Fowler–Nordheim tunneling

Fowler–Nordheim tunneling. It is a quantum mechanical process describing the emission of electrons from a solid material into a vacuum or another material under the influence of a strong external electric field. This phenomenon is a specific type of field electron emission and is a cornerstone theory in the study of electron transport at nanoscale dimensions. The effect is fundamentally governed by the tunnel effect, where electrons penetrate through a potential barrier that would be insurmountable according to classical physics.

Physical principles

The underlying mechanism relies on the principles of quantum mechanics, particularly the wave-like nature of particles. In the presence of a high electric field, typically on the order of volts per nanometer, the Coulomb barrier at a material's surface becomes thin and triangular. Electrons near the Fermi level can then quantum-mechanically tunnel through this narrowed barrier into the vacuum. This process contrasts with other emission mechanisms like thermionic emission, which relies on thermal energy, or photoemission, driven by photon absorption. The probability of tunneling is exquisitely sensitive to the local electric field strength and the material's work function, a fundamental property describing the minimum energy needed to remove an electron. The theory was developed to explain the cold emission of electrons from cold metal surfaces, a phenomenon first systematically studied by Robert Andrews Millikan and his student C. C. Lauritsen.

Mathematical formulation

The quantitative description is given by the Fowler–Nordheim equation, which derives from solving the Schrödinger equation for an electron facing a triangular potential barrier. The original derivation by Ralph H. Fowler and Lothar Wolfgang Nordheim applied the Wentzel–Kramers–Brillouin approximation to calculate the transmission probability. The resulting equation for the current density *J* is *J = A F² / φ exp(-B φ^(3/2) / F)*, where *F* is the electric field, *φ* is the work function, and *A* and *B* are constants. This formulation assumes a free-electron model with a parabolic dispersion relation and a perfectly smooth, flat emitting surface at zero temperature. Later refinements by theorists like John Bardeen and experimentalists accounted for effects such as image potential corrections, leading to the Schottky effect, and deviations from ideal geometry.

Applications in semiconductor devices

This tunneling mechanism is critically exploited in modern microelectronics and nanoelectronics. It is the fundamental conduction mechanism in flash memory cells, where electrons tunnel through a thin silicon dioxide layer to program and erase memory states. It is also central to the operation of tunnel field-effect transistors, which are being researched as low-power successors to conventional MOSFET technology. Furthermore, the phenomenon enables the functioning of field emission display screens and is utilized in the electron sources of advanced scientific instruments like scanning electron microscopes and transmission electron microscopes. The design of these devices requires precise control over gate oxide thickness and material interfaces to manage tunneling currents.

Experimental observation and measurement

Direct evidence for this type of emission is typically obtained by measuring current-voltage characteristics in ultra-high vacuum systems to avoid field ionization of residual gases. The signature is a linear region in a so-called Fowler–Nordheim plot, where ln(*J/F²*) is plotted against *1/F*. Experimental setups often use sharp tungsten or molybdenum tips as cathodes, as pioneered by Erwin Wilhelm Müller in his invention of the field emission microscope. Modern techniques employ atomic force microscopy or scanning tunneling microscopy probes to create and measure emission with nanoscale precision. Challenges in measurement include distinguishing pure Fowler–Nordheim emission from other effects like space-charge limited current or emission from surface states.

Historical development and context

The theoretical foundation was laid in 1928 by the British physicist Ralph H. Fowler and the German physicist Lothar Wolfgang Nordheim, building upon the nascent field of quantum mechanics established by figures like Erwin Schrödinger and Werner Heisenberg. Their work provided the first satisfactory explanation for the field emission experiments conducted earlier in the decade. The theory gained significant practical importance with the post-World War II advancement of solid-state physics and the dawn of the semiconductor industry. The invention of the tunnel diode by Leo Esaki in 1957, for which he received the Nobel Prize in Physics, demonstrated tunneling in p-n junctions, further validating the quantum mechanical principles. Continued refinement of the theory has been integral to the development of the International Technology Roadmap for Semiconductors and the scaling of integrated circuits.

Category:Quantum mechanics Category:Condensed matter physics Category:Electronic engineering