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 Ab Initio Total-Energy Calculations for Extremely Large Systems: Application to the Takayanagi Reconstruction of Si(111)

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 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]; • 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

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

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…. • 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. • 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 Structural Average height adatoms above ideal tetrahedral positions (Å) Average length side triangle 1st layer atoms below adatom (Å) Average height rest atoms above ideal tetrahedral positions (Å) Average length dimers (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.. Increasing charge transfer between adatoms and restatoms Adatoms and restatoms can relax closer to bulk Increase in charge density between dimers – stronger covalent bonds Adatoms Restatoms Atoms in Supercell 3x3 2 0 68 5x5 6 2 200 7x7 12 6 400

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

Ab Initio Total-Energy Calculations for Extremely Large Systems: Application to the Takayanagi Reconstruction of Si(111