Theory of Semiconductor

The materials can be classified on the basis of energy gap between their valence band and conduction band. The valence band is the band consisting of free valence electron and the conduction band is empty band. Conduction takes place when an electron jumps from valence band to conduction band and the gap between these two bands is energy gap. Wider the gap between the bands, higher the energy it requires to shift the electron to conduction band.

In case of conductors, this energy gap is absent or in other words conduction band and valence band overlap each other. Thus electron requires minimum energy to jump from valence band, e.g. Silver, Copper and Aluminium. In insulators, this gap is very large. Therefore, it requires large amount of energy to shift an electron from valence to conduction band. Thus insulators are poor conductors of electricity, e.g. mica, diamond.

Semiconductors have energy gap in between conductors and insulators (~1 eV) and thus require energy more than conductors but less than insulators. They don’t conduct electricity at low temperature but as temperature increases conductivity increases e.g. silicon and germanium. This is the most basic theory of semiconductor.

The materials that are neither conductor nor insulator with energy gap of about 1 eV (electron volt) are called semiconductors. Most common type of materials that are used as semiconductors are germanium (Ge) and silicon (Si) because of their property to withstand high temperature. For Si and Ge energy gap is given as,

1. E_g = 1.21 - 3.6 X 10^-4T eV (for Si)
2. E_g = 0.785 - 2.23 X 10^-4T eV (for Ge)
Where, T = absolute temperature in ^oK
Assuming room temperature to be 300 ^oK, E_g = 0.72eV for Ge and 1.1eV for Si.
At room temperature resistivity of semiconductor is in between insulators and conductors. Semiconductors show negative temperature coefficient of resistivity i.e. its resistance decreases with increase in temperature.
Both Si and Ge are elements of IV group i.e. both elements have 4 valence electrons. Both form covalent bond with neighbouring atom. At absolute zero temperature both behave as insulator i.e. the valence band is full while conduction band is empty but as temperature is raised more and more covalent bonds break and electrons are set free and jump to conduction band.
Energy band diagram of a semiconductor. CB is the conduction band and VB is the valence band. At 0° K, the VB is full with all the valence electrons.



Intrinsic Semiconductors

As per theory of semiconductor, semiconductor in its pure form is called as intrinsic semiconductor. In pure semiconductor number of electrons (n) is equal to number of holes (p) and thus conductivity is very low as valence electrons are covalent bonded. In this case we write n = p = n_i, where n_i is called the intrinsic concentration. It can be shown that n_i can be written
n_i = n₀T^3/2 exp(-VG /2V_T)
Where, n₀ is a constant, T is the absolute temperature, V_G is the semiconductor band gap voltage, and V_T is the thermal voltage.
The thermal voltage is related to the temperature by V_T = kT/q
Where, k is the Boltzmann constant (k = 1.381 × 10 ^{− 23} J/K).
In intrinsic semiconductors conductivity (σ) is determined by both electrons (σ_e) and holes (σ_h) and depends on the carrier density.
 σ_e = neμ_e σ_h = pe_h Conductivity,
σ = σ_e + σ_h = neμ_e + peμ_h = Ne (μ_e + μ_h)
Where n, p = numbers of electrons and holes respectively. μ_h, μ_e = mobility of free holes and electrons respectively N = n = p e = charge on carrier




Extrinsic Semiconductors

As per theory of semiconductor, impure semiconductors are called extrinsic semiconductors. Extrinsic semiconductor is formed by adding a small amount of impurity. Depending on the type of impurity added we have two types of semiconductors: N - type and P-type semiconductors. In 100 million parts of semiconductor one part of impurity is added.



N-type Semiconductor

In this type of semiconductor majority carriers are electrons and minority carriers are holes. N - type semiconductor is formed by adding pentavalent ( five valence electrons) impurity in pure semiconductor crystal, e.g. P. As, Sb.

Four of the five valence electron of pentavalent impurity forms covalent bond with Si atom and the remaining electron is free to move anywhere within the crystal. Pentavalent impurity donates electron to Si that’s why N- type impurity atoms are known as donor atoms. This enhances the conductivity of pure Si. Majority carriers are electrons therefore conductivitry is due to these electrons only and is given by, σ = neμ_e


P-type Semiconductors

In this type of semiconductor majority carriers are holes and minority carriers are electrons. P- type semiconductor is formed by adding trivalent ( three valence electrons) impurity in pure semiconductor crystal, e.g. B, Al Ba.

Three of the four valence electron of tetravalent impurity forms covalent bond with Si atom. This leaves an empty space which is referred to as hole. When temperature is raised electron from another covalent bond jumps to fill this empty space. This leaves a hole behind. In this way conduction takes place. P- type impurity accepts electron and is called acceptor atom. Majority carriers are holes and therefore conductivity is due to these holes only and is given by, σ= neμ_h