Page 22 - Nanotechnology an introduction

Si 1.12 11.9 0.98 0.16 2.5
ZnO 3.35 9.0 0.27 – 1.7
a At 300 K.
b
Of the electron; seeequation (2.12); see also[172].
An even simpler criterion for the optical nanoscale would be simply “invisibility”. This could be determined by the Abbe resolution limit (equation
5.2), or the Rayleigh ratio R(θ) for light of wavelength λ scattered by particles of radius r:

(2.13)

for unpolarized light, where I is the incident irradiance, I is the total intensity of scattered radiation observed at an angle θ and a distance d from
0
s
θ
the volume of scattering V, and the factor f accounts for polarization phenomena. The particles would be deemed to be in the nanoscale if a
suspension of them showed no perceptible turbidity.
In order to establish quantitatively how the band-edge luminescent emission varies with particle diameter, consider a one-electron, one-
dimensional time-dependent Schrödinger equation with energy eigenvalues E

(2.14)

with a potential V(x) = 0 for 0 ≤ x ≤ 2r (within the particle) and V(x) = ∞ for x < 0 and x > 2r. Using a trial wavefunction for the electron,
within the particle we can solve to find . Ψ has to vanish at the particle boundaries, which can happen if the phase ax is an integer
multiple of π, ax = nπ, whence

(2.15)

we can call n the principle quantum number. Note that we have not specified exactly what value of m we should use in equation (2.14). It is an open
e
question whether the effective mass of the electron is the same in a very small particle, already known to be subject to lattice contraction and
possibly other distortions of the crystal structure (Section 2.2). The experimental verification of these formulas is subjected to a number of
difficulties. Firstly it is quite difficult to prepare monodisperse quantum dots. The Stranski–Krastanov mechanism has been made use of for the
preparation of monodisperse GaInAl dots for lasers via semiconductor processing technology (see Section 8.1.2), but most quantum dots are
presently made by a chemical method, by reacting two soluble precursors together in solution (Section 6.1.2), with which it is difficult to get particle
size distributions with a coefficient of variation much better than 10%. The surface atoms may have different electronic properties from the bulk by
virtue of their lower coordination number. This notion was developed by Berry [16] and [17] as the main reason for the difference between bulk and
nanoparticle optical properties. Particles in contact with another condensed phase, such as a suspending liquid, are likely to have additional
electron levels caused by adsorbed impurities; it is extremely unlikely that the particle boundaries correspond to the electron potential jumping
abruptly from zero to infinity.

Jellium
Quantum dots are dielectric nanoparticles, typically I–VII, II–VI or III–V semiconductors. Metal clusters can be modeled as jellium [96]. Each metal
atom contributes a characteristic number ν of electrons to the free electron gas (a corollary of which is that the actual positions of the metal nuclei
can be neglected) filling a uniform charged sphere corresponding to the actual cluster containing N metal atoms. The electronic structure is derived
by solving the Schrödinger equation for an electron constrained to move within the cluster sphere under the influence of an attractive mean field
potential due to the partially ionized metal atoms. The solution yields energy levels organized in shells and subshells with quantum numbers n and l
10
6
6
2
2
14
respectively as with atoms. The order of the energy levels in jellium is a little different, however: 1s , 1p , 1d , 2s , 1f , 2p , etc.—hence even the
first valence shell contains s, p, d and f orbitals. The degeneracy of each level is 2(2l + 1).
A triumph of the jellium model was the correct prediction of “magic” numbers N —corresponding to clusters of exceptional stability. These are
m
clusters with completely filled shells, and the N are the values of N fulfilling the condition
m
(2.16)
where q is the charge of the cluster. Hence (for example) uncharged (q = 0)Cs (ν = 1) clusters will have N = 2, 8, 18, 20, 34, 40, 58, …;
m
will be stable.
Superatoms
The electron affinity of Al is comparable to that of a chlorine atom and KAl is an ionically bound molecule analogous to KCl. Replacing one Al
13
13
atom with C (ν = 4) results in an inert species comparable to argon. An extended periodic table can be constructed from superatoms, defined as
clusters that are energetically and chemically stable [34]. These entities are useful as nanoblocks for assembly into desirable structures (Section
8.3.2).
2.6. Magnetic and Ferroelectric Properties

Ferromagnetism
In certain elements (e.g., Fe, Co, Ni, Gd) and alloys, exchange interactions between the electrons of neighboring ions lead to a very large coupling
between their spins such that, above a certain temperature, the spins spontaneously align with each other. The proliferation of routes for
synthesizing nanoparticles of ferromagnetic substances has led to the discovery that when the particles are below a certain size, typically a few tens
of nanometers, the substance still has a large magnetic susceptibility in the presence of an external field but lacks the remanent magnetism