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Vortices with Magnetic Field Inversion in Noncentrosymmetric Superconductors Julien Garaud, Maxim N

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Vortices with Magnetic Field Inversion in Noncentrosymmetric Superconductors Julien Garaud, Maxim N Vortices with magnetic field inversion in noncentrosymmetric superconductors Julien Garaud, Maxim N. Chernodub, Dmitri E. Kharzeev To cite this version: Julien Garaud, Maxim N. Chernodub, Dmitri E. Kharzeev. Vortices with magnetic field inversion in noncentrosymmetric superconductors. Physical Review B, American Physical Society, 2020, 102 (18), pp.184516. 10.1103/PhysRevB.102.184516. hal-02542879 HAL Id: hal-02542879 https://hal.archives-ouvertes.fr/hal-02542879 Submitted on 9 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Vortices with magnetic field inversion in non-centrosymmetric superconductors J. Garaud,1, ∗ M. N. Chernodub,1, 2, y and D. E. Kharzeev3, 4, 5, z 1Institut Denis Poisson CNRS/UMR 7013, Universit´ede Tours, 37200 France 2Pacific Quantum Center, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia 3Department of Physics and Astronomy, Stony Brook University, New York 11794-3800, USA 4Department of Physics and RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA 5Le Studium, Loire Valley Institute for Advanced Studies, Tours and Orl´eans,France (Dated: March 25, 2020) Superconducting materials with non-centrosymmetric lattices lacking the space inversion symme- try exhibit a variety of interesting parity-breaking phenomena, including magneto-electric effect, spin-polarized currents, helical states, and unusual Josephson effect. We demonstrate, within a Ginzburg-Landau framework describing non-centrosymmetric superconductors with O point group symmetry, that vortices can exhibit an inversion of the magnetic field at a certain distance from the vortex core. In a stark contrast to conventional superconducting vortices, the magnetic-field reversal in the parity-broken superconductor leads to non-monotonic intervortex forces and, as a consequence, to the exotic properties of the vortex matter such as the formation of vortex bound states, vortex clusters, and appearance of metastable vortex/anti-vortex bound states. I. INTRODUCTION Non-centrosymmetric superconductors are supercon- ducting materials whose crystal structure is not symmet- ric under the spatial inversion. These parity-breaking materials have attracted much theoretical [1,2] and ex- perimental [3{5] interest, as they open the possibility to investigate spontaneous breaking of a continuous symme- try in a parity-violating medium (for recent reviews, see [6{8]). The parity-breaking nature of the superconduct- ing order parameter [4,5] in the non-centrosymmetric superconductors leads to various unusual magnetoelec- tric phenomena due to the mixing of singlet and triplet 0.015 0.05 0.1 0.2 0.5 1 3.2 components of the superconducting condensate, correla- tions between supercurrents and spin polarization, to the jBj existence of helical states, and unusual structure of vor- tex lattices. Figure 1. Inversion patterns of the magnetic field B of a vor- Moreover, parity breaking in the non-centrosymmetric tex in a non-centrosymmetric superconductor. The magnetic superconductors also results in an unconventional field forms helicoidal patterns around a straight static vortex. Josephson effect, where the junction features a phase- As the distance from the vortex core increases, the longitudi- shifted relation for the Josephson current [9, 10]. Un- nal (parallel to the vortex core) component of the magnetic conventional Josephson junctions consisting of two non- field may change its sign. The magnetic field may exhibit sev- centrosymmetric superconductors linked by a uniaxial eral sign reversals in the normal plane. In the picture, which ferromagnet were recently proposed as the element of is a result of a numerical simulation of the Ginzburg-Landau a qubit that avoids the use of an offset magnetic flux, theory, the colors encode the amplitude of the magnetic field enabling a simpler and more robust architecture [11]. B, in a normal plane with respect to the vortex line while the In the macroscopic description of such superconducting arrows demonstrate the orientation of the field. states, the lack of inversion symmetry yields new terms arXiv:2003.10917v1 [cond-mat.supr-con] 24 Mar 2020 in the Ginzburg-Landau free energy represented by the so-called the Lifshitz invariants. These terms directly and, at the same time, are invariant under spatial ro- couple the magnetic field B to the supercurrent j and tations. The corresponding Lifshitz invariant featur- thus lead to a variety of new effects that are absent in ing these symmetries is described by a simple, parity- conventional superconductors. The explicit form of the violating isotropic term, γj · B, where the coupling γ de- allowed Lifshitz invariant depends on the point symmetry termines the strength of the parity breaking. This partic- group of the underlying crystal structure. ular structure describes non-centrosymmetric supercon- In this paper, we consider a particular class of non- ductors with O point group symmetry such as Li2Pt3B centrosymmetric superconductors whose macroscopic in- [5, 12], Mo3Al2C[13, 14], and PtSbS [15]. teractions break the discrete group of parity reversals Vortex states in cubic non-centrosymmetric supercon- 2 ductors feature a transverse magnetic field, in addition to II. THEORETICAL FRAMEWORK the ordinary longitudinal field. Consequently, they also carry a longitudinal current on top of the usual trans- We consider non-centrosymmetric superconductors verse screening currents [16{18]. Therefore, as illustrated with the crystal structure possessing the O point group in Fig.1, both the superconducting current and the mag- symmetry. Such materials are described, in the vicin- netic field form a helical-like structure that winds around ity of the superconducting critical temperature, by the the vortex core (for additional material illustrating the Ginzburg-Landau free energy F = R d3x F with the free- helical spatial structure of the magnetic streamlines, see energy density given by (see e.g. [6, 20]): AppendixB, and animations [19] ). The previous theo- retical papers studied vortices in the perturbative regime B2 β F = + kjD j2 + γj · B + (j j2 − 2)2 ; (1) where the coupling to the Lifshitz invariant γ is small, ei- 8π 2 0 ther in the London limit (with a large Ginzburg-Landau where j = 2e Im ( ∗D ); we use =c=1. Here, the sin- parameter) [16, 17], or beyond it [18]. For currently ~ gle component order parameter = j jei' is a complex known non-centrosymmetric materials, these approxima- scalar field that is coupled to the vector potential A of tions are valid since the magnitude of the Lifshitz invari- the magnetic field B = r×A through the gauge deriva- ants, which can be estimated in a weak-coupling approx- tive D ≡ r − ieA, where e is a gauge coupling. The imation, is typically small. We propose here a general explicit breaking of the inversion symmetry is accounted study of vortices, for all possible values of the Lifshitz in- by the Lifshitz invariant term with the prefactor γ, that variant coupling, both in the London limit and beyond. directly couples the magnetic field B and the supercur- rent j = 2ej j2(r' − eA). The current j is the usual We demonstrate that vortices may feature an inver- superconducting current at γ = 0, i.e. in the absence of sion of the magnetic field at distance of about 4λL from parity breaking. The parameter γ can be chosen to be the vortex center. Moreover, for rather high values of the positive without loss of generality. At a nonzero parity- coupling γ, alternating reversals may occur several times, breaking coupling γ, the current gets an additional con- at different distances from the vortex core. Such an in- tribution from the Lifshitz term [20]. The other coupling version of the magnetic field is illustrated, in Fig.1. The constants k and β describe, respectively, the magnitude reversal of the magnetic field, which is in stark contrast to of the kinetic and potential terms in the free energy (1). conventional superconducting vortices, becomes increas- The variation of the free energy (1) with respect to the ingly important for larger couplings of the Lifshitz invari- scalar field ∗ yields the Ginzburg-Landau equation for ant term. This property of field inversion is responsible the superconducting condensate, for other unusual behaviours, also absent in conventional 2 2 type-2 superconductors. Indeed, we show that it leads kD + 2ieγB · D = β(j j − 0) ; (2) to the formation of vortex bound states, vortex clusters, while the variation of the free energy with respect to the and meta-stable pairs of vortex and anti-vortex. These gauge potential A gives the Amp`ere-Maxwell equation: phenomena should have numerous physical consequences B on the response of non-centrosymmetric superconductors r× + γj = kj + 2γe2j j2B : (3) to an external magnetic field. 4π The physical length scales of the theory are the coherence The paper is organized as follows. In Sec. II, we length ξ and the London penetration depth λL, introduce the phenomenological Ginzburg-Landau the- k 1 ory that describes the superconducting state of a non-
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