Correlated electronic structure, orbital-dependent correlations, and Lifshitz transition in tetragonal FeS

Iron-based binary chalcogenide superconductors FeSe and FeS have attracted much recent attention due to their complex orbital-selective correlations and Cooper pairing, offering the minimal model system holding the key properties to understanding the physics of high-$T_c$ superconductors. Here, using density functional plus dynamical mean-field theory method with full self-consistency over the charge density, we study the effect of electronic correlations on the electronic structure, magnetic properties, orbital-dependent band renormalizations, and Fermi surface of the tetragonal phase of bulk FeS. We perform a direct structural optimization of the $P_4/nmm$ crystal structure of paramagnetic FeS, minimizing the total energy of FeS with respect to the lattice constant $a$ and the internal coordinate $z_{\mathrm{S}}$ of atom S. Our results show an anomalous sensitivity of the electronic structure and magnetic properties of FeS (intrinsic to all Fe-based chalcogenide and pnictide superconductors) to fine details of its crystal structure, e.g., to a small variation of the chalcogen coordinate $z_{\mathrm{S}}$. Upon expansion of the lattice volume (which can be realized, e.g., in ${\mathrm{FeS}}_{1-x}{\mathrm{Se}}_x$ and ${\mathrm{FeS}}_{1-x}{\mathrm{Te}}_x$), we observe a remarkable change of the electronic structure of FeS which is associated with a complete reconstruction of the Fermi-surface topology (Lifshitz transition). This behavior is ascribed to a correlation-induced shift of the Van Hove singularity associated with the Fe $xy$ and $xz/yz$ orbitals at the $M$ point across the Fermi level. The Lifshitz phase transition is accompanied by a significant growth of local magnetic moments and emergence of strong orbital-selective correlations. It is seen as a pronounced anomaly (“kink”) in the total energies upon expansion of the lattice, associated with a remarkable enhancement of compressibility. This behavior is accompanied by an orbital-dependent formation of local moments, a crossover from itinerant to localized orbital-selective moment behavior of the Fe $3d$ electrons. While exhibiting weak effective mass enhancement of the Fe $3d$ states $m^{*}/m\sim 1.3–1.4$, correlation effects reveal a strong impact on a position of the Van Hove singularity originating from the Fe $xz/yz$ and $xy$ orbitals at the $M$ point, implying a complex interplay between electronic correlations and band structure effects in FeS. Our results suggest a complex interplay between electronic correlations, magnetism, and lattice degrees of freedom in FeS.

Figure 1

Total energy (upper panel) and instantaneous local moment $\sqrt{〈{\widehat{m}}_z^2〉}$ (lower panel) of paramagnetic FeS as a function of the lattice constant calculated by the charge self-consistent $\text{DFT}+\text{DMFT}$ method at $T=290$ K for the experimental ($z_{\mathrm{S}}^{\mathrm{expt}}$) and optimized ($z_{\mathrm{S}}^{\mathrm{opt}}$) internal position of sulfur. The results for the total energy obtained within nonmagnetic GGA are shown in the upper panel for comparison (dashed curves). The phase transition point is indicated by arrows (upper panel) and by vertical dashed lines (lower panel).

Figure 2

Orbitally resolved Fe $3d$ spectral functions of FeS computed by $\text{DFT}+\text{DMFT}$ at $T=290$ K for the equilibrium lattice volume (black) and those in the expanded-volume phase with experimental $z_{\mathrm{S}}=0.252$ (red) and optimized $z_{\mathrm{S}}=0.127$ (green). The Fermi energy is shown by a vertical dashed line ($E_{\mathrm{F}}=0$ eV).

Figure 3

Band structure of FeS along the $\mathrm{\Gamma }-X-M-\mathrm{\Gamma }$ path obtained by $\text{DFT}+\text{DMFT}$ at ambient pressure ($a=6.90$ a.u., $z_{\mathrm{S}}=0.252$) for $T=290$ K. The dashed curves show the nm-GGA results.

Figure 4

Orbitally resolved quasiparticle mass enhancement $m^{*}/m$ of the Fe $3d$ states in paramagnetic FeS as a function of lattice constant calculated by $\text{DFT}+\text{DMFT}$ with optimized $z_{\mathrm{S}}$ at $T=290$ K. The phase transition point is indicated by a vertical dashed line.

Figure 5

Fermi surface (upper panels) and spectral weight at the Fermi level (lower panels) of tetragonal FeS as computed by $\text{DFT}+\text{DMFT}$ for $a=6.90$ a.u. (left) and $a=8.20$ a.u. (right) with fixed ${\mathrm{z}}_{\mathrm{S}}=0.252$.

Figure 6

Fermi surface (upper panels) and spectral weight at the Fermi level (lower panels) of tetragonal FeS as computed by $\text{DFT}+\text{DMFT}$ for $a=6.90$ a.u. (left) and $a=8.60$ a.u. (right) with optimized ${\mathrm{z}}_{\mathrm{S}}$ (${\mathrm{z}}_{\mathrm{S}}=0.247$ and 0.127, respectively).

Figure 7

Orbitally resolved local spin correlation functions $\chi \left(\tau \right)=〈m_z\left(\tau \right)m_z\left(0\right)〉$ of FeS at the equilibrium (left panels) and those in the expanded phase (right panels) as computed by $\text{DFT}+\text{DMFT}$ at $T=290$ K. Top row shows the result obtained with fixed internal coordinate $z_{\mathrm{S}}$, bottom row corresponds to $z_{\mathrm{S}}$ optimized within $\text{DFT}+\text{DMFT}$.

Figure 8

Orbitally resolved static spin susceptibility $\chi \left(\mathbf{q}\right)$ of FeSe calculated by $\text{DFT}+\text{DMFT}$ at fixed (top panels) and relaxed (bottom panels) internal coordinate $z_{\mathrm{S}}$. Left panels correspond to the equilibrium lattice constant, right panels show the $\text{DFT}+\text{DMFT}$ results for the expanded phase.

Figure 9

Our results for total energy (upper panel) and instantaneous local moment $\sqrt{〈{\widehat{m}}_z^2〉}$ (lower panel) of paramagnetic FeS obtained by $\text{DFT}+\text{DMFT}$ with the AMF double-counting correction. Results for the fixed experimental $z_{\mathrm{S}}^{\mathrm{expt}}=0.252$ and optimized $z_{\mathrm{S}}^{\mathrm{opt}}$ are shown. The $\text{DFT}+\text{DMFT}$ calculations are preformed in the paramagnetic state at an electronic temperature $T=1160$ K. The phase transition point is indicated by arrows (upper panel) and by vertical dashed lines (lower panel).

Figure 10

Total energy (upper panel) and instantaneous local moment $\sqrt{〈{\widehat{m}}_z^2〉}$ (lower panel) of paramagnetic FeS as a function of the lattice constant obtained by $\text{DFT}+\text{DMFT}$ with the “nominal” DC. The $\text{DFT}+\text{DMFT}$ calculations are performed for the experimental position $z_{\mathrm{S}}=0.252$ at an electronic temperature $T=1160$ K. The phase transition point is indicated by arrows (upper panel) and by vertical dashed lines (lower panel). For clarity, the total energy curves corresponding to different $N_d$ were shifted by a constant of 0.2 eV.

Figure 11

Band structure of FeS along the $\mathrm{\Gamma }-X-M-\mathrm{\Gamma }$ path obtained by $\text{DFT}+\text{DMFT}$ for the lower-volume phase (left column) and at the calculated equilibrium lattice constant $a=8.20$ a.u. (right column) using the “nominal” DC with $N_d=7$ (top row) and $N_d=6$ (bottom row). The $\text{DFT}+\text{DMFT}$ calculations are performed at an electronic temperature $T=1160$ K. $z_{\mathrm{S}}$ was fixed to its experimental value $z_{\mathrm{S}}=0.252$. The dashed curves show the nm-GGA results.