Supporting Information. Potential semiconducting and superconducting metastable Si 3 C. structures under pressure

Transcription

1 Supporting Information Potential semiconducting and superconducting metastable Si 3 C structures under pressure Guoying Gao 1,3,* Xiaowei Liang, 1 Neil W. Ashcroft 2 and Roald Hoffmann 3,* 1 State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao, Hebei, , China 2 Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, 14853, USA 3 Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, 14853, USA

2 Further details of computational methods We computed the elastic constant C ij from the stress strain relation 1 and used the Voigt Reuss Hill (VRH) approximation 2 to calculate the isotropic polycrystalline elastic moduli from the corresponding single-crystal elastic constants. Then, we used the Paugh criterion (B/G) 3 to estimate ductility, where B and G is the bulk modulus and shear modulus, respectively. follows: For the tetragonal structure, the mechanical stability criteria 4 are as C ii > 0, (i =1, 3, 4, 6), (C 11 C 12 ) > 0, (C 11 + C 33 2C 13 ) > 0, [2(C 11 + C 12 ) + C C 13 ] > 0. The Vickers hardness calculation is based on H v = [ ( H ) n ν ] 1/ n, where =.,. * e ( n e ) / ν b, N = ( n e ) * = [( Z A ) * / N CA + ( Z B ) * / N CB ], 1

3 ν b = ( d ) 3 / ν [( d ν ) 3 N ν b ], Here, H v is the Vickers hardness, d is a bond length, n is the number of bonds of type composing the actual complex crystal (e.g., Si-C, Si-Si), Ne is the number of valence electrons of type per cubic angstrom, fi is the ionicity of the binary compound in which there are -type bonds, is the number of valence electrons per bond, z or z is the valence electron number of the A or B atom constructing the bond, respectively, ν is the bond volume, and is the bond number of type ν per unit volume Dynamical and electronic properties of Si 3 C To investigate the dynamic stability of the newly predicted crystal structures, we have calculated the phonons of three optimum I-42d, R-3m-2 and R-3m-3 structures and two metastable R-3m-1 and P3m1-1 structures (Figure S3). All structures are dynamically stable, since there were no imaginary modes to be found in phonon calculations. Electronic band structure and density of states of these high-pressure phases are shown in Figure S5, each at a pressure corresponding to its stability region. The density of electronically occupied states at the Fermi level reveals the metallic character of all the new predicted high-pressure phases. The square onset in the density of states at lower energy is consistent with the layered nature of the structures. 2

4 Figure S1. Predicted metastable P3m1-1, R3m, P6 3 /mmc, P-3m1 and P3m1-2 structures for Si 3 C under pressure. Orange, large balls are Si; grey, small balls carbon. 3

5 Figure S2. The crystal structure of R-3m-1 (upper panel), electronic band structure (a), density of states (b) and COHP between Si-Si, and Si-C atoms (c) of R-3m-1 Si 3 C at 50GPa. The horizontal dashed red line in (a), (b) and (c) denotes the Fermi level. We plot the negative value of COHP to follow the convention of the chemical community. From the electronic band structure and density of states, we can see that R-3m-1 is metallic. The bonding between Si1-Si2, Si1-C is strong, as shown in the COHP. Note that as is typical of VASP calculations, the projections of the DOS do not add up to the total DOS. Substantial contributions from both Si and C may be seen throughout the DOS, indicating strong Si-C bonding. 4

6 (a) I-42d 1atm (b) I-42d 50GPa (c) R-3m-1 1atm (d) R-3m-1 300GPa (e) R-3m-2 25GPa (f) R-3m-2 250GPa (g) R-3m-3 100GPa (h) R-3m-3 400GPa 5

7 (i) P3m1-1 50GPa (j) P3m GPa Figure S3. Phonon spectra of Si 3 C: (a) and (b) I-42d at 1atm and 50GPa. (c) and (d) R-3m-1 at 1atm and 300GPa. (e) and (f) R-3m-2 at 25 and 250 GPa. (g) and (h) R-3m-3 at 100 and 400GPa. (i) and (j) P3m1-1 at 50 and 400 GPa. No imaginary frequencies in the phonon spectra for all the structures indicate they are all dynamically stable. 6

8 Figure S4. The calculated integration of -ICOHP for Si(CH 3 ) 4 at 1 atm, diamond and rocksalt SiC, R-3m-2 and R-3m-3 Si 3 C at 50 and 300 GPa, respectively. 7

9 (a) R-3m-1 1atm (b) R-3m-1 300GPa (c) R-3m-2 25GPa 8

10 (d) R-3m-2 250GPa (e) R-3m-3 250GPa (f) R-3m-3 400GPa 9

11 (g) P3m1-1 50GPa (h) P3m GPa Figure S5. Electronic band structure and density of states of Si 3 C: (a) and (b) R-3m-1 at 1atm and 300GPa. (c) and (d) R-3m-2 at 25 and 250GPa. (e) and (f) R-3m-3 at 250 and 400GPa. (g) and (h) P3m1-1 at 50 and 200GPa. All the structures would be metallic. The square onset in the density of states at lower energy is consistent with the layered nature of the structures. 10

12 Figure S6. Theoretically proposed structure for Si 3 C in the literature 12 and the experimental structure for Si 4 C 12 at 1atm. 11

13 Figure S7. Ground-state static enthalpy curves per formula unit as a function of pressure for the I-42d phase and theoretically proposed structure for Si 3 C in the literature, 12 with respect to the I-42d structure relaxed from GGA calculations. 12

14 Figure S8. Enthalpy curves per formula unit as a function of pressure for synthesized Si 4 C with respect to Si+C and SiC+Si by using GGA. 13

15 Table S1. Elastic constants C ij (GPa), bulk modulus B (GPa), shear modulus G (GPa), and the Vickers hardness H v (GPa) of I-42d Si 3 C and β-sic. C 11 C 33 C 44 C 66 C 12 C 13 B G B/G H v Si 3 C(I-42d) β-sic a 30 b a Data from experiment 13 b Data from calculations by Gao et al. 5 As shown in Table S1, the calculated Vickers hardness of β-sic is in good agreement with experimental results, 13 although our result is a little higher than that of a previous theoretical calculation. 5 The calculated Vickers hardness for Si 3 C is 20 GPa, which is much smaller than that of β-sic with 34 GPa. Since I-42d Si 3 C and β-sic possess similar diamondoid structures, the presence of Si-Si bonds in I-42d Si 3 C caused by the increase of silicon content might lead to its smaller hardness. However, the value of B/G for Si 3 C is larger than that of β-sic, showing that the ductility is improved by increased silicon concentration. 14

16 Reference (1) Page, Y. L.; Saxe, P. Symmetry-general least-squares extraction of elastic data for strained materials from ab initio calculations of stress. Phys. Rev. B 2002, 65 (10), (2) Hill, R. The Elastic Behaviour of a Crystalline Aggregate. Proc. Phys. Soc., London, Sect. A 1952, 65 (5), (3) Pugh, S. F. XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals. Philos. Mag. 2009, 45 (367), (4) Watt, J. P.; Peselnick, L. Clarification of the Hashin Shtrikman bounds on the effective elastic moduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries. J. Appl. Phys. 1980, 51 (3), (5) Gao, F.; He, J.; Wu, E.; Liu, S.; Yu, D.; Li, D.; Zhang, S.; Tian, Y. Hardness of covalent crystals. Phys. Rev. Lett. 2003, 91 (1), (6) Gilman, J. J. Why silicon is hard. Science 1993, 261 (5127), (7) Jhi, S. H.; Louie, S. G.; Cohen, M. L.; Ihm, J. Vacancy hardening and softening in transition metal carbides and nitrides. Phys. Rev. Lett. 2001, 86 (15), (8) Gilman, J. J. Flow of covalent solids at low temperatures. J. Appl. Phys. 1975, 46 (12), (9) Phillips, J. C. Ionicity of the Chemical Bond in Crystals. Rev. Mod. Phys. 1970, 42 (3), (10) Siethoff, H. Homopolar band gap and thermal activation parameters of plasticity of diamond and zinc-blende semiconductors. J. Appl. Phys. 2000, 87 (7), (11) He, J.; Wu, E.; Wang, H.; Liu, R.; Tian, Y. Ionicities of Boron-Boron Bonds in B 12 Icosahedra. Phys. Rev. Lett. 2005, 94 (1), (12) Zhang, P.; Crespi, V. H.; Chang, E.; Louie, S. G.; Cohen, M. L. Theory of metastable Group-IV alloys formed from CVD precursors. Phys. Rev. B 2001, 64 (23), (13) Westbrook, J. H.; Peyer, H. C. The Science of hardness testing and its research applications. ASM, Metals Park, Ohio,