What is measured when you measure a resistance?—The Landauer formula revisited

We re-examine the question of what constitutes the physically relevant quantum-mechanical expression for the resistance of a disordered conductor in light of recent experimental and theoretical advances in our understanding of the conducting properties of mesoscopic systems. It is shown that in the absence of a magnetic field, the formula proposed by Büttiker, which expresses the current response of a multi-port conductor in terms of transmission matrices, is derivable straightforwardly from linear response theory. We also present a general formalism for solving these equations for the resistance given the scattering matrix. This Landauer-type formula reduces to g=(e 2 /h)Tr(tt † ), where g is the conductance and t is the transmission matrix, for the two-probe case. It is suggested that this formula provides the best description of the present class of experiments performed in two-probe or multi-probe measuring configurations, and that the subtleties leading to various different Landauer formulae are not relevant to these experiments. This is not because of the large number of channels in real conductors, but is due to the fact that apparently no present experiment probes a “local chemical potential” in the conductor. Certain standard objections to deriving a Landauer-type formula from linear response theory are answered. Applications of this formula to fluctuations in disordered multi-probe conductors are discussed.