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It is generally believed that superconductivity and magnetism are two antagonistic long-range phenomena. However, as was preliminarily highlighted in Stamopoulos et al (2007 Phys. Rev. B 75 014501), and extensively studied in this work, under specific circumstances these phenomena instead of being detrimental to each other may even become cooperative so that their synergy may promote the superconducting properties of a hybrid structure. Here, we have studied systematically the magnetic and transport behavior of such exchange biased hybrids that are comprised of ferromagnetic (FM) Ni80Fe20 and low-Tc superconducting (SC) Nb for the case where the magnetic field is applied parallel to the specimens. Two structures have been studied: FM–SC–FM trilayers (TLs) and FM–SC bilayers (BLs). Detailed magnetization data on the longitudinal and transverse magnetic components are presented for both the normal and superconducting states. These data are compared to systematic transport measurements including I–V characteristics. The comparison of the exchange biased BLs and TLs that are studied here with the plain ones studied in Stamopoulos et al (2007 Phys. Rev. B 75 184504) enable us to reveal an underlying parameter that may falsify the interpretation of the transport properties of relevant FM–SC–FM TLs and FM–SC BLs investigated in the recent literature: the underlying mechanism motivating the extreme magnetoresistance peaks in the TLs relates to the suppression of superconductivity mainly due to the magnetic coupling of the two FM layers as the out-of-plane rotation of their magnetizations takes place across the coercive field where stray fields emerge in their whole surface owing to the multidomain magnetic state that they acquire. The relative in-plane magnetization configuration of the outer FM layers exerts a secondary contribution on the SC interlayer. Since the exchange bias directly controls the in-plane magnetic order it also controls the out-of-plane rotation of the ferromagnets' magnetizations so that the magnetoresistance peaks may be tuned at will. All the contradictory experimental data reported in the recent literature are discussed fairly in the light of our results; based on a specific prerequisite we propose a phenomenological stray-field mechanism that efficiently explains the evolution of the magnetoresistance effect in TLs. Our experiments not only point out the need for a new theoretical treatment of FM–SC hybrids but also direct us toward the design of efficient supercurrent-switch elemental devices.