Modeling van der Waals forces in graphite

Modeling van der Waals forces in
graphite
Tony Carlson
Structure of Graphene
Flat hexagonal sheet
-10
•
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
http://stm2.nrl.navy.mil
www.physik.uni-augsburg.de
Graphite
●
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
together
●
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,
Dispersive
●
?
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
Repulsion
Attraction
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)
●
Compressibility
● 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
stacking
Step 2
- Fit function for C and α, such that
* Equilibrium separation ( Zeq =3.35 Å )
* Exfoliation energy ( 42.6 meV)
Fit results
Quantity
% error
Zeq
Exf
< 0.1
< 0.1
Kc
E2g1 mode
~ 0.5
~ -4.7
Not explicitly fitted to
Lateral Energy Landscape
E2g1 ~ Well curvature