# Spectroscopy of multi-electrode tunnel barriers

## TL;DR

This paper develops and experimentally validates a compact, physics-based model for multi-electrode tunnel barriers in nanoscale electronic devices, crucial for quantum device engineering and understanding electron transport phenomena.

## Contribution

It introduces a new theoretical model for multi-electrode tunnel barriers, validated by transport spectroscopy experiments, capturing voltage-dependent barrier behaviors and effects on electron transport.

## Key findings

- Barrier height exhibits linear and exponential voltage dependence.
- Exponential dependence correlates with electron channel formation.
- Model agrees qualitatively with semi-classical Poisson solutions.

## Abstract

Despite their ubiquity in nanoscale electronic devices, the physics of tunnel barriers has not been developed to the extent necessary for the engineering of devices in the few-electron regime. This problem is of urgent interest, as this is the precise regime into which current, extreme-scale electronics fall. Here, we propose theoretically and validate experimentally a compact model for multi-electrode tunnel barriers, suitable for design-rules-based engineering of tunnel junctions in quantum devices. We perform transport spectroscopy at $T=4$ K, extracting effective barrier heights and widths for a wide range of biases, using an efficient Landauer-B\"uttiker tunneling model to perform the analysis. We find that the barrier height shows several regimes of voltage dependence, either linear or approximately exponential. The exponential dependence approximately correlates with the formation of an electron channel below an electrode. Effects on transport threshold, such as metal-insulator-transition and lateral confinement are non-negligible and included. We compare these results to semi-classical solutions of Poisson's equation and find them to agree qualitatively. Finally, we characterize the sensitivity of a tunnel barrier that is raised or lowered without an electrode being directly above the barrier region.