Bond-Order Potential for MD Simulation: Relaxation of Semiconductor Nanostructures

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