Úvod do tímového projektu

Peter Ballo

Katedra fyziky

Fakulta elektrotechniky a informatiky

Interatomic Potentials

• Before we can start a simulation, we need the model!

• Interactions between atoms, molecules,… are determined by quantum mechanics:

– Schrödinger Equation + Born-Oppenheimer (BO) approximation

– BO: Because electrons T is so much higher (1eV=10,000 K) than true T and they move so fast, we can get rid of electrons and consider interaction of nuclei in an effective potential “surface.” V(R).

– Approach does not work during chemical reactions.

• Crucial since V(R) determines the quality of result.

• But we don’t know V(R).

– Semi-empirical approach: make a good guess and use experimental data to fix it up

– Quantum chemistry approach: works in a real space.

– Ab initio approach: it works really excellent but…

Semi-empirical potentials• Assume a functional form, e.g. 2-body form.• Find some data: theory + experiment• Use theory + simulation to fit form to data.

• What data? – Atom-atom scattering in gas phase– Virial coefficients, transport in gas phase– Low-T properties of the solid, cohesive energy, lattice constant,

bulk modulus.– Melting temperature, critical point, triple point, surface tension,

….• Interpolation versus extrapolation. • Are results predictive?

Some tests

-Lattice constant

-Bulk modulus

-Cohesive energy

-Vacancy formation energy

-Property of an impurity

Lennard-Jones potential V(R) = i<jv(ri-rj) v(r) = 4[(/r)12- (/r)6]

= minimum

= wall of potential

Reduced units:– Energy in – Lengths in

Good model for rare gas atoms

Phase diagram is universal!

(for rare gas systems)

.

Silicon potential• Solid silicon is NOT well described by a pair potential.

• Tetrahedral bonding structure caused by the partially filled p-shell: sp3 hybrids (s+px+py+pz , s-px+py+pz , s+px-py+pz , s+px+py-pz)

• Stiff, short-ranged potential caused by localized electrons.

• Stillinger-Weber (1985) potential fit to:

Lattice constant,cohesive energy, melting point, structure of liquid Si

for r<a

• Minimum at 109o

ri

rk

rj

i

v2(r) (B /r4 – A)e(r a) 1

v3(r) i, j,k e

/(rij a)/(rik a)[cosijk 1/3]2

Metallic potentials• Have a inner core + valence electrons

• Valence electrons are delocalized. Pair potentials do not work very well. Strength of bonds decreases as density increases because of Pauli principle.

• EXAMPLE: at a surface, LJ potential predicts expansion but metals contract

• Embedded Atom Method (EAM) or glue models better.

Daw and Baskes, PRB 29, 6443 (1984).

Embedding function electron density pair potential

• Good for spherically, closed-packed, symmetric atoms: FCC Cu, Al, Pb

• Not so good for BCC.

V (R) atoms F(i)

pairs (rij )

[210]

<110>

BALLO, P., KIOUSSIS, N., LU, G.Materials Research Society Proceedings, Vol.634. : MRS, 2001, s. B3.14.1-7.Boston. USA, 27.11.-1.12.2000.

0 5 10 15 20 25 30 35 40 45-0.01

0.00

0.01

0.02

0.03

Relative grain displacement (CLS cell %)

EGB

(J/m

2 )

0 5 10 15 20 25 30 35 40 45-0.01

0.00

0.01

0.02

0.03

Relative grain displacement (CLS cell %)

EGB

(J/m

2 )

0 5 10 15 20 25 30 35 40 45-0.01

0.00

0.01

0.02

0.03

Relative grain displacement (CLS cell %)

EGB

(J/m

2 )

0 5 10 15 20 25 30 35 40 45-0.01

0.00

0.01

0.02

0.03

Relative grain displacement (CLS cell %)

EGB

(J/m

2 )

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

EGB

(J/m

2 )

(a )

(b )

BALLO, P., KIOUSSIS, M., LU, G. Phys. Rev. B, 64, 024104 (2001).

[210]

<110>

0 1 2 3 4 5 6 70,2

0,4

0,6

0,8

1,0

1,2

1,4

Evf (e

V)

Layer

Vacancy formation energy as a function of the layer number from the interface.

BALLO, P., SLUGEN, V. Phys. Rev. B, 65, 012107 (2002).

BALLO, P., HARMATHA, L. Phys. Rev. B, 68,153201 (2003).

E

EE

E

V

V

V

C

+ 0.511+ 0.469

E

E

E

E

V

V

V

C + 0.752

+ 0.375