Frustrated magnetic interactions in FeSe

The structurally simplest high-temperature superconductor FeSe exhibits an intriguing superconducting nematic paramagnetic phase with unusual spin excitation spectra that are different from typical spin waves; thus, determining its effective magnetic exchange interactions is challenging. Here we report neutron scattering measurements of spin fluctuations of FeSe in the tetragonal paramagnetic phase. We show that the equal-time magnetic structure factor, $\mathcal{S}\left(\mathbf{Q}\right)$, can be effectively modeled using the self-consistent Gaussian approximation calculation with highly frustrated nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) exchange couplings, and very weak further neighbor exchange interaction. Our results elucidate the frustrated magnetism in FeSe, which provides a natural explanation for the highly tunable superconductivity and nematicity in FeSe and related materials.

Figure 1

Crystal structure, phase diagram, and polarized neutron scattering data of FeSe. (a) Crystal structure of FeSe. $J_1, J_2$, and $J_3$ denote the nearest-neighbor, next-nearest-neighbor, and third-nearest-neighbor in-plane exchange couplings, respectively. The black dashed lines represent the orthorhombic (4-Fe) unit cell projected in the $ab$ plane ($a/b$ axis is along the nearest Fe-Fe bond), which is used throughout our presentation. (b) Phase diagram of FeSe. The red star emphasizes that the magnetic interactions are extracted from the inelastic neutron spectrum measured at 110 K. SC, superconductivity. (c)–(e) Polarized inelastic neutron scattering data of FeSe at 110 K. The polarization analysis is based on $x\parallel \mathbf{Q}$ configuration. Scattering intensities in the spin-flip channel with $x, y, z$ polarization (${\mathrm{SF}}_x, {\mathrm{SF}}_y$, and ${\mathrm{SF}}_z$) were measured at $\mathbf{Q}=\left(1,0,0\right)$ (c) and $\mathbf{Q}=\left(1,0,1\right)$ (d). The background signal (BG) is derived from $\mathrm{BG}=\mathrm{S}{\mathrm{F}}_y+\mathrm{S}{\mathrm{F}}_z-\mathrm{S}{\mathrm{F}}_x$. Energy dependence of spin fluctuations $M_y$ and $M_z$ at $\mathbf{Q}=\left(1,0,0\right)$ and (1,0,1) is thus obtained using the relationship $M_y=\mathrm{S}{\mathrm{F}}_z-\mathrm{BG}$ and $M_z=\mathrm{S}{\mathrm{F}}_y-\mathrm{BG}$, with the ${\mathrm{Fe}}^{2+}$ magnetic form factor corrected (e). Error bars indicate one standard deviation. The solid and dashed lines are a guide to the eye.

Figure 2

Determination of magnetic interactions in FeSe. Contour plot of the goodness-of-fit ${\chi }^2$ between calculations and neutron scattering data. The equal-time magnetic structure factor, $\mathcal{S}\left(\mathbf{Q}\right)$, is calculated through the self-consistent Gaussian approximation for the $J_1\text{−}{J}_2\text{−}{J}_3$ Heisenberg model at $k_{B}T/J_1=0.86$. Inelastic neutron scattering data of FeSe were collected at 110 K. The red star denotes the best-fit interaction parameters.

Figure 3

Momentum dependence of calculated and measured equal-time structure factor in FeSe. (a) The equal-time structure factor calculated using SCGA for the optimized parameters $k_{B}T/J_1=0.86, J_2/J_1=0.413$, and $J_3/J_1=0.069$. (b) The energy-integrated intensity, $I\left(\mathbf{Q}\right)={\int }_0^{E^{\prime }}\left(1+e^{-E/k_{B}T}\right)I\left(\mathbf{Q},E\right)dE$, obtained from the measured magnetic intensity, $I\left(\mathbf{Q},E\right)$, of FeSe at $T=110\mathrm{K}$, with the ${\mathrm{Fe}}^{2+}$ magnetic form factor corrected. $E^{\prime }=220\mathrm{meV}$ is the spin excitation energy's upper limit. The wave vector $\mathbf{Q}$ at $\left(q_x,q_y,q_z\right)$ is defined as $\left(H,K,L\right)=\left(q_{x}a/2\pi ,q_{y}b/2\pi ,q_{z}c/2\pi \right)$ in reciprocal lattice units (r.l.u.) of the orthorhombic unit cell, where $a=b=2d_{\mathrm{Fe}-\mathrm{Fe}}$ is twice the nearest Fe-Fe distance in the tetragonal phase.

Figure 4

Calculated and measured equal-time structure factor along the high-symmetry directions in FeSe. (a), (b) Measured equal-time structure factor along the high-symmetry directions and comparison with SCGA calculations for the optimized parameters $k_{B}T/J_1=0.86, J_2/J_1=0.413$, and $J_3/J_1=0.069$. The scan directions are marked by the arrows in the insets. Error bars indicate one standard deviation.