Electronic and Optical Properties of Mg2GeO4 Under Pressure Effect: ab Initio Study

: We report first-principles studies the structural, elastic, electronic, and optical properties of Mg 2 GeO 4 in orthorhombic structure, including pressure dependence of structural parameters, band structures, density of states, and optical constants up to 20 GPa. The calculated structural parameters are in good agreement with the available experimental values at zero pressure. The mechanical stability of Mg 2 GeO 4 has been confirmed by calculation of the elastic constants. And the non-uniform pressure dependence of the lattice parameters may also mean that Mg 2 GeO 4 undergoes anisotropic compression. Meanwhile, the pressure dependence of the electronic band structure, density of states and partial density of states of Mg 2 GeO 4 up to 20 GPa were presented. The band structures show a direct band gap for this compound and the calculated band gaps expend with increasing pressure. Moreover, the evolution of the dielectric function, absorption coefficient ( α(ω) ), reflectivity ( R(ω) ), and the real part of the refractive index ( n(ω) ) at high pressure are also presented. According to our work, we found that the optical properties of Mg 2 GeO 4 undergo a blue shift with increasing pressure.


Introduction
In recent years, the germanate forsterite compounds has attracted tremendous attention to be functional materials owing to their potential technological applications and various unique properties, such as optical catalytic activities, unusual magnetic, and ferroelectric properties [1][2][3][4] .
Among these compounds, Mg 2 GeO 4 is an end-member of the olivine solid solution series and they are regarded as the most important components in the Earth's upper mantle 5 . An understanding of electronic and optical properties in these systems is also fundamental interest in solid-earth geochemistry and geophysics [6][7] .
At room temperature and normal pressure, the Mg 2 GeO 4 has an orthorhombic crystal structure with the space group Pnma, in which four formula units are contained in the unit cell 8 . In the crystal, Ge atoms are coordinated with four O atoms to form GeO 4 tetrahedra, as in other silicate minerals. The GeO 4 tetrahedra are linked together by the Mg atoms lying between them 9 . Due to its unusual characteristics, many researchers have investigated the structural, mechanical and electronic properties of Mg 2 GeO 4 at ambient pressure. B. Reynard et al used Raman spectroscopic diffraction to determine that Mg 2 GeO 4 crystals structural at low pressures 10 . To explore the feasibility of this mechanism, A.
Schubnel et al performed deformation experiments on germanate forsterite compounds under differential stress at high pressure 11 . These studies have demonstrated that Mg 2 GeO 4 structure can metastably exist up to 20 GPa at ambient temperature. Moreover, it is well known that the optical properties of materials rely heavily on their electronic structures.
High pressure is an effective approach which can be used to regulate and control the requisite electronic structure and optical properties of these oxide-based materials. Here, we provide a clear picture about pressure dependence of Mg 2 GeO 4 geometrical structure, electronic band structure and optical properties up to 20 GPa. In this paper, the evolution of the dielectric function, absorption coefficient (α(ω)), reflectivity (R(ω)), and the real part of the refractive index (n(ω)) at high pressure are also presented. The aim of the present study is to investigate the Mg 2 GeO 4 properties of interest in the orthorhombic structure, with emphasis on their dependence on hydrostatic pressure.

Method of calculation
The present calculations were performed with the Vienna ab initio Simulation Package code (VASP), which makes use of the Perdew-Burke-Ernzerhof (PBE) exchange-correlation function applying the projector augmented wave method with GGA-PBE potentials [12][13][14]  Moreover, the different band and electronic density to states can induce different dielectric response. The dielectric function ε(ω) = ε (ω) + iε (ω) is an important function to describe the optical properties. Because the imaginary part ε (ω) is the pandect of optical properties, we use formula (1) to calculate the ε (ω) of Mg 2 GeO 4 . In the formula, the superscript c and ν represent conduction and valence bands, respectively. u is the electric-field vectors of incident light field.
The real part ε (ω) was calculated by the Kramers-Kronig relation (formula (2)). In the relation, p repents the principal value of the integral.
The formula (1) and (2) show the ε (ω) and ε (ω) are the response of the incident light.

Structural properties
We first performed an optimization of the geometry of the lattice parameters within both the GGA and LDA schemes. The optimized results of the primitive cell of Mg 2 GeO 4 in orthorhombic structure are shown in Table I The bulk modulus at ambient pressure and temperature is =104.5 GPa and its derivative is ′ = 9.72. The high bulk modulus shows that the sample is hardly to be compressed under pressure.
In order to study the mechanical stability of Mg 2 GeO 4 , we calculated the second-order elastic constants using the "stress-strain method".
The nine independent elastic constants of orthorhombic Mg 2 GeO 4 at 0 GPa are shown in Table II. For orthorhombic crystal, the mechanical stability requires the elastic constants satisfying the well-known Born stability criteria 18  Hence, for orthorhombic crystals under pressure, the mechanical stability requires that the elastic constants satisfy the following stability criteria: C − P > 0, (C + C + C + 2C + 2C + 2C + 3P) > 0, and

Band structures and density of states
At room temperature and normal pressure, Mg 2 GeO 4 is an insulator.
The effect of pressure on the electronic structure of Mg 2 GeO 4 is an important parameter for better understanding its optical properties [19][20] .

Optical properties
We are interested in the effect of pressure on the optical properties. Figure 8 shows the reflectivity ( (ω)), absorption coefficient ( (ω)), refractive index ( (ω)) and the extinction coefficient ( (ω)) at 0, 10, and 20 GPa with photon energy ranging from 0 to 30 eV. From Fig. 8, we can see that peaks of optical constants shift to higher energies with increasing pressure, which meaning undergo a blue shift. As shown in Fig. 8 (a), the values of R(ω) of a main peak are at 10.74 eV under normal pressure and the reflectivity further decrease rapidly in the high energy region.
With increase pressures, the main peak of R(ω) shifts to a higher energy region from 11.06 eV under a pressure of 10 GPa to 11.26 eV under a pressure of 20 GPa. Notably, as can be seen in Fig. 6  and reflectivity (R(ω)).
Meanwhile, the dielectric function is a crucial physical parameter describing the optical properties, and it reflects the linear response of materials to electromagnetic radiation. As shown in Fig. 9 (b), we notice there is a main peak at 9.09 eV in the ε (ω) spectrum of Mg 2 GeO 4 under no-pressure conditions. This peak takes its origin from the optical transitions between Ge 4p states in the conduction band. When the pressure increases from 0 to 20 GPa, the spectrum exhibits a significant blue shift without any notable shape changes. From Fig. 9 (a), the peaks of ε (ω) follow a similar trend of ε (ω) when applying an increasing pressure on Mg 2 GeO 4 . The zero frequency dialectric constant ε (0) are found to be about 2.04, 2.01, and 1.98 at 0, 10, and 20 GPa, respectively.
Interestingly, it also can be seen from Fig. 9 (b), as the pressure is increased to 20 GPa, the maximum peak of ε (ω) has moved from 9.09 to 9.92 eV. According to the non-uniform pressure dependence of the structural properties, both the energy gap growth and the blue shift of the optical properties are more strongly pronounced in the low pressure range compared to the high pressure region. We believe that the above results can help to offer a theoretical basis for the experiments and applications of alkaline-earth aluminates crystals.