Effect of acoustic-phonon confinement on the phonon-drag thermopower of a two-dimensional electron gas in a semiconductor thin film

We study theoretically the effect of acoustic-phonon confinement on the phonon-drag thermopower $S^g$ of a two-dimensional (2D) electron gas in a quantum well embedded in a semiconductor thin film. The screened interaction of 2D electrons via the deformation and piezoelectric potentials in the film with stress-free surfaces is considered. Analytical expressions are obtained for $S^g$ (due to flexural, dilational, and shear modes) in the Bloch-Gruneisen regime where the nature of confinement is pronounced. Interestingly, $S^g$ dependence on temperature $T$ is the same for both deformation and piezoelectric potential mechanisms, which may be attributed to the similar phonon wave vector dependence of their interaction matrix elements at very low temperatures. We find $S^g\sim T^3$ for flexural modes and $T^5$ for dilatational and shear modes. $S^g$ due to flexural modes, which have quadratic dispersion and high density of states, is found to be dominant over contribution from other modes. $S^g$ in thin films is suppressed due to a smaller phonon mean free path compared to bulk phonons. Herring’s formula relating $S^g$ to phonon limited mobility ${\mu }_p$ is discussed. We find $S^g{\mu }_p\sim T^{-1/2}$ for flexural modes and $T^{-1}$ for dilatational and shear modes showing the invalidity of the Herring’s formula for flexural modes.