Simple linear response model for predicting energy band alignment of two-dimensional vertical heterostructures

The Anderson and midgap models are often used in the study of semiconductor heterojunctions, but for van der Waals (vdW) vertical heterostructures they have shown only very limited success. Using the group-IV monochalcogenide vertical heterostructures as a prototypical system, we propose a linear response model and compare the effectiveness of these models in predicting density functional theory (DFT) band alignments, band types, and band gaps. We show that the DFT band alignment is best predicted by the linear response model, which falls in between the Anderson and midgap models. Our proposed model can be characterized by an interface dipole $\alpha \times (E_{m2}-E_{m1})$, where the linear response coefficient $\alpha =0$ and 1 corresponds to the Anderson and midgap model, respectively, and $E_m$ is the midgap energy of the monolayer, which can be viewed as an effective electronegativity. For group-IV monochalcogenides, we show that $\alpha =0.34$ best captures the DFT band alignment of the vdW heterostructure, and we discuss the viability of the linear response model considering other effects such as strains and band hybridization, and conclude with an application of the model to predict experimental band alignments.

Figure 1

Group-IV monochalcogenides. (a) Side view and (b) top view of the optimized atomic structure of group-IV monochalcogenide heterostructure. The rectangular unit cell with in-plane lattice vectors, $\vec{a}$ and $\vec{b}$ are indicated. (c) Corresponding Brillouin zone with symmetry directions.

Figure 2

Extracting linear response coefficient $\alpha$. DFT vacuum dipole step of heterostructures versus midgap energy difference between monolayers (as within the Anderson picture). Linear fitted line with a slope $\alpha =0.34$ is indicated.

Figure 3

Comparing energy band models. (a) Energy band alignment for two isolated 2D monolayers. According to the Anderson model, the heterostructure is a type II with a band gap of 0.67 eV. The values for CBM, VBM, and midgap for the first monolayer, SiSe (blue color), and the second monolayer, SnS (orange color), are presented. All energies are in eV. (b) Energy band alignment in the midgap model, where midgap energies are aligned and energy bands of the second monolayer are shifted to the lower energies, creates a vacuum dipole step of $e{V}_h$. (c) Band model in the linear response model, where vacuum dipole is obtained by $e{V}_h=0.34\times (E_{m2}-E_{m1})=0.22$ eV. This leads to a type II heterostructure with a band gap of 0.89 eV, in great agreement with the DFT. (d) Energy band diagram for the DFT calculated SiSe-SnS heterostructure with a vacuum dipole step of $e{V}_{h,\mathrm{DFT}}$. DFT band gap is 0.93 eV. The percentages represent the contribution of each layer to the band edges. DFT band structure of isolated (e) SiSe, (f) SnS monolayers, and (g) SiSe-SnS heterostructure. CBM and VBM values are indicated by dots. The midgap energies are illustrated by horizontal dashed lines. All energies are taken with respect to the vacuum level (for the heterostructure, we used the highest vacuum level).

Figure 4

Survey of band gaps across the models. (a) Comparison between the band gaps extracted from the Anderson model and that computed from DFT, $E_{g,\mathrm{DFT}}$. (b),(c) Same as (a) but for the midgap and linear response model, respectively.

Figure 5

Survey of the band alignment types. Comparison of band types for IV-VI heterostructures according to the Anderson, midgap, linear response models, and DFT calculations. Band hybridization strength (H) displayed as a heatmap (top row).

Figure 6

Application of the linear response model. (a) HSE06 calculated band edges of free-standing group-IV monochalcogenides monolayers, relative to the vacuum energy. CBM and VBM values are indicated. An example: (b) Band lineup of SnSe-GeTe in the Anderson and linear response models. SnSe and GeTe band edges are illustrated by blue and orange colors, respectively. Vacuum dipole step of 0.084 eV is computed from linear response model as illustrated, which yields a band gap of 1.139 eV for the heterostructure. All energies are in eV.