Quantum Interference in Multiwall Carbon Nanotubes

Christoph StrunkUniversität Regensburg

Coworkers and Acknowledgements:

B. Stojetz, Ch. Hagen, Ch. Hendlmeier (Regensburg)

L. Forró, E. Ljubovic (Lausanne)

A. Bachtold , M. Buitelaar, Ch. Schönenberger (Basel)

K. Richter, G. Cuniberti (Regensburg)

R. Schäfer (Karlsruhe)

multiwalled carbon nanotubes

S. Ijima, Nature 354, 56 (1991)

26 nm

Outline

Introduction: Electronic structure of carbon nanotubes

Quantum interference

Changing the electron density

Coulomb blockade

Perspectives

sp2-hybridization leads toplanar carbon sheets

2D electronic bandstructure determined by p-orbitals

-bands touch at K-points

kxky

E

*

K’

K

kx

ky

Graphene: a single sheet of graphite

wrapping graphene to nanotubes:

x

y

wrapping vector R determines:

chirality (real space) allowed k-vectors (k-space)

RARB

Density of states

kx

ky

K

K’

K’

K

Metallicbehavior

Semicond.behavior

are MWNTs ballistic conductors at 300 K?

Frank, et al., Science 280, 1744 (1998)

G (2

e²/h

)z-position (nm)

Conductance changes in units of 2e²/h !

Weak localization and universal conductance fluctuations (UCF)

signatures of coherent backscattering in disordered quantum wires:

P r r t A A A Aii

i i ji ji

( , ' , ) *

22

2 r r’

Ai

Aj

r =r’ A+ =A-

Closed loop of time reversed paths:

P r r t A A A( , , )

2 24

enhanced backscattering probability!

Magnetic field breaks time-reversal symmetry:coherent backscattering suppressed by magnetic field:negative magnetoresistance near B=0

reproducible fluctuation pattern specific forimpurity configuration: “magneto-fingerprints”

Weak localization and universal conductance fluctuations (UCF)

signatures of coherent backscattering in disordered quantum wires:

P r r t A A A Aii

i i ji ji

( , ' , ) *

22

2 r r’

Ai

Aj

r =r’

Closed loop of time reversed paths:

P r r t A A A( , , )

2 24

enhanced backscattering probability!

Magnetic field breaks time-reversal symmetry:coherent backscattering suppressed by magnetic field:negative magnetoresistance near B=0

reproducible fluctuation pattern specific forimpurity configuration: “magneto-fingerprints”

A+ =A-

A. Bachtold et al., ‘98

Similar results obtained by many other groups:

Leuven, IBM, Stuttgart, Helsinki …..

How to confirm the presence of elastic scattering ?

200 nm

Au contact

Au contact

Al gate

(native oxide)

MWNT

Induce drastic change of electron density by gate electrode (distance 2-3 nm)

Change number of current carrying subbands

Tune electrochemical potential through charge neutrality point

Induce transition between quasi-1dim and strictly 1dim transport ?

k

E

EF

Doping state of MWNTs

Effect on weak localization ?

Effects of Coulomb interaction ?

Gate sweep

2

-3 -2 -1 0 1

10

15

20

25

R (k

)

U Gate (V)

1.7 K

5 K

10 K

15 K20 K

40 K

low temperaturesuniversal conductance fluctuations (UCFs)

(curves shifted)

-1,0 -0,5 0,0 0,5 1,01,52

1,54

1,56

1,58

G (2

e2 /h)

UGate

(V)

T = 300 K

charge neutrality point ?

high temperaturesshallow minimum in conductance

Universal conductance fluctuations

Ensemble averaging of conductance fluctuations G if L < l

l

Interference of many diffusion paths lead to aperiodic fluctuation pattern in the conductance:

2/32

12

Ll

heGRMS

vary interference pattern by applying electric or magnetic fields

determine phase coherence length l at different temperatures

l > tube diameter (28 nm) l < tube length (400 nm)

Magnetoresistance at different gate voltages

-3 -2 -1 0 1 28

12

16

20

24

28

R (k

)

UGate (V)

magnetic field B perpendicular to tube axis

magnetoresistancetraces taken at various gate voltages (arrows)

select different members within statistical ensemble of magneto-fingerprints

T = 1.7 K

Ensemble averaging

-10 -5 0 5 10

20

40

60

80

100

B (T)

R (k

)

-2.79 V

-2.27 V

-2.22 V

-2.10 V

-2.03 V

-1.15 V

-1.02 V

-0.52 V

-0.4 V 0 V

+0.63 V

+1.34 V

+1.52 V

+1.92 V

+1.87 V

-10 -5 0 5 10

10

12

R (k

)

B (T)

average

weak localization peak survives averaging

UCFs averaged out partially, but not completely

T = 1.7 K (curves shifted)Stojetz et al., New J. Phys. ‘04

Weak localization

2/1

2

222

2

2

31

BeW

lLeGWL

conductance correction due to weak localization:

Fitting WL-theory to data:

T (K) l (nm)

1.7 150 20 80 40 50

1.7 K

20 K

40 K

effective width W~diameter/2 requiredorigin: flux-cancellation effects ?

Phase coherence length

diamonds:UCF measurement

triangles:weak localization

line:prediction for electron-electron dephasing ~T-1/3

elastic mfp: 14 nm

Good agreement of l from WL and UCFsSubstantiation of diffusive transport pictureFurther experiments required to identify origin of disorder

:UCF :WL

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,50,00,51,0

UGate (V)

T=10 K

T=2 K

0,51,0

G (2

e2 /h)

T=60 K

T=5 K

0,51,0

T=300 K

0,6

0,8

1,0

1,2

Measure a larger statistical ensemble:

shallow conductance minimum at 300K

emerging fluctuation pattern at lower T

decrease of correlation voltage Vc

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,50,0

0,5

1,0 30 mK

UGate

(V)

G (2

e2 /h)

0,0

0,5

1,0

500 mK

0,0

0,5

1,01 K

Crossover to Coulomb blockade at lowest T :

decrease of average conductance

Resonant transmission of single channels?

-10 0 10 20-0,4

-0,2

0,0

0,2

0,4

00

0,30

0,50

UGate

(mV)

UD

C (m

V)

G (2e2/h)

T=30 mK

disordered MWNT with irregular Coulomb diamonds:

typical capacitances:

CGate ~ 55 aF C ~ 800 aF

charging energyEc ~ 100 eV ~ 1.2 K

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-2

-1

0

1

2

UD

C (m

V)

0.2

0.4

0.7

0.8

UGate

(V)

G (2e2/h)

T=3K

broad zero bias anomalies remain at higher T:

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-2

0

2

UD

C (m

V)

T=10K U

Gate (V)

0.4

0.6

0.7

0.8G (2e2/h)

T = 3 K T = 10 K

estimated subband spacing ~ 25 meV

gate lever arm EF/UGate ~ 1/10

-2 0 2 4 6-1.0

-0.5

0.0

0.5

1.0

UG

ate

(V)

B(T)

0.5 0.6 0.7 0.8G (2e2/h)

T=10K

-2 0 2 4 60.4

0.6

0.8

A

G (2

e2/h

) 0.4

0.6

0.8

UGate

= 500 mV

UGate

= -200 mV

T = 10 K

Magnetoconductance shows pronounced gate dependence:

Open questionsSource of disorder -extrinsic or intrinsic ?

Strength of disorder?

Effect of Coulomb blockage and number of channels on the shape of the WL-peak?

Gate dependence of Aharonov-Bohm effect in parallel magnetic field?

B