M. Beleggia

Demagnetization factors for elliptic cylinders

The magnetometric (volume averaged) demagnetization factors for cylinders with elliptical cross section are computed using a Fourier-space approach and compared with similar results obtained with a different treatment. The demagnetization factors are given as a series expansion in the eccentricity � of the elliptical cross section, where the terms up to order � 10 are given explicitly as a function of the cylinder aspect ratio. Other simplified expressions, valid in restricted regimes, are also given. Two different series expansions, obtained previously and valid in particular combinations of shape parameters, are recalled and compared with the new results. After the computation of the magnetostatic and exchange-energy terms associated with a vortex closure-domain state in the elliptic cylinder, the single-domain limit, or the critical size below which the structure can support quasi-uniform magnetization, is derived and discussed.

The equivalent ellipsoid of a magnetized body

The equivalent ellipsoid for magnetized bodies of arbitrary shape can be determined by imposing the equality between the demagnetization factors of the two shapes of equal volume. It is shown that the commonsense criterion for mapping two different shapes by imposing the equality of the demagnetization factors for equal aspect ratios often results in large errors. We propose a general method for the rigorous determination of the equivalent ellipsoid. The cases of the exact equivalent ellipsoids for discs, cylinders with elliptical cross section and prisms are worked out and discussed.

On the computation of the demagnetization tensor field for an arbitrary particle shape using a Fourier space approach

A method is presented to compute the demagnetization tensor field for uniformly magnetized particles of arbitrary shape. By means of a Fourier space approach it is possible to compute analytically the Fourier representation of the demagnetization tensor field for a given shape. Then, specifying the direction of the uniform magnetization, the demagnetizing field and the magnetostatic energy associated with the particle can be evaluated. In some particular cases, the real space representation is computable analytically. In general, a numerical inverse fast Fourier transform is required to perform the inversion. As an example, the demagnetization tensor field for the tetrahedron will be given.

Electron-optical phase shift of magnetic nanoparticles I. Basic concepts

The electron-optical phase shift induced in the electron beam due to the interaction with the electromagnetic field of magnetized nanoparticles of defined shape and arbitrary dimensions is calculated, presented and discussed. Together with the computable knowledge of vector potential and magnetic induction, including the demagnetizing field, and with the extension to more realistic geometries which will be presented in part II, this theoretical framework can be employed for the interpretation of transmission electron microscopy experiments on magnetic particles on the nanometre scale.

Demagnetization factors of the general ellipsoid: An alternative to the Maxwell approach

A transparent, exhaustive, and self-contained method for the calculation of the demagnetization tensor of the uniformly magnetized ellipsoid is presented. The method is an alternative to the established Maxwell derivation and is based on a Fourier-space approach to the micromagnetics of magnetized bodies. The key to the success of the procedure lies in the convenient treatment of shape effects through the Fourier representation. The scaled form of the demagnetization factors which depends on two dimensionless aspect ratios is argued to be their natural integral representation. Amongst other advantages, it allows for the immediate implementation of symmetry arguments such that only one of the principal factors needs to be computed. The oblate and prolate ellipsoids of revolution are examined from the same general point of view. The demagnetization factors for these distinct types of spheroid are seen to be related by analytic continuation of well-known Gaussian hypergeometric functions.

Characterization of JEOL 2100F Lorentz-TEM for low-magnification electron holography and magnetic imaging.

We present results that characterize the performance and capabilities of the JEOL 2100F-LM electron microscope to carry out holography and quantitative magnetic imaging. We find the microscope is well-suited for studies of magnetic materials, or for semi-conductor dopant profiling, where a large hologram width ( approximately 1 microm) and fine fringe spacing ( approximately 1.5 nm) are obtained with good contrast ( approximately 20%). We present, as well, measurements of the spherical aberration coefficient Cs=(108.7+/-9.6)mm and minimum achievable focal step delta f=(87.6+/-1.4)nm for the specially designed long-focal-length objective lens of this microscope. Further, we detail experiments to accurately measure the optical parameters of the imaging system typical of conventional holography setup in a transmission electron microscope. The role played by astigmatic illumination in the hologram formation is also assessed with a wave-optical model, which we present and discuss. The measurements obtained for our microscope are used to simulate realistic holograms, which we compare directly to experimental holograms finding good agreement. These results indicate the usefulness of measuring these optical parameters to guide the optimization of the experimental setup for a given microscope, and to provide an additional degree of practical experimental possibility.

Electron-optical phase shift of magnetic nanoparticles II. Polyhedral particles

A method is presented to compute the electron-optical phase shift for a magnetized polyhedral nanoparticle, with either a uniform magnetization or a closure domain (vortex state). The method relies on an analytical expression for the shape amplitude, combined with a reciprocal-space description of the magnetic vector potential. The model is used to construct two building blocks from which more complex structures can be generated. Phase computations are also presented for the five Platonic and 13 Archimedean solids. Fresnel and Foucault imaging mode simulations are presented for a range of particle shapes and microscope imaging conditions.

A Fourier approach to fields and electron optical phase-shifts calculations

The Fourier method is applied to calculate fields and electron optical phase shifts in specimens having long-range electromagnetic fields, like reverse biased p-n junctions or stripe magnetic domains. It is shown that this approach not only allows to take into account rather easily the effect of the fringing fields protruding in the space around the specimen, but also to obtain solutions to interesting models in analytical form.

Quantitative study of magnetic field distribution by electron holography and micromagnetic simulations

The magnetic configuration of a submicrometer Ni88Fe12 permalloy island has been quantitatively mapped by off-axis electron holography. The two main contributions to the electron-optical phase shift, namely the phase shifts induced by the electrostatic and magnetic potentials, including fringing fields, were separated by inverting the specimen of 180° with respect to the electron beam and directly measuring the mean inner potential. A quantitative map of the projected magnetic induction in the sample was thereby retrieved and compared to results of micromagnetic and electromagnetic calculations, providing the minimum-energy configuration and the phase shift, respectively.

Quantitative shadow technique for the investigation of magnetic domain wall widths

Quantitative measurement of domain wall widths in a magnetic thin foil of Nd2Fe14B and a patterned Ni88Fe12 Permalloy element is obtained by the analysis of coherent shadow deformation of an electrostatic biprism in an electron microscope. Information related to the phase gradient in the direction perpendicular to the biprism is extracted by comparing the recorded shadow images with simulations computed according to the experimental electron optical configuration and by varying a single free parameter: the domain wall width. We demonstrate the feasibility of such experiments and the usefulness of the technique for characterization of magnetic features at different length scales.