NATIONAL QUALIFICATIONS CURRICULUM SUPPORT
Physics
Semiconductors and
Band Theory
Support Material
[HIGHER]
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arrangements
for
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Qualifications. Users of all NQ support materials,
whether published by Learning and Teaching
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Acknowledgement
Learning and Teaching Scotland gratefully acknowledges this contribution to the
National Qualifications support programme for Physics.
I gratefully acknowledge the kind guidance and advice I have received from Carol
Trager-Cowan of the University of Strathclyde.
© Learning and Teaching Scotland 2011
This resource may be reproduced in whole or in part for educational purposes by
educational establishments in Scotland provided that no profit accrues at any stage.
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Contents
Electrical conductivity and band theory
4
Summary of band theory
8
Intrinsic semiconductors
9
Extrinsic semiconductors
11
Student Activity 1 – Thermistor investigation
13
Summary of intrinsic and extrinsic semiconductors
14
p–n junctions
15
Photovoltaic cells
17
Student Activity 2 – Photovoltaic cells
18
Light emitting diodes
18
Student Activity 3 – LED threshold voltage
20
Why use LEDs?
21
Summary of p–n junctions, LEDs and photovoltaic cells
21
References and further reading
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SEMICONDUCTORS AND BAND THEORY
Semiconductors and band theory
The purpose of this document is to introduce the new approach that is 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.
Permitted energy levels
Permitted energy bands
Figure 1 Discrete energy levels within an individual atom (left) and bands of permitted
energy levels within a solid (right)
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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).
Conduction band (empty)
Valence band (full)
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 are effectively unavailable for conduction.
Electrons fill the valence band from the lowest level to the high est. 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 atom of that material
at a temperature of 0 K. In insulators and semiconductors, the valence band is
completely filled with electrons. The conduction band is empty (Figure 2).
SEMICONDUCTORS AND BAND THEORY (H, PHYSICS)
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SEMICONDUCTORS AND BAND THEORY
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 could move 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.
http://phet.colorado.edu/en/simulation/conductivity
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 electro ns within atoms,
materials may exhibit a certain range of ‘forbidden’ energy levels (Figure 3).
It is simply not possible for an electron to exist with an energy level that
would place it in this range. This leaves insulators and semiconductors with a
gap between the two bands.
Gap of forbidden energies
Conductor
Insulator
Semiconductor
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.
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SEMICONDUCTORS AND BAND THEORY
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 provide energies that are smaller than 1 eV on an atomic scale. It
would be necessary to expose an insulator to electric fields of the order of
1010 V m–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, the forbidden 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 example if there is some
thermal excitation. As a result, semiconductors exhibit increased conductivity
with increasing temperatures. In many semiconductors, a temperature
increase of 10 K 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 increases in
conductivity as a result of the narrowing of the gap between the conduction
and valence bands.
SEMICONDUCTORS AND BAND THEORY (H, PHYSICS)
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Summary of band theory
 In solids, permitted electron energy levels are organised as bands .
 The valence band contains electrons that can 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 that is empty
in insulators and semiconductors, but partially filled in conductors.
 Only partially filled bands may permit conduction .
 There is a forbidden zone that forms an energy gap between the valence
and conduction bands in insulators and semiconductors.
 That energy 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 the width of the energy
gap.
For further information on this topic, try this high -level simulation:
http://phet.colorado.edu/en/simulation/band-structure
There is also further information from the Hyperphysics website:
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html
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.
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Intrinsic semiconductors
Pure, undoped silicon and germanium are two simple examples of intrinsic
semiconductors (Figure 4). They are both in Group IV of 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 she ll, and
each of these electrons is used in a covalent bond with one of the atom’s four
neighbours.
Figure 4 Two-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 1.1 eV for silicon, and only 0.7 eV for germanium.
This means that a significant number of electrons are available in the
conduction band, even at room temperature (Figure 5).
SEMICONDUCTORS AND BAND THEORY (H, PHYSICS)
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SEMICONDUCTORS AND BAND THEORY
Silicon
Germanium
Conduction band
Electrons
moving
across energy
gap due to
thermal
excitation
Valence band
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 0.7 eV, 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 temperatur es. 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).
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electrons
holes
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 b and. 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 propo rtions 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 that
may be either electrons or holes.
Consider a semiconductor that is doped with a Group III element (Figure 7).
Each atom of the doping agent will have only three electrons in its outer
shell. This is insufficient to form the four covalent bonds with i ts 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.
SEMICONDUCTORS AND BAND THEORY (H, PHYSICS)
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Electron
Hole
Silicon
Group III
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.
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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:
http://www.youtube.com/watch?v=_ATDraCQtpQ&feature=related
http://www.youtube.com/watch?v=FUNnziMmgSQ&feature=related
Student Activity 1 – 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 te mperature for an NTC
thermistor:
A
V
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.
SEMICONDUCTORS AND BAND THEORY (H, PHYSICS)
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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 t o 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 that you can adjust
yourself:
http://phet.colorado.edu/en/simulation/semiconductor
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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).
Depletion zone
p-type
---
+++
-------
+ + ++
n-type
+++
--Junction
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.
SEMICONDUCTORS AND BAND THEORY (H, PHYSICS)
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p-type
n-type
Conduction band
Depletion zone
Valence band
Figure 8b Band theory gives us a model for explaining what happens at a p –n junction.
Holes and electrons diffuse towards the junction in their different bands. At the junction they
combine, producing the depletion zone. This p–n junction is forward biased.
The p–n junction may only allow current to flow if it is forward biased. By
connecting the negative terminal of a power supply to the n -type material, the
junction becomes forward biased. Electrons may be pushed across the
depletion zone if the supply has a sufficient potential difference to overcome
the Coulomb repulsion discussed above. This is typically of the order of 0.7
V. Once the depletion zone is crossed, conduction is easily facilitated by the
majority charge carriers in each of the semiconducting materials.
If the p–n junction is reverse biased, ie the n-type material is connected to the
positive terminal of a power supply, the depletion zone effectively becomes a
greater and greater barrier to conduction. One can imagine the depletion zone
illustrated in Figures 8a and 8b becoming a higher and higher barrier to
conduction as electrons are driven further and further back from the depletion
zone. The junction can only allow a tiny leakage current to flow because of
the intrinsic semiconductor’s electrons and holes. Since this is usually
undesirable, silicon is preferable to germanium because its leakage current is
so much smaller as a result of it having a larger energy gap between its
valence and conduction bands (Figure 5).
Eventually, if the reverse voltage continues to increase, the semiconductor
will break down and may result in damage to the junction.
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Photovoltaic cells
A photovoltaic cell consists of a very thin layer of p -type semiconductor that
is in contact with a layer of n-type material. The conduction electrons are
freed through the action of photons of light. The photons provide sufficient
energy to the electrons to enable them to jump up across the en ergy gap to the
conduction band, leaving behind a hole. The band gap energy for silicon is of
the order of 1.1 eV, and so only photons with at least 1.1 eV of energy can
cause the release of conduction electrons. The wafer of semiconductor is very
thin and so there is a good chance that this process will happen at or very
close to the p–n junction. The electric field produced by the depletion layer at
this junction forces the electron and hole apart, creating a potential
difference, and so a current can flow if the cell is connected to a circ uit. The
p-type layer must be very thin, perhaps 1 m thick, to prevent conduction
electrons from being captured and immobilised by holes.
This animation is useful:
http://solarhorizon.com.au/Flash/LightCurrent.html
Photons interact with
electrons close to p–n
junction
n-type
p-type
Contact
s
Figure 9 Simplified photovoltaic cell in cross-section.
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A solar cell must also have a layer of antireflective coating (not shown in
Figure 9) to improve efficiency because a silicon crystal is so shiny that
without this layer many of the photons would be reflected before they could
cause the release of an electron. Even so, typical efficiencies of solar cells
stand close to 15% and the greatest efficiencies rarely exceed 25%. Finally,
the cell must be coated in glass to reduce damage from the eleme nts.
A compromise must be reached in the choice of materials used to optimise the
performance of the solar cell. By reducing the band gap energy in the
semiconducting material, photons with longer wavelengths and lower
frequencies may be harnessed to free electrons and holes. Although this may
seem desirable and will release more charges, it has the effect of reducing the
strength of the electric field across the junction. It turns out that a band gap
energy of about 1.4 eV is close to ideal, maximising the current and voltage,
and therefore the power, of the cell.
Student Activity 2 – Photovoltaic cells
Students can do investigations into the voltage produced by a photovoltaic
cell as a function of one of the following:
 Irradiance. This investigation may be better suited to a qualitative rather
than quantitative approach because of difficulties in measuring irradiance
accurately. One approach could involve a dimmable light source, another
could be to vary the distance from source to cell.
 Angle of incidence on the solar cell. Use one light source keep the distance
constant at, say, 1.0 m in order to keep the test as fair as possible, then
vary the angle.
 Frequency of radiation. This could employ a range of high-powered lightemitting diodes (LEDs) or filters on a white light source. Again, distance
and power must be kept constant.
Turning this investigation into a look at the inverse square law is another
possibility. Best results will be achieved if the laboratory can be blacked out
for this investigation.
Light-emitting diodes
When a diode is forward biased, electrons from the n-type semiconductor may
move across the junction and combine with holes in the p-type material. The
electrons in the n-type semiconductor move within the higher energy
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conduction band, and as they cross the junction they move briefly into the
empty conduction band of the p-type material. Since the lower energy p-type
valence band is only partially filled, however, the electrons rapidly fall into
an energy level within that valence band. In effect, electrons fall into holes,
and as this happens energy is released in the form of emitted photons (Figure
10). For ordinary diodes, these photons have a relatively low frequency and
long wavelength, which means that they fall outside the visible spectrum.
However, in the construction of LEDs the semiconducting materials may be
engineered in such a way as to result in the photons having sufficiently high
frequency that they fall within the range of visible light.
The frequency of the light emitted from LEDs is controlled by the size of the
energy gap between the conduction and valence bands. A bigger gap will
result in a larger energy change and, in accordance with the relationship E =
hf, a higher frequency of light will be emitted . So, a small energy gap will
result in red light and a much larger energy gap is required for green or blue
light.
n-type
Junction
p-type
Photons emitted as
electrons drop
down to valence
band
Conduction band
(partially filled)
Valence band
(partially filled)
Figure 10 In an LED electrons cross the junction from n to p in the conduction band. Once
on the p side of the junction, they fall back across the energy gap to the valence band. This
releases photons of light.
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Since electrons usually drop from the bottom of the conduction band into the
top of the valence band, light from LEDs tends to be nearly (although not
completely) monochromatic, with a narrow emission spectrum.
To alter the energy gap in an LED, different doping agents are used. They
may typically include indium, gallium and nitrogen to produce blue light,
gallium and phosphorous for green light, and gallium, phosphorous and
arsenic for red light. By using combinations of red, green and blue it is
possible to produce any colour of light and this has led to the advent of LED
televisions. Furthermore, by varying the proportions of the doping agents,
single intermediate colours may also be produced.
Student Activity 3 – LED threshold voltage
There is an approximate correlation between the threshold voltage for an LED
and its colour, since the energy returned in the form of an emitted photon is
approximately the same as the energy required to raise an electron across the
energy gap of the material. This can be investigated in a practical, as
illustrated in Figure 11.
V
+ Supply –
Figure 11 A practical investigation into the correlation b etween energy band gaps and
colours of LEDs can be carried out. Different colours of LEDs can be put into the circuit and
the potential difference required to illuminate them noted. A graph of frequency of light
versus potential difference may be plotted.
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Organic light emitting diodes (OLEDs) use organic polymer layers
sandwiched between two electrodes. When a voltage is applied across the
layers, electrons and holes are generated, which then recombine to emit
photons. The layers may be put into a composite of red-, green- and blueemitting sections so as to produce full-colour displays. Further information
on this topic can be found here:
http://www.chemistry.wustl.edu/~courses/genchem/Tutorials/LED/bands_06.
htm
Why use LEDs?
LEDs have many advantages over other light sources. An LED typically has
an efficiency of around 80%. This is far superior even to energy-saving
compact lights because of the way in which light is produced. This clearly
has major implications for reducing carbon dioxide emissions and fuel bills.
Because of the very low levels of undesirable thermal energy produced, LEDs
may also be expected to have much greater longevity than conventional light
sources. LED and OLED TVs are expensive but with low power usage and
high reliability they are set to become more common in future.
Infrared LEDs are used in remote controls because of their reliability and low
power consumption. Ultraviolet LEDs can be used for detecting counterfeit
notes and even for sterilisation procedures.
LEDs have extremely fast switching speeds, which means that they are
particularly useful for applications where light sources must be pulsed,
strobed or simply switched on and off rapidly with reliability. Additionally,
they work well at low temperatures, are shock resistant and contain fewer
hazardous materials than energy-saving light bulbs.
Summary of p–n junctions, LEDs and photovoltaic cells
 The interface between p-type and n-type material is called the p–n
junction.
 Majority charge carriers diffuse towards the junction and electrons
combine with holes, forming ions.
 This lack of charge carriers results in a depletion zone across the p –n
junction, with positive ions on the n-type side and negative ions on the p type side.
 If the p-type material is connected to the positive terminal of a supply and
the n-type to the negative terminal, then the junction is forward biased.
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 If the potential difference across the junction is sufficient to force
electrons to cross the depletion zone, then the junction will conduct.
 If the terminals are reversed, the junction is reverse biased and cannot
conduct.
 LEDs emit photons of light as electrons ‘fall’ from the conduction band of
the n-type material into holes in the valence band of the p -type material.
 The bigger the energy gap between the bands, the greater the energy, and
therefore the frequency, of the emitted photons.
 Photovoltaic cells use the energy of absorbed photons to separate elect rons
and holes and thus produce a potential difference.
References and further reading
You will find a great deal of further reading on the internet and in specialist
electronics books. The following sources are recommended:
http://www.chemistryexplained.com/Ru-Sp/Semiconductors.html
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html
http://www.educypedia.be/education/solarcellanimations.htm
and many other linked pages associated with this topic from Hyperphysics.
http://en.wikipedia.org/wiki/P-n_junction
In addition, for those with a sense of humour, the somewhat eccentric
http://britneyspears.ac/physics/basics/basics.htm
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