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The number of states Z(k) up to a wave vector k is generally given by
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| Z(k) | = |
Volume of "sphere" in m dimensions
Volume of state |
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 |
The volume Vm of a "sphere" with radius k in m
dimension is |
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æ
ç
ç
ç
ç
è |
Volume of sphere = 4/3 p · k3 |
for m = 3 | | |
| |
| | Vm(k) |
= |
Volume = area of circle = p · k2 |
for m = 2 | | |
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Volume = length = 2k |
for m = 1 |
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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. |
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One obtains the following relations: |
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æ
ç
ç
ç
ç
è |
(E)½ |
for m = 3 | | |
| |
| | Dm(E) |
µ |
const. |
for m = 2 | | |
| |
| | |
(E)–½ |
for m = 1 |
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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. |
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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 will encounter these cases. We will run
across examples later in the lecture course. |
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© H. Föll (Semiconductors - Script)
Exercise 2.1-2: Density of States for Lower Dimensions 
With frame