In the last two posts I've talked a bit about contact resistances, but I haven't said much of anything about how to infer these experimentally.
In some sense, the best, most general way to understand contact voltages is through scanning potentiometry. For example, this paper (pdf - sorry for the long URL) in Fig. 10 uses a conductive AFM tip to look at the local electrostatic potential as a function of position along an organic transistor under bias. When done properly, this allows the direct measurement of the potential difference between, e.g., the source electrode and the adjacent channel material. If you know the potential difference and the current flowing, you can calculate the contact resistance. Even better, this method lets you determine the \( I-V \) characteristic of the contact even if it is non-Ohmic, because you directly measure \(V\) while knowing \(I\). The downside, of course, is that not every device (particularly really small ones) has a geometry amenable to this kind of scanned probe characterization.
A more common approach used by many is the transmission line method. In the traditional version of this, you have a whole series of (otherwise identical) devices of differing channel lengths. You can then plot the resistance of the device as a function of \(L\). For Ohmic contacts and an Ohmic device, the slope of the \(R-L\) plot gives the channel resistance per unit length, while the intercept at \(L \rightarrow 0\) is the total contact contribution. This does not tell you how the contact resistance is apportioned between source/channel and channel/drain interfaces (this can be nontrivial - see the figure I mentioned above, where most of the voltage is dropped at the injecting contact, and a smaller fraction is dropped at the collecting contact). Related to the transmission line approach is the comparison between two- and four-terminal measurements of the same device. The four-terminal measurement, assuming that no current flows in the voltage contacts and that the voltage probes are ideal, should tell you the contribution of the channel. Comparison with the two-terminal resistance measurement should then let you get some total contact resistance. I should also note that, if you know that the channel is Ohmic and that one contact dominates the resistance, you can still use length scaling to infer the \( I-V \) characteristic of the contact even if it is non-Ohmic.
The length scaling argument to infer contact resistances has also been used to great effect in molecular junctions. There, for non-resonant transport, the usual assumption is that the bulk of the molecule (whatever that means) acts as an effective tunneling barrier, so that conductance should fall exponentially with increasing molecular length (assuming the barrier height does not change with molecular length, an approximation most likely to be true in saturated as opposed to conjugated molecules). Thus, one can plot \(log G\) as a function of molecular length, and expect a straight line, with an intercept that tells you something about the contact between the molecule and the metal electrodes. This has been done in molecular layers (see here, for example), and in single molecule junctions (see here, for example). These kinds of contact resistances can then be related, ideally, to realistic electronic structure calculations looking at overlap between electronic states in the metal and those of the linking group of the molecule.
Hopefully these three posts have clarified a little the issue of contact effects in electronic devices - why they are not trivial to characterize, and how they may actually tell you interesting things.