Transit time model for space charge limitted current

Summary form only given. Space-charge-limited (SCL) electron flow describes the maximum current density allowed for steady-state electron beam transport across planar, cylindrical, or spherical diode. Base on a transit time model, it is found that the Langmuir-Blodgett (LB) solutions for the SCL current density, for both cylindrical and spherical diodes, may be approximated by a novel scaling law J<;sub>app<;/sub> = 4/9ϵ<;sub>0<;/sub>√2e/mE<;sub>c<;/sub><;sup>3/2<;/sup>/√D over a wide range of parameters, where E<;sub>c<;/sub> is the surface electric field on the cathode and D is the anode-cathode spacing. It is important to note that the proposed new scaling is very different from the classical LB scaling proposed 80 years ago. Minor empirical corrections to the above scaling yield fitting formulas that are accurate to within 2.5 percent for all 0.002 <; R<;sub>a<;/sub>/R<;sub>c<;/sub> <; 30000, where R<;sub>a<;/sub> and R<;sub>c<;/sub> is, respectively the radius of the anode and cathode. The model is applicable even in the Coulomb blockade regime for a nanogap, where the electron number may be in the single digits. In this regime, there is a threshold voltage (V<;sub>th<;/sub>) equals to one-half of the single electron charging energy. For voltage in the range of 1<; V=V<;sub>th<;/sub> <; 2, there is only 1 electron inside the gap, and the time-average value of the single electron injection may be equal or higher than the classical value, and its transit time frequency is in the THz frequency. The model is versatile that it can be used to study the space charge limited electron injection into a solid.