Van der Pauw method
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The Van der Pauw method, developed by L.J. van der Pauw in 1958[1], is a technique for doing 4-probe resistivity and Hall effect measurements. In essence it provides an easy way to measure:
- Sheet resistance/conductance
- From this one can calculate the resistance/conductance provided the thickness of the sample is known
- Hall voltage
- Sheet carrier density
- From this one can calculate the carrier density provided the thickness of the sample is known
- Hall mobility
The advantages of this method include low cost and simplicity. The Van der Pauw technique can be used on any thin sample of material and the four contacts can be placed anywhere on the perimeter/boundary, provided certain conditions are met:
- The contacts are on the boundary of the sample (or as close to the boundary as possible)
- The contacts are infinitely small (or as close as possible)
- The sample is thin relative to the other dimensions
- There are no isolated holes within the sample
Unfortunately, the original paper with the full proof of the theory is not available online (L.J. van der Pauw, A method for measuring specific resistivity and Hall effect of discs of arbitrary shape, Philips Res. Repts. 13, 1-9, 1958).
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Measuring Sheet Resistance
In order to calculate the sheet resistance, at least 2 measurements must be made, using a current source and a voltage meter. Four contacts are needed on the perimeter of the sample. A current must be forced through two adjacent contacts, and a voltage must be measured across the other two contacts. Let the current passing from contact i to contact j be denoted as:
- <math>I_{ij}</math>
and the voltage measured across contacts k (negative) and l (positive) be give by:
- <math>V_{kl}</math>
Van der Pauw defines in his paper the resistance <math>R_{ij,kl}</math> as:
- <math>R_{ij,kl} = \frac{V_{kl}}{I_{ij}}</math>
Assuming that the contacts are numbered sequentially along the perimeter of the sample, Van der Pauw discovered that the sheet resistance of samples with arbitrary shape can be determined from <math>R_{12,34}</math> and <math>R_{23,41}</math>. The actual sheet resistance ρ is obtained by solving:
- <math>e^{-\pi R_{12,34}/\rho}+e^{-\pi R_{23,41}/\rho}=1</math>
In general, the solution for ρ cannot be expressed explicitly in known functions. See Van der Pauw's paper for details about obtaining a solution.
Methods for more accurate measurements
The reciprocity theorem [2] gives: <math>R_{ij,kl} = R_{kl,ij}</math>. Therefore, it is possible to obtain a more precise value for the resistances (<math>R_{12,34}</math> and <math>R_{23,41}</math>) by 2 additional measurements (<math>R_{34,12}</math> and <math>R_{41,23}</math>) and averaging the results.
We define
- <math>R_{A} = \left( R_{12,34} + R_{34,12}\right) /2</math>
and
- <math>R_{B} = \left( R_{23,41} + R_{41,23}\right) /2</math>
Then, the sheet resistivity ρ is obtained by solving:
- <math>e^{-\pi R_A/R_S}+e^{-\pi R_B/R_S}=1</math>
Where the sheet resistivity <math>\rho</math> is defined as
- <math>\rho = R_S d </math>
where d is the thickness of the sample.
Further improvement of resistance values can be obtained by repeating the resistance measurements after switching polarities of both the current source and the voltage meter, as this will cancel offset voltages as thermoelectric potentials due to the Seebeck effect.
References
- L.J. van der Pauw, A method for measuring specific resistivity and Hall effect of discs of arbitrary shape, Philips Research Reports 13, 1-9 (1958) (unfortunately not online)
External links
- Page at MIT
- van der Pauw technique at KTH Stockholmnl:Van der Pauwmethode