Chapter 1: Fundamentals of Solid-State Electronic Theory

Introduction to Band Theory - Why Do Metals Conduct Electricity While Insulators Do Not?

📖 Reading Time: 30-35 minutes 📊 Difficulty: Intermediate 💻 Code Examples: 0 📝 Exercises: 0

AI Terakoya Top›Materials Science›Materials Properties›Chapter 1

🌐 EN | 🇯🇵 JP | Last sync: 2025-11-16

Learning Objectives

Upon completing this chapter, you will be able to explain:

Fundamental Understanding

Practical Skills

Application Capabilities

1.1 Why Do We Need Band Theory?

1.1.1 Starting with Everyday Questions

Why do copper wires conduct electricity while rubber does not? Both are composed of atoms, so where does this difference originate?

To answer this simple question, physicists established “band theory” in the early 20th century. This theory is one of the most important foundational theories in materials science and can explain:

💡 Historical Background

The foundations of band theory were laid in the 1920s by Felix Bloch and Rudolf Peierls. In the 1930s, Wilson, Brillouin, Wannier, and others developed it further, establishing a universal framework for describing the electronic structure of solids. This theory became the theoretical foundation supporting the semiconductor revolution (invention of the transistor, 1947).

1.1.2 Limitations of Classical Theory

In classical physics (19th century), the Drude model (1900) was proposed, treating electrons as freely moving particles. While this model could explain conductivity to some extent, it had the following critical problems:

PhenomenonDrude Model PredictionExperimental Result
Electronic contribution to specific heatLarge (3/2 Nk_B)Very small (~0.01 Nk_B)
Existence of insulatorsCannot explainClearly exists
Temperature dependence of semiconductorsSame as metalsCompletely different (exponential)

It became clear that to resolve these contradictions, we need to consider quantum mechanics and the periodic structure of crystals.

1.2 Free Electron Model - The Simplest Quantum Description

1.2.1 Quantum Mechanical Basics: Particles Are Waves

According to the fundamental principle of quantum mechanics (de Broglie, 1924), all particles possess wave-like properties:

λ = h / p
(wavelength = Planck’s constant / momentum)

For light particles like electrons, this wavelength becomes comparable to interatomic distances (~several Å), so wave-like properties become prominent.

1.2.2 Electron in a Box: Quantized Energy

Consider an electron confined in a one-dimensional “box” (length L). Solving the Schrödinger equation, energy can only take discrete values:

E_n = (n²π²ℏ²) / (2m L²) (n = 1, 2, 3, …)

Key points:

1.2.3 Extension to Three Dimensions: Actual Metals

Considering a real metal as a box with volume V = L³, electron states are specified by three quantum numbers (n_x, n_y, n_z):

E = (π²ℏ²/2mL²) × (n_x² + n_y² + n_z²)

This equation means states with energy E are distributed on a spherical surface in 3D n-space with radius R = √(n_x² + n_y² + n_z²).

1.2.4 Fermi Energy - The Water Level of the Electron Sea

Electrons in metals are fermions (obeying Pauli’s exclusion principle), so at most 2 electrons (spin ↑↓) can occupy one quantum state.

When N electrons fill from the lowest energy state upward, the energy of the highest energy electron at absolute zero (T=0K) is called the Fermi energy E_F :

E_F = (ℏ²/2m) × (3π²n)^(2/3)

Here, n = N/V is the electron density.

✅ Important Numerical Example

Copper (Cu) :

Room temperature (300K) is only 0.4% of T_F. This is why the electronic specific heat is small. Only a small fraction (~T/T_F) of electrons are thermally excited.

1.2.5 Density of States (DOS)

The function indicating how many quantum states exist in the energy range from E to E+dE is called the density of states D(E). In the free electron model:

D(E) = (V/2π²) × (2m/ℏ²)^(3/2) × √E

Key characteristics:

💡 Pro Tip

The density of states is extremely important for understanding material properties. Many physical quantities such as electrical conductivity, specific heat, and magnetic susceptibility are proportional to D(E_F) (density of states at the Fermi energy).

1.3 Let’s Calculate with Python - Fermi Energy and Density of States

Example 1: Basic Calculation - Fermi Energy

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
# - scipy>=1.11.0

import numpy as np
import matplotlib.pyplot as plt
from scipy import constants

# Physical constants
hbar = constants.hbar  # Reduced Planck constant (J·s)
m_e = constants.m_e    # Electron mass (kg)
e = constants.e        # Elementary charge (C)

def fermi_energy(n):
    """
    Calculate Fermi energy in the free electron model

    Args:
        n (float): Electron density [m^-3]

    Returns:
        float: Fermi energy [eV]
    """
    E_F = (hbar**2 / (2 * m_e)) * (3 * np.pi**2 * n)**(2/3)
    return E_F / e  # Convert J to eV

# Typical metal electron densities
metals = {
    'Li': 4.70e28,   # Lithium
    'Na': 2.65e28,   # Sodium
    'Cu': 8.45e28,   # Copper
    'Ag': 5.85e28,   # Silver
    'Au': 5.90e28    # Gold
}

print("Fermi Energy of Metals")
print("-" * 40)
for metal, n in metals.items():
    E_F = fermi_energy(n)
    T_F = E_F * e / constants.k  # Fermi temperature [K]
    print(f"{metal:3s}: E_F = {E_F:5.2f} eV, T_F = {T_F/1000:5.1f} × 10³ K")

# Output example:
# Fermi Energy of Metals
# ----------------------------------------
# Li : E_F =  4.74 eV, T_F =  55.0 × 10³ K
# Na : E_F =  3.24 eV, T_F =  37.6 × 10³ K
# Cu : E_F =  7.00 eV, T_F =  81.2 × 10³ K
# Ag : E_F =  5.49 eV, T_F =  63.7 × 10³ K
# Au : E_F =  5.53 eV, T_F =  64.2 × 10³ K

Example 2: Visualizing Density of States

def density_of_states(E, V):
    """
    3D free electron density of states

    Args:
        E (array): Energy [J]
        V (float): Volume [m^3]

    Returns:
        array: Density of states [J^-1 m^-3]
    """
    factor = V / (2 * np.pi**2) * (2 * m_e / hbar**2)**(3/2)
    DOS = factor * np.sqrt(E)
    return DOS

# Copper parameters
n_Cu = 8.45e28  # m^-3
E_F_Cu = fermi_energy(n_Cu) * e  # J
V = 1e-6  # 1 mm³

# Energy range (0 to 2E_F)
E = np.linspace(0.01 * E_F_Cu, 2 * E_F_Cu, 1000)
DOS = density_of_states(E, V)

# Plot
plt.figure(figsize=(10, 6))
plt.plot(E / e, DOS * e, 'b-', linewidth=2, label='D(E)')
plt.axvline(E_F_Cu / e, color='r', linestyle='--', linewidth=2, label=f'E_F = {E_F_Cu/e:.2f} eV')
plt.fill_between(E / e, 0, DOS * e, where=(E <= E_F_Cu), alpha=0.3, color='blue', label='Occupied states (T=0K)')

plt.xlabel('Energy (eV)', fontsize=12)
plt.ylabel('Density of States (eV⁻¹ mm⁻³)', fontsize=12)
plt.title('3D Free Electron Model: Density of States', fontsize=14, fontweight='bold')
plt.legend(fontsize=11)
plt.grid(alpha=0.3)
plt.tight_layout()
plt.show()

# Verify total electron count at absolute zero
occupied_DOS = DOS[E <= E_F_Cu]
dE = E[1] - E[0]
N_electrons = 2 * np.sum(occupied_DOS) * dE  # Factor of 2 for spin
print(f"Calculated electron count: {N_electrons:.2e}")
print(f"Expected value (n×V): {n_Cu * V:.2e}")
# These should match

Example 3: Fermi-Dirac Distribution and Temperature Effects

def fermi_dirac(E, E_F, T):
    """
    Fermi-Dirac distribution function

    Args:
        E (array): Energy [J]
        E_F (float): Fermi energy [J]
        T (float): Temperature [K]

    Returns:
        array: Occupation probability [0, 1]
    """
    if T == 0:
        return (E <= E_F).astype(float)
    else:
        k_B = constants.k
        return 1 / (1 + np.exp((E - E_F) / (k_B * T)))

# For copper
E = np.linspace(0, 2 * E_F_Cu, 1000)
temperatures = [0, 300, 1000, 3000, 10000]  # K

plt.figure(figsize=(12, 5))

# (a) Fermi-Dirac distribution
plt.subplot(1, 2, 1)
for T in temperatures:
    f = fermi_dirac(E, E_F_Cu, T)
    label = f'T = {T} K' if T > 0 else 'T = 0 K'
    plt.plot(E / e, f, linewidth=2, label=label)

plt.xlabel('Energy (eV)', fontsize=12)
plt.ylabel('Occupation Probability f(E)', fontsize=12)
plt.title('Fermi-Dirac Distribution', fontsize=14, fontweight='bold')
plt.axvline(E_F_Cu / e, color='k', linestyle='--', alpha=0.5)
plt.legend(fontsize=10)
plt.grid(alpha=0.3)

# (b) Effectively excited electrons (changing portion at T > 0)
plt.subplot(1, 2, 2)
E_range = np.linspace(E_F_Cu - 0.5*e, E_F_Cu + 0.5*e, 500)
for T in [300, 1000, 3000]:
    f = fermi_dirac(E_range, E_F_Cu, T)
    thermal_width = 2 * constants.k * T / e  # Energy width ~k_B T
    plt.plot(E_range / e, f, linewidth=2, label=f'T = {T} K (Δ ≈ {thermal_width*1000:.1f} meV)')

plt.xlabel('Energy (eV)', fontsize=12)
plt.ylabel('Occupation Probability f(E)', fontsize=12)
plt.title('Thermal Broadening near E_F', fontsize=14, fontweight='bold')
plt.axvline(E_F_Cu / e, color='k', linestyle='--', alpha=0.5)
plt.axhline(0.5, color='gray', linestyle=':', alpha=0.5)
plt.legend(fontsize=10)
plt.grid(alpha=0.3)

plt.tight_layout()
plt.show()

# Fraction of excited electrons at room temperature
T_room = 300  # K
thermal_energy = constants.k * T_room
fraction = T_room / (E_F_Cu / constants.k)
print(f"\nAnalysis at room temperature ({T_room} K):")
print(f"Thermal energy k_B T ≈ {thermal_energy/e*1000:.1f} meV")
print(f"Fermi energy E_F = {E_F_Cu/e:.2f} eV")
print(f"Fraction of excited electrons: ~{fraction*100:.2f}%")

⚠️ Important Note

While the free electron model succeeds in many qualitative predictions, it cannot explain:

To understand these, we need to consider the periodic potential of the crystal.

1.4 Band Structure Formation - Effects of Periodic Potential

1.4.1 Why Do Bands Form?

Electrons in crystals move in a periodic potential created by regularly arranged atomic nuclei. This periodicity causes dramatic changes in the electronic energy structure.

💡 Intuitive Understanding

In isolated atoms, electrons have discrete energy levels (1s, 2s, 2p, …). When N atoms come together to form a crystal:

  1. Each level splits into N levels (due to interaction)
  2. Since N ~ 10²³ is very large, levels form a practically continuous “band”
  3. However, band gaps remain between bands originating from different atomic orbitals

1.4.2 Bloch’s Theorem

The solution to Schrödinger’s equation in a periodic potential V(r + R) = V(r) takes a special form called a Bloch function :

ψ_nk(r) = e^(ik·r) u_nk(r)

Where:

Energy depends on k and n: E = E_n(k)

1.4.3 Brillouin Zone and Dispersion Relation

In a simple one-dimensional example (lattice constant a), the wave vector k is physically independent in the first Brillouin zone -π/a ≤ k ≤ π/a.

✅ Important Concept

What happens at the Brillouin zone boundary (k = ±π/a)?

For free electrons, E ∝ k² varies smoothly, but with periodic potential:

1.4.4 Simple Model: Band Structure of 1D Crystal

In the simplest “nearly free electron approximation,” treating the periodic potential as a weak perturbation, the dispersion relation changes as follows:

E_±(k) = (ℏ²k²)/(2m) ± |V_G| × √(1 + (ℏ²k·G)/(m|V_G|)²)

Where V_G is the Fourier component of the periodic potential and G is the reciprocal lattice vector.

Example 4: Plotting 1D Band Structure

def band_structure_1d(k, a, V_G):
    """
    Band structure of 1D nearly free electron model

    Args:
        k (array): Wave vector [m^-1]
        a (float): Lattice constant [m]
        V_G (float): Periodic potential [J]

    Returns:
        tuple: (E_lower, E_upper) Lower and upper bands [J]
    """
    E_free = (hbar * k)**2 / (2 * m_e)  # Free electron

    # Gap at Brillouin zone boundary
    G = 2 * np.pi / a
    gap = 2 * V_G

    # Band structure with perturbation (simplified)
    cos_term = np.cos(k * a)
    E_lower = E_free - V_G * np.abs(cos_term)
    E_upper = E_free + V_G * np.abs(cos_term)

    return E_lower, E_upper

# Parameters
a = 3e-10  # 3 Å lattice constant
V_G = 2 * e  # 2 eV potential

# First Brillouin zone
k = np.linspace(-np.pi/a, np.pi/a, 500)
E_lower, E_upper = band_structure_1d(k, a, V_G)

# Free electron (for comparison)
E_free = (hbar * k)**2 / (2 * m_e)

# Plot
plt.figure(figsize=(10, 7))
plt.plot(k * a / np.pi, E_free / e, 'k--', linewidth=1.5, alpha=0.5, label='Free electron')
plt.plot(k * a / np.pi, E_lower / e, 'b-', linewidth=2.5, label='Lower band')
plt.plot(k * a / np.pi, E_upper / e, 'r-', linewidth=2.5, label='Upper band')

# Highlight band gap region
k_gap = np.pi / a
E_gap_lower = band_structure_1d(np.array([k_gap]), a, V_G)[0][0] / e
E_gap_upper = band_structure_1d(np.array([k_gap]), a, V_G)[1][0] / e
plt.fill_between([-1, 1], E_gap_lower, E_gap_upper, alpha=0.2, color='gray', label='Band gap')

plt.xlabel('Wave vector k (π/a)', fontsize=12)
plt.ylabel('Energy (eV)', fontsize=12)
plt.title('1D Band Structure: Nearly Free Electron Model', fontsize=14, fontweight='bold')
plt.axhline(0, color='k', linewidth=0.5)
plt.axvline(-1, color='k', linewidth=0.5, linestyle=':')
plt.axvline(1, color='k', linewidth=0.5, linestyle=':')
plt.text(0.7, E_gap_upper + 0.5, f'Band gap\n~{(E_gap_upper - E_gap_lower):.2f} eV', fontsize=10, ha='center')
plt.legend(fontsize=11, loc='upper right')
plt.grid(alpha=0.3)
plt.xlim(-1, 1)
plt.tight_layout()
plt.show()

1.5 Revisiting Density of States (DOS) - Calculation from Band Structure

1.5.1 General Definition of Density of States

Given band structure E_n(k), the density of states at energy E is:

D(E) = Σ_n ∫ δ(E - E_n(k)) (dk/(2π)³)

In actual calculations, we compute the area of the constant energy surface E_n(k) = E.

1.5.2 Features of DOS from Band Structure

Unlike free electron D(E) ∝ √E, systems with band structure exhibit:

Example 5: Calculating Density of States from Band Structure

def dos_from_band(E_array, E_band, V):
    """
    Calculate density of states from band structure (histogram method)

    Args:
        E_array (array): Energy axis [J]
        E_band (array): Band energy [J] (k-point array for 1D)
        V (float): Volume [m^3]

    Returns:
        array: Density of states [J^-1 m^-3]
    """
    hist, bin_edges = np.histogram(E_band, bins=E_array, density=False)
    dE = E_array[1] - E_array[0]
    # Normalization for 1D (different for 3D)
    DOS = hist / (dE * V) * 2  # Spin factor 2
    return DOS

# 1D band structure example
k = np.linspace(-np.pi/a, np.pi/a, 10000)
E_lower, E_upper = band_structure_1d(k, a, V_G)

# Energy axis
E_axis = np.linspace(-2*e, 20*e, 500)

# Calculate DOS for each band
DOS_lower = dos_from_band(E_axis, E_lower, 1e-9)
DOS_upper = dos_from_band(E_axis, E_upper, 1e-9)
DOS_total = DOS_lower + DOS_upper

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

# (a) Band structure
ax1.plot(k * a / np.pi, E_lower / e, 'b-', linewidth=2.5, label='Lower band')
ax1.plot(k * a / np.pi, E_upper / e, 'r-', linewidth=2.5, label='Upper band')
ax1.set_xlabel('Wave vector k (π/a)', fontsize=12)
ax1.set_ylabel('Energy (eV)', fontsize=12)
ax1.set_title('(a) Band Structure', fontsize=13, fontweight='bold')
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3)
ax1.set_xlim(-1, 1)

# (b) Density of states
E_plot = E_axis / e
ax2.plot(DOS_total * e, E_plot, 'k-', linewidth=2.5, label='Total DOS')
ax2.fill_betweenx(E_plot, 0, DOS_total * e, alpha=0.3, color='blue')
ax2.set_xlabel('Density of States (arb. units)', fontsize=12)
ax2.set_ylabel('Energy (eV)', fontsize=12)
ax2.set_title('(b) Density of States', fontsize=13, fontweight='bold')
ax2.axhline(E_gap_lower, color='gray', linestyle='--', alpha=0.5)
ax2.axhline(E_gap_upper, color='gray', linestyle='--', alpha=0.5)
ax2.text(0.7 * np.max(DOS_total * e), (E_gap_lower + E_gap_upper) / 2,
         'Band gap\n(D=0)', fontsize=10, ha='center', va='center',
         bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.5))
ax2.legend(fontsize=11)
ax2.grid(alpha=0.3)
ax2.set_ylim(E_plot[0], E_plot[-1])

plt.tight_layout()
plt.show()

1.6 Classification of Metals, Semiconductors, and Insulators

1.6.1 Electrical Properties Determined by Band Occupation

A material’s conductivity is determined by where the Fermi energy E_F is located in the band structure.

Material TypeBand OccupationBand GapConductivity (room temp)Typical Examples
MetalValence band partially occupied
or multiple bands overlapNone (or negative)10⁴-10⁶ S/cmCu, Al, Na
SemiconductorValence band fully occupied
Conduction band emptySmall (0.1-3 eV)10⁻⁶-10² S/cmSi, GaAs, GaN
InsulatorValence band fully occupied
Conduction band emptyLarge (>3 eV)<10⁻¹⁰ S/cmSiO₂, Al₂O₃

1.6.2 Why Do Metals Conduct Electricity?

In metals:

  1. E_F is located within a band
  2. Occupied and empty states coexist near E_F
  3. Under electric field, electrons near E_F easily move to empty states (=current)
  4. Having D(E_F) > 0 is essential

💡 Concrete Example: Sodium (Na)

Na atom has electron configuration [Ne] 3s¹. In the crystal:

1.6.3 Why Don’t Insulators Conduct Electricity?

In insulators (e.g., diamond):

  1. Valence electrons exactly completely fill the valence band
  2. E_F is between the valence band top and conduction band bottom (within the band gap)
  3. The band gap is large (diamond: 5.5 eV)
  4. Room temperature thermal energy (~0.026 eV) cannot excite electrons to the conduction band
  5. → No conductivity (σ < 10⁻¹⁴ S/cm)

1.6.4 Semiconductors - Intermediate Properties

Semiconductors have a structure similar to insulators but with a small band gap (Si: 1.1 eV, GaAs: 1.4 eV), so:

✅ Numerical Example: Si Intrinsic Carrier Density

Silicon (T = 300 K):

By doping (e.g., 10¹⁶ cm⁻³ phosphorus), carrier density can be increased by a million times.

Example 6: Comparing Band Structures of Metals, Semiconductors, and Insulators

def plot_band_occupation():
    """
    Visualize band occupation states of metals, semiconductors, and insulators
    """
    fig, axes = plt.subplots(1, 3, figsize=(15, 6))

    E_range = np.linspace(-3, 3, 100)

    # (a) Metal
    ax = axes[0]
    # Overlapping bands
    ax.fill_between([-1, 1], -2, 0, alpha=0.6, color='blue', label='Valence band (filled)')
    ax.fill_between([-1, 1], 0, 2, alpha=0.3, color='blue', label='Conduction band (partial)')
    ax.axhline(0.5, color='red', linewidth=2, linestyle='--', label='Fermi level (E_F)')
    ax.text(0, 0.7, 'E_F', fontsize=12, color='red', ha='center', fontweight='bold')
    ax.set_ylim(-3, 3)
    ax.set_xlim(-1.5, 1.5)
    ax.set_ylabel('Energy (eV)', fontsize=12)
    ax.set_title('(a) Metal', fontsize=13, fontweight='bold')
    ax.legend(fontsize=9, loc='upper right')
    ax.grid(alpha=0.3)
    ax.set_xticks([])

    # (b) Semiconductor
    ax = axes[1]
    ax.fill_between([-1, 1], -2, -0.5, alpha=0.6, color='blue', label='Valence band (filled)')
    ax.fill_between([-1, 1], 0.5, 2, alpha=0.3, color='lightblue', label='Conduction band (empty)')
    ax.axhline(0, color='red', linewidth=2, linestyle='--', label='Fermi level (E_F)')
    ax.text(0, 0.15, 'E_F', fontsize=12, color='red', ha='center', fontweight='bold')
    # Band gap
    ax.annotate('', xy=(1.2, -0.5), xytext=(1.2, 0.5),
                arrowprops=dict(arrowstyle='<->', color='black', lw=2))
    ax.text(1.5, 0, 'E_g\n~1 eV', fontsize=11, va='center', ha='left')
    ax.set_ylim(-3, 3)
    ax.set_xlim(-1.5, 1.8)
    ax.set_title('(b) Semiconductor', fontsize=13, fontweight='bold')
    ax.legend(fontsize=9, loc='upper right')
    ax.grid(alpha=0.3)
    ax.set_xticks([])
    ax.set_yticks([])

    # (c) Insulator
    ax = axes[2]
    ax.fill_between([-1, 1], -2, -1, alpha=0.6, color='blue', label='Valence band (filled)')
    ax.fill_between([-1, 1], 2, 3, alpha=0.3, color='lightblue', label='Conduction band (empty)')
    ax.axhline(0.5, color='red', linewidth=2, linestyle='--', label='Fermi level (E_F)')
    ax.text(0, 0.7, 'E_F', fontsize=12, color='red', ha='center', fontweight='bold')
    # Band gap
    ax.annotate('', xy=(1.2, -1), xytext=(1.2, 2),
                arrowprops=dict(arrowstyle='<->', color='black', lw=2))
    ax.text(1.5, 0.5, 'E_g\n>3 eV', fontsize=11, va='center', ha='left')
    ax.set_ylim(-3, 3.5)
    ax.set_xlim(-1.5, 1.8)
    ax.set_title('(c) Insulator', fontsize=13, fontweight='bold')
    ax.legend(fontsize=9, loc='upper right')
    ax.grid(alpha=0.3)
    ax.set_xticks([])
    ax.set_yticks([])

    plt.tight_layout()
    plt.show()

plot_band_occupation()

1.7 Fermi Surface - Understanding 3D Band Structure

1.7.1 What is the Fermi Surface?

In 3D k-space (wave vector space, also called reciprocal lattice space), the set of points satisfying E_n(k) = E_F is called the Fermi surface.

Why it’s important:

1.7.2 Free Electron Fermi Surface - Perfect Sphere

In the free electron model with E = ℏ²k²/(2m), k satisfying E = E_F is:

|k| = k_F = √(2mE_F)/ℏ = (3π²n)^(1/3)

That is, the Fermi surface is a sphere of radius k_F (Fermi sphere).

1.7.3 Actual Metal Fermi Surfaces - Complex Shapes

Due to the crystal’s periodic potential, Fermi surfaces distort from spherical:

Example 7: Visualizing 2D Fermi Surface

def fermi_surface_2d():
    """
    Calculate Fermi surface from simple 2D band structure
    """
    # 2D grid
    kx = np.linspace(-np.pi/a, np.pi/a, 200)
    ky = np.linspace(-np.pi/a, np.pi/a, 200)
    KX, KY = np.meshgrid(kx, ky)

    # Simple band structure (tight-binding model)
    t = 2 * e  # hopping parameter
    E = -2 * t * (np.cos(KX * a) + np.cos(KY * a))

    # Fermi surfaces at several Fermi energies
    E_F_values = [-3*t, -2*t, 0, 2*t]

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

    for idx, E_F in enumerate(E_F_values):
        ax = axes[idx]

        # Band structure contour
        contour = ax.contourf(KX*a/np.pi, KY*a/np.pi, E/t, levels=50, cmap='RdBu_r', alpha=0.7)

        # Fermi surface (E = E_F contour line)
        ax.contour(KX*a/np.pi, KY*a/np.pi, E/t, levels=[E_F/t], colors='black', linewidths=3)

        # Brillouin zone boundary
        ax.plot([-1, 1, 1, -1, -1], [-1, -1, 1, 1, -1], 'k--', linewidth=2, alpha=0.5)

        ax.set_xlabel('k_x (π/a)', fontsize=11)
        ax.set_ylabel('k_y (π/a)', fontsize=11)
        ax.set_title(f'E_F = {E_F/t:.1f}t', fontsize=12, fontweight='bold')
        ax.set_aspect('equal')
        ax.grid(alpha=0.3)

        # Colorbar
        plt.colorbar(contour, ax=ax, label='E/t')

    plt.suptitle('Fermi Surface Evolution in 2D Tight-Binding Model',
                 fontsize=14, fontweight='bold', y=0.995)
    plt.tight_layout()
    plt.show()

fermi_surface_2d()

Example 8: 3D Fermi Surface (Simple Example)

from mpl_toolkits.mplot3d import Axes3D

def fermi_surface_3d_sphere():
    """
    3D plot of free electron Fermi surface (sphere)
    """
    # Copper example
    n_Cu = 8.45e28  # m^-3
    k_F = (3 * np.pi**2 * n_Cu)**(1/3)

    # Sphere parameters
    u = np.linspace(0, 2 * np.pi, 100)
    v = np.linspace(0, np.pi, 100)
    kx = k_F * np.outer(np.cos(u), np.sin(v))
    ky = k_F * np.outer(np.sin(u), np.sin(v))
    kz = k_F * np.outer(np.ones(np.size(u)), np.cos(v))

    # Plot
    fig = plt.figure(figsize=(10, 10))
    ax = fig.add_subplot(111, projection='3d')

    ax.plot_surface(kx * 1e-10, ky * 1e-10, kz * 1e-10,
                    color='cyan', alpha=0.6, edgecolor='navy', linewidth=0.1)

    # Brillouin zone boundary (cube)
    G = np.pi / a
    # Draw cube edges
    for i in [-1, 1]:
        for j in [-1, 1]:
            ax.plot([i*G*1e-10, i*G*1e-10], [j*G*1e-10, j*G*1e-10],
                   [-G*1e-10, G*1e-10], 'k--', alpha=0.3)
            ax.plot([i*G*1e-10, i*G*1e-10], [-G*1e-10, G*1e-10],
                   [j*G*1e-10, j*G*1e-10], 'k--', alpha=0.3)
            ax.plot([-G*1e-10, G*1e-10], [i*G*1e-10, i*G*1e-10],
                   [j*G*1e-10, j*G*1e-10], 'k--', alpha=0.3)

    ax.set_xlabel('k_x (Å⁻¹)', fontsize=12)
    ax.set_ylabel('k_y (Å⁻¹)', fontsize=12)
    ax.set_zlabel('k_z (Å⁻¹)', fontsize=12)
    ax.set_title('Fermi Surface of Copper (Free Electron Approximation)',
                 fontsize=14, fontweight='bold')

    # Isotropic aspect ratio
    max_range = G * 1e-10
    ax.set_xlim([-max_range, max_range])
    ax.set_ylim([-max_range, max_range])
    ax.set_zlim([-max_range, max_range])

    plt.tight_layout()
    plt.show()

    print(f"Copper Fermi wave vector: k_F = {k_F:.2e} m⁻¹ = {k_F*1e-10:.2f} Å⁻¹")

fermi_surface_3d_sphere()

1.7.4 Fermi Surface and Transport Phenomena

Geometric properties of the Fermi surface directly affect material properties:

Fermi Surface FeaturePhysical EffectTypical Example
Nearly sphericalIsotropic conductivityCu, Al
EllipsoidalAnisotropic conductivityBi (100x difference by crystal direction)
Nesting
(parallel sections)Charge density waves, superconductivityLayered compounds
Topological changesLifshitz transitionMetals under pressure

1.8 Experimental Verification - Is Band Theory Correct?

1.8.1 Angle-Resolved Photoemission Spectroscopy (ARPES)

The most direct method to measure band structure:

✅ Measurement Precision (Latest ARPES)

This enables observation of superconducting gaps (~1 meV) and charge ordering formation processes.

1.8.2 Optical Measurements - Determining Band Gaps

Band gaps in semiconductors can be precisely measured by optical absorption:

α(ω) ∝ √(ℏω - E_g) (for direct transitions)

Where α is the absorption coefficient and ω is the angular frequency of light.

Example 9: Estimating Band Gap from Optical Absorption

def optical_absorption(photon_energy, E_g, A=1.0, indirect=False):
    """
    Calculate optical absorption coefficient of semiconductor

    Args:
        photon_energy (array): Photon energy [eV]
        E_g (float): Band gap [eV]
        A (float): Proportionality constant
        indirect (bool): True for indirect transition

    Returns:
        array: Absorption coefficient [arbitrary units]
    """
    alpha = np.zeros_like(photon_energy)
    above_gap = photon_energy > E_g

    if indirect:
        # Indirect transition: α ∝ (ℏω - E_g)²
        alpha[above_gap] = A * (photon_energy[above_gap] - E_g)**2
    else:
        # Direct transition: α ∝ √(ℏω - E_g)
        alpha[above_gap] = A * np.sqrt(photon_energy[above_gap] - E_g)

    return alpha

# Typical semiconductor parameters
semiconductors = {
    'GaN': {'E_g': 3.4, 'type': 'direct'},
    'Si': {'E_g': 1.12, 'type': 'indirect'},
    'GaAs': {'E_g': 1.42, 'type': 'direct'},
    'Ge': {'E_g': 0.66, 'type': 'indirect'}
}

# Plot
photon_energy = np.linspace(0, 4, 500)

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

for idx, (name, params) in enumerate(semiconductors.items()):
    ax = axes[idx]
    E_g = params['E_g']
    indirect = (params['type'] == 'indirect')

    alpha = optical_absorption(photon_energy, E_g, A=1.0, indirect=indirect)

    ax.plot(photon_energy, alpha, 'b-', linewidth=2.5)
    ax.axvline(E_g, color='r', linestyle='--', linewidth=2, label=f'E_g = {E_g:.2f} eV')
    ax.fill_between(photon_energy, 0, alpha, where=(photon_energy > E_g), alpha=0.3, color='blue')

    ax.set_xlabel('Photon Energy (eV)', fontsize=11)
    ax.set_ylabel('Absorption Coefficient α (arb. units)', fontsize=11)
    ax.set_title(f'{name} ({params["type"]} gap)', fontsize=12, fontweight='bold')
    ax.legend(fontsize=10)
    ax.grid(alpha=0.3)
    ax.set_xlim(0, 4)
    ax.set_ylim(0, np.max(alpha) * 1.1)

plt.suptitle('Optical Absorption Spectra of Semiconductors', fontsize=14, fontweight='bold')
plt.tight_layout()
plt.show()

# Tauc plot for band gap determination (experimental data analysis method)
fig, ax = plt.subplots(figsize=(10, 6))

for name, params in semiconductors.items():
    E_g = params['E_g']
    indirect = (params['type'] == 'indirect')
    alpha = optical_absorption(photon_energy, E_g, A=1.0, indirect=indirect)

    # Tauc plot: (αhν)^(1/n) vs hν (n=1/2 for direct, n=2 for indirect)
    if indirect:
        y_axis = (alpha * photon_energy)**0.5  # (αhν)^1/2
    else:
        y_axis = (alpha * photon_energy)**2  # (αhν)^2

    ax.plot(photon_energy, y_axis, linewidth=2, label=name)
    ax.axvline(E_g, linestyle=':', alpha=0.5)

ax.set_xlabel('Photon Energy (eV)', fontsize=12)
ax.set_ylabel('(αhν)^n (arb. units)', fontsize=12)
ax.set_title('Tauc Plot for Band Gap Determination', fontsize=14, fontweight='bold')
ax.legend(fontsize=11)
ax.grid(alpha=0.3)
ax.set_xlim(0, 4)
plt.tight_layout()
plt.show()

print("\n💡 How to use Tauc plot:")
print("Extrapolate the rising edge linearly and find the intersection with the x-axis;")
print("that corresponds to the band gap E_g.")

1.8.3 Quantum Oscillations - Precise Measurement of Fermi Surface

The de Haas-van Alphen effect is a phenomenon where magnetization oscillates in a strong magnetic field, enabling precise measurement of Fermi surface cross-sections:

Oscillation period ∝ 1/A
(A: extremal cross-sectional area of Fermi surface perpendicular to magnetic field)

This method can determine Fermi surface topology, effective mass, scattering rate, etc.

Learning Objectives Verification

Having completed this chapter, you can now explain:

Fundamental Understanding

Practical Skills

Application Capabilities

Exercises

Easy (Basic Verification)

Q1: Which is the most appropriate definition of Fermi energy?

a) Average energy of electrons in a metal
b) Energy of the highest energy electron at absolute zero
c) Minimum energy of the conduction band
d) Energy at the center of the band gap

See Answer

Correct: b) Energy of the highest energy electron at absolute zero

Explanation:
Fermi energy E_F is the energy of the highest energy electron when electrons fill from the lowest energy state upward at T=0K. Fermions (electrons) obey Pauli’s exclusion principle, so at most 2 electrons (spin ↑↓) can occupy one state.

Additional note:
Options c) and d) describe semiconductors or insulators, not metals. Option a) gives an average energy lower than E_F (approximately 3E_F/5 at T=0K).

Q2: Copper (Cu) has a Fermi energy of about 7.0 eV. What is the Fermi temperature T_F in K? (k_B = 8.617 × 10⁻⁵ eV/K)

See Answer

Correct: Approximately 81,000 K

Calculation:
T_F = E_F / k_B = 7.0 eV / (8.617 × 10⁻⁵ eV/K) ≈ 81,200 K

Key Point:
Room temperature (300 K) is only 0.4% of T_F. This is why the electronic specific heat is much smaller than classical predictions (3/2 Nk_B). Only electrons near the Fermi energy (within ~kT range) are actually thermally excited.

Q3: Explain the difference in band structure between metals and insulators.

See Answer

Sample Answer:

Key Point:
Whether a material is a metal or insulator is determined not by the shape of the bands but by “how filled the bands are with electrons.” The same band structure with different electron counts will have different properties.

Medium (Application)

Q4: Silicon (Si) has a band gap of 1.12 eV. Can it absorb visible light (wavelength 400-700 nm)? Calculate and explain. (h = 4.136 × 10⁻¹⁵ eV·s, c = 3 × 10⁸ m/s)

See Answer

Conclusion: Can absorb some visible light (blue to violet) but not red

Calculation:

Photon energy E = hc/λ :

Since E_g = 1.12 eV:

Practical Meaning:
This is one reason Si is used as a solar cell material. It covers the main part of sunlight (visible to near-infrared). However, since infrared is transmitted, tandem solar cells combining Si with materials having smaller band gaps like Ge (0.66 eV) or GaAs (1.42 eV) are also being researched.

Q5: In the 3D free electron model, the density of states is D(E) ∝ √E. Qualitatively explain how this leads to the electronic specific heat C_v ∝ T at the Fermi energy.

See Answer

Explanation:

  1. Number of excited electrons : At temperature T, only electrons within an energy range ~k_BT around E_F are thermally excited. This number is proportional to D(E_F) × k_BT.
  2. Energy gained per electron : On average ~k_BT.
  3. Total energy : U ~ [D(E_F) × k_BT] × k_BT = D(E_F) × (k_BT)²
  4. Specific heat : C_v = dU/dT ~ D(E_F) × k_B² × T ∝ T

Important points:

Q6: Modify the code in Example 1 to calculate and plot Fermi energy and Fermi temperature for alkali metals (Li, Na, K, Rb, Cs) from their electron densities. Discuss how E_F changes with increasing atomic number.

See Answer

Code Example:

alkali_metals = {
    'Li': 4.70e28,
    'Na': 2.65e28,
    'K': 1.40e28,
    'Rb': 1.15e28,
    'Cs': 0.91e28
}

names = list(alkali_metals.keys())
E_F_values = [fermi_energy(n) for n in alkali_metals.values()]
T_F_values = [E_F * e / constants.k / 1000 for E_F in E_F_values]

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

ax1.bar(names, E_F_values, color='skyblue', edgecolor='navy', linewidth=2)
ax1.set_ylabel('Fermi Energy (eV)', fontsize=12)
ax1.set_title('Fermi Energy of Alkali Metals', fontsize=13, fontweight='bold')
ax1.grid(axis='y', alpha=0.3)

ax2.bar(names, T_F_values, color='salmon', edgecolor='darkred', linewidth=2)
ax2.set_ylabel('Fermi Temperature (10³ K)', fontsize=12)
ax2.set_title('Fermi Temperature of Alkali Metals', fontsize=13, fontweight='bold')
ax2.grid(axis='y', alpha=0.3)

plt.tight_layout()
plt.show()

Discussion:

Hard (Advanced)

Q7: Derive the density of states for 2D materials (such as graphene) in the free electron model. Discuss the difference from the 3D case D(E) ∝ √E.

See Answer

Derivation:

In 2D, wave vector space is a plane (k_x, k_y). Free electron energy:

E = ℏ²k²/(2m) = ℏ²(k_x² + k_y²)/(2m)

Density of states in k-space (per unit area): 1/(2π)² = 1/(4π²)

States with energy E lie on a circle of radius k = √(2mE/ℏ²).

k-width corresponding to energy width dE: dk = (m/ℏ²k) dE

Number of states: dN = [circumference 2πk] × dk × [1/(4π²)] × 2 (spin)

D(E) = dN/dE = (2πk × dk/dE) / (2π²) = (2m)/(2πℏ²) = constant

Key differences:

DimensionD(E)Feature
1D∝ E^(-1/2)Diverges at low energy
2D= constantEnergy independent
3D∝ E^(1/2)Increases at high energy

Physical meaning:

Q8: Diamond (C) and graphite (C) are made of the same element, yet one is an insulator and the other is a semimetal. From a band theory perspective, explain the structural factors causing this difference.

See Answer

Structural differences:

Electronic structure differences:

Diamond:

Graphite:

Key points:

  1. Hybridization difference (sp³ vs sp²) is decisive
  2. Dimensionality : Graphite’s 2D layer structure promotes π electron delocalization
  3. Interlayer interaction : Isolating one layer of graphite produces “graphene” with a unique Dirac cone-type band structure

Practical implications:

Q9: (Programming task) In the tight-binding model, calculate and plot how bandwidth and band gap change when varying the nearest-neighbor hopping integral t for a 1D chain. Discuss the physical meaning.

See Answer

Code example:

def tight_binding_1d(k, a, t, epsilon_0=0):
    """
    Band structure of 1D tight-binding model

    Args:
        k (array): Wave vector
        a (float): Lattice constant
        t (float): Hopping integral
        epsilon_0 (float): On-site energy

    Returns:
        array: Energy
    """
    return epsilon_0 - 2 * t * np.cos(k * a)

a = 3e-10  # 3 Å
k = np.linspace(-np.pi/a, np.pi/a, 500)
t_values = [0.5*e, 1.0*e, 2.0*e, 4.0*e]  # eV

plt.figure(figsize=(10, 7))

for t in t_values:
    E = tight_binding_1d(k, a, t)
    bandwidth = np.max(E) - np.min(E)
    plt.plot(k*a/np.pi, E/e, linewidth=2.5, label=f't = {t/e:.1f} eV (BW = {bandwidth/e:.2f} eV)')

plt.xlabel('Wave vector k (π/a)', fontsize=12)
plt.ylabel('Energy (eV)', fontsize=12)
plt.title('1D Tight-Binding Model: Effect of Hopping Parameter', fontsize=14, fontweight='bold')
plt.axhline(0, color='k', linewidth=0.5, linestyle='--')
plt.legend(fontsize=11)
plt.grid(alpha=0.3)
plt.tight_layout()
plt.show()

# t dependence of bandwidth
t_range = np.linspace(0.1*e, 5*e, 50)
bandwidth = [4*t for t in t_range]  # BW = 4t (theoretical value)

plt.figure(figsize=(8, 6))
plt.plot(t_range/e, np.array(bandwidth)/e, 'b-', linewidth=3)
plt.xlabel('Hopping parameter t (eV)', fontsize=12)
plt.ylabel('Bandwidth (eV)', fontsize=12)
plt.title('Bandwidth vs Hopping Parameter', fontsize=14, fontweight='bold')
plt.grid(alpha=0.3)
plt.tight_layout()
plt.show()

print("📊 Analysis results:")
print(f"Bandwidth = 4t (larger t gives wider band)")
print(f"Physical meaning: Large t → electron wavefunction strongly overlaps with adjacent sites")
print(f"                → Electrons move easily in crystal (effective mass m* ∝ 1/t)")

Discussion:

Q10: (Advanced problem) Mott insulators are predicted to be metals by band theory but are actually insulators (e.g., NiO, La₂CuO₄). Research why band theory alone cannot explain this phenomenon and briefly summarize the role of electron correlation.

See Answer

What are Mott insulators:

Why band theory fails:

  1. Ignores strong electron correlation
    • Band theory assumes electrons move independently (one-electron approximation)
    • Actually, electron-electron Coulomb repulsion U can be larger than kinetic energy (bandwidth W)
  2. Mott-Hubbard picture
    • Cost to put 2 electrons on one site (U) is too large
    • Electrons localize, placing one per site
    • Moving electrons requires U energy → gap formation

Phase diagram (U vs W):

Success of Dynamical Mean Field Theory (DMFT):

Practical importance:

Conclusion:

Band theory is a powerful framework for understanding many materials, but is insufficient for systems with strong electron correlation (transition metal oxides, f-electron systems, etc.). To understand these, methods considering many-body effects (DMFT, quantum Monte Carlo, density functional theory+U, etc.) are necessary.

Next Steps

In Chapter 1, we learned the basic concepts of band theory as the foundation of solid-state electronic theory. In the next chapter, we will learn methods to calculate the electronic structure of actual materials based on this knowledge.

Chapter 2 Preview: Introduction to Density Functional Theory (DFT)

References

Fundamental Textbooks

  1. Ashcroft, N. W., & Mermin, N. D. (1976). Solid State Physics. Holt, Rinehart and Winston. (Classic masterpiece in solid-state physics)
  2. Kittel, C. (2004). Introduction to Solid State Physics (8th ed.). Wiley. (Ideal as an introductory text)
  3. Marder, M. P. (2010). Condensed Matter Physics (2nd ed.). Wiley. (Modern approach)

Band Theory Details

  1. Harrison, W. A. (1980). Electronic Structure and the Properties of Solids. Freeman. (Detailed tight-binding method)
  2. Yu, P. Y., & Cardona, M. (2010). Fundamentals of Semiconductors (4th ed.). Springer. (Standard semiconductor physics textbook)

Computational Physical Chemistry

  1. Martin, R. M. (2004). Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press. (DFT theory and practice)
  2. Sholl, D., & Steckel, J. A. (2009). Density Functional Theory: A Practical Introduction. Wiley. (DFT introduction)

Online Resources

  1. Materials Project. (2024). Electronic Structure Calculations. https://materialsproject.org
  2. ASE (Atomic Simulation Environment). https://wiki.fysik.dtu.dk/ase/
  3. GPAW Documentation. https://wiki.fysik.dtu.dk/gpaw/

Original Papers (Historically Important)

  1. Bloch, F. (1929). “Über die Quantenmechanik der Elektronen in Kristallgittern.” Zeitschrift für Physik , 52(7-8), 555-600. (Bloch’s theorem)
  2. Wilson, A. H. (1931). “The Theory of Electronic Semi-Conductors.” Proceedings of the Royal Society A , 133(822), 458-491. (Foundation of semiconductor theory)

← Series Index Chapter 2: Introduction to DFT →

Disclaimer

Back to Series Index Next Chapter →