Electrodynamics of Superconductors Hirsch

8/3/2019 Electrodynamics of Superconductors Hirsch

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rXiv:cond-mat/031

2619v4

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Electrodynamics of superconductors

J. E. HirschDepartment of Physics, University of California, San Diego

La Jolla, CA 92093-0319(Dated: December 30, 2003)

An alternate set of equations to describe the electrodynamics of superconductors at a macroscopiclevel is proposed. These equations resemble equations originally proposed by the London brothers

but later discarded by them. Unlike the conventional London equations the alternate equations arerelativistically covariant, and they can be understood as arising from the rigidity of the superfluidwave function in a relativistically covariant microscopic theory. They predict that an internalspontaneous electric field exists in superconductors, and that externally applied electric fields,both longitudinal and transverse, are screened over a London penetration length, as magnetic fieldsare. The associated longitudinal dielectric function predicts a much steeper plasmon dispersionrelation than the conventional theory, and a blue shift of the minimum plasmon frequency for smallsamples. It is argued that the conventional London equations lead to difficulties that are removedin the present theory, and that the proposed equations do not contradict any known experimentalfacts. Experimental tests are discussed.

PACS numbers:

I. INTRODUCTION

It has been generally accepted that the electrodynam-ics of superconductors in the London limit (where theresponse to electric and magnetic fields is local) is de-scribed by the London equations[1, 2]. The first Londonequation

Jst

=nse

2

meE (1a)

describes the colisionless response of a conducting fluid ofdensity ns, i.e. free acceleration of the superfluid carrierswith charge e and mass me, giving rise to the supercur-

rentJs. The second London equation

Js = nse

2

mecB (1b)

is obtained from Eq. (1a) using Faradays law and settinga time integration constant equal to zero, and leads to theMeissner effect. These equations together with Maxwellsequations

E = 1

c

B

t(2a)

B =4

c Js +1

c

E

t (2b)

E = 4 (2c)

B = 0 (2d)

are generally believed to determine the electrodynamicbehavior of superconductors. In this paper we argue that

these equations are not correct, and propose an alternate

set of equations.There is ample experimental evidence in favor of Eq.

(1b), which leads to the Meissner effect. That equationis in fact preserved in our alternative theory. However,we argue that there is no experimental evidence for Eq.(1a), even if it appears compelling on intuitive grounds.In fact, the London brothers themselves in their earlywork considered the possibility that Eq. (1a) may notbe valid for superconductors[3]. However, because of theresult of an experiment[4] they discarded an alternativepossibility and adopted both Eqs. (1a) and Eq. (1b),which became known as the first and second Londonequations.

It is useful to introduce the electric scalar and magneticvector potentials and A. The magnetic field is givenby

B = A (3)

and Eq. (2a) is equivalent to

E = 1

c

A

t(4)

The magnetic vector potential is undefined to within thegradient of a scalar function. The gauge transformation

A A + f (5a)

1

c

f

t(5b)

leaves the electric and magnetic fields unchanged.The second London equation Eq. (1b) can be written

as

J = nse2

mecA (6)
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However the right-hand-side of this equation is not gauge-invariant, while the left-hand side is. From the continuityequation,

J +

t= 0 (7)

it can be seen that Eq. (6) is only valid with a choice of

gauge that satisfies

A =mec

nse2

t(8a)

In particular, in a time-independent situation (i.e. whenelectric and magnetic fields are time-independent) the

vector potential A in Eq. (6) is necessarily transverse,i.e.

A = 0 (8b)

It is currently generally accepted that the London Eq.(6) is valid for superconductors in all situations, time-

independent or not, with the gauge chosen so that Eq.(8b) holds (London gauge), and hence J = 0 fromEq. (6). Together with Eq. (1a) this is equivalent to

assuming that no longitudinal electric fields (i.e. E =0) can exist inside superconductors, as can be seen byapplying the divergence operator to Eq. (1a).

However, the possibility of an electrostatic field insidesuperconductors was recently suggested by the theoryof hole superconductivity[5, 6, 7], which predicts thatnegative charge is expelled from the interior of super-conductors towards the surface. The conventional Lon-don electrodynamics is incompatible with that possibil-ity, however the alternate electrodynamics proposed here

is not. We have already discussed some consequences forthe electrostatic case in recent work[8].

Recently, Govaerts, Bertrand and Stenuit have dis-cussed an alternate Ginzburg-Landau formulation for su-perconductors that is relativistically covariant and hassome common elements with the theory discussed here[9].For the case of a uniform order parameter their equationsreduce to the early London theory[3] that allows for elec-trostatic fields within a penetration length of the surfaceof a superconductor, as our theory also does. However incontrast to the theory discussed here, the theory of Gor-vaets et al does not allow for electric charge nor electricfields deep in the interior of superconductors.

II. DIFFICULTIES WITH THE LONDON

MODEL

Following London[1], let us assume that in additionto superfluid electrons there are normal electrons, givingrise to a normal current

Jn = n E (9)

so that the total current J = Jn + Js satisfies

J

t=

c2

42LE+ n

E

t(10)

with the London penetration depth L given by

1

2L

=4nse

2

mec2. (11)

Applying the divergence operator to Eq. (10) and usingEqs. (2c) and (7) leads to

2

t2+ 4n

t+

c2

2L = 0 (12)

London argued[1] that Eq. (12) leads to a rapid decayof any charge buildup in superconductors, given by thetime scales

11,2 = 2n [(2n)2 (

c

L)2]1/2 (13)

the slower of which he estimated to be 1012

sec, andconsequently that any charge buildup inside the super-conductor can be ignored. He concluded from this thatit is reasonable to assume = 0 inside superconductors,which from the continuity equation and Eq. (6) impliesEq. (8b). The same argument is given in Ref. [10].

However, this numerical estimate is based on assumingfor n a value appropriate for the normal state, and forL its value near zero temperature. Instead, the tem-perature dependence of n and L should be considered.The conductivity of normal carriers n should be propor-tional to the number of normal electrons, that goes tozero as the temperature approaches zero, hence n 0as T 0. On the other hand, the number of super-fluid electrons ns goes to zero as T approaches Tc, henceL as T Tc. These facts lead to a strong temper-ature dependence of the relaxation times in Eq. (13), andin fact to the conclusion that long-lived charge fluctua-tions should exist both when T Tc and when T 0.

Eq. (12) describes a damped harmonic oscillator. Thecrossover between overdamped (at high T) and under-damped (low T) regimes occurs for

2n(T) =c

L(T)(14)

As T Tc, L(T) and n(T) approaches its nor-mal state value at Tc; as T 0, L(T) L(0) andn(T) 0; hence condition Eq. (14) will always be sat-isfied at some temperature between 0 and Tc. For exam-ple, for a superconductor with low temperature Londonpenetration depth L 200A and normal state resistiv-ity 10cm the crossover would be at T /Tc 0.69; ifL 2000A and 1000cm, at T /Tc 0.23. How-ever, no experimental signature of such a cross-over be-tween overdamped and underdamped charge oscillationsat some temperature below Tc has ever been reported forany superconductor.


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For T approaching Tc, the slower timescale in Eq. (13)is

1 4n(T)

(c/L(T))2(15)

so that for T sufficiently close to Tc overdamped chargefluctuations should persist for arbitrarily long times.However, such a space charge would give rise to an elec-

tric field in the interior of the superconductor and hence(due to Eq. (1a)) to a current that grows arbitrarilylarge, destroying superconductivity. No such phenomenahave ever been observed in superconductors close to Tcto our knowledge.

At low temperatures, Eq. (13) implies that under-damped charge oscillations should exist, with dampingtimescale

1 =1

2n(T)(16)

diverging as T 0. The frequency of these oscillationsis

=

c2

2L 422n (17)

and as T 0 it approaches the plasma frequency

p =c

L(18)

Hence at zero temperature plasma oscillations in the su-perconducting state are predicted to exist forever, justas persistent currents. No such persistent charge oscilla-tions have been observed to our knowledge.

Reference [10] recognized this difficulty and suggestedthat the London electrodynamic equations should only

be assumed to be valid in situations in which the fieldsdo not tend to build up a space charge, hence not in theregions close to Tc and close to T = 0 discussed above.On the other hand, the consequences of London equa-tions concerning magnetic fields and persistent currentsare generally believed to be valid for arbitrary tempera-tures, and indeed experiments support this expectation.The fact that the implications of Londons equations con-cerning the behavior of the charge density in supercon-ductors appear to have at best a limited range of validityis disturbing and suggests a fundamental inadequacy ofthese equations.

A related difficulty with Londons equations arisesfrom consideration of the equation for the electric field.Taking the time derivative of Eq. (2b) and using (2a),(2c) and (1a) yield

2 E =1

2LE+

1

c22 E

t2+ 4 (19)

For slowly varying electric fields and assuming no chargedensity in the superconductor

2 E =1

2LE (20)

which implies that an electric field penetrates a distanceL, as a magnetic field does. Indeed, electromagneticwaves in superconductors penetrate a distance L, henceEq. (20) properly describes the screening of transverseelectric fields. However Eq. (20) does not depend on thefrequency and hence should remain valid in the staticlimit; but such a situation is incompatible with the firstLondon equation (1a), as it would lead to arbitrarily large

currents. This then suggests that application of an ar-bitrarily small static or quasi-static electric field shouldlead to destruction of the superconducting state, whichis not observed experimentally.

In a normal metal these difficulties do not arise. Ap-plying the divergence operator to Eq. (9) and using (2c)and (7) yields

t= 4n (21)

which predicts that any charge fluctuation in the in-terior of metals is screened over a very short time (1017secs), and consequently that a longitudinal electric

field cannot penetrate a normal metal. A static uniformelectric field can penetrate a normal metal and it causesa finite current to flow whose magnitude is limited by thenormal state resistivity.

In summary, Londons equations together withMaxwells equations lead to unphysical predictions re-garding the behavior of superconductors in connectionwith space charges and electric fields, and to predictionsthat appear to be contradicted by experiment. How arethese difficulties avoided in the conventional London pic-ture? By postulating that = 0 inside superconductors,and that applied static electric fields do not penetrate thesuperconductor[1, 2, 10]. These are postulates that are

completely independent of Eqs. (1) and (2), for which noobvious justification within Londons theory exists.

III. THE ALTERNATE EQUATIONS

The possibility of an alternative to the conventionalLondon equations is suggested by the fact that takingthe time derivative of Eq. (6) and using Eq. (4) leads to

Jst

=nse2

me( E+ ) (22)

without making any additional assumptions on the gaugeof A. Clearly, Eq. (22) is consistent with the existence ofa static electric field in a superconductor, deriving froman electrostatic potential , which will not generate anelectric current, contrary to the prediction of Eq. (1a).How it may be possible for a static electric field to existin a superconductor without generating a time-dependentcurrent is discussed in ref. [8]. Assuming that Eq. (22)and Eq. (1a) are equivalent, as is done in the conven-tional London theory, is tantamount to making an addi-


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tional independent assumption, namely that no longitu-dinal electric field can exist inside superconductors.

Consequently it seems natural to abandon Eq. (1a)and explore the consequences of Eq. (22) in its full gen-erality. Starting from Eq. (22), the second London equa-tion (1b) also follows, taking the curl and setting thetime integration constant equal to zero as done by Lon-don. However to completely specify the problem we need

further assumptions. Following the early London work[3]we take as fundamental equation Eq. (6) together with

the condition that A obeys the Lorenz gauge:

Js = c

42LA (23a)

A = 1

c

t(23b)

These equations imply that the magnetic vector potentialthat enters into Londons equation (23a) is transverse ina static situation as in Londons case, but has a longitu-

dinal component in a time-dependent situation.Application of the divergence operator to Eq. (23a),together with Eq. (23b) and the continuity Eq. (7) thenleads to

t=

1

42L

t(24)

and integration with respect to time to

(r, t) 0(r) = 42L((r, t) 0(r)) (25)

where 0(r) and 0(r) are constants of integration. Apossible choice would be 0 = 0 = 0. Instead, motivatedby the theory of hole superconductivity[5, 6, 7, 8], wechoose

0(r) = 0 > 0 (26)

that is, a uniform positive constant in the interior ofthe superconductor. Equation (25) then implies thatthe electrostatic potential (r, t) equals 0(r) when thecharge density inside the superconductor is constant, uni-form and equal to 0, hence from Maxwells equations wededuce that 0(r) is given by

0(r) =

V

d3r0

|r r|(27)

where the integral is over the volume of the supercon-ducting body.

In summary, we propose that the macroscopic elec-trodynamic behavior of a superconductor is described byEqs. (23) and a single positive number 0, which togetherwith Eq. (27) determines the integration constants inEq. (25). 0 is a function of temperature, the particularmaterial, and the dimensions and shape of the super-conducting body[8]. In the following we explore someconsequences of this proposal.

IV. ELECTROSTATICS

For a static situation Eq. (25) is

(r) = 0(r) 42L((r) 0) (28)

with 0(r) given by Eq. (27)[11]. Using Poissons equa-tion we obtain for the charge density inside the super-

conductor

(r) = 0 + 2L

2(r) (29)

Inside and outside the superconductor the electrostaticpotential obeys

2((r) 0(r)) =1

2L((r) 0(r)) (30a)

2(r) = 0 (30b)

respectively. Furthermore we assume that no surface

charges can exist in superconductors, hence that both and its normal derivative /n are continuous acrossthe surface of the superconducting body. For given 0,these equations have a unique solution for each value ofthe average charge density of the superconductor

ave =1

V

d3r(r) (31)

In particular, ifave = 0 the solution is (r) = 0 every-where inside the superconductor and (r) = 0(r), with0 given by Eq. (27), valid both inside and outside thesuperconductor.

For the general case ave = 0 the solution to these

equations can be obtained numerically for any given bodyshape by the procedure discussed in ref. [8]. For a spher-ical body an analytic solution exists, and we speculatethat analytic solutions may exist for other shapes of highsymmetry. Quite generally, Eq. (29) implies that forbody dimensions much larger than the penetration depththe charge density is 0 deep in the interior of the super-conductor, and the potential is 0(r). Deviations of(r)from 0 exist within a layer of thickness L of the surface,to give rise to the given ave. In particular, for a chargeneutral superconductor, ave = 0, excess negative chargewill exist near the surface as discussed in ref. [8].

The electrostatic field is obtained from the usual rela-

tionE(r) = (r) (32)

and also satisfies the equation

E(r) = E0(r) + 2L

2 E(r) (33)

with E0(r) = 0(r). Deep in the interior, E(r) =E0(r). Because of Eq. (22), no current is generated bythis electrostatic field.


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If an external electrostatic field is applied, the chargedensity will rearrange so as to screen the external fieldover a distance L from the surface. This is easily seenfrom the superposition principle, since the total electricfield will be the sum of the original field and an added

field E(r) that satisfies

E(r) = 2L2 E(r) (34)

inside the superconductor, and approaches the value ofthe applied external field far from the superconductor.

Equation (34) implies that the additional field E(r) isscreened within a distance L from the surface, just asan applied magnetic field would be screened. Quantita-tive results for general geometries can be obtained by thesame procedure outlined in ref. [8] and will be discussedelsewhere.

For the particular case of a spherical geometry the so-lution to these equations is easily obtained. The electro-static potential for a sphere of radius R and total chargeq is (r) = (r) + 0(r), with

(r) =Q

f(R/L)

sinh(r/L)

r; r < R (35a)

(r) =Q

f(R/L)

sinh(R/L)

R+Q(

1

R

1

r) ; r > R (35b)

and

Q = Q0 q (36a)

Q0 =4

3 R30 (36b)

f(x) = xcoshx sinhx (36c)

0(r) =Q02R

(3 r2

R2); r < R (36d)

0(r) =Q0r

; r > R (36e)

and the electric field and charge density follow from Eqs.(32) and (28). In the presence of a uniform applied elec-tric field Eext the potential is (r) +

(r, ), with

(r, ) = a2Lr2

f(r/L)Eextcos ; r < R (37a)

(r, ) = (

r2 r)Eextcos ; r > R (37b)

with

a = 3R

sinh(R/L)(38a)

= R3[1 32LR2

f(R/L)

sinh(R/L)] (38b)

The induced charge density is easily obtained from Eq.(28). Note that a dipole moment Eext is induced on thesphere and that the polarizability becomes increasinglyreduced compared with the normal metal value = R3

as the ratio of radius to penetration length decreases. ForR >> L eq. (38b) yields = (R L)

3 as one wouldexpect, and for R


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instead of Eq. (40b).The simplicity of eqs. (40) derives from the fact that

the theory is relativistically covariant[3]. This is seen asfollows. We define the current 4-vector in the usual way

J = ( J(r, t),ic(r, t)) (42)

and the four-vector potential

A = ( A(r, t), i(r, t)) (43)

The continuity equation sets the four-dimensional diver-gence of the four-vector J equal to zero,

DivJ = 0 (44)

where the fourth derivative is /(ict), and the Lorenzgauge condition Eq. (23b) sets the divergence of the four-vector A to zero

DivA = 0 (45)

Furthermore we define the four-vectors associated withthe positive uniform charge density 0 and its associatedcurrent J0, denoted by J0, and the associated four-vectorpotential A0. In the frame of reference where the su-perconducting body is at rest the spatial part of thesefour-vectors is zero, hence

J0 = (0,ic0) (46a)

A0 = (0, i0(r)) (46b)

in that reference frame. In any inertial reference frame,A0 and J0 are obtained by Lorentz-transforming Eq.(46) and are related by

2A0 =

4

cJ0 (47)

with the dAlembertian operator

2 = 2

1

c22

t2(48)

according to Maxwells equations, just as the four-vectorsJ and A obey

2A =

4

c

J. (49)

Our fundamental equation is then the relation betweenfour-vectors

2(A A0) =

1

2L(A A0) (50a)

or, equivalently using Eqs. (47) and (49)

J J0 = c

42L(A A0) (50b)

which we propose to be valid in any inertial referenceframe. In the frame of reference at rest with respect tothe superconducting body, J0 and A0 have only time-likecomponents, in another reference frame they will alsohave space-like components. The spatial and time-likeparts of Eq. (50b) give rise to Eqs. (23a) and (25) re-spectively.

Equations (40) can also be written in covariant form.

Eqs. (40c) and (d) are

2(J J0) =

1

2L(J J0) (51a)

and Eqs. (40a,b)

2(F F0) =

1

2L(F F0) (51b)

where F is the usual electromagnetic field tensor andF0 is the field tensor with entries E0 and 0 for E andB respectively when expressed in the reference frame atrest with respect to the ions.

VII. RELATIVISTIC COVARIANCE

The fundamental equation (50) relates the relativemotion of the superfluid and the positive background,with four-currents J and J0, and associated vector po-tentials A and A0 respectively. It is a covariant rela-tion between four-vectors. This means it is valid in anyinertial reference frame, with the four-vectors in differ-ent inertial frames related by the Lorentz transformationconnecting the two frames.

In contrast, the conventional London equation Eq. (6)

with the condition Eq (8b) is not relativistically covari-ant, rather it is only valid in the reference frame wherethe superconducting solid is at rest. It can certainly beargued that the frame where the superconducting solidis at rest is a preferred reference frame, different fromany other reference frame. (In fact, our covariant equa-tions (50) do recognize the special status of that refer-ence frame, as the only frame where the spatial part ofthe four-vector J0 is zero.) Hence it is not a priori obvi-ous that the electrodynamics of superconductors has tobe describable by relativistically covariant equations.

However, imagine a superconductor with a flat surfaceof arbitrarily large extent, and an observer moving paral-lel to this surface. Since in the conventional London the-ory the superconductor is locally charge neutral, the stateof motion of the observer relative to the superconductoris determined by its motion relative to lateral surfaces ofthe body that are arbitrarily far away. According to theconventional London equations this observer should beable to detect instantaneously any change in the positionof these remote lateral surfaces. Since information cannottravel at speeds larger than the speed of light, this wouldimply that Londons equations cannot be valid for timesshorter than the time a light signal takes in traveling from


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the observer to the lateral surface; this time increases asthe dimensions of the body increase, and can become ar-bitrarily large unless one assumes there is some limitingvalue to the possible size of a superconductor. Clearly,to describe the electrodynamics of superconductors withlocal equations whose range of validity depends on thedimensions of the body is not satisfactory. Our covariantequations avoid this difficulty.

Note also that a relativistically covariant formulationdoes not make sense if 0 = 0, as assumed in the earlyLondon work[3]. In that case, the fundamental equation2A = (4/c)J makes no reference to the state of mo-

tion of the solid and would describe the same physicsirrespective of the relative motion of the superfluid andthe solid, in contradiction with experiment.

VIII. THE LONDON MOMENT

The presence of a magnetic field in rotatingsuperconductors[1] presents another difficulty in the con-

ventional London theory. A rotating superconductor hasa magnetic field in its interior given by[12]

B = 2mec

e (52)

with w the angular velocity. This is explained as fol-lows: in terms of the superfluid velocity vs the superfluid

current is Js = nsevs, so that Eq. (6) is

vs = e

mecA (53)

Next one assumes that in the interior the superfluid ro-tates together with the lattice, so that at position r

vs = r (54)

and replacing in Eq. (53) and taking the curl, Eq. (52)results.

The problem with this explanation is that, as discussedearlier, the conventional London equations are not covari-ant, rather they are valid only in the rest frame of thesuperconducting body. However in writing Eq. (54) oneis affirming the validity of Londons equation in a framethat is not the rest frame of the solid, but is a particularinertial frame. To assume the validity of the equationswith respect to one particular inertial frame that is notthe rest frame of the solid but not in other inertial frames

does not appear to be logically consistent and is reminis-cent of the old theories of the aether: one is statingthat the superconductor drags the aether with it if ittranslates but not if it rotates.

IX. LONDON RIGIDITY

In the conventional microscopic theory ofsuperconductivity[2], as well as in the theory of

hole superconductivitiy[6], the superfluid carriers arepairs of electrons with total spin 0. The Schrodingerequation for particles of spin 0 is the non-relativisticlimiting form of the more fundamental Klein-Gordonequation[18]. It is then reasonable to expect that theproper microscopic theory to describe superconductivityshould be consistent with Klein-Gordon theory. It isvery remarkable that the London brothers in their early

work[3], without knowledge of Cooper pairs, suggestedthe possible relevance of the Klein-Gordon equation tosuperconductivity.

In Klein-Gordon theory, the components of the current

four-vector J = ( J(r, t),ic(r, t)) in the presence of the

four-vector potential A = ( A(r, t), i(r, t)) are given interms of the scalar wave function (r, t) by[18]

J(r, t) =e

2m[(

i

e

cA(r, t)) +

(

i

e

cA(r, t))] (55a)

(r, t) =e

2mc2[(i

t e(r, t)) +

(i

t e(r, t))] (55b)

Deep in the interior of the superconductor we have (forsuperconductors of dimensions much larger than the pen-etration depth)

(r, t) = 0 (56a)

(r, t) = 0(r) (56b)

J(r, t) = A(r, t) = 0 (56c)

independent of any applied electric or magnetic fields.We now postulate, analogously to the conventionaltheory[1, 2], that the wave function (r, t) is rigid. Interms of the four-dimensional gradient operator

Grad = (, i

c

t) (57)

what we mean is that the combination

[Grad Grad] (58)

is unaffected by external electric and magnetic fields, aswell as by proximity to the boundaries of the supercon-ductor. With = ns, the superfluid density, thisassumption and Eqs. (55, 56) lead to

J(r, t) = nse

2

mecA(r, t) (59a)


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(r, t) 0 = nse

2

mec2((r, t) 0(r)) (59b)

i.e. the four components of the fundamental equationEq. (50b).

We believe this is a compelling argument in favor ofthe form of the theory proposed here. It is true that

for particles moving at speeds slow compared to thespeed of light the Klein-Gordon equation reduces to theusual Schrodinger equation. However, by the same to-ken the Schrodinger equation satisfied by Cooper pairscan be viewed as a limiting case of the Klein-Gordonequation. In the conventional theory in the frameworkof non-relativistic quantum mechanics, rigidity of thewave function leads to the second London equation. Itwould be unnatural to assume that the same argumentcannot be extended to the superfluid wavefunction in itsrelativistic version, independent of the speed at whichthe superfluid electrons are moving.

X. DIELECTRIC FUNCTION

As discussed in previous sections, the electric potentialin the interior of the superconductor satisfies

2( 0) 1

c22( 0)

t2=

1

2L( 0) (60a)

while outside the superconductor the potential satisfiesthe usual wave equation

2 1

c22

t2= 0 (60b)

If a harmonic potential ext(q, ) is applied, the super-conductor responds with an induced potential (q, ) re-lated to ext by

(q, ) =ext(q, )

s(q, )(61)

and we obtain for the longitudinal dielectric function ofthe superconductor

s(q, ) =2p + c

2q2 2

c2q2 2(62)

with

p =c

L= (

me4nse2

)1/2 (63)

the plasma frequency. For comparison, the dielectricfunction of the normal metal is given by the Linhardtdielectric function[13]

n(q, ) = 1 +4e2

q2

k

fk fk+qk+q k + i

(64)

with fk the Fermi function.Let us compare the behavior of the dielectric functions

for the superconductor and the normal metal. In thestatic limit we have for the superconductor from Eq. (62)

s(q, 0) = 1 +2p

c2q2= 1 +

1

2Lq2

(65)

For the normal metal, the zero frequency limit of theLinhardt function yields the Thomas Fermi dielectricfunction[13]

TF(q) = 1 +1

2TFq2

(66a)

with

1

2TF= 4e2g(F) (66b)

with g(F) the density of states at the Fermi energy. Eqs.(65) and (66) imply that static external electric fields arescreened over distances L and TF for the superconduc-

tor and the normal metal respectively. For free electronswe have,

g(F) =3n

2F(67)

so that

1

2TF=

6ne2

F=

1

2L

3mec2

2F(68)

assuming the density of superconducting electrons ns isthe same as that of normal electrons. Eq. (68) shows thatthe superconductor is much more rigid than the nor-mal metal with respect to charge distortions: the energycost involved in creating a charge distortion to screenan applied electric field is F in the normal metal versusmec2 in the superconductor, resulting in the much longerscreening length in the superconductor compared to thenormal metal.

The same rigidity is manifest in the dispersion relationfor longitudinal charge oscillations. From the zero of thedielectric function Eq. (62) we obtain for the plasmondispersion relation in the superconducting state

2q,s = 2p + c

2q2 (69)

Notably, this dispersion relation for longitudinal modesin the superconductor is identical to the one for trans-verse electromagnetic waves in this medium. In contrast,

the zeros of the Linhardt dielectric function yield for theplasmon dispersion relation[13]

2q,n = 2p +

3

5v2Fq

2 (70)

so that the plasmon dispersion relation is much steeperfor the superconductor, since typically vF 0.01c. Wecan also write Eqs. (69) and (70) as

q,s = p(1 +1

22Lq

2) (71a)


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q,n = p(1 +9

102TFq

2) (71b)

showing that low energy plasmons in superconductors re-quire wavelengths larger than L according to the alter-nate equations, in contrast to normal metals where thewavelengths are of order TF, i.e. interatomic distances.

This again shows the enhanced rigidity of the supercon-ductor with respect to charge fluctuations compared tothe normal metal.

XI. PLASMONS

In the conventional London theory one deduces fromEq. (1a) in the absence of normal state carriers

2

t2+ 2p = 0 (72)

upon taking the divergence on both sides and using thecontinuity equation. The solution of this equation is acharge oscillation with plasma frequency and arbitraryspatial distribution

(r, t) = (r)eipt (73)

In other words, the plasmon energy is independent ofwavevector. This indicates that charge oscillations witharbitrarily short wavelength can be excited in the super-conductor according to London theory. Clearly this isunphysical, as one would not expect charge oscillationswith wavelengths smaller than interelectronic spacings.

Consequently one has to conclude that the perfect con-ductor equation (1a) necessarily has to break down atsufficiently short lengthscales.

Experiments using EELS (electron energy loss spec-troscopy) have been performed on metals in the nor-mal state[14] and plasmon peaks have been observed,with plasmon dispersion relation approximately consis-tent with the prediction of the Linhardt dielectric func-tion Eq. (70). If Londons theory was correct one wouldexpect that in the superconducting state plasmon exci-tations energies should be independent of q, at least forvalues of q1 larger than interelectronic distances.

However instead it is expected from BCS theory thatplasmons below Tc should be very similar to plasmonsin the normal state[15, 16]. This expectation is basedon the fact that plasmon energies are several orders ofmagnitude larger than superconducting energy gaps, andas a consequence within BCS theory plasmons shouldbe insensitive to the onset of the superconducting state.However no EELS experiments appear to have ever beenperformed on superconducting metals to verify this ex-pectation.

In contrast, the counterpart to Eq. (72) with the al-ternate equations is the equation for the charge density

obtained from (40d):

2plt2

+ 2ppl = c22pl (74)

where pl is the difference between the charge densityand its static value obtained from solution of Eq. (29).The right-hand side of this equation gives a rigidity tocharge oscillations that is absent in the London model.From Eq. (74) we obtain the dispersion relation Eq. (68)for plasmons in the superconducting state.

Furthermore the allowed values of the wavevector qwill be strongly constrained in small samples of dimen-sion comparable to the penetration depth. Consider forsimplicity a small superconducting sphere of radius R. Aplasma oscillation is of the form

pl(r, t) = plsinqr

reiwq,st (75)

and because of charge neutrality

V d

3

rpl(r, t) = 0 (76)

we obtain the condition on the wavevector

tan(qR) = qR (77)

The smallest wavevector satisfying this condition is

q =4.493

R(78)

so that the smallest frequency plasmon has frequency

p = p1 + 20.22L

R2

(79)

This shift in the plasmon frequency can be very large forsmall samples. For example, for a sphere of radius R =10L, Eq. (74) yields a 20% blue shift in the minimumplasmon frequency.

The optical response of small samples will also bedifferent in our theory. Electromagnetic waves excitesurface plasmons in small metallic particles, and res-onance frequencies depend on sample shape and itspolarizability[17]. As the simplest example, for a spheri-cal sample the resonance frequency is given by[17]

2M =Q2

M

(80)

with Q the total mobile charge, M its mass and = R3

the static polarizability of a sphere of radius R. Weexpect the polarizability to become smaller in the su-perconducting state as given by eq. (38b), hence ourtheory predicts an increase in surface plasmon resonancefrequency upon entering the superconducting state forsmall samples, which should be seen for example in pho-toabsorption spectra. For example, for samples of radius100L and 10L the decrease in polarizability predicted


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10

by Eq. (38b) is 6% and 27% respectively. The conven-tional theory would predict no such change.

In summary, the conventional theory and our theorylead to very different consequences concerning the behav-ior of plasmon excitations when a normal metal is cooledinto the superconducting state. In the London theoryplasmons are predicted to be completely dispersionless.Within BCS theory, no change with respect to the normal

state is expected either in the plasmon dispersion relationnor in the long wavelength limit of the plasmon frequency. Instead, in our theory the plasmon dispersion shouldbe much steeper than in the normal state. Furthermorethe minimum volume plasmon frequency should becomelarger as the sample becomes smaller, and surface plas-mon resonance frequencies should also become larger forsmall samples.

XII. EXPERIMENTAL TESTS

We do not know of any existing experiments that would

be incompatible with the proposed theory. Here we sum-marize the salient features of the theory that may beamenable to experimental verification.

A. Screening of applied electric field

Our equations predict that longitudinal electric fieldsshould be screened over distances L rather than themuch shorter Thomas Fermi length. This could be testedby measuring changes in capacitance of a capacitor withsuperconducting metal plates, or with a superconduc-tor in the region between plates, upon onset of super-conductivity. Such an experiment was performed by H.London[4] in 1936 but no change was observed. We arenot aware of any follow-up experiment. More accurateexperiments should be possible now.

B. Measurement of charge inhomogeneity

The theory predicts excess negative charge within apenetration depth of the surface of a superconductor anda deficit of negative charge in the interior. It may bepossible to detect this charge inhomogeneity by directobservation, for example by electron microscopy or otherspectroscopic tools.

C. External electric field

For small superconducting samples of non-sphericalshape an electric field is predicted to exist outside thesuperconductor near the surface[8], which should be de-tectable by electrostatic measurements. Associated withit there should be a force between small superconductingparticles leading to the formation of spherical aggregates.

D. Internal electric field

The predicted internal electric field is small on a mi-croscopic scale but extends over macroscopic distances.Perhaps that makes it experimentally detectable.

E. Plasmons

Plasmon dispersion relations should be strongly af-fected by the transition to superconductivity, with plas-mons becoming much stiffer at low temperatures. Vol-ume and surface plasmon frequencies should increase inthe superconducting state for small samples. Small isdefined by the value of the ratio of a typical sample di-mension to L, and effects should be detectable even forthis ratio considerably larger than unity. EELS and op-tical experiments should be able to detect these changes.

F. Polarizability

The polarizability of small samples should be smallerin the superconducting than in the normal state. Theeffect should be largest at low temperatures. For sam-ples small compared to the penetration depth the polar-izability should scale as V5/3 rather than V, with V thevolume.

XIII. DISCUSSION

We have proposed a fundamental reformulation of the

conventional London electrodynamics. The proposedtheory is relativistically covariant and embodied in thesingle equation

2(A A0) =

1

2L(A A0) (81)

with A the four-vector potential, and A0 the four-vectorpotential corresponding to a uniform charge density 0at rest in the rest frame of the superconducting body.

Similarly as the conventional London theory[1], theequations proposed here can be understood as arisingfrom the rigidity of the microscopic wave function ofthe superfluid with respect to perturbations. However,in our case the relevant microscopic theory is the (rela-tivistically covariant) Klein-Gordon theory, appropriatefor spin 0 Cooper pairs, rather than the non-relativisticSchrodinger equation. Rigidity in this framework leadsinescapably to the new equation (25) and hence to thefour-dimensional Eq. (81). Furthermore, rigidity in ourcontext refers to both the effect of external electric andmagnetic fields on the superfluid wave function as wellas to the effect on it of proximity to the surface of thesuperconducting body in the absence of external fields.


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The constant 0 may be viewed as a phenomenologicalparameter arising from integration of equation (24), in-dependent of any microscopic theory. Instead, within thetheory of hole superconductivity 0 is a positive param-eter determined by the microscopic physics[5, 6, 7, 8].The magnitude of 0 does not correspond to an ionicpositive charge but is much smaller. It originates in theabsence of a small fraction of conduction electrons from

the bulk which, as a consequence of the undressing[7]associated with the transition to superconductivity, havemoved outwards to within a London penetration depthof the surface. In reference [8] we estimated the excessnegative charge near the surface for Nb to be one extraelectron per 500, 000 atoms. For a sample of 1cm radiusthis correspond to a deficit of 1 electron per 1011 atomsin the bulk, which gives rise to an electric field of order106V/cm near the surface. This electric field is very smallat a microscopic level, yet it gives rise to very large po-tential differences between different points in the interiorof a macroscopic sample.

The existence of A0 in Eq. (81), originating in thepositive charge 0, breaks charge conjugation symme-try. As discussed earlier, a non-zero 0 is necessary fora meaningful relativistically covariant theory. The pre-diction that 0 is positive for all superconductors followsfrom the fundamental electron-hole asymmetry of con-densed matter that is the focus of the theory of holesuperconductivity[6, 7, 19]. The fact that electron-holeasymmetry is a fundamental aspect of superconductivityis already experimentally established by the fact that themagnetic field of rotating superconductors always pointsin direction parallel, never antiparallel, to the mechan-ical angular momentum[20].

The electrodynamic equations proposed here describeonly the superfluid electrons. At finite temperatures be-

low Tc there will also be a normal fluid composed ofthermally excited quasiparticles. A two-fluid model de-scription of the system at finite temperatures should be

possible and lead to interesting insights.The theory discussed here appears to be simpler than

the conventional London theory in that it requires fewerindependent assumptions. It is also consistent with themore fundamental Klein-Gordon theory, while the con-ventional London theory is not, and it avoids certain diffi-culties of the conventional London theory. We do not be-lieve it contradicts any known experimental facts, except

for the 1936 experiment by H. London[4] which to ourknowledge has never been reproduced. Also, recent re-markable experiments by Tao and coworkers[21] indicatethat the properties of superconductors in the presence ofstrong static or quasistatic electric fields are not well un-derstood. The theory leads to many consequences thatare different from the conventional theory and should beexperimentable testable, as discussed in this paper. Itshould apply to all superconductors, with the magnitudeof the charge-conjugation symmetry breaking parameter0 being largest for high temperature superconductors[8].

Recent experiments indicate that optical properties ofcertain metals in the visible range are affected by the

onset of superconductivity[22]. This surprising couplingof low and high energy physics, unexpected within con-ventional BCS theory, was predicted by the theory ofhole superconductivity[23]. In this paper we find thatphysical phenomena associated with longitudinal plasmaoscillations, also a high energy phenomenon, should alsobe affected by superconductivity. Further discussion ofthe consequences of this theory and its relation with themicroscopic physics will be given in future work.

Acknowledgments

The author is grateful to A.S. Alexandrov and L.J.Sham for stimulating discussions, and to D. Bertrand forcalling Ref. 9 to his attention.

[1] F. London, Superfluids, Dover, New York, 1961.[2] M. Tinkham, Introduction to Superconductivity,

McGraw-Hill, New York, 1996, Ch. 1.[3] F. London and H. London, Proc.Roy.Soc. A149, 71

(1935); Physica 2, 341 (1935).[4] H. London, Proc.Roy.Soc. A155, 102 (1936).[5] J.E. Hirsch, Phys.Lett.A 281, 44 (2001); Phys.Lett. A

309, 457 (2003).[6] J.E. Hirsch and F. Marsiglio, Phys. Rev. B 39, 11515

(1989) ; F. Marsiglio and J.E. Hirsch, Phys. Rev. B41,6435 (1990).

[7] J.E. Hirsch, Phys.Rev.B 62, 14487 (2000); Phys.Rev.B62, 14498 (2000).

[8] J.E. Hirsch, Phys.Rev.B 68, 184502 (2003);Phys.Rev.Lett. 92, 016402 (2004).

[9] J. Govaerts, D. Bertrand and G. Stenuit, Supercond.Sci.Technol. 14, 463 (2001).

[10] G. Rickayzen, Theory of Superconductivity, John Wi-

ley & Sons, New York, 1965.[11] Note that we are using a slightly different convention

than in ref. [8], the definition of the potential differs bythe constant 42L0.

[12] R. Becker, F. Sauter and C. Heller, Z. Physik 85, 772(1933); A.F. Hildebrand , Phys.Rev.Lett. 8, 190 (1964);A.A. Verheijen et al, Physica B 165-166, 1181 (1990).

[13] J.M. Ziman, Principles of the Theory of Solids, Cam-bridge University Press, Cambridge, 1972.

[14] H. Raether, Excitation of Plasmons and InterbandTransitions by Electrons, Springer Tracts in ModernPhysics, Vol. 88, Springer-Verlag, Berlin, 1980.

[15] J. Bardeen and J.R. Schrieffer, Prog.Low Temp. Phys.Vol. III, North Holland, Amsterdam, 1961, p. 251.

[16] G. Rickayzen, Phys.Rev. 115, 795 (1959).[17] W.A. de Heer, Rev.Mod.Phys. 65, 611 (1993).[18] G. Baym, Lectures on Quantum Mechanics, W.A. Ben-

jamin, Reading, 1978, Chpt. 22.


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[19] J.E. Hirsch, Phys.Rev.B 65, 184502 (2002).[20] J.E. Hirsch, Phys.Rev.B 68, 012510 (2003).[21] R. Tao , X. Xu , Y.C. Lan , Y. Shiroyanagi , Physica C

377, 357 (2002); R. Tao, X. Xu X and E. Amr, PhysicaC 398, 78 (2003); Phys.Rev.B 68, 144505 (2003).

[22] H. J. A. Molegraaf, C. Presura, D. van der Marel, P.H. Kes, and M. Li Science 295, 2239 (2002); A.F.

Santander-Syro, R.P.S.M. Lobo, N. Bontemps, Z. Kon-stantinovic, Z.Z. Li and H. Raffy, cond-mat/0111539(2001), Europhys.Lett. 62, 568 (2003).

[23] J.E. Hirsch, Physica C 199, 305 (1992); Physica C 201,347 (1992).
http://arxiv.org/abs/cond-mat/0111539http://arxiv.org/abs/cond-mat/0111539

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