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 
, the charging energy is 
and the dot-electrode coupling (say it’s the same for both electrodes) is 
, and if 
, then one can consider three regimes of dot-electrode coupling: 
(weak coupling), 
(intermediate coupling), and 
(strong coupling). As they say, it’s all about contacts, and ultimately, these three kinds of contacts give rise to different physics. For example, transport can be dominated by Coulomb effects, giving rise to ballistic transport in which the maximum conductance is given by 
(or, more precisely, 
where 
and 
are the transmission and reflection coefficients of the contacts (for carbon nanotubes, this is doubled due to the two-fold degeneracy of the graphene bandstructure). On the other hand, for different dot-electrode coupling, other effects may come into play (such as Kondo / Fabry-Perot / Fano resonance effects). I wrote about the physics of Fabry-Perot interferometers in a previous post; in this case, the nanostructure is the interferometer, with the contacts playing the part of the mirrors. This is nicely illustrated in figure 1b of the Jarillo-Herrero et al. nanotube paper (for 
i.e. ‘normal’ electrodes). The key point of all this is that the electrode-nanostructure coupling is important in devices with superconducting electrodes, as well. In particular, having transparent contacts is key for the measurements done below 
.
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 
) is needed to disrupt this condensate and an energy gap 
is produced in the spectrum of free electron allowed energy states (a ‘gap to excitations’, so to speak).
Anyway, the distance over which the two electrons are coupled (the coherence length 
of the Cooper pair) may be many orders of magnitude larger than the lattice spacing itself i.e. several hundred nanometers (in which case, repulsive Coulomb interactions between the two are quite small). In a superconductor-normal conductor-superconductor geometry, if this coherence length is longer than the length of the conduction channel, these Cooper pairs can phase-coherently tunnel through the normal material, leading to a Josephson supercurrent. For example, in the absence of any potential difference between the superconducting electrodes, this is given by 
through the device, where 
is the phase difference of the wavefunctions in the electrodes (see chapter 6 of M. Tinkham’s book on superconductivity for details). 
is known as the critical current, and although the Josephson supercurrent picks up an oscillatory component when a dc voltage is applied across the device, Ambegaokar and Baratoff showed that 
, where 
is the normal-state resistance of the device. That is to say, this product is a constant, independent of sample geometry.
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 . Can a Josephson supercurrent flow in the nanostructure (the weak link)? How does the discretized nature of the electronic states affect this? It turns out that a dissipationless Josephson supercurrent does flow, due to the ‘proximity effect’ that occurs when the superconducting leads are so transparent so as to allow leakage of Cooper pairs from the superconductor into the nanostructure. Surprisingly, given a low enough temperature, this current can flow in the normal conductor over an arbitrarily long length (see this paper, available for free here), and these experiments are able to study the effects of the discrete quantum dot energy levels on the Ambegaokar-Baratoff relation that governs this supercurrent in a ‘conventional’ Josephson device. They tie in their results with theoretical predictions (such as those of Beenakker and van Houten published here and here), and a key point is that 
is tunable (via the gate voltage) in a controllable manner, which is exciting – see, for example, figure 4a of the Xiang et al. Ge/Si nanowire paper.
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 
– the energy picked up by the electron as it travels from one lead to the other (
is the bias voltage across the nanostructure weak link) – must also be formed, with an equal and opposite momentum. This is what’s known as Andreev reflection, and this process continues (hence the term multiple Andreev reflections) until the particle bouncing back and forth has more than in energy. It can then enter the superconductor as a ‘normal’ electron in the first excited state. This leads to an important relation: if a charged particle is reflected (that is, it traverses the junction) 
times, features will appear in the I-V characteristic curve (for a given gate voltage ) when 
i.e. 
. And indeed, this is what is seen in these experiments, although the reason why MAR is observed for certain values of and isn’t for others seems to be unclear at this point.
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 
junctions made using InAs nanowire or carbon nanotube Josephson junctions. All in all, this seems to be a very exciting field right now, and it’ll be interesting to see how it continues to develop.
for 
? My response to all these was: who cares?!
