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Bond-Order Potential for MD Simulation:Relaxation of Semiconductor Nanostructures
• tight binding and bond order• 4th moment approximation
• parameterization and fit • some examples
Volker Kuhlmann and Kurt ScheerschmidtMax Planck-Institute of Microstructure Physics
Halle - Germany
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accurate atomistic potential
quantum mechanicsof electrons
(slow)
large time and length scales
density functional theory
empirical potential(fast)
tight binding
bond order potential
pair potentialmany-bodycluster expansion
- transferable- few parameter- chemical bonds
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Tight Binding
exact diagonalisation
Slater-Koster integrals:
electronic part(bandstructure)
scaling part(elastic constants)
two-center approximation:
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moment
Bond Order Potential
local density of states
many atom expansion
Greens function:
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2nd moment: contribution negligible
angular function:
normalized moment:
reduced TB parameter:
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4th moment approximation
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new contributions to
torsion angle:
bond terms :
on site term :
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at constant angle of largest contribution
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at constant angle of most pronounced new angular dependence
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Potential energy above Si(100) surface
BOP2 BOP4 BOP4+
maximum
minimum minimumraised
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Parametrization and Fit
7 parameter
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smooth promotion energy
invested energy: promote one electron
Gained energy: form new bonds
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fit via Monte Carlo/ Conjugate gradient
• propose and accept/reject
fitness of set {r}:
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improved 4th moments and promotion energy
for pure carbon systems
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simulation of Si(100) waferbonding with rotational twist
Scheerschmidt and Kuhlmann, Interface Science 12 (2004)
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recursion method and local density of states
• solve Gii recursively:
• LDOS approximated by moments: moments-theorem
• semi-infinite linear chain: ai=a=0 eV bi=b=0.1 eV
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moments expansion of LDOS
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• adjust parameter to recover properties
(Ro,Ucoh,B,C11,…)• s(r) must die out suffic.
before cut off via spline
• must cut off before 2nd nearest neighbors:– # of paths of length 4 (4th moment) = Nbrs^2– 256 paths @ 16Nbrs vs. 16 paths @ 4Nbrs– 6th Moment : 64 vs. 4096
• low slopes (n,m) required by elasticity conflict with cutoff -> make a compromise