Quantum Tunneling Examples

Why This Matters

Quantum tunneling is the phenomenon that makes stars shine, enables atomic-scale imaging, and powers cutting-edge quantum technologies. Particles can pass through energy barriers they classically shouldn't be able to cross. This principle connects directly to wave-particle duality, the uncertainty principle, and probability amplitudes.

Don't just memorize that "tunneling happens in alpha decay" or "STMs use tunneling." You need to understand why tunneling occurs in each case: what's the barrier, what's tunneling through it, and what physical consequence results. Exams will ask you to apply tunneling concepts to unfamiliar scenarios, so focus on the underlying mechanism: the exponential decay of the wavefunction inside a classically forbidden region allows a nonzero probability of transmission.

---

Overcoming Nuclear Barriers

These examples demonstrate tunneling through the Coulomb barrier, the electrostatic repulsion between positively charged nuclei. Without tunneling, nuclear reactions would require impossibly high energies or temperatures.

Alpha Decay in Radioactive Nuclei

An alpha particle inside a nucleus is bound by the strong nuclear force, which creates a deep potential well. Beyond the range of the strong force, the Coulomb repulsion between the alpha particle and the remaining nucleus creates a barrier. Classically, the alpha particle doesn't have enough energy to get over this barrier, but its wavefunction doesn't abruptly stop at the barrier edge. Instead, it decays exponentially through the barrier and emerges with nonzero amplitude on the other side.

• Barrier width determines decay rate. The Geiger-Nuttall law relates half-life to alpha particle energy through tunneling probability. Higher-energy alphas face a thinner effective barrier, so they tunnel more readily and the isotope decays faster.
• Transforms the parent nucleus. Emission reduces atomic number by 2 and mass number by 4, producing a different element.

Nuclear Fusion in Stars

In stellar cores, protons need to get close enough for the strong force to bind them, but Coulomb repulsion pushes them apart. At the Sun's core temperature (~15 million K), the average proton kinetic energy is roughly 1 keV, yet the Coulomb barrier between two protons is around 1 MeV. That's a factor of ~1000 too low for classical barrier crossing. Tunneling bridges this gap.

• Powers stellar nucleosynthesis. The pp-chain and CNO cycle both depend critically on tunneling for their initial steps.
• The Gamow peak determines reaction rates. Fusion probability is a balance between two competing effects: higher proton energy increases tunneling probability, but the Maxwell-Boltzmann distribution means fewer protons have that energy. The product of these two factors peaks at the Gamow energy, which sets the effective reaction rate.

---

Electron Tunneling in Materials

These applications exploit electrons tunneling through thin potential barriers in solids or across vacuum gaps. The tunneling current depends exponentially on barrier width, making these systems extraordinarily sensitive.

Scanning Tunneling Microscope (STM)

A sharp conducting tip is brought within about 1 nm of a surface. A bias voltage is applied, and electrons tunnel across the vacuum gap between tip and surface. Because tunneling current varies exponentially with distance, even a 0.1 nm change in gap width produces a measurable current change. This gives the STM sub-angstrom vertical resolution.

• Probes electronic density of states. The tunneling spectrum reveals information about surface electronic structure, not just geometry.
• Enables atomic manipulation. By adjusting tip voltage and position, individual atoms can be moved on a surface.

Field Emission

When a strong external electric field (on the order of $10^9$ V/m) is applied to a metal surface, it deforms the potential barrier at the surface, making it narrower and more triangular. Electrons that couldn't classically escape now tunnel through this thinned barrier.

• The Fowler-Nordheim equation governs the current: $J \propto E^2 \exp(-B/E)$, where $E$ is the applied field strength and $B$ is a material-dependent constant related to the work function.
• Enables bright, coherent electron sources. Field emission tips are used in electron microscopes and flat-panel displays because they produce well-collimated beams.

Cold Emission

Cold emission is closely related to field emission but emphasizes that the process works at low temperatures. Unlike thermionic emission, where electrons gain enough thermal energy to jump over the barrier, cold emission relies entirely on field-assisted tunneling.

• Sharp tips enhance local fields. Nanoscale tip geometry concentrates the electric field, making cold emission practical at moderate applied voltages.
• Critical for vacuum electronics. Provides electron sources for devices requiring minimal thermal noise or rapid switching.

---

Tunneling in Semiconductor and Superconductor Devices

These technologies harness tunneling across engineered thin barriers to achieve unique electronic properties. Device performance depends on precise control of barrier thickness at the nanometer scale.

Tunnel Diodes

A tunnel diode has an extremely thin depletion region (~10 nm), achieved by heavily doping both the p and n sides. At low forward bias, electrons tunnel directly from the conduction band on one side to available states on the other. As voltage increases further, the bands shift out of alignment and tunneling probability drops, causing the current to decrease even as voltage increases.

• Negative differential resistance is the signature feature. This makes tunnel diodes useful for oscillators and high-speed switching.
• Ultrafast operation. Because tunneling is nearly instantaneous (no transit-time delay like in conventional diodes), tunnel diodes can operate at frequencies exceeding 100 GHz.

Josephson Junctions

A Josephson junction consists of two superconductors separated by a very thin insulating barrier (typically 1-2 nm). In a superconductor, electrons form Cooper pairs that share a macroscopic quantum phase. These paired electrons can tunnel coherently through the barrier, preserving their phase relationship.

• DC Josephson effect: A supercurrent flows with zero applied voltage, driven purely by the phase difference across the junction.
• AC Josephson effect: When a DC voltage $V$ is applied, the junction produces an oscillating current at frequency $f = 2eV/h$. This precise frequency-voltage relationship is used to define voltage standards.

---

Tunneling in Molecular and Biological Systems

Tunneling influences chemistry and biology at the molecular scale, where barrier widths are naturally on the order of atomic dimensions.

Ammonia Inversion

In $\text{NH}_3$, the nitrogen atom sits above the plane of the three hydrogen atoms in a pyramidal shape. There's an equivalent configuration with nitrogen below the plane. A potential energy barrier (the planar configuration) separates these two states. The nitrogen atom tunnels through this barrier, causing the molecule to oscillate between the two pyramidal forms at ~24 GHz.

• Inversion splitting is observable spectroscopically. The tunneling rate creates a characteristic doublet in the microwave spectrum.
• First molecular maser. Ammonia's inversion transition was used in the first maser (1954), demonstrating stimulated emission at microwave frequencies.

Quantum Biological Processes

• Exciton tunneling enhances photosynthesis. Energy transfers between chlorophyll molecules in photosynthetic complexes faster than classical hopping would allow, contributing to >95% energy transfer efficiency.
• Enzyme catalysis may involve tunneling. Proton and hydrogen tunneling could explain anomalously fast reaction rates in certain enzymes, where kinetic isotope effects are larger than classical models predict.

---

Tunneling at Cosmological Scales

Even gravity and spacetime curvature connect to quantum tunneling effects, though these remain theoretical frontiers.

Hawking Radiation from Black Holes

Near a black hole's event horizon, quantum vacuum fluctuations constantly produce virtual particle-antiparticle pairs. Normally these pairs annihilate almost immediately, but at the horizon, one particle can tunnel outward while its partner falls inward. The escaping particle carries away energy, and the black hole loses mass as a result.

• Black holes have a temperature. The Hawking temperature is $T_H = \frac{\hbar c^3}{8\pi G M k_B}$, which decreases with mass. Smaller black holes are hotter and radiate faster.
• Implies black hole evaporation. Over cosmological timescales, black holes lose mass and eventually disappear, raising the information paradox: what happens to information about matter that fell in?

---

Quick Reference Table

---

Self-Check Questions

1. Both alpha decay and nuclear fusion involve tunneling through Coulomb barriers. What determines whether tunneling occurs into or out of a nucleus, and how does this affect the energy requirements?

2. The STM and tunnel diode both rely on electron tunneling. Compare the role of barrier width in each device: how is it controlled, and why does exponential sensitivity matter?

3. Explain why nuclear fusion can occur in stellar cores at temperatures far below what classical physics predicts. What would happen to stellar lifetimes if tunneling didn't exist?

4. Ammonia inversion and photosynthetic energy transfer both demonstrate tunneling at room temperature. What features of these systems make quantum effects observable despite thermal noise?

5. FRQ-style: A student claims that Josephson junctions and tunnel diodes operate on the same principle. Evaluate this claim by identifying one key similarity and two key differences in their tunneling mechanisms.