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 δo( ), and the vertical resistivity (ρy) is equal to the Hall’s conductivity ( δH). 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 δo =δHωC , where ωC 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.
Highlights
Drude had explained the electrical conductivity of metals by treating electrons in the metal as a gas performing diffusive motion [1]. He found that the dc conductivity of metals to be
Hall found that when a current passes in the x-direction of a conductor placed in a transverse magnetic field, the magnetic force forbids the movement of electrons across the y-axis
We further show that under low magnetic field, the horizontal conductivity reduces to the Drude’s conductivity, whereas the vertical component becomes vanishingly small
Summary
Drude had explained the electrical conductivity of metals by treating electrons in the metal as a gas performing diffusive motion [1]. The lateral potential difference divided by the horizontal current defines a transverse resistance that increases linearly with the magnetic field This is known as Hall’s effect [3]. Since the motion of electrons in a conductor is generally twodimensional, a unified approach exhibiting this nature can be formulated using complex numbers Such a formulation is shown recently to lead to interesting properties governing a two-dimensional system [4]. Under high magnetic field the horizontal conductivity is less than the value suggested by Drude’s, while the vertical conductivity reduces to the Hall’s conductivity. In two-dimensions, the Drude’s and Hall’s conductances are equal This shows that the relaxation time of a two-dimensional conductor is about two orders of magnitudes bigger than the one in three dimensions
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