Quantum Tunneling Explained

How Tunneling Works

In classical physics, a ball rolling toward a hill will stop and roll back if it does not have enough energy to reach the top. The hill acts as an energy barrier, and the ball is classically forbidden from appearing on the other side. In quantum mechanics, the situation is fundamentally different. A quantum particle approaching an energy barrier does not simply stop at the barrier edge. Its wave function extends into and through the barrier, decaying exponentially but never reaching exactly zero. If the barrier is thin enough or low enough, there is a nonzero probability that the particle will be found on the other side.

The tunneling probability depends exponentially on two factors: the height of the barrier above the particle energy and the width of the barrier. A particle with energy close to the barrier height has a much higher tunneling probability than one with energy far below it. A thinner barrier allows much more tunneling than a thicker one. For typical atomic-scale barriers, tunneling probabilities can range from nearly 100% to astronomically small, depending on the specific conditions.

The key insight is that the wave function, not the particle itself, is the fundamental object. The wave function obeys the Schrodinger equation everywhere, including inside the barrier. Inside a classically forbidden region, the wave function does not oscillate as it does in allowed regions; instead, it decays exponentially. But as long as the barrier has finite width, the wave function has a nonzero value on the far side, meaning there is a probability of finding the particle there. The particle does not have to go over the barrier; it goes through it.

Tunneling in Nuclear Physics

Quantum tunneling explains radioactive alpha decay, one of the first applications of quantum mechanics to nuclear physics. An alpha particle (two protons and two neutrons) inside a nucleus is trapped by the nuclear force, which creates a potential well. Surrounding this well is a Coulomb barrier created by the electrostatic repulsion between the positively charged alpha particle and the remaining nucleus. Classically, the alpha particle does not have enough energy to escape over this barrier. But quantum mechanically, it can tunnel through.

George Gamow explained alpha decay using quantum tunneling in 1928, just two years after the Schrodinger equation was formulated. His calculation showed that the probability of tunneling depends exponentially on the barrier parameters, which explains why different radioactive isotopes have vastly different half-lives, ranging from fractions of a second to billions of years. The tunneling model quantitatively predicts these half-lives, matching experimental data remarkably well.

Tunneling is also essential for nuclear fusion in stars. At the temperatures inside the sun (about 15 million degrees Celsius), protons have thermal energies far below the Coulomb barrier that repels them from each other. Classically, they should almost never get close enough to fuse. But quantum tunneling allows protons to penetrate the Coulomb barrier with a small but sufficient probability. This is what makes nuclear fusion possible in stellar cores and is the reason the sun shines. Without tunneling, stars could not exist.

Tunneling in Technology

The scanning tunneling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer in 1981 (earning them the 1986 Nobel Prize), exploits tunneling to image surfaces at the atomic level. A sharp metallic tip is brought within a few angstroms of a surface. At this distance, electrons can tunnel between the tip and the surface. The tunneling current depends exponentially on the tip-surface distance, making it exquisitely sensitive to the surface topography. By scanning the tip across the surface and measuring the current, the STM produces images showing individual atoms.

Flash memory in USB drives, solid-state drives, and smartphones uses tunneling to store data. Electrons are pushed through a thin oxide barrier by applying a voltage, trapping them on a floating gate. The trapped charges change the electrical properties of the transistor, representing a stored bit. To erase the data, a reverse voltage is applied, allowing the electrons to tunnel back out. The ability to tunnel through thin barriers on demand is what makes flash memory programmable and erasable.

Tunnel diodes, invented by Leo Esaki in 1957 (earning him the 1973 Nobel Prize), use tunneling to achieve extremely fast switching speeds. In a heavily doped p-n junction, the depletion region is thin enough for electrons to tunnel through. This produces a region of negative differential resistance, where increasing the voltage actually decreases the current, making tunnel diodes useful for high-frequency oscillators and fast digital circuits.

Tunneling in Chemistry and Biology

Proton tunneling plays an important role in many chemical reactions, particularly those involving hydrogen transfer. Because the proton is relatively light, its de Broglie wavelength is large enough for tunneling to be significant even at room temperature. Enzyme-catalyzed reactions in biology often involve proton tunneling, where the enzyme structure positions the proton close enough to the reaction barrier for tunneling to occur efficiently. This has been observed experimentally using kinetic isotope effects: replacing hydrogen with the heavier deuterium reduces the tunneling rate, slowing down the reaction more than classical theory would predict.

DNA mutations may also involve proton tunneling. The hydrogen bonds between base pairs in DNA involve protons that can potentially tunnel between two positions, changing the tautomeric form of the base and potentially leading to mispairing during replication. This is a topic of ongoing research, but it suggests that quantum mechanics may play a role in genetic mutation and evolution at the most fundamental level.

Macroscopic Quantum Tunneling

While tunneling is primarily a nanoscale phenomenon, it has been observed in macroscopic systems under extreme conditions. Superconducting quantum interference devices (SQUIDs) demonstrate macroscopic quantum tunneling of the phase of the superconducting order parameter. Josephson junctions, which consist of two superconductors separated by a thin insulating barrier, exhibit tunneling of Cooper pairs (paired electrons), producing the Josephson effect that is used in the most sensitive magnetometers ever built.

These macroscopic tunneling phenomena are not just curiosities. They are the basis of superconducting quantum computers, where qubits are made from Josephson junctions. The ability of the superconducting phase to tunnel between two states is precisely what creates the two-level quantum system needed for a qubit.

Measuring and Controlling Tunneling

The tunneling rate can be calculated from the Schrodinger equation using several methods. For simple rectangular barriers, an exact analytical formula exists. For more realistic barrier shapes, the WKB (Wentzel-Kramers-Brillouin) approximation provides accurate results. This semiclassical method works by integrating the imaginary momentum of the particle through the classically forbidden region, giving the exponential decay factor that determines the tunneling probability.

In modern nanotechnology, engineers routinely design structures where tunneling is either exploited or suppressed. Quantum dot devices use thin barriers to allow controlled tunneling of individual electrons. Tunnel junctions in superconducting circuits are fabricated with precise oxide thicknesses to achieve desired tunneling rates. As transistors in computer chips shrink toward atomic scales, unwanted tunneling of electrons through the gate oxide becomes a major design challenge, setting fundamental limits on how small classical transistors can be made.

Resonant tunneling occurs when the barrier contains a quantum well, a region where the particle energy matches a bound state of the well. At these resonant energies, the tunneling probability jumps dramatically, sometimes reaching nearly 100% even through barriers that would otherwise be almost impenetrable. Resonant tunneling diodes exploit this effect for high-speed electronics, and the phenomenon is analogous to the resonance conditions in optical Fabry-Perot interferometers.