Physical Characteristics of Hydride Perovskites XZrH₃ (X = Mg, Ca, Sr, and Ba) as Materials for Hydrogen Storage: A First-Principles Investigation

Abstract

In this study, density functional theory (DFT) within the generalized gradient approximation (GGA) is employed to investigate the structural, electronic, mechanical, and thermoelectric properties of perovskite hydrides XZrH₃ (X = Mg, Ca, Sr, Ba). Mechanical stability and ductility are evaluated through the Cauchy pressure, Pugh’s ratio, and Poisson’s ratio, all of which point to ductile behavior with a dominant ionic-bonding character. Electronic structure calculations reveal metallic behavior arising from band overlap at the Fermi level. Equilibrium energy–volume data are fitted with the Murnaghan equation of state, and transport coefficients are extracted using the BoltzTraP package as implemented in WIEN2k. The absence of a band gap and the overlap between valence and conduction bands confirm conductor-like behavior. Lattice thermal conductivity for MgZrH₃, CaZrH₃, SrZrH₃, and BaZrH₃ increases monotonically with temperature. Overall, the results identify MgZrH₃ in particular as a promising candidate for thermoelectric devices and solid-state hydrogen storage, thereby supporting progress toward a sustainable hydrogen economy.

1. Introduction

Anthropogenic climate change, intimately coupled to the world’s present and future energy demand, is arguably the most pressing challenge facing society [1,2]. Because hydrogen offers a high gravimetric energy density, can be produced by diverse renewable routes, is amenable to long-term storage, and serves a wide spectrum of end uses, it is widely viewed as a cornerstone of 21st-century energy systems [3,4,5]. Current technologies already exploit hydrogen in internal combustion engines, gas turbines, and, most prominently, fuel cells [6,7,8,9,10]. However, industrial-scale storage remains problematic. Compressed-gas tanks are limited by safety and practical pressure ceilings [11,12,13], cryogenic liquefaction reduces volume but incurs boil-off losses and additional safety hazards [14,15,16], and physisorption systems demand materials with both very high capacity and fast sorption kinetics [17,18,19]. Reversible metal hydrides also solve some of the issues listed by chemically linking hydrogen, offering higher volumetric capacity and inherent safety advantages [20,21]. Of these, ternary perovskite hydrides (formula structure ABX₃, with H in the X site) are especially promising: their open, tunable crystal form can accept hydride anions interstitially or on lattice sites to facilitate extensive, reversible hydrogen absorption [22,23,24,25,26]. Owing to immense compositional liberty inherited from mineral perovskites, hydride perovskites can be tailored for a wide range of applications [27,28]. New reports have presented gravimetric storage capacity near practical levels, demonstrating their potential to live up to the full potential of a hydrogen-based economy [29,30,31,32,33,34]. State-of-the-art density functional theory (DFT) has become indispensable for probing and optimizing these materials [35,36,37]. Atomistic modeling elucidates their structural stability, electronic and optical response, and key transport parameters such as electrical conductivity, thermal conductivity, and the thermoelectric figure of merit [38,39,40]. Further, the introduction of catalytically active ions into the perovskite lattice can connect hydride chemistry to more effective photo-electrochemical and electrochemical conversion routes [41,42]. Combined, these findings chart a path towards perovskite hydrides that surpass not only the demanding criteria for hydrogen storage but also serve as multifunctional building units in future sustainable-energy technologies [43,44,45,46].

Recent advances in computational materials science have enabled the engineering of compounds with optimized thermoelectric performance, paving the way for cleaner, more efficient waste-heat harvesting technologies. Perovskite-type hydrides have attracted increasing attention due to their favorable electronic structure and potential for hydrogen storage and thermoelectric applications; however, kinetic limitations in hydrogen sorption remain a significant challenge for their practical use [47,48,49]. Similar to many other metal and complex hydrides, slow hydrogen absorption and desorption rates, as well as high activation barriers for hydrogen diffusion, can limit their operational performance [50,51]. Although most studies on perovskite hydrides have primarily focused on structural stability, formation enthalpies, and hydrogen release thermodynamics, recent computational and experimental works have begun to address dehydrogenation pathways, desorption temperatures, and diffusion characteristics, which are indirectly related to kinetic behavior [52,53].

In this work we couple density functional theory within the generalized gradient approximation (GGA) to semiclassical Boltzmann transport calculations, implemented in BoltzTraP2, to probe the electronic and thermoelectric behavior of the cubic hydrides XZrH₃ (X = Mg, Ca, Sr, Ba). The following sections detail the computational protocol, present the temperature-dependent transport coefficients we have obtained, and discuss how these results inform the use of XZrH₃ in renewable-energy systems, combining its thermal-to-electrical conversion with reversible hydrogen storage.

2. Calculation Method

In this study, density functional theory (DFT) was employed to assess the hydrogen-storage capacity and other key physical properties of the perovskite hydrides XZrH₃ (X = Mg, Ca, Sr, Ba). Exchange–correlation effects were treated with the generalized gradient approximation (GGA). Before analyzing electronic and thermoelectric behavior, the equilibrium lattice parameters were obtained by minimizing the total energy as a function of volume, using the Birch–Murnaghan equation of state in the WIEN2k code [54,55]. Self-consistency was deemed achieved when the total-energy change fell below 10⁻⁴ Ry and the total charge difference below 10⁻³ e. The Brillouin zone was sampled with a dense 10 × 10 × 10 k-point mesh to ensure accurate integration. Thermoelectric transport coefficients were subsequently evaluated with BoltzTraP, which interpolates the DFT band structure and solves the semiclassical Boltzmann transport equations within the constant-relaxation-time approximation. This combined workflow yielded a comprehensive picture of the structural stability, electronic structure, and thermoelectric performance of the XZrH₃ series.

In the FP-LAPW framework implemented in the WIEN2k code, the Muffin-Tin radii (RMT) were chosen as follows: X = (2.2–2.5 atomic unit); Zr = 2.3 atomic unit, and H = 1.6 atomic unit, ensuring no sphere overlap. The plane-wave cutoff parameter was set by RK_max = 7.0, which guaranteed the convergence of total energies within 1 meV/atom. The maximum angular momentum inside the Muffin-Tin spheres was set to l_max = 10. The charge density Fourier expansion was limited to G_max = 12 (atomic unit)⁻¹. These parameters ensure high numerical accuracy and full reproducibility of the reported results.

3. Results and Discussion

3.1. Structural Properties

The XZrH₃ (X: Mg, Ca, Sr, Ba) perovskite hydrides’ structural characteristics are covered in this section. These perovskite hydrides are members of the space group Pm-3m (#221) and have a cubic crystal shape. The cubic compound XZrH₃, with X = Mg, Ca, Sr, and Ba, has a three-dimensional crystal structure shown in Figure 1. The atoms within the cubic crystal unit cell are positioned as follows during the crystalline structure optimization process: Figure 1 displays the optimal architectures of the unit cell with hydrogen atoms at the face centers, Zr cations at the center, and X-cations at the corners.

Table 1. Theoretical and optimized lattice parameters, gravimetric capacity Cwt%.

Compound	Lattice Parameter a0 = b0 = c0		Gravimetric Capacity Cwt%
MgZrH3	3.84 Å [56]	Other Study	2.55
	3.8686 Å	In this Study
CaZrH3	3.95 Å [56]	Other Study	2.25
	4.003 Å	In this Study
SrZrH3	4.04 Å [56]	Other Study	1.66
	4.1189 Å	In this Study
BaZrH3	4.24 Å [56]	Other Study	1.31
	4.2460 Å	In this Study

summarizes the lattice constants (Å), optimized and theoretical, of XZrH₃ (X: Mg, Ca, Sr, and Ba). Because of the variations in ionic radii, volume, and valence electrons, the computed lattice constants of the investigated XZrH₃ materials are BaZrH₃ (4.24 Å) > SrZrH₃ (4.04 Å) > CaZrH₃ (3.95 Å) > MgZrH₃ (3.84 Å).

A statistical relationship known as the Murnaghan equation of state accounts for the pressure behavior of materials; specifically, it provides how a material’s volume changes upon applied pressure. It is useful in the crystal structure optimization of a material in DFT calculations. The Murnaghan equation of state (EOS) matches the total energy to the normalized volume to determine the assumed equilibrium a₀. Our calculated and experimental equilibrium a₀ are in good agreement. Using the equation of Birch–Murnaghan, volumes determined as a function of the expected energies are graphed in Figure 2 and listed

Table 2. Calculated optimized parameters a₀, V, E, B, and B’ of XZrH₃ (X = Mg, Ca, Sr, Ba).

Parameter	Lattice Parameter a0 (Å)		Volume Minimum V (Atomic Unit)3	Total Energy E (Ry)	Bulk Modulus B (GPa)	Pressure Derivative B’
	Optimized	Other Work
MgZrH3	3.8686 Å	3.84 Å [56]	390.7144	−7602.641	69.437	14.687
CaZrH3	4.003 Å	3.95 Å [56]	431.9880	−8562.838	92.335	16.231
SrZrH3	4.1189 Å	4.04 Å [56]	471.5657	−13,561.733	58.827	5.960
BaZrH3	4.2460 Å	4.24 Å [56]	516.5612	−23,480.759	61.088	9.197

.

$$E=E_0+{\displaystyle \frac{B_0}{B_0^{\prime }}}\left(V-V_0\right)-{\displaystyle \frac{B_0{V}_0}{B_0^{\prime }\left(1-B_0^{\prime }\right)}}\left[{\left({\displaystyle \frac{V}{V_0}}\right)}^{1-B_0^{\prime }}-1\right]$$

(1)

where parameter B₀, referred to as the bulk modulus or isostatic compressibility modulus, represents the material’s resistance to compression; the pressure derivative of the compressibility modulus, ${\mathrm{B}}_0^{\prime }$, represents the change in the modulus with the applied pressure; and assumes E₀, the ground state energy for the volume V₀ of the unit cell, to be the minimum energy.

The gravimetric hydrogen storage capacity of XZrH₃ (X = Mg, Ca, Sr, Ba) perovskites is evaluated using the formula presented in Equation (2) to determine their suitability for hydrogen storage applications. The gravimetric ratio (Cwt%) quantifies the amount of hydrogen stored relative to the total mass of the host perovskite material. This value is calculated as follows:

$$C_{w\%}= \left({\displaystyle \frac{\frac{H}{M}{m}_H}{m_{Host}+\left(\frac{H}{M}\right)m_H}}\times 100\right)\%$$

(2)

where mH is the molar mass of hydrogen; mHost is the molar mass of the host material; and H/M refers to the hydrogen-to-material atom ratio. The Cwt% values in Table 1 represent the gravimetric hydrogen storage capacity of XZrH₃ (X = Mg, Ca, Sr, Ba) perovskites.

3.2. Electronic Properties

The electronic structures of the XZrH₃ hydrides (X = Mg, Ca, Sr, Ba) are investigated using the primitive unit cell, where the Zr atom occupies a high-symmetry Wyckoff position. To understand the nature of the chemical bonding and the electronic behavior, we computed the electronic band structures along high-symmetry directions of the Brillouin zone shown in Figure 3, as well as the partial densities of states (PDOS) shown in Figure 4. As illustrated in Figure 3, the Fermi level (E_F) is set to zero energy and represented by a horizontal dashed line. For all four hydrides, the valence band maximum and the conduction band minimum intersect at the Fermi level, indicating the absence of a band gap. This clearly indicates that all the compounds we have investigated possess metal-like properties. In the density of states, which we have plotted from −7 eV to +7 eV, it is clear that these compounds possess metal-like properties. In the case of metal hydrides, the presence of free-moving charge carriers, which are not bound to a particular location, will be highly interesting in the storage of hydrogen. Our results match the results given by other people on XAlH₃ [40], as well as in MgCuH₃ and MgCoH₃ [39]. The metal-like behavior of these compounds will similarly be highly interesting in the storage of hydrogen. In the PDOS plot, it is clear that the valence band is a combination of the Zr-4d and H-1s orbitals. The conduction band will have a combination of the X-ns orbitals and the Zr-5s orbitals, with some remaining Zr-4d orbitals. This indicates that there is a hybridization of the Zr-d and X-s orbitals in these compounds. The Zr-d and X-s orbitals have a strong interaction with each other in these compounds. The strong interaction of the Zr-d and X-s orbitals in these compounds may be the reason for the observation of the metal-like behavior in the PDOS plot of the Zr-d and X-s orbitals.

In order to better understand the nature of the charge distribution, the results of the Bader charge analysis will be discussed. It can be seen from the analysis that there is a large transfer of electrons from the electropositive X and Zr atoms to the more electronegative H atoms. These electrons are primarily in the H-1s orbitals, with the remaining donor charge in the Zr-4d and X-(3–6)s orbitals. While the metallic character and charge transfer toward hydrogen atoms suggest favorable electronic conditions for hydrogen bonding, a quantitative assessment of hydrogen reversibility requires explicit calculation of hydrogen desorption energies and diffusion barriers. Such thermodynamic and kinetic analyses are beyond the scope of the present study and will be addressed in future work to provide a more comprehensive evaluation of the storage potential.

The predicted metallic character of XZrH₃ hydrides differs significantly from classical hydrides such as MgH₂ and several simple XH₃ compounds, which are generally wide band gap insulators due to the dominant ionic bonding and complete filling of valence states. In MgH₂, strong Mg²⁺–H⁻ interactions lead to a large electronic band gap and insulating behavior. In contrast, in XZrH₃ perovskite hydrides, the presence of transition-metal Zr introduces partially filled 4d states that strongly hybridize with H-1s orbitals near the Fermi level. This hybridization generates finite electronic states crossing the Fermi level, resulting in metallic conductivity. Similar electronic features have been reported in previous first-principles investigations on Zr-containing and perovskite-type hydrides [57,58,59]. The metallic character may also influence hydrogen transport properties. Compared to purely insulating hydrides, metallic systems exhibit enhanced electronic screening, which can reduce Coulomb interactions and potentially lower migration barriers for hydride ions. However, hydrogen diffusivity remains governed not only by electronic structure but also by lattice dynamics, defect chemistry, and migration pathways [60,61,62,63,64].

3.3. Elastic Properties

The elastic stiffness tensor components are calculated using the finite strain method within the density functional theory (DFT) framework as implemented in the WIEN2k code. After full structural optimization, small symmetry-conserving strains are applied to the equilibrium unit cell, and the corresponding total energy variations are calculated self-consistently. The elastic constants are then obtained from the second-order derivatives of the total energy. For cubic crystals such as XZrH₃, only three independent elastic constants (C₁₁, C₁₂, and C₄₄) are required to fully describe the stiffness tensor. All elastic constants are calculated for the fully relaxed stoichiometric compounds with fixed hydrogen composition corresponding to the XZrH₃ formula. The effect of hydrogen composition variation was not considered in this work and will be investigated in future studies.

The elastic constants C₁₁, C₁₂, and C₄₄ of the XZrH₃ compounds (X = Mg, Ca, Sr, Ba) are key parameters that determine their mechanical behavior, including stiffness, structural stability, and resistance to various deformation modes [65,66]. The mechanical stability of cubic systems requires that the following Born stability criteria be satisfied:

C₁₁ + 2C₁₂ > 0, C₁₁ − C₁₂ > 0, C₁₁ > 0

(3)

For cubic crystals, it is generally observed that C₁₁ > C₁₂ and C₁₁ > C₄₄, reflecting strong resistance to longitudinal deformation compared to shear deformation modes. In the present study on metal hydrides (see

Table 3. Elastic constants of XZrH₃.

Compound	C11 (GPa)	C12 (GPa)	C44 (GPa)
MgZrH3	147.6366	32.4142	20.4028
CaZrH3	96.6624	27.0057	33.3996
SrZrH3	87.4971	40.9646	31.5442
BaZrH3	63.9735	39.3910	9.4161

), particular attention is given to the C₁₁ value, which shows that resistance to uniaxial compression decreases systematically from MgZrH₃ (147.64 GPa) to BaZrH₃ (63.97 GPa). This indicates that MgZrH₃ exhibits the highest stiffness under compressive loading, whereas BaZrH₃ is the most compliant. This progressive softening can be attributed to the increasing atomic radius of the X-site cation (Mg → Ba), which reduces interatomic bonding strength and thus lowers the resistance to axial compression.

When we look at the values of C₁₂, which show how the atoms interact with each other when we apply stress, we see that they are in the middle for all the compounds. These values go up when we use heavier alkaline earth elements, and they are the highest for BaZrH₃.

On the other hand, the values of C₄₄, which show how well the material can resist being deformed, are the highest for CaZrH₃ at 33.3996 GPa. For BaZrH₃, the value of C₄₄ is much lower at 9.4161 GPa. This means that BaZrH₃ is not very good at resisting shear deformation and is more able to change shape without breaking, which is called ductility. Despite these variations, all compounds satisfy the mechanical stability criteria, confirming their thermodynamic stability at small strains [67].

The mechanical performance of the cubic hydrides XZrH₃ (X = Mg, Ca, Sr, Ba) is governed by the interplay between interatomic bond strength, crystal symmetry, and the ionic radius of the alkaline-earth cation. Particular attention is given to the C₁₁ value and the bulk modulus, defined by Equation (4) as

$$B={\displaystyle \frac{C_{11}+{2C}_{12} }{3}}$$

(4)

which measure the resistance to uniaxial and uniform compression, respectively. Both properties decrease systematically across the series, with C₁₁ falling from 147. 63 GPa for MgZrH₃ to 63.97 GPa for BaZrH₃, and the bulk modulus decreasing from 70.82 GPa to 47.58 GPa. This softening reflects lattice expansion and weaker bonding as the X-site cation becomes larger, indicating that MgZrH₃ exhibits the highest stiffness under compressive loading, whereas BaZrH₃ is the most compliant.

The shear modulus, obtained from the Voigt–Reuss–Hill average [68,69] and defined by Equations (5)–(7) as

$$G={\displaystyle \frac{G_V+G_R }{2}}$$

(5)

$$\mathrm{Or} G_V={\displaystyle \frac{C_{11}-{2C}_{12}+{3C}_{44} }{5}}$$

(6)

$$G_R={\displaystyle \frac{{5(C}_{11}-C_{12})C_{44} }{{4C}_{44}{+3(C}_{11}-C_{12})}}$$

(7)

shows a similar but more pronounced decline across the series, with G ≈ 40–42 GPa for Mg-, Ca-, and SrZrH₃, and only 22.29 GPa for BaZrH₃. This indicates that the lattice of BaZrH₃ is much more susceptible to shear deformation compared to the other hydrides.

The overall stiffness can be evaluated through Young’s modulus [70,71], expressed as:

$$E={\displaystyle \frac{9BG}{3B+G}}$$

(8)

This parameter attains its minimum value for BaZrH₃ (57.85 GPa), markedly lower than the values (>97 GPa) found for the lighter congeners.

Ductility is assessed via Pugh’s ratio B/G: materials with B/G > 1.75 are generally ductile, whereas lower ratios signify brittleness. BaZrH₃ exhibits the highest ratio (2.13) and is thus the most ductile; CaZrH₃, with B/G = 1.19, is markedly brittle, while MgZrH₃ (1.72) and SrZrH₃ (1.40) occupy an intermediate regime.

Poisson’s ratio [72,73] is expressed as:

$$\nu ={\displaystyle \frac{(3B-2G) }{3(2B+G)}}$$

(9)

The calculated values show a clear variation among the studied compounds. BaZrH₃ exhibits the highest value (ν ≈ 0.279), which is consistent with ductile metallic bonding. In contrast, CaZrH₃ displays the lowest value (ν ≈ 0.156), indicating a more brittle and covalent-like network.

Although BaZrH₃ exhibits the lowest bulk modulus (≈47 GPa), indicating reduced mechanical stiffness, such moderate softness is not necessarily unfavorable for hydrogen storage applications (

Table 4. Parameters of elastic.

Compound	B (GPa)	G (GPa)	E (GPa)	B/G	v
MgZrH3	70.821	41.122	103.361	1.722	0.237
CaZrH3	50.224	42.066	98.655	1.193	0.155
SrZrH3	56.475	40.213	97.498	1.404	0.193
BaZrH3	47.585	22.294	57.848	2.134	0.278

). Materials used for reversible hydrogen storage undergo repeated lattice expansion and contraction during hydrogen absorption and desorption cycles. A relatively lower bulk modulus may facilitate structural adaptability and reduce internal stress accumulation during cycling. However, very low stiffness could limit resistance to mechanical deformation under external pressure. Therefore, BaZrH₃ represents a trade-off between mechanical flexibility and structural robustness, whereas MgZrH₃ offers higher rigidity and better mechanical resistance.

In summary, MgZrH₃ is the most rigid and least compressible member of the series, whereas BaZrH₃ is the most ductile but also the softest, while CaZrH₃ exhibits pronounced brittleness. These trends arise from the increasing cation size and the consequent weakening of Zr–H and X–H bonds from Mg to Ba. This behavior provides a clear guideline for selecting hydrides with specific combinations of stiffness, ductility, and resistance to deformation [74].

Although phonon dispersion calculations were not performed in the present study, the fulfillment of the Born mechanical stability criteria and the positive elastic constants confirm the structural robustness of the optimized cubic phase at small deformations. A full dynamical stability investigation via phonon spectra will be addressed in future work.

3.4. Thermoelectrical Properties

The transport properties, including electrical conductivity, electronic thermal conductivity, power factor, and thermoelectric figure of merit, are calculated using the BoltzTraP2 code based on the semiclassical Boltzmann transport theory. This approach uses the electronic band structure obtained from density functional theory calculations performed with the WIEN2k code. The band energies are interpolated on a dense k-point grid to accurately determine the transport coefficients as a function of temperature and chemical potential. The calculations are performed within the constant relaxation time approximation, which assumes that the carrier relaxation time remains constant. Consequently, the electrical conductivity and electronic thermal conductivity are presented in normalized form with respect to the relaxation time. This method provides reliable insight into the intrinsic transport behavior of the studied hydrides.

Large fractions of the energy produced by renewable technologies are ultimately lost as low-grade heat. Converting this waste heat directly into electricity requires thermoelectric materials that combine high electrical conductivity and low thermal conductivity [75,76,77]. With this motivation, we assessed the thermoelectric performance of the cubic hydrides XZrH₃ (X = Mg, Ca, Sr, Ba) using density functional theory combined with semiclassical Boltzmann transport theory, as implemented in the BoltzTraP2 code. Within the constant relaxation time approximation, the temperature-dependent electrical conductivity (σ/τ), electronic thermal conductivity (κ_e/τ), power factor, and figure of merit ZT [78,79] are analyzed over a temperature range from 300 K to 900 K.

• Electrical conductivity (σ/τ)

Electrical conductivity reflects both carrier concentration and mobility, which are strongly influenced by the electronic band structure and bonding characteristics of the material. As shown in Figure 5a, the electrical conductivity increases with temperature for MgZrH₃, CaZrH₃, and SrZrH₃, indicating enhanced carrier transport due to thermal excitation and increased occupation of electronic states near the Fermi level. The variation in transport behavior across the XZrH₃ series is closely related to the influence of the alkaline earth cation on the electronic structure, which modifies band dispersion and carrier effective mass.

A high electrical conductivity combined with a favorable Seebeck coefficient contributes to an enhanced power factor and improved thermoelectric performance, provided that thermal conductivity remains sufficiently low. These transport characteristics are governed by the electronic structure near the Fermi level, which determines carrier dynamics and transport efficiency. In metallic hydrides, conduction electrons play an important role in screening electrostatic interactions between metal atoms and hydrogen. This electronic screening weakens the effective Coulomb interaction and stabilizes the metal–hydrogen bonding environment. As a result, it can influence hydrogen binding strength, diffusion, and reversibility within the lattice. Improved electronic screening facilitates hydrogen mobility and contributes to reversible hydrogen absorption and desorption processes. However, hydrogen storage performance is primarily controlled by thermodynamic stability, lattice structure, and hydrogen diffusion barriers, while electronic carriers provide a secondary but supportive role in stabilizing the bonding environment. Overall, the calculated transport properties demonstrate that the electronic structure, modified by alkaline earth substitution, plays a key role in determining both the thermoelectric response and hydrogen-related functional behavior of XZrH₃ hydrides.

• Electronic thermal conductivity (κe)

One of the major indicators to measure the efficiency of thermoelectric materials is electronic thermal conductivity (κ_e). Thermal conductivity of XZrH₃ (X = Mg, Ca, Sr, and Ba) is important to forecast how efficient these materials would be in devices for energy conversion since heat transfer in a material has a direct impact on whether or not it can maintain a useful thermal gradient. The results indicate that for all the components examined, thermal conductivity rises proportionally to temperature (Figure 5b). Enhancement of the vibration of atoms by thermal energy carriers with temperature is the mechanism responsible for this rise in thermal conductivity. At 900 K, κ_t values reach 15.7 W/(K·m) for MgZrH₃, 14.2 W/(K·m) for CaZrH₃, 13.1 W/(K·m) for SrZrH₃, and 11.2 W/(K·m) for BaZrH₃. However, excessively high thermal conductivity constitutes a limitation on thermoelectric devices because of the inclination to dissipate the temperature gradient responsible for current generation quickly. This result implies that the X-element selection can be maximized for the operating temperature of interest. For instance, MgZrH₃ with reduced thermal conductivity at high temperatures may be more suitable for devices being used at high temperatures with a high conversion efficiency. In conclusion, one key consideration in the development of effective thermoelectric materials is creating a compromise between electrical and thermal conductivity. According to this study, XZrH₃ compounds have scalable properties and by properly choosing the alkaline element, thermal properties may be optimized in terms of certain applications to power generation or heat recovery from waste thermal sources. This is because of the increase in atomic vibrations with temperature that improves the transfer of heat through thermal energy carriers. However, since it can quickly drain the temperature gradient necessary for current production, too high a thermal conductivity can be a limiting factor for application in thermoelectric devices. Thus, to produce the highest ZT figure of merit, a suitable thermoelectric material should have high Seebeck coefficient, superior electrical conductivity, and low thermal conductivity. Atomic structure affects the thermal properties of XZrH₃ compounds, as indicated by varying κe of MgZrH₃, CaZrH₃, SrZrH₃, and BaZrH₃.

• Electronic factor of merit (ZT):

The ZT factor of merit, the central parameter for evaluating the efficiency of a thermoelectric material, is defined by the following equation [80,81]:

$$\mathrm{ZT} = {\displaystyle \frac{\mathsf{\sigma }{S}^2\mathrm{T}}{\mathsf{\kappa }}}$$

(10)

where T is the absolute temperature; κ is the overall thermal conductivity; S is the Seebeck coefficient; and σ is the electrical conductivity. A perfect balance of these properties must be achieved to optimize heat-to-electricity conversion, as this equation illustrates.

Figure 5c shows that MgZrH₃ exhibits the highest thermoelectric figure of merit among the studied compounds, reaching approximately 8.0 × 10⁻² at 900 K. In contrast, BaZrH₃ presents slightly lower thermoelectric performance, with a ZT value of approximately 7.2 × 10⁻² at the same temperature. The reduced performance of BaZrH₃ is mainly attributed to its lower power factor and higher electronic thermal conductivity, which limit its thermoelectric efficiency compared to the other compounds.

For SrZrH₃ and CaZrH₃, the thermoelectric figure of merit exhibits a slight decrease in the intermediate temperature range, followed by a significant increase at higher temperatures, reaching values of approximately 8.2 × 10⁻² and 7.9 × 10⁻² at 900 K, respectively. This behavior can be attributed to the temperature dependence of carrier transport properties. At higher temperatures, increased thermal excitation enhances carrier concentration and transport, improving electrical conductivity while maintaining favorable Seebeck coefficient values. As a result, the thermoelectric performance improves significantly in the high-temperature region.

These results indicate that the thermoelectric efficiency of XZrH₃ hydrides improves at elevated temperatures, particularly above 600 K, suggesting their potential suitability for high-temperature thermoelectric applications. The observed trends are primarily governed by the electronic structure near the Fermi level and the associated carrier transport properties.

• Power factor (PF)

The power factor (PF) is a fundamental parameter for assessing the thermoelectric performance of a material, particularly under high-temperature conditions, and is described by the equation [80,81]:

PF = σS²

(11)

This is a measure of the efficiency of a substance in transferring thermal energy to electric energy, where σ is the electrical conductivity and S is the Seebeck coefficient. The larger the value of PF, the greater the prospect of using a substance in thermoelectric conversion. There is a small drop in PF from 300 K to 400 K, after which its variation with temperature in XZrH₃ (X = Mg, Ca, Sr, and Ba) exhibits an intriguing thermal characteristic. A two-step variation in ZT is observed for SrZrH₃ and CaZrH₃, with a slight decrease from 300 K to 500 K (Figure 5d). In all cases, the power factor (PF) increases steadily up to 900 K. At this temperature, values reach 13.8 × 10⁻⁴ W/(K²·m) for MgZrH₃, 13.6 × 10⁻⁴ W/(K²·m) for CaZrH₃, 12.8 × 10⁻⁴ W/(K²·m) for SrZrH₃, and 9.2 × 10⁻⁴ W/(K²·m) for BaZrH₃. The material is suited for heat recovery in an industrial energy system based on performance, as this indicates a substantial improvement in the efficiency at a high temperature. Because the synergy level for electrical conductivity and the thermoelectromotive forces is higher, MgZrH₃ is most suitable. These findings open up for synergistic use in next-generation energy devices and vindicate the promise of XZrH₃ (X = Mg, Ca, Sr, and Ba) hydride perovskites as bifunctional materials with thermoelectric and hydrogen storage capabilities.

4. Conclusions

In this work, cubic hydride perovskites XZrH₃ (X = Mg, Ca, Sr, Ba) were investigated using density functional theory calculations to evaluate their structural, electronic, elastic, and thermoelectric properties. These materials are of significant interest due to their potential applications in hydrogen storage and energy conversion technologies. First-principles simulations were performed to gain detailed insight into their physical properties and underlying mechanisms. The electronic structure analysis revealed that these compounds exhibited metallic behavior, characterized by electronic states crossing the Fermi level, which contributed to their favorable electrical conductivity. This electronic characteristic was beneficial for facilitating charge transport and may also influence hydrogen stability and mobility within the lattice. The calculated elastic constants satisfied the Born mechanical stability criteria, confirming that all studied compounds were mechanically stable in the cubic phase. Furthermore, the calculated transport properties indicated that MgZrH₃ and CaZrH₃ exhibited superior thermoelectric performance compared to SrZrH₃ and BaZrH₃ over the temperature range of 300–900 K. These compounds showed higher electrical conductivity, favorable power factor values, and improved thermoelectric figure of merit. Overall, MgZrH₃ and CaZrH₃ demonstrated the most promising combination of electrical and thermoelectric properties among the studied hydride perovskites.