Investigation of AcXO 3 (X = Al, Ga) perovskites for energy harvesting applications: a DFT approach

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Introduction
Recurrent handling of environmental pollution and global energy crises has undoubtedly attracted researcher's interest for the development of new electrochemical energy storage devices and clean energy resources [1][2][3].For this purpose, the exploration of new materials is one of the main forces accelerating current scientific development and technical innovation.Also, new materials with specific, improved, or innovative functionalities are essential for developing future technologies.Density functional theory (DFT) based theoretical modelling and simulating technology is considered as well developed and opening blueprints for investigating new materials to evaluate various physical characteristics of materials for the applications in innovative technologies [4][5][6][7].
Till up to date, much anticipated ABO 3 type perovskite oxides are the materials of exceptionally versatile class owing to their variety of physical properties and having flexibility to accommodate numerous cations of different sizes and plenty of oxygen atoms [8,9].The benefits of perovskite oxides are found to be staggering with their reported power conversion efficiency (PCE) rising well above 20% for light absorption [10,11].The incorporation of actinide elements with perovskite structure could prove to be a great deal to understand emergent properties that could lead these materials for technological applications [12] [13].In addition, it could be viable to have a computational insight into their various physical properties to guide the experiments towards more specific outcomes [14][15][16].
In this paper, a systematic DFT based investigation of optoelectronic and thermodynamic characteristics of ideally cubic AcAlO 3 and AcGaO 3 perovskite oxides is presented.Both proposed compounds are found to be stable in non-magnetic phase, while retaining the exceptional electronic (wide band gap), optical (high dielectric constant) and temperature-dependent transport properties.To best of our knowledge, there is not any experimental or theoretical work on these compounds published prior to this study.

Method of calculation
The proposed perovskites AcAlO 3 and AcGaO 3 , ideally, have stable cubic structural symmetry having Pm-3m space group.In the structure, Ac, Al/Ga and O atoms were positioned at (0, 0, 0), (½, ½, ½) and (0, ½, ½), respectively.The internal atomic positions were then fully relaxed and optimized excluding the spin polarization (due to non-magnetic nature).The optimized crystal structures of both studied perovskites are displayed in Fig. 1

(b&e).
Presented physical properties of AcXO 3 (X = Al, Ga) are computed by solving Kohn-Sham equations [33].Ground-state energies and geometry optimizations for both perovskites are computed by condensing the lattice parameters within the self-consistent DFT based FP-LAPW method as employed in WIEN2k code [34].The Kohn-Sham equation is given as: Where − ℎ 2

2𝑚𝑚
2 represent the kinetic energy,   () stands for effective potential and   () is for Kohn-Sham orbitals.The value of   () is calculated by using following expression: In this equation,   represents the electron-nucleus interactions, second term corresponds to e --e -interactions named as Hartree potential and last term corresponds to the exchange correlation potential.DFT can deliver precisely calculated energies and related properties if E XC is known.However, we have to approximate the E XC [33,35,36].Therefore, to estimate E XC , Perdew-Burke-Ernzerhof for solids plus generalized-gradient-approximation (PBEsol + GGA) was employed [37].The convergence of basis set was controlled by the wave function expansion.For this purpose, the plane wave cutoff parameter R MT ×K MAX = 7 was chosen.Here, R MT represents the smallest value of muffin tin radius, and K MAX is the maximum value of wave vector.A denser mesh of k-points (10×10×10) was selected to perform the Brillion zone (BZ) integration.The energy convergence criterion was selected 10 -4 Ry and the difference of integrated charge density was chosen as < 10 -4 e/a.u. 3 .The maximum value of angular momentum (l Max ) was selected as constant for outside while a maximum value of l Max was set as 10 for inside the sphere for wave function.
Further, the temperature dependent transport characteristics were calculated by using the BoltzTraP code which is based on Boltzmann's semiclassical transport theory [38].The value of constant relaxation time approximation was set as τ = 5×10 -15 s.The equations involved in computing the different physical characteristics are stated explicitly in the subsequent sections.

Thermodynamical stability
Thermodynamical stabilities of the compounds under investigation are briefly discussed here and are compared with the related literature.The formula for formation enthalpies (ΔH F ) of AcXO 3 (X = Al, Ga) is given as under: Here, E stands for total energy of respective compound or elements.From the perspective of calculated enthalpies, both compounds are found to be stable thermodynamically with ΔH F value of -7.6799 eV for AcAlO 3 and -6.992 eV for AcGaO 3 .This thermodynamical stability can be verified by comparing the AcAlO 3 and AcGaO 3 thermodynamics with Ac 2 O 3 which has been synthesized experimentally in various studied [25][26][27][28][29]. Also, AcCrO 3 and AcFeO 3 has been computationally explored by Munir et al. and reported the thermodynamic stability of relevant compounds [39].

Structural stability
Structural characteristics have a critical role in forecasting the ground state phase along with atomic occupancy at various Pm-3m lattice positions and depict the bonding nature.Our proposed compounds AcXO 3 (X = Al, Ga) are ideally cubic with Pm-3m space group (see Fig. 1(e&b)).Brich Murnaghan's equation of state [40] was solved in order to compute ground states energies of cubic AcXO 3 (X = Al, Ga) in nonmagnetic (NM) and ferromagnetic (FM) phases to determine the stable phase.The Brich Murnaghan's equation is mathematically expressed as: In this equation, V and V 0 represent the thermodynamic and equilibrium volumes, respectively, B 0 is bulk modulus and  0 ′ is the value of its pressure derivative and E 0 is the equilibrium energy.It is observed that optimized cubic structures of both compounds have least energy (stable) in nonmagnetic phase (see Fig. 1).
Furthermore, cubic stability can be determined by computing the Goldschmidt's tolerance factor (τ G ), formula of which is given below [41]: Here, r is respective ionic radii.The calculated τ G values of AcAlO 3 and AcGaO 3 are 0.93 and 0.90, respectively.Governed by these theoretically predicted tolerance factors, both titled perovskite materials are found to be stable in cubic configuration with the value of τ G approaching to unity [41].Several other structural parameters like lattice constants a 0 in (Å), equilibrium volumes V 0 in (a.u.) 3 , B 0 in GPa, its pressure derivative  0 ′ , and ground state energies E 0 in Ryd. are reported in Table 1.Tolerance factor, τG 0.93 0.90

Electronic structures and density of states
Electronic band structures (BS) of materials elucidate a fundamental explanation of material's electrical properties.In this section of results and discussions, we have discussed the BSs of AcAlO 3 and AcGaO 3 calculated along highly symmetric directions of BZ within PBE+GGA scheme.It can be seen from the calculated BSs (see Fig. 2) that AcAlO 3 has direct band gap while AcGaO 3 has indirect band gap.The numerical values of bandgaps (E g ) calculated for AcAlO 3 and AcGaO 3 by PBE+GGA are 3.98 eV (Γ-Γ) and 2.75 eV (R-Γ), respectively, and are also tabulated in Table 2.This reduction of band gap after Ga substitution at Al site in MO 6 octahedra is due to the greater ionic radii of Ga than Al, which eventually lead to less coulomb repulsion resulting in low band gap.For the deeper insight into the electronic properties, the total (T), atomic total DOS and contribution of partially filled Ac-6d, Al-3p/Ga-4p, and O-2p valence states towards the band alignment are computed (see Fig. 3).Among all states, most interesting states are 6d/2p of Ac/O atoms which lie in vicinity of Fermi level (E F ). Coulomb repulsion among electrons of Ac with nearby atoms making the case for degeneracy of d-orbitals that are split into doublet d-e g (d x y 2 ,d z 2 ) states at lower energy and triplet d-t 2g (d xy ,d yz ,d zx ) states pushed towards higher energy.Oxygen, the most electro-negative constituent atom in both titled perovskite oxide systems gain electrons from all metals and hence have maximum contribution in valence band (VB) formation (see Fig. 3).The obtained results of electronic states of nonmagnetic AcXO 3 (X = Al, Ga) are similar to the electronic profiles of recently reported BaPaO 3 , BaUO 3 [42,43], BaNpO 3 [16] perovskites.

Optical spectra
The energy levels of the ground state gradually perturb over incident light, and optical transitions between the occupied and unoccupied states are investigated to describe the interaction of electron with incident photon.Motivated by the prospect of their application in optoelectronic devices, the BS dependent optical properties of titled single perovskite oxides are explored.Frequency (ꞷ) dependent complex dielectric function (DF) is calculated, through which all other optical parameters are extracted.The dielectric tensor Ԑ(ꞷ) describes the polarizability of a material, relating the shape of the VB charge density, and demonstrates the deformability of the electronic contribution.The optical characteristics of a material are predicted by calculating complex dielectric function, which is comprised of two parts, real part Ԑ 1 (ꞷ) and imaginary part Ԑ 2 (ꞷ) and are expressed in a relation given below [44]: The imaginary part Ԑ 2 (ꞷ) which arises form intra and inter band transitions, depicts the probable transitions between occupied and unoccupied energy levels using fixed k-vector over Brillouin zone (BZ).It is mainly due to the inter-band transitions, rather than intra-band changes, that contribute more to the semiconductors.The Ԑ 2 (ꞷ) is mathematically defined as [45]: where q and k denote the wave functions of occupied and unoccupied states, and the first term inside the integration indicates the normalization of momentum matrix element.The real part of dielectric function Ԑ 1 (ꞷ) elucidates the polarization of light and can be computed via Kramers-Kronig relationship which is given as under [46]: where, p is the integral's principal value.
The Ԑ 1 (ꞷ) and Ԑ 2 (ꞷ) plots (see Fig. 4a&b) show the cut off value Ԑ 1 (ꞷ) of 4.0 for AcAlO 3 and 4.2 for AcGaO 3 and turns to peak values of 9.1 at 6.75 eV for AcAlO 3 and 9.35 at 6.15 eV for AcGaO 3 .Above this energy limit, a dip to the negative values is obtained in the corresponding energy ranges of 9 to 9.8 eV for AcAlO 3 while AcGaO 3 remains negative from 9.0 eV to onward (see Fig. 4a & b).These negative Ԑ 1 (ꞷ) values show metallic character of the studied perovskites within corresponding photon energy ranges.The Ԑ 2 (ꞷ) plots for AcAlO 3 indicate that maximum peaks lie within the energy range of 7-10 eV with an onset of saturation edge recorded at 4.2 eV.While with an onset of the first Ԑ 2 (ꞷ) peak edge at 4 eV, the AcGaO 3 compound show highest imaginary part within 6.2 to 9 eV energy range.Afterwards, both perovskites show considerable variation towards minimum Ԑ 2 (ꞷ) value.
The optical absorption α(ꞷ) and optical conductivity σ(ꞷ) can be deducted from Ԑ(ꞷ) and the formulas for both properties are given below [45]: Fig. 4c shows α(ꞷ) plots which is basically the measure of attenuation percentage for intensity of incident photons that is absorbed per unit length in studied oxides.The pattern of α(ꞷ) plot is relatively similar to that of Ԑ 2 (ꞷ) plot (see Fig. 4a,b&c) for both Al and Ga based perovskites.Hence, α(ꞷ) is also related to the presented BSs of both compounds.The onset edge of absorption spectra is at 5 eV for AcAlO 3 and 4 eV for AcGaO 3 .At lower energies, from 0 eV to onset edge, there is zero or negligible absorption, meaning that material is transparent to the incident light with the energy of this range (reflectivity is also minimum in this range as discussed below).The maximum α(ꞷ) peaks are observed in 7-10 eV energy range for considered oxides.The highest absorption is obtained at 9 eV with α = 180×10 4 cm -1 for AcAlO 3 and α = 170×10 4 cm -1 for AcGaO 3 .When light of particular frequency interacts with a material, its electrons gain energy and jump to CB.These free charge carriers make the material conductive for which optical conductivity σ(ꞷ) term is used.The onset edges of calculated σ(ꞷ) (see Fig. 4d) are observed at 4.5 and 4.0 eV energy for Al and Ga based perovskites, correspondingly.The first maximum σ(ꞷ) peaks have the values of 7650 (at 7.1 eV) and 7680 Ω -1 cm -1 (at 6.7 eV) for AcAlO 3 and AcGaO 3 , respectively.This conductivity arises along the highly symmetric (Γ-Γ) direction of the BZ for AcAlO 3 , while (R-Γ) direction of first BZ for AcGaO 3 .The maximum optical conductivity occurs with a value of 11000 Ω -1 cm -1 at 8.9 eV for AcAlO 3 and 7800 Ω -1 cm -1 at 9.0 eV for AcGaO 3 followed by decrement afterwards.This interplay of EM interaction with material and resulted transitions from valence band to conduction band are well correlated with shapes of band structures of studied oxides.
Generally, the relationship between propagation of EM wave through vacuum and in materials can be mathematically expressed as: N(ꞷ) = n(ꞷ) + ik(ꞷ).An analytical glance on n(ꞷ) plot (Fig. 5a) revealed the static n(0) value of 2.0 and 2.1 for AcAlO 3 and AcGaO 3 perovskites, respectively.The calculated static values of n(ꞷ) satisfy the following optical relation n 2 (0)=Ԑ 1 (0).The n(ꞷ) plot for both compounds have pattern similar to that of real part of DF, i.e., Ԑ 1 (ꞷ) (see Figs. 5a & 4c).The principal peak value of n(ꞷ) recorded for AcAlO 3 is 3.3 at 7 eV and for AcGaO 3 is 3.2 at 6.2 eV.Thereafter, the n(ꞷ) plot shows a fluctuated dip all the way towards 10 eV with lowest value of ~1 at 10 eV for both oxides.From the k(ꞷ) spectra (Fig. 5b), it is clear that nature of k(ꞷ) peaks is analogous to that of Ԑ 2 (ꞷ).The onset edges of k(ꞷ) peaks are similar as that of Ԑ 2 (ꞷ), α(ꞷ), and σ(ꞷ).The first maximum peak value of k(ꞷ) observed for AcAlO 3 is 1.49 at 7.2 and 1.55 at 7 eV for AcGaO 3 .The trend in k(ꞷ) fluctuation is similar to that of Ԑ 2 (ꞷ) peaks.
To further understand the response of studied perovskites to incident light, reflectivity R(ꞷ) in terms of percentage is also calculated for AcXO 3 (X = Al, Ga) compounds by using following formula: From the Fig. 5c, we can realize that R(ꞷ) values for AcXO 3 (X = Al, Ga) at 0 eV are 11% and 12.5%, respectively.From 0 to 4.0 eV, the reflection of incident light is negligible.In this energy range, the α(ꞷ) for studied perovskites are of the order of 10 4 cm -1 while reflection is less than 15%.Clearly, this comparison of the absorption and reflectivity curves in the abovementioned range shows that incoming photons of this energy range are mostly transferred.The similar transparent character of actinide series-based perovskite oxides has been interpreted similarly in Ref. [47].Moreover, Table 2 also enlists the cutoff values of various optical parameters i.e., Ԑ(0), n(0) and R(0).

Thermoelectric (TE) response
As a result of the serious environmental changes, organic energy resources are diminishing or decreasing, posing a significant danger to sustainable living.This demands the exploration for novel materials to eliminate such kind of challenges.The scientific community is extensively working to identify the particular alternative resources that are pollution free, financially viable, and highly efficient.It was recently suggested that increase in TE efficiency of a material is dependent on quantum confinement of charge carriers [48].Inspired by this suggestion, researchers used half Heusler alloys together with silicides, clathrates, and skutterudites in an attempt to solve these problems [49].The shortcomings of such material's instability at high temperatures and rarity in abundance make them impractical for widespread commercialization.In this regard, TE oxides are anticipated to overcome these obstacles [12,[50][51][52][53][54].Here, in an attempt to forecast TE characteristics of AcAlO 3 and AcGaO 3 , semiclassical transport theory of Boltzmann was utilized along with constant relaxation time and rigid body approximation as integrated in BoltzTraP code [38].The theory is quite helpful for comprehending different transport coefficients including Seebeck coefficient (S), electrical and thermal conductivities (ϭ/τ & κ/τ), associated power factor (PF), and figure of merit (ZT).
In order to better understand how waste energy is transformed into usable energy, these coefficients have been thoroughly evaluated.For AcAlO 3 , the value of ϭ/τ rises with the upsurge in temperature (see Fig. 6a).It is reported that semiconductor materials have negative temperature coefficient, i.e., their resistance decreases with the linear increase in temperature [55].While electrical conductivity (ϭ/τ) for AcGaO 3 is low at higher temperatures might be due to indirect transition.The maximum ϭ/τ for AcAlO 3 is observed within 450 -800 K temperature range.However, for AcGaO 3 , the maximum ϭ/τ is observed at 200 K and it is lowest at 800 K.The thermal energy is transferred by two mechanisms: first, holes and second, electron drift, which contribute the electronic thermal conductivity (κ e ) while phonon's traveling contributes the lattice thermal conductivity (κ l ), both parts are related as κ = κ e + κ l .However, only κ e are computed due to low contribution of lattice conductivity at high temperature (see Fig. 6b).An increasing trend for AcAlO 3 and AGaO 3 is observed with highest κ e values of 25×10 14 and 25.2×10 14 W/cm.K.s., respectively at 800 K.A single type of carriers-either p-type or n-type-should be used to assure high thermopower since multiple types of carriers cancel each other out and produce lower potential [56].By calculating the seebeck coefficient (S), one may determine a material's thermopower, and S can be computed as follows: Hence, S and effective mass (m*) are directly proportional to each other.The magnitude of the thermoelectric voltage produced by a temperature gradient between two ends of a material is expressed in terms of the seebeck coefficient.For efficient TE material, the S value should be high.The value of S increases sharply at low temperatures for both compounds, however, the slopes become smaller at higher temperatures (see Fig. 6c).AcAlO 3 retains comparatively high S values than AcGaO 3 throughout the whole 200-800 K temperature range.
Further, power factor PF of AcAlO 3 and AcGaO 3 is investigated (see Fig. 6d) which is calculated by using the formula PF=S 2 .At 800 K, the PF reaches to its maximum value of 7.5×10 11 and 6.2×10 11 W/cm.K 2 .s. for AcAlO 3 and AcGaO 3 , respectively.For all the temperature range of 200-800 K, AcAlO 3 has higher PF as compared to AcGaO 3 .This behaviour could be due to higher value of S.
The efficiency of TE materials depends on ZT, also termed as the figure of merit, and is enumerated using following formula: The resultant ZT quantity is unit less.From the Fig. 7, a linearly up surging trend is observed for both proposed oxides within 200 to 800 K.At 800 K, the recorded ZT for AcAlO 3 and AcGaO 3 are 0.23 and 0.20, respectively.This trend is in line with the ϭ/τ, κ e , and S calculated for both titled perovskites as presented in Fig. 6.

Conclusions
We have employed FP-LAPW method which is based on DFT to examine and predict the physical characteristics of AcAlO 3 and AcGaO 3 perovskites.The PBE-GGA has been utilized to estimate the band gaps.The calculated band gap values are 3.98 for AcAlO 3 and 2.75 eV for AcGaO 3 .Optical parameters including absorption coefficient and optical conductivity are also calculated to shed the light on the optical response of the studied materials.Furthermore, temperature dependent transport properties have been explored.Based on presented results, both AcAlO 3 and AcGaO 3 are found to be good UV absorbers and are regarded as potential candidate for UV based electronics and thermoelectric devices.

Table 2 .
PBE+GGA calculated electronic band gap E g (eV) and cut off values of various optical parameters for both AcAlO 3 and AcGaO 3 perovskite oxides.