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Abstract
Using first-principles calculations, we investigate the effect of tube size on optical properties of the zigzag, armchair, and chiral SiC nanotubes. The results indicate that the optical spectra of SiC nanotubes are dependent on the diameter and chirality, and that optical anisotropy is observed for different light polarizations. For a given chirality of SiCNTs, redshifts or blueshifts of the peaks in the dielectric function and energy loss function with increasing tube diameter are possible due to the competition between the size effect and π orbitals overlapping, and the shifts become smaller as the tube diameter increases. The unusual optical properties of semiconducting SiC nanotubes present an opportunity for applications in electro-optical devices.
©2007 Optical Society of America
1. Introduction
Since the discovery of carbon nanotubes (CNT) [1] in 1991, the fabrication of nanometer-scaled one-dimensional materials has attracted considerable interests due to their potential use in both nanotechnology and nanoscale engineering. The successful synthesis of nanotubes made from boron nitride [2–4] has opened up the possibility of the existence of tubular structures made of either noncarbon or part carbon materials. In 2002, Sun et al. [5] synthesized one-dimensional silicon-carbon nanotubes (SiCNTs) via the reaction of silicon (produced by disproportionation reaction of SiO) with multiwalled carbon nanotubes (as templates) at different temperatures. SiCNTs are expected to have advantages over CNTs because they may possess high reactivity of exterior surface facilitating sidewall decoration and stability at high temperature. [6] And SiCNTs are predicted to be more suitable materials for hydrogen storage than pure CNTs at theoretical levels. [7] The structure and stability of
SiC single wall nanotubes have been investigated by ab initio theory in detail. [8] It was found that the SiCNTs with alternating Si-C bonds are energetically preferred over the forms which contain C-C or Si-Si bonds. [8] Zhao et al., using density functional theory (DFT), studied the electronic structure of SiCNTs, suggesting that all SiCNTs are wide band-gap semiconductors. [6] Theoretical studies have also been reported on the structural and electronic properties of native defects [9], substitutional impurities [10], the sidewall hydrogenation [11] in SiCNTs, and SiCNTs decorated by CH3, SiH3, [12] N and NHx (x=1,2) [13] groups. However, there are not theoretical and experimental studies addressing the optical properties of SiCNTs. Because of one-dimensional character, CNTs have unusual optical properties. [14–16] What about the SiCNTs? Therefore, precise predictions for the fundamental optical properties of SiCNTs are strongly desirable. In this work, we investigate the optical properties of the zigzag (n, 0), armchair (n, n), and chiral (n, m) SiCNTs with first-principles calculations.
2. Theoretical method and computational details
The calculations use the total-energy code CASTEP, [17, 18] which employs pseudopotentials to describe electron-ion interactions and represents electronic wave functions using a plane-wave basis set. The coordinates of all the atoms in the SiCNTs were optimized without any symmetry constraint using Broyden-Fletcher-Goldfarb-Shanno [19] scheme, Ultrasoft pseudopotentials [20, 21] and local-density approximation (LDA). The total energy and properties are calculated within the framework of the Perdew-Burke-Ernzerhof generalized gradient approximation (GGA-PBE). [22] The interactions between the ionic cores and the electrons are described by the norm-conserving pseudopotentials. [23] The orbital electrons of C-2s22p2 and Si-3s23p2 are treated as valence electrons. We have used approximately 10 Å vacuum in the lateral directions to avoid artificial tube-tube interaction. This should be a sufficient distance since the basis functions do not overlap. The supercells of the armchair and zigzag SiCNTs contain four layers of atoms along the tube axis (z axis). The unit is periodic in the direction of the tube. Considering the balance of the computational cost and precision, we choose a cutoff energy of 600 eV and a 1×1×4 k-point set mesh for SiCNTs.
The linear response of the system to an external electromagnetic field with a small wave vector is measured through the complex dielectric function ε(ω). ε(ω) is connected with the interaction of photons with electrons. The real part and imaginary part of ε(ω) are often referred to as ε1(ω) and ε2(ω), respectively. ε>2(ω) can be thought of as detailing the real transitions between occupied and unoccupied electronic states. The imaginary part ε2(ω) of the dielectric function ε(ω) is given by equation:
where Ecv(k)=Ec(k)-Ev(k). Here, fc and fv represent the Fermi distribution functions of the conduction and valence band. The term picv(k) denotes the momentum matrix element transition from the energy level c of the conduction band to the level v of the valence band at the kth point in the BZ, and Veff is the effective unit cell volume.
The real part ε1(ω) of the dielectric function ε(ω) follows from the Kramer-Kronig relationship. All the other optical constants may be derived from ε1(ω) and ε2(ω). [24, 25] For example, the loss function L(ω) can be calculated using the following expression:
The loss function L(ω) is an important optical parameter describing the energy loss of a fast electron traversing in the material. The peaks represent the characteristic associated with the plasma oscillation and the corresponding frequencies are the so-called plasma frequencies. [26]
It is noted here that the orbital plots given below are calculated by the DMol3 module based on the optimized geometrical structures from CASTEP code. For the calculations with DMol3, we have used generalized gradient approximation functional in the manner suggested by Perdew-Burke-Ernzerhof, and Double Numerical plus d-functions (DND) basis set.
Fig. 1. Optimized cells of (a) (12,0), (b) (6,6) and (c) (6,3) SiCNTs. Gray and yellow balls represent carbon and silicon atoms, respectively.
3. Results and discussion
We first obtained the equilibrium configurations of SiCNTs with different diameters and chiralities, where the Si and C atoms are placed alternatively without any adjacent Si or C atoms. The optimized cells of zigzag (12,0), armchair (6,6) and chiral (6,3) SiCNTs are shown in Fig. 1. The calculated average Si-C bond length of these tubes is about 1.775 Å in agreement with the theoretical results in Ref. 10 and Ref. 27, but smaller than the results in Ref. 6 and Ref. 8, and shorter than the length of the bulk SiC phases such as cubic(3C), hexagon (4H, 6H), or rhombohedral (15R) structures, [28] etc. After the relaxation, Si atoms move toward the tube axis and C atoms move in the opposite direction, which is in accordance with the results of Ref. 6.
Table 1. Band gaps and static dielectric functions for SiCNTs. The d and ind in parentheses denote direct and indirect band gap semiconductors, respectively.
Before discussing the optical properties, we will simply describe the electronic structures of the SiCNTs. Table 1 lists the calculated band gaps of SiCNTs, and it shows that the zigzag SiCNTs are direct-gap semiconductors, while the armchair and chiral tubes are indirect-gap ones. The experimental gaps should be larger than our calculated ones due to the well known fact that electronic structure calculations within GGA generally underestimate the energy gap. For the zigzag SiCNTs, the band gap increases with the tube diameter, which is due to smaller curvature leading to smaller π-σ hybridizations and larger repulsions between π and π * states. However, for the armchair ones, the band gap increases with the odd or even n. This result is not consistent with the previous study [6], which reported that the band gap of armchair SiCNTs increases monotonically with the tube diameter. This discrepancy may be due to the different optimization algorithms used and thus the different Si-C bond lengths. For a given n, the (n,n) tube has a bigger band gap than the (n,0) tube because it has a larger radius. The ratio between the (n,n) and (n,0) tube radius is √3, independent of n. The orbitals localized at G (0, 0, 0) point of the top most valence band (HOMO) and lowest conduction band (LUMO) of (12, 0), (6, 6), and (6, 3) tubes as the representatives of zigzag, armchair, and chiral tubes are plotted in Figs. 2(a)–2(b). It is found that the HOMO is constructed by an anti-phase p-orbital interatctions between the two neighboring carbon atomes along the tube radius.
Fig. 2. The orbitals localized at G (0, 0, 0) point of (a) the top most valence band (HOMO) and (b) the lowest conduction band (LUMO) of (12, 0), (6, 6), and (6,3) tubes (The absolute values of the isosurfaces of the wavefunctions are 0.03.). The variations of the energies of the top of HOVB and the bottom of LUCB as a function of n for (c) zigzag and (d) armchair SiCNTs.
The smaller is the n value; the larger is the curvature and the stronger is the anti-phase interactions of the SiCNTs. Accordingly, the HOMO energy decreases as the SiCNT radius increases. The LUMO is formed by nonbonding or weak antibonding interactions between Si atoms. Figures 2(c)–2(d) give the plots of the energies of the top of the highest occupied valence band (HOVB) and the bottom of the lowest unoccupied conduction band (LUCB) against the n values from the CASTEP calculations. It is shown that the energy of the top of HOVB decreases as the SiCNT radius increases, whereas the variations of tube radius, except for the smallest n value, have not substantial effects on the energy of the bottom of LUCB, in particular for armchair SiCNTs. These results tell us why the band gap between the top most valence band and lowest conduction band increases with the SiCNT radius.
The band structures, total density of states (DOS), and partial DOS (PDOS) projected on the constitutional atoms for the zigzag (12, 0), the armchair (6, 6) and the chiral (6, 3) tubes are plotted in Fig. 3. Here, we must note that the energy of HOVB is taken as a reference and always set to zero in drawing band structures. The band structures of (12,0) and (6,6) tubes are more oscillating than that of the (6,3) tube, which indicates that the states of (6,3) tube are
more localized. For (12,0) and (6,6) tubes, the degeneracy of energy levels at Z (0, 0, 0.5) point is higher than that of energy levels at G (0, 0, 0) point. The DOS and PDOS are similar for all SiCNTs being studied. The regions below the Fermi level (the Fermi level is set at the top of the valence band) can be divided into two regions. The bottom valence-band region are mainly composed of C-2s states, with a non-negligible contribution from Si-3s,3p states. The top of the valence-band region mainly arises from C-2p states with a small mixing of Si-3s,3p
states. The conduction bands above the Fermi level are mainly due to Si-3p states.
Fig. 3. The band structures and density of states for the zigzag (12, 0), the armchair (6, 6) and the chiral (6, 3) tubes.
The zero frequency dielectric constants εx(0)and εz(0) of SiCNTs are also given in Table I. It is noted here that no scissors-operator shift is applied for the calculations of optical properties of SiCNTs. As the diameter of SiCNTs increases, the zero frequency dielectric constants εx(0)and εz(0) decrease monotonically. The same phenomenon happens in single-walled BN nanotubes. [29] For a given SiC nanotube, εz(0) is larger than εx(0), i.e. the zero frequency dielectric constant along the tube axis is larger than that perpendicular to the tube axis. It appears that the optical anisotropy becomes smaller as the diameter becomes larger. Figure 4 shows the dielectric function and loss function under different polarizations for chiral (6, 3) tube. The peaks under parallel polarization (E‖z) are stronger than the ones under perpendicular polarization (E,z), because the optical transition probability for parallel polarization is about half of that for perpendicular polarization. And the ε2‖ is larger than the ε2⊥at almost all frequencies. The first peak below 5.0 eV of dielectric function and loss function under parallel polarization is red-shifted when compared with the one under perpendicular polarization. The prominent peak at ~12.5 eV of loss function under parallel polarization is slightly blue-shifted when compared with the one under perpendicular polarization.
Fig. 4. (a). The imaginary part of dielectric function, and (b) the loss function under different polarizations for chiral (6, 3) tube. The inset is the dispersion of the loss function at 0 ~11 eV.
Figure 5 shows the imaginary parts of the dielectric functions of armchair and zigzag SiCNTs when the electric field of the light is parallel to the tube axis. For both armchair and zigzag SiCNTs, the spectra can be divided into two regions, namely, the low-energy range from 0 to 5 eV and the high-energy range from 5 to 20 eV. The peak at the low-energy region is stronger than the one at the high-energy region. For armchair SiCNTs, the first peak is located at about 3.1 eV, which is mainly due to the electronic transitions from C-2p bonding orbitals to Si-3p nonbonding orbitals (inter-π band transition). This peak is slightly blueshifted (from 3.05 to 3.22 eV) with increasing tube diameter (or n from 3 to 10). The second peak originated from the π band to π * band transition is at about 7.1 eV, which shifts (6.582→6.930→7.218→7.258→7.263→7.207→7.213→7.158 eV) with the increase of the tube diameter. For zigzag SiCNTs, the first peak due to the inter-π band transitions
is located at about 2.9 eV, which is slightly blueshifted (from 2.80 to 3.02 eV) with the increase of the tube diameter (or n from 8 to 14). The second peak originated from the π band to π * band transition is at about 7.2 eV, which is redshifted as the tube diameter increases. Finally, there is one more point to be noted regarding the spectra in Fig. 5. The amplitudes of all the peaks in these spectra become smaller as the diameters of the nanotubes become larger.
Fig. 5. The imaginary parts of the dielectric functions under parallel polarization: (a) armchair SiCNTs and (b) zigzag SiCNTs.
This implies that, if one desires to attain a larger optical absorption of SiCNTs, one should preferably use those fabricated from small-diameter tubes.
The loss functions under the light polarizations of different directions for armchair and zigzag SiCNTs are shown in Fig. 6. Optical anisotropy can be clearly seen, which is mainly attributed to the less efficient optical excitation for the electric field perpendicular to the tube axis. The peaks under parallel polarization are stronger than the ones under perpendicular polarization. For both armchair and zigzag SiCNTs, there are two weak peaks and one pronounced peak under parallel and perpendicular polarization. The first two weak peaks become apparent as n increases. And the pronounced peak located within the range 10̃15 eV redshifts with increasing tube diameters, which indicates that the plasma frequency become small as the tube diameter increases. Under parallel polarization, the first peak located around 5.0 eV is blueshifted with the increase of the tube diameter. In the case of perpendicular light polarization, the first peak located around 4.5 eV remains where it is as the size of the tube increases. For zigzag SiCNTs, the prominent peak is suppressed when n increases from 8–11 and gains strength when n increases from 11-14 under parallel polarization, while the prominent one becomes intense when n increases from 8–10 and is suppressed as n varies from 10–14 under perpendicular polarization. For armchair SiCNTs, the prominent peak become intense with n from 4–6 and is suppressed with n from 6–10 under perpendicular polarization, while in the case of parallel polarization, the prominent one is suppressed with n from 3–6, gains strength with n from 6–8, and is suppressed again from 8–10. The peaks below the energy of 10.0 eV represent characteristic associated with the oscillation of the plasma of losing π-electrons, and correspond to plasma frequency of losing π-electrons. And π+σ plasmons are mainly responsible for the peaks above 10.0 eV.
Fig.6. . The loss function. (a) Parallell polarization for armchair SiCNTs, (b) Perpendicular polarization for armchair SiCNTs, (c) Parallell polarization for zigzag SiCNTs, (d) perpendicular polarization for zigzag SiCNTs.
From the results of optical properties, we can find that for a given chirality of SiCNTs, the peaks localized at low energy region in the dielectric function are blueshifted and the peaks localized at high energy region in the energy loss function are redshifted with the increase of tube diameter. Furthermore, redshifts or blueshifts of the peaks localized at the other energy region in the dielectric function and energy loss function with the increase of tube diameter are possible due to the competition between the size effect and π orbitals overlapping, [30, 31] which is similar to the optical properties of singer-wall BC2N nanotubes [26], and that the shifts become smaller as the tube diameter increases.
4. Conclusion
In conclusion, the optical properties of SiC nanotubes with different sizes and chiralities are studied systematically using the ab-initio calculations based on the density functional theory. The results show that the optical spectra of SiCNTs are closely related to the diameter and chirality, and that optical anisotropy is observed for different light polarizations. For a given chirality of SiCNTs, redshifts or blueshifts of the peaks in the dielectric function and energy loss function with increasing tube diameter are possible due to the competition between the size effect and π orbitals overlapping. The peaks in energy loss function are due to the collective excitations of π electrons and the high frequency π+σ plasmons, and the collective excitation of π electrons is weaker than that of the high frequency π+σ plasmons. The unusual
optical properties of semiconducting SiC nanotubes present an opportunity for applications in electro-optical devices.
Acknowledgments
The authors are grateful to the National Science Foundation of China (No. 20373073), the National Basic Research Program of China (No. 2004CB720605), and the Fund of Fujian Key Laboratory of Nanomaterials (No. 2006L2005) for financial supports.
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