CHEM01200604009 Sreejith Kaniyankandy - Homi Bhabha ...

11
( r, r ) ( r ) ( r )
(1.13)
e h e e e h
The solution to the problem can be obtained in terms of Bessel functions for the Radial Part
and spherical harmonics for angular part as given below
r
Jl
n,
l
2
a
() r Y
(1.14)
a J
eh ,
nlm ,, lm , 3
l1 n,
l
Where n can take values 0, 1, 2, --- and l=0, 1, 2, ---and m values are given by l m l.
nl
,
is the n th order of J
l
. Applying the boundary condition we arrive at Energies
E
(1.15)
2 2
eh ,
nl ,
nl , 2
2meh
,
a
It is clear from the energy Eigen values that discreteness evolves from continuum of states on
introducing confinement. The shift from the bulk values can be given by
2 2
E
(1.16)
2
2
a
is the reduced mass of electron-hole pair.
The above energy shift arises from confinement of carriers and is positive indicating
that there will be a positive energy shift with confinement or in other words the band gap
increases.
In the above treatment we have neglected coulomb interaction, however this problem can be
solved by looking at scaling of coulomb interaction with radius as compared to confinement
energy. The coulomb interaction scales inversely with radius, r and confinement energy
varies with inverse r 2 . This clearly indicates that coulomb interaction can be approximated as
a correction factor to the above obtained Eigen values. Coulomb interaction acts only as a