JEE Physics Semiconductor Electronics Notes Set 04

Energy Bands and Semiconductors

In a solid the energy levels form bands. A conductor has overlapping bands, an insulator has a large forbidden gap (more than about 3 eV) and a semiconductor has a small gap (silicon 1.1 eV, germanium 0.67 eV). At 0 K a pure semiconductor has a full valence band and an empty conduction band, so it behaves as an insulator; as temperature rises, the number of electron-hole pairs grows roughly as \( e^{-E_g/2kT} \), so its resistance falls, unlike a metal whose resistance rises. For an intrinsic semiconductor \( n_e=n_h=n_i \), and in doped material \( n_en_h=n_i^2 \).

Doping and the p-n Junction

Doping with pentavalent atoms (P, As) gives n-type material with electrons as majority carriers; trivalent atoms (B, Al, In) give p-type material with holes as majority carriers. Both are electrically neutral. Electrons have greater mobility than holes. At a p-n junction, diffusion of carriers leaves a depletion region of fixed ions and a built-in field directed from n to p, which creates a barrier of a few tenths of a volt (about 0.3 V for germanium and 0.7 V for silicon). Forward bias lowers the barrier and gives a large diffusion current; reverse bias raises the barrier and gives a tiny drift current of minority carriers. The ideal diode relation is \( i=i_0(e^{eV/kT}-1) \), with dynamic resistance \( r=dV/di \).

Diode Applications

A diode rectifies alternating current. A half-wave rectifier has ripple frequency equal to the input frequency; a full-wave rectifier doubles it (100 Hz from 50 Hz mains). Special diodes include the Zener diode (voltage regulation in reverse breakdown), photodiode (reverse bias, current sensitive to light), LED (forward bias, emits light of wavelength \( \lambda=\frac{1240}{E_g(\text{eV})} \) nm) and the solar cell.

Transistors

A transistor has emitter, thin base and collector, with \( I_E=I_B+I_C \). In normal (active) operation the emitter-base junction is forward biased and the collector-base junction is reverse biased. The current gains are \( \alpha=\frac{I_C}{I_E}<1 \) and \( \beta=\frac{I_C}{I_B}>1 \), related by \( \beta=\frac{\alpha}{1-\alpha} \). In the common-emitter amplifier the input resistance is \( R_{in}=\frac{\Delta V_{BE}}{\Delta I_B} \), the voltage gain is \( \beta\frac{R_L}{R_{in}} \) and the output is out of phase with the input by \( \pi \); in the common-base amplifier the output is in phase. As a switch, the transistor is cut off or saturated.

Logic Gates

OR gives \( Y=A+B \), AND gives \( Y=A\cdot B \), NOT gives \( Y=\overline A \), NAND is AND followed by NOT, NOR is OR followed by NOT, and XOR gives \( Y=\overline AB+A\overline B \). NAND and NOR are universal gates. Useful laws are \( A+AB=A \) and \( \overline{A+B}=\overline A\cdot\overline B \). An ideal operational amplifier in inverting mode gives \( V_{out}=-\frac{R_f}{R_{in}}V_{in} \).

Worked Examples

If \( \alpha=0.96 \), then \( \beta=\frac{0.96}{0.04}=24 \), so a collector current of 0.6 mA needs a base current of 25 \( \mu \)A.
A silicon junction with a 1 \( \mu \)m depletion layer and 0.7 V barrier has a field of \( 7\times10^5 \) V/m.
An LED with band gap 2.0 eV emits light of wavelength \( \frac{1240}{2.0}=620 \) nm.