I will make an argument now that contact resistances are much maligned, and instead of rigorously trying to avoid worrying about them, we should instead look for opportunities (with well defined, reproducible contact interfaces) when they can actually tell us something. I'll punctuate this with some papers from our own group and areas I happen to know, but that's only because those are the examples that come to my mind.
What happens when you try to inject charge from a metal into a hopping conductor - a material with some energy-dependent density of localized states? Many organic semiconducting polymers are such systems. In this situation, an injected charge carrier faces a competition between diffusion away into the channel by hopping, and an attraction to its own image charge in the metal. The rather odd result is that this contact often tends to be Ohmic (in the sense that the contact voltage is directly proportional to the current), but the contact resistance ends up being inversely proportional to the mobility of the charge in the channel. This is true even when the metal Fermi level lies somewhere in the tail of the band (a situation where you would expect a Schottky contact in a nonhopping semiconductor). We ran into this here, and systematically varied the contact resistance by using surface chemistry to adjust the energetic alignment.
In correlated materials, the situation may seem tantalizing yet hopeless. On the one hand, you know something interesting must happen when charge is injected into the material - carriers in the metal are boring, electron-like quasiparticles, while charge excitations in the correlated system could in principle be very different, with fractional charge or spin-charge separation. On the other hand, depending on the bulk properties and ability to make reproducible contacts, it can be very hard to extract useful information from contact resistances in these systems. We did get lucky, and found that in magnetite conduction in both the high temperature (short range ordered) state and in the low temperature (long range ordered) state seems to be through hopping, similar to the description above. I definitely think that there is a lot more to be done in such materials by using contact effects as a tool rather than avoiding them.
In the world of molecular junctions, often one is in the limit where the device is "all contact", in the sense that the "bulk" is only a couple of nanometers and a few atoms. Next time I'll talk about some great measurements by others in these systems, as part of a discussion on how one can measure contact resistance.