How to reveal metastable skyrmionic spin structures by spin-polarized scanning tunneling microscopy

We predict the occurrence of metastable skyrmionic spin structures such as antiskyrmions and higher-order skyrmions in ultra-thin transition-metal films at surfaces using Monte Carlo simulations based on a spin Hamiltonian parametrized from density functional theory calculations. We show that such spin structures will appear with a similar contrast in spin-polarized scanning tunneling microscopy (SP-STM) images. Both skyrmions and antiskyrmions display a circular shape for out-of-plane magnetized tips and a two-lobe butterfly contrast for in-plane tips. An unambiguous distinction can be achieved by rotating the tip magnetization direction without requiring the information of all components of the magnetization.


I. INTRODUCTION
Magnetic skyrmions were recently observed via neutron diffraction in bulk chiral magnets such as MnSi 1 and in the multiferroic material Cu 2 OSeO 3 2 . Currently, they are attracting an enormous attention due to their stability 3,4 and their displacement speed upon applying electrical currents which makes them suitable for technological applications 5,6 . Real space observation of skyrmions in FeGe and Fe 0.5 Co 0.5 Si thin films has become possible using Lorentz microscopy and magnetic force microscopy 7-10 and more recently in transition-metal films using spin-polarized low-energy electron microscopy 11 and magneto-optical Kerr effect (MOKE) measurements 12 . In ultra-thin films of a few monolayers, the skyrmion diameter can shrink down to a few nanometers and spinpolarized scanning tunneling microscopy (SP-STM) 13,14 is a powerful tool for their observation and manipulation [15][16][17] .
SP-STM is sensitive to the projection of the local magnetization density of states of the sample onto the magnetization direction of the tip 18 and does not allow a direct determination of the three magnetization components in a single measurement. However, in most experimental setups it is not possible to continuously rotate the tip magnetization direction and conclusions have to be drawn from SP-STM experiments performed with only one or two tip magnetization directions.
Such measurements only allow a partial determination of the spin structure 16,19,20 . It is therefore essential to know (i) if the skyrmion ground states and some metastable states can be differentiated via simple SP-STM experiments (based on one or two tip magnetization directions) and (ii) to establish a clear proposal in order to discriminate between the different possible chiral spin structures via SP-STM.

II. METHODS
The occurrence of metastable skyrmionic spin structures is studied in a single atomic layer of Pd in fcc stacking on the fcc monolayer Fe on the Ir(111) surface denoted as Pd(fcc)/Fe/Ir(111). This system has been studied experimentally using SP-STM 16,17 and from first-principles calculations 21,22 which allow to understand the transition from a spin spiral to a skyrmion and a ferromagnetic (FM) phase in an external magnetic field. We numerically solve the spin Hamiltonian using Monte-Carlo (MC) simulations with parameters obtained from density functional theory calculations 21 : with exchange constants J ij , the vector D ij of the Dzyaloshinskii-Moriya (DM) interaction, the magnetocrystalline anisotropy K and an external magnetic field 23 .
We have obtained metastable states in MC simulations by relaxing a super cell of 100×100 spins on a two-dimensional hexagonal lattice starting from a random spin configuration at 1 K under a magnetic field of 20 T with a standard Metropolis algorithm. At this field value we are in the region where the skyrmions are metastable in the ferromagnetic background 21 . (c) Skyrmion density of the spin structure (color contrast). The spin structure was obtained with a super cell of 100×100 spins on a hexagonal lattice at a temperature of 1 K after a relaxation with 10 7 MC relaxation starting from a random spin configuration at a magnetic field of B = 20 T, i.e. in the region of the phase diagram in which the ferromagnetic state is the ground state.
We have simulated SP-STM images of the spin structures obtained from MC using the model described in Ref. 24 . The tunneling current is given by where R T is the tip position, the sum extends over all surface atoms α, the vacuum tail of a spherical atomic wave function is approximated by h(r) = exp (−2κ|r|), and the decay constant is given by κ = 2mφ/h 2 with the work function φ. P S and P T denote the spin-polarization of sample and tip atoms, respectively, and θ α is the angle of the magnetization of atom α with respect to the tip magnetization direction m T . Figure 1(a) shows a simulated SP-STM image with an out-of-plane magnetized tip (P eff = P T P S = 0.4) of the spin structure at z = 8Å from the surface. The image shows a brighter contrast for the FM background with several black spots. All darker spots have a round shape and could correspond to skyrmion spin structures. However, when the tip magnetization is changed from out-of-plane to in-plane ( Fig. 1(b)), the simulated SP-STM image shows two types of contrast compatible with recent observation of skyrmion 17 . The first contrast has a two-lobe pattern with one brighter and one darker side. The lobes can be aligned along the x axis or are rotated with respect to it. The second type of contrast has four lobes and appears seldom. It does not seem to have a preferred alignment.

III. RESULTS
The topological character of different spin structures is given by their winding or skyrmion number: where m is the unit vector of the local magnetization and S can take only integer values. identified in Fig. 1 in separate panels. Fig. 2(a) shows a right-handed skyrmion i.e. S = 1 which is metastable for Pd(fcc)/Fe/Ir(111) at magnetic field values higher than 16 T 21 . For completeness, we also consider a left-handed skyrmion, Fig. 2(b), which has a skyrmion number of S = 1 as well but exhibits an opposite chirality and is unstable. Fig. 2(c) shows an antiskyrmion (S = −1) that is characterized by a change of chirality for two high symmetry directions, i.e. the rotational sense changes from right-to left-handed every 90 • . Fig. 2(d) displays a higher-order antiskyrmion with S = −2 which was recently also reported in Ref. 26 . In that case, the rotational sense changes every 60 • . in this imaging mode. When the tip magnetization changes to in-plane, the images of the righthanded skyrmion (Fig.3(b)), the left-handed skyrmion (Fig. 3(d)) and the antiskyrmion (Fig. 3(f)) 6 are still very similar. The simulated SP-STM images of a skyrmion and an antiskyrmion could only differ by a rotation as seen Fig. 1(b). The only spin structure that can be easily distinguished is the higher-order skyrmion due to the multiple nodes of the contrast (cf. Fig. 3(h)). Note, that the simulated SP-STM images of the right-handed skyrmion for both magnetization directions are in good agreement with the experiments of Romming et al. 16,17 .  Fig 3), we obtain line profiles which are also in quantitative agreement with experimental data 17  When the tip magnetization is switched to in-plane, the line profiles show the same behavior for the right-handed skyrmion, the left-handed skyrmion and the antiskyrmion (Fig. 4(a-c)). Since the contrast of the SP-STM images of the skyrmion and antiskyrmion are only rotated with respect to each other and the corrugation amplitudes are very similar, they can only be distinguished in experiments when both spin structures are present simultaneously. On the other hand, the higherorder skyrmion (Fig. 4(d)) can be easily discriminated due to the presence of multiple nodes of the magnetization density also seen in Fig. 3(h).

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In order to distinguish between the skyrmion and the antiskyrmion spin structures, we propose an experiment based on a 3D vector field available in STM experiments 27 . Such a field enables a rotation of the tip magnetization both within the surface plane as well as from in-plane to out-ofplane. Although all in-plane magnetized tips result in the same contrast i.e. a butterfly with a bright and a dark lobe (as shown in Fig. 3), the behavior of this contrast when the tip changes its direction is different as shown in Fig. 5. For a right-handed skyrmion, the lobes will rotate in phase with the tip magnetization direction (thick black arrows). In the case of an antiskyrmion, a clockwise rotation of the in-plane component of the tip will induce a counterclockwise rotation of the lobes.
Therefore, in-plane rotation of the tip magnetization allows an unambiguous distinction between the right-handed skyrmion and the antiskyrmion. On the other hand, in order to distinguish a left-and right-handed skyrmion the tip magnetization must be rotated from the in-plane to the out-of-plane direction.

IV. CONCLUSION
In conclusion, we have demonstrated that it is non-trivial to distinguish via SP-STM between metastable spin structures at surfaces that differ by their chirality and/or topological charge.
Skyrmions and antiskyrmions exhibit a spherical shape in SP-STM using tips with an out-of-