Electron Transport in Graphitic Systems
Philip Kim
Department of PhysicsColumbia University
SP2 Carbon: 0-Dimension to 3-Dimension
Fullerenes (C60) Carbon Nanotubes
Atomic orbital sp2σ
π
GraphiteGraphene
0D 1D 2D 3D
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
Dirac Fermions in Graphene : “Helicity”
E
κxκy
K
⊥⋅= kvH Feff
rrh σ
E
κxκy
K’
⊥⋅= kvH Feff
rrh *σ
momentumpseudo spin
E
kx
ky
Single Wall Carbon Nanotube
…. since 1991
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
)( +=
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
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
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)
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
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
Quantum Hall Effect in Graphene
Quantization:
4 (n + )Rxy =-1 ___ eh
2
21
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
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%
-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
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)
Quantum Hall Effect in Graphene at High Magnetic Field
B = 45 TT = 1.4 K
Zhang, et al, PRL (2006)
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!
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
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!
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?
Jiang et al. PRL (2007)
Energy Gap Measurement: Cyclotron Resonance
n
n+1Bnve FC hh 2 =ω
-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!
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
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)
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)
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)
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 …
Contacts:
PMMAEBLEvaporation
Graphene patterning:
HSQEBLDevelopment
Graphene etching:
Oxygen plasma
Local gates:
ALD HfO2EBLEvaporation
From Graphene “Samples” To Graphene “Devices”
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
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
-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)
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)
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
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