Quantum mechanics in the home

Physics

By Jim Lesurf

19 January 1991

Quantum mechanics is not confined to the world of particle accelerators and nuclear physics. It also lets us understand the properties of materials. With this knowledge we can turn those properties to our advantage and build the transistors that control millions of devices from computers to wristwatches

IF you are sitting at home reading this you are probably surrounded by thousands of transistors. They are in the TV in the corner, the hi-fi, perhaps even the washing machine or your wristwatch. Each of these is likely to contain a hundred transistors or more. If you own a home computer, it may well contain a million transistors. There are more transistors on the Earth than people.

We can use transistors not only to amplify electrical signals, such as the radio signals picked up by an aerial, but also as electronic switches. Networks of these switches can form logic circuits which control your electronic appliances or manipulate information in your computer. We can understand how transistors work using quantum mechanics. This theory is normally associated with such things as particle accelerators and nuclear physics (see Inside Science number 25). Yet the behaviour of the transistors which surround us can be understood using the same basic ideas.

Energy levels

Electrons in atoms

FIRST, we need to understand how electrons behave in an isolated atom. One of the fundamental discoveries of quantum mechanics was that the electrons in an isolated atom can only have specific amounts of energy, known as energy levels.

The electrons behave just as if they were bad-tempered birds who want to perch on a tree which has only a few branches. Each bird will land on a branch and then drive away any others who try to land next to it. After a while, the branches get filled up with one bird per branch.

The heights of the birds above the ground is determined by the shape of the tree. It is impossible for a bird to perch at a height where there is no branch. In an atom, each electron can occupy one of a number of definite energy levels but cannot stay at energies in between. Unlike birds, however, electrons will always settle at the lowest energy level possible.

But atoms rarely exist in isolation and normally extra energy is arriving from the outside. This has the effect of knocking electrons up into higher levels. Each atom will be moving around, bouncing off others, and occasionally being hit by photons of light. These interactions can give electrons extra energy. The higher the temperature the faster the atoms are moving and the more energy is available from collisions to make the electrons jump higher.

When an electron is knocked up to a higher energy level it leaves an opening behind it. Shortly afterwards, it or another electron above this opening will drop down into it, releasing its extra energy as a photon of light.

The force that binds electrons to the atom鈥檚 nucleus is electromagnetic, because electrons are negatively charged and the nucleus is positive. If a large number of atoms are brought close together, the electromagnetic forces between nearby electrons and nuclei make the atoms stick together and form a solid.

Bands and gaps

Electrons in solids

THE ELECTRONS attached to any particular nucleus will also be influenced by the nearby atoms. The effect this has on energy levels is similar to the trees in a rainforest which are so close together that their branches intertwine to form an unbroken canopy. The closeness of the atoms of a solid make the electron energy levels merge into a series of continuous bands separated by energy gaps.

Under most conditions only two of the energy bands of a solid have much effect upon its behaviour. These two bands are called the valence band and the conduction band. If there were no extra energy around all the electrons would sit in the lowest bands with the valence band being the highest of these filled bands. The electrons in the valence band provide most of the forces which glue together the atoms to form a solid. The conduction band is the lowest empty band 鈥� the one directly above the valence band.

In full bands, the electrons cannot move around, so they are stuck. At higher temperatures, vibrations in the solid knock electrons across the gap into the conduction band. There they are free to move around and conduct electricity.

So the size of the gap between the valence band and the conduction band is what makes a solid a good or bad conductor. If the gap is large, electrons will need a lot of energy to jump across it, so not many will be able to make it and the solid will be a poor conductor. If the gap is small, many more electrons will be able to reach the conduction band and move around. The size of the gap and the pattern of all the electron energy bands in a solid will depend upon the sorts of atoms it is made from and how they have been fitted together.

Broadly speaking, we can divide solids into three types: metals, insulators and semiconductors. In a metal the energy bands spread out and overlap each other and there is no gap between the valence and conduction bands. Only a tiny amount of extra energy is required for an electron to jump up to a free level. The electrons can move around in the material quite easily so metals make good electrical conductors.

In an insulator the gap between the valence and conduction bands is large. It is almost impossible for an electron to obtain the energy needed to jump up to the conduction band in an insulator, so all the bands from the valence band down are virtually full and those above nearly empty. This means the material contains practically no electrons which can move around, and it is not able to conduct very well.

If we warm up an insulator we put some more energy into it. This will improve the chance that an electron can find enough extra energy to jump the band gap and move around. So, in principle we could just heat up an insulator until it became hot enough to become a fairly good electrical conductor. In practice the temperature required is usually so high that the solid will melt, evaporate, or even burst into flames.

Semiconductors are materials with an intermediate gap size. At room temperature there is enough energy vibrating through the lattice to knock a moderate number of electrons over the gap into the conduction band where they are free to move around. As a result, semiconductors have an electrical conductivity between that of a metal and that of an insulator.

In fact, most transistors are made from materials which have quite a large band gap and, in themselves, would be fairly good insulators. They are converted into semiconductors by adding a few new atoms of a type which differs from the main bulk of the solid. So a certain number of atoms in the semiconductor crystal, say silicon, will be replaced with another atom, such as indium or phosphorus. This process of adding some other atoms to alter the behaviour of the material is called doping.

Doping semiconductors

Conducting made easy

THERE ARE two kinds of doping. In the first, you add atoms known as donor atoms which supply extra electrons to the material. These extra electrons can jump into the conduction band much more easily than the electrons from the valence band. Alternatively, we can dope the semiconductor with atoms known as acceptor atoms which have one electron too few. The atoms then grab electrons from the valence band. This leaves behind holes which behave as if they were positive particles moving through the valence band. The current in a semiconductor can therefore be caused by the flow of electrons and holes.

Semiconductors to which donor atoms have been added to provide extra moveable electrons are called n-type semiconductors because the moving charges are negative. If, instead, we add acceptor atoms, the resulting material is called p-type because the apparent moving charges behave as if they are positive. In reality, it is electrons shuffling from atom to atom that makes the holes appear to move.

We can now think about building useful devices out of the semiconductors by joining together different types of the material. The simplest combination is to join a bit of p-type material to a bit of n-type. This forms a pn-junction.

In the n-type material, the donor atoms appear to be positively charged because each of them has lost an electron which is now moving around in the conduction band.

Similarly, acceptor atoms in the p-type semiconductor appear to be negatively charged because they have grabbed an extra electron from thc valence band.

Electrical forces generated by these oppositely charged atoms in the two materials prevent most of the electrons from crossing the pn-junction. If an electron attempts to move from the n-type material to the p-type it will be repelled by the negative acceptor atoms in the p-type material. It will also be attracted back into the n-type material by the positive donor atoms. Work must be done to push the electrons across the junction so the potential energy of the electrons must be higher on the p-type side.

In the diagrams we have been drawing to describe the energy bands, the potential energy of an electron has been represented by its height. This difference in energy potentials on either side of the junction can be shown by putting the valence and conduction bands higher in the p-type material than in the n-type.

Drawn like this an electron will slow down if it moves 鈥渦phill鈥� but a hole slows down if it moves downhill. This is because a hole is an IOU for an electron, in the same way that a bubble in a lake could be an IOU for water. If a bubble is trapped under ice on the surface of the lake it automatically moves to the highest possible point. Work would have to be done to push it downwards to a lower point under the ice.

We can alter the difference in heights between the two sides of the junction by attaching a pair of wires, one to each side, and applying a voltage between the two pieces of material. If we apply a voltage which makes the n-type more positive and the p-type more negative we will simply increase the force which prevents charges moving across the barrier 鈥� a steeper hill for the electrons to climb. The change in energy at the junction will rise and almost no current will be able to flow.

However, if we apply a voltage which makes the p-type side seem less negative and the n-type less positive the energy required to cross the barrier will be reduced. Charges will then be able to move relatively easily across the barrier. So the pn-junction acts as diode, resisting the flow of current in one direction but allowing it in the other.

Bipolar transistors

Semiconductor sandwich

TO MAKE a simple form of transistor we can join three pieces of material to produce a semiconductor sandwich which can be either npn or pnp. Both types of sandwich contain two pn-junctions and make what is called a bipolar transistor. In the following description we will look at how an npn sandwich works but a pnp sandwich works in exactly the same way except it is holes that move around and all the voltages applied will be the opposite sign.

Some of the basic properties of these transistors can be understood by thinking of them as pairs of diodes joined back-to-back. Some electrons will be wandering around in the conduction bands of the two pieces of n-type material and some holes will be moving around in the valence band of the central strip of p-type.

The central strip behaves a bit like a hill placed between two flat plains. Most of the electrons do not have enough energy to get over the top. A few will have the energy to reach the other side but their total number will be small.

By applying a voltage between the two sides of the transistor we can alter the relative heights of the conduction bands either side of the central hill. If we then apply a voltage to the central strip of p-type to reduce the height of the hill so that electrons need less energy to scale it, it will suddenly be very easy for the electrons to move from the negative to the positive side of the sandwich.

However, the voltage applied between the two n-type materials is usually much larger than that applied to the central strip. So it becomes almost impossible for any electrons to move in the other direction, from the positive side up the hill into the p-type.

In an npn transistor the positive side is called the collector, because the electrons collect on that side, and the negative side is called the emitter, since it emits its electrons into the p-type material. The central strip is known as the base.

Once voltages are applied in this way, quite a large number of electrons will move from the emitter, through the base, and into the collector. As a result, quite a large electric current can flow. Of course, to keep things going we have to keep removing electrons from the collector and putting fresh ones into the emitter. There must be a current flowing through the wires connecting emitter and collector to whatever is providing the voltage.

Controlling current

Amplifier and switch

THE SIZE of the current through the transistor can easily be controlled by slight alterations to the voltage between the emitter and the base 鈥� changing the height of the hill the electrons must climb to move from the emitter to the collector.

A small percentage of the electrons flowing through the base will encounter one of the atoms which have a missing electron. So some of them will fall into a hole and fill it up. The base then accumulates more electrons than it started with and will become negatively charged, making it harder for any more electrons to pass. To avoid this we need to use the voltage applied between the base and emitter to remove the extra electrons and return them to the emitter. The resulting flow is called the base current.

For a typical bipolar transistor only around 1 per cent of the electrons which leave the emitter are caught by a hole in the base. So the base current required to keep things going is about a hundred times less than the current through the collector.

Electrical engineers often use bipolar transistors to amplify currents. Applying a small input signal in the form of a varying current to the base will produce a collector current which varies in sympathy with the input but is around 1O times bigger. Such a device is said to have a current gain of 100.

In electronic devices, transistors are often used as switches. If we put no voltage between the base and the emitter, very few electrons will be able to pass from the emitter to the collector. The transistor behaves as an 鈥渙pen鈥� switch, refusing nearly all current flow.

But if we apply a large voltage between the emitter and the base, their energy levels can be made almost identical, removing the hill altogether and allowing a free flow of current. The switch has now been 鈥渃losed鈥�. So by changing the voltage between the emitter and base between two particular values, the flow of current can be rapidly turned on and off and the transistor can be used to process digital signals.

When you look inside most modern electronic equipment you will probably see some small black slabs of plastic, each sitting on a number of metal legs. In each one there is an integrated circuit which may contain many hundreds or thousands of transistors.

Some of these slabs contain patterns of transistors which can take the faint radio signals picked from the air by an aerial, amplify them and change their form to drive loudspeakers to play music. Other circuits can compose the picture on a television screen, or collect the information stored as millions of tiny bumps and dips on a compact disc, or control the spin-wash cycle of a washing machine.

In general, quantum mechanics is presented as the subject which brings us face to face with uncertainty and an unavoidable tendency of the real world to behave unpredictably. Yet our understanding of quantum mechanics allows us to engineer the transistors which control the complex precision of a digital watch or the power of a modern computer.

Doped materials, and what鈥檚 in a hole

Doping puts a quantity of different atoms into the lattice of the semiconductor material. This has the effect of adding some new energy levels which sit somewhere in the energy gap between the valence and conduction bands of the material.

If we add some donor atoms, these provide energy levels just below the bottom of the conduction band. They have an extra electron which is not tightly bound to the atom. As the new energy levels are close to the conduction band, it is easy for the extra electrons to jump up into it. The electrons donated to the conduction band are free to move about, increasing the material鈥檚 ability to conduct electricity.

Alternatively, we can add a few acceptor atoms. These atoms provide some new energy levels a little way above the top of the valence band. The atoms are chosen to be of a type which is hungry for an extra electron. As the energy levels created by the acceptor atoms are only just above the top of the valence band, some of the electrons which would normally sit in the valence band are grabbed by the acceptor atoms, producing a hole.

How the holes produced in the valence band by acceptor atoms move around is not as simple as electrons in the conduction band. When one of the atoms of the material has an electron missing in the valence band it becomes quite easy for an electron from an nearby atom to move across and fill the hole. But this means that the nearby atom now has an electron missing. The hole has simply been transferred from one atom to another in exchange for an electron.

We can think of the hole as a sort of credit note which is passed from atom to atom to indicate the lack of an electron. In a normal atom the negative charges of its electrons balance the positive charge of its nucleus. So an atom with an electron missing will be positively charged. By placing acceptor atoms in a material it appears as if there were a population of positively charged holes which may move around and contribute to the conductivity.

Field-effect transistors

MANY modern transistors operate in a completely different way from the bipolar sandwich. Most digital computers, watches and so on use field-effect transistors, or FETs. These come in more than one flavour, just as bipolar transistors can be pnp or npn, but they all work in much the same way.

One flavour of FETs is made by placing a channel of n-type material inside some p-type material. When we apply a voltage between the ends of the channel, a current will flow. To keep this current going we will have to keep putting fresh electrons in at one end and removing them as they arrive at the other. We therefore call the input end of the channel the source and output end the drain.

However, almost none of this current will be able to move into the surrounding p-type material because of the forces of the charged acceptor and donor atoms. If we apply a voltage to the surrounding material (called the gate) and then raise the potential energy of its bands, the electrons in the n-type channel cannot even get near the boundary between n-type and p-type.

A depletion region is formed around the boundary which is devoid of electrons. This bunches the electrons carrying the current into the centre of the channel, making them behave as if the channel were getting narrower. By varying the voltage on the gate we can alter the thickness of the depletion region and can control how hard it is for current to flow along the channel.

The bipolar transistor required a current to the base to replace electrons which fell into holes. But the FET only requires a very tiny input current because the different types of charge carriers 鈥� electrons and holes in the gate and channel 鈥� are kept apart. The low input current needed for a FET makes them ideal for portable uses, such as in calculators and watches. It also means that they reduce the amount of power wasted as heat in applications such as microcomputers.

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