Quasi-Langevin method for shot noise calculation in single-electron tunneling

It is shown that quasi-Langevin method can be used for the calculation of the shot noise in correlated single-electron tunneling. We generalize the existing Fokker-Plank-type approach and show its equivalence to quasi-Langevin approach. The advantage of the quasi-Langevin method is a natural possibility to describe simultaneously the high (“quantum”) frequency range. 73.40.Gk; 72.70.+m Typeset using REVTEX 1 Correlated single-electron tunneling [1] remains an attractive topic during last decade. Because in the systems of small-capacitance high-resistance tunnel junctions electrons tunnel almost as the classical particles, the most of experiments are well explained within the framework of “orthodox” theory [1] of single-electronics which is based on the classical master equation. The theory of the shot noise in single-electron transistor [2] developed in Refs. [3,4] is also based on the master equation. The shot noise appears due to the randomness of the tunneling events. Despite the classical description of the system, the current in this theory should be treated as a kind of an operator because the current is caused by tunneling events which change the charge state of the system. This theory has been studied in detail in a number of subsequent publications see, e.g., Refs. [5–11]. The first experimental confirmation of the theory has been obtained [12] in 1995, and other experiments are definitely coming because several groups has succeeded in the fabrication of the single-electron transistor operating at relatively high frequencies (hundreds of kHz) at which 1/f noise should become less than the shot noise. Among the most noticeable developments of the shot noise theory in single-electronics after 1991, let us mention the following. It has been shown [9] that the noise due to the processes of Andreev reflection which transfer two electrons per tunneling event can be described by the same formalism with e substituted by 2e. The shot noise in single-electron transistor with large level discreteness has been studied [13]. The shot noise theory has been applied to single-electron systems other than single-electron transistor [14,15]. Besides the noise in the “orthodox” frequency range ω ∼ W/eR (where W is a typical energy and R is a typical resistance), the noise in the “quantum” frequency range ω ∼ W/h̄ has been studied and the matching of the high and low frequency limits of two ranges has been proven [14]; however, the approach unifying both frequency ranges in one formalism has not been found. The existing formalism of the shot noise in single-electron tunneling is of the FokkerPlank type and is based on the deterministic master equation. However, generally more popular method in the study of the noise is the Langevin approach in which the random 2 term is introduced into the evolution equation. Three decades ago the Langevin approach was applied [16] to study the fluctuations in the nonequilibrium electron gas. In the present letter we show that a similar method can be used for the single-electron tunneling. (We call it quasi-Langevin because the regular Langevin method cannot be applied directly.) It is equivalent to existing formalism in the “orthodox” frequency range, but it naturally allows also the calculations in the “quantum” frequency range. Let us start with the generalization of the existing Fokker-Plank type method to an arbitrary system consisting of voltage sources, capacitances and tunnel junctions with sufficiently large resistances (Rj ≫ RK = h/e 2 ≃ 26kΩ). For simplicity we do not consider Ohmic resistances. In this case the dynamics is governed by the matrix master equation