ELECTRON NONLOCALITY IN SEMICONDUCTORS

The Boltzmann equation is the theoretical foundation for analyzing the behavior of most semiconductor devices. This equation treats electron s as point particles traveling along well-defined, semiclassical traje ctories. Electrons are not point particles but are described by wavefu nctions which are non-localized to some extent, and the Boltzmann equa tion can not be used when this non-locality is important. We resolve t his impasse by deriving the quantum mechanical corrections to the Bolt zmann equation. Among these corrections are dispersion terms which det ermine how the electron probability density spreads out from the semic lassical trajectories; i.e. which govern electron non-locality. This e xtends the reach of the Boltzmann, equation to situations where non-lo cality is important, such as tunneling devices, heterostructures and n ano devices. These derived dispersion terms depend only on the band en ergies of the crystal and on imposed electromagnetic fields. More spec ifically, we investigate the general problem of electron transport in crystalline solids. We restrict attention to the situation of greatest physical interest, where the imposed electromagnetic fields vary on l ength scales much larger then the size of a unit cell in the crystal ( typically about 5 Angstrom). This enables us to derive quantum mechani cal corrections to the Boltzmann equation by using singular perturbati on techniques to systematically analyze the Schrodinger equation for e lectrons in the crystal.