For long time, it has been well recognized that transport in mesoscopic systems may relate to their scattering properties as it was originally proposed by Landauer [1, 2]. The single-channel conductance formula has been rst obtained heuristically as proportional to the ratio between transmission and re ection coe cients G = 2e 2 h T R [1]. Main attempts to derive this formula from linear-response theory (LRT) are known in the literature (see Ref. [3] and references therein). However, for long time, the generalization to a multichannel formula has been a great challenge for the physicists. The major di culty is that, in the LRT, the currentow in a sample is obtained as a response to an applied external perturbation inside the reservoirs in accordance to the Kubo viewpoint (KVP), whereas in the Landauer viewpoint (LVP) the current itself is considered as responsible for the induction of a position-dependent potential inside the sample. At this level, one has to note that it has been established heuristically that contacts between the reservoirs and the sample are at the origin of the di erence between the two points of view [4]. In recent works [5], a theory of transport that combines the LVP with the LRT and which does not consider the e ect of the contacts (called thereafter a self-consistent LRT) in order to calculate the relevant part of the density-operator for mesoscopic samples in the absence of an external magnetic eld, has been proposed. The obtained density-operator, necessary to evaluate all physical quantities, was used to derive a new and more plausible four-lead multichannel conductance formula that does not endure any of the inconsistencies cited in the literature for the other known formulae [3, 6]. This self-consistent LRT was recently extended to take into account the existence of a uniform magnetic eld [7, 8]. It has been shown that the density-operator may be written as a sum of two terms [7]: the rst represents the Kubo density-operator ρ [7 9] and the second denes the contribution of self-consistent e ects ρ [7, 8]: ρ = ρ + ρ. (1) Our aim in this work is the use of Eq. (1) to determine quantitatively the magnetoconductance of a four-lead mesoscopic measurement in terms of the scattering matrix elements whenever self-consistent e ects are relevant; that is, whenever the e ect of the contacts is neglected and the induced potential inside the sample is considered.
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