Mobility of charged particles in classical gas

[Sigurdsson]

This is problem 9.2 in solid state physics by Ibach and Luth.

Homework Statement

Using the stationary Boltzmann equation, show that the mobility of the charged particles in a classical gas is given by

$\mu = \frac{e \langle v^2 \tau(\mathbf{k}) \rangle}{m \langle v^2 \rangle}$

Homework Equations

The stationary Boltzmann equation is
$\mathbf{v} \cdot \nabla_{\mathbf{r}} f - \frac{e}{\hbar} \vec{\mathscr{E}} \cdot \nabla_{\mathbf{k}} f = \left( \frac{\partial f}{\partial t} \right)_s = -\frac{f(\mathbf{k}) - f_0(\mathbf{k}) }{\tau(\mathbf{k} ) }$


The mobility is defined as the proportionality between the drift velocity of the electrons and the electric field
$-\frac{v_D}{\mathscr{E} } = \mu$

and the function $f$ is the fermi distribution.

The Attempt at a Solution

Now the first thing I notice is the in the final answer we have the electron mass $m$ but not the effective mass $m^*$. Now I'm going to assume that the distribution does not depend on position so we have
$- \frac{e}{\hbar} \vec{\mathscr{E}} \cdot \nabla_{\mathbf{k}} f = -\frac{f(\mathbf{k}) - f_0(\mathbf{k}) }{\tau(\mathbf{k} ) }$

The term on the left is called drift term. From there we should get the drift velocity. The electrical conductivity is
$\sigma = j/\mathscr{E} = n e \mu$

Where $j, e, n$ are in correct order the current density, electron charge, electron density.

Now I have that
$\sigma = j_x/\mathscr{E}_x = \simeq \frac{e^2}{8 \pi^3 \hbar} \int_{E = E_F} \frac{v_x^2(\mathbf{k}) }{v(\mathbf{k})} \tau(\mathbf{k}) \ df_E$


What is really bugging me is the final answer has the mean value of the velocity and the classical mass of the electron but not the effective mass. Any tips on what happens when I'm working with the Boltzmann equation in classical electron gas?

Cheers

 

[mark726]

,
Thank you for your question. The final answer for the mobility in the classical gas does indeed include the electron mass, not the effective mass. This is because the stationary Boltzmann equation is derived from classical mechanics, which does not take into account the effects of band structure and effective mass. Therefore, the equation only considers the classical mass of the electron.

Additionally, the mean value of the velocity in the final answer is taken over all possible velocities at a given energy, not just the average velocity of the electrons in the system. This is because the mobility is a measure of how quickly the electrons respond to an applied electric field, and therefore it takes into account all possible velocities.

I hope this clarifies any confusion you may have had. Please let me know if you have any further questions.