Low work-function cathodes from Schottky to field-induced ballistic electron emission: Self-consistent numerical approach

A systematic study was done in order to relate the current density $J$-applied field $F$ characteristic variation with three emission mechanisms: thermionic, tunneling, and ballistic. All three are now effective during the field emission from cathodes with work function $\Phi$ less than $2\mathrm{eV}$. The current density is computed using the transmission probability for an electron to cross the barrier between the electron sea of the cathode and the vacuum. The corresponding Schrödinger equation is solved by means of the self-consistent Lippmann-Schwinger equation, with values of the effective potential corrected with the image potential between the cathode and the anode, and resolved by spatial discretization. This method allows computing the exact current within a zero emitted current approximation. It fills the gap left by the former analytical and numerical resolutions. The $\ln (J∕F^2)$ vs $(1∕F)$ plot shows, from the very beginning of the electron emission, three zones for the current variation. The first zone, corresponding to low applied electric fields, is a nonlinear variation of the current specific to thermionic-field emission; it is followed by a second zone having a linear variation with a slope proportional to ${\Phi }^{3∕2}$ characteristic of the conventional Fowler-Nordheim field emission. The third zone, concerning high field values, indicated a current saturation behavior related to a field-induced ballistic emission.

Figure 1

(a) Potential diagram for an electron with the Schottky effect. (CB is the energy at the bottom of the conduction band and TB is the top of the surface barrier under an applied field $F$.) (b) The field-induced ballistic region lies over the curve $F_{\mathrm{bal}}$, it corresponds to situation where the top of the barrier is pulled down under the Fermi level by the Schottky effect.

Figure 2

(Color online) The three emission validity regions for a $1.05\mathrm{eV}$ work function cathode.

Figure 3

(Color online) J-F variations showing a linear behavior specific of a tunneling process. Similar behaviors were obtained from the analytical analysis by Murphy and Good [Eq. (3)] and our numerical calculations.

Figure 4

(Color online) Emission current variation with applied field for a cathode work function $\Phi =1.05\mathrm{eV}$. Open (black) circles correspond to numerical simulation and (red) triangles to data calculated with Murphy and Good relations. The arrows 1 to 5 indicate the J-F values respective to the surface barrier and TED plots having the same number in Fig. 5. Open (pink) squares correspond to field-induced ballistic emission when $F>F_{\mathrm{bal}}=0.075\mathrm{V}∕\mathrm{\AA }$.

Figure 5

(Color online) Surface barriers and total energy distribution spectra for five values of the applied fields. The work function of the cathode is $1.05\mathrm{eV}$.

Figure 6

(Color online) The J-F characteristics showing the convergence, in the field-induced ballistic emission, towards the same saturation current for different cathode temperatures.

Figure 7

(Color online) Boundaries of the field emission region as given by Murphy and Good in $(F,\Phi )$ coordinates and for different cathode temperatures. The validity region is strictly limited to the area over each curve plotted for different temperatures from $300\text{to}2000\mathrm{K}$. (b) is a blow up of (a) for small values of work function and field. [e.g.: for $\Phi =1.05\mathrm{eV}$, the lecture of (b) indicates a linear variation of the Fowler-Nordheim plot only between $0.035<F<0.065\mathrm{V}∕\mathrm{\AA }$ at $300\mathrm{K}$. This FE interval does not exist anymore for a cathode temperature $T=500\mathrm{K}$.]

Figure 8

Comparison of experimental data at $300\mathrm{K}$ and $490\mathrm{K}$ with numerical simulations of J-F done with $\Phi =1.05\mathrm{eV}$ at the same temperatures.