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ELECTRONIC MATERIALS Lecture 10. SEMICONDUCTOR MATERIALS. Applied Electronics Department Technical University of Cluj-Napoca Cluj-Napoca, Cluj, 400027, Romania Phone: +40-264-401412, E-mail: [email protected]

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Applied Electronics DepartmentTechnical University of Cluj-NapocaCluj-Napoca, Cluj, 400027, RomaniaPhone: +40-264-401412, E-mail: [email protected]


According to their electrical conductivity, materials are conventionally classified into three groups: conductors, semiconductors and insulators. Conductors, which include all metals, have high conductivities, semiconductors show intermediate conductivities and insulators have low conductivities. The distinction between semiconductors and insulators is only quantitative, whereas the distinction between semiconductors and metals is more profound.

All characteristic properties of semiconductors are the consequence of a basic physical phenomenon: the existence of certain energy bands in the energy spectrum of electrons.

Semiconductor materials are insulators at absolute zero temperature that conduct electricity in a limited way at room temperature. The defining property of a semiconductor material is that it can be doped with impurities that alter its electronic properties in a controllable way.



Once we know the bandstructure of a given material we still need to find out which energy levels are occupied and whether specific bands are empty, partially filled or completely filled.

Empty bands do not contain electrons. Therefore, they are not expected to contribute to the electrical conductivity of the material. Partially filled bands do contain electrons as well as available energy levels at slightly higher energies. These unoccupied energy levels enable carriers to gain energy when moving in an

applied electric field. Electrons in a partially filled band therefore do contribute to the electrical conductivity of the material.

Completely filled bands do contain plenty of electrons but do not contribute to the conductivity of the material. This is because the electrons cannot gain energy since all energy levels are already filled.


Semiconductors differ from metals and insulators by the fact that they contain an "almost-empty" conduction band and an "almost-full" valence band. This also means that we will have to deal with the transport of carriers in both bands.

To facilitate the discussion of the transport in the "almost-full" valence band of a semiconductor, we will introduce the concept of holes. It is important to understand that one could deal with only electrons if one is willing to keep track of all the electrons in the "almost-full" valence band. After all, electrons are the only real particles available in a semiconductor.

The concepts of holes is introduced in semiconductors since it is easier to keep track of the missing electrons in an "almost-full" band, rather than keeping track of the actual electrons in that band.

Holes are missing electrons. They behave as particles with the same properties as the electrons would have when occupying the same states except that they carry a positive charge.




Group III elemental semiconductors

Boron (B)

Group IV elemental semiconductors

Diamond (C)

Silicon (Si)

Germanium (Ge)

Tin (Sn)

Group V elemental semiconductors

Phosphorus (P)

Arsenic (As)

Antimony (Sb)

Group VI elemental semiconductors

Sulfur (S)

Selenium (Se)

Tellurium (Te)

Group VII elemental semiconductors

Iodine (I)

Group IV compound semiconductors

Silicon carbide (SiC)

Silicon germanide (SiGe)

III-V semiconductors

Aluminium antimonide (AlSb)

Aluminium arsenide (AlAs)

Aluminium nitride (AlN)

Aluminium phosphide (AlP)

Boron nitride (BN)

Boron arsenide (BAs)

Gallium antimonide (GaSb)

Gallium arsenide (GaAs)

Gallium nitride (GaN)

Gallium phosphide (GaP)

Indium antimonide (InSb)

Indium arsenide (InAs)

Indium nitride (InN)

Indium phosphide (InP)


III-V quaternary semiconductor alloys

Aluminium gallium indium phosphide (AlGaInP, also InAlGaP, InGaAlP, AlInGaP)

Aluminium gallium arsenide phosphide (AlGaAsP)

Indium gallium arsenide phosphide (InGaAsP)

Aluminium indium arsenide phosphide (AlInAsP)

Aluminium gallium arsenide nitride (AlGaAsN)

Indium gallium arsenide nitride (InGaAsN)

Indium aluminium arsenide nitride (InAlAsN)

III-V quinary semiconductor alloys

Gallium indium nitride arsenide antimonide (GaInNAsSb)

III-V ternary semiconductor alloys

Aluminium gallium arsenide (AlGaAs, AlxGa1-xAs)

Indium gallium arsenide (InGaAs, InxGa1-xAs)

Aluminium indium arsenide (AlInAs)

Aluminium indium antimonide (AlInSb)

Gallium arsenide nitride (GaAsN)

Gallium arsenide phosphide (GaAsP)

Aluminium gallium nitride (AlGaN)

Aluminium gallium phosphide (AlGaP)

Indium gallium nitride (InGaN)

Indium arsenide antimonide (InAsSb)

Indium gallium antimonide (InGaSb)


I-VII semiconductors

Cuprous chloride (CuCl)

IV-VI semiconductors

Lead selenide (PbSe)

Lead sulfide (PbS)

Lead telluride (PbTe)

Tin sulfide (SnS)

Tin telluride (SnTe)

IV-VI ternary semiconductors

lead tin telluride (PbSnTe)

Thallium tin telluride (Tl2SnTe5)

Thallium germanium telluride (Tl2GeTe5)

II-VI semiconductors

Cadmium selenide (CdSe)

Cadmium sulfide (CdS)

Cadmium telluride (CdTe)

Zinc oxide (ZnO)

Zinc selenide (ZnSe)

Zinc sulfide (ZnS)

Zinc telluride (ZnTe)

II-VI ternary alloy semiconductors

Cadmium zinc telluride (CdZnTe, CZT)

Mercury cadmium telluride (HgCdTe)

Mercury zinc telluride (HgZnTe)

Mercury zinc selenide (HgZnSe)


  • There are two main types of semiconductor materials:

  • intrinsic - where the semiconducting properties of the material occur naturally i.e. they are intrinsic to the material's nature.

  • extrinsic - they semiconducting properties of the material are manufactured, by us, to make the material behave in the manner which we require.

  • Nearly all the semiconductors used in modern electronics are extrinsic. This means that they have been created by altering the electronic properties of the material.

  • Several different semiconducting materials exist, but the most common semiconductor material is Silicon and the two most common methods of modifying the electronic properties are:

  • Doping - the addition of 'foreign' atoms to the material.

  • Junction effects - the things that happen when we join differing materials together.


Semiconductors' intrinsic electrical properties are very often permanently modified by introducing impurities, in a process known as doping. Usually it is reasonable to approximate that each impurity atom adds one electron or one "hole" that may flow freely. Upon the addition of a sufficiently large proportion of dopants, semiconductors conduct electricity nearly as well as metals.

In addition to permanent modification through doping, the electrical properties of semiconductors are often dynamically modified by applying electric fields. The ability to control conductivity in small and well-defined regions of semiconductor material, statically through doping and dynamically through the application of electric fields (like transistors). Semiconductor devices with dynamically controlled conductivity are the building blocks of integrated circuits.

In certain semiconductors, when electrons fall from the conduction band to the valence band (the energy levels above and below the band gap), they often emit light. This photoemission process underlies the light-emitting diode (LED) and the semiconductor laser, both of which are tremendously important commercially. Conversely, semiconductor absorption of light in photodetectors excites electrons from the valence band to the conduction band, facilitating reception of fiber optic communications, and providing the basis for energy from solar cells.



Like all crystals, semiconductor crystals can be obtained by cooling the molten semiconductor material.

However, this procedure yields poly-crystalline material since crystals start growing in different locations with a different orientation. Instead when growing single-crystalline silicon one starts with a seed crystal and dips one end into the melt. By controlling the temperature difference between the seed crystal and the molten silicon, the seed crystal slowly grows. The result is a large single-crystal silicon boule. Such boules have a cylindrical shape, in part because the seed crystal is rotated during growth and in part because of the cylindrical shape of the crucible containing the melt. The boule is then cut into wafers with a diamond saw and further polished to yield the starting material for silicon device fabrication.


This pure silicon ingot is about six inches across and a few feet long. When it's sliced, it will make thousands of silicon wafers, ready for processing.

These wafers have been sawed from a solid ingot of silicon. Each wafer is six inches in diameter. One side of each wafer has already been polished smooth. Note the slight flat spot on the outside edge of each wafer to help hold it still during the polishing process.


Polishing the Wafer Smooth

Before the real work begins, one side of each wafer must be polished absolutely smooth. These wafers will be so smooth after they're finished that you couldn't detect any imperfections on the surface even with a microscope. The process is called chemical-mechanical polishing (CMP). As the name implies, it involves bathing the wafers in special abrasive chemicals and gently grinding any imperfections away.

The wafers need to be smooth and flat because the features that will be projected onto them in the chip "darkroom" are extremely small and close together. In photography, it's important to keep the print lying flat as it develops because any warping or bending will throw the picture out of focus. Each part of the print must be the same distance from the projecting lens, and the principle is the same for chip making. Any variations in the surface of the silicon wafer will make the chip design out of focus, possibly causing shorts and other faults.


Building the Layers

After a wafer is polished, it's time to start building up the layers of material that will become the electronic components on the chip. A wafer is pure silicon, but it takes more than silicon to make a chip work. There are the metal wires, but those will be added later. At this stage, we need to build up a few layers of silicon (which conducts electricity) alternating with a few layers of an insulator (which doesn't conduct electricity). These two types of material will be stacked, like layers of frosting between layers of cake. Later, we'll etch away some of this insulator (most of it, actually), leaving a carefully planned-out three-dimensional pattern of silicon and other materials that channel electricity in exactly the paths we want.

Engineers call this layering process deposition because chemicals are deposited on the supersmooth surface of our wafer. Deposition can be done in a couple of different ways. One way is to put the wafer in an oven along with pure oxygen gas and let the oxygen seep into the top of the wafer. (Oxygen doesn't conduct electricity; it is an insulator.) Another way is to spray the chemicals onto the wafer, called chemical vapor deposition (CVD). After we've built up the first layer of our chip, it will look like in figure.


Layers of material are sprayed or deposited on top of the base silicon wafer. Alternating layers of conducting and insulating material are built up. The number of layers depends on the complexity of the chip. After each pair of conducting/insulating layers is put down, some of it is removed to make intricate wires.


After we coat the wafer with insulating chemicals, there comes a second layer of special chemicals, this one called photoresist. Photoresist is like the coating on black-and-white print film before it's been developed. The stuff reacts to light, which we're depending on to make our chips work.

Once the wafer is coated with photoresist, it's light sensitive and must be kept in the dark so that it doesn't accidentally get "developed" before it's ready. Fortunately for us, the photoresist is sensitive to light we can't see, so there's no need to shut off the lights in the clean room. The wafer still needs to be handled carefully, however, to avoid scratching its delicate and carefully prepared surface.

To develop the wafer, we put it into a machine called a stepper. Usually a robot arm takes a wafer off the top of a stack and places it in the stepper automatically. Mounted just a few inches above the wafer is a copy of the film negative that the chip designers created (which was described in the previous chapter). Engineers call this film the reticle, and it's usually made of quartz instead of normal camera film. Just above the film is a lens and behind that is a bright light. Basically, we're going to project an image of the film down onto our wafer, like projecting slides onto a wall. The only real difference is the scale: The image we project will be tiny, only a fraction of an inch on a side, just big enough to make one copy of our chip.


This wafer has been through several stages of the stepping and etching processes. The chips inscribed on it are clearly visible, making a grid pattern on the wafer's surface. Later, the wafer will be cut apart with a diamond-tipped saw, separating each of the square chips.


These analog components, scattered across two silicon wafers, show the variety of shapes and sizes these components can take.



The energy band diagrams of semiconductors are rather complex. The detailed energy band diagrams of germanium, silicon and gallium arsenide are shown in the next figure. The energy is plotted as a function of the wavenumber, k, along the main crystallographic directions in the crystal, since the band diagram depends on the direction in the crystal. The energy band diagrams contain multiple completely-filled and completely-empty bands. In addition, there are multiple partially-filled band.

Energy band diagram of (a) germanium, (b) silicon and (c) gallium arsenide.


The energy band diagrams are frequently simplified when analyzing semiconductor devices. Since the electronic properties of a semiconductor are dominated by the highest partially empty band and the lowest partially filled band, it is often sufficient to only consider those bands. This leads to a simplified energy band diagram for semiconductors as shown in the figure.

The diagram identifies the almost-empty conduction band by a horizontal line. This line indicates the bottom edge of the conduction band and is labeled Ec. Similarly, the top of the valence band is indicated by a horizontal line labeled Ev. The energy band gap is located between the two lines, which are separated by the bandgap energy Eg.


A simplified energy band diagram used to describe semiconductors. Shown are the valence and conduction band as indicated by the valence band edge, Ev, and the conduction band edge, Ec.



The energy bandgap of semiconductors tends to decrease as the temperature is increased. This behavior can be better understood if one considers that the interatomic spacing increases when the amplitude of the atomic vibrations increases due to the increased thermal energy. This effect is quantified by the linear expansion coefficient of a material. An increased interatomic spacing decreases the average potential seen by the electrons in the material, which in turn reduces the size of the energy bandgap. A direct modulation of the interatomic distance - such as by applying compressive (tensile) stress - also causes an increase (decrease) of the bandgap.


The temperature dependence of the energy bandgap, Eg, has been experimentally determined yielding the following expression for Eg as a function of the temperature, T:

where Eg(0), a and b are the fitting parameters.

Parameters used to calculate the energy bandgap of germanium, silicon and gallium arsenide (GaAs) as a function of temperature are listed in the next table.


Calculate the energy bandgap of germanium, silicon and gallium arsenide at 300, 400, 500 and 600 K.



The bandgap of silicon at 300 K equals:

Similarly one finds the energy bandgap for germanium and gallium arsenide, as well as at different temperatures, yielding:


A plot of the resulting bandgap versus temperature is shown in the next figure for germanium, silicon and gallium arsenide.

Temperature dependence of the energy bandgap of germanium (Ge), silicon (Si) and gallium arsenide (GaAs).



An intrinsic semiconductor, also called an undoped semiconductor or i-type semiconductor, is a pure semiconductor without any significant dopant species present. The presence and type of charge carriers is therefore determined by the material itself instead of the impurities; the amount of electrons and holes is roughly equal.

Intrinsic semiconductors conductivity can be due to crystal defects or to thermal excitation. In an intrinsic semiconductor the number of electrons in the conduction band is equal to the number of holes in the valence band.


A layer of i-type semiconductor is used in PIN diodes.


PIN diode

PIN diode (p-type, intrinsic, n-type diode) is a diode with a wide, undoped intrinsic semiconductor region between p-type semiconductor and n-type semiconductor regions.

PIN diodes act as near perfect resistors at RF and microwave frequencies. The resistivity is dependent on the DC current applied to the diode.

A PIN diode exhibits an increase in its electrical conductivity as a function of the intensity, wavelength, and modulation rate of the incident radiation.

The benefit of a PIN diode is that the depletion region exists almost completely within the intrinsic region, which is a constant width (or almost constant) regardless of the reverse bias applied to the diode. This intrinsic region can be made large, increasing the area where electron-hole pairs can be generated. For these reasons many photodetector devices include at least one PIN diode in their construction, for example PIN photodiodes and phototransistors (in which the base-collector junction is a PIN diode).

They are not limited in speed by the capacitance between n and p region anymore, but by the time the electrons need to drift across the undoped region.


Conduction in Intrinsic Semiconductors

A solid with filled bands is an insulator, but at finite temperature, electrons can be thermally excited from the valence band to the next highest, the conduction band. The fraction of electrons excited in this way depends on the temperature and the band gap, the energy difference between the two bands. Exciting these electrons into the conduction band leaves behind positively charged holes in the valence band, which can also conduct electricity.

An intrinsic (pure) semiconductor's conductivity is strongly dependent on the band gap. The only available carriers for conduction are the electrons which have enough thermal energy to be excited across the band gap, which is defined as the energy level difference between the conduction band and the valence band.


The band structure for intrinsic semiconductors.

In intrinsic semiconductors (like silicon and germanium), the Fermi level is essentially halfway between the valence and conduction bands. Although no conduction occurs at 0K, at higher temperatures a finite number of electrons can reach the conduction band and provide some current.


Conduction in Intrinsic Semiconductors

Semiconductors are the class of elements which have four valence electrons. Two important semiconductors are germanium (Ge) and silicon (Si). Early solid-state electronic devices were fabricated almost exclusively from germanium, whereas modern devices are fabricated almost exclusively from silicon. Gallium arsenide (GaAs) is a semiconductor compound made up of gallium, which has three valence electrons, and arsenic, which has five. This semiconductor is making inroads in digital applications which require extremely high switching speeds and in extremely high-frequency analog applications. However, silicon remains the most useful semiconductor material and is expected to dominate for many years to come.

Semiconductor materials are normally in crystalline form with each valence electron shared by two atoms. The semiconductor is said to be intrinsic if it is not contaminated with impurity atoms.


Two-dimensional illustration of the crystal lattice of an intrinsic semiconductor at T=0K.


The figure shows a two-dimensional view of an intrinsic semiconductor crystal. Each circle represents both the nucleus of an atom and all electrons in that atom except the valence electrons. The links between the circles represent the valence electrons. Each valence electron can be assumed to spend half time with each of two atoms so that each atom sees eight half-time electrons. Compared to a metal, the valence electrons in a semiconductor are tightly bound.

The thermal energy stored in a semiconductor crystal lattice causes the atoms to be in constant mechanical vibration. At room temperature, the vibrations shake loose several valence electrons which then become free electrons. In intrinsic silicon, the number of free electrons is approximately one in 1012 of the total number of valence electrons. The free electrons behave similarly to those in a metal. Under the influence of an applied electric field, they have a mobility and exhibit a drift velocity which produces a conduction current. However, because of the small number of free electrons, the conductivity of an intrinsic semiconductor is much lower than that of a metal.


When an electron is shaken loose from an atom, an electron vacancy is left which is called a hole. The parent atom then becomes an ion. The constant mechanical vibration of the lattice can cause the ion to capture a valence electron from a neighboring atom to replace the missing one. When such a transfer takes place, the position of the hole moves from one atom to another. This is equivalent to a positive charge +q moving about in the semiconductor. (The motion of a hole can be likened to the motion of a bubble in water.) Like free electrons, holes have a mobility and exhibit a drift velocity which produces a conduction current under the influence of an applied electric field. Because of the opposite charge polarity of electrons and holes, they drift in opposite directions under the influence of a field.


Intrinsic semiconductor at T>0K.


Next figure illustrates the drift of free electrons and the drift of holes in an intrinsic semiconductor under the application of an electric field that is directed from right to left for free electrons. When an electron is shaken loose from its valence shell, an electron-hole pair is formed. The force generated by the electric field causes the free electrons to drift to the left. In effect, a hole drifts to the right when a bound valence electron shifts to the left from one atom to another. The movement of holes may be likened to the movement of bubbles of air in water, where the water represents the bound electrons and the bubbles represent the holes. The movement of a bubble in one direction is really the result of a movement of water in the opposite direction.

In summary, the flow of current in the semiconductor is the result of the flow of two components. One component is the flow of free electrons in one direction. The other component is the flow of the absence of bound electrons in the other direction. Because of the opposite charge polarities, the electron current and the hole current add to produce the total conduction current.


Illustration of the drift of free electrons and the drift of holes under the application of an external electric field.



Because hole-electron pairs are continually created by thermal agitation of a semiconductor lattice, it might seem that the number of holes and free electrons would continually increase with time. This does not happen because free electrons are continually recombining with holes. At any temperature, a stable state is reached when the creation rate of hole-electron pairs is equal to the recombination rate. The mean lifetime τn(s) of a free electron is the average time that the electron exists in the free state before recombination. The mean lifetime τp(s) for the hole is defined similarly. In the intrinsic semiconductor, τn is equal to τp because the number of free electrons must be equal to the number of holes. However, the addition of an impurity to the semiconductor lattice can cause the mean lifetimes to be unequal.



This equation defines the conductivity σ of the intrinsic semiconductor. It is given by:


Example 1

A rod of intrinsic silicon is 1 cm long and has a diameter of 1mm. At room temperature, the intrinsic concentration in the silicon is ni = 1.5 × 1016 per m3. The electron and hole mobilities are μe = 0.13m2 V−1 s−1 and μh = 0.05m2 V−1 s−1. Calculate the conductivity σ of the silicon and the resistance R of the rod.



The conductivity is calculated as follows:

The resistance is calculated as follows:



The preceding example illustrates how poor a conductor intrinsic silicon is at room temperature. The conductivity can be increased by adding certain impurities in carefully controlled minute quantities. When this is done, the semiconductor is called a doped semiconductor. There are two classes of impurities that are used. These are donor impurities and acceptor impurities. Typically one impurity atom is added per 108 semiconductor atoms. A semiconductor that is doped with a donor impurity is called an n-type semiconductor. One that is doped with an acceptor impurity is called a p-type semiconductor.


n-Type Semiconductor

An n-type semiconductor is produced by adding a donor impurity such as arsenic, antimony, or phosphorus to an intrinsic semiconductor. Each donor atom has five valence electrons. When a donor atom replaces an atom in the crystal lattice, only four valence electrons are shared with the surrounding atoms. The fifth valence electron becomes a free electron as illustrated in figure. The number of free electrons donated by the donor atoms is much greater than the number of free electrons and holes in the intrinsic semiconductor. This makes the conductivity of the n-type semiconductor much greater that of the intrinsic semiconductor. Because the number of free electrons is far greater than the number of holes, the free electrons are the majority carriers. The semiconductor is called n-type because the majority carriers have a negative charge.


Two-dimensional illustration of the crystal lattice of an n-type semiconductor.


Structure of energy bands of extrinsic semiconductors, doped with donor impurities.


Hole-electron pairs are continually formed by thermal agitation of the lattice in an n-type semiconductor. Because of the large number of donor electrons, there are many more free electrons available for recombination with the holes. This decreases the mean lifetime for the holes which decreases the number of holes in the n-type semiconductor compared to the intrinsic semiconductor. For this reason, the current due to the flow of holes in an n-type semiconductor is often neglected in calculations.

It is important to understand that a donor atom is electrically neutral if its fifth valence electron does not become a free electron in the lattice. If the fifth electron becomes a free electron, the number of protons in the atom is greater than the number of electrons by one. In this case, the donor atom becomes a bound positively charged ion.


p-Type Semiconductor

A p-type semiconductor is produced by adding an acceptor impurity such as gallium, boron, or indium to an intrinsic semiconductor. Each acceptor atom has three valence electrons. When an acceptor atom replaces an atom in the crystal lattice, there are only three valence electrons shared with the surrounding atoms. This leaves a hole as illustrated in next figure. The number of holes created by the acceptor atoms is much greater than the number of free electrons and holes in the intrinsic semiconductor. This makes the conductivity of the p-type semiconductor much greater that of the intrinsic semiconductor. Because the number of holes is far greater than the number of electrons, the holes are the majority carriers. The semiconductor is called p-type because the majority carriers have a positive charge.


Two-dimensional illustration of the crystal lattice of a p-type semiconductor.


Structure of energy bands and conduction of extrinsic semiconductors (doped with acceptor impurities).


Hole-electron pairs are continually formed by thermal agitation of the lattice in a p-type semiconductor. Because of the large number of holes, there are many more holes available for recombination with the free electrons. This decreases the mean lifetime for the free electrons which decreases the number of electrons in the p-type semiconductor compared to the intrinsic semiconductor. For this reason, the current due to the flow of free electrons in a p-type semiconductor is often neglected in calculations.

It is important to understand that an acceptor atom is electrically neutral if the hole created by the absence of its fourth valence electron is not filled by an electron from an adjacent silicon atom.

Once an electron fills the hole, the number of electrons in that atom is greater than the number of protons by one. In this case, the acceptor atom becomes a bound negatively charged ion.


Mass-Action Law

In an intrinsic semiconductor, we have noted that the electron concentration and the hole concentration are both equal to the intrinsic concentration, i.e. n = p = ni. If this were not true, the material would not be electrically neutral. We have seen that adding an n-type impurity to the semiconductor increases n and decreases p. Similarly, adding a p-type impurity increases p and decreases n. It can be shown that the product of n times p is a constant independent of the doping type and the doping level. The product is given by:

This relation is called the mass-action law.


Electrical Neutrality

An intrinsic semiconductor is electrically neutral, i.e. there is no net charge stored. The addition of n-type or p-type impurities does not change this. To state this mathematically, let ND be the number of donor atoms per m3 and NA the number of acceptor atoms per m3. We assume that all donor atoms and all acceptor atoms are ionized so that there are ND bound positive charges per m3 and NA bound negative charges per m3. Each donor ion has a charge +q and each acceptor ion has a charge −q. The total number of negative charges per m3 is equal to the number n of free electrons per m3 plus the number NA of bound acceptor atoms per m3, i.e. n+NA. Similarly, the number of positive charges per m3 is equal to the number p of holes per m3 plus the number ND of bound donor atoms per m3, i.e. p+ND. Because the semiconductor is electrically neutral, the number of positive charges must equal the number of negative charges. This gives the condition:

n + NA = p + ND


In an n-type semiconductor, NA=0 and p<<n so that the above equation can be solved for n to obtain:

n = p + ND≈ ND

The approximation in this equation and mass-action lawcan be used to solve for the hole concentration p to obtain:

Similarly, in a p-type semiconductor, we can write:


Example 2

In the silicon rod of Example 1, the number of silicon atoms per m3 is 5×1028. A

donor impurity is added to the silicon in the concentration of one donor atom per 108 atoms of silicon. Calculate the new resistance of the rod. Assume that each donor atom contributes one free electron.



The donor concentration in the silicon is calculated as follows:

The free electron concentration is n ≈ND=5×1020 electrons per m3. The hole concentration is

holes per m3

Because p << n, we can neglect p in calculating the conductivity. The conductivity is:

The resistance is calculated as follows:

Compared to the intrinsic silicon rod of Example 1, this is smaller by a factor of 24,1.


PN junction

A pn junction is a merger of the semiconductor p- and n-type regions. In a transition layer between the two regions the carrier densities are much lower than in the neutral regions away from the junction. For this reason, this transition layer is also called the depletion region. The simplest pn region is a junction where both sides are homogeneously doped and the transition between the acceptor and donor doping is abrupt. Such a junction is called an abrupt pn junction.

The width of the depletion region of an abrupt pn junction is given by

Here εs is the relative dielectric constant of the semiconductor, ε0 is the permittivity of free space, NA and ND are the acceptor and donor concentrations at the p- and n-side of the junction, respectively, Vbi(≤Eg/q) is the built-in potential, and V is the externally applied bias voltage. V>0 if the p-side is positively biased.


A p-n junction consists of two semiconductor regions with opposite doping type as shown in figure. The region on the left is p-type with an acceptor density Na, while the region on the right is n-type with a donor density Nd. The dopants are assumed to be shallow, so that the electron (hole) density in the n-type (p-type) region is approximately equal to the donor (acceptor) density.

Cross-section of a p-n junction.


We will assume, unless stated otherwise, that the doped regions are uniformly doped and that the transition between the two regions is abrupt. We will refer to this structure as an abrupt p-n junction.

Frequently we will deal with p-n junctions in which one side is distinctly higher-doped than the other. We will find that in such a case only the low-doped region needs to be considered, since it primarily determines the device characteristics. We will refer to such a structure as a one-sided abrupt p-n junction.

The junction is biased with a voltage Va. We will call the junction forward-biased if a positive voltage is applied to the p-doped region and reversed-biased if a negative voltage is applied to the p-doped region. The contact to the p-type region is also called the anode, while the contact to the n-type region is called the cathode, in reference to the anions or positive carriers and cations or negative carriers in each of these regions.


Semiconductor materials

The most commonly used semiconductor material is Silicon. This is an element, it has 14 electrons, and its pure solid form melts at 1420 °C. Used for thousands of years to make ordinary glass, Silicon is a very common element. Silicon turns up in lots of rocks and forms the sand on beaches.

The earliest commercial semiconductor devices mostly used Germanium. This element has 32 electrons per atom and melts at 985 °C. It has now largely fallen into disuse because it is much rarer and more expensive than Silicon and has no real advantages for most purposes.

The second most common modern material is Gallium Arsenide, GaAs. This is a combination of Gallium, an element with 31 electrons per atom, and Arsenic, with 33 electrons per atom. This is a crystalline compound, not an element. Hence we can get an extra degree of control over its properties by varying the relative amount of Gallium and Arsenic.

GaAs has the advantage of making semiconductor devices which respond very quickly to electrical signals. This makes it better than Silicon for doing tasks like amplifying the high frequency (1GHz to 10GHz) signals from TV satellites, etc. The main disadvantage of GaAs is that it is more difficult to make and the chemicals involved are quite often poisonous!


GaAs can be used with signal frequencies up to about 100 GHz. At even higher frequencies more esoteric materials such as Indium Phosphide (InP) may be used. At present, however, the MMWave region (frequencies above about 50 GHz) is only used for special purposes, so most of the electronics in the world thends to be based on Silicon, with some GaAs, and only a few InP devices.

Silicon carbide (SiC) is a ceramic compound of silicon and carbon.

Pure α-SiC is an intrinsic semiconductor with band gaps of 3.28 eV (4H) and 3.03 eV (6H) respectively.

Silicon carbide is used for blue LEDs, ultrafast Schottky diodes, MESFETs and high temperature IGBTs and thyristors for high power switching. Due to its high thermal conductivity, SiC is also used as substrate for other semiconductor materials such as gallium nitride. Due to its wide band gap, SiC-based parts are capable of operating at high temperature (over 350 °C), which together with good thermal conductivity of SiC reduces problems with cooling of power parts. They also possess increased tolerance to radiation damage, making it a material desired for defense and aerospace applications. Its main competitor is gallium nitride. Although diamond has an even higher band gap, SiC-based devices are easier to manufacture due to the fact that it is more convenient to grow an insulating layer of silicon dioxide on the surface of a silicon carbide wafer than it is with diamond.






Silicon Carbide


Silicon Carbide



The Hall effect describes the behavior of the free carriers in a semiconductor when applying an electric as well as a magnetic field. The experimental setup shown in the figure, depicts a semiconductor bar with a rectangular cross section and length L. A voltage Vx is applied between the two contacts, resulting in a field along the x-direction. The magnetic field is applied in the z-direction.


Hall setup and carrier motion for a) holes and b) electrons.


As shown in figure a), the holes move in the positive x-direction. The magnetic field causes a force to act on the mobile particles in a direction dictated by the right hand rule. As a result there is a force, Fy, along the positive y-direction, which moves the holes to the right. In steady state this force is balanced by an electric field, Ey, so that there is no net force on the holes. As a result there is a voltage across the sample, which can be measured with a high-impedance voltmeter. This voltage, VH, is called the Hall voltage. For the sign convention shown in 2.7.8, the Hall voltage is positive for holes.

The behavior of electrons is shown in figure b). The electrons travel in the negative x-direction. Therefore the force, Fy, is in the positive y-direction due to the negative charge and the electrons move to the right, just like holes. The balancing electric field, Ey, now has the opposite sign, which results in a negative Hall voltage.

To calculate the Hall field, we first calculate the Lorentz force acting on the free carriers:


A measurement of the Hall voltage is often used to determine the type of semiconductor (n-type or p-type) the free carrier density and the carrier mobility. Repeating the measurement at different temperatures allows one to measure the free carrier density as well as the mobility as a function of temperature. Since the measurement can be done on a small piece of uniformly doped material it is by far the easiest measurement to determine the carrier mobility. It should be noted that the scattering mechanisms in the presence of a magnetic field are different and that the measured Hall mobility can differ somewhat from the drift mobility. A measurement of the carrier density versus temperature provides information regarding the ionization energies of the donors and acceptor that are present in the semiconductor. While the interpretation of the Hall measurement is straightforward in the case of a single dopant, multiple types of impurities and the presence of electrons and holes can make the interpretation non-trivial.


Exercise 1

A conducting line on an IC chip is 2.8 millimeters (mm) long and has a rectangular cross section 1x4 micrometers (μm). A current of 5 mA produces a voltage drop of 100 mV across the line. Determine the electron concentration given that the electron mobility is 500 cm2/Vs.



The electron concentration can be obtain from conductivity. The conductivity is determined by solving equation:

Then, from equation:

We obtain:


Exercise 2

An intrinsic silicon bar is 3 mm long and has a rectangular cross section 50x100 μm. At 300K, determine the electric field intensity in the bar and the voltage across the bar when a steady current of 1 μA is measured. The resistivity of intrinsic silicon at 300K is 2.30x105 [Ω·cm].



The field intensity can be obtained from the current density and conductivity as:

The voltage across the bar is:

The result obtained in this example indicates that an extremely large voltage is needed to produce a very small current (1μA). This, however, is not surprising since the intrinsic carrier concentration is much closer to that of an insulator than it is to a conductor. Thus intrinsic semiconductor are not suitable for electron devices. The carrier concentration must be increased.


Exercise 3

An n-type silicon sample is 3 mm long and has a rectangular cross section 50x100 μm. The donor concentration at 300K is 5x1014cm-3 and corresponds to 1 impurity atom for 108 silicon atoms. A steady current of 1 μA exists in the bar. Determine the electron and hole concentrations, the conductivity and the voltage across the bar. (Note that this is an n-type sample that has the same dimensions and current as does the intrinsic silicon in exercise 2.)

Intrinsic concentration at 300K is 1.45x1010 cm-3. The electron mobility at 300K is 1500 cm2/Vs.



The electron concentration is:

And the hole concentration is:

As n»p, only electron concentration need to be considered. So, that the conductivity is:

The voltage across the bar is:


The efficacy of using extrinsic semiconductors in electronic devices is readily apparent when the results of exercises 2 and 3 are compared. To produce a small current of 1μA, 1380V must be applied to the intrinsic sample, whereas only 50mV is required for the n-type sample. This reduction of voltage by a factor of 28000 exactly equals the decrease in resistivity (from 2.3x105 to 8.33 Ω·cm). Yet the dramatic increase in the number of free electrons (1.45x1010 to 5x1014 cm-3) occurs when only 1 silicon atom in 100 million is replaced by an impurity atom!

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