Silicon

• Slab calculations
• Bulk-slab comparison
• Parameter checks

Crystal Structures

• Diamond
• Beta-Tin

Elastic Properties

• Energies calculated with pseudopotentials (PP) using Quantum ESPRESSO and FP-LAPW with Elk
• Energies fit to a Murnaghan equation of state

Our Calculations

Method	Core Electron Representation	Diamond				Beta-tin					Transition
		V0 (Å3/atom)	K0 (GPa)	K0′	VT (Å3/atom)	V0 (Å3/atom)	c/a	K0 (GPa)	K0′	VT (Å3/atom)	PT (GPa)
DFT-LDA	Von Barth-Car PP	19.70	94.2	4.16		15.31	0.564	90.3	15.0
	Bachelet-Hamman-Schlüter PP	19.32	98.2	4.30		14.42	0.549	120.6	4.31
	PSLibrary 0.1 Ultrasoft PP	19.68	96.4	4.19		14.80	0.517	118.4	4.01
	FP-LAPW [Elk]	19.67	96.2	6.04
DFT-PBE	Ultrasoft	20.46	88.6	4.31		15.39	0.561	107.5	4.57
	KJPAW	20.52	87.6	4.2	18.89	15.35	0.557	105.1	4.59	14.31	8.68
PBEsol	KJPAW	20.08	92.9	4.15	19.02	15.01	0.553	113.9	4.43	14.36	5.63
DFT-TPSS	Ultrasoft	19.73	74.1	1.00
DFT-PZ	VBC	19.74	91.7	4.07	18.39	15.01	0.557	111.1	4.4	14.15	7.49
	RRKJUS	19.72	94.8	4.13	18.55	14.81	0.554	114.7	4.44	14.07	6.63
SCAN	VAN	19.53	95.1	7.92	17.52	15.29	0.569	119.5	0.79	13.43	16.39
	VAN.2	19.78	87.6	8.5	19.02	15.23	0.571	112.6	2.07	14.7	4.07
	VAN.3	19.61	94.6	7.3	17.56	15.33	0.57	110.1	1.96	13.47	16.12
	VAN.4	19.59	95.4	7.53	17.57	15.33	0.57	110.1	1.96	13.47	16.14
TPSS	VAN	19.59	87.2	8.8	18.2	15.04	0.561	116.8	1.69	13.98	9.04

Reference Calculations

Hennig et al. 2010 PRB

Method	Core Electron Representation	Diamond				Beta-tin					Transition
		V0 (Å3/atom)	K0 (GPa)	K0′	VT (Å3/atom)	V0 (Å3/atom)	c/a	K0 (GPa)	K0′	VT (Å3/atom)	PT (GPa)
LDA		19.72	96.4	4.13	17.86	14.82	0.548	116.0	4.59	13.63	5.8
PBE		20.48	89.0	4.12	18.42	15.36	0.550	103.6	4.52	14.04	8.4
PW91		20.45	89.1	4.14	18.40	15.47	0.551	103.6	4.52	14.11	10.0
WC		20.05	94.2	4.10	18.12	15.11	0.549	112.4	4.41	13.86	6.0
AM05		20.07	93.1	4.08	18.13	14.82	0.546	120.5	4.54	13.67	3.5
PBEsol		20.06	93.9	4.09	18.13	15.02	0.548	115.0	4.52	13.80	4.8
HSE06		20.07	99.1	4.00	18.21	15.10	0.565	117.0	4.35	13.89	13.4
DMC		19.98	98	4.6	18.14(5)	15.2	0.550	107	4.6	13.9(1)	14.0±1.0
EXP		20	99.2	4.11	18.15					13.96	11.3-12.6

Poisson Ratio

• ν = 0.28 (Hopcroft, 2010)
• ν = -slope of y=log(a), x=log(c) (Chou, 1983)

Phonons

Frequencies by Q-Point

Method	Pseudopotential	Phonon Frequencies (THz)
		ΓLO/TO
Experiment
DFPT-LDA	Ultrasoft
	Norm-conserving
DFPT-PBE	Ultrasoft
DFPT-PBEsol	Ultrasoft

Frequency Bands

Method	Pseudopotential	Phonons
DFPT-LDA	Ultrasoft	File:Si.fd-3m.LDA USPP.freq vdos.png
	Norm-conserving	File:Giannozzi1991 Si.LDA vBCNCPP.freq vdos.png<ref name="Giannozzi1991" />
DFPT-PBE	Ultrasoft	File:Si.Fd-3m.PBE.freq vdos.png
DFPT-PBEsol	Ultrasoft	File:Si.Fd-3m.PBEsol.freq vdos.png

Cutoff Energy Calculations

Method	Pseudopotential	Diamond	Beta-Tin	Difference
PBE	RRKJ	File:DiamondCutoff RRKJ.png	File:BetaSn cutoff.png	File:Cutoffdifference RRKJ.png

Kpoint Energy Calculations

Method	Pseudopotential	Diamond	Beta-Tin	Difference
PBE	RRKJ	File:DiamondKpoint RRKJ.png	File:BetaSnKpoint RRKJ.png	File:Kpointdifference RRKJ.png

Equation of State Graphs

Method	Pseudopotential	Diamond	Beta-Tin	MultiPlot
PBE	RRKJ	File:DiamondEOS RRKJ.png	File:BetaSneos RRKJ.png	File:Eos RRKJ.png

References

• Giannozzi, P., de Gironcoli, S., Pavone, P. and S. Baroni, Ab initio calculation of phonon dispersions in semiconductors, Phys. Rev. B, 43, 7231 (1991).
• Hopcroft, M. A., Nix, W. D., and Kenny, T. W., "What is the Young's Modulus of Silicon?", J. Microelectromech. S. 19, 1057-7157 (2010).
• Chou, M. Y., Lam, P. K., and Cohen, M. L., "Ab initio study of structural and electronic properties of beryllium", Phys. Rev. B, 28, 4179 (1983).
• Hennig, R. G., Wadehra, A., Driver, K. P., Parker, W. D., Umrigar, C. J., and Wilkins, J. W., "Phase transformation in Si from semiconducting diamond to metallic β-Sn phase in QMC and DFT under hydrostatic and anisotropic stress", Phys. Rev. B, 82, 014101 (2010).