Single-electron based model of the Child-Langmuir law

Summary form only given. Space-charge-limited (SCL) electron flow describes the maximum current density allowed for steady-state electron beam transport across a planar diode, which is also known as the Child-Langmuir (CL) law. Extensive studies have been done to revise the 1D classical CL law to multi-dimensional models, quantum regime and ultrafast time scale. Recently, some studies were conducted in using ultrafast lasers to have localized photo-field electron emission from metallic nanotips and a finite number of emitted electrons per pulse can be obtained depending on the laser pulse energy. Thus, we are interested to study if the space charge effect of the electrons can be ignored when the electrons are emitted or injected into a gap in the limit of single electron per injection. In comparison to the classical CL law, our model can be considered as a single electron based model, including the effects of single electron charging known as the Coulomb blockade. In this paper, we have extended the 1D classical CL law to the Coulomb blockade regime based on the time of fight method and capacitor model. Coupled equations are solved numerically to obtain the current as a function of applied voltage. It is found that there is a threshold of voltage (Vth) equals to one-half of the single electron charging energy = e/C, where C is the capacitance of the gap. For voltage in the range of 1 <; V/Vth <; 2, there is only 1 electron inside the gap, and the time-average of the single electron injection may be equal or higher than the 1D CL current. Transition to the traditional CL law.