## AQME: Advancing Quantum Mechanics for Engineers

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### Discovery that is Possible through Quantum Mechanics

Nanotechnology has yielded a number of unique structures that are not found readily in nature. Most demonstrate an essential quality of Quantum Mechanics known as quantum confinement. Confinement is the idea of keeping electrons trapped in a small area, about 30 nm or smaller. Quantum confinement comes in several dimensions. 2-D confinement, for example, is restricted in only one dimension, resulting in a quantum well (or plane). Lasers are currently built from this dimension. 1-D confinement occurs in nanowires, and 0-D confinement is found only in the quantum dot.

The study of quantum confinement leads, foremost, to electronic properties not found in today’s semiconductor devices. The quantum dot works well as a first example. The typical quantum dot is anywhere between 3-60 nm in diameter. That’s still 30 to 600 times the size of a typical atom. A quantum dot exhibits 0-D confinement, meaning that electrons are confined in all three dimensions. In nature, only atoms have 0-D confinement; thus, a quantum dot can be described loosely as an ‘artificial atom.’ This knowledge is vitally important, as atoms are too small and too difficult to isolate in experiments. Conversely, quantum dots are large enough to be manipulated by magnetic fields and can even be moved around with an STM or AFM. We can deduce many important atomistic characteristics from a quantum dot that would otherwise be impossible to research in an atom.

Confinement also increases the efficiency of today’s electronics. The laser is based on a 2-D confinement layer that is usually created with some form of epitaxy such as Molecular Beam Epitaxy or Chemical Vapor Deposition. The bulk of modern lasers created with this method are highly functional, but these lasers are ultimately inefficient in terms of energy consumption and heat dissipation. Moving to 1-D confinement in wires or 0-D confinement in quantum dots allows for higher efficiencies and brighter lasers. Quantum dot lasers are currently the best lasers available, although their fabrication is still being worked out.

Confinement is just one manifestation of quantum mechanics in nanodevices. Tunneling and quantum interference are two other manifestations of quantum mechanics in the operation of scanning tunneling microscopes and resonant tunneling diodes, respectively. For more information on the theoretical aspects of Quantum Mechanics check the following resources:

Because understanding quantum mechanics is so foundational to an understanding of the operation of nanoscale devices, almost every Electrical Engineering department (in which there is a strong nanotechnology experimental or theoretical group) and all Physics departments teach the fundamental principles of quantum mechanics and their application to nanodevice research. Several conceptual sets and theories are taught within these courses. Normally, students are first introduced to the concept of particle-wave duality (the photoelectric effect and the double-slit experiment), the solutions of the time-independent Schrödinger equation for open systems (piece-wise constant potentials), tunneling, and bound states. The description of the solution of the Schrödinger equation for periodic potentials (Kronig-Penney model) naturally follows from the discussion of double well, triple well and n-well structures. This leads the students to the concept of energy bands and energy gaps, and the concept of the effective mass that can be extracted from the pre-calculated band structure by fitting the curvature of the bands. The Tsu-Esaki formula is then investigated so that, having calculated the transmission coefficient, students can calculate the tunneling current in resonant tunneling diode and Esaki diode. After establishing basic principles of quantum mechanics, the harmonic oscillator problem is then discussed in conjunction with understanding vibrations of a crystalline lattice, and the idea of phonons is introduced as well as the concept of creation and annihilation operators. The typical quantum mechanics class for undergraduate/first-year graduate students is then completed with the discussion of the stationary and time-dependent perturbation theory and the derivation of the Fermi Golden Rule, which is used as a starting point of a graduate level class in semiclassical transport. Coulomb Blockade is another discussion a typical quantum mechanics class will include.