Evolution of possible Weyl semimetal states across the Mott transition in pyrochlore iridates induced by hole doping

We study possible Weyl semimetals of strongly correlated electrons by investigating magnetotransport properties in pyrochlore $R_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ ($R$ denotes rare-earth ions), choosing three types of $R$ ions to design the exchange coupling scheme between $R4f$ and Ir $5d$ moments: nonmagnetic Eu $\left(4f^6\right)$, isotropic Gd $\left(4f^7\right)$, and anisotropic Tb $\left(4f^8\right)$. In the doping-induced semimetallic state, distinctive features of magnetoresistance and the Hall effect are observed in $R=\mathrm{Gd}$ and Tb compounds due to the effects of the exchange-enhanced isotropic and anisotropic Zeeman fields, respectively, exemplifying the double-Weyl semimetal and the two-in two-out line-node semimetal as predicted by theories. In particular, the Hall angle of an $R=\mathrm{Gd}$ compound is strongly enhanced to 1.5% just above the critical doping for the Mott transition. Furthermore, an unconventional Hall contribution is discerned for a lower doping regime of the $R=\mathrm{Gd}$ compound, which can be ascribed to the emergence of Weyl points with the field-distorted all-in all-out order state. These findings indicate that the hole-doping-induced Mott transition as well as the characteristic $f\text{−}d$ exchange interaction stabilizes versatile topological semimetal states in a wide range of material parameter space.

Figure 1

(a) Phase diagram of $R_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ as a function of rare-earth ionic radius, divalent-ion concentration, and temperature. Here PI stands for paramagnetic insulator, PM stands for paramagnetic metal, AIAO stands for all-in all-out state, and WSM stands for Weyl semimetal. (b) Schematic picture of modulation of electronic band structures as a function of electron correlation $U$ and hole doping $\delta$. Here QBT stands for quadratic band touching.

Figure 2

Temperature dependence of resistivity for ${\left({\mathrm{Eu}}_{1-x}{\mathrm{Ca}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ ($x=0$ and 0.10), ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ ($x=0$ and 0.12), and ${\left({\mathrm{Tb}}_{1-x}{\mathrm{Ca}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ ($x=0$ and 0.15).

Figure 3

Magnetic-field dependence of (a), (c), (e), and (g) longitudinal resistivity and (b), (d), (f), and (h) Hall resistivity for (a) and (b) ${\left({\mathrm{Eu}}_{1-x}{\mathrm{Ca}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ ($x=0.10$), (c) and (d) ${\left({\mathrm{Tb}}_{1-x}{\mathrm{Ca}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ ($x=0.15$), (e) and (f) ${\mathrm{Pr}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$, and (g) and (h) ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ ($x=0.12$). Schematic magnetic configurations of $R$ spins and expected electronic structures are displayed on top of the respective windows.

Figure 4

(a) Schematic magnetic structures of Ir moments (green) and Gd moments (blue). (b) Temperature dependence of resistivity for ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ with various $x$. Arrows indicate the magnetic transition temperature. Also shown is the magnetic-field dependence $H$ of (c)–(f) magnetization $M$ and (g)–(j) Hall resistivity at 2 K in ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ for (c) and (g) $x=0.07$, (d) and (h) $x=0.08$, (e) and (i) $x=0.09$, and (f) and (j) $x=0.12$. Thick black lines are fitting curves with the $H$-linear normal Hall plus the $M$-linear anomalous Hall terms (see the text).

Figure 5

A log-log plot of Hall conductivity versus longitudinal conductivity for ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ with various $x$. The dashed line indicates the relation ${\sigma }_{xy}\propto {\sigma }_{xx}^{1.6}$.

Figure 6

(a) Phase diagram of ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ with contour mapping of resistivity in the plane of temperature and Cd concentration $x$. Black circles denote the magnetic transition temperature $T_{\mathrm{N}}$. Also shown are contour plots of (b) the Hall angle ${\rho }_{yx}/{\rho }_{xx}$ at 14 T and (c) $\mathrm{\Delta }{\rho }_{yx}$/${\rho }_{xx}^{0.4}$ at 4 T, which represent the Hall responses from the DWSM and AIAO WSM, respectively.

Figure 7

Schematic picture of electronic band structures with different magnetic ordering patterns stemming from the quadratic band touching semimetal at zero magnetic field in Fig. 1. Shown from top to bottom are the line-node semimetal with a two-in two-out magnetic state, the Weyl semimetal with a three-in one-out magnetic state, and the double Weyl semimetal with a collinear magnetic state. Shown on the left are the respective magnetic ordering configurations. The middle column displays the distribution of Weyl points and a line node in the Brillouin zone. The red and blue circles denote Weyl points with different chiralities and the purple line denotes a line node. Shown on the right are the schematic band energy levels for respective electronic states. Red and blue squares (circles) denote Weyl points with monopole charges of $\pm 1$ ($\pm 2$).

Figure 8

Temperature dependence of (a), (c), (e), and (g) resistivity and (b), (d), (f), and (h) magnetization for ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$. The Cd concentration is (a) and (b) $x=0.07$, (c) and (d) $x=0.08$, (e) and (f) $x=0.09$, and (g) and (h) $x=0.12$. The vertical dashed line indicates the magnetic transition temperature.

Figure 9

Magnetic-field dependence of (a)–(d) Hall resistivity, (e)–(h) resistivity, and (i)–(l) magnetization for ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$. The data are (a), (e), and (i) $x=0.07$, (b), (f), and (j) $x=0.08$, (c), (g), and (k) $x=0.09$, and (d), (h), and (l) $x=0.12$.

Figure 10

Magnetic-field dependence of Hall resistivity for ${\left({\mathrm{Gd}}_{1-x}{\mathrm{Cd}}_x\right)}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ ($x=0.07$) at temperatures (a) 2 K, (b) 5 K, (c) 10 K, and (d) 20 K. Thick black lines are fitting curves (see the text).