How can this approximate donor ionization concentration be derived?

I'm currently working about doner ionization. I've got carrier density $$n_0 e^\frac{{\mu}-E_c}{k_bT}$$ and product of donor density and fermi distribution, $$\frac{N_d}{1+2e^\frac{{\mu}-E_d}{k_bT}}$$

And with those, somehow the textbook says the ionization density of donor is approximately given by $$n \approx (n_0 N_d)^{1/2}e^\frac{-E_d}{k_bT}$$

i've done some works to derive it, like multiplying forms i got and taking sqrt, but it seems to be "approximate" somewhere i am missing.

How can i get there?

edit: $$n_0 = 2\biggr(\frac{m_ek_BT}{2\pi \bar h^2}\biggr)^{3/2}$$ , factor originated from density of states, etc.

• What is $n_{0}$ here? Commented Oct 15, 2018 at 12:45
• made a note about it Commented Oct 16, 2018 at 12:03

Generally this is covered early on in most semiconductor physics textbooks. For example, in Sze's Physics of Semiconductor Physics this is in chapter 1.

In the intrinsic case,

There, one has the effective density of states in the conduction band is

$$N_{C} = 2({2\pi m_{de} k T \over h^{2}} )^{3 \over 2}M_{C}$$

where $$M_{C}$$ is the number of minima in the conduction band and $$m_{de}$$ is the effective mass for electrons in the conduction band. (For acceptors one flips to holes in the valence band).

For reasonable temperatures (i.e. the Fermi level is several $$kT$$ below the conduction band edge, the overlap integral for the intrinsic case (that is the intrinsic electron concentration) becomes:

$$n = N_{C} \exp (-{{E_{C} - E_{F}} \over kT})$$

One then proceeds to consider the impact of donors on the carrier concentrations. In that case one can derive (still chapter 1 in Sze) that the number of ionized donors is:

$$N_{D}^{+} = N_{D}{1 \over {1 + g \exp ({{E_{F}-E_{D} \over kT}})}}$$

where g is the ground state degeneracy normally taken as 2.

Then, using net neutrality one gets a pretty horrible equation relating the elecyron density in the conduction band with expressions for the ionized donors and the density of holes:

$$n = N_{C} \exp (-{{E_{C} - E_{F}} \over kT}) = N_{D} {1 \over {1 + g \exp ({{E_{F}-E_{D} \over kT}})}} + N_{V} \exp ({{E_{V} - E_{F}} \over kT})$$

At temperatures where the Fermi energy is close to the donor level, one can approximate this as:

$$n = ({N_{D} - N_{A} \over 2 N_{A}}) N_{C} \exp(-E_{d}/kT)$$ where $$E_{d} = E_{C}-E_{D}$$

In the appropriate limits,

$$n = {1 \over \sqrt 2}(N_{D}N_{C})^{1/2}\exp(-E_{d}/2kT)$$

Messy, but laid out well in Sze.

• thanks. i've only got Kittel and Streetman. Heard that Sze does well in details. it helped me a lot. Commented Oct 17, 2018 at 8:47