Electron-phonon interaction in a very low mobility $\mathrm{GaAs}/{\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ $\delta$-doped gated quantum well

The energy relaxation rate for hot electrons in a gated $\mathrm{GaAs}/{\mathrm{Ga}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ $\delta$-doped quantum well has been measured over the temperature range 0.3–3 K. At higher temperatures the loss rate varies as $T^5$ and the magnitude agrees well with that predicted by the standard theory for piezoelectric electron-phonon scattering. At low temperatures the observed dependence changes to $T^4,$ the crossover occuring near $\mathrm{ql}\sim 0.35,$ where q is the average magnitude of the phonon wave vector and l the electron mean free path. This is in agreement with recent theoretical predictions for piezoelectric scattering in the dirty limit. The theory also predicts that the magnitude of the energy-loss rate should depend inversely on the conductivity of the sample. Good agreement is found at higher conductivities, but the measured values show saturation when the conductivity becomes very low.