Nanoscale Superconductivity

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-T_c 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 = k_F) 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 T_c, 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.

Talks Part 2: Imaging Spins

Imaging electrical spin injection/transport in spintronics devices: Scott Crooker (Los Alamos)
This was another cool (and very understandable) talk based on recent work on trying to understand the physical processes involved in spin injection and transport in lateral feromagnet/semiconductor structures. In a seminal paper in 1990, Datta (no relation to me) and Das proposed one of the earliest versions of a spin-FET: that is, a field-effect transistor made from doped silicon (versus the carbon nanotube FETs we make in our lab on a regular basis) with ferromagnetic contacts. The point, of course, is that the functionality of the device is to come not from coupling to the charge of the electron, but to its spin degree of freedom. It’s just a very cool idea, and people have taken it pretty far since then (although I’m not sure that industry will be ’switching’ to spin-based transistors anytime soon. Get it - switching to transistors? Hilarious.) Although many, many proposals currently exist, they all need certain things: a way to electrically inject spin-polarized electrons into the semiconducting channel, a way for these spins to be transported, a way to controllably manipulate the spins (i.e. with an external field), and a way to electrically detect this spin-polarized current. In particular, one of the key ways of confirming this electrical detection is using the ‘Hanle effect’ due to precession and dephasing of the spins in a transverse field.

Although this had been observed in all-metal devices (including the channel), a number of subtleties prevented a similar observation in semiconductor devices, until Crooker et al.’s work. What they did was use scanning Kerr rotation microscopy (using a continuous-wave (cw) probe laser and a sample resting on the cold finger of an optical cryostat, with an applied transverse field) to measure the out-of-plane component of the spin, and sure enough, they were able to obtain a Hanle signal. They extended (and continue to extend) this in a number of ways, comparing their data to a drift-diffusion model, injecting spins optically and seeing how the conductance changes, and even studying the effect of an applied strain (which interestingly leads to a term in the Hamiltonian that looks like a Rashba spin-orbit interaction with E replaced by the strain). A number of questions remain to be answered, but this work represents an interesting step forward.

Further reading…
- S. A. Crooker et al., “Imaging Spin Transport in Lateral Ferromagnet/Semiconductor Structures“, Science 309 2191 (2005), X. Lou et al., “Electrical Detection of Spin Accumulation at a Ferromagnet-Semiconductor Interface“, PRL 96, 176603 (2006), and X. Lou et al., “Electrical Detection of Spin Transport in Lateral Ferromagnet-Semiconductor Devices“, Nature Physics 3, 197 (2007).
- Some of the first work on strain-induced effects: G. L. Bir and G. E. Pikus, “Symmetry and strain-induced effects in semiconductors” (Wiley, 1974) - can be found on BorrowDirect.
- First spin-FET: S. Datta and B. Das, “Electronic analog of the electro-optic modulator“, APL 56, 665 (1990).
- Somewhat related: chapter 2 of D. D. Awschalom, D. Loss and N. Samarth eds., “Semiconductor Spintronics and Quantum Computation” (Springer 2002).

Philosophia Naturalis #8

This post was delayed by a number of ridiculous technical mishaps, but issue number eight of Philosophia Naturalis - the physics blogosphere’s very own blog carnival - is finally here. There were a number of very interesting submissions spanning pretty much everything that is involved in physics and the practice of it, and so I’ve split them up accordingly. Enjoy!

Cool Experiments
Motivated by some recent quantum optics work recording the birth and death of microwave photons in a superconducting resonant cavity by a French group, Chad Orzel has proclaimed this to be “the golden age of experimental quantum optics”. And indeed, it seems to be: two other such experiments include this study of the Hanbury-Brown-Twiss effect, and this more recent realization of the delayed-choice experiment first proposed by John Wheeler in 1978. A closely related experiment is that of the ‘quantum eraser’ proposed by Scully and Drühl in 1981, and this post does an excellent job of summarizing the general principles.

On the opposite end of the size spectrum, Cosmic Variance recently hosted a very interesting discussion on some analysis of cosmic microwave background radiation data from WMAP (NASA’s much-publicized effort to very accurately measure the temperature of the CMB).

Interesting Theory
As tends to be the case, most posts were of a more theoretical bent. The mapping of the E₈ Lie group recently received a good deal of press, including a rather vague article in the New York Times (the gist of which was something along the lines of “a bunch of mathematicians did something really complicated involving a pretty picture, and apparently it has profound implications but we’re not exactly sure what.”) Mark Chu-Carroll and John Baez have taken a different approach, recently posting about the actual math involved and the whole point of the project. And speaking of recent math-y work that has received a good deal of press, these two posts report on this paper by Peter Lu and Paul Steinhardt (who used to be at Penn!) on signatures of quasicrystalline Penrose tilings in medieval Islamic architecture.

Penrose is an excellent segway into two posts by Scott Aaronson. The first poses the question: “what’s the connection between a black hole having an event horizon and its having a singularity? In other words, once you’ve clumped enough stuff together that light can’t escape, why have you also clumped enough together to create a singularity?” (This is related to the Penrose-Hawking theorems of general relativity). The second (or rather, the subsequent comments) deals with possible connections between the brain and quantum computers, something Roger Penrose has discussed in a good deal of depth. (Matt Leifer has a similar post, asking the question: “if quantum computers are more efficient than classical ones then why didn’t our brains evolve to take advantage of quantum information processing?“)

There’s more out there, too: see, for example, this post discussing the much-storied Bayesian theorem and connections to Bell’s inequalities, or this post on ‘biophysical economics’, an economic theory rooted in biological and physical realities first put forth in the 20’s. Something that struck me as being particularly interesting was this post on the use of evolutionary algorithms in lattice QCD simulations. Meanwhile, Ponder Stibbons has been plowing through Huw Price’s book (Time’s Arrow and Archimedes’ Point) on some of the more philosophical questions of physics, with posts on Price’s objection to dynamical explanations of entropy increase (”they can never account for the asymmetry in our observations unless they themselves have asymmetric assumptions”) and a modern-day version of Olbers’ paradox.

And of course, a good deal of very interesting physics (albeit of a different sort) goes into fields of inquiry that some would consider unconventional, like geophysics. These two posts dealing with earthquakes and volcanoes touch on this to a certain extent. The latter is particularly interesting, looking into the various possible triggers for volcanoes (and drawing connections between large earthquakes and volcanic eruptions, motivated by a fictional account of Charles Darwin’s journey on The Beagle).

The Culture of Physics
Speaking of geophysics, Jennifer Oullette has written about a talk at the recent APS March Meeting on large-scale pattern formation in geological systems, citing some work by Meredith Betterton (who gave a talk here at Penn on an unrelated subject recently) on the creation of artificial spiky ice formations. (March Meeting is an event when thousands of physicists get together and tell each other about what they’re working on - held, incidentally enough, in March.) A number of people have posted about various events at March Meeting; see, for example, this other post by Jennifer Oullette, this post by Travis Hime, and this one by Doug Natelson - or see the PhysicsWeb blog.

Having huge meetings and partying like rock stars isn’t everything, though. Among other things, the physics community (just like any other) has its share of scandals, politics, marketplace tactics, things of that sort. Sabine Hossenfelder, for example, has recently blogged about the problems of treating the scientific community as a marketplace, while Julianne Dalcanton’s post on physics’ “cult of genius” definitely touched a nerve among readers. Meanwhile, Clifford Johnson has shared his views on recent events regarding an imprisoned theoretical physics grad student. (And of course, there’s the media aspect of things: John Conway recently picked up on his two previous posts on the search for the Higgs boson to blog about the unexpected media response.)

Communicating Physics
Tommaso Dorigo recently posted about some of the problems associated with the way physicists communicate things to laypeople, dealing specifically with an example from high energy physics (what is a lower limit at 95% confidence level, anyway?). At the end of the day, the physics blogosphere’s rather good with this kind of thing. For example, ‘basic concepts’ posts (like the ones mentioned in this excellent post, or this one - part of a series - on special relativity) do an excellent job. And hey, communicating physics is kind of the whole point of this blog carnival, in a sense. I think that’s where I’ll end things - hopefully it’s been interesting. Thanks to everyone who submitted either their posts or someone else’s.

High-Energy Materials Science

Someone I know recently asked me about this recent New Scientist article, and honestly, I’m not sure what to make of it. To summarize the article, ‘string-net condensation‘ (a model in which “electrons are not really elementary, but are formed at the ends of long ’strings’ of other, fundamental particles”) predicts interesting new phases of matter in certain spin models, and recent experiments on the amusingly-named Herbertsmithite may be a signature of one such phase of matter.

The subject matter is definitely interesting, although I am in no position to comment on the actual science (especially the theory, because I don’t understand it - my very little exposure to renormalization group theory so far has been in the context of statistical mechanics). On the experimental side, there is no doubt that recent work studying material properties of this system (such as looking at the specific heat or the temperature-dependence of the magnetic susceptibility, as well as using inelastic neutron scattering) is yielding some very cool physics. And of course on the theory side of things, it is clear that being able to come up with a unified theory from which electrons and photons are emergent is a Big Deal. The problem I have is that while the New Scientist article makes it sound like these measurements are a clear signature of a new phase of matter as predicted by string-net condensation, I can’t really discern the degree to which the link between the theory and experiments is scientifically rigorous, at least from reading the relevant papers. Rather, the article strikes me as being yet another example of really bad sensationalist journalism (see this post by Doug Natelson for more).

However, I find this to be a nice example of how things like high-energy physics theories of string-net condensation or quantum electrodynamics are becoming increasingly important in the study of materials. The ‘hot’ material for this kind of thing these days is obviously graphene, the “new bridge between condensed matter physics and quantum electrodynamics“. Another surprising example of this, for example, is this recent paper connecting SU(2) Yang-Mills theory in the low-temperature phase to nematic liquid crystals (who’d have thought? Certainly not me - I found it amusing that the only equations in the paper I actually understood were 18-20, the ones relating to liquid crystals).

And of course, this Herbertsmithite stuff is an example of another big thing in condensed matter physics - namely, ‘discovering’ new states of matter (at least theoretically). A lot of interesting physics is coming out of this effort, such as this recent work by Shoucheng Zhang at Stanford.

Quantum Information, et cetera

Today’s been a pretty exciting day. Among other things, we finally used all these abstract concepts relating to modules that we’ve been developing in my algebra class to derive some really neat results: in particular, rational canonical form and Jordan canonical form for matrices that can’t be diagonalized. There’s just something about taking all this seemingly useless theory and deriving something nice (and not so obvious) that you can actually use from it that’s very satisfying. Hey, I may even blog about it at some point. Another interesting math idea today was the subject of a colloquium that I wasn’t able to attend (but found out a good deal about from those who did): can we hear the shape of a drum? (Among other things, the question ties in with work done by our Dean, NASA’s WMAP project/this PRL, and the general notion of inverse problems such as those people deal with in things like MRI). And of course, my abstract algebra recitation session turned into me arguing with my TA (an algebraic geometer) and a computer scientist about why statistical mechanics is The Coolest Thing ever. What can I say? Never get me started on statistical mechanics - it’s just such a gorgeous subject, and I can’t get enough of it.

Anyway, while the math colloquium was going on, I was off at today’s physics colloquium by Prof. Charles Marcus of Harvard University, something I’ve been looking forward to for a good deal of time now. And what a talk it was: although I would have preferred more technical details, the talk catered to a pretty general audience, and it was perhaps the clearest physics talk I’ve been to in a long time. He started off by reviewing the history of computation (from the Antikythera mechanism of 150 BC to the first integrated circuit, 50-ish years ago), noting that quantum information processing is really a new paradigm in this history (to paraphrase, the parallelism in computation implied by the multiplicity of states inherent in quantum mechanics is something that hasn’t really been possible till now) and surveying recent developments in solid-state implementations of controllable qubits from his lab. Most of the relevant papers are on his lab webpage (linked above), and a lot of the technical details are presented in this talk he gave at KITP in 2006, although he did present some very recent data pertaining to this paper. All in all, a very cool talk - I especially like the terminology in this field, what with the ‘Zamboni’ effect and ‘bucket brigades’.

I can’t wait for next month’s colloquium - David Nelson will be speaking!