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Evolution of density of states for Fulde–Ferrell-type superconductors
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) e1097–e1098
Evolution of density of states for Fulde–Ferrell-type superconductors Mario Cuoco*, Paola Gentile, Canio Noce Unita" I.N.F.M. di Salerno, Dipartimento di Fisica ‘‘E. R. Caianiello’’, Universita" di Salerno, Baronissi Salerno I-84081, Italy
Abstract The evolution of the density of states as a function of Zeeman magnetic field is studied assuming that the superconducting order parameter has a spatial modulation in the form of a Fulde–Ferrell–Larkin–Ovchinnikov-type state. The competition between the gain in the paramagnetic energy for the normal electrons and that one of the depaired electrons induced by the modulation of the order parameter, is discussed for a s-wave-type superconductor. r 2003 Elsevier B.V. All rights reserved. PACS: 74.81.g; 74.25.Ha; 74.70.Pq Keywords: Supercondutivity; Ferromagnetism; Depaired electrons
According to the conventional view, superconductivity and magnetic field are incompatible. The fundamental reason is that in an external magnetic field, the order parameter becomes frustrated. This orbital frustration raises the free energy of the superconducting state leading, as the field is increased, to a transition back to the normal state. Nevertheless, a nonuniform superconducting state in the Chandrasekhar–Clogston limit in a strong magnetic field has been predicted by Fulde and Ferrell and independently by Larkin and Ovchinnikov (FFLO state) [1]. They noted that the destructive influence of Pauli paramagnetism on superconductivity can be mitigated by pairing spin-up and spin-down electrons with a nonzero total momentum whose value depends on the magnetic field. In this way, the pairing condition, which requires that opposite spin electrons with equal energy and a given total momentum should be paired, can be fulfilled with improved accuracy over some parts of the Fermi surface. On other parts of the Fermi surface, it may then not be possible to pair electrons at all, but the FFLO state can nonetheless be more stable than the uniform solution. This superconducting state occurs only at temperatures *Corresponding author. Tel.: +39-089-965235; fax: +39089-965275. E-mail address: [email protected] (M. Cuoco).
smaller than Tc C0:56Tc0 where Tc0 is the zero-field superconducting transition temperature [2]. The phase transition is of the first order from FFLO state to the ordinary uniform superconducting state and of second order to the normal metallic state. Although this nonuniform state was predicted many years ago, there has been up to now no unequivocal experimental evidence of its existence. A possible formation of a nonuniform superconducting state has been found in the heavy-fermion compound UPd2 Al3 [3]. In this case, thermal expansion of magnetostriction measurements below the superconducting critical temperature have been utilized to establish a first-order phase transition into a inhomogeneous superconducting state. In the following we shall study within a mean-field approximation a weak-coupling BCS model searching for the FFLO state for s-wave symmetry of the order parameter. Specifically, we shall consider how the density of states and the occupation number of depaired electrons evolve as a function of the Zeeman magnetic field. Since the pairs in the FFLO state have finite momentum, it follows that the phase of the gap function is no longer constant but it varies spatially with the wave vector of the total momentum of the pairs. In the meanfield approximation, by applying to the model Hamiltonian the Bogoliubov rotations, it is possible to get the
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1382
ARTICLE IN PRESS e1098
M. Cuoco et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) e1097–e1098
Fig. 1. Behavior of the occupation number of depaired electrons as function of k, in the first Brillouin zone, for a s-wave superconductor, in the presence of a magnetic field H ¼ 1:1D0 ; where D0 is the zero temperature BCS energy gap. The upper figure represents the BCS case; the lower one refers to the Fulde and Ferrell case.
pair amplitude, lowers the pair amplitude itself with respect to the BCS case and at the same time, the formation of depaired electrons allows for gain magnetic energy in such a way that, in the presence of magnetic field higher than the Pauli limit, the total energy balance gives rise to a state with energy lower than the paramagnetic state. These considerations can be clarified by looking at Fig. 1, where we have represented the evolution of the density of the occupation number of depaired electrons in the momentum space for a twodimensional system, by assuming a near-neighbor hopping. In the upper figure we show the formation of depaired electrons in a BCS superconductor in presence of a magnetic field higher than the Pauli limit. In this case the distribution of depaired electrons spreads in the k-space along the lines jkx j þ jky j ¼ 1; which represent the boundaries of the Fermi surface. If we consider the possibility that the FFLO phase can be realized, some depaired electrons form before the field reaches the Pauli limit; when the magnetic field becomes higher than the Pauli limit, for finite value of the pair momentum q; the number of depaired electrons forming on the Fermi surface is reduced with respect to the BCS case, as represented in the lower figure in Fig. 1. However, as the strength of the field is increased, the number of depaired electrons becomes greater and greater until, for a critical field strength, the system is energetically favored to pass in the normal state. For brevity, no discussion has been included on the modifications induced on the density of states by the assumption of a singlet d-wave order parameter. Results will be presented elsewhere.
References quasi-particle energy dispersions of the fermionic excitations. Contrary to the BCS-case, in the FFLO phase the quasi-particle energies could be negative and consequently the quasi-particle occupation numbers could be nonvanishing, for suitable values of the total pairs momentum. As a consequence, the presence of depaired electrons, whose energies are dependent on the
[1] P. Fulde, R.A. Ferrell, Phys. Rev. 135 (1964) A550; A.I. Larkin, Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47 (1964) 1136 [Sov. Phys. JETP 20 (1965) 762]. [2] H. Shimahara, Phys. Rev. 50 (1994) 12760. [3] K. Gloos, R. Modler, H. Schimanski, C.D. Bredl, C. Geibel, F. Steglich, A.I. Buzdin, N. Sato, T. Komatsubara, Phy. Rev. Lett. 70 (1993) 501.
Journal of Magnetism and Magnetic Materials 272–276 (2004) e1097–e1098
Evolution of density of states for Fulde–Ferrell-type superconductors Mario Cuoco*, Paola Gentile, Canio Noce Unita" I.N.F.M. di Salerno, Dipartimento di Fisica ‘‘E. R. Caianiello’’, Universita" di Salerno, Baronissi Salerno I-84081, Italy
Abstract The evolution of the density of states as a function of Zeeman magnetic field is studied assuming that the superconducting order parameter has a spatial modulation in the form of a Fulde–Ferrell–Larkin–Ovchinnikov-type state. The competition between the gain in the paramagnetic energy for the normal electrons and that one of the depaired electrons induced by the modulation of the order parameter, is discussed for a s-wave-type superconductor. r 2003 Elsevier B.V. All rights reserved. PACS: 74.81.g; 74.25.Ha; 74.70.Pq Keywords: Supercondutivity; Ferromagnetism; Depaired electrons
According to the conventional view, superconductivity and magnetic field are incompatible. The fundamental reason is that in an external magnetic field, the order parameter becomes frustrated. This orbital frustration raises the free energy of the superconducting state leading, as the field is increased, to a transition back to the normal state. Nevertheless, a nonuniform superconducting state in the Chandrasekhar–Clogston limit in a strong magnetic field has been predicted by Fulde and Ferrell and independently by Larkin and Ovchinnikov (FFLO state) [1]. They noted that the destructive influence of Pauli paramagnetism on superconductivity can be mitigated by pairing spin-up and spin-down electrons with a nonzero total momentum whose value depends on the magnetic field. In this way, the pairing condition, which requires that opposite spin electrons with equal energy and a given total momentum should be paired, can be fulfilled with improved accuracy over some parts of the Fermi surface. On other parts of the Fermi surface, it may then not be possible to pair electrons at all, but the FFLO state can nonetheless be more stable than the uniform solution. This superconducting state occurs only at temperatures *Corresponding author. Tel.: +39-089-965235; fax: +39089-965275. E-mail address: [email protected] (M. Cuoco).
smaller than Tc C0:56Tc0 where Tc0 is the zero-field superconducting transition temperature [2]. The phase transition is of the first order from FFLO state to the ordinary uniform superconducting state and of second order to the normal metallic state. Although this nonuniform state was predicted many years ago, there has been up to now no unequivocal experimental evidence of its existence. A possible formation of a nonuniform superconducting state has been found in the heavy-fermion compound UPd2 Al3 [3]. In this case, thermal expansion of magnetostriction measurements below the superconducting critical temperature have been utilized to establish a first-order phase transition into a inhomogeneous superconducting state. In the following we shall study within a mean-field approximation a weak-coupling BCS model searching for the FFLO state for s-wave symmetry of the order parameter. Specifically, we shall consider how the density of states and the occupation number of depaired electrons evolve as a function of the Zeeman magnetic field. Since the pairs in the FFLO state have finite momentum, it follows that the phase of the gap function is no longer constant but it varies spatially with the wave vector of the total momentum of the pairs. In the meanfield approximation, by applying to the model Hamiltonian the Bogoliubov rotations, it is possible to get the
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1382
ARTICLE IN PRESS e1098
M. Cuoco et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) e1097–e1098
Fig. 1. Behavior of the occupation number of depaired electrons as function of k, in the first Brillouin zone, for a s-wave superconductor, in the presence of a magnetic field H ¼ 1:1D0 ; where D0 is the zero temperature BCS energy gap. The upper figure represents the BCS case; the lower one refers to the Fulde and Ferrell case.
pair amplitude, lowers the pair amplitude itself with respect to the BCS case and at the same time, the formation of depaired electrons allows for gain magnetic energy in such a way that, in the presence of magnetic field higher than the Pauli limit, the total energy balance gives rise to a state with energy lower than the paramagnetic state. These considerations can be clarified by looking at Fig. 1, where we have represented the evolution of the density of the occupation number of depaired electrons in the momentum space for a twodimensional system, by assuming a near-neighbor hopping. In the upper figure we show the formation of depaired electrons in a BCS superconductor in presence of a magnetic field higher than the Pauli limit. In this case the distribution of depaired electrons spreads in the k-space along the lines jkx j þ jky j ¼ 1; which represent the boundaries of the Fermi surface. If we consider the possibility that the FFLO phase can be realized, some depaired electrons form before the field reaches the Pauli limit; when the magnetic field becomes higher than the Pauli limit, for finite value of the pair momentum q; the number of depaired electrons forming on the Fermi surface is reduced with respect to the BCS case, as represented in the lower figure in Fig. 1. However, as the strength of the field is increased, the number of depaired electrons becomes greater and greater until, for a critical field strength, the system is energetically favored to pass in the normal state. For brevity, no discussion has been included on the modifications induced on the density of states by the assumption of a singlet d-wave order parameter. Results will be presented elsewhere.
References quasi-particle energy dispersions of the fermionic excitations. Contrary to the BCS-case, in the FFLO phase the quasi-particle energies could be negative and consequently the quasi-particle occupation numbers could be nonvanishing, for suitable values of the total pairs momentum. As a consequence, the presence of depaired electrons, whose energies are dependent on the
[1] P. Fulde, R.A. Ferrell, Phys. Rev. 135 (1964) A550; A.I. Larkin, Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47 (1964) 1136 [Sov. Phys. JETP 20 (1965) 762]. [2] H. Shimahara, Phys. Rev. 50 (1994) 12760. [3] K. Gloos, R. Modler, H. Schimanski, C.D. Bredl, C. Geibel, F. Steglich, A.I. Buzdin, N. Sato, T. Komatsubara, Phy. Rev. Lett. 70 (1993) 501.