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Relaxed continuous random network models
Journal of Non-Crystalline Solids, 15 (1974) 199-214. © North-Holland Publishing Company
RELAXED CONTINUOUS RANDOM NETWORK MODELS (I). Structural characteristics* P. S T E I N H A R D T * * , R. A L B E N and D. W E A I R E Department of Engineering and Applied Science, Yale University, New Haven, Connecticut 06520, USA Received 29 August 1973
Two elastic-energy-relaxed continuous random network (Polk) models for tetrahedrally bonded amorphous semiconductors have been obtained: a 201-atom model built entirely at Yale and a 519-atom model relaxed from a structure built by Polk and Boudreaux which originated at Harvard. In relaxing the coordinates to minimize the total energy the Keating potential was used for the interatomic interactions. The models are analyzed in terms of density, elastic distortion energy, elastic constants, numbers of five-, six- and seven-fold rings, distribution of dihedral angles, and radial distribution functions. We find that, despite their different origins, the models have essentially identical characteristics. Our principal conclusions are as follows: (a) The density of the CRN model is, to within 1%, that of diamond cubic. (b) The bulk modulus is about 3% lower than that for the diamond cubic structure and the shear modulus lies between the two diamond cubic shear moduli. (c) There are, to within -+ 10% (and with corrections for surface effects), 0.38 five-fold, 0.91 six-fold and 1.04 seven-fold rings per atom. (d) For a reasonable value of the bond bending force constant, rms bond length distortions are about 1.0% and bond angle distortions are about 7.0°. (e) The radial distribution function agrees very well with experiment for all four principal peaks.
1. Introduction The reasonableness o f the c o n t i n u o u s r a n d o m n e t w o r k ( C N R ) m o d e l for the structure o f tetrahedraUy c o o r d i n a t e d a m o r p h o u s s e m i c o n d u c t o r s is d e m o n s t r a t e d by Polk [ 1 ], w h o showed that it is possible to c o n n e c t rather rigid plastic and metal tetrahedral m o l e c u l a r units in a structure with little b o n d strain, no unsatisfied bonds and no perceptible long-range order [ 1 - 3 ] . F o r several years, our k n o w l e d g e o f the characteristics of these structures was based on laborious direct m e a s u r e m e n t s o f a single m o d e l o f 440 units built by Polk [ 1 ]. Models built by c o m p u t e r m e t h o d s [4, *Research supported in part by NSF.
**Present address: Dept. of Physics, California Institute of Technology, Pasadena, California 91109.
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P. Steinhardt et al., Relaxed continuous random network models
5] have yet to achieve the small degree of distortion and absence of dangling bonds characteristic of Polk's model. Recently, the coordinates for the original Polk model, expanded to 519 units, were obtained by Polk and Boudreaux [3] and refined so as to equalize nearest neighbor distances. The availability of numerical coordinates made possible a more complete characterization of the model with extremely encouraging initial results. Very recently, the present authors have relaxed the coordinates of ~ e 519-atom model so as to minimize an elastic energy which depends on both bond-length and bond-angle distortions [6]. We have also used our relaxation procedure to rapidly obtain coordinates for a new 201-atom model. In this paper we report a detailed characterization of the two models with regard to density, elastic distortion energy, elastic constants, numbers of five-, six- and seven-fold rings, distribution of dihedral angles, radial distribution function and G(r) (including a comparison with experimental data) [7-10] and contributions of neighbors belonging to rings of different sizes and of eclipsed and staggered bond character to the form of the radial distribution function. We find that the two models are essentially identical in their characteristics, which leads us to conclude that the 'ideal' CRN structure may be a reasonably welldefined concept. Also, when the CRN radial distribution functions are compared with experimental data for amorphous germanium prepared by any of the common methods, the agreement is excellent for all four principal peaks and some more subtle features as well. This agreement is far better than has ever been demonstrated for microcrystalline [11] or amorphonic models [12]. Thus we believe that the analysis of the CRN model which we present below should form the basis for a more complete understanding of the structure of amorphous group IV semiconductors. Thepaper:is divided into three sections. In sect. 2, we describe our procedure for relaxing to an accurate minimum elastic energy structure. In sect. 3, we present our characterization of the 519-atom and the 210-atom relaxed structures. Paper II, by J.F. Graczyk and P. Chaudhari, which is a continuation of this paper, is concemed with the diffraction properties of the models. An analysis of relaxed CRN models has been independently performed by Duffy and Polk [13].
2. Relaxation
procedure
A numerical procedure has been developed which simultaneously relaxes both bond-length and bond-angle distortions in tetrahedrally coordinated continuous random network models by minimizing the Keating elastic energy [ 14] expression:
3 ~{3 l(~t) ~ (rli'rli' + ~d2) 2 V= d162 3__~ l,i ~ (rli'rli-d2)2 +-8
(1)
where t~ and/3 are the bond-stretching and bond-bending force constants, respectively. The first sum in the expression is on all atoms I and their four neighbours specified
P. Steinhardt et al., Relaxed continuous random network models
201
by i; the second sum is on all atoms and pairs of distinct neighbors; and rli is the vector from I to its i'th neighbor. Provided with a table of nearest neighbor relationships (not altered by the relaxation process) and a listing of the approximate equilibrium coordinates before relaxation, the refinement program moves each atom, in turn, to essentially the exact position of equilibrium under the bond stretching and bending forces due to its nearest and next-nearest neighbors. Specifically, for each repositioning of an atom, the force (vector) on that atom and the force derivative (tensor) are calculated, after which a matrix inversion gives the coordinates of a point of zero force under the Keating potential. The coordinates are tested to insure that they are not positions of m ~ i m u m energy, although no direct check prevents their being saddle points of the energy. The repositioning of each atom in the model is repeated over many cycles until convergence to equilibrium is achieved for the entire structure. From starting coordinates within 1/2 of a bond length of the equilibrium positions, angles and bond lengths were found to converge to 1 part in 103 of their final values (as determined by very long iterations on test cases) after 25 cycles for the models considered. The coordinates used in most of the analysis were obtained after 50-100 iteration cycles and are estimated to be accurate to about 0.05% for individual bond lengths and 0.05 ° for bond angles. Thus, our estimates of quantities that are averages over all atoms in the structure should be sufficiently accurate for practical purposes. To analyze the properties of relaxed continuous random network structures, the coordinates for two CRN models were obtained using the numerical procedure described above. A combination of the full relaxation procedure and a partial relaxation procedure, in which a central core of atoms was held f'Lxedwhile the added atoms were relaxed from automatically guessed start positions, was used to build a 201-atom structure from a 21-atom roughly measured seed. To provide the necessary table of neighbor relationships (extended to include the new atoms), a reference hand-built model which paralleled the relaxed structure was constructed from plastic and metal units of the same type used by Polk [1]. Coordinates for a second relaxed random network structure were calculated from the 519-atom model reported in ref. [3]. In that model, bond-length deviations (only) had been minimized in the structure. However, the condition of equal bond lengths represents two constraints per atom (less when a surface is present) while there are three degrees of freedom per atom. Thus, coordinates from the bond-length equalization procedure depend to some extent on the initial measurements of the physical model and have rather larger bond-angle deviations than necessary. By contrast the relaxed coordinates are determined by the form of the elastic energy expression and depend on the properties of the plastic and metal units used to build the model only insofar as these influence the connectivity. In relaxing both models the ratio [J/a was taken as 0.2, which is in a range of plausible values as indicated by measurements of phonon frequencies in diamondcubic silicon and germanium [ 15]. Most properties of the models (except for the absolute distortion energy) were quite insensitive to the particular choice of/~. This is
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fortunate since a simple bending force cannot be expected to accurately describe complex distortions in either silicon or germanium [16].
3. Characteristics of random networks An intensive study has been made of the basic physical characteristics of the two models with the intent of discovering what features are common to all such structures. The characteristics considered are in turn: density, elastic distortion energy, elastic constants, ring statistics, dihedral angle distribution, radial distribution function and corrected G(r), and the contribution of rings and neighbors to different features of the corrected radial distribution function, Finally, we compare the CRN G(r)'s and radial distribution function with experimental results for electron [7] and X-ray [8-10] diffraction for amorphous germanium. 3.1. Density
To find the density of each of the two models required a characterization of density fluctuations and possible special properties of atoms near the surface. Our procedure was to find the radius, r o (measured in bond lengths), of the largest sphere about the center-of-mass which contained only fully bonded atoms and then to plot the average density (number of atoms/sphere volume) for spheres with radii from r o - 1 to ro+l. In fig. 1 we show the result for the 519-atom model. The plot for the 201-atom model was similar except that fluctuations were about twice as large as for the 519atom model. Using for the average nearest neighbor distance the value of 2.45 )~ appropriate for crystalline Ge, densities of 4.40 + 0.10 X 1022 atoms/cm 3 for the 201atom structure and 4.41 + 0.06 X 1022 atoms/cm 3 for the 519-atom structure were determined. The diamond cubic density is 4.42 × 1022 atoms/cm 3. Thus, the density of the CRN model is, to within 1%, that of diamond cubic, a conclusion in agreement with Polk's analyses [1,2]. 3. 2. Distortions and distortion energy
CRN structures have characteristic bond-length and bond-angle deviations from the perfect tetrahedral configuration. In the 201-atom model a root-mean-square bond length deviation of 0.80% and root-mean-square angular deviation of 6.66 ° over the entire model were found. For the relaxed 519-atom network, a root-meansquare bondiength deviation of 1.04% and a root-mean-square angular deviation of 7.10 ° were measured. In comparison, before the full relaxation procedure had been applied to the Polk and Boudreaux coordinates, the root-mean-square bond length deviation was O. 16% and the root,mean-square angular deviation was 9.25 °. Since the stored elastic energy of a structure is approximately proportional to the square of the rms angular deviation, the relaxation process accounts for about a 40% de-
P. Steinhardt et al., Relaxed continuous random network models
4.6xj
40
I
I
I
I
I
I 4.0
I 4.5
I 5.0
I 5.5
203
6.0
Radius
Fig. 1. Average density in atoms/cm 3 for spheres of different radii (in bond lengths) about the center of mass of the 519-atom relaxed model. The bond length was taken as 2.45 A which is the appropriate value for germanium. The dashed line represents the diamond cubic density. The density deficit followed by a density increase and fall off in the region from 5 to 6 bond lengths is a surface effect.
crease in the total stored energy in the unrefined 519-atom model. Using the value = 3.87 × 104 dyn/cm appropriate for germanium, the distortion energy for the 201-atom is found to be 19.7 meV/electron; for the relaxed 519-atom model, 23.1 meV/electron; and, for the unrelaxed 519-atom model, 36.6 meV/electron. By contrast, for a widely used handbuilt periodic model due to Henderson (private communication) the energy for relaxed and unrelaxed coordinates correspond to 77 meV/ electron and 82 meV/electron respectively. The somewhat smaller distortions of the 201-atom model relative to the 519atom model are partially due to its relatively larger number of surface atoms (which are under fewer constraints). If only the 52 atoms within the sphere of fully bonded atoms are considered, the distortions for the 201-atom model become 0.97% for bond lengths and 6.99 ° for bond angles. We cannot eliminate the possibility that distortions continue to grow with model size; however, from our experience we believe (in agreement with Polk [ 1 ] ) that the models (which are already fairly large; the 519-atom model corresponds to a 28 A diameter germanium cluster) can be extended indefinitely without increasing the distortions. For/3/ct = 0.2, a good guess f o r the extrapolated distortions o f an infinite 'Meal' model is 1% f o r bond lengths and 7° f o r bond angles. 3. 3. Elastic constants
The elastic constants of the model were determined by applying different types of strains and allowing fully bonded atoms to relax so as to minimize the energy while surface atoms (those with at least one unsatisfied bond) were held fixed. Strains of - 0 . 0 2 , - 0 . 0 1 , 0 , +0.01 and +0.02 were applied and the curvature of the energy was determined both from the least-squares parabolic fit for all five measurements and from the second difference of the middle three measurements. Results
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P. Steinhardt et aL, Relaxed continuous random network models
Table 1 Elastic constants for the CRN model compared to those for the model without internal relaxation and those for a diamond cubic model with the same force constants (a = 0.387 X 10s dyn/ cm, ~/c~= 0.2) and density (4.216 X 1022 atoms/cma).
CRN model CRN model (no relaxation) Change on permitting relaxation Diamond cubic
B
M
Bulk modulus (X 104 dyn/cm 2)
Shear modulus (x 104 dyn/cm 2)
7.08 ± 0.04 7.26
3.80 ± 0.06 6.03
-2.5% 7.30
-37.5% 2.74 a) 4.56 b)
a) for z2-{/2(x 2 +y2) type shear. b) for xy-type shear. were obtained for both models for uniform compression, extension along x and compression alongy and z, and for x y type shear. For the 201-atom model other types of shears were applied, and tests were made on both models for interior spheres of atoms. The results are summarized in table 1. They are compared with results for a diamond cubic structure with the same density and force constants and for the CRN model without internal relaxation. The error limits give the deviations of the results for the different models and different ways of calculating elastic constants from the energy for different shears. It is seen that the results for the two models and different shears are within a few percent of each other. Thus the models are both quite similar and quite isotropic. The bulk modulus (B) for the CRN is very close to that for diamond, an unexpected result since in the limit of small ~/a it can be proved that B should approach zero for such structures (as it does for a model of Ge III [17]). The explanation for the undramatic result for the CRN with/3/c~ = 0.2 lies in the fact that the relaxed structure has close to the maximum possible volume consistent with almost equal bond lengths. The volume is thus quite insensitive to internal displacements which conserve bond lengths and the softness of such internal displacements does not significantly affect the bulk modulus. 3.4. Ring statistics
A raw count of the numbers of five-, six- and seven-fold rings in the two models yielded the results given in the first two rows of table 2. It was decided to include seven-fold rings (which have not been previously considered in CRN model investigations) since they are in no sense less 'elemental' in these structures than are six-fold rings, and examination of the 201-atom model indicated that they might be important
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P. Steinhardt et al., Relaxed continuous random network models
Table 2 Estimates for the bulk numbers of five-, six- and seven-fold rings per atom for the CRN model compared with diamond cubic. 5-fold 6-fold 7-fold (rings/atom) (rings/atom) (rings/atom) Raw count 201-atom model 519-atom model
0.244 0.264
0.489 0.593
0.507 0.592
With broken surface bond correction 201-atom model 519-atom model
0.39 0.38
0.89 0.93
1.05 1.04
From density of centers of mass 201-atom model 519-atom model
0.3 ±0.1 0.39±0.05
0.8±0.2 0.9±0.1
1.0±0.2 1.1±0.1
From two point extrapolation to zero surface-tovolume ratio
0.32
0.85
0.82
"Best guess" values (± 10%)
0.38
0.91
1.04
Diamond cubic and wurtzite
0
2
0
in characterizing the structure. The raw count is by itself not very enlightening since there are many rings involving surface atoms which cannot be completed just because of the finite size of the model. To correct for finite size effects, three different methods were used and all gave quite similar results. The first method was based on the assumptions that: (1) if surface atoms are classified according to the number of their broken bonds (singlets, doublets, and triplets), the broken bonds of the atoms in each class are typical o f singlet, doublet or triplet bond sets inside the model; and (2) on the average, half the atoms of rings broken due to the surface lie inside the model. Using these assumptions, together with the fact that any ring broken by the surface must connect two surface bonds, the average numbers of rings belonging to the cluster but destroyed by an atom with one, two or three broken bonds were calculated. From this we obtained the following relation between the bulk number of p-fold rings per atom np (p = 5, 6, or 7), the raw count of p-fold rings N p , numbers of surface atoms with i broken bonds S i (i = 1,2, or 3), and the total number of atoms N ( N = 201 or 519): 25
3
3s
3
n5 = N 5 / ( N - S s 1 _ ~_ $ 2 _ ~_$3) ' 7
n 7 = N 7 / ( N - g S 1 - ~-~ $ 2 - ~-$3),
The results for np are given in table 2.
n 6 = N 6 / ( N - ~S 1 6 _ ~_~$2_3o~_$3),3 (2)
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P. Steinhardt et al., Relaxed continuous random network models
1.2 F o
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I 6
I 7
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.c_ 0.6 5-fold 0,4 <>
0.2
l 2
I 3
I 4
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8
Rodius
Fig. 2. Ratio of numbers of centers of five-, six- and seven-fold rings to numbers of atoms in spheres of different radii about the center-of-mass for the 519-atom relaxed model. Vertical marks on the curves indicate the largest radius before surface effects become significant. For the second method the ratios of the numbers of mass centers of p-fold rings to the numbers of atoms within spheres of different radii about the centers-of-mass of the models were calculated and plotted against sphere radius. These ratios are expected to fluctuate widely for small radii and decrease for radii large enough to include the centers-of-mass of rings broken by the presence of the surface. As shown in fig. 2 for the 519-atom model, it was possible to identify radii large enough to contain a significant sampling of rings but small enough to avoid the surface effects. There is considerably more uncertainty for the 201-atom model (not shown). The results for np from this method are also given in table 2. The final method was based on the assumption that the ring statistics for the two models differed only because of the difference in surface-to-volume ratios. Plotting the ring statistics for the two models against the surface-to-volume ratio (N --1/3) and extrapolating to zero yielded the results indicated in table 2. Since all these methods give similar answers we conclude that the two models have essentially identical ring statistics and the 'best guess' values are as given in the table. 3.5. Dihedral angle distribution
The dihedral angle distribution for the relaxed models was found by taking all four-fold paths connecting atoms i, j, k and l, and then making a histogram of the angles between the projections o f r i / a n d rkl on the plane perpendicular to ~'k where r ~ is the displacement vector between the atoms a and/3. Because the two models had a different ntmber of atoms their distributions were scaled for comparison purposes (see fig. 3). Although the two models are entirely distinct, a remarkable similarity is found in their dihedral angle distributions. In particular, for both models there are twice as many 60 ° and 180 ° dihedral pairs as 0 ° pairs, indicating a preference in the structure for staggered band configurations. These reshlts agree with those obtained in ref. [3] for the unrelaxed 519-atom model.
P. Steinhardt et al., Relaxed continuous random network models
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t-
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I 30
o
I 6o
I 90
Dihedral
I Izo
I tsO
iI
s
leO
Angle
Fig. 3. Distributions of dihedral angles for the 519-atom model (full line) and the 201-atom model (dashed line). The scales for the two curves have been adjusted to make them of comparable heights. (There are, of course, more angles in the larger model.)
519-olom
IOO J(r) 5o
0
2
4 Distance
6
8
to
(bond lengths)
Fig. 4. Radial distribution function, J(r), uncorrected for finite size effects, for the 201- and 519-atoms models, The curves are formed by connecting the tops of histogram bars corresponding to a width of 1/50 bond length. Note that the broad peak at 3.1 bond lengths is present in both models. 3. 6. Radial distribution function and G(r ) The radial distribution functions (J(r)'s) for the models were found directly from the coordinates by forming a histogram of the distances between all pairs of atoms and normalizing for the number of atoms (see fig. 4). In order to obtain J(r)'s and G(r)'s corrected for the finite sizes of the models, the J(r)'s were compared with those for a finite sphere of uniform density. For such a sphere, the radial distribution function is given by: N [ 3 {r'~5 9 )2 , Jsph(r) = a [ 1 - 6 \ a / _ ~ _ ( r ) 3 + 3 ( r ]
(3)
208
P. Steinhardt et al., Relaxed continuous random network models
401
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Distance (bond lengths) Fig. 5. Corrected G(r)'s for the 201-atom model and 519-atom relaxed model, both relaxed so as to minimize the Keating elastic energy with #/a = 0.2, and for the unrelaxed 519-atom model as described in ref. [3]. The units of G(r) are atoms/(bond length) 2. The curves are obtained by connecting the tops of histogram bars of width 1/50 bond length. The first peaks are truncated for clarity of presentation. where a is the radius of the sphere and N is the number of atoms. For the comparison with the two random network models, the magnitude of a was adjusted so that the ratio of J(r) to Jspb(r) approached unity as the distance from the origin became large. This agreed to two significant figures with the value of a determined from the density. The value for a was found to be 4.2 bond lengths for the 201-atom model and 5.7 for the relaxed and unrelaxed 519-atom models. Given N, a and the J(r) for each structure, we calculated a corrected G(r) given by [18]:
G(r) : [J(r) - Jsph(r)]/r,
(4)
and obtained the graphs in fig. 5. When the atoms inside the largest sphere about the center-of-mass containing only fully bonded atoms were considered, only a slight sharpening in some of the details in G(r) and an increase in the statistical fluctuations
P. Steinhardt et al., Relaxed continuous random network models
209
occurred. We surmise from this that spurious effects from partially bonded surface atoms are not significant. An inspection of the curves in fig. 5 leads us to make two observations. First, the G(r) curves for relaxed continuous random network models do exhibit significantly sharper features than those for unrelaxed models (compare the relaxed and unrelaxed 519-atom model G(r)'s). Most obvious in this respect is the narrowing of the second neighbor peak, a narrowing due to the decrease of 2.1 degrees in the rms angular deviation after full relaxation. The relaxation procedure also makes more visible the details found between the second and third neighbor peaks in G(r). Second, although the two models were constructed by different methods, their G(r) curves have in common all their prominent features and many subtler features.
3. 7. Decomposition o f the radial distribution function To better understand the prominent features of the CRN G(r)'s, we have performed a decomposition of the radial distribution function into contributions from different types of nearest neighbors. To minimize effects due to finite size without introducing a complicated decomposition procedure, we restrict attention to a radial distribution function (J(r)) for the 519-atom model for pairs of atoms at least one member of which is interior to a sphere containing 316 fully bonded atoms. The analogous J(r) for the 201-atoms model is considerably noisier; however, it indicates that the result below should apply to both models. In fig. 6 the J(r) is compared with separate contributions from second neighbors and third neighbors which are not also second neighbors. (There is no overcounting so the sum of the separate contributions can never exceed the total J(r).) According to this figure, out to a distance of 2.5 bond lengths the features in J(r) are due almost entirely to the second and third nearest neighbors. Fig. 7a is a decomposition of the second neighbor peak into contributions from second neighbors joined to the origin atom by five-, six- (but not five-), and seven- (but not five- or six-) fold rings. As expected from the analysis of diamond cubic structures, the six-fold ring distribution is centered about 1.63 bond lengths. The distribution of second nearest neighbors from five-fold rings, however, is significantly shifted towards the origin, thus creating the left shoulder on the second nearest neighbor peak. Furthermore, the shift is due to bent five-fold rings, since the change in the distance of second nearest neighbors of planar five-fold rings from that of six-fold rings should be negligible small. (Note that this is not found in amorphonic planar five-fold ring structures [ 11 ].) The second nearest neighbors associated with higher (> six-fold) rings contribute slightly to the right shoulder on the second nearest neighbor peak, although third nearest neighbors are observed to have the effect. In fig. 7b the third nearest neighbor distribution is decomposed. Third nearest neighbors of fivefold rings are equivalent to the second nearest neighbors; their distribution has already been exhibited. A six-fold ring third nearest neighbor distribution is observed near the 1.92 third neighbor distance found in the diamond cubic structure. The
210
P. Steinhardt et al., Relaxed continuous random network models
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0.5
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vv
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z.o
z.5
3.0
(band- lengths)
Fig. 6. Contributions of second and third neighbors (reached by traversing two and three bonds respectively) to the radial distribution function centered on a sphere of 316 fully bonded atoms in the 519-atom relaxed model. There are exactly 12 second neighbors. However, since second neighbors which are members of five-fold rings are not counted as third neighbors, there are fewer than the 24 third neighbors found in the diamond cubic structure.
2nd~, 2.0 !
40
(a)
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~
3rd
20
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(bond
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Fig. 7. (a) Decomposition of the second neighbor peak in J(r) into contributions from members of five-fold rings (dashed line), six-fold (but not five-fold) rings (full line), and higher rings (dotted line). The five-fold ring contribution is shifted inward because most five-fold rings are slightly bent. (b) Decomposition of the third neighbor contribution of J(r) into parts from members of six-fold rings (full line), seven-fold (but not six-fold) rings (dotted line), and eightand higher-fold rings (dot-dashed line).
peak is reduced in area from exactly 12 in d i a m o n d to 4.8 in the CRN. This is due to the loss of third nearest neighbors in five-fold rings and the pushing o u t o f neighbors in seven-fold rings. However, the six-fold ring c o n t r i b u t i o n added to that o f the seven-fold rings produces the small peak found b e t w e e n 1.9 and 2.0 b o n d lengths in
P. Steinhardt et al., Relaxed continuous random network models
!
I
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211
I
200
.'".,,.
J(r) I00
•
o 0
I
2
Dis/once
I
I
3
4
5
(bond lengths)
Fig. 8. Comparison of the radial distribution function for the 519-atom CRN model (corrected for finite size) with that for diamond cubic (Gaussian broadened). The broadening factor for the diamond cubic J(r) increases linearly with distance. Note the suppression of the peak at 1.9 and the enhancement of the peak at 2.4 in the CRN model relative to diamond cubic. These effects are explained by the presence of five- and seven-fold rings.
200 d(r)
CRN
I00
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o
0
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Distonce
i
3
4
5
(bond lengths)
Fig. 9. Comparison of the radial distribution function for the 519-atom CRN model (corrected for finite size) with that for wurtzite (Gaussian broadened). A mixture of eclipsed and staggered configurations such as are present in wurtzite seems to explain the broad peak at 3.1. Five- and seven-fold rings, which are not present in wurtzite, seem to be required to explain the behavior in the region from 1.7 to 2.5 bond lengths.
40 /
,
,
!
F
,
,
experiment
0
~ " ~ I
0
I
l
I 2 Distonce
I
----I
3 4 (bond lengths)
Fig. 10. Experimental O(r) for sputtered amorphous germanium from the data of Graczyk and Chaudhari. Units are the same as in fig. 1, with the bond length taken as 2.45 A.
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P. Steinhardt et al., Relaxed continuous random network models
both the model (and experimental [9])J(r)'s. The distribution of 'other' rings is peaked near the 2.52 bond length radial distance associated with 'trans' bonds in completely staggered structure. The seven-fold third nearest contribution is relatively diffuse but does peak up around 2.4, accounting for the inward shift of the 'trans' peak relative to diamond. It might be noted that there are very few third neighbors belonging to six-fold rings in the vicinity of 1.67 distance associated with eclipsed bonds. Essentially all of the eclipsed bonds in the CRN are thus associated with five-fold rings. In figs. 8 and 9 we show a comparison of the finite-size-corrected J(r) for the 519-atom model with broadened J(r)'s for diamond and wurtzite, respectively. The finite size corrected J(r) was obtained by dividin$ the measured J(r) (see fig. 4) by Jsphere(r) and then multiplying the result by 47rr~Po where Po is the density in atoms/(bond length) 3. The diamond and wurtzite J(r)'s are sums of appropriately weighted Gaussian broadened contributions associated with crystal neighbor distances. The broadening factor increased linearly with distance and was chosen so as to give a second peak of the same width in the CRN results. From these comparisons, we may note that a broadening, together with a mixture of eclipsed and staggered configurations such as are present in wurtzite (in a ratio of 1 to 3), is sufficient to explain the broad fourth peak around 3.1 bond lengths in the CRN result. As noted in the previous analysis, five- and seven-fold rings are necessary to account for the positions and relative magnitudes of the features between about 1.7 and 3.5 bond lengths.
4O
~ 3o 0
o20
0
2
4 Dislonce
6
8
i0
(A)
Fig. 11. Comparison of CRN model radial distribution functions (J(r)'s) with a measured J(r) for amorphous germanium (see ref. [9], fig. 14 curve (b)-high-temperature deposited film). The bond length used for the CRN results was taken as 2.47 A. The CRN results are corrected for the finite size of the models by a procedure which dePends only on the predetermined density of the models.
P. Steinhardt et al., Relaxed continuous random network models
I
I
i I
J 2
I
1
I
• 3
4
5
I
I
I
I
6
7
8
9
213
50 ~ 40 E o ~ 30
&
-" 16 0 0
A Distonee
I0
(A)
Fig. 12. Comparison of model radial distribution functions J(r) with experiment for Ge, as in fig. 11, but including a Gaussian broadening of the model J(r)'s as explained in the text, to allow for thermal, instrumental and termination broadening. Heavy line is the experiment, dashed line is the 519-atom model, and thin line is the 201-atom model.
3.8. Comparison with experiment In fig. 10 we reproduce the experimental measurements of Chaudhari and Graczyk [7] for the G(r) of amorphous germanium. This is to be compared with the CRN G(r) results shown in fig. 5. In fig. 11 we show a comparison of the finite-size-corrected J(r) for the 519- and the 201-atom models with the measured J(r) of Temkin et al. [9] for a sample deposited at high temperature. For this comparison the bond length was taken as 2.47 A, which is the value given in ref. [9]. Apart from the choice of bond length, there are no adjustable parameters in these comparisons. Finally, in fig. 12 we show the same comparison as in fig. 11, except that the model J(r)'s have been convoluted with Gaussian broadening functions. For the nearest neighbor peak a Gaussian of width 0.28 A was used. For the rest of the J(r), a Gaussian of width 0.37 A was used. These values are those which were observed by Temkin et al. [9] for a powder sample of crystalline Ge at the same temperature and using the same apparatus. This procedure therefore accounts for termination and instrument broadening and also for thermal broadening, if the interatomic forces are not greatly different in the crystalline and amorphous phases. The agreement between the J(r)'s for the two different CRN models and experiment is seen to be quite good in the region from 0 to 10,8, and indeed presents a formidable challenge to rival models. The second and third prominent peaks are somewhat sharper for the models than for the experimental curve, which may be due to the oversimplified force constant scheme used to relax the models. The small peak near 4.8 A in the model curves does not correspond precisely to the structure evident in the experimental curve, but it is possible that termination effects may have somewhat distorted the experimental curve. Remaining discrepancies are hardly significant and might be accounted for by defects in the 'ideal' network; such defects include cracks and voids [ 19, 20] as well as some number of dangling bonds [21].
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P. Steinhardt et al., R e l a x e d continuous random n e t w o r k models
Acknowledgments We thank D.E. Polk and D. Boudreaux for generously providing us with the coor dinates of their 519-atom model. We are further indebted to J.F. Graczyk and P. Chaudhari for their electron diffraction data and R. Temkin, G.N. Connell and W. Paul for their X-ray diffraction data. We also thank S.C. Moss, M.F. Thorpe and M.G. Duffy for helpful comments. This research was conducted as part of the Yale Department of Engineering and Applied Science Summer Program for College Juniors.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [ 10] [11] [12] [13] [14] [I5] [16] [17] [18] [19] [20] [21]
D.E. Polk, J. Non-Crystalline Solids 5 (1971) 365. 13. TurnbuU and D.E. Polk, J. Non-Crystalline Solids 8-10 (1972) 19. D.E. Polk and D.S. Boudreaux, Phys. Rev. Lett. 31 (1973) 92. N.J. Shevchik and W. Paul, J. Non-Crystalline Solids 8-10 (1972) 381. D. Henderson, in: Computational Solid State Physics, ed. F. Herman (Plenum, New York, 1972) p. 175. P. Steinhardt, R. Alben, M.G. Duffy and D.E. Polk, submitted to Phys. Rev. J.F. Graczyk and P. Chaudhari, Phys. Star. Sol. (b), to be published. R. Temkin, G.N. Connell and W, Paul, Advances in Physics, to be published. G.N. Connell, W. Paul and R. Temkin, Advances in Physics, to be published. W. Paul, R. Temkin and G.N. Connell, Advances in Physics, to be published. R. Grigorovici and R. Manuela, J. Non-Crystalline Solids 2 (1969) 371. M.L. Rudee and A. Howie, Phil. Mag. 25 (1972) 1001. M.G. Duffy and D.E. Polk, to be published. P.N. Keating, Phys. Rev. 145 (1966) 637. See R. Alben, J.E. Smith, J.R., M.H. Brodsky and D. Weaire, Phys. Rev. Lett. 30 (1973) 1141. S.C. Moss, R. Alben, D. Adler and J.P. De Neufville, J. Non-Crystalline Solids 13 (1973/74) 185. R. Alben and D. Weaire, submitted to Phys. Rev. This is only one of several possible ways of doing the finite size correction. See also J.F. Graczyk and P. Chaudhari, Bull. Am. Phys. Soc. II 18 (1973) 420. S.C. Moss and J.F. Graczyk, Phys. Rev. Lett. 23 (1969) 1167. G.S. Cargill III, Phys. Rev. Lett. 28 (1972) 1372. M.H. Brodsky, R.S. Title, K. Weiser and G.D. Petit, Phys. Rev. BI (1970) 2632.
RELAXED CONTINUOUS RANDOM NETWORK MODELS (I). Structural characteristics* P. S T E I N H A R D T * * , R. A L B E N and D. W E A I R E Department of Engineering and Applied Science, Yale University, New Haven, Connecticut 06520, USA Received 29 August 1973
Two elastic-energy-relaxed continuous random network (Polk) models for tetrahedrally bonded amorphous semiconductors have been obtained: a 201-atom model built entirely at Yale and a 519-atom model relaxed from a structure built by Polk and Boudreaux which originated at Harvard. In relaxing the coordinates to minimize the total energy the Keating potential was used for the interatomic interactions. The models are analyzed in terms of density, elastic distortion energy, elastic constants, numbers of five-, six- and seven-fold rings, distribution of dihedral angles, and radial distribution functions. We find that, despite their different origins, the models have essentially identical characteristics. Our principal conclusions are as follows: (a) The density of the CRN model is, to within 1%, that of diamond cubic. (b) The bulk modulus is about 3% lower than that for the diamond cubic structure and the shear modulus lies between the two diamond cubic shear moduli. (c) There are, to within -+ 10% (and with corrections for surface effects), 0.38 five-fold, 0.91 six-fold and 1.04 seven-fold rings per atom. (d) For a reasonable value of the bond bending force constant, rms bond length distortions are about 1.0% and bond angle distortions are about 7.0°. (e) The radial distribution function agrees very well with experiment for all four principal peaks.
1. Introduction The reasonableness o f the c o n t i n u o u s r a n d o m n e t w o r k ( C N R ) m o d e l for the structure o f tetrahedraUy c o o r d i n a t e d a m o r p h o u s s e m i c o n d u c t o r s is d e m o n s t r a t e d by Polk [ 1 ], w h o showed that it is possible to c o n n e c t rather rigid plastic and metal tetrahedral m o l e c u l a r units in a structure with little b o n d strain, no unsatisfied bonds and no perceptible long-range order [ 1 - 3 ] . F o r several years, our k n o w l e d g e o f the characteristics of these structures was based on laborious direct m e a s u r e m e n t s o f a single m o d e l o f 440 units built by Polk [ 1 ]. Models built by c o m p u t e r m e t h o d s [4, *Research supported in part by NSF.
**Present address: Dept. of Physics, California Institute of Technology, Pasadena, California 91109.
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P. Steinhardt et al., Relaxed continuous random network models
5] have yet to achieve the small degree of distortion and absence of dangling bonds characteristic of Polk's model. Recently, the coordinates for the original Polk model, expanded to 519 units, were obtained by Polk and Boudreaux [3] and refined so as to equalize nearest neighbor distances. The availability of numerical coordinates made possible a more complete characterization of the model with extremely encouraging initial results. Very recently, the present authors have relaxed the coordinates of ~ e 519-atom model so as to minimize an elastic energy which depends on both bond-length and bond-angle distortions [6]. We have also used our relaxation procedure to rapidly obtain coordinates for a new 201-atom model. In this paper we report a detailed characterization of the two models with regard to density, elastic distortion energy, elastic constants, numbers of five-, six- and seven-fold rings, distribution of dihedral angles, radial distribution function and G(r) (including a comparison with experimental data) [7-10] and contributions of neighbors belonging to rings of different sizes and of eclipsed and staggered bond character to the form of the radial distribution function. We find that the two models are essentially identical in their characteristics, which leads us to conclude that the 'ideal' CRN structure may be a reasonably welldefined concept. Also, when the CRN radial distribution functions are compared with experimental data for amorphous germanium prepared by any of the common methods, the agreement is excellent for all four principal peaks and some more subtle features as well. This agreement is far better than has ever been demonstrated for microcrystalline [11] or amorphonic models [12]. Thus we believe that the analysis of the CRN model which we present below should form the basis for a more complete understanding of the structure of amorphous group IV semiconductors. Thepaper:is divided into three sections. In sect. 2, we describe our procedure for relaxing to an accurate minimum elastic energy structure. In sect. 3, we present our characterization of the 519-atom and the 210-atom relaxed structures. Paper II, by J.F. Graczyk and P. Chaudhari, which is a continuation of this paper, is concemed with the diffraction properties of the models. An analysis of relaxed CRN models has been independently performed by Duffy and Polk [13].
2. Relaxation
procedure
A numerical procedure has been developed which simultaneously relaxes both bond-length and bond-angle distortions in tetrahedrally coordinated continuous random network models by minimizing the Keating elastic energy [ 14] expression:
3 ~{3 l(~t) ~ (rli'rli' + ~d2) 2 V= d162 3__~ l,i ~ (rli'rli-d2)2 +-8
(1)
where t~ and/3 are the bond-stretching and bond-bending force constants, respectively. The first sum in the expression is on all atoms I and their four neighbours specified
P. Steinhardt et al., Relaxed continuous random network models
201
by i; the second sum is on all atoms and pairs of distinct neighbors; and rli is the vector from I to its i'th neighbor. Provided with a table of nearest neighbor relationships (not altered by the relaxation process) and a listing of the approximate equilibrium coordinates before relaxation, the refinement program moves each atom, in turn, to essentially the exact position of equilibrium under the bond stretching and bending forces due to its nearest and next-nearest neighbors. Specifically, for each repositioning of an atom, the force (vector) on that atom and the force derivative (tensor) are calculated, after which a matrix inversion gives the coordinates of a point of zero force under the Keating potential. The coordinates are tested to insure that they are not positions of m ~ i m u m energy, although no direct check prevents their being saddle points of the energy. The repositioning of each atom in the model is repeated over many cycles until convergence to equilibrium is achieved for the entire structure. From starting coordinates within 1/2 of a bond length of the equilibrium positions, angles and bond lengths were found to converge to 1 part in 103 of their final values (as determined by very long iterations on test cases) after 25 cycles for the models considered. The coordinates used in most of the analysis were obtained after 50-100 iteration cycles and are estimated to be accurate to about 0.05% for individual bond lengths and 0.05 ° for bond angles. Thus, our estimates of quantities that are averages over all atoms in the structure should be sufficiently accurate for practical purposes. To analyze the properties of relaxed continuous random network structures, the coordinates for two CRN models were obtained using the numerical procedure described above. A combination of the full relaxation procedure and a partial relaxation procedure, in which a central core of atoms was held f'Lxedwhile the added atoms were relaxed from automatically guessed start positions, was used to build a 201-atom structure from a 21-atom roughly measured seed. To provide the necessary table of neighbor relationships (extended to include the new atoms), a reference hand-built model which paralleled the relaxed structure was constructed from plastic and metal units of the same type used by Polk [1]. Coordinates for a second relaxed random network structure were calculated from the 519-atom model reported in ref. [3]. In that model, bond-length deviations (only) had been minimized in the structure. However, the condition of equal bond lengths represents two constraints per atom (less when a surface is present) while there are three degrees of freedom per atom. Thus, coordinates from the bond-length equalization procedure depend to some extent on the initial measurements of the physical model and have rather larger bond-angle deviations than necessary. By contrast the relaxed coordinates are determined by the form of the elastic energy expression and depend on the properties of the plastic and metal units used to build the model only insofar as these influence the connectivity. In relaxing both models the ratio [J/a was taken as 0.2, which is in a range of plausible values as indicated by measurements of phonon frequencies in diamondcubic silicon and germanium [ 15]. Most properties of the models (except for the absolute distortion energy) were quite insensitive to the particular choice of/~. This is
202
P. Steinhardt et al., Relaxed continuous random network models
fortunate since a simple bending force cannot be expected to accurately describe complex distortions in either silicon or germanium [16].
3. Characteristics of random networks An intensive study has been made of the basic physical characteristics of the two models with the intent of discovering what features are common to all such structures. The characteristics considered are in turn: density, elastic distortion energy, elastic constants, ring statistics, dihedral angle distribution, radial distribution function and corrected G(r), and the contribution of rings and neighbors to different features of the corrected radial distribution function, Finally, we compare the CRN G(r)'s and radial distribution function with experimental results for electron [7] and X-ray [8-10] diffraction for amorphous germanium. 3.1. Density
To find the density of each of the two models required a characterization of density fluctuations and possible special properties of atoms near the surface. Our procedure was to find the radius, r o (measured in bond lengths), of the largest sphere about the center-of-mass which contained only fully bonded atoms and then to plot the average density (number of atoms/sphere volume) for spheres with radii from r o - 1 to ro+l. In fig. 1 we show the result for the 519-atom model. The plot for the 201-atom model was similar except that fluctuations were about twice as large as for the 519atom model. Using for the average nearest neighbor distance the value of 2.45 )~ appropriate for crystalline Ge, densities of 4.40 + 0.10 X 1022 atoms/cm 3 for the 201atom structure and 4.41 + 0.06 X 1022 atoms/cm 3 for the 519-atom structure were determined. The diamond cubic density is 4.42 × 1022 atoms/cm 3. Thus, the density of the CRN model is, to within 1%, that of diamond cubic, a conclusion in agreement with Polk's analyses [1,2]. 3. 2. Distortions and distortion energy
CRN structures have characteristic bond-length and bond-angle deviations from the perfect tetrahedral configuration. In the 201-atom model a root-mean-square bond length deviation of 0.80% and root-mean-square angular deviation of 6.66 ° over the entire model were found. For the relaxed 519-atom network, a root-meansquare bondiength deviation of 1.04% and a root-mean-square angular deviation of 7.10 ° were measured. In comparison, before the full relaxation procedure had been applied to the Polk and Boudreaux coordinates, the root-mean-square bond length deviation was O. 16% and the root,mean-square angular deviation was 9.25 °. Since the stored elastic energy of a structure is approximately proportional to the square of the rms angular deviation, the relaxation process accounts for about a 40% de-
P. Steinhardt et al., Relaxed continuous random network models
4.6xj
40
I
I
I
I
I
I 4.0
I 4.5
I 5.0
I 5.5
203
6.0
Radius
Fig. 1. Average density in atoms/cm 3 for spheres of different radii (in bond lengths) about the center of mass of the 519-atom relaxed model. The bond length was taken as 2.45 A which is the appropriate value for germanium. The dashed line represents the diamond cubic density. The density deficit followed by a density increase and fall off in the region from 5 to 6 bond lengths is a surface effect.
crease in the total stored energy in the unrefined 519-atom model. Using the value = 3.87 × 104 dyn/cm appropriate for germanium, the distortion energy for the 201-atom is found to be 19.7 meV/electron; for the relaxed 519-atom model, 23.1 meV/electron; and, for the unrelaxed 519-atom model, 36.6 meV/electron. By contrast, for a widely used handbuilt periodic model due to Henderson (private communication) the energy for relaxed and unrelaxed coordinates correspond to 77 meV/ electron and 82 meV/electron respectively. The somewhat smaller distortions of the 201-atom model relative to the 519atom model are partially due to its relatively larger number of surface atoms (which are under fewer constraints). If only the 52 atoms within the sphere of fully bonded atoms are considered, the distortions for the 201-atom model become 0.97% for bond lengths and 6.99 ° for bond angles. We cannot eliminate the possibility that distortions continue to grow with model size; however, from our experience we believe (in agreement with Polk [ 1 ] ) that the models (which are already fairly large; the 519-atom model corresponds to a 28 A diameter germanium cluster) can be extended indefinitely without increasing the distortions. For/3/ct = 0.2, a good guess f o r the extrapolated distortions o f an infinite 'Meal' model is 1% f o r bond lengths and 7° f o r bond angles. 3. 3. Elastic constants
The elastic constants of the model were determined by applying different types of strains and allowing fully bonded atoms to relax so as to minimize the energy while surface atoms (those with at least one unsatisfied bond) were held fixed. Strains of - 0 . 0 2 , - 0 . 0 1 , 0 , +0.01 and +0.02 were applied and the curvature of the energy was determined both from the least-squares parabolic fit for all five measurements and from the second difference of the middle three measurements. Results
204
P. Steinhardt et aL, Relaxed continuous random network models
Table 1 Elastic constants for the CRN model compared to those for the model without internal relaxation and those for a diamond cubic model with the same force constants (a = 0.387 X 10s dyn/ cm, ~/c~= 0.2) and density (4.216 X 1022 atoms/cma).
CRN model CRN model (no relaxation) Change on permitting relaxation Diamond cubic
B
M
Bulk modulus (X 104 dyn/cm 2)
Shear modulus (x 104 dyn/cm 2)
7.08 ± 0.04 7.26
3.80 ± 0.06 6.03
-2.5% 7.30
-37.5% 2.74 a) 4.56 b)
a) for z2-{/2(x 2 +y2) type shear. b) for xy-type shear. were obtained for both models for uniform compression, extension along x and compression alongy and z, and for x y type shear. For the 201-atom model other types of shears were applied, and tests were made on both models for interior spheres of atoms. The results are summarized in table 1. They are compared with results for a diamond cubic structure with the same density and force constants and for the CRN model without internal relaxation. The error limits give the deviations of the results for the different models and different ways of calculating elastic constants from the energy for different shears. It is seen that the results for the two models and different shears are within a few percent of each other. Thus the models are both quite similar and quite isotropic. The bulk modulus (B) for the CRN is very close to that for diamond, an unexpected result since in the limit of small ~/a it can be proved that B should approach zero for such structures (as it does for a model of Ge III [17]). The explanation for the undramatic result for the CRN with/3/c~ = 0.2 lies in the fact that the relaxed structure has close to the maximum possible volume consistent with almost equal bond lengths. The volume is thus quite insensitive to internal displacements which conserve bond lengths and the softness of such internal displacements does not significantly affect the bulk modulus. 3.4. Ring statistics
A raw count of the numbers of five-, six- and seven-fold rings in the two models yielded the results given in the first two rows of table 2. It was decided to include seven-fold rings (which have not been previously considered in CRN model investigations) since they are in no sense less 'elemental' in these structures than are six-fold rings, and examination of the 201-atom model indicated that they might be important
205
P. Steinhardt et al., Relaxed continuous random network models
Table 2 Estimates for the bulk numbers of five-, six- and seven-fold rings per atom for the CRN model compared with diamond cubic. 5-fold 6-fold 7-fold (rings/atom) (rings/atom) (rings/atom) Raw count 201-atom model 519-atom model
0.244 0.264
0.489 0.593
0.507 0.592
With broken surface bond correction 201-atom model 519-atom model
0.39 0.38
0.89 0.93
1.05 1.04
From density of centers of mass 201-atom model 519-atom model
0.3 ±0.1 0.39±0.05
0.8±0.2 0.9±0.1
1.0±0.2 1.1±0.1
From two point extrapolation to zero surface-tovolume ratio
0.32
0.85
0.82
"Best guess" values (± 10%)
0.38
0.91
1.04
Diamond cubic and wurtzite
0
2
0
in characterizing the structure. The raw count is by itself not very enlightening since there are many rings involving surface atoms which cannot be completed just because of the finite size of the model. To correct for finite size effects, three different methods were used and all gave quite similar results. The first method was based on the assumptions that: (1) if surface atoms are classified according to the number of their broken bonds (singlets, doublets, and triplets), the broken bonds of the atoms in each class are typical o f singlet, doublet or triplet bond sets inside the model; and (2) on the average, half the atoms of rings broken due to the surface lie inside the model. Using these assumptions, together with the fact that any ring broken by the surface must connect two surface bonds, the average numbers of rings belonging to the cluster but destroyed by an atom with one, two or three broken bonds were calculated. From this we obtained the following relation between the bulk number of p-fold rings per atom np (p = 5, 6, or 7), the raw count of p-fold rings N p , numbers of surface atoms with i broken bonds S i (i = 1,2, or 3), and the total number of atoms N ( N = 201 or 519): 25
3
3s
3
n5 = N 5 / ( N - S s 1 _ ~_ $ 2 _ ~_$3) ' 7
n 7 = N 7 / ( N - g S 1 - ~-~ $ 2 - ~-$3),
The results for np are given in table 2.
n 6 = N 6 / ( N - ~S 1 6 _ ~_~$2_3o~_$3),3 (2)
206
P. Steinhardt et al., Relaxed continuous random network models
1.2 F o
1.0
I f..
I
I
I
; '. .'" ............. 'l..,. . . : ~ " - .
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I
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rings
I 6
I 7
-2 o.8
g
.c_ 0.6 5-fold 0,4 <>
0.2
l 2
I 3
I 4
I 5
8
Rodius
Fig. 2. Ratio of numbers of centers of five-, six- and seven-fold rings to numbers of atoms in spheres of different radii about the center-of-mass for the 519-atom relaxed model. Vertical marks on the curves indicate the largest radius before surface effects become significant. For the second method the ratios of the numbers of mass centers of p-fold rings to the numbers of atoms within spheres of different radii about the centers-of-mass of the models were calculated and plotted against sphere radius. These ratios are expected to fluctuate widely for small radii and decrease for radii large enough to include the centers-of-mass of rings broken by the presence of the surface. As shown in fig. 2 for the 519-atom model, it was possible to identify radii large enough to contain a significant sampling of rings but small enough to avoid the surface effects. There is considerably more uncertainty for the 201-atom model (not shown). The results for np from this method are also given in table 2. The final method was based on the assumption that the ring statistics for the two models differed only because of the difference in surface-to-volume ratios. Plotting the ring statistics for the two models against the surface-to-volume ratio (N --1/3) and extrapolating to zero yielded the results indicated in table 2. Since all these methods give similar answers we conclude that the two models have essentially identical ring statistics and the 'best guess' values are as given in the table. 3.5. Dihedral angle distribution
The dihedral angle distribution for the relaxed models was found by taking all four-fold paths connecting atoms i, j, k and l, and then making a histogram of the angles between the projections o f r i / a n d rkl on the plane perpendicular to ~'k where r ~ is the displacement vector between the atoms a and/3. Because the two models had a different ntmber of atoms their distributions were scaled for comparison purposes (see fig. 3). Although the two models are entirely distinct, a remarkable similarity is found in their dihedral angle distributions. In particular, for both models there are twice as many 60 ° and 180 ° dihedral pairs as 0 ° pairs, indicating a preference in the structure for staggered band configurations. These reshlts agree with those obtained in ref. [3] for the unrelaxed 519-atom model.
P. Steinhardt et al., Relaxed continuous random network models
I
I
I
!
207
I
t-
I
I 30
o
I 6o
I 90
Dihedral
I Izo
I tsO
iI
s
leO
Angle
Fig. 3. Distributions of dihedral angles for the 519-atom model (full line) and the 201-atom model (dashed line). The scales for the two curves have been adjusted to make them of comparable heights. (There are, of course, more angles in the larger model.)
519-olom
IOO J(r) 5o
0
2
4 Distance
6
8
to
(bond lengths)
Fig. 4. Radial distribution function, J(r), uncorrected for finite size effects, for the 201- and 519-atoms models, The curves are formed by connecting the tops of histogram bars corresponding to a width of 1/50 bond length. Note that the broad peak at 3.1 bond lengths is present in both models. 3. 6. Radial distribution function and G(r ) The radial distribution functions (J(r)'s) for the models were found directly from the coordinates by forming a histogram of the distances between all pairs of atoms and normalizing for the number of atoms (see fig. 4). In order to obtain J(r)'s and G(r)'s corrected for the finite sizes of the models, the J(r)'s were compared with those for a finite sphere of uniform density. For such a sphere, the radial distribution function is given by: N [ 3 {r'~5 9 )2 , Jsph(r) = a [ 1 - 6 \ a / _ ~ _ ( r ) 3 + 3 ( r ]
(3)
208
P. Steinhardt et al., Relaxed continuous random network models
401
.
[
~"
G(r)20~
.
.
IIA
0
4
0
.
zol-atom model
~
2
3
4
,
,
,
5Jg-atom relaxed 20 G(r) 0 I
0 40
t G(E~OF 0
I
I
2
3
4
,
i
,
J~
unrelaxedsJg-otom
IIA
l
2
3
4
5
Distance (bond lengths) Fig. 5. Corrected G(r)'s for the 201-atom model and 519-atom relaxed model, both relaxed so as to minimize the Keating elastic energy with #/a = 0.2, and for the unrelaxed 519-atom model as described in ref. [3]. The units of G(r) are atoms/(bond length) 2. The curves are obtained by connecting the tops of histogram bars of width 1/50 bond length. The first peaks are truncated for clarity of presentation. where a is the radius of the sphere and N is the number of atoms. For the comparison with the two random network models, the magnitude of a was adjusted so that the ratio of J(r) to Jspb(r) approached unity as the distance from the origin became large. This agreed to two significant figures with the value of a determined from the density. The value for a was found to be 4.2 bond lengths for the 201-atom model and 5.7 for the relaxed and unrelaxed 519-atom models. Given N, a and the J(r) for each structure, we calculated a corrected G(r) given by [18]:
G(r) : [J(r) - Jsph(r)]/r,
(4)
and obtained the graphs in fig. 5. When the atoms inside the largest sphere about the center-of-mass containing only fully bonded atoms were considered, only a slight sharpening in some of the details in G(r) and an increase in the statistical fluctuations
P. Steinhardt et al., Relaxed continuous random network models
209
occurred. We surmise from this that spurious effects from partially bonded surface atoms are not significant. An inspection of the curves in fig. 5 leads us to make two observations. First, the G(r) curves for relaxed continuous random network models do exhibit significantly sharper features than those for unrelaxed models (compare the relaxed and unrelaxed 519-atom model G(r)'s). Most obvious in this respect is the narrowing of the second neighbor peak, a narrowing due to the decrease of 2.1 degrees in the rms angular deviation after full relaxation. The relaxation procedure also makes more visible the details found between the second and third neighbor peaks in G(r). Second, although the two models were constructed by different methods, their G(r) curves have in common all their prominent features and many subtler features.
3. 7. Decomposition o f the radial distribution function To better understand the prominent features of the CRN G(r)'s, we have performed a decomposition of the radial distribution function into contributions from different types of nearest neighbors. To minimize effects due to finite size without introducing a complicated decomposition procedure, we restrict attention to a radial distribution function (J(r)) for the 519-atom model for pairs of atoms at least one member of which is interior to a sphere containing 316 fully bonded atoms. The analogous J(r) for the 201-atoms model is considerably noisier; however, it indicates that the result below should apply to both models. In fig. 6 the J(r) is compared with separate contributions from second neighbors and third neighbors which are not also second neighbors. (There is no overcounting so the sum of the separate contributions can never exceed the total J(r).) According to this figure, out to a distance of 2.5 bond lengths the features in J(r) are due almost entirely to the second and third nearest neighbors. Fig. 7a is a decomposition of the second neighbor peak into contributions from second neighbors joined to the origin atom by five-, six- (but not five-), and seven- (but not five- or six-) fold rings. As expected from the analysis of diamond cubic structures, the six-fold ring distribution is centered about 1.63 bond lengths. The distribution of second nearest neighbors from five-fold rings, however, is significantly shifted towards the origin, thus creating the left shoulder on the second nearest neighbor peak. Furthermore, the shift is due to bent five-fold rings, since the change in the distance of second nearest neighbors of planar five-fold rings from that of six-fold rings should be negligible small. (Note that this is not found in amorphonic planar five-fold ring structures [ 11 ].) The second nearest neighbors associated with higher (> six-fold) rings contribute slightly to the right shoulder on the second nearest neighbor peak, although third nearest neighbors are observed to have the effect. In fig. 7b the third nearest neighbor distribution is decomposed. Third nearest neighbors of fivefold rings are equivalent to the second nearest neighbors; their distribution has already been exhibited. A six-fold ring third nearest neighbor distribution is observed near the 1.92 third neighbor distance found in the diamond cubic structure. The
210
P. Steinhardt et al., Relaxed continuous random network models
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peak is reduced in area from exactly 12 in d i a m o n d to 4.8 in the CRN. This is due to the loss of third nearest neighbors in five-fold rings and the pushing o u t o f neighbors in seven-fold rings. However, the six-fold ring c o n t r i b u t i o n added to that o f the seven-fold rings produces the small peak found b e t w e e n 1.9 and 2.0 b o n d lengths in
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P. Steinhardt et al., Relaxed continuous random network models
both the model (and experimental [9])J(r)'s. The distribution of 'other' rings is peaked near the 2.52 bond length radial distance associated with 'trans' bonds in completely staggered structure. The seven-fold third nearest contribution is relatively diffuse but does peak up around 2.4, accounting for the inward shift of the 'trans' peak relative to diamond. It might be noted that there are very few third neighbors belonging to six-fold rings in the vicinity of 1.67 distance associated with eclipsed bonds. Essentially all of the eclipsed bonds in the CRN are thus associated with five-fold rings. In figs. 8 and 9 we show a comparison of the finite-size-corrected J(r) for the 519-atom model with broadened J(r)'s for diamond and wurtzite, respectively. The finite size corrected J(r) was obtained by dividin$ the measured J(r) (see fig. 4) by Jsphere(r) and then multiplying the result by 47rr~Po where Po is the density in atoms/(bond length) 3. The diamond and wurtzite J(r)'s are sums of appropriately weighted Gaussian broadened contributions associated with crystal neighbor distances. The broadening factor increased linearly with distance and was chosen so as to give a second peak of the same width in the CRN results. From these comparisons, we may note that a broadening, together with a mixture of eclipsed and staggered configurations such as are present in wurtzite (in a ratio of 1 to 3), is sufficient to explain the broad fourth peak around 3.1 bond lengths in the CRN result. As noted in the previous analysis, five- and seven-fold rings are necessary to account for the positions and relative magnitudes of the features between about 1.7 and 3.5 bond lengths.
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P. Steinhardt et al., Relaxed continuous random network models
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3.8. Comparison with experiment In fig. 10 we reproduce the experimental measurements of Chaudhari and Graczyk [7] for the G(r) of amorphous germanium. This is to be compared with the CRN G(r) results shown in fig. 5. In fig. 11 we show a comparison of the finite-size-corrected J(r) for the 519- and the 201-atom models with the measured J(r) of Temkin et al. [9] for a sample deposited at high temperature. For this comparison the bond length was taken as 2.47 A, which is the value given in ref. [9]. Apart from the choice of bond length, there are no adjustable parameters in these comparisons. Finally, in fig. 12 we show the same comparison as in fig. 11, except that the model J(r)'s have been convoluted with Gaussian broadening functions. For the nearest neighbor peak a Gaussian of width 0.28 A was used. For the rest of the J(r), a Gaussian of width 0.37 A was used. These values are those which were observed by Temkin et al. [9] for a powder sample of crystalline Ge at the same temperature and using the same apparatus. This procedure therefore accounts for termination and instrument broadening and also for thermal broadening, if the interatomic forces are not greatly different in the crystalline and amorphous phases. The agreement between the J(r)'s for the two different CRN models and experiment is seen to be quite good in the region from 0 to 10,8, and indeed presents a formidable challenge to rival models. The second and third prominent peaks are somewhat sharper for the models than for the experimental curve, which may be due to the oversimplified force constant scheme used to relax the models. The small peak near 4.8 A in the model curves does not correspond precisely to the structure evident in the experimental curve, but it is possible that termination effects may have somewhat distorted the experimental curve. Remaining discrepancies are hardly significant and might be accounted for by defects in the 'ideal' network; such defects include cracks and voids [ 19, 20] as well as some number of dangling bonds [21].
214
P. Steinhardt et al., R e l a x e d continuous random n e t w o r k models
Acknowledgments We thank D.E. Polk and D. Boudreaux for generously providing us with the coor dinates of their 519-atom model. We are further indebted to J.F. Graczyk and P. Chaudhari for their electron diffraction data and R. Temkin, G.N. Connell and W. Paul for their X-ray diffraction data. We also thank S.C. Moss, M.F. Thorpe and M.G. Duffy for helpful comments. This research was conducted as part of the Yale Department of Engineering and Applied Science Summer Program for College Juniors.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [ 10] [11] [12] [13] [14] [I5] [16] [17] [18] [19] [20] [21]
D.E. Polk, J. Non-Crystalline Solids 5 (1971) 365. 13. TurnbuU and D.E. Polk, J. Non-Crystalline Solids 8-10 (1972) 19. D.E. Polk and D.S. Boudreaux, Phys. Rev. Lett. 31 (1973) 92. N.J. Shevchik and W. Paul, J. Non-Crystalline Solids 8-10 (1972) 381. D. Henderson, in: Computational Solid State Physics, ed. F. Herman (Plenum, New York, 1972) p. 175. P. Steinhardt, R. Alben, M.G. Duffy and D.E. Polk, submitted to Phys. Rev. J.F. Graczyk and P. Chaudhari, Phys. Star. Sol. (b), to be published. R. Temkin, G.N. Connell and W, Paul, Advances in Physics, to be published. G.N. Connell, W. Paul and R. Temkin, Advances in Physics, to be published. W. Paul, R. Temkin and G.N. Connell, Advances in Physics, to be published. R. Grigorovici and R. Manuela, J. Non-Crystalline Solids 2 (1969) 371. M.L. Rudee and A. Howie, Phil. Mag. 25 (1972) 1001. M.G. Duffy and D.E. Polk, to be published. P.N. Keating, Phys. Rev. 145 (1966) 637. See R. Alben, J.E. Smith, J.R., M.H. Brodsky and D. Weaire, Phys. Rev. Lett. 30 (1973) 1141. S.C. Moss, R. Alben, D. Adler and J.P. De Neufville, J. Non-Crystalline Solids 13 (1973/74) 185. R. Alben and D. Weaire, submitted to Phys. Rev. This is only one of several possible ways of doing the finite size correction. See also J.F. Graczyk and P. Chaudhari, Bull. Am. Phys. Soc. II 18 (1973) 420. S.C. Moss and J.F. Graczyk, Phys. Rev. Lett. 23 (1969) 1167. G.S. Cargill III, Phys. Rev. Lett. 28 (1972) 1372. M.H. Brodsky, R.S. Title, K. Weiser and G.D. Petit, Phys. Rev. BI (1970) 2632.