Van der Pauw method

The van der Pauw Method is a technique commonly used to measure the resistivity and the Hall coefficient of a sample.

Its strength lies in its ability to accurately measure the properties of a sample of any arbitrary shape, as long as the sample is approximately two-dimensional (i.e. it is much thinner than it is wide), solid (no holes), and the electrodes are placed on its perimeter.

The van der Pauw method employs a four-point probe placed around the perimeter of the sample, in contrast to the linear four point probe: this allows the van der Pauw method to provide an average resistivity of the sample, whereas a linear array provides the resistivity in the sensing direction.

[1] This difference becomes important for anisotropic materials, which can be properly measured using the Montgomery Method, an extension of the van der Pauw Method (see, for instance, reference[2]).

From the measurements made, the following properties of the material can be calculated: The method was first propounded by Leo J. van der Pauw in 1958.

The van der Pauw technique can also be applied to samples with one hole.

In order to reduce errors in the calculations, it is preferable that the sample be symmetrical.

Certain conditions for their placement need to be met: In addition to this, any leads from the contacts should be constructed from the same batch of wire to minimise thermoelectric effects.

For an anisotropic material, the individual resistivity components, e.g. ρx or ρy, can be calculated using the Montgomery method.

) can be found using Ohm's law: In his paper, van der Pauw showed that the sheet resistance of samples with arbitrary shapes can be determined from two of these resistances - one measured along a vertical edge, such as

The benefit of doing this is that any offset voltages, such as thermoelectric potentials due to the Seebeck effect, will be cancelled out.

Combining these methods with the reciprocal measurements from above leads to the formulas for the resistances being and The van der Pauw formula takes the same form as in the previous section.

If any of the reversed polarity measurements don't agree to a sufficient degree of accuracy (usually within 3%) with the corresponding standard polarity measurement, then there is probably a source of error somewhere in the setup, which should be investigated before continuing.

The same principle applies to the reciprocal measurements – they should agree to a sufficient degree before they are used in any calculations.

In general, the van der Pauw formula cannot be rearranged to give the sheet resistance RS in terms of known functions.

The most notable exception to this is when Rvertical = R = Rhorizontal; in this scenario the sheet resistance is given by The quotient

In most other scenarios, an iterative method is used to solve the van der Pauw formula numerically for RS.

Typically a formula is considered to fail the preconditions for Banach Fixed Point Theorem, so methods based on it do not work.

Recently, however, it has been shown that an appropriate reformulation of the van der Pauw problem (e.g., by introducing a second van der Pauw formula) makes it fully solvable by the Banach fixed point method.

Note that centimeters are often used to measure length in the semiconductor industry, which is why they are used here instead of the SI units of meters.

If an external magnetic field is then applied perpendicular to the direction of current flow, then the resulting Lorentz force will cause the electrons to accumulate at one edge of the sample (see part (c) of the figure).

is the charge on an electron, results in a formula for the Lorentz force experienced by the electrons: This accumulation will create an electric field across the material due to the uneven distribution of charge, as shown in part (d) of the figure.

This in turn leads to a potential difference across the material, known as the Hall voltage

, we can say that the strength of the electric field is therefore Finally, the magnitude of the Hall voltage is simply the strength of the electric field multiplied by the width of the material; that is, where

is defined as the density of electrons multiplied by the thickness of the material, we can define the Hall voltage in terms of the sheet density: Two sets of measurements need to be made: one with a magnetic field in the positive z-direction as shown above, and one with it in the negative z-direction.

This is then repeated for I13 and V42, P. As before, we can take advantage of the reciprocity theorem to provide a check on the accuracy of these measurements.

Having completed the measurements, a negative magnetic field is applied in place of the positive one, and the above procedure is repeated to obtain the voltage measurements V13, N, V42, N, V31, N and V24, N. Initially, the difference of the voltages for positive and negative magnetic fields is calculated:

The formula given in the background can then be rearranged to show that the sheet density Note that the strength of the magnetic field B needs to be in units of Wb/cm2 if ns is in cm−2.

For instance, if the strength is given in the commonly used units of teslas, it can be converted by multiplying it by 10−4.

Generally, the material is sufficiently doped so that there is a difference of many orders-of-magnitude between the two concentrations, allowing this equation to be simplified to where nm and μm are the doping level and mobility of the majority carrier respectively.