---
abstract: |
    The theory of the electron-phonon interaction ($EPI$) with strong forward scattering peak ($FSP$) in an extreme delta-peak limit [@Kulic-Zeyher1-2]-[@Dan-Dol-Kul-Oud] is recently applied in [@Lee-Interfacial-supplement]-[@Johnston1-2] for the explanation of high $T_{c}(\sim100$ $K)$ in a monolayer $FeSe$ grown on $SrTiO_{3}$ [@Lee-Interfacial-supplement] and $TiO_{2}$ [@Rebec-FeSe-TiO2] substrates. The $EPI$ is due to a long-range dipolar electric field created by the high-energy oxygen vibrations ($\Omega\sim90$ $meV$) at the interface [@Lee-Interfacial-supplement]-[@Johnston1-2]. We show that in leading order (with respect to $T_{c0}/\Omega$) the mean-field critical temperature $T_{c0}=\left\langle V_{epi}(q)\right\rangle _{q}/4)$ $\sim$ $(aq_{c})^{2}V_{epi}(0)$ and the gap $\Delta_{0}=2T_{c0\text{ }}$ are due to an interplay between the maximal $EPI$ pairing potential $V_{epi}(0)$ and the $FSP$-width $q_{c}$. For $T_{c0}\sim100$ $K$ one has $\Delta_{0}\sim16$ $meV$ in a satisfactory agreement with $ARPES$ experiments. We find that in leading order $T_{c0}$ is * mass-independent* and a very small oxygen isotope effect is expected in next to leading order. In clean systems $T_{c0}$ for $s$-wave and $d$-wave pairing is degenerate but both are affected by non-magnetic impurities, which are *pair-weakening* in the $s$-channel and *pair-breaking* in the $d$-channel.

    The self-energy and replica bands at $T=0$ and at the Fermi surface are calculated and compared with the corresponding results at $T>0$ [@Johnston1-2]. The $EPI$ coupling constant $\lambda_{m}=\left\langle
    V_{epi}(q)\right\rangle _{q}/2\Omega$, which enters the self-energy $\Sigma(k,\omega)$, is *mass-dependent* ($M^{1/2}$) which at $\omega(\ll\Omega)$ makes the slope of $\ \Sigma(k,\omega)(\approx-\lambda
    _{m}\omega)$ and the replica intensities $A_{i}(\sim\lambda_{m})$ *mass-dependent*. This result, overlooked in the literature, is contrary to the prediction of its mass-independence in the standard Migdal-Eliashberg theory for $EPI$. The small oxygen isotope effect in $T_{c0}$ and pronounced isotope effect in $\Sigma(k,\omega)$ and $ARPES$ spectra $A_{i}$ of replica bands in $FeSe$ films on $SrTiO_{3}$ and $TiO_{2}$ is a smoking-gun experiment for testing an applicability of the $EPI-FSP$ theory to these systems. The $EPI-FSP$ theory predicts a large number of low-laying pairing states (above the ground state) thus causing internal pair fluctuations. The latter reduce $T_{c0}$ additionally, by creating a pseudogap state for $T_{c}<T<T_{c0}$.

    Possibilities to increase $T_{c0}$, by designing novel structures are discussed in the framework of the $EPI-FSP$ theory.
address: |
    $^{1}$Institute for Theoretical Physics, Goethe-University D-60438 Frankfurt am Main, Germany\
    $^{2}$Institute of Physics, Pregrevica 118, 11080 Belgrade (Zemun), Serbia\
    $^{3}$Max-Planck-Institut für Festkörperphysik,70569 Stuttgart, Germany
author:
- '$^{1,2}$M. L. Kulić, $^{3}$O. V. Dolgov'
title: 'The electron-phonon interaction with forward scattering peak is a relevant approach to high $T_{c}$ superconductivity in $FeSe$ films on $SrTiO_{3}$ and $TiO_{2}$ '
---

Introduction
============

The scientific race in reaching high temperature superconductivity ($HTSC$) started by the famous Ginzburg’s proposal of an *excitonic mechanism of pairing* *in metallic-semiconducting sandwich-structures* [@Ginzburg-excitons]. In such a system an electron from the metal tunnels into the semiconducting material and virtually excites high-energy exciton, which is absorbed by another electron, thus making an effective attractive interaction and Cooper pairing. However, this beautiful idea has not been realized experimentally until now. In that sense V. L. Ginzburg founded a theoretical group of outstanding and talented physicists, who studied at that time almost all imaginable pairing mechanisms. In this group an important role has played the Ginzburg’s collaborator E. G. Maksimov, who was an “inveterate enemy” of almost all other mechanisms of pairing in $HTSC$ but for the electron-phonon one - see his arguments in [@Dolg-Kirzh-Maks]. It seems that the recent discovery of superconductivity in a $Fe$-based material made of one monolayer film of the iron-selenide $FeSe$ grown on the $SrTiO_{3}$ substrate - further called $1ML$ $FeSe/SrTiO_{3}$, with the critical temperature $T_{c}\sim(50-100)$ $K$ [@Wang; @QJ-1-Lee; @JJ], as well as grown on the rutile $TiO_{2}$ (100) substrate with $T_{c}\sim65$ $K$ [@Rebec-FeSe-TiO2] - further called $1ML$ $FeSe/TiO_{2}$, in some sense reconciles the credence of these two outstanding physicists. Namely, $HTSC$ is realized in a sandwich-structure but the pairing is due to an high-energy ($\sim90-100$ $meV$) *oxygen optical phonon*. This (experimental) discovery will certainly revive discussions on the role of the electron-phonon interaction ($EPI$) in $HTSC$ cuprates and in bulk materials of the $Fe$-pnictides (with the basic unit $Fe-As$) and Fe-chalcogenides (with the basic unit $Fe-Se$ or $Te$, $S$). As a digression, we point out that after the discovery of high $T_{c}$ in $Fe$-pnictides a non-phononic pairing mechanism was proposed immediately, which is due to: $(i$) nesting properties of the electron- and hole-Fermi surfaces and ($ii$) an enhanced (due to ($i$)) spin exchange interaction ($SFI$) between electrons and holes [@Hirschfeld-review]. This mechanism is called the *nesting SFI pairing*. However, the discovery of alkaline iron selenides $K_{x}Fe_{2-y}Se_{2}$ with $T_{c}\sim30$ $K$, and intercalated compounds $Li_{x}(C_{2}H_{8}N_{2})Fe_{2-y}Se_{2}$, $Li_{x}(NH_{2})_{y}(NH_{3})_{1-y}Fe_{2}Se_{2}$, which contain *only electron-like Fermi surfaces*, rules out the nesting pairing mechanism as a common pairing mechanism in Fe-based materials. In order to overcome this inadequacy of the $SFI$ nesting mechanism a pure phenomenological “strong coupling” $SFI$ pairing is proposed in the framework of the so called $J_{1}-J_{2}$ Heisenberg-like Hamiltonian, which may describe the $s$-wave superconductivity, too. However, this approach is questionable since the LDA calculations cannot be mapped onto a Heisenberg model and there is a need to introduce further terms in form of biquadratic exchange [@Hirschfeld-review]. It is interesting, that immediately after the discovery of high $T_{c}$ in pnictides the electron-phonon pairing mechanism was rather uncritically discarded. This attitude was exclusively based on the $LDA$ band structure calculations of the electron-phonon coupling constant [@Boeri-Dolgov], which in this approach turns out to be rather small $\lambda<0.2$, thus giving $T_{c}<1$ $K$.

In the past there were only few publications trying to argue that the $EPI$ pairing mechanism is an important (pairing) ingredient in the $Fe$-based superconductors [@Kulic-Dolgov-Drechsler]-[@Kulic-Ginzburg; @Conf-2012]. One of the theoretical arguments for it, may be ilustrated in the case of 2-band superconductivity. In the weak-coupling limit $T_{c}$ is given by $T_{c}=1.2\omega_{c}\exp\{-1/\lambda_{\max}\},$ where $\lambda_{\max}=(\lambda_{11}+\lambda_{22}+\sqrt{(\lambda_{11}-\lambda_{22})^{2}+4\lambda_{12}\lambda_{21}})/2$. In the nesting $SFI$ pairing mechanism one assumes a dominance of the repulsive inter-band pairing ( $\lambda_{12}$,$\lambda_{21}<0$), i.e. $\left\vert \lambda_{12},\lambda_{21}\right\vert
\gg\left\vert \lambda_{11},\lambda_{22}\right\vert $. Since the intra-band pairing depends on $\lambda_{ii}=\lambda_{ii}^{epi}-\mu_{ii}^{\ast}$, where $\lambda_{ii}^{epi}$ is the intra-band $EPI$ coupling constant and $\mu
_{ii}^{\ast}>0$ is an screened intra-band Coulomb repulsion, then in order to maximize $T_{c}$ the intra-band $EPI$ coupling it is wishful that $\lambda_{ii}^{epi}$ at least compensate negative effects of $\mu_{ii}^{\ast}$ (on $T_{c}$), i.e. $\lambda_{ii}^{epi}\geq\mu_{ii}^{\ast}$. Since in a narrow band one expects rather large screened Coulomb repulsion $\mu_{ii}^{\ast}$ ($\sim0.2$) then the intra-band $EPI$ coupling should be also appreciable. Moreover, from the experimental side the Raman measurements in Fe-pnictides [@Keimer-A1g] give strong evidence for a large phonon line-width of some $A_{1g}$ modes (where the $As$ vibration along the c-axis dominates). They are almost $10$ times larger than the $LDA$ band structure calculations predict. In [@Kulic-Haghighirad] a model was proposed where high electronic polarizability of $As$ ($\alpha_{As^{3-}}\sim12$ Å$^{3}$) ions screens the Hubbard repulsion and also give rise to a strong $EPI$ with A$_{1g}$ (mainly $As$) modes. An appreciable $As$ isotope effect in $T_{c0}$ was proposed in [@Kulic-Haghighirad], where the stable $^{75}As$ should be replaced by unstable $^{73}As$ - with the life-time of $80$ days, quite enough for performing relevant experiments. The situation is similar with $Fe-Se$ compounds, where an appreciable $EPI$ is expected, since $^{78}Se$ is also highly polarizable ($\alpha_{Se^{2-}}\sim7.5$ Å$^{3}$) and can be replaced by a long-living $^{73}Se$ isotope - the half-time $120$ days. Unfortunately these experiments were never performed.

We end up this digression by paying attention to some known facts, that the $LDA$ band structure calculations are unreliable in treating most high $T_{c}$ superconductors, since as a rule $LDA$ *underestimates non-local exchange-correlation effects and overestimates charge screening effects* - both effects contribute significantly to the $EPI$ coupling constant. As a result, $LDA$ strongly underestimates the $EPI$ coupling in a number of superconductors, especially in those near a metal-isolator transition. The classical examples for this claim are: (*i*) the $(BaK)BiO_{3\text{ }}$superconductor with $T_{c}>30$ $K$ which is $K$-doped from the parent isolating state $BaKBiO_{3}$. Here, $LDA$ predicts $\lambda<0.3$ and $T_{c}\sim1$ $K$, while the theories with an appropriate non-local exchange-correlation potential [@Yin-Kutepov-Kotliar-PRB] predict $\lambda_{epi}\approx1$ and $T_{c}\sim31$ $K$; (*ii*) The high temperature superconductors, for instance $YBaCu_{3}O_{7}$ with $T_{c}\sim100$ $K$, whose parent compound $YBaCu_{3}O_{6}$ is the Mott-insulator [@Kulic-PhysRep], [@Ma-Ku-Do-Advances].

After this digression we consider the main subject of the paper - *the role of the* $EPI$* with forward scattering peak (*$FSP$*) in pairing mechanism* of the $1ML$ $FeSe/SrTiO_{3}$ (and also $1ML$ $FeSe/TiO_{2}$) superconductor(s) with high critical temperatures $T_{c}\sim(50-100)$ $K$. In that respect, numerous experiments on $1ML$ $FeSe/SrTiO_{3}$ (and also on$1ML$ $FeSe/TiO_{2}$), combined with the fact that the $FeSe$ film on the graphene substrate has rather small $T_{c}\approx8$ $K$ (like in the bulk $FeSe$), give strong evidence that interface effects, due to $SrTiO_{3}$ (and $TiO_{2}$), are most probably responsible for high $T_{c}$. It turns out, that the most important results in $1ML$ $FeSe/SrTiO_{3}$ (and also $1ML$ $FeSe/TiO_{2}$), related to the existence of quasi-particle *replica bands* - which are identical to the main quasiparticle band [@Lee-Interfacial-supplement], [@Rebec-FeSe-TiO2], [@Replica-bands], can be coherently described by the $EPI-FSP$* *theory. This approach was proposed in seminal papers [@Lee-Interfacial-supplement]-[@Johnston1-2]. The beauty of these papers lies in the fact that they have recognized sharp replica bands in the $\acute{A}RPES$ spectra and related them to a sharp forward scattering peak in the $EPI$. (This is a very good example for a constructive cooperation of experimentalists and theoreticians.) Let us mention, that the $EPI-FSP$ theory was first studied in a connection with $HTSC$ cuprates [@Kulic-Zeyher1-2], while the extreme case of the $EPI-FSP$ pairing mechanism with delta-peak is elaborated in [@Dan-Dol-Kul-Oud] - see a review in [@Kulic-PhysRep]. Physically, this (in some sense exotic) interaction means that in some specific materials (for instance in cuprates and in $1ML$ $FeSe/SrTiO_{3}$) electron pairs exchange virtual phonons with small (transfer) momenta $q<q_{c}\ll k_{F}$ only, and as a result the effective pairing potential becomes long-ranged in real space [@Kulic-PhysRep]. It turns out that this kind of pairing can in some cases give rise to higher $T_{c}$ than in the standard (Migdal-Eliashberg) $BCS$-like theory. Namely, in the $EPI-FSP$ pairing mechanism one has $T_{c}^{(FSP)}=\left\langle V_{epi}(q)\right\rangle
_{q}/4\sim\lambda_{epi}^{(FSP)}/N(E_{F})$ [@Dan-Dol-Kul-Oud] - see below, instead of the $BCS$ dependence $T_{c}^{(BCS)}\sim\Omega e^{-1/\lambda
_{epi}^{(BCS)}}$. Here, $\lambda_{epi}^{(FSP)}$ and $\lambda_{epi}^{(BCS)}$ are the corresponding *mass-independent* $EPI$ coupling constants, where $\Omega$ - is the phonon energy, $N(E_{F})$ - the electronic density of states (per spin) at the Fermi surface. So, even for small $\lambda
_{epi}^{(FSP)}\ll\lambda_{epi}^{(BCS)}$ the case $T_{c}^{(FSP)}>T_{c}^{(BCS)}$ can be in principle realized. We inform the reader in advance, that the $EPI-FSP$ theory predicts also that $T_{c}^{(FSP)}\sim(q_{c}/k_{F})^{d}V_{epi}(0)$, ($d=1,2,3$ is the dimensionality of the system), which means that when $q_{c}\ll k_{F}$ high $T_{c}^{(FSP)}$ is hardly possible in $3D$ systems. However, the detrimental effect of the phase-volume factor $(q_{c}/k_{F})^{d}$ on $T_{c}^{(FSP)}$ can be compensated by its linear dependence on the pairing potential $V_{epi}(0)$. In some favorable materials this competition may lead even to an increase of $T_{c}$. We stress that properties of the superconductors with the $EPI-FSP$ mechanism of pairing are in many respects very different from the standard (BCS-like) superconductors, and it is completely justified to speak about *exotic superconductors*. For instance, the $EPI-FSP$ theory [@Kulic-PhysRep]-[@Dan-Dol-Kul-Oud] predicts, that in superconductors with the $EPI-FSP$ pairing *the isotope effect should be small* in leading order, i.e. $\alpha\ll1/2$ [@Kulic-PhysRep]-[@Dan-Dol-Kul-Oud] - see discussion in the following. This result is contrary to the case of the isotropic $EPI$ theory in standard metallic superconductors, where $\alpha$ is maximal, $\alpha=1/2$ (for $\mu^{\ast}=0$). We point out, that the $EPI-FSP$ pairing mechanism in strongly correlated systems is rather strange in comparison with the corresponding one in standard metals with good electronic screening, where the large transfer momenta dominate and the pairing interaction is, therefore, short-range. As a result, an important consequence of the $EPI-FSP$ pairing mechanism in case of $HTSC$-cuprates is that $T_{c}$ in the $d$-wave channel is of the same order as in the $s$-wave one. Since the residual repulsion is larger in the $s$- than in the $d$-channel ($\mu_{d}^{\ast}\ll\mu_{s}^{\ast}$) this result opens a door for $d$-wave pairing in $HTSC$-cuprates, in spite of the fact of the $EPI$ dominance [@Kulic-Zeyher1-2]-[@Kulic-PhysRep].

In the following, we study the superconductivity in $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$ [@Rebec-FeSe-TiO2]) in the framework of a semi-microscopic model of $EPI$ first proposed in seminal papers [@Lee-Interfacial-supplement]-[@Johnston1-2]. Namely, due to oxygen vacancies: $(i)$ an electronic doping of the $FeSe$ monolayer is realized, which gives rise to electronic-like bands centered at the $M$-points in the Brillouin zone, while the top of the hole-bands are at around $60$ $meV$ below the electronic-like Fermi surface; $(ii)$ the formed charge in the interface orders dipoles in the nearby $TiO_{2}$ layer; $(iii)$ the free charges in $SrTiO_{3}$ screen the dipolar field in the bulk, thus leaving the $TiO_{2}$ dipolar layer near the interface as an important source for the EPI. The oxygen ions in the $TiO_{2}$ dipolar layer vibrate with high-energy $\Omega\approx90$ $meV$, thus making a long-range dipolar electric field acting on metallic electrons in the $FeSe$ monolayer. This gives rise to a long-ranged $EPI$ [@Lee-Interfacial-supplement], [@Johnston1-2], which in the momentum space gives a forward scattering peak - the $EPI-FSP$* pairing mechanism*.

In this paper we make some analytical calculations in the framework of the $EPI-FSP$ theory with a very narrow $\delta$-peak, with the width $q_{c}\ll
k_{F}$, wher $k_{F}$ is the Fermi momentum [@Dan-Dol-Kul-Oud]. Here, we enumerate the obtained results, only: ($\mathbf{1}$) in leading order the *critical temperature* is linearly dependent on the pairing potential $V_{epi}(\mathbf{q})$, i.e. $T_{c0}\approx\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/4$. In order to obtain $T_{c0}\sim100$ $K$ we set the range of semi-microscopic parameters ($\varepsilon_{\parallel}^{eff}$, $\varepsilon_{\perp}^{eff}$, $q_{eff}$, $h_{0}$, $n_{d}$ - see below) entering $\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}$. Furthermore, since $\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}$ is independent of the the oxygen ($O$) mass, then $T_{c0}$ is *mass-independent* in leading order with respect to $T_{c0}/\Omega$. This means, that in $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$) one expects very small $O$-isotope effect ($\alpha_{O}\ll1/2$). Note, in [@Johnston1-2] large $\alpha_{O}=1/2$ is found; ($\mathbf{2}$) the *self-energy* $\Sigma(\mathbf{k},\omega)$ at $T=0$ is calculated analytically which gives: ($i$) the positions and spectral weights of the replica and quasiparticle bands at $T=0$ - all this quantities are *mass-dependent*; ($ii$) the slope of the quasiparticle self-energy for $\omega\ll\Omega$ ($\Sigma(\omega)\approx-\lambda_{m}\omega$) is *mass-dependent*, since $\sim\lambda_{m}\sim M_{O}^{1/2}$; ($\mathbf{3}$) in the $EPI-FSP$ model (without other interactions) the critical temperature for $s$-wave and $d$-wave pairing is degenerate, i.e. $T_{c0}^{(s)}=T_{c0}^{(d)}$. The presence of non-magnetic impurities (with the parameter $\Gamma=\pi n_{i}N(E_{F})u^{2}$) lifts this degeneracy. It is shown, that even the $s$-wave pairing (in the $EPI-FSP$ pairing mechanism) is sensitive to non-magnetic impurities, which are *pair-weakening* for it, i.e. $T_{c0}^{(s)}$ is decreased for large $\Gamma$, but never vanishes. It is also shown that for $d$-wave pairing $T_{c0}^{(d)}$ strongly depends on impurities, which are *pair-breaking*. The curiosity is that in the presence of non-magnetic impurities $T_{c0}^{(d)}$ in the $EPI-FSP$ pairing mechanism is more robust than the corresponding one in the $BCS$ model; ($\mathbf{4}$) the long-range $EPI-FSP$ pairing potential in real space makes a short-range potential in the momentum space. The latter gives rise to numerous low-laying excitation energy (above the ground-state) of pairs, thus leading to strong internal pair fluctuations which reduce $T_{c0}$. At $T_{c}<T<T_{c0}$ a pseudogap behavior is expected.

The structure of the paper is following: in *Section II* we calculate the $EPI-FSP$ pairing potential as a function of semi-microscopic parameters in the model of a dipolar layer $TiO_{2}$ with vibrations qf the oxygen ions [@Lee-Interfacial-supplement]-[@Johnston1-2]. In *Section III* the self-energy effects, such as replica bands and their intensities at $T=0$, are studied. The critical temperature $T_{c0}$ is calculated in *Section IV* in terms of the semi-microscopic parameters ($\varepsilon
_{\parallel}^{eff}$, $\varepsilon_{\perp}^{eff}$, $q_{eff}$, $h_{0}$, $n_{d}$). The range of of these parameters, for which one has $T_{c0}\sim100$ $K$, is estimated, too. In *Section V* the effect of nonmagnetic impurities on $T_{c0}$ are studied, while the effects of internal fluctuations of Cooper pairs are briefly discussed in *Section VI*. Summary of results are presented in *Section VII*.

$EPI-FSP$ pairing potential due to dipolar oxygen vibrations in the $TiO_{2}$layer
==================================================================================

It is important to point out that in $1ML$ $FeSe/SrTiO_{3}$ material, with $1$-monolayer of $FeSe$ grown on the $SrTiO_{3}$ substrate - mainly on the $(0,0,1)$ plane, the Fermi surface in the $FeSe$ monolayer is *electron-like *and* *centered at four M-points in the Brillouin zone - see more in [@Rebec-FeSe-TiO2], [@Sadovskii1-2]. The absence of the (nested) hole-bands on the Fermi surface rules out all $SFI$ nesting theories of pairing. Even the pairing between an electron- and incipient hole-band [@Hirschfeld-incipient] is ineffective since: ($1$) in the $FeSe$ monolayer the top of the hole band lies below the Fermi level around $60-80$ $meV$; ($2$) because of ($1$) the SFI coupling constant is (much) smaller than in the nesting case. This brings into play the interface interaction effects. The existence of *sharp replica bands* in the $ARPES$ spectra at energies of the order of optical phonons with $\Omega
\sim90$ $meV$, implies inevitably that the dominant interaction in $1ML$ $FeSe/SrTiO_{3}$ (and also in $1ML$ $FeSe/TiO_{2}$ [@Rebec-FeSe-TiO2]) is due to $EPI$ with strong forward scattering peak [@Lee-Interfacial-supplement]-[@Johnston1-2]. The physical mechanism for $EPI-FSP$ is material dependent and the basic physical quantities such as the width of the $FSP$, phonon frequencies and bare $EPI$ coupling can vary significantly from material to material. For instance, in $HTSC$-cuprates the effective $EPI-FSP$ potential $V_{epi}(q)$ is strongly renormalized by strong correlations, which is a synonym for large repulsion of two electrons on the $Cu$ ions - the doubly occupancy is forbidden. In that case the approximative $q$-dependence of $V_{epi}(q)$ is given by $V_{epi}(q)\approx\lbrack
1+(q/q_{c})^{2}]^{-2}V_{0,epi}(q)$, $q_{c}\sim\delta/a$, where $V_{0,epi}(q)$ is the bare (without strong correlations) coupling constant, $\delta(\ll1)$ is the hole concentration and $a$ is the $Cu-O$ distance [@Kulic-Zeyher1-2]-[@Kulic-PhysRep]. The prefactor is a vertex correction due to strong correlations and it means a new kind of (anti)screening in strongly correlated materials.

![*Left*: The microscopic structure of the $FeSe/SrTiO_{3}$ interface of the $1ML$ $FeSe/SrTiO_{3}$ structure. $h_{0}$ - distance between the $FeSe$ monolayer and $TiO_{2}$ dipolar layer. $\delta h$ is amplitude of the oxygen vibration in the dipolar layer. *Right*: The perpendicular view on the $FeSe-SrTiO_{3}$ interface in the model with one $TiO_{2}$ dipolar layer at the end of the $SrTiO_{3}$ substrate. The anisotropy of the effective dielectric constant $varepsilon$ is shown. The similar schema holds also for the $1ML$ $FeSe/TiO_{2}$ structure[]{data-label="1"}](fig1.eps){width="8cm"}

The interface in $1ML$ $FeSe/SrTiO_{3}$ can be considered as highly anisotropic material with the parallel and perpendicular (to the $FeSe$ plane) dielectric constants $\varepsilon_{\parallel}^{eff}\gg\varepsilon_{\perp
}^{eff}$. It is assumed [@Lee-Interfacial-supplement]-[@Johnston1-2] that the oxygen from the $TiO_{2}$ dipolar layer - placed at height ($-h_{0}$) from the $FeSe$ plane, vibrate and make dipolar moments $\delta p_{z}=q_{eff}\delta h(x,y,-h_{0})$ perpendicular to the $FeSe$ ($x-y$) plane - see $Fig.1$. It gives rise to a dipolar electric potential $\Phi_{dip}(x,y,-h_{0}-\delta h)$ acting on electrons in the $FeSe$ ($x-y$ plane). Here, $q_{eff}$ is an effective charge per dipole and $\delta h$ is the polar (dominantly oxygen) displacement along the $z$-axis [@Lee-Interfacial-supplement]. Due to some confusion in the literature on the form of $\Phi_{dip}$ [@Lee-Interfacial-supplement] we recalculate it here, in order to know its explicite dependence on the semi-microscopic parameters $\varepsilon_{\parallel}^{eff}$, $\varepsilon_{\perp}^{eff}$, $q_{eff}$, $h_{0}$, $n_{d}$. An elementary electrodynamics approach [@Landau-Electrodynamics] gives for the dipolar potential $\Phi
_{dip}(x,y,-h_{0}-\delta h)$ $$\Phi_{dip}(x,y,-h_{0}-\delta h)=\frac{(\varepsilon_{\parallel}^{eff})^{1/2}}{(\varepsilon_{\perp}^{eff})^{3/2}}(n_{d}q_{eff}h_{0})\times$$$${\displaystyle\iint}
\frac{dx^{\prime}dy^{\prime}\delta h(x^{\prime},y^{\prime},-h_{0})}{\left(
\frac{\varepsilon_{\parallel}^{eff}}{\varepsilon_{\perp}^{eff}}h_{0}^{2}+(x-x^{\prime})^{2}+(y-y^{\prime})^{2}\right)  ^{\frac{3}{2}}},
\label{Phi}$$ where $n_{d}$ is the number of the oscillating $Ti-O$ dipoles per unit $FeSe$ surface. The coefficient in front of the integral is different from that in [@Lee-Interfacial-supplement] - where it is $\varepsilon_{\parallel}^{eff}/(\varepsilon_{\perp}^{eff})^{3/2}(q_{eff}h_{0})$ (probably typos?) and with missed dipole density $n_{d}$. This coefficient does not fulfill the condition $\Phi\sim\varepsilon^{-1}$ in the isotropic case, while $Eq.$(\[Phi\]) does. By introducing $g_{epi}(\mathbf{q})=e\Phi(\mathbf{q})$ the electron-phonon interaction Hamiltonian $H_{epi}=\sum_{\mathbf{q}}e\hat{\Phi}(\mathbf{q})\hat{\rho}(\mathbf{q})$ is rewritten in the form $H_{epi}=\sum_{\mathbf{k},\mathbf{q}}g_{epi}(\mathbf{q})(\hat{b}_{\mathbf{q}}+\hat{b}_{-\mathbf{q}}^{\dagger})\hat{c}_{\mathbf{k}+\mathbf{q}}^{\dagger
}\hat{c}_{\mathbf{k}}$, where $\hat{b}_{-\mathbf{q}}^{\dagger},\hat
{c}_{\mathbf{k}+\mathbf{q}}^{\dagger}$ are boson and fermion creation operators, respectively. The Fourier transformed potential $g_{epi}(\mathbf{q})(=(g_{0}/\sqrt{N})e^{-q/q_{c}})$ is given by $$g_{epi}(\mathbf{q})=\frac{2\pi n_{d}eq_{eff}}{\varepsilon_{\perp}^{eff}}\sqrt{\frac{\hbar}{M\Omega N}}e^{-q/q_{c}}, \label{g-epi}$$ $g_{0}=(2\pi n_{d}eq_{eff}/\varepsilon_{\perp}^{eff})(\hbar/M\Omega)^{1/2}$, $e$ is the electronic charge. Here, the screening momentum $q_{c}=(\varepsilon_{\perp}^{eff}/\varepsilon_{\parallel}^{eff})^{1/2}h_{0}^{-1}$ characterizes the range of the $EPI$ potential, i.e. for $q_{c}\ll k_{F}$ ($k_{F}$ is the Fermi momentum) the $EPI$ is sharply peaked at $\mathbf{q}=0$ - the forward scattering peak ($FSP$), and the potential in real space is long-ranged, while for $q_{c}\sim k_{F}$ it is short-ranged, like in the standard $EPI$ theory. Since we are interested in the $T_{c}$ dependence on the effective parameters $\varepsilon_{\parallel}^{eff},\varepsilon_{\perp
}^{eff},q_{eff},h_{0}$, then an explicit dependence of the potential is important. We shall see below, that in order that this approach is applicable to $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$) $\varepsilon_{\parallel
}^{eff},\varepsilon_{\perp}^{eff}$ must be very different from the bulk values of $\varepsilon$ in the bulk $SrTiO_{3}$ - where $\varepsilon\sim500-10^{4}$, or in the rutile $TiO_{2}$ structure where $\varepsilon<260$ [@Rebec-FeSe-TiO2].

Self-energy effects and $ARPES$ replica bands
=============================================

The general self-energy $\Sigma_{epi}(\mathbf{k}_{F},\omega)$ at $T=0$ in the extreme $FSP$ $\delta$-peak limit with the width $q_{c}\ll k_{F}$) is given by (see Appendix)$$\Sigma_{epi}(\mathbf{k}_{F},\omega)\approx-\lambda_{m}\frac{\omega}{1-(\omega/\Omega)^{2}}, \label{Sigma-epi}$$ where $\lambda_{m}=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/2\Omega$ is the *mass-dependent coupling constant*. Here, the average $EPI$ potential is given by $\left\langle V_{epi}(\mathbf{q})\right\rangle
_{q}=Ns_{c}(2\pi)^{-2}\int d^{2}qV_{epi}(\mathbf{q},0)\approx(1/4\pi
)(aq_{c})^{2}V_{epi}^{0}$, where $s_{c}=2a^{2}$ is the surface of the $FeSe$ unit cell and $a$ is the Fe-Fe distance, and the bare pairing $EPI$ potential is $V_{epi}^{0}=2g_{0}^{2}/\Omega$. The coupling constant $\lambda_{m}$ corresponds to $\lambda_{m}$ used in [@Johnston1-2], where the self-energy effects are studied at $T>0$. It is important to point out that $\lambda_{m}$ is (oxygen) *mass-dependent*, contrary to [@Johnston1-2]. Since $\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}$ is *mass-independent* then $\lambda_{m}\sim\Omega^{-1}\sim M^{1/2}$. In the following we discuss the case when $\mathbf{k}=\mathbf{k}_{F}$, i.e. $\xi(\mathbf{k})=0$. For $\omega\ll\Omega$ one has $\Sigma_{epi}(\mathbf{k},\omega)=-\lambda_{m}\omega$ which means that the slope of $\Sigma_{epi}(\mathbf{k},\omega)$ is *mass-dependent*. The latter property can be measured by $ARPES$ and thus the $EPI-FSP$ theory can be tested. Note, that in the $EPI-FSP$ theory the critical temperature $T_{c0}(=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/4)$ - see details below, is *mass-independent*. Both these results are *opposite to the standard Migdal-Eliashberg theory*, where the self-energy slope is mass-independent and $T_{c0}$ is mass-dependent.

The *quasiparticle and replica bands* at $T=0$ are obtained from $\omega-\Sigma_{epi}(\omega)=0$. In the following we make calculations at $T=0$ and at the Fermi surface $\xi(\mathbf{k}_{F})=0$. The solutions are: ($1$) $\omega_{1}=0$ - the *quasiparticle band*; ($2$) $\omega
_{2}=-\Omega\sqrt{1+\lambda_{m}}$ is the $ARPES$ *replica band*; ($3$) the inverse $ARPES$ replica band $\omega_{3}=\Omega\sqrt{1+\lambda_{m}}$. The single-particle spectral function is $A(\mathbf{k}_{F},\omega,T=0)=\sum
_{i=1}^{3}(A_{i}/\pi)\delta(\omega-\omega_{i})$, where $A_{i}/\pi$ are the spectral weights. For the quasiparticle band $\omega_{1}$ one obtains $A_{1}=(1+\lambda_{m})^{-1}$, while for the replica bands at $\omega_{2}$ and $\omega_{3}$ one has $A_{2}=A_{3}=(\lambda_{m}/2)(1+\lambda_{m})^{-1}$. The ratio of the intensities at $T=0$ of the $\omega_{2}$ replica band and quasiparticle band $\omega_{1}$ is given by$$\frac{A_{2}(\mathbf{k}_{F},\omega,T=0)}{A_{1}(\mathbf{k}_{F},\omega
,T=0)}=\frac{\lambda_{m}}{2}. \label{ARPES-inten}$$ It is necessary to mention that at finite $T(>0)$ this ratio is changed as found in [@Johnston1-2]. In that case $\Sigma_{epi}^{(T)}(\mathbf{k\approx
k}_{F},\omega,T\neq0)\approx\lambda_{m}/(\omega+\Omega)$ which gives the quasiparticle and replica band $\omega_{1}^{(T)}=\Omega(-1+\sqrt
{1+4\lambda_{m}})/2$ and $\omega_{2}^{(T)}=-\Omega(1+\sqrt{1+4\lambda_{m}})/2$ and $(A_{2}/A_{1})_{T}=\lambda_{m}$ [@Steve-Kulic]. This intriguing difference of the $T=0$ and $T\neq0$ results for $(A_{2}/A_{1})$ in the $EPI-FSP$ theory, $(A_{2}/A_{1})_{T}=2(A_{2}/A_{1})_{0}$, is due to the sharpness of the Fermi function $n_{F}(\xi_{\mathbf{k+q}})$ entering in $\Sigma_{epi}(\mathbf{k},\omega)\ $- see $Eq.$(\[Sigma-integ\]) in Appendix [@Steve-Kulic].

We stress that, the $ARPES$ measurements of $A_{2}/A_{1}$ in $1ML$ $FeSe/SrTiO_{3}$ were done at finite temperatures ($T\neq0$) and in the $k=0$ point with $\xi(k=0)\sim-50$ $meV$ which gives $(A_{2}/A_{1})_{T}\approx0.15-0.2$ [@Lee-Interfacial-supplement], [@Replica-bands]. According to the theory in [@Lee-Interfacial-supplement], [@Johnston1-2] one obtains $\lambda_{m}^{(ARPES)}\approx0.15-0.2$. Below we show, that $\lambda_{m}$ can be also extracted from the formula $Eq.$(\[Tco\] ) for $T_{c0}\approx100$ $K$, which gives $\lambda
_{m}^{(T_{c0})}\approx0.18$. The latter value is in a good agreement with $\lambda_{m}^{(ARPES)}$ from $ARPES$ [@Steve-Kulic]. If we put this value in  $Eq.$(\[ARPES-inten\]) one obtains that at $T=0$ and at $k=k_{F}$ one has $(A_{2}/A_{1})\sim0.1$. From this analysis we conclude that the ARPES measurements at $k_{F}$ should give the similar ratio as at $k=0$. The calculated $ARPES$ spectra at $T=0$ $K$ and at $k=k_{F}$ give $\Delta
\omega=\left\vert \omega_{2}-\omega_{1}\right\vert =\Omega\sqrt{1+\lambda_{m}}$ while the experimental value is $\Delta\omega\approx100$ $meV$, which for $\lambda_{m}^{(ARPES)}\approx0.2$ gives the optical phonon energy of the order of $\Omega\approx90$ $meV$.

The superconducting critical temperature $T_{c0}$ and gap $\Delta
_{0}$
=================================================================

In the weak coupling limit ($\lambda_{m}\ll1$) of the Eliashberg equations with $q_{c}v_{F}<\pi T_{c0}$ ($v_{F}$ is the Fermi velocity) the linearized gap equation (near $T_{c0}$) is given by $$\Delta(\mathbf{k},\omega_{n})=T_{c0}\sum_{\mathbf{q}}\sum_{m}\frac
{V_{epi}(q,\omega_{n}-\omega_{m})\Delta(\mathbf{k}+\mathbf{q},\omega_{m})}{\omega_{m}^{2}+\xi^{2}(\mathbf{k}+\mathbf{q})}, \label{Delta}$$ where $\omega_{n}=\pi T_{c0}(2n+1)$, $V_{epi}(\mathbf{q},\Omega_{n})=g_{epi}^{2}(\mathbf{q})(2\Omega/(\Omega_{n}^{2}+\Omega^{2}))$, $\Omega
_{n}=2\pi T_{c0}\cdot n$. For $\Omega\gg\pi T_{c0}$ one has $V_{epi}(\mathbf{q},\Omega_{n})\approx V_{epi}(\mathbf{q},0)=2g_{epi}^{2}(\mathbf{q})/\hbar\Omega$. In the strong $FSP$ limit when $(q_{c}v_{F})\ll(\pi
T_{c0})^{2}$ the highest value of $\Delta(\mathbf{k},\omega_{n})$ is reached at $\mathbf{k}=\mathbf{k}_{F}$ ($\xi(\mathbf{k}_{F})=0$ in $Eq.$(\[Delta\]). The solution $\Delta(\mathbf{k},\omega_{n})$ is searched in the standard BCS-like *square-well* approximation $\Delta(\mathbf{k}_{F},\omega
_{n})\approx\Delta_{0}=const$. In leading order with respect to $(T_{c0}/\Omega)\ll1$ one obtains $T_{c0}$ [@Kulic-PhysRep]-[@Dan-Dol-Kul-Oud] $$T_{c0}\approx\frac{1}{\pi^{2}}\left\langle V_{epi}(\mathbf{q})\right\rangle
_{q}\sum_{-\Omega/\pi T_{c0}}^{\Omega/\pi T_{c0}}\frac{1}{(2m+1)^{2}}
\label{Tco}$$ For $\Omega\gg\pi T_{c0}$ this gives $T_{c0}=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/4\approx(1/16\pi)(aq_{c})^{2}(2g_{0}^{2}/\Omega)$, where $a$ is the $Fe-Fe$ distance. Note, that $T_{c0}$ is *mass-independent (*$\alpha_{O}=0$) - note $\alpha_{O}=1/2$ is found in [@Johnston1-2]. The small isotope-effect can be a smoking-gun experiment for the $EPI-FSP$ pairing mechanism in $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$). From $Eq.$(\[Delta-n-FSP\]) in the Appendix it is straightforward to obtain the *energy gap* $\Delta_{0}=2T_{c0}$. Note, that $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$) is a $2D$ system and $T_{c0}\sim q_{c}{}^{2}$, while in the $d$-dimensional space one has $T_{c0}\sim q_{c}{}^{d}$. This means that the $EPI-FSP$ mechanism of superconductivity is more *favorable in low-dimensional systems* ($d=1,2$) than in the $3D$ one. Since high $T_{c}$ cuprates are also quasi-$2D$ systems, where strong correlations make a long-ranged $EPI$, it means that the $EPI-FSP$ mechanism of pairing may be also operative in cuprates [@Kulic-PhysRep]. Note, that in estimating some semi-microscopic parameters we shall use as a reper-value $T_{c0}\approx100$ $K$, while in real systems $T_{c}\sim(60-80)$ $K<T_{c0}$ is realized. However, $T_{c0}$ is the mean-field value obtained in the Migdal-Eliashberg theory, while in $2D$ systems it is significantly reduced by the phase fluctuations - to the Berezinski-Kosterliz-Thouless value. There is an additional reduction of $T_{c0}$ (which might be also appreciable) in the $EPI-FSP$ systems, which is due to internal pair-fluctuations - see discussion below.

One can estimate the coupling constant $\lambda_{m}=\left\langle
V_{epi}(\mathbf{q})\right\rangle _{q}/2\Omega$ in $1ML$ $FeSe/SrTiO_{3}$ from the value of $T_{c0}$. Then for the reper-value $T_{c0}\sim100$ $K$ one has $\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}\approx33$ $meV$ and $\lambda_{m}^{(T_{c0})}\approx$ $0.18$. Since, $\lambda_{m}^{(ARPES)}\approx\lambda_{m}^{(T_{c0})}$ the consistency of the theory is satisfactory. Note, if one includes the wave-function renormalization effects (contained in $Z(i\omega_{n})>1$) then in the case $(T_{c0}/\Omega)\ll1$ and for the square-well solution $T_{c0}$ is lowered to $T_{c0}^{(Z)}=T_{c0}/Z^{2}(0)$, where $Z(0)\approx1+\lambda_{m}$ [@Dan-Dol-Kul-Oud]. This means, that the nonlinear corrections (with respect to $\lambda_{m})$ in $T_{c0}$ and $\Delta_{0}$ [@Johnston1-2], [@Murta] should be inevitably renormalized by the $Z$-renormalization.

Let us estimate the parameters ($\varepsilon_{\parallel},\varepsilon_{\perp
},q_{eff},h_{0}$) which enter in $T_{c0}$. In order to reach $T_{c0}\sim100$ $K$ (and $\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}=4T_{c0}\sim400$ $K\approx33$ $meV$) then for $aq_{c}\approx0.2$ and $\Omega\approx90$ $meV$ one obtains $g_{0}\approx0.7$ $eV$. Having in mind that $g_{0}=(2\pi
n_{d}eq_{eff}/\varepsilon_{\perp}^{eff})(\hbar/M\Omega)^{1/2}$ and that the zero-motion oxygen amplitude is $(\hbar/M\Omega)^{1/2}\approx0.05$ Å and by assuming that $n_{d}\approx\alpha/s_{c}$, $s_{c}=\tilde{a}^{2}$, $\tilde
{a}=\sqrt{2}a\approx4$ Å, $q_{eff}\sim2e$, $\alpha\gtrsim1$, then in order to obtain $g_{0}\approx0.7$ $eV$ $\varepsilon_{\perp}^{eff}$ must be small, i.e. $\varepsilon_{\perp}^{eff}\sim1$. Since $aq_{c}=(a/h_{0})\sqrt
{\varepsilon_{\perp}^{eff}/\varepsilon_{\parallel}^{eff}}\sim0.2$ and for $(a/h_{0})\sim1$ it follows $\varepsilon_{\parallel}^{eff}\sim30$. Note, that in $SrTiO_{3\text{ }}$the bulk $\varepsilon$ is large, $\varepsilon
\sim500-10^{4}$. So, if $T_{c0}$ in $1ML$ $FeSe/SrTiO_{3}$ is due solely to the $EPI-FSP$ mechanism, then in the model where the oxygen vibrations in the single dipolar monolayer $TiO_{2\text{ }}$are responsible for the pairing potential the effective dielectric constants $\varepsilon_{\perp}^{eff}$, $\varepsilon_{\parallel}^{eff}$ are very *different from the bulk values* in $SrTiO_{3}$ (in $1ML$ $FeSe/TiO_{2}$ one has $\varepsilon\leq260$ [@Rebec-FeSe-TiO2]). This is physically plausible since for the nearest (to the $FeSe$ monolayer) $TiO_{2}$ dipolar monolayer there is almost nothing to screen in the direction perpendicular to $FeSe$, thus making $\varepsilon
_{\perp}^{eff}\ll$ $\varepsilon^{bulk}$. Note, that for the parameters assumed in this analysis and for $T_{c0}\approx100$ $K$ one obtains rather large bare pairing potential $V_{epi}^{0}\approx10$ $eV$ . This means that in the absence of the $FSP$ in $EPI$ and for the density of states of the order $N(E_{F})\sim0.5$ $(eV)^{-1}$ (typical for Fe-based superconductors) the bare coupling constant $\lambda_{epi}^{0}=N(E_{F})V_{epi}^{0}$ would be large, $\lambda_{epi}^{0}\sim5$. We stress that the above theory is also applicable to recently discovered $1ML$ $FeSe/TiO_{2}$ [@Rebec-FeSe-TiO2]. To conclude, the high $T_{c0}$ in $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$) is obtained on the expense of the large maximal $EPI$ coupling $V_{epi}^{0}$ which compensates smallness of the (detrimental) phase-volume factor $(aq_{c})^{2}$.

Effects of impurities on $T_{c0}$
=================================

In clean systems with the $EPI-FSP$ mechanism of superconductivity $T_{c0}$ is degenerate - it is equal in $s$- and $d$-channels. In the following we show, that the $s$-wave superconductivity is also affected by isotropic non-magnetic impurities, i.e. $T_{c0}$ is reduced and the Anderson theorem is violated. This may have serious repercussions on the *s-wave superconductivity* in $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$) where $T_{c0}$ may depend on chemistry. Then by using equation $Eq.$(\[Tc-s-wave\]) from Appendix one obtains $T_{c}^{(s)}$$$\frac{T_{c}^{(s)}}{T_{c0}}=\frac{4}{\pi^{2}\rho}\psi(\frac{1}{2}+\frac{\rho
}{2})-\psi(\frac{1}{2}),$$ where $\rho=\Gamma/\pi T_{c}$, $\Gamma=\pi n_{i}N(E_{F})u^{2}$, $n_{i}$ is the impurity concentration and $u$ is the impurity potential. Let us consider some limiting cases: ($1$) for $\Gamma\ll\pi T_{c}^{(s)}$one has $T_{c}^{(s)}\approx T_{c0}[1-7\zeta(3)\Gamma/\pi^{3}T_{c0}]$; ($2$) for $\Gamma
\gg\pi T_{c}$ one has $T_{c}^{(s)}\approx(\Gamma/2\pi)\exp(-\pi\Gamma
/4T_{c0})$, i.e. $T_{c}^{(s)}$ never vanishes. This means that in the $EPI-FSP$ systems the non-magnetic impurity scattering is *pair-weakening* for the s-wave superconductivity.

In the case of *d-wave superconductivity* the solution of $Eq.$(\[Tc-d-wave\]) in limiting cases is:  ($1$) $T_{c}^{(d)}\approx
T_{c0}[1-2\Gamma/\pi T_{c0}]$ for $\Gamma\ll\pi T_{c}^{(d)}$. We point out that the slope $-dT_{c}^{(d)}/d(\Gamma)=2/\pi$ is smaller than the slope for the standard $d$-wave pairing, where $-dT_{c}/d(\Gamma)=\pi/4$. ($2$)  For $\Gamma>\Gamma_{cr}^{(FSP)}\approx(4/\pi)T_{c0}$ one has $T_{c}^{(d)}=0$, i.e. the effect of non-magnetic impurities is *pair-breaking*. Note, that $\Gamma_{cr}^{(FSP)}>\Gamma_{cr}(=(2/\pi)T_{c0})$. These two results mean that in the presence of non-magnetic impurities the $d$-wave superconductivity which is due to the $EPI-FSP$ pairing is more robust than in the case of the standard $d$-wave pairing. We stress, that the $T_{c}$ dependence on non-magnetic impurities in $1ML$ $FeSe/SrTiO_{3}$ (as well as in$1ML$ $FeSe/TiO_{2}$) might be an important test for the $EPI-FSP$ pairing in this material.

Finally, it is worth of mentioning, that the real isotope effect in $T_{c0}$ of $1ML$ $FeSe/SrTiO_{3}$ (and in $1ML$ $FeSe/TiO_{2}$) might depend on the type of non-magnetic impurities. If their potential is also long-ranged (for instance due to oxygen deffects in the $TiO_{2}$ dipole layer), then there is $FSP$ in the scattering potential, i.e. $u_{imp}^{2}(q)\approx u^{2}\delta(\mathbf{q})$. Then, such impurities affect in the same way $s$- and $d$-wave pairing and they are pair weakening, as shown in [@Dan-Dol-Kul-Oud]. Naimly, one has $(a)$ $T_{c}^{(s,d)}\approx
T_{c0}[1-4\Gamma_{F}/49T_{c0}]$ for $\Gamma_{F}\ll\pi T_{c}$, where $\Gamma_{F}=\sqrt{n_{i}}u$; (b) $T_{c}^{(s,d)}\approx0.88$ $\Omega\exp
(-\pi\Gamma_{F}/4T_{c0})$, for $\Gamma_{F}\gg\pi T_{c}$. There are two important results: $(1)$ There is a nonanalicity in $\Gamma_{F}\sim\sqrt
{n_{i}}$; $(2)$ there is a full isotope effect in the “dirty” limit $\Gamma_{F}\gg\pi T_{c}$, i.e. $\alpha_{O}=1/2$, since $T_{c}^{(s,d)}\sim\Omega\sim M^{-1/2}$. We stress, that if the full isotope effect would be realized experimentally in $1ML$ $FeSe/SrTiO_{3}$ (and in $1ML$ $FeSe/TiO_{2}$), then this does not automatically exclude the $FSP-EPI$ mechanism of pairing, since it may be due to impurity effects. In that case the nonanalicity of $\Gamma_{F}$ in $n_{i}$ might be a smoking-gun effect.

Internal pair fluctuations reduce $T_{c0}$
==========================================

The $EPI-FSP$ theory, which predicts a long-range force between paired electrons, opens a possibility for a pseudogap behavior in $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$). As we have discussed above, the $EPI-FSP$ theory predicts a non-$BCS$ dependence of the critical temperature $T_{c0}$, i.e. $T_{c0}=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/4$. However, this mean-field ($MFA$) value is inevitably reduced by the *phase* and *internal Cooper pair fluctuations* - which are present in systems with long-range attractive forces. Namely, in the $MFA$ the order parameter $\Delta(\mathbf{x},\mathbf{\mathbf{x}}^{\prime})(=V(\mathbf{x-\mathbf{x}}^{\prime})\langle\mathbf{\psi}_{\downarrow
}(\mathbf{x}^{\prime})\mathbf{\psi}_{\uparrow}(\mathbf{x})\rangle)$ depends on the relative (internal) coordinate $\mathbf{r}=\mathbf{x}-\mathbf{\mathbf{x}}^{\prime}$ and the center of mass $\mathbf{R}=(\mathbf{x}+\mathbf{\mathbf{x}}^{\prime}\mathbf{\mathbf{)/2}}$, i.e. $\Delta(\mathbf{x},\mathbf{\mathbf{x}}^{\prime})=\Delta(\mathbf{r},\mathbf{\mathbf{R}})$. In usual superconductors with short-range pairing potential one has $V_{sr}(\mathbf{x-\mathbf{x}}^{\prime})\approx V_{0}\delta(\mathbf{x-\mathbf{x}}^{\prime})$ and $\Delta(\mathbf{r},\mathbf{\mathbf{R}})=\Delta(\mathbf{\mathbf{R}})$. Therefore only the spatial ($\mathbf{\mathbf{R}}$-dependent) fluctuations of the order parameter are important. In case of a long-range pairing potential there are additional pair-fluctuations due to the dependence of $\Delta
(\mathbf{r},\mathbf{\mathbf{R}})$ on internal degrees of freedom (on $\mathbf{r}$). The interesting problem of fluctuations in systems with long-range attractive forces in $3D$ systems was studied in [@Yang] and we sketch it briefly, because it shows that standard and $EPI-FSP$ superconductors belong to different universality classes. The best way to see importance of the internal pair-fluctuations is to rewrite the pairing Hamiltonian in terms of pseudospin operators (in this approximation first done by P. Anderson the single particle excitations are not included) $$\hat{H}=\sum_{\mathbf{k}\sigma}2\xi_{\mathbf{k}}\hat{S}_{\mathbf{k}\sigma}^{z}-(1/2)\sum_{\mathbf{k},\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}(\hat{S}_{\mathbf{k}}^{+}\hat{S}_{\mathbf{k}^{\prime}}^{-}+\hat
{S}_{\mathbf{k}^{\prime}}^{+}\hat{S}_{\mathbf{k}}^{-}), \label{H-pseudo}$$ where $S_{\mathbf{k}\sigma}^{z}=(\hat{c}_{\mathbf{k}\uparrow}^{\dagger}\hat
{c}_{\mathbf{k}\uparrow}-\hat{c}_{-\mathbf{k}\downarrow}^{\dagger}\hat
{c}_{-\mathbf{k}\downarrow}-1)/2$, $\hat{S}_{\mathbf{k}\sigma}^{+}=\hat
{c}_{\mathbf{k}\uparrow}^{\dagger}\hat{c}_{-\mathbf{k}\downarrow}^{\dagger}$ [@Yang]. This is a Heisenberg-like Hamiltonian in the momentum space. In case of the $s$-wave superconductivity with short-range forces $V_{sr}(\mathbf{x-\mathbf{x}}^{\prime})\approx V_{0}\delta
(\mathbf{x-\mathbf{x}}^{\prime})$ one has $V_{\mathbf{k}-\mathbf{k}^{\prime}}=const$ and the pairing potential is *long-ranged in the momentum space*. In that case it is justified to use *the mean-field approximation* $\hat{H}\rightarrow\hat{H}_{mf}=-\sum_{k}\mathbf{h}_{\mathbf{k}}\mathbf{\hat{S}}_{\mathbf{k}}$ with the mean-field $\mathbf{h}_{\mathbf{k}}=-2\xi_{\mathbf{k}}\mathbf{z+}\sum_{\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}\langle S_{\mathbf{k}^{\prime}}^{x}\mathbf{x+}S_{\mathbf{k}^{\prime}}^{y}\mathbf{y\rangle}$ . The excitation spectrum (with respect to the ground state) in this system have a gap, i.e. $E(\mathbf{k})=2\sqrt{\xi_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}}$ where the gap $\Delta_{\mathbf{k}}$ is the mean-field order parameter defined by $\Delta_{\mathbf{k}}=\sum_{\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}\langle\hat{S}_{\mathbf{k}^{\prime}}^{x}\rangle$. In case of the $EPI-FSP$ pairing mechanism the pairing potential is long-ranged in real space and short-ranged in the momentum space. For instance, in $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$) one has $V_{\mathbf{k}-\mathbf{k}^{\prime}}=V_{0}\exp\{-\left\vert \mathbf{k}-\mathbf{k}^{\prime}\right\vert /q_{c}\}$ with $q_{c}\ll k_{F}$, and the excitation spectrum is boson-like $0<E(\mathbf{k})<2\sqrt{\xi_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}}$ (like in the Heisenberg model) with large number of low-laying excitations (around the ground state). This means, that there are many low-laying pairing states above the ground-state in which pairs are sitting. This, so called internal fluctuations effect, reduces $T_{c0}$ to $T_{c}$. For instance, tin $3D$ systems with $q_{c}\xi_{0}\ll1$ [@Yang] one has $T_{c}\approx(q_{c}\xi
_{0})T_{c0}$, where the coherence length $\xi_{0}=v_{F}/\pi\Delta_{0}$ and $\Delta_{0}=2T_{c0}$. It is expected, that in the region $T_{c}<T<T_{c0}$ the pseudogap ($PG$) phase is realized. However, in $2D$ systems, like $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$), there are additionally phase fluctuations reducing $T_{c}$ further to the Berezinskii-Kosterliz-Thouless value. We stress, that recent measurements of $T_{c}$ in $1ML$ $FeSe/SrTiO_{3}$ by the Meissner effect and resistivity ($\rho(T)$) give that $T_{c}^{(\rho
)}<T_{c}^{(M)}$ what may be partly due to these internal fluctuations of Cooper pairs. It would be interesting to study theoretically these two kind of fluctuations in $2D$ systems, such as $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$.

 Summary and Discussion
=======================

In the paper we study the superconductivity in the $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$ sandwitch-structure, which contains one metallic $FeSe$ monolayer grown on the substrate $SrTiO_{3}$, or rutile $TiO_{2}^{(100)}$. It turns out that in such a structure the *Fermi surface is electron-like* and the bands are pockets around the $M$-point in the Brillouin zone. The bottom of the electron-like bands is around $(50-60)$ $meV$ below the Fermi surface at $E_{F}$. The top of the hole-like band at the point $\Gamma$ lies $80$ $meV$ below $E_{F}$ which means that pairing mechanisms based on the electron-hole nesting are ruled out. This holds also for the pairing with hole-incipient bands (very interesting proposal) [@Hirschfeld-review]. The superconductivity in $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$ is realized in the $FeSe$ monolayer with $T_{c}\sim(60-100)$ $K$. The decisive fact for making a theory is that the $ARPES$ spectra show s*harp replica bands* around $100$ $meV$ below the quasiparticle band, what is approximately the energy of the oxygen optical phonon $\Omega\approx90$ $meV$. The analysis of superconductivity is based on the semi-microscopic model - first proposed in [@Lee-Interfacial-supplement], [@Johnston1-2], where it is assumed that a $TiO_{2}$ dipolar layer is formed just near the interface. In that model the oxygen vibrations create a dipolar electric potential, which acts on electrons in the $FeSe$ monolayer, thus making the $EPI$ interaction long-ranged. In the momentum space a forward scattering peak ($FSP$) appears, i.e. $EPI$ is peaked at small transfer momenta ($q<q_{c}\ll
k_{F}$) with $g_{epi}(\mathbf{q})=g_{0}\exp\{-q/q_{c}\}$. Here, this is called the $EPI-FSP$ pairing mechanism. The $EPI-FSP$ theory is formulated first in [@Kulic-Zeyher1-2] for strongly correlated systems, while its extreme case with delta-peak is elaborated in [@Dan-Dol-Kul-Oud] - see also [@Kulic-PhysRep]. This limiting (delta-peak) case makes not only analytical calculations easier, but it makes also a good fit to experimental results [@Lee-Interfacial-supplement], [@Johnston1-2]. In the following, we summarize the main obtained results of the $EPI-FSP$ theory and its relation to the $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$ sandwitch-structures.

$(1)$ - The mean-field critical temperature $T_{c0}$ in the $s$-wave and $d$-wave pairing channels is* degenerate* and given by $T_{c0}=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/4\approx(1/16\pi
)(aq_{c})^{2}V_{epi}^{0}$, where $V_{epi}(\mathbf{q})=2g_{epi}^{2}(\mathbf{q})/\hbar\Omega$ and the maximal pairing potential $V_{epi}^{0}(\equiv V_{epi}(q=0))=(2g_{0}^{2}/\Omega)$. On the first glance this linear dependence of $T_{c0}$ on $V_{epi}^{(0)}$ seems to be favorable for reaching high $T_{c0}$ - note in the $BCS$ theory $T_{c0}$ is exponentially dependent on $V_{epi}^{(0)}$ and very small for small $N(E_{F})V_{epi}^{0}$. However, for non-singular $g_{epi}(\mathbf{q})$ when $g_{epi}(\mathbf{q}=0)$ is finite, $T_{c0}$ is limited by the smallness of the phase-volume effect, which is in $2D$ systems (such as $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$) proportional to $(aq_{c})^{2}\ll1$. In that sense optimistic claims that the $EPI-FSP$ mechanism leads inevitably to higher $T_{c0}$ - than the one in the standard Migdal-Eliashberg theory, are not well founded. This holds especially for $3D$ systems, where $T_{c0}\sim$ $(aq_{c})^{3}$ and $T_{c0}^{(3d)}\ll T_{c0}^{(2d)}$ for the same value of $V_{epi}^{0}$. However, higher $T_{c0}$ (with respect to to the $BCS$ case) can be reached by fine tuning of $aq_{c}$ and $V_{epi}^{0}$. This is probably realized in $HTSC$ cuprates and with certainty in $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$. The weak-coupling theory predicts the superconducting gap to be $\Delta
_{0}=2T_{c0}$ and for $T_{c0}\sim100$ $K$ one has $\Delta_{0}\sim16$ $meV$ what fits well the $ARPES$ experimental values [@Lee-Interfacial-supplement], [@Rebec-FeSe-TiO2]. Note, in order to reach $T_{c0}=100$ $K$ for $aq_{c}\approx0.2$ a very large maximal $EPI$ coupling $V_{epi}^{0}\approx10$ $eV$  is necessary. For $N(E_{F})\sim0.5$ $(eV)^{-1}$ the maximal coupling constant would be rather large, i.e. $\lambda_{epi}^{0}(=N(E_{F})V_{epi}^{0})\approx5$. Note, that $V_{epi}^{0}$ is almost as large as in the metallic hydrogen under high pressure $p\sim20$ $Mbar$, where $T_{c0}\approx600$ $K$ with large $EPI$ coupling constant $\lambda_{epi}^{0}\approx7$ - this important prediction is given in [@Maksimov-Savrasov]. In real $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$ materials the contribution of another pairing mechanism, which exists in the $FeSe$ film in absence of the substrate and is pronounced in the *s-wave channel* with $T_{c0}\approx8$ $K$, triggers the whole pairing to be $s$-wave. The latter only moderately decreases the contribution of the $EPI-FSP$ pairing mechanism. The existence of sharp replica bands in $1ML$ $FeSe/SrTiO_{3}$ and $1UCFeSe/TiO_{2}$ and large value of $V_{epi}^{0}$ imply inevitably that the $EPI-FSP$ pairing mechanism is the main candidate to explain superconductivity in these materials. We stress, that in $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$ high $T_{c0}$ is obtained on the expense of the large maximal $EPI$ coupling $V_{epi}^{0}$, which compensates the small (detrimental) phase volume factor $(aq_{c})^{2}$.

$(2)$ - The semi-microscopic model proposed in [@Lee-Interfacial-supplement], [@Johnston1-2], and refined slightly in this paper, contains phenomenological parameters, such as $n_{d}$ - the number of dipoles per unit cell, $q_{eff}$ - the effective dipole charge, $\varepsilon_{\parallel}^{eff}$, $\varepsilon_{\perp}^{eff}$ - effective parallel and perpendicular dielectric constanty in $SrTiO_{3}$ (and $TiO_{2}$) near the interface, respectively. For $T_{c0}\approx100$ $K$ and by assuming $aq_{c}\approx0.2,$ $q_{eff}\approx2e$, $n_{d}\sim2$ $/unit-cell$ makes $\varepsilon_{\parallel}^{eff}\sim30$, $\varepsilon_{\perp}^{eff}\sim1$. These values, which are physically plausible, are very far from $\varepsilon_{bulk}$ in the bulk $SrTiO_{3}$, where $\varepsilon_{bulk}\sim500-10^{4}$ (and $\varepsilon_{bulk}\leq260$ in the rutile $TiO_{2}$). We point out that our estimation of these parameters is based on the effective microscopic model where the bulk $SrTiO_{3}$ is truncated by a monolayer ($1ML$) made of $TiO_{2}$ [@Lee-Interfacial-supplement]. In reality it may happen that the bulk $SrTiO_{3}$ is truncated by two monolayers ($2ML$) of $TiO_{2}$, as it is claimed to be seen in the synchrotron $x$ray diffraction [@Zou-Bozovic].This finding is confirmed by the $LDA$ calculations in [@Zou-Bozovic], which show that for the $2ML$ $TiO_{2}$ structure: ($\emph{i}$) the electrons are much easier transferred to the $FeSe$ metallic monolayer and ($\emph{ii}$) the top of the hole band is shifted far below the electronic Fermi surface than in the $1ML$ model. If the $2ML$ of $TiO_{2}$ is realized it could be even more favorable for the $EPI-FSP$ pairing, since some parameters can be changed in a favorable way. For instance, the effective charge could be increased, i.e. $q_{eff}^{(2ML)}>q_{eff}^{(1ML)}$and since $T_{c0}\sim
q_{eff}^{2}$ the $2ML$ model may gives rise to higher critical temperature.

$(3)$ - The* isotope effect* in $T_{c0}$* *should be small ($\alpha_{O}\ll1/2$) since in leading order one has $T_{c0}\sim V_{epi}^{0}$, where $V_{epi}^{0}$ is *mass-independent*. This is contrary to [@Johnston1-2] where $\alpha_{O}=1/2$. The next leading order gives $\alpha_{O}\sim(T_{c0}/\Omega)<0.09$. We stress that the small isotope-effect maybe a smoke-gun experiment for the $EPI-FSP$ pairing mechanism.

$(4)$ - In the $EPI$-$FSP$ pairing theory* *the *non-magnetic impurities* affect both $s$-wave and $d$-wave pairing. In the case of $s$-wave* *they are* pair-weakening*, while *for* $d$-wave are* pair-breaking*. However, the non-magnetic impurities with forward scattering peak give in the “dirty” limit ($\Gamma_{F}\gg\pi
T_{c}$) the full isotope effect $\alpha_{O}=1/2$, since $T_{c}^{(s,d)}\sim\Omega\sim M^{-1/2}$. In that case, the nonanalicity of $T_{c}^{(s,d)}$ with respect to the impurity concentration $n_{i}$, would resolve the question - what kind of pairing is realized in $1ML$ $FeSe/SrTiO_{3}$ and $1UCFeSe/TiO_{2}$ - the $EPI-FSP$ or the standard $EPI$.

$(5)$ - In the case of the $EPI$-$FSP$ pairing the superconducting order parameter depends strongly on the internal pair coordinate and of center of mass, i.e. $\Delta=\Delta(\mathbf{r},\mathbf{R})$. The *internal pair fluctuations* reduce additionally the mean-field critical temperature so that in the interval $T_{c}<T<T_{c0}$ a pseudogap behavior is expected.

$(6)$ - The *EPI self-energy* in the normal state at $T=0$ and $\xi(\mathbf{k}_{F})=0$ is given by $\Sigma_{epi}(\mathbf{k},\omega
)\approx-\lambda_{m}\omega/(1-(\omega/\Omega)^{2})$, where $\lambda
_{m}=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/2\Omega$, which for $G^{-1}(\mathbf{k},\omega)=0$ gives the dispersion energy of the quasiparticle band $\omega_{1}=0$ and the replica bands $\omega_{2}$ and $\omega_{3}$. The ratio of the ARPES intensities of the replica band $\omega_{2}$ and the quasiparticle band $\omega_{1}$ at $T=0$ and at the Fermi surface ($k=k_{F}$) is given by $R(T=0,k_{F})=(A_{2}/A_{1})=\lambda_{m}/2$. This means, that for $\lambda_{m}\sim0.2$ the experimental value of $R(T=0,k_{F})$ should be $(A_{2}/A_{1})\approx0.1$. This ratio is slightly smaller than the experimental value $R(T\neq0,k=0)$ measured in [@Lee-Interfacial-supplement], [@Replica-bands].

$(7$) Since the coupling constant $\lambda_{m}$ is *mass-dependent*, $\lambda_{m}\sim M^{1/2}$ then the isotope effect in various quantities, in $1ML$ $FeSe/SrTiO_{3}$ and $1ML$ $FeSe/TiO_{2}$ systems, may be a smoke-gun experiment in favour of the $EPI-FPS$ theory. To remind the reader: $(i)$ $T_{c0}$ is almost *mass-independent*; $(ii)$ the self-energy slope at $\omega\ll\Omega$ is *mass-dependent*, $(-d\Sigma/d\omega)\sim M^{1/2}$; $(iii)$ the $ARPES$ ratio $R$ of the replica band intensities is *mass-dependent*, $R\sim M^{1/2}$.

Concerning the role of $EPI$ in explaining superconductivity in $1ML$ $FeSe/SrTiO_{3}$ there were other interesting theoretical proposals. In [@Gorkov1-2] the $EPI$ is due to the interaction with longitudinal optical phonons and since $\Omega>E_{F}$ the problem is studied in anti-adiabatic limit, where $T_{c}$ is also weakly dependent on the oxygen mass. In [@Cohen] the substrate gives rise to an antiferromagnetic structure in $FeSe$, which opens new channels in the $EPI$ coupling in the $FeSe$ monolayer, thus giving rise for high $T_{c}$. In [@Zi] the intrinsic pairing mechanism is assumed to be due to $J_{2}$-type spin fluctuations, or antiferro orbital fluctuation, or nematic fluctuations. The extrinsic pairing is assumed to be due to interface effects and the $EPI-FSP$ interaction. The problem is studied by the sign-free Monte-Carlo simulations and it is found that $EPI-FSP$ is an important ingredient for high $T_{c}$ superconductivity in this system.

Finally, we would like to comment some possibilities for designing new and complex structures based on $1ML$ $FeSe/SrTiO_{3}$ (or $1ML$ $FeSe/TiO_{2}$) as a basic unit. The first nontrivial one is when a double-sandwich structure with two interfaces is formed, i.e. $SrTiO_{3}/1ML$ $FeSe/SrTiO_{3}$ (or $TiO_{2}/1ML$ $FeSe/TiO_{2}$). Naively thinking in the framework of the $EPI-FSP$ pairing mechanism one expects in an “ideal” case doubling of $T_{c0}$, since phonons at two interfaces are independent. However, this would only happen when the electron-like bands on the Fermi surface due to the two substrates were similar and if the condition $q_{c}v_{F}<\pi T_{c0}$ is kept in order to deal with a sharp $FSP$. However, many complications in the process of growing, such structures may drastically change properties, leading even to a reduction of $T_{c0}$. It needs very delicate technology to control the concentration of oxygen vacancies and appropriate charge transfer at both interfaces. However, eventual solutions of these problems might give impetus for superconductors with exotic properties. For instance, having in mind the above exposed results on effects of non-magnetic impurities on $T_{c0}$, then by controlling and manipulating their presence at both interfaces one can design superconducting materials with wishful properties.

**Acknowledgment** M.L.K. is thankful to Radoš Gajić for useful discussions, comments and advises related to the experimental situation in the field. M.L.K. highly appreciates fruitful discussions with Steve Johnston and Yan Wang on ARPES of the replica bands at finite temperature, and on microscopic parameters of the theory.

Appendix
========

Migdal-Eliashberg equations in superconductors
----------------------------------------------

In the paper we study superconductivity with the $EPI-FSP$ mechanism of pairing by including effects of non-magnetic impurities, too. The full set of Migdal-Eliashberg equations is given for that case. The normal and anomalous Green’s functions are $G_{n}(\mathbf{k},\omega_{n})=$ $-[i\omega_{n}Z_{n}(\mathbf{k})+\bar{\xi}_{n}(\mathbf{k})]/D_{n}(\mathbf{k})$, $F_{n}(\mathbf{k})=-Z_{n}(\mathbf{k})\Delta_{n}(\mathbf{k})/D_{n}(\mathbf{k})$, respectively where $D_{n}(\mathbf{k})=[\omega_{n}Z_{n}(\mathbf{k})]^{2}+\bar{\xi}_{n}^{2}(\mathbf{k})+[Z_{n}(\mathbf{k})\Delta_{n}(\mathbf{k})]^{2}$ ($\omega_{n}=\pi T(2n+1)$). Here, $Z_{n}(\mathbf{k})$ is the wave-function renormalization defined by $i\omega_{n}(1-Z_{n}(\mathbf{k}))=(\Sigma(\mathbf{k},\omega_{n})-\Sigma(\mathbf{k},-\omega
_{n}))/2$, where the self-energy $\Sigma(\mathbf{k},\omega_{n})=\Sigma
_{epi}(\mathbf{k},\omega_{n})+\Sigma_{imp}(\mathbf{k},\omega_{n})$ describes the $EPI$ and impurity scattering, respectively. The energy renormalization is $\bar{\xi}_{n}(\mathbf{k})=\xi(\mathbf{k})+\chi_{n}(\mathbf{k})$, $\chi
_{n}(\mathbf{k})=(\Sigma(\mathbf{k},\omega_{n})+\Sigma(\mathbf{k},-\omega
_{n}))/2$ and $\Delta_{n}(\mathbf{k})$ is the superconducting order parameter.$$Z_{n}(\mathbf{k})=1+\frac{T}{\omega_{n}}\sum_{\mathbf{k}^{\prime},n^{\prime}}\frac{V_{eff}(n-n^{\prime},\mathbf{k}-\mathbf{k}^{\prime})\omega_{n^{\prime}}Z_{n^{\prime}}(\mathbf{k}^{\prime})}{D_{n^{\prime}}(\mathbf{k}^{\prime})}
\label{Z-nk}$$$$\bar{\xi}_{n}(\mathbf{k})=\xi(\mathbf{k})-T\sum_{\mathbf{k}^{\prime},n^{\prime}}\frac{V_{eff}(n-n^{\prime},\mathbf{k}-\mathbf{k}^{\prime})\bar
{\xi}_{n^{\prime}}(\mathbf{k}^{\prime})}{D_{n^{\prime}}(\mathbf{k}^{\prime})}
\label{Ksi-nk}$$$$Z_{n}(\mathbf{k})\Delta_{n}(\mathbf{k})=T\sum_{\mathbf{k}^{\prime},n^{\prime}}\frac{V_{eff}(n-n^{\prime},\mathbf{k}-\mathbf{k}^{\prime})Z_{n^{\prime}}(\mathbf{k}^{\prime})\Delta_{n^{\prime}}(\mathbf{k}^{\prime})}{D_{n^{\prime}}(\mathbf{k}^{\prime})}, \label{Delta-nk}$$ where $V_{eff}(n-n^{\prime},\mathbf{k}-\mathbf{k}^{\prime})=V_{epi}(n-n^{\prime},\mathbf{k}-\mathbf{k}^{\prime})+V_{imp}(n-n^{\prime},\mathbf{k}-\mathbf{k}^{\prime})$, $V_{epi}(n-n^{\prime},\mathbf{k}-\mathbf{k}^{\prime})=-g_{epi}^{2}(\mathbf{k}-\mathbf{k}^{\prime})\mathcal{D}_{ph}(\mathbf{k}-\mathbf{k}^{\prime},\omega_{n}-\omega_{n^{\prime
}})$ and $V_{imp}(n-n^{\prime},\mathbf{k}-\mathbf{k}^{\prime})=\delta
_{nn^{\prime}}n_{imp}u^{2}(\mathbf{k}-\mathbf{k}^{\prime})/T$. Here, the phonon Green’s function in the Einstein model with the single frequency $\Omega$ is given by $\mathcal{D}_{ph}(\mathbf{k}-\mathbf{k}^{\prime},\omega_{n}-\omega_{n^{\prime}})=-2\Omega/(\Omega^{2}+(\omega_{n}-\omega_{n^{\prime}})^{2})$ while the impurity scattering is described in the Born-approximation. Here, $n_{imp}$ is the impurity concentration and $u(\mathbf{k}-\mathbf{k}^{\prime})$ is the impurity potential. To these three equations one should add the equation for the chemical potential $\mu$, i.e. $N=\sum G_{n}(\mathbf{k},\omega_{n};\mu)=const$. However, in the following we study only problems where the (small) change of $\mu$ due to $EPI$ and impurity scattering does not change the physics of the problem. For instance we do not study problems such as $BCS-BEC$ transition, where the equation for $\mu$ plays important role, etc.

Note, that in the case of systems with very *large Fermi energy* $E_{F}$ and with an *isotropic* $EPI$ ($Z_{n}(\mathbf{k})\equiv Z_{n}$, $\bar{\xi}_{n}(\mathbf{k})\rightarrow0$) one integrates over the energy $\xi_{\mathbf{k}^{\prime}}$ by introducing the density of states at the Fermi surface $N(0)$, i.e. $\sum_{\mathbf{k}^{\prime}}(...)\Rightarrow
N(0)\int_{-\infty}^{\infty}(...)d\xi_{\mathbf{k}^{\prime}}$. This leads to standard Migdal-Eliashberg equations.$$Z_{n}=1+\frac{\pi T}{\omega_{n}}\sum_{n^{\prime}}\frac{N(0)V_{eff}(n-n^{\prime})\omega_{n^{\prime}}Z_{n^{\prime}}}{\sqrt{(\omega_{n^{\prime}}Z_{n^{\prime}})^{2}+\Delta_{n^{\prime}}^{2}}} \label{Zn}$$$$Z_{n}\Delta_{n}=\pi T\sum_{n^{\prime}}\frac{N(0)V_{eff}(n-n^{\prime
},Z_{n^{\prime}}\Delta_{n^{\prime}}}{\sqrt{(\omega_{n^{\prime}}Z_{n^{\prime}})^{2}+\Delta_{n^{\prime}}^{2}}} \label{Delta-n}$$ In the case of *strongly momentum-dependent* $EPI-FSP$,$\ $where $V_{epi}(n-n^{\prime},\mathbf{q})$ is finite for $\left\vert \mathbf{q}\right\vert <q_{c}\ll k_{F}$, the Migdal-Eliashberg equations are given by$$Z_{n}(\xi)=1+\frac{T}{\omega_{n}}\sum_{m}\left\langle V_{epi}(n-m,\mathbf{q})\right\rangle _{q}\frac{\omega_{m}Z_{m}(\xi)}{D_{m}(\xi)}, \label{Zn-FSP}$$ $$\bar{\xi}_{n}(\xi)=\xi(\vec{k})-T\sum_{m}\frac{\left\langle V_{epi}(n-m,\mathbf{q})\right\rangle _{q}}{D_{m}(\xi)}\bar{\xi}_{m}(\xi),
\label{Ksi-FSP}$$$$Z_{n}(\xi)\Delta_{n}(\xi)=T\sum_{m}\left\langle V_{epi}(n-m,\mathbf{q})\right\rangle _{q}\frac{Z_{m}(\xi)\Delta_{m}(\xi)}{D_{m}(\xi)}.
\label{Delta-n-FSP}$$

Effects of non-magnetic impurities on $T_{c0}$ in the $EPI-FSP$ theory
----------------------------------------------------------------------

In this paper we study the superconductivity which is due to $EPI-FSP$ of the Einstein phonon with $\Omega^{2}\ll(2\pi T_{c0})^{2}$. In that case $\left\langle V_{epi}(n-m,\mathbf{q})\right\rangle _{q}\approx\left\langle
V_{epi}(0,\mathbf{q})\right\rangle _{q}=\left\langle 2g_{epi}^{2}(\mathbf{q})\right\rangle _{q}/\Omega$ and the contribution to $Z_{n}(\xi)$ is $\sim\lambda_{m}=\left\langle V_{epi}(0,\mathbf{q})\right\rangle _{q}/2\Omega
$. Since in the weak coupling limit one has $\lambda_{m}\ll1$ then we neglect this contribution. Also the non-Migdal corrections can be neglected in this case - see [@Johnston1-2] The effects of non-magnetic impurities on $T_{c0}$ is studied in the standard model with weakly momentum dependent impurity potential $u(\mathbf{k}-\mathbf{k}^{\prime})\approx const$ . In that case $Z_{n}(\xi)$ contains the impurity term only. After the integration of the impurity part over the energy $\xi^{\prime}$ in $Eqs.$(\[Z-nk\]-\[Delta-nk\]) -see [@Allen-Mitrovic], and putting $\xi=0$ (since in that case $\Delta_{n}(\xi=0)$ is maximal) one obtains ($D_{m}(\xi
)\approx\omega_{n}^{2}Z_{n}^{2}$) for the *s-wave pairing* ($\Delta=const$) $$Z_{n}=1+\frac{\Gamma}{\left\vert \omega_{n}\right\vert } \label{Zn-imp}$$$$Z_{n}\Delta_{n}=T_{c}\sum_{m}\left\langle V_{epi}(n-m,\mathbf{q})\right\rangle
_{q}\frac{Z_{m}\Delta_{m}}{D_{m}(0)}+\frac{\Gamma}{\left\vert \omega
_{n}\right\vert }\Delta_{n}. \label{Delta-n-imp}$$ Note, the the second term on the right side cancels the same term on the left side. In the approximation $\Delta_{n}(\xi)\approx\Delta$ one obtains the equation for impurity dependence of $T_{c}^{(s)}(\Gamma)$ for a *s-wave superconductor* $$1=T_{c}^{(s)}\left\langle V_{epi}(0,\mathbf{q})\right\rangle _{q}\sum_{m}\frac{1}{\omega_{n}^{2}Z_{n}}. \label{Tc-s-wave}$$ We point out that in the case of *d-wave superconductivity* $\Delta=\Delta(\varphi)$ is angle dependent on the Fermi surface and changes sign. In that case the last term in $Eq.$(\[Delta-n-imp\]) $\Delta$ should be replaced by $\left\langle \Delta(\varphi\right\rangle )=0$ giving equation for $T_{c}^{(d)}$$$1=T_{c}^{(d)}\left\langle V_{epi}(0,\mathbf{q})\right\rangle _{q}\sum_{m}\frac{1}{\omega_{n}^{2}Z_{n}^{2}}. \label{Tc-d-wave}$$ Note, $Z_{n}$ vs $Z_{n}^{2}$ renormalization for the $s$-wave and $d$-wave superconductivity, respectively.

$EPI-FSP$ self-energy in the normal state
-----------------------------------------

We shall calculate the self-energy at $T=0$. The leading order self-energy (on the Matsubara axis) in the Migdal-Eliashberg theory of $EPI$ is given by $$\Sigma_{epi}(\mathbf{k},\omega_{n})=-T\sum_{\mathbf{q},\Omega_{m}}g_{epi}^{2}(\mathbf{q})\mathcal{D}_{ph}(\mathbf{q},\Omega_{m})G(\mathbf{k}+\mathbf{q},\omega_{n}-\Omega_{m}), \label{Sigma-ME}$$ where $\omega_{n}=\pi T(2n+1)$ and $\Omega_{m}=2\pi mT$, $\mathcal{D}_{ph}(\mathbf{q},\Omega_{m})=-2\Omega/(\Omega^{2}+\Omega_{m}^{2})$, $G(\mathbf{k},\omega_{n})=1/(i\omega_{n}-\xi_{\mathbf{k}})$. By defining $V_{epi}(\mathbf{q},0)=2g_{epi}^{2}(\mathbf{q})/\Omega$  and after summation over $\Omega_{m}$ in $Eq.$(\[Sigma-ME\]) one obtains (note that $V_{epi}(\mathbf{q},0)=V_{epi}(-\mathbf{q},0)$ $$\begin{aligned}
\Sigma_{epi}(\mathbf{k},\omega_{n})  &  =\frac{\Omega}{2}\sum_{\mathbf{q}}V_{epi}(\mathbf{q},0)\label{Sigma-integ}\\
&  \times\left[  \frac{n_{F}(\xi_{\mathbf{k+q}})}{i\omega_{n}-\xi
_{\mathbf{k+q}}+\Omega}+\frac{1-n_{F}(\xi_{\mathbf{k+q}})}{i\omega_{n}-\xi_{\mathbf{k+q}}-\Omega}\right]  .\nonumber\end{aligned}$$ Let us calculate $\Sigma_{epi}$ at $\mathbf{k}_{F}$. Since $\mathbf{qv}_{F}=qv_{F}\cos\theta$ and by taking into account that $n_{F}(\xi
_{\mathbf{k}_{F}\mathbf{+q}})=1$ for $\cos\theta<0$, $n_{F}(\xi_{\mathbf{k}_{F}\mathbf{+q}})=0$ for $\cos\theta>0$ one obtains for $q_{c}v_{F}\ll\Omega$ $$\Sigma_{epi}(\mathbf{k},\omega_{n})=-\lambda_{m}\frac{i\omega_{n}}{1-(\frac{i\omega_{n}}{\Omega})^{2}}, \label{Sigma-final}$$ where $\lambda_{m}=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/2\Omega$ and $\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}=Ns_{c}(2\pi)^{-2}\int
d^{2}qV_{epi}(\mathbf{q},0)$, $s_{c}$ is the surface of the $FeSe$ unit cell. Note, that $V_{epi}(\mathbf{q},0)=2g_{epi}^{2}(\mathbf{q})/\Omega$ and $g_{epi}(\mathbf{q})(=(g_{0}/\sqrt{N})e^{-q/q_{c}})$ so that $N$ disappears from $\lambda_{m}(=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/2\Omega)$.

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