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Structural correlations in GexSe1−x glasses – a neutron diffraction study
Journal of Non-Crystalline Solids 240 (1998) 221±231
Structural correlations in GexSe1ÿx glasses ± a neutron diraction study N. Ramesh Rao a, P.S.R. Krishna b, S. Basu b, B.A. Dasannacharya K.S. Sangunni a, E.S.R. Gopal c b
b,*
,
a Department of Physics, Indian Institute of Science, Bangalore 560 012, India Solid State Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India c National Physical Laboratory, New Delhi 110 012, India
Received 9 February 1997; received in revised form 5 January 1998
Abstract Neutron diraction measurement is carried out on Gex Se1ÿx glasses, where 0.1 6 x 6 0.4, in a Q interval of 0.55±13.8 ÿ1 . The ®rst sharp diraction peak (FSDP) in the structure factor, S(Q), shows a systematic increase in the intensity A and shifts to a lower Q with increasing Ge concentration. The coherence length of FSDP increases with x and becomes maximum for 0.33 6 x 6 0.4. The Monte-Carlo method, due to Soper, is used to generate S(Q) and also the pair correlation function, g(r). The generated S(Q) is in agreement with the experimental data for all x. Analysis of the ®rst four peaks in the total correlation function, T(r), shows that the short range order in GeSe2 glass is due to Ge(Se1=2 )4 tetrahedra, in agreement with earlier reports. Se-rich glasses contain Se-chains which are cross-linked with Ge(Se1=2 )4 tetrahedra. Ge2 (Se1=2 )6 molecular units are the basic structural units in Ge-rich, x 0.4, glass. For x 0.2, 0.33 and 0.4 there is evidence for some of the tetrahedra being in an edge-shared con®guration. The number of edge-shared tetrahedra in these glasses increase with increasing Ge content. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In recent years, structural investigations have been carried out on semiconducting chalcogenide glasses [1±4]. The Ge±Se system has been studied using both diraction [5±9] and spectroscopic [10± 15] techniques. The existence of Ge(Se1=2 )4 tetrahedral units in Gex Se1ÿx glasses has been inferred from Raman scattering studies. The Raman spectrum consists of several vibrational modes, some of * Corresponding author. Present address: Inter University Consortium for DAE Facilities, University Campus, Khandwa Road, Indore 452 017 M.P., India. Tel.: +91-731 462 265; fax: +91-731 462 294.
which have been identi®ed with the normal modes of these tetrahedral units [10±12]. The Raman mode at 217 cmÿ1 , according to Sugai [11] or at 26.8 meV ( 216 cmÿ1 ) according to Sinclair et al. [16] has been subject of debate. This mode, known as the companion mode, A1c , is associated with the intermediate range order (IRO) observed in these glasses. Another structural feature, de®nitely due to IRO, is the `pre-peak' or the ®rst sharp diraction peak (FSDP) in the structure factor S(Q) ÿ1 [17]. These experimentally obaround 1.0 A served features can be explained within the framework of either the constraint or percolation theory [18,19] or the chemically ordered continuous random network (COCRN) model [20,21].
0022-3093/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 7 0 5 - 4
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According to the percolation model, a glass network consists of chemically ordered clusters. The presence of such clusters implies the existence of homopolar bonds even at the stoichiometric composition [12]. Thus, chemical order is broken at this composition. In the case of GeSe2 glass the structure of these molecular fragments is believed to be similar to the layer-like structure of the high temperature polymorph of c-GeSe2 . The FSDP observed in S(Q), is taken as evidence for the presence of such a layer-like fragment. Further, the A1c mode is associated with the breathing mode of Se±Se dimers present in the fragments [18,12]. On the basis of bond energy considerations, the COCRN model predicts that the heteropolar bonds are maximised at any composition x. Thus at x 0.33 the glass network consists of only Ge±Se bonds while for x < 0.33, Se±Se bonds are present in addition to Ge±Se bonds. Similarly, Gerich glasses i.e. x > 0.33 contain Ge±Ge as well as Ge±Se bonds [20]. The origin of the A1c mode and the FSDP are explained on the basis of rings of dierent sizes present in the glass network. Speci®cally, the A1c mode arises because of four membered rings i.e. edge-shared tetrahedra while the FSDP is associated with six membered rings [21]. The FSDP has been observed in all AX2 (where A Si or Ge and X S, Se, or O) type of glasses. Furthermore, when scaled with the bond distance r1 i.e. Qr1 , it lies within 2.2±2.8 for all the glasses [22]. A variety of explanations have been advanced to understand this feature. These include the presence of layer-like morphology [12], random packing of tetrahedra [22] and, recently, as being due to the chemical ordering of voids [23]. In this regard the GeSe2 has been widely studied and recent results of Susman et al. [5] on glass and Penfold and Salmon [6] on liquid suggest that an appreciable number of Ge-tetrahedra are in edgeshared con®guration. Molecular dynamics simulations of molten and glassy GeSe2 also support this picture [24,25]. Thus, in the light of these investigations it is natural to ask; how does the network connectivity in Gex Se1ÿx glasses change as the Ge content is varied? In literature there are reports of X-ray as well as electron diraction studies on Ge±Se glasses, both
on ®lms as well as larger samples [26±28]. The most recent report [28], is on Gex Se1ÿx glasses lying on the Se-rich side of the glass forming region, with x going to 0.3. These data were collected for a maximum momentum transfer vector, Qmax , of 6.6 ÿ1 , thus limiting the resolution in real space. In A the present structural investigation of Gex Se1ÿx glasses, where 0.1 6 x 6 0.4, a much larger Q interval is covered. For the ®rst time, to the best of our knowledge, the Monte-Carlo method (MC) is used to generate the pair correlation function [29± 31] in our samples. This MC method is a departure from the conventional procedure of Fourier transformation of the experimental data modi®ed with a Lorch function [32]. Some preliminary results in these samples have been reported earlier [33,34]. Section 2 gives a brief account of the formalism relevant to the experiment. Sample preparation, data collection and reduction are described in Section 3. An account of the MC method used [29] is presented in Section 4. The results of present investigations are presented in Section 5. Discussion on the intermediate range order, the short range order and the network connectivity is presented in that order in Section 6 followed by conclusions in Section 7.
2. Basic formalism The cross-section, dr/dX, after the data reduction, for a multicomponent system in terms of the partial structure factors, Sij (Q), is given by [5] X rinc dr X 1 ci cj 1=2 bi bj Sij Q ci i ; 4p dX i;j i where ci , bi and rinc are the concentration, coi herent scattering length and incoherent scattering cross-section of the ith component, respectively. The total structure factor, S(Q), can be expressed in terms of the partial structure factors Sij (Q) as 1 X ci cj 1=2 bi bj Sij Qÿdij ; 2 S Q 1 2 hbi i;j P where hbi i ci bi :
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231
The Fourier transformation of S(Q) gives the pair distribution function, g(r), Z1 1 Q S Q ÿ 1 sin Qr dQ; 3 g r 1 2 2p rq 0
where q is the total number density. In literature, three other correlation functions are generally used. The dierential correlation function, D(r), is given by
223
where Wij are the weighting factors de®ning the contributions of each of the partial Nij to Ce . 3. Experimental procedures 3.1. Sample preparation
and the radial distribution function, N(r), is given by
Gex Se1ÿx samples where 0.1 6 x 6 0.4, were prepared by the melt-quench method. Elemental Ge and Se of 5N purity were placed in round ampoules of length 6.0 cm in correct stoichiometric proportion. The ampoules were vacuum sealed at a pressure of 2 ´ 10ÿ5 Torr. They were kept in a horizontal furnace for 24 h at 1233 K. Glasses were obtained by quenching the liquid in a solution of NaOH + ice-water. About 13 g of each composition was prepared and con®rmed to be amorphous with X-ray powder diraction. Neutron diraction studies were carried out on pulverised samples. Infrared absorption spectrum was recorded to ensure the absence of oxygen contamination in the samples: no detectable absorption due to Se±O (904±925 cmÿ1 ) or Ge±O (855 cmÿ1 ) [5] was observed.
N r rT r 4pr2 qg r:
3.2. Experiment and data reduction
2 D r p
Z1 Q S Q ÿ 1 sin Qr dQ;
4a
0
D r 4prqg r ÿ 1:
4b
The total correlation function, T(r), is used for obtaining the peak positions [30] and is given by T r 4prq D r;
5a
T r 4prqg r
5b
6
N(r) is used to derive the average number of atoms lying within a range, r to r + dr, of a given atom. The area of a peak in N(r) gives the average coordination number Ce [5], which in turn is related to the partial coordination numbers Nij expressing the average number of j-type atoms around any i-type atom within the range of integration, Ce
1 X ci bi bj Nij 2 hbi i;j
since ci Nij cj Nji ! b1 2 b b 1 2 N11 2c1 Re Ce c1 N12 hbi hbi2 b2 2 N22 ; c2 hbi Ce W11 N11 W12 N12 W22 N22 ;
7
8 9
Neutron diraction measurements were carried out on the High-Q diractometer at the Dhruva reactor at Trombay, Bombay [35]. Data was collected on samples in vanadium containers in a Qÿ1 . Procedures given by range of 0.55±13.8 A Eglesta [36] were used in carrying out the data reduction, with corrections for background, multiple scattering and inelastic scattering. Vanadium normalisation was used to put the data on an absolute cross-section scale. The peak and valley positions of the present data on GeSe2 glass agree well with that of Susman et al. [5]. The peak heights in the data are generally lower, except for the FSDP, partly because the data of Ref. [5] was collected at 10 K, whereas, the present experiments were carried out at 300 K and partly because of poorer Q-resolution of our data (2.5%) compared to 0.35%; of Ref. [5]. The larger FSDP in the present data is due to its well known anomalous temperature dependence [23].
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4. Monte-Carlo G(r) (MCGR) As the experimental data is collected in a ®nite Q range the upper limit of integral in Eq. (3) has to be replaced by Qmax instead of in®nity. This results in termination errors which show up as spurious oscillations (peaks) in g(r). In the conventional method of analysis one uses a modi®cation function [32] to reduce the eect of termination of data at a ®nite Qmax . This function in turn, while reducing the spurious oscillations, leads to a broadening of the genuine peaks in g(r). To overcome this problem a one-dimensional MonteCarlo method [29] with similar basic principles to the earlier published work of Soper [30,31] was used. We have used a computer program developed by us after testing it successfully with Pencus±Yevick Hard Sphere data. A brief description of the method is given below. In the MC method [29] used, a trial g(r) and its analytically known S(Q) are taken. The g(r) is then modi®ed randomly at any randomly chosen r. For every change in g(r), its S(Q) is modi®ed at every P point analytically. If v2 (v2 {(SExp (Q) ) SCal (Q))2 /r2 }, where r2 is the statistical error in the experimental data) reduces the move is accepted. If it increases, the move is accepted with a probability of exp()Dv2 /T), where Dv2 is the change in v2 and T is a ®ctitious `temperature'. The process is repeated until a suitable v2 is reached (1% in our case). One of the advantages of this method is that a direct estimation of errors in g(r) is possible by repeating the process and arriving at an ensemble of pair correlation [29,31]. For Gex Se1ÿx glasses, the trial g(r) was taken to be zero below r/rc < 1.0 and equal to one for [5]. The condition r/rc P 1.0, where rc 2.0 A g(r) 0 for r/rc < 1.0 was maintained throughout the simulation. In the present investigation g(r) has upto been generated at an interval (Dr) of 0.05 A, The experimental and the simulated rmax of 60 A. ÿ1 ) structure factors for glasses (obtained upto 25 A with x 0.1, 0.2, 0.33 and 0.4 are shown in Fig. 1. As is apparent from the ®gure, the simulation procedure outlined above gives a good ®t to the experimental data. It is interesting to note that the peak and valley positions of the extrapolated region of S(Q) for GeSe2 glass agree very well with
Fig. 1. The total structure factor S(Q) for Gex Se1ÿx glasses. Solid line: S(Q) generated by the Monte-Carlo method, dots: experimental data; solid lines: generated data. The data (both the dots and the solid lines) corresponding to x 0.2, 0.33 and 0.4 are shifted by one unit for sake of clarity.
the measurements of Susman et al. [5]. Soper has shown that in the case of hard spheres where S(Q) is analytically known, the MCGR method could be used to extrapolate S(Q) [29]. Unlike the S(Q) of hard spheres real experimental data have errors and this procedure may be of doubtful value to
Fig. 2. The total correlation function T(r) (generated by the Monte-Carlo method) of Gex Se1ÿx glasses. Each curve was generated using 500 histories. For the sake of clarity the curves corresponding to x 0.2, 0.33 and 0.4 are shifted by one unit. The vertical lines are the error bars for GeSe9 composition. The inset shows r1 vs. x. n correspond to our data. n correspond to other neutron diraction data for x 0 (Ref. [1]), x 0.33 (Ref. [5]) and x 1.0 (Ref. [43]).
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231
extend S(Q) data. The present work shows that even in a real situation like ours the extrapolation is not unreasonable. However, it is not the intention here to suggest that this extrapolation be treated as real structure factor. We have not made use of this extrpolated data in any analysis. In Fig. 2 the T(r)s generated using 500 histories for Errors on T(r) are each x are shown upto 10 A. shown only for GeSe9 , for the sake of clarity. The error for other xs is typically equal to or less than that for GeSe9 . The number densities and the weighting factors, Wij , of all the samples are given in Table 1. 5. Results Fig. 1 shows the measured structure factors of Gex Se1ÿx glasses. We observe that a peak appears ÿ1 . For x 0.1 it appears in S(Q) around 1.0 A only as a shoulder. This peak becomes larger for larger x and also shifts to a smaller Q. Fig. 2 shows the T(r) functions obtained from the measured S(Q) data using MCGR method described above. We observe that the ®rst peak at shows a marginal shift towards a larger r 2.37 A r with increasing x. The inset in Fig. 2 shows earlier results of ®rst peak position (r1 ) as a function of x along with earlier positions, showing good agreement with existing measurements. In
225
the shift towards the case of the peak at r 3.8 A, larger r is larger (3.84±3.95 A). Unlike these two is observed only for peaks, the peak around 3.0 A x 0.2, 0.33 and 0.4. The intensity of this peak increases with x. On the other hand, the peak becomes broader as x increases from around 4.7 A 0.1 to 0.4. This trend is in a direction, opposite to that of the peak at 3.0 A. 6. Discussion 6.1. Intermediate range order (IRO): The peak parameters for the FSDP are obtained by following the procedure outlined by Johnson et al. [1] and are given in Table 2. These parameters indicate that the characteristic length (R 2p/Q) responsible for IRO increases with increasing Ge content. The `coherence length' (L 2p/DQ, where DQ is the full width at half maximum) increases up to x 0.33, indicating that the IRO is more in glasses with x 0.33 and 0.4 as compared to that in glasses with smaller x. Structural studies on Six Se1ÿx glasses also show a similar trend [1]. Neutron diraction study of molten GeSe2 with isotopic substitution [6], X-ray anomalous scattering of g-GeSe2 [37], and molecular dynamics simulation of liquid and glassy GeSe2 [24] show that the IRO arises primarily because of
Table 1 The values of number densities q and the weighting factors Wij used in the the data analysis of Gex Se1ÿx glasses are given below ÿ3 ) x q (A WGe±Se WGe±Ge WSe±Se WSe±Ge 0.1 0.2 0.33 0.4
0.0330 0.0336 0.0328 0.0346
0.204 0.406 0.673 0.804
0.105 0.209 0.346 0.413
0.895 0.791 0.655 0.587
1.836 1.624 1.346 1.206
Table 2 The position (Q1 ) and the full width at half maximum (FWHM, DQ1 ) of the FSDP of Gex Se1ÿx glasses are given below (also given in the table are the values of the coherence length L and the correlation length R of the glasses) ÿ1 ) ÿ1 ) DQ1 (A R 2p/Q1 (A) L 2p/DQ1 (A) x Q1 ( A 0.1 0.2 0.33 0.4
) 1.095 0.988 0.965
) 0.30 0.22 0.22
) 5.74 6.36 6.51
) 20.9 28.6 28.6
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Ge±Ge correlations in the glass. The observed trends in the FSDP with x, in the present investigations, is in conformity with these results. 6.2. Short range order (SRO): The radial distribution functions N(r) of all the glasses obtained by (i) the Monte-Carlo method and (ii) by Fourier transformation of the experimental data i.e., conventional method (Fourier transformation was carried out using Lorch ÿ1 ) are modi®cation function with Qmax 13.8 A shown in Fig. 3(a) and (b), respectively. N(r) of GeSe2 glass is in good agreement with that of Susman et al. [5]. Especially, the peak around 3.0 which is indicative of the presence of edgeA, shared tetrahedra in the glass [5], is noticeable. This feature is not observable in the data of Ha®z et al. [28]. This absence is due to the small Qmax (6.6 ÿ1 ) of their data, thus reducing the resolution in A real space. We have investigated this absence of the above feature by terminating our data at smaller Qmax . In the present work this peak is better resolved in the T(r)s generated using MCGR method (Fig. 3(a)). To obtain information regarding the SRO, a knowledge of the nearest neighbour separation (r1 ) as well as the weighted coordination number (Ce )
of the ®rst peak [32] is essential. To obtain these parameters, a sum of gaussian functions was least Glasses square ®tted to T(r) in the range 2.0±4.2 A. with x > 0.1 could be ®tted with four gaussians while a three gaussian ®t gave better results for x 0.1. The gaussian ®ts for x 0.33 and 0.1, along with the residuals, are shown in Fig. 4(a) and (b), respectively. The ®tted parameters are given in Table 3. The bond angles h (h 2 sinÿ1 (r4 /2r1 ) where r1 and r4 are the peak positions of the ®rst two intense peaks in T(r) which correspond to the Ge±Se and Se±Se distances within a tetrahedron, respectively) and the ratio r4 /r1 for all the glasses are given in Table 3. These are in agreement with the tetrahedral angle (109.47°) and the c/a ratio (1.633) of a regular tetrahedron [6]. It is particularly pleasing to note that these numbers in the case GeSe2 are the closest to the ideal tetrahedron. Thus, the glasses studied, undoubtedly, contain Ge-tetrahedra. The presence of Ge-tetrahedra in this system is in accordance with the Raman scattering results [11], which shows the breathing mode A1 around 201 cmÿ1 for all x. The h and the ratio r4 /r1 for x 0.1 and 0.2 are less compared to that of x 0.33. This decrease can be understood in the following manner. For any composition, the position of a peak in T(r) will
Fig. 3. (a). The radial distribution function N(r) of glasses obtained by the Monte-Carlo method. The curves corresponding to x 0.2, 0.33 and 0.4 are shifted by ®ve units. (b) The radial distribution function N(r) of glasses obtained by the conventional method (Fourier ÿ1 ). The curves corresponding to x 0.2, 0.33 and 0.4 are shifted by ®ve units. transformation was carried out with Qmax 13.8 A
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231
227
Fig. 4. (a). The total correlation function of GeSe9 glass along with the ®tted gaussians. Three gaussians could be ®tted to data upto 4.0 The solid curve corresponds to the generated T(r) while the dotted curves are the ®tted gaussians. Also shown in the ®gure is the A. residue (dashed line). (b) The total correlation function of GeSe2 glass along with the ®tted gaussians. Four gaussians could be ®tted to The solid curve corresponds to the generated T(r) while the dotted curves are the ®tted gaussians. Also shown in the data upto 4.0 A. ®gure is the residue (dashed line).
Table 3 The value of the peak positions, the weighted coordination number and gaussian mean square vibrational amplitudes (r2 ) of the ®rst four peaks alongwith the mean bond angle h and ratio r4 /r1 of all the glasses studied are listed below (as III and IV are unresolved peaks we could not give errors on Ce for these two peaks) x
0.1
0.2
0.33
0.4
I Peak 0.005 (A) r1 (A) Ce 0.18 2 ) r2 (A
2.368 2.45 0.0243
2.378 2.68 0.0231
2.386 2.77 0.0219
2.392 2.92 0.0225
II Peak 0.01 (A) r2 (A) Ce 0.1 2 ) r2 (A
) ) )
2.93 0.146 0.0121
2.96 0.24 0.0144
2.98 0.59 0.0169
III Peak 0.01 (A) r3 (A) Ce 2 ) r2 (A
3.62 1.36 0.0324
3.60 0.38 0.0324
3.58 1.28 0.0289
3.55 1.16 0.0256
IV Peak 0.01 (A) r4 (A) Ce 2 ) r2 (A h (in deg) r4 /r1
3.84 6.28 0.0961 108.3 1.621
3.86 7.78 0.0961 108.5 1.623
3.89 7.36 0.0961 109.2 1.630
3.95 7.94 0.0961 110.5 1.643
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be governed by the relative concentration of various correlations contributing to it. In g-GeSe2 it is well known that the ®rst peak is due to Ge±Se bonds in a tetrahedron [5,6,24]. In pure a-Se the [1] nearest neighbour Se±Se separation is 2.359 A while Ge±Se distance in g-GeSe2 is 2.386 A, given in Table 3. The ®rst peak in glass with x 0.1 (Table 3). At this composition, occurs at 2.368 A where Ge concentration is small, the position of the ®rst peak is largely determined by Se±Se bonds. A marginal shift in the peak position with at x 0.1 to 2.386 A at increasing x, from 2.368 A x 0.33, is due to an increase in the number of Ge(Se1=2 )4 tetrahedral units. The second nearest Se±Se neighbour in pure a [1] while the Se±Se distance Se occurs at r 3.74 A (Table 3). As with within a tetrahedron is 3.89 A the ®rst peak, the observed shift in the peak at towards higher r value can also be unr 3.8 A derstood as being due to an increase in the number of Ge(Se1=2 )4 units with increasing x. The lower value of the ratio r4 /r1 for x 0.1 and 0.2 as compared to x 0.33, is, thus, a re¯ection of the fact that in these two glasses more than one type of correlation contributes to the two peaks. In a-Ge the nearest neighbour Ge±Ge separa [38]. The observed tion is known to be 2.463 A for GeSe2 , shift in the ®rst peak from 2.386 A where only Ge±Se bonds are present, to 2.392 A for Ge2 Se3 glass, indicates that at the latter composition Ge±Ge bonds are also present. The presence of Ge±Ge bonds (for x > 0.33) has been used by Street and Biegelsen [39] to interpret their photoluminescence results. The composition of Ge2 Se3 glass is such that on an average one Ge±Ge bond will be present per Ge site. It is possible that the molecular unit, concomitant with one Ge±Ge bond per Ge site is an ethane-like Ge2 (Se1=2 )6 unit. The higher value of the ratio r4 /r1 and the bond angle h observed for x 0.4 as compared to x 0.33 can be due to the presence of such a molecular unit in the glass. In fact, Lucovsky et al. have used the existence of such a molecular unit to interpret Raman and IR spectra of glass [40]. For the stoichiometric composition GeSe2 , COCRN model predicts that only Ge±Se bonds contribute to the peak at r1 . This would yield 2.69 for the area of this peak. On the other hand, the
model of Phillips predicts Ge±Ge and Se±Se bonds also contribute to the peak at r1 in addition to Ge± Se bonds. The area of ®rst peak calculated using this model is 2.64. The experimentally observed area is 2.77(18). With the accuracies of present work it is not possible to choose between the two models. It is not likely that this question can be answered on the basis of diraction experiments with the presently obtainable accuracies on the areas of peaks in T(r). The observed monotonic increase in the average coordination number, Ce (Table 3), of the ®rst peak is consistent with the COCRN model. The average coordination number of the peak is a weighted sum of the partial coordination numbers (Nij ) due to all the correlations contributing to it. The Ce s and Nij s for the ®rst peak for all the glasses studied are given in Table 4. For x 0.33, assuming NSe±Se and NGe±Ge to be equal to zero in Eq. (9). We obtain a 4.1(2) for NGe±Se (these assumed Nij s are depicted by an asterisk (*) in Table 4). This NGe±Se would imply that on an average each Ge atom has four Se atoms as the nearest neighbours. In other words, in the glass, Ge(Se1=2 )4 units are the building blocks. In the case of non-stoichiometric glasses (i.e., x < 0.33 or x > 0.33), the ®rst peak has contributions from both similar and dissimilar neighbours. For instance, in Se-rich glasses (x < 0.33), both Ge±Se and Se±Se correlations will be present. As already discussed earlier in the text, Ge(Se1=2 )4 tetrahedra are present in glasses with x 6 0.33 i.e., NGe±Se 4. With NGe±Se 4 and assuming NGe±Ge 0, for x 0.1 and 0.2 one obtains NSe±Se 1.8(2) and 1.3(2), respectively. A NSe±Se 1.8(2) for x 0.1 would essentially mean that the structure Table 4 The average coordination no. (Ce ) and the partial coordination nos. (Nij ) of various correlations contributing to the ®rst peak in T(r) are given below Composition (x) 0.1 0.2 0.33 0.4
Ce 2.45 2.68 2.77 2.92
NSe±Se 1.8(2) 1.3(2) 0 0
NGe±Ge
0 0 0 1.0
NGe±Se 4.0 4.0 4.1(2) 3.1(2)
The asterisk (*) refers to the constancy of the Nij values (described in the text).
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231
of this glass consists of Se-chain segments which are cross-linked by Ge(Se1=2 )4 units. The structural sequence Ge±Se±Se±Se±Ge is one probable unit which is commensurate with the NSe±Se . As x increases the Se chain segments shorten and at x 0.2, Ge-tetrahedra are interconnected by two Se atoms on an average i.e., Ge±Se±Se±Ge thus accounting for the observed decrease in NSe±Se with increasing x. However, the presence of such a sequence would imply NSe±Se 1.0, whereas it is observed to be 1.3. This suggests that some of the Ge-tetrahedra may be interconnected with more than one Se atom at this composition. For Ge-rich glasses (x > 0.33) Ge±Ge correlation is present in addition to Ge±Se correlation. At x 0.4, with NGe±Ge 1.0 and assuming NSe±Se 0, one obtains NGe±Se 3.1(2). We suggest that each Ge atom has on an average three Se atoms and one Ge atom as its nearest neighbours giving support to the presence of Ge2 (Se1=2 )6 ethane-like units i.e., Se±Ge±Ge±Se: From the variation observed in the peak positions as well as the changes in the average coordination number of the ®rst shell with x, we infer that the Se chain segments and Ge(Se1=2 )4 units are the building blocks for x < 0.33. For x 0.33, the SRO consists of Ge-tetrahedra, while for x 0.4 there is evidence for Ge2 (Se1=2 )6 molecular units being the building blocks. The results of the present investigations on the SRO in these glasses are in qualitative agreement with the Raman [10] and EXAFS [38] reports. Furthermore, they also agree with Johnson's report on Six Se1ÿx glasses [1], which have a similar SRO. 6.3. Network connectivity: In crystalline GeSe2 , it is known that 50% of Ge-tetrahedra are in an edge-shared (ES) con®guration, assuming at most one ES tetrahedron per Ge site [41]. The remaining tetrahedra are in a corner-shared (CS) con®guration. The peak in T(r) of glass is known to result around 3.0 A
229
from Ge±Ge correlation in the ES con®guration [5]. In glasses with x 0.2, 0.33 and 0.4 one ob The area of this serves a small peak around 3.0 A. peak shows an increase with increasing Ge content, as shown in Fig. 2 and Table 3. Thus, with increasing x, Ge(Se1=2 )4 units are not only interconnected, but also the number of these units in ES con®guration increases. Since, in diraction experiments the errors associated with the area of the peaks after the ®rst one are larger, it is dicult to get a reliable quantitative estimate of the fraction of ES tetrahedra in these glasses. It is known that in c-GeSe2 the distance between a Ge atom in ES con®guration and a Se atom on neighbouring tetrahedra lies between 4.6 6 r 6 5.3 A and the separation between a Ge A atom in CS con®guration and a Se atom lies be 6 r 6 4.8 A [6]. One notices from tween 4.0 A becomes Fig. 2 that the peak around 4.7 A broader with increasing x. The origin of this peak in the case of glasses with x 0.33 and 0.4, where the tetrahedral connectivity is complete, could possibly be associated with the above mentioned Ge±Se correlation. However, an unambiguous explanation of the origin of this peak for x 0.2 and y 0.1 cannot be given as in these glasses tetrahedral connectivity is not complete. Thus, in these glasses, correlation other than the Ge±Se correlation between two neighbouring tetrahedra This would be contributing to the peak at 4.7 A. contribution becomes even more certain for x 0.1, where tetrahedra are dilute i.e., dispersed in Se chains. It is likely that reverse Monte Carlo [42] studies may provide an explanation for the in smaller Ge origin of the peak around 4.7 A content glasses as also a better estimate of ES tetrahedra in these glasses. 7. Conclusions The Monte-Carlo method used to generate the structure factor, S(Q), in Gex Se1ÿx glasses, where 0.1 6 x 6 0.4, gives food ®t to the experimental data. The Ge(Se1=2 )4 tetrahedra are the basic units of short range order in GeSe2 glass. The glass with x 0.1 consist predominantly of Se-chain segments interlinked with tetrahedra. At x 0.2, the
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tetrahedra are either connected through bridging Se atoms or by short Se-chain segments. The GeSe2 glass on the other hand, consists of Getetrahedra which are interlinked by bridging Se atoms. At x 0.4, the glass alloy contains Ge2 (Se1=2 )6 molecular units and the network connectivity is complete. A gradual shift in the ®rst peak in the total correlation function T(r) is consistent with this model. In glasses with x 0.2, 0.33 and 0.4 there is evidence of the presence of edgeshared tetrahedra; there is an increase in ES tetrahedra with x. The intermediate range order also shows a systematic increase with x. The FSDP shifts to a smaller Q and has maximum coherence length for glasses with x 0.33 and 0.4. Acknowledgements A part of this work was carried out under the IUC-DAEF project no. IUC/BOM/54. The authors gratefully acknowledge the help rendered by their colleagues Dr R. Chakravarthy, Dr B.N. Meera, Mr K. Ramesh and Mr A.B. Shinde in carrying out the experiments. One of us (N.R.R.) would like to thank the Solid State Physics Division, BARC, Bombay for the hospitality extended during his stay on the campus. We thank the referees for the suggestions which improved the presentation of our results. References [1] R.W. Johnson, D.L. Price, S. Susman, M. Arai, T.I. Morrison, G.K. Shenoy, J. Non-Cryst. Solids 83 (1986) 251. [2] Y. Sagara, O. Uemura, S. Okuyama, T. Satow, Phys. Stat. Sol. (a) 31 (1975) K33. [3] E.D. Crozier, F.W. Lytle, D.E. Sayers, E.A. Sayers, Can. J. Chem. 55 (1977) 1968. [4] J.E. Griths, M. Malyj, G.P. Espinosa, J.P. Remeika, Phys. Rev. B 30 (1984) 6978. [5] S. Susman, K.J. Volin, D.G. Montague, D.L. Price, J. Non-Cryst. Solids 125 (1990) 168. [6] I.T. Penfold, P.S. Salmon, Phys. Rev. Lett. 67 (1991) 97. [7] O. Uemura, Y. Sagara, D. Muno, T. Satow, J. Non-Cryst. Solids 30 (1978) 155. [8] A.C. Wright, D.L. Price, A.G. Clare, G. Etherington, R.N. Sinclair, Diusion Defect Data 53&54 (1987) 255.
[9] U. Walter, D.L. Price, S. Suman, K.J. Volin, Phys. Rev. B 37 (1988) 4232. [10] P. Tronc, M. Bensoussan, A. Brenac, C. Sebenne, Phys. Rev. B 8 (1973) 5947. [11] S. Sugai, Phys. Rev B 35 (1987) 1345. [12] P.M. Bridenbaugh, G.P. Espinosa, J.E. Griths, J.C. Phillips, J.P. Remeika, Phys. Rev. B 20 (1979) 4140. [13] K. Murase, T. Fukunaga, K. Yakushiji, T. Yoshimi, I. Yunoki, J. Non-Cryst. Solids 59&60 (1983) 883. [14] R.J. Nemanich, S.A. Solin, G. Lucovsky, Solid State Commun. 21 (1977) 273. [15] P. Boolchand, in: D. Adler, B.B. Schwartz, M.C. Steele (Eds.), Physical Properties of Amorphous Materials, Plenum, New York, 1985, p. 221. [16] R.N. Sinclair, A.C. Wright, T.M. Brunier, R.A. Hulme, C.A. Guy, A.C. Hannon, M. Arai, J. Non-Cryst. Solids 150 (1992) 219. [17] S.R. Elliott, Nature 354 (1991) 445. [18] J.C. Phillips, J. Non-Cryst. Solids 34 (1979) 153; 43 (1981) 37. [19] J.C. Phillips, M.F. Thorpe, Solid State Commun. 53 (1985) 699. [20] G. Lucovsky, T.M. Hayes, in: M.H. Brodsky (Ed.), Amorphous Semiconductors, Springer, Berlin, 1979, p. 125. [21] R.J. Nemanich, F.L. Galeener, J.C. Mikkelsen Jr., G.A.N. Connell, G. Etherington, A.C. Wright, R.N. Sinclair, Physica B 117&118 (1983) 959. [22] S.C. Moss, D.L. Price, in: D. Adler, H. Fritzsche, S.R. Ovshinsky (Eds.), Physics of Diordered Materials, Plenum, New York, 1985, p. 77. [23] S.R. Elliott, J. Phys.: Condens. Matter. 4 (1992) 7661. [24] P. Vashishta, R.K. Kalia, G.A. Antonio, I. Ebbsj o, Phys. Rev. Lett. 62 (1989) 1651. [25] P. Vashishta, R.K. Kalia, I. Ebbsj o, Phys. Rev. B 39 (1989) 6034. [26] J. Malaurent, J. Dixmier, J. Non-Cryst. Solids 35&36 (1980) 1227. [27] R.W. Fawcett, C.N.J. Wagner, G.S. Cargill III, J. NonCryst. Solids 8±10 (1972) 369. [28] M.M. Ha®z, F.H. Hammad, N.A. El-Kabany, Physica B 183 (1993) 392. [29] P.S.R. Krishna, S. Basu, N. Ramesh Rao, Solid State Phys. C (India) 37 (1994) 258. [30] A.K. Soper, in: M. Davidovic, A.K. Soper (Eds.), Static and Dynamic Properties of Liquids, Springer, Berlin, 1989, p. 189. [31] A.K. Soper, in: M.W. Jonson (Ed.), Neutron Scattering Data Analysis, IOP Confer. Ser., IOP, Bristol, vol. 107, 1990, p. 57. [32] A.C. Wright, A.J. Leadbetter, Phys. Chem. Glasses 17 (1976) 122. [33] N. Ramesh Rao, K.S. Sangunni, E.S.R. Gopal, P.S.R. Kirshna, R. Chakravarthy, B.N. Meera, B.A. Dasannacharya, Solid State Phys. C (India) 35 (1992) 221. [34] N. Ramesh Rao, K.S. Sangunni, E.S.R. Gopal, P.S.R. Krishna, R. Chakravarthy, B.A. Dasannacharya, Physica B 213&214 (1995) 561.
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231 [35] B.A. Dasannacharya, Physica B 180&181 (1992) 880. [36] P.A. Egelsta, in: D.L. Price, K. Sk old (Eds.), Methods of Experimental Physics B, vol. 23, Acadamic Press, New York, 1987, p. 405. [37] P.H. Fuoss, P. Esienberger, W.K. Warburton, A. Bienenstock, Phys. Rev. Lett. 46 (1981) 1537. [38] W. Zhou, M. Pasesler, D.E. Sayers, Phys. Rev. B 43 (1991) 2315. [39] R.A. Street, D.K. Biegelsen, J. Non-Cryst. Solids 32 (1979) 339.
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[40] G. Lucovsky, F.L. Galeener, R.H. Geils, R.C. Keezer, in: P.H. Gaskell, E.A. Davis (Eds.), Proceedings of the Conference on Structure of Non-crystalline Material, Tayor and Francis, London, 1976, p. 127. [41] G. Von Dittmar, H. Sch afer, Acta Crystallogr. B 32 (1976) 2726. [42] R.L. McGreevy, L. Pustzai, Mol. Simulation. 1 (1988) 359. [43] G. Etherington, A.C. Wright, J.T. Wenzel, J.C. Dore, J.H. Clarke, R.N. Sinclair, J. Non-Cryst. Solids. 48 (1982) 265.
Structural correlations in GexSe1ÿx glasses ± a neutron diraction study N. Ramesh Rao a, P.S.R. Krishna b, S. Basu b, B.A. Dasannacharya K.S. Sangunni a, E.S.R. Gopal c b
b,*
,
a Department of Physics, Indian Institute of Science, Bangalore 560 012, India Solid State Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India c National Physical Laboratory, New Delhi 110 012, India
Received 9 February 1997; received in revised form 5 January 1998
Abstract Neutron diraction measurement is carried out on Gex Se1ÿx glasses, where 0.1 6 x 6 0.4, in a Q interval of 0.55±13.8 ÿ1 . The ®rst sharp diraction peak (FSDP) in the structure factor, S(Q), shows a systematic increase in the intensity A and shifts to a lower Q with increasing Ge concentration. The coherence length of FSDP increases with x and becomes maximum for 0.33 6 x 6 0.4. The Monte-Carlo method, due to Soper, is used to generate S(Q) and also the pair correlation function, g(r). The generated S(Q) is in agreement with the experimental data for all x. Analysis of the ®rst four peaks in the total correlation function, T(r), shows that the short range order in GeSe2 glass is due to Ge(Se1=2 )4 tetrahedra, in agreement with earlier reports. Se-rich glasses contain Se-chains which are cross-linked with Ge(Se1=2 )4 tetrahedra. Ge2 (Se1=2 )6 molecular units are the basic structural units in Ge-rich, x 0.4, glass. For x 0.2, 0.33 and 0.4 there is evidence for some of the tetrahedra being in an edge-shared con®guration. The number of edge-shared tetrahedra in these glasses increase with increasing Ge content. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In recent years, structural investigations have been carried out on semiconducting chalcogenide glasses [1±4]. The Ge±Se system has been studied using both diraction [5±9] and spectroscopic [10± 15] techniques. The existence of Ge(Se1=2 )4 tetrahedral units in Gex Se1ÿx glasses has been inferred from Raman scattering studies. The Raman spectrum consists of several vibrational modes, some of * Corresponding author. Present address: Inter University Consortium for DAE Facilities, University Campus, Khandwa Road, Indore 452 017 M.P., India. Tel.: +91-731 462 265; fax: +91-731 462 294.
which have been identi®ed with the normal modes of these tetrahedral units [10±12]. The Raman mode at 217 cmÿ1 , according to Sugai [11] or at 26.8 meV ( 216 cmÿ1 ) according to Sinclair et al. [16] has been subject of debate. This mode, known as the companion mode, A1c , is associated with the intermediate range order (IRO) observed in these glasses. Another structural feature, de®nitely due to IRO, is the `pre-peak' or the ®rst sharp diraction peak (FSDP) in the structure factor S(Q) ÿ1 [17]. These experimentally obaround 1.0 A served features can be explained within the framework of either the constraint or percolation theory [18,19] or the chemically ordered continuous random network (COCRN) model [20,21].
0022-3093/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 7 0 5 - 4
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According to the percolation model, a glass network consists of chemically ordered clusters. The presence of such clusters implies the existence of homopolar bonds even at the stoichiometric composition [12]. Thus, chemical order is broken at this composition. In the case of GeSe2 glass the structure of these molecular fragments is believed to be similar to the layer-like structure of the high temperature polymorph of c-GeSe2 . The FSDP observed in S(Q), is taken as evidence for the presence of such a layer-like fragment. Further, the A1c mode is associated with the breathing mode of Se±Se dimers present in the fragments [18,12]. On the basis of bond energy considerations, the COCRN model predicts that the heteropolar bonds are maximised at any composition x. Thus at x 0.33 the glass network consists of only Ge±Se bonds while for x < 0.33, Se±Se bonds are present in addition to Ge±Se bonds. Similarly, Gerich glasses i.e. x > 0.33 contain Ge±Ge as well as Ge±Se bonds [20]. The origin of the A1c mode and the FSDP are explained on the basis of rings of dierent sizes present in the glass network. Speci®cally, the A1c mode arises because of four membered rings i.e. edge-shared tetrahedra while the FSDP is associated with six membered rings [21]. The FSDP has been observed in all AX2 (where A Si or Ge and X S, Se, or O) type of glasses. Furthermore, when scaled with the bond distance r1 i.e. Qr1 , it lies within 2.2±2.8 for all the glasses [22]. A variety of explanations have been advanced to understand this feature. These include the presence of layer-like morphology [12], random packing of tetrahedra [22] and, recently, as being due to the chemical ordering of voids [23]. In this regard the GeSe2 has been widely studied and recent results of Susman et al. [5] on glass and Penfold and Salmon [6] on liquid suggest that an appreciable number of Ge-tetrahedra are in edgeshared con®guration. Molecular dynamics simulations of molten and glassy GeSe2 also support this picture [24,25]. Thus, in the light of these investigations it is natural to ask; how does the network connectivity in Gex Se1ÿx glasses change as the Ge content is varied? In literature there are reports of X-ray as well as electron diraction studies on Ge±Se glasses, both
on ®lms as well as larger samples [26±28]. The most recent report [28], is on Gex Se1ÿx glasses lying on the Se-rich side of the glass forming region, with x going to 0.3. These data were collected for a maximum momentum transfer vector, Qmax , of 6.6 ÿ1 , thus limiting the resolution in real space. In A the present structural investigation of Gex Se1ÿx glasses, where 0.1 6 x 6 0.4, a much larger Q interval is covered. For the ®rst time, to the best of our knowledge, the Monte-Carlo method (MC) is used to generate the pair correlation function [29± 31] in our samples. This MC method is a departure from the conventional procedure of Fourier transformation of the experimental data modi®ed with a Lorch function [32]. Some preliminary results in these samples have been reported earlier [33,34]. Section 2 gives a brief account of the formalism relevant to the experiment. Sample preparation, data collection and reduction are described in Section 3. An account of the MC method used [29] is presented in Section 4. The results of present investigations are presented in Section 5. Discussion on the intermediate range order, the short range order and the network connectivity is presented in that order in Section 6 followed by conclusions in Section 7.
2. Basic formalism The cross-section, dr/dX, after the data reduction, for a multicomponent system in terms of the partial structure factors, Sij (Q), is given by [5] X rinc dr X 1 ci cj 1=2 bi bj Sij Q ci i ; 4p dX i;j i where ci , bi and rinc are the concentration, coi herent scattering length and incoherent scattering cross-section of the ith component, respectively. The total structure factor, S(Q), can be expressed in terms of the partial structure factors Sij (Q) as 1 X ci cj 1=2 bi bj Sij Qÿdij ; 2 S Q 1 2 hbi i;j P where hbi i ci bi :
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231
The Fourier transformation of S(Q) gives the pair distribution function, g(r), Z1 1 Q S Q ÿ 1 sin Qr dQ; 3 g r 1 2 2p rq 0
where q is the total number density. In literature, three other correlation functions are generally used. The dierential correlation function, D(r), is given by
223
where Wij are the weighting factors de®ning the contributions of each of the partial Nij to Ce . 3. Experimental procedures 3.1. Sample preparation
and the radial distribution function, N(r), is given by
Gex Se1ÿx samples where 0.1 6 x 6 0.4, were prepared by the melt-quench method. Elemental Ge and Se of 5N purity were placed in round ampoules of length 6.0 cm in correct stoichiometric proportion. The ampoules were vacuum sealed at a pressure of 2 ´ 10ÿ5 Torr. They were kept in a horizontal furnace for 24 h at 1233 K. Glasses were obtained by quenching the liquid in a solution of NaOH + ice-water. About 13 g of each composition was prepared and con®rmed to be amorphous with X-ray powder diraction. Neutron diraction studies were carried out on pulverised samples. Infrared absorption spectrum was recorded to ensure the absence of oxygen contamination in the samples: no detectable absorption due to Se±O (904±925 cmÿ1 ) or Ge±O (855 cmÿ1 ) [5] was observed.
N r rT r 4pr2 qg r:
3.2. Experiment and data reduction
2 D r p
Z1 Q S Q ÿ 1 sin Qr dQ;
4a
0
D r 4prqg r ÿ 1:
4b
The total correlation function, T(r), is used for obtaining the peak positions [30] and is given by T r 4prq D r;
5a
T r 4prqg r
5b
6
N(r) is used to derive the average number of atoms lying within a range, r to r + dr, of a given atom. The area of a peak in N(r) gives the average coordination number Ce [5], which in turn is related to the partial coordination numbers Nij expressing the average number of j-type atoms around any i-type atom within the range of integration, Ce
1 X ci bi bj Nij 2 hbi i;j
since ci Nij cj Nji ! b1 2 b b 1 2 N11 2c1 Re Ce c1 N12 hbi hbi2 b2 2 N22 ; c2 hbi Ce W11 N11 W12 N12 W22 N22 ;
7
8 9
Neutron diraction measurements were carried out on the High-Q diractometer at the Dhruva reactor at Trombay, Bombay [35]. Data was collected on samples in vanadium containers in a Qÿ1 . Procedures given by range of 0.55±13.8 A Eglesta [36] were used in carrying out the data reduction, with corrections for background, multiple scattering and inelastic scattering. Vanadium normalisation was used to put the data on an absolute cross-section scale. The peak and valley positions of the present data on GeSe2 glass agree well with that of Susman et al. [5]. The peak heights in the data are generally lower, except for the FSDP, partly because the data of Ref. [5] was collected at 10 K, whereas, the present experiments were carried out at 300 K and partly because of poorer Q-resolution of our data (2.5%) compared to 0.35%; of Ref. [5]. The larger FSDP in the present data is due to its well known anomalous temperature dependence [23].
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4. Monte-Carlo G(r) (MCGR) As the experimental data is collected in a ®nite Q range the upper limit of integral in Eq. (3) has to be replaced by Qmax instead of in®nity. This results in termination errors which show up as spurious oscillations (peaks) in g(r). In the conventional method of analysis one uses a modi®cation function [32] to reduce the eect of termination of data at a ®nite Qmax . This function in turn, while reducing the spurious oscillations, leads to a broadening of the genuine peaks in g(r). To overcome this problem a one-dimensional MonteCarlo method [29] with similar basic principles to the earlier published work of Soper [30,31] was used. We have used a computer program developed by us after testing it successfully with Pencus±Yevick Hard Sphere data. A brief description of the method is given below. In the MC method [29] used, a trial g(r) and its analytically known S(Q) are taken. The g(r) is then modi®ed randomly at any randomly chosen r. For every change in g(r), its S(Q) is modi®ed at every P point analytically. If v2 (v2 {(SExp (Q) ) SCal (Q))2 /r2 }, where r2 is the statistical error in the experimental data) reduces the move is accepted. If it increases, the move is accepted with a probability of exp()Dv2 /T), where Dv2 is the change in v2 and T is a ®ctitious `temperature'. The process is repeated until a suitable v2 is reached (1% in our case). One of the advantages of this method is that a direct estimation of errors in g(r) is possible by repeating the process and arriving at an ensemble of pair correlation [29,31]. For Gex Se1ÿx glasses, the trial g(r) was taken to be zero below r/rc < 1.0 and equal to one for [5]. The condition r/rc P 1.0, where rc 2.0 A g(r) 0 for r/rc < 1.0 was maintained throughout the simulation. In the present investigation g(r) has upto been generated at an interval (Dr) of 0.05 A, The experimental and the simulated rmax of 60 A. ÿ1 ) structure factors for glasses (obtained upto 25 A with x 0.1, 0.2, 0.33 and 0.4 are shown in Fig. 1. As is apparent from the ®gure, the simulation procedure outlined above gives a good ®t to the experimental data. It is interesting to note that the peak and valley positions of the extrapolated region of S(Q) for GeSe2 glass agree very well with
Fig. 1. The total structure factor S(Q) for Gex Se1ÿx glasses. Solid line: S(Q) generated by the Monte-Carlo method, dots: experimental data; solid lines: generated data. The data (both the dots and the solid lines) corresponding to x 0.2, 0.33 and 0.4 are shifted by one unit for sake of clarity.
the measurements of Susman et al. [5]. Soper has shown that in the case of hard spheres where S(Q) is analytically known, the MCGR method could be used to extrapolate S(Q) [29]. Unlike the S(Q) of hard spheres real experimental data have errors and this procedure may be of doubtful value to
Fig. 2. The total correlation function T(r) (generated by the Monte-Carlo method) of Gex Se1ÿx glasses. Each curve was generated using 500 histories. For the sake of clarity the curves corresponding to x 0.2, 0.33 and 0.4 are shifted by one unit. The vertical lines are the error bars for GeSe9 composition. The inset shows r1 vs. x. n correspond to our data. n correspond to other neutron diraction data for x 0 (Ref. [1]), x 0.33 (Ref. [5]) and x 1.0 (Ref. [43]).
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231
extend S(Q) data. The present work shows that even in a real situation like ours the extrapolation is not unreasonable. However, it is not the intention here to suggest that this extrapolation be treated as real structure factor. We have not made use of this extrpolated data in any analysis. In Fig. 2 the T(r)s generated using 500 histories for Errors on T(r) are each x are shown upto 10 A. shown only for GeSe9 , for the sake of clarity. The error for other xs is typically equal to or less than that for GeSe9 . The number densities and the weighting factors, Wij , of all the samples are given in Table 1. 5. Results Fig. 1 shows the measured structure factors of Gex Se1ÿx glasses. We observe that a peak appears ÿ1 . For x 0.1 it appears in S(Q) around 1.0 A only as a shoulder. This peak becomes larger for larger x and also shifts to a smaller Q. Fig. 2 shows the T(r) functions obtained from the measured S(Q) data using MCGR method described above. We observe that the ®rst peak at shows a marginal shift towards a larger r 2.37 A r with increasing x. The inset in Fig. 2 shows earlier results of ®rst peak position (r1 ) as a function of x along with earlier positions, showing good agreement with existing measurements. In
225
the shift towards the case of the peak at r 3.8 A, larger r is larger (3.84±3.95 A). Unlike these two is observed only for peaks, the peak around 3.0 A x 0.2, 0.33 and 0.4. The intensity of this peak increases with x. On the other hand, the peak becomes broader as x increases from around 4.7 A 0.1 to 0.4. This trend is in a direction, opposite to that of the peak at 3.0 A. 6. Discussion 6.1. Intermediate range order (IRO): The peak parameters for the FSDP are obtained by following the procedure outlined by Johnson et al. [1] and are given in Table 2. These parameters indicate that the characteristic length (R 2p/Q) responsible for IRO increases with increasing Ge content. The `coherence length' (L 2p/DQ, where DQ is the full width at half maximum) increases up to x 0.33, indicating that the IRO is more in glasses with x 0.33 and 0.4 as compared to that in glasses with smaller x. Structural studies on Six Se1ÿx glasses also show a similar trend [1]. Neutron diraction study of molten GeSe2 with isotopic substitution [6], X-ray anomalous scattering of g-GeSe2 [37], and molecular dynamics simulation of liquid and glassy GeSe2 [24] show that the IRO arises primarily because of
Table 1 The values of number densities q and the weighting factors Wij used in the the data analysis of Gex Se1ÿx glasses are given below ÿ3 ) x q (A WGe±Se WGe±Ge WSe±Se WSe±Ge 0.1 0.2 0.33 0.4
0.0330 0.0336 0.0328 0.0346
0.204 0.406 0.673 0.804
0.105 0.209 0.346 0.413
0.895 0.791 0.655 0.587
1.836 1.624 1.346 1.206
Table 2 The position (Q1 ) and the full width at half maximum (FWHM, DQ1 ) of the FSDP of Gex Se1ÿx glasses are given below (also given in the table are the values of the coherence length L and the correlation length R of the glasses) ÿ1 ) ÿ1 ) DQ1 (A R 2p/Q1 (A) L 2p/DQ1 (A) x Q1 ( A 0.1 0.2 0.33 0.4
) 1.095 0.988 0.965
) 0.30 0.22 0.22
) 5.74 6.36 6.51
) 20.9 28.6 28.6
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Ge±Ge correlations in the glass. The observed trends in the FSDP with x, in the present investigations, is in conformity with these results. 6.2. Short range order (SRO): The radial distribution functions N(r) of all the glasses obtained by (i) the Monte-Carlo method and (ii) by Fourier transformation of the experimental data i.e., conventional method (Fourier transformation was carried out using Lorch ÿ1 ) are modi®cation function with Qmax 13.8 A shown in Fig. 3(a) and (b), respectively. N(r) of GeSe2 glass is in good agreement with that of Susman et al. [5]. Especially, the peak around 3.0 which is indicative of the presence of edgeA, shared tetrahedra in the glass [5], is noticeable. This feature is not observable in the data of Ha®z et al. [28]. This absence is due to the small Qmax (6.6 ÿ1 ) of their data, thus reducing the resolution in A real space. We have investigated this absence of the above feature by terminating our data at smaller Qmax . In the present work this peak is better resolved in the T(r)s generated using MCGR method (Fig. 3(a)). To obtain information regarding the SRO, a knowledge of the nearest neighbour separation (r1 ) as well as the weighted coordination number (Ce )
of the ®rst peak [32] is essential. To obtain these parameters, a sum of gaussian functions was least Glasses square ®tted to T(r) in the range 2.0±4.2 A. with x > 0.1 could be ®tted with four gaussians while a three gaussian ®t gave better results for x 0.1. The gaussian ®ts for x 0.33 and 0.1, along with the residuals, are shown in Fig. 4(a) and (b), respectively. The ®tted parameters are given in Table 3. The bond angles h (h 2 sinÿ1 (r4 /2r1 ) where r1 and r4 are the peak positions of the ®rst two intense peaks in T(r) which correspond to the Ge±Se and Se±Se distances within a tetrahedron, respectively) and the ratio r4 /r1 for all the glasses are given in Table 3. These are in agreement with the tetrahedral angle (109.47°) and the c/a ratio (1.633) of a regular tetrahedron [6]. It is particularly pleasing to note that these numbers in the case GeSe2 are the closest to the ideal tetrahedron. Thus, the glasses studied, undoubtedly, contain Ge-tetrahedra. The presence of Ge-tetrahedra in this system is in accordance with the Raman scattering results [11], which shows the breathing mode A1 around 201 cmÿ1 for all x. The h and the ratio r4 /r1 for x 0.1 and 0.2 are less compared to that of x 0.33. This decrease can be understood in the following manner. For any composition, the position of a peak in T(r) will
Fig. 3. (a). The radial distribution function N(r) of glasses obtained by the Monte-Carlo method. The curves corresponding to x 0.2, 0.33 and 0.4 are shifted by ®ve units. (b) The radial distribution function N(r) of glasses obtained by the conventional method (Fourier ÿ1 ). The curves corresponding to x 0.2, 0.33 and 0.4 are shifted by ®ve units. transformation was carried out with Qmax 13.8 A
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231
227
Fig. 4. (a). The total correlation function of GeSe9 glass along with the ®tted gaussians. Three gaussians could be ®tted to data upto 4.0 The solid curve corresponds to the generated T(r) while the dotted curves are the ®tted gaussians. Also shown in the ®gure is the A. residue (dashed line). (b) The total correlation function of GeSe2 glass along with the ®tted gaussians. Four gaussians could be ®tted to The solid curve corresponds to the generated T(r) while the dotted curves are the ®tted gaussians. Also shown in the data upto 4.0 A. ®gure is the residue (dashed line).
Table 3 The value of the peak positions, the weighted coordination number and gaussian mean square vibrational amplitudes (r2 ) of the ®rst four peaks alongwith the mean bond angle h and ratio r4 /r1 of all the glasses studied are listed below (as III and IV are unresolved peaks we could not give errors on Ce for these two peaks) x
0.1
0.2
0.33
0.4
I Peak 0.005 (A) r1 (A) Ce 0.18 2 ) r2 (A
2.368 2.45 0.0243
2.378 2.68 0.0231
2.386 2.77 0.0219
2.392 2.92 0.0225
II Peak 0.01 (A) r2 (A) Ce 0.1 2 ) r2 (A
) ) )
2.93 0.146 0.0121
2.96 0.24 0.0144
2.98 0.59 0.0169
III Peak 0.01 (A) r3 (A) Ce 2 ) r2 (A
3.62 1.36 0.0324
3.60 0.38 0.0324
3.58 1.28 0.0289
3.55 1.16 0.0256
IV Peak 0.01 (A) r4 (A) Ce 2 ) r2 (A h (in deg) r4 /r1
3.84 6.28 0.0961 108.3 1.621
3.86 7.78 0.0961 108.5 1.623
3.89 7.36 0.0961 109.2 1.630
3.95 7.94 0.0961 110.5 1.643
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be governed by the relative concentration of various correlations contributing to it. In g-GeSe2 it is well known that the ®rst peak is due to Ge±Se bonds in a tetrahedron [5,6,24]. In pure a-Se the [1] nearest neighbour Se±Se separation is 2.359 A while Ge±Se distance in g-GeSe2 is 2.386 A, given in Table 3. The ®rst peak in glass with x 0.1 (Table 3). At this composition, occurs at 2.368 A where Ge concentration is small, the position of the ®rst peak is largely determined by Se±Se bonds. A marginal shift in the peak position with at x 0.1 to 2.386 A at increasing x, from 2.368 A x 0.33, is due to an increase in the number of Ge(Se1=2 )4 tetrahedral units. The second nearest Se±Se neighbour in pure a [1] while the Se±Se distance Se occurs at r 3.74 A (Table 3). As with within a tetrahedron is 3.89 A the ®rst peak, the observed shift in the peak at towards higher r value can also be unr 3.8 A derstood as being due to an increase in the number of Ge(Se1=2 )4 units with increasing x. The lower value of the ratio r4 /r1 for x 0.1 and 0.2 as compared to x 0.33, is, thus, a re¯ection of the fact that in these two glasses more than one type of correlation contributes to the two peaks. In a-Ge the nearest neighbour Ge±Ge separa [38]. The observed tion is known to be 2.463 A for GeSe2 , shift in the ®rst peak from 2.386 A where only Ge±Se bonds are present, to 2.392 A for Ge2 Se3 glass, indicates that at the latter composition Ge±Ge bonds are also present. The presence of Ge±Ge bonds (for x > 0.33) has been used by Street and Biegelsen [39] to interpret their photoluminescence results. The composition of Ge2 Se3 glass is such that on an average one Ge±Ge bond will be present per Ge site. It is possible that the molecular unit, concomitant with one Ge±Ge bond per Ge site is an ethane-like Ge2 (Se1=2 )6 unit. The higher value of the ratio r4 /r1 and the bond angle h observed for x 0.4 as compared to x 0.33 can be due to the presence of such a molecular unit in the glass. In fact, Lucovsky et al. have used the existence of such a molecular unit to interpret Raman and IR spectra of glass [40]. For the stoichiometric composition GeSe2 , COCRN model predicts that only Ge±Se bonds contribute to the peak at r1 . This would yield 2.69 for the area of this peak. On the other hand, the
model of Phillips predicts Ge±Ge and Se±Se bonds also contribute to the peak at r1 in addition to Ge± Se bonds. The area of ®rst peak calculated using this model is 2.64. The experimentally observed area is 2.77(18). With the accuracies of present work it is not possible to choose between the two models. It is not likely that this question can be answered on the basis of diraction experiments with the presently obtainable accuracies on the areas of peaks in T(r). The observed monotonic increase in the average coordination number, Ce (Table 3), of the ®rst peak is consistent with the COCRN model. The average coordination number of the peak is a weighted sum of the partial coordination numbers (Nij ) due to all the correlations contributing to it. The Ce s and Nij s for the ®rst peak for all the glasses studied are given in Table 4. For x 0.33, assuming NSe±Se and NGe±Ge to be equal to zero in Eq. (9). We obtain a 4.1(2) for NGe±Se (these assumed Nij s are depicted by an asterisk (*) in Table 4). This NGe±Se would imply that on an average each Ge atom has four Se atoms as the nearest neighbours. In other words, in the glass, Ge(Se1=2 )4 units are the building blocks. In the case of non-stoichiometric glasses (i.e., x < 0.33 or x > 0.33), the ®rst peak has contributions from both similar and dissimilar neighbours. For instance, in Se-rich glasses (x < 0.33), both Ge±Se and Se±Se correlations will be present. As already discussed earlier in the text, Ge(Se1=2 )4 tetrahedra are present in glasses with x 6 0.33 i.e., NGe±Se 4. With NGe±Se 4 and assuming NGe±Ge 0, for x 0.1 and 0.2 one obtains NSe±Se 1.8(2) and 1.3(2), respectively. A NSe±Se 1.8(2) for x 0.1 would essentially mean that the structure Table 4 The average coordination no. (Ce ) and the partial coordination nos. (Nij ) of various correlations contributing to the ®rst peak in T(r) are given below Composition (x) 0.1 0.2 0.33 0.4
Ce 2.45 2.68 2.77 2.92
NSe±Se 1.8(2) 1.3(2) 0 0
NGe±Ge
0 0 0 1.0
NGe±Se 4.0 4.0 4.1(2) 3.1(2)
The asterisk (*) refers to the constancy of the Nij values (described in the text).
N. Ramesh Rao et al. / Journal of Non-Crystalline Solids 240 (1998) 221±231
of this glass consists of Se-chain segments which are cross-linked by Ge(Se1=2 )4 units. The structural sequence Ge±Se±Se±Se±Ge is one probable unit which is commensurate with the NSe±Se . As x increases the Se chain segments shorten and at x 0.2, Ge-tetrahedra are interconnected by two Se atoms on an average i.e., Ge±Se±Se±Ge thus accounting for the observed decrease in NSe±Se with increasing x. However, the presence of such a sequence would imply NSe±Se 1.0, whereas it is observed to be 1.3. This suggests that some of the Ge-tetrahedra may be interconnected with more than one Se atom at this composition. For Ge-rich glasses (x > 0.33) Ge±Ge correlation is present in addition to Ge±Se correlation. At x 0.4, with NGe±Ge 1.0 and assuming NSe±Se 0, one obtains NGe±Se 3.1(2). We suggest that each Ge atom has on an average three Se atoms and one Ge atom as its nearest neighbours giving support to the presence of Ge2 (Se1=2 )6 ethane-like units i.e., Se±Ge±Ge±Se: From the variation observed in the peak positions as well as the changes in the average coordination number of the ®rst shell with x, we infer that the Se chain segments and Ge(Se1=2 )4 units are the building blocks for x < 0.33. For x 0.33, the SRO consists of Ge-tetrahedra, while for x 0.4 there is evidence for Ge2 (Se1=2 )6 molecular units being the building blocks. The results of the present investigations on the SRO in these glasses are in qualitative agreement with the Raman [10] and EXAFS [38] reports. Furthermore, they also agree with Johnson's report on Six Se1ÿx glasses [1], which have a similar SRO. 6.3. Network connectivity: In crystalline GeSe2 , it is known that 50% of Ge-tetrahedra are in an edge-shared (ES) con®guration, assuming at most one ES tetrahedron per Ge site [41]. The remaining tetrahedra are in a corner-shared (CS) con®guration. The peak in T(r) of glass is known to result around 3.0 A
229
from Ge±Ge correlation in the ES con®guration [5]. In glasses with x 0.2, 0.33 and 0.4 one ob The area of this serves a small peak around 3.0 A. peak shows an increase with increasing Ge content, as shown in Fig. 2 and Table 3. Thus, with increasing x, Ge(Se1=2 )4 units are not only interconnected, but also the number of these units in ES con®guration increases. Since, in diraction experiments the errors associated with the area of the peaks after the ®rst one are larger, it is dicult to get a reliable quantitative estimate of the fraction of ES tetrahedra in these glasses. It is known that in c-GeSe2 the distance between a Ge atom in ES con®guration and a Se atom on neighbouring tetrahedra lies between 4.6 6 r 6 5.3 A and the separation between a Ge A atom in CS con®guration and a Se atom lies be 6 r 6 4.8 A [6]. One notices from tween 4.0 A becomes Fig. 2 that the peak around 4.7 A broader with increasing x. The origin of this peak in the case of glasses with x 0.33 and 0.4, where the tetrahedral connectivity is complete, could possibly be associated with the above mentioned Ge±Se correlation. However, an unambiguous explanation of the origin of this peak for x 0.2 and y 0.1 cannot be given as in these glasses tetrahedral connectivity is not complete. Thus, in these glasses, correlation other than the Ge±Se correlation between two neighbouring tetrahedra This would be contributing to the peak at 4.7 A. contribution becomes even more certain for x 0.1, where tetrahedra are dilute i.e., dispersed in Se chains. It is likely that reverse Monte Carlo [42] studies may provide an explanation for the in smaller Ge origin of the peak around 4.7 A content glasses as also a better estimate of ES tetrahedra in these glasses. 7. Conclusions The Monte-Carlo method used to generate the structure factor, S(Q), in Gex Se1ÿx glasses, where 0.1 6 x 6 0.4, gives food ®t to the experimental data. The Ge(Se1=2 )4 tetrahedra are the basic units of short range order in GeSe2 glass. The glass with x 0.1 consist predominantly of Se-chain segments interlinked with tetrahedra. At x 0.2, the
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tetrahedra are either connected through bridging Se atoms or by short Se-chain segments. The GeSe2 glass on the other hand, consists of Getetrahedra which are interlinked by bridging Se atoms. At x 0.4, the glass alloy contains Ge2 (Se1=2 )6 molecular units and the network connectivity is complete. A gradual shift in the ®rst peak in the total correlation function T(r) is consistent with this model. In glasses with x 0.2, 0.33 and 0.4 there is evidence of the presence of edgeshared tetrahedra; there is an increase in ES tetrahedra with x. The intermediate range order also shows a systematic increase with x. The FSDP shifts to a smaller Q and has maximum coherence length for glasses with x 0.33 and 0.4. Acknowledgements A part of this work was carried out under the IUC-DAEF project no. IUC/BOM/54. The authors gratefully acknowledge the help rendered by their colleagues Dr R. Chakravarthy, Dr B.N. Meera, Mr K. Ramesh and Mr A.B. Shinde in carrying out the experiments. One of us (N.R.R.) would like to thank the Solid State Physics Division, BARC, Bombay for the hospitality extended during his stay on the campus. We thank the referees for the suggestions which improved the presentation of our results. References [1] R.W. Johnson, D.L. Price, S. Susman, M. Arai, T.I. Morrison, G.K. Shenoy, J. Non-Cryst. Solids 83 (1986) 251. [2] Y. Sagara, O. Uemura, S. Okuyama, T. Satow, Phys. Stat. Sol. (a) 31 (1975) K33. [3] E.D. Crozier, F.W. Lytle, D.E. Sayers, E.A. Sayers, Can. J. Chem. 55 (1977) 1968. [4] J.E. Griths, M. Malyj, G.P. Espinosa, J.P. Remeika, Phys. Rev. B 30 (1984) 6978. [5] S. Susman, K.J. Volin, D.G. Montague, D.L. Price, J. Non-Cryst. Solids 125 (1990) 168. [6] I.T. Penfold, P.S. Salmon, Phys. Rev. Lett. 67 (1991) 97. [7] O. Uemura, Y. Sagara, D. Muno, T. Satow, J. Non-Cryst. Solids 30 (1978) 155. [8] A.C. Wright, D.L. Price, A.G. Clare, G. Etherington, R.N. Sinclair, Diusion Defect Data 53&54 (1987) 255.
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