Effect of strain on the band gap and effective mass of zigzag single-wall carbon nanotubes: First-principles density-functional calculations

PHYSICAL REVIEW B 77, 1553 8 Effect of strain on the band gap and effective mass of zigzag single-wall carbon nanotubes: First-principles density-functional calculations S. Sreekala, 1, * X.-H. Peng, 1 P. M. Ajayan, and S.. Nayak 1 1 Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 118, USA Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 118, USA Received December 7; revised manuscript received March 8; published 3 April 8 We have studied the behavior of band gap and effective mass of both the electrons and the holes in small diameter zigzag single-walled carbon nanotubes under uniaxial mechanical strain by using first-principles density-functional theory. The band gap of these nanotubes is modified by both compressive and tensile strain and all zigzag single-wall carbon nanotubes show a semiconductor-metal transition with strain. We also find that both compressive and tensile strains have a similar effect on the effective mass of the electrons and holes in these nanotubes. Our studies also show that the response of the changes in band gap and effective mass to the uniaxial strain could be grouped into three categories, depending on their chirality. DOI: 1.113/PhysRevB.77.1553 PACS number s : 73.. f, 73.3.Fg, 71..Tx, 71.15.Mb I. INTRODUCTION Carbon nanotubes CNTs are one-dimensional materials with outstanding mechanical 1 and electronic properties 5 9 and are attractive building blocks for future electronic, optical, and strain sensing 1,11 nanodevices. There is a wealth of literature on the studies of various properties of the carbon nanotubes and also on the application of the nanotube-based devices. One of the most interesting properties of the CNTs is the change in the electrical resistance under mechanical loading. 1 Tombler et al. 7 found a orders of magnitude change in conductance, when an atomic force microscope tip was used to stretch a metallic nanotube suspended across a trench. This effect is also found to be completely reversible. It was suggested by Maiti et al. 13 that for a small gap semiconducting tube, axial strain could induce an increase in the band gap. Recent advances on CNT devices 1,15 use both uniaxial or torsional strain through the entire CNT channel as a potential approach to improve the device performance. These results are vital for flexible electronics, 1 nanoscale pressure sensors, and nanoelectromechanical transducers 17 19 as the CNT s sensitivity to strain exceeds that of silicon and has a gauge factor of 1. Single-walled carbon nanotubes SWCNTs have been the focus of much attention because they provide experimental realizations of quasi-one-dimensional conductors. SWCNTs can be viewed as a rolled up graphene strip and depending on the cut of the strip, they can be zigzag n,, armchair n,n, and chiral n,m nanotubes. Electronically, SWCNTs can behave as either metallic or semiconducting depending on the chirality of their atomic arrangements and diameter. 9,1 3 The different mechanical deformations on the CNTs are torsion, uniaxial straining, and bending. These deformations cause a corresponding change in their electronic structures, 13, 7 which also depends upon the chiral symmetries of the CNTs. The studies by Zhang et al. 8 suggest that a n, nanotube has an elastic limit nearly twice that of an n,n tube of the same radius. Their studies further revealed that the critical elongations above which the nanotube showed plastic deformation are approximately 1% for zigzag nanotubes. 8 There are two threads of thoughts to explain the changes in the electronic structure, namely, the metal to semiconductor transition MST that takes place in the single-walled carbon nanotunes. Park et al. 9 pointed out that the breaking of the mirror symmetry leads to a MST in armchair SWCNTs. For metallic zigzag SWCNTs, all studies agree that the MST is driven by the curvature effect. 9,3 Most deformed single walled carbon nanotubes will show a MST, repeatedly occuring with increasing strains. 5, Since the changes in the electronic structures can be clearly reflected in their transport properties, the effects of deformations on the conductance of CNTs have been studied by several groups. 7,8,5, Although there have been some studies carried out to understand the electron transmission through strained CNTs, it is still interesting and important to have a collective understanding of how the individual single-walled carbon nanotubes will behave under uniaxial strain both bond stretching and contraction. The effective mass of the charge carriers, which is one of the important parameters to understand the transport in the CNTs, has not been studied for all the SWCNTs, the only work being that of Zhao et al., 31 which deals with some chiral and zigzag nanotubes with n=3q+1, with q being an integer. There is practically no studies on the effect of straining on the effective mass of the electron and hole of these nanotubes, which, in turn, would offer a in-depth understanding of the essential physics related to the electromechanical effects in the deformed CNTs, which is the purpose of this paper. We present the results of our detailed study of the effect of axial strain on the band structure and also address the question, namely, whether the chirality plays a role in choosing the minimum tension or compressive strain that causes the first semiconductor to metallic transitions in the zigzag single-walled CNTs. It is generally known that the armchair metallic SWCNTs retain their high symmetry under tensile stretching and contraction, hence the electrical properties should be least sensitive to uniaxial strain. Therefore, we have considered only the zigzag SWCNTs that are sensitive to strain. We have also studied the effective mass of the charge carriers with and without strain and whether the different kinds of strain have different effects on them. This paper is organized as follows. In Sec. II, we present the 198-11/8/77 15 /1553 7 1553-1 8 The American Physical Society

SREEALA et al. details of computational method we have employed, in Sec. III, we present the results along with discussions, and Sec. IV summarizes our conclusions. PHYSICAL REVIEW B 77, 1553 8 Strain= 5% II. COMPUTATIONAL METHODS We have used the first-principles density-functional theory DFT 3 to study the electronic properties of zigzag n, single-walled carbon nanotubes. In particular, we have used generalized gradient corrected approximation of Perdew and Wang 1991 33 and the projector augmented wave representation 3,35 as implemented in the Vienna ab initio simulation VASP code. 3 The plane wave energy cutoff was taken to be E cut =87 ev and the Brillouin zone k-point sampling was done by using the Monkhorst Pack algorithm, 37 and 1 1 k points were chosen for the DFT calculations. In our study, we first construct the initial configuratrion of a given zigzag SWCNTs by simply wrapping its corresponding graphene strip. The C-C bond length is taken as 1. Å and we have employed a supercell geometry, such that the replica of the nanotube along the x and y axis are further apart more than.8 nm, so that the interactions between them are negligible. Along the z axial direction, the dimension of the supercell was taken to be. nm. The equilibrium configuration of the zigzag SWCNTs is then obtained by relaxing the initial configuration. Strain was applied to these CNTs along the axial z direction and allowed to relax along the other two directions. The force on each atom for the strain applied was relaxed to less than.1 ev/å. Simultaneous changes in the band gap during the aplication of strain were assessed. We have calculated the electronic band structure of zigzag SWCNTs of chirality 7, to 17, corresponding to diameters of 5.8 to 13.31 Å under axial tension and compression. In our approach, the axial strain given by = l l /l is initially applied to the zigzag SWCNT, where l is the equilibrium length in the axial direction for the unit cell of the unstrained SWCNT and l is the corresponding length in the strained tube. The equilibrium configuration of the strained tube is then obtained through force minimization along the X and Y directions. The energy gap of the strained tube is determined from the electronic structure calculations of the equilibrium configuration of the strained tube. We can further use the calculated electronic structure to determine other physical quantities such as the effective mass of charge carriers, which is an important physical quantity in the analysis of the electron transport. We have calculated the effective mass m* of both the electrons and holes using the following relation: m* = d E 1 dk k=. 1 It is generally represented in terms of the mass of an electron m. The effective mass of the charge carriers near the top of the valence band corresponds to the effective mass of the holes and that near the bottom of the conduction band is that of electrons. E F f Strain=% Strain=5% FIG. 1. Change in the band structure with the application of uniaxial strain for SWCNT 9,. Negative strain corresponds to compression and positive strain corresponds to tension. III. RESULTS AND DISCUSSIONS A. Metallic to semiconductor transition The zigzag SWCNTs that are represented as n, are metallic, if n is a multiple of 3 and all others are semiconducting in unstrained condition. For simplicity, one can classify all zigzag SWCNTs as n=3q, n=3q+1, and n=3q+ and study the effect of uniaxial strain on each of these cases. 1553-

EFFECT OF STRAIN ON THE BAND GAP AND PHYSICAL REVIEW B 77, 1553 8 1. 1. 1.8. (1,) (13,) (1,) (15,) (11,) (1,) Strain= 5%.. 8 Strain % FIG.. Color online Band gap E E F ev versus strain curves for of SWCNTs 1,, 11,, 1,, 13,, 1,, and 15,. Negative and positive strains correspond to compressive and tensile, respectively. Strain=% 1. Zigzag single walled carbon nanotubes with n=3q We have shown the effect of uniaxial strain on one of the metallic SWCNTs 9, in Fig. 1. Figure 1 shows that at zero strain, the SWCNT 9, shows zero band gap and on the application of strain, both compressive and tensile, the band gap opens, similar to the results obtained by Guo et al. 38 This kind of behavior was also observed in other zigzag SWCNTs such as 1, and 15,, as shown in Fig.. The chirality of these nanotubes can be generalized as n,, where n=3q. The opening of the band gap in this class of systems can be explained as follows. 1 and that are the reciprocal lattice vectors along the circumference and axial direction of the CNTs are quantized quantities. The Brillouin zone BZ that is parallel to passes through the point point where and * bands of the graphene sheet meet of the carbon nanotube; then, the CNT is metallic. On straining the CNT, the BZ does not cut through the point and hence the band gap opens. Strain=5%. Zigzag single walled carbon nanotubes with n=3q+1 We will now discuss the effect of strain on semiconducting SWCNT n,, where n=3q+1. As an example, we have considered 1,. At zero strain, this nanotube shows a band gap of.737 ev. On application of compressive strain, the band gap decreases and becomes near zero at a strain of 5%, as shown in Fig. 3. In contrast, the band gap opens up further on the application of tensile strain. This type of behavior was also observed for SWCNT 13,, as shown in Fig.. Similarly CNTs 1, and 7, show MST in the compressive strain regime, as listed in Table I. From these results, we can conclude that the SWCNTs n, with n=3q+1 will show closing of band gap on application of compressive strains and tensile strains cause opening of the band gap. The closing of the band gap for compressive strain and opening of the band gap for tensile strain of SWCNT 1, can also be explained by using the BZ of the CNT. Compressive strain in SWCNT 1, distorts the carbon hexagon in such a way that it decreases the distance between the point of the CNT and, while the distortion due to the tensile strain increases FIG. 3. Change in the band structure with the application of uniaxial strain for SWCNT 1,. Negative strain corresponds to compressive and positive strain corresponds to tensile strain. the distance between them. Hence, the compressive strain causes the band gap to decrease, whereas the tensile strain opens up the band gap. However, when the strain applied shifts the point of the CNT such that passes through it, then for that particular strain, the SWCNT shows a zero band gap. Also, as shown in Fig., the band gap for the CNTs does not monotonically increase. It reaches a maximum, and then drop down, which is in agreement with experiments. 19,39 1553-3

SREEALA et al. PHYSICAL REVIEW B 77, 1553 8 TABLE I. Strain value for zigzag SWCNTs at which MST occurs. Strain= % Chirality Types of strain when MST occurs Minimium strain % when MST occurs 7, Compressive 8, Tensile 9 9, Zero strain 1, Compressive 5 11, Tensile 1, Zero strain 13, Compressive 1, Tensile 5 15, Zero strain 1, Compressive 17, Tensile Strain=% 3. Zigzag single walled carbon nanotubes with n=3q+ We have plotted the band structure of SWCNT 11, in Fig.. This SWCNT falls in the category n, with n=3q +. At zero strain, the band gap of SWCNT 11, is.9 ev and it decreases on the application of tensile strain. It shows a zero band gap at tensile strain of %. Also, Fig. shows that SWCNT 1, shows MST in the tensile strain regime. Similarly, in CNTs 8, and 17,, the MST occurs in tensile strain regime see Table I. From Fig., one can see that the slope of the curves is very similar in the linear region for all the SWCNTs, even though the metallic to semiconductor transition takes places either in the tensile, compressive, or unstrained condition. When a strain was applied to the SWCNTs, both the lattice parameter and the Brillouin zone line get shifted. Then, the change in band gap as a function of the changing strain d is given by d E E f = 3t 1+, d where is the Poisson ratio of the SWCNTs and t =. ev. For the SWCNTs with n=3q or n=3q+1, the slope is positive and for SWCNTs with n=3q+, the slope is negative. It is interesting to note that, with increase in diameter, the strain at which MST occurs decreases. Hence, for larger diameter SWCNTs n, with n=3q+1, the compressive strain needed to cause MST will decrease and reach zero, which is in agreement with the fact that band gap decreases as a function of diameter in the SWCNTs in the absence of strain. Same is the case with SWCNTs n, where n=3q+ with large diameters, the tensile strain needed for MST decreases. strain=% FIG.. Change in the band structure with the application of uniaxial strain for SWCNT 11,. Negative strain corresponds to compressive and positive strain corresponds to tensile strain. B. Effective mass of the charge carriers Effective mass of the charge carriers is an important parameter for the electrical transport properties of semiconducting materials, as is inversely proportional to the mobility of the charge carriers. Hence it, in turn, influences the diffusivity of the charge carriers and the electrical conductivity of the material. The effective mass varies, depending on the interaction with the core electrons. Hence, the stronger the interaction, the more tightly bound the electrons are to atoms and larger the effective mass. From the shape of the band structure, we have calculated the effective mass of the electrons and holes of the zigzag SWCNTs. In order to calculate the effective mass, we have plotted the conduction and the valence bands as a function of 1553-

EFFECT OF STRAIN ON THE BAND GAP AND PHYSICAL REVIEW B 77, 1553 8 TABLE II. Effective mass of electron and hole of zigzag SWCNTs without strain..9 (a) (13,) Chirality Effective mass of e m Effective mass of hole m 7,..85 8,.8.8 1,.9.87 11,.1.55 13,.9.5 1,.51.19 1,.5. 17,.35.38 k points, which are near the Dirac point. The curves of the energy versus k points are fitted with a second order polynomial fit, say =Ak +Bk+C, and calculate the curvature of the polynomial d. The effective masses of the dk electrons and holes are then calculated by substituting the value of the curvature in Eq. 1. Table II gives the effective mass of the electron in the conduction band and hole in the valence band in terms of the mass of the electron m at zero strain. In the case of SWCNTs 1, to 17,, the effective mass of the electrons is approximately the same as the effective mass of the holes, which is similar to the work of Zhao et al. 31 This is due to the fact that the dispersions of the highest occupied valence band and the lowest unoccupied conduction band in these SWCNTs are nearly symmetric around the point. From Table II, we can also see the effective mass of the charge carriers for the n, SWCNTS, where n=3q+1 decreases with increase in diameter. This is also observed in SWCNTs n, where n=3q+; however, the slopes are different. It should be pointed out that the effective mass of the charge carriers for smaller SWCNTs 7, and 8, is different from the values of the other SWCNTs in that their values are not the same. This is due to the curvature effects. In the case of SWCNT 8,, the conduction band near the point has a larger curvature than the valence band and hence the effective mass of the hole is larger than that of the electrons. Similarly in SWCNT 7,, the valence band near the point has a larger curvature than the conduction band and hence the effective mass of the electrons is larger than that of the holes. As can be seen from Figs. 1, 3, and, the band structure of the zigzag SWCNTs changes with the application of the uniaxial strain, especially the conduction and valency bands. This implies that the effective mass of the electrons and holes would also change upon the straining the nanotubes. We have calculated the changes in the effective mass of the electrons and holes due to the uniaxial strain. This is another study to explore how effective mass varies with strain in SWCNTs. These variations have been plotted in Figs. 5 a 5 c for three SWCNTs 13,, 1,, and 15, as a function of strain. As mentioned before, the band gap variation with strain can be classified into three groups depending on their chirality. The effective mass of the electrons and holes also follow these trends, with minimum effective mass Effective mass (m o ) Effective mass (m o ) Effective mass (m o ).7.5.3 Electron Hole.1 8 % strain.55.5.35.5 (b) (1,) Electron Hole.15 8 % Strain.9.7.5.3 (c) (15,) Electron Hole.1 8 % Strain FIG. 5. a Variation of the effective mass of the electron and hole on SWCNT 13, as a function of strain. b Variation of the effective mass of the electrons and holes of SWCNT 1, as a function of strain. c Variation of the effective mass of the electrons and holes of SWCNT 15, as a function of strain. at the strain where MST occurs. It is also clear from our calculations that the variation of the effective mass for both the electrons and holes to strain is similar, unlike other nanoscale semiconducting materials such as silicon where compression increases the effective mass of electrons and tension increases the effective mass of holes. The difference in the value of the effective mass of the electrons and holes is significant only at the strain at which the nanotube has maximum band gap. These correspond to large value of 1553-5

SREEALA et al. the effective mass, which, in turn, means that the mobility of the charge carriers is less at those strain values. Hence, it can be concluded that different kinds of uniaxial strain has the same effect on the charge carriers suggesting that strained zigzag SWCNTs can be favorably doped to form either n type or p type semiconductors by using appropriate dopants. IV. CONCLUSIONS In summary, we show that the uniaxial strain applied parallel to the axis of the carbon nanotube can significantly modify the band gap and induce semiconductor-metallic transition. For the zigzag tubes n,, the behavior is strongly dependent on whether n=3q, 3q+1,or3q+ and also the minimum strain needed to produce the MST depends on chirality. In the case where n=3q, the band gap is zero at zero strain and band gap opens up on application of both PHYSICAL REVIEW B 77, 1553 8 tensile and compressive strains. In the semiconducting SWCNTs n, where n=3q+1, compressive strain decreases the band gap whereas tensile strain increases it, and finally in the case of SWCNTs n, where n=3q+, the band gap decreases for tensile strain and opens up for compressive strain. We have also calculated the effective mass of both the electrons and the holes at zero strain and also in strained conditions and we find that the strain does not have favorable effect on either of the charge carriers. Also, the effective mass values of the charge carriers are nearly the same, suggesting that they can be doped to both p-type and n-type materials. ACNOWLEDGMENT This work is supported by the NEW York State Interconnect Focus Center IFC, RPI. *sreeks@rpi.edu 1 M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature London 381, 78 199. B. I. Yakobson, C. J. Brabec, and J. Bernholc, Phys. Rev. Lett. 7, 511 199. 3 E. W. Wong, P. E. Sheehan, and C. M. Lieber, Science 77, 1971 1997. J.-C. Charlier, X. Blase, and S. Roche, Rev. Mod. Phys. 79, 77 7. 5 H. Dai, E. W. Wong, and C. W. Lieber, Science 7, 53 199. T. W. Ebbesen, H. J. Lezec, H. Hiura, J. W. Bennett, H. F. Ghaemi, and T. Thio, Nature London 38, 5 199. 7 T. W. Tombler, C. W. Zhou, L. Alexseyev, J. ong, H. J. Dai, L. Lei, C. S. Jayanthi, M. J. Tang, and S. Y. Wu, Nature London 5, 79. 8 E. D. Minot, Y. Yaish, V. Sazonova, J. Y. Park, M. Brink, and P. L. McEuen, Phys. Rev. Lett. 9, 151 3. 9 N. Hamada, S. I. Sawada, and A. Oshiyama, Phys. Rev. Lett. 8, 1579 199 ; L. F. Chibotaru, S. Compernolle, and A. Ceulemans, Phys. Rev. B 8, 151 3. 1 P. Avouris, Acc. Chem. Res. 35, 1. 11 A. Jungen, C. Meder, M. Tonteling, C. Stampfer, R. Linderman, and C. Hierold, Proc. IEEE 93 5. 1 J. Suhr, P. Victor, L. Ci, S. Sreekala, Z. Zhang, O. Nalamasu, and P. M. Ajayan, Nat. Nanotechnol., 17 7 ; P. Victor, L. Ci, S. Sreekala, A. umar, S. esapragada, D. Gall, O. Nalamasu, and P. M. Ajayan, Appl. Phys. Lett. 91, 15311 7. 13 A. Maiti, A. Svizhenko, and M. P. Anantram, Phys. Rev. Lett. 88, 185. 1 P. Avouris, J. Appenzeller, R. Martel, and S. J. Wind, Proc. IEEE 91, 177 3. 15 P. L. McEuen, M. S. Fuhrer, and H.. Park, IEEE Trans. Nanotechnol. 1, 173131. 1 G. Gruner, J. Mater. Chem. 1, 3533. 17 R. J. Grow, Q. Wang, J. Cao, D. Wang, and H. Dai, Appl. Phys. Lett. 8, 931 5. 18 C. Stampfer, T. Helbling, D. Obergfell, B. Schhberle, M.. Tripp, A. Jungen, S. Roth, V. M. Bright, and C. Hierold, Nano Lett., 33. 19 D. Sickert, S. Taeger, I. ühne, W. Pompe, and G. Eckstein, Phys. Status Solidi B 3, 35. J. Cao, Q. Wang, and H. Dai, Phys. Rev. Lett. 9, 1571 3. 1 R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Appl. Phys. Lett., 199. J. W. G. Wildöer, L. C. Venema, A. G. Rinzler, R. E. Smalley, and C. Dekker, Nature London 391, 59 1998. 3 T. W. Odom, J.-L. Huang, P. im, and C. M. Lieber, Nature London 391, 1998. A. Rochefort, P. Avouris, F. Lesage, and D. R. Salahub, Phys. Rev. B, 138 1999. 5 L. Yang, M. P. Anantram, J. Han, and J. P. Lu, Phys. Rev. B, 1387 1999. L. Yang and J. Han, Phys. Rev. Lett. 85, 15. 7 L. Liu, C. S. Jayanthi, M. Tang, S. Y. Wu, T. W. Tombler, C. Zhou, L. Alexseyev, J. ong, and H. Dai, Phys. Rev. Lett. 8, 95. 8 P. Zhang, P. E. Lammert, and V. H. Crespi, Phys. Rev. Lett. 81, 53 1998. 9 C. J. Park, Y. H. im, and. J. Chang, Phys. Rev. B, 15 1999. 3 P. E. Lammert, P. Zhang, and V. H. Crespi, Phys. Rev. Lett. 8, 53. 31 G. L. Zhao, D. Bagayoko, and L. Yang, Phys. Rev. B 9, 51. 3 P. Hohenberg and W. ohn, Phys. Rev. 13, B8 19 ; W. ohn and L. J. Sham, Phys. Rev. 1, A1133 195 ; R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules Oxford University Press, Oxford, 1989. 33 J. P. Perdew and Y. Wang, Phys. Rev. B 5, 13 199. 3 G. resse and D. Joubert, Phys. Rev. B 59, 1758 1999. 35 P. E. Blöchl, Phys. Rev. B 5, 17953 199. 3 G. resse and J. Hafner, Phys. Rev. B 8, 13115 1993 ; G. resse and J. Furthmüller, ibid. 5, 1119 199. 37 H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 197. 1553-

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