CHEM01200604009 Sreejith Kaniyankandy - Homi Bhabha ...

10
In the previous sections the discussion was centered on description of electronic structure in a
Bulk or an Infinite crystal. In an ideal bulk crystal the electronic structure impart a coherence
length or mean free path (a b ) for the charge particle. When the radius of the particle is lesser
than this characteristic length i.e. r< a b , the coherence length of charge carriers will be a
function of radius. Additionally the charge carriers in a semiconductor i.e. electrons and
holes confined within a small radius can also interact with each other by coulomb interaction
and form a composite particle called the exciton. The exciton has a different coherence
length as the reduced mass of exciton is different. How can one treat an Exciton in a confined
solid?
The first attempt at solving this problem was carried out using the simple particle trapped in a
sphere model (3-D Confinement) [1.6, 1.7], where the particle was assumed to be trapped in
a potential well with infinite barriers. The crystal structure of the confined system is assumed
to be unchanged, one can write the wave function by using Bloch function u () r ,as
() r u () r ()
r
(1.10)
In the absence of degeneracy near the zone centre, neglecting coloumb interaction and using
parabolic dispersion we can write the Hamiltonion as
vk
vk
Hˆ
( ) ( )
2m
2
2 2
2 2
e h
Ve re Vh rh
e
mh
(1.11)
The confining potential for electron is given by
0 for | re
| a
Ve( re)
for | re
| a
(1.12)
Similar potential is used for holes.
Since we assumed coulomb interaction is absent we can write the envelope wave function as