# Skyrmion robustness in non-centrosymmetric magnets with axial symmetry: The role of anisotropy and tilted magnetic fields

###### Abstract

We investigate the stability of Néel skyrmions against tilted magnetic fields, in polar magnets with uniaxial anisotropy ranging from easy-plane to easy-axis type. We construct the corresponding phase diagrams and investigate the internal structure of skewed skyrmions with displaced cores. We find that moderate easy-plane anisotropy increases the stability range of Néel skyrmions for fields along the symmetry axis, while moderate easy-axis anisotropy enhances their robustness against tilted magnetic fields. We stress that the direction, along which the skyrmion cores are shifted, depends on the symmetry of the underlying crystal lattice. The cores of Néel skyrmions, realized in polar magnets with C symmetry, are displaced either along or opposite to the off-axis (in-plane) component of the magnetic field depending on the rotation sense of the magnetization, dictated by the sign of the Dzyaloshinskii constant. The core shift of antiskyrmions, present in chiral magnets with D symmetry, depends on the in-plane orientation of the magnetic field and can be parallel, anti-parallel, or perpendicular to it. We argue that the role of anisotropy in magnets with axially symmetric crystal structure is different from that in cubic helimagnets. Our results can be applied to address recent experiments on polar magnets with C symmetry, GaVS and GaVSe.

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## I The Introduction

In magnetic compounds lacking inversion symmetry, the underlying crystal structure induces a specific asymmetric exchange coupling, the so-called Dzyaloshinskii-Moriya interaction (DMI), which can stabilize long-period spatial modulations of the magnetization with a fixed rotation sense Dz64 ; Bak . Within a continuum approximation for magnetic properties, the DMI is expressed by inhomogeneous invariants involving first derivatives of the magnetization M with respect to the spatial coordinates:

(1) |

Depending on the crystal symmetry, certain combinations of these Lifshitz invariants (LI) can contribute to the magnetic energy of the material Bogdanov89 .

In the last few years, a renewed interest in non-centrosymmetric magnets has been inspired by the discovery of two-dimensional localized modulations, commonly called magnetic skyrmions Bogdanov94 ; Yu10 ; Roessler11 . Recently, skyrmion lattice states (SkL) Yu10 ; Yu11 and isolated skyrmions (IS) Romming13 ; Romming15 were discovered in bulk crystals of non-centrosymmetric magnets near the magnetic ordering temperatures Muehlbauer09 ; Wilhelm11 ; Kezsmarki15 and in nanostructures with confined geometries over larger temperature regionsYu10 ; Yu11 ; Du15 ; Liang15 . The small size, topological protection and easy manipulation of skyrmions by electric fields and currents Schulz12 ; Jonietz10 ; Hsu17 generated enormous interest in their applications in information storage and processing Sampaio13 ; Tomasello14 .

The DMI provides not only a unique stabilization mechanism Bogdanov94 , protecting magnetic skyrmions from radial instability, but also governs their internal structure. In Ref. Bogdanov89, by Bogdanov and Yablonskii, three distinct types of skyrmions and two types of antiskyrmions (having opposite topological charge as compared with skyrmions) were predicted to occur in non-centrosymmetric magnets with different crystallographic symmetries.

The first observed type of skyrmions is called Bloch skyrmions, whose internal structure is schematically shown in Fig. 1 (a). It is the most ubiquitous skyrmionic archetype observed in materials with B20 structure like MnSi Muehlbauer09 , FeGe Wilhelm11 or CuOSeO Seki12 . The magnetization in Bloch skyrmions is perpendicular to the radial direction. Since the magnetic energy term corresponding to the DMI is reduced to the isotropic form , the axes of Bloch skyrmions in cubic helimagnets always co-align with the applied magnetic field, irrespective of the direction of the field.

Antiskyrmions were recently observed in Mn-Pt-Sn Heusler materials with acentric D crystal symmetry Nayak . They were shown to exist over a wide temperature interval both as IS and in ordered SkL. Although the antiskyrmions break the cylindrical symmetry and carry a quadrupolar moment of the magnetostatic charges, they are invariant under all symmetry operations of D, including the symmetry (-fold rotation followed by inversion).

Polar magnets with C symmetry, such as GaVS Kezsmarki15 and GaVSe Bordacs17 ; Fujima17 , host Néel skyrmions in which the magnetization rotates radially from the skyrmion center as shown in Fig. 1 (b). The helicity angle (”out-ward” rotation of the magnetization) or (”in-ward” rotation of the magnetization) depends on the sign of the Dzyaloshinskii constant in model (2) , described later, and endows the Néel skyrmions with a magnetic monopole moment on the contrary to Bloch skyrmions with a magnetic toroidal moment Spaldin08 . Moreover, Néel skyrmions are dressed with an electric polarization Ruff15 ; Wang15 and induce internal magnetic charges due to the non-zero divergence of the rotating magnetization vector Bogdanov94 .

In the C and D symmetry classes investigated here, no Lifshitz invariants along the high-symmetry axis, the direction, are present Bogdanov89 . Therefore, only modulated magnetic structures with wave vectors perpendicular to the axis are favoured by the DMI. As it will be shown in this work, in magnetic fields tilted with respect to the high-symmetry axis both the Néel skyrmions and antiskyrmions Nayak shift their cores with respect to the in-plane component of the field instead of coaligning their axes with the field as it happens for Bloch skyrmions in cubic helimagnets, the so-called locking of the skyrmion axes.

LI (1) define a set of competing modulated phases in the phase diagrams specific to the different crystallographic classes. For the C and D symmetry classes, usually only spirals, skyrmions (or antiskyrmions), and collinear spin-structures are identified Nayak ; Kezsmarki15 . There is no competing conical state for fields applied along the high-symmetry axis. Although a transverse conical state arises for fields spanning large angles with the high-symmetry axis Bordacs17 , but this is fundamentally different from the longitudinal conical state in cubic helimagnets, where the -vector co-aligns with the field. The lack of the longitudinal conical state may lead to wide stability regions of skyrmions even in bulk materials with C and D symmetry classes. In contrast in bulk cubic helimagnets, the longitudinal conical phase is present for arbitrary direction of the magnetic field and restricts the region of skyrmion stability to a small pocket near the ordering temperatures (A-phase) Muehlbauer09 ; Wilhelm11 .

In the present paper, we investigate the robustness of Néel skyrmions with displaced cores in oblique magnetic fields and the role of the uniaxial anisotropy (UA). Lacunar spinels GaVS and GaVSe Kezsmarki15 ; Ehlers17 ; Bordacs17 bear uniaxial anisotropy of easy-axis and easy-plane type, respectively, as they undergo a Jahn-Teller transition from a room-temperature cubic structure into a low-temperature rhombohedral polar (C) structure Hlinka16 ; Ruff15 ; Wang15 . From the experimental phase diagrams Kezsmarki15 ; Fujima17 ; Bordacs17 , showing a temperature dependent ratio of , where is the field of the first-order phase transition between spirals and SkL and is the field of SkL to ferromagnetic state transition, one can conclude that the magnitude of effective UA varies with temperature (see also Fig. 7). In particular, the magnetic phase diagrams of GaVS obtained for different orientations of the magnetic field (see e.g. Fig. 2 in Ref. Kezsmarki15, ) exhibit tri-critical points and by their topology resemble the theoretical phase diagrams constructed in Refs. Butenko10, ; Leonov16a, . Therefore in the present paper, we construct the phase diagrams (Fig. 6) on the plane spanned by the axial and in-plane components of the magnetic field for both signs and various values of the UA. We show that the easy-axis anisotropy decreases the region of SkL stability and promotes the formation of ISs, whereas the easy-plane anisotropy enhances the SkL stability. We also discuss the internal structure of modulated states and the way how the theoretical phase diagrams could be set up in correspondence with the experimental ones. We also study the deformation of antiskyrmions in oblique magnetic fields, realized in Heusler compounds with tetragonal crystal structure. This compound family provide a perfect platform for the design of magnetic configurations with variable uniaxial anisotropy Nayak .

## Ii The Model

The magnetic energy density of a non-centrosymmetric ferromagnet with C and D symmetry can be written as the sum of the exchange, the DMI, Zeeman, and the anisotropy energy density contributions, correspondingly:

(2) |

where spatial coordinates are measured in units of the characteristic length of modulated states . is the exchange stiffness, is the Dzyaloshinskii constant. The unit vector along the magnetization is . is the applied magnetic field, . The magnetic field has a tilt angle with the axis, i.e.

(3) |

The UA . In the case of Néel skyrmions, is the effective uniaxial anisotropy which includes the intrinsic uniaxial anisotropy () and the anisotropy due to the stray field energy Bogdanov94 .

The DMI energy density has the following form:

(4) |

where . In both cases, only modulated phases with the propagation directions perpendicular to the polar axis of C or to the tetragonal axis in magnets with D symmetry, i.e. in the -plane of model (2), are favored.

Functional (2) includes only the main energy contributions essential to stabilize modulated states and an additional UA. As the main goal of the paper is to investigate the role of UA in the stability regions of skyrmions with shifted cores, we do not take into account other anisotropic contributions. In particular, besides the single-ion type anisotropy, considered here, we neglect the difference in exchange couplings for two sets of bonds present in lacunar spinels GaVSe Bordacs17 and GaVS Kezsmarki15 as well as the exchange anisotropy. Moreover, we do not capture the influence of thermal fluctuations near the ordering temperatures.

## Iii Results and discussion

### iii.1 Chiral modulations in C magnets

Three magnetic phases obtained as the solutions of the model (2) can be identified as follows:

(i) Spirals. For the case of C magnets the spirals represent cycloids with the rotation plane of the magnetization containing the wave vector and the high-symmetry () axis. For D crystals, the type of the spiral depends on the in-plane orientation of the -vector: it is a helicoid (a Bloch spiral) for and , while cycloids are formed for diagonal directions as can be discerned in the corresponding structure of the antiskyrmion in Fig. 3. The and axes are selected in a way that and planes are the mirror planes of the D symmetry.

(ii) Skyrmions. Skyrmions can exist either in the form of a thermodynamically stable SkL (the corresponding regions are red shaded in the phase diagrams of Figs. 5 and 6) or as metastable IS within the stable homogeneous state (the corresponding regions are blue-shaded in Figs. 5 and 6). Isolated Néel skyrmions in bulk systems with C have not been experimentally observed yet, although they were reported in thin films with interface-induced DMI Romming13 ; Leonov16a . Isolated antiskyrmions in bulk Heusler alloys were reported in Ref. Nayak, .

When the magnetic field is parallel with the high-symmetry (C) axis, i.e. , the period of the cycloid and its anharmonicity is gradually increased with increasing field. When the magnetic field is perpendicular to the high-symmetry axis, i.e. , a transverse conical state emerges, with a uniform magnetization developing parallel to the field, besides the magnetization component rotating in the plane perpendicular to the field. The opening angle of the cone is decreased with increasing field. As the field points in a general oblique angle, the modulated phases are skewed. In particular, skewed cycloids with the q-vectors along axis, when is along the axis, can be expanded by cycloids with magnetization rotating in the plane (period of such a cycloid gradually increases with increasing -component of the field), plus a transverse component oscillating in the plane and a uniform component.

(iii) Angular homogeneous phases. The magnetization of these field polarized ferromagnetic phases is tilted away from the axis in the presence of oblique fields.

### iii.2 The internal structure of skewed skyrmions

(i) Skewed Néel skyrmions. The shape of the Néel skyrmions in tilted magnetic fields, as shown in Fig. 3, deforms in the following way: the cores of the skyrmions are shifted from the center of the unit cell of the SkL, but the lattice retains the stability against transformation into spirals. Whether the skyrmion centers shift along or opposite to depends on the sign of the Dzyaloshinskii constant in Eq. (2): Néel skyrmions with the ”in-ward” sense of the magnetization rotation (Fig. 2 (a)) have a core displacement along , whereas Néel skyrmions with the ”out-ward” magnetization rotation (Fig. 2 (b)) have a core shift opposite to . The effect is easily understood, as the in-plane component of the magnetic field increases the area, where the magnetization points along the field. The corresponding shift of skyrmion cores is marked by blue arrows in Fig. 2 (a) and (b).

Figs. 2 (c) and (d) show the displacement of skyrmion centers as a function of the orientation of the in-plane field component for . The values and , the components of the core shift along the two axis, are relatively small as compared with the period of a spiral in zero magnetic field. The displacement of the core is of the same magnitude, irrespective of the orientation of the in-plane component of the field and does not reflect the six-fold symmetry of the SkL unit cell.

In Ref. Lin15, , it was shown that the skewed skyrmions have anisotropic inter-skyrmion interaction, i.e. the interaction energy between two skyrmions depends on their relative orientation and leads to the distorted triangular lattice. Nevertheless, the skyrmion-skyrmion interaction remains repulsive with the strongest repulsion between skyrmions in the direction where they are elongated Lin15 . This phenomenon singles out the Néel skyrmions from the Bloch skyrmions in B20 compounds: while Boch skyrmions orient their cores along the magnetization of the ferromagnetic host phase, skyrmions in magnets with axially symmetric crystal structure experience an orientational confinement, with their cores nearly fixed to the direction of the high-symmetry axis. Therefore for B20 materials, it is instructive first to make a full analysis of the magnetization processes for homogeneous states Leonov08 and to define the global and local minima of the energy functional (2). Then, the stable (metastable) SkL will orient their axes along the corresponding homogeneous states and form multiple skyrmionic domains with non-trivial topological boundaries between them.

In thin films of B20 materials, the reorientation process of the Bloch skyrmions in the tilted fields is a more delicate issue, leading to an undulated shape of the skyrmion cylinders Wang17 . Such a skyrmion deformation must minimize, in particular, contributions from so called surface twists, modifying the structure of skyrmions near the surface and even leading to their thermodynamical stability (see Ref. Leonov16b, for details), and from the demagnetization effects. In Ref. Yokouchi14, the process of skyrmion core rotation in tilted magnetic fields has been observed in thin epitaxial films of MnSi. The increase of the angle results in the effective increase of the film thickness for skyrmions and, thus, in the appearance of the conical phase which is stable for thicker MnSi-films (in accordance with the theoretical phase diagram constructed in Ref. Leonov16b, ).

(ii) Skewed antiskyrmions. In case of the antiskyrmions the core displacement direction depends on the orientation of the in-plane field component with respect to the coordinate axes. The modulations propagating parallel to these axes are helical, while the modulations propagating along the 45 degree lines are cycloidal (Fig. 3 (b)). The core shift is perpendicular to the field if is coaligned with the coordinate axes, i.e. for (or ), corresponding to points 1,3,5,7 in Fig. 3 (a) and (c). The core shift occurs along and opposite to the field in points 2, 6 and 4,8 of Fig. 3 (a) and (c), respectively.

The distance , by which the skyrmion centers are displaced from their original position (point ) in the oblique magnetic field, depends on the value of the field and the angle : Fig. 3 (a) shows two curves for (solid line) and (dashed line), with in both cases..

### iii.3 Fourier components of skewed SkL

The internal structure of the skyrmion lattice can also be characterized by the behavior of the Fourier components in oblique magnetic fields. Fig. 4 shows field dependence of the coefficients (red line), (blue line), and (green line) for ”stright” skyrmions with (solid lines) and elongated skyrmions with (dotted lines). The Fourier components are introduced according to

(5) |

where is the surface area of the corresponding unit cell for SkL. The ”stright” and elongated skyrmions can be distinguished by the behavior of their coefficients: for these coefficients never become zero as is the case for .

The coefficients and for ”stright” skyrmions become zero for some values of the field and are small in the field region around . In Ref. Wilson14, (see Fig. 10Wilson14 ), it was shown that for the case of B20 magnets at the field the difference between the energy densities of the SkL and the conical phase is minimal. On this ground, it was suggested that the skyrmion lattice is stabilized with respect to the cones exactly around this field value and underlies the phenomenon of A-phase Muehlbauer09 . Analysis of the Fourier components (5) may justify the use of the triple-q ansatz in this field interval Muehlbauer09 . Near the transition into the ferromagnetic state, however, the values of higher harmonics become significant and the ansatz in the form of the triple-q becomes incorrect.

### iii.4 Phase diagrams of states

By comparing the equilibrium energies of spirals, SkL and polarized FM states, we construct the phase diagrams on the planes () of the field components for different values of the uniaxial anisotropy . The areas of magnetic states corresponding to the global minimum of the energy functional are indicated by different colors.

We start the analysis of the phase diagrams with the case. The SkL is thermodynamically stable within a curvilinear triangle (red-shaded region) with vertices , , and Butenko10 ; Bogdanov94 . The line is the line of the first-order phase transition between spirals and SkL. At the line the SkL transforms into the polarized FM state whereas at the line (thin dotted line) the metastable SkL undergoes an elliptical instability toward the spiral state (an elliptical instability of ISs was considered in Refs. Bogdanov99, ; Leonov16a, ). Therefore, the point is a cusp point in which different lines characterizing different processes joint. The point A also corresponds to the threshold angle of an oblique magnetic field, where the thermodynamically stable SkL state still emerges. As a metastable solution, SkL exists even in a perpendicular magnetic field in the range of Fig. 5. For only skewed spirals (gray-shaded region) can be observed culminating in a conical phase at the point B ( for ). The analysis of the type of the phase transition between skewed spirals and FM states as observed along the line has been done in Ref. Laliena16, .

For the easy-axis uniaxial anisotropy ( in Eq. (2), Fig. 6 (a)), the existence range of the skyrmion lattice is the largest for the field co-aligned with the magnetic easy axis (point C in Fig. 6) and decreases with increasing tilting angle . The angle is a function of the uniaxial anisotropy as shown in Fig. 7 (b). Since the skyrmion cores are aligned along the easy axes, therefore, the stronger the anisotropy is, the higher field is needed to shift the skyrmionic cores and the larger can be achieved. At the same time, the UA suppresses the modulated phases. Thus, the temperature or UA-dependence of shows a maximum at , which is realized at T=11K in GaVS.

In Fig. 7 (a) for , we plot the ratio : a red dotted line has been obtained from the theoretical phase diagram in Ref. Butenko10, and has been plotted as a function of (corresponding to upper scale), while the solid black line (with circular markers) is the temperature dependence of as extracted from the experimental data of Refs. Kezsmarki15, ; Bordacs17, for GaVS and GaVSe in (lower scale). By this, we make a correspondence between the experimental phase diagrams obtained for lacunar spinels GaVS and GaVSe and the theoretical phase diagrams in Fig. 6. Value in Fig. 7 (a) corresponds to the three-critical point with and temperature K for GaVSKezsmarki15 . For , skyrmions are suppressed and only cycloids are observed. For K () only the ferromagnetic state is present in the phase diagram. Thus the increasing UA first suppresses the SkL and eventually the cycloids.

In GaVS Kezsmarki15 at the structural transition, the lattice is stretched along one of the four body diagonals. The four distortion directions correspond to the magnetic easy axes of the four structural domains coexisting below 44 K. The magnetic state in each domain depends on the strength and orientation of the magnetic field with respect to the corresponding easy axis. In Ref. Kezsmarki15, the magnetic field was applied along the following directions in the cubic setting: , , and . The experimentally observed is slightly larger than the theoretical value in Fig. 7 (b) which could be explained, e.g., by the exchange anisotropy contribution or the slight rotation of Néel skyrmions neglected in our theory.

For the easy-plane anisotropy ( in Eq. (2), Fig. 6 (b)), the region of the skyrmion stability is significantly enhanced for . Indeed, it was reported recently in GaVSe Bordacs17 ; Fujima17 that SkL remains stable down to zero Kelvin. This corroborates with the prediction by Randeria and co-workers as applied for SkL in polar materials with Rashba-type spin-orbit interaction Rowland16 .

The angular stability of SkL, however, impairs, as the decreases with as discerned in Fig. 7 (d). This is accompanied by the re-entrance phenomenon in some interval of angles : two successive first-order phase transitions from the spiral state to SkL and back take place with increasing magnetic field. In Fig. 6 (b) corresponds to the point A and signifies the onset of the re-entrant spiral state, coincides with the farthest point of the SkL stability region and signifies the limiting angle of the tilted field after which the re-entrance phenomenon disappears. According to Fig. 7 (d) the largest interval is reached for . No experimental observation of the predicted re-entrance phenomenon has been reported yet.

The values of extracted from the experimental data on GaVSe in Ref. Bordacs17, (white circles in Fig. 7 (c)) give the values of in the diapason . Only the temperature range K was taken into account since for some additional unknown phases appear which complicates the assignment of and . These values slightly differ from the values extracted from Ref. Fujima17, (blue and red circles in Fig. 7 (c)). No new phases below K have been reported in Ref. Fujima17, .

The structure of IS surrounded by the canted ferromagnetic phase (blue-shaded regions in Figs. 5 and 6) was recently studied in Ref. Leonov17, . Such IS has asymmetric magnetic structure and exhibit anisotropic inter-skyrmion potential due to the intricate domain-wall region, which connects the core of the IS with the embedding canted ferromagnetic phase. In Ref. Lin15, the shape of isolated skyrmions has been investigated for the field and in the units of Eq. (2), i.e., at the boundary between the skyrmion lattice and the homogeneous phase. With the increasing angle , as seen from the phase diagram, one gets onto the region of the stable skewed spiral. Therefore, a clear coexistence phase of the isolated skyrmion and the skewed spiral can be reached. For even for ISs exist as localized entities within the homogeneously magnetized in-plane phase as also demonstrated for antiskyrmions in Fig. 3 (a).

The theoretical phase diagrams in Fig. 6 reproduce the main aspects of the experimental phase diagrams in Refs. Kezsmarki15, ; Bordacs17, for both GaVSe and GaVS as compared with the isotropic case: i) the ratio of the critical fields required to reach the ferromagnetic state along and perpendicular to the polar c axis (i.e. the ratio ), ii) the extended or reduced angular stability range of SkL for the tilted fields in case of UA of easy-axis or easy-plane type, iii) the suppression or elongation of the stability range of the cycloidal states against fields perpendicular to the polar axis (point B). In particular, the value of the UA reproduces well the experimentally constructed phase diagram in Ref. Bordacs17, and was chosen to match the ratio in Fig. 6 (b) and the temperature value in Fig. 7 (c).

We therefore argue that the role played by the UA is different in chiral B20 magnets and in considered materials with C and D symmetry. In the first case, the UA of the easy-axis type (introduced by the axial deformation of the originally cubic structure) is invoked to suppress the conical phase Butenko10 . SkL becomes thermodynamically stable in some interval of anisotropy parameters although in the rest of the phase diagram SkL remains a metastable state. For the UA of the easy-plane type, only the conical phase is the global minimum of the functional (2). To stabilize SkL in this case, one should find corresponding directions of an applied magnetic field. In particular, cigar-like skyrmions can be stabilized with their axes lying in the easy plane together with the co-aligned magnetic field Wilson14 .

In the case of C and D acentric magnets, the UA modifies correspondingly the regions of the SkL stability promoting either ISs (easy-axis type) or enhancing the stability range of SkL (easy-plane type). The easy-plane anisotropy may also lead to a stability of non-axisymmetric IS with unique properties Leonov17 as compared to ordinary axisymmetric skyrmions Leonov16a .

## Iv Conclusions

In conclusion, we present theoretical studies on the robustness of skyrmions in tilted magnetic fields with special attention on the role of uniaxial magnetic anisotropy, which turns out to be a key factor governing the stability range of the modulated states. Together with earlier complementary studies on the cubic chiral magnets, our results on axially symmetric non-centrosymmetric magnets establish a consistent picture about the role of uniaxial anisotropy in the stability of modulated phases including spiral, skyrmion and antiskyrmion lattices. Our findings can be tested experimentally in non-centrosymmetric magnets lacking the Lifshitz invariants along , e. g. with C, D, and S crystallographic symmetry. We also described the deformations of skyrmions and antiskyrmions, both in lattices and individual ones, when the magnetic field is tilted away from this unique axis, which is the high-symmetry axis in these crystallographic classes.

## V Acknowledgements

The authors are grateful to K. Inoue, A. Bogdanov, Y. Togawa, M. Mostovoy and S. Bordacs for useful discussions. This work was funded by JSPS Core-to-Core Program, Advanced Research Networks (Japan). This work was supported by the Hungarian Research Fund OTKA K 108918.

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