Chapter 3: Spin-Orbit Torque (SOT)

Spin-Orbit Torque (SOT) is the torque exerted on magnetization by in-plane current through spin-orbit interaction in heavy metal/ferromagnet bilayer structures. It is attracting attention as a next-generation technology that overcomes the challenges of STT-MRAM. In this chapter, we learn the physical origins of SOT, the two torque components, switching mechanisms, and SOT-MRAM design principles.

3.1 Physical Origins of SOT

SOT arises mainly from two mechanisms:

Spin Hall Effect (SHE) Origin

When in-plane current flows through a heavy metal (Pt, W, Ta, etc.), the Spin Hall Effect generates spin current perpendicular to the charge current:

where $\theta_{SH}$ is the spin Hall angle, $\hat{\sigma}$ is the spin polarization direction, and $\mathbf{J}_c$ is the charge current density.

Rashba-Edelstein Effect Origin

Interfacial Rashba SOI directly induces spin accumulation from current:

Code Example 3.1: Spin Current Generation via Spin Hall Effect

"""
Visualization of spin current generation via Spin Hall Effect
"""
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

def spin_hall_current(J_c, theta_SH, direction='x'):
    """
    Spin current vector via SHE

    Parameters:
    J_c: Charge current density (A/m²)
    theta_SH: Spin Hall angle
    direction: Charge current direction
    """
    hbar = 1.055e-34
    e = 1.6e-19

    if direction == 'x':
        # J_c || x → J_s || z, σ || y
        J_s_magnitude = theta_SH * (hbar / (2 * e)) * J_c
        return np.array([0, 0, J_s_magnitude]), np.array([0, 1, 0])
    elif direction == 'y':
        # J_c || y → J_s || z, σ || -x
        J_s_magnitude = theta_SH * (hbar / (2 * e)) * J_c
        return np.array([0, 0, J_s_magnitude]), np.array([-1, 0, 0])

# Parameters
J_c = 1e11  # A/m²
theta_SH_values = {'Pt': 0.08, 'W': 0.30, 'Ta': 0.15}

fig = plt.figure(figsize=(14, 5))

for i, (material, theta_SH) in enumerate(theta_SH_values.items()):
    ax = fig.add_subplot(1, 3, i+1, projection='3d')

    # Charge current (red arrow)
    ax.quiver(0, 0, 0, 1, 0, 0, color='red', arrow_length_ratio=0.2, linewidth=3, label='$J_c$')

    # Spin current (blue arrow)
    J_s, sigma = spin_hall_current(J_c, theta_SH)
    scale = theta_SH / 0.30  # Scale relative to W
    ax.quiver(0, 0, 0, 0, 0, scale, color='blue', arrow_length_ratio=0.2, linewidth=3, label='$J_s$')

    # Spin polarization direction (green arrow)
    ax.quiver(0.5, 0, scale/2, 0, 0.5, 0, color='green', arrow_length_ratio=0.3, linewidth=2, label='σ')

    ax.set_xlim(-0.5, 1.5)
    ax.set_ylim(-0.5, 1.0)
    ax.set_zlim(-0.5, 1.5)
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    ax.set_zlabel('z')
    ax.set_title(f'{material} (θ_SH = {theta_SH})', fontsize=12)

plt.suptitle('Spin Current Generation via Spin Hall Effect', fontsize=14)
plt.tight_layout()
plt.show()

3.2 Two Components of SOT: Field-like and Damping-like

SOT can be decomposed into two orthogonal components:

Field-like Torque (FL)

• Acts as effective field $\mathbf{H}_{FL} \propto \boldsymbol{\sigma}$
• Mainly originates from Rashba effect
• Induces magnetization precession

Damping-like Torque (DL)

• Rotates magnetization toward $\boldsymbol{\sigma}$
• Mainly originates from SHE
• Directly contributes to magnetization switching

Code Example 3.2: Visualization of SOT Components

"""
Visualization of field-like and damping-like SOT components
"""
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

def sot_components(m, sigma, tau_FL, tau_DL):
    """
    Calculate two SOT components

    Parameters:
    m: Magnetization direction (unit vector)
    sigma: Spin polarization direction
    tau_FL: Field-like torque strength
    tau_DL: Damping-like torque strength
    """
    FL = tau_FL * np.cross(m, sigma)
    DL = tau_DL * np.cross(m, np.cross(m, sigma))
    return FL, DL

# Magnetization and spin polarization setup
sigma = np.array([0, 1, 0])  # y-direction polarization

# Torque for various magnetization directions
theta = np.linspace(0, 2*np.pi, 36)
phi = np.pi / 4  # 45 degrees from xz plane

fig, axes = plt.subplots(1, 2, figsize=(14, 6))

# Left: Angular dependence of torque magnitude
tau_FL_mag = []
tau_DL_mag = []

for th in theta:
    m = np.array([np.sin(phi)*np.cos(th), np.sin(phi)*np.sin(th), np.cos(phi)])
    m = m / np.linalg.norm(m)
    FL, DL = sot_components(m, sigma, 1.0, 1.0)
    tau_FL_mag.append(np.linalg.norm(FL))
    tau_DL_mag.append(np.linalg.norm(DL))

axes[0].plot(np.degrees(theta), tau_FL_mag, 'b-', linewidth=2, label='Field-like')
axes[0].plot(np.degrees(theta), tau_DL_mag, 'r-', linewidth=2, label='Damping-like')
axes[0].set_xlabel('Magnetization Azimuth (degrees)', fontsize=12)
axes[0].set_ylabel('Torque Magnitude (a.u.)', fontsize=12)
axes[0].set_title('Angular Dependence of SOT', fontsize=14)
axes[0].legend()
axes[0].grid(True, alpha=0.3)

# Right: 3D vector plot
ax2 = fig.add_subplot(122, projection='3d')

m = np.array([1, 0, 1])
m = m / np.linalg.norm(m)
FL, DL = sot_components(m, sigma, 0.5, 0.5)

# Draw vectors
ax2.quiver(0, 0, 0, m[0], m[1], m[2], color='black', arrow_length_ratio=0.1,
           linewidth=3, label='m (magnetization)')
ax2.quiver(0, 0, 0, sigma[0], sigma[1], sigma[2], color='green', arrow_length_ratio=0.1,
           linewidth=3, label='σ (spin)')
ax2.quiver(m[0], m[1], m[2], FL[0], FL[1], FL[2], color='blue', arrow_length_ratio=0.15,
           linewidth=2, label='τ_FL')
ax2.quiver(m[0], m[1], m[2], DL[0], DL[1], DL[2], color='red', arrow_length_ratio=0.15,
           linewidth=2, label='τ_DL')

ax2.set_xlim(-1, 1.5)
ax2.set_ylim(-1, 1.5)
ax2.set_zlim(-0.5, 1.5)
ax2.set_xlabel('x')
ax2.set_ylabel('y')
ax2.set_zlabel('z')
ax2.set_title('SOT Torque Vectors', fontsize=14)
ax2.legend(loc='upper left')

plt.tight_layout()
plt.show()

3.3 Magnetization Switching by SOT

Perpendicular magnetization switching by SOT requires symmetry breaking.

Methods of Symmetry Breaking

Code Example 3.3: SOT Magnetization Switching Simulation

"""
LLG simulation of perpendicular magnetization switching by SOT
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

def llg_sot(m, t, gamma, alpha, H_k, H_x, tau_DL, sigma):
    """
    LLG equation with SOT

    Parameters:
    m: Magnetization direction
    gamma: Gyromagnetic ratio
    alpha: Damping
    H_k: Anisotropy field (z-direction)
    H_x: In-plane symmetry-breaking field
    tau_DL: Damping-like torque strength
    sigma: Spin polarization direction
    """
    m = m / np.linalg.norm(m)

    # Effective field
    H_eff = np.array([H_x, 0, H_k * m[2]])

    # LLG precession and damping terms
    precession = -gamma * np.cross(m, H_eff)
    damping = alpha * np.cross(m, precession)

    # SOT damping-like torque
    sot_DL = tau_DL * np.cross(m, np.cross(m, sigma))

    return precession + damping + sot_DL

# Parameters
gamma = 1.76e11
alpha = 0.05
H_k = 0.8  # PMA (T equivalent)
sigma = np.array([0, 1, 0])  # y-direction spin polarization

# Time settings
t_max = 5e-9
t = np.linspace(0, t_max, 5000)

# Initial state
m0 = np.array([0.01, 0, 0.99995])
m0 = m0 / np.linalg.norm(m0)

# Switching at different in-plane fields
H_x_values = [0, 0.05, 0.1, 0.2]
tau_DL = 5e11  # Fixed

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
axes = axes.flatten()

for ax, H_x in zip(axes, H_x_values):
    sol = odeint(llg_sot, m0, t, args=(gamma, alpha, H_k, H_x, tau_DL, sigma))

    ax.plot(t*1e9, sol[:, 0], 'r-', label='$m_x$', linewidth=1.5)
    ax.plot(t*1e9, sol[:, 1], 'g-', label='$m_y$', linewidth=1.5)
    ax.plot(t*1e9, sol[:, 2], 'b-', label='$m_z$', linewidth=1.5)

    ax.set_xlabel('Time (ns)', fontsize=11)
    ax.set_ylabel('Magnetization Component', fontsize=11)
    ax.set_title(f'$H_x$ = {H_x} T', fontsize=12)
    ax.legend()
    ax.grid(True, alpha=0.3)
    ax.set_ylim(-1.1, 1.1)

plt.suptitle('SOT Switching and Symmetry-Breaking Field', fontsize=14)
plt.tight_layout()
plt.show()

print("H_x=0: No switching due to symmetry")
print("H_x>0: Symmetry broken, deterministic switching possible")

3.4 Evaluation of SOT Efficiency

SOT efficiency is evaluated by effective field per unit current:

Code Example 3.4: SOT Efficiency Measurement Simulation

"""
Harmonic measurement simulation for SOT efficiency evaluation
"""
import numpy as np
import matplotlib.pyplot as plt

def sot_harmonic_response(H_ext, H_DL, H_FL, H_k, phi=0):
    """
    Magnetization tilt and second harmonic response from SOT

    Parameters:
    H_ext: External field
    H_DL: Damping-like effective field
    H_FL: Field-like effective field
    H_k: Anisotropy field
    phi: External field azimuthal angle
    """
    # Equilibrium magnetization tilt (small angle approx)
    theta_0 = H_ext * np.cos(phi) / H_k

    # First harmonic voltage (AMR + PHE)
    V_1omega = np.cos(2*phi) + np.sin(2*phi)

    # Second harmonic voltage (SOT origin)
    V_2omega_DL = H_DL * np.cos(phi) / (H_k - H_ext * np.cos(phi))
    V_2omega_FL = H_FL * np.sin(phi) / (H_k - H_ext * np.cos(phi))

    return V_1omega, V_2omega_DL + V_2omega_FL

# Parameters
H_k = 1.0  # T
H_DL = 0.1  # T (with current)
H_FL = 0.02  # T

# External field scan
H_ext = np.linspace(-0.8, 0.8, 200)

# Response at different azimuthal angles
phi_values = [0, np.pi/4, np.pi/2]
phi_names = ['φ=0° (x-axis)', 'φ=45°', 'φ=90° (y-axis)']

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

for phi, name in zip(phi_values, phi_names):
    V_1, V_2 = sot_harmonic_response(H_ext, H_DL, H_FL, H_k, phi)
    axes[0].plot(H_ext, V_1 * np.ones_like(H_ext), linewidth=2, label=name)
    axes[1].plot(H_ext, V_2, linewidth=2, label=name)

axes[0].set_xlabel('External Field (T)', fontsize=12)
axes[0].set_ylabel('First Harmonic $V_{1ω}$ (a.u.)', fontsize=12)
axes[0].set_title('First Harmonic Response', fontsize=14)
axes[0].legend()
axes[0].grid(True, alpha=0.3)

axes[1].set_xlabel('External Field (T)', fontsize=12)
axes[1].set_ylabel('Second Harmonic $V_{2ω}$ (a.u.)', fontsize=12)
axes[1].set_title('Second Harmonic Response (SOT Origin)', fontsize=14)
axes[1].legend()
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

print("φ=0°: Extract damping-like torque component")
print("φ=90°: Extract field-like torque component")

3.5 SOT-MRAM

SOT-MRAM is next-generation memory that overcomes challenges of STT-MRAM.

Comparison with STT-MRAM

Code Example 3.5: SOT-MRAM Switching Analysis

"""
SOT-MRAM switching characteristics analysis
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

class SOTMRAM:
    def __init__(self, diameter_nm, t_FL_nm, t_HM_nm, M_s, H_k, alpha, theta_SH):
        self.diameter = diameter_nm * 1e-9
        self.t_FL = t_FL_nm * 1e-9
        self.t_HM = t_HM_nm * 1e-9
        self.M_s = M_s
        self.H_k = H_k
        self.alpha = alpha
        self.theta_SH = theta_SH

        self.area = np.pi * (self.diameter/2)**2
        self.volume = self.area * self.t_FL

    def critical_current_density(self):
        """SOT switching critical current density"""
        hbar = 1.055e-34
        e = 1.6e-19
        mu_0 = 4 * np.pi * 1e-7

        J_c = (2 * e * self.M_s * self.t_FL * self.H_k * mu_0) / (hbar * self.theta_SH)
        return J_c

    def switching_time(self, J_ratio=1.5):
        """Approximate switching time"""
        gamma = 1.76e11
        mu_0 = 4 * np.pi * 1e-7
        return 1 / (self.alpha * gamma * mu_0 * self.M_s) * 1 / (J_ratio - 1)

    def power_consumption(self, J_ratio=1.5, rho_HM=1e-6):
        """Write power consumption"""
        J_c = self.critical_current_density()
        J = J_c * J_ratio
        tau_sw = self.switching_time(J_ratio)

        # Heavy metal layer resistance
        L = self.diameter  # Write line length ≈ diameter
        W = self.diameter
        R_HM = rho_HM * L / (W * self.t_HM)

        I = J * W * self.t_HM
        E = I**2 * R_HM * tau_sw
        return E

# Design parameter comparison
diameters = np.linspace(20, 80, 30)

# Pt vs W comparison
materials = {
    'Pt': {'theta_SH': 0.08, 'color': 'blue'},
    'W(β)': {'theta_SH': 0.30, 'color': 'red'},
}

M_s = 1.2e6
H_k = 0.4e6
alpha = 0.02

fig, axes = plt.subplots(1, 3, figsize=(15, 4))

for name, params in materials.items():
    J_cs = []
    tau_sws = []
    E_sws = []

    for d in diameters:
        cell = SOTMRAM(d, 1.5, 5, M_s, H_k, alpha, params['theta_SH'])
        J_cs.append(cell.critical_current_density() / 1e11)
        tau_sws.append(cell.switching_time() * 1e9)
        E_sws.append(cell.power_consumption() * 1e15)

    axes[0].plot(diameters, J_cs, color=params['color'], linewidth=2, label=name)
    axes[1].plot(diameters, tau_sws, color=params['color'], linewidth=2, label=name)
    axes[2].plot(diameters, E_sws, color=params['color'], linewidth=2, label=name)

axes[0].set_xlabel('MTJ Diameter (nm)', fontsize=11)
axes[0].set_ylabel('$J_c$ (×$10^{11}$ A/m²)', fontsize=11)
axes[0].set_title('Critical Current Density', fontsize=12)
axes[0].legend()
axes[0].grid(True, alpha=0.3)

axes[1].set_xlabel('MTJ Diameter (nm)', fontsize=11)
axes[1].set_ylabel('Switching Time (ns)', fontsize=11)
axes[1].set_title('Switching Time', fontsize=12)
axes[1].legend()
axes[1].grid(True, alpha=0.3)

axes[2].set_xlabel('MTJ Diameter (nm)', fontsize=11)
axes[2].set_ylabel('Energy (fJ)', fontsize=11)
axes[2].set_title('Write Energy', fontsize=12)
axes[2].legend()
axes[2].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

print("W enables lower current density switching due to high spin Hall angle")

3.6 Field-Free Switching

Practical SOT-MRAM requires deterministic switching without external magnetic field.

Main Approaches

1. Exchange Bias Method

"""
Field-free SOT switching with exchange bias
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

def llg_sot_exchange_bias(m, t, gamma, alpha, H_k, H_eb, tau_DL, sigma):
    """SOT-LLG with exchange bias"""
    m = m / np.linalg.norm(m)

    # Effective field (PMA + exchange bias)
    H_eff = np.array([H_eb, 0, H_k * m[2]])

    precession = -gamma * np.cross(m, H_eff)
    damping = alpha * np.cross(m, precession)
    sot_DL = tau_DL * np.cross(m, np.cross(m, sigma))

    return precession + damping + sot_DL

# Parameters
gamma = 1.76e11
alpha = 0.05
H_k = 0.8
H_eb = 0.1  # Exchange bias field
sigma = np.array([0, 1, 0])

t = np.linspace(0, 3e-9, 3000)

# Switching with positive and negative current
fig, axes = plt.subplots(1, 2, figsize=(12, 4))

for ax, tau_sign, title in zip(axes, [1, -1], ['Positive Current (+J)', 'Negative Current (-J)']):
    m0 = np.array([0.01, 0, 0.99995 * (-tau_sign)])
    m0 = m0 / np.linalg.norm(m0)
    tau_DL = tau_sign * 8e11

    sol = odeint(llg_sot_exchange_bias, m0, t,
                 args=(gamma, alpha, H_k, H_eb, tau_DL, sigma))

    ax.plot(t*1e9, sol[:, 2], 'b-', linewidth=2)
    ax.axhline(y=0, color='k', linestyle='--', alpha=0.3)
    ax.set_xlabel('Time (ns)', fontsize=11)
    ax.set_ylabel('$m_z$', fontsize=11)
    ax.set_title(f'{title}: Exchange Bias H_eb = {H_eb} T', fontsize=12)
    ax.grid(True, alpha=0.3)
    ax.set_ylim(-1.2, 1.2)

plt.tight_layout()
plt.show()

2. z-Component Spin Polarization Method

Code Example 3.6: Symmetry Breaking via Tilted Anisotropy

"""
Field-free switching with tilted magnetic anisotropy
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

def llg_sot_tilted_anisotropy(m, t, gamma, alpha, H_k, theta_tilt, tau_DL, sigma):
    """SOT-LLG with tilted anisotropy"""
    m = m / np.linalg.norm(m)

    # Tilted anisotropy axis
    n_easy = np.array([np.sin(theta_tilt), 0, np.cos(theta_tilt)])
    H_anis = H_k * np.dot(m, n_easy) * n_easy

    H_eff = H_anis

    precession = -gamma * np.cross(m, H_eff)
    damping = alpha * np.cross(m, precession)
    sot_DL = tau_DL * np.cross(m, np.cross(m, sigma))

    return precession + damping + sot_DL

# Parameters
gamma = 1.76e11
alpha = 0.05
H_k = 0.8
sigma = np.array([0, 1, 0])

t = np.linspace(0, 5e-9, 5000)

# Comparison at different tilt angles
theta_tilts = [0, 5, 10, 15]  # degrees

fig, axes = plt.subplots(2, 2, figsize=(12, 8))
axes = axes.flatten()

for ax, theta_deg in zip(axes, theta_tilts):
    theta_tilt = np.radians(theta_deg)
    m0 = np.array([0.01, 0, 0.99995])
    m0 = m0 / np.linalg.norm(m0)
    tau_DL = 6e11

    sol = odeint(llg_sot_tilted_anisotropy, m0, t,
                 args=(gamma, alpha, H_k, theta_tilt, tau_DL, sigma))

    ax.plot(t*1e9, sol[:, 0], 'r-', label='$m_x$', linewidth=1.5)
    ax.plot(t*1e9, sol[:, 2], 'b-', label='$m_z$', linewidth=1.5)
    ax.axhline(y=0, color='k', linestyle='--', alpha=0.3)
    ax.set_xlabel('Time (ns)', fontsize=10)
    ax.set_ylabel('Magnetization Component', fontsize=10)
    ax.set_title(f'Tilt Angle = {theta_deg}°', fontsize=11)
    ax.legend()
    ax.grid(True, alpha=0.3)
    ax.set_ylim(-1.2, 1.2)

plt.suptitle('Field-Free SOT Switching with Tilted Anisotropy', fontsize=13)
plt.tight_layout()
plt.show()

print("θ_tilt > 0: Deterministic switching possible without external field")

3.7 Latest Research Trends

Code Example 3.7: SOT Efficiency Comparison of New Materials

"""
Performance metrics comparison of SOT materials
"""
import numpy as np
import matplotlib.pyplot as plt

# Latest experimental data (approximate values)
materials = {
    'Pt': {'xi_DL': 0.08, 'xi_FL': 0.01, 'rho': 20},
    'W(β)': {'xi_DL': 0.30, 'xi_FL': 0.02, 'rho': 150},
    'Ta(β)': {'xi_DL': 0.15, 'xi_FL': 0.03, 'rho': 180},
    'Pt/Co bilayer': {'xi_DL': 0.12, 'xi_FL': 0.08, 'rho': 25},
    'WTe₂': {'xi_DL': 0.40, 'xi_FL': 0.05, 'rho': 500},
    'Bi₂Se₃': {'xi_DL': 1.0, 'xi_FL': 0.2, 'rho': 1000},
}

names = list(materials.keys())
xi_DL = [m['xi_DL'] for m in materials.values()]
xi_FL = [m['xi_FL'] for m in materials.values()]
rho = [m['rho'] for m in materials.values()]

# Efficiency / resistivity figure of merit
figure_of_merit = [x / r * 1e5 for x, r in zip(xi_DL, rho)]

fig, axes = plt.subplots(1, 3, figsize=(15, 5))

# SOT efficiency
x = np.arange(len(names))
width = 0.35

axes[0].bar(x - width/2, xi_DL, width, label='ξ_DL (Damping-like)', color='blue', alpha=0.7)
axes[0].bar(x + width/2, xi_FL, width, label='ξ_FL (Field-like)', color='red', alpha=0.7)
axes[0].set_xticks(x)
axes[0].set_xticklabels(names, rotation=45, ha='right')
axes[0].set_ylabel('SOT Efficiency ξ', fontsize=11)
axes[0].set_title('SOT Efficiency', fontsize=12)
axes[0].legend()
axes[0].grid(True, alpha=0.3, axis='y')

# Resistivity
axes[1].bar(names, rho, color='green', alpha=0.7)
axes[1].set_xticklabels(names, rotation=45, ha='right')
axes[1].set_ylabel('Resistivity (μΩ·cm)', fontsize=11)
axes[1].set_title('Resistivity', fontsize=12)
axes[1].set_yscale('log')
axes[1].grid(True, alpha=0.3, axis='y')

# Figure of Merit
axes[2].bar(names, figure_of_merit, color='purple', alpha=0.7)
axes[2].set_xticklabels(names, rotation=45, ha='right')
axes[2].set_ylabel('ξ_DL / ρ (a.u.)', fontsize=11)
axes[2].set_title('Figure of Merit (Efficiency/Resistivity)', fontsize=12)
axes[2].grid(True, alpha=0.3, axis='y')

plt.tight_layout()
plt.show()

print("Topological insulators (Bi₂Se₃) have giant SOT efficiency but high resistivity is a challenge")
print("W systems have good balance of efficiency and resistivity, advancing toward commercialization")

Chapter Summary

Preparation for Next Chapter

In the next chapter, we learn about advanced spintronics topics including antiferromagnetic spintronics, magnetic skyrmions, and 2D magnetic materials.

References

1. Manchon, A., et al. (2019). "Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems." Rev. Mod. Phys., 91, 035004.
2. Miron, I. M., et al. (2011). "Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection." Nature, 476, 189-193.
3. Liu, L., et al. (2012). "Spin-torque switching with the giant spin Hall effect of tantalum." Science, 336, 555-558.
4. Garello, K., et al. (2018). "SOT-MRAM 300MM integration for low power and ultrafast embedded memories." IEEE Symp. VLSI Circuits.