I spent some time several weeks ago (over spring break) reading up on nanoscale superconductivity, Andreev reflection, all that fun stuff, and we talked about a few of the relevant papers at journal club recently. Here are some of my notes, summarized.
The 20th century saw two huge milestones in the study of electronic properties of materials in the discovery of superconductivity, and the recently burgeoning field of nanotechnology. Ever since its discovery in 1911, superconductivity has been the subject of an enormous amount of research in physics, the results of which have often been surprising, new, and very important, to say the least. While few applications of superconducting phenomena may exist (partly because of the temperature scales involved - see this or this for a review of the history of high-Tc superconductivity and where it’s potentially headed), Josephson junctions are a famous example of a phenomenon of superconductivity that has found many applications (for example, SQUIDs or physical realizations of qubits). On the other hand, fueled by a lot of promising work in low-dimensional electron gases, carbon nanotubes, and nanowires, nanotechnology has undoubtedly emerged as one of today’s hottest fields. True, the field is rife with hype, but the physics is pretty interesting too, and attempting to engineer and understand nanoscale electronic devices (like nanoscale FETs) is an exciting prospect.
One is led to ask: what happens to superconductivity at the nanoscale? After all, while nanoscale FETs could potentially push Moore’s Law much further and enable incredibly powerful computers (or so they say), superconducting nanoscale electronic devices would potentially revolutionize electronics in further unimaginable ways. Could the unique phenomena associated with superconductors be coupled with those of nanostructures? For example, transistors are important in digital circuits because they utilize the electronic properties of semiconductors and can thus be used as switches. Nanoscale FETs attempt to scale this down by contacting a nanostructure (the ‘channel’) with metal electrodes and modulating the carrier density in the channel via a gate voltage. Could it be possible to somehow retain this switching behavior of nanoscale FETs, while making the currents which flow in them dissipationless? A number of papers - three from the Delft group, one from Cambridge - dealing with this question came out at the end of last year, demonstrating significant progress towards integrating semiconducting nanostructures (in particular, carbon nanotubes, n-type InAs nanowires, Ge/Si core/shell nanowires heterostructures, and graphene) with superconducting materials.
In particular, the sample geometry is such that the nanostructure acts as a ‘weak link’ (as in a Josephson junction), and thus a supercurrent – that is, a current characterized by zero resistance – is found to flow through it. The geometry is similar to conventional FET geometries: the nanostructure bridges two conductive electrodes (a superconducting metal such as Al) which act as a source and a drain when a suitable voltage is applied across them; the electrodes are typically deposited using optical or electron-beam lithography and microfabrication/etching techniques coupled with metal evaporation techniques. The nanostructure then acts as a conduction channel that can be tuned via the electric field effect of a highly doped Si back gate separated using several hundred nm of SiO2 dielectric (or, in the case of the Ge/Si NW paper, using a top-gate separated using a high k HfO2 dielectric) - that is, a quantum dot.
Prof. Dr. Christian Schönenberger has recently written a very nice review of charge and spin transport in carbon nanotubes that is available here. A number of ideas from the paper are relevant to this post, and are nicely encapsulated in its figure 4. A key property of quantum dots is that their energy levels are discretized. If the separation between these is 
What happens in this regime? Among other things, the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity utilizes the concept of Cooper pairs to develop a quantum mechanical model for superconductivity; here’s an elementary view. Cooper pairs are correlated pairs of conduction electrons of opposite spin. The coupling between the electrons is due to phonons in the crystal lattice, or as one may say, the exchange of ‘virtual’ phonons (since they only exist during the exchange). Simplistically, one can think of a single conduction-band electron attracting nearby positively-charged lattice ion cores, thus creating lattice distortions (phonons) and regions of net positive charge to which another electron will be attracted. Cooper showed in 1956 that the effective interaction between pairs of electrons just outside the Fermi surface (k = kF) is such that this bound state forms, particularly when they have equal and opposite momenta. This was a very surprising result.
Since the net spin of a Cooper pair is zero, it acts as a boson (with a lower energy than the two individual electrons), and when a superconductor is cooled below Tc, the Cooper pairs ‘condense’ (similar to Bose-Einstein Condensation) into a correlated electron state characterized by a macroscopic quantum mechanical phase. A finite amount of energy (equal to the Cooper pair binding energy 
Anyway, the distance over which the two electrons are coupled (the coherence length 
At the end of the day, the key question is whether these phenomena carry over to these superconductor-quantum dot-superconductor systems, in which the length of the nanostructure is made to be smaller than 
When considered in the framework of Blonder-Tinkham-Klapwijk (BTK) theory, the existence of this supercurrent is intimately related to the notion of Andreev reflection (AR) or multiple Andreev reflections (MAR). A schematic of this is above, adapted from W. Belzig, Nature Nanotech. 1, 168 (2006). Since electrons with energy within the BCS energy gap can’t exist in the superconducting electrodes (as I discussed a bit earlier) when the contacts are sufficiently transparent, electrons in the nanostructure weak link can’t simply enter the superconducting electrodes (and vice versa). The only way for electrons of a given to enter the superconducting electrodes is by forming Cooper pairs. Charge/momentum conservation decree that a hole (or electron) of energy 
And there’s way more to talk about. For starters, there are a large number of previous experiments upon which these build, such as this 1999 paper from Stanford on the gate-controlled proximity effect in nanotubes or the initial experiments on proximity-induced superconductivity in nanoscale systems by this French group (in single-walled nanotubes as well as nanotube ropes, DNA, Gd metallofullerenes, altering the effect in nanotubes using organic polymer coatings, and most recently, observing proximity effects in few layer graphene). It isn’t clear to me what the status of some of these papers are or to what extent this kind of stuff has been reproduced, but it seems interesting. And of course, there’s more data in these experiments that I haven’t talked about (for example, further electronic structure can be probed using an external microwave field, giving rise to ‘Shapiro steps’ as outlined in section 6.3.4 of Tinkham’s book; or for example, the observation of a bipolar supercurrent in graphene). There are also other recent papers that I haven’t mentioned, such as this one or this one on tunable 
quite simply - since we’re treating the particles as hard spheres of radius 
, the associated potential is given by 
for 
and 
for 
. Then the 
-sphere partition function is just:
and what 
:![Z_{N} \sim \displaystyle\int_{0}^{L - (N-1)2R} dy_{N} \displaystyle\int_{0}^{y_{N}} dy_{N-1}\textit{...} \displaystyle\int_{0}^{y_{3}} dy_{2} \displaystyle\int_{0}^{y_{2}} dy_{1} = [L - (N-1)2R]^N](http://l.wordpress.com/latex.php?latex=Z_%7BN%7D+%5Csim+%5Cdisplaystyle%5Cint_%7B0%7D%5E%7BL+-+%28N-1%292R%7D+dy_%7BN%7D+%5Cdisplaystyle%5Cint_%7B0%7D%5E%7By_%7BN%7D%7D+dy_%7BN-1%7D%5Ctextit%7B...%7D+%5Cdisplaystyle%5Cint_%7B0%7D%5E%7By_%7B3%7D%7D+dy_%7B2%7D+%5Cdisplaystyle%5Cint_%7B0%7D%5E%7By_%7B2%7D%7D+dy_%7B1%7D+%3D+%5BL+-+%28N-1%292R%5D%5EN+&bg=ffffff&fg=000000&s=-2 "Z_{N} \sim \displaystyle\int_{0}^{L - (N-1)2R} dy_{N} \displaystyle\int_{0}^{y_{N}} dy_{N-1}\textit{...} \displaystyle\int_{0}^{y_{3}} dy_{2} \displaystyle\int_{0}^{y_{2}} dy_{1} = [L - (N-1)2R]^N")
and 
, where in this case the ‘volume’ 
, one finds that![P = \frac{\partial}{\partial L} \left\{ k_{B}TN\ln [L - (N-1)2R] \right\} \simeq \frac{Nk_{B}T}{L-2NR}](http://l.wordpress.com/latex.php?latex=P+%3D+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+L%7D+%5Cleft%5C%7B+k_%7BB%7DTN%5Cln+%5BL+-+%28N-1%292R%5D+%5Cright%5C%7D+%5Csimeq+%5Cfrac%7BNk_%7BB%7DT%7D%7BL-2NR%7D+&bg=ffffff&fg=000000&s=-2 "P = \frac{\partial}{\partial L} \left\{ k_{B}TN\ln [L - (N-1)2R] \right\} \simeq \frac{Nk_{B}T}{L-2NR}")
