Lectures on superconductivity and noise

I forgot to write a blog entry after my previous lecture last Thursday, so today I write about two lectures, concentrating mostly on superconductivity and a bit on fluctuations. The idea of the superconductivity chapter in the book is to introduce the topic enough so that I can describe the properties of superconducting junctions. There is so much physics in them, and especially of present interest: quantum mechanics of circuits, superconducting qubits, coolers, Andreev bound states, SQUIDs and so on. Unfortunately, the fact that Chapter 5 tries to be the minimal introduction to the topic of superconductivity shows up in the treatment of the BCS theory: I should probably have written a bit more about the BCS ground state, and I should definitely have discussed the self-consistency relation in more detail. I did not have such a strong motivation for this, because the topic has been presented in many other books as well, but eventually it would not have taken very much space.

Initially I added the concept of Andreev reflection in order to be able to describe phenomena one needs to understand when dealing with small superconducting systems. Then I realized that it is a good way to provide a microscopic derivation of the Ambegaokar-Baratoff relation for the critical current via the Andreev bound states. Today in the lecture I noticed that the concept is very topical, because people try to realize Majorana fermions with their help. I am not sure if mentioning it affected the interest in the topic, but perhaps it did.

On today’s lecture I reached the discussion of the fluctuation-dissipation theorem. I will not derive it in the lectures, but I will try to make a video derivation so that I don’t have to get stuck in details in the lecture. Let’s see how that will work out.

FD theorem has two limits: that of thermal noise at low frequencies, and the vacuum fluctuations at high frequencies. There was an interesting question about the latter in the lecture: Can they be directly connected with the description of physics of vacuum fluctuations, such as the Casimir effect? I am not quite sure about the static version (Casimir force) – this is typically described merely via the energy of confinement, and the force related to the dependence of this energy on the size of the confinement. Perhaps one could do this by looking at the vacuum fluctuations in a model LC circuit. However, the dynamical Casimir effect (creation of photons from vacuum via the time-dependent confinement) can certainly be described via the vacuum fluctuations present in the second-order correlator. I think it roughly goes so that one inserts the ac driving field with frequency larger than the temperature and looks what happens to the second-order correlator at half that frequency. This should reveal the presence of the photon pairs (if I remember correctly, the photons do not have to come at half the frequency, but only the sum of their frequencies must equal that of the driving).

Scattering and interference

From the title one might get the idea that the lecture today was pretty messed up, with people interfering with my lecturing a lot. Unfortunately, it was pretty smooth with only a couple of (admittedly good) questions. A ballistic lecture with no backscattering is never a good idea, because no information can be inferred from the lecture that way.

Jokes apart, I seem to be always late from my schedule, as I had hoped to get a bit further in the interference chapter. I did discuss at length the treatment of many-probe systems in scattering theory, showed examples of the voltage probe and the comparison between two- and four-probe resistances. Then I discussed the resonant tunneling effect, which is probably somewhat familiar from quantum mechanics I. In fact, I think that the scattering formulation is the best way to describe this effect: it is generic (describes both the quantum dot case and the case of Fabry-Perot cavities in optics), the only differential equation one needs to solve is the one related with the dynamic phase, and it shows the difference in summing the probability amplitudes or probabilities for the particle paths. Moreover, it connects waves with quantized levels.

In the scattering formulation, this along with the four-probe conductance formula allows showing how to get localization-type of scaling of conductance in a disordered wire. Note that this was debated a bit in the 1970’s-1980, as the proper result depends on which quantity is averaged.

Anyway, I only reached mid-way of weak localization, so next time I will have to deal with the negative magnetoresistance related to it, and discuss a bit universal conductance fluctuations. Persistent currents I will probably leave for the self-study, as I want to be able to discuss superconductivity also during the next lecture.

After the lecture I got a question about notation in the book. It seems that in chapter 3 I am using two notations for the number of channels in a lead: M and N. M is the number of modes, N denotes the size of the reflection matrix, but these two are of course the same.

Lecture on scattering theory

Today I continued the discussion of the scattering theory, deriving the Landauer formula, discussing the quantized conductance and the properties of scattering matrices. I was a bit slow, as I would have liked to finish the chapter on scattering, but I failed to show the derivation of the current from the current operator description, or the description of resonant tunneling.

It was first difficult to figure out clicker questions, but then I managed to find a few, with which I was quite satisfied. The first one was to figure out the average rate of electrons passing through a single-mode ballistic channel biased with voltage V. It was actually an exercise on dimensional analysis, as only one of the proposed alternatives had a dimension of rate. Anyway, I would like to teach this point in all courses: dimensional analysis in the sense of figuring out the correct dimensions (say, in 1/s or m etc.) is a very helpful tool in checking a result, or trying to find relevant quantities describing a certain system. Let’s say we want to study the dynamics of electrons in some system. We are hence interested in processes that change the number of electrons in a given region from time t to time t+dt, i.e., we should write a differential equation for this number. This means that the right hand side of this equation should have only quantities that have a dimension of rate. For a given model that we make on a given system, there are typically only a handful of such quantities, and only a few that are independent of each other. More often or not this number of quantities is one or two. In that case you can characterize the full dynamics by seeing what happens if you change the values of these quantities. This is where you get with a simple dimensional analysis!

Lecture on semiclassical theory

On Tuesday I gave a lecture about the semiclassical theory, completing the simplification of the regular Boltzmann equation presented in materials physics books to that in the diffusive limit (diffusion equation for distribution function, written separately for each energy), and finally to the quasiequilibrium limit, where it is enough to write equations for position dependent potential and temperature. I think the message went through quite well, I managed to avoid excessive derivations (which IMHO should not be given too much in the lecture, because they tend to confuse the students from the physics), and I got some good questions.

The diffusive limit can be represented using the circuit theory, and especially in the quasiequilibrium limit it is a convenient tool. I am not sure if this becomes very clear from my book, though, so this should be improved a bit. Anyway, I used its ideas to talk about some basic phenomena in spintronics, like the spin accumulation and the giant magnetoresistance. Let’s see if the students understood it – there were two exercise problems on them.

Today I will talk about scattering theory, which I already started in Tuesday’s lecture.