Anomalous Ferromagnetic Phase in the Gd 1 − x Er x B 4 Series: Crystal Growth, Thermal, and Magnetic Properties

: Rare-earth tetraborides R B 4 are of great interest due to the occurrence of geometric magnetic frustration and corresponding unusual magnetic properties. While the Gd 3+ spins in GdB 4 align along the ab plane, Er 3+ spins in the isomorphic ErB 4 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)B 4 provide an excellent playground for exploring the intricate magnetic behavior in these compounds. Single crystals of Gd 1 − x Er x B 4 ( 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 N é el temperature ( T N ) with increasing Er content. The effective magnetic moment determined from the magnetization measurement agreed with the calculated free ion values for Gd 3+ and Er 3+ , 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 − x Er x B 4 borides.


Introduction
Motivated by their interesting magnetic properties, the rare-earth tetraborides with general formula RB 4 (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 RB 4 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 J 1 and J 2 , 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 Er 3+ ions in ErB4 have a large total angular moment J = 15/2. It shows an antiferromagnetic transition at TN = 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 B [10]. The M × H curve exhibits a plateau region at the midpoint of the saturation magnetization (MS) value. In this plateau phase, half of the Er 3+ 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 MS/3 plateau, but it was not able to reproduce the appearance of the MS/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 Gd 3+ 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 TN = 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 Gd 3+ was determined to be 7.14 ± 0.17 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 Gd1−xErxB4 (x = 0-0.4). There are some noteworthy differences between GdB 4 and ErB 4 . All heavy RB 4 (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 Er 3+ ions in ErB 4 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 µ 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 Er 3+ 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 Gd 3+ magnetic moments in GdB 4 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 GdB 4 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 Gd 3+ was determined to be 7.14 ± 0.17 µ 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 RB 6 and RB 4 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−x Er x B 4 (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 GdB 4 . We have observed an anomalous and highly anisotropic ferromagnetic phase within the geometrically frustrated magnetic system of Gd 1−x Er x B 4 .

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−x Er x B 4 (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−x Er x B 4 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]. GdB 4 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 GdB 4 single crystal is described elsewhere [22]. The grown GdB 4 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−x Er x B 4 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 GdB 4 have been previously investigated and reported in Ref. [22].

Samples and X-ray Diffraction
The flux growth method for the synthesis of Gd 1−x Er x B 4 (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 Figures 3 and 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 Figures 3 and 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−x Er x B 4 (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, alongside our previously reported data for GdB 4 [22] and data from the literature for ErB 4 [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.  Structural refinement of the Gd1−xErxB4 (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, 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.  Structural refinement of the Gd1−xErxB4 (x = 0.2, and 0.4) crystals were carried ou 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 refinemen are listed in Table 1, 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 resul consistent with the smaller ionic radius of Er [24], and, in turn, a drop of the unit cel volume.  Given the different magnetic structures of GdB 4 and ErB 4 , the progressive substitution of Er for Gd in the Gd 1−x Er x B 4 is quite likely to affect the magnetic and thermal properties. We monitored these changes by measuring M(T, H) and C p (T) respectively.

Specific Heat
The zero-field specific heat C p (T) measurements were conducted on all three samples of Gd 1−x Er x B 4 (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 GdB 4 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 0.8 Er 0.2 B 4 and Gd 0.6 Er 0.4 B 4 , respectively. A small bump centered near T ≈ 10 K is observed in GdB 4 , 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 ErB 4 occurs at higher temperatures [26].   [10].
Given the different magnetic structures of GdB4 and ErB4, the progressive substitution of Er for Gd in the Gd1−xErxB4 is quite likely to affect the magnetic and thermal properties. We monitored these changes by measuring M(T, H) and Cp(T) respectively.

Specific Heat
The zero-field specific heat Cp(T) measurements were conducted on all three samples of Gd1−xErxB4 (x = 0, 0.2, and 0.4) for the present investigation, and the obtained Cp(T) data are shown in Figure 6a. The Cp(T) curve for the undoped GdB4 clearly shows a pronounced peak at the Néel temperature TN = 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 Gd0.8Er0.2B4 and Gd0.6Er0.4B4, 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 Cp = Cel + Clatt + Csch + Cm, where Cel, Clatt, Csch, and Cm are the contributions due to the electron system, 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 YB 4 .
Comparable values of a and b have been reported for GdB4 (a = 5.96 × 10 −4 J/mol K 2 and b = 4.89 × 10 −5 J/mol K 4 ) and ErB4 (a = 8.46 × 10 −4 J/mol K 2 and b = 6.82 × 10 −5 J/mol K 4 ) in previous studies [25,26]. Therefore, it is reasonable to assume that the contributions of Cel and Clatt to the heat capacity of the Gd1−xErxB4 series are like those of YB4. Consequently, the specific heat of YB4 was subtracted from the specific heat of the Gd1−xErxB4 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 Csch for a system with n levels, separated by energies and with degeneracy , is given by where R is the ideal gas constant and is given in units of kelvin [28]. The Csch curve for all samples was obtained by fitting the data below TN using the equation C = AT α + Csch, 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 Gd 3+ were used, where 0 = 2, 1 = 2, 2 = 4, 1 = 30 K, and 2 = 75 K. For Er 3+ , the data from Ref. [26] are 0 = 2, 1 = 4, 2 = 6, 3 = 4, 1 = 85 K, 2 = 240 K, and 3 = 700 K. For the fitting procedure, only 1 and 2 for Gd 3+ were free parameters. The parameters for Er 3+ were kept fixed since initial fits indicated their values did not vary significantly. Thus, Cm was obtained by subtracting the fitted Csch, considering the proportional contribution due to the Gd 3+ and Er 3+ .
The different contributions to the total specific heat are shown in Figure 7a, for the x = 0.4 sample. After isolating the Cm curve, its temperature dependence was investigated by fitting to the equation for temperatures up to 0.7TN, as shown in Figure 7b. For GdB4, α = 3, as expected for antiferromagnetic systems. The exponent α decreases to 2.6 The heat capacity of YB 4 at low temperatures can be expressed as C p:YB4 = C el + C latt = aT + bT 3 , where a = 12 × 10 −4 J/mol K 2 and b = 2.1 × 10 −5 J/mol K 4 , values obtained by fitting. Comparable values of a and b have been reported for GdB 4 (a = 5.96 × 10 −4 J/mol K 2 and b = 4.89 × 10 −5 J/mol K 4 ) and ErB 4 (a = 8.46 × 10 −4 J/mol K 2 and b = 6.82 × 10 −5 J/mol K 4 ) 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−x Er x B 4 series are like those of YB 4 . Consequently, the specific heat of YB 4 was subtracted from the specific heat of the Gd 1−x Er x B 4 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 ε n and with degeneracy g n , is given by where R is the ideal gas constant and ε 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 Gd 3+ were used, where g 0 = 2, g 1 = 2, g 2 = 4, ε 1 = 30 K, and ε 2 = 75 K. For Er 3+ , the data from Ref. [26] are g 0 = 2, g 1 = 4, g 2 = 6, g 3 = 4, ε 1 = 85 K, ε 2 = 240 K, and ε 3 = 700 K. For the fitting procedure, only ε 1 and ε 2 for Gd 3+ were free parameters. The parameters for Er 3+ 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 Gd 3+ and Er 3+ . 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 AT α for temperatures up to 0.7T N , as shown in Figure 7b. For GdB 4 , 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 0.6 Er 0.4 B 4 , 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 0.8 Er 0.2 B 4 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 0.6 Er 0.4 B 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 0.6 Er 0.4 B 4 , is possibly due to a metamagnetic transition, occurring exclusively along the c-axis, as suggested by the magnetization data.

Magnetization
The temperature-dependent magnetic susceptibility curves χ = (M/H) × T for Gd 1−x Er x B 4 (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 χ/dT 2 . Consistently with the C p (T) data, T N drops with the Er concentration, as shown in the inset of Figure 8b. 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 Cp(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 Gd0.6Er0.4B4, resulting in a drop in the temperature of the Cp peak at TN, and a significant change in morphology for H//c. In general, for fields in the 0 ≤ H ≤ +9 T range, the value of TN drops by ~15% and ~7% for fields applied parallel and perpendicular to the c-axis, respectively. In contrast, Cp(T) for the Gd0.8Er0.2B4 is nearly insensitive to the magnetic field, with the value of TN dropping but staying within 3% of the zero-field value, for both orientations.
In addition to the main feature at TN, the Cp(T) data for Gd0.6Er0.4B4, 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 Gd0.6Er0.4B4, is possibly due to a metamagnetic transition, occurring exclusively along the c-axis, as suggested by the magnetization data. For both x = 0.2 and x = 0.4 samples, χ (T) exhibits anisotropic behavior up to ~150 K, a temperature considerably higher than TN. In contrast, the undoped GdB4 displays anisotropic χ (T) behavior only below TN. 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 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 GdB 4 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 ErB 4 [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 0.8 Er 0.2 B 4 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 0.8 Er 0.2 B 4 , the magnetic susceptibility (χ) in the PM region follows the Curie-Weiss law [29]

Magnetization
where θ 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 Here, N a is the Avogadro's number, k B the Boltzmann constant, and µ eff the effective magnetic moment, calculated from where g J is the Landé g-factor, J the total quantum number, and µ B the Bohr magneton.
The theoretical values of µ eff calculated for Gd 3+ and Er 3+ are 7.94 and 9.58 µ B , respectively. For the Gd 1−x Er x B 4 solid solutions, the theoretical values of µ eff can be approximately obtained from On the other hand, the experimental values of µ eff can be obtained by fitting the (1/χ) vs. T data to the expression 1 resulting from combining Equations (2) and (3), with N a = 6.022·10 23 mol −1 , k B = 1.381·10 −16 erg/K, and µ B = 9.274·10 −21 emu, as shown in the inset of Figure 8a. The experimental values of µ eff obtained from fits of the experimental data to Equation (6) at temperatures above 100 K, are displayed in Table 2. The µ eff values obtained from the χ(T) data with field along the two crystallographic directions are consistent with the calculated µ eff values for the Gd 3+ and Er 3+ free ions, providing additional evidence for the effective partial substitution of Er for Gd within this series. Table 2. Calculated and experimental values of effective magnetic moment (µ eff ) and Curie-Weiss constant (θ CW ) for Gd 1−x Er x B 4 (x = 0, 0.2, 0.4, and 1.0). µ calc eff values are calculated using Equation (5) and µ exp eff values are obtained from the fittings of (1/χ) vs. T data to Equation (6). The data displayed in Table 2 indicate that the effective magnetic moments µ exp eff for GdB 4 and Gd 0.8 Er 0.2 B 4 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 0.6 Er 0.4 B 4 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 θ 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−x Er x B 4 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.
tions 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 Gd1−xErxB4 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 HC 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 HC 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, Gd0.6Er0.4B4 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/MS = ½, where MS is the saturation magnetization of the Er 3+ ions. The second transition corresponds to the full alignment of the Er 3+ ions along the c-axis. The theoretical value of the saturation magnetization per Er 3+ ion is = = 9 . Based on this value, the saturation magnetization for the Er 3+ ions in the x = 0.4 sample corresponds to 3.6 . Although GdB 4 and ErB 4 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 0.6 Er 0.4 B 4 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 = 1 /2 , where M S is the saturation magnetization of the Er 3+ ions. The second transition corresponds to the full alignment of the Er 3+ ions along the c-axis. The theoretical value of the saturation magnetization per Er 3+ ion is M S = gJµ B = 9 µ B . Based on this value, the saturation magnetization for the Er 3+ ions in the x = 0.4 sample corresponds to 3.6 µ 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 µ B . By subtracting the contribution of the Gd 3+ ions, the experimental saturation magnetization corresponding to the Er 3+ ions is 94% of the theoretical value. In this procedure, as a first approximation, we assume that the Gd 3+ 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 0.8 Er 0.2 B 4 composition, as shown in Figure 9a. Using the same procedure applied for the x = 0.4 composition to subtract the contribution of the Gd 3+ ions, the magnetization attributed to the Er 3+ 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 Er 3+ , 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 ErB 4 [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 0.6 Er 0.4 B 4 with pure ErB 4 , 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 GdB 4 and TbB 4 are the only members of the RB 4 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 GdB 4 . 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.

Conclusions
In conclusion, we carried out a study of the thermal and magnetic properties of fluxgrown Gd 1−x Er x B 4 single crystals. This study revealed detailed magnetic transitions in both x = 0.2 and 0.4 compositions, including the full alignment of Er 3+ 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−x Er x B 4 (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 ErB 4 . 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.

Data Availability Statement:
The data presented in this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.8152048 (accessed on 16 July 2023).