Class‑12 CBSE – Physics
Semiconductor Electronics
Goal – Build a solid conceptual and mathematical foundation.
Exam‑style focus – Derive key equations, understand assumptions, spot pitfalls, and solve complex problems under time pressure.
1. Recap: Energy Bands & Fermi Level
| Concept | Definition | Key Point |
|---|---|---|
| Conduction Band (CB) | Uppermost band where electrons are free to move | Bandgap (E_g) separates CB from Valence Band (VB) |
| Valence Band (VB) | Highest occupied band | Holes are missing electrons |
| Intrinsic Fermi Level (E_{Fi}) | Energy where electron and hole concentrations are equal | (n_i = p_i = \sqrt{N_cN_v},e^{-E_g/2kT}) |
| Doping | Introduces extra electrons (n‑type) or holes (p‑type) | Shifts Fermi level toward CB or VB |
2. Doping & Carrier Concentrations
2.1 Intrinsic Carrier Density
[ n_i = \sqrt{N_cN_v},e^{-E_g/2kT} ]
- (N_c, N_v) – effective density of states in CB and VB.
- Temperature dependence: (n_i \propto T^{3/2} e^{-E_g/2kT}).
2.2 Extrinsic Densities (Non‑degenerate)
[ n \approx N_D,\quad p \approx \frac{n_i^2}{N_D}\quad(\text{n‑type}) ] [ p \approx N_A,\quad n \approx \frac{n_i^2}{N_A}\quad(\text{p‑type}) ]
Tricky Point: The above holds for non‑degenerate doping (i.e. (N_D, N_A \ll 10^{19},\text{cm}^{-3})). For degenerate semiconductors Fermi‑Dirac statistics must be used.
3. p–n Junction
3.1 Built‑in Potential
When a p‑type and n‑type material join, electrons diffuse from n to p and holes from p to n, creating a depletion region. The resulting electric field yields the built‑in potential:
[ V_{bi} = \frac{kT}{q}\ln!\left(\frac{N_A N_D}{n_i^2}\right) ]
- Derivation: Set diffusion and drift currents equal at equilibrium; integrate the electric field across depletion width.
3.2 Depletion Width
Assuming abrupt junction and 1‑D charge neutrality:
[ W = \sqrt{\frac{2\epsilon_s}{q}\left(\frac{N_A+N_D}{N_A N_D}\right) V_{bi}} ]
- (W = W_p + W_n) with (W_p/W_n = N_D/N_A).
Temperature Dependence
(V_{bi}) decreases as (T) rises because (n_i) increases exponentially.
[
V_{bi}(T) = \frac{kT}{q}\ln!\left(\frac{N_A N_D}{n_i(T)^2}\right)
]
3.3 Capacitance of the Depletion Region
[ C_D = \frac{\epsilon_s A}{W} = \sqrt{\frac{q\epsilon_s N_A N_D}{2(N_A+N_D) V_{bi}}} ]
- Useful in C-V measurements to extract doping profiles.
4. Diode Current–Voltage Characteristics
4.1 Shockley Equation
[ I_D = I_S !\left[e^{\frac{qV_D}{n kT}}-1\right] ]
- (I_S = q A \left( D_n \frac{p_0}{L_n} + D_p \frac{n_0}{L_p}\right))
(diffusion saturation current) - (n) – ideality factor (typically 1–2)
4.2 Ideality Factor
- n = 1 : pure diffusion current.
- n = 2 : recombination dominated (Shockley‑Read‑Hall).
- Higher n: series resistance, high‑order processes.
4.3 Series & Shunt Resistances
The real diode equation: [ I_D = I_S!\left[e^{\frac{q(V_D - I_D R_s)}{n kT}}-1\right] + \frac{V_D - I_D R_s}{R_p} ]