Controlling the helicity of magnetic skyrmions in a $\beta$-Mn-type high-temperature chiral magnet

Magnetic helices and skyrmions in noncentrosymmetric magnets are representative examples of chiral spin textures in solids. Their spin swirling direction, often termed as the magnetic helicity and defined as either left handed or right handed, is uniquely determined by the Dzyaloshinskii-Moriya interaction (DMI) in fixed chirality host crystals. Thus far, there have been relatively few investigations of the DMI in metallic magnets as compared with insulating counterparts. Here, we focus on the metallic magnets ${\mathrm{Co}}_{8-x}{\mathrm{Fe}}_x{\mathrm{Zn}}_8{\mathrm{Mn}}_4$ ($0\le x\le 4.5$) with a $\beta$-Mn-type chiral structure and find that, as $x$ varies under a fixed crystal chirality, a reversal of magnetic helicity occurs at $x_{\mathrm{c}}\sim 2.7$. This experimental result is supported by a theory based on first-principles electronic structure calculations, demonstrating the DMI to depend critically on the electron band filling. Thus by composition tuning our work shows the sign change of the DMI with respect to a fixed crystal chirality to be a universal feature of metallic chiral magnets.

Figure 1

(a),(b) Schematics of $\beta$-Mn-type chiral crystal structures as viewed along the [111] direction. Two enantiomers with space group $P{4}_{3}32$ and $P{4}_{1}32$ are shown. Their crystal chirality is defined as left handed (${\mathrm{\Gamma }}_{\mathrm{c}}=-1$) and right handed (${\mathrm{\Gamma }}_{\mathrm{c}}=+1$), respectively. Blue and red circles represent $8c$ and $12d$ Wickoff sites, respectively. The network of $12d$ sites forms a windmill structure with corner-sharing triangles. (c),(d) Schematic configurations of magnetic moments for left-handed helix (${\gamma }_{\mathrm{m}}=-1$) and right-handed helix (${\gamma }_{\mathrm{m}}=+1$). (e),(f) Schematic configurations of in-plane magnetic moments of skyrmions: counterclockwise (CCW) skyrmion and clockwise (CW) skyrmion. Here, an external magnetic field is assumed to be applied into the page, and the magnetic moments at the periphery and the core of skyrmions are pointing into the page and out of the page, respectively. The magnetic configurations of the CCW and CW skyrmions along a radial direction indicated by dashed rectangles are equivalent to left-handed and right-handed helices as shown in (c) and (d), respectively. The corresponding overfocused and underfocused Lorentz transmission electron microscope images are also displayed schematically.

Figure 2

(a)–(c) Temperature ($T$) dependence of the magnetization ($M$) under a magnetic field of 20 Oe for $x=0$, 3, and 4.5, respectively. A blue line shows the data taken in a field-cooling (FC) process and a red line the data collected in a field-warming process after zero-field cooling (ZFC) down to 2 K. A helimagnetic transition temperature $T_{\mathrm{c}}$ (blue triangle) is determined as an inflection point of the steep increase in $M$ on cooling. A reentrant spin-glass transition temperature $T_{\mathrm{g}}$ (yellow triangle) is defined as an inflection point of the sharp increase in $M$ in the warming process after ZFC. $T^{*}\sim 60\mathrm{K}$ indicated with red triangles in (a) and (c) corresponds to an inflection point of gradual decrease in $M$ on cooling below 120 K, which is hardly observed for $x=3$ in (b). (d)–(f) Magnetic field ($H$) dependence of the real part of the ac magnetic susceptibility (${\chi }^{\prime }$) in a field-decreasing process from an induced-ferromagnetic region at several temperatures near $T_{\mathrm{c}}$ for $x=0$, 3, and 4.5, respectively. The phase boundaries between conical and SkX states [$H_{\mathrm{sk}1}$ and $H_{\mathrm{sk}2}$, red triangles in (d)] were determined as peak positions of ${\chi }^{\prime }$. The phase boundary between conical and induced-ferromagnetic states [$H_{\mathrm{c}}$, black triangle in (d)] was determined as an inflection point of ${\chi }^{\prime }$. In $x=4.5$, signatures of the SkX phase become unclear below 120 K, but some anomalies are still discerned, implying a reduced volume fraction of skyrmions. (g)–(i) $H-T$ phase diagram determined by ${\chi }^{\prime }$ for $x=0$, 3, and 4.5, respectively. In $x=3$, the magnetic state at low fields is either a conical state with a very long periodicity or a ferromagnetic multidomain state. In $x=4.5$, the blurred boundaries of the skyrmion phase below 120 K are plotted with dotted red lines.

Figure 3

(a) Small-angle neutron scattering (SANS) intensity as a function of $\left|\mathbf{q}\right|(=q)$ for all the measured samples at 150 K (120 K for $x=4.5$, red circles) and 50 K (40 K for $x=0$, blue circles) under zero magnetic field. These SANS intensities are obtained by averaging over a full azimuthal-angle range of ${360}^{\circ }$ around the origin of the two-dimensional SANS patterns (Supplemental Fig. S2 [38]). Data points are fitted to a Gaussian function denoted with pink lines for 150 K (120 K for $x=4.5$) and light-blue lines for 50 K (40 K for $x=0$). (b) $x$ dependence of helical $q$ (red circles) at 150 K (120 K for $x=4.5$) determined as a peak position of a Gaussian function in (a). For $x=3$, an error bar is appended as the width of the Gaussian function centered at $q=0$. By interpolating the data points (red lines), $q$ is expected to be zero at $x_{\mathrm{c}}\sim 2.7$. The helical periodicity $\lambda =2\pi /q$ is also plotted with black squares to show the divergent behavior around $x_{\mathrm{c}}$. (c) $T-x$ phase diagram determined by magnetization measurements at 20 Oe [see Figs. 2–2 and Supplemental Fig. S3(a) [38] for details]. The paramagnetic (PM)–helimagnetic (HM) transition temperature $T_{\mathrm{c}}$ is shown by blue circles, and the spin glass (SG) transition temperature $T_{\mathrm{g}}$ is denoted by yellow triangles. $T^{*}$ (red asterisks) indicates a temperature corresponding to the inflection point during a gradual reduction in $M$ on cooling, and also to an increase in helical $q$ (see Supplemental Fig. S3 [38]). Two helimagnetic phases (light-blue and pink areas) separated by a nearly ferromagnetic region (white area below $T_{\mathrm{c}}$) are characterized by opposite signs of ${\mathrm{\Gamma }}_{\mathrm{c}}{\gamma }_{\mathrm{m}}$.

Figure 4

(a),(b) Convergent-beam electron diffraction (CBED) patterns observed from a thin-plate specimen of ${\mathrm{Co}}_8{\mathrm{Zn}}_8{\mathrm{Mn}}_4$. (c),(d) Simulated CBED patterns for right-handed crystal with the space group of $P{4}_{1}32$. (e),(f) CBED patterns observed in a thin-plate specimen of ${\mathrm{Co}}_{3.5}{\mathrm{Fe}}_{4.5}{\mathrm{Zn}}_8{\mathrm{Mn}}_4$. (g),(h) Simulated CBED patterns for left-handed crystal with the space group of $P{4}_{3}32$. For all the panels (a)–(h), the incident electron beam direction is along the [120] direction. A red triangle in upper panels (a), (c), (e), and (g) indicates the disconnected intensity at the top part of circular diffraction that makes a distinction with the connected intensity at the bottom part, reflecting broken inversion symmetry. Lower panels (b), (d), (f), and (h) are expanded images for the red dotted-line enclosed areas of (a), (c), (e), and (g), respectively. Blue and green dotted circles in (b), (d), (f), and (h) highlight [002] and [$00\overline{2}$] diffraction signals, respectively. (i)–(l) Lorentz transmission electron microscope (LTEM) images observed for the same thin-plate specimens as used in the CBED measurements. Panel (i)[(j)] is an over- (under-) focused LTEM image for ${\mathrm{Co}}_8{\mathrm{Zn}}_8{\mathrm{Mn}}_4$, measured at 293 K and under a magnetic field of 45 mT applied perpendicular (into the page) to the thin specimen. Panel (k)[(l)] is an over- (under-) focused LTEM image for ${\mathrm{Co}}_{3.5}{\mathrm{Fe}}_{4.5}{\mathrm{Zn}}_8{\mathrm{Mn}}_4$, measured at 110 K and 75 mT.

Figure 5

(a) Dopant concentration ($x$) dependence of $\tilde{D}\equiv (-{\mathrm{\Gamma }}_{\mathrm{c}}{\gamma }_{\mathrm{m}})T_{\mathrm{c}}q$, which is proportional to Dzyaloshinskii-Moriya interaction $D$, in ${\mathrm{Co}}_{8-x}{TM}_x{\mathrm{Zn}}_8{\mathrm{Mn}}_4$. Here, $\mathit{TM}$ = Fe (red circles) and Ru (green squares) correspond to hole doping, while $\mathit{TM}$ = Ni (blue triangles) to electron doping. The values of $T_{\mathrm{c}}$ and $q$ are determined by magnetization and SANS measurements, respectively. See Supplemental Figs. S5 and S6 [38] for details of magnetization and SANS measurements for $\mathit{TM}$ = Ni and Ru. The minus sign is appended to the definition of $\tilde{D}$ due to the following reason: both the experiment and theory consistently show that the right-handed crystal (${\mathrm{\Gamma }}_{\mathrm{c}}=+1$) of the Fe nondoped material exhibits a left-handed helix (${\gamma }_{\mathrm{m}}=-1$). In our theoretical definition, this corresponds to $D>0$; therefore, we included the minus sign to keep the consistency. (b) Theoretically calculated value of $D$ plotted as a function of electron number per unit cell, which is taken as the origin for a hypothetical model compound ${\mathrm{Co}}_8{\mathrm{Zn}}_{12}$, where $8c$ and $12d$ sites are occupied by only Co and Zn, respectively. The calculated band structure and density of states are shown in Supplemental Fig. S7 [38].