Chapter 2: Hall Effect Measurement

The Hall effect is a powerful technique that utilizes the deflection of charge carriers in a magnetic field to determine carrier density, carrier type (electron/hole), and mobility. In this chapter, you will learn the theory of the Hall effect based on the Lorentz force, the relationship between Hall coefficient and carrier density, van der Pauw Hall measurement configuration, multi-carrier system analysis, and the elucidation of carrier scattering mechanisms from temperature dependence, and perform practical Hall data analysis using Python.

2.1 Theory of Hall Effect

2.1.1 Lorentz Force and Hall Voltage

Hall effect (discovered in 1879 by Edwin Herbert Hall) is a phenomenon in which, when a perpendicular magnetic field is applied to a conductor carrying electric current, a voltage is generated in a direction perpendicular to both the current and the magnetic field.

Physical Mechanism:

1. When current $I$ flows through a conductor, carriers (electrons) move in the x-direction at drift velocity $v_x$
2. When a magnetic field $B_z$ is applied in the z-direction, the Lorentz force $\vec{F} = q\vec{v} \times \vec{B}$ acts
3. Electrons ($q = -e$) are deflected in the y-direction and accumulate on one side
4. Charge accumulation generates Hall electric field $E_y$
5. In steady state, Lorentz force and electric field force balance: $eE_y = ev_x B_z$

Derivation of Hall Voltage:

If the sample width is $w$, thickness $t$, and current $I$, the current density is:

If the carrier density is $n$, the relationship between current density and drift velocity is:

From force balance in steady state:

Hall voltage $V_H$ は, 試料幅 $w$ にわたる電場のintegral:

Hall coefficient $R_H$ is defined as:

Therefore, carrier density $n$ can be directly obtained from the Hall coefficient:

2.1.2 Determination of Carrier Type

The sign of the Hall coefficient can determine whether carriers are electrons or holes:

Code Example 2-1: Calculation of Hall Coefficient and Carrier 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

def calculate_hall_coefficient(V_H, I, B, t):
    """
    Calculate Hall coefficient

    Parameters
    ----------
    V_H : float
        Hall voltage [V]
    I : float
        Current [A]
    B : float
        Magnetic field [T]
    t : float
        Sample thickness [m]

    Returns
    -------
    R_H : float
        Hall coefficient [m^3/C]
    """
    R_H = V_H * t / (I * B)
    return R_H

def calculate_carrier_density(R_H):
    """
    Calculate carrier density

    Parameters
    ----------
    R_H : float
        Hall coefficient [m^3/C]

    Returns
    -------
    n : float
        Carrier density [m^-3]
    carrier_type : str
        Carrier type（'electron' or 'hole'）
    """
    e = 1.60218e-19  # Elementary charge [C]
    n = 1 / (np.abs(R_H) * e)
    carrier_type = 'electron' if R_H < 0 else 'hole'
    return n, carrier_type

# Measurement Example 1: n-type Silicon
V_H1 = -2.5e-3  # Hall voltage [V]（Negative:electron）
I1 = 1e-3  # Current [A]
B1 = 0.5  # Magnetic field [T]
t1 = 500e-9  # Thickness [m] = 500 nm

R_H1 = calculate_hall_coefficient(V_H1, I1, B1, t1)
n1, type1 = calculate_carrier_density(R_H1)

print("Measurement example 1: n-type silicon")
print(f"  Hall voltage: {V_H1 * 1e3:.2f} mV")
print(f"  Hall coefficient: {R_H1:.3e} m³/C")
print(f"  Carrier type: {type1}")
print(f"  Carrier density: {n1:.3e} m⁻³ = {n1 / 1e6:.3e} cm⁻³")

# Measurement Example 2: p-type Gallium Arsenide
V_H2 = +3.8e-3  # Hall voltage [V]（Positive:hole）
I2 = 1e-3  # Current [A]
B2 = 0.5  # Magnetic field [T]
t2 = 300e-9  # Thickness [m] = 300 nm

R_H2 = calculate_hall_coefficient(V_H2, I2, B2, t2)
n2, type2 = calculate_carrier_density(R_H2)

print("\nMeasurement example 2: p-type gallium arsenide")
print(f"  Hall voltage: {V_H2 * 1e3:.2f} mV")
print(f"  Hall coefficient: {R_H2:.3e} m³/C")
print(f"  Carrier type: {type2}")
print(f"  Carrier density: {n2:.3e} m⁻³ = {n2 / 1e6:.3e} cm⁻³")

# Visualization of carrier density dependence
n_range = np.logspace(20, 28, 100)  # [m^-3]
R_H_range = 1 / (n_range * 1.60218e-19)

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

# Left: n vs R_H
ax1.loglog(n_range / 1e6, np.abs(R_H_range), linewidth=2.5, color='#f093fb', label='|R_H| = 1/(ne)')
ax1.scatter([n1 / 1e6], [np.abs(R_H1)], s=150, c='#f5576c', edgecolors='black', linewidth=2, zorder=5, label='n-Si (example 1)')
ax1.scatter([n2 / 1e6], [np.abs(R_H2)], s=150, c='#ffa500', edgecolors='black', linewidth=2, zorder=5, label='p-GaAs (example 2)')
ax1.set_xlabel('Carrier Density n [cm$^{-3}$]', fontsize=12)
ax1.set_ylabel('|Hall Coefficient R$_H$| [m$^3$/C]', fontsize=12)
ax1.set_title('Hall Coefficient vs Carrier Density', fontsize=14, fontweight='bold')
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3, which='both')

# Right: Hall voltage vs magnetic field
B_range = np.linspace(0, 1, 100)  # [T]
V_H_n = R_H1 * I1 * B_range / t1 * 1e3  # n-type [mV]
V_H_p = R_H2 * I2 * B_range / t2 * 1e3  # p-type [mV]

ax2.plot(B_range, V_H_n, linewidth=2.5, color='#f5576c', label='n-type (electron, R_H < 0)')
ax2.plot(B_range, V_H_p, linewidth=2.5, color='#ffa500', label='p-type (hole, R_H > 0)')
ax2.axhline(0, color='black', linestyle='--', linewidth=1.5)
ax2.set_xlabel('Magnetic Field B [T]', fontsize=12)
ax2.set_ylabel('Hall Voltage V$_H$ [mV]', fontsize=12)
ax2.set_title('Hall Voltage vs Magnetic Field', fontsize=14, fontweight='bold')
ax2.legend(fontsize=11)
ax2.grid(alpha=0.3)

plt.tight_layout()
plt.show()

2.2 Determination of Mobility

2.2.1 Relationship between Mobility and Hall Coefficient

Mobility $\mu$ is obtained from electrical conductivity $\sigma$ and Hall coefficient $R_H$:

This is directly derived from $\sigma = ne\mu$ and $R_H = 1/(ne)$.

Physical Meaning:

• $\sigma$ is the ease of electrical conduction (depends on both carrier density $n$ and mobility $\mu$)
• $R_H$ depends only on carrier density $n$
• By combining both, mobility $\mu$ can be obtained separately

Code Example 2-2: Calculation of Mobility

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

import numpy as np

def calculate_mobility(sigma, R_H):
    """
    Calculate mobility

    Parameters
    ----------
    sigma : float
        Electrical conductivity [S/m]
    R_H : float
        Hall coefficient [m^3/C]

    Returns
    -------
    mu : float
        Mobility [m^2/(V·s)]
    """
    mu = sigma * np.abs(R_H)
    return mu

# Measurement example: n-type Silicon (from previous example)
R_H = -2.5e-3  # [m^3/C]
sigma = 1e4  # Electrical conductivity [S/m](typical value)

mu = calculate_mobility(sigma, R_H)

e = 1.60218e-19  # [C]
n = 1 / (np.abs(R_H) * e)

print("Electrical properties of n-type Silicon:")
print(f"  Electrical conductivity σ = {sigma:.2e} S/m")
print(f"  Hall coefficient R_H = {R_H:.2e} m³/C")
print(f"  Carrier density n = {n:.2e} m⁻³ = {n / 1e6:.2e} cm⁻³")
print(f"  Mobility μ = {mu:.2e} m²/(V·s) = {mu * 1e4:.1f} cm²/(V·s)")
print(f"\nVerification: σ = neμ = {n * e * mu:.2e} S/m(match)")

# Material comparison
materials = {
    'Si (bulk, n-type)': {'sigma': 1e4, 'R_H': -2.5e-3},
    'GaAs (bulk, n-type)': {'sigma': 1e5, 'R_H': -5e-3},
    'InSb (n-type)': {'sigma': 1e6, 'R_H': -1e-2},
    'Graphene': {'sigma': 1e5, 'R_H': -1e-4}
}

print("\nMaterial comparison:")
print(f"{'Material':<25} {'n [cm⁻³]':<15} {'μ [cm²/(V·s)]':<20}")
print("-" * 60)

for name, props in materials.items():
    n_mat = 1 / (np.abs(props['R_H']) * e)
    mu_mat = calculate_mobility(props['sigma'], props['R_H'])
    print(f"{name:<25} {n_mat / 1e6:.2e}      {mu_mat * 1e4:.1f}")

Output Interpretation:

• Silicon mobility: ~1000 cm²/(V·s) (typical value)
• GaAs is a high-mobility material (~5000 cm²/(V·s))
• InSb has ultra-high mobility (~77,000 cm²/(V·s))
• Graphene has extremely high mobility (~10,000-100,000 cm²/(V·s))

2.3 van der Pauw Hall Measurement Configuration

2.3.1 8-Contact van der Pauw Hall Configuration

van der Pauw Hall measurement is a standard technique that can measure the Hall effect on thin film samples of arbitrary shape. Combined with the sheet resistance measurement learned in Chapter 1, all of $\sigma$, $R_H$, $n$, and $\mu$ can be determined from a single sample.

Measurement Procedure:

1. Sheet Resistance Measurement (no magnetic field, B = 0):
   • Current through contacts 1→2, voltage measurement between 3-4 → $R_{12,34}$
   • Current through contacts 2→3, voltage measurement between 4-1 → $R_{23,41}$
   • Calculate $R_s$ using van der Pauw equation
2. Hall Measurement (apply magnetic field B, e.g., B = +0.5 T):
   • Current $I$ through contacts 1→3, measure voltage $V_{24}^{+B}$ between 2-4
   • Reverse magnetic field (B = -0.5 T), measure $V_{24}^{-B}$ with same current
   • Hall voltage: $V_H = \frac{1}{2}(V_{24}^{+B} - V_{24}^{-B})$
   • Hall coefficient: $R_H = \frac{V_H t}{IB}$
3. Derivation of Electrical Properties:
   • Electrical conductivity: $\sigma = \frac{1}{R_s t}$
   • Carrier density: $n = \frac{1}{eR_H}$
   • Mobility: $\mu = \sigma R_H$

Important: The reason for measuring with reversed magnetic field is to cancel offset voltage (due to thermoelectric effect and inhomogeneity). Hall voltage is an odd function with respect to magnetic field ($V_H(B) = -V_H(-B)$), but offset voltage is an even function, so taking the difference yields pure Hall voltage.

Code Example 2-3: Simulation of van der Pauw Hall Measurement

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

import numpy as np
from scipy.optimize import fsolve

def van_der_pauw_sheet_resistance(R1, R2):
    """Calculate sheet resistance using van der Pauw equation"""
    def equation(Rs):
        return np.exp(-np.pi * R1 / Rs) + np.exp(-np.pi * R2 / Rs) - 1

    R_initial = (R1 + R2) / 2 * np.pi / np.log(2)
    R_s = fsolve(equation, R_initial)[0]
    return R_s

def complete_hall_analysis(R_AB_CD, R_BC_DA, V_24_pos_B, V_24_neg_B, I, B, t):
    """
    Complete van der Pauw Hall analysis

    Parameters
    ----------
    R_AB_CD, R_BC_DA : float
        van der Pauw resistance [Ω]
    V_24_pos_B, V_24_neg_B : float
        at positive and negative magnetic field forHall voltage [V]
    I : float
        Current [A]
    B : float
        Magnetic field magnitude [T]
    t : float
        Sample thickness [m]

    Returns
    -------
    results : dict
        Analysis results (R_s, sigma, R_H, n, mu)
    """
    e = 1.60218e-19  # [C]

    # 1. Sheet resistance
    R_s = van_der_pauw_sheet_resistance(R_AB_CD, R_BC_DA)

    # 2. Electrical conductivity
    sigma = 1 / (R_s * t)

    # 3. Hall voltage (offset removed)
    V_H = 0.5 * (V_24_pos_B - V_24_neg_B)

    # 4. Hall coefficient
    R_H = V_H * t / (I * B)

    # 5. Carrier density
    n = 1 / (np.abs(R_H) * e)
    carrier_type = 'electron' if R_H < 0 else 'hole'

    # 6. Mobility
    mu = sigma * np.abs(R_H)

    results = {
        'R_s': R_s,
        'sigma': sigma,
        'rho': 1 / sigma,
        'V_H': V_H,
        'R_H': R_H,
        'n': n,
        'carrier_type': carrier_type,
        'mu': mu
    }

    return results

# Measurement data example: n-type silicon thin film
R_AB_CD = 1000  # [Ω]
R_BC_DA = 950   # [Ω]
V_24_plus = -5.2e-3  # Voltage at +B [V]
V_24_minus = +4.8e-3  # Voltage at -B [V]
I = 100e-6  # Current [A] = 100 μA
B = 0.5  # Magnetic field [T]
t = 200e-9  # Thickness [m] = 200 nm

results = complete_hall_analysis(R_AB_CD, R_BC_DA, V_24_plus, V_24_minus, I, B, t)

print("van der Pauw Hall Measurement Analysis Results:")
print("=" * 60)
print(f"Measurement Conditions:")
print(f"  R_AB,CD = {R_AB_CD:.1f} Ω")
print(f"  R_BC,DA = {R_BC_DA:.1f} Ω")
print(f"  V_24(+B) = {V_24_plus * 1e3:.2f} mV")
print(f"  V_24(-B) = {V_24_minus * 1e3:.2f} mV")
print(f"  Current I = {I * 1e6:.1f} μA")
print(f"  Magnetic field B = ±{B:.2f} T")
print(f"  Thickness t = {t * 1e9:.0f} nm")
print("\nAnalysis Results:")
print(f"  Sheet resistance R_s = {results['R_s']:.2f} Ω/sq")
print(f"  Electrical conductivity σ = {results['sigma']:.2e} S/m")
print(f"  Resistivity ρ = {results['rho']:.2e} Ω·m = {results['rho'] * 1e8:.2f} μΩ·cm")
print(f"  Hall voltage V_H = {results['V_H'] * 1e3:.2f} mV")
print(f"  Hall coefficient R_H = {results['R_H']:.2e} m³/C")
print(f"  Carrier type: {results['carrier_type']}")
print(f"  Carrier density n = {results['n']:.2e} m⁻³ = {results['n'] / 1e6:.2e} cm⁻³")
print(f"  Mobility μ = {results['mu']:.2e} m²/(V·s) = {results['mu'] * 1e4:.1f} cm²/(V·s)")
print("\nVerification:")
print(f"  σ = neμ = {results['n'] * 1.60218e-19 * results['mu']:.2e} S/m")
print(f"  (Matches calculated value σ = {results['sigma']:.2e} S/m)")

2.4 Multi-Carrier Analysis (Two-Band Model)

2.4.1 Theory of Two-Carrier Systems

In semiconductors, both electrons and holes may contribute to conduction (e.g., narrow bandgap semiconductors, InSb, HgCdTe, etc.). In this case, a simple one-band model is insufficient and a two-band model is required.

Electrical Conductivity (two-carrier):

Hall Coefficient (two-carrier):

Here, $n_e$, $\mu_e$ are the electron carrier density and mobility, and $n_h$, $\mu_h$ are the hole carrier density and mobility.

Physical Interpretation:

• $\mu_h \gg \mu_e$ case, holeがHall効果を支配（$R_H > 0$）
• When $\mu_e \gg \mu_h$, electrons dominate the Hall effect ($R_H < 0$)
• When mobilities are comparable, the sign of $R_H$ depends on the ratio of carrier densities $n_h/n_e$

Code Example 2-4: Fitting of Two-Band Model

# 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 lmfit import Model

def two_band_conductivity(n_e, mu_e, n_h, mu_h):
    """Electrical conductivity of two-band model"""
    e = 1.60218e-19
    sigma = n_e * e * mu_e + n_h * e * mu_h
    return sigma

def two_band_hall_coefficient(n_e, mu_e, n_h, mu_h):
    """Hall coefficient of two-band model"""
    e = 1.60218e-19
    numerator = n_h * mu_h**2 - n_e * mu_e**2
    denominator = (n_h * mu_h + n_e * mu_e)**2
    R_H = numerator / (e * denominator)
    return R_H

# Simulation: InSb（electron・hole coexistence system)
T_range = np.linspace(200, 400, 50)  # Temperature [K]

# Temperature dependence (simplified)
n_e = 1e22 * np.exp(-0.1 / (8.617e-5 * T_range))  # electrondensity [m^-3]
n_h = 5e21 * np.exp(-0.08 / (8.617e-5 * T_range))  # holedensity [m^-3]
mu_e = 7e4 * (300 / T_range)**1.5 * 1e-4  # electronMobility [m^2/(V·s)]
mu_h = 1e3 * (300 / T_range)**2.5 * 1e-4  # holeMobility [m^2/(V·s)]

sigma = two_band_conductivity(n_e, mu_e, n_h, mu_h)
R_H = two_band_hall_coefficient(n_e, mu_e, n_h, mu_h)

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

# Top left: Carrier density
axes[0, 0].semilogy(T_range, n_e / 1e6, linewidth=2.5, label='Electron density n$_e$', color='#f5576c')
axes[0, 0].semilogy(T_range, n_h / 1e6, linewidth=2.5, label='Hole density n$_h$', color='#ffa500')
axes[0, 0].set_xlabel('Temperature T [K]', fontsize=12)
axes[0, 0].set_ylabel('Carrier Density [cm$^{-3}$]', fontsize=12)
axes[0, 0].set_title('Carrier Densities (Two-Band Model)', fontsize=13, fontweight='bold')
axes[0, 0].legend(fontsize=11)
axes[0, 0].grid(alpha=0.3)

# Top right: Mobility
axes[0, 1].loglog(T_range, mu_e * 1e4, linewidth=2.5, label='Electron mobility μ$_e$', color='#f5576c')
axes[0, 1].loglog(T_range, mu_h * 1e4, linewidth=2.5, label='Hole mobility μ$_h$', color='#ffa500')
axes[0, 1].set_xlabel('Temperature T [K]', fontsize=12)
axes[0, 1].set_ylabel('Mobility [cm$^2$/(V·s)]', fontsize=12)
axes[0, 1].set_title('Mobilities (Temperature Dependence)', fontsize=13, fontweight='bold')
axes[0, 1].legend(fontsize=11)
axes[0, 1].grid(alpha=0.3, which='both')

# Bottom left: Electrical conductivity
axes[1, 0].semilogy(T_range, sigma, linewidth=2.5, color='#f093fb')
axes[1, 0].set_xlabel('Temperature T [K]', fontsize=12)
axes[1, 0].set_ylabel('Conductivity σ [S/m]', fontsize=12)
axes[1, 0].set_title('Electrical Conductivity', fontsize=13, fontweight='bold')
axes[1, 0].grid(alpha=0.3)

# Bottom right: Hall coefficient
axes[1, 1].plot(T_range, R_H, linewidth=2.5, color='#f093fb')
axes[1, 1].axhline(0, color='black', linestyle='--', linewidth=1.5)
axes[1, 1].set_xlabel('Temperature T [K]', fontsize=12)
axes[1, 1].set_ylabel('Hall Coefficient R$_H$ [m$^3$/C]', fontsize=12)
axes[1, 1].set_title('Hall Coefficient (Sign Change)', fontsize=13, fontweight='bold')
axes[1, 1].grid(alpha=0.3)

plt.tight_layout()
plt.show()

# Detect sign reversal temperature of Hall coefficient
sign_change_idx = np.where(np.diff(np.sign(R_H)))[0]
if len(sign_change_idx) > 0:
    T_sign_change = T_range[sign_change_idx[0]]
    print(f"\nHall coefficient sign reversal temperature: {T_sign_change:.1f} K")
    print("  → Low temperature: R_H < 0（electrondominant）")
    print("  → High temperature: R_H > 0（holedominant）")

2.5 Temperature-Dependent Hall Measurement

2.5.1 Analysis of Carrier Scattering Mechanisms

Dominant carrier scattering mechanisms can be identified from temperature dependence of mobility:

Matthiessen's Rule (mobility version):

Code example2-5: Temperature-Dependent Hall Measurement analysis

# 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 lmfit import Model

# Acoustic phonon scattering model
def acoustic_phonon_mobility(T, mu0, T0=300):
    """μ ∝ T^(-3/2)"""
    return mu0 * (T0 / T)**(3/2)

# Ionized impurity scattering model
def ionized_impurity_mobility(T, mu1, T0=300):
    """μ ∝ T^(3/2)"""
    return mu1 * (T / T0)**(3/2)

# Matthiessen's rule
def combined_mobility(T, mu0, mu1, T0=300):
    """1/μ_total = 1/μ_phonon + 1/μ_impurity"""
    mu_phonon = acoustic_phonon_mobility(T, mu0, T0)
    mu_impurity = ionized_impurity_mobility(T, mu1, T0)
    mu_total = 1 / (1/mu_phonon + 1/mu_impurity)
    return mu_total

# Generate simulation data
T_range = np.linspace(50, 400, 30)  # [K]
mu0_true = 8000  # Phonon scattering limited mobility (room temperature) [cm^2/(V·s)]
mu1_true = 2000  # Impurity scattering limited mobility (room temperature) [cm^2/(V·s)]

mu_data = combined_mobility(T_range, mu0_true, mu1_true)
mu_data_noise = mu_data * (1 + 0.05 * np.random.randn(len(T_range)))  # 5% noise

# Fitting
model = Model(combined_mobility)
params = model.make_params(mu0=5000, mu1=3000, T0=300)
params['T0'].vary = False  # T0 is fixed

result = model.fit(mu_data_noise, params, T=T_range)

print("Temperature-Dependent Hall Measurement fitting results:")
print(result.fit_report())

# 各Scattering Mechanism contributionを計算
mu_phonon_fit = acoustic_phonon_mobility(T_range, result.params['mu0'].value)
mu_impurity_fit = ionized_impurity_mobility(T_range, result.params['mu1'].value)

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

# Left: Mobility vs temperature
ax1.scatter(T_range, mu_data_noise, s=80, alpha=0.7, edgecolors='black', linewidths=1.5, label='Measured data', color='#f093fb')
ax1.plot(T_range, result.best_fit, linewidth=2.5, label='Fit (Matthiessen)', color='#f5576c')
ax1.plot(T_range, mu_phonon_fit, linewidth=2, linestyle='--', label='Phonon scattering (T$^{-3/2}$)', color='#ffa500')
ax1.plot(T_range, mu_impurity_fit, linewidth=2, linestyle=':', label='Impurity scattering (T$^{3/2}$)', color='#99ccff')
ax1.set_xlabel('Temperature T [K]', fontsize=12)
ax1.set_ylabel('Mobility μ [cm$^2$/(V·s)]', fontsize=12)
ax1.set_title('Temperature-Dependent Hall Mobility', fontsize=14, fontweight='bold')
ax1.legend(fontsize=10)
ax1.grid(alpha=0.3)
ax1.set_yscale('log')

# Right: Scattering rate (1/μ) vs temperature
ax2.scatter(T_range, 1/mu_data_noise, s=80, alpha=0.7, edgecolors='black', linewidths=1.5, label='Data (1/μ)', color='#f093fb')
ax2.plot(T_range, 1/mu_phonon_fit, linewidth=2.5, label='Phonon (1/μ$_{ph}$)', color='#ffa500')
ax2.plot(T_range, 1/mu_impurity_fit, linewidth=2.5, label='Impurity (1/μ$_{imp}$)', color='#99ccff')
ax2.plot(T_range, 1/result.best_fit, linewidth=2.5, label='Total (sum)', color='#f5576c', linestyle='--')
ax2.set_xlabel('Temperature T [K]', fontsize=12)
ax2.set_ylabel('Scattering Rate 1/μ [V·s/cm$^2$]', fontsize=12)
ax2.set_title('Matthiessen Rule: Scattering Rate Analysis', fontsize=14, fontweight='bold')
ax2.legend(fontsize=10)
ax2.grid(alpha=0.3)

plt.tight_layout()
plt.show()

# Interpretation of results
print(f"\nScattering Mechanism analysis:")
print(f"  Phonon scattering limited mobility (room temperature): {result.params['mu0'].value:.1f} cm²/(V·s)")
print(f"  Impurity scattering limited mobility (room temperature): {result.params['mu1'].value:.1f} cm²/(V·s)")
print(f"\nDominant scattering mechanisms:")
print(f"  Low temperature (< 150 K): Impurity scattering（μ ∝ T^(3/2)）")
print(f"  High temperature (> 250 K): Phonon scattering（μ ∝ T^(-3/2)）")

2.6 Complete Hall Data Processing Workflow

Code Example 2-6: Complete Analysis VH → n, μ

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - pandas>=2.0.0, <2.2.0

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

class HallDataProcessor:
    """Complete Hall data processing class"""

    def __init__(self, thickness):
        """
        Parameters
        ----------
        thickness : float
            Sample thickness [m]
        """
        self.t = thickness
        self.e = 1.60218e-19  # Elementary charge [C]
        self.data = {}

    def load_data(self, filename=None, mock_data=True):
        """
        Load measurement data

        Parameters
        ----------
        filename : str or None
            CSV filename（None caseはGenerate mock data）
        mock_data : bool
            Whether to generate mock data
        """
        if mock_data:
            # Generate mock data (temperature dependence)
            T = np.array([77, 100, 150, 200, 250, 300, 350, 400])  # [K]
            I = 100e-6  # Current [A]
            B = 0.5  # Magnetic field [T]

            # van der Pauw resistance (temperature dependent)
            R_AB_CD = 500 + 2.0 * T
            R_BC_DA = 480 + 1.8 * T

            # Hall voltage (temperature dependent, including carrier density change)
            V_pos = -3e-3 * (1 + 0.002 * (T - 300))
            V_neg = +2.9e-3 * (1 + 0.002 * (T - 300))

            self.data = pd.DataFrame({
                'T': T,
                'I': I,
                'B': B,
                'R_AB_CD': R_AB_CD,
                'R_BC_DA': R_BC_DA,
                'V_pos_B': V_pos,
                'V_neg_B': V_neg
            })
        else:
            # Load actual data from CSV
            self.data = pd.read_csv(filename)

        return self.data

    def calculate_sheet_resistance(self):
        """Calculate sheet resistance"""
        from scipy.optimize import fsolve

        def vdp_eq(Rs, R1, R2):
            return np.exp(-np.pi * R1 / Rs) + np.exp(-np.pi * R2 / Rs) - 1

        R_s_list = []
        for _, row in self.data.iterrows():
            R1, R2 = row['R_AB_CD'], row['R_BC_DA']
            R_initial = (R1 + R2) / 2 * np.pi / np.log(2)
            R_s = fsolve(vdp_eq, R_initial, args=(R1, R2))[0]
            R_s_list.append(R_s)

        self.data['R_s'] = R_s_list
        self.data['sigma'] = 1 / (np.array(R_s_list) * self.t)
        self.data['rho'] = 1 / self.data['sigma']

    def calculate_hall_properties(self):
        """Calculate Hall properties"""
        # Hall voltage (offset removed)
        self.data['V_H'] = 0.5 * (self.data['V_pos_B'] - self.data['V_neg_B'])

        # Hall coefficient
        self.data['R_H'] = (self.data['V_H'] * self.t) / (self.data['I'] * self.data['B'])

        # Carrier density
        self.data['n'] = 1 / (np.abs(self.data['R_H']) * self.e)

        # Mobility
        self.data['mu'] = self.data['sigma'] * np.abs(self.data['R_H'])

        # Carrier type
        self.data['carrier_type'] = ['electron' if rh < 0 else 'hole' for rh in self.data['R_H']]

    def plot_results(self):
        """Visualize results"""
        fig, axes = plt.subplots(2, 3, figsize=(18, 10))

        T = self.data['T']

        # Sheet resistance
        axes[0, 0].plot(T, self.data['R_s'], 'o-', linewidth=2.5, markersize=8, color='#f093fb')
        axes[0, 0].set_xlabel('Temperature [K]', fontsize=11)
        axes[0, 0].set_ylabel('Sheet Resistance [Ω/sq]', fontsize=11)
        axes[0, 0].set_title('Sheet Resistance', fontsize=12, fontweight='bold')
        axes[0, 0].grid(alpha=0.3)

        # Electrical conductivity
        axes[0, 1].semilogy(T, self.data['sigma'], 'o-', linewidth=2.5, markersize=8, color='#f5576c')
        axes[0, 1].set_xlabel('Temperature [K]', fontsize=11)
        axes[0, 1].set_ylabel('Conductivity [S/m]', fontsize=11)
        axes[0, 1].set_title('Electrical Conductivity', fontsize=12, fontweight='bold')
        axes[0, 1].grid(alpha=0.3)

        # Hall voltage
        axes[0, 2].plot(T, self.data['V_H'] * 1e3, 'o-', linewidth=2.5, markersize=8, color='#ffa500')
        axes[0, 2].axhline(0, color='black', linestyle='--', linewidth=1.5)
        axes[0, 2].set_xlabel('Temperature [K]', fontsize=11)
        axes[0, 2].set_ylabel('Hall Voltage [mV]', fontsize=11)
        axes[0, 2].set_title('Hall Voltage', fontsize=12, fontweight='bold')
        axes[0, 2].grid(alpha=0.3)

        # Hall coefficient
        axes[1, 0].plot(T, self.data['R_H'], 'o-', linewidth=2.5, markersize=8, color='#99ccff')
        axes[1, 0].axhline(0, color='black', linestyle='--', linewidth=1.5)
        axes[1, 0].set_xlabel('Temperature [K]', fontsize=11)
        axes[1, 0].set_ylabel('Hall Coefficient [m³/C]', fontsize=11)
        axes[1, 0].set_title('Hall Coefficient', fontsize=12, fontweight='bold')
        axes[1, 0].grid(alpha=0.3)

        # Carrier density
        axes[1, 1].semilogy(T, self.data['n'] / 1e6, 'o-', linewidth=2.5, markersize=8, color='#99ff99')
        axes[1, 1].set_xlabel('Temperature [K]', fontsize=11)
        axes[1, 1].set_ylabel('Carrier Density [cm$^{-3}$]', fontsize=11)
        axes[1, 1].set_title('Carrier Density', fontsize=12, fontweight='bold')
        axes[1, 1].grid(alpha=0.3)

        # Mobility
        axes[1, 2].semilogy(T, self.data['mu'] * 1e4, 'o-', linewidth=2.5, markersize=8, color='#ff9999')
        axes[1, 2].set_xlabel('Temperature [K]', fontsize=11)
        axes[1, 2].set_ylabel('Mobility [cm$^2$/(V·s)]', fontsize=11)
        axes[1, 2].set_title('Hall Mobility', fontsize=12, fontweight='bold')
        axes[1, 2].grid(alpha=0.3)

        plt.tight_layout()
        plt.show()

    def save_results(self, filename='hall_results.csv'):
        """Save results to CSV"""
        self.data.to_csv(filename, index=False)
        print(f"Results saved to {filename}")

# Usage example
processor = HallDataProcessor(thickness=200e-9)  # 200 nm
processor.load_data(mock_data=True)
processor.calculate_sheet_resistance()
processor.calculate_hall_properties()

print("Hall measurement data processing results:")
print(processor.data[['T', 'R_s', 'sigma', 'n', 'mu']].to_string(index=False))

processor.plot_results()
processor.save_results('hall_analysis_output.csv')

Code Example 2-7: Uncertainty Analysis of Hall Measurement

# 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_measurement_uncertainty(V_H, delta_V_H, I, delta_I, B, delta_B, t, delta_t):
    """
    Uncertainty propagation for Hall measurement

    Parameters
    ----------
    V_H, delta_V_H : float
        Hall voltageand its uncertainty [V]
    I, delta_I : float
        currentand its uncertainty [A]
    B, delta_B : float
        magnetic fieldand its uncertainty [T]
    t, delta_t : float
        thicknessand its uncertainty [m]

    Returns
    -------
    R_H, delta_R_H : float
        Hall coefficientand its uncertainty
    n, delta_n : float
        Carrier densityand its uncertainty
    """
    e = 1.60218e-19

    # Hall coefficient: R_H = V_H * t / (I * B)
    R_H = V_H * t / (I * B)

    # 不確かさ伝播（partial derivatives）
    # δR_H/R_H = sqrt((δV_H/V_H)^2 + (δt/t)^2 + (δI/I)^2 + (δB/B)^2)
    rel_unc_V_H = delta_V_H / np.abs(V_H)
    rel_unc_t = delta_t / t
    rel_unc_I = delta_I / I
    rel_unc_B = delta_B / B

    rel_unc_R_H = np.sqrt(rel_unc_V_H**2 + rel_unc_t**2 + rel_unc_I**2 + rel_unc_B**2)
    delta_R_H = np.abs(R_H) * rel_unc_R_H

    # Carrier density: n = 1 / (e * R_H)
    n = 1 / (e * np.abs(R_H))

    # δn/n = δR_H/R_H
    delta_n = n * rel_unc_R_H

    return R_H, delta_R_H, n, delta_n, rel_unc_R_H

# measurementexample
V_H = -5.0e-3  # [V]
delta_V_H = 0.1e-3  # voltage計 precision [V]
I = 100e-6  # [A]
delta_I = 0.5e-6  # current source precision [A]
B = 0.5  # [T]
delta_B = 0.01  # Magnetic field precision [T]
t = 200e-9  # [m]
delta_t = 5e-9  # thickness measurement precision [m]

R_H, delta_R_H, n, delta_n, rel_unc = hall_measurement_uncertainty(
    V_H, delta_V_H, I, delta_I, B, delta_B, t, delta_t
)

print("Uncertainty analysis of Hall measurement:")
print("=" * 60)
print("Measured values:")
print(f"  Hall voltage: ({V_H * 1e3:.2f} ± {delta_V_H * 1e3:.2f}) mV")
print(f"  current: ({I * 1e6:.1f} ± {delta_I * 1e6:.2f}) μA")
print(f"  magnetic field: ({B:.2f} ± {delta_B:.3f}) T")
print(f"  thickness: ({t * 1e9:.1f} ± {delta_t * 1e9:.1f}) nm")
print("\nResults:")
print(f"  Hall coefficient: ({R_H:.3e} ± {delta_R_H:.3e}) m³/C")
print(f"  Relative uncertainty: {rel_unc * 100:.2f}%")
print(f"  Carrier density: ({n:.3e} ± {delta_n:.3e}) m⁻³")
print(f"                = ({n / 1e6:.3e} ± {delta_n / 1e6:.3e}) cm⁻³")

# Visualize uncertainty contributions
contributions = {
    'V_H': (delta_V_H / np.abs(V_H))**2,
    't': (delta_t / t)**2,
    'I': (delta_I / I)**2,
    'B': (delta_B / B)**2
}

fig, ax = plt.subplots(figsize=(10, 6))

labels = list(contributions.keys())
values = [np.sqrt(v) * 100 for v in contributions.values()]

bars = ax.bar(labels, values, color=['#f093fb', '#f5576c', '#ffa500', '#99ccff'], edgecolor='black', linewidth=1.5)
ax.set_ylabel('Relative Uncertainty Contribution [%]', fontsize=12)
ax.set_title('Uncertainty Budget for Hall Measurement', fontsize=14, fontweight='bold')
ax.grid(alpha=0.3, axis='y')

# Display values above bars
for bar, val in zip(bars, values):
    height = bar.get_height()
    ax.text(bar.get_x() + bar.get_width()/2., height,
            f'{val:.2f}%', ha='center', va='bottom', fontsize=11, fontweight='bold')

# Add total uncertainty
ax.axhline(rel_unc * 100, color='red', linestyle='--', linewidth=2, label=f'Total: {rel_unc * 100:.2f}%')
ax.legend(fontsize=11)

plt.tight_layout()
plt.show()

2.7 Exercises

Exercise 2-1: Calculation of Hall Voltage (Easy)

Easy Problem:Carrier density $n = 1 \times 10^{22}$ m$^{-3}$, thickness $t = 100$ nm, current $I = 1$ mA, magnetic field $B = 0.5$ T , when, Hall voltage $V_H$ 。

e = 1.60218e-19  # [C]
n = 1e22  # [m^-3]
t = 100e-9  # [m]
I = 1e-3  # [A]
B = 0.5  # [T]

V_H = I * B / (n * e * t)
print(f"Hall voltage V_H = {V_H:.3e} V = {V_H * 1e3:.2f} mV")

Exercise 2-2: Calculation of Carrier Density (Easy)

Easy Problem:Hall coefficient $R_H = -1.5 \times 10^{-3}$ m$^3$/C was measured。Carrier typeとCarrier density。

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

"""
Example: EasyProblem:Hall coefficient $R_H = -1.5 \times 10^{-3}$ m$^

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

e = 1.60218e-19  # [C]
R_H = -1.5e-3  # [m^3/C]

carrier_type = 'electron' if R_H < 0 else 'hole'
n = 1 / (np.abs(R_H) * e)

print(f"Carrier type: {carrier_type}")
print(f"Carrier density n = {n:.3e} m⁻³ = {n / 1e6:.3e} cm⁻³")

Exercise 2-3: Calculation of Mobility (Easy)

Easy Problem:Electrical conductivity $\sigma = 1 \times 10^4$ S/m, Hall coefficient $R_H = -2 \times 10^{-3}$ m$^3$/C , when, Mobility $\mu$ 。

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

"""
Example: EasyProblem:Electrical conductivity $\sigma = 1 \times 10^4$

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

sigma = 1e4  # [S/m]
R_H = -2e-3  # [m^3/C]

mu = sigma * np.abs(R_H)
print(f"Mobility μ = {mu:.2f} m²/(V·s) = {mu * 1e4:.1f} cm²/(V·s)")

Exercise 2-4: Analysis of van der Pauw Hall Configuration (Medium)

Medium Problem:van der Pauwmeasurementで, $R_{\text{AB,CD}} = 950$ Ω, $R_{\text{BC,DA}} = 1050$ Ω, $V_{24}^{+B} = -4.5$ mV, $V_{24}^{-B} = +4.3$ mV, $I = 100$ μA, $B = 0.5$ T, $t = 300$ nm was obtained。$\sigma$, $R_H$, $n$, $\mu$ 。

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

"""
Example: MediumProblem:van der Pauwmeasurementで, $R_{\text{AB,CD}} = 

Purpose: Demonstrate optimization techniques
Target: Intermediate
Execution time: 10-30 seconds
Dependencies: None
"""

import numpy as np
from scipy.optimize import fsolve

def van_der_pauw_Rs(R1, R2):
    def eq(Rs):
        return np.exp(-np.pi * R1 / Rs) + np.exp(-np.pi * R2 / Rs) - 1
    R_init = (R1 + R2) / 2 * np.pi / np.log(2)
    return fsolve(eq, R_init)[0]

# Parameters
R1, R2 = 950, 1050  # [Ω]
V_pos, V_neg = -4.5e-3, 4.3e-3  # [V]
I = 100e-6  # [A]
B = 0.5  # [T]
t = 300e-9  # [m]
e = 1.60218e-19  # [C]

# Sheet resistance
R_s = van_der_pauw_Rs(R1, R2)
sigma = 1 / (R_s * t)

# Hall analysis
V_H = 0.5 * (V_pos - V_neg)
R_H = V_H * t / (I * B)
n = 1 / (np.abs(R_H) * e)
mu = sigma * np.abs(R_H)

print(f"Sheet resistance R_s = {R_s:.2f} Ω/sq")
print(f"Electrical conductivity σ = {sigma:.2e} S/m")
print(f"Hall coefficient R_H = {R_H:.3e} m³/C")
print(f"Carrier density n = {n:.2e} m⁻³ = {n / 1e6:.2e} cm⁻³")
print(f"Mobility μ = {mu:.2e} m²/(V·s) = {mu * 1e4:.1f} cm²/(V·s)")

Exercise 2-5: Analysis of Two-Band Model (Medium)

Medium Problem:$n_e = 1 \times 10^{22}$ m$^{-3}$, $\mu_e = 0.5$ m$^2$/(V·s), $n_h = 5 \times 10^{21}$ m$^{-3}$, $\mu_h = 0.05$ m$^2$/(V·s) , when, Electrical conductivity $\sigma$ とHall Coefficient $R_H$ 。

e = 1.60218e-19  # [C]
n_e = 1e22  # [m^-3]
mu_e = 0.5  # [m^2/(V·s)]
n_h = 5e21  # [m^-3]
mu_h = 0.05  # [m^2/(V·s)]

# Electrical conductivity
sigma = n_e * e * mu_e + n_h * e * mu_h
print(f"Electrical conductivity σ = {sigma:.2e} S/m")

# Hall coefficient
numerator = n_h * mu_h**2 - n_e * mu_e**2
denominator = (n_h * mu_h + n_e * mu_e)**2
R_H = numerator / (e * denominator)
print(f"Hall coefficient R_H = {R_H:.3e} m³/C")

# 見かけのCarrier density
n_apparent = 1 / (abs(R_H) * e)
print(f"見かけのCarrier density: {n_apparent:.2e} m⁻³ = {n_apparent / 1e6:.2e} cm⁻³")
print(f"  （trueelectrondensity {n_e / 1e6:.2e} cm⁻³different from）")

Exercise 2-6: Temperature Dependence Fitting (Medium)

Medium Problem:Mobilityが T = 100 K で 5000 cm$^2$/(V·s), 300 K で 1500 cm$^2$/(V·s) であった。$\mu \propto T^{-\alpha}$ モデルで指数 $\alpha$ 。

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

"""
Example: MediumProblem:Mobilityが T = 100 K で 5000 cm$^2$/(V·s), 300 K

Purpose: Demonstrate data visualization techniques
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

T1, mu1 = 100, 5000  # [K], [cm^2/(V·s)]
T2, mu2 = 300, 1500  # [K], [cm^2/(V·s)]

# μ ∝ T^(-α) より, log(μ) = -α log(T) + const
# log(mu1/mu2) = -α log(T1/T2)
# α = -log(mu1/mu2) / log(T1/T2)

alpha = -np.log(mu1 / mu2) / np.log(T1 / T2)
print(f"指数 α = {alpha:.2f}")
print(f"モデル: μ ∝ T^(-{alpha:.2f})")
print(f"\nInterpretation: α ≈ 1.5 → Acoustic phonon scattering is dominant")

Exercise 2-7: Analysis of Magnetic Field Dependence (Hard)

Hard Problem:Hall voltageがmagnetic field B = 0, 0.2, 0.4, 0.6, 0.8, 1.0 T で V$_H$ = 0, 1.0, 2.1, 3.0, 4.1, 5.0 mV とmeasurementされた（current I = 100 μA, thickness t = 200 nm）。線形Fittingで Hall coefficientを求め, 非線形性を評価せよ。

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

"""
Example: HardProblem:Hall voltageがmagnetic field B = 0, 0.2, 0.4, 0.6

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import linregress

B = np.array([0, 0.2, 0.4, 0.6, 0.8, 1.0])  # [T]
V_H = np.array([0, 1.0, 2.1, 3.0, 4.1, 5.0]) * 1e-3  # [V]
I = 100e-6  # [A]
t = 200e-9  # [m]

# 線形Fitting
slope, intercept, r_value, _, std_err = linregress(B, V_H)

# Hall coefficient
R_H = slope * t / I
print(f"線形Fitting: V_H = {slope * 1e3:.3f} mV/T × B + {intercept * 1e3:.3f} mV")
print(f"Hall coefficient R_H = {R_H:.3e} m³/C")
print(f"決定係数 R² = {r_value**2:.4f}")

# 非線形性評価
V_H_fit = slope * B + intercept
residuals = V_H - V_H_fit
rel_residuals = residuals / V_H_fit[1:] * 100  # 最初の点（B=0）を除く

print(f"\n非線形性評価:")
print(f"  最大残差: {np.max(np.abs(residuals[1:])) * 1e6:.2f} μV")
print(f"  平均相対残差: {np.mean(np.abs(rel_residuals)):.2f}%")

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

ax1.scatter(B, V_H * 1e3, s=100, edgecolors='black', linewidths=2, label='Measured', color='#f093fb', zorder=5)
ax1.plot(B, V_H_fit * 1e3, linewidth=2.5, label=f'Fit: {slope * 1e3:.3f} mV/T', color='#f5576c', linestyle='--')
ax1.set_xlabel('Magnetic Field B [T]', fontsize=12)
ax1.set_ylabel('Hall Voltage V$_H$ [mV]', fontsize=12)
ax1.set_title('Hall Voltage vs Magnetic Field', fontsize=14, fontweight='bold')
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3)

ax2.scatter(B[1:], residuals[1:] * 1e6, s=100, edgecolors='black', linewidths=2, color='#ffa500')
ax2.axhline(0, color='black', linestyle='--', linewidth=1.5)
ax2.set_xlabel('Magnetic Field B [T]', fontsize=12)
ax2.set_ylabel('Residuals [μV]', fontsize=12)
ax2.set_title('Fit Residuals', fontsize=14, fontweight='bold')
ax2.grid(alpha=0.3)

plt.tight_layout()
plt.show()

Exercise 2-8: Uncertainty Propagation (Hard)

Hard Problem:Hall voltage $V_H = (5.0 \pm 0.2)$ mV, current $I = (100 \pm 1)$ μA, magnetic field $B = (0.50 \pm 0.02)$ T, thickness $t = (200 \pm 10)$ nm , when, Carrier density $n$ の不確かさ $\Delta n$ 。どのParametersが最も影響するか評価せよ。

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

"""
Example: HardProblem:Hall voltage $V_H = (5.0 \pm 0.2)$ mV, current $

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: ~5 seconds
Dependencies: None
"""

import numpy as np

V_H, dV_H = 5.0e-3, 0.2e-3  # [V]
I, dI = 100e-6, 1e-6  # [A]
B, dB = 0.50, 0.02  # [T]
t, dt = 200e-9, 10e-9  # [m]
e = 1.60218e-19  # [C]

# n = 1 / (e * R_H) = I * B / (e * V_H * t)
n = I * B / (e * V_H * t)

# Relative uncertainty contribution
rel_V_H = (dV_H / V_H)**2
rel_I = (dI / I)**2
rel_B = (dB / B)**2
rel_t = (dt / t)**2

rel_unc_total = np.sqrt(rel_V_H + rel_I + rel_B + rel_t)
dn = n * rel_unc_total

print(f"Carrier density: n = {n:.3e} m⁻³ = {n / 1e6:.3e} cm⁻³")
print(f"不確かさ: Δn = {dn:.3e} m⁻³ = {dn / 1e6:.3e} cm⁻³")
print(f"相対不確かさ: {rel_unc_total * 100:.2f}%")
print(f"\nUncertainty contributions:")
print(f"  V_H: {np.sqrt(rel_V_H) * 100:.2f}%")
print(f"  I:   {np.sqrt(rel_I) * 100:.2f}%")
print(f"  B:   {np.sqrt(rel_B) * 100:.2f}%")
print(f"  t:   {np.sqrt(rel_t) * 100:.2f}%")
print(f"\nConclusion: Uncertainty in thickness t has the largest impact ({np.sqrt(rel_t) * 100:.1f}%)")

Exercise 2-9: Experimental Planning (Hard)

Hard Problem:未知の薄膜半導体（thickness 100 nm）のCarrier type, density, Mobilityを決定する実験計画を立案せよ。必要なmeasurement, temperaturerange, magnetic fieldrange, データ解析手法を具体的に説明せよ。

2.8 Learning Check

Check your understanding with the following checklist:

Basic Understanding

• Can derive Hall voltage equation from Lorentz force
• Understand the relationship between Hall coefficient $R_H = 1/(ne)$ and carrier density
• Can explain the physical meaning of mobility $\mu = \sigma R_H$
• Understand the measurement procedure of van der Pauw Hall configuration
• Can explain the purpose of reversed magnetic field measurement (offset removal)

Practical Skills

• Hall voltageからCalculate carrier densityできる
• Can completely analyze van der Pauw Hall measurement data ($\sigma$, $R_H$, $n$, $\mu$)
• PythonでComplete Hall Data Processing Workflowcan implement
• Can calculate uncertainty propagation and evaluate measurement accuracy
• temperature依存性データからScattering Mechanismcan estimate

Application Skills

• Can analyze multi-carrier systems with two-band model
• Temperature Dependence of MobilityからScattering Mechanismcan identify
• Can evaluate measurement nonlinearity and estimate causes
• Can plan experiments and determine appropriate measurement conditions

2.9 References

1. Hall, E. H. (1879). On a New Action of the Magnet on Electric Currents. American Journal of Mathematics, 2(3), 287-292. - Hall effect discovery paper
2. van der Pauw, L. J. (1958). A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape. Philips Technical Review, 20(8), 220-224. - van der Pauw Hallmeasurement法の原論文
3. Look, D. C. (1989). Electrical Characterization of GaAs Materials and Devices. Wiley. - 半導体Hallmeasurementの実践的テキスト
4. Putley, E. H. (1960). The Hall Effect and Related Phenomena. Butterworths. - Comprehensive explanation of Hall effect
5. Ashcroft, N. W., & Mermin, N. D. (1976). Solid State Physics (Chapter 2: The Sommerfeld Theory of Metals). Holt, Rinehart and Winston. - Drude model and carrier transport theory
6. Schroder, D. K. (2006). Semiconductor Material and Device Characterization (3rd ed., Chapter 2: Resistivity). Wiley-Interscience. - Hallmeasurement技術の詳細
7. Popović, R. S. (2004). Hall Effect Devices (2nd ed.). Institute of Physics Publishing. - Hall効果デバイスとmeasurement技術

2.10 To Next Chapter

In the next chapter, you will learn the principles and practice of magnetic measurements. You will master magnetization measurements using VSM (Vibrating Sample Magnetometer) and SQUID (Superconducting Quantum Interference Device), M-H curve analysis, magnetic anisotropy evaluation, and integrated measurement techniques using PPMS (Physical Property Measurement System).