${\mathrm{Cr}}_2{\mathrm{S}}_3$ a bipolar semiconducting fully compensated ferrimagnet

The promises of spintronics have longed for the realization of fully compensated states and full spin polarization. Half-metallic/half-semiconducting (HM/HS) and fully compensated materials are suggested as candidates. We use density functional theory to report that the experimentally realized rhombohedral ${\mathrm{Cr}}_2{\mathrm{S}}_3$ is a promising candidate as a bipolar magnetic semiconducting (BMS)/HS fully compensated ferrimagnetic (FCFiM) material. Our calculations demonstrate that the uniquely layered structure of ${\mathrm{Cr}}_2{\mathrm{S}}_3$ produces different interlayer and intralayer $d\text{−}p\text{−}d$ hybridization schemes between Cr atoms. Strong interlayer and weak intralayer antiferromagnetic coupling between different Cr sites make the overall magnetic state a fully compensated structure. The origin of BMS/HS and fully compensated state has been explained by a structural analysis, where magnetic exchange interaction between Cr sites is dependent on bond distance and bond angle of each Cr-centered octahedron. Furthermore, by applying strains perpendicular to the basal plane, distortion of Cr octahedron sites and Cr–Cr distance is altered, resulting in the phase transition of the material both electronically and magnetically from BMS-FCFiM to HS-FCFiM to ferrimagnetic (FiM). These studies enable us to rediscover ${\mathrm{Cr}}_2{\mathrm{S}}_3$ as a novel class of BMS materials.

Figure 1

Schematic representation of the crystal structure. (a) side and (b) top views of a unit cell ${\mathrm{Cr}}_2{\mathrm{S}}_3$. Three different Cr sites are indicated in (a) as I, II, and III. (c) Octahedral arrangement of ${\mathrm{CrS}}_6$. S-atoms in (c) are numbered based on their orientation relative to Cr-atom on the $c$-axis. (i.e., 1–3 above, 4–6 below Cr atom) Cr–S bond distances and S–Cr–S bond angles are indicated by $l_a$, $l_b$, ${\theta }_a$, and ${\theta }_b$.

Figure 2

(a) Electronic band structure of ${\mathrm{Cr}}_2{\mathrm{S}}_3$ with spin-up (red dashed line) and spin-down (blue solid line). (b) The total and element-projected partial density of states. (c) $p$-orbital projected PDOS for S and (d) Cr-site projected PDOS. The Fermi level is set to be zero, marked as a horizontal dashed line.

Figure 3

(a) Schematic representation of exchange interactions between Cr atoms. The relative positions of atoms are paralleled with the axes of the crystal structure shown. Horizontal blue arrows represent spin order for each Cr site. (b) MAE as a function of magnetization angle ${\theta }_m$ for both in-plane $\left(ab\right)$ and out-of-plane $\left(ac\right)$. A geometry for out-of-plane magnetization angle ${\theta }_m$ and lattice $a$ and $c$ are shown (inset).

Figure 4

The magnetic ground state of ${\mathrm{Cr}}_2{\mathrm{S}}_3$ under the $c$-axis uniaxial strain. (a) Relative total energies of four different collinear magnetic configurations (FM, FCFiM, FiM-I, and FiM-II, shown in Fig. S1) as a function of uniaxial strain along the $c$ axis. (b) Bulk elastic response (equivalent $ab$-plane strain) upon the $c$-axis strain. The color for the parabola in (a) and shadings in (b) are meant to match.

Figure 5

Dependence of structural parameters and magnetic moment on $c$-axis strain (ε) for ${\mathrm{Cr}}_2{\mathrm{S}}_3$. (a) Cr–Cr distances. (b) Cr–S–Cr bond angles (c) Magnetic moment m for different sites of Cr atom. (d) Schematic representation of the relative position of Cr sites. (e) The total magnetic moment M per unit cell. Spin orientation for each phase is shown in the inset following relative position shown in (c). (f) Cr – S bond lengths ($l_a$ and $l_b$) in ${\mathrm{CrS}}_6$ octahedral for all three Cr sites. Note that $l_a$ and $l_b$ for Cr(I) are different and are plotted with black solid and dashed lines, respectively. Vertically shaded regions are drawn as a guide to the eye, indicating different magnetic phases.

Figure 6

The dependence of electronic band structure and projected DOS of ${\mathrm{Cr}}_2{\mathrm{S}}_3$ on the various $c$-axis strain. (a)–(e) Band structures for compressive and tensile strains of (a) $-6\%$, (b) $-4\%$, (c) $+1\%$, (d) $+4\%$, and (e) $+6\%$ with spin-up/down (red-dashed/blue-solid). (f)–(j) S-3p PDOS, (k)–(o) Cr-3d site-projected PDOS under different corresponding strains to (a)–(e). The zero of energy is at the Fermi level, marked as a horizontal dashed line. The magnetic ground state is indicated with FCFiM, FiM-I, and FIM-II.

Figure 7

The variation of MAE value of ${\mathrm{Cr}}_2{\mathrm{S}}_3$ under uniaxial $c$-axis strain. Vertical lines at each strain range separate the corresponding magnetic ground states (FiM-II, FCFiM, FiM-I). The horizontal shaded region splits $XY$- and Ising-type spin orderings.