The recent introduction of high-k-metal-gate transistors draws attention to a more accurate modeling of leakage current. Two different models are commonly used, namely the Tsu-Esaki formula and a Quasi-Bound State (QBS) tunneling formalism. The Tsu-Esaki expression relies on the transmission coefficient for the barrier and a supply function, determined by the carrier distributions in the gate and channel regions. The QBS method is based on the electron populations of the discrete subbands in the MOS inversion layer and a finite lifetime. Both approaches neglect the carrier density in the dielectric due to the assumption of hard wall boundary conditions and are thus inconsistent with the non-vanishing current density.
The current spectra of the Tsu-Esaki, the QBS-tunneling model and the Non-Equilibrium Green's Functions (NEGF) formalism are shown in the figure. The NEGF approach clearly shows the distinct resonant states. Furthermore, it captures the influence of both the quasi-bound states and the continuum. Compared to the QBS model, the peaks show a realistic broadening due to the scattering processes modeled by the optical potential. On the other hand, the resonances are completely neglected by the Tsu-Esaki model. While all three models provide similar results for the macroscopic quantity, the tunneling current, there are significant differences in the microscopic quantity, the carrier spectrum. Therefore, any model sensitive to the changes in the current spectrum are affected by these effects. This is especially true for trap-assisted tunneling models which are needed for the characterization of high-k materials.