Graphene: Scratching the Surface

1. Graphene: Scratching the Surface Michael S. Fuhrer Professor, Department of Physics and Director, Center for Nanophysics and Advanced Materials University of Maryland

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

3. 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

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

5. Bonding vs. Anti-bonding γ0 is energy gained per pi-bond ψ “anti-bonding” anti-symmetric wavefunction “bonding” symmetric wavefunction

6. Band Structure of Graphene – Γ point (k = 0) Bloch states: “anti-bonding” E = +3γ0 FA(r), or A B Γ point: k = 0 FB(r), or “bonding” E = -3γ0 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 d 0 Im Re Phase: E2 E1 E3

9. 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!

10. 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

11. Visualizing the Pseudospin 180 degrees 540 degrees 0 π/3 5π/3 4π/3 2π/3 π

12. Visualizing the Pseudospin 0 degrees 180 degrees 0 π/3 5π/3 4π/3 2π/3 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