Semiconductors

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your JEE Advanced exam.

Semiconductors are materials with electrical conductivity between conductors (metals) and insulators. Silicon (Si) and Germanium (Ge) are the most important — they have four valence electrons and form covalent bonds in a diamond crystal lattice.

Band Theory — The Key Framework:

In isolated atoms, electrons occupy discrete energy levels. In a crystal with $N$ atoms, these levels split into bands. Three bands matter:

• Valence band (VB): fully occupied at $T = 0$ — electrons here cannot conduct
• Conduction band (CB): empty or partially filled — electrons here can conduct
• Band gap $E_g$: forbidden energy region between VB and CB

Material	VB	CB	Conductivity
Insulator	Full	Empty	$E_g > 3$ eV: almost zero
Semiconductor	Full	Empty (at $T=0$)	$E_g \approx 0.7-1.1$ eV: small
Conductor	Overlaps or partially filled	—	High

$E_g$ for Si = 1.1 eV; for Ge = 0.67 eV.

Intrinsic (Pure) Semiconductors:

At $T > 0$, some electrons thermally excite from VB to CB, creating equal numbers of free electrons (in CB) and holes (missing electrons in VB). The intrinsic carrier concentration: $$n_i = AT^{3/2}e^{-E_g/(2k_BT)}$$

At 300 K: $n_i \approx 1.5 \times 10^{16}$ m⁻³ for Si.

⚡ JEE Advanced exam tips:

• In intrinsic semiconductor: electron concentration = hole concentration = $n_i$
• Conductivity $\sigma = n_i e(\mu_e + \mu_h)$ where $\mu$ = mobility
• Silicon has smaller $n_i$ than germanium at the same temperature → Si has lower intrinsic conductivity but is preferred for devices because of better thermal stability (larger $E_g$)

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🟡 Standard — Regular Study (2d–2mo)

For JEE Advanced students who want genuine understanding.

Doping — n-type and p-type Semiconductors:

n-type doping (donors): Add Group V elements (P, As, Sb) to Si. Five valence electrons — one extra electron per dopant is loosely bound (donor level just below CB). At room temperature, this electron is free to conduct.

• Majority carriers: electrons
• Minority carriers: holes

p-type doping (acceptors): Add Group III elements (B, Ga, In) to Si. Three valence electrons — one incomplete bond creates a hole that can accept an electron (acceptor level just above VB).

• Majority carriers: holes
• Minority carriers: electrons

The p-n Junction — Key Device:

When p-type and n-type materials are joined:

1. Depletion region: around the junction, no free carriers exist — only ionized dopant atoms
2. Built-in potential $V_{bi}$: due to band bending at the junction; $V_{bi} = \frac{k_BT}{e}\ln\frac{N_AN_D}{n_i^2}$ (for Si at 300 K, $V_{bi} ≈ 0.7$ V)
3. Barrier potential: approximately 0.7 V for Si, 0.3 V for Ge

Forward Bias (p-side connected to positive terminal):

• Depletion region narrows
• Current flows when applied voltage > $V_{bi}$
• Diode equation: $I = I_0\left(e^{eV/(k_BT)} - 1\right)$
• At room temperature, $k_BT/e ≈ 26$ mV

Reverse Bias (n-side connected to positive terminal):

• Depletion region widens
• Only a tiny reverse saturation current ($I_0$) flows
• Breakdown occurs at high reverse voltage (Zener or avalanche breakdown)

Junction Transistor (BJT):

npn transistor: emitter-base is forward biased, collector-base is reverse biased. Current relations: $$I_E = I_B + I_C$$ $$\alpha = \frac{I_C}{I_E} \approx \frac{\beta}{\beta+1}$$ $$\beta = \frac{I_C}{I_B}$$

Common emitter current gain $\beta$ typically ranges from 20 to 200.

⚡ Common student mistakes:

1. Confusing majority and minority carriers in n-type vs p-type
2. Forgetting that holes move opposite to electron flow (conventional current direction is opposite to electron flow)
3. Not understanding that the depletion region has no free carriers but has ionised impurity atoms

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for JEE Advanced mastery of semiconductor physics.

Fermi Level in Semiconductors:

The Fermi level $E_F$ is the energy level at which the probability of occupation is 50%.

In an intrinsic semiconductor: $E_F$ is at the centre of the band gap.

In an n-type semiconductor: $E_F$ is closer to the CB (above the centre of the gap). In a p-type semiconductor: $E_F$ is closer to the VB (below the centre of the gap).

More precisely, for n-type with donor concentration $N_D$: $$E_F - E_i = k_BT \ln\frac{N_D}{n_i}$$ where $E_i$ is the intrinsic Fermi level.

Hall Effect:

When a magnetic field $B$ is applied perpendicular to current $I$ in a conductor, a transverse voltage develops (Hall voltage $V_H$). This confirms that conduction is by negative charges (electrons): $$V_H = \frac{IB}{ned}$$ where $n$ = charge carrier density, $d$ = thickness.

For p-type semiconductors, the Hall voltage is opposite in sign, confirming hole conduction.

Thermistors and Photoconductors:

Thermistor (NTC): resistance decreases exponentially with temperature — used for temperature sensing. $R_T = R_0 e^{B(1/T - 1/T_0)}$ where $B$ is the material constant (typically 3000–5000 K).

Photoconductor/LDR: conductivity increases when light is incident (photons create electron-hole pairs).

LED (Light Emitting Diode):

When forward biased, electrons recombine with holes in the p-region, releasing energy as photons. The photon energy: $$E = hf = \frac{hc}{\lambda} \approx \frac{1.24 \text{ eV·μm}}{\lambda(\mu\text{m})}$$

For GaAs LED ($\lambda ≈ 0.87$ μm): $E ≈ 1.42$ eV. LED colour depends on band gap: infrared ($> 1.78$ μm), red (620–750 nm), green (495–570 nm), blue (450–495 nm).

Zener Diode:

A heavily doped p-n junction designed to operate in reverse breakdown region. The Zener voltage $V_Z$ is controlled by doping level. For $V_Z ≈ 5-10$ V, temperature coefficient is minimal because Zener breakdown (tunnelling) has opposite temperature coefficient to avalanche breakdown.

Solar Cell (Photovoltaic):

A p-n junction under illumination generates electron-hole pairs. If the junction is open-circuited, the photo-generated carriers accumulate, creating a photovoltage ( Voc ). If connected to a load, photocurrent flows.

Transistor Configurations — Detailed:

For common emitter (CE) configuration: $$I_C = \beta I_B$$ $$I_E = (1+\beta)I_B$$ Voltage gain $A_v = \frac{\beta R_C}{r_e}$ where $r_e = \frac{k_BT}{eI_E}$.

For a silicon BJT at 300 K with $I_E = 1$ mA: $r_e ≈ 26$ Ω.

Rectifier Circuits:

Half-wave rectifier: conducts only on positive half-cycle. Output frequency = input frequency. Ripple factor = 1.21.

Full-wave bridge rectifier: conducts on both half-cycles (each diode conducts alternately). Output frequency = $2 \times$ input frequency. Ripple factor = 0.48.

Filter capacitor reduces ripple — the capacitor charges to peak voltage and discharges slowly through the load between peaks.

JEE Advanced Previous Year Patterns:

• p-n junction and diode characteristics: very common
• Transistor configurations and current relations: very common
• Fermi level: common
• LED and solar cell: periodic
• Hall effect: occasionally tested
• Transistor as amplifier: common

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