Comparison of $GW$ band structure to semiempirical approach for an FeSe monolayer

We present the $G_0{W}_0$ band structure, core levels, and deformation potential of monolayer FeSe in the paramagnetic phase based on a starting mean field of the Kohn-Sham density functional theory (DFT) with the Perdew, Burke, and Ernzerhof functional. We find the GW correction increases the bandwidth of the states forming the $M$ pocket near the Fermi energy, while leaving the $\mathrm{\Gamma }$ pocket roughly unchanged. We then compare the $G_0{W}_0$ quasiparticle band energies with the band structure from a simple empirical $+\mathrm{A}$ approach, which was recently proposed to capture the renormalization of the electron-phonon interaction going beyond DFT in FeSe, when used as a starting point in density functional perturbation theory. We show that this empirical correction succeeds in approximating the GW nonlocal and dynamical self-energy in monolayer FeSe and reproduces the GW band structure near the Fermi surface, the core energy levels, and the deformation potential (electron-phonon coupling).

Figure 1

The band structure of the electron pocket near the $M$ point with path taken along the $X\to M\to \mathrm{\Gamma }$ direction at different levels of theory. Different colors indicate contributions from different atomic orbitals (amount of green is proportional to the contribution of $d_{zx}$ and $d_{zy}$, blue to $d_{xy}$ and $d_{x^2-y^2}$, and red to the contribution from $d_{z^2}$, which is negligible). The Fermi level ($E_F$) is set at zero. The $GW$ bands are calculated at the $G_0{W}_0$ level with frequency dependence in the dielectric screening treated within the HL-GPP model.

Figure 2

Band structure of the hole pocket near the $\mathrm{\Gamma }$ point with path taken along the $M\to \mathrm{\Gamma }\to X$ direction at different levels of theory as in Fig. 1. Different colors indicate contributions from different atomic orbitals (the amount of green is proportional to the contribution of $d_{zx}$ and $d_{zy}$, blue to $d_{xy}$ and $d_{x^2-y^2}$, and red to $d_{z^2}$). The Fermi level ($E_F$) is set at zero.

Figure 3

Fermi surface with pockets at $\mathrm{\Gamma }$ and $M$ ($k_x=k_y=0.5$) points at different levels of theory. The $GW$ results are calculated at the $G_0{W}_0$ level with frequency dependence in the dielectric screening treated within the HL-GPP model.

Figure 4

Energy levels for occupied states at the $\mathrm{\Gamma }$ point in monolayer FeSe as calculated at the DFT level, $G_0{W}_0$ level with the frequency dependence in the screening captured within the HL-GPP model, and at the $+\mathrm{A}$ level with a static, semilocal approximation to the self-energy.

Figure 5

Colored lines are showing the band structure of the electron pocket near the $M$ point, exactly as in Fig. 1. The grayed areas show the evolution of the bands if the selenium height in FeSe is increased by 0.15 bohr.