Quantum Tunnelling: Probability, Alpha Decay and the STM

Passing Through Walls

Roll a ball towards a hill without enough energy to reach the top, and it will always roll back. Classical physics says it can never appear on the other side. But electrons, protons and alpha particles sometimes do exactly that: they pass straight through energy barriers they cannot climb. This is quantum tunnelling, and it is not a curiosity. It powers the Sun, explains alpha decay, makes flash memory and some diodes work, and lets scanning tunnelling microscopes image individual atoms. This tutorial explains the classical expectation, why waves can leak through barriers, how the tunnelling probability depends on the barrier's width and height and the particle's mass, and the main applications.

The Classical Picture

Suppose a particle with kinetic energy E approaches a region where its potential energy would be V, with V greater than E. Inside the barrier its kinetic energy would be E − V, which is negative. Since kinetic energy ½mv² can never be negative, classical physics says the particle cannot enter the barrier at all: it is reflected every time, however thin the barrier.

The Quantum Picture: Waves Leak Through

In quantum mechanics, a particle is described by a wavefunction ψ, and the probability of finding it at a place is proportional to |ψ|². The Schrödinger equation describes how ψ behaves.

Outside the barrier, where E > V, the wavefunction oscillates like a wave.
Inside the barrier, where E < V, the solution does not drop instantly to zero. Instead it decays exponentially, like e^(−κx), where κ = √[2m(V − E)] / ħ.
If the barrier is thin enough, the wavefunction has not quite died away when it reaches the far side. There it becomes an oscillating wave again, with a smaller amplitude.

A non-zero wavefunction beyond the barrier means a non-zero probability of finding the particle there: it has tunnelled through. The particle does not lose energy in the process; it emerges with the same energy E it started with.

There is an optical version of the same effect. Light totally internally reflected inside a glass block actually penetrates a tiny distance beyond the surface. If a second block is brought within a fraction of a wavelength, some light jumps the gap ("frustrated total internal reflection").

The Quantum Tunnelling Probability

For a barrier of width L that is not too thin, the probability that a particle tunnels through (the transmission coefficient) is approximately

T ≈ e^(−2κL), with κ = √[2m(V − E)] / ħ.

Because L and √m appear in an exponent, the probability is extremely sensitive to:

the width L of the barrier: wider barriers reduce T dramatically;
the height of the barrier above the particle's energy, V − E;
the mass m of the particle: heavier particles tunnel far less.

Worked Examples

Example 1. An electron with 5 eV of energy meets a barrier 10 eV high, so V − E = 5 eV = 8.0 × 10⁻¹⁹ J. Find κ.

κ = √(2 × 9.11 × 10⁻³¹ × 8.0 × 10⁻¹⁹) / 1.055 × 10⁻³⁴ ≈ 1.14 × 10¹⁰ m⁻¹.

Example 2. Find the tunnelling probability for barrier widths of 0.1 nm, 0.5 nm and 1.0 nm.

L = 0.1 nm: 2κL = 2.29, T ≈ e^(−2.29) ≈ 0.10 (about 1 in 10).
L = 0.5 nm: 2κL = 11.4, T ≈ 1.1 × 10⁻⁵ (about 1 in 100,000).
L = 1.0 nm: 2κL = 22.9, T ≈ 1.1 × 10⁻¹⁰.

Doubling the width from 0.5 nm to 1 nm doesn't halve the probability; it reduces it about 100,000 times. Every extra 0.1 nm reduces it about tenfold.

Example 3. A proton with the same energy meets the same 0.1 nm barrier. Its mass is about 1,836 times the electron's, so κ is √1836 ≈ 43 times larger, and T ≈ e^(−98) ≈ 3 × 10⁻⁴³. Effectively it never tunnels. This is why tunnelling is important mainly for electrons, and for nuclear particles only over the very short distances inside nuclei.

Alpha Decay

Inside a heavy nucleus such as uranium-238, an alpha particle (two protons and two neutrons) is held by the strong nuclear force. Just outside, the electrostatic repulsion from the rest of the nucleus forms a high "Coulomb barrier", around 30 MeV high for uranium. Yet alpha particles escape with only about 4 to 9 MeV of energy, far less than the barrier height.

In 1928 George Gamow explained this: the alpha particle tunnels through the barrier. It strikes the barrier about 10²¹ times per second, and each time it has a tiny chance of getting through. Because the tunnelling probability depends exponentially on the energy, a small difference in alpha energy makes an enormous difference to the half-life:

uranium-238 emits alpha particles of about 4.3 MeV and has a half-life of 4.5 billion years;
polonium-212 emits alpha particles of about 8.8 MeV and has a half-life of about 0.3 microseconds.

Doubling the energy changes the half-life by a factor of around 10²³. This relationship (the Geiger–Nuttall law) was one of the first great successes of quantum mechanics.

Nuclear Fusion in the Sun

In the Sun's core, hydrogen nuclei (protons) must get very close together to fuse, but they repel each other strongly. At the core temperature of about 15 million K, the typical proton energy is only around 1 keV, far below the Coulomb barrier of roughly 1 MeV. Classically, almost no fusion would happen and the Sun would not shine. Tunnelling allows a small fraction of collisions to succeed, which, with the huge number of protons, is enough to power the Sun for about 10 billion years.

Technology That Uses Quantum Tunnelling

Scanning tunnelling microscope (STM): a very sharp metal tip is held about a nanometre above a surface with a small voltage applied. Electrons tunnel across the gap, giving a tiny current. Because the current changes about tenfold for every 0.1 nm change in gap, scanning the tip and keeping the current constant maps the surface with enough precision to see individual atoms. Its inventors, Binnig and Rohrer, won the 1986 Nobel Prize.
Flash memory: USB drives, memory cards and phone storage store each bit as charge trapped on an insulated "floating gate" transistor. Writing and erasing data uses a high voltage to make electrons tunnel through a thin insulating oxide layer.
Tunnel diodes exploit tunnelling across a very thin, heavily doped p–n junction to switch extremely quickly.
A limit on chip size: as transistors shrink, their insulating layers become only a few atoms thick, and electrons tunnel through them, causing leakage currents. This is one of the main challenges in making ever-smaller computer chips.

Summary

Classically, a particle with E less than V can never cross a barrier; quantum mechanically it can, because its wavefunction decays but does not vanish inside the barrier.
Tunnelling probability T ≈ e^(−2κL), with κ = √[2m(V − E)]/ħ.
T falls extremely rapidly with barrier width, barrier height and particle mass.
Tunnelling explains alpha decay and the huge range of alpha half-lives, and allows fusion in the Sun.
STMs, flash memory and tunnel diodes depend on it, and it limits how small transistors can be.