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
(18 states)
sp2 bonding
σ*
2s,2p
(8 states)
π
σ
1s
(2 states)
π orbital (┴ to plane)
derived from pz
σ orbital (in plane)
derived from s, px, py
• σ bonds: exceptional structural rigidity
• 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
unit cell
Benzene
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
unit cell
Benzene
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
unit cell
Benzene
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
p
Ec = Ec0 − *
2mc
2
p
0
Ev = Ev − *
2mv
E = ±vF | p |
“Fermi velocity”
v F = 8 × 10 m / s
5
Theory of Relativity
A 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
Velocity:
∂E
cp
=c
v=
∂p
(mc 2 ) 2 + (cp ) 2
2
Albert Einstein 1879-1955
Massive Particle (e.g. electron)
E = (mc 2 ) 2 + (cp) 2
Nonrelativistic limit (v<<c)
2
p
E ≈ mc 2 +
+ ...
2m
Massless Particle (e.g. photon)
m=0
E =c| p|
v=c
Wave Equation
∂
E= ω ~i
∂t
; p = k ∼ −i ∇
(e.g. ψ = ei ( k i r −ωt ) )
Non relativistic particles: Schrodinger Equation
p2
E=
2m
2
∂ψ
2
⇒ i
=−
∇ψ
2m
∂t
Relativistic particles: Klein Gordon Equation
E 2 = c 2 p 2 + m2c 4
⇒
2
∂
ψ
2
2 2 2
2 4
(
c
m
c )ψ
−
=
−
∇
+
2
∂t
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)
How can you take the square root of px2+py2+m2
without taking a square root?
⎛ m
1 0⎞ ⎜
( px2 + p y2 + m 2 ) ⎛⎜
=
⎟
⎝ 0 1 ⎠ ⎜ px + ip y
⎝
px − ip y ⎞ ⎛ m
⎟⎜
− m ⎠⎟ ⎝⎜ px + ip y
px − ip y ⎞
⎟
−m ⎠⎟
( p + p + m ) I = pxσ x + p yσ y + mσ z
2
x
2
y
“Dirac Matrices” :
2
⎞ ; σ =⎛
σ x = ⎛⎜
⎟
⎜i
y
⎝1 0⎠
⎝
0 1
0
−i ⎞
⎛1
;
=
σ
⎜0
z
0 ⎟⎠
⎝
⎤
∂ψ ⎡ ⎛
∂
∂ ⎞
Dirac
i
= ⎢ −i ⎜ σ x
+ σ y ⎟ + σ z m ⎥ψ
Equation
∂t ⎣ ⎝ ∂x
∂y ⎠
⎦
0⎞
−1⎟⎠
⎛ψ A ⎞
ψ = ⎜ψ ⎟
⎝ B⎠
Low Energy Electronic Structure of
Graphene
A
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 A ⎞
ψ = ⎜ψ ⎟
⎝ B⎠
sublattice A
sublattice B
Emergent Dirac Fermions
Consequences of Dirac Equation
1. The existence of Anti Particles
E = ± (mc ) + (cp )
2 2
anti electron = positron
Massive Dirac Eq. ~ Semiconductor
Gap 2 mec2
Effective Mass m*=me
Anti Particles
~
Holes
2
Consequences of Dirac Equation
2. The existence of Spin
• Electrons have intrinsic angular momentum
J = L+S
Sz = ±
Total a.m.
Orbital a.m.
2
Spin a.m.
N
• Electrons have permanent magnetic
e-
moment (responsible for magnetism)
S
• Interpretation natural for graphene
⎛ψ ↑ ⎞
⎛ψ A ⎞
⎜ψ ↓ ⎟ ∼ ⎜ψ B ⎟
⎝ ⎠
⎝ ⎠
“pseudo spin” ~ sublattice index
Experiments on Graphene
• Gate voltage controls charge n on graphene
(parallel plate capacitor)
• Ambipolar conduction:
electrons or holes
Landau levels for classical particles
4
3
2
1
1 eB
En = (n + )
2 m
for n=0, 1, 2, ....
Landau levels for relativistic particles
2
1
En = ± e v 2F Bn
0
1
2
for n=0, 1, 2, ....
Existence of landau level at 0 is deeply related to spin in Dirac Eq.