BCS theory with the external pair potential

# BCS theory with the external pair potential

Konstantin V. Grigorishin Boholyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine, 14-b Metrolohichna str. Kiev-03680, Ukraine.
###### Abstract

We consider a hypothetical substance, where interaction between (within) structural elements of condensed matter (molecules, nanoparticles, clusters, layers, wires etc.) depends on state of Cooper pairs: an additional work must be made against this interaction to break a pair. Such a system can be described by BCS Hamiltonian with the external pair potential term. In this model the potential essentially renormalizes the order parameter: if the pairing lowers energy of the structure the energy gap is slightly enlarged at zero temperature and asymptotically tends to zero as temperature rises. Thus the critical temperature of such a superconductor is equal to infinity formally. For this case the effective Ginzburg-Landau theory is formulated, where the coherence length decreases as temperature rises, the GL parameter and the second critical field are increasing functions of temperature unlike the standard theory. If the pairing enlarges energy of the structure then suppression of superconductivity and the first order phase transition occur.

BCS model, critical temperature, external pair potential, free energy functional, effective Ginzburg-Landau theory
###### pacs:
74.20.Fg, 74.20.Mn

## I Introduction

Critical temperature and the second critical magnetic field are some of the key characteristics of a superconductor. These parameters depend on an effective coupling constant with some collective excitations ( is a density of states at Fermi level, is an interaction constant), on frequency of the collective excitations and on Ginzburg-Landau (GL) parameter (). The larger coupling constant, the larger the critical temperature. For large we have mahan ; ginz (or in BCS theory). Formally the critical temperature can be made arbitrarily large with increasing of the electron-phonon coupling constant. However in order to reach room temperature such values of the coupling constant are necessary which are not possible in real materials. On the other hand we can increase the frequency due to nonphonon pairing mechanisms as proposed in ginz . However with increasing of the frequency the coupling constant decreases as . The second critical magnetic field can be enlarged in a ”dirty” limit, where the mean free path is much less than a coherence length schmidt . In this case the GL parameter can be , but still the critical field is low near the critical temperature: . Many different types of superconducting materials with a wide variety of electron pairing mechanisms exist, however all they have the critical temperature limited by values , despite the fact that the highly exotic mechanisms have been proposed.

The coupling constant determines a value of the energy gap at zero temperature , in the same time a ratio between the gap and the critical temperature is

 2ΔkBTc=3.52 (1)

for BCS theory. For presently known materials this ratio is typically between 3 and 8 wesche : the relatively large value 8 is considered as evidence for the strong coupling in high-temperature superconductors, while in conventional metallic superconductors the ratio is close to 3.5. Thus increasing the zero temperature gap we enlarge . However the critical temperature can be enlarged by violation the ratio (1) as due to some addition influence on condensate of Cooper pairs as shown in Fig.(1). The simplest examples of such systems are systems where the proximity effect takes place: two superconductors are placed in contact schmidt , or in multi-band superconductors where the interband mixing of order parameters from different bands occurs asker ; grig . In this cases the Cooper pairs are injected into the superconductor (band) from another superconductor (band) with higher critical temperature (stronger interaction) which plays a role of a source of Cooper pairs. As a result in a band with lower critical temperature the ratio (1) can be violated as grig ; litak as illustrated in Fig.(1). However the source is function of temperature with own critical temperature. In a case of the contact of two metal the critical temperature of the source drops, however in a case of the multi-band superconductor the proximity effect supports superconducting state in all bands.

The hypothetical system where is first proposed in loz . In this work the pairing of massless Dirac electrons and holes located on opposite surfaces of thin film of three-dimensional topological insulator is considered. In such a system both electron-hole Coulomb attraction, which leads to pairing, and tunneling (or hybridization) of the electrons between surfaces, which leads to appearance in Hamiltonian the ”sources” of Cooper pairs, act jointly. At the gap gradually tends to some value . Thus the hybridization leads to the smearing of a phase transition into the paired state in close analogy with the behavior of a ferromagnetic in an external magnetic field. This solution means the nonzero energy gap if the electron-hole attraction is switched off or temperature is infinite, which is not physically. Apparently the gap cannot be order parameter, but it can mean presence of uncorrelated pairs. In its turn the systems with ”preformed” Cooper pairs have been considered in works dom ; dzum ; cap . In this regime, the electrons are paired, but they lack the phase coherence necessary for superconductivity. The existence of preformed pairs implies the existence of a characteristic energy scale associated to a pseudogap. In a work grig1 a model of hypothetical superconductivity has been proposed, which demonstrates the principal differences from results of BCS and GL theory due to presence of the source term in the model Hamiltonian. Unlike the previous works in this model the coherent paired state is absent if the electron-electron coupling is absent . This means the electron-electron coupling is the cause of the transition to superconducting state only but not the source term. In this theory the source term determines energy of a Cooper pair in some external field which can be called as the external pair potential by analogy with the terminology in matt1 ; dreiz . In a case of decreasing of Cooper pair’s energy by the potential the energy gap tends to zero asymptotically as temperature increases. Thus the ratio between the gap and the critical temperature is instead of a finite value in BCS theory. In this work the free energy functional in the limit has been obtained. It is shown that the energy gap tends to zero asymptotically as magnetic field increases. Thus the second critical magnetic field is equal to infinity as well as the critical temperature, that is not correct physically, because distance between vortexes cannot be less than diameter of the vortex’s core.

In present work we generalize BCS theory in the sense that the invariant under transformation source of Cooper pairs (the external pair potential) is added to BCS Hamiltonian. We demonstrate that such Hamiltonian describes a hypothetical substance, where an interaction energy between (within) structural elements of condensed matter (molecules, nanoparticles, clusters, layers, wires etc.) depends on state of Cooper pairs: an additional work must be made against this interaction to break a pair. In this model the potential essentially renormalizes the order parameter so that the ratio (1) changes as if the breaking of a Cooper pair increases energy of the molecular structure (or creation of the pair lowers the energy). In another regime (the breaking of a Cooper pair lowers the energy of the structure) the renormalization of the order parameter by the potential suppresses a superconducting state and the first order phase transition occurs. We obtain the free energy of such a system which generalizes Landau free energy for the presence of the external pairing potential. For the case in the limit we formulate the effective GL theory which is obtained in high-temperature limit from the general expression for free energy (unlike the work grig1 where the free energy functional was obtained modifying the ordinary GL expansion). Results of our effective GL theory radically differ from results of the ordinary GL theory: the coherence length decreases as temperature rises, the GL parameter and the second critical field are increasing functions of temperature.

## Ii The model

According to BCS theory an electron-electron attraction leads to the appearance of nonzero anomalous averages and , which are the order parameter (pair potential) of the superconducting state. The order parameter is determined with some self-consistency equation , which reflects the fact, that superconductivity is a many-particle cooperative effect. In this regime the charge is carried by pairs of electrons (current carriers are the pairs with charge ). To break a pair with transfer of its constituents in free quasiparticle states the energy is needed. We can consider the quantity as a work against the effective electron-electron attraction given the fact that the quantity is a collective effect. The superconducting state of a metal and the breaking of a pair are illustrated in Fig.2a.

In our model we consider a hypothetical substance, where an interaction energy between (within) structural elements of condensed matter (molecules, nanoparticles, clusters, layers, wires etc.) depends on state of Cooper pairs: if the pair is broken, then energy of the molecular system is changed by quantity , where and are energies of the system after- and before the breaking of the pair accordingly. Thus to break the Cooper pair we must make the work against the effective electron-electron attraction and must change the energy of the structural elements:

 2|Δ|⟶2|Δ|+υ>0. (2)

We will call the parameter as the external pair potential, since it is imposed on the electron subsystem by the structural elements of a substance, unlike the pair potential , which is result of electron-electron interaction and determined with the self-consistency equation . The parameter can be either or and in the simplest case it is not function of the energy gap , is a trivial case corresponding to BCS theory. Moreover we suppose that does not depend on temperature essentially like parameters of electron-phonon interaction. The condition ensures stability of the Cooper pairs (bound state of the electrons is energetically favorable), otherwise transformation (2) has no sense and such superconducting state cannot exist. If then the breaking of a Cooper pair lowers energy of the molecular structure (or creation of the pair raises the energy). In this case the pairs become less stable. If then the breaking of the pair increases the energy (or creation of the pair lowers the energy). In this case the pairs become more stable. A possible variant of breaking of a Cooper pair in these cases is illustrated in Fig.2b and Fig.2c.

Without going into the details of interaction of the structural elements we can write an effective Hamiltonian which takes into account the effect of the structure on Cooper pairs as some effective external field, like the BCS Hamiltonian is an effective Hamiltonian describing a system of interacting electrons independently of nature of this interaction. The order parameter is a complex quantity , where is a phase, and it is the result of a many-particle self-consistent coherent effect. In the same time the field is an additional parameter imposed on the electron subsystem by the structural elements. It is easy to see that the following transformations of the order parameter correspond to the transformation (2):

 Δ⟶Δ+Δ|Δ|υ2=Δ(1+υ2|Δ|),Δ+⟶Δ++Δ+|Δ|υ2=Δ+(1+υ2|Δ|). (3)

Really, the work to break the pair is

 2 ⎷ΔΔ+(1+υ2|Δ|)2=|2|Δ|+υ|.

However the transformations (3) correspond to the transformation (2) if only. If then the work is . Thus the model based on the transformations (3) can give ”parasitic” solutions where the condition (2) is not satisfied. Such solutions must be omitted.

The Hamiltonian corresponding to the transformations (3) is

 ˆH = ˆHBCS+ˆHυ=∑k% ,σε(k)a+k,σak,σ−λV∑k,pa+p↑a+−p% ↓a−k↓ak↑−υ2∑k[Δ|Δ|a+k↑a+−k↓+Δ+|Δ|a−k↓ak↑] (4)

where is BCS Hamiltonian: kinetic energy + pairing interaction, energy is measured from Fermi surface. Indeed, singling out anomalous averages and introducing the order parameter

 Δ+=λV∑p⟨a+p↑a+−p↓⟩,Δ=λV∑p⟨a−p↓ap↑⟩, (5)

we can rewrite the Hamiltonian in a form

 ˆH=∑k,σε(k)a+k,σak,σ+∑k[Δ+(1+υ2|Δ|)ak↑a−k↓+Δ(1+υ2|Δ|)a+−k↓a+k↑]+1λV|Δ|2. (6)

The combinations and are creation and annihilation operators of a Cooper pair, thus the mean field (3) acts on Cooper pairs. The order parameter is a complex quantity . Due the multipliers and in the energy does not depend on the phase (). Thus both and are invariant under the transformation. The term is similar to ”source term” in matt1 , where it means the injection of Cooper pairs into the system. On the other hand, has a form of an external field acting on a Cooper pairs only, and is energy of a Cooper pair in this field. Eigenfunctions of the operator may not be eigenfunctions of a particle number operator , that reflects the presence of Cooper pairs’ condensate.

In our model we make the following assumptions. A change of the chemical potential at transition to superconducting state can be neglected, so that , because in the model we suppose , that occurs for systems with week electron-electron attraction and high density of carriers, unlike the systems with the strong attraction and low particle density in the BEC regime, so that , where the change of the chemical potential plays important role in formation of the superconducting state lok . In addition we should notice that the coupling constant is a sum of phonon term and Coulomb term: . In metals, as a rule, that corresponds to repulsive electron-electron interaction, however in such systems the pairing is possible as result of the second order processes, which lead to effective attraction regardless of the sign of interaction kirz : , where is a Coulomb pseudopotential. Such mechanism is not considered in our model. In the model we suppose a stronger condition , which can occur in nonmetallic superconductors (for example, in alkali-doped fullerides , where competition between the Jahn-Teller coupling and Hund’s coupling takes place han ; nom ).

In order to find equilibrium value of the order parameter we should obtain system’s energy. Using the BCS wave function , where , we obtain the energy in a form:

 Ws=⟨ˆH⟩=∑k2ε(k)\textslv2(k)−λV∑k,pu(k)\textslv(k)u(p)\textslv(p)−υΔ|Δ|∑ku(k)\textslv(k), (7)

where can be supposed real in the absence of magnetic field and current. The first term is internal kinetic energy, the second term is energy of the electron-electron interaction, the third term is energy of Cooper pairs’ condensate in the external pair field. The functions and have to minimize the energy, that is :

 2ε(k)−1−2\textslv2(k)u(k)\textslv(k)⎡⎣λV∑pu(p)\textslv(p)+υ2Δ|Δ|⎤⎦=0. (8)

There are two different solutions of this equation, which correspond two different physical situations:

1. We can suppose

 Δ=λV∑pu(p)\textslv(p)+υ2Δ|Δ|. (9)

Then we obtain the functions in a form

 \textslv2(k)=12(1−εE)u2(k)=12(1+εE) (10)

as in BCS theory with

 E2=ε2+|Δ|2,\textslv(k)u(k)=Δ2E. (11)

From Eq.(9) we can see that if the coupling constant is then the energy gap is nonzero (if ) that corresponds to results obtained in works loz ; cap . Thus the operator describes the external source of Cooper pairs injecting the pairs into the system. In this case the operator can have a noninvariant form . In a work matt1 this operator and its analogues are used to break down the initial symmetry of a noninteracting system and to produce the desired structure (superconducting, ferromagnetic, solid etc.) - quasiaverages Bogolyubov method. In the other case the operator can have a symmetrical form , where is the order parameter of another superconductor, for example, the boundary (proximity) effect, when a superconductor is placed in contact with a normal metal schmidt or interband mixing of two order parameters belonging to different bands in a multi-band superconductor asker ; grig ; litak occurs.

2. On the other hand, in order to obtain the functions (10) we can suppose

 Δ=λV∑pu(p)\textslv(p), (12)

then

 E2=ε2+|Δ|2(1+υ2|Δ|)2,\textslv(k)u(k)=Δ(1+υ2|Δ|)2E. (13)

The integration domain in the self-consistency equation (12) must be cut off at some characteristic phonon energy ( if , outside ), then we have

 Δ=g∫ℏω−ℏωdεΔ(1+υ2|Δ|)2√ε2+|Δ|2(1+υ2|Δ|)2. (14)

We can see from Eq.(14), that if the coupling constant is then . This means, that only electron-electron coupling is the cause of superconductivity but not the potential . In this case the operator is an external field acting on the Cooper pairs only, and is energy of the pair in this field. As stated above, the potential is called as external pair potential since it is imposed on the electron subsystem by the structural elements of a substance (molecular, clusters etc. if their energy depends on state Cooper pairs), unlike the pair potential , which is result of electron-electron interaction and determined with the self-consistency equation. If we have usual self-consistency equation in BCS theory. In presence of the external pair potential the self-consistency condition has a form - Eq(14). Thus the potential renormalizes the order parameter .

It should be noticed that if we suppose , when , then an uncertainty takes place in the self-consistency equation (14). That is, it would seem, is not solution, unlike BCS theory. The absence of this solution is consequence of the fact that the derivative does not exist at because and the function is non-differentiable in this point (the derivative has a jump discontinuity), unlike BCS theory where . However, it should be noticed that the order parameter is a complex quantity: . If then the phase loses the sense so that . Then we have , hence becomes a solution of the self-consistency equation (14) and it means normal state of a superconductor.

Thus a system described with the Hamiltonian (4) has the energy with two minimums (9) and (14), which correspond to two physically different situations: injection of the Cooper pairs into the system from external source - option 1, and action of the external pair potential caused by dependence of energy of the molecular structure on state of a Cooper pair - option 2. In this work we will consider the model with the external pair potential only. If then the pairing lowers the energy of the molecular structure, that supports superconductivity . If then the pairing of electrons increases the energy of the molecular structure, that suppresses superconductivity , moreover such a critical value of the potential exists, that if then Eq.(14) has not any solutions. As discussed above this region has been supposed as the normal state of a superconductor. Thus the first order phase transition (on the parameter ) occurs - Fig.3. From the figure we can see for all , hence the pairs are stable. In the same time Eq.(14) has solutions at , however it can shown that in this case the condition (2) is not satisfied, that is the ”parasitic” solution occurs in this case.

Superconducting state is energetically favorable if the energy (7) of the superconducting state is less than energy of normal state i.e. . The energy can be determined as . Using Eq.(7) we have:

 Ws−Wn = VνF⎡⎣∫ℏω−ℏω2ε\textslv2(ε)dε−g(∫ℏω−ℏω\textslv(ε)u(ε)dε)2−υΔ|Δ|∫ℏω−ℏω\textslv(ε)u(ε)dε⎤⎦ (15) − VνF⎡⎣∫ℏω−ℏω2ε\textslv20(ε)dε−g(∫ℏω−ℏω\textslv0(ε)u0(ε)dε)2−υ∫ℏω−ℏω\textslv0(ε)u0(ε)dε⎤⎦,

where

 \textslv20(ε)=12(1−εE0),u20(ε)=12(1+εE0),u0(ε)\textslv0(ε)=υ/22E0,E20=ε2+(υ2)2. (16)

This renormalization of the normal state is a consequence of the extrapolation in the point where the self-consistent equation (14) does not have solutions. The expressions for the energies and have not any sense in themselves, only their difference is a physical quantity. However for a case interpretation for and is possible. Let the electron-electron interaction is switched off: . In addition it can be shown that . Then charge is still carried by the pairs of electrons (current carriers are the pairs) because for their breaking it must be made the work . This means that the quasiparticle spectrum has a gap . But this state is not superconducting because the ordering , i.e. long-range coherence, is absent. In our opinion such state can be interpreted as state with a pseudogap. However this state principally differs from the models of pseudogap in high- superconductors dzum ; dzum1 ; dzum2 ; dzum3 ; dzum4 , where preformed uncorrelated pairs are formed due to strong electron-electron interaction, but the phase coherence is possible only at more low temperature or at more large concentration of the pairs. In our model the uncorrelated pairs are pairs in momentum space (unlike bipolarons which are pairs in real space) which exist in noninteracting Fermi system due to the external pair potential. Switching-on of the electron-electron interaction stipulates the phase coherence. The fermionic nature of the pairs for such a system is discussed in Appendix A. For the case such interpretation as ”pseudogap” is impossible, because at the inequality (2) is not satisfied.

The energy (15) is shown in Fig.4. If we have the energy in BCS theory: the energy has symmetric minimums and a local maximum in a point . If then the minimums becomes deeper and are located at larger values of . If then the minimums becomes shallower and are located at smaller values of . At some value of the superconducting state becomes metastable, because the extremum in becomes a local minimum with lower energy. At lager value of the side minimums disappear and we have only one minimum in , that means the superconducting state is suppressed. The minimums of the function corresponding to different quantities of the potential (that is values of the function in points determined with Eq.(14)) are shown in Fig.5. We can see the domain of metastable states, where but Eq.(14) has solutions. The region of the solutions corresponding to the metastable states is shaded in Fig.3. The minimum of the energy (15) can be approximated with the expression:

 (Ws−Wn)min=−VνF[|Δ|22+c⋅υ|Δ|ln(2ℏω|Δ|)], (17)

where is a coefficient of the order of one and . If then we have the energy as in BCS theory. If then it can be at , that corresponds to the metastable states shown in Fig.3 and Fig.5.

## Iii Nonzero temperatures

At nonzero temperatures thermal activated quasiparticles appears with some a distribution function , hence the probability that a pair can be involved in formation of superconducting state is schmidt . Then generalizing Eq.(7) we obtain the free energy in a form:

 Fs = ∑k2ε(k)[u2(k)f(k)+\textslv2(k)(1−f(k))]−λV∑k,pu(k)\textslv(k)u(p)\textslv(p)(1−2f(k))(1−2f(p)) (18) + 2kBT∑k[f(k)lnf(k)+(1−f(k))ln(1−f(k))]−υΔ|Δ|∑ku(k)\textslv(k)(1−2f(k)),

where the first term is internal kinetic energy, the second term is energy of the electron-electron interaction, the third term is contribution of entropy, the last term is energy of Cooper pairs’ condensate in the external pair field. Minimizing the free energy and we obtain , and

 f(k)=1exp(EkBT)+1,E2=ε2+|Δ|2(1+υ2|Δ|)2, (19)

where

 Δ=λV∑pu(p)\textslv(p)(1−2f(p))⟹Δ=g∫ℏω−ℏωdεΔ(1+υ2|Δ|)2Etanh(E2kBT). (20)

A solutions of the self-consistency equation (20) at are shown in Fig.6, where we can see suppression of superconductivity and the first order phase transition. The critical temperature (which is a function of the parameter ) is less than the critical temperature of the superconductor without the external pair potential . At the equation (20) has only one solution for reasons discussed in Section II. From Fig.6 we can see for all temperatures , that is the work to break a Cooper pair (2) is positive, hence the pairs are stable.

If then the pairing lowers the energy of the molecular structure, that supports superconductivity . In this case a solution of Eq.(20) is such that the order parameter is not zero at any temperature - Fig.7. At large temperature , where , the gap is

 |Δ|≈g∫ℏω−ℏωdευ/22√ε2+(υ/2)2tanh(√ε2+(υ/2)22kBT)⟶gℏωυ4kBT. (21)

Formally the critical temperature of such system is equal to infinity (in reality it limited by the melting of the substance). It should be noted that if , then for any a superconducting state does not exist ( always). This means the electron-electron coupling is the cause of the transition to superconducting state only but not the external pair potential.

Superconducting state is energetically favorable if the free energy (18) is less than the free energy of normal state i.e . The free energy can be determined as . Then using Eq.(18) and the method of Section II - Eqs.(15,16), we can write the free energy in a form:

 Fs−Fn = VνF[∫ℏω−ℏω2ε[u2f+\textslv2(1−f)]dε−g(∫ℏω−ℏω\textslvu(1−2f)dε)2 (22) + 2kBT∫ℏω−ℏω[flnf+(1−f)ln(1−f)]dε−υΔ|Δ|∫ℏω−ℏω\textslvu(1−2f)dε] − VνF[∫ℏω−ℏω2ε[u20f0+\textslv20(1−f0)]dε−g(∫ℏω−ℏω\textslv0u0(1−2f0)dε)2 + 2kBT∫ℏω−ℏω[f0lnf0+(1−f0)ln(1−f0)]dε−υ∫ℏω−ℏω\textslv0u0(1−2f0)dε],

where

 f0(ε)=1exp(E0kBT)+1,E20=ε2+(υ/2)2. (23)

If we have the free energy in BCS theory: the energy has symmetric minimums and a local maximum in a point - Fig.9a. At the free energy has only one minimum in - superconducting state is absent. The free energy (22) at different temperatures and is shown in Fig.8. With increasing of temperature the minimums becomes shallower and are located in smaller values of . At some temperature the superconducting states in the minimums becomes metastable, because the extremum in becomes a minimum with lower energy. In the metastable phase the free energy but Eq.(20) has solutions. At higher temperature these minimums disappear, and we have only one minimum in , that means the superconducting state is absent. The region of the solutions corresponding to the metastable states is shaded in Fig.6.

If then the minimums become deeper and they are located in larger values of than in the case - Fig.9b. However the main difference between these cases is that the free energy at has absolute minimums where at all temperatures. This means the system is in superconducting state at any temperature. The minimums of the free energy (22) at are determined with Eq.(21).

Let us consider several important limit cases. Let the external pair potential is absent and temperature is slightly less than the critical temperature , hence the gap is small . Then we can expand the free energy (22) in powers of the order parameter:

 Fs−Fn=VνF[a(T−Tc)|Δ|2+b2|Δ|4], (24)

where and are some coefficient. Thus we have Ginzburg-Landau expansion.

The case in a limit is the most interesting. The energy gap is small in this region. Then we can expand the free energy (22) in powers of and in powers of :

 Fs−Fn=VνF[ℏω2kBT|Δ|2−υg(ℏω2kBT)2|Δ|]. (25)

From condition we obtain that coincides with Eq.(21). It should be noted that in this limit the contribution of the kinetic energy (the first term in Eq.(18)) is proportional to , in the same time the contribution of the entropy (the third term in Eq.(18)) is proportional to . Thus main factor, which resists the pairing at , is the entropy, and the contribution of the kinetic energy can be omitted unlike GL expansion. Substituting the energy gap (21) in the function (25) we have a gain in the free energy for the superconducting state:

 (Fn−Fs)min=VνFg2υ2(ℏω)316(kBT)3=μ0H2cm2V⟹Hcm∝υT3/2, (26)

that determines the thermodynamical magnetic field , which tends to zero asymptotically as temperature rises and is proportional to the potential .

## Iv Free energy functional

At presence of gradient of the order parameter (for example, when a current occurs) electrons are paired with momentums and accordingly. The nonzero momentum of the pairs causes an additional kinetic energy of the condensate, hence the expansion of superconductor’s free energy has a form schmidt :

 Fs−Fn=−αns+β2n2s+nsq24m, (27)

where is the condensate’s density. In this case the density is a function of the momentum, and a critical momentum exists such that . As a rule . Thus we have to consider an operator of the kinetic energy in a form

 ˆHkin=∑k[ε(%k+q2)a+k+q2↑ak+q2↑+ε(−k+q2)a+−k+q2↓a−k+q2↓], (28)

where

 (29)

Using the free energy (22) with the kinetic energy calculated on the operator (28) and passing to the limit at we can obtain a free energy:

 Fs−Fn=V[−A|Δ|+B2|Δ|2+Cq2|Δ|], (30)

where the coefficients are

 A=νFg(ℏω2kBT)2υ,B=νFℏωkBT,C=νFℏωk2F144(kBT)3m2υ. (31)

Then we can obtain an equilibrium value of the gap, value of the free energy in this point and the critical momentum of the condensate accordingly:

 δFδ|Δ|=0⟹|Δ|min=AB(1−CAq2) (32) (Fs−Fn)min=−A22B(1−CAq2)2⟹q2c=AC (33)

In the general case the vector q can run through the whole momentum space (when the pairing potential is spatially inhomogeneous ), then we should sum the free energy (30) over all values of the momentum q. Moreover if the system is in a magnetic field, then the momentum q must be replaced by , where is a Fourier component of a magnetic vector potential , so that is a magnetic induction vector. Then the free energy functional takes a form

 Fs−Fn=V∑q[−A|Δq|+B2|Δq|2+C(q−2eaq)2|Δq|+12μ0ℏ2(q2a2q−(qaq)2)], (34)

where the last term is energy of the magnetic field . Minimizing the free energy functional (34) by the magnetic field we obtain a current in a form

 jq=4eC|Δ|q−8e2C|Δ|aq, (35)

where we can suppose in the local (London) limit. In a case of a singly connected superconductor using gauge transformations Eq.(35) can be reduced to the equation describing Meissner’s effect. The Londons’ equation can be written in a form , where is a superfluid density (the density of correlated pairs). Then

 ns=νFℏωk2F36(kBT)3mυ|Δ|. (36)

Thus the superfluid density is , unlike BCS theory where . At the same time, as in BCS theory, the response of a superconductor to electromagnetic field is determined by the gap which has long-range coherence. If then the pseudogap cannot provide such electromagnetic response. General expression for the superfluid density is given in Appendix A. If we consider a superconductor with an inner cavity, then along a closed path lying within the superconductor around the cavity at a distance from the cavity’s surface larger than magnetic penetration depth we have . Integrating along the path we obtain magnetic flux quantization , where is a magnetic flux quantum. Thus the free energy functional (34) describes basic properties of superconductivity.

Using the transformations and the inverse transformations we can write the functional Eq.(34) in real space, however the functional will have a complicated and inconvenient form due to terms and . Nevertheless we have seen above that the functional (34) describes basic properties of superconductivity, hence the functional can be replaced by an effective GL functional, which has the same symmetry, the same extremes and the same values in these extremes - Fig.10. The effective GL functional has a form

 Fs−Fn=V∑q[−B|Δq|2+B32A2|Δq|4+BCA(q−2eaq)2|Δq|2+12μ0ℏ2(q2a2q−(qaq)2)]. (37)

The extremum , the value of the free energy and the critical momentum (if there are only one q in the sum and ) are

 |Δq|min=AB√1−CAq2,(Fs−Fn)min=−A22B(1−CAq2)2⟹q2c=AC. (38)

Comparing Eq.(38) with Eqs.(32,33) we can see coincidence of extremes at , coincidence of the values and coincidence of the critical momentums . Some different dependence has no basic value. The current are

 jq=4eBCA|Δ|2q−8e2BCA|Δ|2aq=4eACBq−8e2ACBaq,

that coincides with the current (35) in the local limit. This means that the magnetic responses will be identical.

In real space the functional (37) takes a form

 Fs−Fn=∫[−B|Δ(r)|2+B32A2|Δ(r)|4+BCA|(−iℏ∇−2eA)Δ(r)|2+