Short-Range Ordering Structure Found in Pd₈₂Ge₁₈ Amorphous Alloy Determined from Anomalous X-ray Scattering (AXS) Data by Applying Reverse Monte-Carlo (RMC) Simulation Method

Abstract

The topological and chemical short-range ordering (TSRO and CSRO) of Pd₈₂Ge₁₈ amorphous alloys prepared by the single-roll liquid quench method were analyzed using a combination of anomalous X-ray scattering and reverse Monte Carlo (RMC) simulation. The degree of development of TSRO and CSRO in the amorphous structure model made with the RMC simulation (RMC model) was evaluated by comparison with the dense random packing model (DRP model) calculated using metallic and covalent atomic radii. In the RMC model, a shortening of the Pd–Ge nearest neighbor correlation and a decrease in the coordination number around Ge were observed. Voronoi polyhedral analysis was performed to evaluate the local geometrical structure around the Pd and Ge atoms. Although the DRP structure can reasonably approximate the local structure around the Pd atoms, the TTP structure corresponding to the local structure in crystalline Pd2Ge develops around the Ge atoms as a characteristic TSRO. However, the preferential coordination of Pd atoms around Ge, which is a CSRO associated with the TTP structure in the crystalline phase, was not observed. These results indicate that strong CSRO does not necessarily accompany the TTP structures in metal-semimetal amorphous alloys.

Fig. 3 Partial pair distribution functions, g(r)_PdPd, g(r)_PdGe, and g(r)_GeGe calculated from the DRP and RMC models.

1. Introduction

The high glass-forming ability and mechanical strength of amorphous alloys^1–5) make them attractive from a materials science perspective. To elucidate the mechanism of these unique properties, it is necessary to understand the topological and chemical short-range ordering (TSRO and CSRO) found in the amorphous structure,^6–9) however the details of these ordered structures are not fully clear. One of the most notable properties of amorphous materials is their glass-forming ability, which is indicative of their amorphous state. Amorphous metals consisting mainly of transition metal elements can dramatically increase their glass-forming ability by adding approximately 10–20% of semi-metallic and nonmetallic elements. Typical examples are Zr-based alloys with Al (Zr–Cu–Ni–Al)^10,11) and Pd-based alloys with P (Pd–Ni–Cu–P),^12,13) which can produce bulk amorphous alloys on a centimeter scale. Thus, the amorphous structure formed by the addition of semi-metallic and nonmetallic elements is considered one of the keys to high glass-forming ability, and many studies have been conducted to elucidate this structure. Geskel et al. proposed a local structure model, the tricapped trigonal prism (TTP) structure, which explains the features of the radial distribution function (RDF) obtained from the X-ray and neutron diffraction of amorphous alloys of transition metal (TM)–nonmetal (N) systems with deep eutectic points around the TM₈₀N₂₀ composition.^14–16) It has also been reported that the TTP structure develops as a local structure in Pd–Ni–P^17–19) and Pd–Si^20–22) systems, even in three-dimensional structural models created using computational method. However, it has been reported that the TTP structure is not necessarily the main local structure in Ni–P systems and that the local structure is determined by the atomic radius ratio^23,24) and Hosokawa et al. carried out structural analyses of Pd–Ni–P, Pd–Cu–P and Pd–Cu–Ni–P amorphous metals using ND and AXS, they found that P–P nearest-neighbour correlations exist in these amorphous alloys.^25–27) These divergent conclusions of previous reports are suggesting that a quantitative study using state-of-the-art methods is necessary to evaluate the degree of TSRO and CSRO development. We have previously revealed the chemical and density heterogeneity in Zr₈₀Pt₂₀ amorphous alloys using a combination of anomalous X-ray scattering and reverse Monte Carlo methods.^28,29) We believe that the same method is currently the best tool for discussing the existence and morphology of TTP-type local structures, which has been controversial for amorphous structures containing semimetals and nonmetals. For this purpose, we have focused on Pd–Ge amorphous alloys for which anomalous X-ray scattering measurements are possible for all of the constituent elements. In this study, we performed a detailed structural analysis of amorphous Pd₈₂Ge₁₈ metals to clarify the behavior of semi-metallic elements in amorphous structures.

2. Methodology of AXS-RMC Analysis

The analytical procedure for the AXS measurements has been discussed previously, and the procedure for the RMC simulation employed in this study is identical to that described in our previous studies.^28,30–32) Only the essential points are given below for the convenience of discussion. When the incident energy for AXS is set close to the absorption edge, E_abs, using Pd as an example, the anomalous dispersion becomes significant. The resulting variation in the signal intensity Δi_Pd(Q, E₁, E₂) is attributed to a change in the real part of the anomalous dispersion terms of Pd.

  

\begin{align} & \varDelta i_{\textit{Pd}}(Q,E_{1},E_{2})\\ &\quad = \frac{\{I(Q,E_{1}) - \langle f^{2}(Q,E_{1})\rangle\} - \{I(Q,E_{2}) - \langle f^{2}(Q,E_{2})\rangle\}}{c_{\textit{Pd}}\{f'{}_{\textit{Pd}}(E_{1}) - f'{}_{\textit{Pd}}(E_{2})\}W(Q,E_{1},E_{2})}\\ &\quad = \frac{c_{\textit{Pd}}\Re \{f_{\textit{Pd}}(Q,E_{1}) - f_{\textit{Pd}}(Q,E_{2})\}}{W(Q,E_{1},E_{2})}(a_{\textit{PdPd}}(Q) - 1) \\ &\qquad +\frac{c_{\textit{Pd}}\Re \{f_{\textit{Ge}}(Q,E_{1}) - f_{\textit{Ge}}(Q,E_{2})\}}{W(Q,E_{1},E_{2})}(a_{\textit{PdGe}}(Q) - 1) \end{align} (1)

  

\begin{equation} W(Q,E_{1},E_{2}) = \sum_{k = 1}^{2}c_{k}\Re \{f_{k}(Q,E_{1}) + f_{k}(Q,E_{2})\} \end{equation} (2)

where Q and E are the wave vector and incident X-ray energy, respectively, and c_i is the atomic fraction. The atomic scattering factor of the i-th element is given by the equation: $f_{i}(Q.E) = f_{i}^{0}(Q) + f'_{i}(E) + if''_{i}(E)$, where f⁰(Q) is the scattering factor of the atom at an energy sufficiently away from the absorption edge, and f′(E) and f′′(E) are the real and imaginary parts of the anomalous dispersion terms, respectively. In this study, theoretical calculated values³³⁾ of f′(E) and f′′(E) were used for analysis. Equation (2) includes two functions: I(Q.E) and W(Q.E₁, E₂). The function I(Q.E) is the coherent X-ray scattering intensity obtained by ordinary analysis coupled with the generalized Krough-Moe-Normaan method.³⁴⁾ Function W(Q, E₁, E₂) was calculated using eq. (2), where $\Re \{ \} $ denotes the real parts of the values in parentheses. Δi_Pd(Q, E₁, E₂) is a function of two partial structure factors: a_PdPd(Q) and a_PdGe(Q). Similarly, Δi_Ge(Q, E₁, E₂) measured at the Ge K absorption edge contains contributions from a_PdPd(Q) and a_PdGe(Q). Fourier transforms of the functions Δi_Pd(Q, E₁, E₂) and Δi_Ge(Q, E₁, E₂) using the average number density (ρ₀) of the system readily yield the environmental radial distribution functions (RDFs) 4πr²ρ_Pd(r) and 4πr²ρ_Ge(r), respectively.

The detailed parameters for the RMC simulation³⁵⁾ are summarized as follows: The simulation was initiated with a model comprising 25000 atoms (Pd: 20500 and Ge: 4500) for a dense random packing (DRP) structure in a cubic hypercell with a unit size of L = 7.206 nm. The number of atoms and cell size were selected for the model to meet the chemical composition as well as the measured density (11.1 Mg/m³) of the Pd₈₂Ge₁₈ sample. The RMC simulation is carried out by self-made computer program. The atomic positions in the DRP structure were obtained using the algorithm proposed by Clarke et al.³⁶⁾ by employing the Goldschmidt radius of Pd (0.137 nm) and the covalent radius of Ge (0.122 nm).³⁷⁾

3. Experimental

An alloy ingot with a nominal composition of Pd₈₂Ge₁₈ was prepared via conventional arc melting using a mixture of pure Palladium (>99.9%) and Germanium (>99.99%) in a purified Ar atmosphere. Pd₈₂Ge₁₈ amorphous ribbon samples of about 20 µm thickness and 5 mm width were produced by a single roller melt-spinning technique in an Ar atmosphere. The density of the amorphous ribbon sample measured using Archimedes’ method was 11.1 Mg/m³. The obtained ribbons were cut into small pieces of approximately 20 mm in length, closely arranged, and fixed on an aluminum frame with a window of approximately 20 mm in width and 15 mm in height. AXS measurements at the Pd K-absorption edge and Ge K-absorption edge were performed at beamline stations BL02B1 of Super Photon Ring - 8 GeV, Japan Synchrotron Radiation Research Institute, Hyogo, Japan, and BL7C of the Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba, Japan, respectively. The diffraction intensities are measured using a portable germanium semiconductor detector with energy resolution. Elastic scattering, inelastic scattering and X-ray fluorescence intensities are measured at each diffraction angle and only the elastic scattering is extracted in the subsequent analysis. A pair of incident energies, which correspond to 25 and 300 eV below the Pd and Ge K absorption edges, respectively, were used in the present AXS measurements, and the anomalous dispersion terms for Pd and Ge used in this work are listed in Table 1. It should be added that Si (311) and (111) double-crystal monochromators are equipped at the BL02B1 and BL7C stations, respectively.

Table 1 Anomalous dispersion terms f′(E) and f′′(E) used in the present AXS measurements.

4. Results and Discussion

Figure 1 shows the ordinary RDF measured at E = 24.048 keV (Pd K edge −300 eV) and the environmental RDFs around Pd and Ge obtained by the present AXS measurements. The vertical dashed lines in Fig. 1 indicate the nearest-neighbor atomic distances for the Pd–Pd, Pd–Ge, and Ge–Ge pairs estimated from the Goldschmidt radius of Pd and the covalent radius of Ge. The nearest neighbor correlation in the radial distribution function is observed as a single broad peak in the 0.22–0.32 nm region. It is considered to be composed of the nearest neighbor correlations of Pd–Pd, Pd–Ge, and Ge–Ge pairs. In the environmental radial distribution function obtained from Ge-AXS, the first proximity correlation peak is shifted to a lower distance compared with the normal radial distribution function, suggesting that the Pd–Pd correlation disappears and the Ge–Ge and Ge–Pd correlations are emphasized. In the Pd-AXS spectrum, an emphasized Pd–Pd correlation was observed at approximately 0.274 nm, with a hump at approximately 0.240 nm. The atomic correlation indicated by this hump was slightly smaller than the sum of those of Pd and Ge. Previous reports have indicated that the nearest-neighbor distance of Pd–Ge is shorter than that obtained by the simple addition of atomic radii.^38,39) The Pd–Ge correlation in the structure of the intermetallic compound Pd₂Ge⁴⁰⁾ was similarly shortened (0.247 nm), suggesting that the Ge–Pd bonds in the amorphous alloys were covalent. The environmental RDFs obtained from AXS measurements represent the atomic size and chemical bonding features of Pd and Ge, suggesting that the structural model obtained by subsequent RMC-simulation well reproduces the environmental structure around Pd and Ge.

Fig. 1

Ordinary and the environmental RDFs around Pd and Ge obtained by the present AXS measurements. The vertical dashed lines indicate the nearest neighbor atomic distances for each atomic pair.

Figure 2 shows experimentally measured ordinary interference function, Qi(Q) and environmental interference functions for Pd and Ge, QΔi_Pd(Q) and QΔi_Ge(Q). The ordinary and environmental interference functions calculated using the DRP and RMC models are shown in the same figure. In the measured interference functions, there was no obvious pre-peak signal, as observed in amorphous alloys with well-developed medium-range correlations,^29,41–43) suggesting that the ordered structure formed in Pd₈₂Ge₁₈ amorphous alloys is limited to nearest-neighbor atomic correlations.

Fig. 2

Experimentally obtained ordinary interference function, Qi(Q) and environmental interference functions for Pd and Ge, QΔi_Pd(Q) and QΔi_Ge(Q). The calculated values of these interference functions from the DRP and RMC models are also shown.

The ordinary and environmental interference functions calculated using the DRP model can reproduce the rough shape of the measured values; however, they cannot reproduce details such as humps observed in the second and third peaks and the damping of wave vectors higher than 100 nm-1 in the measured data. In contrast, the RMC model reproduces these features almost perfectly. For Qi(Q)_Pd, the oscillations in the RMC model shift from the 3rd peak to a lower wavevector compared to the DRP model, which may correspond to the longer-range Pd–Pd correlations observed in the AXS-RDF. Therefore, by comparing the structures of the DRP and RMC models, it is possible to quantitatively evaluate the degree of TSRO and CSRO development caused by the chemical effects in the actual amorphous structure.

Figure 3 shows the partial pair distribution functions g(r)_PdPd, g(r)_PdGe, and g(r)_GeGe calculated using the DRP and RMC models. The first peak positions of the homologous atomic correlation Pd–Pd and Ge–Ge pairs were shifted towards a slightly long-range direction with respect to the DRP model. In contrast, those of the heteroatomic correlation Pd–Ge were shifted towards the short-range direction. These features reproduced the characteristic features of each atomic correlation inferred from the ordinary and environmental RDFs. The Pd–Ge correlation in the RMC model showed obvious decreases between first and second correlations compared to those in the DRP model, especially at ∼3.4 A. This indicates that the Pd–Ge bond distances tend to take discrete values, such as in crystals, and some TSRO units with regular Pd–Ge atomic correlation distances may exist in the amorphous structures. Based on the distance at which g(r) decreased between the first and second peaks of each atomic correlation in the DRP and RMC models, the cutoff distances, that is, the maximum distances in the first proximity region, were determined to be 0.360, 0.340, and 0.330 nm for the Pd–Pd, Pd–Ge, and Ge–Ge correlations, respectively. Table 2 lists the atomic distances and coordination numbers within the nearest correlation ranges for the DRP and RMC models. The average interatomic distances and the first peak top positions in g(r)s of the RMC model tend to be almost the same or slightly longer than those of the DRP model for the homologous element pairs Pd–Pd and Ge–Ge. In contrast, those for the Pd–Ge hetero element pair are clearly shorter than those of the DRP model and are close to the Pd–Ge nearest-neighbor correlation distance (0.247 nm) in crystalline Pd₂Ge. The total coordination number around the Pd atom is 12.7 in both the RMC and DRP models, on the other hands, the total coordination number around Ge showed distinctly different values for DRP and RMC. The total coordination number around Ge is 10.4 in the RMC model compared to 11.1 in the DRP model, which is close to the 9-coordination value of the TTP structure. These features suggest that the average Ge–Pd bond distances and Ge coordination numbers in the RMC model were closer to those in the crystalline Pd₂Ge phase than those in the DRP model. However, the occupancies of the coordination atoms around Ge were almost the same between the DRP and RMC models, suggesting that no strong CSRO corresponded to the crystalline Pd₂Ge phase in which only Pd atoms were coordinated around the Ge atom. To consider the details of the geometrical arrangement, Voronoi polyhedron analysis was performed. Figure 4 shows typical Voronoi polyhedra around Pd and Ge in the RMC and DRP models. The distribution trend of the Voronoi index observed around Pd was similar between the RMC and DRP models, indicating that the DRP structure could explain the geometrical structure around Pd in the Pd₈₂Ge₁₈ amorphous alloy. The abundance of icosahedral(-like) clusters (0 2 8 2 0), (0 1 10 2 0) and (0 0 12 0 0) around Pd was consistent with previous reports. However, the distribution of Voronoi polyhedra around Ge differed between the DRP and RMC models, with the (0 3 6 0 0) index corresponding to the TTP structure being observed more frequently in the RMC model (Fig. 5). This result clearly indicates that a TTP-type TSRO, similar to the crystalline phase, develops around Ge. In addition, the decrease in (0 2 8 2 0) icosahedral-like structures around Ge and the increase in (0 2 8 0 0) 10-coordinated antiprismatic structures suggest an increase in prismatic, crystal-like local structures around Ge. In amorphous alloys with short- to medium-range-ordered structures, the common neighbor atoms between certain atomic pairs may show a strong chemical orientation,²⁹⁾ in which case CNA analysis^44,45) is of great help in CSRO considerations. The Pd–Ge nearest-neighbor correlations in the TTP structures observed in the crystalline Pd²Ge structures form the characteristic [555]_CN and [444]_CN, where all common neighbor atoms are Pd. The Pd occupancies of the nearest-neighbor common atoms of [555]_CN and [444]_CN in the RMC and DRP models are listed in Table 3. For both CNA indices, the Pd occupancies were almost the same for the RMC and DRP models and were close to the elemental ratios of the Pd₈₂Ge₁₈ amorphous alloy. This result reinforces the fact that the distribution of Pd and Ge in the amorphous structure is not chemically oriented, similar to that in the crystalline phase.

Fig. 3

Partial pair distribution functions, g(r)_PdPd, g(r)_PdGe, and g(r)_GeGe calculated from the DRP and RMC models.

Table 2 Averaged atomic distance, r_ij, first peak top position of g_ij(r), p_ij, averaged coordination number Z_ij, chemical occupancy of coordinate atoms for DRP (the value in the round bracket), and RMC structural model.

Fig. 4

The frequency distribution of typical Voronoi index around (a) Pd and (b) Ge atoms in DRP and RMC structural models.

Fig. 5

Ge-centered TTP structure with (0 3 6 0 0) Voronoi index found in Pd₂Ge crystalline phase.

Table 3 Occupancy of Pd, Z_Pd in the common neighbor atoms for [555]_CN and [444]_CN of Pd–Ge pair in DRP- and RMC-models and crystalline Pd₂Ge.

5. Conclusion

To evaluate the degree of TSRO and CSRO development in the Pd₈₂Ge₁₈ amorphous alloy, an amorphous structure analysis was carried out using a combination of environmental structure analysis by AXS and RMC simulation. The three-dimensional structural model constructed by the RMC simulation and the RMC model can reproduce the environmental interference functions around Pd and Ge extracted by AXS almost perfectly, and the amorphous structural model can fully consider the chemical orientation in the amorphous material. The partial pair distribution function of the RMC model showed a different trend in the Pd–Ge correlations than that of the DRP model, with a shorter distance between the Pd and Ge correlations and a distinct decrease between the 1st and 2nd peaks. The average number of coordinations around Ge in the RMC model was smaller than that in the DRP model and was closer to the 9-coordination of the TTP structure. Furthermore, the (03600) Voronoi index, which indicates the TTP structure, is a major structure around Ge and is observed more frequently in the RMC model than in the DRP model. From these results, it can be concluded that the DRP model can approximate the geometrical structure around Pd in the Pd₈₂Ge₁₈ amorphous alloy; however, the TTP structure develops around the Ge atoms as TSRO, similar to the crystalline phase, which is different from the DRP structure. However, the strong CSRO observed in the TTP structure of the crystal, that is, the preferential coordination of Pd atoms with respect to the ligands around Ge and the common neighbor of the Pd–Ge pair, was not observed in the present RMC model. These results suggest that the shortening of the covalent Pd–Ge bond distances and the coordination numbers around Ge are close to those in the crystalline phase, resulting in the development of the 9-coordinated TTP structure as one of the major TSRO; however, the TTP-type structure in amorphous materials is not necessarily accompanied by a strong CSRO.

Acknowledgment

The authors express their gratitude to Dr. H. Nitani, Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba, Japan, for assistance with the AXS measurements (Proposal No. 2018G145). Synchrotron radiation experiments were performed at BL02B1 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2021B1497). Part of this study was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers JP19K15653, 18H05456, 20H00189 and 21K04707. This work was performed under the GIMRT Program of the Institute for Materials Research, Tohoku University (Proposal No. 202112-CRKEQ-0415).

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