Optimizing Chip Placement on Circular Wafers Using Breadth-First Search (BFS)

Introduction

Semiconductor manufacturing involves fabricating microchips on circular silicon wafers. Since microchips (or dies) are rectangular, an efficient arrangement is needed to maximize yield and minimize waste. This article explores how Breadth-First Search (BFS) can be used to optimize chip placement, defect detection, and exposure path planning in wafer lithography.

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The Challenge of Chip Placement on Circular Wafers

1. Maximizing Yield

The goal is to fit the maximum number of chips on the wafer while ensuring no overlap and minimum wastage at the wafer edges.

2. Avoiding Defects

Some sections of the wafer may contain defects due to process variations. BFS can help identify and isolate defective regions.

3. Optimizing Exposure Paths

Lithography machines expose patterns onto the wafer in a systematic scanning pattern. BFS can be used to find the most efficient exposure sequence, reducing machine movement and time.

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Using BFS for Optimal Chip Placement

Approach:

We model the wafer as a graph, where:

• Each node represents a possible chip position.
• Edges connect adjacent chip positions.
• BFS is used to explore valid placements, starting from the center and expanding outward layer by layer.

Algorithm for BFS Chip Placement:

1. Start from the center of the wafer (optimal die placement begins at the core).

2. Expand outward using BFS, checking valid placements.

3. Ensure no chip extends beyond the circular boundary.

4. Continue placing until no more valid positions are found.

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C++ Implementation:

#include <iostream>
#include <queue>
#include <vector>
#include <cmath>
using namespace std;

struct Point {
    int x, y;
};

void bfs_chip_placement(int wafer_radius, int chip_size) {
    queue<Point> q;
    vector<Point> placements;
    q.push({0, 0});
    vector<vector<bool>> visited(2 * wafer_radius + 1, vector<bool>(2 * wafer_radius + 1, false));

    while (!q.empty()) {
        Point p = q.front();
        q.pop();

        int x = p.x, y = p.y;
        if (x * x + y * y > wafer_radius * wafer_radius || visited[x + wafer_radius][y + wafer_radius])
            continue;

        placements.push_back({x, y});
        visited[x + wafer_radius][y + wafer_radius] = true;

        int directions[4][2] = {{chip_size, 0}, {-chip_size, 0}, {0, chip_size}, {0, -chip_size}};
        for (auto d : directions)
            q.push({x + d[0], y + d[1]});
    }

    cout << "Chip Placements:\n";
    for (auto p : placements)
        cout << "(" << p.x << ", " << p.y << ")\n";
    cout << "Total chip placements on wafer with radius " << wafer_radius << " are " << placements.size() << " with size of " << chip_size << endl; 
}

int main() {
    int wafer_radius = 50;  // Example wafer radius
    int chip_size = 5;      // Example chip size
    bfs_chip_placement(wafer_radius, chip_size);
    return 0;
}

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How This Algorithm Works

1. Graph Representation

• The wafer is modeled as a grid.
• Each valid chip position is treated as a graph node.
• BFS expands chip placement outward, ensuring no overlaps.

2. Boundary Check

• Ensures that no chip placement exceeds the circular wafer boundary.

3. Efficient Traversal

• BFS systematically expands outward, guaranteeing maximum wafer utilization.

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BFS for Defect Detection

Defects in wafers can cause clusters of faulty dies. BFS can be used to identify defective regions and mark them for exclusion.

Approach:

• Treat the wafer as a graph, where each die is a node.
• Use BFS to detect connected defective regions.
• Isolate and prevent defective chip placement.

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BFS for Lithography Exposure Path Optimization

In photolithography, exposure must follow a systematic pattern to reduce laser movement and increase throughput. BFS can optimize the scanning path.

Approach:

• Model the wafer as an exposure grid.
• Use BFS to determine the most efficient exposure sequence.
• Expand in a spiral or concentric pattern for efficiency.

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Conclusion

Breadth-First Search (BFS) is a powerful technique in semiconductor lithography for:

• Chip placement optimization (maximizing wafer utilization).
• Defect detection and isolation (ensuring quality control).
• Efficient lithography exposure paths (minimizing processing time).

By leveraging BFS, manufacturers can reduce waste, improve yield, and optimize wafer production, making it an essential tool in modern chip fabrication.