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chip-module (MCM) substrate. These bump bonds are about80 μm in diameter, which makes them very thick whencompared to other interconnects and vias that range from 0.3μm near active devices to 4 μm for top layers. Therefore, forthe full-chip thermal analysis, we focused on appropriatemodeling of the bump bonds and the nearest area. They areshown as red squares in Figure 1. One such bond isillustrated below, in Figure 2.IC1SiliconILD = Inter LayerDielectricIC SiO2 (ILD)Bump Bond24-26 September 2008, Rome, ItalyThere are many layers in every IC stack and package, andtherefore, Equation 1 is solved in each material with itsindividual set of properties, with data transfer at theinterfaces. The source term (S h ) for Equation 1 may beprovided either as a volumetric heat source of specifiedpower (in W/m 3 ), or as a surface heat source (in W/m 2 ).Fine 3D Thermal Modelof just a small part of a Pad Mesh ~ 590K cells !IC1 SiliconGoldBump BondSmall vias replaced by effectivethermal conductivity (k_eff )k_eff 3D Thermal Model of entire Pad Mesh ~ 2K cellsIC1 Sik_effGoldBump BondMCMSubstrateMCM SiO2 (ILD)Figure 2. A bump bond structure between the IC1 chip (flipped, face down)and the multi-chip-module (MCM) substrate. SEM photo and 3Dmodel/cross-section are shown.Modern IC technologies often use arrays of many smallvias, which we call here a “forest of vias”. Similar structuresare also used for purposes of IC wafer processing steps, suchas chemical mechanical polishing (CMP). The presence ofthousands of such small metallic structures (vias inparticular) in many IC areas does not allow to directly builda 3D computational mesh with all such details resolved. Justone bump bond structure may contain about 10-20 thousandvias, and a fully resolved 3D mesh may contain 1-2 millioncells/nodes. Taking into account that a realistic RF IC maycontain from 40 to 100 bump bonds, it would produce a 3Dmesh of 80-200 million cells, which makes building a fullyresolved model of full chip impossible.To solve this problem, we have developed an approach inwhich we replace the forests of vias areas with blocks ofartificial material of equivalent or effective thermalconductivity, k eff . We select in the IC layout a repeatingsection of “forest of vias” which is important for the ICthermal characteristics (e.g., near a bump bond), and we cutoff this section of layout to build a detailed 3D meshedthermal model of that part. Then, using our 3D finite volumesolver, we compute a detailed, fully resolved 3D thermalsolution which is then used to determine k eff for this block.An example is illustrated in Figure 3, and details aredescribed in Section IV.III. HEAT EQUATION SOLVEDThe full 3D thermal simulation is done with a high-fidelityfinite volume based solver, namely, Heat Transfer Module ofthe CFD-ACE+ Multiphysics suite [5]. It solves the energyequation in a fully conservative manner (total enthalpyequation). In the present steady-state case, it reduces to amore familiar form:∇ • k ∇T+ S =(1)( ) 0hMCM SubstrateFigure 3. Replacing forest of vias near the bump bond with effectivethermal conductivity (k_eff) blocks. The 3D mesh cell count, just for onebond, is reduced over 1000 times!IV. EFFECTIVE THERMAL CONDUCTIVITYThe methodology of computing the value of the effectivethermal conductivity, k eff , for the selected section of IClayout that includes forests of vias, is illustrated in Figure 4.To the fully meshed, detailed 3D model of that section, weapply a surface heat source, S h (as power density, in W/m 2 )on one side of the model, and isothermal boundary condition(BC), typically with applied constant temperature T = 300 K,on the other side of the model. The computed temperaturegradient, ΔT, is then used to calculate the effective thermalconductivity, using the following formula:k eff = S h · h / ΔT (2)where h is the height of the model between the heat sourceand the isothermal BC. This is illustrated in Figure 4, wherethe 3D temperature map was obtained as an accuratenumerical solution on a fully-resolved, detailed 3D mesh,solved by the CFD-ACE+ solver.3D, Meshed Thermal Model Temp. Resultsk eff= S h * h / ΔTSurface HeatSource(Power Density)S h(W/m 2 )ΔT,h (height)Isothermal BC(T = 300 K)Figure 4. Illustration of computing the effective thermal conductivity, keff,for a selected IC layout section (3D model) containing forests of vias.©EDA Publishing/THERMINIC 2008 65ISBN: 978-2-35500-008-9