Ferroelectricity and ferromagnetism in a $\mathrm{VO}{\mathrm{I}}_2$ monolayer: Role of the Dzyaloshinskii-Moriya interaction

Multiferroics with intrinsic ferromagnetism and ferroelectricity are highly desired but rather rare, while most ferroelectric magnets are antiferromagnetic. A recent theoretical work [Tan et al., Phys. Rev. B 99, 195434 (2019)] predicted that oxyhalides $\mathrm{VO}{X}_2$ ($X$: halogen) monolayers are two-dimensional multiferroics by violating the empirical $d^0$ rule. Most interestingly, the member $\mathrm{VO}{\mathrm{I}}_2$ are predicted to exhibit spontaneous ferromagnetism and ferroelectricity. In this work, we extend the previous study on the structure and magnetism of $\mathrm{VO}{\mathrm{I}}_2$ monolayer by using density-functional theory and Monte Carlo simulation. The presence of the heavy element iodine with a strong spin-orbit coupling gives rise to an effective Dzyaloshinskii-Moriya interaction in the polar structure, which favors a short-period spiral magnetic structure.. Another interesting result is that the on-site Coulomb interaction can strongly suppress the polar distortion thus leading to a ferromagnetic metallic state. Therefore, the $\mathrm{VO}{\mathrm{I}}_2$ monolayer is either a ferroelectric insulator with spiral magnetism or a ferromagnetic metal, instead of a ferromagnetic ferroelectric system. Our study highlights the key physical role of the Dzyaloshinskii-Moriya interaction.

Figure 1

(a) The top view (upper) of the $\mathrm{VO}{\mathrm{I}}_2$ monolayer. The dashed rectangle indicates the primitive cell. (b) The side views of a unit cell, where the O–V–I bond angle Φ can characterize the polar distortion. (c) The charge-density profile indicates the occupied $d_{xy}$ orbital. Noting that the $x$ and $y$ directions are defined along the two equivalent V-I bonds in the paraelectric Pmmm phase, and the $z$ direction is along the V–O bond (i.e., the $a$ axis).

Figure 2

(a) The four collinear magnetic orders, which are calculated to extract the values of $J_a$,$J_b$, and $J_{ab}$. (b), (c) The two ${120}^{\circ }$ noncollinear magnetic orders, which are calculated to extract the Dzyaloshinskii-Moriya interactions along $a$-/$b$ direction, respectively.

Figure 3

(a) The MC simulated heat capacity $C$ as a function of temperature for the $\mathrm{VO}{\mathrm{I}}_2$ monolayer. (b) The MC snapshot of spiral-spin order at low temperature. The color bar denotes the $a$ component of normalized spin.

Figure 4

Structure and polarization as a function of $U_{\mathrm{eff}}$, calculated with SOC and without SOC. (a) Lattice constants $a$ and $b$. (b) The O–V–I bond angle Φ. Inset: the top view of a unit cell. (c) Ferroelectric polarization.

Figure 5

(a) Comparison of DOSs of $\mathrm{VO}{\mathrm{I}}_2$ monolayer with/without $U_{\mathrm{eff}}$. Inset: magnified view near the Fermi level. (b) The band gaps of spin-up and spin-down channels as a function of $U_{\mathrm{eff}}$. (c)–(d) Projected DOS (PDOS) of V's $d$ orbitals and O's $p$ orbitals with/without $U_{\mathrm{eff}}$. Two characteristics are clear for the $U_{\mathrm{eff}}=1\mathrm{eV}$ case. First, the Hubbard splitting is enlarged. Second, the occupied $d^1$ bands are broader.

Figure 6

Magnetic coefficients as a function of $U_{\mathrm{eff}}$. (a) The exchange interactions. (b) The $c$ component of ${\mathbf{D}}_b$. (c) The magnetocrystalline anisotropic coefficients.