Online physics lecture notes

Next week is going to be largely internet-free for me, so don't expect much excitement here until early August.  I do have a few things I want to discuss (thermoelectric effects, the great mess that was San Jose State's experiment in MOOCs, etc.), but that will wait.  In the mean time, I wanted to point out some nice lecture notes I'd found lately on the arxiv and elsewhere.  If people have their own favorite notes to point out online, please do so in the comments.

Neri Merhav has produced a couple of nice sets of notes, written from the perspective of trying to teach very physicsy concepts to electrical engineering students.  This past week he put up these notes about statistical mechanics, and previously he had written this set about the connections between information theory and statistical physics.  I found them both very readable.

Doron Cohen's notes on statistical mechanics and mesoscopics are a bit more mathy and closer to notes than a textbook-style discourse.

Not on the arxiv, but Yoshi Yamamoto's online notes regarding noise and noise processes are great.

Printing at the 180 nm scale??

I stumbled across this post, where it is asserted (with no link, and my google-fu is inadequate) that some group or collaboration at Berkeley is working on the ability to make 180 nm critical dimension transistors via printing (and therefore over really large areas, rather than only on dinner-plate-sized Si wafers).  Can a reader out there point me to who is really doing this, or is this a case of a press office distorting things (e.g., taking a layer thickness and claiming it's a lateral feature size)?

Physics is hard - how much should that worry us?

I'm a bit late to the party, but there have been discussions lately about the number of undergrad STEM majors, including physics, with some gnashing of teeth about overall difficulty. For example, a report by the National Bureau of Economic Research has been interpreted as saying that the main reason students bail on science majors is poor grades. That is, students go in, knowing that science will require more work than majoring in something fluffy, but when many receive tough grades even though they work hard, that's too much for them and they change fields. Chad Orzel does his usual thorough job looking into what the study really says, and it does seem true that tough grading drives some people out of STEM pursuits.

Similarly, there is a new report from the National Academy of Sciences called "Adapting to a Changing World: Challenges and Opportunities in Undergraduate Physics Education". I found the content rather disappointing, in the sense that it didn't seem to say much new. We all know that some approaches can be better under some circumstances than traditional lecture. However, many of those are very labor intensive, and I'm sure that my 50 person class would benefit if it were instead five ten-person classes. More to the point, though, the report specifically claims that hard grades are a major factor in the low participation of women and underrepresented groups in the physics major.

So, is physics unnaturally harsh in its grading, to its detriment? Or is this a question of high school preparation on the one hand, and grade inflation in nonscience majors on the other? I lean toward the latter.

(Note that the NSF has proven that science is hard. Also, here is the paper featured in that article - it's actually very interesting.)

(One other note: no one commented on my three part post about the physics of contacts, and the hit rate on those posts was very low. At the same time, in one 15 minute interval last week my post about "whiskey stones" got nearly 500 page views after it was mentioned in an argument about whiskey on reddit. Guess I should write about other things besides physics if I want more readership:-).

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Contacts III: The search for measurements

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.

Contacts, part deux

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.

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Contacts - annoying or an opportunity

In condensed matter physics, often we are interested in the flow of charge from a "source" electrode, through some material (the "channel", probably a different material than the source), and away into a "drain" electrode.  When the source, channel, and drain are all metals, life is simple.  While there might be some mismatch in the electrical conductivities of the materials, in the end an electron can go from some delocalized (extended, wavelike) state in the source, into such a state in the channel, and then into such a state in the drain, smoothly.   Because of the discontinuity in electronic band structure and dielectric properties, there is some reflection at the interface, leading to a contact resistance.  That is, some fraction of the applied voltage, linearly proportional to the applied voltage, is dropped across the source-channel contact, and some across the channel-drain contract.  This is an example of an ohmic contact.

However, the situation can be more complicated.  If the channel is a crystalline semiconductor, the Fermi level of the metal usually winds up sitting somewhere in the band gap.  If the is appropriate band bending takes place, there can then be an energy barrier (a Schottky barrier) for injection if charge from the metal into the semiconductor.   The spatial width of the barrier depends on the level of doping in the semiconductor, with higher doping leading to a narrower (though not necessarily shorter) barrier.   In this case, the current-voltage characteristics of the contact is not Ohmic, and looks instead like a diode, because the applied bias changes the shape of the barrier.  To avoid this in transistors, the regions of the channel where the source and drain contact it are very highly doped.  Still, in this case we are still assuming that the actual electronic states are extended, delocalized things.

The situation gets more complicated when the channel does not have delocalized states near the Fermi level.  

Usually experiments are designed to mitigate contact effects, either by avoiding measurements of the contact voltages (so-called four terminal measurements) or by making the contact contribution negligible compared to the bulk channel.  However, it turns out that sometimes contact effects can provide valuable insights into charge transport properties in the bulk.  I'll write more soon about this.