E c. Contact 2. e +. E v. f. A magnetic field applied parallel to the nanotube axis modifies by a factor where Φ 0 is the magnetic flux inside the nanotube. B. k ┴. DOS. E. S. Roche. Ф / Ф 0 =1/2. Ф / Ф 0 =0.
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A magnetic field applied parallel to the nanotube axis modifies by a factor where Φ0is the magnetic flux inside the nanotube.
In the ballistic regime, we expect a strong magneto-conductance behaviour
depending on the Fermi level position with respect to the Charge Neutrality Point.
is symmetric and centered at B=24T, suggesting a 48T periodic behaviour at higher fields. This result is in agreement with the expected Aharonov-Bohm theory for a ballistic carbon nanotube diameter of 10nm.
B = 26T; Φ = Φ0/2
The formation of Schottky barriers limits the transmission coefficient
Transport is dominated by thermally assisted tunnel processes (WKB).
Different barrier profiles have been tested (triangular, polynomial and logarithmic)
Best agreement with logarithmic barrier profile.
Realistic barrier profile estimations lead to an excellent agreement between theory and experiment.
Magneto transport on individual MWNT
S. Nanot1, B. Lassagne1, B. Raquet1, J.M. Broto1 and W. Escoffier1J.P. Cleuziou2, M. Monthioux2, T. Ondarçuhu2 R. Avrilier3, S. Roche3
Abstract : We report giant quantum flux modulation of the conductance in ballistic multi wall carbon nanotubes threaded by a 55T magnetic field in the high temperature regime (100K). This is the first evidence of the Aharonov-Bohm effect on conductance in ballistic carbon nanotubes. Our experimental data are well reproduced assuming a variable electronic transmission coefficient due to band bending at the contacts, under magnetic field.
Aharonov-Bohm effect in carbon nanotubes
Sample and measurement characteristics
Nanotube synthesisArc-dischargeMWCNT from EPFL
(Ecole Polytechnique Fédérale de Lausanne)
Nanotubes A & B diameters
Sample A: d ~ 9-10 nm
Sample B : d ~ 7-8 nm
Inter-electrodes distance: L ~ 200 nm
Low bias voltage (10mV)
Fermi level depends on gate voltage
Two probe conductance around 1G0
Mean free path: le >> L
Ballistic transport regime
Flux quantum Φ0 = h/e
Ballistic carbon nanotube (72,72) of diameter 10nm at T=100K.
Under magnetic field, an energy gap Ec opens which scales as :
Magneto-conductance of sample A at 103K
Magnetic field up to 55T
CNP = 0
Flat band regime with ohmic contacts
Kubo formalism with determination of the magneto-dependent TDOS and the Fermi level shift.
Good agreement between theoretical predictions and measurements assuming a p-doped CNT with CNP at Vg ~ 10V : maximum of gap opening at Φ0/2.
Problem with the evolution between Φ and Φ0/2 : far from CNP, the conductance decreases two slowly in the theoretical predictions.
Magneto-conductance of sample B at 146K
Magnetic field B up to 35T
Solely DoS effect scenario.
Band bending at the contact sc-CNT/Pd
Semi-analytical model with Landauer formula.
G(Vg) shows quasi-periodic oscillations of period ~3,3V. Such a modulation is Fabry-Perot like interferences, with the nanotube acting as an electronic waveguide between non-perfect contacts. This effect provides support for the ballistic regime.
1 National Pulsed Magnetic Field Laboratory, LNCMP, 143 Avenue de Rangueil 31400 Toulouse (France)
2 Centre for material elaboration and structural studies, CEMES, 29 Rue Jeanne Marvig 31055 Toulouse
3 Commissariat à l’Energie Atomique, DRFMC/SPSMS, 17 rue des Martyrs 38042 Grenoble (France)