Topology for Substrate Routing
in Semiconductor Package Design

Semiconductor devices are at the forefront of innovation in the information technology (IT) industry and play an essential role in driving innovations in areas such as consumer electronics, telecommunications, artificial intelligence or data analysis and security. Although semiconductor devices play such a pivotal role in IT innovation, the integrated circuit (IC) packaging process of semiconductor devices still heavily relies on human expertise. For substrates in, for example, chip-scale packages such as multi-layered Fine Pitched Ball Grid Array (FBGA) packages as illustrated in Fig. 1, most of the design process is about finding the optimal connections between bond fingers, vias and solder balls. Given the variety of types for semiconductor packages, the problem of substrate routing is challenging. As a result, substrate routing problems are often solved with the help of routing methods that are implemented in many computer-aided design (CAD) solutions. In line with recent advances in Electronic Design Automation (EDA), in this work, we outline a new routing method for package substrate design that competes with the performance of other routing methods.

The problem of finding non-intersecting paths that connect a set of start and end points on a plane is one of the oldest problems in computational geometry and graph theory. We know that Dijkstra’s algorithm and the A*-algorithm [9, 14] are examples of graph traversal algorithms, which are used to solve such routing problems. However, substrate routing becomes exponentially more complicated with an increasing number of start and end point pairs.

Most routing algorithms such as Dijkstra’s algorithm, the A*-algorithm and other grid-based Maze Router algorithms [18, 17, 16, 1, 6] are known as geometrical routers. Their disadvantage is that when start and end point pairs are connected sequentially on consecutive shortest paths, it becomes increasingly more likely that there will be not enough clearance left for consecutive connections between pairs of points. This problem with geometrical routers is illustrated in Fig. 2.

In this work, we are interested in a different class of routers known as topological routers [10]. In order to connect fully all points, topological routers aim to find the topological class of the connections first, i.e. the relative positions of paths. After the topological class of the connecting paths is found, with a choice of representation scheme, absolute coordinates are assigned to represent the routing result in real space. This avoids situations where there is a lack of clearance as it is often the case for geometrical routers. For topological routers, paths can always be inserted between already routed paths in order to solve the connection problem. Fig. 2 illustrates this difference between geometrical and topological routers.

Our work is based on the concept of topological routers and proposes a novel topological representation and routing algorithm for substrate routing that competes with the performance of conventional geometrical routers. We make use of topology, more specifically the study of 2-manifolds and polygonal schema [13, 20, 11, 12] in mathematics in order to topologically transform the package substrate into a simpler abstract environment where routing design can be performed more straightforwardly.

In an earlier work [21], we outlined the general principle of our new method for general routing problems. In the current work, we extend our proposal with a focus on the problem of substrate routing in semiconductor chip package design. In particular, we apply our substrate routing method to an explicit example of a Fine Pitch Ball Grid Array (FBGA) package.

Note that our work concentrates on a substrate routing method that finds a fully connected routing solution and does not take into account other metrics such as the wire length or optimal placement of via points.²²2Our routing method can be adjusted to take into account an optimization metric and this will be the subject of upcoming work. Our work also concentrates on signal routing in substrates where the routing problem involves connections between a single start and a single corresponding end point.³³3Multi-pin routing that occurs in power and ground routing or plating lines can be covered in a generalized version of our method, which we plan to cover in future works.

We test our routing method’s performance against geometrical routers and conclude with a summary of results and an actual FBGA package substrate design that was completed using our new routing method.

In this work, we propose based on our earlier work in [21] a new method of solving routing problems that occur during the package substrate design process by using topology. The idea of making use of concepts in topology for designing circuits is not new as shown by the works on rubber-band routing in [10, 8]. These works discuss how certain design features in circuit design can be altered without changing the connections between points, i.e. the topology of the paths, as illustrated in Fig. 3. Moreover, they give an insight into how paths can be bent and moved in such a way that problems of clearance occurring with traditional geometrical routers can be avoided.

Several routing algorithms have been proposed for EDA since the 1990s [17, 16, 8, 6], which are based on the idea of grid-dependent geometrical routers. Moreover, more recent work in EDA considers applications and improvements on geometrical routers in areas such as length matching routing [7], escape routing [4, 2, 3, 22], routing with obstacle avoidance [19] and pin assignment and placement algorithms [15]. In comparison, topological routers have been studied less extensively [10, 8] although, as mentioned above, they have considerable advantages over geometrical routers.

In contrast to the developments made in geometrical routing, our work tries to push forward the development of topological routing. In particular, our work proposes the use of a novel topological transformation to completely transform the substrate routing environment into a topologically equivalent environment. This is a completely new approach for routing in package substrates. Our proposed transformation maps the routing problem to a topologically simpler space where the problem can be solved more straightforwardly. This is the case when in the new environment only relative positions are preserved under the transformation. Given that the transformation is reversible, after all nets are connected, the space with the routing result is transformed back to its original substrate environment.

Such topological transformations and representations that preserve relative positions rather than absolute positions occur extensively in the study of compact 2-manifolds through polygonal schema [13]. These were introduced in mathematics to study the topology of compact 2-manifolds and are particularly useful in representing the homotopy of paths on these manifolds [11]. As a result, polygonal schema appeared also extensively in relation to so-called non-crossing walk problems on compact 2-manifolds [20, 12].

Let us illustrate briefly the concept behind polygonal schema using one of the simplest compact 2-manifolds, the Riemann surface of genus 1, which is also known as a torus or doughnut. The torus can be represented by a rectangle when opposite sides of the rectangle are identified with each other. Any such simple convex polygon together with a boundary gluing pattern shown in Fig. 4 is known as a polygonal schema of the represented 2-manifold. Using the example of the 2-torus, we learn that a rectangle with its opposite boundary sides identified with each other is topologically equivalent to a torus. We can see from this example that even though a torus is 3-dimensional, it can be much more straightforwardly represented by its 2-dimensional polygonal schema.

We claim that a semiconductor package substrate, which usually contains multiple interconnected layers, can be described topologically in terms of polygonal schema. Substrate layers, which are connected by vias, can be separated and individually represented by polygonal schema. Because we split the layers for the topological transformation, each layer has its layer-specific start and end points corresponding to either pins, solder balls or vias. We keep track of which via connects which layers together so that when we reverse the topological transformation, we are able to sew back together the vias between each pair of layers to form the original multi-layered package substrate as shown in Fig. 5. Note that the locations of the via points plays an important role in the overall global routing solution. Since we focus on the problem of finding a fully connected routing solution and consider no other optimization metrics, we refer to future work on optimizing the routing solution using our method.

In the following section, we describe how we make use of the topological transformation specific to our problem and describe a method of how to complete the routing in the topologically transformed routing environment.

Let there be a set $S$ of start points $s_{i}$ and a set $T$ of end points $t_{i}$ with pairwise identification $s_{i}\rightarrow t_{i}$. For our routing problem, we call such a pair a net. These points are on a plane bounded by $B$ as shown in Fig. 6 (a). In order to transform this environment, we introduce trees $R$ consisting of a set of edges $r_{i}$ such that these edges have at their ends either $s_{i}\in S$, $t_{i}\in T$ or $b_{i}\in B$. All points in $S$ and $T$ are each connected to a single tree. Note that a tree $R$ is always connected by exactly one edge with the boundary $B$ at a point $b_{i}$ as shown in Fig. 6 (b). These trees $R$ can be found using a minimum spanning tree algorithm such as Kruskal’s algorithm.⁴⁴4Note that the choice of method for finding the spanning trees may lead to a single tree. Furthermore, the choice will impact the routing result and leads to questions about optimization that will be studied in future work [5]. Such an algorithm needs to be generalized such that each tree $R$ gets connected to the boundary $B$ at a point $b_{i}$ by a single edge $r_{i}$. The start and end points do not need to be connected to $B$ by a single tree $R$. Each point can be connected to the boundary $B$ by separate trees where each tree is separately connected to $B$.

Our proposed topological transformation cuts the plane along all the edges $r_{i}$ such that all points in $S$ and $T$ are now placed on a new boundary that includes the cut-lines along $r_{i}$ as shown in Fig. 7. The cutting process splits some of the points $s_{i}$ and $t_{i}$ to multiple copies if the original points are connected to more than one tree edge $r_{i}$. The boundary points at which trees are attached to the original boundary $B$ are always separated into a pair $b_{i}$ and $b_{i}^{\prime}$. We also notice that during the cutting process the edges $r_{i}$ separate into pairs $r_{i}$ and $r_{i}^{\prime}$.

We pinch the edges $r_{i}$ and $r_{i}^{\prime}$ originating from the trees $R$ in such a way that they are also represented by points on the new boundary $H$ as shown in Fig. 8. As a result, the start points $s_{i}$, end points $t_{i}$, the tree edges $r_{i}$, boundary points $b_{i}$ and their corresponding partners generated by the cutting process are all represented as points on a single combined boundary $H$ as shown in Fig. 8.

The combined boundary $H$ can be deformed into a circle as illustrated in Fig. 9. We call this representation of the original substrate layer the Circular Frame. The order in which the points appear along the circle is the same as they appear when one traverses $H$ in a given orientation as shown in Fig. 9.

The Circular Frame is topologically equivalent to the original substrate layer where the routing is taking place. The advantage of using the Circular Frame representation of the routing problem is that paths connecting pairs of points are represented as straight line segments connecting points on the boundary of the Circular Frame. These points are either start or end points of the original path, points representing $r_{i}$ or $r_{i}^{\prime}$, or points on the original boundary $B$. When a path is connected to $r_{i}$ or $r_{i}^{\prime}$ in the Circular Frame, it corresponds in the substrate layer to a path that crosses the associated tree edge $r_{i}$ as illustrated in Fig. 10. A further advantage of the Circular Frame is that line intersections can be easily detected by going through the ordering of line ends on the boundary of the Circular Frame.

The fact that the topological transformation is reversible enables us to solve the routing problem in the simpler Circular Frame environment and then transform the routing solution back to the original substrate layer environment. This is done by reversing the transformation as illustrated in Fig. 10. Within the Circular Frame, the routing problem is simply a problem of connecting points on the boundary of a circle with non-intersecting straight line segments as illustrated in Fig. 10 (a).

In this section, we outline a method of connecting the nets in the Circular Frame. As noted in the section above, although the Circular Frame is topologically equivalent to the original planar substrate layer bounded by $B$, it simplifies the routing problem to a problem of connecting points on a circle with straight line segments that do not intersect in the interior of the circle. The following section outlines how the Circular Frame simplifies the routing problem.

Starting from a Circular Frame with no points connected, as illustrated in Fig. 9 (b), we can choose to connect the first net, i.e. $s_{1}$ with $t_{1}$. Due to the cutting process of the original routing plane, as shown in Fig. 8, the end point $t_{1}$ is split into 3 copies in the Circular Frame, i.e. $t_{1}$, $t_{1}^{\prime}$ and $t_{i}^{\prime\prime}$. We note that in the Circular Frame, connecting $s_{1}$ to either $t_{1}$, $t_{1}^{\prime}$ or $t_{1}^{\prime\prime}$ is possible. In the actual routing plane, the choice will determine in which direction the connecting path is going to enter the end point $t_{1}$ in the original substrate layer environment.

For the moment, without loss of generality, let us assume that we connect in the Circular Frame $s_{1}$ with $t_{1}^{\prime\prime}$ as illustrated in Fig. 11. Note that any connection between two points in the Circular Frame can be realized in terms of straight line segments that do not intersect in the interior of the Circular Frame. Due to the line segment connecting $s_{1}$ with $t_{1}^{\prime\prime}$, the Circular Frame gets divided into two sections, which we call slices. Fig. 11 shows the two slices $\sigma_{1}$ and $\sigma_{2}$. Each slice has its own boundary with a subset of points from the boundary of the Circular Frame. For our example in Fig. 11, the two slices $\sigma_{1}$ and $\sigma_{2}$ have the points $\{s_{1},t_{1}^{\prime\prime},r_{1}^{\prime},b_{1}^{\prime},b_{1},r_{1},t_{1},r_{2},s_{2},r_{3}\}$ and $\{s_{1},r_{3}^{\prime},s_{2}^{\prime},r_{2}^{\prime},t_{1}^{\prime},r_{4},t_{2},r_{4}^{\prime},t_{1}^{\prime\prime}\}$ each on their respective boundaries. Note that the points that we connected, $s_{1}$ and $t_{1}^{\prime\prime}$, are both shared by the boundary of the two slices. The line segment, which connects $s_{1}$ with $t_{1}^{\prime\prime}$, is precisely the overlap of the two boundaries.

As shown in Fig. 12, the two slices $\sigma_{1}$ and $\sigma_{2}$ are not completely disconnected. We recall that the points $r_{i}$ and $r_{i}^{\prime}$ that represent tree edges in the Circular Frame always come in pairs as explained in Section 3. $r_{i}$ and $r_{i}^{\prime}$ precisely identify the tree edges along which the original substrate layer was cut in order to obtain the Circular Frame as illustrated in Fig. 7. Accordingly, they represent points that need to be pairwise glued together when the Circular Frame is transformed back to the original substrate layer environment. Fig. 12 shows these pairwise connections as dotted lines. The two slices $\sigma_{1}$ and $\sigma_{2}$ in Fig. 12 are connected by the pairs $(r_{2},r_{2}^{\prime})$ and $(r_{3},r_{3}^{\prime})$.

When we now attempt to connect start point $s_{2}$, which is on the boundary of $\sigma_{1}$, with its corresponding end point $t_{2}$, which is on the boundary of $\sigma_{2}$, we have to move between the two slices $\sigma_{1}$ and $\sigma_{2}$. As we noted above, the two slices are connected by the point pairs $(r_{2},r_{2}^{\prime})$ and $(r_{3},r_{3}^{\prime})$. Without loss of generality, by choosing point pair $(r_{2},r_{2}^{\prime})$, $s_{2}$ is connected with $r_{2}$ in $\sigma_{1}$, and then its partner $r_{2}^{\prime}$ is connected with $t_{2}$ in $\sigma_{2}$ as illustrated in Fig. 13. Note that by connecting $s_{2}$ to $t_{2}$ through the point pair $(r_{2},r_{2}^{\prime})$, the original slices $\sigma_{1}$ and $\sigma_{2}$ are each divided into two slices by the two line segments connecting $s_{2}$ with $r_{2}$ and $r_{2}^{\prime}$ with $t_{2}$. As a result, we end up with a total of four slices.

There is also the possibility that more than one path goes through a point pair $(r_{1},r_{1}^{\prime})$ as shown in Fig. 14. In the example in Fig. 14, both $(s_{1},t_{1}^{\prime})$ and $(s_{2},t_{2})$ are connected through the point pair $(r_{1},r_{1}^{\prime})$. In such a situation, one has to make sure that the slice containing the origin point and the slice containing the destination point are in the same order. Let us define the order $o(\sigma,r_{i})$ of $\sigma$ with respect to the point $r_{i}$ as the segment number of $\sigma$ attached to $r_{i}$ in the Circular Frame when one counts anti-clockwise around $r_{i}$ starting from the boundary of the Circular Frame. In analogy, let us define the order $o(\sigma,r_{i}^{\prime})$ of $\sigma$ with respect to the point $r_{i}^{\prime}$ as the segment number of $\sigma$ attached to $r_{i}^{\prime}$ in the Circular Frame when one counts clockwise around $r_{i}$ starting from the boundary of the Circular Frame. For example, in Fig. 14, we note that $o(\sigma_{1},r_{1})=o(\sigma_{5},r_{1}^{\prime})=1$, $o(\sigma_{2},r_{1})=o(\sigma_{4},r_{1}^{\prime})=2$ and $o(\sigma_{3},r_{1})=o(\sigma_{3},r_{1}^{\prime})=3$. Accordingly, anything starting in slice $\sigma_{1}$ can go through $(r_{1},r_{1}^{\prime})$ to slice $\sigma_{5}$, not any other slice. Similarly, we have $o(\sigma_{2},r_{1})=o(\sigma_{4},r_{1}^{\prime})$ and $o(\sigma_{3},r_{1})=o(\sigma_{3},r_{1}^{\prime})$, meaning that anything in slice $\sigma_{2}$ can be connected to $\sigma_{4}$ and anything in slice $\sigma_{3}$ can be connected to $\sigma_{4}$ via the edge point pair $(r_{i},r_{i}^{\prime})$. Note that slice ordering is essential to make sure that when the edge points are glued together, the correct slices recombine with each other to give the original substrate layer as shown in Fig. 14 (c).

Following these rules on connecting points in the Circular Frame, we outline a basic algorithm for connecting all points in Fig. 15. In line 12 of the algorithm in Fig. 15, the closest $r_{k}$ to a given point $p$ in a slice $\pi$ is identified by the smallest number of points one needs to pass in order to go from $p$ to $r_{k}$ along the boundary of $\pi$.

We note that the algorithm in Fig. 15 is one example amongst many possible connection algorithms that one can formulate with the help of the Circular Frame. We plan to present variations of this algorithm in future work. For now, the algorithm in Fig. 15 does not have the aim to find the shortest possible paths between start and end point pairs. Instead, the algorithm in Fig. 15 simply has the aim to achieve full connection for all start and end point pairs. In fact, with the Circular Frame and the algorithm in Fig. 15, complete connection is always guaranteed. This is because the Circular Frame is only encoding the topology of the routing problem as illustrated in Fig. 2 and as a result there is no problem of clearance as it is the case for geometrical routers. Furthermore, in signal routing, paths always connect a single start point $s_{i}$ with a single end point $t_{i}$ and hence there is no possibility that a path completely encircles points that need to be connected by other paths in the Circular Frame.⁵⁵5Note that there is an optimization problem in terms of the choice of geometric sketch one uses to represent the routing solution given by its corresponding topological class. The problem of optimization will be the subject of future work. Fig. 16 illustrates an example where the algorithm in Fig. 15 is applied to solve the connection problem in the Circular Frame.

We call the process of transforming the routing result in the Circular Frame back to the original substrate layer the embedding process. As discussed in the sections above, the Circular Frame can be transformed back to the original substrate layer by glueing together the point pairs $(r_{i},r_{i}^{\prime})$ for all $i$. Under this reverse transformation, the topology of the routing result, i.e. the identified paths connecting start points with corresponding end points, is preserved.

The information about which points correspond to which slices in the Circular Frame and the information about which slices are adjacent to each other is called the topological class $T$ [13] of the routing solution. The geometric sketch [10, 13] of a topological class is a specific embedding of the connecting paths in the routing solution.

An example of such a topological class $T_{i}(P_{i},W_{i},H_{i})$ for a path $\rho_{i}$ is the following information:

• 1.

  The set of points $P_{i}=\{p_{k}^{(i)}\}$ a path $\rho_{i}$ passes starting from $s_{i}$ and ending at $t_{i}$.

• 2.

  The set of orientations $W_{i}=\{w_{k}^{(i)}\}$ a path $\rho_{i}$ takes when it passes points $\{p_{k}^{(i)}\}$. If $\rho_{i}$ passes $p_{k}^{(i)}$ anti-clockwise $w_{k}^{(i)}=+1$, if $\rho_{i}$ passes $p_{k}^{(i)}$ clockwise $w_{k}^{(i)}=-1$, and if $p_{k}^{(i)}=s_{i}$ or $t_{i}$ then $w_{k}^{(i)}=0$.

• 3.

  The set of heights $H_{i}=\{h_{k}^{(i)}\}$ a path $\rho_{i}$ has when it passes points $\{p_{k}^{(i)}\}$. A height $h_{k}^{(i)}=m$ indicates that between $\rho_{i}$ and $p_{k}^{(i)}$ there are $m-1$ other paths passing $p_{k}^{(i)}$. If $p_{k}^{(i)}=s_{i}$ or $t_{i}$ then $h_{k}^{(i)}=0$.

Note that $T_{i}(P_{i},W_{i},H_{i})$ for any path $\rho_{i}$ can be obtained from the Circular Frame of the routing solution.⁶⁶6 We note that one can introduce several other topological classes that encapsulate the routing result in the Circular Frame, for instance including the edges $r_{i}$. We hope to present further versions in future work. In the example in Fig. 16, for path $\rho_{4}$ connecting $s_{4}$ with $t_{4}$, $T_{4}$ is given by $P_{4}=(s_{4},t_{3},t_{2},t_{4})$, $W_{4}=(0,+1,+1,0)$ and $H_{4}=(0,1,1,0)$.

We make use of the rubber-band sketch from [10, 8] in order to represent the topological class of the routing result from the Circular Frame on the original planar environment. Fig. 3 (a) shows the rubber-band sketch of the same topological class represented in Fig. 3 (b). A characteristic feature of the rubber-band sketch is that paths are represented as line segments that can have any angle and the line segments are connected by arcs whenever the path passes a point. Fig. 17 illustrates an example of a topological class and its corresponding rubber-band representation.

For the purpose of this work, which is to present a new topological routing method that results in a topological class of a fully-connected routing result via the Circular Frame, we keep the review on topological classes and the rubber-band sketch representation short and refer to the works in [10, 8].

Let us design experiments to compare the performance of the proposed routing algorithm based on the Circular Frame (CF) with variations of the A*-algorithm (AS).

Let us summarize the results of the experiments in the following section.

In this paper, we have proposed a new method based on our earlier work in [21] for solving substrate routing problems using topology. Our proposed topological transformation of the original routing environment into the Circular Frame has accelerated in experiments the substrate routing process significantly in comparison to grid-based geometrical routers such as the A*-algorithm. Moreover, the Circular Frame representation guarantees for substrate routing problems with start and end point pairs full connection as a topological router.

In addition, Fig. 22 shows a 2-layered substrate for a FBGA package with 200 solder balls and a completed routing design that was obtained using our new Circular Frame algorithm. Our experiments and the positive routing results on real semiconductor package substrates are a clear indication that our new Circular Frame routing method has the potential to significantly improve and at the end fully automate the package substrate routing process.

We note that the Circular Frame routing algorithm can lead to different routing solutions given by topological classes depending on via placement, spanning tree generation and even net ordering during the routing process in the Circular Frame. Moreover, in our work we have given only a single basic routing method based on the Circular Frame representation and depending on other routing algorithms based on the Circular Frame representation, the routing result can differ significantly. Finding the most optimal routing solution based on the Circular Frame representation depends on metrics such as wire length or wire widths and is an optimization problem that we hope to cover in future work.

We are currently testing the Circular Frame routing algorithm on larger FBGA packages and other package designs. Moreover, beyond semiconductor package design, we are applying our routing method on problems related to the design of printed circuit boards (PCB) and the logistics and manufacturing industry. We hope to report on our progress in these areas in future work.

The authors would like to thank Minsoo Kim for suggesting the problem and Seungjai Min and Youngjae Gwon at Samsung SDS for helpful discussions. The authors are also grateful to Joung Oh Yun and Minkyu Jung for helpful guidance during the project, and Chanho Min for collaborating on an earlier project.