Heat Spreading Revisited Effective Heat Spreading Angle

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1 Heat Spreadng Revsted Effectve Heat Spreadng Angle Drk Schwetzer and Lu Chen Infneon Technologes AG Am Campeon -, Neubberg, Germany Abstract There s probably no thermal engneer who has not yet developed hs or her own spreadsheet to calculate the thermal resstance of a layered structure such as the chp / de-attach / lead-frame stack n a power semconductor. The more sophstcated versons of such spreadsheets consder also the ect of heat-spreadng nsde the layers, usually assumng a constant spreadng angle whch s often chosen to be 45. As smple as ths approach s as poor are often the results compared to Fnte Element smulatons or measurements. Heren we propose a defnton of the ectve heat-spreadng angle whch s based on the local varaton of the heat-flux densty along the heat-flow path. Usng ths defnton t s possble to accurately calculate the heat-spreadng angle nsde a gven structure and thus to develop more accurate heat spreadng models e.g. for spreadsheet calculatons. Keywords Heat flux, heat spreadng angle, Rth-JC.. Introducton One of the most often used formulas n thermal engneerng s probably the equaton x R th () ka for the thermal resstance R th of a layer wth thckness x, cross-sectonal area A and thermal conductvty k. Assumng that the heat flow path nsde a layered structure can be modeled by a seres of truncated cones as shown n fgure, thus takng nto account the heat spreadng, the total thermal resstance across ths structure could be approxmated usng x R, () th k A x, A, k beng thckness, cross-secton, thermal conductvty, and specfc heat of the -th slce. In the followng we wll refer to ths model by truncated cone model (TCM). Many spreadsheet calculatons are based on t. However there are several problems wth ths approach:. The model assumes that the heat s spread homogeneously over the area A of each slce whch s not true.. The model mplctly assumes that the temperature on all nterfaces between any two slces s sotherm,.e. that all sothermals are planar and parallel. Otherwse we would not be allowed to apply eq. () to calculate the partal resstance of each slce. Isothermal faces on both sdes of a layer are n fact an (often forgotten about) pre-condton for the applcablty of (). But as any thermal engneer knows the sothermals n real structures are nether planar nor sothermal (fgure ).. The model assumes that the heat spreadng angle s constant wthn each layer whch s not true ether as we wll show below. 4. The model assumes that the heat spreadng s ndependent from the external boundary condtons. As dscussed n [] and [] and wll be shown agan below ths s not the case ether. Therefore the results obtaned usng the truncated cone model and eq. () often devate consderably from FE smulaton results or measured R th values and should be used wth care. Addtonally the heat-spreadng angle wthn each layer s unknown. A value of 45 s often used but mostly based on our ntutve magnaton of the heat spreadng n a materal wth sotropc thermal conductvty; though some authors have nvestgated the applcablty of the 45 heat spreadng angle [, 4]. In ths paper we propose the defnton of an ectve heat spreadng angle whch s based on the local varaton of the heat-flux densty [W/mm ] along the heat flow path. In combnaton wth the truncated cone heat spreadng model (TCM) as n fgure the use of the thus defned heatspreadng angle accurately reproduces the correct R th of the structure. TIM Cold-plate Glue Chp x Fgure : Smple heat spreadng ( truncated cone ) model for the calculaton of the thermal resstance of a layered structure. Fgure : Non-planar sothermals n a real sem-conductor devce /5/$. 5 IEEE 88 st SEMI-THERM Symposum

2 . Effectve heat spreadng angle The concept of the ectve heat spreadng angle s qute smple. Instead of consderng the heat flux dstrbuton n the whole structure we focus on the local heat flux densty along the heat flow path and ask whch heat spreadng angle would result n the observed functonal dependency n a truncated cone model f assumptons + were true. In the followng we consder the path between the pont of maxmum temperature T j on a semconductor chp and the pont of maxmum case temperature T c on the bottom sde of the structure (fgure ). The heat source s located at x and the case surface at x x case. A Fnte Element smulaton wth suffcently fne spatal resoluton along the x-axs reveals the decrease of along ths path (fgure 4). All parameters for that smulaton are shown n table. The heat flux densty decreases quckly throughout the slcon chp, t remans almost constant n the regon of the glue de attach, and t slowly decreases further n the leadframe untl t abruptly drops to zero at the case surface where a fxed temperature boundary condton was appled. Only an area A =.. mm n the center of the.. mm chp surface was heated. The drop n heat flux densty s caused by heat spreadng; therefore t seems reasonable to calculate the spreadng angle based on or the dervatve / thereof. x = Path x = x case T J T C Chp Layer Materal Sze [mmmmmm] Chp Slcon De attach Glue....5 Cu alloy Thermal cond. [W/mK] Table : FE smulaton model parameter: A power of W was dsspated homogenously on an actve area A =..mm on the surface of the chp,.e. the ntal heat flux densty p s W/mm. At case a fxed temperature boundary condton s appled. In the followng we shall assume that the heat s always spread homogeneously over the whole area as n assumpton () of the smple cone heat spreadng model from fgure. I.e. at depth x the heat s spread homogeneously over an ectve area A( whch s gven by p A( A () For a square area A( ts sde length s p a( A( a. (4) Plottng y( = ½ a( versus x we obtan the (upper half of the) heat spreadng profle as shown n fgure 5. The nclnaton of the tangent to that profle wth respect to the horzontal s the ectve heat spreadng angle at that pont. It shall be emphaszed agan that the real heat spreadng profle and angle wll devate somewhat from that n fgure 5 because of. De attach regon Fgure : Path along whch the heat flux densty s montored. [W/mm ] S-Chp Glue de attach Fgure 4: Heat flux densty along the path between juncton and the case montor pont. y( = / a( [mm] S-Chp..4.6 Fgure 5: Effectve heat spreadng profle calculated for the heat flux densty from fgure 4.

3 of the underlyng assumptons () + () whch are not met n realty. Therefore we call them ectve heat spreadng angle and ectve heat spreadng profle. But based on ths profle we can construct a truncated cone heat spreadng model whch wll exactly reproduce the true R th of the layer structure because t correctly reproduces the heat flux densty along the path between T j and T c. The temperature dfference between start and end pont of the heat flow path s unambguously defned by the heat flux densty along that path: T. (5) k(. Calculaton of the ectve heat spreadng angle If the heat flux s spread from an area A( wth crcumference L( to an area A(x + d the area ncreases by da L( dr (6) where dr = tan( ) (see fgure 6). Takng the dervatve of eq. () we obtan da p A (7) and from eq. (6) we know that da dr L( L( tan( ). (8) Therefore we obtan p A tan( ) (9) L( or, usng agan eq. (), A( tan( ) () L( for the ectve spreadng angle at poston x. Ths expresson can be further evaluated for dfferent shapes of the actve area as shown n table to obtan solely as functon of and the dervatve / thereof. In ths concept the ectve heat spreadng angle seems to be a local quantty because t s derved solely from the local heat flux densty. But mplctly depends on the geometry, the materal propertes, and the boundary condtons of the whole structure and the ectve heat spreadng angle therefore reflects all these nfluencng factors. L( A( da Fgure 6: Heat spreadng from area A( to area A(x + d. dr Actve area A( L( Square, a( sde a 4 Crcle, r( radus r Rectangle, aspect rato β = b /a a( ( ) Effectve heat spreadng angle a p tan( ) 4 r p tan( ) a p tan( ) ( ) Table : Formulas for the ectve heat spreadng angle for dfferent shapes of the actve area. Fgure 7 shows the resultng heat spreadng angle ( for our example. Wthn the slcon chp the heat spreadng angle ncreases from about to 8, contrary to the 45 spreadng models. Wthn the glue de attach the heat spreadng angle drops almost to zero. Wthn the leadframe the heat s ntally spread wth an angle of whch narrows down untl the spreadng angle becomes zero at the case sde wth fxed temperature boundary condtons. We wll see n the next secton that the heat spreadng profle looks very dfferent f we apply more realstc coolng condtons at the case sde. We see n the equatons n table that the ectve heat spreadng angle depends not only on the heat flux densty and ts dervatve but also on the dmenson a or r of the actve area: At frst sght seems to ncrease wth the sze of the actve area whch would be wrong snce we can expect that the heat spreadng orgnatng from a larger area s lower. For an nfntely large planar heat source the spreadng angle s n fact zero (one-drectonal heat-flow). [ ] S-Chp Glue de attach..4.6 Fgure 7: Effectve heat spreadng angle calculated for the heat flux densty from fgure 4.

4 But f we keep the total power dsspaton P constant, does also decrease when a or r are ncreased, thus explanng the seemngly contradcton. 4. Influence of boundary condtons and devce geometry In ths secton we apply the concept of the ectve heat spreadng angle to vsualze the nfluence of boundary condtons, materal propertes, and devce geometry on the nternal heat spreadng of a power semconductor devce. For ths purpose we consder agan the three layer structure consstng of chp, de attach, and leadframe, as descrbed n table. 4. Influence of the coolng condton at package case Frst we want to demonstrate the nfluence of the external coolng condton. Fgure 9 shows the ectve heat spreadng profle and angle nsde the structure when dfferent heat transfer cocents are appled to ts bottom (case) surface, namely: 5.,. 4, 5. 4,. 5 W/(m K), and fxed case temperature, the latter correspondng to an nfntely hgh heat transfer cocent. For comparson: datasheet values for heat transfer cocents of lqud cooled cold-plates range n between 5 and W/(m K), whereas wth lqud jet mpngement values as hgh as W/(m K) can be acheved [5]. Whle the coolng condton has almost no nfluence on the heat spreadng nsde slcon de and de attach we see a bg mpact of the boundary condton on the heat spreadng nsde the leadframe. The hgher the heat transfer cocent at the case surface the less the heat s spread nsde the leadframe. For the fxed case temperature boundary condton fnally, whch can only be realzed n smulatons but s nevertheless often used to calculate the juncton to case thermal resstance Rth-JC, the heat spreadng behavor s fundamentally dfferent: Whle for fnte values of the heat transfer cocent the spreadng angle ncreases throughout the leadframe t s much lower for the fxed case temperature condton and even drops to zero at the case surface. These results underlne once more the fact that heat spreadng and consequently also the Rth-JC of a power semconductor are nfluenced by the external boundary condtons (see also []). 4. Influence of de attach materal propertes In the next step we nvestgate the nfluence of the de attach materal, comparng thermally hgh conductve solder de attach (k = 5W/(mK)) to thermally low conductve glue de attach (k =.5W/(mK)). Agan we use the structure from table but ncrease for solder the thckness of the de attach from µm for glue to 5µm for solder. Fgure shows the resultng ectve heat spreadng profle and angle for glue (blue) and solder (red). We can see that the de attach has a bg nfluence on the heat spreadng nsde the slcon chp. For glue de attach the heat s spread out much more nsde the de than t s for solder de attach. A heat barrer, as represented by the thermally low conductve glue obvously causes the heat to spread out more n the precedng layer(s). If the Rth-JC s to be computed based on a truncated cone model (TCM) t s crucal to model the spreadng nsde the slcon chp correctly, especally so for thermally low conductve glue. (a) y( = / a( [mm] [ ] W/(m K) W/(m K)..4.6 Fxed case temp.. 4 W/(m K) 5. 4 W/(m K). 5 W/(m K) Fxed case temp. 5. /. 4 W/(m K) 5. 4 W/(m K) Fgure 9: Effectve heat spreadng profle (a) and spreadng angle for dfferent coolng condtons (heat transfer cocents) at the case surface. Ths s because the heat spreadng nsde the chp determnes the cross sectonal area of the heat flow through the de attach and therefore ts thermal resstance. Despte ts small thckness the de-attach often contrbutes a major part to the total thermal resstance of the devce, snce t s normally the materal wth the lowest thermal conductvty. Not surprsngly the heat spreadng nsde the de attach s hgher for solder than for glue due to ts hgher thermal conductvty. Insde the glue layer the spreadng angle s close to zero. On the other hand the de attach has lttle nfluence on the heat spreadng angle nsde the leadframe whch s only slghtly larger for glue than for solder de attach (fgure b).

5 (a) (a).5.5 y( = / a( [mm].5 glue y( = / a( [mm].5. x. mm. x. mm solder.5.5. x. mm glue 8 7. x. mm 6 6 [ ] 5 4 solder [ ] 5 4. x. mm. x. mm Fgure : Effectve heat spreadng profle (a) and spreadng angle for glue and solder de attach. A heat transfer cocent of. 4 W/(m K) was appled at the case surface. 4. Influence of the sze of actve area Fnally we demonstrate the mpact of the sze of the actve area on the chp over whch the power s dsspated. For that purpose we ncrease the sze of the actve area on the. x. mm chp of our test structure from. x. mm to. x. mm and. x. mm. Fgure compares the resultng ectve heat spreadng profles and spreadng angles, revealng that the sze of the actve area has a major mpact on the heat spreadng. We observe that the heat spreadng angle nsde the chp decreases wth ncreasng sze of the actve area. Fgure : Effectve heat spreadng profle (a) and spreadng angle for dfferent szes of the actve area. A heat transfer cocent of. 4 W/(m K) was appled at the case surface. Once the actve area becomes equal to the chp sze the heat spreadng angle nsde the chp drops to small values however not to zero whch may be surprsng at frst. The subsequent leadframe whch s larger than the chp provdes plenty of room for lateral heat spreadng whch causes the heat flux lnes to bend already nsde the chp. Ths ect can also be observed n fgure where we see non-planar sothermals already n the chp, even so the whole surface of the de was heated homogeneously. The heat spreadng angle nsde the copper leadframe on the other hand s not nfluenced by the sze of the actve area.

6 5. A practcal example In order to demonstrate the applcablty of the ectve heat spreadng concept we use the method to calculate Rth-JC values for four power packages wth exposed de pads of dfferent leadframe thckness (fgure ). The Rth-JC of the same packages had prevously been determned n a Fnte Element smulaton study usng the detaled models shown n fgure. Therefore accurate Rth-JC values are avalable for comparson. In the FE smulatons a fxed case temperature boundary condton had been appled to determne Rth-JC. To gan nsght nto the dependence of the heat spreadng nsde the leadframe on ts thckness we performed a few smulatons wth our test structure (table ) for dfferent values of the leadframe thckness and wth fxed case temperature, the results of whch are shown n fgure. We see that the heat spreadng angle nsde the leadframe starts at values from 5 to 45, dependng on ts thckness, and that t decreases to zero towards the case surface due to the deal coolng boundary condton. Snce we do not want to treat each leadframe thckness separately we decded to approxmate ths behavor by a heat spreadng angle whch starts at an average value of 5 and decreases lnearly to zero towards the case surface. Furthermore we neglect the small amount of heat spreadng nsde chp and de attach, assumng zero heat spreadng n that regon. The resultng truncated cone model wth a 4-slce dscretzaton of the leadframe can be seen n fgure 4. We mplemented ths TCM n a spreadsheet calculator as follows: The bottom sde length a of the -th slce of the leadframe heat spreadng cone wth square cross secton and thckness d s a a d tan, a A () To compute the thermal resstance R th, of each slce we use the area of the center plane d Rth, A, ( a a ). () k A 4 D PAK DPAK (a) y( = / a( [mm] [ ] mm LF.5mm.5mm LF.5mm.9mm LF.9mm.7mm.5.5 LF.7mm TOLL SO8 Fgure : FE models of 4 power packages wth exposed depad of dfferent thckness. Fgure : Effectve heat spreadng profle (a) and spreadng angle for the structure from table, but solder de attach and varyng leadframe thckness. A fxed case temperature boundary condton was appled at package case. De attach 45 = 5 = 6.5 = 7.5 = 8.75 Chp, area A Fgure 4: Truncated cone model used to calculate Rth-JC. d

7 Package LF Thckness Chp sze (a) Rth-JC FEM smulaton Snce we do not assume heat spreadng nsde chp and de attach the correspondng layers need not to be subdvded. The Rth-JC values obtaned by summaton over the R th, values of all slces of ths model are lsted n column n table. For comparson we have also computed the Rth-JC values wth the same TCM assumng a constant 45 heat spreadng angle nsde the leadframe n column (c). Wth respect to the Fnte Element results n column (a) the error of ectve heat spreadng model s much smaller than that of the 45 heat spreadng model whch overestmates the spreadng angle. The thus valdated spreadsheet could now be used to quckly calculate the Rth-JC for a wde range of dfferent power packages wth exposed de pad. However we have to keep n mnd the assumptons for whch the underlyng heat spreadng model has been derved; e.g. n ths case we assumed deal coolng and no heat spreadng nsde the de. For other coolng condtons or devces wth a small actve area on a larger de we would have to adapt the heat spreadng model n our spreadsheet. 6. Concluson We propose the defnton of an ectve heat spreadng angle whch s based on the local heat flux densty along a heat flow path. Based on ths heat spreadng angle and the assocated heat spreadng profle t s n prncple possble to calculate the exact value of the temperature dfference between start and end pont of the path usng a truncated cone heat spreadng model (TCM). For a sngle specfc case there would be lttle motvaton to do so snce the calculaton of tself requres a Fnte Rth-JC Effectve heat spread. model Error (c) Rth-JC spreadng angle 45 Error D PAK % % % % % % %.49-5.% DPAK %.9-9.4% % % % % %. -.9% TOLL % % % % % % % % SSO % % % % %.8 -.9% %. -8.6% Table : Comparson of the Rth-JC values computed usng (a) detaled fnte element models, the ectve heat spreadng model, and (c) the 45 heat spreadng model. Element smulaton whch could as well be used to drectly calculate the temperature dfference. But as demonstrated n the prevous example t s often possble to derve a heat spreadng model for a wde enough range of applcatons to make t worth the ort. We hope that the results presented heren can help to solve the mystery of the correct heat spreadng angle n numerous spread sheet calculators, hopefully resultng n more accurate estmates of Rth-JC and other thermal resstances. References. C.J.M. Lasance, Heat Spreadng Not a trval Problem, Electroncs Coolng, Vol. 4, No., May 8.. Drk Schwetzer, The Juncton-To-Case Thermal Resstance A Boundary Condton dependent Thermal Metrc, Proc. 6 th SEMI-THERM, Santa Clara, pp. 5-57,.. Bruce Guenn, The 45 Heat Spreadng Angle An Urban Legend?, Electroncs Coolng, Vol. 9, No. 4, Nov.. 4. Yasush Koto, Shoryu Okamoto, and Tosho Tommura, Two dmensonal numercal nvestgaton on applcablty of 45 heat spreadng angle, Journal of Electroncs Coolng and Thermal Control, Vol. 4, pp. -, T.A. Shedd, "Fundamental Behavors and Lmts of Impngement Coolng", Proc. th SEMITHERM, San Jose, pp. 79-8, 7.