Does diffusion current in semiconductor always exist? According to what I understand, diffusion current is caused by the change in concentration of charge carriers in semiconductor (free electrons and holes) from higher concentration region to lower concentration region due to thermal energy. However, I'm still unsure that if the diffusion current always exist inside crystal. As the concentration drop in higher region and increase in lower region, finally, the concentration in the whole crystal will be equal so no concentration change. Will the diffusion current still exist after that. 
Moreover, in non-doping semiconductor (ex: pure Si) the density/amounts of holes and electrons the same so is there diffusion current. Suppose that the diffusion is only 1 dimensional and 1 direction. The charge carriers which diffuse contain both hole and electron then the total charge movement is zero that make the current zero, is it true?
 A: In a semiconductor at thermal and diffusive equilibrium, the net current is zero for each of the carrier types, i.e., it is zero both for the electron and hole currents separately.  The total current is the sum of the diffusion current (proportional to the gradient of the density of carriers) and the drift current driven by an external field (the electrostatic field in most of the cases for a semiconductor). In brief, in a semiconductor in thermodynamic equilibrium the diffusion current compensates the drift current for each of the carriers separately resulting in a zero net current.
Note that an intrinsic semiconductor at thermodynamic equilibrium may have diffusion and drift currents different from zero but the total net current still be zero due to the compensation of the drift and diffusion currents (this would be the case of an isolated semiconductor crystal placed inside an electric field).    
A: The diffusion current serves to equalise charge density everywhere. But in any given crystal at a finite temperature above 0 K, there will be some concentration gradients somewhere for both holes and electrons. This leads to diffusion currents. But note that the magnitude of such currents is very small and it is very difficult to measure them. The same thing also happens in conductors but takes place much faster than semiconductors.
A: Semiconductors (doped and not, both) has some charge carriers.
For penta-valent semiconductors, it is electrons, and for tri-valent semiconductors, it is holes.
And, diffusion of charge carriers always takes place. But, the motion of charge carriers is completely random. And, statistically, the motion of charge carriers in every direction is equally distributed making the net current zero.
So, there is no net current caused by the diffusion of charge carriers.
