Multiscale Pseudoatomistic Quantum Transport Modeling for van der Waals Heterostructures

Several electronic and optoelectronic devices have been proposed in recent years based on vertical heterostructures of two-dimensional (2D) materials. The large number of combinations of available 2D materials and the even larger number of possible heterostructures require effective and predictive device-simulation methods, to inform and accelerate experimental research and to support the interpretation of experiments. Here, we propose a computationally effective and physically sound method to model electron transport in 2D van der Waals heterostructures, based on a multiscale approach and quasiatomistic Hamiltonians. The method uses ab initio simulations to extract the parameters of a simplified tight-binding Hamiltonian based on a uniform three-dimensional lattice geometry that enables device simulations using the nonequilibrium Green’s function approach in a computationally effective way. We describe the application and limitations of the method and discuss the examples of two use cases of practical electronic devices based on 2D materials, such as a field-effect transistor and a floating-gate memory, composed of molybdenum disulphide, hexagonal boron nitride and graphene.

Figure 1

(a) A planar representation of the hexagonal lattice with the two centers $A$ (red) and $B$ (yellow), along with the first-nearest-neighbor and second-nearest-neighbor in-plane hopping parameters. (b) A schematic representation of the pseudolattice describing two $A{A}^{\prime }$ stacked layers of the same material, with off-plane nearest-neighbor (next-nearest-neighbor) hopping indicated by dark (light) green lines. (c) The pseudolattice for two $A{A}^{\prime }$ stacked layers of different materials, with off-plane nearest-neighbor (next-nearest-neighbor) hopping indicated by dark (light) blue lines.

Figure 2

The scheme of the multiscale simulation procedure. Ab initio (DFT) simulations are used to extract the parameters of the tight-binding Hamiltonian, which is then used in the self-consistent simulation of quantum transport and electrostatics.

Figure 3

A comparison of the band structures for various materials, as computed by DFT and by the effective model on the real lattice. The band structure on the common lattice for Gr and $h$-BN is reported in Appendix pp1. (a)–(c) Tight binding versus DFT on the real lattice band structures. After the fit, the $h$-BN band gap is artificially widened to match that in the transmission spectrum; for further details, see Appendix pp3. (d) Vertical transmission for bulk Gr (12 layers, linear scale) and (e) for $h$-BN embedded in Gr (4L, four layers; 12 layers in total, logarithmic scale). The dashed line indicates the Dirac-cone energy for graphene.

Figure 4

Three-dimensional (3D) representations of the (a) Gr/h-BN/Gr and (b) h-BN/MoS₂/h-BN heterostructures used for band alignment. The semitransparent planes represent the electrodes, while the central part is the scattering region. (c) The density of states D for each material, after band alignment, as referred to the h-BN valence-band top. The dashed line indicates the Dirac-cone energy for graphene.

Figure 5

(a) A view of the FET, with the core vertical heterostructure (and the substrate) highlighted. (b) The self-consistent scheme: the Fermi level of the semimetallic gate is used to self-consistently determine the potential everywhere along the device. The current is then obtained from the Büttiker-Landauer formula, after solving the NEGF problem.

Figure 6

(a) A view of the floating-gate memory, with the core vertical heterostructure (and the substrate) highlighted. (b) The self-consistent scheme adopted to study the transport problem: the electrostatic potential $\phi (z)$ is computed on the left and imposed on the right, where the current through the barrier is computed via the NEGF formalism. From this current, we calculate the time required to trap a small charge $\mathrm{\Delta }Q$, so that $Q(z_{\mathrm{FG}};t+\mathrm{\Delta }t)=Q(z_{\mathrm{FG}};t)+\mathrm{\Delta }Q$. The new charge changes the electrostatic potential in the device and the procedure is repeated until full equilibrium is reached.

Figure 7

A comparison of the gate-leakage current versus the barrier width in a FET composed of a monolayer $\mathrm{Mo}\mathrm{S}$${}_2$ channel, a monolayer Gr gate, and a multilayer $h$-BN barrier (2L, two layers;...; 12L, 12 layers), for various applied gate-channel voltages $V_{gc}$. A $n=5\times {10}^{12}{\mathrm{cm}}^{-2}$ doping is imposed on the channel.

Figure 8

(a) The time evolution of the charge trapped on the floating gate for different barrier widths, from 10 to 13 $h$-BN layers. In all cases, a 16.3-nm-thick insulator separates the floating and control gates, a $n=2\times {10}^{13}{\mathrm{cm}}^{-2}$ doping is considered for the $\mathrm{Mo}\mathrm{S}$${}_2$ channel, and $V_{gc}$ is fixed at 0.0 V. Note that the initial potential on the floating gate depends on the barrier width. The band structure at different times for the 13–$h$-BN–layer evolution curve (the colored dots) is reported in (b), where matching colors refer to the same time $t$.

Figure 9

A comparison of the bands, centered around the $K$ point, for Gr and $h$-BN as obtained from DFT, the tight-binding best fit on the same lattice parameter as DFT, and the tight-binding best fit on the common lattice. The high-symmetry points in the bottom axis refer to the first Brillouin zone relative to the DFT lattice ($a_{\mathrm{DFT}}=0.246\mathrm{nm}$ for both materials); whereas the primed ones, on the top axis, refer to the first Brillouin zone relative to the common lattice ($a_{\mathrm{VH}}=a_{{\mathrm{Mo}\mathrm{S}}_2}$).

Figure 10

The DFT (thin) versus the tight-binding (thick) LDOS for the two heterostructures $h$-BN/$\mathrm{Mo}\mathrm{S}$${}_2$/$h$-BN and $\mathrm{Gr}/h$-BN/Gr.

Figure 11

The DFT results for a 12-layer $h$-BN structure. As the transmission spectrum presents a wider gap than that predicted by the band structure, the wider value is taken as a reference for the tight-binding modeling of $h$-BN.