Theoretical Analysis of Resonant Tunneling Enhanced Field Emission

In this paper, we develop an exact analytical quantum theory for field emission from surfaces with a nearby quantum well, by solving the one-dimensional time-independent Schr\"odinger equation. The quantum well, which may be introduced by ions, atoms, nanoparticles, etc., is simplified as a square potential well with depth H, width d, and distance to the surface L. The theory is used to analyze the effects of the quantum well (d, H, and L), the cathode properties (work function W and Fermi energy ${E}_{F}$), and dc field F. It is found that the quantum well can lead to resonant tunneling enhanced field emission up to several orders of magnitude larger than that from bare cathode surfaces. In the meantime, the electron-emission-energy spectrum is significantly narrowed. The strong enhancement region is bounded by the conditions eFL + H \ensuremath{\ge} W + C and eFL \ensuremath{\le} W, with e being the elementary charge (positive) and C a constant dependent on dc field F. It is also found that the linear shift of resonance peaks in the electron-emission-energy spectrum with dc field F follows ${\ensuremath{\epsilon}}_{p}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}{\ensuremath{\epsilon}}_{p0}\ensuremath{-}\phantom{\rule{0.1em}{0ex}}eFL$, with ${\ensuremath{\epsilon}}_{p0}$ being approximately the eigenenergies for electrons confined in a square potential well without a dc field. The theory provides insights for the design of high-efficiency field emitters, which can produce a high current and highly collimated electron beams.