A new relation between the hot electron power loss and acoustic phonon limited mobility in Bloch-Gr\"uneisen regime

Expressions for the electron power loss F(T) and mobility {\mu}_p due to acoustic phonon scattering are given in the Bloch-Gr\"uneisen (BG) regime for three- and two- dimensional electron gas in semiconductors and Dirac-fermions. We obtain a simple relation F(T) {\mu}_p = {\eta}ev_s^2, where {\eta} (~1) is a constant, e is the electron charge and v_s is acoustic phonon velocity. It is found to be independent of temperature and electron concentration. This relation is applied to GaAs heterojucntions and graphene, to obtain {\mu}_p from the measured F(T). We propose that, using this relation, the measurements of F(T), in BG regime, which depends exclusively upon acoustic phonons coupling, could serve as a tool to determine the low temperature {\mu}_p, which is otherwise difficult to measure due to the contributions from disorders