Magnetic and Electronic Inhomogeneity in Sm_1−xEu_xB₆

Abstract

While SmB₆ attracts attention as a possible topological Kondo insulator, EuB₆ is known to host magnetic polarons that give rise to large magnetoresistive effects above its ferromagnetic order transition. Here, we investigate single crystals of Sm_1−xEu_xB₆ by magnetic and magnetotransport measurements to explore a possible interplay of these two intriguing phenomena, with a focus on the Eu-rich substitutions. Sm_0.01Eu_0.99B₆ exhibits generally similar behavior as EuB₆. Interestingly, Sm_0.05Eu_0.95B₆ combines a global antiferromagnetic order with local polaron formation. A pronounced hysteresis is found in the magnetoresistance of Sm_0.1Eu_0.9B₆ at low temperature ($T=$ 1.9 K) and applied magnetic fields between 2.3 and 3.6 T. The latter is in agreement with a phenomenological model that predicts the stabilization of ferromagnetic polarons with an increasing magnetic field within materials with a global antiferromagnetic order.

1. Introduction

Hexaborides RB₆ of the cubic structure type CaB₆ [1] form for quite a number of elements R [2,3], primarily including alkaline earths or rare earths (RE), and Y. They typically feature high melting points, high hardness and excellent chemical stability [4]. In addition, this class of compounds is known for a remarkable variety of properties ranging from the low work function of LaB₆ [5] to the antiferromagnetic behavior in PrB₆, NdB₆ and GdB₆ [6], the quadrupolar order in CeB₆ [7] and even superconductivity below 7.1 K in YB₆ [8]. In general, the B-octahedra take up two electrons such that hexaborides formed with trivalent elements are typically highly conductive, while those based on divalent elements exhibit very low charge carrier densities [9]. An interesting exception here is SmB₆ with its intermediate and temperature-dependent valence [10,11,12] of around 2.6 at room temperature.

SmB₆ attracted enormous interest as a correlated topological insulator candidate [13,14,15]. The existence of surface states on SmB₆ is well established [16,17]. Their origin due to non-trivial topology remains—despite overwhelming evidence (see, e.g., [14] and references therein)—controversially discussed [18]. Nonetheless, the surface states, once formed on the highly insulating bulk at low enough temperatures [19], can be considered an electronic inhomogeneity (with respect to the bulk) with potential applications. Indeed, topological insulators with their spin-polarized surface states are considered candidates for next-generation spintronics [20].

Another type of electronic inhomogeneity, also considered useful for applications, is magnetic polarons [21,22,23]. Here, a magnetic polaron corresponds to an entity made up by a charge carrier and local magnetic moments within which strong exchange coupling localizes the charge carrier as well as aligns the magnetic moments locally in a ferromagnetic fashion. Eu²⁺ with its half-filled 4f shell often provides the required exchange interaction [24]. In addition, divalent Eu²⁺ in EuB₆ establishes a very low carrier concentration (as mentioned above) such that separated polarons instead of extended bands may form above the global ferromagnetic ordering transition at around 12 K [25,26,27,28,29]. In general, early indications for developing the concept of polaron formation [30] were provided by magnetotransport measurements [31,32], which are still heavily used in this context. Further evidence for magnetic polarons came from more elaborate techniques, including muon spin-relaxation [33], neutron spectroscopy [34] and scanning tunneling microscopy (STM) [28,35]. The latter gave an approximate extension of the polarons of an order of few a nanometers in layered manganites and EuB₆, respectively.

The crystal structure of intertwined cubes of R and B-octahedra in the hexaborides RB₆ allows for numerous elements to be placed on the R site as well as the formation of solid solutions [4] and the fabrication of high-entropy ceramics [36]. Along this line of thinking, one may ask whether the two types of electronic inhomogeneities discussed above could be combined in a substitution series Sm_1−xEu_xB₆, and possibly be tailored by adjusting x. Here, it should be noted that the substitution of Sm by Eu increases the lattice constant only slightly but changes the carrier density due to the different valencies and simultaneously influences the magnetic order [37]. As a result, an early study of the substitution series focused on the global magnetic and transport properties [38]. At low temperature, insulating behavior was found up to $x\lesssim$ 0.4 and metallic behavior for larger x. Antiferromagnetism prevailed for 0.2 $\lesssim x\lesssim$ 0.95. In contrast, the regions for smaller and larger x resemble the respective end member: SmB₆ is nonmagnetic [39,40] in its ground state (it orders magnetically under pressures above 6 GPa [41]) like many other Kondo insulators, whereas EuB₆ is a ferromagnetic semimetal [42]. A photoelectron spectroscopy study revealed that the Kondo coherence and the related gap survive at least up to $x=0.15$ and up to 50 K [43]. Magnetotransport studies on samples with $x\le$ 0.05 supported the persistence of a gap [44] and the surface states [45]. More recent studies were able to follow the topological coherence encountered in SmB₆ up to $x=0.3$, which, again, is related to the presence of a direct Kondo gap [46,47].

In contrast, we concentrate on the Eu-rich samples and investigate the changes in magnetic and magnetotransport properties upon increasing Sm content in EuB₆. As the ferromagnetic Curie temperature $T_{\mathrm{C}}$ is expected to decrease rapidly with x departing from 1 [38], samples with closely spaced x were prepared. We find that the fate of the polaron formation in Sm_1−xEu_xB₆ as the Eu content is reduced to below $x=1$ does not follow this rapid dependence of $T_{\mathrm{C}}\left(x\right)$. In consequence, and in line with recent observations in a few other antiferromagnetic Eu compounds [48,49,50], we find indications for ferromagnetic (fm) polaron formation in the globally antiferromagnetic (afm) material Sm_0.05Eu_0.95B₆. These results highlight the complexity of the interactions in Eu-based materials.

2. Results

2.1. Magnetic Measurements

The dc susceptibilities $\chi \left(T\right)$ measured for the differently substituted samples Sm_1−xEu_xB₆ at 0.1 T after zero-field and field-cooled conditions are presented in Figure 1. The data of pure SmB₆ can very nicely be compared to earlier results [51,52]. The magnetic properties of Eu upon substitution of Sm become immediately obvious, even for an x as small as 0.2: There is an increase in the low-T susceptibility of almost two orders of magnitude [38]. In comparison, for a similar amount of La-substitution, $\chi \left(T\right)$ does not even increase by a factor of 1.5 [52]. The onset of the afm order in sample $x=$ 0.2 is observed at around 8.6 K, which is in line with [38]. The latter is reduced to 5.8 K for $x=$ 0.9 and 4.3 K for $x=$ 0.95. We note that the magnetization measurements for samples $x=$ 0.9 and 0.95 did not reveal signatures of an fm component at the zero field (besides the increase in the magnitude of $\chi$), as will be discussed below.

For $x=$ 0.99 and 1.0, a qualitatively different behavior is observed: The rising $\chi \left(T\right)$ down to lowest T indicates ferromagnetic behavior. The Curie temperatures $T_{\mathrm{C}}$ of ferromagnets can be estimated from Arrott plots [53], as shown in Figure 2c for sample $x=0.99$. In particular, the temperature dependence of the zero-field susceptibility ${\chi }_0\left(T\right)$ was evaluated, where ${\chi }_0^{-1}\left(T\right)$-values are the intercepts with the $H/M$-axis [54,55]. We estimated $T_{\mathrm{C}}\sim$ 7.3 K for $x=0.99$ and $T_{\mathrm{C}}\sim$ 12.7 K for $x=1.0$. The latter value of the end member EuB₆ is in good agreement with reported ones [26,55,56], while the exemplary $M^2$ vs. ${\mu }_{0}H/M$ data, the dashed lines in Figure 2c, are in accordance with [55]. It is important to note, however, that $T_{\mathrm{C}}$ in EuB₆ is different from the metal-semiconductor transition temperature $T_{\mathrm{M}}$ at which the magnetic polarons percolate [25,27,57,58]. The formation of magnetic polarons in EuB₆ is reported to set in between 25 and 35 K [25,28,56,59].

Magnetization curves $M\left(H\right)$ of Sm_0.01Eu_0.99B₆ were measured at several temperatures; the results are presented in Figure 2a. The saturation value of $M_{\mathrm{sat}}\approx 7.08{\mu }_{\mathrm{B}}/$Eu agrees well with the expected saturation magnetic moment of primarily Eu²⁺, $g{\mu }_{\mathrm{B}}J=$ 7 ${\mu }_{\mathrm{B}}$. Even at small fields, there is no hysteresis seen at $T=$ 1.8 K, in contrast to the data for EuB₆ ($x=1$, $M_{\mathrm{sat}}\approx 7.02{\mu }_{\mathrm{B}}/$Eu), Figure 2b. We note that care has to be taken if $M\left(H\right)$-data are to be inspected on a small field scale: the remnant field of the superconducting magnet of the MPMS was determined before and after each measurement.

The inverse susceptibility $1/\chi \left(T\right)$ of Sm_0.01Eu_0.99B₆ measured at a field of 0.1 T is presented in Figure 3. A Curie–Weiss fit of the data in the range 25 K $\le T\le$ 250 K yields a Curie–Weiss temperature ${\theta }_{\mathrm{CW}}=+7.7\pm 0.4$ K and an effective moment ${\mu }_{\mathrm{eff}}=7.93\pm 0.08{\mu }_{\mathrm{B}}$—the latter in excellent agreement with ${\mu }_{\mathrm{eff}}^{\mathrm{cal}}=7.94{\mu }_{\mathrm{B}}$, which is expected for Eu²⁺. However, there are clear deviations from the Curie–Weiss law below about 20 K, as seen by the difference $∆(1/\chi )$ between the $1/\chi$-data and the extrapolated linear fit (the red markers in Figure 3). We speculate that these deviations result from the impact of magnetic inhomogeneities like polarons, as observed in a number of Eu compounds [49,55,60,61].

We now turn to sample Sm_0.05Eu_0.95B₆. Figure 4a presents the low-temperature ac susceptibility ${\chi }_{\mathrm{ac}}\left(T\right)$ as measured at different applied magnetic fields. Clearly, an afm transition is indicated by a peak in ${\chi }_{\mathrm{ac}}\left(T\right)$ at $T_{\mathrm{N}}=4.9\pm 0.1$ K. As expected, this peak is gradually suppressed by applying a magnetic field. At ${\mu }_{0}H=$ 0.1 T, the peak temperature $T_{\max }\approx$ 4.1 K of ${\chi }_{\mathrm{ac}}\left(T\right)$ is in good agreement with the result shown in Figure 1. However, the peak in ${\chi }_{\mathrm{ac}}\left(T\right)$ in zero field is not as sharp as in typical antiferromagnets (see, e.g., [62]). Nevertheless, in order to exclude the possibility of spin-glass-like behavior, as, for example, observed in Cu_1−xMn_x for $0.01\le x\le 0.063$ [63], we measured the frequency dependence of ${\chi }_{\mathrm{ac}}\left(T\right)$ near $T_{\mathrm{N}}$, as plotted in Figure 4b. A shift of $T_{\max }$ with frequency is not obvious despite the fact that a reduction in ${\chi }_{\mathrm{ac}}\left(T_{\max }\right)$ with increasing frequency is observed. These observations can be viewed as indications of the existence of magnetic polarons (in contrast to Sm_0.1Eu_0.9B₆ discussed below). Additionally, there is a strong reduction in ${\chi }_{\mathrm{ac}}\left(T\right)$ with increasing H well above $T_{\mathrm{N}}$, Figure 4a. A similar behavior in Eu₅In₂Sb₆ was also discussed [49] as an indication of polaron formation.

Ferromagnetic interactions in the sample with $x=0.95$ are corroborated by further analyzing ${\chi }_{\mathrm{ac}}\left(T\right)$. Well above $T_{\mathrm{N}}$, ${\chi }_{\mathrm{ac}}\left(T\right)$ follows a Curie–Weiss law nicely with a clearly positive ${\theta }_{\mathrm{CW}}=+2.7\pm 0.5$ K; see Figure 5a. The value ${\mu }_{\mathrm{eff}}=7.7\pm 0.1{\mu }_{\mathrm{B}}$ obtained from the fit is in good agreement with ${\mu }_{\mathrm{eff}}^{\mathrm{cal}}=7.59{\mu }_{\mathrm{B}}$, which is expected for a mixture of 95% Eu²⁺ and 5% Sm³⁺. A large imaginary component of the susceptibility, ${\chi }_{\mathrm{ac}}^{\prime \prime }\left(T\right)$, is observed near $T_{\mathrm{N}}$, as shown in Figure 5b, which is in line with a magnetically inhomogeneous state. An applied field as small as 50 mT suppresses ${\chi }_{\mathrm{ac}}^{\prime \prime }\left(T\right)$ significantly, indicating an effective suppression of these inhomogeneities. Importantly, ${\chi }_{\mathrm{ac}}^{\prime \prime }\left(T\right)$ is already enhanced at temperatures well above $T_{\mathrm{N}}$, again indicating magnetic inhomogeneities. For comparison, the small ${\chi }_{\mathrm{ac}}^{\prime \prime }\left(T\right)$ for sample $x=$ 0.9 is also shown in Figure 5b.

Here, we recall that sample $x=0.99$ is a ferromagnet, while samples with $x=0.90$ and 0.95 order globally antiferromagnetically. However, the positive value of ${\theta }_{\mathrm{CW}}$ and the behavior of ${\chi }_{\mathrm{ac}}^{\prime \prime }\left(T\right)$ of Sm_0.05Eu_0.95B₆ may again be indicative of polaron formation, as will be discussed below. In contrast, data from the afm sample Sm_0.1Eu_0.9B₆ do not reveal any sign of polaron formation in ${\chi }_{\mathrm{ac}}^{\prime \prime }\left(T\right)$, as shown in Figure 5b. Fitting ${\chi }_{\mathrm{ac}}\left(T\right)$ to a Curie–Weiss law yields ${\theta }_{\mathrm{CW}}=-2.9\pm 0.5$ K and ${\mu }_{\mathrm{eff}}=7.6\pm 0.1{\mu }_{\mathrm{B}}$. As shown in Figure 6a, $T_{\mathrm{N}}=5.7\pm 0.1$ K. The peak in ${\chi }_{\mathrm{ac}}\left(T\right)$ is also suppressed by magnetic fields but much less effectively compared to the sample $x=0.95$. Moreover, there is no noticeable frequency dependence of ${\chi }_{\mathrm{ac}}\left(T\right)$ in Figure 6b, pointing to a homogeneous magnetic state at zero field (in contrast to the observations around 3 T discussed below).

2.2. Magnetotransport Measurements

Figure 7 summarizes the temperature-dependent resistance behavior of our samples. The resistance of Sm-rich samples $x=$ 0 and 0.2 increases upon lowering T, indicating insulating behavior. For pristine SmB₆, two temperature ranges with different values of the hybridization gap $∆$ can be distinguished in resistance $R\left(T\right)\propto exp(∆/k_{\mathrm{B}}T)$ [19,64], where $k_{\mathrm{B}}$ is the Boltzmann constant. The values obtained here, ${∆}_1\sim$2.6 meV at lower T and ${∆}_2\sim$5.2 meV at higher T, are in good agreement with the results in [64]. For $x=$ 0.2, there are also two distinct temperature ranges (white dashed lines in Figure 7) albeit of a narrower extent in T. Interestingly, upon Eu substitution with $x=$ 0.2, the low-T gap ${∆}_1\sim$2.7 meV remains stable, while the high-T gap ${∆}_2\sim$1.9 meV reduces drastically. In addition, $R\left(T\right)$ appears to approach saturation at low T, which is in line with the reported existence of surface states at this substitution level [46,47] but in contrast to similar La, Lu or Ce substitutions [43,44,65].

Higher Eu substitutions, i.e., $x=$ 0.9, 0.95 and 0.99, result in semimetallic behavior (as in EuB₆ [66]). In fact, $\rho \left(T\right)$ of sample $x=$ 0.99 is highly reminiscent of pristine EuB₆ [26,58]: it passes through a minimum at $T^{\ast }\sim$ 35 K before steeply rising to a peak at $T_{\mathrm{M}}\approx$ 8.4 K. $T_{\mathrm{C}}$ (as determined from ${\chi }_{\mathrm{ac}}\left(T\right)$) manifests itself as a small hump in $\rho \left(T\right)$, as seen in the inset of Figure 7. These close similarities to EuB₆ [55,58] encourage us to assign $T^{\ast }$ to the polaron formation and $T_{\mathrm{M}}$ to their percolation. We note, however, that $T_{\mathrm{M}}$ and $T_{\mathrm{C}}$ in sample $x=$ 0.99 are reduced to almost half the values found in EuB₆.

In general, sample $x=$ 0.95 exhibits a $\rho \left(T\right)$-behavior very similar to sample $x=$ 0.99, specifically if plotted as $R\left(T\right)/R(T=$ 200 K), irrespective of the different magnetic order at low T in these two samples. We recall that $T_{\mathrm{N}}=$ 4.9 K for sample $x=$ 0.95. As seen from the inset to Figure 7, $T_{\mathrm{M}}\approx$ 4.2 K is now below $T_{\mathrm{N}}$, and the only subtle hint at the onset of the afm order in $\rho \left(T\right)$ might be a broadened minimum in d$\rho \left(T\right)/$dT. A subtle influence of the afm order on $\rho \left(T\right)$ is suggested by the $\rho \left(T\right)$-behavior observed for sample $x=$ 0.9: A peak of a much less pronounced relative height in $R\left(T\right)/R(T=$ 200 K) is observed at $T_{\mathrm{M}}\approx$ 5.4 K, i.e., very close to $T_{\mathrm{N}}=$ 5.7 K, and hence, likely related to the onset of the afm order.

It is interesting to note that $\rho \left(T\right)$ of sample Sm_0.1Eu_0.9B₆ follows a linear dependence very nicely within the range 45 K $\lesssim T\lesssim$ 220 K; see the yellow dashed line in Figure 7. EuB₆ has a low carrier concentration of the order of 0.01 per unit cell in the fm and paramagnetic regime [42,67], and hence, a scattering model as proposed in [68] may be relevant assuming an only moderately enhanced carrier concentration due to substitution in Sm_0.1Eu_0.9B₆. However, additional measurements on the latter material are needed, e.g., of the Hall effect or the impact of disorder [69], to gain further insight into the linear-in-T resistivity.

One of the striking phenomena related to polaron physics is an enhanced negative magnetoresistance (MR) [21,23,34]. For separated polarons in EuB₆, the charge transport is dominated by magnetic scattering at these fm polarons [70]. Above $T_{\mathrm{M}}$, the more conducting volumes of the fm polarons grow out of the less conductive paramagnetic “background” either upon lowering the temperature or by application of a magnetic field [27,55,58,67]. Upon reaching a high enough number density (in the zero field, this is at $T_{\mathrm{M}}$), the polarons percolate and the resistivity drops, giving rise to a large negative MR. Obviously, at higher magnetic fields, the percolation threshold becomes shifted to higher temperatures.

At first glance, the $\rho \left(T\right)$ of our sample $x=$ 0.99 is qualitatively very similar to pristine EuB₆ [55], with $T_{\mathrm{C}}$, $T_{\mathrm{M}}$ (see

Table 1. Properties of the highly Eu-containing samples Sm_1−xEu_xB₆.

Sample $\mathit{x}$	Magnetic Order	${\mathit{T}}_{\mathbf{C}}$/${\mathit{T}}_{\mathbf{N}}$ K	${\mathit{\theta }}_{\mathbf{CW}}$ K	${\mathit{\mu }}_{\mathbf{eff}}$ ${\mathit{\mu }}_{\mathbf{B}}$	${\mathit{M}}_{\mathbf{sat}}$ ${\mathit{\mu }}_{\mathbf{B}}$	${\mathit{T}}_{\mathbf{M}}$ K
0.9	afm	5.7	$-2.9$	7.6	/	5.4
0.95	afm	4.9	$+2.7$	7.7	6.91	4.2
0.99	fm	7.3	$+7.7$	7.93	7.08	8.4
1.0	fm	12.7	$+15.4$	8.1	7.02	$15.2$ [58]
error		$\pm 0.1$	$\pm 0.5$	$\pm 0.1$	$\pm 0.1$	$\pm 0.2$

) and $\rho \left(T_{\mathrm{M}}\right)$ somewhat reduced. The latter is likely the result of an increased carrier density due to the substitution of Eu by Sm. The MR, however, is notably different: While MR $=\left[\rho \left(H\right)-\rho (H=0)\right]/\rho (H=0)$ is positive for EuB₆ below about 5 K [42,71], i.e., well in the metallic regime, this is not observed for sample $x=$ 0.99; see Figure 8a. The positive MR in the fm regime of EuB₆ has been explained by the subtle interplay of electron and hole carriers in a two-band model, whereas the negative MR above $T_{\mathrm{M}}$ was attributed to spin disorder scattering [72,73,74] near $T_{\mathrm{C}}$ and the percolation of magnetic polarons at intermediate temperatures further above $T_{\mathrm{C}}$ [27,28,58], which are both known to result in large negative magnetoresistance [75]. Therefore, we interpret the large negative MR even at the lowest $T=$ 1.9 K (MR $=-0.47$ at 1 T) also in terms of predominant suppression of spin disorder scattering in our 1% Sm-substituted sample. Strikingly, very similar values at which the MRs saturate are observed for high fields: at 1.9 K, we find MR $=-0.63$, compared to an estimate of $-0.62$ at $T=$ 12 K, the latter being well above $T_{\mathrm{C}}$ and $T_{\mathrm{M}}$, i.e., in the polaron regime. Moreover, for spin disorder scattering, a scaling was suggested [76,77] for small magnetization: MR $=-C{(M/M_{\mathrm{sat}})}^2$. As shown in Figure 9a, this scaling is obeyed in our sample $x=$ 0.99 up to ${(M/M_{\mathrm{sat}})}^2\lesssim$ 0.012 with $C\sim 0.7$. The latter is much smaller than the value reported for EuB₆ ($C=$ 75, [55]), yet close to the prediction 1 $\le C\lesssim$ 5 [76]. We also note that electrons are thought to be responsible for the magnetic ordering via an RKKY interaction [72]. A changing carrier density will affect the RKKY interaction strength. Hence, one may speculate that the substitution of Eu by 1% Sm helps to tip the subtle balance between hole and electron carriers in EuB₆ toward electrons in Sm_0.01Eu_0.99B₆.

The afm sample $x=$ 0.95 also exhibits large negative MR values, reaching almost $-0.5$. Spin disorder scattering in the paramagnetic regime is again indicated by the scaling of MR with $M^2$ in Figure 9b. For this sample, the scaling only works up to ${(M/M_{\mathrm{sat}})}^2\lesssim$ 0.003, which is likely related to the lesser increase in $M\left(H\right)$ with H compared to sample $x=$ 0.99 (see, e.g., the magnetization data at 10 K in Figure 4c and Figure 2a, respectively). Accordingly, C is increased to about 0.94. This points to magnetic inhomogeneities also being present in sample $x=$ 0.95. More striking, however, is the evolution of MR$\left(H\right)$ with temperature in Figure 8b: |MR$\left(H\right)|$ is larger at all fields for $T=$ 5 K compared to the data at 1.9 K. Very likely, this is a consequence of the 5 K-curve taken above but very close to $T_{\mathrm{N}}$. It is a hallmark of polaronic systems to exhibit the largest MR values close to the magnetic ordering temperature [21]; together with the large MR values, we therefore surmise that polarons also form in sample $x=$ 0.95. At 1.9 K, the formation of fm polarons competes with the global afm order and, as a result, a tiny positive MR is observed at small fields, as shown in the inset o Figure 8b, which is also seen in other afm materials with polaron formation [78].

Compared to sample $x=$ 0.95, the MR in sample $x=$ 0.9 is reduced by a factor of approximately two, with the exception of the 1.9 K-curve. Intriguingly, the latter also shows a significant hysteresis near 3 T, as shown in Figure 8c, which may point to fm contributions in this afm material. Here, it should be noted that the MR was measured for field sweeps ${\mu }_{0}H=$ 0 T $\to +7$ T $\to -7$ T $\to +7$ T. The lack of any difference in the up-sweep data (curved blue arrows) indicates that this hysteresis is reproducible and independent of magnetic history. In Ref. [79], a phenomenological model was introduced that predicts a stabilization of fm polarons inside a globally antiferromagnetically ordered phase upon lowering the temperature. As one may expect, such polarons are also energetically favored by the application of a magnetic field. One may, therefore, speculate that the hysteresis in MR results from magnetic inhomogeneities that may form at intermediate fields between the global afm order at low fields and the field-polarized state at high fields. The small slope of MR$\left(H\right)$ at low fields (similar in magnitude to the one observed for afm sample x = 0.2 up to ∼1.3 T, Figure 8d), which increases drastically above the hysteresis, may support such a line of thinking.

In order to scrutinize the possible existence of magnetic polarons in Sm_0.1Eu_0.9B₆ at intermediate fields, the results of magnetic measurements were analyzed in detail. Figure 10a exhibits the difference $∆M=M(H\downarrow )$$-M(H\uparrow )$ of the magnetization measured while sweeping the magnetic field up $(H\uparrow )$ and down $(H\downarrow )$. Clearly, there is some hysteresis in $M\left(H\right)$ (not seen at the scale of Figure 6c) centered around 3 T, as in the MR$\left(H\right)$; see the arrow in Figure 10a. Apparently, the magnetization within the volume of these magnetic inhomogeneities increases the total magnetization only slightly despite introducing significant scattering. Both ${\chi }_{\mathrm{ac}}\left(H\right)$ and ${\chi }_{\mathrm{ac}}^{″}\left(H\right)$ also exhibit tiny humps just above 3 T, as shown in Figure 10b,c. Here, we wish to point out that the accuracy of our measurement setup is, unfortunately, of the order of 0.005 ${\mu }_{\mathrm{B}}$/RE, which may explain the noise of our ${\chi }_{\mathrm{ac}}^{″}$-data as well as their seemingly negative values at certain magnetic fields. Notably, another feature appears in the magnetic data at small fields (below 0.6 T) that is not seen in the MR (cf. inset to Figure 8c) and whose origin is presently unknown.

3. Discussion

The fm order in EuB₆ is very efficiently suppressed by introducing Sm: only 1% Sm reduces $T_{\mathrm{C}}$ to almost half its value; see Table 1. For comparison, Eu_0.8Ca_0.2B₆ still orders ferromagnetically albeit with reduced $T_{\mathrm{C}}$ [71,80]. In contrast, our sample Sm_0.05Eu_0.95B₆ orders antiferromagnetically with $T_{\mathrm{N}}=$ 4.9 K even though the small but positive Curie–Weiss temperature ${\theta }_{\mathrm{CW}}$ indicates the presence of fm interactions. The latter likely supports the formation of fm polarons in our sample $x=$ 0.95. Such polarons are indicated by a large negative MR, which is enhanced near the magnetic ordering temperature. More generally, in EuB₆, the fm and afm interactions appear to be close in energy, and to compete. Hence, only small substitutions of Eu by Sm are then sufficient to change the ground state from ferromagnetic in EuB₆ to antiferromagnetic in $x\le 0.95$ while polaron formation is still feasible.

Indeed, indications of the formation of fm polarons are also found in the MR of sample Sm_0.1Eu_0.9B₆ but only within a field range of approximately 2.3–3.6 T. This is consistent with a recently developed phenomenological model that predicts a stabilization of fm polarons with increasing magnetic field within a material with global afm order if there is an appropriately low charge carrier concentration and a sufficiently strong exchange interaction between the local RE spin and the spin of the charge carriers [23,79].

4. Materials and Methods

The single crystalline samples Sm_1−xEu_xB₆ investigated here were synthesized by the Al-flux technique [81]. The nominal stoichiometries were chosen to allow a closer investigation of the Eu-rich samples: x = 0, 0.2, 0.9, 0.95, 0.99 and 1.0.

Magnetic measurements up to applied fields of 7 T were conducted using magnetic property measurement systems (MPMS3, Quantum Design Inc., San Diego, CA, USA). If not noted, the magnetic ac susceptibility ${\chi }_{\mathrm{ac}}$ was measured with an applied ac field of 10 Oe after cooling in a zero applied field (ZFC). Electronic transport was measured with a standard 4-probe configuration and an ac technique utilizing a physical property measurement system (PPMS, Quantum Design Inc.). In some cases, an external, lock-in-based circuitry was hooked up to the PPMS in an effort to improve accuracy.