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

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

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

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

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

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)

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

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

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

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

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

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

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

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