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?

M. Grundmann, O. Stier, and D. Bimberg, "InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure," Phys. Rev. B 52, 11969 - 11981 (1995).

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?

A. Rochefort, P. Avouris, F. Lesage, and D. H. Salahub, "Electrical and mechanical properties of distorted carbon nanotubes," Phys. Rev. B 60, 13824 - 13830 (1999).

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?

E. B. Tadmor, G. S. Smith, N. Bernstein, and E. Kaxiras, "Mixed finite element and atomistic formulation for complex crystals," Phys. Rev. B 59, 235 - 245 (1999).

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.