Nonlinear magnetic responses at the phase boundaries around helimagnetic and skyrmion lattice phases in MnSi: Evaluation of robustness of noncollinear spin texture

The phase diagram of a cubic chiral magnet MnSi with multiple Dzyaloshinskii-Moriya (DM) vectors as a function of temperature $T$ and dc magnetic field $H_{\mathrm{dc}}$ was investigated using intensity mapping of the odd-harmonic responses of ac magnetization $(M_{1\omega }$ and $M_{3\omega })$, and the responses at phase boundaries were evaluated according to a prescription [J. Phys. Soc. Jpn. 84, 104707 (2015)]. By evaluating $M_{3\omega }/M_{1\omega }$ appearing at phase boundaries, the robustness of noncollinear spin texture in both the helimagnetic (HM) and the skyrmion lattice (SkL) phases of MnSi was discussed. The robustness of vortices-type solitonic texture SkL in MnSi is smaller than those of both the single DM HM and chiral soliton lattice phases of a monoaxial chiral magnet ${\mathrm{Cr}}_{1/3}{\mathrm{NbS}}_2$, and furthermore the robustness of the multiple DM HM phase in MnSi is smaller than that of its SkL. Through magnetic diagnostics over the wide $T-H_{\mathrm{dc}}$ range, we found a new paramagnetic (PM) region with ac magnetic hysteresis, where spin fluctuations have been observed via electrical magnetochiral effect. The anomalies observed in the previous ultrasonic attenuation measurement correspond to the peak positions of out-of-phase $M_{1\omega }$. The appearance of a new PM region occurs at a characteristic magnetic field, above which indeed the SkL phase appears. It has us suppose that the new PM region could be a phase with spin fluctuation like the skyrmion gas phase.

Figure 1

Spatial location of spin phase orders: (a) noncollinear helimagnet, (b) noncollinear chiral soliton lattice (CSL), (c) collinear ferromagnet, and (d) noncollinear skyrmion lattice (SkL).

Figure 2

(a) $T-H_{\mathrm{dc}}$ phase diagram of B20-type chiral magnets such as MnSi and FeGe recognized thus far via a series of studies for bulk crystals [21, 23, 26]. The description of each phase is HM: helimagnetic phase, CM: conical magnetic phase, SkL: skyrmion lattice phase, IM: Intermediate phase, PM: paramagnetic phase, and FFM: forced ferromagnetic region (elsewhere often termed field-polarized region). The anomalies in the in-phase ac magnetic susceptibility (FeGe [22]), ultrasonic attenuation (MnSi [24, 25]), and heat capacity (FeGe [26, 27]) measurements were observed near the broken green line. The second harmonic resistivity in MnSi was observed in a certain region near the broken green line [28]. In MnSi, the boundary between FFM and PM has not been determined experimentally. (b) $T-H_{\mathrm{dc}}$ phase diagram of MnSi determined via the present study. Here the PM region in (a) is divided into two regions (i.e., PM1 and PM2).

Figure 3

$H_{\mathrm{dc}}$ dependence of ac magnetic responses (a) $M_{1\omega }^{\prime }$, (b) $M_{1\omega }^{\prime \prime }$, and (c) $M_{3\omega }$ for one single crystal of MnSi under an ac field with $h=3.9$ Oe and $f=10$ Hz. Each figure shows the corresponding intensity map, where the intensity increases as the color changes from blue to red. The description of each phase has been described in the figure caption of Fig. 2.

Figure 4

Temperature dependence of ac magnetic responses (a) and (b) $M_{1\omega }^{\prime }$, (c) and (d) $M_{1\omega }^{\prime \prime }$, and (e) and (f) $M_{3\omega }$ for MnSi at various $H_{\mathrm{dc}}$ values under an ac field with $h=3.9$ Oe and $f=10$ Hz. Both $H_{\mathrm{dc}}$ and $H_{\mathrm{ac}}$ are applied along the [111] direction.

Figure 5

Temperature dependence of ac magnetic responses (a) $M_{1\omega }^{\prime }$ and (b) $M_{1\omega }^{\prime \prime }$ for MnSi at $H_{\mathrm{dc}}=4.5$ kOe under an ac field with $h=3.9$ Oe and $f=1-100\mathrm{Hz}$. Both $H_{\mathrm{dc}}$ and $H_{\mathrm{ac}}$ are applied along the [111] direction.

Figure 6

Temperature dependence of ac magnetic responses (a) $M_{1\omega }^{\prime }$, (b) $M_{1\omega }^{\prime \prime }$, and (c) $M_{3\omega }$ for MnSi at $H_{\mathrm{dc}}=1.5$ kOe under an ac field with $h=3.9$ Oe and $f=1$, 10, and 100 Hz. Both $H_{\mathrm{dc}}$ and $H_{\mathrm{ac}}$ are applied along the [111] direction.

Figure 7

Intensity maps of $M_{1\omega }^{\prime }$ (a), $M_{1\omega }^{\prime \prime }$ (b), and $M_{3\omega }$ (c) for MnSi under an ac field with $h=3.9$ Oe and $f=10$ Hz obtained from $T$-scanning measurements at a fixed $H_{\mathrm{dc}}$. The anomalies observed by the ultrasonic attenuation in the paper of Komatsubara et al. [24] were plotted using white and red closed squares in all three figures: White and red data are for the low and high temperature sides of the IM phase, respectively. In (b), black closed circles present the peak position of $M_{1\omega }^{\prime \prime }$ and yellow ones present those of the maximal and minimal of $d{M}_{1\omega }^{\prime \prime }/dT$. Purple closed circles present the data position with the experimentally trustworthy level of $M_{1\omega }^{\prime \prime }=8\times {10}^{-4}$ emu/mol that corresponds to approximately 0.1% of the maximum $M_{1\omega }^{\prime }$.

Figure 8

Three-dimensional Intensity maps of $M_{1\omega }^{\prime }$ (a), $M_{1\omega }^{\prime \prime }$ (b), and $M_{3\omega }$ (c) for MnSi under an ac field with $h=3.9$ Oe and $f=10$ Hz obtained from $T$-scanning measurements at a fixed $H_{\mathrm{dc}}$. The physical information is consistent with that of Fig. 7.

Figure 9

Change in dynamical magnetic responses as a function of temperature $\left(T\right)$ at various dc magnetic fields $(H_{\mathrm{dc}}$'s). The descriptions to characterize the phase or region have been described in the figure caption of Figs. 2 and 7.

Figure 10

Overview of robustness of noncollinear spin textures, evaluated via the Klirr factor $M_{3\omega }/M_{1\omega }$. (a) Symmetric helix in which two chiralities, such as left-handed and right-handed helicities, coexist across a domain boundary [51]. Four examples based on the DM vector are presented: (b) Multiple DM helimagnet (HM) (present work). (c) Skyrmion lattice (SkL) (2D lattice of spin vortices, present work). (d) Chiral soliton lattice (CSL), a magnetic superlattice formed in a monoaxial chiral magnet at $H_{\mathrm{dc}}$ [33]. (e) Chiral magnet in which there is only a single type of helicity such as left-handed or right-handed (single DM HM) [33, 52, 53]. For reference, $M_{3\omega }/M_{1\omega }$ for spin glass and cluster glass systems are also shown [44, 49, 50].