Electronic structure theory of strained two-dimensional materials with hexagonal symmetry

We derive electronic tight-binding Hamiltonians for strained graphene, hexagonal boron nitride, and transition-metal dichalcogenides based on Wannier transformation of ab initio density functional theory calculations. Our microscopic models include strain effects to leading order that respect the hexagonal crystal symmetry and local crystal configuration and are beyond the central force approximation which assumes only pairwise distance dependence. Based on these models, we also derive and analyze the effective low-energy Hamiltonians. Our ab initio approaches complement the symmetry group representation construction for such effective low-energy Hamiltonians and provide the values of the coefficients for each symmetry-allowed term. These models are relevant for the design of electronic device applications since they provide the framework for describing the coupling of electrons to other degrees of freedom including phonons, spin, and the electromagnetic field. The models can also serve as the basis for exploring the physics of many-body systems of interesting quantum phases.

Figure 1

The conventions for the honeycomb hexagonal crystal structure: (a) Top view of the crystal lattice with primitive vectors ${\mathit{a}}_i$ with $A\left(B\right)$ basis atoms shown as blue (red) solid circles. In hBN, boron (nitrogen) atoms occupy sublattice $A\left(B\right)$ sites, whereas in TMDCs metal atoms (chalcogen pairs) sit at $B\left(A\right)$ sites. The three thick black arrows labeled by $t_i$ denote the hopping bonds used in strained graphene and hBN up to third-nearest neighbors in Eq. (6). For TMDCs, the hoppings from $M$ sites are denoted by the thick orange arrows instead for Eqs. (10) and (11). (b) Perspective side view of the trilayer structure in TMDC. (c) Brillouin zone in momentum space.

Figure 2

Wannier tight-binding Hamiltonian construction from DFT for the ${\mathrm{WSe}}_2$ monolayer: (a) DFT band structure, blue circles (without spin-orbit coupling) with the 11 $p\text{−}d$ hybrid bands which are relevant for low-energy electronic properties, used to derive the Wannier tight-binding Hamiltonian (red lines). (b) Wannier Hamiltonian results with truncation to limit the range of neighbor coupling terms. (c) Hamiltonian augmented by the atomic spin-orbit-coupling terms (red lines), compared with the full DFT calculation with spin-orbit coupling included (blue circles).

Figure 3

Comparison of the (a) DFT and (b) TBH electronic band structure without spin-orbit coupling for a monolayer ${\mathrm{WSe}}_2$ crystal with isotropic strain. The black dashed lines are the bands from the pristine crystal, whereas the red (blue) solid ones are from the crystal with $-2\%(+2\%)$ isotropic strain. The high-energy bands in the DFT calculations are those beyond the $p\text{−}d$ hybrids included in the TBH basis. The vacuum level is at zero energy. (c) and (d), similar comparison with spin-orbit coupling.

Figure 4

The energy gap between the highest valence band and the lowest conduction band at the K valley for a ${\mathrm{MoS}}_2$ monolayer crystal under isotropic strain (a) without and (b) with spin-orbit coupling corrections included. The blue (red) lines are from DFT (TBH) calculations. The slopes agree well for the strain effects.