SUPPLEMENTARY INFORMATION

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1 Spin-orbit torque magnetization switching controlled by geometry C.K.Safeer, Emilie Jué, Alexandre Lopez, Liliana Buda-Prejbeanu, Stéphane Auffret, Stefania Pizzini, Olivier Boulle, Ioan Mihai Miron, Gilles Gaudin Table of contents: S1. Device fabrication S2. Angular dependence of DW velocity, critical current and critical in-plane field. S3. Dynamic DW deformation S4. DW motion asymmetry at different current densities S5. Imaging of DW motion during switching S6. Switching: Size, speed and nucleation limits NATURE NANOTECHNOLOGY 1

S1. Device fabrication The sample fabrication process includes several steps of electron beam lithography (EBL), electron beam physical vapor deposition (EBPVD) as well as mechanical and chemical

2 S1. Device fabrication The sample fabrication process includes several steps of electron beam lithography (EBL), electron beam physical vapor deposition (EBPVD) as well as mechanical and chemical etching. The Pt(3nm)/Co(0.6nm)/AlOx(2nm) trilayer was deposited by sputtering. On top of this trilayer, two levels of masks were fabricated using EBL followed by the EBPVD deposition. The first mask consists of Ti 10nm and Au 20nm (mask1) in the form of a rectangle. On top of this, we placed a second Ti 5nm mask (mask2) having the form desired for the magnetic device. After a first step of mechanical etching everything outside the rectangle defined by the mask1 was removed. A second step of selective chemical etching was used to remove both AlOx and Co outside the area still covered by Ti (mask 2). A schematic diagram of the different structures obtained after each etching is shown in figure S1. Ti 5nm Au 50nm electrical contacts made by standard EBL and EBPVD were used to inject the electric current. Figure S1. The schematic diagrams of the different structures after each level of etching. a. The structure after the fabrication of the masks above the Pt/Co/AlOx layer. Mask 1 has rectangular shape and mask 2 has the shape of final geometry expected for the magnetic object. b. The structure after the mechanical etching. The rectangular part under the mask 1 was protected from etching. c. The final structure after chemical etching. 2 NATURE NANOTECHNOLOGY

3 SUPPLEMENTARY INFORMATION S2. Angular dependence of DW velocity, critical current and critical in-plane field. Angular dependence of DW velocity. The angular dependence of non-collinear current induced DW motion was shown in the figure 3 of the main text. To confirm that this angular dependence is a characteristic feature of the DW velocities rather than an offset contribution to the DW displacements, we performed additional measurements. We measure the DW velocity following the same procedure as in I.M.Miron et al. 14. For each wire, we measured DW displacements at a constant current density (1.81x10 12 Am -2 ) for three different current pulse lengths. DW displacement varies linearly with pulse length as shown in Figure S2b. The DW velocity is extracted from the slope of the linear dependence. We observe the same trend as the DW displacement curves, confirming that the angular dependence is an intrinsic feature of the DW velocity. Angular dependence of critical current. Another important parameter that characterizes the current induced DW motion is the critical current density. In our experiment the critical current is defined as the current required to displace the DWs over the smallest distance detectable using our microscope (200nm). We have performed measurements for different pulse durations. Ideally one should use continuous current for this experiment, but unfortunately increasing the length of the current pulses also increases the sample temperature, which leads to nucleation of new domains. Therefore we have measured the critical current for 1000 pulses with three different durations (1.8ns, 3.8ns and 6.3ns). The critical current density shown in Figure S3c, mirrors the dependence of the DW velocity on tilt angle of the wire. This result also excludes the tilt formation in the wires: not only the pulses are too short, but also DWs do not move over a sufficiently long distance to form the tilt. Angular dependence of critical in-plane field. The standard method for characterizing the DMI field in the DMI-SOT model is to apply a longitudinal field (parallel to the electric current and perpendicular to the DW). We have performed a similar measurement. In our case the magnetic field is parallel to the electric current while the angle with respect to the DW varies from wire to wire. In the 1D geometry, the usual interpretation of this data is to consider that zero velocity corresponds to a Bloch DW, that is to say the component of the DW magnetization along the current is zero (Figure S3f). NATURE NANOTECHNOLOGY 3

4 Figure S2. The DW displacement and DW velocity a. The variation of DW displacements with respect to the angle of the wire for a current pulse of 3.75 ns, 3.14 ns, and 2.67 ns. Here we plot DW displacements corresponding to single current pulses, calculated by dividing the total DW displacement for each wire by the total number of pulses used for each case. Red curves correspond to positive current, while blue curves stand for negative current. b. DW displacement variation vs. the pulse length for each wire. Note that for I, we do not plot the displacements for wires with φ > 30 because all the displacements were zero. c. DW velocity vs. φ the tilt angle of the wire. Each DW velocity value in this graph is extracted from the slope of the corresponding linear plot from panel b. d. critical current dependence on tilt angle. The inset shows the data corresponding to the 3 pulse durations normalized to the value obtained for the straight wire. the solid line is the result expected from the DMI-SOT model e. DW displacement obtained for 30 pulses of 2,6 ns at a current density of 1.6x10 12 Am -2 as a function of the in-plane field, for five of the wires. The wires tilted at 15 and 30 exhibit faster DW velocity compared to the straight wire independently of the value of H X. The inset shows the value of the interpolated critical field H c required to stop the motion. We observe a significant discrepancy compared to the DMI-SOT model that predicts a simple cosine variation. f. Schematic representation of the effect of H X on the DW structure in the DMI- SOT model. When the external field is sufficient of compensate the internal DMI field, the DW in the straight wire becomes Bloch, its m X component vanishes and the DW motion stops. For the same H X value, the m X component of the DW magnetization in the tilted wires has already changed sign. This means that the critical field required to stop the DW motion in the tilted wires must be smaller than in the straight wire. This is opposite to what we observe. 4 NATURE NANOTECHNOLOGY

5 SUPPLEMENTARY INFORMATION In the non-collinear geometry, this does not mean that the DW has a Bloch structure; it only means that the component of the DW magnetization along the current is zero. Depending on the wire orientation, the SOT will be zero for different degrees of DW distortion. Experimentally, we observe that contrary to expectations from the DMI-SOT model, stopping the DW motion in the tilted wires requires in-plane fields significantly larger than the parallel wire Figure S3e. This is consistent with the observation of asymmetric velocity and the critical current dependence on the tilt angle. S3. Dynamic DW deformation The current induced DW motion within the SOT-DMI model can be illustrated using a graphical construction, as shown in Figure S3. When the current is applied through the DW, the damping-like SOT induces a distortion of the Néel DW structure. The restoring internal field (H DMI ) creates an out of plane torque that displaces the DW. The dissipative torque associated to the DW motion is opposed to the T DL. In steady state, the in-plane torques must cancel each other T = α DL T DMI and the out-of-plane torque dictates the DW velocity v T DMI As explained in the main text, a possible cause of the velocity asymmetry that we observe experimentally may be the DW distortion by the T DL. Both T DMI and T DL depend on θ, the angle between the actual DW magnetization and its static equilibrium position dictated in this case by the DMI TDMI H DMI sin(θ ) and TDL H DL cos(θ ) Here H DMI and H DL are the corresponding effective fields. In this case, the deformation angle is: tan( θ ) H = α H DL DMI and the DW velocity in steady motion is: v sin(θ ) DW H DMI NATURE NANOTECHNOLOGY 5

6 An important specificity of the SOT-DMI model is that a significant DW dynamic distortion in steady motion is synonym of large velocity. If the DW velocity is small, the accompanying DW distortion is also small. Experimentally, even a moderate reduction of the current density, leads to a large decrease of the velocity. This is due to the imperfections in the material structure, which create local pinning centers for the DW and decrease its velocity. The pinning field does not act uniformly along the DW trajectory. Because of the spatial variations of the pinning potential, the DW motion is fast between pinning sites, but is interrupted by long waiting times at each pinning center. As a consequence, the average velocity is largely determined by the density of pinning centers and their depinning time, but only marginally affected by the short periods of fast DW motion. Therefore we can effectively model consider that the DW pinning acts as an internal field opposing the DW motion. If H DL H pinn H DL H pinn the DW deformation becomes tan( θ ) = 0 α H DMI and the velocity v DW 0 Figure S3. a. Schematic representation of the dynamic DW distortion in the SOT-DMI model. The orientation of the DW magnetization (black arrow in the grey area), initially pointing along H DMI, is modified by the presence of T DL. As the DW equilibrium structure is distorted, the effective restoring field (H DMI in this case) exerts a torque pointing out-of the plane. The perpendicular magnetization variation that moves the DW, also produces a dissipative torque αt DMI. In steady state motion the in-plane orientation of the DW magnetization is fixed by the balance of in-plane torques T DL and αt DMI. b. In the presence of forces that oppose the DW motion, such as pinning to defects, the effect of T DL on the DW distortion is inhibited by the effective pinning torque, T Hpinn. Since there is no DW distortion, there cannot be any out-of plane torque and thus the velocity must be zero. 6 NATURE NANOTECHNOLOGY

7 SUPPLEMENTARY INFORMATION S4. DW motion asymmetry at different current densities In Figure 2 of the main text, we have shown the angular dependence of current induced DW motion in the case of circular magnetic bubble domains. The physics behind this effect could be explained by the DW distortion due to the combined action of DMI and SOT. But this DW deformation is expected to be large only when the DW is moving fast. If so, for slow DW motion the angular dependence of circular bubbles should disappear. In order to check this, we performed the same experiment at large and small current densities. For the DW motion shown in the Figure 2, the applied current density was 1.6x10 12 Am -2. We repeated the same experiment for current densities of 1.1x10 12 Am -2 and 2.1x10 12 Am -2 and the corresponding images are shown in Figure S4. For small current density the DW motion was slowed down approximately by three orders of magnitude. Nevertheless, the angular dependence of DW motion is still preserved. This indicates that the physical origin of the velocity variation is certainly more complex than the DMI-SOT model. Figure S4. The Kerr differential images of the DW motion in the case of the bubble domains. The white arrows show the current direction. The dotted lines show the initial DW position. a. The down/up DW motion for current density 2.1x10 12 Am 2. Here the maximum DW velocity was approximately 70m/s b. The up/down DW motion for current density 1.1x10 12 Am -2 and c. Down to up DW motion for the same current. Here the domain wall velocity was very small, approximately 0.1m/s. The images show that the up/down and down/up cases, the asymmetric angular dependence of DW motion is opposite (towards right and left respectively). The maximum DW displacement is always at an angle (approximately 30 ) irrespective to the strength of the current densities. NATURE NANOTECHNOLOGY 7

S5. Imaging of DW motion during switching In Figure 4 of the main text, we have shown images of the switching after applying a series of 8 current pulses.

8 S5. Imaging of DW motion during switching In Figure 4 of the main text, we have shown images of the switching after applying a series of 8 current pulses. Additional to this, we also imaged step by step DW motion during the switching. For this we divided the 8 pulse series into 4 series of 2 pulses and made differential imaging after each series. The corresponding images are shown in Figure S5. Initially we saturated the magnetization of the devices by applying a magnetic field. Then we applied current that induce nucleation and domain wall motion. In all the cases, DW motion was initiated in one of the pins, and DW propagation lead to the magnetization reversal, as described in the figure 3 of the main text. Figure S5: Step by step imaging of the DW motion during the switching a. for u shape b. for s shape. The length of the current pulse used for u shape and s shape was 4.4ns and 5ns respectively. The first raw contains the images of the initial magnetic states of the switches. The black arrows show the direction of applied current. The further consecutive images in each column are corresponding to the step by step DW motion after applying a series of 2 current pulses at each step. These images confirm that the switching occurs according to the nucleation and DW propagation mechanism explained in figure 3 of the main text. S6. Switching: Size, speed and nucleation limits The switching scheme that we propose is based on heat induced domain nucleation and selective DW propagation. The speed of the switching depends on the total length of the device as well as on the DW velocity. The smaller the size of the bit length and larger the DW velocity, the faster will be the switching. Due to the resolution limit of the optical microscope, 8 NATURE NANOTECHNOLOGY

9 SUPPLEMENTARY INFORMATION we chose the 2 µm length for all the straight wire in all the devices. Ideally it should be possible to make switches with smaller size that work using exactly the same principle as long as their length remains larger than the DW width. A second way of improving the switching time is to increase the DW velocity. The maximum DW velocity reported in Pt/Co/AlOx was around 400 m/s. In our experiments the maximum DW velocity achieved was around 100 m/s. This is because the sample resistance limited the maximum current density that we could apply. By using devices with less resistance, it can be possible to improve the DW velocity and thus the switching speed. Figure S6. The current density required for the switching a. The graph showing the variation of pulse length required for switching with respect to the current density. The dark and light blue regions correspond to the saturated and non-saturated initial states of the switches. Note that the switching pulse length window becomes narrow as the current density increases. This is because for large current density, a small change in the pulse length produces large a variation in Joule heating and nucleation becomes easier. b. A schematic diagram of switching from a saturated state. The reversal begins with a nucleation (upper panel) followed by DW propagation through the straight wire and into the second tilted wire, where it stops (lower panel). c. Since a DW is already present, the high current density required for nucleation is not required. Therefore, the object can be switched back with lower current density. This phenomenon increases the switching range (light blue area in panel a). The range of pulse width and height where controlled switching occurs is limited by domain nucleation. As discussed in Figure 3 of the main text, the shape of the switch contains two tilted wires and a straight wire. The switching is only possible if the nucleation takes place on the tilted wires; not on the straight wire. As discussed before, the tilted ones have a pin shape in order to decrease the thermal stability in the narrower regions. Thus the current required to nucleate the DW in the tilted wires is always smaller than that in the straight wire. NATURE NANOTECHNOLOGY 9

10 At the upper limit, when the current becomes large enough to create nucleation in the straight wire, the switching becomes stochastic. The critical current density required for switching depends on the initial magnetic state of the switches. For all the switching experiments that we discussed so far, magnetization was initially saturated using an external magnetic field. But there is also another possible initial state where DW is already present in the tilted wires. In this case no nucleation is required for switching. One such possible situation is schematically shown in Figure S6. To switch our device, we apply a current such that the nucleated DW in one tilted wire propagates through the straight wire and reaches the second tilted wire. Here, because of the opposite tilt, the DW will stop. Now if we want to do a second switching to the opposite direction, since there is already a DW present in the tilted wire, there is no need of nucleation. Instead we only need to apply current to propagate the DW in opposite direction. Since the current required for propagation is smaller than that for the nucleation, the critical switching current becomes smaller. Note that the possibility to have DWs that remain in the pins depends on the relative magnitude of two effects. The retaining force is given by the strength of the pining to defects. The expelling force is derived from the DW energy reduction corresponding to the shortening of the DW as it moves toward the end of the triangle. To obtain saturation of the tilted wires they could be shortened, thereby increasing the pin angle and the resulting expelling force. We chose to work with long tilted wires such that the DWs do not move out by themselves, since we can easily saturate the samples using external field. In order to evidence these different types of behavior, we systematically studied the switching as a function of the length and the intensity of the current pulses. The result shown in figure S6 indicates the existence of three different regions: no switching, switching and nucleation regions. For a particular current density, an increase in the length of the current pulse increases the joule heating. For short pulses, there is not enough heat for the nucleation and thus no switching. When we increase the pulse length, the heating becomes sufficient to nucleate on the tilted wires and the switching begins. The upper limit is given by the pulse length where nucleation occurs in the straight wire. Above, the switching becomes stochastic. The critical current dependence on the initial state is also illustrated in Figure S6a where the switching region is further divided into two. The dark and light blue region corresponds to the switching window for saturated and non-saturated initial. Note that he DW pinning field in the tapered region is found to be almost identical to the propagation field in the straight section H tapered =16mT and H straight =15.5mT. 10 NATURE NANOTECHNOLOGY

SUPPLEMENTARY INFORMATION

In the format provided by the authors and unedited. Room temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures: Supplementary Information Olivier Boulle, Jan Vogel, Hongxin Yang, Stefania