Why does Fermi level has a probability density of 1/2 while it may lie in the forbidden region? I dont understand how there is a continuous probability density function in semiconductors, when there are several regions which are restricted by Energy, i.e. forbidden energies.
Well i know that in derivation through Fermi-Dirac statistics, it is easy to get the value 0.5, but physically i am not getting a feel. Please explain.
 A: Very strictly speaking, for semiconductors


*

*the term forbidden gap is not extremely well defined in a practical sense because due to unavoidable charge defects, lattice imperfections, etc., spurious energy levels between the valence and conduction bands can always exist, and

*the concept of a Fermi level in pure semiconductors does not make much of a physical sense. It is a term borrowed from conductors, such as metals. 


However, Fermi levels/energies do become physically relevant in case small amounts of impurities are introduced -- this process of doping can drastically change the conductivity of the semiconductor. To elaborate, doping can introduce either acceptor or donor levels (resulting in a p- type or n-type semiconductor, respectively) in the forbidden region; refer page 2 of this document for example. This process also shifts the Fermi level to below or above the middle of the bandgap (where the probability in the undoped semiconductor was 0.5) which is critical for the transport of the mobile carriers.
A: This might help you understand why the continuous probability distribution is used over discontinuous density of states to describe equilibrium distributions of electrons and holes. It's all to do with what the Fermi energy means.
The Fermi energy (or Fermi level or chemical potential) is all tied up with the thermodynamics of the electron gas. Specifically the Fermi-level is the free energy of the electron gas: it is the energy at which electrons can be added or removed from the system without also exchanging heat. Moreover, without increasing or decreasing the entropy of the electron gas.
In a solar cell, for example, you want to extract the potential energy of the electrons. When an electron is removed you don't extract the band energy (I.e. the energy of the energetic location on the band diagram, e.g. 1.4eV) you extract only up to the free energy per charge carrier (I.e. The Fermi energy has a lower value ~1.0eV say).
(Strictly speaking holes should be considered too in this example because what you are really extracting is the difference between two quasi Fermi levels of the electron and hole gases. But it gives you something to visualise.).
