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