Magnetism of layered chromium sulfides $M$CrS${}_2$ ($M=\mathrm{Li}$, Na, K, Ag, and Au): A first-principles study

$M$CrS${}_2$ compounds ($M=\mathrm{Li}$, Na, K, Cu, Ag, and Au) with triangular Cr layers show a large variety of magnetic ground states, ranging from 120${}^{\circ }$ antiferromagnetic order of Cr spins in LiCrS${}_2$ to double stripes in AgCrS${}_2$, helimagnetic order in NaCrS${}_2$, and, finally, ferromagnetic Cr layers in KCrS${}_2$. On the base of ab initio band structure calculations and an analysis of various contributions to exchange interactions between Cr spins, we explain this tendency as originating from a competition between antiferromagnetic direct nearest-neighbor $d$-$d$ exchange and ferromagnetic superexchange via Sulfur $p$ states, which leads to a change of sign of the nearest-neighbor interaction, depending on the radius of an $M$ ion. It is shown that other important interactions are the third-neighbor interaction in a layer and interlayer exchange. We suggest that strong magnetoelastic coupling is probably responsible for the multiferroic properties of at least one material in this family, namely, AgCrS${}_2$.

Figure 1

High-temperature rhombohedral crystal structures of KCrS${}_2$ (a), AuCrS${}_2$ (b), and AgCrS${}_2$ (c). Also shown is a distorted CrS${}_6$ octahedron surrounding a Cr ion in the cell center.

Figure 2

Partial densities of states in AgCrS${}_2$ with the FM alignment of Cr moments. Energies are given relative to the Fermi level $E_F$.

Figure 3

Representation of double-stripe [open (blue) arrows] and 90${}^{\circ }$ [filled (red) arrows] magnetic structures within the Cr plane. One underlying (S${}_1$) and one overlying (S${}_2$) sulfur layer are shown as well. Low-temperature intraplane exchanges between the first ($J_{1x}$, $J_{1y}$), the second ($J_{2x}$, $J_{2y}$), and the third ($J_{3x}$, $J_{3y}$) neighbors are shown as curved lines with arrows. In the high-temperature phase $J_{1x}=J_{1y}=J_1$, $J_{2x}=J_{2y}=J_2$, and $J_{3x}=J_{3y}=J_3$.

Figure 4

Calculated [open (black) circles] and fitted [filled (red) circles], using the least-squares method in the Heisenberg model, profiles $E\left(\mathbf{q}\right)$ of magnetic energies for AgCrS${}_2$ for two sets of wave vectors; see the text. Partial contributions in magnetic energy, calculated according to Eqs. (3, 4, 5, 6, 7), are shown as well. The ${\epsilon }_1\left(\mathbf{q}\right)$, ${\epsilon }_2\left(\mathbf{q}\right)$, ${\epsilon }_3\left(\mathbf{q}\right)$, and ${\epsilon }_z\left(\mathbf{q}\right)$ profiles are represented by solid (blue), dashed (green), dashed with one dot (red), and dashed with two dots (magenta) lines. The dispersion curves shown in (a) and (b) are calculated for the high-temperature phase with ferro (a) and antiferro (b) interlayer ordering, whereas those in (c) are calculated for the low-temperature structure and antiferro interlayer ordering. The ${90}^{\circ }$ structure, by which we model the double-stripe magnetic structure observed in AgCrS${}_2$ and AuCrS${}_2$, is marked by the arrow.

Figure 5

Different contributions to $J_1$.

Figure 6

Schematic dependence of different contributions to the nearest-neighbor Cr-Cr exchange $J_1$.

Figure 7

A possible exchange path contributing to the antiferromagnetic exchange of third neighbors $J_3$.