# The probability function of electrons occupying the donor state

1. Nov 23, 2009

### Dale12

In Semiconductor physics and devices: basic principles
[By Donald A. Neamen], chapter 4, section 4.4, it says:
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One postulate used in the derivation of the Fermi-Dirac probability function was the Pauli exclusion principle, which states that only one particle is permitted in each quantum state. The Pauli exclusion principle also applies to the donor and acceptor states.

Suppose we have $$N_i$$ electrons and $$g_i$$ quantum states, where the subscript i indicates the ith energy level. There are $$g_i$$ ways of choosing where to put the first particle. Each donor level has two possible spin orientations for the donor eletron;thus each donor level has two quantum states. The insertion of an electron into one quantum state, however, precludes putting an electron into the second quantum state. By adding one electron, the vacancy requirement of the atom is satisfied, and the addition of a second electron in the donor level is not possible. The distribution function of donor electrons in the donor eneergy states is then slightly different than the Fermi-Dirac funtion.

The probability function of electrons occupying the donor state is

$$n_d=\frac{N_d}{1+\frac{1}{2}\exp{\frac{E_d-E_F}{kT}}}$$

where n_d is the density of electrons occupying the donor level and E_d is the energy of the donor level. The factor 1/2 in this equation is direct result of the spin factor just mentioned. The 1/2 factor is sometimes written as 1/g. Where g is called a degeneracy factor.
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My question are these:

1.how to understand sentences below? since there are two quantum states, why could only one electron be inserted?

Each donor level has two possible spin orientations for the donor eletron;thus each donor level has two quantum states. The insertion of an electron into one quantum state, however, precludes putting an electron into the second quantum state. By adding one electron, the vacancy requirement of the atom is satisfied, and the addition of a second electron in the donor level is not possible.

2.why the factor 1/2 is related to the spin factor? or why it is added before $$\exp{\frac{E_d-E_F}{kT}$$ but not before $$\frac{N_d}{1+\exp{\frac{E_d-E_F}{kT}}}$$ when considering the Fermi-Dirac distribution function.

Thanks for any help!

ps:
Semiconductor physics and devices: basic principles could be read in google books: