# Modeling of Charge Distribution Using Schrodinger-Poisson Equations: Application to Double-Gate Transistor (i.e GAA-SOI Transistor)

I. Introduction

The Gate-All-Around SOI transistor [1], in which the gate oxide and the gate electrode are wrapped around the semiconductor region between the source and drain electrodes, exhibits very attractive features [2,3]. This device has been shown to present excellent Ion/Ioff trade-off, good threshold voltage roll-off reduction and better resisting to short-channel effect and DIBL. In this study, based on the self-consistent solution of Schrodinger and Poisson equations, we attempt to show that the maximum of electrons concentration can be located in the middle of the semiconductor film and that it exist an "optimal" thickness of the Si-film. This is a major divergence with classical models which predict always the maximum of electron concentration at the semiconductor interface and that the reduction of the film thickness is always better against parasitic effects.

II. Device Structure

The GAA-SOI was built in * ATLAS*. In this
work symmetrical gate-SiO2-Si(p)-SiO2-gate structure with uniformly doped
substrate is considered (Figure 1). The main characteristics are: fixed
gate oxide thickness of 25nm, variable Silicon film thickness from 1.5nm
to 20nm and uniform p-type doping of 1e18 cm.

Figure 1. GAA-SOI device structure.

III. Schrodinger Calculation in *ATLAS*

The self-consistent Schrodinger-Poisson model is enabled
by setting the SCHRO parameter of the model statement. With this parameter
set * ATLAS* solves the one dimensional Schrodinger?s equation
along a series of slices in the y direction relative to the device. Each
slice is taken along an existing set of y nodes in the

*device mesh. After the Schrodinger's equation solution is taken, carrier concentration calculated from Schrodinger's equation are substituted into the charge part of the Poisson's equation. The potential derived from solution of Poisson's equation is substituted back to Schrodinger's equation. This solution process (alternating between Schrodinger's and Poisson's equation) is continued until convergence is reached and a self-consistent solution of Schrodinger's and Poisson's equation is obtained. For more details see [4].*

**ATLAS**

IV. Results and Discussion

Figure 2 presents spatial distributions of electrons concentration. The fundamental difference between the classical and Schrodinger calculation lies in the fact that the maximun of electrons concentration is localized at the semiconductor interface for classical calculation whereas it is localized inside the semiconductor film using Schrodinger-Poisson calculation.

Figure 2. Spacial distribution of electrons concentration

using "classical or "Schrodinger" calculation.

Moreover, the quantization of electron energy in the double gate SOI structure depends on the film thickness. At very thin semiconductor films, the energy quantization results mainly from the confinement of electrons in the semicondcutor. Then the discrete energy level can be expressed by the rectangular well approximation.

Figure 3. Conduction Band and discrete Energy

Levels in the semiconductor film.

At larger semiconductor film thickness two additional potential subwells are created at both surfaces which can confined the lowest electron states E1 (Figure 4). To be able to simulate these effects, the 2D Electron Gaz has to be described by the self-consistent resolution of Schrodinger and Poisson equations.

Figure 4. Conduction Band and discrete Energy Levels

in the semiconductor film. Note the two

subwells for the lowest Energy Level E1.

For the reason explained abve, one interesting point is that if the semiconductor film is thin enough, the maximum of electron concentration can be located in the middle of it (Figure 5), supporting the concept of volume inversion [2]. Again the classical theory gives the electron concentration always maximum at the semiconductor surface.

Figure5. Influence of Silicon film thickness

on electron concentration distribution.

On top of that, one can observed that the maximum charge density increases and then decreases as a function of Tsi [5]. Figure 6 and 7 illustrate and explain this aspect. In Figure 6 it is sown that for the thinnest thicknesses, the charge appears to be mainly controlled by the lowest Energy level. Indeed Energy Levels below the Fermi Level are full of electrons whereas Energy Levels above the Fermi Level are empty. E2 in Figure 6 for Tsi=1.5nm is located above the Fermi level and thus empty of electrons.

Figure 6. Discrete energy Levels(E1
E3) as a function of Si film thickness.

Figure 7. First Discrete Energy Levels in the semiconductor

film as a function of Si film thickness.

However, due to the confiment, the distance between the first Energy level and the Fermi level increases when the film thickness decreases, which conducts to a decrease of the charge for a film thickness below 5nm (Figure 7).

We have quantify the total number of electrons in the channel using one of the powerful features of the extract routine available in deckbuild. The key word to use to do that in the EXTRACT statement is 2D.AREA, which allow to perfrom double integral of any quantities present in the structure file. Refer to [6] for more details. We show that it exists an "optimal" Si-film thickness (Figure 7), for which the total number of electrons is maximum. For Na=1e18 this is verified for Tsi=3.5nm.

As a consequence this result has a direct impact on the design of GAA-SOI Transistor or more generally Double Gate Transistor since it has been demonstrated that it exist an optimized film thickness leading to a maximum of electrons concentration. The idea that consist of reducing the film thickness to obtain better performance seems no more valid in this case. We have investigated the influence of doping of the Si-Film and it appears that the "optimized" thickness increases when the doping level in the film decreases (Figure 9). Indeed an "optimized" film thickness is around 7nm for Na=1e17.

V. CONCLUSION

As a conclusion we have demonstrated in this article the importance of the use of Schrodinger-Poisson calculation to evaluate performance of novel devices like Double Gate Transistor (i.e GAA-SOI Transistor) where the dimension (Si film thickness) are very small and thus leads to confinement and quantization. Based on this study we have shown that the optimization of performance of such devices will not be achieved by only reducing the thickness of the film (as predicted by the classical approach) but that it exist on optimum of Si-film thickness that can potentially conduct to better performance.

References

[1] J.P. Colinge et al. IEDM (1990) p. 595

[2] F. Balestra et al. IEEE Electron Device Lett. 8, 410 (1987)

[3] B. Majkusiak et al. IEEE Trans. Electron Devices 45, 1127 (1998)

[4] "

ATLASUser's Manual- Vol 1" p. 3-85[5] S.Monfray et al. ESSDERC (2000) p. 336

[6] "

Interactive ToolsUser's Manual"- Vol 1 p. 5-16