Go To Review Page          Go To Nano Page

Pioneering Effort II: 
L. E. Brus, J. Chem. Phys., 79, 5566 - 5571, (1983)

The title of this pioneering effort is ``A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites''. The author at the outset admits that he will analyze ``an elementary quantum mechanical model'' for crystallites. Specifically the author employs (a) the effective mass approximation (EMA), (b) an electrostatic potential for dielectric polarization, and (c) penetration of carrier outside the crystallite in cases of small effective mass. The Hamiltonian for a the neutral system is

$$- \frac{\hbar^2}{2 m_e^*} \nabla^2 + V(r)$$ H =

V(r) = 0 for r<R
V(r) = V_o for r>R
V_o = EA - Polarization term (see paragraph preceding section C on page 5569)

The issue of polarization is not often brought up in current works. However the case of small effective mass first invoked by the author is (and ought to be) of relevance in any study. Ben Daniel and Duke method is used for this case. Recently, Singh et al. [M. Singh, V. Ranjan, and Vijay A. Singh, Intnl. J. Mod. Phys. B 14, 1753 - 1765] have carried out an asymptotic analysis to elucidate Brus' suggestion that small effective mass may lead to large surface charge densities and to enhanced tunneling.

The author admits that S/V is large, but chooses to ignore the following surface related effects:

1.

sep -0.2cm

2.

Surface shape may change with size.

3.

Rearrangement of atoms on the surface (surface relaxation).

4.

Dangling bonds on the surface.

5.

Foreign atoms (Hydrogen, Oxygen, etc.) on the surface.

The author suggests a heuristic scale to judge if the crystallite is small or large. The carrier is described by a range of wavevectors and its de Broglie wavelength$\lambda$ is given by

$$\lambda \frac{h}{\sqrt{3 m^* k T}}$$ = (1)

$$\approx 100~~ \AA$$

for m^* = m the bare electron mass , T = 300 K. For sizes less than$\lambda$ the system maybe considered as small.

1999-01-04