Silicon Slab Calculations

• Structural Relaxation

Phonons

Method	Pseudopotential	Phonons	Brillouin Zone	Reference
DFPT-PZ a = 5.424 Å	Norm-conserving (TM)	File:Si.001.PZ.freq.Fritsch1995.png	File:Fd-3m.001.BZ.png	Fritsch (1995)
DFPT-PBEsol (incorrect celldm(1))	Ultrasoft	File:Old.Si.1 ML.PBEsol.freq vdos.png
DFPT-PBEsol	Ultrasoft	File:Si.1 ML.PBEsol.freq vdos.png

Tests

Vacuum

• Si.pbe-n-rrkjus_psl.1.0.0.UPF, Ecut wfc/rho = 50/500, startingwfc='atomic', celldm(1)=10.335466, kpoint: 2-2-1-1-1-0, Gamma q-point

→ Minimum vacuum = 6.7180529 bohr

Celldm(3)	1.0	1.2	1.4	1.6	1.8	2.0	2.2	2.4	2.6	2.8	3.0
Energy (Ry)	-91.05486611	-91.02905414	-90.96623064	-90.97347954	-90.97924947	-90.98131755	-90.98193937	-90.98209842	-90.98212589	-90.98212146	-90.98211064
Frequency (THz)	21.399668	14.956531	14.683791	14.821677	14.864115	14.871671	14.875238	14.875516	14.874547	14.876260	14.877172
Celldm(3)	3.0	3.2	3.4	3.6	3.8	4.0	4.2	4.4	4.6	4.8	5.0
Energy (Ry)	-90.98211064	-90.98210091	-90.98209100	-90.98208215	-90.98207546	-90.98207052	-90.98206569	-90.98206092	-90.98205683	-90.98205416	-90.98205135
Frequency (THz)	14.877172	14.875561	14.875605	14.884300	14.876991	14.876809	14.882501	14.878010	14.878940	14.879234	14.870053

K-point Mesh

Mesh Density

• Quantum ESPRESSO using PS Library 1.0 PZ RRKJ US pseudopotential with E_cut,wfc/rho = 30/300 Ry
• Unrelaxed atomic coordinates using PP-relaxed lattice parameter a = 10.708 Bohr and 1.6a of vacuum in a₃ direction
• NC denotes a run that ran but did not complete successfully

Structure	Pseudopotential	nq1	nq2	nq3	Frequency (THz)	Time
Infinite Crystal	Si.pz-n-rrkjus_psl.1.0.0.UPF	4	4	4	14.686825	5010 min
1 Monolayer	*	*	*	*	*	*

• PZ/PBE mesh testing was done with celldm(1)= 10.208773524, celldm(3)=2.0, E_cut,wfc/rho = 30/300

	Total Energy
X/Y Mesh	1 1 1	2 2 1	4 4 1	6 6 1	8 8 1	10 10 1	12 12 1	14 14 1	16 16 1	18 18 1	20 20 1
PBE	NC	-91.15925071	-90.98188148	-90.98096535	-90.98091576	-90.98081652	-90.98085760	-90.98086999	-90.98086058	-90.98086731	-90.98086589
LDA/PZ	-90.82372513	-90.65027160	-90.64955939	-90.64955378	-90.64952796	-90.64956158	-90.64958106	-90.64957675	-90.64956582	-90.64957273	-90.64957249

	Total Energy (Ry)
x y z	* * 1	* * 2	* * 4	* * 6	* * 8
1 1 *	NC	NC	NC	NC	NC
2 2 *	-90.48534325	-90.48716920	NC	-90.48717819	-90.48717698
4 4 *	-90.50777500	-90.50932817	-90.50934385	-90.50934449	-90.50934394
6 6 *	-90.50934449	NC	NC	-90.50718812	not run
8 8 *	NC	not run	not run	not run	NC

Mesh Center

• Quantum ESPRESSO using PS Library 1.0 PZ RRKJ US pseudopotential with E_cut,wfc/rho = 50/500 Ry
• Relaxed atomic coordinates using PP-relaxed lattice parameter a = 10.708 Bohr and 1.6a of vacuum in a₃ direction
• Smallest converged grid parameter was 2 2 2

Mesh Center Point	Total Energy (Ry)	SCF Cycles
0 0 0	-90.48716920	93
1 0 0	-90.52916013	61
0 0 1	-90.48718293	54
1 1 0	-90.48939869	38
1 1 1	-90.48939583	63

• Add 10 10 1 at different k-point grid offsets
• At best k-point grid offset with 2 2 1 and 2 2 2 k-point grids, check total energy change for celldm(3) = 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, etc. until less than 0.0001 Ry
• Then, run all x x 1 at the most stable k-point grid offset until total energy changes less than 0.0001 Ry

Structure

• DFT-PBE

1-ML

Modification	Total Energy (Ry / 8 atoms)	Total Force (Ry / a0 / 8 atoms)	ω [Γ (1-3)] (THz)
Bottom of Cell	-90.9808882393	0.001189	-1.707084	-0.235194	-0.132992
Centered in Cell	-91.05494884	0.000220	-0.317949	-0.195414	0.172351
Built-In Phonon	-91.05494922	0.000241	-0.329813	-0.229082	0.016019
2x2 Supercell	-91.05616393	0.000913

Build In Displacement Vector

Γ-point, lowest frequency mode

Displacement Factor	Total Energy (Ry / 8 atoms)	Total Force (Ry / a0 / 8 atoms)	ω [Γ (1-3)] (THz)
0.0	-91.05490108	0.001967
0.1	-91.054481	0.016412	-0.322887	-0.189111	0.340710
0.2	-91.053070	0.032972	-0.364120	-0.221765	0.371472
0.3	-91.050710	0.049779	-0.308852	-0.191316	0.326616
0.4	-91.047394	0.066933	-0.348059	-0.270962	0.367059
0.5	-91.043114	0.084544	-0.284463	-0.151811	0.320345
0.6	-91.037859	0.102744	-0.345105	-0.153570	0.498687
0.7	-91.031620	0.121659	-1.000608	-0.236214	-0.135593
0.8	-91.024383	0.141412	-1.583491	-0.278128	-0.180348
0.9	-91.016139	0.162127	-2.078362	-0.369162	-0.281689
1.0	-91.006881	0.183921	-2.543886	-0.341077	-0.267751

2-ML

Modification	Total Energy (Ry / 8 atoms)	Total Force (Ry / a0 / 8 atoms)	ω [Γ (1-3)] (THz)
Bottom of Cell	-91.118108725	0.000038	-7.411048	-3.947619	-1.719877
Centered in Cell	-91.11810858	0.0000405	-0.317949	-0.195414	0.172351

Volume

PBE diamond bulk 1102.9319 bohr^3 = 20.4971 ų

Annotated Bibilography

• (reference)
  page 12:
  Figure 4: