A Generalized Tunneling Current Formula for Metal/Insulator Heterojunctions under Large Bias and Finite Temperature

The Fowler-Nordheim tunneling current formula has been widely used in the design of devices based on metal/insulator (metal/semiconductor) heterojunctions with triangle potential barriers, such as the flash memory. Here we adopt the model that was used to derive the Landauer formula at finite temperature, the nearly-free electron approximation to describe the electronic states in semi-infinite metal electrode and the Wentzel-Kramers-Brillouin (WKB) approximation to describe the transmission coefficient, and derive a tunneling current formula for metal/insulator heterojunctions under large bias and electric field. In contrast to the Fowler-Nordheim formula which is the limit at zero temperature, our formula is generalized to the finite temperature (with the thermal excitation of electrons in metal electrode considered) and the potential barriers beyond triangle ones, which may be used for the design of more complicated heterojunction devices based on the carrier tunneling.