A Study of Heavy-Fermion Superconductors via BCS Equations Incorporating Chemical Potential

G. P. Malik^*

Theory Group, School of Environmental Sciences, Jawaharlal Nehru University, New Delhi, India.

DOI: 10.4236/jmp.2015.69128                

Abstract

Heavy-fermion superconductors (HFSCs) are regarded as outside the purview of BCS theory because it is usually constrained by the inequality , where E_F, μ, k_B, and θ_D are, respectively, the Fermi energy, chemical potential, Boltzmann constant, and the Debye temperature. We show that this restriction can be removed by incorporating μ into the equations for T_c and the gap Δ₀ at T = 0. Further, when μ < k_Bθ_D, we curtail the limits of the equations for T_c and Δ₀ to avoid complex-valued solutions. The resulting equations are applied to a prominent member of the HFSC family, i.e., CeCoIn₅, by appealing to ideas due to Born and Karmann, Suhl et al., and Bianconi et al. Since the equations now contain an additional variable μ, we find that 1) the T_c of the SC can be accounted for by a multitude of values of the (μ, λ) pair, λ being the interaction parameter; 2) the λ vs. μ plot has a dome-like structure when μ < k_Bθ_D; 3) the (μ, λ) values obtained in 2) lead to reasonable results for the range of each of the following variables: Δ₀, s, and n, where s is the ratio of the mass of a conduction electron and the free electron mass and n is the number density of charge carriers in the SC.

Keywords

Range of Application of the BCS Theory, Incorporation of Chemical Potential into Equations for T_c and Δ₀, Extended BCS Theory, Heavy-Fermion Superconductors, Ratio of the Mass of a Conduction Electron and the Free Electron Mass, Density of Charge Carriers

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Hewson, A.C. (1993) The Kondo Problem to Heavy Fermions, Cambridge Studies in Magnetism (No. 2). Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511470752
[2]	Steglich, F., et al. (1976) Physical Review Letter, 43, 1892-1896. http://dx.doi.org/10.1103/PhysRevLett.43.1892
[3]	Riseborough, P.S., Schmiedeshoff, G.M. and Smith, J.L. (2008) Heavy Fermion Superconductivity, Springer Series in Solid State Sciences. Springer Verlag, Berlin.
[4]	Coleman, P. (2007) Heavy Fermions: Electrons at the Edge of Magnetism. In: Kronmuller, H. and Parkin, S., Eds., Handbook of Magnetism and Advanced Magnetic Materials, Vol. 1, John Wiley, Chichester, 95-148. http://dx.doi.org/10.1002/9780470022184.hmm105
[5]	Thompson, J.D. (2007) Superconductivity in Heavy-Fermion Materials. Talk Given at the Conference, BCS Theory @ 50, University of Illinois at Urbana-Champaign, 10-13 October 2007.
[6]	Miyake, K., Rink, S.S. and Varma, C.M. (1986) Physical Review B, 34, 6554-6556. http://dx.doi.org/10.1103/PhysRevB.34.6554
[7]	Monod, M.T.B., Bourbonnias, C. and Emery, V. (1986) Physical Review B, 34, 7716-7720. http://dx.doi.org/10.1103/PhysRevB.34.7716
[8]	Scalapino, D.J., Loh, E. and Hirsch, J.E. (1986) Physical Review B, 34, 8190-8192. http://dx.doi.org/10.1103/PhysRevB.34.8190
[9]	Bianconi, A. (2013) Nature Physics, 9, 536-537. http://dx.doi.org/10.1038/nphys2738
[10]	Kresin, V.Z. and Wolf, S.A. (2009) Reviews of Modern Physics, 81, 481-501. http://dx.doi.org/10.1103/RevModPhys.81.481
[11]	Malik, G.P. (2010) International Journal of Modern Physics B, 24, 1159-1172. http://dx.doi.org/10.1142/S0217979210055408
[12]	Malik, G.P. and Malik, U. (2011) Journal of Superconductivity and Novel Magnetism, 24, 255-260. http://dx.doi.org/10.1007/s10948-010-1009-0
[13]	Malik, G.P. (2010) International Journal of Modern Physics B, 24, 3701-3712. http://dx.doi.org/10.1142/S0217979210055858
[14]	Malik, G.P., Chávez, I. and de Llano, M. (2013) Journal of Modern Physics, 4, 474-480. http://dx.doi.org/10.4236/jmp.2013.44067
[15]	Eagles, D.M. (1969) Physical Review, 186, 456-463. http://dx.doi.org/10.1103/PhysRev.186.456
[16]	Malik, G.P. (2014) International Journal of Modern Physics B, 28, Article ID: 1450054, 13 p. http://dx.doi.org/10.1142/S0217979214500544
[17]	Malik, G.P. (2014) International Journal of Modern Physics B, 29, Article ID: 1450238, 14 p. http://dx.doi.org/10.1142/S0217979214502385
[18]	Petrovic, C., Pagliuso, P.G., Hundley, M.F., Movshovich, R., Sarrao, J.L., Thompson, J.D., et al. (2001) Journal of Physics: Condensed Matter, 13, L337-L342. http://dx.doi.org/10.1088/0953-8984/13/17/103
[19]	Kim, J.S., Alwood, J., Stewart, G.R., Searro, J.L. and Thompson J.D. (2001) Physical Review B, 64, Article ID: 134524, 9 p. http://dx.doi.org/10.1103/PhysRevB.64.134524
[20]	Seitz, F. (1940) The Modern Theory of Solids. McGraw-Hill, New York.
[21]	Pines, D. (1958) Physical Review, 109, 280-287. http://dx.doi.org/10.1103/PhysRev.109.280
[22]	Allan, M.P., Massee, F., Morr, D.K., Van Dyke, J., Rost, A.W., Mackenzie, A.P., et al. (2013) Nature Physics, 9, 468-473. http://dx.doi.org/10.1038/nphys2671
[23]	Malik, G.P. and Varma, V.S. (2013) International Journal of Modern Physics B, 27, Article ID: 1350042, 10 p. http://dx.doi.org/10.1142/S0217979213500422
[24]	Suhl, H., Matthias, B.T. and Walker, L.R. (1959) Physical Review Letters, 3, 552-554. http://dx.doi.org/10.1103/PhysRevLett.3.552
[25]	Cracknell, A.P. and Kong, K.C. (1973) The Fermi Surface. Clarendon Press, Oxford.
[26]	Weinberg, S. (1972) Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & Sons Ltd., New York.
[27]	Malik, G.P. ( 2010) Physica B: Condensed Matter, 405, 3475-3481. http://dx.doi.org/10.1016/j.physb.2010.05.026