The Third-Order Elastic Constants and Mechanical Properties of 30° Partial Dislocation in Germanium: A Study from the First-Principles Calculations and the Improved Peierls–Nabarro Model

Huili Zhang,Lumei Tong,Yu Sun,Yunchang Fu,Defang Lu

doi:10.3390/cryst12010004

Abstract

The elastic constants, core width and Peierls stress of 30° partial dislocation in germanium has been investigated based on the first-principles calculations and the improved Peierls–Nabarro model. Our results suggest that the predictions of lattice constant and elastic constants given by LDA are in better agreement with experiment results. While the lattice constant is overestimated at about 2.4% and most elastic constants are underestimated at about 20% by the GGA method. Furthermore, when the applied deformation is larger than 2%, the nonlinear elastic effects should be considered. And with the Lagrangian strains up to 8%, taking into account the third-order terms in the energy expansion is sufficient. Except the original γ—surface generally used before (given by the first-principles calculations directly), the effective γ—surface proposed by Kamimura et al. derived from the original one is also used to study the Peierls stress. The research results show that when the intrinsic−stacking−fault energy (ISFE) is very low relative to the unstable−stacking−fault energy (USFE), the difference between the original γ—surface and the effective γ—surface is inapparent and there is nearly no difference between the results of Peierls stresses calculated from these two kinds of γ—surfaces. As a result, the original γ—surface can be directly used to study the core width and Peierls stress when the ratio of ISFE to the USFE is small. Since the negligence of the discrete effect and the contribution of strain energy to the dislocation energy, the Peierls stress given by the classical Peierls–Nabarro model is about one order of magnitude larger than that given by the improved Peierls–Nabarro model. The result of Peierls stress estimated by the improved Peierls–Nabarro model agrees well with the 2~3 GPa reported in the book of Solid State Physics edited by F. Seitz and D. Turnbull.

Highlights

The second-order elastic constants (SOECs), third-order elastic constants (TOECs), core width and Peierls stress of 30◦ partial dislocation in germanium have been investigated based on the first-principles calculations and the improved Peierls–Nabarro model
The SOECs and TOECs are obtained from the finite-strain continuum elasticity theory [1,2,3,4,26,27]
Larger than that calculated by local density approximation (LDA), and the elastic constants C11, C12, C111, C112, C155 calculated by the generalized gradient approximation (GGA) method are about 20% smaller than those calculated by LDA

Summary

Introduction

The third-order elastic constants (TOECs) are important quantities characterizing nonlinear elastic properties when large stresses and strains are involved [8]. Otherwise, They are the basis for discussion of other anharmonic properties, such as phonon−phonon interactions, thermal expansion, the temperature dependence of elastic properties, etc. Much theoretical and experimental research has been performed on studying the elastic constants of germanium. Several experimental methods, such as neutron scattering, optical interference and ultrasonic wave propagation, are used to measure the elastic constants of germanium, including the second-order elastic constants (SOECs) and TOECs [11,12,13]. Considering the wide applications of germanium, such as infrared detection and imaging, etc., the study of TOECs is still necessary

Results

Discussion

Conclusion