_{1}

Based on Maxwell’s constraint counting theory, rigidity percolation in
Ge_{x}Se_{1-x} glasses occurs when the mean coordination number reaches the value of 2.4. This corresponds to
Ge
_{0.20}
Se
_{0.80} glass. At this composition, the number of constraints experienced by an atom equals the number of degrees of freedom in three dimensions. Hence, at this composition, the network changes from a floppy phase to a rigid phase, and rigidity starts to percolate. In this work, we use reverse Monte Carlo (RMC) modeling to model the structure of
Ge
_{0.20}
Se
_{0.80} glass by simulating its experimental total atomic pair distribution function (PDF) obtained via high energy synchrotron radiation. A three-dimensional configuration of 2836 atoms was obtained, from which we extracted the partial atomic pair distribution functions associated with Ge-Ge, Ge-Se and Se-Se real space correlations that are hard to extract experimentally from total scattering methods. Bond angle distributions, coordination numbers, mean coordination numbers and the number of floppy modes were also extracted and discussed. More structural insights about network topology at this composition were illustrated. The results indicate that in Ge
_{0.20}Se
_{0.80} glass, Ge atoms break up and cross-link the Se chain structure, and form structural units that are four-fold coordinated (the GeSe
_{4} tetrahedra). These tetrahedra form the basic building block and are connected via shared Se atoms or short Se chains. The extent of the intermediate ranged oscillations in real space (as extracted from the width of the first sharp diffraction peak) was found to be around 19.6
?. The bonding schemes in this glass are consistent with the so-called “8-N” rule and can be interpreted in terms of a chemically ordered network model.

Amorphous materials in general and amorphous chalcogenide glasses in particular play an essential rule in technological applications. Examples include infrared detectors, lenses and infrared optical fibers [

Deep understanding of the local structure of amorphous chalcogenides helps understand their remarkable physical and chemical properties and gives more insights about possible combinations to produce and design new useful materials.

In this paper, we focus on rigidity transition in binary chalcogenide glasses. Rigidity theory [_{c}) is less than 3 (the number of degrees of freedom per atom in 3 dimensions), and is considered as stressed-rigid if n_{c} is greater than 3. The network is considered as isostatic when

Simple enumeration of the average number of constraints experienced by an atom in a glassy network can predict its mechanical property, as well as the optimal isostatic composition, in which the network is rigid but stress-free. Rigidity theory has been applied to tetrahedral network glasses with changing composition and it was found that glass formation is optimal if the network is isostatic [

The mean coordination number, _{c}), plays an important role in determining connectivity and rigidity of a network. In the case of a covalently bonded binary alloy with general formula A_{x}B_{1}_{-}_{x}, the value of

In the mean-field approach, one considers a network of N atoms composed of

The number of floppy modes, f, in a network of N atoms equals the difference between the total number of degrees of freedom (3N) and the total number of constraints present in the network, as given by [

where n_{r} is the number of r-fold coordinated atoms. This reduces to:

This number of floppy modes, f, vanishes when

Among all chalcogenide glasses, the covalently bonded Ge_{x}Se_{1}_{-}_{x} system is of special interest. This system can be made as glasses over a wide composition range (_{x}Se_{1}_{-}_{x} glasses occurs at Ge_{0.20}Se_{0.80} where at this composition the value of_{2} glass. Hence, a detailed determination of the local structure of Ge_{0.20}Se_{0.80} glass is essential for understanding the onset of rigidity. Crucial questions whether Ge_{0.20}Se_{0.80} glass forms a chemically ordered or a covalently random network and the possibility of broken chemical order remain subjects of concern.

The purpose of this paper is to build a structural model of the rigidity percolation threshold (Ge_{0.20}Se_{0.80}) glass from which we can extract different structural parameters that may resolve some controversial structural aspects. So, in this work, we study the short- and intermediate-range orders of Ge_{0.20}Se_{0.80} glass using Reverse Monte Carlo (RMC) modeling by simulating its experimental total atomic pair distribution function (PDF). To the best of our knowledge, this is the first RMC modeling done on melt-quenched Ge_{0.20}Se_{0.80} glass through directly simulating its high resolution real-space PDF data, obtained via high energy synchrotron radiation. In the following we give a brief theoretical account about PDF technique and RMC modeling.

The atomic pair distribution function (PDF) technique is a total scattering technique that gives the local structural environment at the atomic scale. PDF technique allows for both the Bragg and diffuse scattering to be analyzed together on equal terms, revealing the short and intermediate range orders of the material [

The atomic PDF,

where

The function

The PDF

where Q is the magnitude of the scattering vector, and

The structure function is related to the coherent part of the total scattering intensity of the material, and is given by [

where

As can be seen from Equations (4)-(6),

Modeling of the PDF data does not presume periodicity. Therefore, PDF technique is particularly useful for characterizing aperiodic distortions in crystals, analysis of nano structures and glasses.

Improper corrections in PDF data reduction result in distortions to

Coordination numbers and partial coordination numbers are extracted through integrating the corresponding peaks in the so-called radial distribution function (RDF), which is related to

Reverse Monte Carlo (RMC) is an important structural modeling method based on experimental data. It began as a method for creating three-dimensional models of liquid structures. It has been developed considerably since then, and its applications have been applied to include crystalline, amorphous structures and magnetic materials [

The general theme of this method is based on building a three dimensional structural configuration of atoms that have their calculated correlation functions consistent to some extent with the experimental ones. In RMC modeling, a set of points (atoms) are placed in a cubical box of edge-length L, with periodic boundary conditions. The types of the atoms in the box, their relative concentrations as well as their number densities are determined to be consistent with the material being modeled.

A set of experimental data, either in Q-space or in real space can be simulated. Starting from a completely random configuration of atoms, an atom is chosen randomly and moved a specific distance. Every time an atom is moved, the correlation functions are calculated from the new configuration and compared with the corresponding experimental correlation functions. If the move increases the agreement between the calculated and experimental data, the move is accepted, otherwise, it is accepted with some probability.

A set of physical structural constraints are inserted in the simulation process so as to improve the fit. These include the distance of closest approach, where no two atoms can come closer to each other by this distance. Coordination number constraints that are consistent with the chemistry of the material may also be inserted in the modeling process. These constraints aim towards improving the fit and making the resulting configuration more and more reasonable.

In RMC modeling, the RMC-calculated total PDF

where

Similarly, for a model of two atom types i and j, the RMC-calculated partial PDF

Here, the

The function to be minimized during each atom move is:

Here the sum is over m experimental points and

Once the model is obtained, many structural parameters can be directly calculated, such as the partial coordination numbers, average coordination number, partial atomic pair distribution functions

It should be noted that the RMC-generated structural models are never unique. This should not be considered as a weakness of the method. RMC modeling is not supposed to give the structure of a given material, it just solves some questions about the structure of the material, and gives more insights about interpretation of the simulated experimental data. What we should look at it in RMC modeling is weather the generated model is useful or not. Does it give more insights into the structure or properties of the material that would not have been obtained without the model?

The Ge_{0.20}Se_{0.80} glass was prepared using conventional melt quenching process. The details of the preparation and characterization process of the glass as well as the X-ray diffraction experiment performed on it are all mentioned in a previous publication [

It should be noted that the use of high-energy X-ray synchrotron radiation (87.005 keV (^{-}^{1}, where Q is the magnitude of the scattering vector, and is given by:

The measured reduced structure function, _{0.20}Se_{0.80} glass are plotted in

The curves in

The high real space resolution of the current data set makes the analysis and the interpretations of the different peaks unambiguous.

In the current RMC modeling process, we followed the following simulation protocol. A set of 2836 atoms were generated randomly inside a box of edge-length of 43.44 Å. This results in an average number density of 0.0346 atoms/Å^{3}, which is comparable with the experimental value of Ge_{0.20}Se_{0.80} glass. From the 2836 atoms, 567 atoms were assigned to represent Ge and the remaining 2269 atoms were assigned to represent Se. These assignments mimics the concentrations of Ge and Se in Ge_{0.20}Se_{0.80}. RMCA program [

The quality of RMC simulation to the experimental total atomic pair distribution function (

The obtained three-dimensional RMC configuration was tested for homogeneity, and structural defects, such as dangling bonds and it was found to be homogenous and reasonable. _{0.20}Se_{0.80} as well as the Ge and Se sub-networks. The calculated PDF from the RMC-generated model

agreement that it is a signature of intermediate range order (IRO) in network glasses [

The resulting RMC configuration was then used to calculate the full set of partial atomic pair distribution functions:

These partial PDFs, when summed up with proper averaging, gives the total atomic pair distribution function (G(r)). The advantage of the obtained RMC model, is that it enabled us to decompose G(r) into three sets of known origin. Structural correlations responsible for each peak in each partial PDF are now very well known and can be easily interpreted.

Many experimental findings [_{4} tetrahedra form the basic building blocks in Ge-Se networks. To test the validity of this assumption, we have extracted the relevant distances from the corresponding partial PDFs shown in _{4} tetrahedra form the basic building blocks in Ge_{0.20}Se_{0.80} glass. Another proof of this fact is extracted from the bond angle distributions, discussed later in this paper.

Two competing structural models were proposed for these glasses. The first model is the chemically ordered network (CON) model [_{x}Se_{1}_{-}_{x} glasses).

In order to extract the partial coordination numbers

and the corresponding peaks in these partials were then integrated.

It is very clear, as can be seen from _{4} tetrahedra, with some amount of Se atoms are necessarily forced to form homopolar Se bonds. Having said that the structure of Ge_{0.20}Se_{0.80} glass is consistent with the CON model does not fully characterize the short range order in this glass, as there are many different bonding configurations at which the GeSe_{4} tetrahedra can link together, as we will see shortly.

Integration of the first peaks in partial RDFs yields that

Hence, the number of floppy modes, as given by Equation (3), vanishes for the Ge_{0.20}Se_{0.80} glass, which indicates that its network is rigid. The above results are also consistent with the “8-N” rule, where we found that Ge

is 4-fold coordinated and Se is 2-fold coordinated.

The structure of amorphous Se consists mainly of Se chains with some few rings [_{0.20}Se_{0.80} glass, Ge atoms break-up and cross link the Se chain structure, and form structural units that are four-fold coordinated (i.e. the GeSe_{4} tetrahedral units). Existence of Se-Se homopolar bonds_{4} tetrahedra, where linkage through a single Se atom (corner-sharing configuration), two Se atoms (edge-sharing configuration), and through short Se chains are all present in this glass.

In _{0.20}Se_{0.80} glass. Here, we calculated the angular distributions of bonds between first neighbour atoms at a maximum radial distance of 3 Å, which was determined from the position of the first minimum after the first PDF peak.

These bond angle distributions have been smoothed for clarity. The smoothing process did not alter their general behavior, and the associated peaks can be seen clearly in the smoothed data. Following is a description of each of these bond angle distribution functions:

・

This distribution spreads over the entire range with no well defined peaks (except a little hump at around 60˚). The general theme of this distribution is flat, which is due to the very little fraction of Ge-Ge homopolar bonds in the first PDF shell.

・

The main peak in this distribution is broad and extends from 85˚ - 125˚ and centered at 105˚. A little hump also occurs at around 60˚. This distribution describes the connectivity between neighbouring tetrahedra. In the high-temperature phase of GeSe_{2} glass (HT-GeSe_{2}), edge-sharing tetrahedra (EST) show angles close to 80˚ and corner-sharing tetrahedra (CST) show angles between 96˚ - 100˚ [_{0.20}Se_{0.80} glass, while the peak at 105˚ is due to CST. Its extension from 85˚ - 125˚ is consistent with the different linkage schemes available for this bond angle as can be seen in the right panel of

・

This distribution has a peak at around 60˚ which is associated with three-fold rings. The little hump seen around 106˚ is related to tetrahedral angles and n-fold rings present in the glass.

・

This distribution has two peaks, the first one occurs at around 60˚ and a second broad peak centered at around 109˚ which is consistent with the ideal value in a perfect tetrahedron (109.5˚), as shown in the right panel of

・

This distribution is relatively similar to that of_{4} tetrahedra.

・

This distribution has two main peaks, the first one is sharp and centered at around 60˚, while the second peak is broad and extends from 95˚ - 120˚, with a maximum at 110˚. As indicated in _{4} tetrahedra in Ge_{0.20}Se_{0.80} glass are ideal. This finding is consistent with the ratio of

Structural information about intermediate range order (IRO) is contained in the peaks beyond the nearest neighbor distances. As can be seen from

GeSe_{4} tetrahedra. Careful analysis of this distribution function shows a small peak at about 3.1 Å, which is the distance of Ge-Ge correlation when the GeSe_{4} tetrahedra share edges. The peak at around 3.6 Å is due to Ge-Ge correlations in corner-sharing configuration.

The first sharp diffraction peak (FSDP) in the reduced structure function is considered as a signature of intermediate range order present in this glass [^{-}^{1}, and so, the periodicity of the associated intermediate ranged oscillations (given by:^{-}^{1}. This width determines the so-called coherence length (given by:

In conclusion, we have used constrained RMC modeling to build a three-dimensional structural model of the Ge_{0.20}Se_{0.80} glass through simulating its experimental X-ray total atomic pair distribution function (PDF_{0.20}Se_{0.80} network is best described by a chemically ordered network, where all atoms are coordinated according to the “8-N” rule, and the number of hetropolar bonds is maximized. The Ge atoms are four-fold coordinated to Se atoms to form GeSe_{4} tetrahedra, and with some Se atoms are necessarily forced to form homopolar Se bonds. The GeSe_{4} tetrahedra are linked together with different configuration schemes, including CST, EST and linkage through short Se chains. The present investigation on Ge_{0.20}Se_{0.80} glass provides structural insights on the network topology at both short and intermediate atomic length scales. Finally, this work shows the power of RMC simulation of experimental data to build a structural model of an amorphous material. Without such a model, much important structural information cannot be obtained.

It is our pleasure to acknowledge Prof. Simon J. L. Billinge and his research group where the current experimental data set was collected. The X-ray diffraction experiment was performed at the 6ID-D beamline in the Midwest Universities Collaborative Access Team (MUCAT) sector at the Advanced Photon Source (APS). Use of the APS is supported by the US DOE, Office of Science, Office of Basic Energy Sciences, under Contract No. W-31-109-Eng-38. The MUCAT sector at the APS is supported by the US DOE, Office of Science, Office of Basic Energy Sciences, through the Ames Laboratory under Contract No. W-7405-Eng-82. We also thank Prof. Punit Boolchand from the University of Cincinnati, and his former graduate student Ping Chen for making up the studied sample.