design of circular heat spreaders on semi-infinite heat sinks in microelectronics device...

452 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING, AND MANUFACTURING TECHNOLOGY—PART A, VOL. 20, NO. 4, DECEMBER 1997

Design of Circular Heat Spreaders on Semi-InfiniteHeat Sinks in Microelectronics Device Applications

Ping Hui, H. S. Tan, and Y. S. Lye

Abstract—Using the rigorous analytical solutions, we discussdesign aspects in terms of thermal resistance and temperatureuniformity for the canonical heat dissipation configuration con-sisting of a circular heat spreader on a semi-infinite heat sink.Regarding the optimized size of the heat spreader, we havefound that the minimum temperature thickness of the spreaderdepends on the thermal conductivity values of the spreader andthe sink, in contrast to the results published previously by otherresearchers. In addition, a new design formulad = 0.44b forthe selection of spreader thicknessd from the spreader radiusb is proposed to replace the commonly used ruled = b/3. Ourresults have confirmed the design ruleb = 5a for the selectionof the spreader radius b from a given heat source radiusa. Tofacilitate the complete design of the heat spreaders, we presenttwo nomographs in the form of contour plots for the normalizedthermal resistance and the normalized temperature uniformity.

Index Terms—Diamond, heat sinks, heat spreaders, IC pack-ages.

NOMENCLATURE

Radius of the input heat source.Radius of the heat spreader.Thickness of the heat spreader.Input heat flux density.Total input heat flux.Thermal resistance.Ratio of spreader’s thermal conductivity over sink’sthermal conductivity.Normalized spreader thickness ().Normalized spreader radius ( ).Normalized thermal resistance (7).Normalized temperature uniformity (8).Temperature difference above the ambient.

I. INTRODUCTION

I N the packaging for high power density microwave devicesand solid-state lasers, heat spreaders are often inserted

between devices and heat sinks in order to maintain thejunction temperature within operable limits and to make thetemperature distribution more uniform in the active region.The essential function of the heat spreaders with larger thermalconductivity than that of the heat sinks is to broaden the heat

Manuscript received May 28, 1997; revised September 3, 1997. This paperwas presented at the 13th Annual IEEE Semiconductor Thermal Measurementand Management Symposium, Austin, TX, January 28–30, 1997.

P. Hui and H. S. Tan are with the School of Electrical and ElectronicEngineering, Nanyang Technological University, Singapore 639798.

Y. S. Lye is with JVC Electronics Singapore Pte. Ltd., Singapore 139954.Publisher Item Identifier S 1070-9886(97)09148-8.

flux from the small areas of the devices to larger regions,followed by further heat transfer to the surroundings throughthe heat sinks (usually made of copper). The structure of theheat spreaders on the heat sinks models the dies on packages,or chips on bases, in the packages of integrated circuits.

Usually, the size of the heat sinks is taken as much largerthan that of the heat spreaders; consequently, the heat sinks canbe approximated as semi-infinite media. The cross sectionalshape of the spreaders can be either circular or rectangular.For a given area, circular spreaders produce the highesttemperature compared with spreaders of other geometricalshapes. As demonstrated by Kadambi and Abuaf using thefinite element method [1], axisymmetric calculations [two-dimensional (2-D)] yield maximum temperature that are withinabout 5% of the exact three-dimensional (3-D) results for allaspect ratios below five and chip-to-base area ratio below 14.Using the rigorous analytical solutions, Hui and Tan [2] havealso shown that for the same cross-sectional area, 2-D circularspreaders give higher maximum temperatures than that for the3-D square spreaders within 1.3%. Thus, the configuration ofa circular heat spreader mounted on top of a semi-infinite heatsink, as shown in Fig. 1, can be viewed as a canonical formfor many practical heat dissipation systems involving spreaderstructures. This 2-D canonical configuration can be utilized topredict the trends, to obtain a upper bound, and to performextensive sensitivity analysis.

Using the surface element method, Beck, Osman, and Lu [3]obtained the spreader thickness for the minimum temperatureat the center point of the top surface of the circular spreaders as

for (1)

where and are the Bessel functionsof the zeroth and the first order, respectively, and3.8314 is the first positive zero of 0. This formulais independent of thermal conductivity values of the heatspreaders and the heat sinks. Borchelt and Lu [4] have appliedthe numerical solution presented in [3] to the design ofchemical vapor deposited (CVD) diamond spreaders. With therigorous analytical solution [5], we have obtained the mini-mum temperature thickness numerically using the GoldenSearch Section method [6] as to be shown in Fig. 5. Our resultsindicate that depends on the thermal conductivity ratio

.The design of a heat spreader is basically to determine the

spreader radius and the spreader thicknessfor a given heatsource radius , thermal conductivity ratio , and specified

1070–9886/97$10.00 1997 IEEE

HUI et al.: DESIGN OF CIRCULAR HEAT SPREADERS 453

Fig. 1. Geometry of a cylindrical heat spreader on a semi-infinite heat sink.

reduction of both the thermal resistance and the temperatureuniformity. We first examine the effects of the spreader’sgeometrical dimensions and thermal conductivity ratio on thethermal resistance and the temperature uniformity. Followingthat, we give design nomographs for the determination of thespreader dimensions.

II. M ATHEMATICAL FORMULAS

Based on the rigorous analytical solutions presented in [5],we give below the temperature distribution on the top surfaceof the spreader for the configuration shown in Fig. 1. For thepurpose of comparison, we also give the surface temperaturedistributions for two special cases of this configuration, i.e.,(1) the semi-infinite medium where the spreader thicknessbecomes zero ( 0); (2) the layer model where the spreaderradius approaches to infinity ( ). The temperaturesolutions are given in terms of the temperature differenceabove ambient , i.e., . In practice, theambient temperature can be taken as the base temperatureof the heat sink because the size of the heat sink is usuallymuch larger than that of the heat spreader. It is noted that theexposed surfaces of the heat spreader and the heat sink areassumed to be adiabatic.

Circular Spreaders

(2)

where are positive zeros of the first order Besselfunction of the first kind , and

. The expansion coefficients are determined by

(3)

where

(4)

Semi-Infinite Medium

(5)where and are the elliptical integrals of the firstkind and the second kind, respectively.

The Layer Model:

(6)

One of the important parameters in the design of heatspreaders is the thermal resistance, which is defined as theratio of the maximum temperature difference above ambient

to the total input heat flux , i.e.,. Another important design parameter is the

temperature uniformity across the heat area [7], [8], whichis characterized by the percentage difference between themaximum temperature at the disk center and the minimumtemperature at the disk edge, i.e., .For the limiting case of the semi-infinite medium, from (5),the maximum temperature and the minimum temperature are

and . Therefore, the corresponding thermalresistance and the temperature uniformity are and

25.0%, respectively. To reflect the change of thethermal resistance and the temperature uniformity caused bythe introduction of a heat spreader between the heat sourceand the heat sink, we use normalized thermal resistanceand normalized temperature uniformitywith respect to theirvalues for the semi-infinite heat sink, defined as

(7)

(8)

III. RESULTS AND DISCUSSION

A. Effect of Spreader Thickness

Fig. 2 presents both and as functions of normalizedthickness , with the normalized radius as parametersand 4, together with the results from (6) for the limiting


Fig. 2. Dependence of and� on the normalized spreader thicknessd=a.

case of an infinite layer spreader. When the thickness of theheat spreader increases from zero, for finite radii,decreasesrapidly from unity till it reaches the minimum temperaturepoint, then goes up linearly, with larger slopes for smaller radii.For the infinite layer spreader, however, reduces sharplyas increases up to three, then approaches slowly andmonotonically to the other limiting value of 0.25for the semi-infinite medium of . Physically, with the initialincrease of the spreader thickness from zero, the flow of heatflux is broadened in the heat spreader since , therebylowering the temperature. As the spreader thickness increasesfurther, the spreading of the heat flux tube is constrainedby the spreader’s finite radius, leading to the increase ofthe temperature. Therefore, the minimum temperature occursonly for spreaders with finite radii. For the layer spreader,since there is no physical restriction on the spreading of theheat flux tube, the normalized thermal resistance decreasesmonotonically to its asymptotic value of 0.25

Also shown in Fig. 2 are the variations of normalizedtemperature uniformity with the spreader thickness. As aresult of less spreading of heat flux, the improvement on thetemperature uniformity is greater for smaller spreader radii. Onthe other hand, the layer model gives the worstand its valueapproaches to unity as the film thickness of the spreader layerbecomes infinite (effectively another semi-infinite medium).Obviously, trade-off between the reduction of the thermalresistance and the reduction of the temperature uniformityhas to be made during the design. It is observed that theintroduction of spreaders between the heat source and the heatsink does improve temperature uniformity. This conclusion isvalid only for 1, as to be shown in Figs. 4 and 5 in theSection III-C on the effect of thermal conductivities.

B. Effect of Spreader Radius

The dependence of and on the normalized spreaderradius is shown in Fig. 3. As goes from one to three,reduces sharply from various values andincreases drasticallyfrom almost zero. With a further increase of , both and

approach gradually to their asymptotic values determinedby (6) for the layer model. The asymptotic values offor

Fig. 3. Dependence of and� on the normalized spreader radiusb=a.

Fig. 4. Dependence of and� on the normalized spreader thicknessd=afor various �12.

are given in parentheses as indicated in Fig. 3. Forexample, for 4, increasing the normalized heat spreaderradius from 10 to infinity only achieves a lowering ofthe normalized thermal resistance by about 14%. For smaller

values, the improvement is even less. In addition, thetemperature uniformity is worse off for the increasing .Thus, we conclude that there exists an effective radius of theheat spreader beyond which the thermal resistance does notreduce significantly with increasing radius of the heat spreader.

C. Effect of Thermal Conductivities

For nominal values of 2 and 3, we show inFigs. 4 and 5, respectively, the effect of thermal conductivityratio on the thermal performance of the con-figuration. When in Fig. 4, is always greater thanone, indicating no occurrence of the spreading of heat flux.In particular for is even larger than unity forvalues of between 0 and 1.8. As exceeds one, theimprovement of the thermal resistance and the temperatureuniformity results from the spreading of heat flux when thespreader is not too thick compared to the radius of the heatsource, say, 10.

From Fig. 4, it seems that the minimum temperature pointis insensitive to the value of . This observation will be


Fig. 5. Dependence of and� on the normalized spreader thicknessb=afor various�12.

Fig. 6. The minimum temperature spreader thickness for various�12 incomparison with Beck’s results.

examined in detail in the next subsection. Based on the resultsshown in Figs. 3 and 5, we have confirmed the design rule

5 for the selection of spreader radiusfor a given heatsource radius [9].

D. The Thickness for the Minimum Temperature

For and given values of and , the spreaderthickness for the minimum temperature can be determinedfrom the rigorous solution (2) numerically using the GoldenSection Search method [6]. The results of as a function of

are presented in Fig. 6 for various values of . Forthe purpose of comparison, Beck’s results [3] are also plottedusing (1) up to the valid range 0.5. For small heat sourcesor large spreader radii, i.e., 0.2, our numerical resultsand Beck’s results indicate the existence of asymptotic valuesfor . However, our results show a slight dependence on

, while Beck’s are not. The asymptotic values forare approximately at for for

for for by our rigoroussolution, and 0.42 for all values given by Becket al. [3]. Itis noted that the difference between our results and Beck’s for

4 is due to Becket al.’s approximate solutions [3], [5].As shown in Fig. 2, the minimum temperature valley broad-

ens when the normalized spreader radius increases orequivalently decreases. For various values and 0.1,

Fig. 7. Spreader thickness where the minimum temperature is reached by90%.

Fig. 8. The spreader thickness for 90% achievement of the minimum tem-perature for different values of�12.

the value of ranges from 0.38 to 0.46. Obviously thecommon practice of choosing the optimized spreader thicknessto be [9] is not consistent with our results. Therefore,we propose a new selection rule for 25 and 0.2.

In the applications of CVD diamond films as heat spreaders,considering the high cost of growing thicker films ( 500

m), it may not be economical to design the thickness of theheat spreader at the minimum temperature point (minimizationis not optimization). We therefore recommend the use ofsmaller spreader thickness where the minimum temperatureis reached by 90%. Fig. 7 gives the design curves.

It is observed that does not change too much forbetween 0.01 and 0.2. However, asreduces from 0.2 inFig. 8, reduces and increases. To compromise betweenthe reduction of the thermal resistance and the reduction ofthe temperature uniformity, we should chooseto be around0.2, corresponding 5. This confirms again the designrule 5 for the selection of spreader radius proposed inthe last subsection.

E. Design Nomographs

To facilitate the design of heat spreaders, we producenomographs of contours of the normalized thermal resistance

and the normalized temperature uniformitywith and


Fig. 9. Design nomograph for spreader radius and thickness with contoursof the normalized thermal resistance .

as coordinates, as shown in Figs. 9 and 10, respectively.These two graphs are for 4, which is more applicablefor diamond heat spreaders on copper heat sinks. Two practicaldesign approaches are illustrated for the use of these twonomographs. The first one is for given spreader thickness,given heat source radius, and specified normalized ther-mal resistance . From Fig. 9, we are able to determinethe physical feasibility of the desired reduction of thermalresistance for the given normalized spreader thickness. Ifit is feasible, we can determine the required spreader radius,and consequently, the normalized temperature uniformity fromFig. 10. For example, for a microwave power diode with radiusof 50 m and a diamond heat spreader thickness of 50m,in order to achieve the normalized thermal resistance of 0.55,from Fig. 9, we can find 0.42, hence 119 m.Subsequently, from Fig. 10, we determine that the normalizedtemperature uniformity to be 0.47.

The other case is for given heat source radiusand thedesired normalized thermal resistance. In this case, thereare many choices on the spreader radiusand the spreaderthickness . Based on the effective spreading area discussed

Fig. 10. Design nomograph for spreader radius and thickness with contoursof the normalized thermal resistance�.

above, we first choose the spreader radius be . Then forthe given , we can determine the spreader thicknessfromFig. 9, and finally, the temperature uniformityfrom Fig. 10.For example, for a typical microwave power diode with radiusaround 50 m and given the normalized thermal resistance

0.45, we can first determine the spreader radius to be250 m from the design formula . Then from Fig. 9with 0.2 and 0.45, we find that 0.71, hence

35.6 m. Subsequently from Fig. 10, we can find thenormalized temperature uniformity to be 0.61. It is noted thatthe thermal resistance can be further reduced by increasing thethickness of the heat spreader. To obtain the maximum thermalresistance reduction, the spreader radius should be around 110

m based on the design formula .

IV. CONCLUSION

Using the rigorous analytical solutions, we have studied thedesign aspects of circular heat spreaders mounted on semi-infinite heat sinks. In contrast to other researchers’ results,we have found that the spreader thickness for the minimumtemperature depends on the thermal conductivity values even


though the dependence is not strong. We have found that thecommonly-used design rule is not accurate. A newrule has been proposed for 2 5 and 0.2.To be cost-effective, we recommend the use of the spreaderthickness where the minimum temperature is reached by 90%,as shown in Fig. 7. Our study has confirmed the design rule

5 for the selection the spreader radius. Finally for4 and for specified reduction of the thermal resistance andtemperature uniformity, the design nomographs for choosingspreader radius and spreader thickness are presented inFigs. 9 and 10 with examples.

REFERENCES

[1] V. Kadambi and N. Abuaf, “Axisymmetric and three-dimensional chip-spreader calculations,” inProc. AIChE Symp. Series, Seattle, WA, 1983,vol. 79, pp. 130–139.

[2] P. Hui and H. S. Tan, “Three-dimensional thermal analysis of a thermaldissipation system with a rectangular diamond heat spreader on a semi-infinite copper heat sink,”Jpn. J. Appl. Phys., vol. 35, pp. 4852–4861,1996.

[3] J. V. Beck, A. M. Osman, and G. Lu, “Maximum temperatures indiamond heat spreaders using the surface element method,”J. HeatTransfer, vol. 115, pp. 51–57, 1993.

[4] E. F. Borchelt and G. Lu, “Applications of diamond made by chemicalvapor deposition for semiconductor laser submounts,” inProc. Process.Packag. Semiconductor Lasers Optoelectron. Devices, SPIE, 1993, vol.1851, pp. 64–77.

[5] P. Hui and H. S. Tan, “Temperature distributions in a heat dissipationsystem using a cylindrical diamond heat spreader on a copper heat sink,”J. Appl. Phys., vol. 75, pp. 748–757, 1994.

[6] W. H. Press, S. A. Teukoisky, W. T. Vetterling, and B. P. Flannery,Nu-merical Recipes: The Art of Scientific Computing, 2nd ed. Cambridge,U.K.: Cambridge Univ. Press, 1996, pp. 390–395.

[7] P. R. Gray, “A 15 W monolithic power operational amplifier,”IEEE J.Solid-State Circ., vol. SC-7, pp. 474–480, Dec. 1972.

[8] K. Board, “Thermal properties of annular and array geometry semicon-ductor devices on composite heat sinks,”Solid-State Electron., vol. 16,pp. 1315–1320, 1973.

[9] M. Seal, “Thermal and optical applications of diamond thin films,”Phil.Trans. Roy. Soc. Lond. A, vol. 432, pp. 313–322, 1993.

Ping Hui received the B.S.E.E., M.S.E.E.,and Ph.D. degrees from Changsha Institute ofTechnology, Changsha, China, Zhejiang University,Hangzhou, China, and Polytechnic University,Farmingdale, NY, respectively.

He worked as a Research Fellow in the Schoolof Electrical and Electronic Engineering at theNanyang Technological University, Singapore,from 1991 to 1994. He has been working as aLecturer in the same institute since 1994. Hisresearch interests include thermal modelling of

microelectronic devices and electromagnetic scattering.

H. S. Tan received the B.Sc., M.S.E.E., and Ph.D.degrees from Imperial College, London, U.K., theMassachusetts Institute of Technology, Cambridge,and McGill University, Montreal, P.Q., Canada,respectively.

Since 1973, he has been Professor of ElectricalEngineering at the University of Malaya, Malaysia.In 1991, he accepted an appointment as Professorand Head of the Division of Electronic Engineeringat the Nanyang Technological University, Singa-pore. His research interests are in electromagnetic

scattering, microwave heating applications, diamond-like-carbon deposition,and microcellular communications.

Y. S. Lye received the B.Eng. degree from NanyangTechnological University, Singapore, in 1995.

Since July 1995, he has been with JVC Elec-tronics Singapore Pte Ltd., Singapore, as a R&DEngineer, working on the design of PCBs in audioproducts (car audio and microcomponents).

design of circular heat spreaders on semi-infinite heat sinks in microelectronics device...

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