Study of superconducting state parameters of ternary metallic glasses through pseudopotential approach

Abstract

A theoretical investigation on the screening dependence of the superconducting state parameters (SSPs) viz. the electron-phonon coupling strength λ, the Coulomb pseudopotential μ∗, the transition temperature T_C, the isotope effect exponent α and the effective interaction strength N₀V of some ternary metallic glasses such as Ti₅₀Be₃₄Zr₁₀, (Mo_0.6Ru_0.4)₇₈B₂₂, (Mo_0.6Ru_0.4)₈₀B₂₀, (Mo_0.4Ru_0.6)₈₀P₂₀, (Mo_0.6Ru_0.4)₇₀Si₃₀, (Mo_0.6Ru_0.4)₈₄B₁₆, (Mo_0.6Ru_0.4)₇₂Si₂₈, (Mo_0.6Ru_0.4)₈₆B₁₄, (Mo_0.6Ru_0.4)₇₆Si₂₄, (Mo_0.6Ru_0.4)₇₈Si₂₂, (Mo_0.6Ru_0.4)₈₀Si₂₀, (Mo_0.6Ru_0.4)₈₂Si₁₈ and (Mo_0.6Ru_0.4)₈₀P₂₀ is reported for the first time using Ashcroft's empty core (EMC) model potential. Five local field correction functions proposed by Hartree (H), Taylor (T), Ichimaru-Utsumi (IU), Farid et al (F) and Sarkar et al (S) are used in the present investigation to study the effect of screening on the aforesaid properties. It is observed that λ and T_C are reasonably sensitive to the selection of the local field correction functions, whereas μ∗, α and N₀V show weak dependences on the local field correction functions. The transition temperature T_C obtained from the H-local field correction function is found to be in excellent agreement with available experimental data. Also, the present results are found to be in qualitative agreement with other earlier reported data, which confirms the existence of the superconducting phase in the above ternary metallic glasses.

Introduction

During the last several years, superconductivity has remained a dynamic area of research in condensed-matter physics because of the continual discoveries of novel materials and the increasing demand for novel devices for sophisticated technological applications. A large number of metals and amorphous alloys are superconductors, with transition temperature T_C ranging from 1–18 K [1–19]. Pseudopotential theory has been used successfully to explain the superconducting state parameters (SSPs) for metallic complexes in many studies [3–19]. In most studies the well-known model pseudopotential has been used in the calculation of the SSPs for metallic complexes. Recently, we have studied SSPs of some metallic superconductors based on various elements of the periodic table using a single parametric model potential [3–19]. The study of the SSPs of ternary metallic glasses may be of great help in determining their applications; the study of the dependence of the transition temperature T_C on the composition of metallic elements has been helpful in finding new superconductors with a high T_C. The application of pseudopotential to a ternary system involves the assumption of pseudoions with average properties, which are assumed to replace the three types of ions in the ternary systems, and a gas of free electrons is assumed to permeate through them. The electron-pseudoion interaction is accounted for by the pseudopotential and the electron–electron interaction is involved using a dielectric screening function. For the successful prediction of the superconducting properties of alloying systems, the selection of a suitable pseudopotential and screening function is essential [3–19].

Therefore, in the present article, we have used the well-known McMillan theory [20] of superconductivity for predicting the SSPs of Ti₅₀Be₃₄Zr₁₀, (Mo_0.6Ru_0.4)₇₈B₂₂, (Mo_0.6Ru_0.4)₈₀B₂₀, (Mo_0.4Ru_0.6)₈₀P₂₀, (Mo_0.6Ru_0.4)₇₀Si₃₀, (Mo_0.6Ru_0.4)₈₄B₁₆, (Mo_0.6Ru_0.4)₇₂Si₂₈, (Mo_0.6Ru_0.4)₈₆B₁₄, (Mo_0.6Ru_0.4)₇₆Si₂₄, (Mo_0.6Ru_0.4)₇₈Si₂₂, (Mo_0.6Ru_0.4)₈₀Si₂₀, (Mo_0.6Ru_0.4)₈₂Si₁₈ and (Mo_0.6Ru_0.4)₈₀P₂₀ ternary metallic glasses for the first time. We have used Ashcroft's empty core (EMC) model potential [21] for studying the electron-phonon coupling strength λ, Coulomb pseudopotential μ∗, transition temperature T_C, isotope effect exponent α and effective interaction strength N₀V for the first time. To determine the impact of various exchange and correlation functions on the aforesaid properties, we have employed five different types of local field correction functions proposed by Hartree (H) [22], Taylor (T) [23], Ichimaru-Utsumi (IU) [24], Farid et al (F) [25] and Sarkar et al (S) [26]. For the first time we have incorporated the more advanced local field correction functions proposed by Ichimaru-Utsumi [24], Farid et al [25] and Sarkar et al [26] with the EMC model potential [21] in the present computation of the SSPs for ternary metallic glasses.

In the present work, the pseudo-alloy-atom (PAA) model was used to explain the electron-ion interactions in alloying systems [3–19]. It is well known that the PAA model is a more meaningful approach for explaining such interactions in alloying systems. In the PAA approach a hypothetical monoatomic crystal is assumed to be composed of PAAs, which occupy the lattice sites and form a perfect lattice in the same way as pure metals. In this model the hypothetical crystal made up of PAAs is assumed to have the same properties as the actual disordered alloy material, and pseudopotential theory is then applied to study various properties of alloy systems and metallic glasses. The complete miscibility in alloy systems is considered as a special case. Therefore, in such alloying systems the atomic matrix elements in the pure states are affected by the characteristics of alloys such as lattice distortion effects and charging effects. In the PAA model, such effects are involved implicitly. In addition to this the PAA model also implicitly takes into account self-consistent treatment. Considering these advantages of the PAA model, we propose the use of the PAA model for the first time to investigate the SSPs of ternary metallic glasses.

Computational methodology

In the present investigation on ternary metallic glasses, λ is computed using the relation [3–19]

Here, m_b is the band mass, M is the ionic mass, Ω₀ is the atomic volume, k_F is the Fermi wave vector, W(q) is the screened pseudopotential and 〈ω²〉 is the averaged square phonon frequency of the ternary metallic glass. 〈ω²〉 is calculated using the relation given by Butler [27]; 〈ω²〉^1/2=0.69θ_D, where θ_D is the Debye temperature of the ternary system.

Using X=q/2k_F and Ω₀=3π²Z/(k_F)³, we obtain equation (1) in the following form,

where Z and W(X) are the valence and the screened EMC pseudopotential [21] of the ternary metallic glass, respectively.

The well-known screened Ashcroft's EMC model potential [21] used in the present computations of the SSPs of ternary metallic glasses is of the form

where r_C is the parameter of the model potential of the ternary metallic glass. The Ashcroft's EMC model potential is a simple one-parameter model potential [21], which has been successfully found for various metallic complexes [5–16]. When used with a dielectric screening functions with a suitable form, this potential has also been found to yield good results for computing the SSPs of metallic complexes [5–16]. Therefore, in the present work we use Ashcroft's EMC model potential [21] with the more advanced IU- [24], F- [25] and S- [26] -local field correction functions for the first time. The model potential parameter r_C may be obtained by fitting, either to some experimental data or to realistic form factors or other data relevant to the properties being investigated. In the present work, r_C is fitted with experimental values of T_C for the ternary metallic glasses [28] for most of the local field correction functions.

The Coulomb pseudopotential μ∗ is given by [3–19]

where E_F is the Fermi energy, m_b is the band mass of the electron, θ_D is the Debye temperature and ε(X) is the modified Hartree dielectric function, which is written as [22]

ε_H(X) is the static Hartree dielectric function [22] and f(X) is the local field correction function. In the present investigation, the local field correction functions due to Hartree [22], Taylor [23], Ichimaru–Utsumi [24], Farid et al [25] and Sarkar et al [26] are incorporated to determine the impact of the exchange and correlation effects.

Hartree screening function [22] is purely static and does not include the exchange and correlation effects. It is expressed as,

Taylor [23] introduced an analytical expression for the local field correction function that satisfies the compressibility sum rule exactly. This is the most commonly used local field correction function and covers the overall features of the various local field correction functions proposed before 1972. According to Taylor [23],

The IU-local field correction function [24] is a fitting formula for the dielectric screening function of degenerate electron liquids at metallic and lower densities and also accurately reproduces the Monte-Carlo results. Furthermore, it satisfies the self-consistency condition in the compressibility sum rule and short-range correlations. The fitting formula is

On the basis the of IU-local field correction function [24], Farid et al [25] proposed a local field correction function of the form

On the basis of equations (8) and (9), Sarkar et al [26] proposed a local field correction function with a simple form

where Q=2X. The parameters A_IU, B_IU, C_IU, A_F, B_F, C_F, D_F, A_S, B_S and C_S are the atomic-volume-dependent parameters of the IU-, F- and S-local field correction functions. The mathematical expressions of these parameters are given in the respective papers proposing the local field correction functions [24–26].

After evaluating λ and μ∗, T_C and α are investigated using the McMillan formula [3–20]

The expression for N₀V is [3–19]

Results and discussion

The values of the input parameters for the ternary metallic glasses under investigation are listed in table 1. To determine the input parameters and various constants for the PAA model [3–19], the following definitions are adopted for the metallic glass with a formula of (A_1−yB_y)_100−xC_x,

where A, B and C are the first, second and third pure metallic components and y and x are the concentration factors of the second and third metallic components, respectively. The input parameters Z and Ω₀ of the pure metallic components are taken from the literature [22], while the data for M are taken from standard references. The values of the Debye temperature θ_D of the ternary glassy alloys are directly obtained from Mizutani [28]. First, we calculate the input parameters in the square parentheses using equations (15)–(17) and the PAA model from the pure metallic data for the elements. Then, the data for the third metallic element is included in the PAA input parameters.

Table 1.

Input parameters and other constants.

Ternary metallic glass	Z	rC (au)	Ω0 (au)3	M (amu)	θD (K)	〈ω2〉2×10−6 (au)2
Ti50Be34Zr10	2.70	0.8798	96.42	36.68	190.00	0.69415
(Mo0.6Ru0.4)78B22	4.72	0.5894	88.65	78.84	280.00	1.50751
(Mo0.6Ru0.4)80B20	4.76	0.5843	89.66	80.59	277.00	1.47538
(Mo0.4Ru0.6)80P20	4.84	0.5908	107.10	85.46	267.00	1.37078
(Mo0.6Ru0.4)70Si30	4.84	0.6692	110.13	77.05	554.70	5.91644
(Mo0.6Ru0.4)84B16	4.85	0.5205	91.68	84.08	301.00	1.74212
(Mo0.6Ru0.4)72Si28	4.86	0.6616	109.44	78.45	552.12	5.86153
(Mo0.6Ru0.4)86B14	4.89	0.5314	92.69	85.82	295.00	1.67335
(Mo0.6Ru0.4)76Si24	4.91	0.6544	108.06	81.25	546.96	5.75248
(Mo0.6Ru0.4)78Si22	4.94	0.6444	107.37	82.64	544.38	5.69834
(Mo0.6Ru0.4)80Si20	4.96	0.6407	106.68	84.04	541.80	5.64445
(Mo0.6Ru0.4)82Si18	4.98	0.6353	105.99	85.44	539.22	5.59083
(Mo0.6Ru0.4)80P20	5.16	0.5746	109.26	84.62	265.00	1.35032

The presently calculated results of the SSPs of ternary glassy alloys are tabulated in table 2 with the other experimental results [28]. It can be seen from table 2 that among all the five screening functions, the screening function due to Hartree (only static, without exchange and correlation) gives the minimum value of the SSPs while the screening function due to Farid et al gives the maximum value. It can also be observed from table 2 that the electron-phonon coupling strength λ increases for (Mo_0.6Ru_0.4)_100−xB_x and (Mo_0.6Ru_0.4)_100−xSi_x as the concentration x of the third metallic element decreases. The increase in λ with x shows a gradual transition from weak coupling behaviour to intermediate coupling behaviour between electrons and phonons, which may be attributed to an increase in the hybridization of sp-d electrons of the third metallic element with increasing concentration. This may also be attributed to the increase in the role of ionic vibrations in the region rich in the third metallic elements. Also, note that for all the ternary metallic glasses, μ∗ lies between 0.11 and 0.14, which is in accordance with McMillan [20], who suggested that μ∗≈0.13 for transition metals. The weak screening effect is reflected in the computed values of μ∗. Table 2 also contains the calculated values of T_C for ternary metallic glasses computed from the various forms of the local field correction functions along with the experimental results [28]. From table 2, note that the present results obtained from the H-local field correction functions are in good agreement with available experimental data [28]. The theoretical data T_C for most of the ternary metallic glasses is not available in the literature. It can also be seen from table 2 that T_C decreases for (Mo_0.6Ru_0.4)_100−xB_x and (Mo_0.6Ru_0.4)_100−xSi_x ternary systems as the concentration of the third metallic elements increases. The same trend was observed from the graphs of the presently computed T_C for (Mo_0.6Ru_0.4)_100−xB_x and (Mo_0.6Ru_0.4)_100−xSi_x plotted against the concentration (x in at.%) of the third metallic component, as displayed in figures 1 and 2. The present results for the SSPs of Ti₅₀Be₃₄Zr₁₀ ternary glass are found to be in fair agreement with the available experimental data [28]. Figure 3 shows the variation of T_C with the valance (Z) of the ternary metallic glasses. All the graphs suggest that the presently reported metallic glasses have a superconducting nature. The computed values of T_C are found to be in a range that is indicates the suitability of further investigating the applications of the ternary metallic glasses for uses such as lossless transmission lines for cryogenic applications. If alloying elements exhibit good elasticity and can be drawn in the form of wires, they may have potential use as superconducting transmission lines at low temperatures of the order of 7 K. The values of α for the ternary metallic glasses are also tabulated in table 2. The computed values of α show a weak dependence on the dielectric screening. Since experimental values of α have not yet been reported, the present data for α may be used for the study of ionic vibrations in the superconductivity of alloying substances. Since the H-local field correction function yields the best results for λ and T_C, it may be observed that values of α obtained from this screening provide the best data for determining the role of ionic vibrations in the superconducting behaviour of systems. The values of N₀V are also listed in table 2. It is observed that the magnitudes of N₀V for the ternary metallic glasses under investigation lie in the range of those of weak-coupling superconductors. The values of N₀V also show a weak dependence on dielectric screening.

Table 2.

Superconducting state parameters of ternary metallic glasses.

Ternary metallic glass		Present results					Expt. [20]
	SSP	H	T	IU	F	S
Ti50Be34Zr10	λ	0.3813	0.5099	0.5319	0.5328	0.4615	0.40
	μ∗	0.1201	0.1290	0.1302	0.1304	0.1251	–
	TC (K)	0.2743	1.2943	1.5441	1.5506	0.8338	0.274
	α	0.2639	0.3512	0.3604	0.3602	0.3305	–
	N0V	0.1940	0.2602	0.2708	0.2711	0.2370	–
(Mo0.6Ru0.4)78B22	λ	0.6321	0.8295	0.8623	0.8655	0.7245	–
	μ∗	0.1131	0.1206	0.1216	0.1218	0.1165	–
	TC (K)	5.4038	10.1946	10.9880	11.0608	7.6583	5.40, 5.42
	α	0.4329	0.4499	0.4519	0.4520	0.4429	–
	N0V	0.3296	0.4041	0.4152	0.4162	0.3666	–
(Mo0.6Ru0.4)80B20	λ	0.6475	0.8507	0.8844	0.8879	0.7422	–
	μ∗	0.1130	0.1205	0.1215	0.1217	0.1164	–
	TC (K)	5.7510	10.6463	11.4495	11.5255	8.0532	5.75
	α	0.4359	0.4520	0.4538	0.4539	0.4453	–
	N0V	0.3364	0.4118	0.4229	0.4240	0.3737	–
(Mo0.6Ru0.4)84B16	λ	0.6554	0.8690	0.9044	0.9091	0.7488	–
	μ∗	0.1142	0.1218	0.1228	0.1230	0.1176	–
	TC (K)	6.4006	11.9833	12.8916	13.0077	8.8574	6.40
	α	0.4356	0.4522	0.4541	0.4543	0.4446	–
	N0V	0.3392	0.4174	0.4289	0.4304	0.3756	–
(Mo0.6Ru0.4)86B14	λ	0.6542	0.8665	0.9017	0.9063	0.7481	–
	μ∗	0.1140	0.1216	0.1226	0.1228	0.1174	–
	TC (K)	6.2534	11.6941	12.5811	12.6900	8.6755	6.25
	α	0.4357	0.4522	0.4541	0.4543	0.4448	–
	N0V	0.3388	0.4167	0.4282	0.4296	0.3754	–
(Mo0.4Ru0.6)80P20	λ	0.6183	0.8252	0.8603	0.8642	0.7187	0.65
	μ∗	0.1156	0.1235	0.1246	0.1248	0.1195	–
	TC (K)	4.6833	9.4181	10.2240	10.3100	6.9723	4.68
	α	0.4260	0.4463	0.4486	0.4488	0.4381	–
	N0V	0.3218	0.4009	0.4128	0.4141	0.3624	–
(Mo0.6Ru0.4)80P20	λ	0.7226	0.9622	1.0026	1.0073	0.8361	0.65, 0.71
	μ∗	0.1147	0.1224	0.1235	0.1237	0.1184	−
	TC (K)	7.3102	12.7635	13.6323	13.7281	9.9639	6.0, 7.31
	α	0.4450	0.4585	0.4600	0.4601	0.4528	−
	N0V	0.3669	0.4479	0.4599	0.4613	0.4078	−
(Mo0.6Ru0.4)70Si30	λ	0.4893	0.6465	0.6731	0.6752	0.5716	−
	μ∗	0.1269	0.1366	0.1379	0.1381	0.1317	−
	TC (K)	3.2042	8.8915	10.0186	10.0944	5.9950	3.20
	α	0.3450	0.3920	0.3973	0.3974	0.3760	−
	N0V	0.2508	0.3212	0.3321	0.3328	0.2895	−
(Mo0.6Ru0.4)72Si28	λ	0.5008	0.6620	0.6892	0.6914	0.5845	−
	μ∗	0.1266	0.1362	0.1375	0.1377	0.1313	−
	TC (K)	3.6039	9.6184	10.7901	10.8723	6.5512	3.60
	α	0.3539	0.3976	0.4025	0.4026	0.3824	−
	N0V	0.2572	0.3282	0.3392	0.3400	0.2959	−
(Mo0.6Ru0.4)76Si24	λ	0.5131	0.6773	0.7050	0.7073	0.5975	−
	μ∗	0.1259	0.1354	0.1367	0.1369	0.1305	−
	TC (K)	4.0523	10.3443	11.5459	11.6318	7.1304	4.05
	α	0.3635	0.4035	0.4080	0.4081	0.3894	−
	N0V	0.2640	0.3354	0.3463	0.3472	0.3026	−
(Mo0.6Ru0.4)78Si22	λ	0.5315	0.7019	0.7306	0.7331	0.6181	−
	μ∗	0.1256	0.1350	0.1362	0.1365	0.1301	−
	TC (K)	4.7547	11.5329	12.7978	12.8935	8.0647	4.75
	α	0.3744	0.4104	0.4145	0.4146	0.3974	−
	N0V	0.2737	0.3461	0.3572	0.3580	0.3125	−
(Mo0.6Ru0.4)80Si20	λ	0.5379	0.7101	0.7390	0.7416	0.6250	−
	μ∗	0.1252	0.1346	0.1359	0.1361	0.1297	−
	TC (K)	5.0087	11.9174	13.1954	13.2931	8.3770	5.0
	α	0.3784	0.4129	0.4168	0.4169	0.4003	−
	N0V	0.2772	0.3497	0.3608	0.3617	0.3158	−
(Mo0.6Ru0.4)82Si18	λ	0.5477	0.7229	0.7523	0.7549	0.6357	−
	μ∗	0.1249	0.1342	0.1355	0.1357	0.1294	−
	TC (K)	5.4019	12.5274	13.8300	13.9319	8.8687	5.40
	α	0.3837	0.4163	0.4200	0.4201	0.4043	−
	N0V	0.2823	0.3552	0.3663	0.3672	0.3209	−

Figure 1.

Transition temperature (T_C in K) of (Mo_0.6Ru_0.4)_100−xB_x versus concentration (x in at.%).

Figure 2.

Transition temperature (T_C in K) of (Mo_0.6Ru_0.4)_100−xSi_x versus concentration (x in at.%).

Figure 3.

Transition temperature (T_C in K) versus valance (in at.%).

Also, the main differences in the local field correction functions play an important role in determining the values of the SSPs of ternary metallic glasses. The H-dielectric function [22] is purely static and does not include the exchange and correlation effects. Taylor [23] introduced an analytical expression for the local field correction function that satisfies the compressibility sum rule exactly. The IU-local field correction function [24] is a fitting formula for the dielectric screening function of degenerate electron liquids at metallic and lower densities, and also accurately reproduces the Monte-Carlo results as well as satisfying the self-consistency condition in the compressibility sum rule and short-range correlations. Therefore, the H-local field correction function [22] gives the best agreement with the experiment based on the EMC model potential and is found to be suitable in the present case. On the basis of the IU-local field correction function [24], Farid et al [25] and Sarkar et al [26] proposed local field correction functions. Hence, the IU- and F-functions represent the same characteristic nature of the SSPs. Also, the SSPs computed from the S-local field correction [26] are found to be in qualitative agreement with the available experimental data [28]. The numerical values of the aforesaid properties are found to be reasonably sensitive to the selection of the local field correction function and show a significant variation upon changing the function. Thus, the use of these more promising local field correction functions has been successfully shown. The theoretical data for SSPs for ternary superconductors are not available in the literature for detailed comparison, but the agreement with other such theoretical values is encouraging, which confirms the applicability of the EMC model potential in explaining the superconducting state parameters of ternary mixtures.

According to the Matthias rules [29, 30] alloys with Z>2 exhibit superconductivity. Hence, the presently studied ternary metallic glasses are superconductors. Also, for (Mo_0.6Ru_0.4)_100−xB_x ternary metallic glasses, when Z increases from Z=4.72 to 4.89 and for (Mo_0.6Ru_0.4)_100−xSi_x ternary metallic glasses when Z increases from Z=4.84 to 4.98, λ changes with lattice spacing α. Similar trends are also observed in the values of T_C for most of the ternary metallic glasses. Hence, a strong dependence of the SSPs of the ternary metallic glasses on the valence Z is found, which is shown in figure 3. Also from the presently computed results of the SSPs of ternary metallic glasses, we observed that for (Mo_0.6Ru_0.4)_100−xB_x, as the atomic volume Ω₀ increases, the SSPs increase, whereas for (Mo_0.6Ru_0.4)_100−xSi_x ternary metallic glasses, as Ω₀ decreases, the SSPs increase.

Conclusions

The H-local field correction used with the EMC model potential provides the best explanation for superconductivity in the ternary systems considered in this study. The values of λ and T_C show an appreciable dependence on the local field correction function, whereas for μ∗, α and N₀V, a weak dependence is observed. The magnitudes of λ, α and N₀V show that the ternary metallic glasses are weak-to-intermediate superconductors. In the absence of theoretical or experimental data for α and N₀V, the presently computed values of these parameters may be considered to form reliable data for these ternary systems, as they lie within the theoretical limits of the Eliashberg-McMillan formulation. The results of comparisons of the presently computed results of the SSPs of the ternary metallic glasses with the available experimental findings are highly encouraging, which confirms the applicability of the EMC model potential and the different forms of the local field correction functions. Such a study on the SSPs of other multicomponent metallic alloys is in progress.