michael s. fuhrer university of maryland graphene: scratching the surface michael s. fuhrer...
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Michael S. Fuhrer University of Maryland
Graphene: Scratching the Surface
Michael S. FuhrerMichael S. FuhrerProfessor, Department of Physics andProfessor, Department of Physics and
Director, Center for Nanophysics and Advanced MaterialsDirector, Center for Nanophysics and Advanced MaterialsUniversity of MarylandUniversity 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 atoms2a
1a
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
Bonding vs. Anti-bondingBonding vs. Anti-bonding
00
0 0
0
EH
ψ “anti-bonding” anti-symmetric wavefunction
“bonding” symmetric wavefunction
022 1
1
2
1
E
011 1
1
2
1
E
γ0 is energy gained per pi-bond
Michael S. Fuhrer University of Maryland
Bloch states:
AB
AB
0
1
1
0
FA(r), or
FB(r), or
“anti-bonding”E = +3γ0
“bonding”E = -3γ0
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
3
4
3
2
0 1
i
i
i
e
e
e
Phase:
Bonding is Frustrated at K pointBonding is Frustrated at K point
3
2
02
ieE
001ieE
3
4
03
ieE
0
03
4
3
20
0
ii
i eeeE
3
2
02
i
eE
001ieE
3
4
03
i
eE
003
4
3
20
0
iii
i eeeeE
Re
Im
E1
E2
E3
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
180 degrees
540 degrees
Michael S. Fuhrer University of Maryland
Visualizing the PseudospinVisualizing the Pseudospin0
π/3
2π/3
π
5π/3
4π/3
0 degrees
180 degrees
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
-80 -60 -40 -20 0 20 40 60 800
5
10
15
20
-34-30-26-22-18-14-10-6-22610141822263034
xy (e
2/h)
QHE at T=2.3K, B=7.94T
Rxx
(k)
Vg (V)
2
14
2
nh
exy
422 vsgg Berry’s phase = π
holes
electr
ons