Sensitivity kernel for local Hall effect measurements: An adjoint finite-element approach

A micro-four-point-probe Hall measurement made in a wafer interior does not return a local value of the transverse conductivity; it returns a spatial average of the Hall conductivity weighted by a sensitivity kernel fixed by the probe geometry and the magnetic flux density. We compute this dual-configuration Hall kernel with an adjoint finite-element method, extending established longitudinal, four-point, and van der Pauw sensitivity formulations to the scanned, bounded-wafer geometry while retaining the full conductivity tensor in the bulk operator. The kernel is antisymmetric about the electrode array and confined to within about one probe pitch, setting the intrinsic resolution of a local Hall readout; because the interior kernel has nearly zero integral, a spatially uniform Hall-conductivity offset is the weakest observable mode, and the baseline-subtracted reading responds to non-uniformity. We then pose and solve the associated inverse problem: from a dual-orientation raster of readings, with the kernel as the forward operator, we reconstruct the zero-mean Hall-conductivity non-uniformity map δσxy(x, y) (its absolute level fixed by a calibrated reference). As in electrical impedance tomography, a regularized reconstruction recovers localized non-uniformities at a resolution between roughly one and two electrode pitches over the accessible noise range, quantified through the model-resolution matrix and noise propagation. Separating carrier-density from mobility variations requires the companion longitudinal channel, which follows from the same construction. This provides a minimally invasive route to wafer-scale Hall-conductivity uniformity metrology.