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Optical properties of BeO nanotubes: Ab initio study
Solid State Communications 156 (2013) 1–7
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Optical properties of BeO nanotubes: Ab initio study Ali Fathalian a,b,n, Rostam Moradian a, Masoud Shahrokhi a a b
Department of Physics, Razi University, Kermanshah, Iran Department of Nano Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran
a r t i c l e i n f o
abstract
Article history: Received 30 August 2012 Accepted 19 November 2012 by P. Hawrylak Available online 4 December 2012
The electronic and the linear optical properties of BeO nanotubes are investigated through the density functional theory. The optical properties are calculated for both parallel and perpendicular electric field polarizations. We found that by increasing diameter of nanotubes the adsorption peaks of imaginary part of dielectric constant shifted to the higher energies while these changes in the armchair nanotubes is very negligible. Optical conductivity of BeO nanotubes in the electric field parallel starts with a gap, confirms that all BeO nanotubes have semiconductor property. The results show that the optical properties of (6,0)@(11,0) BeO nanotube is different with the optical properties of single wall nanotubes. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.
Keywords: A. BeO nanotube B. Density functional theory D. Electronic property D. Optical property
1. Introduction Since the discovery by Iijima [1] of quasi-one-dimensional crystalline structures of carbon atoms generally referred to as carbon nanotubes (CNTs), several unique physical properties have been predicted theoretically and detected experimentally. For example, due to their one-dimensional character, metallic CNTs are quantum wires that may exhibit exotic Luttinger-liquid behavior rather than the usual Fermi liquid behavior in normal metal wires [2]. CNTs can be considered as a layer of graphene sheet rolled up into a cylinder, and the structure of a CNT is completely specified by the chiral vector which is given in terms of a pair of integers (n,m) [3]. Since then, a considerable number of nanotubes, such as BN [4] and SiC [5], have been prepared and some other ones such as BeO [6,7] have been predicted theoretically. Similar to carbon nanotubes, beryllium oxide (BeO) nanotubes are quasi-one-dimensional nanostructures. However, their physical properties are different from carbon nanotubes. Carbon nanotubes exhibit metallic or semiconducting behavior depending on the wrapping angle, while the band gap of BeO nanotube (BeONT) is larger than that of SiC and BN nanotubes (SiCNTs and BNNTs) [4]. Beryllium oxide (BeO) nanotubes, as a new inorganic non-carbon nanotube, has been studied by ab initio calculation and its structure and physical properties have been calculated [4]. The dependence of properties such as atomic relaxation, strain energy, and electronic structure of BeO nanotube on its diameter is also established [8]. BeONTs is predicted to be an
n Corresponding author at: Department of Nano Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran. Tel./fax: þ98 831 427 4556. E-mail address: [email protected] (A. Fathalian).
insulator and BN and SiCNTs are predicted to be semiconductors. Recently, the effect of boron, carbon and nitrogen doping in the electronic structure and magnetization of BeONTs was studied by ab initio band structure calculations [8]. In this paper, the electronic and optical properties of BeONTs with different diameters were studied by using the density-functional theory (DFT) within the generalized gradient approximation (GGA). This paper is organized as follows. The first task is calculation of density of states (DOS) of BeONTs. Then in the next step for this system, all optical spectra have been calculated for both electric fields polarized, parallel and perpendicular to the BeONTs.
2. Computational details All calculations presented in this work are based on density functional theory (DFT) [9] using the all electrons, full potential code WIEN2K [10]. The maximum angular momentum of the atomic orbital basis functions was set to lmax ¼ 10. In order to achieve energy eigenvalues convergence, the wave functional in the interstitial region was expanded in terms of plane waves with a cut-off parameter of RMT Kmax ¼ 8, where RMT denotes the smallest atomic sphere radius and Kmax largest k vector in the plane wave expansion. For the exchange-correlation energy functional, we used the generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) [11–13]. The Muffin-tin radii were set to RMT ¼ 1:44 a.u. for Be and O. Magnitude of largest vector in charge density Fourier expansion or the plane wave cutoff were set to Gmax¼12. The optical spectra were calculated using 238 k-points in the first Brillouin zone and setting Lorentzian broadening with gamma equal to 0.05 eV.
0038-1098/$ - see front matter Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.11.017
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A. Fathalian et al. / Solid State Communications 156 (2013) 1–7
3. Results and discussion 3.1. Single wall BeO nanotubes 3.1.1. Electronic properties We study the electronic properties of pristine BeO nanotubes. The density of states are illustrated in Fig. 1 for (a) armchair,
(b) zigzag and (c) chiral BeONTs. The results show that all BeONTs have semiconductor properties. The ground state of pristine BeONTs is non-magnetic. Our calculations show that the completely occupied valence zone for BeNTs contains two separate bands (Fig. 2). These bands are formed mainly by 2s and 2p orbitals of oxygen with a small admixture of Be 2s, 2p states suggesting the predominant ionic bonding between oxygen and Be atoms [14,15]. Also in Fig. 3, we investigated the density of state of (6,0)@(11,0) nanotube (for example). It can be seen that the double-wall nanotube has semiconductor behavior. The energy gap of double-wall is less than pristine BeO nanotubes because of interaction between layers. 3.1.2. Dielectric function The optical properties of BeO NTs, such as real and imaginary parts of dielectric tensor, optical conductivity, and loss function were calculated. The calculations were performed in non-spin case and RPA approximation. We have calculated dielectric function of BeONTs for parallel and perpendicular polarized light. Dielectric function is a complex quantity that describes the linear response of the system to an electromagnetic radiation. Imaginary part of dielectric function is obtained by calculating momentum matrix elements between the occupied and unoccupied wave functions within selection rules as [16] Z _2 e2 X c v Im efinterg dk/ckn 9pa 9ckn S ð o Þ ¼ ab pm2 o2 n v
c
/ckn 9pb 9ckn SdðEckn Evkn oÞ:
ð1Þ
The absorption spectrum is proportional to the sum over interv band transitions from occupied valence states ðckn Þ to empty cn conduction ðck Þ states over the first Brillouin-zone k points, where Eckn and Evkn are conduction band (CB) and valence band (VB), respectively. The real part of the dielectric tensor component is obtained by the Kramers–Kronig transformation from its corresponding imaginary part: Z 1 0 o Im eab ðo0 Þ 2 Re efinterg ðoÞ ¼ dab þ P : ð2Þ ab p 0 ðo0 Þ2 o2
Fig. 1. (Color online) Illustrates density of states (DOS) for (a) armchair (b) zigzag and (c) chiral BeO nanotubes.
The real and imaginary parts of dielectric function for parallel and perpendicular polarized light are calculated for armchair, zigzag and chiral BeO nanotubes. Figs. 4 and 5 show our calculated imaginary and real parts of dielectric constant for BeO nanotubes. The imaginary part of all the tubes is positive throughout the range of energy. Fig. 4a shows that the armchair BeO nanotubes ((5,5), (6,6), (7,7) and (11,11)) have a sharp point at 9.07–9.4 eV. We found that by increasing the diameter of nanotubes the adsorption peaks of imaginary part of dielectric constant shifted to the higher energies, although these changes are very negligible. It is revealed that the values of dielectric constants for armchair BeO nanotubes are different in the parallel and perpendicular direction. Fig. 4b shows that the imaginary parts of dielectric constant of zigzag BeO nanotubes. The nanotubes (6,0) have a sharp point at 8.75 eV, whereas (9.0), (12,0) and (14,0) BeONTs have a sharp peak value at 9.07 eV, 9.23 eV and 9.38 eV, respectively. We found that by increasing the diameter of Zigzag BeONTs the absorption peaks of imaginary part of dielectric constant shifted to the higher energies. We considered the imaginary part of dielectric constant for two chiral BeO Nanotubes (Fig. 4c). The results show that the nanotubes (4,2) have a sharp peak at 8.75 eV and (5,2) has a sharp peak at 8.95 eV, So the energy for imaginary part in parallel direction corresponds to peak values is increased by increasing the diameter. Since static dielectric constant is the real part of dielectric constant at zero energy, so this could be determined from data of Fig. 5. It is observed in Fig. 5a–c that the static dielectric constant
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Fig. 2. (Color online) Illustrates band structures for (a) zigzag and (b) chiral BeO nanotubes.
nanotubes. First main peak of armchair nanotubes ((5,5), (6,6), (7,7) and (11,11)) is sharp and occurs at energy of 9.5–9.7 eV that is related to p electron plasmon peak (Fig. 6a). Second main peak is broad and occurs around 18.75–19.5 eV that is related to p þ s electron plasmon peak. Some weak peak is observed between these two peaks that are related to weak resonances. Fig. 6b shows that the energy loss function for zigzag BeO nanotubes ((6,0), (9,0), (12,0) and (14,0)). The results show that the first main peak occurs at 9.3–9.7 eV and second main peak is broad and occurs around 18.13–19.38 eV. The energy loss function for two chiral BeO nanotubes is investigated (Fig. 6c). The first main peak occurs at 8.75–9.25 eV and the second main peak is broad and occurs around 19.07 eV. For perpendicular polarized light, the magnitude of the loss function of armchair, zigzag, and chiral BeONTs has decreased. It is observed in Fig. 6 that BeO nanotubes have multiple peaks.
Fig. 3. (Color online) Illustrates density of states for (6,0) and (11,0) single wall nanotubes and (6,0)@(11,0) double wall nanotubes.
for armchair, zigzag and chiral BeONTs for both polarizations. From Fig. 5 it is found that the static dielectric constant for armchair and zigzag BeONTs is not changed by increasing diameter of nanotubes, while for chiral nanotubes by increasing diameter it is increased. 3.1.3. Loss function The energy-loss function in terms of real and imaginary parts of dielectric tensor, eab ðoÞ is calculated, and defined by: e00ij ðoÞ 1 ¼ 0 : ð3Þ Lij ðoÞ ¼ Im eij ðoÞ eij 2ðoÞ þ e00ij 2ðoÞ Energy loss function for BeO nanotubes in both directions of electric field is shown in Fig. 6. The energy peaks under 10 eV are attributed to p electrons while high energy peaks are attributed to p þ s plasmons. The appearance of single peak is corresponding to unique collective excitations while multiple peaks are corresponding to various collective excitations. It is found that there are two main peaks in the electric field parallel to the BeO
3.1.4. Optical conductivity In the next step, the real part of the optical conductivity is calculated by using the following relations: Re sij ðoÞ ¼
o 4p
Im eij ðoÞ:
ð4Þ
According to Eq. (4), the real part of optical conductivity is related to the imaginary part of dielectric function. The real part of optical conductivity for armchair, zigzag and chiral BeO nanotubes is shown in Fig. 7. Optical conductivity of armchair nanotubes in the electric field parallel starts with a gap about 4.65 eV, confirms that armchair BeO nanotubes have semiconductor property (Fig. 7a). Fig. 7b shows that the real part of optical conductivity of zigzag BeO nanotubes. Our results show that the optical conductivity in the electric field parallel starts with a gap about 5.63 eV. The optical conductivity of (4,2) and (5,2) BeO nanotubes is calculated (Fig. 7c). In our study it is found that the optical conductivity in the electric field parallel starts with a gap about 5.15 eV, hence chiral nanotubes have semiconductor behavior.
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Fig. 4. (Color online) Imaginary parts of dielectric function for the electric field parallel and perpendicular to the (a) armchair, (b) zigzag and (c) chiral BeO nanotubes.
3.2. Double wall BeO nanotubes The optical properties of double wall BeO NTs, such as real and imaginary parts of dielectric tensor and loss function, were calculated. Real and imaginary parts of dielectric function for (5,5)@(11,11) double wall nanotube are calculated for both directions of electric field (parallel and perpendicular) are shown in Fig. 8a and b. It can be seen that the values of static dielectric
Fig. 5. (Color online) Real parts of dielectric function for the electric field parallel and perpendicular to the (a) armchair, (b) zigzag and (c) chiral BeO nanotubes.
constant (the values of dielectric constant at zero energy) in the electric field parallel is 1.58, while the static dielectric constant for (5,5) and (11,11) single wall nanotubes is 1.53; this value in the electric field perpendicular for double wall nanotube is almost 1.54. For both directions of electric field, imaginary part of dielectric function is shown in Fig. 8b. The results show that double wall (5,5)@(11,11) BeO nanotubes for electric field parallel and perpendicular have optically semiconductor property. Energy
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Fig. 6. (Color online) The electron energy loss function of BeO nanotubes for the electric field parallel and perpendicular to the (a) armchair, (b) zigzag and (c) chiral BeO nanotubes.
Fig. 7. (Color online) The real part of optical conductivity of BeO nanotubes for the electric field parallel and perpendicular to the (a) armchair, (b) zigzag and (c) chiral BeO nanotubes.
loss function for double wall (5,5)@(11,11) BeO nanotube in both directions of electric field is shown in Fig. 8c. The first main peak of double wall similar to single wall nanotubes is sharp and occurs at energy of 9.5–9.7 eV. Finally, we have calculated the optical properties of double wall (6,0)@(11,0) BeO nanotube. The results show that the values of static dielectric constant in the electric field parallel are 1.65 while the static dielectric constant
for (5,5) and (7,7) single wall nanotubes is 1.54; this value in the electric field perpendicular for double wall nanotube is almost 1.63 that it has increased compared to the single wall nanotubes (Fig. 9a). For both directions of electric field, imaginary part of dielectric function is shown in Fig. 9b. The results show that double wall (6,0)@(11,0) BeO nanotubes for electric field parallel and perpendicular have optically semiconductor property. Energy
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Fig. 8. (Color online) Illustrate (a) The electron energy loss function of (5,5)@(11,11) BeO nanotube for both electric field parallel and perpendicular. (b) Real parts of dielectric function for the electric field parallel and perpendicular to the (5,5)@(11,11) BeO nanotube. (c) Imaginary parts of dielectric function for the electric field parallel and perpendicular to the (5,5)@(11,11) BeO nanotube.
loss function for double wall (6,0)@(11,0) BeO nanotube in both directions of electric field is shown in Fig. 9c. The first main peak of double wall in parallel polarization occurs at energy of 9.06 eV, while the first main peak of (6,0) and (11,0) single wall nanotubes
Fig. 9. (Color online) Illustrate (a) Real parts of dielectric function for the electric field parallel and perpendicular to the (6,0)@(11,0) BeO nanotube. (b) Imaginary parts of dielectric function for the electric field parallel and perpendicular to the (6,0)@(11,0) BeO nanotube. (c) The electron energy loss function of (6,0)@(11,0) BeO nanotube for both electric field parallel and perpendicular.
is sharp and occurs at energy of 9.3–9.7 eV. It is found that the energy of first main peak of double wall in parallel polarization is has decreased compared to the single wall nanotubes. In the electric field perpendicular, the first main peak of double wall occurs at energy of 10.95 eV that is has increased compared to the single wall nanotubes.
A. Fathalian et al. / Solid State Communications 156 (2013) 1–7
4. Conclusions We have calculated the electronic structure and optical properties of BeO nanotubes by density functional theory in the generalized gradient approximation. To investigate optical properties of mBeO nanotubes, the parallel and perpendicular electric field with respect to the BeO nanotubes are considered. It is found that the optical spectra are anisotropic along these two polarizations. The real part of dielectric function is positive for all BeO nanotubes. We found that by increasing diameter of armchair, zigzag and chiral nanotubes the adsorption peaks of imaginary part of dielectric constant shifted to the higher energies, Although these changes in the armchair nanotubes are very negligible. Since static dielectric constant is the real part of dielectric constant at zero energy. We found that the static dielectric constant for armchair and zigzag BeONTs is not changed by increasing diameter of nanotubes while for chiral nanotubes by increasing diameter it is increased. There are two main peaks in the electric field parallel to the BeO nanotubes. Some weak peak is observed between these two peaks that are related to weak resonances. Optical conductivity of armchair, zigzag and chiral nanotubes in the electric field parallel starts with a gap, confirms that all BeO nanotubes have semiconductor property. Finally, we have calculated the optical properties of (5,5)@(11,11) and (6,0)@(11,0) double wall BeO nanotubes. We found that the optical properties of (5,5)@(11,11) double wall nanotube has not significant changes compared to the optical properties of (5,5) and (11,11) single-wall nanotubes. The static dielectric constant
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of (6,0)@(11,0) double wall nanotube for both directions of electric field is has decreased compared to the single wall nanotubes. The results ow that the energy of first main peak of (6,0)@(11,0) in parallel polarization is has decreased compared to single wall nanotubes while it is increased in the electric field perpendicular.
References [1] S. Iijima, Nature (London) 354 (1991) 56. [2] M. Bockrath, D.H. Cobden, J. Lu, A.G. Rinzler, R.E. Smalley, L. Balents, P.L. McEuen, Nature (London) 397 (1999) 598. [3] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College, London, 1998. [4] N.G. Chopra, R.J. Luyken, K. Cherrey, V.H. Crespi, M.L. Cohen, S.G. Louie, A. Zettl, Science 269 (1995) 966. [5] Tomitsugu Taguchi, Naoki Igawa, Hiroyuki Yamamoto, Shin-ichi Shamoto, Shiro Jitsukawa, Physica E 28 (2005) 431. [6] P.B. Sorokin, A.S. Fedorov, L.A. Chernozatonskii, Phys. Solid State 48 (2006) 398. [7] B. Baumeier, P. Kruger, J. Pollmann, Phys. Rev. B 76 (2007) 085407. [8] M.A. Gorbunova, I.R. Shein, Yu.N. Makurin, V.V. Ivanovskaya, V.S. Kijko, A.L. Ivanovskii, Physica E 41 (2008) 164. [9] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (864) (1964). [10] K. Schwarz, P. Blaha, Comput. Mater. Sci. 28 (259) (2003). [11] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [12] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. [13] A. Fathalian, J. Jalilian, Phys. Lett. A 374 (2010) 4695. [14] L.-C. Ma, J.-M. Zhang, K.-W. Xu, Physica B 407 (2012) 784. [15] M.A. Gorbunova, I.R. Shein, Yu.N. Makurin, V.V. Ivanovskaya, V.S. Kijko, A.L. Ivanovskii, Physica E 41 (2008) 164. [16] F. Wooten, Optical Properties of Solids, Academic Press, New York, 1972.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Optical properties of BeO nanotubes: Ab initio study Ali Fathalian a,b,n, Rostam Moradian a, Masoud Shahrokhi a a b
Department of Physics, Razi University, Kermanshah, Iran Department of Nano Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran
a r t i c l e i n f o
abstract
Article history: Received 30 August 2012 Accepted 19 November 2012 by P. Hawrylak Available online 4 December 2012
The electronic and the linear optical properties of BeO nanotubes are investigated through the density functional theory. The optical properties are calculated for both parallel and perpendicular electric field polarizations. We found that by increasing diameter of nanotubes the adsorption peaks of imaginary part of dielectric constant shifted to the higher energies while these changes in the armchair nanotubes is very negligible. Optical conductivity of BeO nanotubes in the electric field parallel starts with a gap, confirms that all BeO nanotubes have semiconductor property. The results show that the optical properties of (6,0)@(11,0) BeO nanotube is different with the optical properties of single wall nanotubes. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.
Keywords: A. BeO nanotube B. Density functional theory D. Electronic property D. Optical property
1. Introduction Since the discovery by Iijima [1] of quasi-one-dimensional crystalline structures of carbon atoms generally referred to as carbon nanotubes (CNTs), several unique physical properties have been predicted theoretically and detected experimentally. For example, due to their one-dimensional character, metallic CNTs are quantum wires that may exhibit exotic Luttinger-liquid behavior rather than the usual Fermi liquid behavior in normal metal wires [2]. CNTs can be considered as a layer of graphene sheet rolled up into a cylinder, and the structure of a CNT is completely specified by the chiral vector which is given in terms of a pair of integers (n,m) [3]. Since then, a considerable number of nanotubes, such as BN [4] and SiC [5], have been prepared and some other ones such as BeO [6,7] have been predicted theoretically. Similar to carbon nanotubes, beryllium oxide (BeO) nanotubes are quasi-one-dimensional nanostructures. However, their physical properties are different from carbon nanotubes. Carbon nanotubes exhibit metallic or semiconducting behavior depending on the wrapping angle, while the band gap of BeO nanotube (BeONT) is larger than that of SiC and BN nanotubes (SiCNTs and BNNTs) [4]. Beryllium oxide (BeO) nanotubes, as a new inorganic non-carbon nanotube, has been studied by ab initio calculation and its structure and physical properties have been calculated [4]. The dependence of properties such as atomic relaxation, strain energy, and electronic structure of BeO nanotube on its diameter is also established [8]. BeONTs is predicted to be an
n Corresponding author at: Department of Nano Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran. Tel./fax: þ98 831 427 4556. E-mail address: [email protected] (A. Fathalian).
insulator and BN and SiCNTs are predicted to be semiconductors. Recently, the effect of boron, carbon and nitrogen doping in the electronic structure and magnetization of BeONTs was studied by ab initio band structure calculations [8]. In this paper, the electronic and optical properties of BeONTs with different diameters were studied by using the density-functional theory (DFT) within the generalized gradient approximation (GGA). This paper is organized as follows. The first task is calculation of density of states (DOS) of BeONTs. Then in the next step for this system, all optical spectra have been calculated for both electric fields polarized, parallel and perpendicular to the BeONTs.
2. Computational details All calculations presented in this work are based on density functional theory (DFT) [9] using the all electrons, full potential code WIEN2K [10]. The maximum angular momentum of the atomic orbital basis functions was set to lmax ¼ 10. In order to achieve energy eigenvalues convergence, the wave functional in the interstitial region was expanded in terms of plane waves with a cut-off parameter of RMT Kmax ¼ 8, where RMT denotes the smallest atomic sphere radius and Kmax largest k vector in the plane wave expansion. For the exchange-correlation energy functional, we used the generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) [11–13]. The Muffin-tin radii were set to RMT ¼ 1:44 a.u. for Be and O. Magnitude of largest vector in charge density Fourier expansion or the plane wave cutoff were set to Gmax¼12. The optical spectra were calculated using 238 k-points in the first Brillouin zone and setting Lorentzian broadening with gamma equal to 0.05 eV.
0038-1098/$ - see front matter Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.11.017
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A. Fathalian et al. / Solid State Communications 156 (2013) 1–7
3. Results and discussion 3.1. Single wall BeO nanotubes 3.1.1. Electronic properties We study the electronic properties of pristine BeO nanotubes. The density of states are illustrated in Fig. 1 for (a) armchair,
(b) zigzag and (c) chiral BeONTs. The results show that all BeONTs have semiconductor properties. The ground state of pristine BeONTs is non-magnetic. Our calculations show that the completely occupied valence zone for BeNTs contains two separate bands (Fig. 2). These bands are formed mainly by 2s and 2p orbitals of oxygen with a small admixture of Be 2s, 2p states suggesting the predominant ionic bonding between oxygen and Be atoms [14,15]. Also in Fig. 3, we investigated the density of state of (6,0)@(11,0) nanotube (for example). It can be seen that the double-wall nanotube has semiconductor behavior. The energy gap of double-wall is less than pristine BeO nanotubes because of interaction between layers. 3.1.2. Dielectric function The optical properties of BeO NTs, such as real and imaginary parts of dielectric tensor, optical conductivity, and loss function were calculated. The calculations were performed in non-spin case and RPA approximation. We have calculated dielectric function of BeONTs for parallel and perpendicular polarized light. Dielectric function is a complex quantity that describes the linear response of the system to an electromagnetic radiation. Imaginary part of dielectric function is obtained by calculating momentum matrix elements between the occupied and unoccupied wave functions within selection rules as [16] Z _2 e2 X c v Im efinterg dk/ckn 9pa 9ckn S ð o Þ ¼ ab pm2 o2 n v
c
/ckn 9pb 9ckn SdðEckn Evkn oÞ:
ð1Þ
The absorption spectrum is proportional to the sum over interv band transitions from occupied valence states ðckn Þ to empty cn conduction ðck Þ states over the first Brillouin-zone k points, where Eckn and Evkn are conduction band (CB) and valence band (VB), respectively. The real part of the dielectric tensor component is obtained by the Kramers–Kronig transformation from its corresponding imaginary part: Z 1 0 o Im eab ðo0 Þ 2 Re efinterg ðoÞ ¼ dab þ P : ð2Þ ab p 0 ðo0 Þ2 o2
Fig. 1. (Color online) Illustrates density of states (DOS) for (a) armchair (b) zigzag and (c) chiral BeO nanotubes.
The real and imaginary parts of dielectric function for parallel and perpendicular polarized light are calculated for armchair, zigzag and chiral BeO nanotubes. Figs. 4 and 5 show our calculated imaginary and real parts of dielectric constant for BeO nanotubes. The imaginary part of all the tubes is positive throughout the range of energy. Fig. 4a shows that the armchair BeO nanotubes ((5,5), (6,6), (7,7) and (11,11)) have a sharp point at 9.07–9.4 eV. We found that by increasing the diameter of nanotubes the adsorption peaks of imaginary part of dielectric constant shifted to the higher energies, although these changes are very negligible. It is revealed that the values of dielectric constants for armchair BeO nanotubes are different in the parallel and perpendicular direction. Fig. 4b shows that the imaginary parts of dielectric constant of zigzag BeO nanotubes. The nanotubes (6,0) have a sharp point at 8.75 eV, whereas (9.0), (12,0) and (14,0) BeONTs have a sharp peak value at 9.07 eV, 9.23 eV and 9.38 eV, respectively. We found that by increasing the diameter of Zigzag BeONTs the absorption peaks of imaginary part of dielectric constant shifted to the higher energies. We considered the imaginary part of dielectric constant for two chiral BeO Nanotubes (Fig. 4c). The results show that the nanotubes (4,2) have a sharp peak at 8.75 eV and (5,2) has a sharp peak at 8.95 eV, So the energy for imaginary part in parallel direction corresponds to peak values is increased by increasing the diameter. Since static dielectric constant is the real part of dielectric constant at zero energy, so this could be determined from data of Fig. 5. It is observed in Fig. 5a–c that the static dielectric constant
A. Fathalian et al. / Solid State Communications 156 (2013) 1–7
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Fig. 2. (Color online) Illustrates band structures for (a) zigzag and (b) chiral BeO nanotubes.
nanotubes. First main peak of armchair nanotubes ((5,5), (6,6), (7,7) and (11,11)) is sharp and occurs at energy of 9.5–9.7 eV that is related to p electron plasmon peak (Fig. 6a). Second main peak is broad and occurs around 18.75–19.5 eV that is related to p þ s electron plasmon peak. Some weak peak is observed between these two peaks that are related to weak resonances. Fig. 6b shows that the energy loss function for zigzag BeO nanotubes ((6,0), (9,0), (12,0) and (14,0)). The results show that the first main peak occurs at 9.3–9.7 eV and second main peak is broad and occurs around 18.13–19.38 eV. The energy loss function for two chiral BeO nanotubes is investigated (Fig. 6c). The first main peak occurs at 8.75–9.25 eV and the second main peak is broad and occurs around 19.07 eV. For perpendicular polarized light, the magnitude of the loss function of armchair, zigzag, and chiral BeONTs has decreased. It is observed in Fig. 6 that BeO nanotubes have multiple peaks.
Fig. 3. (Color online) Illustrates density of states for (6,0) and (11,0) single wall nanotubes and (6,0)@(11,0) double wall nanotubes.
for armchair, zigzag and chiral BeONTs for both polarizations. From Fig. 5 it is found that the static dielectric constant for armchair and zigzag BeONTs is not changed by increasing diameter of nanotubes, while for chiral nanotubes by increasing diameter it is increased. 3.1.3. Loss function The energy-loss function in terms of real and imaginary parts of dielectric tensor, eab ðoÞ is calculated, and defined by: e00ij ðoÞ 1 ¼ 0 : ð3Þ Lij ðoÞ ¼ Im eij ðoÞ eij 2ðoÞ þ e00ij 2ðoÞ Energy loss function for BeO nanotubes in both directions of electric field is shown in Fig. 6. The energy peaks under 10 eV are attributed to p electrons while high energy peaks are attributed to p þ s plasmons. The appearance of single peak is corresponding to unique collective excitations while multiple peaks are corresponding to various collective excitations. It is found that there are two main peaks in the electric field parallel to the BeO
3.1.4. Optical conductivity In the next step, the real part of the optical conductivity is calculated by using the following relations: Re sij ðoÞ ¼
o 4p
Im eij ðoÞ:
ð4Þ
According to Eq. (4), the real part of optical conductivity is related to the imaginary part of dielectric function. The real part of optical conductivity for armchair, zigzag and chiral BeO nanotubes is shown in Fig. 7. Optical conductivity of armchair nanotubes in the electric field parallel starts with a gap about 4.65 eV, confirms that armchair BeO nanotubes have semiconductor property (Fig. 7a). Fig. 7b shows that the real part of optical conductivity of zigzag BeO nanotubes. Our results show that the optical conductivity in the electric field parallel starts with a gap about 5.63 eV. The optical conductivity of (4,2) and (5,2) BeO nanotubes is calculated (Fig. 7c). In our study it is found that the optical conductivity in the electric field parallel starts with a gap about 5.15 eV, hence chiral nanotubes have semiconductor behavior.
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Fig. 4. (Color online) Imaginary parts of dielectric function for the electric field parallel and perpendicular to the (a) armchair, (b) zigzag and (c) chiral BeO nanotubes.
3.2. Double wall BeO nanotubes The optical properties of double wall BeO NTs, such as real and imaginary parts of dielectric tensor and loss function, were calculated. Real and imaginary parts of dielectric function for (5,5)@(11,11) double wall nanotube are calculated for both directions of electric field (parallel and perpendicular) are shown in Fig. 8a and b. It can be seen that the values of static dielectric
Fig. 5. (Color online) Real parts of dielectric function for the electric field parallel and perpendicular to the (a) armchair, (b) zigzag and (c) chiral BeO nanotubes.
constant (the values of dielectric constant at zero energy) in the electric field parallel is 1.58, while the static dielectric constant for (5,5) and (11,11) single wall nanotubes is 1.53; this value in the electric field perpendicular for double wall nanotube is almost 1.54. For both directions of electric field, imaginary part of dielectric function is shown in Fig. 8b. The results show that double wall (5,5)@(11,11) BeO nanotubes for electric field parallel and perpendicular have optically semiconductor property. Energy
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Fig. 6. (Color online) The electron energy loss function of BeO nanotubes for the electric field parallel and perpendicular to the (a) armchair, (b) zigzag and (c) chiral BeO nanotubes.
Fig. 7. (Color online) The real part of optical conductivity of BeO nanotubes for the electric field parallel and perpendicular to the (a) armchair, (b) zigzag and (c) chiral BeO nanotubes.
loss function for double wall (5,5)@(11,11) BeO nanotube in both directions of electric field is shown in Fig. 8c. The first main peak of double wall similar to single wall nanotubes is sharp and occurs at energy of 9.5–9.7 eV. Finally, we have calculated the optical properties of double wall (6,0)@(11,0) BeO nanotube. The results show that the values of static dielectric constant in the electric field parallel are 1.65 while the static dielectric constant
for (5,5) and (7,7) single wall nanotubes is 1.54; this value in the electric field perpendicular for double wall nanotube is almost 1.63 that it has increased compared to the single wall nanotubes (Fig. 9a). For both directions of electric field, imaginary part of dielectric function is shown in Fig. 9b. The results show that double wall (6,0)@(11,0) BeO nanotubes for electric field parallel and perpendicular have optically semiconductor property. Energy
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Fig. 8. (Color online) Illustrate (a) The electron energy loss function of (5,5)@(11,11) BeO nanotube for both electric field parallel and perpendicular. (b) Real parts of dielectric function for the electric field parallel and perpendicular to the (5,5)@(11,11) BeO nanotube. (c) Imaginary parts of dielectric function for the electric field parallel and perpendicular to the (5,5)@(11,11) BeO nanotube.
loss function for double wall (6,0)@(11,0) BeO nanotube in both directions of electric field is shown in Fig. 9c. The first main peak of double wall in parallel polarization occurs at energy of 9.06 eV, while the first main peak of (6,0) and (11,0) single wall nanotubes
Fig. 9. (Color online) Illustrate (a) Real parts of dielectric function for the electric field parallel and perpendicular to the (6,0)@(11,0) BeO nanotube. (b) Imaginary parts of dielectric function for the electric field parallel and perpendicular to the (6,0)@(11,0) BeO nanotube. (c) The electron energy loss function of (6,0)@(11,0) BeO nanotube for both electric field parallel and perpendicular.
is sharp and occurs at energy of 9.3–9.7 eV. It is found that the energy of first main peak of double wall in parallel polarization is has decreased compared to the single wall nanotubes. In the electric field perpendicular, the first main peak of double wall occurs at energy of 10.95 eV that is has increased compared to the single wall nanotubes.
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4. Conclusions We have calculated the electronic structure and optical properties of BeO nanotubes by density functional theory in the generalized gradient approximation. To investigate optical properties of mBeO nanotubes, the parallel and perpendicular electric field with respect to the BeO nanotubes are considered. It is found that the optical spectra are anisotropic along these two polarizations. The real part of dielectric function is positive for all BeO nanotubes. We found that by increasing diameter of armchair, zigzag and chiral nanotubes the adsorption peaks of imaginary part of dielectric constant shifted to the higher energies, Although these changes in the armchair nanotubes are very negligible. Since static dielectric constant is the real part of dielectric constant at zero energy. We found that the static dielectric constant for armchair and zigzag BeONTs is not changed by increasing diameter of nanotubes while for chiral nanotubes by increasing diameter it is increased. There are two main peaks in the electric field parallel to the BeO nanotubes. Some weak peak is observed between these two peaks that are related to weak resonances. Optical conductivity of armchair, zigzag and chiral nanotubes in the electric field parallel starts with a gap, confirms that all BeO nanotubes have semiconductor property. Finally, we have calculated the optical properties of (5,5)@(11,11) and (6,0)@(11,0) double wall BeO nanotubes. We found that the optical properties of (5,5)@(11,11) double wall nanotube has not significant changes compared to the optical properties of (5,5) and (11,11) single-wall nanotubes. The static dielectric constant
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of (6,0)@(11,0) double wall nanotube for both directions of electric field is has decreased compared to the single wall nanotubes. The results ow that the energy of first main peak of (6,0)@(11,0) in parallel polarization is has decreased compared to single wall nanotubes while it is increased in the electric field perpendicular.
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