3.5 Tunneling

Quantum Tunneling Fundamentals

Quantum tunneling is the process by which particles pass through potential energy barriers that classical physics says they cannot cross. In condensed matter physics, tunneling governs how electrons move through thin insulating layers, semiconductor junctions, and nanoscale devices. Understanding tunneling is essential for explaining everything from diode behavior to atomic-resolution microscopy.

This topic covers the physics behind tunneling probability, its role in quantum wells and semiconductors, key device applications (STM, Josephson junctions, single-electron transistors), and the theoretical tools used to analyze tunneling quantitatively.

Potential barrier concept

A potential barrier is a region where the potential energy exceeds the kinetic energy of an incoming particle. Classically, a particle with energy $E$ less than the barrier height $V_0$ simply bounces back. Quantum mechanically, the particle's wave-like nature gives it a finite probability of appearing on the other side.

• The shape of the barrier matters: rectangular, triangular, and parabolic barriers each produce different tunneling probabilities.
• Rectangular barriers are the textbook starting point, but real devices often have triangular barriers (as in field emission) or more complex profiles.

Tunneling probability

The transmission coefficient $T$ quantifies how likely a particle is to tunnel through a barrier. For a simple rectangular barrier of height $V_0$ and width $d$, with a particle of mass $m$ and energy $E < V_0$:

$T \approx e^{-2\kappa d}$

where the decay constant is:

$\kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}$

Key dependencies to remember:

• $T$ drops exponentially with barrier width $d$. Even a small increase in $d$ dramatically reduces tunneling.
• Heavier particles (larger $m$) tunnel far less effectively than lighter ones. This is why electron tunneling is common but proton tunneling is rare in solids.
• A taller barrier (larger $V_0 - E$) also suppresses tunneling.
• In electrical systems, an applied bias voltage effectively reshapes the barrier, changing the tunneling current.

Wavefunction decay in barriers

Inside the classically forbidden region, the wavefunction doesn't oscillate. Instead, it decays exponentially:

$\psi(x) \propto e^{-\kappa x}$

This decaying solution is called an evanescent wave. The wavefunction and its first derivative must be continuous at both barrier boundaries, which is how you solve for the transmission and reflection coefficients. If the barrier is thin enough that the wavefunction hasn't decayed to negligible amplitude by the far side, the particle has a real probability of emerging. That "leaking through" is tunneling.

Tunneling in Quantum Wells

Quantum wells confine particles along one dimension, producing discrete energy levels. They're the building blocks of semiconductor heterostructures and many optoelectronic devices.

Finite vs. infinite wells

An infinite quantum well has perfectly rigid walls: the wavefunction is strictly zero outside the well, and there are infinitely many bound states. A finite well is more realistic.

• In a finite well, the wavefunction extends into the barrier region as an evanescent tail. This means the particle has some probability of being found outside the well.
• Finite wells support only a limited number of bound states.
• Energy levels in a finite well are slightly lower than those of an infinite well with the same width, because the effective confinement region is a bit larger.
• When two finite wells sit close together, their evanescent tails overlap, allowing tunneling between wells. These coupled quantum wells are the basis for superlattice structures and minibands.

Energy levels and wavefunctions

Quantum confinement forces the allowed energies to be discrete rather than continuous. For a well of width $L$:

• The ground state wavefunction has no nodes; the first excited state has one node, and so on.
• Energy spacing increases as $L$ decreases (tighter confinement means larger energy gaps).
• For a symmetric well, wavefunctions alternate between even and odd parity.
• The probability distribution $|\psi(x)|^2$ tells you where you're most likely to find the particle within the well.

Resonant tunneling

Resonant tunneling occurs when the energy of an incoming particle matches one of the quasi-bound states inside a quantum well sandwiched between two barriers (a double-barrier structure). At resonance, the transmission probability spikes dramatically, sometimes reaching unity even though each individual barrier would block most particles.

• This is the operating principle behind resonant tunneling diodes (RTDs), which exhibit negative differential resistance.
• In superlattices (periodic sequences of quantum wells), overlapping resonant states form minibands, enabling controlled electron transport through the structure.

Scanning Tunneling Microscopy

The scanning tunneling microscope (STM) uses quantum tunneling to image conducting surfaces with atomic resolution. It was one of the first direct demonstrations that tunneling is a real, measurable, and technologically useful phenomenon.

Operating principles

1. A sharp metallic tip is brought within a few angstroms of a conducting sample surface.
2. A bias voltage is applied between tip and sample.
3. At these tiny separations, electrons tunnel across the vacuum gap, producing a measurable tunneling current.
4. Because tunneling current depends exponentially on the tip-sample distance, even sub-angstrom changes in height produce detectable current variations.
5. A feedback loop adjusts the tip height to maintain constant tunneling current as the tip scans across the surface. The recorded height adjustments produce a topographic map.

Tip-sample interaction

The tunneling current depends not just on distance but also on the local density of states (LDOS) of the sample at the Fermi level. This means the STM image reflects both geometric and electronic structure.

• Tip sharpness determines lateral resolution. Ideally, tunneling occurs predominantly through a single atom at the tip apex.
• At very small separations, chemical bonding forces and van der Waals interactions between tip and sample can perturb the measurement.
• The tip material and any contamination on it affect imaging contrast.

Atomic resolution imaging

STM routinely achieves lateral resolution sufficient to resolve individual atoms on clean surfaces, with vertical resolution on the picometer scale.

• By reversing the bias polarity, you can probe either filled states (electrons tunneling from sample to tip) or empty states (tip to sample), mapping different aspects of the electronic structure.
• Surface reconstructions, adsorbed molecules, and even standing waves of surface electrons have all been imaged with STM.

Tunneling in Semiconductors

Tunneling plays a central role in modern semiconductor devices, especially as feature sizes shrink into the nanometer regime. It governs breakdown mechanisms, leakage currents, and the operation of several important device types.

Band structure effects

In semiconductors, tunneling involves transitions between energy bands rather than transmission through a simple rectangular barrier.

• The effective mass $m^*$ of carriers in the band structure replaces the free electron mass in tunneling calculations. Lighter effective mass means higher tunneling probability.
• In heterostructures, the band alignment (type I, type II, or type III) determines the shape and height of the tunneling barrier.
• In indirect bandgap materials (like silicon), tunneling between valence and conduction bands requires a phonon to conserve crystal momentum, reducing the tunneling rate.
• Strain engineering can shift band edges and modify tunneling characteristics.

Zener tunneling

Zener tunneling is band-to-band tunneling in a strongly reverse-biased p-n junction. Under high reverse bias, the valence band on the p-side can align energetically with the conduction band on the n-side. Electrons then tunnel directly across the depletion region from valence band to conduction band.

• This mechanism dominates breakdown in heavily doped junctions (where the depletion region is narrow enough for significant tunneling).
• Unlike avalanche breakdown, Zener breakdown is nearly temperature-independent, which is why Zener diodes are used as stable voltage references.

Esaki diode operation

The Esaki (tunnel) diode is a heavily doped p-n junction where the Fermi level sits inside the conduction band on the n-side and inside the valence band on the p-side. Its current-voltage characteristic has three distinct regions:

1. Tunneling region: At small forward bias, electrons tunnel from filled conduction band states on the n-side into empty valence band states on the p-side. Current increases to a peak.
2. Valley region: As bias increases further, the bands misalign and tunneling current drops, producing negative differential resistance (NDR). This is the signature feature of the Esaki diode.
3. Diffusion region: At still higher bias, normal diode diffusion current takes over and current rises again.

The NDR makes Esaki diodes useful in high-frequency oscillators and fast switching circuits.

Josephson Junctions

A Josephson junction consists of two superconductors separated by a thin insulating barrier (typically ~1-2 nm of oxide). Cooper pairs tunnel coherently through the barrier, producing macroscopic quantum effects that have no classical analog.

Superconducting tunneling

• In a normal tunnel junction, single electrons tunnel incoherently. In a Josephson junction, Cooper pairs (bound electron pairs in the superconducting state) tunnel as a coherent quantum entity.
• The tunneling supercurrent depends on the phase difference $\phi$ between the superconducting order parameters on each side: $I = I_c \sin(\phi)$, where $I_c$ is the critical current.
• Andreev reflection occurs at superconductor-normal metal interfaces: an incoming electron is retroreflected as a hole, transferring a Cooper pair into the superconductor. This process is important in junctions with non-insulating barriers.
• The proximity effect extends superconducting correlations a short distance into adjacent normal metals.

Josephson effects

• DC Josephson effect: A supercurrent flows through the junction with zero voltage drop, as long as the current stays below $I_c$.
• AC Josephson effect: When a DC voltage $V$ is applied across the junction, the supercurrent oscillates at a frequency given by $f = \frac{2eV}{h}$. This is an exact relationship, which makes Josephson junctions useful as voltage standards.
• The critical current is sensitive to magnetic fields. A uniform field produces a Fraunhofer-like diffraction pattern in $I_c$ vs. applied flux, analogous to single-slit diffraction in optics.

SQUID devices

A SQUID (Superconducting Quantum Interference Device) is a superconducting loop interrupted by one (RF SQUID) or two (DC SQUID) Josephson junctions.

• The critical current of the loop oscillates as a function of the magnetic flux threading it, with a period of one flux quantum $\Phi_0 = h/2e \approx 2.07 \times 10^{-15}$ Wb.
• This makes SQUIDs the most sensitive magnetometers available, capable of detecting fields below $10^{-15}$ T.
• Applications include magnetoencephalography (mapping brain activity), geophysical surveying, and materials characterization.

Tunneling in Nanostructures

At the nanoscale, quantum confinement enhances tunneling effects and enables control over individual electrons. This regime is where tunneling transitions from a bulk transport mechanism to a tool for manipulating single charges.

Quantum dots

A quantum dot confines electrons in all three spatial dimensions, producing atom-like discrete energy levels. Tunneling between the dot and external leads (source and drain electrodes) is the mechanism by which current flows through the dot.

• The energy level spacing and charging energy are tunable through the dot's size, shape, and material.
• When the charging energy $E_C = e^2/2C$ (where $C$ is the dot's total capacitance) exceeds the thermal energy $k_BT$, adding even one extra electron costs significant energy. This is the Coulomb blockade regime.

Coulomb blockade

Coulomb blockade is the suppression of tunneling onto a small conductor because of the electrostatic energy cost of adding one electron.

• It occurs when $E_C \gg k_BT$, meaning thermal fluctuations can't supply enough energy to overcome the charging penalty.
• The result is a staircase-like I-V curve: current remains blocked until the bias voltage provides enough energy to add the next electron, then jumps in discrete steps.
• Observable in quantum dots, metallic nanoparticles, and single-molecule junctions at low temperatures (or room temperature for very small structures).

Single-electron transistors

A single-electron transistor (SET) is a three-terminal device where a quantum dot is coupled to source and drain electrodes through tunnel junctions, with a capacitively coupled gate electrode.

• The gate voltage shifts the energy levels of the dot, controlling whether an electron can tunnel on or off.
• As the gate voltage is swept, the conductance oscillates periodically. These Coulomb oscillations occur each time the gate brings a new charge state into resonance with the Fermi levels of the leads.
• SETs can detect charge changes smaller than a single electron, making them useful for ultra-sensitive electrometry and as charge sensors in quantum computing architectures.

Applications of Tunneling

Tunneling is not just a textbook curiosity. It underpins several major technologies and continues to enable new device concepts.

Flash memory

Non-volatile flash memory stores data as charge trapped on a floating gate, which is electrically isolated by thin oxide layers.

• Writing and erasing involve Fowler-Nordheim tunneling: a strong electric field across the oxide creates a triangular barrier through which electrons tunnel onto or off the floating gate.
• The tunnel oxide (typically ~8-10 nm of $SiO_2$) must be thin enough for efficient programming but thick enough to retain charge for years without significant leakage.
• As device scaling pushes oxide thickness below ~5 nm, direct tunneling leakage becomes a serious reliability challenge.

Tunnel diodes

Tunnel diodes exploit the negative differential resistance that arises from band-to-band tunneling (as in the Esaki diode discussed above).

• Because tunneling is a majority-carrier process with no minority-carrier storage delay, tunnel diodes switch extremely fast.
• Resonant tunneling diodes (RTDs) offer improved peak-to-valley current ratios compared to Esaki diodes and can operate at frequencies approaching the terahertz range.
• Applications include high-frequency oscillators, mixers, and fast logic circuits.

Quantum cascade lasers

A quantum cascade laser (QCL) uses intersubband transitions within coupled quantum wells rather than interband transitions.

• Electrons cascade through a series of quantum wells, emitting a photon at each stage via resonant tunneling between wells. A single electron can therefore produce many photons.
• QCLs emit in the mid-infrared and terahertz spectral regions, which are difficult to reach with conventional semiconductor lasers.
• Applications include gas sensing and spectroscopy, medical imaging, and free-space communications.

Theoretical Approaches

Several analytical and numerical methods exist for calculating tunneling rates and transmission coefficients. The choice of method depends on the barrier geometry and the level of accuracy required.

WKB approximation

The Wentzel-Kramers-Brillouin (WKB) approximation is a semiclassical method valid when the potential varies slowly on the scale of the particle's de Broglie wavelength. The tunneling probability is:

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

where $x_1$ and $x_2$ are the classical turning points and $\kappa(x) = \sqrt{2m(V(x)-E)/\hbar^2}$.

• Works well for smooth, slowly varying barriers.
• Breaks down near the classical turning points (where $E = V(x)$) and for barriers where the energy is close to the top.
• Despite its limitations, WKB gives quick, physically transparent estimates and is widely used in device modeling.

Transfer matrix method

The transfer matrix method handles arbitrary barrier shapes numerically by slicing the potential into many thin layers of approximately constant potential.

1. For each thin slice, write down the forward and backward propagating (or evanescent) wave amplitudes.
2. Apply continuity of $\psi$ and $d\psi/dx$ at each interface to relate amplitudes across the boundary via a 2×2 transfer matrix.
3. Multiply all the transfer matrices together to get the overall transmission and reflection coefficients.

This method handles multiple barriers, resonant structures, and arbitrary potential profiles with no approximations beyond the discretization.

Bardeen's tunneling theory

Bardeen's approach treats tunneling perturbatively. Instead of solving the Schrödinger equation for the entire barrier region, you solve for the states of each electrode separately, then calculate the tunneling rate using Fermi's golden rule with a tunneling matrix element $M$ that couples states on opposite sides.

• The tunneling current is proportional to $|M|^2$ and to the density of states on both sides.
• This framework is the theoretical foundation for interpreting STM data (Tersoff-Hamann theory is a specific application).
• It extends naturally to many-body treatments that include electron-electron and electron-phonon interactions.

Measurement Techniques

Experimental characterization of tunneling relies on electrical transport measurements that reveal barrier properties and electronic structure.

Current-voltage characteristics

Measuring $I(V)$ across a tunnel junction is the most direct probe of tunneling behavior.

• Tunneling produces characteristically nonlinear I-V curves. At low bias, the current is approximately linear (ohmic), but it grows faster than linearly at higher bias as more states become available for tunneling.
• The differential conductance $dI/dV$ is proportional to the density of states at energy $eV$ relative to the Fermi level. This is the basis of tunneling spectroscopy.
• Temperature dependence helps distinguish tunneling (weakly temperature-dependent) from thermionic emission (strongly temperature-dependent, following an Arrhenius law).

Tunneling spectroscopy

Tunneling spectroscopy uses the energy dependence of the tunneling current to map out the electronic density of states.

• Scanning tunneling spectroscopy (STS): Records $dI/dV$ as a function of bias voltage at each point on a surface, producing spatially resolved maps of the LDOS.
• Superconducting tunneling spectroscopy: Uses a superconducting electrode whose sharp density-of-states features (coherence peaks at the gap edge) provide high energy resolution for probing the gap structure of other superconductors or materials.
• Inelastic electron tunneling spectroscopy (IETS): Discussed below.

Inelastic electron tunneling

In most tunneling, the electron's energy is conserved (elastic tunneling). In inelastic tunneling, the electron excites a vibrational mode, phonon, or magnetic excitation in the barrier region, losing a quantum of energy in the process.

• Inelastic tunneling opens a new conduction channel when the bias voltage exceeds the excitation energy: $eV \geq \hbar\omega$.
• This appears as a step in $dI/dV$ and a peak in $d^2I/dV^2$.
• IETS provides chemical fingerprinting of molecules adsorbed at interfaces, with energy resolution (~0.5 meV at low temperature) that can exceed optical spectroscopy techniques.

Advanced Tunneling Phenomena

Beyond simple barrier transmission, several tunneling effects involve additional degrees of freedom or collective quantum behavior.

Fowler-Nordheim tunneling

Fowler-Nordheim (FN) tunneling describes field emission of electrons from a metal surface under a strong applied electric field (typically $>10^7$ V/cm).

• The strong field bends the vacuum barrier into a triangular shape, making it thin enough near the Fermi level for significant tunneling.
• Image charge effects round off the barrier top, further enhancing emission.
• A Fowler-Nordheim plot of $\ln(I/V^2)$ vs. $1/V$ yields a straight line, which is the standard diagnostic for this tunneling mechanism.
• FN tunneling is the mechanism used in flash memory programming and in field emission electron sources.

Spin-dependent tunneling

When ferromagnetic electrodes are used, the tunneling probability depends on the electron's spin orientation relative to the magnetization of the electrodes.

• In a magnetic tunnel junction (ferromagnet/insulator/ferromagnet), the resistance is lower when the magnetizations are parallel and higher when antiparallel. This is tunnel magnetoresistance (TMR).
• TMR ratios exceeding 600% have been achieved with crystalline MgO barriers, making magnetic tunnel junctions the read sensors in modern hard drives and the storage elements in MRAM.
• Spin-polarized STM uses a magnetic tip to image magnetic domain structures at the atomic scale.

Macroscopic quantum tunneling

Josephson junctions can exhibit tunneling of a macroscopic variable (the superconducting phase difference) rather than a single particle.

• In a current-biased Josephson junction, the phase evolves in a tilted washboard potential. At low temperatures, the phase can tunnel out of a metastable minimum rather than being thermally activated over the barrier.
• The crossover temperature between thermal activation and quantum tunneling is typically in the tens of millikelvin range.
• Macroscopic quantum tunneling is directly relevant to superconducting qubits, where quantum coherence of the phase degree of freedom is the computational resource.
• These experiments also provide a testing ground for understanding the quantum-to-classical transition.