Anomalous (H, T) phase diagram in bilayered superconducting systems A. Buzdin a,b , S. Tollis a , J. Cayssol a, * a Condensed Matter Theory Group, CPMOH, UMR 5798, Universite ´ Bordeaux I, 33405 Talence, France b Institut Universitaire de France, Paris, France Available online 30 March 2007 Abstract We demonstrate that in a superconducting multilayered system with alternating interlayer coupling a new type of nonuniform super- conducting state can be realized under in-plane magnetic field. The Zeeman effect in this state is compensated by the energy splitting between bonding and antibonding levels. At low temperature such compensation mechanism leads to the field-induced superconduc- tivity. We discuss the choice of the optimal coupling between the layers in order to observe the predicted phenomena in experiments. Ó 2007 Elsevier B.V. All rights reserved. There are two mechanisms of the superconductivity destruction by a magnetic field: orbital and paramagnetic effects. Usually it is the orbital effect that is more restric- tive. However in the systems with large effective mass of electrons [1,2] or in low-dimensional compounds, the orbi- tal magnetism is weakened and the superconductivity is essentially suppressed by the paramagnetic effect. In this case Larkin and Ovchinnikov [3], and Fulde and Ferrell [4], predicted the appearance of a nonuniform supercon- ducting state at low temperature. In quasi-two-dimensional superconductors under in-plane magnetic field H the criti- cal field of this so-called FFLO state is given by H FFLO 2D ¼ D 0 =l B [5] when there is a single superconducting plane per unit cell, D 0 being the zero temperature supercon- ducting gap and l B the Bohr magneton. In this paper, we address the situation with different and alternating transfert integrals t 1  t 2 between nearest neighbors atomic planes. Thus the unit cell can be chosen as a superconducting bilayer with tunable internal coupling t 1 , whilst the coupling t 2 between bilayers is always smaller than D 0 . Such a multilayered system has a cristallographic structure similar to those of the high-T c superconductors. Namely we assume that the electrons are confined in the atomic planes with the same zero-field dispersion relation n(p)= p 2 /2m  E F , E F being the Fermi energy. It is further assumed that Cooper pairing occurs within the planes and the whole analysis is performed in the framework of the standard mean field BCS theory. Our model includes both translational and gauge symmetry breaking. Indeed, the superconducting order parameters in adjacent planes are respectively given by D n (r)= De iqÆr+ijna and D n (r)e iv , where n labels the unit cells, a is the lattice period in the direction perpendicular to the planes, and r is the coordinate in the planes. The wave vectors q and j describe respectively the in-plane and the perpendicular modulations. These vec- tors and the superconducting phase difference v must be determined in order to maximize the critical temperature. In the limit t 2 ! 0, the superconducting phase difference v between neighboring layers is either zero or p. In the fol- lowing, we denote U  v (resp. FFLO  v) the uniform (resp. modulated) superconducting state with phase differ- ence v between the planes. 1. Phase diagrams The Gor’kov equations corresponding to this model have been solved self-consistently in [6,7]. For small fields and/or ‘‘high temperatures’’, the superconducting order parameter is uniform and the interlayer phase difference is v = 0. The corresponding part of the (h = l B H, T) phase diagram, i.e. the region h/(2pT) < 0.3 is similar to that of a 0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.206 * Corresponding author. E-mail address: j.cayssol@cpmoh.u-bordeaux1.fr (J. Cayssol). www.elsevier.com/locate/physc Physica C 460–462 (2007) 1028–1030