Combination of optical properties acquired by first principles calculations with the empirical Tauc method for determining the band gap energy of alkaline-earth metal oxides: MO (M = Be,Mg,Ca,Sr,Ba)

Abstract

In this paper, we have investigated the structural, electronic and optical properties of alkaline earth oxides MO (M $=$ Be, Mg, Ca, Sr, Ba) using the full-potential linearized augmented plane wave method (FP-LAPW) within the framework of density functional theory (DFT), as implemented in the Wien2k package within the generalized gradient approximation (GGA) and GGA plus Tran and Blaha modified Becke and Johnson potential (TB-mBJ) for treating the exchange and correlation effects. The main objective of this work is to introduce the first principal calculations used for the determination of the optical parameters of alkaline-earth metal oxide materials before introducing them into the experimental Tauc formula to determine the band gap energy of both direct and indirect bang gap semiconductors. The obtained results are of great interest regarding the convergence of the obtained band gap energies using the combined method compared to several research works.

Keywords

Band gap Tauc method DFT electronic properties optical properties

1. Introduction

Alkaline earth metal oxides MO (M $=$ Be, Mg, Ca, Sr, Ba) have been thoroughly studied as they known to be potential candidates for technological applications [1]. BeO is considered as a good ceramic material due to its excellent mechanical, thermal and chemical properties compared to other alkaline earth oxides [2]. It is used as a moderator and an additive in the recently proposed accident tolerant nuclear fuel [3]. BeO crystallizes in the Wurtzite (B4) structure at ambient conditions [1]. Formichev et al. [4] have found in their experimental work a band gap value of 10.3 eV for the BeO. MgO is also gained more attention by researchers [5, 6], highly recommended in serval industrial and technological applications such as a catalysis ferroelectric and microelectronic devices. It crystallizes in cubic NaCl (B1) structure at ambient conditions with experimental band gap of 7.8 eV [7]. CaO with its rock-salt structure at ambient temperature has a wide band gap of 7.1 eV [8]. Nanocrystalline CaO is used as an absorbent to remove chemical oxygen demand (COD) from pulp and paper mill effluent [9]. Equilibrium properties of SrO in rock-salt structure were calculated within the Tersoff potential model by using Molecular Dynamics simulation [10], the experimental band gap of SrO was found to be 5.9 eV [11]. BaO has an indirect band gap at (X-X) direction with value of 2.10 eV in the cubic NaCl structure within First-principles calculations [12] when the experimental value is equal to 4.28 eV [13].

The Tauc method is a precise experimental method for determining the band gap energy of semiconductors from optical measurements; however, the measured band gap takes in consideration the ambient conditions and the structural defects omnipresent in all synthetized and natural materials, which is not reached by theoretical calculations. The main aim of this study is to calculate the band gap of alkaline earth metal oxides through two methods using both electronicand optical properties within the two approximations GGA and TB-mBJ. The electronic band gap is calculated directly from band structure when the optical band gap is calculated using the absorption coefficient of Tauc’s formula. We speculate that by introducing the optical parameters from the first principal calculations into the Tauc formula we narrow the difference between the simulation and experimental results.

2. Computational method

In this work we have used full potential linearized augmented plane wave (FP-LAPW) method [14] as implemented in the wien2k package [15] to determine the structural and optoelectronic properties of alkaline-earth metal oxides MO (M $=$ Be, Mg, Ca, Sr, Ba). Exchange and correlation effects were treated with the Perdew-Burke-Ernzerhof (PBE) [16] generalized gradient approximation (GGA) [17] and the Tran-Blaha-modified Beck-Johnson (TB-mBJ) functional [18]. We have used $R_{MT}K_{\max}=$ 8.5 as a cutoff parameter where $R_{MT}$ denotes the smallest muffin Tin radius of atoms and $K_{\max}$ is the maximum value of the reciprocal lattice vectors used in plane wave expansion. Self-consistent calculations of total energy were used with a precision of 10 ${}^{-5}$ Ry. The valence wave function inside the spheres are expended up to $l_{\max}=$ 10. The charge density was Fourier expanded up to the value $G_{\max}=$ 12 Ry ${}^{1/2}$ and the core energy cutoff was token as $-$ 60 Ry.

3. Results and discussion

3.1 Structural properties

The optimized lattice parameter of alkaline metal oxides in $B_{1}$ phase for MgO, CaO, SrO, BaO and in $B_{4}$ phase for BeO were determined by computing the total energies for various volumes. Lattice parameter $a_{0}$ , bulk modulus $B_{0}$ and its pressure derivative B’ were obtained by fitting to the Murnaghan equation of states [19]:

$\displaystyle E(V)=\frac{B_{0}V}{B^{\prime}_{0}}\left[{\frac{V_{0}/V^{B^{% \prime}_{0}}}{B^{\prime}_{0}-1}+1}\right]+E_{0}-\frac{V_{0}B_{0}}{B^{\prime}_{% 0}-1}$ (1)

Where

$\displaystyle B_{0}=\frac{\partial^{2}E}{dV^{2}}$ (2)

The lattice parameter of these compounds has a proportional relationship; it is found to be increase with the increase in size of the metal atoms contrary to the bulk modulus, which has a proportionally reversed relationship; it decreases with the increase of the atomic number of metal atoms [1]. The calculated structural parameters of MO (M $=$ Be, Ba, Sr, Mg, Ca) are regrouped together with other available results in Table 1. Our results are in good agreement with other theoretical and experimental results [1, 13, 20, 21, 22, 23, 24].

Table 1

Lattice parameters $a_{o}$ and $c$ , bulk modulus $B_{o}$ and its pressure derivative $B^{\prime}_{0}$ of MO (M $=$ Be, Mg, Ca, Sr and Ba) in comparison with the experimental and theoretical data

	$a_{o}$ (Å)	c (Å)	u	$B_{0}$ (GPa)	$B^{\prime}_{0}$	Ref
BeO	2.70 2.71 2.71 2.67 2.69	4.44 4.39 4.41 4.34 4.37	0.375 0.375 0.377 0.377 0.378	205 206 229.4 /	3.8 / 4.1 /	Our work a b c d
MgO	4.26 4.24 4.17 4.20 4.21 4.2	/	/	150.23 167 164.8 162 160.3 182	4.20 4.0 4.1 / / 3.92	Our work a c e f g
CaO	4.84 4.84 4.76 4.81 4.81 4.86	/	/	104.27 107 119.9 / 114 120	4.37 4.2 3.6 / / 4.4	Our work a c e f g
SrO	5.20 5.21 5.09 5.16 5.15 5.22	/	/	84.08 82 97.2 / 88 101	4.71 4.1 4.5 / / 4.09	Our work a c e f g
BaO	5.57 5.60 5.48 5.52 5.53 5.65	/	/	67.94 75 79.6 / 61–89 81	4.58 4.1 4.7 / / 4.2	Our work a c e f g

a Ref [1]. b Ref [20]. c Ref [21]. d Ref [22] Exp. e Ref [23] Exp. f Ref [13] Exp. g Ref [24].

Table 2

Energy band gap of MO (M $=$ Be, Mg, Ca, Sr, Ba)

Samples	GGA	TB-mBJ	GGA	TB-mBJ	Exp	Others
	Eg (eV)
	(Electronicproperties)		(Optical properties)
BeO	07.43	09.48	07.78	09.72	10.6 ${}^{\text{h}}$	07.2 ${}^{\text{i}}$ ; 07.44 ${}^{\text{j}}$
MgO	04.12	06.85	04.49	07.06	07.83 ${}^{\text{k}}$	04.43 ${}^{\text{i}}$
CaO	03.69	05.46	04.29	05.87	06.93 ${}^{\text{l}}$	04.16 ${}^{\text{i}}$ , 03.45 ${}^{\text{m}}$
SrO	03.79	06.08	03.70	05.34	05.7 ${}^{\text{l}}$	04.01 ${}^{\text{i}}$ , 03.9 ${}^{\text{n}}$
BaO	02.08	03.52	02.48	03.82	04.28 ${}^{\text{l}}$	02.27 ${}^{\text{i}}$ , 02.12 ${}^{\text{o}}$

h Ref [25]. i Ref [1]. j Ref [20]. k Ref [8]. l Ref [13]. m Ref [21]. n Ref [26]. o Ref [27].

Figure 1.

Electronic band structure of MO (M $=$ Be, Mg, Ca, Sr, Ba) within GGA and TB-mBJ.

Figure 2.

Total density of states of MO (M $=$ Be, Mg, Ca, Sr, Ba) within GGA and TB-mBJ.

Figure 3.

Partial density of states of MO (M $=$ Be, Mg, Ca, Sr, Ba) within GGA and TB-mBJ.

3.2 Electronic properties

The electronic band structures of MO (M $=$ Be, Mg, Ca, Sr, Ba) were calculated at equilibrium lattice parameter within GGA and TB-mBJ potentials. In Fig. 1, we have plotted the upper band from the valence region and the lower band from the conduction region within the two approximations GGA and TB-mBJ, where the Fermi energy level is set at zero energy and represented by horizontal dotted line. As a result, it is seen that the two compounds BeO and MgO have a semiconductor behavior with a direct band gap in ( $\Gamma$ - $\Gamma$ ) direction and in (X-X) direction for BaO. Indirect band gap in ( $\Gamma$ -X) direction was observed for CaO and SrO. The predicted band gap values within GGA were found to be 7.43, 4.12, 3.69, 3.79 and 2.08 eV for BeO, MgO, CaO, SrO and BaO respectively, where with TB-mBJ were found to be 9.48, 6.85, 5.46, 6.08 and 3.52 eV respectively. It is also observed that the obtained results from TB-mBJ approximation are better than those obtained from GGA. Our results are regrouped in Table 2 and compared with other available data [1, 8, 13, 20, 21, 25, 26, 27].

Figure 4.

Plots of $({\alpha h\nu})^{2}$ versus $h\nu$ of MO (M $=$ Be, Mg, Ca, Sr, Ba) within GGA and TB-mBJ.

The total and partial density of states of all these metal oxides are represented in Figs 2 and 3 respectively, it is shown that just one part is observed in the conduction band of all the alkaline earth metal oxides, this part mainly dominated for BeO by Be s, p and O s, p states with hybridization of s states of Oxygen with s states of berylium and another hybridization between p states of oxygen and p states of berylium. For MgO, the oxygen p states widely dominate in the conduction band with a little participation of magnesium s, p and d states. In CaO, the conduction band is dominated by the Ca d states with a little contribution of s and p states of oxygen. The DOS plots of SrO and BaO in the conduction band are very similar to the CaO DOS, we can see an important contribution of d states of Sr and Ba with a little contribution of s and p states of oxygen. The DOS plots in the valence band of BeO, MgO and SrO show two regions, the region just below the Fermi level with 5 eV, 4 eV and 1.5 eV of width for BeO, MgO and SrO respectively, the oxygen p states are mainly dominated with a little contribution of s, p and d states of the metal atoms. The lower region of the valence band presents 2 eV, 1.5 eV and 4 eV of width for BeO, MgO and SrO respectively with great contribution of oxygen s states for BeO and MgO whereas SrO represents a contribution of oxygen s states and Strontium p states in the lower region of the valence band. Three regions were observed in the valence band of CaO, the first one near Fermi level is dominated by p states of oxygen with little contribution of Ca s, p, d states where the middle one is dominated by oxygen s states and Ca p states, The lower part is dominated by oxygen s states with Ca p states. We can distinguish four regions in the valence band of BaO with width values of 1.5 eV, 1.5 eV, 0.5 eV and 0.5 eV. Below the Fermi level, an important contribution of oxygen p states with a little of Ba s, p, d states was observed, the second band was dominated by Ba p and O s states, the third one was dominated by Ba p with O s states. The last one is mainly dominated by s states of Ba. In our previous paper [28], electronic properties of SnO ${}_{2}$ were calculated using both GGA and TB-mBJ approximations, the obtained values showed that the TB-mBJ well performed in the results compared to the GGA. The same remark in other works [29, 30].

3.3 Optical properties

It is well known that the calculation of optical properties plays an important role to understand the nature of material. The complex dielectric function $\varepsilon(\omega)$ is given by:

$\displaystyle\varepsilon(\omega)=\varepsilon_{1}(\omega)+i\varepsilon_{2}(\omega)$ (3)

This equation is used to describe the optical properties, where $\varepsilon_{1}(\omega)$ is the real part of the function, while $\varepsilon_{2}(\omega)$ is the imaginary part. The latter is expressed by [31]:

$\displaystyle\varepsilon_{2}(\omega)=\frac{4\pi^{2}}{m^{2}\omega^{2}}\mathop{% \sum}\limits_{V.C}\mathop{\int}\limits_{B.Z}d^{3}k\frac{2}{2\pi}|{eM_{CV}(k)}|% ^{2}\delta[{E_{C}(k)-E_{V}(k)-h\omega}]$ (4)

Where $h=\frac{h}{2\pi}$ , $m$ is the mass of free electrons, $e$ is the charge of free electrons, $\omega$ is the frequency of incident photons, $C$ and $V$ denote the conduction and valence bands respectively, BZ is the first Brillouin zone and $k$ is the reciprocal vector.

Figure 5.

Band gap of MO (M $=$ Be, Mg, Ca, Sr, Ba) as function as the atomic number of the metal atoms within GGA and TB-mBJ.

The absorption coefficient $\alpha(\omega)$ can be calculated using the optical parameters $\varepsilon_{1}(\omega)$ and $\varepsilon_{2}(\omega)$ according to the following expression [32]:

$\displaystyle\alpha(\omega)=\sqrt{2}\omega\left[{\sqrt{\varepsilon_{1}^{2}(% \omega)-\varepsilon_{2}^{2}(\omega)}-\varepsilon_{1}(\omega)}\right]^{2}$ (5)

The optical band gap can be calculated via Tauc’sformula which exhibits the relationship between incident photon energy and absorption coefficient, this formula is presented as [33]:

$\displaystyle\alpha h\nu=A({h\nu-E_{g}})^{n}$ (6)

Where $\alpha$ is the absorption coefficient, $n$ is equal to 1/2 for indirect band gap and equal to 2 for direct band gap, $A$ is a constant given by [34]:

$\displaystyle A=\frac{e^{2}}{\pi\textit{ncm}_{e}^{\ast}h^{2}}({2m_{r}})^{3/2}$ (7)

Where $m_{e}^{\ast}$ and $m_{r}$ are the effective and reduced masses of charge carriers, respectively. $E_{g}$ is the optical band gap which was estimated by extrapolating the straight segment of the plot $({\alpha h\nu})^{2}$ versus $h\nu$ to zero absorption coefficient value.

Figure 4 shows the plots of $({\alpha h\nu})^{2}$ versus $h\nu$ of MO (M $=$ Be, Mg, Ca, Sr, Ba) compounds within both GGA and TB-mBJ approximations. The band gap values within GGA and TB-mBJ respectively were found to be 7.78 eV and 9.72 eV for BeO, 4.49 eV and 7.06 eV for MgO, 4.29 eV and 5.87 eV for CaO, 3.70 eV and 5.34 eV for SrO and finally 2.48 eV and 3.82 eV for BaO. The obtained values are listed in Table 2 for comparison with other theoretical and experimental results.

In one hand, we have found that the TB-mBJ approximation gives band gap values closer to the experimental values compared to the GGA approximation. In the other hand, we have found that the band gap values obtained from the optical properties are very close to the experimental results compared to those obtained from the electronic properties. We note that the energy band gap values calculated from the electronic properties are obtained by the difference between the minimum of the conduction band and the maximum of the valence band.

Regarding to Fig. 5 which represents the variation of band gap of MO (M $=$ Be, Mg, Ca, Sr, Ba) as function of the atomic number of the metal atoms within GGA and TB-mBJ approximations compared with the experimental results from literature, the band gap has a proportionally reversed relationship; it decreases with the increase of the atomic number of the metal atoms, this decrease is validated only with the values obtained from the optical properties using Tauc’s equation. As results, we can saythat the Tauc method based on optical measurements is an accurate method for obtaining satisfactory results of band gap.

4. Conclusion

In this work we have performed abinitio calculations to investigate the structural, electronic and optical properties of alkaline earth metal oxides MO (M $=$ Be, Mg, Ca, Sr and Ba) within GGA and TB-mBJ approximations. Structural results were found to be in good agreement with other theoretical and experimental results. Electronic properties display a direct band gap for BeO, MgO, BaO and indirect band gap for CaO and SrO with underestimated energy band gap. The usual problem of the DFT gap underestimationforced us to use the optical properties to obtain energy band gap values close to the experimental results. The calculated band gap values from these properties are in good agreement with experimental values. Note that the optical properties using Tauc’s formula can be useful by researchers using DFT calculationsto obtain satisfactory results of energy band gap.

Footnotes

Acknowledgments

This work was supported by Directorate General for Scientific Research and Technological Development (DGRSDT) – Ministry of Higher Education and Scientific Research of Algeria.

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