of 46 A ˚ diameter; the corresponding result for a passivated CdS crystallite of the
same size is shown in the lower panel. The two results appear essentially the same,
suggesting the same gross electronic structures for both systems. When we expand
the band gap region, however, some important differences are easily seen, as
shown in the insets to the panels. Evidently, the unpassivated cluster (upper panel,
inset) shows the presence of many states within the band gap region which cannot
be normally seen on the usual scale due to their low intensity. However, these midgap states affect the emission properties of the system, as discussed earlier in Section 11.4. Passivating the surface removes these states as shown in the lower panel.
While experimentally such passivation provides a convenient way to recover the
band gap emission, and therefore the tunability of the emission frequency by controlling the size, as discussed earlier, in the present context it also provides a theoretical handle for proper identification of the band gap by the removal of the unwanted surface states from the mid-gap region.
In Figure 11.11 we show the variation of the band gap with size for the various
A
II B
VI semiconductors calculated within the sp
3 d
5 model [75], along with the results based on the IP-EMA and those from the sp
3 s* model [76]. The Coulomb
term (the third term in Eq. (2)) has been subtracted from each of the calculated
band gaps in order to facilitate comparison with the experimental data [8, 12, 16,
77–84, 86–92, 99] that are also plotted in the same figure. Evidently, EMA grossly
overestimates the experimentally observed shifts in the band gaps in every case.
The results of [75], including the next nearest neighbor interactions (shown by
solid lines), are found to describe the experimental data points considerably better
than the nearest neighbor sp
3 s* model (shown by dashed lines in the figure).
Sapra et al. have also used a similar model to calculate the band gap variation in
the case of InP nanocrystals [93]. They used the sp
3 d
5 orbitals for P atoms and the
sp
3 orbitals for In atoms, as the In d orbitals are much lower in energy than the
TVB and do not influence the bands close to the band gap region. Here, they
needed the InaP, InaIn and PaP interactions in order to obtain a good agreement
of the ab initio and TB fitted band dispersions. The calculated band gap variation is
in excellent agreement with the experimental results.
It has also been possible to calculate the electronic structure of doped semiconductors using the TB method. For example, the variation in the electronic structure
of Ga 1Àx Mn x As has been reported [94] for sizes of particles ranging between 12
and 71 A ˚ for a fixed Mn-doping level of about 6.25%. The authors observed dramatic changes in the electronic structure of the Mn-doped GaAs nanocrystals at
sizes smaller than about 30 A ˚ . Figure 11.12 shows the spin resolved electronic
density of states for (a) 25.0, (b) 18.5 and (c) 12.1 A ˚ Ga 1Àx Mn x As nanocrystals. Also
shown are the As p (dotted lines in the respective panels) and Mn d (panel d) partial density of states. Here, we notice that the two larger sized nanocrystals have
a finite DOS at the Fermi level whereas the 12.1 A ˚ nanocrystals show an insulating behavior. Bulk Ga 1Àx Mn x As is a known half-metallic ferromagnet [95, 96] and
this behavior is observed for the nanocrystals larger than 12 A ˚ i.e. the down-spin
channel is insulating whereas the up-spin channel is metallic. Calculations sug11 Electronic Structure and Spectroscopy of Semiconductor Nanocrystals
392
same size is shown in the lower panel. The two results appear essentially the same,
suggesting the same gross electronic structures for both systems. When we expand
the band gap region, however, some important differences are easily seen, as
shown in the insets to the panels. Evidently, the unpassivated cluster (upper panel,
inset) shows the presence of many states within the band gap region which cannot
be normally seen on the usual scale due to their low intensity. However, these midgap states affect the emission properties of the system, as discussed earlier in Section 11.4. Passivating the surface removes these states as shown in the lower panel.
While experimentally such passivation provides a convenient way to recover the
band gap emission, and therefore the tunability of the emission frequency by controlling the size, as discussed earlier, in the present context it also provides a theoretical handle for proper identification of the band gap by the removal of the unwanted surface states from the mid-gap region.
In Figure 11.11 we show the variation of the band gap with size for the various
A
II B
VI semiconductors calculated within the sp
3 d
5 model [75], along with the results based on the IP-EMA and those from the sp
3 s* model [76]. The Coulomb
term (the third term in Eq. (2)) has been subtracted from each of the calculated
band gaps in order to facilitate comparison with the experimental data [8, 12, 16,
77–84, 86–92, 99] that are also plotted in the same figure. Evidently, EMA grossly
overestimates the experimentally observed shifts in the band gaps in every case.
The results of [75], including the next nearest neighbor interactions (shown by
solid lines), are found to describe the experimental data points considerably better
than the nearest neighbor sp
3 s* model (shown by dashed lines in the figure).
Sapra et al. have also used a similar model to calculate the band gap variation in
the case of InP nanocrystals [93]. They used the sp
3 d
5 orbitals for P atoms and the
sp
3 orbitals for In atoms, as the In d orbitals are much lower in energy than the
TVB and do not influence the bands close to the band gap region. Here, they
needed the InaP, InaIn and PaP interactions in order to obtain a good agreement
of the ab initio and TB fitted band dispersions. The calculated band gap variation is
in excellent agreement with the experimental results.
It has also been possible to calculate the electronic structure of doped semiconductors using the TB method. For example, the variation in the electronic structure
of Ga 1Àx Mn x As has been reported [94] for sizes of particles ranging between 12
and 71 A ˚ for a fixed Mn-doping level of about 6.25%. The authors observed dramatic changes in the electronic structure of the Mn-doped GaAs nanocrystals at
sizes smaller than about 30 A ˚ . Figure 11.12 shows the spin resolved electronic
density of states for (a) 25.0, (b) 18.5 and (c) 12.1 A ˚ Ga 1Àx Mn x As nanocrystals. Also
shown are the As p (dotted lines in the respective panels) and Mn d (panel d) partial density of states. Here, we notice that the two larger sized nanocrystals have
a finite DOS at the Fermi level whereas the 12.1 A ˚ nanocrystals show an insulating behavior. Bulk Ga 1Àx Mn x As is a known half-metallic ferromagnet [95, 96] and
this behavior is observed for the nanocrystals larger than 12 A ˚ i.e. the down-spin
channel is insulating whereas the up-spin channel is metallic. Calculations sug11 Electronic Structure and Spectroscopy of Semiconductor Nanocrystals
392