Download Dirac Electrons in Graphene

Topic 3
Spinors, Fermion Fields, Dirac Fields
Lecture 15
Dirac Electrons in Graphene
This phenomenon was discovered by K.S. Novoselov, A.K. Geim, et al., Two-dimensional gas of
massless Dirac fermions in graphene, Nature 438, 197-200 (2005). Novoselov and Geim were awarded
the 2010 Nobel Prize in Physics for these experiments.
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Spinors, Fermion Fields, Dirac Fields
Lecture 15
A.H. Castro Neto et al., The electronic properties of graphene, Rev. Mod. Phys. 81, 109–162 (2009)
have a comprehensive of this topic. Eqs. (19,20) in their article
−ivF σ · ∇ψ(r) = Eψ(r) ,
1
ψ±K (k) = √
2
−iθk /2
e
±eiθk /2
is the Dirac equation in 3-dimensional spacetime for massless electrons traveling with an effective
speed of light equal to the Fermi velocity vF and complex spinor wavefunctions parametrized by
Berry’s phase θ. The wavevector K is a Dirac point on the Brioullin zone boundary
Altland-Simons discuss a Hubbard model Hamiltonian approximation for electron hopping between
sites of the graphene lattice in §2.2, and the Quantum Hall Effect in more detail in §9.3.
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Supersymmetric Quantum Mechanics of Dirac Electrons
The Fermi sea of filled electron states in a metal is conceptually similar to the sea of filled negative
energy electron states used by Dirac Proc. R. Soc. Lond. A 126, 360-385 (1928) to predict the
existence and properties of positrons.
The basic principles of supersymmetric quantum mechanics were developed by E. Witten more than
30 years ago.
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Spinors, Fermion Fields, Dirac Fields
Lecture 15
These ideas have been applied to the Graphene problem, for example by M. Ezawa in Phys. Lett.
A 372, 924–929 (2008). He refers to G.W. Semenoff, Condensed-Matter Simulation of a ThreeDimensional Anomaly, Phys. Rev. Lett. 53, 2449–2452 (1984), who predicted that graphene might
exhibit unusual Quantum Hall effect (QHE) behavior. This effect was also studied in Carbon nanotubes by H. Ajiki and T. Ando, Electronic States of Carbon Nanotubes, J. Phys. Soc. Jpn. 62
(1993) 1255.
Dirac Electrons
The low-energy band structure of graphene is described by cones located at two inequivalent Brillouin
zone corners
4π
±K = ± √ , 0 ,
3a
where a is the lattice constant. They are called the K and K 0 points. In these cones, the twodimensional energy dispersion relation is linear
E± = ~vF |k ∓ K| ,
for
k ' ±K ,
where vF ≈ 106 m/s ' c/300 is the Fermi velocity.
The electrons near the K, K 0 points are described by a Dirac Hamiltonian in the Weyl representation
0
σ
0
1
±
HD
= vF (±αx px + αy py ) = vF (±σx px + σy py ) γ5 , α =
, γ5 =
,
σ 0
1 0
This is a (2+1)-dimensional system and the electrons are massless.
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Spinors, Fermion Fields, Dirac Fields
Lecture 15
Energy Levels in a Magnetic Field
In an applied magnetic field normal to the graphene layer the energy eigenstates are described by
Landau levels. Because the electrons have spin magnetic moment, the levels are further split by
the Zeeman effect. As the magnetic field is varied, the system exhibits the Shubnikov-de Haas effect
oscillations observed by Novoselov et al. The oscillations can also be studied by varying a gate voltage
across the sheet in the x direction, keeping the magnetic field constant.
The Hamiltonian in an external magnetic field perpedicular to the sheet B = ∇ × A = (0, 0, −B) is
obtained with the minimal substitution
~
p → P = ∇ + eA .
i
A electron in magnetic field undergoes cyclotron motion with angular frequency ωc . This circular
motion can be decomposed into harmonic motion in the x, y directions. The Schrödinger equation
0 Q±
±
±
HD
ψ± (r) = E ± ψ± (r) , HD
=
, Q± = vF (±σx Px + σy Py ) ,
Q±
0
is first transformed to block diagonal form
q
±
HD
=
Q2±
0

0

q
Q2±
and fully diagonalized using harmonic oscillator ladder operators
`B (Px + iPy )
0 a†
0 a
√
Q+ = ~ωc
, Q− = ~ωc
,
a
=
,
a 0
a† 0
2~
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`B =
√
2~
vF
,
ωc
9/30/2013
Topic 3
Spinors, Fermion Fields, Dirac Fields
Lecture 15
where `B is the magnetic length. The Fock space of the ladder operators is
N
1
|N i = √
a† |0i ,
N!
N = 0, 1, 2, · · ·
In the conventional Quantum Hall Effect, the Zeeman splitting of Landau levels can usually be treated
as a small perturbation. In graphene, however, the Zeeman energy turns out to be comparable with
the cyclotron energy, essentially because the electrons are relativistic with β = v/vF = 1. The Pauli
Hamiltonian for the diagonal blocks is
#
"
2
~
∇ + eA ∓ e~σz B .
HP± = Q2± = vF2
i
According to this expression, the intrinsic Zeeman energy is precisely one half of the cyclotron energy
for Dirac electrons, so that two Landau levels mix to form a single energy level.
Consider the K Dirac point with Hamiltonian HP+ point. It is obvious that the up-spin and down-spin
states are simultaneous eigenstates of the Pauli and Dirac Hamiltonians. The up-spin state has a
lower energy than the down-spin state when they belong to the same Landau level. On the other
hand, the direction of the magnetic field is effectively opposite at the K and K 0 points. Therefore,
we can unambiguously make the following identification of quantum numbers of the energy levels
 ±↑

EN
0
0
0
±↓
√
 0
E
0
0 
+↑
+↓
−↓
−↑
±
N


hN |HD |Bi = 
, EN = EN −1 = EN = EN −1 = ~ωc N .
±↑
0
0
−EN
0 
±↓
0
0
0
−EN
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Spinors, Fermion Fields, Dirac Fields
Lecture 15
The energy level diagram is completed by adding the hole (positron-like) states below the Fermi
level. Each energy level above and below the Fermi level is 4-fold degenerate. Each occupied level
contributes one Conductance quantum d2 /~ to the Hall conductivity, which exactly accounts for the
anomalous conductivity plateaux series ν = ±2, ±6, ±10 · · · observed by Novoselov et al.
The graphene system is an interesting case of reduced effective spatial dimensions, and exhibits many
interesting types of quantum statistics.
Forming a monolayer effectively reduces the system from the 3-spatial dimensions of macroscopic
phenomena to a 2-spatial dimensional system.
Applying a magnetic field perpedicular to the graphene sheet forces the electrons into cyclotron
motion in which they move in circular orbits around the magnetic flux lines. In an infinitely large
magnetic field the cyclotron radius becomes infinitely small, and the system becomes effectively
1-spatial dimensional.
Quantum mechanics quantizes the cyclotron orbits resulting in a discrete spectrum of Landau levels.
In each Landau level the effective spatial dimensionality is zero.
Supersymmetry
The 4-fold degenerate energy levels of the Pauli Hamiltonian can be described by supersymmetry in
which the operators Q± are treated as conjugate supercharges
0 A†
0 A
Q+ =
, Q− =
,
A 0
A† 0
which generate the superalgebra
±
{Q± , Q± } = 2HP± ,
HP , Q± = 0 .
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PHY 510
Spinors, Fermion Fields, Dirac Fields
Lecture 15
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Lecture 15
Superpartner Hamiltonians H ±↑ at the K point and H ±↓ at the K 0 point are then defined by
HP+ =
A† A
0
0
AA†
≡
H +↑
0
0
H +↓
HP− =
,
AA†
0
0
A† A
≡
H −↑
0
0
H −↓
.
The eigenstates of the K-point superpartner Hamiltonians H +↑ , H +↓
H +↑ |ψn+↑ i = En+↑ |ψn+↑ i ,
H +↓ |ψn+↓ i = En+↓ |ψn+↓ i ,
can be constructed using the ladder operators and
AH +↑ = AA† A = H +↓ A ,
to obtain
H +↓ A|ψn+↑ i = AH +↑ |ψn+↑ i = En+↑ A|ψn+↑ i ,
H +↑ A† |ψn+↓ i = A† H +↑ |ψn+↑ i = En+↓ A† |ψn+↓ i .
Elementary particles can be classified as bosons or fermions. The wavefunctions of a state with many
identical bosons is symmetric under interchange of any two positions. Wavefunctions of multi-fermion
states are antisymmetric.
In two-dimensional space, quasi-particle states with Anyon statistics are allowed, and have been found
experimentally.
In one spatial dimension, bosons and fermions are essentially equivalent.
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Spinors, Fermion Fields, Dirac Fields
Lecture 15
In quantum mechanics treated as a 0-space dimensional field theory, the distinction between bosons
and fermions cannot be made by interchanging the positions of two particles because every particle
is located at the same spatial point.
Bosonic and fermionic states can be defined using the commutation and anti-commutation properties
of operators that generate symmetry transformations.
In supersymmetric quantum mechanics, the non-degenerate ground state is identified as either bosonic
or fermionic. In the graphene problem, it seems natural to take the ground states of H ± to be
bosonic. The supercharges are fermionic because they obey anticommutation relations. This leads
to the identification of supermultiplets in the graphene spectrum shown in Ezawa’s article.
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