Calculation of Conduction Electron Population for Semi Conductor

• opuktun
In summary, the conversation discusses the formula for the density of electrons in the conduction band of a semiconductor and raises questions about its derivation. The Fermi level is approximated to lie at the center of the band gap in intrinsic semiconductors, but at non-zero temperatures it deviates due to differences in effective masses of holes and electrons. There may be an error in the exponential factor of the formula and integration by parts may have been applied. It is suggested to refer to solid state physics books for clearer derivations.

opuktun

Hi everyone,

From this website (link: http://hyperphysics.phy-astr.gsu.edu/HBASE/solids/fermi3.html#c1 ), we get the following expression:

$$Population Density of Conduction Electron = \int ^{\infty}_{Egap}N(E)d(E) = \frac{2^{5/2}(m \pi k T)^{3/2} exp (-E gap/2kT)}{h^3}$$

My questions are based on the derivation given by the website as follow:-

1. EF = Egap / 2
Is this a defined property of semiconductors?

2. Is there an error in the following expression?
$$N(E)d(E) = \frac{8\sqrt{2} \pi m^{3/2}}{h^3} \sqrt{E-Egap}\ e^{-(E-Egap/2)}$$
I mean originally it has kT in the power of exponential term. Is it mistakenly dropped?

3. Is "integration by parts" applied on the following expression?
$$N(E)d(E) = \frac{8\sqrt{2} \pi m^{3/2}}{h^3} \sqrt{E-Egap}\ e^{-(E-Egap/2)}$$
Somehow, I can't solve the integration.

Thanks.

Formula for density of electrons in conduction band is a good approximation for a classical gas of carriers in intrinsic semiconductors. Only if electron gas is classical you can approximate Fermi - Dirac statistic with Boltzmann one and only if the semiconductor is intrinsic Fermi's level is in the middle of energy gap. In a more general
way Fermi's level depends on doping level, temperature and effective density of states.
Regarding point 2 the exponential factor must be dimensionless.

For more details it's better if you'll see on solid state physics's books. You' ll find clearer derivations than the one you found on the link you posted.

Matteo.

Just to address your first question: in intrinsic semiconductors (no impurities present), the Fermi level is often approximated as lying at the center of the band gap. It doesn't lie exactly at the center at non-zero temperatures, however, because the effective masses of holes and electrons are different. A more precise equation is

$$E_F=E_V+\frac{E_g}{2}+\frac{3}{4}k_B T \ln\frac{m^\star_h}{m^\star_e}$$

Mapes said:
$$E_F=E_V+\frac{E_g}{2}+\frac{3}{4}k_B T \ln\frac{m^\star_h}{m^\star_e}$$

Thanks Mapes for the explanation, I finally understood the r/s.

matteo83 said:
Regarding point 2 the exponential factor must be dimensionless.

For more details it's better if you'll see on solid state physics's books. You' ll find clearer derivations than the one you found on the link you posted.

Thanks matteo. I'm gng to hit those books in my lib. :)