Introduction

Chiral magnets have attracted interest over the last few years due to experimental observation of a new chiral modulated magnetic state, commonly referred to as skyrmion lattice, first in MnSi1 and then in Fe0.5Co0.5Si2 and FeGe3. Skyrmion lattice in chiral magnets can be described as chiral spin structures with a whirling configuration, which can be described mathematically by a combination of three plane-wave functions in space (hence referred to as a “triple-Q” structure). These magnetic skyrmions are stabilized by the antisymmetric Dzyaloshinskii-Moriya (DM) interactions4,5 and are well known for their emergent electromagnetic properties6,7 and topological Hall effect8,9.

Due to the magnetoelastic coupling in chiral magnets, it is known that application of mechanical loads or misfit strains can stabilize the skyrmion lattice2,10,11,12. Moreover, it is found in FeGe that elastic deformation and the deformation of the skyrmion lattice are strongly coupled, which leads to large emergent deformation of skyrmion lattice when elastic stress is applied13,14. To understand such an exotic phenomenon, we first have to discuss the internal elastic field induced by presence of magnetic skyrmions. This induced elastic field should be composed of a homogeneous part, which has been solved in our previous work15 and a periodic part, which is to be discussed in the present work. In the early studies of MnSi, it has already been confirmed that periodic distribution of elastic fields coexists with periodic magnetization in the spin-density-wave phases (e.g., helical and conical phase)16. Besides, it is shown theoretically that the magnitude of periodic strain waves should be considered as independent variables when formulating the free energy of the system, since it results in new terms in the free energy functional17. Existence of the periodic elastic field represents realization of the same mathematical structure as magnetic skyrmions in a different physical field due to multiphysics coupling. Moreover, regarding the nonlinear nature of the magnetoelastic coupling in chiral magnets, additional periodic structures with changed wave vectors may occur in the solution of elastic field. Induced by the magnetic skyrmions, these periodic elastic structures should always accompany the magnetic structures. Hence the solution of the periodic elastic fields may characterize the elastic property of magnetic skyrmions in some way.

The elasticity problem induced by presence of any kind of phase transition strains is called an eigenstrain problem16 in micromechanics. In this case, the eigenstrains refer to the strains that occur due to a change of magnetization during a phase transition from the paramagnetic phase to the skyrmion phase. Since in the skyrmion phase, the magnetization is a periodic vector function in space, we encounter an eigenstrain problem with periodic eigenstrains. The analytical method for solving the elastic fields induced by periodic distribution of eigenstrains was developed long ago18,19, mainly due to its mathematical significance for constructing the Fourier transform-based analytical solution method. Yet, the eigenstrain problem with periodic eigenstrains composed of plane waves with several discrete wave vectors was not treated. Before the discovery of magnetic skyrmions in chiral magnets, it’s not clear such a solution is of any physical significance.

In this paper, we formulate the eigenstrain problem from an extended micromagnetic model for chiral magnets developed upon group theoretical analysis and the Ginzburg-Landau theory15 and then obtain the analytical solution of the problem for two different chiral magnetic phases: the conical phase and the skyrmion phase. We find that appearance of the triple-Q skyrmion lattices is accompanied by formation of three types of triple-Q structures in the displacement field, described by uS1(r) (with the same wave number q as the magnetic skyrmions), uS2(r) (wave number 2q) and uS3(r) (wave number ), as well as formation of three types of triple-Q structures in the stress field, described by (wave number q), (wave number 2q) and (wave number ). By using the values of equilibrium magnetization obtained through free energy minimization at given temperature and magnetic field15, we plot the field configuration of uS1(r), uS2(r), uS3(r) and , , and discuss their variation with temperature and magnetic field. We find that as the applied magnetic field increases, the field configuration of uS1(r) and gradually undergoes a “configurational reversal” process, where the field configuration of the initial state and the final state remains similar, but the direction of every vector in the field is reversed. The phenomenon provides a possibility of developing novel information storage devises and microwave applications. The analytical solution of the periodic elastic fields also lay a foundation for discussion of the emergent elastic behavior of magnetic skyrmions13, which is to be discussed in a subsequent work of ours.

Formulation of the eigenstrain problem in chiral magnets

For chiral magnetics with cubic symmetry, the Helmholtz free energy density contains two terms related to the elastic strains15, which are the elastic energy density

and the magnetoelastic energy density

where C11, C12 and C44 are the elastic stiffness at constant magnetization, εij (i, j = 1, 2, 3) are the elastic strains, are the engineering shear strains, Ms denotes the saturation magnetization, and

In eq. (2), higher order terms of magnetoelastic coupling (e.g., in ref. 15) are omitted for convenience of deduction. According to our previous work15, such a simplification leads to an estimated error in the order of 0.01% when calculating the elastic strains and in .

For a bulk material free from any mechanical loads, the stresses are obtained from when and when . After manipulation we get

where and , the eigenstrains, are related to the magnetization:

In eq. (5), the parameters with a superscript “*” are defined as , , .

Substituting eq. (4) into the equilibrium equation and using the geometrical equations under small-deformation assumption, we have

where

Here resemble components of the body force caused by presence of eigenstrains. Assume M(r) and u(r) to be periodic functions in space, which can be expressed as

where m−q =  and . Here and are complex conjugates of mq and Uq. Through eq. (8), the equilibrium state of the system is determined by minimizing F with respect to mq, Uq and q. When the system is free from external mechanical loads, the solution of Uq can be obtained by solving the eigenstrain problem for chiral magnets as

where

and is solved from . Eq. (9) determines the elastic fields in the material when the magnetic state of the system is determined.

Before moving on to the solution of eq. (6) for different chiral magnetic phases, we give a brief discussion on the situation in the ferromagnetic phase. When the material is stabilized in a ferromagnetic state, the magnetization M is a constant vector inside the material. In this case, the eigenstrains are constants according to their definition given in eq. (5) and we have from eq. (7). We thus obtain a solution of zero stresses for free boundary condition. The physical interpretation of this solution is clear: when the material is free from external loads, it is free to deform and so the total strains equal to the eigenstrains, while the stresses vanish.

Solution for the conical phase

In the conical phase, the magnetization can be written without loss of generality as

where it is assumed that the magnetic field is applied along the z-axis. Of course, the following results change with the direction of the magnetic field. But the method and the form of the solution are exactly the same. Substitution of eq. (11) into eq. (5) yields

where , and

and . The solution of the displacement field can be sought in the following form

where uc dentoes the displacement field which corresponds to constant eigenstrains. By direct integration of eq. (13), we have

where rigid body movements are not considered. Substituting eqs (14, 15, 16) into eqs (6, 7), after manipulation we obtain

where

Eq. (18) can thus be expanded as

The solution for the displacement field can be obtained by combining eqs (16, 17, 20). The total strains and stresses are solved as

and

In eq. (12) the periodic eigenstrains contain a part with a wave number 2q. This wave-number-doubling phenomenon derives from the nonlinear nature of the magnetostriction effect and has been discussed before16. By solving the elastic fields of this eigenstrain problem, it is shown that the magnitude of the periodic part of stresses with wave number q vanishes. To explains the physical origin of this result, one needs to examine the compatibility condition for the eigenstrains . For the three composition of in eq. (12), and satisfy the compatibility condition, while a part of does not satisfy the compatibility condition. The compatible part of eigenstrains directly generates elastic strains, while the incompatible part of eigenstrains is constrained by the elastic body through an internal stress field given in eq. (22).

Using the related parameters for MnSi:

7. , , , 15. , , 15. 15.

it is found that for , .

Solution for the skyrmion phase

In the skyrmion phase, the magnetization vector can be written as

where it is assumed that the external magnetic field is applied along the z-axis and , , . For the magnetization defined in eq. (23), the solution of the displacement field can be sought in the following form

One should notice that and thus terms with , and are merged with terms with , and , respectively. By substituting eqs (23, 24) into eqs (5, 6, 7), the solution for the displacement field can be obtained in the same way as above. We have after manipulation

where

and .

The solution of periodic displacement field in the skyrmion phase is composed of three types of triple-Q structures, defined by

With eq. (38), eq. (24) can be recasted as .

Eqs (25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37) show that the magnitude of the periodic displacement field depends on the magnetization as well as the magnetoelastic effects. To be more specific, in terms of the magnetization, depends on both m3 and mq defined in eq. (23), while and depend merely on ; in terms of the magnetoelastic effects, and depend on K, which represents the dominant magnetoelastic term, while is independent of K.

It is easy to verify that in the skyrmion phase, the eigenstrains do not satisfy the compatibility condition. Similar to explanation given in section 3, the part of eigenstrains that doesn’t satisfy the compatibility will generate periodic elastic stresses. The components of the three triple-Q stress tensors can be obtained as

where

The analytical expressions of can be obtained by substituting eqs (25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38) into eqs (39, 40), which are too lengthy to be expanded here. This result indicates that in bulk materials free from any mechanical loads, appearance of magnet skyrmions is always accompanied by nontrivial periodic stress fields.

Field configuration of the triple-Q elastic structures for MnSi and discussion

For MnSi, the configuration of the three triple-Q displacement structures are plotted in Fig. 1(a–c) using the magnetization obtained through free-energy minimization at temperature 4 K and zero magnetic field using the extended micromagnetic model15. From Fig. 1(a–c), we learn that , and have field configurations that are different from each other and also different from the magnetic skyrmions. However, they all form hexagonal networks of localized fields, while all the three triple-Q displacement structures have only in-plane components. This can be explained from eqs (25, 26, 27, 28, 29, 30, 31, 32, 33, 34), where all the displacement components in z-axis rely only on LO4, LO5 and LO6, which are set to be zero for . In fact, since LO4, LO5 and LO6 represent high order magnetoelastic effects15, the smallness of z-component of the skyrmion induced displacement field is guaranteed for any B20 compound.

Figure 1
figure 1

Configuration of the three triple-Q displacement structures at temperature 4 K and magnetic field 0 T: (a) uS1, (b) uS2, (c) uS3 and (d) uS1 + uS2 + uS3. The arrows represent the direction of the displacement vectors which vary in space and the length of the arrows reflects the magnitude of the displacement vectors.

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From Fig. 1(d), we see the total periodic displacement field denoted by appears to have the same periodicity with , which can be explained as follow. The lattice vectors of the periodic displacement structure , denoted by a1 and a2, satisfy

The periodicity of can generally be described by , where n1 and n2 are arbitrary integers. From eq. (38), we can easily prove that and . Hence and (or M(r)) shares the same lattice vectors and period. For bulk chiral magnets, this period, determined by the strength of the DM interaction and the stiffness of the exchange energy density, is independent of the period of the underlying atomic lattice. This explains directly why magnetic skyrmion materials are incommensurate systems and also provides quantitative description of the deformed lattice configuration of the skyrmion materials.

The maximum achievable displacement of uS1, uS2, uS3 and uS1 + uS2 + uS3 are listed in Table 1, from which we can see . The magnitude of uS1 and uS3 is significantly larger than that of uS2, since the solution of uS1 and uS3 is related to the exchange-interaction-induced magnetoelastic coupling term with coefficient K, which is the dominant term among all magnetoelastic coupling effects (for MnSi, the magnitude of K is at least an order of magnitude larger than the coefficient of other magnetoelastic terms15).

Table 1 Magnitude of maximum achievable displacement for the three triple-Q displacement structures of MnSi and their sum calculated at temperature 4 K and magnetic field 0 T.
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When solving the stress field of , it is found that the stress components defined on (001) plane is generally much larger than other stress components. This can be understood since the periodic in-plane displacements release most of the eigenstrains defined on (100) plane and (010) plane while the other incompatible eigenstrains cause stresses mainly on the (001) plane. The field configuration of is plotted in Fig. 2(a–c), where we can see that is the most significant stress component.

Figure 2
figure 2

Configuration of the components of the three triple-Q stress tensors defined on the (001) plane (or the xy plane) at temperature 4 K and magnetic field 0 T: (a) , (b) and (c) . The unit used in all three figures is MPa.

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Variation of the field configuration with temperature and magnetic field

The periodic elastic fields solved as functions of the equilibrium magnetization, should vary with temperature and magnetic field. When the temperature is increased from 0 K to the critical temperature, it is found that the field configuration of all the triple-Q structures is merely changed, but the magnitude of vectors gradually decreases to zero. This is understood since the components of the magnetization gradually decrease to zero as the temperature approaches the critical temperature.

On the other hand, we find that the field configuration of uS1 and is very sensitive to variation of the magnitude of applied magnetic field while the field configuration of other triple-Q field structures is not. In Figs 3(a–d) and 4(a–d), we plot the variation of the field configuration of uS1 and when the magnetic field gradually increases from 0.1 T to 0.4 T. We find an interesting phenomenon that the field configuration of both uS1 and undergoes a “configurational reversal” when the external magnetic field increases: comparing Fig. 3(a,d) (Fig. 4(a,d)), it is observed that field configuration of uS1 () plotted at applied field 0.1 T and 0.4 T shares similar pattern but with opposite direction of all the field vectors. One should notice that such a reversal of elastic field configuration does not affect the magnetic state of the material (i.e., no magnetic phase transition occurs and the magnetization is not reversed). Through thermodynamic analysis within the mean-field theory15, we already known that the variation of m3 and mq with external magnetic field in the skyrmion phase is insignificant. uS2, uS3, and depend merely on mq and so their field configuration merely changes with external magnetic field. To understand why uS1 and are sensitive to external magnetic field, we first consider the K–dependent terms in uS1 and . It is found after manipulation that the uS1 and depend on K linearly, with a coefficient proportional to , which vanishes at . For MnSi, as the magnetic field increases from 0 T to 0.4 T, the equilibrium value of in the skyrmion phase decreases from 2.59 to 2.15. As a result, the coefficient of the linear term of K changes sign as the magnetic field increases, which is responsible for the configurational reversal of uS1 and .

Figure 3
figure 3

Configuration of uS1 at temperature 4 K and magnetic field (a) 0.1 T, (b) 0.2 T, (c) 0.3 T and (d) 0.4 T.

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Figure 4
figure 4

Configuration of defined on the (001) plane (or the xy plane) at temperature 4 K and magnetic field (a) 0.1 T, (b) 0.2 T, (c) 0.3 T and (d) 0.4 T.

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In Fig. 5(a–d), we plot the variation of the field configuration of uS1 + uS2 + uS3 when the magnetic field gradually increases from 0.1 T to 0.4 T. We find that the configurational reversal of uS1 is smeared by uS2 + uS3, which is almost unchanged for increasing magnetic field. Yet, by comparing Fig. 5(a,d) we see that the location of a “significant outburst” changes from the center in Fig. 5(a) to six adjacent points of the center in Fig. 5(d).

Figure 5
figure 5

Configuration of uS1 + uS2 + uS3 at temperature 4 K and magnetic field (a) 0.1 T, (b) 0.2 T, (c) 0.3 T and (d) 0.4 T.

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In eq. (23), the magnetization function introduced is specified for the case where the external magnetic field is applied along the z-axis. Concerning the magnetoelastic effects described in eq. (2), all terms except KM2εii are anisotropic, which means that the solution of displacement field, as well as the field configurations, all vary with the direction of applied field. In other words, the solution given in eqs (25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39) are specified for the case where the external field is applied along the z-axis.

Condition for occurrence of a configurational reversal of uS1 and for B20 compounds

The analytical solution of uS1 and derived in eqs (38, 39) applies to any B20 compounds. As a result, the coefficient of the linear term of K in uS1 and always vanishes at . The occurrence of a configurational reversal of uS1 and is garanteed if the equilibrium value of in the skyrmion phase at 0 T is larger than 2.31 and the equilibrium value of in the skyrmion phase at some critical magnetic field is smaller than 2.31. After analysing the free-energy minimization process within the extended micromagnetic model15, we find that the equilibrium value of is determined by the Landau expansion terms and the Dzyaloshinskii-Moriya (DM) coupling term , where T denotes the temperature. Neglecting the DM coupling term and substituting eq. (23) into wL, we find that the equlibrium value of in the skyrmion phase equals to 2.23, regardless of the value of α, β and T. Adding the DM coupling term back to the functional, it can be calculated that once , we always have in the skyrmion phase, where q denotes the wave number. Hence the stronger the DM coupling, the larger . For MnSi, at 0 K. On the other hand, application of external magnetic field along the z-axis will inevitably increase m3 and decrease mq, which leads to a decrease of . Concerning the above analysis, we think the occurrence of a configurational reversal of uS1 and with increasing magnetic field is more of a general phenomenon for any B20 compound than a specific issue for MnSi. Even if a configurational reversal does not occur, the variation of uS1 and with external magnetic field should be significant.

Technological interest of elastic triple-Q structures in chiral magnets

Magnetic skyrmions are regarded as one type of possible information carrier, since their motion in materials can be manipulated by small current density6,7,20 and their existence can be manipulated by various approaches21,22,23. Stem from the intrinsic magnetoelastic coupling in chiral magnets, the elastic triple-Q structures always move together with the magnetic skyrmions. This provides the possibility of identifying the existence of magnetic skyrmions by checking the localized elastic state. Moreover, the field configuration of the elastic triple-Q structures having the same periodicity with the skyrmions is sensitive to external magnetic field, which means that we can have opposite elastic state for almost unchanged magnetic state. From the solution obtained in eqs (25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37), the magnitude of the periodic elastic field can be enhanced by increasing the magnetoelastic coefficients and also the size (the wavelength) of an individual skyrmion, which may be achieved by choosing appropriate materials.

Besides data-storage devices, magnetic skyrmions are also promising for developing new microwave applications, since they excite gigahertz collective spin vibration modes when exposed to magnetic microwave14,24,25. Existence of the elastic triple-Q structures provides a variety of options for development of related technology, such as novel magneto-acoustic actuators and sensors. For this purpose, it is of interest to derive the corresponding collective elastic field vibration modes from the known collective spin vibration modes and study their coupling in dynamical conditions.

Conclusion

We obtain the analytical solution of the periodic elastic fields for the eigenstrain problem in chiral magnets. In the skyrmion phase, the nonlinear magnetoelastic coupling leads to formation of three types of triple-Q elastic field structures. For MnSi, the triple-Q displacement structure uS1 and the triple-Q stress structure are found to undergo a configurational reversal when the magnetic field increases from 0 T to 0.4 T. Through thermodynamic analysis, we find that such a configurational reversal is likely to occur for any B20 compound. It will be interesting to experimentally detect the periodic elastic structure and further discuss the intrinsic vibration modes of these elastic triple-Q structures.

Additional Information

How to cite this article: Hu, Y. and Wang, B. Reversible “triple-Q” elastic field structures in a chiral magnet. Sci. Rep. 6, 30200; doi: 10.1038/srep30200 (2016).