An Ellipsoidal Model for Small Multilayer Particles

This paper presents an ellipsoidal model that is constructed for small layered nonspherical particles and methods for constructing “effective” multilayer ellipsoids, the light-scattering properties of which would be close to the corresponding properties of original particles. In the case of axisymmetric particles, prolate or oblate spheroids (ellipsoids of revolution) are implied. Numerical calculations of the polarizability and scattering cross sections of small layered nonspherical particles, including nonconfocal (similar) spheroids, Chebyshev particles, and pseudospheroids, are performed by different approximate and rigorous methods. Approximate approaches involve the use of an ellipsoidal model, in which the polarizability of a layered particle is determined in two ways. In the first case, the polarizability is calculated in the approximation of confocal spheroids, while, in the second case, it is sought as a linear combination of the polarizabilities of embedded spheroids proportionally to the volumes of layers. Among rigorous methods, the extended boundary conditions method and the generalized separation of variables method are applied. On the basis of a comparison of the results obtained with rigorous and approximate approaches, their drawbacks and advantages are discussed.     

Light scattering by multilayer ellipsoids in the Rayleigh approximation

A problem of light scattering by multilayer confocal ellipsoids is solved in the Rayleigh approximation. The electric field of a light wave is assumed constant and a set of Laplace equations with the corresponding boundary conditions is considered. The final expression for the polarizability of a particle is represented in the matrix form (2×2 matrices) in terms of parameters of a nucleus and subsequent layers. Numerical calculations of the scattering and absorption efficiencies of small multilayer spheres obtained using the exact (the generalization of the Mie theory) and approximate solutions well agree with each other.     

Constrained Brownian Motion of a Single Ellipsoid in a Narrow Channel

We study the Brownian motion of a single ellipsoidal particle diffusing in a narrow channel by video-microscopy measurement. The experiments allow us to obtain the trajectories of ellipsoids and measure the diffusion coefficients. It is found that the channel constraints lead to suppression of the particle motion, especially the perpendicular motion to the channel, and the long axis of the particle tends to be parallel to the channel. A stable stratification phenomenon is observed, which is rarely discussed in studies of spherical particles. We also derive an approximate solution of theoretical prediction with the method of reflections, and obtain numerical simulation results using finite element software. They are proven to be effective by comparing them with the experimental results. All of these indicate that the aspect ratio and size of ellipsoid, the width of channel, and the transverse position distinctly affect the Brownian motion of ellipsoids.     

Light scattering by homogeneous and multilayer ellipsoids in the quasi-static approximation

The quasi-static approximation for homogeneous and multilayer ellipsoidal particles is considered in detail. Expressions for the elements of the amplitude scattering matrix, as well as for the absorption and scattering cross sections, are presented for an arbitrary orientation of the particles relative to the incident radiation. The accuracy and the domain of applicability of the approximation are investigated for the elements of the scattering matrix of absorbing particles. Comparison of some approximate and exact methods showed that the quasi-static approximation gives good results (it is substantially more preferable than the Rayleigh and Rayleigh-Gans approximations), if the ratio of the largest size of a particle (or the boundaries of the main layer) to the smallest one does not exceed ～3. A new rule of the effective medium theory for multilayer ellipsoidal particles is proposed, which adequately takes into account their structure.     

Magnetoresistance and noncollinear structures of multilayer ferromagnetic nanoparticles

The distribution of the magnetization over multilayer particles, including three ferromagnetic layers separated by insulating spacers, is studied experimentally and theoretically. Experimental data on the magnetic state of these particles are obtained by measuring their magnetoresistance. For the case of zero applied field, it is shown that a multilayer particle with easy-plane magnetic anisotropy is a noncollinear helical state.     

两种反射膜系的激光热力损伤特性

 在驻波场理论基础上,用表面和界面强吸收理论模型,计算出了激光辐照下两种不同反射膜系中温度场的“多峰”分布特性,分析了激光对薄膜的热力损伤特性、主要破坏模式和阈值条件。结果表明,光学薄膜对激光能量的吸收,主要由驻波场分布及膜层的消光系数决定;膜系表面的杂质吸附层和界面吸收层是膜系的主要能量沉积区,从而在这些界面形成温度场的“多峰”分布,界面因此成为最容易受激光损伤的部分。导致光学薄膜激光损伤的主要模式为膜层熔融和脱落两种。如果膜层中含有杂质,则容易诱导杂质处膜层剥落破坏。     

Ellipsoidal space-times,sources for the Kerr metric

The paper develops a systematic derivation of the Kerr metric and its possible sources in a clear geometric manner. It starts with a concise account of previous attempts at constructing an interior Kerr solution. Then a treatment of stationary-axisymmetric spacetimes, specially fitted to the needs of the following analysis, is presented. A new notion of an ellipsoidal space-time is introduced: it is a space-time in which local rest 3-spaces of some observers split naturally into congruences of concentric and coaxial ellipsoids. It is shown that these 3-spaces are natural spaces to consider the ellipsoidal figures of equilibrium. The investigation is carried out in detail for axially symmetric oblate confocal ellipsoids, but possible generalizations are indicated. The Kerr metric is found to be an ellipsoidal space-time of this special kind. Some remarks concerning an (unfound) explicit interior Kerr solution conclude the paper.     

Modelling self-assembling of colloid particles in multilayered structures

Simulations of particle multilayer build-up in the layer by layer (LbL) self-assembling processes have been performed according to the generalized random sequential adsorption (RSA) scheme. The first (precursor) layer having an arbitrary coverage of adsorption centers was generated using the standard RSA scheme pertinent to homogeneous surface. Formation of the consecutive layers (up to 20) was simulated by assuming short-range interaction potentials for two kinds of particles of equal size. Interaction of two particles of different kind resulted in irreversible and localized adsorption upon their contact, whereas particles of the same kind were assumed to interact via the hard potential (no adsorption possible). Using this algorithm theoretical simulations were performed aimed at determining the particle volume fraction as a function of the distance from the interface, as well as the multilayer film roughness and thickness as a function of the number of layers. The simulations revealed that particle concentration distribution in the film was more uniform for low precursor layer density than for higher density, where well-defined layers of closely packed particles appeared. On the other hand, the roughness of the film was the lowest at the highest precursor layer density. It was also predicted theoretically that for low precursor layer density the film thickness increased with the number of layers in a non-linear way. However, for high precursor layer density, the film thickness increased linearly with the number of layers and the average layer thickness was equal to 1.58 of the particle radius, which is close to the closely packed hexagonal layer thickness equal to 1.73. It was concluded by analysing the existing data for colloid particles and polyelectrolytes that the theoretical results can be effectively exploited for interpretation of the LbL processes involving colloid particles and molecular species like polymers or proteins.     

Analysis of non-linear multilayer waveguides with Kerr-like permittivities

This paper presents a semi-analytical method for calculating the dispersion relationships for stationary non-linear TE waves guided by general multilayer waveguides with Kerr-like permittivities. The non-linear wave equations are solved rigorously for each layer, and the mode index and field distribution are systematically determined so as to satisfy the boundary conditions at all interfaces. To verify the validity of the present method, a non-linear graded-index waveguide is approximated by a number of step layers and the resultant multilayer waveguide is analysed numerically. The convergence properties of mode index and total power flow are presented, and these are compared with direct solutions of the non-linear wave equation by the numerical integration method. The dispersion relationships for a non-linear multiple-quantum-well waveguide have also been investigated under two simple permittivity models. The present method is useful for the analysis of non-linear graded-index waveguides in addition to non-linear multilayer waveguides such as the multiple-quantum-well structure.     

Control of spontaneous emission by complex nanostructures

Spontaneous emission in the presence of complex nanostructures is discussed by use of a calculational scheme that permits us to deal with interfaces of arbitrary shape. Control over the field associated with the emission is shown to be attainable. In particular, decay rates are offered for geometries that lead to focusing and collimation of near- and far-field distributions. Emission from axially symmetric gratings is shown to lead to narrow angular distributions of emission, and focusing at the foci of dielectric ellipsoids is achieved for dimensions comparable with the wavelength. In the latter case the total emission rate for two atoms in an ellipsoidal cavity is shown to be enhanced in a way that deviates from the predictions of the Dicke effect by means of intermediate- and far-field contributions.     

New type of domain walls: Domain walls caused by frustrations in multilayer magnetic nanostructures

Roughness of the interfaces between layers in a multilayer magnetic structure causes frustration of the exchange interaction between spins. Under certain conditions, frustration brings about the formation of domain walls (DWs) of a new type, whose parameters are determined by the competition between different exchange interactions rather than between the exchange and anisotropy energies as is the case with conventional DWs. Such DWs are much sharper than conventional DWs. The conditions under which micro-(nano-) domains arise are considered, and magnetic phase diagrams for ferromagnet-nonmagnetic metal-ferromagnet and ferromagnet-antiferromagnet nanostructures are discussed.     

Permalloy-FeMn exchange-biased multilayers grown on flexible substrates for microwave applications

Permalloy-FeMn multilayers deposited onto flexible substrates oriented for wide-band absorber applications were fabricated using RF sputtering deposition. The ferromagnetic resonance (FMR) frequency was tuned by changing the thickness of the Permalloy layers. Plural FMR frequencies appeared in the multilayer film due to the difference in exchange couple energies at their interfaces. A multilayer thin film with varying thickness of Permalloy layers was also fabricated with the properties of a wide-band absorber. Its range of 1-4 GHz (the absorption width where the reflection loss is less than 10 dB) appears promising for future applications.     

New recursive solution of the problem of scattering of electromagnetic radiation by multilayer spheroidal particles

A new recursive algorithm for the solution of the problem of scattering of light (of an arbitrarily polarized plane electromagnetic wave) by multilayer confocal spheroidal particles is constructed. This approach preserves the advantages of the two approaches proposed earlier by us for single-layer and two-layer spheroids (special choice of scalar potentials and utilization of the basis of wave spheroidal harmonics) and for homogeneous axially symmetric particles (formulation of the problem in terms of surface integral equations, calculation of the potentials inside the particle from the potentials of the incident radiation, and calculation of the potentials of the scattered radiation from the potentials inside the particle). In the case of multilayer particles, the potential inside each shell is a sum of two terms. The first has the properties of the incident radiation (no singularities inside the volume enclosed by the external boundary of the shell), whereas the second term has the properties of the scattered radiation (satisfies the radiation conditions at infinity). Therefore, as the calculation progresses from one layer to the next (from the core to the outer shell), the dimensionality of the reduced linear matrix equations for the unknown expansion coefficients of the scattered field potentials does not increase with respect to the case of a homogeneous spheroid. The algorithm is particularly simple and lucid (as far as possible for such a complex problem). In the case of spherical multilayer particles, the solution can be found explicitly.     

Integrated characterization of multilayer periodic systems with nanosized layers as applied to Mo/Si structures

The potential inherent in integrated characterization of multilayer periodic systems employed in development of extreme-ultraviolet mirrors was demonstrated using the example of Mo/Si structures grown by magnetron sputtering in different technological regimes. An integrated study provided mutually consistent data on the thicknesses and crystal structure of the layers, as well as on the quality of the interfaces. Measurements by atomic force microscopy permitted a comparison of surface roughness of the substrates and the multilayer systems grown on them. An analysis of the power spectral density functions revealed that low-frequency roughness is replicated from the substrate, whereas the high-frequency one can become smoothed out in the course of growth. X-ray diffractometry performed in the thin film mode showed that the Mo layers in the samples studied have different crystal structures, from the amorphous and polycrystalline to the [110]-textured one. An analysis of the transmission electron microscopy data confirmed that there is a difference in the degrees of crystallinity of Mo layers. The thicknesses of individual layers, the period, and the irreproducibility of the thicknesses and the period were determined using X-ray reflectometry. The root-mean-square roughness amplitude of the interfaces was estimated, and the existence of transition layers originating primarily from the Si layer was demonstrated. The study was used to formulate a proper strategy for the analysis of multilayer periodic systems with nanosized layers.     

Stress field and interaction forces of dislocations in anisotropic multilayer thin films

Utilizing Fourier transforms, the elastic field of three-dimensional dislocation loops in anisotropic multilayer materials is developed. Green's functions and their derivatives, obtained first in the Fourier domain and then in the real domain by numerical inversion, are used in integrals to determine the elastic field of dislocation loops. The interaction forces between dislocations and free surfaces or interfaces in multilayer thin films are then investigated. The developed method is based on rigorous elasticity solutions for dislocations approaching to within one to two atomic planes from the interface. For a dislocation in one layer, the interface image force is determined mainly by the elastic moduli and thicknesses of neighbouring layers. When a dislocation approaches an interface between two layers, within 10–20 atomic planes, the image force changes rapidly. Interaction forces are then kept constant up to the interface. The model shows that, when a dislocation crosses an interface from a soft to a hard layer, additional external forces must be applied to overcome an elastic mismatch barrier. The developed method extends the concept of the Kohler barrier in 2D, and shows that the interface force barrier not only depends on the relative ratio of the elastic moduli of neighbouring layers, but also on the 3D shape of the dislocation, the number of interacting adjacent layers, and on layer thicknesses.     

Investigation of roughness cross correlation in a Ni/C multilayer mirror by X-ray diffuse scattering method

The possibility of applying X-ray diffuse scattering for studying roughness in multilayer X-ray mirrors, including the correlation of roughnesses of neighboring interfaces (roughness cross-correlation) is considered. It is shown that the reliability and informativeness of this method can be improved by rejecting the classical experimental schemes and using alternative schemes in which not only the intensity of diffuse scattering itself, but also its dependence on certain experimental parameters (conditions), vary. Such parameters can be the spatial coherence of incident radiation, the direction of the momentum transfer relative to the specular diffraction plane, or the X-ray wavelength. In the framework of this approach, the results of comparative measurements of diffuse scattering from a Ni/C multilayer X-ray mirror prepared by laser ablation are considered for two close values of photon energy: below (8.325 keV) and above (8.350 keV) the K absorption edge for nickel. It is shown that, in view of effective screening of deep layers in the hard photoabsorption mode, this method provides more reliable (as compared to the standard diffuse scattering method) information on the evolution of interfaces between the layers. It is found that the smoothing of roughness in the experimental sample occurs over large spatial scales such as the micrometer scale. Only large-scale defects with a size exceeding 10 µm are replicated well from layer to layer. Possible physical reasons for the observed effect are considered. It is shown that effective smoothing on the micrometer and submicrometer spatial scales is of fundamental importance for preparing multilayer X-ray mirrors with high reflectances.     

[Pd/Co-Nb/Pd/Si]多层膜的结构与磁性

用高频溅射法制备了两套[Pd/Co-Nb/Pd/Si]多层膜,分别用X射线衍射和振动样品磁强计做了结构和磁性测量。随Pd层厚度增加(或Co-Nb层厚度减少),Pd层由非晶态过渡到晶态,并观察到Pd的fcc(111)双峰结构,双峰的位置逐渐从两侧向体材料Pd的fcc(111)峰的位置靠近。双峰来源于Co-Nb层与Pb层、Pd层与Si层的晶格失配度以及靠近这两种界面的Pd原子的极化不同。样品的饱和磁化强度随Pd层增厚(或Co-Nb层增厚)从小于同样成分的Co-Nb合金体材料的饱和磁化强度值单调增大到大于体材料     

The structure of Mo/Si multilayers prepared in the conditions of ionic assistance

The influence of a negative substrate-applied bias potential on the structure of periodic Mo/Si multilayer compositions has been investigated by means of cross-sectional electron microscopy, small-angle X-ray reflectivity, X-ray diffraction and by modeling the small-angle spectra. It is known that the crystalline structure of molybdenum layers is the main source of interface roughness. In the absence of a bias potential application, the interface roughness tends to develop from the substrate towards the surface of a Mo/Si multilayer composition. A negative bias potential (up to -200 V) applied to a substrate during silicon layer deposition leads to smoother interfaces and improves the layer morphology. After increasing the bias potential over -200 V a considerable growth of an amorphous interlayer transition zone can be observed at Si-on-Mo interfaces. By raising the bias potential during the deposition of Mo layers a development of roughness at Mo-on-Si interfaces as well as growing interlayer thicknesses were found. PACS 61.10.Kw; 68.37.-d; 68.65.Ac     

Peculiarities of the ferromagnetic resonance in multilayer CoFeZr-α-Si films

The static magnetic properties and ferromagnetic resonance spectra of multilayer CoFeZr-α-Si films with different numbers and thicknesses of magnetic and nonmagnetic layers have been investigated. It is established that the shape of the ferromagnetic resonance spectrum and the resonant fields H _res depend on the thickness of nonmagnetic layers and their total number. The character of changes in the spectrum makes it possible to estimate the quality of layers and interfaces.