Bistability and hysteresis of intersubband absorption in strongly interacting electrons on liquid helium

We study nonlinear inter-subband microwave absorption of electrons bound to the liquid helium surface. Already for a comparatively low radiation intensity, resonant absorption due to transitions between the two lowest subbands is accompanied by electron overheating. The overheating results in a significant population of higher subbands. The Coulomb interaction between electrons causes a shift of the resonant frequency, which depends on the population of the excited states and thus on the electron temperature $T_e$. The latter is determined experimentally from the electron photoconductivity. The experimentally established relationship between the frequency shift and $T_e$ is in reasonable agreement with the theory. The dependence of the shift on the radiation intensity introduces nonlinearity into the rate of the inter-subband absorption, resulting in bistability and hysteresis of the resonant response. The hysteresis of the response explains the behavior in the regime of frequency modulation, which we observe for electrons on liquid ${}^3$He and which was previously seen for electrons on liquid ${}^4$He.

Figure 1

Sketch of the energy subbands and the electron transitions. The total energy of an electron $E_{\alpha }={\epsilon }_{\alpha }+p^2/\left(2m\right)$ is plotted on the vertical axis. Vertical (red) arrows indicate photon-induced transitions, while horizontal (blue) arrows indicate transitions due to elastic scattering by excitations in helium.

Figure 2

Graphical solution of the energy balance equation (20) for $T=0.4$ K and $n_s=4.0\times {10}^7$ cm${}^2$. The dashed line is the energy loss rate [the left-hand side of Eq. (20)] due to inelastic scattering by helium vapor atoms. The solid lines show the rate of the microwave power absorption [the right-hand side of Eq. (20)] calculated for ${\omega }_F=104.5$ GHz, ${\Omega }_R=2.0$ MHz, and ${\omega }_F-{\omega }_{21}=0.1$ GHz (a) and 0.4 GHz (b). For the case (a), the stable solutions of Eq. (20), which are given by the intersections of solid and dashed lines, are marked by (red) circles.

Figure 3

Relative change of the reciprocal conductivity $-\Delta {\sigma }_{xx}^{-1}/{\sigma }_{xx}^{-1}$ vs the voltage applied to the bottom electrode $V_{\mathrm{B}}$ obtained for SEs on liquid ${}^3$He at $T=0.4$ K, $n_s=4\times {10}^7$ cm${}^{-2}$ and several values of the input power of radiation. The lines a to k correspond to $-$17.5, $-$14.0, $-$11.4, $-$8.8, $-$6.2, $-$4.2, $-$2.3, and $-$0.6 dB of the input power expressed as a fraction (in decibels) of the maximum power $\sim$1 mW.

Figure 4

Relative change of the reciprocal conductivity $-\Delta {\sigma }_{xx}^{-1}/{\sigma }_{xx}^{-1}$ vs $V_{\mathrm{B}}$ obtained at $T=0.4$ K, $n_s=4\times {10}^7$ cm${}^{-2}$, and at the maximum input power (0 dB). The dashed and solid lines correspond to sweeping through the resonance from the low and high sides of $V_{\mathrm{B}}$, respectively, as indicated by the arrows.

Figure 5

Relative change of the reciprocal conductivity $\Delta {\sigma }_{xx}^{-1}/{\sigma }_{xx}^{-1}$ vs $V_{\mathrm{B}}$ obtained at $T=0.2$ K, $n_s=4.2\times {10}^7$ cm${}^{-2}$ and three different values of the input power: $-$6.2, $-$5.4, and $-$4.2 dB (lines a, b, and c, respectively). The dashed and solid lines correspond to the sweeping through the resonance from the low- and high-$V_{\mathrm{B}}$ sides, respectively. A satellite peak that appears on the low-field side of the resonance is indicated by the full triangle.

Figure 6

$\Delta {\sigma }_{xx}^{-1}/{\sigma }_{xx}^{-1}$ vs $V_{\mathrm{B}}$ for $T=0.2$ K, $n_s=8.0\times {10}^6$ cm${}^{-2}$, and $-$2.3 dB of the input power. The dashed and solid lines correspond to the sweeping through the resonance from the low and high $V_{\mathrm{B}}$ sides, respectively, as indicated by arrows. (Inset) Plots obtained under the same conditions, but with the slower by a factor of two rate of sweeping from the lower (opened squares) and the higher (solid squares) $V_{\mathrm{B}}$ side.

Figure 7

Integrated line shape of the absorption signal measured with SEs on liquid ${}^3$He for $T=0.4$ K, $n_s=4\times {10}^7$ cm${}^{-2}$, at different power levels. The graph shows the in-phase (A) and the quadrature (B) components of the measured signal. The lines a to e correspond to $-$27.7, $-$20.6, $-$17.5, $-$14.4, and $-$11.4 dB of the input power attenuation.

Figure 8

Many-electron frequency shift $\Delta {\omega }_{21}$ vs electron temperature $T_e$ obtained from the data shown in Fig. 3 (squares); $\Delta {\omega }_{21}$ is counted off from its value for $T_e=T=0.4$ K. The solid line shows the result of calculations using Eq. (12) for $n_s=4\times {10}^7$ cm${}^{-2}$. The dashed line shows the calculation that excludes the term proportional to $|z_{12}{|}^2$ in Eq. (12).