On modeling quantum point contacts in quantum Hall systems

Abstract

Quantum point contacts (QPC) are a key instrument in investigating the physics of edge excitations in the quantum Hall effect. However, at not-so-high bias voltage values, the predictions of the conventional point QPC model often deviate from the experimental data both in the integer and (more prominently) in the fractional quantum Hall regime. One of the possible explanations for such behaviors is the dependence of the tunneling between the edges on energy, an effect not present in the conventional model. Here we introduce two models that take QPC spatial extension into account: wide-QPC model that accounts for the distance along which the edges are in contact; long-QPC model accounts for the fact that the tunneling amplitude originates from a finite bulk gap and a finite distance between the two edges. We investigate the predictions of these two models in the integer quantum Hall regime for the energy dependence of the tunneling amplitude. We find that these two models predict opposite dependences: the amplitude decreasing or increasing away from the Fermi level. We thus elucidate the effect of the QPC geometry on the energy dependence of the tunneling amplitude and investigate its implications for transport observables.

Figures (14)

• Figure 1: Four-terminal setup of a quantum Hall QPC experiment. The quantum Hall bulk supports a single chiral edge channel going around the sample (we focus on $\nu=1$ integer quantum Hall regime). The edges from the opposite sides of the sample approach each other in one location --- this is the QPC. Four Ohmic contacts are present in the sample, two (sources) are used to inject current into the chiral edge, two (drains) are used to extract the current that has passed through the QPC region. The key object for this paper's consideration is the energy-dependent transmission probability ${\cal T}\left(E\right)$: the probability for an excitation to be transmitted through the QPC without jumping to the other edge channel.
• Figure 2: Point-QPC model. Two counter-propagating chiral edge modes approach each other at a single point, $x=0$, which allows for tunneling of excitations between the edge modes.
• Figure 3: Numerical simulation of a QPC geometry with narrow QPC-defining gates. The figure presents the current density of QH edges in a sample that has four leads and a QPC defined by infinite potential on narrow gates (blue lines). The sample is modeled as 180x180 lattice, and the QPC-defining potential is applied to regions of 1-site width and 70-site length. The red density map represents the current density of two edge states. The edge states go around the gated regions smoothly, so that the tunneling would happen over a finite QPC width of a few magnetic lengths ($W$). This motivates our wide-QPC model shown in Fig. \ref{['fig:wide-QPC_model']}. The distance $L$ between the edges in the QPC determines the size of the gapped system bulk between the edges. This motivated our long-QPC model shown in Fig. \ref{['fig:long-QPC_model']}. The simulation has been performed using KWANT package Groth2014.
• Figure 4: Wide-QPC model. Two chiral edge channels undergo tunneling over a finite region of width $W$.
• Figure 5: The dependence of $\mathcal{T}_{\mathrm{wqpc}}\left(E\right)$ on energy for various values of $\zeta W/v$. The value of the dimensionless parameter $\zeta W/v$ determines $\mathcal{T}_{\mathrm{wqpc}}\left(E=0\right)$. At $\left|E\right|\rightarrow\infty$, the transmission becomes perfect, $\mathcal{T}_{\mathrm{wqpc}}\left(E\right)\rightarrow1$. Oscillatory behavior for $\left|E\right|>\zeta$ strongly depends on $\zeta W/v$.