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

Abstract

In this study, density functional theory (DFT) with Hubbard U correction calculations (DFT+U) was used to examine the ferromagnetic properties of HoM₂ Laves phases (M = {Al, Ni}) under external hydrostatic pressure from 0 GPa to 1.0 GPa. The resulting net magnetic moments of 8.61 µ_B/f.u. (HoAl₂) and 8.12 µ_B/f.u. (HoNi₂) align with values reported in experiments. Additionally, for both alloys, the ferromagnetic behavior remains unchanged under applied pressures from 0 GPa to 1.0 GPa. The study also confirms that the magnetic properties of the alloys are mainly influenced by the 4f electrons, with 3d electrons playing a slightly more significant role in HoNi₂ Laves phases compared to HoAl₂. The contribution of electrons in d and f orbitals to the net magnetic moment of each Laves phase alloy within the specified pressure range was examined. Furthermore, the crystal geometry optimization and electronic specific heat coefficient were calculated as functions of applied pressures up to 1.0 GPa for both ferromagnetically ordered Laves phases.

1. Introduction

Experimental and theoretical research in the field of magnetocaloric-based cooling has grown over the past 25 years, driven by 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, 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 [7,8,9,10,11]. 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 heavy R elements such as Ho and Er [6,7,8,9,10,11,12].

These compounds crystallize into the MgCu₂-type structure (C15, space group: Fd–3m) [12,13], with a lattice parameter of 7.810 Å for HoAl₂ and 7.130 Å for HoNi₂ [14,15], and show Weiss–Curie temperatures T_C of 29 K and 13.4 K, respectively [16,17]. The Laves phases (AB₂) have two lattice-related characteristics: the relationship between the atomic radii of A and B atoms is between 1.05 and 1.68, and for an atomic radius ratio (r_A/r_B) of 1.225, the crystal structures have a higher packing density (around 71%) [12,13,18,19,20,21]. A green hydrogen economy needs hydrogen storage (due to hydriding properties) in the cell unit of Laves C15 phases. Laves intermetallics can be used to store hydrogen interstitially by offering different positions (i.e., three tetrahedral interstices) [13].

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 [16,22]. Additionally, a spin reorientation occurs at T_SR = 20 K. The hard magnetization axis in HoAl₂ is <111> crystal direction [17,22]. The directions of the easy, intermediate, and hard magnetization axes of HoNi₂ are <100>, <110>, and <111>, respectively [15,22,23,24,25].

On the other hand, density functional theory (DFT) is a fundamental computational method in materials science, providing atomistic insights into crystals and molecules. Resolving a material’s electronic structure enables the investigation and prediction of the structure–property relationships and the underlying physicochemical phenomena in solids [26,27,28,29,30].

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 the <001> crystal direction. We systematically explore how the electronic density of states is affected 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 magnetocaloric and barocaloric effects has previously been used [2,3]. The combination of different external fields (e.g., magnetic field and hydrostatic pressure) enables an enhancement of the caloric response by tailoring the magnetic moment (related to the electronic density of states) with the pressure during the magnetic phase transition. Our finding is that the net magnetic moment is modified (drops by 22.5%) when a 0.1 GPa hydrostatic pressure is applied. To our knowledge, the electronic and magnetic properties of HoAl₂ and HoNi₂ under hydrostatic pressure, as determined by ab initio calculations, have not yet been reported.

2. Materials and Methods

The present study was performed using the Cambridge Serial Total Energy Package (CASTEP) within the density functional theory framework using BIOVIA Materials Studio^® 2020 (version 20.1.0.5). For the exchange correlation, the revised Perdew–Burke–Ernzerhof (RPBE) functional was applied as part of the generalized gradient approximation (GGA). It is 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 the calculations [31,32,33]. It is important to note that U corrections within GGA show better accuracy when investigating the magnetic and electronic structures of 4f and 3d compounds (e.g., strongly correlated systems) compared with local density approximation (LDA) or hybrid functionals [34,35]. 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 due to the lack of localized electrons. The spin–orbit coupling calculation was not performed due to high computing times; instead, a Koelling–Harmon relativistic scheme was used for faster calculation [36]. 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 [37,38,39,40]. The energy convergence criterion was set at 1 × 10⁻⁶ eV/atom for self-consistent field cycles. The maximum values of the convergence thresholds were 0.03 eV/Å, 0.05 GPa, and 0.001 Å for force, stress, and displacement, respectively. 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 the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. The Cauchy stress tensor was used as σ_ij = −P δ_ij, where P and δ_ij are the hydrostatic pressure value and the Kronecker delta, respectively. The latter tensor is equivalent to hydrostatic pressure P. The obtained bulk modulus B was obtained by fitting the third-order Birch–Murnaghan state equation.

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 structures 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 16d 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, in the search for accuracy, we used experimental lattice parameters rather than those obtained from minimizing the total energy as a function of volume for the crystalline structures. The lattice parameters a = b = c were 7.810 Å for HoAl₂ and 7.130 Å for HoNi₂ [14,15]. The collinear 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. Diagonal component σ_ij of the stress tensor, bulk modulus B, primitive unit cell volume V_P (i.e., rhombohedral trigonal), compressive stress Δ, and interatomic Al-Al, Ho-Al, and Ho-Ni distances obtained for the HoAl₂ and HoNi₂ Laves phases under applied hydrostatic pressures of 0 GPa ≤ P ≤ 1.0 GPa.

Alloy	HoAl2							HoNi2
P (GPa)	B (GPa)	σij (GPa)	a (Å)	dAl-Al (Å)	dHo-Al (Å)	VP (Å3)	Δ (%)	B (GPa)	σij (GPa)	a (Å)	dNi-Ni (Å)	dHo-Ni (Å)	VP (Å3)	Δ (%)
0.0	40.66	−0.00005	5.652	2.826	3.314	127.675	0.000	49.79	−0.00115	5.262	2.631	3.086	103.066	0.000
0.1	45.68	−0.10067	5.646	2.823	3.311	127.299	−0.294	56.58	−0.10030	5.258	2.629	3.083	102.840	−0.219
0.2	52.56	−0.19968	5.640	2.820	3.307	126.908	−0.600	58.98	−0.19724	5.255	2.628	3.081	102.655	−0.399
0.3	56.69	−0.29771	5.635	2.818	3.304	126.533	−0.894	60.33	−0.30162	5.252	2.626	3.080	102.480	−0.569
0.4	58.00	−0.39932	5.629	2.815	3.301	126.165	−1.182	68.79	−0.40011	5.248	2.624	3.077	102.241	−0.801
0.5	60.58	−0.49793	5.623	2.812	3.297	125.780	−1.484	71.50	−0.50638	5.245	2.623	3.076	102.072	−0.964
0.6	64.07	−0.60178	5.618	2.809	3.294	125.380	−1.797	76.23	−0.60287	5.242	2.621	3.074	101.868	−1.162
0.7	65.22	−0.70149	5.613	2.807	3.291	125.055	−2.052	82.56	−0.69933	5.239	2.620	3.072	101.703	−1.322
0.8	66.65	−0.80276	5.607	2.804	3.288	124.681	−2.345	84.36	−0.80034	5.235	2.618	3.070	101.479	−1.540
0.9	68.37	−0.89994	5.602	2.801	3.285	124.328	−2.621	91.47	−0.89917	5.232	2.616	3.068	101.321	−1.693
1.0	70.89	−0.99911	5.597	2.799	3.282	123.992	−2.884	98.45	−0.99998	5.229	2.615	3.066	101.132	−1.876

displays the lattice parameter and interatomic distances between Al-Al, Ho-Al, and Ho-Ni in the Laves phase 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 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 –13.470 × 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 remain 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 A 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. The obtained bulk modulus B follows an increasing linear dependence with the applied pressure, having slopes dB/dP of 27.93 and 46.44 with intercepts B₀ = 45.06 GPa and 49.41 GPa for HoAl₂ and HoNi₂, respectively. The R² values of the fitting are 0.93 and 0.99 for each alloy. Finally, a measure of lattice constant changes can be deduced from the relative compressive stress ∆ caused by applied pressure; see Table 1. This quantity can be calculated as ∆ = (V_P − V₀)/V₀ × 100%, where ∆V is the volume difference calculated as V_P − V₀. The resulting ∆(P = 1 GPa) values are −2.88% and −1.87% for HoAl₂ and HoNi₂, respectively.

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 [14], M(HoNi₂) = 282.316 g/mol, and ρ(HoNi₂) = 10.33 g/cm³ [41] are 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:

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

(1)

where $h$ and $m_e$ are the Planck constant and the electron mass, respectively. Using Equation (1) and previous values, the Fermi temperature T_F can be calculated through the following formulae:

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

(2)

where $k_B$ is the Boltzmann constant. The electronic coefficient ${\gamma }_e$ of specific heat capacity can be calculated using the following equation:

$${\gamma }_e={\displaystyle \frac{{\pi }^2 k_B N_A}{3 T_F}}$$

(3)

The calculated values of the Fermi energy and its temperature and the electronic coefficient ${\gamma }_e$ using Equations (1)–(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, Ag, and Cu [42].

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 [43], the coefficient ${\gamma }_e$ can be calculated as follows:

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

(4)

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

Table 2. The calculated values of Fermi energy (E_F), Fermi temperature (T_F), and the electronic specific heat capacity coefficient (${\gamma }_{\mathrm{e}}$) obtained by Equations (1)–(3) for both Laves phases (HoAl₂ and HoNi₂). Other pure elements [42] and Laves phases RM₂ with R = (La, Lu, Ho) and M = (Al, Ni) [44,45,46] are included for comparison purposes.

Laves Phases or Elements	N/V (×1028 m−3)	${\mathit{E}}_{\mathit{F}}$ (eV)	${\mathit{T}}_{\mathit{F}}$ (×104 K)	${\mathit{\gamma }}_{\mathit{e}}$ (×10–3 J mol–1 K–2)	Reference
HoAl2	1.678	2.392	2.777	0.984	this work
HoNi2	2.203	2.869	3.331	0.820	this work
K	1.40	2.13	2.47	1.106 a	[42]
Ag	5.86	5.53	6.41	0.426 a	[42]
Cu	8.47	7.06	8.19	0.333 a	[42]
LaAl2	-	-	-	10.6 c	[44]
LuAl2	-	-	-	5.5 c	[44]
LaNi2.2	-	-	-	4.8 c	[44]
LuNi2	-	-	-	4.6 c	[44]
LuAl2 b	-	-	-	5.4 c	[45]
HoAl2 bulk polycrystalline	-	-	-	7.0 a	[46]

^a calculated using Equation (3). ^b c_p(T) data is like HoAl₂. ^c determined from c_p(T) data.

Table 2. The calculated values of Fermi energy (E_F), Fermi temperature (T_F), and the electronic specific heat capacity coefficient (${\gamma }_{\mathrm{e}}$) obtained by Equations (1)–(3) for both Laves phases (HoAl₂ and HoNi₂). Other pure elements [42] and Laves phases RM₂ with R = (La, Lu, Ho) and M = (Al, Ni) [44,45,46] are included for comparison purposes.

Laves Phases or Elements	N/V (×1028 m−3)	${\mathit{E}}_{\mathit{F}}$ (eV)	${\mathit{T}}_{\mathit{F}}$ (×104 K)	${\mathit{\gamma }}_{\mathit{e}}$ (×10–3 J mol–1 K–2)	Reference
HoAl2	1.678	2.392	2.777	0.984	this work
HoNi2	2.203	2.869	3.331	0.820	this work
K	1.40	2.13	2.47	1.106 a	[42]
Ag	5.86	5.53	6.41	0.426 a	[42]
Cu	8.47	7.06	8.19	0.333 a	[42]
LaAl2	-	-	-	10.6 c	[44]
LuAl2	-	-	-	5.5 c	[44]
LaNi2.2	-	-	-	4.8 c	[44]
LuNi2	-	-	-	4.6 c	[44]
LuAl2 b	-	-	-	5.4 c	[45]
HoAl2 bulk polycrystalline	-	-	-	7.0 a	[46]

^a calculated using Equation (3). ^b c_p(T) data is like HoAl₂. ^c determined from c_p(T) data.

Figure 2 shows how the electronic specific heat capacity coefficients, calculated by Equation (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 (0.820 × 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 modeling.

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 [44]. 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 LuAl_2, stating that this alloy shows the same structure as HoAl₂ and a similar ${\gamma }_e$ value [45]. 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 [46]. The DFT and Fermi gas approach of ${\gamma }_e$ values at P = 0 GPa for the Laves phase alloys agree with experimental ones obtained by c_p(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 between −49.00 eV and −48.00 eV with a maximum 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 show that the spin-up channel is shifted to lower energies compared to the spin-down channel while maintaining symmetry between 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 Figure 3a,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 net magnetic moment [43] can be 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. 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 [17]. Notice that the magnetic moment measurements for polycrystalline ribbons of HoAl₂ [14] were carried out using in 9 T Dynacool PPMS-VSM magnetometer along the major axis of the ribbon length (in-plane). The T_C value is 29 K for this HoAl₂ sample. As summarized in Table 3, the net magnetic moments determined for polycrystalline ribbons [14,15] and bulk/massive [47,48] samples are close to that obtained from DFT quantum calculations.

Table 3. Cell parameter a, calculated magnetic moment µ_T, and magnetization of HoAl₂ and HoNi₂ compared with experimental data reported in the literature [14,15,16,17,22,47,48].

Laves Phase	Alloy Type	a (Å)	µT (µB/f.u.)	${\mathit{T}}_{\mathbf{C}}$ (K)	MS (Am2kg−1)	Magnetization Axis	Reference
HoAl2	DFT+U framework	7.810	8.61 a	-	220 a	<001>; intermediate	this work
	single-crystal	7.816 c 7.838	9.18 b 9.15 c	31.5 29.0	235 b 234 c	<011>; easy	[16] [22]
	bulk polycrystalline	7.8024	7.86	27.0	201	close to <001>; intermediate	[47]
	polycrystalline ribbons (20 m/s)	7.8109	7.08 c, e	24.0	181 c, e	close to <001>; intermediate	[14]
HoNi2	DFT+U framework	7.130	8.12 a	-	161 a	<001>; easy	this work
	single-crystal	-	8.52 d	13.4	168 d	<001>; easy	[17]
	bulk polycrystalline	7.1318	8.40	22.0	167	very close to <001>; easy	[47,48]
	polycrystalline ribbons (20 m/s)	7.1497	8.02 c	13.9	159 c	close to <001>; easy	[15]

Experimental data measured at the following temperatures: ^a T = 0 K. ^b T = 4.2 K. ^c T = 2.0 K. ^d T = 1.4 K. ^e determined from Figure 4. The experimental crystal structure cell unit is reported at room temperature.

Table 3. Cell parameter a, calculated magnetic moment µ_T, and magnetization of HoAl₂ and HoNi₂ compared with experimental data reported in the literature [14,15,16,17,22,47,48].

Laves Phase	Alloy Type	a (Å)	µT (µB/f.u.)	${\mathit{T}}_{\mathbf{C}}$ (K)	MS (Am2kg−1)	Magnetization Axis	Reference
HoAl2	DFT+U framework	7.810	8.61 a	-	220 a	<001>; intermediate	this work
	single-crystal	7.816 c 7.838	9.18 b 9.15 c	31.5 29.0	235 b 234 c	<011>; easy	[16] [22]
	bulk polycrystalline	7.8024	7.86	27.0	201	close to <001>; intermediate	[47]
	polycrystalline ribbons (20 m/s)	7.8109	7.08 c, e	24.0	181 c, e	close to <001>; intermediate	[14]
HoNi2	DFT+U framework	7.130	8.12 a	-	161 a	<001>; easy	this work
	single-crystal	-	8.52 d	13.4	168 d	<001>; easy	[17]
	bulk polycrystalline	7.1318	8.40	22.0	167	very close to <001>; easy	[47,48]
	polycrystalline ribbons (20 m/s)	7.1497	8.02 c	13.9	159 c	close to <001>; easy	[15]

Experimental data measured at the following temperatures: ^a T = 0 K. ^b T = 4.2 K. ^c T = 2.0 K. ^d T = 1.4 K. ^e determined from Figure 4. The experimental crystal structure cell unit is reported at room temperature.

Figure 4. Magnetization isotherm at 2 K (a) and temperature dependence of magnetization measured under magnetic fields of 5 mT and 5 T (b) for HoAl₂ melt-spun ribbons fabricated at 8 m/s in a melt-spinner. The short-dashed line in (a) indicates the saturation magnetization extrapolated to a zero magnetic field (reported in Table 3). The magnetic moment per formula unit is calculated from the magnetization values. As pointed out by the vertical arrow in (b), the sample shows T_C = 29 K. More experimental details can be found in reference [14], and the demagnetizing field effect was neglected when measuring along the large side of ribbon flakes.

Figure 4. Magnetization isotherm at 2 K (a) and temperature dependence of magnetization measured under magnetic fields of 5 mT and 5 T (b) for HoAl₂ melt-spun ribbons fabricated at 8 m/s in a melt-spinner. The short-dashed line in (a) indicates the saturation magnetization extrapolated to a zero magnetic field (reported in Table 3). The magnetic moment per formula unit is calculated from the magnetization values. As pointed out by the vertical arrow in (b), the sample shows T_C = 29 K. More experimental details can be found in reference [14], and the demagnetizing field effect was neglected when measuring along the large side of ribbon flakes.

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

Figure 5 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 5a) than in HoNi₂ (Figure 5b). 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 6a 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 the applied pressure increases up to 1 GPa. In contrast, 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 22.5% decrease indicates that DOS is modified after 0.1 GPa, which can be useful in a multicaloric approach during magnetic phase transition [2,3].

Additionally, Figure 6b 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 pressure value.

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 7. 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. Net magnetic moment values calculated using Equation (5) from the obtained DOS at P = 0 GPa and compared with experimental data [16,17]. Electronic quantities with spin up and down, along with the difference in electronic states at the Fermi level, are shown for the ferromagnetically ordered HoAl₂ and HoNi₂ Laves phases along the <001> direction.

Physical Magnitude	Alloy
	HoAl2	HoNi2
nS↑ (EF) (e−/eV)	1.76	14.59
nS↓ (EF) (e−/eV)	−15.10	−9.11
Δn (EF) (e−/eV)	−13.34	5.48
${\mu }_{S\uparrow }$ (µB/f.u.)	31.30	37.08
${\mu }_{S\downarrow }$ (µB/f.u.)	−22.69	−28.96
${\mu }_T^{\mathrm{D}\mathrm{F}\mathrm{T}+\mathrm{U}}$ (µB/f.u.)	8.61	8.12
${\mu }_T^{\mathrm{E}\mathrm{x}\mathrm{p}}$ (µB/f.u.)	9.18 †	8.52 ‡
μs (μB/f.u.)	−0.03	0.11
μp (μB/f.u.)	0.04	−0.17
μd (μB/f.u.)	0.28	0.57
μf (μB/f.u.)	8.79	7.45
μP (μB/f.u.)	9.08	7.96

Experimental magnetic moment values for single crystals are measured in ^† ref [16] and ^‡ ref [17].

. 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 A2. Finally, the contribution of electrons localized at s and p orbitals to the total magnetic moment is almost negligible for both HoAl₂ and HoNi₂.

On the one hand, Figure 7a 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 on the Fermi energy level (at −0.70 eV); see Figure 7b. 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). Notice 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.

Table 4 presents the calculated net magnetic moment computed from the simulated DOS at P = 0 GPa 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} is the f 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 A3 of the Appendix A Section.

Figure 8 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 8a. 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 8b). Meanwhile, the 4f electrons in the spin-down channel, initially localized at −0.70 eV, nearly reach the Fermi level.

Figure 8c,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 8d.

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 the multicaloric approach in solid-state cooling by using magnetocaloric and barocaloric effects. The magnetic moment can be modified by 22.5% under 0.1 GPa in the range of fully reversible loading and unloading regimes in the alloy when the hydrostatic pressure is applied and released, respectively. The HoAl₂ Laves phase is the most affected by the applied pressures, with a compressive stress of −2.88%, while the HoNi₂ exhibits 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.