Investigation of the transport, structural and mechanical properties of half-metallic REMnO₃ (RE = Ce and Pr) ferromagnets

Abstract

Systematic investigation of the ground state structure, which includes elastic and transport properties, of perovskite oxides REMnO₃ (RE = Ce and Pr) has been carried out by first principles calculations. We present the analytical as well as DFT calculated equilibrium lattice constants which show good agreement with experimental data. Three independent elastic constants are emphasised to yield the corresponding mechanical properties, including the elastic moduli (B, G and Y), Poisson's ratio (ν), anisotropy factor (A) and Pugh ratio B/G, for these compounds. These calculations predict the brittle PrMnO₃ as a less hard material than the ductile CeMnO₃ oxide. Post DFT treatment involving Boltzmann's theory is conveniently employed to investigate the thermoelectric properties of these compounds. The analysis of the thermal transport properties specifies the dimensionless figure of merit of 0.24 and 0.19 at room temperature for PrMnO₃ and CeMnO₃, respectively. Their half-metallic nature with efficient thermoelectric parameters, including electrical conductivity, Seebeck coefficient and thermal conductivity, suggest the likelihood of these materials to have a potential application in the design of shape memory devices and imminent thermoelectric materials.

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Introduction

Ferromagnets, which have efficient thermoelectric (TE) performance and half-metallicity, have proven to be a thrust area for researchers and industrial technologists to feed the demands of smart and spintronic materials, and highly competent thermoelectric materials for future devices. Thermoelectric materials, because of their multiple applications, from the reuse of waste heat to Peltier cooling in order to conserve or to reduce fuel consumption in automobiles, are in high demand. The characterization of a material as an efficient thermoelectric is represented by a dimensionless constant, zT = S²σT/κ, called the thermoelectric figure of merit. Where, S is the Seebeck coefficient, σ is electrical conductivity, T is temperature, and κ is the thermal conductivity, which is comprised of the electron, ε, and lattice, λ contributions of the material. Larger values of zT indicate a more pronounced thermoelectric performance in a material and hence higher efficiency in converting waste heat into useful electricity.^1,2 The enhancement of zT can be achieved by the combination of different strategic parameters. The prime idea to achieve this involves the minimization of the lattice contribution towards thermal conductivity.³ The second way of improvement is to maximize the product S²σ, which is defined as the power factor, by varying the carrier concentration with different doping levels.^4,5 Lastly, the band engineering of materials with strongly correlated electrons in order to induce peaks in the DOS is an alternative approach.

From the above perception, perovskite oxides are very attractive materials which display diverse structural phase-transition sequences, metal–insulator transitions, ferroelectricity, superconductivity, piezoelectricity, and half-metallic ferromagnetism (HMF) or thermoelectric properties, thus labeling them as ultimate candidates for multi-functional devices.^6,7 In transition metal ABO₃ perovskites, the flexibility of the structure exposes the possible ways to tailor the electronic, magnetic and orbital states, which in turn alter the bond lengths, angles and distances, and thus guide various parameters such as electronic band gap and transport properties. Owing to strong coulomb repulsion (correlations), the energy and occupancy of the t_2g (triplet) and e_g (doublet) states are strongly affected in transition metal perovskites. This has a strong influence on their physical properties, for example magnetic order and/or orbital order can emerge from the interplay of the correlation phenomenon with charge, spin and orbital degrees of freedom.^8–10

Na_xCoO₂ cobaltates, which have large thermopower and the highest recorded zT value of up to 1.0 at 800 K for oxides, have promoted the study of the TE properties of oxide materials.^6,11 Recently, tuning of the DOS in double perovskites, including Sr₂NbIrO₆, LaSrTiIrO₆, Sr₂TaIrO₆ and Sr₂NbRhO₆, was revealed to yield efficient thermoelectric materials when optimally doped.¹ Also, the n-type doped SrTiO₃ oxide shows large power factors, and more recently CrN which is not an oxide, but very close in terms of electronic structure, having empty d-bands just above the Fermi energy to produce a set of doping (electron) sensitive localized levels, has been found to increase the zT vividly when nanostructured.^12,13 In the high temperature region, oxide materials are more suitable for TE applications due to their stable structure, chemical environment, oxidation resistance and low-cost.^14–16 Therefore, to determine the potential capabilities of oxides to be used for TE applications, it is important to estimate suitable physical parameters. A pint-sized study of rare-earth based magnate perovskites (PrMnO₃ and CeMnO₃) was performed to determine their magnetic and electronic properties, however their thermo-electronic and elastic properties are still unexplored.^17,18 It is noteworthy to mention here that there is no information on the theoretical or experimental understanding of the structural aspects, electronic transport and second order elastic constants (SOECs) of these compounds, which are pivotal to illustrate their mechanical and structural stability, nature, and strength in addition to their usefulness for both fundamental research and technological applications. Hence, a detailed study is needed to go explore their structural, elastic and transport phenomena. Thus, we hereby investigate for the first time the elastic as well as thermoelectric properties of REMnO₃ (RE = Ce and Pr) theoretically along with their structural, magnetic and electronic properties.

Methodology

Geometric optimization and ground state properties, including the lattice constants, bulk modulii, bond lengths and total and partial density of states, of these compounds were determined through the condensing of the lattice parameter, within the self-consistent Density Functional Theory (DFT) by means of the full potential linearized augmented plane wave (FPLAPW) method.¹⁹ Also, the Ernzerhof generalized gradient approximation (GGA)²⁰ along with the on-site Coulomb repulsion (GGA+U)²¹ were employed for the exchange and correlation effects. The Hubbard parameter U (defined as U_eff = (U − J), where U is the on-site Coulomb term and J is the on-site exchange term), is optimized to minimize the atomic forces and simultaneously reproduce the band gap and spin flip energy of the hybrid functional. Hubbard U can also be determined by other methods which are less parametric.²¹ We kept J = 0 and the U_eff value was varied from 0.5 to 1.7 eV throughout the calculations. The appropriate U_eff values for CeMnO₃ and PrMnO₃ were found to be 1.23 eV and 1.36 eV, respectively. The ground states were put under calculations to reproduce band gaps and cohesive energies. To brand the calculated results computationally tractable and allow the simulation of many more chemistries, a modified Becke–Johnson (mBJ) semilocal exchange potential was also implemented to obtain the band structures,²² since this functional has been shown to successfully reproduce the band gap for a wide range of semiconductors and oxides including SrTiO₃.²³ To control the convergence of the basis set, the plane wave cut-off value R_MTK_max = 7 was used, where K_max is the maximum modulus for the reciprocal lattice vector in the plane wave expansion. The sampling of the Brillion zone was done by employing the tetrahedral method over a dense set of 2000 k points. The selected convergence criteria for the structural optimization during the self-consistency cycles considered is the energy difference between conjugate gradient steps better than 0.01 mRy and an integrated charge difference less than 0.0001e/a.u.³ per unit cell. Also, the elastic constants for both compounds were determined by applying a given homogeneous strain with a finite value and thereafter calculating the resulting stress. These calculations were performed by employing the (Charpin) cubic elastic code.²⁴

Structural properties

The present study reports the structural, elastic, magnetic and thermoelectric properties of REMnO₃ (RE = Ce and Pr) compounds. The unit cell of REMnO₃, which is assumed to crystallize in the Pm3̄m (221) space group, contains a single molecule with atomic positions in the elementary cell as: RE 1a (0, 0, 0), Mn 1b (0.5, 0.5, 0.5) and O 3c (0.5, 0.5, 0). To establish a stable ground state, the volume optimization of each compound was performed as given in ref. 25. As seen from the ground state structure optimization curves of both the compounds in Fig. 1, the PrMnO₃ and CeMnO₃ oxides are stable in the ferromagnetic (FM) phase due to their minimized energy values rather than the paramagnetic (PM) or antiferromagnetic (AFM) phases. Consequently, from the details of the structure optimization, the lattice constants, bulks modulus, equilibrium volume and cohesive energies were calculated, as shown in Table 1. Using the GGA and GGA+U approximations, the calculated lattice parameters for CeMnO₃ are 3.8 and 3.74, and PrMnO₃ are 3.88 and 3.85 Å respectively, which display good agreement with the experimental value of 3.82 Å for PrMnO₃ (ref. 26) and the theoretical value of 3.87 Å for CeMnO₃.¹⁷ We also calculated the lattice constants empirically, using the formula:²⁷

a₀ = α + β(r_RE + r_O) + γ(r_Mn + r_O) (1)

where, α (0.06741), β (0.4905) and γ (1.2921) are constants, r_RE is the ionic radii of Ce (1.34 Å) and Pr (1.30 Å), r_Mn is the ionic radius of Mn (0.645 Å) and r_O is the ionic radius of O (1.35 Å).²⁸ Since, GGA overestimates the lattice parameter, equilibrium volume and the total energy of a crystal, the GGA+U method was therefore applied and very reasonable results for the lattice parameter and band gap were obtained, as shown in Tables 1 & 2.

Fig. 1 Calculated double cell optimization curves of REMnO₃ for paramagnetic, ferromagnetic and anti-ferromagnetic phases in the Pm3̄m configuration.

Table 1 Calculated values of the lattice constant, unit cell volume, ground-state energy, and tolerance factor (t) of CeMnO₃ and PrMnO₃

Parameter	Present	Empirical	Others	Expt
CeMnO3
Lattice constant (Å)	3.88	3.96	3.87 (ref. 17)
Volume (a.u.)^3	396.40
Tolerance factor (t)	1.00	0.96
Bond lengths (Å)
Ce–O	2.82
Ce–Mn	3.46
Mn–O	1.99
ΔE (Ry) = ENM − EAFM	0.1027
PrMnO3
Lattice constant (Å)	3.85	3.94	3.88 (ref. 18)	3.82 (ref. 26)
Volume (a.u.)^3	394.46
Tolerance factor (t)	0.99	0.94
Bond lengths (Å)
Pr–O	2.74
Pr–Mn	3.36
Mn–O	1.94
ΔE (Ry) = ENM − EAFM	0.2025

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Table 2 Calculated elastic constants C_ij (GPa), Bulk's modulus B (GPa), Shear modulus G (GPa), Young's modulus Y (GPa), Poisson's ratio υ, anisotropy constant A, Cauchy pressure C′′ and B/G ratio of the cubic CeMnO₃ and PrMnO₃

Parameter	C11	C12	C44	C′′	B	Y	G	υ	B/G	A
CeMnO3	170.71	130.44	63.18	67.26	143.86	124.60	40.01	0.36	3.59	3.13
PrMnO3	346.27	69.43	110.24	−40.81	161.79	31.63	120.67	0.03	1.34	2.49

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Thus, the DFT predicted and analytically calculated lattice constants, as presented in Table 1, are in fairly good agreement with the previously reported results.^17,18,26 An increase in bond length between M and O is observed as we go from Ce to Pr, which is due to the larger size of Pr than Ce. Furthermore, employing the ionic radii and Goldschmidt method,^25,27 the tolerance factor (t) was calculated for the present set of materials. The bond length values are furnished to estimate the tolerance factor of perovskites, which is at least a good approximation to ensure that the cubic perovskite structure is preserved. The empirical as well as the DFT computed values of the tolerance factors are found to be in range of (0.9–1.0) for both compounds, as depicted in Table 1, which agree well with the specific criteria for cubic perovskites.^29,30

Mechanical properties

The mechanical properties of these materials are governed by second-order elastic constants (SOECs), therefore much effort was placed on their investigation and measurement. Efficient and highly-accurate calculations based on DFT have proven to be an alternative to the experimental determination of elastic constants, especially for newly predicted and complicated crystal structures. In the present study, we performed first-principles calculations for the elastic parameters of REMnO₃ single crystals systematically, such as the elastic constant C_ij, and the bulk (B) and shear (G) moduli, which determine the response of the crystal to external forces and obviously play an important role in determining the strength of the materials. Also, the connection of the elastic properties with thermodynamical quantities, including specific heat, thermal expansion, Debye temperature and Gruneisen parameter, endorse their extensive application for describing the elastic and thermal performance of materials. The Viogt–Reuss–Hill method was employed to obtain the bulk and shear modulii. The Voigt bounds³¹ of the bulk modulus B_V and shear modulus G_V for the cubic system are:

Also, the Reuss formulae³² for the bulk and shear moduli are:

Along with these, the bulk and shear (B, G) moduli were determined via the Hill approximation:³³

The Hill's elastic moduli (B) and (G) furnished were used to obtain the Young's modulus (Y) and Poisson's ratio (σ) with the following equations:

The calculated mechanical and elastic properties of both compounds are listed in Table 2. The elastic stability is confirmed from the calculated elastic constants for both compounds, which satisfies the generalized mechanical stability criteria C₁₂ < B < C₁₁, (C₁₁ − C₁₂) > 0, (C₁₁ + 2C₁₂) > 0 and C₄₄ > 0.³⁴ It can be seen from Table 1 that the value of the bulk modulus of PrMnO₃ is larger than CeMnO₃, which indicates that it has a strong resistance towards the volumetric change created by applied pressure. Bulk's modulus (B) is needed to fully characterize the stiffness of a material. The higher the value of B, the stiffer the material. The present calculated B results for both materials demonstrate the stiff nature of PrMnO₃ compared to CeMnO₃.²⁵ Also, hard materials have covalent bonds between constituent atoms and they are characterized by a higher shear modulus value.³⁵ The Poisson's ratio (υ), which reflects the stability of a material against shear is defined as the ratio of transverse strain to the longitudinal strain, and provides information about the nature of the bonding forces. Its critical value is: −1 < υ < 0.5. Since, no real material is found to have a negative value, the inequality thus is replaced with 0 < υ < 0.5. A low Poisson's ratio value specifies a large compression of volume, a high value reflects the good plasticity of the material and, υ = 0.5 means no volume change has occurred. Since a large Poisson's ratio indicates better plasticity, the presently calculated results of ν illustrate that these perovskite compounds have good plasticity. Moreover, a Poisson's ratio value larger than the lower limit value (υ = 0.25) indicates the presence of central interatomic forces.

Furthermore, the Zener's anisotropy factor (A) reveals the degree of elastic anisotropy in a solid.³⁶ For a perfectly isotropic material, A will take a value equal to 1, otherwise (either smaller or greater than unity) the material is anisotropic. The anisotropy factor (A) is presented in Table 2 for both compounds, as calculated by the equation: A = 2C₄₄/(C₁₁ − C₁₂), which clearly indicates that the materials are anisotropic.

The Pugh's ratio B/G simply recognizes the brittle or ductile nature of materials. The critical value which separates ductile and brittle materials is 1.75.³⁷ A low value (B/G < 1.75) characterizes brittleness, whereas a high (B/G > 1.75) ratio is associated with ductility. The calculated results for REMnO₃ are listed in Table 2. These values indicate that the CeMnO₃ compound can be classified as a ductile material, whereas PrMnO₃ is a brittle material at zero pressure.

Another parameter, the Cauchy relation, which is defined as: C′′ = (C₁₂ − C₄₄), also classifies the brittleness or ductility of the material. If the value of C′′ is positive, then the material is ductile on the other hand, a negative value signifies that the material is brittle.³⁵ Thus, the ductile nature of the CeMnO₃ compound can be related to its calculated positive value and the brittle nature of PrMnO₃ is reflected from its calculated negative value.

Henceforward, from the first principle calculations of the elastic and mechanical properties of REMnO₃ compounds, it is concluded that the CeMnO₃ perovskite oxide is ductile, whereas PrMnO₃ is stiffer and brittle in nature.

Densities of states and their TE contribution

We performed electronic structure calculations for PrMnO₃ and CeMnO₃. The equilibrium lattice parameters were fetched to understand the electronic band structure and the contribution of constituent atoms towards the corresponding densities of states (DOS). Since the underestimation of intrinsic band gap by LDA and GGA is inevitable,³⁸ we also performed additional calculations for the electronic structures with the onsite Coulomb interaction (GGA+U) and mBJ methods. The spin dependent band structures, as shown in Fig. 2, and overall DOS versus energy (see Fig. 3) were calculated by the GGA, GGA+U and mBJ methods in order to have an in-depth knowledge of the band structure and energy gap in these materials. The absence of a gap in both spin phases at the Fermi level confirms the metallic behavior of the materials from GGA. On the other hand, when the GGA+U and mBJ methods are applied, the majority-spin band shows metallic character, while the minority spin band clearly displays an indirect semiconducting gap around the Fermi level. However, the band gaps produced by the GGA+U method are somehow reasonable rather than that from the mBJ calculations because the latter predicts slightly larger energy gaps as compared to the earlier results (see Table 3). For the GGA+U calculations, the energy bands cross the Fermi level present a metallic character in the majority spin state, while in the minority spin state half metallic nature is reflected. The valance band maximum (VBM) of CeMnO₃ is located at the X position of 0.19 eV and the lowest of the conduction band minimum (CBM) is at the M symmetry point of 2.57 eV, whereas the VBM of PrMnO₃ is located at the Γ position of 0.32 eV and the CBM is at the M symmetry point of 2.69 eV in the minority-spin channel for both materials. Thus, for the GGA+U calculations, both CeMnO₃ and PrMnO₃ show an indirect band gap of 2.76 eV and 2.69 eV, while for mBJ the corresponding values are 3.43 eV and 3.64 eV, respectively. The calculated band gap values agree well with the previously reported results.^17,18 Thus, high spin polarization at the Fermi level is observed in the minority-spin channel of both compounds, which shows their half-metallic behavior at the equilibrium state. We also plotted the partial density of states DOS in Fig. 4 for Ce/Pr-f & d, Mn-d, and O-p electrons for both materials in order to see the contributions of electrons from different atomic orbitals to the Seebeck coefficient (S). The Ce/Pr-f states contribute more toward the electrical conductivity in the spin up phase and Mn-d states with a little contribution from the O-p states populating the DOS near the Fermi level. Therefore, thermally excited electrons from these orbitals contribute to the value of S. As inferred from the projected density of states (pDOS), the electronic structure of the optimized ground states for both compounds is characterized by strong hybridizations between rare earth RE-f and Mn-3d levels. Small differences on comparing the pDOS corresponding to the 3d and 4f levels of the two different Mn sub-lattices reflect weak disproportionation effects and a small charge ordering.

Fig. 2 Spin dependent band structures of (a) CeMnO₃ and (b) PrMnO₃ calculated by the GGA+U method.

Fig. 3 Spin-polarized total density of states (DOS) of CeMnO₃ and PrMnO₃ at equilibrium lattice constant by GGA, GGA+U and mBJ.

Table 3 Calculated values of total, individual and interstitial spin moments of CeMnO₃ and PrMnO₃

Compound	μCe,Pr	μMn	μO	μInt.	μTotal	Band gap
CeMnO3
GGA	0.95	3.09	0.04	0.38	4.46
GGA+U	1.03	3.54	0.01	0.42	5.00	2.76
TB–mBJ	1.06	3.51	0.05	0.28	6.00	3.43
Others						2.84 (ref. 17)
PrMnO3
GGA	2.17	3.23	0.04	0.42	5.86
GGA+U	2.19	3.37	0.03	0.41	6.00	2.69
TB–mBJ	2.17	3.24	0.04	0.42	5.87	3.64
Others						3.00 (ref. 18)

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Fig. 4 Spin-polarized partial densities of states (PDOS) of CeMnO₃ and PrMnO₃ at equilibrium lattice constant calculated by the GGA+U approach.

The magnetic ground state for both compounds is confirmed by the absolute positive values of the total energy difference, ΔE = E_AFM − E_FM, which is provided by double cell structural optimization.²⁵ The calculated total and individual magnetic moments, using GGA, GGA+U and mBJ for the CeMnO₃ and PrMnO₃ compounds, are listed in Table 3. It is clear from this table that the interstitial magnetic moments by GGA+U are 0.41 μ_B and 0.412 μ_B, respectively, and the calculated total magnetic moments are 5 μ_B (CeMnO₃) and 6 μ_B (PrMnO₃). Similarly, by mBJ the corresponding values of interstitial and total magnetic moments are 0.42 μ_B and 0.41 μ_B and 6 μ_B and 5.87 μ_B for CeMnO₃ and PrMnO₃ compounds, respectively. The total integral magnetic moment, which is a characteristic of half metallic materials, for the present compounds arises from the rare earth (Ce/Pr) and Mn ions with a little contribution from O atoms. From the overall DOS and the total magnetic moments, we can clearly define that the double exchange mechanism is responsible for the ferromagnetic interaction in REMnO₃ perovskites.²⁶ Thus the half metallicity along with 100% spin polarization at the Fermi level and quantized magnetic moments will enable the application of these materials in a new class of spintronic devices, which use the magnetic moments of electrons rather than relying on the transport of electronic charge to store and transmit information. These spintronic devices have potentially revolutionized computing and shape memory devices with enormous data storage capacity.

Also, the magnetic behavior of these materials is confirmed by the post-DFT technique.³⁹ The plots of χ and (χ⁻¹) versus temperature for the REMnO₃ (RE = Ce and Pr) compounds are plotted in Fig. 5. This figure shows a decreasing trend of the susceptibilities with an increase in temperature of the compounds. In the temperature range of 0–250 K, a considerable decrease occurs in the susceptibilities of the compounds, while this change is insignificant from 250 K to 800 K. This decrease in magnetic susceptibility with increasing temperatures is offered by the decline in the alignment of the magnetic moments due to thermal agitation. Saturation in the magnetic susceptibility in both materials is observed beyond 300 K. The Curie Weiss law of ferromagnetism is justified by the positive Weiss constant (θ) for CeMnO₃ at 50 K and PrMnO₃ at about 75 K. The magnitude of the magnetic susceptibility at 50 K for CeMnO₃ is 2.6 (10⁻⁹ emu mol⁻¹) and 3.3 (10⁻⁹ emu mol⁻¹) for PrMnO₃. However, at room temperature the former shows a higher value of 1.4 (10⁻⁹ emu mol⁻¹), whereas the latter exhibits a smaller value of 0.9 (10⁻⁹ emu mol⁻¹). Thus, there is a considerably smaller decrease in the overall susceptibility curve of CeMnO₃ than PrMnO₃.

Fig. 5 Magnetic susceptibility (χ) and inverse of magnetic susceptibility (χ⁻¹) plots versus temperature of the CeMnO₃ and PrMnO₃ compounds.

Thermoelectric properties

In order to examine the TE properties of these cubic perovskites, calculations were performed within the semiclassical Boltzmann theory with a constant relaxation time and the rigid band approach.³⁹ The latter assumes that the band structure does not change with temperature or doping and therefore, it is independent of the chemical potential. The transport properties rely upon the Fourier expansion of the band-energies and their elusive nature towards Brillouin zone (BZ) sampling made us use a dense mesh of 100 000 k-points to reach the convergence for the Fermi-surface integrals involved in the TE quantities presented. Due to the cubic symmetry of the system, the only relevant quantities are σ, zT and thermopower which are presented as the average value of the diagonal components of the TE tensors. The thermoelectric properties, including temperature-induced electrical conductivity (σ/τ), electronic thermal conductivity (κ_e/τ) (where τ is the relaxation time) and Seebeck coefficient (S) are determined for both materials. Since, the transport coefficients are defined by a single kernel function:⁴⁰where, α and β are the tensor indices, ν_α and ν_β are the group velocities, and e and τ_k are the charge of an electron and relaxation time, respectively.

Thereafter, the transport coefficients σ, k and chemical potential (μ) are obtained using the equations given elsewhere.³⁵

The temperature variations of the thermoelectric parameters are presented in Fig. 6(a)–(d). From the plots, it can be seen that both compounds display a similar trend in results. Since, we have adopted the energy independent relaxation time (τ) in all the thermoelectric quantities except S in this calculation, the overall S as a function of temperature (T) for the present materials is therefore calculated via two current models,⁴¹ which are given by the following equation.

where, σ(↑) and σ(↓) are electrical conductivities and S(↑) and S(↓) are Seebeck coefficients in the spin up and down channels, respectively.

Fig. 6 Temperature dependence of (a) electrical conductivity over relaxation time (σ/τ); (b) thermal conductivity (k); (c) Seebeck coefficient (S) and (d) figure of merit (zT) calculated for CeMnO₃ and PrMnO₃.

The temperature variation of electrical conductivity (σ) over relaxation time (τ), as reported in Fig. 6(a), illustrates a linearly increasing trend with temperature for both materials. This linear increase in σ/τ from a very small value of ∼2.18 × 10¹⁷ (Ω ms)⁻¹ for CeMnO₃ and ∼1.2 × 10¹⁷ (Ω ms)⁻¹ for PrMnO₃ at 50 K, which attains a high value of ∼4.75 × 10¹⁸ (Ω ms)⁻¹ and ∼5.45 × 10¹⁸ (Ω ms)⁻¹ at 800 K, respectively, unveils a bright potential for high-temperature TE power generation. The Seebeck coefficient, S, as a function of temperature is negative in the entire temperature range, which signifies the presence of n-type charge carriers, i.e. electrons, in both materials, whereas a positive value suggests the presence of p-type carriers i.e. holes. The sharp decrease in the S value at lower temperatures (∼from −2.8 × 10⁻³ V K⁻¹ at 50 K to 2.5 × 10⁻⁴ V K⁻¹ at 300 K in CeMnO₃ and ∼from −2.0 × 10⁻³ V K⁻¹ at 50 K to 5.2 × 10⁻⁴ V K⁻¹ at 300 K in PrMnO₃) is due to the depopulation of the phonon modes.⁴² Moreover, the dependence of κ with T is observed to be linear. The calculated value of S for CeMnO₃ and PrMnO₃ at room temperature (300 K) is found to be 135 μV K⁻¹ and 150 μV K⁻¹, respectively.

Furthermore, from the measured data of electrical, thermal and Seebeck coefficients in the temperature range of 50–800 K, the temperature dependent thermoelectric figure of merit zT has been estimated. In Fig. 6(d), we show zT as a function of temperature for CeMnO₃ and PrMnO₃. It can be seen from the plot that the zT at 300 K is 0.24 and 0.27 for CeMnO₃ and PrMnO₃, respectively, and it achieves the highest value of 0.40 and 0.47 at a temperature of 500 K. Also, later on the zT value for both systems is almost constant. This is due to the fact that the maximum carrier concentration is reached at the corresponding temperatures. The maximum value of zT is backed by the large magnitude of S with linear like behavior of σ/τ and k. The direct reliance of zT on S (due to continuous increase in the magnitude of S) leads to increased zT values at the higher temperature range. Hence, the present materials can also prove to be excellent candidates for thermoelectric application. It is possible to obtain an increase in efficiency with these perovskites provided that they are subjected to proper chemical doping in order to reduce their thermal conductivity, tuning of carrier concentration to increase their electrical conductivity, and incorporation of nano-scale precipitates and grain refinement.^43,44 Unfortunately, a comparison of our results is not achieved due to the lack of any experimental data regarding these materials, however this work can act as reference and may offer valuable guidance to future experimental investigations.

Conclusion

In summary the investigation of the ground state structural, and elastic and thermoelectric properties of the cubic perovskites CeMnO₃ and PrMnO₃ have been carried out using GGA, GGA+U and mBJ within the frame work of DFT. The calculated relaxed lattice parameters [i.e., equilibrium lattice constants (a), bulk modulus (B), optimized volume (V) and cohesive energies, (E₀)] achieved from the analytical ionic radii methods and DFT calculations are consistent with the experimental results. The theoretically calculated tolerance factors and mechanical properties demonstrate that CeMnO₃ is a ductile material in comparison to the brittle and hard PrMnO₃ compound, and are both stable in the cubic phase. The brittle and stiff nature of PrMnO₃ endorses its application as an incompressible (hard) material. The electrical, Seebeck and thermal coefficients of these compounds were calculated via the Boltzmann theory, and these details were used to determine the temperature dependent figure of merit for their thermoelectric performance. At 300 K, zT is high for the PrMnO₃ compared to CeMnO₃. The present work will provide reference data to future researchers for the utilization of these materials in the field of thermoelectrics.

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