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MICRO-BUMP ASSIGNMENT FOR 3D ICS USING ORDER RELATION TA-YU KUAN, YI-CHUN CHANG, TAI-CHEN CHEN DEPARTMENT OF ELECTRICAL ENGINEERING, NATIONAL CENTRAL UNIVERSITY, TAOYUAN, TAIWAN 1

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OUTLINE Introduction Problem formulation Algorithm Experimental Results 2

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INTRODUCTION 3D ICs need to consider inter-die connection problems. Therefore, 3D ICs have connection issues between silicon-base layers and routing layers. Two popular technologies, through-silicon vias (TSVs) and micro bumps, are adopted widely. The redistributed layer (RDL) is used to solve the alignment problem. By attaching RDLs on the junction between dies, signals can be distributed to appropriate positions on RDLs. 3

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INTRODUCTION 4

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A suitable micro-bump location reduces the wirelength on both upper and lower RDLs. This paper propose a micro-bump assignment algorithm based on the order relation, which is generated by adopting a 45-degree coordinate mapping and considering the relative location of terminals in the upper and lower RDLs simultaneously. INTRODUCTION 5

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PROBLEM FORMULATION The micro-bump location significantly affects the routabilities of the adjacent upper- and lower-RDL. In order to achieve higher routabilities and shorter wirelength in both upper and lower RDLs, it is necessary to avoid detour paths in both RDLs simultaneously. 6

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PROBLEM FORMULATION 7

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ALGORITHM 8

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Since the RDL routing allows 135- degree routing angle, shortest paths of two terminals are inside the X bounding box. 9 We define the X bounding box as the smallest bounding box formed by the 0°/45°/90°/135° line segments that enclose two terminals.

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ALGORITHM 45-Degree Coordinate mapping N- and P-axis are the clockwise 45-degree coordinate rotation on X- and Y-axis, respectively. 10 The terminal orders of a, b, c, and d on N- and P-axis are (c, b, a, d) and (a, d, c, b), respectively. The relative location of terminals on the plane could be verified by the orders on N- and P-axis.

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ALGORITHM 11 As shown in Fig. 5(a), since the relative location of m a and m b is different from that of u a and l b or l a and u b, two nets have edge crossing problem in the lower RDL. As shown in Fig. 5(b), since the relative location of m a and m b is the same as that of u a and l b, two nets have no edge crossing problem in both upper and lower RDLs.

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ALGORITHM 12 Flow of order determination

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ALGORITHM 3Net Picking Since two nets have two possible terminal orders (u a -l b and l a -u b for nets a and b), two candidate micro-bump orders will be generated. It implies that a unique micro- bump order for two nets cannot be determined. Picking three nets is the most suitable number to determine the micro-bump order for non-crossing straight paths. 13

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ALGORITHM 3Net Picking Nets are in the unselected group initially. Nets with shorter Manhattan distances have smaller bounding boxes and less routing resources for their shortest non-crossing paths, these nets require higher priority to avoid crossing problems. As a result, the net with the shortest Manhattan distance in the unselected group is picked as Master. 14

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ALGORITHM 3Net Picking Nets which have the higher detour level are picked as Servants. At each iteration, one Master and two Servants (two highest detour-level nets for Servant-1 and Servant-2) are picked from the unselected group to form a 3-net. The detour level is graded into the Inner Bounding Box (IBB) and the Outer Bounding Box (OBB). Since OBB always incurs detour paths, we define that OBB has higher detour level than IBB does. 15

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ALGORITHM 16 Fig. 7. Net-Xi detours round the X bounding box of Master. (a)IBB. Non-crossing paths for net-Xi inside its Manhattan bounding box are existed. (b) OBB. Non-crossing paths for net-Xi inside its Manhattan bounding box are not existed.

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ALGORITHM 17

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ALGORITHM 18

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ALGORITHM Order duplication The order duplication can be applied on determining the micro- bump orders to avoid crossing problems. Obtain two candidates of micro-bump orders by duplicating the specific terminal orders. 19 The candidate table consists of two kinds of information: (1)micro-bump orders on N- and P-axis and (2) assignable region. A micro-bump order is a duplication of a terminal order. Assignable region defines an ideal region for assigning micro bump.

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ALGORITHM Order duplication 20

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21 ALGORITHM

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Cyclic path checking 22

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ALGORITHM 23 The 45-degree coordinate has two axes, so we need to generate two final micro- bump orders for N- and P- axis, respectively. visit / finish time

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ALGORITHM MICRO BUMP SHIFTING There are two kinds of motion affecting on m 4. The first motion is that m 4 is attracted by its signal straight path. The other motion is that m 4 is repelled by adjacent micro bumps such as m 2 on N-axis, m 1 and m 3 on P-axis. 24

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ALGORITHM 25 The shifting rules are defined as follows: Start the shifting on the N-direction before P-direction. Shift micro bumps which have the positive motion earlier than negative motion. Start the shifting from the first micro bump.

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EXPERIMENTAL RESULTS 26

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EXPERIMENTAL RESULTS 27

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THANKS 28

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