Article pubs.acs.org/JPCA
Extended Hü ckel Theory for Carbon Nanotubes: Band Structure and Transport Properties Andreas Zienert,*,† Jörg Schuster,‡ and Thomas Gessner†,‡ †
Center for Microtechnologies, Chemnitz University of Technology, 09107 Chemnitz, Germany Fraunhofer Institute for Electronic Nano Systems, 09126 Chemnitz, Germany
‡
ABSTRACT: Extended Hückel theory (EHT) is a well established method for the description of the electronic structure of molecules and solids. In this article, we present a set of extended Hückel parameters for carbon nanotubes (CNTs), obtained by fitting the ab initio band structure of the (6,0) CNT. The new parameters are highly transferable to different types of CNTs. To demonstrate the versatility of the approach, we perform self-consistent EHT-based electron transport calculations for finite length CNTs with metal electrodes.
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INTRODUCTION Carbon nanotubes (CNTs) are atomically well-defined onedimensional cylinders of carbon. Their excellent thermal, mechanical, and electrical properties make them an ideal material for a multitude of possible applications,1 including electrical conductors (microelectronic interconnects),2−5 transistors,5−8 and sensors.9−11 Of major importance for the development of CNT-based devices is an accurate, reliable, and computationally cheap description of their electronic properties, taking into account the details of the tubes atomic structure. Among the most successful approaches are density functional theory (DFT) and the tight binding (TB) approximation. The former, being an ab initio method, is computationally expensive, but does not need any experimental input and is sufficiently accurate for the description of CNTs.12,13 The TB approximation is computationally cheap, but a good parametrization of the chemical elements and their interaction within the studied structure is needed. The transferability of sets of TB parameters is often limited to very similar materials. Extended Hückel theory is a well-known semiempirical quantum chemistry method.14 In contrast to TB, where the entire Hamiltonian matrix is composed of empirical parameters, explicit numerical orbitals are employed for the construction of overlap matrices, from which the off-diagonal Hamiltonian matrix elements are constructed via the Hückel principle. Only the shape of those orbitals is determined by a few empirical parameters. The explicit use of orbitals ensures a natural distance dependency of the electronic interaction of atoms and enables (to some degree) the description of interfaces between different materials. The present article demonstrates an extended Hückel theory, tailored for the description of carbon nanotubes. Extended Hückel (EH) parameters for carbon are fitted to reproduce the DFT band structure of a (6,0) CNT. A self-consistent variant of the EHT is used for quantum transport calculations in a model © XXXX American Chemical Society
system, a CNT-based resonant tunneling device, demonstrating the versatility of the new parameters.
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THEORETICAL FRAMEWORK AND NUMERICAL DETAILS Self-Consistent Extended Hü ckel Theory. In EHT, the electronic structure is expanded in a basis of atom centered Slater type orbitals (STOs) φnlm(r) = R nl(r )Ylm(r)
(1)
where n, l, m are the principal, angular, and magnetic quantum numbers. Ylm(r) are the spherical harmonics. We adopt the parametrization by Cerdá, describing the chemical elements by double-ζ spd basis sets (two STOs per atomic orbital).15 Thus, the radial parts Rnl(r) are R nl(r ) = C1R1(nl)(r ) + C2R 2(nl)(r )
(2)
with R i(nl)(r ) =
r n−1 (2n)!
(nl)
(2ηi(nl))n + 1/2 e−ηi
r
(3)
This leaves three adjustable parameters η1, η2, and C1. The parameter C2 is fixed by imposing the normalization of the wave function: C2 = −
p + 2
p2 − C12 + 1 4
(4)
with Received: December 20, 2012 Revised: March 22, 2013
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orbitals of carbon.15 Thus, an EHT of carbon, as described above, needs 10 fitting parameters. Density Functional Theory. The reference band structures of CNTs are calculated by DFT in the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) using the Software package ATK.17,18 The electronic structure is expanded in a double-ζ plus polarization (DZP) basis set of SIESTA-type numerical atom centered orbitals.21 Core electrons are described by standard pseudopotentials.22,23 The one-dimensional Brillouin zone of the CNT is sampled by 100 k-points, while only 25 points are used in the EHT for the fitting. Fitting Procedure. The central part of the present work consists of finding a set of EH parameters for CNTs so that the resulting band structure closely resembles the one obtained by DFT. Thus, a rather complex fitting approach is needed, which can be divided in three tasks. We use the non-self-consistent form of EHT, omitting the term HSCF in eq 7, because there is ij no charge transfer between the atoms in an ideal CNT. First, an EHT band structure calculation is performed for a given set of EH parameters (using the set of Cerdá as initial guess). Second, a measure f fit (see eq 10) for the deviation between the new EHT and the DFT band structure is calculated. Finally, a fitting algorithm is applied, which gives a new set of EH parameters. Those steps are repeated until f fit is minimized. There exists a multitude of robust and fast fitting algorithms, most of which, however, rely on the knowledge of derivatives of f fit with respect to the free parameters. We choose the Simplex algorithm by Nelder and Mead,24 which does not need such information. We have tested various forms of the measure f fit. It turned out that eq 10 produces good results:
4n + 1C1(η1η2)n + 1/2 (η1 + η2)2n + 1
(5)
The elements of the Hamiltonian H and overlap matrix S are calculated from the orbitals via the Hückel principle (orbital i is centered at Ri): ⎧ δij Ri = Rj ⎪ Sij = ⎨ ⎪ φi*(r − R i)φj(r − R j)d3r R i ≠ R j ⎩
(6)
⎧ E ion + H SCF + Evac i = j ij ⎪ i Hij = ⎨ ⎪(Wij + HijSCF)Sij i≠j ⎩
(7)
∫
with Wij =
1 (β + βj)(Eiion + Ejion) 4 i
(8)
and HijSCF =
1 (δVH(R i) + δVH(R j)) 2
(9)
The ionization potentials E are fitting parameters specific for each orbital. All weighting parameters are chosen as β = 2.8.15 The term HSCF is missing in the original non-self-consistent ij EHT. It has been introduced by Stokbro et al. to account for the induced Hartree potential δVH, which follows from an approximate solution of Poisson’s equation.16 We use the selfconsistent EHT as implemented in the software package ATK.17,18 To combine EH parameters of different chemical elements, it must be ensured that their energy scales are properly aligned. In practice, this is done by introducing the parameter Evac, which rigidly shifts the electronic band structure (change of the vacuum level). Values of Evac can be determined by requiring some quantities like work functions or ionization energies to match experimental values. The EH parameter set of Cerdá,19,20 used as a starting point for the present work, is listed in the upper part of Table 1. For the s- and d-orbitals, a single STO is adequate, allowing to set C2 = 0 and η2 = ∞, while a double-ζ basis is used for the pion
f
Eion [eV]
l
C1 [a0−3/2]
(E = −7.3658 eV) (a) Cerdá 2 0 −19.8892 0.76422 2 1 −13.08 0.27152 3 2 −2.04759 0.49066 (b) Present work: EHT* (Evac = −9.4392 eV) 2 0 −21.6084 0.70777 2 1 −14.2883 0.034603 3 2 −5.78956 0.382108 19,20
η1 [a0−1]
η2 [a0−1]
2.0249 1.62412 1.1944
∞ 2.17687 ∞
2.01357 1.192 1.22177
∞ 2.06477 ∞
max Nbands
=
∑ ∑ i = 0 k = k Γ, k z
EiEHT(k) − EiDFT(k) EiDFT(k)
2
2
(10)
The energetic differences between DFT and EHT band structures are summed band by band up to a number Nmax bands chosen such that only bands with E(k) ≲ EF + 7 eV are taken into account. High lying bands are not important, and their accuracy is questionable anyway. Band structures are always calculated with the respective Fermi energy set to zero. Thus, the special form of eq 10 ensures that deviations of the bands near the Fermi energy are weighted more strongly. Only the center (kΓ) and the edge (kz) of the Brillouin zone are included in the fitting. The vacuum level Evac is determined such that the Fermi level, calculated with the new parameters, matches the value obtained from Cerdá’s parameters.
Table 1. EH Parameters of Cerdá, Based on the DFT-GGA Band Structure of Graphite (a); EH Parameters of the Present Work, Based on the DFT-GGA Band Structure of a (6,0) CNT (b)a n
fit
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vac
RESULTS AND DISCUSSION The new set of EH parameters is listed in Table 1b. Note the small value of C1 for the 2p orbitals. This indicates that an equivalent single-ζ spd basis set could be developed, based on our parameters. In the current case, given the small C1, the exact value of η1 plays a minor role for the 2p shell (apart from its contribution to the normalization of the wave function). All following results are obtained with the self-consistent version of EHT for both sets of parameters. Figure 1 shows the band structure of the (6,0) CNT, the one for which the new parameters have been fitted. Compared to the parameters of Cerdá, the new band structure (EHT*) is in much better agreement with the DFT results, especially near the Fermi level.
a
For each of the three orbitals 2s, 2p, and 3d, the values of the ionization potential Eion, the orbital weighting factor C1, and the two Slater coefficients η1 and η2 are given. The values of C2 are fixed by imposing the normalization condition upon the orbitals. The vacuum level Evac ensures consistency among different EH parameter sets by rigidly shifting the band structures. B
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EHT*, as expected. More importantly, the EHT* does not perform worse, which demonstrates the good transferability of the parameter method. This is emphasized in Figure 3 by comparing band structures of four different tubes: the semimetallic (9,0) zigzag CNT, the metallic (10,10) armchair CNT, the semimetallic (8,2) chiral CNT, and the semiconducting (6,2) chiral CNT. In any case, the agreement with DFT is equal or better.
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APPLICATION: ELECTRON TRANSPORT THROUGH A FINITE CNT WITH CNT ELECTRODES To demonstrate the utility of the developed EHT, we perform electron transport calculations using the simple model system Figure 1. Band structures of the (6,0) CNT (see also Figure 5a).
It is well-known that the curvature of small diameter CNTs causes a hybridization of π*- and σ*-states, which results in an overlap of bands at the Fermi energy.12 This curvature induced hybridization is strongly underestimated by the EHT of Cerdá because it has been derived from the band structure of graphite (or graphene). Thus, it performs better for large diameter tubes. The limiting case of graphene, which has zero curvature, is shown in Figure 2. No significant improvement is visible for
Figure 4. Our simple model of a CNT-based conductor consists of a finite-sized (6,0) CNT (conductor C) with semi-infinite CNT electrodes (L and R) of the same geometry. The length of the conductor is LC = 19.89 Å (NC = 5 unit cells). Imperfect contacts (tunnel barriers) are introduced by contact distances d.
depicted in Figure 4. Five unit cells of a (6,0) CNT (total length LC = 19.89 Å) are connected to semi-infinite CNT electrodes via tunnel contacts characterized by a contact distance d = dCC + Δd, where Δd is the deviation from the ideal C−C separation dCC ≈ 1.42 Å. For given bias voltages Vb, assuming ballistic transport (neglect of incoherent scattering), the current through such a device is given by the Landauer−Büttiker formula 2e I(Vb) = − (fL (E , Vb) − fR (E , Vb))T (E , Vb)dE h
∫
(11)
The electron distributions within the left and right electrode are described by Fermi functions f L and f R, shifted so that the difference between their chemical potentials equals Vb. The transmission spectrum T(E,Vb), which is the sum of the transmission probabilities of available conduction channels at energy E for a given Vb, is calculated within the non-equilibrium Green’s function formalism (NEGF) implemented in ATK.17,18
Figure 2. Band structure of graphene. The strong deviations from DFT at large energies are due to the neglect of those bands in the fitting procedure.
Figure 3. Band structures of some CNTs. C
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Figure 5. Band structure (a), DOS (b), T(E) at zero bias (c), and T(E) at Vb = 0.1 V (d) for the (6,0) CNT. The single degenerate π*-states, which result from the finite curvature of the tube, are marked in panel a. In panel d, the Fermi levels of the left and right electrode are indicated.
that the (6,0) CNT is metallic. This is in agreement with the work of Matsuda et al.,13 who have also shown that DFT with the Becke−Lee−Yang−Parr (B3LYP) functional27,28 is able to reproduce the experimental band gaps very well. In Figure 5, the band structure, density of states (DOS), and transmission spectra (for zero bias and low bias Vb = 0.1) of the ideal (6,0) CNT (Δd ≈ 0) are shown. The DOS exhibits characteristic van Hoove singularities at subband edges. The transmission of the ideal CNT at zero bias equals the number of states at a given energy. Both DFT and our new EHT* predict T(E) = 3 over a broad range of energies around EF. This is a consequence of a single degenerate π*-state shifted below EF due to the curvature-induced σ*−π* hybridization. The underestimation of this effect by the EHT of Cerdá results in a very narrow T(E) peak at EF, which is turned into a dip when a finite bias is applied. In Figure 6, the low bias current is shown as a function of the contact distance, measured by the deviation Δd from the ideal case (Figure 5d). The EHT* results are in fair agreement with DFT, especially in the case of good contacts. Some transmission spectra that are used for the I(Δd) curves are depicted in Figure 7. As the contact distance is increased, the device is gradually turned from the regime of quasi-ballistic transport toward resonant tunneling, where the electron flow is governed by discrete states of the finite central CNT, which are shifted and broadened due to the presence of the semi-infinite electrodes. This distance dependency of the influence of the electrodes is automatically taken into account by the EH method. In TB-based formalisms, it must be included manually in the parametrization. From the good agreement between DFT and the new EHT* in Figure 7 it can be concluded that EHT* also improves the description of finite-sized CNTs. The remaining discrepancies may be due to the different treatment of the Hartree potential
Figure 6. Low bias (Vb = 0.1 V) current as a function of the contact distance (d = dCC + Δd).
We choose a low bias voltage Vb = 0.1. In a previous paper, we have stressed the importance of underlying electronic structure theories for quantum transport simulations, comparing TB, Cerdá’s EHT, and DFT.25 Therein, a more detailed explanation of the Green’s function formalism is given, too. Kienle, Cerdá, and Gosh have performed a similar study of electron transport properties of various CNTs.20 They compare the spd parameters (Table 1a) and a more simple sp parametrization within the framework of non-self-consistent EHT. The sp parametrization is recommended for the description of CNTs because it reproduces small band gaps of semimetallic zigzag CNTs. Such tiny band gaps, which are a result of the curvature of the tubes, are observed in scanning tunneling spectroscopy experiments on (9,0), (12,0), and (15,0) CNTs.26 For those CNTs (the (9,0) tube is shown in Figure 3), our parametrization predicts small curvature-induced band gaps, but they are overestimated compared to the experimental values. For the (6,0) CNT, there is no experimental data available. In contrast to the sp parametrization of Kienle et al., DFT and our new EHT* predict
Figure 7. Low bias (Vb = 0.1 V) transmission spectra of the CNT-based model device in Figure 4 for different contact distances. D
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within the self-consistent extension16 of EHT as compared to DFT.
(15) Cerdá, J. I.; Soria, F. Phys. Rev. B 2000, 61, 7965−7971. (16) Stokbro, K.; Petersen, D. E.; Smidstrup, S.; Blom, A.; Ipsen, M.; Kaasbjerg, K. Phys. Rev. B 2010, 82, 075420. (17) Brandbyge, M.; Mozos, J.-L.; Ordejón, P.; Taylor, J.; Stokbro, K. Phys. Rev. B 2002, 65, 165401. (18) Atomistix ToolKit 12.2.0. QuantumWise A/S. www. quantumwise.com. (19) Cerdá, J. I. Extended Hückel Theory (EHT) Parameters for Some Elements and Compounds. http://www.icmm.csic.es/jcerda/ EHT_TB/TB/Periodic_Table.html. (20) Kienle, D.; Cerdá, J. I.; Ghosh, A. W. J. Appl. Phys. 2006, 100, 043714. (21) Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P.; Sánchez-Portal, D. J. Phys.: Condens. Matter 2002, 14, 2745−2779. (22) Troullier, N.; Martins, J. L. M. Phys. Rev. B 1991, 43, 1993− 2006. (23) Kleinman, L.; Bylander, D. M. Phys. Rev. Lett. 1982, 48, 1425− 1428. (24) Nelder, J. A.; Mead, R. Comput. J. 1965, 7, 308−313. (25) Zienert, A.; Schuster, J.; Streiter, R.; Gessner, T. Microelectron. Eng. 2013, DOI: 10.1016/j.mee.2012.12.018. (26) Ouyang, M.; Huang, J.-L.; Cheung, C. L.; Lieber, C. M. Science 2001, 292, 702−705. (27) Becke, A. D. J. Chem. Phys. 1993, 98, 5648−5652. (28) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623−11627.
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CONCLUSIONS We have presented a set of EH parameters for carbon nanotubes based on DFT band structure calculations. The new parameters are well suited for the description of the electronic structure of graphene and all kinds of CNTs (both finite-sized and infinite ones). The simplicity of the EHT (and its self-consistent extension) reduces the simulation time of advanced computations, which need accurate electronic structures as a prerequisite. For the demonstrated quantum transport calculations, EHT is about an order of magnitude faster compared to DFT, while preserving the overall accuracy due to the use of the new parameters. The new EHT will be subsequently used to study quantum transport in CNTs with more realistic metal electrodes. Preliminary results indicate a good performance of the new carbon parameters combined with existing parameters of different metals. This will be the subject of a future publication.
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AUTHOR INFORMATION
Corresponding Author
*(A.Z.) E-mail:
[email protected]. Phone: +49 (0)371 531 37937. Fax: +49 (0)371 531 837937. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS A.Z. acknowledges financial support from the German Research Foundation under the International Research Training Group GRK 1215.
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