Orders of approximation in transport theory for electrons or holes in a scattering medium

In the framework of the free-path method, a transport quantity for free electrons (or holes) either in gases or in semiconductors in the presence of a static and uniform electric field E can be expressed as a sum of terms, each containing as a factor a properly indexed function Vsub(j)(a) = Vsub(j)(eE/m), which can be reduced to a series of integrals over the electron speed v. Such integrals are always finite even when nonphysical divergences appear in the corresponding terms of the expansion obtained by approximating the solution of the Boltzmann equation by truncated Legendre expansions. In a wide variety of cases, the integrals (appearing as terms of our series) can be calculated by series expansion, and Vsub(j)(a) turns out to be given by a series in powers of W2/[v2]<<1, where W is the drift velocity. This ratio, for elastic collisions, can reach an asymptotic value (for high E fields) of the order Σ2 = m/M, where m and M are the masses of the electron and of a scattering centre, respectively. The integrands appearing in the integrals over v can be expanded in a power series of dimensionless terms containing the acceleration a = eE/m. The term in asup(s) produces, after integration over v, a corresponding term containing (W2/[v2])sup(s/2). With the aim of calculating Vsub(j)(a) one can, therefore, speak of approximation of order s if terms including asup(s) are retained in the integrands. This allows convenient truncations and the solution of many transport problems with the desired accuracy. (author)