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

2. Michael S. Fuhrer University of Maryland Carbon and Graphene C - - - - CarbonGraphene 4 valence electrons 1 p z orbital 3 sp 2 orbitals Hexagonal lattice; 1 p z orbital at each site

3. Michael S. Fuhrer University of Maryland Graphene Unit Cell Two identical atoms in unit cell: A B Two representations of unit cell: 1/3 each of 6 atoms = 2 atoms Two atoms

4. Michael S. Fuhrer University of Maryland Band Structure of Graphene Tight-binding model: P. R. Wallace, (1947) (nearest neighbor overlap = γ 0 ) kxkx kyky E

5. Michael S. Fuhrer University of Maryland Bonding vs. Anti-bonding ψ “anti-bonding” anti-symmetric wavefunction “bonding” symmetric wavefunction γ 0 is energy gained per pi-bond

6. Michael S. Fuhrer University of Maryland Bloch states: ABAB ABAB F A (r), or F B (r), or “anti-bonding” E = +3γ 0 “bonding” E = -3γ 0 Γ point: k = 0 Band Structure of Graphene – Γ point (k = 0)

7. Michael S. Fuhrer University of Maryland λ λ λ K K K F A (r), orF B (r), or Phase: K Band Structure of Graphene – K point

8. Michael S. Fuhrer University of Maryland Phase: Bonding is Frustrated at K point 0  Re Im E1E1 E2E2 E3E3

9. Michael S. Fuhrer University of Maryland F A (r), or F B (r), or K 0 π/3 2π/3 π 5π/3 4π/3 “anti-bonding” E = 0! “bonding” E = 0! K point: Bonding and anti-bonding are degenerate! Bonding is Frustrated at K point

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

11. Michael S. Fuhrer University of Maryland Visualizing the Pseudospin 0 π/3 2π/3 π 5π/3 4π/3 180 degrees 540 degrees

12. Michael S. Fuhrer University of Maryland Visualizing the Pseudospin 0 π/3 2π/3 π 5π/3 4π/3 0 degrees 180 degrees

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

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