Self-consistent numerical solution of quantum regime with exchange-correlation effects of space charges for electron field emission in a nano-gap

The quantum effects of space charge on electron field emission have been widely investigated since the last century. When electrons energy and their mean spacing approach the Hartree level and the de Broglie wavelength respectively, the influence of the quantum effects on the field emission current becomes significant. In this work, by developing an in-house software, we self-consistently solve the one-dimensional Poisson-Schrödinger equation together with the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) model for metal-vacuum-metal nanogaps, after considering the anode screening effect, space charge Coulomb potential and exchange-correlation effects simultaneously. Employing the method, the electron field emission characteristics were studied by varying the nanogap spacing (D) and the electric field strength (F), and four different emission regimes including quantum regime (QR), space charge limited regime (SCLR), direct tunnelling regime (DTR) and field emission regime (FER) are defined. The influences of space charge field components on the field emission characteristics and space charge distribution are analyzed for different emission regimes in nanogap. In addition, the impact of using different exchange-correlation functionals (LDA, GGA and meta-GGA) on Jacob’s ladder for describing the quantum effects of space charge on the electron emission current density was analyzed. Finally, electron field emission properties of one-dimensional (1-D) nanogaps consisting of refractory metals (W and Mo) as well as the three-dimensional (3-D) nano-tip are discussed to elucidate the impact of the exchange-correlation effects on the enhanced field emission process at nanoscale.