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Pioneering Effort I: 
Al. L. Efros and A. L. Efros, Sov. Phys. Semicond., 16, 772 - 775 (1982)

This pioneering effort begins with a humble disclaimer that ``a very simple model'' is being used to study light absorption in a semiconductor sphere. The paper is suitably titled: ``Interband Absorption of Light in a Semiconductor Sphere''. The motivation for this work is experimental studies by Ekimov et al. in the erstwhile Soviet Union. However, no comparison of theory with experiment is attempted.

Instead the authors identify three size regimes on the basis of quantum scales. They start with an excitonic Hamiltonian within the effective mass theory.

$$\frac{p_e^2}{2 m_e} + \frac{p_h^2}{2 m_h} - \frac{e^2}{\kappa\mid r_e - r_h \mid}$$ H =

Here the symbols have their usual meaning. The subscript e (h) stands for electron (hole) and $\kappa$ is the dielectric constant of the bulk semiconductor. The oxide coating is modeled by an infinite barrier height. The solution of this Hamiltonian is written in terms of spherical Bessel functions. The three size regimes the authors identify are:

1.

$R \ll \{a_e , a_h\}$. Here R is the radius of the quantum dot (QD) and a_e (a_h) is the Bohr radius of the electron (hole). For this case the magnitude of Coulomb attraction is neglected because the kinetic energy of localization is much larger.

$$\Delta E \frac{\hbar^2}{2 \mu} \left( \frac{n \pi}{R} \right)^2$$ =

where $\mu$ is the reduced mass. This case is termed Strong Confinement.

2.

The heavy hole case. $a_h \ll R \ll a_e$. Here the Coulomb attraction is considered in a novel fashion. An expansion is made of the Hartree potential seen by the hole. One obtains an SHO type potential.

3.

$R \gg \{a_e , a_h\}$. Once again the Coulumb attraction is ignored

$$\Delta E \frac{\hbar^2}{2 M} \left(\frac{n \pi}{R} \right)^2 - E_{Ry}^*$$ =

where M is the total mass m_e + m_h. This case is termed Weak Confinement.

Very interestingly cognizance is taken of the presence of an ensemble of crystallites. Its size distribution is taken to be Lifshitz - Slezov and not Gaussian [G. C. John, Vijay A. Singh and V. Ranjan (Phys. Rev. B 50, 5329-5334 (1994)); Phys. Rev. B. ,58 , 1158-1161 (1998)] or Log normal [ Y. Kanemitsu (Phys. Rep.263 1 - 91 (1995) ].

The distinction between strong and weak confinement regimes was first mentioned here.

1999-01-04