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 Surface reconstructions - Si (111) Total Energy Calculations Geometry optimisation 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 400 6 12 7x7 200 2 5x5 68 3x3 Atoms in Supercell Restatoms Adatoms 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]; 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

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

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. 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

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

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