  We study theoretically the piezoelectric interaction of a surface acoustic
wave (SAW) with a two-dimensional electron gas confined to an isolated quantum
dot. The electron motion in the dot is diffusive. The electron-electron
interaction is accounted for by solving the screening problem in real space.
Since the screening in GaAs/AlGaAs heterostructures is strong, an approximate
inversion of the dielectric function epsilon(r,r') can be utilized, providing a
comprehensive qualitative picture of the screened SAW potential and the charge
redistribution in the dot. We calculate the absorption and the scattering
cross-sections for SAW's as a function of the area of the dot, A, the sound
wave vector, q, and the diffusion coefficient D of the electrons. Approximate
analytical expressions for the cross-sections are derived for all cases where
the quantities A*q^2 and A*omega/D are much larger or smaller than unity; omega
is the SAW frequency. Numerical results which include the intermediate regimes
and show the sample-specific dependence of the cross-sections on the angles of
incidence and scattering of surface phonons are discussed. The weak
localization corrections to the cross-sections are found and discussed as a
function of a weak magnetic field, the frequency, and the temperature. Due to
the absence of current-carrying contacts, the phase coherence of the electron
motion, and in turn the quantum corrections, increase as the size of the dot
shrinks. This shows that scattering and absorption of sound as noninvasive
probes may be advantageous in comparison to transport experiments for the
investigation of very small electronic systems.
