Chapter 14: Semiconductors
In the winter of 1947, in a laboratory at Bell Telephone Laboratories in the United States, two physicists, John Bardeen and Walter Brattain, were working with a small piece of germanium. Germanium is neither a good conductor like copper nor a good insulator like glass. It sits awkwardly in between, and for that reason most engineers of the time had little use for it.
Bardeen and Brattain pressed two closely spaced metal contacts onto the surface of the germanium crystal and passed a small signal into one of them. What came out of the other contact was the same signal, but much stronger. A tiny, cold, solid lump of material had amplified an electrical signal — something that until then needed a bulky, hot, power-hungry vacuum tube. Working with them, William Shockley soon developed a more practical version of the device. It was named the transistor, and the three of them later received the Nobel Prize for it.
That single result changed the direction of technology. The room-sized computers of the 1940s shrank into the phone in your pocket. Today, billions of such switches, carved into silicon, sit inside a chip smaller than your fingernail. Every one of them works because a semiconductor can be persuaded — by heat, by light, or by a carefully added impurity — to change how it conducts.
Diagram
Fig. 14.1 – An opening illustration contrasting a bulky glass vacuum tube with a modern silicon chip held on a fingertip, with a magnified view of a semiconductor crystal lattice.
This chapter will answer each of these questions.
14.1 Introduction
Why Semiconductors Matter
Modern life leans heavily on electronic gadgets. A cell phone, a smart watch, a computer, even an LED lamp — all of them look very different from one another, yet they share one common factor. Inside each of them are semiconductor devices, and it is these devices that make them work.
Semiconductors have made everyday life comfortable and easy. But the real reason physicists and engineers find them so valuable is not simply that they conduct electricity. It is that their electrical properties can be tailored to suit our requirements. A material whose behaviour we can adjust is far more useful than one that behaves the same way no matter what we do to it.
Conductors, Insulators — and Something in Between
Before semiconductors were discovered and understood, the electrical properties of materials were thought to fall into just two classes: conductors and insulators.
Conductors, such as metals, have a very high electrical conductivity. Electrical conductivity, written as \(\sigma\), tells us how easily a material allows electric current to pass through it; its SI unit is siemens per metre (\(\mathrm{S\ m^{-1}}\)). The conductivity of silver, for example, is \(6.25 \times 10^{7}\ \mathrm{S\ m^{-1}}\) — a very large number.
An insulator, or a bad conductor, sits at the opposite extreme. Glass has an electrical conductivity of the order of \(10^{-10}\ \mathrm{S\ m^{-1}}\). That is smaller than silver’s value by roughly seventeen powers of ten, which is why glass is used to stop current, not to carry it.
Silicon does not belong comfortably in either group. The electrical conductivity of silicon is \(1.56 \times 10^{-3}\ \mathrm{S\ m^{-1}}\). This value lies between that of a good conductor and that of a bad conductor. Materials that occupy this middle ground are called semiconductors, and this in-between position is exactly what makes them special, as you can see on the conductivity scale in Fig. 14.2.
Diagram
Fig. 14.2 – A horizontal logarithmic conductivity scale from \(10^{-25}\) to \(10^{8}\) S m\(^{-1}\), marking wood/teflon/glass at the insulator end, silicon and germanium in the middle, and silver/copper at the conductor end.
The Two Handles We Have on a Semiconductor
There are two properties of a semiconductor that we can deliberately control, and both are used in this chapter.
First, a semiconductor can be customised so that its electrical conductivity has the value we want. We are not stuck with the conductivity nature gave it.
Second, the way its electrical conductivity depends on temperature can also be controlled. This is not a small point: in most materials, temperature dependence is a nuisance to be tolerated, whereas in a semiconductor it becomes a property we can put to work.
Table 14.1 lists the electrical conductivity of some commonly used materials, arranged so that you can compare metals, semiconductors and insulators at a glance.
Table 14.1: Electrical conductivities of some commonly used materials
| Material | Electrical conductivity (\(\mathrm{S\ m^{-1}}\)) |
|---|---|
| Silver | \(6.30 \times 10^{7}\) |
| Copper | \(5.96 \times 10^{7}\) |
| Aluminium | \(3.5 \times 10^{7}\) |
| Gold | \(4.10 \times 10^{7}\) |
| Nichrome | \(9.09 \times 10^{5}\) |
| Platinum | \(9.43 \times 10^{6}\) |
| Germanium | \(2.17\) |
| Silicon | \(1.56 \times 10^{-3}\) |
| Air | \(3 \times 10^{-15}\) to \(8 \times 10^{-15}\) |
| Glass | \(10^{-11}\) to \(10^{-15}\) |
| Teflon | \(10^{-25}\) to \(10^{-23}\) |
| Wood | \(10^{-16}\) to \(10^{-24}\) |
Read the table carefully and notice the enormous spread. From teflon at about \(10^{-25}\ \mathrm{S\ m^{-1}}\) to silver at about \(10^{7}\ \mathrm{S\ m^{-1}}\) is a range of more than thirty powers of ten. Germanium and silicon sit in the middle of this vast range, far above the insulators and far below the metals.
In examination questions, conductivity data of this kind is usually tested in one of two ways: either you are asked to place a given material in the correct class (conductor, semiconductor or insulator) from its conductivity value, or you are asked to convert between conductivity \(\sigma\) and resistivity \(\rho\) using \(\rho = 1/\sigma\) and then find a resistance. Both are quick marks, provided the powers of ten are handled carefully.
14.2 Electrical Conduction in Solids
Electrical conduction in a solid takes place by the transport of charge carriers. A charge carrier is simply a particle that carries charge and is free to move through the material — in most solids these are electrons. If nothing inside a solid is free to move, no current can flow through it, no matter how large a voltage you apply.
How well a solid conducts depends on several things together: its temperature, the number of charge carriers available inside it, how easily those carriers can move through the solid (this ease of movement is called mobility), its crystal structure, and the types and nature of the defects present in it.
On the basis of how well they conduct, solids can be placed in three groups. A solid may be a good conductor, a semiconductor, or a bad conductor.
Conductors (Metals)
The best example of a conductor is any metal. Metals contain a large number of free electrons that are available for electrical conduction. A typical metal has about \(10^{28}\) electrons per \(\mathrm{m^{3}}\).
Metals are good conductors of electricity precisely because of this huge supply of free electrons. In a metal, the outermost electron of each atom is not held by any particular atom; it belongs to the metal as a whole and wanders freely through it. Apply even a small electric field and this vast population of electrons begins to drift, producing a large current.
Insulators
Glass, wood and rubber are some common examples of insulators. Insulators have a very small number of free electrons — about \(10^{23}\) per \(\mathrm{m^{3}}\).
Compare this with the \(10^{28}\) per \(\mathrm{m^{3}}\) of a metal. The insulator has about one hundred thousand times fewer free electrons in the same volume. Nearly all of its electrons are locked into bonds and are not free to move, which is why an insulator carries almost no current.
Semiconductors
Silicon, germanium, gallium arsenide, gallium nitride and cadmium sulphide are some of the commonly used semiconductors. The electrical conductivity of a semiconductor lies between the conductivity of a metal and that of an insulator.
Two features make semiconductors extraordinarily useful. First, the number of charge carriers in a semiconductor can be controlled as per our requirement. Second, their structure can also be designed to suit our requirement.
Because of this, such materials are very useful in the electronic industry. They are found in almost every gadget of daily use, such as a cell phone, and in complex systems such as a satellite.
What Conductivity Actually Depends On
It is worth writing down, in one line, everything that decides the conductivity of a solid.
This one relation, \(\sigma = nq\mu\), explains the whole classification above. A metal has a very large \(n\), so \(\sigma\) is large. An insulator has a very small \(n\), so \(\sigma\) is tiny. And in a semiconductor, \(n\) is something we can deliberately change — which is why \(\sigma\) becomes something we can deliberately change.
The Opposite Effect of Temperature
Figure 14.3 shows the temperature dependence of the electrical conductivity of a typical metal and of a semiconductor. The two curves run in opposite directions, and this contrast is one of the most important facts in the chapter.
Diagram
Fig. 14.3 – Two graphs of conductivity versus temperature: (a) for a metal, a falling curve; (b) for a semiconductor, a rising curve that saturates.
When the temperature of a semiconductor is increased, its electrical conductivity also increases. The electrical conductivity of a metal, on the other hand, decreases with an increase in its temperature.
Look again at \(\sigma = nq\mu\) and the reason becomes clear. In a metal, the number of free electrons \(n\) is already fixed and does not grow with heating; but heating makes the atoms of the lattice vibrate more strongly, and these vibrations obstruct the moving electrons. The mobility \(\mu\) therefore falls, and with it the conductivity.
In a semiconductor the situation is reversed. Here \(n\) is small to begin with, and heat supplies the energy needed to free additional carriers. The rise in \(n\) is so steep that it easily overwhelms any fall in mobility, and the conductivity rises. Exactly how heat frees these carriers is explained in the sections that follow.
A Broad Classification of Semiconductors
Semiconductors can be classified broadly into three groups, according to what they are made of.
a. Elemental semiconductors: Silicon, germanium. These are made of a single element.
b. Compound semiconductors: Cadmium sulphide, zinc sulphide, and so on. These are made by combining two or more elements.
c. Organic semiconductors: Anthracene, doped phthalocyanines, polyaniline, and so on. These are carbon-based materials.
Elemental semiconductors and compound semiconductors are widely used in the electronic industry. The discovery of organic semiconductors is relatively new, and they find lesser applications.
Why Silicon and Germanium?
The electrical properties of semiconductors are different from those of metals and insulators because of their unique conduction mechanism. To understand that mechanism we must first look at the atoms themselves.
The electronic configuration of the elemental semiconductors silicon and germanium plays a very important role in their electrical properties. Both are from the fourth group of elements in the periodic table, and both have a valence of four — that is, each atom has four electrons in its outermost shell available for bonding.
Their atoms are bonded to one another by covalent bonds. In a covalent bond, two neighbouring atoms share a pair of electrons, and each shared pair holds the two atoms together. Each silicon or germanium atom uses its four valence electrons to form four such bonds with four neighbouring atoms.
At absolute zero temperature, all the covalent bonds are completely satisfied in a single crystal of pure silicon or germanium. Every valence electron is tied up in a bond; not one of them is free to wander. A pure semiconductor at absolute zero therefore behaves like a perfect insulator — an important idea that we return to in Section 14.4.
The conduction mechanism in a semiconductor can be understood much better with the help of the band theory of solids, which we take up next.
14.3 Band Theory of Solids, a Brief Introduction
We begin with the way electron energies in an isolated atom are distributed. An isolated atom has its nucleus at the centre, surrounded by a number of revolving electrons. These electrons are arranged in different and discrete energy levels.
The word discrete is important here. An electron in an isolated atom cannot have just any energy it likes. It is allowed only certain fixed values of energy — the energy levels — and nothing in between, rather like a person who may stand on any step of a staircase but never float between two steps.
When a solid is formed, a very large number of atoms are packed together. The outermost electronic energy levels in a solid are then occupied by electrons coming from all the atoms of the solid, not from any one atom alone. The sharing of these outermost energy levels, and the formation of energy bands that results from it, can be easily understood by considering the formation of solid sodium.
Why Sodium Is a Convenient Example
The electronic configuration of sodium (atomic number 11) is \(1s^{2},\ 2s^{2},\ 2p^{6},\ 3s^{1}\). The outermost level, 3s, can take one more electron, but in sodium it is only half filled — it holds one electron where it could hold two.
This half-filled outermost level is what makes sodium such a clear example, as you will see shortly.
When solid sodium is formed, the atoms interact with each other through the electrons in each atom. The energy levels are then filled according to Pauli’s exclusion principle.
According to this principle, no two electrons can have the same set of quantum numbers — or, in simple words, no two electrons with similar spin can occupy the same energy level.
Any energy level can therefore accommodate only two electrons — one in the spin up state and the other in the spin down state. According to this principle, there can be two states per energy level.
From Levels to Bands
Figure 14.4 (a) shows the allowed energy levels of an isolated sodium atom by horizontal lines. The curved lines in the same figure represent the potential energy of an electron near the nucleus, due to the Coulomb interaction between the negatively charged electron and the positively charged nucleus.
Diagram
Fig. 14.4 – Potential energy diagram, energy levels and bands for (a) an isolated sodium atom, (b) two sodium atoms brought close, (c) sodium metal.
Now consider two sodium atoms brought close enough to each other that the outer 3s electrons are equally likely to be found on either atom. The 3s electrons from both the sodium atoms now need to be accommodated in the same level.
But a single level has room for only two electrons. Nature’s way out is to split the 3s level into two sub-levels, so that Pauli’s exclusion principle is not violated. Figure 14.4 (b) shows this splitting of the 3s level into two sub-levels.
When solid sodium is formed, the atoms come close to each other — the distance between them is about \(2\) to \(3\ \text{Å}\). Therefore, the electrons from different atoms interact with each other, and also with the neighbouring atomic cores.
The interaction between the outermost electrons is greater, because their orbits overlap, while the inner core electrons remain mostly unaffected. The inner electrons sit deep inside each atom and hardly notice their neighbours.
Each of these energy levels is now split into a very large number of sub-levels — of the order of Avogadro’s number. This is because the number of atoms in a piece of solid sodium is itself of the order of that number, roughly \(10^{23}\).
The separation between these sub-levels is so extremely small that the energy levels appear almost continuous. Such a continuum of energy levels is called an energy band. The bands are named after the levels they came from: the 1s band, the 2s band, the 2p band, and so on.
Figure 14.4 (c) shows these bands in sodium metal. Notice in the figure that the broadening of the valence band and the higher bands is greater; this is because the interaction of these outer electrons is stronger.
Valence Band and Conduction Band
For a sodium atom, the topmost occupied energy level is the 3s level. This level is called the valence level. The corresponding energy band in the solid is called the valence band.
Thus, the valence band in solid sodium is the topmost occupied energy band. Since the 3s level held only one of the two electrons it could hold, the valence band in sodium is half filled. Figure 14.5 shows the energy bands in sodium: the 1s, 2s and 2p bands are completely filled, while the 3s band is only half filled.
Diagram
Fig. 14.5 – Energy bands in sodium, showing the completely filled 1s, 2s and 2p bands and the half-filled 3s band.
When sufficient energy is provided to the electrons in the valence band, they are raised to higher levels. The immediately next energy level that an electron from the valence band can occupy is called the conduction level. The band formed by these conduction levels is called the conduction band.
In sodium, the valence band and the conduction band overlap. There is no separation at all between them, and this single fact is why sodium is a metal.
In a semiconductor or an insulator, matters are different. There is a gap between the bottom of the conduction band and the top of the valence band. This gap is called the energy gap or the band gap, and Fig. 14.6 shows it for a typical solid that is not a good conductor.
Diagram
Fig. 14.6 – Energy bands for a typical solid: a valence band at the bottom, a conduction band at the top, and the energy gap between them.
It is important to remember that this band structure is related to the energy of the electrons in a solid, and it does not represent the physical structure of a solid in any way. The conduction band is not a place located “above” the valence band inside the crystal; it is simply a range of higher energies that an electron may possess.
Getting an Electron Across the Gap
All the energy levels in a band, including the topmost band, in a semiconductor are completely occupied at absolute zero. Every electron sits in the lowest level available to it, and there is no vacancy anywhere in the valence band.
At some finite temperature \(T\), however, a few electrons gain thermal energy of the order of \(kT\), where \(k\) is the Boltzmann constant. This is the energy that random thermal agitation of the lattice hands over to the electrons.
Electrons in the bands below the valence band cannot move to a higher band, since the levels there are already occupied — Pauli’s principle blocks them. Only electrons from the valence band can be excited into the empty conduction band, and this happens only if the thermal energy gained by these electrons is greater than the band gap.
In the case of sodium, no such gap has to be crossed at all. Electrons from the 3s band can gain thermal energy and occupy a slightly higher energy level within the same band, because the 3s band is only half filled — there are empty levels sitting right above the filled ones.
Electrons can also gain energy when an external electric field is applied to a solid. The energy gained from an electric field is much smaller than thermal energy, and hence only the electrons at the topmost energy level can gain such energy and take part in electrical conduction.
Band Structure of Metals, Semiconductors and Insulators
The difference in the electrical conductivities of various solids can now be explained on the basis of their band structure. The band structure of a metal, of a semiconductor and of an insulator is different, and Fig. 14.7 shows a schematic representation of all three side by side.
Diagram
Fig. 14.7 – Band structure of (a) a metal, with overlapping valence and conduction bands, (b) a semiconductor, with a small band gap, and (c) an insulator, with a wide band gap.
For metals, the valence band and the conduction band overlap and there is no band gap at all, as shown in Fig. 14.7 (a). Electrons therefore find it easy to gain electrical energy when some external electric field is applied. They are, as a result, easily available for conduction.
In the case of semiconductors, the band gap is fairly small — of the order of one electron volt or less — as shown in Fig. 14.7 (b). When excited, electrons gain energy and occupy energy levels in the conduction band easily, and can then take part in electric conduction. Ordinary room-temperature heat is enough to push at least a few of them across.
Insulators, on the contrary, have a wide gap between the valence band and the conduction band, as shown in Fig. 14.7 (c). Diamond, for example, has a band gap of about \(5.0\ \mathrm{eV}\). In an insulator, therefore, electrons find it very difficult to gain sufficient energy and occupy energy levels in the conduction band. Room-temperature thermal energy is nowhere near enough, so the conduction band stays empty and no current flows.
The magnitude of the band gap thus plays a very important role in the electronic properties of a solid. It is not a minor detail of the material — it is the single number that decides whether the solid will behave as a conductor, a semiconductor or an insulator.
Table 14.2: Magnitude of energy gap in silicon, germanium and diamond
| Material | Energy gap (eV) at 300 K |
|---|---|
| Silicon | 1.12 |
| Germanium | 0.66 |
| Diamond | 5.47 |
In examinations, this section is most often tested through band diagrams: you may be asked to draw and label the band structure of a metal, a semiconductor and an insulator, to state the band gap of silicon or germanium, or to explain why a given solid conducts as it does. The reasoning always runs the same way — band gap first, conduction second.
14.4 Intrinsic Semiconductor
A pure semiconductor, such as pure silicon or pure germanium, is called an intrinsic semiconductor. The word intrinsic means “belonging to the thing itself” — whatever conduction takes place in such a crystal comes from the host material alone, with nothing added from outside.
The Structure of Pure Silicon
Silicon (Si) has atomic number 14, and its electronic configuration is \(1s^{2}\ 2s^{2}\ 2p^{6}\ 3s^{2}\ 3p^{2}\). Adding up the electrons in the outermost shell — the two in 3s and the two in 3p — gives four. Its valence is therefore 4.
Because it has four valence electrons, each atom of Si forms four covalent bonds with its neighbouring atoms. Every one of its four outer electrons is paired off with an electron from a neighbouring atom, and each shared pair forms one bond.
These four bonds do not lie in a plane. One Si atom is surrounded by four Si atoms placed at the corners of a regular tetrahedron, as shown in Fig. 14.8, with the first atom at the centre. This arrangement repeats itself endlessly in all directions, giving the crystal its rigid, highly ordered structure.
Diagram
Fig. 14.8 – Structure of silicon: one Si atom at the centre of a regular tetrahedron, covalently bonded to four Si atoms at the corners.
Pure Silicon at Absolute Zero
At absolute zero temperature, all the valence electrons are tightly bound to their respective atoms and the covalent bonds are complete. Not a single bond is broken anywhere in the crystal.
Electrons are therefore not available to conduct electricity through the crystal, because they cannot gain enough energy to get into higher energy levels. In the language of Section 14.3, the valence band is completely full, the conduction band is completely empty, and no electron has the energy needed to cross the band gap.
A perfectly pure semiconductor crystal at absolute zero is thus an insulator. This is a result worth remembering, because it shows that a semiconductor is not “naturally” conducting — its conduction has to be caused, either by heat, as we see next, or by an impurity, as we see in Section 14.5.
What Heat Does to the Crystal
At room temperature, however, the picture changes. A few covalent bonds are broken due to thermal agitation — the constant random vibration of the atoms of the lattice — and some valence electrons gain energy from this agitation.
An electron that gains enough energy is no longer tied to its bond. In the language of bands, we say that a valence electron has been moved to the conduction band. It is now free to wander through the crystal and take part in conduction.
But the electron leaves something behind. Where it used to sit, the covalent bond now has an empty place, and the valence band has a vacancy, as shown in Fig. 14.9. Every electron promoted to the conduction band creates exactly one such vacancy.
Diagram
Fig. 14.9 – Creation of a vacancy in the valence band: electrons excited into the conduction band leaving holes behind in the valence band.
Holes: The Second Charge Carrier
These vacancies of electrons in the valence band are called holes. A hole is thus the absence of an electron in the valence band, and it carries an effective positive charge.
The word effective deserves attention. A hole is not a real particle and carries no charge of its own. But the region around a broken bond is left with one electron less than the positive charge of the nuclei there requires, so that region behaves, for every practical purpose, as though a single positive charge were sitting in it.
For an intrinsic semiconductor, the number of holes per unit volume (the number density, \(n_\mathrm{h}\)) and the number of free electrons per unit volume (the number density, \(n_\mathrm{e}\)) are the same:
\[n_\mathrm{h} = n_\mathrm{e}\]
Here \(n_\mathrm{h}\) and \(n_\mathrm{e}\) are both measured in \(\mathrm{m^{-3}}\). The reason for this equality is simple: in a pure crystal, electrons and holes are always born together. One broken bond releases one electron and leaves behind exactly one hole, so the two populations must be equal.
How Current Flows Through a Pure Semiconductor
Electric conduction through an intrinsic semiconductor is quite interesting, because it is unlike conduction in a metal. There are two different types of charge carriers in a semiconductor: one is the electron and the other is the hole, that is, the absence of an electron.
Electrical conduction takes place by the transportation of both carriers, or of any one of the two carriers, in a semiconductor.
When a semiconductor is connected in a circuit, electrons, being negatively charged, move towards the positive terminal of the battery. Holes have an effective positive charge, and so they move towards the negative terminal of the battery. Figure 14.10 shows both these movements taking place at the same time inside the crystal.
Diagram
Fig. 14.10 – Current through a semiconductor: electrons drifting one way and holes drifting the other way, with the battery connected across the crystal.
Thus, the current through a semiconductor is carried by two types of charge carriers which move in opposite directions. Note carefully that the two currents do not cancel: the carriers have opposite charges and move in opposite directions, so their contributions to the conventional current point the same way and add up.
This conduction mechanism, with two independent kinds of carriers, is what makes semiconductors so very useful in designing a large number of electronic devices. A metal offers only electrons; a semiconductor offers electrons and holes, and — as the following sections show — lets us decide which of the two shall dominate.
A common examination trap is worth naming here. Students often assume that a hole is a positively charged particle like a proton. It is not. It has no mass of its own and no charge of its own; it is a vacancy that behaves like a positive charge. Whenever a question asks you to compare electrons and holes, this distinction — real particle versus effective positive charge — is what is being tested.
14.5 Extrinsic Semiconductors
The electric conductivity of an intrinsic semiconductor is very low at room temperature. Only a handful of covalent bonds break by thermal agitation, so only a handful of carriers exist, and the current such a crystal can carry is far too small to be of practical use. Hence no electronic devices can be fabricated using pure semiconductors.
The remedy is remarkable in its simplicity. The addition of a small amount of a suitable impurity to an intrinsic semiconductor increases its conductivity appreciably.
A semiconductor with impurity added to it is called a doped semiconductor or an extrinsic semiconductor. The impurity that is added is called the dopant, and the parent atoms of the crystal are called hosts.
Choosing the Right Impurity
The dopant material is not chosen at random. It is so selected that it does not disturb the crystal structure of the host. The size and the electronic configuration of the dopant should be compatible with that of the host.
The reason is easy to see. The dopant atom has to sit inside the silicon lattice in place of a silicon atom, occupying its position and using its bonds. If the dopant atom were much larger or much smaller, or if its outer shell were completely unlike that of silicon, it would strain and spoil the crystal rather than blend into it.
Silicon or germanium can be doped with a pentavalent impurity — an element with five valence electrons — such as phosphorus (P), arsenic (As), or antimony (Sb). They can also be doped with a trivalent impurity — an element with three valence electrons — such as boron (B), aluminium (Al), or indium (In).
Notice how close these are to silicon’s own valence of four: one electron more, or one electron less. That single extra or missing electron is the whole point of doping.
The addition of pentavalent or trivalent impurities in intrinsic semiconductors gives rise to different conduction mechanisms. This is very useful in designing many electronic devices. Accordingly, extrinsic semiconductors can be of two types: (a) n-type semiconductor, or (b) p-type semiconductor.
n-type Semiconductor
When a silicon or germanium crystal is doped with a pentavalent impurity such as phosphorus, arsenic, or antimony, we get an n-type semiconductor. Figure 14.11 shows the schematic electronic structure of antimony, in which the five valence electrons of the outermost shell can be counted.
Diagram
Fig. 14.11 – Schematic electronic structure of antimony, showing its five valence electrons in the outermost shell.
When a dopant atom with 5 valence electrons occupies the position of a Si atom in the crystal lattice, 4 electrons from the dopant form bonds with the 4 neighbouring Si atoms, and the fifth electron from the dopant remains very weakly bound to its parent atom. Figure 14.12 shows such a pentavalent impurity sitting in a silicon lattice.
Diagram
Fig. 14.12 – A pentavalent impurity (antimony) atom in a silicon crystal, with four bonds formed and the fifth electron left over.
The crystal, in effect, has no bond left to offer this fifth electron. It has nothing to hold on to, and so it is held only loosely by the electrostatic attraction of its own parent nucleus.
To make this electron free even at room temperature, very small energy is required. It is \(0.01\ \mathrm{eV}\) for Ge and \(0.05\ \mathrm{eV}\) for Si.
Compare these figures with the band gaps of Section 14.3 — \(0.66\ \mathrm{eV}\) for germanium and \(1.12\ \mathrm{eV}\) for silicon. The energy needed to release the fifth electron of the dopant is smaller than the band gap by a factor of more than twenty. Room-temperature heat supplies it easily, so at ordinary temperatures practically every dopant atom has already given up its fifth electron to the crystal.
As this semiconductor now has a large number of electrons in the conduction band, and as its conductivity is due to negatively charged carriers, it is called an n-type semiconductor — the “n” standing for negative.
An n-type semiconductor is not, however, made of electrons alone. It also has a few electrons and holes produced by thermally broken bonds, exactly as in an intrinsic crystal.
Therefore the density of conduction electrons (\(n_\mathrm{e}\)) in a doped semiconductor is the sum total of the electrons contributed by the donors and the thermally generated electrons from the host. The density of holes (\(n_\mathrm{h}\)), on the other hand, is only due to the thermal breakdown of some covalent bonds of the host Si atoms — the donors contribute no holes at all.
Some electrons and holes recombine continuously, because they carry opposite charges and attract each other. A free electron falling back into a vacant bond destroys both a free electron and a hole at the same moment.
The net result of all this is clear: the number of free electrons far exceeds the number of holes. Thus, in a semiconductor doped with a pentavalent impurity, electrons (negative charge) are the majority carriers and holes are the minority carriers. This is why it is called an n-type semiconductor.
For an n-type semiconductor, \[n_\mathrm{e} \gg n_\mathrm{h}\]
The free electrons donated by the impurity atoms occupy energy levels which lie within the band gap, very close to the bottom of the conduction band. They can therefore be easily made available for conduction, since only a tiny step in energy separates them from the conduction band. Figure 14.13 shows the schematic band structure of an n-type semiconductor, with these extra donor levels drawn just below the conduction band.
Diagram
Fig. 14.13 – Schematic band structure of an n-type semiconductor, showing donor energy levels lying in the band gap just below the conduction band.
Extrinsic semiconductors are thus far better conductors than intrinsic semiconductors. And the conductivity of an extrinsic semiconductor can be controlled, simply by controlling the amount of impurity added.
The amount of impurity is expressed as part per million, or ppm — that is, one impurity atom per one million atoms of the host.
Features of n-type Semiconductors
These are materials doped with pentavalent impurity (donor) atoms. Electrical conduction in these materials is due to electrons as the majority charge carriers.
- The donor atoms lose electrons and become positively charged ions.
- The number of free electrons is very large compared to the number of holes, \(n_\mathrm{e} \gg n_\mathrm{h}\). Electrons are the majority charge carriers.
- When energy is supplied externally, negatively charged free electrons (majority charge carriers) and positively charged holes (minority charge carriers) are both available for conduction.
p-type Semiconductor
When a silicon or germanium crystal is doped with a trivalent impurity such as boron, aluminium or indium, we get a p-type semiconductor. Figure 14.14 shows the schematic electronic structure of boron, whose outermost shell holds three electrons.
Diagram
Fig. 14.14 – Schematic electronic structure of boron, showing its three valence electrons in the outermost shell.
The trivalent dopant atom has one valence electron less than that of a silicon atom. Every trivalent dopant atom shares its three electrons with three neighbouring Si atoms to form covalent bonds. But the fourth bond, between the silicon atom and its neighbour, is not complete — there is simply no fourth electron available from the dopant to complete it.
Figure 14.15 shows such a trivalent impurity sitting in a silicon crystal, with one incomplete bond beside it.
Diagram
Fig. 14.15 – A trivalent impurity (boron) atom in a silicon crystal, showing the one incomplete covalent bond.
The incomplete bond can be completed by another electron in the neighbourhood, borrowed from a nearby Si atom. Since each trivalent atom can accept an electron in this way, it is called an acceptor impurity.
But the borrowed electron has to come from somewhere. It leaves behind a vacancy in the place it came from, and this vacancy — this absence of an electron — is a hole.
Thus, a hole is available for conduction from each acceptor impurity atom. Holes are the majority carriers and electrons are the minority carriers in such materials.
The acceptor atoms, having taken in an extra electron, are negatively charged, while the majority carriers are holes, which are positively charged. Therefore, an extrinsic semiconductor doped with a trivalent impurity is called a p-type semiconductor — the “p” standing for positive.
For a p-type semiconductor, \[n_\mathrm{h} \gg n_\mathrm{e}\]
These vacancies of electrons are created in the valence band; therefore we can say that the holes are created in the valence band.
The impurity levels of a p-type semiconductor are created just above the valence band, inside the band gap. Electrons from the valence band can easily occupy these levels — the step up is tiny — and in doing so they leave behind holes in the valence band and conduct electricity. Figure 14.16 shows the schematic band structure of a p-type semiconductor.
Diagram
Fig. 14.16 – Schematic band structure of a p-type semiconductor, showing acceptor energy levels lying in the band gap just above the valence band.
Features of p-type Semiconductors
These are materials doped with trivalent impurity (acceptor) atoms. Electrical conduction in these materials is due to holes as the majority charge carriers.
- The acceptor atoms acquire an electron and become negatively charged ions.
- The number of holes is very large compared to the number of free electrons, \(n_\mathrm{h} \gg n_\mathrm{e}\). Holes are the majority charge carriers.
- When energy is supplied externally, positively charged holes (majority charge carriers) and negatively charged free electrons (minority charge carriers) are both available for conduction.
Diagram
Fig. 14.17 – A p-type semiconductor bar connected across a battery, showing electrons at positions 1, 2, 3, 4 hopping into neighbouring holes at x, y and the apparent motion of holes towards the negative terminal.
Charge Neutrality of Extrinsic Semiconductors
An n-type semiconductor has an excess of electrons — but these extra electrons are supplied by the donor atoms, which themselves become positively charged in the process.
Since each atom of the donor impurity is electrically neutral to begin with, the semiconductor as a whole remains electrically neutral. Every free electron released is exactly matched by a positive donor ion left behind in the lattice.
Here, “excess electron” refers to an excess with reference to the number of electrons needed to complete the covalent bonds in a semiconductor crystal — not to an excess of negative charge. These extra free electrons increase the conductivity of the semiconductor.
Similarly, a p-type semiconductor has holes, that is, an absence of electrons in some energy levels. When an electron from a host atom fills this level, the host atom becomes positively charged and the dopant atom becomes negatively charged — but the two charges are equal and opposite, so the semiconductor as a whole is again electrically neutral.
Thus, n-type as well as p-type semiconductors are electrically neutral.
The Product of the Carrier Densities
Always remember, for a semiconductor, \[n_\mathrm{e} \cdot n_\mathrm{h} = n_\mathrm{i}^{2}\]
where \(n_\mathrm{e}\) is the number density of free electrons, \(n_\mathrm{h}\) is the number density of holes, and \(n_\mathrm{i}\) is the intrinsic carrier concentration of that material at the given temperature — all measured in \(\mathrm{m^{-3}}\).
This relation carries a message that students often miss. Doping does not merely add carriers of one kind; it also suppresses the other kind. Raising \(n_\mathrm{e}\) by doping forces \(n_\mathrm{h}\) down, because their product must stay fixed at \(n_\mathrm{i}^{2}\). Physically, the flood of extra electrons in an n-type crystal makes recombination far more likely, so most holes are swallowed up almost as soon as they are born.
Note also that \(n_\mathrm{i}\) depends on temperature. The relation therefore holds at a given temperature, and in numerical problems the value of \(n_\mathrm{i}\) supplied to you is the value at the temperature stated in the question.
14.6 p-n Junction
When n-type and p-type semiconductor materials are fused together, a p-n junction is formed. A p-n junction shows many interesting properties, and it is the basis of almost all modern electronic devices. Figure 14.18 shows a schematic structure of a p-n junction.
Diagram
Fig. 14.18 – Schematic structure of a p-n junction: a block of p-type silicon joined to a block of n-type silicon, with terminals on either end.
It is important to understand at the outset that the two halves are not two separate pieces of material stuck together. They are two differently doped regions of one continuous crystal, and the boundary between them is called the junction. Everything interesting happens in a very thin layer around that boundary.
Diffusion
When n-type and p-type semiconductor materials are fused together, then initially the number of electrons on the n-side of the junction is very large compared to the number of electrons on the p-side. The same is true for the number of holes on the p-side compared with the n-side.
Thus the density of carriers on the two sides is very different, and a large density gradient exists on both sides of the p-n junction. A density gradient simply means that a quantity is crowded in one region and sparse in the adjoining one — and whenever that happens in nature, the crowded particles spread out into the sparse region, just as a drop of ink spreads through still water.
This density gradient causes the migration of electrons from the n-side to the p-side of the junction. On arriving there, they fill up the holes in the p-type material and produce negative ions.
When the electrons from the n-side of the junction migrate to the p-side, they leave behind positively charged donor ions on the n-side. Effectively, holes from the p-side migrate into the n-region.
As a result, in the p-type region near the junction there are negatively charged acceptor ions, and in the n-type region near the junction there are positively charged donor ions.
The extent up to which the electrons and the holes can diffuse across the junction depends on the density of the donor and the acceptor ions on the n-side and the p-side of the junction respectively. Figure 14.19 shows the diffusion of charge carriers across the junction.
Diagram
Fig. 14.19 – Diffusion of charge carriers across a junction: electrons crossing from the n-side to the p-side and holes crossing the other way.
The Depletion Region and the Potential Barrier
The diffusion of carriers across the junction, and the resulting accumulation of positive and negative charges across the junction, builds a potential difference across the junction. This potential difference is called the potential barrier.
The magnitude of the potential barrier for silicon is about \(0.6\) – \(0.7\) volt, and for germanium it is about \(0.3\) – \(0.35\) volt. These two numbers are worth memorising; they reappear throughout the rest of the chapter.
This potential barrier always exists, even if the device is not connected to any external power source. It is built into the junction itself. It prevents the continuous diffusion of carriers across the junction, and a state of electrostatic equilibrium is thus reached across the junction.
Notice why the diffusion must stop. The very charges left behind by diffusion oppose further diffusion — the negative acceptor ions on the p-side repel any further electrons trying to arrive from the n-side. Diffusion therefore builds up its own brake, and settles down when the brake becomes strong enough to hold it.
Free charge carriers cannot be present in a region where there is a potential barrier. The regions on either side of the junction therefore become completely devoid of any charge carriers. This region across the p-n junction, where there are no free charges, is called the depletion layer or the depletion region. Figure 14.20 shows the potential barrier and the depletion layer.
Diagram
Fig. 14.20 – Potential barrier and the depletion layer: the p-region and n-region with a carrier-free region containing fixed positive and negative ions across the junction.
The potential across the junction, and the width of the potential barrier, can be controlled. This is a very interesting and useful property of a p-n junction — and, as the following pages show, it is the property on which every diode-based device rests.
The Electric Field Across the Junction
The n-side near the boundary of a p-n junction becomes positive with respect to the p-side, because it has lost electrons, while the p-side has lost holes.
Thus the presence of impurity ions on both sides of the junction establishes an electric field across this region, such that the n-side is at a positive voltage relative to the p-side. Figure 14.21 shows the electric field thus produced, directed from the n-side towards the p-side across the depletion region.
Diagram
Fig. 14.21 – Electric field across a junction, directed from the positive donor ions on the n-side to the negative acceptor ions on the p-side.
This built-in field is not a detail to be skimmed over. It is the thing an incoming charge carrier must fight against, and it is the reason a p-n junction behaves so differently depending on how a battery is connected to it.
Biasing a p-n Junction
As a result of the potential barrier across the depletion region, the charge carriers require some extra energy to overcome the barrier. Left to themselves they cannot cross it, and no useful current flows.
A suitable voltage therefore needs to be applied to the junction externally, so that these charge carriers can overcome the potential barrier and move across the junction. Applying such an external voltage to a junction is called biasing it. Figure 14.22 shows the two possibilities of applying this external voltage across the junction.
Diagram
Fig. 14.22 – A p-n junction connected in a circuit: (a) forward biased, with a narrow depletion region; (b) reverse biased, with a widened depletion region.
Forward Bias
Figure 14.22 (a) shows a p-n junction connected in an electric circuit where the p-region is connected to the positive terminal, and the n-region is connected to the negative terminal, of an external voltage source.
This external voltage effectively opposes the built-in potential of the junction. The width of the potential barrier is thus reduced.
Also, the negative charge carriers (electrons) in the n-region are pushed towards the junction, being repelled by the negative terminal to which the n-region is joined. A similar effect is experienced by the positive charge carriers (holes) in the p-region, and they too are pushed towards the junction.
Both the charge carriers thus find it easy to cross over the barrier and contribute towards the electric current. Such an arrangement of a p-n junction in an electric circuit is called forward bias.
Reverse Bias
Figure 14.22 (b) shows the other possibility, where the p-region is connected to the negative terminal and the n-region is connected to the positive terminal of the external voltage source.
This external voltage effectively adds to the built-in potential of the junction. The width of the potential barrier is thus increased.
Also, the negative charge carriers (electrons) from the n-region are pulled away from the junction, attracted towards the positive terminal to which the n-region is joined. A similar effect is experienced by the positive charge carriers (holes) in the p-region, and they too are pulled away from the junction.
Both the charge carriers thus find it very difficult to cross over the barrier, and so they do not contribute towards the electric current. Such an arrangement of a p-n junction in an electric circuit is called reverse bias.
The Junction as a One-Way Switch
Therefore, when used in forward bias mode, a p-n junction allows a large current to flow across it. This current is normally of the order of a few milliamperes (\(10^{-3}\ \mathrm{A}\)).
A reverse biased p-n junction, on the other hand, carries a very small current, normally of a few microamperes (\(10^{-6}\ \mathrm{A}\)). The ratio between the two is roughly a thousand to one.
A p-n junction can thus be used as a one way switch, or a gate, in an electric circuit. It conducts easily in forward bias, and acts as an open switch in reverse bias.
Features of the Depletion Region
- It is formed by the diffusion of electrons from the n-region to the p-region. This leaves positively charged ions in the n-region.
- The p-region accumulates electrons (negative charges) and the n-region accumulates the holes (positive charges).
- The accumulation of charges on either side of the junction results in the formation of a potential barrier and prevents the flow of charges across it.
- There are no free charges in this region.
- The depletion region has a higher potential on the n-side and a lower potential on the p-side of the junction.
14.7 A p-n Junction Diode
A p-n junction, when provided with metallic connectors on each side, is called a junction diode, or simply, a diode. The name comes from the fact that it is a device with two electrodes — di-electrodes. Figure 14.23 shows the circuit symbol for a junction diode.
Diagram
Fig. 14.23 – Circuit symbol for a p-n junction diode, with the p-side marked as anode and the n-side as cathode.
In the circuit symbol, the ‘arrow’ indicates the direction of the conventional current — the direction in which the diode is willing to let current pass. The p-side is called the anode of the diode, and the n-side is called the cathode.
When a diode is connected across a battery, the carriers can gain additional energy to cross the barrier, as per the biasing. Everything that was said about biasing a bare p-n junction in Section 14.6 applies to the diode without change; what the diode adds is the two metal contacts that let us wire it into a circuit.
A diode can be connected across a battery in two different ways — forward bias and reverse bias — as shown in Fig. 14.24.
Diagram
Fig. 14.24 – A diode connected across a battery in (a) forward bias and (b) reverse bias.
The behaviour of a diode in the two cases is entirely different. This is because the barrier potential is affected differently in the two cases: the barrier potential is reduced in the forward biased mode, and it is increased in the reverse biased mode.
Why the Two Currents Are So Unequal
Carriers find it easy to cross the junction in forward bias and contribute towards the current, for two reasons acting together. First, the barrier width is reduced. Second, the carriers are pushed towards the junction and gain extra energy to cross it.
The current through the diode in forward bias is therefore large. It is of the order of a few milliamperes (\(10^{-3}\ \mathrm{A}\)) for a typical diode.
When connected in reverse bias, the width of the potential barrier is increased, and the carriers are pushed away from the junction. Only very few thermally generated carriers can then cross the junction and contribute towards the current.
This results in a very small current through a reverse biased diode. The current in a reverse biased diode is of the order of a few microamperes (\(10^{-6}\ \mathrm{A}\)).
The width of the depletion layer decreases with an increase in the applied forward voltage. It increases when a reverse voltage is applied. The reasons for this difference were discussed earlier.
When the polarity of the bias voltage is reversed, the width of the depletion layer changes, and so the ease with which carriers can cross also changes. This results in an asymmetrical current flow through a diode, as shown in Fig. 14.25 — large current on the forward side of the graph, almost none on the reverse side.
Diagram
Fig. 14.25 – Asymmetrical current flow through a diode: forward current in mA on the +I axis, reverse current in µA on the −I axis, with reverse breakdown shown at large reverse voltage.
A diode can thus be used as a one way switch in a circuit. It is forward biased when its anode is connected so as to be at a higher potential than that of the cathode. When the anode is at a lower potential than that of the cathode, it is reverse biased. A diode can also be zero biased, if no external voltage is applied across it.
Forward Biased Diode
a) Forward biased: The positive terminal of the external voltage is connected to the anode (p-side), and the negative terminal to the cathode (n-side), across the diode.
In the case of forward bias, the width of the depletion region decreases and the p-n junction offers a low resistance path, allowing a high current to flow across the junction, as shown in Fig. 14.26.
Diagram
Fig. 14.26 – Decrease in the width of the depletion region in a forward biased diode.
Figure 14.27 shows the I-V characteristic of a forward biased diode — a graph of the current \(I\) through the diode against the voltage \(V\) applied across it.
Diagram
Fig. 14.27 – I-V characteristic of a forward biased silicon diode, with forward voltage on the x-axis, forward current in mA on the y-axis, and the knee marked at about 0.7 V.
Initially, the current is very low, and then there is a sudden rise in the current. The point at which the current rises sharply is shown as the ‘knee’ point on the I-V characteristic curve, and the corresponding voltage is called the ‘knee voltage’.
The knee voltage is about \(0.7\ \mathrm{V}\) for silicon and \(0.3\ \mathrm{V}\) for germanium.
These values should look familiar. They are essentially the potential barrier of the junction, met in Section 14.6. Below the knee voltage the applied voltage has not yet cancelled the barrier, so hardly any carriers get across and the current stays small. Once the applied voltage matches the barrier, the barrier collapses and carriers pour across — hence the sharp knee.
A diode effectively becomes a short circuit above this knee point, and can conduct a very large current. Resistors are therefore used in series with a diode to limit its current flow. If the current through a diode exceeds the specified value, it can heat up the diode due to Joule heating, and this can result in its physical damage.
This is a practical point of some importance. A diode does not protect itself. In any circuit you build, the series resistor is not an optional extra — it is the only thing standing between the diode and its destruction.
Reverse Biased Diode
b) Reverse biased: The positive terminal of the external voltage is connected to the cathode (n-side), and the negative terminal to the anode (p-side), across the diode.
In the case of reverse bias, the width of the depletion region increases and the p-n junction behaves like a high resistance, as shown in Fig. 14.28.
Diagram
Fig. 14.28 – Increase in the width of the depletion region in a reverse biased diode.
Practically, no current flows through it as the reverse bias voltage is increased. However, a very small leakage current does flow through the junction, which is of the order of a few micro-amperes (\(\mu\mathrm{A}\)).
Avalanche Breakdown
When the reverse bias voltage applied to a diode is increased to a sufficiently large value, it causes the p-n junction to overheat. The overheating of the junction results in a sudden rise in the current through the junction.
This happens because the covalent bonds break under these conditions, and a large number of carriers become available for conduction. Once that happens, the diode no longer behaves like a diode — it conducts heavily in the reverse direction, which is precisely what it was supposed to prevent.
This effect is called the avalanche breakdown. The word avalanche is apt: a few high-energy carriers knock loose more carriers, which in turn knock loose still more, and the process runs away, exactly as a few sliding stones can bring down a mountainside. The reverse biased characteristic of a diode is shown in Fig. 14.29, where the current can be seen shooting up at the breakdown voltage.
Diagram
Fig. 14.29 – Reverse biased characteristic of a diode, showing the reverse current rising abruptly at the breakdown voltage \(V_z\).
Diagram
Fig. 14.30 – A zero biased p-n junction diode with its terminals shorted, showing the depletion region and the two equal and opposite carrier flows across the junction.
Static and Dynamic Resistance of a Diode
c) Static and dynamic resistance of a diode: One of the most important properties of a diode is its resistance in the forward biased mode and in the reverse biased mode. Figure 14.31 shows the I-V characteristics of an ideal diode.
An ideal diode offers zero resistance in the forward biased mode and infinite resistance in the reverse biased mode. It is a perfect switch: fully closed one way, fully open the other.
Diagram
Fig. 14.31 – I-V characteristics of an ideal diode: a vertical line along the current axis in forward bias and zero current in reverse bias.
A real diode is not ideal, and its resistance has to be found from its actual characteristic curve. The I-V characteristic of a forward biased diode (Fig. 14.27) is used to define two of its resistances: (i) the static (DC) resistance and (ii) the dynamic (AC) resistance.
i) Static (DC) resistance: When a p-n junction diode is forward biased, it offers a definite resistance in the circuit. This resistance is called the static, or DC, resistance (\(R_g\)) of a diode.
The DC resistance of a diode is the ratio of the DC voltage across the diode to the DC current flowing through it, at a particular voltage.
\[R_g = \frac{V}{I}\]
Here \(V\) is the forward voltage across the diode in volt (V), \(I\) is the forward current through it in ampere (A), and \(R_g\) is the static resistance in ohm (\(\Omega\)).
ii) Dynamic (AC) resistance: The dynamic (AC) resistance of a diode, \(r_g\), at a particular applied voltage, is defined as
\[r_g = \frac{\Delta \mathrm{V}}{\Delta I}\]
Here \(\Delta \mathrm{V}\) is a small change in the voltage across the diode (in V), and \(\Delta I\) is the resulting small change in the current through it (in A). So \(r_g\) answers a different question from \(R_g\): not “what is the ratio of voltage to current here?” but “if I nudge the voltage a little, how much does the current change?”
The dynamic resistance of a diode depends on the operating voltage. It is the reciprocal of the slope of the characteristic curve at that point. Figure 14.32 shows how the DC and the AC resistance of a diode are found out from the curve.
Diagram
Fig. 14.32 – Determining the DC and AC resistance of a diode from its forward I-V characteristic, showing the operating point \((V_0, I_0)\) and the small increments \(\Delta V\) and \(\Delta I\).
14.8 Semiconductor Devices
Semiconductor devices find applications in a wide variety of fields — in communication, in computing, in medicine, in transport, in power generation, and in almost every gadget of daily use.
They have many advantages, which is why they have displaced the older technologies so completely. But they also have some disadvantages, and an engineer must know both. Here we discuss some advantages and some disadvantages of semiconductor devices.
14.8.1 Advantages
- The electronic properties of semiconductors can be controlled to suit our requirement. This is the advantage from which all the others follow: by choosing the dopant and its concentration, we decide the conductivity, the type of majority carrier, and the behaviour of the junction.
- They are smaller in size and light in weight.
- They can operate at smaller voltages (of the order of a few mV) and require less current (of the order of \(\mu\mathrm{A}\) or \(\mathrm{mA}\)). They therefore consume lesser power.
- Almost no heating effects occur, and therefore these devices are thermally stable.
- They have a faster speed of operation, due to their smaller size. A signal has a shorter distance to travel inside a small device, and so the device can respond more quickly.
- Fabrication of ICs is possible. Hundreds of components can be grown into a single small crystal, instead of being made separately and then wired together.
14.8.2 Disadvantages
- They are sensitive to electrostatic charges. A stray static charge picked up from the human body can puncture the extremely thin layers inside a semiconductor device.
- They are not very useful for controlling high power.
- They are sensitive to radiation.
- They are sensitive to fluctuations in temperature. As we saw in Section 14.2, the carrier density in a semiconductor rises steeply with temperature, so the behaviour of the device drifts as it warms.
- They need controlled conditions for their manufacturing. Doping at the level of parts per million is meaningless in a dusty room, so semiconductor devices are made in specially built, ultra-clean facilities.
- Very few materials are semiconductors.
The contrast in that side note is the clearest way to see what semiconductor devices are for. Electric devices are built to move energy about; semiconductor devices are built to move information about. That is why the currents in one are measured in amperes and in the other in microamperes — and why a single chip can hold billions of the latter but not one of the former.
14.9 Applications of Semiconductors and p-n Junction Diode
A p-n junction diode is the basic building block of a large number of semiconductor devices. Almost everything in this section is, at bottom, a junction — or several junctions — put to a particular use.
A semiconductor device can have more than one junction. A transistor, for instance, has two, and more complex devices have many. And the properties of any such device can be controlled by controlling the concentration of the dopants used in each of its regions. This is the practical payoff of everything studied in Section 14.5: by deciding the doping, the designer decides the device.
Some Important Semiconductor Devices
1. Solar cell: It converts light energy into electric energy. It is useful for producing electricity in remote areas, and also for providing electricity for satellites, space probes and space stations, where no other source of power is available for years at a stretch.
2. Photo resistor: It changes its resistance when light is incident on it. The brighter the light falling on it, the more carriers are freed inside it, and so the lower its resistance becomes.
3. Bi-polar junction transistor: These are devices with two junctions and three terminals. A transistor can be a p-n-p or an n-p-n transistor, depending on the order in which the doped regions are arranged. Conduction in it takes place with both holes and electrons — which is what the word bi-polar refers to. Many other types of transistors are designed and fabricated to suit specific requirements, and transistors are used in almost all semiconductor devices.
4. Photodiode: It conducts when it is illuminated with light. It is used as a reverse biased junction whose leakage current depends on the light falling on it.
5. LED (Light Emitting Diode): It emits light when current passes through it. Household LED lamps use similar technology. They consume less power, are smaller in size, have a longer life, and are cost effective.
6. Solid state laser: It is a special type of LED. It emits light of a specific frequency. It is smaller in size and consumes less power.
7. Integrated Circuits (ICs): A small device having hundreds of diodes and transistors, which performs the work of a large number of electronic circuits.
Notice what runs through the whole list. In every one of these devices, light, heat or an electrical signal is used to change the number of charge carriers, or the way they cross a junction — and it is the ability of a semiconductor to have its carrier population altered, which no metal and no insulator possesses, that makes each of them possible.
14.10 Thermistor
A thermistor is a temperature sensitive resistor. Its resistance changes with a change in its temperature. The name itself is a compression of thermal resistor.
An ordinary resistor is designed so that its resistance stays as steady as possible when it warms up; a thermistor is designed for exactly the opposite. Its whole purpose is to react to heat, and to react strongly.
There are two types of thermistors: the Negative Temperature Coefficient (NTC) type and the Positive Temperature Coefficient (PTC) type.
NTC Thermistors
The resistance of an NTC thermistor decreases with an increase in its temperature. Its temperature coefficient is therefore negative, and this is what the name records.
NTC thermistors are commonly used as temperature sensors, and also in temperature control circuits. Since their resistance is a known function of temperature, measuring the resistance amounts to measuring the temperature.
PTC Thermistors
The resistance of a PTC thermistor increases with an increase in its temperature.
PTC thermistors are commonly used in series with a circuit. They are generally used as a reusable fuse, to limit the current passing through a circuit and to protect it against over current conditions — that is, as resettable fuses.
The way such a fuse protects a circuit is worth following through. If the current through the circuit becomes excessive, the PTC thermistor heats up. Its resistance then rises, and the higher resistance itself cuts the current down. Once the fault is removed and the thermistor cools, its resistance drops back to its normal low value and the circuit works again — which is why, unlike an ordinary fuse wire, it does not have to be replaced after every fault.
What Thermistors Are Made Of
Thermistors are made from thermally sensitive metal oxide semiconductors.
Thermistors are very sensitive to changes in temperature: a small change in the surrounding temperature causes a large change in their resistance. This large response to a small stimulus is precisely what makes a good sensor.
They can also measure temperature variations of a very small area, because of their small size. A bulky sensor would average the temperature over the whole region it occupies; a tiny thermistor reports the temperature of the exact point where it is placed.
Both types of thermistors have many applications in industry.
14.11 Summary
Conduction in Solids
Electrical conduction in a solid takes place by the transport of charge carriers. It depends on the temperature, the number of charge carriers, their mobility, the crystal structure of the solid, and the types and nature of the defects present in it.
Solids are of three types electrically: good conductors (metals), semiconductors, and bad conductors (insulators). A typical metal has about \(10^{28}\) free electrons per \(\mathrm{m^{3}}\), while an insulator has only about \(10^{23}\) per \(\mathrm{m^{3}}\).
The electrical conductivity of a solid is \[\sigma = nq\mu\] where \(n\) is the charge carrier density (\(\mathrm{m^{-3}}\)), \(q\) is the charge on the carriers (C), and \(\mu\) is the mobility of the carriers (\(\mathrm{m^{2}\ V^{-1}\ s^{-1}}\)).
Mobility is the measure of the ease with which a carrier moves in a material under an external electric field. It depends on the mass of the carrier, whether the material is crystalline or amorphous, structural defects, and the nature of impurities.
The conductivity of silver is \(6.25 \times 10^{7}\ \mathrm{S\ m^{-1}}\), of glass about \(10^{-10}\ \mathrm{S\ m^{-1}}\), and of silicon \(1.56 \times 10^{-3}\ \mathrm{S\ m^{-1}}\) — a semiconductor lies between a good conductor and a bad conductor.
On heating, the conductivity of a semiconductor increases, while the conductivity of a metal decreases.
Semiconductors are broadly classified as elemental (silicon, germanium), compound (cadmium sulphide, zinc sulphide) and organic (anthracene, doped phthalocyanines, polyaniline).
Band Theory of Solids
Energy bands are formed in a solid because of the small distances between atoms, the resulting interaction amongst electrons, and Pauli’s exclusion principle.
Pauli’s exclusion principle: no two electrons can have the same set of quantum numbers; no two electrons with similar spin can occupy the same energy level. Any energy level can hold only two electrons — one spin up and one spin down.
The valence band is the topmost occupied energy band. The conduction band is the band formed by the conduction levels, immediately above it. The gap between the top of the valence band and the bottom of the conduction band is the energy gap or band gap.
In a metal, the valence and conduction bands overlap and there is no band gap. In a semiconductor, the band gap is small, of the order of one electron volt or less. In an insulator, the band gap is wide; diamond, for example, has a band gap of about \(5.0\ \mathrm{eV}\).
Band gaps at 300 K: silicon \(1.12\ \mathrm{eV}\), germanium \(0.66\ \mathrm{eV}\), diamond \(5.47\ \mathrm{eV}\).
\(1\ \mathrm{eV}\) is the energy gained by an electron while it overcomes a potential difference of one volt; \(1\ \mathrm{eV} = 1.6 \times 10^{-19}\ \mathrm{J}\).
At a finite temperature \(T\), electrons gain thermal energy of the order of \(kT\), where \(k\) is the Boltzmann constant. Only valence-band electrons can be excited across the gap, and only if the energy gained exceeds the band gap.
Intrinsic Semiconductors
An intrinsic semiconductor is a pure semiconductor such as pure silicon or pure germanium. Silicon has the configuration \(1s^{2}\ 2s^{2}\ 2p^{6}\ 3s^{2}\ 3p^{2}\), a valence of 4, and each atom forms four covalent bonds with four neighbours placed at the corners of a regular tetrahedron.
At absolute zero, all covalent bonds are complete and the crystal behaves as an insulator. At room temperature, thermal agitation breaks a few bonds; the freed electron enters the conduction band and leaves a vacancy — a hole — in the valence band.
A hole is the absence of an electron in the valence band and carries an effective positive charge.
For an intrinsic semiconductor, \[n_\mathrm{h} = n_\mathrm{e}\]
Current in a semiconductor is carried by two types of carriers moving in opposite directions: electrons towards the positive terminal, holes towards the negative terminal.
Extrinsic Semiconductors
Doping is the process of adding a small amount of a suitable impurity to an intrinsic semiconductor; it increases the conductivity appreciably. The doped material is an extrinsic semiconductor, the impurity is the dopant, the parent atoms are the hosts. Doping levels are expressed in ppm (one impurity atom per million host atoms).
n-type: doped with a pentavalent impurity — phosphorus (P), arsenic (As) or antimony (Sb). Four of the dopant’s five valence electrons form bonds; the fifth is very weakly bound and needs only \(0.01\ \mathrm{eV}\) (Ge) or \(0.05\ \mathrm{eV}\) (Si) to be freed. The pentavalent dopant is a donor. Donor levels lie in the band gap, close to the conduction band.
- Donor atoms lose electrons and become positively charged ions.
- Electrons are the majority carriers, holes the minority carriers: \(n_\mathrm{e} \gg n_\mathrm{h}\).
p-type: doped with a trivalent impurity — boron (B), aluminium (Al) or indium (In). The dopant forms only three bonds; the fourth bond is incomplete and is completed by an electron from a neighbouring host atom, creating a hole. The trivalent dopant is an acceptor. Acceptor levels lie in the band gap, just above the valence band.
- Acceptor atoms acquire electrons and become negatively charged ions.
- Holes are the majority carriers, electrons the minority carriers: \(n_\mathrm{h} \gg n_\mathrm{e}\).
The mobility of holes is less than that of electrons, because hole transport is really an indirect movement of electrons.
Both n-type and p-type semiconductors are electrically neutral as a whole. Doping changes the number and type of mobile carriers, not the net charge.
For any semiconductor, \[n_\mathrm{e} \cdot n_\mathrm{h} = n_\mathrm{i}^{2}\] where \(n_\mathrm{i}\) is the intrinsic carrier concentration of that material at that temperature.
The p-n Junction
A p-n junction is formed when p-type and n-type materials are fused together. It is the basis of almost all modern electronic devices.
Diffusion: because of the density gradient, electrons migrate from the n-side to the p-side and holes effectively migrate from the p-side to the n-side. This leaves positively charged donor ions on the n-side and negatively charged acceptor ions on the p-side.
Depletion region: the region across the junction that is devoid of free charge carriers. The accumulated charges build a potential barrier, which prevents further diffusion and establishes electrostatic equilibrium. The n-side is at a positive potential relative to the p-side, so an electric field exists across the junction.
Magnitude of the potential barrier: about \(0.6 - 0.7\ \mathrm{V}\) for silicon and about \(0.3 - 0.35\ \mathrm{V}\) for germanium. It exists even with no external source connected.
Forward bias: p-region to the positive terminal, n-region to the negative terminal. The barrier and the depletion width are reduced; carriers are pushed towards the junction; a large current flows, of the order of a few milliamperes (\(10^{-3}\ \mathrm{A}\)).
Reverse bias: p-region to the negative terminal, n-region to the positive terminal. The barrier and the depletion width are increased; carriers are pulled away from the junction; only a very small current flows, of the order of a few microamperes (\(10^{-6}\ \mathrm{A}\)).
A p-n junction therefore acts as a one-way switch: it conducts easily in forward bias and acts as an open switch in reverse bias.
The p-n Junction Diode
A p-n junction with metallic connectors on each side is a junction diode. The p-side is the anode, the n-side is the cathode. The arrow in the circuit symbol shows the direction of conventional current.
I-V characteristic (forward bias): the current is very small at first and then rises sharply at the knee point. The knee voltage is about \(0.7\ \mathrm{V}\) for silicon and \(0.3\ \mathrm{V}\) for germanium. Above the knee the diode behaves almost as a short circuit, so a series resistor is used to limit the current and prevent damage by Joule heating.
I-V characteristic (reverse bias): practically no current flows, except a small leakage current of a few \(\mathrm{\mu A}\), carried by minority carriers. At a sufficiently large reverse voltage the junction overheats, covalent bonds break, and the current rises suddenly — this is avalanche breakdown, and the device no longer behaves like a diode.
An ideal diode offers zero resistance in forward bias and infinite resistance in reverse bias.
Static (DC) resistance: \[R_{g} = \frac{V}{I}\]
Dynamic (AC) resistance: \[r_{g} = \frac{\Delta \mathrm{V}}{\Delta I}\] The dynamic resistance depends on the operating voltage and is the reciprocal of the slope of the characteristic at that point.
Devices and Applications
Advantages of semiconductor devices: properties can be controlled; small size and light weight; operate at small voltages (a few mV) and currents (\(\mathrm{\mu A}\) or mA), so consume less power; almost no heating effects, hence thermally stable; faster operation; ICs can be fabricated.
Disadvantages: sensitive to electrostatic charges; not very useful for controlling high power; sensitive to radiation; sensitive to temperature fluctuations; need controlled manufacturing conditions; very few materials are semiconductors.
Devices built on the junction: solar cell (light energy to electric energy), photo resistor (resistance changes with light), bi-polar junction transistor (two junctions, three terminals, p-n-p or n-p-n), photodiode (conducts when illuminated), LED (emits light when current passes), solid state laser (a special LED emitting light of a specific frequency), and integrated circuits (hundreds of diodes and transistors in one small device).
A thermistor is a temperature sensitive resistor made from thermally sensitive metal oxide semiconductors. An NTC thermistor’s resistance decreases with increasing temperature (used as a temperature sensor and in temperature control circuits); a PTC thermistor’s resistance increases with increasing temperature (used in series as a resettable fuse to protect against over-current).
14.12 Check Your Concepts
The electrical conductivity of a metal decreases when it is heated, but that of a semiconductor increases. Using the relation \(\sigma = nq\mu\), explain the reason for this opposite behaviour.
Silicon and diamond both have a valence of four and are held together by four covalent bonds per atom, yet silicon is a semiconductor and diamond is an insulator. Explain why.
A hole is only the absence of an electron. Explain clearly how such a vacancy can carry an electric current, and state why it is treated as a carrier with an effective positive charge.
Why is it possible to change the conductivity of a semiconductor enormously by adding an impurity in a concentration as small as one part per million, whereas the same impurity would make almost no difference to the conductivity of a metal?
An n-type semiconductor contains a large number of free electrons, and yet it is electrically neutral as a whole. Explain how both of these statements can be true at the same time.
State, with reasons, whether the following are true or false: (i) At absolute zero, a pure semiconductor behaves like an insulator. (ii) In a p-type semiconductor, the acceptor impurity levels lie close to the conduction band.
Explain why the width of the depletion region of a p-n junction decreases in forward bias and increases in reverse bias.
A p-n junction diode conducts a current of a few milliamperes in forward bias but only a few microamperes in reverse bias. Explain the reason for this large difference, in terms of the potential barrier and the charge carriers.
What is meant by the knee voltage of a diode? Why is a resistor connected in series with a diode in a circuit?
Distinguish between the static (DC) resistance and the dynamic (AC) resistance of a diode. Why does a diode have two different resistances while an ordinary resistor has only one?
What is avalanche breakdown? Why does a diode no longer behave like a diode once it has occurred?
The sensitivity of a semiconductor’s conductivity to temperature is listed as a disadvantage of semiconductor devices, and yet it is the working principle of a thermistor. Explain this apparent contradiction.
Distinguish between an NTC thermistor and a PTC thermistor, and give one use of each.
A solar cell, a photodiode and a photo resistor all respond to light. Explain how the three devices differ from one another in the way they respond.
Why is the mobility of holes less than the mobility of electrons in a semiconductor? What practical step is taken during the fabrication of a p-n junction to make the same current flow through the p-region and the n-region?
What is the importance of the energy gap in a semiconductor?
Which element would you use as an impurity to make germanium an n-type semiconductor?
On which factors does the electrical conductivity of a pure semiconductor depend at a given temperature?
Why is the conductivity of an n-type semiconductor greater than that of a p-type semiconductor even when both of these have the same level of doping?
Explain how solids are classified on the basis of the band theory of solids.
Distinguish between intrinsic semiconductors and extrinsic semiconductors.
Explain the importance of the depletion region in a p-n junction diode.
Explain the I-V characteristic of a forward biased junction diode.
14.13 Practice with Numericals
The electrical conductivity of silicon is \(1.56 \times 10^{-3}\ \mathrm{S\ m^{-1}}\). Calculate its resistivity, and hence find the resistance of a silicon bar of length \(2.0\ \mathrm{cm}\) and cross-sectional area \(1.0\ \mathrm{mm^{2}}\).
A conductor has a free electron density of \(9.0 \times 10^{28}\ \mathrm{m^{-3}}\) and an electron mobility of \(3.0 \times 10^{-3}\ \mathrm{m^{2}\ V^{-1}\ s^{-1}}\). Taking the electronic charge as \(1.6 \times 10^{-19}\ \mathrm{C}\), calculate its electrical conductivity.
The band gap of silicon is \(1.12\ \mathrm{eV}\). Express this energy in joules. Taking \(1\ \mathrm{eV} = 1.6 \times 10^{-19}\ \mathrm{J}\), also find the energy in joules required to free the fifth electron of a phosphorus dopant atom in silicon, for which the required energy is \(0.05\ \mathrm{eV}\).
A pure silicon crystal has \(5 \times 10^{28}\) atoms \(\mathrm{m^{-3}}\). It is doped with a \(1\ \mathrm{ppm}\) concentration of arsenic, a pentavalent impurity, each atom of which donates one free electron. Taking \(n_\mathrm{i} = 1.5 \times 10^{16}\ \mathrm{m^{-3}}\), calculate the number density of free electrons and of holes in the doped crystal. Name the type of semiconductor formed and state which are the majority carriers.
A pure semiconductor at a certain temperature has an intrinsic carrier concentration \(n_\mathrm{i} = 2.5 \times 10^{19}\ \mathrm{m^{-3}}\). On doping, the electron concentration becomes \(5.0 \times 10^{22}\ \mathrm{m^{-3}}\). Calculate the hole concentration in the doped crystal, and state whether it is n-type or p-type.
A germanium crystal is doped with indium so that its hole concentration becomes \(3.0 \times 10^{22}\ \mathrm{m^{-3}}\). If the intrinsic carrier concentration of germanium at that temperature is \(2.4 \times 10^{19}\ \mathrm{m^{-3}}\), find the electron concentration in the doped crystal.
An intrinsic semiconductor has a carrier concentration \(n_\mathrm{i} = 1.5 \times 10^{16}\ \mathrm{m^{-3}}\). The mobility of electrons in it is \(0.13\ \mathrm{m^{2}\ V^{-1}\ s^{-1}}\) and that of holes is \(0.05\ \mathrm{m^{2}\ V^{-1}\ s^{-1}}\). Taking \(e = 1.6 \times 10^{-19}\ \mathrm{C}\), calculate the electrical conductivity of the material. (Both carriers contribute, so \(\sigma = n_\mathrm{i}e(\mu_\mathrm{e} + \mu_\mathrm{h})\).)
A silicon diode with a knee (forward) voltage of \(0.7\ \mathrm{V}\) is connected in series with a resistor \(R\) and a \(6\ \mathrm{V}\) battery. Taking the voltage across the forward biased diode to be equal to its knee voltage, calculate the value of \(R\) needed to limit the current to \(20\ \mathrm{mA}\).
A germanium diode has a knee voltage of \(0.3\ \mathrm{V}\). It is connected in series with a \(470\ \Omega\) resistor across a \(5\ \mathrm{V}\) supply, in forward bias. Assuming the diode drops its knee voltage, calculate the current in the circuit.
From the forward characteristic of a diode it is found that a forward voltage of \(0.70\ \mathrm{V}\) gives a current of \(5\ \mathrm{mA}\), while a forward voltage of \(0.72\ \mathrm{V}\) gives a current of \(15\ \mathrm{mA}\). Calculate (i) the static (DC) resistance of the diode at \(0.72\ \mathrm{V}\) and (ii) the dynamic (AC) resistance of the diode between the two points.
In the circuit between points A and B, one branch (ACB) contains a \(40\ \Omega\) resistor in series with an ideal diode, and a second branch (ADB), in parallel with the first, contains a \(60\ \Omega\) resistor. Find the resistance between A and B when the diode is (i) forward biased and (ii) reverse biased.
An NTC thermistor has a resistance of \(12\ \mathrm{k\Omega}\) at \(20\ ^{\circ}\mathrm{C}\) and \(3\ \mathrm{k\Omega}\) at \(60\ ^{\circ}\mathrm{C}\). It is connected directly across a \(6\ \mathrm{V}\) supply. Calculate the current through the thermistor at each temperature, and find the factor by which the current increases when the thermistor is warmed from \(20\ ^{\circ}\mathrm{C}\) to \(60\ ^{\circ}\mathrm{C}\).
A household replaces five incandescent lamps of \(100\ \mathrm{W}\) each with five LED lamps of \(12\ \mathrm{W}\) each, giving the same illumination. If the lamps are used for \(6\) hours a day, calculate the energy saved per day (in kWh) and the energy saved over a \(30\)-day month.
One \(\mathrm{cm^{3}}\) of pure germanium at \(20\ ^{\circ}\mathrm{C}\) contains about \(4.2 \times 10^{22}\) atoms and about \(2.5 \times 10^{13}\) free electrons. Calculate the ratio of the number of germanium atoms to the number of free electrons in this volume, and comment on what this ratio tells you about the conductivity of a pure semiconductor.
A reverse biased diode carries a leakage current of \(5\ \mathrm{\mu A}\), while the same diode in forward bias carries a current of \(8\ \mathrm{mA}\). Calculate the ratio of the forward current to the reverse current, and state what this ratio illustrates about the behaviour of a diode.
14.14 Objective Questions
- Electric conduction through a semiconductor is due to:
- electrons
- holes
- none of these
- both electrons and holes
- The energy levels of holes are:
- in the valence band
- in the conduction band
- in the band gap but close to the valence band
- in the band gap but close to the conduction band
- Current through a reverse biased p-n junction increases abruptly at:
- breakdown voltage
- 0.0 V
- 0.3 V
- 0.7 V
- A reverse biased diode is equivalent to:
- an off switch
- an on switch
- a low resistance
- none of the above
- The potential barrier in a p-n diode is due to:
- depletion of positive charges near the junction
- accumulation of positive charges near the junction
- depletion of negative charges near the junction
- accumulation of positive and negative charges near the junction