- Email: [email protected]
Single wall carbon nanotubes density of states: comparison of experiment and theory
Chemical Physics Letters 370 (2003) 597–601 www.elsevier.com/locate/cplett
Single wall carbon nanotubes density of states: comparison of experiment and theory Pavel V. Avramov
*,1,
Konstantin N. Kudin, Gustavo E. Scuseria
Department of Chemistry, Center for Biological and Environmental Nanotechnology, Rice University, 6100 Main Street, Houston, TX 77005-1892, USA Received 20 November 2002; in final form 17 January 2003
Abstract We study the electronic structure of a variety of single wall carbon nanotubes and report density of states obtained with the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation and hybrid PBE0 approximation of density functional theory using Gaussian orbitals and periodic boundary conditions. PBE gives very good results for metallic tubes but the addition of a portion of exact exchange in the hybrid PBE0 functional worsens the agreement between experiment and theory. On the other hand, the PBE0 hybrid significantly improves the theoretical predictions (compared to PBE) for semiconducting tubes. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The electronic structure of individual single wall carbon nanotubes (SWNT) is currently the focus of considerable theoretical and experimental interest [1–6]. In SWNTs, a small change in atomic structure (diameter and/or chiral angle) may produce significant changes in electronic properties of the tube. Tight binding calculations predict that SWNTs with indices ðn; mÞ such that ð2n þ mÞ= 3 ¼ k (where k is an integer) should be metals, whereas for k non-integer, the tubes should be semiconductors [7–9]. It has also been shown [10] *
Corresponding author. Fax: +713-348-5155. E-mail address: [email protected] (P.V. Avramov). 1 Permanent address: L.V. Kirensky Institute of Physics SB RAS, Academgorodok, Krasnoyarsk 660036, Russia.
that the energy difference between the first two van Hove singularities (VHS), which reflect the onedimensional (1D) band structure of the tubes and appear as sharp peaks in the density of states (DOS), is a smooth function of the tube diameter. The tight binding (TB) model has been widely used in theoretical studies of SWNTs [11]. TB is an empirical method based on p- and r-states without the self-consistent procedure. Commonly, TB models oversimplify the DOS and give, for example, symmetrical p=r–p =r VHS relatively to the Fermi level [12]. Recent experiments employing scanning tunneling microscopy (STM) in combination with scanning tunneling spectroscopy (STS) [1–5] have confirmed theoretical predictions about the shape and positions of van Hove singularities. However, fine effects such as small gaps in the DOS of
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00113-1
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P.V. Avramov et al. / Chemical Physics Letters 370 (2003) 597–601
metallic SWNTs [1] are not reproduced by these TB models [7–9]. Density functional theory (DFT) calculations of various types have been applied to predict the electronic structure of SWNTs [13–15]. The most popular approach for calculations with periodic boundary conditions (PBC) is the local density approximation (LDA) in conjunction with planewaves and pseudopotentials [13]. The TB and DFT calculations yield very different predictions for the band gap and positions of VHS, as well as the asymmetry behavior of the VHS relative to the Fermi level. The LDA approach in combination with Gaussian-type basis functions [15] underestimates the separation between the first van Hove singularities up to 25%. The most reliable experimental technique for studying the electronic structure of individual SWNT is the combination of STM and STS [1–5]. The STS method gives a common experimental picture for the valence and conduction bands. Pieces of nanotube soot are usually dispersed in an organic solvent by ultrasound to unravel the nanotube bundles. A droplet of the dispersion is deposited on an inert surface like Au (1 1 1). Imaging of the SWNT is performed in constant current mode at typical parameters of 50 pA for tunneling current and 0.1 V for bias voltage. In order to determine the experimental DOS of individual nanotubes, current–voltage tunnel spectra are taken by recording the current as a function of the bias voltage, while keeping the STM tip at a fixed position. A set of I–V measurements can then be done in one tube with different positions of the STM tips on the SWNT surface [3]. Although some small differences can be detected in the I–V curves of a nanotube, the conductance dI=dV V or normalized conductance ðV =IÞðdI=dV Þ V curves can be numerically calculated from I–V spectra and have been reported as an experimental DOS for the SWNTs. The normalized conductance ðV =IÞðdI=dV Þ V provides a better measure of the DOS in comparison with the conductance dI=dV V [16]. Resonant Raman scattering [17–19], which is based on analysis of stokes and anti-stokes Raman processes, can probe the electronic dispersion indirectly via absorption from the valence to conduction states.
Even though these experimental results depend on many factors such as defects in the SWNT atomic structure and the interaction between the tube and the surface, it is possible to compare the experimental DOS with those obtained directly from first-principles computations. As with fullerenes like C60 , the Hartree–Fock method fails to describe the experimental energy gap [20]. In C60 , second-order perturbation theory (MP2) can significantly improve results for ionization potentials and electronic affinities [20]. However, the computational cost of MP2 with PBC remains substantial [21]. On the other hand, DFT is significantly faster and gives relatively accurate results for SWNTs [13,15,22,23], especially for metals [24]. In this work, we carry out a systematic comparison of experimental and theoretical DOS of individual nanotubes with different diameters and chiral angles using for the first time hybrid functionals, i.e., those which contain a portion of exact HF exchange. We have calculated the optimized geometries and electronic structures of a number of zigzag, armchair, and chiral SWNTs with the PBE [25] and hybrid PBE0 [26,27] functionals using the development version of the GA U S S I A N suite of programs [28], which allows for DFT calculations with periodic boundary conditions [6] and geometry optimizations using analytic energy gradients. Equilibrium structures (including both atomic positions and lattice dimensions) of the zigzag and armchair SWNTs were obtained with the PBE functional using a 6-31G basis set. For the chiral tubes, PBE/3-21G was used in the geometry optimization because of their large unit cell sizes. The number of atoms in the unit cell varies from 32 for the (8,8) tube (480 basis functions) to 186 for the (13,7) SWNT (2,790 basis functions). Once the geometry was optimized with this procedure, the electronic structure (DOS) was calculated using a polarized 6-31G* basis set consisting of 4s2p1d basis functions on each carbon atom, and 128 points in k-space for the Brillouin zone integration. Plots of the calculated DOS were obtained using a broadening energy parameter of 0.1 eV. The following tubes were explicitly considered: metallic armchair (8,8), metallic zigzags (9,0),
P.V. Avramov et al. / Chemical Physics Letters 370 (2003) 597–601
(12,0), and (15,0), semiconducting zigzag (10,0), metallic chiral (13,7), and semiconducting chiral (14,3). The theoretical DOS are compared with the STS experimental results [1,3,5,29] in Figs. 1–5. In Fig. 1, the STS experimental (solid line, [29]) normalized conductance ðV =IÞðdI=dV Þ V and theoretical DOS obtained with the PBE and PBE0 functionals (dashed lines) for the (10,0) zigzag semiconducting SWNT are presented. The experimental energy difference between the first VHS is 1.1 eV, whereas the theoretical PBE result is 0.8 eV. The experimental energy difference between the second VHS is 3.0 eV but only 1.9 eV with PBE. On the other hand, the hybrid PBE0 functional gives much better results for the first and the second VHS transitions for the (10,0) tube: 1.1 and 2.9 eV, respectively. In Fig. 2a, the STS experimental normalized conductance ðV =IÞðdI=dV Þ V [3] and theoretical
599
Fig. 2. The STS experimental normalized conductance ðV =IÞðdI=dV Þ V [3] (a) and theoretical (b) PBE and PBE0 DOS of semiconducting chiral (14,3) SWNT.
Fig. 3. The STS experimental conductance dI=dV V [1] (solid line) and theoretical (dashed lines) PBE and PBE0 DOS of metallic zigzag (15,0), (12,0) and (9,0) SWNTs.
Fig. 1. The STS experimental normalized conductance ðV =IÞðdI=dV Þ V [29] (solid lines) and theoretical (dashed lines) PBE and PBE0 DOS of semiconducting zigzag (10,0).
in Fig. 2b, DOS with the PBE and PBE0 functionals for the (14,3) chiral semiconducting SWNT are presented. The experimental energy difference between the first VHS is 0.9 eV, whereas the
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Fig. 4. The STS experimental conductance dI=dV V [5] (solid line) and theoretical (dashed line) PBE DOS of metallic chiral (13,7) SWNT.
Fig. 5. The STS experimental conductance dI=dV V [1] (solid line) and theoretical (dashed lines) PBE and PBE0 DOS of armchair (8,8) SWNT.
theoretical PBE value is only 0.6 eV. Here again, PBE0 yields much better results for the first VHS transition: 0.8 eV. In Fig. 3, the STS experimental conductance dI=dV V (solid lines, [1]) and theoretical DOS calculated with the PBE and PBE0 functionals (dashed lines) for metallic zigzag (15,0), (12,0), and (9,0) SWNTs are presented. The experimental energy differences between the first VHS are 1.7, 2.1, and 2.6 eV for the three tubes in Fig. 3, respectively. The calculated PBE values are 1.6, 1.8, and 2.5 eV, in very good agreement with experiment, respectively. For the (9,0) tube, the PBE0 result for the first VHS transition (3.1 eV) is in poor agreement with the experiment (1.7 eV). Similarly
low-quality predictions were obtained with PBE0 for the other metallic tubes in Fig. 3 and are not presented. Also in Fig. 3, the experimental energy differences between the second VHS are 1.9 and 2.4 eV for the (15,0) and (12,0) SWNTs, respectively. There are no experimental data for the (9,0) tube. The theoretical PBE results are 2.1 and 2.4 eV for the (15,0) and (12,0) tubes, respectively. Here again, the agreement between experiment and theory is quite good. It should also be mentioned that in the case of the metallic zigzag nanotubes, the PBE approximation seems to reproduce peculiarities of the experimental DOSlike small secondary gaps (0.1 eV width, energy 0 eV). In Fig. 4, the experimental conductance dI=dV V (solid line, [5]) and theoretical DOS (dashed line) for the (13,7) chiral metallic tube are presented. The experimental energy difference between first VHS is 1.6 eV. The theoretical PBE value is only 0.2 eV smaller (1.4 eV). The energy difference between the second VHS is roughly 2.0 eV in experiment, and 1.9 eV with the PBE functional. Finally, in Fig. 5, the STS experimental conductance dI=dV V (solid line, [1]) and theoretical (dashed lines) DOS for the (8,8) armchair tube are presented. The energy difference between the first VHS in the experimental STS DOS is 1.5 eV, whereas the theoretical PBE result is 2.0 eV. For the second VHS, the experimental energy difference is 2.8 eV and the PBE prediction is 3.3 eV. For this metallic tube, addition of exact exchange in the PBE0 functional worsens the comparison between experiment and theory (see Fig. 5). The energy separations become 2.7 and 4.6 eV, for the first and second VHS, respectively. Based on the comparison of the theoretical data with STS experimental results, we conclude that the PBE functional describes fairly well the density of electronic states of metallic nanotubes and gives good qualitative agreement for the semiconducting ones. Addition of a portion of exact exchange, as implemented in the hybrid PBE0 functional, substantially improves the agreement between experiment and theory for the semiconducting tubes but worsens this agreement for the metallic ones.
P.V. Avramov et al. / Chemical Physics Letters 370 (2003) 597–601
The good agreement between experiment and theory for the DOS of metallic SWNTs is not surprising given that PBE is usually deemed as a good approximation for metals. The PBE functional correctly reproduces the uniform electron gas limit, which is a metal itself. On the other hand, for systems with a finite band gap, PBE (as well as many other functionals including LDA) predicts band gaps which are too narrow. The Hartree–Fock method, which contains 100% exact exchange and 0% correlation, tends to spatially localize spins, and overemphasizes the insulating character of materials. Thus, addition of a portion of exact Hartree–Fock exchange to the functional (as in PBE0) widens these gaps thus bringing the theoretical results in better agreement with the experiment.
Acknowledgements This work was supported by the National Science Foundation and the Welch Foundation. PVA also acknowledges support by Grant No. 31 of the Federal Russian Program ÔIntegrationÕ.
References [1] M. Ouyang, J.-L. Huang, C.L. Cheung, C.M. Lieber, Science 292 (2001) 702. [2] L.C. Venema, J.W. Janssen, M.R. Buitelaar, J.W.G. Wild€ oer, S.G. Lemay, L.P. Kouwenhoven, C. Dekker, Phys. Rev. B 62 (2000) 5238. [3] T.W. Odom, J.-L. Huang, P. Kim, C.M. Lieber, Nature 391 (1998) 62. [4] J.W.G. Wild€ oer, L.C. Venema, A.G. Rinzler, R.E. Smalley, C. Dekker, Nature 391 (1998) 59. [5] P. Kim, T.W. Odom, J.-L. Huang, C.M. Lieber, Phys. Rev. Lett. 82 (1999) 1225. [6] K.N. Kudin, G.E. Scuseria, Phys. Rev. B 61 (2000) 16440. [7] J.W. Mintmire, B.I. Dunlap, C.T. White, Phys. Rev. Lett. 68 (1992) 631.
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[8] N. Hamada, S. Sawada, A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1579. [9] R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 46 (1992) 1804. [10] M.S. Dresselhaus, G. Dresselhaus, P. Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, 1996. [11] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, 1998. [12] J.-C. Charlier, Ph. Lambin, Phys. Rev. B 57 (1998) R15037. [13] X. Blase, L.X. Benedict, E.L. Shirley, S.G. Louie, Phys. Rev. Lett. 72 (1994) 1878. [14] O. G€ ulseren, T. Yildrim, S. Ciraci, Phys. Rev. B 65 (2002) 153405. [15] S. Reich, C. Tomsen, P. Ordej on, Phys. Rev. B 65 (2002) 155411. [16] J.A. Strocio, R.M. Feenstra, Scanning Tunneling Microscopy, Academic Press, New York, 1993. [17] A.M. Rao, E. Richter, S. Bandow, B. Chase, P.C. Eclund, K.A. Williams, S. Fang, K.R. Subbaswamy, M. Mennon, A. Thess, R.E. Smalley, G. Dresselhaus, M.S. Dresselhaus, Science 275 (1997) 187. [18] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 61 (2000) 2981. [19] A. Jorio, A.G.S. Filho, G. Dresselhaus, M.S. Dresselhaus, R. Saito, J.H. Hafner, C.M. Lieber, F.M. Matinga, M.S.S. Dantas, M.A. Pimenta, Phys. Rev. B 63 (2001) 245416. [20] W. Andreoni, Annu. Rev. Phys. Chem. 49 (1998) 405. [21] P.Y. Ayala, K.N. Kudin, G.E. Scuseria, J. Chem. Phys. 115 (2001) 9698. [22] J. K€ urti, G. Kresse, H. Kuzmany, Phys. Rev. B 58 (1998) R8869. [23] P. Delaney, H.J. Choi, J. Ihm, S.G. Louie, M.L. Cohen, Phys. Rev. B 60 (1999) 7899. [24] E. Jaffe, Z. Lin, A.C. Hess, Phys. Rev. B 57 (1998) 11834. [25] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865 (E) 78 (1997) 1396. [26] M. Ernzerhof, G.E. Scuseria, J. Chem. Phys. 110 (1999) 5029. [27] C. Adamo, V. Barone, J. Chem. Phys. 110 (1999) 6158. [28] M.J. Frisch et al., Gaussian Development Version, Gaussian, Inc., Pittsburgh, PA, 2002. [29] T.W. Odom, J.-L. Huang, P. Ki m, C.M. Lieber, J. Phys. Chem. B 104 (2000) 2794.
Single wall carbon nanotubes density of states: comparison of experiment and theory Pavel V. Avramov
*,1,
Konstantin N. Kudin, Gustavo E. Scuseria
Department of Chemistry, Center for Biological and Environmental Nanotechnology, Rice University, 6100 Main Street, Houston, TX 77005-1892, USA Received 20 November 2002; in final form 17 January 2003
Abstract We study the electronic structure of a variety of single wall carbon nanotubes and report density of states obtained with the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation and hybrid PBE0 approximation of density functional theory using Gaussian orbitals and periodic boundary conditions. PBE gives very good results for metallic tubes but the addition of a portion of exact exchange in the hybrid PBE0 functional worsens the agreement between experiment and theory. On the other hand, the PBE0 hybrid significantly improves the theoretical predictions (compared to PBE) for semiconducting tubes. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The electronic structure of individual single wall carbon nanotubes (SWNT) is currently the focus of considerable theoretical and experimental interest [1–6]. In SWNTs, a small change in atomic structure (diameter and/or chiral angle) may produce significant changes in electronic properties of the tube. Tight binding calculations predict that SWNTs with indices ðn; mÞ such that ð2n þ mÞ= 3 ¼ k (where k is an integer) should be metals, whereas for k non-integer, the tubes should be semiconductors [7–9]. It has also been shown [10] *
Corresponding author. Fax: +713-348-5155. E-mail address: [email protected] (P.V. Avramov). 1 Permanent address: L.V. Kirensky Institute of Physics SB RAS, Academgorodok, Krasnoyarsk 660036, Russia.
that the energy difference between the first two van Hove singularities (VHS), which reflect the onedimensional (1D) band structure of the tubes and appear as sharp peaks in the density of states (DOS), is a smooth function of the tube diameter. The tight binding (TB) model has been widely used in theoretical studies of SWNTs [11]. TB is an empirical method based on p- and r-states without the self-consistent procedure. Commonly, TB models oversimplify the DOS and give, for example, symmetrical p=r–p =r VHS relatively to the Fermi level [12]. Recent experiments employing scanning tunneling microscopy (STM) in combination with scanning tunneling spectroscopy (STS) [1–5] have confirmed theoretical predictions about the shape and positions of van Hove singularities. However, fine effects such as small gaps in the DOS of
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00113-1
598
P.V. Avramov et al. / Chemical Physics Letters 370 (2003) 597–601
metallic SWNTs [1] are not reproduced by these TB models [7–9]. Density functional theory (DFT) calculations of various types have been applied to predict the electronic structure of SWNTs [13–15]. The most popular approach for calculations with periodic boundary conditions (PBC) is the local density approximation (LDA) in conjunction with planewaves and pseudopotentials [13]. The TB and DFT calculations yield very different predictions for the band gap and positions of VHS, as well as the asymmetry behavior of the VHS relative to the Fermi level. The LDA approach in combination with Gaussian-type basis functions [15] underestimates the separation between the first van Hove singularities up to 25%. The most reliable experimental technique for studying the electronic structure of individual SWNT is the combination of STM and STS [1–5]. The STS method gives a common experimental picture for the valence and conduction bands. Pieces of nanotube soot are usually dispersed in an organic solvent by ultrasound to unravel the nanotube bundles. A droplet of the dispersion is deposited on an inert surface like Au (1 1 1). Imaging of the SWNT is performed in constant current mode at typical parameters of 50 pA for tunneling current and 0.1 V for bias voltage. In order to determine the experimental DOS of individual nanotubes, current–voltage tunnel spectra are taken by recording the current as a function of the bias voltage, while keeping the STM tip at a fixed position. A set of I–V measurements can then be done in one tube with different positions of the STM tips on the SWNT surface [3]. Although some small differences can be detected in the I–V curves of a nanotube, the conductance dI=dV V or normalized conductance ðV =IÞðdI=dV Þ V curves can be numerically calculated from I–V spectra and have been reported as an experimental DOS for the SWNTs. The normalized conductance ðV =IÞðdI=dV Þ V provides a better measure of the DOS in comparison with the conductance dI=dV V [16]. Resonant Raman scattering [17–19], which is based on analysis of stokes and anti-stokes Raman processes, can probe the electronic dispersion indirectly via absorption from the valence to conduction states.
Even though these experimental results depend on many factors such as defects in the SWNT atomic structure and the interaction between the tube and the surface, it is possible to compare the experimental DOS with those obtained directly from first-principles computations. As with fullerenes like C60 , the Hartree–Fock method fails to describe the experimental energy gap [20]. In C60 , second-order perturbation theory (MP2) can significantly improve results for ionization potentials and electronic affinities [20]. However, the computational cost of MP2 with PBC remains substantial [21]. On the other hand, DFT is significantly faster and gives relatively accurate results for SWNTs [13,15,22,23], especially for metals [24]. In this work, we carry out a systematic comparison of experimental and theoretical DOS of individual nanotubes with different diameters and chiral angles using for the first time hybrid functionals, i.e., those which contain a portion of exact HF exchange. We have calculated the optimized geometries and electronic structures of a number of zigzag, armchair, and chiral SWNTs with the PBE [25] and hybrid PBE0 [26,27] functionals using the development version of the GA U S S I A N suite of programs [28], which allows for DFT calculations with periodic boundary conditions [6] and geometry optimizations using analytic energy gradients. Equilibrium structures (including both atomic positions and lattice dimensions) of the zigzag and armchair SWNTs were obtained with the PBE functional using a 6-31G basis set. For the chiral tubes, PBE/3-21G was used in the geometry optimization because of their large unit cell sizes. The number of atoms in the unit cell varies from 32 for the (8,8) tube (480 basis functions) to 186 for the (13,7) SWNT (2,790 basis functions). Once the geometry was optimized with this procedure, the electronic structure (DOS) was calculated using a polarized 6-31G* basis set consisting of 4s2p1d basis functions on each carbon atom, and 128 points in k-space for the Brillouin zone integration. Plots of the calculated DOS were obtained using a broadening energy parameter of 0.1 eV. The following tubes were explicitly considered: metallic armchair (8,8), metallic zigzags (9,0),
P.V. Avramov et al. / Chemical Physics Letters 370 (2003) 597–601
(12,0), and (15,0), semiconducting zigzag (10,0), metallic chiral (13,7), and semiconducting chiral (14,3). The theoretical DOS are compared with the STS experimental results [1,3,5,29] in Figs. 1–5. In Fig. 1, the STS experimental (solid line, [29]) normalized conductance ðV =IÞðdI=dV Þ V and theoretical DOS obtained with the PBE and PBE0 functionals (dashed lines) for the (10,0) zigzag semiconducting SWNT are presented. The experimental energy difference between the first VHS is 1.1 eV, whereas the theoretical PBE result is 0.8 eV. The experimental energy difference between the second VHS is 3.0 eV but only 1.9 eV with PBE. On the other hand, the hybrid PBE0 functional gives much better results for the first and the second VHS transitions for the (10,0) tube: 1.1 and 2.9 eV, respectively. In Fig. 2a, the STS experimental normalized conductance ðV =IÞðdI=dV Þ V [3] and theoretical
599
Fig. 2. The STS experimental normalized conductance ðV =IÞðdI=dV Þ V [3] (a) and theoretical (b) PBE and PBE0 DOS of semiconducting chiral (14,3) SWNT.
Fig. 3. The STS experimental conductance dI=dV V [1] (solid line) and theoretical (dashed lines) PBE and PBE0 DOS of metallic zigzag (15,0), (12,0) and (9,0) SWNTs.
Fig. 1. The STS experimental normalized conductance ðV =IÞðdI=dV Þ V [29] (solid lines) and theoretical (dashed lines) PBE and PBE0 DOS of semiconducting zigzag (10,0).
in Fig. 2b, DOS with the PBE and PBE0 functionals for the (14,3) chiral semiconducting SWNT are presented. The experimental energy difference between the first VHS is 0.9 eV, whereas the
600
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Fig. 4. The STS experimental conductance dI=dV V [5] (solid line) and theoretical (dashed line) PBE DOS of metallic chiral (13,7) SWNT.
Fig. 5. The STS experimental conductance dI=dV V [1] (solid line) and theoretical (dashed lines) PBE and PBE0 DOS of armchair (8,8) SWNT.
theoretical PBE value is only 0.6 eV. Here again, PBE0 yields much better results for the first VHS transition: 0.8 eV. In Fig. 3, the STS experimental conductance dI=dV V (solid lines, [1]) and theoretical DOS calculated with the PBE and PBE0 functionals (dashed lines) for metallic zigzag (15,0), (12,0), and (9,0) SWNTs are presented. The experimental energy differences between the first VHS are 1.7, 2.1, and 2.6 eV for the three tubes in Fig. 3, respectively. The calculated PBE values are 1.6, 1.8, and 2.5 eV, in very good agreement with experiment, respectively. For the (9,0) tube, the PBE0 result for the first VHS transition (3.1 eV) is in poor agreement with the experiment (1.7 eV). Similarly
low-quality predictions were obtained with PBE0 for the other metallic tubes in Fig. 3 and are not presented. Also in Fig. 3, the experimental energy differences between the second VHS are 1.9 and 2.4 eV for the (15,0) and (12,0) SWNTs, respectively. There are no experimental data for the (9,0) tube. The theoretical PBE results are 2.1 and 2.4 eV for the (15,0) and (12,0) tubes, respectively. Here again, the agreement between experiment and theory is quite good. It should also be mentioned that in the case of the metallic zigzag nanotubes, the PBE approximation seems to reproduce peculiarities of the experimental DOSlike small secondary gaps (0.1 eV width, energy 0 eV). In Fig. 4, the experimental conductance dI=dV V (solid line, [5]) and theoretical DOS (dashed line) for the (13,7) chiral metallic tube are presented. The experimental energy difference between first VHS is 1.6 eV. The theoretical PBE value is only 0.2 eV smaller (1.4 eV). The energy difference between the second VHS is roughly 2.0 eV in experiment, and 1.9 eV with the PBE functional. Finally, in Fig. 5, the STS experimental conductance dI=dV V (solid line, [1]) and theoretical (dashed lines) DOS for the (8,8) armchair tube are presented. The energy difference between the first VHS in the experimental STS DOS is 1.5 eV, whereas the theoretical PBE result is 2.0 eV. For the second VHS, the experimental energy difference is 2.8 eV and the PBE prediction is 3.3 eV. For this metallic tube, addition of exact exchange in the PBE0 functional worsens the comparison between experiment and theory (see Fig. 5). The energy separations become 2.7 and 4.6 eV, for the first and second VHS, respectively. Based on the comparison of the theoretical data with STS experimental results, we conclude that the PBE functional describes fairly well the density of electronic states of metallic nanotubes and gives good qualitative agreement for the semiconducting ones. Addition of a portion of exact exchange, as implemented in the hybrid PBE0 functional, substantially improves the agreement between experiment and theory for the semiconducting tubes but worsens this agreement for the metallic ones.
P.V. Avramov et al. / Chemical Physics Letters 370 (2003) 597–601
The good agreement between experiment and theory for the DOS of metallic SWNTs is not surprising given that PBE is usually deemed as a good approximation for metals. The PBE functional correctly reproduces the uniform electron gas limit, which is a metal itself. On the other hand, for systems with a finite band gap, PBE (as well as many other functionals including LDA) predicts band gaps which are too narrow. The Hartree–Fock method, which contains 100% exact exchange and 0% correlation, tends to spatially localize spins, and overemphasizes the insulating character of materials. Thus, addition of a portion of exact Hartree–Fock exchange to the functional (as in PBE0) widens these gaps thus bringing the theoretical results in better agreement with the experiment.
Acknowledgements This work was supported by the National Science Foundation and the Welch Foundation. PVA also acknowledges support by Grant No. 31 of the Federal Russian Program ÔIntegrationÕ.
References [1] M. Ouyang, J.-L. Huang, C.L. Cheung, C.M. Lieber, Science 292 (2001) 702. [2] L.C. Venema, J.W. Janssen, M.R. Buitelaar, J.W.G. Wild€ oer, S.G. Lemay, L.P. Kouwenhoven, C. Dekker, Phys. Rev. B 62 (2000) 5238. [3] T.W. Odom, J.-L. Huang, P. Kim, C.M. Lieber, Nature 391 (1998) 62. [4] J.W.G. Wild€ oer, L.C. Venema, A.G. Rinzler, R.E. Smalley, C. Dekker, Nature 391 (1998) 59. [5] P. Kim, T.W. Odom, J.-L. Huang, C.M. Lieber, Phys. Rev. Lett. 82 (1999) 1225. [6] K.N. Kudin, G.E. Scuseria, Phys. Rev. B 61 (2000) 16440. [7] J.W. Mintmire, B.I. Dunlap, C.T. White, Phys. Rev. Lett. 68 (1992) 631.
601
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