Coupled dynamical Boltzmann transport equations with long-range electron-phonon and electron-electron interactions in 2D materials

Abstract

We study the interplay between long-range electron-phonon and electron-electron interactions in electrostatically doped two-dimensional semiconductors, including interlayer couplings in van der Waals heterostructures. We evaluate the effects of those interactions on transport properties by writing dynamically coupled Boltzmann equations for the electrons and for the electrodynamically active excitations. We develop a theory with a general validity, and apply it both to simplified parabolic models, and to the realistic BN-encapsulated graphene system which we present in an accompanying paper [arXiv:2604.00678]. We show that dynamical screening effects are of fundamental importance in order to correctly describe the electronic transport properties of two-dimensional materials, and in particular the scattering from polar phonons, whether those come from the semiconductor itself or the surrounding layers.

Figures (14)

• Figure 1: (Upper panels) -Im$\epsilon^{-1}_{\rm tot}$ for a doping level of 0.1 eV, for BN (left) and MoS$_2$ (right). (Lower panels) same but for -Im$\epsilon^{-1}_{\rm el}$. Materials are modeled with an isotropic parabolic band dispersion with parameters given in Tab. \ref{['tab:materials']}. Upwards parabolae drawn as black dashed lines represent the limits of the electron-hole continuum at zero temperature (regions II and III, while in regions I and IV no single electron excitations are possible). The blue dashed horizontal lines are the uncoupled TO modes, while green dashed lines represent the uncoupled plasmons.
• Figure 2: -Im$\epsilon^{-1}_{\rm tot}$ as a function of $\omega$ for given cuts at fixed $q/k_{\rm F}$, for BN. The cuts are presented in the left panel as white vetical lines, with the corresponding $q/k_{\rm F}$ value reported, and the value of -Im$\epsilon^{-1}_{\rm tot}$ is reported in the right panel as black continuous line. Dashed lines represent instead the value of -Im$\epsilon^{-1}_{\rm el}$. The emergent excitations are not of Lorentzian nature.
• Figure 3: Electron-mode coupling in BN-encapsulated graphene. The coupling is computed for an electron in graphene $g^2_{\rm Gr} = g^2_{\rm kk}$ with $k$ the layer index of graphene. The number of BN layers is 1 and 20 on each side for the top and bottom panel, respectively. The electrodynamic active modes are graphene's plasmon and BN's remote LO and ZO phonons. Zones II and IV are the intra- and interband electron-hole continua. Zone I is where the plasmon excitation is undamped. There are no electron-hole excitations in zone III. The limits between those zones (dashed lines) are smeared by temperature ($300$ K here).
• Figure 4: (Upper panels) $-\rm{Im}\epsilon^{-1}_{\rm tot}$ and $-\rm{Im}\epsilon^{-1}_{\rm el}$ and (lower panels) $\mathcal{F}_{\rm el/ph}=-\rm{Im} v\chi^0_{\rm el/ph}/\epsilon_{\rm tot}$ as function of $\omega$, for BN, along those cuts of Fig. \ref{['fig:cuts']} situated at $q/k_{\rm F}=3.18$ (left panels) and 1.04 (right panels). Gray dashed areas are regions where $\mathcal{F}_{\rm ph}>0$: light gray if $\mathcal{F}_{\rm el}<0$ and dark gray if $\mathcal{F}_{\rm el}>0$.
• Figure 5: Phonon content $\mathcal{F}$ of Eq. \ref{['eq:phcontent']} for BN (left panel) and MoS$_2$ (right panel), which shows localization around the phonon mode.