Electronic, magnetic and optical properties of XScO₃ (X=Mo, W) perovskites

Structural and electronic properties

The MoScO₃ and WScO₃ Perovskites in the cubic form with space group is Pm-3m (#221) contain one formula unit and the Mo/W, Sc and O atoms are positioned at 1a (0, 0, 0), 1b (½, ½, ½) and 3c (0, ½, ½) sites of Wyckoff coordinates, respectively.

The lattice constants of XScO₃ structures are optimized using Murnaghan equation of state (Murnaghan, 1944) and the energy vs volume curves are presented in Fig. 1 along with the structure visualization. The energies have been shifted by the minimum energy value which is set to zero in the plot. Since, the structure is cubic only volume optimization is required.

The optimized lattice constants and bulk moduli at the equilibrium minimum energy ground state are shown in Table 1. We notice from the Table that the lattice constant of the perovskites decreases slightly with increasing atomic numbers of the TM. Although, the individual ionic radii of Mo and W are comparable the decrease in size of the compounds can be attributed to the change in electronic density and changes in the occupation of the band orbitals in new molecular environment.

The thermodynamic stability of the perovskites is important for the existence and synthesis of these compounds. We have evaluated the energies of formation E_form for the perovskites to give an indication of their stability and these values are presented in Table 1. The energy of formation is obtained by using Eq. (1) below:

(1) $$E_{form}=E_{XSc{O}_3}^{Bulk}-E_X^{Bulk}-E_{Sc}^{Bulk}-3E_O^{free}$$where, E_form is the formation energy of the perovskite; X = Mo/W; $E_{XSc{O}_3}^{Bulk}$; $E_X^{Bulk}$ and $E_{Sc}^{Bulk}$ are the bulk equilibrium total energies of the XScO₃ compound, the TM element and Sc respectively; and $E_O^{free}$ is the free energy of a single oxygen atom. The E_form obtained for MoScO₃ and WScO₃ are −1.958 and −1.848 Ry respectively. The negative values of formation energy indicate that the perovskites are energetically stable and can be easily synthesized experimentally.

In addition to the thermodynamic stability, the mechanical/structural stability of the perovskites is an essential factor that needs to be evaluated. The dynamic stability of a crystal is measured by its response to deformations and the evaluation of the elastic energy viz the elastic constants. If a system is deformed under equilibrium conditions arbitrarily through a small strain “e”, the change in energy would be as denoted in Eq. (2) below (Mouhat & Coudert, 2014).

(2) $${\mathrm{U}}_{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}={\displaystyle \frac{V}{2}{\sum }_{ij=1}^6{\mathrm{C}}_{\mathrm{i}\mathrm{j}}{\mathrm{e}}_{\mathrm{i}}{\mathrm{e}}_{\mathrm{j}}+\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{s}\left(\mathrm{H}\mathrm{O}\right)}$$

The number of independent elastic constants in a crystal is dictated by symmetry of the system. For cubic symmetry as in the present study only three independent elastic constants C₁₁, C₁₂ and C₄₄ are required to confirm the stability and the above equation is reduced to

(3) $${\mathrm{U}}_{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}={\displaystyle \frac{V}{2}\left({\mathrm{C}}_{11}-{\mathrm{C}}_{12}\right)({{\mathrm{e}}_1}^2+{{\mathrm{e}}_1}^2+{{\mathrm{e}}_1}^2)+{\mathrm{C}}_{12}{\left({\mathrm{e}}_1+{\mathrm{e}}_2+{\mathrm{e}}_3\right)}^2+{\mathrm{C}}_{44}({{\mathrm{e}}_1}^2+{{\mathrm{e}}_1}^2+{{\mathrm{e}}_1}^2)+\mathrm{H}\mathrm{O}}$$

The system will be dynamically stable if U_elast is positive i.e. the deformed system is at a higher energy. This condition restricts the values of elastic constants, which are the necessary stability conditions of Born (Born, 1940); C₁₂ < B < C₁₁; (C₁₁ − C₁₂) > 0; (C₁₁ + 2C₁₂) > 0 and C₄₄ > 0.

We have obtained the second order elastic constant values by performing rigorous strain-energy calculations using the package ElaStic 1.0 (Golesorkhtabar et al., 2013) with the wien2k DFT code. The results for a second order fit are given in Table 1. A look at these values show that the comprehensive Born criteria holds true for both the perovskites and the mechanical dynamical stability of these perovskites is confirmed. Due to lack of previous experimental or theoretical works on these new perovskites we cannot compare our results for the thermodynamic or mechanical stability.

The dynamic stability of a crystal also requires that all the phonon frequencies in the spectra are positive and that there are no imaginary or soft phonon modes. We have investigated the phonons at the gamma point (zone center) by performing a density functional perturbation theory (DFPT) calculation using the Quantum Espresso code (Baroni et al., 2001) with projector augmented wave (PAW) pseudopotentials and the GGA-PBE exchange correlation (Perdew, Burke & Ernzerhof, 1996). Fully relaxed structures and very stringent convergence thresholds of 10⁻²²Ry are used for highly precise and accurate results. The results for both the perovskites show the presence of imaginary phonon frequencies (negative values) at Γ point and indicate dynamic instability as seen from Table S1 of supplementary information. This implies the possibility of lower order symmetry phases and could indicate the presence of interesting ferroelectric order. More detailed experimental and theoretical investigations with full phonon dispersions for cubic as well other lower order structures are required to confirm these results and is a strong motivation for future work on these perovskites. For the sake of comparison by experimentalists we have included the simulated powder diffraction patterns of these perovskites in the supplementary information Fig. S1.

The electronic and optical calculations have been performed using the mBJ exchange functional to obtain highly accurate band structures of the perovskites and very reliable results for all the optical properties under investigation. The MoScO₃ and WScO₃ band structures along the high symmetry points are shown in Figs. 2 and 3 respectively.

As seen from Fig. 2, MoScO₃ shows interesting semi-conducting behavior in the minority spin with an indirect R−Γ bandgap of 3.61 eV which predominates over the direct Γ−Γ gap of 3.64 eV. In fact we see that the difference between valence band maximum (VBM) at the R and Γ points are very close, with just a difference of 0.033 eV.

In Fig. 3 the WScO₃ band structures are depicted and in contrast to MoScO₃ we see a more pronounced half metallic behavior. The minority band shows an indirect R−Γ gap of 3.82 eV and the majority spin band is strongly metallic with valence bands deep in the conduction band. Table 2 presents a summary of the band structure results.

Magnetic properties

Although the TM are paramagnetic and individually do not show any magnetism, the Mo/WScO₃ perovskites are magnetic. The spin polarized total density of states (TDOS) plots is shown in Figs. 4 and 5 for MoScO₃ and WScO₃ respectively.

The TDOS of the minority spin of the compound shows clearly the energy gap, with the Mo DOS dominating the conduction band and oxygen states on the valence side of the TDOS. Meanwhile for the majority spin the TDOS of the compound and Mo and O DOS have peaks around E-Fermi, clearly indicating a metallic behavior for MoScO₃ majority spin state.

The TDOS for WScO3 in Fig. 5 shows a similar trend as in MoScO₃, with minority spin showing the band gap and semiconducting nature, while, the majority TDOS has prominent peaks at E-Fermi belonging to the full compound, Mo and O indicating the metallic behavior and showing consistency with the band structure plots Figs. 2 and 3. In addition, we notice, that the TDOS for both the perovskite compounds in Figs. 4 and 5 shows a marked difference in the spin up and spin down density and this clearly gives rise to the magnetism with a magnetic moment of 2.99μ_B in the two perovskites as shown in Table 3 which lists the total, atom wise and interstitial magnetic moments of the perovskite compounds.

Optical properties

It is highly important to have a very dense grid and an accurate exchange correlation in the FP-LAPW to calculate the complex dielectric function that takes into account the self-energy and local field corrections. The TB-mBJ proves to be an excellent choice with a 30 × 30 × 30 grid for our optical properties calculations to obtain a high degree of accuracy of results. In this section we present the results for the dielectric function, optical conductivity and the important optical constants namely, the refractive index, the reflectivity, the extinction and absorption coefficients.

The complex dielectric function (ε = ε₁ + iε₂) is a function of the amount of light absorbed by the material. The imaginary part of dielectric function, ε₂(ω), which represents absorption behavior, can be calculated from the electronic band structure of solids (Dressel & Gruner, 2002). The real part of dielectric function, ε₁(ω), can be calculated according to Kramers–Kroing relation (Kramers, 1927; Kronig, 1926) which represents the electronic polarization under incident light. Figure 6 shows the real and imaginary plots for the dielectric function for the Mo/WScO₃ perovskites in the photon energy range of 0–14 eV. The complex index of refraction of the medium N is defined as

(4) $$N=\sqrt{\epsilon }=n+ik$$where, n is the usual refractive index and k is the extinction coefficient. Using the values of ε₁ and ε₂ we can obtain the refractive index n, reflectivity, and the extinction and absorption coefficients from the Eqs. (5), (6), (7) and (8).

(5) $$R\left(\omega \right)={\displaystyle \frac{n+ik-1}{n+ik+1}}$$

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

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

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

The refractive index and reflectivity plots are shown in Figs. 7A and 7B respectively in the photon energy range 0–14 eV, which captures all the necessary features. Refractive index is a function of incident frequency and the graphs display maximum peak values around ω = 0, with a decrease in values as we go towards higher frequencies. Both MoScO₃ and WScO₃ have large refractive index values of 10.6 and 5.7 respectively as a result of the high electron densities in these materials (Babu et al., 2012). The refractive index shows an almost constant and optically isotropic behavior at high energies > orange. The reflectivity is a maximum in the visible energy range; beyond this we observe an almost constant region with very small reflectance values in 3–8 and 2–8 eV for MoScO₃ and WScO₃ respectively. This region of the spectra therefore provides good ultraviolet (UV) and higher energy absorption.

In Table 4 we list the energies of the zeros of ε₁(ω) and static values of the refractive index, the reflectivity and absorption coefficients.

The extinction coefficient and energy loss graphs are shown in Fig. 8. The extinction or attenuation coefficient relates to how strongly light of a particular wavelength is absorbed. It is an intrinsic property that depends on the chemical composition of the compound. It is a measure of the diminution of transmitted light due to scattering and absorption in the medium.

The electron energy loss denoted by α(ω) is an important quantity and it represents the energy loss of the electron as it traverses in the medium. This function is shown in Fig. 8B. When the incident photon propagates through a polarizable dielectric or magnetic material and excites some internal degrees of freedom of the material, polaritons are formed (Sun et al., 2016; Rödl & Bechstedt, 2012). The polariton type depends on the type of elementary excitation i.e. phonon, plasmon, magnon or exiton. This is a complicated process and the peaks can be interpreted as a polariton excitation. We observe 4 peak curves for the two compounds, which occur at different characteristic energy values.

The optical conductivities and absorption coefficients are shown in Figs. 9A and 9B respectively.

The plots have 4 maximum peaks at different energy values. First, in the infra red (IR) region; the next peaks appear in the UV and above region. Similar features appear for the absorption coefficients.