Electrical conductivity
Electrical conductivity
The current flowing through a conductor is directly proportional to the potential difference across the end of the conductor i.e.,
Figure
$$I\propto V$$
$$I=\frac{V}{R}$$
$$V=IR\dotsm(1)$$Here R is known as the resistance (electrical)
I=amount of current
The electrical resistance of material depends upon
 Nature of material
 Temperature of material
 Length of material(geometry)
 Area of section of material(geometry)
$$R=\frac{\rho l}{A}\dotsm(2)$$where
\(\rho\)=resistivity of material
L=length
A=area of cross section
Types of carrier:
 In most metals the carrier are electrons so the conduction is said to be electronic conduction.
 In ionic crystal, the carriers are ions and electrons.
 In semiconductors, the carriers are electrons and holes.
The electrical conductivity of material is its ability to transport electric current. It is denoted by \(\sigma\) and it is reciprocal of electrical resistivity of material.
Inside current carrying rod electric field is given by,
E=potential gradient =\(\frac{V}{L}=\frac{potential difference}{length}\)
Using equation (1),
$$\frac{E}{A}=\frac{IR}{L}\times \frac{1}{A}$$
$$\frac{E}{A}=J.\frac{R}{L}$$
$$\frac{E}{A}=\frac{J}{L}.\frac{\rho L}{A}$$
$$J=\sigma E\dotsm(3)$$
Equation (3) is another form of OHM’S law. Here, J is said to be current density. Equation (3) is vector form of OHM’s law.
The value of \(\sigma\)(electrical conductivity) has 27 order of magnitude in different materials. It’s the greatest variable of any physical property of material.
The conductivity of metal is approximately greater than \(10^5(\Omega m)^{1}\) for semiconductor the conductivity lies between, \(10^6 to 10^5 (\Omega m)^1\) for insulator the approximate value of conductivity is always less than \(10^6 (\Omega m)^1\).
Energy band structure
How energy bands are formed in solid?
a) Metals
b) Semiconductor
c) Insulator
Individual atoms (isolated atoms) have discrete energy level for electrons. When two atoms are brought near to each other each energy level split into two energy levels. When three isolated are bring towards each other then each energy level of individual atom splits into three energy levels. When N atoms are bring together to form an object then reach energy level of an isolated atom splits into N energy levels. The gap between successive energy levels is too small called these states. So we called the energy levels of N atoms as energy bands.
The bands are separated by energy gap where electron cannot exist (electrons cannot have this much amount of energy) this energy gap is known as band gap. The highest field energy state at 0K is known as Fermi energy state or Fermi state. The energy of electron in this state is called Fermi energy.
What do you mean by valence band, conduction band and band gap in solid?
The highest energy band where electrons are present at 0k is known as valence band. Thee valence band is formed due to electrons in valence shell of atoms when brought together to form a solid.
Conduction band: A partially filled or empty energy band where the electrons can increase their energy going to higher energy levels within a band when an electric field is applied is known as conduction band. Valence band and conduction band are shown in figure below:
Energy bands in metals
 In metals or conductors, highest occupied band are partially filled or bands overlap.
 Conduction occurs when electrons jumps from valence band to conduction band.
 The electrons that start to move in conduction band are form Fermi level.
 The band gap is too small or zero.
 The energy supplied by electric field or temperature gradient is sufficient to excite electron from valence band to conduction band.
 Examples:The energy bands in metals such as copper, aluminum, magnesium etc.
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: PretenceHall of India, 2003.
Tiley, R.J.D. Understanding solids: The science of Materials. Engalnd : John wiley & Sons , 2004.
1.$$I\propto V$$
$$I=\frac{V}{R}$$
$$V=IR$$
$$R=\frac{\rho l}{A}$$
$$J=\sigma E$$

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