# Circuit theory of charge transport in mesoscopic conductors

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## Abstract

This Thesis is devoted to the circuit theory of mesoscopic transport. The emphasis is put on its extension which provides a method to obtain the complete statistics of the transferred charge. To accomplish this task, several topics have to be combined: a mathematical description of the charge transfer statistics, the scattering approach to mesoscopic transport, and the nonequilibrium Keldysh-Green’s function technique. Although the underlying theory is rather complex, the circuit-theory rules which are obtained at the end are in fact very simple. They resemble Kirchhoﬀ’s laws for conventional macroscopic conductors, with currents and voltages replaced by their mesoscopic counterparts. An important diﬀerence is that the mesoscopic “currents” and “voltages” acquire matrix structure, and that the “current”-“voltage” relation is in general nonlinear. The matrix structure originates from the KeldyshGreen’s function formalism which is needed to account for the many-body quantum state of the electrons in the system. The circuit theory is applicable to multiterminal mesoscopic structures with terminals of diﬀerent types, e.g., normal metals, superconductors, and ferromagnets. The junctions within the structure can be diﬀerent also, e.g., transparent quantum point contacts, tunnel barriers, disordered interfaces, diﬀusive wires, etc. The Thesis is organized as follows. In Chapter I, we provide introductory information on noise. We discuss early experiments on noise in vacuum tubes and the Schottky result which relates the spectral density of current ﬂuctuations and the average current. We summarize some important results on noise in mesoscopic conductors which can be obtained within circuit theory. Chapters II and III are devoted to theoretical prerequisites needed for the circuit theory. In Chapter II, we deﬁne the notion of the cumulant generating function and its relation to statistically independent processes. In Chapter III, we introduce the scattering approach to mesoscopic transport and the method of Keldysh-Green’s functions. The circuit theory is presented in Chapter IV focusing on the extension which provides complete information on the charge transfer statistics. The method is illustrated by calculation of the transmission distribution in 2-terminal junctions, and by studying current cross correlations in a superconductor-beam splitter geometry. In Chapters V – VII we apply the general template of the circuit theory and obtain the charge transfer statistics in several physical systems of interest: a cavity coupled to a superconductor and a normal terminal, several junctions in series, and a voltage driven mesoscopic junction. The knowledge of the charge transfer statistics enables us to identify the elementary charge transfer processes in these systems. The conclusion is given in Chapter VIII.

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