Rafique, S M

Yadav, Jayprakash

English

IACS Institutional Repository

Indian J. Phys. 81(1) 161-165(2007) # Superconducting parameters of metals and alloys: HFP technique Jayprakash Yadav1 and S M Rafique2* JKhunt, Konapatti, Supaul-852 214, Bihar, India 2P G Department of Physics, T M Bhagalpur University, BhagaIpur-812 007, Bihar, India E-mail sm jrafq(«Jyahoo com Abstract : The superconducting parameters of metals and alloys have been studied within the framework of BCS theory via McMillan [1] technique for different metals and alloys using Harrisons First Principles (HFP) pseudopotential. This technique is being used for the first time for this investigation although it is more rigorous and based on sound theoretical background In the present paper we have studied the electron phonon coupling strength A, Coulomb pseudopotential ft*, superconducting transition temperature Tc of MgAI and ZnAI binary alloys The results are compared with available experimental and theoretical data giving a reasonable agreement and providing an insight into the systems under investigation Keywords : Superconductivity, pseudopotential, alloys PACS No. : 74 20.-2 1. Introduction In the present era the study of Superconductivity (SC) of metals and alloys occupies the centre stage in the field of condensed matter physics. The theoretical work in this field got impetus since the proposition of BCS theory which was later developed to suite within the framework of pseudopotentials by McMillan [1], Allen and Cohen [2] and Eliashberg [3]. The theory was extended to binary alloys by Allen and Dynes [4]. Scattered attempts have been made to apply this technique using model potentials or other techniques [6-10]. A study of the existing literature reveals that one of the more rigorous and theoretically sound technique viz, Harrison's first principle (HFP) pseudopotential [11] has not been used for the purposa although it ha9^ been successfully used in the past few decades to explain the atomic, electronic and other properties of metals and alloys [12-15]. This has encouraged us to apply this technique to study the superconducting transition parameters of binary alloys MgAI and ZnAI. Corresponding Author © ^007 IACS 162 Jayprakash Yadav andSM Rafique 2. Formalism The parameters under investigation are the electron-phonon coupling strength (EPCS) A, the Coulomb coupling strength //* and the SC transition temperature Tc represented by n where " (2) L f <*? . (3) ^'nkF [ qe*{q) 7 c «A.exp 0 1.45 1.04Q + A) I A-/T(1 + 0.62A)I (4) In these equations, kB is the Boltzmann's constant, m* is the effective electronic mass, M the atomic mass, Z the valence, /cF the Fermi wave vector, 6D the Debye temperature, q the phonon wave vector, w(k,q) the screened form factor, e*{q) the modified Hartree dielectric function using Vashista-Singwi (V-S) form of exchange correlation [16] and EF the Fermi energy. The input parameters (X) of alloys are evaluated through XAQ « (1 - c)XA + cXB, (5) where A and B are constituent metals, c is the concentration of B and X represents the atomic volume Q0, valence Z, atomic mass Mt Fermi wave vector kf of the species. The Debye temperature of the alloy is given by ( ^ ' M T S O T (6> The electron-phonon coupling strength X involves the screened form factor given by ^(^Q) = (v-'^vJ+varf)/fMQ) + [l-G(Q)K/fMQ) + Wf?l (7) where v*'b, v£ \%and vfq are the valence charge and core electron potential, conduction band-core exchange potential, conduction electron potential and screening potential respectively and WR is the repulsive potential, G(q) is the exchange correlation function which is given by G( r i j )«A[ l -e"a ? f ] where rj = q//fF (8) Superconducting parameters of metals and alloys: HFP technique 163 in the V-S form of exchange-correlation [17]. e*(rj) is the modified Hartree dielectric function given by 6*(f j)-[ l- f l ( l?)][ f i«(l?)- l ]+1 f where £„(77) = 1 + T T M 1 + — - i - / ? 7 4r] 2 + r7 2-77 (9) (10) Is the Hartree dielectric function. The parameter e*(rj) is alao involved in the expression for the Coulomb coupling ju*. 3. Computation The computation of the form factor w(k,q) and the modified Hartree dielectric function r*(q) is carried on through eqs. (7) and (8), respectively. The core energy eigenvalues *•„, needed for the computation of w(/c,g) is taken from Herman-Skillman program [17] and the input parameters are presented in Table 1. From these computed w(k%q) of constituent metals the form factor of the alloys and other constants are obtained through eq. (5) and the Debye temperature {0D)AB is obtained through eq. (6). After these computations we proceed to compute the superconducting parameters through eqs. (1) to (4). The results thus obtained have been presented in Table 1 for comparison with experimental observations. Table 1. SC parameters of binary alloys AB. Alloy (AB) S< MgAI 0.3 MgAI 0.5 ZnAI 0.3 ZnAI 0.5 arameten mb X n* Tc mb A M' Tc mb k M* Tc mb A M* Tr> 5 1.0000 0.2777 0.0773 0.2300 1.0000 0.3022 0.0767 0.4613 1.0000 0.3480 0.0753 0.9702 1.0000 0.3531 0.0752 1.1131 Present 1.2780 0.3550 0.0773 1.2900 1.4650 0.4427 0.0767 3.6400 1.0200 0.3549 0.0753 1.1400 1.2500 0.4414 0.0752 3.1600 0.9180 0.2549 0.0773 0.1000 1.0200 0.3082 0.0767 0.5316 0.9360 0.3257 0.0753 0.6900 0.9800 0.3460 0.0752 0.9900 Expt, 0.84 0.84 0.5-O.84 0.5-0.84 164 Jayprakash Yadav andSM Rafique 4. Result and observation The computed SC parameters A, //* and Tc, alongwith the band mass mb of the alloys MgAI and ZnAI treated as effective mass rrf have been presented in Table 1 for two different concentration c = 0.3 and c = 0.5 (equiatomic composition). Considering the free electron behavior of the constituent metals band mass mb = 1 has been taken initially for the above alloys. Two other band masses have also been evaluated for the binary alloy, as provided in this table. Other theoretical data of A and v* or Tc is not available for comparison. Hence, only the experimental value of Tc has been given in the table. An inspection of Table 1 brings out the fact that for MgAI with c = 0.3 and mb = 1.278 we get A = 0.355 and Tc = 1.29 against the experimental value 0.84 K. With c = 0.5 and mb ~ 1.02 we get A = 0.3082 and Tc = 0.53. These are in qualitative agreement with experiment. For ZnAI with c = 0.3 and mb = 0.936 we get A = 0.3257 and Tc = 0.69 which lies within the experimental range 0.5-0.84 K. However, with c = 0.5 and mb - 0.980 we get A = 0.346 and Tc = 0.99 which is slightly higher than the experimental range. 5. Conclusion Harrison's first principle (HFP) pseudopotential technique has been used for the first time to study the superconducting parameters of binary alloys MgAI and ZnAI with reasonable success. The present investigation shows that the transitin temperature Tc is quite sensitive to the electron-phonon coupling strength A. The Coulomb pseudopotential //* is in general of the order of 0.1 as for pure metals. The accuracy of A and Tc mainly depends upon the correctness of the form factor which, in turn, hinges upon the input parameters such as core energy eigenvalues, Xor-exchange parameter, opw parameter and the form of exchange-correlation function. References [1) W L McMillan Phys. Rev. 167 331 (1968) [2] P B Allen and M L Cohen Phys. Rev. 187 525 (1969) [3) G M Eliashberg Soviet Phys. JETP11 696 (1960); 12 1000 (1961) [4] P B Allen and R C Dynes Phys. Rev. B12 905 (1975) [5J H Khan, V Singh and K S Sharma Indian J. Pure Appl. Phys. 31 628 (1993) [6] V Singh, H Khan and K S Sharma Indian J. Pure Appl. Phys. 32 915 (1994) [7] P P Singh Bull. Mater Sci. 26 131 (2003) [8] O M Gartonde, P Modak, R S Rao and B K Godwal Bull. Mater Sci. 26 137 (2003) [9] S Sharma, Haniph Khan and K S Sharma Indian J. Pure Appl. Phys. 41 301 (2003) [10J AM Vora, M H Patel, P N Gajjarand A R Jani Pramana58 849 (2002) [11] R S Day, F Sun and P H Cutler Phys. Rev. A19 328 (1979) [12] J Hafner Phys. Stat So/. (b)56 579 (1973); 57 101 (1973); 57 497 (1973) Superconducting parameters of metals and alloys: HFP technique 165 [ 13 j J A Morlarty Phys. Rev. B16 2537 (1977) [14] M M Khan and S M Rafique Phys. Stat. Sol. (b)139 K131 (1987) [15] K K Mukerjee, S M Rafique, M M Khan and S Bagchi Phys. Stat. Sol, (b)147 K51 (1988) [16] P Vashishta and K S Singwi Phys. Rev. B6 875 (1972) [17] P Herman and J Skillman Atomic Structure Calculation (Englewood Cliff, N.J.: Prentice Hall) (1963) 

Superconducting Parameters of Metals and Alloys : HFP Technique

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Superconducting Parameters of Metals and Alloys : HFP Technique

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