Skyrmion formation in a bulk chiral magnet at zero magnetic field and above room temperature

We report that in a $\beta$-Mn-type chiral magnet ${\mathrm{Co}}_9{\mathrm{Zn}}_9{\mathrm{Mn}}_2$, skyrmions are realized as a metastable state over a wide temperature range, including room temperature, via field cooling through the thermodynamic equilibrium skyrmion phase that exists below a transition temperature $T_{\mathrm{c}}\sim 400$ K. The once-created metastable skyrmions survive at zero magnetic field both at and above room temperature. Such robust skyrmions in a wide temperature and magnetic field region demonstrate the key role of topology and provide a significant step toward technological applications of skyrmions in bulk chiral magnets.

Figure 1

(a) Schematic figures of a triangular-lattice SkX and a squarelike-lattice SkX in real space. Magnetic moments are indicated by arrows, and their $z$ components $\left(m_z\right)$ parallel to the magnetic field $\left(H\right)$ are represented by colors. The corresponding reciprocal space (SANS) patterns are also indicated in the boxes on the left-hand sides of the real space configurations. (b) Temperature $\left(T\right)\text{−}H$ state diagram of bulk ${\mathrm{Co}}_9{\mathrm{Zn}}_9{\mathrm{Mn}}_2$. The phase boundaries of helical multidomain (dark green area), thermodynamical equilibrium SkX (red area), conical (gray area), and induced-ferromagnetic states (white area below $T_{\mathrm{c}}\sim 396$ K), all indicated with solid symbols, are determined by isothermal ac susceptibility measurements [see Fig. 3] in field-increasing runs after zero field cooling (ZFC) from 400 K to the measurement temperature. The boundaries of the metastable SkX state (yellow area) denoted with open symbols are determined from isothermal ac susceptibility measurements [see Fig. 8] done as field-sweeping runs from 0.04 T to $\pm 0.3\mathrm{T}$ after field cooling (FC) via the equilibrium SkX phase as described by the pink arrow. Note that the displayed metastable SkX state was originally created in the positive-field equilibrium phase and persists down to the negative-field region below 380 K, where the transition to helical state is prevented.

Figure 2

(a) Powder x-ray diffraction profile for the sample of ${\mathrm{Co}}_9{\mathrm{Zn}}_9{\mathrm{Mn}}_2$. Experimental data (black circles) are compared with the simulation (red line) with a $\beta$-Mn-type structure, as illustrated in the inset, with a lattice parameter of 6.$3578\AA$. Peak positions are also indicated by blue bars. In ${\mathrm{Co}}_9{\mathrm{Zn}}_9{\mathrm{Mn}}_2$, the $8c$ sites (blue circles in the inset) are mainly occupied by Co, and the $12d$ sites (red circles in the inset) are randomly occupied by Co, Zn, and Mn. Pictures of single-crystalline bulk samples of ${\mathrm{Co}}_9{\mathrm{Zn}}_9{\mathrm{Mn}}_2$ used for (b) SANS and (c) ac susceptibility measurements. (d) Laue picture of the (110) plane in the single-crystalline sample. The indicated crystal axes are common for (b)–(d).

Figure 3

(a) Schematic illustration of the measurement process. (b) SANS images at 390 K and selected fields. The intensity scale of the color plot is fixed for the three panels. (c) Field dependence of the real part of the ac susceptibility $\left({\chi }^{\prime }\right)$. Since the demagnetization factor is different from that in the SANS measurement, the field values are calibrated as $H_{\mathrm{c}}=2.0H$. The phase boundary (dark green triangle) between the helical and conical states was determined as the inflection point of ${\chi }^{\prime }$. The phase boundaries (red triangles) between the equilibrium SkX and conical states were determined as the peak positions of ${\chi }^{\prime }$. The phase boundary (black triangle) between the conical and induced ferromagnetic states was determined as the inflection point of ${\chi }^{\prime }$. These boundaries and similar data points at different temperatures are plotted as solid symbols in the state diagram of Fig. 1. (d) Field dependence of the SANS intensity integrated over the azimuthal angle region at $\varphi ={90}^{\circ }\pm {45}^{\circ }$ and ${270}^{\circ }\pm {45}^{\circ }$ (light blue region in the inset). Here, $\varphi$ is defined as the clockwise azimuthal angle from the vertical [001] direction.

Figure 4

(a) Schematic illustrations of the measurement processes. The colors of the arrows correspond to the colors of arrows in (b) and (c), and data points in (d). (b) SANS images in the FC (0.04 T) process and in the subsequent zero-field-warming (ZFW) process. The intensity scale of the color plots varies between each panel. (c) SANS images in the ZFC process. The intensity scale of the color plot varies between each panel. (d) Temperature dependence of the SANS intensity for the FC and ZFW processes, integrated over the same azimuthal angle area as for Fig. 3 (light blue region in the inset).

Figure 5

(a) Schematic illustrations of the measurement processes. The colors of the arrows correspond to the colors of arrows in (b) and (c), and data points in (d). (b) SANS images at 300 K and selected magnetic fields in the field-sweeping processes after the FC. The intensity scale of the color plot is fixed for the six panels. (c) SANS images at 300 K and selected magnetic fields in the field-sweeping process after the ZFC. The intensity scale of the color plot is fixed for the three panels. (d) Field dependence of the SANS intensity at 300 K after the FC, integrated over the same azimuthal angle area as for Fig. 3. The intensities in field-sweeping runs from 0.04 to $\pm 0.3$ T and the subsequent returning runs from $\pm 0.3$ T back to 0.04 T are denoted by solid and open symbols with the same colors, respectively.

Figure 6

(a) Schematic illustrations of the measurement processes. The colors of the arrows correspond to the colors of data points in (c) and (d). (b) Schematic view (from the vertical [001] direction) of the experimental configuration. Maintaining the $H\parallel$ [110] geometry, the angle $\omega$ between the neutron beam and the field direction is varied (“rocked”) around the vertical [001] direction. The origin of the rocking angle $\left(\omega =0\right)$ corresponds to the $q$-vector $\parallel$ (110) plane. For rocking curves (intensity versus $\omega )$ in (c) and (d), SANS intensities are integrated over the same azimuthal angle area as for Fig. 3. (c) Rocking curves at selected temperatures for the FC process from 390 to 300 K (blue triangles), the subsequent ZFW process from 300 to 390 K (red squares), and the ZFC process (green circles). (d) Rocking curves at selected fields for the field sweeping from 0.04 to $-0.08$ T after the FC (light blue squares), and the field sweeping from 0 to 0.04 T and to $-0.08$ T after the ZFC (orange circles).

Figure 7

(a) Schematic illustrations of the measurement processes. The colors of the arrows correspond to the colors of arrows in (b) and (c). (b) SANS images at selected temperatures and magnetic fields in the FC (0.04 T) to 300 K, the subsequent field sweep to zero field, and the subsequent returning processes back to 0.04 T and 375 K. The intensity scale of the color plot varies among the five panels. (c) SANS images at 375 K and at 0 and 0.04 T in the field-sweeping process after a ZFC. The intensity scale of the color plot varies between the two panels.

Figure 8

(a) Schematic illustrations of the measurement processes. The colors of the arrows correspond to the colors of data lines in (b). (b) Field dependence of the real part of the ac susceptibility $\left({\chi }^{\prime }\right)$ at 300 K after several cooling processes shown in (a). The results of the field sweepings (not shown) from $\pm 0.3$ T back to 0.04 T (0 T) after FC2 (ZFC) are almost the same as that after FC1. The yellow triangles denote the boundaries of the metastable SkX state. The boundary at the positive field side is determined as the field where the hysteresis of ${\chi }^{\prime }$ (red and blue lines) closes, and the boundary at the negative field side is determined as the field where ${\chi }^{\prime }$ (red line) shows a peak. (c) Field dependence of the SANS intensity for metastable SkX at 300 K after the FC [the same data as in Fig. 5] is shown again for the purpose of comparison with (b). (d) Field dependencies of ${\chi }^{\prime }$ at several different temperatures after the FC process via the equilibrium state. The yellow triangles denote the boundaries of the metastable SkX that are plotted in the state diagram in Fig. 1.

Figure 9

(a) Temporal dependence of the normalized ${\chi }^{\prime }\left(t\right)$ at 0.04 T and several temperatures after a FC process via the equilibrium state (as schematically depicted in the inset). ${\chi }^{\prime }\left(0\right)$ is an initial value (metastable SkX state), and ${\chi }^{\prime }\left(\infty \right)$ is a fully relaxed value (equilibrium conical state) which is assumed to be the value of ${\chi }^{\prime }$ at 0.04 T in the field-decreasing run from 0.3 T. The data points are fitted to stretched exponential functions (dotted lines), $\exp \left\{-{(t/\tau )}^{\beta }\right\}$. The obtained $\beta$ values are in the range of 0.3 to 0.5, indicating a highly inhomogeneous distribution of relaxation times. (b) Temperature dependence of the relaxation time $\tau$ determined by the fits in (a). The experimental data (black symbols) are fitted to a modified Arrhenius law (red line), $\tau ={\tau }_0\exp \left\{a(T_{\mathrm{c}}-T)/T\right\}$, with an assumption of a temperature-dependent activation energy [30]. The obtained parameters are $a=216$ and ${\tau }_0=47$ s.

Figure 10

(a) Schematic illustration of the measurement process. The colors of the arrows correspond to the colors of arrows in (b). (b) Overfocused LTEM images on the (110) plane in the FC process and in the subsequent ZFW process. The defocus values of LTEM images were set to be $+288\mu \mathrm{m}$ at 370 K, $+192\mu \mathrm{m}$ at 350 K, and $+96\mu \mathrm{m}$ at 290 K, respectively. These LTEM images were taken for nearly the same sample position. The insets of the respective panels show the fast Fourier-transformed patterns of the LTEM images taken over a wider sample area. (c) Distribution of in-plane magnetic moments for the area indicated by the red square in (b) at 290 K and zero magnetic field after the FC, as deduced from a transport-of-intensity equation analysis of overfocused and underfocused LTEM images. The color wheel represents the direction and magnitude of the in-plane magnetization.

Figure 11

(a) Schematic illustration of the measurement process. After the FC via the equilibrium SkX phase to 290 K and the subsequent field-decreasing process to zero field, the sample was taken out of the electron microscope, turned upside down, and installed into the electron microscope again. Subsequently, the magnetic field was applied along the same (positive) direction for the electron microscope as before, which was the opposite (negative) direction for the sample. (b) Underfocused LTEM images on the (110) plane at 290 K and selected magnetic fields in the field-sweeping process from zero field to negative fields, and after the above sample reversing procedure. The defocus values of the LTEM images were set to be $-288\mu \mathrm{m}$ for 0 and $-0.049\mathrm{T}$, and $-384\mu \mathrm{m}$ for $-0.078\mathrm{T}$, respectively.