Analytical expressions for the conductance noise measured with four circular contacts placed in a square array
Authors: Leroy G. , Gest J. , Vandamme L. K. J. , van Deursen A. P. J. 
Published: March 2007

In the ideal case, noise measurements with four contacts minimize the contribution of the contact interface. There is a need to characterize conductance noise and noise correction factors for the different geometries provided with four contacts, as already is the case for resistivity measurements with van der Pauw structures. Here, we calculate the noise correction factors for two geometries with a pair of sensors and a pair of current driver electrodes placed in a square array. The first geometry investigated is a very large film compared to the distance L between four circular electrodes, which are placed in a square array far away from the borders of the film. The second is a square-shaped conductive film with side length L and provided with four quarter-circle corner contacts with radius l. The effect of the conductance noise in the film can be observed between current free sensors in a four-point measurement or between current carrying drivers in a two-point measurement. Our analytical expressions are based on approximations to solve the integrals ∬(J∙J̃)2dA and ∬∣J∣4dA for the voltage noise measured across a pair of sensors, SVQ, and across the drivers, SVD, respectively. The first and second integrands represent the squared dot product of the current density and adjoint current density and the modulus of the current density to the fourth power, respectively. The current density J in the samples is due to the current I passing through the driver contacts. The calculated expressions are applicable to samples with thickness t⪡l⩽0.1L. Hence, the disturbances in the neighborhood of the sensors on J and of the drivers on J̃ are ignored. Noise correction factors for two- and four-point measurements are calculated for sensors on an equipotential (transversal noise) with the driver contacts on the diagonal of a square and for sensors next to each other on one side of the square with the drivers next to each other on the other side of the square (longitudinal noise). In all cases the noise between the sensors is smaller and less sensitive to the contact size 2l∕L than the noise between the drivers. The ratio SVQ∕SVD becomes smaller with smaller contact radius l. Smaller sensors give a better suppression of interface noise at the contacts. But overly low 2l∕L values result in overly high resistance between the sensors and too strong a contribution of thermal noise at the sensors. Therefore, equations are derived to calculate the current level needed to observe 1∕f conductance fluctuations on top of the thermal noise. The results from the calculated analytical expressions show good agreement with experimental results obtained from the noise in carbon sheet resistance and numerical results. Transversal noise measurements on a square sample with corner contacts are recommended to characterize the 1∕f noise of the layer. This is due to the increased current densities in the sample compared to the open structure, which result in easier detection of the 1∕f on top of the thermal noise.