Observation of Retardation Effects in the Spectrum of Two-Dimensional Plasmons

Retardation effects, theoretically predicted more than 35 years ago, are observed in the spectrum of two-dimensional plasmons in high-electron-mobility $\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ quantum wells. In zero magnetic field, a strong reduction of the resonant plasma frequency is observed due to the hybridization of the plasma and light modes. In a perpendicular magnetic field $B$, hybrid cyclotron-plasmon modes appear with a very unusual dependence of the frequency on $B$ field. Experimental results are in excellent agreement with the theory.

Figure 1

2D magnetoplasmon spectra at 1.5 K for several samples with various electron densities and different mesa diameters. The retardation parameter $A$ has also been included.

Figure 2

Calculated dependence of the normalized magnetoplasma resonances $\omega /{\omega }_0$ versus ${\omega }_c/{\omega }_0$, i.e., normalized magnetic field, for a disk-shaped sample. Only the lowest $n=1$ modes with angular momenta $L=1,2,3$ are shown. The edge magnetoplasmon modes with negative $B$ dispersion are not shown for $L=2$ and 3. [In Fig. 1a the two pairs of ${\omega }_{\pm }$ are the $n=L=1$ modes for two different samples, not the modes with $L>1$.]

Figure 3

(a) Normalized resonance frequency at $B=0$ ${\omega }_{B=0}/{\omega }_0$ and (b) their slope $|d{\omega }_{\pm }/d{\omega }_c|$ as $B\to 0$ vs the retardation parameter $A={\omega }_0\sqrt{ϵ}R/c$; ${\omega }_0=(2\pi n_s{e}^2/m^{\star }\overline{ϵ}R{)}^{1/2}$. Symbols are experimental points; solid curves correspond to theory.

Figure 4

(a) Experimental data plotted on appropriate axes to determine the characteristic frequency $\Gamma$ and the relation between the wave vector $q$ and the disk diameter $d$, $qd=2.4$, as outlined in the text. (b),(c) The 2D-plasmon-polariton dispersion deduced from experiments for two samples with different density. Symbols are experimental points. Curves are the dispersion of light (solid) and the 2D plasmon dispersion calculated in the quasistatic approximation [Eq. (1)].