Anisotropic Hysteretic Hall Effect and Magnetic Control of Chiral Domains in the Chiral Spin States of ${\mathrm{Pr}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$

We uncover a strong anisotropy in both the anomalous Hall effect (AHE) and the magnetoresistance of the chiral spin states of ${\mathrm{Pr}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$. The AHE appearing below 1.5 K at a zero magnetic field shows hysteresis which is most pronounced for fields cycled along the [111] direction. This hysteresis is compatible with the field-induced growth of domains composed by the 3-in 1-out spin states which remain coexisting with the 2-in 2-out spin ice manifold once the field is removed. Only for fields applied along the [111] direction, we observe a large positive magnetoresistance and Shubnikov–de Haas oscillations above a metamagnetic critical field. These observations suggest the reconstruction of the electronic structure of the conduction electrons by the field-induced spin texture.

Figure 1

Hall resistivity ${\rho }_{xy}=R_{xy}·t$, where $t$ is the sample thickness, for a ${\mathrm{Pr}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ single crystal, as a function of magnetic field $H$ along (a) [111], (b) [110], and (c) [100] and for $T=30$ and 500 mK, respectively. These traces were measured for increasing (blue [medium gray] and green [dark gray] traces) and decreasing (magenta [light gray] and black traces) field scans. (d) Magnetization $M$ as a function of $H$, for increasing and decreasing field scans, and, respectively, for $T=0.06$ and 0.5 K [7]. Notice the absence of hysteresis among the traces acquired at $T=0.5\mathrm{K}$, but its presence in the Hall response, as shown in (a).

Figure 2

Hall resistivity ${\rho }_{xy}$ measured at $T=0.03\mathrm{K}$ as a function of $H$ applied along (a) [111], (b) [110], and (c) [100]. The kink at $B_c$ corresponds to a metamagnetic transition observed only for fields along the [111] direction. A positive Hall response is found to increase and nearly saturate as the field increases, particularly for $H\parallel [110]$ and [100] directions, suggesting the progressive orientation and concomitant saturation of the population of tetrahedrons containing the 2-in 2-out spin configurations with a net moment along the field. As discussed in Ref. 18 the 2-in 2-out state is characterized by a net positive chirality. For fields along the [111] direction, the subsequent increase in $H$ leads to a change in the sign of ${\rho }_{xy}$ which was attributed to the field-induced stabilization of tetrahedra containing the 3-in 1-out spin configuration with opposite chirality and concomitant negative contribution to the Hall response [18]. When $H$ is brought down to zero ${\rho }_{xy}(H=0\mathrm{T})$ displays a negative asymptotic value indicating the presence of chiral domains composed of the 3-in 1-out spin configuration.

Figure 3

(a) Magnetoresistance ratio $[{\rho }_{xx}(H)-{\rho }_{xx}(0)]/{\rho }_{xx}(0)$ for all three principal axes at $T=0.03\mathrm{K}$ and for both increasing and decreasing field sweeps. Inset: Oscillatory component in ${\rho }_{xx}$ obtained by subtracting the background signal after adjusting it to a fourth degree polynomial. Dotted line is a fit to a single Lifshitz-Kosevich oscillatory component with field-dependent frequency and Dingle temperature. For $H\ge 23\mathrm{T}$ the system has reached its quantum limit and no Shubnikov de Haas oscillations are observed in fields up to 35 T. (b) Oscillatory component for the [111] direction superimposed into ${\rho }_{xx}$, as a function of $H^{-1}$ and for several temperatures. Red [medium gray] lines are fits to the Lifshitz-Kosevich formulae $x/\sinh x$, where $x=14.69m^{\star }T/H$, for two values of magnetic field, i.e., 16.8 and 24 T, and from which we extract the respective effective masses $m^{\star }$. Both the geometry of the Fermi-surface and the effective mass are field-dependent.

Figure 4

Zero-field Hall resistivity ${\rho }_{xy}$ as a function of increasing $T$ measured after the field was swept to $\pm 1\mathrm{T}$ (or without crossing $B_c$) at respectively $T=0.075\mathrm{K}$ (magenta [light gray]) and at 0.5 K (green [medium-dark gray]). The raw signal as a function of temperature was measured after the field was slowly ramped down to zero at a rate of $1\mathrm{T}/\mathrm{hr}$. The blue [dark gray] trace corresponds to ${\rho }_{xy}$ measured with increasing $T$ after the sample was cooled from 2 K to 0.05 K under $\pm 8\mathrm{T}$, respectively. In each case, ${\rho }_{xy}$ was obtained by subtracting one trace from the other in order to suppress the resistive component. In a similar manner the red [medium gray] trace was obtained from both curves obtained after the subsequent cooldowns at zero field.