IMPURITIES IN UNCONVENTIONAL
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
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
1 INTRODUCTION ............. ...... 1
2 KONDO EFFECT IN SYSTEMS WITH DENSITY OF STATES
VANISHING AT THE FERMI ENERGY . 6
2.1 Mean-Field Theory of the Single-Impurity 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 TWO-CHANNEL 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
LECH S. BORKOWSKI
Chairman: Peter J. Hirschfeld
Major Department: Physics
We present a self-consistent 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 s-wave 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
s-wave 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 high-Tc superconductors, we
study properties of superconducting states with "extended s-wave" 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
impurity concentrations, properties reflecting the density of states alone will
be similar to those of d-wave states, transport measurements may be shown to
qualitatively distinguish between the two.
We also discuss the effect of two-channel Kondo impurities on supercon-
ductivity. In the strong coupling regime such impurities are pairbreakers, in
contrast to the ordinary Kondo effect. Measurements of Tc-suppression may
help in identifying impurities displaying this more exotic exchange coupling to
the conduction band.
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 impurity-conduction 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 Abrikosov-Gor'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 s-wave superconductors are largely
unaffected by the presence of Kondo impurities due to Anderson's theorem [
2]. Superconductors with an anisotropic order parameter, e.g. p-wave, d-wave
etc., are strongly affected, however, and potential scattering is pair-breaking.
The effect of pair breaking is maximal in s-wave 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 heavy-fermion and high-Tc 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.
The Kondo effect is accompanied by the formation of a narrow many-body
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 conduction-electron 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 (Coqblin-Schrieffer) 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 momemtum-dependent 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 a-Sn, Pbl-xSnxTe at a critical
composition, domain walls of PbTe, PbTe-SnTe heterojunctions and graphite.
In the context of heavy-fermion 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
Schmitt-Rink, Miyake, and Varma [ 6] for strong scattering. Moment formation
was not considered in these theories, but pair breaking still occurs because of
the vanishing of the anomalous one-electron impurity-averaged self-energy. The
strength of the scattering was parametrized in the latter works by a phase shift
So for s-wave 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 single-impurity spin- Kondo effect [ 7], a
"bound-state" 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
s-wave 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 d-wave 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 d-wave 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 , it is of interest to explore whether an s-d
type exchange coupling of conduction electrons to an impurity embedded in an
unconventional superconductor can describe the range of behavior observed.
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, self-consistent 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 large-N "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 s-wave type. Many experiments probe only
the low-lying quasiparticle states of the superconductor, which in the pure sys-
tem are quite similar for both anisotropic s- and d-wave states with nodes.
Nonmagnetic scattering reduces anisotropy of such states without destroying
superconductivity, in contrast to the effect the nonmagnetic impurities have
on unconventional (non-s-wave) superconducting states. In anisotropic s-wave
states having lines of zeroes of the order parameter on the Fermi surface, some-
times called "extended-s" 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 low-T,
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 s-wave superconductors
from d-wave superconductors having very similar quasiparticle excitation spec-
trum are discussed in chapter 4 [ 25, 26]. In chapter 5 we study the effect
of two-channel Kondo impurities on superconductors [ 27]. Conclusions are
presented in chapter 6.
KONDO EFFECT IN SYSTEMS
WITH DENSITY OF STATES
VANISHING AT THE FERMI ENERGY
2.1 Mean-Field Theory of the Single-Impurity 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 low-T 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 low-T 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 low-temperature, low-energy 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
understanding of the problem, Withoff and Fradkin  used poor-man'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 ),
with the power-law density of states over the whole bandwidth. In poor-man's
scaling [ 30] one eliminates high-energy 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)
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 poor-man's scaling confirmed by other approaches, e.g.
the large-N approximation? The slave boson approximation is known to give
correct results in the limit of large orbital degeneracy for the ordinary Kondo
problem at low temperature [20,21]. The simplest version of the Anderson
Hamiltonian to describe a rare-earth impurity is
H = EkcCkm + EfEf im + U ftnfmgfftm'
km m m>m' (2.3)
+ Vk(cfm+ h.c.).
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 spin-orbit 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 f-electrons 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 on-site Coulomb repulsion, U > V2/D,
the Hamiltonian (2.3) can be simplified by the Schrieffer-Wolff transformation
[ 31]. The resultant Hamiltonian is the SU(N) generalization of the Kondo
model (2.1), also called the Coqblin-Schrieffer model [ 32],
H = Ekmcm +J kmfmfmCkm. (2.4)
kmckm+ f t fmc, 24
km kk' mm'
The four-fermion term can be eliminated by the Hubbard-Stratonovich trans-
formation. In the limit N -+ oo, the saddle-point solution is equivalent to a
mean field theory with the boson field playing the role of an (unphysical) order
a = (fmckm), (2.5)
The mean field Hamiltonian now reads
H = ckmckm + : (akmfm + h.c.) + f(nf 1). (2.6)
The auxiliary field and position of the saddle point are obtained by solving two
KH)= 0, (2.7)
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 , 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
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 f-electron Green's function, while equation (2.8) becomes
No f ()N() (2.10)
=- c f( () f ReEm(E))2 + (Imj m(c))2
Figure 2.1 Schematic flow diagram for single-channel finite-r Kondo prob-
lem obtained from the Numerical Renormalization Group (based on Ref. ).
The impurity self-energy 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 power-law 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
In a more realistic treatment, the density of states may be assumed to vary
as a power-law 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 spin-singlet-like 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 Curie-like susceptibility as the singlet
ground state becomes unstable.
2.2 NCA Approximation
A more sophisticated version of the large-N 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
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 many-body 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 Abrikosov-Suhl many-body resonance of width TK,
which controls all low-T 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 [
The self-energies for the boson and fermion propagators are, respectively
(see Figure 2.3 ),
Eo(w + i0+) = NV2 f def(e)N(e)Gm(w + e + iO+), (2.13)
Figure 2.3 Self-energies for the empty impurity f-level (slave boson), E0,
and for the occupied f-level fermionn), Em.
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 .
Here we only show the results for the spin susceptibility. The temperature
dependence of the static spin susceptibility exhibits a crossover from the high-T
free-moment behavior to the low-T 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 . Figure 2.4 captures the essence of the
problem correctly: increasing r leads to lower TK.
3 0.1 F/D=0.25
-6 -4 -2 0
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
2.5-2.8. 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
F/D=0.15 00000 2
SF/D=O. 175 '00000222
-0 A AA
-7 -5 -3
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 10-4, 1.96 x 10-4, 5.77 x 10-4, and 1.31 x 10-3, 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
-7 -5 -3 -1
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 10-3,2.5 x 10-3,4.2 x 10-3, and 6.44 x 103,
o' ,1 '0
-6 -4 -2 0
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 10-3, 1.65 x 10-2,2.6 x 10-2,4.9 x 10-2, and 7.6 x 10-2,
F/D=0.3 oo0 0
r=0.2 o0 0o a
m-AA-AA AAA A AA A
AAAA^ A A A A 1&
00 A A A A AM&AAAAAdA &A
-7 -5 -3 -1
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 10-3, 1.65 x 10-2, 2.6 x 10-2, and 4.9 x 10-2, respectively.
o1 -1.0 W 0000
-8 -6 -4 -2
Figure 2.9 Logarithm of TX for two lowest values of F from each set of data
shown in Figures 2.5-2.8.
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 off-diagonal long range order described
by the order parameter A(k). To analyze the problem we start from the SU(N)
Coqblin-Schrieffer Hamiltonian, already introduced in equation (2.4), and add
a simple BCS-like pairing of electrons on opposite sides of the Fermi sphere,
H = e km mf+ k fm c+ km + [(k)Ccm-k-m + h.c. ,
k,m k,k' m,m' k,m
We now generalize the procedure of Read and Newns  to include super-
conducting correlations in the functional integral representation of (2.15). The
saddle-point approximation to this theory is equivalent in the N -+ oo limit to a
mean-field theory of (2.15) with mean-field 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)
= Im dwf(w) Tr [(o + r)GO(k,w + iO+)Gf(w + iO+)], (2.17)
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 particle-hole 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 self-energy. In the supercon-
ducting state, we must solve the full saddle-point 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)
J-oo 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 large-N limit. Here we adopt the point of view that, since the saddle point
for N = 2 is known to reproduce the correct analytic low-temperature normal-
state behavior  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 s-wave state A(k) = A, model p-wave 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  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 s-wave states.
In Figure 2.10 we plot Jc vs AO/D for all three states.
S1.0 po r
0.0 0.2 0.4 0.6 0.8 1.0
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 self-consistent
Green's functions averaged over impurity positions in the usual way, leading
to Gf7(w) = crT0 ef73, and G-1(w) = UTro ckr3 A(k)r1, where
w = + F (/ (2(k) )2 1/2 (2.21)
S=w + ar/(-D2 + 2). (2.22)
Here a = fiTcN/2wr, and f = n/TcoNO is the scaled impurity concentration.
In general, Gf1 and G-1 will also contain additional off-diagonal renormal-
izations, which vanish in the p-wave case considered here.
An interesting consequence of the self-consistent treatment of impurity
scattering is that the conclusions of WF regarding the existence of a critical
exchange coupling Jc based on a single-impurity 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
N(0) is finite. Since in the polar state any finite impurity concentration f may
be shown to lead self-consistently 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) ~ |w|r 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 cot-le /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 WF-type 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
PHYSICAL PROPERTIES OF MAGNETIC
IMPURITIES IN SUPERCONDUCTORS
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 low-T behavior. This is true for both conventional and
unconventional superconductors. After testing our approach and providing
some new results for the s-wave case we discuss results for unconventional
superconductors. We emphasize that the large-N 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 large-N slave boson technique for the SU(N)
Anderson model describing an N-fold 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 on-site repulsion term
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 BCS-like pairing term of electrons on opposite sides of the Fermi
H = kctkmCkm + Ef Z ftmfm + V [ctkmfmb + h.c.]
k,m m k,m
+ -[A(k)ctkmck-m + h.c.] + A(Z ftmfm + btb),
In the limit Ef -+ -oo, NoV2/Ef =const, Equation (3.1) reduces to the
Coqblin-Schrieffer 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 mean-field amplitude
(b), leads to the two equations
1 f 1
S= -Im 0 dwf (w) Tr(ro + T3)Gf(w + iO+), (3.2)
Ef- = Im I dwf(w) Tr [(Tr + r3)(G (k, w + iO+)Gf(w + iO+)) ,
which determine (b) and ef, the latter being the position of the resonant state.
Equations (3.2) and (3.3) should be solved self-consistently together with the
gap equation (2.20). The full conduction electron Green's function G and
- = = =
->-< = =<== :=
= = -- -+- -- >= := + -+-,z <=
== ->-= + ->-=:=
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 self-consistently from the
i) = w + aC+/(-_2 + 2 A2), (3.6)
= 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 off-diagonal renormalizations are
A(k) = A(k) + aA/(-2 + E2f +A2), (3.8)
= 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, off-diagonal corrections vanish and A(k) = A(k).
This class includes but is not limited to odd-parity superconducting states. The
energy scale F is the renormalized resonance width, F = (b)}rNoV2. The low-
temperature Kondo scale in the large-N slave boson theory is given by TK =
/F2 + I2. Although the width of the actual spectral feature corresponding
to the Abrikosov-Suhl 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 s-wave su-
perconductor has been reviewed by Miiller-Hartmann [ 45]. Abrikosov and
Gor'kov first discussed the pair-breaking effects of magnetic impurities weakly
coupled via exchange interactions to conduction electrons . Shiba [ 46] ex-
tended this approach to treat strong scattering by classical spins, using the
t-matrix approximation, and showed the existence of bound states in the gap.
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 Miiller-Hartmann 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 Abrikosov-Gor'kov limit, TK/Tc -+ 0, and made the remarkable predic-
tion of a reentrant superconducting phase if TK < Tc, subsequently observed
in experiments on Lal-xCexAl2 [ 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 Miiller-Hartmann-Zittartz 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
Yamada-Yosida theory [ 54] to the superconducting state. They obtained an
exponential Tc-suppresssion 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 "pair-weakening" as
opposed to pair-breaking, 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 Miiller-Hartmann 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
which is always less than one [53, 55], in contrast to the high temperature
Not surprisingly, qualitatively similar results were obtained by other early
workers for Kondo and Anderson impurities using a variety of other approaches
. More recent treatments include the use of a self-consistent large-N [ 56],
Monte Carlo [ 57] and NRG methods [ 58]. Schlottman [ 59] has treated
the mixed-valence regime using Brillouin-Wigner 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 low-temperature properties of Kondo-doped 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-
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 quantum-mechanical degrees of freedom leads to deviations
from the classic Abrikosov-Gor'kov prediction for the dependence of Tc on im-
purity concentration . These effects depend sensitively on the low-energy
behavior of the self-energy E(w), which enters the linearized gap equation,
obtained from equation (2.20) near Tc,
\ -2 0 2
0.0 20.0 40.0 60.0 80.0 100.0
Figure 3.2 The critical temperature for an s-wave 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
where B(wn) = (Wn + F)2 + 2 .
The slope of the Tc-suppression 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 s-wave 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 Tc-suppression
is found to occur at TK "- 3Tc, similar to the NCA result . The early
high-temperature theory of Miiller-Hartmann and Zittartz  locates this
maximum at TK/Tc 12, whereas in the more recent Monte Carlo calculation
 for the symmetric Anderson model with finite U the maximum slope of the
Tc-suppression 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  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  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 ,
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 finite-U 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
S= w+r 1/2' (3.13)
(A2 _) 2
A = r. (3.14)
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 Abrikosov-Suhl 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
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 Coqblin-Schrieffer limit for N = 2 and several values of F. In the
high-temperature 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 .
Recently, Shiba et al.  studied the position of bound states in the gap in
an s-wave superconductor using the numerical renormalization group (NRG).
Their results cover both the high- and low-temperature limits reliably. The
0 2 0
Figure 3.3 Conduction electron and impurity spectral functions in the
Coqblin-Schrieffer limit in an s-wave 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.
0.0 1.0 2.0
Figure 3.4 Conduction electron density of states in the Coqblin-Schrieffer
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 /
-1.0 -- -- 0.0
-2.0 -1.0 0.0 1.0 2.0
Figure 3.5 The position and spectral weight of the bound states in the gap
for an s-wave 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. 
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 T-matrix 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/A|I 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 slave-boson theory fails in
the high temperature regime. The slave-boson mean-field amplitude vanishes
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
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
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
C* 1 1/ln(TK/2TcO), see Figure 3.7 Ichinose  and Sakurai [ 64] have
obtained a somewhat different result, C* 1 1/ln2 TK/Tc).
0.0 0.2 0.4 0.6
Figure 3.6 Specific heat jump as a function of Tc/TcO. The result of the
Abrikosov-Gor'kov theory is included for comparison.
0.0 1.0 2.0 3.0
Figure 3.7 The derivative of the specific heat jump evaluated at zero im-
purity concentration as a function of TK/Tc for an s-wave superconductor.
The dash-dotted line shows the asymptotic behavior, 1 1/ln(TK/2Tco). The
dashed line is the asymptotic form at TK/Tc -> oo found in Refs.  and ,
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
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 Coqblin-Schrieffer 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
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 1-impurity 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 p-wave
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 d-wave
symmetry, will have similar properties at low temperatures, since the main
factor determining the low-T 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 low-temperature 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 p-wave-like or d-wave-like
superconductors, the influence of impurity scattering on the critical tempera-
ture is qualitatively different from that of s-wave-like superconductors [4, 69].
For the unconventional states of interest, there are no off-diagonal 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"
The initial Tc-suppression 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.
S\ -2 0 2
0.5 \ \ ogoTK/Tco
100' 5 ITK/TCo=1
0.0 2.0 4.0 6.0 8.0 10.0
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
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 s-wave 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 s-wave case by the continuum in which they are embedded.
In the limit TK -+ oo, the current theory coincides with the results given by
phenomenological T-matrix 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
-2 0 2 0 2
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)
where f = f(w) is the Fermi function. Note that the density of states N(w) is
calculated self-consistently, using equations (3.2), (3.3), and (2.20).
The presence of resonances at low energies leads to pronounced features
in the low-T 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 low-temperature 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 0.0 1.5
0.0 0.5 1.0 1.5
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 (dash-dotted 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
5 / 2
0 0.5 1
0.00 0.05 0.10
Figure 3.11 The low-T 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
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 many-body 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 '
S0.5 o 1
0.00 0.05 0.10 0.15 0.20
Figure 3.12 The low-T part of C/T in the axial state. Note the sharp
resonance in the density of states at low energies (the inset).
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 low-temperature 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
1.0 1 i i-
-2.0 -1.0 0.0 1.0 2.0
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-
been calculated in the phenomenological T-matrix approach mentioned above,
and used to analyze experiments on heavy fermion and high-Tc 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
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
0 0.6 -
0.0 0.2 0.4 0.6 0.8 1.0
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 low-T 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, A-2 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 A-2(n) ~ log(nc/n) close to the critical concentration.
0.8 \ TK=5Tco
0.2 N \
0.0 0.2 0.4 0.6 0.8 1.0
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 d-wave 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.
4.1 Experimental Motivation
Recent high resolution angle-resolved photoemission (ARPES) experiments
on Bi-2212 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 s-wave state; c) s-wave state with gap minima at
While alternative explanations consistent with an order parameter Ak
which reduces the symmetry of the Fermi surface (e.g. d-wave, 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
on anisotropic-s 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
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 Ni-doped YBCO crystals [ 80, 81, 82, 83] have shown
evidence of low-temperature thermodynamic properties reflecting the existence
of a residual density of states. They have been interpreted most often in terms
of a d-wave pairing scenario [73, 84, 85, 86, 87], where the well-known effect
of dirt is to induce a residual density of states for infinitesimal concentrations
. 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 extended-s 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 extended-s 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.
-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 Argonne-U. 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 weak-coupling approach yields the
gap magnitude ratio Ao/Tc = 2.92, see Figure 4.3 .
0.0 0.2 0.4 0.6
Figure 4.3 The ratio AO/Tco as a function of 770.
Note that 7r0 controls the angular range over which the s-wave 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 d-wave 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
Measurements of penetration depth, angle-resolved 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)
0.0 0.5 1.0 1.5
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.
low-energy density of single-particle states N(w), and all properties deriving
Figure 4.4 shows density of states for the state of the type 4.1 c. Note the
low-energy gap feature corresponding to the smaller gap maximum.
Systematic impurity-doping 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 s-wave case
(Figure 4.1 b-c). In the d-wave case, suppression of gap variation in k-space can
only result in an overall suppression of the order parameter magnitude, whereas
in s-wave cases b) and c), it results eventually in the elimination of the nodes.
In many-body language, the crucial point is that the off-diagonal impurity self-
energy E1 is nonzero in s-wave states 4.1 a and 4.1 b, but vanishes in the d-
wave state 4.1 c. In the isotropic s-wave state, a large E1 cancels the diagonal
self-energy E0 , leading to Anderson's theorem , whereas in extended-s
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
low-frequency transport coefficients which in extended-s states for T < Tc
vary relatively weakly with temperature. By contrast, in the d-wave (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 extended-s states here are also valid
qualitatively for any mechanism which generates an off-diagonal self-energy in
a d-wave state as well, e.g. tetragonal symmetry breaking or locally induced
s-wave components due to impurity potentials.
The disorder-averaged 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 particle-hole and spin space. The renormalized quantities are
given by w = w E0(w), k = 4k + E3(w), and Ak = Ak + CE(w), where the
self-energy due to s-wave impurity scattering has been expanded E = Ei7ri.
The relevant self-energies are given in a self-consistent t-matrix approximation
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
phase shift, c. The Born limit corresponds to c > 1, so that Eo FNGo,
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 high-Tc
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 d-wave superconductors is that an
infinitesimal concentration of impurities produces a finite density of states
N(0) > 0 at the Fermi level , leading to temperature dependence character-
istic of the normal state in all transport quantities. Solving the self-consistency
equations at w = 0 for the extended-s 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 .
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 d-wave" scenario
also be explained by extended-s 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 Zn-doped 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 extended-s 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
0.6 / 5
z 0.4 -
0.0 0.2 0.4 0.6 0.8 1.0
Figure 4.5 Residual density of states N(O)/No in extended-s 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 well-developed 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 s-wave 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 d-wave superconductor, the self-energies obtained in the
Born approximation and in the resonant scattering limit are almost equivalent
in the highly anisotropic s-wave system, if 710 < Aavg/0o. This insensitivity to
larger phase shifts arises because of off-diagonal self-energy corrections which
Born \ s
Z 0.5 N/Ao=0.1
0.0 0.5 1.0 1.5
Figure 4.6 Normalized density of states N(w)/No for s- and d-wave 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 t-matrix, c2 Go2 + G12 ~ O(1) for
scattering are shown in Figure 4.7 .
z 0.5 )0.5
0.0 1 1
0.0 0.5 1.0 1.5
Figure 4.7 Normalized density of states N(w)/No for s- and d-wave 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 off-diagonal 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.
In the d-wave case, Ak d= Aodd(k), with 4d = cos 2, yielding a simple
equation for the gap maximum Agd,
1 WD /Dw \
Ad-1 = 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 s-wave 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 s-wave and 1 for the d-wave
state considered. In the d-wave 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 s-wave 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 Tc-suppression at small and moderate concentrations
will be very similar to the one shown above. For large concentrations, such
that Tc/TcO < 1, the Tc-suppression by the magnetic component in the scat-
tering will be the dominant effect, if Fs > 0. However, one may not treat the
impurities as independent from one another when the impurity concentration
4.5 London Penetration Depth
The opening of the energy gap with increasing impurity concentration is
an indelible signature of s-wave 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
temperature-dependent 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 d-wave 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 F-1 in the Born limit and r-1/2 in the resonant
scattering case. In the anisotropic s-wave 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 s-wave
case is easy to distinguish from the d-wave case when plotted against (T/Tc)2 as
also shown in Figure 4.8 The important experimentally relevant signature is
Figure 4.8 Temperature dependence of normalized magnetic penetration
depth (Ao/A(T))2 for s- and d-wave 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
S0.8 \ F/Tc0=0
0.00 0.05 0.10 0.15
Figure 4.9 Temperature dependence of normalized magnetic penetration
depth (Ao/A(T))2 for an order parameter of an extended-s 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 spin-flip scattering of conduction electrons [ 98]. This poses the
most serious obstacle for the direct application of the principle distinguishing
d-wave from anisotropic s-wave systems outlined above, since magnetic scat-
tering will lead to gapless superconductivity as in the usual Abrikosov-Gor'kov
theory. Furthermore, even if a gap remains, strong spin-flip 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 d-wave
case. Here we investigate the competition between the opening of the energy
gap in the s-wave 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 t-matrix approxima-
tion analogous to the one applied to the pure potential scattering case. The
0.15 3 /
< I 0 5 10
0.0 1.0 2.0 3.0
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.
self-energies found in the presence of both types of scattering reduce in the
isotropic s-wave 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
s-wave system are very similar to those obtained in the simpler Born approx-
imation, as discussed above, provided r0r is not large. The self-energies in the
Born approximation are Eg = (FN + FT)Go and El = (-rN + Fr)G1 ,
where rs niJ2S(S+ 1)7rNo. For 770 = 0, the induced gap, 2G, in the s-wave
system may then be shown to vary as QG FN Fr] > 0, but the effects of
self-consistency 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
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 t-matrix 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 . 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 high-Tc materials with
simple defects. Walstedt and co-workers estimated JNo 0.015 for a Zn ion
in YBCO, implying that Zn is a nearly pure potential scatterer in this system
. On the other hand, Mahajan et al.  estimate JNO 0.45. For a 1% Zn
concentration, a magnetic moment of 0.36 pB for Zn in fully oxygenated YBCO
 and a density of states of 1.5/eV , we find F -_ 1 x 10-4 eV. From the
residual resistivities of Zn-doped YBCO crystals , we estimate that a 1% Zn
sample corresponds to a total impurity scattering rate of rmp 1 x 10-2eV,
assuming that the inelastic and elastic contributions to the scattering rate
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  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 d-wave states, as suggested by the
following argument. Any DC transport coefficient L(T) in a system charac-
terized by well-defined single-particle 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 d-wave 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
d-wave transport coefficients.
Impurity-limited transport in the extended s-wave state will be qualita-
tively similar to the d-wave case if the scattering is weak, c > 1. In contrast
to the d-wave case, however, for c -+ 0 (and 70
not occur, since the denominator of the t-matrix, c2 Go + G2 (1 0)2
. A simple low-T estimate accounting for nodal quasiparticle contributions
gives L ~ Lo(1 21?0) as T -+ 0. The exact behavior in this range will be
0.0 0.2 0.4 0.6 0.8 1.0
Figure 4.11 Conductivity of extended-s 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
influenced by self-consistency effects and the leading frequency dependence of
the t-matrix. The resultant temperature dependence will then be intermediate
between the strong and weak scattering limits of d-wave transport coefficients.
In Refs. [ 103, 104], expressions for the complex conductivity of an anisotropic
s-wave superconductor were derived. We do not reproduce these rather lengthy
expressions here, but merely comment that a fully self-consistent numerical
evaluation confirms the qualitative picture of low-frequency transport described
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,
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 extended-s state is nonzero
and generically much larger than a(Tc) in clean high-Tc 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 d-wave conductivity with
resonant scattering due to the vanishing of the off-diagonal self-energy. 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 d-wave 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 linear-T conductivity observed in microwave experiments
on clean YBCO crystals , 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 twin-free samples. We are not aware
of similar experiments on high-quality 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 d-wave states, as can be seen from comparison of Figure 4.12 and Figure 2
of Ref. . Note, however, that the clean limit regime in the d-wave state
is a much more narrow range of scattering rates than in the s-wave states.
0.0 0.2 0.4 0.6 0.8
Figure 4.12 Conductivity of extended s-state 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
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.
of clean extended s-wave state
= 0.1, c = 0 with dZ2_ 2 state,
To illustrate the comparison of these results with those expected for a
d-wave 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 extended-s conductivity for qr0 = 0.25. Even in
the last case the qualitative differences between the s- and d-wave results are
manifest. The dependence of the s-wave 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 d-wave Born result,
since the denominator of the t-matrix becomes irrelevant.
Given the uncertainty regarding the origin of the residual conductivity in
the high-Tc 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 s-wave state implies
a large linear-T 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 s-wave state of the type considered here for any qo0.
TWO-CHANNEL KONDO IMPURITIES
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
two-channel 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 two-channel 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
non-Fermi liquid behavior below the crossover temperature TK.
The experimental realization of the two-channel Kondo effect in uranium
compounds was proposed but remains controversial [ 110, 111, 112]. It was
argued  that in compounds such as UBel3 and Yl-xUxPd3 with x = 0.2,
the U ions at crystal sites with cubic symmetry have stable 5f2 configuration
and J = 4 total angular momentum. The crystal-field 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 time-reversal symmetry involved in scattering by magnetic
impurities in superconductors has important consequences. The qualitative
type of behavior is well-known 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 . 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 on-site 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 10-5D and the impurity occupation
number is nff 0.9.
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 self-consistently 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
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)
Here k = 2 is the number of channels. We calculate Ff in the "elastic" approx-
imation  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 off-diagonal
(NodTc (NodTc' (NodTc
ddn 0 d \ n dn(
S- ImG (wn) F(n) (5.4)
The detailed form of this formula in the NCA approximation for a single-
channel Kondo effect was derived in Ref. . Here we evaluate analogous
expression for the two-channel model. The contribution associated with the
impurity spectral function is the following:
(NodTc) fr dwpf(w)Rl(w), (5.5)
dn 1 2 -oo
T 11 i-
Rl(w) (2 ) Re +, (5.6)
7rW2 1 2 (2 2T-70
Pf(w) = (1 + e-.) dee-,pg()m(E + w), (5.7)
Zf = dee-'[kpo(w) + Npm(E)], (5.8)
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
R2(w)= 1- tanh (5.10)
4w W 2T
(w) =-/ deV(e; w), (5.11)
V(e; w) =4 cosh(w/2T)e-03ReGo(e)po(e)
x [epW/2pm(E w)ReGm(E + e-W/2pm( + +w)ReGm(e w)]
+2sinh(w/T)e- f[(ReGo(E))2 7r2p2(E)]pm(E + w)pm(E w).
The expression (m ++ 0) indicates the interchange of the fermion and boson
indices, m and 0, respectively, in the formula for af(w). The off-diagonal con-
tribution can be separated into a spin-flip and a spin-preserving part, T2SF and
-6.0 -3.0 0.0 3.0 6.0
W/TK (x 10-3)
Figure 5.2 The anomalous part of the conduction electron T-matrix evalu-
ated at T = 4.5 x 10-4TK. The solid (broken) line is the non-spin-flip (spin-flip)
T2, respectively. The inclusion of inelastic processes is not expected to signif-
icantly affect the results because the spin-flip and non-spin-flip contributions
almost cancel [ 115, 116].
At low temperatures these off-diagonal components of the T-matrix have
a peak at w T (see Figure 5.2 ).
To evaluate Tc-suppression 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, G-1(w) = -T0 EkT3 ,A(k)r1, is found from
c 0.10 ii
-3.0 -1.0 1.0 3.0
Figure 5.3 The slope of the initial Tc-suppression for a superconductor
with two-channel Kondo impurities. The separate contributions come from the
diagonal part of the T-matrix, T1, the off-diagonal spin-flip, T2SF, and the off-
diagonal non-spin-flip, 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
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 Tc-suppression for
a superconductor with single channel Kondo impurities, see Figure 5.4 The
qualitative form of the Tc dependence on n in the two-channel case remains
practically unchanged for all values of TK/TcO, however, as illustrated in Figure
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.
Figure 5.4 The slope of the initial Tc-suppression for a superconductor with
single channel Kondo impurities. The separate contributions come from the
diagonal part of the T-matrix, T1, the off-diagonal spin-flip, TSF, and the off-
diagonal non-spin-flip, T2. The solid line is the sum of the three contributions.
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.
I I 1I I
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 Tc-dependence 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 Tc-suppression is not likely to
be a sensitive tool for identifying the multichannel exchange.
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 large-N
mean field approximation  yields the normalized critical coupling NoJc r,
at low r, and NOJc -+ 2, for r -4 oo. The NCA approach also gives NoJc 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 Coqblin-Schrieffer limit, both at finite temperatures and at T = 0. This
would allow a closer comparison of this method with the NRG  results for
the Kondo model with power-law density of states in the conduction band,
N(w) ~ IWjr. In addition to a single-impurity calculation it would be in-
teresting to see the self-consistent 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
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 s-wave 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
high-temperature superconductivity problems.
The single exception to this success is the failure to properly describe the
s-wave 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.  and Sakurai , and that
the theory as it stands is sufficient to describe the Fermi liquid regime every-
where except for very large concentrations in the s-wave case. The difficulty
does not arise in the unconventional case.
In unconventional superconductors, we have shown that the phenomeno-
logical theories of Refs.  and  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 many-body resonance near the Fermi
surface, but lower its weight. Thus the effect of resonant scattering leading to
low-energy 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 Pr-doped YBCO.
The effect of Zn and Ni-doping 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 higher-degeneracy 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
by a missing Cu must be accounted for, as suggested by Poilblanc et al. 
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 high-Tc materials, with
a residual density of states and low-temperature behavior varying qualita-
tively according to the d-wave plus resonant scattering model [16, 118,80,17].
This data stands in apparent contradiction to the well-known effect of small
amounts of potential scatterers on anisotropic s-wave 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 time-reversal 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 Cu-O planes may induce local spin correlations in
the strongly interacting electron system, leading to spin-flip 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
shown, however, that resonant potential scattering does not take place in s-
wave systems, and argued that low-energy resonant spin scattering is much less
likely than in the isotropic case. We have furthermore made a crude estimate
of the importance of spin-flip scattering in Zn-doped YBCO crystals which
indicates these materials are dominated by potential scattering and should
therefore exhibit an induced gap if the superconducting state is s-wave.
We have also shown that, although gapless behavior in thermodynamic
quantities qualitatively similar to d-wave states is to be expected for a range
of impurity concentrations in extended-s 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 d-wave 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 s-states to those where the average of the order
parameter over the Fermi surface is nearly zero. The ARPES experiments
on the BSCCO-2212 material might imply a strongly anistropic s-wave 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 impurity-induced gap
and to test impurity scaling of residual conductivities can help to distinguish s-
and d-wave 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 two-channel Kondo impurities on su-
perconducting states. This type of impurity is known to have a non-Fermi
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. The ini-
tial slope of Tc suppression does vary greatly, however, reaching its maximum
as TK/Tc -*- oo. There is no reentrant behavior of Tc as a function of n.
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T -+ 0 in the two-channel problem. Therefore, below a certain temperature
impurities cannot be treated as being independent from one another. In prin-
ciple the character of the superconducting response in the regime T
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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.
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.
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.
Dean, Graduate school
UNIVERSITY OF FLORIDA
3 1262 0815570 I37
3 1262 08557 0637