Megahertz dynamics in skyrmion systems probed with muon-spin relaxation

We present longitudinal-field muon-spin relaxation (LF $\mu \mathrm{SR}$) measurements on two systems that stabilize a skyrmion lattice (SkL): ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$, and ${\mathrm{Co}}_x{\mathrm{Zn}}_y{\mathrm{Mn}}_{20-x-y}$ for $(x,y)=(10,10)$, (8, 9), and (8, 8). We find that the SkL phase of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ exhibits emergent dynamic behavior at megahertz frequencies, likely due to collective excitations, allowing the SkL to be identified from the $\mu \mathrm{SR}$ response. From measurements following different cooling protocols and calculations of the muon stopping site, we suggest that the metastable SkL is not the majority phase throughout the bulk of this material at the fields and temperatures where it is often observed. The dynamics of bulk ${\mathrm{Co}}_8{\mathrm{Zn}}_9{\mathrm{Mn}}_3$ are well described by $\simeq 2$ GHz excitations that reduce in frequency near the critical temperature, while in ${\mathrm{Co}}_8{\mathrm{Zn}}_8{\mathrm{Mn}}_4$ we observe similar behavior over a wide range of temperatures, implying that dynamics of this kind persist beyond the SkL phase.

Figure 1

Skyrmion phase diagram of bulk materials. Stabilization of the skyrmion phase occurs through various mechanisms: (i) competition between the exchange and Dzyaloshinskii-Moriya interactions, leading to a Bloch (orange) or Néel (blue) skyrmion lattice, (ii) competition between exchange and uniaxial anisotropy (green), (iii) geometric frustration from the interplay of Ruderman-Kittel-Kasuya-Yosida (RKKY) and four-spin interactions (purple). Phase boundaries are taken from Refs. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

Figure 2

(a) Phase diagram of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$, showing conical (C), helical (H), and skyrmion (S) phases, reproduced from Ref. [29]. Orange color: fields at which the SkL is stabilized; blue: fields which only stabilize C order below $T_{\text{c}}$; and pink: H order. (b) Example LF $\mu \mathrm{SR}$ spectra for ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ measured in $B=22\mathrm{mT}$. For clarity, some data are shown with vertical offsets. (c) Extracted values of $\lambda$. (d) Simulations of contributions to $\lambda$ due to (1) critical slowing down of magnetic fluctuations near $T_{\text{c}}$ (black solid line); (2) reduction in frequency of GHz spectral density (orange solid line). The orange dashed line indicates the value of $\lambda$ one would obtain if the SkL was stabilized at those temperatures. Vertical dashed lines indicate the location of the SkL at 22 mT from AC susceptibility. (e) Extracted values of $\lambda$ from a field scan at $T=56.7$ K, with (f) accompanying baseline amplitude (the solid line is a guide to the eye). Dashed lines indicate the location of the SkL at 56.7 K.

Figure 3

Internal magnetic field distributions $\left[S\right(B\left)\right]$ of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ at various $T$ measured by TF $\mu \mathrm{SR}$ for $B_{\text{ext}}=22\mathrm{mT}$, with comparison to simulations. Black lines: measurements performed after ZFC; red lines: after rapid FC ($\approx 17$ K/minute); solid color: simulated distributions for different spin structures.

Figure 4

Extracted values of $\lambda$ from fitting LF $\mu \mathrm{SR}$ measurements of (a) ${\mathrm{Co}}_{10}{\mathrm{Zn}}_{10}$ and (b) ${\mathrm{Co}}_8{\mathrm{Zn}}_9{\mathrm{Mn}}_3$. The dashed lines indicate the average value of $T_{\text{c}}$ in each sample according to the relaxing amplitude. Fits in (b) are described in the main text.

Figure 5

(a) Representative phase diagram of ${\mathrm{Co}}_8{\mathrm{Zn}}_8{\mathrm{Mn}}_4$, showing helical (H), conical (C), and skyrmion lattice (S) phases, as well as a region of coexistence [11]. (b) AC susceptibility measurements at 250 K after ZFC and FC (in 15 mT) on one sample of ${\mathrm{Co}}_8{\mathrm{Zn}}_8{\mathrm{Mn}}_4$, indicating a metastable SkL after FC. Fields measured with $\mu \mathrm{SR}$ in (d) and Fig. S5 are indicated with dashed lines. (c) and (d) Relaxation rate $\lambda$ from LF $\mu \mathrm{SR}$ measurements on two different samples, with $T_{\text{c}}$ indicated. In (d) different field protocols are employed.