Multiband $\mathbf{k}·\mathbf{p}$ theory of monolayer $X\mathrm{Se}$ ($X=\mathrm{In},\mathrm{Ga}$)

We derive the multiband $\mathbf{k}·\mathbf{p}$ Hamiltonian for an electron in monolayer $X\mathrm{Se}$ ($X=\mathrm{In}$, Ga) near the $\mathrm{\Gamma }$ point in the presence of spin-orbit coupling and strain. With four conduction bands and ten valence bands included, the 14-band $\mathbf{k}·\mathbf{p}$ Hamiltonian is capable of describing the nonparabolic energy band dispersion, strong interband mixing, and strain-induced effects, in good agreement with first-principles calculations. This $\mathbf{k}·\mathbf{p}$ theory provides a simple and convenient way to understand and manipulate the optical and transport properties of monolayer $X\mathrm{Se}$ and their nanostructures by strain and external fields. Interestingly, we find that uniaxial strain can even induce an indirect-to-direct band-gap transition in these two-dimensional materials.

Figure 1

(a) Schematic of the primitive cell of an $X\mathrm{Se}$ monolayer ($X=\mathrm{In}$, Ga). (b) and (c) are side and top views, respectively, of the hexagonal structure. The purple and green spheres correspond to $X$ and selenium atoms, respectively. The $x$ and $y$ axes along zigzag and armchair directions specify the coordinates used in the $\mathbf{k}·\mathbf{p}$ Hamiltonian. (d) A sketch of the monolayer InSe electronic bands around the $\mathrm{\Gamma }$ point, omitting the spin-orbit coupling. $E_F$ is the Fermi Energy.

Figure 2

The energy band structures of monolayer InSe in the (a) absence or (b) presence of SOC. (c) and (d): the same as (a) and (b), but for monolayer GaSe. The red solid (black dashed) lines are obtained from the $\mathbf{k}·\mathbf{p}$ Hamiltonian in Eq. (4) (the first-principles calculations).

Figure 3

Splitting of the valence band energy levels at the $\mathrm{\Gamma }$ point due to SOC and uniaxial strain: (a) No SOC and no strain. (b) SOC only. (c) SOC and small strain. (d) SOC and large strain.

Figure 4

Electronic band valence structures obtained from the $\mathbf{k}·\mathbf{p}$ theory under uniaxial strains: (a) $\epsilon =0.09$, (b) $\epsilon =0.18$, (c) $\epsilon =0.27$, and (d) $\epsilon =0.36$.

Figure 5

Projected energy band structure of monolayer InSe with different uniaxial tensile strains along the zigzag direction: ${\epsilon }_{yy}-{\epsilon }_{xx}=$ (a) 0, (b) 3, (c) 6, and (d) $9\%$. The Fermi level is set to be 0 eV. In the projected band structure, the states dominated by $p_x, p_y$, and $p_z$ orbitals are plotted in red, green, and blue. All the results are obtained from the density-functional theory (DFT) with the PBE functional adopted. (See Appendix pp1.) The energy gap is shaded by the gray rectangles, and the values (in eV) obtained with PBE are given.

Figure 6

The electronic band structures under different biaxial strains (a) 1, (b) 2, (c) 3, and (d) $4\%$. The red solid lines correspond to results obtained from our $\mathbf{k}·\mathbf{p}$ model, while the black dashed lines are energy bands for biaxial strained InSe obtained from the DFT with the PBE functional adopted.