Semiconductor Basics

The Energy Band Theory of Crystals

According to the Bohr atomic model, in an isolated atom the energy of any of its electrons is decided by the orbit in which it revolves. Only discrete values of electron energies are possible. An electron cannot have any value of energy but only certain permissible values. No electron can exist at an energy level other than a permissible one.

Energy Band Diagram for an Isolated Hydrogen atom.

When atoms bond together to from a crystal, the simple diagram of Electron Energy is no longer applicable. In a crystal, the orbit of an electron is influenced not only by the charges in its own atom but by nuclei and electrons of every atom in the solid. Since each electron occupies a different position inside the solid, no two electrons can see exactly the same pattern of surrounding charges. As a result, the orbits of the electrons are different.

There are millions of electrons, belonging to the first orbit atoms in the crystal. Each of the them has different energy. Since there are millions of first orbit electrons, their energy levels differs only slightly and forms a band. So first energy band is formed by energy of electrons when they are in 1^st orbit,  second band if formed by energy of electrons in 2^nd orbit. In this way third band, fourth band, etc are formed.

Valence band : The Highest energy Band filled with valence electrons is called Valence band.      
Conduction band : The lowest unfilled allowed energy band next to valence band is called conduction band.

In semiconductors, only valence band and conduction band are important as electron of valence band are loosely bonded as they already have high energy and can transit to conduction band. The Gap between the top of valence band and bottom of conduction band is called forbidden gap or energy band gap or energy gap. No allowed energy level for electrons can exist. 
(For Silicon Eg=1.12 eV, and for Germanium Eg=0.72eV).

Energy Band Diagram for a Silicon Crystal.  Electron Configuration of Silicon atom =2,8,4.

Metals, Insulators and Semiconductors

A material is able to conduct electricity, if it contains free electrons in the conduction band when external electric field if applied to it. The free electrons thus work as charge carriers.

Metals

Metals have a large number of free electrons at room temperature. There is no gap between the valance and conduction band. The conduction band and valence band overlap eachother so valence band energies and conduction band energies are same. It is very easy for valence electron to transit to conduction band and become free electron even without supply of external energy. Metals have very low resistivity and high conductivity.

ρ (Resistivity)     ≈ 10^-2 – 10^-8 Ω m

σ (Conductivity) ≈ 10² – 10⁹ Sm^-1

Insulators

Insulators have completely filled valence band and conduction band is empty. There is a large energy gap (Eg > 5 eV) between valence and conduction bands due to which it is practically impossible for an electron in the valence band to jump to conduction band. Only at very high temperature or very high voltage, an electron can jump the gap which is known as the breakdown of insulator. At room temperature, no electron is present in conduction band. Insulators have very high resistivity and low conductivity.

ρ (Resistivity)     ≈ 10¹¹ – 10¹⁹ Ω m

σ (Conductivity) ≈ 10^-11 – 10^-19 Sm^-1

Semiconductors

Semiconductors have properties intermediate to metals and insulators. There is a finite small energy gap (Eg< 3 eV) between valence and conduction band. Because of the small gap, at room temperature some electrons from valence band can acquire enough energy to cross the energy gap and enter the conduction band. They have resistivity and conductivity intermediate to metals and insulators.

ρ (Resistivity)     ≈ 10^-5 – 10⁶ Ω m

σ (Conductivity) ≈ 10⁵ – 10^-6 Sm^-1

Mobility and Conductivity

The following symbols will be used in derivation.

Mobility

The mobility of a charge carrier is the drift velocity acquired by it in a unit electric field. SI unit (m²/Vs).

Conductivity

The conductivity of any material is due to its mobile charge carriers. It is the reciprocal of resistivity. SI unit (Sm^-1).

Electrons and Holes in Intrinsic Semiconductor

Intrinsic Semiconductor

The pure semiconductors (impurity < 1 part in 10¹⁰) are called intrinsic semiconductor. The charge carriers (electrons and holes) are formed due to thermal excitation. Holes are vacancies of an electron in the bond of a covalently bonded crystal. It acts as a positive charge carrier.

As each free electron creates one hole, so in an intrinsic semiconductor, the number of free electrons (n_e) is equal to number of holes (n_h) is equal to the intrinsic charge carrier.

n_e = n_h = n_i 

Donor and Acceptor Impurities

Doping

In order to increase the conductivity of pure semiconductors a small amount of impurity atoms having valency different from 4 is added to the pure semiconductors. This process is called doping.

Two types of impurity atoms are added to tetravalent atom.

i)                    Trivalent                ii)         Pentavalent

Trivalent	Tetravalent	Pentavalent
B (Boron)	Si (Silicon)	P (Phosphorus)
Al (Aluminium)	Ge (Germanium)	As (Arsenic)
Ga (Gallium)		Sb (Antimony)
In (Indium)

Donor Impurity

When pentavalent atom is added to pure semiconductor, it forms bonds with neighbouring four tetravalent atoms. The fifth electron is not associated with any covalent bond and it remains loosely bounded to parent atom. The thermal energy at room temperature is practically enough to set all such electrons free from their atoms. Due to this, negatively charged electron is free to move in the lattice of semiconductor. A positive charge is acquired by pentavalent atom due to loss of one electron and it becomes positively charged ion. The pentavalent atom becomes positive ion by donating one extra electron from crystal. So, this type of impurity is called Donor Impurity.

Acceptor Impurity

When trivalent atom is added to pure semiconductor, it forms bonds with neighbouring three tetravalent atoms. It does not have any electron to form fourth bond. There is a deficiency of an electron around trivalent atom. The single electron in the incomplete bond has a great tendency to snatch an electron from neighbouring atom. The thermal energy at room temperature is enough to fill the incomplete bonds around all trivalent atoms. Due to this, a vacancy is created in adjacent bond from where the electron is jumped and has positive charge associated with it, hence it is a hole. Also, due to filling of incomplete bond, the trivalent atom becomes negatively charged ion. The trivalent atom becomes negative ion by accepting one electron from crystal. So, this type of impurity is called Acceptor Impurity.

Electrons and Holes in Extrinsic Semiconductor

Extrinsic Semiconductor

A semiconductor doped with some suitable impurity atoms so as to increase its number of charge carriers is called Extrinsic semiconductor.

Extrinsic Semiconductors are of two types:

i)                    n-type semiconductor

 

  

This semiconductor is obtained by doping tetravalent semiconductor with pentavalent impurities. These semiconductors have free electrons obtained by donors and generated by the thermal process while holes are only due to thermal defects. Hence, electrons are majority charge carriers and holes are minority charge carriers. As most of the current is carried by negatively charged electrons, these are known as n-type semiconductors.

In n-type semiconductor n_e ≫ n_h or n ≫ p.

ii)                  p-type semiconductor

This semiconductor is obtained by doping tetravalent semiconductor with trivalent impurities. These semiconductors have holes created by acceptor atoms and generated by the thermal process while electrons are only due to thermal defects. Hence, holes are majority charge carriers and electrons are minority charge carriers. As most of the current is carried by holes which have effective positive charge, these are known as p-type semiconductors.

In p-type semiconductor n_h ≫ n_e or p ≫ n.

Charge Densities, Mobility and Conductivity

In semiconductor, the electrons and holes are the charge carriers.

Charge Density

The number of charge carriers per unit volume is called charge density.

For Intrinsic semiconductor:

Number of free electrons = Number of holes = Number of Intrinsic charge carries → n_e = n_h = n_i

For Extrinsic semiconductor:

i)                    n-type: Number of free electrons ≫ Number of holes → n_e ≫ n_h

ii)                  p-type: Number of holes ≫ Number of free electrons → n_h ≫ n_e

The semiconductor crystal as a whole remains neutral as the charge of additional charge carriers is just equal and opposite to that of ionised atoms in the lattice.

In extrinsic semiconductors, the minority charge carriers are destroyed by meeting majority charge carriers. Hence, by adding impurities, a large number of current carriers of one type are added, which becomes majority charge carriers, indirectly helps to reduce the intrinsic concentration of minority charge.

Thus at room temperature, the density of majority charge carriers is predominantly due to impurity in the extrinsic semiconductor. The electron and hole concentration in a semiconductor in thermal equilibrium is given by: 

n_i² = n_e x n_h

Mobility of Charge Carries in Semiconductor

SI unit of μ (m²/Vs).

Conductivity in Semiconductor

SI unit of σ = Sm^-1

Diffusion

Diffusion is a process by which controlled amount of impurity is introduced into semiconductor. Impurity atoms are introduced onto the surface of a crystal wafer and diffuse into the lattice because of their tendency to move from regions of high to low concentration. Diffusion of impurity atoms into crystal takes place only high temperatures. There are mainly two types of physical mechanisms by which the impurities can diffuse into the lattice.

1. Substitutional Diffusion

At high temperature many atoms in the semiconductor move out of their lattice site, leaving vacancies into which impurity atoms can move. The impurities, thus, diffuse by this type of vacancy motion and occupy lattice position in the crystal after it is cooled. This mechanism is known as Substitutional diffusion.

2. Interstitial Diffusion

In such, diffusion type, the impurity atom does not replace the crystal atom, but instead moves into the interstitial voids in the lattice.

Flick's Law of Diffusion

The diffusion rate of impurities into semiconductor lattice depends on the following :-

•  Mechanism of diffusion
• Temperature
• Physical properties of impurity
• The properties of the lattice environment
• The concentration gradient of impurities
• The geometry of the parent semiconductor

The rate of transfer of solute atoms per unit area of the diffusion flux density (atoms/cm².sec)

N is the concentration of solute atoms

x is the direction of solute flow.

t is the diffusion time, and

D is the diffusion constant (cm²/sec)