### Bruce Guenin, PhD

Associate Technical Editor

The need to accommodate increasing chip power has led to improved package and heat sink designs having much lower thermal resistance values than previously. This has led to challenges in accurately calculating the thermal performance of the package and heat sink as an integral unit on the basis of thermal resistance measurements performed on each of them separately. Subtle heat transfer effects that can be neglected for lower-power packages become critical in the thermal analysis of high-power ones.

## High-Power Package and Heat Sink

A typical configuration for a high power microprocessor or ASIC (Application-Specific Integrated Circuit) package attached to a heat sink is depicted in Figure 1a. This sort of flip-chip design lends itself to effective routing of power and signals between the chip and the PCB (Printed Circuit Board) below and to the efficient transfer of heat to the heat sink above. The package has a lid that is usually fabricated of copper to promote heat spreading. Thermal interface materials (TIMs) assist in the transfer of heat between the die and the lid (TIM1) and the lid and the heat sink base (TIM2). The heat sink base transfers the heat to the heat sink fin structure. Its thermal conductivity and thickness are chosen to suit the required power dissipation. Obviously, higher power requires larger values of these parameters. The fin structure is also affected by the power requirements. Typically, higher power levels require higher conductivity fins with finer spacing, accompanied by a higher volumetric air flow than do lower power ones.

In years past, when power levels were lower, it became customary to calculate the junction temperature of the chip, TJ, using the following expression, assuming a simple series thermal resistance circuit [1, 2]:

TJ = P ∙ (ΘJC + ΘCS + ΘSA) + TA (1)

where P is the dissipated power, ΘJC, ΘCS, and ΘSA are the junction-to-case, case-to-sink, and sink-to-air thermal resistances, respectively, and TA is the ambient temperature. [The term “case” refers to the top surface of the lid.] ΘJC is usually a test value furnished by the package vendor. ΘSA is normally based on the test value provided by the vendor. A complicating factor is that the heat sink vendors usually measure ΘSA with a uniform heat flux applied to the entire bottom of the base. In many applications, the heat sink is wider than the package, with the result that the vendor-supplied ΘSA values are lower than is appropriate for those applications. ΘSA is increased by the thermal resistance for heat to spread to the extremities of the heat sink base from the contact region with the packages. Fortunately, there is a relatively simple and accurate way to calculate this spreading resistance [3]. ΘCS is essentially the thermal resistance of the TIM2 material, calculated for a specified bond-line thickness (BLT) and lateral dimensions bounding the region of significant heat flux. The calculated value of ΘCS is sensitive to the assumed area of this region.

This analysis ignores the presence of a secondary heat flow path to the ambient air through the PCB. This path is usually negligible for high-power packages because of the very low thermal resistance of the primary path to air via the heat sink. In lower power applications, neglecting this secondary path produces a conservative estimate of TJ.

The following analysis explores the effect of the thermal interactions between the package and the heat sink, specific to various applications, on the actual values of ΘJC, ΘCS, and ΘSA.

## Modeling Approach

Finite element analysis was conducted on a simplified solid model of the package and board as depicted in Figure 1b, using a commercial software tool [4]. The model represents only the chip-to-heat sink thermal path. Regarding the heat sink, only its base is explicitly represented in the model. The fin structure is accounted for by the use of an effective heat transfer coefficient, hEFF, applied to the top surface of the heat sink base [5, 6, 7]. All other surfaces are assumed to be adiabatic.

Table 1 lists the specific dimensions and material properties assumed for the package and heat sink. The lid and die widths of 40 mm and 13 mm, respectively, are typical for the sort of applications under consideration. The heat sink was evaluated at three values of base width, wHS : 40, 70, and 100 mm. Three values of thermal conductivity, kHS, were assumed for the heat sink base: 166 and 240 W/mK, representing the lower and upper bounds for aluminum alloys used in heat sinks, and 390 W/mK for pure copper. The heat sink base thickness, tHS, was set at 2 mm for the Al alloy designs and 6 mm for Cu. Values of hEFF , ranged from 50 to 2000 W/m2K, representing low-end heat sinks, with moderate air velocity, up to high-performance, folded-fin designs, at a high value of air velocity. The values of thermal conductivity chosen for TIM1 and TIM2 represent mid-range thermal performance. P is assumed to be 1W and TA to be 0˚C, in order to simplify the calculation of thermal resistances from the calculated temperatures.

## Model Results

Table 2 provides the calculated values of TJ, TCASE, and TSINK and the values of thermal resistances calculated from these temperatures and TA: ΘJA, ΘJC, ΘJS, ΘCS, and ΘSA for each of the 36 cases studied. The following discussion elaborates on each of these thermal metrics.

Junction-to-Air Thermal Resistance

Figure 2 presents the calculated values of ΘJA plotted versus wHS, at specified values of kHS, tHS, and hEFF. The ΘJA values vary from a maximum of 12.3˚C/W to a minimum of 0.73˚C/W. In an application with TA = 30˚C and a maximum value of TJ of 90˚C, they would be associated with maximum power levels of 5W and 82W, respectively.

On further inspection, one sees that the individual curves coalesce into four groupings, one for each of the different values of hEFF. For heat sinks with wHS = the package size, 40 mm, and for hEFF ≥ 1000 W/m2K, there is not much difference in performance for the different base designs. Conversely, at the largest value of wHS of 100 mm, there is an obvious benefit to having a larger thermal conductivity, which grows with increasing hEFF. This points out the need to have the combination of large kHS and tHS in order to derive the full benefit from a high-efficiency fin structure.

### Junction-to-Case and Junction-to-Sink Thermal Resistance

Figures 3a and 3b illustrate the effect of different heat sink parameters, kHS, wHS, and hEFF, on ΘJC and ΘJS, respectively. These graphs demonstrate that the kHS has a much larger effect than either wHS or hEFF. The variation in ΘJC is relatively small, at 0.003˚C/W, or 3% of the mean value. That in ΘJS is larger, at 0.011˚C/W, or 15% of the mean value. Since ΘJS = ΘJC + ΘTIM2, one suspects that the variations in ΘJS are due to changes in the heat flux pattern in the TIM2 layer.

Figures 4a and 4b characterize the variation in the temperature of the lid, TLID, and the heat sink base, THS, as a function of the distance from their geometrical center for heat sinks with a constant value of wHS = 70mm and the three different values of kHS. In each figure, a constant value of hEFF is assumed: a) 50 W/m2K and b) 2000 W/m2K. It is interesting to note that the different values of hEFF serve to change the magnitude of the temperatures, but not their relative profiles. This has relevance when considering the heat flux distribution for these different situations. The heat flux at a particular x,y location is proportional to TLID(x,y) – THS(x,y). For the two Al alloy heat sinks, THS is more highly peaked near the origin than is the copper heat sink. This leads to less efficient extraction of heat from the lid with the Al heat sinks. Hence, at the edge of the lid (at x = 20 mm) TLID – THS. is very small for the Cu heat sink but is finite for the Al ones. This produces an increase of the heat flux near the edge of the lid for the Al heat sinks.

Figure 5 plots TLID(x,0) – THS(x,0) versus x for the 70 mm heat sink over the full range of values of kHS and hEFF. It is striking to observe how consistent the heat flux distribution is for a given value of kHS, independent of hEFF. The flux profile for the Cu heat sink is more concentrated near the center of the lid than that for the other heat sinks. In contrast, the excess heat flux near the edge of the lid for the Al heat sinks serves to distribute the flux over a wider area than with the Cu one. This effect is more pronounced with the lowest conductivity Al heat sink.

### Effect of Heat Transfer Area on the Sink-to-Air Thermal Resistance

The analysis up to this point indicates a variation in the area over which heat is transmitted from the package to the heat sink and correlates it to changes in ΘCS. It is useful, therefore, to explore how the variation in this area will affect the value of ΘSA, the final stage in the heat flow path from the package to the air.

Figure 6 displays the results of FEA simulations in which heat is applied to a heat sink, not by way of a package in contact with it, but by applying a uniform flux over a square area of varying size, ranging from the die width (13 mm) to the package width (40 mm). This range represents the minimum and maximum conceivable area for heat transfer between the package and the heat sink. The graph quantifies the distribution of THS(x) for a 70 mm wide heat sink.

Figures 6a and 6b depict the “corner cases” representing maximum heat sink performance (6a) and the minimum performance (6b) studied for that value of wHS. Specifically, Figure 6a assumes the copper heat sink at hEFF = 2000 W/m2K and Figure 6b assumes the lower conductivity Al alloy heat sink at hEFF = 50 W/m2K. Varying the width of the heat transfer area (HTA) has the expected result on THS,PEAK. As the HTA width is reduced from 40 mm to 13 mm, THS,PEAK increases significantly. It is notable the change in ΘSA is most significant for the copper heat sink with the high value of hEFF. It is approximately equal to ± 0.05 ˚C/W about the mean value of 0.18˚C/W, or nearly ±30%, a significant effect. In contrast, for the Al alloy heat sink with the relatively low value of hEFF, the variation in ΘSA is only ± 0.3˚C/W or ± 5%. This should not be surprising since the low value of hEFF ensures that the convective contribution to the total value of ΘSA will be much larger than the conductive contribution.

It is desirable to develop a metric for quantifying the HTA. The method developed herein is as follows:

• Calculate ΘSA for each of the 36 heat sink-hEFF configuration, applying a uniform heat flux at each of four values of HTA width (13, 20, 30, and 40 mm).

• Perform regression analysis to fit each ΘSA vs HTA width curve to a 3rd-order polynomial. The polynomial can accurately predict the ΘSA at an arbitrary value of HTA width within the specified 13 -40 mm range.

• Use a spreadsheet solver to determine the value of HTA width for each of the 36 configurations to produce a value of ΘSA equal to the value of ΘSA reported in Table 2 for the integrated package + heat sink simulations.

Figure 7 shows the values of HTA width calculated in this fashion plotted versus kHS for all values of wHS and hEFF evaluated. They range from 17 to 24 mm. It shows that both kHS and wHS have a significant effect on the HTA width. The largest value of HTA width occurs for the smallest value of kHS and the largest value of wHS.

The resultant values of HTA width can be used in the application of analytical methods to predict ΘJA. For example, it could be used to calculate ΘCA = ΘTIM2 by providing an area that, when multiplied by the total power, would yield approximately the same value of TLID – TSINK as that obtained using the original FEA approach.

In light of these findings, it is worthwhile to revisit the comments made at the beginning of this article regarding calculating a spreading resistance term to correct the vendor-supplied ΘSA values when the heat sink is larger than the package. In performing these calculations, it is common practice to assume a uniform heat flux from the package into the heat sink over the full package width. It is now clear that applying this assumption to a ΘSA calculation for a high-performance heat sink, such as the one of Figure 6a, will likely generate a significant underestimate of ΘSA.

## Preview of Part 2

Part 2 of this article will address the ultimate objective of this study: to evaluate plausible methods for predicting ΘJA for an integrated package-heat sink configuration on the basis of test results obtained for the package and heat sink separately. It will leverage both the physical insights developed in the course of this analysis as well as the specific values of HTA in this next exercise.

## References

*1. Guenin, B, “Determining the Junction Temperature in a Plastic Semiconductor Package, Part II,” ElectronicsCooling, Vol. 5., No. 3, September, 1999.
2. Guenin, B. “A Funny Thing Happened On The Way To The Heatsink,” ElectronicsCooling, Vol. 11, No. 3, August 2005.
3. Lee, S., “Calculating Spreading Resistance in Heat Sinks,” ElectronicsCooling, Vol. 4., No. 1, January, 1998.
4. ANSYS®, V11.0.
5. Simons, R., “Using an Equivalent Heat Transfer Coefficient to Model Fins on a Fin,” ElectronicsCooling, Vol. 11, No. 2, May 2005.
6. Lasance, C, “Heat Spreading: Not a Trivial Problem,” ElectronicsCooling, Vol. 14, No. 2, May, 2008.
7. Guenin, B. “Packaging Challenges for High Heat Flux Devices,” ElectronicsCooling, Vol. 12, No. 3, August, 2006. l*