Here are a couple of things I find surprising about the electronic transport properties of materials.
1. One cannot simply have materials, particularly metals, that have any value imaginable for a transport coefficient. For example, one cannot make the conductance or the thermopower as large as one wishes by designing some fantastic material.
2. Quantum mechanics determines what these fundamental limits are. Furthermore, the limiting values of transport coefficients are often set in terms of fundamental constants [Planck's constant, Boltzmann's constant, charge on an electron].
The fact that this is profound is indicated by the fact that this was not appreciated until about 25 years ago. A nice clean example is the case of a quantum point contact with N channels. The conductance must be N times the quantum of conductance, 2e^2/h. This result was proposed by Rolf Landauer in 1957 but many people did not believe it until the first experimental confirmation in 1988.
The thermal conductance through a point contact should also be quantised. The quantum of thermal conductance is
1. note that the Wiedemann-Franz ratio is satisfied.
2. this sets the scale for the thermal conductivity of a bad metal.
A paper in Science this week reports the experimental observation of this quantisation.