Conservation in High-field Quantum Transport

TL;DR

The paper addresses nonlinear quantum transport at mesoscopic scales where exact conservation laws constrain dynamics. It develops a conserving microscopic framework based on the quantum Boltzmann equation, explicitly enforcing particle-number conservation and neutrality, and incorporates interband coupling between subbands in a quantum point contact (QPC). The model captures nonlinear high-field conductance enhancements beyond the Landauer limit and predicts threshold behavior with temperature and saturation at strong driving, aligning with experimental observations. This work highlights the necessity of conserving kinetic theory for accurate high-field transport predictions in nanoelectronic devices and motivates further experimental and theoretical exploration.

Abstract

We give a short overview of the role of microscopic conservation in charge transport at small scales, and at driving fields beyond the linear-response limit. As a practical example we recall the measurement and theory of interband coupling effects in a quantum point contact driven far from equilibrium.

Figures (6)

• Figure 1: Operational equivalence of a battery and a time-varying magnetic flux as nonconservative current sources for a generic closed circuit. As power supplies, both have lifetimes that are finite but sufficiently long relative to the time scale for measurement. With both sources notionally concealed, as in (a) and (b), there is no means to discriminate between the battery, (a'), and the changing flux, (b'), using only measurements on the active device. In either case the electromotive force cannot be described by a conservative potential. Such a potential is incapable of sustaining a constant circulating current. WWkk.
• Figure 2: (a) Schematic of a ballistic quantum point contact driven by an external current source and stabilised by bulk resistive metallic leads. Strong metallic screening at the interfaces enforces electrical neutrality in and charge uniformity over most of the device, independent of current $I$ and EMF $E(I)$. The scattering mean free paths, both elastic and dissipative, are bounded by the length scale including not only the impurity-free QPC geometry but also the adjacent regions of strong boundary scattering, where current enters and exits. The three structures together make up the complete physical device, whose operational length $L$ delimits the longest mean free path. (b) An ensemble of identical such structures in series is concpetualised and described on the average. The current source and sink are now removed to the extremes of the ensemble.
• Figure 3: Landauer conductance measured in a QPC 1dp2, adapted from dePicciotto et aldp2. Conductance is in units of the ideal step, $g_0 \approx 77.5\mu$S. The experimental device was almost ideal, showing the low-field conductance rising in its characteristic steps, increasing discretely as each subband becomes newly occupied under the control of a gate voltage. The carriers in each band thus appear to contribute independently.
• Figure 4: QPC conductance $G$ adapted from dePicciotto et al1dp2 showing pronounced deviation of the Landauer steps from standard predictions. (Greyed-out band-threshold region covers the 0.7 anomaly pepper, a quite different effect at onset, not of interest here.) Vertical scale for $G$ is again in Landauer units. Nominally the gate voltage is set to restrict carrier density to the lowest occupied subband while $V_{\rm sd}$, the driving EMF along the channel, systematically increases. The QPC at low field ($V_{\rm sd} \!=\! 0)$ shows near-ideal behaviour, as in Figure \ref{['03']}. At progressively higher EMF, the step size of $G$ increases and indeed exceeds the Landauer limit. This behaviour is not explicable by assuming carriers to move as single particles, within uncoupled bands. What is needed is a fully conserving kinetic description able to deal with dynamical interband cross-talk.
• Figure 5: QPC conductance corresponding the experimental case, Figure \ref{['F4']}, computed within a conserving quantum Boltzmann model coupling two adjacent channel subbands GD2. At substantial driving fields, beyond the linear reguime, more carriers from the lower band gain sufficient energy to cross the energy gap and start to populate the higher band around its threshold, where $G$ is most sensitive to small changes in population. The excited carriers enhance the net conductance, exceeding the Landauer limit, before collisional cross-coupling de-excites the promoted carriers to the lower band. In this approach it is natural for enhancement of $G$ to be progressively greater as the driving voltage $V_{\rm sd}$ ramps up.