Semiconductors and Band Theory

NATIONAL QUALIFICATIONS CURRICULUM SUPPORT

Physics

Semiconductors and

Band Theory

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[HIGHER]

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Acknowledgement

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Contents

Electrical conductivity and band theory4

Summary of band theory8

Intrinsic semiconductors9

Extrinsic semiconductors11

Student Activity 1 – Thermistor investigation13

Summary of intrinsic and extrinsic semiconductors14

p–n junctions15

Photovoltaic cells17

Student Activity 2 – Photovoltaic cells18

Light emitting diodes 18

Student Activity 3 – LED threshold voltage20

Why use LEDs?21

Summary of p–n junctions, LEDs and photovoltaic cells21

References and further reading22

SEMICONDUCTORS AND BAND THEORY (H, PHYSICS)1

SEMICONDUCTORS AND BAND THEORY

Semiconductors and band theory

The purpose of this document is to introduce the new approach thatis being brought to the Higher Physics course, in teaching about semiconductors from the perspective of band theory.

Electrical conductivity and band theory

All solids can be classified as conductors, semiconductors or insulators according to the availability of conduction electrons in their structures. Band theory gives an explanation for these differences in electrical properties and accounts for the availability, or not, of those conduction electrons.

Although individual atoms have certain permitted energy levels for their electrons, as defined by quantum theory, when large groups of atoms are incorporated into a solid mass those energy levels become reorganised in such a way as to result in bands of possible energy levels (Figure 1). This is known as the tight binding approximation.

Figure 1 Discrete energy levels within an individual atom (left) and bands of permitted energy levels within a solid (right)

There are such enormous numbers of electrons in a solid mass that although the bands actually consist of very large numbers of closely packed discrete energy levels, the bands become essentially continuous.There may be several permitted energy level bands, but in particular we consider the two uppermost bands.These are known as the valence band and the conduction band (Figure 2).

Figure 2 Conduction and valence bands in an insulator. These bands contain the only permitted energy levels, and since the valence band is full and the conduction band is empty, no net movement of electrons can occur within the material. Note the gap separating the bands.

The electrons with lower energy levels are described as occupying the valence band.The innermost electrons in an atom are much less influenced by neighbouring atoms, and occupy discrete energy levels.They are sometimes considered to be bound.At higher levels in the valence band electrons can, in fact, move from atom to atom, but only up to the top of the valence band.Since they are permitted only to swap places with other valence electrons in neighbouring atoms, they areeffectively unavailable for conduction.

Electrons fill the valence band from the lowest level to the highest.The top of the valence band for a material is the highest level, which would,in theory, be filled by all the available electrons within an atomof that material at a temperature of 0K.In insulators and semiconductors, the valence band is completely filled with electrons.The conduction band is empty (Figure 2).

The electrons fill energy levels in order because, as fermions, they must obey the Pauli exclusion principle and cannot occupy identical energy levels.The only way that an electron couldmove from one atom to another in an insulator or semiconductor would be to occupy a slightly different energy level in a neighbouring atom.However, all those energy levels are already full.As discussed above, the electrons may effectively swap places, but in order to facilitate conduction, they must leap up to the conduction band.An energy level band must have some space within it (some vacant energy levels) in order for there to be any net movement of electrons within the material.

For a material to be able to conduct electricity it must have electrons in its conduction band or spaces in its valence band.There must be spaces for charges to move into: a partially filled band.

The simulation demonstrates how energy levels differ in conductors, semiconductors and insulators, and the impact the energy levels have on conductivity.

Again, as a result of the wavelike behaviour of electrons within atoms, materialsmay exhibita certain range of ‘forbidden’ energy levels (Figure 3).It is simply not possible for an electron to exist with an energy levelthat would place it in this range.This leaves insulators and semiconductors with a gap between the two bands.

Figure 3: Conduction bands (blue) and valence bands (yellow) for insulators, semiconductors and conductors. Note the energy gaps in insulators and semiconductors, and how in a conductor there is no gap, simply a continuous, partially filled conduction band.

In insulators this zone of forbidden energy levels is very substantial, and separates the valence band and the conduction band significantly.The forbidden zone is of the order of a few electron volts, and is therefore so large that it is not normally practicable for there to be sufficient energy to move electrons across it from the valence band to the conduction band.For example, thermal excitations and conventional electric circuit voltages within a material provideenergies thatare smaller than 1eV on an atomic scale.It would be necessary to expose an insulator to electric fields of the order of 1010Vm–1 in order to give the valence electrons enough energy to jump across the gap to the conduction band, since this could provide energy in the order of a few electron volts on an atomic scale.This is what happens when there is dielectric breakdown.

Contrastingly, conductors only have one band, and the top of this band is only partially filled, permitting electrical conduction.This means that there are plenty of nearby energy levels available for electrons to move into.They can flow easily from one atom to another when a potential difference is applied across the conducting material.

Like insulators, semiconductors have a completely full valence band and so electrons are not able to facilitate conduction at low temperatures.However, for semiconductors, theforbidden energy level zone between the two bands is sufficiently small to make it much easier for significant numbers of electrons to move across this gap and go from the valence band to the conduction band.This can happen if sufficient energy is supplied, for exampleif there is some thermal excitation.As a result, semiconductors exhibit increased conductivity with increasing temperatures.In many semiconductors, a temperature increase of 10K will permit a doubling of the numbers of electrons in the conduction band.

In order to increase the conductivity of semiconductors, small amounts of doping material can be used.This results in significant increasesin conductivity as a result of the narrowing of the gap between the conduction and valence bands.

Summary of band theory

In solids, permitted electron energy levels are organised as bands.
The valence band contains electrons thatcan be considered to be bound to the atom.In insulators and semiconductors the valence band is full.
The conduction band is a region of permitted energy levels thatis empty in insulators and semiconductors, but partially filled in conductors.
Only partially filled bands may permit conduction.
There is a forbidden zonethatforms an energy gap between the valence and conduction bands in insulators and semiconductors.
Thatenergy gap must be jumped if an electron is to move to the conduction band, and this is not normally possible in insulators because the gap is too large.
In semiconductors, the forbidden zone is much smaller and electrons can jump the gap to the conduction zone as a result of thermal excitation.
Doping of semiconductors can significantly reduce thewidth of the energy gap.

For further information on this topic, try this high-level simulation:

There is also further information from the Hyperphysics website:

Many links lead from this, although it should be noted that some (otherwise very useful) resources describe an overlap between valence and conduction bands in metals.This is misleading and should be treated with caution.

Intrinsic semiconductors

Pure, undoped silicon and germanium are two simple examples of intrinsic semiconductors (Figure 4).They are both in Group IVof the Periodic Table, and form a tetrahedral crystalline structure, similar to diamond.Each atom of silicon and germanium has four electrons in its outermost electron shell, and each of these electrons is used in a covalent bond with one of the atom’s four neighbours.

Figure 4Two-dimensional illustration of a crystal of pure undoped Si. If any individual atom of silicon is considered, it can be seen that each of its four valence electrons is used in maintaining covalent bonds with the atom’s neighbours. These electrons are therefore unavailable for conduction.

Since all valence electrons are involved in bonding, pure silicon and germanium may be expected to be good insulators.However, relatively small energies are required to move a valence electron across the energy gap to the conduction band.This is 1.1eV for silicon, and only 0.7eV for germanium.This means that a significant number of electrons are available in the conduction band, even at room temperature (Figure 5).

Figure 5 It is possible for significant numbers of electrons to cross the energy gap in semiconductors.

It must be noted at this stage that although most thermal excitation involves energies much less than even 0.7eV, quantum mechanics clearly shows that there is a small but significant probability of an electron being able to jump the energy level gap, even at relatively low temperatures.As previously discussed, this probability increases rapidly with temperature.

Once an electron jumps up to the conduction band in the crystal lattice, it leaves behind a ‘hole’ in the covalent bond.This hole can enable another neighbouring valence band electron to move into it.As such, a hole behaves rather like a positive charge carrier, even though it is actually a vacancy for an electron.A hole can travel through the crystal lattice of the semiconductor.A helpful analogy might be to consider a queue of cars on a road.If a space appears at the front of the queue, cars may move forward in turn.Each time a car moves forward, it leaves a space behind it, into which the next car may now move.An observer from above might consider that the cars are moving forwards or that the space is moving backwards.

Some semiconductors, like pure silicon or germanium, are known as intrinsic semiconductors.Intrinsic semiconductors must always contain equal numbers of conduction electrons and holes.If an electron can move from its place then it must leave behind a hole (Figure 6).

Figure 6 In intrinsic semiconductors like pure silicon or germanium, every electron that moves up to the conduction band must leave a hole in the valence band. Electrons and holes exist in equal numbers and both contribute to conduction. There are no majority charge carriers in intrinsic semiconductors.

Extrinsic semiconductors

Often, it is more useful to control the properties of a Group IV semiconductor by deliberately introducing very small proportions of a Group III or Group V element.This is known as doping and results in what is known as an extrinsic semiconductor.Extrinsic semiconductors have majority charge carriers thatmay be either electrons or holes.

Consider a semiconductor thatis doped with a Group III element (Figure 7).Each atomof the doping agent will have only three electrons in its outer shell.This is insufficient to form the four covalent bonds with its Group IV neighbours and therefore results in a hole.Countless holes are now built into the semiconductor’s crystal lattice.It may be referred to as a p-type semiconductor as the majority charge carriers are positively charged holes.As a result of the doping process, it will require much less energy to allow charge to flow through the semiconductor and so its conductivity is greatly enhanced.Unlike metals, a p-type semiconductor’s conduction occurs in the valence band.In effect, the doping agent adds an extra energy level just above the valence band, sometimes called an acceptor band.

Figure 7 Introducing small quantities of Group III atoms into a silicon lattice (in practice only around one part in a million) leaves holes built into the valence band.

Technically, there will also still be a small degree of intrinsic behaviour, as electrons leave behind holes, but this is not considered to be significant in comparison with the overwhelming number of majority charge carriers.

A similar process is involved if a Group V element is used for doping.This gives an extra electron, surplus to covalent bonding requirements, for each atom of the doping agent.These electrons are negatively charged and so an

n-type semiconductor has been produced.In an n-type semiconductor, the majority charge carriers are electrons.The conductivity has been greatly enhanced as before, but this time conduction occurs in an extra energy level just below the conduction band, which is sometimes called the donor band.

Therefore, in p-type and n-type semiconductors conduction can occur easily because there is effectively unfilled space within either the valence or the conduction band, respectively.

Semiconductors are crucial to modern life.According to estimates (Sheffield University) 43% of all semiconductor production goes into computers, 23% into consumer products, 13% into communication and 12% into manufacturing.

In Scotland, Silicon Glen (which is a large proportion of the central belt) has been pioneering electronics production since the 1940s, employing around 50 000 people at its peak in 2000.The name ‘Silicon Glen’ reflects the importance of semiconductors to this sector of Scottish industry, whilst making comparisons with California’s Silicon Valley.

One specialist application for semiconductors is the detection of magnetic fields using the Hall effect.You may be lucky enough to have a Hall effect probe in your school.If not, here are some simulations of the effect:

Student Activity1 – Thermistor investigation

Thermistors use semiconductors in order to vary resistance as a function of temperature.Negative temperature coefficient (NTC) thermistors use thermal energy to free up more charge carriers, so an increase in temperature results in a reduction in resistance.

Students can investigate the resistance variation with temperature for an NTC thermistor:

The thermistor can be immersed in a small beaker of hot water (with a thermometer) and the meter readings used to calculate resistance at 5°C intervals as it cools.A plot of resistance versus temperature can then be produced by the students.A similar procedure could be used for an Light Dependent Resistor.

Summary of intrinsic and extrinsic semiconductors

Semiconductors allow conduction by means of negative charge carriers, which are electrons, or positive charge carriers, which are holes.
The energy band gap in semiconductors is small enough that thermal excitation is sufficient for significant numbers of electrons to be able to move up from the valence to the conduction band.
Intrinsic semiconductors, such as pure silicon, will always have equal numbers of holes and electrons since each conduction electron will leave behind a hole.
Semiconductors may be doped with impurities that add either extra electrons or holes to the lattice.
These doped semiconductors now have a majority charge carrier present and are known as extrinsic semiconductors.
Group III doping agents result in p-type extrinsic semiconductors, which contain extra holes.
Group V doping agents result in n-type extrinsic semiconductors, which contain extra electrons.

This link gives you a simulation for a semiconductor thatyou can adjust yourself:

p–n junctions

If a single semiconducting crystal is doped in such a way that one end is
p-type and the other n-type, then some very useful properties come into play.The interface between the p-type and n-type sections is known as a p–n junction.In this boundary region, electrons from the n-type material may diffuse across the boundary and combine with holes from the p-type material, and vice versa.This results in a lack of majority charge carriers in the immediate vicinity of the junction and as such the region is known as the depletion zone.The p–n junction greatly affects the conductivity of the semiconductor as a whole.When electrons from the n-type material diffuse into the p-type material, they form negative ions as they combine with holes.Positive ions are also left behind in the n-type material.Eventually, this process results in there being no further diffusion of electrons or holes as a result of Coulomb attraction and repulsion (Figures 8a and 8b).

Figure 8a In a p–n junction a depletion zone is formed by the diffusion of electrons from the n-type material into the p-type material. As the electrons combine with holes, ions are formed in the depletion zone.