Introduction to Quantum Mechanics for Physicists and Engineers with AQME Nanotechnology has yielded a number of unique structures that are not found anywhere in nature. Most demonstrate an essential quality of Quantum Mechanics known as quantum confinement. The idea behind confinement is all about keeping electrons trapped in a small area. The sizes we’re talking about here for confinement have to be less than 30 nm for effective confinement. Quantum confinement comes in several flavors. 2-D confinement is only restricted in one dimension, and the result is a quantum well (or plane). These are what most lasers are currently built from. 1-D confinement occurs in nanowires. 0-D confinement is found only in the quantum dot.
One is probably wondering why confinement is so important. For one thing, it leads to new electronic properties that are not present in today’s semiconductor devices. Consider the quantum dot. 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. The only things in nature that have 0-D confinement are atoms. So a quantum dot can be loosely described as an ‘artificial atom’. This is vitally important because we can’t readily experiment on regular atoms. They’re too small and too difficult to isolate in an experiment. Quantum dots, on the other hand, 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 like Molecular Beam Epitaxy or Chemical Vapor Deposition. The bulk of modern lasers created with this method are highly functional, but 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 though their fabrication is still being worked out. Confinement is just one manifestation of quantum mechanics in nanodevices. Tunneling and quantum interference are the other two manifestations of quantum mechanics in the operation of, for example, scanning tunneling microscopes and resonant tunneling diodes, respectively.
Because of the importance of understanding quantum mechanics to understand 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 its application to nanodevice research. Within these courses one is first introduced to the concept of particle-wave duality (the photoelectric effect and the double-slit experiment), the solutions of the time-independent Schrodinger equation for open systems (piece-wise constant potentials), tunneling and bound states. The description of the solution of the Schrodinger 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 precalculated bandstructure by fitting the curvature of the bands. The Tsu-Esaki formula is then derived so that having calculated the transmission coefficient one 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 concept 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. Yet another issue that is discussed sometimes in a typical quantum mechanics class is the concept of Coulomb Blockade.
AQME assembles a set of nanoHUB tools that we believe are of immediate interest for the teaching of quantum mechanics class for both Engineers and Physicists. Users no longer have to search the nanoHUB to find the appropriate applications for this particular purpose. This curated page provides a “on-stop-shop” access to associated materials such as homework or project assignments. We invite you to participate in this open source, interactive educational initiative:
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Publicized early in the debate about whether light was composed of particles or waves, a wave-particle dual nature soon was found to be characteristic of electrons as well. The evidence for the description of light as waves was well established at the turn of the century when the photoelectric effect introduced firm evidence of a particle nature as well. On the other hand, the particle properties of electrons was well documented when the DeBroglie hypothesis and the subsequent experiments by Davisson and Germer established the wave nature of the electron.
Solution of the Time-Independent Schrodinger Equation
Piece-Wise Constant Potential Barrier Tool – Open Systems
The Piece-Wise Constant Potential Barrier Tool (PCPBT) allows calculation of the transmission and the reflection coefficient of arbitrary five, seven, nine, eleven and 2n-segment piece-wise constant potential energy profile.
Scanning Tunneling Microscope (STM)
For the case of multi-well structure it also calculates the quasi-bound states so it can be used as a simple demonstration tool for the formation of energy bands. Also, it can be used in the case of stationary perturbation theory exercises to test the validity of, for example, the first order and the second order correction to the ground state energy of the system due to small perturbations of, for example, the confining potential. The PCPBT tool can also be used to test the validity of the WKB approximation for triangular potential barriers.
Tunneling: Limiting Device Miniaturization
Bound States Lab
The Bound States Calculation Lab determines the bound states and the corresponding wavefunctions in a square, harmonic and triangular potential well. Maximum number of eigenstates that can be calculated is 100. Students clearly see the nature of the separation of the states in these three prototypical confining potentials with which we can approximate realistic quantum potentials that occur in nature.
Energy eigenstates of a harmonic oscillator (left panel). Probability density of the ground state that demonstrates purely quantum-mechanical behavior (middle panel). Probability density of the 20th subband where we start to see more classical behavior (right panel) as the well opens.
Energy Bands and Effective Masses =
Periodic Potential Lab
The Periodic Potential Lab solves the time independent Schroedinger Equation in a 1-D spatial potential variation. Rectangular, triangular, parabolic (harmonic), and Coulomb potential confinements can be considered. The user can determine energetic and spatial details of the potential profiles, compute the allowed and forbidden bands, plot the bands in a compact and an expanded zone, and compare the results against a simple effective mass parabolic band. Transmission is also calculated. This Lab also allows the students to become familiar with the reduced zone and expanded zone representation of the dispersion relation (E-k relation for carriers).
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes ranges of energy that an electron is “forbidden” or “allowed” to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice. The band structure of a material determines several characteristics, in particular its electronic and optical properties. The Bandstructure Lab tool enables the study of bulk dispersion relationships of Si, GaAs, InAs. Plotting the full dispersion relation of different materials, students first get familiar with a band-structure of direct band-gap (GaAs, InAs) and indirect band-gap semiconductors (Si). For the case of multiple conduction band valleys one has to determine first the Miller indices of one of the equivalent valleys and from that information it immediately follows how many equivalent conduction bands one has in Si and Ge, for example. In advanced applications, the users can apply tensile and compressive strain and observe the variation in the bandstructure, bandgaps, and effective masses. Advanced users can also study bandstructure effects in ultra-scaled (thin body) quantum wells, and nanowires of different cross sections. Bandstructure Lab uses the sp3s*d5 tight binding method to compute E(k) for bulk, planar, and nanowire semiconductors.
Bandstructure of Si (left panel) and GaAs (right panel).
Real World Applications
Schred calculates the envelope wavefunctions and the corresponding bound-state energies in a typical MOS (Metal-Oxide-Semiconductor) or SOS (Semiconductor-Oxide- Semiconductor) structure and a typical SOI structure by solving self-consistently the one-dimensional (1D) Poisson equation and the 1D Schrodinger equation. The Schred tool is specifically designed for Si/SiO2 interface and takes into account the mass anisotropy of the conduction bands as well as different crystallographic orientations.
This is schematically shown in the figure below.
Right panel – Potential diagram for inversion of p-type semiconductor. In this first notation Î•ij refers to the j-th subband from either the Î”2-band (i=1) or Î”4-band (i=2). Left panel – Constant-energy surfaces for the conduction-band of silicon showing six conduction-band valleys in the < 100> direction of momentum space. The band minima, corresponding to the centers of the ellipsoids, are 85% of the way to the Brillouin-zone boundaries. The long axis of an ellipsoid corresponds to the longitudinal effective mass of the electrons in silicon, , while the short axes correspond to the transverse effective mass, . For < 100> orientation of the surface, the Î”2-band has the longitudinal mass (ml) perpendicular to the semiconductor interface and the Î”4-band has the transverse mass (mt) perpendicular to the interface. Since larger mass leads to smaller kinetic term in the Schrodinger equation, the unprimed lader of subbands (as is usually called), corresponding to the Î”2-band, has the lowest ground state energy. The degeneracy of the unprimed ladder of subbands for < 100> orientation of the surface is 2. For the same reason, the ground state of the primed ladder of subbands corresponding to the Î”4-band is higher that the lowest subband of the unprimed ladder of subbands, The degeneracy of the primed ladder of subbands for (100) orientation of the interface is 4.