

\begin{figure}[b!]
\centering
{\includegraphics[width=12cm]{./figure/iv_1.eps}
}
\caption{Comparison of IV characteristics of the device at various gate voltages between the model and simulations. Different curves are for different gate voltages mentioned by the curves. At lower gate voltages, less than 1.0V, there are no peaks inside the bias window. First peak enters in the bias window at $V_g=1.1V$ so a high current flows. As the gate voltage increases other peaks also contribute to the current.}
\label{fig:iv_1}
\end{figure}

Single electron transistor based on silicon quantum dot of 1nm cross section was simulated in Atlas TCAD using NEGF formalism coupled with Schrodinger equation. The IV characteristics of the device were calculated from simulations as well as from the model. In Fig. \ref{fig:ts_1} the transmission spectra (TS) from simulations and model are shown for different values of \(V_{g}\) and \(V_{ds}\). The lowest unoccupied molecular levels (LUMO) are shown while the highest occupied molecular levels (HOMO) are not shown because they do not contribute to the conduction in this case as they are far below the Fermi energy level (\(E_f=0\)). The HOMO-LUMO gap and the Fermi energy level are also mentioned in the Fig. \ref{fig:ts_1} (a). The LUMO levels are relatively closer to Fermi level and by applying a positive gate voltage we can shift them near Fermi level (Fig. \ref{fig:ts_1} (b)). Then at a small value of \(V_{ds}\) LUMO1 enters inside the bias window and contributes to current (Fig. \ref{fig:ts_1} (c,d,e)).


\begin{figure}[t!]
\centering
{\includegraphics[width=6cm]{./figure/iv_2.eps}
\includegraphics[width=6cm]{./figure/iv_3.eps}
}
\caption{a) IV characteristics of the simulated device at gate voltage equal to 1.8V and for both positive and negative drain to source bias. b) IV characteristics obtained from the previous model without considering the spin and from the current model with the different spectra for spin at \(V_g=2.0V\).}
\label{fig:iv_2}
\end{figure}

The comparison between the IV characteristics is shown in the Fig. \ref{fig:iv_1}. Drain current is plotted as a function of drain to source bias for different values of gate voltage. At low gate voltages (i.e. less than 1 V) there are no transmission peaks inside the bias window so a very small current flows through device because of tunneling of electrons with very low probability. The model provides acceptable estimation of the current as shown in the figure in case of \(V_g=0.7,0.8,0.9,1.0V\). The first peak starts to enter in the bias window at \(V_g=1V\) as shown in the Fig. \ref{fig:ts_1} (c).

The current is modeled very effectively at the gate voltages higher than 1V (Fig. \ref{fig:iv_1}) because the bias window, for the selected range of \(V_{ds}\), includes the transmission peaks. As the model is transmission spectrum based so the results are better when there is a transmission peak inside the bias window. For the sake of clarity, the different curves shown here are selected in such a way that they do not overlap each other. In Fig. \ref{fig:iv_2} (a) the IV characteristics for both positive and negative drain to source bias are shown for \(V_g=2.0V\). This shows that our model is valid for bidirectional electron flow. And in the Fig. \ref{fig:iv_2} (b) the results from the previous model without considering the spin of electron (Model w/o Spin) is compared with the proposed model (Model w Spin) under the same conditions and parameters. We can see that the proposed model provides very accurate results compared to the previous version of the model in \cite{zahir2016ee}.
\begin{figure}[b!]
\centering
{\includegraphics[width=9cm]{./figure/cd.eps}
}
\caption{The current plotted as a function of \(V_{ds}\) and \(V_g\). The shade of greyscale represents the drain current value based on the scale shown in the figure. The three figures are obtained from a) simulations b) model c) model using refined set of applied voltages. }
\label{fig:cd}
\end{figure}

The current as a function of \(V_{ds}\) and \(V_g\) is plotted on grayscale in Fig. \ref{fig:cd}. The magnitude of the current is represented by the shade of the gray according to the scale shown in the figure. The dark regions represent the zero or very small current while the current of higher magnitudes is represented by the light regions. The coulomb blockade diamonds which are unique characteristics of SET can be seen in the form of dark regions. One large blockade diamond and one small one can be observed in the figure. The model successfully predicts the same behavior as obtained from simulations. The resolution of the plots in Fig. \ref{fig:cd} (a,b) is not good and it would take high computational effort to increase it in case of simulations. We have shown the high resolution achieved from the model in Fig. \ref{fig:cd} (c). The time required for the calculations in Fig. \ref{fig:cd} (a,b) is 6346s by the TCAD simulations and 1.57s by the model. The model takes 128s for the calculation with high resolution plot (210 \(V_{ds}\) points and 260 \(V_g\) points) Fig. \ref{fig:cd} (c). All the simulations were performed on the same system with the Intel core i5 1.8 GHz processor and 6 GB of RAM. The proposed model provides sufficiently accurate results while maintaining the efficiency. While other compact models based on macromodeling are also efficient, the proposed model captures the physical information of device with more details.





