Robust noncoplanar magnetism in band-filling-tuned (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x{)}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$

In the metallic pyrochlore ${\mathrm{Nd}}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$, the conducting molybdenum sublattice adopts canted, yet nearly collinear, ferromagnetic order with nonzero scalar spin chirality. The chemical potential may be controlled by replacing ${\mathrm{Nd}}^{3+}$ with ${\mathrm{Ca}}^{2+}$, while introducing only minimal additional disorder to the conducting states. Here, we demonstrate the stability of the canted ferromagnetic state, including the tilting angle of molybdenum spins, in (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x$)${}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ with $x\le 0.15$ using magnetic susceptibility measurements. Mo-Mo and Mo-Nd magnetic couplings both change sign above $x=0.22$, where the canted ferromagnetic state gives way to a spin-glass metallic region. Contributions to the Curie-Weiss law from two magnetic species are separated systematically.

Figure 1

Unconventional Hall effect from momentum-space Berry curvature in (${\mathrm{Nd}}_{0.93}{\mathrm{Ca}}_{0.07}$)${}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$. The total Hall conductivity (violet) is separated into three components: normal Hall effect ${\sigma }_{xy}^{\mathrm{N}}$ (slope of red dashed line), spin-orbit driven anomalous Hall effect ${\sigma }_{xy}^{\mathrm{KL}}$, and the geometrical Hall conductivity ${\sigma }_{xy}^{\mathrm{G}}$ (green line, hatched area). The latter is observed only in the low-field noncoplanar ground state, termed canted ferromagnetic phase (C-FM, orange shading and left inset). ${\sigma }_{xy}^{\mathrm{KL}}$ is present both in C-FM and the field-aligned ferromagnetic state (FA-FM, right inset). Insets show canted (left) and collinear (right) configurations of molybdenum moments on an individual tetrahedron of the pyrochlore structure. Hall effect data adapted from Ref. [8].

Figure 2

Rietveld refinement of powder x-ray diffraction data from crushed single-crystal pieces of (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x$)${}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ obtained at room temperature. Red dots are raw data, cyan is the fit curve, blue is the difference between the former two (shifted by a vertical offset). The green lines indicate expected reflections from the cubic pyrochlore structure. The square root of the scattered intensity is shown. No phase impurities beyond $1\%$ volume fraction were detected in this analysis. $R_B$ is a figure of merit defined in the text.

Figure 3

Cubic lattice constant $a$ for the pyrochlore structure of (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x$)${}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ and its evolution with $x$. A triangle marks a crystal batch for which the nominal Ca content differed from the measured value (see text). The gray shaded line is a guide to the eye. Statistical error bars are smaller than the symbol size. Inset: cubic pyrochlore structure of ${\mathrm{Nd}}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ where pink and gray polyhedra mark local environments of Nd and Mo, respectively [16]. Oxygen atoms are shown in red.

Figure 4

Low-field magnetization $M\left(T\right)$ in selected crystals of (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x$)${}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$, for magnetic field along the $\langle 001\rangle$ magnetic easy-axis. Data was recorded for increasing temperature after cooling in ${\mu }_{0}H=1\mathrm{T}$ (red) and ${\mu }_{0}H=0.01\mathrm{T}$ (violet), respectively. Dashed vertical line indicates ordering transitions $T_C, T_G$ summarized in Fig. 6.

Figure 5

Magnetization isotherms $M\left(H\right)$ for (${\mathrm{Nd}}_{0.93}{\mathrm{Ca}}_{0.07}$) ${}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$. Hysteresis appears at the lowest $T<20\mathrm{K}$, where dashed and solid lines mark increasing and decreasing magnetic field, respectively (c.f. black arrows). Inset: coercive field $H_c$ for external magnetic field $\mathbf{H}$ along the $\langle 001\rangle$ axis, at $T=2\mathrm{K}$.

Figure 6

Phase diagram of (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x$)${}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ (NCMO) at zero magnetic field, with canted ferromagnetic (C-FM), spin-glass metal (SGM), and paramagnetic (PM) regime. Red and blue markers are for Curie temperature $T_C$ and for the onset of glassy order ($T_G$), respectively. Black diamonds and a dashed line indicate the strong departure of field-cooled and high-field-cooled magnetization curves (cf. Fig. 4). Some data points at low $x$ adapted from Ref. [8]. Inset: conducting molybdenum (red spheres) sublattice of NCMO in phase C-FM, stressing the tetrahedron as a key structural motif. The canting angle of magnetic moments (red arrows) is exaggerated.

Figure 7

Curie-Weiss analysis of magnetic susceptibility ${\chi }_{\mathrm{DC}}=M/H$ in (${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_x$)${}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$. (a) ${\chi }_{\mathrm{DC}}$ at elevated field of ${\mu }_{0}H=1\mathrm{T}$ for four selected crystals. (b) Inverse susceptibility retains finite curvature even above room temperature. The magnetic field was along the $\langle 110\rangle$ axis, but the behavior at these larger values of $H$ is nearly isotropic. Inset: derivative of the inverse susceptibility with respect to temperature, with dashed line at right side of panel marking the $T\to \infty$ limit of $C_{\mathrm{HT}}^{-1}\approx d{\chi }_{\mathrm{DC}}^{-1}/dT$ predicted from the free-ion values for the magnetic moments at $x=0.00$.

Figure 8

Curie-Weiss constants $C_{\mathrm{HT}}, C_{\mathrm{Nd}}$ and temperatures ${\theta }_{\mathrm{Mo}}, {\theta }_{\mathrm{Nd}}$ extracted from susceptibility data as shown in Fig. 7. In (a) and (c), the dashed lines are guides to the eye. The shaded lines in (b), and (d) were calculated from Eqs. (6) and (10), respectively.

Figure 9

Canting angle ${\alpha }_{\mathrm{Mo}}$ of molybdenum spins away from the $\langle 001\rangle$ direction. ${\alpha }_{\mathrm{Mo}}$ was calculated from the exchange integrals $J_{dd}, J_{df}$, which in turn originate from ${\theta }_{\mathrm{Mo}}$ and $C_{\mathrm{Nd}}$, respectively. Inset: molybdenum magnetic moments (red arrows) on an individual tetrahedron, with dashed lines marking $\langle 001\rangle$.