
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 V_{m} of a "sphere"
with radius k in m dimension is 




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Volume sphere = 4/3 (p · k)^{3} 
for m = 3 




V_{m}(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 




D_{m}(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 1dim. and 2dim. 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. 


