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
1.4 Å 3.4 Å
Graphene = A single layer of graphite Graphene Electronic Structure Carbon: Z=6 ; 4 valence electrons
3s,3p,3d 2 (18 states) sp bonding π orbital (┴ to plane) derived from p σ* z
2s,2p π (8 states)
σ σ orbital (in plane) derived from s, px, py 1s • σ bonds: exceptional structural rigidity (2 states) • π electrons: allow conduction Hopping on the Honeycomb
π
Textbook QM problem: Tight binding model on the Honeycomb lattice sublattice A Just like CJ’s homework! sublattice B Benzene
unit cell C6H6 Hopping on the Honeycomb
π
Textbook QM problem: Tight binding model on the Honeycomb lattice sublattice A Just like CJ’s homework! sublattice B Benzene
unit cell C6H6 Hopping on the Honeycomb
π
Textbook QM problem: Tight binding model on the Honeycomb lattice sublattice A Just like CJ’s homework! sublattice B Benzene
unit cell C6H6 Electronic Structure Metal • Partially filled band • Finite Density of States (DOS) at Fermi Energy
Semiconductor • Filled Band • Gap at Fermi Energy
Graphene A critical state • Zero Gap Semiconductor • Zero DOS metal Semiconductor Graphene
2 =−0 p =± EEcc * Epv|F | 2mc p2 “Fermi velocity” EE=−0 vv * v810/=× 5 ms 2mv F Theory of Relativity A stationary particle (p=0) has rest energy Emc= 2 A particle in motion is described by the relativistic dispersion relation:
Albert Einstein 1879-1955 E =+()()mc22 cp 2
Velocity:
∂Ecp v ==c ∂p ()()mc22+ cp 2 Massive Particle (e.g. electron) Emccp=+()()22 2
Nonrelativistic limit (v< Massless Particle (e.g. photon) m = 0 E = cp|| v = c Wave Equation ∂ −ω Eipki==−∇ ω ~ ; ∼ (e.g. ψ = eikr()i t ) ∂t Non relativistic particles: Schrodinger Equation p22∂ψ Ei= ⇒ =− ∇2ψ 22mtm∂ Relativistic particles: Klein Gordon Equation ∂2ψ Ecpmc22224=+ ⇒ −=−∇+ 2 ( 22224 c mc )ψ ∂t 2 In order to preserve particle conservation, quantum theory requires a wave equation that is first order in time. Niels Bohr 1885-1958 Paul Dirac 1902-1984 Niels Bohr : “What are you working on Mr. Dirac?” Paul Dirac : “I’m trying to take the square root of something” Dirac’s Solution (1928) 2 2 2 How can you take the square root of px +py +m without taking a square root? −− ⎛⎞⎛⎞m pxy ip m px ipy 22++ 2⎛⎞10 =⎜⎟⎜⎟ ()ppmxy ⎜⎟ ⎝⎠01 ⎜⎟⎜⎟+− +− ⎝⎠⎝⎠pipmpipmxy xy 22++ 2 =σσσ + + ()pxypmIp xxyyz p m − σσ==⎛⎞01 ⎛ 0i ⎞ σ = ⎛ 1 0 ⎞ xy⎜⎟ ; ⎜ ⎟ ; z ⎜ ⎟ “Dirac Matrices” : ⎝⎠10 ⎝i 0 ⎠ ⎝ 0− 1 ⎠ Dirac ∂∂∂ψ ⎡ ⎛⎞⎤ ψ =− + + ψ ψ = ⎛⎞A ii ⎢ ⎜⎟xy z m⎥ ⎜⎟ψ Equation ∂∂∂txy B ⎣ ⎝⎠σσ σ⎦ ⎝⎠ Low Energy Electronic Structure of A Graphene B The low energy electronic states in graphene are described by the Dirac equation for particles with Mass : m=0 “Speed of light” c = vF ψ sublattice A ⎛⎞A ψ = ⎜⎟ψ ⎝⎠B sublattice B Emergent Dirac Fermions Consequences of Dirac Equation 1. The existence of Anti Particles Emccp= ± ()()22+ 2 anti electron = positron Massive Dirac Eq. ~ Semiconductor 2 Gap 2 mec Effective Mass m*=me Anti Particles ~ Holes Consequences of Dirac Equation 2. The existence of Spin • Electrons have intrinsic angular momentum J =+LS =± Sz Total a.m. Orbital a.m. Spin a.m. 2 N • Electrons have permanent magnetic - moment (responsible for magnetism) e S • Interpretation natural for graphene ψ ψ ⎛⎞↑ ⎛⎞A ⎜⎟ψψ ∼ ⎜⎟“pseudo spin” ~ sublattice index ⎝⎠↓ ⎝⎠B Experiments on Graphene • Gate voltage controls charge n on graphene (parallel plate capacitor) • Ambipolar conduction: electrons or holes Landau levels for classical particles 4 1eB 3 =+ Enn () 2 2m 1 for n=0, 1, 2, .... Landau levels for relativistic particles 2 =± 2 1 En eBn vF 0 1 for n=0, 1, 2, .... 2 Existence of landau level at 0 is deeply related to spin in Dirac Eq.