Nonstationary Superconductivity: Quantum Dissipation and Time-Dependent Ginzburg-Landau Equation

Abstract

Transport equations of the macroscopic superfluid dynamics are revised on the basis of a combination of the conventional (stationary) Ginzburg-Landau equation and Schrödinger's equation for the macroscopic wave function (often called the order parameter) by using the well-known Madelung-Feynman approach to representation of the quantum-mechanical equations in hydrodynamic form. Such an approach has given (a) three different contributions to the resulting chemical potential 𝜇_𝑠 for the superfluid component, (b) a general hydrodynamic equation of superfluid motion, (c) the continuity equation for superfluid flow with a relaxation term involving the phenomenological parameters 𝜏_GL and 𝜉_GL, (d) a new version of the time-dependent Ginzburg-Landau equation for the modulus of the order parameter which takes into account dissipation effects and reflects the charge conservation property for the superfluid component. The conventional Ginzburg-Landau equation also follows from our continuity equation as a particular case of stationarity. All the results obtained are mutually consistent within the scope of the chosen phenomenological description and, being model-neutral, applicable to both the low-𝑇_c and high-𝑇_c superconductors.

1. Introduction

In superconductivity, the present efforts of scientists are mainly concentrated on studying the high-𝑇_c superconductors. Yet, there are some theoretical problems in the low-𝑇_c superconductivity left unsolved up to now. One of them (relating to thermoelectricity and having remained mysterious for many years because of great discrepancy between theory and experiments) has recently been explained by using the Fermi liquid description and charge imbalance notion [1].

Another problem left unsolved from the past is associated with the nonequilibrium behavior of superfluid flow when time-dependent situations result in dissipative effects, which are non-existent in the stationary regime. In particular, many efforts were spent previously to convert the usual (stationary) Ginzburg-Landau equation into a time-dependent (nonstationary) form but without effect, unfortunately, because all such attempts resulted in an equation of charge non-conserving form.

Before pursuing this point in more detail, it is useful to consider the similar problem relating to elaboration of the dissipative quantum theory for usual (non-superconductive) systems with damping.

On the classical level, a variational principle was first applied to dissipative systems by Bateman [2] on the basis of doubling the system degrees of freedom. Bateman's classical approach was later imported to quantum field theory [3] and this method has been since then applied to a number of problems. However, for our purposes it is more interesting to consider another approach set forth below.

Over sixty years ago, Caldirola [4] and Kanai [5] derived a quantum-mechanical equation for the wave function 𝜓. They have described the damped motion of a charged particle with charge 𝑒 and mass 𝑚 by introducing a phenomenological dissipation parameter 𝜆. Another form of the nonlinear wave equation, known as the Schrödinger–Langevin equation, was obtained by Kostin [6] when examining the quantum motion of a Brownian particle.

Afterwards, Wagner [7] carried out a comparative study of the Caldirola-Kanai and Schrödinger-Langevin equations on the basis of the hydrodynamic treatment, which was first suggested by Madelung [8]. In accordance with Madelung's picture, the wave function is represented as 𝜓=√𝜌exp(𝑖𝑆/ℏ). In this case, the separation of Schrödinger's equation into the real and imaginary parts yields both the continuity equation, 𝜕𝜌𝜕𝑡+∇⋅(𝜌𝐯)=0,(1) and the hydrodynamic equation of motion,𝜕𝐯𝜕𝑡+(𝐯⋅∇)𝐯=−𝑚𝟏∇(𝑉+𝑄)−𝜆𝐯,(2) where 𝐯=𝟏𝑚∇S,𝑄=−2ℏ𝑚²∇²𝜌^1/2𝜌^1/2.(3)

The quantum hydrodynamic equation (2) takes into account dissipative effects in the ordinary way by introducing a phenomenological relaxation term −𝜆𝐯. Besides the electric potential 𝑉, it contains the so-called quantum potential 𝑄 (both 𝑉 and 𝑄 are measured in 𝑒𝑉). In the classical limit ℏ→0, when the quantity 𝑆 makes sense of the classical action, the quantum potential 𝑄 vanishes.

Expressions (3) are valid only for the Schrödinger-Langevin-Kostin formulation [6], while the Caldirola-Kanai equations [4, 5] produce some extra loss factors for 𝐯 and 𝑄, exp(−𝜆𝑡) and exp(−2𝜆𝑡), respectively.

As is shown by Wagner [7], in the dynamic regime with taking into account both the scalar potential 𝜑=𝑉/𝑒 and the vector potential 𝐀, when (in the SI system)𝐄=−𝜕𝐀𝜕𝑡−∇𝜑,𝐁=∇×𝐀,(4) expression (3) for the quantum potential 𝑄 remains unchanged. However, expression (3) for the hydrodynamic velocity 𝐯 assumes the following form: 𝐯=(∇𝑆−𝑒𝐀)𝑚.(5) Moreover, with allowance for 𝐯, the hydrodynamic equation (2) acquires some additional terms which, however, have no reasonable physical explanation [7].

It should be noted that the quantum potential concept appearing in Madelung's formulation of quantum mechanics [8] was applied by a number of authors to solve different physical problems (e.g., see [9–12]) including quantum dissipation effects in Schrödinger-Langevin-Kostin formulation [13, 14].

As applied to the superfluid dynamics in superconductors, the problem of quantum dissipation is treated in the form of the time-dependent Ginzburg-Landau (TDGL) equation (e.g., see [15]). Generalization of the usual Ginzburg-Landau (GL) stationary theory [16–18] to time-varying situations was the subject of much investigation (e.g., see reviews [19–21]). However, all known versions of the TDGL equation suffer from a general drawback: they have no charge conservation property, that is, do not provide the continuity equation for the superfluid flow (but not for the total flow, superfluid and normal fluid together) (see the text after formula (38)).

As will be shown below, unlike the nonsuperconductive systems with losses subject to (1) and (2), insertion of the quantum dissipation effect in the TDGL equation for superconductors gives rise to the continuity equation for superfluid flow in the form with nonzero right-hand part, whereas the equation of supercarrier motion involves no relaxation term like −𝜆𝐯. This inference results from applying Madelung's treatment to our version of the TDGL equation which results from combining Schrödinger's equation with the conventional Ginzburg-Landau equation.

Section 2 deals with such a phenomenological treatment based on the quantum-mechanical Madelung-Feynman approach to obtain the transport equations for superfluid motion. Our modified version of the TDGL equation provides a basis for deriving the desired forms of the continuity equation, the dynamic equation of motion, and the general expression for the total chemical potential of the superfluid component in dynamic regime.

Section 3 converts the continuity equation for superfluid flow into two forms corresponding to the stationary and time-dependent Ginzburg-Landau equations written, for the modulus of the order parameter.

2. Transport Equations of Macroscopic Superfluid Dynamics

2.1. Madelung-Feynman's Approach to the Quantum-Mechanical Equations in Hydrodynamic Form

Madelung [8] was the first to suggest the hydrodynamic form of quantum theory by decomposing Schrödinger's equation into the real and imaginary parts. As is noted in Section 1, Madelung's approach was first applied [4–7] to the dissipative quantum-mechanical systems described by the microscopic wave function 𝜓=√𝜌exp(𝑖𝑆/ℏ), which gives rise to the customary hydrodynamic equations (1) and (2).

Afterwards, Feynman [22, 23] extended this approach to superconductivity assuming the macroscopic wave function (often called the order parameter) written in the following form Ψ(𝐫,𝑡)=𝑎(𝐫,𝑡)exp[𝑖Θ(𝐫,𝑡)],(6) satisfies the ordinary Schrödinger equation to describe the superconducting condensate. The reasoning for such an assumption is that the macroscopic wave function describes a large number of zero-spin particles (Cooper's pairs), all doing precisely the same thing (strongly correlated motion). So the time rate of change in this function must be exactly the same as the rate for any single particle. This wave function, being complex-valued, is normalized so that |Ψ|²≡𝑎²=𝑁_s/2, where 𝑁_s is the superconducting electron density. The phase Θ≡𝑆/ℏ is associated with the canonical (generalized) momentum of Cooper's pairs 𝐏=2𝑚𝐯_s+𝑒𝐀,(7) by the known relation 𝐏=ℏ∇Θ≡∇𝑆 [17, 18].

Relation (7) is a generalization of expression (5) to Cooper's pairs with charge 𝑞_s=2𝑒 and mass 𝑚_s=2𝑚, having velocity 𝐯_s and concentration 𝑁_s/2.

On these grounds, following Feynman [22, 23] we write Schrödinger's equation for the macroscopic wave function (6) in the conventional form (using the SI system) 𝑖ℏ𝜕Ψ𝜕𝑡=2𝑚1_s−𝑖ℏ∇−𝑞_s𝐀²+𝑈Ψ.(8) Here the potential energy, written as𝑈=𝑞_s𝜑+2𝜇_s0=2𝑒𝜑+𝜇_s0,(9) is determined not only by the electric potential 𝜑 but also by the chemical potential 2𝜇_s0 for Cooper's pairs in the condensate. A physical meaning of the chemical potential 𝜇_s0 will be discussed later (see formula (49)). Its insertion into (9) is an essential feature of our treatment. This point distinguishes between our form (8) of Schrödinger's equation for the macroscopic wave function Ψ and the similar one for the microscopic wave function 𝜓 (with no insertion of the chemical potential into the potential energy (9), as it is usually done in the conventional quantum-mechanical theory).

Substituting the expression (6) for Ψ in Schrödinger's equation (8) and separating the result into the real and imaginary parts, following Madelung [8], yield: (i) from the imaginary part 𝜕𝑁_s𝜕𝑡+∇⋅𝑁^s𝐯_s=0,(10)(ii)from the real part 𝜇_s0+𝜇_s1+𝑒𝜑+𝑚𝑣^2s2+ℏ2𝜕𝜕Θ𝑡=0.(11)

Here, in addition to 𝜇_s0, there is an extra contribution to the total chemical potential 𝜇_s1=−ℏ²8𝑚∇^𝟐𝑁^1/2_s𝑁^1/2_s.(12)

As is seen from expression (12), the additional chemical potential 𝜇_s1 is due to a spatial nonuniformity of the superfluid density 𝑁_s, which takes place, for example, in the vicinity of any vortex line or the NS-interface.

Relation (11) includes also the term 𝑚𝑣²_s/2 caused by superfluid flow with velocity 𝐯_𝑠. Let us apply the gradient operation to (11) by using the known identity [24] ∇𝑚𝑣^2s2=𝑚𝐯^s⋅∇𝐯_s+𝑚𝐯_s×∇×𝐯_s(13) and the total derivative of 𝐯_s with respect to time𝑑𝐯_s𝑑𝑡≡𝜕𝜕𝐯𝑡^s+𝐯^s⋅∇𝐯_s=𝜕𝐯^s𝜕𝑡+∇𝑣2^2s−𝐯^s×∇×𝐯_s.(14)

Employing the following expression for 𝐯_s resulted from (7):𝐯_s=𝐏2𝑚−𝑚𝑒𝐀≡2ℏ𝑚∇Θ−𝑚𝑒𝐀(15) and expression (4) for 𝐁, we obtain∇×𝐯_𝑠=−𝑒𝑚∇×𝐀=−𝑚𝑒𝐁.(16)

After the above transformations applied to (11), we arrive at the following equation of superfluid motion: 𝑚𝑑𝐯^s𝑑𝑡=𝑒𝐄+𝐯^s×𝐁−∇𝜇_s0+𝜇_s1.(17)

A comparison of (2) and (4) with (17) shows that the chemical potential (12), being doubled, can be interpreted, by analogy with formula (3) for 𝑄, as the quantum potential of Cooper's pairs (𝜌_s=𝑞_s𝑁_s/2=𝑒𝑁_s) 𝑄_s≡2𝜇_s1=−ℏ²2𝑚_s∇²𝜌^1/2_s𝜌^1/2_s.(18)

The continuity equation (10) obtained at this stage of our examination does not take into account the dynamic processes of breaking and creating for Cooper's pairs. This point should not be surprising because such processes are necessarily beyond the ordinary Schrödinger equation (8). This equation has been used above as an initial one to obtain (10) and (11) and it describes only the dynamic processes of nondissipative character. In the phenomenological theory of nonstationary superconductivity, the dissipation effects are usually attained by using the time-dependent Ginzburg-Landau (TDGL) equation [18–21]. The TDGL equation is a generalization of the ordinary GL equation (originally obtained [16] for the steady-state nonuniform superconductors) to time-varying situations.

2.2. Combined Approach to the Schrödinger and Ginzburg-Landau Equations

In their pioneer work [16], Ginzburg and Landau began searching phenomenologically a proper stationary form for the free energy functional,ℱ_s=𝑓_sΨ,Ψ^∗,𝐀𝑑𝑉,(19) where its density 𝑓_s depends on the vector potential 𝐀, as well as on the complex order parameter Ψ given in the form of (6). To find Ψ(𝐫), (19) was derived by requiring ℱ_s to be minimum with respect to variation of Ψ^∗, that is, 𝛿ℱ_s/𝛿Ψ^∗=0. Such an operation has led to the usual GL equation [16, 18]𝜕𝑓_s𝜕Ψ^∗≡12𝑚_s−𝑖ℏ∇−𝑞_s𝐀²+𝛼+𝛽||Ψ||²Ψ=0,(20) where the zero equality is required to ensure stationary conditions. The parameters 𝛼 and 𝛽 introduced phenomenologically are often associated with two other measurable temperature-dependent quantities: the equilibrium superfluid density 𝑁^∘_s(𝑇)=2|Ψ^∘|² and the thermodynamic critical magnetic field 𝐻_c(𝑇) by means of the following relations (written in the SI system of units) [18, 25]:𝛼(𝑇)=−2𝜇₀𝐻²_c(𝑇)𝑁^∘_s(𝑇),𝛽(𝑇)=2𝑁||𝛼^∘_s((𝑇𝑇))||.(21) Here (and also below) the superscript ° corresponds to the equilibrium state and 𝜇₀ is the vacuum permeability.

In the literature, there are different versions of the time-dependent generalization of the stationary GL equation (20). A common feature of all the known phenomenological approaches is the assumption that the function Ψ(𝐫,𝑡) tends to its stationary value (corresponding to minimum of the free energy functional ℱ_s) at a rate governed by the slope of the density 𝑓_s(Ψ,Ψ^∗,𝐀) in the functional space, that is, by the value −𝜕𝑓_s/𝜕Ψ^∗ which is now other than zero, unlike (20). This way has led to the following general form of the TDGL equation (e.g., see [15, 19, 20, 26]):𝜕𝜕𝑡−2𝑖ℏ𝜇Ψ=−ℏ1𝛾𝜕𝜕Ψ𝑓^s∗≡−1ℏ𝛾2𝑚1_s−𝑖ℏ∇−𝑞_s𝐀²+𝛼+𝛽||Ψ||²Ψ.(22)

The right-hand part of (22) includes the relaxation parameter 𝛾 introduced phenomenologically and the functional derivative 𝜕𝑓_s/𝜕Ψ^∗ in the form of (20) but not being equal to zero under nonequilibrium conditions. Second term on the left reduces to the following time dependence Ψ^∘(𝑡)=|Ψ^∘|exp(−2𝑖𝜇𝑡/ℏ) in equilibrium when 𝛾→∞.

Microscopic theory of the time-dependent superconductivity meets with serious complications arising from existence of the energy gap and singularity in the density of states at the gap edges. For this reason, the TDGL equation was derived by Gor’kov and Eliashberg [27] only for the gapless regime with predominant effect of paramagnetic impurities. Schmid [28] has obtained the analogous equation valid only near the transition temperature 𝑇_c and proposed that 𝜇 appearing in (22) should be considered as the electrochemical potential. A detailed discussion of the time-dependent problems can be found in a review [20].

All versions of the TDGL equation have the left-hand side similar to that of (22). As is shown below, such a structure of the TDGL equation is not of charge-conservation character because it cannot provide the continuity equation for the superfluid motion (but not for the total, normal and superfluid, motion which always conserves the charge).

Our idea of obtaining another version of the time-dependent GL equation is based on an analogy with the conventional way in deriving the kinetic equation for the distribution function to describe the nonstationary behavior of carriers in semiconductors and normal metals.

As is well-known (e.g., see books [29, 30]), in kinetics theory the time rate of change in the distribution function 𝑓(𝐫,𝑡) is taken to be divided into two parts: (i) the field contribution (𝜕𝑓/𝜕𝑡)_field caused by the coherent action of electromagnetic fields and (ii) the dissipation contribution (𝜕𝑓/𝜕𝑡)_diss caused by the incoherent relaxation processes of electron scattering and known as the collision integral, so that𝜕𝑓𝜕𝑡=𝜕𝜕𝑓𝑡_field+𝜕𝑓𝜕𝑡_diss,(23) discarding the actual expressions for the two terms in (23), which can be found in the literature (see [29, 30]).

In accordance with the line of our reasoning, the similar approach can be phenomenologically applied to the macroscopic wave function Ψ(𝐫,𝑡) of the superconducting state, so that its time rate of change is represented, by analogy with expression (23), as a sum of the similar two contributions:𝜕Ψ𝜕𝑡=𝜕𝜕Ψ𝑡_field+𝜕Ψ𝜕𝑡_diss.(24)

The field contribution is assumed to obey Schrödinger's equation (8) rewritten in the following form:𝜕𝜕Ψ𝑡_field=𝑖1ℏ𝐻Ψ,(25) where, in accordance with (8) and (9), the Hamiltonian operator for Cooper's pairs is𝐻=12𝑚_s−𝑖ℏ∇−𝑞_s𝐀²+𝑞_s𝜑+2𝜇_s0.(26)

The dissipation contribution is taken to satisfy the following equation:𝜕𝜕Ψ𝑡_diss=−ℏ1𝛾𝜕𝜕Ψ𝑓^s∗,(27) whose right-hand part coincides with that of (22), where the functional derivative𝜕𝑓_s𝜕Ψ^∗=12𝑚_s−𝑖ℏ∇−𝑞_s𝐀²+𝛼+𝛽||Ψ||²Ψ(28) has the form of (20) theoretically substantiated by Ginzburg and Landau [16] but different from zero under nonequilibrium conditions.

Substitution of relations (25)–(28) into (24) gives a desired novel version of the nonstationary equation, giving rise to both the usual GL equation (see (62)–(64)) and the TDGL equation (see (67)–(69)):𝜕Ψ𝜕𝑡−𝑖1ℏ2𝑚1_s−𝑖ℏ∇−𝑞_s𝐀²+𝑞_s𝜑+2𝜇_s0=−1Ψℏ𝛾2𝑚1_s−𝑖ℏ∇−𝑞_s𝐀²+𝛼+𝛽||Ψ||²Ψ.(29)

This equation, being time-dependent and space-nonuniform, allows for dissipation processes characterized by the phenomenological parameter 𝛾, as well as by 𝛼 and 𝛽. In comparison with the conventional equation (22) existing now in the literature [15, 19, 20, 26], (29) contains in the left-hand side the additional terms involving the space operator ∇ and the electromagnetic potentials 𝐀 and 𝜑, provided the quantity 𝜇 in (22) means 𝜇_s0. It is these terms that turn out to be of crucial importance in obtaining below the unambiguous and consistent results, including: (i) the continuity equation (36) with nonzero relaxation term instead of (10) and (ii) the superfluid motion equation (52) instead of (17).

Phenomenological parameters 𝛼,𝛽, and 𝛾 are often replaced by others [15, 25]:𝜏_GL(𝑇)=ℏ𝛾(𝑇)||𝛼(𝑇)||≡ℏ𝛾2(𝑇)𝑁^∘s(𝑇)𝜉𝜇₀𝐻²_c(𝑇)_2GL,(𝑇)=ℏ²2𝑚_s||𝛼(𝑇)||≡8ℏ𝑚²𝑁^∘s(𝑇)𝜂𝜇₀𝐻²_c(𝑇)_GL,(𝑇)=𝛽(𝑇)||𝛼(𝑇)||≡𝑁^∘_s2(𝑇)(30) where the last equalities are written by applying formula (21). The quantities 𝜏_GL(𝑇) and 𝜉_GL(𝑇) are known as the temperature-dependent relaxation time and coherence length of the GL theory [17–21, 27].

In the case of employing parameters (30), our initial equation (29) takes the following form:𝑖ℏ𝜕Ψ𝜕𝑡−𝐻Ψ=𝜏𝑖ℏ_GL𝑅Ψ.(31)

Here the Hamiltonian operator 𝐻 (nondissipative) is given by (26) and the relaxation operator 𝑅 (dissipative) on the right is now defined as𝑅=𝜉²_GL∇+2𝑖ℏ𝑒𝐀²−𝜂_GL||Ψ||²+1.(32)

It is easy to see that (31) is converted either into the ordinary GL equation (20) when 𝜏_GL→0 (relaxation processes are dominant) or into Schrödinger's equation (8) when 𝜏⁻¹_GL→0 (relaxation processes are negligible). Something close to our result was remarked by Abrahams and Tsuneto [26].

It is very important to note that the right-hand side of (31) can exactly be reduced to the form obtained on the basis of microscopic theory for gapless superconductors by Gor'kov and Eliashberg [27] (see also works [18–20]) only if the quantity 𝐷_GL≡𝜉²_GL/𝜏_GL=ℏ/4𝑚𝛾 is identified with the normal-state diffusion coefficient 𝐷=𝑣²_F𝜏/3.

The left-hand and right-hand sides of (31) are, respectively, transformed by using the macroscopic wave function in the form of Ψ=√𝑁_s/2exp(𝑖Θ) (see (6)) into the following: 𝑖ℏ𝜕Ψ𝜕𝑡−𝐻Ψ=−2𝜇^s0+𝜇_s1+𝑒𝜑+𝑚𝑣^2s−𝑖ℏ2+ℏ2𝜕𝜕Θ𝑡2𝑁_s𝜕𝜕𝑁𝑡^s+∇⋅𝑁^s𝑖ℏ𝐯_sΨ,(33)𝜏_GL𝑅Ψ=−𝑖ℏ𝜏_GL𝐺+𝜉^2GL2𝑚𝑖ℏ∇⋅𝑁^s𝐯_s𝑁_sΨ.(34) Here we have used formula (7) the relations (𝐯_s⋅∇𝑎)/𝑎=(∇⋅𝑁_s𝐯_𝑠)/2𝑁_s and ∇⋅𝐯_s=0 (see (44)), and also introduced a quantity𝐺=𝜉²_GL8𝑚ℏ²𝑚2𝑣^2s+𝜇^s1+𝑁𝑁^s∘_s−1.(35)

After substituting (33) and (34) into (31) and separating out the imaginary and real parts, we obtain: (i) the continuity equation for superfluid motion 𝜕𝑁_s𝜕𝑡+∇⋅𝑁^s𝐯_s=−2𝐺𝜏_GL𝑁_s,(36)(ii) the relation for chemical potentials 𝜇_s0+𝜇_s1+𝜇_s2+𝑒𝜑+𝑚𝑣^2s2+ℏ2𝜕𝜕Θ𝑡=0.(37)

Here, in addition to the chemical potentials 𝜇_s0 and 𝜇_s1 we have introduced another contribution𝜇_s2=−ℏ4𝛾𝑁_s∇⋅𝑁_s𝐯_s≡−𝜉^2GL𝜏_GL𝑚𝑒∇𝑁⋅_s𝐉^s.(38)

The chemical potential 𝜇_s2, just like 𝜇_s1, is due to the nonuniform spatial distribution of supercarriers but, unlike 𝜇_s1, is solely of dissipative nature. It is interesting to notice that 𝜇_s2 fully coincides with the quantity 2(𝜇_𝑝−𝜇) introduced by Rieger et al. [31] to describe mutual conversion of the superfluid and normal fluid components in the vicinity of NS-interfaces.

It is pertinent to stress that if we employed the conventional TDGL equation (22) instead of our equations (31)-(32) as a basis for the transformations done above, it would lead us to the different result: there would be neither the term ∇⋅(𝑁_s𝐯_s) in (36) nor the terms 𝜇_s1, 𝑒𝜑 and 𝑚𝑣²_s/2 in (37). It is the former fact that allows us to draw the conclusion about the charge-nonconservation character of all the existing versions of the TDGL equation, which have the left-hand side similar to that of (22). The above-mentioned arguments count in favor of our version of the TDGL equation given by (29) or (31)-(32).

If we define the total chemical potential of the superfluid component in a dynamic regime as the sum of the four contributions (see (12) and (38)):𝜇_Σ=𝜇_s0+𝜇_s1+𝜇_s2+𝑚𝑣^2s2=𝜇^s+𝑚𝑣^2s≡𝜇_s02−ℏ²8𝑚∇²𝑁^1/2_s𝑁^1/2_s−𝜉^2GL𝜏_GL𝑚𝑒∇𝑁⋅_s𝐉^s+𝑚2𝑣^2s,(39) then formula (37) may be rewritten in two waysΦ≡𝑒𝜑+ℏ2𝜕𝜕Θ𝑡=−𝜇^s+𝑚𝑣^2s2≡−𝜇^Σ𝜁,(40)_Σ≡𝜁_s+𝑚𝑣^2s2≡𝜇^Σ+𝑒𝜑=−ℏ2𝜕𝜕Θ𝑡,(41) where 𝜁_s=𝜇_s+𝑒𝜑 is the electrochemical potential of the superfluid component.

Equation (40) shows that the total chemical potential 𝜇_Σ always varies oppositely in phase with respect to the so-called gauge-invariant potential Φ, being equal to it in magnitude. Besides Φ, the superconductivity theory deals with another gauge-invariant quantity, namely, the superfluid velocity 𝐯_s [17, 18]. The use of them allows us, following Stephen [32], to introduce the gauge-invariant electromagnetic potentials (marked with subscript g)𝐀_g=−𝑚𝐯^s𝑒,𝜑^𝑔=𝚽𝑒.(42)

In this case, the electric field (4) can be rewritten in the gauge-invariant form: 𝐄=−𝜕𝐀^g𝜕𝑡−∇𝜑^g.(43)

As is known from the classical electrodynamics [33], in order for the gradient term in (43) to be actually the potential field 𝐄_p=−∇𝜑_g with the scalar potential 𝜑_g obeying Poisson's equation, it is necessary to employ the Coulomb gauge (∇⋅𝐀_g=0). Hence, using formulae (16) and (42), we arrive at the following requirements imposed on the superfluid velocity:∇⋅𝐯_s=0,∇×𝐯_s=−𝑒𝑚𝐁≠0.(44)

Formula (44) expresses the property of solenoidality inherent in the superfluid flow. The universal condition ∇⋅𝐯_s=0 ensures that the component of superfluid velocity normal to any boundary vanishes (𝐧⋅𝐯_s=0). In the vicinity of NS-interface there is a conversion of the normal current with 𝐧⋅𝐯_n≠0 into the supercurrent with 𝐧⋅𝐯_s=0 and nonuniform distribution of 𝑁_s.

According to (41), the electrochemical potential 𝜁_s=𝜇_s+𝑒𝜑 is determined by the rate of change in the phase Θ of the order parameter Ψ. One should recall that all the energy values are measured with respect to the unperturbed bottom of the energy band (ℰ^∘_b=0). Then, by definition, in equilibrium 𝜇^∘_s≡𝜁^∘_s−ℰ^∘_b=𝜁^∘_s. There being the potential electric field 𝐄_p=−∇𝜑 with the London gauge ∇⋅𝐀=0, the band bottom suffers an offset from ℰ^∘_b=0, so that ℰ_b=𝑒𝜑 and 𝜇_s≡𝜁_s−ℰ_b=𝜁_s−𝑒𝜑.

Let us clarify a sense of the initial chemical potential 𝜇_s0 appearing in (9) and find its relationship to the Fermi energy ℰ_F. First consider the equilibrium superconducting state with no superfluid flow (𝐯_s=0) when the chemical potentials 𝜇^seq and 𝜇^neq are both equal to ℰ_F. Then the Gibbs free energy per unit volume is 𝒢𝐯_s=0=𝜇^seq𝑁^seq+𝜇^neq𝑁^neq=ℰ_F𝑁.(45) In the presence of the stationary superfluid flow (𝐯_s≠0), the equilibrium is not disturbed, yet both the chemical potentials and electron densities become altered so that 𝜇^∘_s=𝜇^∘_n≠ℰ_F and 𝑁^∘_s+𝑁^∘_n=𝑁^seq+𝑁^neq=𝑁. Then𝒢𝐯_s≠0=𝜇^∘_s𝑁^∘_s+𝜇^∘_n𝑁^∘_n=𝜇^∘_s𝑁.(46) It is evident that the alteration in 𝒢 is brought about by an increase in the kinetic energy of supercarriers𝒢𝐯_s≠0−𝒢𝐯_s=0=𝑚𝑣^2s2𝑁^∘s.(47) From the above reasoning it follows that in the case of stationary superfluid flow,𝜇^∘_s=ℰ_F+𝑚𝑣^2s2𝑁𝑁^∘s=𝜇^∘n.(48) It is this quantity taking account of the superfluid motion that can be identified with the above introduced value 𝜇_s0+𝑚𝑣²_s/2, then 𝜇_s0=𝜇^∘_s−𝑚𝑣^2s2=ℰ^F−𝑚𝑣^2s2𝑁𝑁^∘n.(49)

Equations (48) and (49) are similar to those given by Ginzburg and Zharkov [34, formula (9.8) and (9.9)], the addition to ℰ_F in (48) being associated with the Bernoulli potential, which was observed experimentally [35, 36].

Thus, we can rewrite the total chemical potential (39) in the following form: 𝜇_Σ=𝜇^∘_s+𝛿𝜇_s,(50) where the stationary value 𝜇^∘_s and its perturbation 𝛿𝜇_s are defined as follows (see (12), (38), and (49)): 𝜇^∘_s=𝜇_s0+𝑚𝑣^2s𝛿𝜇_s2,=𝜇_s1+𝜇_s2≡−ℏ²8𝑚∇²𝑁^1/2_s𝑁^1/2_s−𝜉^2GL𝜏_GL𝑚𝑒∇𝑁⋅_s𝐉^s.(51)

The dynamic addition 𝛿𝜇_s to the equilibrium chemical potential 𝜇^∘_s=𝜇^∘_n reflects an appearance of the normal current 𝐉_n=𝑒𝑁_n𝐯_n together with the nonuniform superfluid density 𝛿𝑁_s=𝑁_s−𝑁^∘_s as a result of the time-dependent and space-dependent external actions.

Applying the gradient operation to (37) and carrying out the same procedure, as applied to formula (11), we finally obtain instead of (17) the required hydrodynamic equation of superfluid motion:𝑚𝑑𝐯^s𝑑𝑡=𝑒𝐄+𝐯^s×𝐁−∇𝜇_s.(52) Here, the resulting chemical potential 𝜇_s of the superfluid component consists of three contributions,𝜇_s=𝜇_s0+𝜇_s1+𝜇_s2,(53) defined by (49), (12), and (38), respectively.

In comparison with (17), (52) contains, besides 𝜇_s0 and 𝜇_s1, the chemical potential 𝜇_s2, (38). The latter reflects influence of peculiar dissipative effects specified by the parameter 𝛾 entering originally into the right-hand side of the time-dependent equation (29). This influence differs from that expressed by the dissipative term −𝜆𝐯 in (2) and, being proportional to the gradient ∇𝑁_s (since ∇⋅(𝑁_s𝐯_s)=𝐯_s⋅∇𝑁_s owing to ∇⋅𝐯_s=0), it is of specific diffusion character.

3. Continuity Equation and Time-Dependent Ginzburg-Landau Equation

Let us return to our result in the form of (36) giving the continuity equation for the superfluid flow, which can be rewritten as𝜕𝑁_s𝜕𝑡+∇⋅𝑁^s𝐯_s=−ℛ_diss.(54) Here the relaxation (dissipation) term, in accordance with (35), is equal toℛ_diss=2𝑁^s𝜏_GL𝜉^2GL8𝑚ℏ²𝑚2𝑣^2s+𝜇^s1+𝑁𝑁^s∘_s−1,(55) with 𝜏_GL and 𝜉_GL being phenomenological parameters of the GL theory [17–21, 27] (see formulae (30)).

If the right-hand side of (54) was absent, the universal solenoidality condition ∇⋅𝐯_s=0 for the superfluid flow would imply that the superfluid behaves as an incompressible ideal fluid. However, in effect this is not the case because from (54) it follows that 𝑑𝑁_s𝑑𝑡≡𝜕𝜕𝑡+𝐯^s⋅∇𝑁_s=−ℛ_diss≠0,(56) where the right-hand side is not equal to zero in the general case. This refutes the conclusion about the superfluid incompressibility, which can incorrectly be inferred from the condition ∇⋅𝐯_s=0, as it was done before [26].

It is convenient to represent ℛ_diss in the following form:ℛ_diss=Γ_R𝑁_swithΓ_R≡1𝜏_R=2𝐺𝜏_GL,(57) where, according to formula (35), we have 𝐺=𝜉²_GL8𝑚ℏ²𝑚2𝑣^2s+𝜇^s1+𝑁𝑁^s∘_s≡𝜉_2GL−12𝑚ℏ𝐯^s²−𝜉²_GL∇²𝑁^1/2_s𝑁^1/2_s+𝑁^s𝑁^∘_s−1.(58) The last equality in (58) has been obtained by employing expression (12) for the chemical potential 𝜇_s1.

The quantity Γ_R≡𝜏⁻¹_R=2𝐺/𝜏_GL, appearing in (57) and playing a role of the effective pair-breaking rate, consists of two terms: Γ_R=Γ₁+Γ₂, which correspond to the contributions from two terms in (58).

The first term Γ₁ is related to the contribution of 𝑚𝑣²_s/2 to the quantity 𝐺 and realized in the particular case of spatial uniformity, when 𝑁_s≃𝑁^∘_s and the quantity 𝜇_s1∝∇²𝑁^1/2_s is vanishingly small, so that Γ₁=𝜉^2GL𝜏_GL8𝑚ℏ²𝑚𝑣²_s=2𝐷_GL2𝑚ℏ𝑣^s²,(59) where 𝐷_GL=𝜉²_GL/𝜏_GL.

In the literature (e.g., see [21, 37, 38]), there is a similar quantity Γ=2𝐷(𝑚𝑣_s/ℏ)² involving the normal-state diffusion constant 𝐷=𝑣²_F𝜏/3. If the diffusion constants 𝐷 and 𝐷_GL are identified with each other, the quantity Γ becomes four times smaller than Γ₁ because it includes the electron mass 𝑚 instead of the effective mass 𝑚_s=2𝑚 for Cooper's pairs, as it occurs in the real case. It is known that Γ=𝜏⁻¹_E/2+𝜏⁻¹_s near 𝑇_c [21, 37], whenceΓ₁=4Γ=21𝜏_E+2𝜏_s.(60)

Here 𝜏_E is the energy relaxation time due to the inelastic electron-phonon collisions and 𝜏_s is the elastic spin-flip scattering time due to magnetic impurities (if any), with both being taken for the normal state of a superconductor.

The second term Γ₂ is related to the contributions from 𝜇_s1 and the two last terms in (58) to the quantity 𝐺 and caused by spatial nonuniformity of 𝑁_s≠𝑁^∘_s. This is also due to the pair-breaking processes with participating phonons, whose relaxation time (𝜏_ph≈𝜏_E) is usually much more than the GL relaxation time 𝜏_GL [18, 21, 38].

So in order to ensure 𝜏_E≈𝜏_R≡𝜏_GL/2𝐺≫𝜏_GL, it is necessary that values of 𝐺 should be sufficiently small. The above inequality provides sufficiently fast formation of the quasilocal equilibrium in the condensate (characterized by 𝜏_GL) as compared with the relatively slow pair-breaking processes (characterized by 𝜏_E≈𝜏_R). Such an assertion should be true at any temperature near 𝑇_c when 𝜏_GL≃𝜋ℏ/8𝑘_B(𝑇_c−𝑇) [18, 21, 38]. Consequently, the closer temperature 𝑇 is to 𝑇_c, the smaller should be the quantity 𝐺 (𝐺→0 in the limiting case 𝑇→𝑇_𝑐) to ensure the inequality 𝜏_GL≪𝜏_E≈𝜏_R≡𝜏_GL/2𝐺. This provides fast formation of the quasi-local equilibrium in the condensate under nonstationary physical conditions.

Before rearranging the continuity equation (54), let us consider the extremely limiting case of 𝐺=0 and show that it can be realized, irrespective of temperature, only in the particular case of stationary situations described by the usual (time-independent) GL equation (20).

3.1. Stationary Ginzburg-Landau Equation

A comparison of equations (20) and (29) displays a condition for realizing the GL equation in the form of 𝛾=0 imposed on (29), which reflects the dominant effect of relaxation processes. It is clearly evident from (31) that the condition 𝛾=0 (or 𝜏_GL=0) is equivalent to the equation 𝑅Ψ=0, where the relaxation operator 𝑅 is defined as (32). From expression (34), it follows that the stationary GL equation (20) assumes the form𝑅Ψ≡−𝐺−𝑖𝜉²_GL2𝑚ℏ∇𝜌⋅_s𝐉^sΨ=0.(61)

The quantity (61) must be equal to zero for any arbitrary function Ψ, which is fulfilled if and only if both the real and imaginary terms enclosed in parentheses vanish.

Vanishing the real part of 𝑅 in (61) means 𝐺=0. Formula (58) for 𝐺, being written in terms of the normalized modulus 𝜓=|Ψ|/|Ψ^∘|≡(𝑁_s/𝑁^∘_s)^1/2 of the complex order parameter Ψ, converts the equality 𝐺=0 into the following equation:𝜉²_GL∇²𝜓+1−𝜉²_GL2𝑚ℏ𝑣^s²𝜓−𝜓³=0(62) (𝜓 here and below should not be confused with the microscopic wave function 𝜓 mentioned in Section 1).

The general expression 𝐯_s=−(𝑒/𝑚)𝐀 for the simply connected and vortex-free superconductors [17, 18, 25] allows one to write the following relation:2𝑚𝑣_sℏ=2𝑒ℏ𝐴≡2𝜙𝜋₀𝐴,(63) with 𝜙₀=𝜋ℏ/𝑒 being the quantum of magnetic flux (written in the SI system of units). Then equation (62) assumes the following form:𝜉²_GL∇²𝜓+1−𝜉²_GL2𝑒𝐴ℏ²𝜓−𝜓³=0.(64)

For simply-connected and vortex-free superconductors the order parameter Ψ is always real-valued, then 𝜓=Ψ/Ψ^∘≡(𝑁_s/𝑁^∘_s)^1/2. Consequently, (64) is the ordinary GL equation and it fully coincides with the similar ones existing now in the literature (e.g., see books [17, 18, 25]).

Vanishing the imaginary part of 𝑅 in (61) establishes the following requirement for stationary situations:∇⋅𝐉_s=0.(65)

The requirement (65) removes the chemical potential 𝜇_s2 of relaxation nature from both the chemical potential relation (37) and the force equation (52) because 𝜇_s2∝∇⋅𝐉_s=0, in accordance with (38).

Therefore, in the stationary situation with steady-state superfluid flow described by the GL equation (62) or (64), the superfluid motion is governed by the simplified equation (17), which is due to 𝜇_s2=0, rather than the general equation (52). In (17), there are the uniform chemical potential 𝜇_s0 given by (49) and the nonuniform chemical potential 𝛿𝜇_s=𝜇_s1 determined by the quantum potential 𝑄_s of Cooper's pairs in the form of (18).

Since the superfluid flow is always solenoidal, the requirement (65) by virtue of the solenoidality condition ∇⋅𝐯_s=0 (see formula (44)) yields𝐯_s⋅∇𝑁_s=0.(66)

So under stationary conditions we have ∇𝑁_s⟂𝐯_s, that is, any spatial nonuniformity in 𝑁_s arises only in the plane perpendicular to the supervelocity vector 𝐯_s. Thus, the longitudinal component of the gradient ∇𝑁_s (∇𝑁_s‖𝐯_s) can appear solely in nonstationary situations when the requirement (65) and its consequence (66) fail to occur for solenoidal motion of the superfluid component.

3.2. Time-Dependent Ginzburg-Landau Equation

In the case of nonstationary situations, the general continuity equation (54) for the superfluid flow can be rearranged with the help of (57) and (58) and reduced to the following form:𝐷_GL∇²𝜓−𝑑𝜓𝑑𝑡=−1−𝜓²−𝜉²_GL2𝑚ℏ𝑣^s²𝜏𝜓_GL,(67) where 𝐷_GL=𝜉²_GL/𝜏_GL and 𝑑𝜓/𝑑𝑡=𝜕𝜓/𝜕𝑡+𝐯_s⋅∇𝜓.

Equation (67) is the desired version of the time-dependent Ginzburg-Landau equation which describes the time evolution of the normalized modulus 𝜓=|Ψ|/|Ψ^∘|≡(𝑁_s/𝑁^∘_s)^1/2 of the nonequilibrium order parameter Ψ(𝐫,𝑡).

Expressions (59) and (60) with 𝜏⁻¹_s=0 (no magnetic impurities) yield a ratio of the relaxation times 𝜏_GL𝜏_E=𝜉²_GL2𝑚ℏ𝑣^s²=𝜉²_GL2𝑒𝐴ℏ²≪1.(68) Here the second equality conforms to simply connected and vortex-free superconductors, whereas the inequality physically means relatively fast formation of the quasi-local equilibrium (as compared with the pair-breaking processes) in the condensate under nonstationary conditions.

Application of formula (68) reduces the general TDGL equation (67) to the simplified form 𝐷_GL∇²𝜓−𝑑𝜓𝑑𝑡=−1−𝜓²𝜏𝜓_GL,(69) where we have used the inequality 𝜏_GL≪𝜏_E.

It is evident that the general form (67) of the TDGL equation assumes the special form (62) of the GL equation with 𝜏_GL≪𝜏_E if 𝑑𝜓/𝑑𝑡=0. This is indeed realized under the stationary conditions when (i) 𝜕𝜓/𝜕𝑡=0 and (ii) 𝐯_s⋅∇𝜓≡(4𝑁_s𝑁^∘_s)^−1/2(𝐯_s⋅∇𝑁_s)=0 as a result of the stationary requirement (66). In this case, (69) also changes its form into the known one [17, 18, 25]𝜉²_GL∇²𝜓+𝜓−𝜓³=0,(70) which fully coincides with the stationary GL equations (62) and (64) when employing the inequality (68).

Thus, any time-dependent and space-nonuniform physical situation is governed by the equation of superfluid motion in hydrodynamic form of (52) and the TDGL equation (67) for the modulus of the order parameter. The superfluid motion equation (52) follows from the chemical potential relation (37) and is fully equivalent to this relation. Our version (67) of the TDGL equation, being a consequence of the continuity equation (54), is of charge conservation character. In general, this equation has the form of a nonhomogeneous diffusion equation with nonlinear terms allowing for the superfluid motion and dissipation processes, which assumes the conventional form (62) of the GL equation under stationary conditions.

4. Conclusion

The macroscopic equations of superfluid transport have been derived on the basis of a phenomenological combination of the usual Ginzburg–Landau equation and Schrödinger's equation for the macroscopic wave function (see (24)–(28)), which has resulted in the desired form of (31) to describe nonstationary superconductivity. Application of the Madelung-Feynman approach [8, 22, 23] to the newly derived equation (31) by separating it into the real and imaginary parts has finally provided us with the required equation (52) to describe the superfluid motion in hydrodynamic form. Besides, such an approach has given: (i) four different contributions to the total chemical potential 𝜇_s for superfluid component in the form of (39) and (ii) the continuity equation (36) or (54) for the superfluid flow with the relaxation term in the form of (55) involving the phenomenological Ginzburg-Landau parameters 𝜏_GL and 𝜉_GL.

The hydrodynamic equation (52) of superfluid motion and the novel version (67) of the TDGL equation for the modulus of the order parameter have turned out to be a consequence, respectively, of the chemical potential relation (37) and of the continuity equation (54). This result explains why, unlike semiconductor electronics, in stationary superconductor theory nobody applies the equation of superfluid motion and the continuity equation. Instead of them, the chemical potential relation (37) and the GL equation (62) are commonly used.

Taking into account quantum dissipation effects with the help of the relaxation term −𝜆𝐯 entering into the hydrodynamic equation of motion in the form of (2), as is commonly used for nonsuperconductive media, is not the case for the theory of time-dependent superconductivity. For the latter, the hydrodynamic equation (52) of superfluid motion has taken into account the relaxation processes in terms of 𝜇_s2 given by (38). Again, the dissipative effects have proved to be inserted into the continuity equation (54) or into its output in the form of the time-dependent Ginzburg-Landau equation (67) for the modulus of the order parameter.

All the above results are mutually consistent within the scope of the chosen macroscopic description and, being phenomenological and model-neutral, can be applied to both the low-𝑇_c and high-𝑇_c superconductors.

Acknowledgment

The author wishes to acknowledge the Russian Foundation for Basic Research for financial support (Grant no. 10-03-00845-a).