Applying Magnetic Fields to CarbonApplying Magnetic Fields to CarbonApplying Magnetic Fields to CarbonApplying Magnetic Fields to Carbon----based low based low based low based low

Dimensional Materials: Dimensional Materials: Dimensional Materials: Dimensional Materials:

from from from from AharonovAharonovAharonovAharonov----BohmBohmBohmBohm effects to Landau levelseffects to Landau levelseffects to Landau levelseffects to Landau levels

Stephan RocheStephan Roche

2

2

High Magnetic Field Effects in CNTs

-4 -2 0 2 4 6

10-9

10-8

10-7

10-6

10-5

0T 6T 30T

G, Ω

-1

Gate Voltage, V

B

NHMFL,Tallahassee, Florida,“RRC Kurchatov Institute”, Moscow

B

INTRODUCTIONBasics of sp2 electronic features1.Aharonov-Bohm Effect in CNTs2.

3. Landau levels in CNTs

OUTLINE of the TALK

Theory and original controversies

Magnetic field induced metal-semiconductor transition

Fabry-Perot regimePropagative Landau levels and Fermi level pinning

4

Graphene (ribbons) & Carbon Nanotubes

51 Brillouin Zone

HybridHybrid MolecularMolecular OrbitalesOrbitalesCohesion

Electronic Properties in the vicinity of EF

Effective Model

2 atoms/ cell nearest neighbor orbital overlap1

1

1

2

2

2

1

1

2 1

2

2D2D QuasiQuasi--1D1D

6

Nanotubes: Electronic Properties

Periodic Boundary conditions

Symmetry choice

K

K

K

K’

K’

K’

K

K

K

K’

K’

K’

EF=0

7

Remark :Massless Dirac Fermions in 2D vs 1D

2D Graphene

s=±1

Massless

particles

E(p) = svF |p|

E(p) = s√v2F p

2 +m∗2c4

Dispersion relation

ARPES

Eigenstates (« pseudospin » )

BBAA

=1√2

(e−iθ(p)/2

−e+iθ(p)/2)

|ΨK+

p 〉 = 1√2

(ψK+

p (A)

ψK+p (B)

)

| ⇑〉 ∼(10

)

| ⇓〉 ∼(01

)

EF Q= K±+p/

(in the sublattice basis)

HK+(p) = vFσ.p

Linearization close to Fermi level

Nanotubes (n,n) armchairs

8

• 1D metallic nanotubes : Absence of backscattering,

• 2D graphene : Anti-localization phenomenon,

Sign inversion of quantum correction ~ S.O coupling effects

E. McCann et al., Phys. Rev. Lett. 97, 146805 (2006)

Pseudospin symmetry & backscattering effects

Transition Weak localization

/ Anti-weak localization

F.V. Tikhonenko et al., Phys. Rev. Lett. 100, 056802 (2008)

=0 2d2d2d2d2d2d2d2d1d1d1d1d1d1d1d1d

T. Ando, T. Nakanishi and R. Saito,

J. Phys. Soc. Jpn 67, 2857 (1998)

Short range potentialIntervalley scattering

Long range potentialIntravalley scattering

9

Unconventional Quantum Hall effect

Huge Mobility: 20.000-100.000 cm2/V·s(order of magnitude better than silicon)

Room Temperature and low magnetic field

Integer Quantum Hall effect !σxy (4e2/h)

n (1012 cm-2)

ρxx (kΩ)

-2-4 40 2

5

10

0

1.5

-1.5

-2.5

-3.5

-0.5

2.5

3.5

0.5

12T

Quantisation at νννν=N+1/2

-40 -20 0 20 40

-4

-2

0

2

4

σ xy ,

e2 /h

Vg, V

hexy /2 2=σ

Large inter-Landau level distance

X. Du et al. Nature 2009

1.Aharonov-Bohm Effect in CNTs2.

3.

OUTLINE of the TALK

Theory and original controversies

Magnetic field induced metal-semiconductor transition

11

Aharonov-Bohm effects on the Electronic Spectrum

K’

K

K

K

K’

K’

Wavefunction

Cr

h

K’K

K

K

K’

K’

Landau gauge

12

Periodic Gap-oscillations

Magnetic field driven spectral changes

Van-Hove Singularities modulations

H. Akiji and T. Ando, J. Phys. Soc. Jpn 62, 2470 (1993)

H. Akiji and T. Ando, J. Phys. Soc. Jpn 65, 505 (1996)

S.R., G. Dresselhaus, M. Dresselhaus, R. Saito, PRB 62, 16092 (2000)

13

Magnetotransport in nanotubes : first experiments

I

L

Br

A. Bachtold et al, Nature 397, 673 (1999)

Br

Weak localization (AAS oscillations)

* Negative Magnetoresistance

* -periodic oscillations

VV

Diffusive regime (small elastic mean free path)

14

Weak localization phenomenon

Quantum interference effects

Enhanced return probability to the origin

Increase of quantum resistance

Quantum conductance

O

+ -

P QTime-reversed trajectories interfere constructively

Classical term Interference termP Q

Beyond the diffusive regime

15

WL & Aronov-Altshuler-Spivak oscillations

Theory: Altshuler,Aronov & Spivak JETP 1981

Experiment: Sharvin & Sharvin JETP 1981

NEGATIVE MAGNETORESISTANCENEGATIVE MAGNETORESISTANCENEGATIVE MAGNETORESISTANCENEGATIVE MAGNETORESISTANCE

----periodicperiodicperiodicperiodic oscillationsoscillationsoscillationsoscillations

Switching on Perpendicular BO

+ -P Q

16

B-dependent diffusion coefficient

Average over the spectrum

SR, F. Triozon, A. Rubio, D. Mayou,

PRB 64, 121401 (2001)

Positive MR

-oscillations (AB)

Negative MR

-oscillations (AB)

Metallic tube

(Anderson-type disorder)

Negative MR

-oscillations (type AAS)

17

Magnetoconductance for parallel fields

Tube length: 1.4 µmdiameter: ~ 36 nm

T = 4.5 K

Périodicity en

Ch. Strünk

(regensburg, Germany)

Field dependent

additional features ?Weak localization signatures

Oscillations AAS

Negative MR

0 1 2 3

0,22

0,24

0,26

G (

2e2 /h

)

magnetic flux (h/e)

18

B-dependent bandstructure features

Parallel Magnetic field

Van-hove singularity splitting

Orbital degeneracy breaking

Field-dependent DoS + conductance for

(22-22) 3nm metallic nanotube

S.R., G. Dresselhaus, M. Dresselhaus, R. Saito,

PRB 62, 16092 (2000)

19

Comparison with experiments

Density of states for a (260,260) metallic nanotube (diameter ~ 36 nm)

C Strunk, B Stojetz and SR

Semicond. Sci. Technol. 21 (2006)

S38–S45

Experimental data

For conductance

Systems too disordered

and larger diameter to see

Gap opening…

20

-10 0 10

0

1.0

Vgate (V)

Semiconducting nanotube

Field Effect Transistors

Metallic nanotube

Vgate (V)

-10 0 10

0.15

0

G(e

2 /h)

Conducting nanoscale

interconnects

??

CNT-based device characteristics

21

Basic Principle to engineer a B-modulated FET ?

??

G Fedorov, A Tselev, D Jiménez, S Latil, N Kalugin, P Barbara, D Smirnov, SR,

Nano Lett. 7, 960 (2007)

22

ChiralityChirality dependentdependent effectseffects

Charge transport Charge transport mechanismmechanism

-) Temperature-dependent G reveals that

charge transport is dominated by aTunneling regime through a Schottky barrier (B-dependent

features)

Arrhenius plots

-)Tight-binding calculations for all possible chiralities

(diam ~ 1-2 nm)

Chirality identification

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

K

ε g (a C

-Cγ 0/r

)

φ/φ0

K'

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

ε g (a C

-Cγ 0/

r)

φ/φ0

1.2.

3. Landau levels in CNTs

OUTLINE of the TALK

Fabry-Perot regimePropagative Landau levels and Fermi level pinning

24

An experimental challenge

To fit the Landau radius on the surface the CNT

1²2

2

>>=

= B

eR

l

Rv

B h

302633B(v=1) (T)

MWCNTSWCNTd (nm)

To work on clean CNT for a clear observation of Landau quantization

Be ll >>

In clean MWCNT, le ≈≈≈≈ few 100nm

300nm

Perpendicular Magnetic field

25

band structure calculations under B⊥⊥⊥⊥

Ar

B=5T, v=3.04

B=0T, v=0 Zigzag (510,0), r =20nm

B=10T, v=6.07

Flattening of the 1st subband

Other subbands up-shifted to higher energy

B=20T, v=12.14

eBnEn h2∝

Perpendicular Magnetic field

-- B=0T

-- B=10T

-- B=20T

26

B=10T, v=6.07

Some pictures of Landau states

At high k

Propagating (snake-like)

states at the flanks

BB

At low k

localized states at bottom

and the top of the CNT

27

Ballistic Metallic Tube : Fabry-Pérot Cavity

•Coherent low T regime.

•Fabry-Perot oscillation.

meVLhvE F 5.72/ ≈=δ

2K

12K

120K

-20 -15 -10 -5 0 5 100.4

0.6

0.8

1.0

1.2

1.4

1.6

CNP

G (2

e2/h)

E (meV)

E=7.5 meV(a)

-0.2 -0.1 0.0 0.1 0.20.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Without disorder With disorder (W=0.8 ; le~Lt)

G (2

e2/h)

E (unit of tCC)

(b)

Simulation

(Landauer)

-20 -15 -10 -5 0 5 100.4

0.6

0.8

1.0

1.2

1.4

1.6

CNP

G (2

e2/h)

E (meV)

E=7.5 meV(a)

-0.2 -0.1 0.0 0.1 0.20.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Without disorder With disorder (W=0.8 ; le~Lt)

G (2

e2/h)

E (unit of tCC)

(b)resonance

antiresonance

D2

Diapositive 27

D2 Expérience interférométrique : dépendance de la phase de la fonction d'onde dans une cavité résonante. Oscillations de Fabry-Pérot.DRFMC; 26/03/2008

28

( ) F

B

FB kk

I

vkE −±=± 2

0 2),(

νν h

•Modification of band structure under Bperp

field :

•Maximas of G each time the matching phase

condition is recovered :

0 5 10 15 20 25 30 35

0.6

0.8

1.0

1.2

0 1 2 3 4 5 6

2

4

6

0.0 0.2 0.4 0.6 0.8 1.0 1.2

G (2

e2 /h)

Magnetic field (T)

(b)

I0(2

)

n

10 15 20 25 30 35

-0.6

-0.4

-0.2

0.0

0.2

G (2

e2 /h)

Magnetic field (T)

B

(a)

πνδ pLk B =)(

Magnetic-field induced Modulation of interferences

Maximum of

FP resonance

Minimum of

FP resonance

D4

Diapositive 28

D4 Mécanisme : pente relation dispersion diminue (structure de bande modifiée sous champ) et condition d'accord de phase modifée. Résultats cohérents qualitativement : fonction de Bessel retrouvée en fonction de l'ordre de la résonance. Pente des courbes à un ordre donné dB/dE bon signe et bon ordre de grandeur.Oscillation apréiodiques pilotées par la fonction de Bessel.Vg entrre max et min d'une résonance. Pente bas champ : effet de contact ?DRFMC; 26/03/2008

29B. Raquet, R. Avriller, B. Lassagne, S. Nanot, W. Escoffier, JM Broto, S.R..

Phys. Rev. Lett. 101, 046803 (2008)

With Disorder

Signature of Landau level formation on conductance

30

Acknowledgements

Coworkers

TheoryTheory : R: Réémi mi AvrillerAvriller (UAM)(UAM)

David Jimenez (UAB)

ExperimentsExperimentsBenjamin Lassagne, Sebastien

Nanot, Bertrand Bertrand RaquetRaquet

Georgy Fedorov