## Quantum Dots

Quantum dots are often nanocrystals with all three spatial dimensions in the nanometre range. Sometimes, as is the case of the II-VI materials, such as CdSe or CdS, the nanocrys-tals can be grown from liquid phase solutions at well-specified temperatures. Conversely, they can also be prepared by lithographic etching techniques from macroscopic materials.

1 dimension

o si

Figure 4.7. Density of states function for a 1D electron system, as a function of energy.

Although the word "dot" implies an infinitely small size, in practice dots might have a large number of atoms: 104-106, and still have their three dimensions in the nanometre region, so that the electron de Broglie wavelength is comparable to the size of the dot. In this case, the wave nature of the electron becomes important. Quantum dots are often referred to as artificial atoms, because, as we will see later, the spectrum of the energy levels resembles that of an atom. In addition, at least in theory the energy spectrum can be engineered depending on the size and shape of the dot. In analogy to atoms, we can also define an ionization energy, which accounts for the energy necessary to add or remove an electron from the dot. This energy is also called the charging energy of the dot, in an image similar to the concept of capacitance of a body, in which the addition or subtraction of electric charge is specified by the Coulomb interaction. Therefore the atom-like properties of the quantum dots are often studied via the electrical characteristics. From this point of view, it is very important to remark that even the introduction or removal of one single electron in quantum dots, in contrast to the case of 2D or 1D systems, produces dominant changes in the electrical characteristics, mainly manifested in large conductance oscillations and in the Coulomb blockade effect (Chapter 6).

Let us now study the energy spectrum of quantum dots. The simplest case would be that of a confining potential that is zero inside a box of dimensions ax, ay, and az and infinite outside the box. Evidently the solution to this well-known problem are standing waves for the electron wave function and the energy levels are given by

Figure 4.7. Density of states function for a 1D electron system, as a function of energy.

Energy

Energy

0 dimensions!

Energy

Figure 4.8. Density of states function for a 0D electron system.

In contrast to the 2D and 1D cases, now the energy is completely quantized, and, as in the case of atoms, there is no free electron propagation. However, the levels are frequently degenerate, for instance, if two or three of the dimensions of the box are equal.

The case of a spherical dot in which the potential is zero inside the sphere and infinite outside can also be exactly solved, and the solutions are expressed in terms of the spherical functions. This problem resembles that of a spherically symmetric atom and the energy depends on two quantum numbers, the principal quantum number n, arising from the one-dimensional radial equation, and the angular momentum quantum number l.

Since in the case of quantum dots the electrons are totally confined, the energy spectrum is totally discrete and the DOS function is formed by a set of peaks in theory with no width and with infinite height (Figure 4.8). Evidently, in practice, the peaks should have a finite width, as a consequence, for instance, of the interaction of electrons with lattice phonons and impurities.

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