a good friend who is always willing to helpand give his best suggestions. I’m also so thankful to my fellow studentsJames 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 throughoutmy years of study.xiChapter 1IntroductionSuperconductivity, discovered in 1911 by Onnes, [16] is one of the few macro-scopic quantum phenomena in nature, and is characterized by the completeabsence of DC electrical resistance and expulsion of magnetic fields underparticular temperature and magnetic regimes. The development of BCStheory, the discovery of high temperature cuprate superconductors, andiron-based superconductors have marked several milestones in the historyof superconductivity. Tremendous efforts have been devoted to discovernew superconductors and unveil the secret of superconducting mechanisms.However, there are still a lot of fundamental questions about superconduc-tivity remaining unanswered and under investigation.1.1 Conventional superconductivityAccording to the pairing mechanism, superconductors usually can be dividedinto two branches: conventional and unconventional superconductors. Thesuperconductivity occurring in conventional superconductors can be wellexplained by BCS theory proposed by Bardeen, Cooper, and Schrieffer in1957 [17], for which they received the Nobel Prize in 1972.The BCS theory builds on the assumption that superconductivity emergeswhen the attractive Cooper pair interaction dominates over the Coulomb re-pulsion. A conventional Cooper pair is a weak electron-electron bound pairpossessing zero total spin, zero angular momentum (s-wave) and zero totalmomentum, and the attraction is mediated by the electron-phonon interac-tion. More specifically, the electrons are bound together by their interactionwith the vibrations of the lattice (phonon): one electron in a pair trav-eling through the crystal lattice will distort the lattice by attracting thenuclei towards it, creating a temporary region of excess positive charge inthe 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 1.1demonstrates this simplified process. Cooper pairs behave more like bosons,and can condense into the lowest energy level.11.2. Unconventional superconductivityFigure 1.1: Cooper pair formation via electron-phonon interaction. Picturetaken from [4].Most elemental superconductors, such as Nb, Pb and Sn, are conven-tional. The highest Tc among the family of conventional superconductors is39 K found in magnesium diboride (MgB2) [18].1.2 Unconventional superconductivityUnconventional superconductors refer to all the superconductors that can-not be understood in the context of BCS theory driven by electron-phononinteraction, for example the high-Tc and heavy fermion superconductors.A fundamental difference between conventional and unconventional su-perconductors is the symmetry of the superconducting gap function, ∆(k),which is defined as the energy difference between the ground state energyof the superconductor and the energy of the lowest quasiparticle excitation.For conventional superconductors, the zero total angular momentum of elec-tron pairs (cooper pairs) is caused by isotropic attractive forces betweentwo electrons in all spatial directions, which produces an isotropic super-conducting 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 theelectron correlations caused by the large Coulomb repulsion between elec-trons on a Cu site in the CuO2 planes. The pairing symmetry for hightemperature cuprates is antisymmetric spin singlet state [1], of which fourpossible distinct singlet pairing states are listed in Table 5.2, based on thegroup-theoretic calculations for a square CuO2 plane [19].21.2. Unconventional superconductivityFigure 1.2: The superconducting gap in k -space (momentum-space) for (a)isotropic (s-wave) and (b) dx2−y2 (d-wave) superconductors. The cylindricalFermi surfaces are shown as bold circles. The hatched region denotes thefilled electrons states. For a conventional s-wave superconductor, the energygap 2∆ has the same sign in all directions. However, for a d-wave supercon-ductor, the sign and magnitude of the gap is a function of direction in thekx and ky-plane. Figure taken from Ref [2, 5].1.2.1 Cuprate superconductorsIn 1986, Bednorz and Mu¨ller discovered superconductivity in a ceramic ox-ide La2−xBaxCuO4+δ with a record-breaking transition temperature Tc near29 K [20], soon followed by the discovery of high temperature superconduc-tivity in YBa2Cu3O6+x with Tc = 93 K, above the boiling point of liquidnitrogen [21], and later reached the highest value of 138 K in Tl-dopedHgBa2Ca2Cu3O8+δ [22]. Many other cuprate superconductors also were dis-covered and intensively studied, such as Tl2Ba2CuO6 and Bi2Sr2CaCu2O8,and they form the large family of high temperature superconductors. Allthe materials in this family share a common feature: weakly-coupled stacksof two-dimensional copper oxide planes. It is widely believed that the su-perconductivity occurs within these CuO2 planes.A lot of cuprate superconductors have very high critical temperatureswhich can’t be explained by the BCS theory. A new theory is needed toexplain the unexpected high Tc superconductivity. Although these cupratesuperconductors are the most widely studied family of superconductors dur-ing the past three decades, many of the fundamental questions, such as thepairing mechanism, are still challenging condensed matter physicists.The generic phase diagram for hole-doped cuprate materials is presentedin Figure 1.3. The different phases can be accessed by changing the tem-31.2. Unconventional superconductivityFigure 1.3: The schematic phase diagram of the high Tc cuprate supercon-ductors as a function of temperature and hole-doping. Figure taken from[6].41.2. Unconventional superconductivityWave functionnameGroup-theoreticnotationOrderparameterbasis functionNodess+(s-wave) A1g const.accidental linenodes possibles−(g − wave) A2g xy(x2 − y2) linedx2−y2(d-wave) B1g x2 − y2 linedxy B2g xy lineTable 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 Ne´el temperatureTN , the undoped (p = 0) material is an antiferromagnetic Mott-insulator.As the hole doping increases, this antiferromagnetic phase vanishes rapidlyand superconductivity emerges with critical temperature Tc roughly varyingparabolically with hole doping. An empirical parabolic relationship betweenthe superconducting transition temperature Tc and hole doping level p hasbeen formulated [23, 24] as follows:1− TcTc,max= 82.6(p− 0.16)2 (1.1)The maximum Tc,max is obtained at a doping level of p = 0.16 (optimal dop-ing); lower and higher doping levels are known as underdoped and overdopedregimes, respectively.Another important property of cuprate superconductors is the symmetryof the pairing states. The NMR measurements of the Knight shift showedthat spin susceptibility declines rapidly below Tc, indicating that the super-conducting states are spin-singlet [25, 26]. In 1993, Hardy et al. [7] ob-served the linear temperature dependence of London penetration depth inYBa2Cu3O6.95 (Figure 1.4), a proof for line nodes in ∆(k). However, thesemeasurements could not distinguish between the three allowed symmetrybasis function shown in Table 5.2, all of which have, or could have, linenodes. Subsequently, the phase-sensitive scanning SQUID measurementsyielded convincing evidence for a predominant d-wave pairing symmetry ina wide variety of cuprate systems [1]. The corresponding superconductinggap in d-wave superconductors is shown in Figure 1.2 (b).51.2. Unconventional superconductivityFigure 1.4: The superfluid density (inverse square of the superconductingpenetration depth) as a function of temperature [7]. The linear T depen-dence at low temperature is a consequence of the linear density of statesN(E) (inset) of a dx2−y2 superconductor with a cylindrical Fermi surface.1.2.2 Heavy-fermion superconductorsHeavy-fermion superconductors are another class of unconventional super-conductors, containing rare earth or actinide elements and possessing enor-mous effective mass of charge carriers. The unexpected discovery of su-perconductivity in CeCu2Si2 [27] opened up the door to the heavy fermionnon-BCS superconductivity. In heavy-fermion systems, it is believed thatthe pairing of quasiparticles is not mediated by an electron-phonon inter-action, but by the spin fluctuations of a nearby antiferromagnetic phase[28, 29]. The pairing symmetry is a key piece of information to understandhow this works. Recently, a nodal d-wave character of superconductingpairing state has been demonstrated to exist in CeCoIn5 via high-resolutionSTM techniques [30].1.2.3 Organic superconductorAn organic superconductor refers to a synthetic organic compound exhibit-ing superconductivity below Tc. The first organic superconductor (TMTSF)2PF6was synthesized by Klaus Bechgaard in 1979, with Tc = 1.1 K at an external61.2. Unconventional superconductivitypressure of 6.5 kbar [31]. This discovery inspired the creation of a large fam-ily of related organic compounds, known as Bechgaard salts, which exhibita variety of unique properties. The structure of Bechgaard salts is verydifferent from those of metallic superconductors; a lot of effort has beenpoured into investigating the superconducting mechanism. NMR Knightshift measurements [32] and specific heat measurements [33] on organic su-perconductor (TMTSF)2CIO4 provided strong evidence of spin-singlet nodalsuperconductivity. Nuclear spin relaxation rate measurements showed thesuperconducting pairing is mediated by interstack antiferromagnetic spinfluctuations [34].1.2.4 Iron-based superconductorsIn 2008, the discovery of superconductivity in LaFeAsO1−xFx with a tran-sition temperature Tc ≈ 26 K [35] generated tremendous interest in thesuperconductivity 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 Tcsuperconductivity.To date, the newly discovered iron-based superconductors (FeSCs) haveroughly five classes of structures, with examples being FeTe (the 11 com-pounds), 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 planarlayer of iron atoms joined by tetrahedrally oriented chalcogen (S, Se, Te) orpnictogen (P, As) anions arranged in a stacked sequence separated by alkali,alkaline-earth or rare-earth and oxygen/fluorine ‘blocking layers’. The su-perconducting transition temperature for some iron-based superconductorsis summarized in Table Unconventional superconductivityFigure 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 takenfrom [8].Oxypnictide Tc (K)SmFeAsO∼0.85 55GdFeAsO0.85 53.5NdFeAsO0.89F0.11 52ErFeAsO1−y 45La0.5Y0.5FeAsO0.6 43.1SmFeAs0.9F0.1 43CeFeAs0.84F0.16 41LaO0.9F0.2FeAs 28.5LaO0.89F0.11FeAs 26BaFe1.8Co0.2As2 25.3(a) Oxypnicide materials.Non-oxypnictideTc (K)Sr0.5Sm0.5FeAsF 56Ba0.6K0.4Fe0.2As0.2 38FeSe < 27Ca0.6Na0.4Fe2As2 26NaFeAs 9-25CaFe0.9cO0.1AsF 22(b) Non-oxypnictide materials.Table 1.2: The transition temperature of iron-based superconductors, citedfrom [3].81.2. Unconventional superconductivitySimilar to high-Tc cuprates, the properties of FeSCs change dramati-cally with chemical modification. Most of FeSC parent compounds exhibitan antiferromagnetic spin-density wave (SDW) ground state, and supercon-ductivity is induced upon charge-carrier doping or by applying an externalpressure. In these materials, the superconductivity occurs within FeAs,FeP, FeSe or FeTe crystallographic planes, analogous to the CuO2 planes incuprates.A compilation of experimental phase diagrams is shown in Figure 1.6a forthe Ba-based 122 system [8], widely thought to share the main traits of allFeSCs. In BaFe2As2, it has been shown that the systematic substitution ofeither the alkaline-earth (Ba), transition-metal (Fe) or pnictogen (As) atomwith a different element almost universally shares the same phase diagrampresented in Figure 1.6a. The superconducting phase (SC) is approximatelycentred at the critical doping where the antiferromagnetic (AFM) ordervanishes. The coexistence of AFM and SC phases in a small doping rangeis believed to be an intrinsic property of the typical FeSc phase diagram,whereas, in fluorine-doped 1111-systems such as LaFeAsO1−xFx, the super-conducting phase is completely separated from AFM, as shown in Figure1.6b [36].All the Fe-based compounds have similar electronic band structure con-sisting of hole pockets centred at Γ (0, 0) and electron pockets centred atM (±pi, ±pi) or (±pi, 0) in the Brillouin zone (BZ). Figure 1.7 shows thesquare FeAs lattice and the corresponding Fermi surface (folded and un-folded) for a stoichiometric parent compound. The electron Fermi surfacepocket is strongly connected to the hole Fermi sheets by a reciprocal latticevector, known as Fermi surface nesting. This strong nested structure leadsto the SDW instabilities [40] and suggests the possibility of s±-wave pairingsymmetry mediated by antiferromagnetic spin fluctuations with supercon-ducting gaps having opposite sign between the electron and hole pockets.Although most experimental results favor s±-wave for the majority of mate-rials, a universal consensus regarding the pairing symmetry of the iron-basedsuperconductors has not been reached.A particularly interesting compound among the family of Fe-based su-perconductors 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 ∼ 8K, but its Tc can be enhanced to 37 K by applying high pressure [41]. FeSecompounds have advantageous features for measuring and elucidating itssuperconducting properties: (1) A vapour transport technique allows us togrow clean stoichiometric FeSe crystals with low disorder; (2) FeSe has the91.2. Unconventional superconductivity(a) The phase diagram of the BaFe2As2 system, shown for K [37], Co [38] andP [39] substitutions. The dotted line represents tetragonal (T) to orthorhom-bic (O) structural phase transition obsserved for Co doped samples.(b) The electronic phase diagram of the LaFeAsO1−xFx system obtained byµSR measurements [36].Figure 1.6: The electronic phase diagrams for two different iron-based su-perconductors as a function of chemical substitution.101.2. Unconventional superconductivityFigure 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, respec-tively. (b) Schematic 2D Fermi surface in the 1-Fe BZ whose boundariesare indicated by a green dashed square. (c) Fermi sheets in the folded BZwhose boundaries are now shown by a solid blue square. Figure taken fromref. [9].simplest Fermi surface consisting of a hole pocket and an electron pocket; (3)The sample is superconducting without chemical substitution, which causesdisorder as well as tuning the doping. Therefore, this provides a good oppor-tunity to better understand the superconducting mechanism of this familyof unconventional superconductors.111.2. Unconventional superconductivityFigure 1.8: FeSe lattice structure: simplest tetragonal structure. Figuretaken from [10].12Chapter 2Microwave electrodynamicsof superconductors2.1 Penetration depthA superconductor is characterized by the complete absence of DC resistivityand by the Meissner effect, at temperatures below Tc. These propertieswere first formulated in a phenomenological picture by the London brothers[42]. They proposed that the electromagnetic response of superconductingcarriers with number density ns is governed by the London equation:∂ ~Js∂t=nse2m∗~E (2.1)where ~Js, e and m∗ are respectively the supercurrent density, charge andeffective mass of the superconducting electrons. Combining Equation 2.1with Maxwell’s equations, we can get a frequency independent skin depth,the London penetration depthλL =(m∗µ0nse2)1/2(2.2)λL for a typical superconductor is less than a micron, implying that themagnetic field is almost completely excluded from a bulk sample.2.2 Generalized two-fluid modelThe generalized two-fluid model builds on the London model, by includingthe effects of the normal fluid, and serves as a phenomenological descrip-tion of the high frequency electrodynamics of superconductors.1 The model1Strictly speaking, the two fluid model is most easily justified in the clean limit wherethe quasiparticle spectrum has its spectral weight well below the superconducting gapfrequency.132.2. Generalized two-fluid modelpostulates that the electromagnetic response of a superconductor can be di-vided into two components: a normal fluid (associated with quasiparticlesexcited out of the condensate) and a dissipationless superfluid (associatedwith Cooper pairs in the condensate), with electron densities nn and nsrespectively. In an electromagnetic field, normal electrons are assumed tobehave like electrons in the normal state and therefore dissipate energy byscattering. The total electron density n is temperature independent andequal to the sum of the normal fluid density nn and the superfluid densityns 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 thetemperature decreases below Tc, quasiparticles start to condense into thesuperconducting state, and ns begins to build up. Ultimately, in the cleanlimit where the energy scale of the superconducting gap is larger than thequasiparticle scattering rate, all of the electrons are supposed to condenseinto the superconducting state at T = 0 such that n = ns(T = 0).The total conductivity is the superposition of the normal-fluid and superfluidconductivity σ˜(ω, T ) = σ˜n(ω, T ) + σ˜s(ω, T ). At low temperature below Tc,the microwave properties are dominated by the superfluid. For now we onlyconsider the superfluid conductivity, which can be introduced with a Drudemodel:σ˜s(ω) = σ1s − iσ2s = nse2m∗sτs1 + ω2τ2s− inse2m∗sωτ2s1 + ω2τ2s(2.4)where n/m∗s is the ratio of superconducting electron density over effectivemass and τ is the current relaxation time of the electrons. Since the super-fluid is dissipationless, τs will go to infinity.σ˜s(ω)|τs→∞ = limτs→∞nse2m∗sτs1 + ω2τ2s− i limτs→∞nse2m∗sωτ2s1 + ω2τ2s(2.5)So, the imaginary conductivity component becomes:σ2s(ω) =nse2m∗sω(2.6)142.3. Microwave surface impedanceAs for σ1s, it can been obtained by the Kramers-Kronig2 formula, whichyieldsσ2s(ω) = − 2pi∫ ∞0ωσ1s(ω′)ω′2 − ω2 dω′ (2.7)Combining with the above equation, we get the solutionσ1s(ω) =pinse2m∗sδ(ω) (2.8)which is a delta function.Then, the total conductivity for a superconductor can be written asσ˜(ω, T ) =pins(T )e2m∗sδ(ω)− ins(T )e2m∗sω+ σ˜n(ω, T ) (2.9)A schematic view of real and imaginary component of complex conduc-tivity varying with the temperature and frequency has been shown in 2.1,taken from the reference [11].2.3 Microwave surface impedance2.3.1 General overviewThe complex surface impedance of a metal or superconductor is a measure ofthe absorption and screening characteristics of the material in the presenceof an electromagnetic field. This quantity is directly accessible through mi-crowave experiments and is defined to be the ratio of the tangential electricand magnetic field at the surface of the sample (e.g. Ex/Hy) in an electro-magnetic field.In the case of local electrodynamics ( where the penetration depth is muchlarger than the coherence length, λ ξ ), the microwave surface impedanceZs is related to microwave conductivity σ viaZ˜s(w, T ) ≡ Rs(w, T ) + iXs(w, T ) =√iµ0wσ1(w, T )− iσ2(w, T ) (2.10)where Rs(w, T ) is the surface resistance and Xs(w, T ) is the surface reac-tance. The surface resistance Rs(w, T ) is directly related to the power ab-sorbed by the sample in the electromagnetic field and the surface reactance2The Kramers-Kronig formula relates the real and imaginary parts of σ˜ = σ1 −iσ2 in terms of of the paired expression: σ1(ω) = +2piP∫∞0ω′σ2(ω′)ω′2−ω2 dω′, σ2(ω) =+ 2piP∫∞0ω′σ1(ω′)ω′2−ω2 dω′, where P denotes principal value integration.152.3. Microwave surface impedanceFigure 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 ofthe sample. Equation (2.10) shows the direct link between the microwaveconductivity and measurable surface impedance.2.3.2 Surface impedance of a superconductorFor a superconductor, within the superconducting state well below Tc, nor-mal electrons condense into the superconducting state, and one expects theimaginary part of the conductivity to be dominated by the superconductingcondensate, with σ2  σ1. The surface impedance can be approximately162.3. Microwave surface impedanceexpressed as:Rs(ω, T ) ' 12µ20ω2λ3L(T )σ1n(ω) (2.11)Xs(ω, T ) ' µ0ωλL(T ) (2.12)These approximations are simple but quite accurate for T bellow Tc whenthe measurement frequency is smaller than the quasiparticle scattering rate(ω < 1τ ). As we can see from (2.11) and (2.12), Xs is primarily related to thepenetration depth λL, while Rs is proportional to σ1n - but with a large λ3Ldependence on the penetration depth. We can clearly observe that measure-ments of surface reactance alone provide the penetration depth directly, andthe conductivity σ1n can be extracted from the measured surface resistanceif 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 spec-troscopy. The experiments presented in this thesis focus on the cavity per-turbation measurements.17Chapter 3Experimental techniques:microwave spectroscopy3.1 Cavity perturbation techniqueThe cavity perturbation technique can be used to measure both the changein the magnetic penetration depth and the surface resistance of a supercon-ducting sample. Basically, the sample is inserted into the centre of a resonantcavity, and the small perturbation caused by the sample will provide infor-mation about the penetration depth and surface resistance. A schematicdiagram of a high-Q cylindrical resonator3 operating in the TE011 mode4 isshown in Figure 3.1. In this mode, an electric field node and a magneticfield antinode5 will exist at the centre of the cavity. The red lines inside thecavity of Figure 3.1 represent the microwave magnetic fields which are fairlyuniform in the middle of cavity. The microwaves generated by a synthesizedsweeper are coupled into (TX), and out of (RX), the cavity through smallholes 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 resonancewidth wB can be related to the change of magnetic penetration depth andsurface resistance. The resonance frequency f is determined by the cavitydimensions, and the width of the resonance wB is related to cavity qualityfactor Q:Q = 2pipeak energy storedenergy dissipated per cycle=fwB(3.1)3This is only used for illustration. All experiments done in this thesis were carried outon a loop-gap resonator.4Most real resonant systems have multiple (often an infinite spectrum of) resonantmodes 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 quitejustifiably neglected.5In the loop-gap resonator, the electric field only exists in the gap and the magneticfield only exists within the half-loop cavity.183.1. Cavity perturbation techniqueWithout any sample in the cavity, the energy dissipated per cycle primarilycomes from the dissipation in the cavity walls where the currents flow backand forth. In order to minimize the dissipation and optimize the Q ofthe resonator, the inner surfaces of the copper cavity are coated with aPb0.95Sn0.05 superconducting alloy with a transition temperature Tc ≈ 7K.Figure 3.1: Schematic diagram of cavity perturbation technique. Withinan empty cavity, the resonant frequency is determined by the dimensions ofthe cavity. When a superconducting sample is introduced into the cavity,superfluid screening currents cause the sample to screen external magneticfield and thereby shift the cavity resonant frequency by changing the cavityvolume. Figure provided courtesy of Richard Harris [12].Now consider a thin superconducting platelet sample that is introducedinto a region of the cavity resonator where the magnetic field is fairly uni-form. The screening effects cause the sample to shield microwave magneticfield from its interior within a penetration depth λ. The screening of mag-netic field from the sample decreases the effective volume of the cavity, whichthen shifts the resonance to a higher frequency f + δf . Moreover, the dis-sipation in the sample results in the reduction of the stored energy withinthe resonator, causing a broadening of the resonance.A temperature-dependent surface impedance means that a change ofsample temperature will result in a shift of resonance frequency and change193.1. Cavity perturbation techniquein cavity Q. A complex frequency notation is introduced to formulate thecavity perturbation problem [43]:ω˜0 = ω0 + iω02Q(3.2)where ω0 = 2pif0 is the resonant angular frequency. The relative shifts ofthe complex frequency due to the temperature change are related to thechanges in surface impedance via :∆w˜0w0=∆f0f0+i2∆(1Q) = iΓAsf0(∆Rs + i∆Xs) = iΓAsf0∆Z˜s (3.3)where Γ is the resonator constant and As is the surface area of sample. Adetailed 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 sam-ple is weak enough that the field distribution does not change significantlyelsewhere in the cavity(i.e. no non-perturbative effect). This must remainreliable when we consider the perturbations from temperature changes ofthe sample.For a thin platelet superconductor (c  a, b) in a low-demagnetizationgeometry with magnetic field H ⊥ cˆ, the change in the resonance frequencyand Q of the cavity upon insertion of the sample are given by [44, 45]:δf0f0=Vs2Vc[1−<{tanh(κ˜c/2)κ˜c/2}](3.4a)δ(1Q) =VsVc={tanh((κ˜c/2)κ˜c/2}(3.4b)where Vs is the sample volume, Vc is the effective volume of the resonator,κ˜ =√iωµ0σ is the complex propagation constant of the electromagneticfield inside the sample, and < and = represent the real and imaginary partsof the quantities in curly brackets.For a normal metal with skin depth δ, κ˜ = (1 + i)/δ and for a supercon-ductor with penetration depth λ well below Tc, κ˜ ≈ 1/λ. In the thick limitc 2λ for a superconductor at low temperature, the hyperbolic tanh termapproaches 1 and Equation 3.4a is simplified to:δf0f0' Vs2Vc[1− 2λc](3.5)In our experiments, the sample temperature can be set independently of thecavity 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 temper-ature is changed. The resonant frequency shift ∆f from a base (reference)203.2. Loop-gap resonatortemperature T0 is defined as:∆(δf0) ≡ δf(T )− δf(T0) ' −f0As2Vc∆λ(T ) (3.6)where ∆λ(T ) ≡ λ(T )− λ(T0).The above analysis, however, neglects the thermal expansion effects andthe c-axis penetration depth λc. The sample dimensions change slightly withthe change of the sample’s temperature, which causes the change of samplethickness c, and sample volume Vs. However, the thermal expansion is asmall effect for temperatures below 70 K [5]. In this thesis, all experimentsare carried out on thin samples at temperatures below 20 K and thereforethe effects of thermal expansion are ignored.The procedure to separate the cˆ-axis penetration depth’s contributionwill be discussed in detail in Chapter 4.3.2 Loop-gap resonatorMicrowave surface impedance was measured through cavity perturbation ofa superconducting loop-gap resonator (shown in Figure 3.2 and 3.3) whichoperates at a resonant frequency ∼ 940 MHz. The cavity is comprised of acopper half-loop bolted tightly on one side to a copper support block, with asapphire plate held firmly in the gap between the other side of the loop andthe block. In order to get a high-Q factor in this design, the superconduct-ing joint (shown in Figure 3.3) between the loop-gap and the support baseshould be lossless. To achieve this, screws are used to tighten the loop-gapand the support base which both have been previously electroplated withPb0.95Sn0.05 (Tc ≈ 7 K ). The joint is superconducting while the apparatusis operating at base temperature (∼1.2 K).The quality factor (Q-factor) has been found to slowly decrease with airexposure over time, due to degradation of the plating and superconductingjoints, from initial value (> 106) to a current value of 3×105 at 1.2 K whichis still adequate for the penetration depth measurements, but somewhat lowfor accurate surface resistance measurements.In such an experiment, the sample is mounted on a thin sapphire plateand held fixed in the centre of the resonator, with the magnetic field appliedparallel to the ab-plane of the sample. This orientation produces currentsprimarily running across the broad face of the slab in the ab-plane, with asmall contribution from currents running down the side of the slab in thecˆ-axis direction.213.2. Loop-gap resonatorFigure 3.2: Digital photograph of the loop-gap resonator, with a side viewschematic diagram of the resonator assembly. Dimensions displayed are inmillimetres. Figure provided by courtesy Jake Bobowski [2] based on originalmaterial by Saeid Kamal [13].223.3. Swept-frequency cavity transmission measurementFigure 3.3: Top: Schematic view of the loop-gap assembly with a sampleloaded. There are coupling loops on either side of the resonator. Bottom:Equivalent circuit of loop-gap resonator. Figure provided by courtesy JakeBobowski [2].3.3 Swept-frequency cavity transmissionmeasurementThis implementation of the microwave cavity perturbation method involvesthe measurement of transmission of the microwave cavity as a function offrequency ω. Since the measurements are not sensitive to phase, only themagnitude of total time-averaged power reaching a “square-law” diode de-tector is measured [6]:Pout(ω) =P01 + 4Q2(ω−ω0ω0 )2(3.7)where Q, ω0 and P0 are quality factor, resonant frequency, and maximumpower, respectively.The hardware for the swept-frequency transmission measurement is shownin Figure 3.4. An HP 83620A microwave synthesized sweeper (with a fre-quency range of 10 MHz to 20 GHz) is connected to the input coupling loop233.3. Swept-frequency cavity transmission measurementFigure 3.4: The electronics setup for the swept-frequency cavity transmis-sion measurement. Figure provided by Jordan Baglo [6].of the loop-gap resonator via flexible coaxial cable; the output loop is cou-pled to an HP 423A crystal detector whose near-DC output is proportionalto the incident microwave power. The output low-frequency signal fromthe detector is typically amplified by a factor of 103 through an amplifierand then connected to an input channel of a Tektronix TDS 520B digitaloscilloscope.The experiment control and data acquisition is performed by a com-puter running the graphical programming environment LabVIEW (NationalInstruments). A LabVIEW “Virtual Instrument” (V I) is employed to auto-matically control the sample stage temperature, synthesized sweeper, digitaloscilloscope, and perform the Lorentzian fits to obtain f0 and Q. Given a setof desired measurement temperatures and tolerances, this V I will handle theentire process of collecting and fitting resonance curves at the programmedtemperatures, including tracking appropriate centre frequency, span, andamplitude of the synthesizer sweeps as they change with temperature. Theoutput data file contains thermometry data, together with the Lorentizianfit parameters, for each temperature setpoint [6].24Chapter 4In- and out-of-planemicrowave electrodynamicsof FeSeSince the discovery of superconductivity in LaFeAsO1−xFx with a transitiontemperature Tc ≈ 26 K [35], great effort has been devoted to finding severalother superconducting Fe-based materials [10, 46, 47] and investigating theirpairing mechanisms and symmetry of the superconducting order parameter.In the family of Fe-based superconductors, the FeSe compound possessesseveral advantageous features for measuring and interpreting its physicalproperties. First, vapour transport technique yields clean stoichiometricFeSe crystals with low disorder. Furthermore, FeSe superconductor has thesimplest crystal structure and also the simplest Fermi surface, comprised ofa hole pocket and an electron pocket. Finally, it is superconducting withoutthe added disorder of chemical substitution.Layered superconductors such as the iron-based superconductors and thecuprates display anisotropic transport properties and therefore need to bestudied in all principal crystallographic directions. In the high-Tc cuprates(such as YBa2Cu3O7−δ), it has been established experimentally [14] andtheoretically [48] that the interplane (cˆ-axis) transport properties are quitedifferent from those observed in the ab-plane and that cˆ-axis transport canbe interpreted in terms of a complex hopping process.In contrast, we will show here that for FeSe superconductors the trans-port properties in the cˆ-axis direction are qualitatively similar to those in theab-plane. In both directions, quasiparticles develop long transport lifetimesbelow Tc.4.1 Sample preparationSingle crystals of β-FeSe were grown by the vapor transport technique firstreported by Chareev et al. [49] and Bohmer et al. [50]. The iron and254.2. Cleave methodFigure 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 placedinto 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 seleniumunder vacuum. The sealed ampoule was then placed in a two-zone furnacewith the high temperature end 686 K and the low temperature end 554 K.Millimeter-size FeSe single crystals were obtained after 25 days growth inthis temperature gradient. The critical temperature of the crystals is Tc=8.8K and the transition width is ∆T= 0.5 K.A photograph of a measured FeSe sample is shown in Figure 4.1. Thesample was mounted on a thin sapphire plate.4.2 Cleave methodMicrowave surface impedance was measured through cavity perturbationof a superconducting loop-gap resonator operating at ∼ 1 GHz. In suchan experiment, the sample is mounted on a thin sapphire plate and heldfixed in the centre of the resonator, with the magnetic field applied parallelto the ab-plane of the sample. Figure 4.2 displays a schematic view of thesample measurement geometry. This orientation produces currents primarilyrunning across the broad face of the slab in the ab-plane, with a smallfraction of currents running down the side of the slab in the cˆ-axis direction.264.3. Finite size considerationλaλc caH || bJcabFigure 4.2: Schematic view of sample measurement geometry. Figure pro-vided by Jordan Baglo [6].The current contribution due to cˆ-axis is then decreased by cleaving the slabinto a set of thinner ones, while the ab-plane’s contribution almost stays thesame because the cleaving process only slightly changes the dimensions ofthe ab-plane. Then, cleaved crystals are remeasured with ~H lying in thesame orientation and with negligible change of the sample’s position in thecavity. This technique is particularly reliable because it has no significantchange in demagnetizing factors, and no change in the distribution of ab-plane currents. A simple cartoon shown in Fig.4.3 illustrates the cleavingsequence.4.3 Finite size considerationIn highly anisotropic superconductors such as FeSe, finite-size effects needto be taken into account in interpreting microwave data. In the limit oflocal electrodynamics, the complex screening length δ˜ is related to otherelectrodynamic variables in the following way:Zs(T ) = Rs(T ) + iXs(T ) = iωµ0δ˜ (4.1)σ(T ) = σ1(T )− iσ2(T ) = 1iωµ0δ˜2(4.2)274.4. Experimental results and analysisFigure 4.3: Cleaved samples to measure c-axis penetration depth. S3 sampleis 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 approximatelyequal to penetration depth δ˜ ≈ λ. With the magnetic field lying in the bˆdirection of a thin crystal which has width a in aˆ-axis direction and thicknessc in cˆ-axis direction, aˆ-axis and cˆ-axis penetration depth can be extractedby numerically solving the following equations [51]:δ˜eff =aca+ c4pi2∑odd n<01n2{tanhαnαn+tanhβnβn}(4.3)αn =c2δ˜a1 +(npiδ˜ca)2 12 (4.4)βn =a2δ˜c1 +(npiδ˜ac)2 12 (4.5)where δ˜eff , δ˜a and δ˜c correspond to the effective, aˆ-axis and cˆ-axis complexscreening length.284.4. Experimental results and analysisa)b)Figure 4.4: a) The penetration depth measured at 1 GHz of a thin crystalbefore (S1) and after cleaving it into thinner ones (S2 and S3). Inset showsscaled 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 andS3, is plotted over the full temperature range measured.294.4. Experimental results and analysis4.4 Experimental results and analysis4.4.1 Penetration depthFigure 4.4a shows the temperature dependence of the measured penetrationdepth ∆λ(T ) = λ(T ) − λ(1.2 K) for three pure FeSe crystals: S1, S2 andS3, 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 thesame ab-plane dimensions: 1.040 mm×1.306 mm. As seen in Fig.4.4a, theconsiderable change in ∆λ after cleaving S1 into thinner samples (S2, S3)is attributed to the decrease in cˆ-axis contribution and this indicates thepenetration depth in the cˆ direction is quite large. The inset displays thescaled penetration depth, obtained from the measured ∆λ for S1, S2 and S3multiplied 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 impliesthat the penetration depth within the ab-plane and along the cˆ-axis havenearly identical functional form but with differing magnitude. The penetra-tion depth across the full temperature range measured is shown in Figure4.4b.4.4.2 Surface resistanceFigure 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. Similarto the in-plane surface resistance observed in YBa2Cu3O7−δ superconduc-tors, the broad peak seen in all three samples is caused by a very largepeak in the temperature dependence of microwave conductivity σ1(T ). Thisincrease in conductivity below Tc has been attributed to a rapid increasein quasiparticle lifetime in the superconducting state [52]. The inset showsthe scaled surface resistances, which were obtained in the manner describedabove but with different scaled numbers: 1.00, 1.26 and 2.10 correspondingto S1, S2 and S3. The scaled ∆Rs agree well at low temperature, and fromthe comparison between the measured and scaled ∆Rs, we can still concludethat the temperature dependence of the surface resistance in the cˆ-axis di-rection is quite similar to that in ab-plane. So, for both directions, a largeconductivity peak would be expected to develop below Tc.4.4.3 Microwave conductivityIn order to directly compare the electrical transport properties in the ab-plane with those along the cˆ-axis, the conductivity σ1ab(T ) and σ1c(T ) shown304.4. Experimental results and analysisa)b)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 scalingconstants for S1, S2 and S3 are 1.00, 1.26 and 2.10 respectively. b) The sur-face resistance Rs(T ) plotted in a log-linear scale over the full temperaturerange measured.314.4. Experimental results and analysis0 2 4 6 8 1001020304050   ab  cT(K)ab (m)- (m)-1Figure 4.6: The real part of the microwave conductivity σ1ab(T ) ( ab-plane, red triangles)) and σ1c(T ) (cˆ-axis, blue squares) were extracted frommeasurements of ∆λ and ∆Rs. Surprisingly, σ1c(T ) also has a broad peakat low temperature.in Figure 4.6 were extracted from the data sets in Fig. 4.4a and Fig. 4.5a. Aremarkable feature of this figure is that the cˆ-axis conductivity also exhibitsa large peak at low temperature (near 0.4Tc), which is only observed inthe 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 issuppressed for charge transport in the cˆ-direction, as well as within the ab-plane [59, 60] and that the charge transfer is enhanced by the developmentof long transport lifetime below Tc. However, the cˆ-axis conductivity overallis two orders of magnitude smaller than that in the ab-plane.324.4. Experimental results and analysis0.0 0.2 0.4 0.6 0.8  T/TC2(0)/2(T) ab-plane c-axis(c(0)= 5 m) c-axis(c(0)= 3 m)  Figure 4.7: The superfluid fraction λ2(0)/λ2(T ) for ab-plane (black squares)and cˆ-axis (λc0 = 5µm, red circles; λc0 = 3µm, blue triangles) versus re-duced temperature T/Tc. In both the ab-plane and cˆ-axis direction, thetemperature dependence of the superfluid fraction varies almost linearlywith temperature below Tc.4.4.4 Superfluid densityIn figure 4.7, the superfluid fraction λ2(0)/λ2(T ) for ab-plane and cˆ-axiswere extracted from the measured penetration depth combined with mag-netization measurements of λab(0) = 445 nm [61], and the inferred valueof λc(0) = 3000 nm obtained from Homes’ law [62] based on the measuredconductivity σ1c(0) = 3× 10−2(µΩ m)−1. A possible limit is set by the reddotted line where λc(0) = 5000 nm. Both ab-plane and cˆ-axis superfluiddensity exhibit a linear temperature dependence over a fairly wide tempera-ture range (T/Tc < 0.8). However, further measurements carried out on thesame sample at much lower temperature [63] observed that the superfluiddensity saturates at very low temperature, which indicates the presence ofa non-zero gap minimum. This finding of a small gap is consistent witha recent thermal conductivity study on similarly grown UBC samples [64],334.4. Experimental results and analysisand 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 fromwhether those measurements can differentiate between a gap node and avery small gap. The quality of samples, to some extent, can also affect themeasurement results.34Chapter 5Discussion and comparisonwith YBa2Cu3O6+δIn this thesis, the 1 GHz loop-gap resonator was employed to study the elec-trodynamics of FeSe crystals. The measurements show that the microwaveproperties of the superconducting state in our FeSe crystals, such as penetra-tion depth and surface resistance, have similar behaviors for both in-planeand cˆ-axis direction. In this chapter I will compare these results to thosefound for YBa2Cu3O6+δ, where there is a large difference between in-planeand out-of-plane electrodynamics.In Figure 4.7, the superfluid density displays an approximately lineartemperature 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 existenceof a very small minimum gap [63], rather than the nodal gap in the d-wavepairing state of YBa2Cu3O6+δ [7]. Therefore, a small gap is needed to modelthe exponentially saturating superfluid density at very low temperature [63]and a large one to interpret the superfluid density behaviour all the way upto Tc. This situation can be modelled with a multi-gap model, which is areasonable approach since, FeSe has a complex Fermi surface comprised ofan electron and hole pocket, and therefore mulitband models are needed tointerpret the superfluid density data. Among multiband models, a two-gapmodel [68] is the simplest next step, and was successfully applied to metallicMgB2, which has two separate superconducting gaps. In this model, for twos-wave isotropic gaps, the low-temperature superfluid density λ2(0)/λ2(T )can be expressed as [69]:λ2(0)λ2(T )≈ 1− x√2pi∆S(0)kBTexp[−∆S(0)kBT]− (1− x)√2pi∆L(0)kBTexp[−∆L(0)kBT](5.1)where x = λ2(0)/λ2S(0) is the fractional contribution of the small gap ∆S(0)to the total superfluid density, which is the sum of the contribution fromthe two gaps:35Chapter 5. Discussion and comparison with YBa2Cu3O6+δ1λ2(0)=1λ2S(0)+1λ2L(0)(5.2)Figure 5.1 illustrates the comparison of Equation 5.1 with the superfluidfraction from Figure 4.7. From this fit, we can obtain all of the parametersshown in Table 5.1. As we can see, for both ab-plane and cˆ-axis, the gap∆S(0) is fairly small compared with the gap ∆L(0): a factor of 5 smallerthan the large gap. In reference [63], David Broun uses a more complicatedmodel (two-band extended s-wave model), which has two gaps and one ofthose two gaps is anisotropic, to fit the experimentally determined superfluiddensity. The fit is shown in Figure 5.2 and the fitting parameters are sum-marized in Table 5.2. In such a model, a power law fit at low temperatureis possible, but yields a power law close to T 3, rather than T 2 that arisesfrom a superconducting gap with nodes in the presence of disorder. Thesmall gap in our fit is in good agreement with the minimum of the smallgap (∆min = 0.198 meV ) found by fitting the 202 MHz data to the twogap anisotropic model. The large gap does not agree very well, but that isbecause it’s probably some average of the maximum of the small gap andthe large gap.To further support a multiband model with a non-zero gap minimum, thetwo-band character without nodes is also observed in the thermal conduc-tivity study [64] and penetration depth measurements [65] on high-qualityFeSe crystals, all indicating that the gap magnitude on one pocket of Fermisurface is an order of magnitude smaller than that on the other pocket [64].A number of STM measurements show evidence for the presence of nodesat the Fermi surface by detecting a V-shaped tunneling spectra [66, 67, 70].However, the STM measurements may not be quite low enough in tempera-ture to cleanly resolve the minimum gap. Furthermore, other measurementsof heat capacity [71, 72], lower critical field [73], and µSR [74] studies areconsistent with non-zero gap minimum.x ∆S(0) (meV ) ∆L(0) (meV )ab-plane 17% 0.23± 0.035 1.06± 0.050cˆ axis 26% 0.20± 0.030 0.94± 0.047Table 5.1: Two-gap fit parameters for ab-plane and cˆ axis.The two-gap model generates quantitatively similar fit parameters forthe ab-plane and cˆ-axis, which is expected since the temperature dependence36Chapter 5. Discussion and comparison with YBa2Cu3O6+δTemperature (K)0 1 2 3 4 5λab2(0)/λab2(T) dataTwo-gap modelTemperature (K)0 1 2 3 4 5λc2 (0)/λc2 (T) dataTwo-gap modelFigure 5.1: Equation 5.1 was used to fit the ab-plane and cˆ-axis superfluiddensity. The green dots denote the experimental data and the blue curverepresents the two-gap model fit. a) Two-gap model fit for ab-plane super-fluid density: x = 17%, ∆S(0) = 0.23 meV , ∆L(0) = 1.06 meV . b) Two-gap model fit for cˆ-axis superfluid density: x = 26%, ∆S(0) = 0.20 meV ,∆L(0) = 0.94 meV . 37Chapter 5. Discussion and comparison with YBa2Cu3O6+δFigure 5.2: a) A two-band extended s-wave model is fitted to the superfluiddensity, and the inset shows a polar plot of two gaps (∆1 and ∆2) at zerotemperature, for various values of the DOS parameter [63]. b) The Temper-ature dependence of the rms gap amplitudes on two bands, and the overallgap minimum.∆1(0) (meV ) ∆2(0) (meV ) ∆min(0) (meV )1.68± 0.12 0.52± 0.04 0.198± 0.03Table 5.2: Two-band extended s-wave model fit parameters [63].of cˆ-axis superfluid density is similar to that of ab-plane as seen in Figure4.7. In contrast, in YBa2Cu3O6+δ, the cˆ-axis superfluid density shows atemperature dependence between T 2 and T 3 [14], discernibly different fromthe linear-T dependence in both aˆ and bˆ directions, as shown in Figure 5.3.According to Xiang and Wheatley’s calculations [15, 48], cˆ-axis superfluiddensity would have a large theoretical T 5 dependence which stems from twosources: one T term comes from the d-wave linear density of state (DOS);the other T 4 term is due to the (cos kx − cos ky)4 factor in t2⊥(k‖) [48]. Incontrast, the in-plane superfluid density varies linearly with temperature dueto the nodes in YBa2Cu3O6+δ. Xiang et al. [48] pointed out this strikinglyanisotropic behaviour in the superfluid response of cuprates is caused by theinterplay between the d-wave superconducting order parameter symmetryand the underlaying Cu 3d orbital based electronic structure. The interlayer38Chapter 5. Discussion and comparison with YBa2Cu3O6+δFigure 5.3: The superfluid fraction in all principal crystallographic directionsof YBa2Cu3O6.95 [5]. The cˆ-axis superfluid density is qualitatively differentfrom the behaviour in either direction in ab-plane.39Chapter 5. Discussion and comparison with YBa2Cu3O6+δhopping integral t⊥ is a function of in-plane momentum k‖, and in a perfecttetragonal system, t⊥ is proportional to (cos kx− cos ky)2 according to localdensity approximation (LDA) band structure calculations. In the diagonaldirections (kx = ±ky) of a 2D Brillouin zone, the d-wave gap nodes ∆k‖ =∆(cos kx − cos ky) perfectly coincide with the zero’s of t⊥, and the cˆ-axissuperfluid 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 tem-perature [75]. Experiments [76, 77] have observed strongly warped Fermisurface, which also suggests a much more 3-dimensional process in the cˆdirection.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 superconduc-tors were extensively studied over the last three decades. In 1998, Hosseiniat al. [14] cleaved a thin plate of YBa2Cu3O6.99 crystal, measured the sur-face 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 inFigure 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 conductivitywas attributed to a competition between two temperature dependent quan-tities: the number density of thermally excited quasiparticles nn(T ) thatdeclines with temperature, and their transport scattering time τ(T ) that in-creases with decreasing temperature below Tc [14, 52]. The rapid increase intransport time has been interpreted as a collapse of the inelastic scatteringprocesses that are responsible for the large normal state resistivity of thehigh temperature superconductors.40Chapter 5. Discussion and comparison with YBa2Cu3O6+δFigure 5.4: Surface resistance along the three crystal directions ofYBa2Cu3O6.99. The cˆ-axis surface resistance is different from that of theab-plane [14].However, Rsc(T ) as well as σ1c(T ) are qualitatively different from thoseobserved 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. Aweak rise appears at low temperature, but is thought to be an artefact whichwas caused by the process of cleaving. To circumvent this problem, Hosseiniat al. [15] polished a relatively thick crystal into a thin blade, measuredthe surface resistance and penetration depth of that blade, and successfullyextracted σ1c(T ) as illustrated in Figure 5.7. The absence of a peak inthese cˆ-axis data indicates that charge transport between CuO2 planes isnot influenced by the development of large scattering lifetime below Tc.41Chapter 5. Discussion and comparison with YBa2Cu3O6+δFigure 5.5: Extracted microwave conductivity σ1c for two YBa2Cu3O6.99samples along the cˆ-direction [14].As we discussed in the superfluid density comparison, the cˆ-axis quasi-particle will mainly come from momenta away from the zone diagonal nodes,due to the dependence of the hopping on the in-plane momentum. On theother hand, photoemission experiments [78] have directly shown the quasi-particle lifetime τ is unusually small everywhere on the Fermi surface exceptclose to the zone diagonals. Therefore, the long lifetime quasiparticles havemomenta mainly in the nodal direction in the CuO2 planes and enhance thein-plane conductivity, but not the cˆ-axis conductivity.Similar to YBa2Cu3O6+δ, FeSe also has many quasiparticles below Tcand they are long-lived in the superconducting state, as shown by the peakin σ1(ω, T ) at low temperature. As with superfluid density, the big differencebetween YBa2Cu3O6+δ and FeSe is that FeSe exhibits this phenomenon in alldirections. It seems experimentally that quasiparticles at the gap minimumin FeSe have long lifetime and they also can move freely in the cˆ direction.This is consistent with the absence of strongly anisotropic behaviour in thesuperfluid density of FeSe.42Chapter 5. Discussion and comparison with YBa2Cu3O6+δFigure 5.6: The ab-plane microwave conductivity of a high purityYBa2Cu3O6.99 crystal exhibits a broad, frequency dependent peak causedby the development of long-lived quasiparticles in the superconducting state[11].43Chapter 5. Discussion and comparison with YBa2Cu3O6+δFigure 5.7: The anisotropy of the microwave conductivity of YBa2Cu3O6.95is illustrated by plotting cˆ axis conductivity (open squares) measured at 18GHz [15] along with the ab-plane conductivity (filled squares) taken at 1GHz [15].44Bibliography[1] C. C. Tsuei and J. R. Kirtley. Pairing symmetry in cuprate supercon-ductors. Rev. Mod. Phys., 72:969–1016, Oct 2000.[2] Jake Stanley Bobowski. Precision Electrodynamics of UnconventionalSuperconductors. PhD thesis, University of British Columbia, 2010.[3] Wikipedia. Iron-based superconductor — wikipedia, the free encyclo-pedia, 2016. 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