1.4 The Tunneling problem
The wave function of particles summarizes everything that can be known about a physical system. Therefore, problems in quantum mechanics center on the analysis of the wave function for a system. Using mathematical formulations of quantum mechanics, such as the Schrodiger equation, the wave function can be solved. This is directly related to the probability density of the particle’s position, which describes the probability that the particle is at any given place. In the limit of large barriers, the probability of tunneling decreases for taller and wider barriers.
For simple tunneling-barrier models, such as the rectangular barrier, an analytic solution exists. Problems in real life often do not have one, so “semi-classical†or “quasi- classical†methods have been developed to give approximate solutions to these problems, like the WKB approximation. Probabilities may be derived with arbitrary precision, constrained by computational resources, via Feynman’s path integral method; such precision is seldom required in engineering practice.
1.5 Applications of the concept of quantum tunneling
Tunneling occurs with barriers of thickness around 1-3 nm and smaller, but is the cause of some important macroscopic physical phenomena. For instance, tunneling is a source of current leakage in very-large scale integration (VLSI) electronics and results in the substantial power drain and heating effects that plague high-speed and mobile technology’ its is considered the lower limit on how small computer chips can be made.
1.5.1 Radioactive Decay
Radioactive decay is the process by which atomic nuclear release energy by emitting particles. This is done via the tunneling of a particle out of the nucleus (an electron tunneling into the nucleus in electron capture). This was the first application of quantum tunneling and led to the first approximations.
1.5.2 Cold Emission
Cold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionic emission, where electrons randomly jump from the surface of a metal to follow a voltage bias because they statistically end up with more than the barrier, through random collisions with other particles. When the electric field is very large, the barrier becomes thin enough for electrons to tunnel out of the atomic state, leading to a current that varies approximately exponentially with the electric field. These materials are important for flash memory and for some electron microscopes.
1.5.3 Tunnel Junction
A simple barrier can be created by separating two conductors with a very thin insulator. These are tunnel junctions, the study of which requires quantum tunneling. Josephson junction takes advantage of quantum tunneling and the superconductivity of some semiconductors to create the Josephson Effect. This has applications in precision measurements of voltages and magnetic fields, as well as the multi-junction solar cell.
1.5.4 Quantum Conductivity
While the Drude model of electrical conductivity makes excellent predictions about the nature of electrons conducting in metals, it can be furthered by using quantum tunneling to explain the nature of he electron’s collisions. When a free electron wave packet encounters a long array of uniformly spaced barriers the reflected eave packet interferes uniformly with the transmitted one between all a barriers so that there are cases of 100% transmission. The theory predicts that if positivity charged nuclear form a perfectly rectangle array, electrons will tunnel thorough the metal as free electrons, leading to an extremely high conductance, and that impurities in the metal will disrupt it significantly.
1.6 QUANTUM DOT
A quantum dot is a portion of matter (e.g. semiconductor) whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei Ekimov in a glass matrix and by Louis E. Brus in colloidal solutions. The term “quantum dot†was coined by Mark Reed.
Researchers have studied quantum dots in transistors, solar cells, LEDs and diode lasers. They have also investigated quantum dots as agents for medical imaging and hope to use them as qubits.
Stated simply, quantum dots are semiconductors whose electronic characteristics are closely related to the size and shape of the individual crystal. Generally, the smaller the size of the crystal, the larger the bandgap, the greater the difference in energy between the highest valence band and the lowest conduction band becomes, therefore more energy is needed to excite the dot, and concurrently, more energy is released when the crystal returns to its resting state.
For example, in fluorescent dye applications, this equates to higher frequencies of light emitted after excitation of the dot as the crystal size grows smaller, resulting in a color shift from red to blue in the light emitted. In addition to such tuning, a main advantage with quantum dots is that, because of the high level of control possible over the conductive properties of the material. Quantum dots of different sixes can be gradient multi-layer nanofilm.
1.7 Quantum Confinement in Semiconductors
3D confined electron wave functions in a Quantum Dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more “s- typeâ€. However, in a triangular dot the wave functions are mixed due to confinement symmetry.
In an unconfined (bulk) semiconductor, an electron-hole pair is typically bound within a characteristics length, called the exciton Bohr radius. This is estimated by replacing the positively charged atomic core with the hole in the Bohr formula. If the electron and hole are constrained further, the properties of the semiconductor change. For example the absorption and emission wavelengths of a light shift toward smaller wavelengths. This effect is a form of quantum confinement, and it is a key feature in many emerging electronic structures.
1.8 Production of Quantum Dots
There are several ways to confine excitons in semiconductors, resulting in different methods to produce quantum dots. In general, quantum wires, wells, and dots are grown by advanced epitaxial techniques in nanocrystals produced by chemical methods or by ion implantation, or in nanodevices made by state-of-the-art lithographic techniques.
1.8.1 Viral Assembly
Lee et al. (2002) reported using genetically engineered M13 bacteriophage viruses to create quantum dot bio-composite structures. As a background to this work, it has previously been shown that genetically engineered viruses can recognize specific semiconductor surfaces through the method of selection by combinatorial phage display. Additionally, it is known that liquid crystalline structures of wild-type viruses (Fd, M13, and TMV) are adjustable by controlling the solution concentrations, solution ionic strength, and the external magnetic field applied to the solutions. Consequently, the specific recognition prosperities of the virus can be used to organize inorganic nanocrystals, forming ordered arrays over the length scale defined by liquid crystal formation. Using this information, Lee et al. (2000) were able to create self-assembled, highly oriented, self supporting films from a phage and ZnS precursor solution. This system allowed them to vary both the length of bacteriophage and the type of inorganic material through genetic modification and selection.
