Unification of field emission and space charge limited emission with collisions

Abstract

Electron emission plays a vital role in device design for systems with pressures ranging from vacuum to atmospheric pressure. Nonuniform pressure in vacuum devices and gap sizes below microscale for electronics near atmospheric pressure necessitate further theoretical characterization of the transition between electron emission phenomena. This letter incorporates collisions into analytical equations describing the transition from the Fowler-Nordheim (FN) equation for field emission to space-charge limited emission (SCLE). We recover the Child-Langmuir (CL) law for vacuum, SCLE at high mobility μ, and the Mott-Gurney (MG) law for collisional SCLE at low μ. The exact solutions follow asymptotic solutions for FN at low voltage V, before transitioning to MG at higher V, and, ultimately, to CL independent of μ. We also define a never before seen “triple-point,” where the asymptotic solutions of all three electron emission regimes converge. Fixing V, μ, or gap distance D uniquely specifies the other two parameters to achieve this triple point, which defines a regime where the electron emission mechanism is very sensitive to experimental conditions. The implications on device design are discussed.