The DFT study of the structural, hydrogen, electronic, mechanical, thermal, and optical properties of KXH3 (X = Ca, Sc, Ti, &amp; Ni) perovskites for H2 storage applications

doi:10.21203/rs.3.rs-4438273/v1

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The DFT study of the structural, hydrogen, electronic, mechanical, thermal, and optical properties of KXH₃ (X = Ca, Sc, Ti, & Ni) perovskites for H₂ storage applications

https://doi.org/10.21203/rs.3.rs-4438273/v1

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In this study, we employ density functional theory calculations to comprehensively investigate the structural, electronic, hydrogen storage capacity, mechanical, thermal, and optical properties of KXH₃ (X = Ca, Sc, Ti, & Ni) hydride perovskites, unveiling their potential for H₂ storage applications. The lattice parameters, calculated using the GGA-PBE functional, are found to be 4.482 Å, 4.154 Å, 3.974 Å, and 3.686 Å for KCaH₃, KScH₃, KTiH₃, and KNiH₃, respectively. Interestingly, the electronic structure analysis reveals that while KScH₃, KTiH₃, and KNiH₃ exhibit metallic behavior, KCaH₃ stands out as a semiconductor. Population analysis indicates that these compounds possess a strong potential for hydrogen storage due to their strong bonding and long bond lengths. Furthermore, the investigation of dynamic and mechanical stability suggests that the studied materials are promising candidates for experimental synthesis, as they exhibit both thermodynamic and mechanical stability. Gravimetric analysis reveals promising hydrogen storage capacities of 3.646 wt%, 3.452 wt%, 3.346 wt%, and 3.005 wt% for KCaH₃, KScH₃, KTiH₃, and KNiH₃, respectively. The calculated hydrogen desorption temperatures are 442.40 K for KCaH₃, 518.68 K for KScH₃, 592.47 K for KTiH₃, and 614.82 K for KNiH₃, indicating the suitability of these materials for hydrogen storage applications within practical operating temperature ranges.

Novelty Statement: In this study, we present a comprehensive theoretical investigation of the novel perovskite materials KXH₃(X = Ca, Sc, Ti, Ni), encompassing their structural, electronic, hydrogen storage, mechanical, thermal, and optical properties. To the best of our knowledge, this is the first report providing insights into these unexplored compounds, as no previous theoretical or experimental studies have been conducted on them.

Hydrogen storage

Gravimetric ratio

DFT

Hydride perovskite

Elastic constants

Hydrogen gas (H₂) has emerged as a versatile and indispensable resource in several critical industrial sectors, encompassing oil refining, steel manufacturing, ammonia synthesis, and methanol production, among others [1, 2]. The widespread adoption of hydrogen as a clean energy carrier has been hindered by various challenges, despite its promising potential to drive the transition towards a sustainable energy future. Alarmingly, despite the ambitious targets set by the international community to combat climate change, global energy-related carbon dioxide (CO₂) emissions soared to a staggering 36.4 gigatonnes (GtCO₂) in 2021, underscoring the urgent need for concerted efforts to decarbonize the energy sector. While this figure represents a marginal 0.8% reduction from the pre-pandemic peak of 36.7 gigatonnes of CO₂ emissions recorded in 2019, it highlights the persistent challenge in curbing greenhouse gas emissions from energy production [3]. Hydrogen emerges as a compelling prospective clean energy carrier for the future energy landscape, offering a sustainable, emissions-free solution that holds the potential to enhance energy security and affordability [4]. Hydrogen has great potential as a clean energy carrier, but the insufficient infrastructure for its transportation and storage is a major barrier to its widespread adoption and incorporation into the energy system [5].

Yuan et al.'s research showcased the significant capacity of Ti-decorated porous graphene sheets for hydrogen adsorption and storage. The study showed that the designed materials have a high capacity to adsorb and hold up to four hydrogen molecules on their surface, with an average adsorption energy of 0.486 eV, suggesting positive binding interactions [6]. The study by Elahi et al. revealed that Li-modified WS2 monolayers have significant potential for hydrogen storage, as they can adsorb six hydrogen molecules with an average energy of -0.59 eV, making them suitable for practical use. A lot of study has focused on creating systems that can achieve the challenging hydrogen storage goals set by organizations like the U.S. Department of Energy (DOE) [7]. The DOE's roadmap aims to achieve a system with a gravimetric hydrogen capacity of 5.5% by weight and a volumetric capacity of 0.040 kg H2/L by 2025, allowing hydrogen-powered vehicles to have a predicted driving range of 300 miles [8, 9]. In the past few years, the scientific community has shown increasing interest in studying rare-earth perovskite hydrides with the ABX₃ structure for hydrogen storage applications. These hydride compounds have shown impressive skills as solid-state hydrogen transporters, revealing numerous potential uses. Magnesium-based perovskite hydrides have attracted considerable interest because of their outstanding ability to store hydrogen, making them appealing subjects for further study [10]. Lefevre et al. [11] thoroughly investigated the hydrogen storage capabilities of MgNi₃H₂ and MgCuH₃ by utilizing ab initio calculations. It has also been proposed that KMgH₃ perovskite hydrides could be used as hydrogen storage. Similar hydrogen release properties for KMgH₃ hydride perovskite were also revealed by Ghebouli et al. [12]. The exceptional hydrogen storage capacity of potassium-containing perovskites, particularly KMgH₃, has attracted much attention.

In this article, crystal structure, phonon dispersion curves, gravimetric ratio, and electronic, mechanical, thermal, and optical properties of hydrides perovskite KXH₃ (X = Ca, Sc, Ti, & Ni) have been investigated for the first time for hydrogen storage applications. This manuscript has been divided into four sections: The section-I provides a concise background and introduction to the study. The section-II explains the computational methodology utilized to calculate these compounds' properties. The section-III explains the results and discussion of the studied materials. Finally, the conclusion of this study is presented in section-IV.

The Cambridge Serial Total Energy Package (CASTEP) simulation package is a widely used in the field of materials science calculations based on DFT. These DFT calculations were done using the pseudopotential method and plane-wave basis set [13]. The PBE (Perdew-Bur-ke-Ernzerhof) framework and the GGA (Generalized-Gradient-Approximation) technique demonstrated correlation and exchange potential. Preparing the crystal structure is the first step in studying perovskite materials using CASTEP [14, 15]. This can be done using software such as Materials Studio or VESTA. It is necessary to treat both the inner electrons and the nucleus as a collective core of ions, facilitating the interaction with the outer electrons, in order to achieve robust and rapid convergence of the electron-ion potential. This is done so that the electron-ion potential can converge more quickly. The Brillouin zone for the sampling of k-points was constructed using the Monkhorst-Pack scheme. We have used the 6×6×6 k-points grids for the unit cell of KXH₃ (X = Ca, Sc, Ti, & Ni) perovskites. It has been determined that the valence electronic configurations of K, Ca, Sc, Ti, Ni, and H are 3s² 3p⁶ 4s¹, 3s² 3p⁶ 4s², 3s² 3p⁶ 4s² 3d¹, 3s² 3p⁶ 4s² 3d², 3s² 3p⁶ 4s² 3d⁸ and 1s¹ respectively. To maintain the uniformity and internal consistency of geometry optimization, it is crucial to maintain the overall energy equilibrium of the system. In these calculations, the energy cutoff value was determined to be 517 eV, and the smearing value was set at 0.5 eV. The energy convergence was chosen as 2×10^− 6 eV/atom, while the highest ionic force per atom was set to 5×10^− 2 eV. The maximum stress was applied to 1×10^− 1 GPa, and the highest displacement was set to be 2×10^− 4 Å. Elastic constants were investigated through the Voiget-Russel-Hill (V-R-H) technique [16–19].

3.1 Geometry optimization and structural stability:

Geometry optimization refers to finding the most energetically favorable configuration or arrangement of atoms in a molecule or solid. This involves the calculation of the potential energy surface of the system, which represents the relationship between the atoms' positions and the system's potential energy. Different methods for performing geometry optimization include force-field calculations, DFT, and ab initio methods. The choice of method depends on the system's complexity and the desired accuracy level. KXH₃ (X = Ca, Sc, Ti, & Ni) and most metal hydrides belong to the Pm3m space group (#221). This space group's cubic structure is almost completely packed [20]. The stability of a crystal structure is an important parameter; therefore, we have calculated the energy-volume curve, which is shown in Fig. 1.

Similarly, the thermal stability is checked with the help of phonon spectra, as shown in Fig. 2. The electronic crystal structures of KXH₃ (X = Ca, Sc, Ti, & Ni) hydride perovskites are shown in Fig. 3. From Fig. 3, atoms are located at the positions: K atoms at (0, 0, 0), X (X = Ca, Sc, Ti, & Ni) atoms at (0.5, 0.5, 0.5), and H atoms at (0, 0.5, 0.5) as shown Table 1 [21]. The determined values of lattice parameters and volumes are presented in Table 1, which agree with the previously reported values. Geometry optimization and structural stability are closely related since a system's stability depends on its geometry. By performing geometry optimization calculations, one can obtain the most stable geometry of a system and predict its behavior under different conditions. For the KCaH₃, KScH₃, KTiH₃, and KNiH₃ compositions, the energy optimization curves are illustrated in Fig. 2, where Birch-equation Murnaghan's of states is used to plot the energy released during the compound's creation versus the unit cell volume. Energy-Volume graphs calculate the ground state's energy corresponding to the minimal volume, which gives information on the optimum values of lattice constants.

In our study, we thoroughly examined the stability criteria of the KXH₃ (X = Ca, Sc, Ti, & Ni) hydride perovskite materials. We employed two simple and practical methods to assess the structural stability of these crystal structures. The first method, known as the tolerance factor (τ), and the second, referred to as the octahedral factor (H), provide valuable insights into the formation of stable perovskite crystal structures. For a highly symmetric and stable perovskite crystal structure to exist, these two parameters, denoted as τ and H, should fall within the ranges of (0.813–1.107) and (0.442–0.895), respectively. We calculated these crucial parameters using well-established mathematical relations [22]:

$${\tau }=\frac{{r}_{K}+{r}_{H}}{\sqrt{2}{(r}_{X}+{r}_{H})}$$

……………….

$$H=\frac{{r}_{x}}{{r}_{H}}$$

……………….

In the above equation, (1 & 2) r_K, r_X, and r_H are represents the effective ionic radiuses of the K, X (X = Ca, Sc, Ti, Ni), and H ions. We assume hydrogen's ionic radius (R_H) is 0.140 nm [23]. The calculated values of the tolerance factor are 0.822, 0.983, 0.905, and 1.031 for KCaH₃, KScH₃, KTiH₃, and KNiH₃, respectively. The calculated values of the octahedral factor are 0.714, 0.535, 0614, and 0.492 for KCaH₃, KScH₃, KTiH₃, and KNiH₃, respectively. These calculated values indicate that the crystal structure of KXH₃ (X = Ca, Sc, Ti, & Ni) perovskite materials is stable. We conducted an analysis of the phonon dispersion curves for perovskite hydrides KXH₃ (X = Ca, Sc, Ti, & Ni) in order to ascertain their dynamic stability. The computed graphs of phonon dispersion curves along high symmetry points within the first Brillion zone are shown in Fig. 2. As we can see, no negative frequency is present in the entire Brillion zone of calculated phonon dispersion curves. This confirms that the KXH₃ (X = Ca, Sc, Ti, & Ni) compounds are also dynamically stable.

Table 1

The computed and previously reported lattice parameters, volume, and band gap of KXH₃ (X = Ca, Sc, Ti, & Ni).
Compound	Lattice Parameters (Å)	Volume (Å)³	Band gap (eV)	References
KCaH₃	4.482	92.407	3.31	Present study
KScH₃	4.154	72.251	0.00
KTiH₃	3.974	63.808	0.00
KNiH₃	3.686	50.039	0.00
RbCaH₃	4.532	93.082	3.32	Experimental [24]
LiScH₃	3.864	57.699	0.00	Theoretical [25]
KTiH₃	3.999	63.952	0.00	Theoretical [26]
CaNiH₃	3.699	50.612	0.00	Experimental [24]

3.2 Hydrogen storage properties:

It is imperative to note that the formation energy of a compound is merely one of several factors determining its suitability for hydrogen storage. Other crucial factors encompass the kinetics of hydrogen uptake and release, the stability of the compound under operating conditions, and the capacity of the material to store hydrogen. The formation energy (ΔH_f) of the suggested compounds was calculated employing the following relation [22]:

$${\Delta }{\text{H}}_{\text{f}}\left({\text{K}\text{X}\text{H}}_{3}\right)=\left[{\text{E}}_{\text{tot. }}\left({\text{K}\text{X}\text{H}}_{3}\right)-{\text{E}}_{\text{s}}\left(\text{K}\right)-{\text{E}}_{\text{s}}\left(\text{X}\right)-3{\text{E}}_{\text{s}}\left(\text{H}\right)\right]$$

……………….

In the above equation, E_s (K), E_s(X), and E_s(H) represent the energy of single atom of K, X = (Ca, Sc, Ti & Ni) and H, respectively. E_total shows the total energy of the compound, and N indicates the total number of atoms present in the compound. Our investigation revealed that all of the examined compounds possess a negative value of formation energy, indicating their thermodynamic stability and feasibility for experimental synthesis. The stability order is as follows: KNiH₃ (-80.358 KJ/mol.H₂) > KTiH₃ (-77.437 KJ/mol.H₂) > KScH₃ (-67.792 KJ/mol.H₂) > KCaH₃ (-57.822 KJ/mol.H₂). The gravimetric ratio refers to the weight of hydrogen that can be stored per unit weight of the storage material. Hydrogen storage properties can be described in gravimetric capacity, as it denotes the amount of hydrogen that can be stored per unit mass of the storage medium. This is typically measured in units of weight percent or wt%. The gravimetric capacity of hydrogen storage materials is influenced by various factors, including the type of material used, the temperature and pressure of storage, and the method of hydrogen storage. Several promising materials under development could meet or exceed this target, including metal hydrides, complex metal hydrides, and porous materials. Gravimetric hydrogen storage capabilities of KXH₃ (X = Ca, Sc, Ti, & Ni) perovskite-type hydrides have been investigated by the gravimetric ratio, denoting the quantity of deposited hydrogen per unit mass of the substance, can be computed using the provided equation [22]:

$${C}_{wt\%}=\left(\frac{\frac{H}{M}{m}_{{H}_{2}}}{{m}_{Host}+\left(\frac{H}{M}\right){m}_{{H}_{2}}}\times 100\right)\%$$

……………….

The constituents of the given equation include ${m}_{{H}_{2}}$, signifying the molar mass of hydrogen, ${m}_{Host}$, representing the molar mass of the host material, and H/M, indicating the hydrogen-to-material atom ratio. The H/M is investigated by using the simulation package. Table 2 illustrates the investigated values of gravimetric ratios, which are 3.646 wt% for KCaH₃, 3.452 wt% for KScH₃, 3.346 wt% for KTiH₃, and 3.005 wt% for KNiH₃. The hydrogen desorption temperature of hydrides perovskite must be determined in addition to the gravimetric ratio. We can calculate the hydrogen desorption temperature by using the equation given below [27]:

$$T=-\frac{{\Delta }\text{H}}{{\Delta }\text{S}}$$

……………….

In the above equation, T, ΔH, and ΔS represent the desorption temperature, formation enthalpy, and change in entropy, respectively. The Standard conditions for the dehydrogenation reaction result in a change in entropy of ΔS (ΔH_Hydrogen = 130.7 J mol^− 1 K^− 1). Table 2 illustrates the calculated values of desorption temperature, which are 442.40 K, 518.68 K, 592.47 K, and 614.82 K for KCaH₃, KScH₃, KTiH₃, and KNiH₃, respectively. Notably, this temperature is still higher than the temperature at which decomposition begins for commercial use of proton exchange membrane fuel cells (PEMFC) or for the most cutting-edge automobile engines, which typically operate between 289 and 393 Kelvin and 363 and 377 Kelvin, respectively [28].

Table 2

The computed results of formation enthalpy (ΔH_f), gravimetric ratio (C_wt%), and desorption temperature (T_d) of KXH₃ (X = Ca, Sc, Ti, & Ni)
Compound	ΔH_f (KJ/mol.H₂)	C_wt% (wt%)	T_d (K)
KCaH₃	-57.822	3.646	442.40
KScH₃	-67.792	3.452	518.68
KTiH₃	-77.437	3.346	592.47
KNiH₃	-80.358	3.005	614.82

3.3 Electronic Properties:

To investigate the electronic characteristics of the materials, a thorough investigation of electronic properties was undertaken, encompassing the electronic band structure (Eg), the total density of states (TDOS), and the partial density of states (PDOS) at high symmetry points within the first Brillouin zone using energy scales ranging from − 10 eV to 10 eV. Hydrogen molecules are adsorbed onto a material's surface in the physisorption method, while in the chemisorption method, hydrogen atoms are attached to the material's atoms; both methods are utilized for hydrogen storage. The strength of the interaction between the surface of the material and hydrogen molecules can be evaluated in the physisorption method by analyzing the material's electronic properties. If the density of states exhibits a large value at the Fermi level and the electronic band is near the Fermi level, the material possesses stronger hydrogen adsorption energies. In the chemisorption-based hydrogen storage method, electronic properties elucidate the bonding behavior between the material and the hydrogen atom. The conduction band (CB) and valence band (VB) positions explain the bonding behavior between the material and hydrogen atoms. If the VB and CB overlap, it suggests strong bonding energy between the material and the hydrogen atom.

Table 3

Mulliken electronic population analysis of KXH₃ (X = Ca, Sc, Ti, & Ni):
Compound	Species	s	p	d	Total	Charge	Bond	Population	Bond Length (Å)
KCaH₃	K	2.11	6.31	0.00	8.420	0.58	---	---	---
	Ca	2.35	6.00	0.63	8.97	1.03	H-Ca	0.31	2.24769
	H	1.53	0.00	0.00	1.53	-0.53	---	---	---
KScH₃	K	2.06	5.62	0.00	7.67	0.27	H-K	-0.36	2.98799
	Sc	2.54	7.03	1.59	11.16	-0.16	H-Sc	0.97	2.11283
	H	1.39	0.00	0.00	1.39	-0.39	H-H	-0.04	2.98799
KTiH₃	K	2.11	5.41	0.00	7.53	1.47	H-K	-0.54	2.80099
	Ti	2.60	7.11	2.67	12.38	-0.38	H-Ti	1.09	1.98060
	H	1.36	0.00	0.00	1.36	-0.36	H-H	-0.04	2.80099
KNiH₃	K	2.49	4.94	0.00	7.42	1.58	H-K	-0.56	2.60908
	Ni	0.82	1.04	8.80	10.66	-0.66	H-Ni	1.06	1.84490
	H	1.31	0.00	0.00	1.31	-0.31	H-H	-0.03	2.60908

The studied materials' electronic band structures were computed by utilizing the GGA-PBE technique, as illustrates in Fig. 5. Electronic band structures suggest that KCaH₃ is semiconducting material with a large band gap value of 3.310 eV, while the conduction band and valance band of KScH₃, KTiH₃, and KNiH₃ are overlapping, which suggests that these materials are metallic and strong hydrogen bonding energy. The computed graphs of TDOS for KXH₃ (X = Ca, Sc, Ti, & Ni) are shown in Fig. 6. A Vertical dashed indicates the Fermi level (E_F), which is set at zero and taken as a reference point. The maximum values of TDOS at E_F are 7.45, 6.13, 3.17, and 0.83 states/eV for KCaH₃, KTiH₃, KScH₃, and KNiH₃, respectively. A large value of DOS at E_F shows that these materials are the metallic behavior and best candidates for hydrogen storage applications. The PDOS of the material gives the information of the electronic state in a solid material at any point of energy level. A solid's electronic structure is defined by its energy bands comprising multiple electronic states. The curves help to investigate the involvement of certain atoms or orbitals to such energy bands and also examine the bonding behavior on these curves, giving information regarding hybridization between states.

The calculated graphs of PDOS for KXH₃ (X = Ca, Sc, Ti, & Ni) are represented in Fig. 7. From Fig. 3, all examined materials have flat-going core states, mostly involving the f-orbital, which are ignored. From − 10 to -5 eV energy, the s-state has shown a small contribution. From − 5 to 0 eV, the energy s-state of KXH₃ (X = Sc, Ti & Ni) and the d-state of the d-state of KNiH₃ show maximum contribution. At the fermi level, the s-state of KCaH₃, d-state of KScH₃, and KTiH₃ have shown maximum values, which show strong hydrogen bonding energy. Mulliken atomic population analysis is the computational technique utilized to investigate the bonding behavior, length of bonds, and electronic structures of solids and molecules. The Mulliken analysis gives information about the electron density distribution in the material and can help predict the intensity of the material's interaction with hydrogen molecules. The + ve value of the population shows the nature of the covalent bonding, and the negative value shows the nature of the ionic bonding of the material. Mulliken atomic population analysis for KXH₃ (X = Ca, Sc, Ti, & Ni) is calculated and presented in Table 3. Furthermore, by population analysis, we can also determine the population ionicity (P_i), which gives information about the "percentage of the covalence behavior of the bond," which can be calculated by utilizing the given formula [29]:

$${\text{P}}_{i}=1-{e}^{-\left|\frac{{P}_{c}-P}{P}\right|}$$

……………….

The above equation shows all the studied materials show covalent bonding behavior. In addition, the bond lengths for H-Sc, H-Ti, and H-Ni in KScH₃, KTiH₃, and KNiH₃ are 2.11283 Å, 1.98060 Å, and 1.84490 Å, respectively. The electronic structure of KXH₃ (X = Ca, Sc, Ti, & Ni) perovskite materials enable strong interactions between the hydrogen molecules and the host lattice. A weak chemical bond between the hydrogen atoms and the nearby atoms in the lattice can result in hydrogen storage. This procedure improves hydrogen storage or absorption due to the expansion of interstitial sites. These compounds have the potential to undergo a chemical reaction with hydrogen, resulting in the formation of intermetallics that offer improved storage capabilities.

3.4 Mechanical Properties:

Hydrogen storage materials are employed to store and release hydrogen for diverse applications, including hydrogen-powered vehicles, fuel cells, and energy storage systems. Mechanical stability emerges as a crucial parameter for H₂-storage materials, as it determines their durability and safety during operation. Mechanical stability refers to the ability of a material to resist deformation or fracture under mechanical stress or strain. In the context of hydrogen storage materials, this encompasses resistance to pressure changes, temperature fluctuations, and repeated hydrogen absorption and desorption cycles. To ensure mechanical stability, hydrogen storage materials are often engineered to exhibit high strength, toughness, and resistance to fatigue and corrosion. Materials such as metal hydrides, which can absorb and release substantial amounts of hydrogen, are frequently reinforced with other materials to enhance their mechanical properties. Through the evaluation of mechanical properties, we can discern the strength and bonding behavior of the crystal structure.

Table 4

The computed results of elastic constants (C_ij) and Cauchy’s pressure (C_P) of KXH₃ (X = Ca, Sc, Ti, & Ni):
Compound	C₁₁	C₁₂	C₄₄	C_P
KCaH₃	42.080	7.439	16.893	-9.454
KScH₃	36.349	7.362	20.286	-12.924
KTiH₃	70.994	19.896	35.254	-15.358
KNiH₃	59.339	24.124	52.302	-28.178

Table 5

The computed mechanical parameters Bulk modulus (B), Shear modulus (G), Young modulus (E), Pugh ratio (B/G), Pugh’s modulus (G/B), Poisson ratio (v), Material’s Hardness (H_v), Machinability index (µ_M), and Anisotropic index (A^U):
Compound	B	G	E	B/G	G/B	V	H_V	µ_M	A^U
KCaH₃	18.985	17.062	39.388	1.113	0.899	0.155	13.903	1.124	0.975
KScH₃	17.025	17.729	39.482	0.961	1.042	0.114	17.081	0.840	1.401
KTiH₃	36.929	30.988	72.645	1.192	0.840	0.173	23.355	1.048	1.380
KNiH₃	35.862	33.836	77.223	1.060	0.944	0.142	28.137	0.686	2.970

The microstructure and crystal structure of the material can also influence its mechanical stability. For instance, materials with a highly ordered crystal structure, such as zeolites, can exhibit better mechanical stability than those with a more disordered structure. The prospective mechanical parameters and anisotropic factor of all studied hydride perovskite materials have been investigated. These mechanical properties can be identified by utilizing the stiffness constants. Stiffness constants like C₁₁, C₁₂, and C₄₄ are reduced to only three for a cubic phase but vary widely across other crystal structures. C₁₁, C₁₂, and C₄₄ were investigated using the CASTEP simulation code. The three elastic constants, namely C₁₁, C₁₂, and C₄₄, correspond to the material's resistance to longitudinal deformation, transverse expansion, and hardness, respectively. For mechanical stability, the elastic constants must fulfill the Born stability criteria, which are given as follows:

C₁₁ + 2C₁₂ > 0; C₁₁ − C₁₂ > 0; C₁₁ > 0; C₄₄ > 0 ………………. (7)

Table 4 shows that the calculated values of elastic constants for KXH₃ (X = Ca, Sc, Ti, & Ni) fulfill the abovementioned requirement, demonstrating the mechanical stability of the materials.

Cauchy’s pressure (C_P) can be computed as:

C_P = C₁₂ - C₄₄ ………………. (8)

By using the VRH technique, B is computed as:

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

……………….

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

……………….

$$B=\frac{{B}_{V}+{B}_{R}}{2}$$

……………….

The following relation can determine Young's modulus:

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

……………….

Shear modulus can be computed by the equation given below:

$$G=\frac{{G}_{R}+{G}_{V}}{2}$$

……………….

Where G_v and G_r are:

$${G}_{V}=\frac{1}{3}\left(3{C}_{44}+{C}_{11}-{C}_{12}\right)$$

……………….

$${G}_{R}=\frac{5\left({C}_{11}-{C}_{12}\right){C}_{44}}{3\left({C}_{11}-{C}_{12}\right)+4{C}_{44}}$$

……………….

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

……………….

The computed values of Cauchy’s pressure for KXH₃ (X = Ca, Sc, Ti, & Ni) are represented in Table 4. The negative values illuminate each material's brittleness and angular bonding. The highest bulk modulus value indicates the material is most resistant to volume change. The calculated Bulk modulus values for KCaH₃, KScH₃, KTiH₃, and KNiH₃ are 18.985, 17.025, 36.929, and 35.862, respectively. This shows that KTiH₃ and KNiH₃ are the most resistant to volume change. The computed results of Bulk modulus (B) are represented in Table 5. Shear modulus indicates the material’s resistance toward shape change under applied stress.

The modulus of rigidity is a mechanical characteristic that offers insight into the hardness properties of a material. Pugh’s ratio (B/G) is a significant parameter that more comprehensively elucidates the ductility and brittleness characteristics of a solid material. B/G is the ratio of the material between resistance to fracture and deformation. In more specific terms, the component B pertains to the fracture resistance of the material, while G pertains to its deformation resistance. The literature commonly recognizes a critical threshold of 1.75 as the standard value for distinguishing between the brittleness and ductility characteristics of materials. The B/G value exceeding 1.75 signifies the material's ductile behavior, whereas its value falling below 1.75 implies excessively brittle behavior. Table 5 shows that all materials are less brittle because their B/G values fall below the critical value (1.75). The Poisson ratio is another significant parameter in assessing the plastic properties as well as the brittleness and ductility behavior of solid materials.

In the evaluation of a material's ductility, the Poisson ratio serves as a crucial parameter, with a value exceeding 0.25 indicating ductility while a value below 0.25 indicates potentially brittle behavior [22]. All the studied materials show brittle behavior because their calculated values of Poisson ratio are less than the critical values (0.25), as shown in Table 5. The Poisson ratio is a potential parameter for identifying the ionic or covalent bonding of a material. The critical values for covalent and ionic behavior are 0.1 and 0.25, respectively. The Poisson ratio can provide insights into the nature of a material's bonding, with a value in proximity to 0.1 indicative of covalent bonding while a value near 0.25 suggestive of ionic behavior [22].

From Table 5, it can be seen that the studied materials show covalent behavior. Pugh’s modulus is calculated by the G/B ratio, which is also used to determine the bonding behavior of the material. A critical value for ionic and covalent behavior through Pugh’s modulus (G/B) is 0.6 and 1.1, respectively. If the value of Pugh’s modulus is near about 0.6, the material has ionic bonding. If the value of Pugh's modulus is near about 1.1, the material has covalent bonding. From Table 5, it can be seen that all the studied materials have covalent bonding [22].

$${A}^{U}=\frac{{2C}_{44}}{{C}_{11}-{C}_{12}}$$

……………….

Isotropic crystals have a value of A = 1, whereas anisotropic crystals have 1 < A < 1. The computed values of A are 0.975, 1.401, 1.380, and 2.970 for KCaH₃, KScH₃, KTiH₃, and KNiH₃, respectively—the computed values of A confirming the anisotropic behavior of studied materials.

3.5 Thermodynamic Properties:

The assessment of thermodynamic properties, including longitudinal, transverse, and average velocities, Debye temperature, and melting temperature, is crucial for understanding the hydrogen storage capabilities of materials. Higher Debye temperatures indicate that the solid can better store hydrogen at high temperatures and needs more energy to disrupt. Melting temperature is also related to the Debye temperature. If the Debye temperature is very high, then that material's melting temperature (T_m) and thermal conductivity must also be high. The determined thermodynamic parameters are listed in Table 6.

Table 6

The computed results of density (ρ), longitudinal velocity (v_l), transverse velocity (v_t), average sound velocity (v_m), Debye temperature (θ_D), and melting temperature (T_m) of KXH₃ (X = Ca, Sc, Ti, & Ni):
Compound	ρ (g/cm3)	v_t (km/s)	v_m (km/s)	θ_D (K)	T_m (K)
KCaH₃	1.472	3.407	3.493	405.39	505.57 ± 300
KScH₃	2.001	2.977	3.023	381.46	493.47 ± 300
KTiH₃	2.342	3.638	3.746	492.65	619.86 ± 300
KNiH₃	3.346	3.180	3.250	463.57	610.47 ± 300

Thermodynamic parameters can be evaluated by using the following equations.

Debye temperature ${{\theta }}_{\text{D}}$ can be evaluated [30].

$${{\theta }}_{\text{D}}=\frac{ħ}{{k}_{B}}{\left[\frac{3n{N}_{a}\rho }{4\pi M}\right]}^{\frac{1}{3}} \times {v}_{m}$$

……………….

The quantities denoted as $ħ$, ${k}_{B}$, $n$_, ${v}_{m}, {N}_{a}, \text{a}\text{n}\text{d} V$ represent fundamental physical parameters, namely Planck’s constant, Boltzmann’s constant, number of atoms, average velocity of sound, Avogadro number, and unit cell volume, respectively.

Furthermore, ${v}_{m}$ can be evaluated by using the following relation [30]:

$${v}_{m}=\frac{1}{3}{\left[\frac{2}{{v}_{t}^{3}}+\frac{1}{{v}_{l}^{3}}\right]}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$$

……………….

Here, ${v}_{t}$ and ${v}_{l}$ can be evaluated by using the given relations [30]:

$${v}_{t}={\left[\frac{G}{\rho }\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{a}\text{n}\text{d} {v}_{l}={\left[\frac{3B+4G}{3\rho }\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

……………….

The determined Debye temperatures for KCaH₃, KScH₃, KTiH₃, and KNiH₃ were found to be 405.39 K, 381.46 K, 492.65 K, and 463.57 K, respectively. The melting temperature, which is also referred to as the melting point, denotes the temperature at which a solid material undergoes a phase transition from its solid state to a liquid state. This transition is marked by a state of equilibrium where both the solid and liquid phases of the substance coexist. We can evaluate the melting point of solid material by using the following relation [30]:

T_m(K) = [553(K) + 5.911 (C₁₂)] GPa ± 300K ………………. (21)

The melting temperature values for KCaH₃, KScH₃, KTiH₃, and KNiH₃ are 505.57 K, 493.47 K, 619.86 K, and 610.47 K, respectively. The thermodynamic parameters were determined by utilizing the elastic constants (C₁₁, C₁₂, C₄₄), and the corresponding calculated values are presented in Table 6.

3.6 Optical Properties:

The optical properties of a material are studied to learn more about how light energy interacts with it. Optical properties of the material such as absorption, reflection, conductivity, energy loss function, refractive index, and complex dielectric functions are important parameters of the material used in a couple of applications like hydrogen storage applications, photocatalytic applications, coatings, solar cell devices, and optoelectronic applications. All the optical parameters are based on the complex dielectric function, which can be computed by using the following relation [31]:

ε(ω) = ε₁(ω) + iε₂(ω) ………………. (22)

The real component ε₁(ω) of the dielectric function ε(ω) may be obtained from the imaginary component ε₂(ω) using the Kramer-Kronig relation, and the imaginary component can be obtained by adding an extensive number of unoccupied states. The optical parameters could be calculated precisely by examining the dielectric function and considering the electronic transitions. A detailed description of real part ε₁(ω) and imaginary part ε₂(ω) of complex dielectric function against photon energy range 0 to 20 eV is shown in Fig. 8 (a & b). The real part ε₁(ω) of the dielectric function measures the ability of the material to store the electric charge and also explains the dispersion effects that occur inside a material. Hydrogen storage could be improved by a greater value of ε₁(ω) because it can provide stronger attractive forces between the hydrogen and the relevant material. The static values of zero photon energy of ε_o(ω) are 45.23, 34.74, 03.80, and 03.19 for KScH₃, KTiH₃, KNiH₃, and KCaH₃, respectively. Then ε₁(ω) decreases sharply to zero at 0.78 eV for KScH₃, 5.65 eV for KNiH₃, 7.11 eV for KCaH₃, and 11.14 eV for KNiH₃. The negative values of ε₁(ω) indicate the metallic nature of the material. The highest value of KScH₃ and KTiH₃ shows that stored energy in these materials can be utilized for useful purposes. It also suggests that stored energy can be used for optoelectronic applications. ε₂(ω) explains the adsorptive behavior of the material to incident photons. The maximum calculated values of ε₂(ω) are 22.39 at 0.31 eV, 11.62 at 0.18 eV, 5.22 at 6.89 eV, and 3.33 at 9.08 eV for KScH₃, KTiH₃, KCaH₃, and KNiH₃, respectively. The refractive index describes the material's ability to absorb light at a certain wavelength and the material's transparency to the incident photon. The maximum values of the refractive index n(ω) are 2.78 at 4.36 eV, 2.41 at 5.00 eV, 2.19 at 3.31 eV, and 2.17 at 3.70 eV for KTiH₃, KScH₃, KCaH₃, and KNiH₃, respectively. The maximum values of the extinction coefficient are 2.79, 1.71, 1.58, and 1.08 for KScH₃, KTiH₃, KCaH₃, and KNiH₃, respectively. The calculated graph of the refractive index and extinction coefficient as a function of photon energy is shown in Fig. 8(c & d).

The rate at which hydrogen may be absorbed into the material and dispersed through it is referred to as the absorption coefficient, also known as the hydrogen diffusion coefficient or hydrogen permeability. The calculated graphs of absorption coefficient α(ω) for KXH₃ (X = Ca, Sc, Ti, & Ni) hydride perovskite drawn against 0 to 20 eV photon energy are shown in Fig. 9(a). It is analyzed that all of the materials show zero absorption when no photons hit the surface of the compound. The absorption rate in the materials increases by increasing the photon energy. A high absorption coefficient is preferred for effective hydrogen storage since it allows for fast absorption of hydrogen and a large amount of storage. The composition and crystal structure of hydride perovskite materials are the variables that affect the absorption coefficient. The determined peaks values of absorption coefficient are 20.01×10⁴ cm^− 1, 18.84×10⁴ cm^− 1, 16.81×10⁴ cm^− 1, and 15.28×10⁴ cm^− 1 for KNiH₃, KCaH₃, KTiH₃, and KScH₃, respectively. All studied materials' α(ω) becomes zero at a high photon energy range. Optical conductivity is used to evaluate the mechanism of conduction according to the photoelectric effect, which occurs when high-energy photons (E = ħω) hit a material's surface and cause photoelectron emission. The material's conductivity in hydrogen storage determines how readily it can pass through. Also, studying the breaking of bonds that occur due to incoming radiations with the material's surface is helpful. In hydride perovskite materials, hydrogen diffusion plays a crucial role in determining the rates of hydrogen absorption and release. A substance with a high conductivity may absorb and release hydrogen more quickly because hydrogen can diffuse through it more quickly and effectively.

As a result, materials with high conductivity are chosen for applications that require effective hydrogen storage. For this purpose, the conductivity of KXH₃ (X = Ca, Sc, Ti, & Ni) hydrides perovskite has been determined against the photon energy from the range 0 to 20 eV, as shown in Fig. 9(b). the calculated peak values of conductivity are 4.37 at 6.97 eV for KCaH₃, 4.28 at 5.03 eV for KTiH₃, 3.84 at 11.01 eV for KNiH₃, and 3.71 at 5.03 eV for KScH₃. So, KScH₃ and KTiH₃, both compounds, predict high conductivity and good material for hydrogen storage applications. The reflectivity is determined to investigate the material's behavior with the interaction of incident radiations. Some light that strikes a substance is absorbed, while the remaining is reflected. The proportion of light reflected and the amount of light that incident determines a material's reflectivity.

Reflectivity will affect the temperature of hydride perovskites when exposed to light. The material may heat up due to light absorption, promoting hydrogen desorption. Conversely, the material may stay colder if it reflects light, which can sometimes decrease hydrogen desorption. The maximum reflectivity peaks for KXH₃ (Ca, Sc, & Ti) are observed in the energy range of 5 eV to 8 eV, while the maximum peak of KNiH₃ is observed at 11.61 eV. The calculated graphs of reflectivity against photon energy for KXH₃ (X = Ca, Sc, Ti, & Ni) are presented in Fig. 9 (c). The energy loss function is a mathematical expression that describes the amount of energy lost during the transition of electrons due to scattering or dispersion. The loss function may affect the interaction between hydride ions and metal cations in hydride perovskites, affecting the material's electronic structure. The energy loss function is proportional to the scattering probabilities during inner shell transitions. Hydrogen transport and binding within a material can be modified by the loss function, which will influence the electronic density of states. The calculated graphs of the energy loss function of KXH₃ (X = Ca, Sc, Ti, & Ni) hydride perovskites against photon energy in the range of 0 to 20 eV are shown in Fig. 9(d). The maximum values of energy loss functions are 1.91 at 12.16 eV, 1.37 at 15.09 eV, 1.19 at 13.56 eV, and 0.84 at 15.40 eV for KCaH₃, KTiH₃, KScH₃, and KNiH₃, respectively.

Through this extensive research, we have employed advanced computational techniques, specifically ab initio calculations using the CASTEP simulation package based on the pseudopotential method, to comprehensively analyze and discuss the material properties of the hydride perovskites KXH₃ (X = Ca, Sc, Ti, & Ni). The determined lattice parameters are in good agreement with the previously reported study. The phonon curves show that no imaginary phonon frequency is present in the whole Brillouin zone. The optoelectronic properties reveal that all the materials exhibit metallic behavior except KCaH₃, which shows a semiconducting behavior. The bond lengths for H-Sc, H-Ti, and H-Ni in KScH₃, KTiH₃, and KNiH₃ are 2.11283 Å, 1.98060 Å, and 1.84490 Å, respectively. These compounds can store hydrogen due to their strong bonds and long lengths. The tolerance factor, octahedral factor, and negative value of formation energy suggest that KXH₃ (X = Ca, Sc, Ti, & Ni) is a structurally and thermodynamically stable compound, which can also be experimentally synthesized. The hydrogen storage properties have been investigated, and it shows large gravimetric ratio values such as 3.646 wt%, 3.452 wt%, 3.346 wt%, and 3.005 wt% for KCaH₃, KScH₃, KTiH₃, and KNiH₃, respectively. The highest value of Bulk modulus indicates the material is most resistant to volume change. The calculated Bulk modulus values for KCaH₃, KScH₃, KTiH₃, and KNiH₃ are 18.985, 17.025, 36.929, and 35.862, respectively. The extensive investigation of structural, electronic, optical, mechanical, and thermal properties suggests that KXH₃ (X = Ca, Sc, Ti, & Ni) can be utilized for hydrogen storage applications. However, KCaH₃ emerges as a potential candidate for hydrogen storage applications due to its higher gravimetric ratio.

Acknowledgements:

The authors would like to thank the Researchers Supporting Project Number (RSP2024R35), King Saud University, Riyadh, Saudi Arabia.

Authors Contributions:
Conception and design of study:

Muhammad Awais Rehman, Zia ur Rehman, Abu Hamad

Acquisition of data:

Zia ur Rehman, Muhammad Usman, Abu Hamad

Analysis and/or interpretation of data:

Muhammad Usman, Muhammad Awais Rehman

Drafting the manuscript:

Zia ur Rehman, Muhammad Usman, Muhammad Awais Rehman

Revising the manuscript critically for important intellectual content:

Muhammad Usman, Zia ur Rehman, Abu Hamad

Approval of the version of the manuscript to be published:

Zia ur Rehman, Muhammad Awais Rehman, Abu Hamad

Competing Interests
The authors declare that they have no competing interests related to the research, authorship, or publication of this article. No financial, professional, personal, or other associations that could be perceived as influencing the research or its presentation in this manuscript exist.
Supplementary information:
Not Applicable
Ethical Approval:
All authors confirm that the submitted work is original, hasn't been published elsewhere, and adheres to ethical guidelines.
Data and code availability:

Data and code will be available from the corresponding author on reasonable request.

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No competing interests reported.

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The DFT study of the structural, hydrogen, electronic, mechanical, thermal, and optical properties of KXH₃ (X = Ca, Sc, Ti, & Ni) perovskites for H₂ storage applications

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Abstract

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1. Introduction

2. Computational Methodology:

3. Results and Discussions

3.1 Geometry optimization and structural stability:

3.2 Hydrogen storage properties:

3.3 Electronic Properties:

3.4 Mechanical Properties:

3.5 Thermodynamic Properties:

3.6 Optical Properties:

4. Conclusion

Declarations

References

Additional Declarations

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