Collective antiskyrmion-mediated phase transition and defect-induced melting in chiral magnetic films
Collective antiskyrmion-mediated phase transition and defect- induced melting in chiral magnetic films
J. F. L?ffler
OPEN Magnetic phase transitions are a manifestation of competing interactions whose behavior is critically modified by defects and becomes even more complex when topological constraints are involved. In particular, the investigation of skyrmions and skyrmion lattices offers insight into fundamental processes of topological-charge creation and annihilation upon changing the magnetic state. Nonetheless, the exact physical mechanisms behind these phase transitions remain unresolved. Here, we show numerically that it is possible to collectively reverse the polarity of a skyrmion lattice in a field-induced first-order phase transition via a transient antiskyrmion-lattice state. We thus propose a new type of phase transformation where a skyrmion lattice inverts to another one due to topological constraints. In the presence of even a single defect, the process becomes a second-order phase transition with gradual topological-charge melting. This radical change in the system's behavior from a first-order to a second-order phase transition demonstrates that defects in real materials could prevent us from observing collective topological phenomena. We have systematically compared ultra-thin films with isotropic and anisotropic Dzyaloshinskii-Moriya interactions (DMIs), and demonstrated a nearly identical behavior for such technologically relevant interfacial systems.
? m ? (?xm ? ?ym)dxdy = ? 1,
with m the magnetization unit vector. This signifies that for both Bloch- and N?el-type skyrmions10 the local
magnetic moment rotates by 2? from one end of a skyrmion to another, as described by a variational ansatz11 for a
2? domain wall. The sign of topological charge depends on the magnetization polarity of a skyrmion, but also on
the direction of magnetization winding (vorticity). Objects of the opposite magnetization winding to skyrmions
are called antiskyrmions12?14, and consequently their topological charge is opposite to that of skyrmions with the
On a surface containing more than one skyrmion, the global topological charge is the net sum resulting from
all topological objects in the system. Skyrmions can be arranged in irregular clusters16 or rectangular lattices17,
but mostly in hexagonal lattices, as observed for the chiral magnets MnSi18,19, FeCoSi20 and FeGe21. Skyrmion
lattices (SkLs) are stable close to the Curie temperature in bulk systems with surfaces, and in a much wider
temperature range in thin films21,22. In ultra-thin systems with interface-induced anisotropic DMIs, such as Ir/Co/Pt
multilayers23,24, skyrmions form even at room temperature and in zero external magnetic field due to strong
PMA. Additionally, it has been shown numerically25 and experimentally26 that a new type of magnetic solitons,
namely radial vortices, also occur in the presence of weak in-plane anisotropy. Contrarily, antiskyrmion lattices
have only been observed in bulk systems such as Mn?Pt?Sn27, and some of the current research is focused on
stabilizing antiskyrmions in a wider range of materials, including thin films28. Nevertheless, the wide range of
temperatures and materials in which skyrmionic objects exist, together with the fact that they can be controlled
by relatively small current densities29, makes them promising candidates for future spin-based applications30?33.
Apart from being technologically relevant, skyrmionic spin textures offer the potential for detailed
investigations into topological aspects of magnetism34. Specifically, since skyrmion textures are protected by their
topological constraints, they cannot be continuously unwound into a trivial ferromagnetic (FM) configuration
athermally without a phase transition. The exact mechanism of this kind of phase transitions, however, remains
relatively unexplored so far.
In the following we demonstrate using micromagnetic simulations that skyrmion lattices undergo a magnetic
field-induced phase transition where an antiskyrmion is created for each skyrmion, which results in a transient
Q = 0 state and enables the switching of the lattice polarity. However, in the presence of even a single defect, this
phase transition is replaced by a melting-type mechanism, where topological charge is gradually lost.
We start by analyzing the magnetization profile of a Bloch and a N?el skyrmion in zero magnetic field in thin films
with DMI and PMA. Fig.?1a?c show that the z-component of the magnetization for the two kinds of skyrmions
is identical and can be described precisely by a variational ansatz for a 2? domain wall11, corresponding to a
topological charge of unity, as also calculated from the relaxed spin texture. We later use this ansatz to define and
calculate the radius of skyrmions by fitting it to the z-component of the skyrmion magnetization. In our system,
skyrmions are configured in a hexagonal lattice (see Fig.?1d,e) and the global topological charge is equal to the
number of skyrmions in the lattice, which in our case is Q = 64. Our findings are identical for bulk and interfacial
DMI systems, i.e. for Bloch and N?el skyrmions. We have tested systems of different thicknesses and obtained
qualitatively the same behavior (data not shown).
In order to investigate changes in skyrmion lattices as a function of external stimuli, we applied an external
out-of-plane magnetic field by performing a magnetic-field sweep in the directions both parallel and antiparallel
to the skyrmion-core polarization, which we define as the positive and negative directions of the external field.
We observe that the skyrmions shrink and consequently the net magnetization along the z-axis decreases as the
field is swept antiparallel to the core polarization (Fig.?2a?d, following the opposite direction of the blue arrows in
Fig.?2a,b), as expected23. In a critical field ?|HAP|, the skyrmions are annihilated and the system reaches the
topologically trivial FM state through a first-order phase transition. If the field is swept parallel to the core polarization
(direction of the blue arrows in Fig.?2a,b), however, the resulting behavior is much more complex. Upon
increasing the field parallel to the skyrmion-core polarization, the skyrmions grow to the point where their boundaries
themselves become a 2? domain wall. At this point skyrmions cannot grow anymore so they form a hexagonal
state where the 2? boundaries assume the lattice symmetry, as shown in Fig.?2e. This state is special in that its
topological charge distribution is different to that of the skyrmion lattices in zero field, where the skyrmions are
small and well separated, and all topological charge is located on individual skyrmions. In the hexagonal state, all
spin winding, and therefore all topological charge, is situated on the 2? domain-wall network, meaning that the
topological charge is shared between the skyrmions, i.e., it is effectively delocalized.
From this state onwards there are two possible scenarios at a critical field + HP : (i) the system undergoes a
first-order phase transition from the hexagonal to the FM state; or (ii) the system undergoes another, very
surprising first-order phase transition in which the skyrmion lattice inverts its magnetic polarity (Fig.?2f). This abrupt
metamagnetic-like transition is characterized by a discontinuous change in the total magnetization and skyrmion
radius (arrows from e to f in Fig.?2a,b). As the field is swept further, the skyrmions shrink and the inverted lattice
undergoes a first-order phase transition to the FM state at + HAP , in analogy to the destruction of the lattice in
the antiparallel sweep.
The metamagnetic transition of the skyrmion-lattice inversion entails striking features related to the
topological charge. As illustrated in Fig.?3a,b, the inversion starts by breaking one third of the boundaries between
the skyrmions. Their 2?-domain-wall shape has a topologically constraining character, which induces the
creation of an antiskyrmion for each skyrmion, exactly offsetting the global topological charge. In the next step,
these antiskyrmions are annihilated and new pairs of elliptical skyrmions and antiskyrmions are created from
the remaining boundaries (Fig.?3c), keeping the global topological charge at zero. At the end of the inversion,
all antiskyrmions become annihilated, which restores the global topological charge to its original value. Here,
new cores of inverted skyrmions form from the elliptical skyrmions, which are situated at the vertices of the
skyrmions in the original SkL (Fig.?3d), i.e., the inverted skyrmions have emerged from the 2? boundaries (see
The phenomena described above depend strongly on the intrinsic material parameters. For a DMI strength
between 1 and 2 mJ/m2, comparable to the material values of intrinsically chiral magnets or Co/Pt-based
multilayers, we have investigated the range of PMA (i) for which skyrmion lattices are stable in zero magnetic field; and
(ii) for which the inversion happens. We have?compared bulk and interfacial systems, i.e., systems with isotropic
and anisotropic DMIs respectively, and found a nearly identical behavior. The SkL phase is stable in the range of
Ku = 150?800 kJ/m3 and the skyrmions grow with decreasing PMA and increasing DMI (Fig.?4a), in agreement
with literature7?9. The inversion can only occur if the inverted SkL is stable in the field range|HP| < H < |HAP|.
The values of|HP| and|HAP|, and therefore whether or not the SkL inversion occurs, depend on the material
properties. As shown in Fig.?4b,c,|HP| increases with increasing PMA, but|HAP| decreases with increasing PMA, so that
the criterion limits the occurrence of inversion to materials with low PMA (shaded area in Fig.?4).
This behavior can be explained by a competition of the various interactions in the system. PMA acts to
decrease the size of the skyrmions in order to minimize the area of the spins misaligned with the easy axis. This
is in competition with the Zeeman energy, which acts to increase the size of the skyrmions in order to align them
with the external magnetic field. At low PMA, the skyrmion lattices reach the hexagonal state in a relatively small
magnetic field. Due to the confined skyrmion winding in the hexagonal lattice, the skyrmions cannot transition
into the trivial ferromagnetic state, so they invert their polarization in order to align more area with the magnetic
field. At high PMA, the skyrmion lattices are relatively small in zero magnetic field, and a large magnetic field is
required for them to reach the hexagonal state. In such high magnetic fields, the DMI energy is small compared
to the Zeeman energy, so that the ferromagnetic state is more favorable. Supplementary Fig.?1 shows a further
analysis of the energetics for various PMA and DMI.
The findings described above assume an infinite ideal system. However, real materials, either bulk or thin
films, contain defects that may strongly affect the magnetic state, particularly in multilayers with interfacial DMIs.
It is therefore important to study the effect of defects on the SkL phase and its stability, which is essential for
enabling the functionality of skyrmion-based devices30?33.
In our simulations, we have implemented single and multiple defects of three different kinds: (i) a local
variation of DMI or PMA, (ii) a local distortion in the skyrmion lattice, and (iii) a vacancy in the skyrmion lattice.
We find that while defects do not significantly affect the system?s behavior when the field is swept antiparallel
to the skyrmion-core polarization, they compromise the stability of the SkL phase and dramatically modify the
associated magnetization processes when the field is swept parallel. In fact, the annihilation of the lattice starts
at the defect site, because the skyrmion of the same shape and size as in the rest of the lattice is not stable within
the region of the defect. Thus, due to the high degree of confinement in the system, the skyrmions start twisting,
deforming, and inhomogeneously growing at the defect site. This starts an avalanche-like effect which results
in elliptical instabilities35 and consequently in a gradual loss of topological charge. At high PMA (Fig.?5a?d,i),
where the inversion does not occur even in ideal systems, all topological charge is lost. This is in contrast to low
PMA, where inversion occurs in ideal systems. Here, not all topological charge is destroyed for defective systems
because some skyrmions still undergo inversion (Fig.?5e?i). Figure?5i demonstrates that the stability of skyrmion
lattices is less compromised by defects at low PMA than at high PMA. This, together with our finding that
inversion still occurs at low PMA regardless of having a single defect or multiple defects, suggests that the inversion
may be experimentally observable even for defective systems (see Supplementary Videos?2 and 3).
Importantly, the critical field in which the lattice is destroyed is strongly reduced in defective systems (see
Fig.?5i), and the transition changes from first-order to second-order (see Supplementary Fig.?2), where we observe
gradual lattice-melting behavior. This destabilization of skyrmion lattices due to the presence of defects illustrates
defect-containing material, where the constraints are weaker and the magnetization unwinds at the defect. The
latter destabilizes the skyrmion lattice both at zero and non-zero temperatures.
Our study reveals the complex underlying mechanisms of topological-charge creation and annihilation in thin
magnetic films with perpendicular magnetic anisotropy, and both isotropic (bulk) and anisotropic
(interfacial)?Dzyaloshinskii-Moriya interaction. We have found that upon the application of an external field to ideal
infinite films the skyrmion-lattice phase undergoes a first-order phase transition, either to a topologically
trivial ferromagnetic state or to an inverted skyrmion-lattice phase via the transient formation of antiskyrmions.
The first-order character of both phase transitions in the ideal lattice is due to a delocalization of the
topological charge within a 2? domain-wall network and the consequent collective response to an external field. In the
presence of even a single defect, however, the skyrmion-lattice phase becomes unstable and collapses gradually
through a defect-induced melting process, where the defect site acts as a topological-charge sink. These findings
emphasize the importance of imperfections in materials and their implications on the stability of topologically
non-trivial spin textures, and demonstrate that the consideration of defects is paramount for the analysis of
experimental data. This provides a basis for a much wider scope of experiments on skyrmion lattices, particularly
concerning the development of materials for skyrmion-based devices.
We have performed high-resolution micromagnetic simulations to investigate in detail the phase transition of
skyrmion lattices. We have studied ultrathin films where the skyrmion-lattice phase is stable at temperatures low
enough for our micromagnetic simulations to be valid, as in most cases we did not consider finite-temperature
The total energy density F consists of: (i) ferromagnetic exchange; (ii) perpendicular magnetic anisotropy;
(iii) isotropic (bulk) or anisotropic (interfacial) Dzyaloshinskii-Moriya interaction; (iv) Zeeman coupling to an
external magnetic field; and (v) dipole-dipole interactions:
F = A(? ? m)2 + Dm ? (? ? m) ? Ku(mz)2 ? ?0Msm ? Hext ?
m ? Hdemag ,
where m = M/Ms is the magnetization unit vector with?M the magnetization and Ms the saturation
magnetization, A is the exchange stiffness, D is the strength of the DMI (either bulk or interfacial), Ku is the first-order
uniaxial anisotropy constant, Hext is the external magnetic field, and Hdemag is the local demagnetizing field due to
dipole-dipole interactions. The z-component of the magnetization is perpendicular to the film plane.
We computed the magnetic state by solving the Landau?Lifshitz?Gilbert (LLG) equation of motion:
?tm = ??(m ? Heff) + ?(m ? ?tm),
where ? is the electron gyromagnetic ratio, ? is the dimensionless damping parameter, and Heff = ??mF/?0Ms is
the effective magnetic field in the material consisting of external and internal magnetic fields, which depend on
the material parameters. The LLG simulations have been done with mumax336, a finite-difference GPU-based
For both isotropic and anisotropic DMI systems we have considered a wide range of material parameters that
are valid for many real materials23,37?40: D = 1.0?2.0 mJ/m2 and Ku = 20?800 kJ/m3. The values of A = 8.78 pJ/m
and Ms = 385 kA/m were also taken from real materials, e.g. FeGe37,40, and the damping parameter was defined
as ? = 0.1. While A and Ms were fixed throughout the study, the DMI and PMA strength were varied to obtain
more detailed insight into the physics of skyrmion lattices and the related effects of energetics.
The thin films were discretized in a 480 ? 480 ? 2 mesh (sample dimensions 832nm ? 960 nm ? 4 nm) with
periodic boundary conditions imposed on the operators such as the exchange interactions in x- and y-directions,
which are reflected by the magnetization of the sample. Additionally, different cell sizes (always less than half the
exchange length11 ?ex = 2Aex /?0Ms2 ? 10 nm) were used to verify the numerical stability of the simulations.
The external magnetic field was always applied and swept perpendicular to the film plane, and the quantities
recorded at each field step were: m, M, Q, the total energy density of the system and the individual contributions
Finite-temperature effects were added by aplying a randomly-fluctuating magnetic field Hthem defined as:
Htherm = ?(step)
where ? is a random vector from a standard normal distribution whose value is changed with every time step, kB the
Boltzmann constant, T the temperature, Bsat the magnetic induction, V the cell volume and t = 10?13 s the time step.
All datasets generated and/or analyzed in this study are available from the corresponding authors on request.
They are not made publicly available online due to their memory size.
The authors gratefully acknowledge funding from the Swiss National Science Foundation (Grant No. 200021?
172934) and thank the Royal Society International Exchanges programme (Ref: IE161506).
L.P., M.C. and C.M. conceived the study, and L.P. and M.C. designed the simulations. L.P. performed the
micromagnetic studies with contributions from Y.L. L.P., M.C., C.M. and J.F.L. analyzed and discussed the results
and wrote the manuscript. M.C. and J.F.L. coordinated and supervised the work.
Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-018-34526-0.
Competing Interests: The authors declare no competing interests.
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