Exploring conduction mechanisms in chalcogenide thin films: an experimental and soft computing approach with ANN and GP techniques

Abstract

The article consists of two main sections. In the first part, an experimental study was conducted to investigate the AC conduction of Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Ge₅ (SBG), and Se₈₃Bi₁₇Te₅ (SBT) in thin film forms. Thermal evaporation was utilized to produce films with different thicknesses. The measurements were performed at temperatures ranging from 293 to 393K, film thicknesses ranging from 230.2 to 543.9 nm, and frequencies ranging from 0.1 to 1000 kHz. The AC data were analyzed using the frequency power law. The values of the frequency exponent and their temperature dependence confirmed that the conduction process in the thin films is governed by the hopping mechanism. Additionally, the impact of introducing Te and Ge on the studied properties was also investigated in this study. The second section of the article focuses on estimating the AC conduction of Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Ge₅ (SBG), and Se₈₃Bi₁₇Te₅ (SBT) films using Artificial Neural Networks (ANNs) and Genetic Programming (GP). The experimental data was employed as inputs for training the ANNs and the GP models. The simulation and prediction results obtained from both models were compared with the experimental data and reported. Equations were developed to describe the behavior of the experimental data. The study concluded that the ANN approach is more suitable for simulating and forecasting AC conductivity.

1 Introduction

Scientists have expressed an increasing interest in understanding "Ore forming" materials, also known as chalcogenide glasses, due to the wide range of economic potential for these materials. Chemical substances known as chalcogenides a minimum one chalcogen anion makes up, an electronegativity element (often a non-metal), and a minimum one or more electropositive components (mainly metals). Chalcogenides are categorized in the literature as metallic (Po) Polonium, nonmetal (S) Sulfur and (O) Oxygen, and metalloids or semiconductors (Te) Selenium and Tellurium (Te). Due to their unique material characteristics, like high nonlinearity and wide mid-IR transparency, threshold/memory switching behavior, and infrared transmission, these glasses are a potential material for memory elements and fiber optics applications, among other things. Chalcogenides glasses are a recognized aggregate of inorganic glasses that contain one or more of the chalcogenide elements (S, Se, or Te) as alloy elements. In recent decades, Se-based chalcogenide glasses in particular have drawn a lot of attention because of their possible use in various solid state technologies. The binary SeBi system has undergone Te addition by numerous researchers [1,2,3]. The crystallization of ternary SeBiTe glasses has been researched by numerous researchers [4,5,6].

Modeling the connection between the variables used as inputs and outputs in the processing of materials is known as material modeling [7,8,9,10]. Establishing the relationships between process variables is the first step in optimizing process outputs. The output variable is then optimized using the right optimization technique. Soft computing techniques like artificial neural networks (ANNs), genetic programming (GP), adaptive neuro-fuzzy inference systems (ANFIS), fuzzy logic, genetic algorithms (GA), simulated annealing (SA), particle swarm optimization (PSO), etc. have become the preferred methods for modeling and optimizing materials in modern times due to their superiority over mathematical models. Due to their many benefits, the soft computing optimization algorithms known as metaheuristics, a stochastics optimization tool, have become the dominant optimization technique in research. As mentioned previously, ANNs and GP are very significant soft computing tools in material modeling. In the literature on materials science, ANN and GP have several uses. Amirjan et al. [11], for instance, employed ANN to forecast the electrical conductivity and hardness of copper alloys with Al₂O₃. They created the composites at five distinct sintering conditions, ranging from 15 to 180 min, and five different temperatures, ranging from 725 to 925 °C. They employed four distinct compositions of the composite consisting of varied quantities of Al₂O₃ reinforcement at 1, 1.5, 2, and 2.5 wt percent. The findings demonstrated that the electrical conductivity and hardness of the specimens, which were predicted by the ANN model, agreed well with the experimental data, with an average error is 3% and 5% for the electrical conductivity prediction and hardness respectively. At high temperatures, Hussaini et al. [12] used the formability of austenitic stainless steel 316 to estimate the drawn cup's thickness, they used factors from the metal forming process, like the cup's distance from the center, temperature, and LDR, as inputs to an ANN. Their models performed well. Using ANNs and GP, Palika et al. [13] predicted concrete's compressive strength, it's clear from the results that model ANN predicts compressive strength values that are quite close to the experimentally measured values. The total electrical conductivity and actual and fictitious components of impedance were predicted and simulated by Ali et al. [14] using ANN. A compatibility between the training and prediction findings and the corresponding experimental results was found. Based on the application of ANNs, a mathematical equation was created to describe the experimental outcomes with high accuracy. An ANN model was utilized by Zahran et al. [15] to simulate and predict the Vickers microhardness of Sn-9Zn Binary Eutectic Alloy. The computed results were contrasted with the findings from the experiment. The outcomes support the ANN model's strong capacity for Vickers microhardness simulation and prediction. Also, the ANN model was utilized by Abd El-Rehim et al. [16] to simulate and forecast the mechanical characteristics of Sn-9Zn-Cu solder alloys. The Sn-9Zn-Cu alloy simulation and forecast results from the BP-ANN were in perfect accord with the experimental findings. Among others, Abd El-Rehim, [17] Utilize an ANN model to simulate and forecast the Vickers hardness of AZ91 magnesium alloy. It was possible to compare the ANN results to the experimental findings. The outcomes demonstrated that the ANN model is capable of simulating and forecasting the AZ91 magnesium alloy's response to age-hardening. To date, no artificial intelligence models have been presented for the simulation and prediction of AC conduction of Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Ge₅ (SBG), and Se₈₃Bi₁₇Te₅ (SBT) in thin film forms. To better understand various physics phenomena, there is a need for novel computer science techniques to analyse experimental data. ANN and GP models have become increasingly popular in recent years as successful approaches for determining data correlations and have been successfully used in materials science due to their generality, noise tolerance, and fault tolerance. Building a prediction model using ANN and GP has many benefits, including being simple to handle, not requiring specialised knowledge to find a solution, and being able to adapt to changing conditions. The goal of this study is to compute AC conduction of Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Ge₅ (SBG), and Se₈₃Bi₁₇Te₅ (SBT) in thin film forms using the ANN and the GP simulation models. Each model yielded mathematical equations that described Ac conduction. The results are then compared to the experimental data that is available. Also, we compared the prediction results of ANN and GP models with the experimental data to confirm the accuracy of the models. The ability of the techniques to simulate experimental data with the highest level of accuracy and the prediction of some temperatures and frequencies to obtain the best model encourages them to dominate the modeling technique in casting thin films and the capacity of the model to predict experimental data in case of difficult experimental at different temperatures and frequencies. To achieve our aim, experimental data was used to train these models. The optimal model produces the minimum error. ANN and GP were evaluated on various performance measuring parameters such as coefficient of correlation (R2), Mean Square error (MSE), Mean Absolute Error (MAE), error (${e}_{i}$), Mean error ($\overline{{e }_{i}}$), and standard deviation error (Std error).

2 Experimental details and theoretical programs

2.1 Samples preparation, characterization

In the current investigation, the melt quench method was used to create glass samples of the compositions Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Ge₅ (SBG), and Se₈₃Bi₁₇Te₅ (SBT). Se, Bi, Te, and Ge (Sigma Aldrich) were utilized as starting ingredients because of their high purity (99.99 percent). Five grams of the high-purity compounds listed above in powder form were weighed in accordance with their atomic weights and sealed in quartz ampoules at a vacuum pressure of 10^–5 Torr. After that, an oscillating furnace was used to raise the temperature of the sealed ampoules by 3 °C per minute up to 1273 K. The ampoules were fixed at these temperatures for 18 h while being regularly rocked to homogenies the melt. The melt was then swiftly cooled with ice water to achieve a glassy shape. To obtain bulk samples, break the silica ampoules. A pestle and mortar were used to create a fine powder. On a dry, clean, highly polished glass substrate of varied thicknesses, thin films of the Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Ge₅ (SBG), and Se₈₃Bi₁₇Te₅ (SBT) compositions are created using the thermal evaporation process under a vacuum of 10^–5 Torr at room temperature (230.2–543.9 nm). During deposition, these films' thickness is regulated using a quartz crystal thickness monitor. Various thicknesses of films. Films are sandwiched between two Aluminum electrodes as a lower and upper electrode with an appropriate thickness to measure AC conductivity (500 nm). The silver paste was used to cover the samples' flat tops. Direct measurements of sample impedance, sample capacitance, and loss tangent were made using a programmable automatic RLC bridge (PM 6304 Phillips). On the bridge's screen, each of the examined film samples was represented by parallel capacitance C and resistance R. Applying the formula ${\sigma }_{tot}(\omega )=\frac{1}{z }\frac{L}{A}$, where Z is the total impedance of the sample, $L$ is the film thickness and $A$ is the film cross-section area, yields the total conductivity ${\sigma }_{tot}$.

2.2 Soft computing techniques for modeling material properties

Many artificial intelligence and computational methods used in modeling, optimisation, and operations research have a connection to soft computing approaches. The term "Artificial Intelligence" (AI) is very well-known, and recent developments and improvements in the field have helped it gain acceptance across a number of industries. AI's function is to teach machines to learn from their errors and perform operations more efficiently. The Artificial Neural Network (ANN) and Genetic Programming (GP) in AI are two of its innovations, which perform tasks that are comparable to those carried out by human brain neurons and the processes of natural development, as well.

2.2.1 Artificial neural networks (ANNs)

ANN is a member of the artificial intelligence family that replicates the operations of human brains. It investigates the information that the human brain takes in, processes, and changes. Although there are other ANN variants, the multilayer neural network is the most used [18]. ANN is a mathematical model made up of several linked processing units known as neurons or nodes and was first introduced by Warren McCulloch and Walter Pitts [19, 20]. Biological nervous systems, which transmit information via numerous nerve cells linked to one another by nerve fibers, inspired ANNs. In ANNs, neurons are linked to one another to create a processing network where each neuron gets, changes, and transmits signals similarly to natural nervous systems. A common model of a synthetic neuron is displayed in Fig. 1. Connections are the links that connect neurons. An incoming signal is modified using the connection weights, which are corresponding values for each connection. The activation function, also known as the transfer function and represented by Eqs. 1, is used to calculate each neuron's outgoing signal.

$$y=f\left(\sum_{i=1}^{n}{ x}_{i}{w}_{ji}-{ \theta }_{j}\right)$$

(1)

when ${x}_{i }\left(i=\mathrm{1,2},\dots n\right),$ is the input value, $y$ is the output, the connection weight coefficient ${w}_{ij}$ links the nodes $i$ and $j$, additionally each node has an activation function $f$ and a threshold ${\theta }_{j}.$ The position of the highest increment in the repeatedly raising activation function is indicated by the threshold value mathematically. The function is typically selected so that it includes nonlinearity to a network's work while simply changing the outcome. The most popular transfer functions are the log-sigmoid (Logsig), tan-sigmoid (Tansig), and linear (Purelin) transfer functions. A network training function called Trainlm modifies bias and weight values based on Levenberg-Marquardt optimisation. Despite needing more memory than other algorithms, Trainlm is strongly advised as a first-choice supervised algorithm. The architecture of the network must be chosen as a first step. The simple structure on which this research was focused was the feedforward networks. This model consists of networks with connections that are completely feedforward, meaning that no neuron, even indirectly, receives input from a neuron to which the latter sends its output. The multilayer perceptron (MLP), which is considered remarkably accurate, will be considered, where units are organised into several layers in a certain order. A typical three-layer MLP is depicted in Fig. 2, where neurons are organised in ordered layers and weighted connections are only permitted between layers that are adjacent units or neurons. A single input layer, one or more hidden layers, and a single output layer make up the typical architecture of an artificial neural network. In the architecture of a neural network, each layer has a number of nodes that connect to the layer above it. A weight is assigned to the nodes that can be changed throughout the training process. To strengthen or weaken the signal of the linked neuron, the weights are increased or decreased. A neuron receives any number of inputs, and each one is weighted differently. Once the number of layers and units per layer has been defined for any MLP, the network's weights are adjusted in the final design step to ensure that it generates the desired output when the appropriate input is provided. This procedure is referred to as either learning the network weights or training the ANNs. Network training is a method of iteration that modifies weights and thresholds to keep the network's error to a minimum or to stop at an established training step. As was originally stated, the focus will be on the learning scenario known as supervised training. According to this method, the experimental data given into the ANN paradigm is divided into three parts: 70%training, 15%validation, and 15% testing. Figure 3 depicts the flow chart outlining the steps involved in the ANN training.

Fig. 1

A model of an artificial neuron

Fig. 2

A simple feed− forward ANN, with one hidden layer

Fig. 3

Basic steps of ANN training

2.2.2 Genetic programming overview

The extension to genetic algorithms is genetic programming (GP). The ideas of genetics and natural selection are the foundation of search and optimization method known as GA. A GA enables the evolution of a population (chromosome) made up of numerous individuals to a state that maximises the "fitness" (i.e. minimises the cost function). The GP and GA are similar, however, in contrast to the latter, the GP's solution is a computer programme or an equation, whereas the GA's answer is an assortment of integers. Koza (1992) [21, 22] gives an excellent explanation of a number of GP-related concepts. In GP, a random population of individuals (equations or computer programmes) is produced, their fitness is assessed, and then a subset of these individuals is chosen to act as the "parents." The parents are then forced to produce "offspring" via reproducing, mutating, and crossing over. The process of producing offspring continues (iteratively) until a certain number of offspring are generated in a generation, and then continues until a further certain number of generations are produced. The problem's solution is the product that emerges at the end of the entire process. The GP follows natural genetic processes including reproduction, mutation, and crossover to gradually change one group of humans into another. Each person makes a genetic contribution to the creation of new ones (offspring) that are better adapted to their environment and have a higher chance of surviving. Genetic algorithms and programming are based on this. The mathematical model of the system is expressed by a tree-structured chromosome, which GP manages. Solutions are presented in GP as syntactic trees rather than as code lines. For example (Fig. 4) shows a solution (*, A, B, + ,  − , C, 6, 8) for tree description. The program variables and constants (A, B, C, 6, and 8) are tree leaves. They are known as terminals in GP, whereas arithmetic operations are performed by internal nodes known as functions. (+ ,  − , and *). A GP scheme's fundamental set consists of sets of features and terminals. For example, (*, A, B, + ,  − , C, 6, 8) becomes (A $\times$ B) + (6  − $\frac{C}{8}$). Genetic operators are used to alter trees (reproduction, mutation, and crossover). A tree branch is pointed to by the crossover operator, which exchanges it for another branch to produce new trees. With the mutation operator, a random new branch is created. If the program is judged to be acceptable by the fitness requirements, reproduction refers to an exact duplicate of the program (Fig. 5). Each chromosome has a unique length. Fitness functions are used to choose individuals for crossover, mutation, and reproduction, and to assess how well they can solve the given task. After determining the fitness of each individual, the fitness function estimates how a person fits into the context of a domain problem. The performance of a created computer program (solution) can be estimated using a variety of fitness functions, including the number of hits, relative squared error (RSE), mean squared error (MSE), etc. [23].

Fig. 4

Tree representation of the program ($AB$) + ($6\hspace{0.17em}$− $\frac{C}{8}$)

Fig. 5

Crossover and mutation processes

The processes listed below are carried out by GP when creating computer programs to solve issues (Fig. 6):

Fig. 6

The flow chart of GP

Step 1 A random beginning population of people with terminals and functions relevant to the problem domain is generated.

Step 2 GP implementation iteratively carries out the subsequent actions up until the termination criterion is me.

• i. Each person's fitness value is calculated using a chosen fitness measure.

• ii. Based on their fitness scores, the population as a whole is sorted.

• iii. Genetic processes are used to create the subsequent generation (reproduction, crossover, and mutation).

The termination criterion is verified in step four. If it is, move on to step 3; otherwise, conduct the next iteration.

Step 3: The outcome might be a fix for the issue at hand.

Six statistical parameters were used to determine which model performed the best. These were the following: (1) coefficient of determination (R²) between observed and predicted values (optimal performance is close to 1), (2) mean square error (MSE), (3) Mean Absolute Error (MAE), (4) error (${e}_{i}$), (5) Mean error ($\overline{{e }_{i}}$), and (6) standard deviation error (Std error) which are each mathematically definable as follows [24]:

$${R}^{2}=1-\frac{\sum_{i=1}^{n}{\left({P}_{i}-{M}_{i}\right)}^{2}}{\sum_{i=1}^{n}{\left({M}_{i}-\overline{{M }_{i}}\right)}^{2}}$$

(2)

$$MSE=\frac{1}{n} \sum_{i=1}^{n}{\left({P}_{i}-{M}_{i}\right)}^{2}$$

(3)

$$MAE= \frac{\sum \left|{P}_{i}-{M}_{i}\right|}{n}$$

(4)

$${e}_{i}=\left({P}_{i}-{M}_{i}\right)$$

(5)

$$\overline{{e }_{i}}=\frac{1}{n} \sum_{i=1}^{n}{e}_{i}$$

(6)

$$STd= {\left( \frac{1}{n-1}\sum_{i=1}^{n}{({e}_{i}-\overline{{e }_{i}})}^{2}\right)}^{1/2}$$

(7)

where $n$ denotes the quantity of experimental data, ${M}_{i}$ is the experimental data measured, ${P}_{i}$ denotes the predicted results, and ($\overline{{M }_{i}}$) denotes the average of the experimental data recorded. MSE was employed to provide a numerical assessment of ANN model error (the best performance value is zero).

3 Results and discussions

3.1 Frequency dependence of AC electrical conductivity

Measurements of the AC conductivity ${{\varvec{\sigma}}}_{{\varvec{A}}{\varvec{C}}}$ provide significant information about the conduction mechanism of glassy systems. The phenomenon of the dispersion of conductivity is interpreted by Jonscher’s universal power law [25, 26]:

$${{{\varvec{\sigma}}}_{{\varvec{t}}{\varvec{o}}{\varvec{t}}}={\varvec{\sigma}}}_{{\varvec{A}}{\varvec{C}}}+{{\varvec{\sigma}}}_{{\varvec{D}}{\varvec{C}}}={\varvec{A}}{{\varvec{\omega}}}^{\boldsymbol{\alpha }}+{{\varvec{\sigma}}}_{{\varvec{D}}{\varvec{C}}}$$

(8)

where ${{\varvec{\sigma}}}_{{\varvec{D}}{\varvec{C}}}$ is the DC conductivity, ω is the angular frequency, A is a temperature-dependent constant that determines the strength of the polarizability and $\boldsymbol{\alpha }$ is the frequency exponent that 0 < α < 1 values. The exponent $\boldsymbol{\alpha }$ presents the degree of interaction between mobile ions and the environments surrounding them. Since the σ_DC is small as compared to σ_tot in our samples under study, therefore σ_tot can be considered to be σ_AC. Figure 7a, b, c shows the obtained results of the dependence of AC conductivity as a function of frequency in the range of (1 Hz–1 MHz) for the investigated Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Ge₅ (SBG), and Se₈₃Bi₁₇Te₅ (SBT) film compositions at room temperature as well as different evaluated temperatures over a range (293–393 K) for the studied samples. It appears that the AC conductivity increases linearly with frequency for all films. Moreover, in the high-frequency range, the AC conductivity becomes independent of temperature. This increment of the AC conductivity is attributed to the hopping or tunneling of charge carriers and/or related models as the applied electric field frequency increases. Where the verification of the application of any of them depends on the values of the power $\boldsymbol{\alpha }$ and their temperature dependence. Furthermore, the incorporation of Te and Ge into the binary Se₈₃Bi₁₇ (SB) glass composition results in an increase in AC conductivity. Analyzing the results in more detail reveals that the increase in AC conductivity is more pronounced in films with Te addition compared to those with Ge addition. This observation can be attributed to the chemical bond energies between the constituent elements in the studied composition. The ternary composition with Ge addition exhibits higher bond energy, specifically the bond energy between Se (the host element) and Ge is greater than that between Se and the other additive (Te atoms). Consequently, this explains the significant increase in AC conductivity observed in Se₈₃Bi₁₇Ge₅ (SBG) compared to Se₈₃Bi₁₇Te₅ (SBT), as well as the general trend of higher AC conductivity in the additive compositions compared to the binary (host) composition.

Fig. 7

Plots of the frequency dependence of the Ac conductivity for a Se₈₃Bi₁₇ (SB) b Se₈₃Bi₁₇Ge₅ (SBG) c Se₈₃Bi₁₇Te₅ (SBT) film samples

As mentioned in the researches concerned with the aim of our study, different models have been considered based on two distinct processes namely classical hopping over a barrier and/or quantum–mechanical tunneling, or some variants or combination of the two. Moreover, it has been differently assumed that electrons (or polarons) or atoms are the carriers responsible [27]. These different models are (QMT) The quantum mechanical tunneling model, in which the exponent $\boldsymbol{\alpha }$ is temperature independent with 0.8 value or increases slightly with temperature [28, 29]. The (CBH) correlated barrier hopping model, in which the exponent $\boldsymbol{\alpha }$ decreases with the increase in temperature [30]. The (OLPT) overlapping large-polaron tunneling model, in which the exponent $\boldsymbol{\alpha }$ depends on both temperature and frequency and drops with the increase in temperature to a minimum value and then rises when the temperature increases [31]. The (NSPT) non-overlapping small polaron tunneling model, in which the exponent $\alpha$ is temperature dependent and increases with the increase in temperature [32].

In order to determine the predominant conduction mechanism of the AC conductivity for the present film samples, the appropriate model for the conduction mechanism in the light of the different theoretical models correlating the conduction mechanism of ac conductivity with $\alpha (T)$ behavior is suggested. Figure 8 represents the temperature dependence of the frequency exponent $\alpha$ for Se₈₃Bi₁₇ (SB), 1.39 eV for Se₈₃Bi₁₇Te₅ (SBT), and 0.79 eV for Se₈₃Bi₁₇Ge₅ (SBG).film samples. It is clear that the value of $\alpha$ decreases as the temperature increases for all compositions. This outcome means that the obtained experimental results can be interpreted according to the Correlated Barrier (CBH) Hopping model [33, 34]. According to the (CBH) model, the conduction of electrons occurs through thermal activation as they transfer across a barrier between two defect sites. Each site is associated with a Coulomb potential well. When two neighboring sites are close enough, their Coulomb wells overlap, resulting in an effective barrier (${U}_{M}$). The frequency dependence of the AC conductivity (σ_AC) can be explained within the framework of the CBH model as follows [35]:

$${\sigma }_{AC} =\frac{n{ \pi }^{2}{\varepsilon }_{1}N{N}_{P}\omega {R}^{6}}{24}$$

(9)

where (n) is the number of polarons involved in the hopping process, the product result (NN_P) is proportional to the square of the concentration of the states and ε₁ is the dielectric constant. R is the distance of hopping for conduction $\left(\omega \tau =1\right)$ given as [35]:

$$R =\frac{4n{ e}^{2}}{\pi {\varepsilon }_{1}{\varepsilon }_{0}[{U}_{M}+(kT ln\left(1/\omega {\tau }_{0}\right)]} $$

(10)

where ${\tau }_{0}$ is the relaxation time for the studied material, ${{\varvec{\varepsilon}}}_{1}$ is the dielectric constant (real part) and n is the no. of electrons involved in a hop (n equal to 1 and 2 for the single polaron and bipolaron processes, respectively) and ${U}_{M}$ is considered as a maximum barrier height. The effective barrier ${U}_{M}$ will reduce to $U$ which is given by [35]

Fig. 8

Temperature dependence of the exponent $\boldsymbol{\alpha }$ for the studied film samples

$$U ={U}_{M}-\frac{n{e}^{2}}{\pi {\varepsilon }_{1}{\varepsilon }_{o}R}$$

(11)

Considering, $e$ is the electronic charge and ${\varepsilon }_{o}$ is the permittivity of the free space. The frequency exponent $\alpha$ according to this model computing according to the relation [28, 35]:

$$ \alpha =1-\frac{6kT}{{U}_{M}+(kTln\left(\omega {\tau }_{o}\right))}$$

(12)

where $\mathrm{K}$ is the Boltzmann constant and $\mathrm{T}$ is the absolute temperature. At lower temperatures,${U}_{M}\gg \mathrm{KTln}(\upomega {\uptau }_{0})$ hence the value of $\mathrm{\alpha }$ is approximately given by [35]

$$\alpha =1-\frac{6kT}{{U}_{M}}$$

(13)

$$\beta =1-\alpha =\frac{6kT}{{U}_{M}}$$

(14)

The calculated ${U}_{M}$ values for the film samples are as follows: 1.99 eV for Se₈₃Bi₁₇ (SB), 1.39 eV for Se₈₃Bi₁₇Te₅ (SBT), and 0.79 eV for Se₈₃Bi₁₇Ge₅ (SBG). Interestingly, the ${U}_{M}$ values for the ternary film samples with Te and Ge addition are lower compared to the binary Se₈₃Bi₁₇ (SB) film composition. This suggests that the thermal agitation cannot significantly alter the degree of overlap of the Coulomb potential wells between the considered sites. Consequently, the addition of a third element (Te or Ge) to the binary sample enhances its AC conductivity.

Furthermore, the decrease in disorder and the subsequent increase in conductivity with the addition of a third element can be explained by considering the nature of chalcogenide glasses. In these glasses, the valence band is primarily composed of lone pair orbitals contributed by the chalcogen atoms. The energy levels of these lone pair electrons near electropositive atoms are higher compared to those near electronegative atoms. Consequently, incorporating electropositive elements into the glass matrix can elevate the energy of the lone pair states, causing the valence band to expand within the band gap. The electronegativity values of Selenium, Tellurium, and Germanium are 2.4, 2.1, and 2.01, respectively. Since both Germanium and Tellurium have lower electronegativity values compared to Selenium, replacing Selenium with these elements can increase the energy of certain lone pair states, resulting in an expanded valence band. Consequently, the presence of Germanium and Tellurium reduces the extent of band tailing in the material.

4 Temperature dependence of AC conductivity

In order to explore the effect of temperature on the AC conductivity of Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Te₅ (SBT), and Se₈₃Bi₁₇Ge₅ (SBG), film samples, the variation of lnσ_AC vs. 1000/T for the film samples at different frequency values, is shown in Fig. 9a, b and c over a working temperature range (293–393 K) at different frequency. Since, the AC conductivity exhibits temperature-dependence and obeys the relation [36]

$${\upsigma }_{\mathrm{AC}}={\upsigma }_{0}\mathrm{exp}\left[\frac{-\Delta \mathrm{E}}{\mathrm{kT}}\right]$$

(15)

where σ_o is the pre-exponential constant and ΔE_AC is the activation energy for the AC conductivity. The σ_AC (T) plot as a function of temperature shows semiconductor behavior as the AC conductivity increases by increasing temperature for all studied film compositions. In addition, these spectra, Fig. 9d, reveal that σ_AC increases with Te and Ge addition. This suggests that AC conductivity is a thermally activated process involving various localized states within the band gap or its tails. As the band gap decreases, the material's conductivity increases due to the exponential increase in thermal generation of electron–hole pairs. This phenomenon is attributed to the short-range order in the studied samples and the formation of defects, which in turn increases the density of localized states in the band tails [37]. In other words, the formation of hetero-polar Te-Se and Ge-Se bonds at the expense of the homo-polar Se-Se bond leads to a reduction in band energy, resulting in increased AC conductivity [38, 39]. Figure 10 illustrates that all film compositions exhibit a single slope across the entire frequency range studied. The slope of the straight lines allows for the easy calculation of the AC electrical conduction activation energy (ΔE_AC). Figure 10 presents the values of ΔE_AC for the Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Te₅ (SBT), and Se₈₃Bi₁₇Ge₅ (SBG) film compositions. It is evident from the figure that the activation energy decreases with increasing frequency and the addition of a third element. The decrease in ΔE_AC with increasing frequency can be attributed to the enhanced electronic jumps between localized states due to the increased applied electric field [38]. Moreover, the small value of ΔE_AC confirms that hopping conduction is the dominant mechanism at play.

Fig. 9

Plots of the AC conductivity ${\sigma }_{AC}$ vs. 1000/T for a Se₈₃Bi₁₇ (SB) b Se₈₃Bi₁₇Ge₅ (SBG) c Se₈₃Bi₁₇Te₅ (SBT) film samples and (d) comparison between the studied film samples at different frequency

Fig. 10

The plot of AC activation energy ${\Delta E}_{AC}$ vs. $\mathrm{ln}(\omega )$ for the studied film samples

5 Theoretical analysis

The objective of the present study was to explore the applicability of the suggested models, that is, model 1 (ANN) and model 2 (GP) for the simulation and prediction of AC conductivity using experimental data for samples of Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Te₅ (SBT), and Se₈₃Bi₁₇Ge₅ (SBG). AC conductivity was developed with temperatures at different frequencies and with frequency at different temperatures using two groups of input parameters. The first group of input parameters included Temperatures and frequency (0.5, 1, 5, 10, 50, 250, and 700 kHZ) were applied in artificial neural networks (ANN1, ANN2, and ANN3) and genetic programing (GP1, GP2, and GP3) which used data only from SB, SBG, and SBT samples respectively. The second group of input parameters included $\mathrm{ln}\omega$ and temperatures (293, 313, 333, 353, 373, and 393 K) were applied in artificial (ANN4, ANN5, and ANN6) and genetic programing (GP4, GP5, and GP6) which used data only from Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Te₅ (SBT), and Se₈₃Bi₁₇Ge₅ (SBG) respectively. The output of the two groups was AC conductivity. These techniques were coded and run in a MATLAB software environment.

For model 1, Data collected from the experiments are divided into three sets, namely, 70% training set, 15% testing set, and 15% validation set. The training set is used to train the model. The validation dataset is used to confirm the accuracy of the proposed model. It ensures that the relationship between inputs and outputs, based on the training and test sets is real. Six networks (ANN1-ANN6) were selected and trained separately to simulate AC conductivity. The six proposed supervised feed-forward network of three layers was trained using the Levenberg–Marquardt optimization technique for different epoch numbers and the number of membership functions assigned to each ANNstructure. The best ANNmodel setting will be selected based on a comparison of MSE, R², ${e}_{i}$, $\overline{{e }_{i}}$, and Std error values.

For the best AC conductivity simulation (ANN1—ANN6), we used one hidden layer with 6, 6, 5, 8, 8, and 8 neurons respectively. The function used was tensing in the hidden layer and purlin in the output layer. Figure 11 shows the architecture of the ANN1 developed in this study, including the input layer, hidden layers, and output layer. Also, Table 1 includes Structure performance evaluation of the ANNs implemented in the designated AC conductivity.

Fig. 11

The optimized ANN1 architecture

Table 1 Structure performance evaluation of the ANNs implemented in the designated of AC conductivity

The training, validation, and testing for different ANN configurations (ANN1-ANN6) are shown in Fig. 12a, b, c, d, e, and f respectively. Figure 12 shows how the mean squared error (MSE) reduces during training, validation, and testing as the number of epochs increases. The learning improves with the number of epochs for the training, validation, and testing. Validation occurs before the number of epochs reach. This eventually culminates in the optimal performance during validation at 48, 328, 126, 20, 8, and 148 epochs respectively having approximately zero for MSE, MAE,$\overline{{e }_{i}}$ and Std values. This behavior shows that the network learns well as the number of epochs increase. The testing session shows an acceptable MSE, ${e}_{i}$, $\overline{{e }_{i}}$, and Std error which are very close to zero as well. Also, R² values are very close to 1. Indicating a reasonably good fit of the model. MSE,${e}_{i}$, $\overline{{e }_{i}}$, Std error, and R² values for the training, validation, and testing for different ANN configurations are shown in Table 1. To make the outcomes of the model reproducible, the equation and matrices of the weights and biases for six nets of all layers of the optimized trained model are presented in appendix.

Fig. 12

a–f Variations of the MSE values with number of training (epochs) during the ANN training process for simulation AC conductivity

For model 2, Genetic programing was used to obtain the optimum parameters to simulate and predict AC conductivity. In this regard, The GP is implemented using the experimental data to produce six different models, GP1, GP2, GP3, GP4, GP5 and GP6 that will simulate and predict AC conductivity at the given range of temperatures and frequency. GP divided data collected from the experiments into two sets, namely, 80% training set and 20% testing set. The simulated evolution process used the following evolutionary parameters: the size of the organism population shifts from 350 to 900, the greatest number in a generation 70, the mutation rate 0.01, crossover rate 0.9 and the function genes of addition ( +), subtraction (−), multiplication (*), division (/), square root (sqrt), sin, cos and tan were used (see Table 2). The GP was running until the fitness function reduced to an acceptable level. The fitness function values for GP1, GP2, GP3, GP4, GP5 and GP6 were 3.0099, 4.7377, 3.7088, 4.6856, 4.6398 and 3.2643 respectively, MAE,$\overline{{e }_{i}}$ and Std values are nearly zero. Also The correlation coefficient (R²) of this fit were 0.9995, 0.9983, 0.9991, 0.9949, 0.9985 and 0.9996 respectively which were high values that represents an acceptable fitting (see Table 2). An equivalent equation is generated using the obtained control GP parameters.

Table 2 Shows our obtained optimal GP control parameters

According to GP1 to GP6 the equivalent equation generated are given as:

$$\mathrm{GP}1 ={ J}_{1}-\frac{10}{{x}_{2}}+{J}_{2}$$

(16)

$${J}_{1}={a}_{1}+sin{a}_{2}-0.245{a}_{3}$$

$${J}_{1}={a}_{5}-0.092$$

$${a}_{1}=cos{\left({\left(10-{x}_{2}\right)/ x}_{2}\right)}^{1/2}$$

$${a}_{2}=\mathrm{sin}(10\mathrm{cos}\left(0.6 sin{x}_{2}\right)-0.0799\mathrm{cos}\left(0.94{ x}_{1}-{x}_{2}\right)$$

$${a}_{3}={\left({x}_{2}+0.1\mathrm{tan}\left(\mathrm{sin}\left(2{x}_{2}\right)\right)/\mathrm{tan}\left(10-{x}_{2}\right)+10\right)}^{1/2}$$

$${a}_{4}=-\mathrm{cos}\left(0.932-{x}_{1}^{2}\right)+1.11{\left(\frac{{x}_{2}\left({x}_{1}+\mathrm{sin}\left(10-{x}_{1}\right)\right)}{4.7{x}_{2}+\mathrm{cos}\left(1.05{x}_{2}\right)}\right)}^{\frac{1}{2}}+10$$

$${a}_{5}={a}_{4}/10\sqrt{{x}_{2}}+(12.6-0.0755{{(x}_{2}+10)}^{1/2})$$

$$\mathrm{GP}2=tan \left({k}_{1}\right)-{k}_{2}^{\frac{1}{2}}$$

(17)

$${k}_{1}=s\mathrm{in}(\mathrm{tan}\left(\mathit{sin}\left(0.765-\left({b}_{1}+77.57{)}^{\frac{1}{2}}\right)\right)\right)$$

$${k}_{2}=\mathrm{cos}\left(\frac{0.578}{-{b}_{2}+{b}_{3}+0.95}\right)-0.32{(tanx}_{2}-{x}_{2})$$

$${b}_{1}=({x}_{1}+{x}_{2})/{\left(\frac{\mathrm{tan}\left({x}_{1}+0.707\mathrm{sec}\left({x}_{1}\right)+{x}_{2}\right)}{10-{x}_{1}}+{\left({x}_{2}\right)}^{1/2}\right)}^{1/2}$$

$${{b}_{2}=\mathrm{sin}10-\mathrm{sin}(x}_{1}-tan{x}_{2}+\mathrm{tan}\left(tan{x}_{2}+0.534\right)+10.57{)}^{1/2}$$

$${b}_{3}= {x}_{2}tan{x}_{2}({tanx}_{2}-{x}_{2})/{x}_{1}^{2}\left({tanx}_{2}+{x}_{2}\right)$$

$$\mathrm{GP}3=\mathrm{sin}{ t}_{1}+{t}_{2}$$

(18)

$${t}_{1}=1.58 \left({c}_{1}+\sqrt{{ (x}_{1}-\frac{{x}_{2}}{{c}_{3}})} \right){ x}_{2}^{\frac{1}{4}}$$

$${t}_{2}{=-1.5x}_{1}^{\frac{1}{4}}-{x}_{2}^{\frac{1}{4}}+2.497sin{x}_{2}-11.47$$

$${c}_{1}=\mathrm{sin}(2.34-10/\left(2{x}_{2}+sin{x}_{2}+0.929\right)\mathrm{tan}(cos{x}_{1}/(0.994\sqrt{{x}_{1}}+1.1)$$

$${c}_{2}=\left(\sqrt{{x}_{1}}\left({x}_{1}+0.65\right)+\mathrm{sin}\left(\mathrm{cos}\left(-{x}_{1+}\sqrt{{x}_{1}}+10\right)\right)\right)/{x}_{2}+0.821$$

$${{c}_{3}=x}_{2}-\mathrm{sin}\left(tan10\right)\mathrm{sin}({c}_{2}-{\left({x}_{1}{x}_{2}\right)}^{\frac{1}{2}} )-({x}_{1}-\frac{{x}_{2}}{{x}_{2}+sin{x}_{2}}{)}^{\frac{1}{2}}+3.6{ x}_{2}^{\frac{1}{4}}$$

$$\mathrm{GP}4= sin\left(10{ x}_{1}^{1/4}\right){ f}_{1}^{\frac{1}{2}}-{f}_{2}$$

(19)

$${f}_{1}={\mathrm{d}}_{1}{-\mathrm{d}}_{2}-{\mathrm{d}}_{3}$$

$${\mathrm{d}}_{1}=10-\mathrm{tan}\left(\mathrm{sin}\left(\mathrm{sin}\left(10{x}_{1}^{1/4}-{x}_{1}+0.9663\right)\right)\right)$$

$${\mathrm{d}}_{2}=\left(-tan\left( 3.16cos\left(\frac{10}{{x}_{2}}-cos\left({x}_{1}\right)\right) -\frac{0.83}{cos{x}_{1}}-8.02\right)\right)/(4.622-{x}_{1})$$

$${\mathrm{d}}_{3}=\mathrm{tan}(\mathrm{sin}\left(1.1955\mathit{sin} \{\mathit{sin}\sqrt{0.37\times 2.54{x}_{1}}\}\right)-10)$$

$${f}_{2}=9.9+\frac{0.8304}{116.22-{x}_{1}}$$

$$\mathrm{GP}5=\mathrm{ sin}M+( {x}_{1}+sin\sqrt{{x}_{1}}+10.174 )$$

(20)

$$M=(\sqrt{{k}_{1}+{k}_{2}+{k}_{3}}-10)$$

$${{k}_{1}=(\{x}_{1}+\mathrm{sin}(\mathrm{sin} \left({x}_{1}+\mathrm{sin}[( 0.185+sin\sqrt{{x}_{1}})+\mathrm{sin}(\mathrm{sin}(sin\sqrt{{x}_{1} }\right))]+\mathrm{sin}[\left({x}_{1}+0.174\right)+\mathrm{sin}(cos{x}_{1}+0.174)+0.174)]+sin\sqrt{{x}_{1}})\}+sin\sqrt{{x}_{1}}+sinsin\sqrt{{x}_{1}})$$

$${{k}_{2}=\mathrm{sin}\left(0.174+{x}_{1}\right)+\mathrm{sin}[((\mathrm{cos}(tanx}_{2}-(0.17+sin{x}_{1})/0.174sin{x}_{1})+\mathrm{sin}(\mathrm{sin}({x}_{1}\mathrm{sin}\left(\mathrm{tan}\left(\mathrm{tan}\left({x}_{1}\right)\right)\right))]$$

$${k}_{3}=\mathrm{sin}\left\{\frac{tantan{x}_{1}}{\sqrt{{x}_{2} }\mathrm{tan}{x}_{1}(\mathrm{sin}\left(0.174+\mathrm{sin}\sqrt{{x}_{1}}\right))}+\mathrm{sin}\left(\mathrm{sin}\left(\mathrm{sin}\left(\sqrt{{x}_{1}}\right)\right)\right)+\mathrm{sin}[\mathrm{sin}\left({x}_{2}/0.9\right)+0.174]+sin\sqrt{{x}_{1}})\right\}+0.174)+2sin\sqrt{{x}_{1}})$$

$$\mathrm{GP}6={L}_{1}+\mathrm{cos}\left({L}_{2}\right) (21)$$

$${L}_{1}={h}_{1}+{\left(10/{x}_{2}^{3}\right)}^{1/2}+{x}_{1}+8.45$$

$${L}_{2}=cos\sqrt{{h}_{2}-{h}_{3}}$$

$${h}_{1}=\mathrm{sin}{\left({x}_{1}-sin\sqrt{{x}_{2}}/(\mathrm{sin}\left(\mathrm{cos}\left(\sqrt{cos{x}_{1}\times 0.316sin{x}_{2}}\right)\right))\right)}^{1/2}$$

$${h}_{2}={x}_{1}- \left(sin\sqrt{{x}_{2}}\right)/{\left(2.29/x1\right)}^{1/2}$$

$${h}_{3}=\left(\mathrm{sin}\left(\sqrt{{x}_{1}}-\frac{0.176}{{x}_{1}}+{x}_{2}+\frac{10}{\sqrt{{x}_{2}}}\right)\right)/\left(10 - \frac{{x}_{1}}{3.16}\right)$$

Finally, the comparison between ANN and GP with experimental data is presented in Figs. 13 and 14). According to these figures, GP results show good results for AC conductivity at different frequency for samples of Se₈₃Bi₁₇ (SB) and Se₈₃Bi₁₇Te₅ (SBT) (Fig. 13a, c) with slightly deviation from experimental data for sample of Se₈₃Bi₁₇Ge₅ (SBG) (Fig. 13b). Furthermore, GP results show good results for AC conductivity with temperatures for samples SBG and SBT (Fig. 14a and c) with slightly deviation from experimental data for sample SB (Fig. 14b) but ANN results show good simulation and prediction for all samples at different temperatures and frequency. Also By comparing the values of MSE, MAE,$\overline{{e }_{i}}$, Std, fitness function and correlation coefficient (R²) of the two models (see Tables 1 and 2). So both the ANN and the GP results were a good fit with the experimental data but the ANN model results were very near to the experimentally values for all samples (SB, SBG and SBT) at different frequency and temperatures as compared to GP model results. Moreover, the values of slopes and ΔE_AC for the film compositions of Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Te₅ (SBT), and Se₈₃Bi₁₇Ge₅ (SBG) were calculated using ANN and GP as shown in Table 3. It has been clearly observed that model 1 is more reliable and provides more accurate simulation and prediction for the AC conductivity dataset as well. It can be concluded that ANN model is the most reliable technique for the purpose.

Fig. 13

a–c Deviation between experimental data (symbols) and the simulated and predicted values by using ANN (solid lines) and GP (dash lines) for SB sample

Fig. 14

a–c Deviation between experimental data (symbols) and the simulated and predicted values by using ANN (solid lines) and GP (dash lines) for SB sample

Table3 Comparison of slope and activation energy at different frequency using ANN and GP models

6 Conclusion

In conclusion, we report here the AC conductivity for glassy Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Ge₅ (SBG), and Se₈₃Bi₁₇Te₅ (SBT) in thin film form. The composition, temperature, and frequency dependence of AC conductivity σ_AC (ω) are investigated in the frequency range of 100–1,000,000 Hz and the temperature range of 293–393 K. The analysis of AC conductivity data are attributed to the power law ${\omega }^{2}$, where $\alpha$ has values less than unit and decreases with increasing temperature which is in good agreement with the correlated barrier hopping (CBH) model. The AC activation energy is calculated from the temperature dependence of AC conductivity and studied as a function of frequency. The addition of a third (Te and Ge elements) to the binary of Se₈₃Bi₁₇ (SB) chalcogenide compositions has been found enhancement the AC conductivity values. On the comparative analysis of GP and ANN techniques, used for the simulation and prediction of AC conductivity, it can be concluded that ANN model is the most reliable technique for the purpose. The MSE, MAE,$\overline{{e }_{i}}$ and Std values that were so acquired are low enough to suggest that the estimations are quite accurate and the trained networks produce better outcomes. According to statistics, there is a strong connection between simulated, predicted, and experimental values for the data present in the dataset if a given model yields R² > 0.9. Both of the models can be used for simulation, as shown by the fact that R² for both models is greater than 0.9 in every scenario. Also, slope and activation energy were calculated using two models. As a result, the simulation model employing model 1, or the ANN model, reveals a high level of stability with the employed AC conductivity, which was validated experimentally. As a result, ANN can work well as a simulation and prediction tool.

Appendix

The equation which describes AC conductivity for Se₈₃Bi₁₇ (SB), Se₈₃Bi₁₇Te₅ (SBT), and Se₈₃Bi₁₇Ge₅ (SBG) is given in appendix.

ACconductivity = purelin[net.LW{2,1}tansig(net.IW{1,1}R + net.b{1)) + net.b{2}] (16).

Where.

R is the inputs (T, frequency for group 1 and $\mathrm{ln}\omega$, $T$ for group 2).

IW and LW are the linked weights as follow:

net.IW{1,1} is linked weights between the input layer and hidden layer,

net.LW{1,1} is linked weights between the hidden layer and output layer,

b is the bias and considers as follow:

net.b{1) is the bias of hidden layer,

net.b{2) is the bias of output layer.

The values of weights and bias for ANN1 were given as follows:

$$IW=\left(\begin{array}{cc}-4.0363844121345389& -3.5866542873245844\\ 1.5386796608769016& -1.7618371249908655 \\ 0.031255223115834924& -56.391138188369311\\ 1.5129205066024167& 11.328612934648993\\ 9.8465596394726802& -0.26981293826982383\\ 0.18049131672169411& 6.7734678994908295\end{array}\right)$$

$$LW=\left(\begin{array}{ccccccc} & 0.0059155048& -0.091191638& -13.752900& 0.147764976& 0.04662287238& 0.532235988308\end{array}\right)$$

$$b1=\left(\begin{array}{c}2.8844898172941917\\ -1.5463905807080229\\ -58.217358600951634\\ 1.618186335566782\\ 9.7608029246862973\\ 6.1568483508090415\end{array}\right)$$

$$b2=\left(-13.576343187104296\right)$$

The values of weights and bias for ANN2 were given as follows:

$$IW=\left(\begin{array}{cc}-0.23573190291274565 & 3.6134272439233976\\ -0.044332570656930019& -72.316315136743157\\ -0.36785108812507289& 26.067562546936319\\ -4.6662766104414093& 0.50833131075994131\\ -0.071595151172195143& 3.8583696073517393\\ 12.919200507989929& -3.0548732889788424\end{array}\right)$$

$$LW=\left(\begin{array}{cccccccc} & [-12.996559& -18.841086& & 0.1155407877& 0.01818624& 8.33130658& -0.008337890778\end{array}\right)$$

$$b1=\left(\begin{array}{c}5.9902486831803117\\ -74.272223154486383\\ 2.5745415571009218\\ -2.7935211749595319\\ 5.1455782395622114\\ 0.016998161395118633\end{array}\right)$$

$$b2=\left(-13.317383037602326\right)$$

The values of weights and bias for ANN3 were given as follows:

$$IW=\left(\begin{array}{cc}-1.1964614788021359 & -5.5696069776858836\\ 0.17188996644165586& -1.7984559073323447\\ -0.035083210023468724& 58.963284099002557\\ -2.3152214461143181& -1.1927200973910737\\ 0.071941029745401092& -2.8350297246859499\end{array}\right)$$

$$LW=\left(-0.15334088\begin{array}{cccccc} & 5.3815297831376405& & 14.215762058& 2.1577735336& -3.58593258856\end{array}\right)$$

$$b1=\left(\begin{array}{c}3.2082555995475954\\ -3.2925671169336321\\ 60.856466356344477\\ -5.5655898779032267\\ -3.4954958824531923\end{array}\right)$$

$$b2=\left(-9.4221755241923599\right)$$

The values of weights and bias for ANN4 were given as follows:

$$IW=\left(\begin{array}{cc}-0.67285275030879788 & -3.9988524263618079\\ 3.9478915318496184 & -0.11900043083820215\\ -2.5480786628008856& 0.33924572299714723\\ 2.3690940748803855& -2.8125842538967221\\ -2.2778752396704105& 0.68286222944189912\\ -0.45407400139095311& -0.051559073090047765\\ 2.2958633185022301& -4.0959616727312413\\ -1.0327972020098268& -3.28537076162536\end{array}\right)$$

$$LW=\left(\begin{array}{ccccccccc} & 0.06701825& 0.181457& -0.2026518& -0.023117944& -0.1539017& -1.758953& -0.04536& 0.1721845\end{array}\right)$$

$$b1=\left(\begin{array}{c}3.8763131634895185\\ -3.766060963250859\\ 1.0567823876409825\\ -1.0640123551277594\\ -0.81987701917795153\\ -0.61158973969963404\\ 2.1043557890775166\\ -4.7632616777278614\end{array}\right)$$

$$b2=\left(-0.59384604165837085\right)$$

The values of weights and bias for ANN5 were given as follows:

$$IW=\left(\begin{array}{cc}2.2783171235927169 & 3.2578596216783762\\ 2.4827021421188915& -0.011547810408120206\\ -2.5119375817049923& -3.3223621360320079\\ 1.4712458898126706& 0.16384927313920733\\ -1.755292847649444& 1.2801498285887769\\ -1.7927208526427649& -2.4093017732385924\\ -2.577928404148611& -0.79385296023117902\\ 2.78198064913592 & 3.0866155532460251\end{array}\right)$$

$$LW=\left(\begin{array}{ccccccccc} & 0.0073297& 0.3770939& -0.0009743629& 0.7774563& -0.06578039& 0.0434803& -0.29379& -0.065340\end{array}\right)$$

$$b1=\left(\begin{array}{c}-3.923877127706572\\ -2.7737031550012161\\ 1.3940385314583073\\ 0.15606943393705788\\ -0.86706202770672802\\ -1.4668236345532597\\ -2.6469236583437383\\ 4.1324311098249558\end{array}\right)$$

$$b2=\left(0.12549152307853598\right)$$

The values of weights and bias for ANN6 were given as follows:

$$IW=\left(\begin{array}{cc}4.7928650359960834 & 2.1314399192156581\\ 2.742053153708206& -2.2212430265117495\\ -0.56604704217494883 & 0.59284038194813238\\ 0.80750456017048711& -0.64625798623347153\\ 0.25889013740583539& -3.1569592450255559\\ 0.26462180715214156& 0.23299496795179733\\ -1.2206932126253511& 1.7281267494016024\\ 3.9233073881819096& 0.59081452023058456\end{array}\right)$$

$$LW=\left(\begin{array}{ccccccccc} & 0.008431979& 0.027515268& -1.0565932& 0.477397428& 0.0321473& 2.08108& -0.098143& -0.011551\end{array}\right)$$

$$b1=\left(\begin{array}{c}-4.3324769280078979\\ -2.6453250918355677\\ 1.7405400469416308\\ -0.082896355843215599\\ 0.58340569262326736\\ -0.086211703081668076\\ -1.6992569701588032\\ 3.8105344132058252\end{array}\right)$$

$$b2=\left(1.0932422659976699\right)$$