Theory of Fluctuations in Superconductors

Theory of Fluctuations in Superconductors Anatoly Larkin

Andrei Varlamov

July 28, 2002

Contents 0.1

I

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ginzburg-Landau formalism

10

1 Introduction 2

5

11

Thermodynamics 16 2.1 1.1 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . 16 2.1.1 GL functional . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.2 GL equations . . . . . . . . . . . . . . . . . . . . . . . 18

2.2

2.3

2.4 2.5 2.6 2.7

2.1.3 Heat capacity jump . . . . . . . . . . . . . . . . . . . Fluctuation contribution to heat capacity . . . . . . . . . . . 2.2.1 Zero dimensionality: the exact solution. . . . . . . . . 2.2.2 Arbitrary dimensionality: case T ≥ Tc . . . . . . . . . 2.2.3 Arbitrary dimensionality: case T < Tc . . . . . . . . . Fluctuation diamagnetism . . . . . . . . . . . . . . . . . . . 2.3.1 Qualitative preliminaries. . . . . . . . . . . . . . . . . 2.3.2 Zero-dimensional diamagnetic susceptibility. . . . . . 2.3.3 GL treatment of fluctuation magnetization. . . . . . Fluctuation contribution to heat capacity in magnetic field . Ginzburg-Levanyuk criterion . . . . . . . . . . . . . . . . . . Scaling and renormalization group . . . . . . . . . . . . . . . Effect of fluctuations on superfluid density and critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Superfluid density. . . . . . . . . . . . . . . . . . . . 2.7.2 Fluctuation shift of the critical temperature. . . . . . 2.7.3 Fluctuation shift of the Hc2 (T ) . . . . . . . . . . . . 2.7.4 Fluctuations of magnetic field . . . . . . . . . . . . .

1

. . . . . . . . . . . .

18 19 19 20 22 23 23 26 28 35 35 38

. . . . .

47 47 50 51 51

3 Ginzburg-Landau formalism. Transport 3.1 Time dependent GL equation . . . . . . . . . . . . . . 3.2 Paraconductivity . . . . . . . . . . . . . . . . . . . . . 3.3 General expression for paraconductivity . . . . . . . . . 3.4 Fluctuation conductivity of layered superconductor . . 3.4.1 In-plane conductivity. . . . . . . . . . . . . . . . 3.4.2 Out-of plane conductivity. . . . . . . . . . . . . 3.4.3 Analysis of the general expressions. . . . . . . . 3.5 Paraconductivity of nanotubes . . . . . . . . . . . . . . 3.5.1 Zero magnetic field . . . . . . . . . . . . . . . . 3.5.2 Non-zero magnetic field . . . . . . . . . . . . . 3.6 Magnetic field angular dependence of paraconductivity 3.7 Nonlinear paraconductivity in strong electric field . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

54 55 57 60 62 63 65 65 67 69 71 74 77

4 Fluctuations in vortex structures 4.1 Vortex lattice and magnetic flux resistivity 4.2 Collective pinning . . . . . . . . . . . . . . 4.2.1 Correlation length . . . . . . . . . 4.2.2 Critical current . . . . . . . . . . . 4.2.3 Collective pinning in other systems 4.3 Creep . . . . . . . . . . . . . . . . . . . . 4.4 The melting of the vortex lattice . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

78 78 80 82 83 85 85 88

II

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

Basic notions of the microscopic theory

92

5 Microscopic derivation of the TDGL equation 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Cooper channel of electron-electron interaction . . . . 5.3 Superconductor with impurities . . . . . . . . . . . . . . . 5.3.1 Account for impurities. . . . . . . . . . . . . . . . . 5.3.2 Propagator. . . . . . . . . . . . . . . . . . . . . . . 5.4 Microscopic derivation of the Ginzburg Landau functional

. . . . . .

6 Microscopic theory of fluctuation conductivity of layered perconductor 6.1 Qualitative discussion of different fluctuation contributions 6.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . 6.2 Aslamazov-Larkin contribution . . . . . . . . . . . . . . . 6.3 Contributions from fluctuations of the density of states . . 6.4 Maki-Thompson contribution . . . . . . . . . . . . . . . .

su109 . . 109 . . 112 . . 114 . . 117 . . 119

2

. . . . . .

93 93 94 99 99 101 103

6.5 6.6 6.7

Fluctuations in the ultra-clean case [190] . . . . . . . . . . . . 125 Nonlinear fluctuation effects [14] . . . . . . . . . . . . . . . . . 129 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

III Manifestation of fluctuations in various properties 134 7

[Magnetoconductivity] The effects of fluctuations on magnetoconductivity [110, 141, 169, 170] 136

8 Fluctuations far from Tc 8.1 Fluctuation magnetic susceptibility far from transition [29]. 8.2 Fluctuation magnetoconductivity far from transition [177]. 8.3 Fluctuations in magnetic fields near Hc2 (0) [183]. . . . . . 8.3.1 Conductivity . . . . . . . . . . . . . . . . . . . . . 8.3.2 Magnetization: one loop approximation . . . . . . . 8.3.3 Magnetization: two loop approximation . . . . . . . 8.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . 8.4 The effect of fluctuations on the Hall conductivity[189] . . 9 DOS and tunneling 9.1 Density of states [107]. . . . . . . . . . . . . . . . . . . . 9.2 The effect of fluctuations on the tunnel current [194]. . . 9.3 Fluctuation tunneling anomaly in superconductor above ramagnetic limit . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

142 . 142 . 143 . 144 . 144 . 147 . 148 . 150 . 151

154 . . . 154 . . . 157 pa. . . 160

10 Optical conductivity

161

11 Heat transport 11.1 Thermoelectric power above the superconducting [218, 216] . . . . . . . . . . . . . . . . . . . . . . 11.2 Thermal conductivity. . . . . . . . . . . . . . . . 11.3 Nernst and Ettinghausen effects . . . . . . . . . .

165

12 Sound attenuation

transition . . . . . . . 165 . . . . . . . 169 . . . . . . . 169 170

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 unconventional superconducting systems 176 14 Fluctuations in superconducting nanodrops 14.1 Ultrasmall superconducting grains . . . . . . . . . . . . . . . 14.2 Superconducting drops in the system with quenched disorder (the method of optimal fluctuation). . . . . . . . . . . . . . 14.2.1 The smearing of the superconducting transition by the quenched disorder[270] . . . . . . . . . . . . . . . . . 14.2.2 Formation of the superconducting drops in magnetic fields H > Hc2 (0) . . . . . . . . . . . . . . . . . . . . 14.3 The exponential DOS tail in superconductor with quenched disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Josephson coupled superconducting drops . . . . . . . . . . 14.5 Classical phase transition in granular superconductors . . . . 14.6 Quantum phase transition in granular superconductors . . . 14.6.1 Coulomb suppression of superconductivity in the array of tunnel coupled granula . . . . . . . . . . . . . . . 14.6.2 Superconducting grains in the normal metal matrix . 14.6.3 Phase transition in disordered superconducting film in strong magnetic field . . . . . . . . . . . . . . . . . .

177 . 177 . 180 . 181 . 184 . . . .

188 190 192 195

. 195 . 196 . 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 16.1 Phase slip events in JJ. . . . . . . . . . . . . . . . . . . . . . 16.2 Phase-slip events in 1D systems . . . . . . . . . . . . . . . . 16.3 3. Quantum phase slip events in nanorings. . . . . . . . . . .

206 . 206 . 206 . 209

17 S-I transition 17.1 Quantum phase transition . . . . . . . . . . . . 17.2 3D case . . . . . . . . . . . . . . . . . . . . . . 17.3 2D superconductors . . . . . . . . . . . . . . . 17.3.1 Preliminaries. . . . . . . . . . . . . . . . 17.3.2 Boson mechanism of the Tc suppression. 17.3.3 Fermion mechanism of Tc suppression. .

210 . 210 . 212 . 216 . 216 . 217 . 218

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

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 experiment 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 natural sciences, but also in unnatural ones, such as money counting (Bornholdt, Wagner, 2001). Forty years ago physics still was a natural science, which studied our three dimensional world. So, the majority of physicists 40 years ago knew neither Ising model nor the Onsager solution and believed in the Landau 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 formulated (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 hypothesis of scaling (Patashinskii, Pokrovsky, 1966; Kadanoff, 1966; Gribov, Migdal, 1968; Polyakov, 1968). According to this hypothesis all physical parameters close to phase transition point are determined by a single correlation 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 transition 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 Landau’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 divergent fluctuation contributions appearing from so called parquet diagrams and sum up these contributions. As the result singularity close phase transition in a real three dimensional system was found exactly. It was found that in uni-axial ferroelectrics and ferromagnets with dipole-dipole interactions 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 multiplicative renormalization group. This method is equivalent to that of parquet diagrams summation, but it is simpler and found later applications in different branches of condensed matter theory. At the same time Di Castro and Iona-Lasinio (1969), applied the renormalization 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-component vector the specific heat has more complex singularity depending 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 −  by Wilson and Fisher (1972). Their single-loop renormalization group approach gave the critical indices in the leading order in . Next order corrections, ∼ 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 conquisted 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 demonstrated that superconductivity is a new state of matter. In 193? Landau and Paierls created the theory of intermediate state. In 1950 Ginzburg and Landau applied to superconductivity the ideas of the Landau phenomenological theory of the second order phase transitions. They did not have a clear understanding 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 became the Abrikosov’s explanation of the Shubnikov’s phase where peacefully coexist the superconductivity and the magnetic field. Abrikosov demonstrated 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 demonstrate 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 superconductors 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 matter 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. Almost at the same time in Russia Bogolyubov succeeded to solve the problem of superconductivity by the method of the approximate second quantization 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 → Tc . Both GL and BCS theories are done in the mean field approximation (MFA). Here it is necessary to mention that usually MFA permits 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 (∼ 10−12 K) 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 > Tc , in the normal 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 fluctuations is the mountain mountain-range connecting these summits. We hope 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

Part I Ginzburg-Landau formalism

10

Chapter 1 Introduction A major success of low temperature physics was achieved with the introduction by Landau of the notion of quasiparticles. According to his hypothesis, the properties of many body interacting systems at low temperatures are determined by the spectrum of some low energy, long living excitations (quasiparticles). Another milestone of many body theory is the Mean Field Approximation (MFA), which permitted achieving considerable progress in the theory of phase transitions. Phenomena which cannot be described by the quasiparticle method or by MFA are usually called fluctuations. The BCS theory of superconductivity is a bright example of the use of both the quasiparticle description and MFA. The success of the BCS theory for traditional superconductors was determined by the fact that fluctuations give small corrections with respect to the MFA results. During the first half of the century after the discovery of superconductivity the problem of fluctuation smearing of the superconducting transition was not even considered. In bulk samples of traditional superconductors the critical temperature Tc sharply divides the superconducting and the normal phases. It is worth mentioning that such behavior of the physical characteristics of superconductors is in perfect agreement both with the GinzburgLandau (GL) phenomenological theory (1950) [1] and the BCS microscopic theory of superconductivity (1957)[2]. The characteristics of high temperature and organic superconductors, low dimensional and amorphous superconducting systems studied today, strongly differ from those of the traditional superconductors discussed in textbooks. The transitions turn out to be much more smeared out. The appearance of superconducting fluctuations above critical temperature leads to precursor effects of the superconducting phase occurring while the system is still in the normal phase, sometimes far from Tc . The conductivity, the heat capacity, the diamagnetic susceptibility, the sound attenuation, etc. may increase 11

considerably in the vicinity of the transition temperature. The first numerical estimation of the fluctuation contribution to the heat capacity of a superconductor in the vicinity of Tc was done by Ginzburg in 1960 [3]. In that paper he showed that superconducting fluctuations increase the heat capacity even above Tc . In this way fluctuations change the temperature dependence of the specific heat in the vicinity of critical temperature, where, in accordance with the phenomenological GL theory of second order phase transitions, a jump should take place. The range of temperatures where the fluctuation correction to the heat capacity of a bulk clean conventional superconductor is relevant was estimated by Ginzburg 1 to be  4 δT Tc Gi = ∼ ∼ 10−12 ÷ 10−14 , (1.1) Tc EF where EF is the Fermi energy. The correction occurs in a temperature range δT many orders of magnitude smaller than that accessible in real experiments. In the 1950s and 60s the formulation of the microscopic theory of superconductivity, the theory of type-II superconductors and the search for high-Tc superconductivity attracted the attention of researchers to dirty systems, superconducting films and filaments. In 1968, in the papers of L. G. Aslamazov and A. I. Larkin [5], K.Maki [6] and a little later in the paper of R.S.Thompson [7] the fundaments of the microscopic theory of fluctuations in the normal phase of a superconductor in the vicinity of the critical temperature were formulated. This microscopic approach confirmed Ginzburg’s evaluation [3] for the width of the fluctuation region in a bulk clean superconductor. Moreover, was found that the fluctuation effects increase drastically in thin dirty superconducting films and whiskers. In the cited papers was demonstrated that fluctuations affect not only the thermodynamical properties of superconductor but its dynamics too. Simultaneously the fluctuation smearing of the resistive transition in bismuth amorphous films was experimentally found by Glover [8], and it was perfectly fitted by the microscopic theory. In the BCS theory [2] only the Cooper pairs forming a Bose-condensate are considered. Fluctuation theory deals with the Cooper pairs out of the condensate. In some phenomena these fluctuation Cooper pairs behave similarly to quasiparticles but with one important difference. While for the 1

The expression for the width of the strong fluctuation region in terms of the Landau phenomenological theory of phase transitions was obtained by Levanyuk [4]. So in the modern theory of phase transitions the relative temperature width of fluctuation region is called the Ginzburg-Levanyuk parameter Gi(D) , where D is the effective space dimensionality.

12

well defined quasiparticle the energy has to be much larger than its inverse life time, for the fluctuation Cooper pairs the ”binding energy“ E0 turns out to be of the same order. The Cooper pair life time τGL is determined by its decay into two free electrons. Evidently at the transition temperature the Cooper pairs start to condense and τGL = ∞. So it is natural to suppose from dimensional analysis that τGL ∼ ~/kB (T − Tc ). The microscopic theory confirms this hypothesis and gives the exact coefficient: τGL =

π~ . 8kB (T − Tc )

(1.2)

Another important difference of the fluctuation Cooper pairs from quasiparticles lies in their large size ξ(T ). This size is determined by the distance on which the electrons forming the fluctuation Cooper pair move away during the pair life-time τGL . In the case of an impure superconductor the electron motion is diffusive with the diffusion coefficient D ∼ vF2 τ (τ is the electron √ √ scattering time2 ), and ξd (T ) = DτGL ∼ vF τ τGL . In the case of a clean superconductor, where kB T τ  ~, impurity scattering does not affect any more the electron correlations. In this case the time of electron ballistic motion turns out to be less than the electron-impurity scattering time τ and is determined by the uncertainty principle: τbal ∼ ~/kB T. Then this time has to be used p in this case for determination of the effective size instead of τ : ξc (T ) ∼ vF ~τGL /kB T . In both cases the coherence length grows with the approach to the critical temperature as (T − Tc )−1/2 , and we will write it down in the unique way (ξ = ξc,d ): ξ ξ(T ) = √ , 

=

T − Tc . Tc

(1.3)

The microscopic theory in the case of an isotropic Fermi surface gives for ξ the precise expression:

2 ξ(D)

v2 τ 2 =− F D

  1 ~ 1 ~ 0 1 ψ( + ) − ψ( ) − ψ( ) , 2 4πkB T τ 2 4πkB T τ 2

(1.4)

where ψ(x) is the digamma function and D = 3, 2, 1 is the space dimensionality. In the clean (c) and dirty (d) limits: 2

Strictly speaking τ in the most part of future results should be understood as the electron transport scattering time τtr . Nevertheless, as it is well known, in the case of isotropic scattering these values coincide, so for sake of simplicity we will use hereafter the symbol τ.

13

r r ~vF 3 3 ξc = 0.133 = 0.74ξ0 , (1.5) kB Tc D D r r p ~vF l 3 3 ξd = 0.36 = 0.85 ξ0 l . (1.6) kB Tc D D Here l = vF τ is the electron mean free path and ξ0 = ~vF /π∆(0) is the conventional BCS definition of the coherence length of a clean superconductor at zero temperature. One can see that (1.5) and (1.6) coincide with the above estimations 3 . Finally it is necessary to recognize that fluctuation Cooper pairs can be really treated as classical objects, but these objects instead of Boltzmann particles appear as classical fields in the sense of Rayleigh-Jeans. This means that in the general Bose-Einstein distribution function only small energies E(p) are involved and the exponent can be expanded: n(p) =

1 exp( E(p) ) kB T

−1

=

kB T . E(p)

(1.7)

This is why the more appropriate tool to study fluctuation phenomena is not the Boltzmann transport equation but the GL equation for classical fields. Nevertheless at the qualitative level the of fluctuation Cooper pairs R treatment dD p as particles with the density n(D) = n(p) (2π~)D often turns out to be useful 4 . Below will be demonstrated both in the framework of the phenomenological Ginzburg-Landau theory and the microscopic BCS theory that in the vicinity of the transition E(p) = αkB (T − Tc ) +

 p2 1  2 2 = ~ /ξ (T ) + p2 . ∗ ∗ 2m 2m

(1.8)

3

Let us stress some small numerical difference between our Exp. (1.4) and the usual definition of the coherence length. We are dealing near the critical temperature, so the definition (1.4) is natural and permits us to avoid many numerical coefficients in further calculations. The cited coherence length ξ0 = ~vF /π∆(0) = 0.18~vF /kB Tc , as is evident, was introduced for zero temperature and an isotropic 3D superconductor. It is convenient to determine the coherence length also from the formula for the upper critical field: Hc2 (T ) = A(T )Φ0 /2πξ 2 (T ). A(Tc ) = 1, while its value at T = 0 depends on the impurities concentration. For the dirty case the appropriate value was found by K.Maki 3D [9] Ad (0) = 0.69, for the clean case by L.Gor’kov [327] A2D c (0) = 0.59, Ac (0) = 0.72. 4

This particle density is defined in the (D)-dimensional space. This means that it determines the normal volume density of pairs in 3D case, the density per square unit in 2D case and the number of pairs per unit length in 1D. The real three dimensional density n can be defined too: n = dD−3 n(D) , where d is the thickness of the film or wire.

14

Far from transition temperature the dependence n(p) turns out to be more sophisticated than (1.7), nevertheless one can always write it in the form   m ∗ kB T 2 ξ(T )p n(p) = ξ (T ) f . (1.9) ~2 ~ In classical field theory the notions of the particle distribution function n(p) (proportional to E −1 (p) in our case) and Cooper pair mass m∗ are poorly determined. At the same time the characteristic value of the Copper pair center of mass momentum can be defined and it turns out to be of the order of p0 ∼ ~/ξ(T ). So for the combination m∗ E(p0 ) one can write m∗ E(p0 ) ∼ p20 ∼ ~2 /ξ 2 (T ). In fact the particles density enters into many physical values in the combination n/m∗ . As the consequence of the above observation it can be expressed in terms of the coherence length: n(D) kB T p0 D kB T 2−D = ( ) ∼ 2 ξ (T ), m∗ m∗ E(p0 ) ~ ~

(1.10)

pD 0 here estimates the result of momentum integration. For example we can evaluate the fluctuation Cooper pairs contribution to conductivity by using the Drude formula kB T D−3 2−D ne2 τ ⇒ d ξ (T )(2e)2 τGL () ∼ D/2−2 . (1.11) m∗ ~2 Analogously a qualitative understanding of the increase in the diamagnetic susceptibility above the critical temperature may be obtained from the wellknown Langevin expression for the atomic susceptibility5 : σ=

χ=−

e2 n 2  4e2 kB T D−3 4−D R ⇒ − d ξ (T ) ∼ −D/2−2 . c2 m∗ c 2 ~2

(1.12)

Besides these examples of the direct influence of fluctuations on superconducting properties, indirect manifestations by means of quantum interference in the pairing process and of renormalization of the density of one-electron states in the normal phase of superconductor take place. These effects, being much more sophisticated, have a purely quantum nature, and in contrast to the direct Cooper pair contributions require microscopic consideration. 5

This formula is valid for the dimensionalities D = 2, 3, when fluctuation Cooper pair has the possibility to ”rotate” in the applied magnetic field and the average square of the rotation radius is < R2 >∼ ξ 2 (T ). ”Size” effects, important for low dimensional samples, will be discussed later on.

15

Chapter 2 Thermodynamics 2.1 2.1.1

1.1 Ginzburg-Landau theory GL functional

The complete description of the thermodynamic properties of a system can be done through the exact calculation of the partition function1 : ( !) b H Z = Tr exp − . (2.1) T As discussed in the Introduction, in the vicinity of the superconducting transition, side by side with the fermionic electron states, fluctuation Cooper pairs of a bosonic nature appear in the system. As already mentioned, they can be described by means of classical bosonic fields Ψ(r) which can be treated as ”Cooper pair wave functions“. So the calculation of the trace in (2.1) can be separated into a summation over the ”fast“ electron degrees of freedom and a further functional integration carried out over all possible configurations of the ”slow” Cooper pairs wave functions: Z Z = DΨ(r)Z[Ψ(r)], (2.2) where 

F[Ψ(r)] Z[Ψ(r)] = exp − T



(2.3)

is the system partition function in a fixed bosonic field Ψ(r), already summed over the electronic degrees of freedom. Here it is assumed that the classical 1

Hereafter ~ = kB = c = 1.

16

field dependent part of the Hamiltonian can be chosen in the spirit of the GL approach2 :

F[Ψ(r)] = FN +

Z

  1 b 2 4 2 dV a|Ψ(r)| + |Ψ(r)| + |∇Ψ(r)| . 2 4m

(2.4)

Let us discuss the coefficients of this functional. In accordance with the Landau hypothesis, the coefficient a goes to zero at the transition point and depends linearly on T − Tc . Then a = αTc ; all the coefficients α,b and m are supposed to be positive and temperature independent. Concerning the magnitude of the coefficients it is necessary to make the following comment. One of these coefficients can always be chosen arbitrary: this option is related to the arbitrariness of the Cooper pair wave function normalization. Nevertheless the product of two of them is fixed by dimensional analysis: ma ∼ ξ −2 (T ). Another combination of the coefficients, independent of the wave function normalization and temperature, is α2 /b. One can see that it has the dimensionality of the density of states. Since these coefficients were obtained by a summation over the electronic degrees of freedom, the only reasonable candidate for this value is the one electron density of states ν. Below will be demonstrated that close to Tc the microscopic theory of superconductivity permits to write down the free energy of superconductor in the form of GL type series over the spectrum gap ∆ (r):  Z  B 4 2 2 F = (2.5) A∆ + ∆ + C (∇∆) dV 2 with A = ν ln

T , Tc

(2.6)

7ζ (3) ν. (2.7) 8π 2 T 2 The coefficient C is related with the square of coherence length (1.4): B=

2 C(D) = νξ(D) (T τ )

(2.8)

One could identify the phenomenological order parameter Ψ with the microscopic ∆ : Ψ= 2

√

4mC∆.

For simplicity the magnetic field is assumed to be zero.

17

(2.9)

In this case the precise values for the coefficients α and b can be carried out from the microscopic theory: 8π 2 4mαTc = ξ ; α /b = ν, 7ζ(3) −2

2

(2.10)

where ζ(x) is the Riemann zeta function, ζ(3) = 1.202. One can notice that the arbitrariness in the normalization of the order parameter amplitude leads to the unambiguity in the choice of the Cooper mass introduced in (2.4) as 2m. Indeed, this value enters in (2.10) in the product with α so one of these parameters has to be fixed. In the case of a clean D-dimensional superconductor it is natural to suppose that the Copper pair mass is equal to two free electron masses what results in α(D) =

2.1.2

GL equations

2.1.3

Heat capacity jump

2Dπ 2 Tc . 7ζ(3) EF

(2.11)

As the first step in the Landau theory of phase transitions Ψ is supposed to be independent of position. This assumption in the limit of sufficiently large volume V of the system permits a calculation of the functional integral in (2.2) by the method of steepest descent. Its saddle point determines the equilibrium value of the order parameter  −αT c /b,  < 0 2 e = |Ψ| . (2.12) 0,  > 0 Choosing α in accordance with (2.11) one finds that this value coincides with the superfluid density ns of the microscopic theory [2]. The fluctuation part of the free energy related to the transition is determined by the minimum of the functional (2.4):  2 2 2 FN − α T2bc  V,  < 0 e . (2.13) F = (F[Ψ])min = F[Ψ] = FN ,  > 0 From the second derivative of (2.13) one can find an expression for the jump of the specific heat capacity at the phase transition point:

18

    Tc ∂SN Tc ∂SS ∆C = CS − CN = − = V ∂T V ∂T  2  1 ∂ F α2 8π 2 = − = T = νTc . c V Tc ∂2 b 7ζ(3)

(2.14)

Let us mention that the jump of the heat capacity was obtained because of the system volume was taken to infinity first, and after this the reduced temperature  was set equal to zero.

2.2

Fluctuation contribution to heat capacity

2.2.1

Zero dimensionality: the exact solution.

In a system of finite volume fluctuations smear out the jump in heat capacity. For a small superconducting sample with √ the characteristic size d  ξ(T ) the space independent mode Ψ0 = Ψ V defines the main contribution to the free energy:

Z(0) = =





( a|Ψ0 |2 + 2 d|Ψ0 | exp − T

Z

F[Ψ0 ] d2 Ψ0 exp − T

r

π3V T exp(x2 )(1 − erf (x))|x=a√ V . 2bT 2b

=π

Z

b 2V

|Ψ0 |4 )

!

(2.15)

By evaluating the second derivative of this exact result [11] one can find the temperature dependence of the superconducting granular heat capacity (see Fig. 2.1). It is evident that the smearing of the jump takes place in the region of temperatures in the vicinity of transition where x ∼ 1, i.e. p r 3  7ζ(3) 1 Tc0 ξ0 √ cr = Gi(0) = , ≈ 13, 3 2π EF V νTc V where we have supposed that the granule is clean; Tc0 and ξ0 are the critical temperature and the zero temperature coherence length (see the footnote 3 in Introduction) of the appropriate bulk material. From this formula one can see that even for granule with the size d ∼ ξ0 the smearing of the transition still is very narrow. Far above the critical region, where Gi(0)    1, one can use the asymptotic expression for the erf (x) function and find 19

=

Figure 2.1: Temperature dependence of the heat capacity of superconducting grains in the region of the critical temperature

π . (2.16) α Calculation of the second derivative gives an expression for the fluctuation part of the heat capacity in this region: F(0) = −T ln Z(0) = −T ln

δC(0) = 1/2 .

(2.17)

The experimental study of the heat capacity of small Sn particles in the vicinity of transition was done in [12]

2.2.2

Arbitrary dimensionality: case T ≥ Tc .

It is possible to estimate the fluctuation contribution to the heat capacity for a specimen of an arbitrary effective dimensionality on the basis of the following observation. The volume of the specimen may be divided into regions of size ξ(T ), which are weakly correlated with each other. Then the whole free energy can be estimated as the free energy of one such zerodimensional specimen (2.16), multiplied by their number N(D) = V ξ −D (T ) : π . (2.18) α This formula gives the correct temperature dependence for the free energy for even dimensionalities. A more accurate treatment removes the ln  dependence from it in the case of the odd dimensions. Let us begin with the calculation of the fluctuation contribution to the heat capacity in the normal phase of a superconductor. We restrict ourselves to the region of temperatures beyond the immediate vicinity of transition, where this correction is still small. In this region one can omit the fourth order term in Ψ(r) with respect to the quadratic one and write down the GL functional, expanding the order parameter in a Fourier series: F(D) = −T V ξ −D (T ) ln

F [Ψk ] = FN +

X k

[a +

X
k2 ]|Ψk |2 = αTc  + ξ 2 k2 |Ψk |2 . 4m k

(2.19)

R Here Ψk = √1V Ψ(r)e−ikr dV and the summation is carried out over the vectors of the reciprocal space. Now we see that the free energy functional 20

appears as a sum of energies of the independent modes k. The functional integral for the partition function (2.3) can be separated to a product of Gaussian type integrals over these modes:   YZ k2 2 2 d Ψk exp −α( + Z= )|Ψk | . (2.20) 4mαT c k Carrying out these integrals, one gets the fluctuation contribution to the free energy: X π ln . (2.21) F ( > 0) = −T ln Z = −T k2 α( + 4mαT ) k c The appropriate correction to the specific heat capacity of a superconductor at temperatures above the critical temperature may thus be calculated. We are interested in the most singular term in −1 , so the differentiation over the temperature can be again replaced by that over  :  2  1 ∂ F 1 X 1 . (2.22) δC+ = − = 2 k2 VTc ∂ V k ( + 4mαT )2 c The result of the summation over k strongly depends on the linear sizes of the sample, i.e. on its effective dimensionality. As it is clear from (2.22), the scale with which one has to compare these sizes is determined by the 1 value (4mαTc )− 2 which, as was already mentioned above, coincides with the effective size of Cooper pair ξ(T ). Thus, if all dimensions of the sample considerably exceed ξ(T ), one can integrate over (2π)−3 Lx Ly Lz dkx dky dkz instead of summing over nx , ny , nz . In the case of arbitrary dimensionality the fluctuation correction to the heat capacity turns out to be VD δC+ = V

Z

D

1 ( +

k2 4mαTc

dD k VD (4mαTc ) 2 = ϑ , D D V )2 (2π)D 2− 2

(2.23)

where VD = V, S, L, 1 for D = 3, 2, 1, 0. For the coefficients ϑD it is convenient to write an expression valid for an arbitrary dimensionality D, including fractional ones. For a space of fractional dimensionality we just mention that the momentum in spherical R Dintegration R D−1 coordinates is carried out according D to the rule: d k/ (2π) =µD k dk, where µD =

D 2D π D/2 Γ(D/2 + 1)

(2.24)

and Γ(x) is a gamma-function. The coefficient in (2.23) can be expressed in terms of the gamma-function too: 21

Γ(2 − D/2) (2.25) 2D π D/2 yielding ϑ1 = 1/4, ϑ2 = 1/4π and ϑ3 = 1/8π. In the case of small particles with characteristic sizes d . ξ() the appropriate fluctuation contribution to the free energy and the specific heat capacity coincides with the asymptotics of the exact results (2.16) and (2.17). From the formula given above it is easy to see that the role of fluctuations increases when the effective dimensionality of the sample or the electron mean free path decrease. ϑD =

2.2.3

Arbitrary dimensionality: case T < Tc .

The general expressions (2.2) and (2.4) allow one to find the fluctuation contribution to heat capacity below Tc . For this purpose let us restrict ourselves to the region of temperatures not very close to Tc from below, where fluctuations are sufficiently weak. In this case the order parameter can be e (see (2.12)) and fluctuation ψ(r) written as the sum of the equilibrium Ψ parts: e + ψ(r). Ψ(r) = Ψ

(2.26)

Keeping in (2.4) the terms up to the second order in ψ(r) and up to the e one can find fourth order in Ψ, e 2 + b/2Ψ e4 Y Z a Ψ e = exp(− ) dReψk dImψk × (2.27) Z[Ψ] T k   2 2 1 k k 2 2 2 2 e +a+ e +a+ × exp − [(3bΨ )Re ψk + (bΨ )Im ψk ] . T 4m 4m Carrying out the integral over the real and imaginary parts of the order parameter one can find an expression for the fluctuation part of the free energy: ) ( πTc T X πTc F =− ln + ln . (2.28) e 2 +a + k2 e 2 + a + k2 2 3bΨ bΨ k

4m

4m

Let us discuss this result. It is valid both above and below Tc . The two terms in it correspond to the contributions of the modulus and phase fluctuations e ≡ 0 and these contributions are equal: of the order parameter. Above Tc Ψ 22

e represent just two equiphase and modulus fluctuations in the absence of Ψ valent degrees of freedom of the scalar complex order parameter. Below Tc , the symmetry of the system decreases (see (2.12)). The order parameter modulus fluctuations remain of the same diffusive type as above Tc , while the character of the phase fluctuations, in accordance with the Goldstone theorem, changes dramatically. Substitution of (2.12) to (2.28) results in the disappearance of the temperature dependence of the phase fluctuation contribution and, calculating the second derivative, one sees that only the fluctuations of the order parameter modulus contribute to the heat capacity. As a result the heat capacity, calculated below Tc , turns out to be proportional to that found above: D

δC− = 2 2 −2 δC+ . Hence, in the framework of the theory proposed we found that the heat capacity of the superconductor tends to infinity at the transition temperature. Strictly speaking, the restrictions of the above approach do not permit us to discuss seriously this divergence at the critical point itself. The calculations in principle are valid only in that region of temperatures where the fluctuation correction is small. We will discuss below the quantitative criteria for the applicability of this perturbation theory.

2.3

Fluctuation diamagnetism

2.3.1

Qualitative preliminaries.

In this Section we discuss the effect of fluctuations on the magnetization and the susceptibility of a superconductor above the transition temperature. Being the precursor effect for the Meissner diamagnetism, the fluctuation induced magnetic susceptibility has to be a small correction with respect to the diamagnetism of a superconductor but it can be comparable to or even exceed the value of the normal metal diamagnetic or paramagnetic susceptibility and can be easily measured experimentally. As was already mentioned in the Introduction the temperature dependence of the fluctuation induced diamagnetic susceptibility can be qualitatively analyzed on the basis of the Langevin formula, but some precautions in the case of low dimensional samples have to be made. As regards the 3D case we would like just to mention here that Exp.(1.12), presented in terms of ξ(T ), has a wider region of applicability than the GL one. Namely, the scaling arguments are valid for diamagnetic susceptibility too and one can write the general relation 23

2

χ(3) ∼ −e T ξ(T ) ∼ −χP 

−1/2



1,  & Gi
 1/2−ν Gi

,  . Gi

,

(2.29)

which is valid in the region of critical fluctuations in the immediate vicinity of the transition temperature too. Here, in order to define the scale of fluctuation effects, we have introduced the Pauli paramagnetic susceptibility χP = e2 vF /4π 2 . Moreover, the Langevin formula permits us to extend the estimation of the fluctuation diamagnetic effect to the other side beyond the GL region: to high temperatures T  Tc . The coherence length far from the transition becomes a slow function of temperature. In a clean superconductor, far from Tc , ξ(T ) ∼ vF /T, so one can write χ(3c) (T  Tc ) ∼ −e2 T ξ(T ) ∼ −χP

(2.30)

and see that the fluctuation diamagnetism turns out to be of the order of the Pauli paramagnetism even far from the transition. More precise microscopic calculations of χ(3) (T  Tc ) lead to the appearance of ln2 (T /Tc ) in the denominator of (2.30). In the 2D case Exp.(1.12) is applicable for the estimation of χ(2) in the case when the magnetic field is applied perpendicular to the plane, permitting 2D rotations of fluctuation Cooper pairs in it: n EF < R2 >∼ e2 T ξ 2 (T ) ∼ −χP . (2.31) m T − Tc This result is valid for a wide range of temperatures and can exceed the Pauli paramagnetism by factor ETF even far from the critical point (we consider the clean case here). For a thin film (d  ξ(T )) perpendicular to the magnetic field the fluctuation Cooper pairs behave like effective 2D rotators, and the formula (1.12) still can be used, though one has to take into account that the susceptibility in this case is calculated per unit square of the film. So for the realistic case (from the experimental point of view) of the dirty film, one has just use in (2.31) the expression (1.6) for the coherence length:   e2 T 2 l Tc χ(2d) ∼ ξ (T ) ∼ −χP . (2.32) d d T − Tc χ(2c) (T ) ∼ e2

Let us discuss now the important case of a layered superconductor (for example, a high temperature superconductor). It is usually supposed that the electrons move freely in conducting planes separated by a distance s. Their motion in the perpendicular direction has a tunneling character, with

24

effective energy J . The related velocity and coherence length can be estimated as vz = ∂E(p)/∂p⊥ ∼ J /p⊥ ∼ sJ and ξz,(c) ∼ sJ /T for clean case. In dirty case the anisotropy p into account in the spirit of formula p can be taken (1.6) yielding ξz,(d) ∼ D⊥ /T ∼ sJ τ /T . We start from the case of a weak magnetic field applied perpendicular to layers. The effective area of a rotating fluctuation pair is ξx ()ξy (). The density of Cooper pairs in the conducting layers (1.10) has to be modified for the anisotropic case. Its isotropic 3D value is proportional to 1/ξ(), p that now has to be read as ∼ 1/ ξx ()ξy (). The anisotropy of the electron motion leads to a concentration of fluctuation Cooper pairs in the conducting p layers and hence, to an effective increase of the Cooper pairs density of ξx ()ξy ()/ξz () times its isotropic value. This increase is saturated when ξz () reachesp the interlayer distance s, so finally the anisotropy factor appears in the form ξx ()ξy ()/ max{s, ξz ()} and the square root in its numerator is removed in the Langevin formula (1.12), rewritten for this case χ(layer,⊥) (, H → 0) ∼ −e2 T

ξx ()ξy () . max{s, ξz ()}

(2.33)

The existence of a crossover between the 2D and 3D temperature regimes in this formula is evident: as the temperature tends to Tc the diamagnetic √ susceptibility temperature dependence changes from 1/ to 1/  . This happens when the reduced temperature reaches its crossover value cr = r (ξz (cr ) ∼ s). The anisotropy parameter  πτ , Tτ  1 J2 4ξz2 (0) 4 = (2.34) r= 7ζ(3) 2 , Tτ  1 s T 8π 2 T plays an important role in the theory of layered superconductors 3 . It is interesting to note that this intrinsic crossover, related to the spectrum anisotropy, has an opposite character to the geometric crossover which happens in thick enough√films when ξ(T ) reaches d . In the latter case the characteristic 3D 1/  − dependence taking place far enough from Tc (where ξ(T )  d), is changed to the 2D 1/ law (see (2.32)) in the immediate vicinity of transition ( where ξ(T )  d) [89]. It is worth mentioning that in a strongly anisotropic layered superconductor the fluctuation-induced susceptibility may considerably exceed the normal metal dia- and paramagnetic effects even relatively far from Tc [25, 26]. Let us consider a magnetic field applied along the layers. First it is necessary to mention that the fluctuation diamagnetic effect disappears in 3

We use here a definition of r following from microscopic theory (see Section 6).

25

the limit J ∼ ξz → 0. Indeed, for the formation of a circulating current it is necessary to tunnel twice, so χ(layer,k) ∼ −e2 T

J /T ξz2 sJ √ . ∼ −χP ( ) √ max{s, ξz (T )} vF  max{ , J /T }

In the general case of an anisotropic superconductor, choosing the z axis along the direction of magnetic field H, the following extrapolation of the results obtained may be written χ ∼ −e2 T

ξx2 ()ξy2 () . max{a, ξx ()} max{b, ξy ()} max{s, ξz ()}

(2.35)

This general formula is useful for the analysis of the fluctuation diamagnetism of anisotropic superconductors or samples of some specific shape: granular, quasi-1D, quasi-2D, and 3D. It is also applicable to the case of a thin film (d  ξz ()) placed perpendicular to the magnetic field: it is enough to replace ξz () by d in (2.35). Nevertheless the formula (2.35) cannot be applied to the cases of thin films in parallel fields, wires and granules. In those cases the Langevin formula (1.12) can still be used with the replacement of hR2 i → d2 :   T χ(D) ∼ −χP ξ 2−D dD−1 ∼ D/2−1 . vF The magnetic field dependence of the fluctuation part of free energy in these cases is reduced only to account for the quadratic shift of the critical temperature versus magnetic field. For 3D systems or in the case of a film in a perpendicular magnetic field the critical temperature depends on H linearly, while the magnetic field de∗ ∗ pendent part of the free energy for H  Hc2 (−) (the line Hc2 (−) is mirrorsymmetric to the Hc2 () with respect to y-axis passing through T = Tc ) is proportional to H 2 . This is why the magnetic susceptibility is determined by Eq.(2.35) for weak enough magnetic fields H  Φ0 /[ξx ()ξy ()] = Hc2 ()  Hc2 (0) only. In the vicinity of Tc these fields are small enough.

2.3.2

Zero-dimensional diamagnetic susceptibility.

For quantitative analysis of the fluctuation diamagnetism we start by writing down the GL functional for the free energy (see Exp.(2.4)) in the presence of the magnetic field

F[Ψ(r)] = Fn +

Z

 b 1 dV a|Ψ(r)|2 + |Ψ(r)|4 + | (−i∇−2eA) Ψ(r)|2 + 2 4m 26

B2 H · B + − 8π 4π



.

(2.36)

where A is vector potential. As long as fluctuation effects are comparatively small, the average magnetic field in the metal B may be assumed to be equal to the external field H. Thus we omit the last two terms in (2.36) (see later on). The fluctuation contribution to the diamagnetic susceptibility in the simplest case of a ”zero-dimensional” superconductor (spherical superconducting granule of diameter d  ξ()) was considered by V.Shmidt [11]. In this case the order parameter does not depend on the space variables and the free energy can be calculated exactly for all temperatures including the critical region in the same way as was done for the case of the heat capacity in the absence of a magnetic field. Formally the effect of a magnetic field in this case is reduced to the renormalization of the coefficient a, or, in other words, to the suppression of the critical temperature. This is why one can use the same formula (2.15) for the partition function with the critical temperature Tc shifted by magnetic field as4 : Tc (H) = Tc (0)(1 −

4π 2 ξ 2 < A2 >). Φ20

(2.37)

Here Φ0 = πe is the magnetic flux quantum and < .... > means the averaging over the sample volume. Such a trivial dependence of the properties of 0D samples on magnetic field immediately allows one to understand its effect on the heat capacity of a granular sample. Indeed, with the growth of the field the temperature dependence of the heat capacity presented in Fig. 2.1 just moves in the direction of lower temperatures. In the GL region Gi(0) .  one can write the asymptotic expression (2.16) for the free energy: F(0) (, H) = −T ln

π α( +

4π 2 ξ 2 Φ20

< A2 >)

.

1 In the case of a spherical particle the relation < A2 >= 10 H 2 d2 can be used in full analogy with the calculation of the moment of inertia of a solid sphere. In this way an expression for the 0D fluctuation magnetization valid for all fields H  Hc2 (0) can be found: 4

Let us stress the difference between the H 2 shift of the critical temperature for a zero-dimensional granule and the linear shift in the case of bulk material.

27

2 2

2π ξ 2 d ∂F(0) (, H) 5Φ20 H. M(0) (, H) = − = −T 2 2 ∂H ( + π ξ2 H 2 d2 )

(2.38)

5Φ0

One can see that the fluctuation magnetization turns out to be negative and linear up to some crossover field, which can be called the temperature √ Φ0 dependent upper critical field of the granule Hc2(0) () ∼ dξ() = dξ Hc2 (0)  at which it reaches a minimum. At higher fields Hc2(0) () . H  Hc2 (0) the fluctuation magnetization of the 0D granule decreases as 1/H. In the weak field region H  Hc2(0) () the diamagnetic susceptibility is:   12πT ξ02 1 ξ 1 2 χ(0) (, H) = − ≈ −2 · 10 χP 2 5Φ0 d  d  which coincides with our previous estimate in its temperature dependence but the numerical factor found is very large. Let us underline that the temperature dependence of the 0D fluctuation diamagnetic susceptibility turns out to be less singular than the 0D heat capacity correction: −1 instead of −2 . The expression for the fluctuation part of free energy (2.21) is also applicable to the cases of a wire or a film placed in a parallel field: as was already mentioned above all its dependence on magnetic field is manifested by the shift of the critical temperature (2.37). In the case of the wire in a parallel field one has to choose the gauge of the vector-potential A = 12 H × r yielding 2 2 hA2 i(wire,k) = H32d (the calculation of this average is analogous to that of the moment of inertia of a solid sphere). For a wire in a perpendicular field, or a film in a parallel field, the gauge has to be chosen in the form A =(0, Hx, 0) (to avoid the appearance of currents perpendicular to surface). One can find 2 2 2 2 hA2 i(wire,⊥) = H16d for a wire and hA2 i(f ilm,k) = H12d for a film. Calculating the second derivative of Eq.(2.21) with the appropriate magnetic field dependencies of the critical temperature one can find the following expressions for the diamagnetic susceptibility:

χ(D) () = −2π

2.3.3

  

ξT χP  vF 

√1 , wire in parallel field  √2 , wire in perpendicular field  d ln 1 , film in parallel field 3ξ

.

(2.39)

GL treatment of fluctuation magnetization.

Let us analyze quantitatively, on the basis of the GL functional, the temperature and field dependencies of the fluctuation magnetization. We will carry 28

on the discussion for a layered superconductor. As was already mentioned this system has a great practical importance because of its direct applicability to high temperature superconductors, where the fluctuation effects are very noticeable. Moreover, the general results obtained will allow us to analyze 3D and 2D situations as limiting cases. The effects of a magnetic field are more pronounced for perpendicular orientation, so let us consider first this case. The generalization of the GL functional for a layered superconductor (Lawrence-Doniach (LD) functional [27]) in a perpendicular magnetic field can be written as  XZ
2 1 b 2 ∇k − 2ieAk |Ψl FLD [Ψ] = d r a |Ψl |2 + |Ψl |4 + 2 4m l
+J |Ψl+1 − Ψl |2 , (2.40) where Ψl is the order parameter of the l−th superconducting layer and the phenomenological constant J is proportional to the Josephson coupling between adjacent planes. The gauge with Az = 0 is chosen in (2.40). In the immediate vicinity of Tc the LD functional is reduced to the GL one with the effective mass M = (4J s2 )−1 along c-direction, where s is the inter-layer spacing. One can relate the value of J to the coherence length along the c-direction: J = 2αTc ξz2 /s2 . Since we are dealing with the GL region the fourth order term in (2.40) can be omitted. The Landau representation is the most appropriate for problems related with the motion of a charged particle in a uniform magnetic field. The fluctuation Cooper pair wave function φnkz (r) can be written as the product of a plane wave propagating along the magnetic field direction and a Landau state wave function. Let us expand the order parameter Ψl (r) on the basis of these eigenfunctions: X Ψl (r) = Ψn,kz φnkz (r) exp(ikz l), (2.41) n,kz

where n is the quantum number related with the degenerate Landau state and kz is the momentum component along the direction of the magnetic field. Substituting this expansion into (2.40) one can find the LD free energy as a functional of the Ψn,kz coefficients: X FLD [Ψn,kz ] = αTc  + n,kz

   H 1 n+ +J (1 − cos(k z s)) |Ψn,kz |2 . 2mΦ0 2 (2.42) 29

In complete analogy with the case of an isotropic spectrum the functional integral over the order parameter configurations Ψn,kz in the partition function can be reduced to a product of ordinary Gaussian integrals, and the fluctuation part of the free energy in a magnetic field takes the form:

F (, H) = −

2πSH X T ln Φ0 αTc  + n,k z

H 2mΦ0

n+

πT
1 2

+ J (1 − cos(k z s))

.

(2.43) Here the summation over the degenerate states of each Landau level was performed (S is the sample cross-section) and results in appearance of the number of particle states (2πHS/Φ0 ) with the definite quantum numbers n and kz . The summation over n has to be performed through all occupied states, i.e. the upper limit of the sum is N ∼ 2mΦ0 EF /H. In the limit of weak fields one can carry out the summation over the Landau states by means of the Euler-Maclaurin’s transformation N X n=0

f (n) =

Z

N +1/2

−1/2

i 1 h 0 0 f (n) − f (N + 1/2) − f (−1/2) 24

and obtain πST H 2 F (, H) = F (, 0) + 24mΦ20

Z

π/s

N sdkz 2π

−π/s



1 αTc +J (1 − cos(kz s))



.

(2.44) Here N is the total number of layers. After the momentum integration one gets: F (, H) = F (, 0) +

πV H 2 1 p 2 24mαsΦ0 ( + r)

with the anisotropy parameter defined as5 5

Let us stress the difference between J and J in the two definitions (2.34) and (2.45) of the anisotropy parameter r. The first one was introduced as the electron tunneling matrix element, while the second one enters in the LD functional as the characteristic Josephson energy for the order parameter. Later on, in the framework of the microscopic theory, it will be demostrated that, in accordance with our qualitative definition, r ∼ J 2 , while J turns out to be proportional to J 2 too. In the dirty case it depends on the relaxation time of the electron scattering on impurities: J ∼ αJ 2 max{τ, 1/T }. Hence both definitions (2.34), appearing in the qualitative consideration, and (2.45), following from the LD model, are consistent.

30

4ξ 2 (0) 2J = z2 . αT s The magnetic susceptibility in a weak field turns out [28, 29] to be r=

χ(layer,⊥) = −

ξ2 e2 T p xy . 3πs ( + r)

(2.45)

(2.46)

These results confirm the qualitative estimation (2.33) additionally providing the exact value of the numerical coefficient and the temperature dependence in the crossover region. In the limit r   Exp.(2.46) transforms into the diamagnetic susceptibility of the 3D anisotropic superconductor [30]. For a film of thickness d the integral over kz in Exp.(2.44) has to be replaced by a summation over the discrete kz and when ξz (T )  d only the term with kz = 0 has to be taken into account: 2 e2 T ξ xy χ(f ilm,⊥) = − . (2.47) 3πd  Note that these formulas predict a nontrivial increase of diamagnetic susceptibility for clean metals [29]. The usual statement that fluctuations are most important in dirty superconductors with a short electronic mean free path does not hold in the particular case of susceptibility because here ξ turns out to be in the numerator of the fluctuation correction. Now we will demonstrate that, besides the crossovers in its temperature dependence, the fluctuation induced magnetization is a nonlinear function of magnetic field too, and these nonlinearities, different for various dimensionalities, take place at relatively low fields. This, strong in comparison with the expected scale of Hc2 (0), manifestation of the nonlinear regime in fluctuation magnetization and hence, field dependent fluctuation susceptibility, was the subject of the intensive debates in early seventies [30, 31, 32, 33, 34, 35, 36, 37, 38, 39] (see also the old but excellent review of W.J. Skocpol and M. Tinkham [40]) and after the discovery of HTS [41, 149, 43, 44] ( see also very recent detailed essay of T.Mishonov and E.Penev [45] with references there). We will mainly follow here the paper of Buzdin et al. [46], dealing with the fluctuation magnetization of a layered superconductor, which permits observing in a unique way all variety of the crossover phenomena in temperature and magnetic field. Let us go back to the general expression (2.43) and evaluate it without taking the magnetic field to small. The difficulty in dealing with it consists in the divergence of the sum over Landau levels n. This divergence can be regularized (see [45, 47]), but let us observe that in order to calculate the

31

magnetization we must know the magnetic field dependent part of the free energy only. So a very convenient method to bypass the divergence problem [48] is to calculate the difference F (H) − F (0), turning the sum over Landau states in F (, 0) into an integral and then, in its turn, turning this integral into a sum of integrals over the unit length intervals x ∈ [n − 1/2, n + 1/2]. Then 2πV H F (, 0) = − lim T H→0 Φ0 × ln

αTc  +

Z

π/s

−π/s

H 2mΦ0

∞ Z dkz X 1/2 dx × 2π n=0 −1/2

πT
. x + n + 12 +J (1 − cos(k z s))

and by introducing the dimensionless variable h=

6

H 2 , Hc2 (0) = 2mαTc /e = Φ0 /2πξxy , Hc2 (0)

(2.48)

one can write F (, H) − F (, 0) = (2.49) Z π X Z 1/2 (2n + 1 + 2x)h + r/2(1 − cos z) +  TV = − h dz dx ln . 2 2πsξxy −π n=0 −1/2 (2n + 1)h + r/2(1 − cos z) +  Performing the integrations over z and x in (2.49) and differentiating with respect to h we finally obtain a very convenient general expression for the fluctuation magnetization in a layered superconductor: T M (, H) = Φ0 s

∞  X ϕ(Rn + 1) n ln + ϕ(Rn ) n=0

ϕ(Rn + 1) n + 1/2 ln −p ϕ(Rn + 1/2) (Rn + 1/2)2 − ρ2

)

(2.50)

p with ϕ(x) = x + x2 − ρ2 , Rn = n + /2h + ρ and ρ = r/2h. The sum in (2.50) converges as 1/n2 and it provides a volume magnetization expression that can be compared with experiment. 6

Let us remind that the exact definition of Hc2 (0) contains the numerical coefficient A(0) (see footnote 3).

32

Let us comment on the different crossovers in the M (, H) field dependence analyzing the general formula (2.50). Let us fix the temperature   r. In this case the c-axis coherence length exceeds the interlayer distance (ξz  s ) and in the absence of a magnetic field the fluctuation Cooper pairs motion has a 3D character. Supposing the magnetic field to be not too high (h  r) we may perform an expansion in (2.50) in1/ρ and obtain √ T 2h1/2 M(3) (  r, h  r) = × Φ0 ξz  X  1/2 (n + 1) n + 1 + − 2h n  2    1/2 3 (n + 1 + 3h )  . (2.51) −n n + − q 2h 2 n+ 1 +   2

2h

For weak fields (h  ) the magnetization grows linearly with magnetic field, justifying our preliminary qualitative results. Nevertheless, √ this linear growth is changed to the nonlinear 3D high field regime M ∼ H already in the region of a relatively small fields Hc2 () . H ( . h). The further increase of magnetic field leads to the next crossover in the magnetization field dependence at h ∼ r . However the Exp.(2.51) was obtained in the assumption h  r as an expansion over 1/ρ, and it does not work any more. Handling with the Hurvitz zeta-function the summation in (2.50) for 3D case can be carried out for an arbitrary field [34]: T M(3) (  r, h) = 3 Φ0 s   1 1 ζ − , + 2 2

 1/2 √ 2 h× r     1 1    −ζ , + . 2h 2 2 2h 6h

(2.52)

One can see from this formula that for large fields the magnetization saturates at the value M∞ [49]: M (h  r) → M∞ = −

ln 2 T T = −0.346 , 2 Φ0 s Φ0 s

(2.53)

that is a typical for 2D superconductors. Therefore at h ∼ r we have a 3D → 2D crossover in M (H) behavior in spite of the fact that all sizes of fluctuation Cooper pair exceed considerably the lattice parameters. The effective ”bidimensionalization” of the fluctuations is related to the effect of 33

a strong magnetic field which ”freezes out” the rotations along its direction. Let us stress that this crossover occurs in the region of already strongly nonlinear dependence of M (H) and therefore for a rather strong magnetic field from the experimental point of view in HTS. Fixing the temperature   r in the formula (2.50) one can find the general formula for 2D fluctuation regime [45]:    T 1  1 ln Γ + − ln (2π) M(2) (  r, h) = Φs 2 2h 2  0    1  ψ + −1 , − 2h 2 2h

(2.54)

where Γ (x) is the Euler gamma-function and ψ (x) = d ln Γ(x)/dx is already cited in Introduction digamma-function. Using (2.54) one can directly pass from the linear regime in a weak magnetic field corresponding to (2.31) to the saturation of magnetization (2.53) in strong fields. Near the line of the upper critical field (hc2 () = −) the contribution of the term with n = 0 in the sum (2.50) becomes most important and for the magnetization the expression h

−M (h) ∼ p

(h − hc2 )(h − hc2 + r)

can be obtained. It contains the already familiar for us ”0D” regime (r  1 h−hc2  1), where the magnetization decreases as −M (h) ∼ h−h (compare c2 with (2.38)), while for h − hc2 << r the regime becomes ”1D” and the 1 magnetization decreases slower, as −M (h) ∼ √h−h . c2 Such an analogy is observed in the next orders in Gi too. In the Ref. [50] the analogy was demonstrated for the example of the first eleven terms for the 2D case and nine for the 3D case. Summation of the series of high order fluctuation contributions to the heat capacity by the Pade-Borel method resulted in its temperature dependence similar to the 0D and 1D cases without magnetic field. Nevertheless a considerable difference has not be forgotten: in the 0D and 1D cases no phase transition takes place while in the 2D and 3D cases in a magnetic field a phase transition of first order to the Abrikosov vortex lattice state occurs. In conclusion, the fluctuation magnetization of a layered superconductor in the vicinity of the transition temperature turns out to be a complicated function of temperature and magnetic field, and it evidently cannot be factorized on these variables. The fit of the experimental data is very sensitive to the anisotropy parameter r and allows determination of the latter with 34

Figure 2.2: Schematic representation of the different regimes for fluctuation ∗ magnetization in the (H,T) diagram. The line Hc2 (T ) is mirror-symmetric to the Hc2 (T ) line with respect to a y-axis passing through T = Tc . This line defines the crossover between linear and non-linear behavior of the fluctuation magnetization above Tc . √ Figure 2.3: Fluctuation magnetization of a YBaCO123 normalized on H as the function of temperature in accordance with the described theory shows the crossing of the isofield curves at T = Tc (0) = 92.3K. The best fit obtained for anisotropy parameter r=0.09. In the inset the magnetization curves as the function of magnetic field are reported. a rather high precision [51, 52] . In Fig. 2.3 the successful application of the described approach to fit the experimental data on YBa2 Cu3 O7 is shown [53].

2.4

Fluctuation contribution to heat capacity in magnetic field

—————————————

2.5

Ginzburg-Levanyuk criterion

The fluctuation corrections to the heat capacity obtained above allow us to answer quantitatively the question: where are the limits of applicability of the GL theory? This theory is valid not too near to the transition temperature, where the fluctuation correction is still small in comparison with the heat capacity jump. Let us define as the Ginzburg-Levanyuk number Gi(D) [4, 3] the value of the reduced temperature at which the fluctuation correction (2.23) equals the value of ∆C (2.14): Gi(D)

  2 D D 1 VD −1 4−D 2 ϑD b(4m) 2 Tc . = α V

(2.55)

Substituting into this formula the microscopic values of the GL theory parameters (2.10) one can find 35



7ζ(3)ϑD = 8π 2

Gi(D)



VD V



1 νD Tc ξ D

2  4−D

.

(2.56)

Since νD Tc ∼ νD vF /ξc ∼ pD−1 ξc−1 ∼ a1−D ξc−1 one can convert this formula to F the form     2 7ζ(3)ϑD VD ξc aD−1 4−D Gi(D) ∼ , 8π 2 V ξD where a is the interatomic distance. It is worth mentioning that in bulk conventional superconductors, due to the large value of the coherence length (ξc ∼ 10−6 ÷ 10−4 cm), which drastically exceeds the interatomic distance (a ∼ 10−8 cm), the fluctuation correction to the heat capacity is extremely small. However, the fluctuation effect increases for small effective sample dimensionality and small electron mean free path. For instance, the fluctuation heat capacity of a superconducting granular system is readily accessible for experimental study. Using the microscopic expression for the coherence length (1.4), the GinzburgLevanyuk number (2.56) can be evaluated for different cases of clean (c) and dirty (d) superconductors of various dimensionalities and geometries (film, wire, whisker and granule are supposed to have 3D electronic spectrum): Gi(3) Gi(2) Gi(1)    4 Tc 0.5, (c) , (c) EF 80 ETFc , (c) −2/3 0.27   (Tc τ )−1/3 , (d) wire 1.3 (p2F S) , (d) p l F 1.6 Tc −2/3 , (d) 1.3 EF 2.3 (p2F S) , (c) whisker (pF l)3 , (d) f ilm p2 ld F

Table 1. One can see that for the three dimensional clean case the result coincides with the original Ginzburg evaluation and demonstrates the negligibility of the superconducting fluctuation effects in clean bulk materials. Now let us consider how the presence of a strong enough magnetic field changes the fluctuation region. Let us start from general anisotropic 2D case and then analyze its 2D and 3D limits. It is enough to repeat our previous evaluation of the fluctuation contribution to the heat capacity in the presence of magnetic field using the expression for free energy (2.43). We suppose the magnetic field so strong that the contribution of only the first level of Landau quantization is important: 2πSH F (, H) = − T Φ0

Z

π/s

−π/s

dk ln 2π αTc  +

36

H 2mΦ0

n+

πT
1 2

+ J (1 − cos(k z s)) (2.57)

√

Gi(0)

7ζ(3) √ 1 2π νTc V q ξ3 Tc0 13.3 EF V0

≈

1 δC(, H) = − VTc



∂ 2 F (, H) ∂2



2πH = Φ0 s

δC (3) (  r, H) =



Z

∂ − ∂

π

−π

dy 1 2πH = 2π  (H) + r/2 (1 − cos y) Φ0 s (2.58)

πH 1 1 √ Φ0 s r  (H)3/2

(2.59)

2πH 1 Φ0 s 2 (H)

(2.60)

δC (2) (r  , H) =

Comparing (2.60) with the heat capacity jump (??) and taking into account (2.10) one can find Gi(2) (H) 2πH 1 8π 2 ν2 Tc = , 7ζ(3) Φ0 s 2 (H)

(2.61)

or 7ζ(3) 1 2πH H = 4πGi (0) , (2) 8π 2 ν2 Tc Φ0 s Hc2 (0) s 4πH Gi(2) (H) = cr (H) = Gi(2) . Hc2 (0)

2cr (H) =

(2.62)

(2.63)

The same consideration can be done for the 3D spectrum 8π 2 2πH 1 1 √ 3/2 ν3 Tc = , 7ζ(3) Φ0 s r cr (H) 3/2 cr

(H) =



(2.64)

7ζ(3) 1 1 2 8π 8π ν3 Tc ξ 3



q 2πH 2πξ 2 H ξ 1 8πξ 3 = Gi(3) (0) 8π √ , Φ0 s Φ0 s r (2.65)

Gi(3) (H) =



4πH Hc2 (0)

2/3 q 3

Gi(3) (0).

(2.66)

Let us note that the way in which we introduced the Ginzburg-Levanyuk number evidently is not unique. For instance, often it is introduced [236, 291] 2 comparing the fluctuation energy with the magnetic part of energy Hc2 (0) ξ 3 : Gi(3,m)

16π 3 κ4 Tc2 32e4 λ4 Tc2 = . = 3 Φ0 Hc2 (0) ξ2 37

(2.67)



∂ − ∂



where κ = λ/ξ is the dimensionless parameter of the Ginzburg-Landau theory. Magnetic field penetration depth λ is related to the superfluid density and can be expressed in terms of microscopic parameters (see section?) (2eλ)2 =

m 7ζ(3) 1 = 2πns 128π 3 ν3 Tc2 ξ 2

(2.68)

f (3) differing only twice with respect to our definiwhat gives the value for Gi tion:

Gi(3,m)

1 2 (2eλ)4 Tc2 = = 2 ξ 2



7ζ(3) 1 1 8π 2 8π νTc ξ 3

2

1 = Gi(3) (0) . 2

(2.69)

One more method to define the Gi number is to call in this way the reduced temperature at which the AL correction to conductivity is equal to the normal value of conductivity (as it was done in [13, 14]). Such definition too results in the change of the numerical factor in Gi number: Gi(2,σ) = 1.44Gi(2) (0)

2.6

(2.70)

Scaling and renormalization group

In the above study of the fluctuation contribution to heat capacity we have restricted ourselves to the temperature range out of the direct vicinity of the critical temperature: || & Gi(D) . As we have seen the fluctuations in this region turn out to be weak and neglecting their interaction was justified. In this Section we will discuss the fluctuations in the immediate vicinity of the critical temperature ( || . Gi(D) ) where this interaction turns out to be of great importance. We will start with the scaling hypothesis, i.e. with the belief that in the immediate vicinity of the transition the only relevant length scale is ξ(T ). The temperature dependencies of all other physical quantities can be expressed through ξ(T ). This means, for instance, that the formula for the fluctuation part of the free energy (2.18) with the logarithm omitted is still valid in the region of critical fluctuations7 F(D) ∼ −ξ −D (), 7

(2.71)

The logarithm in (2.18) is essential for the case D = 2. This case will be discussed later.

38

Figure 2.4: Examples of diagrams for the fluctuation contribution to b. the coherence length is a power function of the reduced temperature: ξ() ∼ −ν . The corresponding formula for the fluctuation heat capacity can be rewritten as ∂2F ∼ Dν−2 . (2.72) ∂2 As was demonstrated in the Introduction, the GL functional approach, where the temperature dependence of ξ(T ) is determined only by the diffusion of the electrons forming Cooper pair, ξ() ∼ −1/2 and δC ∼ −1/2 . These results are valid for the GL region ( || & Gi) only, where the interaction between fluctuations can be neglected. In the immediate vicinity of the transition (so-called critical region), where || . Gi, the interaction of fluctuations becomes essential. Here fluctuation Cooper pairs affect the coherence length themselves, changing the temperature dependencies of ξ() and δC(). In order to find the heat capacity temperature dependence in the critical region one would have to calculate the functional integral with the fourth order term, accounting for the fluctuation interaction, as was done for 0D case. For the 3D case up to now it is only known how to calculate a Gaussian type functional integral. This was done above when, for the GL region, we omitted the fourth order term in the free energy functional (2.4). The first evident step in order to include in consideration of the critical region would be to develop a perturbation series in b 8 . Any term in this series has the form of a Gaussian integral and can be represented bya dia

0 gram, where the solid lines correspond to the correlators Ψ(r)Ψ∗ (r ) . The ”interactions ” b are represented by the points where four correlator lines intersect (see Fig. 14.76). This series can be written as 4−D X  Gi(D)  2 n C∼ cn .  n=0 δC ∼ −

For  & Gi it is enough to keep only the first two terms to reproduce the perturbational result obtained above. For  . Gi all terms have to be summed. It turns out that the coefficients cn can be calculated for the space dimensionality D → 4 only. In this case the complex diagrams from Fig. 14.76 (like 8

Let us mention that this series is an asymptotic one, i.e. it does not converge even for small b. One can easily see this for small negative b, when the integral for the partition function evidently diverges. This is also confirmed by the exact 0D solution (2.15).

39

the diagram similar to an envelope) are small by the parameter ε = 4−D and in order to calculate cn it is sufficient to sum the relatively simple ”parquet” type diagrammatic series. Such a summation results in the substitution of the ”bare” vertex b by some effective interaction eb which diminishes and tends to zero when the temperature approaches the transition point. Such a method was originally worked out in quantum field theory [15, 16, 17]. For the problem of a phase transition such a summation was first accomplished in [18]. As the result singularity close to phase transition in a real three dimensional system was found exactly. It was found that in uni-axial ferroelectrics and ferromagnets with dipole-dipole interactions the specific heat has the singularity: C ∼ (− ln |T − Tc |)1/3 . (2.73) Soon this prediction was confirmed experimentally[114]. Let us stress that the Ex.(2.73) was carried out for the uni-axial ferromagneticparamagnetic phase transition with the simple one-component order parameter, what is already unapplicable to the case of superconductor with twocomponent scalar complex order parameter. In the same paper [18] the effect of the order parameter symmetry on the singularity at the critical point of a non-physical 4D system was considered. It was demonstrated that in the case of transition when the order parameter is an n-component vector the specific heat has singularity: (4−n)

C ∼ (− ln |T − Tc |) (n+8) . Instead of a direct summation of the diagrams it is more convenient and physically obvious to use the method of the multiplicative renormalization group. In the case of quantum field theory it was known long ago [19, 20]. This method is equivalent to that of parquet diagrams summation, but it is simpler and found later applications in different branches of condensed matter theory. In the Appendix of [18] Larkin and Khmelnitskii demonstrated the equivalence of the summation of the parquet diagrams and renormalization group approach in the theory of phase transitions for 4D case. At the same time Di Castro and Iona-Lasinio [21] applied the renormalization group to the 3D theory of phase transitions and showed that the fixed point of RG equation at a finite coupling constant leads to the scaling. The next important achievement was done by Wilson and Fisher (1972). In purpose to study the cross-over from the logarithmic laws in four dimensional case to power-laws at any lower dimension they wrote the RG equations for the phase transition in the space of the fractional dimensionality D = 4−. Their single-loop renormalization group approach gave the critical exponents in the leading order in . Next order corrections, ∼ 2 , were calculated by 40

Wilson (1972) and Abrahams and Tzuneto (1973). After these works the scaling hypothesis became the theory. The definitive step in formulation of the modern renormalization group method in the phase transition theory was done by Wilson [22]. Significance of the Wilson’s theory goes far beyond the physics of the phase transitions. He taught all of us that the renormalization of action9 in the path integral should better be done, while before people renormalized the Green functions. After this works of Wilson the renormalization group became a real working tool, which gave jobs to many theorists. The idea of the renormalization group method consists in separating the functional integration over ”fast” (ψ|k|>Λ ) and ”slow”(ψ|k|<Λ ) fluctuation modes. If the cut off Λ is large enough the fast mode contribution is small and the integration over them is Gaussian. After the first integration over fast modes the functional obtained depends on the slow ones only. They can, in their turn, be divided on slow (|k| < Λ1 ) and fast (Λ1 < |k| < Λ), and the procedure can be repeated. Moving step by step ahead in this way one can calculate the complete partition function. As an example of the first step of renormalization, the partition function calculation below Tc can be recalled. There we separated the order parameter e (”slow” mode) and the fluctuation part into the space-independent part Ψ ψ(r) (”fast” mode) which was believed to be small in magnitude. Being in the GL region it was enough to average over the fast variables just once, while in the critical region the renormalization procedure requires subsequent approximations. The cornerstone of the method consists in the fact that in the critical region at any subsequent step the free energy functional has the same form. For D close to 4 this form coincides with the initial free energy GL functional but with the coefficients aΛ and bΛ depending on Λ. We will perform these calculations by the method of mathematical induction. Let us suppose that after the (n − 1)−st step the free energy functional has the form: Z

n e Λn−1 |2 + aΛn−1 |Ψ  bΛn−1 e 1 4 2 e |ΨΛn−1 | + |∇ΨΛn−1 | . + 2 4m

e Λn−1 ] = FN,Λn−1 + F[Ψ

dV

(2.74)

e Λn−1 in the form Ψ e Λn−1 = Ψ e Λn + ψΛn and choosing Λn close enough Writing Ψ to Λn−1 it is possible to make ψΛn so small, that one can restrict the functional to the quadratic terms in ψΛn only and perform the Gaussian integration in 9

F[Ψ] in the discussed below case of phase transition

41

complete analogy with (2.27). The important property of the spaces with dimensionalities close to 4 is the possibility to choose Λn  Λn−1 and still to e Λn . In this case Ψ e Λn can be taken as coordinate independent, have ψΛn  Ψ and one can use the result directly following from (2.27):   1 b Λ n 2 4 2 e Λn | + e Λn | e Λn ] = FN,Λn−1 + dV aΛn |Ψ e Λn | + F[Ψ |Ψ |∇Ψ 2 4m ( X T πTc − ln + e Λn |2 + aΛn + k2 ) 2 (3bΛn |Ψ 4m Λn <|k|<Λn−1 ) πTc ln . (2.75) e Λn |2 + aΛn + k2 bΛn |Ψ Z

4m

e Λn one can get for F[Ψ e Λn ] Expanding the last term in (2.75) in a series in Ψ the same expression (2.74) with the substitution of Λn−1 → Λn . From (2.75) follows that X πTc Fn,Λn = Fn,Λn−1 − T ln , k2 (a + ) Λ n 4m Λ <|k|<Λ n

aΛn = aΛn−1 + 2T

n−1

X

Λn <|k|<Λn−1

bΛn = bΛn−1 − 5T

X

Λn <|k|<Λn−1

bΛn (aΛn + b2Λn (aΛn +

(2.76)

k2 ) 4m

k2 2 ) 4m

.

Passing to a continuous variable Λn → Λ one can rewrite these recursion equations as the set of differential equations: ∂F (Λ) πTc = −T µD ΛD−1 ln ∂Λ (a(Λ) +  D−1  ∂a(Λ) = −2T µD b(Λ)Λ Λ2 ∂Λ 

∂b(Λ) ∂Λ

= 5T µD

(a(Λ)+ 4m ) b2 (Λ)ΛD−1

Λ2 ) 4m

.

,

(2.77)

(2.78)

Λ2 2 (a(Λ)+ 4m )

These renormalization group equations are evidently valid for small enough Λ only, where the transition from discrete to continuous variables is justified. This means at least 42

Λ2 /4m  Tc0 Gi(D) .

(2.79)

in order to move away from the first approximation. Let us recall that in the framework of the Landau theory of phase transitions the coefficient a(Tc0 ) = 0 at the transition point and this can be considered as the MFA definition of the critical temperature Tc0 . The same statement for the function a = a(Λ) in the framework of the renormalization group method can be written as the a(Tc0 , Λ ∼ ξ −1 ) = 0. With the decrease of Λ the effect of critical fluctuations more and more is taken into account and the renormalized value of the critical temperature decreases, being defined by the equation: a(Tc (Λ), Λ) = 0. Finally, after the application of the complete renormalization procedure, one can define the real critical temperature Tc , shifted down with respect to Tc0 due to the effect of fluctuations, from the equation: a(Tc , Λ = 0) = 0.

(2.80)

It is easy to find this shift in the first approximation. Indeed, let us integrate the first equation of (2.78) over Λ in limits [0, ξ −1 ]. The main contribution Λ2 . Being to the integral will be determined by the region where a(Λ)  4m far from the critical point one can assume that the coefficient b = const and then Z a(ξ−1 ) Z 1/ξ −1 a(ξ ) = αδTc = da = −8mT µD b ΛD−3 dΛ. a(0)

0

For the 3D case this gives the shift of the critical temperature δTc due to fluctuations 10 (3)

δTc Tc

∼−

2mb b 1 7ζ(3) 8p =− = − = − Gi(3) . παξ 2πTc α2 ξ 3 16π 3 νTc ξ 3 π

(2.82)

Let us come back to study of properties of the system of equations (2.78). One can find its partial solution at T = Tc in the form: 10 In the same way we can analyze the shift of the critical temperature in 2D case too and obtain: (2)

δTc Tc

= −2Gi(2) ln

1 . 4Gi(2)

(2.81)

As we will show below both 3D and 2D results for δTc coinside with those obtained by the analysis of the effect of fluctuations on superconducting density in the perturbation approach.

43

4−D Λ2 4m[5 + (4 − D)] 5 4−D b(Tc , Λ) = Λ4−D . 2 2 16m T µD [5 + (4 − D)]

a(Tc , Λ) =

(2.83)

These power solutions are correct in the domain of validity of the system (2.78) itself, i.e. for small enough Λ defined by the condition (2.79). Nevertheless, in a space of dimensionality close to 4 (D = 4−ε, ε  1) it is possible to extend their validity up to the GL region and to observe their crossover to the GL results: a(Tc ) = 0; b(Tc ) = b0 = const. Indeed, in this case, due to proportionality of a(Λ) to ε → 0, one can omit it in the denominators of the system (2.78) and to write down the solution for b(Λ) in the form −1

b (Tc , Λ) =

b−1 0

80m2 + T µD (ΛD−4 − ξ 4−D ). (4 − D)

(2.84)

2

80m 4−D We have chosen the constant of integration as b−1 in order 0 − (4−D) T µD ξ to match the renormalization group and GL solutions at the value of Λ = Λmax ∼ ξ −1 . Now let us pass to study of the function a(T, Λ) for the same interesting case of space dimensionality D → 4 for temperatures slightly different ( but still close enough) from Tc , where one can write

a(T, Λ) = a(Tc , Λ) + α(Tc , Λ)Tc . The first term on the right hand side is determined by Eq.(2.83). In order to determine α(Tc , Λ) let us expand the first equation in (2.78) in terms of 

For

b(Tc , Λ)ΛD−1 ∂α(Tc , Λ) = 2T µD
α(Tc , Λ). Λ2 2 ∂Λ a(Tc , Λ) + 4m

(2.85)

Λ2 /4m & α(Tc , Λ)Tc 

(2.86)

we can again use the solution (2.84) for b(Tc , Λ) and omit a(Tc , Λ) in the denominator of (2.85). The constant of integration, appearing in the process of solution of (2.85), is chosen in accordance with the condition that for Λ = Λmax ∼ ξ −1 we match α(Tc , Λ) = α(Tc0 , ξ −1 ) = α0 with the GL theory:  −2/5 80m2 b0 D−4 4−D α(Tc , Λ) = α0 1 + T µD (Λ −ξ ) . (4 − D) 44

(2.87)

The condition (2.86) can be written as Λ & ξ −1 (T ), where ξ(T ) is the generalized coherence length, determined by the equation:
ξ −2 (T ) = 4mα Tc , ξ −1 (T ) Tc . Such a definition is valid at any temperature. For example, far enough from the critical point, in the GL region, α (Tc , ξ −1 (T )) = α0 and one reproduces the result (1.3). Vice versa, in the critical region the main contribution on the right hand side of the Eq. (2.87) results from the second term containing ΛD−4 so, putting ξ −1 (T ) = Λ, one can rewrite the self-consistent equation for ξ(T ) and get ξ(T ) = (4m)(1−D)/2

4−D √ −ν , 20b0 T µD Tc α0

(2.88)

where 2ν = [1 − (4 − D)/5]−1 . As was already mentioned, strictly speaking this result was carried out for ε = 4 − D  1, so it is confident up to the first in ε expansion only: ν = 1/2 + ε/10. Nevertheless extending it to ε = 1 (D = 3) one can obtain ν3 = 3/5. Let us pass to the calculation of the critical exponent of the heat capacity in the immediate vicinity of the transition. For this purpose one can calculate the second derivative of equation (2.77) with respect to : ∂C(Λ) ΛD−1 α2 (Λ) = T 2 µD Λ2 2 ∂Λ (α(Λ)T c + 4m )

(2.89)

The heat capacity renormalized by fluctuations has the value C(Λ = 0) which is the result of integration over all fluctuation degrees of freedom. Carrying out the integration of (2.89) over all Λ . ξ −1 one can divide the domain of integration on the right hand side in two: Λ . ξ −1 (T ) and ξ −1 (T ) . Λ . ξ −1 . In the calculation of the integral over the region ξ > Λ & ξ −1 (T ) Λ2 the inequality α(Λ)T c  4m holds, and the function α(Λ) can be omitted in the denominator. In the numerator of (2.89) one can use for α(Λ) the solution (2.87). In the region Λ . ξ −1 (T ) one has to use the partial solution (2.83) for α(Λ) and can find that the contribution of this domain has the same singularity as that from the region Λ & ξ −1 (T ), but with a coefficient proportional to (4 − D)2 = ε2 , hence negligible in our approximation. The result is:

C(Λ = 0) =

1 α02 [(4mT )2 µD ] 5



4 (4 − D) 5 b0

4/5

4−D 5 ξ 5 (T ). 4−D

Substituting the expression for ξ(T ) one can finally find 45

(2.90)

C = 212/5 α02 [5µD

m2 T 2 1 −α ]5  , b40 (4 − D)

(2.91)

confirming the validity of the scaling hypothesis and the relation (2.72). The critical exponent in (2.91) is α=

(4 − D) ≈ ε/10. 10[1 − (4 − D)/5]

One can see that generally speaking the critical exponents ν and α appear in the form of series in powers of ε. More cumbersome calculations permit finding the next approximations for them in ε = 4 − D. Nevertheless it is worth mentioning that even the first approximation, giving ν3 = 3/5 and α3 = 1/10 for ε = 1, is already weakly affected by the following steps of the expansion in powers of ε [23]. One can notice that the exercise performed in this section has more academic than practical character. Indeed, the results obtained turn out applicable to the analysis of the critical region of a 3D superconductor only if Gi is so small that the theoretical predictions are hardly experimentally observable. Nevertheless we demonstrated the RG method which helps to see the complete picture of the fluctuations manifestation in the vicinity of the critical temperature. In conclusion of this section let us make the following note. Depending on the sign of Gell-Mann-Low function the renormalization group equation can have two different kinds of solutions. In quantum electrodynamics, the coupling constant decreases at long distances (“zero charge”). In quantum chromodynamics, the coupling constant is small at short distances (“asymptotic freedom”), but it increases with increasing the distance (”confinement”). We demonstrated above that the 4D phase transition theory is of the “zerocharge” type and the interaction decreases at long distances. As we already have seen the four dimensional case is special for renormalization group theory: it becomes logarithmic there. It is interesting that the 3D problems with the dipole interaction are equivalent to the solvable 4D problem. The matter of fact that at large distances such interaction depends on the angle between r and d and such angle plays the role of the fourth coordinate necessary to make the situation logarithmic. As it was mentioned above namely such interaction allowed to resolve the problem of phase transition in ferroelectric. The same concerns such complex problem as the pinning of charge density wave where the dipole-dipole interaction helped to resolve the RG equation and to find the exponentially large dielectric susceptibility [261]. As it will be demonstrated below (section ?) the analogous situation 46

turns out in the problem of pinning of the interacting vortices in superconductor. Here the dipole-dipole interaction takes place at the distances r . λ and for strong enough magnetic fields, where such interaction is relevant, the RG approach allows to calculate the exponentially small value of the critical current.

2.7 2.7.1

Effect of fluctuations on superfluid density and critical temperature Superfluid density.

Among the important thermodynamical properties of a superconductor is the magnetic field penetration depth λ(T ). It is evidently asymmetric with respect to the critical temperature, growing when the temperature tends to Tc in the superconducting phase and being infinite in the normal phase. The simple London electrodynamics of a superconductor relates the current density j and λ(T ) with the superfluid density ns : 2e2 1 m A=− A, λ2 (T ) = . (2.92) 2 m 4πλ (T ) 8πns e2 GL theory permits calculating the temperature dependence of λ(T ) in the vicinity of the critical temperature, identifying the superfluid density with the average value of the order parameter e2 = −a ns =Ψ b and, hence, predicting its inverse square root divergence as a function of the reduced temperature: s m b λ(T ) = . (2.93) 2 8πe αTc || j = −ns

Let us study how the appearance of fluctuation Cooper pairs affects the expulsion of the applied magnetic field. The superfluid density goes to zero at the transition point from below and its value determines λ(T ), so the effect of fluctuations could be well pronounced. Let us first convince ourselves that the presence of fluctuation Cooper pairs above Tc does not result in the appearance of a superfluid density. One can calculate the fluctuation part of the supercurrent as [54, 55] hjf l i = −

∂Ff l (A) ∂ ln Z(A) =T = ∂A ∂A 47

= −

R

  F [Ψ(r),A] DΨ(r)DΨ∗ (r) ∂F [Ψ(r),A]) exp − ∂A T   . R DΨ(r)DΨ∗ (r) exp − F [Ψ(r),0] T

This expression can be written in the form of two contributions:

hjf l i = j1 +j2 = −

 2e2

e A |Ψ(r)|2 A=0 − hIm[Ψ(r)∇Ψ∗ (r)]iA . m m

(2.94)

The first one, j1 , just reproduces the London expression with the replacement of ns by h|Ψ(r)|2 i . In the average the GL free energy with A = 0 can be used. The second term, j2 , has a more sophisticated nature. To get it different from zero one has to include the field dependent part of the GL free energy in the process of further averaging in fluctuation fields. Supposing the vector potential A to be weak enough one can expand the Gibbs exponent in it and find e2 j2 = − 2 mT

  Z ∗ ∗ Im[Ψ(r)∇Ψ (r)] dV1 Im[Ψ(r1 )∇Ψ (r1 )]A(r1 )

. (2.95) A=0

In the following we will restrict ourselves to a superconductor of the strongly second order (κ  1). In this case the characteristic scale of the space variation of A(r1 ) is much larger than that of the order parameter, and the vector potential can be taken out of the integral in (2.95). Thinking about the GL region above Tc , one can expand the order parameter in a Fourier series and reduce the averaging to the expression j2 =

 ∂

2e2 X k(A ) |Ψk |2 , m k ∂k

which yields hjf l i = 0, as intuitively expected. Below Tc the situation is more cumbersome. In order to calculate j2 here let us separate the equilibrium value of the order parameter and its real and e imaginary fluctuating parts: Ψ(r) = Ψ+ψ r +iψi . This allows us to calculate the space integral in the expression (2.95): Z X dV1 Im[Ψ(r1 )∇Ψ∗ (r1 )] = 2i kψrk ψi,−k . k

The further functional integration can be carried out in the spirit of the previous calculations for the fluctuation contribution to the heat capacity below Tc , resulting in 48

" # X

 T 2e2 hjf l i = − A |Ψ(r)|2 A=0 − . m (2αTc || + k2 /4m) k e 2 + hψr2 i + hψi2 i + 2Ψ e hψr i . The Let us calculate the remaining h|Ψ(r)|2 i = Ψ averaging of the first two terms does not create any problem:

 1X T ψr2 = , 2 k 2αTc || + k2 /4m

 T X 1 ψi2 = . 2 k k2 /4m

(2.96)

In order to carry out the hψr i term the anharmonic contributions in the GL functional, originating from the fourth order term, have to be taken into account: hψr i = − Finally one finds

 
1 3 ψr2 + ψi2 . e 2Ψ

(2.97)

" # X T 2e2 e2 − 2 hjf l i = − A Ψ m (2αTc || + k 2 /4m) k and the superfluid density takes the form " # X α 2b 1 ns (T )= Tc || − 2 . 2 b α k (2|| + k /4mαTc )

(2.98)

It is seen that in the 3D and 2D cases the sum over momenta in (2.98) is formally divergent at large k. This is not a first time when such problem has arisen and we know how to deal with it: this ultra-violet divergence is related to the restrictions on the applicability of the GL functional for |k| & ξ −1 , so the integral has to be cut off at ξ · |k|= C ∼ 1. For the 3D case the transparent way to cut off the sum is to separate the upper and lower limit contributions of the sum, which have different P −2 physical senses, by adding and substracting the term ∼ |k| . As a result one obtains two contributions: the first one, ||−independent, originates from p the upper limit cut off and the second turns out to be proportional to || and appears from the lower limit. Using the microscopic relations between 8π 2 ν and 4mαTc = ξ −2 one can finally find the GL parameters α2 /b = 7ζ(3) 49

  7ζ(3) p 32mπ 2 7ζ(3) 2 ns3 (T ) = νTc ξ 2||Tc − C3 + 2|| . 7ζ(3) 16π 3 νξ 3 16π 2 νξ 3

(2.99)

In the 2D case the problem of the ultra-violet divergence is less important since the cut off parameter turns out to be involved in a logarithm:   32mπ 2 7ζ(3) C2 2 ns2 (T ) = νTc ξ Tc || − ln . (2.100) 7ζ(3) 16π 3 νξ 2 ||

2.7.2

Fluctuation shift of the critical temperature.

Let us discuss the second term appeared in (2.99). It evidently determines the fluctuation shift of the critical temperature δTc = Tc − Tc0 , where Tc0 is the mean field (BCS) value of the transition temperature. This shift (3)

δTc 7ζ(3) 8p ∼− = − Gi(3) Tc0 16π 3 νTc0 ξ 3 π coincides with that found in the framework of the renormalization group approach (see Exp. (2.82)). In further calculations we will include this shift in || renormalizing the critical temperature to the value of Tc and identifying it with the experimentally observed transition temperature. The next order corrections demonstrate that the substitution of Tc0 → Tc is necessary in the last term of (2.99) too, so for the superfluid density one can finally write s " # 2Gi (3) ns3 (T ) = nBCS 1+8 . s3 || In spite of the slower decrease of the fluctuation correction to the superfluid density when temperature p tends to the critical value, in comparison with the main contribution ( || instead of || ), it becomes important only at || ∼ Gi(3) . The 2D superfluid density (2.100) can be also written in terms of the 2D Ginzburg-Levanyuk number:   C2 mTc || − 2Gi(2) ln (2.101) ns2 (T ) = πGi(2) || and finally expressed in terms of the fluctuations renormalized critical temperature. It is enough to include the cut off parameter in the Tc -shift (compare with the RG estimation (2.81)): (2)

δTc C2 = −2Gi(2) ln , Tc0 Gi(2) 50

which results in

11

: ns2 (T ) =

2.7.3

nBCS s2

  Gi(2) Gi(2) 1−2 ln . || ||

(2.102)

Fluctuation shift of the Hc2 (T )

Dorsey, Ikeda

2.7.4

Fluctuations of magnetic field

In the above consideration we have discussed fluctuations of the order parameter only supposing the magnetic field to be a constant. At the same time the natural question can arise: what is the effect of the usual black-body radiation fluctuations on the superconductor properties? From the very beginning one can expect this effect to be small at low temperature. The order parameter fluctuation effects in bulk materials turned out to be negligible too. In the case of type I superconductors the coherence length ξ can noticeably exceed the magnetic field penetration depth λ. This means that the magnetic field fluctuations will take place on a scale less than the characteristic superconducting one, which can change some intrinsic superconducting properties. In this Section we will show that magnetic field fluctuations, being small in amplitude, can change the order of phase transition of a type I superconductor. The same GL functional formalism permits us to take into account the fluctuations of a magnetic field [65]. Let us suppose that the external magnetic field H = 0, the system is below the critical temperature, the value of e and we neglect its fluctuations. Furthe equilibrium order parameter is Ψ, ther calculations can be done in spirit of the order parameter fluctuation calculations. One can rewrite the expression (2.36) as the functional of the fluctuating vector potential, choosing the gauge O · A =0 and writing it in the form of Fourier series: "

2 e 2 2 2 e 4 + e |Ψ| A2 + k A e 2 + b |Ψ| F[A(r)] = Fn + dV a|Ψ| 2 m 8π ! X e2 |Ψ| e 2 k2 e 0] + = Fs [Ψ, + A2k . m 8π k

Z

11

For sake of simplicity we still call the reduced temperature

51

Tc −T Tc

as .

#

=

The corresponding free energy can be written as: Z

X e 0] + ln DAx DAy exp − 1 F = Fs [Ψ, T k X πT e 0] + T ln 2 e 2 . = Fs [Ψ, e |Ψ| k2 + k m 8π

e 2 k2 e2 |Ψ| + m 8π

!

A2k

!

=

(2.103)

The sum in (2.103) in 3D case evidently strongly diverges. The most divergent term is nothing else than the consequence of the ultra-violet catastrophe and it, as usual, can be simply attributed to the background value of e 2 , contribution to the sum in the free energy. The next, proportional to |Ψ| (2.103) turns out to be divergent too. This divergence we treat in the same way as above: renormalize the critical temperature of the superconducting transition with respect to its mean field value. Regularized in this way free energy can be rewritten finally as √ 16T πe3 e 3 b e 4 2 e |Ψ| + |Ψ| . (2.104) F = Fn + a|Ψ| − m3/2 2 One can see that an unusual cubic term appears in it due to the fluctuations of the electromagnetic field. This contribution, even being small in magnitude, changes the order of the superconducting phase transition. Indeed, putting a = 0 one can see that the minimum of the free energy takes place for a positive order parameter value. This means that already for positive a the free energy has two minima (one at Ψ = 0 and a second for positive Ψ) appropriate to the normal and superconducting states. A first order phase transition takes place when the values of the free energy in both of these minima coincide, i.e. at e a = 128πT 2 e6 /bm3 , or, using the microscopic 2 expressions for λ and α /b, at  6 α2 213 πT 4 e6 ξ(T ) 4 . (2.105) e = = 2 · (2π) Gi(3) b (4αmT )3 λ(T )

From this result one can see that for a type I superconductor this region noticeably exceeds the critical one and the phase transition, due to electromagnetic field fluctuations, turns out to be of first order. For type II superconductors the region of temperatures e , where the electromagnetic field fluctuations are important, turns out to be inside the critical one. This means that these fluctuations have to be taken into account in the equations of the renormalization group which results in the conclusion

52

of a first order phase transition in type II superconductors too. Nevertheless for temperatures  & Gi(3) these effects are negligible and in the further discussion we will not take them into account.

53

Chapter 3 Ginzburg-Landau theory of fluctuations in transport phenomena The appearance of fluctuating Cooper pairs above Tc leads to the opening of a ”new channel” for charge transfer. In the Introduction the fluctuation Cooper pairs were treated as carriers with charge 2e while their lifetime τGL was chosen to play the role of the scattering time in the Drude formula. Such a qualitative consideration results in the Aslamazov-Larkin (AL) pair contribution to conductivity (1.11) (so-called paraconductivity1 ). Below we will present the generalization of the phenomenological GL functional approach to transport phenomena. Dealing with the fluctuation order parameter, it is possible to describe correctly the paraconductivity type fluctuation contributions to the normal resistance and magnetoconductivity, Hall effect, thermoelectric power and thermal conductivity at the edge of transition. Unfortunately the indirect fluctuation contributions are beyond the possibilities of the description by Time-Dependent GL approach and they will be calculated in the framework of the microscopic theory (see Sections 6-8). 1

This term may have different origins. First of all, evidently, paraconductivity is analogous to paramagnetism and means excess conductivity. Another possible origin is an incorrect onomatopoeic translation from the Russian “paroprovodimost’ ” that means pair conductivity

54

3.1

Time dependent GL equation

In previous Sections we have demonstrated how the GL functional formalism allows one to account for fluctuation corrections to thermodynamical quantities. Let us discuss the effect of fluctuations on the transport properties of a superconductor above the critical temperature. In order to find the value of paraconductivity, some time-dependent generalization of the GL equations is required. Indeed, the conductivity characterizes the response of the system to the applied electric field. It can be defined as E = −∂A/∂t but, in contrast to the previous Section, A has to be regarded as being time dependent. The general non-stationary BCS equations are very complicated, even in the limit of slow time and space variations of the field and the order parameter. For our purposes it will be sufficient, following [66, 67, 68, 69, 70, 71, 72], to write a model equation in the vicinity of Tc , which in general correctly reflects the qualitative aspects of the order parameter dynamics and in some cases is exact. Let us keep in mind the GL functional formalism introduced above. If a deviation from equilibrium is assumed, then it is no more possible to derive the GL equations starting from the condition that the variational derivative of the free energy is zero. At the same time, in the absence of equilibrium Ψ begins to depend on time. For small deviations from the equilibrium it is natural to assume that in the process of order parameter relaxation its time derivative ∂Ψ/∂t is proportional to the variational derivative of the free energy δF/δΨ∗ , which is equal to zero at the equilibrium. But this is not all: side by side with the normal relaxation of the order parameter the effect of thermodynamical fluctuations on it has to be taken into account. This can be done by the introduction the Langevin forces ζ(r, t) in the right-hand side of equation describing the order parameter dynamics. Finally, gauge invariance requires that ∂Ψ/∂t should be included in the equation in the combination ∂Ψ/∂t + 2ieϕΨ, where ϕ is the scalar potential of the electric field. By concluding all these speculations one can write the model time-dependent GL equation (TDGL) in the form   δF ∂ + 2ieϕ Ψ = + ζ(r, t). (3.1) −γGL ∂t δΨ∗ with the GL functional F determined by (2.4),(2.36),(2.40). The dimensionless coefficient γGL in the left-hand-side of the equation can be related to pair life-time τGL (1.2): γGL = αTc τGL = πα/8 by the substitution in (3.1) of the first term of (2.4) only2 . 2

It will be shown below that taking into account the electron-hole asymmetry

55

Neglecting the fourth order term in the GL functional, equation (3.1) can be rewritten in operator form as b−1 − 2ieγGL ϕ(r, t)]Ψ(r, t) = ζ(r, t) [L

(3.2)

b and Hamiltonian c with the TDGL operator L H defined as  −1 h i ∂ 2 b 2 b b b c L = γGL + H , H = αTc  − ξ (∇ − 2ieA) . ∂t

(3.3)

We have introduced here the formal operator of the coherence length ξb to have the possibility to deal with an arbitrary type of spectrum. For example, in the most interesting case for our applications to layered superconductors, the action of this operator is defined by the Exp.(2.40). In the absence of an electric field one can write the formal solution of equation (3.2) as b Ψ(0) (r, t) = Lζ(r, t).

(3.4)

The Langevin forces introduced above must satisfy the fluctuation-dissipation (0)∗ (0) theorem, which means that the correlator < Ψp (t0 )Ψp (t) > at coinciding moments of time has to be the same as < |Ψp |2 >, obtained by averaging over fluctuations in thermal equilibrium (see (15.3)). This requirement is fulfilled if the Langevin forces ζ(r, t) and ζ ∗ (r, t) are correlated by the Gaussian whitenoise law 0

0

0

0

< ζ ∗ (r, t)ζ(r , t ) >= 2T ReγGL δ(r − r )δ(t − t ).

(3.5)

To show it let us restrict ourselves for sake of simplicity to the case of A = 0 and calculate the correlator c f∗ Lζ(r b 0 , t) >= < Ψ∗ (r, t)Ψ(r0 , t) >=< ζ ∗ (r, t)L Z Z dp ip(r−r0 ) ∞ dΩ ∗ = 2T ReγGL e L (p, Ω)L(p, Ω). (2π)D −∞ 2π

(3.6)

L(p, Ω) can be found by making the Fourier transform in (3.3): L(p, Ω) = (−iγGL Ω + εp )−1 ,

(3.7)

resulting in leads to the appearance of an imaginary part of γGL proportional to the derivative ∂ ln(ρv 2 τ )/∂E|EF ∼ O(1/EF ). This is important for such phenomena as fluctuation Hall effect or fluctuation thermopower and, having in mind writing the most general formula, we will suppose γGL = πα/8 + iImγGL , where necessary.

56

∗

0

hΨ (r, t)Ψ(r , t)ip = 2T ReγGL

Z

∞

−∞

 1 dΩ = |Ψp |2 , 2 2 2 2π γGL Ω + εp

(3.8)

where εp = αTc ( + ξb2 p2 ).

(3.9)

is the Cooper pair energy spectrum.

3.2

Paraconductivity

Let us try to clear up the reason why the simple Drude formula works so well for complex phenomenon like paraconductivity. For this purpose let us try to derive the Boltzmann master equation for the fluctuation Cooper pair distribution function Z np (t) = hΨ(r, t)Ψ∗ (r0 , t)i exp(−ip(r − r0 ))d(r − r0 ). (3.10) (0)

Let us recall that in the state of thermal equilibrium np =< |Ψp |2 >= T /εp . In order to determine the electric field dependence of np let us write its time derivative using (3.1)   ∂Ψ(r, t) ∗ 0 ∂Ψ∗ (r, t) 0 d(r − r )e < Ψ (r , t) > + < Ψ(r , t) > = ∂t ∂t Z h e 0 (3.11) = d(r − r0 )e−ip(r−r ) 2 [ϕ(r) − ϕ(r0 )] < Ψ(r, t)Ψ∗ (r0 , t) > + i    2 δF 0 2 ∗ 0 + Re Ψ(r, t) (r , t) + Re hζ(r, t)Ψ (r , t)i , γGL δΨ γGL

∂np (t) = ∂t

Z

0

−ip(r−r0 )

where F is determined by (2.36). Expressing the scalar potential by the elecp tric field E one can transform the first term of the last integral into −2eE ∂n . ∂p The term with the variational derivative can be evaluated by means of (2.19) 2 and expressed in the form − γGL εp np . More cumbersome is the evaluation of the last term, containing the Langevin force. To the first approximation it is possible to use here the order parameter Ψ(0) (r, t) (see (3.4)) unperturbed b by the electric field as the convolution Lζ(r, t). In this way, using (3.5) and (3.6) we calculate the last average in (3.11)

57

2 γGL

Z

D E 0 b d(r − r0 )e−ip(r−r ) Re ζ ∗ (r, t)Lζ(r, t) = Z Z ∞ dΩ 2T 2 dp ip(r−r0 ) 4T e Re L(p, Ω) = = εp n(0) p D (2π) γGL γGL −∞ 2π

and obtain the transport equation

∂np ∂np 2 2 + 2eE =− εp np − n(0) =− np − n(0) . p p ∂t ∂p γGL τp

(3.12)

In the absence of a magnetic field εp was determined by (3.9) and the momentum dependent lifetime, corresponding to the Ginzburg-Landau one, can be introduced: τp = γGL /εp =

τGL () . 1 + ξ 2 ()p2

Let us stress the appearance of the coefficient 2 on the right hand side of the equation (3.12). This means that the real Cooper pair lifetime, characterizing its density decay, is τp /2 . The effect of a weak electric field on the fluctuation Cooper pair distribution function in the linear approximation is determined by (0)

n(1) p = −

eEγGL ∂np eT γGL ∂εp = E · . εp ∂p ε3p ∂p

(3.13)

Substituting this formula into the expression for the electric current (2.94) side by side with the Cooper pair velocity vp = ∂εp /∂p one can find X jα = (2evα np ) =σ αβ E β , (3.14) p

where the paraconductivity tensor components are: αβ σ(D) =

X vpα vpβ π 2 e αT . 3 4 ε p p

In the case of isotropic spectrum

58

(3.15)

 2 e 1   √ 3D case,    32ξ   α β X vp vp π αβ e2 1 σ(D) = e2 αT = 2D film, thickness :d  ξ, 3  4 εp 16d2   p     πe ξ 1 1D wire, cross − section :S  ξ 2 . 16S 3/2 (3.16) One can compare this result with that carried out in the Introduction from qualitative consideration, based on the Drude formula. Those simple speculations reflect correctly the physics of the phenomenon but were carried out with the assumption of the momentum independence of the relaxation time τp , taken as τ0 = 8(T π−Tc ) . As we have just seen, in reality τp decreases rapidly with increase of the momentum, the excess ”2” appeared in (3.12) because of the wave nature of fluctuation Cooper pairs; accounting for this circumstance results in the precise coefficients of (3.16), different from (1.11). An especially simple form the paraconductivity results in the 2D case, where, calculated per unit square, it depends on the reduced temperature only: e2 T . (3.17) 16~ T − Tc The coefficient in this formula turns out to be a universal constant and is given by the value ~/e2 = 4.1kΩ. For electronic spectra of other dimensionalities this universality is lost, and the paraconductivity comes to depend on the electron mean free path. Let us compare σ (T ) with the normal electron Drude part σn = ne e2 τ /m by writing the total conductivity σ (T ) =

e2 pF l 1 ( + ). (3.18) ~ 2π~ 16 One sees that at cr = 0.4/(pF l) ∼ Gi(2c) the fluctuation correction reaches the value of the normal conductivity. Let us recall that the same order of magnitude for the 2D Ginzburg-Levanyuk number was obtained above from the heat capacity study. We will discuss the region of applicability of (3.18) below in Section 8.6. It is worth mentioning that the results derived here for paraconductivity are valid with the assumption of weak fluctuations: for the temperature range  . Gi(D) they are not anymore applicable. Nevertheless, one can see that for not very dirty films, with p2F ld  1, a wide region of temperatures Gi(2d)    1 exists where the temperature dependence of conductivity σ=

59

is determined by fluctuations and in this region the localization effects are negligible. The transport equation (3.12) was originally derived many years ago by L.G.Aslamazov and A.I.Larkin [73]. Recently T.Mishonov et al. [74] rederived Eq.(3.12) and solved it for np in the case of an arbitrary electric field.

3.3

General expression for paraconductivity

Unfortunately the applicability of the master equation derived is restricted to weak magnetic fields (H  Hc2 ()). For stronger fields Hc2 () . H  Hc2 (0) the simple evaluation of averages in (3.11) turns out to be incorrect, the density matrix has to be introduced and the master equation loses its attractive simplicity. At the same time, as we already know, namely these fields, quantizing the fluctuation Cooper pair motion, present special interest. This is why in order to include in the scheme the magnetic field and frequency dependencies of the paraconductivity, we return to the analysis of the general TDGL equation (3.1) without the objective to reduce it to a Boltzmann type transport equation. Let us solve it in the case when the applied electric field can be considered as a perturbation. The method will much resemble an exercise from a course of quantum mechanics. To carry out the necessary generality side by side with a formal simplicity of expressions we will introduce some kind of subscript {i} which includes the complete set of quantum numbers and time. By a repeated subscript a summation over a discrete and integration over continuous variables (time in particular) will be supposed. We will look for the response of the order parameter to a weak electric field applied in the form (0)

(1)

Ψkz (r,t) = Ψ{i} + Ψ{i} ,

(3.19)

(0)

where Ψ{i} is determined by (3.4). Substituting this expression into (3.2) and restricting our consideration to linear terms in the electric field we can write (1)

(0)

b−1 ){ik} Ψ = 2ieγGL ϕ{il} Ψ (L {k} {l}

with the solution in the form (1)

b{ik} ϕ{kl} L b{lm} ζ{m} . Ψ{i} = 2ieγGL L

(3.20)

Let us substitute the order parameter (3.19) in the quantum mechanical expression for current 60

h i (0)∗ (1)∗ (0) b{ik} Ψ(1) b j = 2eRe Ψ{i} v + Ψ v Ψ {k} {i} {ik} {k} ,

(3.21)

b r}{ik} . b{ik} = i{H, v

(3.22)

(1)∗ (0)∗ e ∗ b{ik} Ψ(1) b{ik} Ψ(0) Ψ{i} v {k} = (Ψ{k} v {i} ) ,

(3.23)

b{ik} is the velocity operator which can be expressed by means of the where v commutator of r with Hamiltonian (3.3):

The second term of (3.21) can be written by means of a transposed velocity operator (which is Hermitian) as the complex conjugated value of the first one:

which results in

(0)∗

(1)

j = 2Re{Ψ{i} (2eb v{ik} )Ψ{k} } =

(3.24)

∗ b b b∗ v = −8e2 Im{γGL L {ki} b{il} L{lm} ϕ{mn} L{np} ζ{k} ζ{p} }.

b∗ Let us average now (3.24) over the Langevin forces moving the operator L {ki} from the beginning to the end of the trace and using (3.5). One finds b{lm} ϕ{mn} L b{np} L b∗ }. b{il} L j = −16T e2 Re(γGL )Im{γGL v {pi}

(3.25)

b{lm} operator is diagonal (it is Now we choose the representation where the L evidently given by the eigenfunctions of the Hamiltonian (3.3) ): L{m} (Ω) =

1 , ε{m} − iΩγGL

(3.26)

where ε{m} are the appropriate energy eigenvalues. Then we assume that the electric field is coordinate independent but is a monochromatic periodic function of time: ϕ(r,t) = −E β rβ exp(−iωt). (3.27) In doing the Fourier transform in (3.25) one has to remember that the time dependence of the matrix elements ϕ{mn} results in a shift of the frequency variable of integration Ω → Ω − ω in both L-operators placed after ϕ{mn} or, b{lm} for ω : what is the same, to a shift of the argument of the previous L jαω = 16T e2 Re(γGL ) × (3.28) Z dΩ β b α b b∗{i} (Ω)}Eβ , b{il} × <{γGL v L{l} (Ω + ω)[−ir{li} ]L{i} (Ω)L 2π 61

where
=i

β b{li} v

ε{i} − ε{l}

(3.29)

and, carrying out the frequency integration in (3.28), finally write for the fluctuation conductivity tensor (jαω = σ αβ (ω)Eβ ) : αβ

2

σ (, H, ω) = 8e T Re(γGL )

∞ X

(3.30)

{i,l}=0

"

< γGL

β α b{il} b{li} v v

∗ ε{i} (γGL ε{i} + γGL ε{l} − i|γGL |2 ω)(ε{l} − ε{i} )

#

.

This is the most general expression which describes the d.c., galvanomagnetic and high frequency paraconductivity contribution. In the case when we are interested in diagonal effects only, where it is enough to accept γGL as real (γGL = ReγGL = πα/8) omitting its small imaginary part, the last expression can be simplified by means of symmetrization of the summation variables): " # ∞ α α X b b v v π {il} {li} σ αα (, H, ω) = αe2 T < . 2 ε{i} ε{l} (ε{i} + ε{l} − iγGL ω)

(3.31)

{i,l}=0

Let us demonstrate the calculation of the d.c. paraconductivity in the simplest case of a metal with an isotropic spectrum. In this case we choose a plane wave representation. By using εp defined by (3.9) one has ∂εp = 2αTc ξ 2 p. (3.32) ∂p We do not need to keep the imaginary part of γGL , which is necessary to calculate particle-hole asymmetric effects only. Then the fluctuation conductivity calculated from (3.31) coincides exactly with (3.15). b{pp0 } = vp δpp0 , vp = v

3.4

Fluctuation conductivity of layered superconductor

Let us return to the discussion of our general formula (3.31) for the fluctuation conductivity tensor. A magnetic field directed along the c-axis still 62

permits separation of variables even in the case of a layered superconductor. The Hamiltonian in this case can be written as in (2.42), (3.3):   r 2 2 b H = αTc  − ξxy (∇xy − 2ieAxy ) − (1 − cos(kz s) . (3.33) 2 It it is convenient to work in the Landau representation, where the summation over {i} is reduced to one over the ladder of the Landau levels i = 0, 1, 2.. (each is degenerate with a density 2eH per unit square) and integration over the c-axis momentum in the limits of the Brillouine zone. The eigenvalues of the Hamiltonian (3.33) can be written in the form r εn = αTc [ + (1 − cos(kz s)) + h(2n + 1)] = εkz + αTc h(2n + 1), 2

(3.34)

eH where h = 2mαT was already defined by Eq. (2.48). For the velocity operators c one can write

bx,y = v

3.4.1

1 αrs bz = − (−i∇−2ieA)x,y ; v Tc sin(kz s). 2m 2

(3.35)

In-plane conductivity.

Let us start from the calculation of the in-plane components. The calculation of the velocity operator matrix elements requires some special consideration. First of all let us stress that the required matrix elements have to be calculated for the eigenstates of a quantum oscillator whose motion is equivalent to the motion of a charged particle in a magnetic field. The commutation relation for the oscillator’s velocity components is well known (see [75]): eH iαTc = h. (3.36) 2 2m m In order to calculate the necessary matrix elements let us present the velocity operator components in the form of boson-type creation and annihilation operators b a+ , b a with commutation relation [b a, b a+ ] = 1 : r  +  αTc h b a +b a x,y b = v ib a+ − ib a 2m by ] = i [b vx , v

One can check that the correct commutation relation (3.36) is fulfilled. Taking into account that < l|b a|n >=< n|b a+ |l >= 63

√

nδn,l+1

it is seen that the only non-zero matrix elements of the velocity operator are r  √  √ αTc h lδ + nδ l,n+1 n,l+1 x,y √ √ < l|b v |n >= . i lδl,n+1 − i nδn,l+1 2m Using these relations the necessary product of matrix elements can be calculated: αTc h (lδl,n+1 + nδn,l+1 ). (3.37) 2m Its substitution into (3.31) and accounting of the degeneracy of the Landau levels 2eH = 4mαTc h gives for the diagonal component of the in-plane conductivity tensor < l|b vx |n >< n|b vx |l >=

∞ πα2 Tc2 e2 X X (lδl,n+1 + nδn,l+1 ) σ (, H, ω) = h <[ ] (3.38) 4m εl εn (εl + εn − iγGL ω) {n}=0 {l}=0 Z π ∞ s dk X n+1 z 3 2 2 = πe (αTc ) h <[ ]. εn+1 εn (εn+1 + εn − iγGL ω) − πs 2π n=0 xx

Expanding the denominator into simple fractions we reduce the problem to the calculation of the c-axis momentum integral, which can be carried out in the general case by use of the identity: Z 2π dx 1 1 = −√ (3.39) 2 2π cos x − z z −1 0 valid for any complex parameter z 6= 1 with the proper choice of the square root branch. Using it we write the general expression for the in-plane component of the fluctuation conductivity tensor ∞

e2 h X (n + 1) σ (ε, h, ω) = 16s n=0 xx

(

1 1 p + h − ie ω [ + h(2n + 1)][r +  + h(2n + 1)]

1 1 p − h + ie ω [ + h(2n + 3)][r +  + h(2n + 3)]

where ω e=

πω . 16Tc

2h 1 p 2 2 h +ω e [ + h(2n + 2) − ie ω ][r +  + h(2n + 2) − ie ω]

64

(3.40) )

.

3.4.2

Out-of plane conductivity.

The situation with the out-of plane component of paraconductivity turns z b{in} out to be even simpler because of the diagonal structure of the v = αrs − 2 Tc sin(kz s) × δin × δ(kz − kz0 ). Taking into account that the Landau state −2 degeneracy 2eH = hξxy we write " # ∞ α α X b b v v 1 {il} {li} σ zz (, H, ω) = παe2 T < = 2 ε{i} ε{l} (ε{i} + ε{l} − iαTc ω e) {i,l}=0  2 X ∞ Z π 2 s dk πe (αTc )3 sr sin2 (kz s) z = h <[ 2 ]. π 2π 16 ξxy εn (kz )[εn (kz ) − iαTc ω e] n=0 − s The following transformations are similar to the calculation of the in-plane component: we expand the integrand into simple fractions and perform the kz −integration by means of the identity (3.39). The final expression can be written as

2 X  ∞  ∂ h − × (3.41) ∂λ n=0 ( 1 1 p − λ + ie ω ( + h(2n + 1) + λ)( + h(2n + 1) + λ + r) ) 1 p |λ=0 . ( + h(2n + 1) − ie ω )( + h(2n + 1) − ie ω + r)

πe2 σ (, H, ω) = 32s  < zz

3.4.3



sr ξxy

Analysis of the general expressions.

In principle the expressions derived above give an exact solution for the a.c.(ω  T ) paraconductivity tensor of a layered superconductor in a perpendicular magnetic field H  Hc2 (h  1) in the vicinity of the critical temperature (  1). The interplay of the parameters r, , ω h entering into (3.40)-(3.41), as we have seen in the example of fluctuation magnetization, yields a variety of crossover phenomena. 1. The simplest and most important results which can be derived are the components of the d.c. paraconductivity (ω = 0) of layered superconductor in the absence of magnetic field. Keeping ω = 0 and setting h → 0 one can change the summations over Landau levels into integration and find σ xx (ε, h → 0, ω = 0) = 65

e2 1 p , 16s [(r + )]

e2 s σ (, h → 0, ω = 0) = 2 32ξxy zz



  + r/2 −1 . [( + r)]1/2

2. The Aslamazov-Larkin contribution to the magnetoconductivity can be studied by putting ω = 0 and keeping magnetic field as arbitrary. We will not go into details and just report the results (following [169] with some revision of the coefficient in 3D case) hr (3D)

h σ xx

σ xx (, h = 0) −

σ zz

σ zz (, h = 0) −

e2 [8(+r)+3r2 ] 2 h 28 s [(+r)]5/2 e2 s r2 (+r/2) 2 2 [(+r)]5/2 h 28 ξxy

e2 √ 1 4s 2hr pr 3.24e2 s 2 ξxy h

max{, r}  h (2D) e2 1 8s h 7ζ(3)e2 s r 2 2 29 ξxy h2

Table 2. Here it is worth making an important comment. The proportionality of the fluctuation magnetoconductivity to h2 is valid when using the parametrization  = (T − Tc0 )/Tc0 only. As it well known, a weak field shifts the critical temperature linearly, which often makes it attractive to analyze the experimental data by choosing as the reduced temperature parameter h = (T − Tc (H))/Tc (H). In this parametrization one can get a term in the magnetoconductivity linear in h , which previously was cancelled out by the magnetic field renormalization of the critical temperature. So it is important to recognize that the effect of a weak magnetic field on the fluctuation conductivity cannot be reduced to a simple replacement of Tc0 by Tc (H) in the appropriate formula without the field. Vice versa, this effect is exactly compensated by the change in the functional dependence of the paraconductivity in magnetic field, and finally it contains the negative quadratic contribution only. 3. Letting the magnetic field go to zero and considering nonzero frequency of the electromagnetic field one can find general expressions for the components of the a.c paraconductivity tensor. They are cumbersome enough and in the complete form can be found, for instance, in [76]. We recall here the simplified asymptotics for the Reσ in the 2D regime only: xx Reσ(2D) (r  , ω e) = (3.42) " " ##  2 ω e   2 ω e e2 1 2 arctan − ln 1 + = 16s  ω e  ω e 

66

zz Reσ(2D) (r

e2  , ω e) = 8 2s



sr ξxy

2   2  2 1 ω e ln[1+ ]. ω e 

(3.43)

The general formulas (3.40)-(3.41) allow one to study the different crossovers in the a.c. conductivity of layered superconductor in the presence of magnetic field of various intensity. We leave this exercise for the reader having some practical interest in the problem.

3.5

Paraconductivity of nanotubes

Carbon nanotubes are mesoscopic systems with a remarkable interplay between dimensionality, interaction and disorder [77]. Recent experiments found that the electron transport in single-wall nanotubes (SWNT) has a onedimensional ballistic behavior [78]. Therefore it may be theoretically described within the model of one-dimensional interacting electron systems known as Luttinger liquid [79, 80, 81]. At the same time, the multiwall nanotubes (MWNT), which are composed of several concentrically arranged graphite shells, show properties which are consistent with the weak-localization features of the diffusive transport in magnetoconductivity and zero-bias anomaly in the tunneling density of states [82]. Similar properties have been observed in ropes of SWNTs [78, 83]. Very recent experimental works [84, 85] have addressed the problem of superconductivity in carbon nanotubes. In the article by Tang et al. [84] a superconducting behavior was detected in SWNT at a mean-field critical temperature evaluated as Tc =15 K. At the same time a pure superconducting state with zero resistance was not found and the authors attribute this fact to the presence of strong fluctuations which alter severely the superconducting order parameter both below and above Tc . In Ref. [85] ropes of SWNT were studied and a truly superconducting transition was discovered at Tc =0.55 K. The suppression of Tc by a magnetic field applied along the tube was also measured. In the present section we study the paraconductivity and corresponding magnetoconductivity of a carbon nanotube above Tc . In spite of the nanoscale size of the system we assume the validity of the GL formalism for the description of the fluctuation superconductivity and in purpose to justify this a little bit below will evaluate the Ginzburg-Levanyuk number. In order to describe the one-electron spectrum of carbon nanotubes one has to take into consideration that the electron wavelength around the circumference of a nanotube is quantized due to the periodic boundary conditions and only a discrete number of wavelengths can fit around the tube. 67

Along the tube, however, electronic states are not confined and electrons can move ballistically. Because of the circumferential modes quantization, the electron states in the tube do not form a single wide energy band but instead they split into a number of one-dimensional sub-bands with band onsets at different energies. Consequently, we assume the electron spectrum in the form: p2||

n2 (p) = + , 2m|| 2m⊥ R2

(3.44)

where n = −N, ..., N , N = [pF R], pF is the Fermi momentum and R is the nanotube radius. The number N is determined by the value of the chemical potential and the distance between the levels. It defines the number of electrons filling the 2N + 1 electron sub-bands of the nanotube electron spectrum. A typical value for a realistic nanotube is N ∼ 5 [86]. The longitudinal coordinate z and the angular variable ϕ are chosen as the natural coordinate system for the problem under discussion. The linearized time-dependent GL equation (TDGL) for the fluctuation order parameter Ψ (z, ϕ, t) takes the form:   2 ∂Ψ (z, ϕ, t) 2 ∂ 2 1 ∂ 2 b = HΨ (z, ϕ, t) = αTc  − ξq 2 − ξ⊥ ( − 2ieA⊥ ) Ψ (z, ϕ, t) , −γ ∂t ∂z R ∂ϕ (3.45) b where H is the GL Hamiltonian written for the nanotube geometry, A⊥ = 1 Φ HR = eR is the tangent component of the vector potential, ξq = (4mq αTc )−1/2 Φ0 and ξ⊥ = (4m⊥ αTc )−1/2 are the longitudinal and the transversal GL coherence lengths. The latter is supposed to be comparable with the nanotube radius: ξ⊥ ∼ R. The fluctuation order parameter Ψ can be presented as a Fourier series ∞ Z π/a X dqq ψn (qq , t) exp (−inϕ) exp (−iqq z) , (3.46) Ψ (z, ϕ, t) = 2π −π/a n=−∞ and the TDGL equation for the Fourier component ψn (t) is read as:

−γ

"

∂ψn (qq , t) = εn (qq , Φ)ψn (qq , t) = αTc  + ξq2 qq2 + ∂t

2 ξ⊥ R2



n−

2Φ Φ0

2 #

ψn (qq , t). (3.47)

b Here εn (qq , Φ) are the eigenvalues of H. 68

The angular quantization gives rise to rather distinctive critical temperatures corresponding to the different order h parameter  2  i modes. These critical ξ (n) (0) temperatures are: Tc (Φ = 0) = Tc 1 − R⊥2 n2 and the characteristic ξ2

dimensionless temperature difference is ∆ε0 ∼ R⊥2 . As expected, when the temperature decreases the system tends to the mode with n = 0. In nonzero magnetic fields, when the magnetic flux is Φ ∈ ] − Φ0 /2 , Φ0 /2[, the superconducting transition occurs at the ψ0 state. Let us move to the study of the paraconductivity in a small superconducting cylinder at temperatures above the critical one. We are interested here only in the longitudinal diagonal component of (3.31).The appropriate matrix elements of the velocity operator are cz il,pq = vp δpq δil , vpz = v

∂εp = 2αTc ξq2 p. ∂p

(3.48)

The summation over subscript {i} is carried out over the levels of the angular quantization up to the maximal number N and includes the integration over the z-axis momentum. As a result the general formula (3.31) for the longitudinal paraconductivity of a nanotube is read as παe2 T σ (, H) = 2S q

Z

dpq 2π

Z

N X πe2 ξq =  16S n=−N +

N q q bil,pq bli,qp v v dqq X (3.49) = 2π i,l=−N εi (pq ) εl (qq ) [εi (pq ) + εl (qq )]

1 2 ξ⊥ R2



n−

2Φ Φ0

2 3/2

(here S = πR2 is the cross-section area of the nanotube). This formula can be numerically evaluated to obtain the magnetoconductivity. Nevertheless, in order to get a qualitative understanding of the paraconductivity temperature dependence in zero field and its behavior in the presence of a magnetic field at fixed temperature, let us assume N  1 and try to proceed analytically.

3.5.1

Zero magnetic field

In the immediate vicinity of the critical temperature, where ξ⊥ ()  R, only the term with n = 0 in Eq. (3.49) contributes to the paraconductivity resulting in the 1D behavior of paraconductivity (σ q (, 0) ∼ −3/2 ). √ Relatively far from the critical temperature, where ξ⊥ () = ξ⊥ /   R (but still   1), the most convenient way to continue the analysis of the 69

Eq. (3.49) is to isolate the term with n = 0 and to treat the remaining sum separately. In this temperature range one can replace the sum in Eq. (3.49) with an integral to get

2

σ q (, 0) =

2

πe 1 πe ξq 3/2 + ξq 16S  8S



R ξ⊥

 3  

1 q

1+

R2 2 () ξ⊥

h

1+

q

1+

R2 2 () ξ⊥

1

i−q

N2 +

(3.50) One can see that not too far from the transition point R/N  ξ⊥ ()  R the first term in parenthesis in Eq. (3.50) dominates over all other contributions and Eq. (3.50) reproduces the 2D result σ q (, 0) ∼ −1 . Moving away from the critical region and reaching the temperatures where ξ⊥ ()  R/N , one can find the compensation of the leading order contributions of two terms in parenthesis. The account for next approximation in the root expansions gives the contribution of the same singularity in as the first term of Eq. (3.50), but with the enhancement factor 2N + 1. So the system returns to 1D behavior but, in contrast to the immediate vicinity of Tc , in 2N + 1 channels. All the asymptotics of Eq. (3.49) can be presented in a more compact form as: 
ξ⊥ 2 1  ,    3/2 R
2 e2 ξq   R  2 ξ⊥
2 ,    ξ⊥RN . (3.51) σ q (, 0) = ξ  R 2 ⊥ 16R 
 2  2N +1 , ξ⊥ N   R 3/2

The physics of these crossovers is the following. The first one has a geometrical nature: very near to Tc the fluctuation Cooper pairs are so large that they have only one degree of freedom to slide along the tube axis. The first line of Eq. (3.51) exactly reproduces the paraconductivity of a wire with 2 cross-section S  ξ⊥ . In the intermediate regime rotations over the tube surface become possible and the paraconductivity temperature dependence transforms into the 2D one. Finally, relatively far from Tc , where ξ⊥ () ∼ R/N , the last, most nontrivial, crossover 2D → 1D in the fluctuations dimensionality takes place. To recognize its physical sense let us remind that the value R/N characterizes the distance between the electron wavefunction zero’s in the N-th subband. The result of the averaging of the superconducting type electron correlations in confines of the stripes of this size, parallel to the cylinder axis will be evidently nonzero, while the pairing of the one-electron states belonging to different stripes will result in the average out of such contributions. In other words, relatively far from Tc , the value R/N characterizes the width of effective one-dimensional channels of 70

R2 2 () ξ⊥

h

N+

q

N2 +

the Cooper pair motion on cylinder surface and the fluctuation Cooper pairs transport takes place in each of such channels separately. It is why the total longitudinal paraconductivity acquires the degeneracy factor 2N + 1 equal to the number of subbands.

3.5.2

Non-zero magnetic field

We now move to the study of paraconductivity in the presence of a magnetic field applied. Due to the Little-Parks effect [87], the critical temperatures (n) Tc (Φ) are periodic functions of the flux through the tube with period Φ0 . Therefore we can restrict ourselvesto the flux range −Φ0 /2 < Φ < Φ0 /2,   2 ξ2 (0) where Tc (Φ) = Tc 1 − R⊥2 2Φ . Evidently, two different regimes can Φ0 √ take place: a weak-field one when Φ . Φ0 ξR⊥  (which is equivalent to H .  √ √ ) and a strong-field regime when Φ0 ξR⊥   Φ  Φ0 /2. The Hc2 ξR⊥ general formula (3.49) may be rewritten as πe2 σ (, Φ) = ξq 16S q



R ξ⊥

3

1 

2Φ Φ0

2

+

R2 2  ξ⊥

πe2 ξq 3/2 + 16S



R ξ⊥

3 X N n=1



n±

2Φ Φ0

1 2

+

(3.52) In the case of the weak-field regime one can easily see that the main magnetic field dependence comes from the renormalization of the critical temperature, and therefore the three limiting cases are recovered analogously with the preceding discussion of the zero-field case. In result δσ q (, Φ) = σ q (, Φ) − σ q (, 0) = 
ξ⊥ 2 3  ,      5/2 R 2    2
e2 ξq ξ⊥ Φ ξ⊥ 2 R 1 = − 4 ,  2 ξ⊥  R 8 R 4 Φ0 
  3(2N +1) , ξ⊥ N 2   5/2

R√


ξ⊥ N 2 (3.53) R

R

The strong-field regime Φ0 ξ⊥   Φ  Φ0 /2 can be reached (without passing to the next foil of the Little-Parks effect) in the case R  ξ⊥ () only. In such situation the main contribution originates from the first term in Eq. (3.52):  3 e2 Rξq Φ0 q σ (Φ) = 7 3 . (3.54) 2 ξ⊥ Φ 71

R2 2  ξ⊥

3/2 .


2 This result is valid for temperatures   ξR⊥ . In the temperature range

2 ξ⊥ 2    ξ⊥RN  1 (if such interval exists), where in the absence of R the magnetic field the fluctuations have a 2D character,√the effect of the magnetic field is relevant only for fields so high as Φ0 N ξR⊥   Φ  Φ0 /2, but it still is described by the formula (3.54). One can recognize in the effect of the magnetic field on paraconductivity the usual suppression of the effective fluctuation dimensionality, as it happens even in the 3D case. Nevertheless we would like to attract the reader’s attention to the unusually strong suppression of the nanotube paraconductivity in strong magnetic fields. Its comparison with the corresponding paraconductivity of a layered superconductor shows a remarkable difference in the critical exponent: 3 against 1 (see Ref. [169]). This follows from the channel separation in the Cooper pairs motion and hence the effective decrease of their density in the momentum space. A similar effect is observed in superconducting rings. In its 0D (0) (0) regime σring (H) ∼ H −4 (see Ref. [88]) instead of σgran (H) ∼ H −2 as for superconducting granules. Let us discuss the results obtained. In Fig. 1 we have plotted the resistivity of carbon nanotubes calculated from Eq.(3.52), choosing N = 5 for different magnetic field strengths. It can be seen that the simulated behavior is similar to the experimental one reported in Ref. [85] for metallic ropes of nanotubes. Discussing the application of our results to recent experimental data concerning realistic nanotubes [78, 82, 84, 85], it is important to remember that the physics of superconductivity in these systems is still controversial and it is very likely to be qualitatively different for systems like MWNT, the ropes of SWNTs or individual SWNT. Namely, the effect of interactions within multiwall tubes or hopping between neighboring tubes in a rope drives the system away from the one-dimensionality characterizing an individual nanotube. Therefore the physical properties are substantially altered in both the normal and superconducting states depending on whether hopping is effective or not. Nevertheless our considerations are quite general because the model proposed is based on the GL phenomenology which is independent of the specific pairing mechanism leading to the superconductivity. It is clear that an individual nanotube is rather within the one-dimensional limit of Eq. (3.50), ξ⊥ ()  R, while for multiwall tubes or ropes the other regimes may be observed. As it was demonstrated above, the nontrivial geometry of tube leads to a number of possible crossovers in the temperature dependence of the paraconductivity. The crossover closest to the critical temperature has a clear geometric nature and is analogous to the one occurring in thin films of layered superconductors [89]. As the system moves away from the transition 72

Figure 3.1: Theoretical prediction for the temperature dependence of the resistivity of carbon nanotube. Plotted is the resistivity calculated as a sum of normal-state temperature-independent contribution and paraconductivity versus the reduced temperature (T − Tc (0))/Tc (0). The field strengths are Φ/Φ0 = 0, 0.25 and 0.5. point, the coherence length decreases, thus rotations over the tube surface become possible and the system goes into the 2D regime. The last 2D → 1D crossover has an intrinsic origin, it occurs when ξ⊥ () . R/N . The alternative interpretation of different regimes of paraconductivity behavior can be given on the basis of comparison of the characteristic fluctuation Cooper pair ”binding energy”, T − Tc , with the angular quantization energy level structure. Here it is necessary to remind that the fluctuation Cooper pairs above the critical temperature are not condensed with the zero energy, like it happens below Tc , but they are distributed over energy with the rapid decay at ε & T − Tc . When ξ⊥ ()  R (T − Tc  1/2m⊥ R2 ) the binding energy is so small that the electrons occupying only the n = 0 level can be involved in fluctuation pairing. In result the 1D behavior takes place. As T − Tc growths (R/N . ξ⊥ () . R) the electrons from more and more subbands can be involved in pairing (within the same subband) and due to this additional degree of freedom (subband number n) the fluctuation behavior becomes 2D. Finally, when T − Tc exceeds the energy of the last filled level of angular quantization εN = N 2 /2m⊥ R2 ( what means ξ⊥ () . R/N ) all 2N + 1 subbands are involved in pairing and each one presents the independent one-dimensional channel. Indeed, the corresponding formula differs from the one near Tc by a factor 2N + 1 (see Eqs. (3.51) and (3.52)). Dealing with the superconductivity in such objects of the nanoscale size as nanotubes, ultra-small grains it is necessary to recognize that we already reach the limits of classical superconductivity and the variety of principally new phenomena appears[?]. The Ginzburg Landau description of the superconductivity in nanodrops is valid until the energy spacing of dimensional quantization turns to be much less than mean field value of the superconducting gap. In nanotubes this criterion seems to be less sever because of the possibility of the quasi-continues motion along the tube axis. We can evaluate the Ginzburg-Levanyuk number for the nanotube as it was done in [?]: Gi(1) =

1 2/3 (p2F S)

∼

1 π 2/3 N 4/3

1

(3.55)

and to see that in the limit N  1 some room for quasiclassical description 73

of the system still takes place.

3.6

Magnetic field angular dependence of paraconductivity

We have seen above that in the case of a geometry with a magnetic field directed along the c-axis many sophisticated fluctuation features of layered superconductors can be studied in the most general form. Nevertheless even the attempt to explore the d.c. conductivity in a longitudinal magnetic field (directed in ab plane) [90] or, moreover, with the field directed at some arbitrary angle θ with the c-axis leads to the appearance of the a vector potential component in the argument of cos(kz s) and the problem requires a nontrivial calculation of the matrix elements over the Mathieu functions. We already learned that at temperatures very near to the critical one (  r) the 3D fluctuation regime takes place. Here the size of the Cooper pairs along the c-axis is so large that the peculiarities of the layered structure do not play any more role. This means that only small values of kz are important in the kz -integrations, where the cos(kz s) in (2.42) can be expanded and the LD functional is reduced to its traditional GL form with an anisotropic effective mass tensor: (

3 B 4 X 1 F[Ψ] = d r a|Ψ| + |Ψ| + 2 4mµ µ=1  B2 H · B + − . 8π 4π

Z

3

2

  2 1 d + − 2eA Ψ µ i dxµ (3.56)

We will demonstrate below that in this case a scaling approach provides a direct access to the most general results by rescaling the anisotropic problem to the corresponding isotropic one on the initial level of the GL approach [91]. Let us suppose that the external field H is chosen to lie in the y − z plane and makes angle θ with the z− axis. For sake of simplicity and because the oxide superconductors are within high accuracy uniaxial materials, we choose mx = my = m∗ , while m−1 = 2αs2 r (compare with (2.40)). The z 2 effective anisotropy parameter γa = m∗ /mz = 2αs2 rm∗ < 1 is introduced. In (3.56) the anisotropy enters only in the gauge-invariant gradient term, so the simple rescaling of the coordinate axes: x = x e, y = ye, z = γa ze together with ex , A ey , A ez /γa ) will render this term the scaling of the vector potential: A = (A ex /γa , B ey /γa , B ez ) isotropic. The magnetic field evidently is rescaled to B = (B 74

and the last three terms in (3.56), describing the magnetic-field energy, are transformed to "

 2 3  e2 B 1 X ~ d 2e e ez2 )− + ( xy + B de r − A Ψ µ 4m µ=1 i de xµ c γa # e xy · Hxy B ez Hz ) . −2( +B γa

γa δF[Ψ] = 8π

Z

3

In short, we have removed the anisotropy from the gradient term but reintroduced it into the magnetic energy term. In general it is not possible to make both terms isotropic in the Gibbs energy simultaneously. However, depending on the physical question addressed, we can neglect fluctuations in the magnetic field, as was mostly done above. Let us demonstrate how the method works for the example of the d.c. fluctuation conductivity tensor which was calculated above for a magnetic field directed along the c-axis. We restrict our consideration to the 3D region (  r). One can write the scaling relations between the electric field and current components before and after the scaling transformation by means of a conductivity tensor and the anisotropy parameter: jx,y = e jx,y

ex,y Ex,y = E

jz ∼ evz ∼ γae jz ∂ϕ 1 e Ez ∼ ∼ E z. ∂z γa

(3.57)

Now let us rewrite the relations between the current and electric field vectors before and after the scale transformation jα = σαβ Eβ eβ . e jα = σ eαβ E

Comparing them with (3.57) and introducing the operator of the direct scaling transformation Tαβ   1 0 0 Tαβ =  0 1 0  , 0 0 γa eµ and express the conductivity one can write jα = Tαµe jµ , Eα = (T −1 )αµ E tensor as 75

σαβ = Tαµ σ eµρ Tρβ .

Now let us work in the already isotropic coordinate frame. We suppose that initially the magnetic field was directed along the c-axis and now we rotate it in the X-Z plane by the angle θe with respect to the initial direction. The conductivity tensor will be transformed by the usual matrix law:

and

where

T −1 e = RT σ σ eαβ (θ) )µς σςη (0)(T −1 )ηδ Rδβ αµ eµρ (0)Rρβ = Rαµ (T

e = Tαγ σ e δβ = Tαγ RT (T −1 )µς σςη (0)(T −1 )ηδ Rδκ Tκβ , σαβ (θ) eγδ (θ)T γµ

Rαβ



 cos θe 0 − sin θe . 1 0 = 0 sin θe 0 cos θe

Finally the fluctuation conductivity tensor σαβ (θ) in the initial tetragonal system with the magnetic field directed at the angle θ with respect to the c-axis can be expressed by means of the effective transformation operator Mαβ : T e e e ηβ (θ), σαβ (θ) = Mας (θ)σςη (0, H)M

with e = (T −1 )αδ Rδκ Tκβ Mαβ (θ)



 cos θe 0 − γ1 sin θe   = 0 1 0 . γa sin θe 0 cos θe

The angle p θe can be expressed via the renormalized magnitude of the magnetic e = Hc2 + γa2 Hx2 : field H cos θe =

ez H cos θ γa sin θ ; sin θe = p . =p 2 2 2 2 e H cos θ + γa sin θ cos θ + γa2 sin2 θ

In the case of the paraconductivity of a layered superconductor with the magnetic field applied at an arbitrary angle θ the answer can be written in 76

the general form by means of the three diagonal components of conductivity e in the perpendicular field H: e σii (0, H) σαβ (θ) =

cos2

1 × θ + γa2 sin2 θ



σxx cos θ+  +γa4 σzz sin2 θ ×  0 2 σzz γa sin θ cos θ

0 σyy (cos2 θ + γa2 sin2 θ) 0

 2 σzz γa sin2 θ cos θ−  −σxx sin θ   0 2 σzz cos θ

In the simplest case of a longitudinal field θ = 900 :  γa2 σzz (0, γa H) 0 −γa−2 σzz (0, γa H) . 0 σyy (0, γa H) 0 σαβ (900 , H) =  0 0 0 (3.58) 

3.7

Nonlinear paraconductivity in strong electric field

77

Chapter 4 Fluctuations in vortex structures The properties of the vortex state of second type superconductors have been described in details in handbooks devoted to superconductivity (see, for example, [116, 235, 234]) and review articles [236, 239, 240]. It is why the main goal of the present chapter will be only the discussion of the effect of fluctuations on them. We will restrict our consideration by simple estimations which nevertheless permit to recognize the qualitative picture of the phenomena. There are two different types of fluctuations which affect considerably on the properties of the vortex lattice. The first one is quenched disorder(structure fluctuations). The second type of fluctuations changing qualitatively the properties of the vortex lattice are already studied above thermal fluctuations of the order parameter.

4.1

Vortex lattice and magnetic flux resistivity

What is a vortex lattice? In 1949 Onsager introduced the notion of quantum vortices in superfluid helium. In 1957 Abrikosov discovered similar defects in superconductors in magnetic field and showed that the magnetic field Hc1 ≤ H ≤ Hc2 penetrates in superconductor of the second type in the form of the regular vortex lattice. The lower critical field Hc1 is determined by the formation in the system of the first isolated vortex of the normal phase in the superconducting state. The upper one, Hc2 , corresponds to the formation of the first equilibrium superconducting nuclei in the volume of the bulk normal state (neglecting fluctuations). These conclusions were done on the basis of the analysis of the system of GL equations 78

 2 2πi 2 ξ ∇+ A Ψ+ Φ0 2 2 |Ψ0 | λ 2 ∇× (∇

|Ψ|

|Ψ|2 1− |Ψ0 |2

× A) + A = −

!

Ψ = 0,

Φ0 ∇ϕ. 2π

(4.1)

(4.2)

In the defined above conditions this system has the periodic solution in the form of the triangular lattice of vortices with the period a4 =



2 √ 3

1/2 

Φ0 B

1/2

(4.3)

(B = H + 4πM ). The coordinate depending order parameter of the isolated vortex (ξ  a) can be approximately presented as r

Ψ(r, θ) = Ψ0 p

r2

+

2ξ 2

eiθ .

(4.4)

One can see that the amplitude of the order parameter is depressed in the vortex core region of the size ∼ ξ, while its phase changes in 2π when one circulates the vortex. If there is no quenched disorder in superconductor (there are no pinning centers) an applied current causes the drift of the vortex lattice and dissipation appears[245]. Indeed, when the transport current of the density jtr runs through superconductor the Lorentz force FL = jtr Φ0 applied to the unit length of a unique vortex appears. In homogeneous superconductors under the effect of this force the vortices start to move with the velocity vv . The corresponding flow of the magnetic flux induces, in accordance with the Faradey law, the electric field E = (vv /c) B and energy dissipation of the density jtr E = jtr vv B/c. This dissipation is equal to the power losses of the Lorentz force, i.e. FL vv = jtr vv B/c.

(4.5)

The resistivity ρf appearing in the process of the magnetic flux flow in the direction perpendicular to the transport current is called the flux flow resistivity: ρf = E/jtr .

(4.6)

The energy dissipation takes place only in the region of the vortex normal core, thus the value of ρf can be evaluated as the normal metal resistivity

79

multiplied on the square occupied by vortices per unit cross-section [245, 246, 247, ?]: ξ2 B ρf ∼ ρn 2 ∼ ρn . a4 Bc2

(4.7)

Close to Tc more rigorous treatment of the problem is possible. It takes into account the renormalization of the normal excitation spectrum inside the core and for fields not very close to Bc2 results in [240]:   2 B p 1 − T /Tc 1 + . (4.8) ρf ∼ 0.2ρn Bc2 (T ) τε (Tc − T ) Here τε is the energy relaxation time. All reported in this section results were obtained without taking into account the fluctuation effect to which discussion we pass below.

4.2

Collective pinning

In order to have a quiet superconducting life in our laboratory system we should prevent the motion of vortices, i.e. to introduce the dry friction of vortices. So, we should create, either naturally or artificially, the centers which would pin the vortex lattice. Thus to obtain the global superconductivity, we should destroy superconductivity at rare random positions or, in other words, to introduce the quenched disorder. Let us mention that such disorder is static by nature and manifests itself as the fluctuations of the GL functional parameters a, b, c. They appear due to the structure inhomogeneities of the initial crystalline lattice (dislocations, accumulations of impurities, separation of the other phase grains). Usually these fluctuations are small and they weakly affect on the properties of superconductor. Nevertheless even small structural fluctuations change qualitatively the properties of the vortex structure. The matter of fact that being the centers of pinning they break down the Galilean invariance and result in appearance of friction. Let us start our consideration from the case of interaction of isolated vortex with some pinning center. Let us suppose its size to be small in comparison with the vortex core size ξ and the local critical temperature Tc to be lower than the average one. The GL coefficient a can be presented the form a = a0 + δa (r) . In the first order in δa the vortex energy in the field of such pinning center is Z 2 Ep = |Ψ (r)| δa (r0 ) dr0 (4.9) 80

where r is the distance between the center of pinning and the vortex axis. If δa (r0 ) < 0 then the vortex energy is minimal when its axis passes through the pinning center. The displacement of the vortex from this position results in the appearance of the attractive force f = −∂Ep /dr. The problem of summation of such forces induced by randomly distributed pinning centers presents the central problem of the pinning theory[?, 237, 236]. Such average force would be equal to zero if the vortex lattice were incompressible. The presence of strong pinning centers in compressible lattice can cause a plastic deformation of the vortex lattice[241] and give rise to dry friction. However, such type of pinning centers exists only in strongly disordered low temperature superconductors (rigid superconductors) and does not help to resolve the problem for other (for example HTS). The analogous desperate situation was formed after the individual terrorism of the end XIX century, when Lenin told: “This is a wrong way, we shall go another way”. There is another way of pinning, the collective one. A single weak center causes only a weak elastic deformation, but a collective effect of a large number of weak centers destroys the lattice. A common belief is that a weak force leads to only a small distortion of a system. The result of the action of many weak forces applied at many different points of the system may depend on a system: it may lead either to a weak or to a strong distortion. If you accidentally step on the foot of another person, you say: “excuse me” and it’s okay. But what would happen if a lot people step on the feet of other people? The result depends on the system. For example, if while entering this room a lot of people stepped on other peoples’ feet, we wouldn’t be too disturbed if these are our friends. But if the offenders were all strangers, it could lead to disorder. A similar situation occurs in solid state physics. A small concentration of impurities does not destroy the crystalline lattice. The long range order remains, because these impurities belong to the lattice and move with it. If the concentration of impurities is small, then even a large number of them in a macroscopic body does not destroy the crystalline lattice. Nevertheless, there are other types of lattices, such as the Abrikosov lattice of vortices in superconductors. Here impurities (centers of pinning) belong to our laboratory system and do not move together with vortices. Such pinning centers destroy the long-range order in the lattice. This results in appearance of a friction force, a critical current, a hysteresis. Short-range order exists only at distances shorter than the correlation length Lc [242] to which discussion we pass now.

81

4.2.1

Correlation length

How does the destruction of long-range order happen? A single vortex displaced by a weak pinning center transmits its displacement to other vortices by elastic forces. The displacement of vortices u caused by a force f due to a single pinning center, leads to displacements which only slowly decay with the distance r. Writing down the equation of elasticity ∆u = f δ (r) one finds its solution f , (4.10) Cr here C is elastic modulus and we ignore its tensor nature1 . Due to the longrange character of the interaction (4.10) every vortex of the lattice receives information from a million of other vortices: “A pinning-center from another laboratory system has stepped on my foot”. As a result, this vortex displaces strongly, and the long-range order in the vortex lattice disappears at distances L  Lc . At shorter distances L  Lc , the crystalline order is destroyed weakly and all vortices are displaced by the same distance u. One can find the correlation length Lc as follows. Pinning centers affect vortices independently in various directions, so we have to average its square value. Thus the square of the vortex displacement hu2 i in a crystal of size L can be estimated as the square of the typical force multiplied on the number of pinning centers in volume L3 :  2 f 2 hu (L) i ∼ nL3 , CL u=

n here is the density of pinning centers. As the size of the system grows, p hu2 i ∝ L increases. When hu2 i reaches the size of vortex core ξ, vortices can be assumed as uncorrelated. Thus Lc defined by the condition Lc =

C 2ξ2 , nf 2

(A.L., 1970)

can be chosen as the correlation length of the vortex lattice. At large L  Lc h[u (L) − u (0)]2 i ∼ L2ζ 1

Let us recall that in usual crystals impurities moves together with the lattice, the displacement u falls down quicker, u ∼ 1/r2 , and collective effect does not arise.

82

where ζ is so-called static critical exponent. Its calculation still remains an open problem. The order parameter in a superconductor with the regular Abrikosov’s vortex lattice, varies periodically in space. It can be compared with an antiferromagnet. As a result of collective pinning the order parameter ∆ becomes a random function of position in space, similar to the spins in spin glasses. The vortex lattice behaves here like a glass. The glassy phase properties are not yet fully studied. The existence of the long-range order of the Abrikosov’s lattice, the same like in usual crystals, means that the δ-function peaks at the Bragg’s wave vectors has to be observed in the scattering amplitudes of the neutrons and X-rays. Let us remind that presence of small concentration of impurities in usual crystal does not destroy the long-range order, decreases the amplitude of such peaks but does not smear it out. In contrast to usual crystal lattice, the collective pinning in Abrikosov’s one leads to destruction of the long-range order in vortices positions. Nevertheless a weak logarithmic singularity in the form-factor at the Bragg’s wave vectors remains[256, 257, 258].

4.2.2

Critical current

Each region of size ∼ Lc finds its position of equilibrium at a minimum of the random pinning potential. In order to move these regions from those positions a force is needed. This force of a dry friction Ff r in the equilibrium is equal to the Lorentz force jc B, so to evaluate the critical current let us estimate Ff r . Each pinning center produces randomly directed force f. In purpose to find the average pinning force per unit volume one has to calculate the mean square value of these random forces pP √ 2 f n i fi ∼ 3/2 . (4.11) Ff r = L3c Lc Now, comparing this result with the Lorentz force, we obtain for the critical current jc [?, 237] 1 n2 f 4 jc = . (4.12) B ξ3C 3 One can notice that for a hard lattice with a large elastic modulus C a friction is less. This news would not be surprising for the experienced driver: it is well known that strongly pumped tires are less confident in breaking than the weak ones. Let us remind that the formula (4.12) was obtained omitting the tensor nature of the elasticity. In reality, even isotropic crystal is characterized by 83

several constants: by the moduli of compression C11, tilt, C44 and shear, C66 . Being the components of the dynamical matrix of the theory of elasticity, these values, generally speaking are some functions of the quasi-momentum k. Nevertheless this dispersion has to be taken into account only when the correlation length Lc turns out less than the penetration depth λ. When Lc . λ, it is more energetically favorable to deform the vortex line without the deformation of magnetic field lines. Indeed, in the absence of pinning centers the vortex line coincides with the magnetic field one. The introduced pinning center drags the core of the vortex line and in principle, magnetic field line could be deformed together with the vortex too. Nevertheless the magnetic field energy related with the large scale field distortion exceeds the local gain of the elastic energy near the pinning center and the vortices locally deviate from magnetic field lines. In this case the critical current depends exponentially on the concentration of the pinning centers[240]. Even more important that in these conditions the critical current jc grows exponentially when the magnetic field tends to Hc2 (so-called peak effect[259]). When Lc & λ the peak-effect near Hc2 still takes place but it has another origin. The matter of fact that close to Tc the vortex lattice becomes soft and even one weak pinning center can generate plastic deformation and ti pin the lattice[237]. Analogously to the theory of phase transitions the renormalization group (RG) method turns out to be useful in the pinning theory [261]. In contrast to the former the Gell-Mann-Low function here has an opposite sign and the effective charge (pinning force) grows with the distance increase (compare with the confinement in quantum chromodynamics). One can recognize the physical sense of this statement as follows. In the pinning theory the effective charge is equal to the averaged force of disordered pinning centers. We have demonstrated above that this force increases with the distance (see (4.11)) and becomes large at the distance of the order of Lc . Other important difference between the RG method in the theories of pinning and phase transition consists in the object which for the Gell-Mann-Low equation is written down. Instead of the Gell-Mann-Low function depending on the only effective charge (phase transition theory) in the pinning theory the Gell-Mann-Low functional has to be introduced[262]. This functional depends on the correlator of pinning forces which is in its turn the function of the displacement u. Such RG description is especially convenient when Lc . λ, inter-vortex interaction decreases slowly, elastic moduli have an anomalous dispersion and the problem is equivalent to the four-dimensional one. One can recognize this statement appealing to the described above situation with the RG application to the ferroelectric with the long-range dipole-dipole forces, where the angular dependence of the interaction trans84

forms the problem in effectively four-dimensional what permits to resolve the RG equations[?]. The two loop approximation of the functional RG permits to find Lc and jc more precisely [?].

4.2.3

Collective pinning in other systems

Collective pinning is a generic phenomenon which arises in a number of physical situations. For instance, for the charge density wave [261, 265, 266, 267] the role of critical current is played by the critical electric field Ec . The collective pinning phenomenon arises when two rough elastic surfaces are brought into a contact. The renormalization group method allowed to find the force of dry friction [268]. In a similar way the destruction of the long order occurs in ferromagnets [269]. In this problem the magnetic rigidity plays the role of elastic modulus, and the random magnetic field plays the role of the pinning centers.

4.3

Creep

Quenched fluctuations transform the Abrikosov’s lattice in some kind of glass. Omitting thermal and quantum fluctuations one can find that the persistent superconducting currents with the density less than jc can still run in this glass state. Nevertheless it is necessary to remind that the current carrying state in such hard type II superconductor is only metastable and thus tends to decay. The account of thermal and quantum fluctuations shows that under their influence the flux lines can overcome their pinning barriers and seldom jump to the neighbour favorable pinning valley. This leads to a thermally activated motion of vortices known as the creep [243, 244]. The average velocity of such vortex creep is determined by the Arrenius type thermal activation process probability:   U (j) , (4.13) v = vc exp − T where the ”pinning barrier” height U (j) depends on the carrying current. The equation 4.13 determines the nonlinear diffusion equation for the current density   dj jc U (j) ≈ − exp − (4.14) dt τ T which can be solved with the logarithmic accuracy [?]:

85



t U (j) = T ln 1 + t0



,

(4.15)

where t0 , the same as τ,depends on the sample size. The characteristic property of a glass system is the existence of a huge amount of the metastable states separated by energy barriers of different heights. During the time of experiment t the system succeeds to overcome only relatively low barriers of the heights U < T ln t/t0 and still stays in some metastable state separated from the more energy favorable ones by higher barriers. The maximal barrier height depends on the flowing current but the exact form of this dependence is still unknown. One can only say that U (j) → 0 when j → jc and U (j) → ∞ when j → 0. The interpolation formula can be written as:  µ  jc Uc −1 . (4.16) U (j) ' µ j The energy scale for the pinning barrier U is determined by the collective pinning energy Uc . The qualitative character of the dependence of the energy of the vortex system versus some generalized coordinate at different values of flowing current j one can see in the Fig.K( fig 9 from the review in RMP). Evidently that at j > jc no more metastable states take place, barriers do not exist more. When jc − j  jc the height of the barriers is small. The reason of the barrier height growth with the decrease of current one can recognize looking on Fig K+1 (fig.8 RMP). When j = 0 the vortex (or the system of vortices) is located in one of the most profound valleys and the neighbour valleys are higher in energy. The switch of the current means the appearance of the Lorentz force, hence the formation of the common slope in the ”mountains chain” and formation of the low energy valleys. Nevertheless for small values of j such valleys will appear far enough from the initial position of the chosen vortex and the displacement to the new valley only of the long enough its segment can result in the energy decrease. To reach the new valley this long segment shall overcome a lot of high tops. It is why the barrier height growth with the increase of the segment length, i.e. with the current decrease. The time evolution of the screening current can be easily found from (4.15) knowing the explicit form of U (j) (4.16):   −1/µ µT t j (t) = jc 1 + ln 1 + . Uc t0 86

(4.17)

Under the theory of collective creep, the dynamical critical exponent µ is related to that one for static properties (ζ) [?] 2ζ + D − 2 . (4.18) 2−ζ We see that the temporal decay of the transport current is thus determined by the ratio T /Uc , which can be found experimentally by measuring the relaxation of the diamagnetic moment of a sample in the critical state. The activation energy Uc is therefore experimentally accessible quantity and it provides one test for validity of the weak collective pinning theory. Typical experimental results for the activation energy Uc , obtained in magnetic relaxation experiments at low temperatures are in the range Uc ∼ 100 − 1000K[?, ?, ?, ?, ?, ?, ?]. For conventional hard type-II superconductors T /Uc ∼ 10−3 [?, ?] and the creep phenomenon can be observed only for currents close to the critical one. Expanding (4.17) for small T /Uc we reproduce the famous logarithmic time decay of the diamagnetic current [?]:    T t j (t) = jc 1 − ln 1 + . (4.19) Uc t0 In the new oxide superconductors, however, the corresponding decay coefficients turn out to be much larger, reaching the value 10% at T ∼ 0.5Tc . These large logarithmic decay rates are a result of various factors, such as the high temperatures available in an experiment, the small pinning energies Uc , which in turn are a consequence of the small coherence length ξ, and the large anisotropy of oxides. Combining the large decay coefficients with a typical logarithmic time factor ln (t/t0 ) of the order of 20 (waiting time t ≈ 1minute, the characteristic t0 may be evaluated as 10−6 s ) we have to conclude that the experimentally measured current density j has been roughly halfed due to creep, as compared with the critical current density jc even at such low temperatures as T ∼ 10K. Therefore, it is important to realize that the determination of the critical current density in the oxides is always affected by the presence of creep, and the condition jc − j  jc is no longer fulfilled. The expression ”giant creep” was therefore introduced [?] to describe the phenomenon of very large creep rates characteristic for the oxide superconductors. From a fundamental science point of view the case j  jc and, in particular, the limit j → 0 is very interesting too: wishing to probe the thermodynamic state of the vortex structure, we should perturb the system only infinitesimally and record its response. For a truly superconducting state we would expect to observe the vanishing resistivity ρf in the limit j → 0 µ=

87

or, to put it somewhat differently, to see a sublinear ”glassy” response of the vortex structure. In fact,as it is seen from (4.16) barriers U (j) against creep diverge algebraically with vanishing current density j, which implies a strongly subohmic current-voltage characteristics of the form[263]:   µ  Uc jc V ∼ exp − . (4.20) T j This exponential and strongly non-linear current-voltage characteristic means that the glassy properties show up not only in the structure of the vortex lattice, but also in its dynamics [263]. We have considered above the creep generated by thermal fluctuations. At low temperatures more probable turns out the quantum tunneling of vortices under the potential barriers [249, 250, ?, 252, 253, 254]. The probability of such tunneling is determined by the action of the underbarrier action S (j) :   S (j) v = vc exp − . (4.21) ~ Repeating the same speculations as in the case of the thermal creep one can find:    ~ t j (t) = jc 1 − ln 1 + . (4.22) Sc t0

4.4

The melting of the vortex lattice

Without the account for the quenched disorder and thermal fluctuations in a type-II superconductor at the magnetic field H = Hc2 the phase transition of the second order in the Abrikosov’s periodic structure takes place. The question arises: how the character of this transition changes due to fluctuations? Let us start our consideration from the thermal fluctuations only. Similarly to the situation in usual crystals the thermal fluctuations result in the melting of the vortex lattice [255]. This effect turns out to be especially important for HTS. As we already know the region of strong fluctuations is determined by the Ginzburg-Levanyuk number Gi, hence the melting of the vortex lattice one can expect at the temperatures of the order GiTc below critical temperature (magnetic field is supposed to be fixed). The quantitative consideration shows that the large numerical factor appears in our evaluation what results in the vortex lattice melting at temperatures noticeably below the critical one. For instance in normal solids the only energy parameter which can 88

be constructed from the fundamental values is the Bohr energy ∼ 105 K. Neverthless the energy of atomic interactions by numeric reasons turns out one order less than the Bohr energy, while the melting temperature is yet 1-2 orders less than the interaction energy. Another example: the melting of the Vigner crystal takes place at the temperature T = 10−2 e2 /r, where r is the interelectron distance. The melting temperature is determined by the equality of the free energies of solid and liquid phases. Due to mentioned above numerical smallness the lattice free energy can be calculated in the harmonic approximation, while the calculation of the liquid phase free energy is an extremely difficult task. The Gi parameter of conventional superconductors is very small and the melting of the vortex lattice takes place in the immediate vicinity of Tc (or Hc2 ). In this case the calculations of the fluctuation free energy of the vortex liquid can be simplified been done in the approximation of the lowest Landau level (LLL)[285, 286, 287, 288, 289, 290]. It is based on the fact that in a strong magnetic field the momentum perpendicular to the magnetic field direction is quantized and there appears a region where the lowest Landau level plays a dominant role. Such calculations were carried out up to 9-th order in Gi for the 3D case and up to 11-th order for 2D case. Carrying out the summation of these series by the Borel-Pade’ method one can find the free energy of the liquid phase. Comparing it with the free energy of lattice one can find Tm − Tc (H) = yGiH Tc (H) ,

(4.23)

where y = −10 for 2D case and y = −7 for 3D case and GiH = is the Ginzburg-Levanyuk number in magnetic field In the case of high temperature superconductors the Gi number is already not too small. It is why the melting field Hm is not too close to Hc2 and the LLL approximation does not work more. In the general case the melting temperature Tm is determined by the empirical Lidemann criterion which relates the mean-squared thermal displacement uT , introduced as

X 1  u2T = T Ck 2 k

(C is the elastic moduli matrix), with the lattice parameter a : q hu2T i = cL a.

(4.24)

(4.25)

The empirical Lidemann parameter cL belongs to the interval 0.1-0.2. Coming back to the vortex lattice one can say that the smallness of the melting 89

temperature is related with the smallness both of the value of the Lidemann’s parameter and the shear modulus of the triangular lattice. Calculating the elastic moduli of the vortex lattice one p can find from (4.24) the temperature and magnetic field dependence of hu2T i. Finally substituting it to the Lidemann criterion (4.25) one can find the temperature dependence of the melting field Bm (T ) : " √ # p
4 2−1 bm (t) t 2πc2 √ p (4.26) + 1 = √ L. 1 − bm (t) 1 − t Gi 1 − bm (t) (here we use the dimensionless variables bm (t) = Bm (T ) /Bc2 (T ) and t = T /Tc ). In fields close to Bc2 (T ) one can get [1 − bm (t)]3/2 =

t 0.26 √ Gi √ . 2 cL 1−t

(4.27)

Comparing this expression with (4.23) and taking into account that GiH = Gi?? we find that they are equivalent in the case cL = 0.14. In the opposite limit Bm (T )  Bc2 (T ) from the general equation (4.26) one can find 1 − t π 2 c4L Bc2 (T ) . (4.28) t2 8Gi Comparison of this equation with the numerical results of [?, ?] shows their coincidence with cL = 0.25. The mechanism of the influence of thermal fluctuations on the vortex lattice can be easily realized in the assumptions of strongly second type superconductor (κ  1) and relatively week fields B  Bc2 (T ). The large value of κ means that the order parameter disappears in very narrow (of the order of ξ  a4 ) domains only so the vortex lattice reminds the normal crystal one with the only difference that it constructed of lines instead of particles. Temperature tilts of the vortex lines can result in the melting of the lattice analogously to the melting of a crystal lattice due to the thermal oscillations of its atoms. The precise analytical theory of the lattice melting does not exist, but with good accuracy for evaluation the Lidemann criterion is used. In 2D case in the narrow region between the liquid and lattice can exist socalled hexatic phase [?] where the translational invariance does not exist more although still conserves the orientational order in the vortices positioning. Weak quenched disorder changes weakly Tm and the melting latent heat. Nevertheless when the induced by the quenched disorder vortex displacement Bm (t) =

90

becomes comparable with the thermal displacement the phase transition can change its order from the first to the second [?]. Close to the second order liquid-glass transition the scaling theory was developed [264]. One has to remember that this is the transition from vortex liquid to vortex glass, where there are many metastable states. It is why the character of the transition can be dependent on the rate of freezing [?]. In conventional superconductors where Gi is small quenched disorder can smear out the phase transition stronger than thermal fluctuations. In this case the transition has a percolation character and will be discussed in Chapter ?

91

Part II Basic notions of the microscopic theory

92

Chapter 5 Microscopic derivation of the Time-Dependent Ginzburg-Landau equation 5.1

Preliminaries

We have seen above how the phenomenological approach based on the GL functional allows one to describe fluctuation Cooper pairs (Bose particles) near the superconducting transition and to account for their contribution to different thermodynamical and transport characteristics of the system. Now we pass to the discussion of the microscopic description of fluctuation phenomena in superconductors. The development of the microscopic approach is necessary for the following reasons: 1. This description permits microscopic determination of the values of the phenomenological parameters of the GL theory. 2. This method is more powerful than the phenomenological GL approach and permits treatment of fluctuation effects quantitatively even far from the transition point and for magnetic fields strong as Hc2 , taking into account the contributions of dynamical and short wavelength fluctuations. 3. The electron energy relaxation times in metals are relatively large (τε  ~/T ) which causes the electron low frequency dynamics to be sensitive to the nearness to the superconducting transition. This is why the temperature dependence of fluctuation corrections can be determined generally speaking not only by the Cooper pair motion but also by changes in the single-electron properties. 4. There are some fluctuation phenomena in which the direct Cooper pair contribution is considerably suppressed or even absent altogether. Among

93

them we can mention the nuclear magnetic relaxation rate, tunnel conductivity, c-axis transport in strongly anisotropic layered metals, thermoelectric power and heat conductivity where the fluctuation pairing manifests itself by means of the indirect influence on the properties of the single-particle states of electron system. Formally in the above consideration averaging over the superconducting order parameter has been accomplished by means of a functional integration over all its possible bosonic field configurations. In this description we have dealt with the fluctuation Cooper pair related effects only and the method of the functional integration turned out to be simple and effective for their description. In the following Sections we will develop the diagrammatic method of Matsubara temperature Green functions which is more adequate for the description of the properties of a Fermi system of interacting electrons.

5.2

The Cooper channel of electron-electron interaction

Let us start the microscopic description of fluctuation phenomena in a superconductor from the electron Hamiltonian. We will choose it in the simple BCS form1 : H=

X p,σ

+ e E(p)ψep,σ ψp,σ − g

X

p,p0 ,q,σ,σ 0

+ + ψep+q,σ ψe−p,−σ ψe−p0 ,−σ0 ψep0 +q,σ0 .

(5.1)

The momentum conservation law side by side with singlet pairing are already taken into account in the interaction term. Here E(p) is the quasiparticle spectrum of the normal metal; g is the negative constant of electron-electron attraction which is supposed to be momentum independent and different from zero in a narrow domain of momentum space in the vicinity of the Fermi surface where pF −

ωD ωD < |p|, |p0 | < pF + . vF vF

+ ψep,σ and ψep,σ are the creation and annihilation field operators in the Heisenberg representation, so the first term is just the kinetic energy of the 1

We suppose that reader is familiar with the BCS formulation of the theory of superconductivity (see for example, [105]).

94

non-interacting Fermi gas. The interaction term is chosen in the traditional form characteristic for the electron-phonon mechanism of superconductivity2 . For the description of the properties of an interacting electron system with the Hamiltonian (5.1) we will use the formalism of the Matsubara temperature diagrammatic technique. The state of a non-interacting quasiparticle is described by its Green function G(p, εn ) =

1 , iεn − ξ(p)

(5.2)

where εn = (2n + 1)πT is a fermion Matsubara frequency and ξ(p) = E(p)−EF is the quasiparticle energy measured from the Fermi level. The effective electron-electron attraction leads to a reconstruction of the ground state of the electron system which formally manifests itself by the appearance at the critical temperature of a pole in the two particle Green function + L(p, p0 , q) =< Tτ [ψep+q,σ ψe−p,−σ ψep+0 +q,σ0 ψe−p 0 ,−σ 0 ] >,

where Tτ is the time ordering operator and 4D vector notations are used [105]. As it well known the two particle Green function can be expressed in terms of the vertex part [105]. In the case under consideration it is the vertex part of the electron-electron interaction in the Cooper channel L(q, Ωk ), which will be called below the fluctuation propagator. The Dyson equation for L(q, Ωk ), accounting for the e-e attraction in the ladder approximation, is represented graphically in Fig.5.1 . It can be written down analytically as L−1 (q, Ωk ) = g −1 − Π(q, Ωk ),

(5.3)

where the polarization operator Π(q, Ωk ) is defined as a loop of two singleparticle Green functions: X Z d3 p G(p + q, εn+k )G(−p, ε−n ). (5.4) Π(q, Ωk ) = T (2π)3 ε n

2

Fluctuations in the framework of more realistic Eliashberg [103] model of superconductivity were studied by B.Narozhny [104]. He demonstrated that the strong coupling does not change drastically the results of the weak coupling approximation. The critical exponents turn out to be exactly the same as in the framework of the GL theory, which provides an adequate description of paraconductivity in strong coupling superconductors. The robustness of the critical exponents and their dependence in GL region on the space dimensionallity only was stressed in [96] in relation to the discussion of the paraconductivity at the edge of the superconductor-insulator transition.

95

Figure 5.1: The Dyson equation for the fluctuation propagator (wavy line) in the ladder approximation. Solid lines represent one-electron Green functions, bold points correspond to the model electron-electron interaction. Let us emphasize, that the two quantities introduced above, L (p, p0 , q) and L(q), are closely connected with each other. The former being integrated over momenta p and p0 becomes an average of the product of two order parameters: Z  1

dpdp0 L (p, p0 , q) = 2 ∆q ∆∗q , (5.5) g where ∆q is the superconducting gap proportional to the condensate wave function Ψ. Thus, this quantity represents the coefficient in the linear term in the GL equation. In terms of the polarization operator introduced above it can be written as Z Π dpdp0 L (p, p0 , q) ∝ . 1 − gΠ Comparing this equation with Eq.(5.3) for the fluctuation propagator we see that the corresponding expressions are very similar. After analytical continuation to the real frequencies the fluctuation propagator L(q, iΩ) coincides with the quantity defined by Eq.(5.5) (up to a constant). One can calculate the propagator (5.3) using the one-electron Green functions of the normal metal (5.2). For sake of convenience of future calculations let us define the correlator of two one-electron Green functions d3 p G(p + q, ε1 )G(−p, ε2 ) = (2π)3 * + 1 = 2πνΘ(−ε1 ε2 ) |ε1 − ε2 | + i∆ξ(q, p)|(p)=EF

P(q, ε1 , ε2 ) =

Z

(5.6) ,

F.S.

where Θ(−ε1 ε2 ) Ris Heavyside step function, ν is the one-electron density of p states, <>F.S. = dΩ means the averaging over the Fermi surface, 4π ∆ξ(q, p)|(p)=EF = [ξ(q + p) − ξ(−p)]|(p)=EF ≈ (vp q)ξ(p)=0 .

(5.7)

The last approximation is valid not too far from the Fermi surface, i.e. when (vp q)ξ(p)=0  EF . 96

It is impossible to carry out the angular averaging in (5.6) for a general anisotropic spectrum. Nevertheless in the following calculations of fluctuation effects in the vicinity of critical temperature only small momenta vp q  T will be involved in the integrations, so we can restrict our consideration here to this region, where one can expand the integrand in powers of vp q. Indeed, the presence of Θ(−ε1 ε2 ) leaves the difference of the two fermionic frequencies in (5.6) to be of the order of the temperature which permits this expansion. The first term in vp q will evidently be averaged out, so with quadratic accuracy one can find:   h(vp q)2 iF.S. Θ(−ε1 ε2 ) 1−2 . (5.8) P(q, ε1 , ε2 ) = 2πν |ε1 − ε2 | |ε1 − ε2 |2 Now one can calculate the polarization operator

Π(q, Ωk ) = T

X

P(q, εn+k , ε−n ) =

(5.9)

εn

 X = ν n≥0

1 n + 1/2 +

|Ωk | 4πT

−2

h(vp q)2 iF.S. (4πT )2

∞ X n=0



1 

n + 1/2 +

|Ωk | 4πT

 3  .

The calculation of the sums in (5.9) can be carried out in terms of the logarithmic derivatives of the Γ-function ψ (n) (x). It worth mentioning that the first sum is well known in the BCS theory, one can recognize in it the so-called ”Cooper logarithm”; its logarithmic divergence at the upper limit ωD (ψ(x  1) ≈ ln x) is cut off by the Debye energy (Nmax = 2πT ) and one gets:     1 1 |Ωk | ωD 1 |Ωk | Π(q, Ωk ) = ψ + + −ψ + − (5.10) ν 2 4πT 2πT 2 4πT   h(vp q)2 iF.S. 00 1 |Ωk | ψ + . − (4πT )2 2 4πT The critical temperature in the BCS theory is determined as the temperature Tc at which the pole of L(0, 0, Tc ) occurs L−1 (q =0, Ωk = 0, Tc ) = g −1 − Π(0, 0, Tc ) = 0,   2γE 1 Tc = ωD exp − , π νg

97

(5.11)

where γE = 1.78 is the Euler constant. Introducing the reduced temperature  = ln( TTc ) one can write the propagator as    1 |Ωk | 1 h(vp q)2 iF.S. 00 1 |Ωk | −1 L (q, Ωk ) = −ν  + ψ( + ) − ψ( ) − ψ + . 2 4πT 2 (4πT )2 2 4πT (5.12) We found (5.12) for bosonic imaginary Matsubara frequencies iΩk = 2πiT k. These frequencies are necessary for the calculation of fluctuation contributions to any thermodynamical characteristics of the system. In the vicinity of the transition point one can restrict oneself in summations of the expressions with L(q, Ωk ) over Matsubara frequencies to the so-called static approximation, taking into account the term with Ωk = 0 only, which turns out to be the most singular term in   1 . This approximation physically means that the product of Heisenberg field operators ψep,σ ψe−p,−σ appears here like a classical field Ψ, which in the phenomenological approach describes the Cooper pair wave function and in the vicinity of critical temperature is proportional to the fluctuation order parameter. Having in mind namely this GL region of temperatures we restricted ourselves above by the assumption of small momenta vp q  T . In these conditions the static propagator reduces to 1 1 . (5.13) ν  + ξ 2 q2 With an accuracy of a numerical factor and the total sign this correlator coincides with the expression (15.3) for h|Ψq |2 i. By this expression we also have finally obtained the microscopic value of the coherence length ξ for a clean superconductor with an isotropic D-dimensional Fermi surface which was often mentioned previously (compare with (1.5)) L(q, 0) = −

7ζ(3)vF2 . (5.14) 16Dπ 2 T 2 In order to describe the fluctuation contributions to transport phenomena one has to start from the analytical continuation of the propagator (5.12) from the discrete set of Ωk ≥ 0 to the whole upper half-plane of imaginary frequencies. The analytical properties of ψ (n) (x)−functions (which have poles at x = 0, −1, −2...) permit one to obtain the retarded propagator LR (q, −iΩ) by simple substitution iΩk → Ω . For small Ω  T the ψ−functions can be expanded in −iΩ/4πT and the propagator acquires the simple pole form : 2 ξ(D) =

LR (q, Ω) = −

1 1 8T 1 = −1 iπ ν − 8T Ω +  + ξ 2 q2 πν iΩ − τGL + 98

8T 2 2 ξ q π


.

(5.15)

This expression provides us with the microscopic value of the GL relaxation time τGL = 8(T π−Tc ) , widely used above in the phenomenological theory. Moreover, comparison of the microscopically derived (5.15) with the phenomenological expressions (3.3), (3.7) and (3.26) shows that αTc = ν and γGL = πν/8Tc . In evaluating L(q, Ωk ) we neglected the effect of fluctuations on the oneelectron Green functions. This is correct when fluctuations are small, i.e. not too near to the transition temperature. The exact criterion of this approximation will be discussed in the following.

5.3 5.3.1

Superconductor with impurities Account for impurities.

In order to study fluctuations in real systems like superconducting alloys or high temperature superconductors one has to perform an impurity average in the graphical equation for the fluctuation propagator (see Fig. 5.1). This procedure can be done in the framework of the Abrikosov-Gorkov approach [105], which we shortly recall below. Let us start from the equation for the electron Green function in the potential of impurities U (r):   b GE (r, r0 ) = δ(r − r0 ) E − U (r) − H (5.16)

If we solve this equation using the perturbation theory for the impurity potential and average the solution, then the average product of two Green functions, can be presented as series, each term of which is associated with a graph drawn according the rules of diagrammatic technique (see Fig. 5.2). In this technique solid lines correspond to bare Green functions and dashed lines to random potential correlators. We assume that the impurity system random potential U (r) is distributed according to the Gauss δ−correlated law. Then all the correlators can be represented as the products of pair correlators

 hU (r)i = 0, hU (r)U (r0 )i = U 2 δ(r − r0 ),

(5.17)

where the angle brackets denote averaging over the impurity configuration. Equation (5.17) corresponds to the Born approximation for the electron in
2 R teraction with short range impurities, and hU 2 i = Cimp V (r)dr where Cimp is the impurity concentration and V (r) is the potential of the single impurity. 99

Figure 5.2: The equation for the vertex part λ(q, ω1 , ω2 ) in the ladder approximation. Solid lines correspond to bare one-electron Green functions and dashed lines to the impurity random potential correlators. In conductors (far enough from the metal-insulator transition) the mean free path is much greater than the electron wavelength l  λ = 2π/pF (which in practice means the mean free path up to tens of interatomic distances). As is well known [105] for the electron spectra with dimensionality D > 1 the angular integration in momentum space reduces considerably the contribution of the diagrams with intersecting impurity lines what permits to omit them to the leading approximation in (pF l)−1 . In this approximation the one-electron Green function keeps the same form as the bare one (5.2) with the only substitution εn ⇒ εen = εn +

1 sign(εn ), 2τ

(5.18)

where 1/τ = 2πν hU 2 i is the frequency of elastic collisions. Another effect of the coherent scattering on the same impurity by both electrons forming a Cooper pair is the renormalization of the vertex part λ(q, ε1 , ε2 ) in the particle-particle channel. Let us demonstrate the details of its calculation. The renormalized vertex λ(q, ε1 , ε2 ) is determined by a graphical equation of the ladder type (see Fig. 5.2 ). Here after the averaging 1 is associated with over the impurity configurations the value hU 2 i = 2πντ the dashed line. In the momentum representation this, generally speaking, integral equation is reduced to the algebraic one 1 P(q, εe1 , εe2 ), (5.19) 2πντ where P(q, εe1 , εe2 ) was defined above by (5.6). Now one has to perform a formal averaging of the general expression (5.6) over the Fermi surface (h ...iF.S. ). Restricting consideration to small momenta λ−1 (q, ε1 , ε2 ) = 1 −

∆ξ(q, p)||p|=pF  |e ε1 − εe2 |.

(5.20)

the calculation of λ(q, ω1 , ω2 ) for the practically important case of an arbitrary spectrum can be done analogously to (5.8). Indeed, expanding the denominator of (5.6) one can find λ(q, ε1 , ε2 ) =

|ε1 − ε2 | +

|e ε1 − εe2 | . h(∆ξ(q,p)||p|=pF )2 iF.S. Θ(−ε ε ) 1 2 τ |e ε1 −e ε2 |2 100

(5.21)

Figure 5.3: The Fermi surface in the form of a corrugated cylinder It is easy to see that assumed restriction on momenta is not too severe and is almost always satisfied in calculations of fluctuation effects at temperatures near Tc . In this region of temperatures the effective propagator momenta √ GL −1 −1 are determined by |q|ef f ∼ [ξ (T )] = ξ   ξ −1 , while the Green function q-dependence becomes important for much larger momenta q ∼ min{ξ −1 , l−1 }, which is equivalent to the limit of the condition (5.20). The average in (5.21) can be calculated for some particular types of spectra. For example in the cases of 2D and 3D isotropic spectra it is expressed in terms of the diffusion coefficient D(D) : h(∆ξ(q, p)||~p|=pF )2 iF.S.(D) = τ −1 D(D) q 2 =

vF2 q 2 . D

(5.22)

Another important example is already familiar case of quasi-two-dimensional electron motion in a layered metal: ξ(p) = E(pk ) + J cos(pz s) − EF ,

(5.23)

where E(pk ) = p2k /(2m), p ≡ (pk , pz ), pk ≡ (px , py ), J is the effective nearest-neighbor interlayer hopping energy for quasiparticles. We note that J characterizes the width of the band in the c-axis direction taken in the strong-coupling approximation and can be identified with the effective energy of electron tunneling between planes (see (2.34) and footnote 14). The Fermi surface, defined by the condition ξ(p) = 0, is a corrugated cylinder (see Fig. 5.3). In this case the average (5.22) is written in a more sophisticated form: h(∆ξ(q, p)||~p|=pF )2 iF.S. =

1 2 2 b 2, (v q + 4J 2 sin2 (qz s/2)) = τ −1 Dq 2 F

(5.24)

where we have introduced the definition of the generalized diffusion operator b in order to deal with an arbitrary anisotropic spectrum. D

5.3.2

Propagator.

In Section 4, in the process of the microscopic derivation of the TDGL equation, the fluctuation propagator was introduced. This object is of first importance for the microscopic fluctuation theory and it has to be generalized for the case of an impure metal with an anisotropic electron spectrum. This is easy to do using the averaging procedure presented in the previous Section. Formally it is enough to use in equation (5.3) the polarization operator 101

Π(q, Ωk ) averaged over impurity positions, which can be expressed in terms of P(q, εen+k , εe−n ) introduced above: Π(q, Ωk ) = T

X ωn

= T

X ωn

λ(q, εn+k , ε−n )P(q, ω en+k , ω e−n ) = 1 [P(q, εen+k , εe−n )]−1 −

1 2πντ

(5.25)

.

For relatively small q (∆ξ(q, p)||E(p)|=EF  |e εn+k − εe−n | ∼ max{T, τ −1 }) and Ω  T one can find an expression for the fluctuation propagator, which can be useful in studies of fluctuation effects near √ Tc (  1) for the dirty and intermediate but not very clean case (T τ  1/ ). Expanding (5.21) in
2 powers of ∆ξ(q, p)||E(p)|=EF /|2e εn + Ωk | it is possible write LR (q, Ω) in a form almost completely coinciding with Exp.(5.15): LR (q, Ω) = −

1 1 . πΩ ν  − i 8T + ξ 2 (T τ )q2

(5.26)

Let us stress that the phenomenological coefficient γGL turns out to be equal πν to the same value 8T as in clean case, and hence does not depend on the impurity concentration. The only difference in comparison with the clean case is in appearance of a dependence of the natural effective coherence length on the elastic relaxation time. In the isotropic D-dimensional case it can be written as 2 ξ(D) (T τ ) = (4mαT )−1 = η(D) = (5.27)   τ 2 vF2 1 1 1 1 0 1 − ψ( + ) − ψ( ) − ψ( ) D 2 4πT τ 2 4πT τ 2

(we introduced here the parameter η(D) frequently used in the microscopic theory)3 . The generalization of (5.26) for the case of a layered electronic spectrum is evident: 3

Let us recall that its square determines the product of the GL parameter α and the Cooper pair mass entering in the GL functional. In clean case we supposed the letter equal to two free electron masses and defined α in accordance with (2.11). As we just have seen in the case of the impure superconductor ξ depends on impurity concentration and this dependence, in principle, can be attributed both to α or m. For our further purposes it is convenient to leave α in the same form (2.11) as in the case of a clean superconductor. The Cooper pair mass in this case becomes dependent on the electron mean free pass what physically can be attributed to the diffusion motion of the electrons forming the pair.

102

LR (q,Ω) = −

1 1 . πΩ 2 ν  − i 8T + η(2) qk + r sin2 (qz s/2)

(5.28)

One has to remember that the Exp. (5.26) was derived in the assumption of small momenta ∆ξ(q, p)||E(p)|=EF  |e εn+k − εe−n | ∼ max{T, τ −1 }, so the range of its applicability is restricted to the GL region of temperatures  = ln( TTc )  1, where the integrands of diagrammatic expressions have singularities at small momenta of the Cooper pair center of mass. Finally let us express the Ginzburg-Levanyuk parameter for the important 2D case in terms of the microscopic parameter η(2) . In accordance with (2.56) and the definition (5.27): Gi(2) (T τ ) =

1 7ζ(3) . 2 16π mTc η(2) (T τ )

(5.29)

One can see that this general definition in the limiting cases of a clean and dirty metal results in the same values Gi(2c) and Gi(2d) as was reported in Table 1.

5.4

Microscopic derivation of the Ginzburg Landau functional

At the end of this chapter let us demonstrate how the Ginzburg-Landau functional itself can be carried out from the microscopic theory of superconductivity. For this aim we will use the method of functional integration alternative to the diagrammatic technique approach. Let us start again from the BCS Hamiltonian (5.1) and write it in the form: H = H0 +Hint , where

and

  ∇2 + e + U (r) ψe (r) , H0 =ψ (r) − 2m Z e+ (r) A e (r) dr, Hint = − g A e (r) = ψe↓ (r) ψe↑ (r) dr A 103

(5.30) (5.31) (5.32)

(5.33)

is operator of the Cooper pair annihilation in the Heisenberg representation. Our task now consists of the calculation of the partition function  Z β    H Z = T r exp − Hdτ = T r exp − , (5.34) T 0 but in contrast to Exp.(2.1) of Chapter 1, with the microscopic Hamiltonian (5.30) instead of the phenomenological functional (2.4). In the ideal gas g = 0 , and H0 is just the quadratic form of the Heisenberg electron field operators ψe (r) and ψe+ (r) . One can diagonalize it and carrying out the trace operation in (2.1) to calculate the partition function. In purpose to calculate the partition function of the interacting electron gas with Hamiltonian (5.30) one has in advance to present the operator Hint as the quadratic form over the field operators. Let us start from the separation in the form of multiplier of the exponent with Hint in the expression for the partition function. The problem is nontrivial because the operators H0 do not commute Hint . Nevertheless, introducing the operator of the imaginary time chronologization Tτ and the interaction representation for the Hamiltonian H [105] one can present the partition function in the form: 

 Z Z = T r exp −

β

H0 dτ



 Z Tτ exp −

0

β

Hint (τ ) dτ



.

(5.35)

0

Now let us present the Hint in the form of the quadratic form over the field operators by means of the Habbard-Stratonovich transformation. For this purpose let us write  Z exp −

β

Hint (τ ) dτ

0



=

Y x

  e+ (r, τ ) A e (r, τ ) dτ dr , exp g A

(5.36)

Each multiplier in the product (5.36) can be presented in the form of the integral4 ! # 2 |∆(r, τ )| f+ (r) dx . e (r) − ∆(r, τ )A − ∆∗ (r, τ )A e = d2 ∆(r, τ ) exp − g (5.37) 2 Here d ∆x = d (Im∆x ) d (Re∆x ) . The product of all these multipliers is the functional integral over D∆(r, τ )D∆∗ (r, τ ). Finally[292]: e+ (r)A(r)dx e gA

4

Z

"

We omit here the nesuchestvennyi numerical coefficient

104

*

|∆(r, τ )|2 Z = T r Tτ D∆(r, τ )D∆ (r, τ ) exp − − ∆∗ (r, τ )ψe↓ (r) ψe↑ (r) − ∆(r, τ )ψe↓+ (r) ψe↑ g 0 (5.38) e Now the form is quadratic over the operators ψ and the problem is reduced to the description of one electron motion in the fluctuation field ∆(r, τ ) arbitrary depending on the coordinate r and imaginary time τ. The Green function of such problem satisfies the Gorkov equation which in the matrix Nambu representation has the form Z

Z

∗

β

  ∂ ∇2 b = 1. τz − + U (r) − µ−?Re∆(r, τ )τx −?Im∆(r, τ )τy G ∂τ 2m

(5.39)

?check signs.Here τi (i = x, y, z) are the Pauli matrices in the space of Gorkov-Nambu, while F∆ b ∆ = G∆ G , (5.40) F∆+ G+ ∆ D E F∆ (r, τ , r0 , τ 0 ) =? Tτ ψe↓ (r, τ ) ψe↑ (r0 , τ 0 ) is anomalous electron Green function in superconducting state. Calculating the trace over electron filed operators ψe↓ , ψe↑+ and fluctuation fields ∆(r, τ ), ∆∗ (r, τ ) in (5.38) one finds Z Z = D∆(r, τ )D∆∗ (r, τ ) exp {−S [∆(r, τ )]} (5.41) with

−S [∆(r, τ )] =

Z

dr

Z 0

β

"

# n o |∆(r, τ )|2 b−1 − dτ tr ln G , ∆ g

(5.42)

where the trace is supposed over the Nambu-Gorkov indexes. The correctness of (5.41) can be checked by calculation of the variational derivatives of the ln Z over ∆∗ starting from (5.38) and (5.41): D E δ ln Z e e =? T ψ (r, τ ) ψ (r, τ ) = F∆ (r, τ, r, τ ) . (5.43) τ ↓ ↑ δ∆∗ The mean field approximation as usually corresponds to the calculation of the Exp(5.41) by the steepest descend method. The saddle point is determined by the fluctuation field ∆ independent on r and τ. In this case, 105

recalling that the free energy is related to the action as F = T S∆ , one can find X Z
|∆|2 T S [∆] F − =− =T ν dξ ln ε2n + ξ 2 + ∆2 − . V V g ε

(5.44)

n

V is the sample volume. The saddle point condition ∂S/∂∆ = 0 coincides with the BCS self-consistency equation XZ 1 dξ 2 1 = νg . (5.45) εn + ξ 2 + ∆2 ε n

The account for fluctuations at T > Tc (or H > Hc2 ) in the described formalism means the expansion of (5.41) in the series over ∆(r, τ ). The fluctuation field dependent part of action takes the form: Z

b−1 (r1 , τ 1 , r2 , τ 2 ) ∆ (r2 , τ 2 ) dr1 dτ1 dr2 dτ2 −S∆ = ∆∗ (r1 , τ 1 ) L Z 1 b 1 , τ 1 , r2 , τ 2 , r3 , τ 3 , r4 , τ 4 )∆ (r3 , τ 3 ) ∆ (r4 , τ 4 ) dr1 dτ1 dr2 dτ2 dr3 dτ − ∆∗ (r1 , τ 1 ) ∆∗ (r2 , τ 2 ) B(r 2 h i−1 b is the fluctuation propagator: L ˆ ω = g −1 − Π ˆω , In Eq.(14.25), operator L ˆ in the coordinate representation has the form Πω (r, r0 ) = polarization operator Π P T ε Πω (r, r0 ; ε) where Πω (r, r0 ; ε) = Gε (r, r0 )Gω−ε (r, r0 ) and Gε is the Matb in Eq.(14.25) corresponds to subara Green function. Non-linear operator B the product of the four one-electron Green functions of normal metal. Let us remind that the above consideration was carried out without the averaging over the impurities positions (quenched disorder). As we will show below for the problems of the mesoscopic character, where the disorder fluctuations are important themselves, this averaging must be performed only after the physical answer obtained. But in the most of cases such quenched disorder fluctuations are beyond the interests and it is enough to average over impurity positions the action S itself, i.e. the functions L−1 (r1 , τ 1 , r2 , τ 2 ) and B(r1 , τ 1 , r2 , τ 2 , r3 , τ 3 , r4 , τ 4 ). After such averaging the expression for propagator L in the frequency-momentum representation is determined by Eq.(5.3) and Eq.(5.25). The saddle point approximation δS/δ∆(r, t) = 0 results in the timedependent Ginzburg-Landau equation for a gapless superconductor [306] Close to Tc fluctuations are smooth in their coordinate dependence and one can omit the dependence of ∆ (r, τ ) on τ . In result the function B is 106

constant while L−1 can be expanded over q 2 (see (5.26)). The free energy (5.44) can be expanded in the GL type series:  Z  B 4 2 2 F = A∆ + ∆ + C (∇∆) dV (5.47) 2 The coefficients A and B do not depend on quenched disorder (impurities) and can be obtained from the Eq.(5.44) A = ν ln

T , Tc

(5.48)

7ζ (3) ν. (5.49) 8π 2 T 2 The coefficient C follows from the diagrammatically derived expression Eq.(5.27) for fluctuation propagator and is related with the square of coherence length: B=

C = νξ 2 (T τ ) = νη

(5.50)

In purpose to get the match between the developed microscopic and phenomenological theory results Eq.(5.47) and Eq. (2.4) the correspondence between the phenomenological order parameter Ψ and the microscopic fluctuation field intensity ∆ has to be established. Here it is necessary to make the following comment. The choice of the coefficient of this proportionality is the delicate procedure where some arbitrariness takes place. The most popular are two following choices. In the first one the identity ∆ ≡ Ψ is postulated. In this case a = A, b = B, 1/4m = C. Such choice is convenient since ∆ is equal to the value of one-particle spectrum gap in the microscopic theory, but is embarrassing in the case of impure superconductor, where the Cooper pair mass 2m is not universal more and depends on the impurity concentration. The second choice assumes the Copper pair mass to be fixed and equal to two free electron masses. In this case ∆, a and b are determined by the Exp.(2.9) and (5.30). Due to the Habbard-Stratonovich transformation we succeeded above to split the model short range interaction in the Cooper channel, i.e. to take into account the long wave length fluctuations of ∆. In some problems (like [98, 101]) the long range character of the Coulomb interaction, or electric field fluctuations is of the first importance. Account for such interaction means the appearance in Hamiltonian (5.30) the additional term HQ =

ρ (r) ρ (r1 ) , |r − r1 |

107

(5.51)

with ρ (r) = e |Ψ|2 . After the Habbard-Stratonovich transformation the additional term (∇ϕ)2 −Se = − + ϕρ (5.52) 8π appears in action. Strong Coulomb interaction results in the smallness of the Debye radius |rD |2 = e2 ν with respect to the Cooper pair size (coherence length ξ), while the plasma frequency ω 2 = 4πe2 n/m is large in comparison with ∆. This is the reason why in the most part of the problems instead of functional integration over the fields ϕ (r) one can restrict himself by account of the neutrality constrain only.

108

Chapter 6 Microscopic theory of fluctuation conductivity of layered superconductor 6.1

Qualitative discussion of different fluctuation contributions

In Section 4 the direct fluctuation effect on conductivity, related with the charge transfer by means of fluctuation Cooper pairs, was discussed in details. Nevertheless, below in this Section we return to its discussion and will demonstrate its calculation by means of the microscopic theory. This will be done in purpose to prepare the basis for studies of the Aslamazov-Larkin contribution in the variety of physical values like magnetoconductivity near the upper critical field, conductivity far from transition point, fluctuation conductivity in ultra-clean limit, Hall conductivity etc. Microscopic approach permits also to calculate the cited above indirect fluctuation effects like so called DOS and MT contributions. We will start now from their qualitative discussion. The important consequence of the presence of fluctuating Cooper pairs above Tc is the decrease of the one-electron density of states at the Fermi level. Indeed, if some electrons are involved in pairing they can not simultaneously participate in charge transfer and heat capacity as single-particle excitations. Nevertheless, the total number of the electronic states can not be changed by the Cooper interaction, and only a redistribution of the levels along the energy axis is possible [106, 107]. In this sense one can speak about the opening of a fluctuation pseudo-gap at the Fermi level. The decrease of the one-electron density of states at the Fermi level leads to a reduction of 109

the normal state conductivity. This, indirect, fluctuation correction to the conductivity is called the density of states (DOS) contribution and it appears side by side with the paraconductivity (or Aslamazov-Larkin contribution). It has the opposite (negative) sign and turns out to be much less singular in (T − Tc )−1 in comparison with the AL contribution, so that in the vicinity of Tc it was usually omitted. However, in many cases [29, 108, 109, 110, 111, 112], when for some special reasons the main, most singular, corrections are suppressed, the DOS correction becomes of major importance. Such a situation takes place in many cases of actual interest (quasiparticle current in tunnel structures, c-axis transport in strongly anisotropic high temperature superconductors, NMR relaxation rate, thermoelectric power). The correction to the normal state conductivity above the transition temperature related with the fluctuation DOS renormalization for the dirty superconductor can be evaluated qualitatively. Indeed, the fact that some electrons (∆Ne per unit volume) participate in fluctuation Cooper pairing means that the effective number of carriers taking part in one-electron charge transfer diminishes leading to a decrease of conductivity (we deal here with the longitudinal component): DOS δσxx =−

2ns e2 τ ∆Ne e2 τ =− , m m

(6.1)

where ns is the superfluid density coinciding with the Cooper pairs concentration. The latter can be identified with the average value of the square of the order parameter modulus already calculated as the correlator (15.4) with r ∼ ξ. For the 2D case, which is of the most interest to us, one finds: ns =

√ 1 1 7ζ (3) EF 1 K0 ( ) = 4 2 ln , 2 4παξ s π vF sτ 

(6.2)

where we have used the explicit expressions (2.11) and (15.4) for α and ξ. As we will see the corresponding expression for the fluctuation DOS correction to conductivity (6.1) coincides with the accuracy of 2 with the microscopic expression (6.14) which will be carried out below. The third, purely quantum, fluctuation contribution is generated by the coherent scattering of the electrons forming a Cooper pair on the same elastic impurities. This is the so called anomalous Maki-Thompson (MT) contribution [6, 7] which can be treated as the result of Andreev scattering of the electron by fluctuation Cooper pairs. This contribution often turns out to be important in conductivity and other transport phenomena. Its temperature singularity near Tc is similar to that of the paraconductivity, although being extremely sensitive to electron phase-breaking processes and to the type of orbital symmetry of pairing it can be suppressed. Let us evaluate it. 110

The physical origin of the Maki-Thompson correction consists in the fact that the Cooper interaction of electrons with the almost opposite momenta changes the mean free path (diffusion coefficient) of electrons. As we have already seen in the previous Section the amplitude of this interaction increases drastically when T → Tc : gef f =

T 1 1 g ≈ = . ωD = T T − Tc  1 − g ln 2πT ln Tc

What is the reason of this growth? One can say that the electrons scatter one at another in resonant way with the virtual Cooper pairs formation. Or it is possible to imagine that the electrons undergo the Andreev scattering at fluctuation Cooper pairs binding in the Cooper pair themselves. The probability of such induced pair irradiation (let us remind that Cooper pairs are Bose particles) is proportional to their number in the final state, i.e. n(p) (1.7). For small momenta n(p) ∼ 1/. One can ask why such interaction does not manifest itself considerably far from the transition point? The matter of fact that so intensively interacts just small number of electrons with the total momentum q . ξ −1 (T ). In accordance with the Heisenberg principle the minimal distance between such electrons is of the order of ∼ ξ(T ). From the other hand such electrons in purpose to interact have to approximate one another up to the distance of the Fermi length λF ∼ 1/pF . The probability of such event may be estimated in the spirit of the self-intersection trajectories contribution evaluation in the weak localization theory [113, 116]. In the process of diffusion motion the distance between two electrons increases with the time growth in accordance with the Einstein law: R(t) ∼ (Dt)1/2 . Hence the scattering probability Z tmax D−1 λF W ∼ vF dt. RD (t) tmin The lower limit of the integral can be estimated from the condition R(tmin ) ∼ ξ(T ) (only such electrons interact in the resonant way). The upper limit is determined by the phase breaking time τϕ since for larger time intervals the phase coherence, necessary for the pair formation, is broken. In result the relative correction to conductivity due to such processes is equal to the product of the scattering probability on the effective interaction constant: δσ M T /σ = W gef f . In the 2D case δσ M T ∼

Dτϕ e2 ln 2 . 8 ξ (T ) 111

This result will be confirmed below in the frameworks of the microscopic consideration.

6.1.1

Generalities

Let us pass to the microscopic calculation of the fluctuation conductivity of the layered superconductor. We begin by discussing the quasiparticle normal state energy spectrum. While models with several conducting layers per unit cell and with either intralayer or interlayer pairing have been considered [117], it has been shown [118] that all of these models give rise to a Josephson pair potential that is periodic in kz , the wave-vector component parallel to the c-axis, with period s, the c-axis repeat distance. While such models differ in their superconducting densities of states, they all give rise to qualitatively similar fluctuation propagators, which differ only in the precise definitions of the parameters and in the precise form of the Josephson coupling potential. Ignoring the rather unimportant differences between such models in the Gaussian fluctuation regime above Tc (H), we therefore consider the simplest model of a layered superconductor, in which there is one layer per unit cell, with intralayer singlet s-wave pairing. These assumptions lead to the simple spectrum (5.23) and hence to a Fermi surface having the form of a corrugated cylinder (see Fig.5.3). Some remarks regarding the normal-state quasiparticle momentum relaxation time are necessary. In the ”old” layered superconductors the materials were generally assumed to be in the dirty limit (like T aS2 (pyridine)1/2 ). In the high-Tc cuprates, however, both single crystals and epitaxial thin films are nominally in the ”intermediate” regime, with l/ξxy ≈ 2 − 5. In addition, the situation in the cuprates is complicated by the presence of phonons for T ' Tc ' 100K, the nearly localized magnetic moments on the Cu2+ sites, and by other unspecified inelastic processes. In this Section we assume simple elastic intralayer scattering and restrict our consideration to the local limit in the fluctuation Cooper pair motion. This means that we consider the case of not too clean superconductors, keeping the impurity concentration ni and reduced temperature such that the resulting mean-free path satisfies the requirement l < ξxy (T ) = ξ√xyε and the impurity vertex can be taken in the local form (5.21) with h(∆ξ(q, p)|)2 iF.S. determined by (5.24). The phase-breaking time τϕ is supposed to be much larger than τ . The most general relation between the current density j(r,t) and vectorpotential A(r0 , t0 ) is given through the so-called electromagnetic response operator Qαβ (r, r0 , t, t0 ) [105]:

112

Figure 6.1: Feynman diagrams for the leading-order contributions to the fluctuation conductivity. Wavy lines are fluctuation propagators, thin solid lines with arrows are impurity-averaged normal-state Green’s functions, shaded semicircles are vertex corrections arising from impurities, dashed lines with central crosses are additional impurity renormalisations and shaded rectangles are impurity ladders. Diagram 1 represents the Aslamazov-Larkin term; diagrams 2–4 represent the Maki-Thompson type contributions; diagrams 5–10 arise from corrections to the normal state density of states.

j(r,t) =

Z

Qαβ (r, r0 , t, t0 )A(r0 , t0 ) dr0 dt,

Assuming space and time homogeneity, one can take the Fourier transform of this relation and compare it with the definition of the conductivity tensor jα = σαβ Eβ . This permits us to express the conductivity tensor in terms of the retarded electromagnetic response operator 1 [Qαβ ]R (ω) . (6.3) iω The electromagnetic response operator Qαβ (ων ), defined on Matsubara frequencies ων = (2ν + 1)πT, can be presented as the correlator of two exact one-electron Green functions [105] averaged over impurities and accounting for interactions, in our case the particle-particle interactions in the Cooper channel. The appropriate diagrams corresponding to the first order of perturbation theory in the fluctuation amplitude are shown in Fig. (6.1). With each electromagnetic field component Aα we associate the external vertex evα (p) = e ∂ξ(p) . For the longitudinal conductivity tensor elements ∂pα (parallel to the layers, for which α = x, y), the resulting vertex is simply epα /m. For the c-axis conductivity, the vertex is given by σαβ (ω) = −

evz (p) = e

∂ξ(p) = −eJs sin(pz s). ∂pz

(6.4)

Each solid line in the diagrams represents a one-electron Green function averaged over impurities (5.2), a wavy line represents a fluctuation propagator L(q, Ωk ) (5.26), three-leg vertices were defined by the Exp.(5.21). The four-leg impurity vertex, appearing in diagrams 3-4, 9-10 of the Fig. 6.1, is called the Cooperon in the weak localization theory (see, for example,[119]) It is easy to see that it differs from the above three-leg vertex only by the 113

additional factor (2πντ )−1 . We do not renormalize the current vertices: it is well known (see [105]) that this renormalization only leads to the substitution of the scattering time τ by the transport one τtr . We integrate over the internal Cooper pair momentum q and electron momentum p and sum over the internal fermionic and bosonic Matsubara frequencies, with momentum and energy conservation at each internal vertex (fluctuation propagator endpoint) in the analytical expressions for the diagrams presented in Fig. (6.1). After these necessary introductory remarks and definitions we pass to the microscopic calculation of the different fluctuation contributions.

6.2

Aslamazov-Larkin contribution

We first examine the AL paraconductivity (diagram 1 of Fig.(6.1)). Actually this contribution was already studied in the Section 4 in the framework of the TDGL equation but, in order to demonstrate how the method works, we will carry out here the appropriate calculations in the microscopic approach, as was originally done by Aslamazov and Larkin [5]. The AL contribution to the electromagnetic response operator tensor has the form: X Z d3 q AL 2 Bα (q, Ωk , ων )L(q, Ωk ) × (6.5) Qαβ (ων ) = 2e T (2π)3 Ω k

×Bβ (q, Ωk , ων )L(q, Ωk + ων ), where the three Green function block is given by X Bα (q, Ωk , ων ) = T λ(q, εn+ν , Ωk − εn )λ(q, εn , Ωk − εn ) ×

(6.6)

εn

×

Z

d3 p vα (p)G(p, εn+ν )G(p, εn )G(q − p, Ωk − εn ). (2π)3

Expanding G(q − p, Ωk − εn ) over q one find that the angular integration over the Fermi surface kills the first term and leaves nonzero the second term of the expansion only. Then the ξ−integration is performed by means of the Cauchy theorem. The further summation over the fermionic frequency is cumbersome, so we will show it for the example of the simplest case of a dirty superconductor with T τ  1. In this case the main sources of the εn −dependence in (6.6) are the λ-vertices and that originating from the Green functions can be neglected by the parameter T τ  1 (indeed, one can see that εn ∼ T are important in vertices, while in Green functions 114

Figure 6.2: The contour of integration in the plane of complex frequencies. εn & τ −1 only). The remaining summation in (6.6) is performed in the same way as was done in (5.26) and results in: η(2) hvα qβ vβ iF S × (6.7) vF2 " b 2 b 2 8T 1 |Ωk | + ων + Dq 1 |Ωk | + Dq × ψ( + ) − ψ( + )+ πων 2 4πT 2 4πT # b 2 b 2 1 |Ωk+ν | + ων + Dq 1 ||Ωk+ν | + Dq + ψ( + ) − ψ( + ) . 2 4πT 2 4πT

Bα (q, Ωk , ων ) = ν

Now let us return to the general expression for QAL αβ (ων ) and transform the Ωk − summation into a contour integral, using the identity [120] I X 1 z T f (Ωk ) = dz coth f (−iz), 4πi C 2T Ω k

where z = iΩk is a variable in the plane of complex frequency and the contour C encloses all bosonic Matsubara frequencies over which the summation is carried out. In our case the contour C can be chosen as a circle with radius tending to infinity (see Fig. 6.2):

QAL αβ (ων )

Z I e2 d3 q z = dz coth Bα (q, −iz + ων , −iz) × (6.8) 3 2πi (2π) C 2T ×L(q, −iz)Bβ (q, −iz + ων , −iz)L(q, −iz + ων ).

One can see that the integrand function in (6.8) has breaks of analyticity at the lines Imz = 0 and Imz = −iων . Indeed, the fluctuation propagator L(q, Ωk ) and Green function blocks Bα (q, Ωk , ων ) were defined on the bosonic Matsubara frequencies only, while now we have to use them as functions of the continuous variable z. As it is well known from the properties of Green functions in the complex plane z , two analytical functions, related with L(q, Ωk ), can be introduced. The first one, LR (q, −iz) (retarded), is analytic in the upper half-plane (Imz > 0), while the second one, LA (q, −iz) (advanced), has no singularities in the lower half-plane (Imz < 0). As we have seen above the same lines separate the domains of the analyticity of the Green function blocks, so the functions B RR , B RA , B AA analytic in each 115

domain can be introduced (with the appropriate choices of the |Ωk+ν | and |Ωk | signs in the arguments of the ψ− functions, see (6.7)). This means that by cutting the z-plane along the lines Imz = 0 and Imz = −iων we can reduce the calculation of the contour integral in (6.8) to the sum of three integrals along the contours C1 , C2 , C3 which enclose domains of well defined analyticity of the integrand function. The integral along the large circle evidently vanishes and the contour integral is reduced to four integrals along the cuts of the plane in Fig.6.2: I

z Bα (q, −iz)L(q, −iz)Bβ (q, −iz)L(q, −iz + ων ) = 2T C1 +C2 +C3 Z ∞  z R = dz coth L (q, −iz + ων ) BαRR BβRR LR (q, −iz)− 2T −∞ Z ∞−iων  z A RA RA A −Bα Bβ L (q, −iz) + dz coth L (q, −iz) × 2T −∞−iων  RA RA R  Bα Bβ L (q, −iz + ων ) − BαAA BβAA LA (q, −iz + ων ) .

I(q, ων ) =

dz coth

Now one can shift the variable in the last integral to z = z 0 − iων , take into z account that iων is the period of coth 2T and get an expression analytic in iων → ω. In the vicinity of Tc , due to the pole structure of the fluctuation propagators in (6.5), the leading contribution to the electromagnetic response AL(R) operator Qαβ arises from them rather than from the frequency dependence of the vertices Bα , so we can neglect the Ωk - and ων -dependencies of the Green functions blocks and use the expression for Bα (q, 0, 0) valid for small qab only:  η(2) vF2 qα , α = x, y Bα (q) = −2ν 2 . (6.9) sJ 2 sin qz s, α = z vF Detailed calculations demonstrate that this result can be generalized to an arbitrary impurity concentration just by using the expression (5.27) for η(2) . Finally: AL(R) Qαβ (ω)

2e2 = π

Z

d3 q Bα (q)Bβ (q) (2π)3

Z

∞

 z  LR (q, −iz − iω) 2T −∞  +LA (q, −iz + iω) ImLR (q, −iz). dz coth

Being interested here in the d.c. conductivity one can expand the integrand function in ω. It is possible to show that the zeroth order term is 116

cancelled by the same type contributions from all other diagrams (this cancellation confirms the absence of anomalous diamagnetism above the critical temperature). The remaining integral can be integrated by parts and then carried out taking into account that the contribution most singular in  comes from the region z ∼   T : Z Z ∞ 2 e2 d3 q 2 dz  AL R (q, 0, 0) ImL (q, −iz) = σxx = B x,z 2 z 2πT (2π)3 −∞ sinh 2T 2 Z π 2 e2 η(2) d2 q q2   = s (2π)2 (η(2) q2 + )(η(2) q2 +  + r) 3/2  √ e2 1 e2 1/ r,   r . (6.10) = → 1/2 1/,   r 16s [( + r)] 16s where the Lawrence-Doniach anisotropy parameter r [27] was already defined by (2.34). In the same way one can evaluate the AL contribution to the transverse fluctuation conductivity [121, 108, 122]: Z πe2 sr2 d2 q 1 AL   σzz = = (6.11) 2 2 32 (2π) (η(2) q + )(η(2) q2 +  + r) 3/2    p e2 s  + r/2 e2 s r/, f or  r −1 → . = 1/2 32η(2) [( + r)] 64η(2) (r/2)2 , f or  r Note, that contrary to the case of in-plane conductivity, the critical exponent for σzz above the Lawrence-Doniach crossover temperature TLD (for which (TLD ) = r) is 2 instead of 1, so the crossover occurs from the 0D to 3D regimes. This is related with the tunneling (so from the band structure point of view effectively zero dimensional) character of electron motion along the c-axis.

6.3

Contributions from fluctuations of the density of states

In original paper of Aslamazov and Larkin [5] the most singular AL contribution to conductivity, heat capacity and other properties of a superconductor above the critical temperature was considered. The diagrams of the type 5-6 were pictured and correctly evaluated as less singular in . Nevertheless the specific form of the AL contribution to the transverse conductivity of a layered superconductor, which may be considerably suppressed for small 117

interlayer transparency, suggested to re-examine the contributions from diagrams 5-10 of Fig.6.1 which are indeed less divergent in , but turn out to be of lower order in the transmittance and of the opposite sign with respect to the AL one [108, 109]. These, so-called DOS, diagrams describe the changes in the normal Drude-type conductivity due to fluctuation renormalization of the normal quasiparticles density of states above the transition temperature (see Section 8.5). In the dirty limit, the calculation of contributions to the longitudinal fluctuation conductivity σxx from such diagrams was discussed in [123, 112]. Contrary to the case of the AL contribution, the in-plane and out-of-plane components of the DOS contribution differ only in the square of the ratio of effective Fermi velocities in the parallel and perpendicular directions. This allows us to calculate both components simultaneously. The contribution to the fluctuation conductivity due to diagram 5 is X Z d3 q X 5 2 Qαβ (ων ) = 2e T L(q, Ω )T λ2 (q, εn , Ωk − εn ) × k 3 (2π) εn Ωk Z 3 dp vα (p)vβ (p)G2 (p, εn )G(q − p, Ωk − εn )G(p, εn+ν ), (2π)3 and diagram 6 gives an identical contribution. Evaluation of the integrations over the in-plane momenta p and the summation over the internal frequencies εn are straightforward. Treatment of the other internal frequencies Ωk is less obvious, but in order to obtain the leading singular behavior in the vicinity of transition it suffices to set Ωk = 0 [112]. After integration over qz , we have [108, 110]: Z πe2 d2 q 1 5+6   Aαβ κ1 η (2) σαβ = − 2 2s ( + η(2) q2 )( + r + η(2) q2 ) 1/2 |q|≤ξ −1 (2π)   e2 κ1 2 Aαβ ln 1/2 , (6.12) ≈ − 8s  + ( + r)1/2 where Axx = Ayy = 1, Azz = (sJ/vF )2 , Aα6=β = 0 and      2(vF τ )2 1 1 3 1 0 00 κ1 = 2 ψ + − ψ . π η (2) 2 4πT τ 4πT τ 2 In order to cut off the ultra-violet divergence in q we have introduced here a −1/2 cut off parameter qmax = ξ −1 = η(2) in complete agreement with Section 3. Let us stress that in the framework of the phenomenological GL theory we attributed this cut off to the breakdown of the GL approach at momenta as large as q ∼ ξ −1 . The microscopic approach developed here permits to see 118

how this cut off appears: the divergent shortwave-length contribution arising from GL-like fluctuation propagators is automatically restricted by the qdependencies of the impurity vertices and Green functions, which appear at the scale q ∼ l−1 . In a similar manner, the equal contributions from diagrams 7 and 8 sum to Z d2 q 1 πe2 7+8   σαβ = − Aαβ κ2 η(2) 2 2 2s ( + η(2) q )(+r + η(2) q2 ) 1/2 |q|≤qmax (2π)   e2 κ2 2 ≈ − Aαβ ln 1/2 , (6.13) 8s  + (+r)1/2   (vF τ )2 00 1 κ2 = 3 ψ . 2π T τ 2 Comparing (6.12) and (6.13), we see that in the clean limit, the main contributions from the DOS fluctuations arise from diagrams 5 and 6. In the dirty limit, diagrams 7 and 8 are also important, having -1/3 the value of diagrams 5 and 6, for both σxx and σzz . Diagrams 9 and 10 are not singular in  << 1 at all and can be neglected. The total DOS contribution to the in-plane and c-axis conductivity is therefore   e2 2 DOS σαβ = − κ(T τ )Aαβ ln 1/2 , (6.14) 2s  + ( + r)1/2 where

0 00 −ψ 12 + 4πτ1 T + 2πτ1 T ψ 12


 κ(T τ ) = κ1 + κ2 = 2  1 π ψ 2 + 4πτ1 T − ψ 12 − 4πτ1 T ψ 0 12  56ζ(3)/π 4 ≈ 0.691, T τ  1 √ (6.15) → 2 2 8π (T τ ) / [7ζ(3)] ≈ 9.384 (T τ )2 , 1  T τ  1/  is a function of τ T only. As it will be shown below at the upper limit √ T τ ∼ 1/  the DOS contribution reaches the value of the other fluctuation contributions and in the limit of T τ → ∞ exactly eliminates the MakiThompson one.

6.4

Maki-Thompson contribution

We now consider another quantum correction to fluctuation conductivity which is called the Maki-Thompson (MT) contribution (diagram 2 of Fig. 6.1). It was firstly discussed by Maki [6] in a paper which appeared almost 119

simultaneously with the paper of Aslamazov and Larkin [5]. Both these articles gave rise to the microscopic theory of fluctuations in superconductor. Maki found that, in spite of the seeming weaker singularity of diagram 2 with respect to the AL one (it contains one propagator only, while the AL one contains two of them) it can contribute to conductivity comparably or even stronger than AL one. Since the moment of its discovery the MT contribution became the subject of intense controversy. In its original paper Maki found that in 3D case this fluctuation correction is four times larger than the AL one. In 2D case the result was striking: the MT contribution simply diverged. This paradox was, at least at the level of recipe, resolved by Thompson [7]: he proposed to cut off the infra-red divergence in the Cooper pair center of mass momentum integration by the introduction of the finite length ls of inelastic scatterings of electrons on paramagnetic impurities. In the further papers of Patton [124], Keller and Korenman [125] it was cleared up that the presence of paramagnetic impurities or other external phase-breaking sources is not necessary: the fluctuation Cooper pairing of two electrons results in a change of the quasiparticle phase itself and the corresponding phase-breaking time τϕ appears as a natural cut off parameter of the MT divergence in the strictly 2D case. The minimal quasi-two-dimensionality of the electron spectrum, as we will show below, automatically results in a cut off of the MT divergence. Although the MT contribution to the in-plane conductivity is expected to be important in the case of low pair-breaking, experiments on hightemperature superconductors have shown that the excess in-plane conductivity can usually be explained in terms of the fluctuation paraconductivity alone. Two possible explanations can be found for this fact. The first one is that the pair-breaking in these materials is not weak. The second is related with the d−wave symmetry of pairing which kills the anomalous MakiThompson process [126, 127]. We will consider below the case of s-pairing, where the Maki-Thompson process is well pronounced. The appearance of the anomalously large MT contribution is nontrivial and worth being discussed. We consider the scattering lifetime τ and the pair-breaking lifetime τϕ to be arbitrary, but satisfying τϕ > τ . In accordance with diagram 2 of Fig.(6.1) the analytical expression for the MT contribution to the electromagnetic response tensor can be written as X Z d3 q MT 2 L(q, Ωk )Iαβ (q, Ωk , ων ), (6.16) Qαβ (ων ) = 2e T (2π)3 Ω k

where Iαβ (q, Ωk , ων ) = T

X

λ(q, εn+ν , Ωk−n−ν )λ(q, εn , Ωk−n )×

εn

120

(6.17)

d3 p vα (p)vβ (q − p)G(p, εn+ν )G(p, εn )G(q − p, Ωk−n−ν )G(q − p, Ωk−n ) . (2π)3 In the vicinity of Tc , it is possible to restrict consideration to the static limit of the MT diagram, simply by setting Ωk = 0 in (6.16). Although dynamic effects can be important for the longitudinal fluctuation conductivity well above TLD , the static approximation is correct very close to Tc , as shown in [111, 139]. The main q−dependence in (6.16) arises from the propagator and vertices λ. This is why we can assume q = 0 in Green functions and to calculate the electron momentum integral passing, as usual, to a ξ(p) integration: ×

Z

Iαβ (q, 0, ων ) = πν hvα (p)vβ (q − p)iF S × (6.18) X 1 1 1    ×T . εn+ν | + |e εn | b 2 |2εn | + Dq b 2 |e |2εn+ν | + Dq εn In evaluating the sum over the Matsubara frequencies εn in (6.18) it is useful to split it into the two parts. In the first εn belongs to the domains ]−∞, −ων [ and [0, ∞[, which finally give two equal contributions. This gives rise to the regular part of the MT diagram. The second, anomalous, part of the MT diagram arises from the summation over εn in the domain [−ων , 0[. In this interval, the further analytic continuation over ων leads to the appearance of an additional diffusive pole: (reg)

(an)

Iαβ (q, 0, ων ) = Iαβ (q, ων ) + Iαβ (q, ων ) = ν hvα (p)vβ (q − p)iF S  ∞ X 1 1 1     + × 2πT −1 2 2 2εn + ων + τ b b 2ε + Dq 2ε + Dq n=0 n+ν n  −1 X πT 1 1    . + −1 ων + τ n=−ν 2ε 2 2 b b + Dq −2ε + Dq n+ν n The limits of summation in the first sum do not depend on ων , so it is an analytic function of this argument and can be continued to the upper halfplane of the complex frequency by the simple substitution ων → −iω. Then, tending ω → 0, one can expand the sum over powers of ω and perform the summation in terms of digamma-function: (reg)

Iαβ (q, ων ) = ν hvα (p)vβ (q − p)iF S × 121

    ∂ iω ∂ ∂     + const + · (6.19)  2 ∂ Dq ∂ (τ −1 ) b 2 b 2 ∂ Dq " ! !#) b 2 b 2 1 1 ων + Dq 1 Dq ψ + −ψ + . b 2 2 4πT 2 4πT τ −1 − Dq

The values of characteristic momenta q  l−1 are determined by the domain of convergibility of the final integral of the propagator L(q, 0) in (6.16) b 2 with respect to τ −1 . In result (analogously to (6.13) and one can neglect Dq (reg)R

Iαβ

(q, ω → 0) = ν hvα (p)vβ (q − p)iF S × (6.20) ( "  #)
   00 1 ψ iωτ 2 1 1 1 2 const + ψ0 + − ψ0 − . 4 2 4πτ T 2 4πT τ

The appearance of the constant in Qαβ (ων ) was already discussed in the case of the AL contribution and, as was mentioned there, it is cancelled with the similar contributions of the other diagrams [111] and we will not consider it any more. (an) Now let us pass to the calculation of Iαβ (q, ων ). Expanding the summing function in simple fractions one can express the result of summation in terms of digamma-functions

(an)

Iαβ (q, ων ) =

1 ν hvα (p)vβ (q − p)iF S 1 × b 2 4 ων + τ −1 ων + Dq " ! !# b 2 b 2 1 2ων + Dq 1 Dq ψ + −ψ + . 2 4πT 2 4πT

(6.21)

Doing the analytical continuation iων → ω → 0 and taking into account that (an)R in the further q-integration of Iαβ (q, ω → 0), due to the singular at small b 2  T , one can find q propagator, the important range is Dq (an)R

Iαβ

(q, ω → 0) = −

iπωτ ν hvα (p)vβ (q − p)iF S . b 2 16T −iω + Dq

(6.22)

Because of the considerable difference in the angular averaging of the different tensor components we discuss the MT contribution to the in-plane and out of plane conductivities separately. Taking into account that hvx (p)vx (q − p)iF S = −v 2F /2 one can find that the calculation of the regular part of MT diagram to the in-plane conductivity 122

is completely similar to the corresponding DOS contribution and here we list the final result [110] only:   e2 2 M T (reg) =− κ σxx ˜ ln 1/2 , 2s  + ( + r)1/2 where


0 0 00 1 1 −ψ 12 + 4πτ1 T + ψ 12 + 4πT ψ τ 2


 κ ˜ (T τ ) = π 2 ψ 12 + 4πτ1 T − ψ 12 − 4πτ1 T ψ 0 12  28ζ(3)/π 4 ≈ 0.346, f orT τ  1 √ → 2 π / [14ζ(3)] ≈ 0586, f or1  T τ  1/ 

(6.23)

is a function only of τ T . We note that this regular MT term is negative, as is the overall DOS contribution. For the anomalous part of the in-plane MT contribution we have: Z 1 d3 q M T (an) 2 σxx = 8e η(2) T 3 ˆ 2 ][ + η(2) q2 + r (1 − cos qz s)] (2π) [1/τϕ + Dq 2 ! 2 1/2 1/2 e  + ( + r) = ln , (6.24) 1/2 4s( − γϕ ) γϕ + (γϕ + r)1/2 where, in accordance with [7], the infra-red divergence for the purely 2D case (r = 0) is cut off at Dq 2 ∼ 1/τϕ 1 . The dimensionless parameter  2η π 1, T τ  1 √ γϕ = 2 → 3 v F τ τϕ 8T τϕ 7ζ(3)/ (2π T τ ) , 1  T τ  1/  is introduced for simplicity. If r 6= 0 the MT contribution turns out to be finite even with τϕ = ∞. Comparison of the expressions (6.10) and (6.24) indicates that in the weak pair-breaking limit, the MT diagram makes an important contribution to the longitudinal fluctuation conductivity: it is four times larger than the AL contribution in the 3D regime, and even logarithmically exceeds it in the 2D regime above TLD . For finite pair-breaking, however, the MT contribution is greatly reduced in magnitude. We now consider the calculation of the MT contribution to the transverse conductivity. The explicit expressions for vz (p) and vz (q−p) (see Exp. (6.4)), result in hvx (p)vx (q − p)iF S = 12 J 2 s2 cos qz s. We take the limit Jτ << 1 in evaluating the remaining integrals, which may then be performed exactly. 1

The detailed study of the phase-breaking time, its energy dependence and the effect on the MT contribution was done in [129]

123

The regular part of the MT contribution to the transverse conductivity is M T (reg) σzz

Z d3 q cos qz s e2 s2 πr˜ κ(T τ ) = − 4 (2π)3  + η(2) q2 + 2r (1 − cos qz s)  2 e2 sr˜ κ(T τ ) ( + r)1/2 − 1/2 = − 16η(2) r1/2

This term is smaller in magnitude than is the DOS one, and therefore makes a relatively small contribution to the overall fluctuation conductivity. In the 3D regime below TLD , it is proportional to J 2 , and in the 2D regime above TLD , it is proportional to J 4 . For the anomalous part of the MT diagram one can find Z πe2 J 2 s2 τ d3 q cos qz s M T (an) σzz = ˆ 2 ][ + η(2) q2 + r (1 − cos qz s)] 4 (2π)3 [1/τϕ + Dq 2 " Z 2 2 2 γϕ + η(2) q + r/2 πe s dq   − = 2 4( − γϕ ) (2π) (γϕ + η(2) q2 )(γϕ + η(2) q2 + r) 1/2 #  + η(2) q2 + r/2  − ( + η(2) q2 )( + η(2) q2 + r) 1/2   e2 s γϕ + r +  −1 . (6.25) = 16η(2) [( + r)]1/2 + [γϕ (γϕ + r)]1/2 In examining the limiting cases of (6.25), it is useful to consider the cases of weak (γϕ << r, ⇐⇒ J 2 τ τϕ >> 1/2) and strong (γϕ >> r,⇐⇒ J 2 τ τϕ << 1/2) pair-breaking separately2 . For weak pair-breaking, we have  p r/γ ,   γϕ  r 2 es  p ϕ M T (an) → σzz r/, γϕ    r . 16η(2)  r/ (2) , γϕ  r   In this case, there is the usual 3D to 2D dimensional crossover in the anomalous MT contribution at TLD . There is an additional crossover at Tϕ (where Tc < Tϕ < TLD ), characterized by (Tϕ ) = γϕ , below which the anomalous MT term saturates. Below TLD , the MT contribution is proportional to J, but in the 2D regime above TLD , it is proportional to J 2 . 2

Physically the value J 2 τ . characterizes the effective interlayer tunneling rate [110, 128]. When 1/τφ << J 2 τ << 1/τ , the quasiparticles scatter many times before tunneling to the neighboring layers, and the pairs live long enough for them to tunnel coherently. When J 2 τ << 1/τφ , the pairs decay before both paired quasiparticles tunnel.

124

For strong pair-breaking M T (an) σzz

e2 s → 32η(2)



r/γϕ ,   r  γϕ . r / (4γϕ ) , r  min{γϕ , } 2

In this case, the 3D regime (below TLD ) is not singular, and the anomalous MT contribution is proportional to J 2 , rather than J for weak pair-breaking. In the 2D regime, it is proportional to J 4 for strong pair-breaking, as opposed to J 2 for weak pair-breaking. In addition, the overall magnitude of the anomalous MT contribution with strong pair-breaking is greatly reduced from that for weak pair-breaking. Let us now compare the regular and anomalous MT contributions. Since these contributions are opposite in sign, it is important to determine which will dominate. For the in-plane resistivity, the situation is straightforward: the anomalous part always dominates over the regular and the latter can be neglected. The case of c−axis resistivity requires more discussion. Since we expect τϕ ≥ τ , strong pair-breaking is likely in the dirty limit. When the pair-breaking is weak, the anomalous term is always of lower order in J than the regular term, so the regular term can be neglected. This is true for both the clean and dirty limits. The most important regime for the regular MT term is the dirty limit with strong pair-breaking. In this case, when τϕ T ∼ 1, the regular and anomalous terms are comparable in magnitude. In short, it is usually a good approximation to neglect the regular term, except in the dirty limit with relatively strong pair-breaking and only for the out-of-plane conductivity. Finally let us mention that the contributions from the two other diagrams of the MT type (diagrams 3 and 4 of Fig. 6.1) in the vicinity of critical temperature can be omitted: one can check that they have an additional square of the Cooper pair center of mass momentum q in the integrand of q-integration with respect to diagram 2 and hence turn out to be less singular in .

6.5

Fluctuations in the ultra-clean case [190]

When dealing with the superconductor electrodynamics in the fluctuation regime, it is necessary to remember that in the vicinity of the critical temperature the role of the effective size of a√fluctuation Cooper pair is played by the GL coherence length ξGL (T ) = ξ0 / . So, as was already mentioning above, the case of a pure enough superconductor with electron mean free path `  ξ0 has to be formally subdivided into the clean (ξ0  `  ξGL (T )) and 125

ultra-clean (ξGL (T )  `) limits. The nontrivial cancellation of the contributions, previously divergent in T τ (see, for example, (6.15)), will be shown in this Section. This results in a reduction of the total fluctuation correction in the ultra-clean case to the AL term only. We will base on [190] restricting our consideration to the case of a 2D electron system. In terms of the parameter T τ, used in the theory of disordered alloys, three different domains of√the metal purity can √be distinguished: T τ  1 (dirty case), 1  T τ  1/  (clean case) and 1/   T τ (ultra-clean case of nonlocal electrodynamics). The latter case was rarely discussed in the literature [165, 191, 192] in spite of the fact that it becomes of primary importance for metals of very modest√purity, let us say, with T τ ≈ 10. Really, in this case the condition T τ ≥ 1/  , which in terms of the reduced temperature is read as 10−2 ≤   1, practically covers all the experimentally accessible range of temperatures for the fluctuation conductivity measurements. As regards the √ usually considered local clean case (1  T τ  1/ ) for the chosen value T τ ≈ 10, it would not have any range of applicability. Indeed, the equivalent condition for the allowed temperature interval is   (T τ )−2 , and it almost contradicts the 2D thermodynamical Ginzburg-Levanyuk criterion for the mean field approximation applicability (Gi(2) = ETFc  ). Moreover, as we will show below, for transport coefficients the higher order corrections become comparable with the mean field results much p before they are important for thermodynamical quantities, namely at  ∼ Gi(2) [193, 194]. So in practice one can speak about the dirty and the non-local ultra-clean limits only. As we saw above the 2D AL contribution turns out to be completely independent of the electron mean free path ` [5] . The anomalous Maki-Thompson contribution, being induced by the pairing on the Brownian diffusive trajectories [76], naturally depends on T√τ, but in an indirect way. It turns out to be τ −independent up to T τ ∼ 1/ , (see (6.24)) and diverges as T τ ln(T τ ) √ for T τ  1/  [165, 191]. The analogous problem takes place in the case of the DOS contribution: its standard diagrammatic technique calculations lead to a negative correction (6.15) [108] evidently strongly divergent when T τ → ∞. In the derivation of all these results the local form of the fluctuation propagator and Cooperons √ were used. This is why the direct extension of their validity for T τ  1/  → ∞ is incorrect. One can √ notice [190] that at the upper limit of the clean case, when T τ ∼ 1/ , both the DOS and anomalous MT (6.24) contributions turn out to be of the same order of magnitude but of opposite signs. So one can suspect that in the case of a correct procedure of impurity averaging in the ultra-clean case the large negative DOS contribution can be cancelled with the positive anomalous MT one. In the case of a 2D electron spectrum the Cooperon can be calculated exactly for the case of an arbitrary electron mean 126

free path: λ(q, ε1 , ε2 ) =

Θ(−ε1 ε2 )

1− p τ (e ε1 − εe2 )2 + vF2 q 2

!−1

.

(6.26)

One can see that this expression can be reduced to (5.21) in the case of vF q  |e ε1 − εe2 |. Let us stress that this result was carried out without any expansion over the Cooper pair center of mass momentum q and is valid in the 2D case for an arbitrary `q. The fluctuation propagator in the 2D case of an arbitrary mean free path can be written as [190] ∞  X T 1 −1 −[νL(q, Ωk )] = ln + (6.27) Tc n=0 n + 1/2   1 . −q
2 2 q2 vF Ωk 1 1 1  n + 2 + 4πT + 4πT τ + 16π2 T 2 − 4πT τ Near Tc ln TTc ≈  and in the local limit, when only small momenta `q  1 are involved in the final integrations, the Exp. (6.27) can be expanded in vF q/ max{T, τ −1 } and reduces to the appropriate local expression. Let us demonstrate the specifics of the non-local calculations for the example of the Maki-Thompson contribution. We restrict our consideration to the vicinity of the critical temperature, where the static approximation is valid. Using the non-local expressions for the Cooperon and the propagator one can find after integration over electronic momentum: X Z d2 q (M T ) 2 2 2 Q (ωυ ) = −4πνvF e T L(q, 0) × (6.28) 2 (2π) ε h ∼ ∼ n  ∼ i ∼ × M  n ,  n+ν , q + M  n+ν ,  n , q , where M (α, β, q) =

Rq (2α)Rq (α + β) − Θ(αβ)Rq (2α)Rq (2β)

(β − α)2 Rq (2α) − τ1 Rq (2β) − τ1 Rq (α + β)

p and Rq (x) = x2 + vF2 q2 . The analogous consideration of the DOS diagrams 2 and 4 which are the leading ones in the clean case [108] results in similar expressions. One can see that, after analytical continuation over the external frequency ων → −iω and the consequent tending ω → 0, each of the DOS or MT type 127

diagrams is written in the form of a Laurent series of the type C−2 (T τ )2 + C−1 (T τ )+C0 +C1 (T τ )−1 +... and is divergent at T τ → ∞ in accordance with (6.15). Nevertheless the expansion in a Laurent series of the sum of these nonlocal diagrams leads to the exact cancellation of all divergent contributions. The leading order of the sum of the MT and DOS contributions in the limit of T τ  1 turns out to be proportional to (T τ )−1 only and disappears in the ultra-clean limit. So the correct accounting for non-local scattering processes in the ultra-clean limit results in a total quantum correction negligible in comparison with the AL contribution. Nevertheless, its formal independence on impurities concentration (see (3.16)) was re-examined for ultra-clean case too in [189] and there it was demonstrated that this statement is valid in a rigorous sense only in the case of direct current and absence of a magnetic field. Let us recall that the normal Drude conductivity in the ultra-clean case takes the form σ± (ω) = σxx ± iσxy =

e2 nτ /m , 1 − i(ω ∓ ωc )τ

(6.29)

where ωc is the cyclotron frequency. When τ → ∞ the real part of the conductivity vanishes. The analysis of the AL diagram in the ultra-clean case demonstrates that each of the Green functions blocks B acquires the same denominator. As a result the expression for the fluctuation conductivity contains the same Drude like pole but of second order AL(l)

AL σ± (ω)

AL(l)

σxx ± iσxy = (1 − i(ω ∓ ωc )τ )2

AL(l)

(6.30)

(σαβ is the component of the paraconductivity tensor calculated above in the local limit (3.16-(8.35)). The origin of this pole can be recognized by means of the following speculation. The electric field does not interact directly with the fluctuation Cooper pairs, but it produces the effect by interaction with the quasiparticles forming these pairs only. The characteristic time of the change of a quasiparticle state is of the order of τ . Consequently the single-particle Drude type conductivity in an a.c. field has a first order pole, while in the AL paraconductivity it is of second order [189]. In spite of this difference one can see that the AL conductivity, like the Drude one, vanishes at ω 6= 0, τ → ∞ because in the absence of impurities the interaction of the electrons does not produce any effective force acting on the superconducting fluctuations, while the d.c. paraconductivity conserves its usual τ −independent form. It is impossible to distinguish the motion of the electron liquid from the condensate motion in current experiments without additional scattering. 128

The non-local form of the Cooperon and fluctuation propagator have to be taken into account not only for the ultra-clean case but in every problem where relatively large bosonic momenta are involved: the consideration of dynamical and short wavelength fluctuations beyond the vicinity of critical temperature, the effect of relatively strong magnetic fields on fluctuations etc. Recently such an approach was developed in a number of studies [191, 190, 195, 196].

6.6

Nonlinear fluctuation effects [14]

As we have already seen in the temperature region Gi    1 the thermodynamic fluctuations of the order parameter Ψ can be considered to be Gaussian. Nevertheless the example of the previous Section demonstrates that in transport phenomena nonlinear effects, related with the interaction of fluctuations (higher order corrections) can manifest themselves much earlier. It has been found in paper [193], that nonlinear fluctuation phenomena restrict the Gaussian region in the fluctuation p conductivity of a superconducting film to a new temperature scale: Gi(2d)    1 (see also [124, 125, 194, 129, 207]). In this Section we obtainpexpressions for the conductivity in the temperature region Gi(2d) <  < Gi(2d) , where both the perturbation theory works well and the nonlinear fluctuation effects are important. Let us start from the correlator (5.26) which can be expressed by means of the Gi(2d) number: hΨ∗k Ψk i =

1 T 32π 3 T2 = Gi . (2d) 2  ν  + πD 7ζ(3) k2 k + 8T 8T πD

(6.31)

2 The long-wave-length fluctuations with k 2 < kmin = 8T /πD can be considered as a local condensate. They lead to the formation of the pseudogap

∆pg =

"Z

2 k2 .kmin

#1/2 p d2 k ∗ hΨ Ψ i ' T Gi(2d) . k k (2π)2

(6.32)

in the single-particle spectrum of excitations. p Not very close to the transition ( > Gi(2d) ) only excitations with energies E > ∆pg are important. The pseudogap does not play any role for them. Thus, in this region of temperatures it is sufficient to consider fluctuations in the linear approximation only (see [5, 6, 7]). However, in the temperature p region  < Gi(2d) the nonlinear fluctuation contribution of the excitations with energies E < ∆pg becomes essential. 129

To take into account the spatial dependence of the order parameter we will use the results obtained in [208]. It was shown there that the spatial variations of ∆pg act on single-particle excitations in the same way as magnetic impurities do (the analogy between the effect of fluctuations and magnetic impurities was observed in many papers, see for example, [209]). In this case, the total pairbreaking rate Γ can be written as a sum of the pairbreaking rate due to the magnetic impurities and the fluctuation term. Thus, the self-consistent equation for Γ can be written in the following form [208]: Z d2 k hΨ∗k Ψk i 1 Γ= + . (6.33) 1 (2π)2 E + 2 Dk 2 + Γ τs In the region E . Γ, Γ  T  we obtain from Eqs.(6.33), (6.31): Γ ∼ T Gi(2d)


1/2

' ∆pg ,

(6.34)

which coincides with the results obtained in [124, 115]. Let us note, that the pair-breaking rate Γ was found to be of the order of the pseudogap ∆pg . Thus, a wide maximum appears in the density of states at E ∼ ∆pg . As we already saw (6.24), in purely 2D case the Maki-Thompson correction to the conductivity saturates for T  < Γ (where Γ = 8T γϕ /π) and takes the form [14]: T πΓ δσ M T ∼ Gi(2d) ln . (6.35) σn Γ 8T  As p it can be seen from Eqs.(6.34) such a saturation takes place when  < Gi(2d) . Similar results have been obtained in [124, 125, 129], with slightly different numerical coefficients3 . However, its exact value is not very important since in the region T  < Γ the Maki-Thompson correction is less singular than the Aslamazov-Larkin one and can be neglected. The latter does not saturate when T tends to Tc but becomes more and more singular. In the presence of the pseudogap if there is no equilibrium, the fluctuating Cooper pair lifetime increases with respect to the GL one: τf l = aτGL (a > 1). Recall, that analogous changes in the coefficient a in the TDGL equations appear below the transition temperature (see e.g. [66, 72, 210, 211, 212]). The growth of the coefficient a and, consequently, the increase of the fluctuation lifetime is because the quasiparticles require more time to attain thermal equilibrium (the corresponding time we denote as τe ). A rough estimate gives a ∼ ∆pg τe . In the case of weak energy relaxation, τe has to be determined from the diffusion equation taking account of the pseudogap 3

Note, that the numerical coefficient in Eq.(6.35) depends on the definition of Gi(2d) and how the summation of higher order diagrams is made.

130

(see [211, 212, 213]). Note, that in this complicated case the coefficient a becomes a non-local operator. Rough estimates give the following value for 2 the thermal equilibrium transition time τe ∼ (Dkmin )−1 ∼ (T )−1 . Taking into account Eq.(6.32) we obtain from (1.11) for the paraconductivity contribution in the discussed limit of the weak energy relaxation [14] : 3/2

Gi(2d) δσ ∼ . σn 2

(6.36)

Let us discuss now the role of the energy relaxation processes, characterized by a quasiparticle lifetime τε . Nonelastic electron scattering off phonons and other possible collective excitations can decrease τε significantly. These processes together with additional pairbreaking processes (due to magnetic impurities or a magnetic field) lead to a decrease of the nonlinear effects. In view of these processes, one can write the following interpolation formula for the non-linear fluctuation conductivity [14]: δσ ∼  σn  +

3/2

Gi(2d)  1 Γ 1 + T √Gi T τε

(2d)

.

(6.37)

Note that Eqs.(6.36-6.37) are valid only if the parameters Γ and τε are such that the correction to conductivity δσ is larger Aslamazovp than the usual 2 Larkin correction Eq.(3.16). If Γ > T , T τε < Gi(2d) or if T τε /Γ < Gi(2d) , than nonlinear effects are negligible and the usual result (3.16) is valid for all  > Gi(2d) . We see that the paraconductivity can exceed the value of the normal 3/4 conductivity σn in the region Gi(2d) <  < Gi(2d) . Let us recall, that in this region corrections to all the thermodynamic coefficients are still small and the linear theory is well applicable.

6.7

Discussion

Although the in-plane and out-of plane components of the fluctuation conductivity tensor of a layered superconductor contain the same fluctuation contributions, their temperature behavior may be qualitatively different. In fact, fl for σxx , the negative contributions are considerably less than the positive ones in the entire experimentally accessible temperature range above the transition, and it is a positive monotonic function of the temperature. Moreover, for HTS compounds, where the pair-breaking is strong and the MT contribution is in the overdamped regime, it is almost always enough to take into 131

Figure 6.3: The normalised excess conductivity for samples of YBCO-123 (triangles), BSSCO-2212(squares) and BSSCO-2223 (circles) plotted against  = ln T /Tc on a ln-ln plot as described in [138]. The dotted and solid lines are the AL theory in 3D and 2D respectively. The dashed line is the extended theory of [139]. account only the paraconductivity to fit experimental data. Some examples of the experimental findings for in-plane fluctuation conductivity of HTS materials on can see in [130, 131, 132, 133, 134, 135, 136, 137]. fl In Fig. 6.3 the fluctuation part of in-plane conductivity σxx is plotted as a function of  = ln T /Tc on a double logarithmic scale for three HTS samples (the solid line represents √ the 2D AL behavior (1/), the dotted line represents the 3D one: 3.2/ ) [138]. One can see that paraconductivity of the less anisotropic YBCO compound asymptotically tends to the 3D behavior (1/1/2 ) for  < 0.1, showing the LD crossover at  ≈ 0.07; the curve for more anisotropic 2223 phase of BSCCO starts to bend for  < 0.03 while the most anisotropic 2212 phase of BSCCO shows a 2D behavior in the whole temperature range investigated. All three compounds show a universal 2D temperature behavior above the LD crossover up to the limits of the GL region. It is interesting that around  ≈ 0.24 all the curves bend down and follow the same asymptotic 1/3 behavior (dashed line). Finally at the value  ≈ 0.45 all the curves fall down indicating the end of the observable fluctuation regime. Reggiani et al. [139] extended the 2D AL theory to the high temperature region by taking into account the short wavelength fluctuations. The following universal formula for 2D paraconductivity of a clean 2D superconductor as a function of the generalized reduced temperature  = ln T /Tc was obtained4 : e2 fl σxx = f () 16s with f () = −1 ,   1 and f () = −3 ,  & 1. In the case of the out-of-plane conductivity the situation is quite different. Both positive contributions (AL and anomalous MT) are suppressed by the interlayer transparency, leading to a competition between positive and negative terms. This can lead to a maximum in the c-axis fluctuation resistivity which occurs in the 2D regime (in the case discussed Jτ << 1, rκ << 1 and
In Section 7.2 we will demonstrate how such a dependence 1/ ln3 (T /Tc ) appears by accounting for short wavelength fluctuations for the 2D fluctuation susceptibility. 4

132

Figure 6.4: Fit of the temperature dependence of the transverse resistance of an underdoped BSCCO c-axis oriented film with the results of the fluctuation theory [140]. The inset shows the details of the fit in the temperature range between Tc and 110K. γϕ κ > 1) :   1 1 1 − κ ˜− . m /r ≈ (8rκ)1/2 8κ 2γϕ This nontrivial effect of fluctuations on the transverse resistance of a layered superconductor allows a successful fit to the data observed on optimally and overdoped HTS samples (see, for instance, Fig.6.4) where the growth of the resistance still can be treated as the correction. The fluctuation mechanism of the growth of the transverse resistance can be easily understood in a qualitative manner. Indeed to modify the in-plane result (3.17) for the case of c-axis paraconductivity one has to take into account the hopping character of the electronic motion in this direction. If the probability of one-electron interlayer hopping is P1 , then the probability of coherent hopping for two electrons during the fluctuation Cooper pair lifetime τGL is the conditional probability of these two events: P2 = P1 (P1 τGL ). The AL transverse paraconductivity may thus be estimated as σ⊥ ∼ P2 σkAL ∼ P12 12 , in complete accordance with (6.11). We see that the temperature singularity AL of σ⊥ turns out to be stronger than that in σkAL , however for a strongly AL anisotropic layered superconductor σ⊥ is considerably suppressed by the square of the small probability of inter-plane electron hopping which enters in the pre-factor. It is this suppression which leads to the necessity of taking into account the DOS contribution to the transverse conductivity. The latter is less singular in temperature but, in contrast to the paraconductivity, manifests itself in the first, not the second, order in the interlayer transparency DOS σ⊥ ∼ −P1 ln 1ε . The DOS fluctuation correction to the one-electron transverse conductivity is negative and, being proportional to the first order of P1 , can completely change the traditional picture of fluctuations just rounding the resistivity temperature dependence around transition. The shape of the temperature dependence of the transverse resistance mainly is determined by competition of the opposite sign contributions: the paraconductivity and MT term, which are strongly temperature dependent but are suppressed by the square of the barrier transparency and the DOS contribution which has a weaker temperature dependence but depends only linearly on the barrier transparency.

133

Part III Manifestation of fluctuations in various properties

134

In this Section we will demonstrate the applications of the microscopic theory of fluctuations. The limited volume does not permit us to deliver here the systematic review of the modern theory and we restrict ourselves only by presentation of the several representative recent studies. The details one can find in the articles cited in subtitles. It is necessary to underline that the comparison of the results of fluctuation theory with the experimental findings on HTS materials has to be considered sooner in qualitative than quantitative context. Indeed, as is clear now, the superconductivity in the most of HTS compounds has the nontrivial symmetry. Moreover, as was discussed in the previous Section, these compounds are rather clean than dirty. Both these complications can be taken into account (see for example [126, 110]) but this was not done in the majority of the cited papers.

135

Chapter 7 [Magnetoconductivity] The effects of fluctuations on magnetoconductivity [110, 141, 169, 170] The experimental investigations of the fluctuation magnetoconductivity are of special interest first because this physical value weakly depends on the normal state properties of superconductor and second due to its special sensitivity to temperature and magnetic field. The role of AL contribution for both the in-plane and out-of-plane magnetoconductivities was studied above in the framework of the phenomenological approach. The microscopic calculations of the other fluctuation corrections to the in-plane magnetoconductivity conductivity show that the MT contribution has the same positive sign and temperature singularity as the AL one. In the case of weak pair-breaking it can even considerably exceed the latter. The negative DOS contribution, like in the case of the zero-field conductivity, turns out to be considerably less singular and many authors (see e.g. Refs. [151, 152, 153, 154, 157, 155, 156, 158, 159, 160],[161, 162]) successfully explained the inplane magnetoresistance data in HTS using the AL and MT contributions only [163, 164, 165, 166]. Turning to the out-of-plane magnetoconductivity of a layered superconductor one can find a quite different situation. Both the AL and MT contributions turn out to be here of the second order in the interlayer transparency and this circumstance makes the less singular DOS contribution, which remains however of first order in transparency, to be competitive with the main terms [141]. The large number of microscopic characteristics involved in this

136

competition, like the Fermi velocity, interlayer transparency, phase-breaking and elastic relaxation times, gives rise to the possibility of occurrence of different scenarios for various compounds. The c-axis magnetoresistance of a set of HTS materials shows a very characteristic behavior above Tc0 . In contrast to the ab-plane magnetoresistance which is positive at all temperatures, the magnetoresistance along the c-axis has been found in many HTS compounds (BSSCO [142, 144, 145, 146], LSSCO [147], YBCO [148] and TlBCCO [150]) to have a negative sign not too close to Tc0 and turn positive at lower temperatures. We will show how this behavior find its explanation within the fluctuation theory [110, 149]. We consider here the effect of a magnetic field parallel to the c-axis. In this case both quasiparticles and Cooper pairs move along Landau orbits within the layers. The c-axis dispersion remains unchanged from the zero-field form. In the chosen geometry one can generalize the zero-field results reported in the previous Section to finite field strengths simply by the replacement of the two-dimensional integration over q by a summation over the Landau levels Z d2 q H X h X → = (2π)2 Φ0 n 2πη (2) n 2 (let us remind that η(2) = ξxy ). So the general expressions for all fluctuation corrections to the c-axis conductivity in a magnetic field can be simply written in the form [110]: ∞

AL σzz =

e2 sr2 h X 1 3/2 2 64ξxy n=0 {[ + h(2n + 1)][r +  + h(2n + 1)]}

(7.1)

1/h

DOS σzz

e2 srκh X 1 =− 2 8ξxy n=0 {[ + h(2n + 1)][r +  + h(2n + 1)]}1/2 ∞ e s˜ κh X =− 2 4ξxy n=0 2

M T (reg) σzz

 + h(2n + 1) + r/2 {[ + h(2n + 1)][r +  + h(2n + 1)]}1/2

(7.2) !

−1

(7.3) 2

M T (an) σzz =

e sh − γϕ )

∞ X

γϕ + h(2n + 1) + r/2

− {[(γϕ + h(2n + 1)][γϕ + h(2n + 1) + r)]}1/2 !  + h(2n + 1) + r/2 . (7.4) − {[ + h(2n + 1)][r +  + h(2n + 1)]}1/2

2 (ε 8ξxy

n=0

For the in-plane component of the fluctuation conductivity tensor the only additional problem appears in the AL diagram, where the matrix elements 137

of the harmonic oscillator type, originating from the Bk (qk ) blocks, have to be calculated. How to do this was demonstrated in details in Section 4. The other contributions are essentially analogous to their c-axis counterparts: ∞

AL σxx

e2 X = (n + 1) 4s n=0

1

{[ + h(2n + 1)][r +  + h(2n + 1)]}1/2 2 + {[ + h(2n + 2)][r +  + h(2n + 2)]}1/2 ! 1 , {[ + h(2n + 3)][r +  + h(2n + 3)]}1/2

− (7.5)

1/h

DOS σxx

+

M T (reg) σxx

e2 h(κ + κ ˜) X 1 =− , 1/2 2s {[ + h(2n + 1)][r +  + h(2n + 1)]} n=0 (7.6)

and ∞

M T (an) σxx

X e2 h = 4s( − γϕ ) n=0

1

− {[(γϕ + h(2n + 1)][γϕ + h(2n + 1) + r)]}1/2 ! 1 . (7.7) {[ + h(2n + 1)][r +  + h(2n + 1)]}1/2

These results can in principle be already used for numerical evaluations and fitting of the experimental data which was indeed successfully done in a series of papers [148, 167, 150]. The detailed comparison of the cited results with the experimental data [148, 168], especially in strong fields, raised the problem of regularization of the DOS contribution. If in the absence of the magnetic field its ultra-violet divergence was successfully cut off at q ∼ ξ −1 , in the case under consideration the cut off parameter depends on the magnetic field and makes the fitting procedure ambiguous. The solution of this problem was proposed in DOS DOS [169], where the authors calculated the difference ∆σzz = σzz (h, ) − DOS σzz (0, ) applying to formulas (7.2) and (7.6) the same trick which was already used in Section 2 for the regularization of the free energy in magnetic field (Eq. (2.49)). The corresponding asymptotics for all out-of-plane fluctuation contributions are presented in the following table:

138

Figure 7.1: Magnetoconductivity versus temperature at 27 T for an underdoped Bi-2212 single crystal. The solid line represents the theoretical calculation. The symbols are the experimental magnetoconductivity ∆σzz (B q c q I)[170] h DOS ∆σzz M T (reg)

∆σzz

AL −∆σzz

e2 sκ 2 3 25 ξxy e2 s˜ κ

r2 h2 3 26 ξ 2xy [(+r)]3/2 e2 s r 2 (+r/2) 2 2 [(+r)]5/2 h 28 ξxy

M T (an) −∆σzz

min{, r}  γϕ M T (an) −∆σzz γϕ  min{, r}

r(+r/2) 2 h [(+r)]3/2

e2 s 3

27 ξ 2xy

r2 [(+r)]2

h2

√ r e2 s 2 3/2 h 3 27 ξ 2xy γϕ

  h  r q(3D)

max{, r}  h (2D)

√ e2 sκ √ √h r ln 2 8ξxy + +r 2 2 2 M T (reg) −σzz (0, ) − π28eξ2s˜κ rh xy 2s 2 r AL σzz (0, ) − 7ζ(3)e 2 29 ξxy h2

e2 sκ

0.428 16ξ2 r hr xy q e2 s˜ κ 0.428 8ξ2 r hr xy

AL σzz (0, ) M T (an)

σzz

M T (an)

σzz

−

3.24e2 s 2 ξxy

(0, ) − (0) −

pr

h

e2 s 2 32ξxy

3.24e2 s 2 64ξxy

q

r γϕ

pr

h

M T (an)

(0, ) −

3π 2 e2 s max{r, 2 28 ξxy h

M T (an)

(0, ) −

3π 2 e2 s (r+) 2 28 ξxy h

σzz

σzz

Table 3 The procedure described gives an excellent fitting up to very high fields [170] which is shown in Fig . 7.1. Let us start the analysis from the 2D case (r  ). One can see that here the positive DOS contribution to magnetoconductivity turns out to be dominant. It grows as H 2 up to Hc2 () and then crosses to a slow logarithmic DOS DOS asymptote. At H ∼ Hc2 (0) the value of ∆σzz (h ∼ 1, ) = −σzz (0, ) which means the total suppression of the fluctuation correction in such a strong field. The regular part of the Maki-Thompson contribution does not manifest itself in this case while the AL term can compete with the DOS one in the immediate vicinity of Tc , where the small anisotropy factor r can be compensated by the additional 3 in the denominator. The anomalous MT contribution can contribute in the case of small pairbreaking only, which is opposite to what is expected in HTS. In the 3D case (  r) the behavior of the magnetoconductivity is more complex. In weak and intermediate fields the main, negative, contribution to the magnetoconductivity occurs from the AL and MT terms. At H ∼ Hc2 ()(h ∼ ) the paraconductivity is already considerably suppressed by the magnetic fieldpand the h2 − dependence of the magnetoconductivity changes (f l) r tendency to the high field asymptote −σzz (0, ). In this through the h intermediate region of fields (  h  r), side by side with the decrease (∼ pr ) of the main AL and MT contributions, the growth of the still relatively h small DOS term takes place. At the upper limit of this region (h ∼ r) its 139

Figure 7.2: Decomposition of the calculation of total theoretical magnetoconductivity for an underdoped Bi-2212 single crystal at 27 T. The inset shows the regular and anomalous parts of the MT contribution which are too small to be presented in the same scale as the AL and DOS contributions [170]. positive contribution is of the same order as the AL one and at high fields (r  h  1) the DOS contribution determines the slow logarithmic decay of the fluctuation correction to the conductivity which is completely suppressed only at H ∼ Hc2 (0). The regular part of the Maki- Thompson contribution is not of special importance in the 3D case. It remains comparable with the DOS contribution in the dirty case at fields h . r, but decreases rapidly (∼ hr ) at strong fields ( h & r), in the only region where the robust ∆σcDOS (h, ) ∼ ln hr shows up surviving up to h ∼ 1. The temperature dependence of the different fluctuation contributions to the magnetoconductivity calculated for an underdoped Bi-2212 single crystal at the magnetic field 27 T is presented in Fig. 7.2. The formulas for the in-plane magnetoconductivity are presented in the table:   h  r; max{, r}  h h e2 √ 1 e2 [8(+r)+3r2 ] 2 AL AL AL −σxx (0, ) + 4s ; −σxx (0, ) + ∆σxx − 28 s [(+r)]5/2 h 2hr M T (an)

∆σxx (min{, r}  γϕ ) M T (an) ∆σxx (γϕ  min{, r}) DOS ∆(σxx + M T (reg) σxx )

2

(+r/2) 2 − 3 e25 s [(+r)] 3/2 h 2

MT −σxx (0, ) +

e2 1 8s γϕ

− 3 e25 s γ 3/21r1/2 h2

MT −σxx (0, ) +

e2 (κ+˜ κ) (ε+r/2) h2 3 27 s [ε(ε+r)]3/2

2 κ) 0.428 e (κ+˜ 26 s

q

γ

ln √2h+√ϕ2h+r

0.2e2 √1 ; s hr h ; r

√

MT −σxx (0, ) + e2 (κ+˜ κ) 32s

Table 4 Analyzing it one can see that in almost all regions the negative AL and MT contributions govern the behavior of in-plane magnetoconductivity. Nevertheless, similar to the c-axis case, the high field behavior is again deterM T (reg) DOS mined by the positive logarithmic ∆(σxx + σxx ) contribution, which is the only one to survive in strong field. It is important to stress that the suppression of the DOS contribution by a magnetic field takes place very slowly. Such robustness with respect to the magnetic field is of the same physical origin as the slow logarithmic dependence of the DOS-type corrections on temperature. 140

e2 1 8s h

√

3π 2 e2 1 32s h

ln (√+√h+r)

Another important problem which appears in the fitting of the resistive transition shape in relatively strong fields with the fluctuation theory is the much larger broadening of the transition than predicted by the AbrikosovGorkov theory [171]. Kim and Gray [109] explained the broadening of the c-axis peak with increasing magnetic field in terms of Josephson coupling, describing a layered superconductor as a stack of Josephson junctions. In Refs. [289, 71] the self-consistent Hartree approach was proposed for the extension of fluctuation theory beyond the Gaussian approximation. It results in the considerable shift of Tc (H) toward low temperatures with a corresponding broadening of the transition. The renormalized reduced temperature ε˜h is determined according to the self-consistent equation [71]: 1/h

X 1 1 εh = ε˜h − Gi(2) h 4 [(˜ εh + hn)(˜ εh + h(n + 1) + r)]1/2 n=0

(7.8)

The authors of [173], following the procedure proposed by Dorsey and Ullah [71], modified the expressions (7.1)- (7.7) by account for (7.8). In result they succeeded to fit quantitatively both in-plane resistivity transition and the transverse resistivity peak for BSCCO films strongly broadened by applied magnetic field.

141

Chapter 8 Fluctuations far from Tc or in strong magnetic fields As was mentioned above the role of fluctuations is especially pronounced in the vicinity of the critical temperature. Nevertheless for some phenomena they can be still considerable far from the transition too. In these cases the GL theory is certainly unapplicable since the short-wave and dynamical fluctuation contributions have to be taken into account. It can be done in the microscopic approach which we will demonstrate in several examples.

8.1

Fluctuation magnetic susceptibility far from transition [29].

The given above qualitative estimations (2.29)-(2.35) for the fluctuation diamagnetic susceptibility, based on the Langevin formula, demonstrate that even at high temperatures T  Tc it turns to be of the order of χP for clean 3D superconductors and exceeds noticeably this value for 2D systems. In order to develop the microscopic theory [174, 29, 285] let us start from the general expression for free energy in the one-loop approximation: X F =T Tr{ln[1 − gΠ(Ωk , r, r0 )]}, (8.1) Ωk

where g is the effective interaction constant related with the transition critical temperature by (5.11). This approximation corresponds to the ladder one (see (5.3)) for the fluctuation propagator. The polarization operator Π(Ωk , r, r0 ) is determined by expression (5.4) but in the case of an applied magnetic field the homogeneity of the system is lost and Π(Ωk , r, r0 ) depends not on the space variable difference r− r0 but on each separately. Expanding 142

Π(Ωk , r, r0 ) one can express the magnetic susceptibility of a layered superconductor in a weak magnetic field perpendicular to the layers in terms of the derivatives Πx = ∂q∂x Π(q) [29]: Z ∂2F 2 2 X d3 q 3 χ=− = − e T L Πx (Πx Πyy − Πy Πxy ). (8.2) 3 ∂H 2 3 (2π) Ω k

The final expressions for the fluctuation diamagnetic susceptibility in the clean and dirty cases for wide range of temperatures can be written as:

(3) χf l (T )

χP = 3



0.05(ln−2 (ω − ln−2 (T /Tc )), τ −1  T  ωD √ D /Tc )) 2 T τ ln−1 (T /Tc ), Tc  T  τ −1 (2) χf l (T )

0.05 χP = pF s



EF T



1 ln (T /Tc ) 3

(8.3)

(8.4)

Let us stress that these results are valid for the fluctuation diamagnetism of a normal metal with g > 0 too, if by Tc one uses the formal value Tc ∼ 1 EF exp( νg ).

8.2

Fluctuation magnetoconductivity far from transition [177].

Let us discuss the conductivity of the 2D electron system with impurities in a magnetic field at low temperatures. Even in the absence of the field the effects of quantum interference of the non-interacting electrons in their scatterings on elastic impurities already results in the appearance of a nontrivial temperature dependence of the resistance. This result contradicts the statement of the classical theory of metals requiring the saturation of the resistance at its residual value at low temperatures. In a superconductor above the critical temperature this, so-called weak localization (WL), effect is amplified by the Andreev reflection of electrons on the fluctuation Cooper pair leading to appearance of the MT correction to the conductivity. The characteristic feature of both the MT and WL corrections is their extreme sensitivity to the dephasing time τϕ and to weak magnetic fields. Beyond the GL region (T & Tc ) the MT correction is determined by the same diagram 2 of Fig. 6.1 but now the dynamic (Ωk 6= 0) and short-wavelength (q ∼ ξ −1 ) fluctuation modes have to be taken into account. The corresponding calculations were performed in [111, 177] and the result can be written in the form: 143

e2 [α − β(T )] Y (ΩL τ ) , (8.5) 2π 2 where we introduced the effective Larmour frequency for the diffusion motion ΩL = 4DeH with the diffusion coefficient D 1 and the function  x2 1 1 , x1 24 Y (x) = ln x + ψ( + ) = . (8.6) ln x, x  1 2 x δσW L+M T =

The first term in this formula corresponds to the WL contribution (α = 1 if the spin-orbit interaction of the electrons with the impurities is small while in the opposite limiting case α = −1/2), the second describes the MT contribution to magnetoconductivity. The function β[ln(T /Tc )] was introduced in Ref.[177]. At T → Tc β(x) = 1/x and (8.5) reduces to the already studied MT correction in the vicinity of critical temperature. For T  Tc β(x) = 1/x2 and the MT contribution gives a logarithmically small correction to the WL result. Its zero-field value, being proportional to ln−2 (T /Tc ), decreases with the growth of the temperature faster than both the AL contribution (in the dirty case δσAL ∼ 1/ ln(T /Tc )) and the especially slow DOS contribution (δσDOS ∼ ln ln(1/Tc τ ) − ln ln(T /Tc )) (see Ref. [111, 112]). It worth mentioning that for the region of temperatures T  Tc , analogous to Exp.(8.3)- (8.4), the result (8.5) can be applied both to superconducting and normal metals (g > 0), if in place of the critical temperature 1 the formal value Tc ∼ EF exp( νg ) is undermined. The interplay of the localization and fluctuation corrections was extensively studied (see, for example, [178, 179, 180, 181, 182])

8.3 8.3.1

Fluctuations in magnetic fields near Hc2(0) [183]. Conductivity

As one can see from (7.5)- (7.7), in the vicinity of the upper critical field Hc2 (T ) the fluctuation corrections diverge as −1 for the 2D case and as h −1/2 2 h for the 3D case (it is enough to keep just the terms with n = 0 in 1

Comparison of the expressions (2.48), (3.34) and (8.5) relates the Larmour frequency with the dimensionless field: h = ΩL /2Tc introduced in section 2 and the diffusion coefficient with the phenomenological GL constants D = 1/mα. 2 h is the renormalized by the magnetic field reduced temperature h =  + h

144

these formulas). This behavior is preserved in strong magnetic fields too, but the coefficients undergo changes. A case of special interest is T  Tc (which means H → Hc2 (0)) which represents an example of a quantum phase transition [183]. Microscopic analysis of the magnetoconductivity permits us to study the effect of fluctuations in magnetic fields of the order of Hc2 (0), where the GL functional approach is inapplicable. We restrict our consideration to the case of a dirty metal (T τ  1). In this limit |˜ ωn+µ − ω ˜ −n | ≈ τ −1 and the Green function correlator (5.6) can be written in the form   2 −1 2 b P(q, ε1 , ε2 ) = 2πντ θ(−ε1 ε2 ) τ − |ε1 − ε2 | − Dq . (8.7) Expressing Π(q, Ωk ) in terms of P(q, ε1 , ε2 ) by means of (5.25) and using the definition of the critical temperature one can find an explicit formula for the fluctuation propagator

L−1 (q, Ωk ) = g −1 − Π(q, Ωk ) = (8.8) " ! #   b 2 T 1 |Ωk | + Dq 1 = −ν ln +ψ + −ψ . Tc 2 4πT 2 The prominent characteristic of this expression is that it is valid even relatively far from the critical temperature (for temperatures T  min{τ −1 , ωD }) and for |q|  l−1 , |Ωk |  ωD . One can rewrite this expression in a magnetic field along the c applied  b 2 ⇒ ΩL (n+1/2) axis in the Landau representation by simply replacing Dq k

[174] :

L−1 n (qz , Ωk )

  T 1 |Ωk | = −ν ln +ψ + + (8.9) Tc 2 4πT    ΩL (n + 1/2) + 4τ J 2 sin2 (qz s/2) 1 + −ψ . 4πT 2

In the case of arbitrary temperatures and magnetic fields the expression for the AL contribution to the conductivity takes the form:

AL (ων ) σxx

2

= νe T

∞ X X

Bn,m (Ωk + ων , Ωk )Lm (Ωk ) ×

Ωk n,m=0

×Bm,n (Ωk , Ωk + ων )Ln (Ωk + ων ) 145

(8.10)

(we have restricted our consideration to the 2D case). The expression for Bn,m (Ωk , ων ) can be rewritten as i X h√ √ 2πν Bn,m (Ωk + ων , Ωk ) = − √ τ 2 D(2) T n + 1δm,n+1 + nδm,n−1 × eH εi λn (εi + ων , Ωk − εi )λm (εi , Ωk − εi )

(8.11)

with λm (ε1 , ε2 ) =

1 Θ (−ε1 ε2 ) . τ |ε1 − ε2 | + ΩL (m + 1/2)

(8.12)

The critical field Hc2 (T ) is determined by the equation L−1 0 (qz = 0, Ωk = 0) = 0. This is why in the vicinity of Hc2 (T ) the singular contribution to (8.10) originates only from the terms with L0 (0, Ωk ). The frequency dependencies of the functions Bn,m (Ωk + ων , Ωk ) and L1 (Ωk ) are weak although we cannot omit them to get nonvanishing answer. It is enough to restrict ourselves to the linear approximation in their frequency dependencies. If the temperature T  Tc0 the sum over frequencies in (8.11) can be approximated by an integral. Transforming the boson frequency Ωk summation to a contour integration as was done above and making the analytic continuation in the external frequency ων one can get an explicit expression for the d.c. paraconductivity. In the same spirit the contributions of all other diagrams from the Fig. 6.1 which contribute to fluctuation conductivity in the case under discussion are calculated side by side with the AL one. The final answer can be presented in the form:    2e2 πTc0 3γE T Hc2 (T ) δσtot = − ln + + 3π 2 2γT Tc0 H − Hc2 (T )    Tc0 H − Hc2 (T ) +ψ + (8.13) 2γE T Hc2 (T )      Tc0 Hc2 (T ) Tc0 H − Hc2 (T ) 0 +4 ψ −1 , 2γE T H − Hc2 (T ) 2γE T Hc2 (T ) where γE is the Euler constant. Let us consider some limiting cases. If the temperature is relatively high T /Tc0  (H − Hc2 (T )) /Hc2 (T ), we obtain the following formula for the fluctuation conductivity:   2γE e2 T Hc2 (T ) δσ = . (8.14) π 2 Tc0 H − Hc2 (T )

146

If H < Hc2 (0), we can introduce Tc (H) and rewrite Eq.(8.14) in the usual way 3e2 Tc0 δσ = . (8.15) 2γE π 2 T − Tc (H) If H > Hc2 (0), in the low-temperature limit T /Tc0  (H − Hc2 (T )) /Hc2 (T ) we have Hc2 (T ) 2e2 δσ = − 2 ln . (8.16) 3π H − Hc2 (T ) One can see, that even at zero temperature a logarithmic singularity remains and the corresponding correction is negative. It results from all three fluctuation contributions, although the DOS one exceeds the others by numerical factor. Let us recall that in the case of the c-axis conductivity of a layered superconductor, or in granular superconductors above Tc , the DOS contribution exceeds the MT and AL ones parametrically [175].

8.3.2

Magnetization: one loop approximation

Considering thermodynamic properties of a film, we can calculate the free energy directly. In the one-loop approximation, the free energy can be written as [?]   X ˆ F1 = −T , (8.17) Tr ln 1 − g C(Ω) Ω

ˆ where C(Ω) is the cooperon. Using Eqs.(??), (??) and (8.17), one can easily obtain the magnetization M1 = −

ν ΩH 1 ∂F1 = Iα (h, t), V ∂H 2πd Hc2 (0)

(8.18)

where d is the thickness of the film or the interlayer distance, ν = eH/π is the number of states of a Landau level and function Iα (h, t) is defined in Eq.(??). Thus, at low temperature t  h the susceptibility takes the form: χ1 = −

∂M1 e2 v 2 τ = 2 2 F h−1 . ∂H π ~c d

(8.19)

One can see, that the fluctuation susceptibility (8.19) is large compared to the magnetic susceptibility of the normal metal χL even far from the transition: χ1 ∼

1 χL , Gi h

where Gi = (εF τ )−1 is the Ginzburg parameter. 147

(8.20)

Figure 8.1: Diagrams contributing to the free energy in the two-loop approximation. Similar diagrams appear in the derivation of the Ginzburg-Landau equations from the microscopic theory.

8.3.3

Magnetization: two loop approximation

In the previous sections we found the fluctuation correction to the transport and thermodynamic properties of a superconductor in a magnetic field in the first (one-loop) approximation. The purpose of the given section is to find the order of the subleading corrections. This will determine the area of applicability of the results obtained. We shall calculate the magnetization in the two-loop approximation for a dirty superconductor. This correction can be easily found in view of the simplifications described above. In the two-loop approximation, we have to deal with diagrams presented on Fig. 4. The corresponding contribution can be written in the coordinate representation in the following way X Z 3 F2 = T d2 r1 d2 r2 d2 r3 d2 r4 ε,Ω,Ω0

×Kε (r1 , r2 ; r3 , r4 )LΩ (r1 , r2 )LΩ0 (r3 , r4 ),

(8.21)

where Kε is the operator corresponding to the square blocks in the diagrams presented on Fig. 4. This operator is familiar from the usual BCS theory. It has been calculated by Maki [?] and Caroli et al. [305] and has the form: Kε (r1 , r2 ; r3 , r4 ) = ×

πN (0) 2 (

4 Y

1 δ(r1 − r2 )δ(r1 − r3 )δ(r1 − r4 ) 2 |ε| + 12 D∂(k) k=1   2  2  1  × |ε| + D ∂(1) − ∂(3) + ∂(2) − ∂(4) , 8

) (8.22)

where we make use of the Maki’s notations: ∂(k) = −i∇ − 2e(−1)k A(r). In the coordinate representation, the fluctuation propagator can be expanded on the basis of the eigenfunctions in the magnetic field and has the form: Z+∞ ∞ dpy X 0 ∗ LΩ (r, r ) = Ln (Ω)ψnp (r)ψnpy (r0 ), (8.23) y 2π n=0 −∞

148

where Ln (Ω) are matrix elements of the fluctuation propagator in the magnetic field (see Eq.(??)), ψnpy (r) is the eigenfunction for an electron in a magnetic field in the Landau gauge and py is the y-component of the momentum, which determines the orbit’s center. Again, in the vicinity of the transition line we keep the n = 0 term only in Eq.(8.23). From Eqs.(??— 8.23), we obtain the free energy per unit volume F2 πN (0) 2 3 = ν T V 2d

X

!2

L0 (Ω)

X ε

Ω

1 . (|ε| + ΩH /4)3

(8.24)

Thus, the magnetization takes the form: M2 =

ν2 1 ∂Iα2 (h, t) . π 2 dN (0) Hc2 (0) ∂h

(8.25)

At low temperatures t  h we have M2 = −

2ν 2 1 1 1 ln . 2 π dN (0) Hc2 (0) h h

We see, that the second order correction is negative. From Eqs.(8.18) and (8.25) we obtain the ratio    M2 Gi t 1 0 1 h = 2γ 2 − ψ , M1 π h γt 2γ t

(8.26)

(8.27)

where Gi is the Ginzburg parameter. The one-loop approximation is valid unless this ratio becomes of the order of unity. At low temperatures t  h, Eq.(8.27) yields the following condition h  Gi. If t  h, we have h

√

Gi t.

(8.28)

(8.29)

This indicates that at large enough temperatures the fluctuation region becomes wider. These results stand for the kinetic coefficients as well. In the clean case the formulae (8.28) and (8.29) are valid with Gi ∼ Tc0 /εF . However, the explicit calculations are more complicated due to the non-local structure of the K-operator. Let us note, that at an exponentially low temperature some other effects may reveal themselves. In the dirty case, the mesoscopic fluctuations 149

may be important. [?, ?] Really, the upper critical field depends on disorder. The distribution of impurities is random. There are some regions where the concentration of the impurities is such that the upper critical field is smaller than the bulk value. These regions may form superconducting islands weakly coupled one with another. At extremely low temperature the proximity effect and the Josephson coupling can make these mesoscopic fluctuations observable. The effects due to the mesoscopic fluctuations will be considered elsewhere.

8.3.4

Discussion

The central result of the paper is the existence of the logarithmic correction to the conductivity which persists down to zero temperature. This correction is shown to be negative in the dirty case. The minus sign comes from the DOS diagrams as well as from the MT term. The AL contribution is positive but numerically smaller. Let us note, that similar results (negative fluctuation correction to the conductivity) exist for the granular and layered superconductors. [?, ?] In these cases, the AL and MT contributions are parametrically small compared to the DOS term. The fluctuating magnetization exceeds conventional Landau diamagnetism for a very large range of fields. It is shown to be logarithmically divergent as well at T → 0. Let us note, that the singular behavior of the transport and thermodynamic quantities at low temperature is due to the low dimensionality of the system. In the three dimensional √ case the leading correction to the conductivity is not singular δσ3D ∝ h. The results obtained in the present paper can be checked experimentally by measuring the fluctuation conductivity in two-dimensional and quasi-twodimensional systems. The results obtained in the dirty limit can be checked by measuring the magnetoresistance in the dirty superconducting films at low temperatures. In this case, there could be some experimental difficulties connected with the Hc3 -effects that can screen the bulk properties of a film. The edge effects can be excluded, for example, by putting a sufficient amount of magnetic impurities on the edge of the film. Let us mention some recent experiments of Gantmakher et al.[?, ?] In these experiments the magnetic-field-tuned quantum phase transition has been studied in dirty In–O films at low temperatures. It was found that in the vicinity of the transition, the magnetoresistance reaches a maximum. It is possible that the theory developed in the present paper can give an explanation for the observed effects.

150

The clean case may be relevant to high-Tc superconductors [?] and, probably, to the recently discovered two-dimensional organic superconductors. [?] Let us note, that our results assume s-pairing and isotropic Fermi-surface which is not true for high-Tc superconductors. However, it can be shown, that the logarithmic singularity remains for any pairing type (with a coefficient different from our case). It is worth mentioning, that in the overdoped high-Tc superconductors the Ginzburg parameter Gi is small and, thus, the fluctuations are negligible. In the underdoped superconductors the fluctuations are extremely large and they lead to a large pseudogap which makes the conventional Fermi-liquid theory inapplicable. Hence, optimally doped superconductors should be used to check the results obtained.

8.4

The effect of fluctuations on the Hall conductivity[189]

Let us start with a discussion of the physical meaning of the Hall resistivity ρxy . In the case of only one type of carriers it depends on their concentration n and turns out to be independent of the electron diffusion coefficient: ρxy = H/ (en) . The fluctuation processes of the MT and DOS types contribute to the diffusion coefficient, so their expected contribution to the Hall resistivity is zero. For the Hall conductivity in a weak field one can write

σxy =

2 ρxy σxx

=

(n)2 ρxy σxx

+

(n) 2ρxy σxx δσxx

=

(n) σxy



δσxx 1+2 σxx



(8.30)

so, evidently, the relative fluctuation correction to Hall conductivity is twice as large as the fluctuation correction to the diagonal component. This qualitative speculation is confirmed by the direct calculation of the MT type diagram [184]. The AL process corresponds to an independent charge transfer which cannot be reduced to a renormalization of the diffusion coefficient. It contributes weakly to the Hall effect, and this contribution is related to the Cooper pair particle-hole asymmetry. This effect was investigated in a set of papers: [184, 185, 186, 187, 71, 188, 189]. Let us recall that the proper general expression describing the paraconductivity contribution to the Hall conductivity in the general case of arbitrary magnetic fields and frequencies (in the TDGL theory limits) was already carried out above in the phenomenological approach (see Eq. (3.30)). The microscopic consideration of this value can be AL done in the spirit of the calculation of σxx (see (6.8)) and after the analytical continuation results in

151

AL σxy

2 ∞ Z π/s Z 2hν(0) X dkz ∞ z = (n + 1) coth dz × π 2T −π/s 2π −∞ n=0   ∂ ∂ R R R R × ImLn (z) ReLn+1 (z) − ImLn+1 (z) ReLn (z) . (8.31) ∂z ∂z 

where dimensionless magnetic field h was introduced by (2.48). The phenomenological expression (3.30) can be obtained from this formula by carrying out the frequency integration in the same way as was done in the calculation of (6.8) (the essential region of integration is z  T ). R R One can see from (8.31) that if ImLR n (−z) = −ImLn (z) and ReLn (−z) = R ReLn (z) the Hall conductivity is equal to zero, or, in terms of the phenomenological parametrization, the reality of γGL results in a zero Hall effect. Physically it is possible to say that this zero is the direct consequence of electron-hole symmetry. However, from the formula (5.4) one can see that an energy dependence of the density of states or the electron interaction constant g immediately results in the appearance of an imaginary part of γGL . In the weak interaction approximation   ν(0) ∂ ln Tc ImγGL = . (8.32) 2 ∂E E=EF Usually this value is small in comparison with ReγGL by a ratio of the order of Tc /EF . Taking into account the terms of the order of ImγGL in (8.31) and using the explicit form of the fluctuation propagator for layered superconductor (5.28) one can find ImγGL h 2ν(0)

AL σxy = e2 T

×

Z

π/s

−π/s

(8.33)

dkz 1  + r sin2 (kz s/2)) ),   F( 2π  + r sin2 (kz s/2)) 2 2h

where   1 0 F (x) = 4x ψ(x) + xψ (x) − 1 − ψ( + x) . 2 2

(8.34)

For H → 0 the expression for the fluctuation Hall paraconductivity takes the form

152

AL σxy

e2 T = 6s



ImγGL ν(0)



h

 + r/2

e3 Φ0 T = 6πs



ImγGL ν(0)



 + r/2

h. [ ( + r)]3/2 (8.35) One can see that in the 2D case the temperature dependence of the AL fluctuation correction to the Hall conductivity   e3 Φ0 Tc h AL σxy ≈ 12πs EF 2 3/2

[ ( + r)]

turns out to be more singular than the MT one.

153

Chapter 9 The effect of fluctuations on the one-electron density of states and on tunneling measurements 9.1

Density of states [107].

The appearance of non-equilibrium Cooper pairing above Tc leads to a redistribution of the one-electron states around the Fermi level. A semiphenomenological study of the fluctuation effects on the density of states (DOS) of a dirty superconducting material was first carried out while analyzing the tunneling experiments of granular Al in the fluctuation regime just above Tc [197]. The second metallic electrode was in the superconducting regime and its well developed gap gave a bias voltage around which a structure, associated with the superconducting fluctuations of Al, appeared. The measured DOS energy dependence has a dip at the Fermi level 1 , reaches its normal value at some energy E0 (T ), show a maximum at an energy value equal to several times E0 , finally decreases towards its normal value at higher energies. The characteristic energy E0 was found to be of the order of the inverse of the GL relaxation time τGL introduced above. The presence of a depression at E = 0 and of a peak at E ∼ (1/ τGL ) in the DOS above Tc are precursor effects of the appearance of the superconducting gap in the quasiparticle spectrum at temperatures below Tc . The microscopic calculation of the fluctuation contribution to the one-electron DOS can be carried out within the diagrammatic technique [106, 107]. Let us start from the discussion of a clean superconductor. As is well known the one-electron DOS is determined by the imaginary part of the 1

Here we refer the energy E to the Fermi level, where we assume E = 0.

154

Figure 9.1: The one-electron Green function with the first order fluctuation correction. retarded Green function integrated over momentum. This definition permits us to express the appropriate fluctuation correction in terms of the fluctuation propagator: Z 1 dD p 1 (c) δGR (p, E) = − ImRR (E) (9.1) δν (E, ) = − Im D π (2π) π where RR (E) is the retarded analytical continuation of the expression corresponding to the diagram of Fig. 9.1:

R(εn ) = T

dD q L(q, Ωk ) (2π)D

XZ Ωk

Z

dD p 2 G (p, εn )G(q − p, Ωk − εn ). (9.2) (2π)D

The result of the integration of the last expression depends strongly of the electron spectrum dimensionality: for the two important cases of isotropic 3D and 2D electron spectra one finds [107] (c)

δν(3) (E, ) ν(3) (0)

(c)

δν(2) (E, ) ν(2) (0)

√ (4π)3/2 p Tc × = − Gi(3,c) Re q 7ζ(3) −1 τGL − 2iE + κ32 Tc ( ) 1 1/2 , −1/2  −1 −1 τGL − iE + τGL τGL − 2iE + κ32 Tc

(9.3)

(4π)2 Tc2  = − Gi(2,c)  2 (9.4) −1 × 7ζ(3) E + κ22 Tc τGL q   2 + κ 2 T τ −1   E + E c 2 GL E √ , 1− q ln   κ Tc τGL −1 E 2 + κ22 Tc τGL

p where κD = π πD/7ζ(3). In a dirty superconductor the calculations may be carried out in a similar way with the only difference that the impurity renormalization of the Cooper vertices has to be taken into account [106]. The value of the fluctuation dip at the Fermi level can be written in the form: 155

Figure 9.2: The theoretical curve of the energy dependence for the normalised correction to the single-particle density of states vs energy for a clean twodimensional superconductor above Tc .

δν (d) (0) ∼− ν(0)

Gi(3,d) −3/2 , D = 3 . Gi(2,d) −2 , D=2

 p

(9.5)

−1 At large energies E  τGL the DOS recovers its normal value, according to the same laws (9.5) but with the substitution  → E/Tc . It is interesting that the critical exponents of the fluctuation correction of the DOS change when moving from a dirty to a clean superconductor [107]: the analysis of (9.3)-(9.4) gives  p δν (c) (0) Gi(3,c) −1/2 , D = 3 ∼− . (9.6) Gi(2,c) −1 , D=2 ν(0)

Another important respect in which the character of the DOS renormalization differs strongly for the clean and dirty cases is the energy scale at which this renormalization occurs. In the dirty case this energyp turns out to (d) (c) −1 be [106] E0 ∼ T − Tc ∼ τGL , while in the clean case E0 ∼ Tc (T − Tc ) [107]. To understand this important difference one has to study the character of the electron motion in both cases [107]. The relevant energy scale in the dirty case is the inverse of the time necessary for the electron to diffuse over a distance equal to the coherence length ξ(T ). This energy scale coin−1 −2 cides with the inverse relaxation time: t−1 ξ = Dξ (T ) ∼ τGL ∼ T − Tc . In the clean case, the ballistic motion of the electrons gives rise p to a different −1 −1 1/2 −1 characteristic energy scale tξ ∼ vF ξ (T ) ∼ (Tc τGL ) ∼ Tc (T − Tc ). One can check that the integration of (9.3)-(9.4) over all positive energies gives zero: Z ∞

δν(E)dE = 0

(9.7)

0

This “sum rule” is a consequence of a conservation law: the number of quasiparticles is determined by the number of cells in the crystal and cannot be changed by the interaction. So the only effect which can be produced by the inter-electron interaction is a redistribution of the energy levels near the Fermi energy. The sum rule (9.7) plays an important role in the understanding of the manifestation of the fluctuation DOS renormalization in the observable phenomena. As we will see in the next Section the singularity in

156

the tunneling current (at zero voltage), due to the density of states renormalization, turns out to be much weaker than that in the DOS itself (ln  instead of −1 or −2 , see (9.5)-(9.6)). A similar smearing of the DOS singularity occurs in the opening of the pseudo-gap in the c-axis optical conductivity, in the NMR relaxation rate etc. These features are due to the fact that we must always form the convolution of the DOS with some slowly varying function: for example, a difference of Fermi functions in the case of the tunnel current. The sum rule then leads to an almost perfect cancellation of the main singularity at low energies. The main non-zero contribution then comes from the high energy region where the DOS correction has its ‘tail’. Another important consequence of the conservation law (9.7) is the considerable increase of the characteristic energy scale of the fluctuation pseudo-gap opening with respect to E0 : this is eV0 = πT for tunneling and ω ∼ τ −1 for the c-axis optical conductivity.

9.2

The effect of fluctuations on the tunnel current [194].

It is quite evident that the renormalization of the density of states near the Fermi level, even of only one of the electrodes, will lead to the appearance of anomalies in the voltage-current characteristics of a tunnel junction. The quasiparticle current flowing through it may be written as a convolution of the densities of states with the difference of the electron Fermi distributions in each electrode (L and R): Iqp =

1 × (9.8) eRn νL (0)νR (0)  Z ∞ E + eV E tanh − tanh νL (E)νR (E + eV )dE, 2T 2T −∞

where Rn is the Ohmic resistance per unit area and νL (0), νR (0) are the densities of states at the Fermi levels in each of electrodes in the absence of interaction. One can see that for low temperatures and voltages the expression in parenthesis is a sharp function of energy near the Fermi level. Nevertheless, depending on the properties of the DOS functions, the convolution (9.8) may exhibit different properties. If the energy scale of the DOS correction is much larger than T , the expression in parenthesis in (9.8) acts as a delta-function and the zero-bias anomaly in the tunnel conductivity strictly reproduces the anomaly of the density of states around the Fermi level: 157

δG(V ) δν(eV ) = , Gn (0) ν(0)

(9.9)

where G(V ) is the differential tunnel conductance and Gn (0) is the background value of the Ohmic conductance supposed to be bias independent, δG(V ) = G(V ) − Gn (0). This situation, for instance, occurs in a junction with one amorphous electrode [198], where the dynamically screened Coulomb interaction is strongly retarded, which leads to a considerable suppression of the density of states in the vicinity of the Fermi level, within τ −1  T. It is worth stressing that the proportionality between the tunneling current and the electron DOS of the electrodes is widely accepted as an axiom, but generally speaking this is not always so. As one can see from the previous subsection, the opposite situation occurs in the case of the DOS renormalization due to the electron-electron interaction in the Cooper channel: in this case the DOS correction varies strongly already in the scale of E0 ∼ Eker  T and the convolution in (9.8) with the DOS (9.4) has to be carried out without the simplifying approximations assumed to obtain (9.9). We will show that the fluctuation induced pseudo-gap like structure in the tunnel conductance differs drastically from the anomaly of the density of states (9.4), both in its temperature singularity near Tc and in the energy range of its manifestation. Let us first discuss the effect of the fluctuation suppression of the density of states on the properties of a tunnel junction between a normal metal and a superconductor above Tc . The effect under discussion turns out to be most pronounced in the case of thin superconducting films (d  ξ(T )) and layered superconductors like HTS cuprates. In order to derive the explicit expression for the fluctuation contribution to the differential conductance of a tunnel junction with one thin film electrode close to its Tc we differentiate (9.8) with respect to voltage, and substitute the DOS correction given by (9.4). This results in (see [194]): Z ∞ 1 dE δGf l (V, )
(2) (E, ) ∼ = (9.10) 2 E+eV δν Gn (0) 2T −∞ cosh 2T     2 1 ieV 00 √ ∼ Gi(2) ln √ Reψ − . 2 2πT + +r It is important to emphasize several nontrivial features of the result obtained. First, the sharp decrease (−2(1) ) of the density of electron states in the immediate vicinity of the Fermi level generated by fluctuations surprisingly results in a much more moderate growth of the tunnel resistance at zero

158

Figure 9.3: The theoretical prediction for the fluctuation-induced zero-bias anomaly in tunnel-junction resistance as a function of voltage for reduced temperatures  = 0.05 (top curve),  = 0.08 (middle curve) and  = 0.12 (bottom curve). The insert shows the experimentally observed differential resistance as a function of voltage in an Al-I-Sn junction just above the transition temperature voltage (ln 1/). Second, in spite of the manifestation of thepDOS renormali(d) (cl) zation at the characteristic scales E0 ∼ T − Tc or E0 ∼ Tc (T − Tc ), the energy scale of the anomaly developed in the I − V characteristic is much larger: eV = πT  E0 (see Fig. 9.3). In the inset of Fig. 9.3 the result of measurements of the differential resistance of the tunnel junction Al − I − Sn at temperatures slightly above the critical temperature of Sn electrode is presented. This experiment was done [199] with the purpose of checking the theory proposed [194]. The nonlinear differential resistance was precisely measured at low voltages which permitted the observation of the fine structure of the zero-bias anomaly. The reader can compare the shape of the measured fluctuation part of the differential resistance (the inset in Fig. 9.3) with the theoretical prediction. It is worth mentioning that the experimentally measured positions of the minima are eV ≈ ±3Tc , while the theoretical prediction following from (9.10) is eV = ±πTc . Recently similar results on an aluminium film with two regions of different superconducting transition temperatures were reported [200]. The observations of the pseudogap anomalies in tunneling experiments at temperatures above Tc obtained by a variety of experimental techniques were reported in [201, 202, 203, 204, 205]. We will now consider the case of a symmetric junction between two superconducting electrodes at temperatures above Tc . In this case, evidently, the correction (9.10) has to be multiplied by a factor of ”two” because of the possibility of fluctuation pairing in both electrodes. Furthermore, in view of the extraordinarily weak (∼ ln 1/) temperature dependence of the first order correction, different types of high orderp corrections may manifest themselves on the energy scale eV ∼ T − Tc or Tc (T − Tc ). Among them are the familiar AL and MT corrections which take place in the first order of Gi but in the second order of the barrier transparency. Another type of higher order correction appears in the first order of barrier transparency but in the second of fluctuation strength (∼ Gi2 ) [194]. Such corrections are generated by the interaction of fluctuations through the barrier and they can be evaluated directly from (9.8) applied to a symmetric junction . The second 159

order correction in Gi can be written as [194]: (2) δGf l

(0, ) ∼

Z

∞

−∞

dE cosh2

 (2) 2 Gi2(2)
δν (E, ) ∼ 3 E  2T

(9.11)

This nonlinear fluctuation correction turns out to be small by Gi2 but its (1) strong singularity in temperature and opposite sign with respect to δGf l make it interesting. Apparently it leads to the appearance of a sharp maximum at zero voltage in G(V ) with a characteristic width eV ∼ T − Tc in the immediate vicinity of Tc (one can call this peak as the hyperfine structure). This result was confirmed in [206] but to our knowledge such corrections were never observed in tunneling experiments. √ (1) (2) One can see that δGf l and δGf l become of the same order at ∗cr ∼ 3 Gi, i.e. the critical region where nonlinear fluctuations effects become important in the problem under consideration starts much before the thermodynamical criterion cr ∼ Gi. In the next Section we will discuss this early manifestation of nonlinear fluctuation effects in transport phenomena.

9.3

Fluctuation tunneling anomaly in superconductor above paramagnetic limit

As we already have seen the magnetic field due to the break of time reversal symmetry, suppresses both superconductivity and superconducting fluctuations. This effect can be separated into two mechanisms: the effect of magnetic field on the orbital motion coupling associated with Aharonov-Bohm phase and Zeeman splitting of the states with the same spacial wave functions but opposite spin directions. In the bulk systems the suppression of the superconductivity usually is related with the first mechanism in the fields of the order of Hc2 (0) (see (2.48)). The effect of magnetic field on the spin structure of the Cooper pair determines so-called Clogston limit: gL µB Hspin ' ∆,

(9.12)

where gL is the Land´e g-factor, µB is the Bohr magneton. Comparison of Hspin with Hc2 (0) demonstrates that for normal metal the former is far in excess of the latter: Hspin ' EF τ  1, Hc2 (0) and Clogston limit is practically unaccessible, 160

(9.13)

Chapter 10 The effect of fluctuation on the optical conductivity [214] The optical conductivity of a layered superconductor can be expressed by (R) the same analytically continued electromagnetic response operator Qαβ (ω) (see Exp.(6.3)) but in contrast to the d.c. conductivity case, calculated without the assumption ω → 0. Let us recall that the paraconductivity tensor in an a.c. field was already studied in Section 4 in the framework of the TDGL equation [31] and the most interesting asymptotics for our discussion (3.42)-(3.43), valid for ω  T in the 2D regime, were calculated there. The microscopic calculation of the AL diagram [111] shows that in the vicinity of Tc and for ω  T the leading singular contribution to the response AL (R) operator Qαβ arises from the fluctuation propagators rather than from the Bα blocks, which confirms the TDGL results. Nevertheless the DOS and MT corrections can be calculated only by the microscopic method, as was done in [111, 214]. Let us note that the external frequency ων enters in the expression for the DOS contribution to Qαβ (ω) only by means of the Green’s function G(p, ωn+ν ) and it is not involved in q integration. So, near Tc , even in the case of an arbitrary external frequency, we can restrict consideration to the static limit, taking into account only the propagator frequency Ωk = 0, and to get [214]:   2 e2 DOS κ ˆ (ω, T, τ ) Aαβ ln √ Reσαβ (ω) = − √ , 2πs +r+  where the anisotropy tensor Aαβ was introduced in (6.12). Let us stress that, in contrast to the AL frequency dependent contribution, this result has been found with only the assumption   1, so it is valid for any frequency, 161

and impurity concentration. The function κ ˆ (ω, T, τ ) was calculated in [214] exactly but we present here only its asymptotics for the clean and dirty cases:  7ζ(3) −1   2π
2 , ω  T  τ
8 T 2 κ ˆ d ω, T  τ −1 = , T  ω  τ −1 , ω π 2  − πT , T  τ −1  ω ω3 τ   (T τ )2 , ω  τ −1  T  3

π T 2 κ ˆ cl ω, T  τ −1 = . , τ −1  ω  T ω
28ζ(3)   −4 T 3 , τ −1  T  ω ω The general expression for the MT contribution is too cumbersome, so we restrict ourselves here to the important 2D overdamped regime (r   ≤ γϕ ): ( 2 2 1, ω ˜  τϕ−1 e s r   M T (an)(2D) 2 σzz (ω) = 7 8Tc γϕ 2 η(2) γϕ  , ω ˜  τϕ−1 πω   γ1 ln γϕ , ω  τϕ−1 e M T (an)(2D)  ϕ 2 σxx (ω) = . c γϕ 8s  8Tπω , ω  τϕ−1 2

Let us discuss the results obtained. Because of the large number of parameters entering the expressions we restrict our consideration to the most interesting c-axis component of the fluctuation conductivity tensor in the 2D region (above the Lawrence-Doniach crossover temperature). The AL contribution describes the fluctuation condensate response to the applied electromagnetic field. The current associated with it can be treated as the precursor phenomenon of the screening currents in the superconducting phase. As was demonstrated above the characteristic ”binding energy ” of fluctuation Cooper pair is of the order of T − Tc , so it is not surprising that the AL contribution decreases when the electromagnetic field frequency exceeds this value. Indeed ω AL ∼ T −Tc is the only relevant scale for σ AL : its frequency dependence does not contain T, τϕ and τ . The independence from the latter is due to the fact that elastic impurities do not present obstacles for the motion of Cooper pairs. The interaction of the electromagnetic wave with the fluctuation Cooper pairs resembles, in some way, the anomalous skin-effect where the reflection is determined by the interaction with the free electron system. The anomalous MT contribution also is due to fluctuation Cooper pairs, but this time they are formed by electrons moving along self-intersecting trajectories. Being the contribution related with the Cooper pair electric 162

Figure 10.1: The theoretical dependence [214] of the real part of the conductivity, normalized by the Drude normal conductivity, on ω/T , < [σ 0 (ω)] = Re [σ(ω)] /σ n . The dashed line refers to the ab-plane component of the conductivity tensor whose Drude normal conductivity is σkn = N (0)e2 τ vF2 . The solid line refers to the c-axis component whose Drude normal conductin vity is σ⊥ = σkn J 2 s2 /vF2 . In this plot we have put T τ = 0.3, EF /T = 50, r = 0.01,  = 0.04, T τϕ = 4. charge transfer it does not depend on the elastic scattering time but it turns out to be extremely sensitive to the phase-breaking mechanisms. So two characteristic scales turn out to be relevant in its frequency dependence: T − Tc and τϕ−1 . In the case of HTS, where τϕ−1 has been estimated as at least 0.1Tc , for temperatures up to 5 ÷ 10K above Tc the MT contribution is overdamped, it is determined by the value of τϕ and is almost temperature independent. The DOS contribution to Reσ(ω) is quite different from those above. In the wide range of frequencies ω  τ −1 the lack of electron states at the Fermi level leads to the opposite sign effect in comparison with the AL and MT contributions: Reσ DOS (ω) turns out to be negative and this means an increase of the surface impedance, or, in other words, decrease of the reflectance. Nevertheless, the applied electromagnetic field affects the electron distribution and at very high frequencies ω ∼ τ −1 the DOS contribution changes its sign. It is interesting that the DOS contribution, as a one-electron effect, depends on the impurity scattering in a similar manner to the normal Drude conductivity. The decrease of Reσ DOS (ω) starts at frequencies ω ∼ min{T, τ −1 } which for HTS are much higher than T − Tc and τϕ−1 . tot with the most natural choice of parameters The ω-dependence of Reσzz −1 −1 (Tc r  Tc  ≤ τϕ  min{T, τ }) is presented in Fig. 10.1. Let us discuss it referring to a strongly anisotropic layered superconductot tor. The positive AL and MT contributions to σzz , being suppressed by the square of the interlayer transparency, are small in magnitude and they vary in the low frequency region ω ∼ min{T −Tc , τϕ−1 }. The DOS contribution is proportional to the first order of transparency and remains in this region almost invariable. With a further increase of frequency min{T −Tc , τϕ−1 } . ω the AL and MT contributions decay; Reσ⊥ remains negative up to ω ∼ min{T, τ −1 }, then it changes its sign at ω ∼ τ −1 , reaches maximum and rapidly decreases. The following high frequency behavior is governed by the Drude law. So one can see that the characteristic pseudo-gap-like behavior in the frequency dependence of the c-axis optical conductivity takes place: a transparency 163

window appears in the range ω ∈ [T − Tc , τ −1 ]. In the case of the ab-plane optical conductivity the two first positive contributions are not suppressed by the interlayer transparency, and exceed considerably the negative DOS contribution in a wide range of frequencies. tot Any pseudo-gap like behavior is therefore unlikely in σxx (ω): the reflectivity will be of the metallic kind.

164

Chapter 11 Fluctuation effects in heat transport 11.1

Thermoelectric power above the superconducting transition [218, 216]

Thermoelectric effects are difficult both to calculate and to measure if compared with electrical transport properties. At the heart of the problem lies the fact that the thermoelectric coefficients in metals are the small resultant of two opposing currents which almost completely cancel. In calculating the thermoelectric power one finds that the electrons above the Fermi level carry a heat current that is nearly the negative of that carried by the electrons below EF . In the model of a monovalent metal in which band structure and scattering probabilities are symmetric about EF , this cancellation would be exact; in a real metal a small asymmetry survives. Because of their compensated nature, thermoelectric effects are very sensitive to the characteristics of the electronic spectrum, presence of impurities and peculiarities of scattering mechanisms. The inclusion of many-body effects, such as electron-phonon renormalization, multi-phonon scattering, drag effect, adds even more complexity to the problem of calculating the thermoelectric power. Among such effects, there is also the influence of thermodynamical fluctuations on the thermoelectric transport in a superconductor above the critical temperature. This problem has been attracting the attention of theoreticians for more than twenty years, since the paper of Maki [217] appeared, where the logarithmically divergent AL contribution was predicted for the two-dimensional case. So the AL term turns out to be less singular compared with the corresponding correction to conductivity. In every case where the main AL and MT fluctuation corrections are 165

suppressed for some reason, the contribution connected with fluctuation renormalization of the one-electron density of states (DOS) can become important. The analogous situation also occurs in the case of the thermoelectric coefficient [215, 216]. Although the DOS term has the same temperature dependence as the AL contribution [217, 218], it turns out to be the leading fluctuation contribution in both the clean and dirty cases, due to its specific dependence on the electron mean free path. We introduce the thermoelectric coefficient ϑ in the framework of linear response theory as 1 Im[Q(eh)R (ω)] ϑ = lim T ω→0 ω where Q(eh)R (ω) is the Fourier representation of the retarded correlation function of electric J e and heat J h current operators in Heisenberg representation:   Q(eh)R (X − X 0 ) = −Θ(t − t0 )hh J h (X), J e (X 0 ) ii. Here X = (r, t) and hh· · ·ii represents both thermodynamical averaging and averaging over random impurity positions. The correlation function Q(eh)R in the diagrammatic technique is represented by a bubble with two exact electron Green’s functions and two external field vertices, the first, ev, associated with the electric current operator and the second, 2i (εn + εn+ν )v, associated with the heat current operator (εn is fermionic Matsubara frequency) [187]. The first order fluctuation corrections to Q(eh) (ων ) are represented by the same diagrams as for conductivity (see Fig.6.1). The first diagram describes the AL contribution to thermoelectric coefficient and was calculated in [217, 218] with the electron-hole asymmetry factor taken into account in the fluctuation propagator. Diagrams 2-4 represent the Maki-Thompson contribution, neither anomalous nor regular parts of these diagrams contribute to ϑ in any order of electron-hole asymmetry [187, 218]. The contribution from diagrams 5-10 describes the correction to ϑ due to fluctuation renormalization of the one-electron density of states. Evaluating it in the same way as (6.16) but with one heat current vertex one obtains a vanishing result if electron-hole asymmetry is not taken into account. The first possible source of this factor is contained in the fluctuation propagator; it was used in [218] for the AL diagram but for the DOS contribution this correction results in non-singular contributions to ϑ only and can be neglected. Another source of electron-hole asymmetry is connected with expansion of energy-dependent functions in powers of ξ/EF near the Fermi level:   ∂(ν(ξ)v2 (ξ)) 2 2 ν(ξ)v (ξ) = ν(0)v (0) + ξ . (11.1) ∂ξ ξ=0 166

Only the second term in Eq. (11.1) contributes to the thermoelectric coefficient. Performing the integration over ξ , summations over fermionic frequencies and analytical continuation of the result obtained we find that the contribution to the thermoelectric coefficient associated with the DOS renormalization takes the form   1 eTc ∂(νv2 ) 2 DOS √ ], (11.2) ϑ2D = 2 κ∗ (T τ ) ln[ √ 2 4π ν(0)vF ∂ξ + +r ξ=0 where ∗

1 + 8Tπ τ

1 + 4πT − ψ 12 − τ

 κ (T τ ) = − 1 T τ ψ 12 ψ0 4πT τ ( 8π 2 T τ ≈ 9, 4T τ T τ  1 7ζ(3) . = (T τ )−1 T τ  1

1 2




(11.3)

Summing Eq. (11.2) with the AL contribution [218] one can find the total correction to the thermoelectric coefficient in the case of a 2D superconducting film of thickness d :    ϑDOS + ϑAL 1 1 Tc ωD ∗ = −0.17 ln κ (Tc τ ) + 5.3 ln , ϑ0 EF τ pF d T − Tc Tc Assuming ln(ωD /Tc ) ≈ 2 one finds that the DOS contribution dominates the AL one for any value of impurity concentration: κ∗ has a minimum at T τ ≈ 0.3 and even at this point the DOS term is twice as large. In both limiting cases T τ  1 and T τ  1 this difference strongly increases. In practice, although the Seebeck coefficient S = −ϑ/σ is probably the easiest to measure among the thermal transport coefficients, the comparison between experiment and theory is complicated by the fact that S cannot be calculated directly; it is rather a composite quantity of the electrical conductivity and thermoelectric coefficient. As both ϑ and σ have corrections due to superconducting fluctuations, the total correction to the Seebeck coefficient is given by   ∆ϑ ∆σ ∆S = S0 − (11.4) ϑ0 σ0 We see that the fluctuations result in a decrease of the absolute value of the overall Seebeck coefficient as the temperature approaches Tc . The situation is complicated additionally in HTS materials, where the temperature behavior of the background value of the thermoelectric power remains unknown. This does not permit to extract precisely from the experimental data the fluctuation part ∆ϑ to compare it with the theoretical 167

prediction. Nevertheless the very sharp maximum in the Seebeck coefficient experimentally observed in a few papers [219, 220, 221] seems to be unrelated to the fluctuation effects. This conclusion is supported by recent analysis of the temperature dependence of the thermoelectric coefficient close to the transition in Refs. [222]. Discussion. The main results of this Section, valid for   1, can be summarized as follows: (1) Fluctuations lead to a suppression of the spin susceptibility χs , due to the combined effect of the reduction of the single particle density of states arising from the self energy contributions, and of the regular part of the MT process. (2) “Cooperon” impurity interference terms, involving impurity ladders in the particle-particle channel, are crucial for the χs suppression in the dirty limit. (3) The processes which dominate the results in (1) and (2) above have usually been ignored in fluctuation calculations (conductivity, 1/T1 , etc.). The spin susceptibility is unusual in that the AL and the anomalous MT terms, which usually dominate, are absent. (4) For weak pair-breaking (1/τϕ  Tc ), an enhancement of 1/T1 T , coming from the positive anomalous MT term, takes place [225, 191]. (5) Strong dephasing suppresses the anomalous MT contribution, and 1/T1 is then dominated by the less singular DOS and the regular MT terms. Being negative, these contributions lead to a suppression of spectral weight and a decrease in 1/T1 . An intensive controversy took place in recent years in relation to the magnetic field dependence of the fluctuation contribution to 1/T1 . The situation here resembles much the situation with the magnetoconductivity: a positive MT contribution is suppressed by the magnetic field while the magnetic field dependent part of the DOS contribution increases with the growth of the field. But in contrast to the magnetoconductivity, which can be measured extremely precisely, the NMR relaxation rate measurements are much more sophisticated. The result of this delicate competition, depending on many parameters (r, γϕ , τ,), was found in HTS materials to be qualitatively different in experiments of various groups. The absence of a strong positive AL contribution, possible d-pairing, killing the MT contribution [225], small magnitude of the sum of MT and DOS effects even in the case of s-pairing, lack of the precise values of r, γϕ , τ, leading to contradictive theoretical predictions[191, 127, 195, 227], the dispersion in the qua168

lity of samples and experimental methods were the reason of this discussion [127, 228, 229, 230, 44, 231].

11.2

Thermal conductivity.

11.3

Nernst and Ettinghausen effects

169

Chapter 12 Sound attenuation

170

Chapter 13 The effect of fluctuations on NMR characteristics [191] 13.1

Preliminaries

In this Section we discuss the contribution of superconducting fluctuations to the spin susceptibility and the NMR relaxation rate. For both these effects the interplay of different fluctuation contributions is unusual with respect to the case of the conductivity. Like in the case of the optical conductivity, the fluctuation contributions to the spin susceptibility and the NMR relaxation rate can manifest themselves as the opening of a pseudogap already in the normal phase, a phenomenon which is characteristic to HTS compounds. (R) We begin with the dynamic spin susceptibility χ± (k, ω) = χ± (k, iων → ω + i0+ ) where χ± (k, ων ) =

Z 0

1/T

  dτ eiων τ hhTˆτ Sˆ+ (k, τ )Sˆ− (−k, 0) ii.

(13.1)

Here Sˆ± are the spin raising and lowering operators, Tˆτ is the time ordering operator, and the brackets denote thermal and impurity averaging in the (R) usual way. The uniform, static spin susceptibility is given by χs = χ± (k → 0, ω = 0) while the dynamic NMR relaxation rate is given by Z 1 A d3 k (R) = lim (13.2) 3 Imχ± (k, ω) ω→0 ω T1 T (2π) where A is a positive constant involving the gyromagnetic ratio. For non-interacting electrons χ0± (k, ων ) is determined by the usual loop diagram. Simple calculations lead to the well known results for T  EF : 171

χ0s = ν (Pauli susceptibility) and (1/T1 T )0 = Aπν 2 (Korringa relaxation). We will present the fluctuation contributions in a dimensionless form by normalizing to the above results. To leading order in Gi the fluctuation contributions to χ± can be discussed with the help of the same diagrams drawn for the conductivity in Fig. 6.1. It is important to note that the role of the external vertices (electron interaction with the external field) is now played by the Sˆ± (k, τ ) operators. This means that the two fermion lines attached to the external vertex must have opposite spin labels (up and down). Consequently, the Aslamazov-Larkin diagram for χ± does not exist since one cannot consistently assign a spin label to the central fermion for spin-singlet pairing. The next set of diagrams to consider is the Maki-Thompson contribution. While the MT diagrams for χ± appear to be identical to those for the conductivity, there is an important difference in topology which arises from their spin structure. It is easy to see, by drawing the fluctuation propagator explicitly as a ladder of attractive interaction lines, that the MT diagram is a non-planar graph with a single fermion loop. In contrast the MT graph for the conductivity is planar and has two fermion loops. The number of loops, in accordance with the rules of diagrammatic technique [105], affects the sign of the contribution. The diagrams 5 and 6 represent the effect of fluctuations on the singleparticle self energy, leading to a decrease in the DOS. The DOS diagrams 7 and 8 include impurity vertex corrections (note that these have only a single impurity scattering line as additional impurity scattering in the form of a ladder has a vanishing effect). Finally 9 and 10 are the DOS diagrams with the Cooperon impurity corrections.

13.2

Spin Susceptibility[223, 191].

We note that, when the external frequency and momentum can be set to zero at the outset, as is the case for χs , there is no anomalous MT piece (which as we shall see below is the most singular contribution to 1/T1 ). The MT diagram 2 then yields a result which is identical to the sum of the DOS diagrams 5 and 6. In the clean limit (Tc τ  1) the fluctuation contribution is given by fl χs = χs 2 +χs 5 +χs 6 ; all other diagrams turn out to be negligible. In the dirty case (Tc τ  1), the DOS diagrams 5 and 6, together with the regular part of the MT diagram (2), yield the same result as in the clean limit (of the order O(Tc /EF )). One can see, that this contribution is negligible in comparison with the expected dominant one for the dirty case of the order O(1/EF τ ). A thorough study of all diagrams shows that the important graphs in the 172

dirty case are those with the Cooperon impurity corrections MT 3 and 4, and the DOS ones 9 and 10. This is the unique example known to us where the Cooperons, which play a central role in the weak localization theory, give the leading order result in the study of superconducting fluctuations. Diagrams 3 and 4 give one half of the final result given below; diagrams 9 and 10 provide the other half. The total fluctuation susceptibility is χfls = χs 3 + χs 4 + χs 9 + χs 10 . Interesting, that in both the clean and dirty cases χfls /χs (0) can be expressed by the same formula if one expresses the coefficient in terms of the GL number Gi(2) (5.29):   χfls 2 √ = −2Gi(2) ln √ . (13.3) χs (0) + +r It is tempting to explain the negative sign of the fluctuation contribution to the spin susceptibility in Eq. (13.3) as arising from a suppression of the DOS at the Fermi level. But one must keep in mind that only the contribution of diagrams 5 and 6 can strictly be interpreted in this manner; the MT graphs and the coherent impurity scattering described by the Cooperons do not permit such a simple interpretation.

13.3

Relaxation Rate[224, 225, 219, 191].

The calculation of the fluctuation contribution to 1/T1 requires rather more care than χs because of the subtleties of analytic continuation. Let us define the local susceptibility Z K(ων ) = (dk)χ+− (k, ων ). In order to write down the fluctuation contribution to 1/T1 for the case of an arbitrary impurity concentration including the ultra-clean case let us start from the anomalous MT contribution and evaluate it using the standard contour integration techniques Z 1 πν 2 (an)R (ω) = − (dq)L(q, 0)K(q), (13.4) lim ImK ω→0 ω 8 K(q) = 2τ

Z

∞

−∞

dz 1 p × cosh (z/4T τ ) 2 2 2 l q − (z − i) − 1 2

1 p

l2 q 2

− (z +

i)2

. −1

173

(13.5)

We have used the impurity vertices in the general form (6.26). The first simple limiting case for (13.5) is lq  1, when the square roots in the denominator can be expanded and K(q) = 2π/Dq 2 . As we already know from Section 8.4 this corresponds to the usual local approximation and covers the √ domain Tc τ  1/ . Introducing the pair breaking rate γϕ as an infrared cut off one can find: δ (1/T1 )M T (an) 28ζ(3) 1 = Gi(2,d) ln(/γϕ ). 0 4 π  − γϕ (1/T1 )

(13.6)

The other limiting case is the “ultra-clean limit” when√the characteristic qvalues satisfy lq  1. This is obtained when T τ  1/   1. From (13.5) we then find K(q) = 4 ln(lq)/vq, which leads to √ 1 π3 δ(1/T1 )M T (an) p Gi(2,cl) √ ln(T τ ). = 0 (1/T1 ) ε 14ζ(3)

(13.7)

We note that in all cases the anomalous MT contribution leads to an enhancement of the NMR relaxation rate over the normal state Korringa value. In particular, the superconducting fluctuations above Tc have the opposite sign to the effect for T  Tc (where 1/T1 drops exponentially with T ). One might argue that the enhancement of 1/T1 is a precursor to the coherence peak just below Tc . Although the physics of the Hebel-Slichter peak (pile-up of the DOS just above gap edge and coherence factors) appears to be quite different from that embodied in the MT process, we note that both effects are suppressed by strong inelastic scattering. We now discuss the DOS and the regular MT contributions which are important when strong dephasing suppresses the anomalous MT contribution discussed above. The local susceptibility arising from diagrams 5 and 6 can be easily evaluated. The other remaining contribution is from the regular part of the MT diagram. It can be shown that this regular contribution is exactly one half of the total DOS contribution from diagrams 5 and 6. All other diagrams either vanish (as is the case for graphs 7 and 8) or contribute at higher order in 1/EF τ (this applies to the graphs with the Cooperon corrections). The final results can be presented in a unique way for the clean (but not ultra-clean) and dirty cases by means of the Gi(2) number :   δ(1/T1 )DOS 2 √ = −12Gi(2) ln √ . (13.8) (1/T1 )0 + +r The negative DOS contribution to the NMR relaxation rate is evident from the Korringa formula and it sign seems very natural while the sign of 174

the positive Maki-Thompson contribution can generate a questions about its physical origin. Let us consider a self-intersecting trajectory and the motion of the electron along it with fixed spin orientation (let us say ”spin up”). If, after passing a full turn, the electron interacts with the nucleus and changes its spin state and momentum to the opposite value it can pass again along the previous trajectory moving in the opposite direction . Interaction of the electron with itself on the previous stage of the motion is possible due to the retarded character of the Cooper interaction and such a pairing process, in contrast to the AL one, turns out to be an effective mechanism for relaxation near Tc . This purely quantum process opens a new mechanism of spin relaxation, and so contributes positively to the relaxation rate 1/T1 . In the case of the nuclear magnetic relaxation rate calculations, the electron interaction causing nuclear spin flip is considered. If one would try to imagine an AL process of this type he would be in trouble, because the electron-nuclei scattering with spin-flip evidently transforms the initial singlet state of the fluctuation Cooper pair in a triplet-one, which is forbidden in the scheme discussed. So the formally discovered absence of the AL contribution to the relaxation rate is evident enough. It is worth mentioning that the cancellation of the MT and DOS contributions to conductivity found in Section 8.4 is crucial for the fluctuation contributions to the NMR relaxation rate. In fact, the MT and DOS contributions here have the same structure as in the conductivity while the AL contribution is absent. So the full fluctuation correction to the NMR relaxation rate in clean superconductor simply disappears.

175

Part IV Fluctuations in nanostructures and unconventional superconducting systems

176

Chapter 14 Fluctuations in superconducting nanodrops 14.1

Ultrasmall superconducting grains

The recent progress in nanotechnologies permitted to fabricate the superconducting grains of the nanometer scale [283, 284]. Let us remind that in the First chapter we already studied the thermodynamics of small grain basing on the Ginzburg-Landau phenomenology. The natural question arises: how small is the grain size to what one can apply the GL approach? The answer is contained in the value of the Ginzburg-Levanyuk number which can be rewritten in terms of the level spacing for grain δ = 1/νV (let us remind that the density of states is the number of states per unit energy interval and unit volume) and superconducting gap ∆ (0) ∼ Tc : s p 7ζ(3) 1 δ √ Gi(0) = ∼ . (14.1) 2π ∆ (0) νTc V If this value is small thermal fluctuations smear out the phase transition only in the narrow vicinity of the mean field critical temperature. Beyond this region MFA gives the correct result for thermodynamic characterisitcs. What happens when Gi(0) → 1,i.e. level spacing δ becomes comparable with ∆ (0)? The positions of impurities, the shape of the surface and hence, the level spacing are the random values. It is why the answers on the proposed questions depend on the way of averaging over the ensembles of different random grains. Fortunately recently the new experimental technique of the Coulomb blockade was worked out [283, 284], which permits to measure the change in the energy of the grain as one electron is added. Important that nor shape 177

nor impurities locations are changed in such measurements. The small finite levels spacing and superconducting pairing effects can be observed on the background of the large but smooth Coulomb energy. The energy of superconducting pairing in the mean field approximation is determined by the self-consistency equation: 1 = gb

X k

1 q , 2 2 2 (Ek − µ) + ∆

(14.2)

where Ek are the one-electron levels without account for the pairing interaction. If ∆  δ one can substitute the summation by integration and to get the usual equation of the BCS theory. From the equation (14.2) follows that ∆ turns zero when δ ∼ ∆BCS . It is worth noting that the equation (14.2) treats the superconductivity in small grains within the self-consistent mean field approximation for the superconducting order parameter. Although this approximation works well for large systems, one should expect the quantum fluctuations of the order parameter to grow when the level spacing δ reaches ∆. Below will be presented a theory of superconductivity in ultrasmall grains which includes the effects of quantum fluctuations of the order parameter. We show that the corrections to the mean field results which are small in large grains, δ  ∆, become important in the opposite limit, δ  ∆. The account for quantum fluctuations can be done studying some physically observable value. The parameter ∆ ∼ hak a−k i has sense only in the limit V → ∞. In the system of finite number of particles such anomalous average is equal to zero due to the phase fluctuations. The superconducting gap ∆ in the equation (14.2) is not well defined in the presence of quantum fluctuations. Therefore, we must first identify an observable physical quantity which characterizes the superconducting properties of small grains. The most convenient such quantity for our purposes is the ground state energy of the grain EN as a function of the number of electrons N . More precisely, we study the so-called parity effect in ultrasmall grains, which is described quantitatively by parameter 1 ∆P = E2l+1 − (E2l + E2l+2 ). 2

(14.3)

Here El is the ground state energy for a system with l electrons. Such physical value first was studied in the nuclear physics[276, 277]. In the ground state of a large superconducting grain with an odd number of electrons, one electron is unpaired and carries an additional energy ∆P = ∆. This result is well known in nuclear physics and was recently discussed in connection to 178

superconducting grains in Refs. [278, 279]. The parity effect was demonstrated experimentally in Refs. [280, 281], where the Coulomb blockade phenomenon[282] in a superconducting grain was studied. In such an experiment the intervals between Coulomb blockade peaks in which the grain charge is odd shrink by an amount proportional to ∆P . We describe the grain by the following Hamiltonian: X X † † ˆ = H εk a†kσ akσ − gb ak↑ ak↓ ak0 ↓ ak0 ↑ . (14.4) kk0

kσ

Here k is an integer numbering the single particle energy levels εk , the average level spacing hεk+1 − εk i = δ, operator akσ annihilates an electron in state k with spin σ, and gb is the interaction constant. In Eq. (14.4) we assume zero magnetic field, so that the electron states can be chosen to be invariant under the time reversal transformation[?]. We include in Eq. (14.4) only the matrix elements of the interaction Hamiltonian responsible for the superconductivity; the contributions of the other terms are negligible in the weak coupling regime gb/δε  1 we consider. Finally, we did not include in Eq. (14.4) the charging energy responsible for the Coulomb blockade, as its contribution to the ground-state energy is trivial. In the absence of interactions, gb = 0, the parity parameter ∆P can be easily calculated. Indeed, the ground state energy EN is found by summing up N lowest single-particle energy levels. This results in E2l+1 = E2l + εl+1 and E2l+2 = E2l + 2εl+1 . Substituting this into Eq. (14.3), we find that without the interactions ∆P = 0. For weak interactions one can start with the first-order perturbation theory in gb. In this approximation an electron in state k interacts only with an electron with the opposite spin in the same orbital state k. Thus when the “odd” (2l + 1)-st electron is added to the grain, it is the only electron in the state l + 1 and does not contribute to the interaction energy, δE2l+1 = δE2l . The next, (2l + 2)-nd electron goes to the same orbital state and interacts with it: δE2l+2 = δE2l+1 − gb. From Eq. (14.3) we now find gb ∆P = , 2

at gb → 0.

(14.5)

One should note that the result (??) is not quite satisfactory even in the weak coupling case gb/δ  1. Indeed, the low-energy properties of a superconductor are usually completely described by the gap ∆. The interaction constant gb is related to the gap ∆ in a way which depends on a particular microscopic model, so the result (??) cannot be directly compared with experiments. 179

This problem can be resolved by considering corrections of higher orders in gb, which are known[105] to give rise to logarithmic renormalizations of g. In the leading-logarithm approximation the renormalized interaction constant is found [105] as gb g˜ = . (14.6) g b 1 − δ ln DD0 Here D0 is the high-energy cutoff of our model, which has the physical meaning of Debye frequency, and D  D0 is the low-energy cutoff. At zero temperature, D ∼ δε. Taking into account the relation between the gap in a large grain ∆ and microscopic interaction constant, ∆ ∼ D0 e−δ/bg , we find with logarithmic accuracy g˜ = δ/ ln(δ/∆). Finally, substituting the renormalized interaction constant into Eq. (??), we get ∆P =

δ , 2 ln ∆δ

∆  δ.

(14.7)

Unlike the first-order result (??), ∆P is now expressed in terms of experimentally observable parameters ∆ and δ rather than model-dependent interaction constant gb. Let us stress, that in a very small grain with δ  ∆, the mean field gap vanishes, and no parity effect is expected. On the contrary, our result (14.7) predicts that in small grains the parity effect is stronger than in the large ones. This behavior is due to the strong quantum fluctuations of the order parameter which persist even when its mean field value vanishes. The physics of the fluctuations of the order parameter is hidden in the renormalization procedure leading to Eq. (14.6). The case δ  ∆ has been discussed in [272, 279, 273], where it was found that ∆P = ∆ − δ/2. By comparing these two limits Matveev and Larkin concluded that a minimum should appear in ∆P when the level spacing is of the order of ∆. The numerical analysis of the case δ ∼ ∆BCS [274, 275] demonstrated that ∆P is the smooth function of δ,what indicates on the absence of the phase transition.

14.2

Superconducting drops in the system with quenched disorder (the method of optimal fluctuation).

We will study below the phase transition in the macroscopic system of the large number of granula. The situation turns analogous in superconductor 180

with the quenched disorder. The matter of fact that even weak quenched disorder can change the character of the phase transition. Let us mention that such disorder is static by nature and manifests itself as the fluctuations of the GL functional parameters a, b, c. They appear due to the structure inhomogeneities of the initial crystalline lattice (dislocations, accumulations of impurities, separation of the other phase grains). Usually these fluctuations are small and they weakly affect on the properties of superconductor. It was mentioned above that such quenched disorder changes qualitatively the properties of the vortex structure breaking down the Galilean invariance and resulting in the appearance of dry friction (collective pinning). Below we will demonstrate that in the case when quenched disorder is stronger than the thermal or quantum fluctuations it results in the considerable smearing of the phase transition.

14.2.1

The smearing of the superconducting transition by the quenched disorder[270]

Let us consider the superconducting system with quenched disorder which slowly varies the local critical temperature. It can be described by the Ginzburg-Landau functional with the coefficients being some random functions of the coordinates. The effect of such quenched disorder on the vortex pinning was already discussed for temperatures below Tc in the section ?. Here we will suppose T > Tc , i.e. a > 0. The realization of the situation with a = a + δa (r) < 0 in some large enough domain means that the superconducting drop can be formed there. ”Large enough” signifies that the domain must be so large that the proximity effect will not be able to suppress the superconductivity in it. It is why the probability to find such a domain is small. The problem of calculation of such probability (i.e. of the density of superconducting drops) is the particular case of the general the problem of the optimal fluctuation. Supposing the characteristic scale of the disorder much less than ξ, one can assume the distribution functions of such random values as the Gaussian ones. For instance for a = a + δa (r)   Z 1 2 [δa (r)] dr (14.8) P [δa (r)] = C exp − W where W is the phenomenological parameter which can be determined experimentally measuring the creep critical current. Let us define as the superconducting drop the domain with a < 0 and of so large size L that the superconducting gap ∆ in it exceeds the level spacing 181

δ (hence the modulus fluctuations |∆| in it can be neglected). In result the value |Ψ| can be found from the GL equation with the depending on r coefficient a (r) and then to calculate the drops density, i.e. the probability to find the drop at the point r. Let us firstly resolve the problem qualitatively. Due to the proximity effect the order parameter in the drop of the size L can be evaluated as1 :   1 1 2 |Ψ| = −a − , (14.9) b mL2 so, in accordance with (14.8), the probability to find such drop is " # (a − a)2 LD P (a, L) = C exp − W

(14.10)

with D = 3. The same formula can be used for the drop formation in the superconducting film (D = 2) or wire (D = 1). Integrating over all drop sizes one can find

P (Ψ) =

Z

∞

P (a, L) dL =

"

b |Ψ|2 + a + C exp − W

Z

0


2 1 mL2

LD

#

dL. (14.11)

The extreme value of L for any dimensionality is the same: 1

L∼ q

m b |Ψ|2 + a


.

This gives for the probability distribution "
2−D/2 # a + b |Ψ|2 , P (Ψ) ∼ exp − W while for the drops density "

#
2−D/2 T − Tc ρ ∼ exp − α2−D/2 m−D/2 . W

2 The mean value |Ψ| in the drop is equal to 1

(14.12)

(14.13)

(14.14)

Here the coefficient 1 in the second term is chosen for the sake of convenience. One 2 has to recognize that by our evaluation of |∆| we cannot guarantee the coefficient since we suppose it unvariable in all volume of a drop.

182


2−D/2 W α T − Tc |Ψ| ∼ . (14.15) b In the strong magnetic p field the transverse drop size is determined by the magnetic length LH = Φ0 /H . It is why in the formula (14.13)-(14.15) one has to substitute D → D − 2 and W → W L2H . Let us stress that in the obtained formula we can guarantee the functional dependencies only. The numerical factors, even in the exponent, in the demonstrated approach unfortunately remain unknown. In purpose to find the density of superconducting drops more precisely one has to optimize not only the drop size L, but the dependence a (r) itself too. To do this let us expand the solution of the equation   ∇2 2 a (r) − + b |Ψ| Ψ = 0 (14.16) 4m

2

in the series over the eigenfunctions of the linearized Schr˝odinger equation   ∇2 δa (r) − − (En − a) ψn = 0. (14.17) 4m This equation describes the particle motion in the random potential δa (r) with the energy En − a. In the case when a(hence T − Tc ) is large enough this problem is equivalent to the problem of the calculation of the asymptotic tail of the electron density of states in the random potential [271]. The properties of this tail are determined by the drops where the random potential δa (r) = ψ02 (r) with the function ψ0 (r) satisfying the nonlinear Schr˝odinger equation   ∇2 2 − ψ0 (r) − (E0 − a) ψ0 (r) = 0. (14.18) − 4m The density of states is proportional to the density of such drops and equal [271] "

# 2−D/2 |E| (4m)−D/2 , ρ (E) = ρ0 W −(D+1)/2 E D(5−D)/4 exp −37 W

(14.19)

where E = E0 − a < 0.Let us note that in the expansion of Ψ over ψn the only large term is Ψ (r) = Ψ0 ψ0 (r) , 183

(14.20)

where ψ0 (r) is the solution of (14.18). Multiplying the equation (14.18) by ψ0 (r) from left and integration over the space variable one can find R E0 ψ02 (r) dr 2 R |Ψ0 | = , (14.21) b ψ04 (r) dr

Expressing here E0 = E +a. in terms of Ψ0 one can determine the probability to find the drop with the maximal order parameter Ψ (0) . The density of all superconducting drops

#
2−D/2 T T − c α2−D/2 (4m)−D/2 . ρ (T − Tc ) = ρ0 W −(D−1)/2 [α (T − Tc )]D(6−D)/4−2 mD/2 exp −37 W (14.22) Comparison of (14.22) with (14.14) demonstrates that in result of more precise consideration we have succeeded to gain the large numerical factor 37 in the exponent, which suppresses strongly the energy dependence of the fluctuation superconducting drops density. It is necessary to mention that this factor decreases in the presence of magnetic field. "

14.2.2

Formation of the superconducting drops in magnetic fields H > Hc2 (0) .

The applied magnetic field does not suppress the described above mechanism of superconducting drops formation in the medium with the randomly distributed interaction constant g. But in its presence appears the another mechanism, which can prevail on the first one. The matter of fact that the upper critical field Hc2 (0) =? depends on the diffusion coefficient, i.e. on the electron mean free path `tr , i.e. impurity concentration. At some small domain where occasionally the impurity concentration turns out to be higher than the average one, the local Hc2 (0) > Hc2 (0) and the formation of superconducting drop is possible. There are two methods to account for the randomness of the Hc2 (0) and we will describe both of them. The first one is the usual microscopic Abrikosov-Gorkov diagrammatic technique [105], but the impurity averaging has to be applied directly to calculation of the drop formation probability, not to the propagator L (like it was done in Section?). The second method is semi-phenomenological. The averaging is performed here in two stages. In the first one the equation for the propagator L in dirty superconductor with the diffusion coefficient D as the random function of position is derived. At the second stage the averaging over the random δD(r) is performed. Close to 184

transition in the frameworks of the Ginzburg-Landau formalism this means account for the randomness in the coefficient 1/4m of the gradient term. Two different methods correspond to different physical situations. The first one describes the universal fluctuations analogous to the mesoscopic fluctuations in resistance. Besides `tr it does not involve new parameters: the disorder weaker than universal does not exist. The second approach is more general and it can describe the stronger disorder. Such disorder determines the critical current in the collective pinning phenomenon (see section?). One can consider a generic disordered system with a random diffusion coefficient: D(r) = D + δD(r), (14.23) where a short-scale disorder characterized by the Gaussian white noise is introduced 2 δD(r)δD(r0 ) = D d2 δ (r − r0 ) . (14.24) The randomness can be connected with localized and extended defects present in a superconductor. It characterizes the strength of disorder which can be connected with dislocation clusters, grain boundaries in polycrystal samples etc. Deep into the superconducting state, this randomness leads to the collective pinning effects. Thus, phenomenological constant d is directly connected with the pinning properties of a superconductor [236] and can be extracted independently from experiments. For example, the critical current ofh a superconducting film [293] in the collective pinning regime is i Hc2 (0) 2 2 jc /jc0 ≈ d /LHc2 , where jc0 is the depairing current in zero field. H Let us also note that a possible randomness in the BCS interaction constant g would lead to the same effects on the upper critical field. The proposed model can be realized in a system of superconducting grains. Let us consider the superconductor in the vicinity of the BCS upper critical field Hc2 (0) at zero temperature. The transition is controlled by the  dimensionless parameter h = H − Hc2 (0) /Hc2 (0). The region of strong fluctuations in an homogeneous superconductor is determined by the condition h < Gi. We suppose that the external magnetic field is such that h < 1 but lies outside the fluctuation region h > Gi. Due to quenched disorder (structure fluctuations) superconducting drops appear even above Hc2 (0). In this section we are interested by the 2D case so drops will be called below as ”islands”. To find the distribution of such islands, we need to have the BCS type theory in the form not yet averaged over the quenched disorder. It can be written by means of decoupling the interaction term in the BCS Hamiltonian via a Hubbard-Stratonovich field ∆(see section ?). Then, the fermionic degrees of freedom can be integrated 185

out and one gets an effective action for the superconducting order parameter. In the vicinity of the transition, an expansion on the order parameter is possible and we obtain the following action:

−S∆ = 1 + 2

Z

Z

∆∗ (x1 )L−1 (x1 , x2 )∆(x2 )dx1 dx2

∆∗ (x1 )∆∗ (x2 )B({xi })∆(x3 )∆(x4 )

4 Y

(14.25) dxi ,

i=1

where we use x = (r, t) and dx = d2 r dt for brevity. In Eq.(14.25), operator h i−1 ˆ is the fluctuation propagator: L ˆ ω = g −1 − Π ˆ ω , g is the BCS interL ˆ in the coordinate representation has the form action constant, operator Π P Πω (r, r0 ) = T ε Πω (r, r0 ; ε) where Πω (r, r0 ; ε) = Gε (r, r0 )Gω−ε (r, r0 ) and Gε ˆ = Π+δ ˆ Π ˆ is the Matsubara Green function. Let us emphasize that operator Π ˆ consists of a mean part and a random part δ Π which is responsible for the effects under consideration. Non-linear operator B in Eq.(14.25) corresponds to the diagrams calculated close to Hc2 (0) explicitly by Maki [304] and Caroli et al. [305]. In the vicinity of the transition we can neglect the randomness in the ∆4 term. The saddle point approximation δS/δ∆(r, t) = 0 results in the timedependent Ginzburg-Landau equation for a gapless superconductor [306]. When considering the spatial distribution of the islands we can disregard dynamic effects and consider the static form of the Ginzburg-Landau equation. Below the critical field Hc2 (0) its nontrivial solutions describe the superconducting state. If we neglect the randomness of the polarization operator there are no nontrivial solutions for the corresponding mean-field equation above the BCS upper critical field (h > 0). However, the random part of integral operator ˆ possesses the positive and negative eigenvalues δ and for that negative δΠ ones which are greater in absolute value than h the non-trivial solutions appear. They corresponds to the appearance of the local superconducting islands. To find the distribution of the islands, one should find the distribution ˆ function of the eigenvalues for the random operator Π: Z 1 Π(r, r0 ) ψ(r0 )d2 r0 = ( + δ) ψ(r), (14.26) ν where  =  + δ is the dimensionless eigenvalue of the polarization operator and ν (ν = m/2π for isotropic 2D metal) is the density of states per spin at 186

ˆ its spectrum is discrete the Fermi line. In the absence of a randomness in Π, and is parameterized by the Landau level indexes. The random part smears out the eigenvalues. ˆ ) can be introduced The “density of states” ( density of eigenstates of Π as Z ρ() = D {δΠ} δ ( −  [Π]) w [δΠ] , (14.27) where w [δΠ] is the distribution function for the polarization operator which is supposed to be Gaussian with correlator δΠ(r1 , r2 )δΠ∗ (r3 , r4 ). To find the density of states ρ(), we use the optimal fluctuation method [307]. This means that we evaluate the functional integral (14.27) in the saddle-point approximation. Let us note that the problem of finding ρ() is analogous to the problem of density of states of a particle in a random potential[271]. In the presence of an external magnetic field in 2D, the problem is simplified, since the coordinate dependence of the wave functions is dictated by the magnetic field [300]. In the case of large enough δ,exceeding noticeably the smearing of the levels ((δ)2  (δ)2 ), the solution has a form of rare islands. In the vicinity of a circularly symmetric island located at a point ri , the “wave function” can be taken in the following form: ( ) 1 (r − ri )2 ψi (r) = √ exp − . (14.28) 4L2H 2πLH When (δ)2  1 the LLL approximation can be used and in the first order of the perturbation theory one can obtain: Z 1 1 ψi (r1 )δΠ(r1 , r2 )ψi (r2 )d2 r1 d2 r2 . δ = δΠ00 ≡ ν ν The distribution function reads: "

# (δ)2 ρ() ∝ exp − , 2I

(14.29)

where I = δΠ200 /ν 2 . In the case of a dirty metal, correlator I can be calculated with the help of the conventional cross diagram technique [?]. This yields the following estimate for the correlator: I1 ∼ Gi2 and index “1” refers to the first model we consider (weak mesoscopic fluctuations in a dirty metal). In the system with a short-scale randomness in the diffusion coefficient (14.24), we can calculate the correlator using the differential equation for the 187

polarization operator which in the presence of an external magnetic filed has the form: [∂ D(r) ∂ + iε] Π(r, r0 ; ε) = (2πν) δ(r − r0 ), (14.30) where ∂ = −i∇ − 2eA(r). One can solve Eq.(14.30) using a simple perturbation theory with respect to δD. With the help of Eq.(14.24), we get the correlator and find the distribution function which can be written in the form (14.29) with I2−1 = 8π (LH /d)2 . The modulus of the order parameter in a superconducting island is random and parameterized by random variable  (see Eqs.(14.26) and (14.29)). Using the explicit expression (see Ref. [305]) for the non-linear operator B in Eq.(14.25), one can get the following “mean-field” value of the order parameter for a spherically symmetric island: |∆0 | =

√

4π

D√ δ − h LH

(14.31)

and the coordinate dependence of the order parameter is described by ∆i (r) = ∆0 ψi (r), where ψi (r) is defined in Eq. (14.28). Let us note that the typical size of a superconducting island is LH . Substituting δ from Eq.(14.31) to Eq.(14.29) one can find   " 2 # 2  2   1 |∆0 | LH ρ(∆0 ) ∝ exp − h+ . (14.32)  2I  4π D The typical distance between the islands, being proportional to ρ−1/2 , is exponentially large R ∼ LH exp [h2 /4I]. In the case of the universal disorder I ∼ Gi2 and in the region Gi < h the density of such superconducting drops is exponentially small.

14.3

The exponential DOS tail in superconductor with quenched disorder

Now let us discuss the effect opposite to the one discussed in the previous section: appearance of the exponential tail under the gap in superconductor with quenched disorder. In homogeneous superconductor in absence of magnetic field ∆ plays not only the role of order parameter but characterizes the gap in the quasiparticle spectrum too. The presence in the system of the quenched disorder results in the possibility of appearance of the noticeable domains with superconductivity weaker than in avarage. 188

Let us start from the case when the random, fluctuating in space, physical value is the effective constant of electron-electron interaction g [208]   1 1 = + g1 , (14.33) g g where g1  1. Namelly g1 is the random value distributed by the Gauss law:   Z 1 2 D P [g1 (r)] = exp − g1 (r) d r . (14.34) W Close to critical temperature side by side with fluctuations of g1 changes in space the local Tc , or, what is the same, the GL parameter a, like it was considered in section ?. But here we are interested in the case of T = 0 where the GL formalism is unapplicable and one has to operate in the frameworks of the Green functions method. In the case of dirty superconductor the most convenient formulation of the Green function formalism is so-called Uzadel equations[316], written on Green functions already integrated over the energy gbR (ω). We are interested in the density of states which is expressed in terms of the normal Green function g R (ω) = ν sinh θ : Z 1 dD p R 1 ν (ω) = − Im G (p, ω) = − Img R (ω) (14.35) D π π (2π) which is determined by the Uzadel equation ∆ sinh θ − ω cosh θ + D∇2 θ = 0.

(14.36)

In homogeneous superconductor, when ∆ = const, θ (ω − ∆) ω . ν (ω) = √ ω 2 − ∆2

(14.37)

One can present ∆ = ∆ + ∆1 , where ∆1 is small random value ∆1 = ∆g1 . In this case the expansion over ∆1 gives the equation for θ ∆ sinh θ − ω cosh θ − Γ sinh θ cosh θ = 0,

(14.38)

where Γ=

Z

h∆21 ik D d k. Dk 2

(14.39)

In superconductor with paramagnetic impurities the Uzadel equation for θ has the same form (14.38) but with Γ = 1/τs . If Γ  ∆, the gap in the spectrum decreases to 189

"

3 ωc = ∆ 1 − 2



Γ ∆

2/3 #

.

(14.40)

Below this threshold, due to the formation in the domains where g1 and ∆1 are negative of the drops with the superconducting properties suppressed, the tail in DOS appears. The method of the optimal fluctuation gives the expression for the density of such drops and therefore, for the density of quasiparticle states[208]:   !3/2  r 5/4 48π 2D ωc − ω . ρ (ω) ≈ exp − (14.41) 5W 3∆ ∆ In complete analogy with the case of the drop formation above Tc the numerical factor in the exponent is unusually large and it makes the tail almost unobservable. In some special cases the gap in superconductor spectrum depends on the degree of disorder. Such situation is realized in superconductor with magnetic impurities, in SN structures, in dirty 2D superconductors. In these cases the tails in the density of states appear due to the universal fluctuations of the wave functions in random systems. Such fluctuations is convenient to account by the supersymmetry method [317]. Calculating the probability to find the drop with the small gap one has to optimize not only its size but the Green function parametersresponsible for the randomness of the wave functions. It is why the density of states calculated by the supersymmetry method decreases with the energy increase more slowly than in the case of nonuniversal fluctuations. In this case DOS does not depend more on the interaction constant fluctuations but only on the value of the average conductance[318, 319, 320, 321, 298].

14.4

Josephson coupled superconducting drops

As it was already mentioned above the phase transition does not exist in the single drop. Even if the modulus of the order parameter fluctuates there weakly the strong fluctuations undergoes its phase. Nevertheless the superconducting drops array can undergo the phase transition due to the Josephson connection between them. Let us consider the system of superconducting drops in the insulator matrix 2 . The neighbour superconducting drops inter2

Such system may be obtained by the weak oxidation of the drops surface.

190

action energy is determined by the Josephson effect and is equal to E (φi − φj ) = EJ cos (φi − φj ) .

(14.42)

Here [?] |∆| |∆| tanh , (14.43) 2 4e R 2T where φi is the the phase of the condensate wave function in the i−th drop and R is the tunnel resistance between them in the normal state. In the case when superconducting drops are settled in the normal metal matrix, or on the surface of the normal metal film, or they appear in the superconducting medium above Tc as the result of the optimal fluctuation, the interaction between drops appears due R to the proximity effect. This interaction EJ (Rij ) ∼ h∆1 (0) ∆2 (Rij )i ∼ L (k, 0) exp (ikRij ) dk. At large distances it is determined by the logarithmic singularity of the integral and is equal to   2 Rij 2 2 ξ . (14.44) EJ (Rij ) = νD |∆0 | 2π D exp − ξ (T ) Rij EJ =

For the case D = 1 this expression was firstly obtained in [322]. As it was demonstrated there the account for repulsion in normal metal affects weakly on this result. It is interesting to discuss the dependence EJ (Rij ) in the magnetic field. As one can q see from Eq.(8.9) EJ decreases exponentially at the distances R > LH =

Φ0 . H

Nevertheless this statement is correct only for the value

EJ averaged over the impurity positions. The value EJ2 decays with distance only by the power law[?]: EJ2 ∝ R−4 .

(14.45)

Such slow decrease is related with the fact that the square of Josephson energy in its diagrammatic presentation is a block of four one-electron Green functions (see Fig K). The average of a such product contains two Cooperons C and two diffusons D and if the former have charge 2e the latter are neutral and are not affected by the magnetic field. Physically such result means that before the averaging procedure EJ ∼ R−2 , but it contains some quickly oscillating factor. For each realization of disorder, there is a fixed spatial distribution of the superconducting islands. The interaction Hamiltonian for such a system can

191

be obtained from Eq.(14.25) and has the standard form X Hint = Jij cos (φi − φj + Aij ),

(14.46)

ij

where Jij is the Josephson energy of the interaction between islands i and j and Aij is the phase-shift due to the magnetic field. The average value of the Josephson energy is J(R) ∝ R−2 exp [−R/LH ]. Since the typical distance between the islands is exponentially large compared to LH , the average Josephson energy is negligible. However, as it was shown in [?], the variation of the Josephson energy decays as a power law only J 2 (R) ∝ R−4 . The Hamiltonian (14.46) describes a “frustrated” two-dimensional XY model with random bonds. The frustration comes both from the Josephson energy which is random and from the phase difference due to the magnetic field. At zero temperature such a system should show a glassy behavior if there are no effects capable of destroying phase coherence between the islands.

14.5

Classical phase transition in granular superconductors

Let us consider now the peculiarities of phase transition in the system of drops connected by the interaction EJ . In the case when the transition temperature is high enough (the precise condition will be formulated below) this transition is described by the partition function   Z Y EJ Z= dφi exp − cos (φi − φj ) . (14.47) T It is interesting that this is the partition function of the classical XY model since cos (φi − φj ) = cos φi cos φj + sin φi sin φj = nix njx + niy njy ,

(14.48)

where n is unit 2D vector. If the superconducting drops are separated by insulator the interaction of only nearest neighbours takes place. If the Josephson energy EJ is the same for all nearest neighbours the transition in the system happens at Tc ∼ EJ (numerical calculations give Tc =?EJ ).

192

In 3D system below the transition temperature appears non-zero order parameter Ψ ∼ n. In the mean field approximation it is determined from the equation:
R cos φ exp − ETJ n cos φ dφ
n= R . (14.49) exp − ETJ n cos φ dφ At the transition point n → 0,i.e.

1 Tc = EJ cos2 φ = EJ . (14.50) 2 In this case the MFA is applicable only for evaluations by the order of value. Let us mention that EJ depends on temperature (see (14.43)) and in the vicinity of Tc EJ ∼ (Tc0 − T ) ,

(14.51)

where Tc0 is the critical temperature of the bulk superconducting material of which the drop is fabricated. One can notice from (14.50) that Tc < Tc0 , but these two values can be close enough and we still can believe that the system is beyond the critical region. In Eqs.(14.44) and (14.43) the Josephson energy was supposed to be equal for all junctions. Indeed EJ depends on the insulator thickness exponentially and if this value is random the Josephson energy will be random too and with very wide distribution function. In this case to find the value of critical temperature helps the percolation theory [?]. The granula with EJ  T one can assume as noninteracting, while those one for which EJ  T is possible to treat as the unique granule with the homogeneous phase φ.The superconducting transition in such macroscopic system takes place when such strongly interacting granula form the infinite cluster. The distribution function of R and hence EJ in tunnel junctions are poorly known. Nevertheless what is really important that the same percolation picture determines the average macroscopic resistance of the sample in the normal state [?]. In result [?], one can find for Tc Tc ∼ EJ (R)

(14.52)

where now R has to be understood as the average resistance on the square of the granular film in 2D case or the resistance of the layer of thickness
D 0.9 (this is the characteristic size of the typical claster). Here D is L∼D a the granule diameter while a is the interatomic distance. The similar picture takes place in the dirty metallic superconductors, where the drops exist due to the optimal fluctuations. Nevertheless due to the large numerical factor 193

in (14.22) the drops density is extremely small (in absence of magnetic field) and they give the negligeable contribution in the increase of Tc . The fluctuations of the random value δa result in the smearing of the transition [310]. As it follows from the GL equation the fluctuations of the coefficient a below Tc (a < 0) result in the appearance of the random addition to the order parameter Ψ = Ψ + δΨ : Z Z dk δa (r1 ) exp [ik (r − r1 )] δΨ (r) = Ψ dr1 . (14.53) 3 2 |a| + k2 /4m (2π) Substituting in the Ginzburg-Landau equation (??) and averaging over δa with the distribution function (14.8) one can find the equation for the hΨ2 i : 

Z

dk 1 3 (2π) 2 |a| + k2 /4m





Z

dk 1 = bΨ 1 + 3W α Tc0 − T − W 3 (2π) [2 |a| + k2 /4m]2 (14.54) Our analysis demonstrated that due to inhomogeneities the critical temperature increases as Z dk 1 . (14.55) δTc = Tc − Tc0 = W 3 (2π) 2 |a| + k2 /4m 2

Close to the new Tc the order parameter 2

Ψ =α

|Tc − T | b

3/2

1−

7 W (4m) p 8π 2α |Tc − T |

!

.

(14.56)

The heat capacity below the transition Cs − CN = (∆C)0

3 W (4m)3/2 p 1− 8π 2α |Tc − T |

!

.

(14.57)

Thus the presence of the inhomogeneities results in the smearing of the transition in the range of W 2T 3 . (14.58) D3 If this value exceeds Gi, hence the quenched disorder is more important than the thermal fluctuations. In the opposite case the quenched disorder is noticeable only in the narrow region in the vicinity of Tc , where nevertheless it changes the shape of the scaling singularity of the transition [315, 312, 313]. ∆ ∼

194



.

It worth to mention that the correction to heat capacity induced by inhomogeneities has the same temperature singularity as the correction generated by thermal fluctuations. Nevertheless these corrections have the opposite signs and if W > (νT )−1 the inhomogeneities contribution dominates.

14.6

Quantum phase transition in granular superconductors

14.6.1

Coulomb suppression of superconductivity in the array of tunnel coupled granula

In the case when the Josephson energy EJ is small the transition temperature of the array decreases and the quantum effects become of the first importance. The first consequence of it is the appearance of the quantum nature of the phase φ, which must be treated as the operator. This operator does not commute with the operator of the particles number corresponding to one granule. It turns out that instead of the Hamiltonian it is more convenient to deal with the action S [φ ((τ ))] . The Coulomb interaction tends to fix the number of the particles at each granule, i.e.it increases the phase fluctuations. The Coulomb energy of the granula system is defined as EC =

1X Cij Vi Vj , 2 i,j

(14.59)

where Cij is the capacity matrix. Due to the Josephson relation Vi = 2e∂φi /∂τ and the action of the granula system is equal to ! Z β X S= dτ 2e2 Cij φi φj + EJ cos (φi − φj ) . (14.60) 0

i,j

Here, as it was done in the case of the classical transition, we assume the Josephson connection between the nearest neighbours only. This formula was derived microscopically by Efetov [314]. For the sake of simplicity let us restrict our consideration by the case with the diagonal capacity matrix (this case is realized when the plane granula lie at the metallic substrate). In this case

S = SC + SJ =

Z 0

β

dτ

! X φ2 i + EJ cos (φi − φj ) , 2e2 E C i 195

(14.61)

−1

where EC = (2e2 C) . When T > EC the first term does not play any important role and Tc ∼ EJ . For the contact of the special shape the value of EJ (H) can be suppressed applying magnetic filed. The corresponding Tc decreases and at some value of EJ (H) ∼ EC it turns to zero. One can find the values of Tc and EJ in the MFA, as it was done above for the classical case: EJ−1

= C(β) =

Z

β

dτ C (τ ) ,

(14.62)

0

C (τ ) = hcos φ (0) cos φ (τ )i =

R

exp (−SC [φ (τ )]) cos φ (0) cos φ (τ ) Dφ (τ ) R . exp (−SC [φ (τ )]) Dφ (τ )

The average C (τ ) is calculated with the zero action S0 = S (EJ = 0) . At T =0 C (τ ) = exp (−EC τ ) ,

(14.63) (c)

and the Eq.(14.62) gives for the transition point EJ =?EC . Thus at zero (c) temperature the critical value of the Josephson energy EJ of the order of the Coulomb energy exists, such, that below it the superconductivity cannot appear more. In the region of energies less than this one the system becomes the Mott insulator[314].

14.6.2

Superconducting grains in the normal metal matrix

In this subsection we will consider the quantum superconducting transition in the system of superconducting grains separated by the normal metal. For instance, the granins can be placed on the surface containing 2D electron gas. In contrast to the system of grains in the insulator matrix (section Jcsd ?) here EJ ∼ b−D (b is the distance between grains) is determined by the proximity effect (14.44) instead of the Josephson coupling (14.43). Another very interesting and important difference between the grains in metal with respect to insulator matrix is the weakening of the charge quantization effect. The matter of fact that the grain charge can flow through the conducting metal system, so it is not quantized. Nevertheless at low temperatures the traces of this quantization still remain. It is why some critical grain density exists (the critical distance between granula bc ) when the critical temperature of the global transition Tc → 0. The dynamics of the phase ϕ(τ ) of a single SC grain can be described by a simple imaginary-time action [?], 196

GA SA [ϕ] = − 8π

Z Z

β

dτ dτ 0

0

cos[ϕ(τ ) − ϕ(τ 0)] . (τ − τ 0)2

(14.64)

Here GA is the Andreev subgap, normalized to e2 /~, conductance (below will be given its definition). At low values of GA  1 normal-superconducting transport across the interface is suppressed by the usual Coulomb blockade effect governed by the junction’s charging energy 2e2 /Cj . This case was considered in the previous subsection. For large GA one can start from the Gaussian approximation for SA [ϕ(τ )], what means that one expands the cos[ϕ(τ )−ϕ(τ 0)] in (14.64) up to the second order. Then Z
4 ∞ S A + SC = |ω|GA + ω 2 /EC ϕ2ω dω. (14.65) π −∞ For the correlator (14.62) introduced in the previous subsection one can find: 1 1 2 C(τ ) = e− 2 h(ϕ(τ )−ϕ(0)) i . 2

(14.66)

Phase correlator can be calculated within the logarithmic accuracy and for τ > 1/EC GA gives 4 h|ϕ (0) − ϕ (τ ) | i = π 2

Z

∞

−∞

(1 − cos ωτ ) 8 dω ' ln (τ GA EC ) . (14.67) 2 |ω|GA + ω /EC πGA

We have seen (Exp.(14.62)) that the critical concentration of grains is determined from the behavior of Z β C(β) = C(τ )dτ ∝ β 1−4/πGA (14.68) 0

Indeed, if GA > 4/π, C(β → ∞) diverges and for any small EJ at some temperature the Eq.(14.62) is satisfied and the transition superconductor-normal metal takes place. It seems to indicate that at large GA superconductivity is always stable at T = 0, in agreement with [323]. Nevertheess all this speculation is wrong and is nothing else as the artefact of the Gaussian approximation (14.65) . The crucial point is to note that the employed Gaussian approximation breaks down at a finite time scale t∗ , due to downscale renormalization of GA . This renormalization is caused by the periodicity of the action SA [ϕ] as 197

a functional of ϕ(τ ), that is, in physical terms, by the charge quantization. This problem is analogous to the one studied by Kosterlitz [?]. Translating his results to the present case, one gets the renormalization group (RG) equation dGA (ζ)/dζ = −4/π,

(14.69)

with ζ = ln ωd t. This equation is to be solved with the initial condition GA (0) = GA . As a result, at the time scale τ ∗ ∼ EC−1 eπGA /4 the renormalized Andreev conductance GA (t∗ ) decays down to the value of order unity [324]. At longer time scales C0 (τ ) decays approximately as τ −2 , so the integral C(0) ∼ τ ∗ ∼ EC−1 eπGA /4 . This means that the effective charge energy eC = EC e−πGA /4 E

(14.70)

bc ∼ deπGA /4D .

(14.71)

Taking into account that EJ ∼ b−D , and using Eq. (14.62), one can compare eC = EJ and obtain the critical distance between islands at the point of the E superconductor-metal transition [324]:

Let us comment the introduced above notion of Andreev’s conductance GA . It depends on the system properties. Let us start from the small superconductive grains of radius d immersed into a 3D metal with bulk resistivity ρ and the tunneling resistance RT  RN = ρ/4πd. Here GA = RN /RT2 . Now let us consider an array of small superconducting islands (of radius d each) in contact with a thin film of dirty normal conductor with the dimensionless conductance G  1. The distance between neighboring islands is b  d (more precisely, b−2 will be the concentration of islands). Resistance RT of the interface between each island and the film is low: GT = ~/e2 RT  1. Islands are thick enough, to prevent suppression of superconductivity inside them. The corresponding condition for the superconducting gap reads ∆  GT /νVi , where Vi is the island’s volume and ν is the density of states. In a simplified model [324] with sufficiently strong Cooperchannel repulsion in the film, gn ν  GT /4πG . Here GA = G2T /4πG λn is the Andreev subgap conductance in the limit of weak proximity effect, valid under the condition 1  GT  4πλn G [324, 325]. Nevertheless as the most challenging it seems the case of high conductance GT  4πλn G . In this case instead of the unique renorm-groop equation (14.69) one has to write down the system of large number of such equations for each grain [326] . The numerical solution of this system gives √ eC = EC e−2,5π G (14.72) E 198

and in this case the critical grain concentration is given by √ bc ∼ de1,25π G .

(14.73)

In conclusion, in this subsection we presented the theory of quantum superconductor-metal transition in a 2D proximity-coupled array. The critical conductance Gc is non-universal and large compared to e2 /~. Near the quantum critical point the system behaves as a BCS-like superconductor with the effective Cooper attraction constant vanishing at G → Gc .

14.6.3

Phase transition in disordered superconducting film in strong magnetic field

In sections ?? the phase transitions in the systems of artificial granula were studied. Now we will pass to the study how the formation of the superconducting drops due to the quenched disorder in fields H > H c2 (0) changes the character of the traditional Abrikosov’s phase transition. In homegeneous superconductor the transition temperature Tc → 0 and the phase transition becomes quantum in the field H = H c2 (0) =? [327, 328, ?]. As it was demonstrated in section ? for any field less that the Clogstone limit in nonhomogenous superconductor appear the domains with the value of local Hc2 (0) > H c2 (0) , where superconducting drops are formed. Without the account for quantum fluctuations at T = 0 the inter-drop proximity type interaction would result in the appearance of the global superconductivity in any field below the paramagnetic limit. Account for quantum fluctuations, the same as in the case of the artificial superconducting granula, turns Tc to 0 at some finite density of the superconducting drops. As we already know this density exponentially decreases with the growth of magnetic field (see Exp.(14.32)). It is why the quantum phase transition at T = 0 takes place in the finite field which can considerably exceed the value of H c2 (0) . To find the transition point, we use the action (14.25) which describes the dynamics of the superconducting order parameter. Let us present the superconducting order parameter in the following form X ∆(r, t) = |∆0 i | ψi (r) eiφi (t) , (14.74) i

where ψi is defined in (14.28). We consider the islands in which the modulus of the order parameter is fixed by the static mean-field equations (14.31) and only the phase is allowed to fluctuate. Finally, we obtain the following action

199

describing the system of local superconducting islands: (  2 Z 0 X Z 1 ∂φi 0 cos [φi (t) − φi (t )] − S = − dt dt ηi EC ∂t (t − t0 )2 i ) XZ 0 0 0 + dt Jij (t − t ) cos [φi (t) − φj (t ) + Aij ] . (14.75) j6=i

(compare with the Exp.(14.64)). The coefficient in the dissipative term is random and connected with the modulus of the order parameter (14.31) in an island ηi = ν |∆0i |2 /8πeDH. Note that the typical value of the coefficient is large η ∼ G h. We keep the ω 2 term in the action. As we will see below, the effective charging energy appears only as a high-frequency cut-off. With the logarithmic accuracy, the exact value of EC is not important in our problem. Let us integrate out the high-frequency degrees of freedom in the action. First, we consider strong enough magnetic fields so that the network of the superconducting islands is very dilute and the average Josephson energy is exponentially small. In the domain ω  J, only the first two terms in action (14.75) are important. In this case, the action can be written as a sum of single-island actions. The phases in different islands fluctuate independently. With the aid of the single-island action one can integrate out the high-frequency phase fluctuations using the renormalization group developed by Kosterlitz for a spin system with long-range interactions [308]. Since the first term in Eq.(14.75) is not Gaussian, coefficient η gets renormalized when integrating out fast variables. The corresponding renormalization group equations are identical to the ones derived in [301, 308]. The solution of the RG equation for the renormalized “viscosity” coefficient has the following simple form: 1 EC η(ω) = η − 2 ln . (14.76) 2π ω Note that this equation is valid unless η(ω) becomes of the order of unity. This happens at times tc ∼ ωc−1 ∼ EC−1 exp [2π 2 η]. At larger times the phase fluctuates rapidly. If the Josephson interaction between two islands is such that Jij tmin < 1, c then the Josephson term does not affect the phase dynamics at any times and can be treated as a small perturbation. However, there is always a finite probability of finding a pair of islands for which Jij tmin > 1. In this case, c Jij stabilizes the fluctuations of the relative phase [φi (t) − φj (t)] and the corresponding critical time increases tc ∼ EC exp [2π 2 (ηi + ηj )].

200

−4 Recall, that Jij2 ∝ Rij and the probability distribution for random quantity tc is known and determined by Eqs. (14.29) and (14.31). Thus, one finds the probability of finding a pair of strongly correlated superconducting islands:    π2 2 P ∼ exp −2π G h − G I . (14.77) 2

At large fields, this probability is exponentially small. As the external magnetic field decreases, the fraction of the strongly correlated superconducting islands increases and finite size superconducting clusters are formed. At some threshold field, the infinite superconducting cluster appears and it corresponds to the macroscopic superconductivity. We can estimate the location of the transition point as a field at which probability (14.77) is of the order of unity. This yields π2 hc2 = G I. (14.78) 2 Let us mention that result R ∞ (14.78) can be obtained more formally by calculating correlator C = 0 hexp [iφj (t) − iφj (0)]iS dt. At the transition point, the correlator diverges. One can perform the virial expansion with respect to the density of islands in C. As in the theory of liquids and gases and in the theory of spin-glasses with RKKY interactions [309], the virial expansion can not prove the very existence of the transition. However, it determines the transition point if there is one. The transition is defined as a point at which all terms of the virial expansion become of the same order. Comparing the contribution in correlator C from independent islands and the one from pairs, we find Eq. (14.78). In the case of the weak mesoscopic disorder I1 ∼ G−2  and the shift of the −1 upper critical field, if any, is small: hc2 ∼ G . Let us note that this
result ln G may acquire some logarithmic corrections of the order of G−1  which  are, however, beyond the scope of our investigation. Within the logarithmic accuracy, we can not distinguish the mesoscopic effects under consideration from the usual superconducting fluctuations inside the Ginzburg fluctuation region. In the case of strong disorder, the shift of the critical region can be large and it is described by Eq.(??). Phenomenological constant d measures the strength of disorder which can be connected with dislocation clusters, grain boundaries in polycrystal samples etc. Let us emphasize, that the pinning parameters in a superconductor are determined by d. For example, the critical current h of a isuperconducting film [?] in the collective pinning regime is jc /jc0 ≈ Hc2H(0) d2 /L2Hc2 , where jc0 is the depairing current in zero field. Let us also note that a possible randomness in the BCS interaction constant 201

g would lead to the same effects on the upper critical field. At finite temperatures, there is no phase transition in a strict sense. At any fields, a finite, though exponentially small, resistance exists in a twodimensional superconductor if a magnetic field is applied. One can define the upper critical field as a field at which a sharp fall in the resistance takes place. At very low temperatures, hc2 is determined by Eq. (??). As the temperature increases, hc2 (T ) decreases very rapidly. First of all, at a finite temperature,pthe Josephson coupling decays exponentially at the distances larger than D/T . Second, the thermal fluctuations destroy the Josephson coupling at Jij ∼ T . The both effects lead the following estimate of the 2 transition √ temperature Tc (h) ∼ Tc0 exp [−h /4I]. Thus, in a relatively wide region I < h < hc2 (0), the critical temperature depends on the external magnetic field exponentially. The increase of Hc2 at low temperatures has been observed in a number of experiments[297]. The discussed mechanism can give a possible explanation for the effect.

202

Chapter 15 Phase fluctuations in 2D systems 15.1

Phase fluctuations in 2D systems

Let us discuss the results obtained for the superfluid density in the 2D systems. Being in the limits of the GL theory we cannot enter into the critical region, so the result (2.102) is valid for ||  Gi(2) only. At the same time the Berezinski-Kosterlitz-Thouless (BKT) theory of critical fluctuations in 2D systems establishes the value of superfluid density at the critical temperature (Nelson-Kosterlitz jump [58]) as: 4mTc . (15.1) π An interpolation formula for the superfluid density of the 2D system can be written [13] by the unification of (2.102) and (15.1):   Gi(2) mTc || ns2 (T ) = − 2 ln +4 . π Gi(2) || + Gi(2) ns2 (Tc ) =

One can notice that the fluctuation correction to the average value of order parameter hψi = hψr i below Tc is divergent (see (2.97)). This means that the appropriate corrections cannot be supposed to be weak even relatively far from the transition ||  Gi(2) , where the fluctuations of the value h|Ψ(r)|2 i are still small. Neglecting the order parameter modulus fluctuations one can write the effective functional which describes the order parameter phase fluctuations only: Z ns X 2 2 ns Dϕ (∇ϕ)2 = k ϕk . F[ϕ] = 4m 4m k 203

Let us calculate now the average value of the order parameter using this functional without the assumption of weak fluctuations: ! X 4mT

iϕ  hΨi = |Ψ| e = |Ψ| exp − . (15.2) 2 n sk k In the 2D case the sum in (15.2) diverges and hΨi = 0 [59]. Nevertheless a phase transition in such system exists. In order to see this let us study the behavior of the correlation function hΨ∗ (0)Ψ(r)i at large distances. Above the mean field critical temperature the Fourier component < |Ψk |2 > can be easily calculated with the use of Exp. (2.19): DΨk DΨ∗k |Ψk |2 exp {−α( + ξ 2 k2 )|Ψk |2 } 1 R = , ∗ 2 2 2 α( + ξ 2 k2 ) DΨk DΨk exp {−α( + ξ k )|Ψk | } (15.3) ∗ and the correlator hΨ (0)Ψ(r)i takes form: < |Ψk |2 >=

∗

R

hΨ (0)Ψ(r)i (T > Tc0 ) =

X

k

2

|Ψk |



1 exp (ikr) = K0 4παξ 2



r ξ()



, (15.4)

wherepK0 (x) is the modified Bessel function. At large arguments K0 (x  π −x 1) = 2x e and we find that in the normal phase this correlator decreases exponentially at distances r & ξ(). Below the transition point hΨ∗ (0)Ψ(r)i = |Ψ|2 hexp (iϕ(0) − iϕ(r))i = * !+ X = |Ψ|2 exp i [1 − exp (ikr)] ϕk =

(15.5)

k

! X

 1 = |Ψ|2 exp − |1 − exp (ikr) |2 ϕ2k . 2 k The average value hϕ2k i has already appeared in our calculations (for instance, in the Exp.(2.96)) as the phase fluctuation mode below Tc : hϕ2k i = −1 4mT (ns k2 ) . The last integral (sum) in the exponent of (15.5) evidently converges at small k. After the angular integration it can be expressed in terms of an integral of a Bessel function and it logarithmically diverges at the upper limit. This divergence must be cut off at k ∼ξ −1 , where the expression for hϕ2k i is not valid more. As a result below the transition point the correlator takes the form 204

hΨ∗ (0)Ψ(r)i|r|ξ (T < Tc0 ) = mT    − πn s r mT r 2 2 |Ψ| exp − ln = |Ψ| . πns ξ ξ

(15.6)

These two ((15.4) and (15.6)), quite different, asymptotics of the correlator for the low and high temperature limits were obtained for the regions || & Gi(2) . Nevertheless it is clear that must be some point at which the high temperature exponential asymptotic changes to the low temperature power one. This temperature is reasonably called as the point of the superconducting transition. In the 2D case such a transition is called the BKT transition. We do not discuss its properties in this review (see, for instance, the review articles of P.Minnhagen [60, 61]). It is worth mentioning that, in spite of the fluctuation destruction of hψi , in the low temperature phase the observable physical quantity ns is renormalized by fluctuations finitely and is not zero. In the GL region (|| & Gi(2) ) this renormalization turns out to be even weak.

15.2

Kosterlitz-Thouless conductivity

205

Chapter 16 Phase slip events 16.1

Phase slip events in JJ.

Josephson effect thermal fluctuations?????? MQT.

16.2

Phase-slip events in 1D systems

In the case of a 1D wire Eq. (2.98) also gives a small correction to ns . At the same time it is clear that at T 6= 0 the phenomenon of the superconductivity in 1D does not exist. This results from : a) the general statement about the absence of the phase transitions in 1D systems; b) the evident exponential decrease of the correlator (15.5) for a 1D system below the transition point; c) the partition function in 1D can be calculated exactly1 . So, if the phase fluctuations are unable to destroy ns , what is the mechanism for killing superconductivity in 1D systems? It turns out that it is phase-slip events, which also can be called instantons or topological excitations. At temperatures beyond the critical region (|| & Gi(2) ) the probability of such excitations is exponentially small and they cannot be found by the methods of perturbation theory. 1

It is possible to demonstrate [62] that the free energy of a D-dimensional classical field is equal to the ground state energy of the related D − 1-dimensional quantum mechanical system. Indeed, in the 1D case the functional integral turns out to be the same as the Feynman integral for a particle moving in the potential a|Ψ|2 + 2b |Ψ|4 . The ground state energy in this case is an analytical function of a and does not have a singularity at any temperature.

206

There is a general statement of the absence of phase transitions in onedimensional systems. The superconductive phase should be unstable and supercurrents can not persist for a long time and must decay. Phase-slips are the mechanism of the supercurrent decay. Let us consider a closed loop made from a one-dimensional superconducting wire. The magnetic flux through this loop is a constant. If the phase of the wave function is a continuous function of the position, then it is not possible to change the supercurrent in the loop gradually. However, due to fluctuations, there is a finite probability to have the modulus of the order parameter vanish at some point. Thus, the phase is not defined at this point and a phase-slip may occur (a change of the phase along the loop by 2π). Thus, the supercurrent which is determined by the gradient of the phase decays. Let us note, that this reasoning works for one-dimensional systems only. In higher-dimensional systems, it is always possible to connect any two points by a path along which the phase is a continuous function. With the aim of understanding the origin of phase-slip events let us start from a discussion of a small Josephson junction included in a superconducting ring of induction L. The potential energy related with the phase difference at the junction is the same as in a layered superconductor (2.40), so the energy of the ring can be expressed in terms of the magnetic flux Φ 2 2πΦ Φ2 + . (16.1) Er = EJ cos Φ0 2L Besides the ground state with Φ = 0 there are many metastable states with Φ near to nΦ0 in such a system. The energies of such states are also the minima of the Exp. (16.1), but not absolute minima. In the limit of large EJ the energies of these minima are equal to Φ20 n2 /2L . In order to transit from one metastable state to another one with lower energy, the system has to pass through a potential barrier. The heights of these barriers are determined by the maxima of Ex.(16.1) and are equal to 2EJ . The probability of such a ”jump” (so-called ”phase-slip process”) is proportional to exp{−2EJ /T }. The flux flow decay, and respectively, the voltage in the circuit are proportional to this probability: U=

dΦ ∼ exp{−2EJ /T }. dt

2

The Josephson part of the Ginzburg-Landau energy, related to interlayer tunneling, was written as J |Ψn − Ψn−1 |2 (see (2.40)). Supposing Ψn = |Ψ|eiϕn one can rewrite the corresponding energy as 2J |Ψ|2 (1 − cos ϕ) . Applying this result to the superconducting ring with a weak link one can express the phase difference by means of the magnetic field flux: H H ϕ = ∇ϕdl = 2e Adl = 2πΦ/Φ0 .

207

The positions of the maxima and minima of the function (16.1) are determined by the condition ∂Er /∂Φ = 0. Let us consider the mechanism of the phase slip process in the circuit when the superconducting wire does not have a weak link and its free energy can be described by the GL functional (2.4). The minima of this expression are equal to −a2 V /2L+ Φ20 n2 /2L and they correspond to the magnetic flux quantum Φ0 . The transition between the neighboring minima can only occur through the saddle points of the functional (2.4). These saddle points correspond to the unstable solutions of the GL equation b Ψ + Ψ3 + ξ 2 (T )Ψ00 = 0 a The solutions of this equation are continuous in the same way as those corresponding to the minima of the functional, but they can pass through zero at some points where the modulus of the order parameter is equal to zero [63]. In the vicinity of these points the solution of the GL equation takes the form of an instanton: r a x Ψ(x) = − th . b 2ξ(T ) Substituting it into Exp.(2.4) one can find that the energy of such an instanton in the wire of the cross-section S is [64]: √  √ 3/2 2 || 8 2 a2 ξ(T ) = T. ∆F0 = 3 2b 3 Gi(wire) So the probability for the superconducting ring to change its magnetic flux by one flux quantum only implies that, being thermically activated, it passes over such a barrier. It turns out to be exponentially small, and the wire resistance is equal to R ∼ exp{−∆F0 /T }. In the case of a thin film the topological defects carrying the flux quanta are given by fluctuation vortices. The energy of such a vortex is equal to E=

ns λ ln . 4m ξ

In a thin film λ is very large. In the framework of the BKT theory it is assumed to be infinite. Thus the film resistivity below the critical temperature turns out to be zero.

208

16.3

3. Quantum phase slip events in nanorings.

??

209

Chapter 17 Fluctuations near superconductor-insulator transition Quantum phase transitions occur at zero temperature when another parameter is varied (magnetic field, density, etc). In the absence of static disorder, the classical transition with changing the crystalline structure at space dimension D leads to the same singularity as the quantum transition at space dimension D − 1 (Vaks, Larkin, 1965). The static disorder changes the character of singularities even at the classical transitions (Harris, 1974; Khmelnitskii, 1975; Lubensky, 1975). Strong disorder can suppress the transition temperature and gives rise to a quantum phase transition. Disorder enhances the effect of Coulomb repulsion. As the result, the critical temperature Tc of 2D superconductors decreases (Ovchinnikov, 1973; Takagi and Kuroda, 1981; Fukuyama and Mayakawa, 1981) and can even go to zero (Finkelstein, 1987). At this Quantum critical point the conductivity takes a universal value (M.P.A. Fisher 1989).

17.1

Quantum phase transition

It is usually supposed that the temperature of the superconducting transition does not depend on the concentration of non-magnetic impurities (Anderson’s theorem [92, 93]). Nevertheless when the degree of disorder is very high Anderson localization takes place, and it would be difficult to expect that under conditions of strong electron localization superconductivity can exist, even if there is inter-electron attraction. This means that at T = 0 the phase transition takes place with a change of the disorder strength or carrier 210

concentration. Such a transition is called a quantum phase transition since at zero temperature the classical fluctuations are absent. Indeed, one can see from (1.7) that in the limit T → 0 the thermal fluctuation Cooper pairs vanish. In the metallic phase of a disordered system the conductivity is mostly determined by the weakly decaying fermionic excitations, their dynamics yielding the familiar Drude formula (the method which accounts for the fermionic excitations will be referred to as the Fermi approach later on). Inside the critical region the charge transfer due to fluctuation Cooper pairs turns out to be more important. In some approximation, the pairs may be considered as Bose particles. Therefore the approach dealing with the fluctuation pairs will be called below the Bose approach. Let us suppose that at temperature T = 0 the superconducting state occurs in a weakly disordered system. In principle two scenarios of the development of the situation are possible with an increase of the disorder strength: the system at some critical disorder strength can go from the superconducting state to the metallic state or to the insulating state. The first scenario is natural and takes place in the following cases: if the effective constant of the inter-electron interaction changes its sign with the growth of the disorder; if the effective concentration of magnetic impurities increases together with the disorder growth; if the pairing symmetry of superconducting state is nontrivial it can be destroyed even by the weak disorder level. We will study here the second scenario where the superconductor becomes an insulator with disorder increase. This means that at some disorder degree range, higher than the localization edge when the normal phase does not exist any more at finite temperatures, superconductivity can still survive. From the first glance this statement seems strange: what does superconductivity mean if the electrons are already localized? And if it really can take place beyond the metallic phase, at what value of disorder strength and in which way does the superconductivity finally disappears? One has to have in mind that localization is a quantum phenomenon in its nature and with the approach to the localization edge the coherence length of localization ` grows. From the insulator side of the transition vicinity this means the existence of large scale regions where delocalized electrons exist. If the energy level spacing in such regions does not exceed the value of superconducting gap Cooper pairs still can be formed by the delocalized electrons of this region. The problem can be reformulated in other, already familiar, way: how does the critical temperature of the superconducting transition decrease with the increase of the disorder strength? In the previous Sections we have already tried to solve it by discussing the critical temperature fluctuation shift. 211

We have seenpthat the fluctuation shift of the critical temperature is proportional to Gi(3) for a 3D superconductor and to Gi(2) ln{1/Gi(2) } for 2D. This means that the critical temperature is not changed noticeably as long as the Ginzburg-Levanyuk number remains small. So one can expect the complete suppression of superconductivity when Gi ∼ 1 only. For further consideration it is convenient to separate the 3D and 2D cases because the physical pictures of the superconductor-insulator transition for them are quite different.

17.2

3D case

As one can see from Table 1 in the 3D case  the Ginzburg-Levanyuk number Tc remains small at pF l ∼ 1 : Gi(3d) ≈ EF  1. Nevertheless, approaching the edge of localization, the width of the fluctuation region increases [94]. In the framework of the self-consistent theory of localization [95] such growth of the width of the fluctuation region was found in paper [96]. Instead of the cited self-consistent theory let us make some more general assumptions concerning the character of the metal-insulator (M-I) transition in the absence of superconductivity [13]. We suppose that in the case of very strong disorder and not very strong Coulomb interaction the M-I transition is of second order. The role of ”temperature” for this transition is played by the ”disorder strength” which is characterized by the dimensionless value F l. With its decrease the conductivity of the metallic phase decreases g = p2π and at some critical value gc tends to zero as σ = e2 pF (g − gc )κ .

(17.1)

This is the critical point of the Anderson (M-I) transition. We assume that the thermodynamic density of states remains constant at the transition point. The electron motion in metallic phase far enough from the M-I transition has a diffusion character and the conductivity can be related to the diffusion coefficient D = pF l/3m by the Einstein relation: σ = νe2 D. One can say that diffusion like ”excitations” with the spectrum ω(q) = iDq 2 propagate in the system. At the point of the M-I transition normal diffusion terminates and conductivity, together with D, turns zero. In accordance with scaling ideas, the diffusion coefficient can be assumed here to be a power function of q: D(q) ∼ q z−2 , with the dynamical critical exponent z > 2. The anomalous diffusion excitation spectrum in this case would take the form ω ∼ q z . In the insulating phase (g < gc ) some local, anomalous diffusion, confined to regions of the scale `, is still possible. It cannot provide charge transfer 212

through out all the system, so D(q = 0) = 0, but for small distances (q & `−1 ) anomalous diffusion takes place. Analogously, in the metallic phase (g > gc ) the diffusion coefficient in the vicinity of the transition has an anomalous dependence on q for q & `−1 and weakly depends on it for q . `−1 . So one can conclude that the diffusion coefficient for q & `−1 from both sides of the transition has the same q-dependence as for all q in the transition point. It can be written in the form

D(q) =



g ϕ(q`) 3m pF `

z−2

  x, x  1 1, x  1, g > gc , , ϕ(x) =  0, x  1, g < gc

(17.2)

where the dimensionless localization length `, characterizing the spatial scale near the transition, grows with the approach to the transition point like κ 1 (g − gc )− z−2 . (17.3) pF The critical exponent in this formula is found from the Einstein relation in the vicinity of the M-I transition. At finite temperatures, instead of the critical point gc ,a crossover from metallic to insulating behavior of σ(g) takes place. The width of the crossover region is ge − gc ,where ge is determined from the relation D`−2 (e g) ∼ −z EF [pF `(e g )] ∼ T (we have used the second asymptotic of (17.2)). In this region the diffusion coefficient is

`(g) =

  z2 T E F D(T ) ∼ T `2 ∼ 2 (17.4) pF T and it depends weakly on the g − gc . Beyond this region the picture of the transition remains the same as at T = 0. Let us consider now what happens to superconductivity in the vicinity of the localization transition. In the mean field approximation (BCS) the thermodynamic properties of a superconductor do not depend on the character of the diffusion of excitations. This should be contrasted with the fluctuation theory, where such a dependence clearly exists. We will show that the type of superconducting transition depends on the dynamical exponent z. If z > 3, the transition to superconductivity occurs on the metallic side of the localization transition (we will refer to such a transition as S-N transition). If z < 3, the transition to superconductivity occurs from the insulating state directly (S-I transition). Let us study how the superconducting fluctuations affect the transition under discussion. In spirit of the GL approach fluctuation phenomena in 213

the vicinity of the transition can be described in the framework of the GL functional (2.19). The coherence length in the metallic region, far enough from the Anderson transition, was reported in Introduction to be equal ξ 2 = ξc l = 0.42D/T . In the vicinity of the M-I transition we still believe in the diffusive character of the electron motion resulting in the pair formation. The only difference from the previous consideration is the anomalous character of the quasiparticle diffusion. So in order to describe the superconducting fluctuations simultaneously near superconducting (in temperature) and Anderson (in g) transitions let us use the GL functional (2.19) with the k-dependent diffusion coefficient (17.2). The value of Gi can be estimated from the expression for the fluctuation contribution to heat capacity (2.23) taken at  ∼ Gi, where the fluctuation correction reaches the value of the heat capacity jump: Z d3 q T , (17.5) 1∼ ν (T Gi + D(q)q 2 )2 with T ' Tc . Let us approach the M-I transition from the metallic side. If we are far enough from transition, Gi is small and the integral in (17.5) is determined by the region of small momenta D(q)q 2 . T Gi : Gi ∼

T ν 2 D3 (q

= 0)

.

(17.6)

Two scenarios are possible: Gi becomes of the order of 1 in the metallic phase, or it remains small up to the crossover region, where finally reaches its saturation value. In the first case we can use the second asymptotic of (17.2) for D(q) and find: Gi ∼

Tc (pF `)3z−6 . EF

(17.7) 1

One can see that Gi becomes of the order of 1 at pF `M ∼ (EF /T ) 3z−6 . Comparing this value with pF `(e g ) ∼ (EF /T )1/z at the limit of the crossover region we see that for z > 3 the first scenario is realized. Concluding the first scenario discussion we see that the superconducting critical temperature goes to zero at ` = `M , still in the metallic phase, so at T = 0 a superconductornormal phase (S-N) type quantum phase transition takes place. The second scenario takes place for z < 3 when Gi remains small even at the edge of crossover region, reaching there the value Gi ∼



Tc EF

 2(3−z) z

214

 1.

(17.8)

Figure 17.1: Phase diagram in the temperature – disorder plane for a threedimensional superconductor. In the crossover region the diffusion coefficient, and hence Gi, almost do not vary. This is why the temperature of superconducting transition remains almost frozen with further increase of disorder driving the system through the Anderson transition. The abrupt growth of Gi and decrease of Tc take place when the system finally goes from the crossover to the insulating region. In the insulator phase the diffusion coefficient D(q . l−1 ) = 0 and from (17.5) one can find for Gi : r EF 1 Gi ∼ (17.9) Tc (pF `)3/2 Comparing this result with the Table 1 it is easy to see that it coincides with the Ginzburg-Levanyuk number for a zero-dimensional granule of size `  ξ(T ). Hence we see that in the second scenario the Ginzburg number reaches 1  1/3 and, respectively, Tc → 0 at pF `I ∼ ETFc , which is far enough from the MI transition point. This is why in this case one can speak about the realization at T = 0 of a superconductor-insulator (S-I) type quantum phase transition. The scale `I determines the size of the ”conducting” domains in the insulating phase, where the level spacing reaches the order of the superconducting gap. It is evident that in the domain of scale ` . `I superconductivity cannot be realized. In the vicinity of a quantum phase transition one can expect the appearance of non monotonic dependencies of the resistance on temperature and magnetic field. Indeed, starting from the zero resistance superconducting phase and increasing temperature from T = 0, the system passes through the localization region, where the resistance is high, to high temperatures where some hopping charge transfer will decrease the resistance again. The analogous speculations are applicable to the magnetic field effect: first the magnetic field ”kills” superconductivity and increases the resistance, then it destroys localization and decreases it. The phase diagram in the (T, g) plane has the form sketched in Fig. 17.1. For gI = gc − (Tc /EF )3(z−2)/κ , an S-I transition takes place at T = 0. Increasing the temperature from T = 0 in the region 0 . g . gI we remain in the insulating phase with exponential dependence of resistance on temperature. For gI . g . ge− at low temperatures 0 ≤ T < Tc (g) the system stays in the superconducting state which goes to the insulating phase at higher temperatures. In the vicinity of the Anderson transition (e g− . g . ge+ ) the 215

superconducting state goes with growth of the temperature to some crossover metal-insulator state which is characterized by a power decrease of the resistivity with the increase of temperature. Finally at gc . g the superconducting phase becomes of the BCS type and at T = Tc it goes to a metallic phase. The phase diagram in the magnetic field – disorder plane is similar to that in the (T, g) plane with the only difference that at T = 0 there is no crossover region, instead a phase transition takes place.

17.3

2D superconductors

17.3.1

Preliminaries.

As was demonstrated in Section 4, according to the conventional theory of paraconductivity, the sheet conductivity in the vicinity of the superconducting transition is given by a sum of the electron residual conductivity ge2 (Fermi part) and the conductivity of the Cooper pair fluctuations (Bose part) (see (3.18)). This expression is valid in the Ginzburg Landau region when the second term is a small correction to the first one. The width of the critical region can be determined from the requirement of equality of both contributions in (3.18)1 : 1 = 1.3Gi(2d) . (17.10) 16g In accordance with general scaling ideas one can believe that inside the fluctuation region the conductivity should obey the form:    2 σ(T ) = ge f . (17.11) Gi(2d) cr =

Concerning the scaling function f (x), we know its asymptotes in the mean field region (x  1) and just above the BKT transition [56, 57]:  1 + x−1 ,  x1 f (x) = . (17.12) −1/2 exp −b(x − xBKT ) ) , x → xBKT = −4 The BKT transition temperature TcBKT is determined by Exp.(15.1) and one can find its value by comparing the superfluid density ns from (15.1) with that found in the BCS scheme (see (2.101)): TcBKT = Tc0 (1 − 4Gi). 1

(17.13)

f (2d) = It is worth mentioning that this definition of the Ginzburg Levanyuk number (Gi 0.3 agrees with that defined from the heat capacity fluctuations (Gi(2d) = pF l ).

π 8pF l )

216

Here we assumed that the Ginzburg parameter is small, so that the BKT transition temperature does not deviate much from the mean-field BCS transition temperature Tc0 .

17.3.2

Boson mechanism of the Tc suppression.

The classical and quantum fluctuations reduce ns and therefore, suppress TcBKT . At some g = gc ∼ 1, the superfluid density ns , and simultaneously TcBKT , go to zero. In the vicinity of this critical concentration of impurities TcBKT  Tc0 . Thus a wide new window of intermediate temperatures TcBKT  T  Tc0 opens up. In this window, according the dynamical quantum scaling conjecture [97], one finds   T 2 . (17.14) σ=e ϕ TcBKT At T − TcBKT  TcBKT the Berezinski-Kosterlitz-Thouless law (17.11)-(17.12) should hold, so ϕ(x) = f (x) and is exponentially small. In the intermediate region TcBKT  T  TcBCS the duality hypothesis gives ϕ(x) = π/2. Let us derive this relation. We will start from the assumption that in the region TcBKT  T  Tc0 the conductivity is a universal function of temperature which does not depend on the pair interaction type. Being in the framework of the classical approach, let us suppose that in a weak electric field pairs move with the velocity v = F/η, where F = 2eE is the force acting on the pairs. The current density j = 2env =σE, (here n is the pair density), so one can relate the conductivity with the effective viscosity η: σ=4e2 n/η. Let us recall that we are dealing with a quantum fluid, so another, superconducting, view on the problem of its motion near the quantum phase transition exists. One can say that with the increase of Gi the role of quantum fluctuations grows too and fluctuation vortices carrying the magnetic flux quantum Φ0 = π/e are generated. With electric current flow in the system the Lorentz (Magnus) force acts on a vortex: F = jΦ0 . The electric field is equal to the rate of magnetic flux transfer, i.e. to the density of the vortex current: E =Φ0 nv vv =Φ0 nv F/ηv , where nv is the density and ηv is the viscosity of the vortex liquid. As a result E =Φ20 nv j/ηv = j/σ. So one can conclude that for vortices the velocity is proportional to the voltage, and the force is proportional to the current. For Cooper pairs (bosons) the situation is exactly the opposite. The duality hypothesis consists in the assumption that at the critical point the pair and the vortex liquid density flows are equal: nv vv = nv. 217

Comparing these quantities, expressed in terms of the conductivity from the above relations, one can find a universal value for the conductivity at the critical point 2e 2e2 = . (17.15) Φ0 π One can restrict oneself to a less strong duality hypothesis, supposing the product nη = CT δ with a universal δ exponent both for the pair and the vortex liquids, while the constant C for them is different. In this case, based on duality, is possible to demonstrate that δ = 0 and the conductivity is temperature independent up to Tc0 but its value is not universal any more and can vary from one sample to another. To conclude, let us emphasize that in the framework of the boson scenario of superconductivity suppression, the BCS critical temperature is changed insignificantly, while the “real” superconducting transition temperature TcBKT → 0. σ=

17.3.3

Fermion mechanism of Tc suppression.

Apart from the above fluctuation (boson) mechanism of the suppression of the critical temperature in the 2D case, there exists another, fermionic mechanism. The suppressed electron diffusion results in a poor dynamical screening of the Coulomb repulsion which, in turn, leads to the renormalization of the inter-electron interaction in the Cooper channel. and hence to the dependence of the critical temperature on the value of the high-temperature sheet resistivity of the film. As long as the correction to the non-renormalized BCS transition temperature Tc0 is still small, one finds [98, 99, 100]:   1 3 1 ln . (17.16) Tc = Tc0 1 − 12π 2 g Tc0 τ At small enough Tc0 this mechanism of critical temperature suppression turns out to be the principal one. The suppression of Tc down to zero in this case may happen in principle even at g  1. A renormalization group analysis gives [101] the corresponding critical value of conductance  2 1 1 gc = ln . (17.17) 2π Tc0 τ Here we should recall that the typical experimental [102] values of gc are in the region gc ∼ 1 − 2, and do not differ dramatically from the predictions of the boson duality assumption gc = π2 . If one attempts to explain 218

the suppression of Tc within the fermion mechanism, one should assume that ln Tc01 τ > 5. Then, according to Eq. (17.17), gc > 2/π and the boson mechanism is not important. On the contrary, if ln Tc01 τ < 4, then Eq. (17.16) gives a small correction for Tc even for gc = 2/π and the fermion mechanism becomes unimportant. The smallness of the critical temperature Tc compared to the Fermi energy is the cornerstone of the BCS theory of superconductivity and it is apparently satisfied even in high-Tc materials. Nevertheless it is necessary to use the theoretically large logarithmic parameter with care, if one needs ln Tc01 τ to be as large as 4.

219

Chapter 18 Fluctuations in HTS 18.1

The specifics of the D-pairing

18.2

Phase fluctuations in the underdoped phase of HTS.

Review of the existing theories: V.J.EmeryandS.A.Kivelson, N ature374, 434(1995). V.B.Geshkenbein, L.B.Iof f e, A.I.Larkin, P hys.Rev.B55, 3173(1997). P erali, Artemenko, Strinati

220

Chapter 19 Conclusions Several comments should be made in conclusion. As was mentioned in the Introduction the first ”fluctuation boom” took place at the end of 60’s beginning of 70’s, just after the discovery of the fluctuation smearing of the superconducting transition and formulation of the microscopic theory of fluctuations. The discovery of HTS reanimated this interest and, in order to account for the specifics of these layered structures with high critical temperatures, low charge carrier concentration and other particularities, considerable progress in studies of fluctuation phenomena was achieved (see for instance the conference proceedings [232, 233] and the extensive review article [76]). As it is recognized now the optimally or overdoped phases of HTS compounds present an example of a ”bad” Fermi liquid. The accounting for superconducting fluctuations is identical to including of the electron-electron interaction beyond the Fermi-liquid approximation. As a result a lot of anomalies of the normal state properties of such HTS compounds can be explained. The situation was found to be much more sophisticated in underdoped phases where the quasiparticle approach, which from we have started this Chapter, fails. In the fluctuation theory discussed above, as in modern statistical physics in general, two methods have been used mainly: they are the diagrammatic technique and the method of functional (continual) integration over the order parameter. Each of them as we have seen, has its own advantages and disadvantages and in different parts of this review we used the former or the latter. The years of the fluctuation boom coincided with the maximum development of the diagrammatic methods of many body theory in Condensed Matter Theory. This methods turns out to be extremely powerful: any physical problem, after its clear formulation and writing down the Hamiltonian, can be reduced to the summation of some classes of diagrams. The diagram221

matic technique is especially comfortable for problems containing some small parameter. In the theory of superconducting fluctuations such a small parameter exists: as we have seen, it is the Ginzburg-Levanyuk number Gi(D) which is expressed as some powers of the ratio max{Tc , τ −1 }/EF . This is why superconducting fluctuations led to the appearance of the small corrections to different physical values in a wide range of temperatures, and due to this smallness these corrections can be evaluated quantitatively. On the other hand their specific dependence on nearness to the critical temperature T − Tc permits to separate them in experiment from other effects. In those cases when fluctuations are small it is possible to restrict their summation to the ladder approximation only. The diagrammatic technique permits in a unique way to describe the quantum and classical fluctuations, the thermodynamical and transport effects In the description of thermodynamic fluctuations the method of functional integration turned out to be simpler. The ladder approximation in the diagrammatic approach is equivalent to the Gaussian approximation in functional integration. The method of functional integration turns out to be more effective too in the case of strong fluctuations, for instance, in the immediate vicinity of the phase transition. The final equations of the renormalization group carried out by means of functional integrations turn out to be equivalent to the result of the summation of the parquet diagrams series. Nevertheless the former derivation is much more simple. There is one another reason why we have tried to use both methods and even to carry out some results in both ways. In its explosive development of the last decades physics became an ”oral science” . In the process of such direct communication near a blackboard it is difficult to write and to read some cumbersome formulas. The language of diagrams is much more comprehensive: by drawing them the speaker demonstrates that this one is small and that one has to be taken into account for this and that reasons clear for the experienced listener. The success of the diagrammatic technique in some sense is similar to the success of geometry in Ancient Greece, where the science was ”oral” too. This advantage of the diagrammatic technique transforms into its disadvantage when there is no direct communication between the speaker and listener. It is difficult to learn the diagrammatic technique by a textbook on your own, when no one helps you to find the necessary insight on a complex graph. May be because of similar reasons geometry disappeared in Middle Ages when direct communications between scientists was minimal while the ”written” algebra had continued to develop. Operating with Osvald Spengler ”prosymbols” we can say that the diagrammatic technique belongs more to the Ancient Greece culture style with its ”finite body” prosymbol, while 222

functional integration, side by side with the Vikings travels to unknown lands and Leibnitz analysis of infinitesimals, is an evident modern contribution to West-European culture with its ”infinite space” prosymbol. That is why, suspecting that the modern physics in the near future can fall down to a ”New Middle Ages” period, we have carried out some results by means of functional integration instead of the diagrammatic technique.

223

Chapter 20 Acknowledgments In the first place we would like to express our deep gratitude to R.S. Thompson and T. Mishonov, who were the first readers of the manuscript and made a lot of valuable comments. We are grateful to our colleagues and friends G. Balestrino, A. Buzdin, F. Federici, V. Galitski, D. Geshkenbein, A. Koshelev, D. Livanov, Yu.N. Ovchinnikov, A. Rigamonti, G. Savona, collaboration and discussions with whom helped us in writing this work. A.A.Varlamov acknowledges the financial support of COFIN-MURST 2000 and the Scientific Exchange Programme of the University of Minnesota. A considerable part of this work was written during the visits in their frameworks. A.I. Larkin acknowledges the financial support of the NSF Grant No. DRM-0120702.

224

Bibliography [1] V.L.Ginzburg, L.D.Landau, Zh. Exp. & Teor. Fiz. 20, 1064 (1950). [2] J.Bardeen, L.N.Cooper, J.R.Schrieffer, Phys.Rev. 106, 162; 108, 1175; (1957). [3] V.L.Ginzburg, Soviet Solid State 2, 61 (1960). [4] A.P.Levanyuk, Zh. Exp. & Teor. Fiz. 36, 810 (1959). [5] L.G.Aslamazov, A.I.Larkin, Soviet Solid State 10, 875 (1968); Phys. Letters 26A, 238 (1968). [6] K.Maki, Progr. Theor. Phys. (Kyoto) 39, 897 (1968); 40, 193 (1968). [7] R.S.Thompson, Phys. Rev. B1, 327 (1970). [8] R.E.Glover, Phys. Rev. Lett. A25, 542 (1967); J. Low Temp. Phys. 6, (1970). [9] K.Maki, in Superconductivity, edited by R.D.Parks (Dekker, New York), (1969). [10] L.P.Gorkov, Zh. Exp. & Teor. Fiz. 37, 1407 (1959). [11] V.V.Schmidt, in Proceedings of the 10th International Conference on Low Temperature Physics, (VINITI, Moscow), C2, 205 (1967). [12] Tsuboi Taketumi, Suzuki Takao, Journ. of Phys. Soc. of Japan 42, (1977). [13] A.I.Larkin, Ann.Phys.(Leipzig) 8, 785 (1999). [14] A.I. Larkin, Yu. N. Ovchinnikov. Zh. Exp. & Teor. Fiz. 119, 595 (2001); cond-mat/ 0010014 (2000).

225

[15] L.D.Landau, A.A.Abrikosov, I.M.Khalatnikov, DAN USSR. 95, 437, ibid.773, ibid 1177, (1954). [16] I.T.Dyatlov, V.V. Sudakov, K.A.Ter-Martirosyan, Zh. Exp. & Teor. Fiz. 32, 767 (1957). [17] V.V.Sudakov, DAN USSR. 111, 338 (1956). [18] A.I.Larkin, D.E.Khmelnitski, Zh. Exp. & Teor. Fiz. 56, 2087 (1969). [19] M.Gel Man, F.E.Low, Phys.Rev. 95, 1300 (1954). [20] E.S.Stueckelberg, A.Peterman, Helv. Phys. Acta 24, 153 (1951). [21] C. Di Castro, G.Jona-Lasinio, Physics Lett. A29, 322 (1969). [22] K.G.Wilson, Phys.Rev. B4, 3174, ibid. 3184 (1971). [23] A.S.Patashinski, V.L.Pokrovski, The Fluctuation Theory of Phase Transitions, Nauka, Moscow (1982). [24] A.A.Varlamov, L.Yu, Phys.Rev. 44, 7078 (1991). [25] T.Tsuzuki, Journ. Low Temp. Phys. 9, 525 (1972). [26] D.E.Prober, M.R.Beasley, R.E.Schwall, Phys. Rev. B15, 5245 (1977). [27] W.E.Lawrence and S.Doniach, Proceedings of the Twelfth International Conference on Low Temperature Physics, edited by E.Kanda, Academic Press of Japan, Kyoto, p.361 (1971). [28] K.Yamaji, Phys. Lett. A38, 43 (1972). [29] L.G.Aslamazov and A.I.Larkin, Zh. Exp. & Teor. Fiz. 67, 647 (1973). [30] A.Schmid, Phys. Rev. 180, 527 (1969). [31] H.Schmidt, Z. Phys. B216, 336 (1968). [32] R.E.Prange, Phys. Rev. B1, 2349 (1970). [33] B.R.Patton, V.Ambegaokar, J.W.Wilkins, Solid St.Commun. 7, 1287, (1969). [34] J.Kurkijarvi, V.Ambegaokar and G.Eilenberger, Phys. Rev. B5, 868 (1972).

226

[35] J.P. Gollub, M.R.Beasley, R.S.Newbower, M.Tinkham, Phys. Rev.Lett. 22, 1288 (1969). [36] J.P. Gollub, M.R.Beasley, M.Tinkham, Phys. Rev.Lett. 25, 1646 (1970). [37] J.P. Gollub, M.R.Beasley, R.Callarotti, M.Tinkham, Phys. Rev. B7, 3039 (1973). [38] P.A.Lee, S.R.Shenoy, Phys. Rev.Lett. 28, 1025 (1972). [39] S.P.Farrant, C.E.Gough, Phys. Rev.Lett. 34, 943 (1975). [40] W.J.Skocpol and M.Tinkham, Rep.Prog. Phys. 38, 1094 (1975). [41] R.S.Thompson, V.Z.Kresin, Mod. Phys. Lett. B2, 1159 (1988). [42] K.F.Quader, E.Abrahams, Phys. Rev. B38, 11977 (1988). [43] W.C.Lee, R.A.Klemm, D.C.Johnson, Phys. Rev.Lett. 63, 1012 (1989). [44] P.Carretta, A.Lascialfari, A.Rigamonti, A.Rosso, A.Varlamov, Phys. Rev. B61, 12420 (2000). [45] T.M.Mishonov, E.S.Penev, Thermodynamics of Gaussian Fluctuations and paraconductivity in layered superconductors. International Journal of Modern Physics 14, 3831 (2000) (cond-mat N 0004023). [46] A.Buzdin, V.Dorin in ”Fluctuation Phenomena in High Temperature Superconductors”, ed. by M.Ausloos and A.A.Varlamov, NATO-ASI Series, Kluwer, Dordrecht (1997). [47] T.Mishonov, in ”Physics and material Science of High Temperature Superconductors ”, NATO ASI E 181, Edited by R.Kossovski et al., (Kluwer, Dordrecht, 1990). [48] A.Koshelev, Phys. Rev. B50, 506 (1994). [49] R.A.Klemm, M.R.Beasley, A.Luther, Phys. Rev. B8, 5072 (1973). [50] S.Hikami, A.Fujita, A.I. Larkin, Phys. Rev. B44, 10400 (1991). [51] C.Baraduc, A.Buzdin, J-Y. Henry, J.P.Brison and L.Puech, Physica C248, 138 (1995). [52] A.Junod, J-Y.Genoud, G.Triscone, Physica C294, 115 (1998). 227

[53] A.Lascialfari, A.Rigamonti, P.Tedesco, to be published (2001). [54] A.I.Buzdin, B.Vuyichich, Mod. Physics Letters B4, 485 (1990). [55] L.Glazman, A.Koshelev, Soviet JETP 70, 774 (1990); A.Koshelev, private communication, (1999). [56] V.S.Berezinskii, Zh. Exp. & Teor. Fiz. Letters 34, 610 (1971). [57] J.M.Kosterlitz and P.J.Thouless, J. Phys. C6, 1181 (1973). [58] D.R.Nelson, J.M.Kosterlitz, Phys. Rev.Lett. 39, 1201 (1977). [59] P.C.Hohenberg, Phys. Rev. 158, 383 (1967). [60] P.Minnhagen, Rev. Mod. Phys. 59, 1001 (1987). [61] P.Minnhagen in ”Fluctuation Phenomena in High Temperature Superconductors”, ed. by M.Ausloos and A.A.Varlamov, NATO-ASI Series, Kluwer (1997). [62] V.Vaks, A.Larkin, Soviet JETP 22, 678 (1966). [63] W.A.Little, Phys. Rev. 166, 398 (1967). [64] J.S.Langer, V.Ambegaokar, Phys. Rev. 164, 498 (1967). [65] B.I.Halperin, T.C.Lubenski, Shang-keng Ma, Phys. Rev.Lett. 32, 292 (1974). [66] A. Schmid. Phys. Konden. Mater. 5, 302 (1966). [67] C.Caroli, K. Maki, Phys. Rev. 159, 306,316 (1967). [68] E. Abrahams, T. Tsuneto, Phys. Rev. 152, 416 (1966). [69] J.W.F.Woo, E. Abrahams, Phys. Rev. 169, 407 (1968). [70] C. Di Castro, W.Young, Il Nuovo Cimento B62, 273 (1969). [71] S.Ullah, A.T.Dorsey, Phys. Rev. B44, 262 (1991). [72] L.P. Gor’kov, G.M. Eliashberg. Zh. Exp. & Teor. Fiz. 54, 612 (1968). [73] L.G.Aslamazov, A.I.Larkin, Proceedings of the ”1st European conference on condesed matter” European Phys.Soc, Geneva, Switzerland (1971). 228

[74] T.Mishonov, A.Pozheninnikova, J.Indekeu, Phys. Rev. B6?, ? (2002). [75] L.D.Landau, E.M.Lifshitz, Course of Theoretical Physics, vol.3, ”Quantum Mechanics ”, Pergamon Press, Oxford, (1978). [76] A.A.Varlamov, G.Balestrino, D.Livanov, E.Milani, Advances in Physics 48, 655 (1999). [77] C. Dekker, Phys. Today 52, No. 5, 22 (1999). [78] M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L. Balents, and P. L. McEuen, Nature (London), 397, 598 (1999). [79] R. Egger and A. O. Gogolin, Phys. Rev. Lett. 79, 5082 (1997). [80] C. L. Kane, L. Balents, and M. P. A. Fisher, Phys. Rev. Lett. 79, 5068 (1997). [81] R. Egger, Phys. Rev. Lett. 83, 5547 (1999). [82] A. Bachtold, C. Strunk, J. P. Salvetat, J. M. Bonard, L. Forro, T. Nussbaumer, and C. Schoennberger, Nature (London), 397, 673 (1999). [83] B. Reulet, A. Yu. Kasumov, M. Kosiak, R. Deblock, I. I. Khodos, Yu. B. Gorbatov, V. T. Volkov, C. Journet, and H. Bouchiat, Phys. Rev. Lett. 85, 2829 (2000). [84] Z. K. Tang, L. Zhang, N. Wang, X. X. Zhang, G. H. Wen, G. D. Li, J. N. Wang, C. T. Chan, P. Sheng, Science 292, 2462 (2001). [85] M. Kociak, A. Yu. Kasumov, S. Gueron, B. Reulet, I. I. Khodos, Yu. B. Gorbatov, V. T. Volkov, L. Vaccarini, and H. Bouchiat, Phys. Rev. Lett. 86, 2416 (2001). [86] R. Egger and A. O. Gogolin, Phys. Rev. Lett. 87, 066401 (2001). [87] W. A. Little and R. D. Parks, Phys. Rev. Lett. 9, 9 (1962). [88] A. Buzdin, A. Varlamov, accepted to Phys. Rev. Lett. (2002). [89] A. Varlamov, L. Yu, Phys. Rev. B 44, 7078 (1991). [90] K.B.Efetov, Zh. Exp. & Teor. Fiz. 76, 1781 (1979). [91] G.Blatter, V.B.Geshkenbein, A.I.Larkin, Phys. Rev. Lett. 68, 875 (1992). 229

[92] P.W.Anderson, J. Phys. Chem. Sol. 11, 26 (1959). [93] A.A.Abrikosov, L.P.Gorkov, Soviet JETP 8, 1090; 9, 220 (1959). [94] A.Kapitulnik and G.Kotliar, Phys. Rev. Lett. 54, 474 (1985). [95] P.Wollhardt and D. Wolfle, ”Anderson localization ”, edited by Y. Nagaoka and H.Fukuyama, Springer Verlag, Berlin, (1982). [96] L.N.Bulaevski, A.A.Varlamov, M.V.Sadovski, Soviet Solid State Physics 28, 997 (1986). [97] M.P.A.Fisher, Phys. Rev. Lett. 65, 923 (1990). [98] Yu.N.Ovchinnikov, Soviet JETP 36, 366 (1973). [99] S.Maekawa and H.Fukuyama, Journ. of Phys. Soc. of Japan 51,1380 (1982); H.Takagi and Y.Kuroda, Solid State Commun. 41, 643 (1982). [100] A.A.Varlamov, V.V.Dorin, Sov. JETP 64, 1159 (1986). [101] A.M.Finkelstein, Soviet JETP Letters 45, 37 (1987). [102] A.Goldman and N.Markovic, Physics Today, November 1998 and refs.inside. [103] G.M.Eliashberg, Soviet JETP 11, 696 (1960). [104] B.N.Narozhny, Soviet JETP 77, 301 (1993); Phys. Rev. B49, 6375 (1994). [105] A.A.Abrikosov, L.P.Gorkov, I.E.Dzyaloshinski, ”Methods of the quantum field theory in statistical physics ”, Dover Publications, New York (1963). [106] E.Abrahams, M.Redi, and C.Woo, Phys. Rev. B1, 218 (1970). [107] C.Di Castro, C.Castellani, R.Raimondi, A.Varlamov, Phys. Rev. B49, 10211 (1990). [108] L.B.Ioffe, A.I.Larkin, A.A.Varlamov, L.Yu, Phys. Rev. B47, 8936 (1993); Physica C 235-240, 1963 (1994). [109] K.E.Gray, D.H.Kim, Phys. Rev. Lett. 70, 1693 (1993). [110] V.V.Dorin, R.A.Klemm, A.A.Varlamov, A.I.Buzdin, D.V.Livanov, Phys. Rev. B48, 12951 (1993). 230

[111] L.G.Aslamazov, A.A.Varlamov, Journ. of Low Temp. Phys. 38, 223 (1980). [112] B.L.Altshuler, M.Yu.Reizer and A.A.Varlamov, Soviet JETP 57, 1329 (1983). [113] A.I.Larkin, D.E.Khmelnitski, Soviet Phys. Uspekhi, 136, 536 (1982). [114] Ahlers, 1975 [115] W.Breing, M.C.Chang, E.Abrahams, P.Wolfle, Phys.Rev. B 31, 7001 (1985). [116] A.A.Abrikosov, Fundamentals of the Theory of Metals, North-Holland, Elsevier, Groningen (1988). [117] R.A.Klemm and S. H.Liu, Phys. Rev. B49, 6375 (1994). [118] S. H.Liu and R.A.Klemm Phys. Rev. B48, 4080; ibid 10650 (1993). [119] B.L.Altshuler and A.G.Aronov, Electron-Electron Interaction in Disordered Conductors in Efros, A.L. and Pollak, M. (eds.), Elseiver Scientific Publishing (1985). [120] G.M.Eliashberg, Zh. Exp. & Teor. Fiz. 41, 1241 (1961). [121] R.A.Klemm, J. Low Temp. Phys. 16, 381 (1974). [122] C.Baraduc, V.Pagnon, A.Buzdin, J.Henry, C.Ayache, Phys. Lett. A166, 267 (1992). [123] S.Ami, K.Maki, Phys. Rev. B18, 4714 (1978). [124] B.R.Patton, Phys. Rev. Lett. 27, 1273 (1971). [125] J.Keller, V Korenman, Phys. Rev. Lett. 27, 1270 (1971); Phys. Rev. B5, 4367 (1972). [126] S.Yip, Phys. Rev. B41, 2612 (1990). [127] P.Carretta, A.Rigamonti, A.A.Varlamov, D.V.Livanov, Phys. Rev. B54, R9682 (1996). [128] A.Varlamov, Europhysics Letters 28, 347 (1994). [129] M. Reizer, Phys. Rev. B45, 12949 (1992). 231

[130] M.Ausloos, Ch.Laurent, Phys. Rev. B37, 611 (1988). [131] P.P.Frietas, C.C.Tsuei, T.S.Plaskett, Phys. Rev. B36, 833 (1987). [132] M.Hikita, M.Suzuki, Phys. Rev. B41, 834 (1990). [133] M.Akinaga, D.Abukay, L.Rinderer, Mod. Phys. Lett. 2, 891 (1988). [134] A.Poddar, P.Mandal, A.N.Das, B.Ghosh, P.Choudhury, Physica C159 , 231 (1989). [135] D.H.Kim, A.M.Goldman, J.H.Kang, K.E.Gray, R.T.Kampwirth, Phys. Rev. B39, 12275 (1989). [136] G.Balestrino, A.Nigro, R.Vaglio, Phys. Rev. B39, 12264 (1989). [137] G.Kumm, K.Winzer, Physica B165-166 , 1361 (1990). [138] M.R.Cimberle, C.Ferdeghini, D.Marr`e, M.Putti, S.Siri, F.Federici, A.Varlamov, Phys. Rev. B55, R14745 (1997). [139] L.Reggiani, R.Vaglio, A.Varlamov, Phys. Rev. B44, 9541 (1991). [140] G.Balestrino, E.Milani, A.Varlamov, Physica C210 , 386 (1993). [141] G.Balestrino, E.Milani, and A.A.Varlamov, JETP Letters (1995).

61, 833

[142] Y.F.Yan, P.Matl, J.M.Harris and N.P.Ong, Phys. Rev. B52, R751 (1995) ; N.P.Ong, Y.F.Yan, and J.M.Harris, Procedings of CCAST Symposium on High Tc Superconductivity and the C60 Family, Beijing, (1994). [143] W.Lang, Physica C282-287, (1997). [144] K.Nakao, K.Takamaku, K.Hashimoto, N.Koshizuka and S.Tanaka, Physica B201, 262 (1994). [145] K.Hashimoto, K.Nakao, H.Kado and N.Koshizuka, Phys. Rev. B53, 892 (1996). [146] G.Heine, W.Lang, X.L.Wang and X.Z.Wang, J. Low Temp. Phys. 105, 945 (1996). [147] T. Kimura, S. Miyasaka, H. Takagi, K. Tamasaku, H. Eisaki, S. Uchida, M. Hiroi, M. Sera, K. Kobayashi, Phys. Rev. B53, 8733 (1996). 232

[148] J.Axnas, W.Holm, Yu.Eltsev, and O.Rapp, Phys Rev. Lett. 77, 2280 (1996). [149] K.F.Quader, E.Abrahams, Phys. Rev. B38, 11977 (1988). [150] A. Wahl, D. Thopart, G. Villard, A. Maignan, V. Hardy, J.C. Soret, L.Ammor, A.Ruyter, Phys. Rev. B59, 7216 (1999). [151] M.Suzuki and M. Hikita, Phys. Rev. B39, 4756 (1989); Phys. Rev. B44, 249 (1991). [152] A.Matsuda, T.Ishii, K.Kinishita, T.Yamada, Physica C162-164, 371 (1989); A.Matsuda, T.Hirai, S.Komiyama, Solid State Commun. 68, 103 (1988). [153] J.Sugawara, H.Iwasaki, N.Kobayashi, H.Yamane and T.Hirai, Phys. Rev. B46, 14818 (1992). [154] W.Holm, M.Andersson, O.Rapp, M.A.Kulikov and I.N.Makarenko, Phys. Rev. B48, 4227 (1993). [155] W.Lang, G.Heine, W.Kula and R.Sobolewski, Phys. Rev. B51, 9180 (1995). [156] W.Holm, O.Rapp, C.N.L.Johnson and U.Helmersson, Phys. Rev. B52, 3748 (1995). [157] Yu.I.Latyshev, O.Laborde and P.Monceau, Europhysics Letters 29, 495 (1995). [158] W. Lang, Z. Phys. B97, 583 (1995). [159] T.Watanabe, A.Matsuda, Physica C263, 313 (1996). [160] W.Lang, Physica C245, 69 (1995). [161] W. Volz, F.S. Razavi,G. Quiron, H.-U. Habermeier, A.L. Solovjov, Phys. Rev. B55 6631 (1997). [162] C.Sekirnjak, W.Lang, S.Proyer, P.Schwab, Physica C243, 60 (1995). [163] S.Hikami, A.I.Larkin, Mod. Phys. Lett. B2, 693 (1988). [164] A.G.Aronov, S.Hikami and A.I.Larkin, Phys. Rev. Lett. 62, 965 (1989). [165] J.B.Bieri and K.Maki, Phys. Rev. B42, 4854 (1990). 233

[166] J.B.Bieri, K.Maki, and R.S.Thompson, Phys. Rev. B44, 4709 (1991). [167] A.S.Nygmatulin, A.Varlamov, D.Livanov, G.Balestrino and E.Milani, Phys. Rev B53 (1996). [168] A. Wahl, D. Thopart, G. Villard, A. Maignan, Ch.Simon, J.C.Soret, L.Ammor, A.Ruyter, Phys. Rev. B60, 12495 (1999). [169] A.Buzdin, A.Varlamov, Phys. Rev. B58, 14195 (1998). [170] D.Thopart, A. Wahl, Ch.Simon, J.C.Soret, L.Ammor, A.Ruyter, A.Buzdin, A.A.Varlamov, S. de Brion, Phys. Rev. B62, 9721 (2000). [171] A.A.Abrikosov, L.P.Gorkov, Soviet JETP 11, 1781 (1960). [172] R.Ikeda, T.Ohmi, and T.Tsuneto, Journ. of Phys. Soc. of Japan Letters 58, 1377 (1989); ibid. 60, 1051 (1991). [173] D.Livanov, E.Milani, G.Balestrino and C.Aruta, Phys. Rev. B55 R8701 (1997). [174] K.Maki, Phys. Rev. Lett. 30, 648 (1973). [175] I.S.Beloborodov, K.B.Efetov, A.I.Larkin, Phys. Rev. B61, 9145 (2000). [176] L.N.Bulaevski, Zh. Exp. & Teor. Fiz. 66, 2212(1974). [177] A.I.Larkin, Zh. Exp. & Teor. Fiz. Letters 31, 219 (1980). [178] E.Abrahams, J. Stat. Phys. 38, 89 (1985); J.M.B. Lopes dos Santos, E.Abrahams, Phys. Rev. B31, 172 (1985). [179] M.E.Gershenzon, V.N.Gubankov, Yu.E.Zhuravlev, Solid State Commun. 45, 87 (1983). [180] R.Rosenbaum, Phys. Rev. B32, 2190 (1985). [181] N.Mori, Jpn. J. Appl. Phys. Suppl. 26, 1339 (1987). [182] B.I.Belevtsev, Yu.F.Komnik, A.V.Fomin, J. Phys. B58, 111 (1985); Fiz. Nizk. Temp. (USSR) 10, 850 (1984). [183] V.M.Galitski, A.I.Larkin, Phys. Rev. B63, 1745X (2001). [184] H.Fukuyama, H.Ebisawa, and T.Tuzuki, Prog. Theor. Phys. 46,1028 (1971). 234

[185] E.Abrahams, R.E.Prange, and M.E.Stephen, Physica 55, 230 (1971). [186] T.Inoue, S.Miwa, K.Okamoto, M.Awano, Journ. of Phys. Soc. of Japan 46, 418 (1979). [187] A.A.Varlamov and D.V.Livanov, Soviet JETP 71, 325 (1990). [188] A.G.Aronov and A.B. Rapoport, Mod. Phys. Lett. B6,1093 (1992). [189] A.G.Aronov, S.Hikami and A.I.Larkin, Phys. Rev. B51, 3880 (1995). [190] D.V.Livanov, G.Savona, A.A.Varlamov, Phys. Rev. B62, 8675 (2000). [191] M.Randeria and A.A.Varlamov, Phys. Rev. B50, 10401 (1994). [192] I.Aleiner, L.Glazman, A.Rudin, Phys. Rev. B55, 322 (1997). [193] A.I.Larkin, Yu.A.Ovchinnikov, J. Low Temp. Phys. 10, 407 (1973). [194] A.A.Varlamov and V.V.Dorin , Soviet JETP 57, 1089 (1983). [195] M.Eschrig, D.Rainer, J.Souls, Phys.Rev. B59, 12095 (1999). [196] J.Axnaes, J. Low Temp. Phys. 117, 259 (1999). [197] R.W.Cohen, B.Abels, Phys. Rev. 168, 444 (1968). [198] B.L.Altshuler and A.G.Aronov, Soviet JETP 50, 968 (1979). [199] M.Belogolovski, O. Chernyak and A.Khachaturov, Fiz. Nizk. Temp. (USSR) 12, 630 (1986). [200] M.Park, M.S.Isaacson, J.M.Parpia, Phys. Rev. Lett. 75, 3740 (1995). [201] H.J.Tao, Farun Lu, E.L.Wolf, Physica C 282-287, 563 (1997). [202] T.Watanabe et al., Phys. Rev. Lett. 79, 2113 (1997); A.Matsuda, S.Sugita, T.Watanabe, Phys. Rev. B60, 1377 (1999); T.Watanabe, T.Fujii, A.Matsuda, Phys. Rev. Lett. 84, 5848 (2000). [203] M.Suzuki, S.Karimoto, K.Namekawa, Journ. of Phys. Soc. of Japan 67, v.3 (1998). [204] Ch.Renner, B.Revaz, J-Y. Genoud, K.Kadowaki and O.Fischer, Phys. Rev. Lett. 80, 149 (1998). [205] A.M.Cucolo, M.Cuoco, A.A.Varlamov, Phys. Rev. 59, R11675 (1999). 235

[206] M. Yu. Reizer, Phys. Rev. B48, 13703 (1993). [207] K.Maki, R.S.Thompson, Physica C162-164, 1441 (1989). [208] A.I. Larkin, Yu.N. Ovchinnikov. Zh. Exp. & Teor. Fiz. 61, 2147 (1971). [209] A.A.Varlamov, V.V.Dorin and I.E.Smolyarenko, Soviet JETP 2536 (1988).

67,

[210] A. Schmid, G. Schon. J. Low Temp. Phys. 20, 207 (1975). [211] A.I. Larkin, Yu. N. Ovchinnikov. Zh. Exp. & Teor. Fiz. 68, 1916 (1975). [212] A.I. Larkin, Yu.N. Ovchinnikov. In ”Nonequilibrium superconductivity”. Edited by D.N. Langenberg and A.I. Larkin. Elsevier Science Publishers B.V. 1986. [213] A.I. Larkin, Yu.N. Ovchinnikov. Zh. Exp. & Teor. Fiz. 73, 299 (1977). [214] F.Federici, A.A.Varlamov, JETP Letters 64, 497 (1996); Phys. Rev. B57, 6071 (1997). [215] A.Varlamov, D.Livanov, L.Reggiani, Physics Letters (1992).

A165, 369

[216] A.Varlamov, D.Livanov, F.Federici, JETP Letters 65, 182 (1997). [217] K.Maki, J. Low Temp. Phys. 14, 419 (1974). [218] M.Yu.Reizer and A.V.Sergeev, Phys. Rev. B50, 9344 (1994). [219] M.A.Howson et al., Phys. Rev. B41, 300 (1990). [220] N.V.Zavaritski, A.V.Samoilov, A.A.Yurgens, JETP Letters 55, 127 (1992). [221] S.Keshri, P.K.Barhai, Chech. J. Phys., 47, 249 (1997). [222] J.Mosqueira, J.A.Viera, F.Vidal, Physica C229, 301 (1994); J.Mosqueira, J.A.Viera, J.Masa, O.Cabeza, F.Vidal, Physica C253, 1 (1995); [223] T.Maniv, S.Alexander, J. Phys. C10, 2419 (1977). [224] T.Maniv, S.Alexander, Solid State Commun. 18, 1197 (1976). 236

[225] K.Kuboki, H.Fukuyama, Journ. of Phys. Soc. of Japan 58, 376 (1989). [226] J.Heym, J. Low Temp. Phys. 89, 859 (1992). [227] P.Mosconi, A.Rigamonti, A.Varlamov, Applied Magnetic Resonance 19, 345 (2000). [228] V. Mitrovic et al., Phys. Rev. Lett. 82, 2784 (1999). [229] H.Zimmermann, M.Mali, M.Bankey, D.Brinkmann, Physica C185 189, 1145 (1991); D. Brinkmann, Appl. Magn. Resonance 8, 67 (1995). [230] K.Gorny et al., Phys. Rev. Lett. 82, 177 (1999). [231] G.Zheng et al., Phys. Rev. B60, R9947 (1999); G.Zheng et al., Phys. Rev. Lett. 85, 405 (2000). [232] M.Ausloos and A.A.Varlamov, Editors of ”Fluctuation Phenomena in High Temperature Superconductors”, ed. by NATO-ASI Series, Kluwer, Dordrecht (1997). [233] J.Bok, G.Deutcher, D.Pavuna, S.A.Wolf, Editors of ”Gap Symmetry and Fluctuations in High Temperature Superconductors” (Plenum, London) (1998). [234] V.V.Schmidt, The Physics of Superconductors, Springer-Verlag, Berlin-Heidelberg, (1997). [235] M.Tinkham, Introduction to Superconductivity, McGraw Hill, New York, 2nd edition, (1996). [236] G.Blatter, M.V.Feigelman, V.B.Geshkenbein, A.I.Larkin, V.M.Vinokur, Vortices in high-temperature superconductors Reviews of Modern Physics, 66, N 4,(1994). [237] A.I.Larkin, Yu.N.Ovchinnikov, J.Low Temp.Phys. 34, 409(1979). [238] A.I.Larkin, Yu.N.Ovchinnikov, Soviet JETP 38, 854(1974). [239] G.Blatter, V.B.Geshkenbein, in: Handbook in Superconductivity: Physics of conventional and unconventional superconductors, Springer, (2002). [240] A.I.Larkin, Yu.N.Ovchinnikov, in ”Nonequilibrium Superconductivity” Edited by D.N.Langenberg and A.I.Larkin (Elsevier Science, NY/Amsterdam) 237

[241] R.Labusch, Cryst.Latt.Defects 1, 1 (1969). [242] A.I.Larkin, Soviet Phys. JETP 31, 784 (1970). [243] P.W.Andreson, Phys. Rev. Lett. 9, 309 (1962). [244] P.W.Andreson, Y.B.Kim, Rev. Mod.Phys. 36, 39 (1964). [245] J.Bardeen, M.J.Stephen, Phys.Rev 140, A1197 (1965). [246] L.P.Gorkov, N.B.Kopnin, Sov.Phys.JETP, 38, 195 (1973) [247] J.Bardeen, R.D.Sherman, Phys.Rev B12, 2634 (1975). [248] A.I.Larkin, Yu.N.Ovchinnikov, Soviet JETP Letters, ??? [249] L.I.Glazman, N.Ya.Fogel, Fiz.Nizk.Temp. 10, 95(Soviet J. Low Temp. Phys.,10,51) (1984) [250] A.I.Larkin, Yu.N.Ovchinnikov and A.Schmid, Physica B152, 266 (1988) [251] Y.Liu, D.B.Havilland, L.I.Glazman, A.M.Goldman, Phys. Rev. Lett. 68, 2224 (1992). [252] M.P.A.Fisher, T.A.Tokuyasu, and A.P.Young, Phys. Rev. Lett. 66, 2931 (1991). [253] G.Blatter, V.B.Geshkenbein, and V.M.Vinokur, Phys. Rev. Lett. 66, 3297 (1991). [254] B.I.Ivlev, Yu.N.Ovchinnikov, and R.S.Thompson, Phys. Rev. 44, 7023 (1991). [255] D.R:Nelson, Phys. Rev. Lett. 60, 1973 (1988). [256] T.Nattermann, Phys. Rev. Lett. 64, 2454 (1990). [257] S.E.Korshunov, Europhys- Lett. 11, 757 (1993). [258] T.Giamarchi, P. Le Doussal, Phys. Rev. Lett. 72, 1530 (1994) and Phys. Rev. B52, 1242 (1995). [259] A.I.Larkin, Yu.N.Ovchinnikov, Soviet JETP Letters, ??? ZHETP, 37,301 (1978). [260] E.H.Brandt, J.Low Temp.Phys. 26, 709(1977).

238

Pisma v

[261] K.B.Efetov, A.I.Larkin, Zh. Eksp. [Sov.P hys.JET P, 45, 1236] (1977).

Teor.

Fiz.72,

2350,

[262] D.S.Fischer, Phys. Rev. 22, 1190 (1980). [263] M.P.A.Fischer, Phys. Rev. Lett. 62, 1415 (1989). [264] D.S.Fischer, M.P.A.Fischer, D.A.Huse, Phys. Rev. B43, 130 (1991) [265] Lee, Fukuyama, 1978 [266] Gorkov, 1978; [267] Lee, Rice, 1979) [268] Larkin, D.Khmelnitski, Preprint of the Landau Institute, (1979). [269] Imry, Ma (1975) [270] L.B.Ioffe, A.I.Larkin, Soviet JETP,??? (1981). [271] J.Zittarz, J.S.Langer, Phys. Rev. 148, 741 (1966). [272] K.A.Matveev, A.I.Larkin, Phys. Rev. Lett. 78, 3749 (1997). [273] J. von Delft, D.S.Golubev, W.Tichy, A.D.Zaikin, Phys. Rev. Lett. 77, 4962 (1996). [274] A.Mastellone, G.Falci, R.Fazio, [275] S.D.Berger, B.I.Halperin, [276] [277] R.W.Richardson, Phys.Lett. 3, 277 (1963). [278] B. Janko, A. Smith, and V. Ambegaokar, Phys. Rev. B (1994).

50, 1152

[279] D.S. Golubev and A.D. Zaikin, Phys. Lett. A 195, 380 (1994). [280] P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M.H. Devoret, Phys. Rev. Lett. 70, 994 (1993). [281] M.T. Tuominen, J.M. Hergenrother, T.S. Tighe, and M. Tinkham, Phys. Rev. Lett. 69, 1997 (1992).

239

[282] D.V. Averin and K.K. Likharev, in Mesoscopic Phenomena in Solids, edited by B. Altshuler, P.A. Lee, and R.A. Webb (Elsevier, Amsterdam, 1991). [283] D.C.Ralf, C.T.Black, M.Tinkham, Phys. Rev. Lett. 74, 3241 (1995). [284] C.T.Black, D.C.Ralf, M.Tinkham, Phys. Rev. Lett. 76, 688 (1996). [285] A.J.Bray, Phys. Rev. B 9, 4752 (1974). [286] D.J.Thouless, Phys. Rev. Lett. 34, 949 (1975). [287] G.J.Ruggeri, D.J.Thouless, J. Phys. F 6, 2063 (1976). [288] S.Hikami, A.Fujita, Phys. Rev. B 41, 6379 (1990); Prog. Ther. Phys. 83, 443(1990). [289] R.Ikeda, T.Ohmi, T.Tsuneto, J. Phys. Soc. Jpn. 58, 1377(1989). [290] E.Bresin, A.Fujita, S.Hikami, Phys. Rev. Lett. 65, 1949 (1990), ibid, 65, 2921 (1990) [291] [292] A.V.Svidzinski, Inhomogeneous problems of the theory of superconductivity, Nauka, 1982. [293] A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 65, 1704 (1973) [Sov. Phys. JETP 38, 854 (1974)]. [294] L. P. Gor’kov, Zh. Eksp. Teor. Fiz. 37, 833 (1959) [Sov. Phys. – JETP 14, 628 1962)]. [295] K. Maki, Physics 1, 127 (1964). [296] E. Helfand and N. R. Werthamer, Phys. Rev. 147, 288 (1966). [297] See e.g.: S. Okuma et al., J. Phys. Soc. Jpn. 52, 3269 (1983); A. F. Hebrad and M. A. Paalanen, Phys. Rev. B 30, 4063 (1984). [298] B. Spivak and F. Zhou, Phys. Rev. Lett. 74, 2800 (1995). [299] F. Zhou and B. Spivak, Phys. Rev. Lett. 80, 5647 (1998). [300] L. B. Ioffe and A. I. Larkin, Zh. Eksp. Teor. Fiz. 81, 707 (1981) [Sov. Phys. – JETP 54, 378 (1981)]. 240

[301] M. V. Feigel’man and A. I. Larkin, Chem. Phys. 235, 107, (1998). [302] M. V. Feigel’man et al., Phys. Rev. Lett 86, 1869, (2001). [303] V. M. Galitski and A. I. Larkin, Phys. Rev. B 63, 174506 (2001). [304] K. Maki, Phys. Rev. 148, 362 (1966). [305] C. Caroli et al., Solid State Commun. 4, 17 (1966). [306] A. Schmid, Phys. Konden. Mater. 5, 302 (1966); E. Abrahams and T. Tsuneto, Phys. Rev. 159, 416 (1966); L. P. Gor’kov and G. M. Eliashberg, J. Low Temp. Phys. 2, 161, (1970). [307] B. I. Halperin and M. Lax, Phys. Rev. 148, 722 (1966); see also: B. N. Narozhny et al., Phys. Rev. B 62, 14898 (2000). [308] J. M. Kosterlitz, Phys. Rev. Lett. 37, 1577 (1976). [309] D. E. Khmel’nitskii et al., Sov. Phys. JETP 33, 458 (1971). [310] A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 61, 1221 (1971) [Sov. Phys. JETP ?, ? (1972)]. [311] A.B.Harris, J.Phys. C 7, 1671 (1974) [312] Khmelnitski [313] Lubenski [314] K.B.Efetov, Zh. Eksp. Teor. Fiz. 78, 2017 (1980) [ Soviet Phys. JETP 51, 1015 (1980)]. [315] A.B.Harris, J.Phys. C 7, 1671 (1974) [316] K.Usadel, Phys. Rev. Lett. 25, 507 (1970). [317] K.B.Efetov, ???????Oxford University Press, (2000) [318] M.G.Vavilov, P.W.Brower, V.Ambegaokar and C.W.J.Beenakker, Phys. Rev. Lett. 86, 874 (2001). [319] P.M.Ostrovsky, M.A.Skvortsov, M.V.Feigel’man, Phys. Rev. Lett. 87, 027002 (2001); JETP Letters, (2002) [320] A.Lamacraft, B.D.Simons, Phys. Rev.Lett. 85, 4783 (2001).

241

[321] A.Lamacraft, B.D.Simons, Phys. Rev. B64, 014514 (2001). [322] L.G.Aslamazov, A.I.Larkin, Yu.N.Ovhinnikov, ZHETF 55, 323 (1968) [Soviet Phys, JETP, 28, 171 (1969)]. [323] A. Schmid, Phys. Rev. Lett. 51, 1506 (1983); S. A. Bulgadaev, Sov. Phys. JETP Letters 39, 314 (1984). [324] M. V. Feigel’man and A. I. Larkin, Chem. Phys. 235, 107 (1998). [325] M. V. Feigel’man, A. I. Larkin and M. A. Skvortsov, Phys. Rev. B 61, 12361 (2000). [326] M. V. Feigel’man, A. I. Larkin and M. A. Skvortsov, Phys. Rev.Lett. 87, ? (2002). [327] L. P. Gor’kov, Zh. Eksp. Teor. Fiz. 37, 833 (1959) [Sov. Phys. – JETP 14, 628 1962)]. [328] K. Maki, Physics 1, 127 (1964). [329] E. Helfand and N. R. Werthamer, Phys. Rev. 147, 288 (1966).

242