Semiconductor Electronics: Materials, Diodes and Simple Circuits notes: ISC Class 12 Physics

1. Energy bands: conductors, insulators and semiconductors

In a crystal the electrons of neighbouring atoms interact and no two electrons may share a state, so each atomic energy level splits into N closely spaced levels that merge into a band. The band holding the outer electrons is the valence band; the next band up is the conduction band; between them lies the forbidden gap $E_g$ (energy bands are treated qualitatively only in ISC).
Conductor: the bands overlap (or the conduction band is part-filled): many free electrons at all temperatures.
Insulator: valence band full, conduction band empty, large gap (more than about 3 eV): carbon (diamond) 5.5 eV; nothing conducts at room temperature.
Semiconductor: same arrangement, small gap: germanium 0.7 eV, silicon 1.1 eV. At 0 K it is an insulator; at room temperature a few electrons cross the gap, leaving holes; heating lowers the resistance.
A photon can lift an electron across the gap only if $h\nu \ge E_g$ , i.e. $\lambda \le \dfrac{hc}{E_g} = \dfrac{1240}{E_g}$ nm: silicon absorbs up to 1127 nm (infrared); diamond only below 225 nm, so it is transparent to visible light.
Crystal structure is a removed topic and is not examined.

λ = 1240/Eg

Photons with wavelength above these values cannot lift electrons across the gap.

Formulas

Longest wavelength that can lift an electron across the gap
$\lambda_{max} = \dfrac{hc}{E_g} = \dfrac{1240}{E_g(\text{eV})}\ \text{nm}$

Key terms

Forbidden gap: The range of energies between the top of the valence band and the bottom of the conduction band, where a perfect crystal has no electron states.
Valence band: The band of energies occupied by the outer (valence) electrons of the atoms.
Conduction band: The band above the valence band; electrons in it are free to carry current.

2. Intrinsic and extrinsic semiconductors; doping

Intrinsic (pure) semiconductor: thermal energy breaks bonds, giving equal numbers of free electrons and holes. A hole is a vacant bond that behaves as a carrier of charge +e; bonding electrons jump into it, so it drifts the opposite way to the electrons. Total current = electron current + hole current.
Doping (about 1 atom in 10⁷) gives an extrinsic semiconductor. n-type: a pentavalent donor (P, As, Sb) uses four valence electrons in bonds and gives one free electron, becoming a fixed positive ion; electrons are the majority carriers, holes the minority carriers. p-type: a trivalent acceptor (B, Al, Ga, In) leaves one bond short, i.e. a hole, and becomes a fixed negative ion; holes are the majority, electrons the minority.
In energy terms a donor level lies just below the conduction band and an acceptor level just above the valence band (about 0.05 eV in silicon).
Doped crystals remain electrically neutral. The conductivity is $\sigma = e(n_e\mu_e + n_h\mu_h)$ ; doping raises it by many orders of magnitude (one donor in 10⁷ atoms gives about 5 × 10²¹ electrons per m³, against 1.5 × 10¹⁶ per m³ in pure silicon). Warming adds mainly minority carriers.

Formulas

Conductivity of a semiconductor
$\sigma = e\,(n_e \mu_e + n_h \mu_h)$

Key terms

Hole: A vacancy in a covalent bond that behaves as a mobile carrier of positive charge +e.
Donor: A pentavalent impurity that gives a free electron to the crystal and becomes a fixed positive ion.
Acceptor: A trivalent impurity that accepts an electron from a bond, creating a hole, and becomes a fixed negative ion.
Majority carrier: The type of carrier present in far larger numbers: electrons in n-type, holes in p-type.

3. The p-n junction: depletion region and barrier; bias

Joining p-type and n-type material: electrons diffuse from n to p and holes from p to n, and recombine near the junction. This uncovers fixed positive donor ions on the n side and negative acceptor ions on the p side: the depletion region (about 1 μm wide, no mobile carriers). Its field (about 10⁶ V m⁻¹) points from n to p, and the potential difference across it is the barrier potential V₀: about 0.7 V for silicon, 0.3 V for germanium. It stops further net diffusion.
Forward bias (battery + to p): the barrier falls to V₀ − V, the region narrows, majority carriers cross easily and a large current flows. Reverse bias (battery + to n): the barrier rises to V₀ + V, the region widens, and only a tiny reverse current of minority carriers flows.

Formulas

Barrier height with bias
$V_{barrier} = V_0 - V\ \text{(forward)}, \quad V_0 + V\ \text{(reverse)}$

Key terms

Depletion region: The thin layer at a p-n junction containing fixed ions and no mobile carriers.
Barrier potential: The potential difference V₀ across the depletion region that stops further diffusion of the majority carriers.

4. V–I characteristics and simple diode circuits

Forward: almost no current up to the knee voltage (about 0.7 V for Si, 0.3 V for Ge), then a very steep rise. Reverse: a tiny current (nA for silicon) that hardly depends on the voltage, because it is carried by thermally generated minority carriers, until breakdown at a large reverse voltage.
Dynamic resistance $r_d = \dfrac{\Delta V}{\Delta I}$ (a few ohms in the steep region); static resistance is V/I.
Diode circuits: first decide whether each diode is forward biased (p side at the higher voltage by more than the knee) or reverse biased. A conducting silicon diode drops about 0.7 V (germanium 0.3 V); a reverse-biased diode carries no current. Example: Si diode in series with 220 Ω across 6.0 V: I = (6.0 − 0.7)/220 = 24.1 mA.

silicon
germanium

Model curves I = I₀ exp(V/V_T) fixed to give 1 mA at the knee; the knee is about 0.3 V for Ge and 0.7 V for Si.

reverse current of a silicon diode

The reverse current is tiny and almost independent of the voltage (until breakdown).

Formulas

Dynamic resistance
$r_d = \dfrac{\Delta V}{\Delta I}$
Current in a series diode circuit
$I = \dfrac{E - V_{drop}}{R}$

Key terms

Knee voltage: The forward voltage at which the current of a diode begins to rise steeply (about 0.7 V for Si, 0.3 V for Ge).
Dynamic resistance: The ratio ΔV/ΔI of a small change of voltage to the resulting change of current on the forward characteristic.

5. Rectifiers: half-wave and full-wave (centre-tap); the filter

Half-wave: transformer secondary, diode and load in series. The diode conducts in the positive half-cycles only: output pulses at the same frequency as the input (50 Hz), large ripple.
Centre-tap full-wave (two diodes): the secondary ends A and B, equal and opposite with respect to the centre tap C, go to the anodes of D₁ and D₂; the cathodes join to the load, whose other end goes to C. D₁ and D₂ conduct in alternate half-cycles and the current always flows the same way through the load: ripple at twice the input frequency (100 Hz). (The bridge rectifier with four diodes is removed in ISC.)
Roles: transformer — suitable a.c. voltage (and two phases); diode — one-way conduction; filter capacitor — charges at the peaks and discharges slowly into the load; load — what is supplied.
Ripple with a capacitor: $\Delta V \approx \dfrac{I t}{C}$ with $t = \dfrac{1}{2f}$ (full-wave) or $\dfrac{1}{f}$ (half-wave). Peak output = $V_m - 0.7$ V for a silicon diode, with $V_m = \sqrt{2}\,V_{rms}$ .

full-wave output, no capacitor
with a 470 μF capacitor across a 100 Ω load

The capacitor charges at each peak and supplies the load in between: the ripple is small.

Formulas

Ripple frequency
$f_{ripple} = f\ \text{(half-wave)}, \quad 2f\ \text{(full-wave)}$
Ripple voltage (constant-current discharge)
$\Delta V \approx \dfrac{I\,t}{C}$

Key terms

Rectification: The conversion of an alternating voltage into a unidirectional one by a diode.
Ripple: The small variation of the output voltage of a filtered rectifier about its mean value.
Centre tap: A terminal at the middle of the secondary winding, giving two equal voltages of opposite phase.

6. LED, photodiode and solar cell

LED: a p-n junction in forward bias. Electrons and holes recombine in the junction region and release photons of energy $h\nu \approx E_g$ ; the colour is set by the band gap (about 1.9 eV red, 2.3 eV green, 2.7 eV blue). A series resistor sets the current: $R = \dfrac{V - V_f}{I}$ .
Energy saving: an LED turns far more of its input into light than a filament lamp (which wastes most as heat) and lasts much longer, so the same light needs much less electricity, which means less fuel burnt and less pollution.
Photodiode: used in reverse bias; photons with $h\nu \ge E_g$ create pairs in the depletion region and the field sweeps them apart, so the reverse current rises with the light intensity. The small dark current makes the fractional change large. Uses: detectors, light meters, switches.
Solar cell: a large-area p-n junction with a thin p layer, no bias; the junction field separates the pairs made by light, so the cell is itself a source of emf (about 0.5–0.7 V for silicon). Efficiency = electrical output / light power received.

λ ≈ 1240/Eg

A larger band gap gives light of shorter wavelength: red, then green, then blue.

Formulas

Wavelength of an LED
$\lambda \approx \dfrac{hc}{E_g} = \dfrac{1240}{E_g(\text{eV})}\ \text{nm}$
Series resistor for an LED
$R = \dfrac{V - V_f}{I}$

Key terms

LED: A forward-biased p-n junction that emits light by electron–hole recombination.
Photodiode: A reverse-biased p-n junction whose current increases with the intensity of the light falling on it.
Solar cell: A p-n junction that converts light into electrical energy, with no external bias.

7. The Zener diode and the voltage regulator

A Zener diode is a heavily doped junction that breaks down sharply at a chosen reverse voltage $V_Z$ . In reverse bias there is almost no current until $V_Z$ ; then the current rises steeply while the voltage stays almost equal to $V_Z$ . In forward bias it behaves like an ordinary diode.
Regulator: the Zener diode, reverse connected, is in parallel with the load, with a series resistor $R_s$ . In breakdown $V_{out} = V_Z$ . $I = \dfrac{V_{in} - V_Z}{R_s}$ , $I_L = \dfrac{V_Z}{R_L}$ , $I_Z = I - I_L$ .
If $V_{in}$ rises, the extra current flows through the Zener diode and the extra voltage drops across $R_s$ . If $R_L$ falls, $I_L$ rises and $I_Z$ falls by the same amount. Regulation fails when $I_Z$ would have to fall below its minimum (the load takes all the current) or above the rating (power $V_Z I_Z$ too large).

forward: like an ordinary diode
reverse: almost no current
breakdown at V_Z = 5.6 V

In breakdown the voltage stays almost at V_Z while the current changes over a wide range. (Forward and reverse use the same current scale.)

Zener not conducting: potential divider
Zener in breakdown: output held at V_Z

Above 6.6 V the output is steady at 5.6 V: the regulator at work.

Formulas

Zener current
$I_Z = \dfrac{V_{in} - V_Z}{R_s} - \dfrac{V_Z}{R_L}$
Power dissipated in the Zener diode
$P_Z = V_Z I_Z$

Key terms

Zener voltage: The reverse voltage at which a Zener diode breaks down; the voltage it holds across the load.
Voltage regulator: A circuit that keeps the output voltage nearly constant when the input voltage or the load changes.

Exam tips

Draw band diagrams with the insulator gap clearly largest (diamond 5.5 eV, Si 1.1 eV, Ge 0.7 eV); energy bands are qualitative only.
Doped crystals are neutral; donors are pentavalent, acceptors trivalent; name the majority and minority carriers.
In diode circuits decide forward or reverse bias first, then subtract 0.7 V (Si) or 0.3 V (Ge) only for a conducting diode.
Rectifiers: circuit, waveforms and output frequency (f for half-wave, 2f for full-wave); ripple ΔV = It/C.
LED forward, photodiode reverse, solar cell no bias; Zener reverse across the load with a series resistor, I = I_Z + I_L.

1. Energy bands: conductors, insulators and semiconductors

Formulas

Key terms

2. Intrinsic and extrinsic semiconductors; doping

Formulas

Key terms

3. The p-n junction: depletion region and barrier; bias

Formulas

Key terms

4. V–I characteristics and simple diode circuits

Formulas

Key terms

5. Rectifiers: half-wave and full-wave (centre-tap); the filter

Formulas

Key terms

6. LED, photodiode and solar cell

Formulas

Key terms

7. The Zener diode and the voltage regulator

Formulas

Key terms

Exam tips

Cookies