Uniaxial stress control of skyrmion phase

Magnetic skyrmions, swirling nanometric spin textures, have been attracting increasing attention by virtue of their potential applications for future memory technology and their emergent electromagnetism. Despite a variety of theoretical proposals oriented towards skyrmion-based electronics (that is, skyrmionics), few experiments have succeeded in creating, deleting and transferring skyrmions, and the manipulation methodologies have thus far remained limited to electric, magnetic and thermal stimuli. Here, we demonstrate a new approach for skyrmion phase control based on a mechanical stress. By continuously scanning uniaxial stress at low temperatures, we can create and annihilate a skyrmion crystal in a prototypical chiral magnet MnSi. The critical stress is merely several tens of MPa, which is easily accessible using the tip of a conventional cantilever. The present results offer a new guideline even for single skyrmion control that requires neither electric nor magnetic biases and consumes extremely little energy.

M agnetic skyrmions and their crystalline form called skyrmion crystals (SkX; Fig. 1a,b), have been observed in bulk materials [1][2][3][4] , thin films [5][6][7] and nanowires 8,9 of P2 1 3-type chiral magnets such as MnSi, Fe 1 À x Co x Si and Cu 2 OSeO 3 . They have also been observed quite recently in non-P2 1 3-type chiral magnets of GaV 4 S 8 (ref. 10) and Co-Zn-Mn alloys 11 , and even at the interfaces of magnetic thin films 12,13 and nanoscopic-patterned discs 14,15 . As schematically illustrated in Fig. 1a, a skrymion has a particle-like swirling spin configuration. Unlike the conventional spin textures such as helical or conical states (Fig. 1c) that appear in chiral magnets, skyrmions have spin configurations characterized by a topological index called the skyrmion number 16,17 . The nontrivial topology of skyrmions gives rise to emergent electromagnetic responses as exemplified by the topological Hall effect 18 , skyrmion magnetic resonance 19 and thermally induced ratchet motion 20 .
In addition to these intriguing physical responses, skyrmions have been attracting extensive attention for their potential applications in future solid-state memory technology or storage devices 21,22 because of their small size (3-100 nm) 21 and topological stability 16 as well as ultralow electric currents required to drive their motion [23][24][25] . Pursuing arbitrary control of skyrmions as information carriers towards the attempt to develop skyrmion-based electronics (that is, skyrmionics), a number of theoretical simulations have recently been performed, yielding schemes for controlling skyrmions based on electric currents 22,[26][27][28][29][30][31][32][33] , electric fields 32,34 , magnetic fields 32 , thermal stimuli 35 or conversions between a skyrmion and domain walls 36 .
Compared with these theoretical simulations, experimental demonstrations such as the creation 13,37 , deletion 13 , transfer [23][24][25] and rotation 20,38 of skyrmions seem to be limited. Therefore, it remains necessary to establish and expand a framework for skyrmion manipulation.
Herein, we present a new approach for manipulating a SkX in a bulk single crystal of a prototypical chiral magnet MnSi. Although any continuous deformations of the spins cannot connect topologically different skyrmion and conical states, we find that a simple mechanical stress can overcome the energy barrier to switch between these two states in both directions. The critical stress needed to trigger this topological switching is merely several tens of MPa, and the corresponding strain is of the order of 10 À 4 . This perturbative mechanical stimulus is easily achievable using conventional scanning probe techniques, offering a new guideline for skyrmion manipulation, even at the single skyrmion level, that is essentially free of Joule heating.

Results
Experimental set-up for the application of mechanical stress. We continuously applied and released compressive uniaxial stress over broad ranges of temperature and magnetic field strengths using a previously developed special uniaxial stress probe 39 that was improved for the present study. As shown in Fig. 1d,e, longitudinal (r || H) or transverse (r > H) uniaxial stress was introduced, and the longitudinal or transverse AC magnetic susceptibility was simultaneously monitored by coils. Here, r and H represent the compressive stress and magnetic field,  H and Q i represent the magnetic field and propagation vectors, respectively. The colours of the arrows indicate the out-of-plane components of the spins in the skyrmion spin texture, whereas they represent the directions of the in-plane components (perpendicular to the Q-vector) in the conical structure. (d,e) The set-up for AC magnetic susceptibility measurements under (d) longitudinal and (e) transverse uniaxial stresses with respect to an external magnetic field. The (green) cuboids represent a cubic unit cell, and r represents the compressive stress. In the transverse configuration, several ZrO 2 pistons converted longitudinal forces into lateral ones. The magnetic susceptibility was monitored by coils. (f) Typical results for w 0 as a function of the magnetic field at 0 MPa. The red and blue curves correspond to the longitudinal and transverse AC magnetic susceptibilities, which were measured in configurations (d,e), respectively. These data are shown separately by an offset. Alternating magnetic fields (H ac ) were applied by a coil parallel to the r axis. The critical fields corresponding to the phase transitions from the helical phase to the conical phase and from the SkX phase to the induced ferromagnetic phase are represented by H c1 and H c2 , respectively. The SkX phase can be clearly assigned, as represented by the shaded regions between H A1 and H A2 . respectively. In both configurations, a DC magnetic field was applied in the 001 direction, and the uniaxial stress and AC magnetic field were set parallel to each other. Figure 1f presents a typical evolution of the AC magnetic susceptibility of a MnSi single crystal under a magnetic field sweep at 0 MPa for these two configurations. The red and blue curves correspond to the longitudinal and transverse AC magnetic susceptibilities, respectively. As the magnetic field in 001 direction increases, a transition from the multi-domain helical phase to the single-domain conical phase occurs at H c1 , followed by another transition from the conical phase to the induced ferromagnetic phase at H c2 . In the intermediate range between H A1 and H A2 , as represented by the grey regions, the longitudinal susceptibility shows a clear suppression, whereas the transverse susceptibility exhibits a clear enhancement. This behaviour is ascribable to the SkX phase. In the case of proper helices, the susceptibility parallel to the propagation vector Q is expected to be larger than that perpendicular to the Q-vector. The Q direction of the conical phase is parallel to the H direction, whereas that of the SkX phase is orthogonal (Fig. 1b,c). This difference in Q-vector direction qualitatively explains the opposite trends exhibited by the longitudinal and transverse susceptibilities in the SkX and conical phases. The slight discrepancies between the critical fields for the two configurations are attributed to the demagnetizing effect.
Mechanical creation and annihilation of SkX. We measured the uniaxial stress dependence of the magnetic susceptibilities at low temperatures. Figure 2a,b present the AC susceptibilities near 0.2 T with increasing mechanical stresses, which unambiguously demonstrated that the SkX phase is created by a transverse stress and annihilated by a longitudinal stress. At 27.0 K in the absence of stress, there is no SkX phase. On the application of a stress in the transverse configuration (r > H), as shown in the inset of Fig. 2(a), w 0 begins to increase at B0.2 T, indicating the emergence of the SkX phase. At 27.6 K, however, the SkX phase exists at 0 MPa, as indicated by a reduction in w 0 (Fig. 2b) in the longitudinal configuration (r || H). This reduction is continuously suppressed with increasing uniaxial stress up to 200 MPa, indicating that the SkX phase becomes unstable on the application of longitudinal uniaxial stress and that the conical phase survives between H A1 and H A2 . These results demonstrate that even moderate longitudinal or transverse stresses of up to 200 MPa are sufficient to control the competition between the SkX and conical phases. This anisotropy is consistent with a previous report 41 . The slight discrepancy between Fig. 3a,d originates from the demagnetization effect, which depends on the sample shape used in the uniaxial stress experiments (see Methods). The relative stability of the SkX phase can be varied by applying a transverse or longitudinal stress: the SkX phase extends to lower temperatures on the application of a transverse stress (Fig. 3a-c), whereas its temperature range narrows on the application of a longitudinal stress ( Fig. 3d-f,g-i). Although the collapse of the SkX phase is incomplete even at 200 MPa with the application of stress along the 001 axis, the SkX phase disappears throughout the entire H-T region when a longitudinal stress is applied along a magnetically hard axis, that is, the 111 axis (Fig. 3i). A large response also arises in the helical phase in both the transverse and longitudinal configurations, as seen from the colour changes between 0 and 0.1 T. This indicates a stress-induced reorientation of the Q-vector of the helix from the weakly locked o1114 axes to the stress axis. Regardless of whether the configuration is transverse or longitudinal, a uniaxial stress causes a slight decrease in T c at 0 T. This can be attributed to the effect of volume contraction introduced by the compressive uniaxial stress, which is similar to a hydrostatic pressure effect, as confirmed by Ritz et al. 42 .
Stress-driven topological phase transition. The central results of the present paper are presented in Fig. 4, revealing a stresstriggered topological phase transition. Here, we continuously applied and released stresses between 0 and 100 MPa at fixed temperatures and magnetic fields. Shown in Fig. 4a is a colour map of w 0 in the s-H plane at 27.0 K for the transverse configuration. This figure clearly reveals the emergence of the SkX phase in the background of the conical phase at several tens of MPa. The stress-mediated topological phase transition is displayed as susceptibility stress curves in Fig. 4c. The large nonlinear responses correspond to a conical-to-SkX phase transition at a critical stress s c . Incidentally, another large change in w 0 at 0.00 T corresponds to the reorientation of the Q-vector in the helical phase towards the r axis. Fig. 4b represents the s-H phase diagram at 27.6 K for the longitudinal stress condition. The SkX To directly confirm the stress-triggered phase transition, we performed a small-angle neutron scattering (SANS) measurement under tunable uniaxial stress (Fig. 5a). In the same manner as for the acquisition of Fig. 4d, a longitudinal stress was gradually applied at 27.7 K under a magnetic field of 0.19 T. Because the magnetic field was applied along the 001 axis, six-fold Bragg peaks corresponding to the SkX phase appear on the q x -q y plane. We observed two of these six magnetic reflections when the 110 direction was perpendicular to the incident beam (q x ) and another two on the rotation of the sample by þ p/3 about the 001 axis. These observations are consistent with a previous study reporting a weak Q anisotropy along the 110 direction 1 . The Bragg reflections in the conical phase, however, appeared along the 001 axis (|| H). Figure 5b shows a change in the SANS pattern with an increasing longitudinal stress, in which major intensities shift from the q y direction to the q z direction. Figure 5c displays the stress-dependent integrated intensities of these Bragg peaks for the SkX and conical phases. The SANS results provide unambiguous evidence of the stress-driven SkX-to-conical phase transition, consistent with the w 0 measurements.
Phenomenological interpretation. Our study quantitatively revealed a marked effect of mechanical stress on the SkX phase in intrinsic bulk crystal. The critical stress required to create and annihilate the SkX phase near the phase boundary is several tens of MPa. The corresponding strain is estimated to be of the order of 10 À 4 , assuming C 11 B300 GPa (refs 43,44). Previous theoretical studies have suggested that the application of uniaxial strain is an effective means controlling the stability of the SkX phase through modulation of the magnetic anisotropy 45 . Karhu et al. 46 experimentally investigated such a scenario by using epitaxial MnSi films, in which several anisotropic strains were introduced by varying the thickness of the films. Recently, Wilson et al. 47 also revealed that an in-plane tensile epitaxial strain on a MnSi film destabilizes the SkX phase and causes it to transform into the conical phase, consistent with our results. The epitaxial strain in this case is one or two orders magnitude larger than the strain whose effect we have investigated in this study.
Here, we discuss the strain-induced magnetic anisotropy based on the Ginzburg-Landau phenomenology. The magnetoelastic contribution (F me ) to the free energy can be represented in terms of the spin density, S ¼ {S x , S y , S z }, the uniform magnetization m, and the strain tensor e ab (a, b ¼ x, y, z) as follows 48 : e B þ L 1 e xx S y 2 þ e yy S z j j 2 þ e zz S x j j 2 þ L 2 e xx S z j j 2 þ e yy S x j j 2 þ e zz S y 2 The coefficients K and L i (i ¼ 1, 2, 3) originate from exchange coupling and spin-orbit coupling, respectively. The latter three terms give the magnetic anisotropy, whereas the first term gives the isotropic contribution, which is important when a hydrostatic pressure (e B ¼ e xx þ e yy þ e zz ) is applied 42 . Here, we consider a uniaxial stress applied along the z axis (that is, s zz a0, s xx ¼ s yy ¼ 0). Then the energy contributing to the magnetic anisotropy E A can be written as Here, we set L 1 ¼ L 2 ¼ L for simplicity, and neglect the isotropic term ( ; v represents Poisson's ratio (B0.28 for MnSi 43 ). Equation (2) is equivalent to that given in ref. 45, if L(1 þ v)e zz is considered as an effective magnetic anisotropy constant K. Because the coefficient L is positive for MnSi 44,48 , a compressive strain (e zz o0) induces an easy-plane anisotropy perpendicular to the stress axis. As a result, the Q-vector of the proper helix preferentially aligns along the compressive stress direction. In other words, the conical phase (Q || H) should be stabilized by a longitudinal stress, whereas the SkX phase (Q > H) should be stabilized by a transverse stress (Fig. 1b,c). This explains the creation and annihilation of the SkX by transverse and longitudinal stresses, respectively. Recently, Shibata et al. 49 reported that strain also modulates the Dzyaloshinskii-Moriya interaction (DMI), inducing largeshape deformations of skyrmions in FeGe. Theoretical calculations 50 have revealed that a uniaxial strain caused the DMI to be large along the compressive axis. This effectively induces an easy-plane anisotropy, similar to the effect of stressinduced magnetic anisotropy. Therefore, modulation of the DMI may also provide a qualitative explanation of the results of this study. However, a preliminary SANS measurement of MnSi suggests that the magnetic modulation period remains unchanged under the application of a uniaxial stress 42 . Furthermore, our SANS measurement presented in Fig. 5b reveals that the magnetic modulation period in the SkX phase remains almost unchanged on the application of a uniaxial stress; the change is o0.5%. These findings suggest that the effect of strain on the DMI is less relevant in the case of MnSi. Therefore, we interpret the effect of a uniaxial stress on the SkX phase stability in terms of a stressinduced magnetic anisotropy.

Discussion
Advances in recent nanofabrication technologies and nanoelectromechanical devices have made it possible to utilize simple mechanical forces, instead of the conventional electrical or magnetic biases, to write information bits on a minute scale 51,52 . Nanoscopic mechanical imprinting has now been developed using a cantilever-based configuration 51 , which simultaneously satisfies the requirements for high densities of up to 1 Tbit in À 2 and high speeds of up to Gbit s À 1 in a scaled-up device called Millipede memory 52 . The concept of mechanically recordable memory, therefore, can provide an alternative architecture to various types of magnetically or electrically operated devices. Here, we propose that magnetic skyrmions can be a suitable medium for mechanical imprinting on the nanometric scale, providing the basis for a technology that can be operated at an extremely low energy cost.
The small critical stress of several tens of MPa that is needed to create/annihilate the SkX phase opens the possibility of the selective erasing of single skyrmions by simply pushing the tip of a cantilever in scanning probe microscopy (SPM). This scenario corresponds to the longitudinal configuration. By using a conventional sharp tip with a radius of 10 nm, one can introduce highly localized deformations comparable to the size of a single skyrmion in MnSi (20 nm On another front, the mechanical energy needed to erase single skyrmions is estimated to be surprisingly low. This stems from the scalability of the phenomenon; the mechanical energy per bit greatly decreases with the reduction of the bit size down to the size of skyrmions. The energy cost for erasing a skyrmion can be expressed as E cost ¼ s 2 zz pr 2 d= 2C 11 ð Þ, where s zz is the stress, C 11 is the elastic constant, pr 2 is the deformation area and d is the deformation depth. Here, r and d are assumed to be the tip radius and the thickness of the film, respectively. Supposing that s zz ¼ 100 MPa, C 11 B300 GPa (refs 43,44), r ¼ 10 nm and d ¼ 10 nm, one can obtain E cost ¼ 5 Â 10 À 20 J per single skyrmion. Although this is a rough estimate, the mechanical energy required is far smaller than the energy cost of 9 Â 10 À 14 J incurred for writing to state-of-the-art spin-transfer-torque magnetic random access memory 53 or that of 2.25 Â 10 À 17 J for current-induced single skyrmion creation as calculated in ref. 32.
In conclusion, we demonstrated mechanical control of the SkX phase in a bulk MnSi single crystal, which can be achieved with a small critical stress and a low energy cost. This offers a new methodology for manipulating single skyrmions based on the conventional SPM technique. As discussed from a phenomenological perspective, the proposed mechanical approach is applicable in a broad range of skyrmionic systems with magnetoelastic coupling. By combining simple mechanical imprinting with other approaches such as a racetrack-type architecture, one may achieve novel memory or storage functions using skyrmions.

Methods
Sample preparation. Single crystals of MnSi were grown using the Czochralski method. We prepared two types of rectangular samples with dimensions of 4.4 mm Â 2.3 mm Â 2.4 mm and 3.2 mm Â 1.0 mm Â 1.0 mm for w 0 measurements under longitudinal and transverse uniaxial stresses, respectively. Compressive stress was applied along the longest axes, which corresponded to 001 and 110, respectively. The external DC magnetic field was applied along the 001 direction, which was the longer (shorter) direction of the sample in the longitudinal (transverse) configuration. Therefore, the demagnetizing field was larger in the transverse configuration than in the longitudinal one. We also prepared a sample with dimensions of 3.0 mm Â 2.5 mm Â 2.0 mm for a measurement as shown in Fig. 3gi, in which H and r were applied along 111 direction. This direction corresponded to the longest axis of the sample. The surfaces and edges were carefully polished and prepared to be parallel to prohibit breaking from the edge and to introduce homogeneous compressive stress.
v 0 measurements under continuously variable uniaxial stress. We inserted a special uniaxial stress apparatus 39 into a superconducting magnet of a Quantum Design physical property measurement system. The uniaxial stress was externally controlled by a combination of a micrometre and a SiCr spring coil attached to the top of the probe, and the applied force was measured by a load cell. The mechanical stress was arbitrarily controlled simply by rotating the micrometre. The spring coil ensured that the stress scan was moderate and stable. This external tunability is a notable feature that cannot be achieved using a conventional clamp-cell-type pressure apparatus. A longitudinal uniaxial stress was introduced by means of two ZrO 2 pistons sandwiching the sample. A transverse stress was introduced by means of several ZrO 2 pistons that converted longitudinal forces into lateral ones. Corresponding to these two configurations, the longitudinal and transverse AC magnetic susceptibilities were measured through excitation and pick-up coils covering the sample. The frequencies were 361.3 Hz for the longitudinal susceptibility and 420.9 Hz for the transverse susceptibility. We checked that the magnetic field dependence of w 0 at 0 MPa was reproducible before and after the application of 200 MPa.

SANS experiment.
A stress-induced SkX-to-conical phase transition was directly confirmed using a time-of-flight-type SANS and wide-angle neutron scattering instrument, TAIKAN (S. Takata et al., unpublished observations), constructed at the BL15 of the Materials and Life Science Experimental Facility of Japan Proton Accelerator Research Complex (J-PARC) in Japan. The sample used in the SANS experiment had a rectangular shape with dimensions of 3.6 mm Â 2.3 mm Â 2.4 mm. We developed a uniaxial stress insert for the SANS experiment, which was essentially identical to that used in the measurements of the AC susceptibility under an applied longitudinal uniaxial stress, and loaded it into a vertical-field superconducting magnet. The uniaxial stress was applied to the smallest surfaces of the sample. The directions of the incident neutron beam, the crystallographic axes, the applied uniaxial stress and the magnetic field are summarized in Fig. 5a. The SANS patterns shown in Fig. 5b were obtained by subtracting the background that was measured after removing the sample from the uniaxial stress cell. The relatively weak integrated intensity of the SkX phase is attributed to the existence of SkX domains, in which one of the triple Q-vectors is locked in q x or q y direction. The faint peaks along the q z direction at 0 MPa are due to the residual conical phase.