Electrodynamics of Superconductors Has to According to Maxwell’S Equations, Just As the Four-Vectors J a Be Describable by Relativistically Covariant Equations

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However the right-hand-side of this equation is not gauge- so that the total current J~ = J~n + J~s satisfies invariant, while the left-hand side is. From the continuity ~ 2 ~ equation, ∂J c ~ ∂E = 2 E + σn (10) ∂t 4πλL ∂t ∂ρ ∇~ · J~ + = 0 (7) ∂t with the London penetration depth λL given by 2 it can be seen that Eq. (6) is only valid with a choice of 1 4πnse 2 = 2 . (11) gauge that satisfies λL mec Applying the divergence operator to Eq. (10) and using ~ ~ mec ∂ρ ∇ · A = 2 (8a) Eqs. (2c) and (7) leads to nse ∂t ∂2ρ ∂ρ c2 In particular, in a time-independent situation (i.e. when 2 +4πσn + 2 ρ = 0 (12) electric and magnetic fields are time-independent) the ∂t ∂t λL ~ vector potential A in Eq. (6) is necessarily transverse, London argued[1] that Eq. (12) leads to a rapid decay i.e. of any charge buildup in superconductors, given by the time scales ∇~ · A~ = 0 (8b) −1 2 c 2 1/2 τ1,2 =2πσn ± [(2πσn) − ( ) ] (13) It is currently generally accepted that the London Eq. λL (6) is valid for superconductors in all situations, time- the slower of which he estimated to be 10−12sec, and independent or not, with the gauge chosen so that Eq. consequently that any charge buildup inside the super- ~ ~ (8b) holds (London gauge), and hence ∇ · J = 0 from conductor can be ignored. He concluded from this that Eq. (6). Together with Eq. (1a) this is equivalent to it is reasonable to assume ρ = 0 inside superconductors, assuming that no longitudinal electric fields (i.e. ∇~ · E~ 6= which from the continuity equation and Eq. (6) implies 0) can exist inside superconductors, as can be seen by Eq. (8b). The same argument is given in Ref. [10]. applying the divergence operator to Eq. (1a). However, this numerical estimate is based on assuming However, the possibility of an electrostatic field inside for σn a value appropriate for the normal state, and for superconductors was recently suggested by the theory λL its value near zero temperature. Instead, the tem- of hole superconductivity[5, 6, 7], which predicts that perature dependence of σn and λL should be considered. negative charge is expelled from the interior of super- The conductivity of normal carriers σn should be propor- conductors towards the surface. The conventional Lon- tional to the number of ’normal electrons’, that goes to don electrodynamics is incompatible with that possibil- zero as the temperature approaches zero, hence σn → 0 ity, however the alternate electrodynamics proposed here as T → 0. On the other hand, the number of super- is not. We have already discussed some consequences for fluid electrons ns goes to zero as T approaches Tc, hence the electrostatic case in recent work[8]. λL → ∞ as T → Tc. These facts lead to a strong temper- Recently, Govaerts, Bertrand and Stenuit have dis- ature dependence of the relaxation times in Eq. (13), and cussed an alternate Ginzburg-Landau formulation for su- in fact to the conclusion that long-lived charge fluctua- perconductors that is relativistically covariant and has tions should exist both when T → Tc and when T → 0. some common elements with the theory discussed here[9]. Eq. (12) describes a damped harmonic oscillator. The For the case of a uniform order parameter their equations crossover between overdamped (at high T ) and under- reduce to the early London theory[3] that allows for elec- damped (low T ) regimes occurs for trostatic fields within a penetration length of the surface c of a superconductor, as our theory also does. However in 2πσn(T )= (14) contrast to the theory discussed here, the theory of Gor- λL(T ) vaets et al does not allow for electric charge nor electric As T → T , λ (T ) → ∞ and σ (T ) approaches its nor- fields deep in the interior of superconductors. c L n mal state value at Tc; as T → 0, λL(T ) → λL(0) and σn(T ) → 0; hence condition Eq. (14) will always be satisfied at some temperature between 0 and Tc. For exam- II. DIFFICULTIES WITH THE LONDON ple, for a superconductor with low temperature London MODEL penetration depth λL ∼ 200A˚ and normal state resistivity ρ ∼ 10µΩcm the crossover would be at T/Tc ∼ 0.69; if Following London[1], let us assume that in addition λL ∼ 2000A˚ and ρ ∼ 1000µΩcm, at T/Tc ∼ 0.23. How- to superfluid electrons there are normal electrons, giving ever, no experimental signature of such a cross-over be- rise to a normal current tween overdamped and underdamped charge oscillations at some temperature below Tc has ever been reported for J~n = σnE~ (9) any superconductor. 3

For T approaching Tc, the slower timescale in Eq. (13) which implies that an electric field penetrates a distance is λL, as a magnetic field does. Indeed, electromagnetic waves in superconductors penetrate a distance λL, hence 4πσn(T ) τ1 ∼ (15) Eq. (20) properly describes the screening of transverse (c/λ (T ))2 L electric fields. However Eq. (20) does not depend on the so that for T sufficiently close to Tc overdamped charge frequency and hence should remain valid in the static fluctuations should persist for arbitrarily long times. limit; but such a situation is incompatible with the first However, such a space charge would give rise to an elec- London equation (1a), as it would lead to arbitrarily large tric field in the interior of the superconductor and hence currents. This then suggests that application of an ar- (due to Eq. (1a)) to a current that grows arbitrarily bitrarily small static or quasi-static electric field should large, destroying superconductivity. No such phenomena lead to destruction of the superconducting state, which have ever been observed in superconductors close to Tc is not observed experimentally. to our knowledge. In a normal metal these difficulties do not arise. Ap- At low temperatures, Eq. (13) implies that under- plying the divergence operator to Eq. (9) and using (2c) damped charge oscillations should exist, with damping and (7) yields timescale ∂ρ 1 = −4πσnρ (21) τ1 = (16) ∂t 2πσn(T ) which predicts that any charge fluctuation in the in- diverging as T → 0. The frequency of these oscillations terior of metals is screened over a very short time (∼ is 10−17secs), and consequently that a longitudinal electric c2 field cannot penetrate a normal metal. A static uniform ω = − 4π2σ2 (17) λ2 n electric field can penetrate a normal metal and it causes s L a finite current to flow whose magnitude is limited by the and as T → 0 it approaches the plasma frequency normal state resistivity. c In summary, London’s equations together with ωp = (18) Maxwell’s equations lead to unphysical predictions re- λ L garding the behavior of superconductors in connection Hence at zero temperature plasma oscillations in the su- with space charges and electric fields, and to predictions perconducting state are predicted to exist forever, just that appear to be contradicted by experiment. How are as persistent currents. No such persistent charge oscilla- these difficulties avoided in the conventional London pic- tions have been observed to our knowledge. ture? By postulating that ρ = 0 inside superconductors, Reference [10] recognized this difficulty and suggested and that applied static electric fields do not penetrate the that the London electrodynamic equations should only superconductor[1, 2, 10]. These are postulates that are be assumed to be valid ’in situations in which the fields completely independent of Eqs. (1) and (2), for which no do not tend to build up a space charge’, hence not in the obvious justification within London’s theory exists. regions close to Tc and close to T = 0 discussed above. On the other hand, the consequences of London equations concerning magnetic fields and persistent currents III. THE ALTERNATE EQUATIONS are generally believed to be valid for arbitrary temperatures, and indeed experiments support this expectation. The possibility of an alternative to the conventional The fact that the implications of London’s equations con- London equations is suggested by the fact that taking cerning the behavior of the charge density in supercon- the time derivative of Eq. (6) and using Eq. (4) leads to ductors appear to have at best a limited range of validity is disturbing and suggests a fundamental inadequacy of these equations. ∂J~ n e2 s = s (E~ + ∇~ φ) (22) A related difficulty with London’s equations arises ∂t me from consideration of the equation for the electric field. Taking the time derivative of Eq. (2b) and using (2a), without making any additional assumptions on the gauge (2c) and (1a) yield of A~. Clearly, Eq. (22) is consistent with the existence of a static electric field in a superconductor, deriving from 2 ~ 2 ~ 1 ~ 1 ∂ E ~ an electrostatic potential φ, which will not generate an ∇ E = 2 E + 2 2 +4π∇ρ (19) λL c ∂t electric current, contrary to the prediction of Eq. (1a). How it may be possible for a static electric field to exist For slowly varying electric fields and assuming no charge in a superconductor without generating a time-dependent density in the superconductor current is discussed in ref. [8]. Assuming that Eq. (22) 2 ~ 1 ~ and Eq. (1a) are equivalent, as is done in the conven- ∇ E = 2 E (20) λL tional London theory, is tantamount to making an addi- 4 tional independent assumption, namely that no longitu- IV. ELECTROSTATICS dinal electric field can exist inside superconductors. Consequently it seems natural to abandon Eq. (1a) For a static situation Eq. (25) is and explore the consequences of Eq. (22) in its full gen- 2 erality. Starting from Eq. (22), the second London equa- φ(~r)= φ0(~r) − 4πλL(ρ(~r) − ρ0) (28) tion (1b) also follows, taking the curl and setting the time integration constant equal to zero as done by Lon- with φ0(~r) given by Eq. (27)[11]. Using Poisson’s equa- don. However to completely specify the problem we need tion we obtain for the charge density inside the super- further assumptions. Following the early London work[3] conductor we take as fundamental equation Eq. (6) together with 2 2 the condition that A~ obeys the Lorenz gauge: ρ(~r)= ρ0 + λL∇ ρ(~r) (29)

~ c ~ Inside and outside the superconductor the electrostatic Js = − 2 A (23a) potential obeys 4πλL

2 1 ∇ (φ(~r) − φ0(~r)) = 2 (φ(~r) − φ0(~r)) (30a) 1 ∂φ λ ∇~ · A~ = − (23b) L c ∂t

These equations imply that the magnetic vector potential ∇2φ(~r) = 0 (30b) that enters into London’s equation (23a) is transverse in a static situation as in London’s case, but has a longitu- respectively. Furthermore we assume that no surface dinal component in a time-dependent situation. charges can exist in superconductors, hence that both Application of the divergence operator to Eq. (23a), φ and its normal derivative ∂φ/∂n are continuous across together with Eq. (23b) and the continuity Eq. (7) then the surface of the superconducting body. For given ρ0, leads to these equations have a unique solution for each value of the average charge density of the superconductor ∂ρ 1 ∂φ = − (24) ∂t 4πλ2 ∂t 1 L ρ = d3rρ(~r) (31) ave V and integration with respect to time to Z

2 In particular, if ρave = ρ0 the solution is ρ(~r)= ρ0 every- φ(~r, t) − φ0(~r)= −4πλ (ρ(~r, t) − ρ0(~r)) (25) L where inside the superconductor and φ(~r)= φ0(~r), with φ0 given by Eq. (27), valid both inside and outside the where φ0(~r) and ρ0(~r) are constants of integration. A superconductor. possible choice would be φ0 = ρ0 = 0. Instead, motivated For the general case ρ 6= ρ0 the solution to these by the theory of hole superconductivity[5, 6, 7, 8], we ave equations can be obtained numerically for any given body choose shape by the procedure discussed in ref. [8]. For a spher- ρ0(~r)= ρ0 > 0 (26) ical body an analytic solution exists, and we speculate that analytic solutions may exist for other shapes of high that is, a uniform positive constant in the interior of symmetry. Quite generally, Eq. (29) implies that for the superconductor. Equation (25) then implies that body dimensions much larger than the penetration depth the electrostatic potential φ(~r, t) equals φ0(~r) when the the charge density is ρ0 deep in the interior of the super- charge density inside the superconductor is constant, uni- conductor, and the potential is φ0(~r). Deviations of ρ(~r) form and equal to ρ0, hence from Maxwell’s equations we from ρ0 exist within a layer of thickness λL of the surface, deduce that φ0(~r) is given by to give rise to the given ρave. In particular, for a charge neutral superconductor, ρave = 0, excess negative charge 3 ′ ρ0 will exist near the surface as discussed in ref. [8]. φ0(~r)= d r (27) |~r − ~r′| The electrostatic field is obtained from the usual rela- ZV tion where the integral is over the volume of the superconducting body. E~ (~r)= −∇~ φ(~r) (32) In summary, we propose that the macroscopic electrodynamic behavior of a superconductor is described by and also satisfies the equation Eqs. (23) and a single positive number ρ0, which together ~ ~ 2 2 ~ with Eq. (27) determines the integration constants in E(~r)= E0(~r)+ λL∇ E(~r) (33) Eq. (25). ρ0 is a function of temperature, the particular material, and the dimensions and shape of the super- with E~0(~r) = −∇~ φ0(~r). Deep in the interior, E~ (~r) = conducting body[8]. In the following we explore some E~0(~r). Because of Eq. (22), no current is generated by consequences of this proposal. this electrostatic field. 5

If an external electrostatic field is applied, the charge with density will rearrange so as to screen the external field 3R over a distance λL from the surface. This is easily seen a = − (38a) from the superposition principle, since the total electric sinh(R/λL) field will be the sum of the original field and an added ′ field E~ (~r) that satisfies λ2 f(R/λ ) α = R3[1 − 3 L L ] (38b) R2 sinh(R/λ ) ~ ′ 2 2 ~ ′ L E (~r)= λL∇ E (~r) (34) The induced charge density is easily obtained from Eq. inside the superconductor, and approaches the value of (28). Note that a dipole moment αEext is induced on the the applied external field far from the superconductor. sphere and that the polarizability α becomes increasingly 3 Equation (34) implies that the additional field E~ ′(~r) is reduced compared with the normal metal value α = R screened within a distance λL from the surface, just as as the ratio of radius to penetration length decreases. For 3 an applied magnetic field would be screened. Quantita- R >> λL eq. (38b) yields α = (R − λL) as one would 5 2 tive results for general geometries can be obtained by the expect, and for R<<λL, α = R /15λL. same procedure outlined in ref. [8] and will be discussed elsewhere. For the particular case of a spherical geometry the so- V. MAGNETOSTATICS lution to these equations is easily obtained. The electrostatic potential for a sphere of radius R and total charge The magnetostatics for our case is identical to the con- q is φ(r)= φ˜(r)+ φ0(r), with ventional case. From Eq. (23a), (2b) and (2d) it follows that Q sinh(r/λ ) φ˜(r)= L ; r < R (35a) ~ 2 2 ~ B(~r)= λL∇ B(~r) (39) f(R/λL) r giving rise to the usual Meissner effect. The generated screening supercurrent is not sensitive to the presence of Q sinh(R/λ ) 1 1 φ˜(r)= L +Q( − ) ; r > R (35b) the electrostatic field. f(R/λL) R R r and VI. ELECTRODYNAMICS

Q = Q0 − q (36a) Using Eqs. (23) and Maxwell’s equations the following equations result for the electrodynamic behavior of superconductors: 4π 3 Q0 = R ρ0 (36b) 2 ~ 3 2 ~ 1 ~ 1 ∂ B ∇ B = 2 B + 2 2 (40a) λL c ∂t

f(x)= xcoshx − sinhx (36c) 2 ~ ~ 2 ~ ~ 1 ~ ~ 1 ∂ (E − E0) ∇ (E − E0)= 2 (E − E0)+ 2 2 (40b) λL c ∂t 2 Q0 r φ0(r)= (3 − ); r < R (36d) 2 ~ 2R R2 2 ~ 1 ~ 1 ∂ J ∇ J = 2 J + 2 2 (40c) λL c ∂t

Q0 2 φ0(r)= ; r > R (36e) 2 1 1 ∂ (ρ − ρ0) r ∇ (ρ − ρ0)= 2 (ρ − ρ0)+ 2 2 (40d) λL c ∂t and the electric field and charge density follow from Eqs. (32) and (28). In the presence of a uniform applied elec- so that all quantities obey exactly the same equation. ′ ~ tric field Eext the potential is φ(r)+ φ (r, θ), with Eqs. (40a,b,c) with ρ0 and E0 = 0 are also obtained from the conventional London equations only if one im- 2 ′ λ poses the additional assumption that ρ = 0 inside the φ (r, θ)= a L f(r/λ )E cosθ ; r < R (37a) r2 L ext superconductor. For example, the equation for the electric field in London’s case is 1 1 ∂2E~ ′ α ∇2E~ = E~ + +4π∇~ρ (41) φ (r, θ) = ( 2 − r)Eextcosθ ; r > R (37b) 2 2 2 r λL c ∂t 6 instead of Eq. (40b). which we propose to be valid in any inertial reference The simplicity of eqs. (40) derives from the fact that frame. In the frame of reference at rest with respect to the theory is relativistically covariant[3]. This is seen as the superconducting body, J0 and A0 have only time-like follows. We define the current 4-vector in the usual way components, in another reference frame they will also have space-like components. The spatial and time-like J = (J~(~r, t),icρ(~r, t)) (42) parts of Eq. (50b) give rise to Eqs. (23a) and (25) respectively. and the four-vector potential Equations (40) can also be written in covariant form. Eqs. (40c) and (d) are A = (A~(~r, t), iφ(~r, t)) (43) 2 1 (J − J0 )= (J − J0 ) (51a) The continuity equation sets the four-dimensional diver- λ2 gence of the four-vector J equal to zero, L and Eqs. (40a,b) DivJ = 0 (44) 2 1 (F − F0 )= (F − F0 ) (51b) where the fourth derivative is ∂/∂(ict), and the Lorenz λ2 gauge condition Eq. (23b) sets the divergence of the four- L vector A to zero where F is the usual electromagnetic field tensor and F0 is the field tensor with entries E~0 and 0 for E~ and DivA = 0 (45) B~ respectively when expressed in the reference frame at Furthermore we define the four-vectors associated with rest with respect to the ions. the positive uniform charge density ρ0 and its associated ~ J current J0, denoted by 0 , and the associated four-vector VII. RELATIVISTIC COVARIANCE potential A0 . In the frame of reference where the superconducting body is at rest the spatial part of these four-vectors is zero, hence The fundamental equation (50) relates the relative motion of the superfluid and the positive background, J J J0 = (0,icρ0) (46a) with four-currents and 0 , and associated vector po- tentials A and A0 respectively. It is a covariant relation between four-vectors. This means it is valid in any inertial reference frame, with the four-vectors in differ- A0 = (0, iφ0(~r)) (46b) ent inertial frames related by the Lorentz transformation connecting the two frames. in that reference frame. In any inertial reference frame, In contrast, the conventional London equation Eq. (6) A J 0 and 0 are obtained by Lorentz-transforming Eq. with the condition Eq (8b) is not relativistically covari- (46) and are related by ant, rather it is only valid in the reference frame where the superconducting solid is at rest. It can certainly be 2 4π A0 = − J0 (47) argued that the frame where the superconducting solid c is at rest is a preferred reference frame, different from with the d’Alembertian operator any other reference frame. (In fact, our covariant equations (50) do recognize the special status of that refer- 2 2 2 1 ∂ ence frame, as the only frame where the spatial part of = ∇ − 2 2 (48) c ∂t the four-vector J0 is zero.) Hence it is not a priori obvious that the electrodynamics of superconductors has to according to Maxwell’s equations, just as the four-vectors J A be describable by relativistically covariant equations. and obey However, imagine a superconductor with a flat surface 4π of arbitrarily large extent, and an observer moving paral- 2A = − J . (49) c lel to this surface. Since in the conventional London theory the superconductor is locally charge neutral, the state Our fundamental equation is then the relation between of motion of the observer relative to the superconductor four-vectors is determined by its motion relative to lateral surfaces of the body that are arbitrarily far away. According to the 2 A A 1 A A ( − 0 )= 2 ( − 0 ) (50a) conventional London equations this observer should be λL able to detect instantaneously any change in the position or, equivalently using Eqs. (47) and (49) of these remote lateral surfaces. Since information cannot travel at speeds larger than the speed of light, this would J J c A A imply that London’s equations cannot be valid for times − 0 = − 2 ( − 0 ) (50b) 4πλL shorter than the time a light signal takes in traveling from 7 the observer to the lateral surface; this time increases as hole superconductivitiy[6], the superfluid carriers are the dimensions of the body increase, and can become ar- pairs of electrons with total spin 0. The Schrodinger bitrarily large unless one assumes there is some limiting equation for particles of spin 0 is the non-relativistic value to the possible size of a superconductor. Clearly, limiting form of the more fundamental Klein-Gordon to describe the electrodynamics of superconductors with equation[18]. It is then reasonable to expect that the local equations whose range of validity depends on the proper microscopic theory to describe superconductivity dimensions of the body is not satisfactory. Our covariant should be consistent with Klein-Gordon theory. It is equations avoid this difficulty. very remarkable that the London brothers in their early Note also that a relativistically covariant formulation work[3], without knowledge of Cooper pairs, suggested does not make sense if ρ0 = 0, as assumed in the early the possible relevance of the Klein-Gordon equation to London work[3]. In that case, the fundamental equation superconductivity. 2A = −(4π/c)J makes no reference to the state of mo- In Klein-Gordon theory, the components of the current tion of the solid and would describe the same physics four-vector J = (J~(~r, t),icρ(~r, t)) in the presence of the irrespective of the relative motion of the superfluid and four-vector potential A = (A~(~r, t), iφ(~r, t)) are given in the solid, in contradiction with experiment. terms of the scalar wave function Ψ(~r, t) by[18]

e ∗ ~ e J~(~r, t) = [Ψ ( ∇−~ A~(~r, t))Ψ + VIII. THE LONDON MOMENT 2m i c ~ e ∗ Ψ(− ∇−~ A~(~r, t))Ψ ] (55a) The presence of a magnetic field in rotating i c superconductors[1] presents another difficulty in the conventional London theory. A rotating superconductor has e ∗ ∂ a magnetic field in its interior given by[12] ρ(~r, t) = [Ψ (i~ − eφ(~r, t))Ψ + 2mc2 ∂t 2m c ~ e ∂ ∗ B = − ~ω (52) Ψ(−i~ − eφ(~r, t))Ψ ] (55b) e ∂t with ~w the angular velocity. This is explained as fol- Deep in the interior of the superconductor we have (for lows: in terms of the superfluid velocity ~vs the superfluid superconductors of dimensions much larger than the pen- ~ current is Js = nse~vs, so that Eq. (6) is etration depth) e ~ ~vs = − A (53) ρ(~r, t)= ρ0 (56a) mec Next one assumes that in the interior the superfluid rotates together with the lattice, so that at position ~r φ(~r, t)= φ0(~r) (56b) ~vs = ~ω × ~r (54) and replacing in Eq. (53) and taking the curl, Eq. (52) results. J~(~r, t)= A~(~r, t)=0 (56c) The problem with this explanation is that, as discussed earlier, the conventional London equations are not covari- independent of any applied electric or magnetic fields. ant, rather they are valid only in the rest frame of the We now postulate, analogously to the conventional superconducting body. However in writing Eq. (54) one theory[1, 2], that the wave function ψ(~r, t) is ’rigid’. In is affirming the validity of London’s equation in a frame terms of the four-dimensional gradient operator that is not the rest frame of the solid, but is a particular i ∂ inertial frame. To assume the validity of the equations Grad = (∇~ , − ) (57) with respect to one particular inertial frame that is not c ∂t the rest frame of the solid but not in other inertial frames what we mean is that the combination does not appear to be logically consistent and is reminis- ∗ ∗ cent of the old theories of the ’aether’: one is stating [Ψ GradΨ − ΨGradΨ ] (58) that the superconductor ’drags’ the ’aether’ with it if it translates but not if it rotates. is unaffected by external electric and magnetic fields, as well as by proximity to the boundaries of the supercon- ∗ ductor. With Ψ Ψ = ns, the superfluid density, this IX. LONDON RIGIDITY assumption and Eqs. (55, 56) lead to

In the conventional microscopic theory of n e2 J~(~r, t)= − s A~(~r, t) (59a) superconductivity[2], as well as in the theory of mec 8

with fk the Fermi function. 2 Let us compare the behavior of the dielectric functions nse for the superconductor and the normal metal. In the ρ(~r, t) − ρ0 = − 2 (φ(~r, t) − φ0(~r)) (59b) mec static limit we have for the superconductor from Eq. (62) i.e. the four components of the fundamental equation Eq. (50b). ω2 1 ǫ (q,ω → 0)=1+ p =1+ (65) We believe this is a compelling argument in favor of s 2 2 2 2 c q λLq the form of the theory proposed here. It is true that for particles moving at speeds slow compared to the For the normal metal, the zero frequency limit of the speed of light the Klein-Gordon equation reduces to the Linhardt function yields the Thomas Fermi dielectric usual Schrodinger equation. However, by the same to- function[13] ken the Schrodinger equation satisfied by Cooper pairs 1 ǫ (q)=1+ (66a) can be viewed as a limiting case of the Klein-Gordon T F λ2 q2 equation. In the conventional theory in the framework T F of non-relativistic quantum mechanics, ’rigidity’ of the with wave function leads to the second London equation. It 1 2 would be unnatural to assume that the same argument 2 =4πe g(ǫF ) (66b) λT F cannot be extended to the superfluid wavefunction in its relativistic version, independent of the speed at which with g(ǫF ) the density of states at the Fermi energy. Eqs. the superfluid electrons are moving. (65) and (66) imply that static external electric fields are screened over distances λL and λT F for the superconductor and the normal metal respectively. For free electrons X. DIELECTRIC FUNCTION we have, 3n g(ǫF )= (67) As discussed in previous sections, the electric potential 2ǫF in the interior of the superconductor satisfies so that 2 2 2 2 1 ∂ (φ − φ0) 1 1 6πne 1 3m c ∇ (φ − φ0) − = (φ − φ0) (60a) e c2 ∂t2 λ2 2 = = 2 × (68) L λT F ǫF λL 2ǫF while outside the superconductor the potential satisfies assuming the density of superconducting electrons ns is the usual wave equation the same as that of normal electrons. Eq. (68) shows that the superconductor is much more ’rigid’ than the nor- 1 ∂2φ ∇2φ − = 0 (60b) mal metal with respect to charge distortions: the energy c2 ∂t2 cost involved in creating a charge distortion to screen an applied electric field is ǫF in the normal metal versus If a harmonic potential φext(q,ω) is applied, the super- 2 mec in the superconductor, resulting in the much longer conductor responds with an induced potential φ(q,ω) re- screening length in the superconductor compared to the lated to φext by normal metal. The same rigidity is manifest in the dispersion relation φext(q,ω) φ(q,ω)= (61) for longitudinal charge oscillations. From the zero of the ǫ (q,ω) s dielectric function Eq. (62) we obtain for the plasmon and we obtain for the longitudinal dielectric function of dispersion relation in the superconducting state the superconductor 2 2 2 2 ωq,s = ωp + c q (69) ω2 + c2q2 − ω2 Notably, this dispersion relation for longitudinal modes ǫ (q,ω)= p (62) s c2q2 − ω2 in the superconductor is identical to the one for transverse electromagnetic waves in this medium. In contrast, with the zeros of the Linhardt dielectric function yield for the

c me 1/2 plasmon dispersion relation[13] ωp = = ( 2 ) (63) λL 4πnse 2 2 3 2 2 ωq,n = ωp + vF q (70) the plasma frequency. For comparison, the dielectric 5 function of the normal metal is given by the Linhardt so that the plasmon dispersion relation is much steeper dielectric function[13] for the superconductor, since typically vF ∼ 0.01c. We can also write Eqs. (69) and (70) as 2 4πe fk − fk+q ǫn(q,ω)=1+ 2 (64) 1 2 2 q ǫk+q − ǫk − ~ + iδ ωq,s = ωp(1 + λLq ) (71a) Xk 2 9

obtained from (40d): 9 2 2 2 ∂ ρpl 2 2 2 ωq,n = ωp(1 + λT F q ) (71b) + ω ρ = c ∇ ρ (74) 10 ∂t2 p pl pl showing that low energy plasmons in superconductors re- where ρpl is the difference between the charge density quire wavelengths larger than λL according to the alter- and its static value obtained from solution of Eq. (29). nate equations, in contrast to normal metals where the The right-hand side of this equation gives a ’rigidity’ to wavelengths are of order λT F , i.e. interatomic distances. charge oscillations that is absent in the London model. This again shows the enhanced ’rigidity’ of the supercon- From Eq. (74) we obtain the dispersion relation Eq. (68) ductor with respect to charge fluctuations compared to for plasmons in the superconducting state. the normal metal. Furthermore the allowed values of the wavevector q will be strongly constrained in small samples of dimen- sion comparable to the penetration depth. Consider for XI. PLASMONS simplicity a small superconducting sphere of radius R. A plasma oscillation is of the form In the conventional London theory one deduces from sinqr Eq. (1a) in the absence of normal state carriers ρ (~r, t)= ρ eiwq,st (75) pl pl r ∂2ρ and because of charge neutrality + ω2ρ = 0 (72) ∂t2 p 3 d rρpl(~r, t)=0 (76) upon taking the divergence on both sides and using the V continuity equation. The solution of this equation is a Z charge oscillation with plasma frequency and arbitrary we obtain the condition on the wavevector spatial distribution tan(qR)= qR (77) − ρ(~r, t)= ρ(~r)e iωpt (73) The smallest wavevector satisfying this condition is In other words, the plasmon energy is independent of 4.493 q = (78) wavevector. This indicates that charge oscillations with R arbitrarily short wavelength can be excited in the superconductor according to London theory. Clearly this is so that the smallest frequency plasmon has frequency unphysical, as one would not expect charge oscillations 2 with wavelengths smaller than interelectronic spacings. λL ω˜p = ωp 1+20.2 2 (79) Consequently one has to conclude that the ’perfect con- r R ductor’ equation (1a) necessarily has to break down at This shift in the plasmon frequency can be very large for sufficiently short lengthscales. small samples. For example, for a sphere of radius R = Experiments using EELS (electron energy loss spec- 10λ , Eq. (74) yields a 20% blue shift in the minimum troscopy) have been performed on metals in the nor- L plasmon frequency. mal state[14] and plasmon peaks have been observed, The optical response of small samples will also be with plasmon dispersion relation approximately consis- different in our theory. Electromagnetic waves excite tent with the prediction of the Linhardt dielectric func- surface plasmons in small metallic particles, and res- tion Eq. (70). If London’s theory was correct one would onance frequencies depend on sample shape and its expect that in the superconducting state plasmon exci- polarizability[17]. As the simplest example, for a spheri- tations energies should be independent of q, at least for − cal sample the resonance frequency is given by[17] values of q 1 larger than interelectronic distances. However instead it is expected from BCS theory that Q2 ω2 = (80) plasmons below Tc should be very similar to plasmons M Mα in the normal state[15, 16]. This expectation is based on the fact that plasmon energies are several orders of with Q the total mobile charge, M its mass and α = R3 magnitude larger than superconducting energy gaps, and the static polarizability of a sphere of radius R. We as a consequence within BCS theory plasmons should expect the polarizability to become smaller in the su- be insensitive to the onset of the superconducting state. perconducting state as given by eq. (38b), hence our However no EELS experiments appear to have ever been theory predicts an increase in surface plasmon resonance performed on superconducting metals to verify this ex- frequency upon entering the superconducting state for pectation. small samples, which should be seen for example in pho- In contrast, the counterpart to Eq. (72) with the al- toabsorption spectra. For example, for samples of radius ternate equations is the equation for the charge density 100λL and 10λL the decrease in polarizability predicted 10 by Eq. (38b) is 6% and 27% respectively. The conven- D. Internal electric field tional theory would predict no such change. In summary, the conventional theory and our theory The predicted internal electric field is small on a mi- lead to very different consequences concerning the behav- croscopic scale but extends over macroscopic distances. ior of plasmon excitations when a normal metal is cooled Perhaps that makes it experimentally detectable. into the superconducting state. In the London theory plasmons are predicted to be completely dispersionless. Within BCS theory, no change with respect to the normal E. Plasmons state is expected either in the plasmon dispersion relation nor in the long wavelength limit of the plasmon frequency . Instead, in our theory the plasmon dispersion should Plasmon dispersion relations should be strongly af- be much steeper than in the normal state. Furthermore fected by the transition to superconductivity, with plas- the minimum volume plasmon frequency should become mons becoming much stiffer at low temperatures. Vol- larger as the sample becomes smaller, and surface plas- ume and surface plasmon frequencies should increase in mon resonance frequencies should also become larger for the superconducting state for small samples. ’Small’ is small samples. defined by the value of the ratio of a typical sample di- mension to λL, and effects should be detectable even for this ratio considerably larger than unity. EELS and op- XII. EXPERIMENTAL TESTS tical experiments should be able to detect these changes.

We do not know of any existing experiments that would be incompatible with the proposed theory. Here we sum- F. Polarizability marize the salient features of the theory that may be amenable to experimental verification. The polarizability of small samples should be smaller in the superconducting than in the normal state. The effect should be largest at low temperatures. For sam- A. Screening of applied electric field ples small compared to the penetration depth the polarizability should scale as V 5/3 rather than V , with V the Our equations predict that longitudinal electric fields volume. should be screened over distances λL rather than the much shorter Thomas Fermi length. This could be tested by measuring changes in capacitance of a capacitor with XIII. DISCUSSION superconducting metal plates, or with a superconductor in the region between plates, upon onset of super- We have proposed a fundamental reformulation of the conductivity. Such an experiment was performed by H. conventional London electrodynamics. The proposed London[4] in 1936 but no change was observed. We are theory is relativistically covariant and embodied in the not aware of any follow-up experiment. More accurate single equation experiments should be possible now. 2 1 (A − A0 )= (A − A0 ) (81) λ2 B. Measurement of charge inhomogeneity L

with A the four-vector potential, and A0 the four-vector The theory predicts excess negative charge within a potential corresponding to a uniform charge density ρ0 penetration depth of the surface of a superconductor and at rest in the rest frame of the superconducting body. a deficit of negative charge in the interior. It may be Similarly as the conventional London theory[1], the possible to detect this charge inhomogeneity by direct equations proposed here can be understood as arising observation, for example by electron microscopy or other from the ’rigidity’ of the microscopic wave function of spectroscopic tools. the superfluid with respect to perturbations. However, in our case the relevant microscopic theory is the (relativistically covariant) Klein-Gordon theory, appropriate C. External electric field for spin 0 Cooper pairs, rather than the non-relativistic Schrodinger equation. Rigidity in this framework leads For small superconducting samples of non-spherical inescapably to the new equation (25) and hence to the shape an electric field is predicted to exist outside the four-dimensional Eq. (81). Furthermore, rigidity in our superconductor near the surface[8], which should be de- context refers to both the effect of external electric and tectable by electrostatic measurements. Associated with magnetic fields on the superfluid wave function as well it there should be a force between small superconducting as to the effect on it of proximity to the surface of the particles leading to the formation of spherical aggregates. superconducting body in the absence of external fields. 11

The constant ρ0 may be viewed as a phenomenological possible and lead to interesting insights. parameter arising from integration of equation (24), in- The theory discussed here appears to be ’simpler’ than dependent of any microscopic theory. Instead, within the the conventional London theory in that it requires fewer theory of hole superconductivity ρ0 is a positive param- independent assumptions. It is also consistent with the eter determined by the microscopic physics[5, 6, 7, 8]. more fundamental Klein-Gordon theory, while the con- The magnitude of ρ0 does not correspond to an ionic ventional London theory is not, and it avoids certain diffi- positive charge but is much smaller. It originates in the culties of the conventional London theory. We do not be- absence of a small fraction of conduction electrons from lieve it contradicts any known experimental facts, except the bulk which, as a consequence of the ’undressing’[7] for the 1936 experiment by H. London[4] which to our associated with the transition to superconductivity, have knowledge has never been reproduced. Also, recent removed outwards to within a London penetration depth markable experiments by Tao and coworkers[21] indicate of the surface. In reference [8] we estimated the excess that the properties of superconductors in the presence of negative charge near the surface for Nb to be one extra strong static or quasistatic electric fields are not well un- electron per 500, 000 atoms. For a sample of 1cm radius derstood. The theory leads to many consequences that this correspond to a deficit of 1 electron per 1011 atoms are different from the conventional theory and should be in the bulk, which gives rise to an electric field of order experimentable testable, as discussed in this paper. It 6 10 V/cm near the surface. This electric field is very small should apply to all superconductors, with the magnitude at a microscopic level, yet it gives rise to very large po- of the charge-conjugation symmetry breaking parameter tential differences between different points in the interior ρ0 being largest for high temperature superconductors[8]. of a macroscopic sample. Recent experiments indicate that optical properties of The existence of A0 in Eq. (81), originating in the certain metals in the visible range are affected by the positive charge ρ0, breaks charge conjugation symme- onset of superconductivity[22]. This surprising coupling try. As discussed earlier, a non-zero ρ0 is necessary for of low and high energy physics, unexpected within con- a meaningful relativistically covariant theory. The pre- ventional BCS theory, was predicted by the theory of diction that ρ0 is positive for all superconductors follows hole superconductivity[23]. In this paper we find that from the fundamental electron-hole asymmetry of con- physical phenomena associated with longitudinal plasma densed matter that is the focus of the theory of hole oscillations, also a high energy phenomenon, should also superconductivity[6, 7, 19]. The fact that electron-hole be affected by superconductivity. Further discussion of asymmetry is a fundamental aspect of superconductivity the consequences of this theory and its relation with the is already experimentally established by the fact that the microscopic physics will be given in future work. magnetic field of rotating superconductors always points in direction parallel, never antiparallel, to the mechan- ical angular momentum[20]. The electrodynamic equations proposed here describe Acknowledgments only the superfluid electrons. At finite temperatures below Tc there will also be a normal fluid composed of The author is grateful to A.S. Alexandrov and L.J. thermally excited quasiparticles. A two-fluid model de- Sham for stimulating discussions, and to D. Bertrand for scription of the system at finite temperatures should be calling Ref. 9 to his attention.

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