Thermal Influence on Chirality-Driven Dynamics and Pinning of Transverse Domain Walls in Z-Junction Magnetic Nanowires

Abstract

Magnetic nanowires with domain walls (DWs) play a crucial role in the advancement of next-generation memory and spintronic devices. Understanding the thermal effects on domain wall behavior is essential for optimizing performance and stability. This study investigates the thermal chirality-dependent dynamics and pinning of transverse domain walls (TDWs) in Z-junction nanowires using micromagnetic simulations. The analysis focuses on head-to-head (HHW) and tail-to-tail (TTW) domain walls with up and down chirality under varying thermal conditions. The results indicate that higher temperatures reduce the pinning strength and depinning current density, leading to enhanced domain wall velocity. At 200 K, the HHW_down domain wall depins at a critical current density of 1.2 × 10¹¹ A/m², while HHW_up requires a higher depinning temperature, indicating stronger pinning effects. Similarly, the depinning temperature (T_d) increases with Z-junction depth (d), reaching 300 K at d = 50 nm, while increasing Z-junction (λ) weakens pinning, reducing T_d to 150 K at λ = 50 nm. Additionally, the influence of Z-junction geometry and magnetic properties, such as saturation magnetization (M_s) and anisotropy constant (K_u), is examined to determine their effects on thermal pinning and depinning. These findings highlight the critical role of chirality and thermal activation in domain wall motion, offering insights into the design of energy-efficient, high-speed nanowire-based memory devices.

1. Introduction

The dynamics and pinning of domain walls (DWs) in magnetic nanowires have attracted significant attention for spintronic and memory applications [1,2,3,4,5,6]. Magnetic nanowires are particularly promising for high-density data storage due to their ability to manipulate domain walls as carriers of binary information. Transverse domain walls (TDWs), where magnetization aligns linearly within the wall, are critical to the operation of these devices because of their relatively simple structure and predictable dynamics [7,8,9,10,11,12,13]. The control of TDW motion and pinning are two key factors to optimizing device performance, as it determines read/write speeds, power consumption, and thermal stability.

Domain wall motion is primarily driven by spin-transfer torque (STT), where spin-polarized currents interact with local magnetic moments, causing the domain wall to shift along the nanowire [14,15,16,17,18,19,20]. However, pinning effects at constrictions or material defects can hinder smooth domain wall propagation. These pinning effects are particularly relevant in practical devices, where nanoscale limitations and geometric features are unavoidable [6,20,21,22,23,24]. Consequently, understanding how geometric, material properties, and thermal factors influence pinning and depinning processes is essential for device optimization.

Thermal effects add another layer of significant factors to domain wall dynamics. Elevated temperatures can lead to thermally activated depinning, changes in magnetization behavior, and a reduction in pinning strength. At the same time, thermal fluctuations can influence the chirality-dependent behavior of TDWs, altering their speed and stability [25,26,27,28,29]. For instance, down-chirality domain walls may exhibit distinct dynamics under thermal conditions due to differences in wall energy and magnetization configurations.

Previous studies have explored the role of nanowire geometry and magnetic properties in the thermal dynamics and pinning of domain walls, with findings indicating that these parameters strongly influence wall motion and stability [30,31,32,33,34]. However, the interplay between chirality and thermal effects remains underexplored. The novelty of this work lies in its comprehensive and systematic analysis of chirality-dependent transverse domain wall (TDW) behavior under thermal activation within Z-junction geometries. Specifically, we examine four distinct TDW configurations HHW_up, HHW_down, TTW_up, and TTW_down across a wide temperature range and correlate their depinning thresholds with geometrical factors (junction depth d and width λ) and intrinsic material properties (saturation magnetization M_s and uniaxial anisotropy Ku). This multidimensional approach reveals that domain wall chirality plays a pivotal role in determining thermal pinning strength. Furthermore, we introduce and analyze the depinning temperature (T_d) as a metric to characterize the thermal robustness of different domain wall states, providing a new design parameter for high-temperature spintronic devices.

By investigating how TDWs with different chiralities respond to thermal conditions, this study aims to fill a critical gap in the field. Furthermore, understanding the combined effects of thermal energy, geometric constrictions, and material properties will provide a comprehensive framework for designing robust, energy-efficient nanowire-based memory devices.

The present work focuses on the dynamics and pinning of TDWs in nanowires with Z-junction constrictions. Using micromagnetic simulations, we examine how temperature, chirality, and junction dimensions influence domain wall behavior. This research provides insights into optimizing nanowire-based devices for high-temperature environments, with the goal of enhancing performance in terms of speed, stability, and power efficiency.

2. Modeling

Micromagnetic simulations were conducted using the Object-Oriented Micromagnetic Framework (OOMMF) software (version is 1.2a5) to investigate the thermal dynamics of TDWs. A nanowire with dimensions of 300 nm in length, 60 nm in width, and 5 nm in thickness was modeled. The Z-junction, located at the wire’s center, had varying depths (d) and lengths (λ) (Figure 1).

The simulation device was discretized into small computational cells with dimensions of 2.5 nm × 2.5 nm × 5 nm, ensuring that each cell size remains smaller than the material’s exchange length, which is given by $l_{ex}\sqrt{{\displaystyle \frac{A}{2\pi M_s^2}}}$ and calculated to be 5.3 nm. In this study, permalloy was selected as the material, characterized by a saturation magnetization (M_s) of 800 kA/m, α = 0.01 [35,36], an anisotropy constant (K_u) of 0.5 × 10⁵ J/m³ [37], an exchange stiffness (A) of 1.3 × 10⁻¹¹ J/m, and the time sampling ∆T is 0.1 picoseconds [34].

To analyze magnetization dynamics and domain wall motion, OOMMF [38] was used to solve the Landau–Lifshitz–Gilbert (LLG) equation incorporating both adiabatic and non-adiabatic spin-transfer torques with thermal field [39].

$${\displaystyle \frac{d\mathit{m}}{dt}}=-\gamma \mathit{m}\times {(H}_{eff}+{(H}_{th}+\alpha \mathit{m}\times {\displaystyle \frac{d\mathit{m}}{dt}}-\left(\mathit{u}.\nabla \right)\mathit{m}+\beta \mathit{m}\times \left[\left(\mathit{u}.\nabla \right)\mathit{m}\right]$$

(1)

where α represents the Gilbert damping constant, γ is the gyromagnetic ratio, and β is the non-adiabatic spin-transfer torque parameter. The adiabatic spin-transfer torque parameter u is given by:

$$\mathit{u}={\displaystyle \frac{gP\beta {\mu }_B}{2e\alpha M_s}}J$$

(2)

where J is the current density, $g$ is the Lande factor, P is the spin polarization, μ_B is the Bohr magneton, and e is the elementary charge.

The nanowire structure includes a constriction at its center, designed by displacing a section along the x-axis. $H_{th}$ is the thermal field, which accounts for the effect of temperature on the system.

$$H_{th}=\sqrt{{\displaystyle \frac{2\alpha k_{B}T}{\gamma M_{s}V}}\mathsf{\xi }}$$

(3)

where $k_B$ is the Boltzmann constant, $T$ the device temperature, M_s is the saturation magnetization, $V$ is the volume of the computational cell, and $\mathsf{\xi }$ is a random Gaussian-distributed noise term with zero mean and unit variance [40,41].

3. Results and Discussion

In this study, four different transverse domain wall (TDW) configurations were utilized to explore the impact of wall type and chirality on the pinning behavior in Z-junction magnetic nanowires. As shown in Figure 2, panels (a) and (b) represent HHWs, while panels (c) and (d) illustrate TTWs. In Figure 2a, the HHW exhibits upward chirality, characterized by a clockwise rotation of the magnetization across the wall center. In contrast, Figure 2b shows an HHW with downward chirality, where the magnetization rotates counterclockwise. For the TTWs, Figure 2c displays an upward chirality with clockwise rotation, whereas Figure 2d presents a downward chirality with counterclockwise magnetization rotation. The HHWs involve a convergence of magnetization vectors toward the wall center, typically resulting in higher magnetic charge accumulation and stronger pinning. TTWs, on the other hand, involve a divergence of magnetization vectors at the wall, leading to different depinning behavior. In all configurations, the color map indicates the in-plane magnetization direction (red for +x and blue for −x), and black arrows represent the local magnetization vectors. The Z-shaped junction geometry introduces asymmetry that, when combined with the domain wall type and chirality, significantly influences the thermal stability, pinning strength, and dynamic response of the domain walls. These four configurations provide a systematic framework for studying chirality-dependent pinning and are essential for understanding domain wall motion in spintronic memory applications.

Each TDW configuration exhibits a slightly different relaxed energy state due to chirality and head/tail orientation within the Z-junction. The differences in energy are small but significant enough to influence depinning thresholds, as shown in our results. These states can be experimentally accessed by applying tailored magnetic field pulses or spin-polarized currents to induce desired domain wall chirality and orientation, as demonstrated in previous studies on DW control [16,22,30].

3.1. Chirality-Dependent Dynamics at Varying Temperatures

The influence of temperature on domain wall (DW) speed was first investigated prior to the DW reaching the Z-junction, considering all four chirality configurations. The graphs in Figure 3 illustrate the dependence of domain wall speed on J at temperatures of 200 K and 300 K for different domain wall configurations: HHW_down, HHW_up, TTW_down, and TTW_up. In all cases, the domain wall speed increases linearly with current density, indicating a direct relationship between applied current and domain wall motion driven by the spin-transfer torque in the Z-junction device with λ = 20 nm and d = 40 nm.

At a higher temperature of 300 K, the domain wall moves faster compared to 200 K under the same current conditions, suggesting that thermal activation assists in overcoming pinning effects and enhances domain wall motion.

Comparing the different configurations in Figure 3, HHW_down and HHW_up exhibit similar trends, with HHW_up showing slightly higher speeds (Figure 3). TTW_down and TTW_up also follow a similar pattern, with TTW_up demonstrating slightly greater mobility. TTW configurations generally show higher speeds compared to HHW configurations, indicating that transverse domain walls experience less resistance in motion and are more efficient for current-driven propagation. The temperature effect is more pronounced in TTW configurations, where an increase in temperature leads to a more significant enhancement in domain wall speed compared to HHW configurations. These results highlight the role of temperature in facilitating domain wall motion, which is seen as increasing speed and reducing pinning effects, under the same current density. The higher mobility observed in TTW domain walls suggests that they may be more suitable for applications requiring fast domain wall propagation. The linear relationship between domain wall speed and current density further confirms the influence of the spin-transfer torque, making these findings relevant for spintronic and memory applications.

The enhanced speed of down-chirality TDWs is attributed to their lower wall energy, which reduces resistance to thermal fluctuations. Higher temperatures amplify these effects, leading to greater speed differences between chirality.

This linear behavior persisted across the entire temperature range studied, with the slope of the speed-current density curve decreasing slightly at elevated temperatures due to increased thermal noise.

3.2. HHW_up and HHW_down Thermal Pinning

In this study, micromagnetic simulations were conducted to investigate thermal pinning through a Z-junction (λ = 20 nm and d = 40 nm) based on different domain wall chirality. The analysis begins with the HHW_down and HHW_up configurations. The HHW_down domain wall, shown in Figure 4a–c and analyzed in graph (g), exhibits a pronounced response to temperature variations. At 100 K, the domain wall remains pinned for an extended period with minimal depinning. However, as the temperature increases to 200 K, the depinning process accelerates, leading to a noticeable rise in the magnetization component m_x. The gradual transition in depinning behavior across temperatures suggests that HHW_down is more thermally sensitive, enabling easier depinning as thermal energy increases.

In contrast, the HHW_up domain wall, illustrated in Figure 4d–f and examined in graph (h), demonstrates a stronger pinning effect, particularly at lower temperatures. At 200 K, the domain wall experiences significant pinning, taking a longer time to transition compared to HHW_down. However, at 250 K, the depinning process becomes more pronounced, causing an abrupt increase in m_x. The more sudden shift in behavior with increasing temperature indicates that HHW_up retains greater thermal stability, making it more resistant to temperature-induced depinning.

The comparison between graphs shown in Figure 4g,h further emphasizes the distinction between the two domain wall configurations. HHW_down undergoes a gradual transition from pinned to depinned states, while HHW_up exhibits a more abrupt depinning shift at higher temperatures. This suggests that HHW_up is better suited for applications requiring robust domain wall retention, while HHW_down, being more thermally sensitive, depins more easily at elevated temperatures, which could affect the reliability of memory storage in high-temperature environments.

For further clarification of HHW_down and HHW_up, the investigations focus on pinning and depinning behavior across different values of d and λ. As shown in Figure 5a, the depinning temperature (T_d) increases with increasing Z-junction depth (d) for both HHW_down and HHW_up domain walls. The HHW_up domain wall consistently exhibits a higher depinning temperature than HHW_down, indicating that it requires more thermal energy to overcome the pinning barrier. This trend suggests that a deeper Z-junction provides greater stability for domain walls, reinforcing their resistance to thermal depinning.

In Figure 5b, the depinning temperature decreases as the domain wall width (λ) increases, demonstrating that wider Z-junctions experience weaker pinning effects. Once again, HHW_up maintains a higher T_d compared to HHWd_own, confirming that it remains more stable against thermal fluctuations. The observed decrease in T_d with increasing λ suggests that Z-junctions with larger widths are more susceptible to domain wall depinning due to reduced interaction with pinning sites.

Overall, HHW_up consistently requires a higher temperature for depinning than HHW_down, regardless of Z-junction depth or width. Increasing Z-junction depth enhances thermal stability by raising the energy barrier for depinning, while increasing Z-junction width lowers T_d, making the domain walls more prone to depinning. These results highlight the significance of structural control in optimizing the thermal stability of domain walls for memory storage applications.

3.3. TTW_up and TTW_down Pinning

The micromagnetic simulation was also conducted to investigate the thermal domain wall (DW) pinning and depinning in the Transverse Tail-to-Tail (TTW) configuration through the Z-junction nanowire. The results illustrate the thermal pinning and depinning behavior of Transverse Tail-to-Tail (TTW) up and down domain walls. The micromagnetic snapshots Figure 6a–c represent TTW_up, while Figure 6d–f correspond to TTW_down. In both cases, the domain walls interact with the notch, with their evolution influenced by temperature variations. At lower temperatures, the domain walls remain pinned for a longer duration, while at higher temperatures, thermal activation assists in overcoming the pinning barrier, leading to depinning.

The time evolution graphs further highlight these differences. In graph Figure 6g, TTW_up shows a significant difference in depinning behavior between 50 K and 100 K, with the domain wall remaining pinned for a longer period at 50 K but depinning rapidly at 100 K. Similarly, in the graph shown in Figure 6h, TTW_down exhibits a delayed depinning response at 100 K, but depinning occurs much faster at 200 K, suggesting that it requires a higher temperature to overcome the pinning potential.

Comparing both configurations, TTW_down exhibits greater thermal stability, as it remains pinned for longer and requires a higher temperature for depinning. In contrast, TTW_up is more thermally sensitive, depinning more easily as the temperature increases. These findings indicate that TTW_down could be more suitable for applications requiring greater resistance to thermal fluctuations, whereas TTW_up may be more responsive to external thermal activation.

To gain a deeper understanding, further investigations were conducted using different values of the Z-junction depth (d), while keeping the length (λ) fixed. The graphs in Figure 7 illustrate the thermal depinning temperature (Tₐ) as a function of Z-junction depth (d) and length (λ) for up and down Transverse (TTW) domain walls. In Figure 7a, the depinning temperature increases with increasing d for both TTW_down and TTW_up. However, TTW_up consistently exhibits a higher T_d than TTW_down, indicating that it requires more thermal energy to depin. This suggests that increasing d enhances thermal stability, making the domain walls more resistant to thermal depinning due to the stronger pinning effect.

In Figure 7b, the depinning temperature decreases as the domain wall width increases. Wider Z-junction experience weaker pinning effects, making them more susceptible to depinning at lower temperatures. TTW_up maintains a higher T_d compared to TTW_down, confirming that it remains more stable against thermal fluctuations. The observed decrease in T_d with increasing λ suggests that domain walls with larger widths interact less strongly with pinning sites, reducing the energy required for depinning.

Overall, TTW_up exhibits greater thermal stability than TTW_down, requiring a higher temperature for depinning across different nanowire thicknesses and domain wall widths. While increasing thickness enhances thermal stability by raising the energy barrier for depinning, increasing the domain wall width lowers T_d, making the walls more prone to depinning. These results highlight the significance of structural control in optimizing the thermal stability of domain walls for memory storage applications.

3.4. Magnetic Property Pinning and Depinning

The magnetic properties play a significant role in determining the thermal domain wall (DW) pinning and depinning at the Z-junction. In this study, micromagnetic simulations were conducted to investigate how different magnetic properties influence the thermal behavior of DW pinning and depinning, particularly considering different DW chirality.

Figure 8a,b present the micromagnetic configurations of the head-to-head (HHW_up) domain wall with a saturation magnetization of M_s = 800 kA/m. At a device temperature of 100 K, the HHW_up remains pinned at the Z-junction, as seen in Figure 8a. However, as the temperature is increased to 200 K, thermal activation provides sufficient energy for the domain wall to overcome the pinning barrier, leading to depinning and propagation toward the end of the nanowire, as shown in Figure 8b.

The time evolution of the x-component of magnetization (m_x), shown in Figure 8c, further highlights the impact of temperature on domain wall depinning. At 100 K (black curve), the domain wall remains pinned for a longer duration, exhibiting only minor fluctuations in m_x. In contrast, at 200 K (red curve), the domain wall depins rapidly, leading to a sharp increase in m_x, indicating full propagation along the nanowire. This behavior confirms that thermal energy significantly influences the depinning process; higher temperatures facilitate domain wall motion by lowering the energy barrier at the Z-junction. These findings emphasize the importance of saturation magnetization and temperature control in optimizing domain wall stability and motion in magnetic nanowires, particularly for memory and logic applications where precise control of DW dynamics is essential.

This study examines the thermal pinning and depinning behavior of a domain wall at a Z-junction for a material with an anisotropy constant of K_u = 0.5 × 10⁵ J/m³. Figure 9a,b display the micromagnetic configurations at different temperatures, while Figure 9c presents the time evolution of the x-component of magnetization (m_x).

At a temperature of 100 K, as shown in Figure 9a, the domain wall remains pinned at the Z-junction, indicating that the thermal energy at this temperature is not sufficient to overcome the pinning potential. However, when the temperature is increased to 200 K, as illustrated in Figure 9b, the domain wall exhibits significant thermal fluctuations and depins from the junction, subsequently propagating along the nanowire. This behavior suggests that the increased thermal energy at higher temperatures reduces the effective pinning barrier, thereby facilitating the depinning transition. The time evolution of m_x in Figure 9c further confirms this behavior. At 200 K (black curve), the magnetization component increases sharply, indicating a rapid depinning event where the domain wall moves across the junction and stabilizes at the nanowire’s end. In contrast, at 100 K (red curve), the domain wall remains pinned for an extended period, resulting in a lower and more stable m_x value over time. This suggests that at higher temperatures, thermal fluctuations contribute to a reduced pinning effect, resulting in a more pronounced depinning response.

These findings highlight the influence of anisotropy on thermal domain wall stability. With K_u = 0.5 × 10⁵ J/m³, domain walls exhibit temperature-dependent pinning behavior, where lower temperatures promote strong pinning, while higher temperatures facilitate depinning.

The graphs illustrate the dependence of the thermal depinning temperature (T_d) on the saturation magnetization (M_s) and anisotropy constant (K_u) for head-to-head (HHW_up) and HHW_down domain walls at the Z-junction.

In Figure 10a, the depinning temperature increases as the saturation magnetization (M_s) increases for both HHW_up and HHW_down configurations. The HHW_up domain wall consistently exhibits a higher T_d than HHW_down, indicating that it requires more thermal energy to depin. This suggests that increasing M_s enhances the thermal stability of the domain wall by increasing the overall magnetic energy, thereby strengthening the pinning effect at the Z-junction.

In Figure 10b, a similar trend is observed with the anisotropy constant (K_u), where T_d increases with increasing K_u. Again, HHW_up exhibits a higher depinning temperature than HHW_down, indicating that the up-chirality configuration is more resistant to thermal activation. The increase in T_d with K_u suggests that higher anisotropy enhances the domain wall’s stability, making it more resistant to depinning by increasing the energy barrier.

Overall, the results indicate that both M_s and K_u contribute to the thermal stability of the domain wall at the Z-junction. A higher saturation magnetization or anisotropy constant leads to a greater resistance to thermal depinning, with HHW_up exhibiting stronger pinning effects than HHW_down in all cases. These findings highlight the importance of material properties in optimizing domain wall stability for applications in memory and logic devices.

Although this study is based on micromagnetic simulations, the observed behaviors of thermally activated domain wall (DW) pinning and chirality-dependent depinning can be experimentally validated using advanced magnetic imaging techniques. Nanowires with tailored Z-junction geometries and controlled magnetic properties can be fabricated using focused electron beam lithography and thin-film deposition methods. These fabrication techniques provide a feasible experimental platform for testing the simulation results presented in this work and for further exploring the design of chirality-engineered, thermally stable spintronic devices. In particular, magneto-optical Kerr effect (MOKE) microscopy can be employed to track domain wall motion under applied current and temperature, enabling real-time visualization of depinning events. Lorentz transmission electron microscopy (TEM) or magnetic force microscopy (MFM) offers high-resolution imaging of magnetic structures and can confirm the influence of Z-junction geometry on DW behavior.

4. Conclusions

This study explored the impact of temperature, chirality, and structural parameters on the pinning and depinning dynamics of transverse domain walls (TDWs) in Z-junction nanowires. The results reveal that thermal activation significantly influences domain wall motion, reducing the pinning energy barrier and enhancing depinning at higher temperatures. The depinning temperature (T_d) was found to increase with Z-junction depth, reaching 300 K for d = 50 nm, whereas increasing the domain wall width (λ) weakened the pinning effect, lowering T_d to 150 K for λ = 50 nm. Furthermore, domain wall chirality played a crucial role, with HHW_up exhibiting stronger pinning and requiring a higher depinning current density of 1.5 × 10¹¹ A/m², compared to HHW_down, which depinned at 1.2 × 10¹¹ A/m² at 200 K.

The findings indicate that tuning nanowire geometry and material properties, such as saturation magnetization (M_s = 800 kA/m) and anisotropy constant (K_u = 0.5 × 10⁵ J/m³), can effectively control domain wall stability and motion. The results also highlight that TTW domain walls generally exhibit greater mobility compared to HHW configurations, making them more suitable for applications requiring fast domain wall propagation.

Overall, this work provides critical insights into the thermal behavior of domain walls, which is essential for optimizing spintronic devices and magnetic memory technologies. Future studies could focus on non-uniform geometries, spin–orbit coupling effects, and the influence of stochastic thermal fluctuations on domain wall dynamics to further refine control over nanoscale magnetization processes.