An Introduction to Graphene Electronic Structure

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. Carbon and Graphene - - C - - Carbon 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 E kx ky Tight-binding model: P. R. Wallace, (1947) (nearest neighbor overlap = γ0)

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 FA(r), or FB(r), or K K K λ λ λ K Phase:

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

11. Visualizing the Pseudospin 0 π/3 5π/3 4π/3 2π/3 π 30 degrees 390 degrees

12. 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 anti-bonding bonding K’: k||-x K: k||-x K: k||x bonding orbitals anti-bonding orbitals bonding orbitals real-space wavefunctions (color denotes phase) 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. Graphene: Single layer vs. Bilayer Single Layer vs. Bilayer Single layer Graphene Bilayer Graphene

16. Graphene Dispersion Relation: “Light-like” Bilayer Dispersion Relation: “Massive” E E ky ky kx kx Massive particles: Light: Electrons in bilayer graphene: Electrons in 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 Quantum Hall Effect Bilayer: Single layer: Berry’s phase = 2π Berry’s phase = π See also: Zhang et al, 2005, Novoselov et al, 2005.