Abstract

In this work, we determined the structural and electronic properties of Zn_1−xMg_xO wurtzite minerals, where the Mg concentration, x, varies between 0 and 1. These properties were studied using the Quantum Espresso code, which is based on density functional theory (DFT) and pseudopotentials. All calculations were performed using the Generalized Gradient Approximation (GGA) with the exchange-correlation potential in the Perdew-Burke-Ernzerhof (PBE) formalism. The results show that the bandgap energies increase as the molar fractions of Mg increase. They also show that the valence band comprises three regions: a deep region resulting from the contribution of oxygen s states, an intermediate region resulting mainly from the contribution of zinc d states, and finally, a region corresponding to the valence band maximum and resulting essentially from oxygen p states. However, the conduction band consists mainly of zinc s states, magnesium s and p states, and oxygen p states.

Keywords

Wurtzite Zn_1?xMg_xO, Structural and Electronic, PDOS

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1. Introduction

Given their broad optical spectrum, ranging from near-visible UV to deep ultraviolet, oxides such as ZnO and MgO and their alloys have attracted considerable interest as materials for optoelectronic devices [1]-[7]. Furthermore, since the ionic radius of Mg²⁺ (0.57 Å) is similar to that of Zn²⁺ (0.60 Å), Zn can be replaced by Mg to form the alloy [8]. Thanks to their tunable bandgaps (3.3 to 7.8), low growth temperatures (100˚C - 750˚C), and radiation resistance, Zn_1−xMg_xO wurtzite ternary compounds are ideal materials for the development of solar cells [9]. Wurtzite Zn_1−xMg_xO alloys can be synthesized for x values up to 0.66 [10].

Over the past few decades, numerous theoretical studies based on density functional theory, in which the LDA [11] and GGA [12] approximations were used, have been conducted to investigate the electronic and structural properties of ZnO. However, these studies did not yield satisfactory results in terms of accuracy, as they underestimate the bandgap values and incorrectly place the energy levels for the Zn-3d states. To correct the problem of bandgap underestimation, the authors of [13] and [14] proposed using the Hubbard correction, which yielded satisfactory results. It is in this context that, in this work, the structural and electronic properties of wurtzite Zn_1−xMg_xO (x = 0 to 1 in steps of 0.25) were calculated using DFT with the GGA + U functional. However, the x = 1 end corresponds to pure MgO in the wurtzite phase. This phase is treated here as a purely hypothetical reference obtained by calculation for extrapolation purposes.

2. Calculation Method

We then constructed a 2 × 2 × 1 wz-ZnO supercell using the VESTA software, and subsequently used magnesium atoms to replace zinc atoms for each configuration. For each configuration, there are several ways to replace the zinc atoms with magnesium atoms. However, we selected the one with the highest symmetry. Our calculations were performed using the QUANTUM ESPRESO code [15]. It is based on density functional theory, plane waves, and pseudopotentials. We used the generalized gradient approximation (GGA) with the exchange-correlation potential in the Perdew-Burke-Ernzerhof (PBE) formalism [12]. The interaction between ions and electron for all atoms is described by ultrasoft pseudopotentials obtained from the Quantum Espresso website. To strike a balance between computation time and accuracy, the cutoff energy of the plane wave is set to 400 eV. Integration by sampling special points on the Brillouin zone is performed using the Monkhorst-Pack method [16] with a 4 × 4 × 2 k-point grid. The Hubbard correction for Up-O and Ud-Zn is 6 and 10 eV, respectively [17]. The Mg 3s², Zn 3d¹⁰4s², and O 2s²2p⁴ electrons are treated as valence states. In all calculations, the convergence thresholds and maximum force are 1 × 10⁻⁸ Ry and 0.001Ry/atom. The Brodyden-Fletcher-Goldfarb-Shanno (BFGS) minimization algorithm [18] was used for geometric and structural optimization.

The mathematical formalism of DFT was first developed by Kohn and Hohenberg and later by Kohn and Sham. In the Kohn-Sham formalism, the total energy is calculated as follows [11]:

$E\left(n\right)=T_s\left(n\right)+{\displaystyle \int v\left(r\right)n\left(r\right)\text{d}r+\frac{1}{2}}{\displaystyle \iint \frac{n\left(r\right)n\left(r^{\prime }\right)}{\left|r-r^{\prime }\right|}}\text{d}r\text{d}{r}^{\prime }+E_{xc}\left(n\right)$

E_xc is known as the exchange-correlation energy.

The E_xc energy is expressed as follows [19]:

$E_{xc}^{GGA}\left(n\right)={\displaystyle \int {\epsilon }_{xc}^{GGA}}\left[n\left(r\right),\nabla n\left(r\right)\right]d^{3}r$

The Hubbard-U correction is expressed by the following equation [20]:

$E_{GGA+U}\left[n\left(r\right)\right]=E_{GGA}\left[n\left(r\right)\right]+E_U\left[n\left(r\right)\right]-E_{dc}$

where E_GGA is the energy derived from the conventional GGA functional, E_U is the Hubbard energy, and E_dc is the double-counting correction energy.

3. Result and Discussion

3.1. Structural Properties

The lattice parameters of the equilibrium lattice for the unit cells of ZnO, MgO, and Zn_1−xMg_xO (where x ranges from 0.25 to 0.75 in steps of 0.25) were determined by relaxation calculations. For ZnO and MgO wurtzite, we obtained results of a = 3.2433 Å and c = 5.1960 Å, and a = 3.2577 Å and c = 5.0837 Å, respectively. Our values for ZnO are in agreement with theoretical [21]-[23] and experimental [14] [24] [25] results. However, the results found by Wang Zhi-Jun et al. (2009) [26] for a and b are slightly higher than ours. Regarding MgO, our results are consistent with those found theoretically or experimentally in the literature [24] [27]. However, the results reported by [22] [28] are higher than ours. For Zn1-xMgxO (0.25 < x < 0.75) wurtzite, our results were compared with those in [29] and are consistent. However, there is a slight difference between our values and those found in [22] [26] [28] [30]. Furthermore, we observe that the values of a increase with x and that those of c decrease as x increases. The variation of the lattice parameters a and c as a function of the Mg concentration x for Zn_1−xMg_xO in the wurtzite structure is shown in Figure 1 and Figure 2 below.

Figure 1. Variation of the mesh parameter a(x) as a function of the Mg content x.

Figure 2. Variation of the mesh parameter c(x) as a function of the Mg content x.

We observe that this variation is nonlinear. This nonlinearity of a and c as a function of x means that the two lattice parameters deviate from Vegard’s law, which states that the lattice parameters of ternary semiconductor alloys vary linearly with the molar fraction x (i.e., (1 − x) for ZnO and x for MgO in our case). Furthermore, this observation is consistent with numerous experimental and theoretical results that have reported a deviation from Vegard’s law in semiconductor alloys [22] [31] [32].

Therefore, the lattice parameters of Zn_1−xMg_xO can be written in the form [28]:

$a_{\left(Z{n}_{1-x}M{g}_{x}O\right)}=x{a}_{MgO}+\left(1-x\right)a_{ZnO}+bx\left(1-x\right)$

$c_{\left(Z{n}_{1-x}M{g}_{x}O\right)}=x{c}_{MgO}+\left(1-x\right)c_{ZnO}+bx\left(1-x\right)$

where the term b represents the linearity correction due to lattice distortion.

The polynomial fit of our data for a and c for the wurtzite Zn_1−xMg_xO yielded the following expressions:

$a_{{\text{Zn}}_{1-x}{\text{Mg}}_{\text{x}}\text{O}}=3.2423+0.0342\text{x}-0.0170{\text{x}}^2$

$c_{{\text{Zn}}_{1-\text{x}}{\text{Mg}}_{\text{x}}\text{O}}=5.1942+0.0130\text{x}-0.1228{\text{x}}^2$

Our results regarding the mesh parameters a and c are listed in Table 1 below. Data from certain authors in the literature are also included for comparison purposes.

Table 1. Equilibrium lattice parameters of wurtzite Zn_1−xMg_xO.

Continued

^aOur results.

3.2. Electronics Properties

3.2.1. Projected State Density

The properties of semiconductor materials are closely related to the electronic configuration of the atoms. Therefore, to study the distribution of electrons across the various orbitals, the PDOS (projected density of states) are calculated for the hexagonal wurtzite Zn_1−xMg_xO.

The valence band consists precisely of three regions, as shown in Figures 3-7 below.

The deepest region, or the region corresponding to the minimum of the valence band, consists mainly of the s states of the oxygen atom (O_s), with peaks observed between −15 and −17 eV regardless of the Mg concentration. The intermediate region, i.e., the region between the minimum and maximum of the valence band, is essentially composed of the d states of the zinc atom (Zn_d), with peaks located between −8 and −11 eV for all configurations except for x = 1 (MgO). Furthermore, it is observed that an increase in Mg concentration leads to a narrowing of the Zn d-orbital band width. In the case of MgO, this region originates primarily from the p states of the oxygen atom.

The highest peak of the Zn d states for ZnO is located at approximately −9.5 eV, which is higher than the experimental results. Göpel et al. [33] and Powell et al. [34] performed UV photoemission and angle-resolved photoemission measurements, respectively, on vacuum-cleaved wurtzite ZnO, both of which placed the Zn d-band at approximately −7.5 eV. The X-ray photoemission results reported by Ruckh et al. [35] (−8.2 eV), Vesely et al. [36] (−8.5 eV), and Ley et al. [37] (−8.81 eV) are also higher but closer to our result than that of Powell et al. Finally, the shallowest region that is, the one closest to the Fermi level consists mainly of the p states of the oxygen atom. This observation was made for all configurations. In this region, peaks are observed around −1.25 to −6.9 eV. Regarding the conduction band, the main contributions come from the s states of the zinc atom for ZnO (x = 0), with peaks located between 1.5 and 13 eV. For the other configurations (x = 0.25, 0.5, 0.75), the contributions come mainly from the s states of zinc, the p and s states of magnesium, and the p states of oxygen. However, the s states of zinc and the p states of magnesium are clearly dominant. For MgO, the conduction band consists of the s and p states of magnesium and the s and p states of oxygen. However, the main contributions come from the s and p states of magnesium.

The following figures (Figures 3-7) show the PDOS of the Zn_1−xMg_xO.

Figure 3. PDOS of the ZnO.

Figure 4. PDOS of the Zn_0.75Mg_0.25O.

Figure 5. PDOS of the Zn_0.5Mg_0.5O.

Figure 6. PDOS of the Zn_0.25Mg_0.75O.

Figure 7. PDOS of the MgO.

3.2.2. Band Structure

In the literature, the bandgap of the ZnO wurtzite structure calculated by GGA ranges from 0.74 to 0.813 eV [17] [38], which is significantly underestimated compared to the experimental value. This underestimation is due to the fact that the GGA approximation is deficient and underestimates the binding energy of the Zn d states, leading to strong hybridization with the oxygen p states [23]. Thus, strong p-d coupling ultimately resulted in a smaller electronic energy gap than known experimental data. To address this issue, we used the PBE-GGA+U approach. Indeed, the Hubbard term (U) corrects the strong p-d hybridization between Zn and O, thereby shifting the Zn d states. ZnO and MgO (x = 0 and x = 1, respectively) are characterized by a direct bandgap located at the center of the Brillouin zone at the Γ point, where the bandgap values Eg are 3.4 eV and 5.5 eV, respectively. Our result for ZnO is in agreement with the experimental value (Eg = 3.44 eV [35]). However, the value found for MgO (5.5 eV) is larger than the theoretically obtained value (5.328 eV) and smaller than the experimentally measured value (7.67 eV) reported by Taib et al. [39] and Y. Z. Zhu et al. [40], respectively.

The band structures for 0 < x < 1, respectively, are shown in the figures (Figures 8-12) below.

Figure 8. Band structure of ZnO.

Figure 9. Band structure of Zn_0.75Mg_0.25O.

Figure 10. Band structure of Zn_0.5Mg_0.5O.

Figure 11. Band structure of Zn_0.25Mg_0.75O.

Figure 12. Band structure of MgO.

Our results are presented in Table 2 below. There are differences between our results and those found in the literature, which are shown in the table. This difference may be due to differences in the structural model or in the arrangement of the constituent alloys. It is important to note that there are no points of intersection between the Fermi level and the energy bands for any of our configurations. Consequently, Zn_1−xMg_xO (x = 0, 0.25, 0.50, 0.75, and 1) in the hexagonal wurtzite structure can be considered a semiconductor material. In addition, we observe that the band energy of the Zn_1-xMg_xO ternary alloys increases as the Mg concentration (x) increases. Our results and some findings from the literature are summarized in Table 2 below.

Table 2. Gap band values for wurtzite Zn_1−xMg_xO.

The polynomial fit of our data for the band gap of wurtzite Zn_1−xMg_xO yielded the following expression:

$E_g\left(\text{x}\right)=3.4714+1.3885\text{x}+0.5714{\text{x}}^2$

In our calculations, we found a Bowing parameter b = 0.5714, which differs from the value found by A. Djelal et al. [22] and F.Z. Aouacheria et al. [28].

The variation of the band gap (Г - Г) as a function of Mg concentration, x, for hexagonal wurtzite Zn_1−xMg_xO is shown in Figure 13 below.

Figure 13. Change in the band gap as a function of the Mg concentration x.

4. Conclusions

In this study, we presented a comprehensive theoretical analysis of the structural and electronic properties of Zn_1-xMg_xO alloys, which were calculated using DFT + U. We observed a nonlinear behavior for both lattice parameters a and c, indicating a violation of Vegard’s empirical law.

Second, it is well known that the experimental bandgap is underestimated by conventional DFT. The use of the Hubbard correction U_d-Zn and U_p-O allowed for the correct reproduction of the band gap for ZnO, but not for MgO. Furthermore, the valence band generally consists of three main regions, and the conduction band was found to be dominated by the Zn_s, Mg_p, and Mg_s states.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References