energy term V p ðrÞ in the Hamiltonian
H ¼ À
p
2
2m
‘
2 þ V p ðrÞ
ð 4Þ
is the empirically determined pseudopotential
V p ðrÞ ¼
X
G
½V s ðGÞS s ðGÞ þ iV A ðGÞS A ðGÞ expðiGrÞ
ð 5Þ
where V s and V A are the symmetric and antisymmetric form factors, respectively.
They are determined from the optical absorption data. S s and S A are the symmetric
and antisymmetric structure factors, respectively, determined from the crystal
structure of the semiconductor. For a nanocrystal of radius R, a reasonable assumption is to use the bulk pseudopotential V p ðrÞ inside the crystallite and terminate this potential at the edge of the cluster by an infinite potential corresponding
to an infinite potential barrier. The Schro ¨dinger equation
HC n; k ðrÞ ¼ E n ðkÞC n; k ðrÞ
ð 6Þ
is solved at a set of k points in the Brillouin zone. Due to the finite size of the
nanocrystal, the allowed wave vectors are discrete. In the cartesian coordinates, the
values of k are limited to
k ¼
p
ffiffiffiffiffiffi
3R
p ½n x ; n y ; n z
ð 7Þ
The solution at these k-points gives the band structure of the nanocrystals.
Ramakrishna and Friesner [57] calculated the band structure of CdS and GaP
nanocrystals up to 30 A ˚ radius in the zinc-blende phase. The calculated variation
agrees quite well with the experimental data points. The band gap variation curve
in the case of GaP shows a red shifted band gap in the case of very small nanocrystals. This has not yet been experimentally verified.
Recently, Zunger and coworkers [18, 58–63] employed the semi-empirical pseudopotential method to calculate the electronic structure of Si, CdSe [60] and InP [59]
quantum dots. Unlike EMA approaches, this method, based on screened pseudopotentials, allows the treatment of the atomistic character of the nanostructure
as well as the surface effects, while permitting multiband and intervalley coupling.
The atomic pseudopotentials are extracted from first principles LDA calculations
on bulk solids. The single particle LDA equation,
À
1
2
‘
2 þ V
ps
nonlocal ðrÞ þ V LDA ðrÞ
!
C i ¼ e i C i
ð8Þ
11 Electronic Structure and Spectroscopy of Semiconductor Nanocrystals
386
H ¼ À
p
2
2m
‘
2 þ V p ðrÞ
ð 4Þ
is the empirically determined pseudopotential
V p ðrÞ ¼
X
G
½V s ðGÞS s ðGÞ þ iV A ðGÞS A ðGÞ expðiGrÞ
ð 5Þ
where V s and V A are the symmetric and antisymmetric form factors, respectively.
They are determined from the optical absorption data. S s and S A are the symmetric
and antisymmetric structure factors, respectively, determined from the crystal
structure of the semiconductor. For a nanocrystal of radius R, a reasonable assumption is to use the bulk pseudopotential V p ðrÞ inside the crystallite and terminate this potential at the edge of the cluster by an infinite potential corresponding
to an infinite potential barrier. The Schro ¨dinger equation
HC n; k ðrÞ ¼ E n ðkÞC n; k ðrÞ
ð 6Þ
is solved at a set of k points in the Brillouin zone. Due to the finite size of the
nanocrystal, the allowed wave vectors are discrete. In the cartesian coordinates, the
values of k are limited to
k ¼
p
ffiffiffiffiffiffi
3R
p ½n x ; n y ; n z
ð 7Þ
The solution at these k-points gives the band structure of the nanocrystals.
Ramakrishna and Friesner [57] calculated the band structure of CdS and GaP
nanocrystals up to 30 A ˚ radius in the zinc-blende phase. The calculated variation
agrees quite well with the experimental data points. The band gap variation curve
in the case of GaP shows a red shifted band gap in the case of very small nanocrystals. This has not yet been experimentally verified.
Recently, Zunger and coworkers [18, 58–63] employed the semi-empirical pseudopotential method to calculate the electronic structure of Si, CdSe [60] and InP [59]
quantum dots. Unlike EMA approaches, this method, based on screened pseudopotentials, allows the treatment of the atomistic character of the nanostructure
as well as the surface effects, while permitting multiband and intervalley coupling.
The atomic pseudopotentials are extracted from first principles LDA calculations
on bulk solids. The single particle LDA equation,
À
1
2
‘
2 þ V
ps
nonlocal ðrÞ þ V LDA ðrÞ
!
C i ¼ e i C i
ð8Þ
11 Electronic Structure and Spectroscopy of Semiconductor Nanocrystals
386