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Journal Club Theme of Sept. 15 2008: Defects in Solids---Where Mechanics Meets Quantum Mechanics
Defects in solids have been studied by the mechanics community for over five decades, some of the earliest works on this topic dating back to Eshelby. Yet, they still remain interesting, challenging, and often spring surprises—one example being the observed hardening behavior in surface dominated structures (as discussed in past journal club themes by Wei Cai and Julia Greer). In this journal theme, I wish to concentrate on the underlying physics behind defect behavior and motivate the need to combine quantum mechanical and mechanics descriptions of materials behavior. Through this discussion, I hope to bring forth: (i) The need to bridge mechanics with quantum mechanics; (ii) The challenges in quantum mechanical calculations; (iii) How the mechanics community can have a great impact.
(i) The need to bridge mechanics with quantum mechanics:
Defects play a crucial role in influencing the macroscopic properties of solids—examples include the role of dislocations in plastic deformation, dopants in semiconductor properties, domain walls in ferroelectric properties, and the list goes on. These defects are present in very small concentrations (few parts per million), yet, produce a significant macroscopic effect on the materials behavior through the long-ranged elastic and electrostatic fields they generate. But, the strength and nature of these fields as well as other critical aspects of the defect core are all determined by the electronic structure of the material at the quantum-mechanical length-scale. Hence, there is a wide range of interacting length-scales, from electronic structure to continuum, that need to be resolved to accurately describe defects in materials and their influence on the macroscopic properties of materials.
At this point, I wish to stress the importance of both electronic structure (quantum-mechanical effects) and long-ranged elastic fields by presenting some known results on the energetics of a single vacancy. The vacancy formation energy in aluminum computed from electronic-structure (ab-initio) calculations is about 0.7 eV, of which the contribution of elastic effects (atomic relaxations) is less than 10% of the formation energy, rest is electronic effects (quantum-mechanical effects)! In mechanics, these electronic effects are lumped as the core-energy, which is considered an inconsequential constant, and we deal with only elastic effects. On the other hand, computational materials scientists often work with only core energies as they appear to be the major contribution to the total defect energy. In my opinion, both are equally important and neither can be neglected and I will present some evidence to corroborate this claim. Some recent electronic-structure calculations have been performed to investigate the influence of homogeneous macroscopic strain on the energetics of vacancies (some of which are present in Ho et al. Phys. Chem. Chem. Phys. 2007, 9, 4951), where, in one case atomic relaxations are suppressed and the energetics are solely due to electronic effects and another where atomic relaxations are allowed which contain both electronic effects and elastic interactions with macroscopic fields. In the first case, the vacancy formation energy changed from 0.7 eV at no imposed macroscopic strain to 0.2 eV for 0.15 volumetric strain. This suggests that the defect core energy is very strongly influenced by the macroscopic deformation at the core site, and is not an inconsequential constant! This dependence is quantum-mechanical and there is no obvious way to determine this other than resorting to electronic structure (ab-initio) calculations. On the other hand, in the second case, upon relaxing the atoms and accounting for the elastic effects, the contribution for these elastic effects changed from 10% of the total formation energy at no macroscopic deformation to 50% at 0.15 volumetric strain. These results provide strong evidence that both the core of a defect and the long-ranged elastic fields are equally important in understanding the behavior of defects and these are inherently coupled through the electronic structure of the material.
(ii) The challenges in electronic structure calculations:
The basis of all electronic structure calculations is quantum mechanics which has the mathematical structure of an eigen-value problem. Though the physics behind quantum mechanics has been well-known for almost seven decades, the challenge arises from the computational complexity of the resulting governing equations (Schrodinger’s equation). Unfortunately solutions to the full Schrodinger’s equation are intractable beyond a few electrons (<10) making any meaningful computation of materials properties beyond reach. The direction pursued by the computational physics community in the mid-nineteenth century was beautifully summarized by Paul Dirac: “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of quantum mechanics should be developed, which can lead to an explanation of the main features of the complex atomic systems without too much computation”. These approximate methods are what constitute the electronic structure calculations which are widely used in the present day. The starting point of all electronic structure theories for computing ground-state materials properties is a variational principle, something which is very often seen in mechanics. I have written a brief overview (for readers interested in more details) of various electronic structure calculations and the various approximations involved in arriving at these theories:
One of the most popular electronic structure theory that is widely used is the density-functional theory (DFT). It has its roots in the seminal work of Kohn, where he rigorously proved that the ground-state properties of a material system are only a function of the electron-density, which has made electronic structure calculations of materials possible. Albeit many theoretical developments in this field and the advent of supercomputing, the computational complexity of these calculations still restricts computational domains to couple of hundred atoms. Thus, historically it was natural to concentrate on periodic properties of materials. DFT has been very successfully in capturing a wide range of bulk properties which include elastic moduli, band-structure, phase transformations, etc. The interest in periodic properties has resulted in the use of a plane-waves as a basis set to compute the variational problem associated with density functional theory. Such a Fourier space formulation has limitations, especially in the context of defects: it requires periodic boundary conditions, thus limiting an investigation to a periodic array of defects. This periodicity restriction in conjunction with the cell-size limitations (200 atoms) arising from the enormous computational cost associated with electronic structure calculations, limits the scope of these studies to very high concentrations of defects that rarely—if ever—are realized in nature. Thus recently, there is an increasing thrust towards using real-space formulations and using finite-element or a wavelet basis, or a finite-difference scheme. The following three articles are good representations of the use of these methods.
1. J.E. Pask, B.M. Klein, C.Y. Fong, P.A. Sterne, Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach, Phys. Rev. B. 59 12352 (1999).
2. T.A. Arias, Multiresolution analysis of electronic structure: semicardinal and wavelet basis, Rev. Mod. Phys. 71, 267 (1999).
3. C.J. Garcia-Cevera, An efficient real-space method for orbital-free density functional theory, Comm. Comp. Phys. 2, 334 (2006).
(iii) How the mechanics community can have a great impact.
Although the use of real-space formulations seems to provide freedom from periodicity, the computational complexity still restricts calculations to a few hundred atoms. However, an accurate description of defects requires resolution of the electronic structure of the core as well as the long-ranged elastic effects. There have been some multi-scale methods based on embedding schemes that have been proposed which address this problem. One representative article for these methods is the following:
4. G. Lu, E. Tadmor, and E.Kaxiras, "From electrons to finite elements: A concurrent multiscale approach for metals" Phys. Rev. B 73, 024108 (2005).
The philosophy behind these embedding schemes is to embed a refined electronic structure calculation (inside a small domain) in a coarser atomistic simulation using empirical potentials, which in turn is embedded in a continuum theory. Valuable as these schemes are, they suffer from some notable shortcomings. In some cases, uncontrolled approximations are made such as the assumption of separation of scales, the validity of which can not be asserted. Moreover, these schemes are not seamless and are not solely based on a single electronic structure theory. In particular, they introduce undesirable overlaps between regions of the model governed by heterogeneous and mathematically unrelated theories.
I feel there is tremendous potential for the mechanics community to contribute in the development of multi-scale schemes solely based on electronic structure calculations which are seamless, have controlled approximations, assure the notion of convergence, and provide insights into the behavior of defects. To start the discussion and motivate, let me provide an analogy. The electronic structure of a defect in a material has similar structure to a composite problem with a damage zone. Homogenization techniques and adaptive finite-element basis sets are common solutions to such composite problems! With a little care, I believe it is possible for the mechanics community to make a huge impact in electronic structure calculations of defects.