1. An Introduction to Graphene Electronic Structure
Michael S. Fuhrer
Department of Physics and
Center for Nanophysics and Advanced Materials
University of Maryland

2. If you re-use any material in this presentation, please credit:
Michael S. Fuhrer, University of Maryland

3. C - Carbon Graphene Carbon and Graphene Hexagonal lattice;
1 pz orbital at each site
4 valence electrons
1 pz orbital
3 sp2 orbitals

4. Graphene Unit Cell Two identical atoms in unit cell: A B
Two representations of unit cell:
Two atoms
1/3 each of 6 atoms = 2 atoms

5. Band Structure of Graphene
Tight-binding model: P. R. Wallace, (1947)
(nearest neighbor overlap = γ0) kx ky E

6. Band Structure of Graphene – Γ point (k = 0)
Bloch states:
“anti-bonding”
E = +γ0
FA(r), or A B
Γ point:
k = 0
FB(r), or
“bonding”
E = -γ0 A B

7. Band Structure of Graphene – K point
FB(r), or
FA(r), or λ K
Phase:

8. Bonding is Frustrated at K point
“anti-bonding”
E = 0!
FA(r), or
“bonding”
E = 0!
FB(r), or
π/3
2π/3 π
5π/3
4π/3
K point:
Bonding and anti-bonding are degenerate!

9. Band Structure of Graphene: k·p approximation
Hamiltonian: K K’
Eigenvectors:
Energy:
linear dispersion relation
“massless” electrons
θk is angle k makes with y-axis
b = 1 for electrons, -1 for holes
electron has “pseudospin”
points parallel (anti-parallel) to momentum

10. Visualizing the Pseudospin
π/3
2π/3 π
5π/3
4π/3

11. Visualizing the Pseudospin
π/3
2π/3 π
5π/3
4π/3
30 degrees
390 degrees

12. Pseudospin Hamiltonian corresponds to spin-1/2 “pseudospin”
σ || k
σ || -k K K’
Hamiltonian corresponds to spin-1/2 “pseudospin”
Parallel to momentum (K) or anti-parallel to momentum (K’)
Orbits in k-space have Berry’s phase of π

13. Pseudospin: Absence of Backscattering
K’: k||-x
K: k||-x
K: k||x
bonding
orbitals
anti-bonding
orbitals
bonding
orbitals
real-space
wavefunctions
(color denotes
phase)
bonding
anti-bonding
k-space
representation K’ K

14. “Pseudospin”: Berry’s Phase in IQHE
holes
electrons
π Berry’s phase for electron orbits results in ½-integer quantized Hall effect
Berry’s phase = π

15. Single Layer vs. Bilayer
Graphene: Single layer vs. Bilayer
Single layer Graphene
Bilayer Graphene

16. Graphene Dispersion Relation: “Light-like”
Bilayer Dispersion Relation: “Massive” ky kx E ky kx E
Massive particles:
Light:
Electrons in graphene:
Electrons in bilayer graphene:
Fermi velocity vF instead of c
vF = 1x106 m/s ~ c/300
Effective mass m* instead of me
m* = 0.033me

17. Quantum Hall Effect: Single Layer vs. Bilayer
Berry’s phase = π
Berry’s phase = 2π
See also: Zhang et al, 2005, Novoselov et al, 2005.