Electron Transport in Molecules,
Nanotubes and Graphene
Philip Kim
Department of Physics
Columbia University
Rolling Up Graphene: Periodic Boundary Condition
a1
Ch
(n, 0)
Periodic Boundary Condition
r
Ch ⋅ k ⊥ = 2π q
Ch
a2
E(k2D)
(n, m)
+ band
ky
(n, n)
kx
- band
x
Allowed states
Semiconducting nanotube
E
Metallic nanotube
E
ky
ky
k1D
kx
k1D
kx
Tuning Carrier Density by Electric Field Effect
nanotube
Atomic force microscope image of nanotube device
metal
electrode
Vsd
nanotube
SiO
-------
metal
SiO2
++++++++++++++++
Doped Si
2
I
Vg
Si
Induced charge: Cg Vg = e n = e (D.O.S) ∆EF
Semiconducting nanotube
Metallic nanotube
E
E
E
E
EF
EF
k1D
EF
Vg << 0
k1D
Vg ~
>0
k1D
EF
Vg << 0
k1D
Vg ~
>0
Electrical Transport in Nanotube Devices
Semiconducting nanotube
Metallic nanotube
Vsd I
Vg
4
Vsd = 10 mV
Vsd = 10 mV
I (nA)
I (nA)
100
50
2
ON
0
0
-5
0
-5
5
E
EF
EF
k1D
5
E
E
E
0
Vg (V)
Vg (V)
EF
OFF
k1D
EF
k1D
k1D
Controlled Growth of Ultralong Nanotubes
B. H. Hong, J. Y. Lee, T. Beetz, Y. Zhu, P. Kim and S. Kim (2005)
Extraction of Inner Shells from MWNTs
AFM manipulation of long MWNTs
2 µm
2 µm
2 µm
5 µm
(Avouris, IBM)
5 µm
Hong, et al, PNAS (2005)
Intershell and Intrashell Nanotube Devices
Hong, et al, PNAS (2005)
Extremely Long SWNT Field Effect Transistor
gate
source
FET characteristics
drain
Vsd = 20 V
Isd (nA)
40
30
20
10
15 mm
0
-9 -6 -3
100 µm
0
3
6
Vg (V)
100 µm
ρ ~ 10-7 Ω m
9
Electron Transport in Long Single Walled Nanotubes
Multi-terminal Device with Pd contact
7
6
T = 250 K
R (kΩ)
5
4
3
Resistance (kΩ)
400
2
400
200
T0 = 250 K
0
20
40
60
L (µm)
R (kΩ)
100
200
8
ρ = 8 kΩ/µm
7
6
5
R~L
4
30
R ~ RQ
0
2
20
40
60
Length (µm)
10
8
7
6
* Scaling behavior of resistance:
R(L)
M. Purewall, A. Ravi, and P. Kim (2006)
5
2
0.1
4
6 8
2
4
6 8
1
2
10
L (µm)
4
6 8
Ballistic Transport and Mean Free Path
Electron Transport in 1D Channel
le : mean free path
+
- -
+
- L
L
Diffusive transport: le << L
R(L) = ρ L
Ballistic transport: L < le
R(L) = h /N e2 = RQ
# of channel
Resistance of N- 1D Channel:
h
h L
+ 2
R ( L) =
2
Ne
Ne le
For a nanotube, N = 4 ( 2 from spin and 2 from K and K’)
Electron Mean Free Path of Nanotube
Lines are fit to
7
400
T = 250 K
5
175 K
3
110 K
2
200
R (kΩ)
4
R (kΩ)
60 K
30 K
100
9
8
7
6
0
20
R( L) = Rc +
T = 250 K
0
h
= 6.45 kΩ
2
4e
110 K
Non-ideal contact resistance Rc < 2 kΩ
60 K
40
L (µm)
30 K
5
4
3
2
10
9
8
7
6
5
2
10
6
4
2
1
6
4
le ~ 0.5 µm @ RT !!
2
0.1
2
2
3
4 5 6
2
1
3
h
h L
+
4e 2 4e 2 le
175 K
mean free path (µm)
6
4 5 6
2
10
L (µm)
3
4 5 6
100
1
4
6 8
10
2
4
6 8
100
Temperature (K)
2
4
Extremely Long Mean Free Path: Hidden Symmetry ?
Carbon nanotube:
Ga[Al]As HEMT:
le ~ 10 µm @ 1.6 K
le ~ 100 µm @ 1.6 K
le ~ 0.5 µm @ 300 K
le ~ 0.06 µm @ 300 K
•Small momentum transfer backward
scattering must be inefficient.
E
Selection rules by hidden symmetry in graphene?
k1D
EF
right moving
left moving
T. Ando, JPSJ (1998);McEuen at al, PRL (1999)
Electric Field Effect in Mesoscopic Graphite
Simple Yet Efficient Mechanical Extraction
Using Scotch Tape is Essential!!
A Few Layer Graphene on SiO2/Si Substrate
Optical microscope images
AFM Image
1 µm
1.2 nm
0.4 nm
0.8 nm
Transport Single Layer Graphene
Field Effect Resistance
~h/4e2
Single layer graphene device
5000
4000
Cleaved graphite crystallite
20 µm
Rxx (Ω)
< 1 nm
3000
2000
1000
10 nm
0
-80
-60
-40
-20
0
20
Vg (V)
Zhang, Tan, Stormer & Kim (2005), see also Novoselov et al (2005).
40
60
80
Graphene : Dirac Particles in 2D Box
Band structure of graphene
E
Energy
hole
kx'
ky'
electron
ky
kx
Massless Dirac Particles with effective speed of light vF.
r
E ≈ hv F k ⊥′
Graphene v.s. Conventional 2D Electron System
Band structures
Density of States
*
E
me
π h2
h 2k 2
E=
2me*
Conventional 2D
Electron System
N2D(E)
~ h/4e2
h 2k 2
E=−
2 mh*
mh*
π h2
E
Graphene
• Zero band mass
• Strict electron hole symmetry
• Electron hole degeneracy
N2D(E)
2D Gas in Quantum Limit : Conventional Case
Density of States
Landau Levels in Magnetic Field
Quantum Hall Effect in GaAs 2DEG
E
hωc = heB / m *
me*
π h2
EF
DOS
mh*
π h2
s
Graphene
s
• Vanishing carrier mass near Dirac point
• Strict electron hole symmetry
eB
ωc = *
• Electron hole degeneracy
m
Quantum Hall Effect in Graphene
6
4
10
1 __
h
__
6 e2
5
2
1 __
h
__
10 e2
0
0
0
2
4
6
8
B (T)
Quantization:
2
1 __
e
_
Rxy = 4 (n + 2 )
h
-1
T=50 mK
Vg=-2 V
1 __
h
__
6 e2
10
Hall Resistance (kΩ)
1 __
h
__
2 e2
Magnetoresistance (kΩ)
Hall Resistance (kΩ)
15
1 __
h
__
2 e2
15
1 __
h
__
10 e2
5
1 __
h
__
14 e2
0
1 __
h
__
-14 e2
-5
1 __
h
__
-10 e2
1 __
h
__
-6 e2
-10
1 __
h
__
-2 e2
-15
-50
0
50
Vg (V)
T= 1.5K, B= 9T
Relativistic Landau Level and Half Integer QHE
Haldane, PRL (1988)
n=3
n=2
+
_
Energy
Energy
Landau Level
Landau Level Degeneracy
gs = 4
2 for spin and 2 for sublattice
n=1
Quantized Condition
n=0
σxy
DOS
n = -1
gse2/h
n = -2
EF
n = -3
T. Ando et al (2002)
Quantum Hall Effect in Graphene
Mobility
~ 60,000 cm2/V s
T = 1.7 K
B=9T
Nanotube Electrodes for Molecular Electronics
• Nanotubes are inherently small,
yet compatible to microfabrication processes
• Covalent chemistry between electrode and molecules
• Potentially good conduction via π-bonding network
Nanotube Nanogaps
Nanotube device
Narrow (<10nm) trench via e-beam
lithography
Thin PMMA coating
Oxygen Plasma etching creates gaps (0-10 nm) in tubes.
Cut ends likely to be carboxyl-terminated
Hone, Wind, Nuckolls, and Kim Collaboration
Nanotube Nanogaps
~ 5 nm
SEM micrograph
SEM micrograph
AFM micrograph
Process are optimized to Yield of cut tubes: 25% of ~2600 devices
Columbia NSEC (Hone, Wind, Nuckolls, and Kim) Collaboration
Molecular Bridges
Bis-oxazole
Pyridine, EDCI
•Self-assembled
•Covalently bonded
•Conduction through π-back bone
Guo et al., Science (2006)
Does It Work?
Vsd
~ 10-15% of reconnection
out of ~ 100 fully cut tubes
Vg
Semiconducting Nanotube + Molecular Bridge
14
Metallic Nanotube + Molecular Bridge
before cut
before cut
50
10
Vsd = 50 mV
8
6
4
2
after reconnection
after cut
Current (nA)
Current (nA)
12
40
30
20
0
0
-5
0
Vg (v)
5
after reconnection
10
after cut
-5
0
Vg (v)
5
Control Experiments
Pyridine, EDCI
Connection
with diamine bisoxazole
* Pyridine +EDCI without molecules
No connection
* Bis-oxazole without amines
No connection
* Bis-oxazole with Monoamine
No connection
* 1,12 dodecane diamine (insulator)
No connection
Oligoaniline: PH sensing
Pyridine, EDCI
Transport Measurement
Bis-oxazole + metallic SWNT
40
1.6 K
250 K
20
I (nA)
180 K
120 K
50
I (nA)
50 K
10 K
0
-20
1.6 K
0
-40
-800
-400
0
400
800
Vsd (mV)
-50
0
1.0
-2
-4
0.5
-100
-6
-800
-400
0
400
800
0.0
-8
-450
Vsd (mV)
-400
-350
-300
150
200
250
300
350
Temperature Dependence Transport Spectroscopy
20
1.6 K
10
0
-600
-400
-200
0
200
400
600
1.6K
10K
20K
50K
3
2
1
10 K
10
0
0
150
-600
-400
-200
0
200
400
20
200
250
300
Vsd (mV)
600
50 K
10
10
0
8
-600
-400
-200
0
200
400
600
20
10
180 K
0
-600
-400
-200
0
200
Vsd (mV)
400
dI/dV (10−8S)
dI/dV (10−8S)
20
dI/dV (10−8S)
4
250 K
220 K
180 K
6
4
2
600
120 K
30 K
20 K
80 K
50 K
0
-250
-200
-150
Vsd (mV)
Gate Voltage Dependence
0
0.1
dI/dVsd (µS)
Vsd (mV)
500
250
0
-250
-500
-6.0
-5.0
-4.0
Vg (V)
225
0.2
Vsd (mV)
Vg=-9.25 V
-9.23 V
-9.21 V
-9.19 V
-9.17 V
0.1
0.4
200
dI/dV (µS)
dI/dV (µS)
0.3
0.2
175
150
0
0
150
160
170
Vsd (mV)
180
200
125
-9.3
-9.25
Vg (V)
-9.2
Summary
•Transport in long nanotubes:
Subshell extraction in MWNTs
Extremely long mean-free path in SWNTs
•Transport in graphene:
Unusual quantum Hall effect
Graphene nano ribbon devices
Gate dependent Raman spectroscopy
•Nanotube electrode for
single molecular electronics
Acknowledgement
Special Thanks to:
Yuanbo Zhang
Meninder Purewal
Byung Hee Hong
Josh Small
Melinda Han
Barbaros Oezyilmaz
Collaboration:
Stormer, Pinczuk, Heinz,
Nuckolls, Brus, Flynne, Hone,
Funding:
Kim Group 2004 Summer Central Park