Fowler-Nordheim tunneling

Fowler-Nordheim tunneling occurs when a strong electric field (typically >10^9 V/m) is applied to a metal surface, reducing the width of the surface potential barrier to a few nanometers. Electrons near the Fermi level can then tunnel through this barrier into the vacuum, producing a current that depends exponentially on the field strength.¹

The process is distinct from thermionic emission, where electrons are thermally excited over the barrier, and from photoemission, where photons provide the energy. In Fowler-Nordheim tunneling, the electron's energy is unchanged; it tunnels elastically through the barrier.

The current density J in Fowler-Nordheim tunneling is described by the Fowler-Nordheim equation:

J = (A F² / φ) exp(−B φ^(3/2) / F)

where F is the local electric field, φ is the work function, and A and B are constants (A ≈ 1.54×10⁻⁶ A eV V⁻², B ≈ 6.83×10⁹ V m⁻¹ eV⁻³/²).² The exponential dependence on φ^(3/2)/F makes the current extremely sensitive to field and work function variations.

In practice, the equation is often written in a form that includes the image charge correction, which rounds the barrier and slightly reduces its effective height and width.

The phenomenon was first observed in the late 19th century as unexplained electron emission from metals under high voltage. In 1928, Ralph Fowler and Lothar Nordheim applied quantum mechanics to explain it, deriving the first quantitative theory.³ Their work was one of the early successes of quantum tunneling theory, following Gamow's explanation of alpha decay.

Later refinements by Murphy and Good (1956) and Forbes (2006) improved the accuracy of the equation, particularly regarding the treatment of the image potential and the use of the 'pre-exponential' factor.

Fowler-Nordheim tunneling is exploited in several technologies:

• Field emission displays – arrays of sharp tips emit electrons onto phosphors.
• Electron microscopes – cold field emission sources provide high-brightness electron beams.
• Flash memory – electrons tunnel through a thin oxide layer to store charge.
• X-ray tubes – field emission cathodes enable compact, fast-switching devices.

In vacuum microelectronics, Fowler-Nordheim tunneling is also used in miniature vacuum tubes and sensors.

Beyond the standard picture, several subtle points are often overlooked:

• Temperature dependence – At finite temperatures, electrons above the Fermi level contribute, leading to a 'thermal-field' emission regime that bridges Fowler-Nordheim and thermionic emission.
• Field enhancement – The local field at sharp tips or protrusions can be hundreds of times the macroscopic field, enabling emission at lower applied voltages.
• Resonant tunneling – In metal-insulator-metal structures, resonant states in the insulator can enhance tunneling, a phenomenon used in some devices.
• Many-body effects – The simple single-electron picture ignores electron-electron interactions, which can modify the barrier and current.
• Relativistic effects – At extremely high fields, relativistic corrections become important, though rarely reached in practice.

Another niche aspect is the use of Fowler-Nordheim tunneling in vacuum breakdown studies, where it initiates arcs in high-voltage equipment.

Fowler-Nordheim tunneling is typically characterized by measuring the current-voltage (I-V) characteristics of a field emitter. A plot of ln(I/V²) versus 1/V yields a straight line (the Fowler-Nordheim plot), from which the field enhancement factor and emission area can be extracted.

Modern techniques include scanning tunneling microscopy (STM) to probe local emission sites and field emission microscopy (FEM) to image the emission pattern.

Work function

The minimum energy needed to remove an electron from a solid to a point just outside the surface.

Field enhancement factor

The ratio of the local electric field at a tip to the macroscopic applied field.

Image potential

The attractive potential experienced by an electron near a metal surface due to its induced image charge.

Thermionic emission

Emission of electrons from a heated surface over the potential barrier.