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The solution of the Boltzmann transport equation with an external force provides the distribution function from which macroscopic quantities can be derived. The right-hand side of (5.5) describes the changes to the distribution function induced by scattering. The particle's group velocity is determined from the semiconductor band structure as . In the parabolic band approximation, , and the particle's group velocity can be calculated from the effective mass tensor .
The distribution function defines the probability density to find a particle in at a given time . Obviously, such a statistical description can only be appropriate when the number of carriers is large. Extremely down-scaled devices may contain too few carriers to justify this kind of statistical treatment.
Since carriers interact through their electric fields, the distribution function at a particular point in the six dimensional position-momentum (phase) space at a given time can only be determined from the knowledge of in all other points. This would involve a treatment using an particle system and an particle distribution function. However, if the carrier-carrier correlations are weak, the particle distribution function can be contracted to a one-particle distribution function [Harris04]. Alternatively, the influence of other carriers can be treated through the self-consistent electric field [Venturi89] and schemes where the Pauli exclusion principle is included [Bosi76,Lugli85,Yamakawa96,Ungersboeck06b].
A main assumption of the Boltzmann transport equation is that particles can be treated semiclassically, obeying Newton's law. Quantum mechanics enters the equation only through the band structure and the description of the collision term. Since both the position and the momentum of a particle are arguments of the distribution function, apparently the quantum mechanical uncertainty principle is violated. Assuming a spread in particle energy of , one finds that the spread in position is
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