ELECTRONIC STRUCTURE OF MATERIALS

From reality to simulation and back

A roundtrip ticket

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?

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)

.

Morse potential

• Like Lennard-Jones

• Repulsion is more realistic-but attraction less so.

• Minimum neighbor position at r0

• Minimum energy is • Extra parameter “a” can be used to fit a third property:

lattice constant, bulk modulus and cohesive energy.

v(r) [e 2a(r r0) 2e a(r r0)]

dE

dr r00 B V

dP

dV V0

Vd2E

dV 2V0

Various Other Potentials

a) simplest: Hard-sphere

b) Hard-sphere, square-well

c) Coulomb (long-ranged) for plasmas

d) 1/r12 potential (short-ranged)

Atom-atom potentials

• Total potential is the sum of atom-atom pair potentials

• Assumes molecule is rigid, in non-degenerate ground state, interaction is weak so the internal structure is weakly affected. Geometry (steric effect) is important.

• Perturbation theory as rij >> core radius– Electrostatic effects: multipole expansion (if molecules are

charged or have a permanent dipole, …)

– Induction effects (by a charge on a neutral atom)

– Dispersion effects:

• dipole-induced-dipole (C6/r6)

– Short-range effects-repulsion caused by cores: exp(-r/c)

V (R) v(| ri rj |)i j

C6 d A( )B()

Fit for a Born (1923) potential

EXAMPLE: NaCl

• Obviously Zi=1

• Use cohesive energy and lattice constant (at T=0) to determine A and n:• EB=ea/d + er/dn

dEB/dr= –ea/d2 + ner/dn-1

=0

=> n=8.87 A=1500eVǺ8.87

• Now we need a check. The “bulk modulus”. – We get 4.35 x 1011 dy/cm2 experiment is 2.52 x 1011 dy/cm2

• You get to what you fit!

•Attractive charge-charge interaction

•Repulsive part determined by atom core.

V (R) ZiZ j

| ri rj |1

A

| ri rj |n

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 )

Problems with potentials

• Difficult problem because potential is highly dimensional function. Arises from QM so it is not a simple function.

• Procedure: fit data relevant to the system you need to simulate, with similar densities and local environment.

• Use other experiments to test potential if possible.

• Do quantum chemical (SCF or DFT) calculations of clusters. Be aware that these may not be accurate enough.

• No empirical potentials work very well in an inhomogenous environment.

• This is the main problem with atom-scale simulations--they really are only suggestive since the potential may not be correct. Universality helps.

Some tests

-Lattice constant

-Bulk modulus

-Cohesive energy

-Vacancy formation energy

-Property of an impurity

What about relaxation and other monkey-tricks

What are the forces?

• Common examples are Lennard-Jones (6-12 potential), Coulomb, embedded atom potentials.

• They are only good for simple materials.

• The ab initio philosophy is that potentials are to be determined directly from quantum mechanics as needed.

• But computer power is not yet adequate, in general. • But nearing in the future (for some problems).

• A powerful approach is to use simulations at the quantum level to determine parameters at the classical level.

Go ahead on “real” systems

MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN

1,0 1,5 2,0 2,5 3,0 3,5 4,0-0,05

0,00

0,05

0,10

0,15

0,20

0,25

0,30

En

erg

y (e

V)

Distance (A)

GB

[210]

<110>

The interatomic potential used in the simulation is based on the Embedded Atom Method (EAM). For additional simulations we usedAb initio method in combination withthe usual copper pseudopotential.

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