Dynamical Hall Effect in a Two-Dimensional Classical Plasma

Abstract

We have shown that the collective modes of a bounded two-dimensional plasma in a magnetic field exhibit unexpected features which are dynamical manifestations of the Hall effect. In high fields we observed the appearance of a novel magnetically localized one-dimensional wave which propagates along the perimeter of the plasma. PACS numbers: 73.20.Cw, 71.45.Gm, 72.15.Gd The Lorentz force acting on a bounded system of charge carriers in a magnetic field can induce charge accumulation at the boundaries. The Hall resistance, which is a stationary manifestitation of this phenomenon, has traditionally been exploited to measure the charge carrier density. But it has not yet been recognized that the dynamics of a plasma in a magnetic field in the presence of density inhomogeneity (in particular boundaries) has physical aspects which are more general than the Hall resistance. In this Letter we present the first theoretical and experimental results on the dynamical Hall effect in a twodimensional classical plasma. The theoretical results obtained in a linearized frictionless hydrodynamic approach will be compared with measurements of the plasmon mode frequencies of electrons confined on a liquid helium surface in a cylindrical cell placed in a normal magnetic field Hz. The spectrum and its behavior with magnetic field can be seen in Figs. 1 and 2. An unusual feature is the existence of modes whose frequencies decrease with field, in contradiction to the generally used infinite-geometry rule (o 2 = a)p(H = 0) +w c 2 where w p (/f = 0) is the zerofield plasmon frequency and co c = eH/mc is the cyclotron frequency. 1 We shall show that this decrease is one of the dynamical manifestations of the Hall effect.We suppose the electrons of mass m and charge e to be confined to a single region of the plane z = 0 by a set of externally imposed potentials which determine the equilibrium charge distribution n s (r) = n 0 o-(r) where r is a two-dimensional position vector. We will now show that the hydrodynamic problem can be reduced to an electrostatic one. The only force relevant to their dynamics arises from the in-plane component of the (total) electrostatic field described by a scalar potential </> (T,z,t). The equation of motion of the electron beam is thus d\/dt = (e/m)V<£ +OJ C XV where v is the velocity field and <o c = a> c £. If we suppose a time variation </ > = </>(r,z)exp/a^, we can combine the linearized equation of motion rewritten as (0 o (/) < v= (e/m)(V 2 0+o> c x V 2 <£)/(co c 2 -co 2 ) (1) with the linearized continuity equation and a Gauss integral of Poisson's equation to write the boundary con-20 30 40 Frequency (MHz)FIG. 1. Part of the longitudinal spectrum of electrons of density « 0 =1.7xl0 7 cm" 2 obtained at T = 60 mK. The upper-plate, guard-ring, and corresponding electron-surface potentials are respectively -3, -7, and -3 V. The solid and dashed arrows of (a) indicate the calculated mode frequencies with and without the density profile correction. For this density the solid-phase vibration frequency of ...

On-Chip Microwave Quantum Hall Circulator

“…Central to the operation of our device are edge magnetoplasmons (EMPs), resonant modes of the electron gas first observed in the classical response of electrons on the surface of liquid helium [18,19]. Such excitations have since been found to propagate along a quantum Hall edge in response to a capacitively coupled microwave excitation [20][21][22][23][24][25][26][27].…”

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“…When the frequency is lowered, the induced Hall charge in the edge channel tunnels to the bulk 3 The discontinuity is due to the absence of the second derivative ∇ 2 x φ e+ (x, ω) in Eq. (23). The boundary condition at x = x − and Eq.…”

“…In the following we focus on the slower development of the induced potential in the bulk region and calculate the potential at ω < 10 s (4), (11), and (9) become (23) with j b+e+ (x, ω) defined by Eq.(3). Since the Hall potential only changes the sign at (x, y) → (−x, −y) in the boundary condition we consider, the above set of equations is sufficient to obtain the solution.…”

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