Nonlocal electrodynamics of two-dimensional anisotropic magneto-plasmons

TL;DR

This work develops a Madelung-based, nonlocal hydrodynamic framework for anisotropic 2D electron systems, deriving continuity and Euler equations that incorporate the Bohm potential and Fermi pressure while coupling to electrostatics and magnetic fields. It yields a magnetoplasmon dispersion $igl( ext{hbar} abla _{oldsymbol{q}}igr)$ given by $oxed{ ext{hbar}\omega_{oldsymbol{q}}= ext{sqrt}ig((K_{oldsymbol{q}}+V^F+V_{oldsymbol{q}}^{C})K_{oldsymbol{q}}+( ext{hbar}\omega_c)^2ig)}$ with direction-dependent $K_{oldsymbol{q}}$, and provides a nonlocal optical conductivity tensor $oldsymbol{ppa}(oldsymbol{q},oldsymbol{
omega})$ that reduces to Drude in the long-wavelength limit. Application to phosphorene and black phosphorus shows that nonlocality, especially via the Bohm term, aligns with ab initio plasmon dispersion and suppresses hyperbolic surface plasmon-polaritons in Reststrahlen bands, while modifying dipole-induced plasmon wakes and Purcell enhancements. The results highlight the necessity of nonlocal anisotropic models for accurately describing strongly confined polaritons in 2D materials and provide a versatile platform for exploring nonlinear and many-body regimes.

Abstract

We present a hydrodynamic model, grounded in Madelung's formalism, to describe collective electronic motion in anisotropic materials. This model incorporates nonlocal contributions from the Thomas-Fermi quantum pressure and quantum effects arising from the Bohm potential. We derive analytical expressions for the magnetoplasmon dispersion and nonlocal optical conductivity. To demonstrate the applicability of the model, we examine electrons in the conduction band of monolayer phosphorene, an exemplary anisotropic two-dimensional electron gas. The dispersion of plasmons derived from our hydrodynamic approach is closely aligned with that predicted by ab~initio calculations. Then, we use our model to analyze few-layer black phosphorus, whose measured infrared optical response is hyperbolic. Our results reveal that the incorporation of nonlocal and quantum effects in the optical conductivity prevents black phosphorus from supporting hyperbolic surface plasmon polaritons. We further demonstrate that the predicted wavefront generated by an electric dipole exhibits a significant difference between the local and nonlocal descriptions for the optical conductivity. This study underscores the necessity of moving beyond local approximations when investigating anisotropic systems capable of hosting strongly confined plasmon-polaritons.

Figures (6)

• Figure 1: Electronic energy band structure of phosphorene. Panel (a) shows the band structure of extended monolayer phosphorene with doping charge carrier density $n=1\times10^{13}$ cm$^{-2}$. The inset shows the conduction band along the $\mathrm{\Gamma}$--X and $\mathrm{\Gamma}$--Y directions of the first Brillouin zone (BZ) together with the parabolic bands (orange dashed lines) resulting from the fitted effective masses extracted from Fig. \ref{['fig:ab_initio']}. Panel (b) shows the unit cell of the 2D phosphorene crystal seen from above (top) and the side (bottom). Panel (c) shows the first BZ of the 2D crystal.
• Figure 2: Density plot of the absorption obtained from ab initio calculations as a function of the the photon energy $\hbar\omega$ and the photon wavenumber $q$ with zero magnetic field. Panel (a) shows variations of $q$ along the $\mathrm{\Gamma}$--X direction in the Brillouin zone of the electronic-structure problem, while panel (b) is for the $\mathrm{\Gamma}$--Y direction. In both panels, results for the dispersion relation from the hydrodynamic model [Eq. \ref{['eq:magneto_plasmon_disp']}] are superimposed, showing both the local approximation (dashed white line), the additional effect of Bohm's potential (dash-dotted green line), the Fermi pressure (dash-dotted blue line), and the combined effects of Bohm's potential and the Fermi Pressure (solid white line). Maxima in absorption from the first principles calculation at low momenta are used for fitting of the effective masses (circles).
• Figure 3: Panel (a) is a density plot of $\Delta\omega/\omega_0$ -- the relative difference between the nonlocal and local plasmon dispersion of phosphorene -- as function of the electronic density $n_0$ and the plasmon wavenumber $q_y$ in the zigzag direction. Panel (b) shows the magnetoplasmon dispersion relation for phosphorene -- frequency $f=\omega/2\pi$ versus wavenumber $q$ -- comparing the nonlocal hydrodynamic model (solid lines) to the local Drude model (dashed lines) for both $B_z=0$ T (blue coloring) and $B_z=10$ T (red coloring). The effective mass was obtained from the ab initio fitting and $n_0=10^9$ cm$^{-2}$. We consider a direction that makes a 45$^\circ$ angle with either the zigzag or armchair direction. Panels (c-f) show Quiver plots of the plasmon velocity field and colormap of the charge-density fluctuation for phosphorene for $q\ell_0=90$, as a function of the real-space coordinates $x$ and $y$ for different magnetic fields, for both the nonlocal hydrodynamic and the local Drude models. We use the effective mass obtained from the ab initio fitting and $n_0=10^9$ cm$^{-2}$.
• Figure 4: Black phosphorus anisotropic dielectric function $\epsilon(\omega)$, calculated from Eq. \ref{['eq:diel']} for an electronic density $n_0=10^{12}$ cm$^{-2}$. The Reststrahlen band with $\mathop{\mathrm{Re}}\nolimits\{\epsilon_{xx}\} \times \mathop{\mathrm{Re}}\nolimits\{\epsilon_{yy}\}<0$ is highlighted in grey.
• Figure 5: Panels (a) and (b) show density plots of the loss function $\mathop{\mathrm{Im}}\nolimits\{r_{pp}(\mathbf{q},\omega)\}$ as a function of $\mathbf{q}$ for a BP thin film at $f=\omega/2\pi=9.0$ THz calculated using Eq. \ref{['eq:r_pp']}. Panel (a) is for the nonlocal hydrodynamic model, while panel (b) is for the local Drude model. Panel (c) is a zero-contour plot of $\mathop{\mathrm{Im}}\nolimits\{\lambda_1\lambda_2\}$ in wavenumber space, separating positive regimes (in yellow) from negative regimes (in orange). The plot is for a photon frequency $f=9.0$ THz and the eigenvalues $\lambda_{1,2}$ are those of the optical conductivity tensor matrix [Eq. \ref{['eq:total_cond']}].