Coupled spin-charge-phonon fluctuation in the all-in/all-out antiferromagnet ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$

We report on a spin-charge fluctuation in the all-in/all-out pyrochlore magnet ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$, where the spin fluctuation is driven by the conduction of thermally excited electrons/holes and associated fluctuation of Os valence. The fluctuation exhibits an activation energy significantly greater than the spin-charge excitation gap and a peculiar frequency range of ${10}^6–{10}^{10}{\mathrm{s}}^{-1}$. These features are attributed to the hopping motion of carriers as small polarons in the insulating phase, where the polaron state is presumably induced by the magnetoelastic coupling via the strong spin-orbit interaction. Such a coupled spin-charge-phonon fluctuation manifests as a part of the metal-insulator transition that is extended over a wide temperature range due to the modest electron correlation comparable with other interactions characteristic for $5d$-subshell systems.

Figure 1

Schematic diagrams of the band structure and associated electronic/magnetic phases based on optical spectroscopy in ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ [15]. Magnetic transition [with All-In/All-Out (AIAO) spin structure] occurs in the midst of semimetal phase at $T_N$, whereas the semimetal-to-insulator transition occurs in the AIAO-magnet phase at $T^{*}\simeq 210$ K. In this work, the small-polaron band (SPB) is suggested to convey thermally activated carriers for $T_H\lesssim T\lesssim T^{*}$. ${\mathrm{\Delta }}_C\left({\mathrm{\Delta }}_D\right)$ and $E_F$ refer to the indirect (direct) band gap in the insulating phase and the Fermi level, respectively.

Figure 2

Typical ZF-$\mu \mathrm{SR}$ time spectra observed in (a) single-crystalline (sc-) ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ at various temperatures (up to $0.5\mu \mathrm{s})$ and (b) in ${\mathrm{Cd}}_2{\left({\mathrm{Os}}_{1-x}{\mathrm{Re}}_x\right)}_2{\mathrm{O}}_7$ at 50 K for $x=0$, 0.01, and 0.03 (up to $3\mu \mathrm{s})$. Temperature dependence of (c) spontaneous muon precession frequency deduced from the ZF-$\mu \mathrm{SR}$ spectra, (d) the Hall coefficient, (e) resistivity normalized at 300 K, and (f) magnetic susceptibility measured at 2 T, with $T_H$ and $T_N$ representing the characteristic temperatures.

Figure 3

(a) Muon depolarization rate versus $T(>T_H\simeq 150$ K) in sc-${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ under a longitudinal field (LF) of 0, 0.2, and 0.4 T, and (b) muon-Os hyperfine parameter deduced from the data shown in (a) by fits using Eq. (5). Arrows indicate maxima of ${\lambda }_{\parallel }^{*}$ for the respective sets of data, and dashed curves are the least-square fits using Eqs. (5) and (10) in the text. (c), (d) Muon depolarization rate $({\lambda }_{\perp },{\lambda }_{\parallel })$ vs. Re content $x$ at 50 K $\left(<T_H\right)$. (e)–(g) ${\lambda }_{\parallel },{\lambda }_{\perp }$, and spin fluctuation rate $\left(\nu \right)$ in ${\mathrm{Cd}}_2{\left({\mathrm{Os}}_{1-x}{\mathrm{Re}}_x\right)}_2{\mathrm{O}}_7$ for the samples with respective Re content. Dashed curves show the result of least square fit for $\nu$ and ${\lambda }_{\parallel }$ in sc-${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$.

Figure 4

(a) Crystal structure of ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ with the arrow showing the possible muon position with distance $r_{\mathrm{O}\mu }$ from the ${\mathrm{O}}^{\prime }$ site along $\langle 111\rangle$ axis. (b) The total energy of ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ hosting a hydrogen atom in the interstitial position evaluated by the DFT calculation, where the position $\mu \left(1\right)$ corresponding to the 32e site (Wycoff position) exhibits a minimum. A local minimum $\mu \left(2\right)$ is located near those with 48f/96g sites. (c) A contour map of the local magnetic field calculated for the $\langle 110\rangle$ plane, where the Os moment size of $1{\mu }_B$ is assumed. The labels $\mu \left(i\right)$ indicate the local potential minima shown in (b). (d) The fast Fourier transform of the $\mu \mathrm{SR}$ spectra under a transverse field of 6 T, where the bars indicate linewidth expected for the $\mu \left(i\right)$ positions. The signal from silver sample holder (Ag) is also observed at $+94$ ppm (2 K).

Figure 5

(a) Muon-electron hyperfine parameter $|{\delta }_{\mu }|$ per Os moment ${\mu }_{\mathrm{Os}}$ in the paramagnetic phase calculated for the position $r_{\mathrm{O}\mu }$ along the $\langle 111\rangle$ direction. (b) The effective size of ${\mu }_{\mathrm{Os}}$ obtained by dividing experimental ${\delta }_{\mu }$ by $|{\delta }_{\mu }|/{\mu }_{\mathrm{Os}}$ in (a). (c) The local magnetic field $B_{\mathrm{m}}$ per ${\mu }_{\mathrm{Os}}$ calculated along the $\langle 111\rangle$ axis quoted from Fig. 4. (d) The $r_{\mathrm{O}\mu }$ dependence of the calculated $B_{\mathrm{m}}$ obtained as the product of (b) and (c), where $B_{\mathrm{m}}$ coincides with the experimental value (62.1 mT) for $T\le T_H$ at $r_{\mathrm{O}\mu }\simeq 0.094$ nm, which is in good agreement with $r_{\mathrm{O}\mu }$ for $T<T_H$.

Figure 6

Schematic illustrations of magnetoelastic polaron motion showing (a) sequential spin configurations and (b) their energetics. In (a), the ${\mathrm{Os}}^{4+}$ (electron) or ${\mathrm{Os}}^{6+}$ (hole) state is presumed to be in the low spin state $\left(S=0\right)$ of the $t_{2g}$ multiplet (opaque ball), which locally modulates AIAO spin configuration successively in moving from top to the bottom on an Os chain (left to right). In (b), phonons are involved to adjust the magnetoelastic polaron energy $\left(\mathrm{\Omega }\right)$ for tunneling from the Os1 to Os2 site (with an effective matrix $t_{\mathrm{eff}})$. These phonons also cause potential fluctuation $\left(\mathrm{\Gamma }\right)$ to reduce the tunneling probability.