exponential fall/rise time dependence. Figure S48C shows that the average $1/\tau$ increases approximately linearly with $H_{RKKY}$ in agreement with the expectation of the Thiele model (see below). This demonstrates the clear link between the inertial behavior seen in the rise and fall of the SAF skyrmion speed and the RKKY exchange coupling. Phenomenologically, when the SOT pulse triggers the SAF skyrmion motion, the Magnus effect splits the AF coupled FM skyrmions apart since they have opposite core polarity. The RKKY exchange coupling is a recovering force that compensates for this splitting and stores energy, until a steady state is reached which explains the inertia seen in our simulations. We show in the next section that this inertia can be described from the Thiele equations of two skyrmions coupled by the RKKY exchange interaction. Note that a similar analysis was recently published in Ref. (54).

## 1.7 Thiele model of the SAF skyrmions dynamics

To model the dynamics of the SAF skyrmion, we consider the Thiele model, which assumes that the skyrmion spin texture is rigid during its motion (42). We describe the dynamics of the SAF skyrmions as two AF skyrmions (Sk1, Sk2) coupled by the RKKY exchange interactions:

$$F_{SOT}^{1} + G_{1} \times v_{1} - k(R_{1} - R_{2}) - \alpha D_{1}v_{1} = 0$$

$$F_{SOT}^{2} + G_{2} \times v_{2} - k(R_{2} - R_{1}) - \alpha D_{2}v_{2} = 0$$

Here $R_i$ the position of the skyrmion i, $v_i$ the velocity, $D_i$ the dissipative tensor, $G_i$ the gyrovector, $F_{SOT}^i$ the force due to the DL-SOT and $k$ is a spring constant that describes the restoring force between the two skyrmions due to the RKKY exchange interaction $J_{RKKY}$.

We write $d = (R_1 - R_2)$, $F_{SOT} = F_{SOT}^1 + F_{SOT}^2$, $\Delta F_{SOT} = F_{SOT}^1 - F_{SOT}^2$, $v_m = \frac{(v_1+v_2)}{2}$, $\Delta v = v_1 - v_2$. We assume $D_1 = D_2 = D$ and since the two skyrmions have opposite core polarities, $G_1 = -G_2 = Gu_z$. Then

$$F_{SOT} + G \times \Delta v - 2\alpha D v_m = 0 \quad (1)$$

$$\Delta F_{SOT} + 2G \times v_m - 2kd - \alpha D \Delta v = 0 \quad (2)$$

For a Néel skyrmion and a current flowing along x, $F_{SOT} = F_{SOT}u_x$ and assuming $D$ is diagonal with $D_{xx}=D_{yy}=D$, (3) and (4) lead to

$$F_{SOT} - G\Delta v_y - 2\alpha D v_{m,x} = 0 \quad (3)$$

$$G\Delta v_x - 2\alpha D v_{m,y} = 0 \quad (4)$$

$$2Gv_{m,x} - 2kd_y - \alpha D \Delta v_y = 0 \quad (5)$$

$$\Delta F_{SOT,x} - 2Gv_{m,y} - 2kd_x - \alpha D \Delta v_x = 0 \quad (6)$$

### Steady state

In the steady state, the skyrmions have the same speed thus $\Delta v = 0$. This leads to the solution:

$$v_m = \frac{F_{SOT}}{2\alpha D}$$

Note that this result is equivalent to considering a single SAF skyrmion with $G=0$ and a Thiele equation: $F_{SOT} - 2\alpha D v_m = 0$ (Note here that $D$ is the dissipative tensor for 1 skyrmion in the SAF).

$F_{SOT}$ and $D$ write (13) $F_{SOT} \approx \mu_0 M_s t \pi^2 H_{DL} R u_x$ and $D \approx \frac{2\pi M_s t'}{\gamma} (\frac{R}{\Delta} + \frac{\Delta}{R})$ assuming $R \gg \Delta$.

Here $t$ is the film thickness, $t'=t/2$, $R$ the skyrmion radius, $\mu_0 H_{DL}$ the damping like effective field. Thus, the steady state velocity writes for $R \gg \Delta$

$$v_m = v_{m,x} \approx \frac{\pi B_{SOT} J R \gamma}{2\alpha \left(\frac{R}{\Delta} + \frac{\Delta}{R}\right)}$$
This expression was used in Figure S4, red line, of the main paper using the parameters of Table S2.

An expression valid for smaller R can also be obtained using (72) $D \approx \frac{M_s t'}{\gamma} (\frac{2\pi R}{\Delta} + 4\pi \exp(-\frac{R}{2\Delta}))$, which leads to

$$v_m \approx \frac{\pi\mu_0H_{DL}\gamma R}{2\alpha(\frac{R}{\Delta} + 2\exp(-\frac{R}{2\Delta}))}$$

## Inertial dynamics

Combining (3) and (5) and writing $\Delta v_y = \frac{d}{dt} \frac{d}{dy}$

One gets

$$(F_{SOT} + \frac{\alpha D}{2k} \frac{dF_{SOT}}{dt}) - 2\alpha D v_{m,x} = m \frac{dv_{m,x}}{dt} (7)$$

$$\text{Where } m = \frac{G^2}{k} \left(1 + \left(\frac{\alpha D}{G}\right)^2\right)$$

Thus, the RKKY exchange interaction leads to an additional inertial term in the Thiele equation with an inertial mass m and characteristic time constant $\tau = \frac{m}{2\alpha D} = \frac{G^2}{2k\alpha D} (1 + \left(\frac{\alpha D}{G}\right)^2)$. For circular skyrmions with no deformation and small lateral shift d, one can show that $k \approx 2\pi J_{RKKY} (\frac{R}{\Delta} + \frac{\Delta}{R})$, with R the skyrmion radius and $\Delta$ the domain wall width. Thus, the inverse of the relaxation time scales as $\frac{1}{\tau} \sim k \sim J_{RKKY}$, a result in agreement with the micromagnetic simulations. Note that during the review process, Panigrahy et al. reported similar results on the skyrmion inertia in synthetic antiferromagnets (54).

## 1.8 Topological spin Hall torque

We explore numerically the effect of a topological spin Hall torque in an effort to explain the high speed measured for AF skyrmions (22, 73–75). The resulting torque writes (22) $\frac{dm}{dt} = -b_j \lambda^2 N_{xy} \partial_y \mathbf{m}$ with $b_j = \frac{\mu_B J P}{M_s q}$ where P is the spin polarization, q the electron charge, $\mu_B$ the Bohr magneton, J the applied current density and $\lambda$ a parameter describing the amplitude of the topological spin Hall torque (22, 73).

The resulting Thiele force can be expressed as:

$$F_{tst,x} = -\frac{M_s t}{\gamma} b_j \lambda^2 \iint ((\partial_x \mathbf{m} \times \partial_y \mathbf{m}) \cdot \mathbf{m})^2 dx dy$$

The steady state motion of AF skyrmion writes:

$$v_{sts,x} = \frac{F_{tst,x}}{2\alpha D}$$

We perform a numerical integration of the torque to estimate its effect on an AF skyrmion speed. We vary the core radius and domain wall width of the skyrmion and estimate the speed for a range $\lambda^2$ values from 3 to 50 nm$^2$ (22, 73) assuming $J=8 \times 10^{11}$ A/m$^2$ and $P=0.5$. The skyrmion spin texture is expressed as:

$$\theta(r) = 2 \operatorname{atan} \left( \exp \left( \frac{r - R}{\Delta} \right) \right)$$

and the magnetization is expressed in polar coordinates $(r, \theta, \varphi)$:
This expression was used in Figure S4, red line, of the main paper using the parameters of Table S2.

An expression valid for smaller R can also be obtained using (72) $D \approx \frac{M_s t'}{\gamma} (\frac{2\pi R}{\Delta} + 4\pi \exp(-\frac{R}{2\Delta}))$, which leads to

$$v_m \approx \frac{\pi \mu_0 H_{DL} \gamma R}{2\alpha(\frac{R}{\Delta} + 2\exp(-\frac{R}{2\Delta}))}$$

## Inertial dynamics

Combining (3) and (5) and writing $\Delta v_y = \frac{d}{dt} v_y$

One gets

$$(F_{SOT} + \frac{\alpha D}{2k} \frac{dF_{SOT}}{dt}) - 2\alpha D v_{m,x} = m \frac{dv_{m,x}}{dt} (7)$$

$$\text{Where } m = \frac{G^2}{k} \left( 1 + \left( \frac{\alpha D}{G} \right)^2 \right)$$

Thus, the RKKY exchange interaction leads to an additional inertial term in the Thiele equation with an inertial mass $m$ and characteristic time constant $\tau = \frac{m}{2\alpha D} = \frac{G^2}{2k\alpha D} \left(1 + \left(\frac{\alpha D}{G}\right)^2\right)$. For circular skyrmions with no deformation and small lateral shift $d$, one can show that $k \approx 2\pi J_{RKKY} (\frac{R}{\Delta} + \frac{\Delta}{R})$, with $R$ the skyrmion radius and $\Delta$ the domain wall width. Thus, the inverse of the relaxation time scales as $\frac{1}{\tau} \sim k \sim J_{RKKY}$, a result in agreement with the micromagnetic simulations. Note that during the review process, Panigrahy et al. reported similar results on the skyrmion inertia in synthetic antiferromagnets (54).

## 1.8 Topological spin Hall torque

We explore numerically the effect of a topological spin Hall torque in an effort to explain the high speed measured for AF skyrmions (22, 73–75). The resulting torque writes (22) $\frac{dm}{dt} = -b_j \lambda^2 N_{xy} \partial_y \mathbf{m}$ with $b_j = \frac{\mu_B J P}{M_s q}$ where $P$ is the spin polarization, $q$ the electron charge, $\mu_B$ the Bohr magneton, $J$ the applied current density and $\lambda$ a parameter describing the amplitude of the topological spin Hall torque (22, 73).

The resulting Thiele force can be expressed as:

$$F_{tst,x} = -\frac{M_s t}{\gamma} b_j \lambda^2 \iint ((\partial_x \mathbf{m} \times \partial_y \mathbf{m}) \cdot \mathbf{m})^2 dx dy$$

The steady state motion of AF skyrmion writes:

$$v_{sts,x} = \frac{F_{tst,x}}{2\alpha D}$$

We perform a numerical integration of the torque to estimate its effect on an AF skyrmion speed. We vary the core radius and domain wall width of the skyrmion and estimate the speed for a range $\lambda^2$ values from 3 to 50 nm$^2$ (22, 73) assuming $J=8 \times 10^{11}$ A/m$^2$ and $P=0.5$. The skyrmion spin texture is expressed as:

$$\theta(r) = 2 \operatorname{atan} \left( \exp \left( \frac{r-R}{\Delta} \right) \right)$$

and the magnetization is expressed in polar coordinates $(r, \theta, \varphi)$:
$$m = \begin{pmatrix} \sin(\theta(r)) \cos(\varphi) \\ \sin(\theta(r)) \sin(\varphi) \\ \cos(\theta(r)) \end{pmatrix}$$

This leads to a Néel skyrmion spin texture with left handed domain wall. We perform a numerical integration of the force resulting from the topological spin Hall torque for $\Delta=24.5$ nm with a range of $R/\Delta$ ratio from 1 to 5. We compute the speed of the skyrmion and compare it with the DL-SOT speed of an AF skyrmion according to Thiele equation. The results are plotted in Figure S49. We see that the effect of the topological torque is negligible for large skyrmions, in agreement with literature. However, we also see that there is a strong increase in speed as $R/\Delta$ gets close to 1. Moreover, this increase in speed due to the topological torque is proportional to $\lambda^2$ which shows that a large $\lambda^2$ can lead to very high speed even for small skyrmions.
$$m = \begin{pmatrix} \sin(\theta(r)) \cos(\varphi) \\ \sin(\theta(r)) \sin(\varphi) \\ \cos(\theta(r)) \end{pmatrix}$$

This leads to a Néel skyrmion spin texture with left handed domain wall. We perform a numerical integration of the force resulting from the topological spin Hall torque for $\Delta=24.5$ nm with a range of $R/\Delta$ ratio from 1 to 5. We compute the speed of the skyrmion and compare it with the DL-SOT speed of an AF skyrmion according to Thiele equation. The results are plotted in Figure S49. We see that the effect of the topological torque is negligible for large skyrmions, in agreement with literature. However, we also see that there is a strong increase in speed as $R/\Delta$ gets close to 1. Moreover, this increase in speed due to the topological torque is proportional to $\lambda^2$ which shows that a large $\lambda^2$ can lead to very high speed even for small skyrmions.
Supplementary Figures

<figure><img src="image_15.png" /><figcaption>**Figure S6 Thickness dependence of the magnetic properties of the top and bottom Co layer of the SAF stacks (A, C):** Schematic representations of the stacks used to characterize the magnetic properties of the bottom (A) and top layer (C) constituting the SAF stacks. The thicknesses are in nanometer. The Co(t) layer is deposited as a wedge, such that its thickness varies along the 100 mm Si wafer. (B) Magneto-optical Kerr effect (MOKE) hysteresis loops (magnetic field applied perpendicularly to the sample plane) of the stack A (bottom FM layer) measured at different positions along the 100 mm Si wafer. The different positions correspond to different thicknesses of the Co layer. The curves are offset for clarity. (D) Same for the stack in C (top FM layer). The chosen thickness for experiments is highlighted with the thicker line in orange (15.8Å).</figcaption></figure>
Supplementary Figures

<figure><img src="image_15.png" /><figcaption>**Figure S6 Thickness dependence of the magnetic properties of the top and bottom Co layer of the SAF stacks (A, C):** Schematic representations of the stacks used to characterize the magnetic properties of the bottom (A) and top layer (C) constituting the SAF stacks. The thicknesses are in nanometer. The Co(t) layer is deposited as a wedge, such that its thickness varies along the 100 mm Si wafer. (B) Magneto-optical Kerr effect (MOKE) hysteresis loops (magnetic field applied perpendicularly to the sample plane) of the stack A (bottom FM layer) measured at different positions along the 100 mm Si wafer. The different positions correspond to different thicknesses of the Co layer. The curves are offset for clarity. (D) Same for the stack in C (top FM layer). The chosen thickness for experiments is highlighted with the thicker line in orange (15.8Å).</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure S7 Field-induced DW dynamics experiments for magnetic damping measurements** (A) DW velocity as a function of the applied out-of-plane magnetic field measured at room-temperature. The blue dots denote the experimental data and the solid line is a linear fit in the flow regime. The error bars denote the standard deviation. (Inset) Differential pMOKE image shows the expansion of the magnetic domain after applying the applied out-of-plane magnetic field pulse. (B) Magnetic moment vs. the external in-plane field measured by vibrating sample magnetometry. The extracted anisotropy field is $\mu_0H_K = 0.43 \pm 0.05$ T.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure S8 Field-induced domain wall dynamics experiments to determine $H_{DMI}$.** (A) Differential pMOKE images showing the expansion of the circular domain (white contrast) induced by a magnetic field pulse of 57 mT for a constant in-plane field $B_x = 0$ (left) and $B_x = 60$ mT (right) (B) Domain wall velocity vs. in plane field $B_x$ for a perpendicular magnetic field</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure S7** Field-induced DW dynamics experiments for magnetic damping measurements (A) DW velocity as a function of the applied out-of-plane magnetic field measured at room-temperature. The blue dots denote the experimental data and the solid line is a linear fit in the flow regime. The error bars denote the standard deviation. (Inset) Differential pMOKE image shows the expansion of the magnetic domain after applying the applied out-of-plane magnetic field pulse. (B) Magnetic moment vs. the external in-plane field measured by vibrating sample magnetometry. The extracted anisotropy field is $\mu_0H_K = 0.43 \pm 0.05$ T.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure S8** Field-induced domain wall dynamics experiments to determine $H_{DMI}$. (A) Differential pMOKE images showing the expansion of the circular domain (white contrast) induced by a magnetic field pulse of 57 mT for a constant in-plane field $B_x = 0$ (left) and $B_x = 60$ mT (right). (B) Domain wall velocity vs. in plane field $B_x$ for a perpendicular magnetic field.</figcaption></figure>
field pulse $B_z= 57$ mT and a pulse width of 30 ns for up/down and down/up domain wall (black and red curves).

<figure><img src="image_18.png" /><figcaption>**Figure S9:** Second harmonic planar Hall effect measurements of the spin orbit torque (A-B) First (A) and second (B) harmonic Hall resistance vs the in-plane field angle orientation $\phi$ (B) Second harmonic planar Hall resistance (black) and the fit (red) vs the field angle. (C) Field like and (D) Damping like and thermal contributions of the second harmonic resistance as a function of $\phi$. These measurements are performed for an external field of 1.3T.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Figure S10:** Fit to extract the SOT (A) $R_{DL} + R_T$ vs $1/(\mu_0H_{ext} - \mu_0H_k)$ and (B) $10^3R_{FL}/R_{PHE}$ vs $1/(\mu_0H_{ext})$ for the ferromagnetically coupled bilayer system (note the presence of $\mu_0H_k$, the magnetic anisotropy field).</figcaption></figure>

<figure><img src="image_20.png" /></figure>
field pulse $B_z= 57$ mT and a pulse width of 30 ns for up/down and down/up domain wall (black and red curves).

<figure><img src="image_18.png" /><figcaption>**Figure S9:** Second harmonic planar Hall effect measurements of the spin orbit torque (A-B) First (A) and second (B) harmonic Hall resistance vs the in-plane field angle orientation $\phi$ (B) Second harmonic planar Hall resistance (black) and the fit (red) vs the field angle. (C) Field like and (D) Damping like and thermal contributions of the second harmonic resistance as a function of $\phi$. These measurements are performed for an external field of 1.3T.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Figure S10:** Fit to extract the SOT (A) $R_{DL} + R_T$ vs $1/(\mu_0H_{ext} - \mu_0H_k)$ and (B) $10^3 R_{FL}/R_{PHE}$ vs $1/(\mu_0H_{ext})$ for the ferromagnetically coupled bilayer system (note the presence of $\mu_0H_k$, the magnetic anisotropy field).</figcaption></figure>

<figure><img src="image_20.png" /></figure>
<figure><img src="image_21.png" /><figcaption>**Figure S12 Hysteresis cycle of the synthetic ferromagnetic stack.** Kerr signal as a function of the perpendicular magnetic field.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Figure S13 Current induced skyrmion motion in synthetic ferromagnet observed by magnetic force microscopy** A Sequence of MFM images acquired before and after the injection of a current pulse of 4.95 ns with density $J = 6,95 \times 10^{11} A/m^2$. B Sequence of images acquired before and after the injection of a pulse of 5.01 ns with current density of $J = 6,70 \times 10^{11} A/m^2$. The white arrows in the bottom images show the displacement of the skyrmions induced by the current pulse.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Figure S12 Hysteresis cycle of the synthetic ferromagnetic stack.** Kerr signal as a function of the perpendicular magnetic field.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Figure S13 Current induced skyrmion motion in synthetic ferromagnet observed by magnetic force microscopy** A Sequence of MFM images acquired before and after the injection of a current pulse of 4.95 ns with density $J = 6,95 \times 10^{11} A/m^2$. B Sequence of images acquired before and after the injection of a pulse of 5.01 ns with current density of $J = 6,70 \times 10^{11} A/m^2$. The white arrows in the bottom images show the displacement of the skyrmions induced by the current pulse.</figcaption></figure>
<figure><img src="image_23.png" /></figure>

Figure S14 Current induced skyrmion velocity and skyrmion Hall angle in synthetic ferromagnetic stack Skyrmion velocity A and skyrmion Hall angle B as a function of the current density. One square dot corresponds to one event. An external out-of-plane magnetic field of 15 mT is applied.

<figure><img src="image_24.png" /></figure>

Figure S15 Current induced skyrmion velocity and skyrmion Hall angle in synthetic ferromagnetic stack A Average velocity vs current density of moving skyrmions. B Average skyrmion Hall angle vs skyrmion velocity. The error bars stand for the standard deviation. An external out-of-plane magnetic field of 15 mT is applied.
<figure><img src="image_23.png" /></figure>

Figure S14 Current induced skyrmion velocity and skyrmion Hall angle in synthetic ferromagnetic stack Skyrmion velocity A and skyrmion Hall angle B as a function of the current density. One square dot corresponds to one event. An external out-of-plane magnetic field of 15 mT is applied.

<figure><img src="image_24.png" /></figure>

Figure S15 Current induced skyrmion velocity and skyrmion Hall angle in synthetic ferromagnetic stack A Average velocity vs current density of moving skyrmions. B Average skyrmion Hall angle vs skyrmion velocity. The error bars stand for the standard deviation. An external out-of-plane magnetic field of 15 mT is applied.
<figure><img src="image_25.png" /><figcaption>Figure S16 Current induced skyrmion motion in single ferromagnetic layer stack **observed by magnetic force microscopy**. A Sequence of MFM images acquired before and after the injection of a pulse of 5.05 ns with current density of $J = 4,36 \times 10^{11}$ A/m$^2$. B Sequence of images acquired before and after the injection of a pulse of 4.99 ns with current density of $J = 5,14 \times 10^{11}$ A/m$^2$. An external out-of-plane magnetic field of 15 mT is applied. The white arrows in the bottom images show the displacement of the skyrmions induced by the current pulse.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Figure S17 Current induced skyrmion velocity and skyrmion Hall angle in single ferromagnetic layer stack. Skyrmion velocity A and skyrmion Hall angle B as a function of the current density. One square dot corresponds to one event.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Figure S16 Current induced skyrmion motion in single ferromagnetic layer stack observed by magnetic force microscopy A Sequence of MFM images acquired before and after the injection of a pulse of 5.05 ns with current density of $J = 4,36 \times 10^{11} A/m^2$. B Sequence of images acquired before and after the injection of a pulse of 4.99 ns with current density of $J = 5,14 \times 10^{11} A/m^2$. An external out-of-plane magnetic field of 15 mT is applied. The white arrows in the bottom images show the displacement of the skyrmions induced by the current pulse.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Figure S17 Current induced skyrmion velocity and skyrmion Hall angle in single ferromagnetic layer stack Skyrmion velocity A and skyrmion Hall angle B as a function of the current density. One square dot corresponds to one event.</figcaption></figure>
<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

Figure S18 **Current induced skyrmion velocity and skyrmion Hall angle in single ferromagnetic layer stack** A Average velocity vs current density of moving skyrmions. B Average skyrmion Hall angle vs skyrmion velocity. The error bars stand for the standard deviation.

<figure><img src="image_29.png" /></figure>

Figure S19: Observation of the chiral structure of domain walls using XMCD-PEEM magnetic microscopy. (A) XMCD-PEEM image of a multidomain state of the SAF stack (B) Line profile of the magnetic contrast along the white dotted line in (A). The line scans were performed with a certain width in order to get an average signal.

<figure><img src="image_30.png" /></figure>

Figure S20: Observation of the chiral structure of domain walls using XMCD-PEEM magnetic microscopy. (A) XMCD-PEEM image of a multidomain state of the SAF stack. (B) Line profile of the magnetic contrast along the red dotted line in (A) (Domain wall
<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

Figure S18 **Current induced skyrmion velocity and skyrmion Hall angle in single ferromagnetic layer stack** A Average velocity vs current density of moving skyrmions. B Average skyrmion Hall angle vs skyrmion velocity. The error bars stand for the standard deviation.

<figure><img src="image_29.png" /></figure>

Figure S19: Observation of the chiral structure of domain walls using XMCD-PEEM magnetic microscopy. (A) XMCD-PEEM image of a multidomain state of the SAF stack (B) Line profile of the magnetic contrast along the white dotted line in (A). The line scans were performed with a certain width in order to get an average signal.

<figure><img src="image_30.png" /></figure>

Figure S20: Observation of the chiral structure of domain walls using XMCD-PEEM magnetic microscopy. (A) XMCD-PEEM image of a multidomain state of the SAF stack. (B) Line profile of the magnetic contrast along the red dotted line in (A) (Domain wall
perpendicular to the X-ray beam). (C) Line profile of the magnetic contrast along the blue dotted line in (A) (domain wall along the X-ray beam). The line scans were performed with a certain width in order to get an average signal.

<figure><img src="image_31.png" /></figure>

**Figure S21** MFM contrast of domain wall and skyrmions. A MFM image measured at zero applied magnetic field showing domain walls and skyrmions (same as Figure 2C of the main paper). B MFM phase along the black line in A crossing HH/TT and TT/HH domain walls. C MFM phase along the red and blue lines in A crossing two skyrmions with opposite core magnetization.
perpendicular to the X-ray beam). (C) Line profile of the magnetic contrast along the blue dotted line in (A) (domain wall along the X-ray beam). The line scans were performed with a certain width in order to get an average signal.

<figure><img src="image_31.png" /></figure>

**Figure S21** MFM contrast of domain wall and skyrmions. A MFM image measured at zero applied magnetic field showing domain walls and skyrmions (same as Figure 2C of the main paper). B MFM phase along the black line in A crossing HH/TT and TT/HH domain walls. C MFM phase along the red and blue lines in A crossing two skyrmions with opposite core magnetization.
<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

**Figure S22** Micromagnetic simulation of the MFM contrast A Simulation of a MFM contrast of a magnetic domain wall. B MFM contrast along a line scan perpendicular to the domain wall. C-D. Same for a skyrmion.

<figure><img src="image_36.png" /></figure>

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

**Figure S23** Magnetic force microscopy experiments of the synthetic ferromagnetic stack. (A,C) MFM images at zero field. (B,D) Magnetic contrast along the linescan in A and C. The magnetic tip is the same as the one used for the SAF stack.
<figure><img src="image_32.png" /></figure>

500 nm

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

**Figure S22** Micromagnetic simulation of the MFM contrast A Simulation of a MFM contrast of a magnetic domain wall. B MFM contrast along a line scan perpendicular to the domain wall. C-D. Same for a skyrmion.

<figure><img src="image_36.png" /></figure>

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

**Figure S23** Magnetic force microscopy experiments of the synthetic ferromagnetic stack. (A,C) MFM images at zero field. (B,D) Magnetic contrast along the linescan in A and C. The magnetic tip is the same as the one used for the SAF stack.
<figure><img src="image_39.png" /><figcaption>**Figure S24 Comparison of MFM and NV center magnetometry data**
A) MFM image measured at zero external magnetic field, on the same track geometry as the one used for the current induced motion after applying an out-of-plane magnetic field sequence of -1 T then 175 mT. B) Stray field map of the same region as in A measured with scanning NV center magnetometry, at a distance of 36 nm from the sample surface. The field component shown here is the projection onto the NV center quantification axis (polar angle 61°, azimuthal angle 0°). The measurements were performed at zero external magnetic field. C) Simulation of the expected stray field map above a skyrmion of diameter 200 nm, whose profile is calculated using the micromagnetic parameters shown in Table S2.</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>**Figure S25 Nucleation of SAF skyrmions using external magnetic fields**
(A) MFM images of skyrmions in a magnetic stripe at B = 0 at room temperature acquired after applying sequentially out-of-plane magnetic fields of 220 mT and 175 mT. (B) MFM image measured at zero applied field obtained from A after an additional application of an out-of-plane field of 185 mT. The scale bar is 1 µm.</figcaption></figure>
<figure><img src="image_40.png" /><figcaption>**Figure S24 Comparison of MFM and NV center magnetometry data**
A) MFM image measured at zero external magnetic field, on the same track geometry as the one used for the current induced motion after applying an out-of-plane magnetic field sequence of -1 T then 175 mT. B) Stray field map of the same region as in A measured with scanning NV center magnetometry, at a distance of 36 nm from the sample surface. The field component shown here is the projection onto the NV center quantification axis (polar angle 61°, azimuthal angle 0°). The measurements were performed at zero external magnetic field. C) Simulation of the expected stray field map above a skyrmion of diameter 200 nm, whose profile is calculated using the micromagnetic parameters shown in Table S2.</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>**Figure S25 Nucleation of SAF skyrmions using external magnetic fields**
(A) MFM images of skyrmions in a magnetic stripe at B = 0 at room temperature acquired after applying sequentially out-of-plane magnetic fields of 220 mT and 175 mT. (B) MFM image measured at zero applied field obtained from A after an additional application of an out-of-plane field of 185 mT. The scale bar is 1 µm.</figcaption></figure>
<figure><img src="image_41.png" /></figure>

**Figure S26 Distribution of the diameters of the skyrmions** The inset is a MFM image of a typical skyrmion. The diameter *d* of a skyrmion is calculated from the area A inside the border (drawn in white) with $d = \sqrt{\frac{4A}{\pi}}$. The error bars are the sample standard deviation. We only considered here the diameter of skyrmions for which a motion was observed during the current induced motion experiments B Distribution of the variation of the skyrmion diameter before and after the pulse (average 5%, standard deviation of 35%, excluding events with deformation >100%).

<figure><img src="image_42.png" /></figure>

**Figure S27 Velocity vs skyrmion diameter** for current densities between 7.5×10¹¹ A/m² and 8.9×10¹¹ A/m². Blue dots are single events, black dots are averaged values for different skyrmion diameters (error bars stand for standard deviation), the red line is the prediction of the model for J=8.12×10¹¹ A/m², the dotted orange lines corresponds to the prediction of the model for J=7.5×10¹¹ A/m² and 8.9×10¹¹ A/m². The skyrmion diameter corresponds to the one measured before the pulse injection. B Velocity vs current density for skyrmion diameter smaller than 400 nm (red points) and skyrmion with all diameters (black points).
<figure><img src="image_42.png" /></figure>

**Figure S26 Distribution of the diameters of the skyrmions** The inset is a MFM image of a typical skyrmion. The diameter *d* of a skyrmion is calculated from the area A inside the border (drawn in white) with $d = \sqrt{\frac{4A}{\pi}}$. The error bars are the sample standard deviation. We only considered here the diameter of skyrmions for which a motion was observed during the current induced motion experiments B Distribution of the variation of the skyrmion diameter before and after the pulse (average 5%, standard deviation of 35%, excluding events with deformation >100%).

<figure><img src="image_43.png" /></figure>

**Figure S27 Velocity vs skyrmion diameter** for current densities between 7.5×10¹¹ A/m² and 8.9×10¹¹ A/m². Blue dots are single events, black dots are averaged values for different skyrmion diameters (error bars stand for standard deviation), the red line is the prediction of the model for J=8.12×10¹¹ A/m², the dotted orange lines corresponds to the prediction of the model for J=7.5×10¹¹ A/m² and 8.9×10¹¹ A/m². The skyrmion diameter corresponds to the one measured before the pulse injection. B Velocity vs current density for skyrmion diameter smaller than 400 nm (red points) and skyrmion with all diameters (black points).
<figure><img src="image_43.png" /><figcaption>**Figure S28** Example of current pulse used for the current induced skyrmion experiments. Experimental normalized voltage pulse as a function of time (black line) and its Gaussian fit (red line).</figcaption></figure>

<figure><img src="image_44.png" /><figcaption>**Figure S29** Current induced skyrmion motion of isolated skyrmions at zero external magnetic field. A MFM image of an isolated skyrmion before (top) and after (bottom) the injection of a 0.81 ns current pulse with density $J=7.1 \times 10^{11}$ A/m$^2$. The skyrmion moves by around 310 nm. B MFM image of an isolated skyrmions skyrmion before (top), after (middle) the injection of a positive current pulse of 0.84 ns (skyrmion displacement of 310 nm) and density $J= 6.9 \times 10^{11}$ A/m$^2$ and after (bottom) the injection of a 0.81 ns current pulse of density $J=7.1 \times 10^{11}$ A/m$^2$, skyrmion displacement of 450 nm.</figcaption></figure>
<figure><img src="image_44.png" /><figcaption>**Figure S28** Example of current pulse used for the current induced skyrmion experiments. Experimental normalized voltage pulse as a function of time (black line) and its Gaussian fit (red line).</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>**Figure S29** Current induced skyrmion motion of isolated skyrmions at zero external magnetic field. A MFM image of an isolated skyrmion before (top) and after (bottom) the injection of a 0.81 ns current pulse with density $J=7.1 \times 10^{11}$ A/m$^2$. The skyrmion moves by around 310 nm. B MFM image of an isolated skyrmions skyrmion before (top), after (middle) the injection of a positive current pulse of 0.84 ns (skyrmion displacement of 310 nm) and density $J= 6.9 \times 10^{11}$ A/m$^2$ and after (bottom) the injection of a 0.81 ns current pulse of density $J=7.1 \times 10^{11}$ A/m$^2$, skyrmion displacement of 450 nm.</figcaption></figure>
<figure><img src="image_45.png" /></figure>

**Figure S30** Current induced motion of a SAF skyrmion observed by MFM Magnetic force microscopy image before (top) and after (bottom) the injection of a 1.1 ns current pulse of density $8 \times 10^{11}$ A/m$^2$. The scale bar is 1µm.

<figure><img src="image_46.png" /></figure>

**Figure S31** Current induced motion of SAF skyrmions observed by magnetic force microscopy (MFM). A Sequence of images acquired after injection of subsequent pulses of 1.2 ns, 1.3 ns, 1.2 ns, 1.2 ns with respectively current density $J = 6.4 \times 10^{11}$ A/m$^2$, $-5.8 \times 10^{11}$ A/m$^2$, $-7.2 \times 10^{11}$ A/m$^2$, $7.2 \times 10^{11}$ A/m$^2$. The corresponding velocities are 430 m/s, 160 m/s, 340 m/s, 440 m/s respectively. B Sequence of images acquired before and after the injection of a pulse of 0.9 ns with current density of $J = 8,1 \times 10^{11}$ A/m$^2$. The corresponding velocities are 780, 800, 440 m/s respectively.
<figure><img src="image_46.png" /><figcaption>**Figure S30** Current induced motion of a SAF skyrmion observed by MFM Magnetic force microscopy image before (top) and after (bottom) the injection of a 1.1 ns current pulse of density $8 \times 10^{11}$ A/m$^2$. The scale bar is 1µm.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>**Figure S31** Current induced motion of SAF skyrmions observed by magnetic force microscopy (MFM). A Sequence of images acquired after injection of subsequent pulses of 1.2 ns, 1.3 ns, 1.2 ns, 1.2 ns with respectively current density $J = 6.4 \times 10^{11}$ A/m$^2$, $-5.8 \times 10^{11}$ A/m$^2$, $-7.2 \times 10^{11}$ A/m$^2$, $7.2 \times 10^{11}$ A/m$^2$. The corresponding velocities are 430 m/s, 160 m/s, 340 m/s, 440 m/s respectively. B Sequence of images acquired before and after the injection of a pulse of 0.9 ns with current density of $J = 8,1 \times 10^{11}$ A/m$^2$. The corresponding velocities are 780, 800, 440 m/s respectively.</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>**Figure S32** Current induced motion of SAF skyrmions observed by magnetic force microscopy (MFM). A Sequence of images acquired before and after the injection of two subsequent pulses of 0.54 ns and 0.53 ns with current density respectively of J=7.7 ×10¹¹ A/m² and -8.0 ×10¹¹ A/m². The corresponding velocities are 780 m/s, 800 m/s and 440 m/s. B Sequence of images acquired before and after the injection of a pulse of 1,3 ns with current density of J=5,19 ×10¹¹ A/m². The corresponding velocity is 280 m/s.</figcaption></figure>

<figure><img src="image_48.png" /></figure>
<figure><img src="image_48.png" /><figcaption>**Figure S32** Current induced motion of SAF skyrmions observed by magnetic force microscopy (MFM). A Sequence of images acquired before and after the injection of two subsequent pulses of 0.54 ns and 0.53 ns with current density respectively of J=7.7 ×10¹¹ A/m² and -8.0 ×10¹¹ A/m². The corresponding velocities are 780 m/s, 800 m/s and 440 m/s. B Sequence of images acquired before and after the injection of a pulse of 1,3 ns with current density of J=5,19 ×10¹¹ A/m². The corresponding velocity is 280 m/s.</figcaption></figure>

<figure><img src="image_49.png" /></figure>
<figure><img src="image_49.png" /><figcaption>**Figure S34 Back and forth motion of SAF skyrmions induced by current pulses** Magnetic force microscopy image before (top) and after the injection of a positive (middle) and negative (bottom) 0.53 ns current pulse of density 8.85×10¹¹ A/m². Scale bar is 1 µm.</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>**Figure S34 Back and forth motion of SAF skyrmions induced by current pulses** Magnetic force microscopy image before (top) and after the injection of a positive (middle) and negative (bottom) 0.53 ns current pulse of density 8.85×10¹¹ A/m². Scale bar is 1 µm.</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>**Figure S35** MFM images showing current induced motion experiments. The images are the same as the one of Figure 3 of the main text but zoomed out such that a topological defect can be seen on the image. The scale bars are 1 µm.</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>**Figure S36** A Skyrmion velocities vs current density of all the current induced skyrmion motion events. B same as A but plotted in absolute value.</figcaption></figure>
<figure><img src="image_51.png" /><figcaption>**Figure S35** MFM images showing current induced motion experiments. The images are the same as the one of Figure 3 of the main text but zoomed out such that a topological defect can be seen on the image. The scale bars are 1 µm.</figcaption></figure>

<figure><img src="image_52.png" /><figcaption>**Figure S36** A Skyrmion velocities vs current density of all the current induced skyrmion motion events. B same as A but plotted in absolute value.</figcaption></figure>
<figure><img src="image_52.png" /></figure>

**Figure S37** Skyrmion Hall effect in the SAF stack A Skyrmion Hall angle vs current density of all the current induced motion events. B. Average skyrmion Hall angle vs current density. C. Average skyrmion Hall angle vs velocity.

<figure><img src="image_53.png" /></figure>

**Figure S38** Comparison between the velocities measured for positive and negative current pulses. A shows the values averaged over several events (the error bars stand for the standard deviation) and B shows the corresponding velocities distribution.

<figure><img src="image_54.png" /></figure>

**Figure S39:** Micromagnetic simulation of the SAF skyrmions spin texture (a) and (b) show the z-component of magnetization in top and bottom FM layers, respectively. (c) shows
<figure><img src="image_53.png" /></figure>

**Figure S37** Skyrmion Hall effect in the SAF stack A Skyrmion Hall angle vs current density of all the current induced motion events. B. Average skyrmion Hall angle vs current density. C. Average skyrmion Hall angle vs velocity.

<figure><img src="image_54.png" /></figure>

**Figure S38** Comparison between the velocities measured for positive and negative current pulses. A shows the values averaged over several events (the error bars stand for the standard deviation) and B shows the corresponding velocities distribution.

<figure><img src="image_55.png" /></figure>

**Figure S39:** Micromagnetic simulation of the SAF skyrmions spin texture (a) and (b) show the z-component of magnetization in top and bottom FM layers, respectively. (c) shows
a 1D linescan through (a) and (b). (d) shows the linescan using the vector arrows representing the magnetic moments.

<figure><img src="image_55.png" /></figure>

**Figure S40 Maps of topological charge density for different cases.** (a) and (c) represent the topological charge density map for the top and bottom FM layer at t=0ns (static), respectively. Similarly, (c) and (d) represent the topological charge density map for the top and bottom FM layers, of the moving skyrmion for a DC pulse with J=5.6×10¹¹A/m² at t=0.53ns. Note that slightly different parameters were used for these micromagnetic simulations with α=0.216, H<sub>k,top</sub>=12.4 mT, H<sub>k,bottom</sub>=35.4 mT, γ=175.9 GHz/T, D=0.62 mJ/m².
a 1D linescan through (a) and (b). (d) shows the linescan using the vector arrows representing the magnetic moments.

<figure><img src="image_56.png" /></figure>

**Figure S40 Maps of topological charge density for different cases.** (a) and (c) represent the topological charge density map for the top and bottom FM layer at t=0ns (static), respectively. Similarly, (c) and (d) represent the topological charge density map for the top and bottom FM layers, of the moving skyrmion for a DC pulse with J=5.6×10¹¹A/m² at t=0.53ns. Note that slightly different parameters were used for these micromagnetic simulations with α=0.216, H<sub>k,top</sub>=12.4 mT, H<sub>k,bottom</sub>=35.4 mT, γ=175.9 GHz/T, D=0.62 mJ/m².
<figure><img src="image_56.png" /><figcaption>**Figure S41:** Instantaneous velocity vs time for $J=1×10^{11}$ A/m$^2$. Black dashed line shows the average velocity calculated from the displacement of skyrmion.</figcaption></figure>

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /><figcaption>**Figure S42:** Instantaneous velocity and average velocities for varying current densities</figcaption></figure>

(a) Instantaneous velocity for a Gaussian current pulse of varying amplitudes. The right y-axis shows the shape of the injected pulse (dashed pink line). (b) Maximum instantaneous velocity and average velocity as a function of current pulse amplitude. The solid lines show the analytically calculated velocity.

<figure><img src="image_59.png" /></figure>

<figure><img src="image_60.png" /></figure>
<figure><img src="image_57.png" /><figcaption>**Figure S41:** Instantaneous velocity vs time for J=1×10¹¹ A/m². Black dashed line shows the average velocity calculated from the displacement of skyrmion.</figcaption></figure>

<figure><img src="image_58.png" /></figure>

<figure><img src="image_59.png" /><figcaption>**Figure S42:** Instantaneous velocity and average velocities for varying current densities</figcaption></figure>

(a) Instantaneous velocity for a Gaussian current pulse of varying amplitudes. The right y-axis shows the shape of the injected pulse (dashed pink line). (b) Maximum instantaneous velocity and average velocity as a function of current pulse amplitude. The solid lines show the analytically calculated velocity.

<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>
<figure><img src="image_61.png" /><figcaption>**Figure S44** Current induced skyrmion deformation A Time dependence of the average skyrmion diameter during the application of a 0.5 ns current pulse with density $J=8.9×10^{11}$ A/m². The diameter is calculated from the area of the skyrmion. B skyrmion velocity vs current density for a 0.5 ns current pulse (black squares). Skyrmion dimension along the major ("D₁", open blue triangles) and minor ("D₂", blue circles) axes of the skyrmion elliptical shape at the maximum of the pulse. C Angle $\phi$ of the DW magnetization along the x axis (see inset, right DW) for the bottom (black squares) and top (pink triangles) layers. The angle is calculated with respect to the -x axis. Inset: skyrmion magnetization at the maximum of the current pulse during the injection of a 0.5 ns pulse with density $8.9×10^{11}$ A/m².</figcaption></figure>

<figure><img src="image_62.png" /><figcaption>**Figure S45** Comparison of current induced DW and skyrmion dynamics A Current induced DW (black square) and skyrmion (red circles) velocity vs current density. Inset: example of the magnetization pattern of one layer in the SAF during the current pulse injection. B DW angle $\phi$ vs current density of the top (red dots) and bottom (black dots) layers of a DW in a track (open symbols) and a skyrmion (DW position same as Figure S44C, inset).</figcaption></figure>
**Figure S43 Average velocities for short current pulses** (a) Average velocity as a function of the Gaussian pulse width (FWHM) for J=4×10¹¹ A/m² and J=8.9×10¹¹ A/m². (b) Average velocity vs J for two different skyrmion diameters obtained for two different sets of parameters.

<figure><img src="image_62.png" /></figure>

**Figure S44 Current induced skyrmion deformation** A Time dependence of the average skyrmion diameter during the application of a 0.5 ns current pulse with density J=8.9×10¹¹ A/m². The diameter is calculated from the area of the skyrmion. B skyrmion velocity vs current density for a 0.5 ns current pulse (black squares). Skyrmion dimension along the major (“D₁”, open blue triangles) and minor (“D₂”, blue circles) axes of the skyrmion elliptical shape at the maximum of the pulse. C Angle φ of the DW magnetization along the x axis (see inset, right DW) for the bottom (black squares) and top (pink triangles) layers. The angle is calculated with respect to the -x axis. Inset: skyrmion magnetization at the maximum of the current pulse during the injection of a 0.5 ns pulse with density 8.9×10¹¹ A/m².

<figure><img src="image_63.png" /></figure>

**Figure S45 Comparison of current induced DW and skyrmion dynamics** A Current induced DW (black square) and skyrmion (red circles) velocity vs current density. Inset: example of the magnetization pattern of one layer in the SAF during the current pulse injection. B DW angle φ vs current density of the top (red dots) and bottom (black dots) layers of a DW in a track (open symbols) and a skyrmion (DW position same as Figure S44C, inset).
<figure><img src="image_63.png" /><figcaption>**Figure S46 Impact of the RKKY interaction and pulse width on the skyrmion velocity A**

Skyrmion velocity and diameter at max current vs A. RKKY field for a pulse width of 0.5 ns and B. pulse width for an RKKY field of $\mu_0H_{RKKY}=205$ mT for $J=8.9 \times 10^{11}$ A/m$^2$.</figcaption></figure>

<figure><img src="image_64.png" /><figcaption>**Figure S47 Impact of the RKKY interaction and pulse width on the skyrmion dynamics**

(A-D) A Velocity v, B domain wall magnetization angle $\phi$ (bottom layer), dimension of the skyrmion C along the major (D1) and D minor (D2) axis for (i) a 0.5 ns current pulse and a RKKY field $\mu_0H_{RKKY}=205$ mT (red square), (ii) a 0.5 ns current pulse and $\mu_0H_{RKKY}=410$ mT (blue circles) (iii) a 0.1 ns current pulse and $\mu_0H_{RKKY}=205$ mT (black triangles) (E-G)

Magnetization pattern at the maximum of the current pulse corresponding to E (i), F (ii) and G (iii). The angle $\phi$, and dimensions D1 and D2 are measured at the current pulse maximum.</figcaption></figure>
<figure><img src="image_64.png" /><figcaption>**Figure S46 Impact of the RKKY interaction and pulse width on the skyrmion velocity A**

Skyrmion velocity and diameter at max current vs A. RKKY field for a pulse width of 0.5 ns and B. pulse width for an RKKY field of $\mu_0H_{RKKY}=205$ mT for $J=8.9 \times 10^{11}$ A/m$^2$.</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>**Figure S47 Impact of the RKKY interaction and pulse width on the skyrmion dynamics**</figcaption></figure>

(A-D) A Velocity v, B domain wall magnetization angle $\phi$ (bottom layer), dimension of the skyrmion C along the major (D1) and D minor (D2) axis for (i) a 0.5 ns current pulse and a RKKY field $\mu_0H_{RKKY}=205$ mT (red square), (ii) a 0.5 ns current pulse and $\mu_0H_{RKKY}=410$ mT (blue circles) (iii) a 0.1 ns current pulse and $\mu_0H_{RKKY}=205$ mT (black triangles) (E-G)

Magnetization pattern at the maximum of the current pulse corresponding to E (i), F (ii) and G (iii). The angle $\phi$, and dimensions D1 and D2 are measured at the current pulse maximum.
<figure><img src="image_65.png" /><figcaption>**Figure S48 Impact of the RRKY exchange interaction on the inertial dynamics (a-b)**</figcaption></figure>

Time dependence of the skyrmion velocity in response to a 2 ns square pulse (J=1011 A/m²) for different HRKKY interaction fields ((a) and (b) are zoom on the rise and fall of the velocity). (c) inverse of the time constant of the SAF skyrmion displacement computed by fitting the rising and falling edge of the skyrmion speed. The mean and the error bar are plotted, where the error bar is the standard deviation over the data. The mean value takes into account the top and bottom skyrmions and the rise and fall time constants (i.e. 4 data points each). Note that slightly different parameters were used for these micromagnetic simulations with α=0.216, H<sub>k,top</sub>=12.4 mT, H<sub>k,bottom</sub>=35.4 mT, γ=175.9 GHz/T, D=0.62 mJ/m².

<figure><img src="image_66.png" /><figcaption>**Figure S49: Calculation of the topological spin Hall torque** Comparison of the speed of AF skyrmion due to SOT driving (in orange) and the equivalent speed due to the topological spin torque (in blue and violet). We compare the results for two values of the topological spin torque range: $\lambda^2$ = 3 nm and 50 nm². The applied current density is 8×10¹¹ A/m² along the x direction, the domain wall width is fixed at 24.5 nm and we vary the core radius of the skyrmion from 24.5 nm (R/Δ = 1) to 122.5 nm (R/Δ = 5). Note that slightly different parameters were used for these micromagnetic simulations with $\alpha$=0.216, H<sub>k,top</sub>=12.4 mT, H<sub>k,bottom</sub>=35.4 mT, $\gamma$=175.9 GHz/T, D=0.62 mJ/m².</figcaption></figure>

**References**

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<figure><img src="image_66.png" /><figcaption>**Figure S48 Impact of the RRKY exchange interaction on the inertial dynamics (a-b)**</figcaption></figure>

Time dependence of the skyrmion velocity in response to a 2 ns square pulse (J=1011 A/m²) for different HRKKY interaction fields ((a) and (b) are zoom on the rise and fall of the velocity). (c) inverse of the time constant of the SAF skyrmion displacement computed by fitting the rising and falling edge of the skyrmion speed. The mean and the error bar are plotted, where the error bar is the standard deviation over the data. The mean value takes into account the top and bottom skyrmions and the rise and fall time constants (i.e. 4 data points each). Note that slightly different parameters were used for these micromagnetic simulations with α=0.216, H<sub>k,top</sub>=12.4 mT, H<sub>k,bottom</sub>=35.4 mT, γ=175.9 GHz/T, D=0.62 mJ/m².

<figure><img src="image_67.png" /><figcaption>**Figure S49: Calculation of the topological spin Hall torque** Comparison of the speed of AF skyrmion due to SOT driving (in orange) and the equivalent speed due to the topological spin torque (in blue and violet). We compare the results for two values of the topological spin torque range: $\lambda^2$ = 3 nm and 50 nm². The applied current density is 8×10¹¹ A/m² along the x direction, the domain wall width is fixed at 24.5 nm and we vary the core radius of the skyrmion from 24.5 nm (R/Δ = 1) to 122.5 nm (R/Δ = 5). Note that slightly different parameters were used for these micromagnetic simulations with $\alpha$=0.216, H<sub>k,top</sub>=12.4 mT, H<sub>k,bottom</sub>=35.4 mT, $\gamma$=175.9 GHz/T, D=0.62 mJ/m².</figcaption></figure>

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71. A. Baruth, L. Yuan, J. D. Burton, K. Janicka, E. Y. Tsymbal, S. H. Liou, S. Adenwalla, Domain overlap in antiferromagnetically coupled [Co/Pt]NiO[Co/Pt] multilayers. *Appl. Phys. Lett.* **89**, 202505 (2006).

72. L. Berges, Size-dependent mobility of skyrmions beyond pinning in ferrimagnetic GdCo thin films. *Phys. Rev. B* **106** (2022).

73. C. A. Akosa, O. A. Tretiakov, G. Tatara, A. Manchon, Theory of the Topological Spin Hall Effect in Antiferromagnetic Skyrmions: Impact on Current-Induced Motion. *Phys. Rev. Lett.* **121**, 097204 (2018).

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