michael s. fuhrer university of maryland an introduction to graphene electronic structure michael s....
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Michael S. Fuhrer University of Maryland
An Introduction to Graphene Electronic Structure
Michael S. FuhrerMichael S. FuhrerDepartment of Physics andDepartment of Physics and
Center for Nanophysics and Advanced MaterialsCenter for Nanophysics and Advanced MaterialsUniversity of MarylandUniversity of Maryland
Michael S. Fuhrer University of Maryland
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Michael S. Fuhrer, University of Maryland
Michael S. Fuhrer University of Maryland
Carbon and GrapheneCarbon and Graphene
C-
-
--
Carbon Graphene
4 valence electrons
1 pz orbital
3 sp2 orbitals
Hexagonal lattice;1 pz orbital at each site
Michael S. Fuhrer University of Maryland
Graphene Unit CellGraphene Unit Cell
Two identical atoms in unit cell: A B
Two representations of unit cell:
1/3 each of 6 atoms = 2 atoms
Two atoms
Michael S. Fuhrer University of Maryland
Band Structure of GrapheneBand Structure of Graphene
Tight-binding model: P. R. Wallace, (1947)(nearest neighbor overlap = γ0)
2cos4
2cos
2
3cos41)( 2
0
akakakEE yyxF k
kx
ky
E
Michael S. Fuhrer University of Maryland
Bloch states:
AB
AB
0
1
1
0
FA(r), or
FB(r), or
“anti-bonding”E = +γ0
“bonding”E = -γ0
1
1
2
1
1
1
2
1
Γ point:k = 0
Band Structure of Graphene – Band Structure of Graphene – ΓΓ point ( point (kk = 0) = 0)
Michael S. Fuhrer University of Maryland
3
4
3
2
1
i
i
e
e
λλ
λ
K
K
K
0
1FA(r), or
1
0FB(r), or
Phase:
K 2
3a
a3
4K
Band Structure of Graphene – K pointBand Structure of Graphene – K point
Michael S. Fuhrer University of Maryland
0
1FA(r), or
1
0FB(r), or
K
2
3a
a3
4K
0
π/3
2π/3
π
5π/3
4π/3
“anti-bonding”
E = 0!
“bonding”
E = 0!
1
1
2
1
1
1
2
1
K point:Bonding and anti-bonding
are degenerate!
Bonding is Frustrated at K pointBonding is Frustrated at K point
Michael S. Fuhrer University of Maryland
)()()( rrvF FFkσ
kvbe
ibeek Fi
ii
k
k
;2
12/
2/rk
θk is angle k makes with y-axisb = 1 for electrons, -1 for holes
Eigenvectors: Energy:
Hamiltonian:
)(
)(
)(
)(
0
0
rF
rF
rF
rF
ikk
ikkv
B
A
B
A
yx
yxF
electron has “pseudospin”points parallel (anti-parallel) to momentum
K’
K
linear dispersion relation“massless” electrons
Band Structure of Graphene: k·p approximationBand Structure of Graphene: k·p approximation
Michael S. Fuhrer University of Maryland
Visualizing the PseudospinVisualizing the Pseudospin0
π/3
2π/3
π
5π/3
4π/3
Michael S. Fuhrer University of Maryland
30 degrees
390 degrees
Visualizing the PseudospinVisualizing the Pseudospin0
π/3
2π/3
π
5π/3
4π/3
Michael S. Fuhrer University of Maryland
PseudospinPseudospin
K
K’
kvH
ikk
ikkvkvH
tFK
yx
yxFFK
'
0
0σ || k
σ || -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
Michael S. Fuhrer University of Maryland
K’ K
K: k||-x K: k||xK’: k||-x
real-spacewavefunctions(color denotesphase)
k-spacerepresentation
bondingorbitals
bondingorbitals
anti-bondingorbitals
Pseudospin: Absence of BackscatteringPseudospin: Absence of Backscattering
bonding
anti-bonding
Michael S. Fuhrer University of Maryland
““Pseudospin”: Berry’s Phase in IQHEPseudospin”: Berry’s Phase in IQHE
π Berry’s phase for electron orbits results in ½-integer quantized Hall effect
2
14
2
nh
exy
422 vsgg Berry’s phase = π
holes
electr
ons
-80 -60 -40 -20 0 20 40 60 800
5
10
-30-26-22-18-14-10-6-226101418222630
B = 8 TT = 2.3 K
xy (e
2/h) xx (
e2 /h)
Vg (V)
Michael S. Fuhrer University of Maryland
Graphene: Single layer vs. BilayerGraphene: Single layer vs. Bilayer
Bilayer GrapheneSingle layer Graphene
5.2
4.3
w m
l m
6.0
2.6
w m
l m
Single Layer vs. BilayerSingle Layer vs. Bilayer
Michael S. Fuhrer University of Maryland
Graphene Dispersion Relation: Graphene Dispersion Relation: “Light-like”“Light-like”
ky
kx
E
Light: ckE
Electrons in graphene:
kvE F
Fermi velocity vF instead of c vF = 1x106 m/s ~ c/300
Bilayer Dispersion Relation: Bilayer Dispersion Relation: “Massive”“Massive”
ky
kx
E
Massive particles:em
kE
2
22
Electrons in bilayer graphene:
Effective mass m* instead of me
m* = 0.033me
*2
22
m
kE
Michael S. Fuhrer University of Maryland
-40 -30 -20 -10 0 10 20 30 400.00
0.05
0.10
0.15
0.20
0.25 QHE single layer at T=1.34K B=9T
xx(k
-1)
Vg-V
Dirac(V)
-10
-6
-2
2
6
10
xy(e 2/h
)
Quantum Hall Effect: Single Layer vs. BilayerQuantum Hall Effect: Single Layer vs. Bilayer
-30 -20 -10 0 10 20 300.00
0.05
0.10
0.15
0.20
0.25
0.30
xy (e
2/h) xx(k
-1)
Vg-V
Dirac(V)
-8
-4
0
4
8
bilayer QHE at T=1.35K, B=9T
See also: Zhang et al, 2005, Novoselov et al, 2005.
2
14
2
nh
exy
Berry’s phase = π
Single layer:
14 2
nh
exy
Berry’s phase = 2π
Bilayer:
Quantum Hall EffectQuantum Hall Effect