Phonon-Assisted Electron Emission from Individual Carbon Nanotubes

Dec 22, 2010 - individual electrically biased carbon nanotubes (CNTs) both ..... at a lattice temperature T; (2) electron-phonon interaction. Dη ep =...
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Phonon-Assisted Electron Emission from Individual Carbon Nanotubes Xianlong Wei,*,†,‡ Dmitri Golberg,‡ Qing Chen,*,† Yoshio Bando,‡ and Lianmao Peng† †

Key Laboratory for the Physics and Chemistry of Nanodevices and Department of Electronics, Peking University, Beijing 100871, People's Republic of China ‡ International Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki, 305-0044, Japan

bS Supporting Information ABSTRACT: A question of how electrons can escape from one-atom-thick surfaces has seldom been studied and is still not properly answered. Herein, lateral electron emission from a one-atom-thick surface is thoroughly studied for the first time. We study electron emission from side surface of individual electrically biased carbon nanotubes (CNTs) both experimentally and theoretically and discover a new electron emission mechanism named phonon-assisted electron emission. A kinetic model based on coupled Boltzmann equations of electrons and optical phonons is proposed and well describes experimentally measured lateral electron emission from CNTs. It is shown that the electrons moving along a biased CNT can overflow from the one-atom-thick surface due to the absorption of hot forward-scattering optical phonons. A low working voltage, high emission density, and side emission character make phonon-assisted electron emission primarily promising in electron source applications. KEYWORDS: Phonon-assisted electron emission, one-atom-thick surface, carbon nanotube, electron-phonon interaction

single-walled CNTs (SWCNTs) at a high electric field. The presence of hot phonons was also proposed to explain negative differential conductance11 and electroluminescence in suspended SWCNTs.12 Electrons injected from a scanning tunneling microscope tip into a SWCNT were used to excite a specific vibrational mode that created nonequilibrium phonons.13 Here, we uncover another new phenomenon resulting from the interplay between electrons and phonons in CNTs. We found that the electrons moving along a biased CNT can overflow from the oneatom-thick surface due to the absorption of nonequilibrium forward-scattering optical phonons. The phenomenon uncovered a new mechanism of electron emission from a solid surface. Our experiments were performed in situ inside a scanning electron microscope (SEM) equipped with three independent nanomanipulators.14 A chemically etched tungsten probe was mounted to each nanomanipulator to act as an electrical probe. CNT samples were prepared by connecting an individual suspended arc-discharge grown multiwalled CNT (MWCNT) between two probes. A bias voltage of Vpump was applied to the CNT to pump up electrons. And at the same time a positively

H

ow electrons can escape from solid surfaces into vacuum is an important problem from both fundamental and technological viewpoints. To date, the methods to extract electrons from bulk solid surfaces have included evaporating electrons out of a surface through heating (thermionic emission), pulling electrons through a surface barrier using a high electric field (field emission), and liberating them from a surface via bombardment by photons (photoemission), electrons, or ions (secondary emission). In the physical pictures of thermionic emission and field emission, only electrons moving toward the emission surface may escape from a solid.1,2 However, for the materials which are only one atom thick (e.g., graphenes or shells of carbon nanotubes (CNTs)), electrons are strictly confined in the oneatom-thick layer without the velocity component normal to the surface. So it is difficult to understand electron emission from a one-atom-thick surface based on traditional thermionic emission and field emission models. The question how electrons can escape from one-atom-thick surfaces has seldom been studied and is still not properly answered.3,4 CNTs exhibit interesting physical properties resulting from the interplay between electric and vibrational degrees of freedom. The strong coupling of high-energy electron carriers to optical phonons was found to result in differential conductance decrease5-7 and the presence of nonequilibrium optical phonons8-10 in r 2010 American Chemical Society

Received: November 2, 2010 Revised: December 9, 2010 Published: December 22, 2010 734

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Figure 1. Schematic drawing (a) and a SEM image (b) of the experimental setup.

the results obtained from the CNT shown in Figure 1b. The Ipump-Vpump curve shows typical character of a MWCNT. It can be seen from the Icollect-Vpump curve that there is no measurable Icollect when Vpump is less than ∼2.6 V and that Icollect increases exponentially with Vpump when Vpump is larger than ∼2.6 V. The same CNT was measured at two different lengths (Figure 2b), and Icollect-Vpump curves at different lengths show similar behaviors but different threshold voltages. Tunneling dominated electron emission or field emission can be excluded, since Vcollect and probe positions, and thus the collecting electric field, were fixed during the measurement (except a weak decrease caused by a Vpump increase). Furthermore, the local electric field at the CNT surface is estimated to be about 107 V/m, which is 2 orders of magnitude smaller than that needed for field emission from CNT tips, i.e., 109 V/m.15 Figure 3a shows Icollect-Veff-col (Veff-col is defined as VcollectVpump/2) curves measured from the same tube as in Figure 2 at four different Vpump. The Icollect-Veff-col curves exhibit a fast increasing regime and a slowly increasing regime resembling those of thermionic emission. The slowly increasing regime is named as accelerating field regime following that in thermionic emission. For all samples measured, accelerating field regimes of Icollect-Veff-col curves were found to be fitted remarkably well using the formula R ln Icollect ¼ A þ BVeff-col with A, B, and R as fitting parameters (S1 in the Supporting Information). For the curves in Figure 3a, R = 1.0 was obtained. Although, the present linear dependence of ln Icollect on VReff-col seems to agree with the Schottky effect during thermionic emission, where ln Icollect increases linearly with V0.5 eff-col in an accelerating field regime if a plane metallic surface and a classical image force are assumed,16 it meets a difficulty to explain the presently observed electron emission from the side surface of CNTs within the framework of a traditional thermionic emission model. According to the classical thermionic emission model, only the electrons having kinetic energy (associated with velocity component normal to emission surface) higher than the surface barrier can escape from a solid surface.16,17 A SWCNT is a seamless graphene cylinder, and a MWCNT consists of coaxial SWCNTs coupled by weak van der Waals forces.18 A one-atom-thick graphene sheet is an ideal two-dimensional conductor, and its electrons are strictly confined in the basal plane without the velocity component normal to it.18 Therefore electron emission from CNTs cannot be attributed to sufficient kinetic energy of electrons normal to graphene surface higher than the surface

Figure 2. (a) Simultaneously measured Ipump-Vpump curve and IcollectVpump curve for the tube shown in Figure 1b. The inset in (a) is a TEM image of the CNT, showing a triple-walled tube with outer and inner diameters of 16.6 and 14.9 nm (scale bar is 5 nm). (b) Measured (symbols) and calculated (lines) Icollect-Vpump curves of the tube at two lengths. The following values of the parameters were used in the calculations: t2 = 0.0623, Rth = 5.0  107 K/W, WU = 4.8 eV, σ = 0.5 eV/V, and U = 2.91 V for L = 1.56 μm; t2 = 0.0637, Rth = 2.1  107 K/W, WU =4.8 eV, σ = 0.6 eV/V, and U = 3.57 V for L = 4.33 μm. All Icollect-Vpump curves were measured at a fixed Vcollect of 15 V.

biased (Vcollect) third probe was placed close to the middle of the CNT to collect electrons escaping from the tube. Figure 1a shows a schematic drawing of the measurement setup and the corresponding SEM image is depicted in Figure 1b. An electron beam of SEM was blanked during the measurements. The electrical current passing through the CNT (Ipump) and the collected emission current from the CNT (Icollect) were measured at different Vpump and a fixed Vcollect. Figure 2a shows 735

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Figure 4. (a) Diagrams showing the assumed model of electron transport in a SWCNT. (b) Calculated emission density of the tube in Figure 2 with L = 1.56 μm and Vpump = 3.05 V. (c) An energy diagram showing the possible energy traces of electrons transporting along an electrically biased CNT with (solid arrows) and without (dashed arrows) the presence of hot forward-scattering optical phonons. Here only forward scattering of electrons is considered.

Figure 3. (a) Measured (symbols) and calculated (lines) Icollect-Veff-col curves for the tube in Figure 2 with a length of 1.56 μm. (b) Simultaneously measured Ipump and Icollect of a MWCNT when it was failing shell by shell. (c) Calculated emission current (Icollect) (symbols) under different work functions (W) for the tube in Figure 2 at a length of 1.56 μm and under a bias of 3.05 V.

acoustic phonons. The bottleneck of an energy transfer may accumulate enough energy in electrons to enable some electrons to overflow from the one-atom-thick quantum well. To put the above thoughts into practice, we propose a kinetic emission model based on Boltzmann equations to describe the electron emission from a one-atom-thick surface. In this model, electrons near the Fermi level of a SWCNT are confined in a oneatom-thick quantum well with a depth equal to the CNT work function (W) (Figure 4a). When a voltage is applied, electron distribution functions evolve with time according to the Boltzmann equations. We assume that once their energy is higher than the depth of the quantum well, electrons will break through the confinement along the direction normal to the CNT surface and escape laterally from it into vacuum. Emission current density is defined as an electric charge of the electrons escaping from the unit area in unit time and can be written as Z e DðεÞ ð1Þ ð f1 ðε, xÞ þ f2 ðε, xÞÞ dε JðxÞ ¼ Δt W 2

barrier (∼5 eV19). The mismatch of the physical pictures indicates that some other mechanism should be responsible for presently observed electron emission from CNTs. Some clues can be found by tracing the energy transfer in an electrically biased defect-free CNT. Electron carriers first gain energy from an electric field when speeded up by the electric force. Then accelerated electrons are scattered by acoustic phonons and also optical phonons in a high field,5-7 and the energy is further transferred from electrons to phonons through phonon emission. Anharmonic decay of optical phonons completes energy transfer from optical phonons to acoustic phonons.20 However, different from normal conductors, the long lifetime of optical phonons compared to their generation time in CNTs can induce a buildup of nonequilibrium phonons;8-10,21 thus the energy of emitted optical phonons can return to electrons through phonon reabsorption, rather than completely decay to 736

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where e is the electron charge, ε is energy with zero at the Fermi level, x is coordinate along the CNT axis (Figure 1a), Δt is the time step in numerical simulation of electron transport, f1 and f2 are distribution functions of right and left moving electrons, D(ε) is the density of states per unit area (density of states in graphene is assumed). The latter has an analytical expression rffiffiffiffiffi! 8 1 ε 1 π Z1 DðεÞ ¼ 2 pffiffiffi 2 2 pffiffiffiffiffi F , π 3 3a t 2 Z0 Z0

with lacRT = 300 nm6 being the scattering length of acoustic phonons at T0; (2) inelastic backscattering of optical phonons with fixed energies of 0.16 eV (BS1 phonons) and 0.2 eV (BS2 phonons); and (3) inelastic forward scattering of optical phonons with a fixed energy of 0.2 eV (FS phonons).8,23 For inelastic scattering (Figure 4a), both phonon emission and absorption are considered and the collision operator has the following form:23 C i η ¼ γη fgη ðqi - Þfj - ð1 - fi Þ þ ½gη ðqiþ Þ þ 1fj þ ð1 - fi Þ - gη ðqi þ Þfi ð1 - fj þ Þ - ½gη ðqi- Þ þ 1fi ð1 - fj - Þg

where a = 0.142 nm is the C-C length, t = 3.03 eV is the nearest hopping energy, Z0 = 4ε/t   ε 2 ððε=tÞ2 - 1Þ2 Z1 ¼ 1 þ t 4

with gη the phonon distribution function and γη the electronphonon coupling (EPC) constant of η phonons (η = BS1, BS2, FS). EPC constants of electrons near Fermi level are used. According to ref 8, electrons near the Fermi level are only backscattered by the K-A 1 0 (BS1) and Γ-E2g LO (BS2) modes and forward scattered by the Γ-E 2g TO (FS) mode with diameter (d) dependent EPC constant γ = ν0 /ld, where l = 92.0 for K-A 1 0 phonons and l = 225.6 for Γ-E2g phonons. f i( are evaluated at ε ( pω η, with pω η the phonon energy. For backscattering phonons, we have j 6¼ i and a phonon wave vector q i( = -(2ε ( pωη )/pνi ; for forwardscattering phonons, we have j=i and a phonon wave vector q i( = ω η/νi. The temporary evolution of gη is determined by the Boltzmann equation ð3Þ Dt gη þ vη D x gη ¼ D η

and F(π/2, x) is the complete elliptical integral of the first kind.22 Since electrons with an energy higher than W are assumed to escape from CNTs, f1,2(ε>W) are set zero after calculating emission density in each time step during their evolution. Figure 3b shows Icollect measured from a MWCNT when it failed shell by shell (S2 in the Supporting Information). No remarkable change of Icollect with Ipump is observed, which is understandable since all inner shells are screened by the outermost one and only the outmost shell can feel surface barrier decrease caused by Vcollect.16 Therefore, only the electron emission from the outermost shell needs to be considered. Evolution of electron distributions in highly biased CNTs must be treated in the frame of coupled Boltzmann equations for electrons and phonons.21,23 Here, we assume metallic character of the outermost tube shell and linear energy band over the whole energy range of interest with an electron velocity of v0 and neglect all other semiconductor-like subbands (Figure 4a). We follow the analysis of the coupled system of electrons and phonons as in ref 23. The Boltzmann equation of electrons is Dt fi þ vi Dx fi - evi EDε fi ¼ C i

with vη the phonon velocity. The phonon collision operator D η includes two terms: (1) phonon-phonon interaction D η pp ¼ - ðgη - gη 0 ðTÞÞ=τ with τ = 1.5 ps20 the phonon lifetime and gη 0 ðTÞ ¼ 1=½expðpωη =kB TÞ - 1

ð2Þ

the Bose-Einstein distribution at a lattice temperature T; (2) electron-phonon interaction

where vi is the velocity of the right or left moving electrons and E is the electric field along the tube axis. For given Vpump and Ipump E ¼ - ðVpump - Ipump RC Þ=L

D η ep ¼ 2

here L is the tube length and RC is the contact resistance determined by transmission coefficient (t2) at contacts and number (N) of shells carrying current through ! h 2ð1 - t 2 Þ 1þ RC ¼ 4Ne2 t2

i¼1

γη fðgη þ 1Þfi ðεi þ Þ½1 - fj ðεi - Þ

- gη fj ðεi - Þ½1 - fi ðεi þ Þg with εi( = p(νiq ( ωη)/2 for backscattering phonons, and D η ep ¼

The collision operator C i describes evolution of electron distribution due to electron scattering. We consider three kinds of electron scattering here: (1) elastic backscattering of acoustic phonons ν0 ðfj - fi Þ C i ac ¼ lac

Z 2 4γη L X δq, qi fðgη þ 1Þfi ðεÞ½1 - fi ðε - Þ hν0 i ¼ 1 - gη fi ðε - Þ½1 - fi ðεÞg dε

with ε- = ε - pωη and qi = ωη/νi for forward-scattering phonons.23 Joule self-heating of tubes was considered. For given Vpump and Ipump, the lattice temperature was calculated through T ¼ ðT0 þ PRth =2Þ=½1 - PL=ð12k0 T0 AÞ

with lac being the scattering length of acoustic phonons at lattice temperature T. Since acoustic phonons are thermally excited at room temperature (T0 = 300 K) and above,11 lac has the form of lac ¼ lRT ac

2 X

(S3 in the Supporting Information) with Rth the thermal resistance at contacts, κ0 = 2000 W/mK the room temperature thermal conductivity, A the cross-section area of tubes, and P = Ipump(Vpump IpumpRC) the Joule-heating power.

T0 T 737

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We impose equilibrium distributions at boundaries. At the left contact (x = 0): f1 ðε, 0Þ ¼ t 2 f0 ðε - eVpump =2Þ þ ð1 - t 2 Þf2 ðε, 0Þ

equilibrium distribution of corresponding phonons at the lattice temperature during numerical calculations (Figure 4b). It can be seen that emission current is completely suppressed without hot FS phonons while hot BS1 and BS2 phonons only have a minor effect on emission current. Therefore, the absorption of hot forward-scattering phonons is mainly responsible for the electron emission from CNTs. This results from the fact that electrons can only keep increasing energy by absorbing phonons without changing moving direction when they transport along the direction of electric force. Figure 4c shows the possible energy traces of electrons transporting along an electrically biased CNT with and without the presence of hot forward-scattering optical phonons. An electron near the bottom of conduction band is first accelerated by electric force to reach the threshold energy and then scattered by optical phonons.5-7 In the case with the presence of hot forward-scattering optical phonons, an electron can absorb an optical phonon without changing moving direction when it is scattered and can be further accelerated by electric force, which makes it possible for an electron to keep increasing energy until it surpasses the vacuum level. In the case without hot forwardscattering optical phonons, after reaching the threshold energy under electrical drive, an electron can only emit an optical phonon when it is scattered, which makes it return to the bottom of conduction band again and keep staying at the energy levels near the Fermi level forever. Therefore, electron emission from CNTs originates from the absorption of hot forward-scattering optical phonons, which dramatically increases the probability of electron occupying at high-energy states near the vacuum level. The electron emission also benefits from one-dimensional character of CNTs, in which electrons can only either maintain their moving direction (forward scattering) or reverse it (backscattering) after scattering. Assisted by forward-scattering optical phonons, electrons can be pumped up to energy much higher than the bias-voltage injection energy (e(Vpump - IpumpRC)). For the tube at Vpump = 3.05 V, the bias-voltage injection energy is only about 1.4 eV, which is much less than the surface barrier of ∼4.8 eV. So electrons can escape from CNTs under the drive of a pumping voltage much less than that corresponding to its work function. An average emission density of 6.9 A/cm2 and a maximum density of 31.6 A/cm2 were obtained (Figure 4b). More importantly, as compared to field emission from CNT tips under a high external voltage (tens of volts),15 only a low driving voltage of 2-3 V is needed for phonon-assisted emission and the side emission character implies that planar CNT emitters can be fabricated and integrated in a similar way to CNT transistors. Those make phonon-assisted electron emission from CNTs promising in electron source applications. In conclusion, lateral electron emission from a one-atom-thick surface is thoroughly studied for the first time and a new electron emission mechanism different from existing thermionic emission, field emission, photoemission and secondary emission, named as phonon-assisted emission, is discovered in CNTs. It is shown that the electrons moving along a biased CNT can overflow from the one-atom-thick surface due to the absorption of hot forward-scattering optical phonons. The new mechanism retains the exponential law with work function as in thermionic emission and exhibits a maximum emission density near the electrode with a higher potential. Phonon-assisted electron emission is also expected to occur in other low dimensional materials, e.g., graphene nanoribbons, in which a buildup of

ð4Þ

gη ðq > 0, 0Þ ¼ gη 0 ðT0 Þ with f0(ε) = 1/[exp(ε/kBT0) þ 1]the Fermi-Dirac distribution at T0. Symmetric boundary conditions are set at the right contact. We have solved the coupled Boltzmann equations (eqs 2-4) numerically to obtain the steady-state distribution of electrons and phonons at each Vpump and Ipump and calculated corresponding steady-state electron emission density through eq 1. Collected emission current in accelerating field regime is then calculated through ZL JðxÞ dx Icollect ¼ πd 0

with d being the tube diameter. The decrease of work function with external electric field has been well studied and is known as the Schottky effect in thermionic emission.16 The experimentally observed linear increase of ln Icollect with VReff-col at accelerating field regime, as shown in Figure 3a, can be well simulated by the kinetic model by applying a linear decrease of W with VReff-col, that is W = W0 βVReff-col. For the tube in Figure 3a (R = 1.0), the calculated Icollect-Veff-col curves at different Vpump coincide remarkably well with those of experimental measurements with W0 = 5.0 eV and β = 0.017, 0.021, 0.023, 0.023 eV/V for Vpump = 2.85, 2.90, 2.95, 3.00 V. Other parameter values used in the calculation include t2 = 0.062, Rth = 5.0  107 K/W, v0 = 500 km/s, |vBS1| = 5.00 km/s, |vBS2| = 2.95 km/s and vFS = 0.23 A linear increase of ln Icollect with VReff-col resulted from a linear decrease of W with VReff-col, which indicates that ln Icollect decreases linearly with W in our model, that is Icollect µ exp(-λW) with λ as a coefficient. The exponential law is further confirmed by theoretical calculations in Figure 3c. This law is the same as that in the thermionic emission model.16,17 We noticed an increasing trend of β with Vpump as shown above. An increase of β with Vpump indicates that W decreases faster with Vcollect for a larger Vpump, so W will decrease with Vpump under fixed Vcollect. The effect was considered in simulating Icollect-Vpump curves by assuming W = WU - σ(Vpump - U), where WU is the work function at a pumping voltage of U and σ is a coefficient. The calculated Icollect-Vpump curves also coincide remarkably well with those of experimental measurements (Figure 2b). Figure 4b shows emission density J(x) calculated for the tube in Figure 2 at L = 1.56 μm and Vpump = 3.05 V. It can be seen that J(x) increases along the tube axis from left to right, where positive Vpump was applied. This is different from traditional thermionic emission, where the maximum emission density is in the middle portion with the highest lattice temperature.16,17 The profile of J(x) along the tube axis indicates that the population of electrons with energy higher than surface barrier keeps increasing when they transport along the direction of electric force. Optical phonons were significantly driven out of thermal equilibrium with an effective temperature of ∼5700 K for FS phonons and a lattice temperature of 1410 K at Vpump = 3.05 V (S4 in the Supporting Information). To explore the effects of nonequilibrium hot phonons on the electron emission, we also calculated J(x) without absorption of certain hot phonons by setting 738

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nonequilibrium forward-scattering optical phonons can be induced under electrical drive. The newly discovered phononassisted emission is envisaged to be promising in electron source applications and open new research areas in both fundamental and technological aspects just as each previous emission mechanism did.

(19) Shiraishi, M.; Ata, M. Carbon 2001, 39, 1913. (20) Bonini, N.; Lazzeri, M.; Marzari, N.; Mauri, F. Phys. Rev. Lett. 2007, 99, No. 176802. (21) Lazzeri, M; Mauri, F. Phys. Rev. B 2006, 73, No. 165419. (22) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. Rev. Mod. Phys. 2009, 81, 109. (23) Auer, C.; Schurrer, F.; Ertler, C. Phys. Rev. B 2006, 74, No. 165409.

’ ASSOCIATED CONTENT

bS

Supporting Information. Measurement results for the tube in Figure 2 with a length of 4.33 μm, detailed information about shell-by-shell failure of the tube in Figure 3b, deduction of the formula used to calculate lattice temperature and steady distributions of optical phonons for the CNT in Figure 2 with a length of 1.56 μm and under a bias of 3.05 V. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] and [email protected].

’ ACKNOWLEDGMENT X. L. Wei thanks G. M. Zhang at PKU and M. S. Wang at NIMS for discussions. The work at PKU was supported by NSF of China (60925003, 60771005) and the MOST (2011CB933000, 2009AA03Z315), and the work at NIMS was supported by MANA. ’ REFERENCES (1) Dolan, W. W.; Dyke, W. P. Phys. Rev. 1954, 95, 327. (2) Jensen, K. L. J. Appl. Phys. 2007, 102, No. 024911. (3) Cox, D. C.; Forrest, R. D.; Smith, P. R.; Silva, S. R. P. Appl. Phys. Lett. 2004, 85, 2065. (4) Liu, P.; Wei, Y.; Jiang, K. L.; Sun, Q.; Zhang, X. B.; Fan, S. S.; Zhang, S. F.; Ning, C. G.; Deng, J. K. Phys. Rev. B 2006, 73, No. 235412. (5) Yao, Z.; Kane, C. L.; Dekker, C. Phys. Rev. Lett. 2000, 84, 2941. (6) Javey, A.; Guo, J.; Paulsson, M.; Wang, Q.; Mann, D.; Lundstrom, M.; Dai, H. J. Phys. Rev. Lett. 2004, 92, No. 106804. (7) Park, J. Y.; Rosenblatt, S.; Yaish, Y.; Sazonova, V.; Ustunel, H.; Braig, S.; Arias, T. A.; Brouwer, P. W.; McEune, P. L. Nano Lett. 2004, 4, 517. (8) Lazzeri, M.; Piscanec, S.; Mauri, F.; Ferrari, A. C.; Robertson, J. Phys. Rev. Lett. 2005, 95, No. 236802. (9) Oron-Carl, M.; Krupke, R. Phys. Rev. Lett. 2008, 100, No. 127401. (10) Steiner, M.; Freitag, M.; Perebeinos, V.; Tsang, J. C.; Small, J. P.; Kinoshita, M.; Yuan, D. N.; Liu, J.; Avouris, P. Nat. Nanotechnol. 2009, 4, 320. (11) Pop, E.; Mann, D.; Cao, J.; Wang, Q.; Goodson, K.; Dai, H. J. Phys. Rev. Lett. 2005, 95, No. 155505. (12) Mann, D.; Kato, Y. K.; Kinkhabwala, A.; Pop, E.; Cao, J.; Wang, X. R.; Zhang, L.; Wang, Q.; Guo, J.; Dai, H. J. Nat. Nanotechnol. 2007, 2, 33. (13) LeRoy, B. J.; Lemay, S. G.; Kong, J.; Dekker, C. Nature 2004, 432, 371. (14) Wei, X. L.; Chen, Q.; Peng, L.-M.; Cui, R. L.; Li, Y. Ultramicroscopy 2010, 110, 182. (15) Wang, M. S.; Wang, J. Y.; Peng, L.-M. Appl. Phys. Lett. 2006, 88, No. 243108. (16) Herring, C.; Nichols, M. H. Rev. Mod. Phys. 1949, 21, 185. (17) Kittel, C. Introduction to Solid State Physics, 3rd ed.; John Wiley & Sons: New York, 1966. (18) Saito, R.; Dresselhaus, G.; Dresselhaus, M. S. Physical Properties of Carbon Nanotubes; Imperial College Press: London, 1998. 739

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