Geometrically pinned magnetic domain wall for multi-bit per cell storage memory

Spintronic devices currently rely on magnetic switching or controlled motion of domain walls (DWs) by an external magnetic field or a spin-polarized current. Controlling the position of DW is essential for defining the state/information in a magnetic memory. During the process of nanowire fabrication, creating an off-set of two parts of the device could help to pin DW at a precise position. Micromagnetic simulation conducted on in-plane magnetic anisotropy materials shows the effectiveness of the proposed design for pinning DW at the nanoconstriction region. The critical current for moving DW from one state to the other is strongly dependent on nanoconstricted region (width and length) and the magnetic properties of the material. The DW speed which is essential for fast writing of the data could reach values in the range of hundreds m/s. Furthermore, evidence of multi-bit per cell memory is demonstrated via a magnetic nanowire with more than one constriction.

A magnetic domain wall (DW) is a spatially localized change of magnetization configuration in a ferromagnetic material. The motion of DW using spin transfer torque (STT) has attracted great interest in fundamental theoretical studies and promising potential applications, such as high density magnetic storage and logic devices [1][2][3][4][5][6][7][8][9][10][11] . For memory application, several requirements need to be fulfilled for a good functionality. For instance, the non-volatility is desirable for saving the power consumption while scaling down the device size; i.e. increasing the storage capacity, requires low writing and reading currents 12,13 . Magnetic tunnel junction (MTJ) where two ferromagnetic layers separated by a tunnel barrier was the first prototype of magnetic memory devices. Writing and erasing the data on MTJ memory could be done by a polarized current through reversal of the magnetization of a soft layer (called memory layer). Although many of the requirements above can be achieved in an MTJ [14][15][16][17][18][19][20] , the limitation to two states remains an obstacle toward high capacity memory. Multi-level MRAM where two memory layers could be used to store four states was proposed to boost the storage capacity 21,22 . Storing even larger data in one cell is possible by moving DW at different positions within the nanowire. In DW-based memory, the stability and speed of DW have to be controlled and optimized for actual application. Stabilizing DW at desired positions is very important for a good functionality of the storage memory. Although many reports were devoted to study domain wall dynamics and its motion under magnetic field, electric field and/or polarized current [23][24][25][26][27][28][29][30][31][32][33][34] , controlling its position and its stability remains a big challenge [35][36][37][38][39][40] . Creating artificial defects were proposed and investigated to generate a potential that acts as a pinning site for DW 35,36 . Designing pinning sites by lithography is challenging since this requires a high resolution process that is much better than making the nanowire itself. It is much easier to create notches when the nanowire dimension are in the hundreds nanometers and above. However, scaling down the nanowire size to few tens of nanometer with even smaller notches is a tremendous technological challenge. In this work, we propose a new way for pinning DW in magnetic nanowire with adjustable size and position. The method is based on designing portions of a magnetic nanowire with the same size but with a small off-set in either one direction or both. Figure 1(a) shows a proposed nanowire with a single step. For device fabrication, creating notches on a nanowire for pinning DW requires additional nanofabrication process after the nanowire is made. Furthermore, since the dimension of the notches have to be smaller than the width of the nanowire, their positions and size uniformity will be challenging and could be a serious obstacle for their implementation. In our proposed scheme shown in Fig. 1(a), the design could be made in a single process. It is also easy to create an off-set of the two small patterns in either x or y directions prior to the beam exposure on the resist (polymer) which serves as an etching mask. The study demonstrates that DW could be stabilized in the stepped region (constriction) and the pinning current depends on its dimension (step depth d and step length l)

Results
We consider a magnetic nanowire of length L, width W, and thickness t with a stepped region. As the objective was to stabilize DW in this region we also considered the nanowire is made of two parts with an off-set l in the x direction and d in the y direction as illustrated in Fig. 1(a).
Micromagnetic simulation was conducted to study the magnetic DW motions in the nanowire with the proposed scheme [http://math.nist.gov/oommf]. A magnetic material with in-plane anisotropy was considered in this work with a mesh size of 2.5 nm × 2.5 nm × 3 nm. The polarized electric current was flowing along the nanowire in the positive x direction and the magnetization was initially aligned in the opposite direction. In the first part of this study, the effect of stepped region dimension l and d on DW dynamics is investigated while in a second part, we were interested in the correlation between the magnetic properties and DW stability. In all this study, the length L, the width W and the thickness t were fixed to 200 nm, 40 nm and 3 nm, respectively. Also the exchange stiffness A and damping constant α of the material were fixed to 1.0 × 10 −11 J/m and 0.05, respectively. These values are typical for materials such as Co, CoFe or CoFeB alloys. Nanoconstriction dimension and DW dynamics. By varying d and l we were able to stabilize DW at the stepped region for current density below a critical value J c . For instance, it can be seen from Fig. 1(c) that at l = 20 nm, it is not possible to stabilize DW for values of d smaller than 15 nm which means that there are optimal dimensions of the stepped region to favour DW stability. For current density values above J c , a continuous movement of DW from one side to the other was observed. These calculations were carried out for material with M s = 600 kA/m and K u = 1.0 × 10 5 J/m 3 . Figure 2 Fig. 2(b). This means that stabilizing DW within the vicinity of stepped region is possible by selecting the optimal values of d and l for each applied current density. It is worthy to note that for large d [ Fig. 2(a)] or small l [ Fig.2(b)], the velocity of DW motion is reduced and for the optimal values of d and l it starts to oscillate before the pinning occurs.
To elucidate the effect of device geometry on DW stability and its dynamics, the magnetic configuration of a moving DW for two values of step depth d was examined. The current density and length of the step l were fixed to 4.84 × 10 12 A/m 2 and 20 nm, respectively (Fig. 3). The snapshot images were taken at three different positions within the nanowire. For t = 0.2 ns, DW is still at the first half of the nanowire and did not reach the stepped area.   41 . The velocity can be expressed by: where ħ is the reduced Planck constant, e is charge of electron, γ is the gyromagnetic ratio, P is the spin polarization of the current and ε is the non-adiabatic parameter 42 . It is worthy to note that for K u = 3 × 10 4 J/m 3 , a deviation from a linear behavior was clearly seen. This is because although DW could not be stabilized at the pinning region, it takes some time to be released.  We initially started with all the magnetic moments aligned in the negative x direction and it takes some time to observe a creation of DW under spin transfer torque effect. The average velocity is calculated from the time the DW is created until it is pinned at the stepped region (for large K u values) or vanishes at the end of the nanowire (for small K u values).
As K u increases (K u ≥ 3.5 × 10 4 J/m 3 ), a pinned DW could be observed in the middle of nanowire for J < J c which is indicated by the arrow in Fig. 4(a). This transition from a stable to unstable DW is accompanied by a drop in the velocity. This is mainly due to an increase of the time DW takes to be released from the stepped region. For further increase of J, a steady increase of v is revealed. After plotting J c as a function of K u when DW stability is possible [ Fig. 4(b)], we noticed that there is a region where J c increases linearly with K u (K u between 3.5 × 10 4 J/m 3 and 4.75 × 10 4 J/m 3 ). However, as K u becomes larger, more complex behavior of magnetic domains is observed. By looking at the details of magnetic moments configuration, for two values of K u , the shape of DW and its evolvement with time could be imaged as shown in Fig. 5. The DW position within the left side of the nanowire is not shown for simplicity. When DW is created and until it reaches the stepped region we observed a movement of a transverse DW for both values of K u . However, as the DW passed the stepped region, a change in DW configuration was revealed. For K u = 0.5 × 10 5 J/m 3 , the transverse type DW could still be seen until it vanishes at the end of the nanowire. In contrast, for K u = 1.0 × 10 5 J/m 3 , DW starts to bend and an antivortex type DW could be observed. A current density J = 5.5 × 10 12 A/m 2 was used in this calculation and snapshot images were taken times corresponding to desired DW locations based on m x versus time graph. It is noticed that antivortex type DW moves faster than transverse type 43 . Similar to the study conducted on K u effect on DW dynamics shown above (Figs 4 and 5), it was observed that there is a minimum M s value for stabilizing DW at the stepped region.   For M s > 550 kA/m, DW could be stabilized at the center as shown in Fig. 6 for M s = 560 and 580 kA/m. This is an important finding of this study. It is possible to stabilize DW by creating an off-set of the nanowire at desired position. Furthermore, material with larger M s favors a faster DW creation for given current density as indicated by t + for 580 kA/m case. To evaluate the velocity of DW, we considered the time difference between t + and the time when DW is either pinned t p (large M s case) or annihiled t − (low M s case). In the insert of Fig. 6, the velocity v of DW was plotted as a function of M s for J of 2.6 × 10 12 A/m 2 and 2.9 × 10 12 A/m 2 . A continuous increase of v with M s is observed. The bold arrows show the critical M s separating non-pinned and pinned DW ranges. This critical value depends on the current density, device dimension and K u . It is important to mention about the relatively large values of DW velocity obtained from the time dependence of m x which is beneficial for a fast writing of data by a polarized electric current. For a good stability of DW in the proposed device, it is important to optimize the values of d and l. Figure 7 displays the calculated phase diagram for nanowire with M s = 600 kA/m and K u = 0.5 × 10 5 J/m 3 . The stability of DW inside the nanowire could be seen for d larger than 15 nm and l below 25 nm (W was fixed to 40 nm). We also observed damped oscillation for large value of d and l as shown in dashed region of Fig. 7. DW could be stabilized in very narrow range of current density. More interestingly, DW with large amplitude oscillation could be seen in this range.
For a good performance of the memory device, it is also important to store more than 2 bits/cell (i.e. four states) as experimentally demonstrated in current-perpendicular to plane magnetoresistive devices based on magnetization switching 21 Fig. 8. The six states obtained are very stable for the device dimension reported above. For clarity, states 1 and 6 are not shown. We conducted calculations with same values of M s , K u , L and W, except l and d which were fixed both to 20 nm as used is Fig. 6 but we were not able to obtain all the six states shown in Fig. 8(b).

Discussion
We have demonstrated that in magnetic nanowire with a stepped region, DW could be precisely pinned. The depinning current density could be easily adjusted by the constriction dimension and the materials properties. As K u increases, larger J c is required to move a transverse type DW from one state to the other. However, further increase of K u leads to an antivortex type DW with a lower velocity. Similarly, a resonably large M s is needed to pin DW. Its velocity is improved as M s increases. The proposed scheme was extended to multi-step device which showed a clear stability for DW at different positions. The magnitude of DW depinning current and its movement speed could be well tailored by adjusting the gerometry of the device and the materials properties. Optimal values for K u and M s are required for each device.

Methods
We investigated the magnetization dynamics of a stepped nanowire with micromagnetic simulations. The simulations are performed with the object-oriented micromagnetic framework (OOMMF) which was extended to consider the current-induced magnetization dynamics as described by the Landau-Lifshitz-Gilbert equation with additional spin-transfer torque terms: where m is the local normalized magnetization, γ the gyromagnetic ratio, H eff the effective field, α the Gilbert damping factor, and β the nonadiabatic spin-transfer parameter 41,44 . The local effective magnetic field H eff includes the exchange, anisotropy and magnetostatic fields. The vector u is the adiabatic spin torque which has the dimension of velocity and is proportional to the current density according to = where j is the current density, g is the Lande factor, μ B the Bohr magnetron (μ B = 0.927 × 10 −20 emu), e the electron charge, P the polarization rate of the current fixed to 0.6 and the nonadiabatic spin-transfer parameter β to 0.02.