Table Of ContentThe Method of the Correlation Function
in Superconductivity Theory
G. LI~DERS and K.-D. USAD~L
Contents
Frequently Used Symbols . . . . . . . . . . . . . . . . . . . . . . . . . 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
I. General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01
.1 Survey of Microscopic Superconductivity Theory . . . . . . . . . . . . 01
a) Cooperative Pairing . . . . . . . . . . . . . . . . . . . . . . . 01
b) Gorkov's Equations . . . . . . . . . . . . . . . . . . . . . . . 21
2. Thermodynamics. Second Kind Transitions . . . . . . . . . . . . . . 71
a) Phenomenological Thermodynamics . . . . . . . . . . . . . . . . 71
b) Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . 81
c) Second Kind Transitions . . . . . . . . . . . . . . . . . . . . . 20
.3 General Properties of the Kernel. Dens!ty of States . . . . . . . . . . . 23
a) General Properties of K~(r, r') . . . . . . . . . . . . . . . . . . 23
b) Density of States . . . . . . . . . . . . . . . . . . . . . . . . 25
c) Remark on Strong-Coupling Superconductors . . . . . . . . . . . . 27
.4 Calculation of the Kernel from Classical Mechanics . . . . . . . . . . . 28
a) Quantum-Mechanical and Classical Correlation Function ....... 28
b) General Properties of the Kernel Confirmed . . . . . . . . . . . . . 3l
c) Distribution Function in Phase Space . . . . . . . . . . . . . . . 32
d) Electrons in a Crystal . . . . . . . . . . . . . . . . . . . . . . 33
.5 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . 37
)~ General Formulations . . . . . . . . . . . . . . . . . . . . . . 73
b) Restrictions on Scattering Probability and Cross Sections. General
Properties of the Kernel Confirmed . . . . . . . . . . . . . . . . 14
.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 44
a) Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
b) Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
.7 Proofs for Free Electrons . . . . . . . . . . . . . . . . . . . . . . 49
a) No Impurities Present . . . . . . . . . . . . . . . . . . . . . . 50
b) Impurity Averaged Green's Function . . . . . . . . . . . . . . . . 54
c) Impurity Averaged Kernel . . . . . . . . . . . . . . . . . . . . 95
d) Boundary Conditions for Specular Reflexion . . . . . . . . . . . . 16
.8 Proofs for Electrons in a Periodic Potential . . . . . . . . . . . . . . 63
a) General Formulations . . . . . . . . . . . . . . . . . . . . . . 36
b) Magnetic Field and Impurities Absent . . . . . . . . . . . . . . . 65
c) Quasi-Classical Approximation . . . . . . . . . . . . . . . . . . 69
d) Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . 37
regnirps stcarT nredoM scisyhP 65
2 Contents
.9 Paramagnetic Impurities . . . . . . . . . . . . . . . . . . . . . . 87
a) Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
b) Derivation of the Boltzmann Equation . . . . . . . . . . . . . . . 79
II. General Discussion. Approximation Methods . . . . . . . . . . . . . . . 84
.01 Survey of Chapter I . . . . . . . . . . . . . . . . . . . . . . . . 84
a) Gap Equation . . . . . . . . . . . . . . . . . . . . . . . . . 84
b) Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . 58
c) Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 88
.11 Uniqueness, Positiveness, and Boundedness of the Kernel. Transition
Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
a) Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
b) Positiveness and Boundedness . . . . . . . . . . . . . . . . . . . 93
c) Transition Temperature of a Single Metal . . . . . . . . . . . . . . 95
.21 Classical Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 99
a) Clean Metal . . . . . . . . . . . . . . . . . . . . . . . . . . 99
b) Metal with Impurities . . . . . . . . . . . . . . . . . . . . . . 102
.31 Ginzburg-Landau Approximation . . . . . . . . . . . . . . . . . . 401
a) Derivation of the Linearized Ginzburg-Landau Equation for Clean Metals 501
b) Derivation of the Linearized Ginzburg-Landau Equation for Metals
Containing Impurities . . . . . . . . . . . . . . . . . . . . . . 701
c) Generalized Ginzburg-Landau Equation . . . . . . . . . . . . . . 011
.41 Diffusion Approximation . . . . . . . . . . . . . . . . . . . . . . 411
a) Derivation and Discussion of the Diffusion Equation . . . . . . . . 411
b) General Properties of the Kernel . . . . . . . . . . . . . . . . . 711
c) Ginzburg-Landau-like Formulation . . . . . . . . . . . . . . . . 811
d) Elimination of the Cut-Off . . . . . . . . . . . . . . . . . . . . 321
.51 Ginzburg-Landau Approximation: Boundary Conditions . . . . . . . . 521
a) Surfaces, Lowest Order . . . . . . . . . . . . . . . . . . . . . . 621
b) Surfaces, Higher Orders (for Clean Metals only) . . . . . . . . . . . 921
c) Interfaces (for Clean Metals only) . . . . . . . . . . . . . . . . . 531
d) A Variational Principle . . . . . . . . . . . . . . . . . . . . . . 931
.61 Diffusion Approximation: Boundary Conditions . . . . . . . . . . . . 141
a) Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 241
b) Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 541
.71 Single Homogeneous Metal with Isotropic Impurity Scattering ...... 941
a) General Formulations . . . . . . . . . . . . . . . . . . . . . . 941
b) Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 051
c) Perturbation Treatment . . . . . . . . . . . . . . . . . . . . . 351
.81 Electric Current Density . . . . . . . . . . . . . . . . . . . . . . 951
a) General Formulations . . . . . . . . . . . . . . . . . . . . . . 951
b) Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 161
III. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
.91 The Bulk Critical Field He2 . . . . . . . . . . . . . . . . . . . . . 561
a) Ginzburg-Landau and Diffusion Approximation . . . . . . . . . . . 561
b) A General Theorem . . . . . . . . . . . . . . . . . . . . . . . 761
)c Special Examples . . . . . . . . . . . . . . . . . . . . . . . . 071
.02 The Surface Normal Critical Field Hcj - . . . . . . . . . . . . . . . . 571
a) Ginzburg-Landau and Diffusion Approximation . . . . . . . . . . . 571
b) General Theorems . . . . . . . . . . . . . . . . . . . . . . . . 671
c) Going beyond the Approximations . . . . . . . . . . . . . . . . . 771
Contents 3
.12 The Surface Parallel Critical Field 3eH . . . . . . . . . . . . . . . . 281
a) Ginzburg-Landau and Diffusion Approximation . . . . . . . . . . . 381
b) Going beyond the Approximations . . . . . . . . . . . . . . . . . 481
c) General Theorems Concerning the Electric Current Density ...... 191
.22 The Surface Parallel Critical Field, Hell,of Layers . . . . . . . . . . . . 491
a) Ginzburg-Landau and Diffusion Approximation . . . . . . . . . . . 591
b) Going beyond the Approximations . . . . . . . . . . . . . . . . . 891
c) Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
23. Several Metals in Electric Contact . . . . . . . . . . . . . . . . . . 208
a) General Remarks . . . . . . . . . . . . . . . . . . . . . . . . 208
b) Cooper Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 209
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
4 Frequently Used Symbols
Frequently Used Symbols
This Iist does not contain symbols which appear in a single section only. Single letters
are sometimes used in a different meaning than the one given below.
a thickness of a metallic film or layer
Bj boundary of a metal labeled j, in a sample consisting of several metals
1F ~- ~o . F2 /- o ~t Eqs. (17.41) and (17.49), Figs. 3 and 4
~o(G r') Green's function of normal electrons: Section 1
9 BCS coupling constant
go,(r, v; r') distribution function: Section 5, Eq. (10.5)
H~(r). Ho~(r) Section 14a
i(r) electric current density: Eq. (1.32), Section 18
K~(r, r') kernel in the gap equation (2.15), (10.1): Eqs. (2.16)0 (4.39), (10.4)
IK~ operator which corresponds to Ko~(r, r'):Eq. (11.17)
k wave number vector or classical momentum vector, in particular of con-
duction electrons
k parameter in Eq. (19.3)
I electronic mean free path: Eq. (12.15)
/tr transport length: Eq. (13.27)
m electronic mass
m* effective mass: Section 4d
mass tensor, reciprocal mass tensor: Section 4d
Jk, m jk
N, N(r), Nj density of (orbital) electronic states: Section 3 b, Eq. (10.3)
N operator which corresponds to N(r): Eq. (11.34)
n, n(r) impurity concentration (number per volume)
n surface normal unit vector (pointing to the exterior), interface normal
unit vector
p diffuseness of the reflexion of electrons by a surface: Eqs. (6.11), (10.24),
(15.42)
p' further parameter characterizing the reftexion properties of a surface:
Eq. (15.54)
R(r; uoy t, niv ) reflexion kernel: Section 6, Eqs. (10.16), (10.26)
T, T c temperature, transition temperature of a single metal or of metals in contact,
in the absence of a magnetic field: Sections llc, 23b
Tj(r; tuojV , I~kin) transmission kernel: Section 6b, Eq. (10.26)
tj, tj parameters characterizing the transmission properties of an interface:
Eqs. (15.73), (15.77)
volume (interior) of a metal labeled j, in a sample consisting of several
metals
velocity, modulus of ,v usually equal to Fermi velocity
v velocity vector, modulus usually equal to Fermi velocity
v(k) Eq. (4.40)
velocity vector of electron approaching a surface or interface (incoming
~ni~ ~out
electrons) or of electrons leaving a surface or interface (outgoing electrons)
7 Euler's constant: Eq. (11.43)
A(r) pair potential: Section la
distribution function: Eq. (t 1.22)
~(a, b) delta function of the direction of vectors: Eq. (4.14)
Frequently Used Symbols 5
Ejkl totally antisymmetric Ricci tensor, 321~ = -- 1
)n(~ Riemann's zeta function: Eqs. ,)03.31( ,)21.31( )45.31(
~o(T) Eq. )02.71(
G Eq. )41.21(
~(t,) momentum dependent energy, in particular of conduction electrons;
q(k) = 0: Fermi surface
0 scattering angle
eigenvalue of K~: Eq. )8.71(
;t Eq. )61.71(
2(T) Eq.(13.1); Eqs. (13.13) and (13.26) (Ginzburg-Landau approximation);
Eq. ,)13.41( Fig. 2 (diffusion approximation)
# chemical potential, or parameter in Eq. ,)3.91( or Eq. ,)33.41( Fig. 2
~,~,q dimensionless variables: Eqs. ,)31.12( )22.12(
Eq. )2.31(
~o(T) Eq. )11.31(
G Eq. )4.21(
scattering cross section; atr: Eq. (13.22); a~l): Eq. (13.20); tee): Eq. (13.40);
"O
al, Gp: Section 9b
da(O) da(v, )'v
differential scattering cross section
dO ' dO
~o unit of quantized flux: after Eq. (19.9) (symbol not always used with this
meaning)
Eq. ,)82.31( Fig. 1
~(n) digamma function: Eq. )33.31(
odd multiple of ~T: Section 1 b
fD
Debye frequency, cut-off frequency
(D D
dr vectorial line element
d2r vectorial surface or interface element, I n
d3r scalar volume element (> )0
dS surface element of the energy surface, )k(/~ = const, in k space, in particular
of the Fermi surface
dO element of solid angle, in particular with respect to the direction of v
gauge invariant gradient: Eqs. ,)61.5( )6.01(
Introduction
In 1964 de Gennes 1 initiated a method for the theoretical analysis
of second kind transitions between the normal and the superconducting
state of metallic samples. A correlation function played a central part
in this method. De Gennes' method was subsequently further developed
and applied to various situations by him and by others.
In the present review paper a report shall be given on published and
unpublished work done in this direction at G6ttingen during the last few
years. It is hoped that such a systematic presentation might be helpful
to other workers in this field. Though correlation functions play a minor
part in the formulation to be presented in this report, we want to retain
the name ~ of the correlation function". This baptization should
be regarded as an hommage to de Gennes.
The review paper is divided into three chapters. In Chapter I, the
general formulation is first developed on a somewhat intuitive basis
and afterwards as far as possible provedfrom the microscopic theory.
This chapter is mainly based upon papers by the senior author 2-6
which have been published in German. Chapter II is concerned with a
general discussion of the basic formulation and with the derivation 7-
of approximative formulations (mainly Ginzburg-Landau approxima-
tion 8 and diffusion approximation ,1 ,9 10). These approximative
schemes are useful for the analysis of special problems. In Chapter III
finally, the formulations developed in Chapter II are applied to the
calculation of critical magnetic fields for various geometries and to a
sample consisting of superposed layers of different metals.
The method of the correlation function may be characterized as a
semi-microscopic method. It is based upon the microscopic BCS picture
in its more flexible Gorkov version which, therefore, is surveyed in
Section 1 of Chapter I. This scheme is specialized to second kind transi-
tions in Section .2 The quantity of central importance in Gorkov's theory
is the pair potential, A(r). It vanishes in the normal state. It goes con-
tinuously to zero if a second kind transition is approached from the
superconducting side. Gorkov's self-consistency condition for the pair
potential ("gap equation") becomes a linear integral equation for the
pair potential, at a second kind transition. The main goal of the method
of the correlation function is the calculation of the kernel, K~(r, r'), enter-
ing into this integral equation. General properties of the kernel are
summarized in Section .3
Following de Gennes 1, the kernel is expressed by a correlation
function for electrons in Section .4 Arguments are put forward that, not
the quantum mechanical, but the classical correlation function has to
be used in this context. The classical correlation function may be derived
Introduction 7
from a distribution function in phase space. A Boltzmann equation for
the distribution function is suggested in Section .5 Whereas a strictly
classical treatment may be given only if the system is invariant under
time reversal (no magnetic field and no paramagnetic impurities present),
a simple and straightforward extension to the case of a magnetic field
is also suggested in Section .5 The Boltzmann equation should be valid
inside a homogeneous metal. It has to be supplemented by boundary
conditions for the distribution function on external surfaces and, if the
sample consists of several metals, on interfaces. The boundary conditions
follow immediately from the classical picture. They are formulated in
Section .6
The Boltzmann equation which was first suggested on intuitive
grounds is subsequently proved in Sections 7 (free electrons) and 8
(electrons in a periodic lattice) on the basis of the microscopic Gorkov
theory. Also arguments in favour of the special semi-microscopic
boundary condition for specular reflexion will be given. The analysis
is extended to the case of paramagnetic impurities (where the subsystem
of conduction electrons violates time reversal invariance) in Section .9
The reader not interested in the somewhat intricate proofs on the
basis of the microscopic theory is advised to skip Sections 7-9 and to
go immediately over to Chapter II. It is also possible to start reading
with Chapter II since the semi-microscopic formulation (consisting of
Boltzmann equation and boundary conditions), which was put forward
and derived in Chapter I, is summarized in Section 10 of Chapter II.
The early work reported in Chapter I was stimulated by discussions
with .G Eilenberger, J. Hajdu, .W Moormann, and E. .re16hcS
The G6ttingen work is restricted to second kind transitions. An
interesting extension to the general case with A (r) not being infinitesimally
small has been given by Eilenberger 11. It seems, however, an open
question how surfaces and interfaces may be incorporated into this
more general scheme.
eW should like to summarize what we regard the main advantages of
the method of the correlation function:
.1 A formulation usin 9 concepts of classical mechanics, like particle
orbits dna distribution functions, appeals to physical imagination.
.2 The microscopic theory, ni the presence of impurities, requires a
summation of contributions from infinitely many 9raphs. This summation
si done once for ever ni the proof of the Boltzmann equation.
.3 Surfaces and interfaces with stochastic elements seem not amenable
to a microscopic quantum-mechanical treatment, but stochastic elements
are easily incorporated into the semi-microscopic boundary conditions to
be used ni the method of the correlation function.
8 Introduction
As already said above, Chapter II starts in Section 01 with a summary
of the results obtained in the first chapter. Some general aspects of the
theory are discussed in Section .11 This section is partly based upon
work by H. J. Sommers. Classical particle orbits are exhibited in various
ways in Section .21
The general formulation is still rather clumsy if specific problems
shall be solved. Approximative schemes known from the literature are,
therefore, derived in the next sections. The Ginzburg-Landatl approxima-
tion 8 is valid for temperatures close to the transition temperature
(in the absence of a magnetic field). The linearized Ginzburg-Landau
equation, a second order linear differential equation for the pair potential,
A(r), is derived in Section .31 Boundary conditions which supplement
this equation are obtained 12 in Section .51 The diffusion approxima-
tion 9, 10 is, on the other hand, valid for metals with sufficiently short
electronic mean free path. The diffusion equation, a second order diffe-
rential equation for the kernel, K,~(r, r is derived in Section .41 Boundary
conditions are obtained 13 in Section .61 A special method 14 for
samples consisting of a single metal with isotropic impurity scattering
is exposed in Section .81 This method to some extent unites Ginzburg-
Landau and diffusion approximation.
General expressions for the electric current density, which goes
continuously to zero together with the pair potential in the transition,
are derived in Section .81
The derivation of boundary conditions shows that the customary
requirements for the validity of the Ginzburg-Landau approximation
(temperature close to transition temperature) or the diffusion approxima-
tion (electronic mean free path sufficiently short) are not sufficient in
finite samples. Also the linear dimensions of the samples, including radii
of curvature of surfaces and interfaces, have to be sufficiently big. A more
quantitative version of this condition is given in the respective sections.
Chapter III starts in Section 91 with a discussion of the bulk critical
magnetic field, He2. Critical magnetic fields of samples consisting of a
single metal in special geometries are studied in the next three sections:
the surface normal critical field, He(cid:127) of semi-infinite samples with plane
boundary and of layers with two plane parallel boundaries in Section 20;
the surface parallel critical field, H c ,3 of semi-infinite samples with plane
boundary in Section 21; the surface parallel critical field, toH ,I of plane
layers in Section 22. The methods of Chapter II are mainly used to derive
some corrections to the results obtained with the help of the Ginzburg-
Landau and diffusion approximation.
Samples consisting of several metals in electric contact are studied
in Section 23. Actually the analysis will be restricted to two plane super-
posed layers with sufficiently short electronic mean free path so that
noitcudortnI 9
the diffusion approximation is valid. Only the simplest case (so-called
Cooper limit) is studied regarding its transition temperature and its
critical magnetic fields He(cid:127) and lcH "I
Chapter III is to some extent based upon mostly unpublished work
by K. D. Harms, A. Helms, P. Richter, H. Schultens, H. J. Sommers,
and J. Steppeler.
All theoretical work done so far on samples consisting of a single
metal suggests the conclusion that every stochastic element in the orbital
motion of the electrons leads to an enhancement of critical magnetic fields
for second kind transitions. These stochastic elements may be located ni
the interior (impurities which scatter the electrons) and on the surface
(diffuseness of the scattering process). We quote the main examples for
the general statement:
.1 The bulk critical magnetic field, Hc2, for second kind transitions
grows with increasing concentration of impurities (Section 19).
2. Critical magnetic fields of finite samples (like Hc(cid:127) Ho3, and )II~H
become bigger with increasing diffuseness of the electron scattering by
surfaces (Sections 20-22). This influence is the less pronounced the shorter
the electronic mean free path in the metal .si It does not appear in the Ginz-
burg-Landau and diffusion approximations.
No convincing general proof for the enhancement of critical fields by
stochastic elements has so far been found.
Some work reported in this review paper has also been done by others,
in particular by de Gennes 10, 15 himself. But contributions by other
authors 16-21 should also be mentioned. In these papers some of the
G/Sttingen work on the influence of diffusely scattering surfaces on
critical magnetic fields was actually anticipated.
eW hsiw to thank .srM .I sredifL dna .syhP-.lpiD .H .J sremmoS rof their ecnatsissa
ni gnitcerroc eht .sfoorp
I. General Theory
.1 Survey of Microscopic Superconductivity Theory
a) Cooperative Pairin 9
The peculiar behaviour of superconducting metals is believed to be a
consequence of the state of its conduction electrons. In the normal state
of a metal, the electrons, at least approximately, form a free Fermi gas.
But electron pairs appear whenever the metal becomes superconducting
(BCS theory 22). The pairs are Bosons. They condensate in macroscopic
number into a single one-pair state. But the pair picture should not be
strained since the spatial extension of a pair (as given by one of the
correlation lengthes 23 occurring in superconductivity theory) is big
as compared to the average distance between two pairs.
Pairing requires an attractive interaction between electrons which
is due to a polarization of the lattice or, in quantum-mechanical language,
to an exchange of phonons between the two members of a pair 22.
This phonon-mediated attraction is almost pointlike in space and
strongly retarded in time since the electron velocity at the Fermi surface
(the Fermi velocity) is big as compared to the phonon velocity: a virtual
phonon is dropped b'y an electron at some position and does not move
much until it is picked up some time later by another electron. The
repulsive Coulomb interaction between the electrons is shielded and,
therefore, also practically pointlike. The net interaction is attractive in
metals which may become superconducting at sufficiently low temper-
atures.
Pairing alone does not suffice. The pairs have to assemble in a
single one-pair state. This condensation, which is a cooperative pheno-
menon, may be understood as the result of the action of a kind of mean
field. This mean field was introduced by Gorkov 24 into his flexible
version of the microscopic theory of superconductivity. Various names
have been given to it: pair potential, gap function, order parameter.
The pair potential is produced by the electrons already paired. On the
other hand, it forces the pairs into a single one-pair state. There would
be no cooperative pair potential without the primary attractive electron-
electron interaction which is due to the exchange of phonons.
In order to turn to a more quantitative formulation of the theory
we first introduce the function
G~,(r, r'): = <tv*,(r')tp~(r)>, )1.1(
where the angular brackets on the righthand side denote an average
with respect to the grand canonical ensemble. The symbols in the bracket
are, in second quantization language, operators for the creation of an
