CHEM01200604009 Sreejith Kaniyankandy - Homi Bhabha ...

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Substituting this in Schrödinger’s equation we get,
() r u () r e
ikr
(1.5)
k
k
2 2 2
p ( k. p)
k
V u
E u
2m m 2m
nk nk nk
(1.6)
At regions close to the zone center we can write the energies using parabolic approximation
as follows
E
nk
2 2
k
En0
(1.7)
2
m
Effective mass can be written as
1 1 2 | un | k. p| un
|
m m m k E E
2
0 0
2 2 (1.8)
nn n0
n0
It is clear from the above relationship that the effective mass is a combined effect of mass
and potentials experienced by the electron. Effective mass is tensor due to different
dispersions along different symmetry axes.
E
2
1 1
ko
2
ij
kikj
m
(1.9)
1.2.3 Effect of 3-Dimensional Confinement on Electronic
Structure
From the previous sections of this chapter it was assumed that wave function is only
dependent on the periodicity and the potentials experienced by charge carriers. Therefore it is
necessary to ask, how does dimensionality influence wave function?