Fermi level of n-type semiconductor Nd>Na>0

[girlinphysics]

Homework Statement

This isn't actually a homework question, but in my semiconductors textbook, the following equation has been given:

$$E_f = E_g - k_BTln(\frac{n_0}{N_d - N_a})$$

This is for the limiting case Nd>Na>0. I got a little confused as to where that equation has come from.

Homework Equations

$$N_d - N_a$$ (Where Nd = donor level and Na = acceptor level)
Also: $$E_F = E_g - E_d$$

The Attempt at a Solution

I think this means the fermi level is at donor level so I can write:

$$n = n_0e^{\beta(E_f - E_g)} = n_0e^{-\beta E_d}$$

where beta = 1/kT

Is this how they got to that equation? Just by rearranging for Ef?

 

[mark726]

Hello,

Thank you for sharing your question with us. The equation you have mentioned is known as the Fermi-Dirac distribution function, which describes the probability of an energy level being occupied by an electron in a semiconductor. This equation is derived from the principles of quantum mechanics and statistical mechanics and is widely used in the study of semiconductors.

To answer your question, the equation you have mentioned is not derived from the equation E_f = E_g - E_d, but rather from the definition of the Fermi energy, which is the energy level at which the probability of an electron being present is 0.5. This equation takes into account the donor and acceptor levels, as well as the temperature of the system.

To better understand the derivation of this equation, I would suggest referring to your textbook or other resources that explain the Fermi-Dirac distribution function in detail. I hope this helps to clarify your confusion. Keep up the curiosity and good luck with your studies!