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Tetragonal Allotrope of Group 14 Elements
Supporting Information
Zhisheng Zhao,†^ Fei Tian,‡^ Xiao Dong,‡^ Quan Li,§^ Qianqian Wang,†^ Hui Wang,§^ Xin
Zhong,§^ Bo Xu,†^ Dongli Yu,†^ Julong He,†^ Hui-Tian Wang,‡^ Yanming Ma,§∗and
Yongjun Tian†∗
†State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China §State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, China ‡School of Physics and MOE Key Laboratory of Weak-Light Nonlinear Photonics, Nankai University, Tianjin 300071, China
Computational Details
Our crystal structure predictions for C, Si, and Ge through CALYPSO code^1 ,^2
were performed using simulation cell sizes of 2-30 atoms at pressures ranging from 0
to 20 GPa. In a typical CALYPSO run, each generation contains 3 0 structures, and the
first generation was produced randomly, and 60% of the low-enthalpy structures of
each generation were used to produce the next generation. Usually, the structure
searching simulation was stopped after 600~ 900 structures generated (20~ 30
generations). All structures generated during structure search, were locally optimized
using the density functional theory (DFT) within the local density approximation
(LDA) and ultrasoft pseudopotentials, as implemented in the Vienna Ab-initio
Simulation Package (VASP) code.
3
The plane wave cutoff energies of 310, 180, and
200 eV were used for C, Si, and Ge, respectively, which gave well-converged total
energies within 1 meV/atom over the pressure range. In the results of CALYPSO
simulations, a number of low-enthalpy experimentally and theoretically known
structures (e.g., graphite, diamond, nanotubes, BC8, and ST12 structures) were found.
In addition, a novel tetragonal T12 structure was uncovered.
The subsequent structural optimizations and property predictions were performed
using ultrasoft pseudopotentials through the Cambridge Serial Total Energy Package
(CASTEP) code.
4
The plane wave cutoff energies of 310, 180, and 200 eV were used
for C, Si, and Ge, respectively, which were same as those used in VASP code. The
electron-electron exchange interaction was described using the exchange-correlation
function of Ceperley and Alder, as parameterized by Perdew and Zunger (CA-PZ) of
LDA.
5 , 6
A k - point spacing (2π×0.04 Å
− 1
) was used to generate the k - point grid within
the Monkhorst-Pack scheme.
7
The electronic iterations convergence was 5.0e- 7
eV/atom, and the force tolerance was 0.01 eV/Å. The phonon frequencies were
calculated using the linear response theory.
8 , 9
The phase transition paths were
simulated using the variable-cell nudged elastic band (VC-NEB) method.^10
Table S2. The optimized structural data for various metastable C, Si, and Ge phases
at ambient pressure.
Phase Structure S.G. a b Atomic positions C Diamond Fd - 3 m (227) 3.528 8 a (0, 0, 0) Lonsdaleite P 63/ mmc (194) 2.481 4.133 4 f (-1/3, - 2/3, - 0.563) BC8 Ia - 3 (206) 4.419 16 c (0.094, 0.094, 0.094) C 136 Fd - 3 m (227) 9. 96 g (0.942, 0.442, 0.755); 8 a (1, 0.5, 1); 32 e (0.909, 0.409, 0.909) T12 P 42/ ncm (138) 3.388 6. 4 b (0, 1, 0.5); 8 i (-0.164, 0.336, 0.357) Si Diamond Fd - 3 m (227) 5.468 8 a (0, 0, 0) Lonsdaleite P 63/ mmc (194) 3.841 6.360 4 f (-1/3, - 2/3, - 0.563) ST12 P 4321 2 (96) 5.575 6. 8 b (0.164, 0.368, - 0.253); 4 a (0.080, 0.080, - 1/2) R8 R - 3 (148) 9.281 5. 18 f (0.231, - 0.035, 0.244); 6 c (0, 0, 0.285) BC8 Ia - 3 (206) 6.543 16 c (0.102, 0.102, 0.102) Si 136 Fd - 3 m (227) 14. 96 g (0.942, 0.442, 0.755); 8 a (1, 0.5, 1); 32 e (0.909, 0.409, 0.909) T12 P 42/ ncm (138) 5.135 9. 4 b (0, 1, 0.5); 8 i (-0.164, 0.336, 0.357) Ge Diamond Fd - 3 m (227) 5.548 8 a (0, 0, 0) Lonsdaleite P 63/ mmc (194) 3.909 6.457 4 f (-1/3, - 2/3, - 0.563) R8 R - 3 (148) 9.610 5. 18 f (0.230, - 0.037, 0.238); 6 c (0, 0, 0.284) ST12 P 4321 2 (96) 5 .790 6. 8 b (0.172, 0.370, - 0.244); 4 a (0.088, 0.088, - 1/2) BC8 Ia - 3 (206) 6.764 16 c (0.102, 0.102, 0.102) Ge 136 Fd - 3 m (227) 14. 96 g (0.942, 0.442, 0.755); 8 a (1, 0.5, 1); 32 e (0.909, 0.409, 0.909) T12 P 42/ ncm (138) 5.292 9. 4 b (0, 1, 0.5); 8 i (-0.164, 0.336, 0.357)
Table S 3. The calculated equilibrium volume V 0 (Å^3 /atom), band gaps E g (eV), bulk
modulus B 0 (GPa), shear modulus G 0 (GPa), and Vickers hardness Hv (GPa) for
various metastable C, Si, and Ge phases at ambient pressure, in comparison with the
available experimental data in brackets.
Phase Structure V 0 Eg B 0 G 0 Hv C Diamond 5.49 (5.67)^24 4.20 (5.47)^24 454.6 (446)^24 545.
(96±5)^25
Lonsdaleite 5.51(5.66)^26 3.05 454.5 551.0 96. BC8 5.39 2.35 419.6 557.6 93. C 136 6.37 3.60 381.9 435.9 87. Cco-C 8 5.65 (5.56)^27 3.09 433.4 (447)^27 486.9 95. T12 5.75 3.59 424.8 479.3 94. Si Diamond 20.45 (20.02)^26 0.55 (1.11)^28 8 7.9 (99)^29 62.9 13.7 ( 13 )^30 Lonsdaleite 20.33 (20.13)^26 0.36 79.1 61.4 13. ST12 17.31 1.05 69.7 49.8 1 5. R8 17.28 0.16 83.7 66.0 15. BC8 17.51 (18.30)^31 metallic 90.5 67.9 9. Si 136 23.30 (23.01)^32 1.25 (1.9)^33 68.6 (90)^32 47.2 12. T12 20.15 0.66 88.3 50.5 14. Ge Diamond 21.35 (22.65)^26 0.34 (0.67)^28 78.0 (74.37)^34 56.9 12.8 ( 10 )^30 Lonsdaleite 21.36 (22.31)^26 semimetallic 76.1 52.2 11. R8 19.08 (20.58)^35 metallic 73.4 (73)^35 48.9 8. ST12 19.00 (20.47)^36 0.75 (1.5)^37 67.3 (102)^34 53.1 13. BC8 19.34 (20.82)^36 metallic 75.1 53.1 7. Ge 136 24.48 (25.88)^38 0.84 (0.6)^38 66.5 (76)^35 43.2 11. T12 21.81 (21.72)^39 ,^40 0.41 72.6 47.2 12.
Figure S1. Phonon dispersion curves of T12 structured C (a), Si (b), and Ge (c) at
ambient pressure.
0 100 200 300 400 500 (b) Si Frequency (cm
- ) Z A^ M G Z R X G 0 100 200 300 Frequency (cm -1) Z A^ M G Z R X G (c) Ge 0 200 400 600 800 1000 1200 (a) C Frequency (cm -1) Z A^ M^ G^ Z^ R^ X G
Figure S 2. Electronic band structures of T12 structured C (a), Si (b) and Ge (c) at
ambient pressure.
- - 0 1 2 (c) Ge Energy (eV) Z A^ M^ G^ **Z R X G
-** 0 1 2 3 (b) Si Energy (eV) Z A^ **M G Z R X G
0 2 4 6 8 10 Energy (eV) Z A**^ M G Z R X G (a) C
Figure S 4. Transition paths from β - Sn to BC8 and R8 structured Si (Si-III and
Si-XII) at 5 GPa, respectively. The structural image numbers correspond to those in
Fig. 2d.
Note:
BC8- and R8-Si have the similar structures, and R8 structure can be seen as a
reconfiguration formed through the bonding of one pair of unbonded atoms in parent
BC8 primitive cell, so the two phases appear competitive in the phase transition after
image 20 (Fig. 2d and S 4 ), consistent with experimental observation.^31
Figure S 5. Transition paths from β - Sn to T12 structured Si at 5 GPa. The structural
image numbers correspond to those in Fig. 2d.
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