Semiconductor Materials

Almost all solid-state electron devices are made from the class of material known as semiconductors. As the name implies these materials have electrical conductivities (at room temperature) that are midway between those of conductors and insulators. For example:

Material	Example	Conductivity
Conductor	Copper	10^7Sm^-1
Semiconductor	Silicon	10^–4 Sm^-1
Insulator	Quartz	10^–10Sm^-1

The enormous variation in conductivity has to do with the number of electrons per unit volume (carrier concentration) in the material that are available for carrying electric current.

For the understanding of the properties of semiconductor and how these materials can be used to fabricate useful solid-state devices, we need to have a qualitative took at what is referred as the band theory of solid. From this theory we will arrive at some quantitative information which would be used to determine the electrical characteristics of the semiconductor devices

We all know the atomic structure of different elements. The semiconductors we will consider are silicon and germanium (there are other senemiconductors materials made up of two elements - these are known as compound Semiconductors: Gallium Arsenide, GaAs, is the best example). These elements have four valence electrons. The electron configuration of silicon and germanium atoms in the ground state are given below:

Element	Atomic No.	K	L	M	N
		1s	2s 2p	3s 3p 3d	4s 4p 4d 4f
Si	14	2	2 6	2 2
Ge	32	2	2 6	2 6 10	2 2

In isolation electrons at the different orbitals have definite energy and are firmly attached to the parent nucleus. The energy of these electrons can be calculated using appropriate quantum mechanical theories. The most well known example is perhaps the calculation of the energy of electrons associated with the hydrogen atom. The energy levels are quantized and determined by a set of quantum numbers.

In pure silicon (or germanium) single crystal each of the four valence electrons enter into a valence bond with a neighbouring atom. These electrons are shared between two neighbouring atoms. The energy levels of these electrons are no longer at the original discrete energy levels. They broaden out to two energy bands which extend throughout the physical dimensions of the material. The lower energy hand is completely filled with electrons (at 0 K). it is known as the valence band. The higher energy band which is separated from the valence band is empty and is known as the conduction band. This modification of the valence electron energy levels is depicted in the following diagram in which there is only one physical dimension (the x axis). These bands in which electrons can exist are the allowed bands.

Separating the two allowed bands is the forbidden band - a range of energy levels in which no electrons can be present. This is also known as the band gap. The energy range of this gap is known as the band gap energy E_G. This is an important quantity for any semiconductor material.

It is the distribution of electrons between the valence band and the conduction band which determine the electrical properties of a piece of semiconductor. In general we are only interested in the electrons that ae present at or near the top of the valence band and at or near the bottom of the conduction band. The energy (measured with respect to some reference) of electrons at the top of the valence band is referred to as E_V and the energy of electrons at the bottom of the conduction band is referred to as E_G. Thus,

E_G = E_C - E_V.

At room temperature (300 K):

Silicon: E_G = 1.12 eV = 1.12x1.602xI0¹⁹ joules

Germanium: E_G = 0.66 eV

Gallium Arsenide: E_G = 1.42 eV

To complete our brief description of the hand theory model, it is necessary to introduce the idea of effective density of state for the two bands:

N_V, Effective density of state for the valence band, this is the maximum number of electrons per unit volume that can be found in the whole valence hand.

N_C, Effective density of state for the conduction hand, this is the maximum number of electrons per unit volume that can be found in the whole conduction band.

For a given semiconductor. the actual number of electrons per unit volume in each band will depend on a number of different factors, for example the temperature of the material, the amount of group three atoms (atoms with 3 valence electrons) or group five atoms (atoms with five valence electrons) of the periodic table that have been introduced into the semiconductor It turns out that for the understanding of the electrical behaviour of a piece of semiconductor (such as its conductivity or resistivity) we are interested in:

the number of electrons per unit volume in the conduction band (at or near the bottom of the band, that is at E_C), this is denoted by n, and is known as the electron concentration.

the number of empty electron levels per unit volume in the valence band (at or near the top of the band, that is at E_V), this is denoted by p, and is known as the hole concentration.

p and n are known as the carrier concentrations, as they are the charged carriers which are free to move under the influence of an electric field to produce electric current.

It is interesting to note that a transistor, or for that matter any complicated integrated circuit, is made up of a single crystal semiconductor material in which different regions are selectively made to have different electron and hole concentrations.

Another diagram relating atomic energy levels and band energy levels:

Intrinsic Semiconductor and Intrinsic Carrier Concentration, n_i

At 0 K, there is no thermal energy, all electrons are at their lowest allowed energy states, the valence band is completely filled and the conduction band is totally empty. In this situation, we have:

p=n=0

At room temperature some of the valence hand electrons will have sufficient thermal energy to be "excited" into the conduction band. For every electron going into the conduction band, an empty energy level is left behind in the valence band. For a pure semiconductor more commonly known as INTRINSIC SEMICONDUCTOR, (One that contains no impurity atoms which are likely to influence the electron and hole concentrations), the concentration of (conduction band) electrons and concentration of empty energy levels that is concentration of holes must be the same, that is:

In intrinsic semiconductor:

p = n = n_i

where n_i is known as the intrinsic carrier concentration.

At a given temperature, the intrinsic carrier concentration is a function of the band gap energy E_G. Large band gap leads to smaller intrinsic carrier concentration and vice versa.

At 300 K,

Ge: E_G = 0.66eV: n_i = 2.33x10¹³ cm^-3

Si: E_G = 1.12eV: n_i = 1.45x10¹⁰ cm^-3

GaAs: E_G = 1.45eV: n_i = 9.0 x10⁶ cm^-3

The intrinsic carrier concentration is also a strong function of temperature. The intrinsic carrier concentration as a function of temperature is of the form:

T is temperature in kelvin,

E_G is the bandgap energy, and

k is boltzmann constant

= 1.38x10^-23 joulesK^-1 = 8.62x10^-5 eVK^-1

Bond Theory Model

We can also look at the formation of holes and electrons in semiconductor qualitatively by looking at the covalent bonds formed between neighbouring atoms. As we have indicated previously each Si atom (or Ge atom) forms four co-valent bonds with its four nearest neighbouring atoms. This is depicted in a two-dimensional crystal below. As a single crystal material, the atoms are arranged in a regular pattern known as the crystal lattice.

At any temperature above absolute zero, due to the thermal energy imparted onto the crystal lattice, some of the valence bonds are broken releasing one of the two valence electrons in a covalent bond. This electron is then free to move in the crystal lattice under the influence of an electric field. That is: this electron contributes to electrical conduction.

The free electron leaves behind a positively charged nucleus which is fixed in position in the crystal lattice and is capable of attracting another electron to re-form the covalent bond. Such a vacancy is known as a hole in that a valence electron from a neighbouring co-valent bond may be excited to break loose from its parent nuclei and be attracted to this vacancy. Note that in this process, the net effect is the movement of the hole from its original position to a new position. This is equivalent to the movement of the positive charge in a direction opposite to the that of the transfer of the valence electron.

In subsequent calculation of the conductivity or resistivity of the semiconductor we can treat the complex transfer of the broken co-valent bond as the movement of a positively charged patticle, that is the hole. In fact we will treat the hole as a positively charged mobile carrier with magnitude of charge equal to the electronic charge q.

It is clear that whenever a co-valent bond is broken we have simultaneous formation of a free electron and a hole. This is referred to as electron-hole pair generation.

The idea of breaking a co-valent bond is equivalent to the excitation of a valence band electron into the empty conduction band.

In order to understand the behaviour of semiconductor devices, we need to be able to calculate the electron and hole concentrations. For this purpose we have to use some rather complex solid-state physics concepts. We will not be able to derive them in this subject. We will simply accept them as assumptions. However it is necessary to appreciate what these concepts mean and be able to apply them correctly.

Fermi level or Fermi Energy. E_F. and Equilibrium Carrier Concentrations

We have introduced the idea of Effective Density of States in the conduction band and the valence band, N_C and N_V. In order to calculate the actual electron and hole concentrations we need to use the idea of Fermi-Dirac function f(E). It is function that specifies the probability that a given energy level E in the band diagram will be occupied by an electron under equilibrium condition. f(E) is given by:

where

k = boltzmann constant

T = temperature in K

E_F = Fermi energy or Fermi Level, a energy describing the equilibrium properties of the piece of semiconductor. For our purpose it is sufficient to consider this is the energy level at which F(E) = 0.5.

By applying the Etrmi-Dirac function to the energy levels in the conduction band and valence band, it can be shown that for any Piece of semiconductor in equilibrium. the electron concentration is given by:

and the equilibrium hole concentration is.

These two equations are approximations but should he sufficiently accurate for most of the conditions we are likely to encounter in this course.

It is important to state here again that these equations are applicable ONLY when the semiconductor is in equilibrium. By equilibrium we mean essentially that the piece of semiconductor is left isolated with no input of energy from external sources such as light incident on it or voltage applied across the material.

The actual position of the Fermi level will depend on the basic properties of the semiconductor such as E_G, N_C and N_V. We will see subsequently that by introducing what are known as dopant atoms into the crystal lattice, we can drastically change the electron and hole concentrations in the material. This will also change the position of the Fermi level. These two equations are applicable to these doped semiconductors, again only if they are in equilibrium.

Consider the case of the intrinsic semiconductor. As we have defined earlier, for this material:

p = n =n_i

and

or

The Fermi level for intrinsic semiconductor is referred to as the intrinsic Fermi level, E_i. By using equations for p and n for in terms of Fermi level, we get:

From these two equations we get:

For silicon at room temperature (300K),

N_C = 2.8x10¹⁹ cm^-3 and N_V = 1.04x10¹⁹ cm^-3_,

and kT =0.0259 eV.

This gives:

That is we can consider the intrinsic Fermi level to be at the mid-gap position.

In drawing band diagram it is common to include the intrinsic Fermi level. We will subsequently see that when we introduce dopant atoms into the semiconductor, we shift the Fermi level either towards E_C or E_V. We usually look at this Fermi level E_F with respect to the intrinsic level E_i..

Note that E_C and E_V represent the energy of electrons in the (bottom of) conduction band and (top of) valence band. If a piece of semiconductor is left alone in equilibrium, there is no difference in energy as the electrons move from one point to another in the semiconductor. The bands will be 'flat'.

However if we apply a potential difference between two ends of a semiconductor, energy of electrons changes with position. The bands are no longer 'flat':