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arXiv:cond-mat/0111057 · Physical Review B

Counting statistics of tunneling current

Leonid S Levitov, M. Reznikov

Published Sep 13, 2004• 197 citations
arXivHTMLPDF

Abstract

The form of electron counting statistics of the tunneling current noise in a generic many-body interacting electron system is obtained and universal relations between its different moments are derived. A generalized fluctuation-dissipation theorem providing a relation between current and noise at arbitrary bias-to-temperature ratio $eV∕{k}_{B}T$ is established in the tunneling Hamiltonian approximation. The third correlator of current fluctuations ${S}_{3}$ (the skewness of the charge counting distribution) has a universal Schottky-type relation with the current and quasiparticle charge that holds in a wide bias voltage range, both at large and small $eV∕{k}_{B}T$. The insensitivity of ${S}_{3}$ to the Nyquist-Schottky crossover represents an advantage compared to the Schottky formula for the noise power. We discuss the possibility of using the correlator ${S}_{3}$ for detecting quasiparticle charge at high temperatures.

DOI
10.1103/physrevb.70.115305

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A theory of electron counting statistics in quantum transport is presented. It involves an idealized scheme of current measurement using a spin 1/2 coupled to the current so that it precesses at the rate proportional to the current. Within such an approach, counting charge without breaking the circuit is possible. As an application, we derive the counting statistics in a single channel conductor at finite temperature and bias. For a perfectly transmitting channel the counting distribution is Gaussian, both for zero-point fluctuations and at finite temperature. At constant bias and low temperature the distribution is binomial, i.e., it arises from Bernoulli statistics. Another application considered is the noise due to short current pulses that involve few electrons. We find the time-dependence of the driving potential that produces coherent noise-minimizing current pulses, and display analogies of such current states with quantum-mechanical coherent states.
We study fluctuations of electric current in a quantum resistor and derive a general quantum-mechanical formula for the distribution of transmitted charge. For that we introduce a scheme of current measurement that involves a spin 1/21/21/2 coupled to the current so that it precesses at the rate proportional to the current. Our approach allows the study of charge transfer without breaking the circuit. We analyze a single channel conductor and derive electron counting statistics for arbitrary relation between temperature and voltage. For a perfectly transmitting channel the counting distribution is gaussian, both for zero-point fluctuations and at finite temperature. At constant voltage and low temperature the distribution is binomial, i.e., it arises from Bernoulli statistics.
We discuss the counting statistics and current noise associated with the double Josephson quasiparticle resonance point in a superconducting single electron transistor. The counting statistics are in general phase-dependent, despite the fact that the average current has no dependence on phase. Focusing on parameter regimes where the counting statistics have no phase-dependence, we use a general relation first derived by MacDonald in 1948 to obtain the full frequency-dependent shot noise directly from the counting statistics, without any further approximations. We comment on problems posed by the phase-dependence of the counting statistics for the finite-frequency noise.
The theory of charge partition noise of quasiparticles in a superconducting tunnel junction is developed. The charge fluctuations are shown to have a significant contribution from partition noise that arises from the quantum superposition of the electron and hole character of the quasiparticles. These fluctuations are dominant at small bias voltage. The charge fluctuations are compared to the usual Poisson ``shot noise'' of the current. The implications for the design of energy-resolving single-photon detectors are explored.
We study delta-T noise -- excess charge noise at zero voltage but finite temperature bias -- for weak tunneling in 1D interacting systems. We show that the sign of the delta-T noise is generically determined by the nature of the dominating tunneling process (more specifically, its scaling dimension). We clarify the relation between the sign of delta-T noise and the quantum exchange statistics of tunneling quasiparticles. We find that, for infinite systems hosting chiral channels with local interactions (e.g., quantum Hall or quantum spin Hall edges), when the delta-T noise is negative, the tunneling particles are boson-like, revealing their tendency towards bunching. However, the opposite is not true: Boson-like particles do not necessarily produce negative delta-T noise. Importantly, the bosonic nature of particles generating the negative delta-T is not necessarily intrinsic, but can be induced by the interactions. This particularly implies that negative delta-T noise for tunneling between the edge states cannot serve as a smoking gun for detecting "intrinsic anyons". We also establish a connection between the delta-T noise and the temperature derivative of the thermal noise in interacting systems, both governed by the same scaling dimensions. As a demonstration of the above statements, we study tunneling between two interacting quantum spin Hall edges. With bosonization and renormalization-group techniques, we find that many-body interactions can generate negative delta-T noise for both direct tunneling through a point contact and in Kondo exchange tunneling via a localized spin. In both setups, we show that the noise can become negative at sufficiently low temperatures, when interactions renormalize the tunneling to favor boson-like pair-tunneling of electrons. Our findings show that delta-T noise can probe the nature of collective excitations in interacting 1D systems.