Novel edge excitations of two-dimensional electron liquid in a magnetic field.

We investigate the low-energy spectrum of excitations of a compressible electron liquid in a strong magnetic field. These excitations are localized at the periphery of the system. The analysis of a realistic model of a smooth edge yields new branches of acoustic excitation spectrum in addition to the well known edge magnetoplasmon mode. The velocities are found and the observability conditions are established for the new modes. PACS numbers: 71.45.Gm, 73.20.Mf Typeset using REVTEX 1 The dispersion relation for plasmons in a non-restricted two-dimensional electron liquid is well known to have a form ω ∝ k [1–3]. If the liquid has a boundary, an edge mode appears in addition to these bulk excitations. The spectra of the edge and bulk modes differ from each other only by a numerical factor [4]. A magnetic field applied perpendicularly to the plane of the liquid changes the plasmon spectrum drastically. The spectrum of the bulk mode acquires a gap of the width equal to the cyclotron frequency ωc. The only known gapless mode existing in the presence of the magnetic field propagates along the boundary [4–7]. The “chirality” of this edge magnetoplasmon determined by the direction of the magnetic field (i. e., by the sign of the Hall conductivity σxy) was demonstrated explicitly in the time-domain experiments [8]. The solved theoretical models of the edge modes assumed a sharp electron density profile at the boundary [4–7], i. e. width of the boundary strip was assumed to be infinitesimal. The existence of only a single branch of the edge magnetoplasmons follows directly from this assumption. For a realistic shape of a potential confining the electron liquid, the density profile is smooth at the boundary [9–11]. The results of Refs. [4–7] can be extended on this case only under the assumption that the current and charge oscillations forming the magnetoplasmon wave are homogeneous across the boundary strip. However, the latter condition is excessively restrictive. We demonstrate in this paper the existence of other sound-like modes propagating along the edge. The current for each of these modes alternates across the boundary strip, and therefore the new branches could not be predicted on the basis of a “sharp” boundary model. Below we present an exactly solvable model correctly describing all the edge excitations in the strong magnetic field limit. We obtain also the values of the oscillator strengths and the damping of these modes. The new branches become robust and are not destroyed by a finite relaxation time in the achievable region of relatively short wavelengths. The dynamics of the compressible electron liquid is governed by Euler equation and the continuity equation linearized in the velocity of the liquid v (ρ, t) and in the deviation of the concentration δn (ρ, t) from its equilibrium value n0 (ρ): v̇ + ωc (ẑ × v)− e 2 εm ∇ρ ∫ d2ρ1 δn(ρ1) |ρ−ρ 1 | = 0, (1) δṅ+∇ρ (n0v) = 0. (2) Here ρ is radius-vector in the plane XY of the two-dimensional electron liquid, ẑ is the unit vector along Z-axis, and ε is the dielectric constant. The last term in (1) represents Coulomb interaction [12]. In the following we assume that the electron liquid is homogeneous in y direction and occupies half-plane x > 0. Since the system is translationally invariant in the y direction, we will seek the solution of Eqs. (1), (2) in the form v = exp (iky − iωt)w(x); δn(x, y) = exp (iky − iωt) f(x). (3) Substituting Eqs. (3) into the system (1), (2) and eliminating w(x), we find an integral equation for f(x):