A. Thermophysical parameters and sample characterization

Because of the vast number of HEAs that can be designed from about eighty stable elements,^{Reference Cantor10,Reference Miracle and Senkov14} searching for conceptually and technologically interesting compositions by trial and error is clearly inadequate. Therefore, as mentioned in Sec. I, several semi-empirical criteria for the formation of different phases: single phase solid solution, intermetallic compounds and a mixture of intermetallic and solid solution phases, and amorphous phase (a-HEA) have been developed.^{Reference Miracle6–Reference Gao, Yeh, Liaw and Zhang17,Reference Zhang, Zhou, Lin, Chen and Liaw31–Reference King, Middleburgh, McGregor and Cortie33} These criteria are based on thermophysical parameters such as the configurational entropy ΔS _conf, the mixing ΔH _mix or formation enthalpy ΔH _f, the average difference in atomic sizes of the constituents δ (as in the Hume-Rothery and Inoue’s rules, see, e.g., Refs. Reference Guo13 and Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18), etc. (see Gao et al.^{Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk42} for an excellent, concise discussion of all semi-empirical criteria and corresponding parameters).

For example, the oldest criterion,^{Reference Zhang, Zhou, Lin, Chen and Liaw31} a two dimensional ΔH _mix (or ΔH _f) − δ plot, where δ is the atomic size mismatch, shows how HEAs evolve with increasing δ and decreasing ΔH _mix/ΔH _f from solid solution HEAs situated in the region with δ ≤ 6.6% and −15 kJ/mol ≤ ΔH _mix ≤ −5 kJ/mol to intermetallic and a-HEAs at higher δ and lower or similar ΔH _mix.^{Reference Guo13,Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk42} To predict the crystal structure type of solid solution HEAs, the “valence electron concentration” (VEC) criterion was proposed.^{Reference Sheikh, Mao and Guo43} According to this criterion, a bcc phase forms for VEC ≤ 6.87, a mixture of bcc and face-centered cubic (fcc) phase forms for 6.87 ≤ VEC < 8, and an fcc phase appears for VEC ≥ 8. The expressions for all these criteria and the definitions of the corresponding parameters can be found in reviews of the literature^{Reference Miracle6–Reference Gao, Yeh, Liaw and Zhang17,Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk42} as well as in our previous papers.^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36}

In Fig. 1, we show the variation with concentration x of parameters ΔH _mix, δ and ΔS _conf of (TiZrNbCu)_1−xNi_x alloys. Here, the HEA region of x is distinguished from the Ni-rich one (x ≥ 0.35) by a different color. As already noted,^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} the values of the parameters ΔH _f (ΔH _mix) and δ of all alloys are well outside the ΔH _mix − δ region in which solid solution HEA forms. (More recent and sophisticated criteria such as those in Refs. Reference Troparevsky, Morris, Kent, Lupini and Stocks32, Reference King, Middleburgh, McGregor and Cortie33, and Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk42 corroborate this conclusion). Regardless of the reliability of ΔH _mix − δ^{Reference Guo13} and other^{Reference Troparevsky, Morris, Kent, Lupini and Stocks32,Reference King, Middleburgh, McGregor and Cortie33} criteria, this conclusion is consistent with our difficulties in trying to convert our alloys with low x to solid solution HEAs. Furthermore, the contribution of the ideal ΔS _conf to the free energy is considerably smaller than that of enthalpy^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} even at the liquidus temperature T _l so that its effect on phase formation may not be large, as was confirmed by XRD patterns and DSC/DTA results.^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} In particular, all melt-spun ribbons with x > 0 were fully amorphous, whereas their crystalline counterparts, suction cast rods,^{Reference Cunliffe, Plummer, Figueroa and Todd37} and bulk conical samples were multiphase. The inset of Fig. 1 shows that the strong interatomic bonding between TE and TL components, especially between Zr and Ti with Ni, is responsible for the small overall values of ΔH _mix (the values of ΔH _mix in the inset are taken from Ref. Reference Troparevsky, Morris, Kent, Lupini and Stocks32). These strong interactions are likely to give rise to “chemical short range order” (CSRO) in the amorphous phase^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38} and intermetallic compounds in the crystalline state.^{Reference Ristić, Zadro, Pajić, Figueroa and Babić28} Simultaneously, they seem to stabilize amorphous phases since the alloy with x = 0 did not vitrify.^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18} Similarly, the difference in atomic radii⁴⁴ between TE and TL metals is responsible for rather large values of δ which show a maximum at x = 0.35.

FIG. 1.

Thermophysical parameters of (TiZrNbCu)_1−xNi_x alloys versus x and valence electron count, VEC (upper abscissa). Left scale: atomic size mismatch δ, right scale: mixing enthalpy ΔH _mix (first right scale), and configurational entropy ΔS _conf (second scale). The inset: ΔH _mix between constituent elements.^{Reference Troparevsky, Morris, Kent, Lupini and Stocks32} Dashed lines are guide for eye.

Since VEC is proportional to x of our alloys, the upper abscissa enables one to follow the variations of the parameters shown in Fig. 1 as a function of VEC. We note that VEC ≤ 7 for x ≤ 0.25 (HEA region) and reaches 8 for x = 0.5 (Ni-rich region). Thus, provided the correlation between VEC and crystal structure of numerous solid solution HEAs^{Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk42,Reference Sheikh, Mao and Guo43} also applies to TE–TL type of multicomponent amorphous alloys, the transition from the HEA to Ni-rich concentration range in our alloys could coincide with a change in the “atomic short range order” (SRO), from say bcc-like to mixed bcc–fcc-like and finally to fcc-like. Indeed, an approximate analysis of the first maxima in the XRD patterns of amorphous (TiZrNbCu)_1−xNi_x alloys with x ≤ 0.25 indicated bcc-like local atomic arrangements,^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18} whereas a more recent study of the XRD patterns of the alloys with x = 0.35 and 0.5 (Ni-rich concentration region) indicated deviations from the behavior expected for bcc-like atomic arrangements.^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} In particular, the average lattice parameters a calculated from the XRD patterns by assuming bcc-like local atomic arrangements decreased linearly with x for x ≤ 0.25 (as expected from Vegard’s law) but a showed a strong positive deviation for x ≥ 0.35 (Fig. 2 in Ref. Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36). Before discussing in some detail this unusual finding, we will briefly overview the main results of thermal studies and compositional/homogeneity characterization of our samples (given in more detail in Refs. Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18 and Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36).

The detailed description of SEM/EDS studies performed on our as-cast samples in the HEA composition region, which included SEM images and EDS mapping of the distributions of constituent elements, was given in Ref. Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18. Briefly, it was found that the EDS compositions were the same as the nominal ones to within ±1–2 at.% and that the distributions of all constituent elements were random down to the submicrometer scale. We note that suction cast rods of (TiZrNbCu)_1−xNi_x alloys,^{Reference Cunliffe, Plummer, Figueroa and Todd37} having nominally the same compositions as our a-HEAs, solidified to form dendritic microstructure and EDS analysis showed that these dendrites, of size 2–5 µm, consisted mainly of Nb. Indeed, quite often the distribution of constituent elements in HEAs is uneven, and this sometimes occurs even in alloys showing solid solution HEA behavior in their XRD patterns.^{Reference Pickering and Jones7} We performed the same type of SEM/EDS study on our samples with higher Ni content (x ≥ 0.35). These samples also had a random distribution of constituent elements and their EDS compositions agreed well with nominal ones.^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36}

The detailed DSC/DTA study of all the samples^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} confirmed the XRD results that the as-cast ribbons with x ≥ 0.125 were amorphous and provided their glass transition (T _g), crystallization (T _x), melting (T _m), and liquidus (T_l) temperatures. All alloys showed quite complex crystallization behavior with two distinct exothermic maxima for x ≤ 0.25^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36,Reference Cunliffe, Plummer, Figueroa and Todd37} and three maxima in the Ni-rich alloys.^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} For the sake of simplicity, we analyzed only the first crystallization event (allegedly associated with the precipitation of a bcc-HEA phase for x ≤ 0.2^{Reference Cunliffe, Plummer, Figueroa and Todd37}) which determines the stability of the amorphous phase and the width of the supercooled liquid region, ΔT _x = T _x1 − T _g. Thermal stability parameters T _x1 and T _l showed qualitatively the same variations with x, a rapid increase for x ≤ 0.25 and a tendency to saturate in the Ni-rich region (Fig. 4 in Ref. Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36). We note that the observed change in concentration dependence of T _x and T _l coincides with that in the average lattice parameters of a bcc-like atomic arrangement (Fig. 2 in Ref. Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36). Since T _x and T _l are associated with the strength of interatomic bonding, it seems that a change in SRO at elevated Ni-contents affects the interatomic bonding too. The enthalpies of crystallization ΔH _c1 and the widths of supercooled liquid region ΔT _x showed similar variations with x as those of T _x1 and T _l. In particular, ΔT _x increased from about 40 K at low x to about 100 K on the more Ni-rich side. The reduced glass transition temperature T _rg (which is frequently employed as a criterion for glass forming ability, e.g., Ref. Reference Ristić, Zadro, Pajić, Figueroa and Babić28) showed a modest magnitude around 0.52 (thus indicating modest GFA) and rather little variation with x (Fig. 2 in Ref. Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36). Since we experienced some difficulties in preparing the fully amorphous alloys with the lowest and the highest Ni-contents and were unable to vitrify the alloy with x = 0, the GFA criteria T _rg and ΔT _x^{Reference Ristić, Zadro, Pajić, Figueroa and Babić28} may not be applicable to present alloys.^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36}

B. Atomic structure

As already noted in Sec. III.A, the XRD patterns of amorphous alloys, in addition to showing the amorphous nature of a particular alloy, can also provide some insight into the local atomic arrangements, the average atomic volumes, and the average atomic packing fractions (APFs) (e.g., Refs. Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38 and Reference Calvayrac, Chevalier, Harmelin and Quivy45). The corresponding procedures were previously^{Reference Ristić and Babić46,Reference Ristić and Babić47} used by us to correct the N(E _F) of a hypothetical fcc-phase of pure Zr,^{Reference Bakonyi, Ebert and Liechenstein48} by providing a better estimate for the corresponding atomic volume, and were also used to determine the atomic volume of amorphous copper and the average APFs of amorphous Cu–Hf alloys.^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38} We note that in all these cases, the atomic volumes determined by using XRD patterns agreed quite well with those obtained from the experimental mass-density.^{Reference Bakonyi49}

In particular, from the modulus of the scattering vector k _p, corresponding to the first maximum in the XRD pattern,^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} k _p = 4π sin θ/λ (θ is the Bragg angle and λ is the wavelength of the X-ray radiation), one can calculate the average nearest neighbor distance^{Reference Calvayrac, Chevalier, Harmelin and Quivy45}:

(1)$$d = {{7.73} \over {{k_{\rm{p}}}}}\quad .$$

By assuming an approximate crystal structure of the local atomic arrangement, one can calculate the corresponding average lattice parameter a and the average atomic volume, V. In particular, for a bcc-like local atomic structure, a _bcc = 2d/3^0.5 and V _bcc = a ³/2, whereas for the fcc-like atomic SRO, a _fcc = 2^0.5d and V _fcc = a ³/4. The variation of a _bcc of all our alloys with Ni content was shown in Fig. 2 of Ref. Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36, whereas a _bcc for alloys with x ≤ 0.25 (VEC ≤ 7) follows Vegard’s law for a bcc crystal structure quite well (making allowance for somewhat lower mass-density of amorphous alloys^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36}), the a _bcc data for alloys with more Ni (VEC > 7) show a strong upward deviation from the Vegard’s law. Here, in Fig. 2, we show the corresponding V _bcc values for all our alloys as a function of Ni-content x and VEC (upper abscissa). The variation of V _bcc is qualitatively the same as that of a _bcc,^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} and V _bcc tends to “saturate” for x ≥ 0.35 at values which are considerably larger than those predicted by Vegard’s law. The values of V _fcc, obtained from XRD patterns by assuming an fcc-like local atomic structure, for x ≥ 0.35 and also shown in Fig. 2 seem to agree better with the expected, approximately linear, variation of atomic volumes with concentration. This is a rather general feature of metal–metal type amorphous alloys.^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38,Reference Bakonyi49,Reference Ma, Stoica and Wang50} Therefore, the variations of both average lattice parameters^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} and atomic volumes with x (Fig. 2) indicate a change in atomic SRO in alloys with more Ni (x ≥ 0.35, VEC ≥ 7.4).

FIG. 2.

Atomic volumes and APFs of (TiZrNbCu)_1−xNi_x alloys versus x and VEC (upper abscissa). Left scale: Atomic volume V; right scale: packing fractions η. Dashed lines are guide for eye.

From the atomic (or molar) volumes, one can also calculate the average local APFs.^{Reference Ma, Stoica and Wang50} In particular, the APF for an amorphous alloy, η_a, can be calculated from the expression^{Reference Ma, Stoica and Wang50}:

(2)$${\eta _{\rm{a}}} = {{\sum\limits_k {\eta _k^0{x_k}V_k^0} } \over {{V_{\rm{a}}}}}\quad ,$$

where $\eta _k^{\rm{0}}$ is the APF of the k-th constituent of the alloy in its crystalline state, x _k and $V_k^0$ are its molar fraction and molar volume in the crystalline state, respectively, and V _a = M/D, where M is the molar mass and D is the mass density that was calculated from the atomic volumes of amorphous Ti, Zr, Nb, Cu, and Ni in Ref. Reference Bakonyi49. As can be seen in the upper part of Fig. 2, these values of η_a are approximately constant, as could be expected because V _a obeys Vegard’s law, and fairly high, η_a ≈ 0.75 ± 0.01. The APF obtained by using V _bcc values, (thus V _bcc from Fig. 2 instead of V _a) η_bcc, is also shown in the upper part of the same figure. η_bcc is like η_a approximately constant but is a little lower (lower D ^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18}) than η_a for x < 0.25 but rapidly decreases to η_bcc = 0.67 for x = 0.5. Thus, the variations of both V _bcc and η_bcc indicate changes in atomic SRO for x ≥ 0.35.

However, by replacing V _bcc with V _fcc for x = 0.5, one recovers the expected approximately constant APF, specific to amorphous alloys of the metal–metal type.^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38,Reference Ma, Stoica and Wang50} Thus, the variations of all parameters derived from XRD patterns (Fig. 2 and Ref. Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36) indicate a progressive change in local atomic arrangements accompanying the transition from HEA to the concentration region with higher Ni content. This change apparently affects the thermal parameters (Sec. III.A and Ref. Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36) and probably the interatomic bonding in the alloys studied.

However, Guinier’s procedure^{Reference Calvayrac, Chevalier, Harmelin and Quivy45} for the determination of the average distance between the nearest neighbors in amorphous alloys from XRD patterns has been vigorously criticized (e.g., Ref. Reference Hufnagel, Ott and Almer51). Therefore, we recently started SXPD measurements which provide a more direct insight into the possible changes in the atomic SRO, obtained from the structure factors S(Q) and radial distribution functions RDF(r) as well as from a more reliable determination of d. The total X-ray S(Q) was obtained from integrated raw intensity data, I ^raw(Q), using the procedure described elsewhere.^{Reference Egami and Billinge52,Reference Bednarcik, Michalik, Kolesar, Rutt and Franz53} Briefly, I ^raw(Q) was corrected for background (air scattering), self-absorption, fluorescence, and Compton scattering and then scaled and normalized into electron units using the high-angle region method.^{Reference Waseda54} The corrected intensity, I ^cor(Q), was used to calculate S(Q) by applying the Faber–Ziman formalism.^{Reference Faber and Ziman55} All the corrections mentioned above were obtained using the PDFGetX2 program.^{Reference Qiu, Thompson and Billinge56}

The S(Q) curves (not shown) consisted of a strong initial peak, followed by a series of broad, damped oscillations extending up to Q = 16 Å⁻¹, which together rule out any crystallinity in our samples with x ≥ 0.125. The position of the first peak shifted rapidly with x to higher Q for x ≤ 0.25, but the shift slowed down for x = 0.5. A more drastic change occurred in the second, split maximum of S(Q) which is probably more directly related to the atomic SRO than the first one (e.g., Ref. Reference Scudino, Stoica, Kaban, Prashanth, Vaughan and Eckert57). In particular, its asymmetry, the difference in the magnitudes of its first and second part (higher Q), was strongly reduced at x = 0.5 with respect to those observed for x ≤ 0.25.

The radial distribution functions RDF(r) were obtained through sine Fourier transformation of S(Q):

(3)$${\rm{RDF}}\left( r \right) = 4\pi {r^2}\rho \left( r \right) = 4\pi {r^2}{\rho _0} + r{2 \over \pi }\int_0^\infty {Q\left[ {S\left( Q \right) - 1} \right]} \sin \left( {rQ} \right){\rm{d}}Q\quad ,$$

where ρ(r) and ρ₀ are the local and average atomic number densities, respectively, and r is the radial distance.

RDF(r) curves shown in Fig. 3 confirm a strong change in both atomic short- and medium-range order occurring for x = 0.5. In particular, the shape of the first maximum SRO becomes strongly asymmetric on the right side of the peak, while in the second, split maximum, the magnitude of its second part (higher r) is strongly enhanced with respect to that of its first part. From the positions of the first peaks, we determined corresponding interatomic distances and average bcc-atomic volumes which agree well with those obtained from XRD patterns as demonstrated in Fig. 2. From RDF(r), we also calculated the average coordination number, N (the number of atoms in a spherical shell between radii r ₁ and r ₂ around an average atom):

(4)$$N = \int_{{r_1}}^{{r_2}} {{\rm{RDF}}\left( r \right)} {\rm{d}}r\quad .$$

FIG. 3.

Radial distribution functions RDF(r) versus r for (TiZrNbCu)_1−xNi_x alloys. The inset: corresponding coordination numbers N versus x. Dashed line in the inset is guide for eye.

The variation of N for the cut-off r ₂ at the minimum between the first and second maxima in Fig. 3 is shown in the inset of Fig. 3. A strong increase of N is observed at x = 0.5, confirming the change in SRO. We note that N ≥ 13 is usual in multicomponent MGs.^{Reference Scudino, Stoica, Kaban, Prashanth, Vaughan and Eckert57} Since for high r ₂ N may have some contribution from the second atomic shell,^{Reference Hufnagel, Ott and Almer51} we have checked the variations of N for somewhat lower values of r ₂ and obtained qualitatively the same variations with x. In particular, for a cut-off at 0.95 r ₂, we obtained N increasing from 12.4 for x ≤ 0.25 to 12.8 at x = 0.5. Therefore, the initial results of SXPD provide strong support for a change in atomic arrangements at x = 0.5 (VEC = 8). This change is accompanied by an increase in N, which is reminiscent of the proposed correlation between VEC and crystal structure in HEAs.^{Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk42,Reference Sheikh, Mao and Guo43} Due to the correlation between the local atomic order and electronic structure, we expect strong changes in the electronic structure in the region with more Ni as described in Sec. III.C.

C. Electronic structure

As already noted in Sec. I and in several recent papers (e.g., Refs. Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18, Reference Wu, Troparevsky, Gao, Morris, Stocks and Bei25, Reference Ristić, Zadro, Pajić, Figueroa and Babić28, Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36, Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38, and Reference Tian, Varga, Chen, Shen and Vitos58 for metallic systems, regardless whether they are amorphous or crystalline, the electronic structure controls practically all their intrinsic properties (those that are hardly affected by the exact preparation and/or post-processing conditions). The importance of the electronic structure in understanding the properties of alloys probably shows up the best in the case of amorphous TE–TL alloys. Soon after the discovery of these MGs, the PES revealed the split-band structure of these alloys with TL metals having a full or nearly full d sub-band for which the electronic DoS at the Fermi level, N ₀(E _F), is dominated by TE d-states.^{Reference Oelhafen, Hauser and Güntherodt59} Thus, the effect of alloying with TL is approximately described by a dilution of a-TE,^{Reference Ristić, Babić, Šaub and Miljak60} which simplifies the explanation of the linear variations of most properties of these MGs with TL content.^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38,Reference Ristić and Babić46,Reference Ristić, Babić, Šaub and Miljak60} Furthermore, N ₀(E _F) values of TE-rich alloys (determined from LTSH) were higher than those of stable hexagonal close-packed crystalline phases of corresponding TE^{Reference Bakonyi61} and were close to those calculated for hypothetical fcc structures of TEs.^{Reference Bakonyi, Ebert and Liechenstein48} High N ₀(E _F) in TE-rich MGs leads to enhanced superconductivity and magnetic susceptibility^{Reference Ristić and Babić46} but also to weaker interatomic bonding, thus to lower elastic moduli and thermal stability.^{Reference Ristić, Stubičar and Babić62} The combined studies of PES and ab initio calculations were also performed on Zr-based multicomponent MGs and showed that N ₀(E _F) is dominated by TE d-electrons as in binary MGs (e.g., Ref. Reference Hasegawa, Sato, Takeuchi, Soda and Mizutani63). We note that a combination of PES, LTSH, and ab initio theory is the best to fully comprehend the electronic structure. In particular, ordinary PES and LTSH experiments reveal the variation of the total DoS with energy (PES) and give accurate value for N ₀(E _F) (LTSH), but they cannot provide accurate contributions of the alloying elements (pDoS) to these quantities. The theory can in principle provide all these quantities as well as the probable local atomic structure (including CSRO^{Reference Hasegawa, Sato, Takeuchi, Soda and Mizutani63}) but its results are often limited by the rather small size of the sample and approximations involved in a given calculation (e.g., Refs. Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38, Reference Tian, Varga, Chen, Shen and Vitos58, and Reference Hasegawa, Sato, Takeuchi, Soda and Mizutani63).

Accordingly, we performed combined PES^{Reference Babić, Biljaković, Figueroa, Kuršumović, Mikšić Trontl, Pajić, Pervan, Remenyi, Ristić and Starešinić22,Reference Mikšić Trontl64} and LTSH^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} studies of a-(TiZrNbCu)_1−xNi_x alloys with x covering both HEA and regions with higher Ni content. The results of PES in the a-HEA region are illustrated in Fig. 4 and compared with those for binary and ternary Zr-based MGs.^{Reference Zehringer, Oelhafen, Güntherodt, Yamada and Mizutani65} The recorded “ultraviolet photoemission spectrum” (UPS) reflects the DoS within the valence band of the amorphous (TiZrNbCu)_0.8Ni_0.2 alloy. The spectral maximum at 3.5 eV below the Fermi level, associated with the Cu-3d states, is well resolved and separated from the low-energy part of the spectrum. From the comparison with the spectrum of amorphous (Ni_0.33Zr_0.67)_0.85Cu_0.15 and Ni_0.33Zr_0.67 alloys,^{Reference Zehringer, Oelhafen, Güntherodt, Yamada and Mizutani65} shown in the lower part of Fig. 4, it is evident that the low-energy part of the spectrum has contributions from Ni-3d and Zr-4d bands.^{Reference Zehringer, Oelhafen, Güntherodt, Yamada and Mizutani65} It appears that most of the spectral intensity between 1 and 2 eV below E _F comes from the DoS associated with Ni-3d bands.^{Reference Mikšić Trontl, Pervan and Milun66,Reference Calloni, Bussetti, Berti, Yivlialin, Camera, Finazzi, Duò and Ciccacci67} The Ni contribution to the intensity at the Fermi level is not negligible and is likely to increase at high Ni-content.^{Reference Greig, Gallagher, Howson, Law, Norman and Quinn68} The Zr-4d band is expected to contribute to the DoS mainly at the Fermi level^{Reference Takahara and Narita69} as is the Ti-3d band, as judged from the spectrum from pure Ti^{Reference Höchst, Steiner, Reiter and Hiifner70} and an amorphous (Ni_0.33Zr_0.67)_0.85Ti_0.15 alloy.^{Reference Zehringer, Oelhafen, Güntherodt, Yamada and Mizutani65} The Nb-4d band contributes to the spectral intensity at the Fermi level and around 1.2 eV^{Reference Xiang, Liu, Li, Yang, Liu, Fan, Shen, Wang and Liu71} but possibly to a lesser extent than Ni.^{Reference Kübler, Bennemann, Lapka, Rösel, Oelhafen and Güntherodt72} The s–p bands of all constituents span a larger energy range and generally contribute less to the photoemission intensity. Therefore, we could not extract their contribution to the DoS from the spectra of the alloys. The effects of increasing Ni content x and of crystallization of the sample with x = 0.125 on photoemission spectra have also been studied.^{Reference Mikšić Trontl64} The present results add to mounting evidence that PES is an efficient tool for assessing the contributions of individual constituents to the electronic properties of 3d and 4d transition metal-based HEAs. Indeed, the insight obtained from PES will be helpful for the interpretation of LTSH, magnetic susceptibility, and superconductivity data for the same alloys.

FIG. 4.

UPS of (TiZrNbCu)_0.8Ni_0.2 (blue dots^{Reference Mikšić Trontl64}), and for comparison, of (Ni_0.33Zr_0.67)_0.85Cu_0.15 and Ni_0.33Zr_0.67 from Ref. Reference Zehringer, Oelhafen, Güntherodt, Yamada and Mizutani65.

As noted above, LTSH provides a quantitative insight into the DoS at E _F which controls the physical properties of metallic systems. In particular, the Sommerfeld coefficient of the linear term in LTSH is given as^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36,Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38}

(5)$$\gamma = {{{\pi ^2}k_{\rm{B}}^2{N_0}\left( {{E_{\rm{F}}}} \right)\left( {1 + {\lambda _{{\rm{e}} - {\rm{p}}}}} \right)} \over 3}\quad ,$$

where k _B is the Boltzmann constant and λ_e–p represents the electron–phonon enhancement of the DoS at E _F, N ₀(E _F). Since the local atomic arrangements determine the ES of alloys (e.g., Ref. Reference Ristić, Zadro, Pajić, Figueroa and Babić28), we expect a change of N ₀(E _F), thus also γ of our alloys, in going from a-HEAs (x ≤ 0.25) to conventional amorphous alloys with higher Ni content (x ≥ 0.35). In Fig. 5, we compare the variations of γ with TL content in our alloys^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} with those in several binary and ternary a-TE–TL alloys composed from the same TEs and TLs.^{Reference Moody, Ng, Eckern, Schmid, Weber and Wühl73–Reference Karkut and Hake77} In Fig. 5, we plot γ versus total TL content (that of Cu and Ni) in our alloys since in binary TE–TL amorphous alloys, the variations of γ in the alloys with TL = Cu and Ni are very similar, thus it is the total TL content which causes a decrease of γ in alloys containing both Cu and Ni. (If we plotted γ versus Ni content only, as was done in Ref. Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36, this will only shift the data in Fig. 5 toward the left but will not change the overall variation with x.) As can be seen in Fig. 5, in our alloys, γ values initially decrease with x as in other a-TE–TL alloys, but they saturate for x ≥ 0.35, this does not occur in binary alloys at similar TL contents. In particular, γ of binary alloys seems to follow a linear decrease with x throughout the x-range explored (x ≤ 0.70). Possible exceptions are a-Zr–Ni alloys, which seem to show slower variation of γ with x for x > 0.60 (VEC > 7.5).^{Reference Bakonyi61} In these alloys, this change is associated with a strong CSRO effect^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38,Reference Bakonyi49,Reference Bakonyi61,Reference Hasegawa, Sato, Takeuchi, Soda and Mizutani63} and the change in the position of E _F with respect to the two sub-bands in the DoS, as evidenced by Hall effect and thermopower measurements.^{Reference Ivkov, Babić and Jacobs78} Ab initio studies of the atomic and electronic structure of binary amorphous alloys of Ni with Ti, Zr, V, or Nb performed by Hausleitner and Hafner^{Reference Hasegawa, Sato, Takeuchi, Soda and Mizutani63} corroborated these findings. They found an increase in CSRO and a change in local atomic order at high Ni contents. Simultaneously, the d-band of Ni broadened and shifted towards the Fermi energy. These effects could also affect the variation of γ with Ni content in our alloys, but preliminary measurements of the Hall effect in a-HEAs do not seem to indicate such band-crossing^{Reference Kuveždić, Tafra, Basletić, Ristić, Trontl, Pervan, Figueroa and Babić79} for x < 0.5. As regards its magnitude, γ of our alloys with x = 0.35 agrees rather better with those for a-Zr–Cu, Ni alloys than with those for a-Ti–Cu, Ni alloys, in spite of the sizable Ti-content. This may be due to the Nb content in our alloys which reduces both their Zr and Ti contents and, as can be seen in Fig. 5, the addition of Nb strongly suppresses γ in a-Zr–Nb–Ni alloys.^{Reference Karkut and Hake77} Since γ depends on the DoS which is enhanced by the electron–phonon interaction, it cannot be used to prove that the change in γ for x ≥ 0.35 is due to N ₀(E _F). However, in superconducting transition metal alloys, one can use the McMillan expression^{Reference McMillan80} to disentangle N ₀(E _F) from λ_e–p. By using the results of recent measurements of the superconducting transition temperatures, T _c,^{Reference Kuveždić, Tafra, Basletić, Ristić, Trontl, Pervan, Figueroa and Babić79} we have calculated the values of N ₀(E _F) shown in Fig. 6, which confirm a change in N ₀(E _F) for x ≥ 0.35. Thus, our study of the electronic structure is consistent with that of the atomic structure in Sec. III.B, in that a change of electronic structure accompanies the change in local atomic arrangements for x > 0.35. This change in electronic structure exerts strong influence on physical properties, as demonstrated in Sec. III.D.

FIG. 5.

Variation of electronic coefficient γ versus x for (TiZrNbCu)_1−xNi_x^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} and some binary and ternary amorphous alloys. Data for Ti–Cu are from Ref. Reference Moody, Ng, Eckern, Schmid, Weber and Wühl73, for Ti–Ni from Ref. Reference Kanemaki, Suzuki, Yamada and Mizutani74, for Zr–Cu from Ref. Reference Garoche and Bigot75, for Zr–Ni from Ref. Reference Matsuura and Mizutani76, for (Zr_0.8Ti_0.2)_0.78Ni_0.22 and (Zr_0.8Nb_0.2)_0.78Ni_0.22 from Ref. Reference Karkut and Hake77. Dashed lines are guide for eye.

FIG. 6.

Density of states N ₀(E _F) and Pauli susceptibility for (TiZrNbCu)_1−xNi_x alloys versus x and VEC (upper abscissa). Left scale: χ_p and right scale: N ₀(E _F). Dashed lines are guide for eye.

D. Physical properties

Among the physical properties, the paramagnetic Pauli susceptibility of nonmagnetic alloys is directly related to the electronic structure. However, the magnetic susceptibility χ_exp of transition metals and alloys is quite complex and consists of three main contributions^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38,Reference Bakonyi, Ebert and Liechenstein48} :

(6)$${\chi _{\exp }} = {\chi _{\rm{p}}} + {\chi _{{\rm{dia}}}} + {\chi _{{\rm{orb}}}}\quad ,$$

where χ_p is the Pauli paramagnetic contribution of d-electrons and χ_dia and χ_orb are the diamagnetic and orbital paramagnetic contributions, respectively. χ_dia and χ_orb are calculated by adding corresponding contributions from the constituents.^{Reference Bakonyi, Ebert and Liechenstein48,Reference Banhart, Ebert and Voitlander81,Reference Place and Rhodes82} The Pauli paramagnetism of a d-band is enhanced over the free-electron value, $\chi _{\rm{p}}^0 = {\mu _{\rm{0}}}\mu _{\rm{B}}^2{N_0}\left( {{E_{\rm{F}}}} \right)$ (where µ₀ is the permeability of the vacuum and µ_B is the Bohr magneton), by the exchange interaction, namely, ${\chi _{\rm{p}}} = S\chi _{\rm{p}}^0$, where S is the Stoner enhancement factor, which also depends on N ₀(E _F). But in many amorphous TE–TL alloys,^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38} S is nearly constant within their GFR, thus the variation of χ_p with concentration is dominated with N ₀(E _F). In spite of its complex structure, χ_exp in a-TE–TL alloys often varies with composition in qualitatively the same way as γ.^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38} This is probably due to an approximately linear decrease of χ_orb with TL content, which does not affect the overall dependence on composition. This has also been observed in our alloys,^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} therefore it seems to be quite general feature of nonmagnetic a-TE–TL alloys which does not depend on their number of components. Indeed, when the variation of χ_exp with x of our alloys is compared with those for binary amorphous alloys made of the same constituent TE and TL metals,^{Reference Babić, Biljaković, Figueroa, Kuršumović, Mikšić Trontl, Pajić, Pervan, Remenyi, Ristić and Starešinić22} the result is qualitatively the same as that for the corresponding values of γ shown in Fig. 5. In particular, the variation of χ_exp with x for x ≤ 0.35 is close to that for a-Zr–Ni alloys but tends to saturate for x > 0.35 (as does γ in Fig. 5). In Fig. 6, we show that the variations of χ_p = χ_exp − (χ_dia + χ_orb) and N ₀(E _F) in our alloys are qualitatively the same, which shows that the change in the electronic structure for x ≥ 0.35 (VEC = 7.4) affects the magnetic properties of our alloys.

As already noted, LTSH also provides information on the atomic vibrations and interatomic bonding.^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38,Reference Remenyi, Biljaković, Starešinić, Dominko, Ristić, Babić, Figueroa and Davies41} This information is contained in the Debye βT ³ phonon term in C _p. Since β ∝ θ_D⁻³,^{Reference Remenyi, Biljaković, Starešinić, Dominko, Ristić, Babić, Figueroa and Davies41} one can calculate θ_D from the measured β. The variation of θ_D with x in our alloys shown in Fig. 7 is qualitatively the same as those of the thermal stability parameters T _x and T _l^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36}: θ_D increases rapidly for x ≤ 0.35 but tends to saturate in the Ni-rich region. Since the average atomic mass in the alloys studied changes linearly and relatively little with x,^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18} the abrupt change in the variation of θ_D for x > 0.35 is apparently related to a change in interatomic bonding and electronic structure, as was the case with T _x and T _l.^{Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} Careful measurements^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18} have shown that C _p increases faster with temperature than the T ³ law predicted by the Debye model.^{Reference Remenyi, Biljaković, Starešinić, Dominko, Ristić, Babić, Figueroa and Davies41} This indicates the presence of the so-called “BP”, an excess of low-energy vibrational states with respect to that predicted by the Debye model (e.g., Ref. Reference Cheng and Ma83). A preliminary analysis shows a nonmonotonic variation of the magnitude of BP with Ni-content (the inset of Fig. 7) similar to that of θ_D. To our knowledge, this is the first observation of a possible correlation between the ordinary (θ_D) and excess atomic vibrations in amorphous solids. At present, we have no proper explanation for the observed behavior of the BP in our alloys. However, a change in local atomic arrangements with Ni concentration is expected to affect the atomic vibrations, thus the observed variations of both the BP and θ_D seem plausible. At present, there is no commonly accepted explanation of the BP in glassy systems: some researchers emphasize localized phonon modes (like those caused by oscillations of loosely bonded atoms within the cages of surrounding atoms), whereas others ascribe the BP to a smeared van Hove anomaly.^{Reference Remenyi, Biljaković, Starešinić, Dominko, Ristić, Babić, Figueroa and Davies41,Reference Cheng and Ma83,Reference Salčinović Fetić, Remenyi, Starešinić, Kuršumović, Babić, Sulejmanović and Biljaković84} We note however that the size of the BP depends quite strongly on the amount of the quenched-in disorder which complicates the study of its variation with composition.^{Reference Salčinović Fetić, Remenyi, Starešinić, Kuršumović, Babić, Sulejmanović and Biljaković84} It is important to note that in our^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36} and other a-TE–TL alloys (e.g., Ref. Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38) as well as in the crystalline cubic TE-based HEAs,^{Reference Koželj, Vrtnik, Jelen, Jazbec, Jagličić, Maiti, Feuerbacher, Steurer and Dolinšek21} the rule-of-mixtures does not describe θ_D properly. This conclusion holds even in the case of a self-consistent choice of the values of θ_D of the constituent elements.^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18}

FIG. 7.

Phonon coefficients ß and Debye temperatures of (TiZrNbCu)_1−xNi_x alloys versus x and VEC (upper abscissa). Left scale: variation of θ_D; right scale: variation of β. The inset: variation of the magnitude of BP versus x. Dashed lines are guide for eye.

As noted earlier,^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36,Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38,Reference Ristić and Babić46,Reference Hasegawa, Sato, Takeuchi, Soda and Mizutani63,Reference Cheng and Ma83,Reference Ristić, Babić, Stubičar, Kuršumović, Cooper, Figueroa, Davies, Todd, Varga and Bakonyi85} there is a very simple experimental relationship between ES and mechanical properties (including hardness) and thermal stability (represented by T _x and T _l) of the nonmagnetic a-TE–TL alloys. In particular, a decrease in N ₀(E _F) or γ or χ_exp is usually accompanied by increases in Young’s modulus, hardness, Debye temperature, and thermal stability of these alloys. Therefore, a decrease in DoS at E _F reflects in these systems an increase in the interatomic bonding, and accordingly the stiffness and parameters related to atomic vibrations and thermal stability increase too. We note that such a simple relationship between the electronic structure and interatomic bonding is quite common in crystalline alloys and in nonmagnetic a-TE–TL alloys, it probably stems from their simple electronic band structure.^{Reference Ristić, Stubičar and Babić62} In Fig. 8, we compare the variations of E of our relaxed samples that have received a short anneal close to T _g, with total (Cu + Ni) content to those of binary a-TE–TL alloys composed from the same TEs and TLs.^{Reference Marohnić, Guberović, Babić and Morgan86–Reference Ristić, Babić, Pajić, Zadro, Kuršumović, Figueroa, Davies, Todd, Varga and Bakonyi90} Comparing the data in Figs. 5, 6, and 8, we see that a correlation between the electronic structure and E is obeyed by our alloys, too. Furthermore, the approximately linear variation of E with x in binary alloys seems to be replaced by a more complex variation in our multicomponent alloys. This probably reflects a change in the SRO and the electronic structure for x (Cu + Ni) > 0.51. The sensitivity of E to quenched-in disorder and the degree of relaxation introduces considerable uncertainty in the variation of E with x. As already noted,^{Reference Biljaković, Remenyi, Figueroa, Ristić, Pajić, Kuršumović, Starešinić, Zadro and Babić18,Reference Figueroa, Ristić, Kuršumović, Biljaković, Starešinić, Pajić, Remenyi and Babić36,Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38} RoM provides a poor description of E in all a-TE–TL alloys and it also fails to describe the mechanical properties of cubic crystalline TE-based HEAs.^{Reference Maiti and Steurer91}

FIG. 8.

Variation of E versus x for (TiZrNbCu)_1−xNi_x and some binary amorphous alloys. Data for Zr–Cu are from Refs. Reference Marohnić, Guberović, Babić and Morgan86 and Reference Whang, Polk, Giessen, Masumoto and Suzuku87, for Zr–Ni from Refs. Reference Marohnić, Guberović, Babić and Morgan86 and Reference Cohrane, Destry, El Amrani, Altounian, Strom-Olsen, Steeb and Warlimont88, for Ti–Cu from Refs. Reference Ristić, Babić, Stubičar, Kuršumović, Cooper, Figueroa, Davies, Todd, Varga and Bakonyi85, Reference Davis, Chou, Tanner and Ray89, and Reference Ristić, Babić, Pajić, Zadro, Kuršumović, Figueroa, Davies, Todd, Varga and Bakonyi90, and Ti–Ni from Ref. Reference Whang, Polk, Giessen, Masumoto and Suzuku87. Dashed lines are guide for eye.

For the sake of completeness, in what follows we briefly summarize some results of an ongoing study of the electronic transport properties of our alloys.^{Reference Kuveždić, Tafra, Basletić, Ristić, Trontl, Pervan, Figueroa and Babić79} As is usual in a-TE–TL alloys,^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38} the electrical resistivities ρ of a-(TiZrNbCu)_1−xNi_x alloys are high (>150 µΩ cm) and accordingly decrease with increasing temperature over most of the explored temperature range (T ≤ 300 K). The variation of their electrical conductivities seems dominated by weak electron localization effects^{Reference Fukuyama and Hoshino92} over a broad temperature range (10–300 K) as is usual in a-TE–TL alloys.^{Reference Šaub, Babić and Ristić93–Reference Ristić, Babić and Šaub95} All the samples are superconducting, but with a low T _c ≤ 1.6 K. For x ≤ 0.35, T _c decreases approximately linearly with x, as is usual in nonmagnetic a-TE–TL alloys.^{Reference Tafra, Basletić, Ristić, Babić and Hamzić96} However, at higher x = 0.5, T _c tends to become constant, hence it follows the same trend with x as γ and N ₀(E _F), as is usual for disordered transition metal alloys.^{Reference Tafra, Basletić, Ristić, Babić and Hamzić96} A preliminary study of the Hall effect for x ≤ 0.25 shows that the Hall coefficients R _H are positive and change only a little with x, again in accord with the results for a-TE–TL alloys with lower TL contents.^{Reference Ristić, Cooper, Zadro, Pajić, Ivkov and Babić38,Reference Ivkov, Babić and Jacobs78} Thus, the results for all physical properties of our multicomponent alloys studied until now seem similar to those of the corresponding binary alloys. This probably stems from their similar split-band electronic structures (Fig. 3).