PHYSICAL REVIE%' 8 VOLUME 33, NUMBER 6 Structural models of the amorphous alloy Geo 15 MARCH 1986 2oA. so 4oSeo 40 by a random technique Nicolas de la Rosa — Fox, Luis Esquivias, Pilar Villares, and Rafael Jimenez-Garay de Frsica Fundamental, Facultad de Ciencias, Apartado 40, Puerto Real (Cadiz), Spain Departamento (Received 25 February 1985) Three different spherical-shaped models of the amorphous alloy Geo zoAso 40se040, obtained from quenching of the molten mixture of the elements, were constructed. The models were built by computer simulation of an x-ray diffraction experiment. Initial models were created on the basis of the short-range order proposed from the radial distribution function (RDF) interpretation, consistent with the covalent coordination of the forming elements. The refinement has been carried out by the Metropolis Monte Carlo method with some modifications. A two-phase structure, consisting of cross-linked clusters of distorted tetrahedra centered on Ge atoms and structurally independent Se chains, has been considered the most probable. These clusters are linked to each other by means of chains of As and Se atoms. An analysis of its main structural parameters has been achieved. We propose the existence of a certain medium-range order, indicated by the appearance of a first sharp diffraction peak (FSDP} in the experimental interference function. This FSDP is reproduced in the theoretical interference function, calculated from the RDF of the proposed model. I. INTRODUCTION The elements of the Ge-As-Se amorphous alloy system have strong covalent interactions that give rise to a broad glass formation zone in the area with high Se percentage. ' The properties studied (optical, electrical, ' conductivity, 4 microhardness, s microstructure radial distribution function, ' etc.) up to the present day have shown such a diversity that it seems that any apparent relation among them could be expected. This fact may be related to the existence of structural group of heterogeneities which cannot be accounted for by a radial distribution function (RDF) structural analysis, its monodimensional character not permitting a continuous amorphous structural model to be distinguished from another containing separate phases. In this paper, previous results from RDF studies are widened by simulation of tridimensional models which allow statistical analysis of the main structural parameters as well as proposing a real average structure of the material. phite monochromator in the diffracted beam. Four series of data were collected in the interval 6'~ 28~ 110', two in increasing angles in 28 and the other two in reverse. Times were measured to complete 4000 counts in each observation point using a scintillation counter. Mean values of those measured in each observation point were assigned to that position. Averaged values lay within 5% of the average. Data were corrected for background, polarization, and multiple scattering, were normalized using the high-angle method, and then Compton scattering was subtracted. III. RADIAL DISTRIBUTION FUNCTION The RDF is calculated as 4trr p(r)=4trr po+rG(r), where po and p(r) represent, respectively, the mean atomic distribution and the local atomic density. G(r } stands for the Fourier transform of a function of experimental intensities, II. EXPERIMENTAL G(r ) = A bulk glass sample was obtained from raw material with a 99.99% purity. Appropriate proportions of the elements were sealed in a silica glass ampoule under a residual inert He atmosphere and then melted. The molten mixture was rotated inside a furnace at a temperature above 1000'C for three days and quenched in ice water. The solid obtained was ground to a fine powder ( &40 pm) and compacted by pressure into a brick of approximately 20X20)&1 mm from which an x-ray diffraction standard diagram, not showing any crystallinity remains, was obtained. The diffraction intensities were measured on a D500 Siemens diffractometer with conventional equipped Bragg-Brentana geometry and a Mo anode. The Mo Ea line (A. =0.71069 A} was selected by means of a bent gra33 J 0 F(s) sin(sr )ds being F(s ) =si (s ) and i (s ) = I,„gx; x;f; where f;— (2) x; is the atomic fraction of the i element with and I,„represents the resulting intensity i=oe, As, Se values after corrections in electronic units. To avoid spurious oscillations in G(r) due to the lack of high-s (scattering vector modulus) data, theoretical extension of the data was carried out by means of a method based on that described by Shevchick. ' In order to create the tnodel, the RDF of this compound was calculated. The interference function (Fig. 1), " 4094 Qc19S6 The American Physical Society STRUCTURAL MODELS OF Ooo. ooAoo. ooSoo. oo 33 bonds. ' Following the line undertaken in early structural models, ' results of which justify the modifications, we have looked for a coordination increment by creating at random the positions needed to saturate the volume, laying down only geometrical and maximum coordination conditions. Afterwards, the type of atom is assigned to some positions on the basis of the hypothesis formulated in the preliminary analysis of the RDF function. The lesser coordination positions are rejected until there remains the number of atoms that fit in a sphere of radius R. A random process is used to assign the type of atom to the rest of unallocated positions. On the initial model, a theoretical diffraction experiment is carried out and the RDF calculated starting from the hypothesis that thermal vibration effects may be In this way the represented by Gaussian broadening. theoretical RDF may be expressed as ' 1.0 i- K LIJ 0 LLJ O S{A ) FIG. 1. Experimental interference function of the amorphous alloy Geo zoAso AOSe04o, showing the first sharp diffraction peak at s=1, 02 A related with medium-range order correlation lengths. XXC,( 4wr p(r}hr= i.e., F(s } shows the widely studied first sharp diffraction peak (FSDP) at s=1.02 A ', which was not described by Krebs and Welte in a previous paper on the Ge-As-Se amorphous alloy system. Nevertheless, the results of both RDF analyses are in agreement (Table I), being the area under first maximum coherent with a short-range order in which the Ge, As, and Se atoms have, respectively, 4, 3, and 2 as the most probable coordination number, in such led. a way that the 8 %rule is fulfil— IV. OPERATIVE METHOD 0 Spherical models of 10 A radius were built. The spherical shape was selected because the essential feature of the RDF is that it is only an r-dependent function. A sphere of 10 A radius was considered adequate to represent the material from a statistical viewpoint without involving too much computation time. The technique is based on the Metropolis Monte Carlo method, ' similar to that used by Renninger et al. ' but some modifications have been accomplished. The technique takes into account that structural models based in aleatory tetrahedra networks and refined by Monte Carlo simulation have a tendency to be low coordinated and present an excess of dangling TABLE I. RDF characteristics. Krebs and %elte' alloy Position (A) Our results 2.41 1st 3.80 Area (at. ) Estimated error Averaged angle bonding 'Reference 7. 2.74 2.76 +0.1 +0.1 103 104 g (2ma z iz) '" &&& ) e = [«~~(»)+4~r'po]«, (4) where the sum is extended over the l shell of width d, r: C" shell around IJ is the number of the j-type atoms in the I 2 an i-type atom, crI is the mean-square thermal displace- ment. In order to compare rG ~(r } with rG(r ), it was necessary to simulate what would be obtained if a sample with the same shape and size as the model had been employed for the experiment. To achieve this we multiplied the experimental rG(r ) by the function D(r ) = 1.0 —1.5(r/2R )+0.5(r/2R )i proposed by Mason' that represents the probability of finding an r distance in a sphere of R radius. Under these The conditions, both functions have been compared. mean standard deviation serves as a criterion to decide on the model validity e is defined as g I«««»(«} —«Gm~(«) I', (6) M being the number of points where the comparison is accomplished. The method employed consists, essentially, in moving an atom, randomly designed, in an aleatory direction. If the new position keeps the geometrical and coordination conditions and improves the fitting of the calculated rG ~(r) with respect to the experimental one, it is accepted and the process starts again. At first, we have taken 0.3-A displacements which are gradually reduced to 0. 1 A. Smaller displacements would be masked by thermal vibration considered as a constant 0. 1 A) during the refinement process. Afterwards, (cri — the minimum square method has been employed to refine this parameter for each coordination sphere. Calculations were performed on a Digital Equipment Corporation PDP11/23 computer using programs written by O'Anjou and Sanz. " FOX et al. NICOLAS de la ROSA — With the aim of having another criterion about the model validity, calculation of the interference function corresponding to the proposed model was achieved by the Fourier transform of ~(r): 6 si ~(s ) =4m. 0 6 ~(r ) sin(sr )dr . (7) Since this function depends on the model size, an experimental interference function was obtained as si'(s)=4nJ 0 'G(r)D(r)sin(sr)dr . the integration 8.0 5.6 3 2 2. 1014 ' 0.8 -1.6 (8) (7) and (8) must actuoo, we have taken into consideration that the fitting has been performed between 0 and 10 A. So, above 10 A there may exist incorrect correlation lengths which would give rise to uncertain information in the theoretical interference function. For that we have fixed A in the calculation of referred integrals so that the results may be compared. Once both interference functions have been found, the standard deviation (e') is calculated in a similar way as Although ally be carried out until 33 Eqs. 8,„= in R,„=10 8.0 O. 5793 3o2 {b) -1.6 -4. 0 before. V. MODEL DESCRIPTION AND RESULTS 8.0 According to experimental density of the compound, the model is made up of 146 atoms, of which 28 are Ge, 59 As, and 59 Se, if we take into account the material composition. Following the formulated hypothesis for the RDF (Fig. 2) interpretation, ' we first supposed, like Borisova, ' a low probability for the Ge As bond formation. A model built under this condition evolved till a standard deviation of 0. 1065 was reached with an appreciable number of dangling bonds on the Ge atoms (which were initially tetrahedrally coordinated). This result brought us to change our point of view about the probability of a G~As bond formation. In a second model, a Ge atom was assigned to the 28 highest coordination positions, and the uncoordinated or onefold coordinated ones were discarded until a total of 146 positions remained. Finally, the type of atom (As or Se) was assigned to the rest of unallocated positions by an aleatory process. This model reached a standard deviation of 5.6 — 0. 1917 3.2 0.8 -1.6 -4. 0 0 FIG. 3. rG ~(r) and rG(r)D(r), dashed and solid lines, respectively, in some steps of the refinement process. As may be expected, the initial model (a) shows a quasialeatory distribution. After 215 valid movements (b), the model has already developed the first and second coordination spheres. %hen the standard deviation is 0. 1917 (c), all the coordination spheres are clearly defined. Ripples for r high values are due to the small number of atoms involved in the model and also to having considered the mean-square thermal displacement as a constant for all coordination spheres during the refinement process. 0.0607 and, hence, 35.2 may be considered more probable than the former one. Nevertheless, since no bonding conditions have been imposed during the model refinement, low coordinated Ge atoms have appeared, as opposed to the tetrahedral framework that one may imagine for this amorphous system. A third model has been built, keeping in mind these results. Atom movements involving breaking of Gc X bonds are not allowed. In order to show how the calculated result changes as a function of refinement, rG ~(r ) of the initial model and two intermediate steps are plotted (Fig. 3). For the initial model rG ~(r) corresponds to a quasialeatory distribution, as may be expected. In fact, it could serve as a criterion for how aleatory the model is. The initial standard deviation was 2. 1017. After 215 valid — l . 6 s{AI FIG. 2. Experimental radial distribution the amorphous alloy Geo 20Aso~Se04O. values are collected in Table I. function 4mr p(r ) of RDF characteristic TR UCTURAI MOD pl S OF Qeo 33 goAs0. 44 0 ~' -O. 28 -0.84 1.4 -0, 28 -1.4 P h ' {f llbo d) ed. Also the two c ains be 2, 0 in epe units, have beenn drawn {white bon s movements, off tetrahete entation of the framework P . 1.28 esforr w which 1530 aleatory to nu numbers were neces- '. . tho first and account of the very 1TlO oment, hi h der are already we 1ne l s with a stan d ar d deviation Ripp es be or f bec of h S(A ) 'd d dashed lines, respectived h mall atoms involve the refinement tion spheres. This does not allow process ion of therma 1 di 1 an a d equaate representation m f deviation or thisi model was g p ments w ' rahedral ram ew represen a Ge atoms a a earing eit er a ructural umts (A and 8) or joining tetr t D, E, and I'i. Incelusion of some s a 7. S 1.2 4. 8 2. 4 of 0.1917 8 Fourier transform o r e atoiiis atoms does oe not discard t e p theory were on y S1 ir atomic numbers. Ann exchange of the h toms was pe rform di a that . 4 could be considered as a g ' c abated p hase. We would i e 'n the standar ev' 1'mprovement in d when ' t isop eration was ac corn lis h ed . As we know y earlier 'ences, when thee re refinement is very experien ry near the en, a bout a worsening wi tytthata 0 e~ factor, iit seems to point out the Af -D. 4 -2. $ i.0 6.6 hh h (r . Also, the theoretica of this manipu a e ---, 1 o nt with respect to than t h.e u ul"d t nnoticeab e is ]I oof the curved shape a e corresponding to re ard to the FSDP (Fig. 6). For especially with reg [(e') and rG ~ r of the final model, solihd and dasshed lines, respectively, once t e e ment was accomplished. io oi =0.0328j b~™~ FOX et a1. NICOLAS de la ROSA — TABLE II. Atom coordinations. The numbers correspond to the amount of atoms present in the final model v~ith the indicated coordination. Type of atom In order to contrast our results, we have compared them with some standard distances in compounds formed by these elements. The Ge Ge ABD is 2.51 A, which is Alvery near to the 2.54 A obtained for amorphous Ge. though in crystalline GeAsz there are six possible firstneighbor interatomic distances between a Ge atom and an As atom, the mean value of these being 2.45 A, ' it is similar to the 2.44 A of the Ge As ABD in the proposed model. The Ge Se bond length in the hexagonal phase of the GeSe2 is 2.37 A (Ref. 22), whereas in the orthorhombic phase it is 2.59 A (Ref. 23). Although the deviation of the ABD for this bond in the model from the aforesaid distances is around 4%, it is remarkable that our result corresponds exactly with the mean value for both distances. The As As ABD is 4% lower than the 2. 49 A (Ref. standard distance in the amorphous As 2. 51 A (Ref. 24} as well as in the rhombohedric one — 25) but it is nearly twice the As covalent radius. Concerning the As Se bond we found an ABD of 2.41 A, coincident with the mean value of the six possible firstneighbor interatomic distances present in the crystalline Finally, the 2.40 A obtained as ABD for the As2Se3. Se Se bond is in agreement with the value proposed in amorphous Se. 7 — Coordinations 2 Ge 20 As Se 0 0 23 reason we have considered this model to represent with higher probability the average structure of the material, and an analysis of its main structural parameters has been achieved. — — — — A. Coordination Table II collects the resulting coordinations of the atoms. They present dangling bonds that, to a large extent, belong to atoms less than a first-neighbor distance from the sphere's periphery and could be satisfied with In the case of atoms hypothetical external neighbors. with two or more dangling bonds, we admit that these could be due to the small size effect when the atom involved is less than 1.1 A from the boundary by taking into account both the average first-neighbor distance and bonding angle. The model contains 13.50% dangling bonds, which are not in an appropriate situation to be satisfied with hypothetical external neighbors. Nevertheless, the existence of dangling bonds is an inherent consequence of the way chalcogemde glasses are prepared. These defects would give rise to an electronic structure with relatively well-defined energy levels in the mobility gap, leading to a peaked density-of-states distribution and a pinned Fermi level. B. — — 36 — C. Bonding angles Averaged bonding angles are in Table IV. All of them are acceptable, since the first distortion that one may expect in an amorphous material is in the bonding angles, being the standard deviations from the averaged values which are greater than those from the bond length. D. Considerations on the medium-range Even though the model is designed to determine correlation lengths from 10 A downwards, some medium-range order traces have appeared. The FSDP is, to all appearances, the strongest and most universal signature of the In fact, if the Fourier transform medium-range order. „=10 A, it does not of G ~(r) is carried out with reproduce the FSDP totally and, on the other hand, neither does the Fourier transform of G(r) (Fig. 6). From this, the smoothing of the FSDP may be explained from the existence of certain medium-range orders in the model Bonding distances The averaged bonding distances (ABD} of different bonds present in the model are shown in Table III. Standard deviations are less than 8%, which agrees with the distortion that one may expect in the amorphous alloys. The ABD variations with regard to the sum of covalent radii of the elements forming the bond never surpass 4%. R, TABLE III. Averaged bonding distances of different bonds present in the model. Distance (mod) Bond type (A) — — 2.51+0. 19 2.44+0. 14 Ge Ge Ge As — As As — Se As Standard 2.54 Amorphous 2.45 Crystalline 2.41+0.06 2.40+0.06 2, 2.41+0.07 distances (A) 2.37 2.59 2.49 2.51 2.41 2.48+0. 15 order Ge GeAs2 Hexagonal GeSe2 Grthorhombic GeSe2 Amorphous As Rhombohedric As Average in crystalline As2Se3 .40 Amorphous Se Sum of covalent radii (A) 2.44 2.42 2.40 2.36 STRUCTURAL MODELS OF Geo 20Asa 33 TABLE IV. Averaged bonding angles in the model. Angle type Averaged X— X Ge— X— As — X X— X Se— bonding angle 108'+23' 109'+20' 109'+23' whose small size effect hinders its total description. Looking at Fig. 4 we propose Se chains and crosslinked tetrahedral clusters containing around 30 atoms as medium-range order features. This would be in agreement with the result which follows from the topological description of covalent network glasses given by J. C. Phillips. VI. CONCLUSIONS The best structural model obtained for this alloy can be basically described by a distorted tetrahedral framework centered on Ge atoms linked to each other by means of either another Ge atom or As and Se chains. Some of these chains are found as independent structural units that 'J. A. Savage and S. Nielson, Phys. Chem. Glasses 5, 82 (1964). ~T. T. Nang, M. Okuda, and T. Matsushita, Phys. Rev. B 19, 947 (1979). 3P. J. Webber and J. A. Savage, J. Non-Cryst. Solids 20, 271 (1976). 4R. W. Haisty and K. Krebs, J. Non-Cryst. Solids 1, 427 (1969). 5B. D. Michels and G. H. Frischat, J. Mater. Sci. 17, 329 (1982). 6A. Seifert and G. H. Frischat, J. Non-Cryst. Solids 49, 173 {1982). 7H. Krebs and H. Welte, J. Solid State Chem. 2, 182 (1970). N. de la Rosa — Fox, Tesina de Licenciatura, Universidad de Sevilla, 1983. 98. E. Warren, X-Ray Diffraetian (Addison-Wesley, Reading, Mass. , 1969). N. J. Shevchik and W. Paul, J. Non-Cryst. Solids S-IO, 369 (1972). ' A. D'Anjou and F. Sanz, J. Non-Cryst. Solids 28, 319 (1978). ' N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953). A. L. Renninger, M. D. Rechtin, and B. L. Averbach, J. of Non-Cryst. Sohds 16, (1974). View publication stats ~eo ~. . . would connect with the proposed structural heterogeneities to explain the physical behavior of the Ge-As-Se system. The second as well as the third model may be considered as representative of the real average structure of the material. Nevertheless, we think that the third, including our proposition of considering the existence of Se as a separate phase, represents with higher probability the mentioned structure and could serve us as a starting point to create models of other alloys of this system. A small number of dangling bonds has appeared, in agreement with that to be expected in amorphous materials. Averaged bonding distances do not present appreciable deviation from those compounds containing these elements. Finally, the proposed model accounts for a certain medium-range order consisting of cross-linked tetrahedral clusters and free Se chains in the voids of these clusters. ACKNOWLEDGMENT We thank T. Leo and pletion of the paper. P. Peers for their help in the com- 4M, Mateos Mota, Tesis Doctoral, Universidad de Navarra, 1983. F. Sanz, Rev. Iberoam. Crist. Miner. Metalogen. 1, 129 (1978). '6L. Esquivias and F. Sanz, J. Non-Cryst. Solids 70, 221 {1985). G. Mason, Nature 217, 733 (1968). ' A. D'Anjou and F. Sanz (unpublished results). 9A. U. Borisova, Glassy Semiconductors (Plenum, New York, ~5A. D'Anjou and 1981). R. Grigorivici, J. Non-Cryst. Solids 1, 303 (1969}. J. H. Bryden, Acta Crystallogr. 15, 167 (1962). J. Ruska and H. Thurn, J. Non-Cryst. Solids 22, 477 (1967). 23G. R. Kannewurf et aL, Acta Crystallogr. 13, 449 (1960). G. N. Greaves and E. A. Davis, Philos. Mag. 29, 1201 (1974). 25X. Wyckoff, in Crystal Structure (Wiley, New York, 1963). 2sA. A. Vaipolin, Kristallografiya 10, 596 (1966) [Sov. Phys. Crystallogr. 10, 509 (1966)]. ~R. W. Fawett et aL, J. Non-Cryst. Solids. 8-10, 369 {1972). 28J C Phillips J. Non-Cryst. Solids 43, 37 {1981). 29J. C. Phillip, Phys. Status Solidi 8 101, 473 (1980). —