Hall measurements and multicarrier ambiguity

Take positive conventional current density along x, magnetic field along positive z, and open transverse boundaries so that the net y current vanishes. Define $\rho_H=E_y/j_x$ and $R_H=\lim_{B\to0}\rho_H/B$. A single isotropic electron band with charge minus e then has negative Hall coefficient, while a hole band has positive coefficient. Here e denotes a positive charge magnitude. Exchanging voltage leads reverses the measured sign without changing the sample, so wiring polarity belongs in the metadata.

Misaligned transverse contacts pick up longitudinal voltage. For a nonmagnetic sample in its reversible linear-response regime, that contaminant is predominantly even in magnetic field, while the ordinary Hall signal is odd. Half the difference between opposite-field measurements removes the even term. Magnetic hysteresis, anomalous Hall contributions, field-dependent temperature gradients, and nonreciprocal effects require more careful protocols. Antisymmetrizing a trace is a symmetry projection, not a universal separation of ordinary Hall physics from every other odd signal.

For independent electron and hole bands, currents add at the same electric field. Therefore add their conductivity tensors and only then invert the sum. In the weak-field limit, each longitudinal term is density times charge magnitude times mobility. Its transverse correction carries an additional mobility and the charge sign. This extra factor explains why a small high-mobility population can control the Hall voltage while a larger low-mobility population controls much of the longitudinal conductivity.

The resulting Hall numerator compares hole and electron densities weighted by mobility squared. Its denominator contains the square of the total longitudinal weight. A compensated metal with equal electron and hole densities can still have a nonzero Hall coefficient if their mobilities differ. Conversely, a vanishing Hall coefficient does not imply an absence of carriers. The commonly quoted Hall number is an effective quantity obtained by forcing these combined weights into a single-band formula.

At larger fields the separate denominators containing mobility times field produce curvature in the Hall and longitudinal traces. Fitting both observables over a broad field range can constrain a two-band model more strongly than fitting one Hall slope. Nevertheless, limited field range, correlated parameters, extra pockets, anisotropy, and background subtraction can produce several nearly indistinguishable fits. A small statistical fit error within a chosen model is not evidence that the model uniquely identifies the material's carriers.

Other probes constrain different combinations: quantum oscillations measure selected extremal areas, photoemission resolves matrix-element-weighted occupied states near a surface, and optics constrains frequency-dependent charge response. Such comparisons require matching temperature, field, orientation, and sample composition. Hall mobility and the relaxation time inferred from oscillation damping also weight scattering differently. Report density estimates with their assumed band count and mobility model, and retain the raw signed data so another analysis can revisit those assumptions.