Two-dimensional square ternary Cu${}_2$MX${}_4$ ($M$ = Mo, W; $X$ = S, Se) monolayers and nanoribbons predicted from density functional theory

Two-dimensional (2D) materials often adopt a hexagonal lattice. We report on a class of 2D materials, Cu${}_2$$M{X}_4$ ($M$ = Mo, W; $X$ = S, Se), that has a square lattice. Up to three monolayers, the systems are kinetically stable. All of them are semiconductors with band gaps from 2.03 to 2.48 eV. Specifically, the states giving rise to the valence band maximum are confined to the Cu and $X$ atoms, while those giving rise to the conduction band minimum are confined to the $M$ atoms, suggesting that spontaneous charge separation occurs. The semiconductive nature makes the materials promising for transistors, optoelectronics, and solar energy conversion. Moreover, the ferromagnetism on the edges of square Cu${}_2$$M{X}_4$ nanoribbons opens applications in spintronics.

Figure 1

HSE06 band structures of Cu${}_2$MoS${}_4$ in the (a) $P\overline{4}2\mathrm{m}$ and (b) $I\overline{4}2\mathrm{m}$ phases.

Figure 2

Phonon band structures of square (a) Cu${}_2$MoS${}_4$, (b) Cu${}_2$MoSe${}_4$, (c) Cu${}_2$WS${}_4$, and (d) Cu${}_2$WSe${}_4$ monolayers calculated by the finite displacement method.

Figure 3

(a) Total energy of square Cu${}_2$MoS${}_4$ as a function of the lattice parameters. (b) Energy required to exfoliate a single layer from a multilayer structure. Insets:Top and side views demonstrating the exfoliation. (c) Top (left) and side (right) views of singe-layer Cu${}_2$MoS${}_4$. (d) Electron localization function in the (110) plane along Mo-S (upper) and Cu-S (lower) bonds.

Figure 4

Snapshots of square Cu${}_2$MoS${}_4$ at (a) 500 K, (b) 750 K, and (c) 1000 K.

Figure 5

(a) HSE06 band structure of square Cu${}_2$MoS${}_4$. The inset shows the Brillouin zone. (b) Projected density of states. (c) Charge density (0.03 |$e$|/ų) of states giving rise to the VBM (left) and the CBM (right). (d) Band alignment. Black solid and red dashed lines represent the PBE and HSE06 functionals, respectively. The water reduction (H${}^{+}$/H${}_2$) and oxidation (H${}_2$O/O${}_2$) potentials are shown by blue dotted lines.

Figure 6

VBM, CBM, and band gap variations of square Cu${}_2$MoS${}_4$ as a function of the number of layers calculated by PBE in the (a) $P\overline{4}2\mathrm{m}$ and (b) $I\overline{4}2\mathrm{m}$ stackings.

Figure 7

Top and side views of the (a) 5-zCu, (b) 5-zS, (c) 5-zCuS, and (d) 5-lCuS NRs.

Figure 8

(a) Band gaps (as taken from the band structures) of four types of NRs as a function of the width. For the zCu and lCuS NRs, the band gaps of both nondegenerate spin channels are shown. The insets depict the charge density (0.002 |$e$|/ų) of states, giving rise to the VBM and CBM for the 5-zS (bottom) and 5-zCuS (top) NRs. (b) Total magnetic moment (MM, or the difference between the spin majority and the spin minority charges) and energy difference (Δ$E$) between spin-polarized and spin-degenerate zCu and lCuS NRs as a function of the width. The insets show the charge difference between the spin majority and the spin minority densities (0.002 |$e$|/ų) of the 5-zCu (bottom) and 5-lCuS (top) NRs.

Figure 9

Spin-polarized total density of states (left) and corresponding schematic diagrams (right) of the edge states for the (a) 5-zCu, (b) 5-zS, (c) 5-zCuS, and (d) 5-lCuS NRs.