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Reverse Monte Carlo simulation of GexSe100−x glasses
Physica B 405 (2010) 4240–4244
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Reverse Monte Carlo simulation of GexSe100 x glasses A.H. Moharram a,n, A.M. Abdel-Baset b a b
Faculty of Science, King Abdul Aziz University, Rabigh Campus, P.O. Box 334, Saudia Arabia Physics Department, Faculty of Science, Assiut University, Assiut 71516, Egypt
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 April 2010 Received in revised form 8 July 2010 Accepted 9 July 2010
Amorphous GexSe100 x (with x ¼10, 20 and 40 at%) alloys were prepared using the melt–quench technique. Two-dimensional Monte Carlo of the total pair distribution functions (MCGR) have been found and used to assemble the three-dimensional atomic configurations using the reverse Monte Carlo rmc ðrÞ, (RMC) method. The simulations are useful to compute the partial pair distribution functions gGeGe rmc rmc rmc rmc ðrÞ, gSeSe ðrÞ and the partial structure factors Srmc gGeSe GeGe ðrÞ, SGeSe ðrÞ, SSeSe ðrÞ of the studied glasses. The partial pair distribution functions indicate that the basic building units are GeSe4 and Ge2Se6 tetrahedral units in the Se-rich and Ge-rich glasses, respectively. Some of these tetrahedral units are connected by the homopolar units as confirmed by the bond angle distribution functions. The partial structure factors have shown that not only the homopolar Ge–Ge bonds, but also Se–Se bonds are behind the appearance of the first sharp diffraction peak (FSDP) in the total structure factor. & 2010 Elsevier B.V. All rights reserved.
Keywords: X-ray diffraction Chalcogenide glasses Reverse Monte Carlo Short-range order Medium-range order
1. Introduction Chalcogenide glasses present a great potential for application in technological devices, such as optical fibers, memory materials and switching devices, but their use is limited due to several factors. One of them is the difficulty in obtaining information about atomic structures, which define the short-range order (SRO) of the alloy. Although, many kinds of structural studies for amorphous Ge–Se alloys [1–6] have been performed to investigate the short-range order of the system, the glass network structure of these alloys is not yet fully understood. High-resolution diffraction technique with a wide range of the scattering vector (K) is required for structural studies of the amorphous materials. For example, high-energy X-ray diffraction measurements have some advantages [7]: highly structural information due to the wider range of K, smaller correction terms, and easy to compare with neutron diffraction data because of similar transmission method. Unfortunately, the X-ray source used in the present experiment, CuKa target emits radiation with ˚ does not have wide range of the wavelength l ¼1.5418 A, scattering vector. To overcome this problem, two-dimensional Monte Carlo of the total pair distribution function (MCGR) has been found, which is used to assemble three-dimensional atomic configuration using the reverse Monte Carlo (RMC) method.
n
Corresponding author. E-mail addresses: [email protected], [email protected] (A.H. Moharram). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.07.018
Some of the structural intermediate-range order (IRO) information can be obtained from the experimental diffraction data. Reverse Monte Carlo (RMC) simulation [8,9] represents, when used carefully, a powerful tool to extract these information. It assembles three-dimensional atomic configurations using experimental diffraction data implicitly in the simulation. The intimate connection between computational and experimental processes means that the better quality, and higher resolution of the experimental data, the more reliable RMC model of a network structure for vitreous materials. The RMC method is considered as an inverse problem in which the experimental data are enforced to build atomic configurations that have the desired structural and electronic properties. The main point is to set up a generalized function containing as much information as possible, and then optimize the function for generating configurations toward exact agreement with the experimental data. X-ray diffraction study of the glassy GexSe1 x systems [10,11], in a wide concentration range 0rx r0.33, has demonstrated that besides the well-established SRO information, a pre-peak appeared in the total structure factor S(K) at a scattering vector K of about 1.1 A˚ 1. The pre-peak, being clear evidence for existence of the intermediate-range order (IRO), showed a systematic decrease in intensity and shifts towards higher K-values with decrease in Ge concentration. A similar variation in the pre-peak of the structure factor with Ge content was also observed using the neutron diffraction measurement [12]. In the present work, the short- and intermediate-range orders (SRO and IRO) for three Ge10Se90, Ge20Se80 and Ge40Se60 glasses have been studied using Monte Carlo of pair distribution (MCGR) and the reverse Monte Carlo (RMC) simulations.
A.H. Moharram, A.M. Abdel-Baset / Physica B 405 (2010) 4240–4244
pre-peak, the K-resolution is better than that with targets having higher characteristic energies.
2. Reverse Monte Carlo (RMC) Three-dimensional arrangement of N atoms is placed into a cubic cell with periodic boundary conditions. The atomic number density (r) should be the same as the experimental value. The positions of the atoms are chosen randomly. The partial pair distribution function gijcal ðrÞ can be calculated from the initial configuration [8] by gijcal ðrÞ ¼
nij ðrÞ 4pr 2 dr rci
ð1Þ
where ci is the concentration of i-type atoms and nij(r) the number of j-type atoms at a distance between r and r +dr from a central i-type atom, averaged over all atoms as centers. The total pair distribution gcal(r) is calculated from m h i 1 X ci cj fi fj gijcal ðrÞ1 g cal ðrÞ ¼ 2 f i,j ¼ 1
ð2Þ
where m is the number of elements in the sample (m¼2 for the P present work), fi the atomic scattering factor and f ¼ i ci fi ðKÞ. 2 cal The deviation w of the calculated g (r) from the experimental, gexp(r), obtained from the MCGR method, can be computed from the expression
w2 ¼
n X ½g cal ðri Þg exp ðri Þ2 i¼1
s2 ðri Þ
4241
ð3Þ
where n is the number of experimental points and s2(ri) the statistical error. Move one atom at random, and recalculate gcal(r) of the new configuration. If w2 decreases, the change is accepted, while if it increases, the move is accepted with probability of exp(–Dw2/2) where Dw2 is the change in w2. As the number of accepted atom moves increases, w2 will initially decrease until it reaches an equilibrium value. Thus, the atomic configuration corresponding to the equilibrium should be consistent with the experimental functions within the experimental errors. All the results shown in this article are based on the partial and total functions obtained at the equilibrium level. Fourier transformation of the partial pair distribution, gij(r), gives the partial structure factor Z 1 sin Kr Sij ðKÞ ¼ r dr ð4Þ 4pr 2 ðgij ðrÞ1Þ Kr 0 The above function is important for determination of the intermediate-range order parameters and also to specify the bond type responsible for the existence of the FSDP. On the other hand, the short-range order parameters such as the inter-atomic distances, the partial coordination number and the bond angle distribution can be obtained from the gij(r) functions.
4. Results and discussion The experimental X-ray data of the present three Ge10Se90, Ge20Se80 and Ge40Se60 compositions were analyzed using some computer programs. The first step was to correct the observed X-ray intensities through background subtraction followed by absorption and polarization corrections. From the corrected X-ray intensities, the variations in the total structure factor, S(K), with the scattering vector (K ¼4psin y/l, where y is half the scattering) can be obtained [13]. Fig. 1 shows the experimental total structure factors, S(K), of the investigated glasses up to K¼ 6.874 A˚ 1. The existence of the first sharp diffraction peak (FSDP) is clear especially for the Ge40Se60 glass, which is located at K ¼1.0 A˚ 1. FSDP is commonly observed in covalently bonded amorphous materials, which implies the presence of intermediate-range order. Its intensity decreases with decrease in Ge content to the point it appears as a shoulder in the Ge10Se90 composition. Indeed, not only its height changes with Ge content, but also it is shifted to a higher K-values with decrease in Ge content. The pre-peak is known to be much dependent on Ge–Ge correlations and, to a lesser extent, on Ge–Se correlations [14–16]. In the structural analysis using Fourier transformations [17,18], a modification factor was suggested to reduce the effect of termination data at a finite Kmax. This factor in turn, while reduces the spurious oscillations, leads to a broadening of the genuine peaks in g(r). To overcome this problem, an inverse method was applied to determine the pair distribution function from the total structure factor, measured by neutron or X-ray diffraction, using a MCGR program [19]. One of its advantages is that a direct estimation of errors in g(r) is possible by repeating the process until getting a minimum deviation of the simulated values from the experimental S(K) ones, as shown in Fig. 1. The g(r) function obtained from the MCGR method does not require integration process or integration limits, which means that considering Kmax instead of infinity has no effect on the obtained functions [12]. For the present work, the starting value of g(r) is ˚ Fig. 2 shows the zero below r o1.9 A˚ and equals one for r Z1.9 A. conventional (solid circles) of the pair distribution functions, g(r), of the investigated GexSe100 x glasses together with the simulated
3. Experimental technique Glassy chalcogenide of GexSe100 x (with x ¼10, 20 and 40 at%) alloys were prepared using the melt–quench method. The glassy nature of the quenched alloys was confirmed by X-ray diffraction technique. X-ray diffraction experiment was done using a Philips diffractometer (PW-1710). The XRD patterns were recorded at 40 kV and 30 mA, with a graphite monochromator, using the ˚ with scanning speed of 0.041/min. The CuKa line (l ¼ 1.5418 A), experiment was done in the scattering angle range 4r2y r1151 in steps of 0.11, which corresponds to K-vector range 0.284rK r6.874 A˚ 1. Copper source was chosen for two reasons. Firstly, its characteristic energy is lower than Ge K absorption edge energy. Therefore, the scattering comprises no fluorescence X-rays, and the anomalous terms in the atomic factors of Ge and Se are very small. Secondly, for obtaining the precise data of the
Fig. 1. Experimental X-ray structure factors (K) together with the results of MCGR (J) and RMC ( ) simulations of the investigated GexSe100 x glasses.
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A.H. Moharram, A.M. Abdel-Baset / Physica B 405 (2010) 4240–4244
values (full lines) obtained using the MCGR method. As said before, the modification factor while reduces the spurious oscillations, leads to a broadening of the conventional g(r) functions. The broadening is wide enough to cause an overlap between the first and the second peaks. Since the simulation does not depend on the modification factor, the simulation g(r) data of Fig. 2 have shown two resolved peaks providing information for
Fig. 2. Conventional pair distribution functions (K) together with the results of MCGR (J) and RMC ( ) simulations of the investigated GexSe100 x glasses.
Fig. 3. The partial pair distribution functions, gij(r), for the studied GexSe100 x glasses.
the first and the second coordination spheres. Indeed, these three curves represent the backbone for the three-dimensional reverse Monte Carlo (RMC) simulations. The starting point in RMC simulation [19] is to randomly generate the configuration distribution of N ¼4000 atoms inside a cubic box. The cubic edges for the investigated Ge10Se90, Ge20Se80 ˚ respectively. The and Ge40Se60 glasses are 23.8, 23.6 and 23.3 A, portions of Ge or Se atoms in the assumed configuration will vary according to their atomic percentages. Fig. 3 shows the partial pair distribution gGe–Ge(r), gGe–Se(r), gSe–Se(r) functions for the present three glasses. In the first coordination sphere, the zero values of the gGe–Ge(r) in case of the Se-rich (Ge10Se90, Ge20Se80) glasses indicate that only homopolar Se–Se bonds exist in addition to heteropolar Ge–Se bonds. However, the Ge-rich (Ge40Se60) glass shows two homopolar Ge–Ge and Se–Se bonds in addition to heteropolar Ge–Se bonds. Summation of these partial pair distribution functions yields the RMC g(r) curves, which give good fit to the point that cannot be distinguished from the corresponding MCGR data shown in Fig. 2. The average Ge–Ge, Ge–Se and Se–Se bond lengths, as obtained from the refined ˚ RMC model, are 2.52 70.065, 2.43 70.065 and 2.47 70.065 A, respectively. RMC simulation of the final atomic configuration gives the number of neighbors (probability) distributions within the first coordination shell as shown in Fig. 4. In Se-rich (Ge10Se90 and Ge20Se80) glasses, the probability of finding Ge atom in the nearest neighbors of a central Ge atom is almost zero, while it goes up to 74% in case of Ge-rich (Ge40Se60) glass. This means that each Ge atom can be surrounded by four Se neighbors in the former glasses, while its four neighbors in case of Ge-rich glass are three Se atoms plus one Ge atom as given by PGe–Se(n) function of
Fig. 4. The number of neighbors distribution within the first coordination shell obtained from RMC model.
A.H. Moharram, A.M. Abdel-Baset / Physica B 405 (2010) 4240–4244
Fig. 4. The present RMC simulations have indicated that about 97%, 95% of Ge atoms are surrounded by four Se atoms in the Ge10Se90 and Ge20Se80 glasses, respectively. In case of Ge-rich (Ge40Se60) glass, 68% of Ge atoms are connected to three Se atoms and one Ge atom. In a previous study [12], the authors reported that the main structure units in Se-rich and Ge-rich glasses are tetrahedral GeSe4 and Ge2Se6 units, respectively. The present PGe–Se(n) function confirms their conclusion. Based on the YSe–Se–Se(y) and PSe–Se(n) functions, the above tetrahedral structural units seem to be connected by Se–Se bridges, forming small chains and rings. The average partial coordination numbers are, given in Table 1, close to some extent to those reported by other references [20,21]. The corresponding values, calculated using the chemical order network model and listed in the above Table, confirm the
Table 1 The present partial coordination numbers for the studied glasses and those reported by other references [20,21]. The values calculated using the chemical order network model (CONM) are also listed.
Ge10Se90
Ge20Se80
Ge40Se60
Present CONM Ref. [21] Present CONM Ref. [20] Ref. [21] Present CONM
Ge–Ge
Ge–Se
Se–Ge
Se–Se
0.03 0 0.0 0.03 0 0.22 0.22 1.23 1
3.96 4 3.99 4.04 4 3.61 3.61 2.97 3
0.44 0.44 0.44 1.01 1 0.9 0.9 1.98 2
2.68 1.56 1.56 2.26 1 1.71 1.71 1.73 0
Fig. 5. Bond angle distribution functions, Y(y), obtained from RMC simulation of the Ge20Se80 glass.
4243
applicability of the CONM in the structure configuration of the investigated alloys. The presence of Se–Se bridges between the tetrahedral units is a possible reason for high partial coordination number for Se–Se as compared to that reported previously and computed from the chemically ordered network model. Bond angle distribution functions, Y(y), are obtained from the final configuration of the present three glasses using triplets program [19], in which the bond angle for any reference atom can be calculated from Cartesian coordinates of the final positions of the surrounding atoms. The Y(y) functions of the Ge20Se80 glass, as an example, are shown in Fig. 5. The YSe–Ge–Se(y) function presents a main peak around 104.21, which is near the ideal tetrahedral angle of 1091. A small peak appearing at 601 can be attributed to the existence of wrong (homo) bonds. In glassy state, some Ge atoms are, freezing in corner positions of the tetrahedral units instead of Se atoms, responsible for the above small peak. A similar behavior is shown by YSe–Se–Se(y) function, with a main peak at 601 and a small one at 104.21. Selenium atoms occupying the face of a perfect tetrahedron should exhibit internal angles of 601, and the 104.21 angle corresponds to Se–Se–Se angles can be associated with inter-tetrahedral units. The present Y(y) functions confirm that distorted tetrahedral units are formed in the investigated glasses. These structural units seem to be connected by Se–Se bridges, forming small chains and rings, as previously reported [22]. YGe–Ge–Ge(y) function shows broad distribution from 501 to 1501, which indicates that each tetrahedral unit is tetrahedrally connected with other tetrahedral by corner share and edge share. From this figure, one can conclude that most of the Ge atoms surrounded either by four Se atoms or three Se atoms and one Ge atom are forming tetrahedral units.
Fig. 6. The partial structure factors, Sij(r), for the studied GexSe100 x glasses.
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Fig. 7. Atomic Ge10Se90 and Ge40Se60 configurations obtained from RMC modeling. Ge and Se atoms are defined by (K) and () balls, respectively. The images represent 10% from the whole configuration and show different tetrahedral structural units.
Fourier transformation of the partial pair distribution gives the corresponding partial scattering factor. The partial scattering factors as function of the scattering vector for the present three glasses are shown in Fig. 6. These curves are very important especially in the region where the FSDP is located. The SGe–Se(K) function is not included in this figure because it does not have a peak in this region. It was mentioned previously [14,19] that the pre-peak is originated by Ge–Ge correlation, which is linked to GeSe4/2 tetrahedron. On the other hand, Machado et al. [20] have used the reverse Monte Carlo (RMC) simulations and noticed the existence of large number of Se–Se pairs in the first coordination shell suggesting that the tetrahedral units are linked by Se–Se bridges. In the present work, the partial FSDP appeared in the SGe–Ge(K) and SSe–Se(K) indicating that the intermediate-range order is not attributed only to Ge–Ge bonds, but also can be referred to Se–Se bonds. Looking at the main two peaks of the SSe–Se(K) curves, shown in the above figure, the Se–Se bonds do exist for the investigated glasses. Finally, three-dimensional structural configurations obtained using RMC modeling for the Ge10Se90 and Ge40Se60 specimens are shown in Fig. 7. In fact, these images represent 10% of the whole configuration. In other words, the depth of the represented ˚ which can be a good reason for configuration is less than 2.4 A, appearance of very small number of tetrahedral units. In each image, one structural unit is circled and taken outside the picture to clarify its building unit. It is easily seen that Se–Se chains and rings predominate at the Ge10Se90 configuration and does not have any Ge–Ge bonds. On the other hand, Ge40Se60 image shows many Ge–Ge bonds, which are responsible for connecting the tetrahedral units, and consequently causing the intermediaterange order in the Ge-rich glasses. As expected from the above discussion, tetrahedral GeSe4 and Ge2Se6 represent the main structural unit in the Ge10Se90 and Ge40Se60, respectively. 5. Conclusions RMC simulation has been successfully applied to get the partial pair distribution functions and the partial structure factors from
the experimental X-ray diffraction measurements of the investigated glasses. The partial pair distribution functions indicate that the basic building units are GeSe4 and Ge2Se6 tetrahedral units in the Se-rich and Ge-rich glasses, respectively. Some of these tetrahedral units are connected by the homopolar units as confirmed by the bond angle distribution functions. The partial structure factors have shown that not only the homopolar Ge–Ge bonds, but also Se–Se bonds are behind the appearance of the FSDP in the total structure factor.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
I. Petri, P.S. Salmon, Phys. Rev. Lett. 84 (2000) 2413. O. Uemura, Y. Sagara, T. Satow, Phys. Status Solidi (a) 26 (1974) 99. V. Petkov, D. Qadir, S.D. Shastri, Solid State Commun. 129 (2004) 239. Y. Murakami, T. Usuki, M. Sakurai, S. Kohara, Mater. Sci. Eng. A 449–451 (2007) 544. ¨ Phys. Rev. B 39 (1989) 6034. P. Vashishta, R.K. Kalia, I. Ebbsjo, P. Boolchand, X. Feng, W.J. Bresser, J. Non-Cryst. Solids 293–295 (2001) 348. S. Kohara, K. Suzuya, Y. Kashihara, N. Matsumoto, N. Umesaki, I. Sakai, Nucl. Instrum. Methods A 467-468 (2001) 1030. R.L. McGreevy, J. Phys. C 13 (2001) R877. R.L. McGreevy, L. Pusztai, Mol. Simulation 1 (1988) 359. Y. Wang, E. Ohata, S. Hosokawa, M. Sakurai, E. Matsubara, J. Non-Cryst. Solids 337 (2004) 54. S. Hosokawa, Y. Wang, M. Sakurai, J.-F. Be´rar, W.-C. Pilgrim, K. Murase, Nucl. Instrum. Methods Phys. Res. B 199 (2003) 165. N.R. Rao, P.S.R. Krishna, S. Basu, B.A. Dasannacharya, K.S. Sangunni, E.S.R. Gopal, J. Non-Cryst. Solids 240 (1998) 221. A.H. Moharram, A.M. Abdel-Basit, Physica B 358 (2005) 279. P.H. Fuoss, P. Eisenberger, W.K. Warburton, A. Bienestock, Phys. Rev. Lett. 46 (1981) 1537. ¨ Phys. Rev. B 39 (1989) 6034. P. Vashishta, R.K. Kalia, I. Ebbsjo, P.S. Salmon, I. Petri, J. Phys. C 15 (2003) S1509. ¨ A. Szczygielska, A. Burian, J.C. Dore, V. Honkimaki, S. Duber, J. Alloys Compd. 362 (2004) 307. S.R. Elliott, Physics of Amorphous Materials, 2nd ed., Longman, New York, 1990. /http://wwwisis2.isis.rl.ac.uk/rmc http://wwwisis2.isis.rl.ac.uk/rmcS. K.D. Machado, J.C. de Lima, C.E.M. Campos, A.A.M. Gasperini, S.M. de Souza, C.E. Maurmann, T.A. Grandi, P.S. Pizani, Solid State Commun. 133 (2005) 411. D.N. Tafen, D.A. Drabold, Phys. Rev. B 71 (2005) 054206. K.D. Machado, J.C. de Lima, C.E.M. Campos, T.A. Grandi, P.S. Pizani, J. Chem. Phys. 120 (2004) 329.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Reverse Monte Carlo simulation of GexSe100 x glasses A.H. Moharram a,n, A.M. Abdel-Baset b a b
Faculty of Science, King Abdul Aziz University, Rabigh Campus, P.O. Box 334, Saudia Arabia Physics Department, Faculty of Science, Assiut University, Assiut 71516, Egypt
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 April 2010 Received in revised form 8 July 2010 Accepted 9 July 2010
Amorphous GexSe100 x (with x ¼10, 20 and 40 at%) alloys were prepared using the melt–quench technique. Two-dimensional Monte Carlo of the total pair distribution functions (MCGR) have been found and used to assemble the three-dimensional atomic configurations using the reverse Monte Carlo rmc ðrÞ, (RMC) method. The simulations are useful to compute the partial pair distribution functions gGeGe rmc rmc rmc rmc ðrÞ, gSeSe ðrÞ and the partial structure factors Srmc gGeSe GeGe ðrÞ, SGeSe ðrÞ, SSeSe ðrÞ of the studied glasses. The partial pair distribution functions indicate that the basic building units are GeSe4 and Ge2Se6 tetrahedral units in the Se-rich and Ge-rich glasses, respectively. Some of these tetrahedral units are connected by the homopolar units as confirmed by the bond angle distribution functions. The partial structure factors have shown that not only the homopolar Ge–Ge bonds, but also Se–Se bonds are behind the appearance of the first sharp diffraction peak (FSDP) in the total structure factor. & 2010 Elsevier B.V. All rights reserved.
Keywords: X-ray diffraction Chalcogenide glasses Reverse Monte Carlo Short-range order Medium-range order
1. Introduction Chalcogenide glasses present a great potential for application in technological devices, such as optical fibers, memory materials and switching devices, but their use is limited due to several factors. One of them is the difficulty in obtaining information about atomic structures, which define the short-range order (SRO) of the alloy. Although, many kinds of structural studies for amorphous Ge–Se alloys [1–6] have been performed to investigate the short-range order of the system, the glass network structure of these alloys is not yet fully understood. High-resolution diffraction technique with a wide range of the scattering vector (K) is required for structural studies of the amorphous materials. For example, high-energy X-ray diffraction measurements have some advantages [7]: highly structural information due to the wider range of K, smaller correction terms, and easy to compare with neutron diffraction data because of similar transmission method. Unfortunately, the X-ray source used in the present experiment, CuKa target emits radiation with ˚ does not have wide range of the wavelength l ¼1.5418 A, scattering vector. To overcome this problem, two-dimensional Monte Carlo of the total pair distribution function (MCGR) has been found, which is used to assemble three-dimensional atomic configuration using the reverse Monte Carlo (RMC) method.
n
Corresponding author. E-mail addresses: [email protected], [email protected] (A.H. Moharram). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.07.018
Some of the structural intermediate-range order (IRO) information can be obtained from the experimental diffraction data. Reverse Monte Carlo (RMC) simulation [8,9] represents, when used carefully, a powerful tool to extract these information. It assembles three-dimensional atomic configurations using experimental diffraction data implicitly in the simulation. The intimate connection between computational and experimental processes means that the better quality, and higher resolution of the experimental data, the more reliable RMC model of a network structure for vitreous materials. The RMC method is considered as an inverse problem in which the experimental data are enforced to build atomic configurations that have the desired structural and electronic properties. The main point is to set up a generalized function containing as much information as possible, and then optimize the function for generating configurations toward exact agreement with the experimental data. X-ray diffraction study of the glassy GexSe1 x systems [10,11], in a wide concentration range 0rx r0.33, has demonstrated that besides the well-established SRO information, a pre-peak appeared in the total structure factor S(K) at a scattering vector K of about 1.1 A˚ 1. The pre-peak, being clear evidence for existence of the intermediate-range order (IRO), showed a systematic decrease in intensity and shifts towards higher K-values with decrease in Ge concentration. A similar variation in the pre-peak of the structure factor with Ge content was also observed using the neutron diffraction measurement [12]. In the present work, the short- and intermediate-range orders (SRO and IRO) for three Ge10Se90, Ge20Se80 and Ge40Se60 glasses have been studied using Monte Carlo of pair distribution (MCGR) and the reverse Monte Carlo (RMC) simulations.
A.H. Moharram, A.M. Abdel-Baset / Physica B 405 (2010) 4240–4244
pre-peak, the K-resolution is better than that with targets having higher characteristic energies.
2. Reverse Monte Carlo (RMC) Three-dimensional arrangement of N atoms is placed into a cubic cell with periodic boundary conditions. The atomic number density (r) should be the same as the experimental value. The positions of the atoms are chosen randomly. The partial pair distribution function gijcal ðrÞ can be calculated from the initial configuration [8] by gijcal ðrÞ ¼
nij ðrÞ 4pr 2 dr rci
ð1Þ
where ci is the concentration of i-type atoms and nij(r) the number of j-type atoms at a distance between r and r +dr from a central i-type atom, averaged over all atoms as centers. The total pair distribution gcal(r) is calculated from m h i 1 X ci cj fi fj gijcal ðrÞ1 g cal ðrÞ ¼ 2 f i,j ¼ 1
ð2Þ
where m is the number of elements in the sample (m¼2 for the P present work), fi the atomic scattering factor and f ¼ i ci fi ðKÞ. 2 cal The deviation w of the calculated g (r) from the experimental, gexp(r), obtained from the MCGR method, can be computed from the expression
w2 ¼
n X ½g cal ðri Þg exp ðri Þ2 i¼1
s2 ðri Þ
4241
ð3Þ
where n is the number of experimental points and s2(ri) the statistical error. Move one atom at random, and recalculate gcal(r) of the new configuration. If w2 decreases, the change is accepted, while if it increases, the move is accepted with probability of exp(–Dw2/2) where Dw2 is the change in w2. As the number of accepted atom moves increases, w2 will initially decrease until it reaches an equilibrium value. Thus, the atomic configuration corresponding to the equilibrium should be consistent with the experimental functions within the experimental errors. All the results shown in this article are based on the partial and total functions obtained at the equilibrium level. Fourier transformation of the partial pair distribution, gij(r), gives the partial structure factor Z 1 sin Kr Sij ðKÞ ¼ r dr ð4Þ 4pr 2 ðgij ðrÞ1Þ Kr 0 The above function is important for determination of the intermediate-range order parameters and also to specify the bond type responsible for the existence of the FSDP. On the other hand, the short-range order parameters such as the inter-atomic distances, the partial coordination number and the bond angle distribution can be obtained from the gij(r) functions.
4. Results and discussion The experimental X-ray data of the present three Ge10Se90, Ge20Se80 and Ge40Se60 compositions were analyzed using some computer programs. The first step was to correct the observed X-ray intensities through background subtraction followed by absorption and polarization corrections. From the corrected X-ray intensities, the variations in the total structure factor, S(K), with the scattering vector (K ¼4psin y/l, where y is half the scattering) can be obtained [13]. Fig. 1 shows the experimental total structure factors, S(K), of the investigated glasses up to K¼ 6.874 A˚ 1. The existence of the first sharp diffraction peak (FSDP) is clear especially for the Ge40Se60 glass, which is located at K ¼1.0 A˚ 1. FSDP is commonly observed in covalently bonded amorphous materials, which implies the presence of intermediate-range order. Its intensity decreases with decrease in Ge content to the point it appears as a shoulder in the Ge10Se90 composition. Indeed, not only its height changes with Ge content, but also it is shifted to a higher K-values with decrease in Ge content. The pre-peak is known to be much dependent on Ge–Ge correlations and, to a lesser extent, on Ge–Se correlations [14–16]. In the structural analysis using Fourier transformations [17,18], a modification factor was suggested to reduce the effect of termination data at a finite Kmax. This factor in turn, while reduces the spurious oscillations, leads to a broadening of the genuine peaks in g(r). To overcome this problem, an inverse method was applied to determine the pair distribution function from the total structure factor, measured by neutron or X-ray diffraction, using a MCGR program [19]. One of its advantages is that a direct estimation of errors in g(r) is possible by repeating the process until getting a minimum deviation of the simulated values from the experimental S(K) ones, as shown in Fig. 1. The g(r) function obtained from the MCGR method does not require integration process or integration limits, which means that considering Kmax instead of infinity has no effect on the obtained functions [12]. For the present work, the starting value of g(r) is ˚ Fig. 2 shows the zero below r o1.9 A˚ and equals one for r Z1.9 A. conventional (solid circles) of the pair distribution functions, g(r), of the investigated GexSe100 x glasses together with the simulated
3. Experimental technique Glassy chalcogenide of GexSe100 x (with x ¼10, 20 and 40 at%) alloys were prepared using the melt–quench method. The glassy nature of the quenched alloys was confirmed by X-ray diffraction technique. X-ray diffraction experiment was done using a Philips diffractometer (PW-1710). The XRD patterns were recorded at 40 kV and 30 mA, with a graphite monochromator, using the ˚ with scanning speed of 0.041/min. The CuKa line (l ¼ 1.5418 A), experiment was done in the scattering angle range 4r2y r1151 in steps of 0.11, which corresponds to K-vector range 0.284rK r6.874 A˚ 1. Copper source was chosen for two reasons. Firstly, its characteristic energy is lower than Ge K absorption edge energy. Therefore, the scattering comprises no fluorescence X-rays, and the anomalous terms in the atomic factors of Ge and Se are very small. Secondly, for obtaining the precise data of the
Fig. 1. Experimental X-ray structure factors (K) together with the results of MCGR (J) and RMC ( ) simulations of the investigated GexSe100 x glasses.
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values (full lines) obtained using the MCGR method. As said before, the modification factor while reduces the spurious oscillations, leads to a broadening of the conventional g(r) functions. The broadening is wide enough to cause an overlap between the first and the second peaks. Since the simulation does not depend on the modification factor, the simulation g(r) data of Fig. 2 have shown two resolved peaks providing information for
Fig. 2. Conventional pair distribution functions (K) together with the results of MCGR (J) and RMC ( ) simulations of the investigated GexSe100 x glasses.
Fig. 3. The partial pair distribution functions, gij(r), for the studied GexSe100 x glasses.
the first and the second coordination spheres. Indeed, these three curves represent the backbone for the three-dimensional reverse Monte Carlo (RMC) simulations. The starting point in RMC simulation [19] is to randomly generate the configuration distribution of N ¼4000 atoms inside a cubic box. The cubic edges for the investigated Ge10Se90, Ge20Se80 ˚ respectively. The and Ge40Se60 glasses are 23.8, 23.6 and 23.3 A, portions of Ge or Se atoms in the assumed configuration will vary according to their atomic percentages. Fig. 3 shows the partial pair distribution gGe–Ge(r), gGe–Se(r), gSe–Se(r) functions for the present three glasses. In the first coordination sphere, the zero values of the gGe–Ge(r) in case of the Se-rich (Ge10Se90, Ge20Se80) glasses indicate that only homopolar Se–Se bonds exist in addition to heteropolar Ge–Se bonds. However, the Ge-rich (Ge40Se60) glass shows two homopolar Ge–Ge and Se–Se bonds in addition to heteropolar Ge–Se bonds. Summation of these partial pair distribution functions yields the RMC g(r) curves, which give good fit to the point that cannot be distinguished from the corresponding MCGR data shown in Fig. 2. The average Ge–Ge, Ge–Se and Se–Se bond lengths, as obtained from the refined ˚ RMC model, are 2.52 70.065, 2.43 70.065 and 2.47 70.065 A, respectively. RMC simulation of the final atomic configuration gives the number of neighbors (probability) distributions within the first coordination shell as shown in Fig. 4. In Se-rich (Ge10Se90 and Ge20Se80) glasses, the probability of finding Ge atom in the nearest neighbors of a central Ge atom is almost zero, while it goes up to 74% in case of Ge-rich (Ge40Se60) glass. This means that each Ge atom can be surrounded by four Se neighbors in the former glasses, while its four neighbors in case of Ge-rich glass are three Se atoms plus one Ge atom as given by PGe–Se(n) function of
Fig. 4. The number of neighbors distribution within the first coordination shell obtained from RMC model.
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Fig. 4. The present RMC simulations have indicated that about 97%, 95% of Ge atoms are surrounded by four Se atoms in the Ge10Se90 and Ge20Se80 glasses, respectively. In case of Ge-rich (Ge40Se60) glass, 68% of Ge atoms are connected to three Se atoms and one Ge atom. In a previous study [12], the authors reported that the main structure units in Se-rich and Ge-rich glasses are tetrahedral GeSe4 and Ge2Se6 units, respectively. The present PGe–Se(n) function confirms their conclusion. Based on the YSe–Se–Se(y) and PSe–Se(n) functions, the above tetrahedral structural units seem to be connected by Se–Se bridges, forming small chains and rings. The average partial coordination numbers are, given in Table 1, close to some extent to those reported by other references [20,21]. The corresponding values, calculated using the chemical order network model and listed in the above Table, confirm the
Table 1 The present partial coordination numbers for the studied glasses and those reported by other references [20,21]. The values calculated using the chemical order network model (CONM) are also listed.
Ge10Se90
Ge20Se80
Ge40Se60
Present CONM Ref. [21] Present CONM Ref. [20] Ref. [21] Present CONM
Ge–Ge
Ge–Se
Se–Ge
Se–Se
0.03 0 0.0 0.03 0 0.22 0.22 1.23 1
3.96 4 3.99 4.04 4 3.61 3.61 2.97 3
0.44 0.44 0.44 1.01 1 0.9 0.9 1.98 2
2.68 1.56 1.56 2.26 1 1.71 1.71 1.73 0
Fig. 5. Bond angle distribution functions, Y(y), obtained from RMC simulation of the Ge20Se80 glass.
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applicability of the CONM in the structure configuration of the investigated alloys. The presence of Se–Se bridges between the tetrahedral units is a possible reason for high partial coordination number for Se–Se as compared to that reported previously and computed from the chemically ordered network model. Bond angle distribution functions, Y(y), are obtained from the final configuration of the present three glasses using triplets program [19], in which the bond angle for any reference atom can be calculated from Cartesian coordinates of the final positions of the surrounding atoms. The Y(y) functions of the Ge20Se80 glass, as an example, are shown in Fig. 5. The YSe–Ge–Se(y) function presents a main peak around 104.21, which is near the ideal tetrahedral angle of 1091. A small peak appearing at 601 can be attributed to the existence of wrong (homo) bonds. In glassy state, some Ge atoms are, freezing in corner positions of the tetrahedral units instead of Se atoms, responsible for the above small peak. A similar behavior is shown by YSe–Se–Se(y) function, with a main peak at 601 and a small one at 104.21. Selenium atoms occupying the face of a perfect tetrahedron should exhibit internal angles of 601, and the 104.21 angle corresponds to Se–Se–Se angles can be associated with inter-tetrahedral units. The present Y(y) functions confirm that distorted tetrahedral units are formed in the investigated glasses. These structural units seem to be connected by Se–Se bridges, forming small chains and rings, as previously reported [22]. YGe–Ge–Ge(y) function shows broad distribution from 501 to 1501, which indicates that each tetrahedral unit is tetrahedrally connected with other tetrahedral by corner share and edge share. From this figure, one can conclude that most of the Ge atoms surrounded either by four Se atoms or three Se atoms and one Ge atom are forming tetrahedral units.
Fig. 6. The partial structure factors, Sij(r), for the studied GexSe100 x glasses.
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Fig. 7. Atomic Ge10Se90 and Ge40Se60 configurations obtained from RMC modeling. Ge and Se atoms are defined by (K) and () balls, respectively. The images represent 10% from the whole configuration and show different tetrahedral structural units.
Fourier transformation of the partial pair distribution gives the corresponding partial scattering factor. The partial scattering factors as function of the scattering vector for the present three glasses are shown in Fig. 6. These curves are very important especially in the region where the FSDP is located. The SGe–Se(K) function is not included in this figure because it does not have a peak in this region. It was mentioned previously [14,19] that the pre-peak is originated by Ge–Ge correlation, which is linked to GeSe4/2 tetrahedron. On the other hand, Machado et al. [20] have used the reverse Monte Carlo (RMC) simulations and noticed the existence of large number of Se–Se pairs in the first coordination shell suggesting that the tetrahedral units are linked by Se–Se bridges. In the present work, the partial FSDP appeared in the SGe–Ge(K) and SSe–Se(K) indicating that the intermediate-range order is not attributed only to Ge–Ge bonds, but also can be referred to Se–Se bonds. Looking at the main two peaks of the SSe–Se(K) curves, shown in the above figure, the Se–Se bonds do exist for the investigated glasses. Finally, three-dimensional structural configurations obtained using RMC modeling for the Ge10Se90 and Ge40Se60 specimens are shown in Fig. 7. In fact, these images represent 10% of the whole configuration. In other words, the depth of the represented ˚ which can be a good reason for configuration is less than 2.4 A, appearance of very small number of tetrahedral units. In each image, one structural unit is circled and taken outside the picture to clarify its building unit. It is easily seen that Se–Se chains and rings predominate at the Ge10Se90 configuration and does not have any Ge–Ge bonds. On the other hand, Ge40Se60 image shows many Ge–Ge bonds, which are responsible for connecting the tetrahedral units, and consequently causing the intermediaterange order in the Ge-rich glasses. As expected from the above discussion, tetrahedral GeSe4 and Ge2Se6 represent the main structural unit in the Ge10Se90 and Ge40Se60, respectively. 5. Conclusions RMC simulation has been successfully applied to get the partial pair distribution functions and the partial structure factors from
the experimental X-ray diffraction measurements of the investigated glasses. The partial pair distribution functions indicate that the basic building units are GeSe4 and Ge2Se6 tetrahedral units in the Se-rich and Ge-rich glasses, respectively. Some of these tetrahedral units are connected by the homopolar units as confirmed by the bond angle distribution functions. The partial structure factors have shown that not only the homopolar Ge–Ge bonds, but also Se–Se bonds are behind the appearance of the FSDP in the total structure factor.
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