Strength of effective Coulomb interaction in two-dimensional transition-metal halides $M{X}_2$ and $M{X}_3$ ($M=\mathrm{Ti}$, V, Cr, Mn, Fe, Co, Ni; $X=\mathrm{Cl}$, Br, I)

We calculate the strength of the effective on-site Coulomb interaction (Hubbard $U$) in two-dimensional transition-metal (TM) dihalides $M{X}_2$ and trihalides $M{X}_3$ ($M=\mathrm{Ti}$, V, Cr, Mn, Fe, Co, Ni; $X=\mathrm{Cl}$, Br, I) from first principles using the constrained random-phase approximation. The correlated subspaces are formed from $t_{2g}$ or $e_g$ bands at the Fermi energy. Elimination of the efficient screening taking place in these narrow bands gives rise to sizable interaction parameters $U$ between the localized $t_{2g}$ ($e_g$) electrons. Due to this large Coulomb interaction, we find $U/W>1$ (with the bandwidth $W$) in most TM halides, making them strongly correlated materials. Among the metallic TM halides in the paramagnetic state, the correlation strength $U/W$ reaches a maximum in $\mathrm{Ni}{X}_2$ and $\mathrm{Cr}{X}_3$ with values much larger than the corresponding values in elementary TMs and other TM compounds. Based on the Stoner model and the calculated $U$ and $J$ values, we discuss the tendency of the electron spins to order ferromagnetically.

Figure 1

(a) Side and top views of the two-dimensional crystal structure of TM dihalides $M{X}_2$. (b) Side and top views of the two-dimensional crystal structure of TM trihalides $M{X}_3$. The blue and purple spheres exhibit $M$ and $X$ atoms, respectively. The $M$ atoms in (a) form a frustrated triangular lattice, while in (b) their honeycomb structure is a bipartite and hence nonfrustrated lattice.

Figure 2

DFT-PBE (red) and Wannier-interpolated band structures (blue) of non-spin-polarized (a) ${\mathrm{NiI}}_2$ and (b) ${\mathrm{CrI}}_3$. (c) The $e_g$-like MLWFs for Ni atoms of ${\mathrm{NiI}}_2$. (d) The $t_{2g}$-like MLWFs for Cr atoms of ${\mathrm{CrI}}_3$.

Figure 3

Orbital-resolved DOS for the non-spin-polarized $M{\mathrm{I}}_2$ and $M{\mathrm{I}}_3$ TM halides. Each panel shows the DOS projected onto $3d$ states of the $M$ atom as well as on $5p$ states of the I atom. The two distinct groups of $d$ bands correspond to $t_{2g}$ and $e_g$ bands, respectively. The former overlaps with $5p$ states in the late-TM iodides.

Figure 4

(a) Stoner criterion for $M{\mathrm{I}}_2$ and $M{\mathrm{I}}_3$ TM halides. (b) Calculated magnetic moments (in units of ${\mu }_B$) of TM atoms for $M{\mathrm{I}}_2$ and $M{\mathrm{I}}_3$ TM halides.