Part II: Semiconductor Physics | Chapter 4

Band Theory & Semiconductors

From atomic orbitals to energy bands: understanding the quantum physics behind semiconductor behaviour.

Energy Bands in Solids

In an isolated atom, electrons occupy discrete energy levels. When \( N \) atoms form a crystal, the Pauli exclusion principle forces each level to split into \( N \) closely-spaced states, forming energy bands. The gap between the highest filled band (valence band) and the lowest empty band (conduction band) is the bandgap \( E_g \).

Insulators

\( E_g > 4 \) eV. The valence band is completely full and the gap is too large for thermal excitation. Essentially no conductivity.

Semiconductors

\( E_g \approx 0.1\text{–}3 \) eV. Si: 1.12 eV, Ge: 0.67 eV, GaAs: 1.42 eV. Moderate conductivity, controllable by doping.

Metals

No bandgap: conduction and valence bands overlap. The Fermi level lies inside a partially filled band, giving high conductivity.

Intrinsic vs Extrinsic Semiconductors

In a pure (intrinsic) semiconductor, carriers arise only from thermal promotion of electrons across the bandgap. The intrinsic carrier concentration at temperature \( T \) is:

\[ n_i = \sqrt{N_C N_V}\,\exp\!\left(-\frac{E_g}{2k_B T}\right) \]

For silicon at 300 K, \( n_i \approx 1.5 \times 10^{10} \;\text{cm}^{-3} \) — tiny compared to \( \sim 5\times10^{22} \) atoms/cm³.

Doping

Replacing a small fraction of Si atoms with impurities dramatically shifts the carrier concentration:

n-type (donors)

Group-V atoms (P, As, Sb) donate one extra electron. Majority carriers are electrons. With donor density \( N_D \gg n_i \): \( n \approx N_D \).

p-type (acceptors)

Group-III atoms (B, Al, Ga) accept one electron, creating holes. Majority carriers are holes. With acceptor density \( N_A \gg n_i \): \( p \approx N_A \).

Mass-Action Law & Fermi Level

Regardless of doping, the product of electron and hole concentrations at thermal equilibrium is fixed by the intrinsic concentration:

\[ n \cdot p = n_i^2 \]

The Fermi level shifts above midgap for n-type material by:

\[ E_F - E_i = k_B T \ln\!\left(\frac{N_D}{n_i}\right) \]

Drift and Diffusion Currents

Drift Current

Driven by electric field \( \mathcal{E} \). Carriers acquire drift velocity \( v_d = \mu \mathcal{E} \).

\[ J_{\text{drift}} = (n\mu_n + p\mu_p)q\mathcal{E} \]

Diffusion Current

Driven by concentration gradient. Einstein relation: \( D = \mu k_B T / q \).

\[ J_{\text{diff}} = qD_n\frac{dn}{dx} - qD_p\frac{dp}{dx} \]

Python: Fermi-Dirac & Carrier Statistics

Plot the Fermi-Dirac distribution at multiple temperatures, compute intrinsic carrier concentration vs temperature for silicon, and show how the Fermi level shifts with doping concentration.

Band Theory: Fermi-Dirac, ni(T), and Fermi Level Shifts

Python

band_theory.py111 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ---- Fermi-Dirac distribution at multiple temperatures ----
k_B = 8.617e-5   # eV/K
E_F = 0.0        # set Fermi level as energy reference

E = np.linspace(-0.4, 0.4, 1000)   # eV relative to E_F

temps = [100, 200, 300, 500, 1000]
colors = ['#d1fae5', '#a7f3d0', '#6ee7b7', '#34d399', '#10b981']

print('=== Fermi-Dirac Distribution ===')
for T in temps:
    f_at_EF = 1.0 / (1 + np.exp(0.0 / (k_B * T)))
    f_at_01 = 1.0 / (1 + np.exp(0.1  / (k_B * T)))
    print(f'T={T:5d} K: f(E_F)={f_at_EF:.4f}, f(E_F+0.1eV)={f_at_01:.4f}')

# ---- Silicon carrier concentration vs temperature ----
# Si: Eg = 1.12 eV, ni = 1.5e10 cm^-3 at 300 K
# Approximate: ni ~ T^(3/2) * exp(-Eg/(2*k_B*T))
# Prefactor calibrated to match ni(300K) = 1.5e10 cm^-3
Eg_Si = 1.12   # eV
T_300 = 300.0
ni_300 = 1.5e10   # cm^-3
T_arr = np.linspace(200, 600, 500)
prefactor = ni_300 / (T_300**1.5 * np.exp(-Eg_Si / (2 * k_B * T_300)))
ni_arr = prefactor * T_arr**1.5 * np.exp(-Eg_Si / (2 * k_B * T_arr))

print()
print('=== Silicon ni vs Temperature ===')
for T_val in [250, 300, 350, 400, 450, 500]:
    pf = ni_300 / (T_300**1.5 * np.exp(-Eg_Si / (2 * k_B * T_300)))
    ni = pf * T_val**1.5 * np.exp(-Eg_Si / (2 * k_B * T_val))
    print(f'T={T_val} K: ni = {ni:.3e} cm^-3')

# ---- Doping: n-type (Nd donors), check ni^2 = n*p ----
Nd = 1e16   # donors, cm^-3
ni_300_val = 1.5e10
n_n = Nd                       # majority carriers (electrons)
p_n = ni_300_val**2 / n_n      # minority carriers (holes)
print()
print('=== N-type doping (Nd = 1e16 cm^-3) at 300 K ===')
print(f'n (electrons) = {n_n:.2e} cm^-3')
print(f'p (holes)     = {p_n:.2e} cm^-3')
print(f'n*p = {n_n*p_n:.2e} cm^-6  (should equal ni^2 = {ni_300_val**2:.2e})')

# Fermi level shift: delta_EF = k_B*T * ln(Nd/ni)
delta_EF = k_B * T_300 * np.log(Nd / ni_300_val)
print(f'Fermi level shift above midgap: {delta_EF:.4f} eV')

# ---- Plots ----
fig, axes = plt.subplots(1, 3, figsize=(16, 5))
fig.patch.set_facecolor('#0a0a0a')
fig.suptitle('Semiconductor Band Theory & Carrier Statistics', color='#d1d5db', fontsize=13, y=0.98)
for ax in axes:
    ax.set_facecolor('#111827')
    for spine in ax.spines.values():
        spine.set_color('#374151')

# Plot 1: Fermi-Dirac at multiple T
ax = axes[0]
for T, c in zip(temps, colors):
    f_E = 1.0 / (1 + np.exp(E / (k_B * T)))
    ax.plot(f_E, E, color=c, lw=2.0, label=f'T={T} K')
ax.axhline(0, color='#fbbf24', ls='--', lw=1.2, label='E_F')
ax.axvline(0.5, color='#6b7280', ls=':', lw=0.8)
ax.set_xlabel('Fermi-Dirac f(E)', color='#9ca3af')
ax.set_ylabel('E - E_F (eV)', color='#9ca3af')
ax.set_title('Fermi-Dirac Distribution', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8)
ax.set_xlim(-0.05, 1.05)

# Plot 2: ni(T) for Silicon
ax = axes[1]
ax.semilogy(T_arr, ni_arr, color='#34d399', lw=2.5)
ax.axvline(300, color='#fbbf24', ls='--', lw=1.5, label='T=300 K')
ax.axhline(1.5e10, color='#f87171', ls=':', lw=1.2, label='ni=1.5e10 cm-3')
ax.set_xlabel('Temperature (K)', color='#9ca3af')
ax.set_ylabel('ni (cm^-3)', color='#9ca3af')
ax.set_title('Silicon Intrinsic Carrier
Concentration vs T', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8, which='both')

# Plot 3: Fermi level vs doping concentration
ax = axes[2]
Nd_range = np.logspace(13, 18, 500)
delta_EF_range = k_B * T_300 * np.log(Nd_range / ni_300_val)
ax.semilogx(Nd_range, delta_EF_range, color='#34d399', lw=2.2, label='n-type (donors)')
Na_range = Nd_range
delta_EF_p = -k_B * T_300 * np.log(Na_range / ni_300_val)
ax.semilogx(Na_range, delta_EF_p, color='#f87171', lw=2.2, ls='--', label='p-type (acceptors)')
ax.axhline(0, color='#fbbf24', ls=':', lw=1, label='Midgap')
ax.set_xlabel('Doping concentration (cm^-3)', color='#9ca3af')
ax.set_ylabel('E_F - E_i (eV)', color='#9ca3af')
ax.set_title('Fermi Level vs Doping (Si, 300 K)', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8, which='both')

plt.tight_layout(pad=2.0)
plt.savefig('output.png', dpi=140, bbox_inches='tight', facecolor='#0a0a0a')
print()
print('Plot saved to output.png')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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