Effective Field Approximation Research Articles

Plasmonic hemispherical Ag nanoparticles on silicon substrate: A comprehensive study of optical properties

In this work, we demonstrate a comprehensive study of the optical properties of hemispherical Ag nanoparticles (AgNPs) on a single-crystal (c-Si) wafer. The fabricated method involves a galvanic displacement reaction of Ag on c-Si and annealing in oxygen atmosphere at 500 °C. Substrates with different morphological parameters of AgNPs were obtained by varying the volume ratio of Ag in the deposition solution. Using the quasinormal modes (QNMs) theory formalism, the localized surface plasmon resonance (LSPR) multipole positions were determined as a function of the AgNP size and incidence angle. Also, the effective field approximation and QNMs methods were used to calculate the specular reflectance of the disordered hemispherical array on c-Si. Thus, the experimentally obtained LSPR positions in the reflectance spectra were described. Finally, surface-enhanced Raman scattering (SERS) demonstrated reliable detection of brilliant green triphenylmethane dye (1 nM), methyl red dye (10 nM) and hemoglobin from bovine blood (1 mM). The dependence of the SERS enhancement factor on the LSPR position and the absorption band of the analyte was established, the maximum value of which was 1.1 × 107. In summary, this study suggests that the numerical and experimental investigation of the optical properties of hemispherical AgNPs on c-Si contributes to the field of optical sensing.

Phase separation in high-T c cuprates

We develop a minimal non-BCS model for the CuO2 planes with the on-site Hilbert space reduced to only three effective valence centers CuO4 with different charge, conventional spin, and orbital symmetry, combined in a charge triplet, to describe the low-energy electron structure and the phase states of HTSC cuprates. Using the S = 1 pseudospin algebra we introduce an effective spin-pseudospin Hamiltonian which takes into account local and nonlocal correlations, one- and two-particle transport, and spin exchange. The T-n phase diagrams of the complete spin-pseudospin model for the CuO2 planes were reproduced by means of a site-dependent variational approach within effective field approximation typical for spin-magnetic systems. Limiting ourselves to two-sublattice approximation and nn-couplings we arrived at several Néel-like phases in CuO2 planes for parent and doped systems with a single nonzero local order parameter: antiferromagnetic insulator, charge order, glueless d-wave Bose superfluid phase, and unusual metallic phase. However, the Maxwell’s construction shows the global minimum of free energy is realized for phase separated states which are bounded by the third-order phase transition line T*(n), which is believed to be responsible for the onset of the pseudogap phenomenon.

Моделирование диэлектрических характеристик синтактных материалов

Syntactic materials (spheroplastics) having low density due to structural features, depending on the material of the microsphere walls can have a sufficiently high strength and low thermal conductivity, which makes them promising for use as heat insulation materials. By selecting the material of the microsphere walls and the concentration of the components of the spheroplastics, their dielectric characteristics can be significantly changed. In this work, the task of modeling the effective dielectric characteristics of a syntactic material with a polymer binder and filler in the form of hollow glass microspheres is considered, taking into account the presence of technological impurities in the material. A model for calculating the effective permittivity of a sample of a syntactic material was proposed, based on a model of a matrix composite with several types of inhomogeneous or homogeneous inclusions. To calculate, a generalized effective-field approximation for a heterogeneous medium with coated inclusions was used. Model calculations were carried out for syntactic foam with an organosilicon binder polydime-thylsiloxane and hollow microspheres with E-glass shell with some moisture presence in the material. Frequency dielectric characteristics of this material were obtained in the range of 102–1010 Hz. It has been shown that an increase in the volume fraction of hollow microspheres leads to a decrease in the dielectric constant and the tangent of the dielectric loss angle. It has also been shown that the calculated values are in satisfactory accordance with the experimental data obtained at an electromagnetic field frequency of 9.8 GHz.

Dynamic magnetic birefringence in a viscoelastic ferrocolloid.

A model is developed to describe the oscillations of optical anisotropy induced in a viscoelastic ferrocolloid (nanodispersion of magnetic particles) by an AC magnetic field. The viscoelasticity of the matrix (carrier medium) is assumed to obey the Jeffreys rheological scheme, whose advantage is that with the aid of just two viscous parameters and a single one for elasticity it enables one to vary the retarded mechanical response of the carrier from a weakly Maxwellian fluid to a medium with the rheology of a Kelvin gel. As the orientational motion of the particles driven by the AC field is always strongly affected by thermal motion, the occurring process is described with the aid of a kinetic (Fokker-Planck type) equation that combines diffusional and drift terms. On this basis, an exact evolution equation for the macroscopic optical anisotropy of a ferrocolloid is derived that is, however, just one link in an infinite chain of equations for statistical moments. The solution is obtained by applying effective field approximation: reducing the number of moment equations to their minimum and closing the chosen set. This solution is substituted to the scheme of a standard polarimetric set-up, and it is demonstrated how the peculiarities imparted by viscoelasticity should manifest themselves on the intensity of the light transmitted through the set up containing a ferrocolloid sample. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Generalized effective-field approximation for inhomogeneous medium with inclusions in multilayered shells

An approach is proposed for calculating effective physical characteristics of a inhomogeneous medium with several levels of nesting of its microstructure --- the generalized effective-field approximation. With the help of this approach, an expression is obtained for an effective permittivity tensor of an inhomogeneous medium with ellipsoidal inclusions in a multilayered shell, the boundaries of all layers of which are ellipsoids. The proposed approach allows to take into account probabilistic distributions of orientations and forms of inclusions, as well as the presence of several types of inclusions. Two cases of matrix composites are considered: 1) with spherical isotropic inclusions with a multilayered shell; 2) with ellipsoidal anisotropic inclusions with a multilayered shell. For an inhomogeneous medium with homogeneous inclusions, this approximation is shown to produce the same result as the generalized singular approximation. Keywords: inhomogeneous medium, composite, matrix, inclusion, multilayered shell, generalized effective-field approximation, generalized singular approximation, effective permittivity.

Обобщенное приближение эффективного поля для неоднородной среды с включениями в многослойной оболочке

An approach is proposed for calculating effective physical characteristics of a inhomogeneous medium with several levels of nesting of its microstructure - the generalized effective-field approximation. With the help of this approach, an expression is obtained for an effective permittivity tensor of an inhomogeneous medium with ellipsoidal inclusions in a multilayered shell, the boundaries of all layers of which are ellipsoids. The proposed approach allows to take into account probabilistic distributions of orientations and forms of inclusions, as well as the presence of several types of inclusions. Two cases of matrix composites are considered: 1) with spherical isotropic inclusions with a multilayered shell; 2) with ellipsoidal anisotropic inclusions with a multilayered shell. For an inhomogeneous medium with homogeneous inclusions, this approximation is shown to produce the same result as the generalized singular approximation.

Reentrant superconductivity in a hybrid heterostructure with a high transparency barrier

Within the framework of the self-consistent effective field approximation of the time-dependent perturbation theory, an influence of the electron tunneling on the spontaneously induced order parameters in a normal metal–superconductor hybrid structure is considered. For a normal-metal model, which does not take into account electron-electron scattering, as well as electron-phonon coupling, a critical barrier transparency, corresponding to the disappearance of superconductivity in the ground state, was obtained. The presence of incoherent excitations leads to a complex relationship between the effects of ordering, thermal fluctuations, and tunneling. Near the critical barrier transparency, this can stabilize a superconducting state in the certain temperature intervals. As a result, a reentrant superconductivity phenomenon was observed. The studied spectral properties of the hybrid structure reflect the existence of both coherent and incoherent elementary excitations.

Nonlinear Magnetic Response of a Viscoelastic Ferrocolloid: Effective Field Approximation

A viscoelastic ferrocolloid has been probed with the use of a weak alternating magnetic field, which causes rotational oscillations of nanoparticles. The rheology of the dispersion medium has been described using Jeffry’s phenomenological scheme, according to which, colloidal particles are assumed to be magnetically rigid. The Langevin equations that describe the Brownian rotational motion have been used to derive a set of equations for the dynamic magnetization and orientation kinetics of a colloidal ensemble. The solution of the problem has been obtained by the effective field method. The linear and cubic dynamic magnetic susceptibilities, as well as the orientational response of the system, have been found. It has been shown that, in the spectra of the responses of all orders, from linear to cubic ones, the effects of mechanical retardation (stress relaxation) lead to the appearance of specific features, namely, an increase in the medium elasticity is accompanied by variations in the position and height of the main maximum. Expressions have been proposed to relate the rheological parameters of the used Jeffry’s model with the structural characteristics of a polymer solution, i.e., the number of monomers in a macromolecule, the blob size, and the concentration of the physical network nodes.

Compensation point phenomena in a graphene-like Ising nanoparticle decorated with spin-3/2 atoms

The possibility of a graphene-like Ising nanoparticle with high critical temperature or with a compensation point below its critical temperature is examined by decorating spin-3/2 atoms on the graphene-like Ising nanoparticle. The phase diagram and the temperature dependences of magnetization in the decorated nanoparticle are investigated by using the effective-field approximation with correlations. It is clarified how the paramagnetic graphene-like nanoparticle may change to a magnetic ordering nanoparticle by changing the physical parameters of decorated atom.

Effective Thermal Conductivity of Composites with Contact Thermal Resistance between the Inclusions and the Matrix

A method is proposed for predicting the effective thermal conductivity of a matrix composite with several types of spherical inclusions, in the case of contact thermal resistance at the matrix–inclusion boundary. The method is based on generalized effective-field approximation for a inhomogeneous medium containing inclusions that have an outer shell. As an example, calculations are presented for a matrix tribocomposite with two types of inclusions.

Prediction of effective thermal conductivity of multicomponent tribocomposites taking into account contact thermal resistance between inclusions and matrix

A method is proposed for predicting the effective thermal conductivity of a matrix composite with several types of spherical inclusions with contact thermal resistance at the boundary of the matrix and inclusions. The method is based on a generalized effective-field approximation for an inhomogeneous medium with inclusions with a shell. Model calculations were performed for a matrix tribocomposite with two types of inclusions. Keywords: effective thermal conductivity, contact thermal resistance, composite material, matrix, inclusion with a shell, Maxwell—Garnett approximation, generalized effective-field approximation. alekc_sap@mail.ru

On the spin – S random field Ising model

Abstract General Spin-S Ising model under the effect of the random field has been investigated within the effective field approximation. Effect of the bimodal distribution as a discrete, and Gaussian distribution as a continuous random field distribution, on the phase diagrams and order parameter of the system, have been determined. Detailed investigations display many interesting phenomena such as reentrant behavior and small ordered phase which can survive any degree of randomness. Underlying physical mechanisms have been given for the effect of the random field distributions. Especially role of the crystal field has been clarified.

Predicting the Effective Thermal Conductivity of Tribocomposites with Coated Antifrictional Inclusions

By means of a generalized effective-field approximation, formulas are derived for the effective thermal conductivity of a three-component tribocomposite based on a phenylone matrix with spherical strengthening inclusions (alumoborosilicate glass) and inhomogeneous antifrictional inclusions (spherical capsules of glycerin in a Kapton® shell). The thermal conductivity is investigated as a function of the concentrations of inclusions and the ratio of the internal and external radii of the antifrictional inclusions.

Predicting the Optical Properties of Matrix Composites Containing Spherical Inclusions with Metal Shells

Individual small particles consisting of an insulating core and a metal shell exhibit more intricate behavior when exposed to electromagnetic radiation than continuous metal particles. A composite medium containing a large number of such particles is therefore bound to have new optical properties. If the inclusions are small relative to the wavelength of electromagnetic radiation, the optical characteristics of an inhomogeneous medium can be estimated from its effective permittivity. On the basis of the generalized effective field approximation, a formula is derived to calculate the effective dielectric characteristics of a matrix composite containing spherical inclusions with shells. The formula can be considered a generalization of the classical Maxwell–Garnett formula for a matrix medium with inhomogeneous inclusions consisting of anisotropic cores and isotropic shells. Using the formula, the frequency dependences of the real and imaginary parts of the effective permittivity in the 0.282–0.855 μm range of wavelengths are calculated for a composite consisting of an α-quartz matrix and spherical nanoinclusions with α-quartz cores and silver shells. The dependences are obtained for different relative volume fractions of cores in the inclusions and of inclusions in the composite. The frequency dependences of the refractive index, the extinction coefficient of the composite, and the transmittance and reflectance of a thin composite film are calculated for the range of wavelengths indicated above. It is shown that inclusions with metal shells in a matrix composite result in an additional plasmon resonance, compared to a composite containing all-metal inclusions. With a matrix composite, the additional plasmon resonance is observed at a wavelength of 0.33–0.34 μm in the ultraviolet range and is much less intense than the principal plasmon resonance. The additional plasmon resonance is responsible for the very low transmittance of a composite film in the ultraviolet region. At a constant volume fraction of inclusions in the composite, an increase in the volume fraction of the cores in them shifts the principal plasmon resonance to longer wavelengths and lowers its intensity.

A multi-term, multi-harmonic Boltzmann equation model for kinetic behavior in intense microwave and terahertz excited low temperature plasmas

A multi-term (MT), multi-harmonic (MH) decomposition of the Boltzmann equation (BE) is developed to describe electron kinetic behavior in microwave and THz excited low temperature plasmas. In the decomposition of the BE, velocity distribution functions retain an arbitrary time dependence enabling the prediction of electron kinetic behavior from an arbitrary initial condition to a steady-state periodic solution. By exploiting the time-periodic nature of the electron swarm, the MTMH-BE model is not restricted to numerically resolving the electric field cycle. The MTMH-BE model is validated via the Reid ramp model gas and the ionization model gas of Lucas and Salee. Following successful validation, the MTMH-BE model is utilized to elucidate the basic electron kinetic behavior in air at atmospheric pressure. Namely, the error associated with the effective field approximation (EFA) is explored, where it is demonstrated that for atmospheric pressure air, given a microwave frequency of 1 GHz, the EFA may result in more than a factor of two errors in the time-averaged ionization rate. In the second part of this study, the MTMH-BE model is demonstrated as a basic modeling tool for low temperature plasmas. First, the MTMH-BE model is utilized to calculate electron heating profiles from a cold initial condition. The MTMH-BE model is demonstrated to be in excellent agreement with strictly time-dependent kinetic models, including a time-dependent MT-BE model and a Monte Carlo collision model. To highlight the advantage of this work, the MTMH-BE model is used to predict the formative delay time of 95 GHz high power microwave induced breakdown. In this example, the numerical time step utilized in the MTMH-BE model is approximately six orders of magnitude larger than is possible using a strictly time-dependent MT-BE model. Overall, the MTMH-BE model presents a powerful pathway to modeling temporal kinetic behavior in microwave and THz excited low temperature plasmas.

Limitations of the effective field approximation for fluid modeling of high frequency discharges in atmospheric pressure air: Application in resonant structures

We study analytically and demonstrate numerically that the local effective field approximation (LEFA) for plasma fluid modeling of high-frequency (GHz-THz) discharges in atmospheric pressure air is not valid in regions where the time scale for electron energy transfer to heavy particles is less than the time-period of the electromagnetic (EM) wave. Greater than 50% modulation of the electron temperature around its mean value is found for frequencies around and under 10 GHz for atmospheric pressure air discharges. This modulation decreases significantly as the EM wave frequency increases. Fully coupled numerical simulation of a resonant metallic cut-array illuminated by high frequency EM waves demonstrates that the LEFA can lead to significant errors on both temporal and spatial evolution of the plasma, in cases where this modulation is significant. The LEFA for high pressure air discharges is found to be valid when the EM wave frequency is around or higher than 100 GHz. For lower frequencies or when the reduced electric fields are high enough, the Local Energy Approximation should be used for an accurate description of the plasma development. For low gas pressures, the modulation is very low rendering the LEFA valid over a large EM wave frequency range.

Plasma fluid modeling of microwave streamers: Approximations and accuracy

Fluid models of microwave streamers at 110 GHz in atmospheric pressure air predict the formation of filamentary plasma patterns that show a good qualitative agreement with experiments. In order to perform more quantitative comparisons with experiments, in this paper, we study the consequences of different types of approximations that are generally used in the fluid models. We consider here the streamer dynamics before gas heating effects become important, i.e., the first few tens of ns after breakdown at atmospheric pressure. The influence on the results of the local effective field approximation vs. the local mean energy approximation is analyzed in detail. Other approximations that are related to the choice and method of calculation of electron transport parameters are also discussed. It is shown that the local effective field approximation is rather good for a large range of conditions of high frequency breakdown at atmospheric pressure in air while the results may be very sensitive to the choice of transport coefficients.

Ferrimagnetism and compensation temperature in spin-1/2 Ising trilayers

The mean-field and effective-field approximations are applied in the study of magnetic and thermodynamic properties of a spin-1/2 Ising system containing three layers, each of which is composed exclusively of one out of two possible types of atoms, A or B. The A-A and B-B bonds are ferromagnetic while the A-B bonds are antiferromagnetic. The occurrence of a compensation phenomenon is verified and the compensation and critical temperatures are obtained as functions of the Hamiltonian parameters. We present phase diagrams dividing the parameter space in regions where the compensation phenomenon is present or absent and a detailed discussion about the influence of each parameter on the overall behavior of the system is made.

Multiple hysteresis behaviors in spin models: Effect of anisotropy in the exchange interaction

Hysteresis characteristics of the general spin-S S>1) Heisenberg model have been studied within the effective field approximation. Particular emphasis has been paid on the large negative valued crystal field region. Since spin-S Ising model displays maximum number of 2S windowed hysteresis loops in this region, this investigation gives the answer of the question: what is the effect of the anisotropy in the exchange interaction on this multiple hysteresis behavior? As an answer, rising number of windows in the hysteresis loops have been observed.

A generalized effective-field approximation for an inhomogeneous medium with coated inclusions

An approach developed on the basis of a model of a composite with heterogeneous inclusions—the effective-field approximation—is proposed. With the help of this approximation, an expression is obtained for the tensor of effective dielectric characteristics of a heterogeneous medium consisting of inclusions representing a core inside a coating, the outer and inner boundaries of which are ellipsoids. The proposed approach has a large degree of generality and can be used both for multicomponent composites and for polycrystals. It was shown that this approach gives the same result as the generalized singular approximation in the case of the uncoated inclusions.