Research ArticlePHYSICS

Disordered skyrmion phase stabilized by magnetic frustration in a chiral magnet

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Science Advances  14 Sep 2018:
Vol. 4, no. 9, eaar7043
DOI: 10.1126/sciadv.aar7043


Magnetic skyrmions are vortex-like topological spin textures often observed to form a triangular-lattice skyrmion crystal in structurally chiral magnets with the Dzyaloshinskii-Moriya interaction. Recently, β-Mn structure–type Co-Zn-Mn alloys were identified as a new class of chiral magnet to host such skyrmion crystal phases, while β-Mn itself is known as hosting an elemental geometrically frustrated spin liquid. We report the intermediate composition system Co7Zn7Mn6 to be a unique host of two disconnected, thermal-equilibrium topological skyrmion phases; one is a conventional skyrmion crystal phase stabilized by thermal fluctuations and restricted to exist just below the magnetic transition temperature Tc, and the other is a novel three-dimensionally disordered skyrmion phase that is stable well below Tc. The stability of this new disordered skyrmion phase is due to a cooperative interplay between the chiral magnetism with the Dzyaloshinskii-Moriya interaction and the frustrated magnetism inherent to β-Mn.


Magnetic spin systems in solids exhibit a rich variety of ordering patterns, dependent on the microscopic interactions, anisotropy, lattice form, etc. Among the various orders, noncollinear or noncoplanar ones with vector or scalar spin chirality attract considerable attention because of the associated collective properties they may generate, such as multiferroic or emergent electromagnetic responses. Magnetic skyrmions (14), vortex-like, spin-swirling textures characterized by an integer topological charge, are a quintessential example of noncoplanar magnetic structures. Thus far, they have been observed or predicted in various systems (310), with their origins attributed to several microscopic mechanisms, such as competition between the Dzyaloshinskii-Moriya interaction (DMI) and ferromagnetic exchange interaction, magnetic frustration (1113), Fermi surface effect (14), and magnetic dipolar interaction (15). In particular, a finite DMI can arise because of broken inversion symmetry either at interfaces of thin-film layers (5, 6) or in bulk materials with chiral or polar structures (3, 4, 710). In the chiral magnets, the effect of the DMI is to gradually twist otherwise ferromagnetically coupled moments to form a helical ground state with a mesoscale periodicity described by a single magnetic propagation vector (q). An applied magnetic field may induce a triangular-lattice skyrmion crystal (SkX), which is often described as a triple-q structure with the q-vectors displaying mutual angles of 120°. Such SkX states are stabilized by thermal fluctuations and confined to a narrow region near the helimagnetic transition temperature Tc (2, 3).

Recently, DMI-based skyrmions have been observed in Co-Zn-Mn alloys with the β-Mn–type chiral cubic structure (9), where the unit cell contains 20 atoms distributed over two inequivalent crystallographic sites (8c and 12d Wyckoff sites, the inset of Fig. 1A). Co10Zn10, one end member of a solid solution (Co0.5Zn0.5)20−xMnx (0 ≤ x ≤ 20) with the β-Mn structure (16), shows a helical ordering of Co spins (periodicity λ ~ 185 nm) below Tc ~ 460 K, with Tc decreasing as the partial substitution of Mn proceeds. In Co8Zn8Mn4 (Tc ~ 300 K), an SkX state created close to Tc can persist as a metastable state over a very wide temperature and magnetic field region upon a field cooling (FC), accompanied by a lattice-form transformation of the SkX at low temperatures (17, 18). The other end member Mn20 (β-Mn itself) is well known to display no transition to magnetic long range order (19, 20) due to the strong geometrical frustration of antiferromagnetic interactions in the three-dimensional hyper-kagome network of the 12d sites (21). Lightly doped β-Mn alloys have been found to exhibit spin glass states (19, 21) or complex incommensurate antiferromagnetic states with noncoplanar structure (22).

Fig. 1 Temperature–Mn concentration phase diagram of (Co0.5Zn0.5)20−xMnx, schematic SANS patterns, and SANS images at 146 K in Co7Zn7Mn6.

(A) The zero-field magnetic phase diagram of (Co0.5Zn0.5)20−xMnx (0 ≤ x ≤ 20) in the T (temperature) versus x (Mn composition) plane, as determined by magnetization measurements (see fig. S2A). The inset shows a schematic of a β-Mn–type structure (space group: P4132) as viewed along the [111] axis. (B) Schematic of magnetic structures in real space and the corresponding SANS patterns in the H (magnetic field) || neutron beam and H ⊥ neutron beam geometries. The direct-beam spot expected at the center of each pattern is masked out. (i) The helical (H) state forms three domains with single-q || [100], [010], or [001], respectively, resulting in four spots in both geometries. (ii) In the conical (C) state with q || H || [001], only two spots are observed in the H ⊥ beam geometry. (iii) The triangular-lattice SkX state forms two domains with one of the triple-q || [100] or [010] (degenerate preferred q-directions), respectively, resulting in 12 spots in the H || beam geometry and 2 spots in the H ⊥ beam geometry. (iv) In the disordered helical (DH) state, the spots are broadened compared with (i). (v) In the three-dimensionally disordered skyrmion (DSk) state, a spherical q distribution manifests itself as a ring in both geometries. (C) SANS images observed in Co7Zn7Mn6 at 146 K and at 0 T and 0.03 T in the H || beam and H ⊥ beam geometries, respectively. The 0 T and 0.03 T patterns represent (i) helical and (iii) SkX [plus (ii) conical] states, respectively. Twelve spots in the H || beam geometry indicate that the SkX state consists of two kinds of domains with one of the triple-q || [100] or [010]. Note that the intensity scale of the color plots varies between each panel. arb. units, arbitrary units.

Figure 1A shows the T (temperature) − x (Mn composition) phase diagram connecting Co10Zn10 and β-Mn according to (Co0.5Zn0.5)20−xMnx, as revealed in the present study (see fig. S2). A spin glass phase symptomatic of frustrated magnetism is found to exist at low temperatures and over a wide range of x. For 3 ≤ x ≤ 7, the spin glass phase invades the helical phase and displays a typical reentrant spin glass behavior (23, 24), indicating a microscopic coexistence of the two states. To investigate the influence of frustration on the helical and topological spin textures, we focused on Co7Zn7Mn6 (x = 6, indicated with a pink arrow in Fig. 1A) with Tc ~ 160 K and spin glass transition temperature Tg ~ 30 K and performed measurements of small-angle neutron scattering (SANS), magnetization, ac susceptibility, and Lorentz transmission electron microscopy (LTEM; see Materials and Methods). As summarized in the T-H (magnetic field) phase diagram in Fig. 2A, two distinct, equilibrium skyrmion phases are found; one is a conventional SkX phase slightly below Tc, and the other is a novel disordered skyrmion (DSk) phase near Tg. This new phase is quenched as a metastable state down to zero field in the field-decreasing process (Fig. 2B). To substantiate our findings in what follows, we summarize the relation between the real-space magnetic structures and the corresponding SANS patterns in Fig. 1B.

Fig. 2 Temperature-field phase diagrams in Co7Zn7Mn6.

T-H phase diagrams in Co7Zn7Mn6 determined by ac susceptibility and SANS measurements (see figs. S3B, S4, D and E, and S8) (A) in the field-increasing runs after ZFC and (B) in the field-decreasing runs from the induced ferrimagnetic (higher H) region. The helical-conical and conical-ferrimagnetic phase boundaries are indicated by black diamonds and squares, respectively. The phase boundaries of the conventional SkX phase are indicated by green circles. The crossover region around 90 K, below which the helical state becomes disordered, is indicated by gray hatching. Spin glass transition temperatures around 30 K are indicated by black triangles. Pink circles show the magnetic fields where either 12-spot or ring-like SANS patterns are observed. The equilibrium phase (plus metastable state in the case of field-decreasing processes) of DSks is indicated by a red color region. Note that the SkX and conical states and conical and DSk states coexist in broad regions, and only the majority phase is indicated in the phase diagrams.


SANS measurements of conventional skyrmion phase at high temperatures

At high temperatures just below Tc, a conventional SkX state is observed (Fig. 1C). The SANS pattern changes from 4 to 12 spots upon the application of magnetic field in the H || beam geometry at 146 K, indicating the transition from a helical multidomain state to a triangular-lattice SkX state (see fig. S4 for details). The SkX state coexists with a conical state (two horizontal spots in the H ⊥ beam geometry). As the temperature is lowered, the volume fraction of the SkX state relative to the conical state becomes smaller (see figs. S4 and S5 for details). Consequently, the T-H phase diagram above 100 K is characterized by a typical SkX pocket near Tc (green region in Fig. 2A) in an otherwise conical phase background.

SANS and ac susceptibility measurements in a ZFC

Upon zero-field cooling (ZFC), the periodicity λ of the helical state shrinks (from 110 to 73 nm), and the diffraction spots in the SANS pattern broaden significantly below ~90 K, providing evidence for growing magnetic disorder upon cooling (see fig. S3, D and E). In this ZFC process, ac susceptibility smoothly decreases below ~90 K and drops at ~ 30 K, the latter of which corresponds to a spin glass transition, as evidenced by a strong frequency dependence of the transition temperature (see fig. S3, B and C). The disordering of the helical state toward low temperature and the subsequent spin glass transition are likely due to the enhancement of frustrated antiferromagnetic correlations of Mn spins, which couple ferromagnetically with Co spins. The spin glass phase exists over a wide x range with a nearly constant Tg ~ 60 K in the T-x phase diagram (Fig. 1A), suggesting a continuous existence of frustrated antiferromagnetic Mn spin correlations from β-Mn to Co7Zn7Mn6.

SANS measurements at low temperatures

Figure 3 shows SANS results from magnetic field up and down scans at 50 K (see also fig. S6). In the field-increasing run after ZFC (Fig. 3B), the pattern with four broad spots changes to a ring pattern above 0.1 T, and finally, SANS intensity disappears in the induced ferrimagnetic region. While Co spins are forcedly aligned ferromagnetically, Mn spins still preserve an antiferromagnetic arrangement, resulting in a ferrimagnetic structure as a whole. In the subsequent field-decreasing run, a ring pattern appears again and persists to zero field. The azimuthal angle-dependent scattering intensity is plotted against magnetic field in Fig. 3C. Above 0.1 T in the field-increasing run and over the whole field region in the field-decreasing run, intensities for directions close to 〈100〉 and 〈110〉 overlap completely, as expected for a homogeneous ring of scattering. Since ring-like patterns are also observed in the H ⊥ beam geometry at 0.1 T (Fig. 3D) and even after the removal of magnetic field (Fig. 3E), we conclude the scattering intensity to be three-dimensionally distributed over a spherical shell in reciprocal space. This spherical SANS pattern may be explained in terms of two possible scenarios; (i) a three-dimensionally disordered helical state (a topologically trivial state) or (ii) a three-dimensionally disordered skyrmion state (a topologically defined state), as illustrated in Fig. 1B(v). In the following sections, we show that scenario (ii) is more plausible.

Fig. 3 Field dependence of the SANS patterns at 50 K in Co7Zn7Mn6.

(A) Schematic of the measurement process. The field-increasing run from 0 T to 0.3 T after ZFC and the subsequent field-decreasing run from 0.3 T to 0 T are denoted by pink and light blue arrows, respectively. In the schematic phase diagram, we use the following notations: P, paramagnetic; H, helical; C, conical; SkX, skyrmion crystal (green region); F, ferrimagnetic; DH, disordered helical; DSk, disordered skyrmion (red region); and SG, spin glass. (B) SANS images at selected fields in the H || beam geometry. Note that the intensity scale of the color plots varies between each panel. (C) Field dependence of integrated SANS intensities. The intensities for directions close to 〈100〉 over the regions at θ = 0° ± 15°, 90° ± 15°, 180° ± 15°, and 270° ± 15° (red region in the inset) are indicated by red squares. The intensities for directions close to 〈110〉 over the regions at θ = 45° ± 15°, 135° ± 15°, 225° ± 15°, and 315° ± 15° (blue region in the inset) are indicated by blue circles. Here, θ is defined as the clockwise azimuthal angle from the vertical [010] direction. The field-increasing and field-decreasing runs are indicated by closed and open symbols with the same colors, respectively. (D and E) SANS images (D) at 0.1 T in the field-increasing run and (E) at 0 T after the field-decreasing run in both geometries, respectively. In (D) and (E), the data for H ⊥ beam geometry were collected within the same field-sweeping process [shown schematically in (A)], as used for collecting the data shown in (B). Note that the intensity scale of the color plots varies between each panel.

SANS measurements in several warming processes

To distinguish the above two possibilities [(i) and (ii)], we present SANS results obtained during the following two warming processes done subsequently to field sweepings at low temperatures.

First, in the zero-field warming (ZFW) process done after the field-decreasing run at 50 K (see fig. S9), the ring pattern in the H || beam geometry changes to a pattern with four sharp spots (helical multidomain state) above ~90 K. In a subsequent ZFC process returning from 130 K, the observed pattern at 50 K is no longer ring-like but forms four broad spots, indicating the strongly irreversible nature of the low-temperature state.

Second, an important clue to the nature of the low-temperature state is provided by SANS data in the H ⊥ beam geometry in the field warming (FW) process at 0.025 T done subsequently to the field-decreasing run at 60 K (Fig. 4). As shown in Fig. 4B, in this FW process, the ring-like patterns observed in the H ⊥ beam geometry also change to a four-spot pattern above 90 K due to a reduced disorder effect at elevated temperatures. Remarkably, while the intensity nearly parallel to the field (q || H) is slightly stronger than that nearly perpendicular to the field (qH) in the ring-like pattern at 60 K, the two vertical spots (qH) become stronger than the two horizontal spots (q || H) in the four-spot pattern at 90 and 110 K. If the two vertical spots were attributed to an ordered helical domain restored from its low-temperature disordered state, the two horizontal spots (from the conical state) should remain stronger than the two vertical spots under this field-forced condition, which is in contradiction to the actual observation. On the other hand, this characteristic change in intensity distribution is reasonably explained if the two vertical spots at 90 and 110 K arise from two-dimensional skyrmions, which are created with random orientation at low temperatures, persist as a superheated metastable state, and become more ordered with respect to the field direction above the order-disorder crossover region (gray hatching area in Fig. 2). The temperature-dependent intensities for qH and q || H and their ratio are presented in Fig. 4 (C and D). With increasing temperature up to 110 K, the intensity observed from the q || H region decreases and transfers to the qH region. This indicates that the broad intensity distribution around q || H at 60 K can be attributed not only to conical state but also to vertically oriented skyrmions, thus confirming that the skyrmions at low temperatures are of a three-dimensionally disordered form, as illustrated in Fig. 1B(v).

Fig. 4 SANS patterns during the FW process in Co7Zn7Mn6.

(A) Schematic of the measurement process. (B) SANS images at selected temperatures in the H ⊥ beam geometry for the FW process at 0.025 T after a field decrease at 60 K. The intensity scale of the color plots varies between each panel. It is noted that the SANS images for both the FW process and the ZFW process (fig. S9C) were obtained from two rocking scans covering the same scanning region where the cryomagnet is rotated from −5° to 5° around both the vertical and horizontal axes. Pink and yellow dotted circles highlight the SANS intensities allocated to the SkX state and conical state, respectively. (C) Temperature dependence of integrated SANS intensities. The intensities integrated over the regions nearly perpendicular to the field at φ = 0° ± 30° and 180° ± 30° (red region in the inset) are indicated by red squares [I(qH)]. The intensities integrated over the regions nearly parallel to the field at φ = 90° ± 30° and 270° ± 30° (yellow region in the inset) are indicated by yellow triangles [I(q || H)]. Here, φ is defined as the clockwise azimuthal angle from the vertical [010] direction. (D) Temperature dependence of the SANS intensity ratio, I(qH)/I(q || H). The intensity ratio becomes smaller than 1 at 146 K because the volume fraction of a conical state is larger than that of an equilibrium SkX state (see also fig. S4). For (C) and (D), the crossover region, above (below) which both helical and skyrmion states are ordered (disordered), is indicated with gray hatching.

Upon further increase of the temperature above 110 K, the intensity of the two vertical spots becomes weaker, which is interpreted as an eventual changeover of the metastable SkX state into the equilibrium conical state. Therefore, the aforementioned irreversibility is better interpreted in terms of a destruction of a metastable skyrmion state rather than, for example, the loss of a disordered helical state with strongly glassy character.

ac susceptibility measurements in several warming processes

The above interpretation is further reinforced by ac susceptibility data shown in Fig. 5. Figure 5B shows the field dependence of the real part of the ac susceptibility [χ′(H)] at 110 K measured after FC and FW processes at 0.025 T, as illustrated in Fig. 5A. After the FC1 process (blue circles) passing through the conventional SkX phase, χ′(H) shows an asymmetric shape with respect to the sign of H, which is a hallmark signature for a supercooled metastable SkX state surviving predominantly at a positive-field region (17, 25). In the field-returning process from field-polarized ferrimagnetic regions, the conical state exhibits a symmetric shape of χ′(H) with larger values (black line) than the metastable SkX state. After the FW1 process (red line) passing through the low-temperature DSk phase, χ′(H) again shows an asymmetric shape similar to the case of FC1, indicating that the superheated metastable SkX state is realized above the crossover region.

Fig. 5 Field-swept ac susceptibility after several cooling/warming processes in Co7Zn7Mn6.

(A) Schematic of measurement processes for (B). FC1 (blue line) is a FC process passing through the SkX phase: (i) ZFC to 146 K, (ii) field increase to 0.025 T, and (iii) FC to 110 K. FW1 (red line) is a FW process passing through the DSk phase: (i) ZFC to 60 K, (ii) field increase to 0.2 T, (iii) field decrease to 0.025 T, and (iv) FW to 110 K. (B) Field dependence of the real part of the ac susceptibility (χ′) at 110 K after FC1 (blue circles) and FW1 (red line). Black lines show the field-returning runs from high-field ferrimagnetic regions. (C) Schematic of measurement processes for (D). FC2 (yellow line) is a FC process bypassing the SkX phase: (i) ZFC to 120 K, (ii) field increase to 0.025 T, and (iii) FC to 110 K. FW2 (green line) is a FW process bypassing the DSk phase: (i) ZFC to 2 K, (ii) field increase to 0.2 T, (iii) field decrease to 0 T, (iv) ZFW to 100 K, (v) field increase to 0.025 T, and (vi) FW to 110 K. (D) Field dependence of χ′ at 110 K after FC2 (yellow squares) and FW2 (green line). Black lines show the field-returning runs from high-field ferrimagnetic regions. In (B) and (D), the black lines for FC1 and FW1 and for FC2 and FW2 completely overlap with each other.

In contrast to these behaviors, χ′(H) is observed to be symmetric (Fig. 5D), as expected for a helical domain state, after both the FC2 process (yellow squares) that bypasses the conventional SkX phase and the FW2 process (green line) that passes through the highly glassy region at 2 K but bypasses the DSk phase, as illustrated in Fig. 5C. Especially, the latter data indicate that the strongly glassy character at low temperatures is completely suppressed already at 110 K. Therefore, the asymmetric shape of χ′(H) in the FW1 process cannot be ascribed to a helical domain state with the glassy character. These results indicate that a novel equilibrium skyrmion phase, namely, DSk phase, exists at low temperatures in addition to the conventional SkX phase just below Tc and that the metastable SkX state is commonly realized by both superheating and supercooling, respectively.

The asymmetry of χ′(H) in the FW process passing through the DSk phase is diminished as the temperature is increased above the crossover region in a warming field–dependent manner: It is lost at lower temperature as the warming field is decreased, as presented in detail in fig. S10. This behavior is also consistent with the interpretation that the asymmetry represents existence of the metastable skyrmions since they are generally known to become more fragile as the field is reduced (17, 2527). Taking the above SANS and ac susceptibility data in the warming processes into account, it is concluded that the spherical SANS pattern observed at low temperatures originates from (ii) disordered skyrmions rather than (i) the disordered helical state with strongly glassy character.

SANS measurements below the spin glass transition temperature

The DSk phase is found to exist at least below 60 K and above Tg ~ 30 K as an equilibrium state, on the basis of field-swept SANS measurements at different temperatures (see fig. S8), as summarized in the T-H phase diagram in Fig. 2. The DSk phase is not stabilized below Tg after ZFC, as evidenced by the field-dependent SANS measurement at 20 K (fig. S7, A to C). On the other hand, the DSk phase is accessible by a FW process from temperatures below Tg, as demonstrated in fig. S7 (D and F). Once created, the DSks also persist below Tg as a supercooled metastable state in the subsequent FC process.

LTEM measurements

Figure 6 shows real-space observations by LTEM on a thin-plate sample (thickness, ~150 nm). In the field-increasing run at 135 K after ZFC, the transition from a helical state to a conventional SkX state is observed (Fig. 6B). At the lower temperature of 50 K, a disordered helical state is observed as a distribution of elongated objects after ZFC (Fig. 6C). In contrast, under a magnetic field of 0.2 T, a number of closed, dot-like objects, assigned to skyrmions, are observed clearly. It is noted that the magnetic contrast of these skyrmions is weaker than at 135 K (Fig. 6B) despite the fact that the local magnetization becomes larger, implying either a magnetic modulation along, or a slight inclination of the skyrmions from, the normal direction of the sample plate, consistent with the spherical distribution of q-vectors in SANS measurements. While the distribution of skyrmions displays positional disorder, they are identified to be topologically nontrivial as indicated from the in-plane magnetization map derived from transport-of-intensity equation analysis of the LTEM images (Fig. 6D; see Materials and Methods for detailed procedures of the analysis). At 0.4 T in the ferrimagnetic state, the magnetic contrast disappears, and at 0 T after the subsequent removal of the field, the magnetic contrast due to a mixture of DSks and disordered helices is observed. In the subsequent ZFW process, the DSks coalesce to form disordered helices [with a number of topologically nontrivial defects (28, 29)] at 80 K and finally become an ordered helical state at 100 K. These temperature and field variations are fully consistent with the SANS results on the bulk crystal sample. In some materials, it is reported that the equilibrium skyrmion phase tends to expand toward lower temperatures in a thin-plate specimen with a thickness comparable to a skyrmion diameter (4, 7, 8). In the present Co-Zn-Mn alloys, however, the effect of a reduced sample thickness on the stability of the equilibrium SkX phase is relatively small: The SkX is observed only over a temperature range that amounts to approximately 10% of Tc even for thin-plate samples (9). Thus, the present LTEM observation of DSks at 50 K can hardly be explained as an effect of reduced dimensionality alone, but should be regarded as an inherent feature of the material.

Fig. 6 LTEM measurements on a (001) thin-plate sample of Co7Zn7Mn6.

(A) Schematic illustration of the measurement processes. The field-increasing runs at 135 and 50 K after ZFC from room temperature are denoted by pink arrows. The field-decreasing run from 0.4 T to 0 T at 50 K and the subsequent ZFW process are denoted by light blue and red arrows, respectively. (B) Underfocused LTEM images at 135 K and at 0 T and 0.05 T. (C) Underfocused LTEM images at selected fields at 50 K and at selected temperatures in the subsequent ZFW process. Only for the image at 50 K and 0.2 T, the corresponding overfocused image is also shown at right side. The assignment of the LTEM images on each panel is given such as H, SkX, DH, DSk, F, and DSk + DH (mixed state). (D) Color coding of in-plane magnetization (white arrows) deduced from a transport-of-intensity equation analysis for the areas marked with the red dashed square in the underfocused and overfocused images at 50 K and 0.02 T.


Here, we discuss the possible origin of the novel DSk phase. Skyrmion phases are generally believed to be stabilized with the help of either thermal (3) or quantum critical (30, 31) fluctuations. As discussed theoretically in detail in (3), fluctuations at both short (atomic) and long (mesoscopic) length scales play a crucial role. Usually, short wavelength fluctuations exist only around Tc. In the present special case, however, frustrated antiferromagnetic correlations between Mn spins become enhanced at low temperature and, via the ferromagnetic coupling between Mn and Co, short length scale fluctuations of the order of several angstroms can be induced in the Co spin system with a long helical periodicity of 100-nm order (1922). Thus, two kinds of fluctuations with a similar short length scale are found to promote topological phase stability in Co7Zn7Mn6; one is the thermal fluctuations against the helical order that stabilize the conventional SkX phase just below Tc, and the other is frustration-induced fluctuations that stabilize the novel DSk phase at low temperatures just above Tg. With this scenario, the absence of the DSk phase below Tg, where dynamical fluctuations are suppressed due to spin freezing, is reasonably understood. The skyrmion size here is governed by the DMI, this being much larger than the length scale of frustrated antiferromagnetic spin modulation. This makes the DSks observed here distinct from the atomic length scale skyrmion states expected in theory for genuinely frustrated systems with centrosymmetric lattices (1113). Thus, the present study demonstrates a novel mechanism of skyrmion formation in DMI-based helical magnets that exploits frustration to stabilize topological noncoplanar spin structures.


Sample preparation

Polycrystalline samples of (Co0.5Zn0.5)20−xMnx (0 ≤ x ≤ 20) were synthesized from pure Co, Zn, and Mn metals with nominal concentrations. These metals were sealed in evacuated quartz tubes and heated above 1000°C, cooled down to 925°C at the rate of 1°C min−1, annealed for 2 to 4 days, and finally quenched to water. Phase purity with a β-Mn–type crystal structure was confirmed using powder x-ray diffraction for each sample. Single-crystalline bulk samples of Co7Zn7Mn6 were grown by the Bridgman method. The crystal orientation was determined using the x-ray diffraction Laue method (fig. S1C), and the samples were cut along the (100), (010), and (001) planes with rectangular shapes for SANS and ac susceptibility measurements, as shown in fig. S1 (A and B). Because of the difference in shape between the samples used in the ac susceptibility and the SANS measurements, their demagnetization factors are different. To correct for this difference, the magnetic field (H) values for the ac susceptibility measurements are calibrated to be Hc = 2.7*H. For all the figures related to ac susceptibility measurements in the main text and the Supplementary Materials, the calibrated value Hc was used with the notation of H.

For the LTEM measurement, a tiny thin-plate sample with (001) face and thickness of approximately 150 nm was prepared from a single-crystalline bulk sample using a focused ion beam of Ga.

Magnetization measurement

dc magnetization measurements for polycrystalline samples of (Co0.5Zn0.5)20−xMnx were performed using the vibrating sample magnetometer mode of a superconducting quantum interference device magnetometer (MPMS3, Quantum Design).

ac susceptibility measurement

ac susceptibility measurements for a single-crystalline sample of Co7Zn7Mn6 were performed using the ac susceptibility measurement mode of the MPMS3. Both the static magnetic field and the ac excitation field (1 Oe) were applied along a [100] direction. The ac frequency f was fixed to be 193 Hz except for the frequency-dependent measurements.

SANS measurement

SANS measurements on a single-crystalline sample of Co7Zn7Mn6 were performed using the instrument D33 at the Institut Laue-Langevin (ILL), Grenoble, France. Neutrons with a wavelength of 10 Å were collimated over 12.8 m before the sample. The scattered neutrons were counted by a two-dimensional position-sensitive multidetector located 13.4 m behind the sample. The mounted single-crystalline sample was installed into a horizontal field cryomagnet so that the field direction was parallel to the [001] direction. Maintaining the H || [001] geometry, the cryomagnet was rotated (rocked) around the vertical [010] direction in the range from −5° to 5° (from −6° to 12° only for H || beam geometry in the process of ZFC, and field scans at 146, 130, and 100 K) and tilted around the horizontal [100] (or [001]) direction in the range from −5° to 5°. These two scans were performed for both H || beam and H ⊥ beam configurations. All the SANS images shown in the main text and the Supplementary Materials were obtained by summing over the above rotation and tilt rocking scans, and the scales of horizontal axis (qx) and vertical axis (qy) are fixed to be −0.14 nm−1qx ≤ 0.14 nm−1 and −0.14 nm−1qy ≤ 0.14 nm−1.

LTEM measurement

LTEM measurements for a single-crystalline thin-plate sample of Co7Zn7Mn6 were performed with a transmission electron microscope (JEM-2800). A magnetic field was applied perpendicular to the (001) plate, and its magnitude was controlled by tuning the electric current of the objective lens. Because of the thin-plate shape (thickness, ~150 nm), its demagnetization factor and thus magnetic field scale are different from those of the bulk samples. For Fig. 6D, we extracted an in-plane magnetization image using a software package QPt based on the transport-of-intensity equationEmbedded Image(1)

Here, I (xyz) and φ(xyz) represent the intensity and the phase of the electron beam, respectively. The analysis of electron beam intensity based on the underfocused and overfocused LTEM images enables the phase image φ(xyz) to be obtained. According to the Maxwell-Ampère equations, ∇xyφ(xyz) is related to the in-plane component of the magnetization viaEmbedded Image(2)where M, t, and n are the magnetization, the sample thickness, and the unit vector perpendicular to the sample surface, respectively.


Supplementary material for this article is available at

Section S1. Magnetic properties in (Co0.5Zn0.5)20−xMnx

Section S2. ZFC process in Co7Zn7Mn6

Section S3. Field-sweeping processes in Co7Zn7Mn6

Section S4. FW and FC processes across the spin glass transition temperature in Co7Zn7Mn6

Section S5. ZFW and FW processes after field sweepings at low temperatures in Co7Zn7Mn6

Fig. S1. Information of single-crystalline bulk samples of Co7Zn7Mn6.

Fig. S2. Mn concentration (x) dependence of polycrystalline samples of (Co0.5Zn0.5)20−xMnx.

Fig. S3. SANS and ac susceptibility measurements in the ZFC process in Co7Zn7Mn6.

Fig. S4. SANS and ac susceptibility measurements in the field-sweeping process at 146 and 130 K in Co7Zn7Mn6.

Fig. S5. SANS and ac susceptibility measurements in the field-sweeping process at 100 K in Co7Zn7Mn6.

Fig. S6. SANS and ac susceptibility measurements in the field-sweeping process at 50 K in Co7Zn7Mn6.

Fig. S7. SANS and ac susceptibility measurements in the field-sweeping process at 20 K, and the subsequent temperature-sweeping process in Co7Zn7Mn6.

Fig. S8. Field dependence of the SANS intensities in the H || beam geometry at all the measured temperatures.

Fig. S9. SANS and ac susceptibility measurements in the ZFW process after the field-decreasing run at 50 K and a subsequent ZFC process in Co7Zn7Mn6.

Fig. S10. Field-swept ac susceptibility measurements after several warming processes in Co7Zn7Mn6.

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Acknowledgments: We are grateful to N. Nagaosa, W. Koshibae, F. Kagawa, and T. Nakajima for the discussions. Funding: This work was supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aids for Scientific Research (S) (grant no. 24224009) and for Young Scientists (B) (grant no. 17K18355), the Swiss National Science Foundation (SNF) Sinergia network “NanoSkyrmionics” (grant no. CRSII5-171003), the SNF projects 153451 and 166298, and the European Research Council project CONQUEST. Author contributions: Y. Taguchi, Y. Tokura, T.-h.A., and H.M.R. jointly conceived the project. A.K., K.K., and Y. Tokunaga prepared the bulk samples. K.K. performed the magnetization and ac susceptibility measurements. J.S.W., K.K., C.D.D., and R.C. performed the SANS measurements. D.M. and X.Y. prepared the thin-plate sample and performed the LTEM measurements. The results were discussed and interpreted by all the authors. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. The data sets for the SANS experiments done on D33 at the ILL are available through the ILL data portal ( and Additional data related to this paper may be requested from the authors.
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