Spintronics Spin Orbit Torque Magnetic Domain Wall Llgs Dmi Skyrmion Racetrack

# Spintronic Domain Walls and Skyrmions: DMI, SOT Drive, Walker Breakdown, and Thiele Dynamics

## 1. Introduction: Topological Defects in Magnetism

Magnetic skyrmions are nanoscale, topologically protected spin textures characterized by a whirling magnetization pattern that wraps the unit sphere multiple times. Originally predicted theoretically in the 1960s (Bogdanov and Yablonskii) and observed experimentally in 2009 (Müller et al. and Yu et al.), skyrmions represent a rich frontier in condensed matter physics, combining fundamental physics (topological defects, nonlinear dynamics, novel statistics) with practical applications (ultra-dense magnetic storage, low-power spintronic devices).

The emergence of skyrmions requires breaking the discrete (up-down) symmetry of ferromagnetism through an asymmetric exchange interaction—the Dzyaloshinskii-Moriya Interaction (DMI)—which stabilizes non-collinear spin configurations. Unlike domain walls (which separate up and down domains in a planar boundary), skyrmions are topologically distinct: they carry a topological charge Q = ±1, protecting them from annihilation. When driven by spin-transfer torque (STT) or spin-orbit torque (SOT) from electrical currents, skyrmions move with ultralow damping, moving efficiently through materials. This enables skyrmion racetrack memory—a concept where skyrmions are nucleated, moved along a magnetic nanowire, and annihilated to read stored information—promising storage densities exceeding terabits/inch² with power consumption far below conventional magnetic storage.

This article develops skyrmion physics from first principles, covering the micromagnetic energy functional, Dzyaloshinskii-Moriya interaction, topological charge, equations of motion, the skyrmion Hall effect, and practical racetrack memory architectures. A companion Python solver implements 2D micromagnetic simulations of skyrmion nucleation, current-driven motion, and topological charge computation.

## 2. Micromagnetic Energy Functional

Magnetic materials are described at the continuum level by the magnetization field mathbfm(mathbfr) (unit vector) with total energy:

where:

• A is the exchange stiffness (units J/m), determining the energy cost of non-collinear spin arrangements
• K_u is the uniaxial anisotropy constant (units J/m³), favoring magnetization along the z-axis (perpendicular to the film)
• mu_0 M_s is the saturation magnetization times permeability
• H_textext is the applied magnetic field
• E_textDMI is the Dzyaloshinskii-Moriya interaction energy

The exchange term A(nabla mathbfm)^2 penalizes spatial variations; the anisotropy term K_u(1 - m_z^2) favors out-of-plane alignment. These two terms alone produce conventional domain walls and ferromagnetic/antiferromagnetic textures with collinear or planar spin arrangements.

## 3. Dzyaloshinskii-Moriya Interaction (DMI)

The Dzyaloshinskii-Moriya interaction is an antisymmetric exchange term arising from spin-orbit coupling in materials with broken inversion symmetry (e.g., at interfaces or in helimagnetic materials). The DMI energy density is:

or in continuum form:

where D is the DMI strength (units J/m²). The key feature is that DMI is proportional to the curl of magnetization, not the magnitude—it does not depend on |nabla mathbfm| but rather on the helicity (handedness) of the spin texture.

Expanding in component form for perpendicular magnetization mathbfm = (m_x, m_y, m_z):

For a skyrmion (to be defined precisely below), the DMI energy is minimized when the spin texture has a specific helicity (clockwise or counter-clockwise circulation when viewed from above). This stabilizes skyrmions against collapse into single domain walls.

## 4. Topological Charge and Skyrmion Definition

The topological charge (or skyrmion number) quantifies the winding of the magnetization field around 2D space. It is defined as:

which equals the stereographic projection of the magnetization onto a sphere. For a spatially uniform magnetization (all pointing up, mathbfm = hatz), the integrand vanishes everywhere, giving Q = 0. A single domain wall (with magnetization varying from up to down across a boundary) also has Q = 0 because the magnetization sweeps only a semicircle on the sphere. However, a skyrmion (with magnetization circulating both horizontally and vertically from up to down and back up) wraps the sphere completely, yielding Q = ±1, depending on the direction of circulation.

A skyrmion with Q = 1 (or Q = -1) is topologically protected: it cannot be continuously deformed into a uniform state without energy barriers. This topological protection arises because any path from a skyrmion to a uniform state must pass through intermediate textures with cost scaling as ~A × (length scale)². Skyrmions thus exhibit a characteristic energy scale and size.

## 5. Skyrmion Size and Energy

For a circular skyrmion, minimizing the total energy E = int d^2r [A (nabla mathbfm)^2 + K_u(1-m_z^2) - D m_z (partial_x m_y - partial_y m_x)] yields characteristic length scale (skyrmion radius):

and energy:

For typical parameters (A ~ 10 pJ/m, D ~ 1-5 mJ/m², K_u ~ 0.1-1 MJ/m³), skyrmion radii are 10-100 nm and energies are ~1-10 pJ, enabling numerous skyrmions to fit within micrometer-scale devices.

## 6. Landau-Lifshitz-Gilbert Equation and Skyrmion Dynamics

The magnetization dynamics under thermal and field influences are governed by the Landau-Lifshitz-Gilbert (LLG) equation:

where mathbfH_texteff = -fracdelta Edelta mathbfm is the effective field derived from the energy functional, γ is the gyromagnetic ratio, and α is the Gilbert damping parameter. The STT and SOT terms represent spin-orbit transfer torques from currents.

For skyrmion dynamics, a collective coordinate approach (Thiele's equation) simplifies the problem by treating the skyrmion as a rigid object with position mathbfR = (X, Y) and velocity mathbfv = dotmathbfR.

## 7. Thiele Equation and Skyrmion Motion

The Thiele equation describes the equation of motion for a skyrmion's center of mass position:

where:

• mathbfG is the gyrovector (related to topological charge): mathbfG = -4pi mathbfQ M_s V hatz for Q = ±1
• mathbfD is the dissipation tensor (proportional to the skyrmion area and damping α)
• mathbfF is the total force on the skyrmion (from STT, SOT, pinning, etc.)

For a cylindrically symmetric skyrmion, mathbfG = G hatz and mathbfD = D mathbfI. Neglecting damping momentarily, the cross-product term mathbfG times mathbfv causes the skyrmion to move perpendicular to the applied force—a topological effect with no counterpart in particle physics (except for charged particles in magnetic fields). The skyrmion Hall angle is defined as:

for typical parameters. For damping α ~ 0.01-0.1, the skyrmion Hall angle ranges from ~85° to ~45°, meaning a current applied along x produces skyrmion motion with both x and y components.

## 8. Spin-Transfer Torque and Current-Driven Skyrmion Motion

When a spin-polarized current passes through a magnetic material containing skyrmions, the angular momentum transfer (spin-transfer torque) drives skyrmion motion. The adiabatic STT term:

where u ~ (ℏ/2e) j / M_s is the spin wind velocity (proportional to the current density j). For a skyrmion with topological charge Q = 1, the STT force is approximately:

which, combined with Thiele's equation, produces skyrmion drift velocity:

showing that skyrmions move nearly parallel to the applied current (at small angles determined by the damping parameter).

## 9. Spin-Orbit Torque and Three-Terminal Skyrmion Drives

An alternative driving mechanism is spin-orbit torque (SOT), which converts charge current in a heavy metal (e.g., Pt, Ta) into spin current via the spin Hall effect. Unlike STT, which requires magnetization information encoded in a reference magnet, SOT is a geometric effect independent of local magnetization direction. The SOT term:

where hatmathbfz is the spin polarization direction (determined by the heavy metal's spin Hall angle). SOT enables three-terminal control: current in drives skyrmion motion, voltage out measures deflection (via anomalous Hall effect or planar Hall effect).

## 10. Skyrmion Hall Effect and Trajectories

The skyrmion Hall effect—the deflection of skyrmion motion perpendicular to the driving current—arises from the gyrovector mathbfG. For a current applied along x, the Thiele equation predicts:

showing a coupled drift: the skyrmion drifts at angle θ_SkHE from the current direction. Experiments confirm this deflection, with angles ranging from 10° to 45° depending on material and damping.

The skyrmion Hall effect poses challenges for racetrack memory (skyrmions drift toward one side of a wire), but also enables new functionalities: multiple parallel tracks can be demultiplexed by exploiting the different Hall angles for different skyrmion sizes.

## 11. Racetrack Memory Architecture

A skyrmion racetrack memory consists of a ferromagnetic nanowire (typically 50-500 nm wide, micrometers to millimeters long) with perpendicular magnetic anisotropy and DMI. Information is encoded by the presence (1) or absence (0) of a skyrmion at specific positions along the track. The operation sequence is:

1. Nucleation: A short current pulse or field excitation creates a skyrmion near one end of the track.

2. Writing: Multi-bit sequences are written by producing patterns of skyrmions separated by domain walls.

3. Moving: A current pulse drives all skyrmions synchronously along the track (taking advantage of the topological protection to move coherently without scattering).

4. Reading: An electrical transducer detects the presence of a skyrmion via anomalous Hall effect (AHE) in a dedicated sensor region.

5. Erasing: A destructive read or annihilation pulse removes skyrmions.

The key advantage over conventional disk drives is that skyrmion racetrack stores data sequentially along a 1D track, similar to tape drive technology but with nanosecond access times and petabit/inch² density potential.

## 12. Numerical Solver: 2D Micromagnetic Skyrmion Simulation

The following Python code implements a simplified 2D micromagnetic solver using finite differences, computing skyrmion textures, topological charge, and current-driven dynamics:

import numpy as np
import matplotlib.pyplot as plt
from scipy.ndimage import convolve
from matplotlib.patches import Circle

def initialize_skyrmion(grid_size, skyrmion_radius, center):
    """
    Initialize a skyrmion texture using analytical approximation.
    """
    m = np.zeros((grid_size, grid_size, 3))
    x = np.linspace(-grid_size//2, grid_size//2, grid_size)
    y = np.linspace(-grid_size//2, grid_size//2, grid_size)
    X, Y = np.meshgrid(x, y)
    
    # Skyrmion profile: m_z decreases from 1 at r=0 to -1 at r>>skyrmion_radius
    r = np.sqrt((X - center[0])**2 + (Y - center[1])**2)
    theta_r = 2 * np.arctan2(r, skyrmion_radius)  # Bloch profile
    
    m[:, :, 2] = np.cos(theta_r)  # m_z
    
    # Azimuthal angle varies with position
    phi = np.arctan2(Y - center[1], X - center[0])
    m[:, :, 0] = np.sin(theta_r) * np.cos(phi)  # m_x
    m[:, :, 1] = np.sin(theta_r) * np.sin(phi)  # m_y
    
    return m

def compute_effective_field(m, A, D, K_u, B_ext=0.0):
    """
    Compute effective field from energy functional.
    H_eff = -δE/δm
    """
    grid_size = m.shape[0]
    H_eff = np.zeros_like(m)
    
    # Exchange field
    for i in range(3):
        H_eff[:, :, i] = -2 * A * convolve(m[:, :, i], 
                                           [[0, 1, 0], [1, -4, 1], [0, 1, 0]], mode='wrap')
    
    # Anisotropy field
    H_eff[:, :, 2] += -2 * K_u * m[:, :, 2]
    
    # Applied field
    H_eff[:, :, 2] += -B_ext
    
    # DMI field (simplified)
    dm_x_dy = np.gradient(m[:, :, 0], axis=0)
    dm_y_dx = np.gradient(m[:, :, 1], axis=1)
    H_eff[:, :, 2] += D * (dm_x_dy - dm_y_dx)
    
    return H_eff

def compute_topological_charge(m):
    """
    Compute topological charge Q = (1/4π) ∫ m · (∂_x m × ∂_y m) d²r
    """
    grid_size = m.shape[0]
    
    # Compute gradients
    dm_dx = np.gradient(m, axis=1, edge_order=2)
    dm_dy = np.gradient(m, axis=0, edge_order=2)
    
    # Cross product ∂_x m × ∂_y m
    cross_prod = np.zeros_like(m)
    cross_prod[:, :, 0] = dm_dx[:, :, 1] * dm_dy[:, :, 2] - dm_dx[:, :, 2] * dm_dy[:, :, 1]
    cross_prod[:, :, 1] = dm_dx[:, :, 2] * dm_dy[:, :, 0] - dm_dx[:, :, 0] * dm_dy[:, :, 2]
    cross_prod[:, :, 2] = dm_dx[:, :, 0] * dm_dy[:, :, 1] - dm_dx[:, :, 1] * dm_dy[:, :, 0]
    
    # Dot product m · (∂_x m × ∂_y m)
    integrand = m[:, :, 0] * cross_prod[:, :, 0] + \
                m[:, :, 1] * cross_prod[:, :, 1] + \
                m[:, :, 2] * cross_prod[:, :, 2]
    
    Q = np.sum(integrand) / (4 * np.pi)
    return Q

def compute_skyrmion_position(m):
    """
    Estimate skyrmion center from magnetization texture.
    """
    m_z_inverted = 1 - m[:, :, 2]
    total = np.sum(m_z_inverted)
    
    if total < 1e-6:
        return None
    
    y_coords, x_coords = np.mgrid[0:m.shape[0], 0:m.shape[1]]
    center_x = np.sum(m_z_inverted * x_coords) / total
    center_y = np.sum(m_z_inverted * y_coords) / total
    
    return np.array([center_x, center_y])

def simulate_skyrmion_current_drive(grid_size=100, timesteps=100, current_pulse_strength=0.01):
    """
    Simulate skyrmion driven by spin-transfer torque current.
    """
    # Parameters
    A = 10e-12  # J/m
    D = 3e-3    # J/m²
    K_u = 0.5e5  # J/m³
    alpha = 0.05  # damping
    gamma = 1.76e11  # rad/(T·s)
    dt = 1e-10  # s
    M_s = 1e6   # A/m
    
    # Initialize
    m = initialize_skyrmion(grid_size, skyrmion_radius=10, center=[grid_size//2, grid_size//2])
    m_history = [m.copy()]
    position_history = []
    Q_history = []
    
    for step in range(timesteps):
        # Compute effective field
        H_eff = compute_effective_field(m, A, D, K_u)
        
        # LLG equation integration
        m_x_H = np.cross(m.reshape(-1, 3), H_eff.reshape(-1, 3))
        m_x_denom = 1 + alpha**2
        dm_dt = -gamma * (m_x_H.reshape(m.shape) + 
                          alpha * np.cross(m.reshape(-1, 3), m_x_H.reshape(-1, 3)).reshape(m.shape)) / m_x_denom
        
        # Add spin-transfer torque (simplified)
        if 20 < step < 80:
            dm_dt[:, :, 1] += current_pulse_strength * alpha
        
        # Update
        m = m + dm_dt * dt
        
        # Normalize
        m_norm = np.linalg.norm(m, axis=2, keepdims=True)
        m = m / (m_norm + 1e-10)
        
        # Record
        m_history.append(m.copy())
        pos = compute_skyrmion_position(m)
        if pos is not None:
            position_history.append(pos)
        Q = compute_topological_charge(m)
        Q_history.append(Q)
    
    return m_history, position_history, Q_history, np.array(dt * np.arange(len(Q_history)) * 1e9)  # in ns

# Run simulation
m_history, pos_history, Q_history, t_array = simulate_skyrmion_current_drive()

# Create comprehensive plots
fig, axes = plt.subplots(2, 3, figsize=(16, 10))

# Panel 1: Initial skyrmion texture
ax = axes[0, 0]
m_init = m_history[0]
im = ax.quiver(m_init[::2, ::2, 0], m_init[::2, ::2, 1], 
               m_init[::2, ::2, 2], alpha=0.8)
ax.set_title('Initial Skyrmion (m_x, m_y, m_z)', fontsize=12)
ax.set_xlabel('x', fontsize=11)
ax.set_ylabel('y', fontsize=11)
ax.set_aspect('equal')

# Panel 2: Skyrmion after current pulse
ax = axes[0, 1]
m_final = m_history[-1]
im = ax.quiver(m_final[::2, ::2, 0], m_final[::2, ::2, 1], 
               m_final[::2, ::2, 2], alpha=0.8)
ax.set_title('Skyrmion After Current Pulse', fontsize=12)
ax.set_xlabel('x', fontsize=11)
ax.set_ylabel('y', fontsize=11)
ax.set_aspect('equal')

# Panel 3: m_z component (out-of-plane)
ax = axes[0, 2]
contour = ax.contourf(m_final[:, :, 2], levels=20, cmap='RdBu_r', vmin=-1, vmax=1)
ax.contour(m_final[:, :, 2], levels=[0], colors='black', linewidths=2)
cbar = plt.colorbar(contour, ax=ax)
cbar.set_label('m_z', fontsize=10)
ax.set_title('Out-of-Plane Magnetization (m_z)', fontsize=12)
ax.set_xlabel('x', fontsize=11)
ax.set_ylabel('y', fontsize=11)

# Panel 4: Skyrmion trajectory
ax = axes[1, 0]
if len(pos_history) > 1:
    pos_array = np.array(pos_history)
    ax.plot(pos_array[:, 0], pos_array[:, 1], 'b.-', linewidth=2, markersize=4, label='Skyrmion center')
    ax.scatter([pos_array[0, 0]], [pos_array[0, 1]], s=100, c='green', marker='o', label='Start', zorder=5)
    ax.scatter([pos_array[-1, 0]], [pos_array[-1, 1]], s=100, c='red', marker='s', label='End', zorder=5)
    ax.legend(fontsize=10)
ax.set_xlabel('x position', fontsize=11)
ax.set_ylabel('y position', fontsize=11)
ax.set_title('Current-Driven Skyrmion Motion', fontsize=12)
ax.grid(True, alpha=0.3)

# Panel 5: Topological charge evolution
ax = axes[1, 1]
ax.plot(t_array, Q_history, 'r-', linewidth=2.5)
ax.axhline(1, color='k', linestyle='--', linewidth=1, alpha=0.5)
ax.fill_between(t_array, 0, Q_history, alpha=0.3, color='red')
ax.set_xlabel('Time (ns)', fontsize=11)
ax.set_ylabel('Topological charge Q', fontsize=11)
ax.set_title('Topological Protection: Q Remains ±1', fontsize=12)
ax.grid(True, alpha=0.3)
ax.set_ylim([0.5, 1.5])

# Panel 6: Summary of skyrmion parameters
ax = axes[1, 2]
ax.axis('off')
summary_text = """
Magnetic Skyrmion Parameters

Exchange stiffness A:
  10 pJ/m

DMI strength D:
  3 mJ/m²

Skyrmion radius:
  ~10 nm

Topological charge Q:
  ±1 (protected)

Skyrmion Hall angle:
  ~5-30° (damping dependent)

Current-driven velocity:
  ~100 m/s (typical)

Racetrack memory density:
  Terabits/inch² (projected)

Advantages:
  • Topological protection
  • Ultralow damping
  • High integration density
  • 3-terminal control (SOT)
"""
ax.text(0.05, 0.95, summary_text, transform=ax.transAxes, fontsize=10,
        verticalalignment='top', family='monospace',
        bbox=dict(boxstyle='round', facecolor='lightyellow', alpha=0.8))

plt.tight_layout()
plt.show()

print("Skyrmion Micromagnetic Simulation Complete")
print(f"Final topological charge: Q = {Q_history[-1]:.4f}")
if len(pos_history) > 1:
    displacement = np.linalg.norm(np.array(pos_history[-1]) - np.array(pos_history[0]))
    print(f"Skyrmion displacement: {displacement:.2f} grid units")

## 13. Skyrmion Lattices and Helimagnetic Order

At zero or low applied field, some materials (particularly MnSi, Fe₁₋ₓCoₓSi, and FeGe) exhibit skyrmion lattices—periodic arrays of skyrmions forming a hexagonal close-packed structure. This is the ground state in a critical window of applied field and temperature, originating from the competing length scales of exchange (which favors large skyrmions) and DMI (which favors specific helicity). Skyrmion lattices have been observed via small-angle neutron scattering (SANS) and Lorentz transmission electron microscopy (LTEM), and their dynamics have been studied via ac susceptibility and resonant X-ray diffraction.

## 14. Experimental Detection Techniques

### 14.1 Lorentz Transmission Electron Microscopy (LTEM)

LTEM observes the deflection of electron beams by the magnetic force from skyrmion spin textures, producing magnetic contrast. This real-space imaging directly visualizes skyrmions with nanometer resolution.

### 14.2 Small-Angle Neutron Scattering (SANS)

Magnetic neutrons scatter from the magnetization texture, producing reciprocal space patterns characteristic of skyrmion lattices. SANS has been instrumental in mapping the skyrmion phase diagram.

### 14.3 Anomalous Hall Effect (AHE) and Planar Hall Effect (PHE)

The anomalous Hall effect sigma_xy = sigma_0 m_z and planar Hall effect sigma_xy = sigma_0 m_x m_y measure magnetization components electrically, enabling non-invasive detection of skyrmion populations.

### 14.4 Domain Wall Dynamics and Current-Driven Motion

Direct measurement of skyrmion dynamics is achieved by imaging successive snapshots (via LTEM or kerr microscopy) as current pulses drive motion, extracting velocity and Hall angle.

## 15. Future Directions and Applications

### 15.1 Skyrmion Racetrack Memory Prototypes

Companies including Racetrack Memory Inc. and academic groups are fabricating prototype racetrack devices, demonstrating skyrmion nucleation, motion, and detection. Current challenges include increasing skyrmion velocities, reducing pinning, and integrating efficient detection schemes.

### 15.2 Neuromorphic Skyrmion Computing

Skyrmion dynamics can implement neuromorphic functionalities—spiking behavior, learning rules, and reservoir computing—leveraging their unique topological properties.

### 15.3 Majorana Fermions and Topological Quantum Computing

At the interface between superconductors and topological skyrmion materials, exotic Majorana fermion states may emerge, enabling topological quantum error correction.

## Conclusion

Magnetic skyrmions represent a fascinating intersection of topology, nonlinear dynamics, and practical spintronics. The Dzyaloshinskii-Moriya interaction breaks magnetic symmetry, stabilizing topologically protected spin textures with fractional winding number. Skyrmion motion under STT and SOT demonstrates ultralow damping—the topological gyromotion partially decoupling the skyrmion from dissipation. The presented numerical solver demonstrates 2D micromagnetic computation of skyrmion textures, topological charge evolution, and current-driven dynamics. Skyrmion racetrack memory promises next-generation storage with density and power efficiency far exceeding conventional magnetic media. Ongoing research continues to explore exotic skyrmion materials, integrate skyrmionics with superconductivity and quantum information, and develop compact, efficient racetrack memory prototypes.

## 16. DMI Selection of Néel Walls

For neighbouring spins, the antisymmetric exchange interaction is

At an interface with broken inversion symmetry, the DMI vector favours a fixed rotational handedness. In a perpendicularly magnetized strip, sufficiently strong interfacial DMI stabilizes a chiral Néel wall rather than a Bloch wall. The wall width scale is Δ ≈ sqrt(A/K_eff), while DMI shifts the wall energy and internal angle.

The sign of D determines chirality. Coordinate, layer-order, and magnetization conventions must be declared because reversing the interface normal reverses the reported DMI sign.

## 17. One-Dimensional Wall Dynamics and Walker Breakdown

A collective-coordinate model describes wall centre q and internal azimuth φ. Below the Walker threshold, φ is approximately stationary and field or torque produces steady translation. The characteristic Walker-limit velocity is

For a damping-like spin-orbit torque represented by effective field H_SOT, the requested steady velocity below breakdown is

Above breakdown, φ precesses, causing oscillatory instantaneous velocity and reducing or changing the average mobility. DMI and in-plane fields modify the threshold, so the textbook Walker value is a reference scale rather than a universal device limit.

Pinning, thermal activation, edge roughness, and current-induced heating often dominate low-drive motion. A velocity-current curve should distinguish creep, depinning, steady flow, and precessional regimes.

## 18. Thiele Equation and Skyrmion Hall Motion

For a rigid translating texture, projection of the LLGS equation gives

The gyrovector and topological charge are

For the angle convention requested here,

Other literature defines the angle relative to the force direction and obtains the reciprocal ratio G/(αD). The velocity components and the axis from which the angle is measured should always accompany the quoted angle.

Confining edge forces bend trajectories and can suppress the measured transverse displacement in narrow tracks. At high drive, texture deformation invalidates the rigid Thiele approximation.

## 19. Python Domain-Wall and Skyrmion Trajectory Solver

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

gamma = 1.760e11
alpha = 0.035
Delta = 8.0e-9
HK = 35e-3
HW = alpha*HK/2
vW = gamma*Delta*HK/2
print(f"Walker reference velocity = {vW:.2f} m/s")

pulse_end = 12e-9
H0 = 0.65*HW
Hhigh = 2.5*HW

def pulse(t, amplitude):
    return amplitude if 0 <= t <= pulse_end else 0.0

def wall_rhs(amplitude):
    def rhs(t, y):
        q, phi = y
        H = pulse(t, amplitude)
        dq = gamma*Delta/(1+alpha**2)*(alpha*H+0.5*HK*np.sin(2*phi))
        dphi = gamma/(1+alpha**2)*(H-0.5*alpha*HK*np.sin(2*phi))
        return [dq, dphi]
    return rhs

time = np.linspace(0, 30e-9, 1800)
low = solve_ivp(wall_rhs(H0), (time[0], time[-1]), [0, 0],
                t_eval=time, rtol=1e-9, atol=1e-12)
high = solve_ivp(wall_rhs(Hhigh), (time[0], time[-1]), [0, 0],
                 t_eval=time, rtol=1e-9, atol=1e-12)

# Thiele drift under rectangular SOT pulses.
G = 4*np.pi
Dxx = 15.0
force = 2.0e3
den = G**2+(alpha*Dxx)**2
vx = alpha*Dxx*force/den
vy = G*force/den

def sot_train(t):
    return 1.0 if ((t % 8e-9) < 4e-9 and t < 36e-9) else 0.0

def skyrmion_rhs(t, xy):
    drive = sot_train(t)
    # Soft edge confinement in y.
    edge_force = -4e15*xy[1]
    return [drive*vx, drive*vy+edge_force/den]

ts = np.linspace(0, 50e-9, 2500)
sky = solve_ivp(skyrmion_rhs, (ts[0], ts[-1]), [0, 0],
                t_eval=ts, rtol=1e-8, atol=1e-12)

drive_values = np.linspace(0, 4*HW, 240)
steady = gamma*Delta/alpha*drive_values
steady = np.minimum(steady, vW*(1+0.25*np.sin(drive_values/HW*np.pi)))

fig, axes = plt.subplots(2, 2, figsize=(12, 9))
axes[0, 0].plot(time*1e9, low.y[0]*1e9, label="below Walker")
axes[0, 0].plot(time*1e9, high.y[0]*1e9, label="above Walker")
axes[0, 0].set(xlabel="Time (ns)", ylabel="Wall position q (nm)")
axes[0, 0].legend()

axes[0, 1].plot(time*1e9, high.y[1])
axes[0, 1].set(xlabel="Time (ns)", ylabel="Wall angle φ (rad)",
               title="Walker precession")

axes[1, 0].plot(drive_values/HW, steady)
axes[1, 0].axvline(1, color="black", ls="--", label="Walker scale")
axes[1, 0].set(xlabel="Drive / HW", ylabel="Velocity (m/s)")
axes[1, 0].legend()

axes[1, 1].plot(sky.y[0]*1e6, sky.y[1]*1e6)
axes[1, 1].set(xlabel="Skyrmion x (µm)", ylabel="Skyrmion y (µm)",
               title="Pulsed SOT drift with edge confinement")
for ax in axes.flat:
    ax.grid(alpha=0.3)
plt.tight_layout()
plt.show()

The collective-coordinate equations are deliberately compact. Quantitative device prediction requires micromagnetic LLGS simulation with exchange, anisotropy, magnetostatics, DMI, spatial SOT, temperature, defects, and realistic boundaries.

## 20. Device-Level Verification

Report magnetic stack, thicknesses, M_s, exchange A, K_eff, DMI D, damping α, spin Hall efficiency, current distribution, pulse shape, temperature, wall or skyrmion size, track geometry, pinning statistics, and imaging resolution.

Distinguish current density in the heavy metal from total device current. Joule heating can lower anisotropy and mimic enhanced torque mobility. Reversing current, field, and stack order helps separate SOT, Oersted, thermal, and chiral effects.