Microwave response of two-dimensional electron rings

We report microwave absorption studies on rings containing two-dimensional electrons. The absorption is measured on single rings with different widths using an optical luminescence technique. The dispersion of the plasmon modes has been determined in the presence of a perpendicular magnetic field. The transformation of two-dimensional plasmons in the disk geometry into plasmons with one-dimensional character in a narrow ring is addressed. Experimental data are compared with theoretical estimates based on a self-consistent solution of the Poisson and hydrodynamic equations.

Figure 1

Luminescence spectra obtained on a disk geometry with a diameter of $1\mathrm{mm}$ with and without microwave radiation present (upper curves). The corresponding differential spectrum is shown at the bottom. The incident radiation has a frequency of $6\mathrm{GHz}$. The magnetic field is chosen so that a resonance occurs $(B=105\mathrm{mT})$. The electron concentration is $1.4\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 2

Typical magnetic-field dependencies of the absorption amplitude for the indicated microwave radiation frequencies. Data are shown for two ring-shaped mesas with the following dimensions: (a) $D=1\mathrm{mm}$, $d=0.4\mathrm{mm}$ and (b) $D=1\mathrm{mm}$, $d=0.6\mathrm{mm}$. The electron concentration in these rings is $1.25\times {10}^{11}$ (a) and $1.15\times {10}^{11}{\mathrm{cm}}^{-2}$ (b). The observed modes are $n=0$ modes.

Figure 3

The resonance frequencies of $n=0$ magnetoplasma modes in disk and ring geometries. The outer diameter $D$ of the disk (a) and rings (b)–(d) is $1\mathrm{mm}$. The inner diameter $d$ of the rings has been indicated on each panel where it has relevance. The electron densities were as follows: $1.4\times {10}^{11}$ (a), $1.1\times {10}^{11}$ (b), (c), and $1.2\times {10}^{11}{\mathrm{cm}}^{-2}$ (d).

Figure 4

Calculated magnetoplasma frequencies of the modes with quantum numbers $(n,l)=(0,\pm 1)$ and $(0,-3)$ for disk and ring geometries as in Fig. 3 (outer diameter $D=1\mathrm{mm}$ in all cases; $d$ is indicated in panels where it is relevant). The electron densities are as in Fig. 3: $1.4\times {10}^{11}$ (a), $1.1\times {10}^{11}$ (b), (c), and $1.2\times {10}^{11}{\mathrm{cm}}^{-2}$ (d). The substrate thickness was assumed to be $0.4\mathrm{mm}$.

Figure 5

A comparison between the ${\omega }_{0-}$ branch of the disk geometry with the ${\omega }_{0-}$ resonance frequencies of one of the three rings under study: (a) wide ring with $d∕D=0.2$ and $n_s=1.1\times {10}^{11}{\mathrm{cm}}^{-2}$, (b) intermediate ring with $d∕D=0.6$ and $n_s=1.15\times {10}^{11}{\mathrm{cm}}^{-2}$, and (c) narrow ring with $d∕D=0.8$ and $n_s=1.2\times {10}^{11}{\mathrm{cm}}^{-2}$. The outer diameter $D$ equals $1\mathrm{mm}$ for all geometries. The resonance frequencies are normalized to the plasma frequency $f_0$ (see text) of a disk with the same outer diameter $\left(1\mathrm{mm}\right)$ and electron density.

Figure 6

The magnetic-field dependence of the ${\omega }_{0-}$ mode of the narrow ring with $D=1\mathrm{mm}$, $d=0.8\mathrm{mm}$, and the density $n_s=1.2\times {10}^{11}$ (hollow dots). The solid line has the functional form described by Eq. (9). The values of ${\beta }^{\mathrm{expt}}$ and $F_0^{\mathrm{expt}}$ were obtained from a least-squares fit to the data. The inset shows the same data and fit but $f^{-2}$ is plotted versus $B^2$.

Figure 7

The zero-field plasma frequency $f_{B=0}$ for the ring geometry as a function of the aspect ratio $\eta =d∕D$. Frequencies are normalized to the zero-field plasma frequency $f_0$ of a disk with the same outer diameter and density. The electron density of the rings that contributed the experimental data points (solid circles) to this plot varied from $1.1\times {10}^{11}$ to $1.4\times {10}^{11}{\mathrm{cm}}^{-2}$. Open circles are theoretical estimates obtained from the self-consistent solution of the Poisson and hydrodynamic equations as described in Sec. 3. The solid curve corresponds to the resonance frequency predicted from Eq. (7) for $B=0$.