4.1 Perturbation Approach to the Boltzmann Equation
Including the Pauli Principle

In the following a homogenous semiconductor is considered. Then the distribution function and the differential scattering rate are independent on position. It is also assumed that the differential scattering rate is time invariant. With these conditions the time dependent Boltzmann equation (2.60) taking into account the Pauli exclusion principle takes the following form:

$$\frac{\partial f(\vec{k},t)}{\partial t}+\frac{q\vec{E}(t)}{\hbar}\nabla f(\vec{k},t)=Q[f](\vec{k},t),$$ (4.4)

where $\vec{E}(t)$ is an electric field and $q$ is the particle charge. $Q[f](\vec{k},t)$ represents the scattering operator which is given by the following expression:

$$Q[f](\vec{k},t) = \int f(\vec{k}^{'},t)[1-f(\vec{k},t)]S(\vec{k}^{'},\vec{k})\,d\vec{k}^{'}-$$ (4.5)

$$- \int f(\vec{k},t)[1-f(\vec{k}^{'},t)]S(\vec{k},\vec{k}^{'})\,d\vec{k}^{'},$$

where $S(\vec{k}^{'},\vec{k})$ stands for the differential scattering rate. Thus $S(\vec{k}^{'},\vec{k})d\vec{k}$ is the scattering rate from a state with wave vector $\vec{k}^{'}$ to states in $d\vec{k}$ around $\vec{k}$, $f(\vec{k},t)$ is the distribution function, and the factors $[1-f(\vec{k},t)]$ mean that the final state must not be occupied according to the Pauli exclusion principle. As can be seen from (4.6), there are terms $f(\vec{k},t)f(\vec{k}^{'},t)$ which render the equation nonlinear. Only when the condition $f(\vec{k},t)\ll 1$ is valid the factors $[1-f(\vec{k},t)]$ can be replaced by unity and the equation takes the usual linear form.

To linearize (4.4) the electric field is written in the form:

$$\vec{E}(t)=\vec{E}_{s}+\vec{E}_{1}(t),$$ (4.6)

where $\vec{E}_{s}$ stands for a stationary field and $\vec{E}_{1}(t)$ denotes a small perturbation which is superimposed on a stationary field. It is assumed that this small perturbation of the electric field causes a small perturbation of the distribution function which can be written as follows:

$$f(\vec{k},t)=f_{s}(\vec{k})+f_{1}(\vec{k},t),$$ (4.7)

where $f_{s}(\vec{k})$ is a stationary distribution function and $f_{1}(\vec{k},t)$ is a small deviation from a stationary distribution. Substituting (4.8) into (4.6) the scattering operator $Q[f](\vec{k},t)$ takes the form:

$$Q[f](\vec{k},t)= \int (f_{s}(\vec{k}^{'})+f_{1}(\vec{k}^{'},t))[1-f_{s}(\vec{k})-f_{1}(\vec{k},t)]S(\vec{k}^{'},\vec{k})\,d\vec{k}^{'}-$$ (4.8)

$$- \int (f_{s}(\vec{k})+f_{1}(\vec{k},t))[1-f_{s}(\vec{k}^{'})-f_{1}(\vec{k}^{'},t)]S(\vec{k},\vec{k}^{'})\,d\vec{k}^{'}.$$

It should be noted that in spite of the fact that $f_{1}(\vec{k},t)\ll 1$ one should take care when linearizing terms such as $1-f_{s}(\vec{k})-f_{1}(\vec{k},t)$. Especially in the degenerate case it may happen that $1-f_{s}(\vec{k})\ll f_{1}(\vec{k},t)$ because of $[1-f_{s}(\vec{k})]\rightarrow 0$.

Subsections

• 4.1.1 The Zeroth Order Equation
• 4.1.2 The First Order Equation

S. Smirnov: