Absolute deformation potentials of two-dimensional materials

We present ab initio calculations of uniaxial absolute deformation potentials of the valence and the conduction bands in monolayer ${\mathrm{MoS}}_2,{\mathrm{MoSe}}_2,{\mathrm{WS}}_2,{\mathrm{WSe}}_2$, h-BN, and phosphorene. Calculations are performed using both semilocal and hybrid functionals. The absolute positions of the band edges in strained and unstrained materials are determined using the vacuum level as reference. For ${\mathrm{WSe}}_2$, we compare the obtained results with measured shifts of the valence band maximum (VBM) and the conduction band minimum (CBM) induced by uniaxial strain and find a very good agreement. The parameters describing the shifts in the VBM and CBM positions under strain can be used in the modeling of devices such as tunneling field-effect transistors.

Figure 1

Top (a) and side (b) views of the structures of ${\mathrm{MoS}}_2$, h-BN, and phosphorene along with the representations of the Brillouin zones (c) and of the band structures calculated with the PBE functional (d). In (d), the points for which the absolute deformation potentials are calculated are indicated. Band structures are aligned to the vacuum level.

Figure 2

Energies of the VBM at the K and $\mathrm{\Gamma }$ points and of the CBM at the K and $\mathrm{\Lambda }$ points of ${\mathrm{MoS}}_2$ as a function of the uniaxial strain in the $x$ direction. Squares represent calculated values and solid lines are linear fits using values calculated for strains of $+1$ and $-1\%$. The vacuum energy is used as reference.

Figure 3

Energies of the VBM at the K and $\mathrm{\Gamma }$ points and of the CBM at the K and $\mathrm{\Lambda }$ points of the transition-metal dichalcogenides as a function of the uniaxial strain in the $x$ direction. Solid and dashed lines correspond to PBE0 and PBE results, respectively. The vacuum energy is used as reference. The band energies result from a linear extrapolation of the values calculated at $-1$ and $+1\%$ strain, indicated with symbols.

Figure 4

Energies of the VBM at the K and $\mathrm{\Gamma }$ points and of the CBM at the K point of monolayer hexagonal boron nitride as a function of the uniaxial strain in the $x$ direction. Solid and dashed lines correspond to PBE0 and PBE results, respectively. The vacuum energy is used as reference. The band energies result from a linear extrapolation of the values calculated at $-1$ and $+1\%$ strain, indicated with symbols.

Figure 5

Energies of the VBM and of the CBM at the $\mathrm{\Gamma }$ point of the phosphorene as a function of uniaxial strain in the $x$ and in the $y$ direction. Solid and dashed lines correspond to PBE0 and PBE results, respectively. The vacuum energy is used as reference. The band energies result from a linear extrapolation of the values calculated at $-1$ and $+1\%$ strain, indicated with symbols.

Figure 6

Comparison between the energy shifts of the VBM and CBM of ${\mathrm{WSe}}_2$, as a function of uniaxial strain, as extracted from experiments [28] (red and blue squares, respectively) and calculated with the PBE0 functional in the present study.