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

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

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

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

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

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

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!

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

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

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

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

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

“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

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

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

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.