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5 Semiconductor Modeling
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dynamic balance of ballistic particle motion and particle collisions (predomi-
nantly collisions of electrons/holes with the crystal lattice of the semiconductor,
quantized as phonons). Obviously, there is a structural similarity to the gas dy-
namics Boltzmann equation of Chapter 1, the main difference lying in the form
of the collision operator, which in the solid state physics case is predominantly
inelastic and allows only Fermi-Dirac distribution as Fermion equilibria.
Macroscopic Modeling is based on various scaling limits of solutions of the
semiclassical Boltzmann equation. There are so called hydrodynamic semicon-
ductor models, similar to the compressible Euler/Navier–Stokes system of fluid
dynamics, energy transport equations and drift-diffusion systems.
For a review of these models, their interrelation and mathematical properties
we refer to [5] and to the references therein.
Here we want to give some details on the oldest and still most important
semiconductor device model, namely on the drift-diffusion-Poisson (DDP) sys-
tem.
Phenomenologically speaking, the main factors for current flow in semicon-
ductors are diffusion of conduction electrons and holes as well as convection of
charged particles by the electric field in the device. Now let n = n(x, t)denote
the density of (negatively charged) conduction electrons in the doped semicon-
ductor at position x and time t, p = p(x, t) the density of (positively charged)
holes, V = V(x, t) the electrical potential and J n (J p ) the electron (hole) current
density vector fields. Clearly, the functions n and p are nonnegative. Then, after
appropriate non-dimensionalisation and scaling, the electron and hole current
densities in the DD model read:

J n = D n grad n − μ n n grad V
J p = −(D p grad p + μ p p grad V).

Here D n and D p denote the (positive) electron and, resp., hole diffusion coeffi-
cients and μ n and μ p the (positive) electron and hole mobilities. Note that the
first terms in the current densities are the diffusion currents and the second
terms the drift currents, generated by the electrical field E = −grad V.The total
current density is

J = J n + J p .
Continuity equations for both carrier types are assumed to hold:

n t = div J n + R
p t = −div J p + R ,

where R denotes the so called recombination-generation rate, which accounts
for instantaneous generation/annihilation of electron-hole carrier pairs and acts
as a reaction term in the continuity equations. In most applications it is modeled
as a function of the position densities n and p.