Analog Devices and Circuits 1 by 2023

Author:2023
Language: eng
Format: epub
ISBN: 9781394255467
Published: 2023-11-20T00:00:00+00:00

3.3.4.5. Classic limit of the Wigner equation

We discuss the classic limit of [3.48] considering the case where the potential Vi is a linear or quadratic function of the position, namely:

[3.49]

The dotted lines represent the quadratic term. Force F can be at most a linear function of the position. As the even terms of the Taylor series of V cancel out in [3.48], the Wigner potential becomes:

[3.50]

The right hand side of [3.47] becomes:

[3.51]

where we used equality:

[3.52]

Then, the Wigner equation is reduced to the Boltzmann equation, without collision:

[3.53]

Now, let us consider as an initial condition a minimum uncertainty on the wave packet. The Wigner function of such a packet is a Gaussian of the position and momentum. The latter can equally be interpreted as a conventional initial distribution of electrons, provided that force is a constant or linear function of the position; the packet evolves like that of the conventional distribution. Despite the spreading in the phase space, Gaussian devices determine the general shape of the packet. fw remains positive during the evolution. However, strong variations in the field with position introduce interference effects. Near band shifts, the packet quickly loses its shape and negative values appear.

Monte Carlo algorithms can be designed based on the idea that conditions on the right-hand side of the Wigner–Boltzmann equation represent gain and loss of terms for the density of the phase space. To fix the ideas, consider the semi-classic Boltzmann equation.

[3.54]

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.