The Fermi level is the energy separating occupied states (or levels) of the valence band from empty states (levels) of the conduction band at the absolute temperature T=0 Kelvin.
The Fermi level is an energy level characteristic of the statistics (distribution law) which controls the occupation of any energy state by a given particle: an electron or a hole in semiconductors.
Electrons and holes are both fermion particles ( with half values of the spins) and both of them obey the Fermi-Dirac statistics which becomes asymptotically the Boltzman statistics in dilute systems (with few electrons and or holes) or nondegenerate systems. 
The position of the Fermi level with respect to valence and or conduction bands depends on various parameters as the temperature, the effective masses of electrons and holes, and the number of free electrons and holes.
This variation of the Fermi level obeys two condictions :
- the ''mass action law'' which states that the number of particles of each type as well as the overall number of the particles must conserve whatever is their distribution on the available energy levels.
- The neutrality equation which states that the electrical neutrality has to be fulfilled, i.e. the number of negative charges must be counterbalanced excatly by the same number of positive charges. 
 
						

							
Dear student 
The band gap merely refers to the amount of potential energy (in units of eV, electron volts) that is required for the electrons to cross the channel of a semiconductor.
There is no physical distance between the valence band and the conduction band
Good Luck