Time evolution of electron flow in a model diode: Non-perturbative analysis

Abstract

Using a combination of Eulerian and Lagrangian variables we obtain some exact results and good approximation schemes for the time evolution of the electron flow from a no-current state to a final stationary current state in a planar one-dimensional diode. The electrons can be injected externally or generated by the cathode via field emission governed by a current-field law. The case of equipotential electrodes and fixed injection is studied along with a positive anode potential. When the current is fixed externally the approach to the stationary state goes without oscillations if the initial electron velocity is high enough and the anode can absorb the injected flow. Otherwise the accumulated space charge creates a potential barrier which reflects the flow and leads to its oscillations, but our method of analysis is invalid in such conditions. In the field emission case the flow goes to its stationary state through a train of decaying oscillations whose period is of the order of the electron transit time, in agreement with earlier studies based on perturbation techniques. Our approximate method does not permit very high cathode emissivity, though the method works when the stationary current density is only about 10% smaller than the ChildLangmuir limit.

Non-perturbative analysis of space charge limited electron flow in critical regimes

“…Our claim here is that the approach developed in Ref. 8 after an improvement of its precision here can be used for studying the flow behavior in much wider conditions.…”

“…This special situation for t < T 1 , studied in Ref. 8, is much simpler because Eqs. (6) and (7) in this case do not include the unknown function T(t) but only f(t) and j(t).…”

“…For time independent j and for j(t) ¼ af(t), a > 0, we developed in Ref. 8 an approximate method of studying the time evolution of the flow by constructing four exact equations with the help of Eqs. (2) and (3).…”

“…Developed in Ref. 8 is an approximate computational method that was used for approximating the cathode current f(t) by considering separately and consequently the time intervals s 1 , s 2 , s 3 … and matching the BC on their ends for f and its derivative f 0 . The solution for f(t) on the interval 0 < t < T 1 can be found analytically using Eq.…”

“…An approximate method for treating much larger flow variations without perturbation techniques was developed under the same assumption in Ref. 8 for the turning on regime of a one-dimensional (1D) diode.…”