Density of Energy Spectrum of an Electron in the Image-Potential Field and a Trapping Electric Field

Density of energy spectrum of an electron bound by an image-potential field and a trapping field near the surface of a metal is calculated in the quasi-classical approximation. The confinement mechanism realized in the system under consideration leads to a completely discrete energy spectrum of electron motion in the direction perpendicular to the metal surface. The density of energy spectrum is expressed in terms of elliptic integrals the argument of which represents a sigmoid function that transforms into a Heaviside step function when the field is turned off. A dimensionless energy parameter determining intervals characterized by qualitatively different variation of width of classically accessible region of motion is introduced. The density of spectrum asymptotically tends to that in a triangular potential with addition of a Coulomb logarithmic correction at large positive values of the energy parameter. At negative values of the energy parameter, the density of spectrum transforms into the dependence corresponding to a one-dimensional Coulomb potential. Approximate expressions that describe the density of spectrum in terms of elementary functions in a wide range of electron energies and electric-field strengths are derived.