CHAPTER ONE
RESONANT TUNNELING THROUGH QUANTUM DOT ARRAYS
1.1    Introduction
Resonant tunneling through Quantum dot arrays is the quantum-mechanical effect of transition through a classically-forbidden energy state.
Consider rolling a ball up a hill. If the ball is not given enough velocity, then it will not roll over the hill. This makes sense classically. But in quantum mechanics, objects exhibit wavelike behavior. For a quantum particle moving against a potential hill, the wave function describing the particle can extend to the other side of the hill. This wave represents the probability of finding the particle in a certain location, meaning that the particle has the possibility of being detected on the other side of the hill. This behavior is called tunneling; it is as if the particle has ‘dug’ through the potential hill.
As a consequence of the wave-particle of matter, tunneling is often explained using the Heisenberg uncertainty principle. Purely quantum mechanical concept is the focus of the phenomenon, so quantum tunneling is one of the defining features of quantum mechanics and the particle-wave duality of matter.
In 1923, during the early stage of quantum theory, de Broglie introduced a fundamental hypothesis the matter has dualistic nature. Quantum tunneling is one of the defining features of quantum mechanics and the particle-wave duality of matter.
1.2    HISTORY
Quantum tunneling was developed from the study of radioactivity, which was discovered in 1896 by Henri Becquerel. It was examined further by Marie and Pierre Curie, for which the earned a Noble Prize in Physics. Ernest Rutherford and Egon Schweidler studied its nature, which was later verified empirically by Friedrich Kohlraush. The ides of the impossibility of predicting decay were created from their work.
Friedrich Hund was the first to make use of quantum mechanical barrier penetration (i.e quantum tunneling) in 1927 when he was calculating the ground state of a double-well potential. It first application was a mathematical explanation for alpha decay, which was done in 1982 by George Gamow and independent Gurney and Edward condon. The two researchers simultaneously solved the Schrodinger equation for a model nuclear potential and derived a relationship between the half-life of the particle and the energy of emission that depend directly on the mathematical probability of tunneling.
After attending a seminar by Gamow, Max Bron recognized the generality of tunneling. He realized that it was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems. Shortly result of quantum mechanics that applies to many different systems. Shortly thereafter, both groups considered the cases of particles tunneling into the nucleus. Shortly thereafter, both groups considered the cases of particles tunneling into the nucleus. The study of semiconductors and the development of transistors and diode led to the acceptance of electron tunneling in solids in 1957.
Eventually, five Noble prizes in physics were awarded for research involving tunneling in semiconductors and superconductors and for the invention of scanning tunneling microscopy. Tunneling occurs in all quantum systems. It is crucial for nucleon synthesis in stars, and it may
also have played an essential role in the evolution of the early universe. The word of Ivar Giaever and Brian David Josephson led to the BSC theory of superconductivity and prediction of super current. From its beginning, recounted here, quantum tunneling has remained a hot topic, with myriad applications to this day.
1.3    INTRODUCTION TO THE CONCEPT OF TUNNELING THROUGH
QUANTUM DOT ARRAYS
Quantum tunneling is in the domain of quantum mechanics, the study of what happens at the quantum scale. This process cannot be directly perceived, so much of its understand is shape by the macroscopic world, which classical mechanics can adequately explain. Particles in that ream are understood to travel between potential barriers as a ball rolls over a hill: if the ball does not have enough energy to surmount the hills, it comes back down; the two forms of mechanics differ in their treatment of this scenario. Classical mechanics predicts those particles that do not have enough energy to classical surmount a barrier will not be able to reach that other side. In quantum mechanics, these particles can, with a very small probability, tunnel to the other side, thus crossing the barrier.
The reason for this difference comes from the treatment of matter in quantum mechanics as having properties of waves and particles. One interpretation of this duality involves the Heisenberg uncertainty principle, which defines a limit on how precisely the position and the momentum of a particle can be known at the same time. This implies that there are no solutions with a probability of exactly zero (or one), though said solution may approach infinity. Hence, the probability of a given particles existence on the opposite side of an intervening barrier is non-zero, and such particles will appear- with no indication of physically transiting the barrier- on the other a semantically difficult word in this instance side with a frequency proportional to this probability.
An electron wave-packet directed at a potential barrier. Note the dim spot on right that represents tunneling electrons.