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An Introduction to Graphene Electronic Structure. Michael S. Fuhrer Department of Physics and Center for Nanophysics and Advanced Materials University of Maryland. If you re-use any material in this presentation, please credit: Michael S. Fuhrer, University of Maryland. Carbon and Graphene.

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slide1

An Introduction to Graphene Electronic Structure

Michael S. Fuhrer

Department of Physics and

Center for Nanophysics and Advanced Materials

University of Maryland

slide2

If you re-use any material in this presentation, please credit:

Michael S. Fuhrer, University of Maryland

carbon and graphene
Carbon and Graphene

-

-

C

-

-

Carbon

Graphene

Hexagonal lattice;

1 pz orbital at each site

4 valence electrons

1 pz orbital

3 sp2 orbitals

graphene unit cell
Graphene Unit Cell

Two identical atoms in unit cell:

A B

Two representations of unit cell:

Two atoms

1/3 each of 6 atoms = 2 atoms

band structure of graphene
Band Structure of Graphene

E

kx

ky

Tight-binding model: P. R. Wallace, (1947)

(nearest neighbor overlap = γ0)

band structure of graphene point k 0
Band Structure of Graphene – Γ point (k = 0)

Bloch states:

“anti-bonding”

E = +γ0

FA(r), or

A

B

Γ point:

k = 0

FB(r), or

“bonding”

E = -γ0

A

B

band structure of graphene k point
Band Structure of Graphene – K point

FA(r), or

FB(r), or

K

K

K

λ

λ

λ

K

Phase:

bonding is frustrated at k point
Bonding is Frustrated at K point

K

0

π/3

5π/3

4π/3

2π/3

π

“anti-bonding”

E = 0!

FA(r), or

“bonding”

E = 0!

FB(r), or

K point:

Bonding and anti-bonding are degenerate!

band structure of graphene k p approximation
Band Structure of Graphene: k·p approximation

Hamiltonian:

K

K’

Eigenvectors:

Energy:

linear dispersion relation

“massless” electrons

θkis angle k makes with y-axis

b = 1 for electrons, -1 for holes

electron has “pseudospin”

points parallel (anti-parallel) to momentum

slide10

Visualizing the Pseudospin

0

π/3

5π/3

4π/3

2π/3

π

slide11

Visualizing the Pseudospin

0

π/3

5π/3

4π/3

2π/3

π

30 degrees

390 degrees

pseudospin
Pseudospin

σ || k

σ || -k

K

K’

  • Hamiltonian corresponds to spin-1/2 “pseudospin”
    • Parallel to momentum (K) or anti-parallel to momentum (K’)
  • Orbits in k-space have Berry’s phase of π
slide13

Pseudospin: Absence of Backscattering

anti-bonding

bonding

K’: k||-x

K: k||-x

K: k||x

bonding

orbitals

anti-bonding

orbitals

bonding

orbitals

real-space

wavefunctions

(color denotes

phase)

k-space

representation

K’

K

pseudospin berry s phase in iqhe
“Pseudospin”: Berry’s Phase in IQHE

holes

electrons

π Berry’s phase for electron orbits results in ½-integer quantized Hall effect

Berry’s phase = π

graphene dispersion relation light like
Graphene Dispersion Relation: “Light-like”

Bilayer Dispersion Relation: “Massive”

E

E

ky

ky

kx

kx

Massive particles:

Light:

Electrons in bilayer graphene:

Electrons in graphene:

Fermi velocity vF instead of c

vF = 1x106 m/s ~ c/300

Effective mass m* instead of me

m* = 0.033me

quantum hall effect

Quantum Hall Effect: Single Layer vs. Bilayer

Quantum Hall Effect

Bilayer:

Single layer:

Berry’s phase = 2π

Berry’s phase = π

See also: Zhang et al, 2005, Novoselov et al, 2005.