CLARKSON UNIVERSITY. Block-Based Compact Thermal Modeling of Semiconductor Integrated Circuits

Transcription

1 CLARKSON UNIVERSIY Block-Based Compact hermal Modelng of Semconductor Integrated Crcuts A dssertaton By Jng Ba Department of Electrcal and Computer Engneerng Submtted n partal fulfllment of the requrement for the degree of Master of Scence (Electrcal and Computer Engneerng) July, 009 Accepted by the Graduate School Date Dean

2 he undersgned have examned the thess enttled Block-Based Compact hermal Modelng of Semconductor Integrated Crcuts Presented by Jng Ba A canddate for the degree of Master of Scence n Electrcal Engneerng, and hereby certfy that t s worthy of acceptance. Date: Advsor: Examnng commttee: Date: Commttee member: Date: Commttee member: Date: Commttee member:

3 Abstract In recent years, compact thermal models have been wdely used for dfferent levels n electronc systems wth dfferent precsons and effcency to meet the desgn requrement. However, these models are usually developed for a standard IC package or system only to offer overall temperature nformaton of the semconductor chp or system and can not offer more detaled temperature for n IC desgn. hs thess presents a block-based compact thermal model for semconductor ntegrated crcuts based on the conventonal compact thermal models developed n recent years. he developed model has been shown to be very accurate and effcent. It s very useful for any structure that can be constructed prmarly usng buldng blocks. he concept apples qute well to semconductor technology snce desgn of ntegrated crcuts are based on standard blocks stored n the lbrares, such as standard cells and functonal crcut blocks, etc. he developed approach frst dvdes the semconductor ntegrated crcuts nto buldng blocks. Detaled numercal smulatons are performed for each selected block. Interor and boundary nodes are then selected, and the compact thermal models for the selected standard buldng blocks are constructed usng the smulaton results and the selected nodes for the block. Wth the compact thermal models for these selected standard blocks, larger IC crcuts can be constructed and smulated wth desred effcency and accuracy.

4 wo methods n ths work are developed for choosng the boundary nodes. One of the methods offers poor thermal contnuty. However, the other enforces the boundary contnuty more properly and leads to consderably better results compared to the detaled numercal smulaton. It s shown that selecton of boundary nodes should consder both the temperature and heat flux varatons along the boundary. In summary, the developed model offers accurate thermal nformaton n crtcal locatons of semconductor ntegrated crcuts usng a small number of nodes compared to any numercal smulaton. he method s very smple and can be mplemented n semconductor chps wthout any dffculty and provdes a very useful tool for thermal aware chp desgn. v

5 Acknowledgements I would lke to extend my heartfelt grattude to my advsor Dr. Mng-Cheng Cheng for hs patent gudance durng my Master study and research. For me, he s a strct teacher as well as a kndly father. Hs serous atttude and enthusasm n research had motvated all hs advsees, ncludng me. And he was always accessble and wllng to help both n research and n lfe. Wthout hs supervson and nspraton, I cannot complete my work and study smoothly and successfully. I am also grateful to Dr. Hou and Dr. Schllng for beng my dssertaton commttee members and advsors and offerng valuable advce and support to my research and thess. In addton, I would lke to thank my colleagues, Kun Zhang and Yu Zhang, for gvng me valuable helps durng my research and beng my frends n the past two years. I would also thank my boyfrend, Chengx Lu, for encouragng me n my work and wrtng the thess. Fnally, my deepest grattude goes to my parents. Wthout ther support and everlastng love, I would never have the chance and courage to study abroad and complete my MS study. v

6 able of Contents Abstract... Acknowledgements... v able of Contents... v Lst of Fgures... x Lst of ables... xv Chapter Introducton.... Current Researches on Compact hermal Model..... Sngle-temperature (S) Node Model for SOI MOSFE..... Compact hermal Model for Electronc System Self-heatng Effect Objectves of Work Overvew... 7 Chapter General Concepts of Least Squares Method and hermal Modelng Introducton of Least Squares Method Least Squares Method Lnear Least Squares Overdetermned System Modelng and Curve Fttng.... hermal Modelng Concept... 3 v

7 .. Heat Conducton Equaton hermal Crcut Equaton... 4 Chapter 3 hermal Crcut Model Development Basc Concept Matrx Equaton Dervaton Resstance Network Dervaton Data Collecton Factors Affect Data Qualtes Random Numbers Resstance Network Optmzaton Compact Model Smulaton Selectng Boundary Nodes for Compact hermal Model Boundary Nodes Represented by Small Areas Applcaton to the sngle and mult-block nterconnect structures usng the frst approach Boundary Nodes Represented by Lnes Applcaton to a mult-block structure Dscusson of wo Dfferent Methods Chapter 4 Applcaton of Block-based Compact hermal Crcuts to an SOI Inverter Structure and Dvsons of Inverter Smulaton Results of Each Block Heat Flux Influence on Boundary hermal Contnuty for the Devce Block Equal Boundary Dvsons for Node Selecton v

8 4.3. Smulaton Result Unequal Boundary Dvsons for Node Selecton Smulaton Result Chapter 5 Concluson and Future Work References... 8 v

9 Lst of Fgures Fgure - Example of curve fttng for a quadratc functon [7] Fgure - hree-dmensonal thermal crcut model... 6 Fgure 3- Cross secton of three rectangle metal wres embedded n oxde, where dmensons of all rectangles are 0. m Fgure 3- emperature profle for three metal nodes for power generatons on the three nodes equal to (a) mw / m, mw / m and 5 mw / m, and (b) mw / m, 4 mw / m and 0 mw / m. MN ndcates the locaton of the mnmum temperature and MX ndcates the locaton of the maxmum temperature Fgure 3-3 Heat fluxes along the lne of AA for (a) Case and (b) Case Fgure 3-4 Resstors n three cases can be removed: redundant resstors, resstors between two far nodes and resstors across dfferent materals Fgure 3-5 Boundary nodes of two neghborng blocks must be consstent Fgure 3-6 -shape nterconnect block wth small areas as the boundary nodes. Dmensons are gven below: x4 x =0.35 μ m, x y x x x 0.75m, x 0.09 μ m, y 0. μ m, x y y 0.48m, y y 0.8m, y y 0.m and L node 0.04m Fgure 3-7 Illustraton of powers and the thermal ground appled to the basc -shape x

10 nterconnect block and ts resstance network after optmzng usng the frst approach Fgure 3-8 (a) he sngle -shape nterconnect block wth nput powers, 0.5mW and 4mW, appled to Nodes 4 and 6, and Node 8 s selected as the thermal ground. (b) emperature contours from the ANSYS smulaton. MN means the mnmum temperature n the structure and MX s the maxmum temperature Fgure 3-9 Comparson of the emperature at each node of the resstance network derved from the ANSYS and Spce smulatons. Devatons of Spce from ANSYS are ncluded Fgure 3-0 (a) he 6-block -shape nterconnect structure wth 6.5mW appled to one of the metal surfaces. (b) emperature contours for the structure derved from ANSYS smulaton. MN ndcates the locaton of the mnmum temperature and MX ndcates the locaton of the maxmum temperature Fgure 3- emperature profles obtaned from ANSYS and SPICE smulatons along the lnes of (a) AA, (b) BB and (c) CC ndcated n Fg.3-(a) Fgure 3- Boundary nodes of the -Shape nterconnect block n the second approach. L node =0.04 μ m, L lne =0.05 μ m... 4 Fgure 3-3 he structure ndcatng how to apply heat flux for the second approach n the selected block that s shown n the rectangle nsde the larger block. One of the areas n the block needs to be grounded to have a unque thermal soluton... 4 Fgure 3-4 Some possble cases for the drectons of heat fluxes wthn a block. (a) 4 possble heat flux dstrbutons wthn a block when none of boundares s adabatc. (b) 4 more possble heat flux dstrbutons when ether two opposte x

11 sdes are adabatc Fgure 3-5 hermal resstance network for the -shape nterconnect block usng the second approach Fgure 3-6 (a) he 6-block structure usng the -shape nterconnect blocks wth a power of 6.5mW appled to the lower left metal surface. (b) emperature contours for the 6-block structure derved from ANSYS smulaton. MN ndcates the locaton of the mnmum temperature and MX ndcates the locaton of the maxmum temperature Fgure 3-7 emperature profles obtaned from ANSYS and SPICE smulatons along the lnes of (a) AA, (b) BB and (c) CC ndcated n Fg. 4-(a) Fgure 4- (a) op vew of an nverter layout. (b) Cross-secton vew along AA of the nverter usng SOI technology, where the selected blocks are shown. he resstance network wll be extracted for each selected block. (c) he detaled dmensons of the devce for the nverter gven n Fg.4-(b). hermal conductvtes n the moderately doped channel and heavly doped source and dran are taken to be equal kc = ks = kd = 63W /( mk)... 5 Fgure 4- he selected substrate block wth, dmensons and resstance network. he node length s L node = 50nm. he dvson length for Node, 3, 4, 8, 9, 0, 4, 5 and 6 s L seg = 08nm and the dvson length for Node 5, 6, 7,, and 3 s L seg = 5nm Fgure 4-3 he selected Metal--Contact nterconnect block wth dmensons and resstance network x

12 Fgure 4-4 he selected metal- juncton block wth dmensons and thermal resstance network Fgure 4-5 he selected oxde- block wth dmensons and resstance network Fgure 4-6 he selected oxde- block wth dmensons and resstance network Fgure 4-7 (a) Node selecton for the devce block wth equal boundary dvsons where each node s located n the center of the dvson. (b) hermal resstance network for the devce block wth node selecton gven n Fg.4-7(a) Fgure 4-8 (a) he D structure for the nverter cross secton wth 0.9mW and 0.6mW powers appled to left and rght two devces, respectvely. (b) emperature contours derved from ANSYS smulaton. MN ndcates the locaton of the mnmum temperature and MX ndcates the locaton of the maxmum temperature Fgure 4-9 emperature comparson along the channel AA between the block-based compact thermal crcut and ANSYS smulaton wth equal boundary dvsons for the node locatons Fgure 4-0 emperature comparson along the Metal- BB between the block-based compact thermal crcut and ANSYS smulaton wth equal boundary dvsons for the node locatons Fgure 4- emperature comparson along the Metal- CC between the block-based compact thermal crcut and ANSYS smulaton wth equal boundary dvsons for the node locatons Fgure 4- emperature comparson along the boundary nterface DD between the block-based compact thermal crcut and ANSYS smulaton wth equal x

13 boundary dvsons for the node locatons Fgure 4-3 Heat flux along the nterface lne DD shown n Fg.4-8(a) Fgure 4-4 Node selecton and thermal network for the devce block wth unequal boundary dvsons for the node locatons Fgure 4-5 Node selecton and thermal network for the oxde- block wth unequal boundary dvsons for the node locatons Fgure 4-6 he temperature comparson along the channel AA between our model and ANSYS model wth unequally dvded boundary nodes Fgure 4-7 he temperature comparson along the Metal- BB between our model and ANSYS model wth unequally dvded boundary nodes Fgure 4-8 he temperature comparson along the Metal- CC between our model and ANSYS model wth unequally dvded boundary nodes Fgure 4-9 he temperature comparson along the boundary DD between our model and ANSYS model wth unequally dvded boundary nodes Fgure 4-0 Heat flux along the nterface lne DD shown n Fg.4-8(a) x

14 Lst of ables able I Analogy between thermal crcut and electrc crcut [] able II Extracted Resstance values and the dfferences for the dentcal resstor pars for the -shape nterconnect block usng the frst approach able III Extracted thermal resstance values and the dfferences for the dentcal resstor pars for the -Shape nterconnect block usng the second approach able IV Extracted thermal resstance and the dfferences of the dentcal pars of resstors n the network for the substrate block able V Extracted thermal resstance values and the dfferences of the dentcal resstor pars n the thermal resstance network for the Metal--Contact nterconnect block able VI Extracted thermal resstance values and the dfferences of the dentcal resstor pars n the thermal resstance network for the Metal- model able VII Extracted thermal resstance values and the dfferences of the dentcal resstors n the thermal resstance network for the Oxde- block able VIII Extracted thermal resstance values and the dfferences of the dentcal resstors n the thermal resstance network for the Oxde- block... 6 able IX Extracted values of thermal resstances n the thermal resstance network wth equal boundary dvsons for the nodes able X Extracted thermal resstance values n the thermal resstance network for devce xv

15 block wth unequal dvded boundary dvsons for the node selecton able XI Extracted thermal resstance values n the thermal resstance network for oxde- block wth unequal dvded boundary dvsons for the node selecton.74 xv

16 Chapter Introducton Due to aggressve reducton n the devce scale, the number of devces per area ncrease rapdly, as well as the amount of power densty dsspaton. he hgh temperature caused by the hgher power densty dsspaton leads to a decreasng dran current, whch could degrade the devce performance and lfe tme []. In tradtonal crcut desgn, the crcuts are usually desgn wthout careful consderaton of thermal behavor n the chp, because t s an extremely tme consumng process to obtan thermal nformaton n the chp. he desgn has to be tested to nclude thermal effects and may need to be repeated to consder the effects. hs teratve process would obvously prolong the desgn cycle tme and ncrease the workload []. herefore, ncluson of thermal analyss n the desgn process wll substantally shorten the desgn cycle tme for the ntegrated crcuts.. Current Researches on Compact hermal Model Researchers and engneers have recently pad ncreasng attenton to the thermal ssues n mcroelectronc chps because of the rsng power densty n the chps due to the downszng of the devce and nterconnect scales. In ths secton, two major approaches to thermal modelng of ntegrated crcuts are brefly presented.

17 .. Sngle-temperature (S) Node Model for SOI MOSFE he S-node model s a smplfed thermal crcut model wdely used n ndustry. When performng the electro-thermal smulaton n a crcut smulator, such as Spce, only constant temperature can be used n nearly all electronc devces. For example, n BSIMSOI (Berkeley short-channel nsulated-gate slcon-on-nsulator), the devce temperature s assumed to be unform, so the whole slcon flm s treated as a sngle node [,5]. Heat flow from the devce to the substrate can be represented by a thermal resstor R th parallel wth a thermal capactor C th, and the power generated n the channel s descrbed by a power source []. In ths model, the substrate temperature can be taken as a thermal reference o, the average temperature of the SOI channel s the channel temperature so, and the power dsspaton P so s defned as P I V so d d. th R can be extracted from R th so I V d d o, whch wll be ntroduced n next chapter. so can be obtaned from mult-dmensonal numercal smulaton or expermental measurements. he S-node model n BSIMSOI (Berkeley short-channel nsulated-gate SOI) can be ntegrated nto the IC desgn process to acheve thermal aware desgn to shorten the desgn cycle; however the constant temperature assumpton n the model does not represent the realstc temperature gradent along the devce channel. Most of the heat s generated from the area under the gate closer to the dran, leadng to a large temperature gradent n the channel because of the very thn slcon sland []. he assumpton n S-node model overestmates the temperature n the source and dran areas but underestmates that n the

18 gate area. In addton, the constant temperature n the channel also leads to erroneous heat flux to the termnals and an unrealstc temperature estmaton of nterconnects... Compact hermal Model for Electronc System A dfferent model that has attracted much attenton from researchers and engneers s compact thermal model whch was ntroduced by the DELPHI consortum based on the concept of thermal crcut equatons. Its am s to buld a Boundary Condton Independent (BCI) compact thermal model that s able to work under a wde range of boundary condtons wth hgh accuracy [3,4,]. here are three steps n ths procedure. Frstly, smulate a detaled component model wth several sets of boundary condtons chosen by the consortum and collect the smulaton results of each set. hen, generate a thermal resstance network for the component model correspondng to ts specfc juncton nodes and boundary nodes. Fnally, optmze the resstance network based upon the heat flows and temperature dstrbuton of the detaled model [3]. Usng the DELPHI procedure, the crcuts could be modeled n dfferent desgn levels from the transstor level to the system level and hgher, and generate the thermal models for each level [4,0]. he accuracy of the model s depends on the numbers of the nodes selected n the compact model. he more nodes n the model, the more accuracy the result would be. However, ths method s only appled to a sngle IC package [5]. It cannot be assembled together to smulate a larger crcut to offer more detal temperature nformaton. In addton, although the concept of ths method s boundary condton ndependent, t s verfed that no lnear compact model s really boundary condton ndependent [3,4]. In some cases, one 3

19 compact thermal model s only accurate for a specfc heat flux dstrbuton on the sdes of the component [3]. In addton, the boundary condtons used to smulate the model s chosen by long experence to be adequate. hese drawbacks restrct the usage and extenson of the model.. Self-heatng Effect Self-heatng effect s an mportant factor that affects the performance of semconductor devces. Wth the drastc advance n semconductor ndustry, thermal problems caused by the self-heatng effect become much more serous. A typcal case s the SOI technology [6]. In ths thess, a semconductor crcut usng SOI technology s taken as an example to llustrate the thermal model. SOI structure s a knd of solaton technque n semconductor fabrcaton. Its man am s to delectrcally solate the semconductor devces by bured oxde (BOX). he BOX solates the sngle devce from other devces as well as the underlyng substrate whch enables more compact desgn and elmnates latch-up effect. SOI technology can reduce the leakage currents and juncton capactance, furthermore, mprove the speed and reduce the power dsspaton [7,8]. Despte of many advantages of SOI technology, t also has some serous drawbacks whch restrct ts development. One of the drawbacks s the self-heatng effect whch s manly caused by the small conductvty of the BOX. he thermal conductvty of the BOX s about.4w / mk whch s about one hundred tmes less than that of slcon and several hundred tmes smaller than that of copper. As a result, heat generated n the devce channel s 4

20 nhbted from dsspatng to the substrate va the BOX. hs results n strong heat dsspaton, through nterconnects, and enhances the temperature n the chps. As we know that the electrcal resstance of the metal ncreases as the temperature rses, ths wll also result n longer delay tme of n the metal wre and degrade the performance of the whole chp [7]..3 Objectves of Work o obtan thermal dstrbutons and to capture hot spots and hgh temperature gradents n ntegrated crcuts, the fnte element or fnte dfference method s usually used [8]. Although these methods offer detaled and accurate results, they are computatonal ntensve and dffcult to be mplemented n electro-thermal smulaton of the crcuts for cost-effectve chp desgn. In ths thess, a block-based thermal model has been developed based on the compact thermal model that has been appled to thermal smulaton of semconductor chps for many years [,3,4,5,9,0]. he block-based compact thermal model frst parttons a semconductor chp nto standard buldng blocks. A set of standard buldng blocks can be selected for a specfc technology. A compact thermal model for each block s constructed and saved n a lbrary. A compact thermal model for a crcut block or a chp can then be constructed by usng the thermal models of the buldng blocks n the lbrary. he concept of the block-based compact thermal model s studed n ths thess and the procedure for constructng the model of each buldng block s presented. he boundary nodes for the blocks are approprately selected and the analyss s carefully performed to ensure temperature and flux contnuty at the nterfaces between blocks. 5

21 A D SOI semconductor crcut technology s taken as an example to llustrate the thermal crcut modelng method, because SOI structure has more serous heatng problem. he concepts of fnte element method and compact thermal modelng are combned together and modfed. Usng the concept of fnte element method, the structure can be represented by thousands of nodes or elements wth correspondng characterstc and attrbute. However, nstead of meshng the materals nto thousands of nodes, we only select some mportant juncton nodes or locatons wthn the structure, whch could reduce the calculaton tme sgnfcantly. Of course, only the fully detaled model could gve out zero error smulaton result. Selectng only the juncton and boundary nodes would cause a larger error. Hence, an optmzaton problem s brought forward: Fnd a thermal network wth a smaller number of nodes but mnmum overall error. o solve ths problem, several experments and comparsons are made whch wll be ntroduced n Chapter 4. Moreover, we want to buld a model that can solve the boundary problems that one compact thermal model only ft for one specfc heat flux dstrbuton on boundary. Regardng the problem of choosng the boundary condton for the smulaton, our object s to fnd a mathematcal method to apply the boundary condton nstead of experence selecton that would be more accurate and convenent. One core contrbuton of our method s to construct several standard blocks so that desgners can choose the blocks they need and put them together to buld a larger crcut system. hs concept has never been mentoned and fulflled by any nsttutons or researchers. he essental work of our research s to make sure that the boundary condtons between each block are contnuous. wo dfferent methods to choose the boundary nodes are 6

22 compared n ths paper to make sure the heat fluxes on the boundares are contnuous. And the nfluence of the heat flux and temperature on the boundary node selecton s shown..4 Overvew Chapter explans some general concepts of least squares method and thermal modelng used n our work. he basc thermal modelng method s to buld the thermal crcut equaton. hen, the least squares method s used to solve the thermal matrx equaton. In Chapter 3, the method to generate the thermal crcut model for a D structure wll be ntroduced. After achevng the thermal crcut model for a sngle standard block, several standard block can be put together to model a large compact structure. Moreover, two dfferent methods of selectng nodes are presented. In Chapter 4, the thermal crcut model s appled to an SOI nverter. From smulaton results, the heat flux and temperature nfluence on block boundary s presented and thermal crcut model s modfed to get more accurate results. 7

23 Chapter General Concepts of Least Squares Method and hermal Modelng. Introducton of Least Squares Method Least squares method s a commonly used method to solve an overdetermned system, whch could also be nterpreted as a fttng-data method. In statstcal problems, t s often desred to ft a certan model to an expermental data wth mnmum errors, for example n regresson analyss and curve fttng. By usng ths method, the best but not the unque soluton for the system could be obtaned []... Least Squares Method Assume that a data set wth n ponts ncludes x, ), x, ),, x, y ), ( y ( y ( n n where x s an ndependent varable and y s a dependent varable. he functon of ths model could be wrtten as [3] y f x ),,, n. (-) ( After defned the model, the resduals r could be calculated as 8

24 r y fˆ( x ),,, n, (-) where y s the values of the dependent varable and f ˆ( ) s the predcted values x calculated by the model. o make the model best ft the data, values of the parameters,, need to be found. he least squares method s to fnd the parameter values, when the sum, S, of squared resduals s the mnmum, S n r. (-3).. Lnear Least Squares Least squares problems could be dvded nto two categores, lnear least squares and nonlnear least squares. Lnear least squares s used when the parameters of the regresson model s combned lnearly. he lnear regresson model s defned as y f m x, x x mm x j j x j. (-4) o mnmze the error, the gradent of sum, S n n m r y j j x, (-5) j wth respect to parameter values should be equal to zero. herefore, the equaton could be 9

25 wrtten as S j n r r j 0. (-6) Lettng r j j x X j, (-7) the equaton could be rewrtten as S j n y m k X k k X j 0, j,, m. (-8) herefore, n m k n k Xk X j X jy, j,, m. (-9) he equaton could be transformed to matrx form, X Xβ X y [4]. (-0) hs matrx equaton s also soluton of the defned model...3 Overdetermned System In mathematcs, the system s consdered overdetermned, when the number of equatons s larger than that of unknown varables [5]. here are two cases of an 0

26 overdetermned system. For an overdetermned system, n dfferent cases, the system equatons would have only one soluton, nfnte soluton or no soluton. A. Homogeneous Case Consder a lnear system, y f ( m m x ) ( x ) ( x ) ( x ) 0,,,, n. (-) When the system s overdetermned, n m, the varable 0 would m always be the soluton and there s no other soluton. B. Inhomogeneous Case Consder another lnear system, y f ( x ) C,,,, n, (-) ( x ) ( x ) ( x ) mm where C s constant number. he least squares method can be used to fnd the best solutons for the overdetermned systems. As t shown n Secton.., for a system, X y, (-3) the soluton would be X X X y. (-4)

27 ..4 Modelng and Curve Fttng One of common use of least squares method s curve fttng. Curve fttng s fndng a curve whch has the best ft to a seres of data ponts and possbly other constrants [6]. Smlar to the dervaton of lnear least squares, let x be the ndependent varable and f (x) represents an unknown functon of x that we want to approxmate. Assumng there are n observatons, y f ( x ),,, n, (-5) = the lnear combnaton of m bass functons to model f (x) would be f ( m x x) ( x) ( x) m ( ), (-6) where the functon (x) mght be nonlnear n the varable of x. ake a quadratc functon, f, (-7) ( x) x x 3 for example. Fg.- shows the curve fttng usng the least squares method.

28 Fgure - Example of curve fttng for a quadratc functon [7]. herefore, n lnear least squares, the functon doesn t need to be lnear n the terms of x. he parameters are determned to gve the best ft.. hermal Modelng Concept.. Heat Conducton Equaton wrtten as he heat conducton equaton descrbng heat flow n a physcal doman can be c pd (-8) t where s materal densty, c specfc heat, thermal conductvty and p d the space and tme dependent power densty generated n the doman. 3

29 In Eq.(-8), heat flux s gven as H. (-9) herefore, the heat conducton equaton could also be descrbed as H H H c p d. (-0) t x y z.. hermal Crcut Equaton Based on the fnte dfference method, consderng a small volume xyz of the th cell n a doman, the heat flux n the x drecton can be gven n terms of as H x, x, (-a) x x x, and H x, x,. (-b) x x x, he power densty flow to the th cell n the x drecton s wrtten as H H H x x, x, px,. (-) x x hus, the total power flowng to the th cell n the x drecton s H H y z Px, px, xy z x, x,. (-3) Substtutng for H n Eq.(-3), x, / 4

30 5,,,,,,,,,,, thx x x thx x x x x x x x R R z y x x P, (-4) where x s the length of the cell, z y s the cross secton and z y x s the thermal resstance thx R of the cell along the x drecton. o consder 3D heat flow, followng the smlar dervaton for the heat flow n y and z drectons, the power densty flow to the th cell n the y and z drectons are wrtten as y H H y H p y y y y,,, (-5a) and z H H z H p z z z z,,,. (-5b) herefore, the heat conducton equaton gven n Eq.(-0) n a physcal doman becomes, s th s thz z z thz z z thy y y thy y y thx x x thx x x s z y x th P R P R R R R R R P P P P t C,,,,,,,,,,,,,,,,,,,,,,,,,. (-6)

31 z z, R thz, R thy, y, R x, thx, x, R thx, x R thy, R thz, y y, C th, z, P s, Fgure - hree-dmensonal thermal crcut model able I Analogy between thermal crcut and electrc crcut []. Electrc crcut model V V C I s, t R hermal crcut model th C, Ps, t R th Charge, Q C Energy, w J o Voltage, V V emperature, C Current, I s dq dt A C s Power, P s dw dt W J s o A/ cmv hermal conductvty, W cm C Electrc conductvty, Charge flux, J V A m Heat flux, H Capactance, C C / V Electrc resstance, R l A V / A hermal capactance, C hermal resstance, R th th W m xy z C J C o x A x d o C / W p 6

32 he dfferental equaton derved n Eq.(-6) for 3D heat flow to a node s actually analogous to the Krchhoff's Current Law (KCL) descrbng current flow to a crcut node. he 3D heat flow to the th node can therefore be descrbed by n a thermal crcut model shown n Fg.-. herefore, an analogy between the thermal and the electrc crcuts can be made as t shown n able I. From the analogy shown n able I, charge Q n the electrc system s an analogy to energy w n the thermal system, whch means the electrc system represents the conservaton of the charge and thermal system the conservaton of energy. Current (charge flow) I s therefore the counterpart of power (energy flow) P n the crcut system. Voltage V and resstance R n the electrc system are the counterparts of temperature and thermal resstance R th n the thermal system, respectvely. able I also shows that charge flux J drven by the voltage gradent V s smlar to heat flux H by the temperature gradent.wth ths analogy, the thermal crcut can be mplemented n crcut smulaton software, such as SPICE. 7

33 Chapter 3 hermal Crcut Model Development In ths chapter, the procedure to generate the thermal crcut model for a D structure s ntroduced based on the least squares method and the thermal crcut concept descrbed n Chapter. he method frst dvdes a physcal doman nto buldng blocks and develops a compact thermal crcut for each buldng block. After obtanng the thermal crcut model for each sngle buldng block, several buldng blocks can be put together to construct a compact thermal crcut model for a larger structure. he approach can be appled to many dfferent felds whose doman structures can be prmarly constructed usng buldng blocks, such as semconductor chps, battery packs and solar cells, etc. In ths thess, the developed method s only appled to the SOI technology. 3. Basc Concept A compact thermal crcut ncludes a consderably smaller number of nodes than numercal methods, and provdes more effcent approach to thermal smulaton of a physcal doman. he approach s not new and has been wdely appled to thermal smulatons n semconductor chps and mcroelectronc systems [,3,4,7,,6]. Although our approach s dfferent from others, the basc concept s derved from prevous work n the compact thermal 8

34 crcut models [3,4,7,]. In order to capture the mportant thermal nformaton n a semconductor chp, such as hot spots and hgh temperature gradents, t s obvously prohbtve to perform a detaled numercal smulaton due to an enormous number of devces and nterconnects. When applyng the compact thermal crcuts for thermal smulaton of semconductor chps, the mportant questons are therefore the followng. ) Can a thermal crcut wth a small number of nodes for a semconductor chp capture the mportant thermal nformaton for chp desgners? hs ncludes thermal consderaton on relablty and thermal management and thermal effects on electrc characterstcs of devces, metal wres and the whole chp. ) Can thermal contnuty be enforced when placng buldng blocks together to construct a larger structure? 3) How are the nodes selected to mnmze the number of nodes and to acheve the accuracy? hs s also related to computatonal memory space and effcency durng smulaton. hese three questons are eventually ted together and are determned by how the nodes are selected and how the thermal elements are extracted for each block. Accordng to our nvestgaton, nodes are selected at the followng locatons. a. he devce juncton where the devce heat source s located, such as the channel-dran juncton n MOSFEs, 9

35 b. he junctons metal and devces, c. he junctons and corners of metal nterconnects, such as ends of a va and metal wre junctons, d. Areas of hgh temperature gradent, e. Every materal needs to have at least one node n the nteror or on the boundary, Examples wll be gven, and node selecton and element extracton related to the soluton accuracy wll be demonstrated later. 3. Matrx Equaton Dervaton Once the thermal nodes of the physcal doman are selected, thermal elements, ncludng thermal resstances and capactances, need to be extracted. In ths thess only steady-state soluton s consdered, and only extracton of thermal resstance s nterested. For a steady state thermal system, t s only changed n spatal doman not n tme doman. In the steady state, thermal crcut equaton, Eq.(-6), can be rewrtten as Ps,, (3-) Rth because C th 0. t Eq.(3-) can be rewrtten as for each node, 0

36 s n k k k k P R,,,, n,, =, (3-) where k R, s thermal resstance between nodes and k where k k R R,,, and s P, are the power flow nto nodes or physcally power generated by nodes. Because one of the nodes s consdered as the reference node, n - equatons for to n- can be rewrtten as,,,,,3 3,,,3 3, n n n n n n n n n n n n n P R R R P R R R P R R R (3-3) For an n node crcut, there are total n n thermal resstances that need to be extracted. o extract the values of the resstances, Eq.(3-3) can be expressed n terms of R,k as,

37 3,, 3,,3, n n n n n n n n n n n n n n n P P P P R R R R R R R R, (3-4) where n and k R, s the thermal conductance k G, between nodes and k. Eq. (3-4) can also be wrtten as P G =, (3-5) where s an n n n matrx, G s an n n matrx and P s an n matrx. 3.3 Resstance Network Dervaton If there are more than two nodes n a crcut (n > ), the number of unknowns k G n Eq.(3-5) wll be greater than the number of equatons

38 n n n, for n 3. (3-6) o extract the n n unknowns G, k, one can generate node temperatures wth appled power sources at the nodes to establsh n n ndependent equatons for G, k, as ndcated n Eq.(3-5). hs can be acheved usng detaled numercal smulaton. However, numercal fluctuatons may lead to large devaton n the values of G, k. he least squares method ntroduced n Secton. s therefore used to solve an overdetermned system n order to mnmze the statstcal fluctuatons In order to buld an overdetermned system, several detaled numercal smulatons can be performed to obtan several sets of temperature and power P s,. In ths way, several sets of the matrx equaton, Eq.(3-5), can be created wth dfferent matrces and P. We can thus ncrease the number of rows of the matrces and P to form a new matrx equaton gven as, G = P, (3-7), n where s an n m n matrx, P s an n m matrx and m s the number of detaled smulatons performed to generate m set of data of large enough, and P s,. When m s n n m n, (3-8) 3

39 whch means ths system s an overdetermned system. Usng the least squares method, G can be determned when As defned n Eq.(-0), ' and P ' are gven. ' ' G ' P', (3-9) and t can be solved as ' ' ' P' G. (3-0) hs equaton has been mplemented n MALAB to extract G usng the backslash operator. hat s, the least squares soluton to such a system s gven as, G '\P'. (3-) 3.4 Data Collecton o solve G n Eq.(3-7), temperature n each node needed to be generated wth power suppled n each node. In ths work, ANSYS s used to generate temperature data. In ANSYS, dfferent thermal exctaton and boundary condtons can be appled accordng to dfferent thermal crcumstance. For example, f powers or heat fluxes are appled n dfferent areas and lnes, temperature dstrbutons n a selected buldng block can be obtaned and appled to Eq.(3-0) to generate resstance network for each buldng block Factors Affect Data Qualtes here are many factors that wll greatly affect the qualty of the generated data. 4

40 Physcally, to ensure a non-fluctuated value of an extracted resstance, heat flux through the resstance must be large enough durng the smulaton to collect temperature data. herefore, when collectng the data, a suffcent number of cases that enforce heat flux n all dfferent drectons need be provded to make sure each par of neghborng nodes has enough heat exchange. Another factor s caused by dependent equatons n Eq. (3-6) nduced by the collected data. o avod ths, m sets of data generated from smulatons should be ndependent. hs means that dfferent sets of power nputs on each node can not be proportonal wth the same BC s. o ensure good qualty of the collected data, nput sets wth substantally dfferent nput power strengths and very dfferent boundares need to be used. he followng example would show how the heat flux and power affect the temperature profle, and further more, the matrx equatons. he followng two cases have the same structure but dfferent power generaton on each node. Fg.3- shows the structure for the examples that can be vewed as a cross secton of three rectangle metal wres (nodes) embedded n oxde. he thermal conductvty o o of the metal s 40e-6W / m C and that of the oxde s.4e-6w m C /. 5

41 y / μm x / μm 0 3 Fgure 3- Cross secton of three rectangle metal wres embedded n oxde, where dmensons of all rectangles are 0. 0.m. In Case, the bottom of the structure s set to a constant temperature 0 o C and the other three boundares are all adabatc. Powers generated on node, 3 and 4 are 0.0 mw / m, 0.04 mw / m and 0. mw / m, respectvely. In Case, the BC s the same, but powers generated on node, 3 and 4 are double of the prevous case, 0.04 mw / m, 0.08 mw / m and 0. mw / m. hese dfferent cases actually generate the same equaton for the resstances n terms of temperature and nput power, and offer only one set of data nstead of two. Fg.3- shows the temperature profles of the two cases from ANSYS smulaton. As can be seen, two sets of power generatons wth proportonal strengths lead to the temperatures wth same proportonal factor at any locaton. Fg.3-3 shows the heat fluxes along the lne of AA for these two cases, where Case wth half nput power strengths results n half heat flux of Case. 6

42 ( μm) y ( μm) y A A x( μm) y( μm) emperature( o C ) (a) A A x( μm) emperature( o C ) (b) Fgure 3- emperature profle for three metal nodes for power generatons on the three nodes equal to (a) mw / m, mw / m and 5 mw / m, and (b) mw / m, 4 mw / m and 0 mw / m. MN ndcates the locaton of the mnmum temperature and MX ndcates the locaton of the maxmum temperature. 7

43 ( W / ) H / μ m x / μm (a) ( W / ) H / μ m x / μm x / m (b) Fgure 3-3 Heat fluxes along the lne of AA for (a) Case and (b) Case. 8

44 3.4. Random Numbers o avod the possblty of generatng a large number of duplcatng or smlar temperature data sets, a large number of smulatons may need to be performed to guarantee generaton of enough number of completely ndependent sets that wll offer better statstcal results. Random numbers are used n our work to acheve good qualty of the generated temperature data sets. Usng random numbers, a large numbers of nput sets (ncludng power appled to each node and the BC appled to each boundary) can be used approprately to generate heat flux between nodes wth dfferent drectons and magntudes and to nduce dfferent temperature dstrbutons. Another advantage of usng random numbers s that data generaton process can be done automatcally. As prevously dscussed, the thermal crcut matrx equaton may be an overdetermned system, where n n n m or m > n/. Based on the concept of the least squares method, t prefers to have the number of data sets that s much greater than n/. hs s necessary to avod the dependence of some generated data because there mght be certan correlaton between some random numbers. 3.5 Resstance Network Optmzaton After resstance network s solved, the number of the resstance network need to be reduced to ncrease the processng speed and relable extracted resstance values. We call t resstance network optmzaton. 9

45 7 8 9 R 3, R 4,6 3 R,3 Fgure 3-4 Resstors n three cases can be removed: redundant resstors, resstors between two far nodes and resstors across dfferent materals. here are manly three cases needed to be optmzed: a. Redundant resstors. As t shown n Fg.3-4, the resstor R, 3 should be removed, because the heat flux from Node to Node 3 are taken nto account by R, and R, 3. Actually, our study shows that the value of R, 3 sometmes appears to be negatve. b. Resstors between two far nodes. In Fg.3-4, the resstor R 3, 7 should also be removed. In ths case, the removed resstance mght be ten tmes or even hundreds tmes larger than other resstances, whch means the heat flux flows though these two nodes s so small that we can gnore t. c. Resstors across dfferent materals. hs case s often caused by nsuffcent heat flux between two nodes, whch leads to hghly fluctuated resstance value. As t shown n 30

46 Fg.3-4, Node 4 and Node 6 are located n the oxde on each sde of the metal. Because the thermal resstance s hundreds tmes larger than the oxde, t s hard to create a heat flow between these two nodes, and the extracted resstance become very unrelable. herefore, the resstor R 4, 6 should be removed. Once these resstors are removed, other extracted resstance values wll change to accommodate the network modfcaton, and wll actually assst n stablzng the values of other extracted resstances. Resstance network optmzaton s more an experental work whch also plays an mportance role n the overall work. 3.6 Compact Model Smulaton In ths secton, the method how to place several standard blocks together after generatng resstance networks s explaned to create a compact model for a large structure. he key to successfully place dfferent blocks together s to make sure the thermal contnuty on the boundary. hat s, on the boundary, temperature and heat flux must be contnuous, (3-4) and H = H. (3-5) herefore, the boundary nodes of two neghborng blocks should be consstent. As t shown n Fg.3-5, boundary nodes of two neghborng blocks must have the same sze and locaton. For thermal smulaton n a crcut smulator, such as Spce, a subcrcut can be used 3

47 for each block. Once several blocks are placed together, the boundary contnutes are satsfed automatcally. Fgure 3-5 Boundary nodes of two neghborng blocks must be consstent. 3.7 Selectng Boundary Nodes for Compact hermal Model In ths secton, two dfferent approaches to select the boundary nodes are dscussed n D structure. One uses a small area to represent a boundary node, and the other uses the boundary lne Boundary Nodes Represented by Small Areas Frstly, we started wth boundary nodes represented by rectangles. ake a block wth a -shape copper metal wre embedded n slcon doxde, for example. Fg.3-6 shows the method of choosng the shape of the nodes on the block boundares. he thermal conductvtes of copper and slcon doxde are 40W / mk and.4w / mk he dmensons of the block are gven n Fg.3-6., respectvely. 3

48 x x x 3 x 4 y 4 Oxde y 3 L node y metal y Fgure 3-6 -shape nterconnect block wth small areas as the boundary nodes. Dmensons are gven below: x4 x =0.35 μ m, x4 x3 x x 0.75m, x 0.09 μ m, y 0. μ m, y y 0.48m, y y 0.8m, y 3 x y L node y 0.m and 0.04m As ndcated n Fg.3-6, nodes are only assgned n the mportant locatons, such as the metal juncton and the boundares of the block. In ths structure, the juncton of the metal wres s selected as a center node and three termnals of the metal are selected as three boundary nodes. In the oxde, four nodes n the corners are selected as another four boundary nodes. o make sure that the shape and sze of the nodes are unform when buldng a large block usng several standard blocks, the sze of nodes on the sde s half of the center nodes and the sze of those n the corner s /4 of the center nodes. As t presented n Secton 3.4, enough heat flux between each par of neghborng nodes should be guaranteed to generate relable resstance networks, thus the best case s that the power densty s appled on each node. Several approaches are dscussed n ths study. In the frst approach, one can apply proper powers on each node and make one of the nodes as the thermal ground to ensure the energy s conserved wthn the block. hese are then 33

49 mplemented n ANSYS to perform thermal smulatons of the structure. Fnally, export the temperature and power of each node. In ths procedure, random numbers are used to assgn the values of power denstes, and all four the boundares of the block are set to be adabatc except for the boundary nodes. Fg.3-7 shows how the powers are appled to the block and the selected thermal ground n one of the cases. he value of the power should be n a proper range, so that the temperature n the block s reasonable, and can be postve or negatve whch represent the power n and out of the nodes, respectvely. he locaton of the thermal ground can be changed to nclude more sets of the heat flux dstrbutons. When collectng the data, the temperature of each node s the average temperature wthn the small area. he power s calculated by P s, pd, S, because, n ANSYS, only the value of surface power densty s appled to the node Fgure 3-7 Illustraton of powers and the thermal ground appled to the basc -shape nterconnect block and ts resstance network after optmzng usng the frst approach. 34

50 Fg.3-7 also shows the resstance network for -shape nterconnect block after the optmzaton. able II shows the extracted resstance values. he fnal number of resstors s reduced from 36 to 4, and the resstances are more relable after removng the redundant or large value resstances. he structure s symmetrcal about the lne from Nodes to 8. herefore, many dentcal pars of resstors exst n the block, such as R, and R,3 or R,6 and R,4, etc. he resstance dfferences for symmetrcal resstors are less than 4%, as dsplayed n able II. able II Extracted Resstance values and the dfferences for the dentcal resstor pars for the -shape nterconnect block usng the frst approach. Resstors hermal resstance o ( C / W ) Identcal resstors Dfferences (%) R, 367 R,3 3.5 R,4 9. R 3,6.53 R, R, 3.5 R, R,6.69 R, R, R,4.69 R 3, R,4.53 R 4, R 5,6 0.8 R 4, R 6, R 5, R 4,5 0.8 R 5,8 7.7 R 6,9 984 R 4, R 7, R 8, R 8, R 7, Applcaton to the sngle and mult-block nterconnect structures usng the frst approach 35

51 mW mW 3 (a) emperature ( o C) (b) Fgure 3-8 (a) he sngle -shape nterconnect block wth nput powers, 0.5mW and 4mW, appled to Nodes 4 and 6, and Node 8 s selected as the thermal ground. (b) emperature contours from the ANSYS smulaton. MN means the mnmum temperature n the structure and MX s the maxmum temperature. Fnte element thermal smulaton of the -shape nterconnect block s carred out n ANSYS, and ts resstance network wth the extracted resstance values gven n able II s performed n Spce usng the nput powers shown n Fg.3-8. Powers 0.5mw and 4mw are 36

52 /oc appled to Nodes 4 and 6, respectvely, and Node 8 s chosen as the thermal ground. Fg.3-9 shows the smulaton results for the sngle -shape nterconnect block derved from the ANSYS and SPICE. In ths approach, dfferences as large as 5% and 8% are observed at Nodes 7 and 3, respectvely, between the SPICE and ANSYS smulatons, whch may be caused by the large temperature gradents n that area. he average dfference s however below 0% SPICE ANSYS Node Number Fgure 3-9 Comparson of the emperature at each node of the resstance network derved from the ANSYS and Spce smulatons. Devatons of Spce from ANSYS are ncluded. o examne the valdty of the resstance network generated from the frst approach, a sx-block structure s generated, as shown n Fg.3-0(a). he structure conssts of sx -shape nterconnect blocks wth powers appled to the metal surface as ndcated n Fg.3-0(a). Fg.3-0(b) shows the temperature contours of the structure obtaned from ANSYS. 37

53 0.96 y/μm C C B B 6.5mW A A 0 (a) x/μm emperature ( o C) (b) Fgure 3-0 (a) he 6-block -shape nterconnect structure wth 6.5mW appled to one of the metal surfaces. (b) emperature contours for the structure derved from ANSYS smulaton. MN ndcates the locaton of the mnmum temperature and MX ndcates the locaton of the maxmum temperature. 38

54 /oc /oc x/μm (a) AA'-SPICE AA'-ANSYS x/μm (b) BB'-SPICE BB'-ANSYS 39

55 /oc x/μm (c) CC'-SPICE CC'-ANSYS Fgure 3- emperature profles obtaned from ANSYS and SPICE smulatons along the lnes of (a) AA, (b) BB and (c) CC ndcated n Fg.3-(a). Fg.3- shows the temperature profles along the AA, BB and CC lnes ndcated n Fg.3-0(a). From the fgure, the left sde whch s near the power has larger errors than the rght sde where the metal surface s thermally grounded. hs examnaton mples that the errors from the Spce smulaton ncrease when the number of blocks ncreases. he average error for the sx-block structure s about 5%. It should be noted that, when optmzng the resstance network, the surfaces of each block are actually adabatc except for the areas of the selected boundary nodes. he extracted resstance network may not be able to account for thermal contnuty on the nterfaces when placng blocks together Boundary Nodes Represented by Lnes In the second approach, the boundary nodes are represented by lne segments nstead of areas, whle nteror nodes are stll represented by areas. Fg.3- shows how the nodes are 40

56 selected for the -shape nterconnect block n the second approach. Each sde of the block s dvded nto several regons (lne segments), and the selected lne segment for the node s smaller than the regon segment length and placed at the center of the regon segment, as shown n Fg.3-. Boundares of dfferent materals are used to select the regons. 3 0 L node L lne nm 09nm 08nm Fgure 3- Boundary nodes of the -Shape nterconnect block n the second approach. L node =0.04 μ m, L lne =0.05 μ m In ths approach, the power s appled contnuous on the surfaces of the block to ensure thermal contnuty when placng blocks together. o mantan the thermal contnuty at the nterfaces between block, the dstance of between the lne segments for neghborng nodes should not be too large. However, to have accurate temperature representaton, the lne segments need to be small enough. o avod the flux and/or temperature dscontnutes on the boundares, a margn s added outsde the block n the smulaton, as shown n Fg.3-3, for extractng the resstance network. 4

57 Fgure 3-3 he structure ndcatng how to apply heat flux for the second approach n the selected block that s shown n the rectangle nsde the larger block. One of the areas n the block needs to be grounded to have a unque thermal soluton. As t s shown n Fg.3-3, dfferent heat fluxes are appled to three boundares and constant temperatures are appled to the fourth boundary as a thermal ground. For the nterors nodes, proper powers are also appled to ensure enough heat exchanges between two neghborng nodes. Random numbers are used to generate random values of heat fluxes, temperatures and powers. On the boundary, postve or negatve heat fluxes can be generated to nclude the nflux or outflux. he locaton of thermal ground can be changed to nclude more sets of heat flux dstrbutons. 4

58 (a) (b) Fgure 3-4 Some possble cases for the drectons of heat fluxes wthn a block. (a) 4 possble heat flux dstrbutons wthn a block when none of boundares s adabatc. (b) 4 more possble heat flux dstrbutons when ether two opposte sdes are adabatc. Fg.3-4(a) shows some possble heat flux dstrbutons wthn a block wthout any adabatc BC s. In addton, four other flux dstrbutons are shown n Fg.3-4(b) when two opposte sdes are adabatc. When calculatng power appled to the node, the total power appled to each node can be calculated by the ntegral of the heat flux on the boundary over 43

59 the regon lne segment. For example, f the boundary s paralleled to the y drecton, the ntegral of the heat flux s performed along the y drecton over the regon segment. When extractng the resstance value for the network, the node temperature s the average temperature over the node lne segment nstead of the regon lne segment Fgure 3-5 hermal resstance network for the -shape nterconnect block usng the second approach. Fg.3-5 shows the resstance network for the -shape nterconnect block usng the second approach, and able III shows the resstance values. Based on the smulaton results, 78 resstors for 3 nodes are reduced to 8 resstors. he fluctuaton of the extracted resstances s small, and the dfference of the dentcal-par resstors due to the symmetrcal structure s stll less than 4%. 44

60 able III Extracted thermal resstance values and the dfferences for the dentcal resstor pars for the -Shape nterconnect block usng the second approach Resstors hermal resstance o ( C / W ) Symmetrcal resstors Dfferences (%) R, 94.8 R, R, R 3, R, R 3, R,3 8.3 R, R, R, R, R,8 59. R, R 3, R, R 3, R, R 4,6 973 R 5, R 4, R 5,7.500 R 5, R 4,7.500 R 5, R 4, R 6,7.0E+00 R 7, R 6, R 8, R 6, 4.97 R 8, R 7,8.869 R 6, R 7, 5.8 R 7, R 7, R 7, R 7, R 8,0 55. R 6, R 8, 4.88 R 6, R 9, R 0, R 9, R 0, R 0, R 9, R 0, R 9, R, 74. R,3.663 R, R, Applcaton to a mult-block structure o examne the valdty of the second approach, a sx-block structure, shown n Fg.3-6(a), usng the -shape nterconnect block s mplemented n the fnte element ANSYS smulaton. A unform power of 6mW s appled to the lower left metal surface. 45

61 Fg.3-6(b) shows the temperature contours from ANSYS C C B B 6mW A A 0 (a) x/um emperature (oc) (b) Fgure 3-6 (a) he 6-block structure usng the -shape nterconnect blocks wth a power of 6.5mW appled to the lower left metal surface. (b) emperature contours for the 6-block structure derved from ANSYS smulaton. MN ndcates the locaton of the mnmum temperature and MX ndcates the locaton of the maxmum temperature. 46

62 /oc /oc x/μm (a) AA'-SPICE AA'-ANSYS x/μm (b) BB'-SPICE BB'-ANSYS 47

63 /oc x/μm (c) CC'-SPICE CC'-ANSYS Fgure 3-7 emperature profles obtaned from ANSYS and SPICE smulatons along the lnes of (a) AA, (b) BB and (c) CC ndcated n Fg. 4-(a). Fg.3-7 shows the temperature profles along the lnes of AA, BB and CC. Results show that the temperature dfferences between the ANSYS and Spce smulatons usng the second approach are sgnfcantly reduced compared to those of the frst approach. As shown n Fgs.3-7(a) to 3-7(c), the average temperature errors are decreased to 6% usng the second approach. he large errors on these boundary nodes may be caused by the poor thermal contnuty due to the network extracton approach Dscusson of wo Dfferent Methods Compared the two dfferent approaches descrbed n Sectons 3.7. and 3.7.3, the frst approach can easly generate the resstance network, because the powers are drectly appled to each node whch makes the heat flux strong enough between any two neghborng nodes. On the other hand, n the second approach there may not have enough heat flux between 48

64 some neghborng nodes leadng to poor statstcal nformaton to extract those resstance values. However, t has been found that the errors n the second approach are much smaller than that n the frst one, because n the second approach more realstc BC s are enforced n the extracton procedure. herefore, the second approach s used to construct the compact thermal models for more realstc applcatons presented n Chapter 4. 49

65 Chapter 4 Applcaton of Block-based Compact hermal Crcuts to an SOI Inverter In ths chapter, the block-based compact thermal crcut presented n Chapter 3 s appled to an nverter on SOI structure. he nverter layout s dvded nto several standard blocks and the thermal crcut model for each standard block s constructed. Influence of heat flux on the boundares s also dscussed. 4. Structure and Dvsons of Inverter Fg.4-(a) shows the layout for an nverter. Fg.4-(b) dsplays the layout cross-secton vew along AA of the nverter usng SOI technology wth gate oxde thckness t gox nm and gate length L g 65nm. More detal devce structure and dmensons are dsplayed n Fg.4-(c). he smulaton structure used for extract of the network s the D cross-secton vew wth dmensons clearly gven n the structure as shown n Fg.4-(b). 50

66 A A (a).885 μ m Metal t m 360nm Metal Metal W m = 80nm L va 90nm Oxde H m.06m.48 μ m t m 00nm Contact Gate Contact d 455nm H m 500nm Slcon flm t sub m Substrate (b) 5

67 L g = 65nm H g = 50nm L poly = 5nm t sf = 0nm L sf = 35nm t BOX = 80nm (c) Fgure 4- (a) op vew of an nverter layout. (b) Cross-secton vew along AA of the nverter usng SOI technology, where the selected blocks are shown. he resstance network wll be extracted for each selected block. (c) he detaled dmensons of the devce for the nverter gven n Fg.4-(b). hermal conductvtes n the moderately doped channel and heavly doped source and dran are taken to be equal k = k = k = 63W /( mk) c s d hermal conductvtes of the materals nclude: slcon-oxde =.4W /( mk), the lghtly doped slcon substrate k s, sub = 48W /( mk), the moderately doped channel and heavly doped source and dran k = k = k = 63W /( mk), copper nterconnect wres c s k Cu = 40W /( mk), ungsten metal contact k W = 74W /( mk) ; and poly-slcon k poly = 63W /( mk). he D structure gven n Fg.4-(b) s dvded nto 6 blocks wth 7 dfferent buldng blocks. o reduce the workload of generatng dfferent compact thermal crcuts for the blocks, the number of the buldng blocks should be as small as possble. o demonstrate the thermal contnuty on the nterfaces between neghborng blocks, smaller blocks are selected n ths study, and 7 dfferent standard buldng blocks are chosen. hey are the substrate (Fg.4-), Metal--Contact nterconnect (Fg.4-3), -shape nterconnect d k ox 5

68 (Fg.4-4), metal- juncton (Fg.4-5), oxde- (Fg.4-6), oxde - (Fg.4-7) and devce blocks. 4. Smulaton Results of Each Block A. Substrate Block Node selecton and dmensons for the selected substrate block are ncluded n Fg.4-. Snce the substrate bottom s usually mantaned at ambent temperature, the whole bottom s desgnated as one sngle node. Also shown are the selected nodes and the resstance network of substrate after the optmzaton. he dvson length and node length on the top need to be consstent wth those n the blocks above t, as mentoned n Chapter 3, to guarantee the boundary contnuty μ m.885 μ m Fgure 4- he selected substrate block wth, dmensons and resstance network. he node length s L node = 50nm. he dvson length for Node, 3, 4, 8, 9, 0, 4, 5 and 6 s L seg = 08nm and the dvson length for Node 5, 6, 7,, and 3 s L seg = 5nm. 53

69 able IV shows thermal resstances n the network of the selected substrate block. From the result, 0 thermal resstors are reduced to 9 resstors. Because the substrate block s symmetrcal wth respect to the y axs, there are many dentcal resstor pars. As shown n able IV, the maxmum dfference of dentcal resstor par s less than %. able IV Extracted thermal resstance and the dfferences of the dentcal pars of resstors n the network for the substrate block. hermal resstors hermal resstance o ( C / W ) Identcal resstors Dfferences (%) R, 56.9 R,6.6 R, R,5 0.0 R, R,4 0. R, R, R, R, 0.86 R, R, 0.0 R, R, R,9 63. R, R, R, R,7 0.0 R, R, R, R, R, R,4 0. R, R,3 0.0 R, R,.6 R, R 5,6.007 R 3, R 4,5 0.9 R 4, R 3,4 0.7 R 5, R,3 0.4 R 6, R, R 7, R 0, R 8, R 9, R 9, R 8, R 0, R 7, R, R 6, R, R 5,6 0.4 R 3, R 4,5 0.7 R 4, R 3,4 0.9 R 5, R,

70 B. Metal--Contact Interconnect Block 5 nm 5 nm 5 nm nm 3 L 0.48 μ m node = 40nm 00 nm nm nm μ m Fgure 4-3 he selected Metal--Contact nterconnect block wth dmensons and resstance network. he network constructed by the thermal resstors and selected nodes are ndcated by the thck lnes the node numbers, respectvely. wo 40 nm 40nm juncton nodes are located n the center of the juncton. he boundary dvsons for the node selecton are chosen based on the nterfaces of the materals, as can be seen n Fg.4-3, and the node lne segment s centered n each dvson wth length of 50nm. able V shows thermal resstances n the network for the Metal--Contact nterconnect block. Applcaton of the extracton procedure leads to reducton n the number 55

71 of the thermal resstors from 0 to 30 n ths block, and the resstance values are less fluctuated. he maxmum dfference for each dentcal resstor par s less than.5%. able V Extracted thermal resstance values and the dfferences of the dentcal resstor pars n the thermal resstance network for the Metal--Contact nterconnect block. hermal resstors hermal resstances o ( C / W ) Identcal resstors Dfferences (%) R, 96.7 R 4, R, R 5, R,3 7. R 3,4 0. R, R 4,7.098 R, R 4, 0.7 R,9 9.7 R 4, R 3, R,3 0. R 3,9 79. R 3, R 3, R 3, R 4, R, R 4, R,6.098 R 4, R, R 4, 3.46 R, R 5,7.9 R, R 6, R 7, R 7, R 6, R 8,9.374 R 0, 0.40 R 8, 07 R,3.4 R 8, R, R 9, 48. R 0,3 0.6 R 9, R 0, 0.44 R 0,.364 R 8, R 0, 56 R 9, R 0, R 9, 0.6 R,3 40 R 8,.4 R, R 8, R, R 3, R 3, R, R 4, R 5,6 0.4 R 5, R 4,5 0.4 C. Metal- Juncton 56

72 08 nm nm 50 nm 600 nm nm nm 7.5 nm 35 nm Fgure 4-4 he selected metal- juncton block wth dmensons and resstance network. In Fg.4-4, the network constructed by the thermal resstors and selected nodes are ndcated the thck lnes and node numbers, respectvely. he juncton node nsde the metal- juncton block s a 40 nm 40nm square located n the center of the Metal-. he bottom boundary dvsons for the node selecton are chosen based on the metal-oxde nterfaces, as can be seen n Fg.4-4, and the top and sde boundares are equally dvded nto three and four dvsons, respectvely. he node lne segment s centered n each dvson wth the length of 50nm. able VI shows the extracted values of the thermal resstances n the network for the 57

73 Metal- block. Usng the procedure presented n Secton 3.5, 05 thermal resstors are reduced to 30 resstors, and very small fluctuatons of the resstance values are obtaned. he dfferences for most dentcal resstor pars are less than %. able VI Extracted thermal resstance values and the dfferences of the dentcal resstor pars n the thermal resstance network for the Metal- model hermal resstors hermal resstances o ( C / W ) Identcal resstors Dfferences (%) R,.3 R, R, R 3, R, R 3, R,3.7 R, R,4 0.5 R, R,5.4 R, R, R, R, R, R, R 3, R, R 3, R, R 4, R 5,7 0.8 R 5,7 577 R 4,6 0.8 R 6, R 7, R 6, R 7, R 7, R 6, R 7,0 704 R 6, R 8, R 9, R 8, R 0,.07 R 9, R 8, R 9, R 9, R 9, R 9, R 9, R 9, R 9, R 9, R 9, R 0, 90.9 R 8,.07 R, R, R, R, R 3,4 964 R 4, R 4,5 97 R 3,

74 D. Oxde- Block nm 400 nm Fgure nm 09 nm 35 nm he selected oxde- block wth dmensons and thermal resstance network. Fg.4-5 dsplays the oxde- block wth the thermal resstors and selected nodes ndcated by the thck lnes and node numbers, respectvely. he nteror node s a 40 nm 40nm square located n the center of the Oxde- block. he top and bottom boundares are equally dvded nto three dvsons and left and rght boundares are dvded nto four. he node length for each node s 50nm located n the center of each dvson. able VII shows the extracted values of the thermal resstances n the network for the Oxde- block. Usng the procedure for extractng the network, 05 thermal resstors are reduced to 36 resstors. Because the oxde- block and the node locatons and szes are symmetrcal wth respect to both x and y-axes, there are many sets of four dentcal resstors due to ts symmetry. he maxmum dfference for values of the dentcal resstors s less than 4.5%. 59

75 able VII Extracted thermal resstance values and the dfferences of the dentcal resstors n the thermal resstance network for the Oxde- block. hermal resstors hermal resstances o ( C / W ) Dfferences (%) R, R, R, R, R, R, R, R, R 3, R 3, R 3, R 4, R 4, R 5, R 5, R 6, R 6, R 7, R 7, R 8, R 8, R 8, R 8, R 8, R 8,4 570 R 8, R 9, R 9, R 9, R 0, R 0, R 0, R, R, R 3, R 4,

76 E. Oxde on metal- level nm 600 nm Fgure nm 5 nm 455 nm he selected oxde- block wth dmensons and thermal resstance network. Fg.4-6 dsplays the oxde- block wth the thermal resstors and selected nodes ndcated by the thck lnes and node numbers, respectvely. he nteror node s a 40 nm 40nm square located n the center of the Oxde- block. he top and bottom boundares are equally dvded nto three dvsons and left and rght boundares are dvded nto four. he node length for each node s 50nm located n the center of each dvson. able VIII shows the extracted values of the thermal resstances n the network for the Oxde- block. Usng the procedure for extracton of the resstance network, 05 thermal resstors are reduced to 3 resstors. Because the oxde- block and the node locatons and szes are symmetrcal wth respect to both x and y axes, there are many sets of four dentcal 6

77 resstors due to the structure symmetry. he maxmum dfferences for every four symmetrcal resstances are less than.%. able VIII Extracted thermal resstance values and the dfferences of the dentcal resstors n the thermal resstance network for the Oxde- block hermal resstors hermal resstances o ( C / W ) Dfferences (%) R, R, R, R, R, R, R, R 3, R 3, R 4, R 4, R 5, R 5, R 6, R 6, R 7, R 7, R 8, R 8, R 8, R 8, R 8, R 8, R 8, R 9, R 9, R 0, R 0, R, R, R 3, R 4,

78 4.3 Heat Flux Influence on Boundary hermal Contnuty for the Devce Block For a large crcut structure whch s composed of several blocks, heat fluxes along nterfaces of most blocks may change subtly. In these cases, the block boundares can be equally dvded nto several nodes. However, for certan blocks, such as devce blocks, whose boundares f close to a heat source, the heat flux along the nterface(s) may change dramatcally. In these cases, larger errors may occur due to poor thermal contnuty on the nterface(s), and small dvsons near strong flux regons are needed to enforce the boundary thermal contnuty Equal Boundary Dvsons for Node Selecton Fg.4-7(a) shows the D structure of the 455nm400nmdevce block wth the poly-slcon and metal contact. he fgure shows how to select the nodes for the devce block. wo sdes of the devce block are equally dvded nto four segments and the bottom boundary s dvded nto three segments. he node n each dvson segment s located n the center of the dvson wth the length 50nm. For the nteror nodes, only the locatons where the temperature s mportant are needed. Fve nodes are selected along the channel to show the temperature gradent. Accordng to the temperature profle along the channel shown n Fg.4-9, the peak temperature occurs n the channel near the channel-dran juncton that s selected as one of the nodes n the channel. Other selected nodes are shown n Fg.4-7(a). he dmenson for the Node 6, 7 and 8 s 3 nm 0nm. 63

79 nm 3 0 nm nm 0 nm nm 55 nm 0 nm 80 nm 0 3 (a) (b) Fgure 4-7 (a) Node selecton for the devce block wth equal boundary dvsons where each node s located n the center of the dvson. (b) hermal resstance network for the devce block wth node selecton gven n Fg.4-7(a). Fg.4-7(b) shows the resstance network for the devce block after the optmzaton. he optmzaton leads to a reducton n the number of resstors from 3 to 50. able IX 64

80 shows the extracted thermal resstance values of the thermal resstance network. able IX Extracted values of thermal resstances n the thermal resstance network wth equal boundary dvsons for the nodes. hermal resstors hermal resstances o ( C / W ) hermal resstors hermal resstances o ( C / W ) R, 40.8 R 8, R, R 9, R, R 9, 5.64 R, R 9, R, R 9, R, R 9, R, R 9, 9.69 R, R 0, 70.7 R, R, R 3, R,5 48 R 3, R 3, R 3, R 3, R 4, R 3, R 4, 853. R 4,5 760 R 5, R 4, R 5, R 4, R 5, R 4, 80.3 R 5, R 5, R 5, R 5, R 5, R 6,8 0. R 6, R 7, R 6, R 8, R 7, R 9, R 7, R 0, R 8, R, Smulaton Result Smulaton result from the compact thermal model s compared to the result from ANSYS smulaton. In ANSYS, totally,57 elements are used to model the whole layout cross-secton; however, n the compact thermal model, the total number of nodes s reduced 65

81 to 57 and only 5 resstors are left after optmzaton. Fg.4-8(a) presents the D structure of the nverter cross secton wth a 0.9mW power appled to the left devce and 0.6mW power appled to the rght one. In addton, an appled power of 0.mW flows out from the left Metal- boundary and 0.mW flows nto the rght Metal- boundary. Fnally, the bottom of the substrate s set to the thermal ground. Fg.4-8(b) shows the temperature profle for D block derved from ANSYS. D C C B B 0.mW 0.mW A A 0.9mW 0.6mW D (a) 66