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Department of Physics, University of Sofia, Sofia 1164, Bulgaria
Received: 8 May 2017 / Accepted: 31 May 2017 / Published: 6 June 2017
In this paper we give a derivation of a system of equations which generalize the London brothers and Ginzburg–Landau systems of equations, to describe the electrodynamics of s-wave superconductors. First, we consider a relativistically covariant theory in terms of gauge four-vector electromagnetic potential and scalar complex field. We use the first-order formalism to obtain the supplemented Maxwell equations for gauge-invariant electric, magnetic, four-vector fields and the modulus of the superconducting order parameter. The new four-vector field appears in some of the equations as a gauge-invariant super-current, and in other ones, while gauge invariant, as a four-vector electromagnetic potential. This dual contribution of the new four-vector field is the basis of the electrodynamics of superconductors. We focus on the system of equations with time-independent fields. The qualitative analysis shows that the applied magnetic field suppresses the superconductivity, while the applied electric field impacts oppositely, supporting it. Secondly, we consider time-dependent non-relativistic Ginzburg–Landau theory.
The earliest study of the electrodynamics of s-wave superconductors is attributed to the London brothers . They supplemented the Maxwell system of equations with a set of equations to explain the electrodynamics of superconductors, and more particularly, the Meissner–Ochsenfeld effect. The generalized Maxwell–London equations are discussed in  without account for the spontaneous breakdown of the charge symmetry. The quantum-mechanical foundation of these equations was discussed in phenomenological Ginzburg–Landau theory .
There is also an attempt to explain the Meissner–Ochsenfeld effect in a purely classical way  and superconductivity as a limiting phenomena . In , a microscopic justification is given that a superconductor may have an electric field in its interior. The phenomenon is considered as a consequence of hole superconductivity  (see also ).
The main purpose of the present paper is to give a derivation of a system of equations which generalize the London brothers’ and Ginzburg–Landau systems of equations. In the Londons’ theory, the amplitude of the order parameter is not included, which does not permit the different impact of the applied electric and magnetic fields to be obtained. The Ginzburg–Landau theory does not consider the electric field.
The Londons’ system of equations is relativistic covariant . We want to generalize this system, and this is why we first consider a relativistically covariant theory in terms of gauge four-vector electromagnetic potentials and scalar complex field.
The results of the relativistic theory are very important guidance for the non-relativistic one, while the non-relativistic theory has more options.
We focus on the system of equations with time-independent fields. The qualitative analysis shows that the applied magnetic field suppresses the superconductivity, while the applied electric field impacts oppositely, supporting it.
The system of equations is derived from a relativistically non-covariant theory, and shows that the effect of the applied electric field depends on the direction of the field.
The paper is organized as follows: In Section 2, we derive the system of equations to describe the electrodynamics of s-wave superconductors from relativistically covariant theory of superconductivity. For the case when density of Cooper pairs is a constant, the system of equations reduces to the London brothers’ equations. The system of equations derived in the present paper includes an equation for the density of Cooper pairs which shows the different impacts of applied electric and magnetic fields on superconductivity. In Section 3, we use the same technique of calculations to consider time-dependent relativistically non-covariant Ginzburg–Landau theory. The main results are reported and commented in Section 4.
2. Relativistically Covariant Theory of Superconductivity
We begin with a well-known field-theory action  for a relativistically covariant theory of superconductivity
written in terms of gauge four-vector electromagnetic potential “A” and complex scalar field “”, the superconducting order parameter. The parameter
where T is the temperature and is the critical temperature, is positive when the system is a superconductor. We use the standard notations for relativistically covariant systems: , , where is the magnetic permeability and is the electric permittivity of the superconductor. We assume that these parameters do not change their values when the system undergoes normal-to-superconductor transition. The action (1) is invariant under the gauge transformations
where is a real function.
We represent the order parameter in the form
where is a gauge invariant, and the gauge transformation of is
The action (1), rewritten in terms of and , adopts the form
It is convenient to use the action in the first-order formalism
where gauge potential , phase , gauge invariant antisymmetric field , gauge invariant four-vector field and gauge invariant scalar field are assumed to be independent degrees of freedom in the theory.
To derive the system of Maxwell equations for s-wave superconductors, we vary the action (7) with respect to , , , , and . The resulting system of equations reads:
If we set in Equations (10)–(12) the expressions for and from Equations (8) and (9), we obtain the equations of motion following from the action (6). This means that theories with actions (6) and (7) are equivalent.
Alternatively, one eliminates the gauge fields and from Equations (8)–(12) to obtain the system of equations for the gauge invariant fields , and :
Equation (13) follows from Equation (8), while Equation (16) from Equation (9).
Straightforward calculations show that Equation (15) can be obtained from Equation (14) and Equation (13) from Equation (16). The system of independent equations is:
We construct the antisymmetric tensor by means of the electric and magnetic fields in a standard way: , and . In terms of , and Q, the system of equations which describes the electrodynamics of s-wave superconductors is:
It is important to stress that the gauge-invariant vector and scalar Q fields take part in Equations (22) and (24) as a magnetic vector and electric scalar potentials, while in Equation (21) is a supercurrent and in Equation (23) is a density of superconducting quasi-particles. This dual contribution of the new fields is the basis of the electrodynamics of superconductors.
We focus on the system of equations with time-independent fields:
To compare our result with the London equations , we assume that deep inside the superconductor and Q are zero, while is a constant determined from the equation , which follows from Equation (30). The gauge transformation (5) of the phase of the order parameter implies that one can impose the gauge fixing condition . In that case, the gauge-invariant vector is equal to the vector potential , the gauge invariant scalar field is equal to the scalar potential , and the system of Equations (26)–(29) adopts the form
Taking the curl of (31), using Equation (32) and the identity , which follows from this equation, we obtain
Taking the divergence of (34) and using the Equation (33) we obtain
Equations (37) and (39) imply that an electric field penetrates a distance as a magnetic field does .
This approximation is very rough and does not account for the last term in Equation (30), which is responsible for the different impact of applied electric and magnetic fields on the superconductivity. If we apply magnetic field (), the qualitative analysis of Equation (30) shows that the magnetic vector potential effectively decreases the parameter, , where is some average value. Therefore, the Ginzburg–Landau coherence length increases (see Appendix A), which means that applied magnetic field destroys superconductivity. On the other hand, when the electric field is applied (, the electric scalar potential effectively increases the parameter . Hence, the Ginzburg–Landau coherence length decreases. This qualitative analysis permits us to formulate the hypotheses that by applying electric field at very low temperature one increases the critical magnetic field. This result is experimentally testable.
3. Time-Dependent Ginzburg–Landau Theory
A number of authors have discussed the non-relativistic time-dependent generalization of the Ginzburg–Landau theory [11,12,13,14,15,16]. We investigate a model with field-theory action [13,15,16]
where is the electric scalar potential, with gauge transformation (3) , D is the normal-state diffusion constant , and () are effective charge and mass of superconducting quasi-particles. The index k runs .
We follow the same procedure to derive the system of equations which describe the electrodynamics of s-wave superconductors. We represent the order parameter by means of modulus and phase (4), and write the field-theory action, in the first-order formalism, in the form
It is important to stress that is a three-component vector field and . Using a variational principle, we obtain the system of equations
We supplement the system of Equations (42)–(47) with Equation (48):
which is a definition of the new gauge-invariant field Q. In the same way, starting from the system of Equations (42)–(48), we arrive at the Maxwell equations for superconductors in a non-relativistic theory.
The system of equations for time-independent fields is:
There are two important differences between Equations in relativistic theory (26)–(30) and Equations in non-relativistic one (54)–(58). In contrast to Equation (28), in Equation (56) there is no dependence on gauge-invariant scalar field Q. The last Equation (58) depends on the electric potential Q linearly, which makes the impact of the applied electric field on superconductivity quite nontrivial.
The electrodynamics of superconductors is based on the fundamental phenomenon in physics-spontaneous breakdown of gauge symmetry. The system of equations depends on the mechanism of symmetry breaking. The equations in the present paper describe the electrodynamics of s-wave superconductors, while the equations for d-wave or p-wave superconductors look in a different way. The equations are different if the superconductor is a metal or insulator in normal state. If the material is metal above the critical temperature, we have to account for the Ohm law, and we have to supplement Equation (26) with a term proportional to the electric field . The conductivity depends on the density of conducting electrons, and is responsible for the screening of the electric field . The density of normal quasiparticles decreases at low temperature; at very low temperature is zero, there is no quasiparticle current, and the system of Equations (26)–(30) is appropriate to describe the electrodynamics of s-wave superconductors. On the other hand, there are materials which possess superconductor–insulator transition. For these systems, Equations (26)–(30) describe the properties of the material well when an electric field is applied, even near the critical temperature. Therefore, we can apply an electric field near the critical superconductor–insulator transition temperature to study the critical behavior of this type of superconductor. One of the examples is sulfur hydride . It is a superconductor at very high pressure and temperature. Temperature dependence of the resistance, measured at different pressures, shows that the material undergoes a superconductor-insulator transition below 129 GPa.
The shape of the sample is very important. In the present paper, we study systems with half space () occupied by the superconductor. We assume that deep inside the superconductor the electric and magnetic fields approach zero and hence the fields and Q are zero, while is a constant. In this way, we obtain London brothers equations and Equations (37) and (39), which imply that an electric field penetrates as a magnetic field does. This approximation fails when the geometry of the system is with finite size—slab geometry. This is because the electric and magnetic fields are finite within a finite-size sample. Therefore, the vector and scalar fields and Q are not zero, and one cannot neglect them in the last term of Equation (30), and cannot set the density of Cooper pairs equal to constant. Due to this, Equations (37) and (39) cannot be considered even as a rough approximations. They are not a correct description of the electrodynamics of finite-size samples. This makes the electrodynamics of superconductors with finite sizes more difficult to investigate.
The possibility that a superconductor may have an electric field in its interior has been discussed in other theoretical frameworks [1,6,10]. The equations in these papers discuss the charge and electric field distribution in superconductors. They result from assumptions  and phenomenological considerations. This is in stark contrast to the present paper, where the Maxwell equations for s-type superconductors are obtained from first principles. In References [1,6,10], authors consider a system of equations for electric field, magnetic field, scalar electric potential and vector magnetic potential. They consider the density of Cooper pairs equal to constant included in the London penetration depth (38). While the resulting Equation (37) explains the Meissner–Ochsenfeld effect, Equations (37) and (39) do not match the physical reality.
In the present paper, a system of equations based on strong mathematical fundament is derived. They describe the electrodynamics of all s-wave superconductors at low temperature, where there are no normal quasiparticles and electric field can penetrate, and at all temperatures below the critical one when the normal state of the superconductor is insulator.
When the model is relativistically covariant, we obtained that the applied electric field supports the superconductivity. When the model is non-relativistic, the impact of the applied electric field on superconductivity is more complicated.
The equations in relativistic theory (26)–(30) are invariant under the discrete transformation , , and independently under the transformation , . In contrast, the system of equations in non-relativistic theory (54)–(58) are invariant under the discrete transformation of magnetic field and gauge-invariant field , but they are not invariant under the discrete transformation of electric field and gauge-invariant field Q. This means that the effects of the applied electric fields and on superconductivity are different.
It is important to underline that the fields Q and are gauge-invariant. This is why they should be measurable, as the electric and magnetic fields are measurable. The role of these fields is fundamental in superconductivity, but is not investigated.
Conflicts of Interest
The author declares no conflict of interest.
To elucidate the qualitative analysis in Section 2, we consider the system of Equations (26)–(30) for fields which depend on z coordinate only. Then, the system of equations for the fields , , , , and adopts the form
After some calculations, one reduces the system (A1)–(A5) to a system of equations for , and
It is convenient to introduce dimensionless functions and of a dimensionless distance , where
is the Ginzburg–Landau coherence length:
In Equations (A10) , the applied electric field is and the applied magnetic field is . The representations of the electric and magnetic fields by means of and are the following:
The system of Equations (A6)–(A8), rewritten in terms of the new functions, reads:
which satisfies for type I superconductors and for type II ones, and parameters and are
For semi-infinite superconductors, with a surface of superconductor orthogonal to the z-axis, the boundary conditions are:
If neither electric nor magnetic fields are applied, the equation for the dimensionless function
is exactly solvable, and the solution for is
The qualitative analysis in Section 2 shows that applied electric field increases the parameter , where is an average value of the scalar field. Within this approximation, the expression for is
where . When the magnetic field is applied, decreases, , and function reads
The three curves (A17)–(A19) are depicted in Figure A1. Graph (a) shows the function (A17) when neither electric nor magnetic fields are applied, graph (b) shows the function (A18), with , and graph (c) shows the function (A19) with .
Curve a: function (A17); Curve b: function (A18) with ; Curve c: function (A19) with .
Curve a: function (A17); Curve b: function (A18) with ; Curve c: function (A19) with .
The Ginzburg–Landau (GL) coherence length measures the distance over which the superconducting order parameter increases up to the bulk value, measured from the surface of the superconductor (). The applied electric field decreases the GL coherence length, which means that the electric field supports the superconductivity, while the applied magnetic field increases the GL coherence length, which means that the magnetic field destroys the superconductivity.
The numerical solutions of the system of Equation (A12)  supports the qualitative analysis in Section 2 and Appendix A.
London, F.; London, H. The electromagnetic equations of the supraconductor. Proc. R. Soc. Lond.1935, 149, 71–88. [Google Scholar] [CrossRef]
Abrahams, E.; Tsuneto, T. Time variation of the Ginzberg–Landau order parameter. Phys. Rev.1966, 152, 416–425. [Google Scholar] [CrossRef]
Caroli, C.; Maki, K. Motion of the Vortex Structure in Type-II Superconductors in High Magnetic Field. Phys. Rev.1967, 159, 306–316. [Google Scholar] [CrossRef]
Gor’kov, L.P.; Eliashberg, G.M. Eliashberg, Generalization of Ginzburg–Landau Equations for Non-Stationary Problems in the Case of Alloys with Paramagnetic Impurities. Sov. Phys. JETP1968, 27, 328–333. [Google Scholar]
Thompson, R.S. Microwave, Flux Flow, and Fluctuation Resistance of Dirty Type-II Superconductors. Phys. Rev.1970, 1, 327–331. [Google Scholar] [CrossRef]
Hu, C.R.; Thompson, R.S. Dynamic Structure of Vortices in Superconductors. Phys. Rev.1972, 6, 110–130. [Google Scholar] [CrossRef]
Michael, T. Introduction to Superconductivity; Courier Corporation: North Chelmsford, MA, USA, 1975. [Google Scholar]