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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
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
Tight Binding
exact diagonalisation
Slater-Koster integrals:
electronic part(bandstructure)
scaling part(elastic constants)
two-center approximation:
moment
Bond Order Potential
local density of states
many atom expansion
Greens function:
2nd moment: contribution negligible
angular function:
normalized moment:
reduced TB parameter:
4th moment approximation
new contributions to
torsion angle:
bond terms :
on site term :
at constant angle of largest contribution
at constant angle of most pronounced new angular dependence
Potential energy above Si(100) surface
BOP2 BOP4 BOP4+
maximum
minimum minimumraised
Parametrization and Fit
7 parameter
smooth promotion energy
invested energy: promote one electron
Gained energy: form new bonds
fit via Monte Carlo/ Conjugate gradient
• propose and accept/reject
fitness of set {r}:
improved 4th moments and promotion energy
for pure carbon systems
simulation of Si(100) waferbonding with rotational twist
Scheerschmidt and Kuhlmann, Interface Science 12 (2004)
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
moments expansion of LDOS
• 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