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Electronic Properties of Strained Carbon Nanotubes: Impact of Induced Deformations Sasa Dmitrovic, Ivanka Milosevic, Milan Damnjanovic, and Tatjana Vukovic J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b02455 • Publication Date (Web): 22 May 2015 Downloaded from http://pubs.acs.org on May 29, 2015

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The Journal of Physical Chemistry

Electronic Properties of Strained Carbon Nanotubes: Impact of Induced Deformations Saˇsa Dmitrović, Ivanka Miloˇsević, Milan Damnjanović,∗ and Tatjana Vuković NanoLab, Center for Quantum Theoretical Physics, Faculty of Physics, University of Belgrade, 11158 Belgrade, Serbia E-mail: [email protected]

∗

To whom correspondence should be addressed

1 ACS Paragon Plus Environment


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Abstract In order to resolve quantitative mismatch between measurements and the existing theory, we perform systematic theoretical study of the effects of small uniform strain on the electronic properties of single-wall carbon nanotubes. Applied torsion or uniaxial strain induces structural deformations (shifts of the two sublattices, radial and torsional strains induced by the applied uniaxial strain, e.g.) which lead to significantly weaker impact on electronic properties of the strained tube. This damping is more pronounced for torsion. For instance, in tubes with chiral angle close to 30◦ , the band gap change is reduced up to 60%. Dominant attenuating factor is the relative shift of the sublattices along the tube axis, manifesting strong electronic coupling with the longitudinal highenergy Raman mode. Obtained results match better the experimental observation of the shifts of optical transitions energies and the gauge factor in carbon nanotube based piezoresistive sensors, giving a base for further device development.

Keywords Nanotubes, Electromechanical nanodevices, Gauge factor, Symmetry

1

Introduction

Due to their strength, small size, and remarkable electronic and transport properties, 1–3 carbon nanotubes (CNTs) have become basic ingredients of a number of nanodevices. In particular, significant change of their electronic bands due to mechanical deformations makes them suitable as components of torsional pendulum, tunable opto-electronic devices and sensors. 4–10 In the view of recently developed 11 chirality-specific growth of single-walled carbon nanotubes (SWCNTs), a need for better understanding of the strain induced change of the tube electronic properties is strengthened. Quite generally, this is essential for optimal design and modeling of CNT based devices and composites. 12


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Though several well known theoretical studies 13–15 have predicted strong chirality dependent electro-mechanical response of CNTs, they are only qualitatively supported by measurements. Firstly, Heyd et al. 13 studied the effects of uniaxial stress on electronic density of states (DOS) of SWCNTs and found linear gap-strain dependence. Yang et al. 14 showed that band gap alteration due to the axial strain is maximal in the zig-zag tubes, while the impact of twisting is the largest in the armchair tubes. Later on, widely used analytic expressions for DOS and band gap change due to small torsional and axial strains were found 15 within the zone-folding model. Experimental studies 5,16–20 support the qualitative predictions of the Yang and Han formula. 15 However, quantitative comparison has not been accurately made as the structure of the studied nanotube could not have been precisely determined. While the mentioned studies refer to the homogeneous deformations, Nisoli el al. 21 pointed out that relative shifts of two graphene sublattices have to be taken into account in order a number of the effects related to the vibration-electron-mechanical coupling to be correctly interpreted. Finally, recent studies of SWCNTs’ mechanical properties have shown significant chirality and diameter dependent strain coupling. 22–26 However, the impact of such a coupling on the electro-optical properties of SWCNTs has not been fully investigated thus far. The measurements 27 on axially strained SWCNTs, the structure of which was precisely characterized, indicate that the existing theory overestimates induced optical transition energy shifts. Also, gauge factor measurement of the highly pure single chirality carbon nanotube based piezoresistive sensor 28 gave much lower values than predicted by Yang and Han. 15 To realize the full potential of CNT use in piezoresistive sensors, a method for fabricating single chirality samples is needed. Thus, new chirality-specific growth of SWCNTs on solid alloy catalysts 11 has potential to speed up sensors development. However, such a progress also demands more precise theoretical study in order quantitative discrepancy between the measurements and theory to be resolved. Here we thoroughly analyze the effects of the applied strain to the electronic properties of



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nanotubes, including the impact of the induced strains and mutual shifts of the sublattices. To this end, all SWCNTs with diameters from 0.7 nm to 1.3 nm, subjected to small uniform twist and uniaxial strain, are systematically studied. The method, which allows to deal with helical configurations without translational periodicity and to take into account natural torsion of the undeformed SWCNTs, 29–31 both beyond the scope of the standard densityfunctional theory (DFT) based codes, is briefly described in Section 2. The main results, presented in Section 3, clarify experimentally observed weaker electronic response to strain than predicted by simplified model: 15 shifts of Van Hove singularities (VHSs) are notably smaller due to the induced deformations. Impact of each of the later is analyzed and origin of the damping effect is found to come mainly from the relative longitudinal shift of the sublattices. Also, the response of the optical isomers is discussed. Linear fits of the band gap change as a function of the applied strain are given. Finally, the effects of strain-induced deformations on the SWCNT gauge factors are analyzed.

2

Method

In order to find the optimal configuration of SWCNTs subjected to small uniaxial or torsional strain, symmetry-based relaxation procedure, 26 which employs the second generation Brenner-Tersoff interatomic potential, 32 is used. The whole nanotube is generated by the symmetry group from its arbitrary atom 33 C0 . Therefore, configuration of the nanotube (n1 , n2 ) is determined by five continual parameters (Q, f, r0 , φ0 , z0 ): Q and f come from the helical symmetry (CQ |f ), and characterize the symmetry group (together with the order n of the principal axis, equal to the greatest common divisor of n1 and n2 ), while the other three are cylindrical coordinates of the atom C0 . The configuration of the undeformed nanotube is found by optimizing total energy over all relaxation parameters; its symmetry is equivalent to that of the rolled-up structure. 31 Clearly, variation of cylindrical coordinates φ0 and z0 produces mutual shifts of the two triangular sublattices, while the change of the remaining


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parameters Q, f and r0 correspond to torsional, uniaxial and radial strains, respectively. To find configuration of the nanotube exposed to axial strain, εz , or torsion, given by shear angle γ, we calculate the value of the corresponding relaxation parameter 26 (f and Q, respectively), and while keeping it fixed, optimize the remaining four parameters. Therefore, all induced deformations are completely taken into account, making the model realistic to a large extent. To take completely into account hybridization and curvature effects, electronic bands are calculated within sp3 density functional tight-binding model (DFTB) implemented into symmetry based POLSym 36 code, with the full range of interaction potential (up to the fourth level neighboring atoms). Further, in order to avoid buckling, which occurs above certain amount of the applied strain, 34,35 we study the effects of coupling for relatively small deformations: εz ∈ [−5%, 5%] and γ ∈ [−5◦ , 5◦ ]. The POLSym code can also treat incommensurate configurations (helical ones, without translational periodicity) which may appear due to the applied or induced torsion.

3

Results

The response to the applied strain is qualitatively determined by the nanotube’s class p = (n1 − n2 ) (mod 3). Namely, applying either axial strain or torsion, the band gap decreases for p = −1, increases for p = 1, while when p = 0 (quasi-metallic SWCNTs) it firstly decreases to zero and then increases. This has been already predicted 13,15 by the model neglecting the induced deformations. However, as notable in Figure 1, due to deformations coupling, gap alteration in uniaxially strained or twisted SWCNT is remarkably attenuated in the present model. Other important features are analyzed below. We proceed with the analysis of SWCNTs with n1 ≥ n2 , while the optical isomers are discussed in Section 4. Note that, by convention, energy of the highest occupied state is set to zero.



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Figure 1: Axial strain, εz , and torsion, γ, induced band gap change ∆Eg for several types of SWCNTs: results obtained with(out) taking into account the deformation coupling are shown by filled (open) symbols.


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3.1

Axial strain

Unless the tube is armchair, applied axial strain affects its conducting properties. In the armchair nanotubes, the mirror symmetries prevent both the z0 -change and the strain-induced torsion. The deformed tube retains the same symmetry group, only the parameter of the fractional period (f ) changes. Consequently, the non-crossing rule does not apply as the double degenerate bands retain the opposite parities and thus keep to intersect at the Fermi level, 37 only the crossing point kF shifts. Thus, apart from the armchair tubes, the band gap changes for a finite value ∆Eg (see Figure 1, top panel): the absolute value of ∆Eg decreases with the chiral angle θ, while the sign of the band gap change depends on the class p. Also, for small εz gap change remains linear function of the applied strain. From Figure 1 it is evident that impact of the axial strain coupling is the strongest in the zig-zag tube (9, 0): stretching for 5% opens the gap which is smaller for 0.14 eV compared to the value of Eg calculated without the coupling, while in the case of compression for 5%, the gap is smaller for 0.28 eV. Note that symmetry prevents strain-induced torsion and alteration of φ0 in the zig-zag tubes. However, during axial deformation variation of z0 is allowed. Within the studied εz - range, the band-gap change is maximal for the tubes with θ close to zero: among all SWCNTs with diameter smaller than 1.3 nm it is up to 0.45 eV, which is about 30% smaller than when the deformation coupling is neglected. Figure 2 illustrates DOS alteration due to applied axial strain within the energy interval [−2.4, 2.4] eV. Clearly, effects of the coupling are much smaller for the DOS peak at E = 0 and every second, both at the positive and negative side of the energy axis (the even peaks), are almost unaffected by the coupling. However, the energy changes of the odd peaks are more notably pronounced. The shifts depend on the amount of the applied strain and on the tube chirality. According to the model of Yang and Han, 15 peaks in DOS can be divided into two classes: within each class the shifts due to the applied εz are opposite, though equal in magnitude (which straightforwardly follows from the zone-folding method used there). In 7 ACS Paragon Plus Environment


uniaxial strain with coupling

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Figure 2: Effects of the small axial strain εz on the electronic DOS for the tubes (12, 0) and (10, 5). For the tube (12, 0) the results neglecting the deformation coupling are also shown (dashed magenta lines).


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Figure 3: Effects of the small torsion (given by shear angle γ) on the electronic DOS for tubes (7, 7) and (8, 4). In the case of the tube (7, 7) the results neglecting the deformation coupling are also shown (dashed magenta lines).



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our energy scale convention, this means that all even VHSs should remain still, while all the odd VHSs should shift identically. However, the results point out that the uniformity of the odd VHSs is preserved only over the energy range [−1, 1] eV. Outside this interval the irregularity of the shifts increases with |E|. The stretching vs. compression asymmetry of the VHS displacements is also observable. The latter is contrast to the Yang and Han model, which predicts that the peaks shifts are even/odd functions (for p = 0/±1, respectively) of the applied strain εz . For example, for quasi-metallic tube (12, 0) tensile strain of εz = 5% causes the gap opening ∆Eg = 0.42 eV, while the same amount of compression gives ∆Eg = 0.32 eV. In case of the semiconducting tube (10, 5), behavior of ∆Eg is in accordance with the analytic formula of Yang and Han: εz = 5% lowers the gap for 0.23 eV, while εz = −5% increases it for 0.23 eV. However, at higher energies the asymmetry of the VHS shifts is notable. With the chiral angle increase, the asymmetry of the VHSs shifts lowers and in the case of the armchair tubes, becomes negligibly low. Observed shifts of the DOS peak positions imply the characteristic behavior of optical absorption maxima. Stretching shifts the first maximum, i.e. the one at the lowest energy which is usually in infrared domain, towards higher energies for p = 0, 1. On the other hand, compression has the same effect for tubes with p = −1, 0. Shift of this maximum towards lower energies occurs for stretched tubes from the class p = −1 or compressed tubes from the class p = 1. The maxima at higher energies shift alternately: the second shifts opposite to the first, the third one the same as the first one etc. Change in optical absorption spectra is more pronounced in tubes with smaller chiral angle, in full accordance with the experimental evidence. 27 Also, calculated shift of optical transitions in axially strained tubes (19, 13) and (20, 12) are in a very good agreement with the measurements. 27 Undeformed tubes from the classes p = 1 and p = −1 (and with diameters smaller than 1.3 nm) have gaps of 0.5 eV or larger. Therefore, for uniaxial deformation within the range εz ∈ [−5%, 5%] it is not possible to close the gap. However, tubes from these classes have larger variation of energy between the maximum stretching and compression, thus


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enabling larger change of the absorption peak frequencies. Accordingly, such tubes are more suitable for optical applications. Zig-zag tubes with p = ±1 enable maximal adjustment of absorption spectra by applying the axial strain. In a case of the tube (13,0), for instance, good absorption over the entire visible range can be achieved by continuously applying the axial strain from εz = −5.5% to εz = 5.5%.

3.2

Torsional strain

Just like uniaxial strain, torsion induces deformations which slow down the gap size alteration (Figure 1, bottom panel), but the chirality dependence is opposite to that obtained for the axial strain: dEg /dγ increases with θ, i.e. the largest gap alteration occurs in the case of twisted armchair tubes. For example, twisting the tube (9, 9) for γ = 2◦ , induces the gap the size of which is only 40% of the value obtained without including the coupling effects. Note that, as reported previously, 38 the zig-zag tubes also have small gap opening: from Figure 1 it is evident that for these tubes ∆Eg is quadratic function of γ. However, the same effect is found in some other tubes with θ close to zero (tube (9, 1) e.g.). This is in contrast to the predictions of the model of Yang and Han according to which ∆Eg is a linear function of γ and ∆Eg = 0 for zig-zag tubes. Alteration of DOS due to the applied torsional strain is illustrated in Figure 3. Twist breaks the vertical mirror plane symmetry of the achiral tubes, which causes the degeneracy halving of the most of the bands. Consequently, within the simple tight binding model with hopping parameter t0 , each peak at |E| < t0 splits, while in a case of metallic achiral tubes the band gap opens. These effects are clearly illustrated in Figure 3: peaks at ±1 eV of the undeformed tube (7, 7) split under finite shear, γ ̸= 0. For energies t0 ≤ |E| ≤ 3t0 , each fourfold degenerate band which corresponds to the four-dimensional representation k Gm , 37 splits in two double degenerate bands. Consequently, when γ ̸= 0, each Γ point van Hove singularity splits up. Another direct consequence of the symmetry of the achiral tubes is that the tubes twisted for γ or −γ are mirror images of each other. Hence, they all have 11 ACS Paragon Plus Environment


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symmetric response with respect to the direction of torsion: their electronic bands, and consequently, their electronic DOS are identical. Also, for these tubes positions of the even peaks are practically unchanged under the shear strain, which is in accordance with Yang and Han. 15 However, in a case of the chiral tubes, the latter is true only for smaller shear angles and for energies closer to the Fermi level. The impact of applied torsion on the optical absorption spectra depends on the class p and on the chiral angle θ of SWCNT. For the chiral tubes the maxima positions changes the same way as under the axial strain and the twisting effects becomes negligible as θ approaches zero. The splitting of peaks in DOS of armchair tubes yields additional absorption maxima: when torsion is applied each initial peak of undeformed tube splits and the splitting increase with γ. It is clear that torsion enables larger modification of SWCNTs electronic properties than uniaxial strain. Consequently, absorption on the entire range [0, 4] eV can be achieved if tubes with θ > 15◦ are twisted for γ ∈ [−5◦ , 5◦ ]. For tubes with θ ≈ 30◦ different values of γ may yield practically identical spectra: for example, tube (8, 6) for γ = −0.7◦ and γ = −5◦ and energy range bellow 2 eV.

4

Discussions

Here presented results, evidence that induced deformations significantly suppress the impact of the directly applied small (torsional or axial) strain on the electro-optical properties of SWCNTs. In order to get better insight, impact of each of the four relaxation parameters (describing deformations induced by the applied strain) are analyzed. All of them are highly complex functions of tube diameter and chiral angle. 26 Here used method of calculation enables analysis of both qualitative and quantitative aspects of each of the induced strains, separately. We find that the dominant contribution comes from the sublattice shift due to the variation of z0 and that qualitatively different alteration of the electro-optical properties


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of equally axially stretched and axially compressed tubes with small chiral angle is due to the the asymmetry in magnitude of ∆z0 (see top panel in Figure 4). Namely, stretching always induces negative, and compression positive ∆z0 , i.e. ∆z0 decreases with the applied strain. On the other hand, |∆z0 | decrease with θ and it is not symmetrical function of εz : for example, in zig-zag tubes, stretching for εz = 5 % induces ∆z0 ≈ −0.006 ˚ A, while for compression εz = −5% results in ∆z0 ≈ 0.015 ˚ A. Observed effects for uniaxially strained zigzag tubes originate only from the induced change of z0 , since vertical mirror plane symmetry forbids induced torsion and variation of φ0 . Also, non-linear dependence of ∆Eg on γ in tubes with small chiral angle corresponds to torsion induced change of z0 (bottom panel in Figure 4): ∆z0 is nonlinear function of γ but with increase of θ it becomes linear. Having at disposal electronic bands of strained tubes, we have analyzed the charge carriers behavior under the strain. We find that, regardless of the SWCNT chirality, for small applied strain, valence and conducting band around the Fermi level remain symmetric. Consequently, for uniaxially strained or twisted SWCNT effective mass of the both electrons and holes follow the trend of the band gap change with strain. Such a behavior is in contrast to other nano-scale semiconducting materials such as silicon, where compression (tension) increases the effective mass of electrons (holes). The effective mass of charge carriers is minimal for strain where metal-semiconducting transition occurs. If the applied strain is close to the critical value when the gap-opening is maximal, the effective mass of holes is slightly larger than that of the electrons (see Figure 5). Otherwise, the difference is negligible. We have also analyzed the effects of the applied strain on band gap for the SWCNT isomers (n1 , n2 ) and (n2 , n1 ). It is clear that when n1 − n2 = 0

(mod 3) the both isomers

belong to the class p = 0. Otherwise, they belong to opposite classes: p = ±1, but still their band gap changes identically under the axial strain. However, if one isomer is twisted for γ, its counterpart should be twisted for −γ, in order the band gap alterations to be mutually identical. The latter implies that measurement of ∆Eg (γ) can be used to distinguish between the isomers.



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Figure 5: Valence (blue) and conducting (red) bands of tube (8, 0) near the Fermi level for εz = 0 (solid lines), εz = 5% (dotted) and εz = −5% (dashed). 14 ACS Paragon Plus Environment

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Our numerical calculations show that the simple analytic formula of Yang and Han overestimates the magnitude of the band gap change, |∆Eg |. Also, it fails to describe the ”V-shape” response, i.e. ∆Eg proportional to absolute value of strain, of SWCNTs pertaining to the p=0 class (Figure 1). Here we give the numerical fits which comprise the all above reported linear effects of the strain-induced band gap alterations: (

)

bp ∆Eg (εz ) = εz (ap + ) cos 3θ + cp cos3 3θ , R ) ( ep sin 3θ, ∆Eg (γ) = γ dp + R

(1)

where, R is tube radius, γ is shear and ap , bp , cp , dp and ep are the fitting parameters (Table 1). The fits are valid for SWCNTs with diameter larger than 7˚ A and chiral angle within the interval [0, π/6], with exception of the counter-clockwise twisted tubes from class p = 0 for which the fit ∆Eg (γ) is applicable only for π/12 < θ ≤ π/6. Table 1: Fitting parameters of Eqs. 1 for radius R given in angstroms and shear γ in radians. Class: ap [eV] bp [eV˚ A] cp [eV] dp [eV] ep [eV˚ A]

p=0 7.7 sign(εz ) − 0.35 0 1.02 sign(εz ) − 0.36 10−2 (9.87 sign(γ) − 0.22) 10−2 (8.77 sign(γ) + 6)

p = ±1 7.7p + 0.1 −4.65p + 0.65 0 0.119p 0

Finally, damping of the band gap change due to the induced deformations, yields a smaller gauge factor GF = (∆R/R)/εz (here ∆R/R is relative change of resistance) relative to the value stemming from the analytic model. 15 Our calculations agree excellently with the measured gauge factor of the initially 85% pure (6, 5) CNT sample after applying the highstrain electrical breakdown technique: 28 the measured value was 34.1, while our calculations give GF = 37 for tube (6, 5). On the other hand, Monte Carlo simulations, based on the Yang and Han formula (which for a single (6, 5) tube gives GF = 57), predicted, for the sample, the gauge factor of 56.6. 15 ACS Paragon Plus Environment


12 (6,5) (16,0) (14,4) (12,6)

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Figure 6: Calculated relative change of the resistance ∆R/R (top) and gauge factor (bottom) vs. axial strain εz . Results calculated using the analytic formula of Yang and Han 15 are also shown (dotted lines).


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Our results suggest that the electrical breakdown technique 28 can be used to improve purity of a sample and performance of the piezoresistive sensors based on CNT resistor networks. New chirality-specific growth on solid alloy catalysts 11 yields single chirality samples of the tubes (12, 0), (14, 4) or (12, 6) which have chiral angle smaller than the (6, 5) tube. Consequently, their gauge factors are larger, making them more suitable for the piezoresistive sensor applications. Relative change of the room temperature resistance ∆R/R (calculated within the model of transport by thermal activation neglecting tunneling across the depleted region

16

) and gauge factor GF versus the applied strain for the above listed tubes are shown

in Figure 6.

5

Conclusions

Electro-optical properties of the SWCNTs exposed to the external uniaxial stress or torsion are revisited. DFTB based calculations are enriched by implementation of the full symmetry in a way 40 enabling inclusion of all degrees of freedom allowed by cylindrical geometry, and separate analysis of their individual contributions to the response of the system. Notable corrections to the simple zone folding based model 15 are found. These cannot be completely revealed within standard DFT techniques, assuming translational periodicity of SWCNTs, except for uniaxial stress in achiral 41 (exactly periodic) and to some extent in considerably thick nanotubes (approximately periodic, though with large number of atoms per period). Coupling of the deformations yields significant electro-optical response of SWCNTs by diminishing the impact of each of the deformations taken separately. The relative shift of the triangular sublattices along the tube axis is the dominant factor that reduces the alteration of electro-optical properties of SWCNTs (indicating significant electron coupling with the longitudinal high-energy Raman active mode as the damping mechanism), as well as their asymmetry with respect to the sign of the applied deformation. Results also show that the deformation coupling damps the change of the electronic bands: larger strain is needed in



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order to obtain the same quantitative results as when the coupling is neglected. The effect is even more pronounced when the torsion is applied. Consequently, for the tubes the chiral vector of which is far from the zig-zag direction, linear domain of the band gap change with torsion, ∆Eg (γ) is larger. Symmetry gives a qualitative insight to the non-linear dependence of ∆Eg on γ in zig-zag tubes (Figure 1, bottom panel). Namely, calculations show that among all torsion induced deformations, only ∆φ0 depends linearly on γ. Such displacement of the sublattices is related to the transversal high-energy mode which in zig-zag tubes is not totally symmetric (as in chiral and armchair ones). Consequently, the corresponding matrix element of electron-fonon interaction is zero, i.e. linear term in ∆Eg (γ) vanishes . Results also show that uniform strain induces only a small difference of the effective masses of the charge carriers, so that regardless of the applied strain, SWCNTs can be equally successfully doped to either p-type or n-type semiconductors. Our results perfectly match the experimental data of Huang et al. 27 and give complete interpretation of the observed lower electromechanical response with respect to the predictions based on the analytic formula of Yang and Han. This justifies that, as far as the available measurements are considered, the main physical ingredients for the considered processes are taken into account through single particle mean-field picture (e.g. differences with respect to Yang-Han model are interpreted above through the first order electron phonon coupling), and that the fine corrections only may be expected within the many-body approaches 41 going beyond DFT calculations. Also, calculated band gap change of tube (6, 5) resolves the discrepancy between the values of measured and simulated gauge factor. 28 Linear fits of the band gap change with the applied strains are given, enabling thus fast calculations of the gauge factors. The presented results, giving accurate quantitative predictions of the electro-optical response to the deformations, are important for better understanding the performance of CNT based piezoresistive sensors. The electronic transition energy shift due to applied strain can be also used to obtain conditions for resonant Raman scattering: instead of tuning the laser


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energy, the resonant energy can be achieved by continuous deformation of SWCNTs. 39 Bearing in mind that the deformation also shifts the vibrational frequencies, this can facilitate the characterization of SWCNTs as nonresonant scattering signal is too low. In addition, deformation induced large variations of the absorption energy of the tubes of specific chirality can be utilized in various opto-electronic device applications. Electronic properties of SWCNTs are very sensitive to torsion. In particular, entire visible light spectrum can be covered by twisting SWCNT with large chiral angle, which promotes such tubes as perfect tunable light absorbers or emitters.

Acknowledgement This work is supported by Serbian Ministry of Science (Project ON171035).

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Electronic Properties of Strained Carbon Nanotubes - American

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