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The present work deals with a detailed analysis of the small-angle X-ray scattering of nanoporous atomistic models for amorphous germanium. Structures with spherical nanovoids, others with arbitrarily oriented ellipsoidal ones, with monodisperse and polydisperse size distributions, were first generated. After relaxing the as-generated structure, we compute its radial distribution function, and then we deduce by the Fourier transform technique its X-ray scattering pattern. Using a smoothing procedure, the computed small-angle X-ray scattering patterns are corrected for the termination errors due to the finite size of the model, allowing so, for the first time at our best knowledge, a rigorous quantitative analysis of this scattering. The Guinier’s law is found to be valid irrespective of size and shape of the nanovoids over a scattering vector-range extending beyond the expected limit. A weighted combination of the Guinier’s forms accounts for well the nanovoid size distribution in the amorphous structure. The invariance of the Q-factor and its relationship to the void volume fraction are also confirmed. Our findings support then the quantitative analyses of available small-angle X-ray scattering data for amorphous germanium.

Amorphous semiconductors have been the subject of extensive experimental and theoretical investigations. Most of the interest has been focused on the tetrahedrally-coordinated amorphous germanium (a-Ge) due to its simple chemical bonding and composition on the one hand, and its potential applications in the ﬁelds of microelectronics and energy-conversion technologies on the other hand. As with all materials, the microstructure controls the electrical as well as the optical properties, so understanding and controlling the structure of a-Ge is crucial to its technological applications.

Unlike the crystalline phase, there is no experimental technique available that can determine the coordinates of atoms in a-Ge. Direct experimental data about atomic structure in amorphous materials were essentially limited to structure factors derived from X-ray, electrons, or neutron diffraction experiments [

Numerical modeling technique has been widely used to simulate the structure of a-Ge. Most of the effort has been devoted to the analysis of its homogeneous structure; the generally accepted topological model is the socalled continuous random network (CRN). In this model, the main building blocks of the material are the same as in its crystalline counterpart, i.e. tetrahedra of Ge, but unlike in a perfect crystal, these blocks can be randomly oriented and connected, allowing “play” in atomic bondlengths and angles. Several approaches have been proposed for representing a homogenous tetrahedrally coordinated CRN models (see [22-24] and references therein). Most of these networks successfully reproduce the socalled wide-angle scattering data, i.e. for scattering vectors upper than 1 Å^{−1}. Some numerical simulations dealing with nanovoids in a-Si have been also reported in the literature [25-31], but only two, to the best of our knowledge, are interested in the small-angle scattering [25,26]. Biswas et al. [^{5} atoms) to simulate the long-wavelength limit of the structure factor of amorphous silicon. Despite this huge model size, however, the scattering-vector threshold has moved very little (0.05 Å^{−1} vs 0.1 Å^{−1} for 4096-atom model), and the most remarkable is the persistence of the small-angle spurious ripples in the structure factor. The question that arises is then: how to make a good estimate of the scattered intensity free of finite model-size effects?

In the present work we reexamine the structural properties of atomistic models for a-Ge with nanovoids. We pay particular attention to the X-ray small-angle scattering. We consider structural models larger and more realistic than those considered in our previous work. We propose a simple way to get rid of finite model-size effects. A quantitative analysis of the estimated small-angle X-ray scattering is performed and comparison with available experimental data is made.

The present investigations are based on the class of CRN models generated with the Wooten-Winer-Weaire bondswitching algorithm by Barkema and Mousseau [

In order to obtain the equilibrium coordinates of these structures, a relaxation procedure is needed to minimize the deformation energy of the system. In the present work, the anharmonic Keating model proposed by Rücker and Methfessel [

The first sum in this expression is on all atoms i in the supercell and their nearest-neighbors specified by j. The second sum is on all atoms i and pairs of distinct neighbors. and are the vectors connecting atom i with its first-neighbors j and k, respectively. R_{0} = 2.45 Ǻ is the unstrained length of the Ge-Ge bond. The force constants α and β essentially describe the bond-stretching and bond-bending restoring forces, respectively. Their dependence laws are given by [

The force constants α_{0} and β_{0} are treated as empirical parameters; they have been fixed at their crystalline values 42.09 N∙m^{−}^{1} and 5.72 N∙m^{−}^{1}, respectively [

Now, given the equilibrium coordinates of all the atoms of the relaxed network, its structural characteristics can be easily computed such as the pair correlation function, the average bond-length, the average bond-angle, the average coordination number, the macroscopic mass density and the static structure factor. The partial pair correlation function, g(r), is defined by the relationship:

where dn(r) is the average number of atoms lying within the spherical shell of radii r and r + dr centered at an atom taken as the origin, and n_{0} is the macroscopic number density of the model. Given the ﬁnite sizes of our structural models, their computed g(r)s are corrected for the supercell size effects. From the computed g(r) we deduce an important structural characteristic of a-Ge, which is directly obtainable from scattering experiments, namely the reduced scattering intensity, F(k), or the structure factor, S(k), defined as:

where G(r), called the reduced radial distribution function, is defined by:

The integral in Equation (4) reminds us of the onedimensional Fourier sine transform of G(r). This transformation between real and reciprocal spaces is only perfect if G(r) is known for an infinite range of r values; in practice, this is obviously not the case and the data are terminated at some finite. This is equivalent to G(r) being multiplied by a modification function, M(r), where M(r) = 1 for and 0 for; the effective result is thus:

or, in a more condensed form:

where is the Fourier cosine transform of M(r) and * stands for he convolution product. This leads to the introduction of termination errors in the “perfect” Fourier transform F(k), which for the step-function M(r) given above is equivalent to features in F(k) being convoluted with the Fourier cosine transform of M(r), i.e. the sinc function. The effect of this is twofold: a loss in resolution is incurred since wave-numbers in reciprocal space less than are lost, resulting in a features broadening of, and spurious termination ripples at values are introduced (where n is an integer), which decay with increasing k.

Increasing these ripples migrate down but at the expense of an increase in amplitude. The usual procedure used to minimize termination errors employs a damping factor which multiplies the finitely ranged G(r) in the Fourier transform [Equation (6)], replacing a sharp discontinuity at by a smoothly varying function. The most used factor is the Lorch function defined by [

This form reduces the termination ripples but at the expense of signal distortion such as broadening, reduction in the intensity and, most interestingly, downward shift. It should be noted here that these effects are noticeable in the range of small values of k, i.e. in the small-angle scattering range which interests us most in this work. Other sources of noise, always related to the finite size of the atomistic model, contribute to a supplementary contamination of this part of the signal. Indeed, a deviation, even infinitesimal, from the exact value of the macroscopic density of the atomistic model leads to additional spurious ripples along the scattering pattern; in addition, statistical errors arising from the computation of radial distribution function are reflected in the Fourier transform in extra wrinkles too. The reduced scattering intensity can then be reformulated as follows:

where the function B(k) reflects the noise we discussed above. The question one might ask at this level is how to estimate the function F(k) from?

We limit ourselves here to the small-angle scattering range where one expects a monotonically varying intensity. The function B(k) can complicate the task if it had features reflecting correlations in r-space. But this is not the case, fortunately, in our problem. Indeed, the noise sources we discussed above relate to numerical problems and have nothing to do with the structural properties of the atomistic model in question. Thus, we expect that B(k) modulates the function without any distortion. The problem of finding F(k) reduces then to a problem of smoothing the noisy data. So, we propose a least square fitting procedure. It consists to, first, choose a function F(k), then, perform its convolution by and, finally, minimize the root-mean-square deviation between the convoluted function thus obtained and the raw data. It should be noted here that according to Equation (6) the function F(k) must be chosen to be odd. What form do we choose for F(k)?

The choice of the form of F(k) is dictated in fact by that of the X-ray scattering intensity I(k), given that these two quantities are related to each other. Indeed, I(k) is expressed in terms of the reduced correlation function G(r) as follows [

where f(k) is the atomic scattering factor. Now, according to Equation (4), this relation can be rewritten as:

which gives:

A simple phenomenological model describing the dependence of f on the scattering vector for Ge atom is available [

with K_{1} = 31.53, K_{2} = 0.13 and K_{3} = 2.10. For a homogenous structure such as the CRN, it is well established that I(k) decreases slowly toward “zero” as k tends to zero; thus, a polynomial form can be a good representation of the small-angle scattering intensity. Structural heterogeneities such as nanovoids in a homogenous matrix can be treated as nanoscatterers leading to smallangle scattering. In the limit of low concentrations, the small-angle scattering intensity is expected to obey the Guinier’s law [

where R_{G}, called the radius of gyration, represents the average size of electron density fluctuations in the material. For polydispersed widely separated centers, the relation (14) can be generalized as follows:

Theoretically, the Guinier approximation is the asymptotic behavior of I(k) at very small values of k; it is usually valid at k-range such that. Beyond this limit, the expression of I(k) is unknown, but it is expected to be a simple monotonically decreasing function with increasing k. We propose here to reproduce the small-angle scattering intensity I(k) by a weighted sum of Gaussians that can be formulated as follows:

where a_{i} can be positive or negative. The number N of the Gaussians and their characteristics are adjustable parameters. Such form must give the Guinier’s law at very small k-values; moreover, we have verified that it reproduces very nicely simple polynomial functions and consequently the scattering intensity due to atomic centers at k < 1.5 Å^{−1}.

The original CRN of Barkema and Mousseau [

Standard measures of the short-range disorder for our structural models, characterized by the bond length, bond angle and dangling bonds density are given in ^{−1}. These curves were fitted to the model function given by Equation (14) in the k-range extending from to 1.5 Å^{−1}. The results of the best fits are displayed in _{N} and b_{N}, and the values obtained in the preceding step for the first (2N−2) parameters. The procedure is stopped when the rms deviation stabilizes. The best fits shown in

^{(a)}monodispersed spherical nanovoids; ^{(b)}polydispersed spherical nanovoids; ^{(c)}monodispersed arbitrarily oriented ellipsoidal nanovoids.

^{*}Weighted diameter of gyration.

Once we have the estimated function F(k) we can now easily deduce the estimated scattering intensity per atom, I(k), according to the relation (10). ^{2} [

The integration in Equation (17) was truncated to 1 Å^{−1} for reasons which we will come back again later. The Guinier plots of versus k^{2} for some nanoporous models are displayed in

From the last column of

As shown in the 4th column of

Now we turn our attention to the pair correlation functions displayed in

figure shows that the g(r)s of nanoporous models are indistinguishable from that of the CRN. They reproduce very well the overall aspect of the experimentally derived pair correlation function for pure a-Ge over the experimentally accessible range 0 - 10 Å. They show that only the two first neighbor shells, at ~2.46 and ~4 Å, are well defined (firstand second-nearest neighbors, respectively). Beyond 5 Ǻ, the correlation between pairs of atoms becomes relatively much weaker, and virtually disappears from ~12 Å. A careful examination of this figure shows that nanoporous models exhibit a pair correlation function more intense than that of the CRN just over the pair-separation range of the order of the nanovoid radius. This fact is well illustrated in the inset showing a direct

comparison of the g(r)s computed for the CRN and one nanoporous model based on it denoted by NPN7 in

We arrive now to the reduced scattering intensities com-

puted for our structural models and shown in ^{−1}. This figure illustrates well the truncation effects and the noise we discussed in the computation method section. The apparent effects are spurious ripples along the k-range. The use of damping function such as the Lorch function effectively reduces these spurious effects; the ripples disappear almost completely from the wide-angle scattering data, i.e. for k > 1 Å^{−1}, but persist at very small values of k and increase in amplitude as k goes to zero. Moreover, a small downward shift of the small-angle features has been shown but it is more obvious in the computed structure factors as illustrated in the inset of this figure.

From this figure, we note that the wide-angle scatter-

ing data are practically insensitive to the presence of voids in a-Ge network. This is not the case, however, for the small-angle scattering data. Indeed, despite the spurious ripples, one can identify an evolution of the overall aspect of at small k-values by introducing nanovoids of progressive size and concentration in the CRN. One can foresee, in particular, a large feature that grows in the range 0 - 0.5 Å^{−1} with increasing the void size and/or the void volume fraction. This expected smallangle feature, free of truncation errors and noise, has become more evident through our Gaussians fitting procedure whose results are displayed in

Usually in a many-particles scattering problem, interference phenomena are expected to be observed in the scattering pattern. Our simulated scattering patterns displayed in ^{−1} and so not accessible by our finite size structural models. These observations lead us to assert that the estimated reduced scattering intensities F(k) on the 0 - 1 Å^{−1} k-range are devoid of interference phenomena.

We now turn to the discussion of the computed scattering intensities I(k) displayed in ^{−1}, contrary to what is expected. In presence of nanovoids in a-Ge networks, the scattering intensity increases rapidly from 1 Å^{−1} practically and tends to a finite value at k = 0. The latter is found to increase rapidly with increasing the void size and/or the void volume fraction. The SAXS due to nanovoids in a-Ge overlap thus with the scattering by atomic centers of the host matrix, but do not exceed 1 Å^{−1}.

A comparison with available SAXS data has been made in this figure. The inset shows three measured scattering patterns reported by Shevchik and Paul [

Our simulated SAXS patterns adjusting for the effects of truncation and noise are now ready for a rigorous quantitative analysis. The Guinier plots of versus k^{2} are displayed in ^{2}, is performed over the same k-range (0 - 0.447 Å^{−1}) and the results are shown by dashed lines in the same figure. From this comparison, we note that the extent of the range on which the relation is linear depends on the void size; it shrinks with increasing void size. The maximum extent corresponds thus to the smallest void size detectable by SAXS measurements. As we will show in the following, this corresponds practically to the smallest void radius examined in our present work, i.e. ~5 Å. We expect thus a maximum extent of ~0.7 Å^{−1}. From the results of the linear regression fits, and according to Equation (14), the diameters of gyration (2R_{G}) in our nanoporous models with monodisperse distribution of nanovoids were extracted and reported in the 6th column of

It is known that the Guinier approximation is usually valid at k-range such that R_{G}k < 1 [21,34]. Our present simulation shows, however, that this law holds even well beyond this limit. For the largest nanopores examined in our present work, this limit reaches 2. It is worthwhile to note here that several of the small-angle scattering studies on a-Si and a-Ge also fitted data with the Guinier form in the angular region where R_{G}k > 1 (see Refs. [7,11] and references therein). Our findings are therefore a support to these experimental works.

A comparison with available experimental SAXS data has been made in this figure. In Figures 4(b) and (c) we have reported the Guinier plots of the measured SAXS patterns for a-Ge films already shown in ^{−1}. Void radius of gyration of 3.8 Å is deduced for the sputtered a-Ge film. The data of evaporated a-Ge film [see ^{−1} k-range. This nonlinearity is usually attributed to a polydisperse void size distribution in the sample. The data were accounted for theoretically by a distribution of Guinier intensities according to the following relation [

where M(R_{G}) is the volume fraction of the voids with radius of gyration R_{G}. The derived fractional void volume distribution is reported in the inset of ^{2} is shown in the top of _{G}) by fitting the simulated scattering intensity to the empirical one given by Equation (18); the result is shown in the top inset of

Now we conclude our discussion with the computed Q-factors and the ratios displayed graphically in ^{−1}. This choice is now obvious, because in our above discussion we have seen that this value represents the cut-off of the SAXS in a-Ge. As expected, the computed Q-factor is practically independent on the void radius of gyration, but enhances when increasing the void volume fraction. The relationship between the integrated k^{2}-weighted SAXS intensity and the void volume fraction v_{f} has been established theoretically for a simple two-phase system, and is given by [

In this equation V stands for the total volume and Δρ is the difference in electron density of the two phases. We have computed the second member of Equation (19) for our nanoporous models. The results are also displayed in _{f} of the microstructural feature producing the SAXS. In order to go further in the quantitative analysis of the SAXS, an interesting relation coupling the invariant Q and the scattering intensity at zero-angle, I(0), has been derived theoretically for a simple two-phase system, and is given by [

Here v_{c} is a correlation volume that measures the mean size of the heterogeneities. It is interesting to note here that this relation is independent on the volume fraction v_{f} of the heterogeneities in the system. For nanoporous network, v_{c} reduces to the mean volume of the nanopores and thus must scale as.

.

We have presented a detailed analysis of the small-angle X-ray scattering of nanoporous atomistic models for a-Ge. The starting point was the high-quality CRN generated by Barkema and Mousseau with the improved WWW method. Widely separated nanovoids of various sizes were introduced in the preceding CRN, resulting in nanoporous models with void volume fractions that don’t exceed 6%. Each generated structure is, first, relaxed to its minimum strain energy described by the anharmonic Keating potential, then, its pair correlation function is computed, and finally, its X-ray scattering pattern is deduced. Using a Gaussians fitting procedure, the SAXS patterns are corrected for the finite size of the models, allowing so a rigorous quantitative analysis of this scattering. We have shown that at 1 Å^{−1} the scattering by the nanovoids is separated from that due to the atoms of the host matrix. Moreover, we confirm that the Guinier’s famous law is valid whatever the shape of the nanovoids. In addition, its k-range extends beyond the expected limit, in agreement with experimental observations. Furthermore, the invariance of the integrated SAXS intensity, on the one hand, and its relationship to the void volume fraction as well as the nanovoids size, on the other hand, are confirmed by our simulation. As a matter of fact, our present simulation is limited to relatively small and simple-shaped nanovoids, but consistent with the size of the basic CRN model. Our present work deserves to be extended to larger nanovoids with complex shapes in order to get closer to reality, but this obviously requires larger atomistic models.

We are indebted to Professor Normand Mousseau, of the Montréal University at Canada, for providing us its continuous random network model on which based our present study.