Floquet theory of microwave absorption by an impurity in the two-dimensional electron gas

We investigate the dynamics of a two-dimensional electron gas (2DEG) under circular polarized microwave radiation in the presence of dilute localized impurities. Inspired by recent developments on Floquet topological insulators we obtain the Floquet wave functions of this system which allow us to predict the microwave absorption and charge density responses of the electron gas; we demonstrate how these properties can be understood from the underlying semiclassical dynamics even for impurities with a size of around a magnetic length. The charge density response takes the form of a rotating charge density vortex around the impurity that can lead to a significant renormalization of the external microwave field which becomes strongly inhomogeneous on the scale of a cyclotron radius around the impurity. We show that this inhomogeneity can suppress the circular polarization dependence which is theoretically expected for microwave induced resistance oscillations but which was not observed in experiments on semiconducting 2DEGs. Our explanation for this so far unexplained polarization independence has close similarities with the Azbel'-Kaner effect in metals where the interaction length between the microwave field and conduction electrons is much smaller than the cyclotron radius due to skin effect generating harmonics of the cyclotron resonance.

Figure 1

Left panel (a): density of Husimi function at the disk radius $r=r_d$ shown by color (maximum for yellow/white; minimum zero for violet/black), Poincaré section for classical Hamiltonian dynamics of (2) is drawn at the moment of collision with disk and is shown by green dots; here $J=2.7,r_d/{\ell }_B=1,q{E}_{ac}{\ell }_B/\hslash {\omega }_c=0.3$ at energy ${\epsilon }_F/\hslash {\omega }_c\approx 40$. Middle panel (b): classical Poincaré section of the collision map (5) at parameters of panel (a). Right panel (c): geometry of collision.

Figure 2

Absorption power $\mathcal{P}$ (in arbitrary units) is shown as a function of $J=\omega /{\omega }_c$ for $r_d/{\ell }_B=2$ at $q{E}_{ac}{\ell }_B/\hslash {\omega }_c=0.1$ and ${\epsilon }_F/\hslash {\omega }_c=60$. Data are shown for two relaxation times with ${\omega }_c\tau =10,100$ (different signs of $J$ correspond to different signs of circular polarization, ${\omega }_c\tau =10$ and 100 datasets are vertically shifted for clarity). Quantum results are obtained from the master equation on the density matrix; they are compared with classical results from the kinetic equation. An excellent agreement between quantum and classical results is observed for a repulsive potential. For an attractive potential the agreement is only qualitative for the kinetic equation giving the same phase of microwave absorption oscillations as the quantum calculation.

Figure 3

Charge density $\rho (x,y)$ dependence on $x$ (at $y=0$) and on $y$ (at $x=0$) obtained from the quantum master equation and the classical kinetic equation presented in the rotation frame. Here the system parameters are the same as for the repulsive case in Fig. 2 with $J=2.7$ for $r_d/{\ell }_B=2$ at $q{E}_{ac}{\ell }_B/\hslash {\omega }_c=0.1$ and ${\epsilon }_F/\hslash {\omega }_c=40$ with a relaxation time ${\omega }_c\tau =10$.

Figure 4

Relative charge density variation $\rho (x,y)=\delta n/n_0$ (expressed in percent) shown in $(x,y)$ plane in the rotation frame where a microwave field is directed along the $x$ axis. Left panel shows the results of quantum simulations with the master equation; left panel show the results of the classical kinetic equation. The parameters are the same as in Fig. 3.

Figure 5

Variation of the enhancement factor $E_{ee}/E_{ac}$ of the amplitude of effective field acting on an electron due to the density variation in the $(x,y)$ plane (see Fig. 4 left panel) induced by the external microwave field. The results, shown in the rotating frame, are obtained from the quantum master equation and the relations given by Eq. (12).

Figure 6

Dependence of resistivity $R_{xx}$ (in arbitrary units a.u.) on $J=\omega /{\omega }_c$. Left panel: the case of initial external microwave field; right panel: the case of screened renormalized field acting on a length scale $r_{\mathrm{eff}}/R_c=0.3$. The results are obtained from the numerical simulation of classical Hamiltonian dynamics of electrons in the presence of impurities with Gaussian potential $U_{\mathrm{amp}}exp(-r^2{r}_d^{-2})$ with a density $n_i{r}_d^2={10}^{-3}$, an amplitude $U_{\mathrm{amp}}/{\epsilon }_F=1.5$, and a characteristic radius $r_d\omega /v_F=0.1$. Here LH and RH correspond to left and right hand microwave (MW) polarization, respectively.

Figure 7

Relative variation of the time averaged electron charge density $\langle \delta n_e\left(r\right)\rangle /n_e$ induced by a microwave irradiation, as a function of radial distance $r$ from the impurity. The results are shown two microwave fields ${\epsilon }_{ac}=0.1$ and 0.2 (${\epsilon }_{ac}=q{E}_{ac}{\ell }_B/\hslash {\omega }_c$). Other parameters are as in Fig. 3 ($r_d/{\ell }_B=2,E_F/\hslash {\omega }_c=40$). ${\omega }_c\tau =100$.

Figure 8

Geometry of reflection during collision with a disk in case of attractive and repulsive potentials.

Figure 9

Dependence of the average absorbed microwave power $P_{\mathrm{abs}}$ for electrons colliding with an impurity for parameters: $q{E}_{ac}/\left(m\omega v_F\right)=0.013,r_d/R_c=0.13,{\omega }_c\tau =100,U_{\mathrm{disc}}=\pm 2{\epsilon }_F$, a good agreement is observed between results from the kinetic equation and from the collision map (A77).

Figure 10

Dependence of the microwave absorption ${\mathcal{P}}_{\mathrm{abs}}$ on $J$ (arbitrary units) for different values of impurity radius $r_d$ obtained from the quantum master equation Eq. (A92) (full curves) and the classical kinetic equation Eq. (A93) (dashed curves). Here $U_{\mathrm{disc}}=2{\epsilon }_F,q{E}_{ac}{\ell }_B/\hslash {\omega }_c=0.1$ and ${\omega }_c\tau =100$. The semiclassical curves reproduce correctly the results from the quantum master equation. For $r_d/{\ell }_B=0.5$ additional fluctuations appear in the quantum calculation around an average lineshape which is still well described by the semiclassical calculation. We attribute those to interference effects which become more pronounced when the size of the impurity becomes closer to the Fermi wavelength. In a real sample such fluctuations would probably be suppressed by ensemble average leaving only an average value which would correspond to the semiclassical result.

Figure 11

Scaling dependence of the microwave absorption ${\mathcal{P}}_{\mathrm{abs}}$ rescaled by incident microwave power $\propto E_{ac}^2$ on $J$ (arbitrary units) for different amplitudes of microwave field $E_{ac}$. The results are obtained from the quantum master equation Eq. (A92) for a repulsive impurity potential ($U_d=90\hslash {\omega }_c$) at ${\omega }_c\tau =100$ and $0.01\le E_{ac}\le 0.1$. The data show that the scaling $P_{\mathrm{abs}}\propto E_{ac}^2$ is highly accurate. At the lowest microwave powers the data become more noisy and start to show individual resonances; we think that the sharp peaks come from contributions of individual levels trapped around the impurity that lead to sharp resonances due to the high ${\omega }_c\tau =100$ value. At higher microwave power the individual resonances are broadened giving the semiclassical absorption curve.

Figure 12

Dependence of $\delta n_e(x=0,y)/n_e$ on $y/R_c$ for different impurity diameters, results are shown for the kinetic equation calculation.

Figure 13

Dependence of $\delta n_e(x=0,y)/n_e$ on $y/R_c$ for different values of $J=\omega /{\omega }_c$.

Figure 14

Dependence of $\delta n_e(x=0,y)/n_e$ on $y/R_c$ for different values of ${\omega }_c\tau$.