Go To Review Page   Go To Nano Page

A Computational Tour de Force 
S. Ogut, J. R. Chelikowsky, and S. G. Louie, Phys. Rev. Lett. 79, 1770 - 1773 (1997)

Title:``Quantum Confinement and Optical Gaps in Si Nanocrystals''

The quantum confinement (QC) dogma in its primitive form is that the ``band gap of a semiconductor increases with decreasing crystallite size''. What, may one ask, is the firmest theoretical evidence in its favour ? A possible answer is the present work. The authors:

1.

Undertake a first principles calculation employing the local density approximation (LDA).

2.

Redress the band gap underestimation pathology in LDA by implementing an appropriate variant of the GW approximation.

3.

Study the size dependence of the gap and related quantities by considering hydrogen-passivated ``spherical'' clusters of upto 800 (!) silicon atoms.

The authors are careful in distinguishing the various energies. They take pains to make their results understandable by delineating size-dependent scaling laws. Here is a sampler:
 
 

$$E_{g,bulk}^{qp} + \frac{c}{d^{\alpha}}\mbox{\hspace{1cm} $\alpha$\space = 1.2}$$ E_g^qp(d) =

$$E_{g,bulk}^{band} + \frac{c^{\prime}}{d^{\beta}}\mbox{\hspace{1cm} $\beta$\space = 1.1}$$ E_g^band(d) =

$$\Sigma(d) \Sigma_{bulk} + \frac{c^{\prime \prime}}{d^{\gamma}}\mbox{\hspace{1cm} $\gamma$\space = 1.5}$$ =

$$E_g^{band}(d) + \Sigma(d)$$ E_g^qp(d) =

E_g^opt = E_g^qp - E_coul (2)

Here d stands for the diameter, `g' for the gap, `qp' for quasi-particle, `opt' for optical, `coul' for Coulomb and $\Sigma$ for the self energy correction. The Coulomb energy is calculated and found to be smaller than the EMT value and with a softer exponent.
EMA: E_coul = 3.572 / $\epsilon$ d
This paper : $E_{coul} \propto 1/d^{0.7}$

In terms of completeness, computational demand and pedagogy this piece of work is outstanding. Somethings, such as geometry optimization are not attempted and this is understandable given the very large number of atoms. The disappointment is that the authors compare their work with a decade old experimental result and have not followed up with calculations for excited states.

1999-01-04