Theoretical and Practical Validation of Combined BEM/FEM Substrate Resistance Modeling

Transcription

1 Teoretical and Practical Validation of Combined BEM/FEM Substrate Resistance Modeling E. Scrik, P.M. Dewilde and N.P. van der Meijs Delft University of Tecnology, DIMES, Circuits and Systems Group Mekelweg 4, 2628 CD, Delft, Te Neterlands Abstract In mixed-signal designs, substrate noise originating from te digital part can seriously influence te functionality of te analog part. As suc, accurately modeling te properties of te substrate as a noise-propagator is becoming ever more important. A model can be obtained troug te Finite Element Metod (FEM) or te Boundary Element Metod (BEM). Te FEM performs a full 3D discretization of te substrate, wic makes tis metod very accurate and flexible but also slow. Te BEM only discretizes te contact areas on te boundary of te substrate, wic makes it less flexible, but significantly faster. A combination between BEM and FEM can be efficient wen we need flexibility and speed at te same time. Tis paper briefly describes te BEM and te FEM and teir combination, but mainly concentrates on te teoretical validation of te combined metod and te experimental verification troug implementation in te SPACE layout to circuit extractor and comparison wit commercial BEM and FEM tools. 1 Introduction In present-day micro-electronic designs, tere is a tendency to incorporate bot analog and digital circuitry on one cip. In tese so-called mixed-signal designs, substrate noise can ave a significant impact on te functionality of te design. Substrate noise originates from a mecanism as scematically drawn in Figure 1. Because of te resistive nature of te substrate, te ground connection is not ideal, and te ig-frequency switcing activity of digital devices will cause noise-like potential fluctuations in te substrate, wic may seriously affect te beaviour of any sensitive analog devices. To predict any problems in circuit performance caused by substrate noise, tere is an increasing need for accurate models tat describe te noise-propagation beaviour of te substrate [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Suc models are usually obtained troug eiter te Finite Element Metod (FEM) as applied in e.g. [4] or te Boundary Element Metod (BEM) as applied in e.g. [10]. Te FEM performs a full 3D discretization of te substrate, and terefore it is very versatile and flexible, but it is usually also slow. Te BEM, on te oter and, assumes te substrate to consist of uniform lay- Figure 1: Te substrate noise mecanism Figure 2: Doping patterns like trences, buried layers, sinkers and cannel-stoppers (not sown) may influence te noise caracteristics of te substrate. ers and it only discretizes te contact areas on te boundary of te substrate. Terefore it is less flexible, but it can be significantly faster. However, in some modeling problems, a dilemma arises wen neiter te BEM nor te FEM would be te most convenient metod to use. Consider, for example, a substrate were te doping patterns (e.g. cannel-stoppers, buried layers, sinkers and trences as depicted in Figure 2) tat are present in te top layers of te substrate are expected to ave a significant influence on noise-propagation. Wile FEMbased modeling could accurately model suc structures, it can often be too slow. On te oter and, te BEM migt be significantly faster, but it may not be accurate enoug, because te doping patterns do not form a uniform layer. To circumvent tis modeling dilemma, we ave developed a combination of BEM and FEM [11] tat is faster tan FEM but can be more accurate tan BEM based-metodologies. Our approac is to apply te FEM for te specific doping /02/$ IEEE

2 patterns, and te BEM for te underlying substrate, after wic we combine te resulting models. Tis paper introduces te combined BEM/FEM metod and solidifies its teoretical and practical validity. Section 2 gives a brief background on te FEM and te BEM and teir combination, followed by a convergence proof for te combined metod in Section 3. Section 4 presents a few details on te implementation of te metod into te SPACE layoutto-circuit extractor, and sows an application of our metod, togeter wit a successful comparison to commercial BEM and FEM tools. Finally Section 5 states our conclusions. 2 Background In substrate resistance extraction, te aim is to find a resistance network between terminals placed on te boundary of a resistive (i.e. passive) domain. From pysics, we know tat resistance relates potential-differences to currents, wic implies tat we essentially need to solve a static potentialproblem. From a matematical point of view, tis is equivalent to imposing (continuous) boundary conditions on te medium, and solving te Laplace equation (σ Φ) = 0 (1) were Φ is te potential-field and σ is te conductivity of te medium. A convenient property of te resulting field is tat it as minimal energy according to te following energy functional E(Φ(p)) = σ Φ(p) 2 dp (2) Ω were Ω is te domain of interest and dp is te relevant volume integral. We can solve Φ wit different metods. Finite Element Metod Te finite element metod (FEM) subdivides te entire domain into triangular (2D) or tetraedral (3D) elements (see Figure 3a), and treats te elements as linear. Te mutual matematical relations between te potentials at te corners of te elements (te FEM nodes) can ten be expressed very easily by a piecewise linear (or piecewise planar for 3D) function. Troug an incidence strategy wic accumulates te FEM elements, a global system of mutual matematical relations between te FEM nodes is obtained, wic is ten used to minimize te energy functional as described above. In tis case, te energy functional is used as an alternative formulation of te Laplace equation. For an extensive analysis of te FEM, te reader is referred to [12]. However, if we take a sligtly different approac to te FEM, te resistance network can be found witout aving to calculate te field solution. Tis approac is based on te observation tat te matematical relations along te edges in te FEM discretization already represent resistances (see e.g. [13]). In oter words, te FEM discretization is equivalent to a resistance network. Wen tis network is used for subsequent circuit simulation, te minimization of te energy a) b) Figure 3: a) 3D tetraedral FEM discretization; b) BEM discretization of contact areas on top of uniformly layered domain functional will automatically (toug implicitly) be ensured by te circuit simulator itself. Te network tat we find in tis way as many nodes and a sparse structure. However, only a small number of te FEM nodes are actually terminal nodes, wile te rest of te FEM nodes are internal nodes. Terefore, any internal nodes can be eliminated troug Gaussian elimination, or, equivalently, star-delta transformation. Te 3D discretization will automatically incorporate any inomogeneities of te domain into te model, because eac element in te discretization can ave its own material properties assigned. As suc, te FEM is very accurate and flexible. Boundary Element Metod Te Boundary Element Metod (BEM) is based on an integral form of te Laplace equation [10, 14] Φ(p) = k(q)g(p, q)dq (3) S 1 were S 1 S is te entire contact area on te boundary surface S, dq represents te relevant surface integral, k(q) represents te unknown continuous current density, and, G(p, q) is te Green s function. Te Green s function is a fundamental solution to te Laplace equation, and, as suc, it automatically ensures te minimization of te energy functional. Te Green s function encodes te caracteristics of te medium, and it can be interpreted pysically as te potential in point p due to a current injected at point q. For te 3D omogeneous case, te Green s function is G(p, q) = 1 (4) 4πσr were r = (p x q x ) 2 +(p y q y ) 2 (te p z and q z coordinates ave deliberately been omitted, because te contact areas are situated on te top plane of te substrate). Also for uniform, layered media (see layers σ 1 and σ 2 in figure 3b) a Green s function can be determined, but it is not possible to include localized doping patterns as depicted in Figure 2. Te BEM only as to discretize tose parts on te boundary of te domain (see Figure 3b) were Diriclet conditions old (i.e. te contact area S 1 were current can enter / leave te domain), and assumes te remaining part of te boundary

3 to be subject to Neumann conditions. Similar to te caracteristics of te medium, tese boundary conditions can also be encoded by te Green s function. Te discretization usually (but not necessarily) utilizes rectangular panels and assumes te current distribution to be constant on eac panel (i.e. eac panel forms an equipotential region). As suc, te discretization allows a piecewise constant approximation of te continuous current density distribution. Based on te discretization, te Metod of Moments [15] allows us (eiter troug te collocation metod or te Galerkin metod) to find tis approximation from a linear system of equations. In tis case, we define P as te vector of panel potentials, K as te vector of (unknown) panel currents and G as te elastance matrix describing te potential at panel i due to a unit current in panel j (i.e. te Green s function is evaluated for eac panel-pair) P = GK (5) Te BEM ten continues by defining an incidence matrix F relating panels and contacts. By denoting V as te contact potential vector and I as te contact current vector, we can write P = F V (6) It follows tat I = F T K (7) I = F T G 1 F V = Y V (8) were Y is an admittance matrix for te resistive substrate wit te substrate contacts as ports. Tus, we see tat also wit te BEM te result is a resistance network. Compared to te FEM, owever, tis is a full resistance network, were eac node is a port-node tat is connected to every oter node; tere are no internal nodes. Model order reduction sould be used to simplify tis network, and/or a windowing tecnique suc as [10] can be used to a-priori extract a reduced order model. Combining FEM and BEM Wen making use of dedicated, igly optimized solution metods for bot BEM and FEM, te BEM is usually significantly faster, but tis is typically at te expense of flexibility. Terefore, as already mentioned in te introduction, some substrate modeling problems pose a dilemma wen neiter te BEM nor te FEM would be te most convenient metod to use. In suc a case, we can supply te BEM wit some of te flexibility of te FEM, by applying te FEM to te specific doping patterns (recall Figure 2), and te BEM for te underlying substrate [11]. In effect, te BEM and te FEM are ten confined to teir own domains and communicate troug an interface. Because bot te BEM and te FEM result in resistance networks, te metods can be combined by properly aligning te nodes along te interface, after wic te networks can be attaced at te coinciding nodes. Tis requirement can be met by conveniently coosing te BEM mes wit respect to te FEM mes. Figure 4: Dual mesing: te exagonal BEM mes is te dual of te triangular FEM mes SUB FEM BEM interface Figure 5: Side-view of te FEM and BEM areas togeter wit te structure of te resulting resistance models. Due to dual mesing, te nodes on te interface will coincide, and a direct connection between te models can be made. Basically, we want to acieve tat a BEM panel be associated wit every FEM node on te interface. Tis requirement suggests a dual relationsip between te BEM and FEM meses. Different interpretations of duality are possible, but te triangular interface mes of te FEM reminds us of te duality between te Delaunay triangulation and te Voronoi tesselation. Even toug te FEM mes may not necessarily be a true Delaunay triangulation, coosing te BEM mes as te Voronoi tesselation of te FEM nodes will still ensure a one-to-one association between te FEM nodes and te BEM panels (see Figure 4). Near te interface, te resulting networks will ten look as illustrated in Figure 5. An alternative way of combining te metods for capacitance extraction was presented in [16] were te BEM and FEM meses coincided. In tis case te interface nodes did not align and generalized ideal transformers were necessary for proper interaction between bot models and for conservation of energy along te interface. Tis combination metod was proven to converge in [17]. Below, we will give a similar, but sligtly different proof of our metod. 3 Proof Te key beind te proof is te fact tat a field (Φ m ) tat satisfies te Laplace equation and tat satisfies te contin-

4 a) Φ m d) Φ1 Φ 2 Φ 1 b) Φ e) Φ 2 c) Φ 1 Φ 2 Φ FEM f) Φ BEM Figure 6: Different stages in te convergence proof Φ Figure 7: Field penomena at te interface. Te dots along te x-axis represent te coinciding BEM and FEM nodes. Legend: solid line = BEM boundary field, dased line = FEM boundary field uous boundary conditions imposed on te domain will ave minimum energy (illustrated in Figure 6a). Tis is indeed te true, pysical field. Additionally, we observe tat fields tat approximately satisfy te boundary conditions and wose energy lies close to te minimum are actually close to te exact field as well. Wen te boundary conditions are discretized (illustrated in Figure 6b), te field will alter itself to a new field (Φ ) suc tat its energy is minimal, consistent wit te discretized boundary. By making te discretization fine enoug, te approximate field Φ can get arbitrarily close to te exact field Φ m. In oter words E(Φ ) can get ε-close to E(Φ m ) We now divide te domain wit an interface (Figure 6c). If te interface preserves energy (i.e. E(interface) = 0), tere will be an exact and continuous matc between fields Φ 1 and Φ 2 along te interface, suc tat te total field will ave minimal energy, and will be consistent wit te discretized boundary conditions along te outer boundary. Wen discretizing tis interface (Figure 6d), tere will be an exact, but discrete matc between Φ 1 and Φ 2 along te interface, suc tat te total field is still consistent wit te conditions on te outer boundary. However, because te discretized potential distribution on te interface approximates te exact distribution, we know tat te total field may not ave minimal energy. In our combined BEM/FEM metod, te discrete matc between Φ BEM and Φ FEM (see Figure 7) is performed by attacing te coinciding interface nodes. However, discretization of te interface also introduces an anomaly in te field, x because of field discontinuities along te interface, alfway between te coinciding nodes (see Figure 7). Assuming a solution to te field problem for te BEM/FEM discretized case, wic also matces in a discrete way on te internal interface, we will now study te energy contained witin te anomaly. Terefore, we introduce an -tin layer along te interface, by wic we effectively separate te domains to wic te fields belong (Figure 6e). As illustrated in Figure 6f, tere is a FEM field on te left and a BEM field on te rigt of te interface. Figure 7 sows bot fields. A difference field will now exist across te -tin layer. Tis field is defined by taking Φ BEM and Φ FEM in point x on te interface, and do a linear interpolation across te distance : Φ i (k, x) =Φ FEM (x)+ Φ BEM(x) Φ FEM (x) k (9) Here, Φ i stands for te potential inside te -tin layer; k and x, are suc tat k is contained witin te continuous interval [0, 1] and x is te relevant point on te interface. Te result is a piecewise continuous field. We observe tat te mes granularities of te BEM and te FEM meses sould be proportional to te tickness of te -tin layer. In tis way, te derivative of te linear interpolation will be kept in proportion wile 0. As sown in Figure 7, Φ BEM is piecewise constant. Tus, for a mes granularity of, we know, troug a Taylor expansion, tat in a given point p, Φ BEM is O() away from te exact field for te situation of Figure 6e. Similarly, since Φ FEM is piecewise linear, we know tat in te same point p, Φ FEM is O( 2 ) away from te exact field. In terms of energy, we need te derivative of Φ i. Equation (9), we find tis to be: Φ i = Φ i k=1 Φ i k=0 = Φ BEM Φ FEM = O() From = O(1) (10) Because te -tin layer as constant tickness, wecan rewrite te energy integral as follows: Φ i 2 dv = Φ i 2 ds (11) V i S i were V i is te volume of te -tin interface layer and S i is te surface of te interface layer. We finally prove te O() convergence wit lim O(1) ds= 0 (12) 0 S i We ave now seen tat te energy contained witin te anomaly in te interface field, disappears as O(). We observe tat te total field formed by combining te two fields wit teir linear interpolation is continuous and piecewise differentiable. Tus, te total field belongs to te same searcspace as Φ m and Φ. Additionally, we observe tat te total field minimizes energy, because bot te BEM and te FEM minimize energy (see Section 2), and te interface contributes a negligible portion to te total energy.

5 2D FEM interface BEM top A d d B G SUB Figure 9: Basic guard-ring structure Figure 8: Side-view of te 2D FEM and te BEM models, under dual mesing (compare Figure 5) 30 Momentum SPACE Summarizing, te situation is as follows: (1) Te energy of te field obtained in Figure 6d may not be minimal, because te discretized potential distribution on te interface differs from te exact distribution. Neverteless, te energy is ε-close to te exact field in 6c; (2) Te field in 6f as energy necessarily smaller tan tat of 6d because BEM and FEM minimize te energy and te mismatc in te interface contributes a negligible part to te total energy; (3) Te minimum obtained in 6f goes over a searc-space tat is part of te searc-space for 6c. Hence, all tese fields are ε-close to eac oter in energy and ence in point-values. Tis concludes te principle of our proof. 4 Experiments We ave implemented a first prototype of our metod into te SPACE layout-to-circuit extractor [18]. Te prototype (see Figure 8) utilizes a 2D FEM instead of a 3D FEM for te specific doping patterns and a BEM for te underlying substrate. Te 2D FEM is a valid modeling metodology, as long as te FEM domain is tin and as a significantly lower resistivity tan te BEM domain. Additionally, because calculating te BEM for exagonal panels (recall Figure 4) is computationally less efficient, our prototype does not use dual mesing, but constructs independent BEM and FEM meses and combines te networks based on proximity of te FEM nodes and te centers of gravity of te BEM panels. In oter words, eac (rectangular) BEM panel is now associated wit te nearest FEM node. An experimental verification of te convergence of our metod was already done in [11]. Wit te following experiment, we will demonstrate te beaviour of our metod in a more practical context. In mixed signal designs, guard rings are sometimes used to improve te noise isolation between te digital and te analog part of te cip. Tis experiment will sow ow our model beaves for suc structures, but note tat it is specifically not intended as a design optimization or as an evaluation of guard-ring effectivity; it is only intended as an illustration of a possible application of our metod. Te basic setup of tis experiment is sown in Figure 9. It represents two terminals A and B tat bot ave size 2µm V(victim) / V(noise) (db) d (µm) Figure 10: Sensitivity (in db) of te noise victim as a function of guard-ring size. 2µm and tat are 30µm apart (center to center). Terminal B is surrounded by a guard ring tat is grounded at terminal G. Te parameter d controls te size of te guard ring wile keeping it square. Te prototype implementation is suc tat te 2D FEM only discretizes te guard ring. Te BEM wil now see Terminal A and Terminal B as single substrate contacts, and te guard ring as a lumped contact. Te substrate as a uniform resistivity of 10 S/m, and a backplane metallization. Te guard ring as a resistivity of 1000 Ω/sq, a widt of 1 µm and a tickness of 0.5 µm. We calculate te sensitivity of te victim terminal (i.e. terminal B) wit respect to te noise terminal (i.e. terminal A) troug te following expression: ( ) Vvictim S[dB] = 20 log (13) V noise Figure 10 sows te results generated wit bot SPACE and Momentum [19]. We clearly see tat te results are very close to eac oter, and tat te sape of te curve is intuitively correct. Our metod places te FEM domain on top of te substrate, wereas te FEM domain sould actually be embedded in te substrate. Terefore, we ave to determine ow good our metod is as an approximation of te actual situation. We ave done tat by comparing our results to tose of te 3D FEM device simulator Davinci [20]. Te basic layout is sown in Figure 11. It is a similar setup

6 2µ 10µ 5µ A B G Figure 11: Guard-ring structure wit reduced size Table 1: Resistances (in kω), Sensitivity and extraction time for te guard-ring structure R AB R AG R BG S(dB) time(s) SPACE Davinci as in Figure 9, but it as a drastically reduced size because of te limitations involved wit Davinci being a device simulator. Te substrate as a conductivity of 10 S/m and is 5µm tick, but witout backplane metallization. Te guard ring as a conductivity of 1000 S/m, is 1 µm wide and is 0.5 µm tick. For Davinci, we ave assumed an n-type silicon substrate wit doping concentration N d = cm 3 and an n-type guard ring wit N d = cm 3, wic correspond to 10 S/m and 1000 S/m, respectively. Te results can be found in Table 1, wic was obtained using suitable mes settings suc tat te results from bot metods ad converged (Davinci used grid points, were a maximum of was allowed for tis version of Davinci). Te table clearly sows tat te results of te FEM metod and te FEM/BEM metod are reasonably close. Tere can be many reasons for te remaining differences, including te fundamental fact tat te FEM (Davinci) needs a bounded domain (a cuboid) and te BEM (SPACE) assumes a domain laterally extending to infinity. Te table also sows tat our metod can be considerably faster. 5 Conclusions Tis paper describes a teoretical analysis and a practical application of a combined BEM / FEM metod for substrate resistance modeling. Combined BEM/FEM modeling may be efficient in cases were accuracy as well as speed are required, but were a separate BEM would not be accurate enoug and a separate FEM would not be fast enoug. In our teoretical analysis, we ave proven tat our combined BEM/FEM metod converges linearly (O()) wit te discretization at te BEM/FEM interface. As a practical application, we ave used our metod to model te substrateresistances and te noise-sensitivity in a simple guard-ring structure. Our results were confirmed by two commercially available tools: Momentum [19] and Davinci [20]. 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