View source for Silicon Slab Calculations

* [[Silicon Slab Structural Relaxation | Structural Relaxation]]

==Energy Under Strain==

===PBEsol===
[[File:Si.PBEsol.Diamond-1_ML.Strain_vs_Cohesive_Energy.png]]

==Phonons==

{|  class="wikitable"
! Method
! Pseudopotential 
! Phonons
! Brillouin Zone
! Reference
|-
| rowspan="1" | DFPT-PZ <br> a = 5.424 &Aring; 
| Norm-conserving (TM)
| [[File:Si.001.PZ.freq.Fritsch1995.png]]
| [[File:Fd-3m.001.BZ.png]]
| [https://doi.org/10.1016/0039-6028(95)00802-0 Fritsch (1995)]
|-
| rowspan="1" | DFPT-PBEsol <br>(incorrect celldm(1))
| [http://www.quantum-espresso.org/wp-content/uploads/upf_files/Si.pz-n-rrkjus_psl.0.1.UPF Ultrasoft]
| [[File:old.Si.1 ML.PBEsol.freq vdos.png|frameless]]
|
|
|-
| rowspan="1" | DFPT-PBEsol
| [http://www.quantum-espresso.org/wp-content/uploads/upf_files/Si.pz-n-rrkjus_psl.0.1.UPF Ultrasoft]
| [[File:Si.1 ML.PBEsol.freq vdos.png|frameless]]
|
|
|-
|}

==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
{| class="wikitable"
! rowspan="1" | Celldm(3)
! colspan="1" | 1.0
! colspan="1" | 1.2
! colspan="1" | 1.4
! colspan="1" | 1.6
! colspan="1" | 1.8
! colspan="1" | 2.0
! colspan="1" | 2.2
! colspan="1" | 2.4
! colspan="1" | 2.6
! colspan="1" | 2.8
! colspan="1" | 3.0
|-
! rowspan="1" | Energy (Ry)
| -91.05486611
| -91.02905414
| -90.96623064
| -90.97347954
| -90.97924947
| -90.98131755 
| -90.98193937
| -90.98209842
| -90.98212589
| -90.98212146
| -90.98211064
|-
! rowspan="1" | Frequency (THz)
| 21.399668
| 14.956531
| 14.683791
| 14.821677
| 14.864115
| 14.871671
| 14.875238
| 14.875516
| 14.874547
| 14.876260
| 14.877172 
|-
! colspan="12" |
|-
! rowspan="1" | Celldm(3)
! colspan="1" | 3.0
! colspan="1" | 3.2
! colspan="1" | 3.4
! colspan="1" | 3.6
! colspan="1" | 3.8
! colspan="1" | 4.0
! colspan="1" | 4.2
! colspan="1" | 4.4
! colspan="1" | 4.6
! colspan="1" | 4.8
! colspan="1" | 5.0
|-
! rowspan="1" | Energy (Ry)
| -90.98211064
| -90.98210091
| -90.98209100
| -90.98208215
| -90.98207546
| -90.98207052
| -90.98206569
| -90.98206092
| -90.98205683
| -90.98205416
| -90.98205135
|-
! rowspan="1" | Frequency (THz)
| 14.877172
| 14.875561
| 14.875605
| 14.884300
| 14.876991
| 14.876809
| 14.882501
| 14.878010
| 14.878940
| 14.879234
| 14.870053
|-}



'''→ Minimum vacuum = 6.7180529 bohr'''


{|  class="wikitable"
! rowspan="1" | Structure
! rowspan="1" | Pseudopotential
! rowspan="1" | nq1
! rowspan="1" | nq2
! colspan="1" | nq3
! colspan="1" | Frequency (THz)
! rowspan="1" | Time
|-
| Infinite Crystal
| Si.pz-n-rrkjus_psl.1.0.0.UPF
| 4
| 4
| 4
| 14.686825
| 5010 min
|-
| 1 Monolayer
| *
| *
| *
| *
| *
| *
|-}

===K-point Mesh===

====Mesh Density====
* [http://www.quantum-espresso.org/ Quantum ESPRESSO] using [http://people.sissa.it/~dalcorso/pslibrary/index.html PS Library] 1.0 PZ RRKJ US pseudopotential with ''E''<sub>cut,wfc/rho</sub> = 30/300 Ry
* Unrelaxed atomic coordinates using PP-relaxed lattice parameter ''a'' = 10.708 Bohr and 1.6''a'' of vacuum in ''a''<sub>3</sub> direction
* NC denotes a run that ran but did not complete successfully
*
{|  class="wikitable"
|-
|
!colspan="11" | Total Energy
|-
! rowspan="1" | X/Y Mesh
! rowspan="1" | 1 1 1
! rowspan="1" | 2 2 1
! colspan="1" | 4 4 1
! colspan="1" | 6 6 1
! colspan="1" | 8 8 1
! colspan="1" | 10 10 1
! colspan="1" | 12 12 1
! colspan="1" | 14 14 1
! colspan="1" | 16 16 1
! colspan="1" | 18 18 1
! colspan="1" | 20 20 1
|-
| rowspan="1" | '''PBE'''
| NC
| -91.15925071 
| -90.98188148
| -90.98096535
| -90.98091576
| -90.98081652
| -90.98085760
| -90.98086999
| -90.98086058
| -90.98086731
| -90.98086589
|-
| rowspan="1" | '''LDA/PZ'''
| -90.82372513
| -90.65027160
| -90.64955939
| -90.64955378
| -90.64952796
| -90.64956158
| -90.64958106
| -90.64957675
| -90.64956582
| -90.64957273
| -90.64957249
|-}
* PZ/PBE mesh testing was done with celldm(1)= 10.208773524, celldm(3)=2.0, ''E''<sub>cut,wfc/rho</sub> = 30/300
{|  class="wikitable"
|-
|
!colspan="6" | Total Energy (Ry)
|-
! rowspan="1" | x y z
! rowspan="1" | * * 1
! rowspan="1" | * * 2
! colspan="1" | * * 4
! colspan="1" | * * 6
! colspan="1" | * * 8
|-
| rowspan="1" | '''1 1 *'''
| NC
| NC
| NC
| NC
| NC
|-
| rowspan="1" | '''2 2 *'''
| -90.48534325
| -90.48716920
| NC
| -90.48717819
| -90.48717698
|-
| rowspan="1" | '''4 4 *'''
| -90.50777500
| -90.50932817
| -90.50934385
| -90.50934449
| -90.50934394
|-
| rowspan="1" | '''6 6 *'''
| -90.50934449
| NC 
| NC
| -90.50718812
| not run
|-
| rowspan="1" | '''8 8 *'''
| NC
| not run
| not run
| not run 
| NC
|}

====Mesh Center====
* [http://www.quantum-espresso.org/ Quantum ESPRESSO] using [http://people.sissa.it/~dalcorso/pslibrary/index.html PS Library] 1.0 PZ RRKJ US pseudopotential with ''E''<sub>cut,wfc/rho</sub> = 50/500 Ry
* Relaxed atomic coordinates using PP-relaxed lattice parameter ''a'' = 10.708 Bohr and 1.6''a'' of vacuum in ''a''<sub>3</sub> direction
* Smallest converged grid parameter was 2 2 2

{| class="wikitable"
|-
! 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====
{|  class="wikitable"
|-
! Modification
! Total Energy (Ry / 8 atoms)
! Total Force (Ry / a<sub>0</sub> / 8 atoms)
!colspan="3" | &omega; [&Gamma; (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=====

&Gamma;-point, lowest frequency mode

{|  class="wikitable"
|-
! Displacement Factor
! Total Energy (Ry / 8 atoms)
! Total Force (Ry / a<sub>0</sub> / 8 atoms)
!colspan="3" | &omega; [&Gamma; (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====
{|  class="wikitable"
|-
! Modification
! Total Energy (Ry / 8 atoms)
! Total Force (Ry / a<sub>0</sub> / 8 atoms)
!colspan="3" | &omega; [&Gamma; (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 &Aring;<sup>3</sup>

==Annotated Bibliography==

* [https://dx.doi.org/10.1088/1367-2630/8/9/200 Pengpeng Zhang et al 2006 New J. Phys. 8 200] <br />Page 12:"A major difference now is that the π∗ surface band introduces a high density of available electronic states in the Si bandgap, acting therefore as a sink for electrons." <br />Figure 4: "The existence of the surface bands results in a reduced effective bandgap (∼0.35eV in this example)between the bulk VBM of the SiNM and the bottom of the surface π∗ band."

* [https://dx.doi.org/10.1088/0953-8984/25/19/195801 Rodrigo G Amorim et al 2013 J. Phys.: Condens. Matter 25 195801] <br />No bulk nanomembrane comparison on band gap magnitude or surface state contribution to band gap reduction.

* [https://doi.org/10.1021/acsami.0c22360 Dirk Konig et al ACS Appl. Mater. Interfaces 2021, 13, 17, 20479–20488] <br />No immediate relevance, archived here for potential revisiting/posterity.

* [https://cdmf.org.br/wp-content/uploads/2021/09/A-new-multifunctional-two-dimensional-monolayer-based-on-silicon-carbide.pdf Nicolas Martins et al]

* [https://link.springer.com/content/pdf/10.1007/s12633-021-01143-y.pdf Ayda Karbakhshzadeh]