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A Pioneering Indian Effort 
Selvakumar V. Nair, Sucharita Sinha, and K. C. Rustagi, Phys. Rev. B, 35, 4098 - 4101, 15 March 1987-I

The paper is titled: ``Quantum size effects in spherical semiconductor microcrystals''. The electron-hole Hamiltonian is similar to earlier works. The trial wavefunction chosen is a three parameter Hyllerass-type one:

$$\psi (r_1,r_2) \left\{ \begin{array}{ll}N \left[ \frac{\sin (\pi r_1 / R)}{r_1 ...... & {\mbox{for $r_1, r_2 \leq R$ }} \\0 & {\mbox{outside}}\end{array} \right.$$ =

Here R is the radius of the spherical quantum dot (QD). The three variational parameters are $\alpha_1$,$\alpha_2$ and $\beta$ while N is the normalization. The systems under study were CdS and CuCl.

The authors consider the case of weak confinement (e.g. size R > a_B, the bulk exciton radius). One would expect, from the work of Efros & Efros (1982), the energy upshift $\Delta E$ to go as

$$\Delta E \frac{\hbar^2 \pi^2}{2 M R^2}$$ =

where

M = m₁ + m₂

The authors try to extract M from the slope of $\Delta E$ vs. 1 / R² but are unsuccessful. This failure led them to suggest that the QD must be modeled by a finite and not infinite potential depth. The authors also plot the oscillator strength as a function of a_B^* / R using the Henry and Nassau formula. This reveals a sharp rise as R is reduced. Over the years Prof. Rustagi's group has produced a series of papers employing both effective mass theory (EMT) and tight-binding (TB) methods.

1999-01-04