[Paper Review] Higher moments of noise

Bertrand Reulet|arXiv (Cornell University)|Feb 2, 2005

Surface and Thin Film Phenomena3 citations

TL;DR

This paper presents a theoretical and experimental study of higher-order current noise moments, particularly the third moment (skewness), in tunnel junctions, demonstrating its temperature independence and non-Gaussian character. The key contribution is the first experimental measurement of the third moment of voltage fluctuations from room temperature down to 50 mK, revealing distinct behavior in the quantum regime (hf > kB T) compared to the classical regime (hf < kB T).

ABSTRACT

In this article we present simple approaches to the calculation of P(i) (the probability distribution of current fluctuations) in a tunnel junction, and to the effect of the environment on noise measurements in terms of the modification of P. We do not provide rigorous calculations, but simple considerations that bear the essential ingredients of the phenomena. We also discuss the effect of a finite measurement bandwidth. We report experimental results of the first measurement of the third moment of voltage fluctuations in tunnel junctions, from room temperature down to 50mK. Then we discuss extensions of that measurements to finite frequencies and to the study of other systems. We show the first data of the third moment in the regime where the frequency is larger than the temperature. Finally we discuss a new quantity, the "noise thermal impedance", which links the second and third moment.

Motivation & Objective

• To investigate the role of higher-order moments (beyond variance) in characterizing non-Gaussian shot noise in mesoscopic conductors.
• To experimentally measure the third moment of voltage fluctuations in tunnel junctions across a wide temperature range, including the quantum regime.
• To explore the influence of finite measurement bandwidth and environmental coupling on higher-order noise moments.
• To introduce and demonstrate the concept of 'noise thermal impedance' as a generalization of thermal impedance to finite-frequency, non-equilibrium noise.
• To establish a framework linking the second and third moments of current fluctuations via environmental response

Proposed method

• Modeling a tunnel junction as a single-channel system with rare, uncorrelated tunneling events, using rates Γ₊ and Γ₋ for left-to-right and reverse tunneling.
• Deriving the p-th moment of current fluctuations, showing that odd moments (e.g., third) are proportional to the dc current I, while even moments (e.g., second) scale with Γ₊ + Γ₋.
• Using the Fourier transform of the characteristic function to relate moments M_p to cumulants C_p, with C₃ = M₃ and C₄ = M₄ - 3M₂².
• Introducing spectral densities S_I^p(f₁,…,f_{p-1}) for higher-order fluctuations, with the constraint that the sum of frequencies is zero.
• Measuring the third moment experimentally via a diode detection system under ac bias, extracting dS_V²/dV as a proxy for S_V³.
• Defining 'noise thermal impedance' as a finite-frequency extension of thermal impedance, linking environmental response to S_I³ in the hot electron regime

Experimental results

Research questions

• RQ1 How does the third moment of current fluctuations (skewness) behave in tunnel junctions across different temperature regimes?
• RQ2 What is the effect of finite measurement bandwidth on the detection of higher-order noise moments?
• RQ3 How does the environment (e.g., thermal bath, coupling to a diffusive wire) modify the probability distribution P(i) of current fluctuations?
• RQ4 Can the third moment of noise be measured experimentally in the quantum regime (hf > kB T), and how does it differ from the classical regime?
• RQ5 To what extent can the concept of thermal impedance be generalized to finite-frequency, non-equilibrium noise through the 'noise thermal impedance'?

Key findings

• The third moment M₃ = e²Iτ⁻² is temperature-independent, in contrast to the second moment, which scales with T via the fluctuation-dissipation theorem.
• The third moment vanishes at zero bias (V = 0), consistent with time-reversal symmetry, and is an odd function of the dc current.
• Experimental data show a change in slope of the third moment at high voltage in the regime hf > kB T, where the slope becomes zero, suggesting dominance of the ⟨i₀i²⟩ term.
• A preliminary measurement of dS_V²/dV at high frequency in a 100 µm Au wire shows a cutoff at ~10 MHz, consistent with electron-phonon relaxation time τ_e-ph ≈ 16 ns.
• The measured cutoff frequency is inconsistent with diffusion cooling (expected ~800 kHz), indicating electron-phonon scattering dominates the thermalization time.
• The concept of 'noise thermal impedance' is introduced as a generalization of thermal impedance, capable of determining S_I³ in the hot electron regime