electron transport in graphitic systemsykis07.ws/presen_files/23/kim.pdf · 2007. 12. 11. ·...
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Electron Transport in Graphitic Systems
Philip Kim
Department of PhysicsColumbia University
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SP2 Carbon: 0-Dimension to 3-Dimension
Fullerenes (C60) Carbon Nanotubes
Atomic orbital sp2σ
π
GraphiteGraphene
0D 1D 2D 3D
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Graphene : Dirac Particles in 2-dimension
Band structure of graphene (Wallace 1947)
kx
ky
Ener
gy
kx' ky'
E
⊥′≈ kvE F
rh
Zero effective mass particles moving with a constant speed vF
hole
electron
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Dirac Fermions in Graphene : “Helicity”
E
κxκy
K
⊥⋅= kvH Feff
rrh σ
E
κxκy
K’
⊥⋅= kvH Feff
rrh *σ
momentumpseudo spin
E
kx
ky
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Single Wall Carbon Nanotube
…. since 1991
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400
200
0
6040200
Length (µm)
Res
ista
nce
(kΩ
)
T = 250 K
ρ = 8 kΩ/µm
Electron Transport in Long Single Walled Nanotubes
Multi-terminal Device with Pd contact
Purewall, Hong, Ravi, Chandra, Hone, and P. Kim PRL (2007)
* Scaling behavior of resistance:R(L)
5678
10
2
3
4
5678
100
2
3
4
567
0.12 4 6 8
12 4 6 8
102 4 6 8
L (µm)
R(k
Ω)
T = 250 K400
200
0
6040200
R(k
Ω)
L (µm)
R ~ RQ
R ~ L
elL
eh
ehLR 22 44
)( +=
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Modulate Doped GaAs: Pfeiffer et al.Modulate Doped GaAs: Pfeiffer et al.
Electron Mean Free Path of Nanotubes
M. Purewall, B. Hong, A. Ravi, B. Chnadra, J. Hone and P. Kim, PRL (2007)
Room temperature mean free path > 0.2 µm
Mea
n Fr
ee P
ath
(µm
)
1 10 1000.1
1
10
Temperature (K)
sc7
sc1sc2sc3sc4sc5sc6
m1m2m3m4
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Extremely Long Mean Free Path: Hidden Symmetry ?
E
k1D
EF
right moving left moving
• Small momentum transfer backward scattering becomes inefficient since it requires pseudo spin flipping.
Pseudo spin
Low energy band structure of graphene1D band structure of nanotubes
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Pd (under HfO2)
Pd (under HfO2)
Pd (over HfO2)
SWCNT (under HfO2)
HfO2 on SiO2/Si+
Carbon Nanotube Superlattice
20 nm
60 nm
1 µm
-54 -50 -45 -40
4
3
2
1
Back Gate (V)
Top Gate (V)
1 .0 1 .5 2 .00 .0
0 .2
dI/d
V (µ
S)
T o p G a te (V )
Purewal, Zuev, Jarillo-Herrero, Kim (2007)
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Discovery of Grphene
Y. Ohashi (1997), R. Ruoff (1998): Mechanical extraction of graphite
McConville (1986): Epitaxial growth on metal surface
Krishanan (1997): Chemical decomposition
Earlier Work (20th Century)
by ~ 2004
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Vg (V)
ρ(Ω
)
Resistivity vs Gate Voltage
5000
4000
3000
2000
1000
0
-80 -60 -40 -20 0 20 40 60 80
Transport Single Layer Graphene
Cleaved graphite crystallite20 µm
Single layer graphene device
~h/4e2
E
N2D(E)
ρ -1 = e2vF le N2D
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Quantum Hall Effect in Graphene
Quantization:
4 (n + )Rxy =-1 ___ eh
2
21
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EF
σxy
Ene
rgy
gse2/h
Relativistic Landau Level and Half Integer QHE
Quantized Condition
Landau Level Degeneracygs = 4
2 for spin and 2 for sublatticeLandau Level +_
Haldane, PRL (1988)
T. Ando et al (2002)
n = 1
n = 2
n = 3
n = -3
n = 0
n = -1
n = -2
DOS
Ene
rgy
E1 ~ 300 K [B(T)]1/2B = 0
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Room Temperature Quantum Hall Effect
+_
E1 ~ 100 meV @ 5 T
Novoselov, Jiang, Zhang, Morozov, Stormer, Zeitler, Maan, Boebinger, Kim, and Geim Science (2007)
1.02
1.00
0.983.02.5
n (1012 cm-2 )
Rxy
(h/2
e2)
300K45 T
Deviation < 0.3%
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-50 0 50
Vg (V)
Con
duct
ivity
100 e2/h
TC17
TC12
TC145
TC130
Conductivity, Mobility, & Mean Free Path
πµσ neen l
h
2
==
103
104
105
-4 -2 0 2 4
n (1012 cm-2)
Mobility (cm
2/V sec)
TC17
TC12
TC145
TC130
Mobility
0.01 0.1 1 10
Lm
(nm)
100
1000
10
TC17
TC12
TC145
TC130
Mean free path
|n| (1012 cm-2)
Tan at al, PRL (2007)
Scattering Mechanism?
•Ripples•Substrate (charge trap)•Absorption•Structural defects
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STM on Graphene
Atomic resolutionRipples of graphene on a SiO2 substrate
Elena Polyakova et al (Columbia Groups), PNAS (2007)
See also Meyer et al, Nature (2007) and Ishigami et al, Nano Letters (2007)
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Quantum Hall Effect in Graphene at High Magnetic Field
B = 45 TT = 1.4 K
Zhang, et al, PRL (2006)
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energy
Landau Level
Splitting of Landau Levels in High Magnetic Fields
9 T25 T ν = 2, 6, 10, …. +_ +_ +_
Low fields (B < 10 T)
ν = 0, 1, 2, 4, 6, …+_ +_
High fields (B > 20 T)
+_ +_
ν = -2
ν = 2
ν = -6
ν = 6ν = 4
ν = 1ν = 0ν = -1
ν = -4
σxy= -Rxy /(Rxy2+Rxx
2)
Zhang, et al, PRL (2006) Spin & sublattice symmetry lifted!
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Quantum Hall Insulator OR Quantum Hall Ferromagnet?
Low magnetic field
n = 1
n = 2
n = 3
n = -3
n = 0
n = -1
n = -2
DOSLand
au L
evel
Ene
rgy
High magnetic field degeneracy break: two scenarios
Spin & valley degenerate
QHE FerromagnetSpin -> Pseudo Spin
B
QHE InsulatorPseudo Spin -> Spin
B
Normura & Macdonald, PRL 96, 256602 (2006); Abanin, Lee, & Levitov, PRL 98, 156801 (2007);
ε
x
QH edge states
ε
x
QH edge states
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Spin or Pseudo Spin Splitting?
Bp= 20 T, Btot=45 T
Bp=20 T, Btot=30T
Tilted Magnetic Field
6000
4000
2000
0-30 -20 -10 0 10
Vg (V)
Rxx
(Ω)
-6
-4 -2-1
+1
ν = -4
ν = +4ν = +2
ν = -2
ν = 0ν = +1
ν = -1
Magnetic Field (T)
∆E/
k B(K
) ν = 1
ν = 4
ν = -4
150
100
50
0
-50
50403020100
Energy Gap Measurements
~B
~B1/2
Quantum Hall Ferromaget!
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Unusual Nature of ν=0 Quantum Hall States: Many-body Origin?
Magnetic Field (T)
∆E/
k B(K
)
ν = 1
ν = 4
ν = -4
150
100
50
0
-50
50403020100
Landau Level Hierarchy
E1 ~ 2500 K
B= 45 T
∆Eν=+4= ~ 30 K
∆Eν=+2= ~ 900 K
∆Eν=+1= ~ 120 K??
* Signature of enhanced e-e interaction near the Dirac point* What is the nature of ν = 0 state?
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Jiang et al. PRL (2007)
Energy Gap Measurement: Cyclotron Resonance
n
n+1Bnve FC hh 2 =ω
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-3
-1
1
0
2
-2
3n
x ( 2+ 1)
∆En, (n+1)= 2ehvFB ( n+1± n)
En= 2ehvFnB
~100cm-1
Excitonic Transition: Electron-electron interaction??
vF ~ 106 m/sec-
Jiang et al. PRL (2007)
e-e interaction is important!
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Graphene Research at Columbia University• High Mobility Graphene Samples:
Extreme Quantum Limit Transport (Kim +Stormer)• Graphene Devices
Nanostructures, heterostructures, Quantum Interference Devices (Kim)• Spin Transport in Graphene:
Spin Hall Devices, Non-local spin transport devices (Kim)
• Graphene for Optical Studies:Raman Spectroscopy (Kim + Pinczuk)Absorption Spectroscopy (Heinz)
• Graphene spectroscopyIR (Kim+Stormer), Photoemission (Osgood)
•STM on graphene:local electronic structure, molecular assembly on graphene (Kim + Flynn)
•Graphene Organic Chemistry:Edge decoration, covalent doping in graphene (Kim + Nuckolls)
• Graphene Synthesis and Photochemstry:Low temperature synthesis and surface photochemistry (Brus)
• Graphene Intercalation (O’brien)
• Graphene Theory: Hybertsen, Millis, Aleiner, Altshuler
r
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Raman Spectroscopy on Graphene: Gate Voltage Dependence
1560 1580 1600 1620
194 196 198 200 202(meV)
-10V
-40V
-80V
20V50V
Inte
nsity
(a.u
.)
Raman shift (cm-1)
graphite
80V
T=10K
Vg =
Raman G bandGraphene G-mode phonon0,200 =≈ kmeVwG
rh
J. Yan, Y. Zhang, P. Kim and A. Pinczuk (2006)
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Raman Spectroscopy on Graphene: Gate Voltage Dependence
4 0 -1 -99 1 -4Charge Density (1012 cm-2)
-300 -200 -100 0 100 200 300
Fermi Energy (eV)
5
10
15
2Gwh
2Gwh
−
G b
and
wid
th Γ
G(c
m-1
)
Gh(π)e(π*)
222
DMvA
F
ucG =∆ΓFermi Golden Rule:
22 )AeV/(40 &=D
Phonon Decay
1580
1585
1590
1595
-300 -200 -100 0 100 200 300
Fermi Energy (eV)
4 0 -1 -99 1 -4Charge Density (1012 cm-2)
G b
and
Ener
gy w
G(c
m-1
)
e(π*) G
h(π)
G
FFG
ucGG E
MvDA
20
20
ωπωω
hhh +=
Phonon Renormalization:
22 )AeV/(35 &=D
Renormalization
J. Yan, Y. Zhang, P. Kim and A. Pinczuk (2006); See also Ferrari et al (2006)
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Unusual Phonon Softening in Bi-Layer Graphene
T. Ando, J. Phys. Soc. Jpn. (2006)
Phonon softe
ning by resonance
G band Raman Spectrum in Bilyaer Graphene
Yan, Henriksen, Kim and Pinczuk (2007)
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Graphene Electronics
Engineers’ Dreams
Cheianov et al. Science (07) Trauzettel et al. Nature Phys. (07)
Theorists’ Dreams
Graphene Veselago lense Graphene q-bits
and more …
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Contacts:
PMMAEBLEvaporation
Graphene patterning:
HSQEBLDevelopment
Graphene etching:
Oxygen plasma
Local gates:
ALD HfO2EBLEvaporation
From Graphene “Samples” To Graphene “Devices”
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W
Dirac Particle Confinement
Egap~ hvF ∆k ~ hvF/W
1 µm
Gold electrode Graphene
10 nm < W < 100 nm
W
Zigzag ribbons
Graphene nanoribbon theory partial list
Graphene Nanoribbons: Confined Dirac Particles
Wky
π⋅=
2
Wky
π⋅=
3
Wky
π⋅=
1
Wky
π=∆W
x
y
22 )/( WnkvE xF π+±= h
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Scaling of Energy Gaps in Graphene Nanoribbons
W (nm)
E g(m
eV)
0 30 60 901
10
100
P1P2P3P4D1D2
Eg = E0 /(W-W0)
Han, Oezyilmaz, Zhang and Kim PRL (2007)
P1
D2
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-8 -4 0 4 8
75
50
25
0
-25
-50
-75
VLG (V)
V BG
(V)
10-7 10-5 10-3 10-1
G (e2/h)
Top Gated Graphene Nano Constriction
source
Back gateSiO2
drain graphene
Hf-oxide
Top gate
-8 -4 0 4 810-6
10-5
10-4
10-3
10-2
10-1
VLG (V)
G(e
2 /h)
OFF
SEM image of devicesourcedrain top gate
graphene1 µm
30 nm wide x 100 nm long
Oezyilmaz, et al., APL (2007)
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Graphene Quantum Hall Edge State Conduction
EL EL
LG
GLs GLs
Local Gate Region
1 µm
simple model (following Haug et al)
Oezyilmaz, et al., PRL (2007) See also Related work by Williams et al. Science (2007)
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SummarySummary
Graphene Electronic Devices
Strong Correlation in Graphene
Graphitic Carbon Systems
• Band Gap Engineering in graphene nanostructures• Local density control of graphene• Peculiar quantum Hall edge states
• e-e interaction• strongly correlated behavior near the Dirac points
• Zero effective mass, Zero gap• Pseudo spin• Extremely Long Mean Free Path in Nanotubes• Unusual quantum Hall effect in Graphene
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AcknowledgementSpecial Thanks to: Yuanbo Zhang (now at Berkeley)Meninder PurewalMelinda HanYuri ZuevYue ZhaoChul Ho LeeAsher MullokandovDmitri EfetovByung Hee HongNamdong KimBarbaros OezyilmazKirill BolotinPablo Jarrilo-HerreroZhigang Jiang
Funding:
Collaboration: Stormer, Pinczuk, Heinz, Uemura, Venkataraman, Nuckolls, Brus, Flynne, Hone, KS Kim, GC Yi
Kim Group: 2007 Roof top of Pupin Laboratory