Applied Mechanics News

Friday, April 14, 2006

Ramblings on Solid Mechanics-Quantum Mechanics Link

In between writing proposals (—which apparently is what assistant professors mostly do now-a-days!), I read the entry by Professor Zhigang Suo on our identity as mechanicians. I found it very interesting and thought provoking. My senior mechanics colleagues inform me that this topic has, over the time, quite frequently been a subject of debate—as evident in Professor Budiansky’s Timoshenko speech in 1989. Zhigang’s particular blog post led me to think of the “mechanics” in another big mechanics area: “quantum mechanics”. While as a mechanics group we have been very active in seeking inroads into diverse fields (e.g. bio, electronic) it is interesting to note that with a few exceptions we have left untouched this fertile research area of quantum mechnics. There is a lot of “solid mechanics” to be done in “quantum mechanics”. A simple example is how mechanical strain impacts the band structure and hence opto-electronic properties of the exotic quantum dots. Ben Freund from Brown and Harley Johnson from UIUC were perhaps the first mechanicians to foray into this and, despite its extraordinary importance to nanotechnologies, applied and fundamental physics, only a few other mechanicians have since looked further into this. This example is symptomatic of a bigger issue. In recent years, physicists have become very preoccupied with mechanical effects and their coupling to quantum mechanical phenomena (and not just in quantum dots). Many essentially have become elasticians in a different guise; same subject but speaking a different language! –as evidence by numerous elastic effects articles that now appear in physics journals. Sometimes the solid mechanics is handled correctly but, in many instances, it is quite clear that the weight and might of decades of progress in solid mechanics (not readily on the radar screens of physicists) could be brought to bear on some of these problems.

Of course a major hurdle is that most mechanicians are not routinely trained in quantum mechanics. I contend though that if we can learn elasticity, quantum mechanics is yet easier! As an interesting aside, I note a beautiful and relatively lesser known paper by the famous mechanician, J.D. Eshelby (“The Interaction of Kinks and Elastic Waves”—Proc. Royal. Soc. Lond. A, Vol 266, n1325, 222, 1962). In this work he analyzes the movement of a dislocation kink. After a “conventional” mechanics approach---employed in typical Eshelby style to “extract maximal insights with minimal work”, he proceeds to analytically solve the same problem using quantum mechanics. That brief section in his paper, while perhaps obvious to many, was an epiphany moment for me that forever provided a link between the quantum and continuum world.

Although strongly biased (and constrained by my limited knowledge) I list here a few topics (among many others) that I think straddle quantum mechanics and elasticity. Hopefully other blog-members can add to the list or make one of their own fashioned after their own interests.

(1) Mechanical strain effects in quantum confinement in quantum dots, wires and wells—the easiest starting point is the book by Davies, (The physics of low dimensional nanostructures). Extensive work is available on strain effects on quantum dots in archival literature. In the mechanics community, Johnson and Freund have a few articles that are a good starting point. I have a review article on my webpage but that unfortunately only talks about strain calculations and not really the coupling to quantum effects. To my mind, given the all pervasive effects of strain in nanostructures in general, quantum dots are where solid mechanics meets head-on with quantum mechanics.

(3) A somewhat related issue to #1, is spin manipulations using strain. Traditionally, spintronics or possibly making advanced computers using electronic spin has been based on the use of magnetic fields. Two nice articles appeared in Nature that suggest how to manipulate spintronics using elastic strain: Flatte, “Relativity on a chip”, Nature, Vol 427, p 21, January 2004 and, Kato et. al., “Coherent Spin Manipulation without Magnetic Fields in Strained Semiconductors”, Nature, Vol 427, p 50, January 2004.

(3) Despite the passage of almost seventy years (when this topic first emerged), the precise definition of “quantum stress” still remains controversial. Here I am not referring to the so-called virial definition of stress used in empirical molecular dynamics but a quantum notion of stress. There is certainly a lot of scope for mechanicians to weigh in on this matter. The landmark paper on this appears to be, Nielsen, O.H. and Martin, R.M. ” Quantum-mechanical Theory of Stress and Force.” Phys. Rev. B 32(6), 3780-3791, 1985. The paper apart from clarifying several issues also gave rise to some controversies (some of which still remain unresolved as far as I can tell especially the issue of uniqueness of stress). Interested readers my wish to look at a review article I co-authored recently on this topic (available on my website). More recently, Rogers and Rappe have made a nice attempt to provide a geometric definition of the quantum notion of stress: Rogers, C.L. and Rappe, A. M., “Geometric formulation of quantum stress fields.” Phys. Rev. B 65(22), 224117 -224124, 2002

(4) What happens to Helium close to 0 K? At such small temperatures, liquid helium becomes a so-called “super-fluid” i.e. it can flow through narrow pores without resistance. A more exotic “supersolid” phase has been predicted theoretically with some controversial experimental evidence. Elasticity is expected to play a major role although I have not seen anything yet which puts this issue to rest—perhaps a mechanician can oblige? A good starting point and recent reference is: A. T. Dorsey, P. M. Goldbart, and J. Toner, ``Squeezing superfluid from a stone: Coupling superfluidity and elasticity in a supersolid,'' Phys. Rev. Lett. 96, 055301 (2006)


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