Suppose I connect a conductive wire (cross section 1 mm$^2$) to an aluminum object. Since aluminum is highly conductive, electricity will flow smoothly inside the object with little resistance. However, since aluminum is also very reactive, there is a thin layer of highly resistive aluminum oxide on the surface of the object. Wikipedia says this layer is about 4 nm thick (citing this paper). Naively, we can calculate the resistance of the oxide layer using the resistivity of alumina, which is about $10^{14} \,\Omega \cdot \text{cm}$:
$$R=\rho \frac{l}{A} \approx 4\times 10^9 \,\Omega$$
Of course, we don't actually measure such a large resistance. But why not? How exactly does an electric current pass through the oxide layer?
The obvious answer is that electrons simply tunnel through the oxide layer. So let's calculate the tunneling probability. According to this document from MIT OpenCourseWare, the aluminum oxide layer presents a potential barrier of 10 eV. Then the transmission coefficient across a 4 nm layer is given by
$$T \approx e^{-2\left( \sqrt{2 m_e / \hbar^2 \cdot (10\text{ eV})} \right) (4\text{ nm})} = 5.16 \times 10^{-57}$$
This is an extremely small number. In principle, we could now find the actual rate from the density of states and Fermi's golden rule, but it seems likely the result will be a very small current.
It's possible that the parameters that I am using might be incorrect. I checked a few other sources and found widely varying values for the potential barrier and oxide thickness. However, the fact that slightly anodized aluminum with a thicker oxide layer (for example, a few tens of nm) still conducts electricity makes me think that tunneling is not a full explanation, since the tunneling rate decreases exponentially with oxide layer thickness.
Another possible explanation might be electrical breakdown or some other change in the oxide crystal structure, such as melting. But if this is the correct answer, what exactly changes in the oxide layer to make it electrically conductive? Normally, oxides are not conductive because the oxygen atoms scavenge free electrons. Does this stop happening for some reason?
I am willing to accept a good theoretical answer, but I am hoping for experimental evidence if possible.
Best Answer
The native oxide that coats aluminum is slightly porous, and the pores tend to trap tiny amounts of moisture in them. This renders them electrochemically active and ever so slightly conductive. (In fact, for the aluminum oxide layer to grow in thickness in hot environments requires that both the aluminum atoms be capable of diffusing up through the existing oxide to reach oxygen in the atmosphere and the oxygen atoms be capable of diffusion down through the oxide to reach unreacted aluminum underneath the oxide.)
In order to render an oxidized aluminum surface pore-free, the aluminum piece must be baked in an oven with an oxygen atmosphere in it, to close off those pores.
In the absence of porosity in the oxide, the conduction mechanism is Frenkel-Poole emission, where a random thermal fluctuation will occasionally promote a bound electron into the conduction band, where it can then drift under the influence of an external field.