Dear Marva,

I hope, that I understood well your question. The carrier concentration in intrinsic semiconductors ni(T) = p = n is governed by the chemical potential µ.

At T = 0 K, the position of the chemical potential, what is for T = 0 K equal to the Fermi level position, is exactly the middle of the band gap. For T > 0 we have:

µ(T) = Eg/2 + 0,5*k*T*ln(NV/NC)

that means, the energy of the level shifts to higher energy. The ratio of the density of states for the valence band NV and band of conductivity NC determine the strength of the shift. The ratio of the density NV and NC is determined by the ratio of the density of states masses. Look the following table:

GaAs: 6,71

Si : 0,75

Ge : 0,67

Due to the low electron masses in GaAs, the energy level shift is high. The lowest mass ratio has Ge, that means the effect is small.

I hope, my answer is satisfactory.

With Regard

R. Mitdank

Thank you sir for giving this explanation...…….. but we know that the intrinsic fermi level lies in the middle of the band gap which means that the effective densities of states are equal. please explain it

Dear Marva,

lets estimate the shift in the case of GaAs for room temperature:

kT/2 = 13 meV; ln(NV/NC ) = ln(mv/mC)1,5=ln(6,71,5) = 2,8

Therefore the shift is nearly 37 meV. The GaAs gap is 1,4 eV. This is a shift of 2,6%. In the other materials you have a lower value.

You must be able to detect a level shift of 1% or absolutely in the meV range.

Practically, the level remains in the middle.

Regards

R. Mitdank

The instrinsic Fermi energy is *not* in the center of the band gap. This would be only the case if effective electron and hole masses would be equal which is never the case. Consequently, the effective densities (of the bands) are not equal. Rüdiger Mitdank gave the correct equation already:

E_F = E_g/2 + 0.5kT ln(N_V/N_C)

Taking into account that N_C is proportional to the effective electron mass to the power of 3/2 we arrive at

E_F = E_g/2 + 3/4 kT ln(m*_h/m*_e)

Now it is easy to look up effective masses for the different semiconductor materials (density of state masses are the relevant ones!) and check where the Fermi energy is located.

Tabulated effective electron and hole masses should be more similar for Si and Ge than for GaAs, consequently. In other words, the ratio m*h/m*_e is closer to unity for GaAs.

Si Ge GaAs

m*_e (m_0) 1.063 0.553 0.066

m*_h (m_0) 0.689 0.377 0.522

ratio ~0.65 ~0.69 ~7.9

[no guarantee for these values, they are taken from my lectrure notes, but the idea can be understood]

Thank you very much to all respected teachers now my concept is very much clear.