
\sect{Boltzmann equation: Conductivity}

%Baruch's  D03.

Electrons in a metal can be described by a spectrum ${\epsilon
({\bf k})}$, where ${\bf k}$ is the crystal momentum, and a Fermi
distribution $f_{0}({\bf k})$ at temperature ${T}$.
\begin{itemize}
\item [(a)]
Find the correction to the Fermi distribution due to a weak
electric field ${\bf E}$ using the Boltzmann equation and assuming
that the collision term can be replaced by $-[f({\bf k})-
f_{0}({\bf k})]/\tau$ where ${\tau}$ is the relaxation time. Note
that ${d{\bf k}/dt=e{\bf E}/\hbar}$  and the velocity is ${\bf
v}_{{\bf k}}= {\bf \nabla}_{{\bf k}}\epsilon({\bf k})/\hbar$ ,
i.e. in general $d{\bf v}_{k}/dt$ is ${\bf k}$ dependent.
\item [(b)]
Find the conductivity tensor ${\sigma}$  , where ${\bf J}=\sigma
{\bf E}$. In what situation would  ${\sigma}$ be non-diagonal?
Show that $\sigma$ is non-diagonal if the mass tensor
$(\frac{1}{m*})_{i,j}=\frac{1}{\hbar^2}\frac{\partial^2\epsilon({\bf k})}{\partial
k_i\partial k_j}$ is not diagonal.
\item [(c)]
Find  ${\sigma}$ explicitly for  ${\epsilon=\hbar^{2} k^{2}/2m*}$
in terms of the electron density ${n}$.
(${m*}$ is an effective mass).\\
\end{itemize}

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