We have used low-field de Haas–Shubnikov oscillations in the resistivity to determine the electron temperatures of a two-dimensional electron gas subjected to various heating currents. This was done at many values of the electron density from 5.5×${10}^{15}$ to 10×${10}^{15}$ ${\mathrm{m}}^{\mathrm{-}2}$ that were incrementally induced by the persistent-photoconductivity effect. The onset of second-subband occupation is clearly reflected in a large rise in energy-loss rates. The variations of the energy-loss rate with both temperature and electron density are well reproduced by detailed numerical calculations using values of deformation-potential and piezoelectric coupling parameters consistent with bulk measurements. The calculations also show that there is a transition from the low- to high-temperature behavior in the liquid${\mathrm{-}}^4$He temperature range. In this region the energy-loss rates vary roughly as the cube of the temperature, a result often observed experimentally but one which is not of fundamental significance. At the lowest temperatures the behavior should be dominated by the piezoelectric interaction which is characterized by a ${\mathit{T}}^5$ law. The present experimental data show the transition from ${\mathit{T}}^3$ to a higher-power law as the temperature is decreased, in accordance with the theoretical predictions.

• Received 24 September 1990

DOI:https://doi.org/10.1103/PhysRevB.43.9033