Quantum Effects

It is well known that besides the quantum size effect, other quantum effects could be very pronounced in limited systems, like thin films, chains, and particles, when at least one of the dimensions of the systems could be negligible compared to others. When the size of the nanoparticles as well as nanocapsules is reduced to be less than a certain critical value, the systems could be treated as zero (or quasi-zero) dimensional ones. In these cases, the quantum effects become dominant, which originate not only from the discreteness of the energy levels but also from the quantum interference as well as the quantum tunneling in the systems. The cyclic condition and the translation symmetry are lost, when the size of the systems is comparable to the optical wavelength, the de Broglie wavelength, the free path length of electrons, or the coherence length of a Cooper pair. Actually, the breaking down of the translation symmetry leads the wavevector of electrons to not be a "good" quantum number. In the limit case of large nanocapsules, one could treat the wavevector as a "pseudo-good" quantum number and deal with the problem approximately using bulk theory and perturbation theory. In the limiting case of small nanocapsuels, such a concept and the corresponding approximation are not valid anymore [64]. The limited systems could be treated as quantum wells, in which the electrons feel the energy/potential difference between different materials when they move. The energy of electrons moving in a quantum well is quantized according to quantum mechanics. If the electrons move in two or more quantum wells separated by spacers, the quantum well interference between the wells may be either neglected when the thickness of the spacers is thick enough or very pronounced when the wells are close enough. It is easy to find the solution of the quantization of the quantum well states in a single quantum well in any textbook on quantum mechanics [65-67]. Recently, the quantum interference in double quantum wells was studied systematically within the framework of the effective mass model [68]. It has been found that several types of quantum effects (e.g., macroscopic quantum tunnel effects, Coulomb blockade, and quantum tunnelling [69]) occur in the nanoparticles.

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