Strongly Enhanced Hole-Phonon Coupling in the Metallic State of the Dilute Two-Dimensional Hole Gas

We have studied the temperature dependent phonon emission rate $P$($T$) of a strongly interacting ($r_s\geq$22) dilute 2D GaAs hole system using a standard carrier heating technique. In the still poorly understood metallic state, we observe that $P$($T$) changes from $P$($T$)$\sim T^5$ to $P$($T$)$\sim T^7$ above 100mK, indicating a crossover from screened piezoelectric(PZ) coupling to screened deformation potential(DP) coupling for hole-phonon scattering. Quantitative comparison with theory shows that the long range PZ coupling between holes and phonons has the expected magnitude; however, in the metallic state, the short range DP coupling between holes and phonons is {\it almost twenty times stronger} than expected from theory. The density dependence of $P$($T$) shows that it is {\it easier} to cool low density 2D holes in GaAs than higher density 2D hole systems.