FERMI LIQUID SUPERCONDUCTIVITY

Concepts, Equations, Applications

M. ESCHRIG, J.A. SAULS

Department of Physics & Astronomy, Northwestern University,

Evanston, IL 60208, USA

AND

H. BURKHARDT, D. RAINER

Physikalisches Institut, Universitat Bayreuth, D-95440 Bayreuth,

Germany

Lecture Notes for the 1998 NATO Advanced Study Institute on Superconductivity, Albena, Bulgaria, published

in “High-Tc Superconductors and Related Materials: Materials Science, Fundamental Properties, and Some

Future Electronic Applications” (S. L. Dreschsler and T. Mishonov, eds.), vol. 86, pp. 33, Springer, 2001. ISBN

0792368738, 9780792368731.

1. Introduction

The theory of Fermi liquid superconductivity combines two important theo-ries for correlated electrons in metals, Landau’s theory of Fermi liquids andthe BCS theory of superconductivity. In a series of papers published in 1956-58 Landau [30] argued that a strongly interacting system of Fermions canform a “Fermi-liquid state” in which the physical properties at low tempera-tures and low energies are dominated by fermionic excitations called quasi-

particles. These excitations are composite states of elementary Fermionsthat have the same charge and spin as the non-interacting Fermions, andcan be labeled by their momentum p near a Fermi surface (defined by Fermimomenta pf ). Landau further argued that an ensemble of quasiparticles isdescribed by a classical distribution function in phase space, f(p,R; t),and that the low-energy properties of such a system are governed by aclassical transport equation, which we refer to as the Boltzmann-Landau

transport equation. A significant feature of Landau’s theory is that quasi-particles are well defined excitations at low energy, yet their interactionsare generally large and can never be neglected. These interactions lead tointernal forces acting on the quasiparticles, damping of quasiparticles, andgive rise to many of the unique signatures of strongly correlated Fermi liq-uids. The quasiparticle interactions are parametrized by phenomenologicalinteraction functions that determine the interaction energy, internal forcesbetween quasiparticles, and damping terms.

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The modern theory of superconductivity started in 1957 with the publica-tion by Bardeen, Cooper and Schrieffer [8] on the Theory of Superconduc-

tivity, wildly known as the BCS theory. Theorists on both the ‘east’ and‘west’ established within a few years basically a complete “standard theoryof superconductivity” which was finally comprehensively reviewed by lead-ing western experts in Superconductivity edited by Parks in 1969 [44].At about the time Parks’ books were edited two papers by Eilenberger[19] and Larkin & Ovchinnikov [31] were published, which demonstrated,independently, that the complete standard theory of (equilibrium) super-conductivity can be formulated in terms of a quasiclassical transport equa-tion. Somewhat later this result was generalized to non-equilibrium condi-tions by Eliashberg [21] and Larkin & Ovchinnikov [32]. We consider thistheory the generalization of Landau’s theory of normal Fermi liquids tothe superconducting states of metals or superfluid states of liquid 3He. Itcombines Landau’s semiclassical transport equations for quasiparticles withthe concepts of pairing and particle-hole coherence that are the basis of theBardeen, Cooper and Schrieffer theory. We will call this theory alternativelythe ”quasiclassical theory”, as it was coined by Larkin and Ovchinnikov, orthe ”Fermi liquid theory of superconductivity”. Several limits of the quasi-classical theory were known before its general formulation was establishedby Eilenberger, Larkin, Ovchinnikov and Eliashberg. De Gennes [17] hadshown that equilibrium superconducting phenomena for T near Tc couldbe described in terms of classical correlation functions, which may be cal-culated from a Boltzmann equation [38]. In 1964 Andreev [4] developeda set of equations (Andreev equations) which are equivalent to the cleanlimit equations of the quasiclassical theory. In the mid 60’s Leggett [36]discussed in a series of papers the effects of Landau’s interactions on the re-sponse functions for a superconductor. Bardeen, Rickayzen and Tewordt [9]introduced in 1959 a semi-classical transport equation, which correspondsto the long-wavelength, low-frequency limit of the quasiclassical dynam-ical equations. The linear response theory obtained by ξ-integrating theKubo response function [1] is also equivalent to the linearized quasiclas-sical transport equation [48]. These early theories are predecessors of thecomplete quasiclassical theory which provides a full description of super-conducting phenomena ranging from inhomogeneous equilibrium states tosuperconducting phenomena far from equilibrium. The theory is valid at alltemperatures and excitation fields of interest, and it covers clean and dirtysystems as well as metals with strong electron-phonon or electron-electroninteractions.In section 2 we introduce the quasiclassical propagators, quasiclassical self-energies, as well as the set of quasiclassical equations. We briefly discusstheir foundations and interpretation. The quasiclassical propagators are the

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generalization of Landau’s distribution functions to the superconductingstate, and the quasiclassical self-energies describe the effects of quasiparticleinteractions, quasiparticle-phonon interactions and quasiparticle-impurityscattering. For further details of the quasiclassical theory and its derivationwe refer to various review articles [34, 18, 49, 46] and references therein.In sections 3 and 4 we present two recent applications of the quasiclassicaltheory. Section 3 discusses the Josephson effect and charge transport acrossa junction of differently oriented d-wave superconductors, while section 4presents calculations of the electromagnetic response of a two-dimensional”pancake” vortex in a layered superconductor.

2. Quasiclassical theory

2.1. QUASICLASSICAL TRANSPORT EQUATIONS

Derivations of the quasiclassical equations were given by Eilenberger, Larkin,Ovchinnikov and Eliashberg in their original papers [19, 31, 21, 32], by She-lankov [51], and in several review articles [34, 18, 49, 47]. All derivationsstart from a formulation of the theory of superconductivity in terms ofGreen’s functions (G), self-energies (Σ), and Dyson’s equation,

G = G0 + G0ΣG . (1)

We use here the notation of Larkin & Ovchinnikov [34] who introducedNambu-Keldysh matrix Green’s functions and self-energies (indicated bya hacek accent), whose matrix structure comprises Nambu’s particle-holeindex and Keldysh’s doubled-time index [28]. Nambu’s particle-hole matrixstructure is essential for BCS superconductivity since off-diagonal termsin the particle-hole index indicate particle-hole coherence due to pairing.Keldysh’s doubled-time index, on the other hand, is a very convenient toolfor describing many-body systems out of equilibrium. Spin dependent phe-nomena require an additional matrix index for spin ↑ and ↓. The Green’sfunctions depend on two positions (x1, x2) and two times (t1, t2) or, alterna-tively, on an average position, R = (x1+x2)/2, average time t = (t1+t2)/2and, after Fourier transforming in x1 − x2 and t1 − t2, on a momentum p

and energy ǫ.1 All the physical information of interest is contained in theGreen’s functions, whose calculation requires i) an evaluation of the self-energies and ii) the solution of Dyson’s equation. The Fermi-liquid theoryof superconductivity provides a scheme for calculating self-energies andGreen’s functions consistently by an expansion in the small parametersTc/Ef , 1/kfξ0, 1/kf ℓ, ω/Ef , q/kf , ωD/Ef , where Ef and kf are Fermi

1We set in this article h = kB = 1. The charge of an electron is e < 0.

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energy and momentum, Tc and ξ0 the superconducting transition temper-ature and coherence length, ℓ the electron mean free path, ωD the Debyefrequency, and ω, q are typical frequencies and wave-vectors of externalperturbations, such as electromagnetic fields, ultra-sound or temperaturevariations. We follow [49] and assign to these dimensionless expansion pa-rameters the order of magnitude ”small”. To leading order in small thefull Green’s functions, G(p,R; ǫ, t), can be replaced by the ”ξ-integrated”Green’s functions (quasiclassical propagators), g(pf ,R; ǫ, t), and the fullself-energies, Σ(p,R; ǫ, t), by the quasiclassical self-energies, σ(pf ,R; ǫ, t).The quasiclassical propagator describes the state at position R and timet, of quasiparticles with energy ǫ (measured from the Fermi energy) andmomenta p near the point pf on the Fermi surface. The reduction to theFermi surface (p → pf ) is an essential step. It establishes the bridge be-tween full quantum theory and quasiclassical theory.The dynamical equations for the quasiclassical propagators are obtainedfrom Dyson’s equation for the full Green’s functions (1), and one finds (see,e.g., the review [46] and references therein)

[ǫτ3 − σ − v , g]⊗ + ivf ·∇g = 0 , (2)

g ⊗ g = −π21 , (3)

where the ⊗-product is defined by

A⊗ B(ǫ, t) = ei2(∂A

ǫ ∂Bt −∂B

ǫ ∂At ) A(ǫ, t)B(ǫ, t) , (4)

and the commutator is given by

[A, B]⊗ = A⊗ B − B ⊗ A . (5)

Eq. (2) turns in the normal-state limit into Landau’s classical transportequation for quasiparticles. Hence, one should consider eq. (2), which hasthe form of a transport equation for matrices, as a generalization of Lan-dau’s transport equation to the superconducting state. This interpretationbecomes more transparent if one drops the Keldysh-matrix notation, andwrites down the equations for the three components gR,A,K (advanced, re-tarded and Keldysh-type) of the Keldysh matrix propagator separately.

[

ǫτ3 − v(pf ,R; t)− σR,A(pf ,R; ǫ, t), gR,A(pf ,R; ǫ, t)]

◦

+ivf ·∇gR,A(pf ,R; ǫ, t) = 0 , (6)

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(

ǫτ3 − v(pf ,R; t)− σR(pf ,R; ǫ, t))

◦ gK(pf ,R; ǫ, t) (7)

−gK(pf ,R; ǫ, t) ◦(

ǫτ3 − v(pf ,R; t)− σA(pf ,R; ǫ, t))

−σK(pf ,R; ǫ, t) ◦ gA(pf ,R; ǫ, t) + gR(p,f R; ǫ, t) ◦ σK(pf ,R; ǫ, t)

+ivf ·∇gK(pf ,R; ǫ, t) = 0 .

The ◦-product stands here the following operation in the energy-time vari-ables (the superscripts a, b refer to derivatives of a and b respectively).

[a ◦ b](pf ,R; ǫ, t) = ei2(∂

aǫ ∂

bt−∂a

t ∂bǫ)a(pf ,R; ǫ, t)b(pf ,R; ǫ, t) , (8)

and the commutator [a, b]◦ stands for a ◦ b− b ◦ a. An important additionalset of equations are the normalization conditions

gR(pf ,R; ǫ, t) ◦ gK(pf ,R; ǫ, t) + gK(pf ,R; ǫ, t) ◦ gA(pf ,R; ǫ, t) = 0, (9)

gR,A(pf ,R; ǫ, t) ◦ gR,A(pf ,R; ǫ, t) = −π21. (10)

The normalization condition was first derived by Eilenberger [19] for su-perconductors in equilibrium. An alternative, more physical derivation wasgiven by Shelankov [51]. The quasiclassical transport equations (6,7) sup-plemented by the normalization conditions, Eqs. (9-10), are the fundamen-tal equations of the Fermi-liquid theory of superconductivity. The varioussteps and simplifications done to transform Dyson’s equations into trans-port equations are in accordance with a systematic expansion to leadingorders in small.The matrix structure of the quasiclassical propagators describes the quan-tum-mechanical internal degrees of freedom of electrons and holes. Theinternal degrees of freedom are the spin (s=1/2) and the particle-hole de-gree of freedom. The latter is of fundamental importance for superconduc-tivity. In the normal state one has an incoherent mixture of particle andhole excitations, whereas the superconducting state is characterized by theexistence of quantum coherence between particles and holes. This coher-ence is the origin of persistent currents, Josephson effects, Andreev scatter-ing, flux quantization, and other non-classical superconducting effects. Thequasiclassical propagators, in particular the combination gK − (gR − gA),are intimately related to the quantum-mechanical density matrices whichdescribe the quantum-statistical state of the internal degrees of freedom.Non-vanishing off-diagonal elements in the particle-hole density matrix in-dicate superconductivity, and the onset of non-vanishing off-diagonal ele-ments marks the superconducting transition. One reason for the increasedcomplexity of the transport equations in the superconducting state (3 cou-pled transport equations for the 3 matrix distribution functions gR,A,K) in

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( small )0 :

( small )1 :

( small )2 :

+ +

+ · · · +

+ + · · ·

Figure 1. The figure shows the leading order self-energy diagrams that contribute tothe Fermi liquid theory of superconductivity. The diagram in the first row representsan effective potential which affects the shape of the quasiparticle Fermi surface and themass of a quasiparticle. The diagrams in the second row are of 1st order in the expan-sion parameter and represent Landau’s quasiparticle interactions and the quasiparticlepairing interaction (first diagram), the Migdal-Eliashberg self-energy which leads to massenhancement and damping of quasiparticles due to their coupling to phonons (second di-agram), quasiparticle-impurity scattering in leading order in 1/kf ℓ (the third and fourthdiagram are representatives of an infinite series of diagrams whose sum is the T-matrixfor quasiparticle scattering at an impurity, multiplied by the impurity concentration).The diagrams in the third row describe quasiparticle-quasiparticle collisions (first dia-gram) and a small correction to the quasiparticle-phonon interaction term of Migdal andEliashberg.

comparison with the normal state of the Fermi liquid (1 transport equationfor a scalar distribution function) is the fact that the quasiparticle statesin the normal state are inert to the perturbations and to changes in theoccupation of quasiparticle states. Hence, the only dynamical degrees offreedom are here the occupation probabilities of a quasiparticle state. Onthe other hand, quasiparticle states of energy ǫ <∼ ∆ are coherent mixturesof particle and hole states, and react sensitively to external as well as inter-nal forces in the superconducting state. Thus the quasiclassical transportequations describe the coupled dynamics of the quasiparticle states andtheir occupation probability (distribution functions). It is only in limitingcases possible to decouple to some degree the dynamics of states and oc-cupation. An important such case, which admits using scalar distributionfunctions, are low frequency (ω ≪ ∆) phenomena in superconductors, asfirst discussed by Betbeder-Matibet and Nozieres [13] (see also [49]).To conclude the section on the general quasiclassical theory we discussthe quasiclassical self-energies. The set of leading order self-energies ar-

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ranged according to their power in the expansion parameter small is shownin Fig. 2.1. The shaded spheres with connections to phonon lines (wiggly),quasiparticle lines (smooth) or impurity lines (dashed) represent interactionvertices describing quasiparticle-quasiparticle interactions, quasiparticle-phonon interactions in Migdal-Eliashberg approximation [42, 20], and quasi-particle-impurity scattering. The full lines represent quasiparticle propaga-tors (smooth) and phonon propagators (wiggly). The diagrams shown inFig. 2.1 comprise the interaction processes taken into account in the stan-dard theory of superconductivity. We note that the interaction vertices arein leading order inert, i.e. independent of temperature, not affected by per-turbations or changes in quasiparticle occupations and, in particular, notaffected by the superconducting transition. These vertices are phenomeno-logical parameters in the Fermi liquid theory of superconductivity whichmust be taken from experiment.

3. The Effect of Interface Roughness on the Josephson Current

In this section we present an application of the quasiclassical theory toJosephson tunneling in d-wave superconductors. Tunneling experiments insuperconductors probe, in general, the quasiparticle states and the pairingamplitude at the tunneling contact. Such experiments give valuable infor-mation on the quasiparticle density of states and the symmetry of the orderparameter. For superconductors with a single isotropic gap parameter thesuperconducting state is in most cases not distorted by the tunneling con-tact, and one measures basically bulk properties of the superconductor. Atypical example is the Josephson current of an S−I−S tunnel junction. TheJosephson current for traditional s-wave superconductors is well describedby the universal formula of Ambegaokar and Baratoff [2],

IJ(ψ, T ) = Ic(T ) sin(ψ) =π

2|e|RN∆(T ) tanh

(∆(T )

2kBT

)

sin(ψ) , (11)

which holds for isotropic BCS superconductors and a weakly transparent,non-magnetic tunneling barrier of arbitrary degree of roughness. It de-scribes the dependence of the Josephson current on the temperature T ,and the phase difference ψ across the junction. This current-phase relationdepends only on two parameters, the bulk energy gap ∆(T ) which char-acterizes the superconductors, and the normal state resistance RN whichcharacterizes the barrier.The universality of the Ambegaokar-Baratoff relation is lost for junctionsinvolving anisotropic superconductors, in particular superconductors withstrong anisotropies and sign changes of the gap function on the Fermi sur-face. The Josephson current depends in these cases on the orientation of

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the crystals with respect to the tunneling barrier, and on the quality ofthe barrier. Ideal barriers, which conserve parallel momentum, will give,in general, a different Josephson current than rough barriers of the sameresistance RN . The origin of these non-universal effects is scattering atthe tunneling barrier, which may lead to a depletion of the order param-eter in its vicinity [3], to new quasiparticle states bound to the barrier[14, 25, 40, 15, 53], and eventually to spontaneous breaking of time reversalsymmetry ([40, 52, 23] and references therein). All these special features,which reflect the anisotropy of the gap function, react sensitively to barrierroughness which smears out the anisotropy, broadens the bound state spec-trum, changes the order parameter near the barrier, and thus influences theJosephson currents and the tunneling spectra.The quasiclassical theory of superconductivity is particularly well suitedfor studying the effects of roughness of tunneling barriers, surfaces, or in-terfaces. A barrier between two superconducting electrodes is modeled inthe quasiclassical theory by a thin (atomic size) interface which may reflectelectrons with a certain probability or transmit them across the interface.Reflection and transmission may be ideal or to some degree diffuse. We focushere on the effects of interface roughness on the Josephson critical current.In the following we present our model for rough interfaces which combinesZaitsev’s model [54] for ideal (no roughness) interfaces with Ovchinnikov’smodel [43] for a rough surface. We then discuss briefly the interface resis-tance in the normal state and present analytical results for the Josephsoncurrent across a weakly transparent interface. Finally we present our numer-ical results for d-wave pairing which demonstrate the strong dependence ofthe Josephson current of junctions with unconventional superconductors onthe junction quality, and discuss, in particular, the Ambegaokar-Baratoffrelation for the Josephson current of weakly coupled junctions of d-wavesuperconductors.

3.1. MODEL FOR A ROUGH BARRIER

Our quasiclassical model of rough barriers combines two models known fromthe literature, i.e., the ideal interface first discussed by A.V.Zaitsev [54] andthe “rough layer” introduced by Yu.N. Ovchinnikov [43]. The ideal part ofthe interface determines in this model the reflectivity and transmittivity ofthe interface, whereas the rough layers lead to some degree of randomnessin the directions of the reflected and transmitted quasiparticles. Zaitsev’sboundary condition relates the propagators gl,r(pf ,RI ; ǫ, t) on any set offour trajectories with the same momentum parallel to the interface (seeFig.1). These trajectories are characterized by the four momenta, pl

f in,

plf out, p

rf in, p

rf out, at the left (superscript l) and right (superscript r) sides

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S I S’

x=ÿ1 x=1x=0

p lf out

p lf in

pf outr

pf inr

Figure 2. Sketch of our interface model. The ideal interface (I) separates two differentsuperconductors (S,S′), whose Fermi surfaces (curved lines) are shown on the right andleft sides. Parallel momentum and energy are conserved in a transmission and reflectionprocess which fixes, as shown, the four momenta, pl,rf in,out. The interface is coated by

rough layers (dotted area).

of a point RI on the interface (see Fig.1). Zaitsev’s relations are

d l + d r = 0 , (12)

d r (s r)2 = iπ1−R

1 +R

[

s l, s r(

1−i

2πd r)]

, (13)

where R = R(plf in) = R(pr

f in) is the reflection parameter, and d s, s s (s =l, r) are defined as the difference and sum of propagators of quasiparticles onincoming trajectories before reflection (incoming velocity, momentum ps

f in)and on outgoing trajectories after reflection (outgoing velocity, momentumpsf out),

d s(psf in,RI ; ǫ, t) = g s(ps

f out,RI ; ǫ, t)− g s(psf in,RI ; ǫ, t) , (14)

s s(psf in,RI ; ǫ, t) = g s(ps

f out,RI ; ǫ, t) + g s(psf in,RI ; ǫ, t) . (15)

We follow here the notation of A.I. Larkin & Yu.N. Ovchinnikov [34] in-troduced in section 1, and use Nambu-Keldysh matrix propagators, indi-cated by a “hacek”. Parallel momentum is conserved at an ideal interface,

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which fixes, together with energy conservation, the kinematics of interfacescattering as shown in Fig.1. In the normal state a quasiparticle movingtowards the point RI at the interface with a momentum ps

f in can ei-ther be reflected with probability R(ps

f in) or transmitted with probabilityT (ps

f in) = 1−R(psf in) to the outgoing trajectory on the other side of the

interface. Two more channels open in the superconducting state becauseof Andreev scattering. An incoming quasiparticle can be “retroreflected”(velocity reversal) into its incoming trajectory or transmitted into the in-coming trajectory on the other side of the interface. The function R(ps

f in)is considered here a phenomenological model describing the reflection andtransmission properties of the ideal interface without roughness.We model the roughness of an interface by coating it on both sides by athin layer of thickness δ of a strongly disordered metal (see Fig.1). These“rough layers” are distinguished from the adjacent metals only by their veryshort mean free path ℓ. Only the ratio δ/ℓ matters in the limit δ, ℓ → 0,and defines the roughness parameter ρ = δ/ℓ. We include regular elasticscattering as well as pair-breaking scattering (e.g., spin-flip scattering, in-elastic scattering), and introduce two roughness parameters, ρ0 (regularscattering) and ρin (pair-breaking scattering). Vanishing ρ’s correspond toa perfect interface, and ρ = ∞ to a diffuse interface, i.e., a reflected ortransmitted quasiparticle looses its memory of the incoming direction. Inthe quasiclassical equations for the rough layers all but the scattering termscan be dropped, and one obtains the following transport equation in therough layer.

−

[

ρin2π

〈g l,r3 〉± +

ρ02π

〈g l,r〉±, gl,r]

+ i v l,r⊥ ∂xg

l,r = 0 , (16)

where the superscripts l and r distinguish the metals on the left and rightsides of the interface, g3 is the Nambu-Keldysh matrix obtained from gby keeping only the τ3-components of its Nambu submatrices, x is thespatial coordinate perpendicular to the interface, measured in units of δ,

and the dimensionless velocity v l,r⊥ is the perpendicular component of the

quasiparticle velocity normalized by an averaged Fermi velocity, v l,r =

vl,rf /

√

〈|v l,rf |2〉. The left and right rough layers are located between x = −1

and x = +1, and are separated by Zaitsev’s ideal interface at x = 0 (seeFig.1). In general, the quasiclassical propagators in the rough layers dependon pf , RI , ǫ, t, and the spatial variable x which specifies the positionin the infinitesimally thin rough layer. We use Ovchinnikov’s model [43]for the scattering processes in the rough layers. In this model scatteringpreserves the sign of v⊥; particles moving towards the interface are notscattered into outgoing directions and vice versa. This “conservation ofdirection” is indicated in (16) by the indices ± in the scattering terms,

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which stand for averaging over the momenta corresponding to v⊥> 0 andv⊥<0, respectively. We assume equal scattering probability into all statescompatible with Ovchinnikov’s conservation of direction. A reversal of thevelocity may only happen at Zaitsev’s interface which separates the tworough layers.In order to obtain the current-phase relationship of the junction we solve thequasiclassical transport equations in both superconductors and in the roughlayers. The solution in the left superconductor has to match continuouslythe solution at x = −1 in the left rough layer and, equivalently, the solutionin the right superconductor has to match the solution in the right roughlayer at x = 1. At x = 0 the solutions in the rough layers are matched viaZaitsev’s conditions (12, 13). The order parameter has to be determinedself-consistently. To get a finite current we fix the phase difference of theorder parameter across the junction,

ψ(RI) = ψ l(RI)− ψ r(RI) , (17)

where the phases ψl and ψr at the interface point RI are determined by

∆l,r(pl,rf ,RI) = ∆l,r(pl,r

f ) exp (ψl,r(RI)) . (18)

Here, ∆l,r(pl,rf ) are convenient reference order parameters on the left and

right sides. The reference order parameter can be taken real in the casesdiscussed in this paper. Note that the phase difference is measured directlyat the interface at point RI . The current density is obtained by standardformulas of the quasiclassical theory, i.e.,

j(R, t) =

∫

dǫ

4πi

∫

d2pf

(2π)3 | vf |evf tr(τ3g

K(pf ,R; ǫ, t)) (19)

in the Keldysh formulation, and

j(R, t) = T∑

ǫn

∫

d2pf

(2π)3 | vf |evf tr(τ3g

M (pf ,R; ǫn)) (20)

in the Matsubara formulation. The critical Josephson current is obtained asthe maximum supercurrent across the junction in equilibrium. The Fermivelocity, vf (pf ), is a function of the momentum. This function has the sym-metry of the lattice, and will be understood here as a material parameter,to be taken from theoretical models or from experiment, if available.This completes our brief review of the quasiclassical equations for a roughinterface.

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3.2. NORMAL STATE RESISTANCE

In this section we calculate the normal state resistance of our model inter-face, and show that interface roughness does not affect the interface resis-tance. This is a unique feature of Ovchinnikov’s model for a rough layerwhich is very convenient for studying the effects of interface roughness atfixed interface resistance.In the normal state the retarded and advanced propagators gR and gA aretrivial, and given by gR,A = ∓iπτ3, and the Keldysh propagator can beparameterized in terms of a distribution function Φ(pf ,R; ǫ, t),

gK(pf ,R; ǫ, t) = 4πi

(

Φ(pf ,R; ǫ, t) 00 Φ(−pf ,R;−ǫ, t)

)

. (21)

Given Φ one can calculate the current density from

j(R, t) = 2

∫

dǫ

∫

d2pf

(2π)3 | vf |evfΦ(pf ,R; ǫ, t) . (22)

The distribution function in the rough layers can be calculated by solvingthe Landau-Boltzmann transport equation,

v l,r⊥ ∂xΦ

l,r + ρtot(

Φl,r − 〈Φl,r〉±

)

= 0 , (23)

which follows by taking the normal state limit of the general quasiclassicaltransport equation (16). In this normal state limit the scattering rates forelastic and inelastic scattering can be added up to a total scattering rateρtot = ρ0 + ρin. Zaitsev’s boundary conditions (12, 13) at an ideal interfaceturn in the normal state into the following classical boundary conditionsfor the distribution function at the left and right sides of the boundary atx = 0,

Φl(plf out) = R(pl

f in) Φl(pl

f in) + T (prf in) Φ

r(prf in) , (24)

Φr(prf out) = R(pr

f in) Φr(pr

f in) + T (plf in) Φ

l(plf in) . (25)

In addition, the solutions of (23) in the two rough layers have to be matchedat x = ±1 to the physical distribution functions on the left and right sidesof the interface.In order to calculate the interface resistance we apply a voltage V to ourjunction, solve the transport equation (23) subject to the proper matchingconditions at x = 0 and x = ±1, and calculate the current from (22).We assume that the junction is formed by very good conductors separatedby a weakly transparent interface, such that the total resistance of thejunction is dominated by interface scattering, and the voltage drop is to a

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good approximation localized at the interface. This leads to a difference eVof the electrochemical potentials in the two conductors, which drives thecurrent. It is assumed that both conductors are in thermal equilibrium faraway from the interface. This fixes the incoming parts of the distributionfunction on the left and right sides of the rough interface,

Φl(plf in,R; ǫ) = f (ǫ) , Φr(pr

f in,R; ǫ) = f (ǫ+ eV ) , (26)

where f is the Fermi function at temperature T . The outgoing excitationsare not in equilibrium, and their distribution function must in general becalculated from the transport equation (23). This calculation can be skippedin Ovchinnikov’s model. In this model the number of incoming and out-going excitations are conserved separately, and the incoming part of theequilibrium distribution function on the left (right) side of the interface aresolutions of the transport equation (23) at x < 0 (x > 0). This pins downthe total current in terms of the incoming parts of the thermal distributionfunctions far away from the interface. The distribution functions at x = −0(x = +0) are given by the sum of the incoming equilibrium distributionfunction on the left (right) side, the reflected distribution function, and thetransmitted distribution function from the other side. On the right side ofthe interface we have

Φr(prf in, x = +0; ǫ) = f(ǫ+ eV ) , (27)

Φr(prf out, x = +0; ǫ) = R(pr

f in)f(ǫ+ eV ) + T (plf in)f(ǫ) , (28)

and the equivalent formulas hold on the left side.The current density is constant, and can be calculated at any convenientpoint of the junction. We pick x = +0, and obtain by inserting the distri-bution functions (27,28) into (22)

j⊥ = 2e

∫

dǫ(

∫ d2prf out

(2π)3 | vrf |

vrf⊥[R(prf in)f(ǫ+ eV ) + T (pl

f in)f(ǫ)]

+

∫ d2prf in

(2π)3 | vrf |

vrf⊥f(ǫ+ eV ))

. (29)

We now use the relations R(prf in) = 1 − T (pr

f in) , T (plf in) = T (pr

f in),and

d2prf in

(2π)3 | vrf (p

rf in) |

vrf⊥(prf in) = −

d2prf out

(2π)3 | vrf (p

rf out) |

vrf⊥(prf out) , (30)

and obtain

j⊥ =1

RNAV =

(

2e2∫ d2pr

f out

(2π)3 | vrf |

vrf⊥T (plf in)

)

V . (31)

14

Equation (30) is a geometric relation which follows from conservation ofparallel momentum. For clarity we write down in (30) explicitly the mo-mentum dependence of the velocity. The factor in brackets in (31) is theinterface conductance per area, 1/(RNA), where A is the area of the inter-face. The conductance is obviously independent of the scattering rate ρtot,i.e., independent of the roughness of the interface.

3.3. WEAKLY TRANSPARENT INTERFACE

Zaitsev’s boundary conditions (12, 13) at x = 0 can be simplified sub-stantially for a weakly transparent interface (T ≪ 1). We expand theMatsubara propagator gM to first order in the transmission amplitude, T :gM = gM(0) + gM(1) +O(T 2). The solution gM(0) corresponds to the equilibrium

solution of a non-transparent interface. Both sides are decoupled in thiscase, and Zaitsev’s boundary conditions turn into dM l

(0) = dM r(0) = 0. Hence,

the 0th order propagators on the left and right sides of the interface de-scribe superconductors bounded by a rough, non-transparent surface. Thefirst order correction to the difference of the incoming and reflected prop-agator, dM(1) = gM(1)(p

outf , x = 0; ǫn) − gM(1)(p

inf , x = 0; ǫn), can be obtained

directly from (13) in terms of the 0th order propagators, i.e. without havingto solve the quasiclassical transport equation for gM(1). Expansion of (13) to

first order in the transparency T [6] leads to

dM r(1) = −

i

8πT[

sM l(0) , s

M r(0)

]

. (32)

Equation (30) implies that the current density (20) can be written in terms

of the difference, dM , of outgoing and incoming propagators alone, anddoes not depend on the sum. Hence, one can calculate from Eq. (32) theJosephson current in terms of the 0th order quantities sM l

(0) and sM r(0) . The

straight forward calculation leads to the following formula for the current-phase relation.

jJ(ψ, T ) =2e

πkBT

∑

ǫn

∫ d2prf out

(2π)3 | vrf |

vrf⊥T (prf )

×(

[fM l1(0)(p

lf ; ǫn)f

M r1(0)(p

rf ; ǫn) + fM l

2(0)(plf ; ǫn)f

M r2(0)(p

rf ; ǫn)] sin(ψ)

+[fM l1(0)(p

lf ; ǫn)f

M r2(0)(p

rf ; ǫn)− fM l

2(0)(plf ; ǫn)f

M r1(0)(p

rf ; ǫn)] cos(ψ)

)

,(33)

where fM1 and fM2 are the off-diagonal components of the propagatorgM (pf , x = 0; ǫn), defined via gM = fM1 τ1 + fM2 τ2 + gM τ3. In the case

of real reference order parameters ∆l,r (this is a possible choice for an

15

s-wave or d-wave superconductor but, e.g., not for d+ i s) the term propor-tional cos(ψ) vanishes, and we get the standard sinusoidal behavior of theJosephson current jJ [12].We note that the Josephson current vanishes in the limit of strong inelasticroughness, ρin → ∞. The reason is that the off-diagonal components ofgM decay to zero in the dirty layer. Hence, one has gM = −iπsgn(ǫn)τ3 atx = 0, and the current is zero.

3.4. CALCULATION OF THE JOSEPHSON CURRENT

αrα l

−

+−

−

+−

++

Figure 3. The figure introduces the tilt angles, αl and αr, which describe the orientationsof the d-wave order parameters on the left and right sides of the interface.

We are interested here in the dc Josephson current of a junction of twod-wave superconductors, and study first the case of a weakly transparentinterface (T ≪ 1) introduced in Sect.3.3. The results are then comparedwith numerical calculations for a finite T .We present selected numerical results for the Josephson current in S−I−S ′

junctions of layered, tetragonal d-wave superconductors. Of special interestare the effects of changing the orientation of the two crystalline super-conductors [39], and we consider for this reason junctions with interfacesparallel to the c-direction, and the crystals rotated in the a-b plane by thetilt angles αl and αr (see Fig.2). In order to calculate the current we fix thephase difference ψ across the interface, solve the quasiclassical equations,and obtain the order parameter ∆(pf ,R), and the Matsubara propagatorgM (pf ,R; ǫn). The supercurrent across the interface can then be calculatedfrom (20).We neglect the coupling in c-direction, which seems a good first approxima-tion for layered cuprate superconductors, and model the conduction elec-trons of our d-wave superconductors by a cylindrical Fermi surface withradius pf , and an isotropic Fermi velocity vf in the a-b plane. The pairing

16

0.0 0.2 0.4 0.6 0.8 1.0temperature T/Tc

−1.0

−0.5

0.0

0.5

criti

cal c

urre

nt j c(

T)/j

L *1

02

45o

0o

25o

10o

20o

15o

(a)

0.0 0.2 0.4 0.6 0.8 1.0temperature T/Tc

−0.2

0.0

0.2

criti

cal c

urre

nt j c(

T)/j

L

45o

0o

25o

(b)

20o

15o

10o

Figure 4. Temperature dependence of the Josephson critical current jc(T ) of an idealinterface (ρ0 = ρin = 0) for different tilt angles αl = −αr between 0◦ and 45◦. (a) Tunneljunctions (T0 = 0.01), (b) strongly coupled junctions (T0 = 0.50).

interaction V (pf ,p′f ) is taken as purely d-wave, i.e.,

V (pf ,p′f ) = 2V0 cos(2φ) cos(2φ

′) . (34)

The angles φ and φ ′ are the polar angles of the momenta pf and p ′f

in the a-b planes. The critical temperature is determined by the dimen-sionless coupling constant V0 and a cut-off energy ǫc via the BCS relationTc = 1.13ǫce

−1/V0 . In this simple model the superconductors are isotropicin the a-b planes, except for the pairing interaction, which leads to an orderparameter, ∆(pf ) = ∆cos(2φ), with gap zeros and a sign change in (1,1)direction.Our interface is modeled by a reflection probability R(θ), and a roughnessparameter ρ0. The pair-breaking scattering rate ρin is zero in this section.We follow [29] and choose the following one-parameter model for the de-pendence of R on the angle of incidence θ,

R(θ) =R0

R0 + (1−R0) cos2(θ). (35)

It interpolates smoothly between a reflection probability R0 at perpendicu-lar incidence and total reflection (R = 1) at glancing incidence. All resultspresented in this section were obtained by numerical calculations, whichinvolve a self-consistent calculation of the order parameter in the supercon-ducting electrodes and of the scattering self-energy in the two rough layers.

In Figs.3a,b we show the temperature dependence of the Josephson crit-ical current for d-wave superconductors, and ideally smooth (ρ0 = 0) inter-faces. We compare the cases of a very weak transparency (T0 = 1 −R0 =0.01) and a rather large transparency (T0 = 0.50). The critical currents are

17

0.0 0.5 1.0phase ψ/π

−1.5

0.0

1.5

curr

ent d

ensi

ty j(

ψ)/j

L *1

04

T=0.22Tc

T=0.23Tc

T=0.24Tc

T=0.25Tc

T=0.26Tc

(a)

0.0 0.5 1.0phase ψ/π

−0.08

−0.04

0.00

0.04

curr

ent d

ensi

ty j(

ψ)/j

L

T=0.11Tc

T=0.16Tc

T=0.21Tc

T=0.26Tc

T=0.31Tc

T=0.36Tc

(b)

Figure 5. Current-phase relationship jJ(ψ) of an ideal interface (ρ0 = ρin = 0)at different temperatures. (a) T0 = 0.01, (b) T0 = 0.50. The tilt angles are fixed atαl = −αr = 20◦.

calculated for a representative set of tilt angles, αl = −αr = 0◦−45◦. Thecritical current is normalized in all figures by jL = |e|vfNf∆0 which is theLandau critical current at T = 0 of a clean s-wave superconductor with thesame Tc as the d-wave superconductor. Our results for d-wave supercon-ductors differ significantly from the Ambegaokar-Baratoff curve for s-wavesuperconductors. One finds [7]:

a) A strong dependence on the orientation of the crystals.b) The temperature dependence shows an anomalous positive curvature

near Tc. This is a consequence of the increasing width of the region ofa depleted order parameter with increasing coherence length ξ(T ).

c) For intermediate tilt angles the junction characteristics switches withincreasing temperature from having its maximum current at phasedifference ψ = π (π-type) to a maximum at ψ = 0 (0-type).

d) This leads for junctions with a weak transparency to a non-monotonictemperature dependence of the critical current (see also [53]). It de-creases with increasing T at low T, reaches a minimum, and increasesagain before it decreases to 0 when approaching Tc.

e) Junctions with a weak transparency show an anomalous enhancementof the critical current at low T. This is a consequence of the existenceof zero-energy bound states at the interface [25].

f) Strongly coupled junctions show a similar behavior, but the anomaliesare reduced in this case.

The current-phase relation for the junctions of Figs.3a,b is shown in Figs.4a,bfor the specific tilt angles, αl = −αr = 20◦, where the anomaly in jc(T )is most pronounced. The temperatures are chosen below, above and at thecharacteristic temperature at which the junction jumps from 0-type to π-type. The figure demonstrates that the anomalies in the critical current are

18

accompanied by equally significant anomalies in the current-phase relation.At the jump we find a nearly sinusoidal behavior of the critical current witha doubled period, i.e., jJ(ψ) ∼ sin(2ψ). This is a consequence of the finitetransparency of the interface and cannot be understood within the frame-work of equation (33), which is valid only to linear order in the transparencyT .

The influence of interface roughness on the Josephson critical currentis shown in Figs.5a,b. Roughness reduces strongly the magnitude of thecritical current of anisotropic superconductors, even at fixed normal stateresistance. The reduction is in our cases by a factor 5 for αl = 0◦, and bytwo orders of magnitude for αl = 45◦. The strong reduction of the Joseph-son current at diffuse interfaces can be understood in terms of a destruc-tive interference of contributions of different trajectories. This may occur ifquasiparticles moving along different trajectories experience different phasechanges of the order parameter when going from one side [∆s(pf in)] to the

other side [∆s′(pf out)] of the junction. This effect requires anisotropic orderparameters whose phase (e.g., the sign) changes with changing momentumdirection. It is important to note that anomalies such as the transitionsfrom a 0-junction to a π-junction, and the enhancement at low tempera-tures are sensitive to interface roughness, and have disappeared for ρ0 = 2.At ρ0 = 2 we have already reached to a good approximation the rough limit(ρ0 = ∞) of our model. The results for ρ0 > 2 are essentially unchangedcompared to ρ0 = 2, except for αl = −αr = 45◦ where jJ vanishes forρ0 = ∞.Figs.6a,b finally demonstrate the role of the distorted order parameternear the interface. They show the calculated critical currents for a non-selfconsistent, constant order parameter. These approximate results exhibitthe same qualitative features as Figs.3a,5a, but show significantly differentmagnitudes and T-dependences at temperatures above ≈ 0.5 Tc.

0.0 0.2 0.4 0.6 0.8 1.0temperature T/Tc

0.00

0.05

0.10

criti

cal c

urre

nt j c(

T)/j

L *1

02

(a)

0o

10o

15o

20o

25o

30o

35o

45o

0.0 0.2 0.4 0.6 0.8 1.0temperature T/Tc

0.00

0.02

0.04

0.06

criti

cal c

urre

nt j c(

T)/j

L

45o

0o

25o

(b)

20o

15o

10o

30o

35o

Figure 6. The same as in Figs.3a,b but for a rough interface (ρ0 = 2, ρin = 0).

19

0.0 0.2 0.4 0.6 0.8 1.0temperature T/Tc

−1.0

−0.5

0.0

0.5

criti

cal c

urre

nt j c(

T)/j

L *1

02

45o

0o

25o

(a)

20o

15o

10o

0.0 0.2 0.4 0.6 0.8 1.0temperature T/Tc

0.00

0.05

0.10

criti

cal c

urre

nt j c(

T)/j

L *1

02

(b)0

o

10o

15o

20o

25o

30o

35o

45o

Figure 7. The same as in Figs.3a,5a, but with an artificially suppressed depletion of theorder parameter near the interface. A comparison with Figs.3a,5a shows the effect of thedepletion layers.

3.5. AMBEGAOKAR-BARATOFF RELATION

[2] derived a universal relation between the critical Josephson current acrossa junction of weakly coupled isotropic BCS superconductors, the energygap and the junction resistance in the normal state (11). The maximumJosephson current (Ambegaokar-Baratoff limit) is obtained at T = 0:

IcRN =π∆0

2 | e |, (36)

This relation no longer holds for anisotropic superconductors. The IcRN -product is non-universal and depends, in general, on the orientation of theorder parameter and the quality of the interface.The critical Josephson current, Ic = Ajc, is given in section 5 in terms ofthe Landau critical current

jL =| e | vfNf∆0 . (37)

In addition one obtains from (31) and the reflection probability (35) in thelimit of weak transparency the following interface resistance in the normalstate.

RN =3π

4e2ANfvfT0. (38)

By combining (37) and (38) one obtains

RNIc =π∆0

2 | e |×

3

2T0

jcjL

(39)

We factorized the right side of (39) into the Ambegaokar-Baratoff resultand a correction term which is 1 for isotropic BCS superconductors and

20

0.0 0.2 0.4 0.6 0.8 1.0temperature T/Tc

0.0

0.5

1.0

criti

cal c

urre

nt j c(

T)/j

c(0)

s−waved−wave

Figure 8. Temperature depen-dence of the critical current den-sity of a symmetric single grainboundary junction. The experimen-tal data are taken from Mannhartet al. (1988) (10◦ (✷) and 15◦ (◦)YBa2Cu3O7−δ tilt boundary). Thesolid (dashed) line is a theoreticalcurve obtained from our model ford-wave (s-wave) pairing with inter-face parameters T0 = 0.001, ρ0 = 0.5,ρin = 0, αl = −αr = 5◦ (T0 = 0.001,ρ0 = 0, ρin =0.34). Dotted line: Am-begaokar-Baratoff formula [Eq. (1)].

determines, in general, the deviation from the Ambegaokar-Baratoff rela-tion. The correction factor is written in terms of the ratio jc/jL given infigures 3, 5, 6. One can infer from these results that the correction factormay be negative and larger than 1 for ideal junctions with d-wave supercon-ductors (see Fig.3a), and is much smaller than 1 for rough junctions withd-wave superconductors (see Fig.5a). In addition, the correction factor de-pends strongly on the orientation of the order parameter (see Figs.3, 5).In Fig.7 we compare temperature dependent critical current measurementson symmetric single grain boundary junctions by J. Mannhart et al. [39](10◦ and 15◦ YBCO tilt boundaries) with our theoretical calculations. Ourcalculations have shown that the standard Fermi-liquid theory of supercon-ductivity in correlated, anisotropic metals predicts characteristic differencesfor d-wave and s-wave superconductors in the temperature dependence ofthe Josephson current and the current-phase relation. These differences areto some degree washed out by interface roughness, such that the best wayto observe these effects would be experiments on ideally clean grain bound-ary junctions with a weak transparency, and the possibility of continuouslyvarying the tilt angles.

4. Electromagnetic Response of a Pancake Vortex in LayeredSuperconductors

In oder to demonstrate the capacities of the Fermi liquid theory of super-conductivity we apply in this section the quasiclassical theory of supercon-ductivity to a non-trivial dynamical problem. We calculate the responseof the currents in the core of a vortex to an alternating electric field. Anelectric field has two principal effects on a superconductor. It changes thesupercurrents by accelerating the superfluid condensate, and it generates

21

dissipation by exciting “normal” quasiparticles. This two-fluid picture of acondensate and normal excitations is clearly reflected in the optical conduc-tivity of standard s-wave superconductors in the Meissner state [41]. Theconductivity has a superfluid part,

σs(ω) =e2nsm

(

δ(ω) + iP1

ω

)

, (40)

and a dissipative normal part. At T = 0 dissipation starts at the thresholdfrequency for creating quasiparticles, ω = 2∆. At finite temperature thecoupling to thermally excited quasiparticles leads to dissipation at all fre-quencies. The two-fluid picture no longer holds for type-II superconductorsin the vortex state. The response to an electric field consists of contributionsfrom gapless2 vortex core excitations [16], and contributions from excita-tions outside the cores, where the quasiparticle spectrum shows the bulkgap. For high-κ superconductors the electric response at low frequenciesof the region outside the core can be described very well by the Londonequations [37], i.e. by two-fluid electrodynamics. The response of the coreis more complex. In the traditional model of a “normal core” (Bardeenand Stephen [10]) the conductivity in the core is that of the normal state,σcore = σn. The Bardeen-Stephen model [10] is a plausible approximationfor dirty superconductors with a mean free path (ℓ) much shorter than thesize of the core (ℓ≪ ξ0). In this limit the vortex core excitations of Caroliet al. [16] may be considered as a continuum of normal excitations.In 1969 Bardeen et al. [11] published a detailed discussion of the boundstates in the core of a vortex, and argued that the bound states contributesignificantly to the circulating supercurrents in the vortex core. This effectwas confirmed by recent self-consistent calculations of free and pinned vor-tices in clean and medium dirty superconductors [47]. The authors showthat all currents in the core (circulating currents as well as superimposedtransport currents) are carried predominantly by the bound states. Thismeans that the model of a normal core needs to be modified for relativelyclean (ℓ >∼ ξ0) superconductors. Furthermore the response of the boundstates is expected to show dissipative as well as superfluid features.

4.1. QUASICLASSICAL THEORY OF THE ELECTROMAGNETICRESPONSE

A convenient formulation of the quasiclassical theory for our purposes is interms of the quasiclassical Nambu-Keldysh propagator g(pf ,R; ǫ, t), whichis a 4×4-matrix in Nambu-Keldysh space, and a function of positionR, timet, energy ǫ, and Fermi momentum pf . We consider a superconductor with

2We ignore the “mini-gap” of size ∆2/Ef predicted by Caroli et al. [16].

22

random atomic size impurities in a static magnetic field, and an externallyapplied a.c. electric field, E = −1

c∂tδA. In a compact notation the transportequation for this system and the normalization condition read

[(ǫ+e

cvf ·A)τ3 − ∆mf − σi − δv , g]⊗ + ivf ·∇g = 0 , (41)

g ⊗ g = −π21 , (42)

where A(R) is the vector potential of the static magnetic field, B = ∇×A,∆mf (pf ,R; t) the mean-field order parameter matrix, and σi(pf ,R; ǫ, t) isthe impurity self-energy. The perturbation δv(pf ,R; t) includes the externalelectric field and the field of the charge fluctuations, δρ(R; t), induced bythe external field. For convenience we describe the external electric field bya vector potential δA(R; t) and the induced electric field by the electro-chemical potential δϕ(R; t). Hence in the Nambu-Keldysh matrix notationthe perturbation has the form,

δv = −e

cvf · δA(R; t)τ3 + eδϕ(R; t)1 , (43)

and is assumed to be sufficiently small so that it can be treated in linearresponse theory.Equations (41) and (42) must be supplemented by self-consistency equa-tions for the order parameter and the impurity self-energy. We use theweak-coupling gap equations,

∆R,Amf (pf ,R; t) =

∫ +ǫc

−ǫc

dǫ

4πi〈V (pf ,p

′f )f

K(p′f ,R; ǫ, t)〉 , (44)

∆Kmf (pf ,R; t) = 0 , (45)

and the impurity self-energy in Born approximation with isotropic scatter-ing,

σi(R; ǫ, t) =1

2πτ〈g(p′

f ,R; ǫ, t)〉 , (46)

where fK is the off-diagonal part of the 2× 2 Nambu matrix gK , and theFermi surface average is defined by

〈 . . . 〉 =1

Nf

∫ d2p′f

(2π)3 |v′f |

. . . . (47)

The materials parameters that enter the self-consistency equations are thepairing interaction, V (pf ,p

′f ), the impurity scattering lifetime, τ , in addi-

tion to the Fermi surface data pf (Fermi surface), vf (Fermi velocity), and

23

Nf =∫ d2pf

(2π)3|vf |.

In the linear response approximation one splits the propagator and theself-energies into an unperturbed part and a term of first order in the per-turbation,

g = g0 + δg , ∆mf = ∆mf0 + δ∆mf , σi = σ0 + δσi , (48)

and expands the transport equation and normalization condition throughfirst order. In 0th order we obtain

[(ǫ+e

cvf ·A)τ3 − ∆mf0 − σ0 , g0]⊗ + ivf ·∇g0 = 0 , (49)

g0 ⊗ g0 = −π21 , (50)

and in 1st order

[(ǫ+e

cvf ·A)τ3− ∆mf0− σ0 , δg]⊗+ ivf ·∇δg = [δ∆mf + δσi+ δv , g0]⊗ ,

(51)

g0 ⊗ δg + δg ⊗ g0 = 0 . (52)

In order to close this system of equations one has to supplement the trans-port and normalization equations with the self-consistency equations of 0th

and 1st order:

∆R,Amf0(pf ,R) =

∫ +ǫc

−ǫc

dǫ

4πi〈V (pf ,p

′f )f

K0 (p′

f ,R; ǫ)〉, ∆Kmf0 = 0 , (53)

δ∆R,Amf (pf ,R; t) =

∫ +ǫc

−ǫc

dǫ

4πi〈V (pf ,p

′f )δf

K(p′f ,R; ǫ, t)〉, δ∆K

mf = 0 ,

(54)and

σ0(R; ǫ) =1

2πτ〈g0(p

′f ,R; ǫ)〉 , (55)

δσi(R; ǫ, t) =1

2πτ〈δg(p′

f ,R; ǫ, t)〉 . (56)

Finally, the electro-chemical potential, δϕ, is determined by the condition oflocal charge neutrality [24, 5]. This condition follows from the expansion ofcharge density to leading order in the quasiclassical expansion parameters.One obtains δρ(R; t) = 0, i.e.

−2e2Nfδϕ(R; t) + eNf

∫

dǫ

4πi〈TrδgK(p′

f ,R; ǫ, t)〉 = 0 . (57)

24

The self-consistency equations (53)-(56) are of vital importance in the con-text of this paper. Equations (54) and (56) for the response of the quasiclas-sical self-energies are equivalent to vertex corrections in the Green’s func-tion response theory. They guarantee that the quasiclassical theory doesnot violate fundamental conservation laws. In particular, (55) and (56) im-ply charge conservation in scattering processes, whereas (53) and (54) implycharge conservation in a particle-hole conversion process. Any charge whichis lost or gained in a particle-hole conversion process is balanced by the cor-responding gain or loss of condensate charge. It is the coupled quasiparticledynamics and collective condensate dynamics which conserves charge insuperconductors. Neglect of the dynamics of either component, or use ofa non-conserving approximation for the coupling of quasiparticles and col-lective degrees of freedom leads to unphysical results. Condition (57) isa consequence of the long-range of the Coulomb repulsion. The Coulombenergy of a charged region of size ξ30 and typical charge density eNf∆is ∼ e2N2

f∆2ξ50 , which should be compared with the condensation energy

∼ Nf∆2ξ30 . Thus, the cost in Coulomb energy is a factor (Ef/∆)2 larger

than the condensation energy. This leads to a strong suppression of chargefluctuations, and the condition of local charge neutrality holds to very goodaccuracy for superconducting phenomena.Equations (49)-(57) constitute a complete set of equations for calculatingthe electromagnetic response of a vortex. The structure of a vortex in equi-librium is obtained from (49), (50), (53) and (55), and the linear responseof the vortex to the perturbation δA(R; t) follows from (51), (52), (54),(56) and (57). The currents induced by δA(R; t) can then be calculateddirectly from the Keldysh propagator δgK via

δj(R; t) = eNf

∫

dǫ

4πi〈vf (p

′f )Tr

(

τ3δgK(p′

f ,R; ǫ, t))

〉 . (58)

4.2. STRUCTURE AND SPECTRUM OF A PANCAKE VORTEX

We consider an isolated pancake vortex in a strongly anisotropic, layeredsuperconductor, and model the Fermi surface of these systems by a cylin-der of radius pf , and a Fermi velocity of constant magnitude, vf , along thelayers. We also assume isotropic (s-wave) pairing and isotropic impurityscattering. Thus, the materials parameters of the model superconductorare: Tc, vf , the 2D density of states Nf = pf/2πvf , and the mean freepath ℓ = vfτ . The model superconductor is type II with a large Ginzburg-Landau parameter κ >∼ 100, so the magnetic field is to good approximationconstant in the region of the vortex core.We first present results for the equilibrium vortex. The order parameter,

25

−10.0 −5.0 0.0 5.0 10.0R / vf / 2πTc

0.0

0.5

1.0

1.5

2.0

a)

|∆(R

)| /

T c

l= ξ0 /3

l= ξ0

l=10ξ0

−10.0 −5.0 0.0 5.0 10.0R / vf / 2πTc

0.0

0.2

0.4

0.6

b)

|j(R

)| /

2e

N fvfT

c

l=10ξ0

l= ξ0

l= ξ0 /3

Figure 9. Results of self-consistent calculations of the modulus of order parameter(a) and the current density (b) at T = 0.3Tc for superconductors with electron meanfree path ℓ = 10ξ0 (clean) to ℓ = 1

3ξ0 (dirty). R is the distance from the vortex center

measured in units of the coherence length.

∆0(R), is calculated by solving the transport equation (49), the gap equa-tion (53), and the self-energy equation (55) self-consistently. These calcula-tions are done at Matsubara energies (ǫ→ iǫn = i(2n+1)πT ). Details of thenumerical schemes for solving the transport equation self-consistently aregiven elsewhere [22]. Charge conservation for the equilibrium vortex followsfrom the circular symmetry of the currents. Nevertheless, self-consistencyof the equilibrium vortex is important; the equilibrium self-energies (∆0,σ0) and propagators (g0) are input quantities in the transport equation(51) for the linear response. Charge conservation in linear response is non-trivial and requires self-consistency of both the equilibrium solution andthe solution in first order in the perturbation.

Fig.1 shows the order parameter and the current density in the vortexcore of an equilibrium vortex for different impurity scattering rates τ . Asexpected, scattering reduces the coherence length and thus the size of thecore, and has a strong effect on the current density. Numerical results for theexcitation spectrum of bound and continuum states at the vortex togetherwith the corresponding spectral current densities are shown in Fig.2. Thelocal density of states (per spin) and the spectral current density are definedby

N(R, ǫ) = Nfi

4πTr〈τ3

(

gR(p′f ,R, ǫ)− gA(p′

f ,R, ǫ))

〉 (59)

j(R, ǫ) = eNfi

2πTr〈τ3vf (p

′f )(

gR(p′f ,R, ǫ)− gA(p′

f ,R, ǫ))

〉 . (60)

The zero-energy bound state is remarkably broadened by impurity scatter-ing. Its width is well approximated by the scattering rate 1/(vf ℓ), whichis 0.63Tc for ℓ = 10ξ0. The broadening of the bound states decreases with

26

0.0 1.0 2.0 3.0ε/Tc

0.0

1.0

2.0

3.0

4.0a)

N(ε

,R)/

N f

0.0 1.0 2.0 3.0ε/Tc

0.0

1.0

2.0

3.0b)

J x(ε,

R) /

2eN f v

f Tc

Figure 10. Local density of states N(R, ǫ) (a) and local spectral current density jx(R, ǫ)(b) in the vortex core of a superconductor with ℓ = 10ξ0 at T = 0.3Tc. Results are shownfor a series of spatial points on the y-axis, at distances 0, .25πξ0, .5πξ0, . . . , 4πξ0 fromthe vortex center. The thickest full line corresponds to the vortex center, and decreasingthickness indicates increasing distance from the center. Results for the outermost point(4πξ0) are shown as dashed lines. The thin dotted lines show the density of states of thehomogeneous superconductor (a) and the value of the bulk gap (b) respectively.

increasing energy. The results for the spectral current density (Fig.2b) showthat nearly all of the current density of the equilibrium vortex resides inthe energy range of the bound states. This reflects the observation that thesupercurrents in the vortex core are predominantly carried by the boundstates [11, 47]. The physics of the core is dominated by the bound states,in particular also, as we will show, the response of the core to an electricfield.

4.3. DISTRIBUTION FUNCTIONS

The set of formulas for calculating the quasiclassical linear response of asuperconductor is given in a compact notation in (48)-(57). In this sec-tion we transform these formulas into a more suitable form for analyticaland numerical calculations. The central differential equation of the qua-siclassical response theory is the transport equation (51). It comprises 12differential equations for the components of the three 2×2-Nambu matrices,δgR,A,K . The number of differential equations can be reduced significantlyby using general symmetry relations and the normalization conditions of

27

the quasiclassical theory. We focus on the simplifications which follow fromthe normalization equations (50) and (52). We use the projection operatorsintroduced by Shelankov [50],

PR,A+ =

1

2

(

1 +1

−iπgR,A0

)

, PR,A− =

1

2

(

1−1

−iπgR,A0

)

. (61)

Obviously, PR,A+ +PR,A

− = 1. The algebra of the projection operators followsfrom the normalization conditions.

(PR,A+ )2 = PR,A

+ , (PR,A− )2 = PR,A

− ,

PR,A+ PR,A

− = PR,A− PR,A

+ = 0. (62)

A key result is that the Nambu matrices δgR,A,K can be expressed, with thehelp of Shelankov’s projectors, in terms of 6 scalar distribution functions,δγR,A, δγR,A, δxa and δxa, each of which is a function of pf , R, ǫ, t, andsatisfies a scalar transport equation. The distribution functions are definedby

δgR,A (63)

= ∓2πi

[

PR,A+ ⊗

(

0 δγR,A

0 0

)

⊗PR,A− − PR,A

− ⊗

(

0 0−δγR,A 0

)

⊗PR,A+

]

,

and

δga (64)

= −2πi

[

PR+ ⊗

(

δxa 00 0

)

⊗ PA− + PR

− ⊗

(

0 00 δxa

)

⊗ PA+

]

.

where the anomalous response, δga, is defined in terms of δgK , δgR, δgA by

δgK = δgR ⊗ tanh(βǫ/2)− tanh(βǫ/2)⊗ δgA + δga , (65)

The transport equations for the various distribution functions follow from(49) and (51) and one finds [22],

ivf ·∇δγR,A + 2ǫδγR,A

+(γR,A0 ∆R,A − ΣR,A)⊗ δγR,A + δγR,A ⊗ (∆R,AγR,A

0 + ΣR,A) (66)

= −γR,A0 ⊗ δ∆R,A ⊗ γR,A

0 + δΣR,A ⊗ γR,A0 − γR,A

0 ⊗ δΣR,A − δ∆R,A,

ivf ·∇δγR,A − 2ǫδγR,A

+(γR,A0 ∆R,A − ΣR,A)⊗ δγR,A + δγR,A ⊗ (∆R,AγR,A

0 +ΣR,A) (67)

= −γR,A0 ⊗ δ∆R,A ⊗ γR,A

0 + δΣR,A ⊗ γR,A0 − γR,A

0 ⊗ δΣR,A− δ∆R,A,

28

ivf ·∇δxa + i∂tδxa+ (γR0 ∆

R− ΣR)⊗ δxa + δxa ⊗ (∆AγA +ΣA)

= γR0 ⊗ δΣa ⊗ γA0 − δ∆a ⊗ γA0 − γR0 ⊗ δ∆a − δΣa , (68)

ivf ·∇δxa − i∂tδxa+ (γR0 ∆

R− ΣR)⊗ δxa + δxa ⊗ (∆AγA0 + ΣA)

= γR0 ⊗ δΣa ⊗ γA0 − δ∆a ⊗ γA0 − γR0 ⊗ δ∆a − δΣa . (69)

We have used the following short-hand notation for the driving terms inthe transport equations, which includes external potentials, perturbationsand self-energies:

−e

cvf ·Aτ3 + ∆mf0 + σR,A

i0 =

(

ΣR,A ∆R,A

−∆R,A ΣR,A

)

, (70)

δ∆mf + δσR,Ai + δv =

(

δΣR,A δ∆R,A

−δ∆R,A δΣR,A

)

, δσai =

(

δΣa δ∆a

δ∆a −δΣa

)

.(71)

The functions γR,A0 and γR,A

0 in (66)-(69) are defined by the following con-venient parameterization of the equilibrium propagators.

gR,A0 = ∓iπ

1

1 + γR,A0 γR,A

0

(

1− γR,A0 γR,A

0 2γR,A0

2γR,A0 −(1− γR,A

0 γR,A0 )

)

. (72)

After elimination of the time-dependence in (66)-(69) by Fourier transformone is left with four sets of ordinary differential equations along straighttrajectories in R-space. For given right-hand sides these equations are de-coupled, and determine the distribution functions δγR,A, δγR,A, δxa andδxa. On the other hand, the self-consistency conditions relate the right-handsides of (66)-(69) to the solutions of (66)-(69). Hence, equations (66)-(69)may be considered either as a large system of linear differential equationsof size six times the chosen number of trajectories, or as a self-consistencyproblem. We solved the self-consistency problem numerically using spe-cial algorithms for updating the right hand sides. Details of our numericalschemes for a self-consistent determination of the response functions aregiven elsewhere [22].

4.4. RESPONSE TO AN A.C. ELECTRIC FIELD

We consider a pancake vortex in a layered s-wave superconductor and cal-culate the response of the electric current density, δj(R; t), to a small homo-geneous a.c. electric field, δEω(t) = δE exp(−iωt). The results presented inthis section are obtained by solving numerically the quasiclassical transport

29

Figure 11. Snapshots of the time-dependent current pattern in the core of a pancakevortex at successive times, t = 0, 1

12T , 2

12T , 3

12T , 4

12T , 5

12T (upper left pattern to lower

right pattern) . The arrows show the current density induced by an a.c. electric field inx-direction, Ex(t) = δE cos(ωt), of frequency ω = 2π/T = 1.5∆. The data are calculatedin linear response approximation for a clean superconductor (ℓ = 10ξ0) at T = 0.3Tc.Current patterns in the second half-period, t = 1

2T − 11

12T , are obtained from the patterns

in the first half-period by reversing the directions of the currents. The distance betweentwo neighboring points on the hexagonal grid is 0.25πξ0.

equations (66)-(69) together with the self-consistency equations (54), (56),and the condition of local charge neutrality (57). The calculation gives thelocal conductivity tensor, σij(R, ω), defined by

δji(R, t) = σij(R, ω)δEωj (t) . (73)

Figures 11 and 12 show the time development of the current pattern inducedby an oscillating electric field in x-direction with time dependence δEx(t) =δE cos(ωt). Results are given for a medium range frequency (ω = 1.5∆)and a low frequency (ω = 0.3∆). Two features should be emphasized.At medium and higher frequencies the current flow induced by the electricfield is to good approximation uniform in space and phase shifted by π/2(non-dissipative currents). The phase shift of π/2 is the consequence of apredominantly imaginary conductivity at frequencies above ∆, as shownin Fig.5. The current pattern at low frequencies (Fig.4) is qualitativelydifferent. At the vortex center the current is phase shifted by ≈ π/4 inaccordance with the conductivity at ω = 0.3∆, which has about equal real(dissipative) and imaginary (non-dissipative) parts, whereas further awayfrom the center the conductivity becomes more and more non-dissipative.

30

Figure 12. The same as in fig. 11, but for a smaller external frequency, ω = 0.3∆. Thelength of the current vectors is scaled down by a factor 5 compared to those in fig. 11.

Fig.4 shows that the current flow at low frequencies is non-uniform. Thedissipative currents exhibit a dipolar structure with enhanced currents atthe vortex center, and back-flow currents away from the center. On theother hand, non-dissipative currents are approximately uniform.

The frequency dependence of the local conductivity, σxx(R, ω), for R

along the x- and y-axis is shown in figures 13a,b. These figures include,for comparison, the Drude conductivity of the normal state and the con-ductivity of the homogeneous bulk superconductor. A significant feature ofconductivity in the core is the strong increase of Re σ at low frequencies.The conductivity is in this frequency range much larger than the normalstate Drude conductivity and the exponentially small conductivity of thebulk s-wave superconductor. The real part of conductivity scales at lowfrequencies like 1/ω2. Its value at at the vortex center is 69.5e2Nfv

2f/∆

(outside the range of the figure) for ω = 0.1∆. The enhancement of the dis-sipative part at low frequencies is a consequence of the coupling of quasi-particles and collective order parameter modes, and cannot be obtainedfrom non-selfconsistent calculations. Fig.13b shows that the real part ofthe conductivity becomes negative in the region of dipolar backflow on they-axis. This leads locally to a negative time averaged power absorption,< j · E >t < 0, and a corresponding gain in energy, which is compensatedby the strongly enhanced dissipation of energy in the center of the vortex.The dissipative part of the local conductivity at a distance R from thevortex center exhibits pronounced maxima whose frequencies increase with

31

0.0 1.0 2.0 3.0ω/∆

0.0

0.2

0.4

0.6

0.8

1.0a)

ωσ xx

(x,ω

) / 2e

2 Nf v

f2

vortex center

Drude

homogeneous

0.0 1.0 2.0 3.0ω/∆

−0.2

0.0

0.2

0.4

0.6

0.8

1.0b)

ωσ xx

(x,ω

) / 2e

2 Nf v

f2

vortex center

Drude

homogeneous

Figure 13. Frequency dependence of the real part (‘lower’ curves) and imaginary part(‘upper’ curves) of the local conductivity σxx for a superconductor with parametersT = 0.3Tc, ℓ = 10ξ0. For convenience, the conductivities are multiplied by ω. The fullblack curves give the conductivity at the vortex center, and the series of dashed lineswith decreasing intensity the conductivity at increasing distance from the center. Fig.13apresents data at points along the x-axis (in steps of (π/4)ξ0), and Fig.13b at points along

the y-axis (in steps of (√3π/4)ξ0). The dashed grey lines show the normal state Drude

conductivity, and the full grey lines the conductivity of the homogeneous superconductor.

increasing R and are given by 2× the energy at the maxima in the localdensity of states shown in Fig.2a. Hence, these features in the absorptionspectrum must be identified as impurity assisted transitions between cor-responding bound states at negative and positive energies. Impurities arerequired for breaking angular momentum conservation in these transitions.The applied electric field δE(t) induces in the vortex core an internal field−∇δϕ(t), which is of the same order as the applied field. Fig.14 shows thetotal electric field, δEtot(t) = δEω(t) − ∇δϕ(t), in the vortex core. Theinduced field is at low frequencies of dipolar form, and oscillates out ofphase (phase shift π/2) with the applied field. This dipolar field originatesfrom small charge fluctuation in the vortex core. At higher frequencies thedipolar field oscillates with a phase shift of ≈ π, and screens part of theapplied field.We finally discuss the role of self-consistency in our calculation. Our resultswere obtained by iterating the self-consistency equations until the relativeerror stabilized below ≤ 10−10. Fig.15 compares the degree of violation ofcharge conservation in a non-selfconsistent calculation (no iteration) with

32

Figure 14. Local electrical field for a superconductor with parameters T = 0.3Tc,ℓ = 10ξ0, and an external frequency of ω = 0.3∆(T ). Each field pattern is a snapshot attime t varying from 0 to half of time periode in time-steps of 1

12periode. The external

electrical field δEω(t) = δE cos(ωt) is maximal and points in positive x-direction for thefirst picture (t = 0). In the first picture of the second line it is zero. The distance betweentwo points in the grid corresponds to 0.25πξ0.

a) b)

Figure 15. Degree of violation of charge conservation by the dissipative cur-rent flow for a non-selfconsistent calculation (Fig.15a), and a self-consistent calcula-tion (Fig.15b). The largest deviation in the non-selfconsistent calculation amounts toδρ + ∇δj = 2.5e2NfvfδE

ω. The data are obtained for a superconductor with ℓ = 10ξ0at ω = 0.4∆(T ) and T = 0.3Tc.

the self-consistent result. We measure the degree of violation at position R

by D(R) = max t[δρ(R, t) +∇ · δj(R, t)]. Charge conservation is obviouslyfulfilled if D(R) = 0. The degree of violation at a point R is indicatedin Fig.15a,b by the size of the filled circles around the grid points. Thenon-selfconsistent calculation (Fig.15a) results in a D(R), which is muchlarger than the time derivative of the correct charge density, δρ(R, t), ob-tained from a self-consistent calculation with δϕ(R, t) = 0 instead of charge

33

neutrality condition. In Fig.15b we show the violation of charge conserva-tion for our self-consistent calculation. The small remaining D(R) is herea consequence of the finite grid size used in our calculations.

4.5. DISCUSSION

In this section we used the quasiclassical theory to calculate the electro-magnetic response of a pancake vortex in a superconductor with finite butlong mean free path. This complements previous calculations for perfectlyclean systems, which were done self-consistently in the limit ω → 0 [26],and at finite frequencies without a self-consistent determination of the orderparameter [27, 55]). The frequency range of interest in our calculations isof the order of the gap frequency, hω = ∆. We have shown that at low fre-quencies (hω < 0.5∆) the electromagnetic dissipation is strongly enhancedin the vortex cores above its normal state value, and that this effect is a con-sequence of the coupled dynamics of low-energy quasiparticles excitationsbound to the vortex core and collective order parameter modes. The in-duced current density has at low frequencies a dipole-like behaviour, whichresults from an oscillating motion of the vortex perpendicular to directionof the driving a.c. field. The response of the vortex in the intermediate fre-quency range, .5∆ <∼ hω <∼ 2∆, is dominated by bound states in the vortexcore. We find peaks in the local dissipation at twice the bound state ener-gies. At higher frequencies, hω > 2∆, the conductivity approaches that ofa very clean homogeneous superconductor, which is in good approximationgiven by the non-dissipative response of an ideal conductor.

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