graphene: scratching the surface
DESCRIPTION
Graphene: Scratching the Surface. Michael S. Fuhrer Professor, Department of Physics and Director, Center for Nanophysics and Advanced Materials University of Maryland. Carbon and Graphene. -. -. C. -. -. Carbon. Graphene. Hexagonal lattice; 1 p z orbital at each site. - PowerPoint PPT PresentationTRANSCRIPT
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
23cos41)( 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 11
21
E
011 1
12
1
E
γ0 is energy gained per pi-bond
Michael S. Fuhrer University of Maryland
Bloch states:
AB
AB
01
10
FA(r), or
FB(r), or
“anti-bonding”E = +3γ0
“bonding”E = -3γ0
11
21
11
21
Γ point:k = 0
Band Structure of Graphene – Band Structure of Graphene – ΓΓ point ( point (kk = 0) = 0)
Michael S. Fuhrer University of Maryland
34
32
1
i
i
e
e
λλ
λ
K
K
K
01FA(r), or
10FB(r), or
Phase:
K 23a
a3
4K
Band Structure of Graphene – K pointBand Structure of Graphene – K point
Michael S. Fuhrer University of Maryland
34
32
0 1
i
i
i
e
e
e
Phase:
Bonding is Frustrated at K pointBonding is Frustrated at K point
32
02
ieE
001ieE
34
03
ieE
0
034
32
00
iii eeeE
32
02
ieE
001ieE
34
03
ieE
0034
32
00
iii
i eeeeE
Re
Im
E1
E2
E3
Michael S. Fuhrer University of Maryland
01
FA(r), or
10
FB(r), or
K
23a
a3
4K
0π/3
2π/3π
5π/3
4π/3
“anti-bonding”
E = 0!
“bonding”
E = 0!
11
21
11
21
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
kvbeibe
ek 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:
)()(
)()(
00
rFrF
rFrF
ikkikk
vB
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/3π
5π/3
4π/3
180 degrees
540 degrees
Michael S. Fuhrer University of Maryland
Visualizing the PseudospinVisualizing the Pseudospin0
π/3
2π/3π
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)
214
2
nhe
xy
422 vsgg Berry’s phase = π
holes
electr
ons