The phonon mechanism explanation of the superconductivity dichotomy between FeSe and FeS monolayers on STO and other substrates
TThe phonon mechanism explanation of the superconductivity dichotomy between
F eSe and
F eS monolayers on
ST O and other substrates.
Baruch Rosenstein and B. Ya. Shapiro Electrophysics Department, National Yang Ming Chiao Tung University, Hsinchu 30050, Taiwan, R. O. C ∗ Physics Department, Bar-Ilan University, 52900 Ramat-Gan, Israel † It was observed recently (K. Shigekawa et al, PNAS 116, 2470 (2019)) that while monolayer ironchalcigenide
F eSe on SrT iO ( ST O ) substrate has a very high critical temperature, its chemicaland structural ”twin” material
F eS/ST O has a very low T c if any. To explain this the substrateinterfacial phonon model of superconductivity in iron chalcogenides is further developed. The mainglue is the oxygen ion Ω s = 60 mev vibrations longitudinal optical (LO) mode. The mode propagatesmainly in the T iO layer adjacent to the monolayer (and genrally present also in similar highlypolarized ionic crystals like BaT iO , rutile, anatase). It has stronger electron - phonon couplingto electron gas in F eSe than a well known Ω h = 100 mev harder LO mode. It is shown thatwhile (taking into account screened Coulomb repulsion effects) the critical temperature of F eSe on ST O and
T iO is above 65 K , it becomes less than 5 K for F eS due to two factors suppressing theelectron - phonon coupling. The effective mass in the later is twice smaller and in addition thedistance between the electron gas in
F eSe to the vibrating substrate oxygen atoms is 15% smallerthan in
F eS reducinng the central peak in electron-phonon interaction. The theory is extended toother ionic insulating substrates.
PACS numbers: PACS: 74.20.Fg, 74.70.Xa,74.62.-c a r X i v : . [ c ond - m a t . s up r- c on ] F e b Introduction.
Several years ago a group of 2D high T c superconductors ( T c > K ) was fabricated by deposition ofa single unit cell (1UC) layer of F eSe on insulating substrates
SrT iO ( ST O ) [1],
T iO (both rutile[2] and anatase[3])and[4] BaT iO . The 3D parent iron chalcogenide ( Se, S, T e ) are unconventional superconductors (s ± wave symmetry)with modest T c = 5 − K . Band structure is similar to that of iron pnictides suggesting a ”nonconventional” spinfluctuation (SF) pairing mechanism within the F eSe layer[5]. However strong O → O isotope substitution effect[6]in 1UC F eSe/ST O indicates that superconductivity is at least enhanced by the electron - phonon interaction[7–11](EPI). The relevant phonon is the oxygen ions vibrations in the interface layers. The role of the insulating substratetherefore clearly extends beyond the efficient monolayer charging[12].Recently the second monolayer iron chalcogenide,
F eS , on
ST O was synthesized[13] by the topotactic reaction andmolecular-beam epitaxy. In both iron chalcogenides Fermi surface consists of two nearly coincident pockets aroundthe M point of the Brillouin zone (BZ), while the electron pocket at Γ point of the parent material sinks (about80 meV ) below Fermi level[14]. Despite the fact that (i) the bulk T c , (ii) the 2D electron gas (2DEG) including spindynamics, and (iii) EPI in F eSe and
F eS are quite similar, superconductivity in
F eS/ST O was not observed[13] atleast at temperatures above 10 K . This came as a surprise and even was termed by the authors ”a dichotomy” that”strongly suggests that the cross-interface electron–phonon coupling enhances T c only when it cooperates with thepairing interaction inherent to the superconducting layer”. This interpretation rules out theories in which the EPI isthe major cause of the tenfold enhancement of T c in F eSe/ST O .However despite the above superficial observations there are two important differences between the two monolayers.First the ARPES measurement[13] clearly demonstrates that the effective mass m ∗ that is twice larger in F eSe thanin
F eS . In addition the scanning transmission electron microscopy image of
F eS/ST O reveals that distance fromthe 2DEG gas in
F eS to the vibrating substrate oxygen atoms, see Fig.1, is d = 5 . A , larger than the correspondingdistance[15] in F eSe/ST O , d = 4 . A . These two observations are in direct contradiction with statements (iii) abovethat the EPI is similar in two systems. Indeed since the EPI has a central peak in scattering (SCP) that exponentiallydepends on d , one would expect reduced EPI strength λ in F eS . The density of states in 2DEG is m ∗ /π , alsoreducing λ in F eS . On the contrary if the in - plane SF mechanism of pairing is similar and dominant, absence ofsuperconductivity in
F eS/ST O poses a problem for this explanation.In this letter the dichotomy between the iron chalcogenides monolayers
F eSe and
F eS is addressed theoreticallyin the framework of the phonon mechanism. The interfacial phonon is considered as the dominant superconductivity”glue” overcoming (the screened) Coulomb interaction. Systems of various effective masses m ∗ , the 2DEG layer -substrate spacing d and dielectric constant of the substrate material are considered. We conclude that the dichotomybetween superconductivity in F eSe/ST O and
F eS/ST O is resolved within this framework.
Model.
As mentioned above most of the theories of high T c in F eSe monolayers[11, 16] are a variant of theincipient band SF model with the phonon pairing ”boosting” T c from 40 K − K up (we are not aware of a similarconsiderations for the F eS ). The EPI is represented by an interfacial mode of high frequency Ω = 100 mev close tothat of the Fuchs - Kliewer modes (FK), observed via high resolution electron energy loss spectroscopy[17]. The FKare vibrations of the substrate oxygen atoms in the direction z perpendicular to the interface, see the blue arrow inFig.1. The EPI strength λ = 0 . T c =47 K to T c = 65 K . If the spin fluctuations were switched off ( U = 0), one would require at least λ = 0 . ST O and the interface modes has been studied in the framework of the DFT [20]. A simple phenomenological model ofionic crystal allowed us[10] to identify two longitudinal optical (LO) surface modes that have the strongest couplingto 2DEG in a sense that their exchange produces effective attraction of electrons in the lateral ( x − y ) direction.These are the T i − O stretching (along the surface, see black arrow Fig.1) mode comparable in energy of the FK,Ω LOst = 100 mev , and a lower frequency
T i − O − T i bending (still along the interface direction, see dark green arrow)mode Ω
LOb = 60 mev . Their matrix elements with the 2DEG electrons are about the same. All the other modes(including phonons in the
F eSe layer itself) have negligible matrix elements.Since the phononic glue comes mostly from the
T iO substrate separated from the 2DEG by the (minimal) distance d , see Fig. 1, the EPI coupling exhibits the exponential forward scattering peak [9]: g ( k ) ≈ πa e − kd . (1)Here a is the lattice spacing, see Fig.1. The T iO layer generally appears in all the substrates[15] (rutile, anatase, ST O, BaT iO ) as the first interface oxide layer (in addition to ST O ). The phonon exchange generate effective electron
FIG. 1. Interfacial phonon modes. Oxygen ion’s vibrations in the
T iO substrate layer ( T i - silver, O - red). The displacementin derection perpendicular (z axis, blue arrow) to the one unit cell thin F e (brown) - chalcogenite (
Se, S, T e - green) layer areassociated with FK modes. The two modes most relevant for the phonon mediated pairing longitudinal optical modes are the
T i − O stretching mode (shown by black arrow) and the T i − O − T i bending (dark green arrow). The next layer Bi (cyan) - O (dark red) influencing the interfacial phonon frequency is also shown. Direction of the vibration wave is assumed to be alongthe x direction. - electron attraction dynamic ”potential” is V ph k ,n = − (cid:0) Ze (cid:1) M g k ω n + Ω s . (2)Here M and Z (cid:39) .
27 are the oxygen ion mass and the ionic charge respectively[21] and ω n = 2 πT n is the bosonicMatsubara frequency. It was shown in ref.[10] that the lower frequency bending mode (Ω LOb = 60 meV ) leads to larger λ = 0 .
23 than the stretching mode (Ω
LOst = 100 meV ) with λ = 0 .
07. Moreover the bending mode pairing alone is strongenough to mediate high T c above 47 K . This implies that the spin fluctuation contribution to pairing in the presentcase might be subdominant. This statement is not at odds with the understanding that the T c = 8 K superconductivityin bulk F eSe or F eS is due to SF, since there are two major differences between the bulk and 1UC. First the holeband at Γ in bulk disappears below Fermi surface and second the recent spin susceptibility measurement[22] frombulk to monolayer
F eSe signal of the spin is completely different. Therefore it will be neglected in the present work.In view of the exponential SCP, Eq.(1), the EPI pairing in
F eS/ST O is weaker than in
F eSe/ST O since the distancebetween 2DEG and the
T iO layer increases[13] by15%. This alone should reduce the EPI coupling. To describethe electron gas it is sufficient for our purposes to use a parabolic approximation for two M point bands of bothsystems, E k = k / m ∗ − µ . Effective masses are m ∗ F eSe = 3 m e and m ∗ F eSe = 1 . m e respectively, while Fermi energiesare µ F eSe = 60 meV and µ F eS = 30 meV (values for
F eS are deduced from the ARPES measurement[13]). TheFermi momentum k F = √ m ∗ µ is nearly the same. As mentioned above the reduced density of state also suppressesthe EPI pairing. As a result of the two facts for the weaker pairing in F eS/ST O one should take into account thepseudo-potential[23]. Coulomb repulsion in 2DEG (although effectively screened by the dielectric substrate[7] in bothmonolayers), might completely suppress superconductivity.The screened potential within RPA in the presence of the semi - infinite dielectric slab is V C k ,n = v C k ,n − v C k ,n Π k ,n ; v C k ,n = 2 πε ( ω n ) k , (3)where the (Matsubara) dielectric function inside the substrate reads[11]: ε ( ω ) = 12 (cid:26) ε ∞ + ( ε − ε ∞ ) Ω T Ω T + ω (cid:27) . (4)Dielectric constants will be taken as follows. The optical value is rather universal for all the substrates ( ST O , rutile,anatase) ε ∞ = 5 .
5, while the static ε varies from as high as ε = 3000 for SrT iO to ε = 50 for some anatasesamples). The (bulk) transverse mode frequency appearing in Eq.(4) is estimated using the Lydanne-Sacks-Tellerrelation Ω T = Ω LO (cid:112) ε ∞ /ε with Ω LO = 120 meV .The 2D Matsubara polarization function due the two nearly degenerate electron bands is:Π k ,n = − m ∗ π (cid:26) (cid:16)(cid:0) / iω n m ∗ /k (cid:1) − ( k F /k ) (cid:17) / (cid:27) . (5)The sum of two competing contributions the effective electron - electron interaction, V k ,n = V ph k ,n + V C k ,n , determinesthe superconducting properties of these systems.The STM experiments[24] demonstrate that the order parameter is gapped (hence no nodes) and indicate a weaklyanisotropic spin singlet pairing. Therefore we look for solutions for the normal and the anomalous Green’s func-tion of the Gorkov equations (derived for a multi - band system in ref. [10]), in the form (cid:68) ψ ρ k ,n ψ ∗ σ k ,n (cid:69) = δ σρ G k ,n , (cid:68) ψ σ k ,n ψ ρ − k , − n (cid:69) = ε σρ F k ,n ( σ, ρ are spin components). In terms of the gap function,∆ k ,m = T c (cid:88) p ,n V k − p ,m − n F p ,n , (6)linearized gap equation becomes (normal Green’s function not renormalized significantly at weak coupling), − T c (cid:88) l ,m V l ,n − m ( ω em ) + ( E l + q − µ ) ∆ l + q ,m = ∆ q ,n , (7)where fermionic Matsubara frequency is ω em = πT (2 m + 1). The angle (between l and q ) integration can be performedfor an conventional s-wave solution (observed in experiment[24]) leading to a simplified eigenvalue problem: T c m ∗ π (cid:88) m ω em (cid:40) (cid:0) πZe (cid:1) M f ph ( ω em / µ ) (cid:0) ω bn − m (cid:1) + Ω − f C (cid:32) | ω em | µ , (cid:12)(cid:12) ω bn − m (cid:12)(cid:12) µ (cid:33)(cid:41) ∆ m = ∆ n . (8)The integrals (over l ≡ | l | / k F ) for the phonon and Coulomb contributions are defined as, f ph ( z ) = (cid:90) l =0 e − (4 k F d ) l R ( z, l ) ; (9) f C ( y, z ) = π (cid:90) l =0 R ( z, l ) (cid:26) ε (4 E F y ) k F le + 2 m ∗ (cid:18) l − (cid:113) ( l + iy ) − l (cid:19)(cid:27) − ,where R ( z, l ) =Re (cid:0) z /l − i | z | − l (cid:1) − / . K K K K K K K K K K FeSe / STO FeSe X / STOFeS / STO m * / m e d / a FIG. 2. Superconductivity critical temperature as function of effective mass and the distance between the iron chalcogenitelayer and the interface
T iO (where the relevant phonon modes originate). The dielectric constant ε = 3000 is fixed torepresent SrT iO . Critical temperature is obtained when the largest eigenvalue of the matrix of the linear Eq.(8) is 1. This was donenumerically by limiting variable n to | n | < < ε < m e < m ∗ < m e , whilethe distance (in units of the lattice spacing a ) between the conducting layer and the vibrating oxygen atoms is1 < d/a < . Results.
The dependence of the critical temperature on the effective mass m ∗ and the distance between the 1UCiron chalcogenide and underlying T iO interface layer is given in Fig. 2. It explains the dichotomy between a veryhigh T c in F eSe/ST O and a very low T c (10 K or less) in F eS/ST O . An approximate location of the two cases isindicated by two circles. The dielectric constants are fixed on the
ST O values mentioned above. It demonstrates thatboth the reduction of the effective mass and (to a lesser degree) the distance d difference contribute to the suppressionof superconductivity in F eS/ST O . In addition a higher effective mass 1UC strained
F eSe epitaxially grown on
N b : SrT iO /KT aO heterostructures[25] is marked as F eSe X . For T c > K the dependence is approximatelylinear T c [ K ] = 18 m ∗ /m e − d (cid:104) ˚ A (cid:105) + 114.Critical temperature as function m ∗ of 1UC F eSe on ionic substrates with various dielectric constant is shownin Fig.3. The ratio d/a is fixed at 1 .
1. Dependence on the dielectric constant is due to screening of the Coulombinteraction. The pseudo - potential becomes important for low T c . High ε = 3000 ST O and two relatively low ε forms of T iO , rutile and anatase, are shown. Discussion and conclusions.
To summarize the interfacial LO phonon pairing theory in 1UC iron chalcogenides
F eCh ( Ch = Se, S, T e ) on polar insulator (
SrT iO , T iO ) substrates is presented including the Coulomb pseudo-potential effects. The LO modes originates in the T iO layer of the substrate adjacent to the 1UC F eCh . The theorypredicts three following tendencies leading to high critical temperature T c . To achieve high critical temperature one K K K K K K K K K FeSe / STO FeSe X / STOFeSe / rutileFeSe / anatase m * / m e l og ( ϵ ) FIG. 3. Critical temperature as function of effective mass and dielectric constant (logarithmic scale). The distance betweenthe iron chalcogenite layer and the interface
T iO d is fixed at 1 . a . Strongly dielectric material ST O and moderately dielectric
T iO forms rutile ( ε = 300) and anatase ( ε = 300) are marked. requires (i) small spacing between the electron gas inside the F eCh layer and the
T iO interfacial layer maximizingthe strength of the electron - phonon coupling, (ii) high effective mass of the electrons in F eCh maximizing DOS, (iii)large dielectric constant ε minimizing the Coulomb repulsion (pseudo-potential) effects. These three effects explainwhy F eSe/ST O has very high T c , while F eS/ST O has very low T c , if any. In addition it explains relative strengthof pairing in F eSe on BaT iO , rutile and anatase structures of T iO .Let us put the interfacial theory of superconductivity in iron chalcogenides on ionic crystals in a more generalframework of superconductivity in iron based materials. 3D pnictides like F eAs and 3D iron chalcogenides like theparent compounds
F eSe or F eS generally have two features. The superconductivity is not the ”plain” s - waveobserved[24] in 1UC
F eCh/T iO . It changes sign and is explained by the SF multiband model[5]. It is crucial thatin addition to an electron band at M there exists also an electron band at Γ. In addition typically one often observesorbital selective Mott transition further favouring SF pairing (usually s ± ) mechanism. It is not easy to modify thesemodels to the plain s - wave gap typical to low T c metals. The basic idea is still to utilize the hole pocket that is no about100 meV below Fermi surface[5, 11, 16] (the incipient band). Similar problem exists in explaining relatively high ( T c up to 48 K ) superconductivity in several 3D modifications of F eCh . These materials, including[26] metal intercalated(up to 48 K ) F eSe , A x F e − y Se ( A = K, Sb, Li ), and organic intercalations[27] like Li x ( N H ) y ( N H ) − y F e Se ,( Li, F e ) OHF eSe (up to 30 K ) and electric field induced superconductivity (48 K ) in F eSe [28] all exhibit s-wavepairing and only electron bands. It is plausible that bulk phonons also might provide a ”glue” for the s - wave pairing.Therefore the pairing glue for the three groups of superconducting materials might be different. They are interfacephonons for
F eSe/ST O , SF for iron pnictides and parent iron chalcogenides and either SF/3D phonon for intercalatediron chalcogenides.Note that often the T c enhancement in all three kinds of systems is attributed to ”charging”[12] of the conductinglayers by either electric field, intercalation (internal pressure). As the present work demonstrates, since in 2D thepairing depends strongly on density of states (on Fermi level), the charging argument is effective only for 3D electrongas . In 2D DOS depends on effective mass only, D ∝ m ∗ / (cid:126) . Charging mostly shifts the chemical potential and forfixed m ∗ increases DOS only in 3D: D ∝ m ∗ / µ / / (cid:126) . Acknowledgements.
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