Stabilizing spin spirals and isolated skyrmions at low magnetic field exploiting vanishing magnetic anisotropy

Skyrmions are topologically protected non-collinear magnetic structures. Their stability is ideally suited to carry information in, e.g., racetrack memories. The success of such a memory critically depends on the ability to stabilize and manipulate skyrmions at low magnetic fields. The non-collinear Dzyaloshinskii-Moriya interaction originating from spin-orbit coupling drives skyrmion formation. It competes with Heisenberg exchange and magnetic anisotropy favoring collinear states. Isolated skyrmions in ultra-thin films so far required magnetic fields as high as several Tesla. Here, we show that isolated skyrmions in a monolayer of Co/Ru(0001) can be stabilized down to vanishing fields. Even with the weak spin-orbit coupling of the 4d element Ru, homochiral spin spirals and isolated skyrmions were detected with spin-sensitive scanning tunneling microscopy. Density functional theory calculations explain the stability of the chiral magnetic features by the absence of magnetic anisotropy energy.

Decreasing B from +190 mT to 0 mT regenerates the balance between dark and bright areas (cf. supplementary fig. 2c). This indicates that the observed bright and dark contrast areas are due to a local out-of-plane orientation of magnetization. Note that for the small field used here, the spin-polarization direction of the tip is not affected [2]. (1) Supplementary Note 3. Magnetic field dependency of the TAMR contrast The evolution of the periodic stripe pattern under an out-of-plane magnetic field was recorded with non-spin polarized tip as well. supplementary fig. 3a 3c) and +150 mT (supplementary fig. 3h), the magnetization is almost saturated and is aligned parallel with the magnetic field. For both positive and negative magnetic field, an expansion of the bright area is observed. The maximum of the dI/dU signal (bright contrast) corresponds to an out-of-plane magnetization direction (either up or down). An in-plane magnetization gives a minimum of the dI/dU signal (dark contrast). The observed contrast is interpreted as originating from an out-of-plane TAMR effect where the dI/dU varies as a cos 2 with the angle formed between the magnetization and an out-of-plane quantization axis [3][4][5]. Note that in the saturated magnetic state (supplementary fig. 3c and h), dark contrast patches still remain on the island edges. At these locations, the magnetization is twisted. While decreasing the magnetic field toward 0 mT (supplementary fig. 3d and e), the spin spiral state is regenerated.
The voltage dependency of the TAMR effect presented in Fig. 2a in the main text was obtained from the two curves presented in supplementary fig. 3i. dI/dU spectra were recorded with the tip located in the middle of a bright stripe (red curve -out-of-plane magnetization) and in the middle of a dark stripe (dark curve -in-plane magnetization).
The relative variation of the dI/dU signal is converted to the voltage dependency of the TAMR effect ( Fig. 2a -main text) as: As explained in the main text, an inversion of the TAMR occurs at −300 mV. supplementary fig. 3 shows the dI/dU maps at a bias voltage of −250 mV. In that case, the TAMR dI/dU maps recorded at out-of-plane magnetic fields as indicated. The tip was unpolarized and the spin spiral was probed through TAMR (I=1 nA, U =−250 mV, ∆U rms =30 mV, scale bar is 25 nm). (i) -(k) dI/dU spectra recorded on a bright stripe (red curve -local magnetization out-ofplane) and on a dark stripe (dark curve and local magnetization in-plane) with three different tips ((i): I=10 nA, U =600 mV, ∆U rms =10 mV -(j) and (k): I=10 nA, U =1000 mV, ∆U rms =20 mV).
(l) Voltage dependency of the TAMR for the three different tips.
is negative, a maximum of the dI/dU signal corresponds to an out-of-plane magnetization, a minimum of the dI/dU amplitude coincide with an in-plane magnetization. In contrast,

Supplementary Note 4. Evaluation of magnetic interaction parameters
The magnetic interactions, were calculated by exploring total energy of spin-spirals [6] characterized by a constant angle between neighboring magnetic moments θ i = q · r i . Here q is the propagation direction of the spin spiral and r i is the position of the i th magnetic moment. In the case of Co/Ru(0001), the experimental angle between neighboring spins is rather small (θ = 2.4 • ) which correspond to a propagation vector of q = 0.0067 2π/a. This justified to consider the spin spiral as a perturbation from the ferromagnetic (FM) state, i.e.
Where J eff is the magnetic exchange interaction, D ij is the DMI and M j are unit vectors collinear to the magnetic moments at position r i and r j , respectively. The Heisenberg exchange contribution J eff was determined from a fit to the dispersion curve E(q) of the spin spiral when SOC is not included. The value can be approximated to the first neighbor exchange interaction and higher order term are negligible here. We found J eff = 13.1 meV/Co.
We obtained the DMI by fitting the SOC contribution to E(q) shown in supplementary fig. 4(b). We use the DMI model of Levy and Fert [9], which approximates D ij as: E n e r g y ( µe V p e r C o ) R i g h t r o t a t i n g L e f t r o t a t i n g where r i and r j are the position of two magnetic atoms (in Co) with respect to a non-magnetic atom (Ru in our case) and λ D is the strength of SOC of the d-electrons of the non-magnetic atom on the conduction electrons and V (λ D ) is a perturbing potential depending linearly on λ D (for more details see [9]). supplementary fig. 4b shows the total contribution (red line), the Co contribution (black line) and the Ru contribution (green line) to the SOC. In the case of Co/Ru(0001), the DMI is induced by the SOC of Co and not of the Ru substrate in contrast to previous works. In that respect, Co/Ru (0001)  In order to estimate the Curie temperature, we analyse the energy and the magnetization density. We compute the specific heat as: where E is the average energy density, k B is the Boltzmann constant and T is the temperature. We have computed the magnetic susceptibility as: We have modeled the energy and the magnetization density as arctangent function. We can then fit C and χ M with the Lorentzian function: where I is the intensity of the peak and σ is the mean-height width. All the fits were carried out with a linear background. By fitting of the specific heat, we obtain T c = 350 K.
This critical temperature corresponds to the transition between the ordered phase to the disordered phase. Ref. [10] reports on the temperature at which the magnetization vanishes i.e. T M c = 170 K. When the susceptibility χ M is fitted with the Lorentzian curve, we found T M c = 152 K, in good agreement with the experimental value. Supplementary Note 6: Dipole-dipole interaction in Co/Ru(0001) We have estimated the strength of the dipole-dipole interaction. The dipole-dipole energy density is given by:

Supplementary Note 8. Magnetic phase diagram
In order to study the stability of the different phases, we minimized their total energies via spin dynamics simulations. We solve the Landau-Lifschitz-Gilbert equation using the extended Heisenberg model parametrized from our DFT calculation. supplementary fig. 6 shows the energy of the different phases as a function of the applied magnetic field with respect to the energy obtained for the spin spiral with D ij = 0.2 meV/Co. As discussed above, when D ij = 0.2 meV/Co, the ground state of the system at B = 0 T is FM with in-plane magnetization (green dashed line). D = 0.3 meV/Co is enough to stabilize a spinspiral ground state for B lower than ≈150 mT (pink dotted line). When B further increases, the FM state aligns to the perpendicular magnetic field. The out of plane the magnetized FM state becomes more stable (green solid line). In the entire range of applied field, we have calculated the energy of isolated skyrmions (shown as red dashed lines). Isolated skyrmions are always metastable with respect to both the FM and the spin spiral states. The energy density differences are very small. For example, at B = 130 mT, the energy difference between the isolated skyrmion state and the FM state is of the order of 0.5 µeV/Co.

Supplementary Note 9. Determination of the skyrmion radii
As indicated in the main text, any stray field of the tip modifies the skyrmion structure or laterally moves it during scanning. Therefore, to determine the dependency of the skyrmion shape with magnetic field (Fig. 5b in the main text), a bare tungsten tip was used to image the skyrmion with the TAMR. In oder to take advantage of the high sensitivity of TAMR to distinguish between in-plane and out-of-plane oriented area, the definition of the skyrmion radius used here defer from the one use in [11]. We measured the diameter of the dark ring observed in the TAMR dI/dU maps (inset Fig. 5b -main text). The theoretical radius was obtained on the same way (green dot - Fig. 5b -main text). 8 show the theoretical skyrmion profiles of the magnetization orientation from the center to the edge for magnetic field from 200 to 600 mT. The skyrmion radius was determine by calculating the distance between the two opposite in-plane oriented sections i.e. two times the distance to the center represented by the vertical dashed lines. in-plane. This distance is half the skyrmion radius defined throughout this paper.