An Intuitive Equivalent Circuit Model for Multilayer Van Der Waals Heterostructures

Stacks of 2-D materials, known as van der Waals (vdW) heterostructures, have gained vast attention due to their interesting electrical and optoelectronic properties. This paper presents an intuitive circuit model for the out-of-plane electrostatics of a vdW heterostructure composed of an arbitrary number of layers of 2-D semiconductors, graphene, or metal. We explain the mapping between the out-of-plane energy band diagram and the elements of the equivalent circuit. Although the direct solution of the circuit model for an n-layer structure is not possible due to the variable, nonlinear quantum capacitance of each layer, this paper uses the intuition gained from the energy band picture to provide an efficient solution in terms of a single variable. Our method employs Fermi-Dirac statistics while properly modeling the finite density of states (DOS) of 2-D semiconductors and graphene. The approach circumvents difficulties that arise in commercial TCAD tools, such as the proper handling of the 2-D DOS and the simulation boundary conditions when the structure terminates with a nonmetallic material. Based on this methodology, we have developed 2dmatstacks, an open-source tool freely available on nanohub.org. Overall, this paper equips researchers to analyze, understand, and predict experimental results in complicated vdW stacks.