📚 Study Notes – Semiconductors
Program: TS EAMCET
Subject: Physics
Topic: Semiconductors
Semiconductors are the heart of modern electronics – from microchips to solar cells. Understanding their physics is essential for the EAMCET exam and for a career in electronics or materials science.
1. Simple Explanation
A semiconductor is a material whose electrical conductivity lies between that of conductors (metals) and insulators.
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Band theory:
- At absolute zero, all electrons occupy the valence band (filled).
- Above it lies the conduction band (empty).
- The energy gap between them is called the band gap (Eg).
- In semiconductors, Eg is small (~0.1–4 eV), allowing electrons to jump to the conduction band with modest thermal energy or photon absorption.
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Intrinsic vs Extrinsic
- Intrinsic semiconductor: pure crystal (e.g., pure silicon).
- Extrinsic semiconductor: intentionally doped with impurities to increase carriers (n‑type or p‑type).
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Doping:
- Donor atoms (group V) add extra electrons → n‑type.
- Acceptor atoms (group III) create holes → p‑type.
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Carrier concentration and conductivity
- Conductivity ( \sigma = q(n\mu_n + p\mu_p) )
- (q) – elementary charge
- (n, p) – electron and hole concentrations
- (\mu_n, \mu_p) – mobilities
- Conductivity ( \sigma = q(n\mu_n + p\mu_p) )
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Key phenomena
- Hall effect – used to determine carrier type, concentration, and mobility.
- Recombination & generation – essential for devices like diodes and transistors.
2. Key Points (Bullet‑style)
| Concept | What to remember | Typical values |
|---|---|---|
| Band Gap (E_g) | Determines if material is metallic, semiconducting, or insulating. | Si: 1.1 eV, Ge: 0.66 eV |
| Intrinsic Carrier Concentration (n_i) | (n_i = \sqrt{N_c N_v} e^{-E_g/(2kT)}) | Si at 300 K: (1.5 \times 10^{10}) cm⁻³ |
| Effective Density of States | (N_c = 2\left(\frac{2\pi m_e^* kT}{h^2}\right)^{3/2}) (similar for (N_v)) | See typical values in tables |
| Doping Levels | (N_D) (donors) or (N_A) (acceptors). | Usually (10^{14})–(10^{18}) cm⁻³ |
| Carrier Concentration (extrinsic) | (n \approx N_D) (n‑type), (p \approx N_A) (p‑type) | |
| Conductivity | ( \sigma = q(n\mu_n + p\mu_p) ) | Si: ( \mu_n \approx 1350) cm²/Vs, ( \mu_p \approx 450) cm²/Vs |
| Hall Coefficient | ( R_H = \frac{1}{q(n - p)} ) (single carrier) | |
| Depletion Width (in PN‑junction) | ( w = \sqrt{\frac{2\epsilon_s(V_{bi}-V)}{q}\left(\frac{N_D+N_A}{N_D N_A}\right)} ) | |
| Recombination Time | Order of (10^{-9})–(10^{-6}) s (depends on material) |
3. Worked Example
Problem:
A silicon wafer (intrinsic) at 300 K has an electron mobility (\mu_n = 1350) cm²/Vs and hole mobility (\mu_p = 450) cm²/Vs.
If the wafer is doped n‑type with donor concentration (N_D = 5 \times 10^{15}) cm⁻³, calculate:
- Electron concentration (n).
- Hole concentration (p).
- Conductivity (\sigma).
- Resistivity (\rho).
Assume full ionization of donors and negligible intrinsic carriers.
Solution
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Electron concentration
Since donors are fully ionized:
[ n \approx N_D = 5 \times 10^{15}\ \text{cm}^{-3} ] -
Hole concentration
From charge neutrality: ( n = p + N_D ) → ( p = n - N_D ).
Since ( n \approx N_D ), ( p \approx 0 ) (intrinsic holes negligible).
For completeness:
[ p \approx \frac{n_i^2}{N_D} \quad (\text{mass action law}) ] Using ( n_i = 1.5\times10^{10}\ \text{cm}^{-3} ): [ p \approx \frac{(1.5\times10^{10})^2}{5\times10^{15}} = \frac{2.25\times10^{20}}{5\times10^{15}} = 4.5\times10^{4}\ \text{cm}^{-3} ] -
Conductivity
[ \sigma = q(n\mu_n + p\mu_p) ] Elementary charge (q = 1.6\times10^{-19}\ \text{C}).
Plugging in: [ \sigma = 1.6\times10^{-19},[ (5\times10^{15}\times1350) + (4.5\times10^{4}\times450) ]\ \text{S/cm} ] The hole term is negligible, so: