Electrical linear-response theory in an arbitrary magnetic field: A new Fermi-surface formation

Abstract

We develop a novel formulation of dc electrical linear-response theory for a phase-coherent conductor with multiple leads valid in arbitrarily strong magnetic field and for a given impurity configuration and measuring geometry. This formulation is convenient for discussion of the quantum Hall effect and mesoscopic transport phenomena. We express the total current response ${I}_{m}$ through lead $m$ completely in terms of the voltages ${V}_{n}$ applied at the leads, independent of the electric field in the material, i.e., ${I}_{m}=\ensuremath{\Sigma}{n}^{}{g}_{\mathrm{mn}}{V}_{n}$. We show that while the current-density response is not in general expressible as a Fermi-surface quantity, the total transport current determined by these conductance coefficients ${g}_{\mathrm{mn}}$ does depend only on the wave functions (or Green functions) at the Fermi surface as $T\ensuremath{\rightarrow}0$. This yields new and useful Green-function expressions for the ${g}_{\mathrm{mn}}$ and the longitudinal and Hall resistances. When transformed by appropriate applications of scattering theory, these expressions are shown to be equivalent to the relation ${g}_{\mathrm{mn}}={T}_{\mathrm{mn}}$, where ${T}_{\mathrm{mn}}$ is the sum of all the transmission coefficients between leads $m$ and $n$, as first obtained by B\"uttiker on the basis of phenomenological arguments. A brief discussion of the relation between this formula and other proposed Landauer formulas is given. It is noted that the occurrence of the quantum Hall effect is very natural in this formulation and simple conditions on the scattering matrix of the conductor which imply the quantum Hall effect are derived.