To study the physical properties of Ta₂O₅, we calculated the total energy as a function of the volume of the unit cell, completely relaxing the ionic positions and the shape of the unit cell for each one of the calculated volumes. Afterward, the obtained data were adjusted to the fourth-order Birch–Murnaghan equation of state [11], [12] and, based on it, we identified the equilibrium geometry [4]. Total energy, the forces on the ions and the components of the stress tensor in this work were calculated using density functional theory (DFT). For the exchange–correlation functional, we used the generalized gradient approximation (GGA) with PBEsol parametrization [13] and the hybrid functional HSE06 [14], [15]. The Kohn–Sham equations were solved using the projector augmented plane-wave (PAW) method [16] as implemented in the VASP code [17], [18]. The reference PAW atomic configurations were: 5p⁶ 5d⁴ 6s¹ for Ta and 2s² 2p⁴ for O, where only states that are treated as valence electrons were listed. The cut-off energy in the plane-wave expansion was converged to a value of 520 eV.

All the structural parameters for each one of the calculated volumes were optimized, by simultaneously minimizing the atomic forces and the components of the stress tensor, through the conjugate gradient algorithm. The integration in the first Brillouin zone was carried out using a 4x4x8 Monkhorst-Pack k-mesh with a Gaussian broadening of 0.01 eV for relaxation (the original forces were converged to 1 meV / ). Subsequently, a calculation was made at a 4x4x8 -centered k-mesh for the total energy (the energy was converged to 1 meV/unit cell), the charge density, and the dielectric and optical properties.

We calculated the static dielectric constant ε(0) using density functional perturbation theory (DFPT) [19], where (1) is the Sternheimer linear equation, was first solved for , where is the periodic part of the auxiliary wave function; H(k), the effective Hamiltonian of an electron; є _nk, the Kohn-Sham energy; n, the band; k, the Bloch wave vector; and S(k) , the overlapping operator. Once this equation was solved, we determined the first-order response of the wave functions to an externally applied field by solving the linear equation (2), where represents the microscopic optic change of the Hamiltonian due to the variation of the wave functions to first-order, corresponds to the principal axes of the macroscopic dielectric matrix, and is the polarization vector. Finally, with the information obtained by solving (1) and (2), the static dielectric constant can be determined using (3), where Ω is the volume of the primitive cell and ωk are the k-point weights,which are defined in such way that theirsum equals 1. The optical properties associated with inter-band transitions was determined from the calculation of the imaginary part of the dielectric tensor at the long wavelength limit (→0) using the methodology developed for the PAW method [19]. The real part of the frequency-dependent dielectric tensor is obtained from the imaginary part through the Kramers-Kröning relationship. From the knowledge of the dielectric tensor, we have calculated the generalized refractive index (ñ = n + ik) and the transmittance explicitly using the equations in [20]. We did not include the spin-orbit coupling (SOC) because the Tantalum atom, which is the chemical speciesthat introduces relativistic effects, mostly affects the conduction bands. The main contribution to the dielectric function is due to the ionic part, which is obtained using the Sternheimer equation, where the occupied bands are theonly ones to be considered in this calculation. Asa result, the SOC becomes non-significant to estimate the dielectric function in this case.

(1)

(2)

(3)

λ-Ta₂O₅ is an orthorhombic model described by the space group Pbam with two formula unit per unit cell, where four Ta atoms are located in the Wyckoff position (4g), while ten O atoms are distributed over the Wyckoff positions (2a), (4g) and (4h). It has a crystal structure is build up for a set of TaO₆ octahedra where the Ta atom is located approximately in the center of the octahedra and the O atoms, in the corners. In addition, consecutive octahedra form chains that share the corners, as illustrated in Fig 1.

Fig. 1. Fig. 1. Crystal structure of λ -Ta2O5. Purple and red circles represent Ta and O atoms, respectively. Source: Authors’ own work.

At this point, it is very important to mention that the orthorhombic symmetry was preserved during the relaxation process for all the fixed volumes, which allowed us to find the equilibrium geometry. Table 1 presents the most important crystallographic parameters associated with the λ model. Clearly, our results of the calculation with both functionals are in very good agreement with the optimized parameters in the model by Lee et al.

Table 1

The model in this work represents the values given in [10].

The calculated dispersion relation for the λ model is presented in Fig 2. The gap of this orthorhombic model calculated with PBEsol and HSE06 was 2.09 eV and 3.7 eV, respectively, where the HSE06 prediction is in good agreement with the value reported by Lee et al. [10] and the experimental gap reported for the L phase (4.0 eV) [21]. This is because excited state properties are better predicted by explicitly orbital-dependent functionals, which include exact exchange such as HSE06. On the other hand, we found that λ-Ta₂O₅ has a direct energy band gap, where the valence band maximum (VBM) and conduction band minimum (CBM) are located in the high symmetry point Z = (0, 0, 1/2).

Fig. 2.

Left: PBEsol eigenvalues. Right: HSE06 eigenvalues.

The dielectric function is second rank tensor, for an orthorhombic crystal it only has three components different from zero: (ε_xx, ε_yy, ε_zz). Fig. 3 (a-c) shows the diagonal components of the imaginary part of the dielectric tensor () calculated with PBEsol and HSE06 functionals. In this case, it is easy to identify some degree of anisotropy in the dielectric tensor. This situation can be observed from the position of the main absorption peaks among its components, which are slightly located at different energies. Furthermore, the intensity of the main peak of the ε_zz component is greater than the other two components, which have fairly close values. Here, it is important to mention that the main peaks in Fig. 3 are associated with inter-band transitions between O p-states in the VB and Ta d-states in the conduction band.

Fig. 3

Blue and red lines represent PBEsol and HSE06 calculations, respectively.

Once the dielectric tensor was calculated along the three Cartesian directions, we obtained the generalized refractive index (ñ). Fig. 4 shows the refractive index n and the extinction coefficient k. It can be seen from this figure that the value calculated with PBEsol (HSE06) for n is 2.48 (2.02). In the literature, we did not find a reported value of refractive index for this model. However, as in the case of phase B, we can make an indirect comparison with the value reported for an ultra-thin film of Ta₂O₅ (n= 2.15) [23], which is consistent with our calculation. In addition, the extinction coefficient enabled us to observe maximum absorption peaks at 6.1 eV and 11.3 eV (7.8 eV and 14.4 eV) with PBESol (HSE06), which are in the region of strong absorption within the ultraviolet region.

Fig. 4. Fig. 4. a) Refractive index and b) extinction coefficient. Blue and red lines represent PBEsol and HSE06 calculations, respectively. Source: Authors’ own work.

Finally, we calculated the transmittance (Fig 5). From this figure it can be observed that, in the visible region (between 0.1 and 1μm), the behavior of the transmittance is constant with a value of 82% (88%) with PBESol (HSE06). This result allows us to conclude that, in this region, λ-Ta₂O₅ is a transparent semiconductor.

Fig. 5. Fig. 5. Transmittance in the visible region. Blue and red lines represent PBEsol and HSE06 calculations, respectively. Source: Authors’ own work.