What is the relationship between Fermi Energy and Electron Density of States?

[Ben473]

Hello,

Im just an amateur physicsist and was interested in Fermi Energy.
Found a page http://http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi2.html#c1" , that helped.

However, I found the page and site a bit hard to understand in some places.
Could someone please explain to me about Fermi Energy/Level and what the 'electron density of states' are?

Thanks in advance,
Ben

 

[phyzguy]

Because of the Pauli exclusion principle, no two electrons can occupy the same state. The density of states basically tells you how many states exist at a given energy - more precisely the density of states f(E) dE tells you how many states there are for electrons to occupy between the energy of E and E+dE. Normally, like in a metal or a semiconductor, all of the lower energy states are full, so if you add an electron you can't put it into the lowest energy state (referred to as the ground state), you have to add it to the lowest energy state which is not already filled. This state is roughly at the Fermi energy. So the Fermi energy tells you how much energy it takes to add an electron.

Are you just satisfying intellectual curiosity, or do you have a specific application in mind?

 

[NaN089]

Hey Phyzguy, that is a very simpe and clear explanation of the Density of States and the Fermi Energy. I have got a question that popped in my mind after seeing this reply.

Would you please explain the difference between 'density of states of electrons' and the 'density of electrons' of a semiconductor?

 

[phyzguy]

Well, the density of states refers to how many states are available to be filled by electrons, whether or not they are actually filled. The density of electrons refers to the number of electrons actually present. In practice, another difference is that the density of states is usually given in Fourier space (energy or momentum). In other words, the density of states f(E)dE gives the number of electron states with energy between E and E+dE, regardless of where in the semiconductor they are, and regardless of whether there are actually electrons in those states or not. The density of electrons is usually given in real space, so that the density of electrons n(x,y,z)dx dy dz gives the number of electrons in the volume dx dy dz centered at the point (x,y,z), regardless of how much energy or momentum they have.

 

[NaN089]

so the formula will give me the density of electrons and not the density of states of electrons:

n(E_c )= N_c.e^(-((E_c-E_f ))/(k.T))

?? In many books and web pages I found E_a which most probably the energy of acceptors. Is that the BINDING energy of the acceptors? I believe we use this to find the fraction of ionised dopants.

 

[phyzguy]

so the formula will give me the density of electrons and not the density of states of electrons:
n(E_c )= N_c.e^(-((E_c-E_f ))/(k.T))

Yes, this formula gives the density of electrons, not the density of states.

In many books and web pages I found E_a which most probably the energy of acceptors. Is that the BINDING energy of the acceptors? I believe we use this to find the fraction of ionised dopants.

Ea refers to where in the band gap the energy level of the acceptors lies. For common acceptors, this is usually just above the top edge of the valence band. By introducing these levels just above the valence band, many electrons will move into these new levels, creating a large number of vacancies (holes) in the valence band. These holes move in response to an applied electric field, and hence carry current and increase the conductivity of the semiconductor. Similarly, the energy level of common donors is just below the bottom of the conduction band

 

[wpj1223]

'density of states of electrons' is related with the trap.

'density of electrons' is N/V. N is the number. V is the volume.

 

[ravi_nigam]

According to Fermi Distribution function the probability of finding electron at fermi level is 1/2.

In intrinsic semiconductor fermi level lies in forbidden gap where no electron can exist. So probability should be 0.

Pleaze explain me the concept and remove the ambiguity!

 

[johng23]

The Fermi-Dirac distribution function doesn't give the probability of finding an electron, it gives the probability that a state in the energy range from E to E+dE is occupied. You then multiply this by the density of states (number of states between E and E+dE) to get the number of electrons in that energy range. The density of states is zero within the band gap, so there are no electrons there. In this case the position of the Fermi level still tells you about the relative occupation of the valence and conduction bands.

 

[ravi_nigam]

Thanks for clearing my doubts. But I have got more doubts. Hope you will help me clear my doubts.

1. If density of state is 0 then no state should be there. So why we study of probability of occupancy of state in these conditions (semiconductor forbidden gap case).

2. Fermi level is also defined as highest energy level at which electron can exist at 0K then it should be top of valence band but why is it in between conduction band and valence band ( in forbiden gap)? It should be in conduction band or valence band. How to explain this?

3. How we calculate fermi level? Please suggest a very fundamental book; more basic than kittel

 

[ravi_nigam]

The Fermi-Dirac distribution function doesn't give the probability of finding an electron, it gives the probability that a state in the energy range from E to E+dE is occupied. You then multiply this by the density of states (number of states between E and E+dE) to get the number of electrons in that energy range. The density of states is zero within the band gap, so there are no electrons there. In this case the position of the Fermi level still tells you about the relative occupation of the valence and conduction bands.

Thanks for clearing my doubts. But I have got more doubts. Hope you will help me clear my doubts.

1. If density of state is 0 then no state should be there. So why we study of probability of occupancy of state in these conditions (semiconductor forbidden gap case).

2. Fermi level is also defined as highest energy level at which electron can exist at 0K then it should be top of valence band but why is it in between conduction band and valence band ( in forbiden gap)? It should be in conduction band or valence band. How to explain this?

3. How we calculate fermi level? Please suggest a very fundamental book; more basic than kittel