The van der Pauw Technique

In order to determine both the relaxation time (mean free path) and the carrier density (Fermi parameters), a combination of a resistivity measurement and a Hall measurement is needed. We discuss here the van der Pauw technique which, due to its convenience, is widely used in research and industry to determine the resistivity of uniform samples. As originally devised by van der Pauw [5], one uses an arbitrarily shaped (but simply connected, i.e., no holes or nonconducting islands or inclusions), thin-plate sample containing four very small contacts placed on the periphery (preferably in the corners) of the plate. A schematic of a rectangular van der Pauw configuration is shown in Fig. 2.

The objective of the resistivity measurement is to determine the sheet resistance $R_{S}$. Van der Pauw demonstrated that there are actually two characteristic resistances $R_{A}$ and $R_{B}$, associated with the corresponding terminals shown in Fig. 2. $R_{A}$ and $R_{B}$ are related to the sheet resistance $R_{S}$ through the van der Pauw equation

$$\exp(-\pi R_{A}/R_{S})+\exp(-\pi R_{B}/R_{S})=1$$ (2)

which can be solved numerically for $R_{S}$. The bulk electrical resistivity can be calculated using $\rho=R_{S}d$.

To obtain the two characteristic resistances, one applies a dc current $I$ into contact 1 and out of contact 2 and measures the voltage $V_{43}$ from contact 4 to contact 3 as shown in Fig. 2. Next, one applies the current $I$ into contact 2 and out of contact 3 while measuring the voltage $V_{14}$ from contact 1 to contact 4. $R_{A}$ and $R_{B}$ are then

$$R_{A}=V_{43}/I_{12};\; R_{B}=V_{14}/I_{23}.$$

The objective of the Hall measurement in the van der Pauw technique is to determine the sheet carrier density $n_{S}$ by measuring the Hall voltage $V_{H}$. The Hall voltage measurement consists of a series of voltage measurements with a constant current $I$ and a constant magnetic field $B$ applied perpendicular to the plane of the sample. Conveniently, the same sample, shown again in Fig. 3, can also be used for the Hall measurement.

To measure the Hall voltage, a current is forced through the opposing pair of contacts 1 and 3 and the Hall voltage $V_{H}$ ($V_{24}$) is measured across the remaining pair of contacts 2 and 4. Once the Hall voltage is acquired, the sheet carrier density can be calculated via

$$n_{S}=\frac{IB}{e\mid V_{H}\mid}.$$

The bulk carrier density is then $n=n_{S}/d$.