Four-point potential drop measurements for materials characterization

Abstract

The technique of measuring the voltage difference (potential drop) between two of the four electrodes of a four-point probe, in order to determine conductivity or surface resistivity of a test piece, is well established in the direct-current (dc) or quasi-dc regime. The technique finds wide usage in the semiconductor industry for the purpose of measuring surface resistivity of semiconductors, and also in the measurement of conductivity of metals, particularly of ferromagnetic metals for which conductivity cannot be easily measured using eddy-current nondestructive evaluation (NDE). In these applications, the conductivity of the test piece is deduced from an analytic formula that depends on the geometry of the probe and test piece. Such a formula requires, as an input, the measured value of the potential drop. Several analytical expressions exist for a variety of test-piece geometries and probe arrangements. Recently, it has been shown that broadband measurements of the potential drop, known as ‘alternating current potential drop’ (ac PD) measurements, can be used not only to obtain the conductivity of a test piece, but also its linear permeability μ. The beauty of this measurement is that the two parameters are completely decoupled in the quasi-static regime. In fact, μ does not appear in the quasi-static expression for σ. Hence, σ may be obtained from low-frequency ac PD measurements and then μ may be deduced as the frequency increases beyond the quasi-static regime, once σ is known. In this review, both dc and ac solutions that are useful in determining the conductivity of metals and semiconductors, and the permeability of ferromagnetic conductors, are summarized. In particular, flat test pieces with arbitrary thickness are considered. At the next level of complexity, a solution for a half-space coated with a surface layer is given, along with a discussion of the use of the four-point potential drop method for determining thickness of a surface layer, such as exists as the result of a surface-hardening process. Recent literature on the topic of surface crack sizing using dc and ac PD is briefly summarized and simple formulas for determining the depth of a long surface crack from ac PD measurements are given. The review also includes four-point potential drop data measured on surface-hardened steel rods and points toward how those data may be used to determine the depth of surface hardening.