Training effect of exchange-bias bilayers in the domain-state model: realistic interfaces, angular dependence, and the identity of the training mechanism

Exchange bias (EB) — the shift of a ferromagnet's hysteresis loop induced by contact with an antiferromagnet — degrades over consecutive field cycles, a phenomenon called the training effect. The canonical atomistic domain-state-model study of this effect (Biternas, Nowak, and Chantrell, Phys. Rev. B 80, 134419 (2009)) left four issues open: the quantitative microscopic characterization of interfacial antiferromagnetic (AFM) spin populations during training, the influence of imperfect interface structure — the model's FM/AFM interface is perfect and flat, without any roughness — on training, the angular dependence of training and loop asymmetry beyond the single fixed field angle, and whether a single training mechanism operates across the variety of experimental EB systems. The interfacial spin-population question was addressed by the same authors for the ideal flat interface (Phys. Rev. B 82, 134426 (2010); Phys. Rev. B 89, 184405 (2014)), and interfacial mixing has been characterized for granular bilayers (Jenkins, Chantrell, and Evans, Phys. Rev. B 103, 104419 (2021)), but the interface-structure and angular-dependence questions remain open. The open problem is to complete the microscopic theory of the training effect: extend the domain-state model beyond the perfect flat interface to imperfect interface structure, in particular interface roughness, and to the full angular range between applied field and anisotropy axes, and establish whether a single microscopic training mechanism is unique across the variety of classes of experimental exchange-bias systems or whether the mechanism is system-dependent.

Background

Exchange bias (EB) is the phenomenon in which a ferromagnet (FM) in contact with an antiferromagnet (AFM), cooled in an applied magnetic field below its blocking temperature (the temperature below which the AFM spin configuration freezes on the measurement timescale), acquires a unidirectional anisotropy: its hysteresis loop is shifted along the field axis by an exchange-bias field $H_{\mathrm{EB}}$. Upon cycling the field through consecutive hysteresis loops, the shift and the coercivity typically degrade gradually — the training effect — which is a reliability concern for EB-based spintronic devices such as read sensors and magnetic random-access memory reference layers. The leading microscopic picture for crystalline, defect-bearing AFMs is the domain-state model (DSM): the AFM is diluted with nonmagnetic defects, and the random sublattice imbalance of the dilution acts like a random field on the AFM, so that after field cooling the AFM breaks into a metastable domain structure carrying a net irreversible magnetization that biases the FM. The DSM was established by Miltényi et al. (Phys. Rev. Lett. 84, 4224 (2000)) and worked out theoretically by Nowak et al. (Phys. Rev. B 66, 014430 (2002)). Simulations treat a classical Heisenberg spin Hamiltonian (unit vectors with FM exchange in the FM layer, AFM exchange in the diluted AFM layer, uniaxial or biaxial magnetocrystalline anisotropy, and FM–AFM exchange across one interfacial monolayer) using heat-bath Monte Carlo dynamics (a single-spin-flip algorithm in which trial moves are accepted with a Boltzmann-type probability, allowing equilibration of metastable spin configurations). Biternas, Nowak, and Chantrell (Phys. Rev. B 80, 134419 (2009)) studied the temperature dependence of training in this model for different AFM thicknesses and anisotropy symmetries. They found two regimes — athermal training (low temperature: the bias drops essentially after the second loop and then stays constant) and thermal training (higher temperature: gradual decay well fit by an empirical power law in the loop number) — and concluded that training is caused by the rearrangement of AFM interfacial spins that become frozen during cooling and evolve toward equilibrium as cycling proceeds. Two technical conventions of that study matter here: the FM/AFM interface was modeled as perfect and atomically flat (no roughness or intermixing, with random site dilution of the AFM bulk), and the external field was applied at a fixed $20^\circ$ angle to the AFM easy axis, an angle deliberately chosen so that uniaxial systems exhibit training (at larger angles training disappears in that model). The paper left several issues unresolved: the in-depth analysis of the interfacial AFM spin populations during the first reversal and subsequent cycles (explicitly deferred to a later publication); the influence of imperfect interface structure on training — the model's FM/AFM interface is stated to be perfect, "without any roughness"; the systematic angular dependence of the training effect and of the asymmetry of the hysteresis loop between its two branches; and the question of whether a single microscopic training mechanism can be unique across the huge variety of experimental EB systems, given that different material classes (diluted single-crystal-like AFMs, epitaxial systems, polycrystalline small-grain AFMs such as IrMn (an iridium-manganese antiferromagnetic alloy used widely in spintronic devices), granular composites) show qualitatively different training behavior.

Problem Statement

Within the atomistic domain-state model of exchange-biased bilayers (a classical Heisenberg FM layer exchange-coupled to a diluted AFM with uniaxial or biaxial anisotropy), the training effect — the evolution of the exchange-bias field $H_{\mathrm{EB}}$, the coercivity, the asymmetry between the two branches of the hysteresis loop, and of the interfacial stable versus unstable AFM spin populations over consecutive hysteresis cycles — has been characterized only for a perfect, atomically flat interface with random site dilution and for a single field angle ($20^\circ$ to the easy axis, at which uniaxial systems train; at larger angles training vanishes in that model). The open problem consists of three separately assessable parts, to be reported quantitatively as the per-cycle exchange-bias field $H_{\mathrm{EB}}(n)$, together with the coercivity and the branch asymmetry, as a function of the roughness parameter $p$ and of the field angle: (A-i) interface roughness — determine quantitatively how training depends on imperfect interface structure beyond the perfect, atomically flat interface, the interface roughness being modeled as a diluted FM–AFM interfacial monolayer in which a fraction $p$ of the interfacial sites is made nonmagnetic at random, with uncorrelated placement (the roughness parameter $p$ is varied from the flat-interface limit $p = 0$), following the imperfect-interface domain-state setting of Spray and Nowak (Ref. 6); (A-ii) angular dependence — determine quantitatively how training depends on the angle between the applied field and the anisotropy axes over the full angular range, including the interplay with the anisotropy symmetry and the onset/disappearance of training with angle; and (B) mechanism uniqueness — using the published training behavior of the granular-composite, polycrystalline small-grain, and diluted single-crystal-like classes of experimental exchange-bias systems, these three classes being the required evidence base (data from further classes such as epitaxial systems may additionally be used), establish whether a single microscopic training mechanism is unique across those classes or whether the training mechanism is system-dependent. A complete answer must address all three parts; addressing only one or two of them does not resolve the objective. The stable/unstable interfacial spin-population analysis for the ideal flat, randomly diluted interface (Biternas et al., Phys. Rev. B 82, 134426 (2010); Phys. Rev. B 89, 184405 (2014)) and the characterization of intermixing effects in granular IrMn/CoFe (Jenkins et al., Phys. Rev. B 103, 104419 (2021)) are already published and constitute the baseline against which any answer must position itself, but they do not by themselves complete the objective.

References

1. A. G. Biternas, U. Nowak, R. W. Chantrell, "Training effect of exchange-bias bilayers within the domain state model", Phys. Rev. B 80, 134419 (2009). DOI: 10.1103/PhysRevB.80.134419, https://doi.org/10.1103/PhysRevB.80.134419 (full text also at https://kops.uni-konstanz.de/handle/123456789/9297)
2. A. G. Biternas, R. W. Chantrell, U. Nowak, "Behavior of the antiferromagnetic layer during training in exchange-biased bilayers within the domain state model", Phys. Rev. B 82, 134426 (2010). DOI: 10.1103/PhysRevB.82.134426, https://doi.org/10.1103/PhysRevB.82.134426 (Konstanz repository record at https://kops.uni-konstanz.de/handle/123456789/866)
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