Band-structure effects in field-emission energy distributions in tungsten

A calculation of the total energy distribution of field-emitted electrons from the (100) and (110) planes of tungsten is presented. The potential inside the metal is represented by a superposition of atomic potentials in the muffin-tin approximation as evaluated by Mattheiss. The crystal is terminated abruptly at the surface and the potential barrier outside the surface is represented by a straight line. Using the mathematical apparatus developed by low-energy-electron-diffraction theorists we reproduce the band structure of tungsten and calculate the tunneling probability by matching the wave function outside the metal to a superposition of Bloch waves, including a number of evanescent waves, inside the metal. The total energy distribution from the (100) plane exhibits a sharp peak at 0.29 eV below the Fermi level which suggests that the experimentally observed hump at about the same energy is due to band-structure effects. We show that this peak in the total-energy-distribution curve is due to electron states associated with certain sections of the constant-energy surface and we argue that spin-orbit interaction is not likely to remove this peak. The calculated distribution from the (110) plane does not show any marked deviation from the free-electron behavior in agreement with the experimental observations. For the (100) plane the ratio of the exact total energy distribution to its corresponding value evaluated in the free-electron approximation for the (100) plane is considerably smaller than unity away from the peak in the total-energy-distribution curve. For the (110) plane it is of the order of unity.