Supplemental Information
Temperature-dependent hardness of zinc-blende structured covalent materials
Xing Feng1, Jianwei Xiao1, Bin Wen1, , Jijun Zhao2, Bo Xu1,†, Yanbin Wang3, Yongjun Tian1
1Center for High Pressure Science, State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China
2 Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Ministry of Education), Dalian University of Technology, Dalian 116024, China
3Center for Advanced Radiation Sources, University of Chicago, Chicago, Illinois 60439, USA.
E-mail address: wenbin@ysu.edu.cn (Bin Wen) Tel: +86 13933969655
†E-mail address: bxu@ysu.edu.cn (Bo Xu) Tel: +86 13903332531
Part I. Molecular Dynamics method
The dislocation motion in diamond can be considered as a process of kink-pairs nucleation and migration, hence its activation energy is depending on kink-pair formation energy and kink migration energy barrier. To obtain activation energy for shuffle-set 0° perfect and glide-set 90° partial dislocation motion, a series of dislocation kink-pair structure models with different kink-pair widths are built. These structure model contains about 115,200 atoms, and their x, y, z axes are redefined along matrix’s 112 , 110 , and111directions of diamond. The constructed structure is optimized by using LAMMPS program [1], and C-C bonding interactions are described by LCBOP (Long-range Bond-order Potential for Carbon) potential [2]. Periodic boundary condition is only imposed along they y axis and free surface was imposed in x and z directions. After optimized, kink-pair formation energy (Wf) different kink-pair widths was obtained according to system energy variation, and the kink migration energy barrier (Wm) was obtained using Nudged Elastic Band (NEB)
method [3]. Based on these calculated kink formation energy at different kink-pair widths and the corresponding kink migration energy barrier, activation energy for shuffle-set 0° perfect and glide-set 90° partial dislocation motion was obtained by finding maximal total energy (addition of kink formation energy and kink migration energy barrier) with respect to kink-pair width [4].
Part II. First-principles method
First-principles calculations for these zinc-blende structured covalent materials are carried out in the framework of the density functional theory (DFT) [5, 6] with the projector-augmented wave (PAW) [7] method, implemented in Vienna ab initio simulation package (VASP) [8-10]. The generalized gradient approximation (GGA) in the form of Perdew-Burke-Ernzerhof (PBE) [11] is used for the exchange-correlation potential. The plane-wave cutoff energy for all crystals is 500 eV, and the k-points is taken to be 15×15×15 using the Monkhorst-Pack method. Forces on the ions are calculated according to the Hellmann-Feynman theorem, and the convergence thresholds for total energy and ionic force component are set to 1×10−6 eV and 0.001 eV/Å, respectively. The Debye frequencies are obtained as the maximum of phonon frequencies calculated by the finite displacement method implemented in the PHONOPY code [12].
To calculate the generalized stacking fault energy (GSFE) surfaces of shuffle-set and glide-set planes in diamond, an orthogonal lattice consisting of 15 atomic layers along the [111] direction and a 15 Å vacuum layer is used. The x, y and z axes of the orthogonal lattice are parallel to the [110], [112] , and [111] of diamond, respectively. This orthogonal lattice is divided into two halves along the slip plane, and these two parts of the lattice were displaced relative to each other. The GSFE surfaces of the shuffle-set and glide-set planes can be obtained by plotting the excess energy with
atomic displacement.
Based on first-principles calculations, three independent elastic constants, i.e., C11, C12, and C44
for zinc-blende structured covalent materials are calculated. According to the Voigt-Ruess-Hill approximations [13], the elastic moduli can be obtained based on the results of elastic constants. For cubic crystals, the Voigt and Reuss bulk modulus (BV, BR) and Voigt and Reuss shear modulus (GV, GR) can be given as
11 12
( 2 ) / 3
BV C C , (S1)
1
11 12
(3 6 )
BR S S , (S2)
11 12 44
( 3 ) / 5
GV C C C , (S3)
1
11 12 44
5(4 4 3 )
GR S S S , (S4) whereSijare the elastic compliance constants (i.e.,Sij Cij1). Finally, the elastic moduli can be approximated by Hill’s average, for bulk modulus B(BV BR) / 2, and for shear modulus
( V R) / 2
G G G . Further, the Poisson’s ratio is given by
3 2
=2(3 )
B G
B G
. (S5)
Fig. S1. The generalized stacking fault energy surface of glide-set and shuffle-set plane in diamond. (a) The generalized stacking fault energy surface of glide-set plane in diamond. (b) The generalized stacking fault energy surface of shuffle-set plane in diamond.
Fig. S2. Activation energy for six types of dislocations in diamond as a function of applied stress.
Fig. S3. Temperature-dependent elastic constants of diamond, Si and Ge. (a) temperature-dependent c11, c12 and c13 of diamond. (b) temperature-dependent c11, c12 and c13 of Si. (c) temperature-dependent c11, c12 and c13 of Ge. (d) the optimized temperature-dependent lattice constant of diamond, Si and Ge. (e) the temperature-dependent Poisson’s ratio of diamond, Si and Ge.
( f ) t h e t e m p e r a t u r e - d e p e n d e n t s h e a r m o d u l u s o f d i a m o n d , S i a n d G e .
Fig. S4. Calculated temperature dependent Vickers hardness for polar covalent materials of cubic BN and SiC in comparison with experimental data (a Ref. [14], b Ref. [15], c Ref. [16] )
Fig. S5. Variation of ln(
m bbvD
) for glide-set and shuffle-set dislocations for selected zinc-blende structured covalent materials. In particular, =104 -1s , ms 0.3 10 8m2 and14 2
0.3 10
g
m m
, b = 100 nm, andbandvDis Burgers vector and Debye frequency for different zinc-blende structured covalent materials.
Fig. S6. (a) Effect of mobility of dislocation density on hardness of diamond. (b) Effect of strain rate on hardness of diamond. (c) Effect of grain size on hardness of diamond by considering the Hall-Petch effect.
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