Origin of layer dependence in band structures of two-dimensional materials

We study the origin of layer dependence in band structures of two-dimensional (2D) materials. We find that the layer dependence, at the density functional theory (DFT) level, is a result of quantum confinement and the nonlinearity of the exchange-correlation functional. We use this to develop an efficient scheme for performing DFT and $GW$ calculations of multilayer systems. We show that the DFT and quasiparticle band structures of a multilayer system can be derived from a single calculation on a monolayer of the material. We test this scheme on multilayers of ${\mathrm{MoS}}_2$, graphene, and phosphorene. This new scheme yields results in excellent agreement with the standard methods at a fraction of the computation cost. This helps overcome the challenge of performing fully converged $GW$ calculations on multilayers of 2D materials, particularly in the case of transition-metal dichalcogenides, which involve very stringent convergence parameters.

Figure 1

(a) Bilayer ${\mathrm{MoS}}_2$ in AB stacking, red spheres represent Mo atoms and yellow spheres S atoms. (b) Planar averaged charge density of bilayer ${\mathrm{MoS}}_2$ (black solid line), planar averaged charge density of layer “a” (red dash-dot line), layer “b” (blue dashed line) and sum of layer a and b (magenta line not seen). The shaded regions indicate the position of the ${\mathrm{MoS}}_2$ layers in the simulation cell. (c) Planar averaged DFT potential of the layer a, $V_{\mathrm{tot}}^a$ (red dash-dot line), layer b, $V_{\mathrm{tot}}^b$ (blue dashed line), the sum $V_{\mathrm{tot}}^a+V_{\mathrm{tot}}^b$ (solid magenta line), and bilayer, $V_{\mathrm{tot}}^{\mathrm{bi}}$ (black solid line). (d) Difference $V_{\mathrm{tot}}^{\mathrm{bi}}-(V_{\mathrm{tot}}^a+V_{\mathrm{tot}}^b)$ (black solid line) and $V_{xc}[{\rho }^a+{\rho }^b]-(V_{xc}\left[{\rho }^a\right]+V_{xc}\left[{\rho }^b\right])$ (violet dashed line).

Figure 2

(a), (b), and (c) plot the DFT band structure of monolayer, bilayer, and trilayer ${\mathrm{MoS}}_2$, respectively. (d), (e), and (f) plot the DFT band structures of monolayer, bilayer, and trilayer graphene. The black solid line indicates the result from separate DFT calculations on these systems. The red dashed line shows the result obtained by a single-shot diagonalization of the constructed Hamiltonian Eq. (1) for these systems in the basis of the wave functions of the constituent layers. The blue dashed lines are the eigenvalues obtained by considering only quantum confinement in these systems.

Figure 3

Modulus squared wave functions of monolayer ${\mathrm{MoS}}_2$, integrated out along [010] lattice vector direction. $z$ is the out-of-plane direction and $x$ the [100] direction. Top: CBM wave functions at K, $\mathrm{\Lambda }$, and $\mathrm{\Gamma }$ points, respectively. Bottom: VBM wave functions at K, $\mathrm{\Lambda }$, and $\mathrm{\Gamma }$ points, respectively.

Figure 4

(a) and (b) Convergence of the constructed $\langle {\psi }_{CBM}|{\mathrm{\Sigma }}^{\mathrm{bi}}|{\psi }_{CBM}\rangle$ and $\langle {\psi }_{CBM}|{\mathrm{\Sigma }}^{tri}|{\psi }_{CBM}\rangle$ with the number of bands for the CBM at $\mathrm{\Lambda }$ (green crosses). The horizontal line is the converged result from full calculations. The blue circles show the slow convergence of the full calculation with the usual procedure. (c), (d), and (e) plot the quasiparticle band structure of monolayer, bilayer and trilayer ${\mathrm{MoS}}_2$, respectively. The black solid line indicates the result from separate $GW$ calculations. The red dashed line shows the result obtained using the scheme described in the text.

Figure 5

(a), (b), and (c) DFT charge density, potential, and band structure of bilayer ${\mathrm{MoS}}_2$ at the equilibrium interlayer spacing of 6.0 Å, respectively. (d), (e), and (f) For interlayer spacing of 7.5 Å. (g), (h), and (i) For interlayer spacing of 9.0 Å.

Figure 6

(a) Band gap of bilayer ${\mathrm{MoS}}_2$ as a function of interlayer spacing evaluated using the constructed $\mathrm{\Sigma }$ for various approximations to $\mathrm{\Delta }\mathrm{\Sigma }$. The dashed line marks the monolayer band gap within $GW$. (b) VBM and CBM of the bilayer as a function of interlayer spacing, evaluated using $\mathrm{\Delta }\mathrm{\Sigma }=\mathrm{\Delta }{\mathrm{\Sigma }}_{GW}^{800}$. The dashed line marks the $GW$ VBM and CBM of a monolayer.