A generalization of the demagnetizing tensor for nonuniform magnetization

Newell, Andrew J.; Williams, Wyn; Dunlop, David J.

doi:10.1029/93jb00694

Cited by 217 publications

(179 citation statements)

References 17 publications

Supporting

Mentioning

174

Contrasting

Unclassified

Order By: Relevance

“…Indeed, the FL magnetization is expected to be more non-uniform -and consequently easier to switchin the low-field P state because of the mutual dipolar coupling field pointing antiparallel to the magnetizations in the P state, but parallel in the AP state. Within the diagonal tensor approximation of the mutual dipolar coupling, the tensor components are found to be significantly smaller than the values predicted by the formalism developed by Newell et al 21 or by the simplified version using for the in-plane components of the mutual dipolar coupling tensor the corresponding components of the self-demagnetizing tensor, as is commonly practiced when modeling flip-flop switching in MRAM cells. 36,37 The coupling between e.g.…”

Section: A Materials and Geometry Parametersmentioning

confidence: 67%

“…Using the formulae in Ref. 21 it can be shown that the dipolar coupling constants obey similar relations, N y ml /N x ml = L x /L y and N z ml = −(N x ml + N y ml ), and for symmetry reasons, N ml = N lm . The remaining components N x F 1 , N x F 2 , and N x 12 are kept as free parameters to be extracted from the experiment, although they can be calculated by means of Ref.…”

Section: Reduction Of Number Of Free Parametersmentioning

confidence: 93%

“…For the given pillar geometry, N ml is diagonal, too, as can easily be shown using the formulae for the tensor components in Ref. 21. The diagonal components will be referred to as the mutual dipolar coupling constants N x ml , N y ml , and N z ml .…”

Section: Eigenexcitations Of Coupled Three-layer Systemmentioning

confidence: 98%

“…Although this approximation is expected to be satisfying for the static demagnetizing field, it is rather crude for the dynamical part, since the dynamical magnetization is non-uniform unless k = 0. The fields resulting from mutual dipolar coupling are given by analogous expressions where the (self-)demagnetizing tensors of trace 1 are replaced by the mutual demagnetizing tensors 21 N ml of trace 0 (l, m ∈ {F, 1, 2}, l = m). For the given pillar geometry, N ml is diagonal, too, as can easily be shown using the formulae for the tensor components in Ref.…”

Section: Eigenexcitations Of Coupled Three-layer Systemmentioning

confidence: 99%

“…The remaining components N x F 1 , N x F 2 , and N x 12 are kept as free parameters to be extracted from the experiment, although they can be calculated by means of Ref. 21. On the basis of previous measurements on MTJ stacks, the number of free parameters can be further reduced: In Ref.…”

Section: Reduction Of Number Of Free Parametersmentioning

confidence: 99%

See 4 more Smart Citations

Quantized spin-wave modes in magnetic tunnel junction nanopillars

et al. 2010

View full text Add to dashboard Cite

We present an experimental and theoretical study of the magnetic field dependence of the mode frequency of thermally excited spin waves in rectangular shaped nanopillars of lateral sizes 60 × 100, 75 × 150, and 105 × 190 nm 2 , patterned from MgO-based magnetic tunnel junctions. The spin wave frequencies were measured using spectrally resolved electrical noise measurements. In all spectra, several independent quantized spin wave modes have been observed and could be identified as eigenexcitations of the free layer and of the synthetic antiferromagnet of the junction. Using a theoretical approach based on the diagonalization of the dynamical matrix of a system of three coupled, spatially confined magnetic layers, we have modeled the spectra for the smallest pillar and have extracted its material parameters. The magnetization and exchange stiffness constant of the CoFeB free layer are thereby found to be substantially reduced compared to the corresponding thin film values. Moreover, we could infer that the pinning of the magnetization at the lateral boundaries must be weak. Finally, the interlayer dipolar coupling between the free layer and the synthetic antiferromagnet causes mode anticrossings with gap openings up to 2 GHz. At low fields and in the larger pillars, there is clear evidence for strong non-uniformities of the layer magnetizations. In particular, at zero field the lowest mode is not the fundamental mode, but a mode most likely localized near the layer edges.

show abstract

Section: A Materials and Geometry Parametersmentioning

confidence: 67%

Section: Reduction Of Number Of Free Parametersmentioning

confidence: 93%

Section: Eigenexcitations Of Coupled Three-layer Systemmentioning

confidence: 98%

Section: Eigenexcitations Of Coupled Three-layer Systemmentioning

confidence: 99%

Section: Reduction Of Number Of Free Parametersmentioning

confidence: 99%

See 3 more Smart Citations

Quantized spin-wave modes in magnetic tunnel junction nanopillars

et al. 2010

View full text Add to dashboard Cite

show abstract

?Local? Demagnetization in a Rectangular Ferromagnetic Prism

Aharoni

2002

phys. stat. sol. (b)

View full text Add to dashboard Cite

A new kind of demagnetizing factors is suggested, that express the local field in a ferromagnetic sample. An analytic expression is given for the case of a general rectangular prism and is compared with a generalization of the ballistic demagnetizing factor. These two factors are found to be nearly the same in the particular case of thin magnetic films, and differ significantly only when the three sides of the prism are of the same order of magnitude.In the early days of research in magnetism, the magnetization was measured by pulling the sample through a coil and measuring the induced voltage. This technique could not reveal the distribution of the magnetization in a non-ellipsoidal sample and could measure only its average. Two kinds of averages were then defined. One is an averaging over the whole sample, and reflects the measurement of a small specimen with a large coil. The demagnetization it yields is known in the literature as the magnetometric demagnetization. The second kind is averaging the magnetization over the midplane of the sample, as is done with short search coils. It leads to the so-called ballistic (or fluxmetric) demagnetization. The demagnetizing factors for both cases are defined by certain integrals [1,2], that can be evaluated in a closed form for a saturated rectangular prism. Specific formulae with different notations, have been published for the general prism, giving the magnetometric [3, 4] and the ballistic [5, 6] demagnetizing factors. There are also results for both factors [7] in the special case of an infinite prism, magnetized transversely, and for the magnetometric demagnetizing factor [8] in a prism with a square cross section. It is also possible to obtain the demagnetizing factors from the calculation [9] of the magnetostatic coupling between two rectangular prisms of equal size by setting their separation to zero.More recently, however, it has become possible to measure the local variations of the magnetization, for example by a small hall probe. A comparison of such measurements with the theoretical demagnetizing factors [6] showed clearly that the averaging causes a loss of information, because the actual magnetization processes, such as the nucleation or elimination of domains, depend on the local field. This result is not surprising, since it has been noted [10] a decade ago, and later emphasized [11] with all the fine details, that numerical computations of the magnetostatic energy must include averaging the field over the discretization unit. Using instead the value of the demagnetizing field at the centre of that unit, leads to considerably different results. A realistic study of hysteresis in composite materials also seems to involve [12] the local variations of the demagnetization in the sample.A uniform and homogeneous ferromagnetic particle in the shape of a rectangular prism is considered here, and the origin of a Cartesian coordinate system is defined at

show abstract

Numerical Micromagnetics: Finite Difference Methods

Miltat¹,

Donahue

2007

Handbook of Magnetism and Advanced Magnetic Materials

View full text Add to dashboard Cite

Micromagnetics is based on the one hand on a continuum approximation of exchange interactions, including boundary conditions, on the other hand on Maxwell equations in the nonpropagative (static) limit for the evaluation of the demagnetizing field. The micromagnetic energy is most often restricted to the sum of the exchange, demagnetizing or self‐magnetostatic, Zeeman, and anisotropy energies. When supplemented with a time evolution equation, including field induced magnetization precession, damping and possibly additional torque sources, micromagnetics allows for a precise description of magnetization distributions within finite bodies both in space and time. Analytical solutions are, however, rarely available. Numerical micromagnetics enables the exploration of complexity in small size magnetic bodies. Finite difference methods are here applied to numerical micromagnetics in two variants for the description of both exchange interactions/boundary conditions and demagnetizing field evaluation. Accuracy in the time domain is also discussed and a simple tool provided in order to monitor time integration accuracy. A specific example involving large angle precession, domain wall motion as well as vortex/antivortex creation and annihilation allows for a fine comparison between two discretization schemes with as a net result, the necessity for mesh sizes well below the exchange length in order to reach adequate convergence.

show abstract

A generalization of the demagnetizing tensor for nonuniform magnetization

Cited by 217 publications

References 17 publications

Quantized spin-wave modes in magnetic tunnel junction nanopillars

Quantized spin-wave modes in magnetic tunnel junction nanopillars

?Local? Demagnetization in a Rectangular Ferromagnetic Prism

Numerical Micromagnetics: Finite Difference Methods

Contact Info

Product

Resources

About