6.3 Periodic Boundaries

Certain elements of integrated circuits like busses or memory cells make use of periodic structures [116,117]. As example a part of an interconnect bus is shown in Fig. <6.22>. Often the capacitance between the wires must be extracted which requires the calculation of the electric field, where the wires are connected alternately to 0 V and to $1$ V [118]. An appropriately fine resolution of the simulation area is important for the accuracy and leads unfortunately for such domains as in Fig. <6.22> to the generation of a huge number of nodes. Therefore the simulation process will demand a lot of memory, and the simulation duration can be unacceptably long. Considering the periodicity of the structure of Fig. <6.22> in direction of the x axis there is a smart way to solve the problem by investigating only one geometrical period of the structure. A possible geometrical period is the area shown in Fig. <6.23>. The electrodes for the capacitance extraction consist of one interconnect line connected to 1 V and two parts of the neighbor lines connected to 0 V, respectively, which can be seen in the top view in Fig. <6.24>. The interconnect bus of Fig. <6.22> can be created by periodic spatial continuation of the area of Fig. <6.23> along the x axis. Therefore it is not necessary to simulate the whole area of Fig. <6.22>. It is sufficient to simulate a single cell of periodicity as in Fig. <6.23>. The simulation has been performed by our software Smart Analysis Programs [119]. It is based on the finite element method using tetrahedral meshes. The algorithm for the linear algebraic equations arising from the finite element discretization is based on the iterative conjugate gradient method which uses an incomplete Cholesky pre-conditioning technique to speed up the iteration process [120,121,122,123]. This application is a typical electro-static problem described by the Laplace equation (4.12) for the electric potential $\varphi$ . The corresponding numerical analysis is based on scalar finite elements and was discussed in Chapter 4. The solution of the Laplace equation in the dielectric is completely extracted from the data defined on the dielectric boundary. Usually the electrodes are modeled by Dirichlet boundary conditions for the electrostatic potential and the outer boundary of the simulation domain by homogeneous Neumann conditions which can be implemented with the finite element method in a quite natural way. Homogeneous Neumann conditions on a planer surface effect the field in the simulation area in such way, as if the simulation area would be mirrored at the respective boundary face. Such ``mirroring conditions'' can be exploited for the simulation of symmetric structures and periodic structures which exhibit symmetry. However for general periodic structures proper boundary conditions must be implemented which require special treatment.

In this application two faces $\mathcal{A}_{1p} \subset \partial\mathcal{V}$ and $\mathcal{A}_{2p} \subset \partial\mathcal{V}$ are defined as periodic boundary, if:

• Each node from $\mathcal{A}_{1p}$ is uniquely mapped to another node from $\mathcal{A}_{2p}$ and vice versa.
• If $\vec{r}_{1i}$ is the position pointer to a simulation point of $\mathcal{A}_{1p}$ and $\vec{r}_{2i}$ is the position pointer to the corresponding simulation point of $\mathcal{A}_{2p}$ then $\varphi(\vec{r}_{1i}) = \varphi(\vec{r}_{2i})$ for each point of $\mathcal{A}_{1p}$ and $\mathcal{A}_{2p}$ .
• Each node of $\mathcal{A}_{1p}$ has its own neighbor nodes and the neighbor nodes of the corresponding node from $\mathcal{A}_{2p}$ .

Although each two corresponding periodic points are separated in the space, due to the periodic condition, they should behave as if they were attached to each other.

If due to the discretization $m$ points are created, the electric potential $\varphi$ in dielectric $\mathcal{V}_D$ is approximated as (4.13) with (4.22) by the sum

$$\varphi\approx\tilde{\varphi}=\sum_{j=1}^{n}c_j\lambda_j(\vec{r})+\sum_{j=n+1}^{m}c_j\lambda_j(\vec{r}).$$ (6.26)

In (6.26) the unknown nodes are numbered from $1$ to $n$ . The Dirichlet (known) nodes are numbered from $n{+}1$ to $m$ . Thus the finite element method leads to a linear equation system for the first $n$ coefficients $c_j$ (the unknown ones with $j\in[1;n]$ ).

In general the mesh generation software does not order the simulation nodes as in (6.26). To implement the desired node ordering a supplemental auxiliary index array with the length $m$ is allocated. This additional index array is used by the assembling procedure. The first $n$ entries of this index array refer to the nodes in $\mathcal{V}_{D}$ without the nodes on the Dirichlet boundary. The remaining entries refer to the nodes on the Dirichlet boundary (from $n{+}1$ to $m$ ). The additional index assignment of the simulation nodes gives advantages to the implementation of the periodic boundary conditions. Each two corresponding points of the plains $\mathcal{A}_{1p}$ and $\mathcal{A}_{2p}$ get the same index in the additional index array. Thus, they are assembled to the same row in the linear equation system. Due to the element-by-element processing of the simulation volume each periodic point has not only its neighbor nodes but it is also connected to the neighbor nodes of the corresponding periodic point.