Abstract: Electrons of a metallic nano-sheet tunneling into vacuum via the electric field is studied. Ignoring the thickness of the sheet, the boundary surfaces of the cathode and anode are mapped into two planes respectively through a conformal transformation, thereby, the electrostatic potential barrier is obtained rigorously and the Schroedinger equation is solved analytically. Analytical expressions for the electron wavefunction and the emission current density are given.

Key words: field electron emission, quantum tunneling, conformal transformation