Quantum Theory of Graphene, Study notes of Quantum Mechanics

Download Quantum Theory of Graphene and more Study notes Quantum Mechanics in PDF only on Docsity! Quantum Theory of Graphene • Graphene’s electronic structure: A quantum critical point • Emergent relativistic quantum mechanics: The Dirac Equation • Insights about graphene from relativistic QM Insights about relativistic QM from graphene • Quantum Hall effect in graphene Allotropes of elemental carbon Graphene = A single layer of graphite 3.4 Å1.4 Å Hopping on the Honeycomb Textbook QM problem: Tight binding model on the Honeycomb lattice π sublattice A sublattice B unit cell Just like CJ’s homework! Benzene C6H6 Hopping on the Honeycomb Textbook QM problem: Tight binding model on the Honeycomb lattice π sublattice A sublattice B unit cell Just like CJ’s homework! Benzene C6H6 Metal • Partially filled band • Finite Density of States (DOS) at Fermi Energy Semiconductor Graphene A critical state • Filled Band • Gap at Fermi Energy Electronic Structure • Zero Gap Semiconductor • Zero DOS metal Massive Particle (e.g. electron) 2 2 2( ) ( )E mc cp= + Nonrelativistic limit (v<<c) 2 2 ... 2 pE mc m ≈ + + Massless Particle (e.g. photon) 0m = | |E c p= v c= Wave Equation Non relativistic particles: Schrodinger Equation ~ ; E i p k i t ω ∂= = − ∇ ∂ ∼ 2 2 2 2 2 pE i m t m ψ ψ∂= ⇒ = − ∇ ∂ ( )(e.g. )i k r te ωψ −= i Relativistic particles: Klein Gordon Equation 2 2 2 2 2 4 2 2 2 2 2 4 2 ( )E c p m c c m c t ψ ψ∂= + ⇒ − = − ∇ + ∂ In order to preserve particle conservation, quantum theory requires a wave equation that is first order in time. Niels Bohr : “What are you working on Mr. Dirac?” Paul Dirac : “I’m trying to take the square root of something” Paul Dirac 1902-1984Niels Bohr 1885-1958 Consequences of Dirac Equation 1. The existence of Anti Particles 2 2 2( ) ( )E mc cp+±= anti electron = positron Massive Dirac Eq. ~ Semiconductor Gap 2 mec2 Effective Mass m*=me Anti Particles ~ Holes Consequences of Dirac Equation 2. The existence of Spin • Electrons have intrinsic angular momentum • Electrons have permanent magnetic moment (responsible for magnetism) • Interpretation natural for graphene J L S= + Total a.m. Orbital a.m. Spin a.m. 2zS = ± N S e- B Aψ ψ ψ ψ↓ ↑⎛ ⎞ ⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ∼ “pseudo spin” ~ sublattice index Experiments on Graphene • Gate voltage controls charge n on graphene (parallel plate capacitor) • Ambipolar conduction: electrons or holes