[Paper Review] Conservation, Dissipation, and Ballistics: Mesoscopic Physics beyond the Landauer-Buettiker Theory

Frederick Green, Mukunda P. Das|arXiv (Cornell University)|Mar 18, 2005

Quantum and electron transport phenomena22 references3 citations

TL;DR

This paper demonstrates that the Landauer-Büttiker (LB) theory of mesoscopic transport and noise fails to conserve particle number and charge, violating fundamental microscopic sum rules. By applying conserving sum rules from quantum kinetics, the authors derive the Landauer conductance formula from first principles, showing that LB's phenomenological assumptions are unnecessary and inconsistent with conservation laws, especially in quantum point contacts where noise data remain unexplained by LB theory.

ABSTRACT

The standard physical model of contemporary mesoscopic noise and transport consists in a phenomenologically based approach, proposed originally by Landauer and since continued and amplified by Buettiker (and others). Throughout all the years of its gestation and growth, it is surprising that the Landauer-Buettiker approach to mesoscopics has matured with scant attention to the conservation properties lying at its roots: that is, at the level of actual microscopic principles. We systematically apply the conserving sum rules for the electron gas to clarify this fundamental issue within the standard phenomenology of mesoscopic conduction. Noise, as observed in quantum point contacts, provides the vital clue.

Motivation & Objective

• To identify and correct the foundational failure of the Landauer-Büttiker (LB) theory in conserving particle number and charge in mesoscopic systems.
• To demonstrate that the standard LB approach to noise in quantum point contacts (QPCs) violates the perfect-screening and compressibility sum rules.
• To establish that the Landauer conductance formula can be derived from first principles using quantum kinetics, without relying on LB's phenomenological assumptions.
• To reconcile strong electron correlations and openness in mesoscopic devices with microscopic conservation laws.
• To provide a consistent, microscopically conserving description of mesoscopic fluctuations that explains experimental QPC noise data previously unaccounted for.

Proposed method

• Systematically apply conserving sum rules for the electron gas to analyze transport and noise in mesoscopic systems.
• Use a strictly kinetic derivation of the Landauer conductance formula based on microscopic quantum kinetics, avoiding LB’s phenomenological assumptions.
• Construct a model of charge distribution in a ballistic wire where the chemical potential varies linearly with position under an applied voltage.
• Derive the mean-square number fluctuation ΔN using the LB methodology, then compare it to the correct compressibility sum rule.
• Formally prove in the appendix that the LB noise formula violates both the perfect-screening sum rule and the compressibility sum rule.
• Show that the LB expression for ΔN depends explicitly on the transmission coefficient 𝒯 and voltage, contradicting the requirement that compressibility κ be independent of current.

Experimental results

Research questions

• RQ1 Why does the Landauer-Büttiker theory fail to conserve particle number and charge in mesoscopic transport, despite its widespread use?
• RQ2 How can the Landauer conductance formula be derived without relying on phenomenological assumptions, using only microscopic quantum kinetics?
• RQ3 What is the role of conservation laws in reconciling open boundary conditions with strong electron correlations in quantum point contacts?
• RQ4 Why does the LB noise formula fail to reproduce the compressibility sum rule, and what are the physical consequences of this failure?
• RQ5 How can a microscopically conserving theory explain experimental QPC noise data that remain unexplained by the standard LB approach?

Key findings

• The Landauer conductance formula can be derived from first principles using quantum kinetics, proving that the LB phenomenological assumptions are not foundational.
• The LB noise formula violates the perfect-screening sum rule, as shown by the non-zero change in carrier number under a voltage bias, which contradicts gauge invariance.
• The LB expression for the mean-square number fluctuation ΔN depends explicitly on the transmission coefficient 𝒯 and voltage, violating the requirement that compressibility κ be independent of current.
• The compressibility sum rule requires that ∂κ/∂I = 0 for all I, but the LB formula fails this condition even at equilibrium, invalidating its physical consistency.
• The LB model predicts a non-zero change in carrier number N(V) − N(0) proportional to eV, which contradicts the perfect-screening sum rule and implies a violation of charge conservation.
• The failure of the LB theory to satisfy the compressibility sum rule explains why it could not account for experimental QPC noise data for nearly a decade.