Phase-Sensitive Determination of the Pairing Symmetry in Sr 2 RuO 4

Transcription

1 Pennsylvania State University Graduate School Eberly College of Science Phase-Sensitive Determination of the Pairing Symmetry in Sr 2 RuO 4 A Thesis in Physics by Karl D. Nelson Copyright 2004, Karl D. Nelson Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy December 2004

2 The thesis of Karl D. Nelson has been reviewed and approved by the following: Ying Liu Associate Professor of Physics Thesis Advisor Chair of Committee Nitin Samarth Professor of Physics Vincent Crespi Professor of Physics Darrell Schlom Professor of Materials Science and Engineering Jayanth Banavar Professor of Physics Head of the Department of Physics Signatures are on file in the Graduate School.

3 iii Abstract Sr 2 RuO 4, an unconventional superconducting material with T c = 1.5 K, was predicted to possess a superconducting state with an order parameter of odd-parity symmetry. Numerous experiments have given support to the prediction, but none of these experiments has been sufficiently definitive to allow certainty about the symmetry of the order parameter. Direct evidence of odd-parity symmetry can be obtained from a phase-sensitive experiment which measures interference in the critical current between two coupled Josephson junctions. Odd-parity symmetry in Sr 2 RuO 4 is established here by a phase-sensitive measurement using Josephson junctions on single crystals of Sr 2 RuO 4, with the junctions prepared on opposite sides of the crystal. Measurements of the critical current versus external magnetic flux for these samples show quantum interference between the junctions, with the critical current at a minimum when zero external flux is applied. For junctions on opposite sides, this result is possible only for a material with odd-parity symmetry. The phase-sensitive measurements are difficult because of a suppression of superconductivity at the surface of Sr 2 RuO 4 crystal which is demonstrated by single-particle tunneling measurements. Tunneling spectra for Au Sr 2 RuO 4 junctions show no features due to superconductivity in Sr 2 RuO 4. However, metal alloys such as Hg-In and Au-In which wet the Sr 2 RuO 4 surface are shown to form Josephson junctions with finite supercurrent despite the normal surface layer, making possible the phase-sensitive experiment. In addition, previous analysis of experiments on Sr 2 RuO 4 has assumed weak spin orbit coupling in the material. A large product of critical current and normal state resistance (I c R n ) for Au-In Sr 2 RuO 4 junctions has been measured and is shown to be an indication of strong spin orbit coupling near the Sr 2 RuO 4 surface. The full text is always available for free at

4 iv Table of Contents List of Figures vii List of Tables viii 1. Introduction Pairing Symmetry Superconducting Order Parameter Single-Particle Tunneling Tunneling Spectroscopy Josephson Effect Background Normal-State Properties of Sr 2 RuO Crystal Structure Electronic Structure Transport Properties Superconducting Properties of Sr 2 RuO Sensitivity to Disorder Spin-Susceptibility Muon Spin Resonance Thermodynamic Properties The 3 Kelvin Phase Josephson-Effect-Based Experiments Josephson Coupling between a Spin-Singlet and a Spin-Triplet Superconductor Phase-Sensitive Experiment in High-T c Superconductors Experimental Design Other Experimental Configurations Sample Preparation and Characterization Crystal Growth Cutting and Cleaving Etching Polishing

5 v 4.5. Sample Support Evaporation Masking Rotating Au 0.5 In 0.5 Films Electrical Lead Attachment Surface Characterization Summary Measurement Technique Refrigeration He Evaporation Refrigerator Dilution Refrigerator Shielding Transport Measurements Computer Programs Sr 2 RuO 4 Tunnel Junctions The Normal Surface Region Hg-In Junctions Junction Preparation Josephson Coupling Andreev Surface States Au-In Junctions Strength of Josephson Coupling between Au 0.5 In 0.5 and Sr 2 RuO Introduction Experimental Results Discussion Phase-Sensitive Measurements Asymmetry and Induced Flux Background Field and Trapped Flux Experimental Results Sample Inductance

6 vi 8.5. Control Experiments Conclusion References Index

7 vii List of Figures Figure Semiconductor model diagrams for an N IS junction Figure Schematic of the pair wave functions in a superconductor insulator superconductor junction Figure Crystal structure of Sr 2 RuO Figure Fermi Surface of Sr 2 RuO Figure Schematic of a Au 0.5 In 0.5 Sr 2 RuO 4 GLB SQUID Figure Picture of a Sr 2 RuO 4 single-crystal rod Figure AFM image of a rough-polished, etched Sr 2 RuO 4 surface Figure Optical image of a fine-polished, etched Sr 2 RuO 4 surface Figure AFM image of a polished Sr 2 RuO 4 surface Figure Conductance of Au SrTiO 3 Sr 2 RuO 4 ab-plane junction Figure Conductance of Au Al 2 O 3 Sr 2 RuO 4 a-axis junction Figure R(T ) for a Hg-In Sr 2 RuO 4 junction Figure Critical current for a Hg-In Sr 2 RuO 4 junction Figure Figure Figure Figure A series of I(V, T ) for an Hg-In Sr 2 RuO 4 junction showing Josephson coupling up to 1.2 K Conductance of a Hg-In Sr 2 RuO 4 junction at several temperatures Temperature dependence of the energy scale in Hg-In Sr 2 RuO 4 junctions determined from the Andreev reflection spectra Tunneling spectrum for a Au 0.5 In 0.5 Sr 2 RuO 4 junction showing two gaps Figure I(V ) with finite I c from a Au 0.5 In 0.5 Sr 2 RuO 4 junction Figure R(T ) for Au 0.5 In 0.5 Sr 2 RuO 4 junction AuIn# Figure Conductance di/dv for Au 0.5 In 0.5 Sr 2 RuO 4 junction AuIn# Figure I c (T ) for several In and Au 0.5 In 0.5 Sr 2 RuO 4 junctions Figure I c (H) for several In and Au 0.5 In 0.5 Sr 2 RuO 4 junctions Figure Equivalent circuit for an asymmetric SQUID Figure Resistance vs. temperature for SQUIDs GLB A and GLB B

8 viii Figure Resistance vs. temperature for SQUID GLB C Figure Voltage vs. current for SQUID GLB A Figure Quantum interference in I c (H) and R(H) for GLB A Figure Temperature progression of I c (H) for GLB A Figure Temperature progression of I c (H) for GLB B Figure Temperature progression of I c (H) for GLB C Figure Figure Figure Figure List of Tables Magnetic field at the minimum I c (H) vs. I min c Temperature progression of I c (H) for same-side SQUID SS A H(I max c ) for same-side SQUIDs SS A, B, and C Temperature progression of I c (H) for corner junction CJ A Table Table Table Basis functions for the triplet irreducible representations of point group D 4h Zero-temperature superconducting parameters for highpurity Sr 2 RuO 4 crystals (T c = 1.49 K) Measured I c R n and calculated AB limit for several In Sr 2 RuO 4 and Au 0.5 In 0.5 Sr 2 RuO 4 junctions

9 Chapter 1. Introduction Interest in the superconductor Sr 2 RuO 4 comes in large part from the fact that it exhibits novel physical properties, making Sr 2 RuO 4 a tool for advancing our understanding of superconductivity. Careful measurements of these properties will lead to a more complete understanding of superconductivity. In the original theory of Bardeen, Cooper, and Schrieffer 1 (BCS), superconductivity is a phase-coherent state of paired electrons, coupled by a weak phonon-mediated attraction into pairs with zero total spin and angular momentum. The theory successfully explained most of the experimental observations on superconductors available at the time. With the discovery of superconductivity at 40 K in (La 1 x Ba x ) 2 CuO 4, 2 a group of materials known as high-t c superconductors was created. These materials showed that the zero spin (singlet) and zero orbital angular momentum (s-wave) symmetry is not applicable to all superconductors. Researchers have determined that the pairs in these materials have two quanta of orbital angular momentum, or d-wave symmetry. 3,4 Some have also proposed that the attraction between electrons is not due to the electron phonon interaction. Although pairing with total spin S = 1 (spin-triplet) has been demonstrated in superfluid 3 He, triplet pairing has not previously been proven to exist in any superconductor. The results of the phase-sensitive experiment described in Chapter 8 show that spin-triplet superconductivity does exist in Sr 2 RuO 4. 1 J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. 108, 1175 (1957). 2 J.G. Bednorz and K.A. Müller, Z. Phys. B 64, 189 (1986). 3 D.A. Wollman et al., Phys. Rev. Lett. 71, 2134 (1993). 4 C.C. Tsuei et al., Phys. Rev. Lett. 73, 593 (1994). 1

10 Pairing Symmetry All known superconductors have electron pairs called Cooper pairs, which can be described by a single wave function. The pairing symmetry is the symmetry of this wave function. When excluding the effects of the crystal lattice, the wave function symmetry can be characterized by the orbital angular momentum. The orbital part of the wave function can then be written in terms of the spherical harmonics ψ l = l m= l b l my l m (1.1.1) where l and m are the quantum numbers for the angular momentum and its z component, respectively. Using atomic spectroscopic terminology, the pairing symmetry for l = 0 is s wave, for l = 1 it is p wave, l = 2 is d wave, and so on. The addition of the spin angular momentum of two paired electrons (s = 1/2) results in a total spin of either S = 0 or S = 1. Using the eigenstates of the single-particle spin operators s 2 and s z (represented by and ), the spin part of the pair wave function can be written χ 0 = 1 2 ( ) for the singlet state. When S = 1, sz can be -1, 0, or 1, so χ 1 can be any combination the three states, 1 2 ( + ), and : the triplet state. The Fermi statistics of the identical electrons making up a pair requires that the pair wave function Ψ ls = ψ l χ S be odd under permutation of the electrons. This imposes a restriction on the spin angular momentum part of the wave function for a state with a given l. For even ψ l (l = 0, 2,...), χ S must be odd (S = 0), forming spin-singlet pairing states. For l = 1, 3,..., χ S must be even (S = 1), resulting in a spin-triplet state. The lattice in a real superconducting material imposes additional restrictions on the symmetry of the pairing states. The complete Hamiltonian including the crystal potential does not have rotational symmetry, so the eigenstates cannot be described by the angular momentum quantum numbers. In addition, mixing of the orbital and spin angular momentum, called spin orbit coupling, prevents the wave functions from being factored into orbital and spin parts. When a material enters the superconducting state from its normal state, some symmetries will be broken. The basis functions which define the symmetry group of the lattice may also be used to form the basis of the orbital wave function. A given symmetry group may

11 3 have several sets of basis functions which apply to it, forming its irreducible representations. If a particular set of basis functions has the full symmetry of the lattice, it is the identity representation. The spherical harmonics used in section 1.1 are the identity representation of an isotropic medium. The D 4h symmetry group which applies to a tetragonal crystal such as Sr 2 RuO 4 allows five triplet states. The basis function which apply for both weak and strong spin orbit coupling are shown in Table The first four have only one basis function (a one-dimensional representation) and the fifth has two (a two-dimensional representation). The two-dimensional representation for Sr 2 RuO 4 is supported by several experiments (section 2.2.2). Table 1.1.1: Basis functions for the five triplet irreducible representations of the tetragonal point symmetry group D 4h. After Ref. 5. Name (Γ) Dimension Basis function(s) when spin orbit coupling is: Weak Strong A 1u or Γ 1 1 k x k y k z (k 2 x k 2 y) aẑk z + b(ˆxk x + ŷk y ) A 2u or Γ 2 1 k z a(ˆxk y + ŷk y )(k 2 x k 2 y) + b(ˆxk y ŷk x ) B 1u or Γ 3 1 k x k y k z a(ˆxk x ŷk y ) + bẑk z (k 2 x k 2 y) B 2u or Γ 4 1 k z (k 2 x k 2 y) a(ˆxk y + ŷk x ) + bẑk x k y k z E u or Γ 5 2 k x and k y ẑk x and ẑk y When spin orbit coupling is strong, the eigenstates are doubly degenerate even though they are not the eigenstates of electron spin. 5 These pseudospin eigenstates can still be classified as singlet and triplet states provided the labels are interpreted in terms of the pseudospin. 5 The basis for these eigenstates is fixed with respect to the crystal axes. 5 V.P. Mineev and K.V. Samokhin, Introduction to Unconventional Superconductivity (Gordon and Breach Science Publishers, Australia, 1999).

12 4 The terms s-wave or p-wave and singlet or triplet apply strictly only to wave functions in free space. Therefore, the symmetry properties under spatial inversion are the most accurate way to describe these two symmetries in a superconducting crystal. Spinsinglet states are odd-parity states with respect to inversion, while spin-triplet states are even-parity states Superconducting Order Parameter The order parameter describing the phase transition into the superconducting state is the magnitude of the wave function of the superconductor. The b l m in Eq. (1.1.1) can be considered the order parameter for that superconducting state. It is clear that the order parameter for a singlet state has one component, but the triplet state has three, corresponding to the three possible values of m. For a completely isotropic superconductor, the order parameter would be a constant. Otherwise, the order parameter is a function of momentum k. For singlet states, a single complex function of momentum called ( k) specifies its k dependence. For a triplet pairing, a vector d( k) gives the three components of the order parameter. The three components of d = (d x, d y, d z ) are defined such that for a pair wave function ( ) χ = ( d x + id y ) + d z + + (dx + id y ). The order parameter d = 0 ẑ (k x ± ik y ) (1.2.1) for Sr 2 RuO 4 is supported by several experiments if ẑ is the c axis of the crystal. This order parameter has d x = d y = 0 and is called an s z = 0 only pairing state Single-Particle Tunneling The magnitude of ( k) or d( k) corresponds to the energy gap of the superconductor. The gap is the minimum excitation energy for a direction k in momentum space. The k dependence of the gap is an important characteristic of its symmetry.

13 Tunneling Spectroscopy A useful tool to probe the gap of a superconductor is single-electron tunneling spectroscopy. The tunneling spectrum is simply the dependence of the junction conductance on the voltage across the junction, or the di(v )/dv curve. The voltage dependence of the tunneling conductance is proportional to the density of electron states in the materials near the surface. In the BCS model, 1 the density of states is N(E) = N(0)E (E 2 2 ) 1/2 for E > and N(E) = 0 for E <, so there is a large increase in conductance when the applied voltage is larger than the gap voltage. The following semiconductor model is useful for visualizing how the density of states can be determined from the conductance. Note that the energy is related to the voltage by the electron charge e. Using the semiconductor model shown in Fig , the proportionality between the density of states and the conductance can be visualized. The energy is displayed on the vertical axis and the density of states (DOS) for the two materials is on the horizontal axis. Fig a shows a Normal metal Insulator Superconductor junction (N IS) at T = 0, with a voltage ev < applied and Fig b shows a voltage ev >. Fig c is at a finite temperature. At T = 0 (Fig a and b), the current is finite only for ev >. At finite temperature (Fig c), thermal excitations make states accessible even for ev <, and when ev >, the tunneling results from the four possibilities indicated by the arrows. The result is that the density of states is convolved with the derivative of the Fermi thermal distribution f(e) 1 1+e E/k B T to give the conductance: G NS di dv = G N S (E) NN N N (0) ( ) f (E + ev ) de (1.3.1) (ev ) where N S (E) is the density of states in the superconducting state and N N (0) is the density of states of the superconductor when it is in the normal state. The normal metal is assumed to have a constant density of states which is taken into account by the constant G NN, the conductance of the junction when both materials are in the normal state. Using this expression, the density of states can be extracted from a measurement even at finite temperature. In the limit T 0, and when the conditions given earlier apply, the conductance is directly proportional to the density of states because lim T 0 f(e+ev ) (ev ) = δ(ev ).

14 6 (a) E (b) E (c) E ev + ev + ev + N I S N I S N I S Figure 1.3.1: Semiconductor model diagrams for an NIS junction, with N(E)/N(0) = 1 in the E metal and N(E)/N(0) = (the BCS density of states) in the superconductor. (a) T = 0 (E 2 2 ) 1/2 and ev < ; here I = 0. (b) T = 0 and ev > with finite tunneling probability (indicated by the arrow) leading to finite I. (c) T > 0 with thermally excited states shown schematically. Finite tunneling probabilities to unoccupied states both above and below the Fermi level are indicated by the arrows. The Fermi level E F is indicated by a horizontal dotted line, and an applied voltage by an offset in the Fermi level. The solid line shows the density of states N(E), with the positive direction away from the insulator in both materials. The BCS form of the density of states is shown in the superconductor. The shading indicates occupied states, including states occupied because of the thermal energy. Unoccupied states (holes) below the Fermi level, are the unshaded areas within the mostly shaded region Josephson Effect As first predicted by Josephson, 6 tunneling through a junction between two superconductors can occur at zero voltage if the Cooper pair wave functions of the two materials overlap sufficiently at the barrier. Josephson tunneling is the transfer of Cooper pairs between two superconductors. As shown schematically in Fig , the amplitude of the pair wave function decays rapidly outside the superconducting material but remains non-zero close to the interface. The overlapping of the wave functions in the tunnel barrier maintains a coherent state between the two superconductors, so the phases of the wave functions on both sides have a definite relationship. The difference in phase φ caused by the presence of the tunnel barrier determines the pair current through the junction, described by the current phase 6 B.D. Josephson, Phys. Lett. 1, 251 (1962).

15 7 relation. In the simplest case, the current dependence is given by the Josephson relation I = I c sin( φ), where the critical current I c is the most current the junction can support before a voltage develops. S Ψ L I Ψ R S Figure 1.4.1: Schematic of the amplitude of the pair wave functions in a superconductor insulator superconductor junction showing the finite amplitude the wave functions in the insulator. After Ref. 7. The symmetry of the order parameters of each superconductor has important consequences for the symmetry of the Josephson effect. 5 The Josephson current can be obtained using a variational technique from the free energy of the surface between the two superconductors. This free energy can be written using the order parameter η and a second function g(ˆn) which contains the symmetries in such a way that g(ˆn)η is invariant under the symmetry transformations which apply to the superconductor, where ˆn is the interface normal. Using these functions, the form of the free energy is F = A g 1 (ˆn 1 ) g 2 (ˆn 2 ) (η1η 2 + η 1 η2) ds I where 1 and 2 refer to the two superconductors and I ds is an integral over the interface. The Josephson coupling may be zero for triplet order parameters due to the impossibility of forming a scalar quantity (the free energy) from any combination of the normal vector ˆn and the vector order parameter η. 5 This is the case for the weak spin orbit coupling basis vectors in Table When there is strong spin orbit coupling, it will always be possible to form a scalar combination of ˆn and the allowed η. 5 7 A. Barone and G. Paterno, Physics and Applications of the Josephson Effect (John Wiley & Sons, New York, 1982).

16 8 The Josephson effect can be used as a probe of several properties of a superconductor. For example, as shown in Chapter 6, the value of product between the critical current of a junction and its normal-state resistance provides information about the strength of spin orbit coupling in Sr 2 RuO 4. The symmetry of the order parameter can also be detected through the Josephson effect. A correct choice of orientations ˆn (Chapters 3 and 8) for two separate but coupled junctions gives information about the phase of the order parameter through interference between the junctions.

17 Chapter 2. Background 2.1. Normal-State Properties of Sr 2 RuO 4 Although Sr 2 RuO 4 was first synthesized in 1959, 8,9 superconductivity in Sr 2 RuO 4 was not discovered until 1994 when the material was cooled below 1 K by Y. Maeno et al.. 10 Rice and Sigrist soon proposed that the superconducting state had triplet pairing based mainly its normal-state properties. 11 The properties of the normal state are described here, with a focus on properties important for the spin-triplet or odd-parity superconducting state. More details can be found in a recent review Crystal Structure Sr 2 RuO 4 is a tetragonal crystal. The structure is that of a layered perovskite with metalcentered oxygen octahedra. This structure is similar to that of the high-temperature superconductor (La,Ba) 2 CuO 4 (Fig ). Instead of the copper-oxygen planes of the high-t c superconductors, there are ruthenium-oxygen planes in Sr 2 RuO 4. The lattice constants (at 4 K) are a = b = Å and c = Å. The layered structure of Sr 2 RuO 4 allows the crystal to cleave easily between SrO layers. The exposed surfaces are called the ab planes and are perpendicular to the c axis (see Fig ). In contrast, there is no definite direction for the crystal to break perpendicular the RuO 2 planes so the exact orientation of a surface parallel to the c axis cannot be known without X-ray characterization. For convenience, any direction perpendicular to the c axis may be referred to as the a direction. Similarly, any surface parallel to the c axis is called an ac face. 8 J.J. Randall and R. Ward, J. Am. Chem. Soc. 81, 2629 (1959). 9 F. Lichtenberg et al., Appl. Phys. Lett. 60, 1138 (1992). 10 Y. Maeno et al., Nature 372, 532 (1994). 11 T.M. Rice and M. Sigrist, J. Phys. Cond. Matter 7, L643 (1995). 12 A.P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003). 9

18 10 Sr 2 RuO 4 (La,Ba) CuO 2 4 O Ru Sr O Cu La,Ba c b a Figure 2.1.1: The crystal structure of Sr 2 RuO 4 and that of high-t c superconductor (La,Ba) 2 CuO 4 for comparison. For Sr 2 RuO 4, a = b = Å and c = Å. For La 1.75 Ba 0.25 CuO 4 at 300 K, a = 3.78 Å and c = Å. The figures are scaled to make the c axes equal Electronic Structure A detailed examination of the Fermi surface of Sr 2 RuO 4 has been made using the method of quantum oscillations. 13,14 The results of these measurements show that the Fermi surface has three almost-cylindrical sheets (with very little z dependence). Two concentric sheets centered in the Brillouin zone (the β and γ bands) have quasiparticles that are electron-like and a third sheet (the α band) in the corners of the Brillouin zone has hole-like quasiparticles (Fig ). The effective mass of the quasiparticles in the bands are m α = 3.3m e, m β = 7.0m e, and m γ = 16.0m e. 12 This large effective mass, even larger than predicted by a band-structure calculations, is one of the arguments for spin-triplet superconductivity in Sr 2 RuO A.P. Mackenzie et al., Phys. Rev. Lett. 76, 3786 (1996). 14 C. Bergemann et al., Phys. Rev. Lett. 84, 2662 (2000).

19 11 α M X γ β Γ Figure 2.1.2: A diagram of the Fermi surface of Sr 2 RuO 4 centered at the Γ point Transport Properties The highest superconducting transition temperature of Sr 2 RuO 4 is 1.5 K. Above this temperature, Sr 2 RuO 4 shows strongly anisotropic transport properties. At low temperatures, the electrical resistivity along the c axis can be times larger than along the planes. The resistivity in both directions is proportional to T 2 below about 20 K, consistent with a Fermi liquid state. At about 120 K, the c-axis resistivity has a maximum and decreases slowly with increasing temperature, indicating a crossover to a different (incoherent) conduction mechanism along the c axis. The specific heat below about 15 K can be decomposed into a phonon part proportional to T 3 and an electron part proportional to T. The linear temperature dependence of the electronic specific heat is also a characteristic of a Fermi liquid.

20 Superconducting Properties of Sr 2 RuO 4 The superconducting transition temperature (T c ) for Sr 2 RuO 4 depends on the level of impurities in the sample (section 2.2.1), with the highest T c about 1.5 K. For such high-purity crystals, the intrinsic superconducting parameters are listed in Table Table 2.2.1: Zero-temperature superconducting parameters for high-purity Sr 2 RuO 4 crystals (T c = 1.49 K): Upper critical field (H c2 ), lower critical field (H c1 ), coherence length (ξ), penetration depth (λ), and Ginzburg-Landau parameter (κ). After Ref. 15. Parameter in ab plane along c axis µ 0 H c2 (0) (T) µ 0 H c1 (0) (T) ξ(0) (Å) λ(0) (Å) κ(0) Sensitivity to Disorder A striking feature of Sr 2 RuO 4 is that the superconductivity is strongly suppressed by disorder, whether non-magnetic impurities 16 or structural defects, 17 consistent with Sr 2 RuO 4 being a non-s-wave superconductor. The T c of Sr 2 RuO 4 is suppressed to zero by as little as 50 parts per million (ppm) of aluminum and 100 ppm of silicon impurity. 16 The detailed dependence of T c on the residual resistivity is in good agreement with a model based on non-s-wave pairing. 15 T. Akima, S. NishiZaki, and Y. Maeno, J. Phys. Soc. Japan 68, 694 (1999). 16 A.P. Mackenzie et al., Phys. Rev. Lett. 80, 161 (1998). 17 Z.Q. Mao et al., Phys. Rev. B 60, 610 (1999).

21 Spin-Susceptibility The most-often-cited evidence for triplet symmetry in Sr 2 RuO 4 is the NMR Knight shift result in which the Knight shift value remained constant as the Sr 2 RuO 4 was cooled below its transition temperature, when measured with the field in the direction of the crystal plane. 18 The Knight shift is a measure of the spin susceptibility of the electrons or Cooper pairs, when the orbital contribution is properly removed. In conventional superconductors, the Cooper pairs form in a singlet (S = 0) state. For these materials we therefore expect a drop in spin-susceptibility below T c as the total spin is reduced by the formation of pairs. On the other hand, for triplet superconductors, the formation of pairs results in no change of spin-susceptibility for fields perpendicular to d and a reduction when the field is parallel to d. Knight-shift measurements on Sr 2 RuO 4 with the field aligned with the ab plane have shown that the spin-susceptibility is a constant. It has been argued 12 that nothing but triplet pairing could account for this. Indeed, triplet pairing is the simplest (and in that sense, most likely) explanation for the result. Unfortunately, other superconductors without triplet pairing have been observed to have a constant Knight shift, so the Knight shift alone cannot be considered definitive evidence. For example, a Knight shift which remains constant across T c has been measured in both vanadium 19 and YBa 2 Cu 3 O t δ for the Cu(2) site. 20 An independent experimental technique has also given results consistent with a constant magnetic susceptibility in Sr 2 RuO 4. Polarized neutrons scattering from a lattice in a magnetic field have peaks in the Bragg diffraction due both to the periodicity of the lattice and to the periodicity of the magnetization density. By reversing the polarization of the neutrons, the component of the diffraction due to the magnetization can be isolated. 21 The results of this experiment show that the total magnetic susceptibility of the Sr 2 RuO 4 crystal is unchanged when it enters the superconducting state, 21 confirming the Knight-shift result. However, this result is subject to the same reservations as the Knight-shift result. 18 K. Ishida et al., Nature 396, 658 (1998). 19 R.J. Noer and W.D. Knight, Rev. Mod. Phys. 36, 177 (1964). 20 S. E. Barrett et al., Phys. Rev. B 41, 6283 (1990). 21 J.A. Duffy et al., Phys. Rev. Lett. 85, 5412 (2000).

22 Muon Spin Resonance A µsr experiment measures the spin lattice relaxation rates of polarized positive muons injected into the crystal. The polarization evolves under the influence of the magnetic field in the vincinity of the muon, and the probability for the direction of the positron emitted when the muon decays is peaked in the direction of the muon spin at the time of the decay. By detecting many decay events, the polarization of the muon spin as a function of time can be established. The difference in the relaxation rate due to fields as small as 0.1 G was detected. For Sr 2 RuO 4, the measured relaxation rate is significantly increased when it enters the superconducting state, indicating a local field has appeared. 22 This strongly suggests broken time-reversal symmetry, associated with a complex order parameter Thermodynamic Properties The temperature dependence of many thermodynamic properties of Sr 2 RuO 4 has been measured, revealing a power-law dependence in each case. For example, the temperature dependence of the electronic specific heat C e is proportional to T This can be interpreted as follows: the specific heat decreases less rapidly than exponentially, due to unpaired electron states below the gap energy. States within the gap exist due to nodes in gap function. Nodes are a characteristic of unconventional (but not necessarily triplet) order parameters. This interpretation can also be applied to these other properties showing power-law temperature dependence: the 101 Ru spin lattice relaxation rate (1/T 1 T 3 ), 24 the (London) penetration depth (λ T 2 ), 25 ultrasound attenuation (α T 3 ), 26 and the thermal conductivity (κ T 2 ). 27 However, the theory of orbital-dependent superconductivity 28 gives an alternative intepretation for each of these results which does not require nodes in the gap function. 22 G.M. Luke et al., Nature 394, 558 (1998). 23 S. NishiZaki et al., J. Phys. Soc. Jpn. 69, 572 (2000). 24 K. Ishida et al., Phys. Rev. B 56, R505 (1997). 25 I. Bonalde et al., Phys. Rev. Lett. 85, 4775 (2000). 26 C. Lupien et al., Phys. Rev. Lett. 86, 5986 (2001). 27 K. Izawa et al., Phys. Rev. Lett. 86, 2653 (2001). 28 D.F. Agterberg, T.M. Rice, and M. Sigrist, Phys. Rev. Lett. 78, 3374 (1997).

23 The 3 Kelvin Phase As a result of the crystal-growing process (section 4.1), Sr 2 RuO 4 crystals contain small domains of pure ruthenium embedded throughout. The interface between the ruthenium and the Sr 2 RuO 4 crystal has an enhanced T c with an onset as high as 3 K, 29 giving it the name of the 3 K phase. The mechanism for this T c enhancement is not yet understood, as the superconducting transition temperature of ruthenium is only 0.5 K. However, below 1.5 K, the pairing of the 3 K phase is the same as in the bulk (1.5 K) phase Y. Maeno et al., Phys. Rev. Lett. 81, 3765 (1998). 30 Z.Q. Mao, K.D. Nelson, R. Jin, Y. Liu, and Y. Maeno, Phys. Rev. Lett. 87, (2001).

24 Chapter 3. Josephson-Effect-Based Experiments 3.1. Josephson Coupling between a Spin-Singlet and a Spin- Triplet Superconductor A quasi-classical analysis obtained by matching wave functions at the boundary of a Josephson junction between a singlet and a triplet superconductor, and assuming an interface potential with translational symmetry, gives the Josephson current as 31 I s = ( ) R(c 12 s 21)I d (ˆn ) k FS (3.1.1) where c 21 is the part of the transmission amplitude which depends on the spin orbit interaction and s 21 is the part which is independent of spin, and and d are the wave functions for the s-wave and odd-parity superconductors, respectively. The vector ˆn is the interface normal and k is the component of Cooper pair momentum which is conserved in the transmission through the interface, and FS is an average over the Fermi surface. This expression reveals that the Josephson current depends on the relative orientations of ˆn and d. The vector order parameter d is tied to the crystal orientation by spin orbit coupling. As a result, the orientation of the crystal face with respect to the crystal axes will have a discernible effect on the Josephson effect in a junction made on that face. For example, the directional dependence of the Josephson coupling is different for each of the basis vectors in Table 1.1.1, leading to selection rules in the Josephson coupling between Sr 2 RuO 4 and a conventional superconductor for different tunneling directions. In an experiment to test for these selection rules, large numbers of Josephson junctions of Sr 2 RuO 4 and a conventional superconductor (indium) in various orientations were checked for a critical current. The result was that Josephson current exists in the direction of the planes, but there is no Josephson current along the direction of the c axis. 32 states in Table 1.1.1, this result is compatible only with the Γ 5 31 A. Millis, D. Rainer, and J.A. Sauls, Phys. Rev. B 38, 4504 (1988). 32 R. Jin et al., Europhys. Lett. 51, 341 (2000). Of the five irreducible representation. 16

25 17 A phase-sensitive configuration which would show the effects of p-wave symmetry was proposed by Geshkenbein, Larkin, and Barone (GLB). 33 The principle is applicable more generally to odd parity, where the wave function is an odd function of k. They consider the phase difference across two Josephson junctions of a SQUID on parallel but opposite faces of an odd-parity crystal. A minimum in free energy for this arrangement is found when the phase difference across one junction is φ but φ + π across the other. A consequence of the net change in phase of π for the loop is that a spontaneous flux of Φ = Φ 0 /2 will form if the inductance of the SQUID is sufficient. Φ 0 is the superconductor flux quantum, Φ 0 = G cm 2. I have prepared samples to be used in a direct measurement of the flux which will be performed in another laboratory, with the results to be reported elsewhere. The experiments described in Chapter 8 take advantage of a second consequence of the net π phase, in which interference between the junctions is manifested in the critical current of the SQUID. In a conventional SQUID, the interference between the two junctions is such that the maximum I c is observed when the total flux enclosed by the SQUID is zero. For a SQUID with an intrinsic π phase change, the state where no external flux is imposed on the SQUID corresponds to a minimum critical current. Theoretically, the Josephson coupling between even- and odd-parity materials could be identically zero for various reasons. As shown by Eq. (3.1.1), if the spin orbit term were zero, the difference in parity would prevent Josephson coupling. Additionally, current through junctions on two identical opposite faces could be disallowed by symmetry for certain forms of the order parameter, or could be zero if the junction planes happen to be parallel to the order parameter d. These issues have led to some confusion about whether the GLB configuration could work. 34 However, for an order parameter with a two-dimensional representation, which is supported by experimental results, 32 Josephson coupling in the GLB orientation is not forbidden by the symmetry. 35 As shown in Chapter 7, sufficient spin orbit coupling exists in Sr 2 RuO 4 junctions to permit Josephson currents between different-parity materials. 33 V.B. Geshkenbein, A.I. Larkin, and A. Barone, Phys. Rev. B 36, 235 (1987). 34 A.J. Leggett, Phil. Mag. B 74, 509 (1996). 35 V.B. Geshkenbein and A.I. Larkin, JETP Lett. 43, 395 (1986).

26 Phase-Sensitive Experiment in High-T c Superconductors For the high-t c materials, the symmetry was suspected to be d-wave so the GLB proposal was modified to take account of the fact that the π phase change for that symmetry occurs between the ˆx and ŷ directions rather than the ˆx and ˆx directions. The results of the I c (H) interference, 3 the measurement of the enclosed flux, 4 as well as many subsequent experiments 36 have been able to confirm the d-wave symmetry for the high-t c cuprates, which non-phase-sensitive experiments could not do satisfactorily Experimental Design To implement the GLB proposal it would be convenient to grow a crystal, or perhaps a thin film, with parallel ac faces. Unfortunately, the best Sr 2 RuO 4 films so far are not superconducting and Sr 2 RuO 4 crystals do not grow with facets. Instead, a method to make a GLB SQUID must be designed using large crystals, without the benefit of lithographic techniques. The design used here calls for polishing two sides of a Sr 2 RuO 4 single crystal such that the polished sides are parallel and both are perpendicular to the crystal planes. The orientation with respect to the a axis is not important and can only be determined by X-ray analysis, which was not performed. The Sr 2 RuO 4 crystal between the two faces is part of the superconducting loop. The conventional (s-wave) junction material is to be deposited on each of the parallel faces, but must also form a continuous link from one junction to the other to complete the loop. This is accomplished by thermally depositing a superconducting film on the crystal while rotating the crystal. The resulting film continuously connects the two junctions. The superconducting film used here is Au 0.5 In 0.5 alloy for the reasons described in Chapter 6. An insulating film (SiO) is used as a spacer between the Sr 2 RuO 4 crystal and the film on the side of the crystal between the two junctions. The result is shown schematically in Fig The area enclosed by the SQUID loop includes the area defined by the insulating film as well as the portion of the superconducting material penetrated by the flux. The area of the SQUID determines the period of the oscillations. 36 C.C. Tsuei and J.R. Kirtley, Rev. Mod. Phys. 72, 969 (2000).

27 19 d I+ V+ Au SiO 150 nm 0.5 In 0.5 h λ ab 100 nm c b a λ c λ 3 =λ= ab 2 λ c λ 1 =2.7 λ λ 2 =0.86 w λ λ f >λ In =40nm Sr 2 RuO 4 λ f HH V I Figure 3.3.1: Schematic showing the SQUID used in the phase-sensitive measurements, implementing the GLB configuration. The dimensions w, h, and d for specific samples are given in section 8.3. The magnetic field H is applied along the direction indicated by the arrow. The shaded area represents the flux penetration governed by the penetration depths which are estimated to be λ µm, λ µm, and λ f 0.07 µm Other Experimental Configurations The corner-junction configuration used for the d-wave experiments (section 3.2) makes a convenient independent verification of the phase-sensitive experiment because the results for both of the even-parity states are known. The corner junction has two faces at ninety degrees to each other, meeting at their shared corner. The junction material is deposited on both faces at once, forming a single junction comprising two crystal faces. Besides having a different relative orientation of the faces, this is not a SQUID. The two faces join at a point and there is no enclosed area other than the penetration depths of the materials. This makes the oscillation period much

28 20 larger compared to the induced flux, so the asymmetry is not as much of a factor in this configuration. For d-wave symmetry, the phase changes by π after a 90 rotation, resulting in the familiar minimum at H = 0 in the I c (H) interference pattern. For s-wave symmetry, I c (H) will be at its maximum at H = 0 (as it will be for any junction orientation since this symmetry is spherically symmetric). For odd parity, the change in phase depends on the specifics of the order parameter. For the simplest order parameter consistent with the phasesensitive result ( d k x ±ik y ), the phase changes continuously with direction and will change by π/2 between two faces at 90 to each other. This will cause neither a minimum nor a maximum for I c (H) at H = 0. A SQUID with both junctions on the same crystal surface can also be used to confirm that the effects observed in the GLB and corner-junction configurations are due to the intrinsic phase difference revealed by the difference in orientation between the surfaces, and not to some other effect. For this SQUID configuration, the maximum of I c (H) will occur at H = 0 because the phase change at each junction is the same.

29 Chapter 4. Sample Preparation and Characterization In this chapter I focus on preparation of samples for the different types of measurements using Sr 2 RuO 4. Detailed descriptions are given which may be of help if the experiments are to be reproduced Crystal Growth I worked exclusively with single crystals of Sr 2 RuO 4. Superconducting films of Sr 2 RuO 4, which could otherwise be useful, are not yet available. Sr 2 RuO 4 crystals are grown without contact with other materials using the floating zone method. 37 The starting materials SrCO 3 (to supply strontium) and RuO 2 (to supply ruthenium) are combined in a starting rod with slightly more Ru than the stoichiometric ratio of 2:1. The higher vapor pressure of Ru will result in higher loss rate for Ru during the growth process. The crystal is grown in a 90% Ar, 10% O 2 environment. The rod is placed in an image furnace where it is held by its ends at the focal point of two elliptical mirrors which focus intense heat at a small portion of the rod. Only this portion melts, and the liquid is held in place by surface tension. Starting at one end, the rod is moved slowly through the focal point so that it gradually cools and crystallizes on one side of the focal point as more material is melted from the other. The solubility of impurities is typically higher in the liquid than in the solid. As a result, more impurities remain in the liquid portion, and tend to be drawn to the end of the rod. Single-crystal rods as large as several centimeters by three to four millimeters in diameter (Fig ) can be grown in this way, with impurities such as aluminum and silicon below 50 ppm. The impurity levels for barium may be higher, but this impurity affects superconductivity in Sr 2 RuO 4 less strongly than the others. Sr 2 RuO 4 rods grown in this way have the c axis of the tetragonal crystal approximately perpendicular to the length of the rod. 37 Z.Q. Mao, Y. Maeno, and H. Fukazawa, Mat. Res. Bull. 35, 1813 (2000). 21

30 22 10mm Figure 4.1.1: Picture of a Sr 2 RuO 4 single-crystal rod grown by the floating-zone technique. Approximately actual size. Courtesy of Z.Q. Mao. (See also Ref. 37.) 4.2. Cutting and Cleaving A large single-crystal rod must be cut into small pieces before preparing individual samples. First, the crystal must be encased in a supporting material to prevent it from cleaving. I used a hard wax, Southbay Technology part number MWH135 both to mount the crystal on a cutting platform and to provide support. I used a wire saw, inches in diameter, with either embedded diamond abrasive, or an Al 2 O 3 abrasive slurry, to cut 1 2 mm slices from the rod. The resulting oval disc is large enough for four to six samples. Polishing is most easily done with a disc of this size, before further cutting (see section 4.4). The oval discs are typically shorter in the direction of the c axis. For my samples, it was convenient cut the discs again, in half along the short axis. Smaller pieces are even more susceptible to cleaving so after encasing in wax, they were cut with the wire saw in a plane perpendicular to the c axis. This orientation seems to result in the least amount of cleaving during cutting. For making samples using the ab plane of the crystal, cleaving worked better than cutting. Sr 2 RuO 4 will cleave easily between the SrO planes (see Fig ), making a clean, flat ab surface. A razor blade or scalpel blade can be used to control the location of the cleave. Starting with either whole discs, cut half-discs, or broken fragments, place the blade parallel to the plane, touching only one corner of the crystal. The corner of the blade should rest stationary on the cutting surface. Then, in a sharp motion, lower the blade, pivoting on the stationary corner. This initiates a cleave at the crystal blade contact point, separating the pieces before the blade comes between them. Only minor damage is done to the crystal (only at the contact point).

31 Etching A common surface-preparation technique in the fabrication of tunnel junctions on single crystals is etching. This may remove contaminated surface layers or smooth irregularities on the surface. However, I was unable to find an etchant for Sr 2 RuO 4. This material is inert and stable, with no interaction with common acids or bases. No etchant for Sr 2 RuO 4 has been reported in literature. A common etchant for high-t c materials is elemental bromine (Br) diluted with an alcohol. I found that this solution had an effect on the crystal, although the etching turned out to be detrimental rather than helpful. A solution of 4% Br, 96% methanol by volume made pits on the crystal surface after five minutes. Extending the time in the etchant enlarged the pits up to a point, but large portions of the surface were never affected. One crystal disintegrated after sitting in a 10% Br solution for 72 hours, but there was no intermediate state where an evenly etched surface was formed. Figures and show two examples of a bromine-etched surface with the resulting pits. In Fig , ruthenium inclusions are visible as bright spots, unaffected by the etchant, showing that the pits are not due to selective etching of the ruthenium. 1.6 µm µm Figure 4.3.1: AFM image of a roughpolished Sr 2 RuO 4 surface etched in a 4% Br, 96% methanol solution for 15 minutes. Figure 4.3.2: Optical image of a finepolished Sr 2 RuO 4 surface etched in a 4% Br, 96% methanol solution for 5 minutes.

32 Polishing Because of the lack of a satisfactory etched surface, all samples used in the measurements were made on cleaved or polished surfaces without etching. Polished surfaces were used when making junctions on the ac face of the crystal, for which polishing is necessary. Early junctions (for example, Hg-In junctions) were polished with sandpaper. After flattening the surface with 400 grit paper, I polished with 1200 grit sandpaper mounted on a rotating disc. I glued the crystal to the edge of a microscope slide and pressed it against the disc. The result was neither completely flat (corners were rounded) nor completely smooth. Much better results came from polishing with a polishing fixture and abrasive slurries or films. The polishing fixture I used was the smallest available from Southbay Technology, where I also got all my abrasives. The following recipe turned out to be most successful: (1) Mount the crystal using hard mounting wax, encasing it completely to provide support. (2) Polish until flat using 9 µm abrasive solution on hard glass polishing surface. (Al 2 O 3 or diamond abrasive works here.) (3) Polish on hard glass with 3 µm abrasive until all scratches from 9 µm polish are gone (check with microscope). (4) Polish on 1 µm diamond abrasive film lubricated with water until a high shine or mirror-like finish is achieved. (5) Polish on 0.5 µm diamond film, dry, several strokes. (6) Polish with 0.1 µm diamond abrasive solution on Southbay Technology Multitex polishing cloth until no scratches are visible at 500 magnification (less than 30 seconds do not over-polish). Sonicate crystal and fixture in water between each step to remove all abrasive. Step 6 was sometimes unnecessary if step 5 produced a satisfactory surface. Atomic force microscope (AFM) scans of surfaces polished in this way showed surface roughness on the order of 5 nm. In some instances, samples needed to be particularly thin along the a direction. When polishing a face perpendicular to the plane, Sr 2 RuO 4 cleaves easily, probably because the abrasive penetrates between layers and initiates a cleave. The thinner the material, the more difficult it is to prevent it from cleaving. The best solution I found was to provide more support than even the hard mounting wax can provide. First, as most cleaving happens at the edges, start with entire oval discs so there is enough sacrificial material to leave a sufficiently large piece at the end. Then trace the outline of the disc on a piece of Teflon and cut a hole just smaller than the outline. A combination of a sharp scalpel and a tiny rotary

33 25 tool works best for this. When the hole is matched to the Sr 2 RuO 4 disc, heat the Teflon on a hot plate to both soften and slightly expand it. Then, push the Sr 2 RuO 4 into the hole with a flat surface. When the Teflon cools, it squeezes the crystal and gives more support during polishing. Mounting a Teflon-supported crystal for polishing is a bit more involved than mounting a bare crystal. The Teflon must be trimmed to expose the crystal on both side on the one side, so it can be glued to the polishing jig, and on the other to allow the abrasive to touch the surface. Standard mounting wax can be used as glue, but it goes between the crystal and the jig and does not encase the crystal in this case. As the crystal becomes thinner with polishing, I trim the Teflon more and more to keep the crystal protruding. The Teflon is too soft to be removed by the abrasive during polishing. The result of this method of crystal support is a polished crystal with a typical size of 1 mm or smaller, with mm of cleaved material flaked off of each edge during the polishing, but with a large center portion remaining intact Sample Support Samples are mounted on a substrate to support the crystal and the gold and copper measurement leads for easier handling. I used thin quartz wafers (for their relatively high thermal conductivity), scored and broken into about 0.25 cm 2 pieces. I glued the crystals to the substrate with a thermally conductive, electrically insulating phenolic (GE varnish 7031, thermal conductivity W/m/K at 1 K). 38 Following air drying for 30 minutes, I sometimes baked the varnish on a hot plate at 400 C until dark brown, making it hard but brittle. If I didn t need to work with the crystal after mounting, the baking was not necessary. However, films needed to be deposited on the crystals after mounting, which sometimes required a firmer mount than the unbaked varnish could provide (see section 4.6) varnishts.html.

34 Evaporation In our evaporation set-up, the substrate and the crystal are 20 cm from the source. The thickness of the condensed film is monitored by a quartz crystal oscillator near the substrate. The materials used in sample preparation should be as pure as possible to avoid uncertainty due to impurities. I used evaporation sources which were all % (6N) pure. Junctions are named by the orientation of the film or by the direction of current flow. ab-plane junctions are also called c-axis junctions (i.e., the current is along the c axis). Junctions on the ac surface are called both ac-face junctions and in-plane junctions (the current through the junction flows along the planes). Note, however, that a planar junction is a junction on either plane with macroscopic contact area. This is to distinguish it from a point contact junction. For these planar junctions, I needed to define a junction area as well as evaporate the material. The size of the junctions did not require anything more sophisticated than a shadow mask. I used two types: glues or resins, which I painted on the crystal, or PTFE tape with which I wrapped the crystal, exposing only the junction area Masking As explained in section 4.8, it was sometimes necessary to mask out tiny areas of exposed ruthenium inclusions on the surface of the crystal. To maximize the remaining junction area, I wanted to mask these inclusions with as small a mask as possible. For this, an epoxy resin was ideal. Because there is no solvent to evaporate, I could paint with a very tiny brush (size ) and make small spots, whereas a solvent-based glue would dry on the brush before it could be painted. Finding and identifying the ruthenium is addressed in section 4.8. For the smallest spots, rather than a brush, I found that a 2 mil wire worked best. After dipping the wire in epoxy, touch it to the ruthenium inclusion. A micromanipulator might be nice for this task but is unnecessary. I controlled the wire tip and eliminated most of the shaking of my hands by attaching it to a stick which pivoted on a stationary surface. My hands were also supported on the surface and I moved the stick as one would a pen. The pivot was much closer to the sample than to my fingers, so the remaining shaking was minimized by the mechanical advantage of the lever.

35 27 When the size of the area to be masked was larger, it was often more convenient to use a solvent-based glue. I used Krylon clear acrylic spray coat, sprayed in a dish and painted on with a brush. This method was also used to cover the sides of the crystal. I started to paint far away from the area to remain exposed and slowly pushed the brush towards that area. This way, I was able to avoid having a drop spread farther than expected when it touched the surface. An alternative masking method uses PTFE (Teflon) tape to define a junction. Although more difficult, I came to prefer this method because there are no chemicals involved. Cut the PTFE tape into strips 1 mm 3 cm and wipe them clean with isopropyl alcohol. Wrap the PTFE pieces in such a way that a rough approximation of the junction area is left exposed. Then, under a binocular microscope, poke and push the tape with tweezers until only the proper areas are exposed. Push the tape flat against the crystal and wrap with more layers of tape to keep the mask in place Rotating To make a continuous film covering more than one side of a crystal, as needed for SQUID samples, the crystal must be rotated during the evaporation. I used a small DC motor mounted in the evaporation chamber, powered via electrical feed-throughs. I rotated the sample at about two revolutions per second. Limited tests suggested that rotating the sample much faster than this could be a factor in producing junctions without Josephson coupling; however, other factors may also have been involved. Because of the rotation, the thickness of the film was smaller than that measured by the quartz crystal oscillator by a factor of π. The rotating sample was first mounted on its substrate in such a way that the three sides of the crystal which should receive the film pointed in turn toward the source. The shadowing from the substrate likely reduced the thickness of the film on two sides by an additional factor < 2. The reported thicknesses do not include this factor.

36 Au 0.5 In 0.5 Films Many samples use an Au-In alloy for the evaporated superconducting electrode. The T c of Au-In alloy is below that of pure indium and below that of Sr 2 RuO 4. The benefit is that the alloy will wet the surface better the pure indium and not be oxidized in air. To make the Au-In alloy, evaporate alternating layers of gold and indium with each layer no thicker than about 100 Å. At room temperature, the two metals interdiffuse over this distance to make an alloy. 39 The thickness of layers corresponds to the ratio of the metals in the alloy. For 1250 Å Au 0.5 In 0.5 films, I evaporated 10 layers each of 50 Å Au and 75 Å In (or, when rotating, 157 Å Au and 236 Å In) because an indium layer will be roughly 3/2 times thicker than a gold layer with the same number of atoms Electrical Lead Attachment In order to make transport measurements of the samples, electrical contact must be made to the crystal. My method of attaching leads was to affix 1 mil (25 µm diameter) gold wires to the sample using a conductive epoxy (epoxy containing silver flakes, commonly called silver epoxy ). Electrical contact between the silver epoxy and the oxide crystals was not automatic, however. Although the mechanism is not clear, I found that baking the crystal and the epoxy improved the electrical contact. The baking helps even after the epoxy compound has cured. For the best contact, however, the epoxy and crystal should be baked while curing. I baked on a hot plate at 40% power for five minutes, where the plate surface reached about 300 C after two minutes and held for the remaining three. The crystal probably did not reach this temperature because it was sitting on a glass microscope slide and had convection air currents cooling it. Making contact with silver epoxy to metals (e.g., the thin films forming the other electrode) required no baking. In fact, the baking procedure could actually be harmful, for example by causing oxidation of the films. The baked leads had to be put in place before film deposition. Also note that if these leads were to be on the bottom of the crystal, they were applied before mounting to the substrate when the bottom surface became coated with GE varnish. 39 Yu. Zadorozhny and Y. Liu, Phys. Rev. B. 66, (2002).

37 29 My method of attaching leads used two different pieces of wire in order to provide strain relief. First, attach a wire of soft metal such as gold to the sample, and glue it to the substrate as well. The soft metal will be less likely to come off during a thermal cycle. Then, attach a copper wire to the substrate and to the gold wire with silver epoxy. The copper wire can be soldered to copper pads on the sample block. The copper wire will break off at the substrate, protecting the sample if there is an accident Surface Characterization Although most researchers working on Sr 2 RuO 4 have been aware of a eutectic Ru Sr 2 RuO 4 system which occurs when crystals are grown with excess ruthenium, and the T c enhancement that occurs at the interface between the two materials (section 2.2.5), it is often not realized that ruthenium inclusions occur in all but the most ruthenium-deficient crystal rods (those with T c below 0.9 K). While the presence of a small amount of ruthenium may be unimportant for bulk studies, measurements made on the surface especially tunneling measurements will be affected by the presence of the ruthenium inclusions. When cutting or cleaving a small piece of crystal from the large rod for study, luck may yield a piece without ruthenium but in general this will not be the case. Ruthenium inclusions are as small as about 1 micron in diameter but may be as large as 10 microns by 2 microns in crystals which are grown especially for the purpose of creating the eutectic system. Large inclusions in ruthenium-rich samples are easy to see under a small microscope and this may have contributed to the smaller ones being overlooked. The smallest inclusions are not visible unless the surface is carefully polished and examined. This observation, along with the following method for systematically finding ruthenium inclusions, is a significant contribution of this research. The method is not fancy. Simply polish the crystal to a reflective surface using the recipe in section 4.4 and examine under a microscope with coaxial illumination and a magnification of at least about 150. The polished surface should be perpendicular to the line of sight so that the reflection from the coaxial illumination is collected by the objective. Under these conditions, the difference in reflectivity of the Sr 2 RuO 4 and ruthenium surfaces will give enough contrast to detect the ruthenium.

38 30 Note, however, is that the polishing technique is important. The reason for the various polishing surfaces given in section 4.4 is to make sure the ruthenium inclusions are polished along with the crystal. At certain stages, certain polishing surfaces, such a glass or the Multitex polishing cloth will tear the ruthenium from the surface and leave small pits behind. Some ruthenium remains in the pits, but it is often invisible since the ruthenium surface is no longer perpendicular to the illumination. Figure shows an AFM picture of a polished surface with a ruthenium inclusion and a pit with a small amount of ruthenium remaining. Figure 4.8.1: AFM image of a polished Sr 2 RuO 4 ac surface. In the upper left is a ruthenium inclusion with small portions of the recess containing it on three sides. Below and to its right is a pit from which the ruthenium has come out. The taller spot in the middle of the pit shows that some ruthenium remains. If a crystal has been carefully polished to remove scratches at each polishing stage, it is almost certain that any remaining blemishes are pits which still contain a small amount of ruthenium.