Critical behavior of the magnetic Weyl semimetal PrAlGe

Noncentrosymmetric PrAlGe is considered to be a magnetic Weyl semimetal that breaks both time-reversal and inversion symmetry, in which the Weyl nodes can be moved by magnetization. However, the magnetic interaction of this system remains poorly understood. In this work we perform a systematical investigation on magnetic properties and critical behaviors in single-crystal PrAlGe. The temperature, field, and angle dependence of magnetization reveal a strong magnetic anisotropy along the $c$ axis and absolute isotropic characteristic in the $ab$ plane. By fitting to the magnetic entropy change $\left[\mathrm{\Delta }{S}_M(T,H)\right]$ around the critical temperature $T_C=16$ K, critical exponents $\beta =0.136\left(6\right), \gamma =1.801\left(7\right)$, and $\delta =14.2\left(6\right)$ are obtained. Based on the obtained critical exponents, $\mathrm{\Delta }{S}_M(T,H)$ and $M(T,H)$ curves are scaled into universality curves under the framework of universality principle. The critical behaviors and exponents suggest that the magnetic interaction in PrAlGe is of a two-dimensional Ising type, revealing a uniaxial magnetic interaction along the $c$ axis. However, the ordering moments are tilted from the $c$ axis, which causes antiferromagnetism in the $ab$ plane. The clarification of the magnetic interaction and magnetic structure is of significance for unveiling its interplay with topological properties in this system.

Figure 1

(a) Crystal structure of PrAlGe. (b) XRD pattern of the single-crystal PrAlGe (inset gives the rocking curve). (c) EDX spectrum for single-crystal PrAlGe.

Figure 2

Temperature-dependent magnetization with (a) $H//ab$ and (b) $H//c$ (inset magnifies the phase transition region for $H//c$).

Figure 3

(a) Field dependence of isothermal magnetization $\left[M\right(H\left)\right]$ up to 30 T at $T=1.8$, 14, 16, and 18 K with $H//ab$ and $H//c$. (b) 3D plot of angle-dependent magnetization $\left[M\right(\phi \left)\right]$ at $T=2$ K under selected field: in-plane $M\left(\phi \right)$ ($xy$ plane) with $H$ rotated within the $ab$ plane; out-of-plane $M\left(\phi \right)$ ($xz$ plane) with $H$ rotated from the $ab$ plane to the $c$ axis.

Figure 4

(a) Field dependence of isothermal initial magnetization $\left[IM\right(H\left)\right]$ around $T_C$ with $H//c$. (b) Temperature dependence of magnetic entropy change $[-\mathrm{\Delta }{S}_M\left(T\right)]$ under different field with $H//c$ (inset shows definitions of the generated parameters of a typical fitting).

Figure 5

Field-dependent parameters of $\mathrm{\Delta }{S}_M\left(T\right)$ for $H//c$: (a) $-\mathrm{\Delta }{S}_M^{\text{max}}$, (b) ${\delta }_{\text{FWHM}}$, (c) $RCP$, and (d) modified Arrott plot (MAP).

Figure 6

Isothermal magnetization at $T_C$ up to high magnetic field of 30 T (inset shows that on log-log scale with the fitting line).

Figure 7

(a) Normalization of $\mathrm{\Delta }{S}_M\left(T\right)$ vs rescaling temperature $\theta$ (inset shows field dependence of the reference temperature $T_r$). (b) Scaling of $\mathrm{\Delta }{S}_M\left(T\right)$ under the obtained critical exponents.

Figure 8

(a) Scaling of $M\left(H\right)$ curves around the phase transition temperature into $m\left(h\right)$. (b) $M{H}^{-1/\delta }$ vs $\varepsilon H^{-1/(\beta +\gamma )}$.