Elastic effects on relaxation volume tensor calculations

B. Puchala, M. L. Falk, and K. Garikipati

Phys. Rev. B 77, 174116 – Published 23 May 2008

Abstract

Relaxation volume tensors quantify the effect of stress on diffusion of crystal defects. Continuum linear elasticity predicts that calculations of these parameters using periodic boundary conditions do not suffer from systematic deviations due to elastic image effects and should be independent of the supercell size or symmetry. In practice, however, calculations of formation volume tensors of the $⟨ 110 ⟩$ interstitial in Stillinger–Weber silicon demonstrate that changes in bonding at the defect affect the elastic moduli and result in system-size dependent relaxation volumes. These vary with the inverse of the system size. Knowing the rate of convergence permits accurate estimates of these quantities from modestly sized calculations. Furthermore, within the continuum linear elasticity assumptions, the average stress can be used to estimate the relaxation volume tensor from constant volume calculations.

Received 11 February 2008

DOI:https://doi.org/10.1103/PhysRevB.77.174116

Authors & Affiliations

B. Puchala^1,*, M. L. Falk¹, and K. Garikipati²

¹Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA
²Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

^*bpuchala@umich.edu

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Vol. 77, Iss. 17 — 1 May 2008

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Images

Figure 1
Structure of the relaxed Si $⟨ 110 ⟩$ dumbbell interstitial with displacements scaled by $3 \times$ for clarity. In configuration (A), the dipole tilts and breaks the symmetry about the (110) plane, resulting in a slightly lower energy than in configuration (B). Both (A) and (B) maintain symmetry about the $(1 \bar{1} 0)$ plane.Reuse & Permissions
Figure 2
In the large-size limit, the formation energy of an isolated $⟨ 110 ⟩$ interstitial (A) in Stillinger–Weber silicon converges to $E_{SiI, ⟨ 110 ⟩, (A)}^{f, \infty} = 4.7091 eV$ for all measurements: FBC; PBC at constant pressure (PBC CP); PBC at constant volume equal to the reference volume with the energy correction [Eq. (12)] for elastic stress (PBC CV); PBC at constant volume equal to the reference volume (PBC CV uncorrected).Reuse & Permissions
Figure 3
In the large-size limit, the trace of the formation volume of an isolated interstitial in Stillinger–Weber silicon converges to $tr [{\bar{\bar{V}}}_{SiI, ⟨ 110 ⟩, (A)}^{f, \infty}] = 13.677 Å^{3}$ for all boundary conditions: FBC; PBC at constant pressure (PBC CP); PBC at constant volume equal to the reference volume with the volume correction [Eq. (11)] for elastic stress (PBC CV).Reuse & Permissions
Figure 4
The formation energy converges in the large-size limit to $E_{SiI, ⟨ 110 ⟩, (A)}^{f, \infty} = 4.7091 eV$ , with the error decreasing as the inverse of the system size, $N$ , for all boundary conditions: FBC; PBC at constant pressure (PBC); PBC at constant volume equal to the reference volume with the energy correction [Eq. (12)] for elastic stress (PBC CV); PBC at constant volume equal to the reference volume (PBC CV uncorrected).Reuse & Permissions
Figure 5
(a) The convergence of the formation volume to $tr [{\bar{\bar{V}}}_{S i I, ⟨ 110 ⟩, (A)}^{f, \infty}] = 13.677 Å^{3}$ depends on [(b) and (c)] the convergence of the moduli. The (b) bulk modulus and (c) shear modulus of systems with a $⟨ 110 ⟩$ interstitial converge to the values ( $K_{inf} = 1.0826 \times 10^{11} Pa$ and $C_{44, inf} = 6.0256 \times 10^{10} Pa$ ) in defect-free Stillinger–Weber silicon in the large-size limit. The under coordination of surface atoms makes the convergence much slower in the FBC than in the periodic boundary case with either constant pressure (PBC CP) or constant volume (PBC CV).Reuse & Permissions
Figure 6
The formation volume converges to $tr [{\bar{\bar{V}}}_{SiI, ⟨ 110 ⟩, (A)}^{f, \infty}] = 13.677 Å^{3}$ with error decreasing as $1 / N$ , the system size, shown above with ○. Knowing this convergence rate, we can estimate a converged value from the two largest systems calculated using Eq. (19). This estimate, which is shown above with $\times$ , is as converged at $N = 1728$ as the $N = 21 952$ calculation.Reuse & Permissions

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