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Analysis of quasi-planar defects using the Thomas--Fermi--von Weizsäcker model

Dharamveer Kumar, Amuthan A. Ramabathiran

TL;DR

This work analyzes quasi-planar defects in crystals within the Thomas–Fermi–von Weizsäcker (TFW) density-functional framework, proving that the relative energy between a defective crystal and a perfect crystal is finite and governed by a well-posed variational principle. By leveraging thermodynamic-limit theory and locality estimates, the authors show convergence of the electron density and electrostatic potential in the defect setting, with perturbations decaying exponentially away from the defect core. A key outcome is the identification of a surface-energy functional whose minimizer is the defect response, yielding a rigorous variational formulation for defect energies and a controlled path to computing generalized stacking fault energies. The paper also includes numerical exploration of the TFWD (Dirac exchange) model, indicating that the locality properties extend beyond convex TFW, and discusses implications for numerical methods and future extensions to dislocation cores and more complex DFT models.

Abstract

We analyze the convergence of the electron density and relative energy with respect to a perfect crystal of a class of volume defects that are compactly contained along one direction while being of infinite extent along the other two using the Thomas--Fermi--von Weizsäcker (TFW) model. We take advantage of prior work on the thermodynamic limit and stability estimates in the TFW setting, and specialize it to the case of quasi-planar defects. In particular, we prove that the relative energy of the defective crystal with respect to a perfect crystal is finite, and in fact conforms to a well-posed minimization problem. In order to show the existence of the minimization problem, we modify the TFW theory for thin films and establish convergence of the electronic fields due to the perturbation caused by the quasi-planar defect. We also show that perturbations to both the density and electrostatic potential due to the presence of the quasi-planar defect decay exponentially away from the defect, in agreement with the known locality property of the TFW model. We use these results to infer bounds on the generalized stacking fault energy, in particular the finiteness of this energy, and discuss its implications for numerical calculations. We conclude with a brief presentation of numerical results on the (non-convex) Thomas-Fermi-von Weizsäcker-Dirac (TFWD) model that includes the Dirac exchange in the universal functional, and discuss its implications for future work.

Analysis of quasi-planar defects using the Thomas--Fermi--von Weizsäcker model

TL;DR

This work analyzes quasi-planar defects in crystals within the Thomas–Fermi–von Weizsäcker (TFW) density-functional framework, proving that the relative energy between a defective crystal and a perfect crystal is finite and governed by a well-posed variational principle. By leveraging thermodynamic-limit theory and locality estimates, the authors show convergence of the electron density and electrostatic potential in the defect setting, with perturbations decaying exponentially away from the defect core. A key outcome is the identification of a surface-energy functional whose minimizer is the defect response, yielding a rigorous variational formulation for defect energies and a controlled path to computing generalized stacking fault energies. The paper also includes numerical exploration of the TFWD (Dirac exchange) model, indicating that the locality properties extend beyond convex TFW, and discusses implications for numerical methods and future extensions to dislocation cores and more complex DFT models.

Abstract

We analyze the convergence of the electron density and relative energy with respect to a perfect crystal of a class of volume defects that are compactly contained along one direction while being of infinite extent along the other two using the Thomas--Fermi--von Weizsäcker (TFW) model. We take advantage of prior work on the thermodynamic limit and stability estimates in the TFW setting, and specialize it to the case of quasi-planar defects. In particular, we prove that the relative energy of the defective crystal with respect to a perfect crystal is finite, and in fact conforms to a well-posed minimization problem. In order to show the existence of the minimization problem, we modify the TFW theory for thin films and establish convergence of the electronic fields due to the perturbation caused by the quasi-planar defect. We also show that perturbations to both the density and electrostatic potential due to the presence of the quasi-planar defect decay exponentially away from the defect, in agreement with the known locality property of the TFW model. We use these results to infer bounds on the generalized stacking fault energy, in particular the finiteness of this energy, and discuss its implications for numerical calculations. We conclude with a brief presentation of numerical results on the (non-convex) Thomas-Fermi-von Weizsäcker-Dirac (TFWD) model that includes the Dirac exchange in the universal functional, and discuss its implications for future work.
Paper Structure (24 sections, 20 theorems, 151 equations, 7 figures, 1 algorithm)

This paper contains 24 sections, 20 theorems, 151 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Background for TFW Models
  3. Thermodynamic Limits
  4. Comparison theorems
  5. Problem Setting and Main Results
  6. Thermodynamic limit analysis of electron density and surface energy
  7. Convergence of electron density
  8. Convergence of energy difference on $\Gamma_\infty$
  9. Minimization principle for the energy difference on $\Gamma_\infty$
  10. Uniqueness of the minimizer
  11. Discussion and Conclusion
  12. Approximate evaluation of relative energies of quasi-planar defects
  13. Stacking fault energy in bicrystals
  14. Extension of the TFW model with Dirac Exchange
  15. Conclusion
  16. ...and 9 more sections

Key Result

Theorem 2.1

[Theorem 3.1 in NO17] Let $m_1, m_2 \in \mathcal{M}^M$ be two given nuclear densities, and let $(u_1, \phi_1)$ and $(u_2, \phi_2)$ be the corresponding unique ground state solutions, respectively. Then for any $y \in \mathbb{R}^3$ there exist positive constants $C_1$ and $C_2$ such that The constants $C_1, C_2$ depend only on the uniform bounds of $m_1$ and $m_2$.

Figures (7)

  • Figure 1: Typical geometry of quasi-planar defects studied in this work. The defective region can be thought of as a thin film that is sandwiched between two perfect crystals of the same material and identical orientation. Periodicity is assumed along the in-plane directions of the defect. Note that we do not assume that the nuclear distribution in the defective region is homogeneous.
  • Figure 2: Procedure to generate a supercell with periodic nuclear density $m_L$ from a given nuclear density $m \in \mathcal{M}_{xy}$. The original system is sliced into boxes with dimension $L$ along the out-of-plane direction of the planar defect and periodically tiled over $\mathbb{R}^3$.
  • Figure 3: Procedure to generate a thin film with nuclear density $\Tilde{m}_H$ from a given nuclear density $m \in \mathcal{M}_{xy}$. The original system is sliced into a box with dimension $H$ along the out-of-plane direction of the planar defect.
  • Figure 4: Perfect crystal modeled by a smooth Dirac comb and perturbed crystal with perturbations on one of the comb's fingers.
  • Figure 5: Comparison of TFW and TFWD models for the nuclear distributions shown in Figure \ref{['fig:tfw_vs_tfwd_smooth_dirac']}. Both the difference in square root density and the difference in potential are shown. The results indicate that the TFWD response shares the decay properties of the TFW model, despite being non-convex.
  • ...and 2 more figures

Theorems & Definitions (32)

  • Theorem 2.1
  • Proposition 2.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Corollary 4.2.1
  • proof
  • Corollary 4.2.2
  • proof
  • ...and 22 more