Coupled quasimonopoles in chiral magnets

TL;DR

This paper investigates the existence and stability of coupled magnetic monopoles, specifically Bloch points, in chiral magnets, revealing their potential as stable localized states within crystals and confined geometries.

Contribution

It introduces the concept of coupled Bloch points as stable localized states in chiral magnets, expanding understanding of magnetic singularities.

Findings

Coupled Bloch points can form stable localized states.

Stability analyzed in bulk and confined geometries.

Potential implications for nanomagnetism applications.

Abstract

Magnetic singularities, also known as magnetic monopoles or Bloch points, represent intriguingphenomena in nanomagnetism. We show that a pair of coupled Bloch points may appear as alocalized, stable state in cubic chiral magnets. Detailed analysis is presented of the stability of suchobjects in the interior of crystals and in geometrically confined systems.

Equations

$$E=-J\sum _{⟨ij⟩}{\mathbf{n}}_i\cdot {\mathbf{n}}_j-\sum _{⟨ij⟩}{\mathbf{D}}_{ij}\cdot [{\mathbf{n}}_i\times {\mathbf{n}}_j]-{\mu }_{\mathrm{s}}{\mathbf{B}}_{\mathrm{ext}}\sum _i{\mathbf{n}}_i,$$

Full text

Point singularities in magnets Feldtkeller_65 ; Doring_68 are topologically non-trivial objects known as Bloch points (BPs) Malozemoff_79 or hedgehogs and recently have become commonly referred to as quasi-monopoles or monopoles Milde ; Kanazawa_16 . Within the classical continuum micromagnetic approach, such a singularity is a point where the magnetization vector $\mathbf{M}(\mathbf{r})$ suffers a discontinuity, but there is, nevertheless, no divergence in the energy Doring_68 . Contrary to the hypothetical Dirac monopole, the magnetic flux through the surface surrounding a magnetic singularity is equal to zero. Nevertheless, noticeable similarity between these objects does exist. For instance, similar to the Dirac string stretching from a magnetic monopole, a magnetic singularity can be considered as an origin of a skyrmion string Milde representing strings-vortices in the magnetization vector field. Typically, skyrmion strings are observed as singularity-free magnetic textures stretching throughout the entire sample between free edges. Of particular interest are systems where one or both ends of the string behave differently and instead of being stretched to a free surface of the sample are coupled to point singularities. The scenario in which one end of a skyrmion string ends at a surface of the crystal but the other ends with a point singularity within the sample has been discussed earlier for isotropic chiral magnets Schutte . This kind of texture may become stable and underlies a new type of particle-like state –- the chiral bobber Rybakov_15 . It has recently been observed by means of off-axis electron holography Zheng_18 . Such objects composed of a point singularity coupled to a continuous vector field can be classified as so-called hybrid solitons Rybakov_15 . Another possible scenario corresponds to the case when both ends of the string represent point singularities with opposite topological charge. The stability of such a magnetic texture has been assumed to require the singularities to be located in the vicinity of opposite surfaces of the sample Leonov_18 . The mechanism of stabilization of such a configuration is similar to that of a coupled pair of chiral bobbers situated near opposite surfaces of the film Rybakov_16 .

A more sophisticated mechanism of stabilization of such a configuration has been discussed in Ref. Liu_Binding_2018 , where the particular case of a thin disk of a chiral magnet is discussed. The disk is covered by additional magnetic layers characterized by strong perpendicular magnetic anisotropy and coupled to a chiral magnet via exchange interaction. The introduction of such artificial interfaces serve to pin the spins on the top and bottom surfaces of the chiral magnet to be pointed perpendicular to the interface. Such a method of artificial confinement is analogous to the stabilization of exotic textures in liquid crystals squeezed between glass plates where the surfaces are treated to align the molecules along the normal Smalyukh_2010 .

We present in this Letter results showing that the stability of such a coupled pair of quasi-monopoles does not in general require geometric confinement but can be achieved even within the interior of bulk chiral magnets favoring skyrmions Bogdanov_89 , such as MnSi Yu_15 , FeGe Yu_11 , Fe1-xCoxSi Yu_10 and some other Si- and Ge-based metallic alloys and insulating magnets such as Cu2OSeO3 Seki_2012 and alloys of $\beta$-Mn-type Co-Zn-Mn Tokunaga_15 .

The energy of the simulated system is described by an extended Heisenberg model for an isotropic chiral magnet defined by the following Hamiltonian Han :

[TABLE]

where $\mathbf{n}_{i}=\mathbf{\mu}_{i}/\mu_{\mathrm{s}}$ is the unit vector of the magnetic moment at lattice site $i$, the $\braket{ij}$ denote unique nearest-neighbour pairs, $J$ is the Heisenberg exchange constant and $\mathbf{D}_{ij}$ is the Dzyaloshinskii-Moriya vector defined as $\mathbf{D}_{ij}=D\mathbf{r}_{ij}$ with the scalar DMI constant $D$ and the unit vector $\mathbf{r}_{ij}$ pointing from site $i$ to site $j$, $\mathbf{B}_{\mathrm{ext}}$ is an external magnetic field. In order to keep the generality of the results presented below, as well as consistency with earlier studies, the size of the simulated domain is always given in terms of reduced units of distances with respect to $L_{\mathrm{D}}=2\pi aJ/D$ – the lowest period of a spin spiral in the continuous limit ($J\!\gg\!D$), where $a$ is the cubic lattice constant and reduced units of external magnetic field are given with respect to $B_{\mathrm{D}}={D^{2}}/(\mu_{\mathrm{s}}J)$ – the critical field at which the bulk system reaches the field polarized ferromagnetic state. We do not take into account the influence of the demagnetization fields and thus describe the phenomena at the level of the basic (minimal) model. It is known that this model (1) describes well fundamental properties of chiral magnets and even has predictive power, while the addition of dipolar interactions can help achieve closer quantitative agreement with experiments, see for instance Rybakov_15 and Zheng_18 .

The magnetic texture of the configuration composed of two coupled singularities is illustrated in Fig. 1. Because the isosurface $n_{z}\!=\!0$ of the corresponding 3D spin texture has a nearly ellipsoidal shape with the singular points located on opposite poles, we refer to this object as magnetic globule (MG). Because of an opposite topological charge of the BPs, the MG can, to a certain extent, be considered as a dipole. Note that due to the presence of discontinuities, such an object can not be classified by the Hopf invariant Bott_Tu .

Despite the fact that the MG as a whole represents a topologically trivial object, a sufficiently large energy barrier is present under certain conditions to prevent its collapse via annihilation of the BPs. For instance, as shown in Fig. 1, a cluster of MGs can be stabilized due to confinement effects. In this case, the interactions between the MGs makes such a configuration stable. Below, we show that a single MG can also be stabilized in the interior of a crystal due to its coupling to textures representing defects or distortions in the ground state.

For a moderately strong external magnetic field, $B_{\mathrm{ext}}<B_{\mathrm{D}}$, the global energy minimum for (1) corresponds to the conical state – a helical spin spiral with $\mathbf{k}\parallel\mathbf{B}_{\mathrm{ext}}$ and magnetization tilted towards the direction of $\mathbf{B}_{\mathrm{ext}}$. However, in real crystals the magnetic texture is not perfectly ordered but rather contains various defects, distortions or inclusions. As has been shown earlier, a skyrmion tube (SkT) represents one such stable inclusion within the conical phase Rybakov_16 ; Leonov_16 . Figure 2a shows an isolated SkT within the conical phase, reflected by the spiraling distortion of the corresponding isosurfaces of $\mathbf{n}\parallel\mathbf{B}_{\mathrm{ext}}$ and $\mathbf{n}\perp\mathbf{B}_{\mathrm{ext}}$. The states shown in Fig. 2b-d represent stable configurations that include an MG of varying size coupled to the SkT. The minimum energy path (MEP) between these configurations is shown in Fig. 2e. It was calculated using the geodesic nudged elastic band (GNEB) method bessarab_method_2015 implemented in the Spirit framework spirit . The MEP clearly shows the presence of an energy barrier between the intermediate configurations, indicating that the MG attached to the SkT indeed represents a metastable state that cannot be destroyed by small excitations. The distance $d_{\mathrm{BP}}$ between the poles of the elongated MGs is quantized with a pitch of $\sim L_{\mathrm{D}}$, while the lowest energy state always corresponds to the state with the smallest $d_{\mathrm{BP}}\lesssim L_{\mathrm{D}}$ as in b, which may slightly vary with the value of applied field. The energy of such a coupled state increases approximately linearly with $d_{\mathrm{BP}}$ and tends to the energy of two coupled infinitely long skyrmion tubes. It has been shown recently that such coupled states appear due to the attractive interaction between skyrmion tubes at $B_{\mathrm{ext}}<B_{\mathrm{D}}$ Leonov_16 ; Loudon_2017 ; Du_18 . A single SkT can host a number of MGs coupled to it as shown in Fig. 3a and in Supplementary Fig. S2.2.

Edge dislocations represent another type of inclusion that can appear within a conical phase. They belong to the family of native defects emerging in strip-like states in magnetism dussaux2016 and beyond harrison2000 . We find that an edge dislocation can provide stability for the MG in the interior of the crystal as shown in Fig. 3b. In order to simulate such a single half plane defect in a crystal, special boundary conditions are applied to the simulated domain  supplement . We find that the range in magnetic field strength where an MG coupled to an edge dislocation is stable includes at least from $0.31B_{\mathrm{D}}$ to $0.83B_{\mathrm{D}}$.

The results presented above clearly show that inclusions and inhomogeneities typically occurring in the conical phase, such as SkTs and edge dislocations, can provide stability to MGs in the interior of crystals. Therefore, one can expect that MGs play a significant role in the thermodynamic and transport properties of helimagnets. However, in order to directly observe nanoscale textures in an experimental setup, such as X-ray or transmission electron microscopy (TEM), the sample needs to be small enough to be transparent to X-ray photons or high energy electrons. Below we show that for samples of finite size, the stability of MGs is naturally provided by coupling with edge modulations inherent in chiral magnets Meynell_14 ; Du_15 ; Jin_17 .

Figure 4a shows a diagram of magnetic states calculated for a long nanowire in an external magnetic field applied along its main axis. We find a surprisingly large range of applied field strength and nanowire width where a single MG remains stable. When the applied field approaches $B_{\mathrm{ext}}\!=\!B_{\mathrm{D}}$ the conical phase is suppressed. Similar to chiral bobbers, the saturation of the conical phase at high magnetic field limits the stability of the MG and leads to its collapse. Similar to the case of an MG attached to an edge dislocation in a bulk crystal, the upper bound for the stability of the MG is $B_{\mathrm{ext}}\approx 0.86B_{\mathrm{D}}$ and remains nearly constant for $W/L_{\mathrm{D}}>2$, but decreases as the wire is made thinner. There is also a lower limiting field below which the MG disappears via the escape of the BPs from the system through the free edges. The value of the lower bounding field decreases gradually with increasing width of the wire, $W$. We estimate the critical width of the nanowire to be $W\approx 10L_{\mathrm{D}}$, above which the MG may remain stable even at zero magnetic field.

In the whole range where the MG is stable its energy is higher than that of the conical phase. Within this range, there is a region where the energy density of a single SkT along the wire is lower than that of the conical phase, meaning that it represents the ground state of the system Charilaou . Outside this region the SkT remains stable up to high $B_{\mathrm{ext}}$, see the gray curve in Fig. 4a. Note, in the spin lattice model the critical field of the skyrmion collapse is defined by the ratio of internal coupling parameters, $J/D$ while in the continuum limit the collapse does not occur even if $B_{\mathrm{ext}}\!\rightarrow\!\infty$ Doring .

Fig. 4b shows the dependence of the total energy of the SkT and MG as a function of the length $L$ of the wire with respect to the ground state which for the chosen parameters corresponds to the conical phase with edge modulations. Since the SkT occupies the whole volume of the wire, its energy increases linearly with $L$. On the other hand, the energy of the MG as a 3D-localized state remains approximately constant. The presence of an MG introduces local distortion to the ground state. In the calculations with periodic boundary conditions, it leads to the fact that for finite $L$ the energy of the MG may somewhat deviate from the self-energy of the MG when $L\rightarrow\infty$ (dashed line). The distortions introduced by the SkT and MG in the nanowire are visualized in Fig. 4c,d, where edge modulations have distinctly different periodicity. Noticeably, such behavior of the self-energy of the SkT and MG applies to a broad range of parameter values. In particular, for any $B_{\mathrm{ext}}$ and any $W/L_{\mathrm{D}}$ where the MG is stable, and the ground state is the conical phase with edge modulations (the blue region in Fig. 4a), there is always a certain critical $L$ above which the MG is lower in energy than the SkT. This can lead to a transition from SkT to MG.

To illustrate the possibility of such a transition, we calculated the MEP from a homogeneous SkT (I) to a single MG (IV) and then to the conical state with edge modulations (V). In Fig 4e and f we show only those parts of MEP which remain approximately invariant as the length of the nanowire is varied The path between states (III) and (IV) depends on the length of the nanowire and is therefore not shown. There is a barrier on this stretch of path, and then the energy gradually drops while SkTs with one end, which are located above and below MG, move to the opposite ends of the wire (see also Supplementary Fig. S2.3). According to the above, the energy difference and thereby energy gain of the transition between state (I) and (IV) depends on the length of the wire and for a sufficiently long wire, $L\!\gg\!L_{\mathrm{D}}$, and appropriate value of $B_{\mathrm{ext}}$ the condition $E_{\mathrm{(I)}}\!\gg\!E_{\mathrm{(IV)}}$ can be satisfied.

The energy barrier of $\sim 5J$ for the collapse of the MG is on the same order of magnitude as the energy barrier for the chiral bobber estimated with the same method in Ref. Rybakov_15, . On the other hand, the height of the energy barrier is not the only factor determining the stability of localized states bessarab_lifetime_2018 . The pre-exponential factor in the Arrhenius expression for the lifetime, reflecting the relative entropy of the transition state and the initial state also needs to be evaluated. The estimation of the lifetime of the metastable states depicted in Fig. 4e is a topic of future studies.

In conclusion, we have shown that in a wide class of isotropic chiral magnets a three-dimensionally localized state composed of a pair of Bloch points can be stable. By calculating minimum energy paths, we have shown that a finite energy barrier protects such magnetic globules from collapsing. In the interior of a crystal, an MG remains stable when it is located near a defect line of the conical phase or is coupled to a skyrmion tube. We have shown that for a wide range in magnetic field strength and parameter values, in both in extended and confined systems, magnetic globules can be stable as isolated objects.

In order to experimentally confirm the existence of MGs, the most promising avenue seems to be the partial or full reconstruction of the 3D magnetic configuration inside a crystal using some magnetic imaging technique, such as off-axis electron holography Jin_17 ; Midgley_09 , the X-ray magnetic circular dichroism Col_14 or X-ray vector nanotomography Donnelly_17 . According to recent studies, it is expected that stable magnetic textures containing singularities may also be confirmed with magnetoresistive measurements Du_14 ; Du_15 ; redies_distinct_2018 .

Acknowledgments

G. P. M. and H. J. acknowledge funding from the Icelandic Research Fund (grant no. 185405-051). The work of F. N. R. was supported by the Swedish Research Council Grant No. 642-2013-7837 and by Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. The work of N. S. K. was supported by Deutsche Forschungsgemeinschaft (DFG) via SPP 2137 “Skyrmionics” Grant No. KI 2078/1-1.

S2. Boundary conditions for the bulk system

Ordinary periodical or open boundary conditions cannot be applied in the case presented in Fig. 3 of the main text due to the broken symmetry of the spin texture. This can be clearly seen in Fig. S2.1. Both the $x$- and $z$-direction cannot be treated with periodic boundary conditions.

In order to calculate physically correct states of a bulk crystal, but with non-periodical boundary conditions, a special scheme was applied. In $y$-direction an ordinary periodical boundary conditions are applied. In the $x$-direction, the spins at the boundary layers are pinned according to the analytical solution for the conical state at the corresponding applied magnetic field, $\theta=\arccos(B_{\mathrm{ext}}/B_{\mathrm{D}})$, $\varphi=2\pi z/L_{\mathrm{D}}+\varphi_{0}$. In order to enforce the stability of the dislocation within the simulated domain, the values of $\varphi_{0}$ on the left and right boundary of the domain ($x_{\mathrm{min}}$ and $x_{\mathrm{max}}$) are set to be different with $\Delta\varphi_{0}=\pi$. The infinite continuation in $z$-direction is achieved by the following method. In ordinary periodical boundary conditions the upward neighbors for spins in the endmost upper layer, $z=z_{\mathrm{max}}$ are the spins in the endmost lower layer with $z=z_{\mathrm{min}}$. Contrary to this, in our implementation, the spins in the layer with $z=z_{\mathrm{max}}-L_{\mathrm{D}}$ play the role of upward neighbors for the endmost upper layer. Correspondingly for the endmost lower layer, $z=z_{\mathrm{min}}$ the downward neighboring spins are in the layer with $z=z_{\mathrm{min}}+L_{\mathrm{D}}$. This enables the calculation of a single defect line in the strip-like phase, as shown in Fig. 3 in the main text. From the above it is clear that the size of the simulated domain along the $z$-axis, $L_{z}$, has to be a multiple of $L_{\mathrm{D}}$. Moreover, $L_{z}$ should be large enough to host at least few periods of the spin spiral and material parameters have to be chosen such that $L_{\mathrm{D}}=N\cdot a$, where $N$ is an integer and $a$ is the lattice parameter. In our simulations we used $L_{z}=8L_{\mathrm{D}}$ and $L_{\mathrm{D}}=32a$, meaning $256$ lattice sites along the $z$-axis. For consistency, we used the same number of lattice sites in the other directions, $L_{x}=L_{y}=L_{z}$.

S4. Supplementary figures