The Effect of Applied Hydrostatic Pressures in Ferromagnetic Ordered HoM2 [M = (Al, Ni)] Laves Phases: A DFT Study

1. Introduction

Experimental and theoretical research in the field of magnetocaloric-based cooling has grown over the past 25 years due to the higher energy efficiency and environmentally friendly nature of this cooling technology compared to the conventional one based on the expansion and compression of gases [1,2,3,4,5,6]. In the last few years, the interest in developing magnetic refrigerators for hydrogen liquefaction has encouraged important efforts to find, synthesize, and assess the magnetocaloric response of many families of rare earth (R) based alloys due to the significant magnetocaloric response of many compounds below the precooled reference temperature of 77 K [7,8,9]. Among all the alloy systems investigated, the cubic Laves phases in the RM₂ systems with M = {Al, Ni} stand out due to the remarkable magnetocaloric properties, especially for the heavy R elements as Ho, and Er [6,7,8,9,10].

Density functional theory (DFT) is a fundamental computational method in materials science, providing atomistic insights into crystals and molecules. By resolving a material’s electronic structure, it enables the investigation and prediction of structure–property relationships and the underlying physicochemical phenomena in solids [11,12,13,14,15]. These compounds crystallize into the MgCu₂-type structure (C15, space group: Fd–3m) [10], with a lattice parameter of 7.810 Å for HoAl₂ and 7.130 Å for HoNi₂ [16,17], and show Weiss-Curie temperatures T_C of 29 K and 13.4 K, respectively [18,19]. Two lattice-related characteristics of Laves phases (AB₂) are that the relationship between the atomic radii of A and B atoms is between 1.05 and 1.68, and that for an atomic radius ratio (r_A/r_B) of 1.225, the crystal structures have a higher packing density (around 71%) [10,20,21,22,23,24]. Green hydrogen economy needs the hydrogen storage due to hydriding properties in cell unit of Laves C15 phases. Laves intermetallics can be used to store interstitially the hydrogen by offering different positions (i.e., 3 tetrahedral interstices) [20].

Furthermore, HoAl₂ exhibits magnetic anisotropy characterized by the easy magnetization axis being <110> for temperatures below 20 K; above this temperature, the easy axis shifts to the <100> intermediate direction [18,25]. Additionally, a spin reorientation occurs at T = 20 K. The hard magnetizations axis in HoAl₂ is <111> crystal direction [19,25]. The directions of the easy, intermediate, and hard magnetization axes of HoNi₂ are <100>, <110>, and <111>, respectively [17,25,26,27,28].

The present work investigates, through Density functional theory with Hubbard U correction calculations (DFT+U), the effect of hydrostatic pressure on the electronic and magnetic properties of the ferromagnetically ordered HoAl₂ and HoNi₂ Laves phases. The spin polarization calculations are performed along <001> crystal direction. We systematically explore how electronic density of states are affected the in the HoAl₂ and HoNi₂ alloys. The theoretical calculations indicate significant changes in the electronic structure under a small hydrostatic pressure of 0.1 GPa in HoNi₂. A multicaloric approach in the solid-state cooling technology based on the magneto-caloric and baro-caloric effects has been used [2,3]. The combination of different external fields (e.g., magnetic field and hydrostatic pressure) allows an enhancement of caloric response by tailoring the magnetic moment (related to electronic density of states) with the pressure during the magnetic phase transition. Our finding is that the net magnetic moment can be modified (drops a 22.5 %) after a 0.1 GPa hydrostatic pressure is applied. To our knowledge, the electronic and magnetic properties from ab-initio calculations under hydrostatic pressure in HoAl₂ and HoNi₂ have not yet been reported to date.

2. Materials and Methods

The present study was performed using Cambridge Serial Total Energy Package (CASTEP) within density functional theory framework, using BIOVIA-Materials Studio^®. For the exchange-correlation, the revised Perdew-Burke-Ernzerhof (RPBE) functional was applied as part of the generalized gradient approximation (GGA). It’s known that the GGA method fails to correctly describe the localized 4f and 3d electrons; therefore, the DFT+U (U-Hubbard) correction was introduced into calculations [29,30,31]. It is important to note that U corrections within the GGA approximation show better accuracy to investigating the magnetic and electronic structures of 4f and 3d compounds compared to the local density approximation (LDA) or hybrid functionals [32,33]. The U values of localized electrons were 2.50 eV and 6.0 eV for Ni and Ho atoms, respectively. The U value was set to 0 eV for Al atoms because the lack of localized electrons. To calculate the electronic density of states (DOS), a 13 × 13 × 13 k-mesh generated by the Monkhorst-Pack scheme was used to integrate the Brillouin zone. For the plane-wave propagation along the crystal, a cut-off energy of 500 eV was applied [34,35,36,37]. The charge convergence criteria was set at 1 × 10⁻⁶ eV for self-consistent field cycles. During the geometric optimization process, the compressive external stress was applied along the a, b, and c axes to consider the effect of external hydrostatic pressure using Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm.

The stoichiometric HoM₂ with M = {Al, Ni} Laves phases crystallize in a cubic MgCu₂-type structure with a space group Fd–3m. Figure 1 schematically shows the crystalline and magnetic structure and primitive cell of HoM₂ (M = {Al, Ni}) Laves phases. In the AB₂ structure, the A atom (i.e., Ho) occupies the 8a Wyckoff site at (0 0 0), while the B atoms (i.e., Al and Ni) occupy the 16a Wyckoff site at (5/8 5/8 5/8) positions. For Al, Ni, and Ho, the electronic configuration is described as [Ne] 3s² 3p¹, [Ne] 3s² 3p⁶ 3d⁸ 4s², and [Ne] 3s² 3p⁶ 3d¹⁰ 4s² 4p⁶ 4d¹⁰ 5s² 5p⁶ 4f¹¹ 6s², respectively. It is important to note that for simulating HoAl₂ and HoNi₂ Laves phases, we used experimental lattice parameters instead of those obtained from minimizing the total energy as a function of volume for the crystalline structures. The used lattice parameters a = b = c were 7.810 Å for HoAl₂ and 7.130 Å for HoNi₂ [16,17]. The ferromagnetic ordering of each compound was modeled assuming that only the rare earth atoms, specifically Ho at 8a positions, possess a magnetic moment aligned along the <001> direction. Zero magnetic moment was assumed for Al and Ni atoms, which are located at 16a positions.

3. Results and Discussion

3.1. Electronic Properties

Table 1 displays the lattice parameter and interatomic distances between Al-Al, Ho-Al, and Ho-Ni in the Laves phases HoM₂ with M = {Al, Ni}, under applied hydrostatic pressures from 0 GPa to 1.0 GPa. It is worth noting that HoAl₂ is more sensitive to external pressure than the HoNi₂ alloys. The structural stability of both HoAl₂ and HoNi₂ remains unchanged across the entire range of applied pressures. Their corresponding formation energy E_f values at P = 0 GPa are –9.485 × 10³ eV (HoAl₂) and –13.47 × 10³ eV (HoNi₂). For non-zero pressures, the overall magnetic behaviors do not exhibit significant variation due to the minimal compaction in their crystal structures. All formation energy values remained nearly constant with a virtually negligible increase (less than 1%) under external pressures up to 1.0 GPa; details are shown in Figure A1 in the Appendix section. Additionally, substituting Al atoms with Ni atoms leads to a reduction in lattice parameters, resulting in an increase in bulk modulus at applied pressures around 1.0 GPa, as detailed in Table 1.

3.2. Determination of Electronic Coefficient in Specific Heat Capacity

Considering that each atom donates one electron to the Fermi gas in the solid, the free-electron number densities ($N/V$) for HoAl₂ and HoNi₂ are 1.678 × 10²⁸ m⁻³ and 2.203 × 10²⁸ m⁻³, respectively. On the other hand, M(HoAl₂) = 218.894 g/mol, ρ(HoAl₂) = 6.08 g/cm³ [16], M(HoNi₂) = 282.316 g/mol, and ρ(HoNi₂) = 10.33 g/cm³ [38] were used.

Bearing in mind that in metals theory at P = 0 GPa and T = 0 K, the Fermi energy E_F can be calculated as follows:

$$E_F={\displaystyle \frac{h^2{k_B}^2}{2m_e}}{\left({\displaystyle \frac{3N}{8\pi V}}\right)}^{2/3}$$

(1)

where $h$ and $k_B$ are the Planck and Boltzmann constants, respectively, and $m_e$ is the electron mass. Using equation (1) and previous values, the Fermi temperature T_F can be calculated through the formulae:

$$T_F={\displaystyle \frac{E_F}{k_B}}$$

(2)

The electronic coefficient ${\gamma }_e$ of specific heat capacity can be calculated using the following equation:

$${\gamma }_e={\displaystyle \frac{{\pi }^2{k}_B{N}_A}{3T_F}}$$

(3)

The calculated values of Fermi energy and its temperature, and the electronic coefficient${\gamma }_e$ using the equations (1), (2), and (3) are listed in Table 2. The N/V, E_F and T_F values agree with those obtained for other elements such as K (i.e., 1.40 × 10²⁸ m⁻³; 2.13 eV; 2.47 × 10⁴ K), Ag (i.e., 5.86 × 10²⁸ m⁻³; 5.53 eV; 6.41 × 10⁴ K), and Cu (i.e., 8.47 × 10²⁸ m⁻³; 7.06 eV; 8.19 × 10⁴ K) [39].

) obtained by the equations (1), (2), and (3) for both Laves phases (HoAl₂ and HoNi₂).

Another way to determine the electronic heat capacity coefficient ${\gamma }_e$ for alloys is through DFT quantum calculations. The Einstein-Debye model states that $c_p\left(T\right)={\gamma }_{e}T+{\beta }_{ph}{T}^3$ at temperatures T << T_D, where T_D is the Debye temperature, and the terms ${\gamma }_{e}T$ and ${\beta }_{ph}{T}^3$ represent the electronic and phonon contributions to the specific heat capacity, respectively. From the Sommerfeld approximation [40], the coefficient ${\gamma }_e$ can be calculated as follows:

$${\gamma }_e={\displaystyle \frac{{\pi }^2{k_B}^2}{3}}\mathsf{\Delta }n\left(E_F\right)$$

(4)

where $\mathsf{\Delta }n\left(E_F\right)$ is the density of electrons per eV at the Fermi level. This expression applies when external hydrostatic pressure is considered.

Figure 2 shows how the electronic specific heat capacity coefficients, calculated by eq. (4), change with increasing external hydrostatic pressures up to 1.0 GPa for the studied Laves phases. For HoAl₂, the obtained ${\gamma }_e$ value at P = 0 GPa is 9.43 × 10⁻³ J mol⁻¹ K⁻². This result differs from that obtained through the electron gas model in metals (Table 2); the values tend to slightly increase, reaching an average of 10.02 × 10⁻³ J mol⁻¹ K⁻² as the applied pressure increases. Conversely, the initial value for the HoNi₂ alloys at 0 GPa (i.e., 3.87 × 10⁻³ J mol⁻¹ K⁻²) is like the value calculated using the gas model in metals; see Table 2 for details (1.231 × 10⁻³ J mol⁻¹ K⁻²). Later, it increases noticeably to 8.02 × 10⁻³ J mol⁻¹ K⁻² with an applied pressure of 0.1 GPa and then remains nearly constant at an average value of 8.05 × 10^-3 J mol^-1 K^-2. At non-zero pressures, the obtained values for HoAl₂ and HoNi₂ are closer to each other, as calculated by DFT+U modelling.

Von Ranke et al. reported electronic specific heat capacity coefficients of 10.6 × 10⁻³, 5.5 × 10⁻³, 4.8 × 10⁻³ and 4.6 × 10⁻³ J mol^-1 K^-2 for LaAl₂, LuAl₂, LaNi_2.2, and LuNi₂, respectively [41]. De Oliveira and colleagues reported a ${\gamma }_e$ experimental value of 5.4 × 10⁻³ J mol^-1 K^-2 from the c_p(T) curve for non-ferromagnetic LuAl₂ stating that this alloy shows the same structure as HoAl₂, and a similar ${\gamma }_e$ value [42]. Campoy et al. experimentally determined a value of ${\gamma }_e$ = 7.0 × 10⁻³ J mol^-1 K^-2 for a bulk polycrystalline HoAl₂ alloy from c_p(T) data [43]. The DFT and Fermi gas approach of ${\gamma }_e$ values at P = 0 GPa for the Laves phases alloys agree with experimental ones obtained by cp(T) data.

3.3. Electronic Density of States

3.3.1. Total Density of Electronic States at P = 0 GPa

Figure 3 shows the total density of electronic states (DOS) obtained in ferromagnetically ordered crystal structures HoM₂ with M = {Al, Ni} Laves phases. The electronic structure for both compounds, s and p orbitals, is nearly symmetric and localized at deeper energies compared to d and f orbitals, which are at the Fermi level. The s orbitals are localized among –49.00 eV and –48.00 eV with a maximum of electronic density of 3.92 e⁻/eV at –48.59 eV for HoAl₂. When the post-transition metal (Al) is replaced by a transition metal (Ni), the localization of s orbitals shifts toward higher energies (i.e., between –47.43 eV and –46.19 eV) with a similar electronic density of 3.87 e⁻/eV. The p orbitals in HoAl₂ are localized in the energy range of –24.30 eV ≤ E – E_F ≤ –22.62 eV, while they are positioned between –23.00 eV and –21.32 eV for HoNi₂. Our calculations showed that the spin-up channel is shifted to lower energies compared to the spin-down channel while maintaining symmetry between them for both alloys. It is important to note that substituting Al with Ni reduces the maximum value in the DOS of p bands from 9.53 e⁻/eV to 5.95 e⁻/eV, and the DOS curve tends to flatten into a double peak, see the Figures 3(a)-3(b).

The d and f orbitals are very close to the Fermi level. Specifically, d bands range from –5.2 eV to 10.0 eV, while f bands range from –5.3 eV to 3.0 eV. Both the DOS of d and f orbitals exhibit a notable asymmetry. Additionally, f bands are the most populated in both compounds. Therefore, the localized f electrons continue to be responsible for ferromagnetic order in HoM₂ (M = {Al, Ni}) Laves phases. Replacing the post-transition metal Al with Ni leads to symmetry collapse of d orbitals, causing hybridization among d and f bands. This suggests that the magnetic behavior of HoAl₂ arises from localized electrons in f orbitals, and both itinerant and localized electrons in d and f orbitals contribute to ferromagnetism in HoNi₂.

The total magnetic moments along the <001> c-axis obtained for ferromagnetically ordered HoAl₂ and HoNi₂ Laves phases are 8.61 µ_B/f.u. and 8.12 µ_B/f.u., respectively. These values, derived from DOS, match the single crystalline data previously reported in scientific literature: 9.15 µ_B/f.u. to 9.18 µ_B/f.u. for HoAl₂ [18,25] and 8.52 µ_B/f.u. for HoNi₂ [19]. As summarized in Table 3, the net magnetic moment determined for polycrystalline ribbons [16,17], and bulk/massive [44,45] samples is close to that obtained from DFT quantum calculations.

3.3.2. Total Electronic Density of States at 0 GPa < P ≤ 1.0 GPa

Figure 4 shows the calculated DOS for ferromagnetically ordered HoAl₂ and HoNi₂ Laves phases under external hydrostatic pressures from 0 GPa to 1.0 GPa. For both alloys, a small increase in the external pressure applied to the crystal structure causes a shift to higher energies of the s and p orbitals. The shift is more evident in HoAl₂ (Figure 4(a)) than in HoNi₂ (Figure 4(b)). Meanwhile, d and f orbitals move closer to the Fermi energy level. For HoNi₂, a significant redistribution occurs in the electronic population of f-orbitals at non-zero pressure; the maximum of the spin-up channel increases from 14.26 e⁻/eV to 23.48 e⁻/eV, while the initial splitting of the spin-down channel disappears, and a maximum electronic density of –14.35 e⁻/eV is observed.

Figure 5(a) shows the calculated total magnetic moment versus hydrostatic pressures in the range of 0 GPa ≤ P ≤ 1.0 GPa for ferromagnetically ordered HoM₂ (M = {Al, Ni}) crystal structures. For HoAl₂, the total magnetic moment remains nearly constant (around a mean value of 8.49 µ_B/f.u.) as applied pressure increases up to 1 GPa. On the contrary, for HoNi₂, the total magnetic moment at P = 0 GPa (8.12 µ_B/f.u.) decreases to 6.29 µ_B/f.u. at 0.1 GPa and then remains nearly constant for a non-zero applied pressure. This is a 22.5 % that DOS was modified after a 0.1 GPa, that can be useful in multicaloric approach during the magnetic phase transition [2,3]. Additionally, Figure 5(b) displays the calculated magnetic moment associated with d and f orbitals for both alloys. It is important to note that the magnetic order mainly results from localized f electrons. The itinerant electrons contribute only minimally to the magnetic moment. Moreover, for HoNi₂, the magnetic contribution of 4f electrons decreases from 7.45 µ_B/f.u. to 6.31 µ_B/f.u. under an applied pressure of 0.1 GPa and then remains nearly constant at a non-zero value pressures.

3.3.3. Electronic Partial Density of States at P = 0 GPa

The obtained partial density of electronic states (PDOS) corresponding to d and f orbitals is shown in Figure 6. Once a transition metal like Ni replaces the post-transition metal Al in the crystalline structure, the electronic population of 3d electrons increases, while the density of electrons in 4f orbitals decreases, resulting in a broadened peak of the DOS curve. The contributions per orbital to the total magnetic moment are listed in Table 4. At P= 0 GPa, electrons localized at d orbitals contributed less to the net magnetic moment in the crystal structure HoAl₂ (i.e., 0.28 µ_B/f.u.) compared to HoNi₂ (i.e., 0.57 µ_B/f.u.). Therefore, the magnetic behavior of the ferromagnetically ordered HoAl₂ crystal structure is just due to unpaired electrons localized at f orbitals. When Al is fully replaced by Ni, the number of unpaired electrons in the d orbital increases; as a result, the magnetic moment of HoNi₂ arises from 3d and 4f electrons. Moreover, s and p orbitals shift to higher energies when Al is replaced by Ni, and p orbitals tend to decrease their maximum population, resulting in a broadened and flattened peak band. Partial DOS for s and p orbitals is shown in Figure A3. Finally, the contribution of electrons localized at s and p orbitals to the total magnetic moment is almost negligible for both alloys, HoAl₂ and HoNi₂.

On the one hand, Figure 6(a) shows that the 4d and 3d electrons of Ho in HoAl₂ are located very close to the Fermi level with a very low density of states. In contrast, the 4f electrons of Ho in the spin-up channel are situated at –4.77 eV, far from the Fermi level, while the 4f electrons in the spin-down channel are practically at the Fermi level (–0.7 eV); see Figure 6(b). For HoNi₂, the 3d electrons in both spin-up and spin-down channels are near the Fermi level (at –1.75 eV). Conversely, the 4f electrons of Ho at the spin-up channel are localized at – 4.03 eV far away from the Fermi level; in the case of the spin-down channel, 4f electrons are very close to the Fermi level (from – 1.16 eV to 0 eV). Noticing that the amount of Ho 4f electrons in HoAl₂ has larger peaks in the density of states compared to the broad, lower peaks of Ho 4f electrons in HoNi₂. Finally, the Ni 3d electrons have a higher density of states compared to the Ho 4d and 3d electrons.

3.3.4. Electronic Partial Density of States at 0 GPa ≤ P ≤ 1.0 GPa

For ferromagnetically ordered HoM₂ (M = {Al, Ni}) Laves phases, the applied external hydrostatic pressure induces a shift in the energies of the s, p, d, and f bands. This shift is more noticeable for the s and p bands in HoAl₂ than in HoNi₂. The PDOS obtained for s and p orbitals are shown in Figure A4 of the Appendix section.

Figure 7 displays the PDOS for the d and f orbitals under external hydrostatic pressures ranging from 0 GPa to 1.0 GPa. In the ferromagnetically ordered HoAl₂ Laves phase, the applied pressures (0 GPa to 1.0 GPa) cause a slight rearrangement of the 3d and 4d Ho spin-up and spin-down channel bands. See details in Figure 7 (a). The 4f electrons of Ho in the spin-up channel, initially localized far from the Fermi level at –4.77 eV, move slightly closer to the Fermi level as pressure increases, reaching –4.34 eV at 1.0 GPa (Figure 7 (b)). Meanwhile, the 4f electrons in the spin-down channel, initially localized at –0.70 eV, nearly reach the Fermi level.

Figures 7(c) and 7(d) illustrate that for HoNi₂, the 3d electrons of Ni in both spin-up and spin-down channels do not undergo significant changes as the external pressure increases from 0 GPa to 1.0 GPa. Furthermore, the 4f electrons of Ho, a rare earth element, in both spin-up and spin-down channels show an increase in their maximum electronic density under non-zero applied pressure. It is important to note that the initial splitting of the spin-down channel tends to disappear. External pressure causes a redistribution of the electronic population across the orbitals for both alloys, but the most affected orbitals are the 4f orbitals of HoNi₂. The maximum population of electrons with spin-up at f orbitals of HoNi₂ increases from 14.26 e⁻/eV to 23.48 e⁻/eV, and their electronic density of the spin-down channel rises to –14.35 e⁻/eV near the E_F level, as shown in Figure 7(d).

The net magnetic moment [40] was calculated using the following equation:

$${\mu }_T={\int }_{E_1}^{E_F}{n}_{S\uparrow }\left(E\right)dE-{\int }_{E_2}^{E_F}{n}_{S\downarrow }\left(E\right)dE$$

(5)

where E₁ and E₂ represent the starting energies of electronic states for spin-up and spin-down channels, respectively. Table 4 presents the calculated net magnetic moment computed from the simulated DOS for ferromagnetically ordered HoAl₂ and HoNi₂ Laves phases, along with the contribution of electrons at s, p, d, and f orbitals to the total magnetic moment. When Al is fully replaced by Ni, the contribution of d electrons to the total magnetic moment slightly increases. Furthermore, the main contributor to magnetic behavior in ferromagnetically ordered HoM₂ with M = {Al, Ni} are the f electrons.

4. Conclusions

This study examined how hydrostatic pressures from 0 GPa to 1.0 GPa affect the crystal stability, electronic properties, and magnetic properties of ferromagnetically ordered HoAl₂ and HoNi₂ Laves phases through DFT+U calculations using the RPBE exchange-correlation functional within the GGA framework. All calculations were performed along the <001> direction, and the net magnetic moment obtained remains nearly constant for ferromagnetically ordered HoAl₂, with values of 8.61 µ_B per formula unit along the intermediate magnetization axis. The ferromagnetically ordered HoNi₂ Laves phases experience a reduction in their initial magnetic moment from 8.12 µ_B/f.u. to 6.29 µ_B/f.u. in the easy magnetization axis when subjected to non-zero applied pressure. The latter HoNi₂ Laves phase is useful for multicaloric approach in solid state cooling by using magneto-caloric and baro-caloric effects. The magnetic moment can be modified a 22.5 % under 0.1 GPa in the range of fully reversible loading and uploading regimes in the alloy when the hydrostatic pressure is applied and released, respectively. The most affected by the applied pressures was the HoAl₂ Laves phase, with a compressive stress of –2.88%, while the HoNi₂ exhibited a compressive stress of –1.87%. The interatomic distances change very little within the pressure range studied. The ferromagnetic order persists, displaying a reorganization of the electronic states, with the f orbitals of HoNi₂ Laves phases being the most affected. The stability of the ferromagnetically ordered HoAl₂ and HoNi₂ Laves phases' crystal structures remains unaffected across the entire range of applied pressures, and the formation energy stays constant up to 1.0 GPa.