Lippens and Lannoo [66] first used the TB method to calculate the variation of
band gap with size for CdS and ZnS nanocrystals. They used the sp
3 s* orbital
basis with only the nearest neighbor interactions, as suggested by the work of Vogl
et al. [67] for bulk semiconductor materials. The calculations were carried out for
clusters ranging in size from 17 atoms (3 shells) to 2563 atoms (15 shells). The
surface dangling orbitals which give rise to non-bonded states in the mid-gap region, were removed from the Hamiltonian, as these states make the determination
of the band gap difficult. Direct diagonalization for smaller nanocrystals up to 83
atoms and the recursion method for the larger crystallites were used to obtain the
TVB and BCB as a function of size. Interestingly, the TVB and BCB do not exhibit
a smooth monotonic shift with size. They show a step-like decrease in the energy
position of the band edge as the size increases. As alternate shells are composed of
cations and anions, the surface shell whose dangling orbitals have been removed is
either purely anionic or purely cationic. This change of character of the outermost
shell has different effects on determining the TVB and BCB, leading to the steplike variations in these quantities. The subtraction of the TVB energy from that of
the BCB gives the band gap of the crystallite. To compare with the experimental
results the exciton binding energy needs to be subtracted from the value of the
bare band gap obtained from the TB model excluding any excitonic effect. This
binding energy is given by the Coulomb term in the EMA equation, Eq. (2). With
the sp
3 s* model, Lippens and Lannoo were able to get a better description of the
band gap variation with size compared to the IP-EMA model. However, the results
of the sp
3 s* nearest neighbor TB model underestimate the band gap at all sizes.
Following the success of the TB model proposed by Lippens and Lannoo, several
researchers have used this scheme. Hill and Whaley employed a time-dependent
TB approach to determine the electronic structure of CdS and CdSe nanocrystals
[68, 69]. They compared the effect of passivating the CdSe surface with oxygen and
removing the dangling orbitals. They also studied the effects of terminating the
nanocrystal with a cationic and an anionic surface. Perez-Conde et al. [70, 71] used
the sp
3 s* orbital basis TB method with spin–orbit coupling for CdSe and CdTe
nanocrystals. The resulting band gap variation with size obeys a 1/D
1:42 function
rather than the 1/D
2 relation expected on the basis of EMA. Recently Allan et al.
[72] calculated the quantum confinement effects for zinc blende III–V and group
IV semiconducting nanocrystals using the nearest neighbor, sp
3 d
5 s* TB model
with the spin–orbit interaction included. Such a model gives a much better description of the bulk semiconductor band structure and thus explains the experimental results for band gap variation more effectively than the sp
3 s* model.
All the TB models discussed so far are based on the nearest neighbor interactions. Recently, Sapra et al. proposed [73] the sp
3 d
5 tight-binding model with
cation–anion nearest neighbor and anion–anion next nearest neighbor (NNN) interactions for the A
II B
VI semiconductor compounds with A ¼ Zn, Cd, Hg and
B ¼ S, Se, Te. The model was chosen after a careful analysis of the bulk band
structures of these compounds obtained from the linearized muffin tin orbital
(LMTO) method as described earlier in this section. These calculations were car11 Electronic Structure and Spectroscopy of Semiconductor Nanocrystals
390
band gap with size for CdS and ZnS nanocrystals. They used the sp
3 s* orbital
basis with only the nearest neighbor interactions, as suggested by the work of Vogl
et al. [67] for bulk semiconductor materials. The calculations were carried out for
clusters ranging in size from 17 atoms (3 shells) to 2563 atoms (15 shells). The
surface dangling orbitals which give rise to non-bonded states in the mid-gap region, were removed from the Hamiltonian, as these states make the determination
of the band gap difficult. Direct diagonalization for smaller nanocrystals up to 83
atoms and the recursion method for the larger crystallites were used to obtain the
TVB and BCB as a function of size. Interestingly, the TVB and BCB do not exhibit
a smooth monotonic shift with size. They show a step-like decrease in the energy
position of the band edge as the size increases. As alternate shells are composed of
cations and anions, the surface shell whose dangling orbitals have been removed is
either purely anionic or purely cationic. This change of character of the outermost
shell has different effects on determining the TVB and BCB, leading to the steplike variations in these quantities. The subtraction of the TVB energy from that of
the BCB gives the band gap of the crystallite. To compare with the experimental
results the exciton binding energy needs to be subtracted from the value of the
bare band gap obtained from the TB model excluding any excitonic effect. This
binding energy is given by the Coulomb term in the EMA equation, Eq. (2). With
the sp
3 s* model, Lippens and Lannoo were able to get a better description of the
band gap variation with size compared to the IP-EMA model. However, the results
of the sp
3 s* nearest neighbor TB model underestimate the band gap at all sizes.
Following the success of the TB model proposed by Lippens and Lannoo, several
researchers have used this scheme. Hill and Whaley employed a time-dependent
TB approach to determine the electronic structure of CdS and CdSe nanocrystals
[68, 69]. They compared the effect of passivating the CdSe surface with oxygen and
removing the dangling orbitals. They also studied the effects of terminating the
nanocrystal with a cationic and an anionic surface. Perez-Conde et al. [70, 71] used
the sp
3 s* orbital basis TB method with spin–orbit coupling for CdSe and CdTe
nanocrystals. The resulting band gap variation with size obeys a 1/D
1:42 function
rather than the 1/D
2 relation expected on the basis of EMA. Recently Allan et al.
[72] calculated the quantum confinement effects for zinc blende III–V and group
IV semiconducting nanocrystals using the nearest neighbor, sp
3 d
5 s* TB model
with the spin–orbit interaction included. Such a model gives a much better description of the bulk semiconductor band structure and thus explains the experimental results for band gap variation more effectively than the sp
3 s* model.
All the TB models discussed so far are based on the nearest neighbor interactions. Recently, Sapra et al. proposed [73] the sp
3 d
5 tight-binding model with
cation–anion nearest neighbor and anion–anion next nearest neighbor (NNN) interactions for the A
II B
VI semiconductor compounds with A ¼ Zn, Cd, Hg and
B ¼ S, Se, Te. The model was chosen after a careful analysis of the bulk band
structures of these compounds obtained from the linearized muffin tin orbital
(LMTO) method as described earlier in this section. These calculations were car11 Electronic Structure and Spectroscopy of Semiconductor Nanocrystals
390