The relaxation time approximation in the Boltzmann equation

5.2 The relaxation time approximation in the Boltzmann equation

At time \(t = 0\) we will switch of all external forces of the system. By scattering the system reaches the thermodynamic equilibrium state again, which will be described in linear approximation by

\begin{equation*} \left(\frac{\partial f}{\partial t}\right) = \left(\frac{\partial f}{\partial t}\right)_{scat} = - \frac{f(\vec{r}, \vec{k}, t)-f_0(\vec{r}, \vec{k})}{\tau(\vec{k})} \end{equation*}

(5.12)

\(f_0(\vec{r}, \vec{k})\) is the equilibrium distribution function.
The relaxation time \(\tau(\vec{k})\) describes how fast the system reaches thermodynamic equilibrium again.
The solution of the relaxation process is:

\begin{equation*} f(\vec{r}, \vec{k}, t)-f_0(\vec{r}, \vec{k}) = \left[ f(\vec{r}, \vec{k}, 0)-f_0(\vec{r}, \vec{k})\right] e^{-\frac{t}{\tau(\vec{k})}} \end{equation*}

(5.13)

The essence for the following calculation is that this relaxation time does not depend on the external forces (This is a very strong assumption; it does not hold e.g. in the space charge region or for ”injection level spectroscopy”).
For steady state \(\left(\frac{\partial f}{\partial t}\right) = 0\) we get

\begin{equation*} -\left(\frac{\partial f}{\partial t}\right)_{field} = \left\langle \vec{\nabla}_r f, \vec{v} \right\rangle + \frac{1}{\hbar} \left\langle \vec{\nabla}_k f, \vec{F}_a \right\rangle =- \frac{f(\vec{r}, \vec{k})-f_0(\vec{r}, \vec{k})}{\tau(\vec{k})} \end{equation*}

(5.14)

This is the fundamental equation for the description of stationary processes in relaxation time approximation.
For small perturbations we evaluate in a series:

\begin{equation*} f(\vec{r}, \vec{k}) = f_0(\vec{r}, \vec{k}) + f^{(1)}(\vec{r}, \vec{k}) + f^{(2)}(\vec{r}, \vec{k}) + ... \end{equation*}

(5.15)

and consider only the linear terms leading to

\begin{equation*} \left\langle \vec{v} , \vec{\nabla}_r f_0(\vec{r},\vec{k}) + \vec{\nabla}_r f^{(1)}(\vec{r},\vec{k}) \right\rangle + \frac{1}{\hbar} \left\langle \vec{F}_a , \vec{\nabla}_k f_0(\vec{r},\vec{k}) + \vec{\nabla}_k f^{(1)}(\vec{r},\vec{k}) \right\rangle =- \frac{f^{(1)}(\vec{r}, \vec{k})}{\tau(\vec{k})} \end{equation*}

(5.16)

Since the gradients \(\vec{\nabla}_r\) and \(\vec{\nabla}_k\) depend already linearly on the perturbation the derivations of \(f^{(1)}\) are of second order \(f^{(2)}\) and are therefor neglected. We find:

\begin{equation*} \vec{\nabla}_r f_0(\vec{r},\vec{k}) = \vec{\nabla}_r \left(\frac{1} {1+e^{\frac{E(\vec{k})-\mu(\vec{r})}{k T(\vec{r})}}} \right) = - \frac{\partial f_0}{\partial E} \left(\vec{\nabla}_r\mu + (E-\mu) \frac{\vec{\nabla}_r T}{T} \right) \end{equation*}

(5.17)

and

\begin{equation*} \vec{\nabla}_k f_0(\vec{r},\vec{k}) = \vec{\nabla}_k \left(\frac{1}{1+e^{\frac{E(\vec{k})-\mu(\vec{r})}{k T(\vec{r})}}} \right) = \frac{\partial f_0}{\partial E} \vec{\nabla}_k E(\vec{k}) = \frac{\partial f_0}{\partial E} \hbar \vec{v} \end{equation*}

(5.18)

For an electrical field

\begin{equation*} \vec{F}=q\vec{E} \end{equation*}

(5.19)

we finally get

\begin{equation*} - \frac{f^{(1)}(\vec{r}, \vec{k})}{\tau(\vec{k})} = \frac{\partial f_0}{\partial E} \left\langle \left(q\vec{E} - \vec{\nabla}_r \mu - (E - \mu) \vec{\nabla}_r\ln(T) \right),\vec{v}\right\rangle \label{linear_f_solution} \end{equation*}

(5.20)

The three terms on the right hand side describe

the ohmic law
particle diffusion
heat transport phenomena

An overview of the above described an other time consuming processes is discussed in the semiconductor script.