Semiconductors & Semiconductor Electronics
Class 12 (CBSE) – Physics
(Advanced MIT‑style notes from first principles)
1. Introduction & Motivation
| What is a semiconductor? | A material whose conductivity lies between that of a conductor and an insulator. |
| Why are they special? | Band theory gives a natural explanation for temperature‑dependent conductivity, carrier statistics, and the operation of electronic devices (diodes, transistors, MOSFETs). |
| Key questions | How does the band structure give rise to electrons & holes? What determines the carrier concentration? How does external bias change the internal electrostatics? |
2. Quantum‑Mechanical Foundations
2.1 Periodic Potential & Bloch’s Theorem
In a crystal, the potential (V(\mathbf r)) is periodic:
[
V(\mathbf r+\mathbf R)=V(\mathbf r),\qquad \mathbf R\in\text{lattice}
]
Bloch’s theorem states that eigenfunctions of the Schrödinger equation can be written as
[
\psi_{\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{\mathbf k}(\mathbf r),\quad u_{\mathbf k}(\mathbf r)=u_{\mathbf k}(\mathbf r+\mathbf R)
]
where (\mathbf k) is a wave‑vector in the first Brillouin zone.
2.2 Energy Bands & Band Gaps
The periodic potential splits the free‑electron energy spectrum into allowed bands and forbidden band gaps.
- Valence band (VB): fully occupied at (T=0).
- Conduction band (CB): empty at (T=0).
- Band gap (E_g): energy difference between CB edge and VB edge.
Typical semiconductors: (E_g\approx 0.7-1.5~\text{eV}).
3. Carrier Statistics
3.1 Fermi‑Dirac Distribution
The probability that a state of energy (E) is occupied is
[
f(E)=\frac{1}{1+e^{(E-E_F)/k_BT}}
]
where (E_F) is the Fermi energy, (k_B) Boltzmann’s constant, (T) the absolute temperature.
3.2 Density of States (DOS)
For a parabolic band in 3‑D: [ g_c(E)=\frac{1}{2\pi^2}\left(\frac{2m_c^\ast}{\hbar^2}\right)^{3/2}\sqrt{E-E_c}\qquad (E>E_c) ] [ g_v(E)=\frac{1}{2\pi^2}\left(\frac{2m_v^\ast}{\hbar^2}\right)^{3/2}\sqrt{E_v-E}\qquad (E<E_v) ] where (m_c^\ast,m_v^\ast) are the effective masses for electrons and holes.
3.3 Intrinsic Carrier Concentration
For an intrinsic (pure) semiconductor, the electron concentration (n_i) and hole concentration (p_i) are equal: [ n_i=p_i=\int_{E_c}^{\infty} g_c(E)f(E)dE ] Using the effective density of states (N_c,N_v) and Maxwell–Boltzmann approximation (f(E)\approx e^{-(E-E_F)/k_BT}) (valid when ((E-E_F)\gg k_BT)): [ n_i=p_i=N_c e^{-(E_c-E_F)/k_B T} = N_v e^{-(E_F-E_v)/k_BT} ] Hence, [ n_i^2 = N_c N_v e^{-E_g/k_B T} \quad\Longrightarrow\quad n_i = \sqrt{N_c N_v},e^{-E_g/(2k_BT)} ] Edge case – degenerate semiconductors: when doping pushes (E_F) into a band, the Maxwell–Boltzmann approximation fails and the full Fermi–Dirac integral must be used.
4. Doping & Extrinsic Semiconductors
| n‑type | p‑type | |
|---|---|---|
| Donor concentration | (N_D) | – |
| Acceptor concentration | – | (N_A) |
| Majority carriers | electrons | holes |
| Minority carriers | holes | electrons |
4.1 Charge Neutrality
In equilibrium: [ n + N_A = p + N_D ] Usually (N_D,N_A) are much larger than the intrinsic carrier concentration, so (n\approx N_D) (for n‑type) or (p\approx N_A) (for p‑type).
4.2 Carrier Concentration Expressions
For n‑type (degenerate or non‑degenerate): [ n \approx N_D