In this paper, a full-wave boundary integral equation formulation is proposed that accurately characterizes 3-D interconnects. By extending a recently presented framework for the resistance and inductance calculation that employs a 3-D differential surface admittance operator, the total impedance is computed. The inclusion of capacitive effects is validated and demonstrated over a broad frequency range

De Zutter, Daniël

Huynen, Martijn

Vande Ginste, Dries

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Broadband Full-Wave BIE Impedance Characterization of 3-D Interconnects

Ghent University Academic Bibliography

Broadband Full-Wave BIE ImpedanceCharacterization of 3-D InterconnectsMartijn Huynen, Danie¨l De Zutter, Dries Vande GinsteElectromagnetics Group/IDLab, Department of Information TechnologyGhent University/imec, Ghent, Belgiummartijn.huynen@ugent.be, daniel.dezutter@ugent.be, dries.vandeginste@ugent.beAbstract—In this paper, a full-wave boundary integral equa-tion formulation is proposed that accurately characterizes 3-Dinterconnects. By extending a recently presented framework forthe resistance and inductance calculation that employs a 3-Ddifferential surface admittance operator, the total impedance iscomputed. The inclusion of capacitive effects is validated anddemonstrated over a broad frequency range.Index Terms—3-D surface admittance, boundary integral equa-tion (BIE), interconnect modeling.I. INTRODUCTIONThe proliferation of electronic devices in a wide varietyof applications has driven electronic circuits and systems toincreasing complexity and diminishing size. These evolutionsbring about a set of challenges for the design engineer,especially in terms of signal and power integrity. To assessand tackle these problems, electromagnetic solvers are widelyutilized. However, the models at the core of such solvers areoften based on approximations. Improved, full-wave modelsare thus increasingly important to obtain reliable simulationresults.Electromagnetic full-wave solvers come in all shapes andsizes and can be classified in various ways. A commondistinction is made based on the discretization approach ofthe method, viz., volume or surface discretization. The for-mer category houses, among others, the well-known finiteelement method (FEM). By meshing the entire 3-D volume,this method provides a versatile tool for a wide range ofapplications. However, volume discretization requires a verylarge number of (small) mesh elements to properly representphenomena such as the skin and proximity effect in goodconductors, leading to computationally intractable systems.By only meshing the boundary of the structure, surfacediscretization schemes, such as boundary integral equation(BIE) methods, result in considerably smaller systems atthe cost of a dense system matrix. Nevertheless, taking intoaccount good conductors remains a challenging feat. A popularclass of approximate techniques employs surface impedancesto model conductors [1]. Other methods actually directly solvethe full-wave inner problem in the conducting medium [2]but suffer from numerical inaccuracies and computationalcomplexity due to the behavior of the Green’s function [3].An approach that solves this troublesome problem utilizes adifferential surface admittance operator [4], constructed usingthe eigenmodes of the conductor’s shape.R+j R−jejFig. 1. Rooftop function of edge ej with its support Sj = R+j ∪R−j .In this paper, we extend our novel framework [5] employ-ing a 3-D differential surface admittance for characterizinginterconnect structures to include capacitive effects. This isachieved by constructing a set of matrix equations analogousto the approach followed in [6] for the electric field integralequation (EFIE). This way, for the first time, a complete broad-band surface equivalent representation of good conductors isobtained.II. FORMULATION OF THE METHODConsider N disjunct volumes Vi, each filled with a (dif-ferent) homogeneous, nonmagnetic material defined by itswavenumber ki, embedded in a homogeneous, nonmagneticbackground medium with wavenumber k0. Applying theequivalence principle to each Vi, we can replace the innermaterial by that of the background medium by introducing anunknown surface current density js on the boundary surface Siwhich ensures that the fields outside Vi, i.e., (e0,h0), remainunchanged. This approach gets rid of the material discontinu-ities and results in a problem that is generally more suitedto be solved with a BIE method as the Green’s functionof the restored background medium G0 (r, r′) is easier toconstruct/evaluate. In particular, we use the EFIE:e0 = −jωa−∇φ, (1)with a the magnetic vector potential and φ the electric scalarpotential (ejωt dependence).Sticking to the BIE approach, we mesh all the boundarysurfaces Si that support the current densities. These surfacemeshes can be triangular with the corresponding RWG basisfunctions but as we focus on cuboids in the examples, wedemonstrate the approach for rectangular meshes with theassociated rooftops functions bj for every edge ej as depictedin Fig. 1. Every rooftop is defined on two adjacent rectanglesR+j and R−j , which together form the support Sj of bj .Expanding e0 and js into these rooftop functions, collectingthe expansion coefficients in the vectors E and I, respectively,and testing (1) with the same basis functions, leads to thematrix equation (for more details, see [5]):GE = −jωLI− TV, (2)with the following definitions(G)ij=∫Si∩Sjbi · bj dS (3)(L)ij= µ0∫Si∫SjG0 (r, r′)bi · bj dS dS′ (4)(V)i =∫RiφAidS, (5)with Ri being a rectangle in the mesh with area Ai. Eachelement in V represents the average potential on a singlerectangle. The incidence matrix T links the edges and patchesof the mesh as follows:(T)ij=1, if Rj is R+i of bi−1, if Rj is R−i of bi0, otherwise.(6)Note that the Gramian matrix G is block diagonal with everyblock the Gramiam matrix of Vi. Matrix L, on the other hand,is dense due to the global nature of the Green’s function.At this point, we introduce the discretized differential sur-face admittance operator [7] to obtain a relation between Eand I, viz., GI = YE, and as such eliminate E from (2):(GY−1G+ jωL)I− TV = 0. (7)An additional connection between I and V is now con-structed by introducing the expression for φ, leading to V =KQ where (Q)i is the charge per rectangle Ri and(K)ij=10∫Ri∫RjG0 (r, r′)AiAjdS dS′. (8)To eliminate the newly introduced Q, we invoke charge con-servation on every rectangle. In other words, by discretizing∇ · js + jωρs = 0 on every Ri, we getTTI+ jωQ = S, (9)where S represents external current sources. Combining (7)and (9) results in a composite matrix equation:[GY−1G+ jωL −TTTjωK−1] [IV]=[0S]. (10)Solving this equation, either directly or iteratively, resultsin the currents on the edges of the mesh and the voltages onthe rectangles. It can be utilized to determine, amongst otherproperties, the impedance response of the structure, as will bedemonstrated in Section III.So far we have assumed that our structure consists of Ndisjunct volumes (all cuboids in the presented examples). Inorder to simulate relevant structures, however, the variouswl5202.5PECFig. 2. Copper plate (σ = 5.8 · 107 S/m) with a feed stub above a perfectelectric conducting (PEC) half-space. The thickness of the conductor is 1µm.All other dimensions are given in µm on the diagram.TABLE ICAPACITANCE IN fF OF THE STRUCTURE IN FIG. 2l [µm] w [µm] α = l/w Cpp Yuan/Trick [8] Eq. (10)50 50 1 9.208 12.035 12.647111.80 22.36 5 9.208 12.742 13.52422.36 111.80 0.2 9.208 12.742 13.462cuboids should be combined into more complex shapes. Weeffectuate this connection by equating the voltage on adjacentrectangles of two cuboids and by introducing an additionalunknown current between those two patches. In the presentedmatrix formalism, this boils down to the addition of an extrarow and column per rectangle pair with two nonzero entrieseach. The column contains a 1 and −1 to include the extracurrent in the charge conservation equation on each rectanglewhile the additional row ensures the equality of both voltages.III. NUMERICAL EXAMPLESA. Validation exampleConsider the geometry depicted in Fig. 2. A copper plate(σ = 5.8 · 107 S/m) of dimension l × w with a thickness of1µm is fed by a stub of 20µm× 5µm and suspended 2.5µmabove a perfect electric conducting ground plane. Initiallywe choose l = w = 50µm and use the simple parallelplate formula (Cpp = A/d) for both cuboids to estimate thetotal capacitance of this structure (see Table I). However, thisignores the fringing fields and will as such underestimate thetrue capacitance. Various analytical expressions to estimatethis effect can be found for 2-D and 3-D capacitors. Wewill employ the simple expressions for 2-D conductors witha finite thickness as found in [8] and apply them along twodimensions for an improved estimation. All results are col-lected in Table I. It is apparent that the parallel plate formulaseverely underestimates the total capacitance. Employing theanalytical formulas results in a 30 % increase while our full-wave solution computes a capacitance that is 37 % larger. Theadditional increase found with the integral equation solutionis due to the corners and the connection of both blocks.According to the parallel plate formula, the capacitanceonly depends on the area of the plate and not on its shape.However, the analytical formulas contradict this and suggestthat fringing effects do depend on the exact shape. Therefore,we investigate configurations with the same, fixed area forvarious aspect ratios α = l/w. The difference between thecalculated and nominal capacitance of 9.208 fF is character-ized by the fringing factor γ such that C = γ Cpp. For some0.1 1 101.31.41.51.6 Yuan/Trick [8]Eq. (10)Aspect ratio α = l/wFringingfactorγFig. 3. Fringing factor φ for various aspect ratios α = l/w.540556051111550wFig. 4. Two copper plates of variable width w with a feeding structure. Alldimensions shown are in µm. The impedance is measured over the 1µm gap.specific values of α the capacitances are included in Table I.From Fig. 2 we see that γ is minimum for a square plate andincreases the more the plate is elongated in either direction.Note as well that the capacitance for reciprocal aspect ratios isslightly different according to the simulation results. Two suchvalues are included in Table I and show that for longer platesthe capacitance is marginally larger than for wider plates.This phenomenon can be attributed to the feed stub, whichis aligned along the length of the plate, and a such introducesa small asymmetry.B. Application exampleThe second example, depicted in Fig. 4, represents a copperparallel plate capacitor with variable width w and a 1µmseparation between the plates, fed via two traces that forma rectangular loop. All dimensions are annotated on the figureand are given in µm. The total impedance of this structureis measured over the 1µm gap for a broad frequency range.In Fig. 5 the absolute value of this impedance is shown forvarious values of w (including results from CST MWS shownwith markers) and for a reference planar loop. This referenceloop has the same shape and dimensions, i.e., 60µm×40µm,as the structure in Fig. 4 but lacks the parallel plates (for asimilar structure, see [5]). For low frequencies, we see thatthe structure behaves as a capacitance which increases forlarger widths w. The response of the planar loop, on the108 109 1010 1011 1012100101102103104105w = 5µmw = 15µmw = 25µmw = 45µmloopFrequency [Hz]|Z|[Ω]Fig. 5. Absolute value of the impedance of the structure in Fig. 4 for variousvalues of w with CST MWS results (blue markers) and the impedance of areference loop (black).other hand, indicates inductive behavior with a non-negligibleresistive component that levels out the response for the lowestfrequencies. Around 150 GHz, the structure undergoes a seriesresonance, whose precise frequency shifts with w as thecapacitance changes. For even higher frequencies (around700 GHz), the structure endures a second (parallel) resonance.This phenomenon is caused by the capacitance of the gapwhere the impedance is measured. The planar loop exhibitsthe same resonance albeit at a lower frequency of 630 GHzdue the absence of the parallel plate capacitance.IV. CONCLUSIONSA full-wave BIE method employing a differential surfaceadmittance operator to extract the total impedance of 3-Dinterconnects was put forward. The formulation circumventsthe cumbersome evaluation of the Green’s function in goodconductors and thus leads to accurate broadband results asdemonstrated in the presented examples.REFERENCES[1] T. B. A. Senior and J. L. Volakis, Approximate boundary conditions inelectromagnetics. IET, 1995.[2] U. R. Patel, S. Sharma, S. Yang, S. V. Hum, and P. Triverio, “Full-wave electromagnetic characterization of 3D interconnects using a surfaceintegral formulation,” in 26th Conference on Electrical Performance ofElectronic Packaging and Systems, San Jose, CA, 2017.[3] J. Peeters, I. Bogaert, and D. De Zutter, “Calculation of MoM interactionintegrals in highly conductive media,” IEEE Trans. Antennas Propag.,vol. 60, no. 2, pp. 930–940, Feb 2012.[4] D. De Zutter and L. Knockaert, “Skin effect modeling based on adifferential surface admittance operator,” IEEE Trans. Microw. TheoryTechn., vol. 53, no. 8, pp. 2526–2538, Aug 2005.[5] M. Huynen, D. De Zutter, and D. Vande Ginste, “Rigorous full-waveresistance and inductance computation of 3-D interconnects,” IEEEMicrow. Wireless Compon. Lett., vol. 28, no. 6, pp. 455–457, June 2018.[6] Z. G. Qian and W. C. Chew, “Fast full-wave surface integral equationsolver for multiscale structure modeling,” IEEE Trans. Antennas Propag.,vol. 57, no. 11, pp. 3594–3601, Nov 2009.[7] M. Huynen, M. Gossye, D. De Zutter, and D. Vande Ginste, “A 3-Ddifferential surface admittance operator for lossy dipole antenna analysis,”IEEE Antennas Wireless Propag. Lett., vol. 16, pp. 1052–1055, 2017.[8] C. P. Yuan and T. N. Trick, “A Simple Formula for the Estimation of theCapacitance of Two-Dimensional Interconnects in VLSI Circuits,” IEEEElectron Device Lett., vol. 3, no. 12, pp. 391–393, 1982.

Broadband full-wave BIE impedance characterization of 3-D interconnects

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Broadband full-wave BIE impedance characterization of 3-D interconnects

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