\documentclass[a4paper,12pt,compress]{article}
\usepackage[top=1in, bottom=1in, left=1in, right=1in]{geometry}
\usepackage{amsmath}
\usepackage{gensymb}
\usepackage{graphicx}
\usepackage{textcomp}
\usepackage{csquotes}
\usepackage{epsfig}
\usepackage{epstopdf}
\usepackage{color}
\usepackage{sectsty}
\usepackage{mathtools}
\usepackage[justification=centering]{caption}
\usepackage{caption}
\usepackage[onehalfspacing]{setspace}
\usepackage[toc,page]{appendix}
\setcounter{section}{0}
\setcounter{secnumdepth}{3}
\setcounter{tocdepth}{3}
\usepackage{multicol}

\begin{document}
	\pagenumbering{roman}
	%\bibliographystyle{}
	\bibliographystyle{agsm}
	\pagenumbering{arabic} \cleardoublepage
	\linespread{1}
	\begin{center}
	\section*{Understanding the Bi-Axial Strain Influence on Mn$_3$Si$_2$Te$_6$: A DFT Study of Electronic, Magnetic, Optical, and Thermoelectric Properties}~\\
	\end{center}
\textit{\textbf{ Mazhar Haleem Awan$^a^{*}$}}\\
mazharhaleem7@gmail.com; +923149888807\\
Department of Physics, Abbottabad University of Science and Technology, Abbottabad,
KPK, Pakistan. \\
~\\
~\\

\begin{abstract}

The increasing focus on sustainable and renewable energy sources has boosted the demand for double perovskites in applications such as solar cells and thermoelectric energy converters. In the recent study, understanding the bi-axial strain influence on Mn$_3$Si$_2$Te$_6$ (MST): DFT study of electronic, magnetic, optical, and thermoelectric properties  were systematically analyzed by using the WIEN2K code. The electronic nature was found to be highly sensitive to external electric fields under biaxial strain engineering. Experimentally, MST show the metallic nature while in DFT study of strain engineering the material become half-semimetal at $5\%$ tensile strain with indirect bandgap of 0.732 eV at the $\Gamma - \kappa$ symmetry point with SCF (self-consistent field) calculation and more accurate the indirect bandgap of 1.24 eV under TB-mBJ approximation. It was observed that, the compressive strain reduces the bandgap, while tensile strain increases it. The optical property predictions indicate that MST exhibits absorption in the ultraviolet and visible regions, with a blue shift under tensile strain and a red shift under compressive strain. In the absorption coefficient of SOC (Spin orbit coupling), the black line show the maximum absorption in xx plane in between visible region of 1.6 eV to 3.2 eV. The thermoelectric performance was evaluated over the temperature range of 300-800 K of MST compound under strain engineering were investigated by BoltzTraP Code including features i.e Seebeck coefficient, electrical conductivity, thermal conductivity, power factor, and figure of merit (ZT). The Seebeck coefficient and electrical conductivity generally decrease with increasing temperature, while thermal conductivity typically increases. The thermoelectric performance was evaluated over the temperature range of 300-800 K using the figure of merit (ZT). The ZT values approached unity, and the ultralow lattice thermal conductivity enhances the significance of these double perovskites for use in thermoelectric generators. Under strain engineering of MST material, the highest value of ZT in n-type region at 0.723 while lowest value of ZT in p-type region at 0.716 by computationally anaylsis. 
~\\

		\noindent\textbf{Keywords:} Biaxial Strain (compressive-tensile), DFT, Electronic, Spin orbit coupling (SOC), Optical, Thermoelectric applications.
	\end{abstract}

	\section*{Introduction}
	In the current era, our need is to address the energy issue that has arisen as a result of population expansion and continuous depletion of natural resources such as timber, oil, and natural gas. In order to solve this energy crisis and to fulfill the demand in the recent period of time the fabrication of novel TE devices with a specific purpose, such as  to develop a low cost, environmentally friendly and renewable energy sources to overcomes the future energy problems and other technological obstacles. Efforts to find sustainable energy sources, such as solar energy, biomass energy, and wind energy, among others, have been made on a constant basis. However, they showed certain limitations such as wind-force to drive a windmill, sunlight for full day, tidal for coastal area only, and earth quick safety for geothermal plant etc. The need for energy in today's society is growing, but because there are not many traditional energy sources available. Therefore, non-conventional energy sources are gaining popularity. Nowadays, manufacture of movable solar device's is preferred since it is relatively easy and a serious issue of environmental pollution \cite{d1,d2}. A significant fraction of the total energy used is discharged as waste heat by several contemporary organisations and processing facilities. Some of this waste heat may be converted into energy through the use of TE power production technology \cite{d3}. The development of novel materials is required to address future technical issues like power shortages. As a consequence, researchers constantly create fresh formulations and modify the properties of various substances to provide improved features. Like inorganic material, which includes a large variety of material compositions, provide a specific function in various applications. They are used in all dimensions, from nano to bulk level, because to their fascinating features, which include the structural composition, magnetic behaviour, electrical, optical, and TE aspects \cite{d4}. TE effect is the direct conversion of thermal fluctuation to electrical voltage through the use of a thermocouple, without the help of any mechanically active components. Because of their low efficiency, TE devices have only been used in a few applications, e.g microprocessor cooling, power generation, refrigeration and heat recovery. However, thermoelectricity has significant potential for certain applications such as laboratory resources, space probes and lastly the medicinal purposes in which the availability, dependability, and predictability are more important than cost and efficiency, thermoelectricity holds immense potential. The term “thermoelectic effect” is divided into three categories: Seebeck, Peltier, and Thomson \cite{d5}. The thermo-electricity performance of Na${_2}$RhO${_2}$ single layer transition-metal material has recently been improved. This efficiency has a high merit of Seebeck coefficient at around (S, 100 mVK$^{^-1}$ at 300 Kelvin) TE materials with strong electrical conductivity but weak heat conductivity. The efficiency is calculated using the fundamental formula ZT = S${^2}$T$\sigma$k$^{-1}$, where `S' stands for the double times Seebeck coefficient, `T' (temperature), `$\sigma$' (electrical conductivity), and `k' for thermal conductivity are listed in Table 1.1. In general, heavily doped semiconductors with carrier concentrations ranging from 10$^{19}$ to 10$^{21}$ are good options for high ZT materials. TE material for semi-conductors devices are, such as (Bi${_2}$Te${_3}$/Sb${_2}$Te${_3}$, PbTe/PbSeTe and PbSnTe/PbSnSeTe), with occuring high efficiency Z (K$^{-1}$) at $\sim$ 2 x 10$^{-3}$, reduce thermal conductivity $\sigma$ (1/$\Omega$cm) $\sim$ 10$^{3}$ and increased Seebeck coefficient S (µVK$^{-1}$) at $\sim$ 200 with this factor ZT = S${^2}\sigma$T/k \cite{d6}. In comparison of CrI${_3}$ with Cr${_2}$(Si,Ge)${_2}$ Te${_6}$, later material are intrinsic ferromagnetic (FM) semiconductors this distinct nature exist due to Si-Si (Ge-Ge) dimers \cite{d7}. During experimental analysis for mono-layer of CrI${_3}$ and bi-layer of Cr${_2}$Si${_2}$Te${_6}$ (CST) and it was observed after first principal calculations that in CST  material (FM) were remain even down to single layer \cite{d8}.  Structural composition of these materials, in ab plane Cr atoms form a honeycomb lattice long with Si or Ge which present in hexagonal central while octahedral covered with Cr atoms. These materials composed of complex multi-layer structure with numerous chemical bonds such as ionic, covalent, metallic bonds were exist between  intralayer Cr-Te,  Si/Ge-Te and  Si-Si/Ge-Ge respectively, while other included inter-layer vander waals (vdW) force. In Cr${_2}$(Si,Ge)${_2}$ Te${_6}$ due to high bonding low lattice was determined \cite{d9}. Other feature such as strong spin phonon coupling and short-range in plane FM correlations were fascinatingly hold on at room temperature even through above 50K out-of plane correlation vanish \cite{d10}. Three dimensional (3D) image disclosed that both MST and CST are analog of each other. Structural composition of MST material layer showed that it comprise of MnTe${_6}$ octahedra which link with ab plane (Mn${_1}$ site) that connect through its edge and along with Si-Si dimers, structure analogous with CST. In MST orders ferrimagnetically (FIM) showed below Tc $\approx$ 78K, this data were obtained via recent neutron diffraction experiment \cite{d11}. The magnetic order in MST materials is due to the Mn${_1}$ and Mn${_2}$ sub-lattices that has anti-parallel alignment with an easy-plane anisotropy. Structural modification of Mn-Te-Mn material through proton irradiation result in turn down it magnetization as a result of its exchange interaction \cite{d12}. Such phenomenon were observed in MST material when replacing Te by Se as a consequence magnetization anisotropy slightly decreases along with it T$_c$ values reduces to 67K \cite{d13}. The electronic structure, effective masses, optical characteristics, and photocatalytic properties of Janus MoSe have been extensively studied through strain engineering, utilizing density functional theory (DFT) analysis. This research has enabled the identification of photocatalytic applications based on the material’s valence and conduction band edge potentials. Thermoelectric (TE) materials, which convert temperature gradients directly into electrical energy, are particularly useful for applications such as waste heat recovery and electronic cooling. The primary aim of this work, to investigate the impact of biaxial strain on the structural, electronic, magnetic, and optical properties of pristine and strained Mn-based silicate telluride (MST) materials, and to explore their thermoelectric properties under different conditions.

\section*{Comptutational Detail}
	In our computational calculation, the FP-LAPW+Io method and the WIEN2k technical code are both used \cite{d14}. We used a more precise exchange correlation potential flavour to approximate the generalised gradient for determining structural, electrical, optical, and thermoelectric characteristics (WC-GGA) \cite{d15}. WIEN2K is predicated on a distinct set of assumptions. The WIEN2K tool can also compute the properties of various materials. The approach used in the WIEN2K code is Plane Wave+Local Orbital. BoltzTraP package provides theoretically robust and efficient approaches for calculating quasi-particle energy analytically. The Boltzmann transport equation (BTE) has found widespread use in a wide range of fields, including thermoelectrics, transparent conductors, intermetallic phases, and superconductors. To calculate the TE properties such as Seebeck coefficient (S), electrical conductivity ($\sigma$), and thermal conductivity ($\kappa$) for strain and unstrain MST using BoltzTraP.A code \cite{d16}. The electrical structure, optical, and transport characteristics of the materials are calculated using the first principle technique and the computational calculations are based on FP-LAPW method, as applied in WIEN2k code \cite{d17, d18}. FP-LAPW is concerned with localised valence electron configurations in transition metals and related compounds. To quantify the various properties of the materials features, we use the more precise exchange-correlation potential (Wu-GGA) flavour as a generalised gradient approximation \cite{d19}. To obtain the volume optimization curve, a minimum of 6 $\times$ 6 $\times$ 4 k-points were used in a single cell. In the first brilliouin zone (BZ), self-consistent field (SCF) was conducted with 8 $\times$ 8 $\times$ 4 k-points with no change in RMT value. To calculate the Electronic, Optical and Thermal properties denser k-mesh is used with 8 $\times$ 8 $\times$ 6 k-points to give more relevant findings.
\section*{Result and Discusion}
\section{Structural Properties}

The structural properties of any material play a vital role in determining its physical and chemical characteristics. In this work, the bulk material MST is investigated using DFT under strain engineering. The structural features of the MST single perovskite material are explored in the FM condition. In fig 1(a), the crystal configuration of MST in hexagonal phase with space group of $P\bar{3}1c$. Mn(1) atoms are located at the 2c Wyckoff positions (1/3,2/3,1/4), Si atoms are at the 4e Wyckoff positions(0,0,0.166164), Mn(2) atoms are at the 4f Wyckoff positions (1/3,2/3,0.506227) and finally, Te atoms are at the 12i Wyckoff positions (0.353306,0.010486,0.878003). In fig 1(a) the structural arranagements are seen with different colors like green for manganese (Mn), yellow for Silicon (Si) and red with Tellurium (Te). The structural stability of the MST material was evaluated using the energy-volume curve, with forces initially computed using the WC-GGA functional. The system was relaxed until the forces acting on the atoms were minimized. The optimized lattice constants, as presented in Table 1, were determined from the energy versus volume optimization curve for MST using the Birch-Murnaghan equation of state \cite{d20}. 

\begin{equation}
E_{total}(V) = \frac{B_{0}V}{B^{'}(B^{'}-1)} B (1- \frac{V_{0}}{V} +(\frac{V_{0}}{V}){B'} -1 ] 	 
\end{equation} ~\\

Where in equ 1 B$_{o}$, B$^{'}$, V$_{o}$ and E(V) are the bulk modulus of first derivative of unit cell volume and energy at equilibrium ground states respectively. In fig 1(b) is the optimization curve obtaining by WIEN2K code \cite{d17} and showing the evolution of total enegy as a function of the unit cell volume of hexagonal phase of MST. The equilibrium lattice parameter was computed from structural optimization with the help of equ 1. The E-V curve is generated by varing the lattice constant and calculating the minimal energy point is connected to the lowest point of the curve, where the arrow is set its mean our structure is dynamically stable at point -2.5 minimum energy in the fig 1(b). All structural parameter of MST compound are taken from the experimental work in table 1 \cite{d21}, respectively. The FM material (MST) taken under the reduction of $5\%$ RMT (reduce muffin tin ) at initial stage to find the new parameter and the nature of the material employed, spin($\uparrow$) and spin($\downarrow$) behavior play's a significant role in the spin polarized volume optimization.

\begin{figure*}[h]
\centering
\includegraphics[width=1.0\textwidth,clip]{Fig1.png}
\textbf{\caption{(a) Hexagonal crystal structure of MST (b) The E-V curve of MST which correspond to the minimum energy point.}}
\end{figure*}

\begin{table} [h!]
\centering
\caption{ Provides the State, Lattice constant ($\AA$), C/A Ratio ($\AA$), Volume ($\AA^{3}$), Bulk's Modulus (GPa), First derivative w.r.t to pressure for the hexagonal phase of the MST perovskite under space group $P\bar{3}1c$ in spin-polarized state. }
\begin{tabular}{c c c c c c c }
\hline\hline
\textbf{Compound} & \textbf{State} & \textbf{a, c} & \textbf{C/A} & \textbf{Volume} & \textbf{B} & \textbf{B$'$} \\
\text{MST} &  & $\AA$ &  & $\AA^{3}$ & {GPa} &  \\
\hline\hline
Present & FM & 7.09, 14.04 & 1.981 & 704.31 & 100 & 5.00 &  \\
\hline
Experimental \cite{d21} & FM & 7.03, 14.24 & 2.025 & 610.41 & - & - &  \\
\hline
\end{tabular}
\end{table}

\section{Electronic properties of Mn$_3$Si$_2$Te$_6$}
	Analyzing the electronic structures of compounds offers insights into the properties of solids by providing a deeper understanding of their macroscopic characteristics. The total density of states (DOS) and band structure are key tools for this analysis. The band structure shows how electrons' energy levels relate to their momentum, indicating whether a material acts as a conductor, semiconductor, or insulator. DOS, on the other hand, describes the number of electronic states at each energy level, helping to determine electronic and thermal properties. Together, these tools reveal the electronic configuration and behavior of solids. To analyze the spin polarised band structures and assoicated densities of states to investigate the electronic characteristics of MST.

\subsection{Band Structure}

By analysing the effects of strain engineering with regard to the Fermi level (E$_f$), our computed result of the band structure demonstrates that MST exhibits metallic behaviour in spin $\uparrow$ while spin $\downarrow$ semiconductive behavior. In Fig 2, by applying biaxial strain from $-3$ to $10\%$ with increasing parametric ratio's of a = b, c simultaneously. The metallic behavior show in between -3 to $4\%$ (spin $\uparrow$ $\&$ $\downarrow$ region), at $5\%$ (tensile strain) in spin $\downarrow$ region the in-direct bandgap of 0.732 eV between $\Gamma - K$ high symmetry point shown in Fig 3 clearly during computing SCF (self-consistent field) calculation. Further application of strain reduces the bandgap due to the dominance of strong vdW forces. The bandgap is further increasing when applying TB-mBJ appromixation in Fig 4, the maxmimum tune of the bandgap is around 1.24 eV between $\Gamma - K$ at $4\%$ (tensile strain) in spin $\downarrow$ region.  The bandgap energy (eV) vs strain ($\%$) graph in fig 5, showing the nature of both potential in Wc-GGA and Wc-GGA+TB-mBJ appoximation. The blue portion for scf calculation having different values from negative region to positive region, at -3 to +4 $\%$ there in no bandgap showing and while after reaching 4 to 5 $\%$ huge difference is to be noted at 0.732 eV and further its decreasing by same applying the strain at different values(6$\%$ = 0.727, 7$\%$ = 0.712 and finally at 10$\%$ = 0.652 ) and in yellow portion from -1 to 0 $\%$ first bandgap is to be taken at 1.11eV and its futher increases to be reach at 4$\%$ maximum bangap is around 1.24 eV and futher its decreasing gradually at 10$\%$ = 1.09 eV. 

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth,clip]{Fig2.eps}
\textbf{\caption{Tunning the bandgap under applying stress-strain from -ive 3 to +ive 10 values with SCF}}
\end{figure*}

\begin{figure*}[h!]
\centering
\includegraphics[width=0.8\textwidth,clip]{Fig3.eps}
\textbf{\caption{In-direct gap show under 5$\%$ tensile strain }}
\end{figure*}

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth,clip]{Fig4.eps}
\textbf{\caption{Tunning the bandgap under applying stress-strain from -ive 3 to +ive 10 values with TB-mBJ }}
\end{figure*}

\begin{figure*}[h!]
\centering
\includegraphics[width=1.0\textwidth,clip]{Fig5.eps}
\textbf{\caption{The bandgap comparison graph of MST in both approximation while in WC-GGA and also in WC-GGA+TB-mBJ }}
\end{figure*}
\cleardoublepage
 
\subsection{T-DOS and P-DOS}

To estimated the density of state (DOS) w.r.t T-DOS and P-DOS to further demonstrate its ferromagnetic character of Mn, Si, and Te Atom's with its orbital's contribute in both spin $(\uparrow)$ and spin $(\downarrow)$ states. In Fig 6, T-DOS Mn atom has major contribution around 34 eV at y-axis in valence band region in spin $(\uparrow)$ state, Te atom has minor peak of 4 eV near VBM across fermi level. While in spin $(\downarrow)$, Mn atom also show major contribution in CBM of 22 eV at -ive y-axis in between -5 to +5 eV on x-axis across the fermi level. The P-DOS of Mn atom describes the major and minor contribution of the orbitals both in (VB $\&$ CB) region such as Mn-d, Mn-DZ2, Mn-DX2Y2+DXY and DXZ+DYZ for both spins.  In spin $(\uparrow)$ state, the dominent role of the Mn-d hybrid orbitals at 8.9 eV of the valence band region, secondly Mn-DX2Y2+DXY orbital at 5 eV, thirdly Mn-DXZ+DYZ orbitals at 3.5 eV and minorly contribution of Mn-DZ2 at 1.9 eV in the y-axis. While in the spin $(\downarrow)$ state, the dominent role of the Mn-d orbital at -5 eV, secondly Mn-DX2Y2+DXY orbital at -2.8 eV, third one is Mn-DZ2 orbital at -2.7 eV and last one is Mn-DXZ+DYZ orbital at -1.3 eV in the conduction band region. The P-DOS of Si atom in spin $(\uparrow)$ valence band region the dominent states are s,p and PX+PY with different energy like 0.21 eV, 0.29 eV and 0.19 eV respectively,  while in the spin $(\downarrow)$ conduction band region s,p and PX+PY with energy of -0.15 eV, -0.26 eV and -0.22 eV. The P-DOS of Te atom in spin $(\uparrow)$ valence band region the dominent peak is p,PZ at 0.48 eV, 0.26 eV while in spin $(\downarrow)$ conduction band region p,PZ orbitals with energy of -0.51 eV and -0.21 eV, respectively.


\begin{figure*}[h!]
\centering
\includegraphics[width=0.8\textwidth,clip]{Fig6.eps}
\textbf{\caption{TDOS and PDOS of MST under 4$\%$ tensile strain with WC-GGA+TB-mBJ approximation }}
\end{figure*}
\cleardoublepage

\section{Optical Properties of Mn$_3$Si$_2$Te$_6$}

The optical properties which includes the absorption coefficient Fig 7(a), Fig (b) real $\&$ Fig (c) imaginary part of dielectric function, Fig (d) refractive index n and Fig (e) extinction coefficient of MST material. The optical properties are calculated by using the complex dielectric function $\epsilon(\omega)$ = $\epsilon_1(\omega)$ + $\epsilon_2(\omega)$ at large range of photon energies. The band structure depend on imaginary part of the dielectric function $\epsilon_2(\omega)$ and also describe it’s absorptive mode. According to the frequency dependent $\epsilon_2(\omega)$ for cubic crystal's symmetry is represented by the relation is:
\begin{equation}
\epsilon_2(\omega) = [\frac{4}{\pi \omega^{2}} \sum_{nn'} \int |P_{nn'},(k)^2|\frac{dS_{k}}{\nabla \omega_{nn'}k}]
\end{equation}

In above equation $S_{k}$ represent the surface energy with a constant value, $\omega_{nn’}$ represent the energy variant among the two energy state and finally $|P_{nn’}, (k)|$ identify the dipole matrix component between the end and starting states. The real part of the dielectric constant $\epsilon_{1}(\omega)$ are obtained by using the Kramers-Kronig dispersion relation:
\begin{equation}
\epsilon_1(\omega) = 1 + \frac{2}{\pi} P \int_{0}^{\infty} \frac{\omega' \epsilon_2 (\omega')}{\omega'^2} - \omega^2 d\omega' 
\end{equation}

According to the relation above, P is regarded as a primary component of the Cauchy's integral. In this case, the interbond and intrabond transitions' key contributing factor is $\epsilon(\omega)$. Metals are divided into transitions such as direct and indirect transitions. Other optical properties including reflectivity, refractive index, and optical conductivity may be determined using $\epsilon_1(\omega)$ and $\epsilon_2(\omega)$. Kramer's Kronig method were used for the investigation of the optical properties such as Dielectric Fucntion Real and Imaginary, Conductivity, Absorption coefficient of the MST over the energy range from 0 to 10 eV. These properties defines under the Infra Red, Visible and Ultra Violet Region. Infra Red from 0 to 1.5 eV, Visible from 1.6 to 3.1 eV and Ultra Violet is 3.2 eV to 6 eV in the graphical representation of Figure 7. The spin orbit coupling (SOC) is perform due to addition of spin $(\uparrow)$ and $(\downarrow)$ nature.   

\subsection{Absorption Coefficient}

The absorption coefficient $\alpha(\omega)$ for material MST is calculated from the dielectric function in below relation:

\begin{equation}
\alpha(\omega) = \sqrt{2\omega}(\sqrt{\epsilon_1(\omega)^2 + \epsilon_2(\omega)^2}} - \epsilon_1(\omega))^{1/2}
\end{equation}

The absorption characteristics show that the absorption spectrum is energy dependant and exhibits observable increasing and decreasing behaviour. The several peaks are caused by the electric stimulation from VB to CB. The oscillatory behaviour is also brought on by many-body interactions, which depend on the strength of the transition. In Fig 7(a) displays the absorption coefficient for the three regions of infrared, visible region, and ultra violet. The black and red line's in the infrared region, no peak show due to larger bandgap of 1.24 eV after while in the visible region the maximum peak show on 24 eV at soc$_{xx}$ and 18 eV at soc$_{zz}$ plane. The incident light with minimum energy ~ 1.24 eV, to absorb the energy and emits photon in the visible range. To jump the electron from $VB_{max}$ to $CB_{min}$. The absorption coefficient play the major role for the production of heat into electricity also known as thermoelectric device applications. 

\subsection{Dielectric Funtion Real and imginary part}

In Fig 7(b) defining the Real Part of Dielectric Function $\epsilon_1(\omega)$. The graph representation is divided into three part (Infra Red, Visible and Ultra Violet) Region. The black,red line in the IR region at -8 eV, 4 eV, visible region black line is dominent at 5.4 eV and red at 3.4 eV and in the ultra violet region the down fall of it with photonic energy along x axis. In Fig 7 (c), the downfall of black line from 7 eV on y-axis and then go straight through photonic energy along x-axis start from 0.2 eV to 1.24 eV and then rise in the visible region maximum at 4.1 eV and futher rise in the ultra region. The red line from 0 to 1.24 eV touches the x-axis in IR region, then rises at 1.92 eV in the visible region and so on in the ultra vilot region.


\subsection{Refractive Index n, Extension cofficient k}

In Fig 7 (d,e) are the refractive index n and extension cofficient k of the MST material. In Fig 7 (d) the dominent peak is the black line in the visible region near at 2.7 eV, red at 2.2 eV. In Fig 7 (e), the black line in the visible region, first peak show at 0.6 eV (black) and 2nd peak show at 0.4 eV (red). The Below Equ 4 and 5 are the refractive index n and Extension cofficient k. 

\begin{equation}
\centering
n(\omega) = \frac{1}{\sqrt{2}} \left[ \left( \epsilon_1(\omega)^2 + \epsilon_2(\omega)^2 \right)^{1/2} + \epsilon_1(\omega) \right]^{1/2}

\end{equation}

\begin{equation}
\centering
k(\omega) = \frac{1}{\sqrt{2}} \left[ \left( \epsilon_1(\omega)^2 + \epsilon_2(\omega)^2 \right)^{1/2} - \epsilon_1(\omega) \right]^{1/2}

\end{equation}

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9\textwidth]{Fig7.eps}
\end{center}
\caption{Optical Properties of the absorption coefficient, real and imaginary part of dielectric function, refractive index and extinction coefficient of the MST material.}
\end{figure}

\section{Thermoelectric properties of Mn$_3$Si$_2$Te$_6$}    
Thermoelectric properties are essential in heat transfer and have diverse applications due to their capacity for efficient energy storage, offering solutions to various challenges. The utilization of thermoelectric materials shows great potential in addressing energy issues faced by society.

\subsection{Thermal Properties}

Thermoelectric (TE) materials have been the focus of extensive research due to their diverse applications in engineering, physics, and chemistry. Their efficiency and recent advancements in novel TE materials have attracted significant interest, leading to their widespread use. The Seebeck effect, a key principle in TE devices, enables cooling and power generation from waste heat, making it a promising solution for addressing global energy and environmental challenges \cite{d22,d23,d24}. The good TE material’s posses high value of a dimensionless figure of merit (ZT = S$^{2}T\sigma/\kappa$), which includes the double times of Seebeck coefficient `S', Temperature `T', `$\sigma$' is the electrical conductivity and finally `$\kappa$' is the thermal conductivity, which includes both electronic and lattice thermal conductivity ($\kappa = \kappa_{e} + \kappa_{ph}$) of a material, respectively. For materials to achieve high rates of efficiency, they needed to have great electrical, poor thermal, and a high Seebeck factor `S'. A high degree of efficiency is implied by the greatest value of ZT = 1, which has been empirically demonstrated to be considerable quantities. However, several compounds with ZT values greater than 1 have been discovered recently. Strain engineering, which enables the phonon scattering to be improved and the thermal conductivity to subsequently be lowered, is one of the most efficient and widely used methods for enhancing TE characteristics \cite{d25,d26,d27}. For adding the spin $(\uparrow) and (\downarrow)$ states, spin orbit coupling (SOC) calculation was performed in all TE (electrical, thermal conductivity, Seebeck coefficient, Power factor and Figure of merit) properties as a function of a chemical potential \cite{d28}. In Fig 8(a) is the seebeck effect, 8(b) is the electrical conductivity, 8(c) is the thermal conductivity, 8(d) is the ZT (figure of merit) and 8(e) is the PF (power factor) all the possible values are mentioned in table 2 for MST material. In table 2, the higher value of ZT is 0.716 (p-type) at 300K and lower value of ZT is 0.573 (p-type) at 800K while for other case the higher value of ZT is 0.723 (n-type) at 300K and lower value of ZT is 0.406 (n-type) at 800K. So, MST material is best use for thermoelectric application's for both p and n-type case. In the fig 9 shows that's, putting the phononic part in the value of ZT, the figure of merit is reduce at 0.55 at 300K and PF (power factor) is also be affected. The values of ZT, PF are mentioned in Table 3 for both p and n-type materials.  

\begin{table}[h!]
\caption{Summary of TE properties of MST with different temperature of 300$-$800K.} \centering
\begin{tabular}{c c c c c c c c}
\hline 
\textbf{T} [K] & \textbf{S} [mV/K] & \boldmath $\sigma/\tau$ [1/\Omega ms] & \boldmath $\kappa_{e}/\tau$ [W/mKs] & \textbf{ZT} & \textbf{PF} [W/m$K^{2}$s]  
\\
 [0.4ex] \hline 
\\
 &  &  & \textbf{p-Type} &  &  \\
\hline
\textbf{300} & 0.298 & 3.714 & 0.0591 & 0.716 & 10.432 \\
\textbf{400} & 0.249 & 3.582 & 0.2268 & 0.633 & 14.381 \\
\textbf{500} & 0.224 & 3.478 & 0.5393 & 0.605 & 18.153 \\
\textbf{600} & 0.209 & 3.402 & 0.9967 & 0.597 & 21.336 \\
\textbf{700} & 0.195 & 3.358 & 1.6372 & 0.589 & 23.634 \\
\textbf{800} & 0.181 & 3.331 & 2.4143 & 0.573 & 24.872 \\
\hline
\\
 &  &  & \textbf{n-Type} &  &  \\
\hline
\textbf{300} & 0.305 & 4.022 & 0.0591 & 0.723 & 5.464\\
\textbf{400} & 0.247 & 3.973 & 0.2268 & 0.632 & 8.143 \\
\textbf{500} & 0.213 & 3.937 & 0.5393 & 0.555 & 10.661\\
\textbf{600} & 0.191 & 3.919 & 0.9967 & 0.496 & 12.857 \\
\textbf{700} & 0.173 & 3.888 & 1.6372 & 0.448 & 14.625 \\
\textbf{800} & 0.157 & 3.842 & 2.4143 & 0.406 & 15.643 \\
\hline
\end{tabular}
\label{table:nonlin}
\end{table}


\begin{figure*}[h]
\centering
\includegraphics[width=1.0\textwidth,clip]{Fig8.eps}
\textbf{\caption{The TE properties of MST material under applying $4\%$ tensile strain with factors of `S' coefficient, `$\sigma/\tau$', `$\kappa/\tau$', ZT and PF.}}
\end{figure*}

\begin{table}[h!]
\caption{Summary of TE properties of MST with different temperature of 300$-$800K.} \centering
\begin{tabular}{c c c c }
\hline\hline 
\textbf{T} [K] & \textbf{ZT} &  \textbf{PF} [W/m$K^{2}$] 
\\ [0.5ex] \hline \hline
\\
 &  \textbf{p-Type} &  \\
\hline
\textbf{300} & 0.555 & 0.000809 \\
\textbf{400} & 0.493 & 0.001118 \\
\textbf{500} & 0.469 & 0.001410 \\
\textbf{600} & 0.457 & 0.001661 \\
\textbf{700} & 0.451 & 0.001838 \\
\textbf{800} & 0.447 & 0.000193 \\
\hline
\\
 &  \textbf{n-Type} &  \\
\hline
\textbf{300} & 0.563 & 0.000429 \\
\textbf{400} & 0.490 & 0.000633 \\
\textbf{500} & 0.433 & 0.000828 \\
\textbf{600} & 0.386 & 0.001005 \\
\textbf{700} & 0.347 & 0.001137 \\
\textbf{800} & 0.314 & 0.001216 \\
\hline
\end{tabular}
\label{table:nonlin}
\end{table}
\cleardoublepage

\begin{figure*}[h]
\centering
\includegraphics[width=1.0\textwidth,clip]{Fig9.eps}
\textbf{\caption{Figure of Merit and Power Factor After putting phononic part.}}
\end{figure*}

\section{Conclusion}    

The impact of biaxial strain on the structural, magnetic, electronic, and thermoelectric characteristics of the MST compound under strain engineering has been carefully studied. In experimental paper the nature of the MST material is metallic with zero band gap reported, the features of polaronic transport and very less amount of thermal conductivity which is enhanced by applying magnetic field (B). The experimental values of ZT is very small 0.01 around 300K but relatively constant thermopower PF. In this work, applying biaxial external field (strain engineering, tensile and compressional strain) to enhanced the value of band gap from −3$\%$ to +10$\%$ to check the thermoelectric characteristics of the MST material by using the BoltzTraP package. Our calculation revealed that, at 5$\%$ tensile strain MST material shows the indirect bandgap of semiconductor which is 0.732 eV between $\Gamma - K$ High symmetry points. The approximation of Wc-GGA+TB+mBJ also show a good bandgap at 4$\%$ tensile strain which is 1.24 eV and also a best application for thermoelectric devices. The addition of electronic and phononic part, the value of ZT is 0.555 (p-type) and ZT is 0.564 (n-type) at 300K temperature.   
\cleardoublepage
    
\begin{thebibliography}{28}
\addcontentsline{toc}{chapter}{Bibliography}
		
				
\bibitem{d1} T. Jinushi, M. Okahara, Z. Ishijima, H. Shikata, and M. Kambe, \enquote{Development of the high performance thermoelectric modules for high temperature heat sources} Mater. Sci. Forum. \textbf{534}, 1521--1524, (2007).

\bibitem{d2} A. Redko, O. Redko, and R. DiPippo, \enquote{9 Industrial Waste Heat Resources} Int. J. Energy Res. \textbf{8}, 329--362, (2020).

\bibitem{d3} R. Aridi, J. Faraj, S. Ali, T. Lemenand, M. Khaled \enquote{Thermoelectric power generators: state-of-the-art, heat recovery method, and challenges} Electr. \textbf{2}, 359--386, (2021).

\bibitem{d4} H. Alqurashi, B. Hamad, \enquote{Magnetic structure, mechanical stability and thermoelectric properties of VTiRhZ (Z= Si, Ge, Sn) quaternary Heusler alloys: First-principles calculations} J. Appl. Phys. \textbf{127}, 1--11, (2021).

\bibitem{d5} M. A. Zoui, S. Bentouba, J. G. Stocholm and M. Bourouis \enquote{A review on thermoelectric generators: Progress and applications} Energies, \textbf{13}, 3601--3606, (2020).

\bibitem{d6} S. Sharma and U. Schwingenschlogl \enquote{Thermoelectric response in single quintuple layer Bi$_2$Te$_3$} ACS Energy Lett. \textbf{1}, 875--879, (2016).

\bibitem{d7} X. Zhang, Y. Zhao, Q. Song, S. Jia, J.Shi and W.Han \enquote{Magnetic anisotropy of the single-crystalline ferromagnetic insulator Cr$_2$Ge$_2$Te$_6$} Jpn. J. Appl. Phys. \textbf{55}, 1--16, (2016).

\bibitem{d8} C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang and Z.Q. Qiu \enquote{Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals} Nature, \textbf{546}, 265--269, (2017).

\bibitem{d9} X. Tang, D. Fan, L. Guo, H. Tan, S. Wang, X. Lu, X. Cao, G. Wang, and X. Zhou \enquote{Achiev-ing higher thermoelectric performance for p-type Cr$_2$Ge$_2$Te$_6$ via optimizing doping} Appl. Phys. Lett. \textbf{113}, 263902--263918, (2018).

\bibitem{d10} T. J. Williams, A. A. Aczel, M .D. Lumsden, S .E .Nagler, M. B.Stone, J. Q. Yan, and D.Mandrus \enquote{Magnetic correlations in the quasi-two-dimensional semiconducting ferromag-net CrSiTe$_3$} Phys. Rev. B \textbf{92}, 144404--144413, (2015).

\bibitem{d11} A. F. May, Y. Liu, S. Calder, D. S. Parker, T. Pandey, E. Cakmak, H. Cao, J. Yan, and M. A. McGuire, \enquote{Magnetic order and interactions in ferrimagnetic Mn$_3$Si$_2$Te$_6$} Phys. Rev. B \textbf{95}, 174440--174449, (2017).

\bibitem{d12} L. M. Martinez, H. Iturriaga, R. Olmos, L. Shao, Y. Liu, T. T. Mai, C. Petrovic, A. R. Hight Walker and S. R. Singamaneni \enquote{Enhanced magnetization in proton irradiated Mn$_3$Si$_2$Te$_6$ van der Waals crystals}  Appl. Phys. Lett. \textbf{116}, 172404--172419, (2020).

\bibitem{d13} A. F. May, H. Cao, and S. Calder \enquote{Magnetic properties of ferrimagnetic Mn$_3$Si$_2$Se$_6$} J. Magn. Magn. Mater. \textbf{511}, 166936--166947, (2020).

\bibitem{d14} M. Yazdanmehr, S. J. Asadabadi, A. Nourmohammadi, M. Ghasemzadeh, M. Rezvanian \enquote{Electronic structure and band gap of Al2O3 compound using Mbj exchange potential} Nanoscal. Resear. Lett. \textbf{7}, 1--10, (2012).

\bibitem{d15} P. Blaha , K. Schwarz, G. K. H Madsen, D. Kvasnicka, \enquote{wien2k} APW+lo method \textbf{60}, 1--7, (2001).

\bibitem{d16} G. K. Madsen, and D. J. Singh, \enquote{BoltzTraP. A code for calculating band-structure dependent quantities} Comput. Phys. Commun., \textbf{175}, 67--71, (2006).

\bibitem{d17} K. Schwarz, P. Blaha, G. K. H Madsen, \enquote{Electronic structure calculations of solids using the WIEN2k package for material sciences} Comput. Phys. Commun. \textbf{147}(1-2), 71--76, (2002).

\bibitem{d18} P. Blaha, K. Schwarz, P. Sorantin, and S. B. Trickey, \enquote{Full-potential, linearized augmented plane wave programs for crystalline systems} Comput. Phys. Commun., \textbf{59}, 399--415, (1990).

\bibitem{d19} J. P. Perdew, K. Burke, and M. Ernzerhof, \enquote{Generalized gradient approximation made simple} Phys. Rev. Lett, \textbf{77}, 3865--3876, (1996).

\bibitem{d20} F. Birch, \enquote{Finite elastic strain of cubic crystals} Physical review \textbf{71}, 809, (1947).

\bibitem{d21} Y. Liu, Z. Hu, M. Abeykoon, E. Stavitski, K. Attenkofer, E. D Bauer, C. Petrovic, \enquote{Polaronic transport and thermoelectricity in Mn$_3$Si$_2$Te$_6$ single crystals} Phys. Rev. Lett. \textbf{103}, 245117--245122, (2021).

\bibitem{d22} J. Recatala-Gomez, A. Suwardi, I. Nandhakumar, A. Abutaha, and K. Hippalgaonkar, \enquote{Toward accelerated thermoelectric materials and process discovery} ACS Appl. Energy Mater. \textbf{3}, 2240--2257, (2020).

\bibitem{d23} S. LeBlanc, S.K. Yee, M.L. Scullin, C. Dames, and K.E. Goodson, \enquote{Material and manufacturing cost considerations for thermoelectrics} Renew. Sust. Energ. Rev.\textbf{32}, 313--327, (2014).

\bibitem{d24} A.M. Reyes, J.L. Ponce-Ruiz, E.S. Hernndez, and A.R. Serrato, \enquote{Mechanical, optical and thermoelectric properties of Janus BiTeCl monolayer} ACS omega, \textbf{6}, 18364−18369, (2021).

\bibitem{d25} A. Soni, Y. Shen, M. Yin, Y. Zhao, L. Yu, X. Hu, Z. Dong, K. A. Khor, M. S. Dresselhaus, and Q. Xiong, \enquote{Interface driven energy filtering of thermoelectric power in spark plasma sintered Bi_{2}Te$_{2.7}$Se$_{0.3}$ nanoplatelet composites} Nano Lett. \textbf{105}, 031910--031915, (2012).

\bibitem{d26} W. Li, R. Yang, and M. Sun, \enquote{Superior thermoelectric properties of bulk and monolayer fullerene networks} J. Mater. Chem. A \textbf{11}, 3949--3960, (2023).

\bibitem{d27} A. H. Reshak, S. A. Khan, and S. Auluck, \enquote{Thermoelectric properties of a single graphene sheet and its derivatives} J. Mater. Chem., \textbf{2}, 2346−2352, (2014).

\bibitem{d28} M.Z. Mohyedin, M.F.M Taib, A. Radzwan, A. Shaari, M. Mustaffa, B.U. Haq, and M.Z.A Yahya, \enquote{First principles study of the effect of spin-orbit coupling on thermoelectric properties of Bismuth telluride} Comput. Theor. Chem, \textbf{1182}, 112844−112851, (2020).


\end{thebibliography}
\end{document}