Chapter 2: Basic Principles of Spintronics

In this chapter, we explore the core physical phenomena of spintronics in detail. Starting from spin-polarized currents, we delve into the mechanisms of GMR and TMR effects, spin injection and accumulation, and the recently prominent Spin Hall Effect, building understanding through theory and Python code.

2.1 Spin-Polarized Currents and the Two-Current Model

In ferromagnetic metals, the density of states for spin-up (↑) and spin-down (↓) electrons differ at the Fermi level. This asymmetry causes the current to be spin-polarized.

Two-Current Model

The two-current model, proposed by Mott, treats current as a parallel combination of two independent spin channels:

The current in each spin channel is determined by its resistance:

In ferromagnets, the majority spin (parallel to magnetization) resistance $R_\uparrow$ is smaller than the minority spin (antiparallel) resistance $R_\downarrow$.

Code Example 2.1: Visualizing the Two-Current Model

"""
Visualization of spin-dependent conduction using the two-current model
"""
import numpy as np
import matplotlib.pyplot as plt

def two_current_model(V, R_up, R_down):
    """
    Calculate currents using the two-current model

    Parameters:
    V: Applied voltage
    R_up: Spin-up resistance
    R_down: Spin-down resistance

    Returns:
    I_up, I_down, I_total, polarization
    """
    I_up = V / R_up
    I_down = V / R_down
    I_total = I_up + I_down
    polarization = (I_up - I_down) / I_total
    return I_up, I_down, I_total, polarization

# Parameters
V = 1.0  # Voltage (a.u.)

# Spin asymmetry for different materials
asymmetry_ratios = np.linspace(1, 10, 50)  # R_down / R_up

# Calculate
polarizations = []
for ratio in asymmetry_ratios:
    R_up = 1.0
    R_down = ratio * R_up
    _, _, _, P = two_current_model(V, R_up, R_down)
    polarizations.append(P)

# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Left: Resistance illustration
ax1.bar(['R↑\n(majority)', 'R↓\n(minority)'], [1, 5],
        color=['red', 'blue'], alpha=0.7)
ax1.set_ylabel('Resistance (relative)', fontsize=12)
ax1.set_title('Spin-Dependent Resistance in Ferromagnets', fontsize=14)

# Right: Spin polarization vs asymmetry
ax2.plot(asymmetry_ratios, polarizations, 'b-', linewidth=2)
ax2.set_xlabel('Resistance Asymmetry (R↓/R↑)', fontsize=12)
ax2.set_ylabel('Current Spin Polarization', fontsize=12)
ax2.set_title('Resistance Asymmetry and Spin-Polarized Current', fontsize=14)
ax2.grid(True, alpha=0.3)
ax2.set_ylim(0, 1)

plt.tight_layout()
plt.show()

# Example
print("=" * 50)
print("Example: Fe case (R↓/R↑ ≈ 5)")
I_up, I_down, I_total, P = two_current_model(1.0, 1.0, 5.0)
print(f"I↑ = {I_up:.3f}, I↓ = {I_down:.3f}")
print(f"I_total = {I_total:.3f}")
print(f"Spin polarization P = {P:.1%}")
print("=" * 50)

2.2 Giant Magnetoresistance (GMR) in Detail

The GMR effect is observed in trilayer structures of ferromagnet/non-magnetic metal/ferromagnet (FM/NM/FM). The resistance changes depending on the magnetization configuration of the two ferromagnetic layers.

Physical Mechanism

Parallel configuration: Majority spin electrons experience low resistance in both FM layers; minority spin electrons experience high resistance in both. Overall low resistance.

Antiparallel configuration: Each spin experiences high resistance in one FM layer. Overall high resistance.

GMR ratio is defined as:

Using spin polarization $P$, this can be written (Jullière formula):

Code Example 2.2: GMR Field Dependence Simulation

"""
Simulating field dependence of GMR (spin valve structure)
"""
import numpy as np
import matplotlib.pyplot as plt

# Parameters
H_forward = np.linspace(200, -200, 500)
H_backward = np.linspace(-200, 200, 500)
Hc1, Hc2 = 50, 150  # Coercivities (free and pinned layers)
R_P, R_AP = 100, 180  # Resistances (Ω)

# Forward sweep (positive to negative)
R_sweep_pos = []
state1, state2 = 1, 1
for h in H_forward:
    if h < Hc2 and state2 == 1:
        state2 = -1
    if h < -Hc1 and state1 == 1:
        state1 = -1
    R_sweep_pos.append(R_P if state1 == state2 else R_AP)

# Backward sweep (negative to positive)
R_sweep_neg = []
state1, state2 = -1, -1
for h in H_backward:
    if h > -Hc2 and state2 == -1:
        state2 = 1
    if h > Hc1 and state1 == -1:
        state1 = 1
    R_sweep_neg.append(R_P if state1 == state2 else R_AP)

# Plot
plt.figure(figsize=(12, 6))
plt.plot(H_forward, R_sweep_pos, 'b-', linewidth=2, label='Field decreasing')
plt.plot(H_backward, R_sweep_neg, 'r--', linewidth=2, label='Field increasing')
plt.xlabel('External Field H (Oe)', fontsize=12)
plt.ylabel('Resistance R (Ω)', fontsize=12)
plt.title('GMR Characteristics of Spin Valve', fontsize=14)
plt.axhline(y=R_P, color='g', linestyle=':', alpha=0.5, label=f'R_P = {R_P}Ω')
plt.axhline(y=R_AP, color='orange', linestyle=':', alpha=0.5, label=f'R_AP = {R_AP}Ω')
plt.legend()
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

gmr_ratio = (R_AP - R_P) / R_P * 100
print(f"GMR ratio: {gmr_ratio:.1f}%")

2.3 Tunnel Magnetoresistance (TMR)

The TMR (Tunnel Magnetoresistance) effect is observed in Magnetic Tunnel Junctions (MTJ), where a thin insulating layer (tunnel barrier) is sandwiched between two ferromagnets.

Jullière Model

In the 1975 model by Jullière, the TMR ratio is determined by the spin polarizations of both FM layers:

With ideal half-metals ($P = 1$), TMR becomes infinite!

Code Example 2.3: TMR Ratio Calculation

"""
TMR ratio calculation based on Jullière model
"""
import numpy as np
import matplotlib.pyplot as plt

def tmr_julliere(P1, P2):
    """
    Calculate TMR ratio using Jullière model
    TMR = 2*P1*P2 / (1 - P1*P2)
    """
    return 2 * P1 * P2 / (1 - P1 * P2)

# Range of spin polarization
P_values = np.linspace(0.01, 0.99, 100)
TMR_values = [tmr_julliere(P, P) * 100 for P in P_values]

# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

ax1.plot(P_values * 100, TMR_values, 'b-', linewidth=2)
ax1.set_xlabel('Spin Polarization P (%)', fontsize=12)
ax1.set_ylabel('TMR Ratio (%)', fontsize=12)
ax1.set_title('Jullière Model: TMR vs Spin Polarization', fontsize=14)
ax1.set_yscale('log')
ax1.grid(True, alpha=0.3)

# Material comparison
materials = {
    'Fe': 0.45, 'Co': 0.42, 'Ni': 0.33,
    'CoFe': 0.50, 'CoFeB': 0.56, 'Half-metal\n(ideal)': 0.99
}

mat_names = list(materials.keys())
mat_TMR = [tmr_julliere(p, p) * 100 for p in materials.values()]

bars = ax2.bar(mat_names, mat_TMR, color='steelblue', alpha=0.7)
ax2.set_ylabel('Predicted TMR Ratio (%)', fontsize=12)
ax2.set_title('TMR Ratio by Material (Jullière)', fontsize=14)
ax2.set_yscale('log')

for bar, tmr in zip(bars, mat_TMR):
    ax2.text(bar.get_x() + bar.get_width()/2, bar.get_height(),
             f'{tmr:.0f}%', ha='center', va='bottom', fontsize=10)

plt.tight_layout()
plt.show()

2.4 Spin Injection and Accumulation

Spin injection is the phenomenon of driving spin-polarized current from a ferromagnet into a non-magnetic material. The injected spins form spin accumulation in the non-magnetic material.

Spin Diffusion Equation

The spin accumulation $\mu_s = \mu_\uparrow - \mu_\downarrow$ in a non-magnetic material follows the spin diffusion equation:

The solution at distance $x$ from the interface is:

Code Example 2.4: Spatial Distribution of Spin Accumulation

"""
Simulating spatial distribution of spin accumulation
"""
import numpy as np
import matplotlib.pyplot as plt

def spin_accumulation(x, mu_s0, lambda_s):
    return mu_s0 * np.exp(-np.abs(x) / lambda_s)

# Parameters
lambda_s_values = {
    'Cu': 350e-9, 'Al': 650e-9, 'Ag': 200e-9, 'Py': 5e-9
}

x = np.linspace(-500e-9, 500e-9, 1000)
mu_s0 = 1.0

# Plot
fig, ax = plt.subplots(figsize=(12, 6))

for material, lambda_s in lambda_s_values.items():
    mu_s = spin_accumulation(x, mu_s0, lambda_s)
    ax.plot(x * 1e9, mu_s, linewidth=2,
            label=f'{material} (λs = {lambda_s*1e9:.0f} nm)')

ax.axvline(x=0, color='k', linestyle='--', alpha=0.5, label='FM/NM interface')
ax.set_xlabel('Distance from Interface (nm)', fontsize=12)
ax.set_ylabel('Spin Accumulation μs (normalized)', fontsize=12)
ax.set_title('Spin Accumulation in Non-Magnetic Metals', fontsize=14)
ax.legend()
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

2.5 Spin Hall Effect

The Spin Hall Effect (SHE) is a phenomenon where charge current causes spin separation perpendicular to the current flow due to spin-orbit interaction. It's important for spin current generation without external magnetic fields.

Inverse Spin Hall Effect

The Inverse SHE converts spin current into charge current (voltage), used for electrical detection of spin currents:

Code Example 2.5: Spin Hall Angle Comparison

"""
Comparison of spin Hall angles across materials
"""
import numpy as np
import matplotlib.pyplot as plt

materials = {
    'Pt': 0.08, 'W (β)': -0.33, 'Ta (β)': -0.15,
    'Au': 0.0035, 'Cu': 0.001, 'Pd': 0.01, 'Bi2Se3': 0.5
}

sorted_mats = dict(sorted(materials.items(), key=lambda x: abs(x[1]), reverse=True))
names = list(sorted_mats.keys())
sha = list(sorted_mats.values())
colors = ['red' if s > 0 else 'blue' for s in sha]

fig, ax = plt.subplots(figsize=(12, 6))
bars = ax.barh(names, sha, color=colors, alpha=0.7)
ax.axvline(x=0, color='k', linestyle='-', linewidth=0.5)
ax.set_xlabel('Spin Hall Angle θSH', fontsize=12)
ax.set_title('Spin Hall Angle by Material (Room Temp.)', fontsize=14)
plt.tight_layout()
plt.show()

Code Example 2.6: Spin Current Generation via SHE

"""
Spin current generation via Spin Hall Effect
"""
import numpy as np
import matplotlib.pyplot as plt

def spin_current_she(J_c, theta_sh, t, lambda_s):
    J_s = theta_sh * J_c * (1 - 1/np.cosh(t / lambda_s))
    return J_s

J_c = 1e11  # A/m²
theta_sh = 0.1
lambda_s = 2e-9

thicknesses = np.linspace(0.1e-9, 20e-9, 100)
J_s_values = [spin_current_she(J_c, theta_sh, t, lambda_s) for t in thicknesses]

plt.figure(figsize=(10, 6))
plt.plot(thicknesses * 1e9, np.array(J_s_values) / 1e9, 'b-', linewidth=2)
plt.xlabel('Pt Thickness (nm)', fontsize=12)
plt.ylabel('Spin Current Density Js (GA/m²)', fontsize=12)
plt.title('Spin Current Generation via SHE', fontsize=14)
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

Chapter Summary

Exercises

Problem 1 (Easy)

In the two-current model with $R_\uparrow = 2\Omega$ and $R_\downarrow = 8\Omega$, calculate the total resistance.

Problem 2 (Medium)

Calculate TMR ratio using Jullière model for an MTJ with $P_1 = P_2 = 0.6$.

Problem 3 (Medium)

For a metal with $\lambda_s = 400$ nm, what percentage of spin accumulation remains at 1 μm from the interface?

Problem 4 (Hard)

With Pt spin Hall angle $\theta_{SH} = 0.1$ and $J_c = 10^{11}$ A/m², calculate the maximum spin current density.

References

1. Jullière, M. (1975). "Tunneling between ferromagnetic films." Physics Letters A, 54(3), 225-226.
2. Valet, T., & Fert, A. (1993). "Theory of the perpendicular magnetoresistance." Physical Review B, 48(10), 7099.
3. Sinova, J., et al. (2015). "Spin Hall effects." Reviews of Modern Physics, 87(4), 1213-1260.