observation of room-temperature magnetic skyrmions and ... · 1 supplementary information...

1

Supplementary Information

Observation of room temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic

ferromagnets

Seonghoon Woo,1 Kai Litzius,2, 3 Benjamin Krüger,2 Mi-Young Im,4,5 Lucas Caretta,1 Kornel Richter,2 Maxwell Mann,1 Andrea Krone,2 Robert M. Reeve,2 Markus Weigand,6 Parnika Agrawal,1 Ivan Lemesh,1 Mohamad-Assaad Mawass,2,6 Peter Fischer,7,8 Mathias

Kläui,2,3* Geoffrey S. D. Beach1*

1Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Institut für Physik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany 3Graduate School of Excellence Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany 4Center for X-ray Optics, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA 5Daegu Gyeongbuk Institute of Science and Technology, Daegu 711-873, Korea 6Max Planck Institute for Intelligent Systems, 70569 Stuttgart, Germany 7Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA 8Department of Physics, University of California, Santa Cruz, California 94056, USA S1. Effective medium model of multilayer stack.

The micromagnetic simulation of multilayer structures can be very demanding in

terms of computational resources since a large number of simulation cells in the thickness

direction must be used. For this reason we simulate the system as an effective medium

where the full film is treated as a single homogeneous magnetic layer, which is apt for

strongly magnetostatically coupled layers as used here. We derive here a list of scaling

laws that must be fulfilled by the effective medium in order to reproduce the static and

Observation of room-temperature magneticskyrmions and their current-driven dynamics in

ultrathin metallic ferromagnets

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2

dynamic behaviors of a multilayer stack. We then show that the effective medium model

agrees quantitatively with full three-dimensional micromagnetic simulations that

explicitely compute the dipolar interlayer interactions in the multilayer stack.

S1.1 Scaling laws for effective medium model.

In the following a variable with a prime denotes a quantity in the effective medium.

Figure S1. Sketch of the layers in the experimental film and in the effective medium. The

experimental film consists of layers made of a ferromagnet (FM) and two heavy metals

(HM1 and HM2). The effective medium consists of an effective ferromagnetic material

only.

The experimental film can be described by two thicknesses: the thickness mt of one

magnetic layer and the rt of one repetition. That is, moving by a distance rt

perpendicularly to the film plane we find the same material as shown in Fig. S1. In the

effective medium the thickness of the magnetic layer mt is the same as rr tt = since

the film of the effective medium should be homogeneously magnetic.


3

The magnetic moment of the film should be the same in the effective medium and the

experimentally used film. We find

.= msms tMtM (1)

The time evolution of the magnetization M

is given by the Landau-Lifshitz-Gilbert

(LLG) equation

,)(1)(1

= 22 HMMM

HMdt

Md

s

(2)

with the gyromagnetic ratio , the effective magnetic field H

, the Gilbert damping ,

and the saturation magnetization sM . We assume that ),,( iyxM

depends on the

position ),( yx parallel to the film and the repetition i only. With the normalized

magnetization sMMm /=

this equation reads

.)(1)(1

= 22 HmmHmdt

md

(3)

For the effective medium the LLG equation reads

.)(1)(1

= 22 HmmHmdt

md

(4)

The effective medium should show the same dynamics of the magnetization as the

experimental film. We therefore obtain mm = . By comparing equations (3) and (4) we

find HH

= .

The magnetic field is proportional to the functional derivative of the energy E of the

system. The field explicitly reads

,1=0 m

E

MH

s

(5)


4

where 0 is the permeability of vacuum. For an energy density that depends only on

the magnetization and its first derivatives with respect to space the effective magnetic

field can be written as

.1=0 sM

z

mzy

myx

mxmH

(6)

From this we find

ss MM

=

(7)

Employing equation (1) one obtains

.== mmr ttt (8)

Thus, the energy density has to scale inversely with the thickness of the magnetic layer.

We now insert all types of energies present in the system into equation (8). For the

energy of the external field extH

we find

.'='= 000 mextsrextsmexts tHmMtHmMtHmM (9)

Thus the external field acting on the effective medium is the same as the experimental

field. From the energy density of the exchange energy

,=

=

222

222

222

z

m

y

m

x

mtA

z

m

y

m

x

mtA

z

m

y

m

x

mAt

r

r

m

(10)


5

we find the same scaling for the exchange constant A . This constant scales as

rm tAtA /= . For the strength of the DMI we find

.== zzrzzrzzm mmmmtDmmmmtDmmmmDt

(11)

This yields the scaling rm tDtD /= for the strength D of the DMI.

The scaling of the dipolar interaction is the most complex one, due to its long range

character. However, we can show that, in the limit of small rt , the above scaling of the

saturation magnetization leads to the correct behavior of the dipolar energy. The energy

of the dipolar interaction can be written as

0M s2

8 dS1 dS2i=0

N

j=0

N

0

t m dz1 0

t m dz2

m(x1, y1, i)

m(x2, y2, j)

| r |33[r m(x1, y1, i)][

rm(x2, y2, j)]

| r |5

.

(12)

Here, we integrate over the plane S of the film and the position z within the current

magnetic film that is the distance between the position and the lower boundary of the

magnetic film. The sums are over all repetitions of the film layers, where the

magnetization ),,( iyxm depends on the positions x and y in the film plane.

Perpendicular to the film we assume that the magnetization depends on the number of the

repetition i only with no dependence on z . The distance r between the two positions is

given by

.)()()()(= 212121 zzryx ezzetjieyyexxr

(13)

In the limit of a thin magnetic layer the absolute value can be expressed as

.)()()(|| 22221

221 rtjiyyxxr

(14)

This approximation becomes imprecise for the interaction of two points that are closer

than roughly the thickness of the magnetic layer. However, at these short distances the


6

interaction will be dominated by the exchange interaction and DMI, justifying our

approximation.

Using the abbreviations ),,(= 111 iyxmm and ),,(= 222 jyxmm

equation (12) reads

.

||))()()2()((

3||

||)))(((

3

||)))(((

3

||)))(((

3

||)(

3||

||))((

3

||)))(((

3

||))((

3

||)(

3||

8

5

22121

222,1,

32,1,

521212,1,

521212,1,

521212,1,

5

2212,1,

32,1,

521212,1,

521212,1,

521212,1,

5

2212,1,

32,1,

20100=0=

21

20

r

zzzztjitjimm

r

mmr

zztjiyymmr

zztjixxmmr

zztjiyymmr

yymm

r

mmr

yyxxmmr

zztjixxmmr

yyxxmmr

xxmm

r

mm

dzdzdSdSM

rrzzzz

ryz

rxz

rzy

yyyy

xy

rzx

yx

xxxx

mtmtN

j

N

i

s

(15)

There are terms of different order in 21 zz . Since this is the only dependence on 1z

and 2z we can easily calculate the integrals over 1z and 2z . For nzz )( 21 we get

.)(1)(=)(1)(=)( 212010121020212010

nnmtmtnnmtmtnmtmtzzdzdzzzdzdzzzdzdz (16)

From this we find

0.=)()1)((1 212010

nmtmtn zzdzdz (17)


7

Thus all terms with odd powers of 21 zz vanish in equation (17). The only term that is

quadratic in 21 zz is small compared to the respective term that is constant in 21 zz .

Finally, the only terms that are left are terms that are constant in 21 zz . Equation

(15) can thus be written as

.||

)(3

||

||))((

3

||))((

3

||))((

3

||)(

3||

||))((

3

||))((

3

||))((

3

||)(

3||

8

5

222,1,

32,1,

5212,1,

5212,1,

5212,1,

5

2212,1,

32,1,

521212,1,

5212,1,

521212,1,

5

2212,1,

32,1,

0=0=21

220

r

tjimm

r

mmr

tjiyymmr

tjixxmmr

tjiyymmr

yymm

r

mmr

yyxxmmr

tjixxmmr

yyxxmmr

xxmm

r

mm

dSdStM

rzzzz

ryz

rxz

rzy

yyyy

xy

rzx

yx

xxxx

N

j

N

i

ms

(18)

In vector notation this becomes

,||

)],,()][,,([3

||),,(),,(

8 52211

32211

0=0=21

220

a

aa

a

N

j

N

i

ms

r

jyxmriyxmr

r

jyxmiyxmdSdS

tM

(19)

with the distance

.)()()(= 2121 zryxa etjieyyexxr

(20)


8

This energy remains constant when changing from the experimental to the effective film

description. Thus, the energy density scales inversely with the thickness of the magnetic

layer, as required.

For small distances we assume the the system is exchange dominated and the

magnetization at both points is the same. The dipolar interaction of the z components

zm of the magnetization then reads

.||

)(3||8 5

221

2

3

2

201021

20

r

zzm

r

mdzdzdSdS

M zzmtmts

(21)

This can be written as

,)(

)3(

)(

18 5

221

2

221

32

212

201021

220

zzr

zz

zzrdzdzdSdS

mM

ii

mtmtzs

(22)

with the in plane distance 221

221 )()(= yyxxr i . Carrying out the integration over

2z one finds

.)(

)(8 3

21

2

13

21

2

11021

220

zr

z

tzr

tzdzdSdS

mM

imi

mmtzs

(23)

The integration over 1z yields

.114 22221

220

mii

zs

trrdSdS

mM

(24)

The remaining in-plane integrations can be written in center of mass and relative

coordinates. This reads


9

,114 22

2

00

220

miiii

Rzs

trrdrdrdYdX

mM

(25)

with )/2(= 21 xxX and )/2(= 21 yyY . Here, we wrote the integration over the relative

coordinates in polar coordinates. This expression is valid if the distance to the edge of the

film is larger than R . It can be written as

,12 220

220

mi

ii

Rzs

tr

rdrS

mM (26)

where S is the area of the film. Carrying out the last integration one gets

.2

2222

0mm

zs ttRRSmM

(27)

For thin films we can use a series expansion in Rt m/ . This reads

.22

2220

R

ttS

mM mm

zs (28)

The second term scales like the above result for points with a large distance. Here, the

energy remains constant during the scaling. In contrast, the first term exhibits a different

scaling where the energy does not remain constant. However, this term can be included in

the uniaxial anisotropy as follows.

The anisotropy energy reads

.2mz StKm (29)

One can see that this energy has the same dependence on the magnetization as the first

term in equation (28). The sum of both terms can be written as

mzeff StmK 2 (30)


10

where we defined an effective anisotropy

2

=2

0 seff

MKK

(31)

From equation (8) we find for the scaling of this effective anisotropy

rzeffrzeffmzeff tmKtmKtmK 2'22 '='= (32)

and thus rmeffeff ttKK /=' . The effective anisotropy constant effK must therefore be

scaled inversely with the thickness.

The spin Hall effect was modeled by a Slonczewski-like field that can be written in

the explicit LLG as

)()(1

= 20

SH

pmpmmadt

mdj

(33)

Here, is the damping parameter, Jz eep

= the normalized cross product of z- and

current-direction, 0 the gyromagnetic ratio and 02

=

deM

Ja

S

Hj

the spin Hall

parameter. This parameter again is composed of the spin Hall angle H , the electron

charge e, the current density J and the effective thickness of the material d .

The latter has to be chosen correctly to represent the scaling of our system. While J

should stay constant in all layers, H , d and SM have to represent the thickness of the

magnetic material. As the scaling of SM has already been derived above, d and H

have to be defined so that the total ja fulfills this criterion. Let repHH n =' the spin


11

Hall angle times the number of repetitions, than we find rr

mm t

t

ttd ==

1

. So the final

equation reads

02

=

rS

Hrepj teM

Jna

(34)

From these calculations we can conclude that the effective medium exhibits the same

magnetization dynamics as the experimental multilayer if we have the following scaling

r

m

m

m

eff

eff

s

s

t

t

t

t

D

D

K

K

A

A

M

M=

'====

' (35)

while all other quantities remain the same.

S1.2 Micromagnetic simulations comparing effective medium model and full three-

dimensional model.

Here we compare the results of the effective medium model to those of a full

micromagnetic simulation that treats explicitely the dipolar interactions between the

individual layers. For the full three-dimensional multilayer simulation, we used

parameters representative of the experimental sample, with 15 Co layers (thickness

tm=0.9 nm), separated by nonmagnetic layers (thickness tnm=7.2 nm). The simulation cell

size was 4nm 4nm in the x-y plane, and 0.9 nm in the thickness direction. The value of

tnm in the simulation was slightly larger than the experimental value (7.0 nm) to allow for

an integral number of cells along the thickness direction. The material parameters for the

Co layer in the simulation are: Ms = 6105 A/m, A = 1.010-11 J/m, and Ku = 3.76105

J/m3, which corresponds to the measured hard-axis anisotropy field 0Hk = 0.5 T.


12

Figures S2(a),(b) show the domain structure for a 400 nm diameter disk, after

relaxing from an initial state comprised of four parallel stripe domains. The simulations

used D = 1.5 mJ/m2; for D = 0, only the uniformly magnetized state is stable, indicating

that the dipolar interactions alone are insufficient to create a multidomain state.

The image in Fig. S2(a) corresponds to the middle Co layer in full 3D simulation.

The same domain structure is found in all 15 Co layers due to the strong magnetostatic

coupling between layers. Figure S2(b) shows the corresponding result for a 2D

micromagnetic simulation of the same multilayer, treated as a single uniform layer where

all material parameters have been scaled according to Eq. 35. The domain structures and

average domain widths are nearly identical for the two simulations.

Figure S2. Micromagnetically computed domain structure in a 400 nm diameter disk for

a 15-repeat magnetic multilayer using a full 3D simulation (a) and a 2D simulation under

the effective medium approximation (b). Grayscale image in (c) shows the difference in

the two images.

Figure S2(c) shows a grayscale image depicting the difference in the two simulated

domain images, showing that variations in the computed domain wall positions between


13

the full 3D model and the 2D model are on the order of the domain wall thickness, and

much smaller than the experimental resolution.

We conclude that the effective medium model accurately captures all energy terms,

including magnetostatic (dipole) interaction energies in the multilayer structure, and

allows for accurate 2D micromagnetic simulations of the domain structures in multilayer

films without the need to explicitly model the individual layers.

S2. Determination of domain wall energy and DMI constant from domain width

data.

S2.1 Analytical treatment using uniform medium approximation

Figure 1C of the main text shows the field dependence of the widths d and d of up

(parallel to Bz) and down (antiparallel to Bz) domains in a continuous Pt/Co/Ta multilayer

film. We approximate the multilayer film as a single layer with uniform average

properties1,2, as justified in Section S1. In this case the domain wall (DW) surface energy

density DW can be determined from these data using well-known domain spacing

model3-5. For periodic stripe domains in a film of thickness t and saturation

magnetization Ms, the low-field domain period ddd is given by3-5

1

32

2

20

/2exp/2111nodds

DW dntdntnt

d

tM

. (36)

At high field the antiparallel domains approach a terminal width min,d given by5

2min,

2min,

2min,2

0

/1ln//1ln2 tdtdtd

tMs

DW

. (37)


14

Taking d 480 nm and min,d 100 nm (estimated from MTXM images), Eqs. (36) and

(37) yield DW = 1.50.2 mJ/m2 and DW = 1.10.2 mJ/m2, respectively, normalized to

the nominal Co cross section. The value DW = 1.30.2 mJ/m2 reported in the main text

represents the mean of these two values.

As described in the main text, we estimate the DMI constant |D| using the relation

DAK effuDW ,4 , with A the exchange stiffness and effuK , the effective uniaxial

anisotropy constant. We find |D| 1.30.2 mJ/m2, assuming A = 110-11 J/m.

One can compare |D| to the critical value |cD | required to stabilize homochiral Néel

DWs6,

dc KD 4|| . (38)

Here, effuKA ,/ is the DW width and dK is the magnetostatic anisotropy energy

density of the DW, which prefers a Bloch configuration. dK can be written in terms of

the demagnetizing factor Nx along the DW normal, 2/20 sxd MNK . The calculated

DW width 8 nm is much smaller than the film thickness, so that the thick-film limit

applies7 and hence 1xN . Thus, /2|| 20 sc MD 0.2 mJ/m2, so that cDD and

homochiral Néel textures are expected.

We note that although the estimated value of |D| depends on the assumed value of

A, the conclusion cDD is very robust. If A is increased by a factor of ten one finds

cDD / increases from 8 to 9. Decreasing A by a factor of ten gives cDD / 4.


15

S2.2 Comparison to full analytical model of multilayer including dipolar interlayer

coupling.

In the analysis of the previous section, we have treated the multilayer film as a

single uniform magnetic layer, with all energy terms, including the dipolar (magnetostatic)

energy, scaled by a factor rm tt / , as described in Section S.1. The calculations in Section

S.1 show that this effective medium model properly accounts for all energy terms, in

particular the dipolar interaction energy between individual magnetic layers, as long as

rt (the repeat thickness) is small.

Draaisma and de Jonge8 have analytically computed the domain structure and

hysteresis loop characteristics in multilayer films by explicitly including the dipolar

coupling energy between the discrete magnetic layers separated by nonmagnetic

interlayers. As shown by Draaisma and de Jonge8 the explicit multilayer solution for

domain spacing converges to the Koey and Enz3-5 solution for a single layer film (with

effective-medium scaled material parameters), except for the range 10/1 nmt ,

where tnm is the nonmagnetic spacer thickness, and 2// 20 sDW M is a critical

lengthscale. The value of DW obtained in the previous subsection using the effective

medium approximation gives 5.0/ nmt . Hence, our treatment of the multilayer film

as a single uniform magnetic layer with volume-averaged properties is analytically well-

justified, consistent with the numerical results in Section S.1.

S2.3 Micromagnetic calculations of domain spacing to determine DMI strength.


16

The analytical results for the value of the DMI in the prior subsection were

confirmed using full micromagnetic simulations of the domain spacing as a function of

DMI. Here, the equilibrium stripe labyrinth spacing in a disk with a diameter of 2 µm

was simulated for |D| ranging from 0.5 mJ/m2 to 4.0 mJ/m2. From the results in Figure S3

it can be seen that the width of the observed stripe domains decreases with increasing

DMI. For a DMI strength |D| of 1.25-1.5 mJ/m2 we find a good agreement between the

simulated and the measured magnetization pattern. The |D| obtained by the

micromagnetic treatment agrees well with the result |D| 1.30.2 mJ/m2 obtained

through the analytical domain spacing models above.

Figure S3. Simulation of the labyrinth domain spacing in Pt/Co/Ta as a function of DMI.

(a) The determined spacing for different magnitudes of DMI constant |D| ranging from

0.5 to 4 mJ/m2. (b) The results of the micromagnetic simulations showing a varying stripe

domain width for different values of the DMI.

S3. Field-driven domain expansion measurements.


17

In order to compare the pinning strengths in Pt/CoFeB/MgO and Pt/Co/Ta and to

verify the existence of DMI-stabilized Néel domain walls in these materials, magnetic

bubble domain expansion was studied using a wide-field polar magneto-optical Kerr

effect (MOKE) microscope. Here we examined both Pt(3nm)/Co(0.9nm)/Ta(4nm) and

Pt(4.5nm)/CoFeB(0.7nm)/MgO(2nm).

Figure S4 shows the domain wall velocity versus out-of-plane field Bz. for

Pt/Co/Ta and Pt/CoFeB/MgO films. The data follow the creep law

Tk

Evv

B

aexp0

with the activation energy scaling as 4/1 za BE , expected for thermally-activated

motion in a two-dimensional disorder potential9. The creep velocity in Pt/CoFeB/MgO is

significantly higher than in Pt/Co/Ta, confirming much weaker disorder in the former, as

expected due to the amorphous nature of the CoFeB layer.

Figure S4. Domain wall creep velocity versus out-of-plane field, showing creep scaling,

for Pt/CoFeB/MgO and Pt/Co/Ta single-layer films.


18

Figure S5. Up-oriented (magnetized along +z) bubble domain expansion under out-of-

plane field Bz, for (a),(b) Pt/Co/Ta and (c),(d) Pt/CoFeB/MgO single-layer films. Panels

(a),(c) show circular domain expansion when no in-plane field is applied. Figures (b),(d)

show asymmetric domain expansion when Hx=-2000 Oe is applied, oriented to the left.

The arrow schematics in panels (a), (c) show the domain wall moment orientations

consistent with the observed asymmetry of the domain expansion under in-plane field

application. The circled dots indicate the domain magnetization direction and are

positioned at the domain nucleation point.

Figure S5 shows representative MOKE images of bubble domains nucleated and

expanded by an out-of-plane field Bz in the creep regime. It has been shown previously

that magnetic bubble domains bounded by DMI-induced Néel domain walls expand


19

asymmetrically under in-plane field application10,11. This is due to the increased

(decreased) domain wall energy where the in-plane field is antiparallel (parallel) to the

domain wall magnetization, leading to slower (faster) creep velocity along the in-plane

field axis. As seen in Fig. S5, in the absence of an in-plane field, the domains expand

circularly in both Pt/Co/Ta and Pt/CoFeB/MgO, as expected. With an in-plane field Bx =

-200 mT applied, a clear asymmetry in the expansion is observed. The same asymmetry

is observed for both Pt/Co/Ta and Pt/CoFeB/MgO, indicating that in both cases the

domain walls are Néel with the same chirality. For the “up” domains shown, the

asymmetric expansion implies radially-inward pointing magnetic moments within the

domain wall, shown schematically in Figs. S5(a),(c), corresponding to left-handed Néel

domain walls, as expected from the Pt interface present in both sample stacks. This

chirality is the same as inferred from the direction of spin Hall current-driven motion of

skyrmions in the main text.

S4. Calculation of magnetic field profile from microcoil.

The current density in the microcoil was calculated by solving Poisson's equation

on a rectangular mesh using a finite differences approach, with a cell size 5 nm in the

lateral directions and 100 nm in the perpendicular direction. The geometry (Fig. S6 has

an inner radius of 1.5 µm, an outer radius of 2.5 µm, an opening angle of 140°, a

thickness of 100 nm, and the straight wire on the left and right side has a width of 1 µm.

The current density distribution is shown in Fig. S6 for a current of 0.06 A,

corresponding to the maximum magnitude of current in the coil at Vpp = 10 V


20

Figure S6. Simulation of the microcoil. (a) current flow through the coil visualized by

the local current density. (b) out of plane field Bz generated by the current flow. The field

in the center of the coil is about +10 mT and the maximum field that was applied at a

position where part of the sample is located is about +60 mT. Simulations correspond

Vpp= 10 V.

The Oersted field generated by this current density is calculated by using the

MicroMagnum software and is shown in Fig. S6. For the calculation of the field, a cube

with a length of 15 µm, a width of 5 µm, and a height of 100 nm with the same mesh as

for the calculation of the current density was used. Current that flows outside this region

is neglected as its contribution to the Oersted field at the position of the sample is small.

S5. Skyrmion size versus applied magnetic field.

The skyrmion diameter in these materials is not fixed, but depends on out-of-

plane field12, as well as geometrical constraints and interactions (dipolar and topological)


21

with neighboring skyrmions. In Fig. 3 of the main text, it is seen that several skyrmion

lattice periodicities can be generated in a geometrically-constrained disk, and that the

skyrmion size depends on the number of skyrmions in the disk. We performed

micromagnetic simulations to verify that several skyrmion lattice configurations are

metastable in a disk, using the experimental materials parameters for Pt/Co/Ta. Fig. S7

shows two relaxed states of a 2 m diameter disk containing the same periodicities as

observed in Fig. 3 of the main text. We find that both a 3x3 skyrmion lattice (Fig. S7 (a))

and a 4x4 skyrmion lattice (Fig. S7 (b)) constitute metastable states.

Figure S7. Micromagnetic simulations of metastable skyrmion lattice states in a disk and

corresponding experimental images. Panels (a), (b) show STXM images of skyrmion

lattices in a 2 m diameter Pt/Co/Ta disk, reproduced from main text Figs. 3(a) and 3(d).

Panels (c), (d) show simulations of metastable skyrmion lattices in a 2 m diameter

Pt/Co/Ta disk that reproduce the experimentally-observed periodicities.


22

Due to repulsive interactions between skyrmions, the skyrmion size decreases when more

skyrmions are present in the disk, and the skyrmion size varies depending on the local

coordination, so that multiple sizes can be found in a given disk, as observed

experimentally (Fig. 3 of main text).

The skyrmion size also depends on applied out-of-plane field. Although the

skyrmions in the main text have diameters ~200-250 nm, this diameter can be

significantly reduced by an applied field oriented antiparallel to the core magnetization.

We measured the field-dependence of the skyrmion diameter for a skyrmion in a 2 m

diameter Pt/Co/Ta disk (Fig. S8(a)). Skyrmions were generated using bipolar field pulses

after initializing the disk into a stripe domain pattern, as described in the main text. The

out-of-plane field Bz was then incremented, antiparallel to the skyrmion magnetization,

and STXM images acquired at each field step. With increasing Bz, the skyrmions

decrease in size, and eventually annihilate. We find a range of annihilation fields, which

is consistent with a local variation of material parameters that also leads to the variation

in wall velocities and pinning that we observe experimentally and in the micromagnetic

simulations in Subsection S6.

We have determined the field-dependence of the skyrmion diameter for the

skyrmion with the largest annihilation field in Fig. S8(a). The diameter was determined

by fitting linescans of the XMCD signal across the STXM image, averaging 4 adjacent

linescans for improved signal to noise. The data were fitted to a Gaussian, and the full

width at half maximum (FWHM) was used to determine the skyrmion diameter after

deconvolution to account for the 90nm FWHM of the STXM beam profile.


23

Figure S8 Field-dependence of skyrmion diameter. (a) STXM images of skyrmions in a

2 m diameter Pt/Co/Ta disk, for several values of out-of-plane field Bz antiparallel to the

skyrmion core magnetization. (b) Linescan of XMCD signal across the skyrmion outlined

in (a), with Gaussian fit. (c) Diameter of the skyrmion outlined in (a) versus Bz. Line is

guide to the eye.


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Figure S8(b) shows a representative linescan for the skyrmion indicated by the

dotted circle in Fig. S8(a), measured at a field of Bz = 7 mT. The skyrmion diameter is

plotted versus Bz in Fig. S8(c). It can be seen that the low-field skyrmion diameter of

~200-250 nm is decreased to <50nm at high fields. This diameter is comparable to the

skyrmion sizes computed micromagnetically for thin films in Ref. 12, as well as the

skyrmion size in bulk helimagnets such as FeCoSi13 with a skyrmion diameter of ~90 nm,

and depending on the materials properties smaller skyrmions can be achieved. We note

that the results are consistent with results from Ref. 14 where a similar dependence of the

skyrmion diameter on the field was found.

S6. Micromagnetic simulations of current-driven skyrmion dynamics.

S6.1: Skyrmion motion due to spin currents including acceleration and deceleration

In this section we present simulations of current-driven motion of magnetic

skyrmions in a track with special focus on the acceleration and deceleration period when

the current is switched on and switched off. As shown in figure S9, the skyrmions move

during these periods at a certain angle with respect to the longitudinal axis of the track.

This angle depends on the used material, the applied magnetic field and current density

and its absolute value is similar for acceleration and deceleration. The latter marks a

major difference between the dynamics of a classical magnetic vortex domain wall and a

skyrmion and can significantly increase the longitudinal distance a skyrmion travels due

to a current pulse. This does not allow for the experimental determination of the


25

equilibrium velocity of a skyrmion on the track, since it is not possible to measure the

angle and duration of speed up or slow down period by static imaging as used here.

Comparing the distance a skyrmion travels in our simulations before reaching its

maximum velocity (about 100 nm) and the experimentally measured displacements for a

single pulse (about 50 nm - 200 nm; averaged), it is very likely that the latter does not

reflect the equilibrium velocities, resulting in a discrepancy of the absolute velocities as

discussed also further below.

The track was modelled using a 3000 x 300 nm2 grid made of an effective

medium (see section S1). In this track, a Neél skyrmion was placed and relaxed to its

ground state, stabilized by an out-of-plane bias field Bz = -20 mT. A current pulse was

applied (current density j = 41010 A/m2, pulse length 21 ns), leading to a Slonczewski-

like spin Hall torque that moves the skyrmion along the track. For opposite chiralities, the

skyrmions move in opposite directions showing that the DMI sign governs the motion

direction. The experimental observation that skyrmions are displaced along conventional

current flow direction (main text Figs. 4,5) corresponds to the case of left-handed

chirality (D<0).

The snapshots (Fig. S9) show that initially the skyrmion moves not only along the

current flow but also towards the edge, where the skyrmion is repelled due to its

topological nature from the DMI. After switching off the current, the relaxation process

moves the skyrmion back to the track center, however the skyrmion also continues to

move in the current flow direction during this process.


26

Figure S9. (a) Simulated trajectory of a skymrion driven along a 300 nm wide magnetic

track by a current pulse flowing along the +x direction. Here, trajectories are shown for

Neél skyrmions with both chiralities (D > 0 and D < 0), which move in opposite

directions. As the skymrion approaches the edge of the track (+/- 150 nm), it is repelled

and follows along the edge. Solid symbols denote the period during which current is on,

and open symbols denote period after current is turned off. j denotes charge current.

Acceleration and deceleration periods result in straight trajectories towards the edge,


27

respectively back towards the wire centre. (b), (c) The path of a skyrmion moving in a

track for the case D < 0, for (b) j>0 and (c) j<0. During acceleration, the skyrmion

travels to the edge of the track, then moves along the edge before relaxing back into the

middle of the track as soon as the current pulse ends. Note the decrease in skyrmion

diameter as it approaches the edge of the track (d) Plot of equilibrium velocity (while

moving along the edge of the track) as a function of current density.

S6.2: Influence of disorder-induced magnetic property variations

As observed in the experiment, we find non-constant velocities and therefore we

also examined the influence of variations of the magnetic properties on the dynamics of

the skyrmions in a magnetic track. While the exact micro-structure of the multilayer stack

and the resulting variations in the magnetic properties is not known, the most likely

parameters to vary are those that rely on the spin orbit interaction at the interfaces that

exhibit atomic scale variations. These were modelled by random spatial fluctuations of

the anisotropy and DMI constant with a cell size of 2 2 nm2, as well as by an

exponentially damped cos(x) cos(y) pattern of different length scales around the initial

position of the skyrmion. The exact function used was

∙ 1 exp cos 2 cos 2 (39)

Here, ∈ 0, 1 is the pinning strength, the coordinates of the starting position of the

skyrmions and the length scale of the pinning.


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Figure S10. Skyrmion velocity at a current density j = 31010 A/m2 in the presence of

disorder, plotted as a function of the amplitude of random variation of anisotropy

constant (K) and DMI parameter (D).

Figure S10 shows the average velocity as a function of disorder strength,

computed separately for the case of random variations in K and random variations in

DMI. Here the driving current density was fixed at j = 31010 A/m2. We found no

significant changes of dynamics due to a varying anisotropy (up to 100% variation),

which is in good agreement with other studies15. The simulations of a random variation in

DMI yield however a very strong impact on the dynamics. We use a random fluctuation,

equally applied to the DMI vector components Dx and Dy. Already a random fluctuation

of just 15% at a cell size of 2 nm influences notably the dynamics of a skyrmion moving

in the magnetic track due to the spin orbit torque. The skyrmion follows a path of lowest

energy, resulting in a random walk around the zero-defect trajectory, and we find that a


29

skyrmion becomes completely pinned for fluctuations of >20%. Taking into

consideration the high sensitivity of the DMI to the interface structure in the multilayer

system of our samples, we expect a significant impact due to DMI variations, which can

thus explain finite critical current and lower velocities seen in our measurements as

compared to the expected velocity for simulations in a perfect sample. We note that these

simulations are performed at zero temperature; in the case that thermal fluctuations at

finite temperature are included, the pinning transition in Fig. S10 would correspond to a

transition from flow to thermally-activated creep motion, which explains the low

experimental velocities in the vicinity of the depinning transition.

S6.3: Current-induced annihilation of pinned skyrmions

In Figs. 4,5 of the main text, we note that pinned skyrmions can be annihilated by

current injection. In addition, as seen in the STXM images in Fig. 4, a mobile skyrmion

can coalesce with a pinned one, becoming pinned itself, when driven by current. We

have shown in the prior sections that short-lengthscale disorder plays an important role in

skyrmion dynamics. In this section, we present micromagnetic simulations that

demonstrate that strong pinning can also lead to the experimentally-observed annihilation

of skyrmions by current.

The simulations here were carried out by using material parameters appropriate

for Pt/Co/Ta, in a rectangular track with 3000x300 nm2 using a cell size of 2x2 nm2. The

skyrmions were stabilized by an out-f-plane field Bz = -20 mT (antiparallel to the

skyrmion core magnetization). Figure S11 shows series of snapshots of current-driven


30

skyrmion motion in different scenarios, with the dc current density and time step between

frames indicated at the bottom of each image sequence.

Figures S11a,b examine the behaviour when a current-driven skyrmion

approaches a pinned skyrmion, in analogy to the experimental observation in Fig. 4 of the

main text. Here, the skyrmion was pinned by a single cell with fixed magnetization,

which assured that the skyrmion remained strongly fixed in position. At lower current

densities (Fig. S11(a)) a current-driven skyrmion moving in the vicinity of the pinned

skyrmion is repelled and passes around the pinned skyrmion. However, at higher current

densities, the driving force from the current can be large enough to overcome skyrmion-

skyrmion repulsion. In this case, the approaching skyrmion and the pinned skyrmion first

begin to shrink in size as they are driven closer together. The mobile skyrmion then

abruptly merges with the pinned skyrmion, after which only a single skyrmion remains,

pinned at the same site as was the original pinned skyrmion. This behavior reproduces

qualitatively the experimental behavior seen in Fig. 4(b) of the main text. We note that

similar results were also obtained analytically and numerically in Ref. 16.

We also examined current-induced annihilation of individual skyrmions, and find

that random disorder can facilitate such annihilation. In Fig. S11(c), a random variation

(30%) of the DMI constant, fluctuating on the scale of the cell size, was used in order to

simulate pinning as discussed in the previous subsection (see also Fig. S10). In the image

sequence in Fig. 11(c), an initialized skyrmion driven by current becomes pinned in a

region with locally lower DMI constant, which reduces its stability. The skyrmion then

decreases slightly in size, and eventually annihilates within a few ns. This behavior


31

matches qualitatively the current-induced annihilation of pinned skyrmions observed in

Fig. 5 of the main text for pinned skyrmions in Pt/Co/Ta.

Figure S11 Annihilation of pinned skyrmions by current. (a) Snapshots of the behavior of

a current-driven skyrmion approaching a strongly-pinned skyrmion at relatively low

current density. (b) Same situation as in (a), but at a higher current density, where the


32

propagating skyrmion merges with and becomes pinned at the same location as the

original pinned skyrmion. (c) Current-induced annihilation of a pinned skyrmion, in the

case of random fluctuations of the DMI constant on the length scale of the cell size. The

current density and time step between frames is indicated below each image sequence.

The arrows denote the direction of motion of the skyrmion, and the lengths of the arrows

are scaled approximately with the skyrmion velocity.

In conclusion, we find that short-lengthscale fluctuations in interfacial energy

terms, which would be expected in sputtered multilayer films with the atomically-

disordered interfaces, can lead to a reduction in current-driven skyrmion velocities,

pinning, and current-induced merging and annihilation, in qualitative agreement with

experiments.

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7. Mougin, A., Cormier, M., Adam, J. P., Metaxas, P. J. & Ferré, J. Domain wall mobility, stability and Walker breakdown in magnetic nanowires. Europhys. Lett. 78, 57007 (2007). 8. Draaisma, H. J. G. & de Jong, W. J. M. Magnetization curves of Pd/Co multilayers with perpendicular anisotropy. J. Appl. Phys. 62, 3318 (1987). 9. Metaxas, P. et al., Creep and flow regimes of magnetic domain-wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy. Phys. Rev. Lett. 99, 217208 (2007). 10. Je, S.-G. et al. Asymmetric magnetic domain-wall motion by the Dzyaloshinskii-Moriya interaction. Physical Review B 88, 214401 (2013). 11. Hrabec, A. et al. Measuring and tailoring the Dzyaloshinskii-Moriya interaction in perpendicularly magnetized thin films. Physical Review B 90, 020402 (2014). 12. Sampaio, J., Cros, V., Rohart, S., Thiaville, A. & Fert, A. Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures. Nature Nanotechnology 8, 839-844 (2013). 13. Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901-904 (2010). 14. Moreau-Luchaire, C. et al. Skyrmions at room temperature: From magnetic thin films to magnetic multilayers. Preprint at http://arxiv.org/abs/1502.07853 (2015). 15. Büttner, F. et al. Dynamics and inertia of skyrmionic spin structures. Nat. Phys. 11, 225 (2015). 16. Müller, J. & Rosch, A. Capturing of a magnetic skyrmion with a hole. Physical Review B 91, 054410 (2015).


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