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Ab initio calculation of vibrational frequencies of AsO glass
Journal of Non-Crystalline Solids 356 (2010) 428–433
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Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Ab initio calculation of vibrational frequencies of AsO glass Ahmad Nazrul Rosli, Noriza Ahmad Zabidi, Hasan Abu Kassim, Keshav N. Shrivastava * Department of Physics, University of Malaya, Kuala Lumpur 50603, Malaysia
a r t i c l e
i n f o
Article history: Received 13 May 2008 Available online 29 December 2009 Keywords: Raman scattering Chalcogenides Ab initio Raman spectroscopy
a b s t r a c t We have used the density-functional theory to make models containing arsenic and oxygen atoms. The structures are optimized for the minimum energy of the Schrödinger equation. In this way, we obtain the bond distances and angles of the stable structures. We obtain the vibrational frequencies of each cluster. The calculated vibrational frequencies are compared with those found in the experimental Raman spectra. The values of the vibrational frequencies calculated for AsO2 ; AsO4 ðT d Þ; AsO2 (rectangular), AsO2 (triangular) and AsO3 (pyramidal) agree with those found in the Raman spectra of vitreous aresenic oxide, indicating that these clusters are really present in the arsenic oxide glass. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Recently, it was pointed out by Phillips [1] that anomalous properties exist over a narrow range of compositions in the molecular glasses. It was shown earlier that long relaxation times occur in glasses [2,3]. In fact, some of the phonon frequencies can become soft and approach towards zero [4]. There is a phase transition in the rigidness of glasses [5]. The random network of atoms is considered to be important for the understanding of glasses. In recent years, we have made clusters of atoms by using densityfunctional theory. The clusters of Ge atoms, S atoms and I atoms were made. Similarly, the clusters of Gex Iy ; Gex Sy ; Gex Sy Iz were made [6]. It was found that the Raman frequencies of Ge0:25 S0:75 Iy glass were the same as those calculated for a few clusters of atoms, indicating that the glass is made of random collection of clusters. In some cases, there was a linear arrangement of atoms, the frequencies of which were the same as found in the experimental data. This means that linear arrangement of atoms exists along with the clusters which give strength to the glass. Therefore, clusters of atoms along with linear molecules describe the structure of the glassy state. We repeated the calculation of clusters of Gex Px S12x glass in as far as the structural parameters such as the bond lengths and the vibrational frequencies are concerned and found that clusters along with linear molecules are present in the glass [7]. The example of Gex Se1x also showed [8] that clusters such as Ge4 Se3 ; Ge2 Se2 ; Ge3 ; GeSe3 and Ge3 Se2 are present in the glass. Therefore, we believe that cluster calculations are important for the understanding of the glassy state. The density-functional theory has been developed by Kohn and Sham [9]. It is possible to write the Schrödinger equation in the form of electron density which depends on the electron coordi* Corresponding author. Tel.: +603 7967 7140; fax: +603 7967 4146. E-mail address: [email protected] (K.N. Shrivastava). 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2009.11.041
nates. The derivative of the Schrödinger equation with respect to density leads to another equation which can be numerically integrated. The form of the Schrödinger equation as a function of density requires approximations. The local density approximation (LDA) is very popular and will be used in the present work. It is also possible to take the generalized gradient approximation (GGA) in which account of the nonlocal atoms is taken by introducing a linear gradient. In this way, we can calculate the energies in the GGA. There are no adjustable parameters in this calculation. The Coulomb interactions between electrons and nuclei are used. This kind of first principles calculations are very useful in predicting electronic properties. Our calculations of vibrational frequencies appear to be in reasonable agreement with the experimental data [6–8]. The first principles calculations of the band structures of AgSbS, AgSbSe and AgSbTe are found to be useful for improving the thermoelectric properties [10], Similarly, it is found that DFT gives good results for the geometry optimization of PbTe [11]. In SiO, it is found that Si atoms switch bonds from one O atom to another leading to expansion of SiO rings. The calculation is performed for multimillion to billion atoms by means of automated computer simulation of SiO nanometer size particles [12]. Arsenic is found in several different valencies. The arsenic trioxide As2 O3 dissolves in alkaline solutions to make arsenites. It reacts with oxidizing agents to form As2 O5 . Albanesi et al. [13] have found that As2 O3 occurs in two structural forms which separate in a phase transition. The cubic arsenolite has eight As4 O6 molecules, per unit cell, each made of AsO3 pyramids linked together by oxygen atoms. The layered claudetite also has AsO3 pyramids. The As2 O5 has both the AsO4 tetrahedra as well as AsO6 octahedra linked by corners so that there is a network of clusters of atoms. The tight binding approximation has been discussed for both the tetrahedral as well as the octahedral configurations by Passeggi et al. [14]. The vibrational excitations of arsenic oxide have been reported [15–18] in vitreous and disordered
A.N. Rosli et al. / Journal of Non-Crystalline Solids 356 (2010) 428–433
phases. The polarized Raman spectra with second-order scattering are reported by Galeener et al. [16]. In this paper, we report our calculation of vibrational frequencies of several clusters of arsenic and oxygen atoms. We optimize the configuration of clusters for which the energy of the Schrödinger equation is a minimum. We have performed the calculation for 26 different clusters made of As and O atoms. In all of the cases we obtained the optimum geometry as well as the vibrational frequencies. Our calculations are based on the kinetic energy of the electrons and that of the nuclei and all of the Coulomb interactions with no adjustable parameters. We compare the values calculated from the first principles with those from the experimental Raman spectra. The values calculated for the clusters are found to be in reasonable agreement with those measured. Therefore, we find that there are clusters in the vitreous As2 O3 . The glassy state is therefore made from the random network of clusters.
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(x) 2. Clusters We use the density-functional theory in the local density approximation (LDA). The Kohn–Sham equations are solved and vibrational frequencies are deduced. The bond lengths and angles are obtained for the minimum energy [9]. The computer program 3 is the DMol of Accelrys. It is the ‘‘windows” version run on duo pentium. We can choose any of the following wave functions. The ‘‘min” selects only one Slater orbital on each atom which is occupied. The double numeric (DN) means at least two atomic orbitals for each one occupied in the free atom. The ‘‘DNP” means DN with polarization. The DND chooses double-numerical+d-DNP basis except that no p wave functions are used on hydrogen. There are several options but the DN seems to give values which agree with the experimental data. The DN wave functions are used throughout in most of the calculations performed by us. (i) As–O. This is a diatomic cluster. In the optimized configuration the distance between the two atoms is found to be 1.698 Å. There is a single vibrational frequency belonging to the bond stretching. The vibrational frequency is 881:4 cm1 . (ii) AsO2 . In this molecule for the optimized geometry, the bond angle is found to be 122.18°. The As–O bond length is found to be 1.719 Å. The vibrational frequencies (intensities) are 219.8 (19.9), 753.7 (13.2) and 754:1ð1:59Þ cm1 ðkm=molÞ. (iii) AsO2 (linear). It is possible to optimize this molecule in a linear form so that O–As–O angle is 180°. The As–O bond length is found to be 1.734 Å. The vibrational frequency is calculated to be 753:8 cm1 . (iv) AsO2 (triangle). It is also possible to optimize AsO2 in the shape of a triangle. The As–O distance is 1.896 Å and O–O distance is 1.590 Å. The vibrational frequencies (intensities) are found to be 535.8 (11), 544.7 (11.8) and 855:1ð7:4Þ cm1 ðkm=molÞ. (v) AsO3 (triangular). In this cluster the three oxygen atoms are at the corners of a triangle and As is in the centre. In the optimized configuration, the As–O distance is 1.662 Å and the vibrational frequencies(intensities) are 156.3 (3.5), 159.5 (3.3), 237.1 (24.7), 819.1 (2.2), 820.2 (2.1) and 826:9ð0:02Þ cm1 ðkm=molÞ. The vibrational spectrum calculated for this cluster from the first principles is given in Fig. 1. (vi) AsO3 (pyramidal). The O–O distance is 1.865 Å in the O3 triangle. The As atom sits on top of the triangle to form a pyramid. The As–O distance is 2.250 Å. It has strong vibrations at 280.3 (5.28), 532.1 (0.91){2} and
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(xii)
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701:8ð11:8Þ cm1 ðkm=molÞfdegeneracyg. The vibrational spectrum calculated from the first principles is shown in Fig. 2. AsO4 T d . This is a tetrahedral molecule. The As–O distance is 1.759 Å. The dominant mode occurs at 335:1 cm1 and it is triply degenerate. As2 O. This molecule has As–O–As angle of 71.36° and bond length is 1.971 Å. The vibrational frequencies (intensities) are 195.4 (6.2), 305.7 (6.09), 605:8ð22:9Þ cm1 ðkm=molÞ. The calculated spectrum for this model is shown in Fig. 3. As2 O (linear). We find that with O in the centre and two As on both sides to form a straight line is stable. The As–O bond distance is 1.759 Å. There is a doubly degenerate vibration at 233.2 (20.8){2} and a singly degenerate oscillation at 981:6ð22:2Þ cm1 ðkm=molÞ. As2 O2 (rectangular). The As–O bond length is 1.871 Åand the angle of the rectangle is 100.5°. There are two strong frequencies at 296.5 (39.7) and 489:7ð48:2Þ cm1 ðkm=molÞ. As2 O2 (pyramidal). In this cluster, the As–As distance is 2.348 Å, the O–O distance is 1.952 Å and the As–O distance is 2.517 Å. The vibrational frequencies (intensities) are 328.9 (15.3), 376.3 (9.6), 384.5 (6.3), 573.8 (52.3), 643:4ð19:5Þ cm1 ðkm=molÞ. The computed vibrational spectrum is shown in Fig. 4. As2 O3 (linear). In this system all of the atoms are in a straight line with oxygen atoms at both ends. The end bond length is 1.721 Åand the inner bond length is 1.968 Å. The end correction is thus clearly seen in the structure. This system has the positive vibrational frequencies (intensities) fdegeneraciesg as 31.6 (2.66){2} , 327.3 (159), 782:4ð240Þ cm1 ðkm=molÞ. As2 O3 (bipyramidal). In this system, first a triangle is made with three oxygen atoms and then one As is put on top of the triangle and another is put symmetrically on the other side to form a bipyramid. The O–O distance in the triangle is 2.496 Åand the As–O distance is 1.927 Å. There are two strong vibrations at 375.7 (39.7){2}, 561.1 (154), 562.6 (34.9){2}. The later two are not well resolved.
Fig. 1. The vibrational spectrum of AsO3 (triangular) calculated from the first principles.
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Fig. 2. The vibrational spectrum of AsO3 (pyramidal) calculated from the first principles. Fig. 4. The vibrational spectrum of As2 O2 (pyramidal) calculated from the first principles.
Fig. 3. The vibrational spectrum of As2 O calculated from the first principles.
(xiv) As2 O4 (bipyramidal). First we make a square with four oxygen atoms and then put one As on top and another symmetrically in the bottom of the square. The O–O distance is 2.260 Åand the As–O distance is 2.036 Å. The vibrational frequencies (intensities) fdegeneraciesg are found to be 138.4 (97.8){2} and 443.9 (774){2}.
(xv) As3 O (triangular). The three As atoms are on the corners of a triangle and one O is in the centre. The As–O distance is 1.868 Å. The vibrational frequencies found are 131.8 (4.3), 671.5 (12.3) and 693:9ð10:6Þ cm1 ðkm=molÞ. (xvi) As3 O (pyramidal). In this model the three As form a triangle with O–O distance of 2.477 Åand one As atom is on top to form a pyramid. The As–O distance is 2.151 Å. The vibrational frequencies (intensities) are 235.9 (0.7), 300.6 (2.95) and 491:5ð17:7Þ cm1 ðkm=molÞ. (xvii) As3 O2 (linear). In this cluster all of the five atoms are in a straight line with As on two ends. The bond distance near the ends is 1.753 Åand towards the centre it is 1.919 Å. There are strong oscillations at the frequencies (intensities) of 23.9 (0.91){2}, 238.8 (5.5), 291.6 (44.3){2}and 760.2 (131). (xviii) As3 O3 (ring). In this cluster, three As and three O atoms are arranged, alternately, so as to form a ring. The As–O bond length is 1.830 Åobtained by using double numeric (DN) wave functions. Hence, the ring diameter is 3.66 Å. The rings and holes are often formed in the thin films of arsenic oxide glass. Hence, it is often relevant to experimental conditions. The vibrational spectrum calculated for this cluster is shown in Fig. 5. The calculated frequencies (intensities) fdegeneraciesg are as follows: 88.94 (0.01){2}, 151.5 (47), 183.4 (7.9){2}, 363.7 (30.5){2}, 696:2ð0:4Þf2g cm1 ðkm=molÞ. (xix) As3 O4 (linear). In this cluster, all of the atoms are in a straight line with oxygen atoms at the ends. The As always have two oxygen atoms as neighbors. There is clearly the effect of ends so that at the end, the bond length is 2.076 Å which successively reduces upon going towards the centre. So it becomes 1.853 and
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Fig. 5. The vibrational spectrum of As3 O3 (ring) calculated from the first principles.
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1.714 Åat the centre. There are several weak oscillations which correspond to internal motion of the atoms and there is a strong vibration at 807:5 cm1 which is due to the uniform stretching or contracting motion. The various frequencies (intensities) fdegeneraciesg are found to be 17.47 (1){2}, 160.2 (18.5), 373.2 (39.5), 807:5ð620Þ cm1 ðkm=molÞ. As4 O (square). The four As atoms are located on the corners of a square with perfect 90° angles. The O atom is located in the centre of the square. The As–O distance is 2.008 Å. Its strong vibration occurs at 252:1 cm1 . As4 O (pyramidal). The four As atoms form a square of As–As square edge length 2.484 Å. One O atom is on top position with As–O distance of 2.323 Å. It has a weak doubly degenerate oscillation at 225.9 (0.26), a medium strength oscillation at 243.1 (2.92) and a strong oscillation at 397:7ð11:3Þ cm1 ðkm=molÞ. As4 OðT d Þ. In this cluster the O atom is in the centre of a tetrahedron with As–O distance 1.884 Å with a strong triply degenerate oscillation at 721:7 cm1 . As4 O2 (bipyramidal). The four As atoms form a square with As–As distance of 2.554 Åand two O atoms sit on top and bottom positions with As–O distance 2.262 Å. There is a weak vibration at 239:2ð2:13Þf2g cm1 ðkm=mol:Þ and a strong oscillation at 379:6 cm1 . As4 O3 (linear). In this cluster the As and O atoms are arranged alternately with As atoms at the two ends. The distances are 1.781 Å at the end, 1.852 at one position earlier than end and 1.753 Åat the centre. The oscillation frequencies (intensities) {degeneracies} are found to be 157.03 (2.8){2}, 180.8 (0.03), 310.9 (65.1){2}, 594.5 (45) and 774:3ð57Þ cm1 ðkm=molÞ. The calculated spectrum is shown in Fig. 6. This is an example of strong internal vibrations within the system.
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Fig. 6. The vibrational spectrum of As4 O3 (linear) calculated from the first principles.
(xxv) As4 O4 (cube). In a simple cubic geometry, there are two As and two O atoms in a plane and then in the next plane As and O are interchanged so that the two layers form a cube. The As–O bond length is 2.041 Å. The vibrational spectrum shows triply degenerate oscillations at the frequencies (intensities) fdegeneraciesg of 181.05 (2.5){3}, 298.3 (7.2){3} and 440.1 (31.2){3}. The calculated spectrum is shown in Fig. 7. (xxvi) As4 O4 (ring). The As and O atoms are arranged alternately to form a ring with eight sides. The bond length is 1.807 Å. This ring has several vibrational modes the frequencies (intensities) of which are 130.1 (0.43), 130.4 (0.58), 137.5 (68), 243.4 (51), 245.1 (50), 274.8 (0.54), 306.6 (3.68), 307.05 (3.57), 370.8 (0.33), 801.2 (7) and 801.8 ð8:4Þ cm1 ðkm=molÞ. The calculated vibrational spectrum is shown in Fig. 8. 3. The Raman spectra The Raleigh light is usually having the same polarization as that of the incident light. In the Raman spectra, the lines need not have the same polarization as that of the incident light. In fact, there is a change in the polarization of light upon scattering. The experimental observation can be taken such that the polarization of the scattered light is the same as that of the incident light. It is also possible to record the spectrum of the scattered light so that the measured polarization is at an angle of 90° from that of the incident light. When the spectra are recorded in the same direction of polarization as that of the incident light, they are called HH spectra and when the recorded spectra are at a polarization of 90° from that of the incident light, we call them HV spectra. The first principles analysis of the Raman spectra of vitreous silica along with the interpretation of the vibrational density of states is given by Umari and Pasquarello [19]. They are able to calculate the Raman spectra
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Fig. 9. The Raman spectra of glassy arsenic oxide obtained experimentally by using argon ion laser at 514.5 nm. The HH and HV are the two polarizations available in the scattered light for which the spectra are recorded.
Table 1 The calculated and measured band frequencies in units of cm1 , in vitreous arsenic oxide along with their cluster identification.
Fig. 7. The vibrational spectrum of As4 O4 (cube) calculated from the first principles.
S. No.
Frequency (experimental)
Frequency (calculated)
Clusters
1 2 3 4 5 6
220 (weak) 342 485 552 700 (weak) 850 (weak)
219.8 335.1 489.7 544.7 701.8 855.1
AsO2 AsO4 ðT d Þ As2 O2 (rectangular) AsO2 (triangle) AsO3 (pyramidal) AsO2 (triangle)
part of the inverse dielectric constant, Imð1=Þ gives the LO modes. The transverse optical (TO) modes as well as the longitudinal optical (LO) modes require the translational symmetry, whereas the glassy state is made of many clusters of varying sizes and shapes so that the translational symmetry is broken. A typical Raman spectrum of vitreous As2 O3 obtained by using 514.5 nm excitation from the argon ion laser is shown in Fig. 9. There are very strong peaks at 342, 485 and 552 cm1 and weak lines at 220, 700 and 850 cm1 . The identification of these bands and comparison with calculated values is shown in Table 1. Apparently, all of the experimentally found frequencies can be identified with the help of large number of computed frequencies. It is found that the vitreous state of As2 O3 is made of several clusters of different valencies. It is possible that second-order Raman scattering at 2m is also present in the data but the six frequencies experimentally found in the oxide of arsenic are identified as arising from the clusters. From the vibrational frequencies calculated for a large number of clusters, we find that most of the experimentally found frequencies occur for a small number of atoms per cluster. In the present case of arsenic oxide, the experimentally measured frequencies occur when the number of atoms in the cluster is 5 or less than 5 such as 3 or 4. The Coulomb interactions vary with inverse distance. Hence, the observation of frequencies for a small number of atoms is related to the Coulomb interactions. It is obvious that As occurs in several different valencies. Whereas the valencies 3 and 5 are commonly known 2, 4, 6 and 8 are also present in the glass. Fig. 8. The vibrational spectrum of As4 O4 (ring) calculated from the first principles.
4. Conclusions directly from the linear response theory. Galeener et al. [16] have obtained the reflectivity of the glassy state of As2 O3 from which they have deduced the complex dielectric constant, the imaginary part of which gives the peaks in TO modes while the imaginary
We have performed the first principles calculations of the vibrational frequencies in a large number of clusters containing As and O atoms. We have identified that the clusters, AsO2 (triangular),
A.N. Rosli et al. / Journal of Non-Crystalline Solids 356 (2010) 428–433
AsO3 (pyramidal), As2 O2 (rectangular) and AsO4 ðT d Þ are actually present in the glassy state. The calculated values of the frequencies are in reasonable agreement with those found from the experimental Raman spectra of the glassy samples. Therefore, we have found a method to identify new clusters and molecules found in the minerals. It may be noted that the vibrational frequencies are sensitive to the size of the clusters due to the Coulomb interactions. Therefore, the glassy state is built from the random arrangement of clusters of atoms. References [1] [2] [3] [4] [5]
J.C. Phillips, Phys. Rev. Lett. 88 (2002) 216401. J.C. Phillips, Rep. Prog. Phys. 59 (1996) 1133. J.C. Phillips, J. Non-Cryst. Solids 43 (1981) 37. M.F. Thorpe, J. Non-Cryst. Solids 57 (1983) 355. Y. Wang, J. Wells, D.G. Georgiev, P. Boolchand, K. Jackson, M. Micoulaut, Phys. Rev. Lett. 87 (2001) 185503.
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[6] V.R. Devi, M.B. Madhavi, E.L. Srihari, K.N. Shrivastava, P. Boolchand, J. NonCryst. Solids 351 (2005) 489. [7] H.A. Kassim, I.A. Jalil, N. Yusof, V.R. Devi, K.N. Shrivastava, J. Non-Cryst. Solids 353 (2007) 111. [8] N.A. Jamali, H.A. Kassim, V.R. Devi, K.N. Shrivastava, J. Non-Cryst. Solids 354 (2008) 1744. [9] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [10] K. Hoang, S.D. Mahanti, J.R. Salvador, M.G. Kanatzidis, Phys. Rev. Lett. 99 (2007) 156403. [11] K. Hoang, S.D. Mahanti, P. Jena, Phys. Rev. B 76 (2007) 115432. [12] Yi-Chun Chen, Z. Lu, Ken-Ichi Nomura, W. Wang, R.K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. Lett. 99 (2007) 155506. [13] E.A. Albanesi, S.J. Sferco, I. Lefebvre, G. Allan, G. Hollinger, Phys. Rev. B 46 (1992) 13260. [14] M.C.G. Passeggi Jr., I. Vaquila, S.J. Sferco, Phys. Rev. B 50 (1994) 2090. [15] G.N. Papatheodorou, S.A. Solin, Phys. Rev. B 13 (1976) 1741. [16] F.L. Galeener, G. Lucovsky, R.H. Geils, Phys. Rev. B 19 (1979) 4251. [17] E.J. Flynn, S.A. Solin, G.N. Papatheodorou, Phys. Rev. B 13 (1976) 1752. [18] S.A. Solin, G.N. Papatheodorou, Phys. Rev. B 15 (1977) 2084. [19] P. Umari, A. Pasquarello, J. Phys.: Condens. Matter 15 (2003) S1547.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Ab initio calculation of vibrational frequencies of AsO glass Ahmad Nazrul Rosli, Noriza Ahmad Zabidi, Hasan Abu Kassim, Keshav N. Shrivastava * Department of Physics, University of Malaya, Kuala Lumpur 50603, Malaysia
a r t i c l e
i n f o
Article history: Received 13 May 2008 Available online 29 December 2009 Keywords: Raman scattering Chalcogenides Ab initio Raman spectroscopy
a b s t r a c t We have used the density-functional theory to make models containing arsenic and oxygen atoms. The structures are optimized for the minimum energy of the Schrödinger equation. In this way, we obtain the bond distances and angles of the stable structures. We obtain the vibrational frequencies of each cluster. The calculated vibrational frequencies are compared with those found in the experimental Raman spectra. The values of the vibrational frequencies calculated for AsO2 ; AsO4 ðT d Þ; AsO2 (rectangular), AsO2 (triangular) and AsO3 (pyramidal) agree with those found in the Raman spectra of vitreous aresenic oxide, indicating that these clusters are really present in the arsenic oxide glass. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Recently, it was pointed out by Phillips [1] that anomalous properties exist over a narrow range of compositions in the molecular glasses. It was shown earlier that long relaxation times occur in glasses [2,3]. In fact, some of the phonon frequencies can become soft and approach towards zero [4]. There is a phase transition in the rigidness of glasses [5]. The random network of atoms is considered to be important for the understanding of glasses. In recent years, we have made clusters of atoms by using densityfunctional theory. The clusters of Ge atoms, S atoms and I atoms were made. Similarly, the clusters of Gex Iy ; Gex Sy ; Gex Sy Iz were made [6]. It was found that the Raman frequencies of Ge0:25 S0:75 Iy glass were the same as those calculated for a few clusters of atoms, indicating that the glass is made of random collection of clusters. In some cases, there was a linear arrangement of atoms, the frequencies of which were the same as found in the experimental data. This means that linear arrangement of atoms exists along with the clusters which give strength to the glass. Therefore, clusters of atoms along with linear molecules describe the structure of the glassy state. We repeated the calculation of clusters of Gex Px S12x glass in as far as the structural parameters such as the bond lengths and the vibrational frequencies are concerned and found that clusters along with linear molecules are present in the glass [7]. The example of Gex Se1x also showed [8] that clusters such as Ge4 Se3 ; Ge2 Se2 ; Ge3 ; GeSe3 and Ge3 Se2 are present in the glass. Therefore, we believe that cluster calculations are important for the understanding of the glassy state. The density-functional theory has been developed by Kohn and Sham [9]. It is possible to write the Schrödinger equation in the form of electron density which depends on the electron coordi* Corresponding author. Tel.: +603 7967 7140; fax: +603 7967 4146. E-mail address: [email protected] (K.N. Shrivastava). 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2009.11.041
nates. The derivative of the Schrödinger equation with respect to density leads to another equation which can be numerically integrated. The form of the Schrödinger equation as a function of density requires approximations. The local density approximation (LDA) is very popular and will be used in the present work. It is also possible to take the generalized gradient approximation (GGA) in which account of the nonlocal atoms is taken by introducing a linear gradient. In this way, we can calculate the energies in the GGA. There are no adjustable parameters in this calculation. The Coulomb interactions between electrons and nuclei are used. This kind of first principles calculations are very useful in predicting electronic properties. Our calculations of vibrational frequencies appear to be in reasonable agreement with the experimental data [6–8]. The first principles calculations of the band structures of AgSbS, AgSbSe and AgSbTe are found to be useful for improving the thermoelectric properties [10], Similarly, it is found that DFT gives good results for the geometry optimization of PbTe [11]. In SiO, it is found that Si atoms switch bonds from one O atom to another leading to expansion of SiO rings. The calculation is performed for multimillion to billion atoms by means of automated computer simulation of SiO nanometer size particles [12]. Arsenic is found in several different valencies. The arsenic trioxide As2 O3 dissolves in alkaline solutions to make arsenites. It reacts with oxidizing agents to form As2 O5 . Albanesi et al. [13] have found that As2 O3 occurs in two structural forms which separate in a phase transition. The cubic arsenolite has eight As4 O6 molecules, per unit cell, each made of AsO3 pyramids linked together by oxygen atoms. The layered claudetite also has AsO3 pyramids. The As2 O5 has both the AsO4 tetrahedra as well as AsO6 octahedra linked by corners so that there is a network of clusters of atoms. The tight binding approximation has been discussed for both the tetrahedral as well as the octahedral configurations by Passeggi et al. [14]. The vibrational excitations of arsenic oxide have been reported [15–18] in vitreous and disordered
A.N. Rosli et al. / Journal of Non-Crystalline Solids 356 (2010) 428–433
phases. The polarized Raman spectra with second-order scattering are reported by Galeener et al. [16]. In this paper, we report our calculation of vibrational frequencies of several clusters of arsenic and oxygen atoms. We optimize the configuration of clusters for which the energy of the Schrödinger equation is a minimum. We have performed the calculation for 26 different clusters made of As and O atoms. In all of the cases we obtained the optimum geometry as well as the vibrational frequencies. Our calculations are based on the kinetic energy of the electrons and that of the nuclei and all of the Coulomb interactions with no adjustable parameters. We compare the values calculated from the first principles with those from the experimental Raman spectra. The values calculated for the clusters are found to be in reasonable agreement with those measured. Therefore, we find that there are clusters in the vitreous As2 O3 . The glassy state is therefore made from the random network of clusters.
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(viii)
(ix)
(x) 2. Clusters We use the density-functional theory in the local density approximation (LDA). The Kohn–Sham equations are solved and vibrational frequencies are deduced. The bond lengths and angles are obtained for the minimum energy [9]. The computer program 3 is the DMol of Accelrys. It is the ‘‘windows” version run on duo pentium. We can choose any of the following wave functions. The ‘‘min” selects only one Slater orbital on each atom which is occupied. The double numeric (DN) means at least two atomic orbitals for each one occupied in the free atom. The ‘‘DNP” means DN with polarization. The DND chooses double-numerical+d-DNP basis except that no p wave functions are used on hydrogen. There are several options but the DN seems to give values which agree with the experimental data. The DN wave functions are used throughout in most of the calculations performed by us. (i) As–O. This is a diatomic cluster. In the optimized configuration the distance between the two atoms is found to be 1.698 Å. There is a single vibrational frequency belonging to the bond stretching. The vibrational frequency is 881:4 cm1 . (ii) AsO2 . In this molecule for the optimized geometry, the bond angle is found to be 122.18°. The As–O bond length is found to be 1.719 Å. The vibrational frequencies (intensities) are 219.8 (19.9), 753.7 (13.2) and 754:1ð1:59Þ cm1 ðkm=molÞ. (iii) AsO2 (linear). It is possible to optimize this molecule in a linear form so that O–As–O angle is 180°. The As–O bond length is found to be 1.734 Å. The vibrational frequency is calculated to be 753:8 cm1 . (iv) AsO2 (triangle). It is also possible to optimize AsO2 in the shape of a triangle. The As–O distance is 1.896 Å and O–O distance is 1.590 Å. The vibrational frequencies (intensities) are found to be 535.8 (11), 544.7 (11.8) and 855:1ð7:4Þ cm1 ðkm=molÞ. (v) AsO3 (triangular). In this cluster the three oxygen atoms are at the corners of a triangle and As is in the centre. In the optimized configuration, the As–O distance is 1.662 Å and the vibrational frequencies(intensities) are 156.3 (3.5), 159.5 (3.3), 237.1 (24.7), 819.1 (2.2), 820.2 (2.1) and 826:9ð0:02Þ cm1 ðkm=molÞ. The vibrational spectrum calculated for this cluster from the first principles is given in Fig. 1. (vi) AsO3 (pyramidal). The O–O distance is 1.865 Å in the O3 triangle. The As atom sits on top of the triangle to form a pyramid. The As–O distance is 2.250 Å. It has strong vibrations at 280.3 (5.28), 532.1 (0.91){2} and
(xi)
(xii)
(xiii)
429
701:8ð11:8Þ cm1 ðkm=molÞfdegeneracyg. The vibrational spectrum calculated from the first principles is shown in Fig. 2. AsO4 T d . This is a tetrahedral molecule. The As–O distance is 1.759 Å. The dominant mode occurs at 335:1 cm1 and it is triply degenerate. As2 O. This molecule has As–O–As angle of 71.36° and bond length is 1.971 Å. The vibrational frequencies (intensities) are 195.4 (6.2), 305.7 (6.09), 605:8ð22:9Þ cm1 ðkm=molÞ. The calculated spectrum for this model is shown in Fig. 3. As2 O (linear). We find that with O in the centre and two As on both sides to form a straight line is stable. The As–O bond distance is 1.759 Å. There is a doubly degenerate vibration at 233.2 (20.8){2} and a singly degenerate oscillation at 981:6ð22:2Þ cm1 ðkm=molÞ. As2 O2 (rectangular). The As–O bond length is 1.871 Åand the angle of the rectangle is 100.5°. There are two strong frequencies at 296.5 (39.7) and 489:7ð48:2Þ cm1 ðkm=molÞ. As2 O2 (pyramidal). In this cluster, the As–As distance is 2.348 Å, the O–O distance is 1.952 Å and the As–O distance is 2.517 Å. The vibrational frequencies (intensities) are 328.9 (15.3), 376.3 (9.6), 384.5 (6.3), 573.8 (52.3), 643:4ð19:5Þ cm1 ðkm=molÞ. The computed vibrational spectrum is shown in Fig. 4. As2 O3 (linear). In this system all of the atoms are in a straight line with oxygen atoms at both ends. The end bond length is 1.721 Åand the inner bond length is 1.968 Å. The end correction is thus clearly seen in the structure. This system has the positive vibrational frequencies (intensities) fdegeneraciesg as 31.6 (2.66){2} , 327.3 (159), 782:4ð240Þ cm1 ðkm=molÞ. As2 O3 (bipyramidal). In this system, first a triangle is made with three oxygen atoms and then one As is put on top of the triangle and another is put symmetrically on the other side to form a bipyramid. The O–O distance in the triangle is 2.496 Åand the As–O distance is 1.927 Å. There are two strong vibrations at 375.7 (39.7){2}, 561.1 (154), 562.6 (34.9){2}. The later two are not well resolved.
Fig. 1. The vibrational spectrum of AsO3 (triangular) calculated from the first principles.
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Fig. 2. The vibrational spectrum of AsO3 (pyramidal) calculated from the first principles. Fig. 4. The vibrational spectrum of As2 O2 (pyramidal) calculated from the first principles.
Fig. 3. The vibrational spectrum of As2 O calculated from the first principles.
(xiv) As2 O4 (bipyramidal). First we make a square with four oxygen atoms and then put one As on top and another symmetrically in the bottom of the square. The O–O distance is 2.260 Åand the As–O distance is 2.036 Å. The vibrational frequencies (intensities) fdegeneraciesg are found to be 138.4 (97.8){2} and 443.9 (774){2}.
(xv) As3 O (triangular). The three As atoms are on the corners of a triangle and one O is in the centre. The As–O distance is 1.868 Å. The vibrational frequencies found are 131.8 (4.3), 671.5 (12.3) and 693:9ð10:6Þ cm1 ðkm=molÞ. (xvi) As3 O (pyramidal). In this model the three As form a triangle with O–O distance of 2.477 Åand one As atom is on top to form a pyramid. The As–O distance is 2.151 Å. The vibrational frequencies (intensities) are 235.9 (0.7), 300.6 (2.95) and 491:5ð17:7Þ cm1 ðkm=molÞ. (xvii) As3 O2 (linear). In this cluster all of the five atoms are in a straight line with As on two ends. The bond distance near the ends is 1.753 Åand towards the centre it is 1.919 Å. There are strong oscillations at the frequencies (intensities) of 23.9 (0.91){2}, 238.8 (5.5), 291.6 (44.3){2}and 760.2 (131). (xviii) As3 O3 (ring). In this cluster, three As and three O atoms are arranged, alternately, so as to form a ring. The As–O bond length is 1.830 Åobtained by using double numeric (DN) wave functions. Hence, the ring diameter is 3.66 Å. The rings and holes are often formed in the thin films of arsenic oxide glass. Hence, it is often relevant to experimental conditions. The vibrational spectrum calculated for this cluster is shown in Fig. 5. The calculated frequencies (intensities) fdegeneraciesg are as follows: 88.94 (0.01){2}, 151.5 (47), 183.4 (7.9){2}, 363.7 (30.5){2}, 696:2ð0:4Þf2g cm1 ðkm=molÞ. (xix) As3 O4 (linear). In this cluster, all of the atoms are in a straight line with oxygen atoms at the ends. The As always have two oxygen atoms as neighbors. There is clearly the effect of ends so that at the end, the bond length is 2.076 Å which successively reduces upon going towards the centre. So it becomes 1.853 and
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Fig. 5. The vibrational spectrum of As3 O3 (ring) calculated from the first principles.
(xx)
(xxi)
(xxii)
(xxiii)
(xxiv)
1.714 Åat the centre. There are several weak oscillations which correspond to internal motion of the atoms and there is a strong vibration at 807:5 cm1 which is due to the uniform stretching or contracting motion. The various frequencies (intensities) fdegeneraciesg are found to be 17.47 (1){2}, 160.2 (18.5), 373.2 (39.5), 807:5ð620Þ cm1 ðkm=molÞ. As4 O (square). The four As atoms are located on the corners of a square with perfect 90° angles. The O atom is located in the centre of the square. The As–O distance is 2.008 Å. Its strong vibration occurs at 252:1 cm1 . As4 O (pyramidal). The four As atoms form a square of As–As square edge length 2.484 Å. One O atom is on top position with As–O distance of 2.323 Å. It has a weak doubly degenerate oscillation at 225.9 (0.26), a medium strength oscillation at 243.1 (2.92) and a strong oscillation at 397:7ð11:3Þ cm1 ðkm=molÞ. As4 OðT d Þ. In this cluster the O atom is in the centre of a tetrahedron with As–O distance 1.884 Å with a strong triply degenerate oscillation at 721:7 cm1 . As4 O2 (bipyramidal). The four As atoms form a square with As–As distance of 2.554 Åand two O atoms sit on top and bottom positions with As–O distance 2.262 Å. There is a weak vibration at 239:2ð2:13Þf2g cm1 ðkm=mol:Þ and a strong oscillation at 379:6 cm1 . As4 O3 (linear). In this cluster the As and O atoms are arranged alternately with As atoms at the two ends. The distances are 1.781 Å at the end, 1.852 at one position earlier than end and 1.753 Åat the centre. The oscillation frequencies (intensities) {degeneracies} are found to be 157.03 (2.8){2}, 180.8 (0.03), 310.9 (65.1){2}, 594.5 (45) and 774:3ð57Þ cm1 ðkm=molÞ. The calculated spectrum is shown in Fig. 6. This is an example of strong internal vibrations within the system.
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Fig. 6. The vibrational spectrum of As4 O3 (linear) calculated from the first principles.
(xxv) As4 O4 (cube). In a simple cubic geometry, there are two As and two O atoms in a plane and then in the next plane As and O are interchanged so that the two layers form a cube. The As–O bond length is 2.041 Å. The vibrational spectrum shows triply degenerate oscillations at the frequencies (intensities) fdegeneraciesg of 181.05 (2.5){3}, 298.3 (7.2){3} and 440.1 (31.2){3}. The calculated spectrum is shown in Fig. 7. (xxvi) As4 O4 (ring). The As and O atoms are arranged alternately to form a ring with eight sides. The bond length is 1.807 Å. This ring has several vibrational modes the frequencies (intensities) of which are 130.1 (0.43), 130.4 (0.58), 137.5 (68), 243.4 (51), 245.1 (50), 274.8 (0.54), 306.6 (3.68), 307.05 (3.57), 370.8 (0.33), 801.2 (7) and 801.8 ð8:4Þ cm1 ðkm=molÞ. The calculated vibrational spectrum is shown in Fig. 8. 3. The Raman spectra The Raleigh light is usually having the same polarization as that of the incident light. In the Raman spectra, the lines need not have the same polarization as that of the incident light. In fact, there is a change in the polarization of light upon scattering. The experimental observation can be taken such that the polarization of the scattered light is the same as that of the incident light. It is also possible to record the spectrum of the scattered light so that the measured polarization is at an angle of 90° from that of the incident light. When the spectra are recorded in the same direction of polarization as that of the incident light, they are called HH spectra and when the recorded spectra are at a polarization of 90° from that of the incident light, we call them HV spectra. The first principles analysis of the Raman spectra of vitreous silica along with the interpretation of the vibrational density of states is given by Umari and Pasquarello [19]. They are able to calculate the Raman spectra
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Fig. 9. The Raman spectra of glassy arsenic oxide obtained experimentally by using argon ion laser at 514.5 nm. The HH and HV are the two polarizations available in the scattered light for which the spectra are recorded.
Table 1 The calculated and measured band frequencies in units of cm1 , in vitreous arsenic oxide along with their cluster identification.
Fig. 7. The vibrational spectrum of As4 O4 (cube) calculated from the first principles.
S. No.
Frequency (experimental)
Frequency (calculated)
Clusters
1 2 3 4 5 6
220 (weak) 342 485 552 700 (weak) 850 (weak)
219.8 335.1 489.7 544.7 701.8 855.1
AsO2 AsO4 ðT d Þ As2 O2 (rectangular) AsO2 (triangle) AsO3 (pyramidal) AsO2 (triangle)
part of the inverse dielectric constant, Imð1=Þ gives the LO modes. The transverse optical (TO) modes as well as the longitudinal optical (LO) modes require the translational symmetry, whereas the glassy state is made of many clusters of varying sizes and shapes so that the translational symmetry is broken. A typical Raman spectrum of vitreous As2 O3 obtained by using 514.5 nm excitation from the argon ion laser is shown in Fig. 9. There are very strong peaks at 342, 485 and 552 cm1 and weak lines at 220, 700 and 850 cm1 . The identification of these bands and comparison with calculated values is shown in Table 1. Apparently, all of the experimentally found frequencies can be identified with the help of large number of computed frequencies. It is found that the vitreous state of As2 O3 is made of several clusters of different valencies. It is possible that second-order Raman scattering at 2m is also present in the data but the six frequencies experimentally found in the oxide of arsenic are identified as arising from the clusters. From the vibrational frequencies calculated for a large number of clusters, we find that most of the experimentally found frequencies occur for a small number of atoms per cluster. In the present case of arsenic oxide, the experimentally measured frequencies occur when the number of atoms in the cluster is 5 or less than 5 such as 3 or 4. The Coulomb interactions vary with inverse distance. Hence, the observation of frequencies for a small number of atoms is related to the Coulomb interactions. It is obvious that As occurs in several different valencies. Whereas the valencies 3 and 5 are commonly known 2, 4, 6 and 8 are also present in the glass. Fig. 8. The vibrational spectrum of As4 O4 (ring) calculated from the first principles.
4. Conclusions directly from the linear response theory. Galeener et al. [16] have obtained the reflectivity of the glassy state of As2 O3 from which they have deduced the complex dielectric constant, the imaginary part of which gives the peaks in TO modes while the imaginary
We have performed the first principles calculations of the vibrational frequencies in a large number of clusters containing As and O atoms. We have identified that the clusters, AsO2 (triangular),
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AsO3 (pyramidal), As2 O2 (rectangular) and AsO4 ðT d Þ are actually present in the glassy state. The calculated values of the frequencies are in reasonable agreement with those found from the experimental Raman spectra of the glassy samples. Therefore, we have found a method to identify new clusters and molecules found in the minerals. It may be noted that the vibrational frequencies are sensitive to the size of the clusters due to the Coulomb interactions. Therefore, the glassy state is built from the random arrangement of clusters of atoms. References [1] [2] [3] [4] [5]
J.C. Phillips, Phys. Rev. Lett. 88 (2002) 216401. J.C. Phillips, Rep. Prog. Phys. 59 (1996) 1133. J.C. Phillips, J. Non-Cryst. Solids 43 (1981) 37. M.F. Thorpe, J. Non-Cryst. Solids 57 (1983) 355. Y. Wang, J. Wells, D.G. Georgiev, P. Boolchand, K. Jackson, M. Micoulaut, Phys. Rev. Lett. 87 (2001) 185503.
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[6] V.R. Devi, M.B. Madhavi, E.L. Srihari, K.N. Shrivastava, P. Boolchand, J. NonCryst. Solids 351 (2005) 489. [7] H.A. Kassim, I.A. Jalil, N. Yusof, V.R. Devi, K.N. Shrivastava, J. Non-Cryst. Solids 353 (2007) 111. [8] N.A. Jamali, H.A. Kassim, V.R. Devi, K.N. Shrivastava, J. Non-Cryst. Solids 354 (2008) 1744. [9] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [10] K. Hoang, S.D. Mahanti, J.R. Salvador, M.G. Kanatzidis, Phys. Rev. Lett. 99 (2007) 156403. [11] K. Hoang, S.D. Mahanti, P. Jena, Phys. Rev. B 76 (2007) 115432. [12] Yi-Chun Chen, Z. Lu, Ken-Ichi Nomura, W. Wang, R.K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. Lett. 99 (2007) 155506. [13] E.A. Albanesi, S.J. Sferco, I. Lefebvre, G. Allan, G. Hollinger, Phys. Rev. B 46 (1992) 13260. [14] M.C.G. Passeggi Jr., I. Vaquila, S.J. Sferco, Phys. Rev. B 50 (1994) 2090. [15] G.N. Papatheodorou, S.A. Solin, Phys. Rev. B 13 (1976) 1741. [16] F.L. Galeener, G. Lucovsky, R.H. Geils, Phys. Rev. B 19 (1979) 4251. [17] E.J. Flynn, S.A. Solin, G.N. Papatheodorou, Phys. Rev. B 13 (1976) 1752. [18] S.A. Solin, G.N. Papatheodorou, Phys. Rev. B 15 (1977) 2084. [19] P. Umari, A. Pasquarello, J. Phys.: Condens. Matter 15 (2003) S1547.