Úvod do tímového projektu Peter Ballo Katedra fyziky Fakulta elektrotechniky a informatiky
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