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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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Electrodynamics of Superconductors Has to According to Maxwell’S Equations, Just As the Four-Vectors J a Be Describable by Relativistically Covariant Equations Electrodynamics of superconductors J. E. Hirsch Department 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 macroscopic level 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 are relativistically covariant, and they can be understood as arising from the ’rigidity’ of the superfluid wave function in a relativistically covariant microscopic theory. They predict that an internal ’spontaneous’ electric field exists in superconductors, and that externally applied electric fields, both longitudinal and transverse, are screened over a London penetration length, as magnetic fields are. The associated longitudinal dielectric function predicts a much steeper plasmon dispersion relation than the conventional theory, and a blue shift of the minimum plasmon frequency for small samples. It is argued that the conventional London equations lead to difficulties that are removed in the present theory, and that the proposed equations do not contradict any known experimental facts. Experimental tests are discussed. PACS numbers: I. INTRODUCTION these equations are not correct, and propose an alternate set of equations. It has been generally accepted that the electrodynam- There is ample experimental evidence in favor of Eq. ics of superconductors in the ’London limit’ (where the (1b), which leads to the Meissner effect. That equation response to electric and magnetic fields is local) is de- is in fact preserved in our alternative theory. However, scribed by the London equations[1, 2]. The first London we argue that there is no experimental evidence for Eq. equation (1a), even if it appears compelling on intuitive grounds. In fact, the London brothers themselves in their early ∂J~ n e2 work considered the possibility that Eq. (1a) may not s = s E~ (1a) ∂t me be valid for superconductors[3]. However, because of the result of an experiment[4] they discarded an alternative describes the colisionless response of a conducting fluid of possibility and adopted both Eqs. (1a) and Eq. (1b), density ns, i.e. free acceleration of the superfluid carriers which became known as the ’first’ and ’second’ London with charge e and mass me, giving rise to the supercur- equations. ~ rent Js. The second London equation It is useful to introduce the electric scalar and magnetic 2 vector potentials φ and A~. The magnetic field is given nse ∇×~ J~s = − B~ (1b) by mec ~ ~ ~ is obtained from Eq. (1a) using Faraday’s law and setting B = ∇× A (3) a time integration constant equal to zero, and leads to the and Eq. (2a) is equivalent to Meissner effect. These equations together with Maxwell’s equations 1 ∂A~ E~ = −∇~ φ − (4) 1 ∂B~ c ∂t ∇×~ E~ = − (2a) c ∂t The magnetic vector potential is undefined to within the gradient of a scalar function. The gauge transformation 4π 1 ∂E~ A~ → A~ + ∇~ f (5a) arXiv:cond-mat/0312619v4 [cond-mat.str-el] 19 Apr 2004 ∇×~ B~ = J~ + (2b) c s c ∂t 1 ∂f ~ ~ φ → φ − (5b) ∇ · E =4πρ (2c) c ∂t leaves the electric and magnetic fields unchanged. The second London equation Eq. (1b) can be written ~ ~ ∇ · B = 0 (2d) as are generally believed to determine the electrodynamic n e2 J~ = − s A~ (6) behavior of superconductors. In this paper we argue that mec 2 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 sat- isfied 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 resistiv- ity 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.
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