- Email: [email protected]
First-principles calculations of optical properties of Titanium nanochains
Computational Materials Science 77 (2013) 224–229
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
First-principles calculations of optical properties of Titanium nanochains Mahmoud Jafari ⇑, Hamid reza Hajiyani, Zeinab Sohrabikia, Habibeh Galavani Department of Physics, K.N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 13 October 2011 Received in revised form 29 December 2012 Accepted 14 April 2013 Available online 24 May 2013 Keywords: Titanium Nanochains Nanowires Optical properties
a b s t r a c t In this paper, optical properties of linear, dimerized, zigzag and ladder nanowires were reported using the full potential linearized augmented plane wave plus local orbital method (FLAPW + LO) and by incorporating the generalized gradient approximation (GGA). Also, the super cell approach, infinite and free standing periodic nanowires were applied. Optical properties such as dielectric function, energy loss function, optical conductivity and reflectivity of nanochains were investigated. Moreover, the results of nanochains were compared with the findings of the present author’s previous works on the bulk and nanowires (NWs). The properties, especially reflectivity and static dielectric function, changed significantly with configuration. The static dielectric function increased slightly from 1.9 in the dimerized nanowire to 3.06 in the linear one. The negative value of the real dielectric function occurred in some frequencies of the dimerized nanochains. Furthermore, equilibrium structure and inter-atomic distance were calculated, the results of which were in agreement with those of the previous works. In addition, nanowire reflectivity spectra showed a number of distinct peaks, which could be attributed to the plasmon resonance. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction In nanotechnology, nanochains have fundamental, unique and practical properties. Fabrication of nanoarrays, nanowires (NWs) and nanochains is of great importance due to their applications in sensors and optoelectronics [1,2]. Because of the strong dependence of electrical, magnetic and optical properties of nanochains on the geometry and configuration of nanowires [3], many pieces of research have been carried out on this topic. In experimental terms, metallic, nonmetallic and compound nanowires and nanochains have been fabricated both chemically and physically [4]. In the chemical fabrication, Ag nanochains have been synthesized by the method involving aqueous chloroaurate ion [AuCl4] with sodium borohydride [NaBH4] in the presence of an amino acid [5] or by the modified citrate reduction process [6]. Modern methods have made it feasible to investigate low-dimensional nanowires in order to reveal the effect of low dimensionality on the electrical properties. Recently, the growth of nanochains has been tested on the substrates of different shapes and a range of materials have been successfully fabricated. Ag nanochains can be grown inside organic nanotubules [7] and other experiments have demonstrated the possibility of growing Ti nanochains on SWNT [8,9]. Short suspended nanowires have been fabricated via sweeping the metallic surface by the tip of a scanning tunneling microscope [10], in which synthesis of nonmetallic nanochains was ⇑ Corresponding author. Tel.: +98 21 22853308; fax: +98 21 22853650. E-mail address: [email protected] (M. Jafari). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.04.029
achieved like a metallic nanochain. Si nanochains were grown using the hydrothermal chemical method in [11,12] or via a selforganized vapor–liquid–solid process in [13]. During this period, theoretical calculations of electronic, magnetic and optical properties of nanochains were carried out using methods such as molecular dynamics, semiempirical tight binding or ab initio density functional theory. The electrical and magnetic properties of metallic nanochains such as Co [14], Fe [15], Ti [16], Au [17] and Ag [18] have been also reported. Titanium, as a member of the 3d transition metal group, has shown an exceptional behavior in both of its bulk and nanowire forms. Ti nanowires (TiNWs) have been manufactured by several different methods [19,20]. TiNWs were deposited on paralytic graphite by electrochemical methods in [21] or they were synthesized on sapphire single crystals in [22]. In addition, Zhang et al. reported the fabrication of Ti NW on anodic aluminum oxide [23]. Many calculations have been carried out on the simulation of nanochains. At first, some simulations were investigated using the Monte Carlo approach by Wang et al. [24]. After that, Li et al. studied different structures of Ti nanowires and found the zigzag arrangement of atoms the most stable [25]. Zhu et al. calculated the magnetic moment of linear Ti nanochains by an ab initio approach [26]. They also calculated the optimized length of linear nanochains and found Ti nanochains had 0 0.61 mb/atom, separated by an equilibrium bond length of 2.2 Å A. The magnetic moments of zigzag and linear nanochains were later reported in [27,28] which showed that all the 3d transition metals in both linear and zigzag structures had stable and metastable ferromagnetic states (FM). It was also shown that geometry of the
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
225
In this article, the imaginary and real parts of dielectric function, energy loss function, optical conductivity and, finally, reflectivity were calculated for four types of linear, dimerized, zigzag and ladder nanochains. These results were compared with those of the author’s previous work, optical properties of a, b and x bulk phases of Titanium [33] and also optical properties of b [1 1 0] and x [1 0 0] nanowires [34]. 2. Computational method Fig. 1. The structure and atomic position of zigzag, dimerized, ladder and linear nanochains, respectively. d is the length of unit cell. Atomic indication (1, 2) is the target atom that is necessarily for atomic simulation.
nanochains played a key role in the magnetic moments of nanochains and that the zigzag nanowires were the most stable configuration with 0.571 mb/atom for the FM state zigzag nanochains. Ataca et al. put emphasis on studying infinite periodic linear, dimerized and zigzag nanochains as well as the finite short segments [28]. Recently, the magnetic moment of zigzag, linear, dimerized and ladder nanowires has been studied and the optimized interatomic distance has been calculated [29]. The optical properties of solids [30] are of wide interest because of their diverse applications in industrial and optoelectronic devices [31,32].
Table 1 Distances d, atomic component and the number of symmetric and optimized parameters for different structures of Ti nanochains. Structure
d (Å)
Coordinate (Å) x/a, y/b, z/c
RMT (Å)
Number of sym
kPoint
Rmt Kmax
Linear
2.23, 2.11a, 2.10b 4.82, 4.74a, 4.87b
0, 0, 0
1.92
16
1500
8
0, 0, 0
1.74
8
700
8.5
2.21
2
500
8
1.95
4
1000
7.5
Dimerized
Zigzag
2.6, 2.41a,b
Ladder
a
2.27, 2.14 , 2.63b
0, 0, 0, 0,
0, 0.4 0, 0 0.17, 0.50 0, 0
0, 0.21, 0 a b
Pseudo potential results from Ref. [29]. Pseudo potential results from Ref. [25].
All the calculations were performed within the framework of density functional theory [35,36] using the full-potential linearized augmented plane wave plus local orbital (FLAPW + LO) in the Wien2k package [37]. Exchange and correlation effects were treated using the generalized gradient approximation (GGA) [38,39] with the formal Perdew–Burke–Ernzerhof0 (PBE) parameterization scheme [40–42]. A distance of about 10 Å A was used between the nanowires in order to eliminate their interaction. The charge convergence of 104 e was used for optimizing the parameters in the SCF cycle. Fig. 1 shows the structure and atomic position of zigzag, dimerized, ladder and linear nanochains. The required atomic coordinates for simulating nanowires are listed in Table 1. Cohesive energy Ec was obtained using the following equation Ec = Ea–ET/N, in which Ea is ground state energy of the free constituent atom, ET is the total energy per unit cell of the given nanochains structure and N is the number of atom in the unit cell. The unit cell includes one atom for linear nanochains and two more for the other three. It is worth mentioning that Ec increased with increase in the number of the nearest neighbors from linear to zigzag structure while there was an increase in the number of bonds. The number of the nearest neighbors was two for linear and dimerized and three for ladder and zigzag structures. Fig. 2 demonstrates the variation of cohesive energy with the length of unit cell (d). Present results were smaller than those in the previous works [25,29]; this difference in Ec values could depend on the different method used in the previous reports. Full potential calculation was used in the current work whereas previous studies [25,29] used pseudopotential calculation. The optimized atomic coordinates of nanochains were calculated, which were in good agreement with the findings of the previous reports [26,27,29]. Moreover, in order to obtain reliable calculations, a number of convergence tests were performed to optimize the input parameters. The parameters, which had to be tested, mainly included k-points and Rmt Kmax.
Fig. 2. The variation of cohesive energy with the length of unit cell (d).
226
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
As can be seen in Table 1, an optimized number of k-points were used; while this value was only 700 for the dimerized nanochains, it reached 1500 points for the linear nanochains. In the combination of spherical harmonic inside non-overlapping spheres surrounding the atomic sites, the so-called muffintin (MT) spheres and, in a Fourier series in the interstitial region in the MT sphere, the basic function of Rmt Kmax were expanded for the Eigen-values convergence. Kmax is the maximum modulus for the reciprocal lattice vector and Rmt is the average radius of the MT sphere. The optimized amount of Rmt Kmax is listed in Table 1. The MT radii were used (as listed in Table 1) for each nanochains. As can be observed, RMT had the minimum and maximum values of 1.74 and 2.21 for dimerized and ladder nanochains, respectively. 3. Results and discussion Optical properties of both metals and insulators were investigated theoretically and experimentally and electromagnetic radiation was used for probing the surface as a function of wave vector ! of k and frequency of x. In this work, optical properties such as dielectric function, optical conductivity, energy loss function and reflectivity of the zigzag, linear, dimerized and ladder nanowires were studied. In theoretical terms, by solving the Schrödinger equation in the presence of an electromagnetic field and by ignoring the nonlinear effect using an ab initio approach, a description of the optical properties was obtained. 3.1. Dielectric function !
The direct dependence of the dielectric function on k and x provided a unified description of the charge response of both met! als and insulators. If the dielectric function e ( x , k ) was separated ! ! ! into real and imaginary parts, as e (x, k ) = e1 (x, k ) + ie2 (x, k ), the dielectric function of the crystal for x > 0 could be expressed by the Hamiltonian [30]: 2 ! 4pe2 X ! ! e2 ðx; k Þ ¼ 2 2 hl; k jpx j k ; mi f ! ð1 f ! ÞdðE ! E ! hxÞ m x k;l;m lk mk m; k l; k !
!
where l, m denotes the band indices, k is the wave vector, jl; k i is the ! Eigen state with the wave vector k and l and E ! is the correspondl; k ing Eigen value of the wave function. Also, f ! is the occupation l; k number and x indicates the component of the momentum transfer. ! The k dependence of the equation was negligible owing to its nor-
mal frequency. At normal frequencies from infrared to ultraviolet, the wave vector was small compared with the dimension of the ! Brillouin zone. This was also the justification for neglecting the k dependence of the dielectric function, which can be written as e (x). The real part of the dielectric function was obtained by inserting the above equation in the Kamer–Kronig [30] relation as follows: !
e1 ðx; k Þ ¼ 1 þ
2 f ! ð1 f ! Þ 4pe2 X ! ! lk mk hl; k jp jk ; mi x m2 x2 k;l;m ðE ! E ! hxÞ m; k l; k
As observed in Fig. 3a–d, the imaginary part of the dielectric function was depicted for linear, dimmer, zigzag and ladder nanochains, respectively. A noteworthy aspect here was the conspicuous differences in the intensities of the peaks. The intensity of the main peak reached around 7.3 eV in linear nanochains while it was equal to 8.55 eV and 8.6 eV in dimerized and ladder structures, respectively. The main peak in the zigzag configuration reached the substantial value of 9.81 eV. In Fig. 4, the real part of dielectric functions of nanochains is illustrated. Overall, e1 (x) leveled off to the defined values of e1 (1) after some fluctuations. Among all types of nanochains, just the dimerized one had a negative dielectric function in some frequencies. In this configuration, e1 (x) was negative from 3.08 eV to 3.27 eV and reached its minimum value of 0.45 eV at around 3.16 eV. e1 (x) in x ? 0 is the static dielectric constant and the calculated values are given in Table 2. Among the nanochains, the linear one had the largest static dielectric constant value of 3.06. e1 (0) increased by 1.16 eV and 0.19 eV from the dimerized to the linear nanochains and from zigzag to ladder nanochains, respectively. The quantity of e1 (1) is listed in Table 2 for various nanochains. The dielectric function leveled off to zero at around 6 eV and 8 eV for the zigzag and dimerized and for ladder nanochains, respectively. The optical dielectric constant, the maximum and minimum and limits of the real part of dielectric constant are presented in Table 2. 3.2. Dielectric energy loss function and optical conductivity While neglecting retardation and surface effects, the electronenergy-loss function I (x) could be given by: !
IðxÞ ¼ Im
1 !
eðk ; xÞ
¼
e2 ðk ; xÞ !
e22 ðk ; xÞ þ e22 ðk; xÞ
Fig. 3. Imaginary part of dielectric function, e2 (x), for (a) linear, (b) dimerized, (c) zigzag and (d) ladder nanochains.
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
227
Fig. 4. Real parts of dielectric function, e1 (x), for (a) linear, (b) zigzag, (c) dimerized, and (d) ladder nanochains, respectively. Inset 1: Real parts of dielectric function of bulk Titanium as a function of wavelength obtained from Ref. [33]. Inset 2: Real parts of dielectric function of b [1 1 0] Titanium nanowires obtained from Ref. [34].
Table 2 The optical and extreme limitation of real part of dielectric constant. Structure
Linear Dimmer Zigzag Ladder Bulk Nanowire a b c d
Full Full Full Full
Real part of dielectric boundary condition in addition to frequency
e1 (0)
e1max, x
e1min, x
ei (1)
3.06 1.90 2.47 2.66 144.7a 63.4b 14.0c 14.3d
3.06, 0.80 2.54, 1.83 2.89, 1.0 3.38, 0.86 144.7a, 0.0 52.0b, 0.7 14.0c, 0.0 14.3d, 0.0
0.07, 3.38 0.45, 3.16 0.47, 5.40 0.40, 5.69 9.0a, 4.7 7.0b, 3.9 0.15c, 4.8 0.6d, 6.2
0.92 0.92 0.84 0.85 0.0a 0.0b 1.0c 0.80d
potential potential potential potential
results results results results
for for for for
x phase from Ref. [33]. b phase from Ref. [33]. b {1 1 0} TiNWs from Ref. [34]. x {1 0 0} TiNWs from Ref. [34].
With hx as the loss energy, as can be seen in Fig. 5, all types of nanochains had a remarkable variation in frequency, which was less than 10 eV. For the nanochains, the curve showed a significant
peak due to plasmon resonance electrons. With this energy, the real part of the dielectric function Re e (x) coming from the high frequency side passed through zero. The sub-peak near the plasmon peaks could be justified by the vanishing real part of the dielectric function. The loss function I (x) for bulk and nanowire is shown on insets: 1 and 2, respectively. As can be seen there, the two main peaks on the bulk phase, inset 1, changed to the one sharp peak on the nanowires phase, inset 2. Moreover, this one sharp peak changed to some peaks on the diameter reductions of nanochains. The amount of I (x) for linear nanochains was similar to the one for dimerized nanochains in three main peaks, except for two shoulders. Zigzag nanochains had four main peaks similar to the ladder, regardless of some noises. The real part of the optical conductivity was related to the imag! inary part of e (x) in the following way: Rerðx; k Þ ¼ 4xp Ime2 ðk; xÞ. The optical conductivity was determined according to the imaginary part of the dielectric function. In Fig. 6, this characteristic is calculated for different frequencies.
Fig. 5. The energy loss for (a) linear, (b) zigzag, (c) dimerized, and (d) ladder nanochains. Inset 1: The energy loss function of bulk Titanium as a function of wavelength obtained from Ref. [33]. Inset 2: The energy loss function of Titanium nanowires as a function of wavelength obtained from Ref. [34].
228
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
In the dimerized configuration, the highest conductivity with the amount of 0.99 appeared at 3.03 eV. For three other configurations, conductivity fluctuation was observed. The highest conductivity for linear, zigzag and ladder nanochains happened at 3.27, 3.94 and 5.43 eV, respectively.
Table 3 Static reflectivity of nanochains. Linear R0 a b
3.3. Reflectivity In this section, the reflectivity and its frequency dependence were calculated from the dielectric function. The reflective index n (x) and extinction coefficient of k (x) were obtained from the imaginary and real parts of e (x) as follows:
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2 e21 þ e22 þ e1 5 ; n¼4 2
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2 e21 þ e22 e1 5 ; k¼4 2
0.07
Dimmer 0.03
Zigzag 0.05
Ladder 0.06
b [1 1 0]a
x
0.4
0.42
ab
bb
xb
6.5
6.5
0.8
[1 0 0]a
Full potential results for nanowire phase from Ref. [34]. Full potential results for bulkphase from Ref. [33].
number of distinct peaks was observed against the NWs (inset 1). These peaks were attributed to the plasmon resonance behavior. As illustrated in Fig. 7, the peaks had less width and more intensity compared with NWs and nanochains structures. All reflection coefficients were stabilized in order to limit a definite value around 6 eV. The interesting parameter of R (x ? 0) is illustrated in Table 3.
2
R¼
ðn 1Þ2 þ k 2
4. Conclusions
2
ðn þ 1Þ þ k
The optical reflectivity of the Ti nanochains is illustrated in Fig. 7. In a work previously conducted by the present author, a featureless spectrum was observed in the bulk phase (inset 2); the
Optical properties like dielectric function, energy loss function, optical conductivity and reflectivity of linear, dimerized, zigzag and ladder nanochains were calculated using the LAPW method within
Fig. 6. Optical conductivity, r (x), of (a) linear, (b) dimerized, (c) zigzag, and (d) ladder nanochains as a function of wavelength.
Fig. 7. The optical reflectivity of (a) linear, (b) zigzag, (c) dimerized, and (c) ladder nanochains as a function of wavelength. Inset 1: The optical reflectivity of Titanium nanowires obtained from Ref. [34]. Inset 2: The optical reflectivity of bulk Titanium obtained from Ref. [33].
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
GGA. The main peak of the dielectric function in all the configurations was less than 4 eV with the energy of about 10 eV. Moreover, for the static dielectric constants of nanochains, it was observed that the linear structure had the biggest one. Also, the results were presented for the energy-loss function and it was observed that all types of nanochains remarkably varied in frequency, which was less than 8 eV. The highest conductivity for linear, zigzag and ladder nanochains happened at 3.27, 3.94, 5.43 eV, respectively. It was also observed that the change from nanowires to nanochains could lead to more intensive peaks in optical reflectivity. Moreover, increased Ec was observed while moving from linear to dimerized, from dimerized to ladder and then from ladder to zigzag structures. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
W.L. Barnes, A. Dereux, T.W. Ebbesen, Nature 424 (2003) 824. Z.Y. Tang, N.A. Kotov, Adv. Mater 17 (2005) 951. Y.N. Xia, N.J. Halas, MRS. Bull. 30 (2005) 338. N.R. Jana, L. Gearhart, C.J. Murphy, Chem. Commun. 617 (2001). L. Polavarapu, Q.-H. Xu, Nanotechnology 19 (2008) 07561. L. Zhiguo, Z. Yuagang, G. Songling, Appl. Surf. Sci. 255 (2009) 5827. S.B. Suh, B.H. Hong, P. Tarakeshwar, S.J. Youn, S. jeong, K.S. kim, Phys. Rev. B 67 (2003) 241402. Y. Zhang, H. Dai, Appl. Phys. Lett. 77 (2000) 3015. Y. Zhang, N.W. Franklin, R.J. Chen, H. Dai, Chem. Phys. Lett. 331 (2000) 35. G. Rubio, N. Agrait, S. Vieira, Phys. Rev. Lett. 76 (1996) 2302. H. Kohno, I. Kikuo, K. Oto, J. Electron Microsc. 54 (2005) i15. Y.W. Chen, Y.H. Tang, L.Z. Pei, C. Guo, L.W. Lin, Mater. Sci. Eng., B 138 (2007) 189. L.Z. Pei, H.S. Zhao, H.Y. Yu, E-j Surf. Sci. Nanotechnol. 6 (2008) 171. A. Dallmeyer, C. Carbone, W. Eberhardt, C. Pampuch, O. Rader, W. Gudat, P. Gambardella, K. Kern, Phys. Rev. B 61 (2000) R5133.
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
[38] [39] [40] [41] [42]
229
C. Ederer, M. Komelj, M. Fahnle, Phys. Rev. B 68 (2003) 052402. J. Hong, R.Q. Wu, Phys. Rev. B 67 (2003) 020406(R). A. Delin, E. Tosatti, Phys. Rev. B 68 (2003) 144434. T. Nautiyal, S.J. Youn, K.S. Kim, Phys. Rev. B 68 (2003) 033407. W. Gousheng, A. Brian, T. Min, C. Aicheng, Electrochem. Commun. 11 (2009) 736. B.J. Inkson, G. Dehm, T. Wagner, J. Microsc. 214 (2004) 252. I. Mukhopadyay, W. Freyland, Langmuir 19 (2003) 1951. A. Nakamura, K. Matsunaga, T. Yamamoto, Y. Ikuhara, Appl. Surf. Sci. 241 (2005) 38. L.s. Zhang, P.x. Zhang, Y. Fang, J. Colloid Interface Sci. 311 (2007). B. Wang, S. Yin, G. Wang, J. Zhao, J. Phys. Condens Matter. 13 (2001) L403. A.Y. Li, R.Q. Li, Z.Z. Zhu, Y.H. Wen, Physica E 30 (2005) 138. Zhi-Li Zhu, Jin-hua Gu, Jia Yu, Physica B 387 (2007) 190. J.C. Tung, G.Y. Guo, Phys. Rev. B 76 (2007) 094413. C. Ataca, S. Cahangirov, E. Durgun, Y.R. Jang, S. Ciraci, Phys. Rev. B 77 (2008) 214413. A. Bala, T. Nautiyal, S. Auluck, J. Magn. Magn. Mater. 321 (2009) 1856. G. Grosso, Solid State Phys., Academic press, 2000. K. Schwarz, P. Blaha, G.K.H. Madsen, Comput. Phys. Commun. 147 (2002) 71. G.V. Rayanor, Rep. Prog. Phys. 15 (1952) 173. M. Jafari, H.R. Hajiyani, Comput. Mater. Sci. 50 (2011) 2549–2553. H.R. Hajiyani, M. Jafari, J. Magn. Magn. Mater. (2011), http://dx.doi.org/ 10.1016/j.jmmm.2011.08.003. P. Hohenberg, W. Kohn, Phys. Rev. B 136. 864 (1964). P. Blaha, K. Schwarz, P. Sorantin, S.B. Rickey, Comput. Phys. Commun. 59 (1990) 399. P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2K, an augmented plane wave plus local orbitals pogram for calculating crystal properties, Vienna University of Technology, Vienna, Austria, 2001. J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. J.P. Perdew, S. Kurth, A. Zupan, P. Blaha, Phys. Rev. B 82 (1999) 2544.
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
First-principles calculations of optical properties of Titanium nanochains Mahmoud Jafari ⇑, Hamid reza Hajiyani, Zeinab Sohrabikia, Habibeh Galavani Department of Physics, K.N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 13 October 2011 Received in revised form 29 December 2012 Accepted 14 April 2013 Available online 24 May 2013 Keywords: Titanium Nanochains Nanowires Optical properties
a b s t r a c t In this paper, optical properties of linear, dimerized, zigzag and ladder nanowires were reported using the full potential linearized augmented plane wave plus local orbital method (FLAPW + LO) and by incorporating the generalized gradient approximation (GGA). Also, the super cell approach, infinite and free standing periodic nanowires were applied. Optical properties such as dielectric function, energy loss function, optical conductivity and reflectivity of nanochains were investigated. Moreover, the results of nanochains were compared with the findings of the present author’s previous works on the bulk and nanowires (NWs). The properties, especially reflectivity and static dielectric function, changed significantly with configuration. The static dielectric function increased slightly from 1.9 in the dimerized nanowire to 3.06 in the linear one. The negative value of the real dielectric function occurred in some frequencies of the dimerized nanochains. Furthermore, equilibrium structure and inter-atomic distance were calculated, the results of which were in agreement with those of the previous works. In addition, nanowire reflectivity spectra showed a number of distinct peaks, which could be attributed to the plasmon resonance. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction In nanotechnology, nanochains have fundamental, unique and practical properties. Fabrication of nanoarrays, nanowires (NWs) and nanochains is of great importance due to their applications in sensors and optoelectronics [1,2]. Because of the strong dependence of electrical, magnetic and optical properties of nanochains on the geometry and configuration of nanowires [3], many pieces of research have been carried out on this topic. In experimental terms, metallic, nonmetallic and compound nanowires and nanochains have been fabricated both chemically and physically [4]. In the chemical fabrication, Ag nanochains have been synthesized by the method involving aqueous chloroaurate ion [AuCl4] with sodium borohydride [NaBH4] in the presence of an amino acid [5] or by the modified citrate reduction process [6]. Modern methods have made it feasible to investigate low-dimensional nanowires in order to reveal the effect of low dimensionality on the electrical properties. Recently, the growth of nanochains has been tested on the substrates of different shapes and a range of materials have been successfully fabricated. Ag nanochains can be grown inside organic nanotubules [7] and other experiments have demonstrated the possibility of growing Ti nanochains on SWNT [8,9]. Short suspended nanowires have been fabricated via sweeping the metallic surface by the tip of a scanning tunneling microscope [10], in which synthesis of nonmetallic nanochains was ⇑ Corresponding author. Tel.: +98 21 22853308; fax: +98 21 22853650. E-mail address: [email protected] (M. Jafari). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.04.029
achieved like a metallic nanochain. Si nanochains were grown using the hydrothermal chemical method in [11,12] or via a selforganized vapor–liquid–solid process in [13]. During this period, theoretical calculations of electronic, magnetic and optical properties of nanochains were carried out using methods such as molecular dynamics, semiempirical tight binding or ab initio density functional theory. The electrical and magnetic properties of metallic nanochains such as Co [14], Fe [15], Ti [16], Au [17] and Ag [18] have been also reported. Titanium, as a member of the 3d transition metal group, has shown an exceptional behavior in both of its bulk and nanowire forms. Ti nanowires (TiNWs) have been manufactured by several different methods [19,20]. TiNWs were deposited on paralytic graphite by electrochemical methods in [21] or they were synthesized on sapphire single crystals in [22]. In addition, Zhang et al. reported the fabrication of Ti NW on anodic aluminum oxide [23]. Many calculations have been carried out on the simulation of nanochains. At first, some simulations were investigated using the Monte Carlo approach by Wang et al. [24]. After that, Li et al. studied different structures of Ti nanowires and found the zigzag arrangement of atoms the most stable [25]. Zhu et al. calculated the magnetic moment of linear Ti nanochains by an ab initio approach [26]. They also calculated the optimized length of linear nanochains and found Ti nanochains had 0 0.61 mb/atom, separated by an equilibrium bond length of 2.2 Å A. The magnetic moments of zigzag and linear nanochains were later reported in [27,28] which showed that all the 3d transition metals in both linear and zigzag structures had stable and metastable ferromagnetic states (FM). It was also shown that geometry of the
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
225
In this article, the imaginary and real parts of dielectric function, energy loss function, optical conductivity and, finally, reflectivity were calculated for four types of linear, dimerized, zigzag and ladder nanochains. These results were compared with those of the author’s previous work, optical properties of a, b and x bulk phases of Titanium [33] and also optical properties of b [1 1 0] and x [1 0 0] nanowires [34]. 2. Computational method Fig. 1. The structure and atomic position of zigzag, dimerized, ladder and linear nanochains, respectively. d is the length of unit cell. Atomic indication (1, 2) is the target atom that is necessarily for atomic simulation.
nanochains played a key role in the magnetic moments of nanochains and that the zigzag nanowires were the most stable configuration with 0.571 mb/atom for the FM state zigzag nanochains. Ataca et al. put emphasis on studying infinite periodic linear, dimerized and zigzag nanochains as well as the finite short segments [28]. Recently, the magnetic moment of zigzag, linear, dimerized and ladder nanowires has been studied and the optimized interatomic distance has been calculated [29]. The optical properties of solids [30] are of wide interest because of their diverse applications in industrial and optoelectronic devices [31,32].
Table 1 Distances d, atomic component and the number of symmetric and optimized parameters for different structures of Ti nanochains. Structure
d (Å)
Coordinate (Å) x/a, y/b, z/c
RMT (Å)
Number of sym
kPoint
Rmt Kmax
Linear
2.23, 2.11a, 2.10b 4.82, 4.74a, 4.87b
0, 0, 0
1.92
16
1500
8
0, 0, 0
1.74
8
700
8.5
2.21
2
500
8
1.95
4
1000
7.5
Dimerized
Zigzag
2.6, 2.41a,b
Ladder
a
2.27, 2.14 , 2.63b
0, 0, 0, 0,
0, 0.4 0, 0 0.17, 0.50 0, 0
0, 0.21, 0 a b
Pseudo potential results from Ref. [29]. Pseudo potential results from Ref. [25].
All the calculations were performed within the framework of density functional theory [35,36] using the full-potential linearized augmented plane wave plus local orbital (FLAPW + LO) in the Wien2k package [37]. Exchange and correlation effects were treated using the generalized gradient approximation (GGA) [38,39] with the formal Perdew–Burke–Ernzerhof0 (PBE) parameterization scheme [40–42]. A distance of about 10 Å A was used between the nanowires in order to eliminate their interaction. The charge convergence of 104 e was used for optimizing the parameters in the SCF cycle. Fig. 1 shows the structure and atomic position of zigzag, dimerized, ladder and linear nanochains. The required atomic coordinates for simulating nanowires are listed in Table 1. Cohesive energy Ec was obtained using the following equation Ec = Ea–ET/N, in which Ea is ground state energy of the free constituent atom, ET is the total energy per unit cell of the given nanochains structure and N is the number of atom in the unit cell. The unit cell includes one atom for linear nanochains and two more for the other three. It is worth mentioning that Ec increased with increase in the number of the nearest neighbors from linear to zigzag structure while there was an increase in the number of bonds. The number of the nearest neighbors was two for linear and dimerized and three for ladder and zigzag structures. Fig. 2 demonstrates the variation of cohesive energy with the length of unit cell (d). Present results were smaller than those in the previous works [25,29]; this difference in Ec values could depend on the different method used in the previous reports. Full potential calculation was used in the current work whereas previous studies [25,29] used pseudopotential calculation. The optimized atomic coordinates of nanochains were calculated, which were in good agreement with the findings of the previous reports [26,27,29]. Moreover, in order to obtain reliable calculations, a number of convergence tests were performed to optimize the input parameters. The parameters, which had to be tested, mainly included k-points and Rmt Kmax.
Fig. 2. The variation of cohesive energy with the length of unit cell (d).
226
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
As can be seen in Table 1, an optimized number of k-points were used; while this value was only 700 for the dimerized nanochains, it reached 1500 points for the linear nanochains. In the combination of spherical harmonic inside non-overlapping spheres surrounding the atomic sites, the so-called muffintin (MT) spheres and, in a Fourier series in the interstitial region in the MT sphere, the basic function of Rmt Kmax were expanded for the Eigen-values convergence. Kmax is the maximum modulus for the reciprocal lattice vector and Rmt is the average radius of the MT sphere. The optimized amount of Rmt Kmax is listed in Table 1. The MT radii were used (as listed in Table 1) for each nanochains. As can be observed, RMT had the minimum and maximum values of 1.74 and 2.21 for dimerized and ladder nanochains, respectively. 3. Results and discussion Optical properties of both metals and insulators were investigated theoretically and experimentally and electromagnetic radiation was used for probing the surface as a function of wave vector ! of k and frequency of x. In this work, optical properties such as dielectric function, optical conductivity, energy loss function and reflectivity of the zigzag, linear, dimerized and ladder nanowires were studied. In theoretical terms, by solving the Schrödinger equation in the presence of an electromagnetic field and by ignoring the nonlinear effect using an ab initio approach, a description of the optical properties was obtained. 3.1. Dielectric function !
The direct dependence of the dielectric function on k and x provided a unified description of the charge response of both met! als and insulators. If the dielectric function e ( x , k ) was separated ! ! ! into real and imaginary parts, as e (x, k ) = e1 (x, k ) + ie2 (x, k ), the dielectric function of the crystal for x > 0 could be expressed by the Hamiltonian [30]: 2 ! 4pe2 X ! ! e2 ðx; k Þ ¼ 2 2 hl; k jpx j k ; mi f ! ð1 f ! ÞdðE ! E ! hxÞ m x k;l;m lk mk m; k l; k !
!
where l, m denotes the band indices, k is the wave vector, jl; k i is the ! Eigen state with the wave vector k and l and E ! is the correspondl; k ing Eigen value of the wave function. Also, f ! is the occupation l; k number and x indicates the component of the momentum transfer. ! The k dependence of the equation was negligible owing to its nor-
mal frequency. At normal frequencies from infrared to ultraviolet, the wave vector was small compared with the dimension of the ! Brillouin zone. This was also the justification for neglecting the k dependence of the dielectric function, which can be written as e (x). The real part of the dielectric function was obtained by inserting the above equation in the Kamer–Kronig [30] relation as follows: !
e1 ðx; k Þ ¼ 1 þ
2 f ! ð1 f ! Þ 4pe2 X ! ! lk mk hl; k jp jk ; mi x m2 x2 k;l;m ðE ! E ! hxÞ m; k l; k
As observed in Fig. 3a–d, the imaginary part of the dielectric function was depicted for linear, dimmer, zigzag and ladder nanochains, respectively. A noteworthy aspect here was the conspicuous differences in the intensities of the peaks. The intensity of the main peak reached around 7.3 eV in linear nanochains while it was equal to 8.55 eV and 8.6 eV in dimerized and ladder structures, respectively. The main peak in the zigzag configuration reached the substantial value of 9.81 eV. In Fig. 4, the real part of dielectric functions of nanochains is illustrated. Overall, e1 (x) leveled off to the defined values of e1 (1) after some fluctuations. Among all types of nanochains, just the dimerized one had a negative dielectric function in some frequencies. In this configuration, e1 (x) was negative from 3.08 eV to 3.27 eV and reached its minimum value of 0.45 eV at around 3.16 eV. e1 (x) in x ? 0 is the static dielectric constant and the calculated values are given in Table 2. Among the nanochains, the linear one had the largest static dielectric constant value of 3.06. e1 (0) increased by 1.16 eV and 0.19 eV from the dimerized to the linear nanochains and from zigzag to ladder nanochains, respectively. The quantity of e1 (1) is listed in Table 2 for various nanochains. The dielectric function leveled off to zero at around 6 eV and 8 eV for the zigzag and dimerized and for ladder nanochains, respectively. The optical dielectric constant, the maximum and minimum and limits of the real part of dielectric constant are presented in Table 2. 3.2. Dielectric energy loss function and optical conductivity While neglecting retardation and surface effects, the electronenergy-loss function I (x) could be given by: !
IðxÞ ¼ Im
1 !
eðk ; xÞ
¼
e2 ðk ; xÞ !
e22 ðk ; xÞ þ e22 ðk; xÞ
Fig. 3. Imaginary part of dielectric function, e2 (x), for (a) linear, (b) dimerized, (c) zigzag and (d) ladder nanochains.
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
227
Fig. 4. Real parts of dielectric function, e1 (x), for (a) linear, (b) zigzag, (c) dimerized, and (d) ladder nanochains, respectively. Inset 1: Real parts of dielectric function of bulk Titanium as a function of wavelength obtained from Ref. [33]. Inset 2: Real parts of dielectric function of b [1 1 0] Titanium nanowires obtained from Ref. [34].
Table 2 The optical and extreme limitation of real part of dielectric constant. Structure
Linear Dimmer Zigzag Ladder Bulk Nanowire a b c d
Full Full Full Full
Real part of dielectric boundary condition in addition to frequency
e1 (0)
e1max, x
e1min, x
ei (1)
3.06 1.90 2.47 2.66 144.7a 63.4b 14.0c 14.3d
3.06, 0.80 2.54, 1.83 2.89, 1.0 3.38, 0.86 144.7a, 0.0 52.0b, 0.7 14.0c, 0.0 14.3d, 0.0
0.07, 3.38 0.45, 3.16 0.47, 5.40 0.40, 5.69 9.0a, 4.7 7.0b, 3.9 0.15c, 4.8 0.6d, 6.2
0.92 0.92 0.84 0.85 0.0a 0.0b 1.0c 0.80d
potential potential potential potential
results results results results
for for for for
x phase from Ref. [33]. b phase from Ref. [33]. b {1 1 0} TiNWs from Ref. [34]. x {1 0 0} TiNWs from Ref. [34].
With hx as the loss energy, as can be seen in Fig. 5, all types of nanochains had a remarkable variation in frequency, which was less than 10 eV. For the nanochains, the curve showed a significant
peak due to plasmon resonance electrons. With this energy, the real part of the dielectric function Re e (x) coming from the high frequency side passed through zero. The sub-peak near the plasmon peaks could be justified by the vanishing real part of the dielectric function. The loss function I (x) for bulk and nanowire is shown on insets: 1 and 2, respectively. As can be seen there, the two main peaks on the bulk phase, inset 1, changed to the one sharp peak on the nanowires phase, inset 2. Moreover, this one sharp peak changed to some peaks on the diameter reductions of nanochains. The amount of I (x) for linear nanochains was similar to the one for dimerized nanochains in three main peaks, except for two shoulders. Zigzag nanochains had four main peaks similar to the ladder, regardless of some noises. The real part of the optical conductivity was related to the imag! inary part of e (x) in the following way: Rerðx; k Þ ¼ 4xp Ime2 ðk; xÞ. The optical conductivity was determined according to the imaginary part of the dielectric function. In Fig. 6, this characteristic is calculated for different frequencies.
Fig. 5. The energy loss for (a) linear, (b) zigzag, (c) dimerized, and (d) ladder nanochains. Inset 1: The energy loss function of bulk Titanium as a function of wavelength obtained from Ref. [33]. Inset 2: The energy loss function of Titanium nanowires as a function of wavelength obtained from Ref. [34].
228
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
In the dimerized configuration, the highest conductivity with the amount of 0.99 appeared at 3.03 eV. For three other configurations, conductivity fluctuation was observed. The highest conductivity for linear, zigzag and ladder nanochains happened at 3.27, 3.94 and 5.43 eV, respectively.
Table 3 Static reflectivity of nanochains. Linear R0 a b
3.3. Reflectivity In this section, the reflectivity and its frequency dependence were calculated from the dielectric function. The reflective index n (x) and extinction coefficient of k (x) were obtained from the imaginary and real parts of e (x) as follows:
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2 e21 þ e22 þ e1 5 ; n¼4 2
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2 e21 þ e22 e1 5 ; k¼4 2
0.07
Dimmer 0.03
Zigzag 0.05
Ladder 0.06
b [1 1 0]a
x
0.4
0.42
ab
bb
xb
6.5
6.5
0.8
[1 0 0]a
Full potential results for nanowire phase from Ref. [34]. Full potential results for bulkphase from Ref. [33].
number of distinct peaks was observed against the NWs (inset 1). These peaks were attributed to the plasmon resonance behavior. As illustrated in Fig. 7, the peaks had less width and more intensity compared with NWs and nanochains structures. All reflection coefficients were stabilized in order to limit a definite value around 6 eV. The interesting parameter of R (x ? 0) is illustrated in Table 3.
2
R¼
ðn 1Þ2 þ k 2
4. Conclusions
2
ðn þ 1Þ þ k
The optical reflectivity of the Ti nanochains is illustrated in Fig. 7. In a work previously conducted by the present author, a featureless spectrum was observed in the bulk phase (inset 2); the
Optical properties like dielectric function, energy loss function, optical conductivity and reflectivity of linear, dimerized, zigzag and ladder nanochains were calculated using the LAPW method within
Fig. 6. Optical conductivity, r (x), of (a) linear, (b) dimerized, (c) zigzag, and (d) ladder nanochains as a function of wavelength.
Fig. 7. The optical reflectivity of (a) linear, (b) zigzag, (c) dimerized, and (c) ladder nanochains as a function of wavelength. Inset 1: The optical reflectivity of Titanium nanowires obtained from Ref. [34]. Inset 2: The optical reflectivity of bulk Titanium obtained from Ref. [33].
M. Jafari et al. / Computational Materials Science 77 (2013) 224–229
GGA. The main peak of the dielectric function in all the configurations was less than 4 eV with the energy of about 10 eV. Moreover, for the static dielectric constants of nanochains, it was observed that the linear structure had the biggest one. Also, the results were presented for the energy-loss function and it was observed that all types of nanochains remarkably varied in frequency, which was less than 8 eV. The highest conductivity for linear, zigzag and ladder nanochains happened at 3.27, 3.94, 5.43 eV, respectively. It was also observed that the change from nanowires to nanochains could lead to more intensive peaks in optical reflectivity. Moreover, increased Ec was observed while moving from linear to dimerized, from dimerized to ladder and then from ladder to zigzag structures. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
W.L. Barnes, A. Dereux, T.W. Ebbesen, Nature 424 (2003) 824. Z.Y. Tang, N.A. Kotov, Adv. Mater 17 (2005) 951. Y.N. Xia, N.J. Halas, MRS. Bull. 30 (2005) 338. N.R. Jana, L. Gearhart, C.J. Murphy, Chem. Commun. 617 (2001). L. Polavarapu, Q.-H. Xu, Nanotechnology 19 (2008) 07561. L. Zhiguo, Z. Yuagang, G. Songling, Appl. Surf. Sci. 255 (2009) 5827. S.B. Suh, B.H. Hong, P. Tarakeshwar, S.J. Youn, S. jeong, K.S. kim, Phys. Rev. B 67 (2003) 241402. Y. Zhang, H. Dai, Appl. Phys. Lett. 77 (2000) 3015. Y. Zhang, N.W. Franklin, R.J. Chen, H. Dai, Chem. Phys. Lett. 331 (2000) 35. G. Rubio, N. Agrait, S. Vieira, Phys. Rev. Lett. 76 (1996) 2302. H. Kohno, I. Kikuo, K. Oto, J. Electron Microsc. 54 (2005) i15. Y.W. Chen, Y.H. Tang, L.Z. Pei, C. Guo, L.W. Lin, Mater. Sci. Eng., B 138 (2007) 189. L.Z. Pei, H.S. Zhao, H.Y. Yu, E-j Surf. Sci. Nanotechnol. 6 (2008) 171. A. Dallmeyer, C. Carbone, W. Eberhardt, C. Pampuch, O. Rader, W. Gudat, P. Gambardella, K. Kern, Phys. Rev. B 61 (2000) R5133.
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
[38] [39] [40] [41] [42]
229
C. Ederer, M. Komelj, M. Fahnle, Phys. Rev. B 68 (2003) 052402. J. Hong, R.Q. Wu, Phys. Rev. B 67 (2003) 020406(R). A. Delin, E. Tosatti, Phys. Rev. B 68 (2003) 144434. T. Nautiyal, S.J. Youn, K.S. Kim, Phys. Rev. B 68 (2003) 033407. W. Gousheng, A. Brian, T. Min, C. Aicheng, Electrochem. Commun. 11 (2009) 736. B.J. Inkson, G. Dehm, T. Wagner, J. Microsc. 214 (2004) 252. I. Mukhopadyay, W. Freyland, Langmuir 19 (2003) 1951. A. Nakamura, K. Matsunaga, T. Yamamoto, Y. Ikuhara, Appl. Surf. Sci. 241 (2005) 38. L.s. Zhang, P.x. Zhang, Y. Fang, J. Colloid Interface Sci. 311 (2007). B. Wang, S. Yin, G. Wang, J. Zhao, J. Phys. Condens Matter. 13 (2001) L403. A.Y. Li, R.Q. Li, Z.Z. Zhu, Y.H. Wen, Physica E 30 (2005) 138. Zhi-Li Zhu, Jin-hua Gu, Jia Yu, Physica B 387 (2007) 190. J.C. Tung, G.Y. Guo, Phys. Rev. B 76 (2007) 094413. C. Ataca, S. Cahangirov, E. Durgun, Y.R. Jang, S. Ciraci, Phys. Rev. B 77 (2008) 214413. A. Bala, T. Nautiyal, S. Auluck, J. Magn. Magn. Mater. 321 (2009) 1856. G. Grosso, Solid State Phys., Academic press, 2000. K. Schwarz, P. Blaha, G.K.H. Madsen, Comput. Phys. Commun. 147 (2002) 71. G.V. Rayanor, Rep. Prog. Phys. 15 (1952) 173. M. Jafari, H.R. Hajiyani, Comput. Mater. Sci. 50 (2011) 2549–2553. H.R. Hajiyani, M. Jafari, J. Magn. Magn. Mater. (2011), http://dx.doi.org/ 10.1016/j.jmmm.2011.08.003. P. Hohenberg, W. Kohn, Phys. Rev. B 136. 864 (1964). P. Blaha, K. Schwarz, P. Sorantin, S.B. Rickey, Comput. Phys. Commun. 59 (1990) 399. P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2K, an augmented plane wave plus local orbitals pogram for calculating crystal properties, Vienna University of Technology, Vienna, Austria, 2001. J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. J.P. Perdew, S. Kurth, A. Zupan, P. Blaha, Phys. Rev. B 82 (1999) 2544.