Chapter 4: Electrical and Magnetic Properties

This chapter covers Electrical and Magnetic Properties. You will learn essential concepts and techniques.

What You Will Learn in This Chapter

Classical Theory of Electrical Conduction: Drude Model

Free Electron Approximation

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

Basic Equations of Drude Model

Equation of motion for electrons in an electric field $\mathbf{E}$:

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

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

Electrical Conductivity

Current density $\mathbf{J}$ is:

Therefore, electrical conductivity is:

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

Relationship with mobility $\mu$:

Drude Model Simulation

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0


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")

# Mobility and temperature dependence for semiconductor (Si n-type)
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 Si n-type Semiconductor', 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 Hall Effect

When a magnetic field perpendicular to a current-carrying conductor is applied, the Lorentz force causes charge separation, resulting in a potential difference (Hall voltage) in the transverse direction.

Hall Coefficient

The Hall electric field $E_y$ is:

The Hall coefficient $R_H$ is:

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

Carrier Density Measurement:

Mobility Measurement:

Hall Effect Measurement Simulation

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0


import numpy as np
import matplotlib.pyplot as plt

def hall_effect_simulation(n, mu, B_range, thickness=1e-3):
    """
    Simulate 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 of Si n-type 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"Hall coefficient sign: {'Negative (electrons)' if R_H < 0 else 'Positive (holes)'}")
print(f"Hall voltage at 1T magnetic field: {V_H[np.argmin(np.abs(B_range - 1.0))] * 1e3:.3f} mV")

Fundamentals of Magnetism

Origins of Magnetic Moment

The magnetic moment of atoms and molecules has two contributions:

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

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

Magnetization and Susceptibility

$M$ 、 ：

susceptibility$\chi$ ：

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

Classification of Magnetism

Ferromagnetism （Weiss）

Ferromagnetism 、「」。Weiss 、「」$H_{\text{eff}}$：

$\lambda$ Weissconstant（constant）。、CurieTemperature$T_C$：

$T < T_C$ 。

Prediction of Magnetism by DFT Calculations

Spin-Polarized DFT Calculations

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

Electron density is decomposed into spin components:

Spin density (magnetization density):

Magnetic moment:

Spin-Polarized Calculation Settings in VASP

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

def create_magnetic_incar(system_name='Fe', initial_magmom=2.0):
    """
    Generate 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]

# Magnetic moment output
LORBIT = 11         # Atom- and orbital-projected magnetic moments

# Parallelization
NCORE = 4
"""
    return incar_content

# FerromagnetismFe（BCC） calculation
incar_fe = create_magnetic_incar('Fe BCC', initial_magmom=2.2)
print("=== Fe Ferromagnetismcalculation INCAR ===")
print(incar_fe)

# AntiFerromagnetismMnO（rocksalt） calculation
# Set initial spins of Mn atoms alternately
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  # Mn4（）+ O4（Magnetism）

LORBIT = 11
NCORE = 4
"""

print("\n=== MnO AntiFerromagnetismcalculation INCAR ===")
print(incar_mno)

Visualization of Spin Density

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0


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():
    """
    Simulate spin density around Fe atom
    """
    x = np.linspace(-3, 3, 50)
    y = np.linspace(-3, 3, 50)
    X, Y = np.meshgrid(x, y)

    # Approximate spin density with 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('FeAround Fe AtomSpin Density（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('FeAround Fe AtomSpin Density（3D）', fontsize=14, fontweight='bold')

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

# Calculate 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"(Experimental Fe: approximately 2.2 μB)")

Spin-Orbit Interaction (SOC)

Origin of SOC

$\mathbf{S}$$\mathbf{L}$。：

$\lambda$ constant、$Z$ （$\lambda \propto Z^4$）。

Physical Effects of SOC

• Magnetic anisotropy: Energy depends on magnetization direction
• Magnetic circular dichroism (MCD): Difference in absorption for circularly polarized light
• Rashba effect: Spin splitting in systems with broken inversion symmetry
• Topological insulators: Band inversion by SOC

SOC Calculation Settings in VASP

# VASP calculation settings including spin-orbit interaction

def create_soc_incar(system_name='Pt', include_soc=True):
    """
    Generate 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 precision required for SOC calculations
ISMEAR = 1
SIGMA = 0.2

#  + SOC
ISPIN = 2
"""

    if include_soc:
        incar_content += """LSORBIT = .TRUE.    # Enable spin-orbit interaction
LNONCOLLINEAR = .TRUE.  # Magnetism（ ）
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("=== Pt + SOC calculation INCAR ===")
print(incar_pt_soc)

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

Band Splitting by SOC

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0


import numpy as np
import matplotlib.pyplot as plt

# Band structure simulation with and without SOC
def simulate_soc_band_splitting():
    """
    Band Splitting by SOC
    """
    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 at k=π/2Energy
idx = len(k) // 4
splitting = E_soc_up[idx] - E_soc_down[idx]
print(f"\n=== Band Splitting by SOC ===")
print(f"Splitting at k=π/2: {splitting:.3f} eV")

Fundamentals of Superconductivity

Superconducting Phenomenon

criticalTemperature$T_c$、。1911、Kamerlingh OnnesHg。

BCS Theory (1957)

Microscopic theory by Bardeen, Cooper, and Schrieffer. Electrons form "Cooper pairs" through attractive interaction mediated by phonons (lattice vibrations).

Superconducting gap:

$T=0$K at：

Representative Superconductors

Temperature Dependence of Superconducting Gap

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0


import numpy as np
import matplotlib.pyplot as plt

def superconducting_gap(T, Tc):
    """
    BCSTemperature Dependence of Superconducting Gap

    Parameters:
    -----------
    T : array
        Temperature [K]
    Tc : float
        criticalTemperature [K]

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

    # BCS theory approximation formula
    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

# Tc for 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 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()

# Verification of 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