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%TCIDATA{Created=Thu Sep 18 08:04:32 1997}
%TCIDATA{LastRevised=Fri Sep 14 10:50:36 2001}
%TCIDATA{Language=American English}

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{\LARGE \ }{\Large Physics 861 -- Fall 01}

{\Large Problem set 2 - Due Tuesday, Sep. 18}
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\noindent {\bf 1. }

Problem 1, page 53, of Ashcroft and Mermin.

\medskip 

\noindent {\bf 2. }

Solve the Schr\"{o}dinger equation for an electron in crossed electric and
magnetic fields. Take the magnetic field along $z$ and the electric field
along $x$. Use the Landau gauge.

\bigskip

\noindent {\bf 3. Two-band formulas}

\medskip

(a) Recall the elementary derivation of the Drude formula for the
conductivity, $\sigma =ne^{2}\tau /m$. What if there are several types of
charge carriers? Suppose that there are $n_{k}$ carriers of mass $m_{k\text{%
, }}$charge $e_{k}$, collision time $\tau _{k}$ and show that $\sigma
=\sum_{k}\sigma _{k}$, where $\sigma _{k}$ is the Drude conductivity when
only the carriers of type $k$ are present.

(b) Recall that the Drude theory predicts that the Hall coefficient is $-1/ne
$ [SI], or $-1/nec$ [Gauss], and the conductivity is unchanged by the
application of a magnetic field. Find the Hall coefficient $R=E_{y}/j_{x}B$
when there are two types of carriers.

Your task is to obtain AM's Eq. (12.73), starting from equations like AM's
(1.19) for each carrier. It is probably best to introduce $\rho
_{k}=1/\sigma _{k}$ and $\omega _{k}\tau _{k}/\sigma _{k}=-R_{k}B$ [SI] $%
=-R_{k}B/c$ [Gauss] from the beginning.

The experimental constraint of the Hall measurement is that $j_{y}=0$, which
allows for ``counterflow'' when there are two or more carrier types. That is
why the Hall phenomena are so complex when more than one carrier type is
present.

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\noindent {\bf Alternate. }

Problem 2, page 53, of Ashcroft and Mermin.

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