Large electron-phonon interactions from FeSe phonons in a monolayer

We show that electron-phonon coupling can induce strong electron pairing in an FeSe monolayer on a SrTiO3 substrate (experimental indications for superconducting Tc are between 65 and 109K). The role of the SrTiO3 substrate in increasing the coupling is two-fold. First, the interaction of the FeSe and TiO2 terminated face of SrTiO3 prevents the FeSe monolayer from undergoing a shear-type (orthorhombic, nematic) structural phase transition. Second, the substrate allows an anti-ferromagnetic ground state of FeSe which opens electron-phonon coupling channels within the monolayer that are prevented by symmetry in the non-magnetic phase. The spectral function for the electron-phonon coupling (alpha2F) in our calculations agrees well with inelastic tunneling data.

these methods relies on a simplified deformation potential approximation, as in [22] since electron-phonon coupling matrix elements are difficult to obtain.
Here we show that making the potential on the iron atoms slightly more repulsive for electrons renormalizes the bands near the Fermi level and selects a ground state of FeSe consistent with most experimental data. More specifically, in this method (GGA+A), we empirically 1 replace the potential V r ( ) GGA within the semi-local density approximation (GGA) with The idea here is to mitigate empirically the fact that the GGA exchange-correlation potential is not the self energy without the second term in equation (1). We find that the detailed form of the dimensionless function > f r ( ) 0 is irrelevant for the computed physical properties of FeSe, as long as f(r) is peaked on the Fe atom (placed at r i ) and the extent of f(r) is comparable with the size of the iron atom d-orbital 2 . Next, for a fixed f(r), we tune the parameter A from 0 up to A c (>0) until 3 one of the properties of FeSe (here, occupied bandwidth of the M-point electron pocket) agrees with experimental data (compare blue and green curves in figure 1). Remarkably, using = A A c improves other salient properties of FeSe as well. For example, the gap (δE M in table 1) at the bottom of the M pocket, and the energy of the Γ band just below the Fermi level are improved in the GGA+A, as well as the peak positions in the density of states at 4 and 6 eV below the Fermi level 4 . Magnetic A A c (blue), and experimental results. We fit the different experimental data to a parabola (light [3], medium [4], and dark green [5]). Table 1. A comparison of the magnetic moment on the iron atom (μ), shear angle α (measured between the primitive unit cell vectors a and b), top of the Γ band (E Γ ) and bottom of the M band (occupied bandwidth, E M ) relative to the Fermi level, and the band splitting at the M point (δE M ) in GGA, GGA+A using = A A c , and from experiments [1,[3][4][5]17]. Parameter A is tuned to A = A c so that occupied bandwidth of the M-point electron pocket (E M ) agrees with experimental data. However, using A = A c significantly improves other properties of FeSe as well.

Bulk
Monolayer on SrTiO 3  1 This approach is similar in spirit to the empirical pseudopotential method from [23] and the semi-empirical method from [24].  4 In GW calculations from [21] these same peaks near 4 and 6 eV were found to agree well with the experiment.
properties are improved as well. Using the experimental crystal structure from [25] in both cases, the GGA+A predicts bulk FeSe to be nonmagnetic as in experiment, while GGA predicts large antiferromagnetically aligned magnetic moments μ on the iron atoms (favored by 0.5 eV per two Fe atoms over the non-magnetic ground state). Finally, the crystal structure is improved in the GGA+A case. A slight shear present in the experimental structure as in [1] (α <°90 ) remains in the GGA+A approach after the structural relaxation, while it disappears in the GGA calculations (α =°90 ).
In these and subsequent calculations we fixed the doping of FeSe monolayer to the level of 0.09 electrons per one Fe atom (as found in ARPES experiments). In the experiment, this doping likely occurs due to presence of oxygen vacancies in the SrTiO 3 substrate.
Our focus here is on the electron-phonon coupling and superconductivity in monolayer FeSe. The underlying origin of the success of the GGA+A is an interesting open question and is left for future studies. We only note here two points in favor of GGA+A. First, portion of the electron self-energy Σ ′ E r r ( , , ) that is missing in the semi-local density approximation is typically large only when | − ′| r r is comparable to the bond length [27], just as for the case of the form of f(r). Second, agreement between GGA+A and experiment is improved not only in monolayer FeSe studied here, but also in bulk KCuF 3 , LaNiO 3 , (La,Sr) 2 CuO 4 , SrTiO 3 (see supplement 5 ), and (Ba,K)Fe 2 As 2 [28].
Equipped with a better FeSe band structure and ground state than obtained from a standard GGA calculation, we are now in a position to compute the electron-phonon coupling strength in the FeSe monolayer. First we discuss the crystal structure of FeSe used in the electron-phonon calculation. Bulk FeSe consists of stacked, weakly interacting, layers of FeSe. Below 90 K these layers are observed to be slightly sheared as shown in figure 2(a) and discussed in [1] (shear is also present in GGA+A calculation, but not in GGA). This shear (nematic) distortion is conventionally described as primitive-tetragonal to base-centered-orthorhombic structural phase transition.
Since the FeSe layers in bulk are only weakly interacting, we expect that the tendency towards a shear distortion will be present even in an isolated single layer of FeSe. This is indeed what we find in the case of monolayer FeSe. Even if we epitaxially constrain the isolated monolayer FeSe unit cell to a cubic SrTiO 3 lattice, it still undergoes a local shear-like structural transition shown in figure 2(b) (again, only in GGA+A, not in GGA).
However, once FeSe is placed on a TiO 2 terminated SrTiO 3 substrate, we find that the interaction of Ti and Se atoms together with the epitaxial strain is able to stabilize FeSe to a nearly square arrangement (see figure 2(c) and supplement (see footnote 5)). A small remnant of the structural distortion present in FeSe is responsible for the electronic gap (δE M ) at the M point shown in figure 1 and in table 1. (An additional smaller component of the gap results from a built-in electric field between FeSe and SrTiO 3 , as discussed in [19].) In addition, in the FeSe monolayer on SrTiO 3 , an antiferromagnetic checkerboard ground state is preferred by 0.11 eV (per unit cell with two Fe atoms) within GGA+A over the non-magnetic one, despite the fact that the opposite is the case for bulk FeSe.
The main effect of the SrTiO 3 on the FeSe is the structural stabilization described above of a non-sheared and antiferromagnetic ground state. Selection of this ground state then affects the electronic and magnetic properties of FeSe, but only indirectly through the fact that FeSe is in this particular state. The direct effect of the SrTiO 3 on the electronic structure of an FeSe monolayer near the Fermi level is negligible. For example, relaxing the structure of FeSe on SrTiO 3 and then removing SrTiO 3 atoms from the calculation does not affect the electronic structure near the Fermi level (see figure 1 in the supplement (see footnote 5)). Therefore to speed up the calculation of the electron-phonon coupling, we perform calculations on an isolated FeSe layer, without  (1) from A c to A 0.9 c and confirm that the electron-phonon matrix elements are not affected by this simplification by carrying out full calculation (see table 1 in the supplement (see footnote 5)).
We use state-of-the-art Wannier interpolation technique from [29] and the Quantum-Espresso package described in [30] to calculate the electron-phonon coupling in the FeSe monolayer with a very fine grid in the Brillouin zone (40 × 40). We obtained the superconducting transition temperature T c by solving the Eliashberg equation [31,32] as described in [33]. Figure 3 shows the calculated Eliashberg spectral function α ω F ( ) 2 of the FeSe monolayer. We focus our analysis on two groups of phonons for which the electron-phonon coupling is the largest. The first group of phonons (labeled 1 in figure 3) corresponds to phonons with frequency close to 10 meV, and the second group (labeled 2) to phonons with 20 meV (in GGA those frequencies are 15 and 25 meV, respectively).
While phonons 1 contribute to about two-thirds of the total electron-phonon coupling strength λ, they contribute to about half of the integrated α ω F ( ) 2 spectral function (since they have a lower frequency). The atomic displacement character of the two groups of phonons is different. Phonons 1 correspond to a branch of phonons that involve transverse, mostly in-plane displacements of atoms (these phonons cause bulk FeSe to undergo a shear phase transition), while phonons 2 correspond to an out-of-plane transverse displacement of Fe atoms. Furthermore, phonons 1 and 2 couple different parts of the electron Fermi surface at M. Phonons 1 couple mostly at parts of the reciprocal space where the Fermi surface (electron M pocket) crosses the M-Γ line and the least where it crosses the M-X line. The opposite is true for phonons 2. However, since both phonons contribute about equally to α F 2 the total electron-phonon coupling (1 and 2 taken together) is nearly constant on the entire M pocket Fermi surface.
Hence the importance of the SrTiO 3 substrate for increasing the superconducting transition temperature within the phonon mechanism in FeSe is two-fold. First, it prevents phonons 1 from becoming unstable and induce a structural phase transition (as in bulk FeSe). Second, SrTiO 3 keeps FeSe in the checkerboard magnetic phase which allows coupling of phonons from groups 1 and 2. In the non-magnetic case, the coupling of these phonons is zero by symmetry [34]. Calculations in [35,36] also found a significantly smaller electron-phonon coupling in the non-magnetic phase than in the magnetic phase. We also note that at this time, there is no direct experimental measurement of magnetic order in FeSe monolayer on SrTiO 3 . However, the measured ARPES band structure is most closely resembled to that of the band structure of FeSe with an antiferromagnetic checkerboard order, both in our GGA+A calculation and in previous work [19,37]. Nevertheless, it is possible that the true ground state of FeSe monolayer consists of fluctuating antiferromagnetic moments on iron atoms. Treatment of electron-phonon coupling in such a state from first-principles goes well beyond the scope of this work.
Comparing α ω F ( ) 2 in GGA and GGA+A (figure 3), we find two reasons for an increased coupling in GGA+A. First, preference for a shear distortion in GGA+A increases the electron-phonon matrix elements of phonons 1 (see figure 3 in the supplement (see footnote 5)). Second, the bottom of the electron M pocket E M is closer to the Fermi level in GGA+A than in the GGA. Therefore, owing to this band renormalization (narrowing of the occupied bandwidth), the density of states at the Fermi level in GGA+A is larger than in GGA (see table 2 here and figure 3 in the supplement (see footnote 5)). Since λ is proportional to the density of states, it is therefore increased in GGA+A.
However, as discussed earlier, we calculated the electron-phonon coupling λ within GGA+A with a reduced value of parameter A from equation (1). Taking into account calculated density of states (1.5 eV −1 ) with = A A 0.9 c and = A A c (1.8 eV −1 ) we conservatively estimate that the value of λ at = A A c is λ = 1.6. Next we use the Eliashberg theory and obtain a conservative estimate of the superconducting transition temperature T c of 26 K (with μ = * 0.0) and 21 K (with μ = * 0.1). This estimate is significantly closer to experiment than a standard GGA result (0.1-1.5 K).
This range of estimated transition temperatures (21-26 K) is close to the value found across the families of bulk iron-based superconductors. Now we discuss possible reasons for an even larger T c in the case of an FeSe monolayer on SrTiO 3 (65-109 K).
When λ is large, transition temperature is proportional to [38] ωλ ∼ T .
( 2 ) c 0.5 Here ω is the averaged phonon frequency and λ is the Brillouin zone averaged electron-phonon coupling strength. Therefore one possibility to get larger T c is to further increase λ. It is at least plausible that this could happen for phonons 1, since their contribution to λ is increased when FeSe is approaching the shear-like structural phase transition. The second possibility is to increase the average frequency ω by pairing electrons with high frequency modes (phonons or some other bosons) in addition to phonons 1 and 2. One possibility are magnetic fluctuations [8]. The role of magnetism for superconductivity in FeSe is additionally enriched by the fact that, in the nonmagnetic phase, certain electron-phonon interaction channels are forbidden by symmetry. In addition, structural and magnetic order parameters are strongly coupled in FeSe. For example, bulk orthorhombic FeSe prefers a nonmagnetic state, while a cubic FeSe monolayer on SrTiO 3 prefers an antiferromagnetic state.
Another tempting possibility suggested in [3] is to pair FeSe electrons to a high-frequency (80 meV) phonon in the SrTiO 3 substrate. This coupling was experimentally determined to be large near the origin of the phonon Brillouin zone ( ∼ q 0). Adding experimentally estimated values of the electron-phonon coupling from [3] to our calculated α ω F ( ) 2 increases the estimated superconducting transition temperature to 47 K (assuming μ = * 0.1), even closer to the experimentally determined value (65-109 K).
In closing, we note that the experimentally inferred superconducting T c is nearly the same for an FeSe monolayer on TiO 2 terminated SrTiO 3 [2,6], BaTiO 3 [39], as well as 2% strained SrTiO 3 [40]. This observation is consistent with our structural stabilization mechanism since in all three cases interaction between Ti atoms in the TiO 2 layer and Se atoms in FeSe is likely the same. However, when a FeSe monolayer is placed on a substrate with a different bonding environment, such as SiC in [41][42][43] the superconducting T c is only 2-9 K. Another indication for the importance of structural stabilization comes from [1]. This study found that bulk FeSe doped with only 2% of iron stays tetragonal (non-sheared) even well below 90 K. This loss of preference for shear is accompanied with loss of superconductivity ( < T 0.5 c K), again consistent with our finding that keeping FeSe close to a shear (orthorhombic, nematic) structural phase transition increases the electron-phonon coupling strength. Another indication of contribution from electron-phonon mechanism is described in [44] on iron isotope effect measurement. . The density of states within the Eliashberg theory calculated using GGA+A and the STM measurement from [45]. The energy is measured relative to the superconducting gap Δ.