Band-gap modulations of double-walled carbon nanotubes under an axial magnetic field

Carbon 45 (2007) 1905–1910 www.elsevier.com/locate/carbon

Band-gap modulations of double-walled carbon nanotubes under an axial magnetic field A. Latge´ b

a,*

, D. Grimm

a,b

a Instituto de Fı´sica, Universidade Federal Fluminense, 24210-340 Nitero´i, RJ, Brazil IFW Dresden, Leibniz Institute for Solid State Research, P.O. Box 270116, D-01171 Dresden, Germany

Received 9 January 2007; accepted 12 April 2007 Available online 4 May 2007

Abstract We address here a theoretical study of electronic and transport properties of commensurate double-wall carbon nanotubes (DWCNTs). A single band tight binding hamiltonian is considered and the magnetic field is theoretically described by the Peierls approximation. Weak intershell interactions between a set of neighboring atoms on the walls of the inner and outer tubes are considered. Taking advantage of the real space description, arbitrary configurations of the relative position of the two tubes are taken into account. We study the possibility of Aharonov–Bohm effects in DWCNTs when a magnetic field is applied along the axial direction. The field intensity is found to be more effective than the relative interatomic positions in providing conductance gap modulations. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction An increasing interest has been addressed to the synthesis and study of DWCNTs, motivated mainly by the possibility they offer on triggering their mechanical, transport, and electronic properties. Dynamical and static friction measurements have been studied on two concentric tube systems [1] and different oscillator realizations have also been proposed [2,3]. Huge changes of stress transmission depending on the transmitting medium using Raman measurements of DWCNTs were recently reported [4], showing the importance of molecular organization at the external wall of the tube. DWCNTs have been used as channels of field-effect transistors revealing singular transport properties. The semiconducting ones have been proved to exhibit ambipolar characteristics different from the p-type behavior of single wall carbon nanotubes (SWCNTs) [5]. DWCNTs may be achieved by peapod-derived methods [6], synthesized by pulsed arc discharge processes and low pressure catalytic chemical vapor deposition [7], among *

Corresponding author. Fax: +55 21 26295887. E-mail address: [email protected]ff.br (A. Latge´).

0008-6223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2007.04.019

others processes. X-ray diffraction analysis of the structural transformation from SWCNTs to DWCNTs, via C-60 peapods [8] determined the intertube spacing between inner and outer tubes as being 0.36 ± 0.01 nm, similar to the distance of 0.35 for AB-stacked graphite [9]. Also, the studies indicate that the tubes are loosely coupled to each other. Interlayer potential energies for different chirality pairs of tubes composing the double-walled system have been calculated [10,11] adopting the van-der-Waals potential. Within tight-binding approaches, the interwall interaction between the atoms of different walls has been usually assumed to decay exponentially [12–14] with the interatomic distance. Commensurate DWCNTs are widely studied nanosystems where the ratio of the two unit cell lengths along the common tube axis is a rational number. Examples are the (n, n)@(2n, 2n), and (m, 0)@(2m, 0). Previous theoretical works on DWCNs [15,16] have discussed the stability of the electronic structure of the coupled system based on symmetry arguments. The consideration of orientational disorder into the theoretical description, was shown to be essential for inducing opening of energy gaps. This is in contrast to the electronic structure of identical metallic nanotubes arranged close packed in bundles

A. Latge´, D. Grimm / Carbon 45 (2007) 1905–1910

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which exhibit a pseudogap at the Fermi level [17] in the ordered phase. Aperture of energy gaps in SWCNTs may be achieved via the application of a magnetic field, too, leading further to the well known Aharonov–Bohm (AB) phenomena. We address here a theoretical investigation on the transport behavior of commensurate DWCNT, motivated mainly by experimental reports on ballistic transport for SWCNs [18] and multiwalled tubes [19]. In particular, Kajiura et al. [20] were able to measure the conductance dependence on the length of DWCNT structures, by submerging then into liquid mercury. They concluded that the electronic transport in the double-walled systems is quasi-ballistic-like at room temperature, and that the mean free path is of the order of the length of the tubes. We include the effects of a magnetic field which is known to change the gap energy of pristine tubes in oscillatory patterns (AB effect). Besides analyzing the quantum interference phenomena, we investigate how the relative positions of the atoms displayed in the inner and outer walls of both the tubes affect the conductance of the DWCNT. 2. Theory A single p-band tight binding model is adopted to calculate local electronic densities of states (LDOS) and conductances of commensurate armchair DWCNTs, neglecting curvature effects. A short piece of the (5, 5)@(10, 10) structure is depicted in Fig. 1. The relative axial (Dz) and angular (D/) distances between both the tubes are marked in the figure. The adopted Hamiltonian is written as  X X þ H ¼ E0 cþ cþ i;t ci;t þ co i;t cj;t þ cj;t ci;t i;t

þ

X

i;j;t

c1 ðrij Þðcþ i;1 cj;2

þ cþ j;2 ci;1 Þ;

ð1Þ

i;j

where E0 is the energy site, rij ¼ j~ ri ~ rj j gives the atomic distance between different i and j carbon atoms, and t = 1, 2 labels the isolated tubes. The intrawall hopping energy between first neighbors co is assumed of the order Δφ

Δz

of 2.7 eV, whereas for the intershell coupling c1(rij), two schemes are adopted. In the first one we use a simplified picture considering a distance independent coupling, where each atom laying in the inner tube interacts with three first neighbors on the outer tube, with co/8 for the closer apart pair of atoms and co/10 for the other two next intertube atomic neighbors. Otherwise, a more realistic description for the interlayer atomic interactions is achieved by using [13,21,22] c1 ðrij Þ ¼ ceðrij dÞ=d ;

assuming c = co/8, and considering for the damping factor ˚ [21,22]; the intertube distance d is taken d a value of 0.45 A ˚ as 3.5 A. The variations allowed for the interatomic hopping energies c1 on the atoms of the outer tube are illustrated with different color intensities in Fig. 1. The reference atom of the inner tube is shown in black. Within this second picture we are able to incorporate the relative displacement between inner and outer tubes, allowing continuous Dz displacements along the axial direction as well as azimuthal rotations, D/, as indicated in Fig. 1. Both shifts are given in units of percentages of hexagons, taking as reference the graphitic-like stacking for which an inner tube atom lies in the center of a hexagon of the outer wall tube, corresponding to the situation Dz = DU = 50%. Different non-symmetric configurations for the relative positions of both the tubes, besides the standard C5, S5, and D5h [12], may be described. This is certainly an advantage of a real-space scheme allowing to vary continuously the relative atomic position between in and outer tubes, in a straightforward way. Otherwise, looking at a k-space description, one expects that the intertube interactions act at the electronic band structure changing the slope of the linear bands that cross at the Fermi level in pristine SWCNTs. Simulations indicate, that the energy barriers for dislocations and rotations of the inner tube are relatively low, allowing a quite easy gliding of the two tubes with respective to each other [23,24]. One needs then to investigate how these relative movements affect the number of transport channels next to the Fermi level. Here we adopt the Green function formalism, and use decimation techniques [25,26] to solve matricial Dyson equations for the propagators. The DWCNT system is represented by a structured linear chain in which each site has been dressed by the on-layer interaction as e i ðwÞ ¼ ½1  Gi ðwÞT ii 1 Gi ðwÞ; G

Fig. 1. Schematic view of a piece of (5, 5)@(10, 10) DWCNT, showing the relative displacement between the intertube atomic positions along the axial direction (Dz) and in the azimuthal angle (D/). The gray scale refers to different intensities for the allowed interatomic hopping energies on the outer wall related to the black atom in the inner tube.

ð2Þ

ð3Þ

with Gi(w) being the undressed local Green function given by 1/(w  Ei), w is the energy, and Tii is the in-layer hopping energy matrix. The corresponding matricial elements denote the hopping energy along a single ring of both internal and external tubes, as well as the intertube connections within a chain layer. The decimation scheme consists in eliminating, at each step of an iterative procedure, half the sites of the dressed linear-chainlike. Doing that we successively renormalize both the local Green function (locators) and the hopping energy between neighbor layers

A. Latge´, D. Grimm / Carbon 45 (2007) 1905–1910

until a pre fixed convergence criteria is attained. The LDOS at site i is directly obtained from the converged locator, using the standard relation qi(w) = 1/p Imag(Gii(w)). Usually, an additional imaginary part (104) is added to the energy to guarantee convergence of the dizimation procedure. This model calculation has been proved quite adequate to deal with systems presenting local potential fluctuations in the microscopic scale. The conductance is calculated using the Landauer formula [22,25,27]. The Peierls approximation [28,25] is adopted to take into account the effects of a magnetic field threading the DWCNT along the axial direction. In this picture, phase factors DGR;R0 are added to the hopping energies, depending on the local atomic neighborhood determined by the vector R and R 0 and on the intensity of the magnetic flux /, which is written in terms of the quantum flux /o = h/e. The matrix elements of the hamiltonian are written as H i;j ¼ ceie=hDGR;R0 . One should notice that the flux within each one of the tubes composing the DWCNT is not the same: in the studied commensurate system, the internal flux is one fourth the external one. 3. Results and discussions In what follows, the conductance is expressed in terms of the usual quantum conductance (2e2/h). For a better comparison with the SWCNT case, the DWCNT conductance results have been divided by two, denoting here the conductance per tube. LDOS and conductance results of the (5, 5)@(10, 10) DWCNT, with a magnetic field applied parallel to the tube axis of are shown in Fig. 2 for magnetic fluxes equal to 0, 0.2, and 0.4 ///o. The distance independent scheme for the interatomic hopping between the two tubes is considered. The plotted LDOS corresponds to the mean value over the internal and outer atoms of a circumferential single layer of the DWCNT system. In the case of zero field and energies close to the Fermi level, both the LDOS and conductance are quite similar to the sum of the corresponding individual tubes (5, 5) and (10, 10).

0.15

Changes are evident as the field is turned on, mainly close to the Fermi energy and at those energies at which new quasi 1-dimensional bands appear, associated with the entrance of new transport channels. Important electronic transitions may be expected due to the magnetic field such as it happens for the pristine tubes. Fig. 3 shows explicitly the dependence of the conductance with the magnetic flux threading the outer tube, for values of the Fermi energy w = wr varying from 0.0 to 0.1, that may achieved, for instance, via doping processes in the leads or external electric fields. The typical conductance plateau corresponding to two transport channels, found in the proximities of the zero Fermi energy when the magnetic flux is zero, is gradually changed for non zero fields, exhibiting reductions and suppressions for particular magnetic flux intensities. Differently to the case of a single pristine tube, one does not obtain an immediate aperture of a conductance gap as the magnetic field is turned on. Now we consider the interlayer–atomic interactions described by the exponential decay given by Eq. (2). The conductance for different configuration of Dz and D/ is calculated for the (5, 5)@(10, 10) DWCNT. The dependence of the LDOS and the conductance with the magnetic flux and Fermi energy is shown in Fig. 4 in a two-dimensional grayscaled picture (color online). In Fig. 4a Dz = D/ = 0%, and in Fig. 4b–d graphite-like Dz = D/ = 50% is used. Integer numbers of quantum conductance channels (0, 1, 2, and 3) are marked in the conductance maps. The energy shifts of the van Hove singularities, as the field increases, are clearly exhibited in the LDOS map by the bright (red) regions. A small electron–hole asymmetry on the conductance results is present, destroying the complete energyinversion symmetry, due to the hopping energy distribution associated with the atomic interactions between the two tubes. Although many other configurations have been explored, the conductance results do not exhibit a significant dependence on the intertube deviations Dz and D/, in the plotted range of energy, as previously pointed by Tomanek [16]. Fig. 4c exhibits the conductance dependence on the magnetic field of two non-interacting (5, 5) and (10, 10) SWCNTs close to the zero Fermi level. Due to cylindrical symmetries, perfect pristine SWCNTs exhibit

φ/φ =0.4 φ/φ =0.2

8 w = 0.1 γ

0.05

7

conductance (2e /h)

0.00

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

5 4

φ/φ =0

3

φ/φ =0.4

φ/φ =0.2

2

6 5

w = 0.05 γ

4 3

w = 0.01 γ

2

1 0

conductance (2e /h)

LDOS (a.u.)

φ/φ =0

0.10

1907

1 -0.3

-0.2

-0.1

0.0 0.1 energy (γ )

0.2

0.3

Fig. 2. LDOS and conductance of a (5, 5)@(10, 10) DWCNT, for different values of magnetic flux and following the simple model of intralayer interactions.

0 0.0

w = 0.0

0.5

1.0 magnetic flux (φ/φ )

1.5

2.0

Fig. 3. Conductance as a function of the magnetic flux for the (5, 5)@ (10, 10) DWCNT at different Fermi energies, wr.

A. Latge´, D. Grimm / Carbon 45 (2007) 1905–1910

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3

0.2 1

0.1

energy (γ )

gap size (γ )

2

0

0.0

0.30

γ = γ /8 γ = γ /5

0.25

independent

0.20 0.15 0.10

-0.1

0.05 -0.2

0.00 0.0 0

1

3 0 1 magnetic flux (φ/φ )

1

0.00

2

3

0.6

0.9

1.2

1.5

1.8

magnetic flux (φ/φo)

Fig. 5. Gap size dependence on the magnetic flux (outer tube) for the (5, 5)@(10, 10) DWCNT, for different intertube energy hoppings. Squares and dots symbols are used for the results concerned to the exponential decay with c = co/8 and co/5, respectively, and the triangles are the results adopting the environment independent model.

2

2

0.05 energy (γ )

2

0.3

1

0

0

-0.05

0.00

0.05

0.10

0.00

0.05

0.10

magnetic flux (φ/φ )

Fig. 4. Gray-scaled LDOS (a) and conductance (b) maps for a (5, 5)@(10, 10) DWCNT in function of the magnetic flux threading the outer tube (x-axis) and the Fermi energy (y-axis). Conductance maps in the energy region next to the zero Fermi level [as marked in (b)] for two (5, 5) and (10, 10) tubes, disconnected (c) and interacting DWCNT system (d). A DWCNT with (a) Dz = D/ = 0% and (b–d) graphite-like Dz = D/ = 50% is used. The marked numbers are the corresponding transport channels.

AB effects with the corresponding gap sizes oscillating with a period equal to one quantum flux. The different diameters of the two tubes composing the DWCNT lead to a picture with two oscillating periods, in our case one being four times the other. As can be observed in Fig. 4d for the interacting DWCNT system, even a weak tube–tube interaction lifts up the band degeneracies resulting in four states. Contrarily to the pristine case, the energy gap around the zero Fermi level remains null, until a sufficient large magnetic field is able to open it. Upon rising the magnetic flux, the gap size of the interacting DWCNT approaches the noninteracting two tube system, indicating a more important contribution of the field than of the tube–tube interaction in this high field regime. To highlight the effect of the magnetic field on the conductance, we show in Fig. 5 the evolution of the conductance gap size with the flux intensity threading the external tube, taking into account the two described model (distance independent and exponential with c = co/8 and co/5). The complete suppression of the transport is achieved in the exponential model for smaller magnetic flux values, as compared to the results using distance independent values for the hopping energy. The initial lack of periodicity may be viewed as a manifestation of the robustness

of the coupling between the two tubes, preserving the metallic behavior (null gap) of the DWCN independent of the applied magnetic field, contrarily to what happens to a pristine tube with a threading magnetic flux. As the magnetic field increases the gap size oscillates with a periodicity given by a quantum of flux. The main difference between the two schemes is the critical magnetic flux intensity at which the gap starts opening. In particular, for the distance independent model the energy gap appears at a critical magnetic flux of the order of 0.35 /o. Also, the gap does not close at one quantum flux in contrast to the exponential model results. As the intertube interactions are more intense in the used environment independent model (triangle results in the figure), the lack of periodicity is even more noticeable. This general feature may be considered surprising since our conductance results without magnetic flux have not exhibited a significant dependence on the intertube deviations, as also discussed for incommensurate double walled systems [14,29]. In that sense we should believe that a perturbation theory in which the weak coupling between the tubes is considered as a small perturbation should not lead to great changes in the well-known periodic Aharonov–Bohm oscillation for the gap size as a function of the magnetic flux. Contrarily, our results clearly indicate that as the magnetic field starts to increase, the metallic states of the tube are preserved until a critical flux. One important thing to take into account is the fact that in the case of DWCNTs the effects of the magnetic flux on the angular momenta are different in the outer and inner tubes. It has been discussed that the presence of a magnetic flux in such systems makes linear bands split into parabolic bands, producing different energy spacing in the conduction and valence bands. The low energy states, close to the E = 0 are certainly hybridized due to the intertube interactions and the effects of a magnetic flux are supposed to be stronger for the outer tubes than for the inner ones [12]. Apparently, the competition between interlayer interactions, even being a small energy interaction, and the extra confinement imposed by

A. Latge´, D. Grimm / Carbon 45 (2007) 1905–1910

magnetic fields are an important characteristic of the cylindrical systems that may be conveniently manipulated.

1.8

energy (eV)

1909

1.5 1.2 0.9

4. Conclusions

0.6

r = 5a 3 /2π

0.3

r = 10a 3 /2π a = 2.46 A

0.0 0

2

4

6

8 magnetic flux (φ/φ )

10

r = 5.0 A r = 10.0 A 2

4

6

8 magnetic flux (φ/φ )

r = 7.0 A r = 10.0 A 0

2

4

6

8 magnetic flux (φ/φ )

10

Fig. 6. Energy spectra of different coupled one dimensional rings with internal and external radii given by rb and ra, respectively.

the magnetic field, may be responsible for such different behavior and justify the maintenance of the metallic states for low magnetic flux. Further analysis should be done to better understand the recovering of the oscillatory behavior. Inclusion of spin-magnetic field interactions [12] affect the results for low values of magnetic flux, reducing even more the system gap size. For comparing this modulation gap effect with other structures presenting cylindrical symmetry, we analyze a system composed of two coaxial decoupled ideal zerodimensional rings, with external and internal radii ra and rb, respectively. It may be viewed as an extreme limit of a finite decoupled DWCNT, composed of a single layer. The differences and similarities with the DWCNT bandgap modulations are discussed. The corresponding energy spectrum for such a system under a magnetic field is given by   2 2 h2  r2 /a /b h2 ð/a  /b Þ r2 E¼ l þ þ þ ; ð4Þ 2 2 2mr2 /o r2a rb 2m /o r2a r2b with l being the magnetic momentum quantum number, /a and /b the magnetic fluxes crossing through the two rings, and r the radius determined by 1=r2 ¼ 1=r2a þ 1=r2b . The parabolic contribution (second term in the right side of the equation) destroys the perfect AB oscillation achieved for a single idealized ring, due to the difference between inside and external fluxes. Moreover, the double ring system will always behaves as a semiconducting system (non-zero energy gap) under the effects of an axial magnetic field. Results for different cases are shown in Fig. 6: in the left panel the wire radii are considered equal to the studied (5, 5)@(10, 10) DWCNT, in the second example the radii are smaller, but the ratio is preserved (equal to 1/2), and in the right panel rb/ra = 7/10. Although the energy spectra exhibit the periodic oscillations, the gap size is an increasing function of the magnetic field. Finite GaAs semiconducting coupled rings shows also deviations from the periodic AB oscillations [30] that depend on the relative wire widths and confinement magnitudes. For the DWCNT, we found a mixed effect underlying the magnetic field response: a double-ring-like behavior until the first quantum of flux and a typical AB feature for higher fields, similar to what happens with a pristine SWCNT. Apparently, the double annular structure is revealed up to this field range. Although exhibiting significant differences between them, the band-gap modulations under axial

In summary, we have presented a theoretical discussion of the dependence of conductance gap sizes of double walled carbon nanotubes on applied magnetic fields. We have studied the possibility of gap modulation occurrence depending on the intertube interactions as well as on the relative positions between outer and inner tubes. Within the adopted approach, the results indicate that the conductance is weakly sensitive to the relative tube positions. Pseudogaps in the LDOS are achieved more efficiently by imposing an axial magnetic field on the DWCNT system. Differently from the case of SWCNT, the double walled system does not immediately exhibit an energy gap opening when a magnetic field is turned on, preserving the metallic nature up to a finite magnetic flux. The dependence of the gap size on the field intensity shows a periodicity for higher magnetic intensities indicating that a DWCNT may behave similarly to a SWCNT in this higher field regime. Acknowledgements We acknowledge the partial financial support of CAPES/PROBRAL grant 197/05, grants from Instituto de Nanotecnologia, and the Brazilian agency CNPq. A.L. thanks the hospitality of the Leibniz Institute IFW-Dresden, where part of this work was done and D.G. the DFG PI 440/3. References [1] Cumings J, Zettl A. Low-friction nanoscale linear bearing realized from multiwall carbon nanotubes. Science 2000;289(5479):602–3. [2] Legoas S, Coluci VR, Braga SF, Coura PZ, Dantas SO, Galva˜o SD. Molecular-dynamics simulations of carbon nanotubes as Gigahertz oscillators. Phys Rev Lett 2003;90(5):55504–7. [3] Wong LH, Zhao Y, Chen G, Chwang AT. Grooving the carbon nanotube oscillators. Appl Phys Lett 2006;88(18):183107-1–3. [4] Puech P, Flahaut E, Sapelkin A, Hubel H, Dunstan DJ, Landa G, et al. Nanoscale pressure effects in individual double-wall carbon nanotubes. Phys Rev B 2006;73(23):233408-1–4. [5] Shimada T, Sugai T, Ohno Y, Kishimoto S, Mizutani T, Yoshida H, et al. Double-wall carbon nanotube field-effect transistors: ambipolar transport characteristics. Appl Phys Lett 2004;84(13):2412–4. [6] Bandow S, Takizawa S, Hirahara K, Yudasaka M, Iijima S. Raman scattering study of double-wall carbon nanotubes derived from the chains of fullerenes in single-wall carbon nanotubes. Chem Phys Lett 2001;337(1–3):48–54. [7] Gru¨neis A, Rummeli MH, Kramberger C, Barreiro A, Pichler T, Pfeiffer R, et al. High quality double wall carbon nanotubes with a defined diameter distribution by chemical vapor deposition from alcohol. Carbon 2005;44(15):3177–82. [8] Abe M, Kataura H, Kira H, Kodama T, Suzuki S, Achiba Y, et al. Structural transformation from single-wall to double-wall carbon nanotube bundles. Phys Rev B 2003;68(4):41405-1–4. [9] Saito Y, Yoshikawa T, Inagaki M, Tomita M, Hayashi T. Growth and structure of graphitic tubules and polyhedral particles in arcdischarge. Chem Phys Lett 1993;204(3–4):277–82.

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