# Semiconductor Materials

A semiconductor is a material that has a conductivity somewhere between the extremes of an insulator and a conductor.

E**nergy Band Diagram**

E_{g} = 1.1 eV(Si), E_{g} = 0.67 eV(Ge), E_{g} = 1.41 eV (Ga As)

# Types of Semiconductor

**Intrinsic Material : **A perfect semiconductor crystal with no impurities or lattice defect is called an intrinsic semiconductor. The electron-hole pair is the only way of charge carrier in intrinsic materials.

For intrinsic material.

Where, n_{i} = intrinsic carrier concentration.

Under steady state carrier concentration the rate of recombination (r_{i}) of EHP (Electron-Hole Pair) is equal to rate of generation (g_{i}) of EHP. so,

At any temperature,

where, n_{o} and p_{o} are equilibrium concentration of electrons and holes.

**
Extrinsic Material: **The characteristics of semiconductor materials can be altered significantly by the addition of certain impurity atoms into the relatively pure semiconductor material by

*doping process*. A semiconductor material that has been subjected to the doping process is called an

*extrinsic material*.

Pure semiconductor material without any impurity is known as an intrinsic semiconductor material.

n**-type Materials**

The *n*-type is created by introducing impurity elements that are (penta valent) such as antimony, arsenic and phosphorous. These diffused impurities are called *donor atoms*.

With doping E_{g} of Si becomes 0.05 eV from 1.1 eV and E_{g} of Ge becomes 0.01 eV from 0.67 eV

p**-type Materials**

The *p*-type material is formed by doping a pure germanium or Si crystal with elements that are trivalent like Boron, Gallium and Indium.

The diffused impurities are called *acceptor atoms*.

If a valence electron acquires sufficient KE to break its covalent bond and fills the void created by a hole, then a vacancy or hole will be created in the covalent bond that released the electron.

In *n*-type material, the electron is called the *majority carrier* and the hole the *minority carrier*.

In *p*-type material holes are majority carriers and electron are the minority carries.

Fermi - level

f(E_{f}) = =

Fermi-level in the energy level at which it has probability of of being occupied by electron.

# Electron and Hole Concentration at equilibrium

The concentration of electron in conduction band is :

Where, N_{C} = Effective density of states in conduction band

N_{c} = 2

Where,

m_{n}^{*} = effective mass

h = planck constant

k = Boltzman constant

The concentration of holes in valence band is

N_{v} = effective density of states in valence band.

N_{v} = 2

where

m_{p}^{*} = effective mass of hole.

Intrinsic electron and hole concentration are : —

,

Intrinsic concentration is also given by

Another convenient way of writing electron concentration and hole concentration at equilibrium is :

n_{o} =

p_{o} =

# Temperature Dependence of Carrier Concentration

n_{i} (T) = 2

From above equation it is clear that the intrinsic concentration depends on temperature. As temperature increases the intrinsic concentration increases as .

Mobility

Mobility of charge carrier is drift velocity per unit electric field. It defines how fast the charge travel from one place to another and is given by.

μ =

where,

V_{d} = drift velocity

E = Electric field

The electron and hole mobility for Ge and Si is given as under :

# Hall Effect

Hall effect states that if a specimen (metal or semiconductor) carrying current I each placed in a transverse magnetic field B, an electric field intensity E is induced in a direction perpendicular to both I and B.

Hall Voltage V_{H} = Ed volts

ρ → Charge Density

- By Hall Experiment

OR

σ = Conductivity Of Given Specimen.

**
Optical Absorption: **The photon with energy hv > E

_{g}is absorbed in the semi- conductor. A photon with energy less than E

_{g}is unable to excite an electron from the valence band to the conduction band. If a beam hv > E

_{g}fall on a semiconductor, there will be some predictable amount of absorption. The ratio of transmitted to incident light intensity depend on photon wavelength and thickness of sample.

= I(*x*)

⇒ I(*x*) = I_{0 }e^{–}^{}^{x}

where,

I(*x*) = intensity of photon remaining

I_{0} = Intensity of photon beam at *x* = 0

= absorption coefficient (cm^{–1})

# Diffusion of Carriers

When excess carriers are created non- uniformly in semi-conductor, the electron and hole concentration vary with position in the sample. Any such spacial variation in n and p calls for a net motion of carriers from a region of high carrier concentration to region of low carrier concentration. This type of motion is called diffusion.

The diffusion current crossing a unit area (the current density) is the particle flux density multiplied by charge of carrier.

J_{n} (d_{eff}) = q Dn

J_{p} (d_{eff}) = –qDp

J_{n}, J_{p} = electron and hole current density respectively.

D_{n}, D_{p} = electron and hole diffusion coefficient.

# Diffusion and Drift of Carrier

and J(x) = J_{n}(*x*) + J_{p}(*x*)

Einstein Relation

** **

where,

D = Diffusion coefficient

μ = Mobility

k = Boltzman constant

Diffusion and Recombination

, = Rate of electron and hole build up.

, = Carrier lifetime for electron and hole respectively.

L_{n} & L_{p} are electron and hole diffusion length respectively.