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aPhys. Status Solidi C 9, No. 1, 22–25 (2012) / DOI 10.1002/pssc.201084156

Electron-phonon and electron-defect scattering rates in semiconducting zigzag carbon nanotubes Kannan Thiagarajan* and Ulf Lindefelt

Department of Information Technology and Media, Mid Sweden University, Sundsvall 85170, Sweden

Received 22 October 2010, revised 1 March 2011, accepted 12 March 2011 Published online 12 August 2011

Keywords carbon nanotubes, electron-phonon scattering, electron-defect scattering rate, tight-binding approximation * Corresponding author: e-mail kannan.thiyagarajan@miun.se, Phone: +46 70 668 5878, Fax: +46 60 148456

Electron-phonon and electron-defect scattering rates have been calculated within a tight-binding approach for semi-conducting zigzag carbon nanotubes. The scattering rates for (5,0), (7,0), (10,0), (13,0) and (25,0) nanotubes have been investigated. The electron-phonon scattering rate shows both diameter and temperature dependence, and the dependence of the electron-defect scattering rate on

nanotube diameter is similar to that of the electron-phonon scattering rate. Backscattering and forward scat-tering for electrons interacting with defects occur with equal probability at all energies. The importance of elec-tron-defect scattering relative to electron-phonon scatter-ing depends very much on the energy of the electron.

© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction The electron transport properties of single-walled car-

bon nanotubes are of fundamental importance to the devel-opment of carbon-based nanotechnology. The electronic properties vary from semiconducting to metallic with each tube’s fundamental indices n and m. Theoretical modelling should play a key role in the development of carbon nano-tube-based electronic devices as it does for conventional electronic devices. The electronic properties depend very much upon the interaction of the electrons with phonons, vacancies, impurities etc. The effect of electron-defect scattering in combination with the electron-phonon scatter-ing is the main theme of this article. The electron-defect in-teraction in this work primarily constitutes electron scatter-ing against isolated mono-vacancies.

2 A brief outline of the theory The electrons and their interaction with phonons and

vacancies are here described within the tight-binding ap-proximation. Following the ideas in Ref. [1], the electron tight-binding Hamiltonian elH for the nanotube is based on one π -orbital per C-atom, unit overlap matrix and with a nearest-neighbour interaction (or transfer) integral t

given by t = -3.033 eV, which is common standard for car-bon nanotubes.

As to the phonon modes, we have used the well-established fourth nearest-neighbour force constant model by Saito et al. [1] to calculate the phonon dispersion rela-tions and the polarization vectors. However, rather than us-ing the parameter values given in Ref. [1] (p. 169), we have used the force constant parameters suggested by Zimmermann et al. [2]. Our reason for choosing the pa-rameters of Ref. [2] is that the force constant parameters described in Ref. [1] lead to four linear acoustic dispersion relations ( qω ∝ ), whereas the parameters given in Ref. [2] lead to two linear modes and two quadratic ( 2qω ∝ ) acoustic phonon modes. These quadratic phonon modes are referred to as flexure modes, and have been discussed in the literature [2, 3].

To model the electron-phonon Hamiltonian, we have used the method given by Pietronero et al. [4], which is based on the tight-binding model for electrons. For a sum-mary of the model, and for other details concerning the calculation of the electron-phonon scattering rates, we re-fer the reader to Ref. [5].

The calculation of the scattering rates for electrons in-teracting with isolated vacancies is also modelled within

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the tight-binding scheme. To treat the scattering from seve-ral identical scattering centres, it is standard procedure in electron transport physics to consider the scattering from one isolated centre and then multiply the transition proba-bility per unit time with the number of such centres [6]. We have followed that procedure here. Furthermore, we assume only on-site interaction with the parameterization given in Ref. [7]. Inclusion of nearest neighbour off-site in-teractions does not change the results by more than a few percent at most. The resulting scattering is elastic. If the initial state ( , )m k is degenerate, for instance, if the states

1( , )m k and 2( , )m k have the same energy (two-fold degen-erate electronic energy bands are very common in carbon nanotubes), we explicitly form an average value of the scattering rate over the two degenerate initial states,

1 2

1 1 1 1

( ) 2 ( ) ( )m m mk k kτ τ τ= +

⎛ ⎞⎜ ⎟⎝ ⎠

(1)

It turns out that the scattering rate in Eq. (1) is independent of the exact value of the position of the vacancy in the lat-tice, as is also expected from symmetry arguments.

3 Results and discussions In this section, we first discuss briefly the phonon scat-

tering rate for a (13,0) carbon nanotube, and then move on to analyze the electron-vacancy scattering rate and how it combines with the electron-phonon scattering rate. Pho-nons can cause the electron to scatter within the conduction band or the valence band. It can also cause scattering from conduction to valence band (and vice versa). However, since we are considering one-phonon processes, phonon-assisted scattering between the conduction and valence bands is not possible in the semiconducting carbon nano-tubes considered in this work because the band gap is al-ways greater than the maximum phonon energy available for a scattering process to occur (just below 0.2 eV). Therefore, we only consider scattering within the two low-est conduction subbands. Furthermore, since we have not included the possibility of electric-field induced transitions, we have limited the energy of the electron to the energy in-terval 0 – 0.75 eV for a (13,0) tube. The two lowest con-duction subbands for a (13,0) tube are shown in Fig. 1. The second conduction subband minimum is about 0.469 eV above the bottom of the first conduction subband. In a (10,0) tube, the lowest conduction subbands have similar shapes, but the second conduction subband is about 0.626 eV above the minimum of the first conduction subband.

We define the scattering as the process in which the electron with initial wave vector k > 0 ends up in a state with wave vector k’ after interaction with a phonon of wave vector q (k’ = k + q). If k’ > 0, then the scattering is defined as forward scattering, and otherwise backward scattering. In Fig. 2, we show scattering rates for scattering within the first conduction subband (i.e., intraband scatter-ing) and scattering from the first to the second conduction

subband (i.e., interband scattering) at T = 300 K for a (13,0) tube. We show the contributions from intraband forward and backward scattering by red and blue solid lines, respectively, and similarly, interband forward and backward scattering by red and blue dotted lines, respec-tively. The total scattering rate for electrons in band 1 as a function of energy is thus the sum of these contributions.

Figure 1 The two lowest conduction subbands in the central part of the Brillouin zone for a (13,0) nanotube. The Brillouin zone boundaries are at -10.7373 ű . The zero energy corresponds to the bottom of the lowest conduction subband.

Figure 2 Forward (solid red line) and backward (solid blue line) electron-phonon scattering rates versus electron energy for intra-band transitions (subband 1→subband 1) in a (13,0) nanotube. Forward (dashed red line) and backward (dashed blue line) elec-tron-phonon scattering rates versus electron energy for interband transitions (subband 1→subband 2) in a (13,0) nanotube.

We notice there are a number of peaks with relatively

smooth regions in between. The peaks reflect the high den-sity-of-states of the electron’s final state around the band minima. At about 26 meV, the first major phonon emission peak occurs and it corresponds to a Raman active radial breathing mode labelled RBM [2]. The occurrence of this

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24 K. Thiagarajan et al.: Electron-phonon and electron-defect scattering rates in carbon nanotubes

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peak coincides with the energy of the radial breathing mode reported in the literature [1-3]. With increasing en-ergy of the electron, the peaks corresponding to emission of high energy phonons can be seen [5]. After the onset of the emission peak at 0.196 eV (this energy value is essen-tially equal to the maximum phonon energy in all carbon nanotubes), the scattering rate arising from transitions within the first conduction subband flattens out and is al-most a constant throughout the energy interval 0.2-0.75 eV. At about 0.274 eV, the high-energetic phonons are ab-sorbed so the electrons could transit from the first conduc-tion subband to the second conduction subband. As we in-crease the electron energy, phonons with less energy are absorbed. The onset of the interband phonon emission processes starts at about 0.474 eV with the peak labelled IE1. With further increase in energy of the electron, emis-sion of high-energetic phonons occurs. All the peaks in Fig. 2 can be attributed to the phonon emission processes ex-cept the one labelled as IA1, which is associated with ab-sorption of a low-energy acoustic phonon [5]. In Fig. 3 we compare the scattering rates for electrons due to the inter-action with phonons for (5,0), (7,0), (10,0), (13,0) and (25,0) nanotubes at room temperature. Noting that the tube diameter is directly proportional to the index n of the zig-zag nanotube ( , 0)n , the diameter dependence of the elec-tron-phonon scattering rate is clearly visible in Fig. 3.

Figure 3 The total electron-phonon scattering rate for a (5,0) tube (solid blue line), a (7,0) tube (dashed red line), a (10,0) tube (dotted black line), a (13,0) tube (dash-dotted magenta line) and a (25,0) tube (solid green line). Lattice temperature is 300K. Only transitions from the lowest conduction subband are included in this figure.

The phonon modes associated with absorption and emission peaks, the relative contributions of phonon emis-sion, phonon absorption, forward scattering and backscat-tering at room temperature, the effect of lattice temperature on phonon scattering and similarities between different tubes have been analyzed in detail in an earlier work [5], to which the interested reader is referred.

We show in Fig. 4 the elastic scattering rates for elec-trons interacting with vacancies, with a density of vacan-cies corresponding to 1 vacancy per 1000 C-atoms, for (5,0), (7,0), (10,0), (13,0) and (25,0) nanotubes. The peaks in the scattering rates occur when the energy of the elec-tron equals the energy separation between the two lowest subbands, so that an electron in the first conduction sub-band can make a transition to the minimum of second con-duction subband. The peak in the scattering rate reflects the high density-of-states around the minimum of the sec-ond conduction subband. We also find that backscattering and forward scattering for electrons interacting with va-cancies are equally probable at all energies. It is worth no-ticing that since the electron-vacancy scattering is elastic, an electron in the lowest conduction subband with an en-ergy below the minimum of the second conduction band (0.469 eV in case of (13,0), or 0.626 eV in case of (10,0)) can only backscatter (i.e., reverse the direction of motion) or remain in the same state (unchanged velocity), since the electron wave vector is forced to remain unchanged if not backscattered (cf. Fig. 1). These two scattering results thus occur with equal probability. For higher energies, forward scattering can occur with final electron velocities either smaller or larger than the initial velocity, depending on the electron energy and band.

Figure 4 The electron-vacancy scattering rate for a (5,0) tube (solid blue line), a (7,0) tube (dash-dotted red line), a (10,0) tube (dashed black line), a (13,0) tube (dotted magenta line) and a (25,0) tube (solid green line). The vacancy concentration is 1 va-cancy per 1000 C-atoms.

In Fig. 4, where we have plotted the scattering rate as a function of energy for different nanotubes but with a fixed value of the number Cn of C atoms per vacancy ( 1000Cn = in this case), a clear dependence on nanotube diameter is seen. For a given value of Cn , the scattering rate behaves approximately as 21 1n n nτ ∝ = as a func-tion of the zigzag nanotube index n. The first factor n comes from the proportionality of the scattering rate to the

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linear density of vacancies, which increases as n when Cn is kept constant, whereas the factor 21 n comes from the normalization of the electronic wave functions in the tran-sition matrix element. Moreover, if we assume a tempera-ture independent vacancy concentration, this scattering rate is temperature independent.

In Fig. 5 we show the combined electron-vacancy and electron-phonon scattering rate for a (10,0) tube at room temperature for four different vacancy concentrations: no vacancies, one vacancy per 400 C-atoms (corresponding to one vacancy per 10 unit cells, i.e., one vacancy per 4.3Å), one vacancy per 2000 C-atoms (corresponding to one va-cancy per 50 unit cells, i.e., one vacancy per 21.6 Å), and one vacancy per 4000 C-atoms (corresponding to one va-cancy per 100 unit cells, i.e., one vacancy per 43.2 Å). Here we have combined the electron-phonon and electron-vacancy scattering rates in an approximate way using Mathiessen’s rule [8].

Figure 5 Combined electron-phonon and electron-defect scatter-ing rate as a function of electron energy for a (10,0) tube at T = 300 K.

At low energies (E < 0.17 eV), the electron-vacancy scattering has a relatively large effect on the total scatter-ing rate, whereas at higher energies (up to about 0.62 eV, i.e., before the onset of interband transitions between the lowest and second lowest conduction subbands) the elec-tron-vacancy scattering plays, relatively speaking, a less important role. To the right of 0.62 eV there is again an energy window about 0.17 eV wide, in which the electron-vacancy scattering is clearly important in comparison to electron-phonon scattering. This latter energy window cor-responds to transitions (phonon emissions) from the lowest to the second lowest conduction subband involving pho-nons with less than the very highest phonon energies. From about 0.79 (= 0.62 + 0.17) eV, where also the phonons with the very highest energies can participate in these in-terband transitions, the relative importance of the electron-defect scattering rate seems to diminish somewhat again. From this we conclude that the transport properties at low

fields can, if the density of defects is higher than around one defect per 5000 C-atoms, say, be affected in a substan-tial way by the electron-vacancy scattering. 4 Conclusions

The electron-phonon and the electron-vacancy interac-tions have been modelled using the tight-binding method, and the corresponding scattering rates have been calculated using Fermi’s Golden rule. It is found that (1) both the electron-vacancy and the electron-phonon scattering proc-esses are clearly dependent on nanotube diameter, (2) backscattering and forward scattering for electrons inter-acting with defects occur with equal probability at all ener-gies, and (3) the combined electron-phonon and electron-defect scattering rates indicates that vacancies have a strong influence on the scattering rate in the energy inter-val 0-0.17 eV from the bottom of lowest conduction sub-band or when the electron, still in the lowest conduction subband, has an energy in the interval 0-0.17 eV measured from the energy minimum of the second conduction sub-band. For other electron energies, the relative importance of electron-defect scattering compared to electron-phonon scattering seems to be clearly smaller.

Acknowledgements We gratefully acknowledge financial support from the Swedish knowledge foundation (KK – Stiftel-sen).

References [1] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physi-

cal Properties of Carbon Nanotubes (Imperial College Press, London, 1998).

[2] J. Zimmermann, P. Pavone, and G. Cuniberti, Phys. Rev. B 78, 045410 (2008).

[3] G. D. Mahan and G. S. Jeon, Phys. Rev. B 70, 075405 (2004).

[4] L. Pietronero, S. Strässler, H. R. Zeller, and M. J. Rice, Phys. Rev. B 22, 904 (1980).

[5] K. Thiagarajan and U. Lindefelt, accepted in September 2010 for publication in J. Comput. Theor. Nanosci.

[6] M. Lundstrom, Fundamentals in Carrier Transport, 2nd ed. (Cambridge University Press, 2000), chap. 2.

[7] U. Lindefelt, Phys. Rev. B 72, 153405 (2005). [8] N. W. Ashcroft and N. D. Mermin, Solid State Physics

(Holt-Saunders International Editions, Philadelphia, 1981), chap. 16.

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