Optical Gain Through Metallic Electro-Optical Effects

TL;DR

This work addresses non-reciprocal optical gain in biased 2D metals by exploiting intraband (Drude) dynamics modified through metallic electro-optic effects. It develops a unified Boltzmann transport and scattered-wave framework that incorporates Berry curvature dipole (BCD) and magnetoelectric tensor (MET) contributions to the optical conductivity, and analyzes TE-mode resonances under bias. A key finding is that a resonant TE mode—and thus substantial gain—appears only when both BCD- and MET-induced EO responses are present, with gain tunable via the bias direction, anisotropy, and the coupling magnitudes D0 and G0; achieving the exact resonance requires careful balancing of the real and imaginary parts of σ^{yy}_{eff}. The results outline material-design strategies for intraband-based optical amplification, suggesting pathways in Moiré or anisotropic 2D heterostructures to realize large BCD and MET and enable terahertz photonic applications through TE-mode engineering.

Abstract

Optical gain is a critical process in today's semiconductor technology and it is most often achieved via stimulated emission. In this theoretical study, we find a resonant TE mode in biased low-symmetry two-dimensional metallic systems which may lead to optical gain in the absence of stimulated emission. We do so by first modeling the optical conductivity using Boltzmann non-equilibrium transport theory and then simulating the scattering problem using a scattered-wave formalism. Assuming that the system may possess a Berry curvature dipole (BCD) and a non-zero Magnetoelectric tensor (MET), we find that the optical conductivity has a non-trivial dependence on the direction of the applied bias, which allows for probing the TE mode. After analyzing the system with one of each of the effects, we find that the resonant TE mode is only accessible when both effects are present. Further studies are necessary to find materials with a suitably large BCD and MET, in order to realize the predictions within this study.

Figures (5)

• Figure 1: Schematic and definition of parameters of the scattering formalism alongside the directions of the non-zero components of the $\textbf{G}$ tensor and $\textbf{D}$ dipole of a 2D metal at $z = 0$. While the $\textbf{G}$ tensor is fixed in the out-of-plane direction, the direction of $\textbf{D}$ ($\textbf{E}_0$) in the $xy$ plane is determined by the angle $\theta$ ($\phi$), measured with respect to the $y$ axis. The 2D metal is in between dielectrics with relative permittivity $\epsilon_1$ ($z < 0$) and $\epsilon_2$ ($z > 0$). We assume that the plane of incidence is the $xz$ plane, containing the incidence wavevector $\textbf{k}_{\textrm{inc}}$.
• Figure 2: Normal scattering in the presence of only $\textbf{D}$, showing that TE-polarized waves and $\mathbf{D}\perp \mathbf{E}_{\textrm{0}}$ with a specific orientation maximizes optical gain. (a) Schematic including only $\mathbf{D}$ and $\mathbf{E}_{\textrm{0}}$. The optical field is assumed to be normally incident ($\rho = 0^{\circ}$); thus, we also fix the direction of $\mathbf{D}$ to point along +$y$ ($\theta = 0^{\circ}$). (b) Transmittance as a function of $\textbf{E}_{\textrm{0}}$ direction with respect to the $y$-axis at the optimal polarization ($T = T_{\textrm{max}}$). (c)-(d) Transmittance of a normally incident-wave with the polarization determined by the tilt ($\Psi$) and ellipticity ($\chi$) angles. In panels (b)-(d), $\rho = 0^{\circ}$, $\theta = 0^{\circ}$, $\omega/(2\pi) = 0.5$ THz, $\epsilon_1 = \epsilon_2 = 1$, ${D}_{\textrm{0}} = 40$ nm, and ${E}_{\textrm{0}} = 8 \times 10^4$ V$\cdot$m$^{-1}$.
• Figure 3: Increasing $\xi$ does not guarantee more optical gain with only $\textbf{D}$ present. (a) Reflection coefficient of an TE-polarized wave as a function of $\xi$. (b) Transmittance of an TE-polarized wave as a function of $\xi$, where $\tilde{\gamma} = 10^{11}\textrm{rad/s}$. (c) Transmittance as a function of $\xi$ and $\sigma^{yy}_{\textrm{Drude}}$ at the optimal polarization; $\tilde{\sigma}^{yy}_{\textrm{Drude}}$ is fixed and given by Eq. \ref{['eq:drude']}. In all panels, $\tilde{\xi} = 15.3$ THz; $\phi = 270 ^{\circ}$; and the values of $\rho$, $\omega$, $\epsilon_1$, and $\epsilon_2$ are the same as in Fig. \ref{['fig:just']}.
• Figure 4: Exploring how optical gain in the presence of only $\textbf{G}$ depends on $\phi$, $\Gamma$, $\sigma_{\textrm{Drude}}^{yy}$, and polarization. $\phi = 270^{\circ}$ and TE-polarized light once again maximize gain. (a) Schematic including only $\mathbf{G}$ and $\mathbf{E}_{\textrm{0}}$. The optical field is now incident at an angle $\rho \neq 0$ and the coordinate axis is oriented such that the optical field is incident in the $xz$-plane. (b) Transmittance of optimally- and TE-polarized light with respect to $\textbf{E}_0$ direction, with $G_0 = 200$ m$^2 \cdot$V$^{-1}\cdot$s$^{-1}$ and ${E}_{\textrm{0}} = 8 \times 10^4$ V$\cdot$m$^{-1}$. (c) Transmittance of optimally-polarized light as a function of $\sigma^{yy}_{\textrm{Drude}}$ and $\Gamma$, where $\tilde{\Gamma} = 0.053$ and $\phi = 270^{\circ}$. In panels (b) and (c), $\rho = 45^{\circ}$; the values of $\omega$, $\epsilon_1$, $\epsilon_2$, and $\tilde{\sigma}^{yy}_{\textrm{Drude}}$ are the same as in Fig. \ref{['fig:just']}.
• Figure 5: TE-resonance condition may be accessed by tuning $E_0$, $D_0$, and $G_0$. (a)-(c) Transmittance as of optimally-polarized light as a function of ${\sigma}^{yy}_{\textrm{Drude}}$ and in-plane bias magnitude $E_0$, with $\tilde{E}_{\textrm{0}} = 8 \times 10^4$ V$\cdot$m$^{-1}$ and $D_0 = 40$ nm. The value of $\tilde{\sigma}^{yy}_{\textrm{Drude}}$ is the same as in Fig. \ref{['fig:just']}. (d) Natural log of transmittance as a function of $D_0$ and $G_0$, where $\tilde{D}_0 = 40$ nm, $\tilde{G}_0 = 200$ m$^2 \cdot$V$^{-1}\cdot$s$^{-1}$, $E_0 = 8 \times 10^4$ V$\cdot$m$^{-1}$, $\rho = 45^{\circ}$, and ${\sigma}^{yy}_{\textrm{Drude}}/\tilde{\sigma}^{yy}_{\textrm{Drude}}=1$. The points which correspond to the values of $D_0$ and $G_0$ used in panels (a)-(c) are marked in white in panel (d). In all panels, $\phi = 270^{\circ}$ and $\theta = 0^{\circ}$. Values of $\omega$, $\epsilon_1$, and $\epsilon_2$ are the same as in Fig. \ref{['fig:just']}.