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1 SUPPLEMENTARY INFORMATION doi: /nature Ising-Macrospin model The Ising-Macrospin (IM) model simulates the configuration of the superlattice (SL) by assuming every layer is a single spin (macrospin) and the magnetization of this layer is only allowed to point along the easy-axis, i.e. up or down (Ising). The inputs to the model are the coercivity H c (Oe) and thickness t (nm) of each layer and the coupling J (Oe nm) [1] between the layers. Based on these three parameters the switching field of each layer can be calculated. First we calculate the total coupling field J = J tot /t, where J tot is the total coupling field in Oe nm on a layer from its nearest neighbors (two contributions for the bulk of the SL and one for the layers at the SL edges) and depends on the relative alignment of the layer with its nearest neighbors. The switching field can then be calculated by adding (subtracting) H c from J when the external applied field is aligned parallel (anti-parallel) with the layer. Hence, when the external applied field is larger than this value the layer switches. This process continues for every new configuration and applied field value. This IM model is valid for J /H k < 1 where J = J tot /t is the total exchange field of a magnetic layer and H k is its effective perpendicular magnetic anisotropy field. This ensures that the magnetization reversal proceeds via spin-flip processes (magnetization always points along the easy-axis; Ising) [2, 3, 4]. To validate the use of the IM model for our SL we need to determine the effective perpendicular magnetic anisotropy field H k. In the following we determine H k =2K u /M s 4πM s [1] by measuring the saturation magnetization M s and perpendicular magnetic anisotropy K u for individual layers used in the SL. The samples consist of a single magnetic layer Ta(2)/Pt(20)/CoFeB(X) /Pt(2) with X = 0.6, 0.7 and 0.8 nm, grown on square 1.00x1.00 ± 0.02x0.02 cm 2 silicon substrates. In Fig. S1 the magnetic moment µ measured by VSM as a function of magnetic field applied perpendicular to the sample plane is shown. A square hysteresis loop with 100% remanence is observed indicating a well-defined effective perpendicular magnetic anisotropy. Furthermore, the coercivity H c varies by less than 10% for the different magnet layer thicknesses and is considered constant for all layers in the IM modeling. The measured moment as function of magnetic layer thickness is shown in Fig. S1b (left axis) which increases linearly with thickness, as expected. From the total volume of the samples M s can be determined. This is shown in Fig. S1b (right axis) and is constant at 1290 ± 70 emu/cm 3. The error bar is 1

2 RESEARCH SUPPLEMENTARY INFORMATION Figure S1: Determining the effective perpendicular magnetic anisotropy a. Hysteresis loops obtained by VSM on 1x1 cm 2 Ta(2)/Pt(20)/ CoFeB(X)/Pt(2) samples with X = 0.6, 0.7 and 0.8 nm. b. Saturated total moment (left axis) and calculated saturation magnetization M s as a function of magnetic layer thickness. c. Residual perpendicular magnetization fraction M z for three different thickness samples as a function of applied field for different applied field angles α relative to the surface normal as shown in the inset of d. The full lines are a fit to a Stoner-Wohlfarth model. d. Effective perpendicular anisotropy field H k as a function of magnetic layer thickness

3 SUPPLEMENTARY INFORMATION RESEARCH dominated by the inaccuracy in determining the volume of the FM layer. We have measured K u of the layers using the anomalous Hall effect (AHE) as a probe of the residual perpendicular component of the magnetization M z as a function of magnetic field H applied at various angle(s) α to the surface normal, as shown in the inset of Fig. S1d [5, 1]. The M z component versus field for different angles is fitted using a Stoner-Wohlfarth model with only one fitting parameter; the uniaxial anisotropy K u. Using a least squares fitting routine the complete data set for each sample is fitted and the obtained K u is listed in Fig. S1d. H k is calculated and plotted as function of layer thickness in Fig. S1d. The perpendicular magnetic anisotropy originates from the Pt/CoFeB interface and decreases linearly with t as the interface contribution decreases linearly with increasing layer thickness. Finally, the use of the IM model for the SL as shown in Fig. 1 of the main text can be justified by determining the largest ratio J /H k in the used SL. This is found for M2: JM2 = (2J 0/t 0 )/H k = 2635/11500 = 0.23, i.e. J /H k < 1. 2 Rearranging of the injector layers In Fig. S2a the hysteresis loop of a tri-layer sample is shown together with a fit using the IM model, the used parameter are shown in the inset. This tri-layer is prepared in the same way as the injector in the SL shown in Fig. 2b of the main text. In Fig. S2b the configuration after every switch is shown as indicated by the numbers in the red balloons. Starting from negative saturation the first layer to switch is M2 as it is the layer feeling the highest total coupling as explained in the main text. For positive field the first switch (2) - (3) indicated by (!) is the rearranging of all three layers. The switching field of this rearrangement is not well described by the IM model and is predicted to happen at higher field. For increasing positive field the field at which M1 and M3 will switch in the IM model is given by H = H c + J/t = 1413 Oe. As M1 and M3 switch at the same field M2 s optimal alignment is now, with M1 and M3 switched, anti-parallel with the applied field direction and a full tri-layer rearrangement is found. We anticipate that the reversal taking place at lower fields as predicted by the IM model in the experimental system is due to domain nucleation and domain-wall propagation processes happening in three neighboring layers simultaneously. As the reversal commences the expansion of nucleated 3

4 RESEARCH SUPPLEMENTARY INFORMATION Figure S2: Rearranging of the injector layers a. Hysteresis curve of a tri-layer sample identical to the injector region of the SL presented in Fig. 2a of the main text. The full line is a fit using the IM model, the number in the balloons correspond to the magnetic configuration as presented in b. The transition indicated by the exclamation mark (!) is not well reproduced by the IM model. domains via domain walls propagating through the individual but neighboring layers interact due to the large coupling between the layers. The complex interactions governing these mechanism could result in a reduction of the reversal field predicted by the IM model [6]. The configurations after the switch, however, correspond again to full-layer switches as can be deduced from the M/M s levels. Note that the failure of the IM model to reproduce the rearranging of the injector layers does not interfere with our soliton injection/propagation schemes as the fields used to propagate the soliton are lower than the value where the rearranging takes place. The exact mechanism of reversal will be addressed in future experiments. 3 Optimizing the ratchet scheme One of the challenges for large scale memories is the distribution in material properties after processing stacks in large arrays of storage cells. For instance, the switching field distribution impeded the successful large scale integration of the original field-written MRAM. It took novel and clever concepts like 5 4

5 SUPPLEMENTARY INFORMATION RESEARCH toggle writing [7], and more recently, spin-transfer-torque [8] based MRAM to overcome these problems. For the field-driven ratchet scheme presented in the main text similar obstacles could be encountered. We would like to stress, however, that in the case of any new large scale memory one has to decide on a specific device/material design and operating mechanism before one can address specific technical issues such as switching field distributions for implementation. For instance, we use field-driven soliton propagation to illustrate the basic operation of the ratchet but, speculatively, other schemes might use a spin-transfer-torque [8] or spin-orbit-torque (e.g. Rashba [9, 10], Spin-Hall effect [11]) mechanism to propagate solitons. In the following, we will illustrate the versatility of the ratchet scheme by optimizing for switching field distributions considering field-driven soliton propagation. We stress again that we in no way imply that this is the only possible propagation scheme. Here, we specifically address the spread in coercivity (H c ) commonly encountered when patterning perpendicularly magnetized materials. This is especially valid in multilayers with ferromagnetic coupling, as is dramatically illustrated in the work by Thompson et al. in [Pd/Co] 8 multilayers [12]. In this work all layers are highly ferromagnetically coupled and so interact strongly with each other at the moment of switching. However, we note that as the soliton propagates through our anti-ferromagnetically coupled multilayer the energetic environment of the magnetic layer in the middle of the soliton (i.e. the layer which needs to switch in order to advance the soliton position by one click) is essentially the same energetic environment as an isolated single layer. Hence, the scaling of the switching field distribution of the soliton layers is expected to be very similar to the coercivity scaling of the free layer of perpendicular STT-MRAM devices, currently an intensely investigated topic. To estimate the expected distribution in coercivity when patterning a whole wafer to 50 nm pillars we take the value by Gajek et al. [13], who reported a 15% spread in coercivity (= 1 standard deviation σ) of perpendicularly magnetized CoFeB based 50 nm MTJ cells (similar to the magnetic material used in our study). Furthermore, to ensure a low bit error rate (BER) we impose a5σ splitting (a BER of 10 6 requires a 5σ separation) between switching fields, i.e. between different magnetization-configurations (states) during the soliton propagation. Let us first consider the ratchet scheme parameters of the stack presented in the main text in light of these requirements. Fig. S3a shows the switching field distributions using a 15% spread on the coercivity i.e. H c = 230 Oe 5

6 RESEARCH SUPPLEMENTARY INFORMATION a Switching probability (arb. u.) H c = 230 Oe J 1 = 650 Oe nm J 2 = 180 Oe nm t 1 = 0.7 nm t 2 = 0.8 nm H p2 H p2 H p1 H p1 H b H p = 2.4 H p1-2 = 11 Overlap H b = 13 = 35 Oe b Switching probability (arb. u.) H c = 230 Oe J 1 = 660 Oe nm J 2 = 115 Oe nm t 1 = 0.6 nm t 2 = 0.8 nm = 35 Oe H p2 H p = 6.7 H p1-2 = 6.7 H p1 H p1 H p2 H b H b = H z (Oe) H z (Oe) Figure S3: Switching field distributions calculated using a 15% spread in coercivity and parameters as used in the main text (see inset). a. Switching field distribution for the ratchet scheme parameters used in the main text. b. Switching field distribution using parameters to ensure a 5σ splitting between all relevant switching fields. with σ = 35 Oe (normal distribution). This figure essentially shows the same information as given in Fig. 1 of the main text with the addition of the switching field distribution. We have now added and labeled the switches H p1 and H p2, these correspond to the switching of the bottom layer of the soliton at stage one and two of soliton propagation, respectively. The switching of these bottom layers during propagation destroys the ratchet behavior when H p1 <H p1 or H p2 >H p2. We have listed the equations governing all the switching fields here: H p1 = H c +(J 1 J 2 )/t 2, H p1 = H c +(J 1 J 2 )/t 1, H p2 = H c +(J 1 J 2 )/t 1, H p2 = H c +(J 1 J 2 )/t 2, H b = H c +(J 1 + J 2 )/t 2. (1) The relevant splittings for the ratchet scheme between these switching fields are expressed as: 7 6

7 SUPPLEMENTARY INFORMATION RESEARCH H p = H p1,2 H p1,2 = (J 1 J 2 )(t 1 t 2 )/(t 1 t 2 ), H p1 2 = H p1 H p2 = 2H c +(J 1 J 2 )(t 1 t 2 )/(t 1 t 2 ), H b = H b H p1 = 2J 2 /t 2. (2) As discussed before, a 5σ splitting between the switching fields is required. We can directly see from Fig. S3a that the tails of the switching distribution of H p1 and H p1 as well as H p2 and H p2, which are identical and expressed by H p (see Eqs. 2) are overlapping (as expected since the splitting between these fields H p is only 2.4σ). In a large array of elements this would severely increase the BER. Here we use the same statistics for inter- and intra-stack H c distributions. In Fig. S3b we show the switching field distribution for a different set of parameters which fulfill the 5σ criterion for all splittings simultaneously. All these parameters are experimentally viable and are not too different from the parameters used in the main text. In the following, we will illustrate how we have derived these parameters. In Fig. S4a we generalize the situation. The red (blue) regions indicate where the 5σ criterion is met (not met) for all three switching field splittings simultaneously as a function of the reduced parameters J 1 /J 2 and H c /J 2, for t 1 =0.7, t 2 =0.8nm. The area where the 5σ criterion is fulfilled (red) cuts out a triangular shape which widens with increasing J 1 /J 2 and H c /J 2 ratios and is cut off at H c /J 2 =3.3 where H b /σ < 5. To interpret this graph we plot the three switching field splittings divided by 1σ in Fig. S4b for the parameters used in the main text (i.e. J 1 /J 2 = 650/180 = 3.6 as indicated by the dashed line in Fig. S4a). All three switching field splittings meet the 5σ criterion for 68 <H c < 110 Oe, corresponding to where the dashed line in Fig. S4a crosses the red area projected on the H c /J 2 axis, i.e <H c /J 2 < Note that for H c < 32 Oe the third of the three operating margins discussed in the main text; H p1 H p2 > 0 (ensuring the soliton propagates only one layer per field step) is violated and the ratchet scheme breaks down. This shows that for a certain range of H c the 5σ criterion is met for all switching field distributions simultaneously, for this set of parameters. Hence, by tuning the parameters J 1, J 2, t 1 and t 2 one can find the optimal parameter set. Whilst this sounds straightforward, the optimal conditions will be dictated by the materials system used, dipole field considerations, spread in J, etc. For instance, in our material system we cannot increase the 7

8 RESEARCH SUPPLEMENTARY INFORMATION a J 1 /J 2 c J 1 /J t 1 = 0.7 nm t 2 = 0.8 nm = 15% of H C >5 < H c /J 2 b Switching field splitting/sigma (-) H c /J 2 d 12 t 1 = 0.6 nm t 2 = 0.8 nm = 15% of H C >5 <5 Switching field splitting/sigma (-) >0 H p H p1-2 H b J 1 = 650 Oe nm J 2 = 180 Oe nm t 1 = 0.7 nm t 2 = 0.8 nm = 15% of H C H C (Oe) > H C (Oe) 5 H p H p1-2 H b J 1 = 660 Oe nm J 2 = 115 Oe nm t 1 = 0.6 nm t 2 = 0.8 nm = 15% of H C Figure S4: Switching field splittings assessment as a function of reduced units and coercivity. a. 5σ criterion assessment (red > 5σ, blue < 5σ) as a function of reduced parameters for t 1 =0.7 nm and t 2 =0.8 nm. b. Switching field splitting divided by 1σ as a function of coercivity for the parameters used in the main text. c. 5σ criterion assessment (red > 5σ, blue < 5σ) as a function of reduced parameters for t 1 =0.6 nm and t 2 =0.8 nm. d. Switching field splitting divided by 1σ as a function of coercivity for the parameters used in the inset. 8

9 SUPPLEMENTARY INFORMATION RESEARCH total coupling field J above H k whilst ensuring Ising-Macrospin behavior. In our surface dominated perpendicular anisotropy material system the thickest layer has the lowest H k, e.g. for a 0.8 nm layer we found H k = 8160 Oe which would allow a maximum total coupling of J 1 + J 2 = J tot = = 6528 Oe nm. The minimal CoFeB thickness that we can use is 0.4 nm, for thinner layers we reach the percolation limit where H k decreases rapidly. Furthermore, one needs to take into account thermal stability, which is reduced with smaller thickness as well as dipole interactions at small lateral sizes. These are all technical issues which also depend on the specific application, device design and driving mechanism as discussed before. Thus, to compensate for a σ = 15% switching field distribution at a coercivity of H c = 230 Oe, as used in the main text, we can use Fig. S4a to select J 1 and J 2 values that satisfy the 5σ criterion. Fig. S4a is however, constructed for t 1 = 0.7 nm and t 2 = 0.8 nm and from Eqs. 2 it can be derived that the difference between the thicknesses t 1 and t 2 has a large effect on H p and H p1 2. To illustrate what happens when the difference between t 1 and t 2 is increased we have plotted in Fig. S4c the situation for t 1 = 0.6 nm and t 2 = 0.8 nm, i.e. only t 1 is decreased by 0.1 nm. Fig. S4c shows that the range of J 1 /J 2 where the criterion is satisfied has decreased compared to Fig. S4a (note the different J 1 /J 2 axis scale between Fig. S4a and c), allowing for smaller J 1 /J 2 ratios. This allows us to minimize the total coupling J tot = J 1 + J 2 to ensure Ising-Macrospin behavior as discussed before. By choosing H c /J 2 = 2 and J 1 /J 2 =5.7, as illustrated by the dashed lines in Fig. S4c, we have a large range along both axes where the 5σ criterion is fulfilled. With H c = 230 Oe this leads to J 2 = 115 Oe nm and J 1 = 660 Oe nm. In Fig. S4d we plot the three switching field splittings for this set of parameters. As can be seen for this particular example we have optimized the switching field splittings at H c = 230 Oe by simply decreasing t 1 and J 2 with a slight increase in J 1 compared to the parameters used in the main text. This brings us back to Fig. S3b where we showed the switching field probability for this set of parameters which satisfy the 5σ criterion for all three switching field distributions simultaneously. The maximum simultaneous switching field splittings for a 15% spread on H c is intrinsically limited to 6.7σ, as can be seen from Fig. S4b and d, and is found when H c = (J 1 J 2 )(t 1 t 2 )/(t 1 t 2 ), i.e. when the coercivity is equal to the field-pressure discussed in the main text. This can be understood by the competing H p and H p1 2 as seen in Fig. S4b and d. The lower the spread on H c the higher maximum switching 9

10 RESEARCH SUPPLEMENTARY INFORMATION field splittings can be obtained and scales as 1/spread, e.g. for a 10% spread in H c a maximum switching field splitting of 10σ can be obtained. This specific example shows the flexibility of the ratchet scheme, where we have concentrated on how the parameters can be optimized to compensate for switching field distributions. Note that when different driving mechanisms or even combination of driving mechanism are used the requirements for some of the switching field distribution splittings could be relieved or even become irrelevant. For example, if spin-transfer-torque is used, which preferentially affects antiparallel aligned layers, and by setting the current direction to affect the top soliton layer, the H p criteria becomes irrelevant as there is no torque on the bottom soliton layer allowing us to disregard the H p1,2 switches in Fig. S3. Note that different driving mechanisms will introduce different technical challenges which will need to be addressed. We believe that the flexibility of the ratchet scheme presented above will greatly help to address the technical challenges faced in large scale memory integration. Moreover, the principles can be equally applied to architectures for logic operations, etc. 4 Effect of dipolar fields In multilayered structures magnetized perpendicular to the plane, dipolar fields create a long-range ferromagnetic interaction between layers which could potentially disrupt the nearest neighbor antiferromagnetic exchange coupled scheme. As the lateral width of an element decreases, dipolar couplings between neighboring layers are expected to increase [14]. In the following, we will discuss what effect dipole interactions will have when lateral dimensions shrink and present calculations where we have taken the full dipole field into account. The effect of dipolar fields on a given layer is different depending on whether the layer is part of a soliton or a bulk layer. Let us first consider a layer inside an infinite bulk domain. This bulk layer has first nearest neighbors which are anti-parallel aligned, these create a destabilizing dipole field on the bulk layer under consideration. The second neighbors of this bulk layer create a stabilizing field etc. Hence, the resulting stray field on a bulk layer is an alternating sum of contributions from layers further and further away. Let us now consider the case of one of the two layers forming an isolated 10

11 SUPPLEMENTARY INFORMATION RESEARCH soliton in an infinite stack. In that case, the contributions from first neighbors cancel out (where we ignore the different interlayer distances on either side of the layer). The contributions from all the other neighbors cancel in a similar manner. In brief, dipolar fields mostly affect the bulk switching field H b while modifying the propagation fields H p1 2 and H p1 2 to a lesser extent. In the case where multiple solitons are present in a stack, the near cancelation of the stray field felt by soliton layers will be affected by the change of phase at the neighboring soliton. Only the field from the first few neighboring layers will cancel out. However, the resulting stray field is small since it involves contributions from layers further away. The same is true of bulk layers in the vicinity of a soliton: only the first few layers on either side will contribute. The change of phase occurring at the soliton will cause the contribution from layers placed further away to cancel out with their counterparts on the other side of the bulk layer considered. When a random data sequence is stored in a stack, solitons are a random multiple of four layers apart. Furthermore, as a data sequence propagates through a stack during a complete field cycle, it goes through four different configurations where the distances between solitons vary. Upward solitons move two layers up during the first negative half field cycle, via a state where they point downward one layer above their original position, followed by the same process where downward solitons move two layers up during the second positive half field cycle (similar to Fig. 2c). The distances between and orientations of solitons vary for these four configurations and therefore the dipolar couplings also vary. In the following, we consider a 100 layer stack which holds a data sequence of 25 data bits, i.e. 1 data bit per 4 layers. Dipolar fields have been included by analytically computing the interaction energy between each layer represented as a square prism [15]. Furthermore, we define H as the limiting (smallest) switching field splitting at any of the four stages of soliton propagation in the bulk as discussed before. Hence, the larger H the better. Note that since dipole fields are now included the relevant switching field splittings do not follow the simple forms as given in Eqs. 2. In Fig. S5a and b, H divided by H c as a function of lateral width (w) is shown for different data sequences. Fig. S5a is composed using the parameters of the main text and Fig. S5b is composed using a different set of parameters as given in the inset. In the limit of large w (zero dipole interactions) the results are identical to those discussed in the former section. This limit is represented by the 11

12 RESEARCH SUPPLEMENTARY INFORMATION a H /H c (-) Single soliton Random sequence Soliton every 4 layers H c = 230 Oe J 1 = 650 Oe nm J 2 = 180 Oe nm t 1 = 0.7 nm t 2 = 0.8 nm b H /H c (-) H c = 1 koe J 1 = 4.5 koe nm J 2 = 1 koe nm t 1 = 0.65 nm t 2 = 0.80 nm w (nm) w (nm) Figure S5: Smallest switching field splitting H divided by H c as a function of lateral width w when dipole fields are included for three different data sequences; single soliton (black symbols), random data sequence (blue), soliton every 4 layers (red). a. For the ratchet parameters as used in the main text (see inset). b. For parameters as given in the inset. solid black line. For example, in Fig. S5a this limit is H /H c = 0.36, i.e. H = 84 Oe, which corresponds to H p /σ =2.4 at H c = 230 Oe in Fig. S3a and Fig. S4b, as discussed before. For the single soliton data sequence we find that with decreasing w, H /H c is nearly constant. For the parameters of the main text this is true down to w = 85 nm, where it drops dramatically. This shows that the ratchet scheme presented in the main text works down to 85 nm when an single soliton is considered. When multiple solitons are present, indicated by the blue and red symbols, H starts to drop at larger w. In the case of the ratchet parameters of the main text (a) this effect is the most dramatic for the case of a soliton every four layers (red) and the ratchet scheme breaks down at w 1100 nm. Hence, the scheme presented in the main text has to be adapted for high density devices. In Fig. S5b we consider the parameters as given in the inset. For this set of parameters the ratios H c /J 2 and J 1 /J 2 are chosen to maximize the switching field splitting (i.e. H /H c = 1) in the limit of large w, as explained in the previous section. Furthermore, we choose H c = 1000 Oe, t 1 =0.65 nm and t 2 = 0.8 nm. This set of parameters shows that the rachet scheme works down to much smaller w as compared to Fig. S5a whilst keeping sufficient switching 12

13 SUPPLEMENTARY INFORMATION RESEARCH field splittings required to compensate for switching field distributions (e.g. to compensate for a σ = 15% of H c one needs H /H c = 0.75). Note that these parameters are by no means fully optimized, for example, they lead to large absolute switching fields to operate the ratchet. By combining the flexibility of the ratchet parameters and by using different materials there is an enormous room for improvement. For instance, the dipole fields scale linearly with M s, hence, by using a material with low M s and high H k dramatic improvements can be made. Interestingly, the length scale of the dipole interaction also depends on w, i.e. for small w the dipole field is large but also shorter ranged compared to intermediate w. The former illustrates that this is a complex problem and requires a detailed study (in progress) to explain the trends and dependencies, which is, however, beyond the scope of this study. Moreover, the exact required criteria, as emphasized in the previous section, depends on the soliton driving mechanism, specific device design and application (memory/storage/logic). 5 3D scaling In conventional 2-dimensional memory, scaling is straightforward: the smaller the lateral size of the elements, w, the higher the areal density of data storage. When extending the storage into the 3 rd dimension the added space needs to be taken into account. Let us start with conservatively estimating what the areal data-storage density would be when the lateral size of the devices is limited to w = 100 nm due to limited achievable aspect ratio of current lithography processes. It has been shown that at a lateral length scale of w = 100 nm using perpendicularly magnetized CoFeB, similar to that used in our study, the thermal stability criterion of 60 k B T is met (guaranteeing data stability for 10 years) with well-behaved material parameters [13]. Many lithography processes are limited by their maximum aspect ratio r, i.e. the ratio of the height to width of the structure. At a 100 nm lateral size an aspect ratio of 4 is easily achieved and dedicated processes are readily available to reach aspect ratios of [16]. Here we assume r = 10, which would give a pillar height of 1000 nm. Using the parameters of the soliton ratchet presented in the main text we define a magnetic layer thickness of t m 0.6 nm and an interlayer spacer thickness t s 2.1 nm, consisting of 0.9 nm of Ru plus two Pt layers of 0.6 nm (see Fig. S6). Furthermore, we set a data coding density of one data bit every n = 4 repeat periods allowing us to store 13

14 RESEARCH SUPPLEMENTARY INFORMATION 92 bits per pillar (370 magnetic layers). We can now model the data storage density as follows. Using the aspect ratio r, the height of each pillar is wr and there are wr/(t m + t s ) repeat units of magnetic / non-magnetic layers in each pillar (see Fig. S6). The effective areal density D is then given by: D = 1 4w r (t m + t s )n Tbits/In 2, (3) where the factor is given by the conversion from bits/m 2 to Tbits/In 2. This is a factor of approximately 15 times higher than today s solid state drives (Flash 0.1 Tbits/In 2 ). Naturally, shrinking the lateral dimension w and increasing the aspect ratio r will drastically increase the data-storage density. In STT-MRAM (the archetypal 2-dimensional magnetic memory technology), the smallest lateral size is set by a combination of lithographic limits and the thermal stability limit. The latter can be modeled approximately by U = K u V = K u w 2 t m (4) where U is the energy barrier separating the two stable data states, K u is the magnetic anisotropy energy density of the magnetic layer and V is the volume of the magnetic layer. V can be expressed as the product of the width squared of the magnetic structure (see Fig. S6), w 2, and its thickness, t m. U is usually set to around 60 k B T (k B is Boltzmann s constant and T is temperature) to guarantee data stability for 10 years. In a 3-dimensional architecture, reducing the lateral size of the element allows more elements to be fitted into a given area, but it may also reduce the maximum height of the structure, thus reducing the number of bits that can be stored vertically. What matters in a 3-dimensional architecture is the effective areal density, which is the areal density of a single layer multiplied by the number of bits stored vertically. The effective areal density may not be maximized by shrinking w as far as possible but rather by choosing some intermediate value that gives good in-plane packing while retaining a large number of bits vertically and assuring thermal stability. Now let us apply the thermal stability criterion. For a given anisotropy and element lateral size w, the magnetic thickness must be no thinner than t m = 60K BT K u w 2 (5)

15 SUPPLEMENTARY INFORMATION RESEARCH Figure S6: Definition of the in-plane and out-of-plane dimensions used in the calculations Eliminating t m gives the effective areal density as D eff = r 4n( 60k BT K uw + t sw) As explained qualitatively in an earlier paragraph, this expression is not maximized by taking w to zero. Setting the derivative of the denominator with respect to w to zero gives a maximum effective areal density when (6) and w = 60kB T K u t s (7) t m = t s (8) The optimized effective areal density D eff,o in Tbits/In 2 is then D eff,o = r Ku (9) 8n 60k B Tt s So high effective areal densities are generated by using spacer and magnetic layers that are as thin as possible, even though this increases the re- 15

16 RESEARCH SUPPLEMENTARY INFORMATION quired lateral size of the elements to guarantee thermal stability. Fortunately, this condition of thin and wide elements is also that which minimizes magnetostatic interactions between elements. High anisotropy materials also give high effective areal density. From Eq. 7 we can now determine the value of w which maximizes the areal density while keeping an energy barrier of 60 k B T. Using t s = t m =0.7 nm and an effective perpendicular anisotropy of K eff = K u 1/2µ 0 M 2 s =1.6 MJ/m 3 1/2µ 0 ( ) 2 = 550 kj/m 3 (using K u =1.6 MJ/m 3 from Fig. S2 for the 0.7 nm CoFeB layer) we find that the areal density is maximized at w 25 nm. Magnetostatic calculations suggest that the materials and thicknesses presented in this paper would lead to short-range dipolar coupling being too strong to sustain operation at w = 25 nm. As illustrated in the section where we discuss the inclusion of dipolar fields, the ratchet scheme still works when the parameters are tuned, further development of materials might open this size range. The w = 25 nm would give an optimized areal density (Eq. 9) of D eff,o 1.1 r Tbits/In 2. Hence, a pillar with aspect ratio of r = 4, which as discussed before is easily achieved with current lithography technology, gives D eff,o 4.4 Tbits/In 2, which is 44 times higher than current solid state drives. If a w of 25 nm cannot be achieved using the presented ratchet scheme it may be possible to construct a scheme using in-plane magnetized materials. For in-plane magnetized materials the dipole coupling between neighboring layers is anti-ferromagnetic, in contrast to the ferromagnetic dipole coupling in perpendicularly magnetized layers. This could then allow soliton ratchet schemes based on dipole coupling. For this scenario we consider a soft in-plane magnetized material such as CoFeB where K u = 5 kj/m 3 might be expected (equivalent to an anisotropy field of 100 Oe, using M s = 1000 emu/cm 3 ). Assuming the layers to be no thinner than 3 nm and a similar ratchet scheme to that in the main text, Eq. 7 tells us that the effective areal density would be optimized for w = 129 nm. This is considerably larger than state-of-the-art lithography and opens up a number of novel routes for fabrication of very high aspect ratio structures. In particular, interference lithography (e.g. [17]) and electrochemical deposition from liquid into deep pores (e.g. [18]) could be feasible. Assuming a soliton every 4 repeat units (i.e. n = 4) and a total depth of pore of 50 µm (i.e. r = 388), Eq. 8 gives an effective areal density of 20 Tbits/In 2. This is a factor of approximately 30 times higher than today s hard disk drives and approximately 200 times higher than today s solid state drives. 16

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