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
Outline
1) Surface reconstructions - Si (111)
2) Total Energy Calculations
3) Geometry optimisation
4)Results
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
Si (111)-(7x7) – Takayanagi
Reconstruction
Adatoms Restatoms







Atoms in
Supercell
3x3
2
0
68
5x5
6
2
200
7x7
12
6
400
In practice see a 7x7 structure
1959 - Observed experimentally by LEED
1985 - 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
Total Energy Calculations – DFT.

Ground state density function no(r) minimises total energy
functional E [n];
Eno  Eo

Minimise E[n] wrt variations in n to give Kohn-Sham
equations
1
(   2  Veff ) j   j
2

Solve self consistently to give wavefunction and
hence ground state density;

n r     i ri
2
i

We can now calculate the total energy
Tricks in the Calculation – Supercells.

Calculation scales as N3; want as few atoms as
possible – use a supercell
Tricks in the Calculation – Supercells.

Calculation scales as N3; want as few atoms as
possible – use a supercell
Tricks in the Calculation – Supercells.

Calculation scales as N3; want as few atoms as
possible – use a supercell
Tricks in the Calculation – Supercells.

Calculation scales as N3; want as few atoms as
possible – use a supercell
Tricks in the Calculation – Supercells.

Calculation scales as N3; want as few atoms as
possible – use a supercell
Tricks in the Calculation – Supercells.

Calculation scales as N3; want as few atoms as
possible – use a supercell
Tricks in the Calculation – Supercells.

Vacuum gap width? Cell Dimensions? Initial
atomic configuration?
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)
Problems – Pseudopotentials.



Many plane waves
needed near to ion cores
PSEUDOPOTENTIAL;
weak effective potential
Outside critical radius
Vpseudo Z/r
pseudo 
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
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
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
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
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
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
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
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.
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.
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.
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.
Results – Energy and Structure.
Surface Energies
3x3
5x5
7x7
Energy per unit cell (eV)
10.765
29.205
56.509
Energy per surface atom (eV)
1.196
1.168
1.153
Average height
Average
rest atoms
length dimers
above ideal
(Å)
tetrahedral
positions (Å)
Structural
Average height
adatoms above
ideal tetrahedral
positions (Å)
Average length
side triangle 1st
layer atoms below
adatom (Å)
3x3
0.554
3.566
…
2.455
5x5
0.521
3.600
0.226
2.451
7x7
0.508
3.618
0.201
2.442
Results – Charge Density..
Adatoms Restatoms



Atoms in
Supercell
3x3
2
0
68
5x5
6
2
200
7x7
12
6
400
Increasing charge transfer between adatoms and
restatoms
Adatoms and restatoms can relax closer to bulk
Increase in charge density between dimers –
stronger covalent bonds
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