Nearly every question from this chapter is conceptual or graph-based, and it is one of the highest-yield chapters in Class 12 Physics because the number of formulas is tiny while the number of "recognise the situation" facts is large. Master the band picture, the p–n junction, and the four special diodes, and you can usually score full marks here without heavy calculation.
Energy Bands: Why Some Solids Conduct and Others Don't
In an isolated atom electrons occupy sharp, discrete energy levels. Bring $10^{23}$ atoms together into a crystal and each level splits into a huge number of closely spaced levels, which merge into continuous energy bands. The highest band that is completely filled at absolute zero is the valence band; the next allowed band above it is the conduction band; the energy gap between the top of one and the bottom of the other is the forbidden gap $E_g$, in which no electron state exists.
An electron can carry current only if it has empty states available immediately above its own energy. That single idea explains the whole classification:
- Metals (conductors): either the conduction band is partly filled, or the valence and conduction bands overlap, so $E_g \approx 0$. Countless empty states sit right next to occupied ones, so even a feeble field produces a large current. Resistivity $\sim 10^{-8}$–$10^{-6}\ \Omega,\text{m}$.
- Insulators: $E_g$ is large (roughly above 3 eV, e.g. diamond ~5.5 eV). Thermal energy at room temperature ($kT \approx 0.026$ eV) cannot promote electrons across such a gap, so the material is effectively non-conducting. Resistivity $\sim 10^{11}$–$10^{19}\ \Omega,\text{m}$.
- Semiconductors: $E_g$ is small — 1.1 eV for silicon, 0.72 eV for germanium. At 0 K a pure semiconductor behaves like an insulator, but at room temperature a small fraction of covalent bonds break, sending electrons into the conduction band.
Every electron that jumps up leaves behind a vacancy in the valence band called a hole, which behaves like a mobile particle of charge $+e$. Conduction in semiconductors therefore has two channels: electron drift in the conduction band and hole drift in the valence band (physically, a valence electron hopping into the vacancy makes the vacancy move backwards). Because the number of carriers grows roughly exponentially with temperature, a semiconductor's resistance falls as it is heated — a negative temperature coefficient of resistance, opposite to that of a metal.
Intrinsic and Extrinsic Semiconductors
A chemically pure semiconductor crystal is called intrinsic. Here electrons and holes are created only in pairs by thermal breaking of bonds, so $$n_e = n_h = n_i$$ where $n_i$ is the intrinsic carrier concentration. For silicon $n_i \sim 10^{16}\ \text{m}^{-3}$ at room temperature, against about $10^{29}\ \text{m}^{-3}$ atoms — hence the very poor conductivity. Intrinsic material is nearly useless for devices, so we deliberately add impurities, a process called doping. The dopant concentration is tiny (parts per million) yet can raise conductivity by many orders of magnitude. Doped material is extrinsic.
For tetravalent Si or Ge, two dopant families are used:
- Pentavalent dopants (As, P, Sb, Bi): four of the five valence electrons form covalent bonds; the fifth is loosely bound and needs only ~0.01–0.05 eV to enter the conduction band. Its donor level lies just below the conduction band. Result: an n-type semiconductor in which electrons are majority carriers and holes are minority carriers, $n_e \gg n_h$.
- Trivalent dopants (In, B, Al, Ga): they can complete only three bonds, leaving one bond short — effectively a hole. The acceptor level lies just above the valence band. Result: a p-type semiconductor with holes as majority and electrons as minority carriers, $n_h \gg n_e$.
Two points are frequently tested. First, both n-type and p-type crystals are electrically neutral — the extra electron in n-type is balanced by the extra proton in the donor nucleus. Calling n-type "negatively charged" is wrong. Second, in thermal equilibrium the law of mass action holds: $$n_e n_h = n_i^2$$ so doping that raises one carrier concentration necessarily lowers the other. For example, if doping makes $n_e = 100,n_i$, then $n_h = n_i/100$.
The p–n Junction and Its I–V Characteristic
A p–n junction is made by doping one region of a single crystal p-type and the adjacent region n-type. Two processes then occur at the boundary:
- Diffusion: majority holes drift from p to n and majority electrons from n to p, because of the steep concentration gradients.
- Drift: the departing carriers leave behind immobile ionised donors (positive, on the n side) and acceptors (negative, on the p side). This forms a depletion region (typically a fraction of a micrometre wide) containing an internal electric field pointing from n to p, which pushes minority carriers back across.
Equilibrium is reached when the drift current exactly cancels the diffusion current, and the built-in barrier potential $V_B$ (about 0.7 V for Si, 0.3 V for Ge) appears across the depletion layer. No net current flows in an open-circuited junction, and no external voltmeter can read $V_B$ directly.
Forward bias (p-side to battery +, n-side to –): the applied field opposes the internal field, the barrier height and depletion width both shrink, and majority carriers cross the junction in large numbers. Current stays negligibly small until the applied voltage exceeds the knee (cut-in) voltage (~0.7 V for Si, ~0.3 V for Ge), then rises very steeply. The diode's dynamic resistance $r_d = \Delta V/\Delta I$ is small, of the order of a few ohms, and the resistance is non-linear, so Ohm's law does not apply to a diode.
Reverse bias (p to –, n to +): the barrier and depletion width increase, majority-carrier flow is blocked, and only a tiny reverse saturation current (µA in Ge, nA in Si) flows, carried entirely by minority carriers. Since the minority-carrier supply is set by temperature, not voltage, this current is almost independent of the reverse voltage. Beyond the breakdown voltage the current increases abruptly; an ordinary diode is usually destroyed here, but a Zener diode is designed to work in this region.
The full characteristic is described by the diode equation $I = I_0\left(e^{eV/k_BT}-1\right)$, and the essential takeaway is unidirectional conduction: the diode is a closed switch when forward biased and an open switch when reverse biased. Note that current is plotted in mA on the forward axis but µA on the reverse axis, which is why the graph looks so asymmetric.
Diode as a Rectifier
Rectification converts ac to dc using the diode's one-way conduction.
Half-wave rectifier: a single diode in series with the load and the secondary of a transformer. During the half-cycle that forward biases the diode, current flows through the load; during the other half-cycle the diode blocks and the output is zero. The output is a train of pulses at the same frequency as the input (50 Hz for mains input). Efficiency is low because half of the input is discarded.
Full-wave rectifier: uses a centre-tapped transformer and two diodes whose anodes go to the two ends of the secondary, with the load between the centre tap and the common cathode point. In each half-cycle one diode conducts while the other is reverse biased, but current through the load flows in the same direction both times. Key facts:
- Output pulses occur in both half-cycles, so the ripple frequency is twice the input frequency (100 Hz for a 50 Hz supply).
- Both diodes never conduct simultaneously.
- The output is still pulsating dc, not steady dc.
To smooth the output, a filter is used: a capacitor in parallel with the load charges at the peaks and discharges slowly through the load during the gaps, so a large capacitance (large $RC$ compared with the pulse period) gives a smaller ripple. An inductor in series performs the complementary job of opposing current changes. The remaining unwanted ac component is the ripple, and the fraction of dc obtained is expressed by the rectifier efficiency.
Special-Purpose Diodes
Zener diode. A heavily doped p–n junction with a very thin depletion layer, giving an extremely strong field ($\sim 10^6$ V/m) at modest reverse voltage. Breakdown therefore occurs by the Zener effect — the field itself tears valence electrons out of host bonds — at a sharp, well-defined reverse voltage $V_z$. Beyond breakdown the voltage across it stays essentially constant while the current changes greatly, which makes it an ideal voltage regulator: connect it in reverse bias across the load, with a series resistance to absorb the excess voltage and limit the current. If the input voltage rises, the extra voltage is dropped across the series resistor and the Zener current increases, while the load voltage remains fixed at $V_z$. Circuit symbol has a bent (Z-shaped) bar.
Photodiode. A junction with a transparent window, operated in reverse bias and illuminated. Photons with $h\nu > E_g$ generate extra electron–hole pairs, increasing the reverse current in proportion to light intensity. Reverse bias is used because the fractional change in the small minority-carrier current is large and easy to measure against the tiny reverse saturation current — in forward bias the light-generated current would be swamped by the much larger majority-carrier diffusion current, making the light's effect hard to detect.
Light-emitting diode (LED). A heavily doped p–n junction made from a direct-bandgap compound semiconductor (GaAs, GaP, or their alloys), operated in forward bias. Injected electrons and holes recombine near the junction, and the released energy (≈ $E_g$) escapes as a photon rather than as heat, provided the bandgap corresponds to a visible or near-visible wavelength. The emitted colour is fixed by $E_g$ (roughly $\lambda \approx 1240/E_g(\text{eV})$ nm in round numbers), so different compositions give red, green or blue LEDs. LEDs need only a modest turn-on voltage (about 1.5–3.5 V) and are far more efficient than incandescent bulbs for a given light output, but have very low reverse breakdown voltages and must never be driven beyond their rated forward current.
Solar cell (photovoltaic cell). Essentially a photodiode run with no external bias, built to convert light directly into electrical energy. Light generates electron–hole pairs near the junction, and the built-in field of the depletion region separates them before they can recombine, sending electrons to the n-side and holes to the p-side — the cell itself becomes the source, unlike a photodiode which needs an external battery. Design choices reflect this: the top layer is thin and lightly doped so light reaches the junction, the junction area is kept large to intercept more photons, and an anti-reflective coating cuts surface losses.
Junction Transistor: Structure and Action
A bipolar junction transistor (BJT) is two back-to-back p–n junctions sharing a thin, lightly doped middle region, the base, sandwiched between a heavily doped emitter and a moderately doped collector. In normal (active-mode) operation of an n-p-n transistor, the emitter–base junction is forward biased and the collector–base junction is reverse biased. Electrons injected from the emitter diffuse across the thin base — very few recombine there, since the base is both thin and lightly doped — and are swept into the collector by its reverse-biased field. The collector current therefore nearly equals the emitter current, with only a small base current making up the difference: $I_E = I_B + I_C$ and $I_C \approx I_E \gg I_B$.
Current gain is defined two ways: $\alpha = I_C/I_E$ (common base, always just under 1) and $\beta = I_C/I_B$ (common emitter, typically 20–500), related by $\beta = \alpha/(1-\alpha)$. In the common-emitter configuration — the one almost always examined — a transistor works either as a switch (cut-off: both junctions reverse biased, negligible current, open switch; saturation: both junctions forward biased, maximum current, closed switch) or as an amplifier (active region: small changes in base current or voltage drive much larger, phase-inverted changes in collector current and output voltage).
Common Mistakes and Exam Traps
Reversing which bias goes with which special diode. A Zener diode and a photodiode are both used in reverse bias (for regulation and for sensitivity respectively), while an LED and an ordinary rectifying diode work in forward bias — mixing these up is the single most common error in this chapter.
Assuming n-type or p-type material carries a net charge. Both are electrically neutral; doping only changes which carrier is in the majority, never the overall charge of the crystal.
Believing resistivity always falls on heating. True for semiconductors and insulators, where more carriers are freed, but the opposite of what happens in a metal, where heating shortens the relaxation time and raises resistivity — the negative-temperature-coefficient idea belongs only to semiconductors.
Forgetting $I_E = I_B + I_C$ and mixing up $\alpha$ with $\beta$. A common numerical trap gives one gain and one current and expects the other via $\beta = \alpha/(1-\alpha)$ or $I_C = \beta I_B$; assuming $\alpha \approx \beta$, or skipping straight past the algebra, gives a wrong numerical answer even when the underlying concept is understood.