Four-probe resistivity and contact effects

Consider a homogeneous diffusive bar with uniform cross-sectional area $A$ and voltage-contact separation $L$. Assume the current density is nearly uniform between the voltage contacts, transport is linear, and the relevant resistivity tensor component lies along the bar. Resistance scales with length and inverse area; resistivity removes those geometric factors. The thickness of a thin film is especially important because an uncertain conducting thickness can dominate the uncertainty even when the voltage ratio is measured precisely.

A thin homogeneous isotropic sheet of arbitrary simply connected shape can instead use the van der Pauw construction. Four small ohmic contacts lie on its boundary, and two positive characteristic resistances determine the sheet resistance through an implicit equation. The familiar factor of pi divided by log two applies when those characteristic resistances are equal. Holes, anisotropy, finite contacts, or inhomogeneous parallel layers require additional treatment; a convenient sample shape does not remove these assumptions.

An ideal voltmeter draws no current. Consequently the series resistance of its sense leads and their contacts causes no voltage drop. Current-contact resistance still contributes to the compliance voltage and dissipated power, but is excluded from the sensed voltage if the probes sample the intended interior region. A real voltage input has finite impedance, so excessive source impedance, leakage, or poor contacts can load the circuit. Nonohmic injection can also distort the current distribution and invalidate a simple bar analysis.

Stable thermoelectric and amplifier offsets can be removed by reversing the applied current while preserving a fixed voltage-polarity convention. The odd part in current isolates the linear resistive response, while the even part estimates the offset. This subtraction works only if the offset remains stable over the reversal sequence. Drift, current-dependent heating, and nonlinear transport can survive it. Recording several positive and negative currents makes these failures visible instead of hiding them inside one ratio.

Joule heating scales quadratically with current in the linear regime. A low measured sample voltage can coexist with substantial heating at a resistive current contact. Compare current levels and sweep directions, allow thermal settling, and monitor whether the inferred resistance changes with excitation. Reciprocity comparisons can diagnose inconsistent wiring or sample inhomogeneity, subject to reversing magnetic field when the appropriate Onsager relation requires it. No universal current threshold applies to samples with different cooling and contact conditions.

Uncertainty in a geometric conversion is often shared across an entire temperature sweep. A thickness error rescales every inferred resistivity rather than creating independent noise at each temperature. Report the raw resistance and sample dimensions as well as the converted quantity. A resistance drop can suggest a transition, but a connected low-resistance filament may short the voltage probes without occupying most of the sample. Bulk thermodynamic or magnetic evidence is needed before assigning a bulk phase fraction.