Drude, Hall and Maximal Conductivities: A Unified Complex Model

Abstract

By adopting a complex formulation of Ohm’s law, we arrive at combined equations connecting the conductivities of conductors. The horizontal resistivity is equal to the inverse of Drude’s conductivity δ o ( ), and the vertical resistivity (ρ y ) is equal to the Hall’s conductivity ( δ H ). At high magnetic field, the horizontal conductivity becomes exceedingly small, whereas the vertical conductivity equals to Hall’s conductivity. The Hall’s conductivity is shown to represent the maximal conductivity of conductors. Drude’s and Hall’s conductivities are related by δ o =δ H ω C , where ω C is the cyclotron frequency, and is the relaxation time. The quantization of Hall’s conductivity is attributed to the fact that the magnetic flux enclosed by the conductor is carried by electrons each with h/e , where h is the Planck’s constant and e is the electron’s charge. The Drude’s conductance is found to be equal to Hall's conductance provided the magnetic flux enclosed by the conductor is a multiple of h/e .