A Study of Heavy-Fermion Superconductors via BCS Equations Incorporating Chemical Potential
Author(s)
G. P. Malik*
Affiliation(s)
Theory Group, School of Environmental Sciences, Jawaharlal Nehru University, New Delhi, India.
Theory Group, School of Environmental Sciences, Jawaharlal Nehru University, New Delhi, India.
ABSTRACT
Heavy-fermion superconductors (HFSCs) are regarded as outside the purview of BCS theory because it is usually constrained by the inequality
, where EF, μ, kB, and θD 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 Tc and the gap Δ0 at T = 0. Further, when μ < kBθD, we curtail the limits of the equations for Tc 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 Tc 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 < kBθD; 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.
Heavy-fermion superconductors (HFSCs) are regarded as outside the purview of BCS theory because it is usually constrained by the inequality
, where EF, μ, kB, and θD 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 Tc and the gap Δ0 at T = 0. Further, when μ < kBθD, we curtail the limits of the equations for Tc 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 Tc 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 < kBθD; 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 Tc 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
Range of Application of the BCS Theory, Incorporation of Chemical Potential into Equations for Tc 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
Cite this paper
Malik, G. (2015) A Study of Heavy-Fermion Superconductors via BCS Equations Incorporating Chemical Potential. Journal of Modern Physics, 6, 1233-1242. doi: 10.4236/jmp.2015.69128.
Malik, G. (2015) A Study of Heavy-Fermion Superconductors via BCS Equations Incorporating Chemical Potential. Journal of Modern Physics, 6, 1233-1242. doi: 10.4236/jmp.2015.69128.
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
[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