Asymptotic Expansion of the Mean Velocities of the Electrons

With special attention on semiconductors and low-dimensional materials, this research article explores the asymptotic increase of mean electron velocities in several quantum transport systems. We derive analytical formulas faithfully describing the asymptotic behavior of electron velocities under various temperature and electric field regimes by using perturbative approaches and non-equilibrium Green's function techniques. By means of a unified scaling theory that considers the transition between diffusive and ballistic transport, our theoretical framework reconciles apparently contradicting experimental results. Experimental results from graphene and silicon-based devices together with numerical simulations grounded on the Boltzmann transport equation verify the analytical predictions. Our results show that with different scaling exponents at various asymptotic conditions, electron mean velocities follow a non-trivial power-law dependence on applied field strength. These findings offer vital new perspectives for the design of next-generation electronic systems in which exact control of electron flow is critical.