Cool Papers 1: General

I’ve come across a number of pretty cool papers in the past few months. Some of them deal with particular phenomena (stay tuned for possible upcoming posts on molecules at surfaces, biomimetics, phononics, crystallization, nanoparticles, wetting phenomena, computational physics, etc. etc. - at some point), and so are probably better off getting their own blog posts. Here are a few papers that didn’t fall into specific categories…

1. Frictional Anisotropy on a Quasicrystal Surface
Along with ~10 other things, a subject that I’ve recently become interested in is nanoscale mechanics, broadly defined. A key experimental tool in this field is the use of local probes to push or pull on things controllably. Miquel Salmeron’s STM group at Berkeley does work on this and related subjects, and I finally got around to reading this paper of theirs from a few years back.

The idea is conceptually very simple: while friction unsurprisingly depends on commensurability (that is, if two surfaces in contact are structurally ‘complementary’, they will ‘lock in’ to each other and hence have high friction between them - an idea that apparently dates back to da Vinci), trying to think about friction using just this notion is unrealistic. For starters, most contacting surfaces are probably incommensurate, and other factors - such as periodicity(?) - contribute, as well.

This paper nicely singles out the role of periodicity by looking at different directions along Al-Ni-Co quasicrystal surfaces using STM (to image the surface and hence distinguish the periodic and aperiodic directions of atom ordering) and AFM (to measure the probe tip-surface friction along these directions) in ultra-high vacuum. The AFM friction data can be modeled using a classical model relevant to the experimental situation (the Derjaguin-Muller-Toporov or DMT model, which I need to learn more about), enabling key parameters to be derived from the measurements.

In particular, the authors find a larger friction force (8x) along the periodic direction than along the aperiodic direction. Unsurprisingly, they ascribe this to differences in energy dissipation via electron or phonon excitation+propagation along the different directions, although it is unclear to what extent each kind of excitation plays a role. Perhaps similar local-probe measurements of a different kind (e.g. ones sensitive to electrical versus mechanical properties) might be useful… At the end of the day, I like this paper because it is an elegant example of using a unique microstructure, in which just one variable (here periodicity) changes in ways that are well understood, to study something interesting as a function of just that variable.

2. Liquid Crystals and the Origins of Life
Noel Clark gave a great talk about this work here at Penn not too long ago. I won’t write too much about this since Randy has a nice description of it over at the condmat journal club.

Here’s the executive summary: according to extensions of Onsager’s rigid-rod model for the formation of liquid crystal phases, individual molecules must be sufficiently anisotropic (i.e. the aspect ratio has to be above a certain minimum) to form a liquid crystal (LC). Surprisingly, the authors of this paper observed LC phases consisting of single-stranded (ss) DNA molecules too short to satisfy this criterion. Optical and x-ray measurements indicate that this results from end-to-end stacking of duplexes of complementary short ss-DNA molecules (known as ‘living polymerization’) into larger rods that satisfy the Onsager criterion, even at low temperatures (in concentrated phases of duplexes separated from the isotropic phase of unpaired ss-DNA molecules).

This autocatalytic behavior is like positive feedback, in a sense, and is why this work is so interesting from a biological point of view: it provides a mechanism by which the right molecules can be ’selected’ out from a ’soup’, and ‘evolve’ into larger ones as part of an RNA world. It’s an interesting idea - definitely one that’s gotten a lot of press, it seems - and while this work doesn’t provide much hard evidence for it, I’ll be interested to see what it stimulates.

3. Suprafroth!
This is a very interesting paper out recently on the arxiv, I think to be published in Nature Physics. While I don’t understand all the details, I like this particularly because it’s a nice combination of ideas from soft- and hard-condensed matter physics, like electronic liquid crystals.

The authors used magneto-optical imaging, which I need to learn more about, to image the flux pattern of superconducting lead (a type-I superconductor). Turns out that the magnetic field on the edge of a disc-shaped sample of lead is larger than the actual applied field, and for large enough magnetic field some flux can penetrate the sample. This leads to a phase intermediate between the normal and superconducting phases, possessing a froth-like magnetic structure - specifically, the froth cell boundaries are superconducting, while the interiors are normal metal. This shows up very clearly in the magneto-optical images (see figures in the paper).

The nice thing is that, unlike ‘conventional’ froths, mass-transport processes like drying or drainage are not present here (as the authors point out, “this superconducting froth involves only electrons”). This means that the froth structure can be tuned reversibly using the applied magnetic field or temperature, and the nice magneto-optical images allow for quantitative analysis of the froth structure as a function of just these parameters.

This is philosophically similar (loosely speaking) to paper #1 - the friction measurements of quasicrystals: again, it is a very nice example of using a unique microstructure (here, a froth structure that doesn’t suffer from irreversible processes, and can be controlled by magnetic field or temperature) to study something interesting (here, the structure and dynamics of froths) as a function of just the variables that you can control.

4. Universality in Conference Registration
This is a cute correspondence recently sent to Nature Physics describing an intriguing social application of statistical mechanics.

The authors used registration data from two physics conferences (# of registrants as a function of time to the deadline), saw that they matched up remarkably well (after rescaling), and came up with a simple model to capture the observed phenomenon in which the ‘pressure’ felt by potential attendees to register varies inversely with respect to the time to the deadline. Also, incorporating a Boltzmann-like factor (instead of uniform probability to register over the period of time) leads to a prediction that agrees well with # of payments as a function of time to the deadline data.

Of course, there are a number of assumptions and fitting parameters floating around here, and I’m not entirely sure this work will change the world of physics, but I always find things like this fun.

‘Hard’ measurements, ’soft’ materials

So it’s been what, a little less than two months since I last posted? I tend to work on many projects at once - some are ones I’ve been plugging away at for a while, while others are “let’s see what happens” experiments that I work on when I get the time, motivated by some half-brained idea. In particular, I’ve made significant progress on a project of the latter category, and the month-and-a-half has been spent making samples, furiously taking and analyzing data, trying to figure out what it means/delving through the literature, &c. - and of course, effectively disrupting any prospects of sleep or studying for pesky standardized tests. And making headway on my other projects, too. The good news is that I, for one, find the data pretty exciting.

(Oh, and moving to my sweet new apartment, which apparently scores a very respectable 98/100 on the walkability scale. Not too bad, especially given the relatively low rent.)

Anyway, when I haven’t been concentrating on my research, I’ve been reading up on things like organic semiconductors and STM modification of molecules (I suppose what one could call ‘hard’ condensed matter measurements of ’soft’ materials, although admittedly some of my own research falls into this genre). I find people like Paul Chaikin, Heinrich Jaeger and George Gruner particularly fascinating since they seem to be actively doing this kind of research in addition to hard condensed matter physics of the more ‘traditional’ kind (superconductivity/correlated electron systems…). I wonder how many other PIs do this kind of thing?

And of course, two new additions to the reading list: “charge transfer on the nanoscale: current status“, and “electrostatic modification of novel materials” - both hefty reviews of topics relevant to this post.

Also: Heinzel’s book on mesoscopic physics is a new addition to my list of the greatest books of all time - in particular, its clarity is unmatched by many other books I’ve come across on the subject.

A few thoughts

Clearly blogging has slowed down now that I’m back into the swing of research. Here are a few minor non-research things that have transpired…

Free coffee? While attempting to read a thesis by a professor here, I came across an interesting line in the acknowledgments in which he thanked “the labours of the coffee and tea pickers whose efforts kept me awake long enough to produce this document”. Here’s a thought: athletes and celebrities receive inordinate amounts of free stuff - and of course, money - to endorse certain products (I presume). Why can’t physicists and other scientists do the same? For example, if Red Bull or La Colombe ran full-page ads in Nature along the lines of “Ed Witten drinks Red Bull - do you?” or “Andrew Wiles: turning La Colombe coffee into theorems”, I’m sure their sales would increase significantly. (I venture that no other single demographic consumes more caffeine.) And of course, they could give the individual/individual’s department free coffee and/or funding in return. It’s a win-win situation.

De Gennes dies: There’s not much I can say that hasn’t already been said (see this NYT article, for example). I’ve had the pleasure of delving into two of his books, the seminal Physics of Liquid Crystals - note to self: learn more about the connections between superconductors and liquid crystals - and the perhaps lesser-known Petit Point: A Candid Portrait on the Aberrations of Science. The latter is a rather interesting book, with very short chapters describing fictional characters based on scientific individuals. The sole reviewer of the book on Amazon claims to be able to identify Benoit Mandelbrot, Brian Josephson and Bernd Matthias in the various characters; my own hunch is that the chapter on “Chazot” is autobiographical in nature (the last line, “…in the end, Chazot’s real vocation is perhaps to give talks to high school students”, pretty much gives it away).

Blog-related: Henry Cate of the Why Home School blog is kicking off a carnival of space, which is a great idea (don’t know what a blog carnival is? See here.) Here are the archives, here is this week’s carnival, here’s the announcement, and most importantly - here’s how to submit a post for inclusion. Go for it!

And, in other news, Arunn of n0noscience and Rod of Perfectly Reasonable Deviations have both tagged me as being a ‘thinking’ blogger, which is a wonderful honor. I’m supposed to link to five other blogs that make me think, but it’s tough; the best I can do is link to the list of blogs I follow when I can since they’re all interesting.

Information theory: Cover and Thomas’ Elements of Information Theory (2nd ed.) is a really, really good book. Sadly I haven’t been able to read as much of it as I’ve wanted to, but it’s been a fascinating fusion of mathematics, physics, and computer science.

Optical Phase Conjugation

I just posted about superconducting effects in nanoscale systems, and in particular, the phenomenon of Andreev reflection, and I forgot to mention something cool I came across a while back that I recently re-read: this paper by Carlo Beenakker (although it’s listed on arxiv, the pdf doesn’t seem to be of the actual paper; I read it in chapter 4 of this excellent book on mesoscopic physics, although some googling brings up a full pdf version here, which may or may not last.) Beenakker uses “the analogy between Andreev reflection and optical phase-conjugation to answer the question: why does a metal-superconductor junction have a resistance?” Apart from being a very clear and interesting way of looking at this process, the paper’s particularly relevant to me since we recently covered optical phase conjugation (by ‘degenerate’ four-wave mixing) in my modern optics class.

Simplistically, phase conjugation is a nonlinear process by which an electromagnetic wave is reflected as (or alternatively , which is why it is often referred to as being a time-reversal process). This is analogous to Andreev reflection for a number of reasons (the ‘pump’ photons in the four-wave mixing process are like Cooper pairs, the pump frequency is like the Fermi energy, and the excitation energy corresponds to the frequency difference between the pump beams and the incident ‘probe’ beam). If the analogy did fully hold, one would expect the normal metal to be disorder/resistance-free, just as a disordered medium appears transparent when back by a phase-conjugated medium - the phase-conjugated light gets rid of aberrations due to inhomogeneities. The point is that the analogy fails because of the extra phase shifts involved in Andreev reflection processes, which explains in a sense why normal metal-superconductor junctions aren’t fully transparent. I’m not sure if there’s any more explanatory power that can be extracted from this analogy, but it’s definitely a cool way of tying together these two processes.

References:
- Where I first learned about optical phase conjugation: section 7.2 of R. W. Boyd, Nonlinear Optics 2nd ed. (Elsevier, 2003).
- The original four-wave mixing phase conjugation paper: A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing“, Optics Letters 1 16 (1977).

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.