How are resistivity and tunneling related?

Question

If we consider a sandwich with three nanometric layers: conductor-insulator-conductor and apply voltage (lower than breakdown voltage) from both sides tunneling will occur. Is tunneling dependent on resistivity of the insulator? Intuitively one can think that tunneling would be lower in good insulators, but is it really that simple?

Answer 1

I don't think it's quite that simple. The resistivity may be related to the barrier potential energy separating the two reservoirs, which directly modulates tunneling in the obvious way (higher energy = less tunneling). But there is one barrier which classically you'd describe with an infinite resistance (a Dirac $\delta$-function potential) which admits tunneling just fine because it's also infinitesimally thin.

Answer 2

Resistivity is a measure of the band gap of the insulator and its density. For tunneling to have a possibility to happen one energy level from the left side of the sandwich must match with another energy level from the right side of the sandwich. The conclusion is that resistivity doesn't affect tunnelling.

Answer 3

The pitfall here is applying macroscopic concept of resistivity to microscopic phenomena, like tunneling.

Macroscopic resistivity, the one appearing in the Ohm's law, $I=V/R, R =\rho L/A$ is a result of averaging over a physically small volume, i.e., a volume containing macroscopic number of atoms, but very small, to be considered infinitesimal for the practical purposes. Note that this also applied to the permittivities, permeabilities and other quantities in macroscopic electrodynamics.

This resistivity owes largely to scattering of electrons by impurities and phonons, inherently involving thermalization/thermal averaging, which in quantum terms is associated with decoherence.

Microscopic resistance
None of the reasoning for the macroscopic resistivity applies to the microscopic world. We are looking here at coherent phenomena, i.e. phenomena on the time and space scales where thermal averaging is not relevant. Importantly, the resistance of a sample is no more reducible to its size, as dictated by the classical formula $R=\rho l/A$. Indeed, it is easy to see that the transmission through a barrier is not scaling linearly with its length. On thus usually speaks of resistance/conductance of the whole sample, rather than resistivity. A host of quantum phenomena, such as universal conductance fluctuations, conductance quantization, quantum Hall effect, etc. are thus not explainable in terms of classical resistivity, but only using the appropriate quantum theory.

The theory of resistance on quantum level is known as Landauer-Büttiker formalism.

Superlattices
Sandwich structures composed of many layers of semiconductors with alternating band gap are known as superlattices - they are characterized by the band splitting into subbands, due to the effective periodic potential of the alternating layers. Thus, some phenomena that can be observed in bulk semiconductors and insulators only at extremely high voltages (like Zener tunneling or negative differential conductance) are observed in superlattices at rather moderate conditions.

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