remove this misconception about fermi level

i have read following 2 definitions about fermi level;
1. Fermi level is the highest energy level that is occupied by electron at 0K.
2. probability of occupancy of fermi level at temperature T>0k is 0.5.

My query is
In intrinsic semiconductor. fermi level lies in the energy gap, where no electron can occupy any state.
But according to the defination stated above, the fermi level will have probability 1 at 0K and 0.5 at T>0K.

Where liws the misconception.

>> Fermi level is the highest energy level that is occupied by electron at 0K.

Let's change this to "Fermi level is the highest energy level that CAN BE occupied by electron at 0K, IFF a valid enery level is present at that energy."

So, in the case of metals, where fermi level is indeed the highest energy level before the electrons leave the metal (photoelectric type of effect)....it is obviously true...


In the case of Semiconductors, fermi level lies in the bandgap or the "forbidden band"...no allowable energy level is present there so no electrons are found there, rather electrons are only found BELOW it (in valence band at 0 K), so i can say that it is the highest energy level...that could have been occupied by an electron at 0K if allowable.



>>But according to the defination stated above, the fermi level will have probability 1 at 0K

NO. the definition doesn't say this. It just says that the highest possible energy level at 0K is fermi level.....all energy levels above fermi level are EMPTY at 0K and all electrons reside in energy level below the fermi level.....thus the probability of occupancy of fermi is still 0.5





HOPE THIS HELPS!

as far as my teacher said these details about fermi level is just a theoretical calculation based on transitions..... even in the case of metals there wont be a energy level at fermi level.... if u refer to millman C.halkias u can see there a derivation that the fermi level is exactly in the middle of the valence and conduction band.....



One main thing related to probability is that it is mostly theoretical.... for example try the head or tail experiment with a coin.... u will rarely get the practical probability to be 0.5....

Hi
Fermi level is just the probability function f(E)which tells bout the probability of the e- state to b occupied by e-.
but it does not tell alone if there is e- present or not in that e- state, there is an e- density function g(E).
multiplication of f(E) and g(E) decides if there is e- present or not.
since in semiconductors g(E) is 0 for forbidden states
so no e- there.

Added after 1 minutes:

Hi
Fermi level is just the probability function f(E)which tells bout the probability of the e- state to b occupied by e-.
but it does not tell alone if there is e- present or not in that e- state, there is an e- density function g(E).
multiplication of f(E) and g(E) decides if there is e- present or not.
since in semiconductors g(E) is 0 for forbidden states
so no e- there.

kapil Tyagi

to be precise, i wld like to add:
at 0 K, Fermi level is actually the top collection of energy level which an electron can occupy!
(Reason: pauli's exclusion principle)

for rest: i agree with zeeshanzia84..

ya but some prob in it that they not define what the sys can it

The probability for an electron to exist at an energy level is assuming there is a state in the energy level to allow an electron occupied. If there is no state at that enery level, any electron can be there stably, thus, no electron can be found there with meaningful probability. Note that the multiplication constant is deternimed by putting Integral(probability_density_function * state_density_function) * dE equal to 1, where the integration is performed along the whole energy range. And, the Fermi level is effective in considering a equilibrium-state condition, less effective in a steady-state condition. Strictly speaking, the Fermi level do not exist in its definition point of view in a steady-state condition.

Sorry, my mistyping.

any electron can be there stably -> no electron can be there stably