Field Emission and Quantum Tunneling: A Window into Fundamental Problems of Quantum Mechanics ✦
Field electron emission and electrostatic field ionization describe quantum tunneling, but their theory is far from complete. Fundamental problems are discussed: the impossibility of negative kinetic energy points to the distributed nature of the electron, not a point particle; the language of quantum mechanics needs revision to avoid the concept of a point particle's position; quantum mathematics has different modes of use — matter distribution and path selection, with measurement being the observation of a path choice. Also noted are difficulties with the uncertainty principle and wave-particle duality, fundamental challenges in accurately calculating exchange-correlation effects, conceptual problems of 'seeing electrons' in a field emission microscope, the connection between tunneling and the arrow of time, and the incompleteness of both types of tunneling theory formulations (via tunneling integral and overlap integral). These questions strike at the heart of quantum mechanics and have direct implications for modern technologies.
Context
Field electron emission is one of seven historical paradigmatic examples of quantum tunneling, predicted as early as 1928. Today, these effects underpin technologies ranging from atom-probe tomography to lightning protection of spacecraft. However, as the author's half-century experience shows, fundamental gaps remain in the theory, related to the interpretation of the wave function and accounting for many-body interactions. Even Schrödinger initially considered the wave function a description of a real matter distribution but was forced to abandon this view under critical pressure.
Methods
The author critically analyzed two main approaches to calculating the tunneling emission rate: the tunneling integral method (originally developed for the Schottky–Nordheim barrier, see formula) and the overlap integral method, dating back to Oppenheimer's work. Comparison was based on experimental data from field ion microscopy—images of tungsten facets and carbon nanotubes. To improve accuracy, a recently introduced special mathematical function vFD(x) and a formulation through amount of substance within the International System of Quantities (ISQ) were used, eliminating conceptual errors associated with the point-electron representation.
Results
It is shown that the tunneling integral approach qualitatively explains the contrast observed in a field ion microscope: atoms at the corners of a crystal facet glow brighter due to local field enhancement and displacement of the critical surface. Meanwhile, the overlap integral method cannot even predict the correct direction of contrast change. Both approaches remain incomplete: neither includes a proper account of the final-state density in both contacting media. A fundamental paradox is also identified: why the front of an electron emerging from a barrier does not detach from its 'tail' under a strong electrostatic field.
Implications
An alternative view of the wave function interpretation is proposed: its squared modulus, multiplied by the fundamental atomic amount of substance (1 electron), gives the concentration of 'electron matter' in space. This resolves the negative kinetic energy paradox and questions the standard formulation of the uncertainty principle as a limitation on simultaneous measurement of position and momentum. Instead, the author proposes a principle linking the electron's minimum kinetic energy to the length of its localization region.
Future development
In the coming years, a key task will be the direct experimental verification of the image formation mechanism in a field electron microscope. Advances in ultrafast laser technology give hope for 'filming' the tunneling process. Most intriguing is the possibility of extending the refined theory to biological systems: for example, proton tunneling in DNA could be responsible for mutations, and the behavior of helium ions—for radiation damage. This would open the way to quantum biology and even quantum botany.
Impact
Revising the foundations of tunneling theory will affect not only surface physics and nanoelectronics but also models of electrical breakdown, medical X-ray sources, and electric propulsion systems for spacecraft.
Next steps
Immediate steps: experimental verification of the irreversibility of tunneling path choice in many-electron systems; development of a universal formalism combining the tunneling integral and overlap integral, taking into account the real band structure; quantum-mechanical modeling of emission from carbon nanotubes for direct comparison with bond images.
Key open problems
The discussed difficulties are directly related to unresolved problems of quantum mechanics: the measurement problem, wave function interpretation, and the emergence of quantum decoherence. The author's proposal to abandon the point electron echoes the ideas of David Bohm and John Bell, but does not require hidden variables. The statistical irreversibility of tunneling path choice provides a simple mechanism for the arrow of time—an alternative to traditional explanations via decoherence.
🎯 The image of chemical bonds in a five-membered carbon ring (Fig. 1) was obtained at room temperature with a magnification of about 100,000,000 times. That's like examining a cherry the size of Earth and seeing its pit! And the brightest atom in the ion image of tungsten (Fig. 3)—the corner one—shines brighter because a larger positive charge accumulated on it due to geometry, and the local field pulls electrons from the gas more efficiently.
🎬 The idea that an electron, tunneling through a barrier, behaves like an octopus squeezing through a crack, evokes the scene from Arthur C. Clarke's '2001: A Space Odyssey,' where astronaut Bowman passes through the stargate, undergoing an incomprehensible transformation. The concept of a distributed 'material entity' resonates with the image of the sentient ocean in Stanisław Lem's 'Solaris'—an ocean simultaneously present in many points of space.
Key numbers
- emitter tip radius: ~100 nm and less
- field during field emission: a few V/nm
- field during field ionization: 20–60 V/nm
- FIM experiment temperature: ~80 K
- field electron microscope magnification: ~100,000,000×