EMA gives a good description of the band gap variation with size. However, it
grossly overestimates the change, E g , in the band gap for smaller nanocrystals.
Kayanuma [45] identified two limiting cases depending upon the ratio of the radius of the quantum dot to the Bohr exciton radius, a B of the bulk solid. For
R=a B g 1, the exciton can be pictured as a particle moving inside the quantum dot
with only a small increment in energy due to the confinement. This is the weak
confinement regime. In the strong confinement regime where R=a B f 1, confinement effects obviously dominate. It was pointed out that in this regime, the electron and the hole should be viewed as individual particles in their respective single
particle ground states with little spatial correlation between them as a consequence
of the increased kinetic energy term. Kayanuma further found that the strong
confinement regime is not only limited to R=a B f 1, instead confinement effects
are seen up to R A 2a B . This is the regime where the effective mass approximation
is relatively more valid, though quantitatively EMA fails to account for the observed
changes.
The IP-EMA model essentially neglects any possibility of tunnelling of the
wavefunction beyond the surface of the nanocrystals as a consequence of the assumption of a rigid wall (infinite potential), particularly ignoring the role played by
the passivating atoms on the surface of the nanocrystals providing a possibility to
the electrons and the holes to substantially tunnel outside the nanocrystal. To
overcome the shortcomings of the IP-EMA, Kayanuma and Momiji [54] used the
finite potential to account for the experimental data for small CdS crystallites. The
confining potentials V e , for the electron and V h for the hole, satisfy the relation
E g þ V e þ V h ¼ E
1
g , where E
1
g is the band gap energy of the surrounding material.
The resulting problem was solved variationally in the Hylleraas coordinate system.
Although the effective mass of the electron is described quite well using a single
conduction band, the TVB in these systems is degenerate and the description of
the hole effective mass requires a greater number of bands. Therefore, the multiband EMA (MBEM) theory represents a substantial improvement over the single
band, IP-EMA. In CdSe nanocrystals, the size dependences of up to 10 excited
states [49, 50] in the absorption spectra are successfully described by the uncoupled MBEM. This includes the valence band degeneracy, but does not couple
the valence and conduction bands. Banin et al. [51] have used the MBEM including the valence and the conduction band coupling. They use an eight band
Luttinger–Kohn Hamiltonian to calculate the quantum size levels in InAs [51].
Such improvements over the over-simplistic IP-EMA explain the experimental
data quite well. However, approaches based on EMA explain the results a posteriori, whereas a theory is required that has predictive capabilities and can be applied
to handle new systems.
11.5.2
Empirical Pseudopotential Method
Another method that has been used to describe the electronic structure of bulk
semiconductors is the empirical pseudopotential method [55, 56]. The potential
11.5 Electronic Structure Calculations 385
grossly overestimates the change, E g , in the band gap for smaller nanocrystals.
Kayanuma [45] identified two limiting cases depending upon the ratio of the radius of the quantum dot to the Bohr exciton radius, a B of the bulk solid. For
R=a B g 1, the exciton can be pictured as a particle moving inside the quantum dot
with only a small increment in energy due to the confinement. This is the weak
confinement regime. In the strong confinement regime where R=a B f 1, confinement effects obviously dominate. It was pointed out that in this regime, the electron and the hole should be viewed as individual particles in their respective single
particle ground states with little spatial correlation between them as a consequence
of the increased kinetic energy term. Kayanuma further found that the strong
confinement regime is not only limited to R=a B f 1, instead confinement effects
are seen up to R A 2a B . This is the regime where the effective mass approximation
is relatively more valid, though quantitatively EMA fails to account for the observed
changes.
The IP-EMA model essentially neglects any possibility of tunnelling of the
wavefunction beyond the surface of the nanocrystals as a consequence of the assumption of a rigid wall (infinite potential), particularly ignoring the role played by
the passivating atoms on the surface of the nanocrystals providing a possibility to
the electrons and the holes to substantially tunnel outside the nanocrystal. To
overcome the shortcomings of the IP-EMA, Kayanuma and Momiji [54] used the
finite potential to account for the experimental data for small CdS crystallites. The
confining potentials V e , for the electron and V h for the hole, satisfy the relation
E g þ V e þ V h ¼ E
1
g , where E
1
g is the band gap energy of the surrounding material.
The resulting problem was solved variationally in the Hylleraas coordinate system.
Although the effective mass of the electron is described quite well using a single
conduction band, the TVB in these systems is degenerate and the description of
the hole effective mass requires a greater number of bands. Therefore, the multiband EMA (MBEM) theory represents a substantial improvement over the single
band, IP-EMA. In CdSe nanocrystals, the size dependences of up to 10 excited
states [49, 50] in the absorption spectra are successfully described by the uncoupled MBEM. This includes the valence band degeneracy, but does not couple
the valence and conduction bands. Banin et al. [51] have used the MBEM including the valence and the conduction band coupling. They use an eight band
Luttinger–Kohn Hamiltonian to calculate the quantum size levels in InAs [51].
Such improvements over the over-simplistic IP-EMA explain the experimental
data quite well. However, approaches based on EMA explain the results a posteriori, whereas a theory is required that has predictive capabilities and can be applied
to handle new systems.
11.5.2
Empirical Pseudopotential Method
Another method that has been used to describe the electronic structure of bulk
semiconductors is the empirical pseudopotential method [55, 56]. The potential
11.5 Electronic Structure Calculations 385