# Solution to Exercise 2.1-2: Density of States for Lower Dimensions

 Calculate the density of states for a one-dimensional semiconductor ("quantum wire") and for the two-dimensional case. Draw some conclusions from the results.
The number of states Z(k) up to a wave vector k is generally given by
Z(k)  =  Volume of "sphere" in m dimensions
Volume of state
The volume Vm of a "sphere" with radius k in m dimension is
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The density of states D(E) follows by substituting the variable k by E via the dispersion relation E(k) and by differentiation with respect to E
One obtains the following relations

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The consequences can be pretty dramatic. Consider, e.g. the concentration of electrons you can get in the three case for E » 0 eV, i.e close to the band edge.
The question, of course, is: Are there 1-dim. and 2-dim. semiconductors? The answer is: yes - as soon as the other dimensions are small enough we encounter these cases. We will run across examples later in the lexture course.

Exercise 2.1-2: Density of States for Lower Dimensions

© H. Föll (Semiconductor - Script)