Semiconductor Electronics: Materials, Devices and Simple Circuits: Class 12 Physics · Chapter 14

14.1 Introduction

Open up a phone charger and you will find a small transformer, a few diodes and a capacitor. The diodes turn the alternating mains supply into the one-way current that a battery needs. Inside the phone, a chip smaller than a fingernail holds billions of transistors, each switching on and off millions of times a second. All of these are semiconductor devices: tiny pieces of specially prepared crystal in which we decide how many charge carriers there are and which way they move.

For the first half of the twentieth century the same jobs were done by vacuum tubes, also called valves. In the simplest one, the vacuum diode, a metal cathode is heated until electrons boil off its surface, and a second electrode, the plate or anode, collects them across an evacuated gap (Fig. 14.1). A wire-mesh grid placed between the two made the triode, and further electrodes made the tetrode and the pentode; the voltages on these electrodes steered the stream of electrons. The gap had to be a vacuum, because electrons crossing air would keep colliding with gas molecules and lose their energy.

Since electrons leave only the hot cathode, a tube can pass current in one direction only, like a one-way valve in a water pipe; that is where the name valve comes from. Tubes did their job, but they were bulky and fragile, wasted power heating their cathodes, needed supplies of a hundred volts or more, and burnt out after a few thousand hours. ENIAC, a famous room-sized computer of 1945, used about 18 000 of them, and keeping it running meant constantly replacing failed tubes.

In the 1930s physicists working out the quantum theory of solids explained why some crystals behave so differently from metals, and by about 1940 they had found that a junction between two kinds of such crystal could act as a gate for charge. In a solid-state device nothing has to be boiled out of hot metal or sent across an empty space: the charge carriers are already inside the crystal, and their number can be changed by a modest voltage, by warming, or by light falling on it. Take away the heater and the glass and what remains is tiny, cool and sturdy, a device that runs on a few volts, draws little power and can work for decades. That is why even the bulky picture tubes of old television sets, themselves vacuum devices, have vanished in favour of flat LCD and LED panels run by chips.

This chapter builds the ideas in three stages. First, what makes a material a semiconductor and how its charge carriers behave (Section 14.2 to Section 14.4). Second, what happens where p-type material meets n-type material, and how that junction acts as a diode (Section 14.5 and Section 14.6). Third, how diodes turn ac into dc (Section 14.7). The transistor, a three-terminal device built from two junctions, rests on exactly the same ideas.

14.2 Classification of metals, conductors and semiconductors

On the basis of conductivity

In Chapter 3 of Part I you described how well a material conducts by its resistivity $\rho$, or by its conductivity $\sigma =1\mathrm{/}\rho$, measured in siemens per metre ($\mathrm{S}{\mathrm{m}}^{-1}$). Across ordinary solids, $\rho$ spans more than twenty-five powers of ten (Fig. 14.2). Three broad groups stand out:

• Metals have very low resistivity, typically ${10}^{-8}$ to ${10}^{-6}\mathrm{\Omega }\mathrm{m}$, so $\sigma$ is about ${10}^6$ to ${10}^8\mathrm{S}{\mathrm{m}}^{-1}$. (Many textbook tables quote a wider band for metals, up to ${10}^{-2}\mathrm{\Omega }\mathrm{m}$.)
• Semiconductors lie in between, roughly ${10}^{-5}$ to ${10}^6\mathrm{\Omega }\mathrm{m}$, with $\sigma$ from about ${10}^5$ down to ${10}^{-6}\mathrm{S}{\mathrm{m}}^{-1}$.
• Insulators have very high resistivity, roughly ${10}^{11}$ to ${10}^{19}\mathrm{\Omega }\mathrm{m}$, so $\sigma$ is only ${10}^{-11}$ to ${10}^{-19}\mathrm{S}{\mathrm{m}}^{-1}$.

Treat these limits as signposts rather than fences: some materials fall outside them. Resistivity on its own is also not a safe test. A semiconductor really gives itself away by how its resistivity changes. Warm it, and its resistivity falls steeply, the opposite of what happens in a metal. Shine light on it, or add a trace of the right impurity, and its resistivity can drop by many powers of ten. Explaining this needs a deeper picture, the energy-band model below.

Semiconductors come in several families:

Most devices today use silicon, germanium or inorganic compounds. Since about 1990, however, devices made from organic semiconductors and semiconducting polymers have opened up new fields, polymer electronics and molecular electronics; the 2000 Nobel Prize in Chemistry honoured the discovery that suitably treated polymers can conduct. In this chapter we stay with the elemental semiconductors silicon and germanium. The ideas carry over, with small changes, to most compound semiconductors.

On the basis of energy bands

In the chapter on atoms you saw that an electron in an isolated atom can have only certain sharp energies. A crystal changes this. Its atoms are packed a few tenths of a nanometre apart, close enough for the outer orbits of neighbours to overlap, so an outer electron no longer belongs to one atom: it is pulled by many nuclei and pushed by many other electrons. Each electron's surroundings differ slightly from every other's, so where the isolated atom had one sharp energy, a crystal of $N$ atoms has $N$ slightly different energies. With $N$ around ${10}^{23}$ they are so closely spaced that together they form a practically continuous energy band.

Why, then, is the lower band of silicon exactly full at absolute zero? It comes down to counting. A silicon atom (outer shell $n=3$) or a germanium atom (outer shell $n=4$) brings four outer electrons into the crystal, two s and two p, but its outer shell has eight places: two s states and six p states, counting both spin directions. So $N$ atoms bring $4N$ electrons for $8N$ places. As Fig. 14.3 shows, the $8N$ places first merge into one broad band and then, at the spacing found in the real crystal, regroup into two bands of exactly $4N$ places each, with a gap between them. At absolute zero the $4N$ electrons settle into the lower band and fill it completely, leaving the upper band empty.

The energy gap between the two bands is the band gap:

Why should the gap decide whether a solid conducts? An electric field speeds up an electron by giving it a little extra energy, and that needs an empty state just above the one the electron occupies. In a completely full band there is no such state, so the electrons of a full band cannot carry a current, however many of them there are. Conduction needs electrons in a band with empty states nearby, or vacancies in a band that is otherwise full. Three situations arise (Fig. 14.4):

1. Metals. The conduction band overlaps the valence band, or the highest occupied band is only partly filled. Plenty of electrons sit right next to empty states, so even a weak field drives a large current: the resistivity is low.
2. Insulators. The gap is wide, $E_g>3\mathrm{e}\mathrm{V}$; for diamond it is about $5.4\mathrm{e}\mathrm{V}$. At room temperature the typical thermal energy is only $kT\approx 0.026\mathrm{e}\mathrm{V}$, so practically no electron is ever lifted across. The conduction band stays empty and no current flows.
3. Semiconductors. The gap is narrow, $E_g<3\mathrm{e}\mathrm{V}$: about $1.1\mathrm{e}\mathrm{V}$ for silicon and $0.7\mathrm{e}\mathrm{V}$ for germanium. At room temperature a small number of electrons do get across. They move freely in the conduction band, and the vacancies they leave let the remaining valence electrons move as well. The conductivity is modest, and it climbs steeply with temperature as more electrons cross.

14.3 Intrinsic semiconductor

Silicon and germanium crystallise in the same arrangement as carbon atoms in diamond, the diamond lattice. Every atom sits at the centre of a tetrahedron whose four corners are its nearest neighbours. Each atom has four valence electrons, and each of them pairs up with one electron from a different neighbour. Such a shared pair holds two atoms together: it is a covalent bond. So every atom is held in place by four covalent bonds, and every valence electron is tied up in one of them.

The real lattice is three-dimensional, but a flat sketch shows the bonds more clearly: atoms on a square grid, each joined to its four neighbours by a bond holding two electrons (Fig. 14.5). At very low temperatures every bond is intact. No electron is free to move, so the crystal cannot conduct: a perfect semiconductor at $0\mathrm{K}$ is an insulator.

Warm the crystal and its atoms jiggle about their places. Every so often the jiggling passes one bonding electron at least $E_g$ of energy, and it tears loose to roam the crystal as a free electron. What it leaves behind is a bond with a gap in it, a hole (होल). The gap is not neutral. The atoms around it were neutral before the electron left, so after losing a charge $-e$ that spot carries $+e$. In band language, an electron has been lifted from the valence band into the conduction band, leaving an empty state below.

In a pure crystal every free electron is born together with its own hole, so the numbers per unit volume are equal. A semiconductor so pure that its carriers come only from thermal excitation is called an intrinsic semiconductor (नैज अर्धचालक). For it,

Two quite separate motions are now possible. The freed electron roams through the conduction band like any conduction electron. Independently, the hole travels by relay: an electron from a neighbouring bond slips sideways into the gap, which completes that bond and opens a new gap where the electron came from. The electron moved one step; the hole moved one step the other way (Fig. 14.5, step 3). Relay after relay, the gap wanders through the crystal as if it were a positive particle.

Apply an electric field and both kinds of carrier drift. Free electrons, being negative, move against the field. Holes behave as positive charges and move along it, towards the negative terminal. A negative charge moving one way and a positive charge moving the other way are both conventional current in the direction of the field. Writing $I_e$ for the part carried by electrons and $I_h$ for the part carried by holes, the total is

The number of pairs is not fixed once and for all. A pair is born whenever a bond breaks, and it dies whenever a roaming electron drops into a hole and remakes the bond, a process called recombination. Like a town whose population stays steady because births and deaths balance, a crystal at constant temperature settles where pairs form and vanish at equal rates, and $n_i$ is the population at that balance. Warm the crystal and the birth rate rises, so the balance moves to a larger $n_i$. That is why the resistance of a pure semiconductor falls as it warms, whereas a metal's rises.

14.4 Extrinsic semiconductor

Pure silicon is a poor conductor at room temperature. It has about $1.0\times {10}^{16}$ free electrons per cubic metre among $5.0\times {10}^{28}$ atoms, so only about one atom in $5\times {10}^{12}$ has lost an electron. Worse, $n_i$ changes rapidly with temperature, so a device made of pure silicon would behave differently on a cold morning and a hot afternoon. Useful devices need far more carriers, in a number we can choose.

The cure is surprisingly gentle. Replace a few silicon atoms in every million, or fewer, by atoms of a carefully chosen element, and the conductivity can rise a million times or more. This deliberate contamination is called doping; the added atoms are dopants, and a crystal treated this way is an extrinsic, or doped, semiconductor (बाह्य अर्धचालक). The dopant must fit: its atoms should be close in size to silicon's, so that each one simply takes the place of a silicon atom in an otherwise perfect lattice.

Silicon and germanium belong to group 14 and have four valence electrons. Useful dopants come from the neighbouring groups: pentavalent atoms, with five valence electrons, such as phosphorus (P), arsenic (As) and antimony (Sb) from group 15; and trivalent atoms, with three, such as boron (B), aluminium (Al), gallium (Ga) and indium (In) from group 13. The two families produce two quite different kinds of semiconductor.

n-type semiconductor

Put a phosphorus atom in place of a silicon atom (Fig. 14.6). Its five valence electrons are one too many for the four bonds on offer, so four go into bonds and one is left over. What holds the spare electron? Only the leftover charge $+e$ of the phosphorus atom, so the pair resembles a hydrogen atom, but a very feeble one: the surrounding silicon weakens the attraction so much that the electron's orbit spreads over many atoms. Pulling it away takes only about $0.05\mathrm{e}\mathrm{V}$ in silicon and $0.01\mathrm{e}\mathrm{V}$ in germanium, against $13.6\mathrm{e}\mathrm{V}$ for real hydrogen and $1.1\mathrm{e}\mathrm{V}$ (silicon) or $0.7\mathrm{e}\mathrm{V}$ (germanium) to break a bond. Room-temperature thermal energy, about $0.026\mathrm{e}\mathrm{V}$, is enough to shake almost every spare electron loose, leaving a fixed $\mathrm{P}{\phantom{X}}^{+}$ ion behind. An impurity that gives the crystal an electron in this way is a donor (दाता).

In silicon doped to typical levels, the donor electrons outnumber the thermally freed ones by many powers of ten. The holes fare even worse than in pure silicon: surrounded by a sea of donor electrons, each one is quickly found and filled, so the hole concentration drops below $n_i$. The electrons are therefore called the majority carriers and the holes the minority carriers, and the material is an n-type semiconductor, n for its negative majority carriers:

p-type semiconductor

Boron does the opposite. With only three valence electrons it can complete three of its four bonds; the fourth is left one electron short, which means a hole sits right beside the boron atom (Fig. 14.6, step 3). Very little energy is needed for an electron from a nearby Si–Si bond to drop into that gap. When it does, the boron atom is fully bonded but carries one electron more than its protons can balance, so it becomes a fixed $\mathrm{B}{\phantom{X}}^{-}$ ion, while the gap has moved off into the crystal as a free hole. An impurity that takes an electron from the lattice in this way, creating one hole per atom, is an acceptor (ग्राही).

Everything now runs in mirror image. Acceptor holes vastly outnumber thermal ones, and the few thermally freed electrons are soon mopped up by the abundant holes. Holes are the majority carriers and electrons the minority carriers, and the material is a p-type semiconductor, p for its positive majority carriers:

A doped crystal is still electrically neutral. In n-type material every extra free electron is matched by a fixed donor ion such as $\mathrm{P}{\phantom{X}}^{+}$; in p-type material every extra hole is matched by a fixed acceptor ion such as $\mathrm{B}{\phantom{X}}^{-}$. Doping changes which charges are free to move, not the total charge.

Energy bands of doped semiconductors

In band language (Fig. 14.7), a donor's spare electron occupies a donor level $E_D$ inside the gap, only about $0.05\mathrm{e}\mathrm{V}$ below $E_C$ in silicon. Set that beside $kT\approx 0.026\mathrm{e}\mathrm{V}$ and it is no surprise that, at room temperature, nearly every donor electron has been lifted into the conduction band; in n-type silicon the conduction band is populated almost entirely by them. An acceptor adds an acceptor level $E_A$ just above $E_V$. Valence electrons hop up into these levels just as easily, attaching themselves to the acceptors and leaving holes in the valence band.

Band diagrams have one quirk. Up means more energy for an electron; a hole is a missing electron, so a hole's energy increases downwards. Electrons that gain energy rise; holes that gain energy sink. That is why you may read that a hole 'falls' from the acceptor level into the valence band. It is the same event as an electron climbing from the valence band to the acceptor level.

14.5 p-n junction

Almost every semiconductor device, from the diode in a charger to the transistors in a processor, the LED in a torch and the solar cell on a roof, is built around one structure: a p-n junction, the boundary between a p-type region and an n-type region in the same crystal. To understand those devices, we first need to see what happens at this boundary when it is made, and then what an applied voltage, called a bias, does to it.

14.5.1 p-n junction formation

A junction cannot be made by pressing two pieces together (Example 14.3 explains why); it has to be grown inside one crystal. A common recipe starts from a thin slice, or wafer, of p-type silicon. Donor atoms are driven into part of it, by diffusion from the surface at high temperature or by firing them in as a beam of ions, until they outnumber the acceptors there and that part becomes n-type. The plane where the doping changes over is called the metallurgical junction.

Two kinds of carrier motion now compete (Fig. 14.8). The first is diffusion: carriers spread from where they are crowded to where they are scarce, the way a drop of ink spreads through still water. The p-side is crowded with holes and the n-side with electrons, so holes diffuse from p to n and electrons from n to p. Both motions amount to conventional current from p to n, called the diffusion current.

Diffusion cannot go on for long, because every crossing leaves something behind. A donor atom whose electron has gone is a positive ion locked into the lattice; an acceptor atom whose hole has gone is a negative ion, equally fixed. The carriers that crossed also meet, and recombine with, the majority carriers waiting on the far side. The result is a thin slab straddling the junction that has been emptied of mobile carriers and holds only bare ions, negative on the p-side and positive on the n-side. This slab is the depletion region (अवक्षय क्षेत्र), typically about $0.1\mu \mathrm{m}$ thick, only a few hundred atomic spacings.

Bare ions mean unbalanced charge, and unbalanced charge means a field. Here it points from the positive ions to the negative ones, from n to p, and it acts on every mobile carrier inside the slab. Minority carriers that stray in, electrons from the p-side or holes from the n-side, are swept across. This field-driven motion is drift, and its current, the drift current, flows from n to p, against the diffusion current.

The process limits itself, like a crowd that builds its own fence. The more carriers diffuse, the more ions are uncovered, the stronger the field becomes and the larger the drift current pushing back. Growth stops when the two currents are equal. Both continue to flow, but in a junction left to itself the net current is zero.

Seen as a potential, the uncovered charge has lifted the n-side above the p-side by an amount $V_0$. An electron trying to diffuse from n to p has to climb this step, and so does a hole trying to go from p to n, which is why $V_0$ is called the barrier potential (रोधिका विभव). It is a few tenths of a volt: roughly $0.6$ to $0.8\mathrm{V}$ for silicon junctions and $0.2$ to $0.4\mathrm{V}$ for germanium. Small as that is, it sits across such a thin layer that the field inside is huge (Example).

14.6 Semiconductor diode

Add metal contacts to the two ends of a p-n junction and you have a semiconductor diode, a two-terminal device (Fig. 14.9). The terminal on the p-side is the anode and the one on the n-side is the cathode. In circuit diagrams a diode is drawn as a triangle pointing into a bar. The triangle points the way conventional current flows easily, from p to n, when the diode is forward biased; the bar marks the n-side. On a real diode a painted band marks the cathode end.

A battery connected across the diode adds its voltage $V$ to the barrier or takes it away, depending on which way round it is connected. With no battery the junction is in the equilibrium of Section 14.5.1: barrier $V_0$ and no net current.

14.6.1 p-n junction diode under forward bias

Join the p-side to the positive terminal of a battery and the n-side to the negative terminal, and the diode is forward biased (अग्रदिशिक बायस). Where does the battery's voltage go? The diode behaves like three resistors in series: the p-region, the depletion region and the n-region. The middle one, stripped of carriers, is by far the largest, so nearly all the voltage drops across it. There it acts against the built-in step: the barrier shrinks from $V_0$ to $V_0-V$, fewer uncovered ions are needed to hold it, and the depletion region narrows (Fig. 14.10).

Why does the current then grow so dramatically? Picture the carriers at the foot of the barrier as athletes of mixed ability, all trying to clear a wall. Most can jump only a little; very few can jump high. With a tall wall almost nobody gets over. Lower it slightly and only the rare strong jumpers clear it, so the current stays small. Lower it further and the number getting over does not just grow, it multiplies, because each step down brings in a much larger group of ordinary jumpers. That is why the forward current rises roughly exponentially with $V$.

What does a carrier do once it is over? An electron arriving on the p-side is now a minority carrier in a sea of holes; a hole arriving on the n-side is likewise outnumbered. This is called minority carrier injection. If you could count minority carriers along the diode, you would find them heaped up just outside the depletion region and thinning out with distance (Fig. 14.11), because they diffuse away from the edge and are gradually lost to recombination. These two diffusion streams, injected electrons spreading into the p-region and injected holes spreading into the n-region, together make up the forward current, typically milliamperes or more.

14.6.2 p-n junction diode under reverse bias

Now turn the battery round, so that the n-side is positive: the diode is reverse biased (पश्चदिशिक बायस). The voltage still lands almost entirely on the depletion region, but this time it adds to the built-in step. The barrier grows to $V_0+V$, more ions must be uncovered to support it, and the depletion region widens. For majority carriers the wall is now far too high, and the diffusion current practically disappears.

A trickle of current survives, carried by minority carriers. Any electron that wanders up to the junction from the p-side, or any hole from the n-side, is caught by the field and carried across. Think of a waterfall: the water pouring over each second is set by the river upstream, not by the height of the cliff. In the same way this current is set by how many minority carriers reach the junction each second, not by the battery. Even a small reverse voltage collects them all, and a larger one cannot collect carriers that are not there, so the reverse current hardly changes with voltage. It is also tiny: microamperes in germanium diodes, nanoamperes in silicon ones. (The same drift flows under forward bias too, but there it is lost beside the injected carriers.) Heating creates more minority carriers, so the reverse current grows with temperature.

Every diode has its limit. Beyond a certain reverse voltage, the breakdown voltage $V_{br}$, the reverse current stops being tiny and shoots up, so that a slightly larger voltage drives a very large current. That current heats the junction, and if nothing else in the circuit holds it below the maximum the manufacturer specifies, the diode is ruined. A forward-biased diode pushed beyond its rated current fails in the same way. Ordinary diodes are therefore kept well short of breakdown. (Zener diodes are special diodes designed to work in breakdown; they are used to hold a voltage steady.)

The graph of current against voltage, the V–I characteristic, is measured with the circuit of Fig. 14.12. A rheostat or a potentiometer sets the voltage across the diode, which a voltmeter connected across it reads, while a meter in series reads the current. Forward currents are milliamperes, so a milliammeter serves. To measure the reverse half, the diode is turned round and a microammeter takes the milliammeter's place, since the currents are a thousand or more times smaller.

The result looks like Fig. 14.13. On the forward side, almost nothing happens at first; then, over a narrow range of voltage, the current takes off and climbs almost exponentially, so that a few hundredths of a volt more can double it. The voltage where this take-off happens is called the cut-in (or threshold) voltage: around $0.7\mathrm{V}$ for silicon diodes and $0.2\mathrm{V}$ for germanium ones. On the reverse side the current stays tiny and nearly level, the reverse saturation current, all the way to breakdown. The two halves are almost always drawn to very different current scales.

A single number such as $V\mathrm{/}I$ cannot describe a device like this. On the steep forward part it is tens of ohms, on the reverse part many megaohms, and it changes from point to point. In a circuit what usually matters is how much the current changes when the voltage changes a little, so for a diode we use the slope of the characteristic at the working point, the dynamic resistance:

For an ohmic resistor the characteristic is a straight line through the origin, so $\mathrm{\Delta }V\mathrm{/}\mathrm{\Delta }I$ and $V\mathrm{/}I$ are the same number everywhere along it. Check this with the simulation below, then go back to the junction explorer, where $r_d$ changes by a factor of a million or more as you move along the curve.

14.7 Application of junction diode as a rectifier

Mains electricity alternates, but batteries, LED lamps and most electronic circuits need a voltage that never reverses. The characteristic in Fig. 14.13 shows how a diode can help: it passes current readily one way and hardly at all the other way. Place a diode between an ac source and a load, and charge reaches the load only during the half cycles that forward-bias the diode. A circuit that uses diodes in this way to turn ac into one-way current is called a rectifier (दिष्टकारी).

Half-wave rectifier

The simplest rectifier is a single diode in series with the load $R_L$, fed from the secondary terminals A and B of a transformer that provides a convenient voltage (Fig. 14.16). What happens depends only on which way the secondary voltage points:

So the load receives a row of positive humps separated by gaps, one hump per input cycle. Its voltage still varies, but it never changes sign, which is what rectified means; and because only half of every cycle reaches the load, the circuit is called a half-wave rectifier. Its tiny reverse current while blocking can be ignored. There is also a design rule hidden in the second row of the table: while the diode blocks, the whole peak secondary voltage is across it in the reverse direction, so its breakdown voltage must be comfortably larger than that peak.

Full-wave rectifier

Using both halves of the cycle takes two diodes and a transformer whose secondary has a connection at its middle, the centre tap C (Fig. 14.17). Diode $D_1$ runs from end A of the secondary and diode $D_2$ from end B, with their p-sides (anodes) on A and B; their cathodes are joined, and the load runs from that joint back to C. Measured from C, ends A and B always have equal and opposite voltages, so the two diodes take turns:

Either way the current through $R_L$ has the same direction, so the load receives a hump in every half cycle and the output pulses come at twice the input frequency. Each diode uses only half of the secondary winding and works only half the time, but no part of the input cycle is wasted, so this full-wave rectifier makes far better use of the supply than the half-wave circuit.

Filtering the output

A rectified output flows one way but is still lumpy, rising from zero to the peak and back a hundred times a second for full-wave rectification of the mains. Its varying part is called the ripple. Smoothing it is the job of a filter: most simply a capacitor connected across the output, in parallel with $R_L$, though an inductor in series with $R_L$ can do the same job.

Think of the capacitor as a water tank that a tap fills in bursts while a hole in its base drains it steadily (Fig. 14.18). Each rectified pulse is a burst from the tap that tops the capacitor up to the peak voltage. Between pulses the diode is off and the load drains the capacitor, so its voltage sags. A big tank barely notices the gaps; a small one nearly empties. In circuit terms the rate of draining is governed by the time constant $\tau =R_{L}C$, and the ripple is small when $\tau$ is much longer than the time between pulses. Power supplies therefore use large capacitors, and the output of such a capacitor-input filter stays close to the peak voltage. It is by far the most common filter in practice.