The electron-phonon interaction with forward scattering peak is a relevant approach to high Tc superconductivity in FeSe films on SrTiO3 and TiO2

The theory of EPI with strong forward scattering peak (FSP)[1-3] is recently applied in [4-5] in studying high Tc(~100 K) in a FeSe grown on SrTiO3 [4] and TiO2 [6]. The EPI is due to a long-range dipolar electric field created by the high-energy oxygen vibrations (E0~90 meV) at the interface [4-5]. Besides obtaining new results we also correct some misleading results from the recent literature. We show that the mean-field critical temperature Tc0 is an interplay between the maximal pairing potential and the FSP-width qc. For Tc0~100 K the gap (G) is G~16 meV in agreement with ARPES experiments. We find that in leading order Tc0 is mass-independent and there is small oxygen isotope effect in next to leading order. In clean systems Tc0 for s-wave and d-wave pairing is degenerate but both are affected by non-magnetic impurities. The non-magnetic impurities are pair-weakening in the s-channel and pair-breaking in the d-channel. The normal state self-energy at the Fermi surface gives rise to the ARPES quasiparticle band at E=0 and to a replica band at Er=-E0(1+l(m))^1/2, respectively. The EPI coupling l(m), which enters the self-energy, is mass-dependent - the fact overlooked in the literature, makes at low energies the slope of the self-energy mass-dependent. The smallness of the oxygen isotope effect in Tc0 and its presence in the self-energy in FeSe films on SrTiO3 and TiO2 is a smoking-gun experiment for the application 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 Tc0 additionally, by creating a pseudogap state for Tc


I. INTRODUCTION
The scientific race in reaching high temperature superconductivity (HT SC) started by the famous Ginzburg's proposal of an excitonic mechanism of pairing in metallic-semiconducting sandwich-structures [7]. In such a system an electron from the metal tunnels into the semiconducting material and virtually excites highenergy 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 HT SC but for the electron-phonon one -see his arguments in [8]. It seems that the recent discovery of superconductivity in a F e-based material made of one monolayer film of the iron-selenide F eSe grown on the SrT iO 3 substratefurther called 1M L F eSe/SrT iO 3 , with the critical temperature T c ∼ (50 − 100) K [9], as well as grown on the rutile T iO 2 (100) substrate with T c ∼ 65 K [6] -further called 1M L F eSe/T iO 2 , in some sense reconciles the credence of these two outstanding physicists. Namely, HT SC is realized in a sandwich-structure but the pairing is due to an high-energy (∼ 90 − 100 meV ) oxygen optical phonon. This (experimental) discovery will certainly revive discussions on the role of the electron-phonon interaction (EP I) in HT SC cuprates and in bulk materials of the F e-pnictides (with the basic unit F e − As) and Fe-chalcogenides (with the basic unit F e − Se or T e, S). As a digression, we point out that after the discovery of high T c in F e-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 (SF I) between electrons and holes [10]. This mechanism is called the nesting SFI pairing. However, the discovery of alkaline iron selenides K x F e 2−y Se 2 with T c ∼ 30 K, and intercalated compounds Li x (C 2 H 8 N 2 )F e 2−y Se 2 , Li x (N H 2 ) y (N H 3 ) 1−y F e 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 SF I nesting mechanism a pure phenomenological "strong coupling" SF I pairing is proposed in the framework of the so called J 1 − J 2 Heisenberg-like Hamiltonian, which may describe the swave 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 [10]. 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 [11], which in this approach turns out to be rather small λ < 0.2, thus giving T c < 1 K.
In the past there were only few publications trying to argue that the EP I pairing mechanism is an important (pairing) ingredient in the F e-based superconductors [12]- [14]. 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 = 1.2ω c exp{−1/λ max }, where λ max = (λ 11 + λ 22 + (λ 11 − λ 22 ) 2 + 4λ 12 λ 21 )/2. In the nesting SF I pairing mechanism one assumes a dominance of the repulsive inter-band pairing ( λ 12 ,λ 21 < 0), i.e. |λ 12 , λ 21 | ≫ |λ 11 , λ 22 |. Since the intra-band pairing depends on λ ii = λ epi ii − µ * ii , where λ epi ii is the intra-band EP I coupling constant and µ * ii > 0 is an screened intra-band Coulomb repulsion, then in order to maximize T c the intra-band EP I coupling it is wishful that λ epi ii at least compensate negative effects of µ * ii (on T c ), i.e. λ epi ii ≥ µ * ii . Since in a narrow band one expects rather large screened Coulomb repulsion µ * ii (∼ 0.2) then the intra-band EP I coupling should be also appreciable. Moreover, from the experimental side the Raman measurements in Fe-pnictides [15] 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 [13] a model was proposed where high electronic polarizability of As (α As 3− ∼ 12Å 3 ) ions screens the Hubbard repulsion and also give rise to a strong EP I with A 1g (mainly As) modes. An appreciable As isotope effect in T c0 was proposed in [13], 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 F e − Se compounds, where an appreciable EP I is expected, since 78 Se is also highly polarizable (α Se 2− ∼ 7.5Å 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 exchangecorrelation effects and overestimates charge screening effects -both effects contribute significantly to the EP I coupling constant. As a result, LDA strongly underestimates the EP I 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 superconductor with T c > 30 K which is Kdoped from the parent isolating state BaKBiO 3 . Here, LDA predicts λ < 0.3 and T c ∼ 1 K, while the theories with an appropriate non-local exchange-correlation potential [16] predict λ epi ≈ 1 and T c ∼ 31 K; (ii) The high temperature superconductors, for instance Y BaCu 3 O 7 with T c ∼ 100 K, whose parent compound Y BaCu 3 O 6 is the Mott-insulator [2], [17].
After this digression we consider the main subject of the paper -the role of the EP I with forward scattering peak ( F SP ) in pairing mechanism of the 1M L F eSe/SrT iO 3 (and also 1M L F eSe/T iO 2 ) superconductor(s) with high critical temperatures T c ∼ (50 − 100) K. In that respect, numerous experiments on 1M L F eSe/SrT iO 3 (and also on1M L F eSe/T iO 2 ), combined with the fact that the F eSe film on the graphene substrate has rather small T c ≈ 8 K (like in the bulk F eSe), give strong evidence that interface effects, due to SrT iO 3 (and T iO 2 ), are most probably responsible for high T c . It turns out, that the most important results in 1M L F eSe/SrT iO 3 (and also 1M L F eSe/T iO 2 ), related to the existence of quasi-particle replica bands -which are identical to the main quasiparticle band [4], [6], [18], can be coherently described by the EP I − F SP theory. This approach was proposed in seminal papers [4]- [5]. The beauty of these papers lies in the fact that they have recognized sharp replica bands in theÁRP ES spectra and related them to a sharp forward scattering peak in the EP I. (This is a very good example for a constructive cooperation of experimentalists and theoreticians.) Let us mention, that the EP I − F SP theory was first studied in a connection with HT SC cuprates [1], while the extreme case of the EP I − F SP pairing mechanism with delta-peak is elaborated in [3] -see a review in [2]. Physically, this (in some sense exotic) interaction means that in some specific materials (for instance in cuprates and in 1M L F eSe/SrT iO 3 ) electron pairs exchange virtual phonons with small (transfer) momenta q < q c ≪ k F only, and as a result the effective pairing potential becomes long-ranged in real space [2]. 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 EP I − F SP pairing mechanism one has T are the corresponding mass-independent EP I coupling constants, where Ω -is the phonon energy, N (E F ) -the electronic density of states (per spin) at the Fermi surface. So, even for small λ can be in principle realized. We inform the reader in advance, that the EP I − F SP theory predicts also that T (F SP ) c ∼ (q c /k F ) d V epi (0), (d = 1, 2, 3 is the dimensionality of the system), which means that when q c ≪ k F high T (F SP ) c is hardly possible in 3D systems. However, the detrimental effect of the phase-volume factor 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 EP I − F SP mechanism of pairing are in many respects very different from the standard (BCSlike) superconductors, and it is completely justified to speak about exotic superconductors. For instance, the EP I − F SP theory [2]- [3] predicts, that in superconductors with the EP I −F SP pairing the isotope effect should be small in leading order, i.e. α ≪ 1/2 [2]- [3] -see discussion in the following. This result is contrary to the case of the isotropic EP I theory in standard metallic superconductors, where α is maximal, α = 1/2 (for µ * = 0). We point out, that the EP I − F SP 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 EP I − F SP pairing mechanism in case of HT SCcuprates 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 (µ * d ≪ µ * s ) this result opens a door for d-wave pairing in HT SC-cuprates, in spite of the fact of the EP I dominance [1]-[2].
In the following, we study the superconductivity in 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 [6]) in the framework of a semi-microscopic model of EP I first proposed in seminal papers [4]- [5]. Namely, due to oxygen vacancies: (i) an electronic doping of the F eSe 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 T iO 2 layer; (iii) the free charges in SrT iO 3 screen the dipolar field in the bulk, thus leaving the T iO 2 dipolar layer near the interface as an important source for the EPI. The oxygen ions in the T iO 2 dipolar layer vibrate with high-energy Ω ≈ 90 meV , thus making a long-range dipolar electric field acting on metallic electrons in the F eSe monolayer. This gives rise to a long-ranged EP I [4], [5], which in the momentum space gives a forward scattering peak -the EP I − F SP pairing mechanism.
In this paper we make some analytical calculations in the framework of the EP I − F SP theory with a very narrow δ-peak, with the width q c ≪ k F , wher k F is the Fermi momentum [3]. Here, we enumerate the obtained results, only: (1) in leading order the critical temperature is linearly dependent on the pairing potential V epi (q), i.e. T c0 ≈ V epi (q) q /4. In order to obtain T c0 ∼ 100 K we set the range of semi-microscopic parameters (ε ef f , ε ef f ⊥ , q ef f , h 0 , n d -see below) entering V epi (q) q . Furthermore, since V epi (q) q is independent of the the oxygen (O) mass, then T c0 is mass-independent in leading order with respect to T c0 /Ω. This means, that in 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 ) one expects very small O-isotope effect (α O ≪ 1/2). Note, in [5] large α O = 1/2 is found; (2) the self-energy Σ(k, ω) 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 massdependent ; (ii) the slope of the quasiparticle self-energy . The presence of non-magnetic impurities (with the parameter Γ = πn i N (E F )u 2 ) lifts this degeneracy. It is shown, that even the s-wave pairing (in the EP I − F SP pairing mechanism) is sensitive to non-magnetic impurities, which are pair-weakening for it, i.e. T (s) c0 is decreased for large Γ, but never vanishes. It is also shown that for d- c0 strongly depends on impurities, which are pair-breaking. The curiosity is that in the presence of non-magnetic impurities T (d) c0 in the EP I − F SP pairing mechanism is more robust than the corresponding one in the BCS model; (4) the long-range EP I − F SP 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 EP I − F SP pairing potential as a function of semi-microscopic parameters in the model of a dipolar layer T iO 2 with vibrations qf the oxygen ions [4]- [5]. 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 (ε ef f , The range of of these parameters, for which one has T c0 ∼ 100 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. It is important to point out that in 1M L F eSe/SrT iO 3 material, with 1-monolayer of F eSe grown on the SrT iO 3 substrate -mainly on the (0, 0, 1) plane, the Fermi surface in the F eSe monolayer is electron-like and centered at four M-points in the Brillouin zone -see more in [6], [19]. The absence of the (nested) hole-bands on the Fermi surface rules out all SF I nesting theories of pairing. Even the pairing between an electron-and incipient hole-band [20] is in- (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 ARP ES spectra at energies of the order of optical phonons with Ω ∼ 90 meV , implies inevitably that the dominant interaction in 1M L F eSe/SrT iO 3 (and also in 1M L F eSe/T iO 2 [6]) is due to EP I with strong forward scattering peak [4]- [5]. The physical mechanism for EP I − F SP is material dependent and the basic physical quantities such as the width of the F SP , phonon frequencies and bare EP I coupling can vary significantly from material to material. For instance, in HT SC-cuprates the effective EP I − F SP 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 . The prefactor is a vertex correction due to strong correlations and it means a new kind of (anti)screening in strongly correlated materials.
The interface in 1M L F eSe/SrT iO 3 can be considered as highly anisotropic material with the parallel and perpendicular (to the F eSe plane) dielectric constants ε ef f ≫ ε ef f ⊥ . It is assumed [4]- [5] that the oxygen from the T iO 2 dipolar layer -placed at height (−h 0 ) from the F eSe plane, vibrate and make dipolar moments δp z = q ef f δh(x, y, −h 0 ) perpendicular to the F eSe (x − y) plane -see F ig.1. It gives rise to a dipolar electric potential Φ dip (x, y, −h 0 − δh) acting on electrons in the F eSe (x − y plane). Here, q ef f is an effective charge per dipole and δh is the polar (dominantly oxygen) dis-placement along the z-axis [4]. Due to some confusion in the literature on the form of Φ dip [4] we recalculate it here, in order to know its explicite dependence on the semi-microscopic parameters ε ef f , ε ef f ⊥ , q ef f , h 0 , n d . An elementary electrodynamics approach [21] gives for the where n d is the number of the oscillating k+q are boson and fermion creation operators, respectively. The Fourier transformed potential g epi (q)(= (g 0 / √ N )e −q/qc ) is given by characterizes the range of the EP I potential, i.e. for q c ≪ k F (k F is the Fermi momentum) the EP I is sharply peaked at q = 0 -the forward scattering peak (F SP ), and the potential in real space is long-ranged, while for q c ∼ k F it is short-ranged, like in the standard EP I theory. Since we are interested in the T c dependence on the effective parameters ε ef f , ε ef f ⊥ , q ef f , h 0 , then an explicit dependence of the potential is important. We shall see below, that in order that this approach is applicable to 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 ) ε ef f , ε ef f ⊥ must be very different from the bulk values of ε in the bulk SrT iO 3 -where ε ∼ 500 − 10 4 , or in the rutile T iO 2 structure where ε < 260 [6].

REPLICA BANDS
The general self-energy Σ epi (k F , ω) at T = 0 in the extreme F SP δ-peak limit with the width q c ≪ k F ) is given by (see Appendix) where λ m = V epi (q) q /2Ω is the mass-dependent coupling constant. Here, the average EP I potential is is the surface of the F eSe unit cell and a is the Fe-Fe distance, and the bare pairing EP I potential is V 0 epi = 2g 2 0 /Ω. The coupling constant λ m corresponds to λ m used in [5], where the self-energy effects are studied at T > 0. It is important to point out that λ m is (oxygen) mass-dependent, contrary to [5]. Since V epi (q) q is mass-independent then λ m ∼ Ω −1 ∼ M 1/2 . In the following we discuss the case when k = k F , i.e. ξ(k) = 0. For ω ≪ Ω one has Σ epi (k, ω) = −λ m ω which means that the slope of Σ epi (k, ω) is mass-dependent. The latter property can be measured by ARP ES and thus the EP I − F SP theory can be tested. Note, that in the EP I − F SP theory the critical temperature T c0 (= V epi (q) 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 massdependent.
The quasiparticle and replica bands at T = 0 are obtained from ω − Σ epi (ω) = 0. In the following we make calculations at T = 0 and at the Fermi surface ξ(k F ) = 0. The solutions are: (1) ω 1 = 0 -the quasiparticle band ; (2) ω 2 = −Ω √ 1 + λ m is the ARP ES replica band ; (3) the inverse ARP ES replica band ω 3 = Ω √ 1 + λ m . The single-particle spectral function is A(k F , ω, T = 0) = 3 i=1 (A i /π)δ(ω − ω i ), where A i /π are the spectral weights. For the quasiparticle band ω 1 one obtains A 1 = (1 + λ m ) −1 , while for the replica bands at ω 2 and ω 3 one has A 2 = A 3 = (λ m /2)(1 + λ m ) −1 . The ratio of the intensities at T = 0 of the ω 2 replica band and quasiparticle band ω 1 is given by It is necessary to mention that at finite T (> 0) this ratio is changed as found in [5]. In that case Σ We stress that, the ARP ES measurements of A 2 /A 1 in 1M L F eSe/SrT iO 3 were done at finite temperatures (T = 0) and in the k = 0 point with ξ(k = 0) ∼ −50 meV which gives (A 2 /A 1 ) T ≈ 0.15 − 0.2 [4], [18]. According to the theory in [4], [5] one obtains λ Below we show, that λ m can be also extracted from the formula Eq.(6 ) for T c0 ≈ 100 K, which gives λ (Tc0) m ≈ 0.18. The latter value is in a good agreement with λ (ARP ES) m from ARP ES [22]. If we put this value in Eq.(4) one obtains that at T = 0 and at k = k F one has (A 2 /A 1 ) ∼ 0.1. From this analysis we conclude that the ARPES measurements at k F should give the similar ratio as at k = 0. The calculated ARP ES spectra at T = 0 K and at k = k F give ∆ω = |ω 2 − ω 1 | = Ω √ 1 + λ m while the experimental value is ∆ω ≈ 100 meV , which for λ (ARP ES) m ≈ 0.2 gives the optical phonon energy of the order of Ω ≈ 90 meV .

Ω/πTc0
−Ω/πTc0 For Ω ≫ πT c0 this gives T c0 = V epi (q) q /4 ≈ (1/16π)(aq c ) 2 (2g 2 0 /Ω), where a is the F e − F e distance. Note, that T c0 is mass-independent ( α O = 0) -note α O = 1/2 is found in [5]. The small isotope-effect can be a smoking-gun experiment for the EP I − F SP pairing mechanism in 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 ). From Eq. (15) in the Appendix it is straightforward to obtain the energy gap ∆ 0 = 2T c0 . Note, that 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 ) is a 2D system and T c0 ∼ q c 2 , while in the d-dimensional space one has T c0 ∼ q c d . This means that the EP I−F SP mechanism of superconductivity is more favorable in lowdimensional 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 EP I, it means that the EP I − F SP mechanism of pairing may be also operative in cuprates [2]. Note, that in estimating some semi-microscopic parameters we shall use as a reper-value T c0 ≈ 100 K, while in real systems T c ∼ (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 EP I − F SP systems, which is due to internal pair-fluctuations -see discussion below.
One can estimate the coupling constant λ m = V epi (q) q /2Ω in 1M L F eSe/SrT iO 3 from the value of T c0 . Then for the reper-value T c0 ∼ 100 K one has V epi (q) q ≈ 33 meV and λ (Tc0) m ≈ 0.18. Since, the consistency of the theory is satisfactory. Note, if one includes the wave-function renormalization effects (contained in Z(iω n ) > 1) then in the case (T c0 /Ω) ≪ 1 and for the square-well solution T c0 is lowered to T . This means, that the nonlinear corrections (with respect to λ m ) in T c0 and ∆ 0 [5], [23] should be inevitably renormalized by the Z-renormalization.
Let us estimate the parameters (ε , ε ⊥ , q ef f , h 0 ) which enter in T c0 . In order to reach T c0 ∼ 100 K (and V epi (q) q = 4T c0 ∼ 400 K ≈ 33 meV ) then for aq c ≈ 0.2 and Ω ≈ 90 meV one obtains g 0 ≈ 0.7 eV . Having in mind that g 0 = (2πn d eq ef f /ε ef f ⊥ )( /M Ω) 1/2 and that the zero-motion oxygen amplitude is ( /M Ω) 1/2 ≈ 0.05 A and by assuming that n d ≈ α/s c , s c =ã 2 ,ã = √ 2a ≈ 4 A, q ef f ∼ 2e, α 1, then in order to obtain g 0 ≈ 0.7 eV ε ef f ⊥ must be small, i.e. ε ef f ⊥ ∼ 1. Since aq c = (a/h 0 ) ε ef f ⊥ /ε ef f ∼ 0.2 and for (a/h 0 ) ∼ 1 it follows ε ef f ∼ 30. Note, that in SrT iO 3 the bulk ε is large, ε ∼ 500 − 10 4 . So, if T c0 in 1M L F eSe/SrT iO 3 is due solely to the EP I − F SP mechanism, then in the model where the oxygen vibrations in the single dipolar monolayer T iO 2 are responsible for the pairing potential the effective dielectric constants ε ef f ⊥ , ε ef f are very different from the bulk values in SrT iO 3 (in 1M L F eSe/T iO 2 one has ε ≤ 260 [6]). This is physically plausible since for the nearest (to the F eSe monolayer) T iO 2 dipolar monolayer there is almost nothing to screen in the direction perpendicular to F eSe, thus making ε ef f ⊥ ≪ ε bulk . Note, that for the parameters assumed in this analysis and for T c0 ≈ 100 K one obtains rather large bare pairing potential V 0 epi ≈ 10 eV . This means that in the absence of the F SP in EP I and for the density of states of the order N (E F ) ∼ 0.5 (eV ) −1 (typical for Fe-based superconductors) the bare coupling constant λ 0 epi = N (E F )V 0 epi would be large, λ 0 epi ∼ 5. We stress that the above theory is also applicable to recently discovered 1M L F eSe/T iO 2 [6]. To conclude, the high T c0 in 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 ) is obtained on the expense of the large maximal EP I coupling V 0 epi which compensates smallness of the (detrimental) phase-volume factor (aq c ) 2 .

V. EFFECTS OF IMPURITIES ON Tc0
In clean systems with the EP I − F SP mechanism of superconductivity T c0 is degenerate -it is equal in sand d-channels. In the following we show, that the swave superconductivity is also affected by isotropic nonmagnetic impurities, i.e. T c0 is reduced and the Anderson theorem is violated. This may have serious repercussions on the s-wave superconductivity in 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 ) where T c0 may depend on chemistry. Then by using equation Eq.(18) from Appendix one obtains T where ρ = Γ/πT c , Γ = πn i N (E F )u 2 , n i is the impurity concentration and u is the impurity potential. Let us consider some limiting cases: This means that in the EP I − F SP systems the nonmagnetic impurity scattering is pair-weakening for the s-wave superconductivity.
In the case of d-wave superconductivity the solution of Eq. (19) in limiting cases is: c . We point out that the slope −dT > Γ cr (= (2/π)T c0 ). These two results mean that in the presence of non-magnetic impurities the dwave superconductivity which is due to the EP I − F SP 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 1M L F eSe/SrT iO 3 (as well as in1M L F eSe/T iO 2 ) might be an important test for the EP I − F SP pairing in this material.
Finally, it is worth of mentioning, that the real isotope effect in T c0 of 1M L F eSe/SrT iO 3 (and in 1M L F eSe/T iO 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 T iO 2 dipole layer), then there is F SP in the scattering potential, i.e. u 2 imp (q) ≈ u 2 δ(q). Then, such impurities affect in the same way s-and d-wave pairing and they are pair weakening, as shown in [3]. Naimly, one has (a) T There are two important results: (1) There is a nonanalicity in Γ F ∼ √ n i ; (2) there is a full isotope effect in the "dirty" limit Γ F ≫ πT c , i.e. α O = 1/2, since T (s,d) c ∼ Ω ∼ M −1/2 . We stress, that if the full isotope effect would be realized experimentally in 1M L F eSe/SrT iO 3 (and in 1M L F eSe/T iO 2 ), then this does not automatically exclude the F SP − EP I mechanism of pairing, since it may be due to impurity effects. In that case the nonanalicity of Γ F in n i might be a smoking-gun effect.

VI. INTERNAL PAIR FLUCTUATIONS REDUCE Tc0
The EP I − F SP theory, which predicts a long-range force between paired electrons, opens a possibility for a pseudogap behavior in 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 ). As we have discussed above, the EP I − F SP theory predicts a non-BCS dependence of the critical temperature T c0 , i.e. T c0 = V epi (q) q /4. However, this mean-field (M F A) 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 M F A the order parameter ∆(x, ) depends on the relative (internal) coordinate r = x − x ′ and the center of mass R = (x + x ′ )/2, i.e. ∆(x, x ′ ) = ∆(r, R). In usual superconductors with short-range pairing potential one has V sr (x − x ′ ) ≈ V 0 δ(x − x ′ ) and ∆(r, R) = ∆(R). Therefore only the spatial (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 ∆(r, R) on internal degrees of freedom (on r). The interesting problem of fluctuations in systems with long-range attractive forces in 3D systems was studied in [24] and we sketch it briefly, because it shows that standard and EP I − F SP 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) [24]. This is a Heisenberg-like Hamiltonian in the momentum space. In case of the s-wave superconductivity with short-range forces V sr (x − x ′ ) ≈ V 0 δ(x − x ′ ) one has V k−k ′ = const and the pairing potential is long-ranged in the momentum space. In that case it is justified to use the mean-field approxi-mationĤ →Ĥ mf = − k h kŜk with the mean-field The excitation spectrum (with respect to the ground state) in this system have a gap, i.e. E(k) = 2 ξ 2 k + ∆ 2 k where the gap ∆ k is the mean-field order parameter defined by In case of the EP I − F SP pairing mechanism the pairing potential is long-ranged in real space and short-ranged in the momentum space. For instance, in 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 2 ) one has V k−k ′ = V 0 exp{− |k − k ′ | /q c } with q c ≪ k F , and the excitation spectrum is boson-like 0 < E(k) < 2 ξ 2 k + ∆ 2 k (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 ξ 0 ≪ 1 [24] one has T c ≈ (q c ξ 0 )T c0 , where the coherence length ξ 0 = v F /π∆ 0 and ∆ 0 = 2T c0 . It is expected, that in the region T c < T < T c0 the pseudogap (P G) phase is realized. However, in 2D systems, like 1M L F eSe/SrT iO 3 (and 1M L F eSe/T iO 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 1M L F eSe/SrT iO 3 by the Meissner effect and resistivity (ρ(T )) give that T 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 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 2 .

VII. SUMMARY AND DISCUSSION
In the paper we study the superconductivity in the 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 2 sandwitchstructure, which contains one metallic F eSe monolayer grown on the substrate SrT iO 3 , or rutile T iO . It turns out that in such a structure the Fermi surface is electron-like and the bands are pockets around the Mpoint in the Brillouin zone. The bottom of the electronlike bands is around (50 − 60) meV below the Fermi surface at E F . The top of the hole-like band at the point Γ 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) [10]. The superconductivity in 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 2 is realized in the F eSe monolayer with T c ∼ (60 − 100) K. The decisive fact for making a theory is that the ARP ES spectra show sharp replica bands around 100 meV below the quasiparticle band, what is approximately the energy of the oxygen optical phonon Ω ≈ 90 meV . The analysis of superconductivity is based on the semi-microscopic model -first proposed in [4], [5], where it is assumed that a T iO 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 F eSe monolayer, thus making the EP I interaction long-ranged. In the momentum space a forward scattering peak (F SP ) appears, i.e. EP I is peaked at small transfer momenta (q < q c ≪ k F ) with g epi (q) = g 0 exp{−q/q c }. Here, this is called the EP I − F SP pairing mechanism. The EP I − F SP theory is formulated first in [1] for strongly correlated systems, while its extreme case with deltapeak is elaborated in [3] -see also [2]. This limiting (delta-peak) case makes not only analytical calculations easier, but it makes also a good fit to experimental results [4], [5]. In the following, we summarize the main obtained results of the EP I − F SP theory and its relation to the 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 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 = V epi (q) q /4 ≈ (1/16π)(aq c ) 2 V 0 epi , where V epi (q) = 2g 2 epi (q)/ Ω and the maximal pairing potential V 0 epi (≡ V epi (q = 0)) = (2g 2 0 /Ω). On the first glance this linear dependence of T c0 on V (0) epi seems to be favorable for reaching high T c0 -note in the BCS theory T c0 is exponentially dependent on V (0) epi and very small for small N (E F )V 0 epi . However, for non-singular g epi (q) when g epi (q = 0) is finite, T c0 is limited by the smallness of the phase-volume effect, which is in 2D systems (such as 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 2 ) proportional to (aq c ) 2 ≪ 1. In that sense optimistic claims that the EP I −F SP 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 ∼ (aq c ) 3 for the same value of V 0 epi . However, higher T c0 (with respect to to the BCS case) can be reached by fine tuning of aq c and V 0 epi . This is probably realized in HT SC cuprates and with certainty in 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 2 . The weak-coupling theory predicts the superconducting gap to be ∆ 0 = 2T c0 and for T c0 ∼ 100 K one has ∆ 0 ∼ 16 meV what fits well the ARP ES experimental values [4], [6]. Note, in order to reach T c0 = 100 K for aq c ≈ 0.2 a very large maximal EP I coupling V 0 epi ≈ 10 eV is necessary. For N (E F ) ∼ 0.5 (eV ) −1 the maximal coupling constant would be rather large, i.e. λ 0 epi (= N (E F )V 0 epi ) ≈ 5. Note, that V 0 epi is almost as large as in the metallic hydrogen under high pressure p ∼ 20 M bar, where T c0 ≈ 600 K with large EP I coupling constant λ 0 epi ≈ 7 -this important prediction is given in [25]. In real 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 2 materials the contribution of another pairing mechanism, which exists in the F eSe film in absence of the substrate and is pronounced in the s-wave channel with T c0 ≈ 8 K, triggers the whole pairing to be s-wave. The latter only moderately decreases the contribution of the EP I − F SP pairing mechanism. The existence of sharp replica bands in 1M L F eSe/SrT iO 3 and 1U CF eSe/T iO 2 and large value of V 0 epi imply inevitably that the EP I − F SP pairing mechanism is the main candidate to explain superconductivity in these materials. We stress, that in 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 2 high T c0 is obtained on the expense of the large maximal EP I coupling V 0 epi , which compensates the small (detrimental) phase volume factor (aq c ) 2 .
(2) -The semi-microscopic model proposed in [4], [5], and refined slightly in this paper, contains phenomenological parameters, such as n d -the number of dipoles per unit cell, q ef f -the effective dipole charge, ε ef f , ε ef f ⊥ -effective parallel and perpendicular dielectric constanty in SrT iO 3 (and T iO 2 ) near the interface, respectively. For T c0 ≈ 100 K and by assuming aq c ≈ 0.2, q ef f ≈ 2e, n d ∼ 2 /unit − cell makes ε ef f ∼ 30, ε ef f ⊥ ∼ 1. These values, which are physically plausible, are very far from ε bulk in the bulk SrT iO 3 , where ε bulk ∼ 500 − 10 4 (and ε bulk ≤ 260 in the rutile T iO 2 ). We point out that our estimation of these parameters is based on the effective microscopic model where the bulk SrT iO 3 is truncated by a monolayer (1M L) made of T iO 2 [4]. In reality it may happen that the bulk SrT iO 3 is truncated by two monolayers (2M L) of T iO 2 , as it is claimed to be seen in the synchrotron xray diffraction [26].This finding is confirmed by the LDA calculations in [26], which show that for the 2M L T iO 2 structure: (i) the electrons are much easier transferred to the F eSe metallic monolayer and (ii) the top of the hole band is shifted far below the electronic Fermi surface than in the 1M L model. If the 2M L of T iO 2 is realized it could be even more favorable for the EP I − F SP pairing, since some parameters can be changed in a favorable way. For instance, the effective charge could be increased, i.e. q ef f the 2M L model may gives rise to higher critical temperature.
(3) -The isotope effect in T c0 should be small (α O ≪ 1/2) since in leading order one has T c0 ∼ V 0 epi , where V 0 epi is mass-independent. This is contrary to [5] where α O = 1/2. The next leading order gives α O ∼ (T c0 /Ω) < 0.09. We stress that the small isotope-effect maybe a smokegun experiment for the EP I − F SP pairing mechanism.
(4) -In the EP I-F SP 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 (Γ F ≫ πT c ) the full isotope effect α O = 1/2, since T (s,d) c ∼ Ω ∼ M −1/2 . In that case, the nonanalicity of T (s,d) c with respect to the impurity concentration n i , would resolve the question -what kind of pairing is realized in 1M L F eSe/SrT iO 3 and 1U CF eSe/T iO 2 -the EP I − F SP or the standard EP I.
(5) -In the case of the EP I-F SP pairing the superconducting order parameter depends strongly on the internal pair coordinate and of center of mass, i.e. ∆ = ∆(r, 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 ξ(k F ) = 0 is given by Σ epi (k, ω) ≈ −λ m ω/(1 − (ω/Ω) 2 ), where λ m = V epi (q) q /2Ω, which for G −1 (k, ω) = 0 gives the dispersion energy of the quasiparticle band ω 1 = 0 and the replica bands ω 2 and ω 3 . The ratio of the ARPES intensities of the replica band ω 2 and the quasiparticle band ω 1 at T = 0 and at the Fermi surface (k = k F ) is given by R(T = 0, k F ) = (A 2 /A 1 ) = λ m /2. This means, that for λ m ∼ 0.2 the experimental value of R(T = 0, k F ) should be (A 2 /A 1 ) ≈ 0.1. This ratio is slightly smaller than the experimental value R(T = 0, k = 0) measured in [4], [18]. (7) Since the coupling constant λ m is mass-dependent, λ m ∼ M 1/2 then the isotope effect in various quantities, in 1M L F eSe/SrT iO 3 and 1M L F eSe/T iO 2 systems, may be a smoke-gun experiment in favour of the EP I − F P S theory. To remind the reader: (i) T c0 is almost mass-independent ; (ii) the self-energy slope at ω ≪ Ω is mass-dependent, (−dΣ/dω) ∼ M 1/2 ; (iii) the ARP ES ratio R of the replica band intensities is mass-dependent, Concerning the role of EP I in explaining superconductivity in 1M L F eSe/SrT iO 3 there were other interesting theoretical proposals. In [27] the EP I is due to the interaction with longitudinal optical phonons and since Ω > E F the problem is studied in anti-adiabatic limit, where T c is also weakly dependent on the oxygen mass. In [28] the substrate gives rise to an antiferromagnetic structure in F eSe, which opens new channels in the EP I coupling in the F eSe monolayer, thus giving rise for high T c . In [29] 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 EP I − F SP interaction. The problem is studied by the sign-free Monte-Carlo simulations and it is found that EP I − F SP 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 1M L F eSe/SrT iO 3 (or 1M L F eSe/T iO 2 ) as a basic unit. The first nontrivial one is when a double-sandwich structure with two interfaces is formed, i.e. SrT iO 3 /1M L F eSe/SrT iO 3 (or T iO 2 /1M L F eSe/T iO 2 ). Naively thinking in the framework of the EP I − F SP 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 < πT c0 is kept in order to deal with a sharp F SP . 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. Z n ∆ n = πT n ′ N (0)V ef f (n − n ′ , Z n ′ ∆ n ′ (ω n ′ Z n ′ ) 2 + ∆ 2 n ′ (12) In the case of strongly momentum-dependent EP I − F SP , where V epi (n − n ′ , q) is finite for |q| < q c ≪ k F , the Migdal-Eliashberg equations are given by B. Effects of non-magnetic impurities on Tc0 in the EP I − F SP theory In this paper we study the superconductivity which is due to EP I − F SP of the Einstein phonon with Ω 2 ≪ (2πT c0 ) 2 . In that case V epi (n − m, q) q ≈ V epi (0, q) q = 2g 2 epi (q) q /Ω and the contribution to Z n (ξ) is ∼ λ m = V epi (0, q) q /2Ω. Since in the weak coupling limit one has λ m ≪ 1 then we neglect this contribution. Also the non-Migdal corrections can be neglected in this case -see [5] The effects of non-magnetic impurities on T c0 is studied in the standard model with weakly momentum dependent impurity potential u(k − k ′ ) ≈ const . In that case Z n (ξ) contains the impurity term only. After the integration of the impurity part over the energy ξ ′ in Eqs.(8-10) -see [30], and putting ξ = 0 (since in that case ∆ n (ξ = 0) is maximal) one obtains (D m (ξ) ≈ ω 2 n Z 2 n ) for the s-wave pairing (∆ = const) Z n = 1 + Γ |ω n | (16) (17) Note, the the second term on the right side cancels the same term on the left side. In the approximation ∆ n (ξ) ≈ ∆ one obtains the equation for impurity dependence of T We point out that in the case of d-wave superconductivity ∆ = ∆(ϕ) is angle dependent on the Fermi surface and changes sign. In that case the last term in Eq.(17) ∆ should be replaced by ∆(ϕ ) = 0 giving equation for T Note, Z n vs Z 2 n renormalization for the s-wave and dwave superconductivity, respectively.