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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Volume sphere = 4/3 (p · k)3 for m = 3
       
Vm(k)  =  Volume = area circle = p · k)2    for m = 2
     
Volume = length = 2k for m = 1
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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(E)½ for m = 3
       
Dm(E)  µ  const.    for m = 2
     
(E)–½ for m = 1
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.
 

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© H. Föll (Semiconductor - Script)