Nonequilibrium electrons in double quantum well structures:  A Boltzmann equation approach

A theoretical study of hot electrons in double quantum well is presented. We consider a system of coupled kinetic equations that describe nonequilibrium population in two lowest subbands, and find an analytical solution of these equations. All previous treatments of the electron distribution functions are based on the two extreme limits ${\tau }_0/{\tau }_{\mathrm{ee}}\ll 1$ or ${\tau }_0/{\tau }_{\mathrm{ee}}>~1,$ where ${\tau }_0$ and ${\tau }_{\mathrm{ee}}$ refer to electron-LO-phonon $(e-\mathrm{LO})$ and electron-electron $(e-e)$ scattering times, respectively. In our approach, the distribution functions are investigated for the whole range of ${\tau }_0/{\tau }_{\mathrm{ee}}$ ratios. The outcome of kinetic equation for subband distribution functions provides a comprehensive description of the lasing process in the intersubband lasers. Our calculations show that the lower subband distribution function above LO-phonon threshold is always strongly nonequilibrium and deviates far from thermal distributions while in the region below the LO-phonon threshold it could be close to a Maxwellian. Using these distribution functions we calculate spectral density of gain at various temperatures.