Theory of exchange bias

The exchange coupling of ferromagnetic and antiferromagnetic films across their common interface causes a shift in the hysteresis loop of the ferromagnet, called exchange bias. The shift can be useful in controlling the magnetization in devices, such as spin valves which sense changing magnetic fields through the giant magnetoresistance effect. Read heads based on this effect are used in magnetic disk storage. The details of the coupling are poorly understood, making it difficult to optimize the performance of the effect in devices. We have investigated of series of models of these systems to try to answer some of the outstanding issues. What is the nature of the coupling at the interfaces? What is the origin of the temperature dependence? What is the origin of the large coercivity found in these systems?

Interfacial Coupling We have developed a model that describes polycrystalline exchange bias systems. The model describes independent antiferromagnetic grains coupled to a ferromagnetic layer with a uniform magnetization. Each grain can order in one of two states, which are degenerate in the absence of coupling to the ferromagnet. This degeneracy is broken by the coupling, which winds up partial domain walls in the antiferromagnet as the ferromagnetic magnetization is rotated. The model depends on several parameters that are not that well known. We have made a virtue of this liability in two ways. First, we have computed enough different properties of the systems so that comparison with enough measurements should overconstrain the model. Second, we have determined the behavior in many different limits which has allowed us to draw general conclusions. For example, our model allowed us to determine that the recently proposed "spin-flop" coupling mechanism does does contribute to the loop shift. Disorder at the interface must be present.

Partial domain walls in an antiferromagnet coupled to a ferromagnet.

The figure above shows the partial domain walls in the antiferromagnet (white and gray balls) ordered in the two different antiferromagnetic states for a particular direction of the ferromagnetic magnetization (green balls).

Temperature dependence Additional insight into these systems can come from comparing models with measurements based on other techniques beside measurements of the properties directly of interest. One such technique is ferromagnetic resonance (FMR), which is used to determine the anisotropy of magnetic samples. In FMR measurements, anisotropy terms lead to decreases in the resonance field in the easy directions of the anisotropy and increases in the hard directions. Variations, characteristic of the unidirectional anisotropy that gives rise to the loop shift, are observed in FMR experiments on exchange biased films, but are superimposed on an isotropic negative shift in the resonance field. Since an isotropic shift corresponds to all directions being "easier," the shift must arise from hysteretic processes and can be interpreted in terms of a rotatable anisotropy.

Unidirectional anisotropy and rotatable anisotropy as a function of temperature
for model exchange biased layers.

$r= \frac{{\sqrt N} J_{\rm int}}{N\, \sigma_0 a^2}$

$b=\frac{N\, a^2 \sigma_0}{k\, T_{\rm N}}$

If the antiferromagnetic state does not change as the ferromagnetic magnetization is rotated, that grain contributes to the unidirectional anisotropy. Other grains, in which the antiferromagnetic order irreversibly switches, contribute to the rotatable anisotropy. The next figure shows the temperature dependence of the unidirectional anisotropy (red) and the rotatable anisotropy (green). The temperature dependence is presented for several system parameters, the ratio of the zero temperature domain wall energy to the Néel temperature of the antiferromagnet, b, and the ratio of the interfacial coupling to the domain wall energy, r. We have found that there are two important contributions to the temperature dependence of these quantities, the natural temperature dependence of the materials properties and the thermal instabilities in the antiferromagnetic grains due to their small size.

Coercivity In the experiments described above, the magnetization is saturated in one direction and then rotated. A more common measurement is a hysteresis loop, in which the magnetization is saturated in one direction and the the field is linearly reversed. As well as a shift of the hysteresis loop, these measurements always show an increase in the width of the hysteresis loop, i.e. increased coercivity. The grains that contribute to the rotatable anisotropy also contribute to the hysteresis; if the antiferromagnetic state irreversibly switches, work is done, contributing to the area of the hysteresis loop. An additional source of coercivity arises from the random orientation of the antiferromagnetic grains.

Spin configurations during reversal for a model exchange biased layer.

This figures shows the spin configurations at several points on a zero-temperature hysteresis loop. As the magnetic field is lowered from its saturated value, the magnetization (given by an arrow) above each grain tilts up (blue) or down (red) depending on the orientation of the antiferromagnetic grain. The competing rotation directions lead to energy barriers that give rise to coercivity.

Coercivity in Exchange-Bias Bilayers
Exchange Bias Relaxation in CoO-Biased Films
Temperature Dependence of Exchange Bias in Polycrystalline Ferromagnet-Antiferromagnet Bilayers
Model for Exchange Bias in Polycrystalline Ferromagnet-Antiferromagnet Bilayers
Ferromagnetic Resonance Studies of NiO-couples Thin Films of Ni₈₀Fe₂₀
Ferromagnetic Resonance Linewidth in Thin Films Coupled to NiO

Mark Stiles

Robert McMichael - NIST

Online: July 1999
Last Updated: February 2008

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