D. From Boltzmann Distribution to Drift-Diffusion Current Equations

Here the relation was used, which is valid for a parabolic energy band. Note that the charge has to be taken with the proper sign of the particle (positive for holes and negative for electrons). A general definition of current density is given by

(D.2) |

where the integral on the right hand side represents the first moment of the distribution function. This definition of current can be related to (D.1). After multiplying both sides by and integrating over v one gets

since the function is odd in , and its integral is therefore zero. Thus, one has from (D.3)

(D.4) |

Integrating by parts yields

(D.5) |

and

(D.6) |

can be written, where is the average of the square of the velocity defined as

(D.7) |

Because of the

The drift-diffusion equations are derived introducing the mobility and replacing with its average equilibrium value , therefore neglecting thermal effects. The diffusion coefficient (Einstein's relation) is also introduced, and the resulting drift-diffusion current is

where q is used to indicate the absolute value of the electronic charge. Although no direct assumptions on the non-equilibrium distribution function was made in the derivation of (D.8), the choice of equilibrium (thermal) velocity means that the drift-diffusion equations are only valid for very small perturbations of the equilibrium state (low fields). The validity of the drift-diffusion equations is

R. Minixhofer: Integrating Technology Simulation into the Semiconductor Manufacturing Environment