Quantum Theory of Graphene

Graphenes electronic structure:A quantum critical point

Emergent relativistic quantum mechanics:The Dirac Equation

Insights about graphene from relativistic QMInsights about relativistic QM from graphene

Quantum Hall effect in graphene

Allotropes of elemental carbon

1.4

3.4

Graphene = A single layer of graphite

Graphene Electronic StructureCarbon: Z=6 ; 4 valence electrons3s,3p,3d (18 states)

sp2 bonding

*2s,2p (8 states)

orbital ( to plane) derived from pz

orbital (in plane) derived from s, px, py

1s (2 states)

bonds: exceptional structural rigidity electrons: allow conduction

Hopping on the Honeycomb Textbook QM problem: Tight binding model on the Honeycomb latticesublattice A sublattice B unit cell

Just like CJs homework!Benzene C6H6

Hopping on the Honeycomb Textbook QM problem: Tight binding model on the Honeycomb latticesublattice A sublattice B unit cell

Just like CJs homework!Benzene C6H6

Hopping on the Honeycomb Textbook QM problem: Tight binding model on the Honeycomb latticesublattice A sublattice B unit cell

Just like CJs homework!Benzene C6H6

Electronic StructureMetal Partially filled band Finite Density of States(DOS) at Fermi Energy

Semiconductor Filled Band Gap at Fermi EnergyGraphene A critical state

Zero Gap Semiconductor Zero DOS metal

Semiconductor

Graphene

p2 Ec = Ec0 * 2mc 2 p 0 Ev = Ev * 2mv

E = vF | p |Fermi velocity

v F = 8 10 m / s5

Theory of RelativityA stationary particle (p=0) has rest energy

E = mc

2

A particle in motion is described by the relativistic dispersion relation:

E = (mc ) + (cp)2 2

2

Albert Einstein 1879-1955

Velocity:

E cp =c v= p (mc 2 ) 2 + (cp ) 2

Massive Particle (e.g. electron)E = (mc 2 ) 2 + (cp) 2Nonrelativistic limit (v