First-Principles Calculations to Investigate Structural, Electronic, Optical and Magnetic Properties of Pyrochlore Oxides Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) for Energy Applications

Abstract

Three newly designed pyrochlore oxides, Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr), are analyzed for their magnetic, optical and electronic properties using ab-initio calculations within the context of density functional theory (DFT). We can refer these compounds as direct bandgap materials because there is a very slight difference between the height of bands at the Γ- and M-point. It is observed that bandgap engineering can be performed by replacing Hf with Sn and Zr. It is observed from total density of states (TDOS) plots that shape and height of curves is not the same in spin up and spin down channels, showing significant magnetic moment in these compounds. It is evident from magnetic properties that a major portion of total magnetic moment ($m_{tot}$) comes from Eu-atoms. In all compounds, the magnetic moment of O, Hf, Sn and Zr atoms is negative, whereas the magnetic moment of Eu-atoms is positive, showing their antiparallel arrangement. In both spin channels, significant absorption of the incoming photons is also shown by these compounds in the ultraviolet (UV) region. We can conclude on the basis of $R\left(\omega \right)$ that these compounds can be utilized in applications such as anti-reflecting coatings. These compounds are potential candidates for photovoltaic applications, such as solar cells, due to efficient absorption of incoming photons in visible and UV regions.

1. Introduction

Pyrochlore oxides with the general formal A₂B₂O₇ have attracted huge attention from scientists due to their interesting thermal, piezoelectric, ferroelectric and dielectric properties since their first discovery in the early 1960s [1,2,3]. Pyrochlores are promising candidates for broad applications in technological [4,5,6,7,8] and industrial fields such as luminescence powders [9,10], probes and sensors [11,12,13], thermal barrier coatings [14,15] and solid oxide fuel cells [16]. Generally, pyrochlore oxides of the A₂B₂O₇ type crystallize into FCC (face centered cubic) crystals with the space group $Fd\overline{3}m$ (S. G. # 227). In ideal pyrochlore structures, rare earth element and transition metal occupies the A-site (16d) and B-site (16c), respectively. The anions take 8b and 48f Wyckoff positions to form an ordered structure. The large and small size cations are placed at the A- and B-site, respectively, in the crystal unit cell. The cation at the A- (positioned within a distorted polyhedron) and B-site (positioned within a distorted octahedron) are 8- and 6-fold coordinated with oxygen atoms, respectively [17,18]. The structural stability of pyrochlore oxides of the A₂B₂O₇ type is significantly influenced by the ionic radius of the A- and B-site cations. In these type of pyrochlores, the stable and suitable ordered cubic phase exists when the radii ratio of A- and B-site cations $\left(\raisebox{1ex}{$r_A$}\!\left/ \!\raisebox{-1ex}{$r_B$}\right.\right)$ are between 1.46 and 1.78. The crystal geometry transforms to monoclinic from the cubic phase when this value $\left(\raisebox{1ex}{$r_A$}\!\left/ \!\raisebox{-1ex}{$r_B$}\right.\right)$ is greater than 1.78. On the other hand, a disordered (anion deficient fluorite structure) structure is formed, which conforms with the $Fm\overline{3}m$ space group. In defective fluorite structures, A- and B-site ions are randomly distributed at the 4a (cation sites) Wyckoff position, and the oxygen vacancies are disordered at 8c (anion sites) Wyckoff positions [19,20]. In addition, this temperature- and pressure-influenced phase transformation can significantly affect physical properties (such as conductivity and compressibility) of the crystal [21,22].

Augilar et al. [23] reported that the efficiency of solar cells can be enhanced by using a mixture of Tm₂Ti₂O₇ and TiO₂ for the photoelectrode while studying Tm₂Ti₂O₇ for the dye-sensitized solar cells. N. Ullah et al. [24] reported that Y₂M₂O₇ (M = Ti, V and Nb) pyrochlores are semiconducting compounds with excellent optical properties while studying their optical and electronic properties. Magnetic properties of platinum base pyrochlore oxides are also reported in the literature [25,26]. For industrial applications (such as thermoelectric and photovoltaic devices), developing suitable materials is the primary goal of researchers in the field of renewable energy. Exceptional chemical and physical properties are a prerequisite for these materials. For catalysis and electrochemistry, Re-X-O ternary systems exhibit numerous desired properties. Here, Re and X are used for rare earth elements and platinum group metals, respectively [27]. For phase equilibrium and thermodynamic properties, various ternary Re-X-O have extensively been investigated at high temperatures [28]. For numerous technological applications, La₂Pd₂O₇ is a promising candidate from this family [29]. Ca_1-xBi_xPd₃O₄ (n-type semiconductor) and Ca_1-xLi_xPd₃O₄ (p-type semiconductor) are both fascinating candidates for thermoelectric applications. Their low mobility due to large effective mass is the primary obstacle for their usability in thermoelectric (TE) devices [30]. Attfield et. Al. [31] used a simple modeling technique along with powder diffraction to analyze crystalline structures of pyrochlores such as La₄PdO₇ and La₂Pd₂O₅. Various types of energy production methods are available in the literature [32,33].

Pyrochlore oxides are effective photon absorbers as well as promising candidates for TE device applications due to appropriate TE characteristics. In this study, electronic, optical and magnetic properties of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are investigated using first-principles-based DFT calculations. Ground state properties (electronic, optical and magnetic) of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are studied using the full-potential linearized augmented plane wave (FP-LAPW) method to obtain an insight of their prospective applications in photovoltaic, spintronic and TE devices. The DFT technique is used to calculate optoelectronic, structural and magnetic properties of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) to evaluate their potential for various industrial applications such as solar cells and memory devices.

2. Results and Discussion

In this section, we discuss the investigated results for electronic, optical and magnetic properties of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr). Their potential for clean energy production using photovoltaic devices has been explored. For optoelectronic applications, usability of these pyrochlore oxides was investigated using their optoelectronic characteristics. The magnetic properties are key parameters to explore the potential of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) for spintronic applications such as memory devices. Based on the presented results, it is established that Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) has huge potential for spintronic and optoelectronic devices.

2.1. Electronic Properties

In this section, we discuss the investigated electronic properties (density of states (DOS)/band structure) for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr). The role of various inter-band/intra-band electronic transitions is important to determine the adequacy of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) for prospective optoelectronic devices. In the first Brillouin zone, high symmetry points were used to present energy band structures (from −3.0 to 5.0 eV) with the Fermi level ($E_g$) at 0 eV. An energy range of −6.0 and 6.0 eV was used to present DOS spectra that give vital information regarding the feasible transitions between valance/conduction bands.

2.1.1. Energy Band Structure

For solid materials, the allowed and forbidden electronic energy levels are represented by band structures. In crystalline materials, the energies of the crystal orbitals are presented using 2D band structures. A band structure plot, sometimes called a “spaghetti diagram”, can quickly reveal the nature of materials such as insulating, semi-metallic or metallic. The nature of the energy bandgap (direct or indirect) for semiconductor materials as well as magnitude of the bandgap can also be established using band structure plots. The difference of the energies of valance band maxima (VBM) and conduction band minima (CBM) is known as the energy bandgap. Moreover, the carrier mobility can also be determined using the curvature of these bands. Energy band structures for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are calculated along high symmetry points R→Γ→X→M→Γ. The calculated band structure plots for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are presented in Figure 1. We can refer to these compounds as direct bandgap materials because there is very slight difference between the height of bands at the Γ- and M-point. The values of energy bandgaps for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇ were 1.78, 1.85 and 1.80 eV, respectively, in the spin up channel. However, in the spin down channel, the values of energy bandgaps were 4.02, 2.42 and 3.98 eV for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively. It was observed that bandgap engineering could be performed by replacing Hf with Sn and Zr.

2.1.2. Density of States

The number of various energy states that electrons can occupy at a specific energy level, or the number of electronic states per unit volume per unit energy, are known as the density of states (DOS). This function defines bulk properties of conductive solids such as paramagnetic susceptibility, specific heat and other transport phenomena. In semiconductors, energy band spacing can be determined using DOS calculations, as well as the general distribution of states as a function of energy. The calculated total density of states (TDOS) plots for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are presented in Figure 2. We can note from TDOS plots that the shape and height of curves are not the same in spin up and spin down channels, showing significant magnetic moment in these compounds. It can be noted that TDOS of Eu₂Hf₂O₇ are greater compared to Eu₂Sn₂O₇ and Eu₂Zr₂O₇, which means that Eu₂Hf₂O₇ has a greater probability for electronic transitions between valance and conduction bands.

The partial density of states gives information about the most probable electronic transitions between VB and CB from various sub-orbitals of the atoms. The calculated partial density of states (PDOS) plots for Eu₂Hf₂O₇ are presented in Figure 3. We can split the valance band of Eu₂Hf₂O₇ into two segments: (i) −6.0 to −4.0 eV and (ii) −4.0 to 0 eV. In segment (i), all compounds were approximately silent in the spin down channel; however, significant contributions in the spin up channel from Eu-atoms are evident from Figure 3. From PDOS plots, significant contributions from Eu[$4f^7$] orbitals are evident in the spin up channel, along with minor contributions from O[$2p^4$] orbitals. In segment (ii), O-atoms were actively participating in VB, whereas minor contributions from Hf-atoms are evident in both spin channels from Figure 3. From PDOS plots, significant contributions from O[$2p^4$] and Hf[$5d^2$] orbitals are evident in both spin channels, along with minor contributions from Hf[$5p^6$] and Hf[$4f^{14}$] orbitals. After that, there is a forbidden region for electrons known as the energy bandgap until 1.78 eV in the spin up channel and 4.02 eV in the spin down channel. A notable peak at 1.78 eV due to the hybridization of O, Eu and Hf-atoms was present in the spin up channel. In this peak, major contributions came from Eu[$4f^7$] orbitals, along with minor contributions from O[$2p^4$] and Hf[$5d^2$] orbitals.

The calculated PDOS plots for Eu₂Sn₂O₇ are presented in Figure 4. We can split the valance band of Eu₂Sn₂O₇ into two segments: (i) −6.0 to −4.0 eV and (ii) −4.0 to 0 eV. In segment (i), all compounds were approximately silent in the spin down channel; however, significant contributions in the spin up channel from Eu-atoms are evident from Figure 4, along with minor contributions from Sn-atoms. From PDOS plots, significant contributions from Eu[$4f^7$] orbitals are evident in the spin up channel, along with minor contributions from Sn[$5s^2$] orbitals. In segment (ii), O-atoms were actively participating in VB, whereas minor contributions from Sn-atoms are evident in both spin channels from Figure 4. From PDOS plots, significant contributions from O[$2p^4$] and Sn[$5p^2$] orbitals are evident in both spin channels, along with minor contributions from Sn[$4d^{10}$] orbitals. After that, there is a forbidden region for electrons known as the energy bandgap until 1.85 eV in the spin up channel and 4.42 eV in the spin down channel. A notable peak at 1.85 eV due to the hybridization of O and Eu-atoms was present in the spin up channel. In this peak, major contributions came from Eu[$4f^7$] orbitals, along with minor contributions from O[$2p^4$] orbitals.

The calculated PDOS plots for Eu₂Zr₂O₇ are presented in Figure 5. We can split the valance band of Eu₂Zr₂O₇ into two segments: (i) −6.0 to −4.0 eV and (ii) −4.0 to 0 eV. In segment (i), all compounds were approximately silent in the spin down channel; however, significant contributions in the spin up channel from Eu-atoms are evident from Figure 5. From PDOS plots, significant contributions from Eu[$4f^7$] orbitals are evident in the spin up channel, along with minor contributions from O[$2p^4$] orbitals. In segment (ii), O-atoms were actively participating in VB, whereas minor contributions from Zr-atoms are evident in both spin channels from Figure 5. From PDOS plots, significant contributions from O[$2p^4$] and Zr[$4d^2$] orbitals are evident in both spin channels, along with minor contributions from Zr[$4p^6$] orbitals. After that, there is a forbidden region for electrons known as the energy bandgap until 1.80 eV in the spin up channel and 3.98 eV in the spin down channel. A notable peak at 1.80 eV due to the hybridization of O, Eu and Zr-atoms was present in the spin up channel. In this peak, major contributions came from Eu[$4f^7$] orbitals, along with minor contributions from O[$2p^4$] and Zr[$4d^2$] orbitals.

2.2. Magnetic Properties

In this section, we discuss the calculated magnetic properties for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) to obtain an insight of prospective spintronic applications such as memory devices. The magnetic moment in Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) derives from the notable anisotropy present in occupied electronic states in spin up and spin down channels. In

Table 1. The calculated magnetic moments for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr).

Compound	$\mathbf{Magnetic} \mathbf{Moment} \left({\mathit{\mu }}_{\mathit{B}}\right)$
Eu2Hf2O7	$m_{int}$	$m_O$	$m_{Hf}$	$m_{Eu}$	$m_{tot}$
	0.31875	−0.37218	−0.00304	24.05657	24.00011
Eu2Sn2O7	$m_{int}$	$m_O$	$m_{Sn}$	$m_{Eu}$	$m_{tot}$
	0.33198	−0.38122	−0.00204	24.04723	24.00003
Eu2Zr2O7	$m_{int}$	$m_O$	$m_{Zr}$	$m_{Eu}$	$m_{tot}$
	0.32337	−0.3775	−0.00344	24.05768	24.00012

, we present calculated values of total/partial magnetic moments for aforesaid compounds. It is evident from Table 1 that a major portion of total magnetic moment ($m_{tot}$) comes from Eu-atoms. All compounds have nearly equal values of total magnetic moments. In all compounds, the magnetic moment of O, Hf, Sn and Zr atoms is negative, whereas the magnetic moment of Eu-atoms is positive, showing their antiparallel arrangement. Therefore, Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are promising candidates for prospective spintronic applications.

2.3. Optical Properties

In this section, we have discuss the investigated optical characteristics for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) to obtain an insight of the prospective optoelectronic applications such as solar cells. For targeted materials, results obtained from band structure calculations can be used to calculate their optical properties. To examine how a material responds to incident radiation, it is crucial to analyze its optical characteristics. Electronic transitions between the unoccupied and occupied states are the result of interactions between electric field and photons. Frequency-dependent complex dielectric function $\epsilon \left(\omega \right)$ is the basic linear optical parameter that can be used to explain a crystal’s response to the incident electromagnetic radiations. Equation (1) can be used to formulate $\epsilon \left(\omega \right)$ [34].

$$\epsilon \left(\omega \right)={\epsilon }_1\left(\omega \right)+i{\epsilon }_2\left(\omega \right)$$

(1)

where ${\epsilon }_1\left(\omega \right)$ and ${\epsilon }_2\left(\omega \right)$ are, respectively, real and imaginary parts of the dielectric function. The electronic structure of crystalline materials can be explained using the most effective tool known as the optical absorption spectra. The real ${\epsilon }_1\left(\omega \right)$ and imaginary ${\epsilon }_2\left(\omega \right)$ part of dielectric function and other dependent parameters (on ${\epsilon }_1\left(\omega \right)$ and ${\epsilon }_2\left(\omega \right)$) are investigated theoretically using an equilibrium constant of 0–14 eV. Equation (1) can be used to formulate ${\epsilon }_2\left(\omega \right)$:

$${\epsilon }_{2 }\left(\omega \right)=\frac{V{e}^2}{2\pi m^2{\omega }^2}\int d^{3}k{\sum }_{n,n^{\prime }}\langle k,n|p|k,{n^{\prime }\rangle }^{2}f\left(kn\right)\left[1-f\left(k{n}^{\prime }\right)\right]\delta \left(E_{kn}-E_{k{n}^{\prime }}-\omega \right)$$

(2)

The first Brillouin zone is used to perform the integration in the above equation. The following dipole matrix elements are used to specify matrix elements for direct transitions between VB and CB:

$$M_{cv}=\langle u_{ck}|\delta .\nabla |u_{vk}\rangle$$

(3)

where vector potential of the electric field is represented by d. The imaginary part ${\epsilon }_2\left(\omega \right)$ of the complex dielectric function and Kramer’s Kroning relation (KKR) is used to formulate ${\epsilon }_1\left(\omega \right)$ using Equation (4) [35]:

$${\epsilon }_{1 }\left(\omega \right)=1+\frac{2}{\pi }{\int }_0^{\infty }\frac{{\epsilon }_{2 }\left(\omega \right){\omega }^{\prime }d{\omega }^{\prime }}{{\omega }^{\prime 2}-{\omega }^2}$$

(4)

The calculated spectra of ${\epsilon }_1\left(\omega \right)$ and ${\epsilon }_2\left(\omega \right)$ for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) using the GGA+U approximation are shown in the Figure 6. The major peaks around 6.0 eV in the spectra of ${\epsilon }_1\left(\omega \right)$ after static values show that the maximum dispersion of the incoming photons by these materials occurs at this value. We can spot in the spectra of ${\epsilon }_1\left(\omega \right)$ that the curves of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) enter the negative region at approximately 10 eV. The materials show a metallic behavior in the negative region, and values at which the curves cross the zero energy axis is known as the plasmon frequency. Values of the ${\epsilon }_1\left(\omega \right)$ at zero frequency ($\omega =0$) are known as static values of the dielectric function ${\epsilon }_1\left(0\right)$. The values of ${\epsilon }_1\left(0\right)$ and $E_g$ are inversely related with each other using Equation (5), known as Penn’s model [36]:

$${\epsilon }_{1 }\left(0\right)=1+\left(\frac{h{\omega }_p}{E_g}\right)$$

(5)

Calculated values of ${\epsilon }_1\left(0\right)$ in the spin up channel are approximately 2,48, 2.53 and 2.61 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively. However, calculated values of the ${\epsilon }_1\left(0\right)$ for the spin down channel are approximately 2.11, 2.13 and 2.15 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively.

An important parameter that explains the optical transitions between VB and CB is known as the imaginary part ${\epsilon }_2\left(\omega \right)$ of the dielectric function. The calculated spectra of ${\epsilon }_2\left(\omega \right)$ for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) using the GGA+U approximation are shown in Figure 6. Notable peaks in the visible region are due to the presence of an intermediate band in the spin up channel. In both spin channels, significant absorption of the incoming photons is also shown by these compounds in the UV region. Maximum values of ${\epsilon }_2\left(\omega \right)$ occur around 6.0 eV for all compounds and in both spin channels. Inter-band transitions between the VB and CB are responsible for these points. The selection rule states that the allowed transitions are only those having change in the angular momentum $\Delta l=\pm 1$. From Figure 6, it is evident that initially, there is no peak in ${\epsilon }_2\left(\omega \right)$ plots and originates from some finite energy values known as threshold energies. In the spin up channel, the values of the threshold energy for ${\epsilon }_2\left(\omega \right)$ are 1.8 eV for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr). However, for the spin down channel, the threshold values of ${\epsilon }_2\left(\omega \right)$ are 3.97, 2.78 and 3.98 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively. These compounds are potential candidates for photovoltaic applications such as solar cells due to efficient absorption of incoming photons in visible and UV regions.

Calculated values of the ${\epsilon }_1\left(\omega \right)$ and ${\epsilon }_2\left(\omega \right)$ are used to calculate other optical parameters such as the absorption coefficient $\alpha \left(\omega \right)$, real optical conductivity $\sigma \left(\omega \right)$, refractive index $n\left(\omega \right)$, extinction coefficient $K\left(\omega \right)$, reflectivity $R\left(\omega \right)$ and energy loss function $L\left(\omega \right)$. Deep knowledge of these optical parameters is required while designing efficient optical devices, i.e., ultraviolet (UV) photodetectors, semiconductor lasers and optically transparent electrodes. The following equations are used to calculate the aforementioned optical parameters [37]:

$$\tilde{n}\left(\omega \right)=n\left(\omega \right)+ik\left(\omega \right)$$

(6)

$$n\left(\omega \right)=1+\frac{2}{\pi }P{{\displaystyle \int }}_0^{\infty }\frac{k\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$

(7)

$$k\left(\omega \right)=-\frac{2}{\pi }P{{\displaystyle \int }}_0^{\infty }\frac{n\left({\omega }^{\prime }\right)-1}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$

(8)

$$n\left(\omega \right)={\left[\frac{\sqrt{\left({\epsilon }_1^2\omega \right)+{\epsilon }_2^2\left(\omega \right)}}{2}+\frac{{\epsilon }_1\left(\omega \right)}{2}\right]}^{\frac{1}{2}}$$

(9)

$$k\left(\omega \right)={\left[\frac{\sqrt{{\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)}}{2}+\frac{{\epsilon }_1\left(\omega \right)}{2}\right]}^{\frac{1}{2}}$$

(10)

$$n\left(0\right)=\sqrt{{\epsilon }_1\left(0\right)}$$

(11)

$$R\left(\omega \right)=\left|\frac{\tilde{n}\left(\omega \right)-1}{\tilde{n}\left(\omega \right)+1}\right|=\frac{{\left(1+n\right)}^2+k^2}{{\left(1-n\right)}^2+k^2}$$

(12)

$$\sigma \left(\omega \right)=\frac{\omega }{4\pi }{\epsilon }_2\left(\omega \right)$$

(13)

$$\alpha \left(\omega \right)=\sqrt{2\omega }{\left[\sqrt{{\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)}-{\epsilon }_1\left(\omega \right)\right]}^{\frac{1}{2}}$$

(14)

$$k\left(\omega \right)=\frac{c}{2\omega }\alpha \left(\omega \right)$$

(15)

$$L\left(\omega \right)=-Im\left(\frac{1}{\epsilon \left(\omega \right)}\right)=\frac{{\epsilon }_2\left(\omega \right)}{{\epsilon }_1^2\left(\omega \right)+{\epsilon }_2^2\left(\omega \right)}$$

(16)

Handsome knowledge of $n\left(\omega \right)$ is essential before opting an optical material for technological device applications. A potential photovoltaic compound must have some fundamental properties such as high optical conductivity, high absorption coefficient, low emissivity and high refractive index. The calculated spectra of $n\left(\omega \right)$ for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) using the GGA+U approximation are shown in the Figure 7. The major peaks around 6.0 eV in the spectra of $n\left(\omega \right)$ after static values show that the maximum dispersion of the incoming photons by these materials occurs at this value. We can spot in the spectra of $n\left(\omega \right)$ for the spin up channel that the curves of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) become less than unity at approximately 9.0 eV. Similarly, we can spot in the spectra of $n\left(\omega \right)$ for the spin down channel that the curves of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) become less than unity at approximately 10.0 eV. Values of the $n\left(\omega \right)$ at zero frequency ($\omega =0$) are known as static values of refractive index $n\left(0\right)$. The values of $n\left(0\right)$ and $E_g$ are related with each other using Equation (17) [38]:

$$n\left(0\right)=\sqrt{1+\left(\frac{h{\omega }_p}{E_g}\right)}$$

(17)

Calculated values of $n\left(0\right)$ for the spin up channel are 1.53, 1.57 and 1.61 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively. However, calculated values of the $n\left(0\right)$ for the spin down channel are 1.49, 1.49 and 1.52 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively.

The calculated spectra of $K\left(\omega \right)$ for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) using the GGA+U approximation are shown in Figure 7. We can note that ${\epsilon }_2\left(\omega \right)$ and $K\left(\omega \right)$ have an analogous profile. Notable peaks in the visible region are due to the presence of an intermediate band in the spin up channel. In both spin channels, significant absorption of the incoming photons is also shown by these compounds in the UV region. Maximum values of $K\left(\omega \right)$ occur around 8.0 eV for all compounds and in both spin channels. From Figure 6, it is evident that initially there is no peak in $K\left(\omega \right)$ plots and originates from some finite energy values known as threshold energies. In the spin up channel, the values of the threshold energy for $K\left(\omega \right)$ is 1.9 eV for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr). However, threshold values of $K\left(\omega \right)$ are 3.97, 2.78 and 3.98 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively. We can conclude on the basis of these results that the compounds under study are potential candidates for photovoltaic applications such as solar cells due to efficient absorption of incoming photons in the visible region.

The ratio of reflected to incident photons for a material is known as reflectivity $R\left(\omega \right)$. The calculated spectra of $R\left(\omega \right)$ for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) using the GGA+U approximation are shown in Figure 8. It is evident from Figure 8 that these materials are weak reflectors of incident photons, as they reflect approximately 25% of incident photons from 0 to 12 eV. Values of the $R\left(\omega \right)$ at zero frequency ($\omega =0$) are known as static values of reflectivity $R\left(0\right)$. Calculated values of $R\left(0\right)$ for the spin up channel are 0.050, 0.052 and 0.058 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively. However, calculated values of the $R\left(0\right)$ for the spin down channel are 0.041, 0.43 and 0.49 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively. We can conclude on the basis of these results that these compounds can be utilized in applications such as anti-reflecting coatings.

Energy loss function $L\left(\omega \right)$ plays an important role while investigating optical properties of the materials. The entire energy range is covered by $L\left(\omega \right),$ consisting of elastically scattered and non-scattered electrons that are responsible for electronic transitions between VB and CB. Energy lost by the electrons passing through the material with sufficient energy is described by $L\left(\omega \right)$. Inter-band transitions are responsible for the major peaks in the spectra of $L\left(\omega \right)$. The calculated spectra of $L\left(\omega \right)$ for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) using the GGA+U approximation are shown in Figure 8. The major peaks specify the values of plasmon resonance, and major peaks in the spectra of $L\left(\omega \right)$ occur at approximately 13.1, 12.7 and 13.2 eV for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively, for the spin up channel. However, for the spin down channel, major peaks occur at approximately 12.8, 13.1 and 12.9 eV for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively.

The calculated spectra of real optical conductivity $\sigma \left(\omega \right)$ for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) using the GGA+U approximation are shown in the Figure 9. Notable peaks in the visible region are due to the presence of an intermediate band in the spin up channel. In both spin channels, significant absorption of the incoming photons is also shown by these compounds in the UV region. Maximum values of $\sigma \left(\omega \right)$ occur around 7.0 eV for all compounds and in both spin channels. From Figure 9, it is evident that initially, there is no peak in $\sigma \left(\omega \right)$ plots and originates from some finite energy values known as threshold energies. In the spin up channel, the values of the threshold energy for $\sigma \left(\omega \right)$ are 1.9 eV for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr). However, threshold values of $\sigma \left(\omega \right)$ are 2.0, 2.88 and 3.2 for Eu₂Hf₂O₇, Eu₂Sn₂O₇ and Eu₂Zr₂O₇, respectively. It is a well-known fact that excitations will be produced when photon energy is greater than $E_g$. We can conclude on the basis of these results that the compounds under study are potential candidates for photovoltaic applications such as solar cells due to efficient absorption of incoming photons in the visible and UV regions.

3. Materials and Methods

The electronic, optical and magnetic properties of pyrochlore oxides Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are calculated within the framework of DFT using the generalized gradient approximation (GGA), taking into account the effect of the Hubbard potential (GGA+U) as implemented in Wein2k [38]. An exceedingly precise, full potential linearized augmented plane wave (FP-LAPW) technique [39] was used for the calculations of optical and electronic properties of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr). Relaxed structures were used to calculate ground state properties of the aforementioned compounds. The unit cell was divided into two regions while working with the FP-LAPW technique: (a) atomic Sphere (Muffin-tin) and (b) interstitial region (IR).

Cut-off values for the plane wave basis set were assumed to be $R_{MT}\times K_{MAX}=7.0$ ($R_{MT}$ is smallest muffin-tin radius and $K_{MAX}$ is maximum length of fermi wave vector, respectively). The value of $l_{MAX}$ equal to 10 (angular momentum with maximum length) was taken to expand the wave function. Forces on the atoms in the unit cell were minimized up to ${10}^{-3}$ Ry/a.u to obtain relaxed geometry. Self-consistent field calculations were terminated when forces on atoms became less than ${10}^{-3}$ Ry/a.u. The value of −6.0 Ry was taken to ensure no leakage of charge from the core state. The crystalline structures for Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are shown in Figure 10.

4. Conclusions

The electronic, optical and magnetic properties of Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) were investigated using FP-LAPW-based ab-initio calculations. Band structure plots reveal that these compounds are direct bandgap semiconductors. The presence of the intermediate band in the spin up channel plays a vital role in electronic transitions between VB and CB. To obtain insight into the active involvement of various atoms and electronic states in the electronic transitions of these materials, partial/total densities of states were also examined. The shape and height of the curves were different for the spin up and spin down channels in the TDOS spectrum, which shows that these compounds have strong magnetic properties. Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) possess nearly the same values of magnetic moments. We discovered that band structures and DOS support the computed optical properties. Based on the calculated $n\left(\omega \right)$, we can say that Eu₂Tm₂O₇ (Tm = Hf, Sn, Zr) are optically active compounds, as their refractive index is between 1.0 to 2.0. It is concluded based on $R\left(\omega \right)$ results that these compounds are very weak reflectors of incident photons. These compounds reflect approximately 25% of the incident photons. The optical conductivity $\sigma \left(\omega \right)$ shows that these materials absorb the maximum of the incident photons in visible and UV regions. These compounds are promising candidates for potential optoelectronic applications such as solar cells working in visible and UV regions.