Galvanomagnetic Transport in Magnetic Sensors Reinterpreted Using Two Instead of Three Kinetic Coefficients

This paper pertains to the topic of galvanomagnetic phenomena arising in semiconductor-based magnetic sensors and to the operation of such devices. An alternative formulation of the nonlinear galvanomagnetic transport equation is derived. Traditionally this equation has involved terms for isotropic conduction, the Hall effect, and the planar Hall effect, each modulated by its own magnetic-flux-density-dependent kinetic coefficient. It is shown that the first and third terms can be jointly expressed using a single symmetric transport coefficient matrix, while the Hall effect is expressed using an antisymmetric matrix. Thereby, conduction in samples of cubic crystal structure exposed to a magnetic field becomes spheroidally anisotropic. Conduction along the magnetic field vector remains unchanged, while that in the perpendicular plane is reduced. In four-contact magnetic sensors, output signal components due to the symmetric transport matrix are canceled by the current-spinning method, leaving only the Hall voltage caused by the Hall effect. The inverse transport equation relying on the resistivity tensor is fully analogous to the conductivity-based formulation. Graphical interpretations of resulting phenomena are derived in both resistivity and conductivity-based pictures. The former is suited for thin devices such as planar Hall sensors intended for quantifying the out-of-plane component of the magnetic flux density, while the latter is relevant for columnar devices, i.e., idealized versions of vertical Hall sensors designed for determining in-plane components. The conclusions of this work are subject to the assumptions of a parabolic, isotropic conduction band and an energy-dependent relaxation time in the relaxation time approximation of the Boltzmann equation.