Supplementary material for "Field-dependent Size and Shape of Single Magnetic

Skyrmions"

Niklas Romming,

Andr Kubetzka, Christian Hanneken, Kirsten von Bergmann, and Roland Wiesendanger

Department of Physics, University of Hamburg, 20355 Hamburg, Germany

(Dated: March 5, 2015)

METHODS

In SP-STM experiments the tunnel current contains

structural, electronic, and magnetic information [1].

Throughout this work, the magnetic contribution to the

SP-STM images was calculated by subtracting an ap-

propriate reference image. For Fig. 1(c),(d), this is an

image of the same area at the same magnetic eld, with-

out skyrmions present, i.e. ferromagnetic alignment of

the spins [2]. The canted tip direction was taken into

account by a subsequent z shift. For Fig. 2, Fig. S2 and

Supplementary Movie 1, the reference image was calcu-

lated as the mean between the frames recorded at highest

positive and negative magnetic elds, i.e. an image with-

out magnetic contribution.

To account for the polar angle T of the tip magneti-zation, the t to 1D height proles (z) was done using

z(x) = C cos (T (x, c, w)) , (S1)

where C is a scaling factor and is given by Eq. (1). Forthe 2D t to SP-STM images, Eq. (2) results in:

z(x, y) = C ( sin (, c, w) sin T

(cosT x

+ sinT y

)+ cos (, c, w) cos T

), (S2)

where T is the azimuthal angle of the tip. For the anal-ysis of dI/dU signals z can be substituted by dI/dU .The saturation magnetization MS was estimated byusing values for the atomic magnetic moments from DFT

calculations [3] and assuming a uniform distribution of

the magnetization in the volume V of the PdFe bilayer:

MS =

V=

3B32 a

2 t, (S3)

where a = 2.715 and t = 4.08 are the DFT nearest-neighbor distance and bilayer thickness, respectively.

Given that the magnetic moment from DFT is correct

and that it is homogeneously distributed over the PdFe

bilayer, the error bars for A, D, K can be estimated inthe following way. Both the standard deviations of the

experimental values of c(B) and w(B) and their scat-tering around the tted curves are within less than 5 %[4]. The error bars of A, D, K were obtained by apply-ing the tting procedure to four sets of c(B) and w(B),which were articially increased or decreased by 5 %.The maximal deviation of the material parameters dur-

ing these ts was chosen as the reasonable error bar for

each parameter A, D and K.The micromagnetic simulations [5, 6] were performed

on a disc with a diameter of 50 nm and a thickness of

4.08, using a lateral cell size of 1 1. A centralskyrmion was relaxed at every eld step with a Runge-

Kutta evolver at T = 0 K.

nromming@physnet.uni-hamburg.de

[1] R. Wiesendanger, Rev. Mod. Phys. 81, 1495 (2009).

[2] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel,

B. Wolter, K. von Bergmann, A. Kubetzka, and

R. Wiesendanger, Science 341, 636 (2013).

[3] B. Dupe, M. Homann, C. Paillard, and S. Heinze, Nat.

Commun. 5, 4030 (2014).

[4] For |Bz| > 2.5T, a small jiggling of the skyrmion positionat a fast time scale could be the reason for a systematic

overestimation of w in the SP-STM-images. Due to thelarger error bar of these values, they only have a small

inuence on the error weighted tting procedure.

[5] S. Rohart and A. Thiaville, Phys. Rev. B 88, 184422

(2013).

[6] http://math.nist.gov/oommf, release 1.2a5.

[7] A. Bogdanov and A. Hubert, Phys. Status Solidi B 186,

527 (1994).

[8] A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138,

255 (1994).

[9] H.-B. Braun, Phys. Rev. B 50, 16485 (1994).

[10] A. Kubetzka, O. Pietzsch, M. Bode, and R. Wiesendan-

ger, Phys. Rev. B 67, 020401 (2003).

2FIG. S1. Comparison of proposed skyrmion prole with numerical calculations. (a) Circles: magnetization proles of an

isolated skyrmion for dierent values of piD/(4AK

)in zero magnetic eld (graphically extracted from ref. [7]), solid red

lines: ts with Eq. (1). Arrow points to a deviation of analytical and numerical prole for piD/(4AK

)= 0.5. (b) Circles:

magnetization proles for zero eective anisotropy for dierent magnetic elds (graphically extracted from ref. [8]), solid red

lines: ts with Eq. (1). Insets show original graphs from refs. [7, 8].

FIG. S2. Additional validation of material parameters A, D, K. (a)-(c) Magnetic contribution to SP-STM dI/dU images atadditional eld values. The magnetic eld is ramped from B = 3T to B = +3T during data acquisition (SupplementaryMovie 1 shows full data set). The qualitative dierences of (a) and (c) demonstrate strong hysteresis of the PdFe layer at

T = 4.2K. (d)-(f) Black circles: proles along rectangles in (a)-(c), respectively. Red lines in (d), (f): ts with Eq. (1). Fromthe set of A, D, K, obtained from the evolution of the size and shape of skyrmions with an external magnetic eld, it ispossible to analytically calculate the values c and w for the 360 domain wall prole [9, 10]. Note the good agreement betweencalculated and measured values c and w. Solid line in (e) shows a t with a sine curve of period 6.6 nm. The zero eld spinspiral period (e) agrees with the expected value [5] for A, D, K determined from the skyrmion evolution in higher magneticelds.

Supplementary material for "Field-dependent Size and Shape of Single Magnetic Skyrmions"MethodsReferences