However, a full NEGF treatment can be highly demanding and in some cases almost impossible to implement due to numerical burden. As a concrete example, let us examine Fig. 1. The schematic shows the well-known band diagram for gate induced drain leakage. When the gate voltage is lowered at high drain voltage, a tunneling current can flow from the bulk to the drain through band-to-band tunneling. There are two energy regions of interest: (i)E1; the two ends of which are cut off by the band edges and (ii)E2; which is the region underneath the valence band in the body region but above E1.
In a ballistic scenario, the region E2 is devoid of electrons and all the band-to-band current flows in the region E1. This is also what is calculated in NEGF. However, in reality, electron-electron interaction quickly fills up E2 according to the Fermi statistics, and so a tunneling current should also flow in this region.Especially, for direct and low band gap materials such as III-V, this current can be a significant part of the total tunneling current. The problem is that it is numerically extremely challenging, if not impossible, to include electron electron interactions in the NEGF formalism, and therefore to account for the current that flows in E2.
To account for this problem, in this work, we have taken a two-pronged approach. First, we show that a Wentzel-Kramers-Brillouin (WKB) approximation calculated using the imaginary part of the full band-structure provides a very close agreement with NEGF results (typically within a few percent). This is especially accurate at low current levels where the Laplace solution of the potential provides a very good estimate for the self-consistent potential of the problem. We then combine this with a mode-space calculation of the currentvoltage characteristics. The mode-space approach is particularly valid when the device is uniform along the transport direction as it is the case for double gate or most of the FinFET structures. The mode space approach demands much smaller numerical burden as each mode is solved independently. This also means that in a standard approach, it cannot account for band-to-band tunneling. Our strategy is to combine the WKB approximation with the mode space approach so that the ordinary current is calculated using the mode space while the band-to-band tunneling current is calculated using WKB.
We first show that this combined strategy provides current-voltage characteristics that show excellent agreement with full, real-space, NEGF calculations, including the band-to-band tunneling in the E1 region. We then add the additional current that flows in E2 by assuming a Fermi distribution of the electrons in that region. Our work thus provides a very fast, yet accurate approach to calculate currentvoltage characteristics in direct bandgap semiconductors using the NEGF method.