Chapter 4: Electrical and Magnetic Properties

What You Will Learn in This Chapter

Classical Theory of Electrical Conduction: The Drude Model

Free Electron Approximation

This is the simplest model, which treats the valence electrons in a metal as "freely moving particles". Electrons move freely through the lattice of atomic nuclei, and scattering occurs due to lattice vibrations (phonons) and impurities.

Basic Equation of the Drude Model

The equation of motion for an electron in an electric field $\mathbf{E}$:

• $m^*$: effective mass of the electron
• $\mathbf{v}$: drift velocity
• $\tau$: relaxation time (mean time between collisions)
• $-e$: electron charge

In the steady state ($d\mathbf{v}/dt = 0$):

Electrical Conductivity

The current density $\mathbf{J}$ is:

Therefore, the electrical conductivity is:

• $n$: carrier density [m⁻³]
• $e$: electron charge ($1.602 \times 10^{-19}$ C)

Relationship with the mobility $\mu$:

Simulating the Drude Model

import numpy as np
import matplotlib.pyplot as plt

# Physical constants
e = 1.602e-19  # Electron charge [C]
m_e = 9.109e-31  # Electron mass [kg]

def calculate_conductivity(n, tau, m_star=1.0):
    """
    Calculate electrical conductivity

    Parameters:
    -----------
    n : float
        Carrier density [m^-3]
    tau : float
        Relaxation time [s]
    m_star : float
        Effective mass (in units of electron mass)

    Returns:
    --------
    sigma : float
        Electrical conductivity [S/m]
    mu : float
        Mobility [cm^2/Vs]
    """
    m_eff = m_star * m_e
    sigma = n * e**2 * tau / m_eff  # Conductivity [S/m]
    mu = e * tau / m_eff * 1e4  # Mobility [cm^2/Vs]
    return sigma, mu

# Typical metal (Cu)
n_Cu = 8.5e28  # [m^-3]
tau_Cu = 2.7e-14  # [s]
sigma_Cu, mu_Cu = calculate_conductivity(n_Cu, tau_Cu, m_star=1.0)

print("=== Electrical Properties of Copper (Cu) ===")
print(f"Carrier density: {n_Cu:.2e} m^-3")
print(f"Relaxation time: {tau_Cu:.2e} s")
print(f"Electrical conductivity: {sigma_Cu:.2e} S/m")
print(f"Mobility: {mu_Cu:.1f} cm^2/Vs")

# Temperature dependence of mobility in a semiconductor (n-type Si)
temperatures = np.linspace(100, 500, 50)  # [K]
# Temperature dependence of mobility (simplified model: μ ∝ T^-3/2)
mu_Si_ref = 1400  # [cm^2/Vs] at 300K
T_ref = 300
mu_Si = mu_Si_ref * (temperatures / T_ref)**(-1.5)

plt.figure(figsize=(10, 6))
plt.plot(temperatures, mu_Si, linewidth=2, color='#f093fb')
plt.axhline(y=1400, color='red', linestyle='--', label='Room-temperature value (300K)')
plt.xlabel('Temperature [K]', fontsize=12)
plt.ylabel('Mobility [cm²/Vs]', fontsize=12)
plt.title('Temperature Dependence of Mobility in n-type Si', fontsize=14, fontweight='bold')
plt.legend()
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('mobility_temperature.png', dpi=300, bbox_inches='tight')
plt.show()

Hall Effect and Carrier Measurement

Principle of the Hall Effect

When a magnetic field is applied perpendicular to a current-carrying conductor, the Lorentz force deflects the charges, producing a transverse potential difference (Hall voltage).

Hall Coefficient

The Hall field $E_y$ is:

The Hall coefficient $R_H$ is:

• For holes: $R_H > 0$
• For electrons: $R_H < 0$

Measuring carrier density:

Measuring mobility:

Simulating a Hall Effect Measurement

import numpy as np
import matplotlib.pyplot as plt

def hall_effect_simulation(n, mu, B_range, thickness=1e-3):
    """
    Simulate the Hall effect

    Parameters:
    -----------
    n : float
        Carrier density [m^-3]
    mu : float
        Mobility [m^2/Vs]
    B_range : array
        Magnetic field range [T]
    thickness : float
        Sample thickness [m]
    """
    e = 1.602e-19

    # Hall coefficient
    R_H = 1 / (n * e)  # [m^3/C]

    # Assume constant current density
    J = 1e6  # [A/m^2]

    # Hall voltage
    V_H = R_H * J * B_range * thickness  # [V]

    # Hall resistance
    R_Hall = V_H / (J * thickness**2)  # [Ω]

    return V_H, R_Hall, R_H

# Example: n-type Si semiconductor
n_Si = 1e22  # [m^-3]
mu_Si = 0.14  # [m^2/Vs] = 1400 cm^2/Vs

B_range = np.linspace(-2, 2, 100)  # [T]
V_H, R_Hall, R_H = hall_effect_simulation(n_Si, mu_Si, B_range)

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

# Hall voltage vs magnetic field
ax1.plot(B_range, V_H * 1e3, linewidth=2, color='#f093fb')
ax1.axhline(y=0, color='black', linestyle='-', linewidth=0.5)
ax1.axvline(x=0, color='black', linestyle='-', linewidth=0.5)
ax1.set_xlabel('Magnetic field [T]', fontsize=12)
ax1.set_ylabel('Hall voltage [mV]', fontsize=12)
ax1.set_title('Magnetic Field Dependence of Hall Voltage', fontsize=14, fontweight='bold')
ax1.grid(True, alpha=0.3)

# Hall resistance vs magnetic field
ax2.plot(B_range, R_Hall, linewidth=2, color='#f5576c')
ax2.axhline(y=0, color='black', linestyle='-', linewidth=0.5)
ax2.set_xlabel('Magnetic field [T]', fontsize=12)
ax2.set_ylabel('Hall resistance [Ω]', fontsize=12)
ax2.set_title('Magnetic Field Dependence of Hall Resistance', fontsize=14, fontweight='bold')
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('hall_effect.png', dpi=300, bbox_inches='tight')
plt.show()

print("=== Hall Effect Measurement Results ===")
print(f"Carrier density: {n_Si:.2e} m^-3")
print(f"Hall coefficient: {R_H:.2e} m^3/C")
print(f"Sign of Hall coefficient: {'negative (electrons)' if R_H < 0 else 'positive (holes)'}")
print(f"Hall voltage at 1T field: {V_H[np.argmin(np.abs(B_range - 1.0))] * 1e3:.3f} mV")

Fundamentals of Magnetism

Origin of the Magnetic Moment

The magnetic moment of atoms and molecules has two contributions:

1. Orbital magnetic moment: arising from the orbital motion of the electron $$\mathbf{\mu}_L = -\frac{e}{2m_e}\mathbf{L}$$
2. Spin magnetic moment: arising from the intrinsic angular momentum (spin) of the electron $$\mathbf{\mu}_S = -g_s \frac{e}{2m_e}\mathbf{S}$$

Here, $g_s \approx 2$ is the g-factor. Using the Bohr magneton $\mu_B$:

Magnetization and Magnetic Susceptibility

The magnetization $M$ is the magnetic moment per unit volume:

The magnetic susceptibility $\chi$ is:

• $\chi > 0$: paramagnetism (magnetization along the field direction)
• $\chi < 0$: diamagnetism (magnetization opposite to the field)

Classification of Magnetism

Mean-Field Theory of Ferromagnetism (Weiss Theory)

In ferromagnets, an "exchange interaction" acts to align neighboring spins parallel to each other. Weiss introduced an "effective field" $H_{\text{eff}}$ felt by each spin:

$\lambda$ is the Weiss constant (molecular field constant). Solving the self-consistent equation yields the Curie temperature $T_C$:

Spontaneous magnetization arises for $T < T_C$.

Predicting Magnetism with DFT Calculations

Spin-Polarized DFT Calculations

In DFT calculations of magnetic materials, spin-up (↑) and spin-down (↓) electrons are treated separately (spin-polarized calculation).

The electron density is decomposed into spin components:

Spin density (magnetization density):

Magnetic moment:

Setting Up Spin-Polarized Calculations in VASP

# INCAR file to set up a spin-polarized calculation in VASP

def create_magnetic_incar(system_name='Fe', initial_magmom=2.0):
    """
    Generate a VASP INCAR file for magnetic materials

    Parameters:
    -----------
    system_name : str
        System name
    initial_magmom : float
        Initial magnetic moment [μB/atom]
    """

    incar_content = f"""SYSTEM = {system_name} magnetic calculation

# Electronic structure
ENCUT = 400
PREC = Accurate
LREAL = Auto

# Exchange-correlation
GGA = PE

# SCF convergence
EDIFF = 1E-6
NELM = 100

# Smearing (for metals)
ISMEAR = 1          # Methfessel-Paxton
SIGMA = 0.2

# Spin-polarized calculation
ISPIN = 2           # Enable spin polarization
MAGMOM = {initial_magmom}  # Initial magnetic moment [μB]

# Output of magnetic moments
LORBIT = 11         # Atom- and orbital-projected magnetic moments

# Parallelization
NCORE = 4
"""
    return incar_content

# Calculation setup for ferromagnetic Fe (BCC)
incar_fe = create_magnetic_incar('Fe BCC', initial_magmom=2.2)
print("=== INCAR for Ferromagnetic Fe Calculation ===")
print(incar_fe)

# Calculation setup for antiferromagnetic MnO (rocksalt)
# Set alternating initial spins on the Mn atoms
incar_mno = """SYSTEM = MnO antiferromagnetic

ENCUT = 450
PREC = Accurate
GGA = PE

EDIFF = 1E-6
ISMEAR = 0
SIGMA = 0.05

# Spin-polarized calculation
ISPIN = 2
MAGMOM = 4.0 -4.0 4.0 -4.0 0 0 0 0  # 4 Mn (alternating) + 4 O (nonmagnetic)

LORBIT = 11
NCORE = 4
"""

print("\n=== INCAR for Antiferromagnetic MnO Calculation ===")
print(incar_mno)

Visualizing the Spin Density

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# Generate dummy spin density data (in practice, read from VASP output)
def generate_spin_density_data():
    """
    Generate a mock spin density around an Fe atom
    """
    x = np.linspace(-3, 3, 50)
    y = np.linspace(-3, 3, 50)
    X, Y = np.meshgrid(x, y)

    # Approximate the spin density with a Gaussian distribution
    spin_density = 2.2 * np.exp(-(X**2 + Y**2) / 2)

    return X, Y, spin_density

X, Y, spin_density = generate_spin_density_data()

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

# Contour plot
contour = ax1.contourf(X, Y, spin_density, levels=20, cmap='RdBu_r')
ax1.contour(X, Y, spin_density, levels=10, colors='black', linewidths=0.5, alpha=0.3)
fig.colorbar(contour, ax=ax1, label='Spin density [μB/ų]')
ax1.set_xlabel('x [Å]', fontsize=12)
ax1.set_ylabel('y [Å]', fontsize=12)
ax1.set_title('Spin Density around an Fe Atom (2D)', fontsize=14, fontweight='bold')
ax1.set_aspect('equal')

# 3D surface
from matplotlib import cm
ax2 = fig.add_subplot(122, projection='3d')
surf = ax2.plot_surface(X, Y, spin_density, cmap=cm.coolwarm, alpha=0.8)
ax2.set_xlabel('x [Å]', fontsize=10)
ax2.set_ylabel('y [Å]', fontsize=10)
ax2.set_zlabel('Spin density [μB/ų]', fontsize=10)
ax2.set_title('Spin Density around an Fe Atom (3D)', fontsize=14, fontweight='bold')

plt.tight_layout()
plt.savefig('spin_density.png', dpi=300, bbox_inches='tight')
plt.show()

# Calculate the magnetic moment (numerical integration)
dx = X[0, 1] - X[0, 0]
dy = Y[1, 0] - Y[0, 0]
total_moment = np.sum(spin_density) * dx * dy

print(f"\n=== Magnetic Moment Calculation Results ===")
print(f"Integrated magnetic moment: {total_moment:.2f} μB")
print(f"(Actual Fe: about 2.2 μB)")

Spin-Orbit Coupling (SOC)

Origin of SOC

This is the effect in which the electron spin $\mathbf{S}$ interacts with the orbital angular momentum $\mathbf{L}$. It arises from relativistic effects:

$\lambda$ is the spin-orbit coupling constant, which increases rapidly with atomic number $Z$ ($\lambda \propto Z^4$).

Physical Effects of SOC

• Magnetic anisotropy: the energy depends on the direction of magnetization
• Magnetic circular dichroism (MCD): difference in absorption of circularly polarized light
• Rashba effect: spin splitting in systems with broken inversion symmetry
• Topological insulators: SOC-induced band inversion

Setting Up SOC Calculations in VASP

# VASP calculation setup including spin-orbit coupling

def create_soc_incar(system_name='Pt', include_soc=True):
    """
    Generate an INCAR file for SOC calculations

    Parameters:
    -----------
    system_name : str
        System name
    include_soc : bool
        Whether to enable SOC
    """

    incar_content = f"""SYSTEM = {system_name} with SOC

ENCUT = 400
PREC = Accurate
GGA = PE

EDIFF = 1E-7        # High accuracy required for SOC calculations
ISMEAR = 1
SIGMA = 0.2

# Spin polarization + SOC
ISPIN = 2
"""

    if include_soc:
        incar_content += """LSORBIT = .TRUE.    # Enable spin-orbit coupling
LNONCOLLINEAR = .TRUE.  # Noncollinear magnetism (spin directions are free)
GGA_COMPAT = .FALSE.    # Recommended for SOC calculations
"""

    incar_content += """
LORBIT = 11
NCORE = 4
"""
    return incar_content

# Pt (heavy element, SOC important)
incar_pt_soc = create_soc_incar('Pt bulk', include_soc=True)
print("=== INCAR for Pt + SOC Calculation ===")
print(incar_pt_soc)

# For comparison without SOC
incar_pt_no_soc = create_soc_incar('Pt bulk', include_soc=False)
print("\n=== INCAR for Pt (without SOC) Calculation ===")
print(incar_pt_no_soc)

SOC-Induced Band Splitting

import numpy as np
import matplotlib.pyplot as plt

# Band structure simulation with and without SOC
def simulate_soc_band_splitting():
    """
    Visualize SOC-induced band splitting with a mock model
    """
    k = np.linspace(-np.pi, np.pi, 200)

    # Without SOC (degenerate)
    E_no_soc = np.cos(k) + 0.5 * np.cos(2*k)

    # With SOC (split)
    lambda_soc = 0.3  # SOC strength
    E_soc_up = E_no_soc + lambda_soc * np.abs(np.sin(k))
    E_soc_down = E_no_soc - lambda_soc * np.abs(np.sin(k))

    return k, E_no_soc, E_soc_up, E_soc_down

k, E_no_soc, E_soc_up, E_soc_down = simulate_soc_band_splitting()

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

# Without SOC
ax1.plot(k/np.pi, E_no_soc, linewidth=2, color='blue', label='Degenerate band')
ax1.axhline(y=0, color='black', linestyle='--', linewidth=0.5)
ax1.set_xlabel('k [π/a]', fontsize=12)
ax1.set_ylabel('Energy [eV]', fontsize=12)
ax1.set_title('Without SOC', fontsize=14, fontweight='bold')
ax1.legend()
ax1.grid(True, alpha=0.3)

# With SOC
ax2.plot(k/np.pi, E_soc_up, linewidth=2, color='red', label='Spin up')
ax2.plot(k/np.pi, E_soc_down, linewidth=2, color='blue', label='Spin down')
ax2.axhline(y=0, color='black', linestyle='--', linewidth=0.5)
ax2.set_xlabel('k [π/a]', fontsize=12)
ax2.set_ylabel('Energy [eV]', fontsize=12)
ax2.set_title('With SOC', fontsize=14, fontweight='bold')
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('soc_band_splitting.png', dpi=300, bbox_inches='tight')
plt.show()

# Splitting energy at k=π/2
idx = len(k) // 4
splitting = E_soc_up[idx] - E_soc_down[idx]
print(f"\n=== SOC-Induced Band Splitting ===")
print(f"Splitting at k=π/2: {splitting:.3f} eV")

Fundamentals of Superconductivity

The Superconducting Phenomenon

Below a critical temperature $T_c$, the electrical resistance drops to zero. It was discovered in Hg by Kamerlingh Onnes in 1911.

BCS Theory (1957)

The microscopic theory by Bardeen, Cooper, and Schrieffer. Electrons acquire an attractive interaction mediated by phonons (lattice vibrations) and form "Cooper pairs".

Superconducting gap:

Gap at $T=0$ K:

Representative Superconductors

Temperature Dependence of the Superconducting Gap

import numpy as np
import matplotlib.pyplot as plt

def superconducting_gap(T, Tc):
    """
    Temperature dependence of the superconducting gap from BCS theory

    Parameters:
    -----------
    T : array
        Temperature [K]
    Tc : float
        Critical temperature [K]

    Returns:
    --------
    Delta : array
        Superconducting gap [meV]
    """
    k_B = 8.617e-5  # Boltzmann constant [eV/K]

    # Approximate formula from BCS theory
    Delta_0 = 1.76 * k_B * Tc * 1000  # [meV]

    Delta = np.zeros_like(T)
    mask = T < Tc
    Delta[mask] = Delta_0 * np.tanh(1.74 * np.sqrt((Tc - T[mask]) / T[mask]))

    return Delta

# T_c of various superconductors
materials = {
    'Al': 1.2,
    'Nb': 9.2,
    'MgB₂': 39,
    'YBCO': 92
}

T = np.linspace(0.1, 100, 500)

plt.figure(figsize=(10, 6))

for name, Tc in materials.items():
    Delta = superconducting_gap(T, Tc)
    plt.plot(T, Delta, linewidth=2, label=f'{name} ($T_c$={Tc}K)')

plt.xlabel('Temperature [K]', fontsize=12)
plt.ylabel('Superconducting gap Δ(T) [meV]', fontsize=12)
plt.title('Temperature Dependence of the Superconducting Gap', fontsize=14, fontweight='bold')
plt.legend()
plt.grid(True, alpha=0.3)
plt.xlim(0, 100)
plt.tight_layout()
plt.savefig('superconducting_gap.png', dpi=300, bbox_inches='tight')
plt.show()

# Verify Δ_0 / k_B T_c (1.76 in BCS theory)
for name, Tc in materials.items():
    k_B = 8.617e-5
    Delta_0 = 1.76 * k_B * Tc * 1000
    ratio = Delta_0 / (k_B * Tc * 1000)
    print(f"{name}: Δ₀/(kB·Tc) = {ratio:.2f}")

Summary

Exercises

References

1. Ashcroft, N. W., & Mermin, N. D. (1976). "Solid State Physics". Harcourt College Publishers.
2. Kittel, C. (2004). "Introduction to Solid State Physics" (8th ed.). Wiley.
3. Blundell, S. (2001). "Magnetism in Condensed Matter". Oxford University Press.
4. Tinkham, M. (2004). "Introduction to Superconductivity" (2nd ed.). Dover Publications.
5. Bardeen, J., Cooper, L. N., & Schrieffer, J. R. (1957). "Theory of Superconductivity". Physical Review, 108, 1175.
6. VASP manual: Magnetism and SOC - https://www.vasp.at/wiki/index.php/Magnetism