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Instability of skyrmions in magnetic fields
Loic Mougel,1 Patrick M. Buhl,2 Ryohei Nemoto,3 Timofey Balashov,1 Marie Hervé,1, 4 Julian Skolaut,1 Toyo
Kazu Yamada,3, 5 Bertrand Dupé,6, 7 and Wulf Wulfhekel1
1)
Physikalisches Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe,
Germany2)
Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
3)
Department of Materials Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522,
Japan
4)
Institut des Nanosciences de Paris, Sorbonne University, CNRS-UMR7588, 75005 Paris,
France5)
Molecular Chirality Research Centre, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522,
Japan
6)
Fonds de la Recherche Scientifique (FNRS), B-1000 Bruxelles, Belgium
7)
Nanomat/Q-mat/CESAM, Université de Liège, B-4000 Sart Tilman, Belgium(Dated: June 16, 2020)

In this combined experimental and theoretical work, we report on the evolution of the skyrmion radius andtheir destruction in the system Co/Ru(0001), when an out-of-plane magnetic field is applied. At low fields,
skyrmions are metastable and display an elliptical instability in which along the short axis, the spin texture
approaches that of the spin-spiral phase and the long axis expands in order to go back to the spin-spiralground state. At high fields, we observe round skyrmions of finite size up to the collapse field Bc , where
they are destroyed and the topological charge is annihilated. We estimate Bc via numerical methods basedon magnetization dynamics simulations parametrized by density functional theory (DFT) calculations andcompare it to experimental scanning tunneling microscopy (STM) observations obtained at ≈30 mK.
Magnetic skyrmions are localized topologically protected structures, in which the magnetization field winds
around a magnetization core, such that no continuoustransformation allows the conversion into a topologicallytrivial, homogeneously magnetized ferromagnetic (FM)state1–3 . The energetic stability of the skyrmions originates from the unique sense of rotation of the magnetization due to the chiral Dzyaloshinskii-Moriya interaction(DMI) which occurs when structural inversion symmetryis broken and spin-orbit coupling is present4 . Due to thisprotection, skyrmions are considered as magnetic quasiparticles and are discussed as ideal information carriersin spintronics5–12 .
When a magnetic field is applied in the opposite direction to the magnetic moment of the skyrmion center, the Zeeman energy leads to a compression of theskyrmion13–15 . When the magnetic field reaches a critical value Bc , the core of the skyrmion will reverse its
magnetization and align to the FM background. Thismechanism is called the collapse of the skyrmion. In thecontinuum limit, the collapse of skyrmions occurs when
their radii shrink down to zero16 , while on a finite latticeof magnetic atoms, the radius reaches a finite value before the collapse14,16 . Skyrmion collapse with its abruptchange of the topology is associated with an energetic
barrier and is usually studied as being thermally activated within the framework of transition state theory17 .
Transition state theory requires the exploration of highenergy states, the so-called saddle points, via the nudged
elastic band (NEB)18 . In magnetism, NEB has been
extended to take into account the constant size of themagnetic moments and is known as the geodesic nudged
elastic band (GNEB)19,20 . This method has been used

extensively to study the thermally activated probability of skyrmion collapse in strong ferromagnets such as
Co/Pt(111)14 and in ultrathin magnetic films such as
Pd/Fe/Ir(111)21–23 . Recently, more approaches for thermally activated skyrmion collapse have been proposed
via Langevin magnetization dynamics simulation24 .
Although the thermally activated transition of single
skyrmions to a ferromagnetic state has been extensively
studied, the exploration of the transition by ramping up
the external magnetic field at low temperatures — which
involves the exploration of the saddle point — remains
unexplored. This low temperature transition is particularly important in spin-polarized STM experiments
which are done at cryogenic temperatures on ultra-thinfilms25,26 . In Pd/Fe/Ir(111) the transition between theskyrmion lattice phase and the FM phase as a function of
the external magnetic field has been measured at 4 K13,25 .
The skyrmion radius decreases as the magnetic field increases. However, in Pd/Fe/Ir(111), the large value of
Bc did not allow the exploration of the saddle point13 .
Compared to the experiment, the saddle point configuration is easy to explore in calculations since the magnetic
configuration can be constrained. At this point, the magnetic moments are expected to align in-plane to create avortex whose size is only several unit cells20 . This vortex
unwinds the skyrmion and cancels the topological stabilization. Recently, the saddle point configurations were
explored theoretically and it was found that even whenthe magnetic interactions are described in the continuum
limit, a skyrmion of finite size could be expected as thesaddle point configuration27 .
Here, we study the skyrmion collapse field Bc of isolated skyrmions in Co islands of single atomic thickness

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2
on Ru(0001). In a previous work26 , we have combined
spin-polarized STM measurements and DFT calculations
to explore the ground state properties of Co/Ru(0001)
and showed that skyrmions were stabilized by a vanishing effective anisotropy in combination with a small DMI.
This particular ratio of magnetic interactions makes
Co/Ru(0001) a prototypical soft ferromagnet in whichisolated skyrmions have been stabilized at down to zeroexternal magnetic fields. We experimentally determine
Bc for skyrmion annihilation at minimal thermal activation using a dilution refrigerated STM with T ≈30 mK28 .
We numerically estimate Bc based on ab-initio parameters and find that it agrees well with experimental observations.
The ground state of Co/Ru(0001) in zero external magnetic field is a spin spiral of unique rotational sense anda periodicity of 37 nm26 . When a magnetic field B isapplied normal to the surface, the spin spiral successively transforms into isolated skyrmions with radii depending on the external magnetic field. In our previous work, the spin structures were studied using tunneling anisotropic magneto-resistance (TAMR) and tunneling magneto-resistance (TMR). Here, we exclusively
used the TAMR signal to determine the shape of theskyrmions to avoid magnetic dipole interactions between
the tip and the sample and tracked the skyrmion dimensions as a function of magnetic field as shown Fig. 1.
Note that in these maps of the differential conductivity
dI/dV recorded at -350 mV, areas of an in-plane magnetization appear bright (higher dI/dV ) opposed to areas with an out-of-plane magnetization appearing black(lower dI/dV )26,29,30 . Although the Ru substrate becomes superconducting below a critical temperature of≈500 mK, all measurements presented were obtained inmagnetic fields above the superconducting upper criticalfield of ≈5 mT, i.e. in the normal state of the substrate31 .
In our experiments, skyrmions were formed first applying 250 mT. Then, the B was reduced to the remanenceof the superconducting magnet (≈7 mT) and metastableskyrmions remained. After this formation step, B was
increased again to the indicated magnitudes to study theevolution of the skyrmion dimensions as function of B.
For fields below Be =230 mT, the skyrmions are foundto be not round but elongated as seen in the left inset
of Fig. 1. This is due to their tendency to reform thespin-spiral state at low fields via an instability against
elongation. Thus, we determined the largest and smallest
axis of the elongated skyrmion indicated in orange andgreen lines in the left inset and open circles in the figure.
At fields higher than 230 mT, the skyrmions are roundobjects (see right inset of Fig. 1). The dI/dV signal varies
as cos2 (θ), with θ the azimutal angle. The function used
to fit the radius of the skyrmion is then32 :cos2 π σρ if ρ ≤ σ,(1)
dI/dV =
0 otherwise
where ρ is the distance from the center of the skyrmion,and σ is the radius of the skyrmion. The distance from

Figure 1. Theoretical (blue filled circles) values of σ/2 ofthe skyrmion versus applied magnetic field in Co/Ru(0001)
together with the experimental values for round skyrmions
B > Be ≈ 230 mT as illustrated in the right inset. At B < Be ,the skyrmions are elongated and both the radii along the long(orange) and short (green) axis are plotted as illustrated in
the left inset. The black dashed curve is a fit to a/B. At
fields in the shaded green area, annihilation of skyrmions was
experimentally observed. The insets are differential conductance maps (white level proportional to dI/dV )of 130x130 nm, V = −350 mV, I = 2 nA, modulation: U =60 mV. left inset: B = 0 mT, right inset: B = 400 mT.

the center of the skyrmion to the maximum brightness inthe dI/dV maps thus corresponds to half the skyrmion
radius. We fitted the dI/dV contrast as shown in theright inset of Fig. 1 via eq. (1) to obtain the skyrmion
radii for different magnetic fields. The radii of the roundskyrmions as a function of the magnetic field are shown as
red dots in Fig. 1. At lower fields, the scatter in the data
is explained by the mentioned elliptical instability andthe skyrmions’ sensitivity to the near spin-spiral phase.
As the field increases, these effects become marginal andthe radius of the different skyrmions converges to thesame value.
At higher field, analytical models predict the radius ofisolated skyrmions to scale as 1/B 33,34 as indicated by a
fit to the high field experimental data (black dashed line).
At fields below Be , this relation necessarily breaks down,
as the ground state is not a single, infinitely extended
skyrmion35 but the spin-spiral phase. Note that for theelongated skyrmions, the short axis half radius saturates
around 12 nm corresponding to a local periodicity of thespin-spiral of ≈ 48 nm, i.e. at similar dimensions as theperiodicity of the spin spiral26 . The long axis expands
further in order to transform into the spin-spiral phase.
Due to the neighboring spin spirals in the sample andthe finite sample size, the long axis, however, does notdiverge to infinity. In other words a kinetic barrier togo back to the spin-spiral phase allows for metastableskyrmions at vanishing fields. At fields larger than Be
the round skyrmions are compressed and the radii keep
decreasing but stay finite until B = 700 mT. At this field,
the collapse region is reached and individual skyrmionsdisappear by annihilating the topological charge (and do
not reappear upon lowering the field again). When thefield reaches B = 800 mT all skyrmions have disappeared
which shows that the collapse field Bc was reached at
Bc = 750±50 mT. As all measurements are made at
T ≈30 mK, this collapse is not due to thermal activation.
Moreover, during ramping the field, the tip was retracted
such that an influence of the tunneling current on theskyrmion collapse was excluded.
The experimental measurements were compared withatomistic simulations utilizing the Spirit code36 which
describes the energy of the magnetic texture as:This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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Figure 2. Energy density distribution with respect to the FM
state at B = 750 mT. (a) and (b): The skyrmion was relaxedwith an easy axis in-plane anisotropy of 13 µeV and without
DDI: (a) MAE contribution and (b) DDI contribution. (c)skyrmion relaxed with both DDI and an easy axis anisotropy
out-of-plane of 60 µeV. (d), (e) and (f): radial energy density
distribution showing the MAE (d), the DDI (e) and bothcontribution (f). The integral of (a) and (c) are of comparable
magnitude with 13.8 meV and 11.8 meV, respectively.

E=−

X

Jij Mi Mj −

ij

−

X

Dij (Mi × Mj )ij

Ms M i B −

i

−

X

X

κM2i,zi

X µ0 M 2 Mi Mj rij2 − (Mi rij ) (Mj rij )
s

ij

4π

rij5

,

(2)

where Jij is the magnetic exchange interaction, Dij isthe Dzyaloshinskii-Moriya vector, κ is the magnetocrystalline anisotropy (MAE) and B is the external magnetic
field. The magnetic interaction parameters, obtained by
DFT in a previous publication26 , are J1 = 13.1 meV,
D1 = 0.2 meV, κ = −0.013 meV and Ms = 1.8 µB . Using
these parameters, the experimental setup was modelled
by a periodic superlattice containing 1000×1000 magnetic moments on a hexagonal lattice with the Ru lattice
parameter of a = 0.27 nm. For each magnetic field H an
isolated skyrmion was relaxed to its minimum of energy
at which the skyrmion radius was fitted with eq. (1).
As in Ref.[30], the radius of the skyrmion without thedipole-dipole interaction (DDI) was calculated as a function of the field (blue dots in Fig. 1). Compared to theexperiment, the radius decreases more steeply with increasing field but follows the same trend and matches wellat higher fields. This discrepancy at lower fields is to be
expected as the experimental skyrmions are more densely
packed which leads to additional skyrmion compression
compared to the simulations. The atomistic simulations
predict a collapse field of Bc = 820±10 mT which is ingood agreement with the experimental value. Considering that all theoretical parameters were obtained from
DFT, the agreement between the atomistic simulations
and the experiments is extremely good. This is particularly remarkable as tiny modifications of the anisotropy
on the scale of the DFT-accuracy result in drastic modifications of the skyrmion radii and Bc in this fragile regimeof atomistic parameters.
Additionally including the often neglected DDI demonstrates the delicate balance of atomistic parameters inthis soft magnet. First, the DDI was evaluated in amagnetic configuration that was minimized neglecting it(Fig. 2 a,b,d,e). The energy gain of the skyrmion compared to the ferromagnetic state through the DDI is morethan 5 times the gain of the MAE. Nevertheless, qualitatively both distributions do not differ much except for

Figure 3. GNEB calculation for B = 0.55 mT (blue) and B =0.75 mT (orange) with zoom-in between 485 and 545 meV.
The right inset shows the magnetic field dependence of the
GNEB energy barrier. The four left insets depict the distributions of the z-component of the magnetization including
zooms of the saddle-point core.

the non-vanishing energy gain at the core in case of the
DDI. This is consistent with the common 2D interpretation of the DDI as shape anisotropy which can be absorbed into an effective magnetic anisotropy37 . Due tothe relatively large magnitude of the DDI in this fragile
system, the magnetic ground state changes drasticallywhen the DDI is included. The background magnetization tilts in-plane in the experimentally investigatedfield range and skyrmions in an out-of-plane backgroundare only viable in a small region around 1.5T. In orderto regain the previous agreement with experiment, theMAE as smallest parameter with large theoretical uncertainty and similar impact was tuned to κ =0.06 meV
which approximately counters the energy gain throughthe DDI. While the shape of the energy gain from the effective anisotropy shifts towards the core when including
the DDI and modifying κ (Fig. 2 c,f), the previous dependence of the skyrmion radius on the field is recovered(not shown). Hence, the experimental results can also
be described when including DDI, although a reasonable
but arbitrary modification to the MAE is necessary.
Since DDI is not required to study the stability ofskyrmions in ultra-thin films, we have explored the energy barrier without explicitly considering DDI in thevicinity of Bc via GNEB simulations19 on a 400 × 400
grid as shown in Fig. 3. The simulations were initializedwith the relaxed skyrmion profile for a given magnetic
field. Along the reaction path, the skyrmion shrinks until it reaches the saddle point which corresponds to the
maximum of energy. The energy maximizes as the half
radius reaches about 2 nm after which the core reverses
and the ferromagnetic state is quickly attained (towards
the right side of the figure). In that respect, even at thesaddle point, we confirm numerically that in soft ferromagnets a finite radius skyrmion solution exists as predicted in Ref.[27].
When the external magnetic field is increased, the energy barrier decreases drastically (right inset of Fig. 3)
reaching 0.5 meV at 800 mT. For all the magnetic fields,
the skyrmion solution at the saddle point has a finite radius. The initial flat section of the energy evolution easily
allows variation of the skyrmion radius in the vicinity of
Bc while the very small barrier at Bc prohibits direct observation of the skyrmion collapse. Still, with the minimal experimental skyrmion half radius of ≈3.1 nm thedirect vicinity of the collapse is observed to a good approximation and all further compression leads to vanishing of the barrier (or tunneling) on timescales too small
for observation.
To conclude, the collapse path of a skyrmion was stud-This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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4
ied in the prototypical soft ferromagnet Co/Ru(0001). As
expected, the radius of the skyrmion decreases monotonically as the magnetic field increases up to the collapse
field Bc . We have reproduced the experimental skyrmion
radii and computed Bc based on atomistic simulation parameterized by DFT calculation. Both experimental andcomputational method confirm that its radius does notreach zero even at the saddle point and that the DDI canbe correctly approximated by an effective anisotropy in
2D ultra-thin films.
The data that support the findings of this study areavailable from the corresponding author upon reasonable
request.
B. Dupé and P. Buhl acknowledge useful discussion with J.V. Kim and U. Ritzmann. W. Wulfhekel
and L. Mougel acknowledge funding by the Deutsche
Forschungsgemeinschaft (DFG) under the grant WU
349/15-1 and WU 349/16-1. B. Dupé and P. Buhl acknowledge funding by the DFG under the grant DU
1489/3-1.

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collapse region

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EMAE

5

position [nm]
0

(b)

5

EDDI/4

hard axis
relaxed without DDI
hard axis
relaxed without DDI

(d)
(e)
(f)

EDDI
EDDI

EMAE

5

position [nm]055EMAE 50sum
0
50

position [nm]05Esky EFM [ eV](a)5easy axis
relaxed with DDI

Esky EFM [ eV](c) EMAE + EDDI
0

10

15

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700

600

500

400

300

100

0
10

Ebarrier[meV]

60 meV

800

30

20

00.5 0.6 0.7 0.8

200

H[T]

B=0.55T
B=0.75T
normalized GNEB reaction coordinate