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IMPURITIES IN UNCONVENTIONAL SUPERCONDUCTORS By LECH S. BORKOWSKI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1995 ACKNOWLEDGEMENTS I would like to express my deep gratitude to Peter Hirschfeld for overseeing my research. He was enthusiastic and generous on both scientific and personal levels. I am very happy I had a chance to work with him. I am grateful to the members of my dissertation committee, Pradeep Ku mar, Kevin Ingersent and Khandker Muttalib for stimulating discussions and for posing thoughtful questions. I would like to thank Bohdan Andraka for discussions on heavy fermion experiments and their interpretation. Yoshio Ki taoka kindly explained results of some of the NMR experiments of the Osaka group. Some of the calculations in chapter 4 were done in collaboration with Bill Putikka. I am also grateful to Selman Hershfield, Avi Schiller, Hans Kroha, Matthias Hettler, Martti Salomaa for discussions. I am greatly indebted to Grzegorz Harati for collaboration in our work on 3He and for being a good and understanding friend. I would like to thank Peter WSlfle and his Institut in Karlsruhe for hospitality during my stay there. All of this would be impossible without the love and support of my family  thank you. This work was supported in part by the Department of Sponsored Research of the University of Florida, the National Science Foundation, Institut fiir The orie der Kondensierten Materie at the Universitdt Karlsruhe, and the Institute for Fundamental Theory at the University of Florida. TABLE OF CONTENTS ABSTRACT . . . v CHAPTERS 1 INTRODUCTION ............. ...... 1 2 KONDO EFFECT IN SYSTEMS WITH DENSITY OF STATES VANISHING AT THE FERMI ENERGY . 6 2.1 MeanField Theory of the SingleImpurity Problem 6 2.2 NCA Approximation . .. 11 2.3 Kondo Impurity in a Superconductor ... 16 3 PHYSICAL PROPERTIES OF MAGNETIC IMPURITIES IN SUPERCONDUCTORS .......... 24 3.1 Introduction . . 24 3.2 Results for Conventional Cuperconductors ... 29 3.2.1 Suppression of the Critical Temperature .. 29 3.2.2 Position of Bound States . .. 31 3.2.3 Specific Heat Jump .. ... 35 3.2.4 Penetration Depth . .... 35 3.3 Unconventional Superconductors . 37 3.3.1 Suppression of the Critical Temperature .. .39 3.3.2 Density of States ... . 39 3.3.3 Specific Heat . . .. 40 3.3.4 Specific Heat Jump . .. 43 3.3.5 Penetration Depth . 44 4 ANISOTROPIC CONVENTIONAL SUPERCONDUCTORS VS. UNCONVENTIONAL SUPERCONDUCTORS 48 4.1 Experimental Motivation . .. 48 4.2 The Model ...................... .49 4.3 Density of States .................... .54 4.4 Critical Temperature. .. . .56 4.5 London Penetration Depth . .... 58 4.6 Effect of Spin Scattering . .... .59 4.7 Microwave Conductivity . .... 64 5 TWOCHANNEL KONDO IMPURITIES IN SUPERCONDUCTORS .. . .69 6 SUMMARY AND CONCLUSIONS . .... 78 REFERENCES . .. ... 84 BIOGRAPHICAL SKETCH ................... 90 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy IMPURITIES IN UNCONVENTIONAL SUPERCONDUCTORS By LECH S. BORKOWSKI August 1995 Chairman: Peter J. Hirschfeld Major Department: Physics We present a selfconsistent theory of superconductors in the presence of Kondo impurities. The impurity degrees of freedom are treated using the large N slave boson technique, leading to tractable equations describing the interplay between the Kondo effect and superconductivity. We show that for a single impurity in a superconductor with density of states N(w) ~ wlr, there exists a critical coupling Jc below which the Kondo effect does not occur. However, for r < 1 or N = 2 any finite concentration of impurities drives Jc 0. The technique is tested on the swave case and shown to give good results compared to other methods for TK>Tc. We calculate low temperature thermodynamic and transport properties for various superconducting states, including isotropic swave and representative anisotropic model states with line and point nodes on the Fermi surface. The theory provides support for phenomenological models of resonant impurity scattering in heavy fermion systems. Motivated by some recent experiments on highTc superconductors, we study properties of superconducting states with "extended swave" symmetry. In the presence of impurities, thermodynamic properties of such states exhibit gapless behavior, reflecting a residual density of states. While for a range of v impurity concentrations, properties reflecting the density of states alone will be similar to those of dwave states, transport measurements may be shown to qualitatively distinguish between the two. We also discuss the effect of twochannel Kondo impurities on supercon ductivity. In the strong coupling regime such impurities are pairbreakers, in contrast to the ordinary Kondo effect. Measurements of Tcsuppression may help in identifying impurities displaying this more exotic exchange coupling to the conduction band. CHAPTER 1 INTRODUCTION The problem of a magnetic impurity in a superconductor has been exten sively studied, but is not completely solved because of the difficulty of treating the dynamical correlations of the coupled impurityconduction electron sys tem together with pair correlations. Generally, the behavior of the system can be characterized by the ratio of the Kondo energy scale in the normal metal to the superconducting transition temperature, TK/Tc. For TK/Tc < 1, conduction electrons scatter from classical spins and physics in this regime can be described by the AbrikosovGor'kov theory [ 1]. In the opposite limit, TK/Tc > 1, the impurity spin is screened and conduction electrons undergo only potential scattering. In this regime swave superconductors are largely unaffected by the presence of Kondo impurities due to Anderson's theorem [ 2]. Superconductors with an anisotropic order parameter, e.g. pwave, dwave etc., are strongly affected, however, and potential scattering is pairbreaking. The effect of pair breaking is maximal in swave superconductors in the in termediate region, TK ~ Tc, while in the anisotropic case it is largest for TK/Tc 4 oc. The interest in the latter class of systems comes from studies of the heavyfermion and highTc superconductors, exhibiting many properties that can be explained by an unconventional order parameter. The word "un conventional" here means that there are broken symmetries in addition to the U(1) symmetry broken in classic superconductor. 2 The Kondo effect is accompanied by the formation of a narrow manybody resonance of width TK near the Fermi level in the impurity spectral density Nf(w). It is reasonable to expect that the opening of a gap A in the con duction electron density of states leads through hybridization processes to a similar gap in Nf(w), which destroys the Kondo effect if sufficiently large. Withoff and Fradkin [ 3] pointed out that the two problems of impurity spins coupled to baths of conduction electrons with (a) constant density of states and (b) with a fully developed gap represent two extreme members of a family of problems given by specifying a generalized conductionelectron density of states N(w) = Ciwlr, WI < D, and 0 < r < oo. Making use of renormaliza tion group arguments as well as explicit calculations for the large degeneracy SU(N) Kondo (CoqblinSchrieffer) model, they showed, for r > 0, the exis tence of a critical coupling Jc below which impurities are decoupled from the conduction band and no Kondo effect occurs. The potential physical examples of this phenomenon include unconventional superconducting states with line and point zeroes in the momemtumdependent gap function, corresponding to densities of states N(w) varying as w and w2, respectively. Such states are possibly realized in the heavy fermion superconductors UPt3, UBe13, URu2Si2, CeCu2Si2, and UPd2Al3. Other examples of systems which have a gapless ex citation spectrum under certain conditions are aSn, PblxSnxTe at a critical composition, domain walls of PbTe, PbTeSnTe heterojunctions and graphite. In the context of heavyfermion superconductivity, theories of impurity scattering in such states have been given by Ueda and Rice [ 4] for the case of weak potential scattering, and by Hirschfeld, Vollhardt, and Wlfle [ 5], and SchmittRink, Miyake, and Varma [ 6] for strong scattering. Moment formation 3 was not considered in these theories, but pair breaking still occurs because of the vanishing of the anomalous oneelectron impurityaveraged selfenergy. The strength of the scattering was parametrized in the latter works by a phase shift So for swave potential scattering of electrons at the Fermi surface. One of the principal results of these treatments was that in the resonant scattering limit, J0 * 7r/2, corresponding to the singleimpurity spin Kondo effect [ 7], a "boundstate" resonance was found to form in the superconducting density of states., N(w), leading to gapless effects in thermodynamic properties. These are the analogs of the bound states found in discussions of Kondo effects in swave superconductors [ 8]. While this work was begun in anticipation of applications to heavy fermion superconductors, recent measurements of penetration depth [ 9], photoemis sion [ 10, 11], and Josephson tunneling on YBCO and BSSCO [ 12, 13, 14, 15] have provided evidence that the copper oxide superconductors may be un conventional, possibly dwave as well. This conclusion remains controversial, however, and experimental tests to distinguish conventional from unconven tional pairing are of great current interest. One would like, for example, to develop an understanding of the effect of doping Zn and Ni impurities in the CuO2 planes. Simple models of Zn and Ni acting as strong and weak potential scatterers in a dwave superconductor, respectively, are consistent with some experiments at low doping levels [ 16, 17, 18], but inconsistent with other mea surements [ 19]. Since in some cases Zn impurities appear to possess a magnetic moment at higher temperatures [19], it is of interest to explore whether an sd type exchange coupling of conduction electrons to an impurity embedded in an unconventional superconductor can describe the range of behavior observed. 4 In this work we focus initially on basic thermodynamic and transport prop erties of superconductors doped with Kondo impurities. Our aim is to develop a tractable, selfconsistent scheme for the calculation of all basic properties of superconductors using methods known to successfully describe the most diffi cult aspect of the problem, namely the dynamics of the Kondo impurity. For this reason we have adopted the largeN "slave boson" approach of Barnes [ 20], Coleman [ 21], and Read and Newns [ 22], as this approach is well known to provide a good description of the spectral properties of the Kondo impurity for sufficiently low temperatures. We emphasize that this description is adequate for most of our purposes (TCTK) even in the case of spin degen eracy N = 2, since the impurity spectral resonance is located at the proper position, i.e. exactly at the Fermi level in this approximation. Impurity doping experiments in high temperature superconductors mo tivated us to consider also the effect of nonmagnetic impurities on strongly anisotropic superconductors of the swave type. Many experiments probe only the lowlying quasiparticle states of the superconductor, which in the pure sys tem are quite similar for both anisotropic s and dwave states with nodes. Nonmagnetic scattering reduces anisotropy of such states without destroying superconductivity, in contrast to the effect the nonmagnetic impurities have on unconventional (nonswave) superconducting states. In anisotropic swave states having lines of zeroes of the order parameter on the Fermi surface, some times called "extendeds" states, scattering on such impurities removes the nodes of the order parameter at certain critical concentration, and induces an energy gap at higher concentrations. This gap should be visible in lowT, 5 low energy experiments. If the gap is too small to be observed in thermody namic experiments, one should try transport measurements, e.g. microwave conductivity or thermal conductivty. In chapter 2 we address questions related to the conditions for the oc curence of the Kondo effect in systems with the density of states vanishing at the Fermi level [ 23]. In chapter 3 we calculate basic properties for both con ventional and unconventional superconductors doped with Kondo impurities [ 24]. Methods of distinguishing strongly anisotropic swave superconductors from dwave superconductors having very similar quasiparticle excitation spec trum are discussed in chapter 4 [ 25, 26]. In chapter 5 we study the effect of twochannel Kondo impurities on superconductors [ 27]. Conclusions are presented in chapter 6. CHAPTER 2 KONDO EFFECT IN SYSTEMS WITH DENSITY OF STATES VANISHING AT THE FERMI ENERGY 2.1 MeanField Theory of the SingleImpurity Problem An antiferromagnetic exchange interaction between a magnetic moment of an impurity and a single channel of conduction electrons leads [ 28, 29] at low temperature to the formation of a spin singlet state. The temperature scale over which the crossover to the lowT state with compensated impurity spin occurs is called the Kondo temperature TK. The resistivity per impurity as a function of temperature becomes logarithmically divergent when the crossover is reached and saturates at low T, remaining finite at T = 0. The lowT state is a Fermi liquid with renormalized Fermi liquid parameters. [28,29] It is well known that any nonzero conduction electron density of states at the Fermi level N(EF) leads to the Fermi liquid ground state in the Kondo problem. What happens in systems with N(e) vanishing at e = F? Is the ground state of such system containing a dilute ensemble of Kondo impurities metallic or not? If the low energy density of states is described by a power law, N(e) _ CIler, how does the lowtemperature, lowenergy physics depend on J and r? Examples of systems in which the density of states vanishes with a power law near EF are unconventional superconductors and semimetals. To have a basic 7 understanding of the problem, Withoff and Fradkin [3] used poorman's scaling for the spin' Kondo model H = k (cl fckT + ct k4) + J Z(ctck't c Ckt') Sz k k,k (2.1) + J (cCtck'S +c C4ck'tS ), k,k' with the powerlaw density of states over the whole bandwidth. In poorman's scaling [ 30] one eliminates highenergy states near band edges, reducing the width of the band to D 6E. The new effective coupling constant Jeff is given by J' = J + N(D)J'6E/D + ... To restore the original number of states, 2CDl+r/(1 + r), the unit of length is scaled by (D/D')l+r. The coupling constant is changed by (D'/D)1+r. To restore the cutoff all quantities with units of energy are multiplied by D/D'. The renormalized coupling constant JR is then Jr = J' J + J(JCDr r)8E/D. (2.2) D ) The fixed points are J = 0, J = oo, and Jc = r/(CDr). The last fixed point is unstable. For J < Jc the effective J flows to 0, and at J > Jc the coupling constant flows to oo. In the ordinary Kondo problem with N(E) = const., there are only two fixed points: the unstable one at J = 0 and the stable one at J = oo. This is then an indication of qualitatively different physics in the problem with r = 0. Is the result of the poorman's scaling confirmed by other approaches, e.g. the largeN approximation? The slave boson approximation is known to give correct results in the limit of large orbital degeneracy for the ordinary Kondo 8 problem at low temperature [20,21]. The simplest version of the Anderson Hamiltonian to describe a rareearth impurity is H = EkcCkm + EfEf im + U ftnfmgfftm' km m m>m' (2.3) + Vk(cfm+ h.c.). km The electrons are labelled by their magnetic quantum number m = j, ...,j, with j the total angular momentum, and the impurity degeneracy is N = 2j+1. Here we ignore spinorbit interaction and crystal fields. An electron on one of the degenerate f orbitals at the impurity site has energy Ef. The energy U of the Coulomb repulsion between two impurity felectrons may be large, e.g. 5 10 eV in the lanthanides. The impurity electrons hybridize with the conduction band with amplitude Vk. Here we assume Vk = V = const., independent of k. When the bare impurity level Ef lies deep below the Fermi energy, such that IV2/EuI < D, and for large onsite Coulomb repulsion, U > V2/D, the Hamiltonian (2.3) can be simplified by the SchriefferWolff transformation [ 31]. The resultant Hamiltonian is the SU(N) generalization of the Kondo model (2.1), also called the CoqblinSchrieffer model [ 32], H = Ekmcm +J kmfmfmCkm. (2.4) kmckm+ f t fmc, 24 km kk' mm' The fourfermion term can be eliminated by the HubbardStratonovich trans formation. In the limit N + oo, the saddlepoint solution is equivalent to a mean field theory with the boson field playing the role of an (unphysical) order parameter, [22] a = (fmckm), (2.5) km 9 The mean field Hamiltonian now reads H = ckmckm + : (akmfm + h.c.) + f(nf 1). (2.6) km km The auxiliary field and position of the saddle point are obtained by solving two selfconsistent equations KH)= 0, (2.7) and =0. (2.8) For a constant conduction electron density of states, N(w) = No = 1/2D, Iwl < D, equations (2.7) and (2.8) were solved by Read and Newns [22], leading to a Lorentzian impurity spectral density centered at ef, of width F = 7rNoo2. The Kondo temperature in their approach is the width of this low energy resonant level, given by TK = v/2 + 2 = Dexp(1/NNoJ). Equation (2.7) can be written as 1 r N = 0 def(e)N(E)Nf(e), (2.9) where Nf(e) = lmGf(e+iO+) is the impurity density of states, and G (w) = w Ef Em(w) is felectron Green's function, while equation (2.8) becomes No f ()N() (2.10) = c f( () f ReEm(E))2 + (Imj m(c))2 0.5 0.0 NoJ Figure 2.1 Schematic flow diagram for singlechannel finiter Kondo prob lem obtained from the Numerical Renormalization Group (based on Ref. [26]). The impurity selfenergy Em is E(e) = P N() i7r2N(E). (2.11) The critical coupling obtained from the solution of equations (2.9) and (2.10) corresponds to a = Ef = 0 at T = 0. From equation (2.10) the critical coupling is Jc r/CDr. Both a and Ef vanish as powerlaw functions of J JcI near Jc, a o (J Jc)1,/r, (J Jc)1/r. This result comes from the work of Withoff and Fradkin and was later confirmed by us in a numerical solution of equations (2.9) and (2.10). The energy scale for the formation of the singlet is given near the transition by TK = (r(J Jc)/CJJc)1/r). Recent Numerical Renormalization Group calculations confirm the mean field result for r < 1 but differ substantially at larger r [ 33]. The qualitative form of the flow diagram obtained within NRG is shown in Figure 2.1 There is a critical exponent, re = 0.5, such that for r > re the Kondo effect does not occur. 11 In a more realistic treatment, the density of states may be assumed to vary as a powerlaw over a small energy scale A << D, N(w) = Clwlr, and to be constant otherwise, N(w) = C, for A < jwl < D. We find that the critical coupling obtained in this case is independent of r, at least for r > 1, Jc  2D/ln(2D/A). It was shown for the problem with a small gap, N(w) = 0 for Iwj < A, that Quantum Monte Carlo and Numerical Renormalization Group approach yield similar results [ 34, 35, 36, 37, 38]. The transition associated with the critical ratio TK/A might be observable provided we can tune TK/A. However, it is not a phase transition in the usual sense it can occur only at T = 0. At any finite T the system will not exhibit singularities of the thermodynamic functions but it will have instead a smooth crossover from the impurity spinsingletlike characteristics to a free moment behavior with increasing A. One of the quantities of interest to experimental studies of such a transition is the spin susceptibility. In the ordinary, single channel Kondo problem, the static susceptibility reaches the limit X > const./TK, or TX  0, as T + 0, indicating the screening of the impurity spin by conduction electrons. In a system with a gap in the conduction electron spectrum, the formation of a singlet state is less advantageous energetically and X(T = 0) becomes finite as the gap size increases or electron density decreases. Measurements of X at finite T should indicate a crossover to a Curielike susceptibility as the singlet ground state becomes unstable. 2.2 NCA Approximation A more sophisticated version of the largeN approach, the noncrossing approximation (for review of the method see Refs. [ 39] and [ 40]) leads to the same conclusions and allows for an extension of the theory into the high Nf(w) S TK Ef f, Figure 2.2 Schematic impurity density of states for the Anderson model; p is the Fermi level and Ef is the position of the bare impurity level. The Kondo temperature is the width of the low energy manybody resonance. temperature regime, T > TK. In NCA one sums all diagrams in the projected perturbation expansion with no crossing conduction electron lines. Here we do the calculations for the Anderson model, H = Ekc mckm + Ef f fm + V [mfmbt + h.c.] k,m m k,m (2.12) A fm + btb (2.12) where A is a Lagrange multiplier enforcing the constraint nf+nb = Ym ftmfm+ b+b = 1, preventing double occupancy of the impurity site. The impurity density of states for the Anderson model in the Kondo regime, Ef > V2/D is shown schematically in Figure 2.2 The narrow feature at the Fermi level is the AbrikosovSuhl manybody resonance of width TK, which controls all lowT thermodynamic and transport properties. The NCA is known to reproduce all qualitative features of the Anderson model spectral function on the scale shown, and fails only below an energy TNCA T2/D [ 41,40]. The selfenergies for the boson and fermion propagators are, respectively (see Figure 2.3 ), )f Eo(w + i0+) = NV2 f def(e)N(e)Gm(w + e + iO+), (2.13) Joo o0 Zm Figure 2.3 Selfenergies for the empty impurity flevel (slave boson), E0, and for the occupied flevel fermionn), Em. and Em(W + i0+) = V2 J de(1 f(e))N(c)Go(w + iO+). (2.14) At T = 0 and for integer r equations (2.13) and (2.14) can be transformed into a system of differential equations and can be solved by a generalization of the method used by Inagaki [ 42] and later by Kuramoto and Kojima [ 43]. The solution of NCA equations at finite temperature for r = 0 is discussed in detail e.g. in a review by Bickers [40]. Here we only show the results for the spin susceptibility. The temperature dependence of the static spin susceptibility exhibits a crossover from the highT freemoment behavior to the lowT strong coupling regime. The levelling off of the r = 0 curve in Figure 2.4 indicates a crossover to the low temperature Fermi liquid state, as confirmed by the Bethe Ansatz [ 44] and NRG solutions of the problem [29]. Figure 2.4 captures the essence of the problem correctly: increasing r leads to lower TK. 4 %0.2 3 0.1 F/D=0.25 II 32 r= 1 0 Ii 6 4 2 0 logo1(T/D) Figure 2.4 The spin susceptibility for r = 0 (circles), 0.1 (triangles), 0.2 (rhombuses); N = 2, F/D = 0.25, Ef/D = 0.67. The Kondo tempera ture is TK/D = (r/7D)l/Nexp(7rEf/Nr), and for r = 0, r/D = 0.25, it is approximately equal to TK/D 4.2 x 103. In Figures 2.5 2.6 2.7 and 2.8 we show logx(T, w = 0) for r = 0.05, 0.1, 0.15, 0.2, and several values of hybridization. We clearly see a decrease of TK with decreasing F in each set of data. Figure 2.9 shows log(TX) for two lowest values of F from each of the Figures 2.52.8. In the case of r = 0.2 the transition from the screened to the finite moment regime is rather convincingly demonstrated. For r = 0.05,0.1, and 0.15, the values of F seem not to be far from the critical ones, although the transition is not reached. How does Fc obtained in NCA compare to the critical coupling obtained in the poor man's scaling and in the mean field approach? Remembering that in the CS limit J = V2/E] = rF/rEf, we find that NOJc,NCA(r = 0.2) _ 0.16, where we assumed Fc 0.32 for r = 0.2. F/D=o.14 2 a r=0.0500 Ao 0 r=0 AAA F/D=0.15 00000 2 SF/D=O. 175 '00000222 0.2 /D=0.2 0 A AA X0.1 a 0.0 ^a 7 5 3 logio(T/D) Figure 2.5 The spin susceptibility for r = 0 and r = 0.05; N = 2, Ef/D = 0.67. For r = 0 and F = 0.14,0.15,0.175, and 0.2, the Kondo temperature is TK/D 1.15 x 104, 1.96 x 104, 5.77 x 104, and 1.31 x 103, respectively. The critical coupling NOJc for r = 0.2, in both the mean field and the NRG approach is approximately 0.2. 0 0.2 F/D=0.275 OI N I~L Oo _____ 0.0 7 5 3 1 logio(T/D) Figure 2.6 The spin susceptibility for r = 0 and r = 0.1; N = 2, Ef/D = 0.67. For r = 0 and F/D = 0.2,0.225,0.25, and 0.275, the Kondo tem perature is TK/D 1.31 x 103,2.5 x 103,4.2 x 103, and 6.44 x 103, respectively. m0.2 /D=0.4 I3 o' ,1 '0 0.0 AMA 6 4 2 0 log o(T/D) Figure 2.7 The spin susceptibility for r = 0 and r = 0.15; N = 2, Ef/D = 0.67. For r = 0 and r/D = 0.3, 0.35, 0.4, 0.5, and 0.6, the Kondo temperature is TK/D = 9.3 x 103, 1.65 x 102,2.6 x 102,4.9 x 102, and 7.6 x 102, respectively. 18 F/D=0.3 oo0 0 r=0.2 o0 0o a y r=0 A mAAAA AAA A AA A F/D=0.35 o 0 A AAAA^ A A A A 1& F/D=0.4 S0.2 /D=0.5 00 S0.1 0 0.0 00 A A A A AM&AAAAAdA &A 7 5 3 1 logio(T/D) Figure 2.8 The spin susceptibility for r = 0 and r = 0.2; N = 2, Ef/D = 0.67. For r = 0 and F/D = 0.3, 0.35, 0.4, and 0.5, the Kondo temperature is TK/D = 9.3 x 103, 1.65 x 102, 2.6 x 102, and 4.9 x 102, respectively. o1 1.0 W 0000 0o0 1.5 0<0.35 2.0 o 2r=0.2 o 2.5 8 6 4 2 logio(T/D) Figure 2.9 Logarithm of TX for two lowest values of F from each set of data shown in Figures 2.52.8. 2.3 Kondo Impurity in a Superconductor One of the obvious applications of the theory discussed in the previous section is the problem of a Kondo impurity in a superconductor. A supercon ductor differs from a normal metal not only through the gap in the density of states, but also through the existence of offdiagonal long range order described by the order parameter A(k). To analyze the problem we start from the SU(N) CoqblinSchrieffer Hamiltonian, already introduced in equation (2.4), and add a simple BCSlike pairing of electrons on opposite sides of the Fermi sphere, H = e km mf+ k fm c+ km + [(k)Ccmkm + h.c. , k,m k,k' m,m' k,m (2.15) We now generalize the procedure of Read and Newns [22] to include super conducting correlations in the functional integral representation of (2.15). The saddlepoint approximation to this theory is equivalent in the N + oo limit to a meanfield theory of (2.15) with meanfield amplitude a = (J/N) Ekm(Cmfm) and Lagrange multiplier ef implementing the average constraint. It leads to the two equations 1 / 1 N = Im J dwf (w) Tr[(o + 3)G (w + iO+)], (2.16) and = Im dwf(w) Tr [(o + r)GO(k,w + iO+)Gf(w + iO+)], (2.17) J oo where GO denotes the conduction electron Green's function in the pure super conductor and Gf is the full impurity Green's function, given by G 1(w) = wro Ef3 Ef f(w). (2.18) Both are matrices in particlehole space spanned by the Pauli matrices ri, G ( G ) =and Gf =( j F ) (2.19) and Ef(w) = o2 Zk G(k, w) is the impurity selfenergy. In the supercon ducting state, we must solve the full saddlepoint equations (2.13) and (2.14) together with Dyson's equations for Gf and G as well as the gap equation A(k) = dwf(w) 1 Vkk, Tr [(71 if2)G(k, w)]. (2.20) Joo k , Let us focus first on the case of a single impurity. We also consider the case of small N, despite the fact that Eqs (2) and (3) are strictly valid only in the largeN limit. Here we adopt the point of view that, since the saddle point for N = 2 is known to reproduce the correct analytic lowtemperature normal state behavior [7] of the f resonance, including its position at Ef = 0, the N = 2 theory will provide a good starting point for a description of TK > T regime. For the superconducting order parameter we take for simplicity the usual isotropic swave state A(k) = A, model pwave states with lines ["polar", A(k) = Aokz] and points of nodes ["axial", A(k) = AO(k +iky)] on the Fermi surface, with densities of states varying at low energies w < AO as 0, w and w2, respectively. The critical coupling Jc in this case [3] is now defined to be that J for which a = ef = 0 is the only solution of Eqs. (2) and (3), with G(k, w) replaced by GO(k, w). We note that to show that a and ef always scale to zero together at the transition for N < oo requires a careful analysis of impurity bound states in Af(w), which occur in the gap and outside the band edges. It follows from this analysis and from Eq. (3) that Jc is independent of N. For the (unphysical) case AO = D we recover the WF r = 1 result, Jc/D = 1, for the polar state, while for the axial state we find Jc/D = 1.44. This differs slightly from the WF r = 2 result Jc/D = 1.33, as the axial density of states deviates from pure w2 behavior at larger energies. In the physical limit A0 < D, we obtain Jc 2D/ln(2D/Ao) for axial, polar and isotropic swave states. In Figure 2.10 we plot Jc vs AO/D for all three states. 2.0 i S 1.5 coaxial S1.0 po r 0.5 0.0 ' 0.0 0.2 0.4 0.6 0.8 1.0 Ao/D Figure 2.10 Critical coupling Jc/D for one impurity as a function of the order parameter amplitude. To study the case of finite impurity concentrations, we calculate selfconsistent Green's functions averaged over impurity positions in the usual way, leading to Gf7(w) = crT0 ef73, and G1(w) = UTro ckr3 A(k)r1, where w = + F (/ (2(k) )2 1/2 (2.21) and S=w + ar/(D2 + 2). (2.22) Here a = fiTcN/2wr, and f = n/TcoNO is the scaled impurity concentration. In general, Gf1 and G1 will also contain additional offdiagonal renormal izations, which vanish in the pwave case considered here. An interesting consequence of the selfconsistent treatment of impurity scattering is that the conclusions of WF regarding the existence of a critical exchange coupling Jc based on a singleimpurity analysis are modified. It is clear from physical considerations or from Eqs. (2) and (3) that the Kondo effect occurs for all J > 0 whenever the density of states at the Fermi level 23 N(0) is finite. Since in the polar state any finite impurity concentration f may be shown to lead selfconsistently to N(0) > 0, as also found in Refs. [4,5,6], the transition discussed by WF does not take place. This may easily be seen by solving the equations for a& and C at w = 0, with ImC(0) > 0. A closer analysis shows that Jc = 0 for all superconducting states with density of states N(w) ~ wr for w < A0, r < 1. In the axial state (r = 2), and indeed for any state with 1 < r < oo, the cases N = 2 and N > 2 are qualitatively different. If N > 2, a critical concentration is required to create a gapless state N(0) > 0, and thus drive Jc 0. However, when the bound state is located exactly at the Fermi surface (N = 2, c = 0), we find again a finite density of states N(0) for any finite concentration. These results are also in accord with earlier studies [5,6], where the phenomenological phase shift 60 is crudely given here by cotle /F. Our results suggest that the transition discussed by WF might be ob servable in ordinary superconductors doped with Kondo impurities with spin degeneracy N > 2, e.g., Ce. For N = 2 we have shown that effective Kondo temperature in the superconducting state is reduced but never vanishes. Never theless, in relatively clean systems deviations in thermodynamic properties per impurity from those of the pure superconductor may be qualitatively similar to what one might expect from a WFtype analysis if the effective TK is driven to zero. The theory presented here provides an easily tractable framework to calculate such properties, as well as transport coefficients, in the superconduct ing state. Furthermore, it improves upon phenomenological theories [5,6] by including a Kondo impurity description of the energy dependence of scattering phase shifts. CHAPTER 3 PHYSICAL PROPERTIES OF MAGNETIC IMPURITIES IN SUPERCONDUCTORS 3.1 Introduction In this chapter we focus on basic thermodynamic and transport properties of superconductors doped with Kondo impurities. We would like to examine these properties both near Tc and at T fully developed. The presence of bound states within the superconducting gap strongly modifies the lowT behavior. This is true for both conventional and unconventional superconductors. After testing our approach and providing some new results for the swave case we discuss results for unconventional superconductors. We emphasize that the largeN description is adequate for most of our purposes (TcTK)) even in the case of spin degeneracy N = 2, since the impurity spectral resonance is located at the proper position, i.e. exactly at the Fermi level in this approximation. As in chapter 2, we use the largeN slave boson technique for the SU(N) Anderson model describing an Nfold degenerate band of conduction electrons, ckm, m = 1, ...N with energy Ek hybridizing through matrix element V with a localized impurity state fm. In general this Hamiltonian contains a term with the Coulomb repulsion U between two electrons present at the impurity simultaneously. In many compounds U is large and for the purpose of studying the low temperature physics we will assume U = co. The onsite repulsion term 25 is then absent, but a constraint is added to ensure that the system remains in the physical part of the Hilbert space. The conduction band is assumed to have a constant density of states in its normal state, N(w) = 1/2D = No. We also include a BCSlike pairing term of electrons on opposite sides of the Fermi sphere, H = kctkmCkm + Ef Z ftmfm + V [ctkmfmb + h.c.] k,m m k,m (3.1) + [A(k)ctkmckm + h.c.] + A(Z ftmfm + btb), k,m m In the limit Ef + oo, NoV2/Ef =const, Equation (3.1) reduces to the CoqblinSchrieffer Hamiltonian with pairing studied in section 2.3. Here we have chosen the more general form (3.1) to study deviations from single oc cupancy, nf 1, although we do not attempt to explore the fully developed mixed valent regime. The mean field approximation to this model, with meanfield amplitude (b), leads to the two equations 1 f 1 S= Im 0 dwf (w) Tr(ro + T3)Gf(w + iO+), (3.2) and Ef = Im I dwf(w) Tr [(Tr + r3)(G (k, w + iO+)Gf(w + iO+)) , (3.3) which determine (b) and ef, the latter being the position of the resonant state. Equations (3.2) and (3.3) should be solved selfconsistently together with the gap equation (2.20). The full conduction electron Green's function G and G F  = = = >< = =<== := = =  +  >= := + +,z <= Gf == >= + >=:= Ff Figure 3.1 Dyson equations for the conduction electron and impurity Green's functions. Double lines are full Green's functions; single lines are bare Green's functions (V = 0). the impurity Green's function Gf are calculated from the diagrams shown in Figure 3.1 yielding G(w)1 = G(w)1 E (w) = jT Ek3 A (k)7, (3.4) Gfr(w)1 = G(wL)1 f(w) = D0ro EfT3 A/TI. (3.5) The Green's functions are now averaged over impurity positions in the usual way. The renormalized frequencies are calculated selfconsistently from the Dyson equations, i) = w + aC+/(_2 + 2 A2), (3.6) and = w L+ FK(/(2(k) (3.7) Here a = NiXTco/2r = n/NoTco is the scaled impurity concentration, and (...) is a Fermi surface average. The offdiagonal renormalizations are A(k) = A(k) + aA/(2 + E2f +A2), (3.8) and = F(k A(k)/ (2(k) 2) 1/2 (3.9) In superconductors with order parameters where the Fermi surface average in equation (3.9) vanishes, offdiagonal corrections vanish and A(k) = A(k). This class includes but is not limited to oddparity superconducting states. The energy scale F is the renormalized resonance width, F = (b)}rNoV2. The low temperature Kondo scale in the largeN slave boson theory is given by TK = /F2 + I2. Although the width of the actual spectral feature corresponding to the AbrikosovSuhl resonance is modified below Tc, in what follows we will normalize all quantities with respect to this TK, evaluated from equations (3.2) and (3.3) with A = 0 at T = 0. In the regime of principal interest, TK > Tc, corrections to this definition are small in any case. The early history of the problem of a Kondo impurity in an swave su perconductor has been reviewed by MiillerHartmann [ 45]. Abrikosov and Gor'kov first discussed the pairbreaking effects of magnetic impurities weakly coupled via exchange interactions to conduction electrons [1]. Shiba [ 46] ex tended this approach to treat strong scattering by classical spins, using the tmatrix approximation, and showed the existence of bound states in the gap. 28 At finite concentration of impurities the bound states were found to form an impurity band whose width and center scaled with impurity concentration and exchange strength. These early works neglected the dynamical screening of the localized spin by the conduction electron gas. These effects were incorporated by MiillerHartmann and Zittartz [ 47], adopting an equation of motion decou pling scheme previously used by Nagaoka [ 48] to calculate the dynamical spin correlations in the normal state. This approach correctly reproduced results in the AbrikosovGor'kov limit, TK/Tc + 0, and made the remarkable predic tion of a reentrant superconducting phase if TK < Tc, subsequently observed in experiments on LalxCexAl2 [ 49, 50, 51]. The failure of the decoupling scheme used to capture the correct crossover to Fermi liquid behavior in the normal state as T + 0 invalidated the MiillerHartmannZittartz approach in the low temperature regime TK > Tc, however. The physics of the Fermi liquid regime, TK/Tc + oo, was studied by Matsuura, Ichinose and Nagaoka [ 52] and by Sakurai [ 53] by extending the YamadaYosida theory [ 54] to the superconducting state. They obtained an exponential Tcsuppresssion with increasing impurity concentration n, Tc  Tco exp(pn/A), where A is the BCS dimensionless coupling constant, and p is a constant of order unity. This is commonly referred to as "pairweakening" as opposed to pairbreaking, since the effective superconducting coupling constant is reduced due to correlations on the impurity site. The exponential form breaks down for concentrations sufficiently close to a critical fic, for which Tc = 0. In this regime the reentrant behavior found by MiillerHartmann and Zittartz does not occur. A further characteristic signature of the Fermi liquid regime is the reduced specific heat jump, C* (AC/ACo)/(Tc/Tco))1T=To 29 which is always less than one [53, 55], in contrast to the high temperature regime. Not surprisingly, qualitatively similar results were obtained by other early workers for Kondo and Anderson impurities using a variety of other approaches [45]. More recent treatments include the use of a selfconsistent largeN [ 56], Monte Carlo [ 57] and NRG methods [ 58]. Schlottman [ 59] has treated the mixedvalence regime using BrillouinWigner perturbation theory. Out of these efforts has evolved a qualitatively consistent picture of the effect of Kondo impurities on the superconducting transition [ 60], but little understanding of the lowtemperature properties of Kondodoped superconductors because of the difficulty of the calculations involved. In the next section we show that the current theory reproduces the known effects of Kondo impurities on the critical temperature, specific heat jump and bound states spectrum of an s wave superconductor. 3.2 Results for Conventional Cuperconductors 3.2.1 Suppression of the Critical Temperature The simplest and most direct effect of impurity scattering on a supercon ductor is the suppression of the critical temperature. Scattering from impuri ties with internal quantummechanical degrees of freedom leads to deviations from the classic AbrikosovGor'kov prediction for the dependence of Tc on im purity concentration [1]. These effects depend sensitively on the lowenergy behavior of the selfenergy E(w), which enters the linearized gap equation, obtained from equation (2.20) near Tc, 1 \,/T.=O.1 1 20 \ 2 0 2 0.0 0.0 20.0 40.0 60.0 80.0 100.0 n/NoTco Figure 3.2 The critical temperature for an swave superconductor as a func tion of impurity concentration. The inset shows the slope of this dependence evaluated at Tc = Tco. ln(Tc/Tco) = 2rTc1 1 (3.10) n(1 +/B(wn)) n + 1/2 n>O n>0 where B(wn) = (Wn + F)2 + 2 . The slope of the Tcsuppression evaluated at n = 0 is therefore (1 dTc N FTcO (3 Tco di J, = 27r n> (n + 1/2)B(wn) In an insert to Figure 3.2 we have plotted a numerical evaluation of equation (3.11) for an swave superconductor. Note the curve is drawn with a broken line for small TK/Tco to reflect the fact that the slave boson mean field theory is expected to break down there. The maximum of the Tcsuppression is found to occur at TK " 3Tc, similar to the NCA result [56]. The early hightemperature theory of MiillerHartmann and Zittartz [47] locates this maximum at TK/Tc 12, whereas in the more recent Monte Carlo calculation 31 [57] for the symmetric Anderson model with finite U the maximum slope of the Tcsuppression is at TK Tc. Unfortunately a direct quantitative comparison with the latter work is not possible, as the simulation is not performed in the fully developed Kondo regime. The present theory predicts an exponential decrease of Tc at small concen trations, in agreement with other theories of the Fermi liquid regime [53,55], Tc Tcoexp 2 (TK/27rTco) Tc1 2n(K/c)) (3.12) _TK \ TK2 where I is the digamma function. In Refs. [ 61] and [53] the initial suppres sion of Tc is proportional to In2(TK/TcO). A full evaluation of equation (3.11) for arbitrary concentrations and various values of the ratio TK/Tc is shown in Figure 3.2 It is interesting to note that the theory reproduces the reentrant behavior characteristic of the high temperature regime [45] although we do not expect the theory to be accurate in this case (dashed line). As is evident in the Figure the current theory predicts no critical concen tration nc for which Tc = 0. This is a subtle point discussed by Sakurai [53], who suggests that a failure to include the dynamics of magnetic scattering by states close to the Fermi surface can lead to such an effect. Such processes are included in the finiteU perturbation theory through Coulomb vertex correc tions to the impurity averaged pair correlation function. In our U = oo theory, such vertex corrections arise first in leading order 1/N corrections due to the exchange of slave bosons, whose dynamics are neglected in this work. We ex pect that effects arising from the absence of these fluctuations in the theory will be quantitatively small for TcTK, except for impurity concentrations so large such that Tc < Tco. 3.2.2 Position of Bound States For conventional superconductors, A(k) = A, equations (3.7) and (3.9) are simply S= w+r 1/2' (3.13) (A2 _) 2 and A = r. (3.14) (2 2)1/2 (,2 W2) It is obvious from equation (3.14) that a finite gap in the conduction electron spectrum induces a gap in the impurity spectrum, Nf (w) = ImGf (w + iO+). When the AbrikosovSuhl resonance, which develops at temperatures below TK, falls in the superconducting gap, bound states appear in the conduction electron spectrum. These peaks are placed symmetrically relative to the cen ter of the gap and their spectral weight depends on TK/Tc and on impurity concentration. The density of states for N = 2 and N = 4, exhibiting pronounced peaks at the bound state positions, is shown in Figure 3.3 Figure 3.4 shows N(w) in the CoqblinSchrieffer limit for N = 2 and several values of F. In the hightemperature regime the bound states emerge from the edges of the gap and move towards the center of the gap as TK/Tc increases [ 62]. In the low temperature limit, the bound states disappear into the gap edges again [61]. Recently, Shiba et al. [58] studied the position of bound states in the gap in an swave superconductor using the numerical renormalization group (NRG). Their results cover both the high and lowtemperature limits reliably. The 0 2 0 w/A w/A Figure 3.3 Conduction electron and impurity spectral functions in the CoqblinSchrieffer limit in an swave superconductor for N = 2 and N = 4. The solid and dashed lines correspond to TK = Tc, and TK = 1OTc, respec tively. The concentration of impurities in all cases is i = 0.4. 2.0 11 2.0 4.0 1.0 z/ 0.0 i 0.0 1.0 2.0 )/AO Figure 3.4 Conduction electron density of states in the CoqblinSchrieffer limit for N = 2 and several values of hybridization F; TK = F in the CS limit. The impurity concentration is n = 0.04. 1.0 A  1.0 0o / 0.8 0.5 oo / 0 0 A /C o A / CD o /0.6 0 \, 0.0 / 0.: 0. 0 / \/ 0.4 / o/ 0.0.0 0.2 1.0   0.0 2.0 1.0 0.0 1.0 2.0 oloo91TK/Tc Figure 3.5 The position and spectral weight of the bound states in the gap for an swave superconductor with Kondo impurities. The solid (dashed) line is the location (spectral weight) of bound states. Only one of the bound states is indicated here, the other one is located at positive energies, symmetrically with respect to the gap center. Circles (triangles) refer to the position (spectral weight) obtained from an NRG calculation by Shiba et al. [52] Monte Carlo study by Jarrell et al. [ 63] confirms the overall dependence of wB on TK/Tc. Bound states correspond to the poles of the Tmatrix for conduction elec trons, T(w) = V2Gf(w) and are given by Sr2 2 1/,2 rll Iw = TK + w2 (2_wI2 2 (3.15) SA2 2 (2 2)1/2 We compare the solution of equation (3.15) with the NRG result in Figure 3.5 There is a good agreement for TK > Tc. For TK > Tc the position of bound states wB/A is a quadratic function of TK/Tc, IwB/AI 1 2A2/T2K, and agrees with NRG. The discrepancy between the NRG result and our cal culation for TK < Tc is not surprising since the slaveboson theory fails in the high temperature regime. The slaveboson meanfield amplitude vanishes 35 around T ~ TK and the theory is unable to describe the crossover to temper atures above TK. In Figure 3.5 we also show the spectral weight of the bound states in the gap which again agree with the NRG calculation for TK > Tc [58]. 3.2.3 Specific Heat Jump To calculate the specific heat jump we use equations (3.4), (3.6) and (3.10), (3.11), and expand the gap equation near Tc in terms of A/T, Tc ___1 1 1 i A 2 T '= (n+ 1/2)(1 +a/B(wn)) nn+1/2 2 2rT n>O n>O where bl (1 + 2aFwn/B2(wn)E(wn)) ( n>O (n + 1/2)3E3(w) and E(wn) = 1 + a/B(wn). After expanding the free energy to fourth order in A we obtain 8 2NOTc 4 a Tc n +r) 3 ) Cs(Tc) Cn(Tc) 872NT 1 47cTc(wn +F)) (3.18) bln> (B(wn + a)2 ) As can be seen in Figure 3.6 the dependence of AC/ACo on Tc/Tco, for TK C* 1 1/ln(TK/2TcO), see Figure 3.7 Ichinose [55] and Sakurai [ 64] have obtained a somewhat different result, C* 1 1/ln2 TK/Tc). 0.0 0.2 0.4 0.6 Tc/Too 0.8 1.0 Figure 3.6 Specific heat jump as a function of Tc/TcO. The result of the AbrikosovGor'kov theory is included for comparison. 1.1 1.0  b 0.9 0.8  0.7 0.0 1.0 2.0 3.0 logioTK/Tco 4.0 5.0 Figure 3.7 The derivative of the specific heat jump evaluated at zero im purity concentration as a function of TK/Tc for an swave superconductor. The dashdotted line shows the asymptotic behavior, 1 1/ln(TK/2Tco). The dashed line is the asymptotic form at TK/Tc > oo found in Refs. [49] and [58], C* 1 1/ln2(TK/Tc). 3.2.4 Penetration Depth If an electromagnetic wave of frequency w is normally incident on plane su perconducting surface, the current response may be written j(q, Q) = K(q, Q)A(q, 0), where A is the applied vector potential. The pen etration depth, A = K(0, 0) can be calculated following the deriva tion of Skalski et al. [ 65], (A = mc 2 1/2 1/2 (3.19) n ((1 + (Pn/A)2)1/2 + Or/AA)(1 + (n/a)2) where A = w2 + Tk + 2wnFn/ A2 + c. The zero temperature penetration depth is given in the strong coupling limit by A(0) a ( rF) N To F2 1 + 1 f+Z o (3.20) AL(0) 2T2 4A 16 A T (3.20) which for N = 2 is equal to 1.071 in the CoqblinSchrieffer limit. A numerical calculation of the penetration depth shows that the derivative 9(A(0)/AL(0))/an, evaluated at n = 0, is a weakly varying function of TK/TcO for 1 < TK/TcO < oo. The penetration depth is much less affected by impurity doping on the high temperature side, where d(A(0)/AL(0))/809n=O reaches values much smaller than those in the strong coupling regime. We now see that the superfluid density at T = 0 does not scale with the Tc suppression in the Fermi liquid regime. While the slope of the initial Tc suppression goes to 0 in the limit TK/Tc0  oo, the slope of the initial increase of the T = 0 penetration depth remains large. 38 3.3 Unconventional Superconductors The problem of an Anderson impurity in an unconventional superconductor is of considerable interest for several reasons. First, it serves as a testing ground for the ideas of Withoff and Fradkin, who argued that in the analogous Kondo problem with power law conduction electron density of states, N(w) = Clwlr, there existed a critical coupling below which impurities are effectively decoupled from the conduction band [3, 66]. We showed in the preceding chapter that in an unconventional superconductor a critical coupling indeed exists for the 1impurity problem, but that for a finite density of impurities there is always a finite density of conduction electron states at the Fermi level, provided N = 2 or if r < 1. This is consistent with phenomenological studies of potential scattering in unconventional superconductors [5,6]. This does not exclude the possibility of observing some vestige of this transition as the Kondo temperature becomes quite small, however. Secondly, Kondo impurities in unconventional superconductors have been proposed as analogues of defects in Kondo lattices [ 67], with the argument that a vacancy in a Kondo lattice may induce a relative phase shift close to 7r/2. We chose to do our calculations for the axial, A(k) = Ao(kx + iky), and polar, A(k) = Aokz, states for simplicity. These two states, nominally pwave pairing states over a spherical Fermi surface in 3D, are quite generally rep resentative of two classes of order parameters. States with order parameters vanishing at points on the Fermi surface, like the axial state, have low temper ature properties associated with a quasiparticle density of states N(w) w2, whereas the states with lines of nodes, like the polar state, correspond to N(w) , w. More complicated order parameters, such as those having a dwave 39 symmetry, will have similar properties at low temperatures, since the main factor determining the lowT properties is the order parameter topology, i.e. whether there are points or lines of zeros of the order parameter. In the case of points (lines) of nodes, the lowtemperature specific heat of pure supercon ductors is proportional to T2 (T). The deviation of the penetration depth AA from its zero temperature value A(0) along the main axes of symmetry of the order parameter is either ~ T2 or ~ T4 in the axial state, according to the direction of current flow [ 68], whereas for the polar state it is either ~ T or ~ T3. The presence of strong impurity scattering complicates the picture and these power laws do not hold in general. 3.3.1 Suppression of the Critical Temperature Due to the absence of the Anderson theorem for pwavelike or dwavelike superconductors, the influence of impurity scattering on the critical tempera ture is qualitatively different from that of swavelike superconductors [4, 69]. For the unconventional states of interest, there are no offdiagonal corrections to the superconducting Green's function, A = A. In this case Tc for either type of state is determined by In(Tc/Tco) = 27rTc E 1 1 (3.21) n(T/T) = 2 n( + a/B(wn)) + a/B(wn) n>n + 1/2" n>O n>O The initial Tcsuppression is then given by (1 dTN F( + ) .22) Tco di! nO 4r2 n>O (n + 1/2)2B(wn)' and approaches NF2/8T2, in the limit TK/TcO + oo, see Figure 3.8 Note that un/A + aF/AT2 as Tc + 0 and as a consequence the critical concen tration nc is finite. 1.0 10 0.3 T.Io/T. \\ S\ 2 0 2 0.5 \ \ ogoTK/Tco 100' 5 ITK/TCo=1 0.0 0.0 2.0 4.0 6.0 8.0 10.0 n/NoTco Figure 3.8 Critical temperature vs. impurity concentration for unconven tional states considered in this work. The initial slope at Tdo is shown in the inset. 3.3.2 Density of States The effect of Kondo impurities on the density of states is shown in Figures 3.9 and 3.10. As in the swave case, the resonant states move towards the edges of the gap as TK/Tc increases, except in the special case N = 2, where they are pinned at the gap center. In all cases the impurity bands are broadened relative to the swave case by the continuum in which they are embedded. In the limit TK + oo, the current theory coincides with the results given by phenomenological Tmatrix treatments [5,6] for N = 2. 3.3.3 Specific Heat To obtain the specific heat we differentiate the entropy with respect to temperature C, = TdS/dT. The entropy of conduction electron quasiparticles renormalized by impurities is given by N=2 N=4 0z5 2 0 2 0 2 w/AO cw/Ao Figure 3.9 Conduction electron and impurity spectral functions in the Kondo limit in a polar state for N = 2 and N = 4. The solid and dashed lines correspond to TK = Tc, and TK = 10Tc, respectively. The concentration of impurities is n = 0.4. S = kB dwN(w)[flnf + (1 f)ln(1 f)], (3.23) Jo where f = f(w) is the Fermi function. Note that the density of states N(w) is calculated selfconsistently, using equations (3.2), (3.3), and (2.20). The presence of resonances at low energies leads to pronounced features in the lowT specific heat, as shown in Figure 3.11 For TK > Tc, N = 2, these features are identical to those predicted by the phenomenological theory of Refs. [5,6, 70]. For smaller TK/Tc, the resonance sharpens, as is evident from, e.g., Figure 3.10 In this case resonances in the lowtemperature specific heat may be quite dramatic. Figure 3.12 shows the temperature dependence of C/T for N = 2 in the axial state for the case when the resonance is very narrow, 1.5 1.0  0.5 0.0 1.5 1.0 0.5 1.5 0.0 1.5 0.0  0.0 0.5 1.0 1.5 W/AO Figure 3.10 Density of states for conduction electrons in axial (at top) and polar (at bottom) state for N = 2 and TK/Tco = 0.3 (solid line), 1 (dashed line), and 20 (dashdotted line). The inset shows the impurity spectrum for TK/TcO = 20 in both states. The scaled impurity concentration is n = 0.2. F < Tc and just above the Fermi surface, Ef < Tc. Such pronounced features are possible only when the bare impurity level is close to the Fermi surface, and the hybridization is weak. They will be sharper for superconducting states with larger exponent r in the unperturbed density of states N(w) C Clwlr at low w. These anomalies may be observed experimentally at sufficiently low temperatures. In this context, it is interesting to note that a sharp peak in C/T has been observed in the heavy fermion superconductor UPt3 at 18 mK [ 71]. This peak is present at roughly the same position also in the normal state at magnetic fields B > Bc2, as might be expected in a situation where the Kondo temperature is significantly smaller than the critical temperature. If such an interpretation of the measurement of Schuberth et al. in these terms is correct, we would expect the size of the peak to scale with other measures 0.20 polar TK=5Tco .n=0.01nc " 0.005 n 0 0.005 0.10 5 / 2 0.0025 0.001 0 0 0.5 1 0.00 0.00 0.05 0.10 T/To Figure 3.11 The lowT part of C/T in the polar state for N = 2 in the Kondo limit. The inset shows C/T for n = 0.01nc over the full temperature range. of the defect concentration, such as Tc the size of the specific heat jump, in different samples. We note, however, that fields of order 1 Tesla would normally destroy a manybody resonance of the usual magnetic type. 3.3.4 Specific Heat Jump The specific heat jump at Tc can be found by the method already mentioned in the preceding section, Cs(Tc) Cn (Tc)= Nr Tc ( G(Wn) bl ((n + 1/2)E(wn) + a/B(wn))2 ' (3.24) 1.0 axial 2 n=0. 15nc zz o 0 S0.5 o 1 II 0.0 0.00 0.05 0.10 0.15 0.20 T/Tc Figure 3.12 The lowT part of C/T in the axial state. Note the sharp resonance in the density of states at low energies (the inset). where bl= b + cH(wn) n>o [(n + 1/2)E(wn) + a/B(wn) with b = 4/5 (3/5), and c = 2/3 (1/3), for the axial (polar) state, and H(wn) = ao [2(wn + r)2/B(w) 1] /B(n) G(wn) = 2awn(wn + r)(n + 1/2 + r/2rTc)/B2(wn) + ar/27rTcB(n) . The derivative of the specific heat jump, C*, is shown in Figure 3.13 . 3.3.5 Penetration Depth We now discuss the lowtemperature response functions of the supercon ductor in the presence of Kondo impurities, which have not to our knowledge been previously calculated in the strongly interacting Fermi liquid regime of interest. The effect of Kondo impurities on the electromagnetic response has 5.0 4.0 b3.0 \ Polar axial 1.0 1.0 1 i i 2.0 1.0 0.0 1.0 2.0 log oTK/Tc Figure 3.13 The derivative of the specific heat jump evaluated at zero impurity concentration as a function of TK/Tc for the axial and polar super conducting states. been calculated in the phenomenological Tmatrix approach mentioned above, and used to analyze experiments on heavy fermion and highTc supercon ductors with impurities, but no microscopic theory is available. The London penetration depth is obtained from A = [ K(0,0) where K(0, 0) is the electromagnetic response kernel. The kernel is given by the linear response formula 3e2T 00 K' (0, Om) =2 dE Tr(kikjG(k, un)G(k,wn am))k. (3.26) 2mc _o n In the static limit, ,m > 0 the kernel becomes 6me2T dQ A2(k) K (0, 0) = 6r24 ki f I (A2) + 2)3/2 (3.27) Smc J 4x (A12(k+ )3/2 1.0 n/nc=0 polar 0.8 TK=5Tco _. ."TK=5Too 0.05 \ 0 0.6  N I.Q.25 e 0.4 0.50 0.2  _0.75 j 0.0 0.0 0.2 0.4 0.6 0.8 1.0 T/T= Figure 3.14 The inverse square of the penetration depth vs. temperature at several concentrations of impurities in the polar state. where the integral represents the angular average. Here we specialize again to the N = 2 case. The temperature dependence of (All)2 in the polar state, which corresponds to the component of the su perfluid density within the plane containing the line of zeroes of the order parameter, is shown in Figure 3.14 The lowT behavior changes from linear to quadratic upon doping. A similar result was obtained earlier [68, 72] within the phenomenological theories. The results for Tk Tc are qualitatively very similar to those shown. Finally, A2 at T = 0 along the main axes of symmetry for the axial and the polar state as a function of impurity concentration is pre sented in Figure 3.15 We note that the largest component of the penetration depth scales in the case of line nodes as A(n)/AL(O) 1 n1/2 at low con centrations [ 73], and A2(n) ~ log(nc/n) close to the critical concentration. 1.0 0.86 0.8 \ TK=5Tco 0.2 N \ N\ 0.0 0.2 0.0 0.2 0.4 0.6 0.8 1.0 n/nc Figure 3.15 The inverse square of the T = 0 penetration depth in the two principal directions as a function of impurity concentration for the polar (full lines) and the axial state (dashed lines). Similar suppression of the superfluid density was calculated for a dwave order parameter within a phenomenological model in Ref. [ 74] It is clear that both the concentration and temperature dependence of the penetration depth components may be important tests of gap anisotropy. CHAPTER 4 ANISOTROPIC CONVENTIONAL SUPERCONDUCTORS VS. UNCONVENTIONAL SUPERCONDUCTORS 4.1 Experimental Motivation Recent high resolution angleresolved photoemission (ARPES) experiments on Bi2212 have been interpreted in terms of a highly anisotropic order param eter with a large gap in the (ir, 0) direction and two lines of nodes near a small gap in the (7r, r) direction [ 75, 76]. + + a) b) c) Figure 4.1 Order parameters plotted over tetragonal Fermi surface: a) dx2_y2 state. b) extended swave state; c) swave state with gap minima at Fermi surface. While alternative explanations consistent with an order parameter Ak which reduces the symmetry of the Fermi surface (e.g. dwave, see Figure 4.1 a) have been put forward [ 77], it is interesting to consider the consequences of the simpler suggestion that Ak has the full symmetry of the crystal, but changes sign (see Figure 4.1 b). Fehrenbacher and Norman [ 78] have consid ered the effects of potential scattering on such a state, following earlier work 49 on anisotropics states with nodes [ 79], and shown that small concentrations of impurities can lead to "gapless" behavior in the density of states (i.e., with residual density of states N(0) > 0), followed by the opening of an actual induced gap in the quasiparticle spectrum as the concentration is increased further. The prediction of a range of impurity concentrations over which "gapless" behavior is predicted is important because microwave and NMR experiments, particularly on Zn and Nidoped YBCO crystals [ 80, 81, 82, 83] have shown evidence of lowtemperature thermodynamic properties reflecting the existence of a residual density of states. They have been interpreted most often in terms of a dwave pairing scenario [73, 84, 85, 86, 87], where the wellknown effect of dirt is to induce a residual density of states for infinitesimal concentrations [69]. The intriguing possibility raised by the ARPES experiment [75,76] is that many of the observations of "gapless" behavior can be equally well explained by an anisotropic state with extendeds symmetry. There are some difficulties associated with this interpretation even if it can successfully account for the microwave and NMR data. The most important of these is the set of SQUID experiments [12,13,14,15] indicating that the order parameter Ak changes sign under a 7r/2 rotation. While the extendeds order parameter discussed here indeed changes sign over the Fermi surface, the symmetry of the expected tunneling currents is not, within the simplest theory, consistent with the ob servations reported [ 88, 89]. Furthermore, the existence of a small gap in the (Tr, 7r) direction in BSCCO has been questioned by at least one other photoe mission group claiming similar angular and energy resolution [ 90]. We do not address any of these discrepancies in this work. 30 25 Ao=35meV 2 770o=0.25 20 E15  10  5\A 10 0 10 20 30 40 50 60 Figure 4.2 ARPES determination of BSCCO energy gap as function of angle around Fermi surface. Data from Ref. 1. Solid line: Ak = A0(I cos 2 0 0), with r7o = 0.25. 4.2 The Model In the absence of definitive answers to the above questions, we assume the plausibility of the ArgonneU. Illinois argument and investigate the conse quences of assuming the order parameter symmetry identified in Refs. [75,76] within a generalized BCS model. While these authors pointed out that their data were consistent with an order parameter over a realistic BSCCO Fermi surface with symmetry Ak  cos kx cos ky, we work here with an even simpler model with cylindrical Fermi surface, in which case a rather good fit to the data can be obtained by assuming Ak Ao(I cos2(0I 770), with AO=35meV and 770 = 0.25, as shown in Figure 4.2 With this set of parameters, a BCS weakcoupling approach yields the gap magnitude ratio Ao/Tc = 2.92, see Figure 4.3 . 5.0 4.0 0 <( 3.0 2.0 0.0 0.2 0.4 0.6 77o Figure 4.3 The ratio AO/Tco as a function of 770. Note that 7r0 controls the angular range over which the swave order param eter is negative. The case o70 = 0 corresponds to a somewhat pathological state of type c), while 70 = 2/7r corresponds to equal positive and negative weights, (Ak) = 0, where (...) is a Fermi surface average. When I0 = 0, the energy of quasiparticle excitations is identical to that of the dwave state dx2_y2. Other choices of basis functions may result in a somewhat better overall fit and avoid the cusp at 0 = r/4, but we do not expect these details to affect our qualitative conclusions. Measurements of penetration depth, angleresolved photoemission, thermal conductivity, and nuclear magnetic relaxation experiments provide evidence for gap nodes, but do not determine if the order parameter changes sign. This is because experiments of this kind are normally assumed to measure properties sensitive to the order parameter Ak only through the Bogoliubov quasiparticle spectrum, Ek = I A 2. Since Ek does not depend on the sign of the order parameter, all three states shown in Figure 4.1 have similar (linear) 2.0 1.5  1.0  z 0.5  0.0 0.0 0.5 1.0 1.5 o/Ao Figure 4.4 Density of states for the superconducting state with an extended s order parameter, Ak = A0(I cos 21 r?0), with r70 = 0.25. lowenergy density of singleparticle states N(w), and all properties deriving directly therefrom. Figure 4.4 shows density of states for the state of the type 4.1 c. Note the lowenergy gap feature corresponding to the smaller gap maximum. Systematic impuritydoping studies of transport properties can in principle help to identify the order parameter symmetry, however. Unconventional states like the dZ2_y2 state (Figure 4.1 a) possess nodes on the Fermi surface for symmetry reasons, whereas their existence is "accidental" in the swave case (Figure 4.1 bc). In the dwave case, suppression of gap variation in kspace can only result in an overall suppression of the order parameter magnitude, whereas in swave cases b) and c), it results eventually in the elimination of the nodes. In manybody language, the crucial point is that the offdiagonal impurity self energy E1 is nonzero in swave states 4.1 a and 4.1 b, but vanishes in the d wave state 4.1 c. In the isotropic swave state, a large E1 cancels the diagonal 53 selfenergy E0 [1], leading to Anderson's theorem [2], whereas in extendeds states this cancellation is only partial. A large El furthermore prevents the formation of a scattering resonance at the Fermi surface, leading to clean limit lowfrequency transport coefficients which in extendeds states for T < Tc vary relatively weakly with temperature. By contrast, in the dwave (or other unconventional state where El = 0), resonant scattering leads to transport properties which vary strongly in temperature for T < Tc [ 91,5,6]. In this sense the characteristics attributed to extendeds states here are also valid qualitatively for any mechanism which generates an offdiagonal selfenergy in a dwave state as well, e.g. tetragonal symmetry breaking or locally induced swave components due to impurity potentials. The disorderaveraged matrix propagator describing any of the states a)c) above is written CO + 13 + AkZl g(k, wn) = 2 (4.1) where the ri are the Pauli matrices and Ak is a renormalized unitary order parameter in particlehole and spin space. The renormalized quantities are given by w = w E0(w), k = 4k + E3(w), and Ak = Ak + CE(w), where the selfenergy due to swave impurity scattering has been expanded E = Ei7ri. The relevant selfenergies are given in a selfconsistent tmatrix approximation [5,6] by rGo FG1 E0 = E 12 (4.2) c2 + G12 G2' ,2 + Gi2_ G02' where F nin/(7rNo) is a scattering rate depending only on the concentration of defects ni, the electron density n, and the density of states at the Fermi level, NO, and we have defined Ga = (i/27rNo)EkTr[rag]. The strength of an individual scattering event is characterized by the cotangent of the scattering 54 phase shift, c. The Born limit corresponds to c > 1, so that Eo FNGo, 1 1 while the unitarity limit corresponds to c = 0. We have defined the normal state impurity scattering rate as TN F/(1 + c2); note that in the highTc cuprates the total scattering rate at Tc includes inelastic scattering and is expected to be much larger for clean samples. 4.3 Density of States A crucial feature of the physics of dwave superconductors is that an infinitesimal concentration of impurities produces a finite density of states N(0) > 0 at the Fermi level [69], leading to temperature dependence character istic of the normal state in all transport quantities. Solving the selfconsistency equations at w = 0 for the extendeds wave state under consideration leads im mediately to the conclusion that such "gapless" behavior is possible only for a range of scattering rates F < Fc, however [78]. This is illustrated in Figure 4.5 A low frequency expansion in the gapless regime yields Tc/Tco r70(1 + c2). Can the same experiments which seem to fit the "dirty dwave" scenario also be explained by extendeds states? The difficulty is how to fix the actual impurity scattering rate, F, given the known concentration. One way is to attribute the additional extrapolated T + 0 resistivity to impurity scattering, such that the elastic and inelastic rates add incoherently. If one attempts such an analysis for Zndoped YBCO crystals using results from, e.g., Chien et al. [ 92], one finds that F/Tc 0.3 0.5 per 1% Zn. Since Zn doping studies of YBCO indicate gapless behavior up to several per cent Zn, it would appear that an extendeds picture is plausible for YBCO only if one assumes 770 close to the critical value 2/7r for which (Ak) = 0. We are not aware of similar 1.0 .. o=2/7T 0.8  c=0 0.6 / 5 o 0.6.5 z 0.4  0.0.4 0.2 _0.25 0.0 0.0 0.2 0.4 0.6 0.8 1.0 F/Tco Figure 4.5 Residual density of states N(O)/No in extendeds state vs. nor malized scattering rate F/TcO for c = 0. Dashed line: 770 = 0.25 obtained from fit to data of Ding et al. doping studies in single crystal BSCCO, but assuming for the moment that Zn scatters equally strongly in this material, we see from the dashed line in Figure 4.5 that a welldeveloped gap Q2G in the excitation spectrum should be induced in BSCCO by a few per cent Zn doping. Note that the results shown in Figure 4.5 are not sensitive to changes in the scattering phase shift 60. In the special case, 770 = 0, a simple estimate shows that for small scattering rates, QG F (rN in Born limit). In the dirty limit F * oo, the swave superconductor becomes isotropic with a BCS density of states N(w) = Re w/ /w2 Aavg2, as shown in Figure 4.6 In contrast to a dwave superconductor, the selfenergies obtained in the Born approximation and in the resonant scattering limit are almost equivalent in the highly anisotropic swave system, if 710 < Aavg/0o. This insensitivity to larger phase shifts arises because of offdiagonal selfenergy corrections which 2.0 Bon Born \ s 1.5  0 5 Z 0.5 N/Ao=0.1 0.5  0.0  Born d 1.5 0. 0.5  0.0  0.0 0.5 1.0 1.5 W/Ao Figure 4.6 Normalized density of states N(w)/No for s and dwave order parameters vs. reduced frequency w/Zo, shown for various potential scattering rates FN/AO in the Born approximation; r70 = 0. prevent the occurence of poles in the tmatrix, c2 Go2 + G12 ~ O(1) for all c scattering are shown in Figure 4.7 . 57 2.0 =2.0 s 1.5 o 5 z 1.0  0.5 FN/Ao=0.1 0.5 z 0.5 )0.5 0.0  c=O d 1.5 1.0 0.5 0.15 '0.025 0.0 1 1 0.0 0.5 1.0 1.5 w/Ao Figure 4.7 Normalized density of states N(w)/No for s and dwave order parameters vs. reduced frequency w/Ag, shown for various potential scattering rates F/A0 in the unitarity limit, c = 0; 7I0 = 0. 4.4 Critical Temperature We first solve the Dyson equation for the renormalized propagator (4.1) together with the gap equation. The order parameter Ak is related as usual to the offdiagonal propagator as A(k) = T n k' Vkk'Tr(71/2)g(k', wn), where VkkI Vd,sDd,s(k)d,(s k) is the phenomenological pair interaction assumed. 58 In the dwave case, Ak d= Aodd(k), with 4d = cos 2, yielding a simple equation for the gap maximum Agd, 1 WD /Dw \ Ad1 = dw tanh Re d (4.3) 0 2 V2 Ak2 where Ad = VdNo, and (...) represents an angular average over the cylindrical Fermi surface. In the swave case, on the other hand, it is convenient to put Ak = Aavg + Alk, where Aavg = (Ak) is the gap average over the Fermi surface. When impurities are added to the system, it is easy to check that Ak = Aavg + Ak, determined by 1 WD W sAk/AO As= / dw tanh Re (4.4) 0J 2 2 244) Note this is effectively an equation for Aavg(w) since the angular variation Alk is given. When 770 = 0 the initial slope of Tc suppression, dTc/dFN = X7r/4, where X = [(s2) (s)2]/( s)2 is 1 8/7r2 for the swave and 1 for the dwave state considered. In the dwave case the critical temperature continues to drop rapidly to zero at a critical concentration of n? = r2NOTcO/2e7, whereas the decrease becomes more gradual as the gap is smeared out in the swave case, finally varying [ 93, 94, 95] as Tc ~ Tcd[1 xln(1.154rN/7rTco)]. The initial suppression of Tc depends also on the magnitude of 7r0 and the magnitude of the spin exchange scattering rate Fs. For small rmo and Fs F the qualitative form of Tcsuppression at small and moderate concentrations will be very similar to the one shown above. For large concentrations, such that Tc/TcO < 1, the Tcsuppression by the magnetic component in the scat tering will be the dominant effect, if Fs > 0. However, one may not treat the 59 impurities as independent from one another when the impurity concentration is large. 4.5 London Penetration Depth The opening of the energy gap with increasing impurity concentration is an indelible signature of swave superconductivity. It will obviously give rise to activated behavior for T < QG in a wide range of thermodynamic properties, of which we have chosen to discuss only one for purposes of illustration, the temperaturedependent magnetic penetration depth. For the model states and Fermi surface under consideration, this may be expressed as Ao = dw tanhW Re (4.5) A(T) 2 27r (L2 3/2 where AO is the pure London result at T = 0. The penetration depth in a dwave superconductor (Figure 4.8 b) is known to vary as A(T) f XA + c2T2 at the lowest temperatures [68, 96, 97,73], over a temperature range which widens with increasing impurity concentration. The coefficient c2 decreases, as F1 in the Born limit and r1/2 in the resonant scattering case. In the anisotropic swave case with T0Q > 0 there is a gradual change in the temperature dependence of the penetration depth at low tem peratures. The deviation from the linear behavior of A(T) grows weakly with increasing impurity concentration, see Figure 4.9 As mentioned earlier the energy gap opens up at Fc/TcO "z 0(1 + c2). For rq0 < 1 the corresponding activated behavior in the anisotropic swave case is easy to distinguish from the dwave case when plotted against (T/Tc)2 as also shown in Figure 4.8 The important experimentally relevant signature is N < r< % 0.1 (T/Tc)2 Figure 4.8 Temperature dependence of normalized magnetic penetration depth (Ao/A(T))2 for s and dwave order parameters (r0Q = 0)vs. reduced temperature (T/Tc)2, shown for various potential scattering rates F/Tco in the unitarity limit, c = 0. of course not simply the exponential behavior, but the increase in the activation gap with impurity concentration. 4.6 Effect of Spin Scattering A simple defect like a vacancy or Zn ion in the CuO2 plane may not behave simply as a potential scatterer, as assumed above. In the presence of large local 1 0.5 0.8  1 0.6 2 0.4 N 0.8 1 0.2 0.6 0.4 e< 0 0.2 1.0 ?7o=0.25 0.9 c= S0.8 \ F/Tc0=0 0.8 e< 0.7 0.6 0.5 0.5 0.00 0.05 0.10 0.15 (T/Tc)2 Figure 4.9 Temperature dependence of normalized magnetic penetration depth (Ao/A(T))2 for an order parameter of an extendeds type, with 70 = 0.25, in the unitarity limit, c = 0. Coulomb interactions, a magnetic moment may form around the defect site, giving rise to spinflip scattering of conduction electrons [ 98]. This poses the most serious obstacle for the direct application of the principle distinguishing dwave from anisotropic swave systems outlined above, since magnetic scat tering will lead to gapless superconductivity as in the usual AbrikosovGor'kov theory. Furthermore, even if a gap remains, strong spinflip scattering may lead to bound states within it [66, 99, 100,46] which may give rise under the proper circumstances to a residual density of states N(w 0) as in the dwave case. Here we investigate the competition between the opening of the energy gap in the swave state due to potential scattering and gapless behavior due to magnetic scattering. To this end we add a term JS a to the Hamiltonian, where S is a classical spin representing the impurity and a is the conduction electron spin density, and study the system in an average tmatrix approxima tion analogous to the one applied to the pure potential scattering case. The 0.20 0.6 0.3 / 0.15 3 / 0 < I 0 5 10 ". 0.10 FN/FN=0.1 0.05 0.2 0.5 0.00 0.0 1.0 2.0 3.0 rN/Tco Figure 4.10 Induced energy gap normalized to clean gap maximum, IG/AO, vs. potential scattering rate FN/TcO for different ratios, rF/FN, of magnetic to potential scattering rates, ro0 = 0. selfenergies found in the presence of both types of scattering reduce in the isotropic swave case to those given by several authors [46,99, 101]. We find that until the dimensionless exchange JNo becomes of 0(1), the results for the swave system are very similar to those obtained in the simpler Born approx imation, as discussed above, provided r0r is not large. The selfenergies in the Born approximation are Eg = (FN + FT)Go and El = (rN + Fr)G1 [1], where rs niJ2S(S+ 1)7rNo. For 770 = 0, the induced gap, 2G, in the swave system may then be shown to vary as QG FN Fr] > 0, but the effects of selfconsistency rapidly become important as the concentration is increased. In Figure 4.10 we plot QG as a function of the impurity concentration through the parameter FN for various assumptions about the scattering charac ter of the impurity ion, where the quantity FN/FN specifies the relative amount of magnetic scattering. The destruction of the induced gap takes place because 63 the system becomes insensitive to large amounts of potential scattering, but magnetic impurities continue to break pairs even at large concentrations. The gap is nevertheless found to persist into the very dirty limit even for systems where the magnetic scattering is nearly as strong as the potential scattering. For weak spin scattering, the bound state in the tmatrix approximation is found to lie at w > ~2G, just below the average order parameter /avg deep in the continuum, and thus plays no role. Stronger spin scattering does not change this qualitative behavior at low concentrations until JNo0 1 when the bound state lies at the Fermi level in the classical spin approximation [46]. In this case the Kondo effect, neglected here, also becomes important. It is known from other analyses [ 102] that the bound state lies near the Fermi level, and will therefore give rise to a residual density of states N(w + 0), only when TK Tc. For any other ratio of TK/Tc, the bound state will lie at an energy corresponding to an appreciable fraction of the average gap in the system, and hence be irrelevant for our purposes. Clearly a quantitative estimate of the relative size of F and TF' is required to decide whether spin scattering plays a role in real highTc materials with simple defects. Walstedt and coworkers estimated JNo 0.015 for a Zn ion in YBCO, implying that Zn is a nearly pure potential scatterer in this system [18]. On the other hand, Mahajan et al. [19] estimate JNO 0.45. For a 1% Zn concentration, a magnetic moment of 0.36 pB for Zn in fully oxygenated YBCO [19] and a density of states of 1.5/eV [19], we find F _ 1 x 104 eV. From the residual resistivities of Zndoped YBCO crystals [92], we estimate that a 1% Zn sample corresponds to a total impurity scattering rate of rmp 1 x 102eV, assuming that the inelastic and elastic contributions to the scattering rate 64 add incoherently. This suggests that potential scattering must dominate the total elastic rate, Fr, < F. On the other hand, the large value of JNo 0.45 deduced for a Zn ion [19] would mean that the Kondo effect may be important, and that we cannot completely rule out the possibility that a bound state sits very close to the Fermi level. Other experimental results do not support such high values for the magnetic exchange interaction of Zn in YBCO. NMR studies of the Osaka group [80,82] led to the conclusion that F, < F, and confirm our estimates of Fs. 4.7 Microwave Conductivity Electrical and thermal conductivities, as well as sound attenuation, will be considerably different in extended s and dwave states, as suggested by the following argument. Any DC transport coefficient L(T) in a system charac terized by welldefined singleparticle excitations will vary with temperature roughly as L(T) ~ N(w ~ T)r(w T), where N(w) is the density of states and r(w) the relaxation time. In the clean dwave case, resonant scattering gives 7(w) = 2EO"(w) ~ N(w)1 up to logarithmic corrections, yielding L(T) ~ T2. In the Born limit, c > 1, similar arguments yield L ~ const. for dwave transport coefficients. Impuritylimited transport in the extended swave state will be qualita tively similar to the dwave case if the scattering is weak, c > 1. In contrast to the dwave case, however, for c + 0 (and 70 not occur, since the denominator of the tmatrix, c2 Go + G2 (1 0)2 [79]. A simple lowT estimate accounting for nodal quasiparticle contributions gives L ~ Lo(1 21?0) as T + 0. The exact behavior in this range will be F/TC=O. 1 0 (70 b b 0.2 0.5 o ____(Tc) 0.0 0.2 0.4 0.6 0.8 1.0 T/Tc Figure 4.11 Conductivity of extendeds state, AO/Tc=2.92, F/Tc=0.1, c = 0, o0=0.1,0.25, and 0.5 (solid lines). Also shown, T/Tc = 0.01, r0 = 0.25 (dashed line). influenced by selfconsistency effects and the leading frequency dependence of the tmatrix. The resultant temperature dependence will then be intermediate between the strong and weak scattering limits of dwave transport coefficients. In Refs. [ 103, 104], expressions for the complex conductivity of an anisotropic swave superconductor were derived. We do not reproduce these rather lengthy expressions here, but merely comment that a fully selfconsistent numerical evaluation confirms the qualitative picture of lowfrequency transport described above. Some typical results are shown in Figure 4.11 for a model in which in elastic scattering has been neglected entirely. The limiting conductivity as T Tc is therefore the impurity Drude result, oa = ne2/(2mFN), which for FN < Tc is much larger than the actual conductivity in the cuprates at the transition, a(Tc), indicated roughly in the figure. At low temperatures T < Tc, 66 inelastic scattering may be neglected and the results displayed are valid. The most important qualitative feature of the results is that for g0 < 2/7r, the effective limiting value of the conductivity in the extendeds state is nonzero and generically much larger than a(Tc) in clean highTc systems, such that a = a(T + 0)/a(Tc) > 1. This residual conductivity diminishes as 770  2/7r, when the result should be qualitatively similar to the dwave conductivity with resonant scattering due to the vanishing of the offdiagonal selfenergy. Note, however, that for generic values of ro0, e.g. that apparently appropriate for BSCCO, the residual conductivity scales inversely with the impurity scattering rate F, in contrast to the resonant dwave case where the residual conductivity is independent of F to leading order. As shown in Figure 4.11 the temperature dependence of the conductivity for most of the low temperature range may mimic a linear behavior. This is intriguing, given the linearT conductivity observed in microwave experiments on clean YBCO crystals [81], but the large residual conductivity predicted (unless r0r is close to 2/7i) would seem to be inconsistent with recent studies indicating that a(T + 0) is very small in twinfree samples. We are not aware of similar experiments on highquality BSCCO single crystals. One final point of interest is that the form of a(T) is only weakly dependent on disorder, as shown in the figure; of course the overall conductivity scale a depends inversely on impurity concentration. The results in the Born approximation are qualitatively similar for both s and dwave states, as can be seen from comparison of Figure 4.12 and Figure 2 of Ref. [104]. Note, however, that the clean limit regime in the dwave state is a much more narrow range of scattering rates than in the swave states. 1.0 0 b b 0.5 0.0 0.0 0.2 0.4 0.6 0.8 T/Tc Figure 4.12 Conductivity of extended sstate for several values of external frequency 2 in the Born approximation, AO/Tc = 2.92, F/TcO = 0.1, 7r0 = 0.25. 0.0 0.2 0.4 0.6 T/Tc Figure 4.13 Comparison of conductivity (dashed line), 70 = 0.25, Ao/Tc = 2.92, F/Tc A/Tc = 2.14, T/Tc = 0.1, c = 10, 1, and 0. '0" o(Tc) 0.8 1.0 of clean extended swave state = 0.1, c = 0 with dZ2_ 2 state, 68 To illustrate the comparison of these results with those expected for a dwave superconductor, we plot in Figure 4.13 the conductivity in a d 2_y2 state in the resonant (c = 0) and Born (c = 10) limits, together with an example of intermediate strength scattering (c = 1) chosen to give the same effective a(T + 0) as the extendeds conductivity for qr0 = 0.25. Even in the last case the qualitative differences between the s and dwave results are manifest. The dependence of the swave result on the scattering phase shift and impurity concentration are found to be quite weak, except in the case c > 1, for which the result is qualitatively similar to the dwave Born result, since the denominator of the tmatrix becomes irrelevant. Given the uncertainty regarding the origin of the residual conductivity in the highTc materials it is perhaps useful to give estimates for other transport coefficients. For example, since the coherence factors are essentially the same for the electronic thermal conductivity nel/T as for o(T), it is straightforward to see that the large residual conductivity in the extended swave state implies a large linearT term in the thermal conductivity of order (1 2)70) YV2T/FN, where vF is the Fermi velocity and 7 is the normal state linear specific heat coefficient. Such a term has not been observed below 2K in single crystal BSCCO [ 105] but there are data at somewhat higher temperatures which are consistent with &Q = Kel(T O)Tc/fel(Tc)T > 1 [ 106]. The linear term in K at low T in YBCO appears to be quite small [ 107]. Very recently the authors of Ref. [ 108] have pointed out that the "bump" in BSCCO gap function near 0 = 7r/4 may be a superlattice effect. In this case available experimental data on BSCCO seem unlikely to be compatible with an anisotropic swave state of the type considered here for any qo0. CHAPTER 5 TWOCHANNEL KONDO IMPURITIES IN SUPERCONDUCTORS In recent years models of fermion systems that cannot be described by a Landau Fermi liquid model received considerable attention. One of those is the twochannel Kondo impurity. In this model conduction electrons described by the flavor a and spin a interact with impurity spin S = 1/2. It was pointed out by and Nozieres and Blandin [ 109] that this model may follow from a realistic description of magnetic impurities in a metal when the orbital structure of the impurity is taken into account. In the twochannel model the impurity spin is overscreened by two exactly degenerate channels of conduction electrons. The new effective spin is 1/2. This spin couples antiferromagnetically to two other electrons from the two channels, and this process continues as T is lowered. The effective coupling of the conduction band grows in this process and saturates at large, finite values at an intermediate fixed point. The NRG approach shows that the size of the electron cloud around the impurity diverges as 1/T when T + 0. The excitation spectrum of this system is not a Fermi liquid. The low temperature magnetic susceptibility and the specific heat coefficient, Yimp Cimp/T, are logarithmically divergent. Other properties also exhibit nonFermi liquid behavior below the crossover temperature TK. The experimental realization of the twochannel Kondo effect in uranium compounds was proposed but remains controversial [ 110, 111, 112]. It was argued [110] that in compounds such as UBel3 and YlxUxPd3 with x = 0.2, 69 70 the U ions at crystal sites with cubic symmetry have stable 5f2 configuration and J = 4 total angular momentum. The crystalfield split J = 4 multiple would have a nonmagnetic F3 doublet as its ground state. In a simplified model this doublet is coupled to an excited doublet in an 5f1 configuration, with J = 5/2 and F7 symmetry, via hybridization with conduction electron J = 5/2, F8 partial waves. A canonical transformation [31,110] yields an effective interaction between the F3 doublet and two channels of conduction states of r8 symmetry. Here we would like to offer another test of the multichannel Kondo effect. The breaking of timereversal symmetry involved in scattering by magnetic impurities in superconductors has important consequences. The qualitative type of behavior is wellknown in the single channel problem to depend on the ratio TK/TcO, where TK is the Kondo temperature and Tco is the transition temperature of a pure superconductor. The divergence of magnetic correlation length and the existence of a residual magnetic moment as T + 0 suggest that in the multichannel case one should expect a very different type of interplay between exchange and pairing interactions. One may use a sensitivity of superconducting correlations to multichannel exchange interaction as an additional criterion to characterize multichannel Kondo behavior and distinguish it from the single channel case. In this paper we study a simplified SU(N)xSU(k) version of the full multichannel problem in the NCA approximation [ 113]. Here, N is the orbital degeneracy of the impurity and k is the number of conduction electron channels coupled to the impurity. Although this approach ignores details of anisotropic exchange it remains in the same universality class [113]. In particular the exponents for Figure 5.1 The leading order diagram for the anomalous impurity propa gator. The dashed lines are the fermion propagators and the wavy and solid lines are boson and conduction electrons respectively. the temperature dependence of the susceptibilities agree with those obtained from the conformal field theory [110, 114]. Our model includes a BCS pairing of electrons in the conduction band. In general the pairing may be either of channel singlet or triplet type. We assume the latter possibility. The quantities we study in this work are not significantly different for the channel singlet state. In the limit of large onsite Coulomb repulsion U and for temperatures T <( U, the model has a simplified form, H = Ekc cLka +E E f f + V E [C qfaba + h.c.] k,a,a C k,a,cT + 5 [A^kc L ka + h.c. + A (z f + E 1 (51) knc,, k,a,aa \ va a The indices a and a refer to channel and spin, respectively. The boson be trans forms according to the conjugate representation of SU(k). In the limit A * oo the unphysical states with the impurity occupation nf > 1 are projected out. The position of the bare impurity level is assumed to be Ef = 0.67D, and r = 7rN(0)V2 = 0.15D, where D is half of the band width. For this parameter set the Kondo temperature is TK = 4.5 x 105D and the impurity occupation number is nff 0.9. 72 The pairing correlations in the conduction band lead to a nonzero anoma lous impurity propagator, Ff,,(7r) = (Trfa(r)ba(T)fli(O)ba(O0)) (see Figure 5.1 ). This propagator, including the internal conduction electron line is evalu ated selfconsistently at finite impurity concentrations. Although Ff 1/N2, and in NCA only the contributions 0(1/N) are retained, Ff is of order O(A) near Tc and must be included in the calculation of the superconducting transi tion temperature. There are no anomalous contributions to boson and fermion selfenergies, Eo(w + iO+) = NV2 dEf(E)N(e)Gm(w + e + iO+), (5.2) Em(w + iO+) = kV2 0 de(1 f ())N(e)Go(w E + iO+). (5.3) oO Here k = 2 is the number of channels. We calculate Ff in the "elastic" approx imation [56] in which the internal anomalous conduction electron propagator, see Figure 5.1 is evaluated at the external frequency. The slope of the ini tial Tc suppression is obtained by including both diagonal and offdiagonal contributions, (NodTc (NodTc' (NodTc ddn 0 d \ n dn( S ImG (wn) F(n) (5.4) Sn> AIW The detailed form of this formula in the NCA approximation for a single channel Kondo effect was derived in Ref. [56]. Here we evaluate analogous 73 expression for the twochannel model. The contribution associated with the impurity spectral function is the following: (NodTc) fr dwpf(w)Rl(w), (5.5) dn 1 2 oo where T 11 i Rl(w) (2 ) Re +, (5.6) 7rW2 1 2 (2 2T70 Pf(w) = (1 + e.) dee,pg()m(E + w), (5.7) f oo /CO Zf = dee'[kpo(w) + Npm(E)], (5.8) oo where 3 is the digamma function and po and Pm are the slave boson and the impurity fermion spectral function, respectively. The second term in equation (5.4) has the following form: (NodTc N 00 n ) = 2 r dw[o f(w) (m +)]R2(), (5.9) dn 2 oo with T 2T R2(w)= 1 tanh (5.10) 4w W 2T (w) =/ deV(e; w), (5.11) Sf Joo V(e; w) =4 cosh(w/2T)e03ReGo(e)po(e) x [epW/2pm(E w)ReGm(E + eW/2pm( + +w)ReGm(e w)] +2sinh(w/T)e f[(ReGo(E))2 7r2p2(E)]pm(E + w)pm(E w). (5.12) The expression (m ++ 0) indicates the interchange of the fermion and boson indices, m and 0, respectively, in the formula for af(w). The offdiagonal con tribution can be separated into a spinflip and a spinpreserving part, T2SF and 74 0.08 0.04 T2SF + 3 0.00  I 0.04 2 0.08 6.0 3.0 0.0 3.0 6.0 W/TK (x 103) Figure 5.2 The anomalous part of the conduction electron Tmatrix evalu ated at T = 4.5 x 104TK. The solid (broken) line is the nonspinflip (spinflip) contribution. T2, respectively. The inclusion of inelastic processes is not expected to signif icantly affect the results because the spinflip and nonspinflip contributions almost cancel [ 115, 116]. At low temperatures these offdiagonal components of the Tmatrix have a peak at w T (see Figure 5.2 ). To evaluate Tcsuppression we use the gap equation, 1 = Vs dwNo(w) tanh(w/2Tc), (5.13) where No(w) is the density of conduction electron states in the normal metal and VS is the pairing potential. The full conduction electron Green's function, averaged over impurity positions, G1(w) = T0 EkT3 ,A(k)r1, is found from 0.20 c 0.10 ii T2SF o  0.00 T2 0.10 ' 3.0 1.0 1.0 3.0 logloTK/Tco Figure 5.3 The slope of the initial Tcsuppression for a superconductor with twochannel Kondo impurities. The separate contributions come from the diagonal part of the Tmatrix, T1, the offdiagonal spinflip, T2SF, and the off diagonal nonspinflip, T2. The solid line is the sum of the three contributions. the Dyson equations, D = w nV2Gf(C), and A = A + nV2Ff(j), where n is impurity concentration. As shown in Figure 5.3 the slope No(O)dTc/dn, at n = 0, remains finite as TK/Tc oo, indicating finite pairbreaking in the strong coupling regime. We include for comparison the NCA result for the initial Tcsuppression for a superconductor with single channel Kondo impurities, see Figure 5.4 The qualitative form of the Tc dependence on n in the twochannel case remains practically unchanged for all values of TK/TcO, however, as illustrated in Figure 5.5. Both results are a consequence of finite magnetic moment of the impurity at all temperatures. The maximum slope is reached for TK/Tco oc, in contrast with the single channel Kondo problem where maximum occurs for TK Tco. 0.3 c 0.1 o 0 z 0.1 0.3 3.0 1.0 1.0 logloTK/Tc 3.0 Figure 5.4 The slope of the initial Tcsuppression for a superconductor with single channel Kondo impurities. The separate contributions come from the diagonal part of the Tmatrix, T1, the offdiagonal spinflip, TSF, and the off diagonal nonspinflip, T2. The solid line is the sum of the three contributions. 1.0 0 0.5 0.0 0.0 10.0 n/NoTco 20.0 5.5 The superconducting transition temperature as a function of concentration. The inset shows the result for a single channel Kondo with TK Tc evaluated with Ef = 0.67D, and F = 0.15D. T2 \ ...T2 I I 1I I Figure impurity impurity 77 Similar results for (dTc/dn)n=o were obtained in a recent Monte Carlo calcula tion [115,116]. If the Kondo scale is known from other experiments, measure ments of Tc as a function of impurity concentration may help in identifying the multichannel behavior. For TcO have found a reentrant or exponential dependence of Tc on concentration. If it were possible to vary the ratio TK/TcO in the strong coupling regime, increased pairbreaking with growing TK/TcO would imply multichannel behavior while the opposite would be true for a single channel coupling. In unconventional superconductors, for which Ek F(k, w) = 0, both single and multichannel ex change result in qualitatively similar form of the Tcdependence on n in the strong coupling regime and the differences which may exist for TK Tc may be harder to identify. Finally, let us note that the breaking of the spin or channel symmetry which takes the system away from the multichannel fixed point should reduce pairbreaking especially in the strong coupling limit. Given the subtleties of this analysis we conclude that Tcsuppression is not likely to be a sensitive tool for identifying the multichannel exchange. CHAPTER 6 SUMMARY AND CONCLUSIONS We have studied metals, mainly superconductors, in the presence of mag netic and nonmagnetic impurities. In systems with power law density of states near the Fermi level and antiferromagnetic exchange interaction between im purity and conduction electrons the single impurity calculation in the largeN mean field approximation [3] yields the normalized critical coupling NoJc r, at low r, and NOJc + 2, for r 4 oo. The NCA approach also gives NoJc r, for r disagree at larger r. There is an upper critical rc = 0.5 for the orbitally non degenerate single channel Kondo problem, above which the Kondo effect does not occur at all, regardless of the strength of the coupling. The conclusions reached for the same model with nonzero impurity concentration in the mean field approximation are quite different: Jc = 0 for any nonzero concentration. It would be desirable to perform the NCA calculations from chapter 2 in the CoqblinSchrieffer limit, both at finite temperatures and at T = 0. This would allow a closer comparison of this method with the NRG [33] results for the Kondo model with powerlaw density of states in the conduction band, N(w) ~ IWjr. In addition to a singleimpurity calculation it would be in teresting to see the selfconsistent NCA treatment for finite concentration of impurities. One would like to know whether in NCA the critical coupling for finite concentrations is Jc = 0, as in the mean field approach. Also, it would 79 be of interest to study the breakdown of the Fermi liquid with increasing r and to use NCA to calculate dynamical quantities. In chapter 3 we have presented a slave boson theory of Kondo impurities in superconductors which has the advantages of being applicable in the Fermi liquid regime and being relatively easy to use in calculating quantities of exper imental interest at all temperatures in the superconducting state. The theory has been shown to reproduce all of the qualitative features of the physics found by previous theories for large TK, and is asymptotically in quantitative agree ment with "exact" NRG calculations for the swave case. It is furthermore capable of going considerably beyond currently available theories in that prac tical calculations of superconducting response functions are possible at low temperatures in the superconducting state, and can be easily generalized to the unconventional states of great current interest in the heavy fermion and hightemperature superconductivity problems. The single exception to this success is the failure to properly describe the swave superconductor at very large impurity concentrations, in that the the ory predicts an infinite critical concentration. In practical terms this is quite academic, as all independent impurity analyses will break down due to inter impurity interaction effects long before any putative critical concentration is achieved. Nevertheless, this formal shortcoming of the theory exists and must be addressed. Preliminary analysis of fluctuations about the saddle point has convinced us that the Gaussian fluctuations to the scattering amplitude arising from the slave boson dynamics will be sufficient to induce a critical concentra tion, in analogy to the works of Matsuura et al. [52] and Sakurai [53], and that 80 the theory as it stands is sufficient to describe the Fermi liquid regime every where except for very large concentrations in the swave case. The difficulty does not arise in the unconventional case. In unconventional superconductors, we have shown that the phenomeno logical theories of Refs. [5] and [6] will be reproduced by the microscopic theory presented here in the limit N = 2 and TK/Tc + oo. The deconden sation of the slave boson amplitude prevents extension of the theory into the high temperature regime, but qualitatively it is clear that the effect of lowering the Kondo temperature is to sharpen the manybody resonance near the Fermi surface, but lower its weight. Thus the effect of resonant scattering leading to lowenergy gapless effects in superconducting thermodynamic and transport properties is reduced. Impurities with larger orbital degeneracy may lead to similar resonances away from the Fermi level, possibly similar to those observed by Maple et al. [ 117] in specific heat experiments on Prdoped YBCO. The effect of Zn and Nidoping on superconducting properties of YBCO suggests that these two dopants may be described within a traditional Kondo type picture in conjunction with the theory discussed here. Ni, with spin one, may possibly be treated as a higherdegeneracy scatterer below its Kondo temperature. As we have seen, such an impurity acts as a weak scatterer compared to the N = 2, TK > Te resonant scattering case, a possible model for Zn. On the other hand, recent NMR measurements appear to suggest that a moment forms around the Zn site. Within an isolated impurity picture, this would suggest that TK < Tc, where the present theory is not applicable. A more plausible explanation, however, is that the local spin correlations induced 81 by a missing Cu must be accounted for, as suggested by Poilblanc et al. [98] Further experimental and theoretical work on this problem is clearly essential. There is by now a considerable body of experimental data supporting the picture of gapless superconductivity in the cuprate highTc materials, with a residual density of states and lowtemperature behavior varying qualita tively according to the dwave plus resonant scattering model [16, 118,80,17]. This data stands in apparent contradiction to the wellknown effect of small amounts of potential scatterers on anisotropic swave superconductors, namely the smearing of energy gap anisotropy. This continues to hold even for ex tremely anisotropic systems with nodes, as illustrated by the simple theory presented here for a representative order parameter. We believe that this data strongly suggests that the pairing is unconventional in these materials, but the analysis presented in chapter 4 does not as it stands allow one to distinguish among possible candidate unconventional states (e.g,, dX2_ 2 and dxz) without further quantitative comparison. It should be noted that timereversal break ing unconventional states with a gap will become gapless in the presence of pure potential scattering. As we have briefly discussed in chapter 4, the major difficulty inherent in such an analysis is the possibility that even an apparently "inert" impurity such as Zn or a vacancy in the CuO planes may induce local spin correlations in the strongly interacting electron system, leading to spinflip scattering. Ruling out gapless superconductivity induced by magnetic scattering then becomes a quantitative problem. Gapless behavior in films suggests that a resonant scat tering mechanism of some type must be present in order to induce a significant residual density of states with comparatively little Tc suppression. We have 82 shown, however, that resonant potential scattering does not take place in s wave systems, and argued that lowenergy resonant spin scattering is much less likely than in the isotropic case. We have furthermore made a crude estimate of the importance of spinflip scattering in Zndoped YBCO crystals which indicates these materials are dominated by potential scattering and should therefore exhibit an induced gap if the superconducting state is swave. We have also shown that, although gapless behavior in thermodynamic quantities qualitatively similar to dwave states is to be expected for a range of impurity concentrations in extendeds states, transport properties are quite different in the two states. In particular, residual T 4 0 conductivities (a and K/T) are expected to be large and to scale inversely with impurity concentra tion, in contrast to the resonant dwave case. Such experiments can therefore be used in conjunction with Josephson measurements to settle the question of order parameter symmetry. Existing transport data on YBCO single crystals appear to restrict possible sstates to those where the average of the order parameter over the Fermi surface is nearly zero. The ARPES experiments on the BSCCO2212 material might imply a strongly anistropic swave order parameter with ri0 0. The existing transport measurements, however, do not support this interpretation. Further measurements on the latter system, particularly systematic doping studies to search for an impurityinduced gap and to test impurity scaling of residual conductivities can help to distinguish s and dwave state. A careful determination of the temperature dependence of n over the 1K 30K range in BSCCO would be of great value in this regard. In chapter 5 we studied the effect of twochannel Kondo impurities on su perconducting states. This type of impurity is known to have a nonFermi 83 liquid ground state in a normal metal. Since the impurity never loses its mag netic moment due to overscreening by conduction electrons, even in the strong coupling regime, it is not surprising that we find qualitatively similar type of Tc dependence on impurity concentration for all values of TK/Tc. 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Luk, Ph.D. Thesis, Ohio State University, 1992. [116] K.H. Luk, M. Jarrell, and D.L. Cox, Phys. Rev. B 50, 15 864 (1994). [117] S. Ghamaty, B.W. Lee, J.J. Neumeier, and M.B. Maple Phys. Rev. B 43, 5430 (1991). [118] J.A. Martindale, K.E. O'Hara, S.M. DeSoto, C.P. Slichter, T.A. Friedmann, and D.M. Ginsberg, Phys. Rev. Lett. 68, 702 (1992). BIOGRAPHICAL SKETCH Lech S. Borkowski was born in 1963 in Ketrzyn, Poland. He attended Szkola Podstawowa Nr 1 and Liceum Og6lnoksztalc4ce in Ketrzyn. He studied chemistry at the Nicolaus Copernicus University in Torufi and Warsaw Uni versity of Technology for one year and a half. Later he studied physics at the Wroclaw University of Technology and received his M.S. degree there in 1987. He then went to the United States and studied at the Virginia Polytech nic Institute and State University in Blacksburg, Virginia, before transferring to the University of Florida, Gainesville, where he did his research with the Condensed Matter Theory Group at the Physics Department, under the su pervision of Prof. Peter Hirschfeld. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of philosophy. Peter J. Irschfeld, Chair Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pradeep Kumar Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Kevin Inge Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Khandker A. Muttalib Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully ade uate, in scope and quality, as a dissertation for the degree of Doctor of Phj hophy SJ mes E. Keesling J professor of Mathematics This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1995 Dean, Graduate school LL: 1780 1994 73, UNIVERSITY OF FLORIDA 3 1262 0815570 I37 3 1262 08557 0637 