Modeling van der Waals forces in graphite

Tony Carlson

Structure of Graphene• Flat hexagonal sheet• Bond length 1.42 Å=1.42x10-10 m

• SP2 covalent bonds• Pz orbital out of planePaula 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 )

● The dispersion term is present in polar systems and is almost always dominant

Dispersion is Quantum Mechanical

● Spontaneously induced dipoles● Purely quantum mechanical effect proved by London (1927)● General distance dependence

Fritz London

Donald McQuarrie and John Simon, Physical chemistry: a molecular approach, Viva Books, 2005.

Lenard Jones (6-12) Potential

Steep repulsion due to the Pauli exclusion principle

Well depth = bond strength

Minima – corresponds to bond length

Energy vs. Bond length

Repulsion

Attraction

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

D. Porezag, Th. Frauenheim, and Th. Kohler, Construction of tight-binding-like potentials on the basis of density-functional theory: Application to carbon,Physical Review B 51 (1995), no. 19, 12947-12957.

J.C. Slater and G.F. Koster, S implified lcao method for the periodic potential problem, Physical Review 94 (1954), 1498-1524.

Weakness of this TB model

Why does it only predict repulsion?

● Interlayer binding non-existent in graphite

No minima at Zeq

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 = 3.35 Å● Exfoliation energy (well depth) = 42.6 meV● Compressibility = 2.97 x 10-12 cm2 dyne-1

● Phonon Frequency = 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 2

- Fit function for C and α, such that * Equilibrium separation ( Zeq =3.35 Å )

* Exfoliation energy ( 42.6 meV)

Step 1

- inflate/deflate Pz orbitals to get 17 meV between AAA/ABA stacking

Fit results

Quantity % error

Zeq < 0.1Exf < 0.1

Kc ~ 0.5E2g1 mode ~ -4.7

Not explicitly fitted to

Lateral Energy Landscape

E2g1 ~ Well curvature