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# Journal Club Theme of September 2007: Quantum Effects in Solid Mechanics

Since the early 1990s, when **quantum dots** and quantum wires began to attract the attention of physicists, and when **carbon nanotubes** were discovered, mechanics related issues have begun to emerge as important in understanding properties of nanostructures. These structures were first considered useful mostly for their electronic or optical applications, yet deformation has been seen to play an important role in their functional characteristics. Advances in modeling also have begun to link electronic structure with mechanical properties of materials at larger length scales, particularly when microstructural or crystallographic effects influence bulk behavior.

The question of how* solid mechanics, through the effects of deformation, connects to the quantum mechanical behavior of electrons in a solid* arises from some of these considerations. The usual way to approach the problem is to note that a crystalline material can be characterized by an electronic energy band structure, from which many electrical, optical (and even thermal or mechanical) properties can be derived. When the crystal is strained or otherwise mechanically perturbed, for example, through the presence of defects, the bandstructure changes. If one can write down a constitutive relation for this coupling, a complete model may be possible. There are several complications to consider, however. For example, in confined geometries, electrons are not always best described by bulk bandstructure, because their properties may be dominated by quantization effects due to the boundaries. Also, the connection between bandstructure and deformation is a two-way coupled problem, in general, so that deformation may affect bandstructure, and changes in electronic structure may also induce strain. With these complications in mind, one must confront many subtle issues in building models for quantum effects in solid mechanics. For example, what level of accuracy is necessary to understand deformation at this scale? Is continuum mechanics sufficient, or should one adopt an atomistic approach, and if so, is an electronic structure method necessary? Can quantum mechanical effects be understood using a homogenization technique such as an effective mass model, or should one consider every electron in the solid? These issues impact a wide range of interesting problems in applied physics.

Three simple questions are posed here to better frame the discussion for this edition of the journal club. The issue of quantum effects in solid mechanics is important for each of these questions, and central to the journal articles selected for discussion. Each of the three representative articles has had significant scholarly impact in the short time since it was published. Readers may have differing opinions about the depth of the physics or mechanics in each of the three papers. Hopefully this will be a thread for discussion in the journal club. Here are the questions, and the papers chosen to represent them:

*1. How does strain affect optical properties at the nanoscale?*

In this work the authors compute the full strain tensor field arising from lattice mismatch between an idealized quantum dot and the underlying substrate. They then compute the effect of the strain distribution on the optical transition energies and find very good agreement between their model and the results of photoluminescence and optical absorption data from experiments. Many other authors have subsequently taken up issues addressed in this paper, including a number of mechanicians interested in accurately calculating strain distributions for such small scale structures.

*2. How does strain affect electrical conduction at the nanoscale?*

As in the first paper, the authors here consider the coupling between electronic structure and mechanical behavior in a simple, idealized nanometer scale structure. Here they model electrical conduction in deformed single-wall carbon nanotubes, using a molecular mechanics approach to study deformation. There have been numerous studies on this topic in the time since this paper appeared, including many analyses (atomistic, continuum, and multi-scale) of deformation in carbon nanotubes. But, like the first paper, this study is significant because it treats the problem of *coupling* between quantum mechanics and deformation.

*3. How does electronic structure influence elasticity?*

This work is different in spirit than the first two papers. Here the authors present details of a quasicontinuum method applicable for complex crystalline materials (such as silicon) where one is interested in connecting the underlying electronic structure with the macroscopic mechanical behavior. Their approach extends the method first presented by Tadmor, Ortiz, and Phillips to include an atomistic formulation based on tight-binding, which is one of the simplest atomic scale approaches that explicitly accounts for electrons. This work opened the door to many studies that have followed, including recent work linking density functional theory atomistics to continuum models via the quasicontinuum methodology.

Note: A student trained in elasticity and continuum mechanics needs to be familiar with basic solid state physics and quantum mechanics to really get into this area. Some quantum mechanics and solid state texts are fairly accessible to solid mechanicians, but for a reader with a mechanics of materials perspective, the book by **Phillips (Crystals, Defects and Microstructures, Cambridge, 2001)** is an excellent place to start.

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## Comments

## quantum mechanics meets solids mechanics

Harley, thanks for leading this discussion on this interesting topic. Indeed, quantum dots are where solid mechanics meets "the other big mechanics", quantum mechanics, head-on. I have some general comments below.

I am biased of course but I think that this research topic is likely to be of active interest for decades due to its wide range of applications from next generation lasers, solar energy, chemical and bio sensors, lighting, biological labels, quantum and optical computing among many others. Although there is an important role to be played by mechanicians in this research topic very few have ventured beyond the self-assembly and formation of quantum dots and into the actual coupling between quantum mechanics and solid mechanics. I hope your discussion thread will spur further interest. Perhaps, during some point this month it would be a good idea to compile a supplementary list of "tutorial" papers and documents that can provide a clear and facile path for mechanicians to get involved in this interesting field. In due course, I will provide a few such references as well.

The first paper that you mention is of course quite famous. Many years ago, after reading it, I spent quite some time revisiting strain calculations in small quantum dots in particular examining the accuracy of oft-used classical elasticity. It turns out that corrections to it, if any, are not too significant and certainly do not exhibit any qualitative differences.

Many mechanicians are getting interested in energy related topics and quantum dots are likely to be the basis for both next generation photovoltaics as well as thermoelectrics. Do you think "strain engineering" is likely to play a major role in these?

## nonlinear effects

Pradeep and Harley:

Thanks for getting the discussion started. One issue that has always bothered me: in many of these works where there has been optomechanical coupling, a small strain formulation is assumed. This bothers me because relaxation due to free surfaces for semiconducting materials is often on the order of percent, particularly when dealing with quantum dots and nanowires.

Is there any systematic work exploring the effectiveness of the optomechanical coupling when the strain is clearly in the finite deformation regime? Or how corrections to the linear(ized) approximation can be made? And how nonlinear strain impacts the observed bandgap modification?

## nonlinear effects

Thanks, Harold, for bringing up several interesting points.

I think there are two big questions here. First, do small strain formulations give us the right deformation fields, particularly when relaxation near surfaces is so big (which is different, of course, from another potentially big issue -- that there may be atomistic surface effects that a continuum model can't capture)? Second, are strain and bandstructure shifts linearly related, as is usually assumed? The second question has not been considered as carefully as the first question, as far as I know.

The origin of the linear assumption is that the strain effect is considered to be a perturbation to the band structure, which leads to the standard deformation potential theory framework (the bandgap modification that you mention). I think there's generally confidence that strain of a few percent will have a linear effect that can be modeled using deformation potential theory. But I don't know of any "systematic" study of this effect. There are possibly other problems near free surfaces (atomistic, in origin, for example) that may lead to bigger errors than the linear perturbation assumption. A practical strategy is to just resort to atomistic approaches, like tight-binding or density functional theory, which may be possible if the structures are small enough. Then you can get the full, nonlinear coupling, with atomistic treatment of the surface mechanics and the surface physics.

## Further on the nonlinear

Further on the nonlinear effects; as Harley has pointed out there are two issues: (i) small strain assumption and the (ii) validity of the typically adopted linear strain-band structure coupling. If I recall correctly there are some papers that have revisited the small strain assumption, one group from England (Faux and co-workers) and other from Poland (Majewski and co-workers).

As Harley alludes to, the second assumption of linear strain-electronic structure dependence is somewhat less studied. I collaborated on a work in which we studied, using

ab initiomethods, scaling of strain-energy gap coupling in Si clusters. We find that below roughly 5 nm, linear strain-energy gap relation is suspect (see Figure 4 in this paper). This paper is available from my website (# 30, Peng et. al., 2006). In a more recent collaborative work (with Harley), the nonlinear coupling between strain and electronic structure is even more evident (#36, Zhang et.al. 2007). However, I think we have barely scratched the surface...a thorough study of this issue would be quite interesting.Interestingly, many years ago there was a paper by Zhang from NREL which re-examined the standard multiband kp model and suggested that (in presence of inhomogeneous strain) gradients of strain be included in the constitutive relation between strain and electronic structure. His derivation is quite rigorous and implies that this effect would be important for small quantum dots.

## photovoltaics and thermoelectrics

Thanks, Pradeep, for your comments.

Yes, I agree that mechanicians may be able to make contributions in the area of quantum dots for next generation photovoltaics and thermoelectrics -- at least I hope we can. There are many important mechanics problems related to growth and fabrication of quantum dots for these applications. The role of mechanics in band engineering via strain effects, which is what I think you may be referring to, may also become important. One can imagine, potentially, strain tunable quantum dots for photovoltaics and thermoelectrics.

## QM and SM: What I would like to have, eventually...

Dear Harley,

I make reference to the following passage in Pradeep's comment:

Pradeep said: Perhaps, during some point this month it would be a good idea to compile a supplementary list of "tutorial" papers and documents that can provide a clear and facile path for mechanicians to get involved in this interesting field. In due course, I will provide a few such references as well.

I very much like this idea. Further, I would like to add to the request.

At the end of the month, I would like it if someone could

systematizeall the comments and thoughts generated here, from the following two viewpoints: (i) The kind ofmodeling abstractionsthat have been used in the papers studied here (and other related papers), for each of QM and SM. (ii) The specific assumptions and techniques used in handlingboundary conditions, and the lessons or insights these hold for further model-making, computational models included.A systematization of the comments and discussions here, from the above two angles, would be interesting to have.

## Quantum stability of ultrathin metal films and alloys

This month's journal club theme reminds me of my earlier posting in this group: http://imechanica.org/node/546. Briefly, the energy associated with the confined motion of the conduction electrons within an ultrathin metal film could dictate the morphological/mechanical stability of the film and the preferred growth mode (tentatively termed "electronic growth" in an earlier paper by Zhang, Niu, and Shih, Phys. Rev. Lett. 80, 5381 (1998)).

In a more recent development along this line, it was demonstrated both theoretically and experimentally that the stability of ultrathin metal ALLOYS could also be tuned in the quantum regime and with atomic-scale precision, together with their transport (in the specific case of PbBi alloys, their superconducting) properties. See Ozer, Jia, Zhang, Thompson, and Weitering, Science 316, 1594 (2007).

It is conceivable that quantum size effects could also play an important role in defining another essential aspect of the metallic thin films and alloys, namely, the frictional force on such films, as the electronic degrees of freedom of the substrate could participate more effectively in the energy dissipation. But this is only a speculation, as to date this frictional aspect of quantum metal films is largely unexplored.

## quantum effect on stability

I am familiar with this very interesting work, Zhenyu, and some of the subsequent work you did on this problem with Zhigang Suo. (

Phys. Rev. B,58, 5116-5120, 1998) One could refer to this generally as quantum mechanical coupling to surface energetics -- as opposed to coupling to deformation -- which would also connect to your idea about the quantum effect resulting in a frictional force. This is very fascinating.## Dynamical quantum behaviours

Researches on the quantum effcts of a passing shock wave are interesting. The quantum effcts may be used to control the behaviour of energetic materials under high speed impact.

## Dynamic quantum behaviors?

Hi Henry,

Do these effects have to do with quantum mechanics of electrons? Or is there an analogy with electronic structure? What do you mean by "quantum"?

## Piezoelectric Effects

Hi Harley et al.:

Perhaps another important class of mechanical-electronic/electrical coupling is the piezoelectric effects, and one recent significant work enabling the community to calculate such effects based on first principles was the formulation of David Vanderbilt and coworkers within the Berry phase approach. See, for example:

Spontaneous polarization and piezoelectric constants of III-V nitridesAuthor(s): Bernardini F, Fiorentini V,

Vanderbilt DSource: PHYSICAL REVIEW B 56 (16): 10024-10027 OCT 15 1997

Abstract: The

spontaneous polarization, dynamical Born charges, and piezoelectric

constants of the III-V nitrides AlN, GaN, and InN are studied ab initio

using the Berry-phase approach to polarization in solids. The

piezoelectric constants are found to be up to ten times larger than in

conventional III-V and II-VI semiconductor compounds and comparable to

those of ZnO. Further properties at variance with those of conventional

III-V compounds are the sign of the piezoelectric constants (positive

as in II-VI compounds) and the very large spontaneous polarization.