It is difficult to measure the conductivity of small samples of semiconductor. First, they are brittle and thus difficult to machine. Second, contacts to the material are resistive. With Van der Pauw's theorem, however, it is possible to measure the conductivity of a sample in the form of a thin plate of arbitrary shape with four contacts around the periphery without interference from the contact resistances. We deduce this theorem for the case of a semi-infinite plate.
(a) Imagine an infinite thin plate of thickness $s$ and conductivity $\sigma$. $A$. current $2 I$ flows into point $A$ in Fig. 8-22(a).
Show that, in the plate, $E=I / \pi \sigma r s$, where $r$ is the radial distance to $A$.
(b) Now consider three points $B, C, D$ as in Fig. 8-22(a), on a line going through $A$. Show that
$$
V_{C}-V_{D}=\frac{I}{\pi \sigma S} \ln \frac{a+b+c}{a+b}
$$
If we cut the plate along the line and remove the lower half, the above equation applies to the upper half, if $I$ is now the current at $A$ flowing into the upper half. So we now have a semi-infinite plate as in Fig. $8-22(b)$.
(c) Now suppose that a current $I$ comes out of $B$ as in Fig. 8-22(c). Calculate $V_{C}-V_{D}$ again; then superpose cases (b) and (c) to obtain Fig. $8-22(d)$. Show that, for case $(d)$
$$
\frac{V_{C}^{\prime}-V_{0}^{\prime}}{I}=\frac{1}{\pi \sigma s} \ln \frac{(a+b+c) b}{(a+b)(b+c)}
$$
This ratio has the dimensions of a resistance; call it $R_{A B, E D} .$ The contact resistances at $A$ and $B$ are unimportant because only the current $I$ between $A$ and $B$ is significant. The contact resistances at $C$ and $D$ are also unimportant if their sum is much smaller than the resistance of the voltmeter that measures $V_{C}-V_{D}$
(d) Show that, with currents as in Fig. 8-22(e),
$$
R_{k C, D A}=\frac{1}{\pi \sigma S} \ln \frac{(a+b)(b+c)}{a c}
$$
(e) You can now derive Van der Pauw's theorem:
$$
\exp \left(\pi_{D S} R_{A B, C D}\right)+\exp \left(\pi \sigma S R_{B C, D A}\right)=1
$$
The only unknown is $\sigma .$ This result, in fact, applies to a lamella of any shape, with contacts $A, B, C, D$ around the periphery.