Hall Effect in the Antiferromagnetic State of Ho_0.8Lu_0.2B₁₂

Detailed measurements of the Hall effect in the Ho_0.8Lu_0.2B₁₂ antiferromagnetic compound (Néel temperature T_N = 5.75 K) in magnetic fields up to 80 kOe oriented in the (110) plane at temperatures of 1.9–6.6 K are performed. It is established with the contribution separation procedure that the anisotropic positive contribution, which is responsible for the double inversion of the sign of the Hall resistance, dominates in the antiferromagnetic state of Ho_0.8Lu_0.2B₁₂ in fields of 30–50 kOe. A sharp decrease in the amplitude of the isotropic negative contribution at the transition to the antiferromagnetic phase is found. The nature of the detected anomalies in the antiferromagnetic metal with dynamic charge stripes and spin-wave component of the magnetic structure is discussed.

1 INTRODUCTION

Dynamic charge stripes in RB₁₂ metals with strong electron correlations are responsible for anomalies in the thermodynamic and transport properties [1, 2]. It was recently shown that the Hall effect in the LuB₁₂ nonmagnetic metal [8] and in the paramagnetic phase of Ho_xLu_{1 – x}B₁₂ antiferromagnetic compounds [9, 10] becomes multicomponent in the presence of the structural (cooperative Jahn–Teller effect [3–7]) and electronic (dynamic stripes in the $\langle 110\rangle$ direction [2–7]) instabilities in the fcc lattice of RB₁₂ rare-earth dodecaborides (R is Ho, Er, Tm, Yb, or Lu). In this case, in addition to the almost isotropic normal negative component with the Hall coefficient R_H(T), the anomalous Hall effect (AHE) is observed in Ho_xLu_{1 – x}B₁₂, which determines the anisotropy of the Hall resistance in the diamagnetic (LuB₁₂ [8]) and paramagnetic (Ho_xLu_{1 – x}B₁₂ [9, 10]) states. Since dynamic stripes in RB₁₂ are high-frequency (~240 GHz, see [4]) oscillations of the electron density, they suppress the indirect exchange interaction through conduction electrons (Ruderman–Kittel–Kasuya–Yosida mechanism) between the nearest magnetic rare-earth ions in the 〈110〉 direction, leading to the appearance of complex multicomponent magnetic phase diagrams for RB₁₂ antiferromagnetic (AFM) compounds [2, 11–15]. In particular, according to [11], numerous various magnetically ordered phases are observed on the H–T phase diagrams in the AFM state of Ho_0.8Lu_0.2B₁₂ for three different directions [001], [110], and [111] of the external magnetic field H (see I–IX in Fig. 1). At the same time, the X phase is detected on the H–φ phase diagram on the (110) plane in the field range H ≤ H_N (H_N is the Néel field) at the deviation of the vector H from the $\langle 100\rangle$ and $\langle 111\rangle$ directions (see the circular magnetic phase diagrams in [11]).

Fig. 1.

(Color online) H–T magnetic phase diagrams of Ho_0.8Lu_0.2B₁₂ for the (a) [001], (b) [110], and (с) [111] directions of the normal n obtained from magnetoresistance, heat capacity, and magnetization data [11]. The vertical dashed straight lines mark the Néel temperature T_N and temperatures at which the angular curves of the Hall resistance were detected. Roman numerals mark magnetically ordered phases and P means the paramagnetic phase.

Since the radial and angular phase boundaries in Fig. 1 and circular diagrams in [11] correspond to sharp change in the magnetoresistance and to the appearance of features of the heat capacity and magnetic susceptibility in the AFM state of Ho_0.8Lu_0.2B₁₂, anomalies in the Hall effect caused by these phase transitions should also be expected. Furthermore, it is expected that the appearance of the complex AFM structure consisting of 4f (localized magnetic moments) and 5d (spin-density wave) components [11] should significantly modify the contributions of the normal and anomalous Hall effects to the Hall resistance. For this reason, it is of interest to analyze features of the Hall effect in the Ho_0.8Lu_0.2B₁₂ AFM compound with dynamic charge stripes. To this end, we performed for the first time measurements with the separation of contributions to the Hall effect in the AFM state of Ho_0.8Lu_0.2B₁₂ with the magnetic phase diagram in the form of a Maltese cross [11].

2 EXPERIMENTAL METHOD

We studied high-quality Ho_0.8Lu_0.2B₁₂ single-crystal single-domain samples grown at the Frantsevich Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, by crucible-free induction zone melting in an inert argon atmosphere [16]. The transverse magnetoresistance and Hall effect in samples with the normal to the surface n || [001] placed in the external magnetic field $H \leqslant 80{\kern 1pt}$ kOe were measured in the temperature range of 1.9–300 K by the conventional dc five-contact method with commutation. The used measuring cell ensured the step-by-step rotation of the single-crystal sample by the angle $\varphi = \angle \mathbf{n},\mathbf{H}$ in the $(\bar {1}10)$ plane perpendicular to the measuring current I || $[\bar {1}10]$; see the inset of Fig. 1.

3 RESULTS AND DISCUSSION

Figure 2 presents the field dependences of the (a) magnetoresistance Δρ/ρ and (b) reduced Hall resistance R_H = ρ_xy/H = ρ_H/H in the magnetic field directed along the normal n || [001] (conventional scheme of the measurement of the Hall effect with opposite directions of the magnetic field ±H || n, arrows next to the lines mark phase transitions). It is seen that all curves Δρ/ρ(H) (Fig. 2а) and ρ_H(H)/H (Fig. 2b) in the AFM state (H < H_N) exhibit a wide maximum corresponding to the additional positive component in the Hall effect and magnetoresistance, whereas orientational phase transitions are detected in low fields. We note that the additional positive contribution in fields of 15–55 kOe at low temperatures exceeds the absolute value of the negative component, leading to the double inversion of the sign of the Hall resistance in the magnetic field (see Fig. 2b). The transition to the paramagnetic state (see, e.g., data for T = 6.6 K in Fig. 2b) is accompanied by a sharp decrease in the amplitude of changes in ρ_H(H)/H, but the absolute values of negative ρ_H(H)/H values in high fields decrease slightly with decreasing temperature.

Fig. 2.

(Color online) Field dependences of the (a) magnetoresistance Δρ/ρ and (b) reduced Hall resistance ρ_H/H in the temperature range of 1.9–6.6 K for Ho_0.8Lu_0.2B₁₂ with the normal n || [001] to the surface of the sample. The data for ρ_H/H are obtained in the traditional scheme of measurements with opposite directions of the magnetic field ±H || n (see the main text). Arrows next to the lines mark the orientational phase transitions and AFM–Р transition at the Néel field H_N.

Figures 3a, 3b, 4a, and 4b show the angular dependences ρ_H(φ) in fields up to 80 kOe for the normal n || [001] at temperatures of 2.1 K (Figs. 3a, 3b) and 4.2 K (Figs. 4a, 4b) in the paramagnetic (Figs. 3a, 4a) and AFM (Figs. 3b, 4b) states. The measured angular curves ρ_H(φ) are corrected by the subtraction of a small (less than 2%) magnetoresistance contribution caused by the nonequipotential location of the Hall contacts to the sample (see, e.g., [8]). Since the angular dependences of the magnetoresistance Δρ/ρ(φ, H₀) at fixed H₀ values were used in [11] to obtain circular sections of the H–φ magnetic phase diagram (see Figs. 3с and 4с), it is of interest to compare the positions of features of the Hall resistance ρ_H(φ, H₀) with phase boundaries. In low fields H < 10 kOe corresponding to phase I in Figs. 3с and 4с, curves are described with a high accuracy by the dependence ${{\rho }_{{\text{H}}}}(\varphi ) = \rho _{{\text{H}}}^{0}\cos \varphi$ (see, e.g., the curve for H₀ = 10 kOe in Fig. 4b), which was observed in [10] in the range H < 40 kOe in the paramagnetic phase of Ho_0.8Lu_0.2B₁₂. It was shown in [10] that the angular dependence of the Hall resistance in the paramagnetic state of Ho_0.8Lu_0.2B₁₂ becomes much more complex with increasing field and can be represented as the sum of two terms

$${{\rho }_{{\text{H}}}}(\varphi ) = \rho _{{\text{H}}}^{0}\cos \varphi + \rho _{{\text{H}}}^{{{\text{an}}}}(\varphi ),$$

(1)

where the first term is the negative isotropic component and the second term is the positive anisotropic component with a complex anharmonic angular dependence (see also Figs. 3a and 4a). Pronounced features (see Figs. 3b and 4b) appear on the ρ_H(φ) curves in the AFM phase with an increase in H in the range of 10–60 kOe in the angular interval corresponding to the phases II–III (near [001] in Figs. 3c and 4с) and the phases IV–V–VI (near [110] in Figs. 3с and 4c). The character of features in the phases II–III implies the common genesis with Hall anomalies observed in the paramagnetic phase near [001] (see Figs. 3a and 4a), but their structure is much more complex. Singularities of ρ_H(φ) in the phases V–VI have a smaller amplitude and are observed near the [110] direction corresponding to $\rho _{{\text{H}}}^{0}\cos \varphi$ = 0 (see Figs. 3b and 4b). As seen in Figs. 3b and 4b, Hall curves in the phase VIII (near [111] in Figs. 3c and 4с) are described with a high accuracy by the harmonic dependence $\rho _{{\text{H}}}^{0}\cos \varphi$, which allows one to separate the isotropic and anisotropic contributions in the Hall effect in terms of Eq. (1). Figures 3d and 4d show the anisotropic anharmonic contribution $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ in the angular range of 90°–270° with correspondence to various magnetically ordered phases on the H–φ phase diagram (denoted by Roman numerals in Figs. 3c and 4с) for temperatures of 2.1 and 4.2 K, respectively. We emphasize that, in addition to sharp features of $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$, which correspond to phase boundaries and were previously found in measured magnetoresistance [11], additional anomalies are observed in the phases III and VI. Their positions correspond to the field directions (i) along (H || $\langle 110\rangle$) and (ii) across (H || $\langle 001\rangle$) the dynamic stripes and (iii) to the singularities near the directions $\langle 001\rangle$ ± 15° (see Figs. 3d and 4d). One of the possible explanations of such sharp changes in $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ in the directions $\langle 001\rangle$ ± 15° can be additional phase boundaries on the H–φ phase diagram, which can also be observed on the angular curves of the magnetoresistance [11]. We note that, using the approach developed for LuB₁₂ [8], two components corresponding to the stepwise $({{\Delta }_{{{\text{step}}}}})$ and smooth $({{\Delta }_{{{\text{smooth}}}}})$ changes on the dependences $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ can be identified in Figs. 3d and 4d in the immediate vicinity of the $\langle 001\rangle$ direction (see the notation in Figs. 3d and  4d). The sum of these components $\rho _{{\text{H}}}^{{{\text{an}}}}(0) =$ ${{\Delta }_{{{\text{step}}}}} + {{\Delta }_{{{\text{smooth}}}}}$ determines the amplitude of the anisotropic AHE both on the angular curves for φ = 0 (H || $\langle 001\rangle$ in Figs. 3d and 4d) and on the field dependences measured in the conventional experimental scheme in the field directed along the normal (±H || n || [001], see Fig. 2b). The analysis within the scope of Eq. (1) makes it possible to estimate the reduced amplitudes of contributions to the Hall effect for the normal n || [001] in the AFM state (T = 2.1 and 4.2 K in Fig. 5) and in the paramagnetic phase (T = 6.5 K). Large symbols in  Fig. 5b show the amplitudes $\rho _{{\text{H}}}^{{{\text{an}}}}({\mathbf{H}}\parallel \langle 001\rangle )$ = ${{\Delta }_{{{\text{step}}}}} + {{\Delta }_{{{\text{smooth}}}}}$, and small symbols present the component ${{\Delta }_{{{\text{smooth}}}}}({\mathbf{H}}\parallel \langle 001\rangle )$. The right vertical axis in Fig. 5a shows the reduced charge carrier density $n{\text{/}}{{n}_{{\text{R}}}} \approx {{(e\rho _{{\text{H}}}^{0}{\text{/(}}H{{n}_{{\text{R}}}}))}^{{ - 1}}}$, where n_R = 0.95 × 10²² cm^–3 is the concentration of rare-earth ions and e is the elementary charge, which corresponds to the Hall coefficient ${{R}_{{\text{H}}}}(H) = \rho _{{\text{H}}}^{0}(H){\text{/}}H$. It is seen that the transition to the AFM state leads to a noticeable (almost by a factor of 2, Fig. 5а) suppression of the amplitude of the isotropic negative contribution $\rho _{{\text{H}}}^{{\text{0}}}{\text{/}}H$, which is accompanied by the appearance and field-induced amplification of the anisotropic positive contribution $\rho _{{\text{H}}}^{{{\text{an}}}}{\text{/}}H$ (Fig. 5b). In turn, the maximum amplitude of the AHE contribution $\rho _{{\text{H}}}^{{{\text{an}}}}{\text{/}}H$ determined from angular measurements is more than twice in absolute value as large as the component $\rho _{{\text{H}}}^{{\text{0}}}{\text{/}}H$ (see Figs. 3b, 3d, and 5). As a result, the positive anisotropic AHE appearing in the AFM state is responsible for the double inversion of the sign of the Hall coefficient in Ho_0.8Lu_0.2B₁₂ (see Fig. 2b).

Fig. 3.

(Color online) (Points) Angular dependences of the Hall resistance ρ_H(φ) and (thin lines) approximating lines $\rho _{{\text{H}}}^{0}\cos \varphi$ at a temperature of 2.1 K for the normal n || [001] in the (a) paramagnetic and (b) antiferromagnetic states. The anisotropic contribution $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ in the Р phase is shown by thick lines in panel (a) and the anisotropic contribution $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ in the AFM phase is shown in panel (d) in the region of the strongest changes. The vertical dashed lines in panels (b) and (d) indicate the phase boundaries; the corresponding phases are marked on the H–φ circular magnetic phase diagram [11] shown in panel (c). The separation of $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ into the contributions ${{\Delta }_{{{\text{step}}}}}$ and ${{\Delta }_{{{\text{smooth}}}}}$ is also shown in panel (d) (see the main text).

Fig. 4.

(Color online) (Points) Angular dependences of the Hall resistance ρ_H(φ) and (thin lines) approximating lines $\rho _{{\text{H}}}^{0}\cos \varphi$ at a temperature of 4.2 K for the normal n || [001] in the (a) paramagnetic and (b) antiferromagnetic states. The anisotropic contribution $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ in the Р phase is shown by thick lines in panel (a) and the anisotropic contribution $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ in the AFM phase is shown in panel (d) in the region of the strongest changes. The vertical dashed lines in panels (b) and (d) indicate the phase boundaries; the corresponding phases are marked on the H–φ circular magnetic phase diagram [11] shown in panel (c). The separation of $\rho _{{\text{H}}}^{{{\text{an}}}}(\varphi )$ into the contributions ${{\Delta }_{{{\text{step}}}}}$ and ${{\Delta }_{{{\text{smooth}}}}}$ is also shown in panel (d) (see the main text).

Fig. 5.

(Color online) Field dependences of the reduced amplitudes of the (a) isotropic $\rho _{{\text{H}}}^{0}{\text{/}}H$ and (b) anisotropic $\rho _{{\text{H}}}^{{{\text{an}}}}$/H contributions (see Eq. (1)) for Ho_0.8Lu_0.2B₁₂ with the normal n || [001] at temperatures of 2.1, 4.2, and 6.5 K. In panel (b), large symbols correspond to the AHE $\rho _{{\text{H}}}^{{{\text{an}}}}$/H, small symbols present the component ${{\Delta }_{{{\text{smooth}}}}}$, and the colored shaded regions demonstrate the contribution ${{\Delta }_{{{\text{step}}}}}{\text{/}}H = \rho _{{\text{H}}}^{{{\text{an}}}}{\text{/}}H - {{\Delta }_{{{\text{smooth}}}}}{\text{/}}H$ (see text and notation in Figs. 3d and 4d).

The formation of dynamic charge stripes in the paramagnetic phase of Ho_xLu_{1 – x}B$_{{12}}$ compounds is considered as the main reason for the appearance of the anisotropic positive component both in the magnetoresistance [2, 11, 12, 17] and in the Hall effect [10]. In this context, a sharp increase in the contribution $\rho _{{\text{H}}}^{{{\text{an}}}}{\text{/}}H$ in the AFM state detected in this work should also be attributed to the dominant effect of stripes on the charge transport in Ho_xLu_{1 – x}B₁₂. It was shown in [11] that the spin-wave component of the magnetic structure (spin polarization of 5d states of conduction electrons) with the amplitude nonmonotonically depending on the external magnetic field strength (see also [18, 19]) is formed in the AFM phase in Ho_0.8Lu_0.2B₁₂. It can be assumed that the nonmonotonic behavior of $\rho _{{\text{H}}}^{0}(H){\text{/}}H$ in this one-electron metal is due to the transition of a large fraction of charge carriers to the spin-density wave state. The enhancement of the spin-density wave in the field is maximal near 30–50 kOe (Fig. 5а). Synchronous changes also occur in the filamentary structure of fluctuating charges, which is stabilized in the magnetic field perpendicular to the stripes, leading to the enhancement of the AHE (Fig. 5b). The field-induced suppression of the regions of the 5d spin polarization in the spin-density wave near H_N [11, 20, 21] and the related enhancement of magnetic-center-associated spin fluctuations are responsible for the enhancement of scattering and, as a result, a sharp change in both components in the Hall effect.

4 CONCLUSIONS

Field, angular, and temperature measurements of the Hall resistance of Ho_0.8Lu_0.2B₁₂ single-crystal samples with dynamic charge stripes have been performed. The results have been analyzed with the separation of contributions in the antiferromagnetic state at T = 1.9–6.6 K in magnetic fields up to 80 kOe for the normal n || [001]. It has been established that the Hall effect in AFM phases is described by the sum of the negative isotropic normal and positive anisotropic AHE components; the AHE dominates in fields of 30–50 kOe and is responsible for the double inversion of the sign of the Hall coefficient. Reasons are presented for the interaction of the spin density wave and charge stripes with the external magnetic field, which finally results in the AHE and a significant renormalization of the isotropic contribution to the Hall effect.