#
Electronic Origin of T_{c} in Bulk and Monolayer FeSe

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{++}as a shorthand for the four-tier QSGW+DMFT+BSE.

^{++}has high fidelity because QSGW captures non-local dynamic correlation particularly well in the charge channel [12,24], but cannot adequately capture effects of spin fluctuations. DMFT does an excellent job at the latter, which are strong but mostly controlled by a local effective interaction given by Hubbard parameters U and J. These are calculated within the c-RPA [25] from the QSGW Hamiltonian using an approach [12] similar to that of Ref. [26]. That it can well describe superconductivity in a parameter-free manner has now been established in several Hund’s materials [21,22]. In FeSe, we have also shown [27] that it reproduces the main features of neutron structure factor [28,29].

^{++}ability. We found a dense spectrum of triplet and singlet superconducting instabilities therein [21]. Lacking at that time the ability to calculate U, J, we borrowed values from an earlier LDA+DMFT study [30]. Later we were able to build our own c-RPA calculated from QSGW (QSGW+cRPA), and found U, J to be approximately 2/3 of the original values. Our initial study concluded that the triplet instability was slightly stronger than the B${}_{1g}$ − d${}_{{x}^{2}-{y}^{2}}$ singlet in unstrained Sr${}_{2}$RuO${}_{4}$. Redoing the procedure with U, J calculated from QSGW+cRPA [31] the conclusions did not qualitatively change, except the singlet eigenvalue became larger than the triplet. Thus the original study [21] did not predict the correct ground state. From a prior study we discovered that the superconducting instability can be sensitive to the exact choices of U, J. Here we show small changes in J have a dramatic effect on spin susceptibility and superconductivity in FeSe. For that reason we recalculate QSGW+cRPA for for all cases we report here, on bulk and layered variants of FeSe. This is a step forward in making the calculations as ab-initio as possible, and as we discuss later in the paper, this provides us with a remarkable unified understanding of both bulk and layered variants of FeSe, the correlations and superconducting instabilities therein.

- Present the mathematical formulation and implementation of different susceptibilities in spin, charge and superconducting channels;
- List the structural and Hubbard parameters used for different materials, and present ab initio results bulk FeSe;
- By treating J and the Se height as free parameters, show how they affect correlations and superconductivity in bulk FeSe;
- Show results for a standing monolayer of FeSe (M-FeSe);
- Show results for a monolayer of FeSe on SrTiO${}_{3}$ (M-FeSe/STO);
- Interpret these results to explain what controls T${}_{c}$.

## 2. Methodology

## 3. Results and Discussion

#### 3.1. Structural Parameters and c-RPA Estimates for U and J

#### 3.2. Fermi Surfaces and Spectral Functions

#### 3.3. Bulk FeSe

^{++}calculations, how excursions in J between 0 and 1 eV affect the single-particle scattering rate $\Gamma $, and the inverse Z factor, a measure of mass or bandwidth normalisation.

#### 3.4. Excursion in Fe-Se bond length in Bulk FeSe

#### 3.5. Free Standing Monolayer of FeSe

#### 3.6. Monolayer of FeSe/SrTiO${}_{3}$

^{++}calculation, the hole pockets of ${d}_{xz,yz}$ character survive although are significantly narrowed. This is a discrepancy with ARPES, which finds no such pockets. However, their small size suggests that they should be sensitive to electron-phonon coupling that can further renormalise them and push them down.

^{++}spectral function with ARPES. We have recently developed the ability to incorporate the electron-phonon self-energy into QSGW

^{++}via a field-theoretic technique. While initial results are very preliminary, they suggest that much of the discrepancy we see with ARPES in bulk FeSe, originates from this interaction. It is perhaps not surprising, given that the electron-phonon interaction is expected to be stronger when pockets are small. These new findings, however, are beyond the scope of this work.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Fermi surfaces are shown in the $\Gamma $-M (z = 0) plane for both bulk FeSe and monolayer FeSe/STO from DFT + DMFT, and QSGW + DMFT. In each case the the U and J are used from respective C-RPA calculations.

**Figure 2.**The QSGW band structure and QSGW+DMFT spectral functions $A(k,\omega )$ are shown on a section of the $\Gamma $-M path for: (

**a**) bulk-FeSe (J = 0.6 eV); (

**b**) bulk FeSe with reduced Se height above the Fe plane (h = 1.27 Å); (

**c**) M-FeSe, a free standing monolayer of FeSe; (

**d**) M-FeSe/STO. In all four panels, the Fe-${d}_{xy}$ state calculated by QSGW is depicted in blue and the Fermi energy ${E}_{F}$ is at 0. Note the strongly marked incoherence in (

**a**,

**d**). In all cases DMFT narrows the width of ${d}_{xy}$ relative to QSGW as is typical of narrow-band d systems [11,39], but incoherence is highly sensitive to the position of ${d}_{xy}$. In (

**a**,

**d**) ${d}_{xy}$ is proximate to ${E}_{F}$ and a high degree of incoherence is present. while in (

**b**,

**c**) ${d}_{xy}$ is pushed far below ${E}_{F}$: and the system has properties similar to a normal Fermi liquid. Panel (

**e**) shows the imaginary part of the spin susceptibility $\chi \left(\omega \right)$, at the AFM nesting vector ${\mathbf{q}}^{\mathrm{AFM}}$ = (1/2, 1/2, 0) $2\pi /a$ for the four geometries (

**a**):orange, (

**b**): red, (

**c**): purple, (

**d**): brown. Shown also in green is Im$\chi \left(\omega \right)$ for bulk FeSe with J = 0.68–the highest ${T}_{c}$ found among parameterised hamiltonians. The more intense Im $\chi (\omega \to 0)$ is, the larger the superconducting instability. Panel (

**f**) shows how the leading eigenvalue $\lambda $ of the linearised particle–particle ladder BSE equation treating J as a free parameter (blue circles). The extreme sensitivity to J is apparent. Shown also are $\lambda $ for the four ab initio calculations (

**a**–

**d**), using the colour scheme in panel e. For (a) B-FeSe and (

**d**), FeSe/STO, ${d}_{xy}$ falls near ${E}_{F}$ and $\lambda $ approximately coincides with the blue line; (

**b**,

**c**) do not.

**Figure 3.**We compute the net local moment and its evolution with J. Orbitally resolved single-particle scattering rate ($\Gamma $) and mass enhancement m${}_{DMFT}$/m${}_{QSGW}$ for Bulk FeSe with varying Hund’s coupling strength.

**Figure 4.**Energy and momentum resolved spin susceptibility Im$\chi (q,\omega )$ shown for (

**a**) bulk FeSe (B-FeSe) (J = 0.6 eV), (

**b**) bulk FeSe with increased Hund’s correlation (J = 0.68 eV), (

**c**) reduced Fe-Se height (${h}_{Se}$ = 1.27 Å), (

**d**) 0.15 electron doped bulk FeSe (a section on uniformly electron doped FeSe is included in the SM), (

**e**) free standing monolayer of FeSe, M-FeSe [38] (

**f**) M-FeSe/STO [38]. The q-path (H,K,L = 0) chosen is along (0,0) − ($\frac{1}{2},0$) − ($\frac{1}{2},\frac{1}{2}$) − (0,0) in the Brillouin zone corresponding to the two Fe-atom unit cell. The intensity of the spin fluctuations at ($\frac{1}{2},\frac{1}{2}$) is directly related to the presence of the Fe-${d}_{xy}$ state at Fermi energy and its incoherence. The more incoherent the A(k,$\omega $) is the more intense is the Im $\chi (\mathbf{q}=(\frac{1}{2},\frac{1}{2},0),\omega )$.

**Figure 5.**(

**a**,

**b**) Inverse Z factors and scattering rates $\Gamma $ for Fe 3${d}_{xy}$ (filled symbols) and 3${d}_{yz}$ (empty symbols) orbitals in three configurations of bulk FeSe (▽, ○, □) ▽ is the ab initio result (h = 1.463 Å, J = 0.60 eV); while □ changes J to 0.68 eV; ○ changes h and J to 1.27 Å and 0.69 eV. Shown also are an isolated FeSe monolayer with h = 1.39 Å and J = 0.71 eV (×), a monolayer on STO with h = 1.40 Å and J = 0.67 eV (+). Correlation is sensitive to changes in ${l}_{Fe-Se}$ and J. (

**c**–

**f**) leading eigenvalue $\lambda $ of the superconducting instability calculated at 290 K drawn against various measures of correlation: $\lambda $ is approximately proportional to ${\Gamma}_{xy}$ and ${\Gamma}_{yz}$ (

**c**), and it is monotonic in $1/{Z}_{xy}$ and $1/{Z}_{yz}$ (

**d**), and also in the strength of Im$\chi [q\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}(1/2,1/2),\omega \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}15\phantom{\rule{0.166667em}{0ex}}\mathrm{meV}]$ (

**e**) and suppression of the dispersion of the paramagnon branches (

**f**). The graded intense purple background separates the most strongly correlated systems with large $\lambda $ from the weakly correlated systems with small $\lambda $ in weaker purple background.

**Table 1.**Structural parameters, chalcogen height ${h}_{Se}$ and computed U and J for the correlated many body Hamiltonian from our QSGW+c-RPA implementation. We also list the bare electronic bandwidths from QSGW which show how electronic correlations enhance in M-FeSe/STO in comparison to the bulk as the bands get narrower. References indicate where the structural inputs were taken.

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**MDPI and ACS Style**

Acharya, S.; Pashov, D.; Jamet, F.; van Schilfgaarde, M.
Electronic Origin of T* _{c}* in Bulk and Monolayer FeSe.

*Symmetry*

**2021**,

*13*, 169. https://doi.org/10.3390/sym13020169

**AMA Style**

Acharya S, Pashov D, Jamet F, van Schilfgaarde M.
Electronic Origin of T* _{c}* in Bulk and Monolayer FeSe.

*Symmetry*. 2021; 13(2):169. https://doi.org/10.3390/sym13020169

**Chicago/Turabian Style**

Acharya, Swagata, Dimitar Pashov, Francois Jamet, and Mark van Schilfgaarde.
2021. "Electronic Origin of T* _{c}* in Bulk and Monolayer FeSe"

*Symmetry*13, no. 2: 169. https://doi.org/10.3390/sym13020169