Supporting information. The Unusual and the Expected in the Si/C Phase Diagram. Guoying Gao, N. W. Ashcroft and Roald Hoffmann.

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1 Supporting information The Unusual and the Expected in the Si/C Phase Diagram Guoying Gao, N. W. Ashcroft and Roald Hoffmann Table of Contents Computational Methods...S1 Hypothetical Structures for Si 3 C and SiC 3 at 1atm,,,,,,,,..S3 Phonon Spectra of the Predicted Structures for Si 3 C and SiC 3 at 1atm..S5 Total Density of States and band structures of the Predicted Structures for SiC 3 at 1atm with PBE and HSE Hybrid Functionals...S7, S8 Structural Information for the Predicted Structures for Si 3 C and SiC 3, and Comparison of Theoretical and Experimental Lattice Constants and Bond Lengths at 1atm... S9, S10 References..S11 S1

2 Computational Methods We searched for Si 3 C, SiC, SiC 3 ground-state structures using Crystal structure AnaLYsis by Particle Swarm Optimization methodology, 1,2 as implemented in the CALYPSO code. 3 This method has been successfully utilized to predict high pressure structures of Li, Mg, and Bi 2 Te 3. 4,5,6 We performed structure searches with system sizes containing up to 8 formula units (f.u.) per simulation cell in their ground-state and at 1 atm. Each generation contain 30 structures, 60% of which are generated by particle swarm optimization, and the others by random choice. We followed generations (depending on the size of the system) to achieve the converged structure. The underlying ab initio structural relaxations were carried out with density functional theory using the Perdew-Burke-Ernzerhof exchange-correlation functional 7 as implemented in the VASP code. 8 The frozen-core all-electron projector-augmented wave method 9 was adopted, with 2s 2 2p 2 (cutoff radius 1.5a 0 ) and 3s 2 3p 2 (cutoff radius 1.9a 0 ) treated as valence electrons for C and Si, respectively. An energy cutoff of 600 ev and appropriate Monkhorst-Pack 10 k meshes were chosen to ensure that enthalpy calculations were well converged to better than 1 mev/formula unit (f.u.). The phonon calculations were carried out by using a supercell approach 11 within the PHONOPY code. 12 S2

3 Hypothetical structures for Si 3 C and SiC 3 at 1atm. Figure S1 (a) and (b) are P-4m2 and R-3m structures for Si 3 C built from diamond structure with one Si substituting for one C or α-silicon structure with one C substituting for one Si. The orange (large) and grey (small) spheres represent Si and C, respectively. (a) Cmcm S3

4 (b) Cm (c) Cmcm Figure S2. Three hypothetical SiC 3 structures at 1atm (a) Cmcm, (b) Cm, and (c) Cmcm. S4

5 Phonon spectra of I-42d, P-4m2, R-3m and C222 1 for Si 3 C and Cm, R-3m, Cmc2 1, P-4m2, Cmmm, Cmcm (4+8), Cmcm_(6), and Cm_(5+6+7) for SiC 3 at 1atm. Figure S3. Phonon dispersion curves of I-42d, P-4m2, R-3m and C Si 3 C at 1atm. S5

6 Figure S4. Phonon dispersion curves of Cm, R-3m, Cmc2 1, P-4m2, Cmmm, Cmcm_4+8, Cmcm_6, and Cm_5+6+7-SiC 3 at 1atm S6

7 Total electronic density of states and band structures for Cm, Cmc2 1, Cmmm, R-3m and P-4m2-SiC 3 at 1atm with PBE and HSE hybrid functionals, respctively. Figure S5 Total electronic densities of states for Cm, Cmc2 1, Cmmm, R-3m and P-4m2-SiC 3 at 1atm with PBE and HSE hybrid functionals, respectively. From the total DOS for various structures of SiC 3, Figure S5, both Cm and Cmc2 1 have a very low DOS (0.02states/eV/electron) at the Fermi level, which may not support SiC 3 as a good superconductor at 1atm. We can see clearly the 2-dimentional characterization at lower energy in Cmmm and benzenoid net in R-3m, indicated by the very sharp peak at low pressure, although it is not obvious in Cm and Cmc2 1. The calculated band gaps with HSE hybrid functional increase for Cmmm and R-3m, but metallic character for Cm and Cmc21 still remains. S7

8 Figure S6 Band structure for Cm, Cmc2 1, Cmmm, and R-3m -SiC 3 at 1atm with PBE functional. S8

9 Structural information for the predicted structures for Si 3 C and SiC 3 at 1atm, and a comparison of theoretical and experimental lattice constants and bond lengths. Table S1: Lattice parameters of the predicted various structures for Si 3 C and SiC 3 at 1atm. The angles between b and c, a and c, and a and b are defined as α, β, and γ, respectively, and given in degrees. Si 3 C SiC 3 Space group I-42d C222 1 R-3m P-4m2 Cm Cmc2 1 Cmmm Lattice parameters(å, ) a= c= a= b= c= a= c= γ=120 a= c= a= b= c= β= a= b= c= a= b= c= Atomic coordinates(fractional) Si: 8d ( , 0.25, 0.125) Si: 4a (0, 0.5, 0.25) C: 4b (0, 0.5, 0.75) Si: 8c (0.1698, , ) Si: 4b (0, , 0.25) C: 4a ( , 0.5, 0.5) Si: 6c (0,0, ) C: 18h ( , , ) Si: 1a (0,0,0) C: 1c (0.5, 0.5, 0.5) C: 2g (0.5, 0, ) Si: 2a ( , 0, ) Si: 2a ( , 0, ) C: 2a ( , 0, ) C: 2a ( , 0, ) C: 2a ( , 0, ) C: 2a ( , 0, ) C: 2a ( , 0, ) C: 2a ( , 0, ) Si: 4a (0, , ) C: 4a (0, , ) C: 4a (0, , ) C: 4a (0, , ) Si: 4g ( , 0, 0) C: 4k (0, 0, ) C: 8o ( , 0, ) S9

10 Table S2: Comparison of the calculated and experimental lattice constants and bond lengths for C, Si and SiC, respectively. Lattice constants(å) Bond lengths(å) PBE Exp. PBE Exp. C (C-C) a a Si (Si-Si) a a SiC(Si-C) a a a Heyd, J.; Peralta, J. E.; Scuseria, G. E.; Martin, R. L. J. Chem. Phys. 2005, 123, Table S3. Enthalpies of various structures predicted for SiC 3 at 1atm. Structure Enthalpy (Hartree/Si 3 C or SiC 3 ) Si 3 C I-42d C R-3m P-4m SiC 3 Cm Cmc Cmmm S10

11 References: (1) Wang, Y.; Lv, J.; Zhu, L.; Ma, Y.; Phys. Rev. B 2010, 82, (2) Lv, J.; Wang, Y.; Zhu, L.; Ma, Y. J. Chem. Phys. 2012, 137, (3) Wang, Y.; Lv, J.; Zhu, L.; Ma, Y. Comput. Phys. Commun. 2012, 183, 2063 ; (4) Li, P.; Gao, G.; Wang, Y.; Ma, Y. J. Phys. Chem. C 2011, 114, (5) Lv, J.; Wang, Y.; Zhu, L.; Ma, Y. Phys. Rev. Lett. 2011, 106, (6) Zhu, L.; Wang, H.; Wang, Y.; Lv, J.; Ma, Y.; Cui, Q.; Ma, Y.; Zou, G. Phys. Rev. Lett. 2011, 106, (7) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett.1996, 77, (8) Kresse, G.; Furthmüler, J. Phys. Rev. B 1996, 54, (9) Blöchl, P.E. Phys. Rev. B 1994, 50, (10) Monkhorst, H.J.; Pack, J. D. Phys. Rev. B 1976, 13, (11) Parlinski, K.; Li, Z.Q.; Kawazoe, Y. Phys. Rev. Lett. 1997, 78, (12) Togo, A.; Oba, F.; Tanaka, I. Phys. Rev. B 2008, 78, S11