1 Equilibrium Diffusion at the Silicon/Silicon-Dioxide Interface

Here we use the vacancy-only model (Section 3.3.3) in combination with the segregation model (Section 3.5.3) to simulate the behavior of phosphorus dopants in the vicinity of the silicon/silicon-dioxide interface.

The extrinsic phosphorus diffusivity is modeled according to [6] assuming that phosphorus atoms diffuse with neutral and negatively charged vacancies,

$$D_{P}=\Bigl[3.85 e^{-3.66 \text{eV}/kT} + 4.44 \Bigl(\frac{n... ....2\Bigl(\frac{n}{n_i} \Bigr)^2 e^{-4.37 \text{eV}/kT}\Bigr] \text{cm}^2/s.$$ (210)

The extrinsic concentration of the electrons $n$ is calculated using (3.13).

The considered MOSFET structure is displayed on Figure 3.3. An initial phosphorus profile is calculated by means of implatation with a dosis $3\cdot 10^{12}$ cm$^{-2}$ and energy $10$ keV (Figure 3.4). The ion-implantation simulation is carried out with the simulator MCIMPL-II [39,40].

Figure 3.3: MOSFET structure

Figure 3.4: The MOSFET structure with a implanted dopant profile of phosphorus with an energy of 10 keV. Phosphorus concentration is given in cm$^{-3}$.

[fillstyle=slopes,slopesteps=10000,slopecolors=0 0.0 0.0 1.0 1000 0.0 0.0 1.0 3000 0.0 1.0 0.0 5000 0.0 1.0 0.0 6000 0.85 0.917 0.160 6200 0.85 0.917 0.160 9000 1.0 0.0 0.0 7,gradangle=90,swapaxes=true,linestyle=none](-3.0,0.4)(2.0,0.0) $\parbox{5cm}{ \vspace*{2.2cm} $1.28 \cdot 10^{13}$\ [6mm] $9.60 ... ... 10^{12}$\ [6mm] $3.20 \cdot 10^{12}$\ [6mm] $1.00 \cdot 10^{6}$\ [6mm] }$

Figure 3.5: Mesh applied for the simulation. Displayed is one-forth of the MOSFET structure.

Figure 3.6: Behavior of normalized a) flux error and b) maximal concentration gradient during the diffusion with the applied segregation model and without it.

a)

b)

Figure 3.7: Penetration of phosphorus into the silicon-dioxide. Phosphorus concentration is given in cm$^{-3}$.

[fillstyle=slopes,slopesteps=10000,slopecolors=0 0.0 0.0 1.0 1000 0.0 0.0 1.0 3000 0.0 1.0 0.0 5000 0.0 1.0 0.0 6000 0.85 0.917 0.160 6200 0.85 0.917 0.160 9000 1.0 0.0 0.0 7,gradangle=90,swapaxes=true,linestyle=none](-3.0,0.4)(2.0,0.0) $\parbox{5cm}{ \vspace*{2.2cm} $1.28 \cdot 10^{13}$\ [6mm] $9.60 ... ... 10^{12}$\ [6mm] $3.20 \cdot 10^{12}$\ [6mm] $1.00 \cdot 10^{6}$\ [6mm] }$

The mesh used for the simulation is presented in Figure 3.5. Since we have expected higher gradients of dopant concentration in the vicinity of silicon/silicon-dioxide interface the simulation mesh is refined in this area with increased number of points in the direction ortoghonal on the silicon/silicon-dioxide interface.

After a diffusion time of $15$ min the dopants have spread deeper into the silicon bulk and penetrated into silicon-dioxide, as can be seen from the Figure 3.7.

The characteristic pileup in the vicinity of the silicon/silicon-dioxide interface can be observed on the basis of the behavior of maximal phosphorus concentration flux and flux error, calculated on the basis of discussion presented in Section 2.9. A pileup effect, as part of the segregation of dopants on the silicon/silicon-dioxide interfaces, is regularly observed in experiments [41,42,43].

For comparison the simulation is carried out with and without the segregation model. The diffusion conditions at the silicon/silicon-dioxide interface produce from the very beginning distinctively higher concentration gradients. The curves presented in Figure 3.6a are normalized to the maximum concentration gradient in the case where no segregation model is applied. The presence of the silicon/silicon-dioxide interface causes a more than 2.4 times larger phosphorus concentration gradient. However, due to a carefully chosen mesh (Figure 3.5) no significant increase of the flux error is observed.