Philip Kim

Department of Physics

Columbia University

Physics of Graphitic Carbon

Nanotstructures

Graphene : Dirac Particles in 2-dimension

Band structure of graphene

kx

ky

Ener

gy

kx' ky'

E

Zero effective mass particles moving with a constant speed vF

hole

electron

- -

Pseudo Spin in Graphene Lattice

‘A’ sublattice

‘B’ sublattice

+ +

pzA(r) + ei! pz

B(r)

Relative amplitude of sublattice wavefunctions

! =0: bonding ! = !: antibonding

k p perturbation theory .

! Spinor

ei!

1 pz

A(r)

pzB(r)

DiVincenzo and Mele, PRB (1984); T. Ando, JPSJ (1998);McEuen at al, PRL (1999)

!k = tan-1(ky / kx)

= eik r .

ei!"

1

Dirac Fermions in Graphene : “Helicity”

E

"x "y

K

E

"x "y

K’

momentum pseudo spin

E

kx

ky

Single Wall Carbon Nanotube

…. since 1991

Modulate Doped GaAs:

Pfeiffer et al.

Modulate Doped GaAs:

Pfeiffer et al.

Cabon Nanotube Mean Free Path

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 F

ree

Pat

h (

µm

)

1 10 100 0.1

1

10

Temperature (K)

sc7

sc1 sc2 sc3 sc4 sc5 sc6

m1 m2 m3 m4

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

T. Ando, JPSJ (1998);McEuen at al, PRL (1999)

Low energy band structure of graphene

1D 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

Purewal, Zuev, Jarillo-Herrero, Kim (2008)

Kouwenhoven PRL (1992)

Nanotube Electronics: Challenges

Pros:

High mobility High on-off ratio

High critical current density

Con:

Controlled growth

Artistic dream (DELFT)

IBM, Avouris group

Nanotube Ring Oscillators

graphene

Mechanical Extraction of Graphene

Graphite debris on Si wafer

Developed by Novoselov et al. Science (2004)

1 µm

AFM Image

A Few Layer Graphene on SiO2/Si Substrate

0.8 nm

0.4 nm

1.2 nm

Optical microscope images

Identifying # of layers

Transport: Novoselov et al, Zhang et al

Raman Spectroscopy: Ferrari et al, Eklund et al,…

Photoemission: Rotenberg et al, Lanzara et al.

TEM: Meyer et al

STM: Flynn et al, Williams et al, Cromie et al.,…

Vg (V)

# ("

)

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 crystallite 20 µm

Single layer graphene device

~h/4e2

E

N2D(E)

# -1 = e2vF le N2D

15

10

5

0

8 6 4 2 0

6

4

2

0

Hall

Re

sis

tance (

k"

)

Ma

gneto

resis

tance (k"

) B (T)

Quantum Hall Effect in Graphene

__ __ h

e2

1

2

__ __ h

e2

1

6

__ __ h

e2

1

10

T=50 mK

__ __ h

e2

1

2

-15

-10

-5

0

5

10

15

-50 0 50

Vg (V)

Hall

Resis

tance (

k"

)

T= 1.5K, B= 9T

__ __ h

e2

1

6

__ __ h

e2

1

10

__ __ h

e2

1

14

h 1 __ __ e2 -14

e2

__ __ h 1

-10

__ __ h

e2

1

-6

__ __ h

e2

1

-2 Quantization:

4 (n + ) Rxy =

-1 __ _ e

h

2

2 1

Vg=-2 V

Novoselov et al, Zhang et al (2005)

Graphene:

kx' ky'

E

B=0

Berry’s Phase and Magnetoresistance Oscillations

Landau orbit near the Fermi level

Pseudo Spin

Graphene

$ ~ $% exp[2!(&B/&o)]

$ ~ $% exp[2!(&B/&o) – S! /

h] !#

kF

S = h/2

! = 2!#

ky

kx

&o = h/e

'F = &o kF2 /4(

#

&B = &o BF /B

Magnetic flux in cyclotron orbit

Rx

x

2DEG

1/BF

1/B Quantum Hall States

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

0.98 3.0 2.5

n (1012 cm-2 )

Rxy (

h /

2e

2)

300K

45 T

Deviation < 0.3%

Vg (V)

Cond

uctivity

100 e2/h

TC17

TC12

TC145

TC130

Conductivity, Mobility, & Mean Free Path

n (1012 cm-2)

Mobility

(cm

2/V s

ec)

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)

Graphene Mobility

n (1012 cm-2)

Mobility

(cm

2/V s

ec)

TC17

TC12

TC145

TC130

Graphene Mobility

Modulate Doped GaAs:

Pfeiffer et al.

GaAs HEMT

Tan et al. PLR (2007)

STM on Graphene

Atomic resolution Ripples of graphene on a SiO2 substrate

Elena Polyakova et al (Columbia Groups), PNAS (2007)

Scattering Mechanism?

•!Ripples

•!Substrate (charge trap)

•!Absorption

•!Structural defects See also Meyer et al, Nature (2007) and Ishigami et al, Nano Letters (2007)

Toward High Mobility: Suspending Samples

graphene

HF etching

-> critical pointing drying

SEM image of suspended graphene

AFM image of suspended graphene You should not apply to high gate voltage, otherwise…

Collapsed graphene devices…

Cleaning Graphene Surfaces: Annealing

0 5 10 -10 -5 0

2

4

3

1

Vg (V)

Res

isti

vit

y ("

)

17B: best unsuspended sample

µ ~ 20,000 cm2/Vsec

S452: suspended sample

S452: after current annealing

Unsuspended (300 nm SiO2): Cg = 115 aF/µm2

Suspended (~150 nm SiO2): Cg ~ 50 aF/µm2

T = 4 K

Sample quality can be improved

after self-heating annealing:

current density ~ 108 A/cm2

(Developed by Bachtold et al, APL (2007))

S452 17B

5 µm

Drude Mobility

unsuspended best

before annealing

after annealing

Density ( 1012 cm-2)

Mobil

ity (

cm2/V

sec

)

µ ~ 200,000 cm2/Vs

@ 2X1011 cm-2

n = 2X1011 cm-2

SdH Oscillations

after annealing

before annealing

onset of SdH

µ! > 1/BSdH ~ 100,000 cm2/Vs

Characteristics of Suspended Samples: Mobility

Charge Inhomogeneity near the Charge Neutrality Point:

Unsuspended samples

n (1012 cm-2)

) (

e2/h

)

0 1 -1 0

10

Conductivity near zero density Unsuspended Samples

Martin et al. Nature Phys. (2008)

Charge inhomogeneity near zero density

Mobility

Versus

Inhomogeneirty

(unsuspended)

*WDirac

Toward High Mobility Samples

Mobility VS. Inhomogeneirty

Suspended samples

n (1010 cm-2)

# (

n)

/ #

(n=

0)

Normalized Resistivity

$WDirac

$WDirac

Density Inhomogeneity < 1010 cm-2 is possible!

best unsuspended

Suspended & annealed

B29.00011 Bolotin et al: Monday

Graphene Electronics

Engineer Dreams

Cheianov et al. Science (07) Trauzettel et al. Nature Phys. (07)

Theorist Dreams

Graphene Veselago lense Graphene q-bits

and

more …

Contacts:

PMMA

EBL

Evaporation

Graphene patterning:

HSQ

EBL

Development

Graphene etching:

Oxygen plasma

Local gates:

ALD HfO2

EBL

Evaporation

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

W

Wide (> 1µm) Graphene

10-5

10-4

10-3

10-2

10-1

-30 -20 -10 0 10 20 30

30mK 4K 77K 290K

Vg (V)

Con

du

ctiv

ity ("

-1)

Graphene Ribbon Devices

W

Dirac Particle Confinement

Egap~ hvF / W

200K 100K 10K 1.7K

10-4

10-5

10-6

10-7

10-8

Conduct

ance

("

-1)

60 40 20 0 Vg (V)

W = 75 nm

200K 100K 10K 1.7K

10-4

10-5

10-6

10-7

10-8

Conduct

ance

("

-1)

60 40 20 0 Vg (V)

W = 53 nm

10-4

10-5

10-6

10-7

10-8

Conduct

ance

("

-1)

60 40 20 0 Vg (V)

200K 100K 10K 1.7K

W = 32 nm

Con

du

ctan

ce (

µS

)

Ribbon Width (nm)

Vg = 0 V

T = 300 K

1 µm

Gold electrode Graphene

10 nm < W < 100 nm

W

Scaling of Energy Gaps in Graphene Nanoribbons

W (nm)

Eg (

meV

)

0 30 60 90 1

10

100

P1 P2 P3 P4 D1 D2

Eg = E0 /(W-W0)

Han, Oezyilmaz, Zhang and Kim PRL (2007)

-8 -4 0 4 8

75

50

25

0

-25

-50

-75

VLG (V)

VB

G (V

)

10-7 10-5 10-3 10-1

G (e2/h) The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

Top Gated Graphene Nano Constriction

source

Back gate

SiO2

drain graphene

Hf-oxide

Top gate

-8 -4 0 4 8

10-6

10-5

10-4

10-3

10-2

10-1

VLG (V)

G (

e2/h

)

OFF

SEM image of device

source drain top gate

graphene

1 µm

30 nm wide x 100 nm long

L = 2 µm

W=150 nm

Vsd (V)

I sd (

µA

)

100

50

0

-50

-100

-4 -2 0 2 4

$Vg = -30 V

-20 V -15 V -10 V -5 V 0 V

-25 V

Graphene High Bias Regime

High Current Density: > 108 A/cm2

High mobility: > 10,000 cm2/Vsec

Saturation Velocity > 2 X 107cm/sec

Low On-off Ratio: ~ 102

Room Temperature, Air

H29.00009 Han et al : Tuesday morning

Graphene Nanostructures out of Mechanical Extraction

Manchester DELFT Stanford

Columbia

ETH

Harvard

Challenges:

•!Better Growth -> higher quality samples

•! Controlled Edges

Acknowledgement

Special Thanks to:

Yuanbo Zhang (now at Berkeley)

Meninder Purewal

Melinda Han

Yuri Zuev

Yue Zhao

Chul Ho Lee

Asher Mullokandov

Dmitri Efetov

Byung Hee Hong

Namdong Kim

Barbaros Oezyilmaz (now at NSU)

Kirill Bolotin

Pablo Jarrilo-Herrero (now at MIT)

Zhigang Jiang

Funding:

Collaboration:

Stormer, Pinczuk, Heinz, Uemura,

Venkataraman, Nuckolls, Brus, Flynne,

Hone, KS Kim, GC Yi

Kim Group: 2007

Roof top of Pupin Laboratory