Enhancing superconductivity: Magnetic impurities and their quenching by magnetic fields

– Magnetic fields and magnetic impurities are each known to suppress superconductivity. However, as the field quenches (i.e. polarizes) the impurities, rich consequences, including field-enhanced superconductivity, can emerge when both effects are present. In superconducting wires and thin films this field-spin interplay is investigated via the EilenbergerUsadel scheme. Non-monotonic dependence of the critical current on the field (and therefore field-enhanced superconductivity) is found to be possible, even in parameter regimes for which the superconducting critical temperature decreases monotonically with increasing field. The present work complements that of Kharitonov and Feigel’man (JETP Lett., 82 (2005) 421), which predicts regimes of non-monotonic behavior of the critical temperature. Introduction. – In their classic work, Abrikosov and Gor’kov [1] predicted that unpolarized, uncorrelated magnetic impurities suppress superconductivity, due to the de-pairing effects associated with the spin-exchange scattering of electrons by magnetic impurities. Among their results is the reduction, with increasing magnetic impurity concentration, of the critical temperature TC, along with the possibility of “gapless” superconductivity in an intermediate regime of the impurity concentrations. It was soon recognized that other de-pairing mechanisms, such as those involving the coupling of the orbital and spin degrees of freedom of the electrons to a magnetic field, can lead to equivalent suppressions of superconductivity [2–5]. Conventional wisdom holds that magnetic fields and magnetic moments each tend to suppress superconductivity [6]. Therefore, it seems natural to suspect that any increase in a magnetic field, applied to a superconductor containing magnetic impurities, would lead to additional suppression of the superconductivity. However, very recently, Kharitonov and Feigel’man [7] have predicted the existence of a regime in which, by contrast, an increase in the field applied to a superconductor containing magnetic impurities leads to a critical temperature that first increases with magnetic field but eventually behaves more conventionally, decreasing with the field and, ultimately, vanishing at a critical value of the field. More strikingly, they have predicted that, over a certain range of concentrations of magnetic impurities, a magnetic field can actually induce the normal state to become superconducting. c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2006-10218-2 944 EUROPHYSICS LETTERS The Kharitonov-Feigel’man treatment focuses on determining the critical temperature via the linear instability of the normal state. The purpose of the present letter is to address properties of the superconducting state itself, most notably the critical current and its dependence on temperature and externally applied magnetic field. The approach that we shall take is to derive the (transport-like) Eilenberger-Usadel equations [8, 9], starting from the Gor’kov equations. We account for the following effects: potential and spin-orbit scattering of electrons from non-magnetic impurities, and spin-exchange scattering from magnetic impurities, along with the orbital and Zeeman effects of the magnetic field. In addition to obtaining the critical current, we shall recover the Kharitonov-Feigel’man prediction for the critical temperature, as well as the dependence of the order parameter on temperature and magnetic field. In particular, we shall show that not only are there reasonable parameter regimes in which both the critical current and the transition temperature vary non-monotonically with increasing magnetic field, but also there are reasonable parameter regimes in which only the low-temperature critical current is non-monotonic, even though the critical temperature varies monotonically with field. We believe the present theory is applicable to explaining certain recent experiments on superconducting wires [10]. Let us pause to give a physical picture of the relevant de-pairing mechanisms. First, magnetic impurities cause spin-exchange scattering of the electrons (including both spin-flip and non-spin-flip terms, relative to an arbitrary spin quantization axis), and therefore lead to the breaking of Cooper pairs [1]. Next, consider the effects of magnetic fields. The associated vector potential scrambles the relative quantum phases of the partners of a Cooper pair, as they move diffusively in the presence of impurity scattering (viz., the orbital effect), which suppresses superconductivity [2,3]. On the other hand, the magnetic field polarizes the magnetic impurity spins, which decreases the rate of exchange scattering, thus diminishing this contribution to de-pairing [7]. In addition, the Zeeman effect associated with the effective field (coming from the magnetic field and the impurity spins) splits the energy of the up and down spins in the Cooper pair, thus tending to suppress superconductivity [6]. But strong spin-orbit scattering tends to weaken the de-pairing by the Zeeman effect [5]. Thus, we see that the magnetic field produces competing tendencies: it causes de-pairing via the orbital and Zeeman effects but it mollifies the de-pairing caused by magnetic impurities. This competition can manifest itself through the non-monotonic behavior of observables such as the critical temperature and critical current. In order for these manifestations to be observable, the magnetic field needs to be present throughout the samples, a scenario readily accessible in wires and thin films. Model. – The full Hamiltonian is H = H0 +Hint +HZ. We take the impurity-free part of the Hamiltonian to be of the BCS form [6,11] H0 = ∫ dr −1 2m ψ† α ( ∇− ie c A )2 ψα + V0 2 ∫ dr ( 〈ψ† αψ β〉ψβψα + ψ† αψ β〈ψβψα〉 ) − μ ∫ dr ψ† αψα, (1) where ψ† α(r) creates an electron having mass m, charge e, position r and spin projection α, A is the vector potential, c is the speed of light, μ is the chemical potential, V0 is the BCS pairing interaction, and 〈· · · 〉 denotes the appropriate thermal average. Throughout this letter we shall put = 1 and kB = 1. Assuming the superconducting pairing is spin-singlet, we introduce the complex order parameter ∆ via −V0〈ψαψβ〉 = iσ αβ∆, V0〈ψ† αψ β〉 = iσ αβ∆, (2) where σ αβ are the Pauli matrices. We assume that the electrons undergo potential (u1) scattering from impurities, located at a set of random positions {xi}, spin-exchange (u2) T.-C. Wei et al.: Enhancing superconductivity 945 scattering from magnetic impurities, located at positions {xj}, and spin-orbit scattering (vso) from a third set of impurities (or defects) located at positions {zk}, as well as being Zeemancoupled to the applied magnetic field: these effects are included via Hint = ∫ dr ψ† αVαβψβ , with Vαβ being given by