Modeling van der Waals forces in graphite



Modeling van der Waals forces in graphite
Modeling van der Waals forces in
Tony Carlson
Structure of Graphene
Flat hexagonal sheet
Bond length 1.42 Å=1.42x10 m
SP2 covalent bonds
• Pz orbital out of plane
Paula Bruice, "Organic Chemistry", Prentice Hall, 2003
covalent bond
AFM Images of Graphene
Graphite is simply stacked graphene layers
Nealy 100% ABA stacking
Interlayer separation
Zeq = 3.35 Å
Stacking Type
Van der Waals dispersion forces hold graphite
Who is this “van der Waals” ?
'a' and 'b' are constant depending on the gas
'a' ~ intermolecular interaction strength
'b' ~ molecular size
Whats the nature of this attraction?
Three distinct contributions, Orientational, Inductive,
Dipole - Dipole
( Keesom energy )
Dipole – Induced Dipole
( Debye energy )
Dispersion is Quantum Mechanical
Spontaneously induced dipoles
● Purely quantum mechanical
effect proved by London (1927)
● General distance dependence
Fritz London
The dispersion term is present in polar systems
and is almost always dominant
Donald McQuarrie and John Simon, P hysical chemistry: a molecular approach, Viva Books, 2005.
Lenard Jones (6-12) Potential
Energy vs. Bond length
Steep repulsion due to the
Pauli exclusion principle
Well depth = bond strength
Minima – corresponds to bond length
Graphite Interlayer energy
Graphene layers are basically closed shell
systems (no covalent bonding between layers)
● Energy between layers is a balance between
Attractive dispersion forces
Corrugated repulsive overlap forces
Energy vs. interlayer separation similar to LJ pot.
Super Quick Quantum Overview
Schroedinger Equation
Linear Combination of Atomic Orbitals
Rayleigh-Ritz variation to minimize E
Eigenvalue problem:
- Solutions
1) eigenvectors c
2) eigenvalues E
Each element of the matrix H is a triple integral over 3 space of two orbitals......
Parameterization via ab-initio methods
Key to TB: parameterize these integrals as a
function of distance and orbital orientation
● Done with ab-initio density functional theory
J.C. Slater and G.F. Koster, S implified lcao method for the periodic potential problem, Physical Review 94 (1954), 1498-1524.
D. Porezag, Th. Frauenheim, and Th. Kohler, C onstruction of tight-binding-like potentials on the basis of density-functional theory:
Application to carbon,Physical Review B 51 (1995), no. 19, 12947-12957.
Weakness of this TB model
Interlayer binding non-existent in graphite
No minima
at Zeq
Why does it only predict repulsion?
How do we address this?
Add an empirical dispersion term to the total energy
Proposed form of potential – motivated by London's derivation
Two free parameters to fit ( C and α )
Interlayer Bonding in Graphite
Experimental data related to interlayer energy
Equilibrium spacing
● Exfoliation energy (well depth)
● Phonon Frequency
= 3.35 Å
= 42.6 meV
= 2.97 x 10-12 cm2 dyne-1
= 1.26 THz (E2g1 shear mode)
Ab-initio data related to interlayer repulsion
Energy difference between AAA/ABA stacking
= 17 meV
Fitting dispersion term
Step 1
- inflate/deflate Pz orbitals to get 17 meV between AAA/ABA
Step 2
- Fit function for C and α, such that
* Equilibrium separation ( Zeq =3.35 Å )
* Exfoliation energy ( 42.6 meV)
Fit results
% error
< 0.1
< 0.1
E2g1 mode
~ 0.5
~ -4.7
Not explicitly fitted to
Lateral Energy Landscape
E2g1 ~ Well curvature

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