# The electron–phonon interaction with forward scattering peak is dominant in high T c superconductors of FeSe films on ${{\rm{SrTiO}}}_{3}$ (TiO2)Academic research paper on "Physical sciences"

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## Academic research paper on topic "The electron–phonon interaction with forward scattering peak is dominant in high T c superconductors of FeSe films on ${{\rm{SrTiO}}}_{3}$ (TiO2)"

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The electron-phonon interaction with forward scattering peak is dominant in high T c

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superconductors of FeSe films on (TiO2

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The electron-phonon interaction with forward scattering peak is dominant in high Tc superconductors of FeSe films on SrTiO3 (TiO2)

M L Kulic12 and O V Dolgov34

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, D-70569 Stuttgart, Germany

4 P.N. Lebedev Physical Institute, RAS, Moscow, Russia

E-mail: kulic@th.physik.uni-frankfurt.de and o.dolgov@fkf.mpg.de Keywords: forward scattering, FeSe, electron-phonon interaction

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Abstract

The theory of the electron-phonon interaction (EPI) with strong forward scattering peak (FSP) in an extreme delta-peak limit (Kulic and Zeyher 1994 Phys. Rev. B 49 4395; Kulic 2000 Phys. Rep. 38 1-264; Kulic and Dolgov 2005 Phys. Status Solidi b 242 151; Danylenko et al 1999 Eur. Phys. J. B 9 201) is recently applied in (Lee etal 2014 Nature 515 245; Rademaker etal 2016 New J. Phys. 18 022001; Wang etal 2016 Supercond. Sci. Technol. 29 054009) for the explanation of high Tc 100 K) in a monolayer FeSe grown on SrTiO3 (Lee etal2014Nature 515 245) andTiO2 (Rebec etal2016 arXiv:1606.09358v1) substrates. The EPI is due to a long-range dipolar electric field created by high-energy oxygen vibrations (W ~ 90 meV) at the interface (Lee etal 2014 Nature 515 245; Rademaker etal 2016 New J. Phys. 18 022001; Wang etal 2016 Supercond. Sci. Technol. 29 054009). In leading order (with respect to To/W) the mean-field critical temperature Tc0 = (Vp (q) )q/4) ~ (aqc)2Vepi (0) andthegap A0 = 2Tc0 are due to an interplay between the maximal EPI pairing potential Vepi (0) and the FSP-width qc. For Tc0 ~ 100 K one has A0 ~ 16 meV in a satisfactory agreement with ARPES experiments. In leading order Tc0 is mass-independent and a very small oxygen isotope effect is expected in next to leading order. In clean systems Tc0 for s-wave and d-wave pairing is degenerate but both are affected by nonmagnetic 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 experimental results at T > 0 (Rademaker etal 2016 New J. Phys. 18 022001; Wang etal 2016 Supercond. Sci. Technol. 29 054009). The EPI coupling constant Am = {Vepi (q) )q/2W is mass-dependent (M1/2) and at w (^W) makes the slope of the self-energy S (k, w)(»-Am w) and the replica intensities At (~ Am) mass-dependent. This result, overlooked in the literature, is contrary to the prediction of the standard Migdal-Eliashberg theory for EPI. The small oxygen isotope effect in Tc0 and pronounced isotope effect in S (k, w) and ARPES spectra Ai of the replica bands in FeSe films on SrTiO3 and TiO2 is a smoking-gun experiment for validity of the EPI-FSP theory to these systems. The EPI-FSP theory predicts a large number of low-laying pairing states, thus causing internal pair fluctuations. The latter reduce Tc0 additionally, by creating a pseudogap state for Tc < T < Tc0. Possibilities to increase Tc0, by designing novel structures are discussed in the framework of the EPI-FSP theory.

1. 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 [7]. 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.

© 2017 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

However, this beautiful idea has not been realized experimentally until now. In that sense Ginzburg founded a theoretical group of experts, who studied at that time almost all imaginable pairing mechanisms. In this group an important role has played the Ginzburg's collaborator Maksimov, who was an 'inveterate enemy' of almost all other mechanisms of pairing in HTSC except of the electron-phonon one—see the arguments in [8]. 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 SrTiO3 substrate—further called 1ML FeSe/SrTiO3, with the critical temperature Tc ~ (50-100) K [9], as well as grown on the rutile TiO2 (100) substrate with Tc ~ 65 K [6]—further called 1ML FeSe/TiO2, in some sense reconciles the credence ofthese two outstanding physicists. Namely, HTSC is realized in a sandwich-structure but the pairing is due to an high-energy (~90-100 meV) oxygen opticalphonon. 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 Tc in Fe-pnictides a non-phononic pairing mechanism was proposed immediately, which is based on: (i) nesting properties ofthe electron- and hole-Fermi surfaces and (ii) an enhanced (due to (i)) spin exchange interaction (SFI) between electrons and holes [10]. This mechanism is called the nestingSFIpairing. However, the discovery of alkaline iron selenides KxFe2—y Se2 with Tc ~ 30 K, and intercalated compounds Lix (C2H8N2) Fe2—y Se2, Lix(NH2)y (NH3)1—y Fe2Se2, which contain only electron-like Fermi surfaces, rules out the nesting pairing mechanism as a common pairing mechanism in Fe-based superconductors. 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 J1 — J2 Heisenberg-like Hamiltonian, which is able to describe s-wave superconductivity, too. However, this approach is questionable, since the LDA calculations cannot be mapped onto a Heisenberg model and it is necessary to introduce further terms in form of a biquadratic exchange [ 10] .It is interesting, that immediately after the discovery of high Tc 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 A < 0.2, thus giving T < 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 [12-15]. One of the theoretical arguments for it, maybe illustrated in the case of 2-band superconductivity. In the weak-coupling limit Tc is given by Tc = 1.2 w exp { — 1/Amax}, where Amax = (A11 + A22 + V(A11 — A22)2 + 4A12 A21)/2. In the nesting SFI pairing mechanism one assumes a dominance ofthe repulsive inter-band pairing (A12,A21 < 0),i.e. |A12, A21| ^ | An, A22|. Since the intra-band pairing depends on Aii = Aj|pi — m*, where Aepi is the intra-band EPI coupling constant and m* > 0 is a screened intra-band Coulomb repulsion, then in order to maximize Tc the intra-band EPIAepi should at least compensate negative effects of m* (on Tc), i.e. Aepi ^ m*. Since in a narrow band one expects rather large screened Coulomb repulsion m* (~0.2) then the intra-band EPI coupling should be also appreciable. Moreover, from the experimental side the Raman measurements in Fe-pnictides [16] give strong evidence for a large phonon line-width of some A1g modes (where As vibrations along the c-axis dominate). They are almost 10 times larger than the LDA band structure calculations predict. In [13] a model was proposed where the high electronic polarizability of As («as3— ~

12 A3) ions screens the Hubbard repulsion and gives rise to a strong EPI with A1g (mainly As) modes. An appreciable As isotope effect in Tc0 was proposed in [13], where the stable 75As should be replaced by an unstable, for instance 73As—with the life-time of 80 days, quite enough for performing relevant experiments. The situation is similar in Fe-Se compounds, where an appreciable EPI is expected, since 78Se is also highly polarizable («Se2— ~ 7.5 A) and can be replaced by a long-living 73Se 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 Tc 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) BiO3 superconductor with Tc > 30 K which is K-doped from the parent isolating compound BaKBiO3. Here, LDA predicts A < 0.3 and Tc ~ 1K, while the theories with an appropriate non-local exchange-correlation potential [17] predict Aepi » 1 and Tc ~ 31 K; (ii) the HTSC, for instance YBaCu3O7 with Tc ~ 100 K, whose parent compound YBaCu3O6 is the Mott-insulator and cannot be desribedbyLDA [2,18].

After this digression we consider the main subject of the paper—the role ofthe EPI with FSP in pairing mechanism ofthe 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2 superconductors with high critical temperatures Tc ~ (50-100) K. In that respect, numerous experiments on 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2, combined with the fact that the FeSe film on the graphene substrate has rather small Tc » 8 K (like in the bulk FeSe), give strong evidence that interface effects, due to SrTiO3 (and TiO2), are most probably responsible for high Tc. It

turns out, that the most important results in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2)—related to the existence of quasi-particle replica bands and which are identical to the main quasiparticle band [4,6,19], can be coherently described by the EPI-FSP 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 the ARPES spectra and related them to a sharp FSP in the EPI. (This is a very good example for a constructive cooperation of experimentalists and theorists.) Let us mention, that the EPI-FSP theory was first studied in a connection with HTSC cuprates [1], while the extreme case of the EPI-FSP pairing mechanism with delta-peak is elaborated in [3]—see a review in [2]. Physically, this (exotic) interaction means that in some specific materials, for instance, in cuprates and in 1ML FeSe/SrTiO3, electron pairs exchange virtual phonons with small momentum transfer q < qc ^ kF,only. 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 a higher Tc than in the standard (Migdal-Eliashberg) BCS-like theory. Namely, in the EPI-FSP pairing mechanism one has Tc(FSP) = <Vepi (q) )q/4 ~ \%SP)/N (EF) [3]—see below,

instead of the BCS dependence Tc(BCS) ~ We V^ '. Here, AepSP) and l^PP"1 are the corresponding mass-independent EPI coupling constants, where O—is the phonon energy, N (EF)—the electronic density of states (per spin) at the Fermi surface. So, even for small ^ the case Tc(FSP) > Tc(BCS) can be in principle realized. We inform the reader in advance, that the in the case of a non-singular FSP what happens in 1 ML FeSe/ SrTiO3 (and 1 ML FeSe/TiO2)—see below, where the EPI-coupling ge2pi (q) is finite at q = 0—non-singular

coupling, the EPI-FSP theory predicts also that TcFSP ~ (qc/kF)dV¡pi (0), (d = 1,2,3 is the dimensionality of the system), which means that for qc ^ kF high Tc is hardly possible in 3D systems. However, the detrimental effect of the phase-volume factor (qc/kF)d on Tc(FSP) can be compensated by its linear dependence on the pairing potential Vepi(0). In some favorable materials this competition may lead even to an increase of Tc. Note, that Tc(,Fmg ~ (qc /kF)d holds only for the non-singular EPI coupling, while for the singular one with qd—gP (q = 0) = const—studiedin [3], one has in the next to leading order Tc(FsSgP)(qc) » Tc0 (qc = 0) [1 - 7Z (3) qc vF/4^2Tc0],i.e. Tc> sg (qc = 0) is finite. The different behavior of TfSPg and Tc(FsSgP) isdueto interplay of the phase volume (qd- 1dq) effects and the analytical behavior of the long-range EPI forces. We stress that properties of 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 [2,3] predicts, that in superconductors with the EPI-FSP pairing the isotope effect should be small in leading order, i.e. a ^ 1/2 [2,3]—see discussion in the following. This result is contrary to the case of the isotropic EPI theory in standard metallic superconductors, where a is maximal, a = 1/2 (for j* = 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 Tc 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 (jm* ^ j*) this result opens a door for d-wave pairing in HTSC-cuprates, in spite of the fact of the EPI dominance [1,2].

In the following, we study the superconductivity in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2 [6]) in the framework ofasemi-microscopic model of EPI first proposed in seminal papers [4,5]. 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 TiO2 layer; (iii) the free charges in SrTiO3 screen the dipolar field in the bulk, thus leaving the TiO2 dipolar layer near the interface as an important source for the EPI. The oxygen ions in the TiO2 dipolar layer vibrate with a high-energy W » 90 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 [4,5], which in the momentum space gives a FSP—the EPI-FSP pairingmechanism.

In this paper we make some analytical calculations in the framework of the EPI-FSP theory with a very narrow ¿-peak, with the width qc ^ kF, where kF 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:pi (q), i.e. Tc0 » <Vepi (q) )q/4. In order to obtain Tc0 ~ 100 K we set a range of semi-microscopic parameters (eeff, ef, qef, h0, nd —see below) entering < V¡pi (q) )q. Furthermore, since < V¡pi (q) )q is independent of the the oxygen (O) mass, then Tc0is mass-independent in leading order with respect to Tc0/W. This means, that in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2) one expects very small O—isotope effect (a0 ^ 1/2). Note, in [5] alarge aO = 1/2 is predicted; (2) the self-energy S (k, w) 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 w ^ W (S (w) » — 1m w) is mass-dependent, since ~1m ~ MO/2; (3) in the

EPI-FSP model (without other interactions) the critical temperature for s-wave and d -wave pairing is degenerate, i.e. Tc(0) = T^-1. The presence of non-magnetic impurities (with the impurity concentration ni and the parameter G = pniN (EF) u2) 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. Tc(0) decreases for large r, but never vanishes. It is also shown, that for the d-wave pairing T^-* strongly depends on impurities, which are pair-breaking. The curiosity is that in the presence of non-magnetic impurities T^-1 in the EPI-FSP pairing mechanism is more robust than the corresponding one in the BCS model; (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 energies (above the ground-state) of pairs, thus leading to strong internal pair fluctuations which reduce Tc0. At Tc < T < Tc0 a pseudogap behavior is expected.

The structure of the paper is following: in section 2 we calculate the EPI-FSP pairing potential as a function of semi-microscopic parameters in the model of a dipolar layer TiO2 with vibrations of the oxygen ions [4,5]. In section 3 the self-energy effects, such as the replica bands and their intensities at T = 0, are studied. The critical temperature Tc0 is calculated in section 4 in terms of the semi-microscopic parameters (ef, sef, qeff, h0, nd). The range ofofthese parameters, for which one has Tc0 ~ 100 K, is estimated, too. In section 5 effects of nonmagnetic impurities on Tc0 are studied, while effects ofinternal fluctuations ofCooper pairs are briefly discussed in section 6. Summary of results are presented in section 7.

2. EPI-FSP potential due to O-vibrations in the TiO2layer

It is important to point out that in 1ML FeSe/SrTiO3, with 1-monolayer of FeSe grown on the SrTiO3 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 [6,20]. 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 holeband [21] 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 the sharp replica bands in the ARPES spectra at energies of the order of optical phonons with W ~ 90 meV, implies inevitably that the dominant interaction in 1ML FeSe/SrTiO3 (and also in 1ML FeSe/TiO2 [6]) is due to EPI with strong FSP [4,5]. The physical mechanism for EPI-FSP is material dependent and the basic physical quantities such as the width of the FSP, phonon energies and the bare EPI coupling constant can vary significantly from material to material. For instance, in HTSC-cuprates the effective EPI-FSP potential Vepi(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 Vepi(q) is given by Vepi (q) » [1 + (q/qc)2]—2 V0,epi (q), qc ~ 6ja, where V0,epi (q) is the bare (without strong correlations) coupling constant, 6 (^1) is the hole concentration and a is the Cu-O distance [ 1,2]. 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 1ML FeSe/SrTiO3 can be considered as a highly anisotropic material with parallel and perpendicular (to the FeSe plane) dielectric constants ef ^ e!®. It is assumed [4,5] that the oxygen ions in the TiO2 dipolar layer—placed at height (— h0) from the FeSe plane, vibrate and make oscillating dipolar moments 6pz = qeff 6h (x, y, — h0) perpendicular to the FeSe (x-y) plane—see figure 1. It gives rise to a dipolar electric potential Fdip (x, y, —h0 — 6h) acting on electrons in the FeSe (x-y plane). Here, qeff is an effective charge per dipole and 6h is the polar (dominantly oxygen) displacement along the z-axis [4]. Due to some confusion in the literature on the precise form of Fdip [4] we recalculate it here, in order to know its explicit dependence on the semi-microscopic parameters ef, e^, qeff, h0, nd. An elementary electrodynamics approach [22] gives for the dipolar potential

Fdip (x, y, 6h) (eeff)^2 f dx'dy6h (x', y', —h0)

«dSeff ho (ef )3/2

r dx 'dy'dh(x', y1, -h0) (!)

J j eeff \f '

[ho2 + (x - x')2 + (y - y')2J

where 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 [4]—where it is e||ff /(eff )3^2 (qeff h0) and with missed dipole density nd. This coefficient does not fulfill the condition F ~ e-1 in the isotropic case, while equation (1) does. By introducing gepi (q) = eF (q) the EPI Hamiltonian Hepi = £q eF (q) p (q) is rewritten in the form Hepi = £k,q gepi (q)

(bq + b-q) c^+q Ck, where b-q, c^+q are boson and fermion creation operators, respectively. The Fourier transformed potential gepi (q) (=(g0j) e-q/qc) is given by

Figure 1. Left: the microscopic structure of the FeSe/SrTiO3 interface of the 1ML FeSe/SrTiO3 structure. h0—distance between the FeSe monolayer and TiO2 dipolar layer. Sh is amplitude of the oxygen vibration in the dipolar layer. Right: the perpendicular view on the FeSe-SrTiO3 interface in the model with one TiO2 dipolar layer at the end of the SrTiO3 substrate. The anisotropy of the effective dielectric constant ^e^L is shown. The similar schema holds also for the 1ML FeSe/TiO2 structure.

g ,(q) = 2p"deqeff rn e-q/q c (2)

^(q) ef \ MWN ' (2)

g0 = (2pnd eqeff /efff) (H/MW)1/2, e is the electronic charge. Here, the screening momentum qc = (£ef / £\L )^2h0-1 characterizes the range of the EPI potential, i.e. for qc ^ kF (kF is the Fermi momentum) the EPI is sharply peaked at q = 0—the FSP (FSP), and the potential in real space is long-ranged, while for qc ~ kF it is short-ranged, like in the standard EPI theory. Since we are interested in the Tc dependence on the effective parameters e|jff, eL, qeff, 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/SrTiO3 (and 1ML FeSe/TiO2) ef ef must be very different from the bulk values of s in the bulk SrTiO3—where e ~ 500-104, or in the rutile TiO2 structure where e < 260 [6].

3. Self-energy effects and ARPES replica bands

The general self-energy Sepi (kF, w) at T = 0 in the extreme FSPS-peak limit with the width qc ^ kF) is given by (see appendix)

Sepi (kF> w) » - Am"-W , (3)

1 — (w/ W)2

where Am = (Vepi (q) )q/2 W is the mass-dependent coupling constant. Here, the average EPI potential is given by (Vepi (q) )q = NSc (2p)—2 J d2qVepi (q, 0) » (1/4p )(aqc)2Ve0pi, where Sc = 2a2 is the surface of the FeSe unit cell and a is the Fe-Fe distance, and the bare pairing EPI potential is Ve0pi = 2g2/W. The coupling constant Am corresponds to Am used in [5], where the self-energy effects are studied at T > 0. It is important to point out that Am is (oxygen) mass-dependent, contrary to [5]. Since (Vepi (q) )q is mass-independent then Am ~ W-1 ~ the following we discuss the case when k = kF, i.e. X (k) = 0.For w ^ W one has Sepi (k, w) = — Am w which means that the slope of Sepi (k, w) 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 Tc0 (=( Vepi (q) )q 14)—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 Tc0 is mass-dependent.

The quasiparticle and replica bands at T = 0 are obtained from w — Sepi (w) = 0. In the following we make calculations at T = 0 and at the Fermi surface X (kF) = 0. The solutions are: (1) w1 = 0—the quasiparticle band; (2) w2 = — W^j 1 + Am is the ARPES replica band; (3) the inverse ARPES replica band w3 = W^J 1 + Am .The single-particle spectral function is A (kF, w, T = 0) = ^3= 1(A^p) S (w — wj), where A{/p are the spectral weights. For the quasiparticle band wx one obtains A1 = (1 + Am)— 1, while for the replica bands at w2 and w3 one has A2 = A3 = (A^2)(1 + Am )—1. The ratio of the intensities at T= 0ofthe w2 replica band and quasiparticle band wx is given by

A2 (kF, w, T = 0) = Am (4)

A1 (kF, w, T = 0) 2'

It is necessary to mention that at finite T (> 0) this ratio is changed as found in [5]. In that case ST (k » kF, w, T ^ 0) » Am/(w + W) which gives the quasiparticle and replica5 band w(T) = W(—1 + yj 1 + 4Am)/2 and w(2T) = —W (1 + yj 1 + 4Am)/2 and (A2/A1)T = Am. This intriguing difference ofthe T = 0 and T ^ 0 results for (A2/A1) in the EPI-FSP theory, (A2/A1)T = 2 (A2/A1)0, is due to the sharpness ofthe Fermi function nF (£k+q) entering in Sepi (k, w)—see equation (22) in appendix (see footnote 5).

We stress that, the ARPES measurements of A2 /A1 in 1ML FeSe/SrTiO3 were done at finite temperatures (T ^ 0) andinthe k = 0pointwith £ (k = 0) ~ —50 meV which gives (A2/A1)T » 0.15-0.2 [4,19].According to the theory and experiments in [4,5] one obtains Am™ » 0.15-0.2. Below we show, that Am can be also extracted from the formula equation (6) for TcQ » 100 K,whichgives A5Tc0) » 0.18. The latter value is in a good agreement with A from ARPES (see footnote 5). If we put this value in equation (4) one obtains that at T = 0 andat k = kF one has (A2/A1) ~ 0.1. From this analysis we conclude that the ARPES measurements at kFand T = 0 should give the similar ratio as at k = 0. The calculated ARPES spectra at T = 0 K and at k = kF give Aw = |w2 — w1| = Wy/1 + Am, while the experimental value is Dw » 100 meV.For A<ARPES) » 0.2 it gives, that the optical phonon energy is ofthe order of W » 90 meV.

4. The superconducting critical temperature Tc0 and gap A0

In the weak coupling limit (Am ^ 1) of the Eliashberg equations with qc vF < pTcQ (vF is the Fermi velocity) the linearized gap equation (near Tc0) is given by

, Vepi (q, wn — wm) A (k + q, wm) q m w m + £2 (k + q)

A n NT V^V^ epi \H Vk T q, ^mJ ,CN A(k, wn) = Tc0 -2-—---, (5)

where wn = pTcq (2n + 1), V^i (q, Wn) = ge2pi (q)(2W/(Wn + W2)), Wn = 2pTcq ■ n.For W2 > (pTcq)2 one has Vepi (q, Wn) » Vepi (q, 0) = 2ge2pi (q)^HW. In the strong FSP limit when (qc vF)2 ^ (pTcQ)2 the highest value of A (k, wn) is reached at k = kF (£ (kF) = 0 in equation (5). The solution A (k, wn) is searched in the standard square-well approximation A (kF, wn) » AQ = const .In leading order with respect to (TcQ/W) ^ 1 one obtains TC0 [2,3]

1 fi/pTco 1

*»P12^(q)>, Trjm+v■ (6)

For W2 > (pTcq)2 this gives %q = (Vepi (q) )q/4 » (1/16p )(aqc)2 (2gQ2/W), where a is the Fe-Fe distance. Note, that Tc0is mass-independent (aO = 0), while aO = 1/2 isfoundin [5]. The small isotope-effect can be also a smoking-gun experiment for the EPI-FSP pairing mechanism in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2). From equation (16) in the appendix it is straightforward to obtain the energy gap A0 = 2^. Note, that 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2) is a 2D system and TcQ ~ qc2, while in the d-dimensional space one has TcQ ~ qc d. This means, that the EPI-FSP mechanism of superconductivity is morefavorable in low-dimensional systems (d = 1,2) than in the 3D one. Since high Tc cuprates are also quasi- 2D systems, where strong correlations make a long-ranged EPI, it means that the EPI-FSP mechanism of pairing maybe also operative in cuprates [2]. Note, that in estimating some semi-microscopic parameters we shall use as an etalon-value TcQ » 100 K, while in real systems Tc ~ (60-80) K < TcQ is realized. However, Tc0 is the mean-field value obtained in the Migdal-Eliashberg theory, while in 2D systems it is significantly reduced by phase fluctuations— to the Berezinski-Kosterliz-Thouless value. There is an additional reduction of Tc0 (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 Am = (Vepi (q) )q/ 2W in 1ML FeSe/SrTiO3 from the value of Tc0. Then for the etalon-value TcQ ~ 100 K onehas (Vepi (q) )q » 33 meV and A(n[co) » 0.18. Since, A<Arpes) » A(n[co) the consistency of the theory is satisfactory. Note, if one includes the wave-function renormalization effects (contained in Z (iwn) > 1) then in the case (TcQ/W) ^ 1 and for the square-well solution for A (kF, wn), Tc0is loweredto TcP = TcQ/ Z2 (0),where Z (0) » 1 + Am [3]. This means, that the nonlinear corrections (with respect to Am) in Tc0 and A0 [5,23] should be inevitably renormalized by the Z-renormalization.

Let us estimate the parameters (e||, e±, qeff, hQ) which enterin Tc0. In order to reach TcQ ~ 100 K (and (Vepi (q) )q = 4Tc0 ~ 400 K » 33 meV) then for aqc » 0.2 and W » 90 meV one obtains gQ » 0.7 eV. Having in mind that g = (2pnd eqeff/ef )(H/M W)1/2 and that the zero-motion oxygen amplitude is (H / MW)1/2 » 0.05 A and by assuming that nd » a/ sc, sc = a2, a = V2 a » 4 A , qeff ~ 2e, a > 1,thenin

5 It turns out that in the first arXiv version of our paper we overlooked the fact that in [4,5] they study the problem at finite temperature T ^ 0 while our study is limited to T = 0. This means that both, the T ^ 0 and T=0, results are correct.

order to obtain g0 » 0.7 eV thevalueof ef must be small, i.e. ee]ff ~ 1.Since aqc = (aj h0)^j ef /e^1 ~ 0.2

andfor (a/h0) ~ 1 itfollows eff ~ 30. Note, that in SrTiO3 thebulk s islarge, e ~ 500-104. So, if Tc0in 1MLFeSe/SrTiO3 is due solely to the EPI-FSP mechanism, then in the model where the oxygen vibrations in the single dipolar monolayer TiO2 are responsible for the pairing potential the effective dielectric constants ef, e||ff arevery different from the bulk values inSrTiO3 (in 1MLFeSe/TiO2 onehas e ^ 260 [6]). This is physically plausible since for the nearest (to the FeSe monolayer) TiO2 dipolar monolayer there is almost nothing to screen in the direction perpendicular to FeSe, thus making eff ^ eeff ^ ebulk .Note, that for the parameters assumed in this analysis and for Tc0 » 100 K one obtains rather large bare pairing potential Ve0pi » 10 eV .This means that in absence of the FSP in EPI and for the density of states of the order N (EF) ~ 0.5 (eV)—1 (typical for the Fe-based superconductors) the bare coupling constant A°pi = N (EF) Ve0pi would be large, A°pi ~ 5. We stress that the above theory is also applicable to recently discovered 1ML FeSe/TiO2 [6].To conclude, the high Tc0in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2) is obtained on the expense of the large maximal EPI coupling V°pi, which compensates smallness ofthe (detrimental) phase-volume factor (aqc)2.

5. Effects of impurities on Tc0

In clean systems with the EPI-FSP mechanism of superconductivity Tc0 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. Tc0 is reduced and the Anderson theorem is violated. This may have serious repercussions on the s-wave superconductivity in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2) where Tc0 may depend on chemistry. Then by using equation (19) from appendix one obtains Tc(s)

To p2r

(i+2)- y (i)

where p = T/pTc, T = pnjN (EF) u2, ni is the impurity concentration and u is the impurity potential. Let us consider some limiting cases: (1) for T < pTc(s) one has Tc(s) » Tc0 [1 — 7( (3) T/p3Tc0]; (2) for T ^ pTc one has T(s) » (T/2p) exp (—pT/4T0),i.e. Tc(s) never vanishes. This means that in the EPI-FSP systems the nonmagnetic impurity scattering is pair-weakening for the s-wave superconductivity.

In case of d-wave superconductivity the solution of equation (20) in limiting cases is: (1) T(d) » Tc0 [1 — 2T/pTc0] for T < pTc(d). We point out that the slope — dTc(d)/d (T) = 2/p is smaller than the slope for the standard d-wave pairing, where —dTc/d (T) = p/4. (2) For T > T(cFSP) » (4/p) Tc0 onehas Tc(d) = 0, i.e. the effect of non-magnetic impurities is pair-breaking. Note, that TcFSP) > Tcr (=(2/p) T0). These two results mean, that in the presence ofnon-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 Tc dependence on non-magnetic impurities might be an important test for the EPI-FSP pairing in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2).

Finally, it is worth of mentioning, that the real isotope effect in Tc0 of 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2) might depend on the type of non-magnetic impurities. If their potential is also long-ranged—for instance, due to oxygen defects in the TiO2 dipole layer, then there is a FSP in the scattering potential, i.e. u^p (q) » u2S (q). Then, such impurities affect in the same way s-and d-wave pairing and they are pair weakening, as shown in [3]. Namely, one has (a) Tc(s,d) » Tc0 [1-4 TF/49Tc0] for TF < pTc,where TF = Jnju; (b) Tc(s,d) » 0.88W exp (—pTF/4T0),for TF ^ pTc .There are two important results: (1) There is a nonanalicity of TF ~ „Jnj in ni; (2) there is a full isotope effect in the 'dirty' limit TF ^ pTc,i.e. ao = 1/2,since T(s,d) ~ W ~ M—1/2. We stress, that ifthe full isotope effect would be realized experimentally in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2),then this does not automatically exclude the FSP-EPI mechanism of pairing, since it maybe due to impurity effects. In that case the nonanalicity of TF in ni might be a smoking-gun effect.

6. Internal pair fluctuations reduce Tc0

The EPI-FSP theory, which predicts a long-range force between paired electrons, opens a possibility for a pseudogap behavior in 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2). As we have discussed above, the EPI-FSP theory predicts a non-BCS dependence ofthe critical temperature To,i.e. Tc0 = (Vepi (q) )q/ 4. However, this mean-field (MFA) value is inevitably reduced by the phase and internal Cooper pair fluctuations—which are characteristic for systems with long-range attractive forces. Namely, in MFA the order parameter A (x, x')(=V (x — x') (y (x') y (x))) depends on the relative (internal) coordinate r = x — x' and on the center ofmass R = (x + x')/ 2,i.e. A (x, x') = A (r, R). In usual superconductors with short-range pairing potential one has Vsr (x — x') » V0 6 (x — x') and A (r, R) = A (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 A (r, R) on internal degrees of freedom (on r). The interesting problem of the internal fluctuations in systems with long-range attractive forces in 3D systems was studied in [24]. 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 the pseudospin operators (in this approximation first done by P. Anderson the single particle excitations are not included)

H = £24 Sks — (1/2) £Vk—k (Sk+Sk— + S+S—, (8)

ks k,k'

where Sjs = (c^c^ — c—c— ^ — 1)/2, S+ = c^ c—[24]. This is a Heisenberg-like Hamiltonian in the momentum space. In case of the s-wave superconductivity with short-range forces VSr (x — x') » VQ S (x — x') one has Vj—j = const and the pairing potential is long-ranged in the momentum space. In that case it is justified to use the mean-field approximation H ^ Hmf = —^k hkSk with the mean-field hj = — 2£kz + Vj—k' (Skx + Sky). The excitation spectrum (with respect to the ground state) in this system has a gap, i.e. E (k) =

2yj£k + Ajj where the gap Aj is the mean-field order parameter defined by Aj = Vj—j (Sk'). However, 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/SrTiO3 (and 1ML FeSe/TiO2) one has Vj—j = VQ exp { —|k — k'|/qc} with qc ^ kF, and the excitation spectrum is boson-like 0 < E (k) < 2^+ Ajj (like magnons 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. The internal fluctuations effect reduces Tc0 to Tc. For instance, in 3D systems with qc £0 ^ 1 [24] one has Tc » (qc £0) TcQ ^ TcQ, where the coherence length £0 = vF/pA0 and A0 = 2Tc0. It is expected, that in the region Tc < T < TcQ thepseudogap (PG) phase is realized. However, in 2D systems,like 1ML FeSe/SrTiO3 (and 1ML FeSe/TiO2), there are additionally phase fluctuations reducing Tc further to the Berezinskii-Kosterliz-Thouless value. We stress, that recent measurements of Tc in 1ML FeSe/SrTiO3 by the Meissner effect and resistivity (p (T)) give that Tc(p) < Tc(M) what maybe 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/SrTiO3 and 1ML FeSe/TiO2.

7. Summary and discussion

In the paper we study the superconductivity in the 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2 sandwitch-structure, which contains one metallic FeSe monolayer grown on the substrate SrTiO3, or rutile TiO(2100) . 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 EF. The top of the hole-like band at the point r lies 60-80 meV below EF 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 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2 is realized in the FeSe monolayer with Tc ~ (60-100) K. The decisive fact for making a theory is that the ARPES spectra show sharp replica bands around 100 meV below the quasiparticle band, what is approximately the energy of the oxygen optical phonon W » 90 meV .The analysis of the superconductivity is based on the semi-microscopic model— first proposed in [4,5], where it is assumed that a TiO2 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 < qc ^ kF) with the non-singular EPI coupling gepi (q) = gQ exp { — q/qc}. The EPI-FSP theory is formulated first in [1] for strongly correlated systems, while its extreme case with delta-peak 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 the experimental results [4,5]. In the following, we summarize the main obtained results ofthe EPI-FSP theory and its relation to the 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2 sandwitch-structures.

(1)—The mean-field critical temperature Tc0 in s-wave and d-wave pairing channels is degenerate and given by TcQ = (Vepi (q) )q/ 4 » (1/16p)(aqc)2VeQpi, where V^i (q) = 2ge2pi (q)/HW and the maximal pairing potential

VeQpi (°Vepi (q = 0)) = (2gQ2/W).On the first glance this linear dependence of Tc0on Ve(Q seems to be favorable for reaching high Tc0—note in the BCS theory Tc0 is exponentially dependent on Ve(p) and very small for small N (Ef) VeQpi. However, for non-singular gepi (q) when gepi (q = 0) is finite, Tc0 is limited by the smallness ofthe phase-volume effect, which is in 2D systems (such as 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2) proportional to (aqc)2 ^ 1. In that sense some optimistic claims that the EPI-FSP mechanism leads inevitably to higher Tc0—

than the one in the standard Migdal-Eliashberg theory, are not well founded for non-singular EPI potentials. This holds especially for 3D systems, where Tc0 ~ (aqc)3 and T^-* ^ Tc(0d) for the same value of V°?pi. However, higher Tc0 (with respect to to the BCS case) can be reached by fine tuning of aqc and Vpi. This is probably realized inHTSCcupratesandwith certainty in 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2. The weak-coupling theory predicts the superconducting gap to be Ao = 2Tc0 and for Tc0 ~ 100 K one has Ao ~ 16meV,what fits well the ARPES experimental values [4,6]. Note, in order to reach Tc0 = 100 K for aqc » 0.2 a very large maximal EPI coupling Ve0pi » 10 eV is necessary. For N (EF) ~ 0.5 (eV)—1 the maximal coupling constant would be rather large, i.e. A°pi (=N (EF) Ve0pi) » 5. Note, that V°pi is almost as large as in the metallic hydrogen under high pressure p ~ 20 Mbar, where Tc0 » 600 K with large EPI coupling constant A0pi » 6—this important prediction is given in [25]. In real 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2 materials the contribution of another pairing mechanism, which exists in the FeSe film in absence ofthe substrate and is pronounced in the s-wave channel with Tc0 » 8 K, triggers the whole pairing to be s-wave. The latter only moderately lowers the contribution of the EPI-FSP pairing mechanism. The existence of the sharp replicabands in 1ML FeSe/SrTiO3 and 1UCFeSe/TiO2 and large value of V°pi imply inevitably that the EPI-FSP pairing mechanism is a serious candidate to explain superconductivity in these materials. We stress, thatin 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2 high Tc0 is obtained on the expense of the large maximal EPI coupling V°pi, which compensates the small (detrimental) phase volume factor. Other possibilities to increase Tc0 is to search for exotic systems with singular EPI potential, with(aqc)2g2. q ^ 0) ^ ¥, which might be illusory.

(2)—The semi-microscopic model proposed in [4,5] and refined partly in this paper, contains phenomenological parameters, such as nd—the number of dipoles per unit cell, qeff—an effective dipole charge, ef, ef —effective parallel and perpendicular dielectric constants in SrTiO3 (and TiO2) near the interface, respectively. For Tc0 » 100 K and by assuming aqc » 0.2, qeff » 2e, n¿ ~ 2/unit-cell makes ef ~ 30,

ef ~ 1. These values, which are physically plausible, are very far from s in the bulk SrTiO3, where e ~ 500-104 (and e ^ 260 in the rutile TiO2). We point out that our estimation of these parameters is based on an effective microscopic model where the bulk SrTiO3 is truncated by a monolayer (1ML) made of TiO2 [4]. In reality it may happen that the bulk SrTiO3 is truncated by two monolayers (2ML) of TiO2, as it is claimed to be seen in the synchrotron x-ray diffraction [26] .This finding is confirmed by the LDA calculations in [26], which show that for the 2ML TiO2 structure: (i) electrons are much easier transferred to the FeSe metallic monolayer and (ii) the top of the hole band is shifted far below the electronic Fermi surface than in the 1ML model. If the 2ML of TiO2 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. qe(2ML) > qeffML) and since Tc0 ~ qf the 2ML model may give rise to higher critical temperature.

(3)—The isotope effect in Tc0 should be small (aO ^ 1/2) since in leading order one has Tc0 ~ Vp and aO = 0,where V°piis mass-independent. This is contrary to [5] where aO = 1/2. The next leading order gives ao ~ (Tc0/W) < 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, for non-magnetic impurities with a FSP in the scattering potential the full isotope effect aO = 1/2 is restored in the 'dirty' limit (TF ^ pTc), since Tc(5,d) ~ W ~ M-1/2. In that case, the nonanalicity of Tc(5,d) with respect to the impurity concentration ni, might resolve the question—what kind of pairing is realized in 1ML FeSe/SrTiO3 and 1UCFeSe/TiO2 —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. A = A (r, R).The internal pair fluctuations reduce additionally the mean-field critical temperature so that in the interval Tc < T < Tc0 a pseudogap behavior is expected.

(6)—The EPI self-energy in the normal state at T = 0and X (kF) = 0 is given by Sepi (k, w) » — Am w/ (1 — (w/ W)2),where Am = (V¡pi (q) )q/ 2W,whichfor G—1 (k, w) = 0 gives the dispersion energy of the quasiparticle band w1 = 0 and the replica bands w2 and w3. The ratio of the ARPES intensities of the replica band w2 and the quasiparticle band w1 at T = 0 and at the Fermi surface (k = kF) is given by R (T = 0, kF) = (A2/A^ = Am/2. This means, that for Am ~ 0.2 the experimental value of R (T = 0, kF) shouldbe

(A2/A1) » 0.1. This ratio is slightly smaller than the experimental value R (T ^ 0, k = 0) ~ 0.15-0.2 measured in [4,19].

(7)—Since the coupling constant Am is mass-dependent, Am ~ M1/2 then the isotope effect in various quantities, in 1ML FeSe/SrTiO3 and 1ML FeSe/TiO2 systems, maybe a smoke-gun experiment in favor of the EPI-FPS theory. To remind the reader: (i) Tc0 is almost mass-independent; (ii) the self-energy slope at w ^ W is mass-dependent, (—d£/dw) ~ M1/2; (iii) the ARPES ratio R ofthe replica band intensities is mass-dependent,

R ~ M1/2. Note, the properties (i)-(iii) are just opposite to the corresponding ones in the BCS- andMigdal-Eliashberg-theory for the standard EPI mechanism of pairing.

Concerning the role of EPI in explaining superconductivity in 1ML FeSe/SrTiO3 there were other interesting theoretical proposals. In [27] the EPI is due to an interaction with longitudinal optical phonons and since W > EF the problem is studied in the anti-adiabatic limit, where Tc is also weakly dependent on the oxygen mass. In [28] 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 Tc. In [29] the intrinsic pairing mechanism is assumed to be due to J2-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 Tc superconductivity in this system.

Finally, we would like to comment some possibilities for designing new and complex structures based on 1ML FeSe/SrTiO3 (or 1ML FeSe/TiO2) asabasicunit. The first nontrivial one is when a double-sandwich structure with two interfaces is formed, i.e. SrTiO3/1ML FeSe/SrTiO3 (or TiO2/1ML FeSe/TiO2). Naively thinking, in the framework of the EPI-FSP pairing mechanism one expects, in an 'ideal' case, doubling of Tc0, 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 qc vF < pTc0 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 Tc0. It needs very delicate technology to control the concentration of oxygen vacancies and appropriate charge transfer at both interfaces. However, eventual solutions ofthese problems might give impetus for superconductors with exotic properties. For instance, having in mind the above exposed results on effects ofnon-magnetic impurities on Tc0, then by controlling and manipulating their presence at both interfaces one can design superconducting materials with wishful properties.

Acknowledgments

The authors are thankful to Rados Gajic for useful discussions, comments and advises related to the experimental situation in the field. We highly appreciate fruitful discussions with Steve Johnston and Yan Wang on ARPES of the replica bands at finite temperature, and on the microscopic parameters of the theory. The authors are thankful to Michael V Sadovskii for discussions on various theoretical aspects of the Fe-based superconductivity.

Appendix

A.1. Migdal-Eliashberg equations in superconductors

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 Gn (k, wn) = — [iwnZn (k) + Xn (k)]/ Dn (k), Fn (k) = — Zn (k) An (k)/ Dn (k), respectively, where Dn (k) = [wnZn (k)]2 + X." (k) + [Zn (k) An(k)]2 (wn = pT (2n + 1)).Here, Zn (k) is the wave-function renormalization defined by iwn (1 — Zn (k)) = (S (k, wn) — S (k, —wn))/2, where the self-energy S (k, wn) = Sepi (k, wn) + Simp (k, wn) describes the EPI and impurity scattering, respectively. The energy renormalization is Xn (k) = X (k) + xn (k), Xn (k) = (S (k, w„) + S (k, — w,))/ 2 and An (k) isthe superconducting order parameter

Zn (k) = , + T E Vf - ,D "If" n-Zn"k'), (9)

Wn k',„' Dn' (k')

X, (k) = X (k) — T E Vf <" — D ^ k) Xn V) , (1

k',n' Dn'(k )

z,(k)a,(k) = tE — "', kn- jj)Zn<k''An(k'), d 1)

k'n D" (k )

where Veff (n — n', k — k') = Vepi (n — n', k — k') + Vmp (n — n', k — k'), Vepi (n — n', k — k') = —g2. (k — k')Pph(k — k', wn — wn') and Vmp(n — n', k — k') = Snn'niU2(k — k')/T. Here, thephonon Green's function in the Einstein model with the single frequency O is given by £>ph(k — k', wn — wn') = —2W/ (W2 + (wn — wn' )2), while the impurity scattering is described in the Born-approximation. Here, ni isthe impurity concentration and u (k — k') is the impurity potential. To these three equations one should add the

equation for the chemical potential i.e. N = £ Gn (k, wn; m) = const. However, in the following we study only problems where a (small) change of ^ 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 ^ plays an important role, etc.

Note, that in the case of systems with very large Fermi energy EF and with an isotropic EPI (Zn (k) ° Zn, Xn (k) ^ 0) one integrates over the energy Xk' by introducing the density of states at the Fermi surface N(0), i.e. £k'(•••) ^ N (0) J*°° (...) d£k'. This leads to standard Migdal-Eliashberg equations

Z = 1 + pT £ N (0) Veff (n - n') Wn'Zn' (1

Wn n' yj(Wn'Zn' )2 + D2n' 7 D N (0) Veff (n — n', Zn' An'

Zn An = pT£----. (13)

n' V(Wn'Zn ')2 + A2'

In the case of strongly momentum-dependent EPI-FSP, where Vepi (n — n ', q) is finitefor |q| < qc ^ kp,the Migdal-Eliashberg equations are given by

Zn(X) = 1 + T E(Vepi(n — m, q))q WmZm(X) , (1

Wn m Dm (X)

Xn(X) = X(k) — T£(Vepi(n~Z, q))qXm(X), (1

m Dm(X)

Zn (X) An (X) = T £<Vepi (n — m, q) )qZmDAm^). (1

m Dm(X)

A.2. Effects of non-magnetic impurities on Tc0 in the EPI-FSP theory

In this paper we study the superconductivity which is due to EPI-FSP of the Einstein phonon with W2 ^ (2pTc0)2. In that case (Vepi (n — m, q)) » (Vepi (0, q) )q = (2\$^ (q) )qjW and the contribution to Zn (X) is ~Am = (Vepi (0, q) )q/ 2W. Since in the weak coupling limit one has Am ^ 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 Tc0 is studied in the standard model with weakly momentum dependent impurity potential u (k — k') » const .In that case Zn (X) contains the impurity term only. After the integration of the impurity part over the energy X' in equations (9)-(11)—see [30], and for X = 0 (since in that case An (X = 0) is maximal) one obtains (Dm (X) » wn Z2) for the s-wave pairing

Zn = 1 + , (17)

Zn An = Tc ^(Vepi (n — m, q) )q m m + -—-An. (18)

m Dm (0) |Wn|

Note, the the second term on the right side cancels the same term on the left side. In the square-well approximation An (X) » A one obtains an equation for impurity dependence o f Tc(s)(r) for the s-wave superconductor

1 = Tc(s)(Vepi(0, q))q£-2-, d

m wnZn

We point out, that in the case of the d-wave superconductivity A = A (j) is angle dependent on the Fermi surface and changes sign. In that case the last term in equation (18) for A should be replaced by (A (j) = 0 giving equation for T(d)

1 = Tc(d) (Vepi (0, q) )q £-2^2 . (20)

m WnZn

Note, Zn versus Zn renormalization for the s-wave and d-wave superconductivity, respectively. A.3. 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

Sepi(k, Wn) = — T £ ge2pi (q)©ph(q, Wm)G(k + q, Wn — Wm), (21)

where wn = pT(2n + 1) and Wm = 2pmT, X>ph(q, Wm) = —2fi/(W2 + fi^n), G (k, wn) = 1/(iwn — Xk). In the following we assume that W ^ T and neglect the bosonic distribution function nB (W) » 0.Bydefining

Vpi (q, 0) = 2ge2pi (q)/W and after summation over Wm in equation (21) one obtains (note that

Vpi (q, 0) = Vpi(-q, 0)

"F (4+q) , 1 - nF(Xk+q)

Sepi (k, W") = W ^Vepi (q, 0) X 2 q

iwn - Xk+q + W iW" - Xk+q - W

Let us calculate Sepi at kF and at T = 0. Since qvF = qvF cos 0 and by taking into account that nF (£kF+q) = 1 for cos 0 < 0, nF (£kF+q) = 0 for cos 0 > 0 one obtains for qc vF ^ W

Sepi (k, wn) = - Am-W.—TT, (23)

1 - (W)

where Am = (Vpi (q) )q/2W and (Vpi (q) )q = Nsc (2p)-2 J d2qVepi (q, 0), sc is the surface oftheFeSe unit cell.

Note, that Vepi (q, 0) = 2ge2pi (q)/W and gepi (q)(=(g0/VN) e-q/qc) so that N disappears from

Am (=(Vepi (q) )q/ 2W). After the analytical continuation iwn ^ w + id oneobtains Sepi (k, w) inequation (3).

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