Anomalous Ferromagnetic Phase in the Gd_1−xEr_xB₄ Series: Crystal Growth, Thermal, and Magnetic Properties

Abstract

Rare-earth tetraborides RB₄ are of great interest due to the occurrence of geometric magnetic frustration and corresponding unusual magnetic properties. While the Gd³⁺ spins in GdB₄ align along the ab plane, Er³⁺ spins in the isomorphic ErB₄ are confined to the c–axis. The magnetization in the latter exhibits a plateau at the midpoint of the saturation magnetization. Therefore, solid solutions of (Gd, Er)B₄ provide an excellent playground for exploring the intricate magnetic behavior in these compounds. Single crystals of Gd_1−xEr_xB₄ (x = 0, 0.2, and 0.4) were grown in aluminum flux. X-ray diffraction scans revealed single-phase materials, and a drop in the unit cell volume with increasing Er content, suggesting the partial substitution of Er at the Gd sites. Heat capacity measurements indicated a systematic decrease of the Néel temperature (T_N) with increasing Er content. The effective magnetic moment determined from the magnetization measurement agreed with the calculated free ion values for Gd³⁺ and Er³⁺, providing further evidence for the successful substitution of Er for Gd. The partial substitution resulted in an anomalous ferromagnetic phase below T_N, exhibiting significant anisotropy, predominantly along the c-axis. This intriguing behavior merits further studies of the magnetism in the Gd_1−xEr_xB₄ borides.

1. Introduction

Motivated by their interesting magnetic properties, the rare-earth tetraborides with general formula RB₄ (R = rare earth) have been studied for many years [1,2,3]. These compounds are metallic conductors and show antiferromagnetic (AF) ordering, except for R = Pr, which is ferromagnetic (FM) [2]. The indirect coupling between the magnetic ions is of the Ruderman–Kittel–Kasuya–Yosida type (RKKY) [2]. The crystal structure is tetragonal belonging to the symmetry group P4/mbm. Due to the nature of the crystal structure, these compounds exhibit strongly anisotropic magnetic and electrical properties [4,5].

The magnetic sublattice of R ions in RB₄ consists of 2d orthogonal R–R dimers in the ab-plane, forming squares and triangles [6]. The bond length between the rare earth nearest-neighbor dimmer (NN) is very close to the next-nearest neighbor (NNN). Therefore, one can presume that the corresponding magnetic interactions J₁ and J₂, as shown in Figure 1, are also close to each other. If the magnetic interaction between the rare-earth ions is antiferromagnetic, it is likely that the system should exhibit geometrically frustrated magnetic interactions, consistently with the theoretical approach described in the Shastry–Sutherland lattice (SSL) [6,7,8].

There are some noteworthy differences between GdB₄ and ErB₄. All heavy RB₄ (R = Tb, Dy, Ho, Er, Tm) display strong Ising-like anisotropies, resulting in the rare-earth magnetic moments being oriented preferably along the c-axis or along the ab-plane. The Er³⁺ ions in ErB₄ have a large total angular moment J = 15/2. It shows an antiferromagnetic transition at T_N = 15.4 K, with an easy axis orientation along the c direction [9], and the magnetic moment was previously determined to be 8.2 ± 0.6 ${\mu }_B$ [10]. The M × H curve exhibits a plateau region at the midpoint of the saturation magnetization (M_S) value. In this plateau phase, half of the Er³⁺ magnetic moments flip in the field direction, and it is suggestive of the formation of ferromagnetic and antiferromagnetic stripe structures, like domain [11]. The attempt to theoretically describe the system in terms of an effective spin-1/2 Shastry–Sutherland model under strong Ising anisotropy could show the M_S/3 plateau, but it was not able to reproduce the appearance of the M_S/2 phase. The inclusion of interactions of longer ranges seems to be a necessary ingredient to reproduce the experimentally observed results [12].

In contrast, the Gd³⁺ magnetic moments in GdB₄ align perpendicularly to the c-axis, i.e., along the basal ab-plane [13]. No plateau is observed in M × H curves, and the antiferromagnetic ordering temperature for GdB₄ is T_N = 42 K [2,14]. Spherical neutron polarimetry revealed that the magnetic spins order non-collinearly (see Figure 1), in a structure with the Shubnikov magnetic space group P4/m’b’m’. The magnetic moment of Gd³⁺ was determined to be 7.14 ± 0.17 ${\mu }_B$, quite close to the free ion value [13].

Many recent papers have focused on the potential applications of rare earth borides. For example, the strong geometrical frustration and competing exchange interactions in these materials indicate their promising application as materials with enhanced magnetocaloric effect [15]. Composite ceramics of RB₆ and RB₄ have been proposed to be used as a new type of high-performance electromagnetic wave-absorbing material that could help to diminish interference and electromagnetic pollution [16]. Also, their good thermal conductivity, resistance to oxidation, and hardness make them suitable for use as coatings for cutting tools, turbine blades, and other high-wear components, as well as for high-temperature applications [17].

The perturbation of the intricate balance between competing exchange interactions within a geometrically frustrated magnetic system can generate novel electronic and magnetic states, resulting from the interaction between frustrated spins and lattice, orbital, and charge degrees of freedom [18]. Hence, considering the rich phase diagram exhibited by rare-earth borides, investigations into these materials, as well as the effects produced by different kinds of doping, remain subjects of significant interest [19]. Therefore, the goal of this work is to probe the thermal and magnetic properties of Gd_1−xEr_xB₄ (x = 0–0.4).

Single crystals were grown from aluminum flux, and analyzed by means of Laue X-ray images, and powder X-ray diffraction (XRD). Measurements of magnetization [M(T, H)] and heat capacity [C_p(T, H)] were carried out with magnetic fields applied parallel and perpendicular to the c-axis. These measurements permitted monitoring the fast evolution of the magnetic properties upon the partial substitution of Er for Gd in GdB₄. We have observed an anomalous and highly anisotropic ferromagnetic phase within the geometrically frustrated magnetic system of Gd_1−xEr_xB₄.

2. Materials and Methods

Stoichiometric amounts of high purity Gd (Merch 99.9%), Er (Merch 99.9%), and B (Alfa Aesar 99.99%) corresponding to the Gd_1−xEr_xB₄ (x = 0.2, and 0.4) compositions and high-purity aluminum shots (Alfa Aesar 99.99%) were placed in 70 mL alumina crucibles, in 5–95% amounts by weight, respectively. The alumina crucibles were loaded on a vertical tube furnace under flowing ultra-high pure argon gas, heated and maintained at 1500 °C for one hour, cooled slowly to 1000 °C, and fast-cooled to ambient temperature by turning the furnace off. The Gd_1−xEr_xB₄ crystals were separated from the flux by dissolving the aluminum in a saturated NaOH solution. Laue photographs were taken by back-reflection methodology to assess the quality of the single crystals and determine the crystallographic orientation [20]. The open source software QLaue (Beta) was used to simulate the diffraction spots [21]. GdB₄ single crystals were previously grown and characterized using the same methodology, as detailed elsewhere [22].

Ambient temperature X-ray powder diffraction was carried out on a few crystals crushed in an agate mortar, using the CuK_α radiation of a D-8 Discovery diffractometer in the 15 ≤ 2θ ≤ 120° range. The characterization of the GdB₄ single crystal is described elsewhere [22]. The grown GdB₄ crystals had plate-like polyhedral morphology; the largest dimension could reach ≈ 1.5 mm, and the larger facets corresponded to the (110) and (001) planes.

Specific heat C_p(T, H) and magnetization M(T, H) measurements of the Gd_1−xEr_xB₄ crystals were obtained with a physical property measurement system (PPMS) from Quantum Design. The C_p(T, H) data of x = 0, 0.2, and 0.4 compositions were collected in the 2–100 K temperature range in applied magnetic fields up to 9 T applied both parallel and perpendicular to the c-direction. The M(T, H) data of x = 0.2 and 0.4 samples were collected in the 2–300 K temperature range in applied magnetic fields up to 9 T applied both parallel and perpendicular to the c-direction. The magnetic properties of GdB₄ have been previously investigated and reported in Ref. [22].

3. Results and Discussion

3.1. Samples and X-ray Diffraction

The flux growth method for the synthesis of Gd_1−xEr_xB₄ (x = 0.2 and 0.4) crystals yielded platelets with typical dimensions of ≈ 1.5 mm × 1.5 mm × 0.5 mm, as shown in Figure 2. The X-ray Laue images displayed in Figure 3 and Figure 4 indicate that the larger facets are perpendicular to the crystallographic c-direction. The orientation of the crystals was determined from the Laue images using the QLaue software. The lack of distortion or smearing in the diffraction spots of Figure 3 and Figure 4 is suggestive of high crystallinity, with the absence of defects or twining. The crystal morphology was suitable for assembling and measuring the physical properties parallel and perpendicular to the c-axis, as discussed in the following sections.

Structural refinement of the Gd_1−xEr_xB₄ (x = 0.2, and 0.4) crystals were carried out using the General Structure Analysis System - II (GSAS-II) software based on the Rietveld methodology [23], and the results are shown in Figure 5. The XRD patterns do not show the presence of additional phases, and the lattice parameters resulting from the refinement are listed in

Table 1. Crystal structure data and refinement for Gd_1−xEr_xB₄ (x = 0, 0.2, 0.4, and 1.0).

x	$\mathit{a}, \mathit{b} \left(\mathit{\AA }\right)$	$\mathit{c} \left(\mathit{\AA }\right)$	${\mathit{\chi }}^2$
0 1	7.1421 (2)	4.0467 (2)	1.7
0.2	7.1298 (2)	4.0377 (2)	2.9
0.4	7.1133 (7)	4.0264 (5)	4.1
1.0 2	7.071 (3)	4.000 (1)	−

¹ Data from Ref. [22]. ² Data from Ref. [10].

, alongside our previously reported data for GdB₄ [22] and data from the literature for ErB₄ [10]. These compounds crystallize in a tetragonal structure at room temperature, P4/mbm (No. 127). The (001) Bragg reflection shifts towards higher 2θ values upon the partial substitution of Er for Gd, as shown in the inset of Figure 5b, a result consistent with the smaller ionic radius of Er [24], and, in turn, a drop of the unit cell volume.

Given the different magnetic structures of GdB₄ and ErB₄, the progressive substitution of Er for Gd in the Gd_1−xEr_xB₄ is quite likely to affect the magnetic and thermal properties. We monitored these changes by measuring M(T, H) and C_p(T) respectively.

3.2. Specific Heat

The zero-field specific heat C_p(T) measurements were conducted on all three samples of Gd_1−xEr_xB₄ (x = 0, 0.2, and 0.4) for the present investigation, and the obtained C_p(T) data are shown in Figure 6a. The C_p(T) curve for the undoped GdB₄ clearly shows a pronounced peak at the Néel temperature T_N = 41.8 K, indicating a transition from a paramagnetic (PM) to an antiferromagnetic (AFM) phase. Upon the partial substitution of Er for Gd, the magnetic ordering feature shifts to lower temperatures, reaching 31.8 and 26.8 K for Gd_0.8Er_0.2B₄ and Gd_0.6Er_0.4B₄, respectively. A small bump centered near T ≈ 10 K is observed in GdB₄, a feature frequently observed in other lanthanide compounds and attributed to the Schottky contribution to the heat capacity [25]. The occurrence of this feature remains in the x = 0.2 sample but is much more suppressed in the x = 0.4 crystal. Previous studies have shown that the Schottky contribution to the heat capacity in ErB₄ occurs at higher temperatures [26].

At low temperatures, the heat capacity can be approximated by the relation C_p = C_el + C_latt + C_sch + C_m, where C_el, C_latt, C_sch, and C_m are the contributions due to the electron system, phonon, Schottky anomaly, and magnetic subsystem, respectively. To probe the effect of the partial change of Er for Gd in the sample magnetism, the magnetic contribution C_m for each composition was estimated, as depicted in Figure 7b. The phonon contribution C_latt was estimated using the method described by Stout and Catalano [27], which relies on measuring the heat capacity C_p of a nonmagnetic isomorph, which in this case was YB₄.

The heat capacity of YB₄ at low temperatures can be expressed as C_p:YB4 = C_el + C_latt = aT + bT³, where a = 12 × 10⁻⁴ J/mol K² and b = 2.1 × 10⁻⁵ J/mol K⁴, values obtained by fitting. Comparable values of a and b have been reported for GdB₄ (a = 5.96 × 10⁻⁴ J/mol K² and b = 4.89 × 10⁻⁵ J/mol K⁴) and ErB₄ (a = 8.46 × 10⁻⁴ J/mol K² and b = 6.82 × 10⁻⁵ J/mol K⁴) in previous studies [25,26]. Therefore, it is reasonable to assume that the contributions of C_el and C_latt to the heat capacity of the Gd_1−xEr_xB₄ series are like those of YB₄. Consequently, the specific heat of YB₄ was subtracted from the specific heat of the Gd_1−xEr_xB₄ series, enabling the determination of the specific heat associated with the Schottky and magnetic anomalies.

The contribution to the specific heat due to the Schottky anomaly C_sch for a system with n levels, separated by energies ${\epsilon }_n$ and with degeneracy $g_n$, is given by

$$C_{sch}=R\frac{g_1{\left(\frac{{\epsilon }_1}{T}\right)}^2{e}^{-\frac{{\epsilon }_1}{T}}+g_2{\left(\frac{{\epsilon }_2}{T}\right)}^2{e}^{-\frac{{\epsilon }_2}{T}}+\cdots }{g_0+g_1{e}^{-\frac{{\epsilon }_1}{T}}+g_2{e}^{-\frac{{\epsilon }_2}{T}}+\cdots }-R{\left(\frac{g_1\left(\frac{{\epsilon }_1}{T}\right)e^{-\frac{{\epsilon }_1}{T}}+g_2\left(\frac{{\epsilon }_2}{T}\right)e^{-\frac{{\epsilon }_2}{T}}+\cdots }{g_0+g_1{e}^{-\frac{{\epsilon }_1}{T}}+g_2{e}^{-\frac{{\epsilon }_2}{T}}+\cdots }\right)}^2,$$

(1)

where R is the ideal gas constant and ${\epsilon }_n$ is given in units of kelvin [28]. The C_sch curve for all samples was obtained by fitting the data below T_N using the equation C = AT ^α + C_sch, where AT ^α represents the magnetic specific heat contribution, with α = 3 for AFM systems. As initial parameters for the Schottky anomaly, the data from Ref. [25] for Gd³⁺ were used, where $g_0=2$, $g_1=2$, $g_2=4$, ${\epsilon }_1=30 \mathrm{K}$, and ${\epsilon }_2=75 \mathrm{K}$. For Er³⁺, the data from Ref. [26] are $g_0=2$, $g_1=4$, $g_2=6$, $g_3=4$, ${\epsilon }_1=85 \mathrm{K}$, ${\epsilon }_2=240 \mathrm{K}$, and ${\epsilon }_3=700 \mathrm{K}$. For the fitting procedure, only ${\epsilon }_1$ and ${\epsilon }_2$ for Gd³⁺ were free parameters. The parameters for Er³⁺ were kept fixed since initial fits indicated their values did not vary significantly. Thus, C_m was obtained by subtracting the fitted C_sch, considering the proportional contribution due to the Gd³⁺ and Er³⁺.

The different contributions to the total specific heat are shown in Figure 7a, for the x = 0.4 sample. After isolating the C_m curve, its temperature dependence was investigated by fitting to the equation $A{T}^{\alpha }$ for temperatures up to 0.7T_N, as shown in Figure 7b. For GdB₄, α = 3, as expected for antiferromagnetic systems. The exponent α decreases to 2.6 and 2.3 for the samples with x = 0.2 and 0.4, respectively, approaching the 1.5 value, expected for ferromagnetic systems. This is an indication of a change in the system’s ordering, due to the possible competition between magnetic anisotropies.

In order to probe the competition between magnetic anisotropies, we have performed measurements of C_p(T, H) in magnetic fields H up to 9 T. Shown in Figure 6b is the effect of a magnetic field H = 5 T applied along the two different crystallographic directions in Gd_0.6Er_0.4B₄, resulting in a drop in the temperature of the C_p peak at T_N, and a significant change in morphology for H//c. In general, for fields in the 0 ≤ H ≤ +9 T range, the value of T_N drops by ~15% and ~7% for fields applied parallel and perpendicular to the c-axis, respectively. In contrast, C_p(T) for the Gd_0.8Er_0.2B₄ is nearly insensitive to the magnetic field, with the value of T_N dropping but staying within 3% of the zero-field value, for both orientations.

In addition to the main feature at T_N, the C_p(T) data for Gd_0.6Er_0.4B_4, shown in Figure 6b, presents a second feature at low temperatures, centered near 10 K, when a 5 T magnetic field is applied parallel to the c-axis. This peak, observed only in Gd_0.6Er_0.4B₄, is possibly due to a metamagnetic transition, occurring exclusively along the c-axis, as suggested by the magnetization data.

3.3. Magnetization

The temperature-dependent magnetic susceptibility curves χ = (M/H) × T for Gd_1−xEr_xB₄ (x = 0.2 and 0.4), taken in 0.5 T magnetic fields parallel or perpendicular to the c-axis, are shown in Figure 8. The maximum value of χ occurs approximately at the same T_N determined from the C_p(T) measurements of Figure 6. The value of T_N was taken from the minimum of the second derivatives $d^2\chi /d{T}^2$. Consistently with the C_p(T) data, T_N drops with the Er concentration, as shown in the inset of Figure 8b.

For both x = 0.2 and x = 0.4 samples, χ (T) exhibits anisotropic behavior up to ~150 K, a temperature considerably higher than T_N. In contrast, the undoped GdB₄ displays anisotropic χ (T) behavior only below T_N. Notably, the partial substitution of 20% Er seems to be sufficient to induce this anisotropic behavior, reminiscent of the behavior in ErB₄ [9]. To gauge the magnitude of the anisotropy at 4 K, we calculated the ratio of the susceptibilities perpendicular and parallel to the c-axis and obtained 2.3 and 2.8 for x = 0.2 and 0.4, respectively. On the other hand, the susceptibilities parallel to the c-axis are higher near T_N, exhibiting a magnitude of 1.7 and 2.8 times greater than the perpendicular susceptibility for the x = 0.2 and x = 0.4 samples, respectively. The magnetic susceptibility data for Gd_0.8Er_0.2B₄ show that the substitution of 20% of Er favors the c-axis for spins alignment since χ tends to lower values with decreasing temperature, while the perpendicular susceptibility shows a small temperature dependence.

As shown in the inset of Figure 8a for Gd_0.8Er_0.2B₄, the magnetic susceptibility ($\chi$) in the PM region follows the Curie–Weiss law [29]

$$\chi =\frac{C}{T-{\theta }_{\mathrm{CW}}},$$

(2)

where ${\theta }_{\mathrm{CW}}$ is the Curie–Weiss constant, which is usually associated with the magnetic interactions between PM ions, expressed in units of temperature, and $C$ is the Curie constant, given by

$$C=\frac{N_a{\mu }_{\mathrm{eff}}^2}{3k_B}.$$

(3)

Here, $N_a$ is the Avogadro’s number, $k_B$ the Boltzmann constant, and ${\mu }_{\mathrm{eff}}$ the effective magnetic moment, calculated from

$${\mu }_{\mathrm{eff}}=g_J{\left[J\left(J+1\right)\right]}^{1/2}{\mu }_B,$$

(4)

where $g_J$ is the Landé g-factor, $J$ the total quantum number, and ${\mu }_B$ the Bohr magneton. The theoretical values of ${\mu }_{\mathrm{eff}}$ calculated for Gd³⁺ and Er³⁺ are 7.94 and 9.58 ${\mu }_B$, respectively.

For the Gd_1−xEr_xB₄ solid solutions, the theoretical values of ${\mu }_{\mathrm{eff}}$ can be approximately obtained from

$${\mu }_{\mathrm{eff}}={\left[\left(1-x\right){\mu }_{{\mathrm{eff}}_{\mathrm{Gd}}}^2+x{\mu }_{{\mathrm{eff}}_{\mathrm{Er}}}^2\right]}^{1/2}.$$

(5)

On the other hand, the experimental values of ${\mu }_{\mathrm{eff}}$ can be obtained by fitting the $\left(1/\chi \right)$ vs. $T$ data to the expression

$$\frac{1}{\chi }=\frac{8\left(T-{\theta }_{\mathrm{CW}}\right)}{{\mu }_{\mathrm{eff}}^2},$$

(6)

resulting from combining Equations (2) and (3), with $N_a$ = 6.022·10²³ mol⁻¹, $k_B$ = $1.381·{10}^{-16}$ erg/K, and ${\mu }_B$ = 9.274·10⁻²¹ emu, as shown in the inset of Figure 8a. The experimental values of ${\mu }_{\mathrm{eff}}$ obtained from fits of the experimental data to Equation (6) at temperatures above 100 K, are displayed in

Table 2. Calculated and experimental values of effective magnetic moment (${\mu }_{\mathrm{eff}}$ ) and Curie–Weiss constant (${\theta }_{\mathrm{CW}}$ ) for Gd_1−xEr_xB₄ (x = 0, 0.2, 0.4, and 1.0). ${\mu }_{\mathrm{eff}}^{\mathrm{calc}}$ values are calculated using Equation (5) and ${\mu }_{\mathrm{eff}}^{\exp }$ values are obtained from the fittings of $(1/\chi )$ vs. $T$ data to Equation (6).

x	${\mathit{\mu }}_{\mathit{e}\mathit{f}\mathit{f}}^{\mathit{c}\mathit{a}\mathit{l}\mathit{c}} \left({\mathit{\mu }}_{\mathit{B}}\right)$	${\mathit{\mu }}_{\mathit{e}\mathit{f}\mathit{f}}^{\mathit{e}\mathit{x}\mathit{p}} \left({\mathit{\mu }}_{\mathit{B}}\right)$ $(\mathit{H}\parallel \mathit{c})$	${\mathit{\theta }}_{\mathit{C}\mathit{W}} \left(\mathit{K}\right)$ $(\mathit{H}\parallel \mathit{c})$	${\mathit{\mu }}_{\mathit{e}\mathit{f}\mathit{f}}^{\mathit{e}\mathit{x}\mathit{p}} \left({\mathit{\mu }}_{\mathit{B}}\right)$ $(\mathit{H}\perp \mathit{c})$	${\mathit{\theta }}_{\mathit{C}\mathit{W}} \left(\mathit{K}\right)$ $(\mathit{H}\perp \mathit{c})$
0	7.94	7.98	−70	7.94	−66
0.2	8.29	8.20	−39	8.26	−57
0.4	8.63	8.34	−15	8.72	−48
1.0	9.58	9.27 *	+11.24 *	9.50 *	−23.26 *

* Data from Ref. [30].

. The ${\mu }_{\mathrm{eff}}$ values obtained from the $\chi \left(T\right)$ data with field along the two crystallographic directions are consistent with the calculated ${\mu }_{\mathrm{eff}}$ values for the Gd³⁺ and Er³⁺ free ions, providing additional evidence for the effective partial substitution of Er for Gd within this series.

The data displayed in Table 2 indicate that the effective magnetic moments ${\mu }_{\mathrm{eff}}^{\exp }$ for GdB₄ and Gd_0.8Er_0.2B₄ are only slightly different for the two orientations of the magnetic field, suggesting that the magnetic anisotropy in these two compositions is very small. Upon normalization with respect to parameters associated with the c-axis, the difference between these values amounts to 0.5% and 0.7% for x = 0 and x = 0.2, respectively. On the other hand, there is a marked difference of 4.6% between the two values for the x = 0.4 composition, a value close to the increased magnetic anisotropy in the Gd_0.6Er_0.4B₄ crystal.

The magnetic anisotropic behavior of χ(T) is also noted in the Curie–Weiss constant for the x = 0.2 and 0.4 samples. The difference between ${\theta }_{\mathrm{CW}}$ values obtained along directions parallel and perpendicular to the c-axis also increase with x. These differences are 5.7%, 46%, and 220% for crystals with x = 0, 0.2, and 0.4, respectively.

The occurrence of appreciable magnetic anisotropy in the Gd_1−xEr_xB₄ series is also noticeable in the isothermal magnetization curves for x = 0.2 and 0.4 samples, as shown in Figure 9, provided that the magnitudes of magnetization differ significantly when measured along two distinct crystallographic directions. Also, a careful examination of Figure 9b clearly shows an increase in anisotropy with the Er content.

Although GdB₄ and ErB₄ are antiferromagnetic systems that exhibit magnetization curves without hysteresis, the x = 0.2 and 0.4 compositions show anomalous remnant magnetization behavior. As shown in Figure 9b, these samples display an appreciable coercive field at 5 K, with H_C values of ~0.46 T and ~1.53 T for x = 0.2 and 0.4, respectively. This anomaly is also significantly anisotropic, i.e., it is more pronounced for H applied along the c-axis. In the x = 0.2 sample, a minor hysteresis is observed along the direction perpendicular to the c-axis, with an H_C value of ~0.025 T. In addition, for the x = 0.4 sample, the coercive field and remnant magnetization are negligibly small along the (001) plane.

As the applied magnetic field along the c-axis increases, Gd_0.6Er_0.4B₄ displays two magnetic transitions for H ≤ 9 T, as shown in Figure 9a. The first (near 4.5 T) is suggestive of a metamagnetic transition, characterized by a magnetization plateau state occurring at M/M_S = ½, where M_S is the saturation magnetization of the Er³⁺ ions. The second transition corresponds to the full alignment of the Er³⁺ ions along the c-axis. The theoretical value of the saturation magnetization per Er³⁺ ion is $M_S=gJ{\mu }_B=9 {\mu }_B$. Based on this value, the saturation magnetization for the Er³⁺ ions in the x = 0.4 sample corresponds to 3.6 ${\mu }_B$.

Immediately after a magnetic field of 5.6 T, the transition to the total magnetization of the system reaches a value close to 3.46 ${\mu }_B$. By subtracting the contribution of the Gd³⁺ ions, the experimental saturation magnetization corresponding to the Er³⁺ ions is 94% of the theoretical value. In this procedure, as a first approximation, we assume that the Gd³⁺ ions contribute linearly to the system’s magnetization with H, corresponding to the initial slope of the virgin magnetization curve up to the considered magnetic field of 5.6 T.

Only one field-induced transition is observed for the Gd_0.8Er_0.2B₄ composition, as shown in Figure 9a. Using the same procedure applied for the x = 0.4 composition to subtract the contribution of the Gd³⁺ ions, the magnetization attributed to the Er³⁺ ions amounts to 99% of the theoretical value of the expected saturation magnetization. These results reveal that the transition to the plateau state only takes place in the x = 0.4 sample, while the transition to the field-induced paramagnetic state, corresponding to the full alignment of the magnetic moments of Er³⁺, occurs for both x = 0.2 and x = 0.4 compositions.

In contrast, M × H curves with a field applied perpendicular to the c-axis do not show field-induced transitions in fields up to 9 T, as shown in Figure 9. This finding is consistent with the specific heat data of Figure 6b, where a metamagnetic transition is detected as a second peak occurring at low temperatures for a 5 T magnetic field also applied along the c-axis. When the same magnitude of a magnetic field is applied perpendicular to the c-axis, only a single maximum is observed in the C_p × T curve, corresponding to the transition to the AF state. Similar findings are observed for magnetic fields up to 9 T applied perpendicular to the c-axis. According to studies conducted in ErB₄ [9], two field-induced transitions occur for fields applied parallel and perpendicular to the c-axis. At 1.5 K, the plateau state is maintained between 2 and 4 T along the c-axis and between 11.5 and 13.0 T along the a-axis. Therefore, if these magnetic transitions exist along the direction perpendicular to the c-axis, they were not observed in our lightly Er-doped materials, as our experimental measurements were conducted up to a maximum applied magnetic field of 9 T. When comparing the M × H curves along the c-axis for Gd_0.6Er_0.4B₄ with pure ErB₄, the transition to the plateau state in the x = 0.4 sample occurs at relatively higher magnetic fields, and this state is maintained within a narrow range of 4.5 to 5.3 T.

It is interesting to note that GdB₄ and TbB₄ are the only members of the RB₄ family that have the easy axis along the ab-plane. In the case of GdB_4, Gd’s 4f shell is half filled, with L = 0, so the ions are in the s-state, and the anisotropy in the ordered phase comes mainly from the exchange interaction [31], which is small. It is suggested that this small magnetic anisotropy is the reason why no plateaus are observed in the M × H curves [32]. Therefore, doping with Er seems to be a convenient way to induce anisotropy in the magnetic properties of GdB₄. Our findings suggest that 40% of the substitution of Er for Gd is already enough to provoke the appearance of a plateau phase in the M × H curves.

4. Conclusions

In conclusion, we carried out a study of the thermal and magnetic properties of flux-grown Gd_1−xEr_xB₄ single crystals. This study revealed detailed magnetic transitions in both x = 0.2 and 0.4 compositions, including the full alignment of Er³⁺ magnetic moments for the magnetic field applied along the c-axis, and a metamagnetic transition corresponding to a plateau phase observed in the x = 0.4 sample.

While GdB4 and ErB4 compounds exhibit antiferromagnetic behavior with reversible magnetization isotherms and no magnetic hysteresis, the partial substitution of Er for Gd in the GdB4 lattice induced an anomalous ferromagnetic phase below the ordering temperature TN of the materials. This ferromagnetic phase exhibited significant anisotropy, with a pronounced manifestation along the c-axis. It is worth noting that the c-axis and the [110] direction correspond to the easy magnetization axes of ErB4 and GdB4 compounds, respectively.

Despite the lower Er content in the Gd_1−xEr_xB₄ (x = 0.2 and 0.4) samples, we observed moderately high values for the coercive field and remnant magnetization along the c-axis, which coincides with the easy axis for ErB₄. These intriguing features are likely due to the competing anisotropies. To further elucidate the origins of these interesting behavior, a comprehensive investigation of the physical properties of these single crystals is currently underway.