Why a Missing Electron Behaves Exactly Like a Particle of Its Own

Every electronic circuit needs some device capable of a controlled flow of electrons, and before the transistor's invention in 1948, this job fell to vacuum tubes (also called valves): the vacuum diode (two electrodes, an anode and a heated cathode), the triode (three electrodes, adding a control grid), and even four- and five-electrode tetrodes and pentodes. Inside a vacuum tube, a heated cathode supplies electrons, and varying the voltage between electrodes controls their flow through a hard vacuum -- vacuum is essential, since colliding with ordinary air molecules would rob the electrons of their energy before they arrived. These devices worked, powering the first century of electronics, but at a real cost: vacuum tubes are bulky, consume substantial power, typically require high operating voltages (around 100 V), and have a limited working life and comparatively low reliability. The seeds of an alternative were already sprouting in the 1930s, when physicists realised that certain solid-state semiconductor materials, and the junctions between them, could control both the number and direction of charge flow entirely within the solid itself -- no heated cathode, no evacuated glass envelope required. A semiconductor device can be small, low-power, low-voltage, long-lived, and highly reliable, which is exactly why solid-state devices have displaced vacuum tubes almost everywhere, right down to the now-obsolete cathode ray tube television and computer monitor, replaced by solid-state liquid crystal displays. The earliest practical use of this idea actually predates any formal understanding of how it worked: a naturally occurring crystal of galena (lead sulphide) with a fine metal point pressed against it was already being used as a radio wave detector, decades before anyone could properly explain why it worked.

Solids can be sorted into metals, semiconductors and insulators purely by their resistivity: metals sit at roughly 10⁻² to 10⁻⁸ Ω m (very low resistivity, very high conductivity), insulators at roughly 10¹¹ to 10¹⁹ Ω m (extremely high resistivity), and semiconductors squarely in between, at roughly 10⁻⁵ to 10⁶ Ω m -- with Si and Ge as the two elemental semiconductors this chapter focuses on, alongside compound semiconductors like GaAs and CdS. A deeper explanation comes from energy bands: inside a solid crystal, so many atoms sit so close together that their outer electron orbits overlap, splitting what would be sharp, single-atom energy levels into continuous bands -- the valence band (holding the valence electrons) and, above it, the conduction band, separated by an energy gap, Eg. Three distinct cases follow directly from the size of this gap. In a metal, the conduction and valence bands either overlap completely or the conduction band is already partly filled, so electrons move into it essentially for free, giving metals their very low resistivity. In an insulator, the gap is large (Eg > 3 eV), far too wide for ordinary thermal energy to bridge, so the conduction band stays completely empty and no conduction is possible. In a semiconductor, the gap is small (Eg < 3 eV) -- small enough that at room temperature, a modest number of valence electrons do acquire enough thermal energy to leap across into the conduction band, giving semiconductors their intermediate, temperature-sensitive conductivity.

Silicon and germanium both crystallise in a diamond-like lattice, each atom bonded to four nearest neighbours -- and since Si and Ge each have exactly four valence electrons, every atom shares one electron with each of its four neighbours, and takes a share of one electron back from each of them in return, forming four covalent bonds that hold the whole lattice rigidly together. At absolute zero, every one of these bonds is intact, and the crystal behaves as a perfect insulator: no free charges exist anywhere. As temperature rises, thermal energy becomes available to jostle these bound electrons, and occasionally, a bond breaks -- one electron gains enough energy to escape entirely, becoming a free, mobile conduction electron, and leaving behind an empty slot in the bond it abandoned. This vacant slot, called a hole, carries an effective positive charge (since the local neighbourhood has lost one negative electron) and behaves, remarkably, as if it were itself a genuine, independently mobile positive particle: a bound electron from a neighbouring, still-intact bond can hop over to fill the vacancy, which makes the hole appear to have moved one step in the opposite direction -- and repeating this many times over lets a hole drift steadily through the crystal, exactly as if it were a real, positively charged particle, even though no actual particle is making that whole journey. In an intrinsic (undoped) semiconductor, free electrons and holes are always created in exactly equal numbers, ne=nh=ni, and the total current is the sum of both contributions, I=Ie+Ih -- a genuinely unusual situation, since ordinary conductors only ever have one kind of mobile charge carrier.

An intrinsic semiconductor's conductivity, however real, is far too low at room temperature and far too sensitive to temperature changes to build reliable devices from directly -- the practical fix is doping: deliberately adding a carefully chosen impurity, the dopant, at a level of just a few parts per million, chosen to be close enough in atomic size to the host lattice that it slots into the crystal structure without distorting it. Doping tetravalent Si or Ge with a pentavalent element (five valence electrons, such as arsenic, antimony or phosphorus) leaves four of the dopant's electrons bonded normally to its four silicon neighbours, with the fifth left only weakly attached to its own parent atom -- weakly enough that it takes just about 0.01 eV to free it in germanium, or about 0.05 eV in silicon, dramatically less than the roughly 0.7 eV or 1.1 eV needed to break an ordinary intrinsic bond in each material respectively. This pentavalent dopant, called a donor, contributes one extra free electron for every atom incorporated, and, since virtually every donor atom ionises at room temperature while only a tiny fraction of intrinsic bonds ever break, electrons become the overwhelming majority carriers (ne>>nh) -- an n-type semiconductor. Doping with a trivalent element instead (three valence electrons, such as indium, boron or aluminium, called an acceptor) leaves one bond short an electron, creating a hole directly, without needing any thermal energy at all -- making holes the majority carriers (nh>>ne) instead, a p-type semiconductor. In both cases, the electron and hole concentrations still obey a fixed relationship, ne nh=ni², and the crystal as a whole remains exactly electrically neutral throughout.

A p-n junction cannot be made by simply pressing a p-type slab against an n-type slab: even the flattest real surface is far rougher, at the atomic scale, than the roughly 2-3 angstrom spacing between neighbouring atoms in a crystal, so true continuous atomic contact is impossible -- instead, a single continuous crystal wafer is doped differently in two adjoining regions. Once formed, the sharp concentration difference across the junction (holes far more numerous on the p-side, electrons far more numerous on the n-side) drives diffusion: holes diffuse from p to n, electrons diffuse from n to p, purely down their own concentration gradients, exactly as any concentrated substance spreads into a region where it is scarce. But each carrier that diffuses across leaves something behind: an electron leaving the n-side leaves behind an immobile, positively-ionised donor atom, and a hole leaving the p-side leaves behind an immobile, negatively-ionised acceptor atom -- so a layer of fixed positive charge builds up on the n-side of the junction, and a layer of fixed negative charge builds up on the p-side, together forming the depletion region (only about a tenth of a micrometre thick), so named because the carriers that once occupied it have been depleted away. This fixed charge sets up an electric field pointing from the positive n-side layer toward the negative p-side layer, which drives a second process, drift, pushing any electron that wanders into the field on the p-side back toward the n-side, and any hole on the n-side back toward the p-side -- exactly opposite in direction to the diffusion current. As the depletion region grows, its field strengthens, and drift current grows to match diffusion current, until the two balance exactly: a stable junction, a fixed barrier potential V0 across it, and zero net current at equilibrium.

Connecting a battery's positive terminal to the p-side and negative terminal to the n-side forward biases the junction, and since this applied voltage runs opposite to the junction's own built-in barrier potential V0, it directly reduces the effective barrier height to (V0-V) -- the depletion layer narrows, and it becomes progressively easier for carriers to cross. With only a small applied voltage, the barrier drops only slightly, and just the small handful of carriers already sitting in the highest energy states have enough energy to cross, so current stays small; increasing the applied voltage further keeps lowering the barrier, letting more and more carriers cross, so forward current rises rapidly, eventually growing large enough (typically measured in mA) to swamp the tiny currents involved at equilibrium. Electrons crossing from n to p, and holes crossing from p to n, arrive as minority carriers on their new side -- a process called minority carrier injection -- and this locally elevated minority-carrier concentration then diffuses further into the bulk material, sustaining a steady current for as long as the forward bias is maintained. Plotting current against voltage reveals a genuinely nonlinear curve: current stays nearly negligible below a characteristic threshold, or cut-in, voltage (about 0.2 V for germanium, about 0.7 V for silicon), then rises sharply, almost exponentially, for any further small increase in voltage beyond that point.

Reversing the battery, positive terminal to the n-side and negative to the p-side, reverse biases the junction instead: the applied voltage now runs the same direction as the built-in barrier, adding to it rather than opposing it, so the effective barrier height grows to (V0+V), the depletion layer widens, and diffusion across the junction is suppressed almost entirely. A small drift current still flows, though, carried by the minority carriers already wandering near the junction (electrons on the p-side, holes on the n-side), which the junction's own field sweeps across regardless of barrier height -- this reverse saturation current is small (typically microamps), and stays nearly constant however much the reverse voltage increases, since it is limited by how many minority carriers happen to be available near the junction, not by the size of the barrier itself. Push the reverse voltage far enough, though, past a critical breakdown voltage Vbr, and the current increases sharply and dramatically -- if left unchecked by an external circuit, this can destroy the junction through overheating. The forward and reverse resistances of a real diode differ enormously: a typical silicon diode's forward (dynamic) resistance, rd=ΔV/ΔI, might be only about 10 Ω, while its reverse resistance can reach 10⁷ Ω or more -- a roughly million-fold asymmetry that makes a diode behave, for most practical purposes, as a genuinely one-way valve for current.

A diode's near-total refusal to conduct in reverse, paired with its easy forward conduction, is exactly the property needed to rectify an alternating voltage -- convert it from a signal that regularly reverses direction into one that flows only one way. The simplest version, a half-wave rectifier, places a single diode in series with a load: whenever the input's polarity forward-biases the diode, current flows and a pulse of output voltage appears across the load; whenever the polarity reverses, the diode blocks it almost completely, and the output drops to essentially zero -- so the output is a series of one-directional pulses, occurring at exactly the same frequency as the input (50 Hz in, 50 Hz worth of pulses out), each pulse only half of the original wave's full cycle. A full-wave rectifier improves on this using two diodes and a centre-tapped transformer: one diode conducts during one half of the input cycle, the other conducts during the opposite half, so together they deliver an output pulse during every half-cycle of the input, doubling the output frequency to twice the input frequency (50 Hz in, 100 Hz worth of pulses out) and roughly doubling the average output as well, making it the more efficient of the two circuits.

A rectifier's raw output, though genuinely one-directional, still isn't steady, constant DC -- it rises and falls in a series of humps, tracing the shape of half-sinusoids. A capacitor connected across the output, in parallel with the load, smooths this out: as the rectified voltage rises toward each peak, the capacitor charges up to match it; once the rectified voltage starts falling again, the capacitor, now charged higher than the momentary output, discharges through the load instead, holding the output voltage up and filling in what would otherwise be a deep dip -- before the next pulse arrives and recharges it to the peak once more. How well this smooths the output depends on the time constant, set by the product of the capacitance C and the load resistance RL: a larger capacitor discharges more slowly between pulses, holding the voltage nearer its peak value throughout, and producing an output closer to true, steady DC -- which is exactly why capacitor-input filters favour large capacitance, and why this simple diode-plus-capacitor combination sits at the heart of virtually every power supply converting household alternating current into the steady direct current that every solid-state device, the very devices that replaced the bulky vacuum tube this chapter opened with, actually needs to run.

To trace out a p-n junction diode's V-I characteristics, two circuits are built side by side, differing only in which way the diode is turned. In the forward-bias circuit, the diode's p-side is wired toward the battery's positive terminal (through a rheostat that varies the voltage), and a milliammeter in series tracks the sizeable current that flows as a voltmeter across the diode reads the rising forward voltage. In the reverse-bias circuit, the very same diode is turned around so its n-side now faces the positive terminal instead, and because so little current gets through, a far more sensitive microammeter replaces the milliammeter to catch the tiny reverse leakage current. Plotting both sets of readings together produces the full V-I curve: a diode that conducts eagerly in one direction and almost refuses to in the other.