Band alignment of two-dimensional semiconductors for designing heterostructures with momentum space matching

We present a comprehensive study of the band alignments of two-dimensional (2D) semiconducting materials and highlight the possibilities of forming momentum-matched type I, II, and III heterostructures, an enticing possibility being atomic heterostructures where the constituent monolayers have band edges at the zone center, i.e., $\mathrm{\Gamma }$ valley. Our study, which includes the group IV and III-V compound monolayer materials, group V elemental monolayer materials, transition-metal dichalcogenides, and transition-metal trichalcogenides, reveals that almost half of these materials have conduction and/or valence band edges residing at the zone center. Using first-principles density functional calculations, we present the type of the heterostructure for 903 different possible combinations of these 2D materials which establishes a periodic table of heterostructures.

Figure 1

Illustration of type I, II, and III heterostructures and their various devices, where red (blue) indicates conduction (valence) bands. (a) Laser utilizing type I heterostructure. (b) High electron mobility transistor, (c) carrier separation in solar cell, and (d) interlayer excitons utilizing type II junction. (e) Tunneling field effect transistor based on type III junction. Specifically, when material A and material B merge, the resulting heterostructure is type I if ${\mathrm{VBM}}_A<{\mathrm{VBM}}_B<{\mathrm{CBM}}_B<{\mathrm{CBM}}_A$; is type II if ${\mathrm{VBM}}_A<{\mathrm{VBM}}_B<{\mathrm{CBM}}_A<{\mathrm{CBM}}_B$; and type III if ${\mathrm{VBM}}_A<{\mathrm{CBM}}_A<{\mathrm{VBM}}_B<{\mathrm{CBM}}_B$.

Figure 2

(a) Stable crystalline structures (planar and buckled) of group IV and group III-V monolayer compounds and their hexagonal BZ with high-symmetry points are shown. The relevant bond lengths, lattice parameter, and buckling distance are indicated by $a$, $d$, and $\mathrm{\Delta }$, respectively. (b) Comparative band alignment of group IV and group III-V monolayer compounds where CBM (shown in red) and VBM (shown in blue) values obtained from PBE and HSE06 calculations are shown with bar and line plots, respectively. The positions of CBM and VBM on the BZ are indicated. Note that the vacuum energies were set to zero while calculating the band diagrams. Structures having buckled (b), puckered (w), 2H, 1T, and distorted 1T phases are indicated with the corresponding indices, where no index is used for planar phases.

Figure 3

(a) Stable crystalline structures (buckled and puckered) of group V elements and their corresponding hexagonal and rectangular BZ with high-symmetry points are shown. The relevant bond lengths and lattice parameters are indicated by $a$ and $d$, respectively. (b) Same as Fig. 2, but for group V monolayers.

Figure 4

(a) Stable crystalline structures (1T and 2H phases) of TMDs and their corresponding hexagonal BZ with high-symmetry points are shown. The relevant bond lengths and lattice parameters are indicated by $a$ and $d$, respectively. Note that the structures of ${\mathrm{ReS}}_2$ and ${\mathrm{ReSe}}_2$ are in a distorted 1T phase. (b) Same as Fig. 2 but for TMDs.

Figure 5

(a) Stable crystalline structure of TMTs and its corresponding rectangular BZ with high-symmetry points are shown. The relevant bond lengths and lattice parameters are indicated by $a$ and $d$, respectively. (b) Same as Fig. 2 but for TMTs.

Figure 6

Periodic table of heterostructures. Type I, II, and III heterostructures are represented by green, red, and blue boxes, respectively. The lower left and upper right of regions of the diagonal line present results calculated by PBE and HSE06, respectively. Two-colored boxes, which correspond to the same energy within the error range of our DFT study, indicate equal chances of two different types of heterostructures for the corresponding materials.

Figure 7

(a) Geometrical parameters and (b) the electronic band structure of the ${\mathrm{MoSe}}_2-{\mathrm{WSe}}_2$ heterostructure. In the band structure, bands projected to ${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$ layers are indicated with red (dark) and green (light) circles, respectively. The Fermi level is shown with blue dashed line.