Conductivity of a Semiconductor – formula

Earlier we have discussed the basic properties, examples, types and applications of semiconductor materials. We became to know that semiconductors are not excellent conductors of electricity at room temperature. That means a pure semiconductor is not suitable for conducting current and hence it cannot be used to design electronic devices. But there is a solution. In this article, I’m going to tell you how to increase the conductivity of a semiconductor and will derive a formula for the conductivity of semiconductors.

1. On what parameters does the conductivity of semiconductors depend?
2. Relation between Conductivity and Mobility
3. How to increase the conductivity of a Semiconductor?
4. The formula of electrical conductivity of a Semiconductor

On what parameters does the conductivity of semiconductors depend?

The conductivity of a Semiconductor depends on the followings –

• The temperature of the semiconductor.
• Doping concentration of the semiconductor.
• Energy Band gap.

Relation between Conductivity and Mobility

Do you know that Mobility and conductivity are related to each other? The mobility of a carrier indicates how fast it moves under the application of a unit amount of electric field and the electrical conductivity of a substance indicates how well the substance carries electricity. Now, the faster the movement, the greater the conductivity. Yes, electrical conductivity is proportional to mobility.

Suggested Article: Mobility of electrons and holes in Semiconductor

If $\small\color{Blue}\mu_{n}$ and $\small\color{Blue}\mu_{p}$ are the mobility of free electrons and holes respectively and $\small\color{Blue}\sigma$ is the conductivity of an intrinsic semiconductor of carrier concentration ni, then

The relation between the Conductivity and Mobility in semiconductor is, $\small\color{Blue}\sigma = en_{i} (\mu_{n}+\mu_{p})$……..(1).

For p-type semiconductors, $\small\color{Blue}\mu_{p} > \mu_{n}$. Then the conductivity $\small\color{Blue}\sigma = ep \mu_{p}$. Here p is the concentration of holes.

For n-type semiconductors, $\small\color{Blue}\mu_{n} > \mu_{p}$. Then the conductivity $\small\color{Blue}\sigma = en \mu_{n}$. Here n is the concentration of free electrons.

How to increase the conductivity of a Semiconductor?

The conductivity of an intrinsic semiconductor is very low due to its less number of charge carriers. However, one can increase its electrical conductivity in two ways –

1. By increasing its temperature.
2. By doping impure atoms in it.

We cannot effort to increase its temperature by more than 50 degrees. Because we need to use these materials physically and it’s very difficult for humans to work at such a high temperature. So, we need to choose the second option.

By doping impure atoms in an intrinsic semiconductor, its conductivity can be increased significantly even at room temperature. This is the most common method of increasing the conductivity of semiconductors. After doping, the semiconductor will no longer be intrinsic. It will be called an extrinsic semiconductor in that situation. Thus the electrical conductivity of extrinsic semiconductors is greater than that of intrinsic semiconductors.

The formula of electrical conductivity of a Semiconductor

Here I’m going to write the formula for the electric conductivity of a semiconductor. I’ll not provide any detailed calculations of these equations. They can be derived by using Fermi-Dirac statistics.

We take an intrinsic semiconductor at a temperature of T Kelvin. It has free electrons and holes equally.

The concentration of free electrons is, $\small {\color{Blue} n = 2(\frac{2\pi m^{*}_{e}KT}{h^{2}})^{\frac{3}{2}}e^{-{\frac{(E_{C}- E_{F})}{KT}}}}$………..(2)

And the concentration of holes is, $\small {\color{Blue} p = 2(\frac{2\pi m^{*}_{h}KT}{h^{2}})^{\frac{3}{2}}e^{-{\frac{(E_{F}- E_{V})}{KT}}}}$………….(3)

Here $\small{\color{Blue}m^{*}_{e}}$ and $\small{\color{Blue}m^{*}_{h}}$ are the effective masses of electrons and holes respectively, $\small{\color{Blue}E_{C}}$, $\small{\color{Blue}E_{V}}$ and $\small{\color{Blue}E_{F}}$ are the energies of lower conduction band, upper valence band and Fermi level respectively.

According to Mass action law, n.p = ni2

Then, the intrinsic concentration of the semiconductor is, $\small {\color{Blue} n_{i} = 2(\frac{2\pi KT}{h^{2}})^{\frac{3}{2}}(m^{*}_{e} m^{*}_{h})^\frac{3}{4}e^{-{\frac{(E_{C}- E_{V})}{2KT}}}}$

or, $\small {\color{Blue} n_{i} = 2(\frac{2\pi KT}{h^{2}})^{\frac{3}{2}}(m^{*}_{e} m^{*}_{h})^\frac{3}{4}e^{-{\frac{E_{g}}{2KT}}}}$…….(4)

Where $\small{\color{Blue}E_{g} = E_{C}-E_{F} }$ is the band gap of the semiconductor.

Now, from the equation-(1) we get,

The expression for conductivity of a semiconductor is $\small {\color{Blue} \sigma = 2e(\frac{2\pi KT}{h^{2}})^{\frac{3}{2}}(m^{*}_{e} m^{*}_{h})^\frac{3}{4} (\mu_{n}+\mu_{p}) e^{-{\frac{E_{g}}{2KT}}}}$…….(5)

This can be written as ${\color{Blue} \sigma = AT^{\frac{3}{2}} e^{-{\frac{E_{g}}{2KT}}}}$……….(6)

Where A is the constant. This is the general equation of the conductivity of semiconductors. Equation-(1) shows that conductivity depends on carrier concentration (doping) and equation-(6) shows that it depends on temperature and band gap also.

This is all from this article on the conductivity of semiconductors and its formula. If you have any doubts on this topic you can ask me in the comment section. In the next articles, I’ve discussed Semiconductor-made electronic devices like a P-N Junction diode, Bipolar Junction Transistor, OP-AMP, etc. Click on the Next Article to read about P-N Junction diode.

Thank you!