Simulating Nanoscale Processes in Solids Using Dft and the Quasicontinuum Method

A framework is proposed for the investigation of chemical and mechanical properties of nanostructures. The methodology is based on a two-step approach to compute the electronic density distribution in and around a nanostructure, and then the equilibrium configuration of its nuclei. The Electronic Problem embeds interpolation and coupled cross-domain optimization techniques through a process called electronic reconstruction. In the second stage of the solution, the Ionic Problem repositions the nuclei of the nanostructure given the electronic density in the domain. The new ionic configuration is the solution of a nonlinear system based on a first-order optimality condition when minimizing the total energy associated with the nanostructure. The overall goal is a substantial increase in the dimension of the nanostructures that can be simulated by using approaches that include accurate DFT computation. This increase stems from the fact that during the solution of the Electronic Problem expensive DFT calculations are limited to a small number of subdomains. For the Ionic Problem, computational gains result from approximating the position of the nuclei in terms of a reduced number of representative nuclei following the quasicontinuum paradigm. ∗Address all correspondence to this author. PARADIGM OF THE PROPOSED APPROACH Nanostructures have dimensions in the range of 1 ∼ 100 nm and typically contain 102 ∼ 108 atoms. Applying the wellestablished Kohn-Sham DFT method [1] for nonperiodic structures of 60 atoms has led to simulations that can take up to three months to complete. When long range interactions are ignored and pseudo-potentials are used, ab-initio simulations have been carried out for nonmetallic structures with up to 1,500 atoms [2]. The approach that enabled the increase in the number of atoms belongs to the family of so-called O(N) methods [3], which scale as N with the dimension of the problem (in this case the number of electrons). This work is not concerned with fundamental electronic structure computation methods. Acknowledging the smalldimension constraint placed on the problem by the existing Density Functional Theory (DFT)-based methods, the goal of the proposed work is to use techniques that, by closing the spatial scale gap, render electronic structure information at the nanoscale. This electronic structure information is then used to investigate the chemical and mechanical properties of the material. In the context of mechanical analysis of nanostructures, 1 Copyright c © 2005 by ASME the methodology proposed follows in the steps of the quasicontinuum work proposed in [4–6]. Specifically, this is an extension of the work in [5, 6], because rather than considering a potential-based interatomic interaction that has a limited range of validity and is difficult to generalize to inhomogeneous materials, the methodology proposed uses ab-initio methods to provide for the particle interaction. At the same time it is a generalization of the method proposed in [4] because rather than considering each mesh discretization element to be part of a periodic and uniformly deformed infinite crystal, the proposed method treats in a generic optimization framework any structure (nonperiodic and inhomogeneous) once the electronic density distribution is available. The electronic structure computation is approached herein as the solution of a constrained minimization problem [7] min ρ E[ρ,ρA] (1a) Z ρ(r)dr = Ne (1b) where Ne represents the number of electrons present in the system. The solution to this problem depends parametrically on the nuclear density ρA, ρ = ρ(ρA), a consequence of the BornOppenheimer assumption. Subsequently, the computation of the ground state of the entire system as the solution of the optimization problem min ρA E[ρ(ρA),ρA] (2) From a geometric perspective two assumptions are made in order to close the gap between the subatomic-level representation of the electron density, and the nanoscale scale associated with the structures investigated: (a) there is a near regularity in the atomic compositions of the material, and (b) almost everywhere in the nanostructure the solution to the Ionic Problem results in only small deformations. The assumption (a) is referred to as near-periodicity and is the vehicle that carries first-principles computation results from micro to macro scale. This work does not build on the periodicity assumption, it merely assumes that the material displays close to periodic structure. As explained later, the near-periodicity assumption enables the use of interpolation for electronic structure reconstruction. In regards to the second assumption; i.e., small deformations, in order to formally quantify this concept, the nanostructure is considered to occupy or be contained inside an initial reference configuration D0 ⊂R3. The structure undergoes a change of shape described by a deformation mapping Φ(r0, t) ∈ R3. This deformation mapping gives the location r in the global Cartesian reference frame of each point r0 represented in the undeformed material frame. As indicated, the mapping might depend on time t. The variable t does not necessarily represent the time contemporary with the structure under consideration. In fact, in a quasi-state simulation framework, this variable might be an iteration index of an optimization algorithm that solves Eq.(2) in the case ρA is made of nuclear point charges. The components of the deformation gradient are introduced as