Physica B 329–333 (2003) 1473–1474 Interplay of ferromagnetism and superconductivity: domain structure Edouard B. Sonin Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel Abstract In superconducting ferromagnets, where superconductivity and ferromagnetism coexist at the same bulk, the equilibrium domain structure is absent in the Meissner state, but it does exist in the spontaneous vortex phase. In a superconductor–ferromagnet bilayer superconductivity and ferromagnetism are separated in space, but affect each other via magnetic fields. The superconducting layer shrinks the size of domains in the ferromagnetic layer by a numerical factor, in contrast to superconducting ferromagnets, where superconductivity can strongly increase the domain size. r 2003 Elsevier Science B.V. All rights reserved. Keywords: Josephson effect; Flux lines; YBa 2 Cu 3 O 7 ; Specific heat Coexistence of superconductivity and ferromagnetism has been revealed experimentally in various materials. Up to now the theory mostly addressed macroscopically uniform structures [1], whereas ferromagnetic materials, even ideally uniform ones, inevitably have a domain structure. The present work addresses the domain structure in superconducting ferromagnets, where super- conductivity and ferromagnetism coexist at the same bulk, and in a superconductor-ferromagnet bilayer, where superconductivity and ferromagnetism are sepa- rated in space, but mutually affect each other via long- range magnetic fields. In order to find the distribution of the magnetic field H and the magnetic induction B ¼ H þ 4pM (M is the spontaneous magnetization) one should solve the equa- tions of magnetostatics and London electrodynamics. The magnetic induction B ¼ H þ 4pM is divergence-free, =  B ¼ 0; but the magnetic field H is not: =  H ¼ 4pr M ; where r M ¼=  M is the magnetic charge. We consider a domain structure in standard geometry [2]: a slab of the thickness d along the anisotropy easy axis and infinite in other directions. The magnetic charges appear only at boundaries of the slab (Fig. 1). Without an external magnetic field in a single-domain structure (Fig. 1a), B ¼ 0 and there exists an uniform magnetostatic field H ¼4pM and high magnetostatic energy H 2 =8pBM 2 in the entire sample. For a stripe domain structure one can find the exact solution for H using the method of complex variables [3]:  H x þ iH y ¼ 4M ln tan pw 2l  ln tan pðw  id Þ 2l   ; ð1Þ where w ¼ x þ iy: Inside the domain bulk HE0; except for the area Bl 2 near the slab boundary (Fig. 1b). The equilibrium period l ðl 5d Þ is determined by minimiza- tion of the sum of the magnetostatic energy and the energy of domain walls [2]: l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi aK M 2 dd r ; ð2Þ where K is the anisotropy energy, d is the wall thickness, and a is a numerical factor. In a superconducting ferromagnet in the Meissner state B must vanish in the bulk. This is compatible only with a single-domain structure, and the equilibrium domain structure is impossible in the Meissner state. Domain walls may appear only in metastable states. If 4pM > H c1 ; the Meissner state is absent, and the superconducting ferromagnet is in the mixed state even in zero external field (H ¼ 0). This is the spontaneous vortex phase with nonzero magnetic induction B ¼ B 0 ð4pMÞ in the bulk [1]. Here H c1 is the lower critical E-mail address: sonin@cc.huji.ac.il (E.B. Sonin). 0921-4526/03/$-see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02392-X