Eliashberg theory with the external pair potential

TL;DR

This paper extends the Eliashberg theory by incorporating an external pair potential into a model with electron-phonon and Coulomb interactions, deriving generalized equations and analyzing high-temperature behavior.

Contribution

It introduces a novel Eliashberg framework with an external pair potential and derives generalized equations considering Coulomb effects.

Findings

Order parameter tends to zero at high temperature

Coulomb pseudopotential influences critical temperature

Effective Ginzburg-Landau theory formulated

Abstract

Based on BCS model with the external pair potential formulated in a work Grigorishin (2017) [1], analogous model with electron-phonon coupling and Coulomb coupling is proposed. The generalized Eliashberg equations in the regime of renormalization of the order parameter are obtained. High temperature asymptotics and effect of Coulomb pseudopotential on them are investigated: as in the BCS model the order parameter asymptotically tends to zero as temperature rises, but the accounting of the Coulomb pseudopotential leads to existence of critical temperature. The effective Ginzburg-Landau theory is formulated for such model.

Equations

$$|\Delta (T\to \infty )|=\frac{g\omega \upsilon }{4T},$$

$$G=i\frac{i{\epsilon }_n+\xi }{(i{\epsilon }_n{)}^2-{\xi }^2-|\Delta {|}^2{\left(1+\frac{\upsilon }{2|\Delta |}\right)}^2}$$

$$F=i\frac{\Delta \left(1+\frac{\upsilon }{2|\Delta |}\right)}{(i{\epsilon }_n{)}^2-{\xi }^2-|\Delta {|}^2{\left(1+\frac{\upsilon }{2|\Delta |}\right)}^2},$$

$$\Delta =gT\sum _{n=-\infty }^{\infty }{\int }_{-\omega }^{\omega }d\xi iF({\epsilon }_n,\xi )⟹1=g{\int }_{-\omega }^{\omega }d\xi \frac{1+\frac{\upsilon }{2|\Delta |}}{2\sqrt{{\xi }^2+|\Delta {|}^2{\left(1+\frac{\upsilon }{2|\Delta |}\right)}^2}}\tanh \frac{\sqrt{{\xi }^2+|\Delta {|}^2{\left(1+\frac{\upsilon }{2|\Delta |}\right)}^2}}{2T}.$$

$$V({\epsilon }_2-{\epsilon }_1,{\text{p}}_2-{\text{p}}_1)=u_{ph}^2\frac{2{\omega }_{{\text{p}}_2-{\text{p}}_1}}{({\epsilon }_2-{\epsilon }_1{)}^2-{\omega }_{{\text{p}}_2-{\text{p}}_1}^2},$$

$$V({\epsilon }_2-{\epsilon }_1=0,{\text{p}}_2-{\text{p}}_1=2p_F)=-u_{ph}^2\frac{2}{{\omega }_{2p_F}}\equiv -\lambda .$$

$$G=i\frac{i{\epsilon }_{n}Z(p)+\tilde{\xi }}{{\left(ipZ(p)\right)}^2-{\tilde{\xi }}^2-|\tilde{W}(p){|}^2}$$

$$F=i\frac{\tilde{W}(p)}{{\left(ipZ(p)\right)}^2-{\tilde{\xi }}^2-|\tilde{W}(p){|}^2},$$

$$\tilde{W}(p)=W(p)+\frac{W(p)}{|W(p)|}\frac{\upsilon }{2}.$$

$$S(\text{p},ip)=-\int \frac{d^{3}q}{(2\pi {)}^3}T\sum _{iq}{V}_{eff}(\text{q},iq)iG(\text{p}+\text{q},ip+iq)$$

$$W(\text{p},ip)=-\int \frac{d^{3}q}{(2\pi {)}^3}T\sum _{iq}{V}_{eff}(\text{q},iq)iF(\text{p}+\text{q},ip+iq).$$

$$Z(2n+1)=1+\frac{1}{2n+1}\sum _{m\ne n}\tilde{\Lambda }(2m+1)V(n-m)$$

$$Z(2n+1)\Delta (2n+1)=\pi T\sum _{m\ne n}\tilde{\Phi }(2m+1)\left[V(n-m)-{\mu }_c\right],$$

$$\tilde{\Lambda }(2n+1)=\frac{{\epsilon }_n}{\sqrt{{\epsilon }_n^2+|\tilde{\Delta }(2n+1){|}^2}}$$

$$\tilde{\Phi }(2n+1)=\frac{\tilde{\Delta }(2n+1)}{\sqrt{{\epsilon }_n^2+|\tilde{\Delta }(2n+1){|}^2}},$$

$$\tilde{\Delta }(2n+1)=\Delta (2n+1)+\frac{\upsilon }{2}\frac{\Delta (2n+1)}{|\Delta (2n+1)|}\frac{1}{Z(2n+1)}.$$

$$V(m-n)=\frac{g}{1+{\left(\frac{2\pi T}{{\omega }_0}\right)}^2(m-n{)}^2},$$

$$Z(2n+1)=1+\frac{2g}{|2n+1|}\sum _{l=1}^{|n|>1}\frac{1}{1+{\left(\frac{2\pi T}{{\omega }_0}\right)}^2{l}^2}=Z(-2n-1).$$

$$Z(2n+1)\Delta (2n+1)=\sum _{m\ne n}\frac{\Delta (2m+1)}{|2m+1|}V(n-m).$$

$$\Delta (1)$$

$$\Delta (-1)$$

$$T_c=\frac{{\omega }_0}{2\pi }\sqrt{g-1}.$$

$$Z(2n+1)\Delta (2n+1)=\sum _{m\ne n}\left[\frac{\Delta (2m+1)}{|2m+1|}V(n-m)+\frac{\upsilon }{2}\frac{\Delta (2m+1)}{|\Delta (2m+1)|Z(2m+1)}\frac{V(n-m)}{|2m+1|}\right].$$

$$\Delta (1)$$

$$\Delta (-1)$$

$$|\Delta |=\frac{\upsilon }{2}\frac{V(1)}{1-V(1)}\approx \frac{\upsilon g{\omega }_0^2}{8{\pi }^2{T}^2}.$$

$$\Delta (1)$$

$$\Delta (-1)$$

$$T_c=\frac{{\omega }_0}{2\pi }\sqrt{\frac{g}{1+{\mu }_c}-1}.$$

$$\Delta (1)$$

$$\Delta (-1)$$

$$|\Delta |=\frac{\upsilon }{2}\left[\frac{g}{1+{\left(\frac{2\pi T}{{\omega }_0}\right)}^2}-{\mu }_c\right],$$

$$\Delta (T_c)=0⟹T_c=\frac{{\omega }_0}{2\pi }\sqrt{\frac{g}{{\mu }_c}-1}.$$

$$|\Delta |=\frac{\upsilon g{\omega }_0^2}{4{\pi }^2{T}_c^3}\left(T_c-T\right).$$

$$F_s-F_n=V\sum _{\text{q}}\left[-A|{\Delta }_{\text{q}}|+\frac{B}{2}|{\Delta }_{\text{q}}{|}^2+C{\left(\text{q}-2e{\text{a}}_{\text{q}}\right)}^2|{\Delta }_{\text{q}}|+\frac{1}{2{\mu }_0{\hslash }^2}\left(q^2{a}_{\text{q}}^2-(\text{q}{\text{a}}_{\text{q}}{)}^2\right)\right],$$

Full text

Eliashberg equations, electron-phonon interaction, Coulomb pseudopotential, external pair potential, critical temperature, effective Ginzburg-Landau theory

I Introduction

In a work grig a hypothetical substance has been considered, where interaction between (within) structural elements of condensed matter (molecules, nanoparticles, clusters, layers, wires etc.) depends on state of Cooper pairs: an additional work $\upsilon$ 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 enlarges energy of the structure then suppression of superconducting (SC) order parameter and the first order phase transition occur, if the pairing lowers energy of the structure then the energy gap is slightly enlarged at zero temperature and at large temperature $T\gg T_{c}$ the gap asymptotically tends to zero as temperature rises:

[TABLE]

where $g$ is a constant of electron-electron coupling via phonons, $\omega$ is the phonon frequency, $\upsilon>0$ is the external pair potential (EPP). It should be noted that if $g=0$ then for any $\upsilon$ the SC state does not exist ($\Delta=0$ always). This means that only the electron-electron coupling is cause of transition to SC state, but EPP is not. Thus in such system the critical temperature is infinity formally as illustrated in Fig.(1). In this model normal $G$ and anomalous $F$ propagators have forms:

[TABLE]

where $\varepsilon_{n}=\pi T(2n+1)$. The self-consistency condition for the order parameter is

[TABLE]

We can see that the quasiparticle spectrum has a gap $\upsilon/2$ ($\upsilon>0$) even when $\Delta=0$. But this state is not SC because the ordering $\left\langle aa\right\rangle,\left\langle a^{+}a^{+}\right\rangle$ is absent. Such state can be interpreted as state with a pseudogap due the strong fluctuations of the phase $\phi(\textbf{r},t)$ of the order parameter $\Delta=|\Delta|e^{i\phi}$ so that $\langle e^{i\phi(\textbf{r},t)}\rangle=0$ grig3 . In a case $\upsilon>0$ at large temperature $T\gg T_{c}$ the gap $\Delta(T)$ tends to zero asymptotically (1) as temperature rises. Based on bipolaron model of superconductivity dzum , in grig has been demonstrated that the size of a Cooper pair is much larger than the mean distance between the Cooper pairs (the pairs are strongly overlapped) even for hypothetical room temperatures, that is the Cooper pairs have fermionic nature.

The above results are based on BCS theory, however real electron-electron interaction is due to the exchange of virtual phonons. The corresponding potential is described with an expression

[TABLE]

where, $u_{ph}$ is an electron-phonon coupling constant, $\varepsilon_{1},\varepsilon_{2}$ are energetic parameters of interacting electrons and $\textbf{p}_{2}-\textbf{p}_{1}$ is transmitted momentum. Since the electrons interact with small total momentum $\textbf{p}_{2}+\textbf{p}_{1}=0$, then we can assume that the transmitted momentum is $\sim 2p_{F}$, at the same time, near the Fermi surface we have $\varepsilon_{2}\sim\varepsilon_{1}\sim 0$. Therefore the expression (5) can be reduced to

[TABLE]

Thus the real interaction is replaced with an effective point attraction, which is nonzero in the layer of width $2\omega_{2p_{F}}\sim 2\omega_{D}$ (Debay frequency) near Fermi surface sad . In other words, the BCS approximation neglects retardation $\varepsilon_{2}-\varepsilon_{1}$ in el.-phon. interaction (the field of lattice deformation is supposed without inertia). On the contrary, the Eliashberg model mahan ; ginz is based on the full interaction (5). Besides el.-phon. interaction the screened Coulomb interaction $V_{c}$ between electrons takes place which has width $\sim\epsilon_{F}$. In metals, as a rule, $\lambda<V_{c}$ 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 suppression of the direct Coulomb interaction ginz ; kirz . A stronger condition $\lambda-V_{c}>0$ can occur in nonmetallic superconductors (for example, in alkali-doped fullerides $\texttt{A}_{n}\texttt{C}_{60}$, where competition between the Jahn-Teller coupling and Hund’s coupling takes place han ; han1 ; nom ; grig4 ).

Our goal is to develop model of superconductivity with EPP using the electron-electron interaction in a form (5) and accounting the Coulomb repulsion. In the section II we develop the Eliashberg theory with EPP and investigate high temperature asymptotics in the absence of the Coulomb pseudopotential and in the presence of one. In the section III, based on effective Ginzburg-Landau theory for the BCS model with EPP developed in grig , we formulate the effective GL theory based on the Eliashberg theory.

II Generalization of Eliashberg equations

Let us take into account the fact that the electron-electron interaction is the result of the exchange of virtual phonons and of screened Coulomb interaction: $V_{eff}(\textbf{q},i\omega)=u_{ph}^{2}(\textbf{q})iD(\textbf{q},i\omega)+V_{c}(\textbf{q})$, where $u_{ph}(\textbf{q})$ is an electron-phonon coupling parameter, $D(\textbf{q},i\varepsilon)=\frac{i2\omega(\textbf{q})}{(\varepsilon_{n}-\varepsilon_{m})^{2}+\omega^{2}(\textbf{q})}$ is a phonon propagator. Eliashberg equations, unlike BCS equations, describe the decrease of effectiveness of the interaction at phonon energies $\omega(\textbf{q})\ll T$ (thermal phonons are perceived by electrons as static impurities ginz ): as a result $T_{c}\propto\overline{\omega}\sqrt{g}$ for the el.-phon. model unlike $T_{c}\propto\overline{\omega}g$ in BCS theory for $g\gg 1$. Moreover, renormalization of electron’s mass takes place: $m=m_{0}\left(1+g\right)$ at low temperatures $T\ll\overline{\omega}$, on the contrary at high temperature $T\gg\overline{\omega}$ the renormalization is negligible $m=m_{0}\left(1+\frac{\mathrm{const}}{T^{2}}\right)$ ginz .

Like Eqs.(2,3) the normal $G$ and anomalous $F$ propagators take forms:

[TABLE]

where $p\equiv[\textbf{p},ip=i\pi T(2n+1)]$ and

[TABLE]

The self-energies are determined by the self-consistency conditions:

[TABLE]

The self energy $S(\textbf{p},ip)$ can be broken up into symmetric and antisymmetric parts: $S(p)=S_{e}(\textbf{p},ip)+ipS_{o}(\textbf{p},ip)$, where $S_{e}$ and $S_{o}$ are both even functions of frequency $ip$. Then renormalization coefficient $Z$ for a single-particle Green function are $Z(\textbf{p},ip)=1-S_{o}(\textbf{p},ip)$. Accounting of $S_{e}$ renormalizes the chemical potential only, that does not influence on the quasiparticles’ spectrum. Thus we have $ip-\xi-S(p)=ipZ(p)-\widetilde{\xi}$, where $\widetilde{\xi}=\xi+S_{e}(p)$. The functions $Z(p)$ and $W(p)$ are functions of $(\textbf{p},ip)$. For isotropic s-wave superconductor we can suppose $|\textbf{p}|\approx p_{F}$ and the main dependence of $G(p)$ and $F(p)$ on p is through the factor $\widetilde{\xi}\approx\xi$. Then, using method of mahan , Eqs.(10,11) can be written in a form of Eliashberg equations:

[TABLE]

where $\mu_{c}=V_{c}\nu_{F}$ is a Coulomb pseudopotential ($\nu_{F}$ is electron density on Fermi surface), $V(m)=2\int\omega d\omega\frac{\alpha^{2}F(\omega)}{\omega^{2}+\varepsilon_{m}^{2}}$ is a phonon interaction. Note that $V(0)=g$, where $g$ is the dimensionless strength of the electron-phonon interaction. The gap function is $\Delta(2n+1)=\frac{W(2n+1)}{Z(2n+1)}$,

[TABLE]

where from Eq.(9) we have

[TABLE]

In Eqs.(12,13) we have excluded the terms with $m=n$ in the summation, because this term corresponds to elastic scattering of the quasiparticles on thermal phonons, that does not make contribution to quasiparticle’s mass $m=m_{0}Z$ and to SC order parameter $\Delta$. Justification of this fact is given in Appendix A.

For simplicity we will consider the Einstein model of phonons: all of the phonons have the same frequency $\omega_{0}$ and $\alpha^{2}F(\omega)=\omega_{0}g\delta(\omega-\omega_{0})/2$. Then

[TABLE]

We can see that efficiency of the el.-el. interaction through phonons exchange decreases with increasing temperature. Let us consider particular cases of Eqs.(12,13):

1. $\upsilon=0$, $\mu_{c}=0$. Near $T_{c}$ we have $|\Delta|\ll T$, then Eq.(13) takes a form:

[TABLE]

The asymptotic limit $g\gg 1$ can be found in a simple way. Assuming that $\frac{2\pi T}{\omega_{0}}$ becomes very large so that $V(l)$ becomes increasingly small for values $|l|>2$. The gap equation (19) can be solved by using only a matrix of dimension two: it is necessary to retain only the gap components $\Delta(1)$ and $\Delta(-1)$ mahan , and the renormalization parameter is $Z(1)=Z(-1)=1$. Then:

[TABLE]

Setting the determinant equal to zero gives the critical temperature:

[TABLE]

2. $\upsilon>0$, $\mu_{c}=0$. Let temperature is high and the gap is small: $T\gg\upsilon\gg|\Delta|$, then Eq.(13) takes a form:

[TABLE]

Analogously to previous case we use only a matrix of dimension two:

[TABLE]

Setting the determinant equal to zero at assumption $|\Delta(1)|=|\Delta(-1)|$ (since the interaction is symmetrical $V(1)=V(-1)$) gives that the energy gap $\Delta$ does not vanish at any temperature:

[TABLE]

However, unlike result of BCS theory (1), the gap tends to zero faster (as $1/T^{2}$) that is consequence of the lower effectiveness of the el.-phon. interaction for low phonon energies $\omega(\textbf{q})\ll T$.

3. $\upsilon=0$, $\mu_{c}\neq 0$. As in previous consideration we suppose $g\gg 1$. For $g>\mu_{c}$ the interaction $V(l)-\mu_{c}$ is attractive for small values for $l$ but it becomes repulsive for large values of $l$. For such values of $m$ that $V(n-m)-\mu_{c}<0$ we suppose $\Delta(2m+1)=0$. Thus, as in previous cases, it is necessary to retain only the gap components $\Delta(1)$ and $\Delta(-1)$, then Eq.(13) has a form:

[TABLE]

Setting the determinant equal to zero gives critical temperature:

[TABLE]

Thus it must be $g>1+\mu_{c}$ for such a solution. However in real materials as a rule the relation $g<\mu_{c}$ occurs and the pairing of electrons can be possible due to reduction of the Coulomb repulsion by Tolmachev’s logarithm: $g>\mu_{c}^{*}=\mu_{c}/\left(1+\mu_{c}\ln\frac{\varepsilon_{F}}{\overline{\omega}}\right)$ which is result of the second order processes ginz ; kirz .

4. $\upsilon>0$, $\mu_{c}\neq 0$, temperature is hight and the gap is small $T\gg\upsilon\gg|\Delta|$. Then Eq.(13) takes a form (we suppose $T\gg\bar{\omega}$ as in previous cases):

[TABLE]

From these equations we find the gap like we have done in Eqs.(II):

[TABLE]

from where we get the critical temperature:

[TABLE]

Thus for such a solution it must be $g>\mu_{c}$, that is discussed in Appendix B. We can see that accounting of the Coulomb pseudopotential leads to existence of critical temperature unlike the result (24) where the gap tends to zero asymptotically. The critical temperature (29) is determined by the coupling constants $g,\mu_{c}$ and the frequency $\omega_{0}$ only, like ordinary superconductor (26), but it does not depend on EPP $\upsilon$. However there is a principal difference of Eq.(29) from Eq.(26): if we suppose $\mu_{c}=0$ then $T_{c}(\upsilon=0)\sim\sqrt{g}$ but $T_{c}(\upsilon>0)=\infty$ and the gap (28) passes into the asymptotic (24): $|\Delta|\propto 1/T^{2}$.

The expression (28) can be expanded in the vicinity of $T_{c}$:

[TABLE]

Thus in this model the gap at $T\rightarrow T_{c}$ linearly depends on the temperature difference $\left(T_{c}-T\right)$ unlike ordinary mean field theory without EPP where the temperature dependence of an order parameter is $\left(T_{c}-T\right)^{1/2}$. Temperature dependencies of the energy gap for different parameters are shown in Fig.2.

III Effective Ginzburg-Landau theory

In a work grig the effective Ginzburg-Landau theory for the BCS model with EPP has been formulated. Corresponding free energy functional has a form:

[TABLE]

where the last term is energy of the magnetic field $\frac{1}{2\mu_{0}}\left(\texttt{curl}\textbf{A}\right)^{2}$, the coefficients are

[TABLE]

From the free energy functional we can obtain an equilibrium value of the gap, value of the free energy in this point and the critical momentum of a Cooper pair accordingly:

[TABLE]

If $q=0$ then we obtain Eq.(1) for the equilibrium value of the gap. Functional (31) can be written in real space using Fourir transformation, however the functional will have a complicated and inconvenient form due to terms $|\Delta_{\textbf{q}}|$ and $q^{2}|\Delta_{\textbf{q}}|$. Then according to grig the functional (31) can be replaced by an effective GL functional, which has the same symmetry, the same extremes and the same values in these extremes. The effective GL functional has a form:

[TABLE]

Unlike BCS theory with EPP, accounting of the Coulomb pseudopotential leads to the gap in a form (28) that provides existence of critical temperature (29). In order to account this facts we should to write the functional (31) in a form

[TABLE]

where the coefficient $\Xi$ is such to obtain the gap (30):

[TABLE]

Then the critical momentum of a pair is

[TABLE]

and the gain in free energy (at $q=0$) is

[TABLE]

Thus phase transition to superconducting state in the Eliashberg theory with EPP is the second order phase transition like in ordinary GL theory. Following the above method we can write the effective GL functional:

[TABLE]

Basic characteristics of a superconductor (coherence length $\xi$, magnetic penetration depth $\lambda$, GL parameter $\kappa$, thermodynamical critical field $H_{cm}$, the first $H_{c1}$ and the second $H_{c2}$ critical fields) for ordinary GL theory at $T\rightarrow T_{c}$, effective GL theory based on BCS theory with EPP at $T\rightarrow\infty$ and effective GL theory based on Eliashberg theory with EPP at $T\rightarrow T_{c}$ are presented in the following table:

[TABLE]

We can see from the table that in the effective GL theory based on the Eliasberg approach the temperature dependencies of the basic characteristics of a superconductor is similar to the ordinary GL theory unlike the approach based on BCS theory. In particular, this model restores ordinary temperature dependence of the coherence length $(T_{c}-T)^{-1/2}$ after the BCS model with EPP where it decreases as $1/\sqrt{T}$ at large $T$ that corresponds extremely small value of order of interatomic distances. However the effective free energy functional (40) has an extraordinary form due temperature dependence of the gap as $\Delta\propto(T_{c}-T)$ unlike the ordinary GL theory where $\Delta\propto(T_{c}-T)^{1/2}$. We cannot write the effective functional as $-(T_{c}-T)^{2}a\Delta^{2}+b\Delta^{4}$ (that gives the desired temperature dependence of the gap too) because we will obtain the free energy as $F_{min}\propto(T_{c}-T)^{4}$, that means the phase transition to superconducting state will not be second-order phase transition.

IV Summary

Based on BCS model with EPP formulated in grig in this work we have proposed analogous model with electron-phonon coupling and Coulomb coupling. We have obtained the generalized Eliashberg equations (12,13,14,15,16) for the case of the external pair potential. Only electron-electron coupling is the cause of the SC ordering, but not EPP, however the potential essentially renormalizes the order parameter. Solving these equations for the case $\upsilon>0$ (that is the pairing lowers the energy of the molecular structure, that supports superconductivity) we have obtained the following asymptotic solutions.

If electron-phonon interaction is present only (the Coulomb pseudopotential is absent $\mu_{c}=0$) then the energy gap $\Delta$ does not vanish at any temperature, however the gap tends to zero faster (as $1/T^{2}$ - Eq.(24)) than the result of BCS theory (as $1/T$ - Eq.(1)) that is consequence of decreasing of efficiency of the el.-el. interaction through phonons exchange with increasing temperature. On the other hand the accounting of the Coulomb pseudopotential $\mu_{c}\neq 0$ leads to existence of critical temperature (29), which is much higher than one in pure material. The gap at $T\rightarrow T_{c}$ linearly depends on temperature difference $T_{c}-T$ - Eq.(30), unlike the ordinary mean field theory without EPP where the temperature dependence of the order parameter is $\left(T_{c}-T\right)^{1/2}$. It should be noted that equilibrium uncorrelated pairs are present at $T>T_{c}$ even, unlike ordinary superconductors (which are described by BCS and Eliashberg theories).

Based on a free energy functional for BCS model with EPP obtained in grig we have written free energy functional (36) for our model which takes into account above-mentioned critical temperature and the linear dependency of the order parameter on the temperature difference $T_{c}-T$. Following grig we have obtained the effective Ginzburg-Landau functional, which has the same symmetry, the same extremes and the same values in these extremes as in the initial functional. The temperature dependencies near $T_{c}$ of the basic characteristics of a superconductor (coherence length, magnetic penetration depth, GL parameter, thermodynamical critical field, the first and the second critical fields) recovers to the temperature dependencies as in the ordinary GL theory after the BCS model with EPP.

Appendix A Scattering on thermal phonons

Let us consider Eq.(12) with the symmetrical term $m=n$ and when $\Delta=0$, $\upsilon=0$:

[TABLE]

Then we have

[TABLE]

where $V(0)=2\int\omega d\omega\frac{\alpha^{2}F(\omega)}{\omega^{2}}\equiv g$, $V(l)=2\int\omega d\omega\frac{\alpha^{2}F(\omega)}{\omega^{2}+(2\pi Tl)^{2}}$. We can see that at $T\rightarrow\infty$ (this means $T\gg\overline{\omega}$) we have $V(l\neq 0)\rightarrow 0$. Thus $Z(T\rightarrow\infty)=1+g$. However it must be $Z(T\rightarrow 0)=1+g$ and $Z(T\rightarrow\infty)=1$ ginz : electron’s mass $m_{0}$ is renormalized due el.-ph. interaction as $m=m_{0}Z$ (an electron is being followed by cloud of virtual phonons), but at high temperatures ($T\gg\overline{\omega}$) the renormalization is negligible, that underlies the experimental method of finding of the constants $g$.

Let us consider the term with $m=n$ in Eqs.(10,41). This term corresponds to elastic interaction because the energetic parameters of electron and phonon do not change $\varepsilon_{n}=\varepsilon_{m}$ but the momentum changes as $\textbf{p}\rightarrow\textbf{p}-\textbf{q}$. Let us consider elastic scattering of an electron on impurities of concentration $\rho$ using diagrammatics for disordered systems sad - Fig.(3). The self-energy has a form:

[TABLE]

Thus the elastic impurities do not influence upon effective mass of quasi-particles but they stipulate the damping of quasi-particles $\gamma\textrm{sgn}\varepsilon_{n}$ (the mean free time and the free length are determined as $\tau=\frac{1}{2\gamma},\quad l=v_{F}\tau$). The self-energy (10) with the symmetrical term $m=n$ only, using (41), takes a form:

[TABLE]

Comparing Eq.(44) and Eq.(43) we can see that elastic scattering of the quasiparticles on thermal phonons is equivalent to the elastic scattering on impurities, and it does not influence upon effective mass of quasi-particles but stipulates the damping of quasi-particles. Hence the term with $m=n$ must be omitted in the equation for $Z$.

Let us consider Eq.(13) when $T\gg\upsilon\gg|\Delta|$ (the first term can be omitted):

[TABLE]

For $m=n$ we have nonzero gap at $T\rightarrow\infty$:

[TABLE]

This gap, like the renormalization factor $Z$ at $m=n$, is result of scattering on thermal phonons. Above we have seen that this scattering stipulates the damping of quasi-particles but it does not make contribution to the effective mass. It can be assumed, that this scattering cannot lead to coherent assemble of Cooper pairs, so that for the gap (46) we have $\left\langle\Delta(\textbf{r},t)\right\rangle=|\Delta(\textbf{r})|\left\langle e^{i\phi(\textbf{r},t)}\right\rangle=0$ although it can be $|\Delta(\textbf{r})|\neq 0$, i.e. the superconducting ordering is destroyed by phase fluctuations as described in a work grig3 . Thus the term with $m=n$ must be omitted in the equation for $\Delta$.

Appendix B The ratio between $g$ and $\mu_{c}$

The electron-electron interaction in a metal has a form ginz :

[TABLE]

where $\varepsilon(q,\omega)$ is a dielectric permittivity, $\omega(q)$ is frequency of remormalized phonons $\omega^{2}(q)\approx\Omega^{2}/\varepsilon(q,0)$, $\Omega^{2}=\frac{4\pi n(Ze)^{2}}{M}$ is frequency of bare phonons in the jelly model (plasmons in gas of ions of charge $Ze$, mass $M$ and with concentration $n$). The first term is a screened Coulomb interaction, the second term is an interaction via phonons. The renormalized el.-phon. coupling constant $g$ is connected with the bare constant $\lambda^{2}(q)$ by a following expression:

[TABLE]

The Coulomb coupling constant $\mu_{c}$ is

[TABLE]

Comparing these equations we can see that to be $g>\mu_{c}$ it is necessary the bare el.-phon. interaction exceeds the bare Coulomb interaction $V_{c}(q)=\frac{4\pi e^{2}}{q^{2}}$:

[TABLE]

In the jelly model $\lim_{q\rightarrow 0}\frac{\lambda^{2}(q)}{V_{c}(q)}=1$ ginz ; schr . This means that $g>\mu_{c}$ can occur without Tolmachev’s reduction $\mu_{c}^{\ast}=\mu_{c}/\left(1+\mu_{c}\ln\left(\frac{\varepsilon_{F}}{\omega_{0}}\right)\right)\ll\mu_{c}$. Thus SC phase can be in the materials with narrow conduction band $\varepsilon_{F}\sim\omega_{0}$, for example, in alkali-doped fullerides han ; han1 ; nom ; grig4 .

Acknowledgements.

This research was supported by theme grant of department of physics and astronomy of NAS of Ukraine: ”Noise-inducing dynamics and correlations in nonequilibrium systems”, N 0120U101347.