This is the typical graph describing how the density of states in a semiconductor depends on dimensionality. (Im aware there is a mistake in the 1D and 0D). I have troubles interpreting the 2D and 1D. For the 2D case even though the DOS function is independent of the Energy, the graph is a step function. And or the 1D case, there are quantized energy levels, however according to the graph between these each 2 of them the energy the DOS decreases by (1/2), how is this possible if energy values are dicrete? Thanks :)
For the 2D case, the DOS is independent of energy, but the red curve is a quantum well case, not a 2D case. In a quantum well, there is a finite thickness d in the z direction. The wavenumber in that direction must be (2 pi nz / d) , where nz is an integer. The energy must be high enough to reach nz=1, then it steps when it can cover nz=2, etc.. I also assume the 1D case is a quantum wire case, not really 1D.
As CHS pointed out, the 2D DoS is for an engineered quantum well, which will always have more than one subband (think particle in a 2D box). Once the electrons achieve enough energy, they can "jump" into the next available discrete subband, however once in that subband they have a constant plateau of energy. Hence the staircase shape.
There is a labeling mistake in your plot. The Delta functions are actually 0D (quantum dot) DoS. Check out the description here: https://www2.warwick.ac.uk/fac/sci/physics/current/postgraduate/regs/mpags/ex5/bandstructure/
I doubt the absolute value of the Delta functions holds any meaning for a qualitative graph like this one.