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