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Superconductivity of amorphous Mg0.70Zn0.30−xGax alloys
Physica C 468 (2008) 937–943
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Superconductivity of amorphous Mg0.70Zn0.30xGax alloys Aditya M. Vora * Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch 370 001, Gujarat, India
a r t i c l e
i n f o
Article history: Received 17 February 2008 Received in revised form 4 April 2008 Accepted 7 April 2008 Available online 14 April 2008 PACS: 61.43.Dq 71.15.Dx 74.20.z 74.70.Ad Keywords: Pseudopotential Superconducting state parameters Ternary amorphous alloys
a b s t r a c t The screening dependence theoretical investigations of the superconducting state parameters (SSP) viz. electron–phonon coupling strength k, Coulomb pseudopotential l*, transition temperature TC , isotope effect exponent a and effective interaction strength NOV of five Mg0.70Zn0.30xGax (x = 0.0, 0.06, 0.10, 0.15 and 0.20) ternary amorphous alloys viz. Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 have been 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 screening influence on the aforesaid properties. It is observed that the electron–phonon coupling strength k and the transition temperature TC are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential l*, isotope effect exponent a and effective interaction strength NOV show weak dependences on the local field correction functions. The transition temperature TC obtained from H-local field correction function is found in an excellent agreement with available experimental data. Quadratic TC equation has been proposed, which provide successfully the TC values of ternary amorphous alloys under consideration. Also, the present results are found in qualitative agreement with other such earlier reported data, which confirms the superconducting phase in the ternary amorphous alloys. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction During last several years, the superconductivity remains a dynamic area of research in condensed matter physics with continual discoveries of novel materials and with an increasing demand for novel devices for sophisticated technological applications. A large number of metals and amorphous alloys are superconductors, with critical temperature TC ranging from 1 to 18 K. Even some heavily doped semiconductors have also been found to be superconductors [1–19]. The pseudopotential theory has been used successfully in explaining the superconducting state parameters (SSP) for metallic complexes by many workers [3–18]. Many of them have used well known model pseudopotential in the calculation of the SSP for the metallic complexes. Recently, Vora et al. [3–10] have studied the SSP of some metals, In-based binary alloys, alkali–alkali binary alloys and large number of metallic glasses using single parametric model potential formalism. The study of the SSP of the ternary alloy based superconductors may be of great help in deciding their applications; the study of the dependence of the transition temperature TC on the composition of metallic elements is helpful in finding new superconductors with high TC. The application of * Tel.: +91 2832 256424. E-mail address: [email protected] 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.04.005
pseudopotential to ternary amorphous alloys involves the assumption of pseudoions with average properties, which are assumed to replace three types of ions in the ternary systems, and a gas of free electrons is assumed to permeate through them. The electron– pseudoion is accounted by the pseudopotential and the electron– electron interaction is involved through a dielectric screening function. For successful prediction of the superconducting properties of the alloying systems, the proper selection of the pseudopotential and screening function is very essential [3–18]. The possible technological applications of metals with high superconducting transition temperature TC have generated great interest in the study in the groups IIA, IIB and IIIB simple metals, Mg, Zn, Ga. In all of which exhibit high TC’s [20]. Therefore, in the present article, we have used well known McMillan’s theory [19] of the superconductivity for predicting the SSP of five Mg0.70 Zn0.30xGax (x = 0.0, 0.06, 0.10, 0.15 and 0.20) ternary amorphous alloys viz. Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20 Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 for the first time. We have used Ashcroft’s empty core (EMC) model potential [21] for studying the electron–phonon coupling strength k, Coulomb pseudopotential l*, transition temperature TC, isotope effect exponent a and effective interaction strength NOV for the first time. To see the impact of various exchange and correlation functions on the aforesaid properties, we have employed here five different
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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]. We have incorporated for the first time more advanced local field correction functions due to IU [24], F [25] and S [26] with EMC model potential in the present computation of the SSP for ternary amorphous alloys. In the present work, the pseudo-alloy-atom (PAA) model was used to explain electron–ion interaction for alloying systems [3– 10]. It is well known that the pseudo-alloy-atom (PAA) model is a more meaningful approach to explain such kind of interactions in alloying systems. In the PAA approach a hypothetical monoatomic crystal is supposed to be composed of pseudo-alloy-atoms, 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 PAA is supposed to have the same properties as the actual disordered alloy material and the pseudopotential theory is then applied to studying various properties of an alloy and metallic glass. The complete miscibility in the alloy systems is considered as a rare 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 it also takes into account the self-consistent treatment implicitly. Looking to the advantage of the PAA model, we propose a use of PAA model for the first time to investigate the SSP ternary amorphous alloys. 2. Computational methodology The well known screened Ashcroft’s empty core (EMC) model potential [21] used in the present computations of the SSP of ternary amorphous alloys is of the form, WðXÞ ¼
2pZ 2
XO X 2 kF eðXÞ
cosð2kF XrC Þ;
ð1Þ
here rC is the parameter of the model potential of ternary amorphous alloys. The Ashcroft’s empty core (EMC) model potential is a simple one-parameter model potential [21], which has been successfully found for various metallic complexes [7–18]. When used with a suitable form of dielectric screening functions, this potential has also been found to yield good results in computing the SSP of metallic complexes [7–18]. Therefore, in the present work we use Ashcroft’s empty core (EMC) model potential with more advanced Ichimaru–Utsumi (IU) [24], Farid et al. (F) [25] and Sarkar et al. (S) [26] local field correction functions for the first time. The model potential parameter rC may be obtained by fitting either to some experimental data or to realistic form factors or other data relevant to the properties to be investigated. In the present work, rC is fitted with experimental TC of the ternary amorphous alloys for most of the local field correction functions. In the present investigation for Mg0.70Zn0.30xGax ternary amorphous alloys, the electron–phonon coupling strength k is computed using the relation [3–18] k¼
mb X0 4p2 kF Mhx2 i
Z
2kF
q3 jWðqÞj2 dq:
ð2Þ
0
Here mb is the band mass, which is taken unity for the sake of simplicity, M is the ionic mass, X0 is the atomic volume, kF is the Fermi wave vector, W(q) the screened pseudopotential and hx2 i is the phonon frequency, of the ternary amorphous alloys, which is calculated using the relation given by Butler [27], hx2 i1=2 ¼ 0:69hD , where hD is the Debye temperature of the ternary amorphous alloys. Using X = q/2kF and XO ¼ 3p2 Z=ðkF Þ3 , we obtain Eq. (2) as, k¼
12mb Z Mhx2 i
Z 0
1
where Z and W(X) are the valence and the screened EMC pseudopotential [21] of the ternary amorphous alloys, respectively. The BCS theory gives a relation T C hD expð1=Nð0ÞVÞ for the superconducting transition temperature TC in terms of the Debye temperature hD. The electron–electron interaction V consists of the attractive electron–phonon-induced interaction minus the repulsive Coulomb interaction. The notation is used k ¼ Nð0ÞV e—ph . And the Coulomb repulsion N (0)VC is called l, so that Nð0ÞV ¼ k l , where l* is a ‘‘renormalized” Coulomb repulsion, reduced in value from l to l=½1 þ l lnðxP =xD Þ. This suppression of the Coulomb repulsion is a result of the fact that the electron–phonon attraction is retarded in time by an amount Dt 1=xD whereas the repulsive screened Coulomb interaction is retarded by a much smaller time, Dt 1=xD where xP is the electronic plasma frequency. Therefore, l* is bounded above by 1= lnðxP =xD Þ which for conventional metals should be 60.2. Values of k are known to range from 60.10 to P2.0. Also, The parameter l* is assigned a value in the range 0.10–0.15, consistent with tunneling and with theoretical guesses. Calculations of l or l* are computationally demanding and are not yet under theoretical control. Calculations of k are slightly less demanding, are under somewhat better theoretical control, and have been attempted for many years. Prior to 1990, calculations of k generally required knowing the phonon frequencies and eigenvectors as input information, and approximating the form of the electron–ion potential. McMillan [19] and Hopfield [28] pointed out that one could define a simpler quantity, g ¼ Nð0ÞhI2 i with R1 hx2 i ¼ 2k 0 dXa2 FðXÞ. The advantage of this is that g and hI2 i are purely ‘‘electronic” quantities, requiring no input information about phonon frequencies or eigenvectors. Gaspari and Gyorffy [29] then invented a simplified algorithm for calculating g, and many authors have used this. These calculations generally require a ‘‘rigid ion approximation” or some similar guess for the perturbing potential felt by electrons when an atom has moved. Given g, one can guess a value for hx2 i (for example, from hD). In the weak coupling limit of the electron–phonon interaction, the fundamental equations of the BCS theory should be derived from the Eliashberg equations. This conversion is possible upon some approximation of the phonon frequency jxj P xD with xD denote the Debye frequency [30]. Morel and Anderson [31] are given the relation of the transition temperature k l ¼ k l=½1 þ l lnðEF =xl Þ, which is nearly equal to the factor 6 for monovalent, bivalent and tetravalent met2 als. Where EF ¼ kF is the Fermi energy and x is the phonon frequency of the metallic substances. The effect of phonon frequency is very less in comparison with the Fermi energy. Hence, the overall effect of the Coulomb pseudopotential is reduced by the large logarithmic term. Therefore, Rajput and Gupta [32] have introduced the new term 10hD in place of the phonon frequency xl from the Butler’s [27] relation for the sake of simplicity and ignoring the lattice vibrational effect, which is generated consistent results of the Coulomb pseudopotential. The parameter l* represents the effective interelectronic Coulomb repulsion at the Fermi surface [30]. Hence, in the present case, we have adopted relation of the Coulomb pseudopotential given by Rajput and Gupta [32]. Therefore, the Coulomb pseudopotential l* is given by [3–18,32]
X 3 jWðXÞj2 dX;
ð3Þ
l ¼
mb pkF mb 1 þ pk ln F
R1
0
dX eðXÞ
EF 10hD
R
1 dX 0 eðXÞ
:
ð4Þ
As it is evident from Eq. (4) that, which was originally derived by Bogoliubov et al. [30], the Coulomb repulsion parameter l* is essentially weakened owing to a large logarithmic term in the denominator. Here, e(X) the modified Hartree dielectric function, which is written as [22] eðXÞ ¼ 1 þ ðeH ðXÞ 1Þð1 f ðXÞÞ:
ð5Þ
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eH (X) is the static Hartree dielectric function [22] and f(X) the local field correction function. In the present investigation, the local field correction functions due to H [22], T [23], IU [24], F [25] and S [26] are incorporated to see the impact of exchange and correlation effects. The Hartree screening function [22] is purely static, and it does not include the exchange and correlation effects. The expression of it is, f ðqÞ ¼ 0:
The Ichimaru–Utsumi (IU) local field correction function [24] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self consistency condition in the compressibility sum rule and short range correlations. The fitting formula is
ð8Þ
On the basis of Ichimaru–Utsumi (IU) local field correction function [24], Farid et al. (F) [25] have given a local field correction function of the form ( ) 2 þ Q 4 Q2 4 4 2 2 : ln f ðqÞ ¼ AFQ þ BFQ þ C F þ ½AFQ þ DFQ C F ð9Þ 2 Q 4Q Based on Eqs. (8), and (9), Sarkar et al. (S) [26] have proposed a simple form of local field correction function, which is of the form f ðqÞ ¼ AS f1 ð1 þ BS Q 4 Þ expðC S Q 2 Þg;
ð10Þ
where Q = q/kF. The parameters AIU, BIU, CIU, AF, BF, CF, DF, AS, BS and CS are the atomic volume dependent parameters of IU, F and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [24–26]. After evaluating k and l*, the transition temperature TC and isotope effect exponent a are investigated from the McMillan’s formula [3–19] hD 1:04ð1 þ kÞ exp ; ð11Þ TC ¼ k l ð1 þ 0:62kÞ 1:45 " # 2 1 hD 1 þ 0:62k 1 l ln a¼ : ð12Þ 2 1:45T C 1:04ð1 þ kÞ The expression for the effective interaction strength NOV is studied using [3–18] NO V ¼
k l : 10 1 þ 11 k
Superconductors
Z
rC (au)
Xo (au)3
M (amu)
hD (K)
hx2 i2 106 (au)2
Mg0.70Zn0.30Ga0.00 Mg0.70Zn0.24Ga0.06 Mg0.70Zn0.20Ga0.10 Mg0.70Zn0.15Ga0.15 Mg0.70Zn0.10Ga0.20
2.00 2.06 2.10 2.15 2.20
1.0446 1.0523 1.0551 1.0870 1.0776
139.73 141.49 142.67 144.14 145.61
36.63 36.89 37.06 37.28 37.50
308 307 307 294 294
1.82409 1.81226 1.81226 1.66203 1.66203
ð6Þ
Taylor (T) [23] has introduced an analytical expression for the local field correction function, which 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 (T) [23], " # q2 0:1534 f ðqÞ ¼ 2 1 þ : ð7Þ 2 4kF pkF
f ðqÞ ¼ AIU Q 4 þ BIU Q 2 þ C IU ) ( 2 þ Q 8AIU 4 Q2 4 2 : Q C IU ln þ AIU Q þ BIU þ 2 Q 3 4Q
Table 1 Input parameters and other constants
ð13Þ
3. Results and discussion The values of the input parameters for the five Mg0.70Zn0.30xGax (x = 0.0, 0.06, 0.10, 0.15 and 0.20) ternary amorphous alloys viz. Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15-
Ga0.15 and Mg0.70Zn0.10Ga0.20 under investigation are assembled in Table 1. To determine the input parameters and various constants for PAA model [3–10], the following definitions fA + gB + hC are adopted, Z ¼ fZ A þ gZ B þ hZ C ;
ð14Þ
M ¼ fMA þ gMB þ hMC ;
ð15Þ
XO ¼ f XOA þ gXOB þ hXOC ;
ð16Þ
where A, B and C are denoted the first, second and third pure metallic components. f, g and h the concentration factor of the first, second and third metallic components. The input parameters such as Z, X0 and M of the pure metallic components are taken from the literature [9]. The values of the Debye temperature hD of the ternary glassy alloys are directly obtained from [20]. The presently calculated results of the SSP are tabulated in Table 2 with the other such experimental findings [20]. The graphical representation of the model potential parameter rC with the concentration (x) of Ga (in at.%) are plotted in Fig. 1. Also, the graphical analyses of the SSP of Mg0.70Zn0.30xGax ternary systems are also plotted in Figs. 2–6. The calculated values of the electron–phonon coupling strength k for five Mg0.70Zn0.30xGax ternary amorphous alloys, using five different types of the local field correction functions with EMC model potential, are shown in Table 2 with other experimental data [20]. The graphical nature of k is also displayed in Fig. 2. It is noticed from the present study that, the percentile influence of the various local field correction functions with respect to the static H-screening function on the electron–phonon coupling strength k is 27.66–51.07%, 27.38–50.18%, 27.21–49.70%, 26.66– 47.76% and 26.62–47.83% for Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20, respectively. Also, the H-screening yields lowest values of k, whereas the values obtained from the F-function are the highest. It is also observed from Table 2 and Fig. 2 that, k goes increasing from the values of 0.3631 ? 0.5884 as the concentration ‘x’ of ‘Ga’ is increased from 0.0 ? 0.20. The increase in k with concentration ‘x’ of ‘Ga’ shows a gradual transition from weak coupling behaviour to intermediate coupling behaviour of electrons and phonons, which may be attributed to an increase of the hybridization of sp–d electrons of ‘Ga’ with increasing concentration (x). This may also be attributed to the increase role of ionic vibrations in the Ga-rich region. The present results are found in qualitative agreement with the available experimental data [20]. The computed values of the Coulomb pseudopotential l*, which accounts for the Coulomb interaction between the conduction electrons, obtained from the various forms of the local field correction functions are tabulated in Table 2. It is observed from Table 2 that for all five Mg0.70Zn0.30xGax ternary amorphous alloys, l* lies between 0.13 and 0.16, which is in accordance with McMillan [19], who suggested l 0:13 for simple and non-simple metals. The graphs of l* versus concentration (x) for different local field correction functions are plotted in Fig. 3, which shows the weak dependence of l* on the local field correction functions. The percentile influence of the various local field correction functions with respect to the static H-screening function on l* for the ternary
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Table 2 Superconducting state parameters Mg0.70Zn0.30xGax amorphous alloys Amorphous alloys
SSP
Present results H
T
IU
F
S
Mg0.70Zn0.30Ga0.00
k l* TC (K) a NOV
0.3631 0.1432 0.1112 0.0059 0.1653
0.5176 0.1570 1.3120 0.2331 0.2452
0.5471 0.1589 1.7254 0.2566 0.2593
0.5485 0.1592 1.7395 0.2565 0.2598
0.4635 0.1518 0.7266 0.1857 0.2193
0.297 0.111, 0.1125
Mg0.70Zn0.24Ga0.06
k l* TC (K) a NOV
0.3676 0.1427 0.1303 0.0194 0.1686
0.5216 0.1564 1.3858 0.2414 0.2478
0.5507 0.1583 1.8041 0.2635 0.2615
0.5521 0.1586 1.8163 0.2633 0.2620
0.4682 0.1512 0.7896 0.1978 0.2223
0.302 0.130, 0.11
Mg0.70Zn0.20Ga0.10
k l* TC (K) a NOV
0.3711 0.1424 0.1461 0.0364 0.1710
0.5252 0.1560 1.4527 0.2475 0.2499
0.5543 0.1579 1.8784 0.2687 0.2636
0.5556 0.1582 1.8899 0.2684 0.2640
0.4721 0.1509 0.8420 0.2062 0.2247
0.305 0.146
Mg0.70Zn0.15Ga0.15
k TC (K) a NOV
0.3815 0.1412 0.1950 0.0857 0.1784
0.5345 0.1546 1.5787 0.2651 0.2557
0.5627 0.1564 1.9963 0.2832 0.2688
0.5637 0.1567 2.0033 0.2828 0.2691
0.4832 0.1495 0.9721 0.2313 0.2318
0.317 0.195
k l* TC (K) a NOV
0.3980 0.1412 0.2931 0.1359 0.1888
0.5577 0.1546 1.9873 0.2888 0.2678
0.5873 0.1564 2.4689 0.3046 0.2812
0.5884 0.1567 2.4777 0.3042 0.2815
0.5039 0.1495 1.2668 0.2595 0.2433
0.333 0.293, 0.306
l*
Mg0.70Zn0.10Ga0.20
Experiment [20]
1.09
0.6
1.085 0.55
1.08 0.5
1.07
0.45
H T IU F S Expt.
λ
Potential Parameter (rC)
1.075
1.065 0.4
1.06 0.35
1.055
1.05
0.3
1.045 0.25 0
1.04 0
0.05
0.1
0.15
0.2
Concentration (x)
0.05
0.1
0.15
0.2
Concentration (x) Fig. 2. Variation of electron–phonon coupling strength (k) with Ga-concentration x (at.%).
Fig. 1. Variation of the model potential parameter rC with Ga-concentration x (at.%).
amorphous alloys is observed in the range of Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and
Mg0.70Zn0.10Ga0.20, respectively. Again the H-screening function yields lowest values of l*, while the values obtained from the Ffunction are the highest.
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A.M. Vora / Physica C 468 (2008) 937–943
0.16
0.344
0.294
0.155
0.244
α
μ
∗
0.194 0.15
0.144
H T IU F S
0.145
0.094 H T IU F S
0.044
-0.006
0.14 0
0.05
0.1
0.15
0
0.2
0.05
Concentration (x)
0.1
0.15
0.2
Concentration (x)
Fig. 3. Variation of Coulomb pseudopotential (l*) with Ga-concentration x (at.%).
Fig. 5. Variation of isotope effect exponent (a) with Ga-concentration x (at.%).
2.5 2
H T IU F S Expt
2
TC (K) = 5.4722x - 0.2262x + 0.1149
0.35 H T IU F S
0.3
NOV
TC (K)
1.5
1
0.25
0.5
0.2
0 0
0.05
0.1
0.15
0.2
Concentration (x) Fig. 4. Variation of transition temperature (TC) with Ga-concentration x (at.%).
0.15 0
0.05
0.1
0.15
0.2
Concentration (x) Table 2 contains calculated values of the transition temperature TC for five Mg0.70Zn0.30xGax ternary amorphous alloys computed
Fig. 6. Variation of effective interaction strength (NOV) with Ga-concentration x (at.%).
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A.M. Vora / Physica C 468 (2008) 937–943
from the various forms of the local field correction functions along with the experimental findings [20]. From Table 2 it can be noted that, the static H-screening function yields lowest TC whereas the F-function yields highest values of TC. The present results obtained from the H-local field correction functions are found in good agreement with available experimental data [20]. The theoretical data of TC for five Mg0.70Zn0.30xGax ternary amorphous alloys is not available in the literature. It is also observed that the static H-screening function yields lowest TC whereas the F-function yields highest values of TC. The calculated results of the transition temperature TC for Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 ternary amorphous alloys deviate in the range of 80.56–100.00%, 75.51–130.03%, 72.54–124.47%, 62.40–106.08% and 94.89–95.58% from the experimental findings, respectively. The variation of the computed values of the transition temperature TC for Mg0.70Zn0.30xGax ternary amorphous alloys with the atomic concentration (x) of ‘Ga’, using five different types of the local field correction functions with EMC potential are shown in Fig. 4. The graph also includes the experimental values due to van den Berg et al. [20]. It is seen that TC is quite sensitive to the local field correction functions, and the results of TC by using Hscreening are in best agreement with the experimental data for the Mg0.70Zn0.30xGax ternary amorphous alloys under investigation, as the relevant curves for H-screening almost overlaps the experimental curves. It is also observed that the static H-screening function yields lowest TC whereas the F-function yields highest values of TC. It is also seen from the graphical nature, TC increases considerably with increasing Ga-concentration (x). The composition dependence can be described by polynomial regression of the data obtained for H-screening for different values of the concentration ‘x’, which yields T C ðKÞ ¼ 5:4722x2 0:2262x þ 0:1149:
ð17Þ
The graph of the fitted TC equation is displayed in Fig. 4, which indicates that TC increases almost considerably with increasing ‘Ga’ content with a slope dTC/dx = 0.2262. Wide extrapolation predicts a TC = 0.1149 K for the hypothetical case of ‘amorphous pure Mg0.70Zn0.30Ga0.00 alloy. The presently computed values of TC are found in the range, which is suitable for further exploring the applications of the ternary amorphous alloys for usage like lossless transmission line for cryogenic applications. While alloying elements show good elasticity and could be drawn in the form of wires as such they have good chances of being used as superconducting transmission lines at low temperature of the order of 7 K. The values of the isotope effect exponent a for five Mg0.70Zn0.30xGax ternary amorphous alloys are tabulated in Table 2. Fig. 5 depicts the variation of a with Ga-concentration (x) increases. The computed values of the a show a weak dependence on the dielectric screening, its value is being lowest for the Hscreening function and highest for the F-function. The negative value of a is observed in the case of Mg0.70Zn0.30Ga0.00 ternary alloy, which indicates that the electron–phonon coupling in these metallic complexes do not fully explain all the features regarding their superconducting behaviour. Since the experimental value of a has not been reported in the literature so far, the present data of a may be used for the study of ionic vibrations in the superconductivity of alloying substances. Since H-local field correction function yields the best results for k and TC, it may be observed that a values obtained from this screening provide the best account for the role of the ionic vibrations in superconducting behaviour of this system. The values of the effective interaction strength NOV are listed in Table 2 and depicted in Fig. 6 for different local field correction
functions. It is observed that the magnitude of NOV shows that the five Mg0.70Zn0.30xGax ternary amorphous alloys under investigation lie in the range of weak coupling superconductors. The values of NOV also show a feeble dependence on dielectric screening, its value being lowest for the H-screening function and highest for the F-screening function. The variation of present values of NOV show that, the ternary amorphous alloys under consideration fall in the range of weak coupling superconductors. The main difference of the local field correction functions are played in important role in the production of the SSP of Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 ternary amorphous alloys. The Hartree (H) dielectric function [22] is purely static and it does not include the exchange and correlation effects. Taylor (T) [23] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. The Ichimaru–Utsumi (IU) local field correction function [24] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also satisfies the self consistency condition in the compressibility sum rule and short range correlations. On the basis of Ichimaru–Utsumi (IU) local field correction function [24], Farid et al. (F) [25] and Sarkar et al. [26] have given a local field correction function. Hence, F-function represents same characteristic nature. Also, the SSP computed from Sarkar et al. [26] local field correction are found in qualitative agreement with the available experimental data [20]. The effect of local field correction functions plays an important role in the computation of k and l*, which makes drastic variation on TC, a and NOV. The local field correction functions due to IU, F and S are able to generate consistent results regarding the SSP of Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 ternary amorphous alloys as those obtained from more commonly employed H- and T-functions. Thus, the use of these more promising local field correction functions is established successfully. The computed results of a and NOV are not showing any abnormal values for the five Mg0.70Zn0.30xGax ternary amorphous alloys.
4. Conclusions Lastly we concluded that, the H-local field corrections when used with EMC model potential provide the best explanation for superconductivity in the Mg0.70Zn0.30xGax ternary systems. The values of the k and the TC show an appreciable dependence on the local field correction function, whereas for the l*, a and NOV a weak dependence is observed. The magnitude of the k, a and NOV values shows that, the ternary amorphous alloys are weak to intermediate superconductors. Quadratic TC equation has been proposed, which provide successfully the TC values of the ternary amorphous alloys under consideration. In the absence of experimental data for a and NOV, the presently computed values may be considered to form reliable data for these ternary systems, as they lie within the theoretical limits of the Eliashberg–McMillan formulation. The comparisons of presently computed results of the SSP of the Mg0.70Zn0.30xGax ternary amorphous alloys with available experimental findings are highly encouraging, which confirms the applicability of the EMC model potential and different forms of the local field correction functions. Such study on SSP of other multi component metallic alloys is in progress.
References [1] A.V. Narlikar, S.N. Ekbote, Superconductivity and Superconducting Materials, South Asian Publishers, New Delhi–Madras, 1983.
A.M. Vora / Physica C 468 (2008) 937–943 [2] P.B. Allen, in: C.P. Poole Jr. (Ed.), Handbook of Superconductivity, Academic Press, New York, 1999, p. 478. [3] P.N. Gajjar, A.M. Vora, A.R. Jani, Mod. Phys. Lett. B 18 (2004) 573. [4] A.M. Vora, M.H. Patel, P.N. Gajjar, A.R. Jani, Pramana-J. Phys. 58 (2002) 849. [5] P.N. Gajjar, A.M. Vora, M.H. Patel, A.R. Jani, Int. J. Mod. Phys. B17 (2003) 6001. [6] P.N. Gajjar, A.M. Vora, A.R. Jani, Indian J. Phys. 78 (2004) 775. [7] Aditya M. Vora, Comp. Mater. Sci. 40 (2007) 492. [8] Aditya M. Vora, Physica C 450 (2006) 135. [9] Aditya M. Vora, Physica C 458 (2007) 21. [10] Aditya M. Vora, Physica C 458 (2007) 43. [11] S. Sharma, K.S. Sharma, H. Khan, Czech. J. Phys. 55 (2005) 1005. [12] S. Sharma, K.S. Sharma, H. Khan, Supercond. Sci. Technol. 17 (2004) 474. [13] S. Sharma, H. Khan, K.S. Sharma, Phys. Stat. Sol. (b) 241 (2004) 2562. [14] S. Sharma, H. Khan, Solid State Phys. 46 (2003) 635. [15] S. Sharma, H. Khan, K.S. Sharma, Ind. J. Pure Appl. Phys. 41 (2003) 301. [16] M. Gupta, P.C. Agarwal, K.S. Sharma, L. Dass, Phys. Stat. Sol. (b) 211 (1999) 731. [17] M. Gupta, K.S. Sharma, L. Dass, Pramana-J. Phys. 48 (1997) 923. [18] R. Sharma, K.S. Sharma, Supercond. Sci. Technol. 10 (1997) 557.
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[19] W.L. McMillan, Phys. Rev. 167 (1968) 331. [20] R. van den Berg, H. von Löhneysen, A. Schröder, U. Mizutani, T. Matsuda, J. Phys. F: Met. Phys. 16 (1986) 69. [21] N.W. Ashcroft, Phys. Lett. 23 (1966) 48. [22] W.A. Harrison, Elementary Electronic Structure, World Scientific, Singapore, 1999. [23] R. Taylor, J. Phys. F: Met. Phys. 8 (1978) 1699. [24] S. Ichimaru, K. Utsumi, Phys. Rev. B 24 (1981) 7386. [25] B. Farid, V. Heine, G. Engel, I.J. Robertson, Phys. Rev. B 48 (1993) 11602. [26] A. Sarkar, D. Sen, H. Haldar, D. Roy, Mod. Phys. Lett. B 12 (1998) 639. [27] W.H. Butler, Phys. Rev. B 15 (1977) 5267. [28] J.J. Hopfield, Phys. Rev. 186 (1969) 443. [29] G.D. Gaspari, B.L. Gyorff, Phys. Rev. Lett. 28 (1972) 801. [30] N.P. Kovalenko, Yu.P. Krasny, U. Krey, Physics of Amorphous Metals, WileyVCH, Berlin, 2001. [31] P. Moral, P.W. Anderson, Phys. Rev. 125 (1962) 1263. [32] J.S. Rajput, A.K. Gupta, Phys. Rev. 181 (1969) 743.
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Physica C journal homepage: www.elsevier.com/locate/physc
Superconductivity of amorphous Mg0.70Zn0.30xGax alloys Aditya M. Vora * Parmeshwari 165, Vijaynagar Area, Hospital Road, Bhuj–Kutch 370 001, Gujarat, India
a r t i c l e
i n f o
Article history: Received 17 February 2008 Received in revised form 4 April 2008 Accepted 7 April 2008 Available online 14 April 2008 PACS: 61.43.Dq 71.15.Dx 74.20.z 74.70.Ad Keywords: Pseudopotential Superconducting state parameters Ternary amorphous alloys
a b s t r a c t The screening dependence theoretical investigations of the superconducting state parameters (SSP) viz. electron–phonon coupling strength k, Coulomb pseudopotential l*, transition temperature TC , isotope effect exponent a and effective interaction strength NOV of five Mg0.70Zn0.30xGax (x = 0.0, 0.06, 0.10, 0.15 and 0.20) ternary amorphous alloys viz. Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 have been 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 screening influence on the aforesaid properties. It is observed that the electron–phonon coupling strength k and the transition temperature TC are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential l*, isotope effect exponent a and effective interaction strength NOV show weak dependences on the local field correction functions. The transition temperature TC obtained from H-local field correction function is found in an excellent agreement with available experimental data. Quadratic TC equation has been proposed, which provide successfully the TC values of ternary amorphous alloys under consideration. Also, the present results are found in qualitative agreement with other such earlier reported data, which confirms the superconducting phase in the ternary amorphous alloys. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction During last several years, the superconductivity remains a dynamic area of research in condensed matter physics with continual discoveries of novel materials and with an increasing demand for novel devices for sophisticated technological applications. A large number of metals and amorphous alloys are superconductors, with critical temperature TC ranging from 1 to 18 K. Even some heavily doped semiconductors have also been found to be superconductors [1–19]. The pseudopotential theory has been used successfully in explaining the superconducting state parameters (SSP) for metallic complexes by many workers [3–18]. Many of them have used well known model pseudopotential in the calculation of the SSP for the metallic complexes. Recently, Vora et al. [3–10] have studied the SSP of some metals, In-based binary alloys, alkali–alkali binary alloys and large number of metallic glasses using single parametric model potential formalism. The study of the SSP of the ternary alloy based superconductors may be of great help in deciding their applications; the study of the dependence of the transition temperature TC on the composition of metallic elements is helpful in finding new superconductors with high TC. The application of * Tel.: +91 2832 256424. E-mail address: [email protected] 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.04.005
pseudopotential to ternary amorphous alloys involves the assumption of pseudoions with average properties, which are assumed to replace three types of ions in the ternary systems, and a gas of free electrons is assumed to permeate through them. The electron– pseudoion is accounted by the pseudopotential and the electron– electron interaction is involved through a dielectric screening function. For successful prediction of the superconducting properties of the alloying systems, the proper selection of the pseudopotential and screening function is very essential [3–18]. The possible technological applications of metals with high superconducting transition temperature TC have generated great interest in the study in the groups IIA, IIB and IIIB simple metals, Mg, Zn, Ga. In all of which exhibit high TC’s [20]. Therefore, in the present article, we have used well known McMillan’s theory [19] of the superconductivity for predicting the SSP of five Mg0.70 Zn0.30xGax (x = 0.0, 0.06, 0.10, 0.15 and 0.20) ternary amorphous alloys viz. Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20 Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 for the first time. We have used Ashcroft’s empty core (EMC) model potential [21] for studying the electron–phonon coupling strength k, Coulomb pseudopotential l*, transition temperature TC, isotope effect exponent a and effective interaction strength NOV for the first time. To see the impact of various exchange and correlation functions on the aforesaid properties, we have employed here five different
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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]. We have incorporated for the first time more advanced local field correction functions due to IU [24], F [25] and S [26] with EMC model potential in the present computation of the SSP for ternary amorphous alloys. In the present work, the pseudo-alloy-atom (PAA) model was used to explain electron–ion interaction for alloying systems [3– 10]. It is well known that the pseudo-alloy-atom (PAA) model is a more meaningful approach to explain such kind of interactions in alloying systems. In the PAA approach a hypothetical monoatomic crystal is supposed to be composed of pseudo-alloy-atoms, 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 PAA is supposed to have the same properties as the actual disordered alloy material and the pseudopotential theory is then applied to studying various properties of an alloy and metallic glass. The complete miscibility in the alloy systems is considered as a rare 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 it also takes into account the self-consistent treatment implicitly. Looking to the advantage of the PAA model, we propose a use of PAA model for the first time to investigate the SSP ternary amorphous alloys. 2. Computational methodology The well known screened Ashcroft’s empty core (EMC) model potential [21] used in the present computations of the SSP of ternary amorphous alloys is of the form, WðXÞ ¼
2pZ 2
XO X 2 kF eðXÞ
cosð2kF XrC Þ;
ð1Þ
here rC is the parameter of the model potential of ternary amorphous alloys. The Ashcroft’s empty core (EMC) model potential is a simple one-parameter model potential [21], which has been successfully found for various metallic complexes [7–18]. When used with a suitable form of dielectric screening functions, this potential has also been found to yield good results in computing the SSP of metallic complexes [7–18]. Therefore, in the present work we use Ashcroft’s empty core (EMC) model potential with more advanced Ichimaru–Utsumi (IU) [24], Farid et al. (F) [25] and Sarkar et al. (S) [26] local field correction functions for the first time. The model potential parameter rC may be obtained by fitting either to some experimental data or to realistic form factors or other data relevant to the properties to be investigated. In the present work, rC is fitted with experimental TC of the ternary amorphous alloys for most of the local field correction functions. In the present investigation for Mg0.70Zn0.30xGax ternary amorphous alloys, the electron–phonon coupling strength k is computed using the relation [3–18] k¼
mb X0 4p2 kF Mhx2 i
Z
2kF
q3 jWðqÞj2 dq:
ð2Þ
0
Here mb is the band mass, which is taken unity for the sake of simplicity, M is the ionic mass, X0 is the atomic volume, kF is the Fermi wave vector, W(q) the screened pseudopotential and hx2 i is the phonon frequency, of the ternary amorphous alloys, which is calculated using the relation given by Butler [27], hx2 i1=2 ¼ 0:69hD , where hD is the Debye temperature of the ternary amorphous alloys. Using X = q/2kF and XO ¼ 3p2 Z=ðkF Þ3 , we obtain Eq. (2) as, k¼
12mb Z Mhx2 i
Z 0
1
where Z and W(X) are the valence and the screened EMC pseudopotential [21] of the ternary amorphous alloys, respectively. The BCS theory gives a relation T C hD expð1=Nð0ÞVÞ for the superconducting transition temperature TC in terms of the Debye temperature hD. The electron–electron interaction V consists of the attractive electron–phonon-induced interaction minus the repulsive Coulomb interaction. The notation is used k ¼ Nð0ÞV e—ph . And the Coulomb repulsion N (0)VC is called l, so that Nð0ÞV ¼ k l , where l* is a ‘‘renormalized” Coulomb repulsion, reduced in value from l to l=½1 þ l lnðxP =xD Þ. This suppression of the Coulomb repulsion is a result of the fact that the electron–phonon attraction is retarded in time by an amount Dt 1=xD whereas the repulsive screened Coulomb interaction is retarded by a much smaller time, Dt 1=xD where xP is the electronic plasma frequency. Therefore, l* is bounded above by 1= lnðxP =xD Þ which for conventional metals should be 60.2. Values of k are known to range from 60.10 to P2.0. Also, The parameter l* is assigned a value in the range 0.10–0.15, consistent with tunneling and with theoretical guesses. Calculations of l or l* are computationally demanding and are not yet under theoretical control. Calculations of k are slightly less demanding, are under somewhat better theoretical control, and have been attempted for many years. Prior to 1990, calculations of k generally required knowing the phonon frequencies and eigenvectors as input information, and approximating the form of the electron–ion potential. McMillan [19] and Hopfield [28] pointed out that one could define a simpler quantity, g ¼ Nð0ÞhI2 i with R1 hx2 i ¼ 2k 0 dXa2 FðXÞ. The advantage of this is that g and hI2 i are purely ‘‘electronic” quantities, requiring no input information about phonon frequencies or eigenvectors. Gaspari and Gyorffy [29] then invented a simplified algorithm for calculating g, and many authors have used this. These calculations generally require a ‘‘rigid ion approximation” or some similar guess for the perturbing potential felt by electrons when an atom has moved. Given g, one can guess a value for hx2 i (for example, from hD). In the weak coupling limit of the electron–phonon interaction, the fundamental equations of the BCS theory should be derived from the Eliashberg equations. This conversion is possible upon some approximation of the phonon frequency jxj P xD with xD denote the Debye frequency [30]. Morel and Anderson [31] are given the relation of the transition temperature k l ¼ k l=½1 þ l lnðEF =xl Þ, which is nearly equal to the factor 6 for monovalent, bivalent and tetravalent met2 als. Where EF ¼ kF is the Fermi energy and x is the phonon frequency of the metallic substances. The effect of phonon frequency is very less in comparison with the Fermi energy. Hence, the overall effect of the Coulomb pseudopotential is reduced by the large logarithmic term. Therefore, Rajput and Gupta [32] have introduced the new term 10hD in place of the phonon frequency xl from the Butler’s [27] relation for the sake of simplicity and ignoring the lattice vibrational effect, which is generated consistent results of the Coulomb pseudopotential. The parameter l* represents the effective interelectronic Coulomb repulsion at the Fermi surface [30]. Hence, in the present case, we have adopted relation of the Coulomb pseudopotential given by Rajput and Gupta [32]. Therefore, the Coulomb pseudopotential l* is given by [3–18,32]
X 3 jWðXÞj2 dX;
ð3Þ
l ¼
mb pkF mb 1 þ pk ln F
R1
0
dX eðXÞ
EF 10hD
R
1 dX 0 eðXÞ
:
ð4Þ
As it is evident from Eq. (4) that, which was originally derived by Bogoliubov et al. [30], the Coulomb repulsion parameter l* is essentially weakened owing to a large logarithmic term in the denominator. Here, e(X) the modified Hartree dielectric function, which is written as [22] eðXÞ ¼ 1 þ ðeH ðXÞ 1Þð1 f ðXÞÞ:
ð5Þ
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eH (X) is the static Hartree dielectric function [22] and f(X) the local field correction function. In the present investigation, the local field correction functions due to H [22], T [23], IU [24], F [25] and S [26] are incorporated to see the impact of exchange and correlation effects. The Hartree screening function [22] is purely static, and it does not include the exchange and correlation effects. The expression of it is, f ðqÞ ¼ 0:
The Ichimaru–Utsumi (IU) local field correction function [24] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also, satisfies the self consistency condition in the compressibility sum rule and short range correlations. The fitting formula is
ð8Þ
On the basis of Ichimaru–Utsumi (IU) local field correction function [24], Farid et al. (F) [25] have given a local field correction function of the form ( ) 2 þ Q 4 Q2 4 4 2 2 : ln f ðqÞ ¼ AFQ þ BFQ þ C F þ ½AFQ þ DFQ C F ð9Þ 2 Q 4Q Based on Eqs. (8), and (9), Sarkar et al. (S) [26] have proposed a simple form of local field correction function, which is of the form f ðqÞ ¼ AS f1 ð1 þ BS Q 4 Þ expðC S Q 2 Þg;
ð10Þ
where Q = q/kF. The parameters AIU, BIU, CIU, AF, BF, CF, DF, AS, BS and CS are the atomic volume dependent parameters of IU, F and S-local field correction functions. The mathematical expressions of these parameters are narrated in the respective papers of the local field correction functions [24–26]. After evaluating k and l*, the transition temperature TC and isotope effect exponent a are investigated from the McMillan’s formula [3–19] hD 1:04ð1 þ kÞ exp ; ð11Þ TC ¼ k l ð1 þ 0:62kÞ 1:45 " # 2 1 hD 1 þ 0:62k 1 l ln a¼ : ð12Þ 2 1:45T C 1:04ð1 þ kÞ The expression for the effective interaction strength NOV is studied using [3–18] NO V ¼
k l : 10 1 þ 11 k
Superconductors
Z
rC (au)
Xo (au)3
M (amu)
hD (K)
hx2 i2 106 (au)2
Mg0.70Zn0.30Ga0.00 Mg0.70Zn0.24Ga0.06 Mg0.70Zn0.20Ga0.10 Mg0.70Zn0.15Ga0.15 Mg0.70Zn0.10Ga0.20
2.00 2.06 2.10 2.15 2.20
1.0446 1.0523 1.0551 1.0870 1.0776
139.73 141.49 142.67 144.14 145.61
36.63 36.89 37.06 37.28 37.50
308 307 307 294 294
1.82409 1.81226 1.81226 1.66203 1.66203
ð6Þ
Taylor (T) [23] has introduced an analytical expression for the local field correction function, which 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 (T) [23], " # q2 0:1534 f ðqÞ ¼ 2 1 þ : ð7Þ 2 4kF pkF
f ðqÞ ¼ AIU Q 4 þ BIU Q 2 þ C IU ) ( 2 þ Q 8AIU 4 Q2 4 2 : Q C IU ln þ AIU Q þ BIU þ 2 Q 3 4Q
Table 1 Input parameters and other constants
ð13Þ
3. Results and discussion The values of the input parameters for the five Mg0.70Zn0.30xGax (x = 0.0, 0.06, 0.10, 0.15 and 0.20) ternary amorphous alloys viz. Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15-
Ga0.15 and Mg0.70Zn0.10Ga0.20 under investigation are assembled in Table 1. To determine the input parameters and various constants for PAA model [3–10], the following definitions fA + gB + hC are adopted, Z ¼ fZ A þ gZ B þ hZ C ;
ð14Þ
M ¼ fMA þ gMB þ hMC ;
ð15Þ
XO ¼ f XOA þ gXOB þ hXOC ;
ð16Þ
where A, B and C are denoted the first, second and third pure metallic components. f, g and h the concentration factor of the first, second and third metallic components. The input parameters such as Z, X0 and M of the pure metallic components are taken from the literature [9]. The values of the Debye temperature hD of the ternary glassy alloys are directly obtained from [20]. The presently calculated results of the SSP are tabulated in Table 2 with the other such experimental findings [20]. The graphical representation of the model potential parameter rC with the concentration (x) of Ga (in at.%) are plotted in Fig. 1. Also, the graphical analyses of the SSP of Mg0.70Zn0.30xGax ternary systems are also plotted in Figs. 2–6. The calculated values of the electron–phonon coupling strength k for five Mg0.70Zn0.30xGax ternary amorphous alloys, using five different types of the local field correction functions with EMC model potential, are shown in Table 2 with other experimental data [20]. The graphical nature of k is also displayed in Fig. 2. It is noticed from the present study that, the percentile influence of the various local field correction functions with respect to the static H-screening function on the electron–phonon coupling strength k is 27.66–51.07%, 27.38–50.18%, 27.21–49.70%, 26.66– 47.76% and 26.62–47.83% for Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20, respectively. Also, the H-screening yields lowest values of k, whereas the values obtained from the F-function are the highest. It is also observed from Table 2 and Fig. 2 that, k goes increasing from the values of 0.3631 ? 0.5884 as the concentration ‘x’ of ‘Ga’ is increased from 0.0 ? 0.20. The increase in k with concentration ‘x’ of ‘Ga’ shows a gradual transition from weak coupling behaviour to intermediate coupling behaviour of electrons and phonons, which may be attributed to an increase of the hybridization of sp–d electrons of ‘Ga’ with increasing concentration (x). This may also be attributed to the increase role of ionic vibrations in the Ga-rich region. The present results are found in qualitative agreement with the available experimental data [20]. The computed values of the Coulomb pseudopotential l*, which accounts for the Coulomb interaction between the conduction electrons, obtained from the various forms of the local field correction functions are tabulated in Table 2. It is observed from Table 2 that for all five Mg0.70Zn0.30xGax ternary amorphous alloys, l* lies between 0.13 and 0.16, which is in accordance with McMillan [19], who suggested l 0:13 for simple and non-simple metals. The graphs of l* versus concentration (x) for different local field correction functions are plotted in Fig. 3, which shows the weak dependence of l* on the local field correction functions. The percentile influence of the various local field correction functions with respect to the static H-screening function on l* for the ternary
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A.M. Vora / Physica C 468 (2008) 937–943
Table 2 Superconducting state parameters Mg0.70Zn0.30xGax amorphous alloys Amorphous alloys
SSP
Present results H
T
IU
F
S
Mg0.70Zn0.30Ga0.00
k l* TC (K) a NOV
0.3631 0.1432 0.1112 0.0059 0.1653
0.5176 0.1570 1.3120 0.2331 0.2452
0.5471 0.1589 1.7254 0.2566 0.2593
0.5485 0.1592 1.7395 0.2565 0.2598
0.4635 0.1518 0.7266 0.1857 0.2193
0.297 0.111, 0.1125
Mg0.70Zn0.24Ga0.06
k l* TC (K) a NOV
0.3676 0.1427 0.1303 0.0194 0.1686
0.5216 0.1564 1.3858 0.2414 0.2478
0.5507 0.1583 1.8041 0.2635 0.2615
0.5521 0.1586 1.8163 0.2633 0.2620
0.4682 0.1512 0.7896 0.1978 0.2223
0.302 0.130, 0.11
Mg0.70Zn0.20Ga0.10
k l* TC (K) a NOV
0.3711 0.1424 0.1461 0.0364 0.1710
0.5252 0.1560 1.4527 0.2475 0.2499
0.5543 0.1579 1.8784 0.2687 0.2636
0.5556 0.1582 1.8899 0.2684 0.2640
0.4721 0.1509 0.8420 0.2062 0.2247
0.305 0.146
Mg0.70Zn0.15Ga0.15
k TC (K) a NOV
0.3815 0.1412 0.1950 0.0857 0.1784
0.5345 0.1546 1.5787 0.2651 0.2557
0.5627 0.1564 1.9963 0.2832 0.2688
0.5637 0.1567 2.0033 0.2828 0.2691
0.4832 0.1495 0.9721 0.2313 0.2318
0.317 0.195
k l* TC (K) a NOV
0.3980 0.1412 0.2931 0.1359 0.1888
0.5577 0.1546 1.9873 0.2888 0.2678
0.5873 0.1564 2.4689 0.3046 0.2812
0.5884 0.1567 2.4777 0.3042 0.2815
0.5039 0.1495 1.2668 0.2595 0.2433
0.333 0.293, 0.306
l*
Mg0.70Zn0.10Ga0.20
Experiment [20]
1.09
0.6
1.085 0.55
1.08 0.5
1.07
0.45
H T IU F S Expt.
λ
Potential Parameter (rC)
1.075
1.065 0.4
1.06 0.35
1.055
1.05
0.3
1.045 0.25 0
1.04 0
0.05
0.1
0.15
0.2
Concentration (x)
0.05
0.1
0.15
0.2
Concentration (x) Fig. 2. Variation of electron–phonon coupling strength (k) with Ga-concentration x (at.%).
Fig. 1. Variation of the model potential parameter rC with Ga-concentration x (at.%).
amorphous alloys is observed in the range of Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and
Mg0.70Zn0.10Ga0.20, respectively. Again the H-screening function yields lowest values of l*, while the values obtained from the Ffunction are the highest.
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A.M. Vora / Physica C 468 (2008) 937–943
0.16
0.344
0.294
0.155
0.244
α
μ
∗
0.194 0.15
0.144
H T IU F S
0.145
0.094 H T IU F S
0.044
-0.006
0.14 0
0.05
0.1
0.15
0
0.2
0.05
Concentration (x)
0.1
0.15
0.2
Concentration (x)
Fig. 3. Variation of Coulomb pseudopotential (l*) with Ga-concentration x (at.%).
Fig. 5. Variation of isotope effect exponent (a) with Ga-concentration x (at.%).
2.5 2
H T IU F S Expt
2
TC (K) = 5.4722x - 0.2262x + 0.1149
0.35 H T IU F S
0.3
NOV
TC (K)
1.5
1
0.25
0.5
0.2
0 0
0.05
0.1
0.15
0.2
Concentration (x) Fig. 4. Variation of transition temperature (TC) with Ga-concentration x (at.%).
0.15 0
0.05
0.1
0.15
0.2
Concentration (x) Table 2 contains calculated values of the transition temperature TC for five Mg0.70Zn0.30xGax ternary amorphous alloys computed
Fig. 6. Variation of effective interaction strength (NOV) with Ga-concentration x (at.%).
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A.M. Vora / Physica C 468 (2008) 937–943
from the various forms of the local field correction functions along with the experimental findings [20]. From Table 2 it can be noted that, the static H-screening function yields lowest TC whereas the F-function yields highest values of TC. The present results obtained from the H-local field correction functions are found in good agreement with available experimental data [20]. The theoretical data of TC for five Mg0.70Zn0.30xGax ternary amorphous alloys is not available in the literature. It is also observed that the static H-screening function yields lowest TC whereas the F-function yields highest values of TC. The calculated results of the transition temperature TC for Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 ternary amorphous alloys deviate in the range of 80.56–100.00%, 75.51–130.03%, 72.54–124.47%, 62.40–106.08% and 94.89–95.58% from the experimental findings, respectively. The variation of the computed values of the transition temperature TC for Mg0.70Zn0.30xGax ternary amorphous alloys with the atomic concentration (x) of ‘Ga’, using five different types of the local field correction functions with EMC potential are shown in Fig. 4. The graph also includes the experimental values due to van den Berg et al. [20]. It is seen that TC is quite sensitive to the local field correction functions, and the results of TC by using Hscreening are in best agreement with the experimental data for the Mg0.70Zn0.30xGax ternary amorphous alloys under investigation, as the relevant curves for H-screening almost overlaps the experimental curves. It is also observed that the static H-screening function yields lowest TC whereas the F-function yields highest values of TC. It is also seen from the graphical nature, TC increases considerably with increasing Ga-concentration (x). The composition dependence can be described by polynomial regression of the data obtained for H-screening for different values of the concentration ‘x’, which yields T C ðKÞ ¼ 5:4722x2 0:2262x þ 0:1149:
ð17Þ
The graph of the fitted TC equation is displayed in Fig. 4, which indicates that TC increases almost considerably with increasing ‘Ga’ content with a slope dTC/dx = 0.2262. Wide extrapolation predicts a TC = 0.1149 K for the hypothetical case of ‘amorphous pure Mg0.70Zn0.30Ga0.00 alloy. The presently computed values of TC are found in the range, which is suitable for further exploring the applications of the ternary amorphous alloys for usage like lossless transmission line for cryogenic applications. While alloying elements show good elasticity and could be drawn in the form of wires as such they have good chances of being used as superconducting transmission lines at low temperature of the order of 7 K. The values of the isotope effect exponent a for five Mg0.70Zn0.30xGax ternary amorphous alloys are tabulated in Table 2. Fig. 5 depicts the variation of a with Ga-concentration (x) increases. The computed values of the a show a weak dependence on the dielectric screening, its value is being lowest for the Hscreening function and highest for the F-function. The negative value of a is observed in the case of Mg0.70Zn0.30Ga0.00 ternary alloy, which indicates that the electron–phonon coupling in these metallic complexes do not fully explain all the features regarding their superconducting behaviour. Since the experimental value of a has not been reported in the literature so far, the present data of a may be used for the study of ionic vibrations in the superconductivity of alloying substances. Since H-local field correction function yields the best results for k and TC, it may be observed that a values obtained from this screening provide the best account for the role of the ionic vibrations in superconducting behaviour of this system. The values of the effective interaction strength NOV are listed in Table 2 and depicted in Fig. 6 for different local field correction
functions. It is observed that the magnitude of NOV shows that the five Mg0.70Zn0.30xGax ternary amorphous alloys under investigation lie in the range of weak coupling superconductors. The values of NOV also show a feeble dependence on dielectric screening, its value being lowest for the H-screening function and highest for the F-screening function. The variation of present values of NOV show that, the ternary amorphous alloys under consideration fall in the range of weak coupling superconductors. The main difference of the local field correction functions are played in important role in the production of the SSP of Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 ternary amorphous alloys. The Hartree (H) dielectric function [22] is purely static and it does not include the exchange and correlation effects. Taylor (T) [23] has introduced an analytical expression for the local field correction function, which satisfies the compressibility sum rule exactly. The Ichimaru–Utsumi (IU) local field correction function [24] is a fitting formula for the dielectric screening function of the degenerate electron liquids at metallic and lower densities, which accurately reproduces the Monte-Carlo results as well as it also satisfies the self consistency condition in the compressibility sum rule and short range correlations. On the basis of Ichimaru–Utsumi (IU) local field correction function [24], Farid et al. (F) [25] and Sarkar et al. [26] have given a local field correction function. Hence, F-function represents same characteristic nature. Also, the SSP computed from Sarkar et al. [26] local field correction are found in qualitative agreement with the available experimental data [20]. The effect of local field correction functions plays an important role in the computation of k and l*, which makes drastic variation on TC, a and NOV. The local field correction functions due to IU, F and S are able to generate consistent results regarding the SSP of Mg0.70Zn0.30Ga0.00, Mg0.70Zn0.24Ga0.06, Mg0.70Zn0.20Ga0.10, Mg0.70Zn0.15Ga0.15 and Mg0.70Zn0.10Ga0.20 ternary amorphous alloys as those obtained from more commonly employed H- and T-functions. Thus, the use of these more promising local field correction functions is established successfully. The computed results of a and NOV are not showing any abnormal values for the five Mg0.70Zn0.30xGax ternary amorphous alloys.
4. Conclusions Lastly we concluded that, the H-local field corrections when used with EMC model potential provide the best explanation for superconductivity in the Mg0.70Zn0.30xGax ternary systems. The values of the k and the TC show an appreciable dependence on the local field correction function, whereas for the l*, a and NOV a weak dependence is observed. The magnitude of the k, a and NOV values shows that, the ternary amorphous alloys are weak to intermediate superconductors. Quadratic TC equation has been proposed, which provide successfully the TC values of the ternary amorphous alloys under consideration. In the absence of experimental data for a and NOV, the presently computed values may be considered to form reliable data for these ternary systems, as they lie within the theoretical limits of the Eliashberg–McMillan formulation. The comparisons of presently computed results of the SSP of the Mg0.70Zn0.30xGax ternary amorphous alloys with available experimental findings are highly encouraging, which confirms the applicability of the EMC model potential and different forms of the local field correction functions. Such study on SSP of other multi component metallic alloys is in progress.
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