Perpendicular Shape Anisotropy (PSA) in Nanomagnets

• PSA is a phenomenon where engineered nanomagnet geometries harness demagnetizing fields to promote perpendicular magnetization, enabling robust magnetic anisotropy.
• The effect scales with the aspect ratio, as high-aspect-ratio nanopillars flip demagnetizing factors to favor an out-of-plane easy axis over traditional in‐plane alignment.
• PSA improves device performance by decoupling thermal stability from interfacial effects, supporting scalable, field-free STT-MRAM and SOT-MRAM applications.

Perpendicular shape anisotropy (PSA) refers to the engineered use of geometry-induced magnetostatic energy in nanostructured ferromagnets to stabilize magnetization perpendicular to the plane, providing an intrinsic source of perpendicular magnetic anisotropy (PMA) that is robust and tunable at nanometric scale. Unlike conventional PMA, which arises mainly from interfacial spin–orbit coupling, PSA exploits the spatial distribution of demagnetizing fields (dipolar interactions) in high-aspect-ratio structures—typically nanopillars whose thickness approaches or exceeds their lateral dimension. This approach has enabled a new class of spintronic devices, particularly spin-transfer-torque magnetic random access memory (STT-MRAM), with scalable data retention down to single-digit nanometers (Watanabe et al., 2017, Perrissin et al., 2018, Meneguolo et al., 19 Mar 2025).

1. Fundamental Physical Basis of PSA

Classical magnetostatics defines the energy of a uniformly magnetized body through demagnetizing (shape) factors, $N_{x}$, $N_{y}$, and $N_{z}$ (with $N_{x}+N_{y}+N_{z}=1$ for a closed shape), which govern the preference of magnetization orientation due to internal dipolar fields. For a body magnetized along direction $i$, the demagnetizing energy per unit volume is $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$, where $M_s$ is the saturation magnetization and $\mu_0$ the permeability of free space.

The anisotropy energy density favoring a particular axis is

$$K_{\text{shape}} = \frac{1}{2}\mu_0 M_s^2 (N_\text{hard} - N_\text{easy}).$$

For a thin film ($t \ll D$), $N_{y}$0 and $N_{y}$1, hence in-plane orientation is favored. For a tall pillar ($N_{y}$2), $N_{y}$3, producing a positive $N_{y}$4 favoring perpendicular orientation (Meneguolo et al., 19 Mar 2025, Watanabe et al., 2017, Perrissin et al., 2018).

The total perpendicular anisotropy includes PSA and any interfacial or bulk contributions: $N_{y}$5 where $N_{y}$6 is the interfacial PMA (e.g., at FeCoB/MgO), and $N_{y}$7 any bulk uniaxial anisotropy (Perrissin et al., 2018).

2. Demagnetizing Factors and PSA Scaling

The demagnetizing factors depend critically on geometry—as parameterized by the aspect ratio $N_{y}$8 for a cylinder of thickness $N_{y}$9 and diameter $N_{z}$0. Standard analytical or numerical formulas allow calculation of $N_{z}$1 and $N_{z}$2 (Watanabe et al., 2017, Meneguolo et al., 19 Mar 2025). For example, Sato & Ishii give $N_{z}$3 (Perrissin et al., 2018). The sign of $N_{z}$4 flips from positive (easy-plane) for $N_{z}$5 to negative (easy-axis) for $N_{z}$6, enabling design of PSA.

Direct determination via electron holography has confirmed the large $N_{z}$7 in high-aspect-ratio pillars (e.g., $N_{z}$8 NiFe, $N_{z}$9) (Almeida et al., 2022).

Geometry	$N_{x}+N_{y}+N_{z}=1$0	$N_{x}+N_{y}+N_{z}=1$1	$N_{x}+N_{y}+N_{z}=1$2 Direction
Thin disk ($N_{x}+N_{y}+N_{z}=1$3)	$N_{x}+N_{y}+N_{z}=1$4	$N_{x}+N_{y}+N_{z}=1$5	Easy-plane ($N_{x}+N_{y}+N_{z}=1$6)
Pillar ($N_{x}+N_{y}+N_{z}=1$7)	$N_{x}+N_{y}+N_{z}=1$8	$N_{x}+N_{y}+N_{z}=1$9	Easy-axis (along $i$0)

3. Device Architectures and Experimental Realizations

PSA has been implemented in a variety of nanostructured systems, especially for STT-MRAM and SOT-MRAM applications. In STT-MRAM, the standard approach employs a thick ferromagnetic storage layer (e.g., Co, NiFe) in a nanopillar geometry on top of an interfacial PMA stack (e.g., MgO/FeCoB/MgO), achieving aspect ratios $i$1 and thus harnessing PSA (Perrissin et al., 2018, Meneguolo et al., 19 Mar 2025). Devices with $i$2 and $i$3 routinely yield thermal stability factors $i$4, with the stability maintained down to $i$5 by tuning $i$6 (Perrissin et al., 2018).

In SOT-MRAM, PSA assists deterministic, field-free switching by introducing a symmetry-breaking internal field through shape engineering—usually by patterning elliptical or polygonal nanomagnets with misaligned principal axes or tailored sidewall profiles (Wang et al., 2019, Chouhan et al., 2 Apr 2025). For instance, an elliptical pillar with its major axis at an angle to the current injection direction generates a built-in in-plane anisotropy field, breaking inversion symmetry and enabling deterministic spin-orbit-torque switching.

Laterally patterned films and stripes with tailored thickness profiles have shown that PSA can be lithographically controlled even beyond sub-micron scale, impacting coercivity, effective anisotropy, and switching characteristics (Kopnov et al., 2022).

4. PSA-Driven Switching Dynamics and Micromagnetic Modes

Magnetization dynamics in PSA structures are modeled by the Landau–Lifshitz–Gilbert (LLG) equation, including spin-transfer (STT) or spin-orbit torque (SOT) terms. For out-of-plane anisotropy dominated by PSA, the macrospin approximation remains valid at low aspect ratio, but for $i$7–$i$8 reversal proceeds via domain-wall nucleation and complex three-dimensional modes such as “flower” states or edge-localized resonances (Meneguolo et al., 19 Mar 2025, Caçoilo et al., 2020). Simulations show:

• Macrospin-like coherent switching for AR < 1.
• Buckling or non-uniform coherent modes at $i$9.
• Domain-wall nucleation regimes for AR > 1.5, propagating along the pillar thickness (Caçoilo et al., 2020).
• 3D inhomogeneities (“flower” states) at pillar ends and excitation of high-frequency edge modes (tens of GHz) that can accelerate switching (Meneguolo et al., 19 Mar 2025).

Field-like STT components become significant in thick pillars, necessitating self-consistent spin-transport treatments.

5. Impact on Device Performance and Scalability

The principal effect of PSA is to decouple data retention (thermal stability, $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$0) from interfacial PMA, enabling small-diameter ($E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$1 nm) MRAM elements with $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$2 (Watanabe et al., 2017, Perrissin et al., 2018, Almeida et al., 2022). Key performance and scaling implications:

• $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$3 scales linearly with the storage layer thickness once PSA dominates, enabling tunable retention at fixed lateral size (Caçoilo et al., 2020, Meneguolo et al., 19 Mar 2025).
• Coercivity and energy barriers remain robust even at elevated temperatures, provided $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$4 is accounted for (Zhang et al., 2021, Almeida et al., 2022).
• Write current and switching time increase at large aspect ratio, and reversal can shift from fast, coherent to slow, domain-wall mediated as $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$5 increases (Caçoilo et al., 2020).
• Lowering the resistance-area product (RA) is mandatory to keep write voltage within breakdown limits as thickness increases (Caçoilo et al., 2020).
• Inclusion of low-damping bulk materials (e.g., Co, NiFe) reduces switching current, as $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$6 is no longer constrained by thin-interface spin pumping (Perrissin et al., 2018).
• Core-shell (dipolar-compensated) PSA architectures can suppress stray field cross-talk in dense arrays without sacrificing $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$7, improving packing density and write speed (Caçoilo et al., 2023).

6. PSA in Field-Free SOT Devices via Symmetry Breaking

PSA is exploited for field-free deterministic SOT switching by designing nanomagnet geometries exhibiting nonzero in-plane anisotropy fields that break inversion symmetry. In elliptical or triangular nanomagnets, the in-plane difference in demagnetizing factors ($E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$8) introduces a built-in symmetry-breaking field, enabling deterministic switching solely by current pulses, without external fields (Wang et al., 2019, Chouhan et al., 2 Apr 2025). Micromagnetic simulations and experiments confirm that precise control over the magnitude and direction of this in-plane field enables robust, fast, and reliable switching windows.

7. Temperature Dependence, Material Selection, and Optimization Guidelines

The PSA contribution to anisotropy scales as $E_{\text{shape},i} = \frac{1}{2}\mu_0 M_s^2 N_i$9 and displays strong temperature dependence, particularly as $M_s$0 approaches the Curie point. Accurate modeling requires inclusion of $M_s$1 scaling via Bloch or Kuz’min laws, and interfacial anisotropy $M_s$2 via Callen–Callen scaling (Zhang et al., 2021). At the device level:

• Aspect ratio $M_s$3 is a tunable parameter for targeting $M_s$4; typically $M_s$5–$M_s$6 gives a practical balance between stability and write energy (Perrissin et al., 2018, Caçoilo et al., 2020).
• Coherent switching is preserved for $M_s$7–$M_s$8; for larger $M_s$9, minimum-energy-path simulations (string method) or micromagnetic modeling are required.
• Use of core-shell structures or flux compensation can mitigate dipolar stray fields, recovering high-density packing (Caçoilo et al., 2023).
• Lateral width and lithographic edge profile engineering can locally modulate $\mu_0$0 for logic or domain-wall devices (Kopnov et al., 2022).

PSA-enabled devices thus offer a geometric degree of freedom for optimizing MRAM and spintronic logic across a range of materials, device sizes, and operational regimes. The interplay of demagnetizing factors, temperature, material choice, damping, and device geometry provides a comprehensive design space for next-generation, robust, and miniaturized spintronic memory (Watanabe et al., 2017, Perrissin et al., 2018, Almeida et al., 2022, Caçoilo et al., 2023, Meneguolo et al., 19 Mar 2025, Chouhan et al., 2 Apr 2025).