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The temporal evolution of quantal Joule heating of 2D electrons in GaAs quantum well placed in quantizing magnetic fields is studied using a difference frequency method. The method is based on measurements of the electron conductivity oscillating at the beat frequency $f=f_1-f_2$ between two microwaves applied to 2D system at frequencies $f_1$ and $f_2$. The method provides $direct$ access to the dynamical characteristics of the heating and yields the inelastic scattering time $\tau_{in}$ of 2D electrons. The obtained $\tau_{in}$ is strongly temperature dependent, varying from 0.13 ns at 5.5K to 1 ns at 2.4K in magnetic field $B$=0.333T. When temperature $T$ exceeds the Landau level separation the relaxation rate $1/\tau_{in}$ is proportional to $T^2$, indicating the electron-electron interaction as the dominant mechanism limiting the quantal heating. At lower temperatures the rate tends to be proportional to $T^3$, indicating considerable contribution from electron-phonon scattering.

Dynamics of quantal heating in electron systems with discrete spectra