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1. INTRODUCTION
For many years, semiconductors have been used in a wide variety of applications that have allowed developments, at a fairly accelerated rate, in several fields of science and engineering. In this context, wide-bandgap semiconductors have drawn great interest in recent years due to their applications in some areas of knowledge, such as electronics, communications and energy [1]-[3]. One of the semiconductors that offers great potential for this type of applications is Ta2O5. Its properties include high dielectric constant, low dielectric loss and good temperature stability. As a result, it can be used, for instance, as a dielectric resonator in microwave communication, where miniaturization critically depends on the dielectric constant of the ceramic component of the device. This indicates that a large dielectric constant, in principle, means a small size for the resonator [2].
Ta2O5 has a set of crystallographic phases that depend on temperature and pressure. To describe some of these phases, there are experimental models which can adequately explain the different chemical and physical properties associated with this compound [4]. Among all the known phases of this semiconductor, the modification of L-Ta2O5 without oxygen vacancies, reported in different experimental and theoretical studies, awakens important interest [5]-[9]. Lee et al. [10] proposed a model to describe it based on its orthorhombic structure with two formula unit per unit cell, called 𝜆 phase, where its local structure corresponds to chains of distorted octahedral TaO6. In this work, we have adopted such model to study the structural, electronic and optical properties of the low-temperature modification. We indirectly compared our results with experimental and theoretical values reported in other studies.
2. COMPUTATIONAL METHODS
2.1 Crystal and electronic structure calculations
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: 5p 6 5d 4 6s 1 for Ta and 2s 2 2p 4 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 e V.
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 /Å). Subsequently, a calculation was made at a 4x4x8 Γ-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.
2.2 Calculation of 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 |𝛁
k
ũ
nk
⟩, where ũnk is the periodic part of the auxiliary wave function;
H
(
k
), the effective Hamiltonian of an electron; ϵ
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 |ζ
nk
⟩ to an externally applied field by solving the linear equation (2), where ∆
H
SCF
(
k
) 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 their sum 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 
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 species that 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 the only ones to be considered in this calculation. As a result, the SOC becomes non-significant to estimate the dielectric function in this case.
3. RESULTS AND DISCUSSION
3.1 Crystal structureFig 1
Source: Authors’ own work.
Fig. 1 Crystal structure of 𝜆-Ta2O5. Purple and red circles represent Ta and O atoms, respectively.
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.
3.2 Electronic structure
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).
Source: Authors’ own work.
Fig. 2 Band structure of 𝜆-Ta2O5. Left: PBEsol eigenvalues. Right: HSE06 eigenvalues.
A detailed analysis of the band structure allowed us to determine that the valence band (VB) is a hybridization of Ta d-states and O p-states, where the latter present the greatest contribution to VBM. On the other hand, the CBM is dominated by Ta d-states. Finally, we can observe from Fig. 2 that HSE06 produces a rigid band shift of the CB to higher energies with respect to VBM, while the VB presents a slightly larger width compared to PBEsol. The topology of the band structure is basically the same for both functionals.
3.3 Dielectric tensor
The dielectric function is second rank tensor, for an orthorhombic crystal it only has three components different from zero: (εxx, εyy, εzz) . 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 εzz 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.
Source: Authors’ own work.
Fig. 3 Non-zero components of the imaginary part of dielectric tensor of -𝜆-Ta2O5. Blue and red lines represent PBEsol and HSE06 calculations, respectively.
Table 2 presents the calculated electronic (ε∞) and ionic contribution (ε0) to the static dielectric tensor with PBEsol parametrization. Using these results, we were able to calculate the directional average and obtain the static dielectric constant (ε(0)= ε∞+ ε0), which equals 51. In this case, our results cannot be directly compared with experimental data, since the experimental value for the static dielectric constant of this type of structures has not been reported. However, we can make an indirect comparison with the calculation performed by Clima et al. [22] for the orthorhombic phase proposed by Ramprasad [7], which reports the following values for the contributions to the static dielectric constant: ε∞= 5.92 and ε0 = 36. By comparing these values with our calculation, we find that the ε∞ component is in good agreement with the value reported by Clima et al.; however, our calculated ε0 is much higher than in the model described by said authors. This is because the ionic contribution of the dielectric constant is strongly dependent on the crystallographic system. Therefore, an orthorhombic model composed of pentagonal bipyramids and distorted octahedra Clima et al.) is very different from a Lambda model defined by distorted octahedra only.
3.4 Optical properties
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.
Source: Authors’ own work.
Fig. 4 a) Refractive index and b) extinction coefficient. Blue and red lines represent PBEsol and HSE06 calculations, respectively.
Finally, we calculated the transmittance (Fig 5). From this figure it can be observed that, in the visible region (between 0.1 and 1 𝜇 m), 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, 𝜆Ta2O5 is a transparent semiconductor.
4. CONCLUSIONS
In this work, we studied different physical properties of an orthorhombic 𝜆-Ta2O5 model without oxygen vacancies. Based on our calculations, we were able to conclude that the VB is a hybridization of O p and Ta d-states, while a large percentage of the CB is composed of Ta d-states. Furthermore, under this model Ta2O5 presents a direct gap, where VBM and CBM are located in a high symmetry point Z = (0, 0, 1/2). In addition, we found that the static dielectric constant, 51, is in good agreement with values reported in other works. As a result, this novel material can be used to manufacture filters and resonators. Finally, we calculated the transmittance and stablished that, in the visible region, this semiconductor described by the 𝜆 model is a transparent material.
