Theory of Fluctuations in Superconductors

243 Pages·2005·0.989 MB·English

by  Anatoly Larkin, Andrei Varlamov

Theory of Fluctuations in Superconductors
Anatoly Larkin Andrei Varlamov
July 28, 2002
Contents
0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
I Ginzburg-Landau formalism 10
1 Introduction 11
2 Thermodynamics 16
2.1 1.1 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . 16
2.1.1 GL functional . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 GL equations . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 Heat capacity jump . . . . . . . . . . . . . . . . . . . . 18
2.2 Fluctuation contribution to heat capacity . . . . . . . . . . . . 19
2.2.1 Zero dimensionality: the exact solution. . . . . . . . . . 19
2.2.2 Arbitrary dimensionality: case T ≥ T . . . . . . . . . . 20
c
2.2.3 Arbitrary dimensionality: case T < T . . . . . . . . . . 22
c
2.3 Fluctuation diamagnetism . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Qualitative preliminaries. . . . . . . . . . . . . . . . . . 23
2.3.2 Zero-dimensional diamagnetic susceptibility. . . . . . . 26
2.3.3 GL treatment of fluctuation magnetization. . . . . . . 28
2.4 Fluctuation contribution to heat capacity in magnetic field . . 35
2.5 Ginzburg-Levanyuk criterion . . . . . . . . . . . . . . . . . . . 35
2.6 Scaling and renormalization group . . . . . . . . . . . . . . . . 38
2.7 Effect of fluctuations on superfluid density and critical tempe-
rature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.7.1 Superfluid density. . . . . . . . . . . . . . . . . . . . . 47
2.7.2 Fluctuation shift of the critical temperature. . . . . . . 50
2.7.3 Fluctuation shift of the H (T) . . . . . . . . . . . . . 51
c2
2.7.4 Fluctuations of magnetic field . . . . . . . . . . . . . . 51
1
3 Ginzburg-Landau formalism. Transport 54
3.1 Time dependent GL equation . . . . . . . . . . . . . . . . . . 55
3.2 Paraconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 General expression for paraconductivity . . . . . . . . . . . . . 60
3.4 Fluctuation conductivity of layered superconductor . . . . . . 62
3.4.1 In-plane conductivity. . . . . . . . . . . . . . . . . . . . 63
3.4.2 Out-of plane conductivity. . . . . . . . . . . . . . . . . 65
3.4.3 Analysis of the general expressions. . . . . . . . . . . . 65
3.5 Paraconductivity of nanotubes . . . . . . . . . . . . . . . . . . 67
3.5.1 Zero magnetic field . . . . . . . . . . . . . . . . . . . . 69
3.5.2 Non-zero magnetic field . . . . . . . . . . . . . . . . . 71
3.6 Magnetic field angular dependence of paraconductivity . . . . 74
3.7 Nonlinear paraconductivity in strong electric field . . . . . . . 77
4 Fluctuations in vortex structures 78
4.1 Vortex lattice and magnetic flux resistivity . . . . . . . . . . . 78
4.2 Collective pinning . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Correlation length . . . . . . . . . . . . . . . . . . . . 82
4.2.2 Critical current . . . . . . . . . . . . . . . . . . . . . . 83
4.2.3 Collective pinning in other systems . . . . . . . . . . . 85
4.3 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4 The melting of the vortex lattice . . . . . . . . . . . . . . . . 88
II Basic notions of the microscopic theory 92
5 Microscopic derivation of the TDGL equation 93
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 The Cooper channel of electron-electron interaction . . . . . . 94
5.3 Superconductor with impurities . . . . . . . . . . . . . . . . . 99
5.3.1 Account for impurities. . . . . . . . . . . . . . . . . . . 99
5.3.2 Propagator. . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Microscopic derivation of the Ginzburg Landau functional . . 103
6 Microscopic theory of fluctuation conductivity of layered su-
perconductor 109
6.1 Qualitative discussion of different fluctuation contributions . . 109
6.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Aslamazov-Larkin contribution . . . . . . . . . . . . . . . . . 114
6.3 Contributions from fluctuations of the density of states . . . . 117
6.4 Maki-Thompson contribution . . . . . . . . . . . . . . . . . . 119
2
6.5 Fluctuations in the ultra-clean case [190] . . . . . . . . . . . . 125
6.6 Nonlinear fluctuation effects [14] . . . . . . . . . . . . . . . . . 129
6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
III Manifestation of fluctuations in various proper-
ties 134
7 [Magnetoconductivity] The effects of fluctuations on magne-
toconductivity [110, 141, 169, 170] 136
8 Fluctuations far from T 142
c
8.1 Fluctuation magnetic susceptibility far from transition [29]. . . 142
8.2 Fluctuation magnetoconductivity far from transition [177]. . . 143
8.3 Fluctuations in magnetic fields near H (0) [183]. . . . . . . . 144
c2
8.3.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . 144
8.3.2 Magnetization: one loop approximation . . . . . . . . . 147
8.3.3 Magnetization: two loop approximation . . . . . . . . . 148
8.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.4 The effect of fluctuations on the Hall conductivity[189] . . . . 151
9 DOS and tunneling 154
9.1 Density of states [107]. . . . . . . . . . . . . . . . . . . . . . . 154
9.2 The effect of fluctuations on the tunnel current [194]. . . . . . 157
9.3 Fluctuation tunneling anomaly in superconductor above pa-
ramagnetic limit . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10 Optical conductivity 161
11 Heat transport 165
11.1 Thermoelectric power above the superconducting transition
[218, 216] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.2 Thermal conductivity. . . . . . . . . . . . . . . . . . . . . . . 169
11.3 Nernst and Ettinghausen effects . . . . . . . . . . . . . . . . . 169
12 Sound attenuation 170
13 Spin susceptibiity and NMR 171
13.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.2 Spin Susceptibility[223, 191]. . . . . . . . . . . . . . . . . . . . 172
13.3 Relaxation Rate[224, 225, 219, 191]. . . . . . . . . . . . . . . . 173
3
IV Fluctuations in nanostructures and unconven-
tional superconducting systems 176
14 Fluctuations in superconducting nanodrops 177
14.1 Ultrasmall superconducting grains . . . . . . . . . . . . . . . . 177
14.2 Superconducting drops in the system with quenched disorder
(the method of optimal fluctuation). . . . . . . . . . . . . . . 180
14.2.1 The smearing of the superconducting transition by the
quenched disorder[270] . . . . . . . . . . . . . . . . . . 181
14.2.2 Formation of the superconducting drops in magnetic
fields H > H (0). . . . . . . . . . . . . . . . . . . . . 184
c2
14.3 The exponential DOS tail in superconductor with quenched
disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
14.4 Josephson coupled superconducting drops . . . . . . . . . . . 190
14.5 Classical phase transition in granular superconductors . . . . . 192
14.6 Quantum phase transition in granular superconductors . . . . 195
14.6.1 Coulombsuppressionofsuperconductivityinthearray
of tunnel coupled granula . . . . . . . . . . . . . . . . 195
14.6.2 Superconducting grains in the normal metal matrix . . 196
14.6.3 Phase transition in disordered superconducting film in
strong magnetic field . . . . . . . . . . . . . . . . . . . 199
15 Phase fluctuations in 2D systems 203
15.1 Phase fluctuations in 2D systems . . . . . . . . . . . . . . . . 203
15.2 Kosterlitz-Thouless conductivity . . . . . . . . . . . . . . . . . 205
16 Phase slip events 206
16.1 Phase slip events in JJ. . . . . . . . . . . . . . . . . . . . . . . 206
16.2 Phase-slip events in 1D systems . . . . . . . . . . . . . . . . . 206
16.3 3. Quantum phase slip events in nanorings. . . . . . . . . . . . 209
17 S-I transition 210
17.1 Quantum phase transition . . . . . . . . . . . . . . . . . . . . 210
17.2 3D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
17.3 2D superconductors . . . . . . . . . . . . . . . . . . . . . . . 216
17.3.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . 216
17.3.2 Boson mechanism of the T suppression. . . . . . . . . 217
c
17.3.3 Fermion mechanism of T suppression. . . . . . . . . . 218
c
18 Fluctuations in HTS 220
18.1 The specifics of the D-pairing . . . . . . . . . . . . . . . . . . 220
4
18.2 Phase fluctuations in the underdoped phase of HTS. . . . . . . 220
19 Conclusions 221
20 Acknowledgments 224
0.1 Preface
In any business, it is important to have certain corner stones, meaning the
results, which rise no doubts in their correctness. Such a corner stone for the
physics of critical phenomena was provided by Onsager in 1944 by his exact
solution of 2D Ising model.
We now live in an artificial world and it is often easier to make an expe-
riment on a two-dimensional object than a three-dimensional one. Moreover,
the theory of phase transitions for Ising model is now used not only in natu-
ral sciences, but also in unnatural ones, such as money counting (Bornholdt,
Wagner, 2001). Forty years ago physics still was a natural science, which stu-
died our three dimensional world. So, the majority of physicists 40 years ago
knew neither Ising model nor the Onsager solution and believed in the Lan-
dau theory. Landau himself was among the first who clearly understood that
the Onsager solution gives an example of what is happening close to critical
points in real materials, violating the orthodox Landau theory. Therefore,
the Onsager solution signified the problem of determining the singularities
at the phase transition points.
The physicists started to climb the peak of phase transitions from two
directions. The first direction was to utilize Onsager’s exact solution to find
the general laws of phase transitions. The hypothesis of universality was for-
mulated (Vaks, Larkin, 1965; Kadanoff, 1966). According to this hypothesis
all physical systems are divided into classes of a different symmetry of the
order parameter, and the critical behavior for all the systems belonging to
the same class are, essentially, identical. Even more important was the hy-
pothesis of scaling (Patashinskii, Pokrovsky, 1966; Kadanoff, 1966; Gribov,
Migdal, 1968; Polyakov, 1968). According to this hypothesis all physical pa-
rameters close to phase transition point are determined by a single correla-
tion length which increases when the system approaches the phase transition
point. The scaling hypothesis enabled Halperin and Hohenberg (1969) to
find singularities of kinetic coefficients.
5
The second trail to the peak began from Landau theory and included
systematic analysis of the fluctuation corrections to it. Lee and Yang (1958)
considered a weakly non-ideal Bose-gas, taking the interaction into account
by means of perturbation theory. They have found that a first order tran-
sition occurs in Bose-gas. It became clear pretty soon that this result was
an artifact of the perturbation theory, which does not work close the phase
transition point (Landau and his group). The fluctuation corrections to Lan-
dau’s theory not very close to the transition point, where these corrections
are small was found by Levanyuk (1959).
The methods of quantum field theory allowed to segregate the most di-
vergent fluctuation contributions appearing from so called parquet diagrams
and sum up these contributions. As the result singularity close phase tran-
sition in a real three dimensional system was found exactly. It was found
that in uni-axial ferroelectrics and ferromagnets with dipole-dipole interac-
tions the specific heat has the logarithmic singularity (Larkin, Khmelnitskii,
1969). This paper had two important appendices of a methodical nature. In
the first appendix was obtained the same result using the method of multipli-
cative renormalization group. This method is equivalent to that of parquet
diagrams summation, but it is simpler and found later applications in dif-
ferent branches of condensed matter theory.
At the same time Di Castro and Iona-Lasinio (1969), applied the renor-
malization group to the theory of phase transitions and showed that the fixed
point of RG equation at a finite coupling constant leads to scaling.
In the second appendix of (Larkin, Khmelnitskii, 1969) the effect of the
symmetry of the order parameter on the singularity at the transition point of
a non-physical 4D system was considered. For the order parameter being an
n-componentvectorthespecificheathasmorecomplexsingularitydepending
on the number of the order parameter components n.
Next step in study of the cross-over from the logarithmic laws in four
dimensional case to power-laws at any lower dimension was performed for
dimension D = 4 − (cid:15) by Wilson and Fisher (1972). Their single-loop re-
normalization group approach gave the critical indices in the leading order
in (cid:15). Next order corrections, ∼ (cid:15)2, were calculated by Wilson (1972) and
Abrahams and Tzuneto (1973).
After these works the scaling hypothesis became a theory. Papers by
Wilson (1971) signified the passage through the summit. Significance of the
Wilson’s theory goes far beyond the physics of the phase transitions. He
taught all of us that the renormalization of action in the path integral should
better be done, while before people renormalized the Green functions. After
that, the renormalization group became a real working tool, which gave jobs
to many theorists.
6
Another important summit of the statistical physics which was conquis-
ted in the second part of XX century is the theory of superconductivity.
Between the discovery by Kamerling-Onnes in 1911 of the phenomenon of
superconductivity [?] and the formulation in 1957 of the microscopic theory
[?] passed almost half a century. Let us remind that the creation of the
theory of superfluidity in liquid helium took only four years.
First theories of superconductivity were phenomenological. So in 1934
brothers F. and ? London write down the equations for the electrodynamics
of superconductor which explained the Meissner-Oxenfeld effect: complete
expelling of the magnetic flux from bulk superconductor. This discovery was
especially important in superconductivity and namely this effect demonstra-
ted that superconductivity is a new state of matter. In 193? Landau and
Paierls created the theory of intermediate state. In 1950 Ginzburg and Lan-
dau applied to superconductivity the ideas of the Landau phenomenological
theory of the second order phase transitions. They did not have a clear un-
derstanding about the physical origin of the order parameter of such phase
transition and just supposed that it is described by some charged field. This
permitted them to derive the equations which described almost all known
at the moment phenomena in superconductors. According to De Gennes
the Ginzburg-Landau theory is the bright example of manifestation of the
physical intuition. The greatest success of the Ginzburg-Landau theory be-
came the Abrikosov’s explanation of the Shubnikov’s phase where peacefully
coexist the superconductivity and the magnetic field. Abrikosov demonstra-
ted that the magnetic field can penetrate in superconductor in the form of
vortices ordering in the perfect lattice.
In purpose to create the theory of superconductivity was enough to de-
monstrate how the gap appears in the excitations spectrum. Such gap does
not exist in the spectrum of ideal electron gas. Perturbation account for the
interaction too did not result in the gap opening. But such approximation
was not really justified, the interelectron interaction in metals is not small.
So it was necessary to refuse from the ideas of the perturbation approach
and to work out the theory of strongly interacting electron system in metal.
At this time the similar problems aroused in the high energy physics and
they had been successfully resolved in the frameworks of the new Feynman
diagrammatic technique. These ideas were transferred to statistical physics
and soon the methods of the quantum field theory were applied to problems
of condensed matter theory.
At the beginning of 50-es the isotopic effect was discovered in super-
conductors and it became clear that for the theory of superconductivity is
important not only the electron system but the phonon one too. Migdal
constructed the theory of strong electron-phonon interaction but even in this
7
way did not succeed to find the gap in the excitation spectrum. The mat-
ter of fact that in all these efforts the important phenomenon of the Bose
condensation had been lost. It was difficult to involve it in the theoretical
models: electrons in metals obey the Fermi statistics and, in view of their
strong Coulomb repulsion, it seemed was no way to unify them in composed
Bose particles.
In 1957 the young student of ???? University L.Cooper found that it is
enough to have the week attraction between the particles of the degenerated
Fermi liquid to get the formation of the bounded states, called now Cooper
pairs. Soon after this discovery Bardeen, Cooper and Schrieffer proposed the
microscopic theory of superconductivity as the theory of the Cooper pairs
Bose condensation.
AlmostatthesametimeinRussiaBogolyubovsucceededtosolvethepro-
blem of superconductivity by the method of the approximate second quan-
tization and a little bit later Gorkov proposed the solution of the problem
in the frameworks of the Green functions formalism. This method turned
out very effective and it gave to Gorkov possibility to demonstrate that the
phenomenological Ginzburg-Landau equations follow from the BCS theory
in the limit T → T . Both GL and BCS theories are done in the mean field
c
approximation (MFA). Here it is necessary to mention that usually MFA per-
mits to get only the qualitative picture of phenomenon. Fortunately in the
case of superconductivity this method works quantitatively. It was Ginzburg
who demonstrated(1960) that in clean bulk superconductors the fluctuation
phenomena become important only in very narrow (∼ 10−12K) region in
the vicinity of the transition temperature. Aslamazov and Larkin (1968)
demonstrated that the fluctuation region in dirty superconducting films is
determined by the resistance per film unit square and could be much wider
than in bulk samples. Even more important that they demonstrated the
presence of fluctuation effects even beyond the critical region and not only in
thermodynamic but in kinetic characteristics of superconductor too. They
have discovered the phenomenon which today is called paraconductivity: the
decrease of the resistance of superconductor still at T > T , in the nor-
c
mal phase. Simultaneously this phenomenon was experimentally observed
by Glover and was found in perfect agreement with the Aslamazov-Larkin
theory. Since this time the variety of fluctuation effects were discovered and
investigated in different phenomena and has been studied today.
The theory of phase transitions and the theory of superconductivity are
two summits of the statistical physics. The theory of superconducting fluc-
tuationsisthemountainmountain-rangeconnectingthesesummits. Wehope
that the book presented would serve as the guide along this mountain-range
The first two parts are written in details and can serve as the textbook (first
8
in phenomenology, the second in microscopic theory). Two last parts sooner
can be considered as the hand book and the guide in numerous recent works
of the theory of fluctuations.
9