Generalizing demagnetizing factors to synthetic antiferromagnets

Magnetostatic interactions are important however difficult to tackle due to their long range. In homogeneous systems the concept of demagnetizing factors, assuming uniform magnetization, has long and widely been used. The anisotropy of demagnetizing factors is directly related to the anisotropy field, coercive field and thermal stability for elements small enough to remain reasonably single-domain. We extended this concept to synthetic antiferromagnets (two stacked layers magnetized in opposite directions), which are widely used as elementary bricks in spintronic devices, and deliver analytical formulas and a discussion of the resulting physics. Synthetic antiferromagnets (SAF,resp.ferrimagnets, SyF) consist of a stack of two thin ferromagnetic films of moments of same (resp. different) magnitude, strongly coupled antiferromagnetically, e.g. thanks to an interlayer RKKY interaction (FIG1). SyFs are widely used to provide spin-polarized layers displaying an overall weak moment. One benefit is to minimize cross-talk of neighboring (e.g. memory bits) or stacked (e.g. in a spin-valve) elements through stray-field coupling. SyFs are also used to decrease the Zeeman coupling with external fields, e.g. to increase coercivity in so-called reference layers.
Fig. 1:Geometry and notations of a prismatic SyF element comprising two ferromagnetic layers F1 and F2
Non-volatility is a key aspect of spintronic devices. Non-volatility is related to the phenomenon of hysteresis and remanence. These are made possible by magnetic anisotropy that defines two or more easy directions of magnetization (states), separated with an energy barrier preventing fluctuations (eg thermally-activated) from one state to another. In the sub-micrometer-sized flat elements used in spintronic devices, magnetic anisotropy is most often achieved thanks to the anisotropy of the magnetostatic energy in dots patterned on purpose with an elongated shape. This is the so-called shape effect, quantified in single elements thanks to the use of so-called demagnetizing coefficients. We made use of simple algebra, widely used for single elements, to derive analytical formulas for demagnetizing coefficients [RHO1954] and applied them to SyFs. This yields for the magnetostatic energy a quadratic form with magnetizations M1 and M2 :

\epsilon_{d}=K_{d,1}N_{z}(a,t_{1},c)V_{1}+K_{d,2}N_{z}(a,t_{2},c)V_{2}+2\sqrt{K_{d,1}K_{d,2}}N_{m}(a,t_{1},s,t_{2},c)\sqrt{V_{1}V_{2}}

Indexes 1 or 2 refer to the layer, K_{d}} to the dipolar constant (1/2) M_{s^{2}}, V to volume, t to layer thickness, s for spacer thickness, N_{z}} to usual demagnetizing coefficients, a and c to lateral dimensions, and we provided an analytical expression for the cross-term (coupling coefficient) N_{m}}. As an example, let us outline two facts deriving from these formulas : (1) The dipolar energy does not vanish upon perfect compensation of moments between the two layers ; it is only slightly reduced with respect to the self-demagnetizing energy of a single element (20-30% at most for usual dimensions used e.g. MRAM dots). So does the anisotropy of dipolar energy and hence the energy barrier separating the two stable states. A practical consequence is thus that a SyF is not expected to be much more vulnerable to thermal excitations than a single layer. (2) In two dimensions magnetostatic interactions are short-ranged, contrary to the three-dimensional case. We made use of this fact to expand to first order in thickness over lateral dimensions the above equation, which then boils down to a single line integral around the edge of the elements, times the upper bound for the line magnetostatic energy E_{\lambda}. While the former has simple analytic forms for geometric elements of common use, the latter is approached as (1/2) K_{dt^{2}} with high accuracy for SAF elements. This provides a reliable way for the fast evaluation of the magnetostatic energy of flat SAF elements. Details and application to experimental literature may be found in Ref.FRU2011.

Contributors

  • Institut Néel : O. Fruchart
  • Spintec, Grenoble, France : B. Diény

References

    [RHO1954] P.Rhodes, G.Rowlands, Demagnetizing energies of uniformly magnetized rectangular blocks,Proc. Leeds Phil. Liter. Soc. 6 (1954) 191.

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Corresponding author  : Olivier Fruchart

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