Spin excitations in the skyrmion host ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

We have used inelastic neutron scattering to measure the magnetic excitation spectrum along the high-symmetry directions of the first Brillouin zone of the magnetic skyrmion hosting compound ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. The majority of our scattering data are consistent with the expectations of a recently proposed model for the magnetic excitations in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$, and we report best-fit parameters for the dominant exchange interactions. Important differences exist, however, between our experimental findings and the model expectations. These include the identification of two energy scales that likely arise due to neglected anisotropic interactions. This feature of our work suggests that anisotropy should be considered in future theoretical work aimed at the full microscopic understanding of the emergence of the skyrmion state in this material.

Figure 1

INS intensity (circles) measured at EIGER for constant $\mathbf{Q}$ as a function of energy transfer, along the line $\mathrm{\Gamma }-Z-R$ [$\left(222\right)-\left(22\frac{5}{2}\right)-\left(\frac{5}{2}\frac{5}{2}\frac{5}{2}\right)$]. (a) Scans measured at $T=70$ K, above $T_{\mathrm{c}}$, where all peaks are the result of lattice excitations (phonons); thick solid lines represent the best fit to a function comprised of a number of temperature-dependent Gaussian peaks, and thin lines represent the individual phonon peaks in each scan. (b) Scans measured at $T=1.5$ K and fit to the same function used to describe the 70 K data (with phonon parameters fixed and phonon intensity rescaled to account for thermal population effects) plus an additional Gaussian to account for magnetic inelastic scattering; thick and thin lines are as in (a), and medium thickness (green) lines represent the magnetic scattering peaks. Adjacent scans are offset by an amount proportional to $q$.

Figure 2

Magnetic excitation dispersion as measured on EIGER (circles) and TASP (squares) overlaid on model intensity calculations for the set of exchange parameters reported in Ref. [22] (magenta) and our best-fit exchange parameters (green), both detailed in Table 1. (a),(b) The high-energy dispersion around $\left(221\right)$ and $\left(220\right)$, respectively. (c)–(e) The low-energy dispersion around $\left(222\right), \left(002\right)$, and $\left(220\right)$, respectively. All points represent fit peak positions from constant-$\mathbf{Q}$ energy scans, with vertical bars indicating the full width at half maximum of each fit peak, which are mainly dominated by the instrumental resolution. Black points were included in our fitting routine, while gray points were excluded.

Figure 3

A sketch of the magnetic unit cell of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. The unit cell contains 16 ${\mathrm{Cu}}^{2+}$ ions located in two symmetry-inequivalent sites. The ${\mathrm{Cu}}_1$ and ${\mathrm{Cu}}_2$ sites for a network of coupled tetrahedra are, respectively, represented by red and blue circles. Four of the five model exchange couplings defined in the text are indicated on the left. The fifth, $J_{\mathrm{o}..\mathrm{o}}$, couples opposite ${\mathrm{Cu}}_1$ and ${\mathrm{Cu}}_2$ sites across the alternating ${\mathrm{Cu}}_1-{\mathrm{Cu}}_2$ hexagon in the unit cell, and is indicated on the right.

Figure 4

Sum of the squared difference in energy (SSE) between measured and calculated peak positions as a function of (a) strong or (b)–(d) weak exchange parameters. The SSE for a point is encoded in its shade, with black indicating small SSE and white indicating large SSE. Overlaid with the SSE maps are constant-SSE contour lines. (a) Map of the SSE calculated from only the high-energy dispersion at the indicated ($J_{\mathrm{s}}^{\mathrm{AF}},J_{\mathrm{s}}^{\mathrm{FM}}$) points. (b)–(d) Maps are minimum-value projections of the SSE calculated from only the low-energy dispersion at points in a ($J_{\mathrm{w}}^{\mathrm{AF}},J_{\mathrm{w}}^{\mathrm{FM}},J_{\mathrm{o}..\mathrm{o}}$) grid. For the indicated values of $J_{\mathrm{w}}^{\mathrm{AF}}$ and $J_{\mathrm{w}}^{\mathrm{FM}}$, (b) shows the minimum SSE independent of $J_{\mathrm{o}..\mathrm{o}}$; similarly, (c) and (d) show the minimum SSE independent of $J_{\mathrm{w}}^{\mathrm{AF}}$ and $J_{\mathrm{w}}^{\mathrm{FM}}$, respectively.

Figure 5

Inverse dc magnetic susceptibility data (symbols) from Ref. [24]. Dashed lines represent the $S=1$ (low-temperature) and $S=1/2$ (high-temperature) limits of the model magnetic susceptibility, as described in the text. The solid line—given by $\chi \left(T\right)={\sum }_S{\chi }_S\{1-{(-1)}^{2S}tanh[(T-t)/w]\}/2$ where $S=\frac{1}{2},1$—smoothly transitions between the two limits via coincident antisymmetric step functions with step parameters fit to $t=242\left(2\right)$ K and $w=78\left(2\right)$ K.

Figure 6

Representative scans showing features not predicted by the model in Ref. [22]. (a) Scans along the line $\mathrm{\Gamma }-R$ [$\left(002\right)-\left(\frac{1}{2}\frac{1}{2}\frac{5}{2}\right)$] performed on EIGER (circles) with $k_{\mathrm{f}}=2.662$ Å${}^{-1}$. (b) Scans along $R-\mathrm{\Gamma }-M$ [$\left(\frac{5}{2}\frac{5}{2}\frac{1}{2}\right)-\left(220\right)-\left(\frac{5}{2}\frac{5}{2}0\right)$] performed on TASP (squares) with $k_{\mathrm{f}}=1.55$ Å${}^{-1}$. Open symbols are constant-$\mathbf{Q}$ energy scans performed at the indicated $\mathbf{Q}$ points and $T=1.5$ K; filled symbols are data measured at $T=60$ (circles) or 70 K (squares) rescaled by the ratio of their Bose thermal population factors and that at $1.5$ K; solid lines are fits to the $1.5$ K data and dashed lines are an estimate of the nonmagnetic background.