Transmission line model for two-dimensional materials and van der Waals heterostructures

Van der Waals heterostructures (VdWHs) composed of stacked two-dimensional (2D) materials have attracted significant attention in recent years due to their intriguing optical properties, such as strong light-matter interactions and large intrinsic anisotropy. In particular, VdWHs support a variety of polaritons---hybrid quasiparticles arising from the coupling between electromagnetic waves and material excitations---enabling the confinement of electromagnetic radiation to atomic scales. The ability to predict and simulate the optical response of 2D materials heterostructures is thus of high importance, being commonly performed until now via methods such as the transfer-matrix-method, or Fresnel equations. While straightforward, for complex structures these often yield long and complicated expressions, limiting intuitive and simple access to the underlying physical mechanisms that govern the optical response. In this work, we demonstrate the adaptation of the transmission line model approach for VdWHs, based on expressing the VdWH constituents by distributed electrical circuit elements described by their admittance. Since the admittance carries fundamental physical meaning of the material response to electromagnetic fields, the approach results in a one-dimensional system of propagating voltage and current waves, offering a compact and physically intuitive formulation that simplifies algebraic calculations, clarifies the conditions for existence of physical solutions, and provides valuable insight into the fundamental physical response. To demonstrate the robustness and advantages of this approach, we derive the transmission line analogs of bulk to monolayer 2D materials and show how the approach can be used to compute the reflection/transmission coefficients, polaritonic dispersion relations, and electromagnetic field distributions in a variety of 2D material VdWHs, and compare them to experimental measurements yielding very good agreement. This method provides a valuable tool for exploring and understanding the optical response of layered 2D systems.