LECTURE-3

Electrons and Holes in semiconductors: Silicon crystal structure, Donors and acceptors in the band model

Electrons and holes in semiconductors

As pointed out before, semiconductors distinguish themselves 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 we will introduce the concept of holes in a semiconductor. It is important for the

reader to understand that one could deal with only electrons (since these are the only real particles available in a semiconductor) if one is willing to keep track of

all the electrons in the "almost-full" valence band.

The concepts of holes is introduced based on the notion that it is a whole lot easier to keep track of the missing particles in an "almost-full" band, rather than keeping

track of the actual electrons in that band. We will now first explain the concept of a hole and then point out how the hole concept simplifies the analysis.

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

positive charge. This definition is illustrated further with the figure below which presents the simplified energy band diagram in the presence of an electric field.

Characteristics

Physical

Silicon crystallizes in a diamond cubic crystal structure

Silicon Crystal Structure

The above illustration shows the arrangement

after Kittel of the silicon atoms in a unit cell, with the numbers indicating the height of the atom

above the base of the cube as a fraction of the cell dimension.

Silicon crystallizes in the same pattern as diamond, in a structure which Ashcroft and Mermin call "two interpenetrating face-centered cubic" primitive lattices. The

lines between silicon atoms in the lattice illustration indicate nearest-neighbor bonds. The cube side for silicon is 0.543 nm. Germanium has the same diamond

structure with a cell dimension of .566 nm.

Silicon is a solid at room temperature, with relatively high melting and boiling points of 1414 and 3265 °C, respectively. It has a greater density in a liquid state than a solid state. It does not contract when it freezes like most substances, but

expands, similar to how ice is less dense than water. With a relatively high thermal conductivity of 149 W·m−1·K−1, silicon conducts heat well.

In its crystalline form, pure silicon has a gray color and a metallic luster. Like germanium, silicon is rather strong, very brittle, and prone to chipping. Silicon, like carbon and germanium, crystallizes in adiamond cubic crystal

structure, with a lattice spacing of 0.5430710 nm (5.430710 Å).[10] The outer electron orbital of silicon, like that of carbon, has four valence electrons. The 1s, 2s, 2p and 3s subshells are completely filled while the 3p subshell contains

two electrons out of a possible six. Silicon is a semiconductor. It has a negative temperature coefficient of resistance, since the number of free charge carriers increases with temperature. The electrical resistance of single crystal silicon significantly changes under the application of

mechanical stress due to the piezoresistive effect.

Intrinsic semiconductor

An intrinsic semiconductor, also called an undoped semiconductor or i-type semiconductor, is a pure semiconductor without any significant dopant species present. The number of charge carriers is therefore determined by the properties of the material itself instead of the amount of impurities. In intrinsic semiconductors the number of excited electrons and the number of holes are equal: n = p.

The electrical conductivity of intrinsic semiconductors can be due to crystallographic defects or electron excitation. In an intrinsic semiconductor the

number of electrons in the conduction band is equal to the number of holes in the valence band. An example is Hg

0.8Cd 0.2Te at room temperature.

An indirect band gap intrinsic semiconductor is one in which the maximum energy of the valence band occurs at a different k (k-space wave vector) than the minimum energy of the conduction band. Examples include silicon and germanium. A direct band gap intrinsic semiconductor is one where the maximum energy of the valence band occurs at the same k as the minimum energy of the conduction band. Examples include gallium arsenide.

A silicon crystal is different from an insulator because at any temperature above absolute zero, there is a finite probability that an electron in the lattice will be knocked loose from its position, leaving behind an electron deficiency called a "hole". If a voltage is applied, then both the electron and the hole can contribute to a small current flow.

The conductivity of a semiconductor can be modeled in terms of the band theory of solids. The band model of a semiconductor suggests that at ordinary temperatures there is a finite possibility that electrons can reach the conduction band and contribute to electrical conduction.

The term intrinsic here distinguishes between the properties of pure "intrinsic" silicon and the dramatically different properties of doped n-type or p-type semiconductors

To understand how diodes, transistors, and other semiconductor devices can do what they do, it is first necessary to understand the basic structure of all semiconductor devices. Early semiconductors were fabricated from the element Germanium, but Silicon is preferred in most modern applications.

The crystal structure of pure silicon is of course 3-dimensional, but that is difficult to display or to see, so the image to the left is often used to represent the crystal

structure of silicon. For you physics types, silicon (and germanium) falls in column IVa of the Periodic Table. This is the carbon family of elements. The essential

characteristic of these elements is that each atom has four electrons to share with adjacent atoms in forming bonds.

While this is an oversimplified description, the nature of a bond between two silicon atoms is such that each atom provides one electron to share with the other. The two electrons thus shared are in fact shared equally between the two atoms.

This type of sharing is known as a covalent bond. Such a bond is very stable, and holds the two atoms together very tightly, so that it requires a lot of energy to break

this bond.

For those who are interested, the actual bonds in a 3-dimensional silicon crystal are arranged at equal angles from each other. If you visualize a tetrahedron (a pyramid

with three points on the ground and a fourth point sticking straight up) with the

atom centered inside, the four bonds will be directed towards the points of the tetrahedron.

Now we have our silicon crystal, but we still don't have a semiconductor. In the crystal we saw above, all of the outer electrons of all silicon atoms are used to

make covalent bonds with other atoms. There are no electrons available to move from place to place as an electrical current. Thus, a pure silicon crystal is quite a

good insulator. In fact, it is almost glass, which is silicon dioxide. A crystal of pure silicon is said to be an intrinsic crystal.

Electrons and holes

In an intrinsic semiconductor such as silicon at temperatures above absolute zero, there will be some electrons which are excited across the band gap into the conduction band and which can support current flow. When the electron in pure silicon crosses the gap, it leaves behind an electron vacancy or "hole" in the regular silicon lattice. Under the influence of an external voltage, both the electron and the hole can move across the material. In an n-type semiconductor, the dopant contributes extra electrons, dramatically increasing the conductivity. In a p-type semiconductor, the dopant produces extra vacancies or holes, which likewise increase the conductivity. It is however the behavior of the p-n junction which is the key to the enormous variety of solid-state electronic devices.

Donors and acceptors in the band model.

An extrinsic semiconductor is a semiconductor that has been doped, that is, into which a doping agent has been introduced, giving it different electrical properties than the intrinsic (pure) semiconductor.

Doping involves adding dopant atoms to an intrinsic semiconductor, which changes the electron and hole carrier concentrations of the semiconductor at thermal equilibrium. Dominant carrier concentrations in an extrinsic semiconductor classify it as either an n-type or p-type semiconductor. The electrical properties of extrinsic semiconductors make them essential components of many electronic devices.

Semiconductor doping

Semiconductor doping is the process that changes an intrinsic semiconductor to an extrinsic semiconductor. During doping, impurity atoms are introduced to an

intrinsic semiconductor. Impurity atoms are atoms of a different element than the atoms of the intrinsic semiconductor. Impurity atoms act as either donors or acceptors to the intrinsic semiconductor, changing the electron and hole concentrations of the semiconductor. Impurity atoms are classified as donor or acceptor atoms based on the effect they have on the intrinsic semiconductor.

Donor impurity atoms have more valence electrons than the atoms they replace in the intrinsic semiconductor lattice. Donor impurities "donate" their extra valence electrons to a semiconductor's conduction band, providing excess electrons to the intrinsic semiconductor. Excess electrons increase the electron carrier concentration (n0) of the semiconductor, making it n-type.

Acceptor impurity atoms have fewer valence electrons than the atoms they replace in the intrinsic semiconductor lattice. They "accept" electrons from the semiconductor's valence band. This provides excess holes to the intrinsic semiconductor. Excess holes increase the hole carrier concentration (p0) of the semiconductor, creating a p-type semiconductor.

Semiconductors and dopant atoms are defined by the column of the periodic table in which they fall. The column definition of the semiconductor determines how many valence electrons its atoms have and whether dopant atoms act as the semiconductor's donors or acceptors.

Group IV semiconductors use group V atoms as donors and group III atoms as acceptors.

Group III-V semiconductors, the compound semiconductors, use group VI atoms as donors and group II atoms as acceptors. Group III-V semiconductors can also use group IV atoms as either donors or acceptors. When a group IV atom replaces the group III element in the semiconductor lattice, the group IV atom acts as a donor. Conversely, when a group IV atom replaces the group V element, the group IV atom acts as an acceptor. Group IV atoms can act as both donors and acceptors; therefore, they are known as amphoteric impurities.

Intrinsic semiconductor Donor atoms Acceptor atoms

Group IV semiconductors

Silicon, Germanium Phosphorus, Arsenic

Boron, Aluminium

Group III-V semiconductors

Aluminum phosphide, Aluminum arsenide,

Gallium arsenide, Gallium nitride

Selenium, Tellurium,

Silicon, Germanium

Beryllium, Zinc, Cadmium,

Silicon, Germanium

The two types of extrinsic semiconductor

N-type semiconductors Band structure of an n-type semiconductor. Dark circles in the conduction band are electrons and light circles in the valence band are holes. The image shows that the

electrons are the majority charge carrier.

N-type semiconductors have a larger electron concentration than hole concentration. The phrase 'n-type' comes from the negative charge of the electron.

In n-type semiconductors, electrons are the majority carriers and holes are the minority carriers. N-type semiconductors are created by doping an intrinsic

semiconductor with donor impurities (or doping a p-type semiconductor as done in the making of CMOS chips). A common dopant for n-type silicon is phosphorus.

In an n-type semiconductor, the Fermi level is greater than that of the intrinsic semiconductor and lies closer to the conduction band than the valence band.

P-type semiconductors

Band structure of a p-type semiconductor. Dark circles in the conduction band are electrons and light circles in the valence band are holes. The image shows that the holes are the majority charge carrier

As opposed to n-type semiconductors, p-type semiconductors have a larger hole concentration than electron concentration. The phrase 'p-type' refers to the positive charge of the hole. In p-type semiconductors, holes are the majority carriers and electrons are the minority carriers. P-type semiconductors are created by doping an intrinsic semiconductor with acceptor impurities (or doping a n-type semiconductor). A common p-type dopant for silicon is boron. For p-type semiconductors the Fermi level is below the intrinsic Fermi level and lies closer to the valence band than the conduction band.

Use of extrinsic semiconductors

Extrinsic semiconductors are components of many common electrical devices. A semiconductor diode (devices that allow current in only one direction) consists of p-type and n-type semiconductors placed in junction with one another. Currently, most semiconductor diodes use doped silicon or germanium.

Transistors (devices that enable current switching) also make use of extrinsic semiconductors. Bipolar junction transistors (BJT) are one type of transistor. The most common BJTs are NPN and PNP type. NPN transistors have two layers of n-

type semiconductors sandwiching a p-type semiconductor. PNP transistors have two layers of p-type semiconductors sandwiching an n-type semiconductor.

Field-effect transistors (FET) are another type of transistor implementing extrinsic semiconductors. As opposed to BJTs, they are unipolar and considered either N-channel or P-channel. FETs are broken into two families, junction gate FET (JFET) and insulated gate FET (IGFET).

Other devices implementing the extrinsic semiconductor:

Lasers Solar cells Photodetectors Light-emitting diodes Thyristors

To allow our silicon crystal to conduct electricity, we must find a way to allow some electrons to move from place to place within the crystal, in spite of the covalent bonds between atoms. One way to accomplish this is to introduce an

impurity such as Arsenic or Phosphorus into the crystal structure, as shown to the right. These elements are from column Va of the Periodic Table, and have five outer electrons to share with other atoms. In this application, four of these five

electrons bond with adjacent silicon atoms as before, but the fifth electron cannot form a bond. This electron can easily be moved with only a small applied electrical voltage. Because the resulting crystal has an excess of current-carrying electrons,

each with a negative charge, it is known as "N-type" silicon.

This construction does not conduct electricity as easily and readily as, say, copper or silver. It does exhibit some resistance to the flow of electricity. It cannot

properly be called a conductor, but at the same time it is no longer an insulator. Therefore, it is known as a semiconductor.

While this effect is interesting, it still isn't particularly useful by itself. A plain carbon resistor is easier and cheaper to manufacture than a silicon semiconductor

one. We still don't have any way to actually control an electrical current.

But wait a moment! We obtained a semiconductor material by introducing a 5-electron impurity into a matrix of 4-electron atoms. (For you physics types, we're only looking at the outer electrons that are available for bonding -- electrons in inner shells are not included in the process or in this discussion.) What happens if we go the other way, and introduce a 3-electron impurity into such a crystal? Suppose we introduce some Aluminum (from column IIIa in the Periodic Table) into the crystal, as shown to the left? We could also try Gallium, which is also in column IIIa right under aluminum. Now what?

These elements only have three electrons available to share with other atoms. Those three electrons do indeed form covalent bonds with adjacent silicon atoms, but the expected fourth bond cannot be formed. A complete connection is impossible here, leaving a "hole" in the structure of the crystal.

Experimentation shows that there is an empty place where an electron should logically go, and often an electron will try to move into that space to fill it. However, the electron filling the hole had to leave a covalent bond behind to fill this empty space, and therefore leaves another hole behind as it moves. Yet another electron may move into that hole, leaving another hole behind, and so forth. In this manner, holes appear to move as positive charges through the crystal. Therefore, this type of semiconductor material is designated "P-type" silicon.

By themselves, P-type semiconductors are no more useful than N-type semiconductors. The truly interesting effects begin when the two are combined in various ways, in a single crystal of silicon. The most basic and obvious combination is a single crystal with an N-type region at one end and a P-type

region at the other. A crystal with two regions as described is known as a semiconductor diode, and is the topic of the next page.