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The number of states
Z(k) up to a wave vector k is
generally given by |
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= |
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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Volume sphere = 4/3 (p · k)3 |
for m = 3 |
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| Vm(k) |
= |
Volume = area circle = p · k)2 |
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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(E)½ |
for m = 3 |
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| Dm(E) |
µ |
const. |
for m = 2 |
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(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 encounter these cases. We will
run across examples later in the lexture course. |
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© H. Föll (Semiconductor - Script)
Exercise 2.1-1 Free Electron Gas with Constant Boundary Conditions
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