1. Ab Initio Total-Energy Calculations for Extremely Large Systems: Application to the Takayanagi Reconstruction of Si(111)
Phys. Rev. Lett., Vol. 68, Number 9, p1351, March 1992.
I. Štich, M. C. Payne, R. D. King-Smith, J-S. Lin (Cavendish Laboratory, University of Cambridge)
L. J. Clarke
(Edinburgh Parallel Computer Centre, University of Edinburgh)
Chris Eames

2. Outline Surface reconstructions - Si (111) Total Energy Calculations
Geometry optimisation
Results

3. Si (111) – Adatoms and Restatoms
T4; on Top of a second layer atom with 4 nearest neighbours. Lower energy than H3
Problems;
Adatoms pull surface atoms closer together to form better bonds (in terms of lengths and angles) – compressive strain
Adatom still has 1 dangling bond – charge imbalance
1 in 4 surface atoms not bonded to adatom – restatom
No dangling bonds – electrons on adatom and restatom paired by charge transfer
Restatom relaxes into bulk - tensile strain to balance compressive strain

4. Si (111)-(7x7) – Takayanagi Reconstruction
400 6 12
7x7
200 2
5x5 68
3x3
Atoms in Supercell
Restatoms
Adatoms
In practice see a 7x7 structure
Observed experimentally by LEED
D.A.S. model resolves structure problem
Dimers formed by atom pairing in subsurface layer
Adatoms; 12 in unit cell locally arranged in 2x2 pattern
Stacking fault in one half of unit cell
Can also get a 3x3 and 5x5 reconstruction

5. Total Energy Calculations – DFT.
Ground state density function no(r) minimises total energy functional E [n];
Minimise E[n] wrt variations in n to give Kohn-Sham equations
Solve self consistently to give wavefunction and hence ground state density;
We can now calculate the total energy

6. Tricks in the Calculation – Supercells.
Calculation scales as N3; want as few atoms as possible – use a supercell

7. Tricks in the Calculation – Supercells.
Calculation scales as N3; want as few atoms as possible – use a supercell

8. Tricks in the Calculation – Supercells.
Calculation scales as N3; want as few atoms as possible – use a supercell

9. Tricks in the Calculation – Supercells.
Calculation scales as N3; want as few atoms as possible – use a supercell

10. Tricks in the Calculation – Supercells.
Calculation scales as N3; want as few atoms as possible – use a supercell

11. Tricks in the Calculation – Supercells.
Calculation scales as N3; want as few atoms as possible – use a supercell

12. Tricks in the Calculation – Supercells.
Vacuum gap width? Cell Dimensions? Initial atomic configuration?

13. Tricks in the Calculation – Plane Waves.
Periodic supercells; plane wavis basis set for wavefunctions
K-point sampling – in this work one k-point sampled in the Brillouin zone for 5x5 and 7x7, 4 k-points for 3x3
Mathematically simple – easily cast into matrix form (solve Kohn-Sham by diagonalisation)
Cutoff energy – in this case 95.2 eV (7 Ry)

14. Problems – Pseudopotentials.
Many plane waves needed near to ion cores
PSEUDOPOTENTIAL; weak effective potential
Outside critical radius

15. Function Minimisation – Steepest Descents.
Search energy landscape for structure with minimum energy
Pick a point
Pick a search direction (in this case that of local gradient)
Make a step and find minimum along line
Pick a point….
Not most efficient way

16. Function Minimisation – Steepest Descents.
Search energy landscape for structure with minimum energy
Pick a point
Pick a search direction (in this case that of local gradient)
Make a step and find minimum along line
Pick a point….
Not most efficient way

17. Function Minimisation – Steepest Descents.
Search energy landscape for structure with minimum energy
Pick a point
Pick a search direction (in this case that of local gradient)
Make a step and find minimum along line
Pick a point….
Used for Ion relaxation

18. Function Minimisation – Steepest Descents.
Search energy landscape for structure with minimum energy
Pick a point
Pick a search direction (in this case that of local gradient)
Make a step and find minimum along line
Pick a point….
Used for Ion relaxation

19. Function Minimisation – Steepest Descents.
Search energy landscape for structure with minimum energy
Pick a point
Pick a search direction (in this case that of local gradient)
Make a step and find minimum along line
Pick a point….
Used for Ion relaxation

20. Function Minimisation – Steepest Descents.
Search energy landscape for structure with minimum energy
Pick a point
Pick a search direction (in this case that of local gradient)
Make a step and find minimum along line
Pick a point….
Used for Ion relaxation

21. Function Minimisation – Steepest Descents.
Search energy landscape for structure with minimum energy
Pick a point
Pick a search direction (in this case that of local gradient)
Make a step and find minimum along line
Pick a point….
Used for Ion relaxation
Convergence when forces <0.1eV/Å

22. Minimisation – Conjugate Gradients
To avoid searching in directions that have been searched before, pick a set of n Conjugate Directions
In each search direction take only one step and make that step just the right length to line up with the minimum
After n steps the function will be minimized over all searched directions.

23. Minimisation – Conjugate Gradients
To avoid searching in directions that have been searched before, pick a set of n Conjugate Directions
In each search direction take only one step and make that step just the right length to line up with the minimum
After n steps the function will be minimized over all searched directions.

24. Minimisation – Conjugate Gradients
To avoid searching in directions that have been searched before, pick a set of n Conjugate Directions
In each search direction take only one step and make that step just the right length to line up with the minimum
After n steps the function will be minimized over all searched directions.

25. Minimisation – Conjugate Gradients
To avoid searching in directions that have been searched before, pick a set of n Conjugate Directions
In each search direction take only one step and make that step just the right length to line up with the minimum
After n steps the function will be minimized over all searched directions.
Used for electronic minimisation.

26. Results – Energy and Structure.
1.153
1.168
1.196
Energy per surface atom (eV)
56.509
29.205
10.765
Energy per unit cell (eV)
7x7
5x5
3x3
Surface Energies
2.442
0.201
3.618
0.508
7x7
2.451
0.226
3.600
0.521
5x5
2.455 …
3.566
0.554
3x3
Average length dimers (Å)
Average height rest atoms above ideal tetrahedral positions (Å)
Average length side triangle 1st layer atoms below adatom (Å)
Average height adatoms above ideal tetrahedral positions (Å)
Structural

27. Results – Charge Density..
400 6 12
7x7
200 2
5x5 68
3x3
Atoms in Supercell
Restatoms
Adatoms
Increasing charge transfer between adatoms and restatoms
Adatoms and restatoms can relax closer to bulk
Increase in charge density between dimers – stronger covalent bonds

28. Conclusions and Summary.
7x7 is the lowest energy structure
Evidence of saturation across series 3x3 – 7x7 indicating 9x9 etc. energetically unfavourable
Charge density plots show as adatom/restatom ratio increases so does charge transfer
This allows relaxation into bulk to decrease strain