Tunneling Applications: First Encounters

Tunneling appears in many parts of physics, but the same word does not always mean the same detailed model. Sometimes it means transmission through an open barrier. Sometimes it means splitting between two bound configurations. Sometimes it means phase-coherent many-body transport.

This page gives first-encounter models for common applications without pretending that the one-dimensional rectangular barrier is the full theory of any of them.

Application Map

Application	First model	Reusable tunneling idea	Important missing physics
Alpha decay	Barrier penetration	Exponentially small escape probability through a forbidden region	Nuclear structure, alpha preformation, Coulomb plus centrifugal barrier
Scanning tunneling microscopy	Vacuum barrier	Current changes roughly as $e^{-2\kappa s}$ with tip-sample separation $s$	Local density of states, tip geometry, bias window, surface electronic structure
Josephson effect	Weak link between superconductors	Coherent tunneling depends on a phase difference	Cooper pairing, broken $U(1)$ phase, electromagnetic environment
Ammonia inversion	Double-well splitting	Symmetric and antisymmetric bound states split by tunneling	Molecular rotations, vibrations, electric fields, spectroscopy
Tunnel diode	Band-to-band tunneling	Filled states can tunnel into empty states across a narrow junction	Semiconductor band structure, doping, Fermi levels, transport

The table is a guide to what the wave-mechanics model contributes and where it stops.

Shared Exponential Lesson

For a one-dimensional barrier with $0\lt E\lt V(x)$ over a forbidden interval, WKB gives the leading estimate

$$T \sim \exp\left[ -2\int_{x_1}^{x_2} \kappa(x)\,dx \right],$$

where

$$\kappa(x) = \frac{\sqrt{2m(V(x)-E)}}{\hbar}.$$

The rectangular-barrier result is the constant-$\kappa$ special case:

$$T\propto e^{-2\kappa a}.$$

This exponential dependence is the portable lesson. Prefactors, dimensionality, available final states, many-body coherence, and environmental coupling are application-specific.

Alpha Decay

In alpha decay, an alpha particle is initially confined by nuclear forces but has an energy below the effective Coulomb barrier outside the nucleus. Classically it cannot escape. Quantum mechanically, the alpha wavefunction penetrates the forbidden region and can emerge outside.

The first model is WKB barrier penetration. The decay rate has the schematic form

$$\Gamma \sim f\,P_{\mathrm{escape}},$$

where $f$ is an attempt frequency scale and

$$P_{\mathrm{escape}} \sim \exp\left[ -2\int_{r_1}^{r_2} \frac{\sqrt{2\mu(V_{\mathrm{eff}}(r)-E)}}{\hbar}\,dr \right].$$

Here $\mu$ is the alpha–daughter reduced mass. The effective barrier includes the Coulomb repulsion between the alpha particle and the daughter nucleus. For nonzero angular momentum it also includes a centrifugal term.

The main qualitative result is the extreme sensitivity of lifetime to the barrier integral. Small changes in alpha energy can produce large changes in half-life. This is the tunneling origin of the Geiger–Nuttall pattern.

What the first model omits:

• the probability that an alpha cluster is already formed inside the nucleus;
• detailed nuclear many-body structure;
• angular-momentum and spin selection effects;
• corrections to the simple WKB prefactor.

Use Barrier Penetration and Tunneling for the reusable WKB method and Gamow Factor for the evaluated Coulomb action, finite-radius correction, and Geiger–Nuttall connection.

Scanning Tunneling Microscopy

Scanning tunneling microscopy uses electron tunneling between a conducting tip and a conducting sample separated by a small vacuum gap. A crude first model treats the gap as a rectangular barrier of width $s$ and height set by an effective work function $\Phi$.

For an electron in the vacuum gap,

$$\kappa \approx \frac{\sqrt{2m\Phi}}{\hbar}.$$

The tunneling current has the leading separation dependence

$$I(s)\propto e^{-2\kappa s}.$$

This exponential dependence is why STM can be sensitive to sub-angstrom changes in tip height. It is also why feedback control is essential: small mechanical changes can produce large current changes.

What the first model omits:

• the sample’s local density of electronic states;
• the tip wavefunction and tip shape;
• the applied bias voltage and energy window;
• surface states, finite temperature, and material-specific band structure.

The safe first statement is not “STM directly photographs atoms.” It measures a tunneling current controlled by electronic states and tip-sample geometry; under suitable conditions that current maps atomic-scale structure.

Scanning Tunneling Microscopy and Spectroscopy continues from this barrier estimate to Bardeen and Tersoff–Hamann theory, feedback and setpoint effects, local spectroscopy, quasiparticle interference, gap imaging, and atomic manipulation.

Josephson Effect

A Josephson junction consists of two superconductors separated by a weak link, often a thin insulating barrier. Single-particle barrier tunneling is not enough to explain the Josephson effect. The essential new ingredient is a coherent superconducting phase on each side.

Let

$$\phi=\phi_L-\phi_R$$

be the phase difference between the two superconductors. The DC Josephson relation is

$$I_s=I_c\sin\phi,$$

where $I_c$ is the critical current. With a voltage $V$ across the junction, the phase evolves as

$$\hbar\frac{d\phi}{dt}=2eV.$$

The Josephson energy scale is often written

$$E_J=\frac{\hbar I_c}{2e}.$$

The connection to elementary tunneling is that a weak link permits coherent coupling between states on the two sides. The major difference is that the tunneling object is the superconducting condensate phase, not merely a single particle described by a one-dimensional wavefunction.

What the first model omits:

• BCS pairing and the superconducting gap;
• quasiparticle tunneling;
• electromagnetic circuit dynamics;
• capacitance, dissipation, noise, and device geometry.

Josephson physics is therefore an application of tunneling, but not just a rectangular-barrier exercise.

Josephson Effect is the detailed canonical home for gauge-invariant phase, microscopic current scales, AC locking, magnetic interference, RCSJ dynamics, SQUIDs, and circuit applications.

Ammonia Inversion

Ammonia has two equivalent pyramidal configurations: the nitrogen atom can lie on either side of the plane formed by the three hydrogens. A one-dimensional inversion coordinate gives a double-well potential.

If $\lvert L\rangle$ and $\lvert R\rangle$ denote states localized in the two wells, tunneling mixes them into approximate symmetric and antisymmetric states:

$$\lvert +\rangle \approx \frac{1}{\sqrt2} \left( \lvert L\rangle+\lvert R\rangle \right), \qquad \lvert -\rangle \approx \frac{1}{\sqrt2} \left( \lvert L\rangle-\lvert R\rangle \right).$$

The splitting

$$\Delta E=E_- - E_+$$

sets an inversion frequency

$$\nu=\frac{\Delta E}{h}.$$

This is bound-state tunneling, not scattering transmission. The molecule is not flying through a barrier and appearing on the other side; its energy eigenstates are delocalized combinations of two classically distinct configurations.

What the first model omits:

• rotation-vibration coupling;
• electric dipole selection rules;
• isotopic changes such as replacing hydrogen by deuterium;
• detailed molecular potential-energy surfaces.

Use Double-Well Potential and Double Delta Potential for the reusable two-well mechanics.

Tunnel Diode

A tunnel diode is a heavily doped p-n junction in which the depletion region is narrow enough for electrons to tunnel between bands. The first wave-mechanics image is barrier transmission across a thin forbidden region. The semiconductor reality is richer because the initial and final states live in bands with Fermi occupation factors.

At certain biases, occupied states on one side align with empty states on the other side, producing strong tunneling current. As the bias changes, that overlap can decrease, so the current can fall even while voltage rises. This produces negative differential resistance.

The first model contributes:

• exponential sensitivity to barrier width;
• the need for available final states;
• the idea that transmission can occur without thermal activation over the barrier.

What the first model omits:

• band bending and self-consistent electrostatics;
• density of states in the valence and conduction bands;
• Fermi factors and temperature;
• scattering, disorder, and device contacts.

The tunnel diode is therefore a tunneling device, but its current-voltage curve is a band-structure and transport result, not just the rectangular-barrier formula.

What Carries Across Applications

The transferable ideas are:

• forbidden-region amplitudes decay rather than vanish;
• finite barriers allow coupling between regions;
• tunneling probabilities often depend exponentially on action, width, mass, or barrier height;
• phases matter when coherent paths or condensates interfere;
• current requires both transmission probability and available states.

The non-transferable shortcut is to use the same one-dimensional formula everywhere. The rectangular barrier is a laboratory for intuition; real applications decide which coordinate tunnels, what counts as the barrier, which states are available, and whether coherence survives.

Common Mistakes

• Saying tunneling means energy is temporarily not conserved.
• Treating every tunneling application as a single-particle rectangular barrier.
• Ignoring final-state availability in electronic devices.
• Confusing Josephson pair tunneling with ordinary single-electron tunneling.
• Treating ammonia inversion as scattering through an open barrier.
• Claiming STM measures only topography rather than electronic structure and geometry.
• Forgetting that alpha-decay rates also depend on nuclear structure and not only the Coulomb barrier.

Where This Is Used

• Quantum Tunneling gives the common conceptual mechanism.
• Rectangular Barrier Tunneling gives the exact constant-barrier model.
• Barrier Penetration and Tunneling gives the smooth-barrier WKB estimate used in alpha decay and field emission.
• Gamow Factor specializes the WKB action to a repulsive Coulomb tail.
• Double-Well Potential gives the bound-state tunneling structure behind molecular inversion.
• Resonant Transmission gives the first model for energy-selective tunneling through compound structures.
• Wave Packets and Scattering explains how stationary transmission coefficients become physical packet probabilities.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, 2005.
• M. Tinkham, Introduction to Superconductivity, 2nd ed., Dover, 2004.
• C. J. Chen, Introduction to Scanning Tunneling Microscopy, 2nd ed., Oxford University Press, 2008.
• S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2006.

Exercises

1. In the STM first model, the current scales as $I(s)\propto e^{-2\kappa s}$. By what factor does the current change if the tip-sample separation increases by $\Delta s$?

Solution

The ratio is

$$\frac{I(s+\Delta s)}{I(s)} = \frac{e^{-2\kappa(s+\Delta s)}}{e^{-2\kappa s}} = e^{-2\kappa\Delta s}.$$

Even a small positive $\Delta s$ can strongly suppress the current when $\kappa\Delta s$ is not small.

2. Why does ammonia inversion belong to bound-state tunneling rather than scattering tunneling?

Solution

In ammonia inversion, the relevant states are molecular bound states in a double-well potential. The symmetric and antisymmetric energy eigenstates are superpositions of configurations localized on the two sides of the inversion coordinate. The splitting $\Delta E$ controls oscillation or spectral frequency.

There is no incoming beam, outgoing transmitted flux, or asymptotic scattering region. Therefore the natural model is double-well level splitting, not a reflection/transmission coefficient.

3. State one reason why Josephson tunneling cannot be reduced to single-particle rectangular-barrier tunneling.

Solution

The Josephson effect depends on the coherent phase difference between two superconducting condensates. The current relation

$$I_s=I_c\sin\phi$$

involves a many-body superconducting phase and Cooper-pair charge $2e$. A single-particle rectangular-barrier model can suggest weak-link coupling, but it does not contain BCS pairing, the condensate phase, or circuit dynamics.

4. Why can a tunnel diode current decrease as voltage increases?

Solution

The current depends not only on barrier transmission but also on the overlap between occupied initial states and empty final states in the bands. As the bias changes, that overlap can decrease even while the voltage increases. The result can be negative differential resistance, where $dI/dV\lt 0$ over a bias range.