The electrical conductivity of solid-state matter is a fundamental physical property and can be precisely derived from the resistance measured via the four-point probe technique excluding contributions from parasitic contact resistances. Over time, this method has become an interdisciplinary characterization tool in materials science, semiconductor industries, geology, physics, etc, and is employed for both fundamental and application-driven research. However, the correct derivation of the conductivity is a demanding task which faces several difficulties, e.g. the homogeneity of the sample or the isotropy of the phases. In addition, these sample-specific characteristics are intimately related to technical constraints such as the probe geometry and size of the sample. In particular, the latter is of importance for nanostructures which can now be probed technically on very small length scales. On the occasion of the 100th anniversary of the four-point probe technique, introduced by Frank Wenner, in this review we revisit and discuss various correction factors which are mandatory for an accurate derivation of the resistivity from the measured resistance. Among others, sample thickness, dimensionality, anisotropy, and the relative size and geometry of the sample with respect to the contact assembly are considered. We are also able to derive the correction factors for 2D anisotropic systems on circular finite areas with variable probe spacings. All these aspects are illustrated by state-of-the-art experiments carried out using a four-tip STM/SEM system. We are aware that this review article can only cover some of the most important topics. Regarding further aspects, e.g. technical realizations, the influence of inhomogeneities or different transport regimes, etc, we refer to other review articles in this field.

https://iopscience.iop.org/article/10.1088/0953-8984/27/22/223201/pdf

The 100th anniversary of the four-point probe technique: the role of probe geometries in isotropic and anisotropic systems

An important aspect of the large expansion in the development and production of solid-state devices has been the demand for more sophisticated techniques for determining the electrical properties of semiconductors, especially silicon and the III-V compounds. A very wide range of measurement techniques now exists and it is the purpose of this article to review those techniques which are in widespread use or which show promise for future application, and at a time for continuing innovation in this area, to indicate present trends and material problems which may arise in the near future.

https://iopscience.iop.org/article/10.1088/0034-4885/41/2/001/pdf

The electrical characterisation of semiconductors

Organometal halide perovskites are under intense study for use in optoelectronics. Methylammonium and formamidinium lead iodide show impressive performance as photovoltaic materials; a premise that has spurred investigations into light-emitting devices and photodetectors. Herein, the optical and electrical material properties of organometal halide perovskites are reviewed. An overview is given on how the material composition and morphology are tied to these properties, and how these properties ultimately affect device performance. Material attributes and techniques used to estimate them are analyzed for different perovskite materials, with a particular focus on the bandgap, mobility, diffusion length, carrier lifetime, and trap-state density.

https://tspace.library.utoronto.ca/bitstream/1807/78834/1/The%20Electrical%20and%20Optical%20Properties_Tspace.pdf

The Electrical and Optical Properties of Organometal Halide Perovskites Relevant to Optoelectronic Performance.

Various four-terminal cross sheet resistor test structures were analyzed to determine the effect of the contact arm width and length on the measured sheet resistance. A nine-point finite difference approximation to the Laplace equation was used with a six-resistor equivalent circuit to solve for the sheet resistance measurement error. The error indicates the difference between the true sheet resistance and the sheet resistance calculated from the van der Pauw formula. The analysis demonstrates that many novel designs are possible. In particular, the Greek-cross sheet resistor is a valid van der Pauw test structure if the arm length is greater than the arm width. This test structure is important in that it allows the accurate measurement of the sheet resistance of a very small region whose width is limited only by the fabrication technology.

A numerical analysis of various cross sheet resistor test structures

Permeability and Hysteresis Measurement

Effects on van der Pauw's resistivity and Hall coefficient measurement due to finite size contacts with selected shapes on a square sample were investigated. For the sheet resistivity measurement, correction factors for the apparent measured values at zero magnetic field were determined from both electrolytic tank experiments and computerized over-relaxation calculations. For the Hall coefficient, correction factors for the effect of voltage shorting due to current electrodes and for the effect of current shorting due to Hall electrodes were calculated (by use of a fast-convergent over-relaxation technique) through a range of Hall angle from tan θ = 0·1–0·5. The current shorting contribution to the correction factor at zero magnetic field was also closely estimated by use of an electrolytic tank. In the symmetrical structures studied the Hall errors introduced by the voltage and current electrodes were approximately equal. The study shows that contacts of appreciable size relative to that of the sample can be a good approximation to van der Pauw's infinitesimal contact. Thus, one can utilize the simplicity and other advantages of finite size ohmic contacts for these measurements in normal semiconductor materials evaulation and still obtain precise data by using the appropriate correction factors determined in this paper.

B.J. Smith

Papers

Contact size effects on the van der Pauw method for resistivity and Hall coefficient measurement