MICROWAVE FREQUENCY VORTEX DYNAMICS OF. THE HEAVY FERMION SUPERCONDUCTOR CeCoIn 5

Transcription

1 MICROWAVE FREQUENCY VORTEX DYNAMICS OF THE HEAVY FERMION SUPERCONDUCTOR CeCoIn 5 by Natalie Murphy B.Sc., Trent University, 2010 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE DEPARTMENT OF PHYSICS FACULTY OF SCIENCE c Natalie Murphy 2012 SIMON FRASER UNIVERSITY Fall 2012 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing." Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review, and news reporting is likely to be in accordance with the law, particularly if cited appropriately.

2 APPROVAL Name: Degree: Title of Thesis: Natalie Murphy Master of Science Microwave Frequency Vortex Dynamics of the Heavy Fermion Superconductor CeCoIn 5 Examining Committee: Dr. Patricia Mooney, Professor (Chair) Dr. David M. Broun, Senior Supervisor Associate Professor Dr. J. Steven Dodge, Supervisor Associate Professor Dr. Erol Girt, Supervisor Associate Professor Dr. George Kirczenow, Internal Examiner Professor Date Approved: August 29th 2012 ii

3 Partial Copyright Licence

4 Abstract Magnetic fields penetrate superconductors as quantized tubes of magnetic flux, or vortices. A transport current passed through such a superconductor exerts a transverse force on the vortices. The dissipation resulting from vortex motion is characterized by a vortex viscosity, and microwave techniques provide a powerful means of accessing this in the presence of pinning. A novel microwave spectroscopy apparatus has been set up that allows sensitive measurements of the dynamical properties of vortices to be made at temperatures down to 80 mk, in magnetic fields up to 9 T, and at frequencies from 2.25 to 25 GHz. A comprehensive study has been carried out at 2.5 GHz on the heavy fermion superconductor CeCoIn 5. Surprising new behaviour is revealed in the vortex viscosity, which, at low fields, exhibits clear signatures of d-wave quasiparticle physics. This suggests that delocalized excitations outside the vortex cores are predominantly responsible for vortex dissipation in CeCoIn 5. iii

5 Acknowledgments I would like to give a big thanks to everyone who made this thesis a reality and most of all thank you all for your patience with my countless questions. David Broun is the most inspiring academic I have met and I consider myself lucky to have had the chance to work and learn in his lab. Wendell Huttema is a great baseball player and knows a thing or two about what goes on in the lab. Sonia Milbradt was a helpful friend to traverse the day-to-day Masters life with from the beginning and a coffee buddy who will be missed. Colin Truncik is the master of analysis and an intense workout buddy. Ricky is the best TA around and has a wisdom that only time can build. AJ Koenig is the bestest new seat friend and makes lots of pretty pictures on my demand. Eric Thewalt put lots of work into the design of the in-field apparatus. Ken Van Wieren, Vic Allen, Ken Myrtle, and Bryan Gormann are always willing to give a hand with machining and are always willing to grab the BBQ for me. Thank you. Lastly I would like to thank my mother for helping me believe that I can do anything that I set my mind to. iv

6 Contents Approval Abstract Acknowledgments Contents List of Tables List of Figures ii iii iv v viii ix 1 Introduction Superconductivity Zero Electrical Resistivity Meissner Effect London Theory BCS Theory Ginzburg Laudau Theory and Vortices Vortices Type I vs. Type II Superconductors Meissner-State Electrodynamics Generalized Two-Fluid Model Complex Conductivity Surface Impedance v

7 CONTENTS vi Rs and Xs Vortex-state Electrodynamics Bardeen Stephen Model Introduction to Vortex Viscosity Pinning Gittleman Rosenblum Model Coffey Clem Model The Waldram Model Interpretation of our Experiment Experimental Apparatus Microwave Cavity Perturbation Cavity Perturbation Techniques Surface Impedance Vortex Dynamics and Cavity Perturbation In-Field Microwave Cavity Perturbation Apparatus The Sample Stage Dilution Refrigeration CeCoIn CeIn 3 Parent Compound of CeCoIn Structure and Fermi Surface of CeCoIn Superconducting Properties of CeCoIn Evidence For d-wave Superconductivity Quasiparticle Dynamics Results on CeCoIn Vortex Viscosity in Ortho-II YBCO Zero-Field Microwave Conductivity of CeCoIn Vortex Dynamics of CeCoIn Surface Impedance Pinning

8 CONTENTS vii Vortex Viscosity Flux-Flow Resistivity Conclusions 64 Bibliography 66

9 List of Tables 3.1 Vortex unit cell magnetic field and frequency limits Summary of model parameters viii

10 List of Figures 1.1 London penetration depth Ginzburg Laudau vortex profile B T phase diagram of type I and type II superconductors Vortex free body diagram Induced vortex electric fields Pinning potential Free-flow and the pinning limit Base mode magnetic field profile Cavity perturbation - f 0 and f B Illustration of in-field experiment In-field microwave cavity apparatus Sample stage illustration Photos of experimental set-up He cycle in the dilution refrigerator Thermal profile of the in-field apparatus Crystal structure of CeIn 3 and CeCoIn Pressure temperature phase diagram of CeIn B c2 (T ) of CeCoIn Alloy series phase diagram of CeXIn Heat capacity of CeCoIn Superfluid density of CeCoIn Penetration depth and superfluid density of CeCoIn ix

11 LIST OF FIGURES x 5.8 Frequency dependence superfluid density of CeCoIn Quasiparticle conductivity of CeCoIn 5 and YBCO Vortex viscosity of Ortho-II YBCO Flux-flow resistivity of Ortho-II YBCO Zero-field complex conductivity of CeCoIn Surface resistance of CeCoIn Surface resistance on a log scale Surface reactance of CeCoIn Depinning frequency Pinning constant Vortex viscosity Vortex viscosity at low magnetic fields Vortex viscosity plotted against B Flux-flow resistivity of CeCoIn Flux-flow resistivity divided by magnetic field of CeCoIn

12 Chapter 1 Introduction This thesis describes a set of experiments that use microwave techniques to reveal the dynamical properties of vortices in the heavy fermion superconductor CeCoIn 5. I will introduce superconductivity, zero-field and vortex state electrodynamics, then present specifics of the experiment and particular sample properties, ending with the presentation of new data on the vortex dynamics of CeCoIn 5. In this chapter, I will introduce superconductivity and some of the theories that describe its properties, with a particular focus on showing how vortices emerge from flux quantization. 1.1 Superconductivity Superconductivity is a macroscopic quantum mechanical phenomenon discovered in 1911 by Heike Kemerlingh Onnes [1], in Leiden, Netherlands. His lab had recently discovered how to liquify helium, making experiments at low-temperatures possible. At the time there were two competing theories on the behaviour of metals at lowtemperatures. Some scientists believed conduction would completely stop at absolute zero, corresponding to an infinite resistivity. On the other hand, some thought the electrical resistivity would slowly decrease until it reached zero. Onnes was astonished to find that when measuring a sample of mercury the electrical resistivity suddenly dropped to zero this was the discovery of superconductivity. 1

13 CHAPTER 1. INTRODUCTION Zero Electrical Resistivity The first defining characteristic of superconductivity is zero DC resistivity below a characteristic critical temperature, T c, of the material. An interesting consequence of this is that, for a loop of superconducting wire, electrical current will persist indefinitely without a power source Meissner Effect In 1933, Meissner and Ochsenfeld observed that superconductors also expel magnetic field below T c [2]. This is the second defining characteristic of superconductivity and is referred to as the Meissner effect. In reality the magnetic filed is not completely expelled, but manages to penetrate a small distance into the surface of the superconductor. This will be discussed in the following section on London Theory, which was the first theory to explain the Meissner effect. It is important to note that superconductivity is the combination of zero DC resistivity (perfect DC conductivity) and the Meissner effect. If a sample exhibited perfect conductivity alone then there would be no electric field inside the sample and, by Faraday s law no change in the magnetic flux. In particular, if the sample were cooled through T c in a weak external field there would be no expulsion of flux. The Meissner effect sets superconductivity apart because the expulsion of flux to create the B = 0 state indicated that superconductivity is a distinct thermodynamic phase. Type I vs. Type II Superconductors Superconductors are classified into two groups depending on their response to external applied field. For type I superconductors the Meissner state is destroyed above the critical field B c (T ). For type II superconductors the Meissner state is broken at the lower critical field, B c1 (T ), to allow partial penetration of the magnetic field, in a configuration known as the mixed state or vortex state. In this state tubes of normal metal run throughout the material these are the vortex cores, discussed in more detail in Section 1.4. Superconductivity is then fully destroyed at the upper critical field, B c2 (T ). See Section for more details on the mixed state and B T phase diagrams.

14 CHAPTER 1. INTRODUCTION 3 I will take this opportunity to address the convention for B and H used throughout this thesis, starting with a quote from Griffiths [3], pg. 271: "Many authors call H, not B, the "magnetic field." Then they have to invent a new word for B: the "flux density," or magnetic "induction" (an absurd choice, since that term already has at least two other meanings in electrodynamics). Anyways, B is indisputably the fundamental quantity, so I shall continue to call it the "magnetic field," as everyone does in the spoken language. H has no sensible name: just call it "H." " To be precise, the relationship between B and H is defined as follows (for linear media): B = µ 0 (H + M), (1.1) where M is the magnetization. Since CeCoIn 5 (the material studied here) is a strong type II superconductor M 0 for all applied fields, thus B µ 0 H. (1.2) For the duration of this thesis, when B is mentioned I am referring to µ 0 H, and I will use B E to explicitly denote the external magnetic field. 1.2 London Theory In 1935, shortly after the discovery of the Meissner effect, Fritz and Heinz London devised the first phenomenological theory of superconductivity [4]. In essence, London theory relates supercurrents and electromagnetic fields to describe the two principal properties of superconductivity. In this theory the assumption is made that the supercurrent can be described by a single superfluid wavefunction Ψ = n s e iθ(r), where n s is the density of superconducting electrons and θ(r) is the phase. Note that the physical meaning of the wavefunction was not understood until the microscopic BCS theory was developed in 1957 (Section 1.3). The London brothers showed that the supercurrent J s can be written in terms of the phase θ and magnetic vector potential A as: ( ) ΛJ s = 2e θ + A, (1.3)

15 CHAPTER 1. INTRODUCTION 4 where Λ = me n se and is called the London parameter, m 2 e being the mass of an electron, n s the density of superconducting electrons, and e the electron charge. Using the Schrödinger equation we can get an equation for the rate of change of the phase of the superfluid wavefunction, θ t = 2µ, (1.4) where µ is the electrochemical potential. This tells us that to change the phase of the wavefunction takes energy. Taking the time derivative of Equation 1.3 and using Equation 1.4 produces the first London equation: t (ΛJ s) = A ( µ ) t + E eff, (1.5) e where E eff is defined as the effective driving field of the electrons. This equation is also known as the acceleration equation as it indicates free acceleration of super-electrons in an electric field. In other words it describes perfect conductivity: after a short pulse of electric field the system will be left with a non-decaying current. Taking the curl of Equation 1.3 produces the second London equation: (ΛJ s ) = B. (1.6) Combining this with Ampere s law reveals the screening equation underlying the Meissner effect, where λ L = 2 B = 1 B. (1.7) λ 2 L Λ µ 0 = me µ 0 n se is the London penetration depth. 2 For a specific case in which the applied magnetic field, B 0, is pointing in the z-direction (parallel to the sample surface) and x is the direction perpendicular to the sample surface (directed into the sample) the solution to the screening equation is B z (x) = B 0 e x/λ L, as seen in Figure 1.1. Thus the magnetic field decays exponentially as it enters the sample, with a characteristic length scale of λ L. Given that a typical value for λ L in conventional superconductors is on the order of 50 nm explains why flux exclusion seems complete in macroscopic samples, thus providing a description of the Meissner state.

16 CHAPTER 1. INTRODUCTION 5 z λ L B0 x y Figure 1.1: Illustration of the London penetration depth, λ L. Solving the screening equation, Equation 1.7, yields B z (x) = B 0 e x/λ L, shown as the dotted line in the figure. Thus magnetic field decays within the sample with the length scale of λ L blue arrows indicate the magnetic field strength inside and outside the sample. Flux Quantization The London theory also reveals that flux can be trapped by a superconducting circuit. We can learn more about this by taking the line integral of Equation 1.3. Since the phase must be single valued, the change on going around the loop must be a multiple of 2π, so θ dl = n2π, where n is an integer. This leads to the condition for fluxoid quantization: (A + ΛJ s ) dl = 2e 2πn = n h 2e = nφ 0, (1.8) where Φ 0 is the superconducting flux quantum, Φ 0 = h. The left hand side of Equation 1.8 2e is thus quantized and known as the fluxoid. Note that magnetic flux itself is not always quantized; this is only the case for geometries containing contours along which J s = 0 (inside the bulk of a superconductor, for instance). In that case, the magnetic flux Φ becomes Φ = Bds = A dl (1.9) = nφ 0. (1.10)

17 CHAPTER 1. INTRODUCTION 6 Because of flux quantization, any loop of superconductor has the potential to trap flux. For type II superconductors, in the mixed state, when flux is able to penetrate the superconductor, each unit cell in the vortex lattice contains carries exactly one superconducting flux quantum, hence the name quantized flux lines. 1.3 BCS Theory The first microscopic theory of superconductivity was developed by Bardeen, Cooper, and Schrieffer in 1957, and is known as BCS theory [5]. This theory explains the supercurrent as a superflow of paired electrons known as Cooper pairs. A Cooper pair has integer spin, and thus behaves as a composite boson. At T c the system undergoes a type of Bose Einstein condensation in which all the Cooper pairs condense into the same quantum state the BCS ground state. All the pairs in this ground state share the same pair momentum, and similarly to London theory, can be described by the center-of-mass wavefuntion. The BCS ground state is the most energetically favourable state, and opens up a BCS energy gap, k, at the Fermi surface. At T = 0, 0 = max[ k (T = 0)] k B T c. (1.11) ( 0 = 1.76k B T c in weak-coupling BCS theory.) Note, k shrinks as T T c until it fully closes at T = T c and superconductivity is lost. At finite temperatures, thermal excitations break some of the Cooper pairs to form normal quasiparticles. This leads to the following energy spectrum for the quasiparticles, Ek 2 = ε 2 k + 2 k, (1.12) where ε k is the corresponding excitation energy in the normal state. While the original BCS theory was devised for the case of conventional superconductors, for which an electron-phonon-electron interaction mediates the formation of Cooper pairs, it is possible to extend the theory to use other pairing mechanisms. This isotropic phonon pairing interaction in conventional superconductors leads to an isotropic energy gap ( k is independent of k), known as an s-wave energy gap. On the other hand, for highly anisotropic pairing interactions, such as pairing mediated by antiferromagnetic spin fluctuations, the generalized BCS theory results in pairing states with different symmetry. These

18 CHAPTER 1. INTRODUCTION 7 are called unconventional superconductors. Examples of this are the d-wave energy gaps of the cuprate superconductors, and CeCoIn 5. One characteristic property of the d-wave superconducting state is the existence of nodes in the energy gap. Quasiparticle excitations near the nodes dominate the low temperature properties. 1.4 Ginzburg Laudau Theory and Vortices Ginzburg Laudau theory is a phenomenological theory of superconductivity developed in 1950, which describes situations in which superconductivity is non-uniform, such as near vortices in the mixed state of type II superconductors [6]. This theory combines Landau s theory of second-order phase transitions with a Schrödinger-like wave function. The superconducting phase transition can be expressed in terms of a complex order parameter, Ψ, reminiscent of the center-of-mass wavfunction of both London and BCS theories. As an order parameter, Ψ is zero above T c and is related to the density of superconducting pairs below T c, such that Ψ 2 = n p, where n p is the density of superconducting pairs. The free energy of the system can be expressed in terms of Ψ, and the second-order phase transition, from a normal metal to a superconductor, results from minimizing this Ginzburg Laudau free energy with respect to variations in the order parameter and vector potential. Ginzburg Laudau theory contains two characteristic length scales: penetration depth, m e β λ = 4µ 0 e 2 α, (1.13) and coherence length, ξ = 2 2m e α, (1.14) where α and β are temperature dependent parameters. The penetration depth reduces to the London penetration depth λ L as Ginzburg and Landau define n p = α/β. The coherence length, on the other hand, is not defined in London theory, and in BCS theory, ξ is regarded as the size of the Cooper pairs. In Ginzburg Laudau theory ξ is the characteristic distance over which spatial changes in the order parameter, Ψ, occur. This helps us describe vortices, as will be shown in the next section.

19 CHAPTER 1. INTRODUCTION Vortices As mentioned above, type II superconductors are a category of superconductors where, in addition to being in the Meissner state for some range of magnetic fields, they also allow magnetic field to partially penetrate, in the form of quantized flux lines, for a higher range of applied B-field. Ginzburg Laudau theory informs us that the order parameter Ψ is suppressed over a distance ξ near the flux line. λ and ξ together enable us to obtain a picture of what is happening in the vicinity of a flux line, or vortex, as shown in Figure 1.2. The vortex core can be approximately treated as normal within the radius r = ξ. Screening supercurrents flow around the normal core up to the characteristic length λ, setting the scale over which magnetic fields decay around the vortex core. Br r Figure 1.2: Magnetic field B ( r) and superconducting order parameter Ψ(r) profile of a vortex, as given by Ginzburg Laudau theory. Ψ(r) sets the size of the vortex core dropping off with a length scale ξ, known as the coherence length. For type II superconductors, like CeCoIn 5, λ is much greater then ξ (as shown in the figure), and since B ( r) drops off with the length scale λ the magnetic field penetrates the vortex unit cell almost uniformly Type I vs. Type II Superconductors Normal/Superconducting Boundaries It turns out that the sign of the boundary energy between normal and superconducting material determines whether or not vortices form in a material. If the boundary energy is positive the boundary will be in tension and thus stable, resulting in a type I superconductor

20 CHAPTER 1. INTRODUCTION 9 (Meissner state only). On the other hand, if the boundary energy is negative the system is unstable and will try to maximize the area of the normal/superconducting boundary. This leads to a finely divided structure of normal regions inside the superconductor, or vortex lattice (most often a triangular lattice). The amount of magnetic flux associated with each vortex is set by the flux quantum, from Section 1.2. The sign of the boundary energy depends on the relative size of the two length scales from Ginzburg Landau theory. The Ginzburg Landau parameter is defined at κ = λ/ξ and has the following values for type I and type II superconductors. κ < positive boundary energy Type I superconductivity κ > negative boundary energy Type II superconductivity Field Temperature Phase Diagrams As mentioned above a type I superconductor consists of only the Meissner state as field is increased up to B c (T ). This results in the B T phase diagram shown in Figure 1.3a. On the other hand, the phase diagram of a type II superconductor contains the mixed state, where magnetic flux is able to partially penetrate the sample in the form of vortices, or quantized flux lines. The resulting B T phase diagram is shown in Figure 1.3b. In this thesis, we are interested in studying type II superconductors, using vortex dynamics. To do so we had to incorporate an external magnetic field into our experiment, in order to drive our samples into the mixed state.

21 CHAPTER 1. INTRODUCTION 10 E E (a) (b) Figure 1.3: B T phase diagram of type I superconductors (a) and type II superconductors (b). Note the critical field lines in each case: B c (T ), which separates the two different states for a type I superconductor: B c1 (T ), the lower critical field separating the Meissner and mixed state for type II superconductors; and B c2 (T ), the upper critical field separating the mixed and normal state. From Zhou [7]; used with permission.

22 Chapter 2 Meissner-State Electrodynamics The goal of this chapter is to derive a two-fluid model of the zero-field electrodynamics of a superconductor, and then show how measurements of surface impedance can access this information. 2.1 Generalized Two-Fluid Model The two-fluid model attempts to describe a superconductor by saying there are two types of charge carriers that coexist in the superconducting material: normal electrons and superelectrons. The normal electrons form a normal fluid, which conducts with resistance, and the superelectrons form a superfluid, which conducts with no resistance and carries no entropy. These two set of electrons conduct in parellel. For a DC measurement the superfluid acts as a short circuit, and ρ DC = Complex Conductivity We can use the ideas from the two-fluid model to split the conductivity into a superelectron component and normal or quasiparticle component: σ tot = σ s + σ qp, (2.1) where σ s is the conductivity from the superelectrons and σ qp is the conductivity from the quasiparticles. 11

23 CHAPTER 2. MEISSNER-STATE ELECTRODYNAMICS 12 Superconducting Conductivity To derive the conductivity of the superconducting electrons we start with the London acceleration equation, Equation 1.5, t (ΛJ s) = E eff, (2.2) where Λ = me n se = µ 2 0 λ 2. We convert to the frequency domain and use phasor notation, with the following convention for time-harmonic fields: x(t) = R { xe iωt }, and thus / t iω. iωλj s = E eff (2.3) J s = 1 iωλ E eff (2.4) Here we are implicitly assuming that the electrodynamics are local, leading to a current response of the form J = σe. Thus, Equation 2.4 implies the superconducting conductivity, σ s, has the following form: Quasiparticle Conductivity σ s = 1 iωλ = 1 iωµ 0 λ 2 = n se 2 iωm e. (2.5) For normal electrons we start from a force equation, incorporating a damping force that uses the relaxation time approximation, F d = m e v qp /τ: v qp m e = ee m ev qp, (2.6) t τ where v qp is the velocity of the quasiparticles. Again using the phasor form for / t iω we rearrange the force equation to obtain an expression for v qp, v qp = ee [ ] 1. (2.7) m e iω + 1/τ We want to get an expression of the form, J = σe, so we start from J n = n n ev qp, J n = n n ev qp = n [ ] ne 2 1 E. (2.8) iω + 1/τ m e This leads to the following equation for the quasiparticle conductivity, σ qp = n [ ] ne 2 1. (2.9) m e iω + 1/τ

24 CHAPTER 2. MEISSNER-STATE ELECTRODYNAMICS 13 Total Conductivity The total conductivity can now be expressed as, σ tot = σ s + σ qp (2.10) = n se 2 + n [ ] ne 2 1. (2.11) iωm e m e iω + 1/τ This form of the conductivity allows us to interpret our complex conductivity data in terms of the quasiparticle and superconducting electron properties. The complex conductivity is commonly written as σ tot = σ 1 iσ 2, where σ 1 and σ 2 are real. Therefore, the real part of the conductivity, σ 1, tells us explicitly about the normal quasiparticles, and σ 2 predominantly contains information about the superconducting electrons. 2.2 Surface Impedance Starting from Maxwell s equations, transforming into the frequency domain (using the phasor notation described above), assuming B = µ 0 H, neglecting displacement currents, and assuming local electrodynamics, J = σe gives, E = B t = iωµ 0H, (2.12) H = J = σe. (2.13) Taking the curl of Equation 2.13 and substituting in Equation 2.12 yields the following screening equation, 2 H = iωµ 0 σh, (2.14) 2 H = 1 δ2 H, (2.15) where δ 2 = 1 1 iωµ 0, or alternatively σ =. Here δ is the known at the complex skin σ iωµ 0 δ2 depth. Surface impedance is defined as Z s = E H, (2.16)

25 CHAPTER 2. MEISSNER-STATE ELECTRODYNAMICS 14 where refers to the field component parallel to the sample surface. Let us consider a specific geometry such that E = E y and H = H z, where we are concerned with field propagation into the sample in the x-direction. Solving the screening equation, Equation 2.15, for the H-field we obtain, H z (x) = H 0 e x/ δ. (2.17) Equation 2.13 expresses the corresponding electric field as, E = 1 σ H z, and this curl gives E purely in the y-direction, 1 σ H z = 1 σ ˆx ŷ ẑ x y z 0 0 H z = 1 σ This leads to the following expression for the electric field, Now we can solve for the surface impedance directly, ( 0, H ) z x, 0. E y = 1 σ δ H z. (2.18) Z s = E H = E y H z = 1 σ δ (2.19) = iωµ 0 δ (2.20) iωµ0 = σ. (2.21) Rearranging Equation 2.21 one can obtain an expression for complex resistivity, ρ, as a function of Z s (using the definition ρ = 1 σ ), where Z s (B, T ). ρ = Z2 s iωµ 0, (2.22) Rs and Xs Starting with Z s (as derived above) in terms of the complex conductivity, σ = σ 1 iσ 2, iωµ0 Z s =, (2.23) σ 1 iσ 2

26 CHAPTER 2. MEISSNER-STATE ELECTRODYNAMICS 15 and manipulating this expression, assuming σ 2 expression for surface impedance: = 1/ωµ 0 λ 2, we arrive at the following ( ) 1/2 1 Z s = iωµ 0 λ. (2.24) 1 + iσ 1 /σ 2 In the limit σ 1 σ 2, the binomial expansion applies, giving: [ Z s = iωµ 0 λ 1 i σ 1 + O [ (σ 1 /σ 2 ) 2] ]. (2.25) 2 σ 2 Substituting in σ 2 again, yields: and thus, Z s = R s + ix s, (2.26) Z s 1 = 2 ω2 µ 2 0λ 3 σ 1 + iωµ 0 λ, (2.27) R s 1 = 2 ω2 µ 2 0λ 3 σ 1, (2.28) X s = ωµ0 λ. (2.29) We see that the surface reactance provides a very direct measure of the penetration depth in situations where σ 1 σ 2. Also, we see that R s σ 1, which at first is somewhat surprising, but arises because the superelectrons set the magnitude of the electric field at the sample surface when σ 2 σ 1.

27 Chapter 3 Vortex-state Electrodynamics One of the more difficult aspects of microwave spectroscopy experiments in the vortex state is the interpretation of the data. Theoretical models help us explain the microscopic details of what our experiment is probing within the sample, and introduce parameters for us to study and compare. In the following chapter I will give a brief outline of the most relevant models, and I will finish with a section summarizing the specific analysis carried out here. 3.1 Bardeen Stephen Model The Bardeen Stephen model was the first and remains the best known model of the microscopic origin of vortex dissipation [8]. This pioneering model was originally designed to explain DC measurements of flux-flow resistivity, in which pinning effects have been overcome (see Section 3.3 for more details). A great simplification made by Bardeen and Stephen is treating the vortex core as a cylinder of normal metal, with a sharp boundary to the surrounding superconducting background at radius r ξ. For simplicity, a single vortex was modelled. See Figure 3.1 where a cross section of a vortex unit cell is shown. Note the supercurrent that flows around the vortex to screen (and trap) the magnetic flux. When a transport current is passed through a superconducting sample in the mixed state it applies a Lorentz force that acts to move the vortex perpendicular to both the external applied magnetic field and the transport current, as shown in Figure 3.1. Moving vortices imply a time rate of change of magnetic flux though the sample, and electric fields are 16

28 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 17 Figure 3.1: Illustration of free flux-flow in a vortex unit cell. The vortex core is represented by the dark grey circle and is surrounded by the superconducting background (light grey). The magnetic field is directed out of the page, producing the supercurrent flow, J S, around the vortex, shown in blue. J T is the transport current being applied across the sample, and v is the resulting vortex velocity. generated. The induced vortex electric field has a dipole form outside the vortex core and is uniform inside the core, see Figure 3.2. The dipole shape of the electric field is derived from the London Equation, Equation 1.5, which states the time rate of change of the vortex screening currents, J s (or J fl in Figure 3.1), is proportional to the effective electric field. The key result of Bardeen Stephen theory is that an essentially uniform electric field inside the core drives the transport current directly through the normal core of the vortex, causing dissipation. The theory comes to a very simple and logical equation for flux-flow resistivity, ρ ff = ρ n B E B c2, (3.1) where ρ n is the normal resistivity of the material, B E is the applied field, and B c2 is the upper critical field. Note, B E B c2 = f n is the normal fraction of the vortex unit cell. This also allows us to define a superconducting fraction, f s = 1 f n. The vortex cores, which are responsible for the increase in dissipation in the mixed state, are embedded in a superconducting medium. At the low frequencies, the impedance of the superconducting medium is so small that it can be ignored. At frequencies in the microwave range, however, this is no longer true, and this background impedance must be taken into account in any analysis of microwave data.

29 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 18 V L J T Figure 3.2: Illustration of the induced vortex electric fields. Note the similarities to a dipole electric field. From Zhou [7]; used with permission. 3.2 Introduction to Vortex Viscosity For this section I will continue to consider the case where the pinning force is zero, and I will work within a single vortex unit cell. As we move away from DC, we become concerned with vortex dynamics and benefit from writing down an equation of motion for the vortex. The primary forces are a drag force and an elastic restoring force. It is very useful to represent the drag force by a vortex viscosity η, which gives the linear coefficient of friction per unit length of vortex. For a vortex moving at a transverse velocity v (v fl in Figure 3.1) the viscous force (per unit length of flux line) is, F l = ηv. (3.2) From this we can obtain the power dissipation per unit length, P l = F l v (3.3) P l = ηv 2. (3.4) It is useful to relate η to flux-flow resistivity, which is defined in the usual manner, as the coefficient relating electric field E to current density j, E = ρ ff j. This gives a power

30 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 19 dissipation per unit volume in terms of ρ ff : P v = j E (3.5) P v = ρ ff j 2. (3.6) Relating these expressions for power dissipation (by taking into account the area of the vortex lattice unit cell, A = Φ 0 ) we obtain, B P l = P v A (3.7) ηv 2 = ρ ff j 2 Φ 0 B ρ ff = ηv2 j 2 (3.8) B Φ 0. (3.9) We can simplify this relation by using the vortex force equation in the steady state. For now we are ignoring pinning, so there are only two forces acting on each vortex: The Lorentz force due to the transport current (F l ); and the viscous force introduced above. In the steady state these forces must be equal and opposite, Φ 0 j = ηv, (3.10) where Φ 0 j is the Lorentz force per unit length written in terms of the superconducting flux quantum Φ 0 and transport current j, and assuming the transport current is being applied perpendicular to the applied magnetic field. Rearranging Equation 3.10 gives the following equation for v: v = Φ 0j η. (3.11) Substituting Equation 3.11 into Equation 3.9 results in a simple relation between ρ ff and η: ρ ff = BΦ 0 η. (3.12) Note, this relation holds in the absence of pinning, but serves to illustrate the connection between the two quantities. Each of these quantities gives us different information about the microscopics of our sample. ρ ff is useful for comparison to DC transport measurements, and we will show that the vortex viscosity has close connections to the zero-field quasiparticle dynamics of d-wave superconductors.

31 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 20 Figure 3.3: Illustration of pinning potentials: (a) harmonic pinning approximation, and (b) applied AC transport current, as used in our experiments. Note that by using careful power selection during the experiment we ensure a linear pinning response for ease of the interpretation. 3.3 Pinning Vortices can be pinned in place by defects in a sample. A particular pinning site in a sample is a place in space where it is more energetically favourable for a vortex to sit. Generally, the condensation energy is lower at the pinning site. That is, it is less costly to suppress superconductivity to form the vortex core. While the physics of pinning is of great fundamental interest and practical importance, in these experiments we are using microwave frequencies to overcome pinning, and are mostly concerned with how pinning affects the interpretation of viscosity data. The effect of pinning is often approximated by a sinusoidal potential running through the sample. See Figure 3.3 where a vortex is represented by a point particle in a sinusoidal potential. Figure 3.3(a) shows the harmonic approximation we use to represent a single pinning site. In a DC measurement, when the flux lines are pinned in place, the transport current J T (DC) will flow around the vortex cores, staying in the superconducting medium. This results in no dissipation. As J T (DC) is increased the current will reach a critical value (know as the critical current density, a property set by the material). The vortices are pushed out of their pinning sites, and are now free to flow through the sample. In reference to the sinusoidal potential the vortices are given enough energy to go up and over the barrier. For

32 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 21 a DC measurement in this regime, pinning can be ignored; once in free flow, the force from the pinning sites should average to zero. AC measurements, on the other hand, must take pinning into consideration. With an AC transport current J T (AC) the vortices are not being forced out of their local pinning sites. Instead the vortices are being shaken back and forth in their potential wells, Figure 3.3(b). The rest of the models that I will describe account for pinning forces. It is important to note, in our experiments we make sure we remain in the linear response regime, which greatly simplifies the analysis. Note, in most situations, the interaction of vortices with the surface of the superconductor is likely to give rise to pinning effects. Although not necessarily stronger than pinning sites inside the bulk of the sample, surface pinning is important for two reasons: for a vortex running parallel to the surface, the pinning potential will act along the entire length of the vortex, rather than at isolated points; and in our experiments, vortex dynamics are inferred from measurements of surface impedance (Section 2.2 and Section 4.1.2) a surface sensitive probe. This is most relevant for the interpretation of our pinning constant and depinning frequency data. However, since we are mainly interested in vortex viscosity, surface pinning might actually work to our advantage, since the simplest situation to interpret is one in which all vortex matter visible to the microwave fields experiences the same pinning potential. 3.4 Gittleman Rosenblum Model The Gittleman Rosenblum model is the first and simplest model for extracting flux-flow resistivity and pinning constant, α p, from microwave experiments [9]. Gittleman and Rosenblum take a different approach from Bardeen and Stephen (Section 3.1) when it comes to vortex representation. Here vortices are treated as massless point objects, ignoring microscopic structure, and pinning is treated as resulting from harmonic potentials (as discussed in Section 3.3). Adding pinning to the vortex equation equation of motion (Equation 3.10) yields Φ 0 j = ηv + α p x. (3.13) Because we are using an AC transport current, it makes sense to work in the frequency

33 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 22 domain. Using the phasor representation introduced previously to solve Equation 3.13 in the frequency domain gives ρ eff = BΦ 0. (3.14) η + αp iω Gittleman and Rosenblum introduced a quantity called the depinning frequency, ω p = α p /η. The effective resistivity Equation 3.14, can then be rewritten as ρ eff = BΦ 0/η. (3.15) 1 i ωp ω The depinning frequency is the frequency at which viscous and elastic forces are equal in magnitude. When ω ω p vortices are essentially pinned in place and ρ ff 0. When ω ω p vortices are free to flow (pinning is essentially zero) and ρ eff ρ ff (DC) (ρ ff defined by Bardeen and Stephen Equation 3.12) hence the name depinning frequency. This model provides a good starting point for the analysis of our microwave measurements as it allows for the extraction of both ρ ff (DC) and α p. However, it doesn t provide the complete picture, as it continues to ignore the superconducting background the vortices are embedded in. 3.5 Coffey Clem Model The Coffey Clem model [10] is a refined version of the Gittleman Rosenblum model (Section 3.4); vortices are again treated as point particles. Coffey and Clem specifically model the AC case, dealing with the exponential attenuation of AC fields at the surface of the superconductor. The experimentally accessible quantity is surface impedance Z s, or equivalently the complex skin depth δ. The Coffey Clem model solves the AC penetration problem at the surface of a superconductor in the presence of vortices. The main contribution of this model is to show how to include the finite impedance of the superconducting background. The end result is very simple: at low temperatures, and for B E B c2, ρ eff ρ v + ρ s, (3.16) where ρ s = iωµ 0 λ 2 L is the effective complex resistivity of the superconducting medium in the absence of vortices, and ρ v = ρ eff 1 iω p/ω is the vortex contribution to the resistivity, exactly

34 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 23 the same as in the Gittleman Rosenblum model (Equation 3.15), but derived independently. We will motivate this result further below, and show how it must be extended as B E B c The Waldram Model The (unpublished) Waldam model [11] is a high frequency version of the Bardeen Stephen model. Like Gittleman Rosenblum and Coffey Clem, Waldram deals with microwave experiments, but instead of using the point particle representation of a vortex Waldram follows the Bardeen Stephen model a vortex is treated as a cylinder of normal metal surrounded by a superconducting background. Waldram focuses on solving for the profiles of the electric field and transport current inside the vortex lattice unit cell. The results provide useful insight into the form of the transport current, which we are able to apply to our analysis. Waldram defines two limits: the free-flow limit, ω ω p (Figure 3.4 a); and the pinning limit, ω ω p (Figure 3.4 b). Most relevant to our analysis is the high frequency limit, ω ω p, in which Waldram shows that the transport current density is uniform throughout the vortex unit cell. This knowledge greatly simplifies the analysis to follow. 3.7 Interpretation of our Experiment In this section I will describe how we combine the models defined above to analyse the data from our experiment. At low fields, B E B c2, the Coffey Clem result applies: ρ eff = ρ s + ρ v, (3.17) where ρ v = ρ ff = BΦ 0. (3.18) 1 i ωp η + αp ω iω In the low field regime, we don t need to make any assumptions about the magnitude of the measurement frequency related to the depinning frequency because the area fraction of the normal core, f n, is very small compared to the area fraction of superconducting background, so f s 1. Another way to put this is, in the low-field limit vortices are small,

35 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 24 Js JT (a) (b) Figure 3.4: Illustration of the flux free-flow limit (a) and the pinning limit (b) as described by Waldram. The vortex core is represented by the grey cylinder; the transport current, J T is shown in red; the supercurrent, J s, is shown in blue; the velocity of the vortex, v, is shown in yellow; and the magnetic field in the sample, B, is shown in green. so we do not need to worry about the small amount of superconductor being excluded by the vortex core; transport current streamlines in Table 3.1 (b) are almost identical to the field lines in Table 3.1 (c). The simple expressions above are not expected to be accurate outside the low field regime. The reason is that as B E becomes comparable to B c2, the vortex core starts to occupy an appreciable fraction of the unit cell, Table 3.1 (d). The main effect of this is simply volume exclusion, and can be taken into account by taking the shrinking size of the superconducting regions into account. To see this clearly, consider the high frequency limit, ω ω p. In this case, from Waldram s work, we know the vortex is essentially in free flux flow Figure 3.4 (a), and that the transport current density is uniform Table 3.1 (f). The exact solution for the power dissipation can then be found. Calculation of the power dissipation (per unit length of flux line) for the case of uniform

36 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 25 Magnetic Field Vortex size ω ω p ω ω p B E B c2 (a) (b) (c) B E B c2 (d) (e) (f) Table 3.1: Illustration of vortex core size relative to vortex unit cell size for the low- and high-field limits, showing transport current profiles for frequency limits in both cases. transport current proceeds as follows: P l = ρ(r)jt 2 (r)ds (3.19) = JT 2 ρ(r)ds (3.20) = JT 2 [ρ s A s + ρ n A n ] (3.21) where ρ s is the resistance of the superconducting material outside the core, A s is the area of superconducting background, ρ n is the resistivity of the normal material in the vortex core, and A n is the area of normal vortex core. This gives us a refined definition for the effective resistivity where we weight the superconducting and normal contributions by their area fraction, ρ eff = f s ρ s + f n ρ n. (3.22) It is important to note that for our analysis we absorb f n into the definition of ρ v : ρ v = f n ρ n in the Bardeen-Stephen case. This results in the following equation of the effective resistivity, ρ eff = f s ρ s + ρ v, (3.23) which agrees with Coffey-Clem at low fields where f s is approximately 1. Thus we are justified in using this model over our entire magnetic field range. The resistivity we measure using cavity perturbation is ρ eff, but ρ v is what interests us as it directly gives the pinning constant α p and vortex viscosity η. We define ρ s using

37 CHAPTER 3. VORTEX-STATE ELECTRODYNAMICS 26 Equation 2.22 as and ρ s = Z2 s (B = 0) iωµ 0, (3.24) f s = 1 f n (3.25) = 1 B B c2, (3.26) which allows the vortex component to the resistivity, ρ v, to be isolated. ρ v = ρ eff f s ρ s (3.27) We can then interpret this data using the models described above Equation 3.18, ρ v = ρ ff 1 i ωp ω = BΦ 0. Table 3.2 provides a summary of the model parameters we calculate and η+ αp iω present in the results section of this thesis. The model we use is somewhat heuristic, but we justify its use for the following reasons: it is simple; it gives the correct crossover behaviour as B E B c2, that is ρ eff ρ n ; and it is exact at B E B c2 since in this limit ω p 0. Thus, at high magnetic fields the system will always satisfy the frequency constraints for which it was defined, ω ω p, Table 3.1 (f). Vortex viscosity Pinning constant Depinning frequency DC flux-flow resistivity { η = BΦ o R } 1 ρ v { α p = ωbφ 0 I ω p = αp η ρ 1 ff = R { 1 I{ = ω R{ 1 } 1 ρ v } 1 ρ v } ρv } ρv Table 3.2: Summary of model parameters.

38 Chapter 4 Experimental Apparatus In this chapter I will introduce the concept of microwave cavity perturbation and describe how it is used to obtain surface impedance data. I will describe our experiment set-up, particularly my own personal contribution to its design and construction. Finally, I will outline the general operation of a dilution refrigerator, describing each stage and how it is used to cool the sample and microwave cavity, maintaining each at a separate temperature. 4.1 Microwave Cavity Perturbation Cavity perturbation is a powerful technique for measuring electrical transport properties of a superconductor, at high frequencies and below its critical temperature, T c. It also benefits from being a contactless measurement, as attaching contacts to millimeter sized samples is difficult and can damage the sample itself. The experimentally accessible quantity is the complex surface impedance, Z s. When carried out correctly, cavity perturbation allows real and imaginary parts of Z s to be measured. This is very important in the current experiment, as it enables us to distinguish between viscous and elastic forces on vortices, which occur 90 o out of phase. Complex surface impedance can then be interpreted in terms of microscopic quantities, which are model-dependent, see Chapter 3. Examples of these derived quantities include: complex conductivity, vortex viscosity, depinning frequency, pinning constant, and the equivalent DC flux-flow resistivity. 27

39 CHAPTER 4. EXPERIMENTAL APPARATUS Cavity Perturbation Techniques The simplest example of a cavity perturbation apparatus is a hollow cylindrical cavity constructed from a highly conductive material, for example, high purity copper or a superconducting material. The resonator configuration used in our experiment is more complex than this, as it includes a dielectric resonator (discussed below), but the following characteristics remain the same. Alternating EM fields, in the microwave range, form standing waves inside the cavity. Different standing waves are generated at different frequencies. These frequencies depend on the cavity geometry, and we refer to them as different resonator modes. Figure 4.1 shows the T 011 mode of a basic cylindrical cavity, with and without a sample. We place our sample in the center of our cavity for the analysis we preform it is necessary to have an approximately uniform magnetic field profile and an electric field node there. For that reason, the mode shown in Figure 4.1 is the lowest frequency mode that we would use in a measurement, and is the frequency mode we use for all data presented here. Figure 4.1: Magnetic field lines of a cross section of a cylindrical cavity for the T 011 mode. (a) empty cavity, no sample, and (b) with a sample in the cavity. The sample size has been enlarged for clarity. The presence of the sample leads to changes in resonant frequency f 0 and resonant bandwidth f B. From Zhou [7]; used with permission. A resonant system can be characterized by its free-decay response. This can be measured in the time domain, by stimulating the resonance and then recording the complex resonance frequency of the decaying oscillations. However, we prefer to work in the fre-

40 CHAPTER 4. EXPERIMENTAL APPARATUS 29 quency domain, where we obtain the same information from the steady-state resonator response as we sweep a microwave driving signal across the resonance. The quantities we infer from this measurement are the resonant frequency f 0, and the resonant bandwidth f B. The key point is that f 0 and f B will be affected by the presence of a sample in the cavity (see Figure 4.1), and we can convert changes in these quantities to useful information about the sample. The resonant frequency, f 0, is affected by the cavity size, and the resonant bandwidth, f B, is determined by the energy dissipation of the cavity. I will now explain in more detail how f 0 and f B change on the insertion of a sample into the cavity. If we place a perfectly conducting sample into a cavity and apply an AC magnetic field, the field will be excluded from the entirety of the sample. This reduces the effective volume of the cavity. As mentioned earlier, f 0 is determined by the size of the cavity, and a smaller cavity volume will correspond to an increase in f 0. There will be no change in f B, as the perfectly conducting sample does not introduce dissipation. Now consider the same configuration but with a superconducting sample with finite penetration depth. The external magnetic field is now excluded from a smaller volume, resulting in an increase in the effective size of the cavity and a decrease in f 0. In addition, if dissipative processes are also occurring in the sample, there will be an increase in energy dissipation in the cavity, resulting in an increase in the bandwidth f B. Figure 4.2 shows an illustration of these effects. In Section I will show how the surface impedance of a sample relates to f 0 and f B. Knowing the surface impedance, we can then infer modelspecific physical parameters of a sample as described in Chapter 3. It is not initially obvious that the surface screening currents in a real superconductor will cause dissipation, and in a DC measurement they would not. For a DC measurement the screening current would be entirely composed of supercurrent (the superelectrons and normal electrons conduct in parallel), thus no dissipation would occur. On the other hand, in an AC measurement we induce finite electric fields at the sample surface, so the screening current will consist of both super and normal current. Thus, there will be energy dissipation associated with this process. Changing the temperature and external magnetic field will also affect f 0 and f B. These are the effects that are most interesting to us as they allow us to study the temperature and field dependence of sample properties. As temperature and field changes present weak perturbations to the resonator fields, we will see below that the cavity perturbation approx-

41 CHAPTER 4. EXPERIMENTAL APPARATUS 30 Figure 4.2: Illustration of shifts in the resonant frequency f 0 and resonant bandwidth f B of a cavity when changing sample temperature or applied magnetic field. From Thewalt [12]; used with permission. imation is most accurate under these conditions. This will be significant for our analysis. Resonant frequency and bandwidth can also tell us about the quality of a resonator cavity itself. In general, the less dissipation in the resonator the more precise the measurement we take. This is quantified in the resonator quality factor, Q = f 0 /f B. Optimizing the Q of our resonator was very important in its design, and is further discussed in Section Surface Impedance As mentioned previously, surface impedance, Z S, is the experimental accessible quantity in cavity perturbation techniques, and is defined by the ratio of the tangential components of the electric field, E, and magnetic field, H, at the surface, Z s = E H. (4.1) Z s is a complex quantity, Z s = R s + ix s, (4.2) where R s is the surface resistance and X s is the surface reactance. In general surface impedance is a tensor quantity. We address this in the experiments reported here by carefully choosing field polarization with respect to the crystal axes, so as to only induce screen-

42 CHAPTER 4. EXPERIMENTAL APPARATUS 31 ing currents in the basal plane of the sample. It is also important to note that our measurements of surface impedance are independent of microscopic theories and assumptions, in contrast to derived quantities such as complex conductivity and vortex viscosity. As introduced above, changes in f 0 and f B give us information about our sample. More specifically they are directly related to the imaginary and real parts of the surface impedance, respectively. If the microwave fields are uniform in the vicinity of the sample the surface impedance is obtained from the following cavity perturbation relation [13] [14] [15]: ( fb R s + i X s = Γ 2 i f 0 ), (4.3) where Γ is the resonator constant, and f B, f 0, R s and X s are all functions of sample temperature and external magnetic field, as outlined below. R s - the absolute surface resistance. X s f B - the shift in the surface reactance with respect to B = 0 and T = T 0, where T 0 is a reference temperature. - the shift in resonant bandwidth compared to the empty cavity. f 0 - the shift in resonant frequency with respect to B = 0 and T = T 0. Surface impedance can then be used to extract electrodynamic properties of our sample, as discussed in Chapter 3. In the local electrodynamic limit the surface impedance is directly related to the complex resistivity and conductivity by the relationship ρ = Z2 s iωµ 0 = 1 σ. (4.4) This is one example of a quantity derived from Z s in a well justified, but model-dependent way Vortex Dynamics and Cavity Perturbation In an attempt to combine the ideas from Chapter 3 with cavity perturbation an illustration of our in-field experiment on the sample scale is shown in Figure 4.3. We are only probing a small surface sheath of the sample with cavity perturbations. However, for any strength

43 CHAPTER 4. EXPERIMENTAL APPARATUS 32 of external magnetic field there are many vortices present along the surface and we are safe to assume our measurements translate to bulk properties of the sample vortex dynamics. 4.2 In-Field Microwave Cavity Perturbation Apparatus The apparatus described in this section is novel as it is the first to allow cavity perturbation measurements at millikelvin temperatures in an external applied magnetic field. Some challenges with carrying out this new in-field cavity perturbation experiment included: obtaining a high Q cavity without the use of superconducting cavity walls; fitting the microwave system inside the center of a superconducting magnet; and integrating the new apparatus with the existing dilution fridge experiment. Figure 4.4 is an illustration of our in-field apparatus. Many of the ideas for the experimental design were modelled on a previous, higher temperature, experiment in the Broun group, described in Huttema et al. [15]. Optimizing the cavity quality factor Generally speaking we want to construct a cavity with a Q as large as possible to improve the frequency and bandwidth resolution of our measurements. An empty ideal cavity with perfectly conducting walls would have no dissipation and therefore zero f B, corresponding to Q =. If a sample was then inserted into this ideal cavity the resulting finite bandwidth would be solely due to the dissipation of the sample. In reality, even without a sample in the resonator cavity there is dissipation from cavity walls, sample holder, and any other material inside the cavity. The lower this background dissipation, the more precise our measurements. A common technique to reduce the dissipation from the cavity walls is to coat the inner surface of the cavity in a superconducting material. To expose our sample to an external applied field (to drive the sample into the vortex state, Section 1.4), so we are unable to use superconducting cavity walls. Instead, we use the displacement currents in a low-loss dielectric to confine the electromagnetic fields. We place a cylindrically shaped dielectric resonator in the centre of our cavity. The dielectric resonator has a hole bored through its centre, along the cylinder axis, to allow the sample to be placed at the centre of the resonator (see Figure 4.4). This resonator is made of TiO 2, also known as rutile, which

44 CHAPTER 4. EXPERIMENTAL APPARATUS 33 Figure 4.3: Illustration of the in-field experiment. The sample is shown in black, with many vortices (light grey) threading throughout it. These vortices are forces into the sample by the static external magnetic field, B E, shown in blue. Cavity perturbation measurements are done using an AC magnetic field, B AC, shown in red. The AC field very weak and thus only penetrates a distance λ into the sample. This distance is shown in the illustration by the red surface of the sample. In this thin surface many vortices are present and the screening currents that flow here act as the transport current, J T, which in our case shakes the vortices back and forth.

45 CHAPTER 4. EXPERIMENTAL APPARATUS 34 Microwave lines Sample stage Silicon sample holder Sample Weak thermal link Copper resonator enclosure Rutile resonator Superconducting magnet Figure 4.4: Illustration of the novel high-field system used for millikelvin microwave cavity perturbation. The copper enclosure is held at a constant temperature while the combination of sample stage, weak thermal link, and sample holder allow the sample temperature to vary under our control.

46 CHAPTER 4. EXPERIMENTAL APPARATUS 35 has a very high dielectric constant, ε r > 120 and low dielectric loss. Both the shape and high dielectric constant of the rutile resonator concentrate the microwave EM field in the vicinity of the sample, effectively increasing the size of the cavity while weakening the fields at the cavity walls that cause background dissipation. This allows us to measure at lower frequencies while maintaining a high filling factor. For the data presented here on the base mode, GHz, we are able to obtain a quality factor on the order of one million. The sample stage and weak thermal link, see Figure 4.4, are also important components of this new experiment and were the main focus of my efforts. They will be discussed in detail in the next section The Sample Stage At the beginning of this project I inherited a partially complete design of the high-field system for millikelvin microwave spectroscopy from a previous student in the group, Eric Thewalt. The focus of his work was to optimize the design of the dielectric resonator and enclosure so that it could fit into the tight space constraints of the bore of the superconducting solenoid, Figure 4.4. While most of this work was complete, designs had not yet been carried out for important auxiliary components such as: Weak thermal link Temperature-controlled sample stage High thermal conductivity / low dielectric loss sample holder Copper 50 mk sample stage Sample stage removable guide posts Thermometry wiring Room temperature sample mounting block The design, construction, and assembly of these components was my responsibility. Figure 4.5 shows a labelled illustration of the sample stage, and Figure 4.6 shows some photos of the in-field apparatus, including a picture of just the sample stage on its own (Figure 4.6a).

47 CHAPTER 4. EXPERIMENTAL APPARATUS 36 Copper 50 mk sample stage Weak thermal link Intermediate thermal stage Vespel supports and thermal isolation Temperature-controlled stage Silicon sample hotfinger Sample Figure 4.5: Illustration of the temperature-controlled sample stage, which allows our sample to be heated to 55 K without overwhelming the cooling power of our dilution refrigerator and maintains a base temperature of 80 mk. Weak thermal link As an element of the sample stage the weak thermal link design determines the temperature range accessible in the experiment, in this case 55 K down to 80 mk. Keep in mind that, during such a temperature sweep, the resonator enclosure must be maintained at a constant temperature. Ideally we would like to have the largest temperature range possible while still being able to reach base temperature in a reasonable amount of time. The weak thermal link allows us to sweep through this temperature range without overwhelming the cooling power of the dilution fridge. Stainless steel was chosen as the material for the weak thermal link because of its low thermal conductivity combined with its tensile strength and machinablility. The existing resonator and sample stage components constrained the maximum length of the weak thermal link. Calculations were carried out to determine the outer diameter and wall thickness for the stainless steel tube that would allow us to reach a sample temperature of 50 K without overwhelming the cooling power of our dilution fridge. The end result was a 0.600" free length of 3/16" diameter stainless steel rod, with a wall thickness of 0.005".

48 CHAPTER 4. EXPERIMENTAL APPARATUS (a) 37 (b) Figure 4.6: Photographs of the sample stage (a) and in-field apparatus with sample stage mounted in place (b). Note, the sample stage and hotfinger are machined with precision to ensure the placement of the sample is as close to the center of the resonator as possible. The hotfinger is 1 mm by 1 mm in dimension, and the largest diameter of the copper enclosure is 2.25 inches. Temperature-controlled sample stage The sample heater and thermometer wiring must be outside the resonator, away from the microwave fields. The solution to this is the combination of a temperature-controlled sample stage and a dielectric sample holder, also known as the hotfinger [16]. The temperaturecontrolled sample stage is where the sample heater and thermometer are placed, as well as where the hotfinger is mounted. This stage is precisely machined to ensure the sample will be accurately centred. As we will be conducting our experiment in strong magnetic fields and at low temperatures, an important design consideration was the nuclear heat capacity of the material used in the temperature-controlled sample stage. We originally considered making the

49 CHAPTER 4. EXPERIMENTAL APPARATUS 38 temperature-controlled sample stage from copper. After carrying out a calculation of thermal response at 9 T for copper and obtaining a time constant of approximately 3 hours we decided to find a material with a much lower nuclear heat capacity in strong magnetic fields and low temperatures. As can been seen in the nuclear spin heat capacity plot of pg. 220 of Pobell [17], silver was the obvious choice, as its Schottky anomaly is at a lower temperature (outside our window of operation). Silicon sample hotfinger Our apparatus benefits from not needing to solder contacts to our samples, but we do have to place the sample in the center of the resonator cavity, so a long, thin sample hotfinger is used. The hotfinger sets the sample temperature, so it must change temperature with the sample and have a high thermal conductivity to transfer heat. We have previously used sapphire for this purpose, but sapphire contains paramagnetic impurities that act as free spins, and impart a systematic background that has a complicated dependence on field, temperature and microwave power level. In contrast, high purity silicon Si (obtained from a company in Denmark called Topsil Semiconductor Materials A/S) has all the properties we need; it is electronic spin free and carrier free, with a high thermal conductivity. 4.3 Dilution Refrigeration To reach temperatures below the boiling point of liquid helium, 4.2 K, and even below the temperatures of pumped 4 He and 3 He, 1 K and 0.25 K respectively, we use a dilution refrigerator. The concept of dilution refrigeration was first presented by Heinz London in 1951, however putting these concepts into practice and carrying them out to maximum efficiency remains a complex and intricate task. In our lab we use an Oxford Instruments Kelvinox MX-40 continuous-cycle dilution refrigerator. Dilution refrigeration uses a mixture of rare and expensive 3 He and more abundant 4 He. The cooling power comes from the fact that the free energy of a 3 He atom in a 3 He- 4 He mixture is larger than the free energy of a 3 He atom in a pure phase of 3 He. This difference results in a latent heat of mixing as 3 He atoms transfer from a pure state into a mixed state,

50 CHAPTER 4. EXPERIMENTAL APPARATUS 39 which provides the cooling power at low temperatures. We therefore need a pure phase of 3 He and a mixed phase of 3 He- 4 He in close proximity to each other. Conveniently, below 0.87 K a 3 He- 4 He mixture undergoes a spontaneous transition into a 3 He-rich phase and a 3 He-poor phase. As the temperature of the system approaches zero the 3 He-rich phase becomes pure 3 He and the dilute mixture of 3 He and 4 He contains a 6.5 percent concentration of 3 He. In fact, 3 He can only dissolve in 4 He up to a 6.5 percent concentration of 3 He. If more 3 He is present in the system having two phases is more energetically favourable. This is exploited to create a constant flow of 3 He in the dilution fridge, to sustain a constant cooling power in the system [18]. There is one last property of 3 He and 4 He that we take advantage of in a continuouscycle dilution refrigerator: the vapour pressure of 3 He is much greater than that of 4 He. This makes it possible to distill only 3 He from the 3 He- 4 He mixture. This distillation is the driving force behind the 3 He cycle. We maintain a constant pressure in the still (the stage of the dilution refrigerator in which 3 He is pumped) to in force 3 He vaporization only. As this 3 He vaporizes the 3 He concentration in the 3 He-poor phase is reduced but, as mentioned previously, this concentration is forced to be constant. The system compensates by pulling 3 He from the 3 He-rich phase, which results in a latent heat of mixing and provides cooling power. Figure 4.7 shows a basic illustration of the workings of the dilution refrigerator including the 3 He cycle just discussed. The coldest section of the dilution fridge is the mixing chamber, in our case held at 50 mk. The mixing chamber is where the phase separation of pure 3 He and dilute 3 He takes place, and where heat is being removed from the experiment to allow 3 He to move from the 3 He-rich phase to the 3 He-poor phase. Simultaneously, the heat load from the experiment is forcing dilute 3 He up into the still, where we pump off 3 He vapour to drive the continuous cycle. The still temperature is regulated, using a small heater, at 0.7 K to create an optimal balance between maximizing the vaporation rate of 3 He, which increases flow and cooling power, and minimizing heat conduction into the lower temperature stage. It should be noted that the still is where the initial cooling power comes from when first starting up circulation in the dilution fridge. The now-warm 3 He vapour passes through a mechanical pump at room temperature, which compresses the gas before returning it to the recondenser. The incoming 3 He is cooled to 4.2 K using the helium bath, at which time the recondenser is able to condense

51 CHAPTER 4. EXPERIMENTAL APPARATUS 40 Figure 4.7: Illustration of the continuous cycle of 3 He in the dilution refrigerator. Each stage of the dilution fridge is labelled and an approximate temperature is shown. We utilize the available cooling power not only at the coldest part of the fridge, the mixing chamber, but at all the stages of the dilution fridge. From Thewalt [12]; used with permission. the 3 He gas back into a liquid. The recondenser is held at 1.5 K as it is thermally connected to the 1-K pot, where 4 He is constantly being pumped on. (The 1-K pot, not shown in figure, has a lot of cooling power and is thus responsible for removing the majority of the heat from the still warm, 4.2 K, 3 He liquid.) Before the newly condensed 3 He liquid enters the mixing chamber again it first passes through the heat exchanger, where it is further cooled by the outgoing 3 He-poor mixture. The now cold 3 He liquid enters the mixing chamber, completing the cycle and supplying a constant cooling power for the experiment. As a result of this process, each stage of the dilution refrigerator has its own source of cooling power. We utilize this cooling power throughout the experiment to intercept heat that would otherwise be conducted down to the cold sample something that was well planned out in the addition and design of the in-field experiment to the existing dilution refrigerator system. Figure 4.8 shows a colour coded diagram of the novel high-field system illustrating the thermal profile of the apparatus along with thermal isolation used.

52 CHAPTER 4. EXPERIMENTAL APPARATUS 41 Sample 0.05 K (mixing chamber) Weak thermal link Vespel supports and thermal isolation Intermediate thermal 0.25 K (heat exchanger) Temperature-controlled stage Silicon sample holder Sample Temperature range K Steel vacuum canister (immersed in liquid 4 K Copper enclosure and rutile 1.5 K (connected to 1-K pot) Vacuum Figure 4.8: The thermal profile of the in-field experimental design. Components are labelled with their nominal temperature, and in brackets is the stage of the dilution refrigerator from which they obtain their cooling power. To ensure we have a constant background signal for all our measurements the entire copper enclosure, resonator included, is held at a constant temperature of 1.5 K. The cooling power for this comes from the 1-K pot, not shown, connected to the copper enclosure using a high purity silver wire, also not shown. A small heater is placed on the copper enclosure itself, which allows us to regular the temperature precisely. Vespel SP-22, a physically strong material with a low thermal conductivity, is used to connect parts of the experiment which are expected to be held at different temperatures, providing proper thermal isolation.

53 Chapter 5 CeCoIn 5 This chapter introduces the physics of CeCoIn 5, focusing on its superconducting properties. CeCoIn 5 is a heavy fermion material with a critical temperature, T c = 2.25 K. I will briefly review the most convincing evidence for d-wave superconductivity, and then discuss zerofield measurements of quasiparticle dynamics, which are relevant to the interpretation of the vortex dynamics data in Chapter 6. CeCoIn 5 is of interest as it is an analogue to the more complex high-t c cuprate superconductors. Both are d-wave superconductors; making a comparison of their vortex dynamic behaviour would be very helpful in the understanding of the underlying physics in both materials, and is what motivates our measurements. 5.1 CeIn 3 Parent Compound of CeCoIn 5 CeCoIn 5 can be thought of as a layered variant of CeIn 3, with a quasi two-dimensional Fermi surface. CeIn 3 has a cubic structure shown in Figure 5.1a (and therefore is threedimensional), and is weakly antiferromagnetic (AFM) at ambient pressures, with a Néel temperature of 10 K. A temperature pressure phase diagram for CeIn 3 is shown in Figure 5.2 (original work on this was done by Mathur et al. [19]). The Ce atoms form local magnetic moments, and the resulting AFM order (in CeIn 3 ) takes the form shown in the inset of Figure 5.2. Because this AFM state is weak, only arising at 10 K, it can be suppressed using hydrostatic pressure. Pressure forces atoms closer together, increasing orbital overlap. This favours kinetic energy over potential energy, and 42

54 CHAPTER 5. CECOIN5 43 (a) (b) Figure 5.1: (a) Crystal structure of CeIn3. (b) Crystal structure of CeCoIn5, showing interleaved CeIn3 and CoIn2 layers. 10 Temperature K s/c Pressure GPa Figure 5.2: Pressure temperature phase diagram of CeIn3, showing how the suppression of AFM order leads to a small window of superconductivity near the quantum critical point. The inset shows the AFM form of the material where the Ce atoms are the local magnetic moments. Data from Grosche et al. [20].