Hall coefficient ($R_H=E_y/(J_x{B}_z)$)

The Hall coefficient relates a transverse electric field to longitudinal current density and transverse magnetic field, with oriented axes as in the displayed formula. Its SI unit is ${\mathrm{m}}^3/C$. For one isotropic Drude model species it is $1/(nq)$; multiple charge carriers need not follow that formula.

In a one-carrier Drude model, the Hall coefficient is $R_H=1/(nq)$. Combining its inferred electron density with atomic density ${\rho }_m{N}_A/M$ gives the displayed number of mobile electrons per atom. The molar mass and mass density must be supplied independently. Multiple bands, anisotropic carrier motion and Hall factors limit this simple interpretation; field reversal helps distinguish the transverse response from voltage offsets.

For one positive carrier population, define $r_H$ by $R_H=r_H/(ep)$ with $e>0$. Different statistical averages of energy-dependent scattering affect hall conductivity and longitudinal electrical conductivity, so $r_H$ need not be one. The measured Hall mobility is ${\mu }_H=\sigma R_H=r_H{\mu }_{\mathrm{drift}}$. Multiple charge carriers require a more general model even before such a correction is made.

In a one-carrier isotropic Drude model, $R_H=1/(nq)$ and an inferred carrier density is $n_H=1/(e|R_H|)$. Dividing by the atomic number density gives an effective Hall count per atom. A noninteger result need not mean a literal fractional count of valence electrons: multiple energy bands, anisotropic Fermi surfaces and different charge carrier mobilities can change the Hall coefficient. In a two-carrier isotropic approximation,Its sign identifies the dominant Hall response rather than necessarily the most numerous charge carrier.

For isotropic electrons and electron holes with positive mobilities, adding their weak-field electrical conductivity tensors and imposing zero transverse total current givesThe Hall coefficient changes sign at $p{\mu }_h^2=n{\mu }_e^2$. Carrier abundance alone does not fix the sign; mobilities enter quadratically. The formula assumes ${\mu }_{e}B\ll 1$, ${\mu }_{h}B\ll 1$, independent semiclassical carriers, and no anomalous Hall contribution.

For fully ionized uncompensated acceptors, $p-n=N_A$ and $np=n_i^2$. If $r_{\mu }={\mu }_e/{\mu }_h>1$, the weak-field two-carrier Hall coefficient is positive when $p>r_{\mu }{n}_i$, giving the displayed sign threshold. A nearly constant Hall coefficient requires a stronger plateau condition. When $N_A\gg n_i$, its relative correction to $1/(e{N}_A)$ is approximately $-(1+r_{\mu }{)}^2(n_i/N_A{)}^2$. Thus specifying only that the coefficient is approximately constant does not define an exact minimum dopant concentration without a tolerance and mobility assumptions.