Mathematical Modeling of Dielectric Permeability and Volt-Ampere Characteristics of a Semiconductor Nanocomposite Conglomerate

Abstract

Mathematical computer models of the permittivity of silicon-based nanostructures upon interaction with electromagnetic radiation in a wide frequency range have been developed. To implement computer models for studying the electrophysical properties of the structures under study, algorithms and a set of programs have been developed. The results of the study of materials will not only provide fundamental information about the physical effects occurring in composite nanostructures but will also be useful for solving problems related to calculations for given electrophysical problems. For a nanocomposite based on ceramics and semiconductor oxides of zinc grains, resonant bursts of permittivity are observed within a wavelength of 300–400 nm; it has been found that this is due to the presence of electronic polarization of the nanocomposite core. The paper presents the results of modeling the current-voltage characteristics of a nanocomposite based on ceramics and semiconductor grains of zinc oxide. The obtained results show that the geometrical parameters, such as the number of layers and sample width, affect the CVC of the nanocomposite, and the operating point of the CVC shifts. This may be of interest in the development of materials with desired electrical characteristics for the creation of varistors.

1. Introduction

The use of modern information technologies in scientific research, in particular in applied tasks [1,2] related to the creation of new functional materials with specified electrophysical properties, opens up wide opportunities [3]. Specialized software and computer models [4] make it possible to explore structures, the study of which by traditional experimental methods can be expensive [5], time-consuming [6], and in some cases completely impossible. Given the high demand, both on the part of scientific researchers and engineers in the field of materials science in applied applications for the operational analysis of the electrophysical properties of nanostructures, the development of such is an urgent task.

The development and research of new nanocomposite materials with specified electrical characteristics is of interest to a wide range of specialists. The rapid development of electronics associated with the use of semiconductor structures has made it possible to obtain valuable knowledge about the phenomena and physical processes occurring in solids.

Works [7,8] are devoted to the study of the electrophysical properties of nanocomposites and the design of such materials. Such materials exploit the simultaneous properties of two or more phases contained in the composites. While the individual properties of each phase can provide unique functionality on their own, very often the interrelationship between different types of material provides a synergistic interaction that can enhance the performance of composites or even bring about new effects not exhibited by each of the phase components separately. Mathematical modeling is an effective tool for predicting the properties of new materials [9,10].

Micro- and nano-electromechanical systems (MEMS, NEMS), which combine electronic and mechanical components on a nanometer scale (up to 100 nm), deserve special attention. The work in [11,12] is devoted to the study of the mechanical behavior of MEMS, NEMS and nanocomposites.

One of the more urgent technical tasks is the creation of varistors with improved characteristics based on polycrystalline nanocomposites [13]. As a rule, such materials are conglomerates of semiconductor grains, the bulk properties of which are linear over a wide range of voltages, and the contacts between grains have a nonlinear volt–ampere characteristic (VAC). To establish the optimal modes of the technology for obtaining a nanocomposite with specified properties, data on the VAC of both contacts between conglomerate grains and on the VAC of the material itself are needed [14]. The VAC of contacts between grains can be quite accurately determined experimentally (for example, by photolithography) [15]. The purpose of this work is to determine the electrical characteristics of the resulting semiconductor nanocomposite conglomerate, based on the data on the electrical characteristics of the substances that make up the nanocomposite, depending on the parameters of external influences. The following tasks were included in the study. The dielectric permittivity of the nanocomposite in the range of wavelengths of ultraviolet, visible, infrared radiation was calculated. The task was also to determine the VAC of the nanocomposite itself based on the known structure of the material, and the VAC of contacts between conglomerate grains.

Computer models and a software package for studying the dielectric constant of silicon-based composite nanostructures also have to be developed. Such structures are used as photodetectors, sensitive sensors for elements of modern microcircuits, etc. The process of computer modeling is divided into several stages. An enlarged diagram of the modeling process is shown in Figure 1.

2. Materials and Methods

At the first stage, the development of a computer 3D model of the investigated nanostructure is carried out. To implement the structural model of nanocomposites, there are a number of specialized software packages, the most popular of which are: OpenBabel, GAMESS, Jmol, ChemSketch, Avogardo, Gabenit, Gaussian, PyMOL, RasMol, etc. Development of a 3D model of a silicon-based nanocomposite was carried out in the Avogadro software package, which is a powerful tool for molecular modeling, quantum chemistry, bioinformatics, materials science and other related scientific fields. The results of computer 3D modeling are shown in Figure 2. Silicon conglomerates with an ordered structure are marked in blue. The conglomerates are interconnected by zinc chains. Such structures do not exist in nature in an explicit form and have unique physical properties.

To develop a mathematical model of the structure under study, the effective medium theory [7], which has experimental confirmation [8], was used. The essence of the model is that an ensemble of nanoclusters is considered as a kind of new medium with effective characteristics. The main advantage of this approach is that, within the framework of the effective medium model for analyzing the interaction of electromagnetic radiation with a nanocomposite, there is no need to solve Maxwell’s equations at each point in space. It is important to take into account the limitations imposed on the described theory. The condition for the applicability of the effective medium model is the small size of the particles that make up the structure, as well as the distance between them, in comparison with the optical wavelength in the medium. The size of the conglomerate is about 10 nm, and the size of the wavelength of external radiation is from 300 to 2500 nm, so the use of this theory is acceptable. A system of equations based on the modified Odelevsky formula is used as a model of the dielectric constant:

$$\{\begin{array}{c}\epsilon =a+\sqrt{a^2+\frac{{\epsilon }_1{\epsilon }_2}{2}};\\ a=\frac{(3V-1){\epsilon }_1+2(2-3V){\epsilon }_2}{4};\\ {\epsilon }_2={\epsilon }_1[1+\frac{6V({\epsilon }_1-{\epsilon }_3)}{3({\epsilon }_1-{\epsilon }_2)(\pi k-6)}],\end{array}$$

(1)

where $\epsilon$ is the dielectric constant of the entire nanostructure; ${\epsilon }_1$, ${\epsilon }_2$, ${\epsilon }_3$ are the dielectric permeability of the 1st, 2nd and 3rd materials, respectively, included in the nanocomposite; $k$ is the distance between the axes of the cylindrical chains; and $V$ is the volume concentration of the dispersion phase.

3. Results

The simulation results for the structure with the initial parameters from [16] and $V$ = 0.1 are shown in Figure 3. All calculations were carried out using the software package “Mathematical modeling and multi-criteria analysis of nonlinear properties of composite materials based on an effective medium” [17]. This software package makes it possible to study the electrodynamic properties of heterogeneous media based on a number of models. The analysis of properties is carried out on the basis of the data entered into the system on the parameters of the medium: the volume fractions of the components, the complex permittivity of substances, and the shape, structure and orientation in space of particles of inclusions. The modularity of the development allows expansion of the functionality; in particular, an important asset is the ability to add new models of heterogeneous media and new numerical methods. At the moment, more than 20 models (Rayleigh, Lichtenecker, Lorentz-Lorentz, Maxwell-Garnett, Bruggeman, Odelevsky, etc.) have been implemented in software for various morphologies of composites. When developing a library of numerical methods, algorithms for finding complex roots of polynomials were programmatically implemented. The programming language C++ was used to implement the software package.

Resonant bursts of permittivity are observed in the wavelength range of 300–400 nm, which indicates the presence of electronic polarization of the studied nanocomposite: the frequency of the external field coincides with one of the frequencies of the electron shells, as a result of which absorption maxima appear (the real part of the permittivity) and difference loss maxima (imaginary part of the dielectric constant).

The results obtained can be of practical interest for studying the electrical characteristics of silicon-based nanostructures, and will also be useful for the design and synthesis of materials with predetermined electrical properties.

Consider a nanocomposite, which is a conglomerate of semiconductor grains (for example, ceramics with semiconducting zinc oxide grains). Figure 4 shows a snapshot of a conglomerate of ZnO grains [18] and the structure of the Si–ZnO nanocomposite.

To obtain the VAC characteristic of a semiconductor conglomerate, an equivalent circuit was developed, which is a grid of parallel-series connected linear and nonlinear elements (Figure 5).

The Kirchhoff equations for individual circuit nodes were used as a mathematical model:

$$G\times \overline{U}-{\overline{I}}_0-\overline{K}(\overline{U})=0$$

(2)

where G is the matrix of the conductance of the circuit, $\overline{U}$ is the vector of nodal potentials, ${\overline{I}}_0$ is the wind of currents from constant sources, $\overline{K}$ is the vector of currents of nonlinear elements. The system of Equation (2) was solved according to the Levenberg—Marquardt algorithm, which turned out to be more efficient in comparison with the Newton—Raphson—Kantorovich method applied to the system of nonlinear equations used in [19].

VAC characteristic of intergranular contacts was set in the form:

$$I=I_0(e^{\alpha \left|U\right|}-1)signU$$

(3)

Parameter ranges: $I_0\in [5\cdot {10}^{-6},5\cdot {10}^{-5}A]$, $\alpha \in [0.1;1B^{-1}]$, $U\in [-5;5B]$.

Specific conductivity was set in the form:

$$\sigma =I\frac{h}{U}\frac{1}{2}B\sqrt{n}$$

(4)

where h is the number of nanocomposite layers, B is the sample width, n is the number of conglomerate layers.

Figure 6 shows the results of modeling the VAC characteristic for a nanocomposite based on ceramics and semiconducting zinc oxide grains.

The results obtained show that geometric parameters, such as the number of layers and the width of the sample, affect the VAC of the nanocomposite, and the operating point of the VAC is shifted. In the area of stress [3; 4 V] there is an intersection of the graphs for the values $B\sqrt{n}$ = 3/2, 9/4. The departure of the main charge carriers leads to the fact that their concentration at the transition boundary decreases. At the same time, a large layer of immobile ions is formed at the boundary of two semiconductors and the width of the potential barrier increases, which leads to an increase in the internal field of the junction. In the future, due to a decrease in concentration at the p–n junction, the diffusion of electrons from the depth of the semiconductor to the near-contact layer will begin, compensating for its loss of charge carriers. This process is also assisted by an external electric field, which acts in the direction of the diffusion movement of electrons.

4. Conclusions

The study shows the possibilities of mathematical modeling in the study of the electrical characteristics of a nanocomposite conglomerate. A computational experiment, unlike a full-scale one, is easily controlled, does not require high financial costs, and is often faster than a full-scale experiment. By no means do we diminish the importance of the natural experiment, since it is this that must finally confirm the results. Nevertheless, mathematical modeling can make it possible to effectively plan a full-scale experiment and significantly reduce its costs and time. As a result of mathematical modeling, resonant bursts of permittivity were found in the range of 300–400 nm. This may be of interest when creating reflective and absorbing screens. The displacement of the operating point of the I–V characteristics may be of interest in the development of materials for the creation of varistors. The results obtained can contribute to the creation of new materials with desired electrophysical properties, which can be used in various fields, including radar and electronics.