The effect on the electron transport of the confinement of the electrons to a narrow\ud channel in GaAs/GaAlAs heterojunctions and Si MOSFETs is reflected in quantities like\ud the thermopower (S) which is sensitive to the transport of both heat and charge. The\ud calculations described here confirm that in these systems S is dominated by phonondrag\ud (Sg) at temperatures (T) around 1-10K and reveals more sensitivity than previously\ud imagined.\ud Simple models and the Boltzmann transport formalism have been investigated. The\ud formalism enhances the predictions of the simple models and reproduces the simple S.\ud formulae in appropriate limits. Amplification of S9 in quasi-2D arises from the loss of the\ud momentum conservation constraint across the channel at small widths b.\ud Earlier calculations were numerically inaccurate and greatly overestimate -S9 by ignoring\ud screening. An effective multi-subband screening dielectric function is defined which\ud reduces to the single subband approximation at small b and low electron density (n). Nondegeneracy\ud has also been included. It is an important consideration despite the low temperatures\ud of most of the data. The treatment of electron confinement has been improved\ud and the temperature dependence of the polarizability investigated. It is unimportant in\ud the current experimental systems but significant at lower n and higher T.\ud The piezoelectric scattering mechanism has been introduced and dominates S. in the\ud heterojunction for T <1K. A dominant 2D wavevector component has been defined for\ud the phonon population at given T which is very helpful in understanding S9. A correction\ud for the energy dependence of the electron relaxation-time is necessary and demonstrates\ud the dependence of S. upon the dominant electron scattering mechanism.\ud The calculations of S. in the quantum-limit and boundary scattering regime now explain\ud the measured S in heterojunctions and peaks in -Sg/T3 in the MOSFET up to an\ud accuracy better than 10% without adjustable parameters
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