Computational energy gap estimation for strontium titanate photocatalyst using extreme learning machine method

Abstract

Strontium titanate is a functional ceramic material with unique optical properties and chemical stability which facilitate its wider applicability as photocatalyst for solving energy crisis as well as environmental challenges, oxide thin film substrate and for manufacturing special oxygen sensors (high temperature) among others. Efficient utilization of strontium titanate as photocatalyst and opto-electronic device requires extension of light harvesting capacity beyond visible region through foreign material incorporation by doping mechanisms which is experimentally demanding and time consuming. This work models band gap of strontium titanate magnetic photocatalyst through extreme learning machine (ELM) using the crystal distortion (as a result of dopant incorporation) and crystallite size as predictors. The developed ELM-based models with sigmoid (Sig) and triangular basis (Tranba) activation function perform excellently better than the existing stepwise regression algorithm (SRA) model in the literature using different performance measuring parameters which include the coefficient of correlation (CC), mean absolute error and root mean square error. In the training developmental stage, the developed Sig-ELM model outperforms the developed Tranba-ELM and the existing SRA (2021) model with performance improvement of 2.96% and 67.37%, respectively, on the basis of CC performance metric, while similar performance improvement was obtained during testing phase of model development using different performance metrics. The superiority in the performance of the present models as compared to the existing model strengthens the potentials of the developed model in adjusting and extending light harvesting ability of strontium titanate semiconductor for various technological and industrial applications.

Keywords:

• extreme learning machine
• strontium titanate
• crystallite size
• lattice parameter
• energy gap

1. Introduction

Perovskite materials have received increasing attention lately as photocatalysts due to their suitable/adjustable energy gap coupled with superior chemical as well as physical properties (Z. Liu et al., 2018; Mohamed & Hernando, 2018; Tasleem & Tahir, 2020). Strontium titanate stands out among other perovskite in photocatalysis applications due to the adjustable nature of its band gap, excellent thermal stability, insulation resistance, dielectric properties and high energy storage ability among others (Olatunji, 2021). The semiconductor is characterized with excellent physical and chemical properties coupled with structural stabilization and chemical composition which translate to excellent thermal conductivity, corrosion resistance, light resistance and ease of incorporating other materials through doping (Duan et al., 2017). However, inability of the semiconductor to absorb and utilize sunlight energy beyond visible region due to its wide energy gap calls for serious concerns in addressing environmental challenges and energy crisis which are the main central goals of photocatalysis applications (Abedi et al., 2023). Opto-electronic and photocatalytic applications of strontium titanate semiconductor call for the need to engineer and tune energy gap to fit the desired applications (Q. Liu et al., 2011). Limited light absorption capacity of strontium titanate, weak absorption at the surface and rapid recombination of electrons and holes are among the major drawbacks of the semiconductor for many applications. Foreign material doping techniques have been widely employed in the literature for energy gap tailoring and engineering (Anitha & Devi, 2020). Metal and non-metal incorporation reveals photocatalytic enhancement under visible light irradiation with strong potential in introducing new energy levels which consequently narrows the energy band gap (Ur et al., 2021). Visible light harvesting capacity of strontium titanate semiconductor can also be tuned through transition metal elements doping for energy gap engineering (S. Wang et al., 2022). The conduction and valence band of strontium titanate are represented by Ti 3d and O 2p, respectively. The potential of oxygen vacancies in creating defect levels near the conduction band has been extensively demonstrated (Kan et al., 2005; Tan et al., 2022). Dopant orbital hybridization with 2p oxygen states of the semiconductor results into energy gap narrowing with enhanced photocatalytic activity. Formation of valence band by dopant energy level which is located at the valence band top due to 2p orbital further lowers the energy gap of the semiconductor (Srtio et al., 2012). This work employs novel extreme learning machine (ELM) approach in adjusting energy gap of strontium titanate semiconductor for energy harvesting capacity enhancement.

Strontium titanate photocatalyst is a metal oxide with known highest electron mobility. Contraction in strontium titanate semiconductor crystallography occurs when strontium cations are replaced and substituted with ions of reduced ionic radii. Similarly, substitution of Ti⁴⁺ with cations of reduced ionic radii further leads to lattice contraction (Yang et al., 2009). Oxygen vacancy creation in strontium titanate semiconductor completely modifies physical and chemical features of the semiconductor for many technological applications such as photocatalysis and nuclear power (Kumar et al., 2021). Presence of oxygen vacancies has potentials to alter photo-absorption capacity of the semiconductor for photocatalytic activity enhancement (Owolabi et al., 2022). The number of electron spins that are unpaired at defect levels controls the intrinsic magnetic intensity of un-doped oxide as well as the absorption photon energies. Hence, the impurity band associated with oxygen vacancy creation extends the applicability of strontium titanate semiconductor opto-magnetic devices to visible light region range. Strontium titanate crystallizes in cubic perovskite structure with pm3m space group. The crystal structure contains Ti⁴⁺ ions with six-fold O²⁻ ions coordination, while TiO₆ octahedra surround every strontium Sr²⁺ ions. This means that there are 12 O²⁻ ions co-coordinating each of strontium Sr²⁺ ions. Covalent bonding exists due to hybridization of Ti-3d states with O-2p states within octahedra TiO₆, while O²⁻ and Sr²⁺ ions show the characteristics of ionic bonding. Therefore, strontium titanate has a mixed covalent and ionic bonding feature (Ur et al., 2021). This unique chemical bonding feature translates to interesting properties exhibited by strontium titanate for various industrial and technological applications. This work establishes a relationship between the crystallite size, lattice distortions and energy gap of strontium titanate using ELM.

ELM algorithm belongs to a class of intelligent method within the domain machine learning tools with potentials to draw patterns connecting descriptors with the desired target using feed-forward networks of neurons with characteristic single layer (Sulaiman et al., 2022; An et al., 2017). Random selection of input weights and biases circumvent major challenges of single-layer feed-forward neural networks that adjust the weights iteratively with consequent slow convergence, local solution convergence and high degree of sensitivity to learning rate (Hua et al., 2022; Pi & Lima, 2021). Higher speed and efficiency demonstrated by ELM-based model further strengthen the algorithms for deployment to pattern acquisition in real-life problems in material science and for engineering materials properties to perfectly fit the need for specific applications (Gao et al., 2022; Gong et al., 2021; Li et al., 2020; Models, 2020; Y. Wang et al., 2020). The energy band gaps estimated by the proposed ELM-based model are closer to the experimental values in comparison with the existing model due to the empirical structural background of ELM-based models in addressing non-linear pattern acquisition between crystallite size, lattice parameter and energy band gap.

2. Mathematical background and computational strategy

Mathematical description of ELM and computational methodology are contained here. Description of dataset employed for pattern acquisition is also presented.

2.1. ELM background

ELM addresses single-hidden layer networks of neurons through random selection of its input weights and biases (Pang et al., 2020; Zhao & Chen, 2022). As such, higher efficiency and fast learning speed are achieved by the algorithm (Z. F. Liu et al., 2020). Implementation of empirical risk error minimization principle and computation of output weights using Moore-Penrose inverse approach are among the key factors in ELM algorithm which strengthen the establishment of mathematical connection between descriptors (crystallite size and lattice parameter) and target (energy band gap) (Shamsah & Owolabi, 2020). Consider a network of a single hidden layer is to be employed for modeling energy gap ${\xi }_g$ of $\mathit{\gamma }$ number of semiconductor samples $\left({\varphi }_j,{{\xi }_g}_{\mathit{j}}\right)$ where $\mathit{\varphi }$ stands for lattice parameter and crystallite size of the semiconductor samples. The input descriptors are defined as ${\varphi }_j={\left[{\mathit{\varphi }}_{\mathit{j}1},{\varphi }_{\mathit{j}2},\dots ,{\varphi }_{\mathit{jm}}\right]}^{\mathit{T}}\in {\mathbb{R}}^m$, while the output descriptors are defined as ${{\xi }_g}_{\mathit{j}}={\left[{{\xi }_g}_{\mathit{j}1},{{\xi }_g}_{\mathit{j}2},\dots ,{{\xi }_g}_{\mathit{jn}}\right]}^{\mathit{T}}\in {\mathbb{R}}^n$. $\mathit{\lambda }$-hidden layer nodes of the network with a single hidden layer are expressed as presented in equation (1)(1) $\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)={\xi }_j,\mathit{i}=1,\dots ,\mathit{M}$(1) (Pi & Lima, 2021).

(1) $\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)={\xi }_j,\mathit{i}=1,\dots ,\mathit{M}$(1)

where $\mathit{\psi }\left(\mathit{\varphi }\right)$, ${\eta }_j={\left[{\mathit{\eta }}_{\mathit{j}1},{\eta }_{\mathit{j}2},\dots ,{\eta }_{\mathit{jm}}\right]}^{\mathit{T}}$ and ${\beta }_j$, respectively, stand for the activation function, input and output weights, while ${\delta }_j$ represents the hidden layer unit bias. The dot product between ${\eta }_j$ and ${\varphi }_i$ is expressed as ${\eta }_j\bullet {\varphi }_i$. ELM algorithm aims at optimizing equation (2)(2) $\sum _{\mathit{i}=1}^M||{{\mathit{\xi }}^{\mathit{pred}}}_{\mathit{j}}-{\xi }_j||=0$(2) for error minimization so that ${\beta }_j$, ${\varphi }_i$ and ${\delta }_j$ satisfy equation (3)(3) $\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)={\xi }_j,\mathit{i}=1,\dots ,\mathit{M}$(3) (Han & Ghadimi, 2022).

(2) $\sum _{\mathit{i}=1}^M||{{\mathit{\xi }}^{\mathit{pred}}}_{\mathit{j}}-{\xi }_j||=0$(2)

(3) $\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)={\xi }_j,\mathit{i}=1,\dots ,\mathit{M}$(3)

Matrix presentation of equation (3)(3) $\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)={\xi }_j,\mathit{i}=1,\dots ,\mathit{M}$(3) is expressed in equation (4)(4) $\mathit{H\beta }={\xi }^{\mathit{pred}}$(4)

(4) $\mathit{H\beta }={\xi }^{\mathit{pred}}$(4)

where $\mathit{\beta }$ = output weight, ${\xi }^{\mathit{pred}}$ = predicted energy gap and $\mathit{H}$ = hidden layer node (output) which can be expressed as presented in equation (4)(4) $\mathit{H\beta }={\xi }^{\mathit{pred}}$(4)

(5) $\mathit{H}\left({\eta }_1,..,{\eta }_{\lambda },{\delta }_1,..,{\delta }_{\lambda },{\varphi }_1,..,{\varphi }_{\lambda }\right)={\left[\begin{array}{c}\psi \left({\eta }_1\bullet {\varphi }_1+{\delta }_1\right)\\ \mathit{\psi }\left({\eta }_1\bullet {\varphi }_M+{\delta }_1\right)\end{array}\begin{array}{c}\cdots \\ \cdots \end{array}\begin{array}{c}\mathit{\psi }\left({\eta }_{\lambda }\bullet {\varphi }_1+{\delta }_{\lambda }\right)\\ \mathit{\psi }\left({\eta }_1\bullet {\varphi }_M+{\delta }_{\lambda }\right)\end{array}\right]}_{\mathit{Mx\lambda }}$(5)

The mathematical expressions of output weight and predicted energy gaps in matrix form are shown in equation (6)(6) $\mathit{\beta }={\left[\begin{array}{c}{\beta }_1^T\\ ⋮\\ {\beta }_{\lambda }^T\end{array}\right]}_{\mathit{\lambda xn}}$(6) and equation (7)(7) ${\xi }^{\mathit{pred}}={\left[\begin{array}{c}{\xi }_1^T\\ ⋮\\ {\xi }_M^T\end{array}\right]}_{\mathit{Mxn}}$(7) , respectively.

(6) $\mathit{\beta }={\left[\begin{array}{c}{\beta }_1^T\\ ⋮\\ {\beta }_{\lambda }^T\end{array}\right]}_{\mathit{\lambda xn}}$(6)

(7) ${\xi }^{\mathit{pred}}={\left[\begin{array}{c}{\xi }_1^T\\ ⋮\\ {\xi }_M^T\end{array}\right]}_{\mathit{Mxn}}$(7)

Training of ELM algorithm by lattice parameters and the crystallite size for semiconductor energy gap determination involves determination of $\widehat{{\eta }_j}$, $\widehat{{\delta }_j}$ and $\widehat{{\beta }_j}$ such that equation (8)(8) $||\mathit{H}\left(\widehat{{\eta }_j},\widehat{{\delta }_j}\right),\widehat{{\beta }_j}-\mathit{\xi }||={min}_{\mathit{\eta },\mathit{\delta },\mathit{\beta }}||\mathit{H}\left({\eta }_j,{\delta }_j\right),{\beta }_j-{\xi }^{\mathit{pred}}||$(8) is satisfied.

(8) $||\mathit{H}\left(\widehat{{\eta }_j},\widehat{{\delta }_j}\right),\widehat{{\beta }_j}-\mathit{\xi }||={min}_{\mathit{\eta },\mathit{\delta },\mathit{\beta }}||\mathit{H}\left({\eta }_j,{\delta }_j\right),{\beta }_j-{\xi }^{\mathit{pred}}||$(8)

Equation (9) minimizes the loss function among $\mathit{j}=1,\dots .,\mathit{\lambda }$ available neurons,

(9) $\mathit{\chi }=\sum _{\mathit{i}=1}^M{\left(\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)\right)}^2$(9)

Finally, the output weight is computed from equation (9)(9) $\mathit{\chi }=\sum _{\mathit{i}=1}^M{\left(\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)\right)}^2$(9) using Moore-Penrose inverse matrix ($H^{\ast }$) of hidden layer output node $\mathit{H}$ as presented in equation (10)(10) $\hat{\beta }=H^{\ast }{\xi }^{\mathit{pred}}$(10) .

(10) $\hat{\beta }=H^{\ast }{\xi }^{\mathit{pred}}$(10)

2.2. Data description and computational methodology adopted for building ELM-based models

Patterns and connections joining the energy gap of strontium titanate semiconductors with the lattice parameter after dopant inclusion and crystallite size were extracted from 39 different compounds of strontium-titanate-based materials. The details and experimental description of the dataset are explicitly expressed in the literature (Anitha & Devi, 2019; Atkinson et al., 2019; Jia et al., 2010; Kang & Park, 2013; Kumar et al., 2021; Nunocha et al., 2021; Padmini & Ramachandran, 2019; Xu et al., 2014). The ELM model that predicts energy gap of doped strontium titanate semiconductor was developed within MATLAB computing environment. Randomization of dataset was initialized and subsequently followed by dataset separation into training and testing phase in the ratio of 4:1. Randomization process facilitates efficient, even and uniform distribution of data points in such a way that the trends of the pattern acquired during the training phase are generalizable for model testing and validation. The parameters that influence and control the precision as well as the accuracy of ELM model include the number of hidden node and the choice of activation functions. The number of hidden node was optimized using a grid search approach (X. Liu et al., 2021; Yan et al., 2022) for each of the selected activation function among the pools of the available functions such as sine function, sigmoid function and triangular basis function. Since limited number of parameters control the precision of ELM-based model, the choice of evolutionary (Karagoz & Yildiz, 2017; Pholdee et al., 2017; Yildiz & Yildiz, 2018; Sabri et al., 2013; Samala & Kotapuri, 2018) and manual search becomes inconsequential. Stepwise procedures detailing the computational methodology are itemized as follows:

Step 1: Initialization of seedling for random number generation and preservation: Input weights and biases were generated randomly with the aid of Mersenne Twister generator contained within MATLAB computing environment. This approach ensures reproducibility of the random number and guarantees the stability of the developed model.

Step 2: Hidden layer optimization through grid search approach: Number of hidden layer was varied between 1 and 100 for every choice of activation function. The functions explored as activation function include the sine function, triangular basis function and sigmoid function.

Step 3: Hidden layer output matrix calculation: The elements of the matrix contained in the hidden layer were computed using equation (5)(5) $\mathit{H}\left({\eta }_1,..,{\eta }_{\lambda },{\delta }_1,..,{\delta }_{\lambda },{\varphi }_1,..,{\varphi }_{\lambda }\right)={\left[\begin{array}{c}\psi \left({\eta }_1\bullet {\varphi }_1+{\delta }_1\right)\\ \mathit{\psi }\left({\eta }_1\bullet {\varphi }_M+{\delta }_1\right)\end{array}\begin{array}{c}\cdots \\ \cdots \end{array}\begin{array}{c}\mathit{\psi }\left({\eta }_{\lambda }\bullet {\varphi }_1+{\delta }_{\lambda }\right)\\ \mathit{\psi }\left({\eta }_1\bullet {\varphi }_M+{\delta }_{\lambda }\right)\end{array}\right]}_{\mathit{Mx\lambda }}$(5) coupled with the dataset assigned for training phase of the model development.

Step 4: Weights (output) computation: Calculation of output weights was computed using equation (11)(11) ${\xi }^{\mathit{pred}}=\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)$(11) using a least square solution approach.

Step 5: Future generalization and predictive strength assessment: Parameters employed for generalization assessment include the correlation coefficient (CC), mean absolute error (MAE) and root mean square error (RMSE). These parameters were computed for both training and testing dataset, while final generalization assessment was made using parameters associated with testing set of data.

Step 6: Varying activation function and hidden nodes for optimization: For each of the activation function, Step 2 to Step 5 were repeated for each of the chosen hidden number of hidden nodes. The best of the models was chosen on the basis of lowest error (RMSE and MAE) and highest CC during testing phase. The biases, input and output weights associated with the best model were saved for future implementation. The computational flow chart of the developed ELM model is presented in Figure 1.

Figure 1. ELM flow chart for strontium titanate energy gap prediction.

3. Result and discussion

The results of the developed ELM models are presented in this section. The model weights are also presented for easy implementation of the developed models in determining energy gaps of any strontium-titanate-based compounds. Comparison of the estimated energy gaps using the developed ELM-based models and the existing model is also presented.

3.1. Model equation and the associated weights

The expression governing the implementation of the developed ELM model for estimating energy gap of strontium titanate semiconductor is presented in equation (11)(11) ${\xi }^{\mathit{pred}}=\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)$(11) .

(11) ${\xi }^{\mathit{pred}}=\sum _{\mathit{j}=1}^{\lambda }{\beta }_j\mathit{\psi }\left({\eta }_j\bullet {\varphi }_i+{\delta }_j\right)$(11)

where ${\xi }^{\mathit{pred}}$ is the predicted energy gap, $\mathit{\psi }$ = activation function (sigmoid and triangular basis), $\mathit{\beta }$ = output weight, $\mathit{\varphi }$ = lattice parameter and crystallite size predictors, $\mathit{\delta }$ = random biases obtained randomly and $\mathit{\eta }$ = input weight initiated randomly. The weights such input, output and biases are presented in Table 1 for easy implementation and reproducibility of the developed ELM models.

Table 1. Input weights (${\eta }_1$ and ${\mathit{\eta }}_2$), output weights ($\mathit{\beta }$), number of nodes (j) and biases ($\mathit{\delta }$) of the developed ELM based

	j	${\eta }_1$	${\eta }_2$	$\mathit{\delta }$	$\mathit{\beta }$		j	${\eta }_1$	${\eta }_2$	$\mathit{\delta }$	$\mathit{\beta }$
Sig-ELM	1	0.309729	−0.09175	0.832908	−12388	ELM-Transig	1	0.259576	0.625267	0.15103	1.37E+10
	2	−0.39638	0.982596	0.458742	7.96E+10		2	−0.75515	−0.36187	0.715341	−3.3E+10
	3	−0.83007	−0.73423	0.474384	8E+10		3	0.570117	0.421422	0.793687	1.82E+10
	4	−0.13061	0.714598	0.760087	−2.2E+11		4	−0.00694	−0.41149	0.979255	4.12E+10
	5	0.012012	0.76412	0.607873	4.22E+10		5	−0.83424	0.387565	0.752627	6.26E+08
	6	0.757412	−0.82876	0.867175	−9.4E+08		6	−0.20388	0.628254	0.840566	2.81E+09
	7	−0.3608	0.433048	0.438081	112496.8		7	0.37889	−0.2164	0.899193	−136630
	8	−0.35988	−0.97792	0.839741	1.51E+11		8	0.32102	0.962072	0.190551	1.92E+10
	9	−0.87213	−0.26723	0.636048	−4731264		9	−0.30369	0.266302	0.389305	15403037
	10	−0.18776	0.747577	0.303582	6.1E+10		10	0.1609	−0.41035	0.114882	−9.9E+10
	11	−0.63221	0.769237	0.716578	7.83E+09		11	−0.47814	−0.31394	0.139826	−9.9E+09
	12	0.163063	0.947345	0.852875	−1.1E+11		12	−0.68126	0.91274	0.404175	−4.3E+10
	13	−0.09569	−0.0852	0.289846	31094.65		13	0.639222	0.847457	0.751158	1.92E+10
	14	−0.57299	−0.43663	0.172557	2.52E+08		14	−0.21692	0.544656	0.35505	−1.3E+11
	15	−0.96638	0.017139	0.591505	−117.392		15	−0.28842	0.72389	0.047195	3.18E+09
	16	0.983179	−0.13317	0.972203	−442.31		16	0.380623	0.935846	0.578133	1.93E+10
	17	0.952592	0.329074	0.428705	−4.8E+07		17	0.80983	0.009464	0.750085	−12670.3
	18	−0.75855	0.873059	0.081395	−1E+10		18	−0.52947	−0.55109	0.839513	8.73E+09
	19	0.753279	−0.10264	0.175146	4701.491		19	−0.5979	0.914338	0.749444	3.57E+09
	20	0.977016	0.760571	0.74116	−5.3E+10		20	−0.92085	−0.77921	0.712343	−1.9E+10
	21	−0.49819	0.624788	0.074054	18325167		21	−0.03424	0.678336	0.411824	1.35E+10
	22	0.677292	−0.61804	0.407401	−6.5E+07		22	−0.66271	−0.1384	0.146353	15042584
	23	0.044229	0.321633	0.442468	6852254		23	−0.87051	0.806891	0.897383	−1.4E+11
	24	0.050179	0.693941	0.910608	2.05E+11		24	0.8397	0.860839	0.192726	1.93E+10
	25	−0.82309	0.305317	0.05529	−62119.2		25	0.282613	−0.14181	0.83266	6325.554
							26	−0.62598	0.396554	0.210727	−2.8E+09
							27	0.742857	−0.10867	0.597708	−18.1699
							28	−0.43537	0.541663	0.936043	5.59E+10
							29	0.015597	0.197461	0.456866	9751751
							30	0.993882	−0.75763	0.840442	90385586
							31	0.626317	−0.21599	0.566459	23430.67
							32	−0.0081	−0.16898	0.352167	−1357301
							33	0.126215	0.218616	0.219677	−1.1E+08
							34	0.51521	−0.23759	0.073497	412628.9
							35	0.611512	−0.73036	0.492	−2.5E+10
							36	−0.73469	0.422929	0.973539	3.12E+08
							37	−0.29775	0.605696	0.326564	−7E+10
							38	0.848973	0.376079	0.135757	2.01E+10
							39	−0.68345	−0.55551	0.309407	−1.6E+10
							40	−0.31469	0.890348	0.285782	1.43E+10

3.2. Performance of the existing and developed ELM models

The generalization and predictive strength of the developed models are assessed and evaluated using three performance measuring parameters which include the CC, RMSE and MAE. The performance of the developed Sig-ELM and Tranba-ELM models in comparison with the existing stepwise regression algorithm (SRA) (Olatunji, 2021) is presented in Figure 1 for the training model phase using CC for performance capacity determination. The developed Sig-ELM model performs better than the developed Tranba-ELM and SRA (2021) (Olatunji, 2021) model with performance improvement of 2.96% and 67.37%, respectively, while the developed Tranba-ELM model performs better than the existing SRA (2021) (Olatunji, 2021) model with improvement of 66.37%. Training phase comparison on the basis of MAE is presented in Figure 2 in which the developed Sig-ELM model outperforms Tranba-ELM and SRA (2021) (Olatunji, 2021) model with improvement of 10.15% and 53.84%, respectively. The developed Tranba-ELM shows performance superiority of 48.62% over the existing SRA (2021) (Olatunji, 2021) model.

Figure 2. Training phase coefficient of correlation-based performance comparison between the developed ELM-based models and existing model.

Similar training phase comparison is presented in Figure 3 using RMSE performance yardstick. In this comparison, Sig-ELM model developed in this work outperforms Tranba-ELM and SRA (2021) (Olatunji, 2021) models with improvement of 8.18% and 48.99%, respectively, while performance improvement of 44.45% is obtained upon comparing the superiority of Tranba-ELM over SRA (2021) (Olatunji, 2021) model. The testing phase of model development using CC performance measuring parameter is presented in Figure 4, where the developed Sig-ELM model outperforms Tranba-ELM and SRA (2021) (Olatunji, 2021) model with improvement of 1.28 % and 67.76%, respectively, while Tranba-ELM demonstrated superiority over SRA (2021) (Olatunji, 2021) model with improvement of 68.17%.

Figure 3. Training phase mean absolute error-based performance comparison between the developed ELM-based models and existing model.

Figure 4. Training phase root mean square error-based performance comparison between the developed ELM-based models and existing model.

Using testing MAE performance yardstick presented in Figure 5, the developed Sig-ELM model outperforms Tranba-ELM and SRA (2021) (Olatunji, 2021) model with improvement of 25.10% and 61.93%, respectively, while Tranba-ELM shows improvement of 49.17% over the existing SRA (2021) (Olatunji, 2021) model.

Figure 5. Testing phase coefficient of correlation-based performance comparison between the developed ELM-based models and existing model.

Figure 6 presents the testing comparison on the basis of RMSE, while the developed Sig-ELM model shows improvement of 23.64% and 64.30%, respectively, over Tranba-ELM and SRA (2021) (Olatunji, 2021) model. Tranba-ELM further shows performance improvement of 53.24% over the existing SRA (2021) (Olatunji, 2021) model.

Figure 6. Testing phase mean absolute error-based performance comparison between the developed ELM-based models and existing model.

The details of the performance of each of the model are presented in Table 2, while comparison with measured band gap and their associated errors is presented in Table 3.

Table 2. Performance comparison of the present ELM-based models and improvement over the existing model

Models		Training		Testing
	CC	MAE	RMSE	CC	MAE	RMSE
Sig-ELM	0.8702	0.1970	0.3049	0.9150	0.1509	0.1912
Tranba-ELM	0.8445	0.2192	0.3321	0.9268	0.2014	0.2504
SRA (2021) (Olatunji, 2021)	0.2840	0.4266	0.5978	0.2950	0.3963	0.5355
% improvement of Sig-ELM over Tranba-ELM	2.9606	10.1473	8.1801	1.2839	25.1014	23.6429
% improvement of Sig-ELM over SRA	67.3664	53.8354	48.9904	67.7625	61.9320	64.2980
% improvement of Tranba-ELM over SRA (2021)	66.3707	48.6219	44.4460	68.1711	49.1740	53.2434

Table 3. Estimates of the developed Sig-ELM and Tranba-ELM model as compared with the prediction of the existing SRA (2021) (Olatunji, 2021) model

Measured energy gap (eV)	ELM-Sig (this work) eV	Error	ELM-Transig (this work) eV	Error	SRA (2021) eV (Olatunji, 2021)	Error
3.20 (Jia et al., 2010)	2.50	0.70	2.50	0.70	3.10	0.10
3.14 (Padmini & Ramachandran, 2019)	3.14	0.00	3.14	0.00	2.83	0.31
3.08 (Anitha & Devi, 2019)	3.08	0.00	3.08	0.00	3.17	0.09
3.02 (Pradhan et al., 2021)	3.16	0.14	3.51	0.49	3.19	0.17
1.90 (Jia et al., 2010)	2.52	0.62	2.55	0.65	3.11	1.21
3.08 (Anitha & Devi, 2019)	3.08	0.00	3.08	0.00	2.89	0.19
2.93 (Kumar et al., 2021)	3.06	0.13	2.92	0.01	3.16	0.23
4.10 (Nunocha et al., 2021)	4.27	0.17	4.19	0.09	3.18	0.92
3.10 (Anitha & Devi, 2019)	2.52	0.58	2.54	0.56	3.23	0.13
3.87 (Xu et al., 2014)	3.65	0.22	3.37	0.50	3.43	0.44
3.12 (Atkinson et al., 2019)	3.10	0.02	3.12	0.00	3.16	0.04
2.09 (Atkinson et al., 2019)	2.09	0.00	2.09	0.00	3.06	0.97
3.15 (Padmini & Ramachandran, 2019)	3.09	0.06	3.38	0.23	2.96	0.19
1.80 (Jia et al., 2010)	2.53	0.73	2.56	0.76	3.11	1.31
3.78 (Kumar et al., 2021)	3.78	0.00	3.78	0.00	2.97	0.81
3.17 (Padmini & Ramachandran, 2019)	3.19	0.02	3.18	0.01	3.04	0.13
3.56 (Xu et al., 2014)	3.65	0.09	3.37	0.19	3.43	0.13
3.32 (Kumar et al., 2021)	3.43	0.11	3.35	0.03	3.17	0.15
3.53 (Xu et al., 2014)	3.48	0.05	3.17	0.36	3.39	0.14
3.14 (Anitha & Devi, 2019)	3.08	0.06	2.96	0.18	3.29	0.15
2.20 (Jia et al., 2010)	2.52	0.32	2.54	0.34	3.11	0.91
3.25 (Atkinson et al., 2019)	3.26	0.01	3.25	0.00	3.17	0.08
4.10 (Nunocha et al., 2021)	3.90	0.20	4.06	0.04	3.18	0.92
3.23 (Kang & Park, 2013)	3.28	0.05	3.25	0.02	3.06	0.17
3.04 (Pradhan et al., 2021)	3.03	0.01	3.04	0.00	3.14	0.10
3.10 (Jia et al., 2010)	3.10	0.00	3.10	0.00	3.11	0.01
2.11 (Atkinson et al., 2019)	1.99	0.12	2.12	0.01	3.16	1.05
2.90 (Jia et al., 2010)	2.51	0.39	2.53	0.37	3.11	0.21
3.20 (Jia et al., 2010)	2.60	0.60	2.70	0.50	3.11	0.09
4.11 (Nunocha et al., 2021)	3.93	0.18	4.02	0.09	3.18	0.93
1.90 (Jia et al., 2010)	2.51	0.61	2.53	0.63	3.11	1.21
3.56 (Xu et al., 2014)	3.66	0.10	3.31	0.25	3.41	0.15
3.59 (Xu et al., 2014)	3.65	0.06	3.37	0.22	3.43	0.16
2.94 (Atkinson et al., 2019)	3.15	0.21	3.07	0.13	3.18	0.24
3.15 (Padmini & Ramachandran, 2019)	3.04	0.11	3.09	0.06	2.76	0.39
3.17 (Anitha & Devi, 2019)	3.23	0.06	2.82	0.35	3.12	0.05
4.10 (Nunocha et al., 2021)	4.10	0.00	4.10	0.00	3.16	0.94
4.12 (Nunocha et al., 2021)	3.79	0.33	3.94	0.18	3.18	0.94
3.12 (Kumar et al., 2021)	2.83	0.29	2.66	0.46	3.18	0.06
		0.19		0.22		0.42

4. Conclusion

ELM algorithm has been employed for determining energy gap of doped strontium titanate semiconductor using lattice parameter and crystallite size as predictors to the model. The two developed ELM-based models with different activation function (sigmoid and triangular basis) are compared with the existing SRA (2021) model in the literature through computation of CC, RMSE and MAE during training and testing phase of model development. The developed Sig-ELM model performs better than the developed Tranba-ELM and SRA (2021) model with performance improvement of 2.96% and 67.37%, respectively, using training CC, while the developed Tranba-ELM model performs better than the existing SRA (2021) (Olatunji, 2021) model with improvement of 66.37%. The developed models further show superior performance using various performance measuring parameters for both training and testing data samples. The demonstrated outstanding performance of the developed ELM-based models coupled with ease of implementation would enhance energy harvesting capacity of strontium titanate semiconductor through extension of absorption capacity of the semiconductor beyond visible region where photocatalysis and other opto-electronic applications can be fully utilized and effectively optimized.

Disclosure statement

No potential conflict of interest was reported by the author(s).