Reduction of fully screened magnetoplasmons in a laterally confined anisotropic two-dimensional electron system to an isotropic one

TL;DR

The paper addresses magnetoplasmon modes in fully screened, anisotropic 2D electron systems with elliptical Fermi surfaces. By formulating Maxwell’s equations with a dynamical anisotropic Drude conductivity, the authors show that electromagnetic retardation can be incorporated as an effective mass renormalization, allowing a coordinate transformation that maps the problem to an isotropic system with $m^* = \sqrt{m_x^* m_y^*}$ and $\omega_c^* = eB_z/(c\sqrt{m_x^* m_y^*})$. They obtain analytical results for a gated disk at zero magnetic field using elliptic coordinates and Mathieu functions, giving $\omega_{n,m}=v^*\alpha_{n,m}/R$ with $\alpha_{n,m}$ from zeros of derivatives of modified Mathieu functions; with a magnetic field, a Mathieu-function basis yields a solvable linear system and magnetodispersion, revealing mode splitting and anticrossings due to anisotropy and clarifying edge versus bulk magnetoplasmon behavior. The framework advances understanding of anisotropic 2D plasmonics in realistic materials and informs the design of sub-terahertz to terahertz devices in systems such as phosphorene and related 2D materials by clarifying how confinement, anisotropy, and retardation shape plasmon spectra.

Abstract

We investigate the properties of natural two-dimensional (2D) magnetoplasma modes in laterally confined electron systems, such as 2D materials, quantum wells, or inversion layers in semiconductors, with an elliptic Fermi surface. The conductivity of the system is considered in a dynamical anisotropic Drude model. The problem is solved in the fully screened limit, i.e., under the assumption that the distance between the two-dimensional electron system and the nearby metal gate is small compared to all other lengths in the system, including the wavelength of plasmons. Remarkably, in this limit plasma oscillations in an anisotropic 2D confined system are equivalent to plasma oscillations in an isotropic 2D electron system obtained by some stretching, even when the electromagnetic retardation is taken into account. Moreover, accounting for electromagnetic retardation leads only to a renormalization of the effective masses of carriers, somewhat like in relativity. As an example, we reduce the equations describing plasmons in a gated disk with an anisotropic two-dimensional electron gas to the equations describing oscillations in an isotropic ellipse. Without a magnetic field, we solve them analytically and find eigenfrequencies. To find a solution in a magnetic field, we expand the current of plasma oscillations in the complete set of Mathieu functions. Leaving the leading terms of the expansion, we approximately find and analyze magnetodispersion for the lowest modes.

Figures (4)

• Figure 1: Schematic view of the system under consideration. The 2D ES is located in the plane $z=0$ between two dielectrics with permittivity $\varepsilon_{-}$ and $\varepsilon_{+}$. A perfectly conducting metal gate is under the two-dimensional electron gas. The system is placed in a magnetic field $\bm{B}$ being perpendicular to the 2D ES plane.
• Figure 2: Dependence of the eigen frequency of plasma oscillations in the disk on the ratio of effective masses in the absence of a magnetic field. The dashed lines correspond to different anisotropic systems (in the quasi-stationary limit $m_x^*=m_x,m_y^*=m_y$): the AlAs/AlGaAs quantum well with $m_x=1.1$, $m_y=0.2$Shayegan2006, phosphorene with $m_x=1.12$, $m_y=0.17$Qiao2014, a monolayer (1L) of TiS$_3$ with $m_x=1. 47$, $m_y=0.41$Dai2015, and a monolayer of ReS$_2$ with $m_x=0.25$, $m_y=0.13$Lin2015. Accounting for electromagnetic retardation will decrease the ratio $m_x^*/m_y^*$, therefore the vertical dashed lines will shift to the left.
• Figure 3: Charge density of seven lowest plasma modes in an isotropic (top) and anisotropic (bottom) 2D ES. The effective masses for the anisotropic system were taken for phosphorene: $m_x^*=m_x=1.12$, $m_y^*=m_y=0.17$.
• Figure 4: Dependence of the frequency of plasma oscillations in the disk on the magnetic field. The dots indicate the phosphorene magnetodispersion ($m_x=1.12$, $m_y=0.17$) Qiao2014. Solid lines correspond to magnetodispersion in isotropic systems. The colors of the dots and solid lines indicate the different subsystems from which they are derived. The dashed line shows the frequencies $\omega=\omega_c^*$.