scholarly journals Formulation of the Boltzmann Equation as a Multi-Mode Drift-Diffusion Equation

VLSI Design ◽  
1998 ◽  
Vol 8 (1-4) ◽  
pp. 539-544
Author(s):  
K. Banoo ◽  
F. Assad ◽  
M. S. Lundstrom
Keyword(s):  
Diffusion Coefficient ◽  
Boltzmann Equation ◽  
Diffusion Equation ◽  
Momentum Space ◽  
Bulk Silicon ◽  
Rigorous Solution ◽  
Drift Diffusion ◽  
The Boltzmann Equation ◽  
And Diffusion ◽  
Multi Mode

We present a multi-mode drift-diffusion equation as reformulation of the Boltzmann equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.

scholarly journals The Nonclassical Boltzmann Equation and DIffusion-Based Approximations to the Boltzmann Equation

2015 ◽  
Vol 75 (3) ◽  
pp. 1329-1345 ◽  
Author(s):  
Martin Frank ◽  
Kai Krycki ◽  
Edward W. Larsen ◽  
Richard Vasques
Keyword(s):  
Boltzmann Equation ◽  
The Boltzmann Equation ◽  
And Diffusion

scholarly journals On the Theory of Electron Diffusion in Electrostatic Fields in Gases

1974 ◽  
Vol 27 (2) ◽  
pp. 195 ◽  
Author(s):  
HR Skullerud
Keyword(s):  
Diffusion Coefficient ◽  
Boltzmann Equation ◽  
Diffusion Equation ◽  
Diffusion Tensor ◽  
Lateral Diffusion ◽  
Mean Square ◽  
Electrostatic Fields ◽  
The Mean ◽  
Lateral Diffusion Coefficient

The motion of electrons in a gas in the presence of large electron density gradients has been studied theoretically, starting from the two-term expansion of the Boltzmann equation. The effects of material boundaries have not been considered. An electron swarm released as a b-function in space and with an equilibrium energy distribution is found initially to develop as a spheroid with dimensions determined by the lateral diffusion coefficient. It subsequently passes through a stage involving a slowly decaying pear-shaped deformation, before ultimately becoming an ellipsoid with dimensions determined by the longitudinal and lateral components of the diffusion tensor. Numerical values cited in the literature for the long-term deviations from the mean square widths predicted by the diffusion equation have been found to be in error by factors of 10 or more.

Analytical Solution for Boltzmann Collision Operator for the1-D Diffusion equation

2021 ◽  
Vol 9 (9) ◽  
pp. 1514-1517
Author(s):  
Wasif Almady
Keyword(s):  
Boltzmann Equation ◽  
Analytical Solution ◽  
Diffusion Equation ◽  
Collision Operator ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
The Boltzmann Equation ◽  
Boltzmann Collision Operator ◽  
The One

Abstract: In this paper, we have presented the analytical solution of the collision operator for the Boltzmann equation of onedimensional diffusion equation using the analytical solution of the one-dimensional Navier Stoke diffusion equation. Keywords: Boltzmann equation; analytical collision operator; one-dimensional diffusion equation.

On Nonlocal Constitutive Relations, Continued Fraction Expansion for the Wave Vector Dependent Diffusion Coefficient

1977 ◽  
Vol 32 (7) ◽  
pp. 678-684 ◽  
Author(s):  
Siegfried Hess
Keyword(s):  
Diffusion Coefficient ◽  
Boltzmann Equation ◽  
Wave Vector ◽  
Continued Fraction ◽  
Constitutive Relations ◽  
Transport Coefficients ◽  
Mean Free Path ◽  
Upper And Lower Bounds ◽  
Continued Fraction Expansion ◽  
The Boltzmann Equation

Abstract Nonlocal constitutive relations which involve wave vector dependent transport coefficients can be derived from the Boltzmann equation. Diffusion of a Lorentzian gas is treated as an illustrative example. Transport-relaxation equations obtained from the Boltzmann equation with the help of the moment method lead to a continued fraction expansion for the wave vector dependent diffusion coefficient D(k). Rapidly converging upper and lower bounds on D(k)/D(0) are found which are meaningful for all values of lk where l is a mean free path and k is the magnitude of the wave vektor k. Also some remarks on a frequency and wave vector dependent diffusion coefficient are made.

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