Experimentally Motivated Order of Length Scales Affect Shot Noise

TL;DR

The paper investigates how the relative ordering of internal charge and thermal equilibration lengths, $l^{\text{ch}}_{\text{eq}}$ and $l^{\text{th}}_{\text{eq}}$, with geometric scales $L_A$ and $L_Q$, governs shot-noise signatures at conductance plateaus in quantum Hall QPCs. By classifying three thermal regimes—No, Hybrid, and Full—and deriving expressions for the current fluctuations $\delta^2 I_1$, $\delta^2 I_2$, and $\delta^2 I_c$ under full charge equilibration, the study reveals universal patterns in the Fano factors $F_1$, $F_2$, and $F_c$ that differ between particle-like and hole-like edge states. The cross-correlation $F_c$ emerges as a key diagnostic for identifying the correct thermal regime, with distinct behaviors for co- and counter-propagating edge modes and for diffusive versus ballistic heat transport. The findings provide a framework for interpreting shot-noise experiments and point toward future work on non-Abelian states, graphene, and interface-based QPC devices to further illuminate edge hydrodynamics in topological systems.

Abstract

Shot noise at a conductance plateau in a quantum point contact (QPC) can be explained by considering equilibrations at the quantum Hall edges. The indication from recent experiments is that the charge equilibration length is much shorter than the thermal equilibration length. We discuss how this discovery gives rise to different thermal equilibration regimes in the presence of full charge equilibration. In this work, we classify these distinct regimes via dc current-current correlations (electrical shot noise) at definite experimentally found (or possible) QPC conductance plateaus for the edges of integer, particle-like, and hole-like filling fractions in a two dimensional electron gas. Our analyses show that distinct universal features arise among the different thermal equilibration regimes for the edges of particle-like and hole-like states.

Figures (5)

• Figure 1: A schematic picture of the device and measurements, which we use throughout the paper. (a) We show a Hall bar with filling $\nu$ in a QPC geometry. We have four contacts as a source $S$, a ground $G$, and two drains $D_1, D_2$. A dc current $I_S$ is injected by $S$, which is biased by a dc voltage $V_{\text{dc}}$, and the measurements are performed at $D_1, D_2$. We describe the chirality of charge propagation by circular arrows. (b) Transport measurements are carried out by measuring currents $I_1, I_2$ at $D_1, D_2$ respectively. The transmission and reflection coefficients are $t=I_1/I_S$ and $r=I_2/I_S$ respectively, where $t+r=1$. A QPC transmission plateau can be observed at a transmission $t=t'$ while measuring $t$ (correspondingly $r$) as a function of the QPC constriction (from a fully open to a fully pinched off QPC). (c) At a transmission plateau ($t=t'$), the current reaching $D_1, D_2$ can be noisy (leading to $\delta^2_{I_{1/2/c}}(t=t') \neq 0$) and there can be different mechanisms to it. The functional dependence of $\delta^2_{I_{1/2/c}}(t=t')$ is schematically shown while changing $V_{\text{dc}}$ and the slope of the linear region is proportional to the corresponding Fano factor $F_{1/2/c}$. The curvature in $\delta^2_{I_{1/2/c}}(t=t')$ shows up due to the temperature dependence.
• Figure 2: An effective modeling of the QPC geometry (c.f. \ref{['Cartoon']}(a)) while we consider equilibration. We describe the chirality of charge propagation of each filling by the circular arrows. (a) We schematically show the QPC geometry, where $\nu$ and $\nu_i (<\nu)$ are the bulk and QPC filling factors respectively and $L_\text{A}, L_\text{Q}$ are the geometric lengths with $L_\text{Q} \ll L_\text{A}$. (b) Effective model of the QPC geometry with edge equilibration for bulk filling $\nu$ and QPC filling $\nu_i$. We show three distinct boundaries: the boundary of $\nu$ with vacuum as "outer", the boundary of $\nu_i$ with vacuum as "upper", and the boundary between $\nu$ and $\nu_i$ as "line".
• Figure 3: We show the generation of hotspots (red circles) resulting in noise spots (green circles) while considering the equilibration for both the $\nu=$ particle-like and $\nu=$ hole-like filling fractions and the QPC has filling $\nu_i (<\nu)$. The chirality of charge propagation of each filling is depicted by the circular arrows. (a) We have $\nu$ as the particle-like filling fraction. The nature of heat transport is ballistic ("B") in the "outer" segment and can be "B" or diffusive ("D") or anti-ballistic ("AB") in the "line" and "upper" segments. We have two hot spots as $H_1, H_2$, which result in two noise spots as $M, N$. We note that the nature of heat transport in the "line" and "upper" segments (pink shaded lines) determine whether we have a constant noise or length-dependent noise. (b) We have $\nu$ as the hole-like filling fraction. The nature of heat transport can be "B" or "D" or "AB" in the "outer" segment, can be "B" or "AB" in the "line" segment, and is "B" in the "upper" segment. We have two hot spots as $H_1, H_2$, which result in four noise spots as $M, N, O, P$. As there can be "D" or "AB" heat transport in the "outer" segment, hence we have additional noise spots $O, P$. We note that the nature of heat transport in the "outer" segment (cyan shaded lines) determine whether we have a constant noise or a length-dependent noise.
• Figure 4: A Hall bar device with bulk filling $\nu$ and QPC filling $\nu_i(< \nu)$ with a source $S$ (biased by a dc voltage $V_{\text{dc}}$), a ground contact $G$, and two drains, $D_1$ and $D_2$. The geometric lengths are $L_{\text{A}}, L_{\text{Q}}$. We denote the segment between the vacuum and $\nu$ as "outer", between the vacuum and $\nu_i$ as "upper", and between $\nu$ and $\nu_i$ as "line". The arrow shows the fully equilibrated charge propagation along each segment while the circular arrow depicts the chirality of each filling. Hot spots $H_1,H_2$ (red circles) are created due to the voltage drops which are responsible for the creation of noise spots $M,N,O,P$ (green circles) PhysRevB.101.075308.
• Figure 5: We assume charge to be fully equilibrated leading to ballistic transport, moving "downstream" along each segment of the device. Each circular arrow shows the chirality of charge propagation of the respective filling ($\nu>\nu_i$). The direction opposite to charge flow ("upstream") is taken as antiballistic. Heat is also fully equilibrated and we show here the case when the heat transport is ballistic in "outer", antiballistic in "line", and ballistic in "upper" segments, as shown by the arrows. Voltage drops occur at the hot spots $H_1,H_2$ (red circles), giving rise to the noise spots $M,N,O,P$ (green circles) PhysRevB.101.075308.