# Simulation of current-driven magnetisation switching in nanopillars with Perpendicular Shape Anisotropy

Natalia Boscolo Meneguolo, Olivier Fruchart, Jean-Christophe Toussaint, Mouad Fattouhi, Liliana Buda-Prejbeanu, Ioan-Lucian Prejbeanu, Daria Gusakova

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## To cite this version:

Natalia Boscolo Meneguolo, Olivier Fruchart, Jean-Christophe Toussaint, Mouad Fattouhi, Liliana Buda-Prejbeanu, et al. Simulation of current-driven magnetisation switching in nanopillars with Perpendicular Shape Anisotropy. Physical Review B, 2025, 112 (1), pp.014448. 10.1103/dl2x-qv7t hal-04999028v2

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Submitted on 31 Jul 2025

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# Simulation of current-driven magnetization switching in nanopillars with perpendicular shape anisotropy

Natalia Boscolo Meneguolo,¹,*, Olivier Fruchart,¹ Jean-Christophe Toussaint,² Mouad Fattouhi,¹

Liliana D. Buda-Prejbeanu,¹ Ioan-Lucian Prejbeanu,¹ and Daria Gusakova ¹,†

¹Université Grenoble Alpes, CEA, CNRS, Grenoble INP, SPINTEC, 38000 Grenoble, France

²Université Grenoble Alpes, CNRS, Institut Néel, 38000 Grenoble, France

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(Received 14 April 2025; revised 24 June 2025; accepted 7 July 2025; published 28 July 2025)

The perpendicular shape anisotropy spin transfer torque magnetic random-access memory (PSA-STT-MRAM) is a recent concept proposed to maintain the thermal stability of standard MRAM at small diameters, considering thick vertical pillars as the free layer. To explore the specific physics of such systems with a three-dimensional nature, we have performed simulations combining a micromagnetic model coupled self-consistently with spin-dependent transport equations. The 3D shape induces flower states at the upper and lower surfaces. In addition, the fieldlike component of STT is found to be larger than in thin elements such as in standard MRAMs, suggesting that it needs to be considered. The combination of both effects leads to the efficient excitation of edge ferromagnetic resonance modes, playing a key role in magnetization reversal. These results highlight features of 3D nanomagnetic systems, largely disregarded so far. These must be considered to reliably describe PSA-STT-MRAM, but more broadly in the rising field of 3D nanomagnetism.

DOI: 10.1103/dl2x-qv7t

## I. INTRODUCTION

Over recent years, the interest in 3D nanomagnetism has steadily grown, with both fundamental and applied interests [1]. On one hand, 3D nanoobjects allow the existence and tailoring of unique magnetic properties related to curvature and finite size: effective anisotropy and Dzyaloshinskii–Moriya interaction, static or dynamic chiral effects, etc. On the other hand, these may find practical applications that take advantage of the effects specific to 3D nanomagnetism [2]: increased stability against thermal excitation, nano-oscillators with tunable frequency, flexible nanomembranes with tunable magnetic properties, magnetic interconnects able to merge memory and logic units [3,4], etc. The third dimension also opens prospects for spintronics [5]. An example is networks of magnetic nanowires [6,7], which may be advantageous when thinking about 3D integrated magnetic devices for neuromorphic applications and in-memory computation, thanks to their enhanced connectivity.

The perpendicular shape anisotropy spin transfer torque magnetic random-access memory (PSA-STT-MRAM) is another example making use of 3D concepts for applied spintronics [8,9]. The main objective of this concept for MRAMs is to boost the thermal stability of the free layer of the magnetic tunnel junction (MTJ, the unit cell of the MRAM) at diameters down to a few nanometers, providing a scalable solution to the foreseen bottleneck of standard perpendicular MTJs below typically 25 nm. This is achieved thanks to the increase in thickness of the MTJ free layer, taking the form of a vertical nanopillar, therefore, a 3D

nanomagnetic system. Unfortunately, some drawbacks stem from the increase of the free layer thickness. We withstand, namely, to the high writing critical voltage and slow writing time. Indeed, in an MTJ, the spin-transfer torque is mostly an interfacial effect, and therefore moderately efficient to switch the magnetization of a 3D system. To address this problem, slicing the free-layer pillar into a multilayer stack has been proposed [10]. However, while this solution indeed reduces the current required for writing, it relies on the standard fabrication route of deposition plus aggressive etching routes [8,9], so it suffers from large device-to-device distributions. Also, the required physical vapor deposition cannot be implemented in microelectronics vias, which would provide a natural route for high-density networks [1]. Therefore, simulation of current-driven magnetization switching in nanopillars remains relevant to identify their dynamics and identify routes to improve it. Such simulations have already been conducted [8,11], in which the so-called adiabatic component of spin-transfer torque was implemented as an interfacial term, such as is routinely done in conventional thin free layers of MRAM [12].

In this article, we refine these approaches by evaluating in more detail the underlying microscopic physics, considering two ingredients relevant to properly describe a 3D spintronic-nanomagnetic system. First, we employ a finite-element method (FEM) micromagnetic solver, to faithfully describe the curved shape of a nanopillar [13]. Second, we evaluate explicitly and consider self-consistently the spin accumulation in the entire nanopillar. This allows us to explicitly derive the spin accumulation in the entire 3D structure, as well as the noteworthy contribution of the fieldlike (FL) spin torque to the physics of the system. The structure of this article is the following: In Sec. II, we describe the methods, presenting the main aspects of the *feeLLGood* micromagnetic

<footnote>*Contact author: natalia.boscolomeneguolo@cea.fr</footnote>

<footnote>†Contact author: daria.gusakova@cea.fr</footnote>
solver: the FEM used for the meshing procedure, the physics underlying the modeling of the magnetization dynamics, focusing, namely, on the spin accumulation computation. In Sec. III, we report the main simulation results, starting from the specificities related to the mesh size, continuing with the magnetization behavior at the edges of the pillar and finishing with spin accumulation and the link between the reversal mode and magnitude of the injected current density. In Sec. IV, we draw conclusions and perspectives.

## II. METHODS

### A. Finite element method micromagnetic solver

There are a number of ways to discretise the specimen in micromagnetic simulations, e.g., the finite-difference method (FDM), the finite-volume method (FVM) and the FEM [14,15]. In the present work, we use FeeLLGood, a finite-element 3D micromagnetic solver developed over the past two decades at Institut Néel and SPINTEC in Grenoble [13]. The FEM method is very flexible and suited to discretize complex and curved shapes. In practice, the specimen volume is meshed with GMSH, an open source 3D finite element mesh generator [16]. We chose a mesh of unstructured type, meaning that it has not a uniform pattern. More precisely, the discretization is obtained by first meshing the surface through the boundary recovery algorithm [17], while the mesh points inside the volume are added through a 3D version of the Delaunay algorithm [18].

### B. Zhang-Li-like STT

The description of the dynamics of the system results from solving the Landau-Liftshits-Gilbert (LLG) equation:

$$ \frac{\partial \mathbf{m}}{\partial t} = -\mu_0 \gamma (\mathbf{m} \times H_{\text{eff}}) + \alpha \left( \mathbf{m} \times \frac{\partial \mathbf{m}}{\partial t} \right). \quad (1) $$

In Eq. (1), $\gamma > 0$ is the gyromagnetic ratio, $\mu_0$ is the vacuum magnetic permeability, $\mathbf{m}$ the unitary magnetization (the value of the magnetization normalized by the spontaneous magnetization $M_s$), $H_{\text{eff}}$ is the effective field, and $\alpha$ is the Gilbert damping factor. More information can be found in Refs. [19–21]. In addition to the LLG equation (1), we also consider the impact of the spin torques (STT) in the system by taking into account an extra term to (1):

$$ \mathbf{T}_{\text{STT}} = \frac{1}{\tau_{\text{sd}}} \frac{\mathbf{s}}{M_s} \times \mathbf{m}. \quad (2) $$

In Eq. (2), $\tau_{\text{sd}}$ is the sd electrons interaction characteristic time, while $\mathbf{s}$ represents the spin accumulation. $T_{\text{STT}}$ accounts for the mutual interaction between magnetization and the spins of conduction electron. Originally, the STT torque description has been simplified for a bilayer, exploiting the so-called LLGS equation (where S stands for Slonczewski). This corresponds to the formalism introduced by Slonczewski in 1996 [22]:

$$ T_{\text{STT}} = -\gamma \mu_0 \eta_{\text{STT}} \frac{\hbar J}{2|e| M_s t} \mathbf{m} \times (\mathbf{m} \times \mathbf{m}_{\text{pol}}) \\ + \gamma \mu_0 \eta'_{\text{STT}} \frac{\hbar j}{2|e| M_s t} (\mathbf{m} \times \mathbf{m}_{\text{pol}}). \quad (3) $$

In Eq. (3), $\eta_{\text{STT}}$ represents the STT efficiency, whose value can be estimated from the tunnel magnetoresistance of the devices, $\hbar$ is the reduced Planck’s constant, $e$ is the electron charge, $M_s$ is the value of the spontaneous magnetization of the material, $j$ is the current density injected in the system, $t$ is the thickness of the magnetic free layer, $\mathbf{m}$ is the unit vector for magnetization, and $\mathbf{m}_{\text{pol}}$ is the unit vector for magnetization in the reference layer assumed to be fixed, separated by a tunnel barrier from the free layer considered. There are two spin torque terms on the right-hand side of Eq. (3). The first one is the so-called dampinglike component (DL, also called in plane), that drives magnetization toward the local effective field. The second term is the FL component (also called out of plane), which induces magnetization to precess around the local effective magnetic field. Usually, in 2D systems, the FL component is neglected, since it has been shown to not play a major role in the reversal mechanism [12].

Another conceptual way to describe the STT is to consider four contributions in total: two of them related to the change in time of magnetization and the other two related to its variations in space. The time-related contributions can be accounted for by redefining the gyromagnetic ratio $\gamma$ and the Gilbert damping factor $\alpha$. The space-related terms instead describe two main interactions: First, the ballistic motion and all the adiabatic processes of conduction electrons; second, the nonadiabatic spin torque that are consequences of the spatial mistracking of spins between conduction electrons and local magnetization. This description is the Zhang-Li one, introduced in 2004 by Zhang and Li [23]. FeeLLGood relies on this Zhang-Li-like description of the STT.

Nevertheless, we highlight that we are considering a simplified model for spin-polarized tunneling effect. The purpose of the present report is indeed to outline the main physical phenomena taking place in nanomagnetic structures during the magnetization reversal. However, if willing to be more precise, other effects could be additionally considered [24]. For example, the type of tunnel barrier [25] or the applied switching bias [26] could affect the STT.

### C. Spin accumulation

Spin-transfer torques result from spin accumulation, i.e., an out-of-equilibrium imbalance of up and down spins of the conduction electrons. There can be different implementations of STTs. When dealing with standard MRAM with perpendicular magnetization, which have an ultrathin layer, one generally assumes that the spin accumulation is uniform across the free layer. Indeed, as already mentioned in the Introduction, usually the efficiency of the DL coupling can be computed from experimentally obtained current vs field switching diagrams [12]. However, this approach may be challenged when dealing with 3D structures. An ersatz consists of considering an effective interfacial STT, e.g., decaying away from the interface in an effective exponential way [11]. However, this choice can also be questionable, not considering the precessional effects of the spins over this length scale. For this reason, here we consider the full spatial variation of spin accumulation in the nanopillar, which is calculated at each iteration of the solving protocol by coupling the LLG equation (used for the determinations of the dynamics of the
system) to diffusive transport equations. In this picture, the spin accumulation **s** induces an additional contribution to the effective magnetic field acting on the local magnetization.

The system of equations that have to be solved to determine the spin accumulation **s** is the following [27]:

$$ \mathrm{div} \, \mathbf{j} = -\mathrm{div} \, C_0 \, \mathbf{grad} \, \mathbf{V} = 0, \quad (4) $$

$$ \frac{\partial Q_{\alpha k}}{\partial x_k} = - \frac{s_{\alpha}}{\tau_{sf}} - \frac{(\mathbf{s} \times \mathbf{m})_{\alpha}}{\tau_{sd}}. \quad (5) $$

In Eq. (5), the $\alpha$ index refers to the spin space while $k$ index refers to the real space and $k, \alpha \in \{x, y, z\}^2$. Equation (4) expresses the total current **j** resulting from the local electric field $\mathbf{E} = -\mathbf{grad} \, \mathbf{V}$, $C_0$ being the electric conductivity. In our convention, positive current **j** means positive charge flowing from the reference layer to the free layer, which means electrons flowing from the free layer through the tunnel barrier, and vice versa. Additionally, spin has the meaning of magnetic momentum, i.e., it is parallel to the corresponding layer magnetization. Equation (5) is a conservation equation, expressing the grounds for the variation of local spin accumulation, resulting either from the spatial variation of the spin-polarized current $Q_i^s$ on the left-hand side or spin flip and spin precession on the right-hand side. In this context, $i$ refers to the spatial direction ($i \in \{x, y, z\}$), $\tau_{sf}$ is the spin-flip characteristic time, and $\tau_{sd}$ is the sd coupling characteristic time. The spin-polarized current is defined as

$$ Q_{\alpha k} = - \frac{\mu_B P C_0}{|e|} u_{\alpha} E_k - D_0 \frac{\partial s_{\alpha}}{\partial x_k}. \quad (6) $$

In our simulations, we simplify the picture by considering that $P$ has the same value as $\beta$, a parameter used to express the polarization of conduction electrons injected in the ferromagnet [28,29], known as the polarizability rate.

When describing spin accumulation, we have to consider the surroundings and boundary conditions for the magnetic elements. On the side of the tunnel junction, we set the boundary condition of a uniform areal density of current. On the surface between the free layer and the electrode, we describe the spatial variation of spin accumulation in the nonmagnetic metal contact, with continuity of both the current density and chemical potential at the interface [30]. In this framework, to hold Eqs. (4) and (5) requires no spin accumulation discontinuity at the interface and an equal normal component of the spin-polarized current (the needed stimulus to trigger the magnetization reversal). The length of the nonmagnetic electrode is set to 60 nm, and a uniform voltage is applied as boundary condition of the electrode.

We will see in Sec. III that considering spin accumulation instead of interfacial STT indeed leads to drastic quantitative difference in the case of a 3D nanomagnetic object.

### D. Material parameters

The magnetic material of the free layer is FeCoB and that of the metal is Cu. Table I lists the parameters used in the simulations.

<table><caption>TABLE I. Material parameters used in the simulations. If not stated otherwise, the parameters apply both to the magnetic system (MS) and the nonmagnetic electrode (NM).</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Saturation magnetization of MS</td><td>M<sub>s</sub></td><td>1 MA/m</td></tr><tr><td>Exchange constant</td><td>A<sub>ex</sub></td><td>1.5 × 10<sup>-11</sup> J/m</td></tr><tr><td>Uniaxial anisotropy</td><td>K<sub>u</sub></td><td>0.96 × 10<sup>6</sup> J/m<sup>3</sup></td></tr><tr><td>sf diffusive length</td><td>l<sub>sf</sub></td><td>5 nm</td></tr><tr><td>sd exchange length in MS</td><td>l<sub>sd,MS</sub></td><td>1 nm</td></tr><tr><td>sd exchange length in NM</td><td>l<sub>sd,NM</sub></td><td>0 nm</td></tr><tr><td>Electric conductivity</td><td>C<sub>0</sub></td><td>4.0 × 10<sup>6</sup> S/m</td></tr><tr><td>Number of atoms</td><td>N<sub>0</sub></td><td>1.344 × 10<sup>47</sup>/(J m<sup>3</sup>)</td></tr><tr><td>Polarizability rate</td><td>β</td><td>0.57</td></tr><tr><td>Polarizability ratio</td><td>P</td><td>0.57</td></tr></tbody></table>

### III. RESULTS

In this section, we highlight simulation and physical features specifically related to the consideration of a 3D nanosystem, which do not show up when considering the thin free layer of a standard MRAM.

#### A. Meshing

A usual concern in micromagnetics is the choice of the meshing size. To investigate this aspect in the present situation, we consider the demagnetizing factors $N_i$ ($i \in \{x, y, z\}$). These are important micromagnetic parameters as they determine the internal field for macrospins, and thus their precession frequency. In addition, the stability parameter $\Delta$ of a PSA-MTJ is directly related to the energy barrier of the system $E_b$, itself linked to $N_i$. Starting from the values of the simulated demagnetizing energies $E_d$ for a uniformly magnetized state, the demagnetizing factors $N_i$ are computed as

$$ N_i = \frac{2E_{d,i}}{\mu_0 M_s^2 V} \quad \text{where } i \in \{x, y, z\}. \quad (7) $$

Since the sum of the three demagnetizing factors along the main directions is equal to one ($N_x + N_y + N_z = 1$) and given that for a cylinder $N_x = N_y$, we may restrict the study of the numerical evaluation of $E_d$ in one direction. In principle, the results should be the same no matter the considered direction. Nevertheless, we computed ($N_x - N_z$) twice, starting either from $E_{d,x}$ or $E_{d,z}$. The results are reported in Table II for different vertical aspect ratios $AR = t/D$ of the magnetic cylinder, with $t$ the thickness of the free layer and $D$ its diameter. Furthermore, in Table II are reported also the values of ($N_x - N_z$), taking into account the analytical values of the demagnetizing coefficients known from literature [31,32].

The relative error in Table II is computed as

$$ \epsilon_{xz} = \frac{(N_x - N_z) |_{E_{d,x}} - (N_x - N_z) |_{E_{d,z}}}{(N_x - N_z) |_{E_{d,x}}}. \quad (8) $$

The relative error for the cylinder with AR 1 is striking. It reaches about 70%, while it drops sharply for larger AR. This can be understood as 1 is an AR for which the magnetization orientation in the system is close to the transition from out of
<table><caption>TABLE II. Demagnetizing factors of cylinders of different aspect ratios calculated from the simulated value of either $E_{d,x}$ or $E_{d,z}$ for uniform magnetization, with subsequent relative error $\epsilon_{xz}$ in the $(N_x - N_z)$ difference value. The average size of the edge of a tetrahedron constituting the mesh is of 4 nm for all five cases. Additionally, the analytical value for the demagnetizing factors difference is reported [32].</caption><thead><tr><th rowspan="2">AR</th><th colspan="3">|N<sub>x</sub> - N<sub>z</sub>|</th><th rowspan="2">ε<sub>xz</sub></th></tr><tr><th>Analytic</th><th>E<sub>d,x</sub></th><th>E<sub>d,z</sub></th></tr></thead><tbody><tr><td>0.2</td><td>0.52</td><td>0.55</td><td>0.49</td><td>11%</td></tr><tr><td>1</td><td>0.032</td><td>0.02</td><td>0.065</td><td>70%</td></tr><tr><td>1.5</td><td>0.158</td><td>0.14</td><td>0.161</td><td>15%</td></tr><tr><td>2</td><td>0.227</td><td>0.2</td><td>0.245</td><td>18%</td></tr><tr><td>3</td><td>0.308</td><td>0.296</td><td>0.308</td><td>4%</td></tr></tbody></table>

plane to in plane for lower values, so small errors translate into large differences. The error drops to 26% upon halving the average length of the edge of tetrahedron unit cells to 2 nm. This issue of large relative error is specific for a 3D nano-object, as for the flat shape of standard p-MRAM the *x* and *z* demagnetizing coefficients are very different one from another, so errors in the difference are low. This shows that one needs to pay extra attention to mesh-size refinement in 3D.

## B. Flower state

Figure 1 shows the micromagnetic state in a pillar with vertical aspect ratio 1 after relaxation, taking uniform magnetization as the starting point. There are deviations of magnetization from the uniform state at the edges of the structure for both the upper and lower surfaces, which is called a flower state [33]. This situation results from the divergence of dipolar fields at edges and corners [34], whose cost in terms of dipolar energy is decreased by rotating magnetization toward the diagonal to the edge, at the expense of exchange energy [35]. In our case, the angular deviation from the *z* axis at the very edge is about 9°, while at the center of both surfaces magnetization is aligned with the *z* axis. The

<figure><img src="image_4.png" /><figcaption>FIG. 1. (a) Representation of the FEM mesh used to discretize a pillar with 20 nm height and 20 nm diameter. (b) Relaxed micromagnetic state. The arrows represent the direction of the local magnetic moments. To highlight the flower state, a color scale indicates the value of the *z* component of the unit magnetization.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 2. Spin accumulation in the magnetic layer and metal contact for a free layer of 20 nm. both thickness and diameter. The current density injected through the tunnel barrier is $j = -5 \times 10^{12} \text{ A/m}^2$. The vertical axis represents the cross-sectional average of the *z* component of spin accumulation. The situations for the parallel state of uniformly magnetized polarizer and free layer are depicted in blue and red, respectively. The inset illustrates the geometry of the parallel (PP) and antiparallel (AP) configurations with positive charge current, gray arrows representing the magnetization direction in the reference layer and the black arrows the magnetization direction in the free layer.</figcaption></figure>

occurrence of the flower state is specific to a 3D system, with a size larger than the dipolar exchange length. In the thin free layer of an MRAM, the dipolar effects from top and bottom surfaces overlap and cancel each other at the length scale of the layer thickness, leaving it in a uniform state. We will see in Sec. III D that this has a direct and specific effect on current-driven magnetization dynamics.

The question if the flower state survives temperature effects and interfacial effects [such as the interfacial perpendicular magnetic anisotropy (iPMA) or the Dzyaloshinskii-Moriya interaction (DMI)] could spontaneously rise. We thoroughly discuss those points in the Supplemental Material [36]. However, we may already anticipate that both aspects enhance or reverse the flower state into a leaf state, with an expected impact on the incubation time of the magnetization reversal (deeply analyzed in Sec. III D).

## C. Spin accumulation

Figure 2 illustrates the simulated spin accumulation *s* in the magnetic and the metal parts for a structure with aspect ratio AR = 1. The blue and red curves represent the spin accumulation for the parallel and antiparallel states, respectively. As mentioned in the Methods section, the so-called spins are parallel to the associated magnetic moment, and electrons flow in the direction opposite to the charge current. Figure 2 is drawn for negative current, i.e., electrons are injected from the polarized layer into the free layer. Several aspects are worth mentioning. First, spin accumulation occurs not only at the tunnel barrier but also at the interface with the normal metal. In a metal/ferromagnet/metal stack, this would induce an antidampinglike torque at the interface where
<figure><img src="image_6.png" /><figcaption>FIG. 3. STT components versus the angle of magnetization in the free layer, with respect to the *z* axis. Black stands for a 1-nm-thick system, while blue stands for a 20-nm-thick system. FL torques are shown with full lines, DL torques with dotted lines.</figcaption></figure>

the electrons enter the ferromagnet due to the accumulation of spins antiparallel to magnetization and a DL torque at the interface where the electrons leave the ferromagnet due to the accumulation of spins parallel to magnetization. This picture is consistent with the generalized Landau-Lifshitz-Gilbert-Slonczewski equation [37] if the gradient of magnetization close to the interfaces is considered: the flow of electrons promotes a larger angular deviation at the interface through which they enter the mostly uniformly magnetized material and a lower angle at the interface through which they exit the material. Second, STT is not localized at interfaces but decays progressively away from these at the length scale of the spin diffusion length, such as described by the Valet-Fert model for giant magnetoresistance in the current-perpendicular-to-plane geometry [30]. This length scale is a few nanometers, comparable to magnetic length scales such as the dipolar and exchange length and therefore liable to efficiency couple to nonuniform magnetization distributions such as the flower state described previously. Third, and as a consequence of the previous statement, the height of the pillar has a direct impact on the value of spin accumulation, depending on whether it is smaller or larger than the spin diffusion length. The former case is that of standard MRAMs with a thin free layer, for which the Valet-Fert-like accumulations with opposite signs overlap and largely cancel each other, leaving the polarization of the injected current as the main driving force for spin accumulation, independent of the magnetization direction in the free layer. For the latter case, which is the situation of study here, the magnitude of spin accumulation at the tunnel barrier depends strongly on the direction of magnetization of the free layer.

Another difference between the model of effective STT and the one of spin accumulation is that the ratio of DL and FL torques is a direct output of the latter. To disentangle both, we compute the amplitude of STT versus the misalignment angle of magnetization with the *z* axis while keeping the polarizer fixed (Fig. 3). The results for a 2D-like system (cylinder of 1 nm × 20 nm) and a 3D-like one

<figure><img src="image_7.png" /><figcaption>FIG. 4. *z* component of the unit magnetization vs time for different values of injected electric current densities. The symbols refer to the time positions at which the snapshots reported in Fig. 6 were taken. The inset is a close-up view of the magnetization evolution during the first 0.4 ns, highlighting oscillations.</figcaption></figure>

(cylinder of 20 nm × 20 nm) are numerically computed, taking into account the value of the torque in reduced units [see Eq. (2)]. Figure 3 shows that while in the case of the 2D system the DL component dominates over the FL, as has been pointed out already [12], the two torques have a similar magnitude for the 3D system. How to link quantitatively the interfacial and bulk descriptions of STT requires further investigation, which lies beyond the scope of the present work. Nevertheless, the effects mentioned above already show qualitatively what is specific in a 3D nanomagnetic system and expected to give rise to peculiar magnetization dynamics, which we explore in the next paragraph.

### D. Current-driven magnetization reversal

If not stated otherwise, simulations are conducted for a cylinder having a diameter of 20 nm and a thickness of 20 nm, and a Gilbert damping factor $\alpha = 0.02$. The starting point of the simulation is uniform magnetization with 1° misalignment with respect to the *z* axis, and abrupt flow of electric current from the start. Some of the values for the density of injected current are unrealistically high for an experimental current through a tunnel barrier. However, covering a large range of stimulus allows us to highlight the various possible physics of 3D systems and their subsequent potentialities.

Figure 4 shows the evolution of the *z* component of the average unit magnetization $m_z$ versus time for five different values of injected current densities. $m_z = \pm 1$ means uniform magnetization aligned with the *z* axis upward and downward, respectively. Let us define two quantities to analyze the timescales of magnetization reversal. A first timescale appearing in Fig. 4 is the time needed before a sizable change is noticeable. We define it arbitrarily when $m_z$ reaches 0.9, which we call $\tau_i$. It is sometimes called the incubation time, which for a macrospin reflects the gradual rise of the precession angle under the influence of the antidamping torque. It is usually strongly dependent on initial misalignment of the
<figure><img src="image_8.png" /><figcaption>FIG. 5. Evolution of the characteristics inverse times $1/\tau_i$ (black curve) and $1/\tau_s$ (blue curve) with the absolute value of the injected STT current. Two different scales are applied for the two curves.</figcaption></figure>

various magnetization vectors, and on thermal and numerical noise. A second timescale is the time needed for the magnetic moment to go from mostly up to mostly down. We define the time for $m_z$ to go from +0.9 to -0.9, which we call $\tau_s$. It is sometimes called the switching time.

Figure 5 shows that up to a current density $|j|$ of circa $10^{12}$ A/m$^2$, $\tau_i$ vary inversely linearly with the injected current. This behavior is expected [38] and observed [39] for the spin-transfer switching of a macrospin when the current exceeds the threshold for switching, simply reflecting the rate of transfer of angular momentum from spin accumulation to magnetization [22]. The low slope is predicted to reflect in a logarithmic manner the initial misalignment angle $\theta_0$. $\tau_s$ also varies inversely linearly with injected current, however, with a larger slope, reflecting the larger reference angle, with cosine equaling 0.9.

However, two features suggest that the magnetization dynamics is not fully grasped by the macrospin picture. First, the inset of Fig. 4 shows that from the very first stages $m_z$ oscillates during the incubation stage. The magnitude of oscillations increases sharply and nonlinearly with the current density, and the frequency is more than one order of magnitude larger than the macrospin gyroscopic precession Fig. 7. Second, a crossover occurs at large current for the variation of inverse $\tau_i$ with applied current. In this regime, $m_z$ departs linearly from saturation from the start of simulation, which again does not fit the standard picture of incubation of a macrospin. Note that as Fig. 4 does not display a plateau at midreversal, one can probably rule out a mechanism of nucleation-propagation of the domain wall [11,40]. In the following paragraph, we examine the magnetization distributions during reversal to understand the microscopic reasons for these features.

Figure 6 illustrates the magnetization distribution at the surface of the magnetic cylinder during the reversal process, for different values of injected current. This series highlights a crossover for nearly coherent reversal at low current density to incoherent at larger current density, with a threshold value comparable to the crossover of $1/\tau_i$ in Fig. 5. Devia-

<figure><img src="image_9.png" /><figcaption>FIG. 6. Top, side, and bottom views of magnetization at the surface of a cylindrical volume of 20 nm thickness and 20 nm diameter over the current-driven switching process. Red corresponds to upward pointing magnetization while blue corresponds to downward pointing magnetization. From top to bottom, the results for increasing injected current densities are represented. The different snapshots are taken for increasing time steps, starting from 0 ns. The time steps are not the same in the four cases but are spread over the switching time required by the specific configuration. The time steps at which the snapshots are taken are highlighted with symbols on the curves of Fig. 4.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 7. Frequency of oscillation of $m_z$ derived from the inset of Fig. 4, expressed in GHz and in the equivalent effective field for gyrotropic motion $H_{\text{eff}}$.</figcaption></figure>
tions from uniform magnetization appear first at the edges of the interface with the tunnel junction, an effect most clearly visible for current density $-3 \times 10^{12}$ A/m$^2$ and higher. The linear decrease of $m_z$ versus time observed for the highest current densities in Fig. 4 is explained by this nonuniform excitation mode, as the angle of precession can reach up to several tens of degrees and involve a macroscopic fraction of the volume of the cylinder. The excitation of these modes can be understood, as spin accumulation is largest at this interface and its effect is most efficient at the edges of the pillar due to the flower state. The efficient pumping of energy at these locations is confirmed by the animations of the simulations of full magnetization evolution, revealing the rise of precession of magnetization turning the micromagnetic state periodically in flowerlike and leaflike configurations with intermediate curling configurations with opposite chiralities (see Supplemental Material [36]), with a frequency linked and thus explaining the oscillations of $m_z$ (inset of Fig. 4). The frequency of precession $f_{\text{edge}}$ is in the range 40 GHz to 45 GHz (Fig. 7). This is fully inconsistent with the Kittel FMR formula of a macrospin in its own demagnetizing field, related to $N_x - N_z$ [41]:

$$f = \frac{\gamma_0}{2\pi} \sqrt{(N_x - N_z)M_s}, \qquad (9)$$

which amounts to 1.23 GHz. This discrepancy and the localization of the nonuniform mode at the edges of the cylinder makes clear that the local effective field involved is not the demagnetizing field, but a combination of dipolar and exchange field involved in the flower state. The oscillation frequencies convert to $\approx 1.3$ MA/m in terms of effective induction field. Given the fact that the radius of the cylinder is close to 4 times the dipolar exchange length, this is exactly the range of frequency expected from the interplay of magnetostatics and exchange at this scale. Examining the evolution over time for a given current density is also interesting. It is clear that a breaking of symmetry is needed to reverse a macrospin, and this is what happens for the low current density. A breaking of symmetry is also required even in the case when an edge mode is the way that the system gets efficiently excited for the large current densities. Indeed, magnetization reversal via a perfectly-symmetry curling state requires the nucleation of a Bloch point on the axis of the cylinder [42], which is an energetically costly process [43]. Examining the edge modes over time, we notice that the breaking of symmetry takes the form of the mode changing progressively from a high azimuthal one to a low-azimuthal mode.

The change of azimuthal mode is associated with a transient state with low amplitude of oscillation before oscillations are recovered; however, with a slightly larger frequency increase, see, for instance, Fig. 4 for current density $-3 \times 10^{12}$ A$^2$/m.

Let us finally discuss the dependence of $f_{\text{edge}}$ with applied current, which decreases by about 6 GHz from low current to $-5 \times 10^{12}$ A/m$^2$, so, amounting to 1.2 GHz m$^2$/A. Two phenomena may explain this dependence. The first possibility is the nonlinearity of precession [44] due to the angle reaching several tens of degrees shortly before reversal for the largest current density. However, there is no clear grad-

ual change of frequency during the increase of amplitude of oscillation for a given current density. The second possibility is the effective field arising from the FL component of STT, the decrease being consistent with the expected antidamping effect. We may link this to the macrospin oscillations observed in the case of a damping torque. For a cylinder displaying 20 nm thickness and returning to equilibrium at zero injected current to 2.35 GHz for an injected current of $-6.79 \times 10^{11}$ A/m$^2$, i.e., amounting to $\approx 0.3$ GHz m$^2$/A. Both figures are therefore not consistent. In the end, there may be a contribution of both effects, nonlinearity and FL torque. The nonlinearity of precession is also expected to induce a nonlinear effect on the effective FL torque, as it is the exchange between spin accumulation and magnetization that connects the DL and FL terms in our approach [see Eq. (2)].

To conclude this section, edge modes are efficiently excited in 3D nanosystems, due to the combination of the flower state and larger FL torque. This leads to pathways for magnetization reversal which may deviate from the macrospin picture at large driving currents.

## IV. CONCLUSIONS

We explored the details of a spin-transfer-torque switching mechanism of 3D nanocylinders connected with a magnetic tunnel junction and highlighted the physical phenomena specific to the 3D situation compared with thin elements such as found in standard MRAM. First, we showed that the explicit consideration of spin accumulation is necessary, being sizable in the vicinity of both interfaces, with the tunnel barrier and the metal lead. In addition, the FL term is comparable with the DL one, contrary to the 2D case. This, combined with the flower state naturally occurring in a 3D element, makes the STT very efficient at the edges of the cylinder and excites edge precession modes with large magnitude and high frequency. These play a key role in the reversal, inducing, in particular, a quick and linear decrease of longitudinal magnetic moment versus time instead of the long incubation stage in a flat cell, which is liable to accelerate magnetization reversal. We believe that our results are meaningful when optimizing perpendicular-shape-anisotropy MRAM cells and, beyond, should be applicable to many situations in the emerging field of 3D spintronics.

## ACKNOWLEDGMENTS

The authors would like to thank Mairbek Chshiev, Christophe Thirion, Ricardo Sousa and Nuno Caçoilo for fruitful discussions. This work has been partially supported by the ANR Project No. M-bed-RAM ANR-23-CE24-0016.

## DATA AVAILABILITY

The data that support the findings of this article are not publicly available. The data are available from the authors upon reasonable request.
<figure><img src="image_11.png" /><figcaption>FIG. 8. Relaxed state of a nanomagnetic cylinder when the environmental temperature is assumed to be 50 K.</figcaption></figure>

# APPENDIX

## 1. Thermal effects

Thermal fluctuations have a significant impact on the physics of magnetization reversal in nanomagnetic systems. It is common knowledge that their impact increases for low-dimensionality or low-volume systems, decreasing the energy barriers preventing switching. This leads to the effect of superparamagnetism. In MRAM, the decrease of energy barriers shows up for thin devices with diameters typically below the usual ones for MRAM [8,9]. As expected, energy barriers are boosted by the much increased volume. Thanks to this structural change, PSA-STT-MRAMs exhibit much greater resistance to thermal events. As thoroughly showed in Ref. [45], thermal fluctuations mainly affect the incubation time in the total magnetization reversal mechanism of PSA-STT-MRAMs, promoting symmetry breaking. However, no unwanted switching at zero applied voltage is observed, thanks to the thermal retention provided by the perpendicular shape anisotropy. The quantitative imaging of magnetization in the free layer of a PSA MRAM directly confirmed the reduced impact of thermal energy in these. As far as the flower state is concerned, as it arises in a reduced volume compared to the entire pillar, it is indeed relevant to question the role of temperature. To do so, we performed additional simulations using a finite-difference effective-STT code, which considers thermal noise with the Langevin equation, magnetization being assumed to be in contact with a heat bath at temperature *T*. We considered an environmental temperature of 50 K, which is expected to correctly describe the experimental situation at 300 K, as the Langevin approach with macrospins overestimates the impact of temperature [46,47]. The results are shown in Fig. 8.

Figure 8 suggests that the flower state remains despite thermal noise. To confirm this quantitatively, let us examine the average magnetization at both a central cross section and an outer surface cross section over a time interval of 5 ns. We find that the normalized *z* component of the magnetization is 0.993 at the center plane of the cylinder and 0.985 at the current-injection surface. In more detail, let us examine the sign of the average $m_x$ component at two points located on opposite sides of a diameter at the outer surface. The extracted

<figure><img src="image_12.png" /><figcaption>FIG. 9. Two points along the same diameter on the bottom surface of the investigated cylinder. The average value of the *x* component of the unitary magnetization is extracted for both of them on a time interval of 10 ns.</figcaption></figure>

values are averaged over a 10 ns time interval. In Fig. 9(b), the value is -0.07 (vs -0.11 at 0 K) while in Fig. 9(b) the value is of 0.12 (vs 0.11 at 0 K). This demonstrates that the tilt of magnetization at the edges of the cylinder are preserved on average, i.e., the flow state survives thermal excitations. Due to the rotational symmetry of the system's physics, it is sufficient to perform this check along a single diameter rather than all possible ones.

## 2. Interfacial effects

Various effects may arise at the interfaces between the layers that constitute the structure. Two main types of contributions can be considered: one due to iPMA and another due to the DMI. Both effects can influence the flower state observed in the free layer under relaxed conditions.

The role of iPMA has already been considered in nanocylinders, e.g., in Sec. 2.2 of Ref. [45], where iPMA is introduced at the interface between the tunnel barrier and the free layer. Using the same finite-difference effective-STT code as for the thermal analysis, let us introduce in our system an interfacial anisotropy of $K_s = 1.4$ mJ/m$^2$. This value is typical for a FeCoB/MgO interface [8]. As could be expected, iPMA stiffens magnetization along the perpendicular direction. This decreases the magnitude of the flower state, however, there is still a deviation of magnetization from the *z* axis, highlighted by a change in the color of the magnetization vectors at the edge of the structure in Fig. 10.

On the other hand, the role of DMI in the flower state of a nanocylinder has not been reported yet, to our knowledge. In Ref. [48], we find that the largest value reported for the DMI at an MgO/FeCoB interface is 1.13 mJ/m$^2$. First, let us note that the sign of the interaction is positive. This implies that the rotation of the magnetization is counterclockwise at the edges of the domain [49], i.e., in the same orientation as that of the flower state at the current-injection surface resulting from the dipolar energy contribution. Thus, we would expect that
<figure><img src="image_13.png" /><figcaption>FIG. 10. Simulation of a nanocylinder without (a) and with (b) iPMA applied to the bottom surface with interfacial anisotropy of $K_s = 1.4$ mJ/m$^2$.</figcaption></figure>

the DMI enhances the flower state at the bottom surface, while disrupting it at the top surface.

To check this claim, we performed a simulation with MUMAX, which allows one to account for DMI at the interfacial level. The sign of the DMI coefficient is defined with respect to the outward normal of surfaces versus the *z* direction (positive if parallel and negative if antiparallel). The results are reported in Fig. 11. The flower state is enhanced at the surface where current is injected, i.e., for DMI >0 [Fig. 11(b)], while it is instead decreased and even inverted when DMI <0 [Fig. 11(c)]. Surprisingly, at first, there is also an impact at the surface opposite to the one where the DMI is applied.

<figure><img src="image_14.png" /><figcaption>FIG. 11. Simulation using MUMAX considering no DMI contribution (a), a positive contribution only at the bottom interface (b) and a negative DMI contribution at bottom interface (c). The DMI constant is taken to be 1.13 mJ/m$^2$ [48].</figcaption></figure>

We believe that this occurs because of the high value of the DMI coefficient considered and the height of the nanopillar being only a few times the dipolar exchange length, so both interfaces are coupled via exchange. The magnitude of the effect at the opposite interface is dramatic, the flower state being suppressed or greatly enhanced depending on the sign of DMI at the other interface. As we have shown that the flower state plays a direct role in the incubation time and dynamical mechanism, in a practical situation with a given material stack it appears crucial to consider whether DMI occurs at the interfaces.

To conclude, iPMA and DMI may decrease or enhance the flower state. We expected that this should have a direct impact on the incubation time.

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