Efficient computational design of two-dimensional van der Waals heterostructures: Band alignment, lattice mismatch, and machine learning

We develop a computational database, website applications (web-apps), and machine-learning (ML) models to accelerate the design and discovery of two-dimensional (2D) heterostructures. Using density functional theory (DFT) based lattice parameters and electronic band energies for 674 nonmetallic exfoliable 2D materials, we generate 226 779 possible bilayer heterostructures. We classify these heterostructures into type-I, -II, and -III systems according to Anderson's rule, which is based on the relative band alignments of the noninteracting monolayers. We find that type II is the most common and type III the least common heterostructure type. We subsequently analyze the chemical trends for each heterostructure type in terms of the Periodic Table of constituent elements. The band alignment data can also be used for identifying photocatalysts and high-work-function 2D metals for contacts. We validate our results by comparing them to experimental data as well as hybrid-functional predictions. Additionally, we carry out DFT calculations of a few selected systems ($\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{Se}}_2, \mathrm{Mo}{\mathrm{S}}_2/\text{h-BN}$, and $\mathrm{Mo}{\mathrm{Se}}_2/\mathrm{Cr}{\mathrm{I}}_3$), to compare the band-alignment description with the predictions from Anderson's rule. We develop web-apps to enable users to virtually create combinations of 2D materials and predict their properties. Additionally, we use ML tools to predict band-alignment information for 2D materials. The web-apps, tools, and associated data will be distributed through the jarvis-heterostructure website. Our analysis, results, and the developed web-apps can be applied to the screening and design applications, such as finding alternative photocatalysts, photodetectors, and high-work-function (WF) 2D-metal contacts.

Figure 1

Flow chart for the computational design of 2D heterostructures. The DFT-calculated inputs for monolayers A and B are CBM, VBM, and the energy level of the vacuum.

Figure 2

(a) DFT-calculated DOS for a monolayer of $\mathrm{Mo}{\mathrm{S}}_2$ in vacuum, normalized to the Fermi energy of the monolayer. (b) The same quantity for $\mathrm{W}{\mathrm{Se}}_2$. (c) The combined DOS of the bilayer $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{Se}}_2$ heterostructure in vacuum, normalized to the Fermi level of the heterostructure. (d) $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{Se}}_2$ band alignment diagram, with energy levels for each material calculated in vacuum and aligned relative to the vacuum energy level. (e)–(h) The same quantities for $\mathrm{Mo}{\mathrm{S}}_2/\text{h-BN}$; (i)–(l) the same quantities for $\mathrm{Mo}{\mathrm{Se}}_2/\mathrm{Cr}{\mathrm{I}}_3$.

Figure 3

Distribution of type-I, type-II, and type-III heterostructures. (a) Combination of all possible heterostructure types for a 2D material. Each point in the plot represents a heterostructure. Each point on the $x$ and $y$ axes represents a 2D material. Only the lower diagonal is plotted because it is a symmetric matrix. (b) A pie chart of the type distribution for the three heterostructures.

Figure 4

Periodic Table trends of percentage probability that an element is a part of the type-I, type-II, and type-III heterostructures.

Figure 5

Examples of type-I, type-II, and type-III heterostructure band alignments relative to vacuum energy. (a) $2\text{H-Mo}{\mathrm{S}}_2/\mathrm{Re}{\mathrm{S}}_2$, (b) $\mathrm{Mo}{\mathrm{S}}_2/\mathrm{W}{\mathrm{Se}}_2$, and (c) $\mathrm{Sn}{\mathrm{Se}}_2/\mathrm{WS}{\mathrm{e}}_2$ interface. The dotted lines show the ${\mathrm{H}}^{+}/{\mathrm{H}}_2$ and ${\mathrm{O}}_2/{\mathrm{H}}_2\mathrm{O}$ levels that must be bridged by the interface CBM/VBM for water-splitting photocatalysis.

Figure 6

Band alignment of the nonmetallic 2D materials with respect to ${\mathrm{H}}^{+}/{\mathrm{H}}_2$ and ${\mathrm{O}}_2/{\mathrm{H}}_2\mathrm{O}$ levels. VBMs are drawn in red while CBMs are green.

Figure 7

WF distribution of 2D metals from the JARVIS database.

Figure 8

Snapshot of the jarvis-heterostructure app showing two sub-apps.

Figure 9

(a)–(c) Histogram of the JARVIS 2D materials data for CBM and VBM, relative to the vacuum level, as well as WF. (d)–(f) Parity plots comparing the DFT-calculated values for these properties on the 10% test set, to the values predicted with the ALIGNN ML method. The green line at $y=x$ is a guide for the eye.

Figure 10

Parity plot comparing the VBM and CBM calculated with OptB88vdW, to those calculated with the PBE (a), (b) and HSE06 (c), (d) functionals. The Pearson coefficient (PC) is a measure of correlation defined as covariance of two variables divided by both their standard deviations.

Figure 11

Histogram of the difference in effective mass between the two materials making up each 2D heterostructure in our study. The effective mass is in units of electron rest mass.

Figure 12

Convergence plot showing training set and validation set performance as a function of the number of training epochs, for the (a) ALIGNN CBM model, (b) VBM model, and (c) WF model.