This paper examines quantum tunneling, a quantum-mechanical phenomenon in which a particle crosses an energy barrier it classically could not overcome. Beginning with the theoretical foundations — including wave functions, the Schrödinger equation, and potential energy barriers — the paper explains how tunneling differs from classical particle behavior. It then surveys key applications, such as alpha decay, field emission, flash memory, and scanning tunneling microscopes. The paper devotes particular attention to Leo Esaki's discovery of the tunnel diode at Sony in 1957, detailing its operating principles, negative resistance region, and potential uses in high-frequency electronics, space applications, and computing.
Quantum tunneling is a quantum-mechanical phenomenon in which a particle moves against a potential energy barrier and appears on the other side of that barrier. More precisely, the wave function describing the particle extends to the other side of the barrier. Because wave functions are the means by which the location of a particle is determined, it is assumed that the particle has effectively crossed to the other side. This behavior is called tunneling because there is no other adequate description: the particle has not penetrated the barrier, nor does it possess enough energy to break through it.
This phenomenon can occur only in the nanoscopic realm. A person, for example, could not tunnel through the chair upon which they are sitting — at least not without an exorbitant amount of time involved. Observing the phenomenon requires a nanoscopic microscope.
In classical mechanics, negative energies are not permitted within the barrier of a nucleus, and a particle could never enter a region where its total energy is less than zero. Consider a marble sitting in a spherical depression: if the marble has less total energy than is required to overcome the potential energy at the rim of the depression, it will remain trapped indefinitely, unable to escape without external assistance.
In the quantum case, however, a particle has some exponentially declining probability of entering a region where the total energy would be negative in the classical sense. Particles do not normally appear within their negative-energy region, but they do have a finite probability of appearing on the other side of a potential energy barrier — a feat that would be impossible under classical physics.
Solutions to the time-independent Schrödinger equation behave differently depending on the sign of the total energy (E − V). Where total energy is positive, solutions are harmonic, as is the case inside an infinite square well. Where total energy is negative, the value of k becomes complex and the solution takes the form of a negative exponential in x.
Schrödinger equation concepts state that a particle with energy (E) less than the height (U₀) of a barrier cannot penetrate the classically forbidden region inside the barrier. However, for penetration to occur, the wave function associated with a free particle at the barrier must be continuous and must show an exponential decay inside the barrier. If the wave function is also continuous on the far side of the barrier, there is a finite probability that the particle will tunnel through.
As a particle approaches the barrier, a free-particle wave function describes it. When it reaches the barrier, it must satisfy the Schrödinger equation. In a real or imaginary application, part of this function would be appropriate depending on the boundary conditions and the potential that has been set. In general, physicists speak of free particles operating within some kind of boundary, subject to conditions defined by the surrounding potential.
The phenomenon of a particle appearing on the other side of a potential energy barrier has important practical applications, particularly in semiconductor electronics, where tunneling devices control currents.
By 1928, George Gamow had solved the theory of alpha decay via tunneling. It takes enormous energy for a particle confined to the nucleus to escape the strong nuclear potential, and an even greater amount to split the nucleus. Quantum mechanics showed that a particle can tunnel through the potential barrier and escape. Gamow created a model potential and derived the relationship between the half-life of a particle and the energy of the emission required to escape. Concurrently, Ronald Gurney and Edward Condon independently solved the problem of alpha decay via tunneling, and both groups recognized that particles could also tunnel into the nucleus.
Max Born recognized the generality of quantum-mechanical tunneling after attending a seminar by Gamow. He realized that tunneling was a general result of quantum mechanics, not restricted to nuclear physics, and could be applied to many different systems. Today, tunneling theory is applied even to early cosmology and the origin of the universe.
Other important applications include the cold emission of electrons and the physics of semiconductors and superconductors. Field emission phenomena, explainable by quantum tunneling, are critical to flash memory. Tunneling is also a source of significant current leakage in very-large-scale integration (VLSI) electronics and accounts for substantial power drain and heating effects in high-speed and mobile technology.
A particularly striking application is the scanning tunneling microscope, which can image objects too small to resolve with conventional optical microscopes. These instruments overcome the limitations of conventional microscopy — such as optical aberrations and wavelength constraints — by scanning the surface of an object with tunneling electrons.
The tunnel diode, also known as the Esaki diode, is an extremely fast-operating diode capable of functioning into the microwave region (GHz). Leo Esaki discovered the underlying effect and was awarded the Nobel Prize in Physics in 1973 for this work. The diode was developed as a byproduct of efficiency improvements at Sony Laboratories, where engineers were upgrading the 2T5 transistor used in Sony radios. They developed a smaller diode, designated the 2T7, and set up factories for mass production. However, the new, smaller diode performed less efficiently than the older model, and experiments were conducted to determine why — theoretically, it should have been more efficient.
"Esaki's accidental discovery at Sony in 1957"
"P-n junction, negative resistance, and Zener effect"
"Future uses, radiation resistance, and computing potential"
Always verify citation format against your institution’s current style guide requirements.