Ab initio study of the effective Coulomb interactions and Stoner ferromagnetism in $M_2\mathrm{C}$ and $M_2{\mathrm{CO}}_2 MX-\mathrm{enes} (M=\mathrm{Sc},\mathrm{Ti},\mathrm{V},\mathrm{Cr},\mathrm{Fe},\mathrm{Zr},\mathrm{Nb},\mathrm{Mo},\mathrm{Hf},\mathrm{Ta})$

The $\mathit{MX}$-enes are favorable two-dimensional materials for a wide variety of applications such as electronic and spintronic devices. The quantum confinement effects arising from reduced dimensionality and the presence of narrow $d$ bands suggest the importance of electron correlation effects in these materials. We calculate the strength of the effective screened Coulomb interaction $U$ for pristine $M_2\mathrm{C}$ and functionalized $M_2{\mathrm{CO}}_2 (M=\mathrm{Sc},\mathrm{Ti},\mathrm{V},\mathrm{Cr},\mathrm{Fe},\mathrm{Zr},\mathrm{Nb},\mathrm{Mo},\mathrm{Hf},\mathrm{Ta}) \mathit{MX}$-enes by employing ab initio calculations in conjunction with the random-phase approximation. The calculated $U$ values lie between 1.9 and 5.2 eV and depend on the ground-state electronic structure, $d$-electron number, and chemical functionalization. The $U$ for the $3d$-based systems shows oscillation when $M$ is varied, which can be explained by the $3d$ partial density of states around the Fermi energy. For nonmetallic functionalized $\mathit{MX}$-enes $M_2{\mathrm{CO}}_2$, the reduced dimensionality and the existence of a band gap result in a small screening of the Coulomb interaction. The long-range Coulomb interactions for both $M_2\mathrm{C}$ and $M_2{\mathrm{CO}}_2$ are also reported. For metallic systems, the interactions turn out to be local, and the nonlocal part is strongly screened due to $d$ states, making metallic $\mathit{MX}$-enes correlated systems. Furthermore, on the basis of the calculated effective Coulomb parameters $U$ and $J$ we discuss the appearance of ferromagnetism at pristine and functionalized $\mathit{MX}$-enes within the Stoner model.

Figure 1

(a) Top and side views of the two-dimensional crystal structure of $M_2\mathrm{C}$ layer. (b) The $d$-like MLWFs for Cr atoms of ${\mathrm{Cr}}_2\mathrm{C} \mathit{MX}$-ene. The $e_g$-like and $t_{2}g$-like MLWFs are shown separately. The blue and tiny dark circles exhibit $M$ and C atoms, respectively.

Figure 2

Top and side views of three types of the considered thinnest functionalized $\mathit{MX}$-enes $M_2{\mathrm{CO}}_2$. (a) Two oxygen atoms are located on top of the hollow sites of three neighboring C atoms (such as ${\mathrm{Cr}}_2{\mathrm{CO}}_2, {\mathrm{Fe}}_2{\mathrm{CO}}_2, {\mathrm{Mo}}_2{\mathrm{CO}}_2$, and ${\mathrm{Zr}}_2{\mathrm{CO}}_2$). (b) Two oxygen atoms are located above the two transition metals (such as ${\mathrm{Ti}}_2{\mathrm{CO}}_2, {\mathrm{V}}_2{\mathrm{CO}}_2, {\mathrm{Nb}}_2{\mathrm{CO}}_2, {\mathrm{Ta}}_2{\mathrm{CO}}_2$, and ${\mathrm{Hf}}_2{\mathrm{CO}}_2$). (c) One oxygen atom is located above the transition metal and the other oxygen is located on top of a hollow site (such as ${\mathrm{Sc}}_2{\mathrm{CO}}_2$).

Figure 3

(a) Total and orbital-projected non-spin-polarized band structures for ${\mathrm{Cr}}_2\mathrm{C} \mathit{MX}$-ene based on DFT-PBE. The Fermi level is set to the zero energy. (b) The same as (a) for spin-polarized calculation.

Figure 4

(a) Non-spin-polarized DFT-PBE and Wannier-interpolated band structure of ${\mathrm{Cr}}_2\mathrm{C} \mathit{MX}$-ene. (b) and (c) The same as (a) for spin-polarized calculation.

Figure 5

(a) Bare Coulomb interaction $V$ and (b) non-spin-polarized partially screened interaction $U$ for $d$ electrons of $M_2\mathrm{C} \mathit{MX}$-enes. The corresponding bare and partially screened on-site Coulomb interaction parameters for functionalized $\mathit{MX}$-enes $M_2{\mathrm{CO}}_2$ are also presented in (c) and (d). Points with $\times$ denote $U$ parameters of spin-polarized calculation for magnetic systems (Cr- and Fe-based materials).

Figure 6

Projected density of states (PDOS) for the $M_2\mathrm{C} \mathit{MX}$-enes. Each panel shows the DOS projected onto the $M$-site cation (red) and onto the C atom (blue). Red dashed lines are DOS projected onto the Cr and Fe sites in the non-spin-polarized calculation.

Figure 7

Projected density of states (PDOS) for the $M_2{\mathrm{CO}}_2$ functionalized $\mathit{MX}$-enes. Each panel shows the DOS projected onto the $M$-site cation (red), onto the C atom (blue), and the O atom (green). Red dashed lines are DOS projected onto the Cr and Fe sites in the non-spin-polarized calculation.

Figure 8

Stoner criterion of different $M$ atoms in $M_2\mathrm{C}$ and $M_2{\mathrm{CO}}_2 \mathit{MX}$-enes.