4.2 Propaedeutic Example

The idea behind this example is to test the heuristic error estimator described in Section 4.1 on a bottom-of-the-line case. The chosen diffusion problem is one-dimensional in its nature but calculated on a three-dimensional test structure, and with proper chosen boundary conditions, an analytical solution can be given. This helps to modify the error estimator and calculate the distance between the piecewise calculated gradient field and the analytical one, as part of the error estimation. Afterwards an anisotropic Hessian refinement is applied to see, if a finer mesh really reduces the estimated error. Finally, the modified error estimator is compared to the original one given in Section 4.1.

The balance of quantity $C$ in a bounded domain $\Omega$ in general form is given by

$$\frac{\partial C}{\partial t} + \vec{\nabla} \cdot \vec{J} = S,$$ (4.6)

where $\vec{J}$ gives the flux, $S$ is called production rate within $\Omega$ , and ${\partial C}/{\partial t}$ gives the increase per time of $C$ within $\Omega$ . If $\vec{J}$ is given by Equation (4.1), then Equation (4.6) forms a parabolic partial differential equation,

$$\frac{\partial C}{\partial t} - D \nabla^2 C = S,$$ (4.7)

where $D$ denotes the diffusion coefficient. In a first approximation the diffusion coefficient $D$ is given by the so-called Arrhenius law,

$$D = D_0 \exp \Big(-\frac{E_{act}}{\mathrm{k_B}T}\Big),$$ (4.8)

where the pre-exponential factor $D_0$ is a material dependent parameter, $\mathrm{k_B}{}$ is Boltzmann's constant, $T$ the temperature, and $E_{act}$ the activation energy.

For the manufacturing cycle of semiconductor devices two conditions of the source term free Equation (4.7) with $S=0$ , are important:

• Constant surface concentration

  $C(0,t)=C_{surf}$

• Constant dose (drive in diffusion)

  $C_d(t) = \int\limits_0^{\infty} C(x,t) \; dx = \operatorname{const}$ .

The following deals with the case of ``constant dose'', which is also referred as so-called drive in diffusion.

For a one-dimensional case under consideration of the following conditions:

$$C_d(t) = \int\limits_0^{\infty} C(x,t) \; dx = \operatorname{const} \quad \underset{x \rightarrow \infty}{\lim} \; C(x,t) = 0$$ (4.9)

a one-dimensional solution of the source term free balance equation, see Equation (4.7) with $S=0$ , is given by:

$$C(x,t) = \frac{C_d}{\sqrt{\pi D t}}\exp \Big( - \frac{x^2}{4 D t} \Big).$$ (4.10)

The numerical representation of time continuous problems based on FE methods yields also a discretization in time. Therefore the ``continuous time'' is split into small slices, named time steps.
In the following a to unity scaled quantity as temporary result of a drive in diffusion simulation at the time step $t=t_k$ is under examination regarding the error estimator discussed in Section 4.1. Figure 4.2 shows the corresponding graph and the scaled norm of the second derivative of the one-dimensional test case which was applied to a three-dimensional test structure depicted in Figure 4.3(a).

Figure 4.2: One-dimensional scalar data test profile and norm of the second derivative (scaled to unity). This profile is used in $x$ direction in the mesh structure shown in Figure 4.3(a). The mesh adaptation according to the second derivative can be seen in Figure 4.4.

Due to the fact, that an analytical solution of the parabolic partial differential equation is available, the error estimator can be modified in such a way, that the element gradient can directly be compared with the gradient of the analytical solution, which is given by:

$$\frac{\partial C(x,t)}{\partial x} = -\frac{C_d x}{2 \sqrt{\pi} (Dt)^{3/2}}\exp \Big( - \frac{x^2}{4 D t} \Big).$$ (4.11)

As the sine qua non of this propaedeutic example and the exceeding case of a one-dimensional problem calculated on a three-dimensional test structure the error estimator given by Equation (4.5) can be rewritten at time $t=t_k$ , with respect to Equation (4.11) as:

$$\mu_{T}:=\int_{T}\Vert\vec{G}(\xi,\eta,\zeta)-\frac{\partial C(x,t_k)}{\partial x}\Vert _{2} dV.$$ (4.12)

The coloration of Figure 4.3(b) gives the normalized estimated error according to the error estimator presented in Section 4.1 with the modification expressed through Equation (4.12). Regions with red colored tetrahedrons indicate that a high error value was calculated. One can clearly observe that in regions with high curvature, i.e. with high second derivatives which can be seen as measure for the curvature of the initial profile (cf. Figure 4.2), a higher error is located. This note gives rise to the idea that the Hessian refinement method described in Section 3.4 can produce a finer anisotropic mesh in the region of higher estimated error. The difference now is that only those tetrahedra with an error higher than $60 \%$ of the maximum error are used for refinement and others are untouched. This means that not the whole structure is involved in the refinement process and the refinement is kept local.

The anisotropic refinement based on the Hessian matrix of the profile takes place only in regions of high curvature as shown in Figure 4.4. The anisotropy is mostly restricted to the $x$ -direction of the test structure while other directions are not influenced. One can clearly observe that the refinement at $x=0.5$ is almost zero, which is forced through the second derivative of the profile which is exactly zero at $x=0.5$ . On the left upper end of the structure the mesh granularity in $x$ -direction shows the most dense mesh, because of a high curvature of the profile and, therefore, a high second derivative which yields to a strong dilation of the anisotropic metric. The region between $x=0.75$ and $1.25$ shows a high curvature too. According to the norm of the second derivative, this forming is not as strong as in the region around $x=0$ and, therefore, has less influence on the refinement procedure.

The error estimator was also applied to the refined structure given in Figure 4.4 and compared to the previous results calculated on the mostly regular, coarse mesh. The results are shown in Figure 4.5 where the gray-scaled bars reflect the coarse structure, whereas the refined one is given in red. The error was normalized to the maximum error of the coarse structure (see Figure 4.3(b)) and divided into ten error classes from $0.1$ (low error) to $1.0$ (high error), respectively. Since the number of tetrahedrons changed after the refinement, the amount regarding the error class is given in $\%$ . The error estimation is performed with Equation (4.12) for each element of the domain. A clear shift towards lower error classes can be observed for the refined structure. Within the two lowest classes an increase of elements of approximately $10\%$ was reached. All other classes are lowered and the maximum error class, carrying the highest error of the coarse structure, vanished completely.

The modified error estimator given by Equation (4.12) was also compared to the original one discussed in Section 4.1 where a piecewise linear gradient field was constructed of a piecewise constant gradient field, see Equation (4.5). The difference between those two estimators is smaller than $2.5 \%$ of the maximum error value. It was also observed that for regions of ``smooth'' gradient fields the difference falls beyond $1 \%$ . This observation gives rise to the assumption that the heuristic error estimator described in Section 4.1 detects excellently regions of high gradient variations.

Figure 4.3: Figure 4.3(a) shows the initial coarse mesh constellation with the one-dimensional test profile depicted in Figure 4.2. Due to the exceptional case that for this illustration also an analytical solution exists, the modified error estimator given by Equation (4.12) can be used to obtain error values for the initial mesh structure depicted in Figure 4.3(b).

Figure 4.4: The Hessian refinement method was used to obtain an anisotropic refinement in regions of high second derivatives of the initial attribute, indicated through a modified gradient recovery based error estimator.

Figure 4.5: Histogram of the error distribution before (gray scale) and after refinement (red). The error was normalized to the maximum error and divided into ten error classes from $0.1$ (low error) to $1.0$ (high error). Since the amount of tetrahedrons changes after the refinement, the amount regarding the error class is given in $\%$ . The error estimation is performed with Equation (4.5) for each element over the domain and a clear shift from higher to lower error classes has been achieved by the refinement.