[Paper Review] Shot Noise in Schottky's Vacuum Tube

Christian Schönenberger, S. Oberholzer+2 more|arXiv (Cornell University)|Dec 31, 2001

Mechanical and Optical Resonators2 references3 citations

TL;DR

This paper demonstrates that shot noise in Schottky’s vacuum tube is fundamentally classical, arising from the classical Boltzmann distribution of electron states in the cathode, not quantum effects. Despite Schottky’s original derivation predating quantum mechanics, the analysis confirms that quantum diffraction effects are negligible under typical operating conditions, and the observed noise perfectly matches the classical Schottky formula $ S = 2e|I| $, with quantum contributions being irrelevant to within 80 decimal places in realistic parameters.

ABSTRACT

In these notes we discuss the origin of shot noise ('Schroteffekt') of vacuum tubes in detail. It will be shown that shot noise observed in vacuum tubes and first described by W. Schottky in 1918 is a purely classical phenomenon. This is in pronounced contrast to shot noise investigated in mesoscopic conductors which is due to quantum mechanical diffraction of electron waves.

Motivation & Objective

• To resolve the longstanding debate on whether shot noise in vacuum tubes is classical or quantum in origin.
• To analyze the statistical origin of current fluctuations in thermionic emission using classical and quantum statistical mechanics.
• To determine under what physical conditions quantum effects (e.g., tunneling) could contribute to shot noise in vacuum tubes.
• To compare the classical and quantum contributions to shot noise and show their equivalence in the limit of full transmission.
• To validate the classical nature of shot noise using historical experimental data from Hull and Williams (1925).

Proposed method

• Derives the general expression for shot noise in a two-terminal conductor using the Landauer-Büttiker formalism, incorporating Fermi-Dirac statistics and transmission probabilities.
• Applies the noise formula to vacuum diodes, distinguishing between classical and quantum contributions to the noise power spectral density.
• Evaluates the transmission probability $ T acksimeq ig[1 + e^{-2 au heta/ar{h}ar{ heta}_0}ig]^{-1} $ using the WKB approximation for electron emission over a potential barrier.
• Calculates the Fano factor $ F = S / (2e|I|) $ to compare measured noise with theoretical predictions, using experimental parameters from Hull and Williams (1925).
• Assesses the dominance of classical vs. quantum contributions by evaluating $ ar{h}ar{ heta}_0 / k_B heta $, showing it is extremely small in real vacuum tubes.
• Uses the Poisson equation to model the space-charge region near the cathode and its effect on electron emission and current flow.

Experimental results

Research questions

• RQ1 Is shot noise in vacuum tubes a classical or quantum phenomenon, given that Schottky derived it before quantum mechanics?
• RQ2 What is the relative contribution of classical electron statistics (Boltzmann tail) versus quantum tunneling to shot noise in vacuum tubes?
• RQ3 Under what physical conditions could quantum effects (e.g., tunneling) become significant in vacuum tube shot noise?
• RQ4 Why does the classical Schottky formula $ S = 2e|I| $ hold so precisely in experiments, even though quantum mechanics underlies electron emission?
• RQ5 How do historical measurements of shot noise (e.g., Hull and Williams, 1925) support or contradict a classical origin of the noise?

Key findings

• Shot noise in vacuum tubes is predominantly classical, originating from the classical occupation of electron states in the cathode via the Boltzmann distribution.
• Quantum tunneling effects are negligible in typical vacuum tubes, as $ ar{h}ar{ heta}_0 / k_B heta o 0 $, making the transmission probability $ T o 1 $, so quantum corrections are smaller than $ 10^{-80} $.
• The classical shot noise formula $ S = 2e|I| $ is recovered when all transmission probabilities are 1, confirming the classical origin of the noise.
• Even in the quantum regime with small $ T_n $, the noise still yields $ S = 2e|I| $, but this is due to quantum diffraction, not classical statistics.
• Historical data from Hull and Williams (1925) show $ F = S / (2e|I|) o 1 $, confirming the classical nature of the noise in the saturation regime.
• The suppression of shot noise at high temperatures due to space-charge effects does not alter the classical origin of the noise, as the Fano factor remains consistent with classical predictions.