  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.
