Hall Effect Anisotropy in the Paramagnetic Phase of Ho0.8Lu0.2B12 Induced by Dynamic Charge Stripes

A detailed study of charge transport in the paramagnetic phase of the cage-cluster dodecaboride Ho0.8Lu0.2B12 with an instability both of the fcc lattice (cooperative Jahn–Teller effect) and the electronic structure (dynamic charge stripes) was carried out at temperatures 1.9–300 K in magnetic fields up to 80 kOe. Four mono-domain single crystals of Ho0.8Lu0.2B12 samples with different crystal axis orientation were investigated in order to establish the singularities of Hall effect, which develop due to (i) the electronic phase separation (stripes) and (ii) formation of the disordered cage-glass state below T*~60 K. It was demonstrated that a considerable intrinsic anisotropic positive component ρanxy appears at low temperatures in addition to the ordinary negative Hall resistivity contribution in magnetic fields above 40 kOe applied along the [001] and [110] axes. A relation between anomalous components of the resistivity tensor ρanxy~ρanxx1.7 was found for H||[001] below T*~60 K, and a power law ρanxy~ρanxx0.83 for the orientation H||[110] at temperatures T < TS~15 K. It is argued that below characteristic temperature TS~15 K the anomalous odd ρanxy(T) and even ρanxx(T) parts of the resistivity tensor may be interpreted in terms of formation of long chains in the filamentary structure of fluctuating charges (stripes). We assume that these ρanxy(H||[001]) and ρanxy(H||[110]) components represent the intrinsic (Berry phase contribution) and extrinsic (skew scattering) mechanism, respectively. Apart from them, an additional ferromagnetic contribution to both isotropic and anisotropic components in the Hall signal was registered and attributed to the effect of magnetic polarization of 5d states (ferromagnetic nano-domains) in the conduction band of Ho0.8Lu0.2B12.


Introduction
Numerous fundamental studies on strongly correlated electron systems (SCES) such as manganites [1][2][3][4], high-temperature superconducting (HTSC) cuprates [5][6][7][8], iron-based superconductors [9][10][11][12][13], chalcogenides [14], etc., have allowed for the discovery of a diversity of physical phenomena universal to SCES. Indeed, all these systems are characterized by a complexity of phase diagrams induced by strong phase separation due to structural or electronic instability [15]. The spatial electronic/magnetic inhomogeneity turns out to be directly related to simultaneously active spin, charge, orbital, and lattice degrees of freedom, which are considered as factors responsible for the appearance of high-temperature superconductivity in cuprates, as well as for the emergence of colossal magnetoresistance in manganites [5,[16][17][18]. In particular, there are two possible mechanisms of the formation H-T magnetic phase diagrams of the AF state of Ho0.8Lu0.2B12 in different field directions, (e) polar plot of the field-angular magnetoresistance dependence in the paramagnetic state and (f,g) angular H-φ magnetic phase diagrams (color shows the magnetoresistance amplitude) of the AF state of Ho0.8Lu0.2B12 at low temperatures (reproduced from [35]).
Let us name the main factors that determine the appearance of spatial inhomogeneity, leading to symmetry lowering in fcc rare earth (RE) dodecaborides. Firstly, the cooperative dynamic Jahn-Teller effect in the rigid boron sub-lattice with covalent bonds that leads to lifting of degeneracy of the highest occupied molecular orbitals (HOMO) in В12 octahedrons and produces static structural distortions with related splitting EJT~500-1500 K (~50-150 meV) of HOMO [36]. Secondly, reaching the Ioffe-Regel limit near TE~150 K (~15 meV) causes a development of vibrational instability, which leads to an increase in the density of phonon states at T~TE [37]. Thirdly, order-disorder transition to the cage-  [35]).
Let us name the main factors that determine the appearance of spatial inhomogeneity, leading to symmetry lowering in fcc rare earth (RE) dodecaborides. Firstly, the cooperative dynamic Jahn-Teller effect in the rigid boron sub-lattice with covalent bonds that leads to lifting of degeneracy of the highest occupied molecular orbitals (HOMO) in B 12 octahedrons and produces static structural distortions with related splitting E JT~5 00-1500 K (~50-150 meV) of HOMO [36]. Secondly, reaching the Ioffe-Regel limit near T E~1 50 K (~15 meV) causes a development of vibrational instability, which leads to an increase in the density of phonon states at T~T E [37]. Thirdly, order-disorder transition to the cage-glass state at T*~60 K (~5-6 meV) causes random displacements of RE ions from central symmetric positions in B 24 cuboctahedra, which form a rigid covalent boron framework [37]. Fourthly, well below T* large amplitude vibrations of neighbored RE ions in trigonal planes (transverse to the axis of the ferrodistortive JT effect [36]) produce periodic changes of hybridization between the 5d(RE) and 2p(B) states in the conduction band. This leads to the emergence of high-frequency charge density fluctuations with frequencies ν S~2 40 GHz [38] (denoted also as dynamic charge stripes, see Figure 1a) along one of the <110> direction in the fcc structure [39][40][41][42]. Stripe patterns are formed at characteristic temperature T S~h ν S /k B~1 5 K (~1 meV). These structural and electronic instabilities initiate nanoscale phase separation and inevitably cause strong charge transport anisotropy in external magnetic field both in the nonmagnetic reference LuB 12 [37] and in magnetic RB 12 [41][42][43][44][45][46]. In particular, the above features of the crystal and electronic structure of RB 12 have a decisive effect on the characteristics of charge transport when Lu ions (with a filled f -shell, 4f 14 configuration) are partially replaced by magnetic Ho (4f 10 ) ions in Ho x Lu 1-x B 12 compounds.
The spatial inhomogeneity of fluctuating electron density is the origin for the strong anisotropy of magnetic phase diagrams in these systems (see, e.g., Figure 1b-d for the best quality single-domain crystals of Ho 0.8 Lu 0.2 B 12 and [35,[43][44][45][46]). Indeed, strong magnetic anisotropy is observed, for instance, in Ho x Lu 1-x B 12 with a high concentration of magnetic ions both in the paramagnetic (P) state (see the color plot in Figure 1e demonstrating the anisotropy of magnetoresistance) and on the angular antiferromagnetic (AF) phase diagrams, which reveal a Maltese cross-symmetry (see Figure 1f,g, [35,43,44] and also [46] for TmB 12 ). It is worth noting that, like in the nonmagnetic reference compound LuB 12 , strong charge transport anisotropy is observed in the paramagnetic state of Ho x Lu 1-x B 12 , ErB 12 [45], and TmB 12 [46] (see, for example, Figure 1e and [47]) and attributed to interaction of electron density fluctuations (stripes) with external steady magnetic field (for recent review see [48] and references therein).
Until now, studies of electron transport in Ho x Lu 1-x B 12 have mainly focused on transverse magnetoresistance (see, e.g., [35,43,47]). Nevertheless, the recent study of LuB 12 [49] and the initial short research on Ho 0.8 Lu 0.2 B 12 (Ref. [50]) have demonstrated a significant anisotropy of the Hall effect due to an anomalous positive anisotropic contribution that appeared below T*~60 K. Thus, it is of great interest to study in detail the effect of electronic phase separation on the off-diagonal component of the resistivity tensor in model magnetic compound Ho 0.8 Lu 0.2 B 12 with dynamic charge stripes. As a continuation of the short study conducted in [50], this work presents results and detailed analyses of the normal and anomalous contributions to the Hall effect in the paramagnetic phase of Ho 0.8 Lu 0.2 B 12 . We investigated both the angular and magnetic field dependences of Hall resistivity in detail and determined the anisotropic component of the resistivity tensor for this model system with electronic phase separation (dynamic charge stripes). The observed complex angular behavior of the anisotropic Hall resistivity is attributed to interaction of the filamentary structure of fluctuating charges with the external magnetic field. The arguments presented here favor both intrinsic and extrinsic mechanisms of the anomalous Hall effect formation.

Temperature Dependences of Resistivity and Hall Resistivity
In conventional Hall effect experiments the Hall coefficient is calculated as R H = ρ H /H = ((V H (+H) − V H (-H))/(2I))·d/H, where d is the sample thickness, I the excitation current and V H (±H) the voltage measured on Hall probes in two opposite directions of external magnetic field. Taking into account the complex field dependence of Hall effect in parent compound LuB 12 [49], and the different origin of the detected anomalous contributions to Hall resistivity [51], the term "reduced Hall resistivity" for ρ H /H is used below in the present study instead of the Hall coefficient R H . Figure 2 shows the temperature dependences of resistivity ρ(T) at H = 0 and 80 kOe, as well as the reduced Hall resistivity ρ H (T)/H in Ho 0. 8 [37] and to the formation of stripes at T S~1 5 K (see also discussion below). In the zero-magnetic field, the ρ(T) curves measured for all three Ho 0.8 Lu 0.2 B 12 samples correspond to metallic conductivity with the same RRR value ρ(300 K)/ρ(4.2 K) = 13.47 ( Figure 2a) [37] and to the formation of stripes at TS~15 K (see also discussion below). In the zero-magnetic field, the ρ(T) curves measured for all three Ho0.8Lu0.2B12 samples correspond to metallic conductivity with the same RRR value ρ(300 K)/ρ(4.2 K) = 13.47 ( Figure 2a) Figure 2b show the results of ρH/H(T) measurements in the scheme with two opposite orientations of H. Significant differences between the ρH/H(T) dependences for different field directions appear below T*~60 K, while the curves for samples with , which is very similar to resistivity anisotropy. Thus, the temperature dependences of ρH/H(T) allow to identify some anisotropic positive component of the Hall signal, which appears in Ho0.8Lu0.2B12 in strong magnetic fields. It is worth noting that temperature-lowering results in the increase in anisotropy for both Hall resistivity and MR components ( Figure 2).   [37] and to the formation of stripes at T S~1 5 K (see discussion below), and denote the temperatures 2.1 K, 4.2 K, and 6.5 K at which Hall resistivity was studied in more detail.   [37] and to the formation of stripes at TS~15 K (see discussion below), and denote the temperatures 2.1 K, 4.2 K, and 6.5 K at which Hall resistivity was studied in more detail.     . Solid line indicates the changes of the magnetic moment of Ho 5 I8-multiplet splitting by CEF in HoB12 (see [52]).

Field Dependences of Hall Resistivity and Magnetization
The AF-P phase transition at TN = 5.75 K can be clearly recognized on the temperature dependence of magnetic susceptibility measured at H = 100 Oe (see Figure 4). Above TN, the low field magnetic susceptibility χ(T) may be described approximately by a Curie-Weiss type dependence where NHo = 0.95•x(Ho)•10 22 cm −3 , and μeff~10 μB are the concentration and the effective magnetic moment of Ho-ions, correspondingly (μB and kB denote Bohr magneton and Boltzmann constant), θp ≈ −14 K is the paramagnetic Curie temperature corresponding to AF exchange between magnetic dipoles. χ0 ≈ −1.78•10 −3 μB/mole/Oe is the temperature-independent additive combination of (i) diamagnetic susceptibility of the boron cage and (ii) Pauli paramagnetism and Landau diamagnetism of conduction electrons. Fitting of χ(T) by Equation (1) with temperature-dependent μeff(T) indicates that within experimental accuracy the susceptibility follows the Curie-Weiss dependence with magnetic moment, which is only slightly below the total moment μeff ≈ 10.6 μB of Ho 3+ 4fshell in the range 80-300 K. As the population of excited magnetic states of the Ho 3+ 5 I8 multiplet (that is split by crystalline electric field (CEF) [52]) declines significantly in the range 8-80 K, μeff decreases moderately (to 9.5 μB; see Figure 4, right scale). Thus, even at TN, the value of μeff noticeably exceeds the magnetic moment of the Г51 ground state triplet μeff (Г51) ≈ 8 μB (solid line in Figure 4, right scale). The difference (∆μeff~1.5 μB, Figure 4) may be related to ferromagnetic correlations, which develop in this SCES below T*~60 K. Note that below 25 K, various short-range ordering features including ferromagnetic components were previously observed in magnetic RB12 [53][54][55].
As can be seen from Figure 3a  The AF-P phase transition at T N = 5.75 K can be clearly recognized on the temperature dependence of magnetic susceptibility measured at H = 100 Oe (see Figure 4). Above T N , the low field magnetic susceptibility χ(T) may be described approximately by a Curie-Weiss type dependence where N Ho = 0.95·x(Ho)·10 22 cm −3 , and µ eff~1 0 µ B are the concentration and the effective magnetic moment of Ho-ions, correspondingly (µ B and k B denote Bohr magneton and Boltzmann constant), θ p ≈ −14 K is the paramagnetic Curie temperature corresponding to AF exchange between magnetic dipoles. χ 0 ≈ −1.78·10 −3 µ B /mole/Oe is the temperatureindependent additive combination of (i) diamagnetic susceptibility of the boron cage and (ii) Pauli paramagnetism and Landau diamagnetism of conduction electrons. Fitting of χ(T) by Equation (1) with temperature-dependent µ eff (T) indicates that within experimental accuracy the susceptibility follows the Curie-Weiss dependence with magnetic moment, which is only slightly below the total moment µ eff ≈ 10.6 µ B of Ho 3+ 4f -shell in the range 80-300 K. As the population of excited magnetic states of the Ho 3+ 5 I 8 multiplet (that is split by crystalline electric field (CEF) [52]) declines significantly in the range 8-80 K, µ eff decreases moderately (to 9.5 µ B ; see Figure 4, right scale). Thus, even at T N , the value of µ eff noticeably exceeds the magnetic moment of the Γ 51 ground state triplet µ eff (Γ 51 ) ≈ 8 µ B (solid line in Figure 4, right scale). The difference (∆µ eff~1 .5 µ B , Figure 4) may be related to ferromagnetic correlations, which develop in this SCES below T*~60 K. Note that below 25 K, various short-range ordering features including ferromagnetic components were previously observed in magnetic RB 12 [53][54][55].
As can be seen from Figure 3a  state of fcc metals (as HoxLu1-xB12) with intense charge carrier scattering in the disordered cage-glass phase.

Angular Dependences of Hall Resistivity in the Paramagnetic State of Ho0.8Lu0.2B12.
To reveal the nature of the strong anisotropy of ρH/H Hall resistivity (Figures 2b and 3a-c) as well as to separate different contributions to Hall effect, it is of interest to study the angular dependencies of Hall resistivity ρH(φ) in Ho0.8Lu0.2B12 for different configurations of magnetic field with respect to principal crystallographic directions. Here we present precision measurements of Hall resistivity ρH(φ,T0,H0,n) angular dependencies performed at 2.1-300 K in magnetic field up to 80    ρ H (ϕ) = ρ H const + ρ H0 ·cos(ϕ + ϕ sh ) + ρ H an (ϕ) (2) where ρ H const is an angle independent component, ρ H0 is the amplitude of the isotropic cosine-like contribution to Hall resistivity f cos (ϕ) = ρ H0 ·cos(ϕ + ϕ sh ), ϕ sh is the phase shift, and ρ H an (ϕ) = ρ H0 an ·g(ϕ) the anisotropic contribution to Hall resistivity (see Figure 5). The approximation of ρ H (ϕ) within the framework of Equation (2) for two crystals with normal directions n||[001] and n||[110] was carried out in two intervals ∆ϕ = 90 ± 35 • and ∆ϕ = 270 ± 35 • where the cosine-type behavior is almost perfect. By analogy, ρ H (ϕ) curves for samples with n||[111] and n||[112] were approximated in same intervals ∆ϕ = 90 ± 35 • and ∆ϕ = 270 ± 35 • (near the zeros of angular dependencies), but without reference to certain crystallographic directions (see Figure S2 in Supplementary Materials). As a result, the ρ H0 (T, H, n) and ϕ sh (T, H, n) parameters of the isotropic contribution f cos (ϕ) in (2) were found directly from this approximation. The anisotropic contribution ρ H an (T, H, n) at fixed direction H||n was determined as an average of the sum of absolute ρ H an (ϕ) values found for n at ϕ = 0 • , ϕ = 180 • , and ϕ = 360 • in the rotation experiment. As can be seen from the analysis of angular ρ H an (ϕ) dependencies undertaken below, the proposed approach reveals significant limitations and inaccuracies inherent in Hall effect measurements according to the conventional field-sweep scheme. Taking into account that ρ H const and ϕ sh ≈ 3-5 • lead only to small corrections in determining the ρ H0 = ρ H0 (T, H, n) and ρ H an = ρ H an (T, H, n) amplitudes in Equation (2), the experimentally measured Hall resistivity is discussed below as a sum of isotropic and anisotropic contributions  Figure S2 in Supplementary Materials). Figure 6 shows the result of approximation by Equation (2) of the measured Hall resistivity ρ H (ϕ) at T = 6.5 K in fixed fields up to 80 kOe for the studied four crystals. It is seen that the anisotropic contribution ρ H an (ϕ) appears just above 40 kOe having the largest amplitude for sample n||[001], it decreases by a factor of 2 for n||[110] and goes to zero for n||[111] and n||[112] samples ( Figure 6). Indeed, below 40 kOe the experimental data (shown by symbols) and the cosine fits (thin solid lines) coincide with a good accuracy indicating the absence of an anisotropic component ρ H an (ϕ) in the low field region (see also Figure S3 Figure 6). This can lead to large errors in determining the amplitude ρ H0 of the main contribution from conventional ±H field-sweep measurements. In our opinion, this finding allows us to explain the different behavior of reduced Hall resistivity ρ H (T)/H for various field directions (see Figure 2) and shed light on the shortcoming of conventional ±H field-sweep scheme commonly used for the Hall effect studies. Thus, avoiding incorrect conclusions, at low temperatures and in magnetic fields H > 40 kOe, an isotropic ordinary contribution to the Hall effect is assumed, and one common ρ H0 value found from the analysis of Equation (2) for n||[001], which was used for these four studied crystals with different n directions. At the same time, in fields H ≤ 40 kOe at T < T S~1 5 K, the ρ H /H(T) curves in intervals ϕ = 90 ± 35 • and ϕ = 270 ± 35 • differ only slightly from cosine curves ( Figure 6); therefore, approximation by Equation (2) was carried out with individual parameters of the harmonic contribution for each of the four samples. (see, e.g., Figure 6). This can lead to large errors in determining the amplitude ρH0 of the main contribution from conventional ±H field-sweep measurements. In our opinion, this finding allows us to explain the different behavior of reduced Hall resistivity ρH(T)/H for various field directions (see Figure 2) and shed light on the shortcoming of conventional ±H field-sweep scheme commonly used for the Hall effect studies. Thus, avoiding incorrect conclusions, at low temperatures and in magnetic fields H > 40 kOe, an isotropic ordinary contribution to the Hall effect is assumed, and one common ρH0 value found from the analysis of Equation (2) for n||[001], which was used for these four studied crystals    Similar behavior of Hall mobility was observed in the range 80-300 K previously for various LuB12 crystals; an α ≈ 7/4 exponent was detected for crystals with large values of It can be seen in Figure 8a that the Arrhenius-type dependence 1/R H (T)~e·n 0 ·exp(−T a /T) is valid above the cage-glass transition at T*~60 K, and that the estimated activation temperatures T a = 14.4 ± 0.6 K and T a = 16.7 ± 0.3 K detected for H||n||[001] and H||n||[110], correspondingly, are close to T S~1 5 K. As the detected initial concentration n 0 = (1.61-1.64)· 10 22 cm −3 coincides within experimental accuracy for these two field directions, the reduced Hall concentration changes in the range n e /n R = 1.2-1.6 ( Figure 8a) in accordance with the results of previous Hall effect measurements in RB 12 (R = Lu, Tm, Ho, Er) [56]. Similar behavior of Hall mobility was observed in the range 80-300 K previously for various LuB 12 crystals; an α ≈ 7/4 exponent was detected for crystals with large values of RRR ≡ ρ(300 K)/ρ(6 K) = 40-70 and α ≈ 3/2 was estimated for LuB 12 with a small RRR~12 [49]. The α = 3/2 exponent is typical for scattering of conduction electrons by acoustic phonons (deformation potential). The increase of α values up to 7/4 in best LuB 12 crystals was interpreted [49] in terms of charge carriers scattering both by quasi-local vibrations of RE ions and by boron optical phonons [57] in the presence of JT distortions and rattling modes of RE ions [58][59][60]. In the case of Ho 0.8 Lu 0.2 B 12 crystals with RRR~10 the further decrease of α value from 3/2 to 5/4 could be attributed to the emergence of strong magnetic scattering in dodecaboride with Ho 3+ magnetic ions. Note that the inequality ω c τ < 1 (see Figure 8b), which is fulfilled in the entire temperature range 1.9-300 K in fields up to 80 kOe, corresponds to the low field regime of charge carriers in Ho 0. 8  of RE ions [58][59][60]. In the case of Ho0.8Lu0.2B12 crystals with RRR~10 the further decrease of α value from 3/2 to 5/4 could be attributed to the emergence of strong magnetic scattering in dodecaboride with Ho 3+ magnetic ions. Note that the inequality ωсτ < 1 (see Figure 8b), which is fulfilled in the entire temperature range 1.9-300 K in fields up to 80 kOe, corresponds to the low field regime of charge carriers in Ho0.8Lu0.2B12. This indicates that the results of Hall effect measurements should be insensitive to the topology of the Fermi surface and depending mainly on the features of disorder and inhomogeneity of the crystals studied.

Analysis of Contributions to Hall Resistivity
Obviously, the Hall effect in Ho0.8Lu0.2B12 is strongly modified by the positive anisotropic contribution ρH an /H. Figure 9a   The amplitude ρH an /H variation with temperature is shown in Figure 9c,d in the logarithmic plot. It is worth noting that two phenomenological relations were used in [49,55,61] for the Hall effect analysis in strongly correlated electron systems with a filamentary structure of conducting channels. Among these, a hyperbolic dependence (3) of Hall resistivity was observed previously in SCES CeCu6-xAux [55] and Tm1-xYbxB12 [61]. The authors of [55,61] argued in favor of a transverse even component of the Hall signal associated with the formation of stripes (see also [38]) on the surface and in the bulk of the crystal, similar to the chains of nanoscale stripes detected in the normal phase of HTSC [62]. Equation (4a) was applied to discuss the temperature induced destruction of the coherent state of stripes in LuB12 [49]. Below, we use the analysis based on Equations (3) and (4a) to highlight quantitatively the changes caused by various orientations of the external magnetic field. Indeed, the approximation by Equation (3) Figs. 9c, 9d). Then, the analysis based on Equation (4a) provides very similar values Tа H ≈ 13.7-13.8 K for these two field directions (see Figs. 9c, 9d, red curves). Note that the estimation of Tа H agrees both with the characteristic temperature of stripe chains formation TS~15 K [48] from one side, and with the activation energy Ea/kB =Ta~14-16 K detected above in the Arrhenius type approximation of the main contribution to the Hall effect (see Figure 8a) from the other. It can be seen from Figure 9c The amplitude ρ H an /H variation with temperature is shown in Figure 9c,d in the logarithmic plot. It is worth noting that two phenomenological relations were used in [49,55,61] for the Hall effect analysis in strongly correlated electron systems with a filamentary structure of conducting channels. Among these, a hyperbolic dependence (3) of Hall resistivity was observed previously in SCES CeCu 6-x Au x [55] and Tm 1-x Yb x B 12 [61]. The authors of [55,61] argued in favor of a transverse even component of the Hall signal associated with the formation of stripes (see also [38]) on the surface and in the bulk of the crystal, similar to the chains of nanoscale stripes detected in the normal phase of HTSC [62]. Equation (4a) was applied to discuss the temperature induced destruction of the coherent state of stripes in LuB 12 [49]. Below, we use the analysis based on Equations (3) and (4a) to highlight quantitatively the changes caused by various orientations of the external magnetic field. Indeed, the approximation by Equation (3) in the range 5-40 K results in values T E ≈ 132 K and C* ≈ 8.9·10 −4 cm 3 /C for the sample with n||[001], while for n||[110], the parameter T E ≈ 135 K is almost the same and C* ≈ 4.3·10 −4 cm 3 /C turns out to be about half the size (see green curves in Figure 9c,d). Then, the analysis based on Equation (4a) provides very similar values T a H ≈ 13.7-13.8 K for these two field directions (see Figure 9c,d, red curves). Note that the estimation of T a H agrees both with the characteristic temperature of stripe chains formation T S~1 5 K [48] from one side, and with the activation energy E a /k B =T a~1 4-16 K detected above in the Arrhenius type approximation of the main contribution to the Hall effect (see Figure 8a) from the other. It can be seen from Figure 9c [111], where the magnetoresistance is minimal. Parameters T a ρ = 13.3-14.8 K found from this approximation in the same range T ≤ 10 K turn out to be close to T a H ≈ 13.3-13.8 K values and also to T S~1 5 K [48], as well as to the activation temperature T a~1 4-16 K (see Figure 8a).

Multicomponent Analysis of the Contributions to Anomalous Hal Effect (AHE) in the Regime of Ferromagnetic Fluctuations
Previous measurements of the Hall effect in Ho 0.5 Lu 0.5 B 12 in P-phase (T > T N ≈ 3.5 K) were carried out in the conventional field-sweep scheme with two opposite field directions ±H||n [63]. Ordinary and anomalous components of the Hall effect observed in [63] were described by the general relation which is usually applied to AHE in ferromagnetic metals [51,64], where R H0 and R S = const(T) are the ordinary and anomalous Hall coefficients, respectively. According to [51], the ferromagnetic AHE regime represented by Equation (5) corresponds to the intrinsic scattering mechanism. Since short-range order effects are observed in the paramagnetic phase of magnetic RB 12 in the temperature range at least up to 3T N [53,60,65] (see also Figure 4), and that a ferromagnetic component was detected in the magnetically ordered state of HoB 12 in magnetic fields above 20 kOe, it is of interest to perform the analysis within the framework of Equation (5) of the ordinary and AHE components in the vicinity of T N . In this case, relying on the above results of angular measurements of Ho 0.8 Lu 0.2 B 12 ( Figures 5-7, 9 and 10), one should use isotropic ρ H0 (T, H, n) and anisotropic ρ H an (ϕ, T, H, n) components of the Hall signal separated within the framework of Equation (2) (see . It is worth noting that the analysis performed in [43] within the framework of Equation (5) is applied to the field dependences of Hall resistivity ρ H (H) obtained in the conventional field-sweep ±H || n scheme, leading obviously to mixing of contributions ρ H0 (H) and ρ H an (H). As a result, the coefficients R H0 and R S were determined in [43] for the total averaged Hall resistivity, which contains both the isotropic ρ H0 (T, H, n) and anisotropic ρ H an (ϕ, T, H, n) components. Actually, each of the angular contributions ρ H0 and ρ H an (ϕ) is characterized by two independent coefficients R 0 and R S , which generally differ in sign. Below, we develop the analysis, considering two ferromagnetic components included in the AHE. To keep generality, we analyze our Hall effect data for all three principal directions of external magnetic field (H||[001],  H||[110], and H||[111]), despite the fact that for H||[111], the intrinsic AHE is found to be practically negligible (see Figures 7 and 10).
Thus, for a full description of the Hall effect in Ho 0.8 Lu 0.2 B 12 , we used the following relations: where ρ H 0 /B = ρ H0 /B(H, T 0 , n) and ρ H an /B = ρ H an /B(H, T 0 , n) are the reduced amplitudes of contributions ρ H0 and ρ H an (ϕ) (see Figure 7), which depend on temperature and direction of the normal vector n to the sample surface; R H 0 and R H an are components of the ordinary Hall effect, connected with magnetic induction B, and R M 0 , R M an are the coefficients of anomalous Hall effect determined by magnetization M and related to the ferromagnetic component (see Figure 3d-f). Obviously, there are two independent ordinary contributions to Hall resistivity R H 0 ·B and R H an ·B, as well as two independent anomalous (ferromagnetic) components R M 0 ·4πM and R M an ·4πM, which differ for various H directions. Note that the last two components R H an ·B and R M an ·4πM are responsible for the observed Hall effect anisotropy. Figure 11 shows the linear approximation within the framework of Equation (6)  This approximation was found to be valid at T = 6.5 K in the interval of small 4πM/B values (i.e., in high magnetic fields, in more detail see Figure S4 in Supplementary Materials), where the estimated parameters R H 0 and R H an are extracted as cutoffs, and R M 0 and R M an are the slopes of corresponding straight lines in Figure 11. Since the parameters R H 0 , R H an , R M 0 , R M an depend weakly on temperature (see Figure S4 in Supplementary Materials), the temperature-averaged values of these coefficients are summarized in Table 1.
Molecules 2023, 28 x FOR PEER REVIEW 17 of 27 Note that the last two components RH an •B and RM an •4πM are responsible for the observed Hall effect anisotropy. Figure 11 shows the linear approximation within the framework of Equation (6) of the reduced amplitudes ρH0/B and ρH an /B vs 4πM/B in the range T0 = 2.1-6.5 K for Ho0. 8Lu0.2B12 crystals with n||[001], n||[110], and n||[111]. Linear fits are acceptable for directions n||[001] and n||[110] at temperatures of 2.1 K and 4.2 K for both the isotropic ρH0/B and anisotropic ρH an /B contributions. This approximation was found to be valid at T = 6.5 K in the interval of small 4πM/B values (i.e., in high magnetic fields, in more detail see Figure S4 in Supplementary Materials), where the estimated parameters RH 0 and RH an are extracted as cutoffs, and RM 0 and RM an are the slopes of corresponding straight lines in Figure 11. Since the parameters RH 0 , RH an , RM 0 , RM an depend weakly on temperature (see Figure S4 in Supplementary Materials), the temperature-averaged values of these coefficients are summarized in Table 1.  The analysis based on Equation (6) allows us to conclude that the values of coefficients RH an and RM an , which are characteristics of the anisotropic component, turn out to be practically equal to zero for H||[111] (Figure 11f). It can be seen that the Hall coefficient RH 0~− 7.5•10 −4 cm 3 /C remains practically invariant for any H direction (see Table 1 and Figure S4 in Supplementary Materials), confirming that this ordinary negative component of   (Figure 11f). It can be seen that the Hall coefficient R H 0~− 7.5·10 −4 cm 3 /C remains practically invariant for any H direction (see Table 1 and Figure S4 in Supplementary Materials), confirming that this ordinary negative component of Hall signal is isotropic. On the contrary, the value of anisotropic positive contribution R H an changes significantly from~6.8·10 −4 cm 3 /C for H||[001] to 2.3·10 −4 cm 3 /C in H||[110] passing through zero for H||[111] ( Table 1). As a result, in Ho 0.8 Lu 0.2 B 12 for strong field H||[001], the anisotropic positive component R H an B·g(ϕ,n), which is proportional to magnetic induction, turns out to be comparable in absolute value with the negative isotropic component R H0 B of the ordinary Hall effect. Note that the values of coefficients R M 0 , R M an of the anomalous (ferromagnetic) contributions, which are proportional to magnetization, dramatically exceed the ordinary parameters R H 0 , R H an that agrees with the result [63] for Ho 0.5 Lu 0.5 B 12 .
It is very unusual that a significant isotropic anomalous positive contribution R M 0~2 5:27·10 −4 cm 3 /C appears in the paramagnetic phase of Ho 0.8 Lu 0.2 B 12 , and it may be attributed to the isotropic ferromagnetic component in the Hall signal. We propose that the R M 0 term depends on the regime of ferromagnetic fluctuations detected in low field magnetic susceptibility above T N (Figure 4). On the contrary, the anomalous negative contribution R M an varies strongly in the range (−)(21:65)·10 −4 cm 3 /C depending on M direction (see Table 1), and this component appears in strong magnetic field and at low temperatures ( Figure 11). To summarize, AHE in the paramagnetic state of Ho 0.8 Lu 0.2 B 12 is proportional to magnetization and is determined both by the positive contribution R M 0 ·4πM·cos(ϕ) and by the strongly anisotropic negative component R M an ·4πM·g(ϕ). These two contributions compensate each other in the vicinity of n||[110] (dynamic charge stripe direction [35,48]).
When discussing the nature of multicomponent Hall effect in Ho 0.8 Lu 0.2 B 12 , it is worth noting the complicated multi-q incommensurate magnetic structure in the Neel state. Magnetic ordering is characterized by propagation vector q = (1/2±δ, 1/2±δ, 1/2±δ) with δ = 0.035 and detected in [53,66,67] for Ho 11 B 12 in the neutron diffraction experiments at low temperatures in low (H < 20 kOe) magnetic field. It was also found for HoB 12 [53,66,67] that as the strength of external magnetic field increases above 20 kOe, the 4q-magnetic structure transforms into a more complex one, in which, apart from the coexistence of two AF 4q and 2q components, there additionally arises some ferromagnetic order parameter. Then, a strong modulation of the diffuse neutron-scattering patterns was observed in HoB 12 well above T N [53,67] with broad peaks at positions of former magnetic reflections, e.g., at (3/2, 3/2, 3/2), pointing to strong correlations between the magnetic moments of Ho 3+ ions. These diffuse scattering patterns in the paramagnetic state were explained in [53,67] by the appearance of correlated 1D spin chains (short chains of Ho 3 + -ion moments placed on space diagonals <111> of the elementary unit), similar to those detected in low dimensional magnets [68]. It was found that these patterns can be resolved both well above (up to 70 K) and below T N , where the 1D chains seem to condense into an ordered antiferromagnetic modulated (AFM) structure [53,67,69]. The authors [53] discussed the following scenario for the occurrence of long-range order in HoB 12 : Far above T N , strong interactions lead to correlations along [111], they are essentially one-dimensional and would not lead to long-range order at finite temperature. As T N is approached, the 1D-correlated regions grow in the perpendicular directions, possibly due to other interactions. Cigar-shaped AFM-correlated regions were proposed in [53] that become more spherical when T N is approached. Within this picture, the ordering temperature is located in the point where spherical symmetry is reached. Only then 3D behavior sets in, and HoB 12 exhibits longrange AFM order [53]. The refinement of Ho 11 B 12 crystal structure was done with high accuracy in the space group Fm3m, but also small static Jahn-Teller distortions were found in RB 12 compounds [36,41]. However, the most important factor of symmetry breaking is the dynamic one [36,41], which includes the formation both of vibrationally coupled Ho-Ho dimers and dynamic charge stripes (see [36,48,52] for more details). As a result, twofold symmetry in the (110) plane is conserved as expected for cubic crystal, but the charge stripes and Ho-Ho-coupled vibrations suppress the exchange between nearest neighbored Ho-ions. This result in the emergence of complicated phase diagrams in the AF state with a number of different magnetic phases separated by radial and circular boundaries (Maltese Cross type of angular diagrams in RB 12 [35,42,44,46]). In this scenario AF magnetic fluctuations develop well above T N in HoB 12 along trigonal axis [111], and dynamic charge stripes along <110> suppress dramatically the RKKY indirect exchange between Ho magnetic moments [44] provoking the formation of cigar-shaped AFM-correlated regions proposed in [53]. In our opinion, these effects are responsible both for the emergence of filamentary structures of fluctuating charges in these nonequilibrium metals and for the formation of spin polarization in the conduction band. This also results in the appearance of a complicated multicomponent Hall effect including two (isotropic and anisotropic) anomalous contributions.

Mechanisms of AHE in Ho 0.8 Lu 0.2 B 12
Returning to commonly used classification [51,70], it is necessary to distinguish between the intrinsic and extrinsic AHE. Intrinsic AHE is related to the transverse velocity addition due to Berry phase contribution in systems with strong spin-orbit interaction (SOI), while the extrinsic AHE associated with scattering of charge carriers by impurity centers. However, AHE also arises in noncollinear ferromagnets, in which a nonzero scalar chirality S i (S j × S k ) = 0 leads to the appearance of an effective magnetic field even in the absence of SOI [71], and in magnetic metals with a nontrivial topology of spin structures in real space [72][73][74][75][76]. When interpreting experimental data, a problem of identifying the actual mechanisms of AHE arises [51]. Among the extrinsic AHE, skew scattering, for which the scattering angularly depends on the mutual orientation of the charge carrier spin and the magnetic moment of the impurity, predicts a linear relationship between the anomalous component of the resistivity tensor ρ an H~ρxx and usually corresponds to the case of pure metals (ρ xx < 1 µOhm·cm) [77]. In the range of resistances ρ xx = 1-100 µOhm·cm the intrinsic AHE dominates, which is due to the effect of the Berry phase (ρ an H~ρxx 2 ) [78]. The contribution to scattering due to another extrinsic AHE, side jumping [79] with a similar scaling (ρ an H~ρ 2 xx0 , where ρ xx0 is the residual resistivity of the metal) is usually neglected [51]. In the "dirty limit" (ρ xx > 100 µOhm*cm), an intermediate behavior is observed with a dependence of the form ρ an H~ρxx β and with an exponent β = 1.6-1.8, which is associated usually with the transition to hopping conductivity [51].
When identifying the AHE mechanism in Ho 0.8 Lu 0.2 B 12 with a small residual resistivity (ρ xx0~1 µOhm·cm, Figure 2a), it is not possible to follow the traditional classification as no presence of the itinerant AHE with an asymptotic ρ an xy~ρxx 2 , or a skew scattering regime (ρ an H~ρxx ) was found here. However, in order to correctly compare these diagonal and off-diagonal components of the resistivity tensor, it is possible to extract the corresponding anomalous contributions. As can be seen from Figure 2, for H||n||[111], the anisotropic anomalous component of Hall signal is negligible and, as a result, the reduced Hall resistivity measured in angular experiments in direction H||n consists of the isotropic contribution only. In addition, according to conclusions made in [43,47], at low temperatures in paramagnetic state the magnetoresistance of Ho 0.8 Lu 0.2 B 12 consists of isotropic negative and anisotropic positive contributions, the latter being close to zero in the direction H||n||[111].
In this situation, for estimating the anisotropic components ρ an H and ρ an xx , e.g., for the n||[001] sample, it suffices to find the difference  components of the resistivity tensor appear in the interval T < T S~1 5 K and turn out to be related to each other as ρ an H~ρ an xx 0.83 (Figure 12), which does not favor skew scattering (β~1) [77]. Note that the exponent β for H||[110] is twice as small as that for H||[100] in Ho 0.8 Lu 0.2 B 12 , and these regimes are observed in adjacent ρ an xx intervals changing one to another at ρ an xx~0 .1 µOhm·cm (Figure 12). Such a different behavior in charge transport parameters for two different magnetic field directions suggests that the AHE is caused by another scattering mechanism, which, in particular, may result from the influence of external magnetic field on dynamic charge stripes directed along <110> (see Figure 1a).
Molecules 2023, 28 x FOR PEER REVIEW 20 of 27 Note that the exponent β for H||[110] is twice as small as that for H||[100] in Ho0.8Lu0.2B12, and these regimes are observed in adjacent ρ an xx intervals changing one to another at ρ an xx~0.1 μOhm•cm (Figure 12). Such a different behavior in charge transport parameters for two different magnetic field directions suggests that the AHE is caused by another scattering mechanism, which, in particular, may result from the influence of external magnetic field on dynamic charge stripes directed along <110> (see Figure 1a). In this scenario the appearance of two types of AHE in Ho0.8Lu0.2B12 may be interpreted as follows. The first mode of AHE associated with charge scattering in the interval T < TS~15 K is detected when magnetic field is applied along charge stripes (H||[110]), and the regime appears due to the formation of a large size cluster (long chains) in the filamentary structure of fluctuating charges. The second mode of AHE is induced by the order-disorder transition at T*~60 K and corresponds to the magnetic field applied transverse to vibrationally coupled dimers of rare-earth ions (H||[001]  <110 > ). In the latter case, when the carrier moves in transverse magnetic field along a complex path, the intrinsic AHE is expected to be influenced by the Berry phase in real space [51,78], but instead, the ρ an H~ρ an xx 1.7 scaling is observed. This unusual behavior seems to be a challenge to the contemporary AHE theory and has to be clarified in future studies.

AHE Anisotropy and Dynamic Charge Stripes
The above analysis of Hall effect contributions in Ho0.8Lu0.2B12, based (i) on measurements in the conventional field-sweep ±H||n scheme (Figures 2, 3a-  In this scenario the appearance of two types of AHE in Ho 0.8 Lu 0.2 B 12 may be interpreted as follows. The first mode of AHE associated with charge scattering in the interval T < T S~1 5 K is detected when magnetic field is applied along charge stripes (H||[110]), and the regime appears due to the formation of a large size cluster (long chains) in the filamentary structure of fluctuating charges. The second mode of AHE is induced by the order-disorder transition at T*~60 K and corresponds to the magnetic field applied transverse to vibrationally coupled dimers of rare-earth ions (H||[001] ⊥ <110>). In the latter case, when the carrier moves in transverse magnetic field along a complex path, the intrinsic AHE is expected to be influenced by the Berry phase in real space [51,78], but instead, the ρ an H~ρ an xx 1.7 scaling is observed. This unusual behavior seems to be a challenge to the contemporary AHE theory and has to be clarified in future studies. . In this situation, it is more convenient to control the projection of the external field H onto the normal vector H n = (H·n) = H 0 ·cosϕ, which is used to determine the amplitude of contributions in the corresponding n direction (see inset in Figure 2a). As can be seen from Figure 10 For ϕ = 0, one should also expect zero values of anomalous contributions to the Hall signal.

AHE Anisotropy and Dynamic Charge Stripes
In [44,[47][48][49]58], it was found that the angular dependence of magnetoresistance (MR) in RB 12 is determined by scattering of carriers on dynamic charge stripes. As a result, the maximum positive values of MR are observed for H||[001] perpendicular to the direction of these electron density fluctuations, while the minimum MR is observed for H||[111] (see Figure 13b). To clarify the nature of these anomalies in angular AHE curves, one can restore the angular dependence of the AHE in the entire range of 0-360º and compare the obtained curve with the related MR data. This analysis can be performed by relying on experimental ρ H /B(ϕ) curves measured at T = 2.1 K in magnetic field H = 80 kOe for four different crystals when H is rotated in the same plane (110). Since both the ordinary and anomalous components of Hall signal can be described by cosine dependence ρ H (ϕ) = ρ H0 ·cos(ϕ − ϕ 1 ) + ρ H an ·g(ϕ), the representation of experimental data shown in Figure 13a in the form of ρ H (ϕ)/(H·cos(ϕ − ϕ 1 )) allows us to separate the isotropic and anisotropic contributions from Hall experiments.
Molecules 2023, 28 x FOR PEER REVIEW 21 of 27 = H0•cosφ, which is used to determine the amplitude of contributions in the corresponding n direction (see inset in Figure 2a). As can be seen from Figure 10 In [44,[47][48][49]58], it was found that the angular dependence of magnetoresistance (MR) in RB12 is determined by scattering of carriers on dynamic charge stripes. As a result, the maximum positive values of MR are observed for H||[001] perpendicular to the direction of these electron density fluctuations, while the minimum MR is observed for H||[111] (see Figure 13b). To clarify the nature of these anomalies in angular AHE curves, one can restore the angular dependence of the AHE in the entire range of 0-360º and compare the obtained curve with the related MR data. This analysis can be performed by relying on experimental ρH/B(φ) curves measured at T = 2.1 K in magnetic field H = 80 kOe for four different crystals when H is rotated in the same plane (110). Since both the ordinary and anomalous components of Hall signal can be described by cosine dependence ρH(φ) = ρH0•cos(φ − φ1) + ρH an •g(φ), the representation of experimental data shown in Figure  13a in the form of ρH(φ)/(H•cos(φ − φ1)) allows us to separate the isotropic and anisotropic contributions from Hall experiments.   The averaged envelope (indicated by yellow shading in Figure 13a) was obtained after removing the particular portions of the related angular dependences with singularities associated with division by small values (zeros of cosine), and then averaging the data of these four angular Hall signal dependences. In this case, in accordance with the data in Figure 2b, in H||<111> directions on the resulting envelope curve ρ H (ϕ)/(H·cos(ϕ − ϕ 1 )), the maximum negative values of about −6 × 10 −4 cm 3 /C correspond to isotropic ordinary component ρ H0 /H of Hall effect, which is independent on magnetic field direction. The positive anisotropic component reconstructed from the data of four measurements (the yellow shading in Figure 13a) provide changes of the ρ H (ϕ)/(H·cos(ϕ − ϕ 1 )) in the range (3.2:6)·10 −4 cm 3 /C. Despite the fact that the initial ρ H an (ϕ) dependence is an odd function (see Figures 5,6,9 and 10), the result of its division by the odd cos(ϕ − ϕ1) allows one to obtain the real anisotropic even amplitude of Hall effect and compare it with MR. The location of its extrema coincides with the positions of anomalies on the MR curve ( Figure 13b). Indeed, the maximum positive contribution to AHE appears synchronously with the MR peak along <001>, while for <110>, a small (if compared with the anomaly along <001>) positive AHE component is recorded (Figure 13b) simultaneously with a small amplitude singularity of MR. We also note that two spatial diagonals <111> on the anisotropic contribution ρ H an g(ϕ)/(H·cos(ϕ − ϕ 1 )) seem to be equivalent and show no hysteretic features. The observed behavior of the Hall effect in Ho 0.8 Lu 0.2 B 12 agrees very well with symmetry lowering of the fcc structure of Ho 0.8 Lu 0.2 B 12 due to static and dynamic Jahn-Teller distortions [36].
Finally, the comparison of the angular dependences of MR and Hall effect in Ho 0.8 Lu 0.2 B 12 ( Figure 13) shows that, along with the normal isotropic contributions to the diagonal (negative MR) and off-diagonal (the ordinary Hall coefficient of negative sign) components of the resistivity tensor, anomalous anisotropic positive components appear both in the MR and in Hall effect at low temperatures. These components reach (i) maximal values in the direction of magnetic field transverse to dynamic charge stripes (H||[001]) and (ii) zero values for H||[111]. This anisotropy arises simultaneously with the transition to the cage glass state at T*~60 K and seems to be related to the formation of vibrationally coupled pairs of rare-earth ions displaced from their centrosymmetric positions in B 24 cavities of the boron sublattice [48]. A significant increase of this anisotropy is detected at temperatures T < T S~1 5 K upon the formation of large size clusters (long chains) in the filamentary structure of fluctuating electron density (stripes). Taking into account that, according to the results of room temperature measurements of the dynamic conductivity of LuB 12 , about 70% of charge carriers participate in the formation of the collective mode (hot electrons) [80], a redistribution of carriers between the nonequilibrium and Drude components should be expected with decreasing temperature.
Apparently, the activation behavior of the Hall concentration of charge carriers observed in the range 60-300 K (T a~1 4-17 K, Figure 8a) may be attributed to the involvement of additional conduction electrons in the collective mode. When vibrationally coupled dimers of rare-earth ions are formed below T*~60 K, short and disordered chains of stripes oriented along <110> appear in magnetic field, initiating the emergence of intrinsic AHE ( Figure 12). We propose that the AHE in Ho 0.8 Lu 0.2 B 12 is caused by a transverse addition to velocity due to the Berry phase contribution [51], which arises for carriers moving in a complex filamentary structure of the electron density in magnetic field applied transverse to dynamic stripes. During the formation of large size clusters in the structure of stripes (interval T < T S~1 5 K) in field orientation along the stripes H||[110], no intrinsic AHE is expected. A skew scattering contribution, for which the scattering angle depends on the mutual orientation of the charge carriers spin and the magnetic moment of the impurity, may become noticeable with a linear relationship ρ an H~ρ an xx between these components of the resistivity tensor ( Figure 12). We propose that some geometric factors are responsible for the reduction of β exponent in these two AHE regimes. Approaching the AF transition above T N , on-site 4f -5d spin fluctuations in the vicinity of Ho 3+ ions lead to magnetic polarization of the 5d states of the conduction band, which gives rise to ferromagnetic fluctuations in Ho 0.8 Lu 0.2 B 12 (Figure 4). These produce ferromagnetic nanoscale domains (ferrons), and, as a result, initiate the appearance of the ferromagnetic contribution to AHE ( Figure 11 and Table 1). We emphasize that such a complex multicomponent AHE in model Ho 0.8 Lu 0.2 B 12 metal with a simple fcc lattice turns out to be due to the inhomogeneity and complex filamentary structures of the electron density in the matrix of this SCES with dynamic Jahn-Teller lattice instability and electron phase separation.

Experimental Details
Ho 0.8 Lu 0.2 B 12 single-domain crystals were grown by crucible-free induction zone melting in an inert argon gas atmosphere (see, e.g., [34]). Magnetization was measured with the help of a SQUID magnetometer MPMS Quantum Design in fields up to 70 kOe in the temperature range 1.9-10 K. The external magnetic field was applied along the principal crystallographic axes -H||[001], H||[110], and H||[111]. Resistivity, magnetoresistance, and Hall effect were studied on an original setup using the standard DC five-probe technique with excitation current commutation. The angular dependences of transverse magnetoresistance and Hall resistivity were obtained using a sample holder of original design, which enables the rotation of the vector H located in the plane perpendicular to fixed current direction I||[110]⊥H with a minimum step ϕ step = 0.4 • (see the schematic view on the inset of Figure 2a). Measurements were carried out in a wide temperature range 1.9-300 K in magnetic fields up to 80 kOe, the angle ϕ = nˆH (n-normal vector to the lateral sample surface) varied in the range ϕ = 0-360 • . The measuring setup was equipped with a stepper motor, which enabled an automatic control of sample rotation, similar to that one used in [35]. High accuracy of temperature control (∆T ≈ 0.002 K in the range 1.9-7 K) and magnetic field stabilization (∆H ≈ 2 Oe) was ensured, respectively, by LLC Cryotel, (Moscow, Russia) TC 1.5/300 temperature controller and Cryotel SMPS 100 superconducting magnet power supply in combination with a CERNOX 1050 thermometer and n-InSb Hall sensors.

Conclusions
In the paramagnetic phase of the cage-cluster high-boron boride Ho 0.8 Lu 0.2 B 12 with a cage-glass state below T*~60 K and electronic phase separation (dynamic charge stripes), magnetotransport was studied at temperatures 1.9-300 K in magnetic fields up to 80 kOe. Field and angular measurements of resistivity and Hall resistivity were performed on single-domain crystals of the Ho 0.8 Lu 0.2 B 12 model metal allowed to separate and analyze several different contributions to the Hall effect. It was shown that, along with the negative ordinary isotropic component of Hall resistivity, an intrinsic AHE of a positive sign arises in the cage-glass state in field direction H||[001], which is perpendicular to the charge stripe chains. This AHE corresponds through the relation ρ an H~ρ an xx 1.7 to the anomalous components of the resistivity tensor. It was also found that at temperatures T < T S~1 5 K, where long chains prevail in the filamentary structure of fluctuating charges (stripes), a contribution to AHE of the form ρ an H~ρ an xx 0.83 becomes dominant when H||[110]. We propose that these two components are intrinsic (a transverse addition to velocity due to the contribution of the Berry phase) and extrinsic (from the skew scattering mechanism) [51], respectively, and exhibit some decrease of exponents from integers due to the geometric factor. In the paramagnetic phase near Neel temperature, on-site 4f -5d spin fluctuations in the vicinity of Ho 3+ ions were found to induce spin-polarized 5d states (ferromagnetic nanoscale domains-ferrons) in the conduction band, which result in the appearance of an additional ferromagnetic contribution to AHE, as observed both in the isotropic and anisotropic components of Hall effect. Detailed measurements of the angular dependences of Hall resistivity and MR with vector H rotation in the (110) plane, perpendicular to the direction of stripes, made it possible to separate the negative isotropic and positive anisotropic contributions to AHE and MR, and to explain them in terms of charge carriers scattering by dynamic charge stripes.

Supplementary Materials:
The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/molecules28020676/s1, Figure S1: The direct match between (a) the data obtained in the conventional field-sweep scheme of Hall effect measurements with two opposite directions of ±H||n and (b) the data extracted in the experiment with step-by-step rotation of the sample around I|| [1][2][3][4][5][6][7][8][9][10] with a fixed H direction in the transverse plane; Figures S2 and S3: The approximation of the angular dependences of Hall resistivity ρH(ϕ) for H = 80 kOe in the temperature range 2.1-30 K for n|| [111] and n||[112] by Equation (2); Figure S4: Coefficients RH0, RHan of the ordinary and RM0, RMan of the anomalous Hall effect (Equation (6)) depending on temperature for three vectors n||[001], n|| [110], and n||[111]; Figure S5: The X-ray powder analysis; Figure S6: Typical X-ray Laue back reflection patterns; Figures S7-S9: Verification of the single-domain crystals by the rocking curve control; Figure S10: The results of the microprobe analysis of the Ho 0.8 Lu 0.2 B 12 single crystals; Table S1: The corrections due to demagnetizing factors and values of the demagnetization factor, depending on the type of experiment; Table S2

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.