Layer-independent ferromagnetic insulators in a new structural phase of ${\mathrm{Cr}}_2{\mathrm{S}}_3$

Ferromagnetic insulators (FMIs) are crucial for sensing and storage, but they are currently rare. The magnetism of the widely studied ${\mathrm{CrI}}_3$ and ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ is layer dependent; thus, obtaining FMIs from these materials is challenging because the film thickness must be carefully controlled. Based on the generalized gradient approximation (GGA), $\mathrm{GGA}+U$, and HSE06 hybrid functional calculations, this work predicts a new structural phase of ${\mathrm{Cr}}_2{\mathrm{S}}_3$ $\left(\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3\right)$ and further predicts it as an intrinsic and layer-independent FMI. The material is dynamically metastable. It crystallizes a rhombohedral lattice, stacking the ${\mathrm{Cr}}_2{\mathrm{S}}_3$ quintuple layers (QLs) along the [111] direction through van der Waals interactions. The in-plane lattice constant is 3.4 Å, and each Cr atom carries a magnetic moment of $3.0{\mu }_B$. $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ behaves as an intrinsic FMI, irrespective of the layer number, but its indirect energy gap can be tuned from 1.23 to 1.97 eV by changing the film thickness according to the HSE06 calculations. In-plane compressive strain facilitates the ferromagnetism and decreases the energy gap, while tensile strain leads to antiferromagnetism and decreases the energy gap. In the 1QL case, 0.5% in-plane tensile strain causes the ferromagnetic-antiferromagnetic phase transition. The origin of the ferromagnetism and the strain-induced magnetic transition of $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ are discussed and attributed to the dominant ferromagnetic part of the superexchange.

Figure 1

Two existing structures of ${\mathrm{Cr}}_2{\mathrm{S}}_3$ in (a) rhombohedral phase and (b) trigonal phase. (c) The new rhombohedral phase of ${\mathrm{Cr}}_2{\mathrm{S}}_3$ predicted in this work and (d) its hexagonal presentation. In (a), (b), and (d), top and bottom panels correspond to top and side views, respectively. The lattice constants are obtained from $\mathrm{GGA}+U$ ($U=3.0$ eV) calculations with vdW corrections. A, B, and C indicate the stacking sequence of atoms along the $c$ direction. For clarity, the three structures are called $\alpha$, $\beta$, and $\gamma$ phases, and the corresponding materials are referred to as $\alpha$-, $\beta$-, and $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$, respectively. Phonon spectra of (e) one quintuple layer (1QL) and (f) bulk $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ calculated using the DFPT method.

Figure 2

Calculated band structures of FM 1QL $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ within (a) GGA, (b) $\mathrm{GGA}+U$, and (c) HSE06 hybrid functional frameworks. Blue and red lines indicate the majority- and minority-spin bands, respectively. Density of states projected on (d) Cr and (e) S atoms of 1QL $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ within the $\mathrm{GGA}+U$ framework.

Figure 3

[(a)–(d)] Considered magnetic orders and (e) the exchange coupling parameters of 1QL $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$. Red and green symbols denote Cr atoms of different heights, and arrows in (a) through (d) indicate the magnetization direction.

Figure 4

(a) Energy difference $\mathrm{\Delta }E$ and (b) band gaps of 1QL $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ at different in-plane strain. (c) Energy difference $\mathrm{\Delta }E$ and (d) band gaps of $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ slabs containing different QLs.

Figure 5

Exchange coupling parameters of 1QL $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ under in-plane strains. Atomic positions used in (a) were fully relaxed, while the fractional coordinates in (b) were fixed at those of the unstrained structure. (c) Structures are constructed with regular (${\mathrm{CrS}}_6$) octahedrons. The results located at point “1” in (c) denote the octahedrons with 2.55 Å Cr-S bond length. The parameters are fitted from the $\mathrm{GGA}+U$ ($U=3.0$ eV) energies.

Figure 6

The schematic figures of the paths contributing to the interlayer magnetic coupling. ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ are two types of energy difference. Energy levels of the $d$ orbitals with spin up and spin down are denoted with red and blue lines. The $p$ orbitals right below the Fermi level are denoted by black lines. The solid (dashed) arrows denote the electrons before (after) virtual hopping. The virtual hopping process is labeled by curved arrows. [(a)–(h)] The paths included in the ${90}^{\circ }$ NN coupling ($J_2$). The local coordinates are defined with the $x,y$ axis along two perpendicular Cr-S(Se) bonds in the Cr-S(Se)-Cr plane. [(i)–(n)] The paths included in the ${180}^{\circ }$ NNN coupling ($J_3$).

Figure 7

Calculated energy band structures of FM $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ bulk within (a) GGA, (b) $\mathrm{GGA}+U$, and (c) HSE06 hybrid functional frameworks.

Figure 8

Projected density of states of FM $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ bulk within [(a), (b)] $\mathrm{GGA}+U$ and [(c), (d)] HSE06 frameworks.

Figure 9

(a) DFT (density functional theory) energy of $\alpha$-, $\beta$-, and $\gamma \text{−}{\mathrm{Cr}}_2{\mathrm{S}}_3$ under isotropic strain. [(b), (c)] Free energy of the three phases of ${\mathrm{Cr}}_2{\mathrm{S}}_3$. The data were obtained based on the $\mathrm{GGA}+U$ ($U=3.0$ eV) calculations.

Figure 10

The evolution of the energy levels after the tension is applied. The left panel is the configuration without strain and the right panel is that with tensile strain. The dashed lines in the right panel denote the energy levels before the strain is applied. The tension causes the enlargement of (${\mathrm{CrS}}_6$) units, which means the crystal-field splitting ${\mathrm{\Delta }}_0$ decreases. The unoccupied $e_g$ orbitals get close to the Fermi level, leading to the decrease of the charge transfer energy ${\mathrm{\Delta }}_2$. On the contrary, the charge transfer energy ${\mathrm{\Delta }}_1$ increases.