Phase-shift analysis of two-dimensional carrier-carrier scattering in GaAs and GaN: Comparison with Born and classical approximations

We study Markovian two-dimensional (2D) carrier-carrier $\mathrm{}(c-c)\mathrm{}$ scattering in GaAs and GaN quantum well systems. We evaluate the phase shifts of scattered partial waves by numerically solving the Schrödinger equation for the 2D $c-c$ collision problem. The output of this “phase shift analysis” is exact quantum results (in Markovian limit) for the 2D $c-c$ scattering cross section and 2D $c-c$ scattering rate. We compare these results with the results of the Born approximation (with the “Fermi-golden-rule based” theory), and find that the Born approximation can strongly overestimate the exact results. In particular, the 2D $c-c$ scattering rate is overestimated by a factor of 1 to 15 in the GaAs quantum well and by a factor of 2 to 100 in the GaN quantum well, if conditions typical for a quantum-well-laser device or for a quantum well excited by a short laser pulse are considered. Our study is performed for a statically screened intercarrier interaction, but we expect the dynamic screening to further deteriorate the Born approximation: We show that the weaker the screening the worse the Born approximation in a 2D system. We also analyze the 2D $c-c$ collision as a classical event, and find a very good agreement with the phase-shift analysis in the case of weak screening. For unscreened 2D $c-c$ collisions the hierarchy of exact, Born, and classical cross sections is derived analytically, and it is shown that the Born approximation breakdown is due to the carrier two dimensionality.