Note on Energy bands in semiconductors

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Energy bands in semiconductors

  1. In semiconductors valence band is filled at 0k. No more electrons can be added to valence at 0k due to Paulie’s exclusion principle.
  2. When electric field is applied or temperature gradient is applied electrons forms valence band jump to empty conduction band and holes are created in valence band and conduction electrons are present in conduction band.
  3. In semiconductors the band gap is a little bit longer than that of conductor and less than that of insulator. The approximate value of band gap for narrow band gap semiconductor is less than 2ev and greater than 0ev.
  4. For example, in case of silicon band gap is 1.1ev at room temperature but that of Germenium is 0.67ev.

fig: energy bands
fig: energy bands(en.wikibooks.org)

  1. The probability that an electron reaches the conduction band is proportional to \(e^\frac{-E_g}{K_B T}\) with increase in temperature the probability to find electron in conduction band increase.
  2. Both electrons in conduction band and holes in valence band participate in the electrical conduction. In metals only electrons participate in conduction.
  3. There are different ways for excitation of electron to conduction band
  4. Temperature ii. Electric field iii. Absorption of radiation
  5. In semiconductor, the band between two atoms is covalent bond whereas for metals the bond between two atoms is metallic bond.

Energy band in insulator

  1. In insulator, the valence band is completely filled whereas conduction band is completely empty at room temperature at 0k.
  2. Large amount of energy is required to jump electron from filled valence band to empty conduction band.
  3. The band gap is greater than 2ev.
  4. At room temperature, electrons couldn’t reach from valence band to conduction band. So insulators are bad conductor of heat and electricity.
fig: energy bands
fig: energy bands

5. Holes also exist in metal and at semiconductor but absent in insulator.

6. In insulator valence electrons are tightly bound or shared with other atoms due to ionic and covalent bonding.

Mobility

What do you mean by mobility? Derive the relationship between electrical conductivity and mobility?

The motion of electrons in metal is random thermal motion. So net current in the absence of electric field is always zero. When external field is applied electron experience electric field in the opposite direction of applied electric field.i.e.

$$F=eE\dotsm(1)$$Due to this force a constant acceleration is produced,

$$a=\frac{F}{m}=\frac{-eE}{m}\dotsm(2)$$

During acceleration of electron, electron experiences collision with ions, electrons, photons and its speed cannot increase linearly. These obstacles for motion of electron behaves like frictional force known as electrical resistance.

The average velocity of electron between successive collisions remains constant known as drift velocity. The magnitude of drift velocity per unit applied electric field is known as mobility. It is denoted by \(\mu_e\)i.e.

$$\mu_e=\frac{|V_d|}{E}\dotsm(3)$$

The current density J in terms of carrier concentration ‘n’ drift velocity is given by,

$$J=neV_d$$

$$\therefore |V_d|=\frac{J}{ne}\dotsm(4)$$

From (3) and (4),

$$\mu_e=\frac{J}{neE}$$

According to OHM’s law

$$J=\sigma E$$

Where \(\sigma\)=electrical conductivity

$$\therefore \mu_e=\frac{\sigma}{ne}$$

$$\sigma=ne\mu_e\dotsm(7)$$

Equation (7)gives the relation between electrical conductivity (\(\sigma\)), number density for electron (n) and mobility (\(\mu_e\)).

Mobility and carrier density for semiconductor and metals:

\(n_{metal}>>n_{semiconductor}\)

\(\mu_{metal}<\mu_{semiconductor}\)

\(\Rightarrow \sigma_{metal}>\sigma_{semiconductor}\)

The conductivity of metal is greater than the conductivity of semiconductor due to the presence of high concentration of carrier density.

What is the source of resistivity in a conductivity material ? State the Matthiessen rule and explain the different contributions.

The resistivity of material is the property of material which resist or oppose the motion of charge carrier. The charge carrier experience resistance for the motion due to different reasons. According to Matthieness rule, the total resistivity of metal as conducting material is,

\(\rho_{total}=\rho_{thermal}+\rho_{impurity}+\rho_{deformation}\)

Where

\(\rho_{thermal}\)=resistivity due to the lattice vibration. Elastic waves in lattice is quantize lattice waves and known as phonon scattered electron and oppose the motion of electron in metal.

References:

Callister, W.D and D.G Rethwisch. Material Science and Engineering. 2nd. New Delhi: Wiley India, 2014.

Lindsay, S.M. Introduction of Nanoscience . New York : Oxford University Press, 2010.

Patton, W.J. Materials in industry . New Delhi : Prentice hall of India, 1975.

Poole, C.P. and F.J. Owens. Introduction To Nanotechnology. New Delhi: Wiley India , 2006.

Raghavan, V. Material Science and Engineering. 4th . New Delhi: Pretence-Hall of India, 2003.

Tiley, R.J.D. Understanding solids: The science of Materials. Engalnd : John wiley & Sons , 2004.

1.$$F=eE$$

$$a=\frac{F}{m}=\frac{-eE}{m}$$

$$\mu_e=\frac{|V_d|}{E}$$

$$J=neV_d$$

$$\sigma=ne\mu_e$$

\(\rho_{total}=\rho_{thermal}+\rho_{impurity}+\rho_{deformation}\)

\(n_{metal}>>n_{semiconductor}\)

\(\mu_{metal}<\mu_{semiconductor}\)

\(\Rightarrow \sigma_{metal}>\sigma_{semiconductor}\)

.

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