The Transmission Line Model for 2D Materials and van der Waals Heterostructures

TL;DR

The paper introduces a physically informed transmission line model (TLM) for van der Waals heterostructures, mapping each 2D-layer constituent to distributed admittance and treating light–matter interactions as propagating voltage and current waves. This framework yields compact analytic expressions for reflection, dispersion, and field distributions, and unifies the treatment of bulk-to-monolayer transitions, surface polaritons, hyperbolic polaritons, and polariton hybridization. The authors validate the approach with experimental reflection measurements on hBN-encapsulated TMDs and show excellent agreement with transfer-matrix calculations, while providing new analytic insights into GEPs, HPhPs, and mode anti-crossings in symmetric and asymmetric structures. The TLM offers a transparent, scalable tool for understanding and designing layered 2D-material systems with complex polaritonic optical responses. This advance enables rapid physical intuition and streamlined calculations for polaritons in VdWHs, facilitating device design and fundamental studies of light–matter interactions in 2D materials.

Abstract

Van der Waals heterostructures (VdWHs) composed of 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 TMM, or Fresnel equations. While straight forward, 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 for VdWHs, based on expressing its 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 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 this, we derive the transmission line analogs of bulk to monolayer 2D materials and show it can be used to compute the reflection/transmission coefficients, polaritonic dispersion relations, and electromagnetic field distributions in a variety of 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.

Figures (5)

• Figure 1: Transmission line model for VdWHs. (a) A VdWH configuration constructed by layers of bulk, few-layer, or monolayer 2D materials. (b) Transmission line model of the VdWH in (a), where bulk layers are modeled as a transmission line and thin layers as a parallel admittance. (c) Transmission line model of the $i$th layer showing the transverse fields as analog to the voltage and current, propagating through forward and backward waves. The $z$ dependent admittance at two different locations in the layer are marked. (d) A thin layer in the VdWH modeled as a parallel admittance between transmission lines.
• Figure 2: Transmission line model for excitonic reflection from a $\mathrm{WS_2}$ and $\mathrm{WSe_2}$ VdWHs. (a) An illustration of the hBN/monolayer TMD/hBN/substrate VdWH. (b) A transmission line scheme for the configuration in (a). Measured reflection contrast (blue points) and TLM fit black lines) for (c) the $\mathrm{WS_2}$ VdWH on a gold mirror at $T=4K$, and(d) $\mathrm{WSe_2}$ VdWH on $\mathrm{SiO_2}$/Si substrate at $T=80K$.
• Figure 3: Transmission line model for GEP. (a) An illustration of the hBN/BLG/hBN VdWH. (b) Transmission line schemes for the configuration. (c) Dispersion relation of GEP in the configuration, calculated from Eq. \ref{['Eq GEP DR']} (dashed black line) compared to TMM simulation (colormap).
• Figure 4: Transmission line model for HPhP. (a) An illustration of the dielectric/hBN/dielectric VdWH. (b) Transmission line schemes for the configuration in (a). (c) Dispersion relation of HPhP in the configuration, calculated from Eq. \ref{['Eq HPhP DR']} (dashed black line for even modes and dashed red line for odd modes) compared to TMM simulation (colormap).
• Figure 5: Transmission line model for hybridized polaritons. (a) An illustration of the dielectric/hBN/BLG/hBN/dielectric VdWH. (b) Transmission line schemes for the configuration in (a). (c) Dispersion relation of the hybridized polaritons in the configuration in the symmetric structure, (dashed purple line for even modes and dashed red line for odd modes) compared to TMM simulation (colormap). Even modes of HPhP for hBN with thickness of $d_b+d_t$ (black lines) and exciton energies of BLG (dashed white lines) are also plotted. (d) Dispersion relation of the hybridized polaritons in the configuration in the asymmetric structure, (dashed red line for the low momentum modes and dashed purple line for the high momentum modes) compared to TMM simulation (colormap). Modes of HPhP for hBN with thickness of $d_b+d_t$ (black lines), odd modes of HPhP for hBN with thickness of $2d_t$ (dashed orange lines), and exciton energies of BLG (dashed white lines) are also plotted.