نانوشیمی و نانوتکنولوژی وکاربرد

Nanoscale Materials as Intermediate between Atomic and Bulk Matter

Nanoscale materials frequently show behavior which is intermediate between that

of a macroscopic solid and that of an atomic or molecular system. Consider, for

instance, the case of an inorganic crystal composed of few atoms. Its properties

will be different from those of a single atom, but we cannot imagine that they will

be the same as those of a bulk solid. The number of atoms on the crystal’s surface,

for instance, is a significant fraction of the total number of atoms, and therefore

will have a large influence on the overall properties of the crystal. We can easily

imagine that this crystal might have a higher chemical reactivity than the corre-

sponding bulk solid and that it will probably melt at lower temperatures. Consider

now the example of a carbon nanotube, which can be thought of as a sheet of

graphite wrapped in such a way that the carbon atoms on one edge of the sheet are

covalently bound to the atoms on the opposite edge of the sheet. Unlike its indi-

vidual components, a carbon nanotube is chemically extremely stable because the

valences of all its carbon atoms are saturated. Moreover, we would guess that

carbon nanotubes can be good conductors because electrons can freely move along

these tiny, wire-like structures. Once again, we see that such nanoscopic objects

can have properties which do not belong to the realm of their larger (bulk) or

smaller (atoms) counterparts. However, there are many additional properties spe-

cific to such systems which cannot easily be grasped by simple reasoning. These

properties are related to the sometimes counterintuitive behavior that charge car-

riers (electrons and holes) can exhibit when they are forced to dwell in such struc-

tures. These properties can only be explained by the laws of quantum mechanics

Quantum Mechanics

A fundamental aspect of quantum mechanics is the particle-wave duality, intro-

duced by De Broglie, according to which any particle can be associated with a

matter wave whose wavelength is inversely proportional to the particle’s linear

momentum. Whenever the size of a physical system becomes comparable to the

wavelength of the particles that interact with such a system, the behavior of the

particles is best described by the rules of quantum mechanics [7]. All the infor-

mation we need about the particle is obtained by solving its Schro¨dinger equation.

The solutions of this equation represent the possible physical states in which the

system can be found. Fortunately, quantum mechanics is not required to describe

the movement of objects in the macroscopic world. The wavelength associated with

a macroscopic object is in fact much smaller than the object’s size, and therefore

the trajectory of such an object can be excellently derived using the principles of

classical mechanics. Things change, for instance, in the case of electrons orbiting

around a nucleus, since their associated wavelength is of the same order of mag-

nitude as the electron-nucleus distance.

We can use the concept of particle-wave duality to give a simple explanation of

the behavior of carriers in a semiconductor nanocrystal. In a bulk inorganic semi-

conductor, conduction band electrons (and valence band holes) are free to move

throughout the crystal, and their motion can be described satisfactorily by a linear

combination of plane waves whose wavelength is generally of the order of nano-

meters. This means that, whenever the size of a semiconductor solid becomes

comparable to these wavelengths, a free carrier confined in this structure will

behave as a particle in a potential box [8]. The solutions of the Schro¨dinger equa-

tion in such case are standing waves confined in the potential well, and the en-

ergies associated with two distinct wavefunctions are, in general, different and

discontinuous. This means that the particle energies cannot take on any arbitrary

value, and the system exhibits a discrete energy level spectrum. Transitions be-

tween any two levels are seen as discrete peaks in the optical spectra, for instance.

The system is then also referred to as ‘‘quantum confined’’. If all the dimensions of

a semiconductor crystal shrink down to a few nanometers, the resulting system is

called a ‘‘quantum dot’’ and will be the subject of our discussion throughout this

chapter. The main point here is that in order to rationalize (or predict) the physical

properties of nanoscale materials, such as their electrical and thermal conductivity

or their absorption and emission spectra, we need first to determine their energy

level structure.

For quantum-confined systems such as quantum dots, the calculation of the

energy structure is traditionally carried out using two alternative approaches. One

approach has just been outlined above. We take a bulk solid and we study the evo-

lution of its band structure as its dimensions shrink down to a few nanometers.

This method will be described in more detail later (Section 2.4). Alternatively, we

can start from the individual electronic states of single isolated atoms as shown in

Section 2.3 and then study how the energy levels evolve as atoms come closer and

start interacting with each other.