Isolated zero field sub-10 nm skyrmions in ultrathin Co films

Due to their exceptional topological and dynamical properties magnetic skyrmions - localized stable spin structures on the nanometre scale - show great promise for future spintronic applications. To become technologically competitive, isolated skyrmions with diameters below 10 nm that are stable at zero magnetic field and room temperature are desired. Despite finding skyrmions in a wide spectrum of materials, the quest for a material with these envisioned properties is still ongoing. Here we report zero field isolated skyrmions with diameters smaller than 5 nm coexisting with 1 nm thin domain walls in Rh/Co atomic bilayers on the Ir(111) surface. These spin structures are characterized by spin-polarized scanning tunnelling microscopy and can also be detected using non-spin-polarized tips due to a large non-collinear magnetoresistance. We demonstrate that sub-10 nm skyrmions are stabilised in these ferromagnetic Co films at zero field due to strong frustration of exchange interaction, together with Dzyaloshinskii-Moriya interaction and a large magnetocrystalline anisotropy.

. Rh grows on the bare Ir(111) both as isolated islands and from the Co step edges. For the Rh/Ir areas a two-stage / signal is found, see blue and brown circles, which correspond to the two possible stackings. In one Rh-stacking the islands have irregular edges and a more dendritic shape (indicated by light blue circles in the Figure), while in the other Rh-stacking the edges are more regular and the island shape is more compact (brown circles in the Figure). Because the former stacking connects smoothly to the fcc-Co/Ir(111) we assign it to fcc-Rh/Ir(111). The other Rh/Ir(111) areas are thus in hcp stacking. Because the hcp-Rh/Ir(111) connects smoothly to one type of Rh/Co islands on the adjacent lower Ir terrace (indicated by green dots in the figure), we conclude that this type of island is the hcp-Rh/fcc-Co/Ir(111). Correspondingly, the fcc-Rh/Ir connects smoothly to the fcc-Rh/Co areas on an adjacent lower Ir terrace (red dots).

Supplementary Figure 2 | Lateral variations of topography and /
signal. a, STM topography colourized with height signal and b, STM topography colourized with spin-resolved / map; the same area is shown in Fig. 2b (top) in the main text. The positions of the domain walls are visible in the topography due to the non-collinear magnetoresistance effect. Also, within the magnetic domains of the Rh/Co areas the topography exhibits lateral variations of the apparent height on the order of about 7 pm. We attribute this inhomogeneity to some intermixing of the two materials (measurement parameters: U = -400 mV, I = 800 pA, B = 0 T, T = 4.2 K, Cr-bulk tip).

Supplementary Figure 3 | Field dependence of domain walls and skyrmions in Rhhcp/Co/Ir(111)
. Spin-resolved differential conductance maps of a sample area with several hcp-stacked Rh islands on Co/Ir(111) (measurement parameters: U = -250 mV, I = 800 pA, T = 4.2 K, Cr-bulk tip). a, In zero magnetic field several domain walls are imaged via the NCMR; in addition, there is an isolated skyrmion in the bottom left corner (see box). b, Upon the application of an out-of-plane magnetic field of 1.5 T the majority of the domain walls are annihilated, while the skyrmion seems unaffected. c, After the field is brought back to zero, some domain wall movement is observed.
Supplementary Figure 4 | Simulated TMR and NCMR signals: line sections across a skyrmion. a, Atomic spin configuration of a (right-rotating) skyrmion, using w = 1 nm and c = 2 nm. b,c,d, Simulated TMR line sections across the skyrmion for a magnetic tip with a magnetization direction of 0°, 45° and 90° relative to the out-of-plane direction (see insets) using = 0.1. Here, the height signal is shown, which is nevertheless equivalent to a current or ⁄ signal as it is calculated from the simulated constant height current in linear approximation. Panel b and c qualitatively reflect the line sections in Fig. 2c, as there the NCMR signal is small. e, Simulated NCMR line section across the skyrmion for

Supplementary Note 1: Details on the STM simulations.
The magnetic skyrmions observed in the ⁄ maps of Fig. 1 are well reproduced by simulated STM images. Input for such a simulation is the magnetic state of the sample, which is modelled by a 2D arrangement of spins located at the atomic positions . The signal, i.e. the current or the ⁄ intensity, is assumed to decay exponentially with distance from each atom. Then, for a given tip-sample distance 0 , the signal for each point of the STM image is calculated as the sum of the contributions from all spins [3] within a certain distance from the tip position , with a typical cut-off distance of 3 0 . The resulting ⁄ signal at the tip position is then: ℏ 2 is the decay constant and is the sample work function. indicates the set of spin indices within the cut-off distance. ( ⁄ ) contains the bias voltage dependent contributions to the differential conductance from the non-spin-polarized collinear magnetic state ( ⁄ ) 0 , the tunnel magnetoresistance (TMR), and the non-collinear magnetoresistance (NCMR): where is the tip magnetization direction, and and are the pre-factors determining the strength of TMR and NCMR, respectively. Here is the spin-polarization at the particular bias voltage and the signal intensity scales with the projection of tip and local sample magnetization due to the TMR.
is an empirical prefactor to capture the NCMR contribution, which is modelled to be proportional to ̅, the average angle between the direction of the spin and those of its six nearest neighbours [4,5]. Examples of simulated profiles across a skyrmion for each of these two contributions and their combination is shown in Supplementary Figure 4.
The domain walls were modelled according to a standard 180° domain wall, cos = tanh( /2 ), of width w, the magnetic skyrmions were modelled by a circular standard 180° domain wall of width w and radius c [1]. The skyrmion diameter is defined as the distance between opposite in-plane magnetizations. The work function was set to = 4.8 eV and the tip sample distance was 0 = 0.8 nm.
For the simulations of the two oppositely magnetized skyrmions in Fig. 2

Supplementary Note 2: Determination of magnetic interaction parameters via DFT.
We obtain the exchange constants of the Heisenberg model by calculating the energy dispersion of homogeneous, flat spin spirals via DFT as implemented in the FLEUR code [9,10]. Spin spirals can be described by the spin spiral vector determining the propagation direction of the spiral and the angle between two adjacent spins along the spiral. The -vector is a vector in reciprocal space and is chosen along the high symmetry directions of the hexagonal Brillouin zone (BZ): ̅ − ̅ − ̅ . The magnetic moment of atom at position is given by = [cos( ⋅ ) sin , sin( ⋅ ) sin , cos ]. Here, is the opening angle for conical spin spirals and in the case of flat spin spirals it is 90°.
Neglecting spin-orbit coupling (SOC), left and right rotating spin spirals are energetically degenerate. Since there is no preferred magnetization direction without SOC, the generalized Bloch theorem can be applied to self-consistently calculate the energy differences between different spin spiral states ( ≠ 0) and the ferromagnetic state (FM state, = 0) within the chemical unit cell. The resulting energy dispersion ( ) is mapped to the Heisenberg model = is the unit vector of the magnetic moment of atom .
In Fig. 4a of the main text, the energy dispersion close to the ̅ -point is very flat. At the same time the energy difference between the FM ( ̅ ), the row-wise antiferromagnetic ( ̅ ) and the Néel ( ̅ ) state (with angles of 120° between neighbouring moments) is large, which requires a fit including ten neighbours to describe the behaviour of the full energy dispersion. The resulting values for 1 … 10 are presented in Supplementary Tables 1 and 2 for Rhfcc/Co/Ir(111) and Rhhcp/Co/Ir(111), respectively.
When an energy dispersion behaves like 2 for → 0 as in the standard micromagnetic model, it is possible to fit this range with an effective nearest neighbour interaction eff . However, in Rh/Co/Ir(111) the resulting eff varies significantly with the fitting region around the FM state between 1 and 10 meV/Co atom. In order to get a reasonable value for eff , we evaluate domain wall widths from spindynamics simulations where we included all atomistic magnetic parameters beyond nearest neighbours of Supplementary Tables 1 and 2. We determine the effective nearest neighbour exchange interaction eff = 2 /6 2 [11], where is the uniaxial magnetocrystalline anisotropy and the lattice constant of the Ir(111) plane, which completes the magnetic parameters in Supplementary Tables 1 and 2. The resulting energy dispersion is presented in Supplementary Figure 6, and it becomes apparent that they differ strongly from the calculated DFT values. Consequently, the standard micromagnetic model with an exchange stiffness based on a 2 energy dependence is insufficient to describe the dispersion around the ground state (cf. Fig. 4b in the main text). Although Co is a strong FM and normally described very well by the conventional micromagnetic model, here the next order of the Taylor expansion ( 4 ) contributes significantly. We recover the 2 energy dependence close to the ̅ -point upon interchanging the Co and the Rh layers, i.e. for Co/Rh/Ir(111) (see Supplementary Figure 7). This also indicates that intermixing of Co and Rh atoms at the interface (which is 100% for Co/Rh/Ir(111)) will lower the exchange frustration thereby supporting the FM state.
The degeneracy of left and right rotating spin spiral states is lifted upon including SOC. Two additional energy contributions will occur due to the presence of SOC: the magnetocrystalline anisotropy energy (MAE) and the antisymmetric exchange interaction, known as the Dzyaloshinskii-Moriya interaction (DMI). The DMI requires a broken inversion symmetry, which is given by the interface and the surface of all films. If SOC is included, the generalized Bloch theorem cannot be used to calculate spin spirals in the chemical unit cell. However, since SOC is typically small compared to the total energy in the system, we treat SOC in first-order perturbation theory starting from self-consistent spin spiral calculations [12]. This approximation has been checked previously and the deviations due to using first-order perturbation theory are up to 20% depending on the spin spiral period [13]. The resulting energy contribution due to SOC is mapped onto the DMI, where is the DMI vector which determines the strength and the sign of the DMI. There are two possibilities to fit the model to the DFT values. We can determine the gradient of the SOC contribution around the ground state, since the sine behaviour of the DMI is linear around → 0. The result of this procedure is given by eff in Supplementary Tables 1 and 2. However, the behaviour of the SOC contributions of Fig. 4c in the main text and Supplementary Figure 6 is not represented by a simple sine. Instead, we have to include 7 neighbours to achieve a good description of the DMI leading to a frustration of the DMI similar to the exchange interaction.
The second energy contribution due to SOC is the magnetocrystalline anisotropy energy (MAE). After selfconsistent scalar-relativistic calculations, we apply SOC in the out-of-plane (⊥) and in-plane (∥) direction and use the force theorem [14] to obtain the MAE which is defined as the total energy difference between the two magnetization directions: = ⊥ − ∥ . We restrict ourselves to the uniaxial anisotropy, which is a good approximation for our ultrathin film systems.