DFT Study of Hydrostatic Pressure Effects up to 1.0 GPa on the Electronic and Magnetic Properties of Laves Phases ErAl₂ and ErNi₂

Abstract

This study employs DFT+U calculations to investigate the ferromagnetic properties of ErAl₂ and ErNi₂ Laves phases under an external hydrostatic pressure P (0 GPa ≤ P ≤ 1.0 GPa). The calculated magnetic moments per formula unit for both crystalline structures align with experimentally reported values: 4.40 μ_B/f.u. in the hard magnetization <001> axis for ErAl₂ and 5.56 μ_B/f.u. in the easy magnetization <001> axis for ErNi₂. The DFT results indicate that the magnetic moment remains unchanged up to 1 GPa of hydrostatic pressure, with no structural instabilities observed, as evidenced by a nearly constant formation energy for ErAl₂ and ErNi₂ alloys. The simulations confirm that the magnetic behavior of ErAl₂ is primarily driven by the electrons localized in the f orbitals. In contrast, for ErNi₂, both d and f orbitals significantly contribute to the total magnetic moment. Finally, the electronic specific heat coefficient was calculated and reported as a function of hydrostatic pressure up to P = 1.0 GPa for each Laves phase.

1. Introduction

A renewed interest has recently emerged in studying rare-earth-based Laves phases due to their excellent magnetocaloric properties, which make them suitable as magnetic refrigerants for hydrogen magnetocaloric liquefaction [1,2,3,4,5,6,7,8,9]. This solid-state cooling technology is environmentally friendly and more energy-efficient than conventional refrigeration methods that rely on the Joule–Thomson effect [1,3,6,7,8,9].

In stoichiometric AB₂-type rare-earth-based Laves phases, A represents the rare earth element, while B denotes a transition or post-transition metal. The relative ratio of elements A and B, r_A/r_B, ranges from 1.05 to 1.68, with a higher packing density achieved at an ideal atomic radius ratio of 1.225, which corresponds to a space-filling of 71% in the crystal structure [10,11]. Rare-earth-based Laves phases exhibit three typical crystal structures known as C15 (cubic), C14 (hexagonal), and C36 (hexagonal).

The present work examines the effect of hydrostatic pressure on the electronic structure and magnetic properties of the ferromagnetic ErM₂ compounds, where M is either Al or Ni. These compounds crystallize into a MgCu₂-type cubic structure (C15) [12,13,14], with lattice parameters a = b = c of 7.770 Å and 7.113 Å, respectively [15,16]. The MgCu₂-type crystal structure exhibits a valence electron concentration per atom (e/a) of less than 1.8 (i.e., e/a < 1.8), which contributes to the stability of the C15-type Laves phases. ErAl₂ and ErNi₂ display ferromagnetic order, with Curie temperatures (T_C) of 12.5 K and 6.8 K, respectively [15,16]. They have r_A/r_B ratios of r_Er/r_Al = 1.245 and r_Er/r_Ni = 1.162. The total magnetic moment per formula unit in both compounds originates from the rare earth element (Er), which possesses unpaired electrons [10,11,17].

Ab initio calculations employing density-functional theory (DFT) are essential tools for investigating the relationship between electronic structure and the physicochemical properties of materials [18]. In this study, these calculations have been utilized to examine the correlation between electronic structures and the magnetic properties of the two previously mentioned ferromagnetically ordered compounds [17,19,20,21,22,23,24]. The application of first-principles methods for computing electronic structures and performing energy calculations to address the inherent challenges of Laves phase investigations is an emerging field [25].

In this work, we examine the effect of hydrostatic pressure on the electronic structure and magnetic properties of the ferromagnetically ordered ErAl₂ and ErNi₂ Laves phases along the <001> direction using the spin-polarized DFT+U method with Hubbard U potential. We highlight the role of the Al and Ni orbitals in the electronic structure. To our knowledge, no in-depth investigations have used DFT methods to analyze the magnetic properties along the <001> axis in these two Laves phases. We underline that the magnetic moment strongly depends on the crystal direction in which the calculations are performed (i.e., easy, intermediate, and hard magnetization axes).

2. Computational Method

For the spin-polarized first-principles electronic structure calculations at T = 0 K, we used the experimental lattice parameters reported for the ErAl₂ and ErNi₂ Laves phases [15,16]. The MgCu₂-type cubic structure (space group Fd-3m (227)) was constructed for both compounds, and the electron configurations of Al, Ni, and Er were defined as [Ne] 3s² 3p¹, [Ne] 3s² 3p⁶ 3d⁸ 4s², and [Ne] 3s² 3p⁶ 3d¹⁰ 4s² 4p⁶ 4d¹⁰ 5s² 5p⁶ 4f¹² 6s², respectively. The selected lattice parameters for constructing the crystalline structures were a = b = c = 7.770 Å for ErAl₂ [15] and a = b = c = 7.113 Å for ErNi₂ [16]. The A and B atoms in the AB₂ structure occupy the 8a and 16d Wyckoff positions (i.e., at (0 0 0) and (5/8 5/8 5/8)). The ferromagnetic order for both compounds was simulated under the assumption that only the rare earth element (i.e., Er), located at the 8a site, possesses the magnetic moment, which is oriented along the <001> direction. Al and Ni, occupying the 16d Wyckoff position, were presumed to have zero magnetic moments [11,12]. The Erbium spin magnetic moment value was set to +3 μ_B, while the spin magnetic moment values for Al or Ni were assigned as 0. The spin quantization axis is <001>. During energy calculations, there were neither any constraints on crystal (volume is fully relaxed) nor on magnetic structures considering. Figure 1a illustrates the crystal structure exhibiting ferromagnetic order used for collinear spin-polarized quantum simulations. The easy magnetization direction corresponds to the crystalline direction in the <111>-axis for ErAl₂ [26] and in the <001>-axis for ErNi₂ [16,27,28]. For ErAl₂, our calculations were performed along the hard magnetization axis (i.e., the c-axis) [27].

The geometric optimization of the crystalline structures of ErAl₂ and ErNi₂ Laves phases was performed using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm [29,30]. The primitive cell (i.e., rhombohedral trigonal with lattice parameters a = b = c and angles α = β = γ = 60°) of the ErM₂ (M = Al, Ni) crystal structure, depicted in Figure 1b, was used for energy calculations. During the geometric optimization process, external compressive stress was applied along the a, b, and c axes, which are perpendicular to the yz, xz, and xy atomic planes, respectively. This external stress is equivalent to applying hydrostatic pressure P. The diagonal components of the stress tensor are σ_ij = −P δ_ij, where P is the hydrostatic pressure, and δ_ij is the Kronecker delta. The bulk modulus B and its derivative were obtained by fitting the calculated third-order Birch–Murnaghan equation of state.

Spin-polarized calculations were done using the revised Perdew–Burke–Ernzerhof (RPBE) exchange–correlation functional within a Generalized Gradient Approximation (GGA) framework. To determine the electronic density of states (DOS), the Brillouin zone was integrated using a k-mesh of 13 × 13 × 13. A cutoff energy of 500 eV was applied for plane-wave propagation through the crystal structure [31,32,33,34]. The charge convergence for self-consistent field cycles was set at 1 × 10⁻⁶ eV. The GGA method alone is inadequate for describing the localized 4f and itinerant 3d electrons; thus, the Hubbard parameter U was introduced to model the ferromagnetic ordering of these 4f and 3d electrons [18,35,36]. The U values for localized electrons were 2.50 eV for Ni atoms and 6.0 eV for Er atoms. The U value due to the lack of localized electrons was set to 0 eV for Al atoms. It is crucial to emphasize that U corrections within the GGA framework have shown better accuracy than other LDA or hybrid functionals for investigating the magnetic behavior and electronic structure of 4f and 3d compounds [18,35,36,37,38]. All calculations were performed along the <001> axis using density functional theory with U corrections (DFT+U) implemented in the Cambridge Serial Total Energy Package (CASTEP) code [32].

3. Results and Discussion

3.1. Electronic Structure at Normal Pressure

Figure 2a,b illustrates the calculated density of states (DOS) for ErAl₂ and ErNi₂ without hydrostatic pressure. In both cases, the s and p orbitals are localized in deeper energy zones (with peak energy values of around −47 and −22 eV, respectively), and their DOS is almost symmetric in shape (as shown in more detail in Figure 3). Notice in Figure 3 the unfolding of the asymmetric p orbitals when Ni replaces Al in the C15-type crystal structure, which evidences the effect of substituting the post-transition metal (Al) with the transition metal (Ni). The electronic states corresponding to the electrons in s orbitals lie at (E − E_F) energies between −48 and −46 eV, while those in p orbitals are at −23 < E − E_F < −20 eV, see Figure 2a. The latter indicates that in the ferromagnetically ordered ErAl₂ and ErNi₂ Laves phases, the p electrons contribute minimally to the net magnetic moment. In contrast, the d and f orbitals are localized around the Fermi energy level (E − E_F = 0 eV). For ErAl₂, 4d and 3d orbitals are a broad band with few electronic states; meanwhile, the Er 4f orbitals are a well-defined peak band. Conversely, for ErNi₂ the Ni 3d electronic band is remarkable (compared with Er 4d and 3d states, see Figure 2b) making it difficult to disjoin from Er 4f states.

For ErAl₂, the spin-up and spin-down total DOS profiles of d orbitals show remarkable symmetry, like that of the s and p orbitals. This symmetry explains their minimal contribution to the total magnetic moment, as Figure 2b illustrates. Conversely, Figure 2b shows that in ErNi₂, the spin-polarized total DOS profiles of d and f orbitals are highly asymmetric, highlighting their significant role in the material’s magnetic behavior. This also indicates that the hybridization between electrons in these two orbitals is more substantial for this compound than for ErAl₂, which has fewer unpaired d electrons than ErNi₂. The hybridization of d and f orbitals occurs at energies near the Fermi level (i.e., in the −10 < E − E_F < 0 eV range). The partial DOS is obtained from total DOS by decomposing into spin-up and spin-down angular momentum channels. The spin-up p-band is shifted compared to spin-down p-band in partial DOS; see Figure 3. In ErAl₂, the spin-up p-band has lower energies, and the spin-down p-band is in slightly higher energies. In contrast, the opposite occurs in energies for ErNi₂.

Table 1. Magnetic moment values obtained from the calculated DOS using Equation (1), the spin-polarized electron numbers of n_S↑ and n_S↓, and the electronic state difference Δn(E_F) at the Fermi level determined for the ferromagnetic Laves phases ErAl₂ and ErNi₂ along the <001> axis.

Compound	nS↑ (EF) (e−/eV)	nS↓ (EF) (e−/eV)	Δn(EF) (e−/eV)	${\mathit{\mu }}_{\mathit{S}\uparrow }$ (µB/f.u.)	${\mathit{\mu }}_{\mathit{S}\downarrow }$ (µB/f.u.)	${\mathit{\mu }}_{\mathit{T}}^{\mathit{D}\mathit{F}\mathit{T}+\mathit{U}}$ (µB/f.u.)	${\mathit{\mu }}_{\mathit{T}}^{\mathit{E}\mathit{x}\mathit{p}}$ (µB/f.u.)	Magnetization Axis	Reference
ErAl2	1.31	−2.24	−0.93	30.19	−25.80	4.40	4.40 4.75 †	hard	[40] [26]
ErNi2	0.78	−6.20	−5.42	36.46	−30.40	5.56	5.30 ‡ 6.23	easy	[27] [40]

^† Single crystal with T_C = 14 K [26]; ^‡ Single crystal with T_C = 7 K [27].

shows the total magnetic moment value obtained through DOS for the ErAl₂ and ErNi₂ Laves phases. To determine the net magnetic moment, we used the following expression:

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

(1)

where E₁ and E₂ denote the initial energies of the electronic states with up and down spins, respectively [39]. The saturated magnetic moment obtained was 4.40 μ_B/f.u. for ErAl₂ and 5.56 μ_B/f.u. for ErNi₂. From these magnetic moment values, the estimated saturation magnetization is approximately 111 Am²/kg for ErAl₂ and 109 Am²/kg for ErNi₂, both of which align with the experimental values at T = 2 K and μ₀H = 5 T reported in the scientific literature for polycrystalline samples [15,16].

For ErAl₂, the DFT values correspond to a spontaneous magnetic moment of zero internal magnetic field (4.20 µ_B/f.u.) along the <001> hard magnetization direction in single crystals [26,40]. The obtained value (4.40 µ_B/f.u. at T = 0 K) is consistent with the reported spontaneous magnetic moment (4.20 µ_B/f.u. at T = 4.2 K) by Purwins et al. [26,40], extrapolated from an intermediate magnetic field to the zero internal magnetic field (from 0 T to 8 T). At high magnetic fields (from 10 T to 14 T), the saturation magnetic moment rises to 5.3 µ_B/f.u. in a single crystal [26,40].

Detailed experimental magnetic measurements by Gignoux and Givord [27] for the ErNi₂ single crystal at T = 1.5 K reported a spontaneous magnetic moment of 5.0 µ_B/f.u. at μ₀H = 0 T along the <001> easy magnetization axis. The extrapolated magnetic moment is 5.3 µ_B/f.u. at zero internal magnetic field. Consequently, when the magnetic field increases to 13 T, the saturation magnetic moment rises slowly and nearly linearly to 5.9 µ_B/f.u. Our spontaneous magnetic moment calculated through DFT+U is 5.56 µ_B/f.u. at T = 0 K and agrees with the magnetic experiments along the <001> easy magnetization axis in ErNi₂.

Figure 4a,b illustrates the partial density of states (PDOS) for the d and f orbitals of both compounds, calculated without considering hybridization. Figure 4a shows that the shape of the d orbitals changes from almost symmetrical to slightly asymmetrical in their spin-polarized DOS profile, when the post-transition metal (Al) is replaced by the transition metal (Ni). There is a remarkable increase in electron populations per state that indicates how electrons at d orbitals became more important in magnetic behavior as the Al is substituted by the Ni element. The maximum n(E_F) value for the d orbitals increases from 0.38 e⁻/eV in ErAl₂ to 9.13 e⁻/eV in ErNi₂. Consequently, hybridization between the d and f orbitals occurs in the crystal structure of ErNi₂. In contrast, in ErAl₂ the Al atoms lack unpaired electrons in the d orbitals; therefore, the d and f orbitals are distinguished in the electronic structure, as depicted in Figure 2b and Figure 4b. This behavior results in a reduced contribution of electrons localized in the d orbitals of Al atoms to the net magnetic moment in the electronic structure of ErAl₂ compared to that of ErNi₂, as shown in

Table 2. Magnetic moment values of the s, p, d, and f orbitals determined from PDOS calculations for ErAl₂ and ErNi₂ Laves phases.

Laves Phase	μs (μB/f.u.)	μ p (μB/f.u.)	μd (μB/f.u.)	μf (μB/f.u.)	μP (μB/f.u.)
ErAl2	−0.005	−0.002	−0.001	4.940	4.932
ErNi2	−0.058	−0.249	1.737	4.701	6.131

. It is remarkable to note that for both alloys in the spin-up channels, the 4f electrons of rare earth elements are far from the Fermi level, located at −3.41 eV (ErAl₂) and −3.57 eV (ErNi₂). In contrast, the spin-down channels of the 4f bands are close to the Fermi level, and for both compounds they are located at −0.74 eV; see Figure 4b for more details. Figure 4a illustrates the 4d and 3d bands (with green lines) of Er atoms in ErAl₂, which are localized from −1.8 eV to the Fermi level, and the electronic configuration of Al atoms has no d-electrons, while the 3d bands (large peaks in purple lines) corresponding Ni atoms are the majority electronic states compared the states of 4d and 3d bands of Er atoms in ErNi₂, as illustrated in the Figure 4a. They are localized between −5 eV and 0 eV for both spin-up and spin-down branches in ErNi₂. The hybridization in ErNi₂ basically comes from 3d and 4f states. The contribution of d electrons is nearly negligible for ErAl₂, while 32% of the net magnetic moment in ErNi₂ arises from d electrons, as illustrated in Figure 2b and Figure 4a. For the ErAl₂ and ErNi₂ compounds, the f orbitals are the most populated, providing the most significant contribution to the net magnetic moment.

Table 2 illustrates the contribution of various orbitals to the net magnetic moment per formula unit (f.u.) derived from partial electronic density of states (PDOS) calculations. The magnetic moment value is negative for electrons localized in the s, p, and d orbitals in ErAl₂, indicating that the magnetic moment is oriented opposite to the applied field (which aligns with the spin-up band orientation). A similar situation occurs for ErNi_2, where the magnetic moments of the s and p electrons are also oriented against the applied field. On one hand, the high symmetry of the electronic structure of the s and p electrons results in a minimal contribution to the net magnetic moment. On the other hand, the asymmetry in the spin-polarized partial DOS of f-electrons in ErAl₂ is responsible for the elevated magnetic moment. The asymmetry in the partial DOS for ErNi₂, observed in both the d and f electrons, contributes to a substantial magnetic moment. For ErAl₂ the magnetic contribution of electrons at f orbitals in PDOS represent almost the 99.8% of total magnetic moment in TDOS. On the contrary, for ErNi₂ the contribution of electrons at d and f orbitals are approximately 25.8% and 69.7%, respectively, due to the rise of electronic population at d orbitals when the post-transition metal (Al) is changed by a transition one (Ni). The contribution of electrons at s and p orbitals is negligible (i.e., 0.8% and 3.7%, respectively) and opposite to the applied field. In RM₂ Laves phases when M is a transition metal, the magnetic behavior comes basically from d and f electrons [10,27].

3.2. Electronic Structure at Moderate Hydrostatic Pressures

Figure 5a–d shows the calculated density of states (DOS) for the ferromagnetic ground state of ErAl₂ and ErNi₂ Laves phases under external hydrostatic pressures ranging from 0 GPa to 1.0 GPa. First, it is essential to note that the s and p orbitals remain nearly symmetric across all hydrostatic pressures and exhibit low electronic occupation compared to the d and f orbitals (refer to Figure 6a–d for additional details on the partial density of states, PDOS). Conversely, in both compounds, the p electronic bands are more responsive to pressure than the s bands, shifting slightly toward higher energies while maintaining their symmetric behavior. The electronic population is redistributed per eV for the p and f electronic bands. In both ErAl₂ and ErNi₂, the s bands maintain their shape. In contrast, the p band in ErAl₂ splits into a double peak at P = 1.0 GPa, whereas in ErNi₂, a double peak is observed within the range of 0 GPa ≤ P ≤ 1.0 GPa. At P = 1.0 GPa, ErNi₂ exhibits a double peak for the spin-up p-band, while the spin-down p-band remains nearly a single peak. Notably, f orbitals are the most electronically populated within the studied pressure range. They shift far from the Fermi level while retaining their asymmetry in the DOS profile as the pressure approaches to 0.9 GPa in ErAl₂. Conversely, in ErNi₂, the f, d, and p orbitals start to disrupt the DOS symmetry at 1.0 GPa (with each spin-up band shifting to higher energy and the spin-down band remaining at lower energy), as shown in Figure 5d. This indicates the presence of Pauli paramagnetism, which stems from the shift of the spin-up band relative to the spin-down band. When the external hydrostatic pressure rises to 1 GPa, the spin-up 4f bands in ErAl₂ alloy move significantly away from the Fermi level compared to P = 0 GPa. They shift from −3.41 eV at 0 GPa to −3.95 eV at 1 GPa for spin-up bands and from −0.74 eV at 0 GPa to −1.55 eV at 1 GPa for spin-down bands, as illustrated in Figure 6b. Conversely, the 4f bands for the ErNi₂ alloy shifted closer to the Fermi level for both spin-up and spin-down branches at pressures up to 0.9 GPa. An exception occurs at an external pressure of 1 GPa, where the band splits into several others; see Figure 6d. Figure 6a shows that in ErAl₂, the 4d and 3d bands move closer to the Fermi level as the applied external pressures increase to 1 GPa, reaching −1.33 eV and −0.98 eV for the bottom of the spin-up and spin-down branches, respectively. The same occurs for both spin-up and spin-down Ni 3d bands (most electronic states) of ErNi₂; they move closer to the Fermi level when the applied external pressure increases from 0 GPa to 1 GPa, as shown in Figure 6c.

Figure 7a illustrates the relationship between the total magnetic moment and the applied hydrostatic pressure for the ErAl₂ and ErNi₂ Laves phases. Notably, up to 1.0 GPa, the calculated magnetic moment for the ferromagnetic ErAl₂ and ErNi₂ Laves phases remains nearly constant, with mean values of 4.33 μ_B/f.u. for ErAl₂ and 5.73 μ_B/f.u. for ErNi₂. Figure 7b illustrates how the contributions of electrons from the d and f orbitals to the net magnetic moment vary with pressure in both compounds. In ErAl₂, the electrons localized in the f orbitals account for a nearly constant in the entire net atomic magnetic moment across the full pressure range. The 4f magnetic moment for ErAl₂ is smoothly decreased when P is going from 0 GPa to 0.2 GPa. The contribution from electrons localized in the 3d orbitals decreases dramatically to zero at 0.1 GPa. As the external hydrostatic pressure increases from 0.1 to 1.0 GPa, the Ni 3d-electron bands do not contribute more the magnetic behavior in ErNi₂. As a result, the electrons in the f orbitals are the main contributors to the total magnetic moment in the ferromagnetic ErNi₂ Laves phase; the 4f magnetic moment is increased when P is 0.1 GPa. Finally, the large polarization of Er 4f electrons is not further pressure-dependent for both alloys.

The formation energy was calculated by employing the followed formulae:

$$E_f=E_T-{\sum }_{i=1}{n}_i{E}_i$$

(2)

where E_T is the total energy of the system, n_i is the number of atoms of the element “i” and E_i is the reference energy of the element “i”. The formation energy for the two studied Laves phases as a function of the applied hydrostatic pressure P, within the range of 0 GPa ≤ P ≤ 1.0 GPa, is illustrated in Figure 8. As indicated by the figure, the stability of the crystalline structures ErAl₂ and ErNi₂ remains unaffected by the external hydrostatic pressures applied. The formation energy, E_f, is approximately −1.04 × 10⁴ eV for ErAl₂ and −1.44 × 10⁴ eV for ErNi₂. The primitive cubic lattice parameter, obtained from geometric optimization, decreases from a₀ = 5.555 Å at 0 GPa to 5.526 Å at 1.0 GPa for ErAl₂. In comparison, the lattice parameter for ErNi₂ decreases from a₀ = 5.103 Å at 0 GPa to 5.085 Å as the pressure increases from 0 to 1.0 GPa. The relative compressive stress caused by this lattice reduction can be calculated using Δ = (V_P − V₀)/V₀ × 100%, where ΔV is the volume difference V_P − V₀. Δ reaches values of −1.56% and −1.01% at P = 1.0 GPa for ErAl₂ and ErNi₂, respectively.

Table 3. Unit cell volume (V_P) of the rhombohedral trigonal structure, bulk modulus B, the diagonal component σ_ij of the stress tensor, and compressive stress (Δ) as a function of the applied hydrostatic pressure for the ErAl₂ and ErNi₂ phases (0.0 GPa ≤ P ≤ 1.0 GPa).

Alloy	ErAl2							ErNi2
P (GPa)	a (Å)	dAl-Al (Å)	dEr-Al (Å)	VP (Å3)	Δ (%)	B (GPa)	σij (GPa)	a (Å)	dNi-Ni (Å)	dEr-Ni (Å)	VP (Å3)	Δ (%)	B (GPa)	σij (GPa)
0.0	5.5551	2.777	3.257	121.218	0.000	63.76	–0.00184	5.1030	2.551	2.991	93.949	0.000	128.13	–0.00631
0.1	5.5530	2.776	3.256	121.090	–0.105	64.53	–0.06960	5.1023	2.550	2.990	93.853	–0.101	128.56	–0.10767
0.2	5.5503	2.775	3.254	120.908	–0.255	64.87	–0.16420	5.1010	2.549	2.989	93.737	–0.225	129.90	–0.22623
0.3	5.5455	2.772	3.251	120.590	–0.517	65.76	–0.31888	5.0999	2.548	2.988	93.653	–0.314	129.89	–0.31054
0.4	5.5434	2.771	3.250	120.456	–0.628	65.96	–0.38991	5.0973	2.547	2.987	93.560	–0.413	130.11	–0.40776
0.5	5.5412	2.770	3.249	120.310	–0.748	66.14	–0.47108	5.0956	2.546	2.986	93.464	–0.516	130.82	–0.50732
0.6	5.5374	2.768	3.247	120.062	–0.953	66.98	–0.59076	5.0939	2.546	2.985	93.366	–0.620	131.39	–0.61002
0.7	5.5345	2.767	3.245	119.873	–1.108	67.08	–0.69311	5.0921	2.545	2.984	93.270	–0.722	131.91	–0.71256
0.8	5.5315	2.765	3.243	119.677	–1.270	67.56	–0.79821	5.0904	2.544	2.983	93.182	–0.816	132.08	–0.80705
0.9	5.5275	2.763	3.242	119.324	–1.561	67.96	–0.88224	5.0888	2.543	2.982	93.102	–0.901	132.88	–0.89113
1.0	5.5259	2.762	3.240	119.318	–1.566	68.04	–0.99389	5.0853	2.542	2.981	92.994	–1.015	133.12	–1.00668

summarizes the compressive stress values in the crystal structure up to 1.0 GPa and the bulk modulus B, which slightly increase with the hydrostatic pressure dependence and the stress tensor diagonal component σ_ij (i = j = 1, 2, 3). The mean value of ErNi₂ bulk modulus (<B> ≈ 131 GPa) is almost twice as large as that of ErAl₂ bulk moduli (<B> ≈ 66 GPa); see Table 3. As previously mentioned, based on pressure spin-polarization DFT calculations, the cubic crystalline structures remain stable within the applied range of compressive stress.

In ErAl₂ and ErNi₂, the d_Al-Al and d_Ni-Ni interatomic distances (between two Al or Ni atoms) decrease by 0.54% and 0.35% under an applied pressure of 1 GPa (see Table 3). Concurrently, the interatomic distances d_Er-Al and d_Er-Ni (between Er atoms and Al or Ni atoms) diminish by 0.52% and 0.33%, respectively. Note that ErAl₂ exhibits greater sensitivity to pressure than the ErNi₂ alloy. The interatomic distances change very little with the applied pressure for both Laves phases. This maintains the same ferromagnetic environment for each atom, with only the electronic states adjusting due to the applied hydrostatic pressure. Consequently, the values of the saturation spontaneous magnetic moment remain nearly constant for each Laves phase.

3.3. Determination of Electronic Coefficient in Specific Heat Capacity

From metals theory, in the absence of hydrostatic pressure, the Fermi energy E_F at T = 0 K can be calculated as

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

(3)

where N/V represents the electron density and m_e is the mass of the electron. If one atom contributes one electron to the Fermi gas in the solid, the ErAl₂ and ErNi₂ compounds exhibit electron densities N/V of 1.687 × 10²⁸ m³ and 2.208 × 10²⁸ m³, respectively (assuming mass densities of 6.20 × 10³ kg/m³ [13,41] and 10.44 × 10³ kg/m³ [42], along with molar masses of 221.223 g mol⁻¹ and 216.468 g mol⁻¹, respectively). Our calculations of the Fermi energy E_F and the temperature $T_F=E_F/k_B$ at T = 0 K yield 2.401 eV and 2.788 × 10⁴ K for ErAl_2, and 2.873 eV and 3.336 × 10⁴ K for ErNi₂. The obtained values align with those reported for other pure metals such as Al (18.1 × 10²⁸ m³, 11.7 eV, 13.6 × 10⁴ K), Au (5.9 × 10²⁸ m³, 5.55 eV, 6.43 × 10⁴ K), and Na (2.65 × 10²⁸ m³, 3.26 eV, 3.78 × 10⁴ K), among others [43]. Thus, the electronic or Sommerfeld coefficient can be calculated using the following formula:

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

(4)

where N_A is the Avogadro constant. The obtained γ_e values from Equation (4) are 1.470 × 10⁻³ J mol⁻¹ K⁻² for ErAl₂ and 1.229 × 10⁻³ J mol⁻¹ K⁻² for ErNi₂.

Based on the DFT simulations, we can also calculate the γ_e coefficient for metals and alloys using the Einstein–Debye model:

$$c_p\left(T\right)={\gamma }_e T+{\beta }_{ph} T^3$$

(5)

where $c_{p, e}={\gamma }_e T$ and $c_{p, ph}={\beta }_{ph} T^3$ denote the electronic and phonon contributions to the specific heat capacity at constant pressure.

In the low-temperature limit (that is, far from the Debye temperature, i.e., T << T_D), the electronic coefficient γ_e in the Sommerfeld approximation can be correlated with the electronic density of states Δn(E_F) at the Fermi level [39] using the expression:

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

(6)

where k_B is the Boltzmann constant. This expression remains valid within the framework of hydrostatic pressure. Figure 9 presents the results obtained from DFT calculations using Equation (6) for the electronic specific heat coefficient of both alloys. At P = 0 GPa, the γ_e coefficient for ErAl₂ is 2.1 × 10 ³ J mol⁻¹ K⁻², which is in reasonable agreement with previous values calculated from the electron gas model in metals (i.e., 1.470 × 10⁻³ J mol⁻¹ K⁻²). In contrast, the γ_e coefficient for ErNi₂ at P = 0 GPa reaches 13.0 × 10⁻³ J mol⁻¹ K⁻², almost 10 times the value obtained from the electron gas model (i.e., 1.229 × 10⁻³ J mol⁻¹ K⁻²). As shown in Figure 9, the γ_e coefficient remains nearly constant for both alloys up to a hydrostatic pressure of 1.0 GPa.

4. Conclusions

Through DFT calculations utilizing the revised Perdew–Burke–Ernzerhof (RPBE) exchange–correlation functional within a Generalized Gradient Approximation (GGA) framework and incorporating Hubbard U correction, we investigated the effects of hydrostatic pressure of up to 1.0 GPa on the stability of the crystal structure, as well as on the electronic and magnetic properties of the cubic Laves phases ErAl₂ and ErNi₂. The main findings can be summarized as follows: (a) The net spontaneous magnetic moment per formula unit remained constant throughout the entire pressure range considered, with values of 4.40 μ_B/f.u. for ErAl₂ along the <001> hard magnetization axis and 5.56 μ_B/f.u. for ErNi₂ along the <001> easy magnetization axis. (b) The cubic MgCu₂-type crystal structure retains its stability up to 1 GPa. At this pressure, the compressive stress affecting the crystalline structure is −1.56% for ErAl₂ and −1.01% for ErNi₂. The interatomic distances change very little with the applied pressure, which keeps virtually the same ferromagnetic environment for each atom, while a rearrangement of the electronic states (mainly in the f, d, and p orbitals) occurs due to the applied hydrostatic pressure. (c) The electronic contribution to the specific heat aligns in order of magnitude with the anticipated data from the metal gas model and remains stable up to 1.0 GPa. (d) The d-f hybridization lowers electronic states and diminishes the spontaneous magnetic moments.