Abstract
Using a simple setup, the linear thermopower is measured in etched one-dimensional (1D) constrictions with large subband energy spacings, allowing the temperature of the two-dimensional electron gas on one side of the constriction to be determined as a function of the applied heating power. As tested by the Cutler-Mott relation, the 1D constriction acts as a reliable electron thermometer for lattice temperatures in the range ${T}_{L}=4.2--13\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The power dissipated per electron, measured as a function of electron temperature, reveals that at ${T}_{L}=4.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ the acoustic phonon scattering is dominated by ${T}^{5}$ behavior due to an unscreened deformation potential. At higher lattice temperatures the scattering can be described by an interpolation between the Gr\"uneisen-Bloch and equipartition regimes.
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