Optical properties of nickel nanoparticles in systems with a statistically inhomogeneous structure

The analysis and further development of the experimental-analytical approach for determining the values of the optical parameters of nanosized particles for a system with a statistically inhomogeneous structure is presented. An improved experimental-analytical method for determining the complex specific electric polarizability of nanosized particles in systems with a statistically inhomogeneous structure is based on spectrophotometric and electron-microscopic measurements on two-dimensional structures and application of analytical solution of Rosenberg’s spectrophotometric equations with the consideration of analysis of particle size distribution. Experimental spectrophotometric and electron-microscopic studies of nickel islet films with weight thickness from 0.3 nm to 2.0 nm deposited by high-vacuum sputtering on quartz substrates in the range of the spectrum 0.2 ÷ 1, 1 μm are performed. Islands nickel film presented morphological microstructure in the form of monolayers isolated from each other nanoislands spherical surface nickel concentration (0.8 ÷ 2.0) · 1012 cm-2 and an average particle diameter 2.5 ÷ 7 nm. The optical characteristics, namely, the complex specific electric polarizability, optical electric conductivity of nanosized nickel particles are determined in the spectral range 0.2 ÷ 1.1 μm with the help of the improved experimental-analytical method. A significant increase to one order of magnitude of the absolute values of complex specific electric polarizability of nickel particles with a decrease in their size and in comparison with absolute values of complex specific electric polarizability of model spheres with refractive index and absorption index of nickel in macroscopic volumes was revealed. It is established that in the spectral dependences of optical electric conductivity of nickel nanoparticles in the range 0.2 ÷ 1.1 μm there is a band that diminishing size is shifted to a high-frequency region. In this case, the value of optical electric conductivity of nickel nanoparticles monotonically decreases with a decrease in the size by 1-2 orders of magnitude. Comparison of the obtained values of optical electric conductivity of particles in the range of the considered interval with the values of optical electric conductivity of macroscopic samples of nickel in the studied range of the spectrum shows that in particles the value of optical electric conductivity is 2-3 orders of magnitude smaller than that of bulk metals. It is shown that in the nanoparticles of nickel absorption of “Drude” type in the near infrared region the spectrum is suppressed. The reason for this phenomenon may be the change of the mechanism of low-frequency electromagnetic response in nanosized metal particles in comparison with macroscopic metals. The research results are of interest for the development and optimization of nanostructured systems with the inclusion of nanosized nickel particles and functional devices based on them with given electromagnetic characteristics. Ref. 22, fig. 4.


INTRODUCTION
Nanocomposite systems based on nanosized metal particles are of particular interest for integrated technology, optoelectronics, informatics, and energy [1]- [3].The prospects of using nickel nanoparticles to create: electromagnetic filters, miniature laser emitters, light modulators, photosensitive materials, protective and thermoregulatory coatings, high-speed optical devices, magnetic memory elements, high-sensitivity optical sensors and increasing the efficiency of renewable energy sources were noted in [4]- [6].
These applied and fundamental aspects cause considerable attention to the study of the optical properties of nickel nanoparticles.As a rule [1], [7], when solving problems of developing nanocomposite systems and optimizing their characteristics, the optical parameters of nickel nanoparticles are identified with the optical parameters characteristic for nickel in macroscopic volumes.It is obvious that in this case the optical parameters of nickel nanoparticles should not depend on their size.In the literature, information on experimental optical parameters (complex specific electric polarizability α = α1 ˗ iα 2 , com-
plex permittivity ε = ε 1 ˗ iε 2 and optical electric conductivity σ) of metal nanoparticles is extremely limited [8], [9].The complexity of such experimental studies is that the electromagnetic parameters of nanosized particles are not accessible to direct measurements.They can be determined on the basis of experimental values of various characteristics of the electromagnetic response of ensembles of particles, such as islet metal films or composite metal-dielectric structures.
So in [10] it was established by solving the inverse problem of the well-known Maxwell-Garnett theory [2], [11] that in the spectral range 0.5 ÷ 6 μm in islet nickel films the values of σ (and, hence, ε2) of an individual nanoparticle-island fall with a decrease in its diameter D 0 (beginning with D 0 < 25 nm).It should be noted that in [10] the values of D 0 were estimated indirectly by the weight thickness of the films, and σ are given in relative units.
We note that in the optical range of the spectrum (0.2 ÷ 1.1 μm), the reliability of the available experimental data on the dispersion and size changes in the electromagnetic parameters of particles raises serious doubts: since they were obtained without accurate consideration of the particle size and microstructure of the system, by crude methods involving primitive model representations of their electromagnetic response.An analytical solution of the inverse problem of the spectrophotometric system of equations of the Rosenberg theory was found in [12] for the optical properties of two-dimensional systems (monolayers or island films) [11].The method for determining the real α 1 and imaginary α 2 part of the complex specific electric polarizability of an individual nanoparticle in a system with a statistically homogeneous structure was developed and justified on the basis of the obtained analytical solution [12].As a result of the application of this method to nickel islet films with a statistically homogeneous structure on glass substrates, the values α 1 , α 2 of nickel spherical particles in the spectral range of wavelengths λ = 0.4 ÷ 1.1 μm were determined [9].
It should be noted that in [8], [9] the microstructure of the nanodispersed system and the particle sizes are estimated formally under the condition of a normal particle size distribution.This approach is fully justified for systems with a statistically homogeneous structure, for example, for islet films with a weight thickness of up to 1 nm and a low concentration of nano-islands and depends on the type of substrate [13].For systems with a statistically heterogeneous structure, which, as a rule, is typical in most island metal films [13], determining the particle size without taking into account the statistical analysis of their microstructural parameters leads to a large error [14].
However, with further development of the experimental-analytical approach, we improved the method for determining the values of α nanoscale particle for systems with a statistically inhomogeneous structure on the basis of spectrophotometric and electron-microscopic measurements on two-dimensional systems and the application of Rosenberg's spectrophotometric equations taking into account the analysis of particle size distribution.It was shown in [14] that the use of an improved experimental-analytical method made it possible to increase the accuracy of obtaining experimental values of α 1 and α 2 nanosized particles.
In addition, in order to study the nature of the dimensional changes in the optical parameters of nickel nanoparticles, it is of interest to expand the spectral range of their experimental values.For this, it is necessary to determine the values of α 1 and α 2 of nickel nanoparticles in an islet nickel film on a fused quartz substrate with a transparency band at electromagnetic radiation frequencies from near infrared to near ultraviolet regions of the spectrum.We note that nanoscale islet nickel films on quartz substrate have a statistically inhomogeneous structure with a significant size fluctuation [15].Therefore, it is expedient to determine the experimental values of the optical characteristics of nanosizeds nickel particles using advanced experimental-analytical techniques for systems with a statistically heterogeneous structure.
The present work is devoted to the determination of the experimental spectral and size dependences of the optical parameters α, σ of nanosized nickel particles in systems with a statistically inhomogeneous structure in the spectral range λ = 0.2 ÷ 1.1 μm.

II. METHODOLOGY OF EXPERIMENT
An experimental analytical approach is used to analyze the electromagnetic properties of composite nanostructured systems and to determine their electromagnetic parameter which establishes the relationship between the electromagnetic parameters of the individual components of the composite system and its response to external electromagnetic effects.The general methodology for analyzing the electromagnetic parameters of the components of a nanostructured system in the framework of the approach under consideration consists of an experimental and an analytical block.The block of experimental metrological support is based on the use of standard methods for studying the microstructure and electromagnetic parameters of materials.The analytical block is based on solving the inverse problem of the equations systems of a mathematical model describing the electromagnetic properties of the corresponding nanostructured system.
To determine the experimental optical parameters of nickel nanoparticles in a system (nanoscale islet film), the model of the electromagnetic response of a two-dimensional nanostructured system based on the use of Rosenberg's spectrophotometric equations taking into account the analysis of the particle size distribution is preferred for use in the optical spectral range [14].
The model establishes the relationship between the energy characteristics (transmission coefficient T, the reflection coefficient from the film side R and the substrate R'), the phase shift in the reflection and transmission of radiation in the system, with the microstructure parameters (concentration, particle size) and the electromagnetic parameter of the nanoparticle (electric polarizability) using Rosenberg's spectrophotometric equations [11].
The model is considered under the following assumptions: • the normal angle of the electromagnetic wave λ falling on the surface of the nanostructured system; • the surface concentration of particles N 0 with diameter D 0 is sufficiently large, so that the distance between them is much smaller than the length of the electromagnetic wave; • particles are spherical and randomly distributed on the surface of substrate with refractive index n 0 .The orientation of their dipole moments is arbitrary; • the effective electric field in the film is determined by the sum of the external field of the incident wave and the wave scattered by all the particles, taking into account the separation of the wave scattered by the layer into a straight and reverse wave; • the electromagnetic properties of an individual particle are determined by the specific (volume) electric polarizability of the particle α, which is characterized by the ratio , where a and b are dimensionless parameters [11].

Specific electric polarizability is determined by the
, where V is the volume of a particle; α′ is the electric polarizability of a particle.Spectrophotometric systems of Rosenberg's equations have the form: (1) ( where The system of equations (1) connects T and R of a nanocomposite system with effective complex polarization of the entire ensemble of nanoparticles of the system P = P 1 + iP 2 and their average size.The system of equations (2) expresses P in terms of the specific electric polarizability α of the nanoparticle, the concentration and size of the nanoparticle.
The microstructure of a nanostructured system is characterized by parameters: particle size distribution function, particle size, concentration which are determined taking into account the nanoparticle size distribution statistics based on the Pearson's consensus criterion χ 2 [16].
The determination of T and R from the known values of the electromagnetic parameters is called the direct problem of spectrophotometry of a two-dimensional nanocomposite system.The solution of the direct problem of the spectrophotometric equations ( 1) and ( 2) does not present fundamental difficulties and is unambiguous.The inverse problem of the Rosenberg's spectrophotometric equations is the calculation of the specific electric polarizability of a nanoparticle according to known values of T, R. We obtained analytical solutions of the inverse problems of systems of equations ( 1), ( 2) [12].
The choice of the physical solution from two mathematical roots (Р 1 < 0, Р 2 > 0 and Р 1 < 0, Р 2 < 0) of the solution of the system of equations ( 1) and ( 2) was carried out on the basis of the generally accepted principleaccording to the conditions of the anomalous dispersion of the effective permittivity ε eff = ε 1eff ˗ iε 2eff of a nanostructured system near the dipole (Rayleigh) resonance band in a separate nanoparticle.In accordance with [1], ε 1eff > 0, ε 2eff > 0 in the region λ > λ R and ε 1eff < 0, ε 2eff > 0 for λ < λ R , and λ R is the resonant wavelength.In order to find the conditions for the anomalous dispersion of Р 1 , Р 2 near λ R , it is expedient to use the relation [11] between ε eff and P.
It follows from the singularities of the dispersion ε eff in a neighborhood of λ R that the physical meaning in the domain λ ≤ λ R has a mathematical root with the values P 1 < 0, P 2 > 0, and in the range λ > λ R the root P 1 < 0, P 2 < 0 (λ R is the point of the stitching of mathematical solutions).It should be noted that in the spectrophotometric systems of Rosenberg's equations the concept of ε eff for two-dimensional systems is not introduced.At the same time, comparing the results of investigations [1], [8], [9] of the optical properties of three-and two-dimensional systems, it can be concluded that the obtained conditions for the anomalous dispersion of P 1 and P 2 near λ R are also preserved in the case of two-dimensional systems.The resonance wavelength λ R is determined from the conditions [1]: where ε a is the permittivity of the environment surrounding a particle.
Typically [1], when assessing λ R on expressions (3) is used approximation ε = ε m and ε m is determined by the values of the refractive indices n and k of the absorption of metals in macroscopic amounts.The data on nickel n and k available in the literature [17], [18] indicate that the dipole resonance in its particles (in the ε = ε m approximation) should take place in the far ultraviolet region.This serves as the basis for choosing in the investigated range λ ≥ 0.2 μm as the physical solution of the root with P 1 < 0, Р 2 < 0 and the corresponding values α 1 , α 2 .
To determine the optical characteristics of nanoparticles by experimental and analytical methods in experimental software made a series of experimental models of islet nanoscale nickel films and held their spectrophotometric and electron microscopic studies.
Nanoscale islet films of nickel with effective thicknesses of 0.5 ÷ 2.0 nm were obtained by thermal evaporation of nickel (with an initial purity of 99.99%) in a vacuum of 10 -6 mmHg at a rate of 0.5 nm/s on thoroughly quartz substrates heated to 650 K.The effective thickness of the films and the rate of their deposition were controlled by the quartz resonator method.After spraying and cooling in vacuum to room temperature, the samples were exposed to air.
Coefficients of transmission of T and reflection R films were measured in the range of lengths of waves λ = 0.2 ÷ 1.1 microns with an accuracy of 3% at a normal hade of light for a film on the spectrophotometer SF-16 with a special prefix for light reflection measurement.
Microscopic studies of the structural phase composition of nickel films were performed using an electron microscope BS-613.Electron microscopic images showed that nanoscale nickel films possess an island-like statistically inhomogeneous structure, and the shape of the islands is close to spherical.
The parameters of the microstructure of nickel island films: the particle size distribution function, the average particle size D0 of the island particles, the average surface concentration of N 0 were determined by analysis of the distribution of nickel nanoparticles in size using the Pearson criterion.With an increase in the effective thickness of samples of nanosized films, the values of D 0 grew from 2.5 nm to 7 nm, and N 0 dropped, respectively, within the interval (0.8 ÷ 2.0) • 10 12 cm -2 .
In addition, in order to compare the size and dispersion characteristics α obtained by the experimental-analytical method with the results given in [5], it is necessary to determine the values of the optical electric conductivity σ of the nickel nanoparticles.Optical electric conductivity of spherical particles is related to α by means of the relation: where ε a is the permittivity of the environment surrounding a particle.Formula (4) follows from expression [1], [11]: where f is the particle shape factor.
The permittivity and the optical constants of the film (index of refractive n and absorption k) are related by the relations [1]: The value of ε a according to [19] was estimated on the basis of the expression   = (1 +  0 ) 2 ⁄ , where ε 0 is the permittivity of the substrate.
The permittivity of the substrate was determined from the measured values of the transmittance of the substrate T 0 , taking into account its dispersion dependence from the relationship  0 = , where n 0 is the refractive index of substrate [20].

III. RESULTS AND DISCUSSION
The obtained experimental dispersion dependences of the real α 1 and imaginary α 2 parts of the specific complex electric polarizability of ultradisperse nickel particles are shown in Fig. 1, and the dimensional ones in Fig. 2 For comparison, in Fig. 1 shows the dispersion dependences of α 1Ni and α 2Ni of a model spherical particle, with properties of "bulk" (in macroscopic volumes) of nickel.α 1Ni and α 2Ni calculated from the formulas (5)(6)(7)(8) and the data on n and k for nickel in the macroscopic volumes contained in [18].
From the above results it can be seen that dispersion changes of α 2 of nickel nanoparticles are monotonic, and their dispersion curves qualitatively coincide in the interval λ = 0.2 ÷ 0.9 μm with the corresponding dependences for model particles with the properties of "bulk" nickel.
The experimental values of α 1 and α 2 grow to one order of magnitude with decreasing particle size, and in comparison with the absolute values of α 1 and α 2 of the model sphere with refractive index n and absorption index k of nickel in macroscopic samples.
From Fig. 2 it follows that when the size of the nickel nanoparticle decreases from 7 nm to 2.5 nm, α 1 and α 2 increase by more than three times in the blue region of the spectrum, and approximately twice in the red region.
It should be noted that in the range λ ≥ 0.9 μm, the optical constants n and k of "bulk" nickel were determined in [18] with a step λ ∼ 0.1 μm.This probably explains the differences between curves 1-5 and 6 (Fig. 1) for the spectral dependences of α 2 for λ ≥ 0.9 μm.More detailed studies [17] of the dispersion of optical conductivity of continuous (d ∼ 60 nm) chemically pure nickel films indicate the detection of a weak band near ħω ∼ 1.4 eV (λ ∼ 0.88 μm), which is attributed to the long-wavelength interband absorption of nickel.
The presence of this band is confirmed by the spectral dependences of the optical conductivity σ of the nickel particles are shown in Fig. 3.The optical conductivity was determined from the experimental values of α 1 and α 2 obtained on the basis of relation (4).In Fig. 3 also shows the dispersion dependences σ of nickel in macroscopic volumes, contained in [18].
A comparison of the dimensional and dispersion dependences σ obtained by us and in [5] showed that they are qualitatively similar.In both cases (except for small deviations caused by the long-wave band in the dispersion σ), a decrease in σ with a decrease in the size of the nanoparticles is observed.
The significant difference in the dispersion σ of bulk nickel and its particles attracts attention in the spectral range λ = 0.2 ÷ 1.1 μm.As is known [21], the absorption of metals in macroscopic volumes is due to intraband transitions of conduction electrons.In accordance with the Drude model [21], σ should increase with decreasing ħω, which is observed on the dispersion relation σ for bulk nickel.
From Fig. 3, we can conclude that in nickel particles, the "Drude" type absorption in the near infrared region of the spectrum is suppressed.The reason for this phenomenon may be a change in the mechanism of the low-frequency electromagnetic response in ultradisperse metallic particles in comparison with macroscopic metals.
From Fig. 3 it follows that the values of σ of the particle fall monotonically with decreasing D0 by 1-2 orders of magnitude.Comparison of the obtained values of σ particles within the considered range D 0 with the values of σ macroscopic samples of nickel, silver in this spectral range shows that in the particles the values of σ are 2-3 orders of magnitude smaller than in bulk metals.We note that, in order of magnitude, the σ of nickel and silver particles are close to each other, and the degree of decay σ with decreasing D 0 depends little on the type of metal [22].
It seems necessary to compare the results of studies of nickel island nanoparticles on a quartz substrate using the experimental-analytical method developed for systems with a statistically inhomogeneous structure with optical parameter values for nickel nanoparticles in the "island film on a glass substrate" system, obtained earlier in [9].A comparative analysis of the spectral and size dependences α 1 , α 2 (Fig. 1, Fig. 2) with the corresponding values for nickel islands nanoparticles for systems with a statistically homogeneous structure [9] shows that they are qualitatively identical.In quantitative terms, the differences between the experimental values of α 1 , α 2 for both indicated cases reach 30%.
It should be noted that such significant quantitative discrepancies are due to the following aspects: • In [9], the microstructure of the nanodispersed system and the particle sizes are estimated formally from the corresponding histogram of the particle size distribution for the pre-selected samples.This approach is completely justified for systems with a statistically homogeneous structure, which is typical, for example, for islet films with an effective thickness of up to 1 nm and a low concentration of nanoislands.
• For systems with a statistically heterogeneous structure, which, as a rule, is typical for most island metal films [13] determining the particle size without taking into account the statistical analysis of their microstructural parameters leads to a large error [14].Copyright (c) 2017 Мачулянський О. В.  • The results of electron microscopic studies have shown that islet nickel films on glass substrates have a statistically homogeneous structure, and on quartz substrates it is statistically inhomogeneous.
• Experimental information on the optical characteristics of nanoparticles indicates the determining effect of the actual distribution function of nanoparticles on their properties.The development of the experimental-analytical method for determining the values of the complex specific electric polarizability of nanoparticles, taking into account the analysis of the statistics of their distribution by size, made it possible to increase the accuracy of obtaining experimental values of the optical parameters.
Fig. 4 shows the dimensional dependences of α1 and α 2 of nickel nanoparticles in a statistically inhomogeneous system and shows the possible interval of the scatter of the values of α determined for different hypotheses of the form of the particle size distribution functions.Thus, the accuracy of determining the values of α 1 and α 2 of nanoparticles increases to 30% with using of experimentalanalytical method, taking into account the size distribution function of the nanoparticles for the investigated systems, by estimating the particle size (D 0 < 7 nm) with a credible probability of 0.95.

CONCLUSIONS
The analysis and further development of the experimental-analytical approach for determining the values of optical parameters of nanosized particles in a system with a statistically inhomogeneous structure is presented.
Experimental-analytical method for determining the complex specific electric polarizability of nanosized par-ticles in systems with a statistically inhomogeneous structure is based on spectrophotometric and electron-microscopic measurements on two-dimensional structures and using an analytical solution of the Rosenberg's spectrophotometric equations, taking into account the analysis of particle size distribution, is proposed.
Experimental spectrophotometric and electron microscopic studies of islet nickel films with weight thicknesses from 0.3 nm to 2.0 nm deposited by high-vacuum deposition on quartz substrates in the spectral range λ = 0.2 ÷ 1.1 μm are performed.The island nickel films represented a morphological microstructure in the form of monolayers of isolated nickel nanoislands with a surface concentration N0 = (0.8 ÷ 2.0) • 10 12 cm -2 and an average particle diameter D 0 = 2.5 ÷ 7 nm.Optical characteristics of nanodimensional nickel particles, namely, complex specific electric polarizability, optical electric conductivity, using an improved experimental-analytical method in the spectral range λ = 0.2 ÷ 1.1 μm are determined.A significant increase to one order of magnitude of the absolute values of α 1 , α 2 nickel particles with a decrease in their size and in comparison with the absolute values of α 1 and α 2 model spheres with refractive indices n and absorption k of nickel in macroscopic volumes was found.

Fig. 1
Fig. 1 Spectral dependences of the real α 1 (a) and imaginary α 2 (b) parts of the complex specific electric polarizability of the nanoparticles of nickel at different values of the diameter D 0 of the nanoparticles of nickel: 1 -D 0 = 2.6 nm; 2 -D 0 = 3.4 nm; 3 -D 0 = 5 nm; 4 -D 0 = 5.6 nm;5 -D 0 = 7 nm; 6 -for a model sphere with values of electromagnetic parameters characteristic of nickel in a macroscopic volume[18]

Fig. 4
Fig. 4 The scatter of the values of the real (a) and imaginary (b) parts of the complex specific polarizability of nickel nanoparticles (on their dimensional dependencies) determined for different hypotheses of the type of particle size distribution functions at λ = 0.4 μm