11.7 Quantum tunneling

Fundamentals of Quantum Tunneling

Quantum tunneling is the process by which a particle passes through an energy barrier it classically shouldn't be able to cross. In classical physics, a ball rolling toward a hill without enough energy simply bounces back. In quantum mechanics, there's always a small but real probability the particle ends up on the other side. This phenomenon is central to processes ranging from radioactive decay to the operation of flash memory in your phone.

Wave-Particle Duality

Tunneling only makes sense once you accept that particles also behave as waves. A particle like an electron isn't just a tiny dot; it's described by a wavefunction that spreads out in space. The double-slit experiment demonstrates this: even single electrons fired one at a time produce an interference pattern, confirming their wave nature.

Because the particle has wave-like properties, its wavefunction doesn't abruptly stop at a barrier. Instead, it penetrates into the barrier region and can emerge on the other side. That's the core reason tunneling happens.

Probability in Quantum Mechanics

Classical mechanics tells you exactly where a particle will be at any time. Quantum mechanics replaces that certainty with probabilities.

• The wavefunction $\Psi(x,t)$ encodes everything about a particle's quantum state
• The probability density $|\Psi(x)|^2$ gives the likelihood of finding the particle at position $x$
• Born's rule states that you square the wavefunction's amplitude to get this probability

Because the wavefunction can be nonzero inside and beyond a barrier, there's a nonzero probability of finding the particle on the far side.

Potential Energy Barriers

A potential energy barrier is a region where the potential energy $V(x)$ exceeds the particle's total energy $E$. Classically, the particle is forbidden from entering this region.

In quantum mechanics, the wavefunction doesn't vanish inside the barrier. Instead, it decays exponentially. If the barrier is thin enough or low enough, the wavefunction still has measurable amplitude on the far side. That surviving amplitude corresponds to a real probability of transmission.

Tunneling Effect Explanation

Classical vs. Quantum Behavior

Think of it this way: a classical tennis ball thrown at a concrete wall always bounces back. But an electron encountering a thin energy barrier has a chance of appearing on the other side, even though it never had enough energy to climb over the top.

• Classical prediction: 100% reflection if $E < V_0$
• Quantum prediction: mostly reflection, but a small transmission probability that depends on the barrier's height, width, and the particle's energy

This is where classical physics breaks down. At atomic scales, the wave nature of matter dominates, and deterministic predictions give way to probabilities.

Wavefunction Penetration

When a particle's wavefunction reaches a barrier, it doesn't just stop. Inside the barrier, the wavefunction takes the form of a decaying exponential rather than an oscillating wave. The amplitude drops off as $e^{-\kappa x}$, where $\kappa = \sqrt{2m(V_0 - E)/\hbar^2}$.

If the barrier is narrow enough, the wavefunction hasn't fully decayed by the time it reaches the other side. Beyond the barrier, it resumes oscillating (though with reduced amplitude), meaning the particle has a real chance of being found there.

Transmission Coefficient

The transmission coefficient $T$ quantifies the tunneling probability. It's defined as the ratio of the transmitted probability current to the incident probability current.

Key dependencies:

• $T$ decreases exponentially as barrier width $L$ increases
• $T$ decreases as the difference $(V_0 - E)$ grows (taller barrier relative to particle energy)
• $T$ increases as particle energy $E$ approaches the barrier height $V_0$
• Lighter particles tunnel more readily than heavier ones ($T$ depends on mass $m$)

Mathematical Framework

Schrödinger Equation Application

The time-dependent Schrödinger equation governs how the wavefunction evolves:

$i\hbar\frac{\partial}{\partial t}\Psi(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi(x,t) + V(x)\Psi(x,t)$

For steady-state tunneling problems, you use the time-independent form. To solve a tunneling problem with a rectangular barrier:

1. Divide space into three regions: before the barrier (Region I), inside the barrier (Region II), and after the barrier (Region III)
2. Write the general wavefunction solution in each region. In Regions I and III, the solutions are oscillating plane waves. In Region II (where $E < V_0$), the solution is a combination of growing and decaying exponentials.
3. Apply boundary conditions: the wavefunction and its first derivative must be continuous at both edges of the barrier
4. Solve for the coefficients to find the ratio of transmitted to incident amplitude, which gives you $T$

Barrier Penetration Probability

For a rectangular barrier of width $L$ and height $V_0$, with a particle of mass $m$ and energy $E < V_0$, the approximate transmission coefficient is:

$T \approx e^{-2L\sqrt{2m(V_0 - E)/\hbar^2}}$

Reading this equation:

• The exponent is negative, so $T$ is always less than 1
• Wider barriers (larger $L$) make $T$ exponentially smaller
• A bigger gap between $V_0$ and $E$ also suppresses tunneling
• Heavier particles (larger $m$) tunnel far less readily, which is why tunneling matters for electrons and protons but is negligible for baseballs

As $E \to V_0$, the exponent approaches zero and $T$ approaches 1.

WKB Approximation

Real barriers aren't always neat rectangles. The Wentzel-Kramers-Brillouin (WKB) approximation handles barriers with arbitrary shapes, as long as the potential varies slowly compared to the particle's wavelength.

The WKB transmission probability is:

$T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x) - E)}\, dx\right)$

Here, $x_1$ and $x_2$ are the classical turning points (where $V(x) = E$). The integral sums up the "difficulty" of tunneling across the entire barrier profile. This is the go-to method for problems like alpha decay, where the Coulomb barrier has a curved shape.

Experimental Observations

Alpha Decay

Alpha decay was one of the first phenomena explained by quantum tunneling. Inside a nucleus, an alpha particle (two protons and two neutrons, essentially a $^4\text{He}$ nucleus) is trapped by the strong nuclear force. But outside the nucleus, the Coulomb repulsion creates a potential barrier.

Classically, the alpha particle doesn't have enough energy to escape. Quantum mechanically, it tunnels through the Coulomb barrier with some small probability. The Geiger-Nuttall law captures the result: nuclei that emit higher-energy alpha particles have dramatically shorter half-lives, because even a small increase in energy greatly increases the tunneling probability.

Scanning Tunneling Microscope (STM)

The STM is one of the most direct applications of tunneling. A sharp conducting tip is brought within about 1 nm of a surface. A voltage is applied, and electrons tunnel across the vacuum gap between the tip and the surface.

The tunneling current depends exponentially on the tip-sample distance. Even a 0.1 nm change in distance produces a measurable change in current. This extreme sensitivity allows the STM to map surfaces with atomic resolution and even move individual atoms.

Field Emission

When a strong electric field is applied to a metal surface, electrons can tunnel through the work function barrier (the energy needed to escape the surface). This is field emission, and it occurs without heating the metal.

• Described quantitatively by the Fowler-Nordheim equation, which relates emission current to the applied electric field
• Used in electron microscopes and certain types of flat-panel displays
• The stronger the field, the thinner the effective barrier, and the greater the tunneling current

Applications in Technology

Tunnel Diodes

Tunnel diodes are semiconductor devices with heavily doped p-n junctions so thin that electrons tunnel directly across the junction. Their distinctive feature is negative differential resistance: over a certain voltage range, increasing the voltage actually decreases the current.

This property makes tunnel diodes useful for:

• High-frequency oscillator circuits
• Fast switching applications (switching times on the order of picoseconds)
• Microwave signal amplification

Flash Memory Devices

Every USB drive and solid-state drive relies on quantum tunneling. Flash memory stores data by trapping electrons on an electrically isolated floating gate.

• Writing data: A high voltage drives electrons through a thin oxide layer onto the floating gate via Fowler-Nordheim tunneling. The trapped charge shifts the transistor's threshold voltage, encoding a "0" or "1."
• Erasing data: A reverse voltage pulls the electrons back off the floating gate, again through tunneling.
• The oxide layer must be thin enough for tunneling to occur but thick enough to retain charge for years without a power source.

Quantum Computing Implications

Tunneling shows up in quantum computing in both helpful and harmful ways:

• Josephson junctions, which are the basis of superconducting qubits, rely on Cooper pairs tunneling across a thin insulating barrier
• Quantum annealing algorithms deliberately exploit tunneling to escape local energy minima when solving optimization problems
• Unwanted tunneling events can cause decoherence, disrupting the fragile quantum states that qubits depend on

Tunneling in Nature

Nuclear Fusion in Stars

The Sun's core temperature is about 15 million K. That sounds extreme, but it's actually not hot enough for protons to overcome their mutual Coulomb repulsion classically. Quantum tunneling bridges the gap, allowing protons to fuse at temperatures far below what classical physics would require.

Without tunneling, stellar fusion wouldn't occur, stars wouldn't shine, and heavier elements wouldn't be synthesized. Tunneling probability is a key input in models of stellar evolution and nucleosynthesis.

Quantum Biology Examples

Tunneling may play a role in several biological processes, though some of these are still active areas of research:

• Photosynthesis: Electrons tunnel between molecular complexes during energy transfer, contributing to the remarkably high efficiency of photosynthetic systems
• Enzyme catalysis: Proton tunneling can speed up certain biochemical reactions beyond what classical transition-state theory predicts
• DNA mutations: Proton tunneling between base pairs could occasionally shift hydrogen bonds, potentially causing point mutations during replication

Hawking Radiation

Stephen Hawking proposed that black holes aren't perfectly black. Near the event horizon, quantum fluctuations constantly produce virtual particle-antiparticle pairs. Occasionally, one particle tunnels outward while the other falls in. The escaping particle carries energy away from the black hole, causing it to slowly lose mass over time.

This process connects quantum mechanics with general relativity and remains theoretical (the predicted radiation is far too faint to detect from astrophysical black holes), but it's a profound result that suggests black holes have a finite lifetime.

Limitations and Considerations

Energy Dependence

Tunneling probability is highly sensitive to the particle's energy relative to the barrier height. As $E$ approaches $V_0$, tunneling probability rises sharply. In multi-barrier systems, resonant tunneling can occur: at specific energies, the transmission probability spikes to nearly 100%, even though each individual barrier would normally block most particles. This energy-filtering effect is exploited in resonant tunneling diodes.

Barrier Width Effects

The exponential dependence on barrier width means tunneling is only significant over very short distances. For electrons tunneling through a typical oxide barrier, a few nanometers is the practical limit. Double the barrier width, and the tunneling probability can drop by many orders of magnitude.

This sharp distance dependence is what gives the STM its extraordinary resolution, and it's also why flash memory oxide layers must be precisely controlled during manufacturing.

Quantum Tunneling Time

How long does tunneling actually take? This is a surprisingly controversial question. Several definitions have been proposed:

• Büttiker-Landauer time: based on the interaction time with the barrier
• Larmor time: measured using spin precession in a magnetic field within the barrier
• Dwell time: the average time a particle spends in the barrier region

Some experiments suggest that tunneling can appear to occur faster than the time light would take to cross the same distance. This doesn't violate relativity (no usable information travels faster than light), but it highlights how counterintuitive tunneling remains even at the measurement level.

Advanced Concepts

Resonant Tunneling

When two or more barriers are placed in series with a quantum well between them, something striking happens. At certain discrete energies, the transmission probability jumps to nearly 1. The particle's wavefunction forms a standing wave in the well that constructively interferes, creating a resonance.

• Resonant tunneling diodes (RTDs) exploit this for extremely fast switching
• Quantum cascade lasers use sequences of quantum wells and barriers to produce mid-infrared and terahertz radiation

Tunneling in Multiple Dimensions

Real systems aren't one-dimensional. In 2D and 3D, a particle can tunnel along different paths through a potential landscape, and these paths can interfere with each other. This matters for:

• Interpreting STM images of surfaces, where the tunneling current depends on the local electronic structure in all three dimensions
• Understanding molecular-scale tunneling in nanostructures and at interfaces

Tunneling of Composite Particles

Atoms and molecules aren't point particles. They have internal structure (vibrations, rotations, electronic states) that can couple to the tunneling process. A molecule tunneling through a barrier may need to rearrange its internal configuration along the way, which changes the effective tunneling probability.

This is especially relevant in chemistry, where proton transfer in hydrogen bonds often proceeds via tunneling. It also matters in molecular spectroscopy, where tunneling splittings reveal information about the shape of potential energy surfaces.