Aharonov-Bohm Conductance Modulation in Ballistic Carbon Nanotubes

PRL 98, 176802 (2007)

PHYSICAL REVIEW LETTERS

week ending 27 APRIL 2007

Aharonov-Bohm Conductance Modulation in Ballistic Carbon Nanotubes B. Lassagne,1 J-P. Cleuziou,2 S. Nanot,1 W. Escoffier,1 R. Avriller,3 S. Roche,3 L. Forro´,4 B. Raquet,1 and J.-M Broto1 1

Laboratoire National des Champs Magne´tiques Pulse´s, UMR5147, Toulouse, France Centre d’Elaboration des Mate´riaux et d’Etude Structurales, UPR8011 Toulouse Cedex 4, France 3 Commissariat a` l’Energie Atomique, DSM/DRFMC/SPSMS/GT, Grenoble Cedex 9, France 4 Institute of Physics of Complex Matter, Ecole Polytechnique Fe´de´rale de Lausanne, Lausanne, Switzerland (Received 17 November 2006; published 24 April 2007) 2

We report on magnetoconductance experiments in ballistic multiwalled carbon nanotubes threaded by magnetic fields as large as 55 T. In the high temperature regime (100 K), giant modulations of the conductance, mediated by the Fermi level location, are unveiled. The experimental data are consistently analyzed in terms of the field-dependent density of states of the external shell that modulates the injection properties at the electrode-nanotube interface, and the resulting linear conductance. This is the first unambiguous experimental evidence of Aharonov-Bohm effect in clean multiwalled carbon nanotubes. DOI: 10.1103/PhysRevLett.98.176802

PACS numbers: 73.63.Fg, 73.23.Ad

When the cross section of mesoscopic conductive rings or cylinders is threaded by an external magnetic field, the electronic wave functions accumulate an additional phase factor given by the enclosed magnetic flux in 0 unit (0 h=e the quantum flux). This spectacular quantum phenomenon, referred to as the Aharonov-Bohm (AB) effect, modulates the transmission coefficients of the system [1] and is responsible for the 0 periodic oscillations of the magnetoresistance. In this context, carbon nanotubes (CNTs) stand as systems of particular interest, since they can be viewed as mesoscopic cylinders made from one to several concentric rolled graphene sheets [2]. The CNT band structure is straightforwardly derived from that of graphene, by imposing periodic boundary condition along the nanotube circumference, which limits the number of allowed k vectors [2]. This quantification condition can be further tuned by the field induced AB quantum phase, with a marked dominant feature of a periodic energy gap modulation of maximum width of Eg 1:25=dnmeV (d is the nanotube diameter) at the charge neutrality point (CNP) [3,4], initially considering a metallic CNT at zero field. Over the past few years, meaningful efforts have been undertaken to evidence magnetic field effects on the electronic band structure [5–14]. In the Coulomb blockade regime, the 0 flux modulation of the energy spectrum has been experimentally observed for multiwalled carbon nanotubes (MWCNTs) at low temperatures [9]. However, the experimental energy gap was found to be 10 times smaller than the predicted one [3]. Optical magnetospectroscopy experiments under 45 T on an assembly of single wall carbon nanotubes (SWCNTs) [8] have confirmed a red shift along with a splitting of the van Hove singularities (vHSs), as predicted theoretically [3,4]. Furthermore, AB interference and beating were reported in ballistic SWCNTs [15]. On the other hand, the interpretation of transport experiments on individual MWCNTs in the low bias regime suffer from a long-standing controversy between weak localization (WL) and AB phenomena [5,7]. 0031-9007=07=98(17)=176802(4)

Recently, it has been shown that the conductivity of diffusive CNTs is influenced by a subtle energy dependent competition between band structure effects and quantum interferences in the WL regime [11,12,14]. In this Letter, we report on large quantum flux modulations of the conductance in ballistic MWCNTs mediated by electrostatic gating. Evidence is given for an unambiguous manifestation of the AB modulation of the electronic band structure. We infer an energy gap opening for the external shell in agreement with the simple tight-binding calculation. A quantitative interpretation of experimental data is achieved by computing the electronic transmission properties within the Landauer-Bu¨ttiker theoretical framework [16], and assuming a field-dependent band bending profile at the nanotube-metal interface. Our devices are made of individual arc discharge MWCNTs deposited on doped Si=SiO2 100 nm wafer and connected with Pd electrodes using standard electron beam lithography. The distance between electrodes is Lt 200 nm. The magnetoconductance is measured under pulsed magnetic field applied parallel to the CNT axis. The electrostatic doping is controlled by a backgate voltage (Vg ) while low bias voltages in the millivolt range are applied on the devices. MWCNTs with moderate diameters of the order of 7–10 nm have been investigated [17] as a compromise between unwanted structural disorder (prominent for large diameter nanotubes) and our maximum available magnetic field (60 T) to observe almost an entire quantum flux threading the tube. In the following, we focus on the experimental magnetoconductance measured under 55 T on a 10 nm diameter MWCNT (atomic force microscopy estimation). Similar results have been obtained on smaller diameter nanotubes (7 nm) with a 35 T magnet. For these measurements (not shown here), only a partial opening of the energy gap with a maximum applied magnetic flux of 0 =3 was achieved. Preliminary transport experiments in zero magnetic field illustrate the ballistic regime of our MWCNTs. The two-

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© 2007 The American Physical Society

probes conductance is around 1G0 at 300 K (where G0 2e2 =h is the quantum of conductance) and slightly increases as the temperature is reduced. From the positive dR=dT slope, a phonon scattering length of about 1 m at 300 K is inferred. The differential conductance at 100 K is weakly backgate dependent between 20 V, with a magnitude of variation less than 20%. Measurements below 40 K obtained on small diameter MWCNTs (7 nm) show quasiperiodic oscillations of dI=dVVg centered at 0:9G0 . The main period Vg is consistent with Fabry-Perot interferences [18], supporting the ballistic regime. Note that contact effects may explain the measured halved quantized conductance [19]. Figure 1 shows the magnetoconductance GB; Vg curves for selected gate voltages between 10 V obtained at 100 K. While the magnetoconductance remains negative over the full magnetic field range, its shape and magnitude are strongly modified by the gate-induced electrostatic doping. Indeed, when Vg 10 V, the applied magnetic field induces a huge decrease of the conductance by a factor ’ 6. It falls off at 0:13G0 and exhibits a U-like curve with a plateau between 7 and 40 T. By diminishing the gate voltage from 10 V to 10 V, the magnitude of the magnetoconductance is steadily reduced. The GB curves appear symmetric and centered at B 24 T, suggesting an oscillatory behavior with a period of 48 T. This constitutes strong experimental evidence of a quantum flux modulation of the conductance, as a similar period of approximately 50 T would be expected for a CNT with diameter roughly equal to 10 nm. GEF ; =G0



PRL 98, 176802 (2007)

FIG. 1 (color online). Left-hand panel: Magnetoconductance GVg ; B at 100 K with magnetic field parallel to the tube axis. Selected gate voltages (in volts) are shown. Right-hand panel: 3D representation of GB; Vg at 100 K.

To further substantiate the experimental data, magnetoconductance calculations are performed for a 72; 72 nanotube (with diameter of the order of 10 nm), limiting first the analysis to single-shell transport properties. The flux modulation of the energy gap (Eg ) and the field induced shift of the vHS E i are given by [4] i Eg 30 r and E sin (1) 0 r ; i n with 0 20 ac-c =d, where 0 is the energy overlap integral between carbon atoms, ac-c , the nearest carboncarbon distance, and n the chirality index. r equals =0 if 0 0 =2, and r 1 =0 if 0 =2 0 . At low bias voltage, the conductance can be written as follows [16]:

Z Ei Z Eg =2 Z E1 X Z Ei TF EdE Ni0 TF EdE 2 TF EdE 2 TF EdE Ni E i

i

E i1

Z Ei1 X Z Ei TF EdE Ni0 TF EdE ; Ni i

E i

E i

where TF E TE@fE EF =@E with TE the coefficient transmission, F the Fermi-Dirac distribution at T 100 K, Ni and Ni0 are the numbers of conduction channels, and EF is the Fermi energy shift with respect to the CNP. We first consider a pristine nanotube in a flat band regime into which electrons enter by thermal activation over the energy gap. Figure 2(a) shows the flux dependence of the conductance at 100 K for various Fermi level shift from the CNP (EF 0) to slightly below half of the first vHS (EF 62 meV). For energies close to the CNP, the conductance drops once the field induced energy gap matches the Fermi energy of the electrodes. By moving the Fermi energy away from the CNP, the conductance drop requires higher fields and its magnitude is drastically reduced. The simulated magnetoconductance depicts reasonably well the overall shapes of the experimental curves (Fig. 1). This strongly supports a solely density of states (DOS) effect scenario under high magnetic field mediated by the backgate voltage. Figure 2(b)

E 1

Eg =2

(2)

shows a direct comparison between the simulated and the experimental magnetoconductance for different backgate voltages. The electronic transmission coefficient TE per contact is set to 0.7 to adjust the zero magnetic field calculated conductance to the experimental value. Deviations from unity are assumed to result from a nonperfect coupling with the Pd electrode. The only fitting parameter to account for the field dependence of the conductance is the Fermi energy shift. A fair agreement, especially at Vg 10 V, is achieved if one considers a p-doped MWCNT where the intrinsic doping is presumably due to the Pd contact. Note that we do not assign the CNP at 10 V to a particular dependence of the zero-field conductance versus Vg . In the following, this particular value of the backgate voltage will set the reference voltage for the CNP. For clarity, we define V~g Vg 10 V at the CNP. Note that for the curve at the CNP, a constant conductance of 0:13G0 is added in parallel. This residual conductance is ascribed to the intershell conductance at

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PRL 98, 176802 (2007)

(a)

(b)

FIG. 2 (color online). (a) Theoretical magnetoconductance GEf ; at 100 K for a 72; 72 diameter metallic nanotube in the flat band regime. The curves are calculated for different EF ranging from the CNP to slightly below the half value of the first vHS. (b) Comparison between the theoretical G curves and the experimental ones under V~g 0, 10, 16, and 20 V. The fitting parameter EF is equal to 0, 20, 42, and 46 meV, respectively.

100 K since the metallic outer shell is turned off for this particular value of Vg [20]. When the tube is progressively p doped by decreasing the backgate voltage down to V~g 20 V, the experimental magnetoconductance gradually deviates from the simulated curves even if the overall diminution of the magnetoconductance is fairly well accounted. Significant discrepancies become apparent under V~g 16 and 20 V [Fig. 2(b)]. Far from the CNP, the flat band regime is certainly not valid. Indeed, the combination of the field induced gap opening with the shift of the Fermi energy mediated by V~g leads to the formation of Schottky barriers (SBs) at the contacts due to the gateinduced band bending [21,22]. In the following, a refinement of Eq. (2) is proposed, considering a semianalytical description of the magnetic field dependence of the SB profile and its corresponding electronic transmission coefficient. The SB parameters are mainly its height (depending on the metal and nanotube work functions as well as the nanotube energy gap) and its penetration width. For simplicity, we assume that the barrier height is half the gap of the nanotube, as proposed by Heinze and co-workers [22]. Note that, according to this model, no bend curvature is expected at the CNP and electrons enter the nanotube by thermal activation as mentioned earlier. When a gate voltage is applied, SBs form at the interfaces and drive the current by thermally assisted tunnel processes. The determination of the band bending has been recently performed in the frame of self-consistent calculations [21–23]. A logarithmic dependence of the band profile following [22] is used, where the potential energy Ux near the contact is written as 0 lnxC0 lnLx Eg C (3) Ux EF Lx 1 : xx0 0 2 ln x ln C 0

The parameter x0 defines the thickness of the profile at the contacts, L sets the penetration width of the SB


[Fig. 3(b)], and C defines the length unit. In the WKB formalism, the transmission coefficient of the barrier is R p given by TE exp2 xx21 f2m Ux E=@2 gdx, where x1 and x2 are, respectively, the entry and exit point of the barrier, and m is the effective mass. To model the magnetic field modulation of the electronic transmission coefficient, we consider its effect on the energy gap, the vHS locations [Eq. (1)], and the effective mass m . From a straightforward calculation, it can be shown that m r 4@2 r =0 a2c-c 9n. A ballistic and coherent regime is assumed between both SBs formed at the source-drain electrodes, whereas incoherent electronic tunneling at each interface is described through the WKB approximation. Fabry-Perot oscillations due to quantum interferences between the electronic wave functions reflecting back and forth into the cavity are neglected, since resonant states are smeared by the thermal broadening at T 100 K. In this context, Eq. (2) is computed with a total electronic transmission coefficient Ttot defined by Ttot E; TE; = 2 TE; [16]. Figure 3(a) shows the experimental flux dependence of the conductance, together with the calculated one, that

FIG. 3 (color online). (a) Experimental magnetoconductance for selected V~g values and their corresponding theoretical curves at 100 K (black dashed curves). The curves are shifted by 0:2G0 for clarity. (b) Sketch of the Schottky barrier at the contact with the parameters used in the model. (c) Corresponding Schottky barrier profiles at 0 =2 for the different backgate voltages and based on parameters of Table I. (d) 2D experimental graph of GVg ; (left-hand graph) and the theoretical one based on Eq. (3) and Table I (right-hand graph).

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PRL 98, 176802 (2007)

TABLE I. Schottky barrier features and Fermi energy shift EF at 0 =2. V~g V 20 10 10 6 4 0

Lnm

x0 nm

EF meV

3 5 25 60 84 Flat band

1 1 1.2 5 10 Flat band

140 5 110 5 80 5 40 5 30 5 0

includes band bending effects described by the two parameters L and x0 . The Fermi energy shift for each V~g values is fixed by the expected backgate coupling defined by EF eV Cg =Celec eV~g , where Celec and Cg are the electrochemical and the geometrical capacitance, respectively. It corresponds to EF meV 7 0:5103 V~g . An excellent agreement is obtained between the experimental GB; V~g curves and the theoretical ones over the full backgate voltage range [Figs. 3(a) and 3(d)]. The extracted band bending parameters are summarized in Table I, and the corresponding profiles at 0 =2 are depicted in Fig. 3(c). For small Fermi energy shift (below 80 meV), an extension of the SB over several tens of nanometers is inferred. This value confirms self-consistent calculations for which the penetration width is predicted to be in the range of the oxide layer thickness (100 nm in our case) [23]. Under larger Fermi energy shift, the barrier becomes much thinner, of the order of the nanotube diameter, and the electron transmission is mainly dominated by tunneling mechanisms at the contact. In earlier transport experiments [5,11,12,14], magnetoconductance oscillations were found to be dominated by h=2e periodic Aronov-Al’tshuler-Spivak oscillations due to coherent electron backscattering around the circumference and universal conductance fluctuations. None of these disorder-induced quantum phenomena are observed in our experiments. The few scattering centers which may contribute to the measured 1G0 conductance in zero field [24], in addition to contact effects, do not prevent a clear manifestation of the AB modulation of the DOS in the metallic bands. Interestingly, many experiments supporting quantum interference phenomena in the WL regime were based on larger diameter MWCNTs connected to gold electrodes. In these systems, larger density of defects as well as stronger charge transfer due to gold contacts can be expected. As a result, a smaller energy separation between the subbands and a larger Fermi level shift would be consistent to the stronger activation of massive higher subbands, more sensitive to elastic disorder and localization effects [25]. In conclusion, our study clearly reveals the AharonovBohm effects on the magnetoconductance modulations of


ballistic MWCNTs, with clear signature of field-dependent Schottky barrier features that develop at the metalnanotube interface. When the metallic external shell conductance is switched off by the magnetic field, in close analogy with the electrical breakdown of the shell, a weak contribution of the inner shells (G0 =10 at 100 K) is observed. Sample preparations have been performed in the LAAS technological platform. This work has been supported by the French Ministry of Research under program ACI NR044 ‘‘NOCIEL’’ and ACI TRANSNANOFILS for financial support. The work in Lausanne is supported by the Swiss National Science Foundation and its NCCR Nanoscale Science.

[1] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959); R. A. Webbs et al., Phys. Rev. Lett. 54, 2696 (1985). [2] R. Saito, G. Dresselhaus, and M. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College, London, 1998). [3] H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 1255 (1993); 65, 505 (1996); H. Ajiki and T. Ando, Physica (Amsterdam) 201B, 349 (1994); 216B, 358 (1996); W. Tian and S. Datta, Phys. Rev. B 49, 5097 (1994). [4] S. Roche et al., Phys. Rev. B 62, 16 092 (2000). [5] A. Bachtold et al., Nature (London) 397, 673 (1999). [6] A. Fujiwara et al., Phys. Rev. B 60, 13 492 (1999). [7] J. O. Lee et al., Phys. Rev. B 61, R16 362 (2000); C. Scho¨nenberger and A. Bachtold, Phys. Rev. B 64, 157401 (2001); J. Kim et al., Phys. Rev. B 64, 157402 (2001). [8] S. Zaric et al., Science 304, 1129 (2004). [9] U. C. Coskun et al., Science 304, 1132 (2004). [10] E. D. Minot et al., Nature (London) 428, 536 (2004). [11] G. Fedorov et al., Phys. Rev. Lett. 94, 066801 (2005). [12] B. Stojetz et al., Phys. Rev. Lett. 94, 186802 (2005). [13] B. Lassagne et al., J. Phys. Condens. Matter 18, 4581 (2006). [14] C. Strunk, B. Stojetz, and S. Roche, Semicond. Sci. Technol. 21, S38 (2006). [15] J. Cao et al., Phys. Rev. Lett. 93, 216803 (2004). [16] S. Datta, Electronic Transport in Mesoscopic System (Cambridge University, Cambridge, England, 1998). [17] J.-M. Bonard et al., Adv. Mater. 9, 827 (1997). [18] W. Liang et al., Nature (London) 411, 665 (2001); J. Kong et al., Phys. Rev. Lett. 87, 106801 (2001). [19] P. Poncharal et al., J. Phys. Chem. B 106, 12 104 (2002); N. Mingo and J. Han, Phys. Rev. B 64, 201401 (2001). [20] B. Bourlon et al., Phys. Rev. Lett. 93, 176806 (2004). [21] F. Le´onard and J. Tersoff, Phys. Rev. Lett. 83, 5174 (1999); 84, 4693 (2000). [22] S. Heinze et al., Phys. Rev. Lett. 89, 106801 (2002). [23] T. Nakanishi, A. Bachtold, and C. Dekker, Phys. Rev. B 66, 073307 (2002). [24] H. Choi et al., Phys. Rev. Lett. 84, 2917 (2000). [25] F. Triozon et al., Phys. Rev. B 69, 121410 (2004).

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