Inheritance factor on the physical properties in metallic glasses

As one of the important systems in condensed matter physics and material science, metallic glasses (MGs) have attracted intense attention over the past decades owing to their distinctive properties, e.g. high strength, good corrosion resistance, and excellent soft magnetic properties [1]. Since their discovery in 1960 [2], thousands of MG compositions have been developed based on a wide range of elements [3]. In parallel with experimental research, many theoretical works have sought to establish the relationships between the structure and physical properties of MGs [4–8]. However, theoretical predictions of properties remain challenging due to their disordered structures and metastable states. Consequently, the design of alloys with specific properties has largely been the result of trial-and-error efforts [9]. Material genetic engineering can greatly improve and accelerate the efficiency of the development of new materials [10, 11]. Therefore, revealing the key inheritance factor(s) that determine the properties of MGs is an urgent need for directing the development of new MGs.

Recently, Ma and Wang et al [12–14] demonstrated that a variety of MGs inherit their elastic modulus from the base components. Furthermore, in subsequent studies, other physical properties of MGs have been revealed to be related to the corresponding properties of the principal elements. Examples of this phenomenon include Gd-, Ho-, Dy-, Er-, and La-based MGs with promising magnetocaloric properties [15–17], Nd-, Pr-, and Sm-based MGs showing hard magnetic behavior [18], Ce-based MGs that display heavy fermionic behavior [19]; and superconductive MG systems exploiting the superconductive properties of La, Fe, Zr, Hf, and Cu [20–23]. While, for other MGs [24], it is not always the case. Many Cu-based MGs show elastic moduli that are markedly different from those of their base components (see table S1). For example, the bulk modulus of the Cu₆₀Zr₂₀Hf₁₀Ti₁₀ glass is quite different from that of its base element Cu and is closer to the bulk modulus of Zr [13]. This underlines that the key inheritance factors determining MG properties are still unclear. The above results raise the following questions: what is the key factor that determines property inheritance in MGs? Revealing the physical parameters of the components that determine these properties is highly important for understanding the inheritance phenomenon and for accelerating the design of MGs with tailored properties.

In this work, we reveal the inheritance factor of physical properties from the perspective of electronic structure by analyzing a total of 90 MG compositions. We have found a universal scaling law that can be rationalized with the fundamental thermodynamics' principles [25, 26] and Fermi sphere-Brillouin zone interactions [27, 28]. Based on this scaling law, we provide a close correlation between the mechanical properties and a certain component that dominates the electronic density of states (EDOSs) at the Fermi surface (E_F) in MGs. These findings guide the design of novel MGs with desirable properties during material genetic engineering.

Cu₆₄Zr₃₆ and Ni₄₅Ti₂₀Zr₂₅Al₁₀ (at. %) MGs with nominal compositions were prepared by arc-melting of a mixture of pure Cu (99.99%), Zr (99.99%), Ni (99.99%), Ti (99.99%) and Al (99.99%) in a highly purified argon atmosphere. Glassy ribbons were produced from these master alloys by rapid quenching onto a single copper wheel at a speed of 45 m s⁻¹. Fe₈₀P₁₃C₇ and Fe₇₄Mo₆P₁₃C₇ (at. %) MGs were prepared by torch-melting a mixture of pure Fe (99.9 mass %), Mo (99.9 mass %), C (99.95 mass %), and Fe₃P (99.5 mass %) in a clear fused-silica tube under a high-purity argon atmosphere by a torch. The as-prepared master alloy ingots were fluxed in a fluxing agent composed of B₂O₃ and CaO with a mass ratio of 3:1 at an elevated temperature for 4 h under a vacuum with a pressure of ∼10 Pa. After fluxing, each ingot was re-melted and cast into a copper mold with an inner diameter of 1.0 mm under a high-purity argon atmosphere.

The glassy nature of the obtained samples was ascertained by x-ray diffraction (D8 Advance, Bruker, Germany) using Cu K_α radiation and by differential scanning calorimetry (DSC-404, NETZSCH, Germany) with a heating rate of 0.67 K s⁻¹. The bonding states were evaluated by x-ray photoelectron spectroscopy (XPS) using a Kratos AXIS ULTRA^DLD instrument with a monochromic Al K_α x-ray source (hν= 1486.6 eV). The power was 120 W and the x-ray spot size was set to 700 × 300 μm. The pass energy of the XPS analyzer was set at 20 eV. The base pressure of the analysis chamber was greater than 5 × 10⁻⁹ Torr. All of the spectra were calibrated using the binding energy of C 1 s (284.8 eV) as the reference, and the etching conditions of beam energy of 2 kV, extractor current of 100 μA, and raster size of 4 mm. Oxygen-free surfaces of glassy samples were obtained by Ar-ion sputtering. Young's modulus and shear modulus were determined using a Quasar RUSpec resonant ultrasound spectrometer. The low-temperature specific heats of the studied amorphous alloys were evaluated using a physical property measurement system (Model-9, Quantum Design Systems, USA). The specific heat measurements were performed in triplicate and the final data were obtained by averaging the three results. The relative error for the specific heat measurements is less than 2%.

Molecular dynamics calculations were performed using the Vienna ab initio simulation package [29]. All simulations were carried out in the canonical ensemble with the temperature controlled by the Nose thermostat [30]. Cubic supercells containing 100 atoms were used, and the dimensions of the cells were estimated according to the alloy densities at room temperature. The cells were first melted and equilibrated at 2000 K, and then gradually cooled first to 1000 K and then to 300 K at a cooling rate of 1.67 × 10¹⁴ K s⁻¹ with a time step of 3 fs. A 5 × 5 × 5 Monkhorst-Pack mesh was used for k-point sampling, and the EDOSs were calculated considering spin polarization.

Figures 1(a)–(d) show the comparisons of Young's moduli (E) and shear moduli (G) between the Cu₆₄Zr₃₆, Ni₄₅Ti₂₀Zr₂₅Al₁₀, Fe₈₀P₁₃C₇ and Fe₇₄Mo₆P₁₃C₇ glassy samples (the glassy nature of samples was shown in figure S1) and their components, respectively. In figures 1(a) and (b), the values of elastic moduli are close to those of a single element, but this element is not the major component. For instance, for the Ni₄₅Ti₂₀Zr₂₅Al₁₀ glass, E is 114 GPa, which is quite close to that of pure polycrystalline Ti (116 GPa) but is different from that of Ni (200 GPa) and Zr (98 GPa). This is unusual because the minor component rather than the major component appears to be responsible for the properties of MGs. An examination of figures 1(c) and (d) show that the elastic moduli are close to the corresponding values of their major component Fe, but minor changes in the Mo component (e.g. through microalloying) affect the values of the elastic moduli. E of Fe₈₀P₁₃C₇ glass can be dramatically enhanced from 137 to 174 GPa, and G increases from 49 to 65 GPa when 6 at. % Mo was added. This phenomenon seems contradictory but is curious. It is well known that the interatomic bonding controls the elastic moduli of MGs [31] and the characteristics of interatomic bonds are mostly determined by the valence electrons (d, p, and/or s) [32]. To reveal the mechanism of inheritance in MGs, the XPS valence-band spectra, and spin-polarized total and partial DOS of the valence electrons in the corresponding MGs were obtained as shown in figures 1(a¹)∼(d¹), respectively. The EDOS at E_F of Cu₆₄Zr₃₆ and Ni₄₅Ti₂₀Zr₂₅Al₁₀ MGs are dominated by the d electrons of Zr and Ti, respectively, as shown in figures 1(a¹) and (b¹), in good agreement with previous results [33]. As shown in figures 1(c¹) and (d¹), Fe contributes to most of the EDOS in the Fe-based MGs at E_F, while the Mo d electrons also make a significant contribution to the EDOS at E_F in Fe₇₄Mo₆P₁₃C₇ (figure S2). It is observed that the modulus inheritance and electronic structure are correlated, namely, the elastic moduli of MGs are mainly inherited from the elements that dominate the EDOS at the E_F in MGs.

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To further explore the underlying physics of inheritance, the relationship between the elastic modulus and structural parameters was investigated. In previous studies, the elastic modulus of MGs was found to have a clear relationship to the glass transition temperature T_g and molar volume V [7, 34]. The inherent relationship of E with T_g and V of MGs in a universal scaling equation is given by [25, 26]:

$$\begin{align}E = 3R\left( {\frac{2}{{{\varepsilon _{\text{E}}}}}} \right)\frac{{{T_{\text{g}}} - {T_0}}}{V}\end{align} \tag{ 1 }$$

where R is the gas constant, T₀ is room temperature, _E is the room temperature elastic limit of bulk MGs (BMGs) that is approximately 2% [35]. Since E = σ_y/_E and E= 2.61 G [24], we can thus obtain a universal scaling relationship between σ_y, G, and (T_g–T₀)/V.

Recently, fractal concepts have been introduced to describe atomic structure [36, 37]. These models have been invoked to explain the widely observed noncubic power laws (D) correlating the positions of the first sharp diffraction peak, q₁, with V [37], that is:

$$\begin{align}V\sim { }{\left( {1/{q_1}} \right)^{\text{D}}}.\end{align} \tag{ 2 }$$

From equations (1) and (2), a strong dependence of the mechanical properties on the structural vector q₁ can be obtained as follows:

$$\begin{align}E = \frac{{6R\left( {{T_{\text{g}}} - {T_0}} \right)q_1^{\text{D}}}}{{{C_1}{\varepsilon _{\text{E}}}}}\end{align} \tag{ 3 }$$

where C₁ is a constant. The above relationship is generally applicable to different types of MGs, independent of their chemical compositions and bonding types. Thus, different types of MGs should follow a similar underlying mechanism of deformation and fracture. Generally, an increase in T_g and q₁ will result in a stronger MG and vice versa. In figure 2, we plotted the correlations between the mechanical properties σ_y (a), E (b), and G (c) and q₁ for 57 MGs (the data are summarized in table S2) as solid lines. It is observed that equation (3) fits the experimental results well, verifying the existence of an intrinsic correlation. The value of the exponent D extracted from the fitting (2.45) is consistent with D= 2.3 [37] and 2.5 [38] that have been measured for MGs in previous work. This indicates that although the fractal atomic packing in MGs has not been proven [39], the conclusions obtained by using this fractal model are acceptable and are in good agreement with the experimental results.

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Furthermore, many studies [27, 28, 41] have shown that MGs are stabilized when the assumed spherical Fermi surface coincides with the diffused pseudo-Brillouin zone boundary, which can be expressed as q₁ = 2k_F, where k_F is the Fermi wave vector. This is also known as Fermi surface-Brillouin zone interaction or the Hume-Rothery stabilization mechanism [42]. All binary and some ternary MGs compositions have been precisely verified experimentally to satisfy the stability criterion [43]. Then, a universal correlation between E and k_F can be obtained as:

$$\begin{align}E = \frac{{{2^{\text{D}}} \cdot 6R\left( {{T_{\text{g}}} - {T_0}} \right)k_{\text{F}}^{\text{D}}}}{{{C_1}{\varepsilon _{\text{E}}}}}.\end{align} \tag{ 4 }$$

On the one hand, k_F can be given by ${k_{\text{F}}} = 2\pi {( {\frac{{3\rho }}{{8\pi }}} )^{1/3}}{ }$ [28], where ρ is the valence electron density. The correlation between E and ρ can then be described by:

$$\begin{align}E = \,C\left( {{T_{\text{g}}} - {T_0}} \right){\rho ^{{\text{D}}/3}}\end{align} \tag{ 5 }$$

where C is a constant. It is noted that equation (5) is based on the assumption that q₁ represents the effect of both atomic spacing V and valence electron density ρ. Fortunately, MGs can be considered metallic materials with isotropic electron clouds. Despite the lack of long-range periodicity in their atomic structures, MGs can be considered virtually isotropic (i.e. a close-to-spherical symmetry) even down to the atomic level [40]. Thus, the 2k_F = q₁ relationship is valid, as can be further confirmed in figure 3. As shown in figure 3(a), equation (5) fits the experimental data well (the data are also summarized in table S3), demonstrating that σ_y of MGs is intrinsically linked with ρ, which is also consistent with previous works [44]. A remarkable agreement was also observed between the calculated and measured E and G values of various MGs, as shown in figures 3(b) and (c), respectively. Thus, equation (5) demonstrates that in MGs, Young's modulus is directly related to the valence electron density.

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On the other hand, EDOS at the Fermi level n(E_F) and k_F is also related to each other by ${k_{\text{F}}} = \frac{{n\left( {{E_{\text{F}}}} \right)}}{{{\pi ^2}{\hbar ^2}mV}}$ [45]. Then, the relationship between E and n(E_F) can be expressed as:

$$\begin{align}E = C\left( {{T_{\text{g}}} - {T_0}} \right){\left[ {\frac{{n\left( {{E_{\text{F}}}} \right)}}{V}} \right]^{\text{D}}}.\end{align} \tag{ 6 }$$

Equation (6) indicates that the mechanical properties of MGs depended on the EDOS at E_F. A straightforward way to obtain experimental information regarding the EDOS at E_F is to measure the low-temperature specific heat C_p . The EDOS at E_F is directly related to the linear term temperature coefficient of the electronic specific heat [46], γ = (1/3)π² k_B ²(1 + λ_ep) n(E_F), where k_B is the Boltzmann constant and λ_ep accounts for the enhancement of γ due to the electron-phonon interaction. Since the electron-phonon enhancement turns out to be of minor importance [47], the composition dependence of γ already reflects the band structure at E_F well. The relationship between E and γ can be approximately expressed as:

$$\begin{align}E = A\left( {{T_{\text{g}}} - {T_0}} \right){\left[ {\frac{\gamma }{V}} \right]^{\text{D}}} + B\end{align} \tag{ 7 }$$

where A and B are constants. This indicates that the mechanical properties should be related to γ and n(E_F) values of MGs. To confirm this correlation, the C_p values of Cu₆₄Zr₃₆, Ni₄₅Ti₂₀Zr₂₅Al₁₀, Fe₈₀P₁₃C_7, and Fe₇₄Mo₆P₁₃C₇ BMGs are presented in figure 4(a). C_p can be separated into the linear electronic contribution (γT) and the cubic Debye's contribution (δT³) below 20 K [48], δ= 12π⁴ R/(5θ_D ³), where θ_D is the Debye temperature. The relationship between C_p and T can be fitted using C_p /T= γ+ δT² with different γ and δ. The results of this fit are shown in figure 4(b) and demonstrate the good linear relationships between C_p /T and T² for these MGs. The θ_D values of these MGs derived from the δ coefficients and those of their components are compared in figure 4(c). Similar to EDOS and moduli inheritance, the θ_D of these MGs are also inherited from one of their components. The γ values of MGs, together with their σ_y, E, G, V, and T_g are summarized in table 1.

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The relationships between σ_y, E, and G vs γ/V are shown by the solid lines plotted in figure 5. It is observed that equation (7) fits the experimental results very well, verifying the existence of a correlation between the mechanical properties and γ values. Generally, two characteristics of the electron-density distribution, namely bond directionality and charge polarizability, determine the strength and modulus [31]. In principle, the bonding characteristics mainly depend on the EDOS at E_F. The γ values are determined by all the electrons contained in the components and their hybridization [59, 60]. If the effects of covalent bonds and electron hybridization are not considered, the γ of a MG shows a correlation with a weighted average of the coefficient of the electronic specific heat γ_i for the constituent element, γ= Σγ_{i ×} f_i , where f_i denotes the atomic percentage of the constituent element. It indicates that the component with the largest γ_i × f_i that dominates the EDOS at E_F has a decisive influence on the physical properties of MGs. The minor changes in atom components (e.g. through microalloying) may significantly change the EDOS of an MG and accordingly influence the physical properties. As shown in figure 6, γ_Cu = 0.7 mJ mol⁻¹ K² is much smaller than γ_Zr= 2.8 mJ mol⁻¹ K², which means that the EDOS at E_F of Cu–Zr glassy systems is mainly dominated by Zr rather than by Cu. Moreover, γ_Ce = 1620 mJ mol⁻¹ K² is much larger than γ_La = 10 mJ mol⁻¹ K². In (Ce_x La_100−x )-based MGs (x = 10 ∼ 80) [61], even though the Ce content is very low, the EDOS value at E_F of (Ce₁₀La₉₀)₆₈Al₁₀Cu₂₀Co₂ glass is mainly determined by Ce [62]. Correspondingly, the elastic moduli of (Ce₁₀La₉₀)₆₈Al₁₀Cu₂₀Co₂ MG are closer to those of Ce rather than to those of La (the data are also summarized in table S4). These results further confirm that EDOS at E_F or the γ_i value is an inheritance factor of physical properties in MGs, i.e. the basic physical properties of MGs are inherited from a certain component with the largest γ_i × f_i . The properties of the MGs can be judged preliminarily through the γ_i values in figure 6 and their atomic percentage f_i .

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Although we have mainly focused on the inheritance factor of elastic moduli in MGs, the other physical properties such as magnetocaloric effects, Debye temperature, and superconducting temperature also follow the inheritance, as examples in table 2. The physical properties of the MGs are dominated by the major contribution composition to the γ (the EDOS at E_F). Thus, the γ_i values in figure 6 can be used to design tailored properties of MGs by high-throughput computational simulation and/or machine learning.

It was found that elastic properties of MGs are primarily determined by their base pure polycrystalline components [12], and usually increased by about 30% after crystallization [67]. In principle, trends similar to the physical properties inheritance and correlations observed here should also exist for intermetallic compounds; however, due to a large number of dislocations and/or defects in MGs after crystallization, the factors affecting mechanical properties are more complex, and it is difficult to determine quantitative relationships between EDOS and mechanical properties. Furthermore, due to covalent bonds and sp-d hybridization, the assignment of valence electrons is complicated in most alloys containing transition metals, metalloids, and/or rare earth [59, 60]. The data scatter observed for metalloid glasses, such as Fe-, Co-, Ni-, Pd-, and Pt-based MGs, may be considered due to the perturbations stemming from the above-mentioned effects. In particular, Fe-based MGs usually contain non-metallic elements such as B, Si, P, and C [68]. These elements will form covalent bonding with Fe and generate sp-d hybridization [59]. This results in that the electron's contribution γ_i to the parameter γ is no longer a weighted average. Therefore, the scattering of Fe-based MGs is far more than that of others. Fortunately, the transition metals can be treated as if they have one free electron per atom [27], and the presence of the Fermi sphere-Brillouin zone interaction has been both theoretically and experimentally confirmed in transition metal and rare earth MGs [28, 69]. The E, G, and σ_y are complex material parameters that may be influenced by the 'micro-structure', fracture and deformation mode, preparation, loading conditions, specimen size, and aspect ratio geometry. It is important to mention that the T_g and molar volume V depends on the thermal history, such as aging, rejuvenation, and processing conditions. The structural heterogeneity and atomic density fluctuation froze during rapid quenching are also expected to result in the fluctuation of free-volume regions, valence electron density, and low-temperature specific heat. In the free-volume regions, where mechanical coupling to the surrounding atomic environment is weak, inelastic relaxation becomes possible by local atom rearrangements, without affecting the surrounding atomic environment significantly [25, 70]. Thus, these sites are the preferred regions to initiate the scattering of data. If these effects can be ruled out, the theoretical and experimental values of MGs will likely show better agreement. Namely, considering the large phase space of possible metallic glass compositions and their multicomponent nature, their exploration as the result of trial-and-error efforts will be very time-consuming. By contrast, if it is known that the EDOS at E_F is controlled by a certain element, then only calculations for the properties of that element are possible to predict the properties of MGs. Thus, this work may help to design novel MGs with specific properties.

In summary, we have successfully derived a universal scaling law based on fundamental thermodynamics' principles and Fermi sphere-Brillouin zone interactions. The linearity between mechanical properties and glass transition temperature unambiguously demonstrates that the linear elastic properties of MGs are dominated by structural vector and/or valence electron density. It was found that the EDOS at E_F is an inheritance factor of physical properties in MGs, i.e. the elastic moduli of MGs are inherited from the components of the MGs that dominate the EDOS at E_F. This work can be used to design novel MGs with specific properties and will have a significant impact on material genetic engineering.

We thank Professor Q Li for his assistance in sample preparations. We acknowledge Y C Wang and W B Zhang for their assistance with simulations. We also thank W Y Li for their assistance with data collection. This work was supported by the National Natural Science Foundation of China (Nos. 51871237 and 52171165). Additional support was provided through the European Research Council under the Advanced Grant 'INTELHYB—Next Generation of Complex Metallic Materials in Intelligent Hybrid Structures' (No. ERC-2013-ADG-340025).

W M Yang, J W Li, H S Liu, and J T Huo conceived and designed the research and analysis. H Y Li contributed to data collection. J Y Mo and M Z Li performed the MD simulations. W M Yang, H S Liu, S Lan, X-L Wang, M Z Li, J Eckert, and J T Huo performed the data analysis and wrote the paper with assistance from all authors. All authors contributed to the analysis of the results and the discussion in the manuscript.

The authors declare no competing financial interests.