Last lectures…

I seem to have some difficulties in keeping up constant blogging. Now after three more lectures (on Coulomb blockade, on superconducting junctions, and on graphene), I have reached already Chapter 10 in the book. I skipped Ch. 8 on quantum dots in order to be able to cover the other chapters on time, and I have skipped also some sections in each of the other chapters as well: I did not talk about cross correlations, full counting statistics or heat current noise related to Chapter 6, skipped dynamical Coulomb blockade in Chapter 7, SINIS physics in Chapter 9, and multilayer graphene and nanoribbons in Chapter 10. On some other years I will probably make some other choices. Anyway, I tried to motivate supercurrent in the superconducting junctions as a dual effect to the Coulomb blockade, and superconducting qubits as elements interpolating between two extreme situations.

For graphene, the main point was to show the origin of the Dirac dispersion, and illustrate the difference to ordinary metals. This is not discussed properly enough in the book, but I tried to fill in the gap in the lectures by discussing the main distinctions between metals, insulators and graphene-type semimetals. When I wrote the graphene chapter (2010 or so), the emphasis in the research community had been on theory related to the Dirac spectrum. This is visible in the graphene chapter, as the main electronic effects that I describe are those arising from the merger of high-energy (Dirac) physics with graphene electron transport. I could have obviously added much more, related to the forms of scattering in graphene, the absence of weak localization, and its recovery with intervalley scattering, and so on. By the way, there is an interesting point that one learns from graphene: when I talk about (relativistic) “high-energy physics”, graphene shows that I get this type of physics only for low energies. It therefore shows that it could be that Dirac equation in fact emerges as a low-energy theory from some high-energy physics (similar to the microscopic model of graphene). Also, some of the symmetries in graphene (such as the isotropicity of the dispersion around the valleys) are emergent, not present in the full microscopic picture. At the same time, this discussion reveals a kind of a chicken-and-egg problem: in the way how it is presented in the book, the Dirac dispersion results from a specific tight-binding model. However, we do not need exactly that model, but the low-energy physics is pretty robust as long as the microscopic model preserves some symmetries (hexagonal lattice, A-B symmetry). Therefore, it is difficult to say which one is more fundamental, the Dirac equation or the underlying microscopic theory. It probably simply depends on the viewpoint. In fact this dilemma is studied in the momentum space topological classification of media, but about that I might write later…

Next week will be the last lecture on nanoelectromechanics and optomechanics, and it will be given by Francesco Massel.

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.

Lecture on Boltzmann theory

I noticed before yesterday’s lecture on Boltzmann theory, that I need to at least remind what the Fermi Golden Rule is (I made a video of the derivation), and to explain again the relation between the chemical potential, Fermi energy and the electrostatic potential. This took so much time that I barely reached the diffusive limit of the Boltzmann equation. It is a pity: it would have been better to show how one can, with consequent simplifications, always assuming that scattering breaks a certain symmetry (or conservation law), reach an increasingly simplified equation describing the state of the particles. This seems to be a “general know-how” of researchers in the field, but I had not seen it explained before, so I wanted to include it in my book. Eventually one gets only the diffusion equation for the (electro)chemical potential, which is what you would get from Ohm’s law.

Even that simplest limit, however, can be interesting if we allow for a spin-dependent potential, and in some portions of the structure spin-dependent conductivities arising from spin-dependent densities of states (in ferromagnets). The behavior of the charge current/resistance in this system is at the heart of spintronics. In my book I only discuss collinear magnetizations and fields. The non-collinear case would also be interesting, because it would allow discussing the spin Hanle effect, and possibly the spin transfer torque that will be used in future devices of spintronics. Perhaps in the next edition of the book I will devote an entire chapter for spintronics… If there will ever be the next edition, though.

So now I have the dilemma whether to try to use part of the project lecture tomorrow for finishing the Boltzmann lecture, or whether to delay my schedule and give the rest of the lecture on Tuesday. Probably I will do the latter, and then later catch up, so that the projects get going better.

Projects

Last Thursday the students got the projects that they will work on during the fall term. The following projects were chosen (I include also here the guidelines and references that I gave to them):

i) Thermoelectric devices and measuring techniques – Sec. 2.8 + references 
- Figure out: Mott law, figure of merit, efficiency, Onsager relations, tunneling thermopower of a gapped system, 3-omega method
– Suggested references:http://www.nature.com/nature/journal/v413/n6856/abs/413597a0.html,
http://arxiv.org/abs/1109.1009http://www.sciencemag.org/content/321/5895/1457

ii) Quantum Hall effect, Landau levels and electron optics (pair project) 
– Explain (integer) QHE and Landau levels; explain how QPCs can be used as beam splitters, and how one can do interference experiments with electron wave packets; what is the relation of scattering theory to all this?
– Electron optics: http://www.nature.com/nature/journal/v422/n6930/full/nature01503.htmlhttp://www.sciencemag.org/content/339/6123/1054.full

iii) Primary thermometry with shot noise and Coulomb blockade – Sec. 6.2, 7.6.1 + references 
– Explain the difference between primary and secondary thermometry, describe the challenges
– Mostly references cited in the book

iv) Spin qubits – Sec. 8.5.4 + references 
– Explain the qubit states, their read-out and manipulation, and explore the recent results
– http://www.sciencemag.org/content/309/5744/2180http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.026801http://www.nature.com/nnano/journal/v9/n9/full/nnano.2014.153.htmlhttp://www.sciencemag.org/content/339/6124/1174.abstract
v) Superconducting qubits (pair project or alone one realization) – Sec. 9.4.3 + references 
– Take one realization and explain the qubit states, their read-out and manipulation, possible practical problems and recent results
- Phase/flux/charge qubit references from the book; transmon (or Xmon) qubits: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.080502http://www.sciencemag.org/content/339/6124/1169.full.pdfhttp://journals.aps.org/prl/abstract/10.1103/PhysRevLett.107.240501

vi) Rashba and Dresselhaus spin-orbit interaction (part of the pair project) – references 

vii) Topological insulators, quantum spin Hall effect  (part of the pair project) – references 
– Explain 2D and 3D TIs, topological charges (Chern numbers) and bulk-surface correspondence, chirality of the edge/surface states, quantized conductance in 2D TIs, quantum spin Hall effect
– References: http://www.sciencemag.org/content/318/5851/766.abstracthttp://www.nature.com/nature/journal/v452/n7190/full/nature06843.html

viii) Nanoelectromechanical mass/force detection – beginning of Ch. 11 + references 
- Explain some basics of elastic theory regarding the resonances
– Explain the detection scheme, what determines the accuracy?
http://www.nature.com/nnano/journal/v7/n5/full/nnano.2012.42.html + references in the book (bottom of p. 203), http://www.nature.com/nnano/journal/v7/n5/full/nnano.2012.66.html

ix) Qubits and mechanical resonators – Sec. 11.4 + references 
- Explain the idea of “macroscopic quantum mechanics”, and show how this could be studied with mechanical resonators; explain the scheme of measuring superpositions in vibration states
- References: http://www.nature.com/nature/journal/v464/n7289/abs/nature08967.htmlhttp://www.nature.com/nature/journal/v494/n7436/full/nature11821.htmlhttp://www.nature.com/nphys/journal/v8/n5/abs/nphys2262.html

Two people took superconducting qubits. They will each focus on a different specific realization. It is a pity nobody chose spin torque – I would have liked to learn a bit more about it :-) Indeed, my motivation is to learn about the latest developments related to these topics via the student projects, although of course the primary goal is that the students get some kind of an idea about the different systems studied in the field. I haven’t done this before related to this course, so let’s see how it goes. In fact, I did something very similar when I was tutoring a course in quantum computing, I think the year was 2001. The lecturer of the course was Mikio Nakahara, but I was probably more aware of the different realizations. Via the students’ projects we compiled a booklet introducing the different suggested realizations of quantum computing. As far as I understand, part of this work then influenced later Mikio’s book on quantum computing – half of it is on the physical realizations.

Formalism…

Today I discussed the formal prerequisites of the course, described in Appendix A of the book. Unfortunately I spent so much time in second quantization that I did not manage to explain Fermi Golden Rule, or magnetic field, and the time spent for chemical potential/Fermi energy was a bit short.

On preparing for the lecture I noticed that in the magnetic field section I should have explained a bit more how the Lagrangian connects with the Lorenz force, as the connection is not entirely trivial. Also, I noticed that the step between eqns (A.52) and (A.53) is a bit non-trivial.

I made a few clicker questions on the topics: For example, the task was to calculate the commutator between bosonic and fermionic number operators. In both cases it vanishes… This seems indeed to be a bit non-trivial point for the students.

Besides, I made a few videos explaining the second quantization. If there are requests for this, I can publish both the videos and my clicker questions on this website.

Blogging about the course

I will lecture a course based on the book during Fall 2014. The first lecture is in fact today. The public course page is here.

My idea is to give lectures on the main topics of each chapter, and in addition give special projects to the students on various aspects of nanoelectronic systems. The list of projects is here, and the relation to the book chapters/sections explained with each project:

Possible projects

i) Spintronic devices: spin valve (CPP and CIP), racetrack memory, spin torque RAM (pair project) – Sec. 2.7 + references
ii) Spin torque theory – Landau-Lifshitz-Gilbert equation – references
iii) Thermoelectric devices and measuring techniques – Sec. 2.8 + references
iv) Thermoelectric properties of quantum dot systems – Sec. 2.8, Ch. 8 + references
v) Quantum Hall effect, Landau levels and electron optics (pair project) – references
vi) Green’s function modeling of nanodevices – Complement 3.1 + references
vii) Cross correlations and Bell’s inequalities measured in small conductors – Sec. 6.5 + references
viii) Primary thermometry with shot noise and Coulomb blockade – Sec. 6.2, 7.6.1 + references
ix) Coulomb blockade used for charge detection, rf-SET – Ch. 7, references
x) Charge pumping and SI standard for current – Sec. 7.6.2 + references
xi) Spin qubits – Sec. 8.5.4 + references
xii) Kondo effect (in quantum dots) – Sec. 8.4 + references
xiii) Superconducting qubits (pair project or alone one realization) – Sec. 9.4.3 + references
xiv) Circuit quantum electrodynamics (quantization of electronic circuits) – references
xv) Graphene pn junctions: Klein tunneling and other physics – Sec. 10.3
xvi) Multilayer graphene structures – Sec. 10.2
xviia) Rashba and Dresselhaus spin-orbit interaction (part of the pair project) – references
xviib) Topological insulators, quantum spin Hall effect  (part of the pair project) – references
xviii) Nanoelectromechanical force detection – beginning of Ch. 11 + references
xix) Optomechanics – Sec. 11.3 + references
xx) Qubits and mechanical resonators – Sec. 11.4 + references
xxi) Weak localization and weak antilocalization; normal conductors and graphene – Sec. 4.2
xxii) Suggest your own project