# Modeling effects of random rough surface on conductor loss at microwave frequencies

код для вставкиСкачать© Copyright 2006 X iaoxiong Gu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Modeling Effects of Random Rough Surface on Conductor Loss at Microwave Frequencies Xiaoxiong Gu A dissertation subm itted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of W ashington 2006 Program Authorized to Offer Degree: Electrical Engineering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3241904 Copyright 2006 by Gu, Xiaoxiong All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. 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University of W ashington G raduate School This is to certify th a t I have examined this copy of a doctoral dissertation by Xiaoxiong Gu and have found th a t it is complete and satisfactory in all respects, and th a t any and all revisions required by the final examining committee have been made. Chair of the Supervisory Committee Leung Tsang Reading Committee: Leung Tsang Yasuo Kug; au Ding Date: (l / ^ 4> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree th a t the Library shall make its copies freely available for inspection. I further agree th a t extensive copying of this dissertation is allowable only for scholarly purposes, consistent w ith “fair use” as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to Proquest Information and Learning, 300 N orth Zeeb Road, Ann Arbor, MI 48106-1346, 1-800-521-0600, to whom the author has granted “the right to reproduce and sell (a) copies of the m anuscript in microform an d /o r (b) printed copies of the manuscript made from microform.” Signature. D a te Peo H*h Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. University of W ashington A b stra ct Modeling Effects of Random Rough Surface on Conductor Loss at Microwave Frequencies Xiaoxiong Gu Chair of th e Supervisory Committee: Professor Leung Tsang Electrical Engineering The roughness of the interfaces between m etal and dielectric layers in high-speed interconnect and microelectronic package often causes significant additional power absorption at microwave frequencies which is detrim ental to signal and power in tegrity. To quantify the roughness effect on power loss, we use a random rough sur face model w ith root-mean-square height, correlation function and spectral density to characterize the roughness. Analytic models based on small perturbation m ethod of second order are developed for two-dimensional and three-dimensional problems. Our formulation takes into account both dielectric and conductive media. The simi larities w ith and differences from M organ’s classical result and th e H am m erstad and Bekkadal formula are discussed. Results are compared and verified w ith numerical method of moments and T -m atrix method. We also propose and dem onstrate pro cedures of estim ating the roughness-induced absorption enhancement factor by ex tracting power spectral densities from measured surface profile data. Comparing the measured propagation loss with the roughness-corrected sm ooth-problem loss yields excellent agreement. Additionally, we apply multiple scattering equations to study the scattering and absorption of electromagnetic waves on a conducting plane surface with a random distribution of hemispherical bosses. We derive m ultipole solutions up Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to third order to analyze close range interactions between nearby bosses. Surface cur rent and absorption enhancement factors are further computed for different embossed surfaces. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page List of Figures .................................................................................................................. iii List of T a b l e s ..................................................................................................................... vi .......................................................................................... 1 C hapter 1: Introduction Chapter 2: Power Absorption of Random Rough Surface of Conductor with Constant Magnetic-Field Boundary Condition in Two-Dimensional P ro b le m .................................................................................................... 9 2.1 In tro d u c tio n ....................................................................................................... 9 2.2 Random Rough Surface in 2-D Problem ................................................... 10 2.3 2-D Analytic Perturbation M e t h o d ............................................................ 11 2.4 2-D Numerical MoM Approach .................................................................. 13 2.5 2-D Numerical T-m atrix M ethod ............................................................... 15 2.6 Numerical R e su lts ............................................................................................. 17 C hapter 3: Modeling Absorption of Random Rough Interface between Con ductor and Dielectric Medium in Two-Dimensional Problem . . 22 3.1 In tro d u c tio n ....................................................................................................... 22 3.2 Two-Media Analytic Small P erturbation M ethod .................................. 23 3.3 Numerical Approach Using T-m atrix M ethod ....................................... 31 3.4 Results and D isc u ssio n ................................................................................... 32 3.5 C onclusion.......................................................................................................... 38 C hapter 4: 4.1 Power Absorption of Random Rough Surface of Conductor with Constant Magnetic-Field Boundary Condition in Three-Dimensional P ro b le m .................................................................................................... 40 In tro d u c tio n ....................................................................................................... i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 4.2 Random Rough Surface in 3-D Problem ................................................ 41 4.3 3-D Analytic Small Perturbation M e th o d ................................................. 42 4.4 Results and D isc u ssio n .................................................................................. 48 4.5 C o n clusion.......................................................................................................... 50 C hapter 5: Modeling Absorption of Random Rough Interface between Di electric and Conductive Medium in Three-Dimensional Problem 52 5.1 In tro d u c tio n ...................................................................................................... 52 5.2 Derivation of 3-D SPM2 F o r m u l a .............................................................. 53 5.3 Numerical Approach Using T-m atrix M ethod ....................................... 60 5.4 Results and D isc u ssio n .................................................................................. 64 5.5 C onclusion.......................................................................................................... 67 Chapter 6: Estim ation of Roughness-Induced Power Absorption from Mea sured Surface Profile D a t a 71 6.1 In tro d u c tio n ...................................................................................................... 71 6.2 Estim ating Power Absorption with PSD E x t r a c ti o n ............................. 72 6.3 Results and D isc u ssio n .................................................................................. 74 6.4 C onclusion.......................................................................................................... 78 Bibliography ..................................................................................................................... 82 Appendix A: Scattering and Absorption of Electromagnetic Waves on a Plane w ith Hemispherical B o sse s................................................................... 89 A .l In tro d u c tio n ...................................................................................................... 89 A.2 Multipole Solution of Hemispherical BossS c a tte rin g .............................. 91 A.3 Results and D isc u ssio n .................................................................................. 96 A.4 C onclusion.......................................................................................................... 98 ii Reproduced with permission of the copyright owner. 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LIST OF FIGURES Figure Number Page 2.1 Example of random rough surface models in 2-D problem: (a) Gaussian 11 correlation function; (b) Exponential correlation function................... 2.2 Power absorption ratio as a function of frequency from 2-D SPM2: Gaussian correlation function (I — h) w ith varying RMS height. . . . 18 2.3 Power absorption ratio from 2-D SPM2: Gaussian surface (I = 1.5h). 19 2.4 Power absorption ratio from 2-D SPM2: Exponential surface (I = 1.5 h ) ................................................................................................................ . 20 2.5 2-D SPM2 vs. 2-D MoM: Gaussian surface (h = 1.2/iim, I = 2h). . 2.6 2-D MoM vs. 2-D T-m atrix (h = 1.2/xm, I = 2h) 2.7 Power absorption ratio from 2-D MoM: Exponential surface (I = 2/jm). 21 3.1 A plane wave impinging on a rough surface with incident angle 0*. . 23 3.2 Power absorption ratio as a function of frequency from SPM2: Gaus sian correlation function (h = 1/irn) w ith varying correlation length. . 33 .................................. 20 21 3.3 Power absorption ratio: surface w ith Gaussian correlation function (h = 0.75/xm).................................................................................................... 34 3.4 Power absorption ratio: surface w ith exponential correlation function (h = lf im ) .......................................................................................................... 35 3.5 SPM2 versus T-m atrix: surface w ith Gaussian correlation function (h = QASjim).................................................................................................... 36 3.6 A surface profile and m agnitude of surface magnetic fields.................. 3.7 Two-media SPM2 vs. one-medium SPM2: h = 1/rm and I = 2^m. . 3.8 Emissivity and absorptivity versus incident angle: surface w ith Gaus sian correlation function (h = 2.4cm, I = 12.0cm), / = 5.0GHz, and er/eo = 15.57 + 3.71i...................................................................................... 39 4.1 Visualization of rough surfaces in three-dimensional problem: (a) Gaus sian correlation function; (b) Exponential correlation function.......... 41 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 . 38 4.2 3-D SPM2: Gaussian correlation function versus exponential correla tion function 49 4.3 3-D SPM2 versus 2-D SPM2 for Gaussian correlation function.... 50 4.4 3-D SPM2 versus 2-D SPM2 for exponential correlation function. . . 51 5.1 A three-dimensional random rough interface between dielectric and conductor.............................................................................................................. 54 5.2 Power absorption ratio: surface w ith Gaussian corrrelation function (h = 1 fan) w ith varying correlation length.................................................. 65 5.3 Power absorption ratio: surface w ith Gaussian correlation function (h = 0.75 p m) w ith varying correlation length............................................ 66 5.4 Power absorption ratio: surface with exponential correlation function (h = 1 fan) with varying correlation length.................................................. 67 5.5 3-D SPM2 versus 2-D SPM2: surface w ith Gaussian correlation func tion (h = 1 //m, I — 2 /im )................................................................................. 68 5.6 3-D SPM2 versus 2-D SPM2: surface with exponential correlation func tion {h = 1 /tm, I = 2 n m )................................................................................. 69 5.7 3-D SPM2 versus T-m atrix: surface w ith Gaussian correlation function (I — 3.0 fan).......................................................................................................... 69 5.8 3-D SPM2 versus T-m atrix: surface w ith Gaussian correlation function (I = 3.5 fim).......................................................................................................... 70 6.1 One-dimensional power spectral density Wi£>(k): E xtracted versus syn thetic model.......................................................................................................... 75 6.2 Correlation function C(p): Extracted versus synthetic model................. 76 6.3 Two-dimensional power spectral density W 2 D{kp)'- E xtracted versus synthetic model................................................................................................... 77 6.4 Absorption enhancement factor: Extracted versus synthetic model. . 78 6.5 Correlation function C(p): Extracted versus synthetic m odel................. 79 6.6 Two-dimensional power spectral density W 2 D{kp)■ E xtracted versus synthetic m odel................................................................................................... 80 6.7 Absorption enhancement factor: Extracted versus synthetic model. . 80 6.8 Surface visualization: AFM measured surface (left) versus synthetic model (right)........................................... 81 6.9 A ttenuation constant: measured loss versus estim ated loss...................... 81 A .l A flat plane with a random distribution of hemispherical bosses. . . . 91 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E |) for 2 bosses in X axis. A.2 Boundary condition of electric field (|n x A.3 Boundary condition of m agnetic field (|n • H |) for 2 bosses in X axis. A.4 Boundary condition of electric field (|n x A.5 Boundary condition of m agnetic field (|n • H |) for 2 bosses in Y axis. 103 A.6 D istribution of surface current on 2 bosses in X axis................................ 104 A.7 D istribution of surface current on 2 bosses in Y axis................................ 104 A.8 D istribution of surface current on 5-boss surface....................................... 105 A.9 D istribution of surface current on 200-boss surface................................... 105 101 E\) for 2 bosses in Y axis. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .100 .102 LIST OF TABLES Table Number Page A .l Absorption and enhancement factor for different embossed surfaces vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 ACKNOWLEDGMENTS I wish to express my sincere appreciation to a number of individuals who helped me in the course of conducting this research. First, I am indebted to my advisor, Prof. Leung Tsang for his constant advice, encouragement, and dem and for excel lence. His diligence and expertise in carrying out the research showed me the quality of a real academician and served as my eminent role model. The rest of my supervi sory committee deserves acknowledment: Prof. Yasuo Kuga, Prof. Vikram Jandhyala, Prof. Kung-Hau Ding, and Prof. Fumio Ohuchi, for their guidance during the prepa ration of this document. I am also truly grateful to Dr. Henning Braunisch for his valuable technical consultations and th e proofreading of my dissertation. The following researchers and engineers deserve my gratitude for their mentorship while I was interning at Intel and IBM Research from 2003 to 2005: Dr. Mohiuddin Mazumder, Dr. Christian Schuster, and Jeff Loyer. Their helpful inquiries, insightful suggestions, and critical comments give new light to the research from a pragm atic perspective. My gratitude also goes to the following Intel engineers for their help with the correlation study with loss measurement: Alejandra Camacho-Bragado, Przemyslaw M itan, Daniel Montes, Grace Hu, and Zhichao Zhang. I would like to thank the following people for communicating the initial idea of us ing hemispherical bosses to estim ate the surface roughness induced losses and current flow on a transm ission line: Edin Sijercic, Olufemi Oluwafemi, Anusha Moonshiram, Gary Brist and Stephen Hall from Intel, and Dr. Paul Huray from the University of South Carolina. While working in the lab, I shared the rewarding experiences w ith my fellow vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. colleagues, Dr. Chung-Chi Huang, Chong Jin Ong, Ding Liang, and Boping Wu. They have all influenced me in positive ways. The following institution should also be recognized for its financial support and equipment contribution: the Intel Corporation, which funded my research position from 2003 to 2006. viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DEDICATION TO M Y WIFE, YAN, FOR HER L O V E A N D SUPPORT. TO M Y P A R E N T S FOR TEACHING M E H O W TO THINK. ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1 INTRODUCTION The surface of conductors are roughened through chemical and topological tre a t ment to promote adhesion of the conductor layer to dielectric layer in high-speed printed circuit board (PCB), interconnect waveguide structure and microelectronic package based on organic materials. Since the speed of interconnects has been rapidly increasing to the multi-GHz region, th e roughness of the surface can have significant effects on power and signal integrity [1], [2]. Capacitance change due to surface roughness has been analyzed in the past by Zhu and W hite [3], [4] in which stochastic integral equations were formulated and solved to compute the mean value and the variance of capacitance of two-dimensional and three-dimensional interconnects with random surface roughness. However, it is of more im portance to model frequencydependent loss in conductors for proper assessment of both signal distortion and delay in high-speed interconnects in order to meet multi-GHz bandw idth requirement of sig nal transmission. It has been shown by measurement th a t the practical topological features of conductor surfaces may have peak to valley distances in th e order of mi crometers [5], [6]. Such roughness can cause significant effects on conductor loss at microwave frequencies, due to the skin effect in classical electrodynamics. At high fre quencies when the skin depth is less th an the hight difference between th e peaks and valleys, the current is concentrated ju st below the conductor surface and flows up the peak and down the valley. The effective p ath length of the current flow increases, and thus th e effective conductor resistance increases. Experiments by Tanaka [7] demon strated the increase of effective resistivity of different copper foils by as much as 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 to 70 percent in the multi-GHz region due to surface roughness. It is th e emphasis of this dissertation to analyze and quantify the roughness effect on conductor loss which is a dom inant loss mechanism separate from dielectric loss in the microwave frequency range of interest [8] [9]. One key application of the predictive capability developed in this study is in com puter-aided design (CAD) of insertion loss lim ited interconnects in high-performance computing platforms. It can also be instrum ental for guiding package and board substrate technology development, i.e., for making the tradeoff between thermomechanical reliability (adhesion), electrical performance (loss), and cost. Presently, the common results of quantifying th e im pact of conductor surface roughness on ohmic loss are due to M organ’s classical paper [10] and the Ham m erstad and Bekkadal formula [11]. In [10], Morgan solved a two-dimensional (2-D) quasistatic eddycurrent problem for a periodic ridge structure where the surface height varies only in one horizontal direction. Morgan computed the power absorption enhancement factor which determines the additional losses due to surface geometry. Ham m erstad and Bekkadal [11] fitted M organ’s results by an empirical formula which is presently the most common model of quantifying the im pact of conductor surface roughness on ohmic loss. Wu [12] analyzed the increased factor of surface resistance and reactance of superconductors using M organ’s square-grooves model. Close-form expressions were derived in [12] similar to the H am m erstad and Bekkadal formula. However, Morgan et al. only used a periodic ridge structure as rough surface which may misrepresent the roughness-induced loss. The underlying physics is focused on the analysis of th e interactions of electro magnetic waves w ith the rough surface. Lord Rayleigh [13] was th e first to study the properties of rough surfaces when he solved the problem of the reflection of a plane acoustic wave from a sinusoidally corrugated surface. Rice [14] combined perturba tion theory and the Rayleigh m ethod to calculate th e reflection of electromagnetic waves from a slightly rough surface with random roughness spectrum . For the studies Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 of rough surface effect on wave propagation in waveguide and interconnects, research by Sanderson [15] shows th a t the Rayleigh-Rice perturbation technique gives good results for periodic surface roughness w ith small slopes. He and Rytov et al. [16] further speculate th a t this may also be true for random or non-periodic roughness. Proekt [17] also analyzed the increase of resistance of conductors due to surface rough ness using perturbation method. However, his results are based on M organ’s periodic triangle-groove structure and the solution of surface field is limited to zero-order and first order which lead to very small change (less th an two percent) of the conductivity due to surface roughness. O ther studies of rough surface effects on conductor loss include a series of works published by Molina and M aradudin, [18], [19], [20], Holloway and K uester [21], [22] where the emphasis was based on the proper enforcement of boundary conditions at the rough interface to avoid expensive numerical com putation of electromagnetic fields. Most research to date on the roughness effect on conductor loss is limited to two-dimensional problems. Twersky [23], Biot [24], [25] and W ait [26] proposed a three-dimensional rough surface model in which the roughness was represented by hemispherical bosses on a perfectly conducting plane. However, in their theory only the bosses may have finite conductivity while losses in the plane are absent. Further more, although the hemispherical boss model gives light to investigate the scattering of electromagnetic field and current distribution on th e rough surface in 3-D prob lems, such embossed plane surfaces have limited capability of resembling accurately the real physical rough surface occurring on the interconnect structures. In this dissertation we use a random rough surface model w ith correlation functions and spectral densities to characterize the surface roughness. The spectral densities can be further reconstructed from measurement of height profiles. To quantify the conductor loss due to surface roughness, we have applied small perturbation m ethod (SPM) and developed two-dimensional and three-dimensional models analytically and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 numerically. The similarities with and differences from Morgans classical result and the H am m erstad and Bekkadal formula are discussed in detail in this dissertation. In C hapter 2, we use the model of random rough surfaces in two-dimensional prob lems where the surface height only varies in one horizontal direction. A random rough surface is characterized by root-m ean-square (RMS) height, correlation length, and correlation function. Compared with the existing M organ’s periodic rough structure, our random rough surface model provides a b etter resemblance w ith th e surfaces oc curring in m etal interconnects in 2-D problems. The characteristics of random rough surfaces can also be extracted quantitatively by measuring surface height profiles. The absorption due to these profiles is calculated by three methods: the analytic small perturbation m ethod of Rayleigh and Rice to second order (SPM2, where 2 stands for second order), the numerical m ethod of moments (MoM), and th e system operator transfer m atrix (T-m atrix) method. The result of absorption based on SPM2 is in term s of the spectral density of the random rough surface. The use of SPM2 is needed because absorption deals w ith power and SPM2 conserves energy to second order. In C hapter 3, we extend the two-dimensional SPM2 approach to two-media studies including both th e dielectric region and conductor region rather th a n enforcing a constant magnetic field as the boundary condition. The effects of a random rough surface between dielectric and lossy conductive medium on power absorption are analyzed by considering incident plane waves impinging on the interface. Similar to the previous chapter, the rough interface is modeled by th e characteristics of RMS height, correlation length, and correlation function. The absorption is calculated by SPM2 and the numerical T -m atrix method. The two m ethods agree w ith each other w ithin the regimes of validity, i.e., for rough surfaces w ith small slope. We show th a t for highly conductive mediums such as copper, th e absorption based on SPM2 can be further simplified because the surface roughness scale is much less than the inverse of the wave number in the dielectric region. As a result, the power absorption enhancement factor has the same form as described in th e one-medium Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 SPM2 m ethod w ith M organ’s boundary condition. For low conductive medium, the absorption depends on the incident angle of the inpinging plane wave. We use air-soil interface as an example and dem onstrate th a t two-media SPM2 obeys the condition of energy conservation, i.e., the absorptivity is equal to the emissivity of the rough interface between the two media. In C hapter 4, we make an initial effort to extend th e SPM2 approach from an ide alized two-dimensional problem to a more realistic three-dimensional problem where the surface height varies in both horizontal directions. As in the studies of 2-D prob lem, the rough surface model of 3-D problem also includes th e RMS height, correlation length and correlation function. To reduce the complexity of the formulation, in this chapter, we first take into account only the conductor region by considering M organ’s constant magnetic field as the boundary condition. We obtain a closed-form of power absorption enhancement factor after deriving zero-order, first-order and second-order solutions of electric field in the conductor region. The formula of enhancement fac tor in the 3-D problem with M organ’s condition is reduced to the previous form in the 2-D problem if the surface happens to be uniform in one horizontal direction. However, the 3-D enhancement factor formula illustrates less power absorption than the 2-D case which cannot be physically interpreted. The reasons, as we found out later, lie in the fact th a t M organ’s boundary condition is only a valid approxim ation for two-dimensional problems. For three-dimensional problems, the magnetic field on the interface between two media are no longer constant. Consequently, we have to consider both dielectric and conductor media in the SPM2 formulation. In C hapter 5, we further study the effects of a random rough surface on the power absorption between a dielectric and conductive medium in a three-dimensional con figuration where th e surface height varies in both horizontal directions. Different from C hapter 4, here we take into account both a dielectric and a highly conductive medium and derive a closed-form 3-D SPM2 formula of power absorption enhance ment factor due to rough interface between th e two media. This work is an extension Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 of previous work, in which we studied wave scattering at the interface between two dielectric regions for remote sensing applications [27]. The simplification here takes advantage of an assumption th a t the conductivity of the lower conducting medium is much greater th an th a t of the dielectric medium. In typical dielectric-metal layers occurring on interconnect and package, the m agnitude of wave num ber in the conduc tor is a few thousand times larger th a n th a t in the dielectric medium at microwave frequencies. The SPM2 and numerical T-m atrix m ethod are used in th e two-media three-dimensional problem. The absorption depends on the RMS height, correlation length and correlation function of the random rough surface. Results show th a t the T-m atrix m ethod agrees with SPM2 for rough surfaces with small slope. We further compare the three-dimensional results to the previous two-dimensional results and show significant difference. The power absorption enhancement factor exhibits satu ration for the Gaussian correlation function, b u t not for the exponential correlation function. In C hapter 6, we describe a methodology of extracting th e power spectral density (PSD) of th e rough surface from height measurement. For isotropic rough surfaces, i.e., the correlation function has no angle dependency in th e horizontal plane, ex tracting the surface spectrum can be further simplified by utilizing fast Fourier-Bessel transform. We com pute the additional power loss due to surface roughness by putting the extracted PSD in the 3-D SPM2 formula of enhancement factor. Results demon strate good correlation between the measured loss and the estim ated loss up to 20 GHz. We also show th a t it is possible to choose a random rough surface model to approxim ate the measured surface and predict the roughness effect on power loss. Rough surface problems are unlike deterministic boundary value problems in elec trom agnetics which have well defined geometries. For a rough surface problem, we first need to characterize the geometry of the surface roughness. There are at least three type of characterizations: (1) random rough surface w ith th e height function described by a random process with spectral density and correlation function, (2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 periodic grating, and (3) hemispherical bosses. The m ethod of solution depends on th e characterization of the rough surface. The random rough surface model is the most common and has been used to represent solid surfaces in optics, land surfaces and ocean surfaces in acoustic and microwave scatter ing. One im portant feature of this characterization is th a t th e param eters of charac terization, i.e., the correlation function or the spectral density, can be quantitatively extracted from the measured heights as a function of position. T he characterizations of periodic grating and hemispherical bosses can be treated as special cases of using the random rough surface model since the spectral densities can be com puted in the same manner. Using the random rough surface model, researchers have developed an alytic methods such as small perturbation m ethod and Kirchoff m ethod which have domains of validity. For rough surfaces th a t have m oderate to large RMS heights and large slopes, numerical m ethods of exact solutions have also been developed such as the m ethod of moments (MoM) using RWG basis functions, e.g., in [28], [29], [30]. The solutions based on MoM are also enhanced by fast com putational methods such as sparse m atrix canonical grid m ethod (SMCG) and multilevel UV m ethod [31]. Be cause of its capability of accurately characterizing rough surfaces, th e random rough surface model has been preferred by the acoustics, microwaves and optics community in studying the wave scattering by solid and liquid surfaces in th e past three decades [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. The second characterization of periodic grating is not justifiable in real life because it is a gross assumption to represent a real life surface by a grating. The th ird characterization is based on hemispherical bosses. Since 1950’s, Twersky has done extensive studies in wave scattering by discrete scatters [42], [43], [44], To make the theory and model applicable to rough surfaces, he assumed th a t the rough surface consists of hemispherical bosses [23]. In the works of Biot [24] and W ait [26], the rough surface is modeled as a distribution of perfectly conducting hemispherical bosses on an underlining flat perfectly conducting plane. The model was studied in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 1950’s and 1960’s. However, since 1960’s, the interests in this characterization have declined since it is difficult to represent real life surfaces by hemispheres. In th e Appendix, we use the hemispherical bosses for characterization. We apply Foldy-Lax m ultiple scattering equations [35] to study th e scattering of electromagnetic waves on the embossed plane. We derive a multipole solution up to th e order of three to analyze close range interactions between nearby bosses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 Chapter 2 POW ER ABSORPTION OF RANDOM ROUGH SURFACE OF CONDUCTOR WITH CONSTANT MAGNETIC-FIELD BOUNDARY CONDITION IN TWO-DIMENSIONAL PROBLEM 2.1 In tro d u c tio n The roughness of th e interfaces between layers, especially in microelectronic packaging based on organic materials, is often used to facilitate the adherence of th e copper structures to the dielectrics. Since the speed of interconnects is rapidly increasing to the multi-GHz region, the roughness of the surface can have significant effects on signal integrity. Existing commercial software tools do not allow users to model the surface roughness of the substrates accurately. Presently, th e common results are due to M organ’s classical paper [10] and the Ham m erstad and Bekkadal formula [11]. There are recent analyses [21] the results of which are consistent w ith M organ’s. However, in M organ’s analysis and in these other analyses, a periodic rough surface is used, often w ith rectangular grooves. The Ham m erstad and Bekkadal formula is -Rj,rough — J— — = * a , sm ooth , 2 1 -1— 7T arctan 1.4 ( 2 . 1) The absorption ratio in (2.1) depends only on the root mean square (RMS) height h of the rough surface profile, besides the skin depth 8. In this chapter, we use the model of random rough surfaces. Random rough surface is characterized by RMS height, correlation length, and correlation function. The advantage of using a random rough surface model is the resemblance w ith th e surfaces occurring in copper interconnects. Furtherm ore, by measuring surface profiles, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 random rough surface characteristics can be extracted quantitatively. The absorption due to these profiles is calculated by three methods: the analytic small perturbation m ethod of Rayleigh and Rice to second order (SPM2, where 2 stands for second order) and th e numerical m ethod of moments (MoM). The result of absorption based on SPM2 is in term s of the spectral density of the random rough surface. The use of SPM2 is needed because absorption deals w ith power and SPM2 conserves energy to second order. 2.2 R a n d o m R ou gh Surface in 2-D P ro b le m For a 2-D problem of random rough surface, the height function f (x) is treated as a stationary Gaussian random process. The two point ensemble average of the random process is i f ( x l ) f { x 2)) = h2C ( \ x i - x 2\) (2.2) where h2C (x) is the correlation function. Two common correlation functions are the Gaussian correlation function with C (x) = exp (—x 2/ l 2) and exponential correlation function w ith C (x) = exp (—\x\ /I), where I is th e correlation length. The expo nential correlation profile appears significantly rougher th a n th a t for the Gaussian correlation function, as shown in Figure 2.1. In generating the roughness profiles [45], we use the spectral density function W (kx) which is the Fourier transform of the correlation function. The spectral density of the Gaussian correlation function is given by W (kx) = by W (kx) = exp {—k 2l2/ 4) and th a t of the exponential correlation function 2 v/i:(i+fc2 i2) [4^]- Note th a t because th e spectral density of the expo nential correlation function decays slowly w ith increasing kx, th e surface contains multi-scale roughness. Besides these two common correlation functions which are used throughout this dissertation, there are alternative suggested forms for correla tion function including the Gaussian-exponential-combined correlation function [47], the Lorentzian [48], [49], the Gaussian cosine function [50], the 1.5-Power correlation function [47], and the Staras function [51]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.1: Example of random rough surface models in 2-D problem: (a) Gaussian correlation function; (b) Exponential correlation function. 2.3 2-D A n a ly tic P e r tu rb a tio n M e th o d Consider a 2-D problem w ith a random rough surface profile z = / (x). We use exp (—jcot) as the phasor notation. Let ip be the magnetic field th a t is in the y direction. Then (2.3) where k\z = y / k \ — and k\ = Here 5 is the skin depth (5 — and <7 is the conductivity of the conductor, y is its magnetic permeability, and cu = 2-irf is the angular frequency. We use a second order small perturbation m ethod, setting ip {kx) = ip{0) [kx) + ip{1) (kx) + ipW (kx) . (2.4) Following Morgan [10], we assume th a t the magnetic field on th e surface z — f (x) is a constant H q. Hence (2.5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 Balancing equation (2.3) to second order, we obtain f t Q ( k x) = H 0S ( k x) (2.6) ^ -jh H o F fa ) (2.7) H 0 J ~ d k ' xF ( k x - k ' x) F( h' x) ( - k 1tflz + ^ ( 2 .8 ) (kx) = ^(fc*) = where F (kx) is the Fourier transform of / (x) and 6 (kx) is th e dirac delta function. The power absorbed by the conductor, for a given w idth w in y direction and length L in x direction, is obtained from = £ « » / d w Calculating the power absorbed to second order leads to Pn = w — Re { { j d x ^ - ^ j j V’(0) (kx) + dkx e x p ( ~ j k xx) + j k lzf (x)\ (2 . 1 0 ) (kx) ( ~ j k x)} ^ rO C I [1 ____ dkx e x p ( j k xx ) [ l - j k ? J ( x ) \ __ V>(0)* ( K ) + ^ (1)* (k'x) } ' — o o W + — Re{{ f dx dkx e x p ( - j k xx) ^ (0) (kx) + v>(1) (kx) + ^ / z f (x) - k h f 2 (x) (kx) ( j k u ) } OO dA£ exp (Jk'xx) 1 - jk[*J (x) •OO 1+ j h K f f 2 (x) L (A4) + ^ (1)* (*4) + ^ (2)* (*4)1}}. Taking the ensemble average of (2.10), using the property (F (kix) F* (k2x)) = 6 {klx - k2x) W (kix) (2 .11) and simplifying the results gives wL , „ ,2 L 2h2 7 J " dk,W (fe) licv/t'= ■ki Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 2 . 12 ) 13 The first term in ( 2 .1 2 ) is the absorption for a sm ooth surface, Pa,sm ooth- The ratio of rough to sm ooth surface absorption is (Pa) = 1+ 2 ( 2 .1 3 ) ik ‘ w (W -Pa,sm ooth A 2-D SPM2 m ethod was also used by Sanderson [15] and his result for internal surface displacement is expressed in term s of spectral density. However, (11) in [15] has an integrand th a t asymptotically approaches W S2 ^ as kx becomes large. Thus the integral is convergent even for the exponential correlation function. 2.4 2-D N u m erica l M o M A p p ro a ch To use MoM, we solve the well-known surface integral equation [45] f TrVh ( O + [ dstpi (f) n -V g i (f, f') — dsgi (f, f') h ■Vipi (f) = 0 2 Js Js (2-14) where tpi = H 0 on the surface z = f (x) and g i ( f ' , f " ) = - -J 4 2)( k i \ f ' - f " \ ) . (2.15) The first integral in (2.14) is taken as th e principle value w ith an infinitesimally small piece subtracted out from the domain of integration. We apply the periodic boundary condition with the period L. It is a valid approx im ation to random rough surface scattering provided th a t the period contains many peaks and valleys and many correlation lengths, i.e., L » I [45]. Using MoM, the m atrix equation is Y , A ^ nu ln = (2.16) n n In the above equation, the m atrix elements are as follows. For m n, OO A $ n = K i o ( f m , f n ) A x + ^ 2 K w (f m, f n + qLx) A x q= —oo <jAo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.17) 14 OO B ^ l = - K i m {rm, rn) A x + ^ A x [ - K 1N0 (rm, rn + q L x )] (2.18) q= —oo q+Q where fm is the center of the m -th patch in the MoM discretization and 7 is Euler’s constant. For m = n, ( 2 . 19 ) lm m OO q= 0 (2.20) B mn {1) where K w (f\f") = gi {f", f' ) K im (f',f") = ti'-V'gi (2.21) (fiV). (2 .22) Note th a t the m atrix elements include an infinite number of sum m ations because of the use of periodic G reen’s functions. The unknowns to solve correspond to the normal derivative of magnetic field on the surface. After the surface integral equation is solved, the ratio of power absorption is calculated by (2.23) a,sm ooth where the integration is over the length L in th e x direction. It can be shown th a t as 8 —» 0 0 , the ratio in (2.23) approaches unity. In the numerical implementation, we take L — 10/ and surface discretization is chosen as A x = rriin { ^}. To calculate the average power absorption, we use a M onte-Carlo simulation ap proach. We generate a large number of realizations of rough profiles. Solving the MoM equation we then calculate the absorption ratio for every realization and the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 average absorption is computed. For those simulation results shown in Section III, 600 realizations are used. 2.5 2-D N u m erica l T -m a trix M e th o d The governing equation for z' > f (xr) is given by / ds [fa (r) n ■Vg i (F, f') — gi (F, r') n ■V f a (F)] = 0 (2.24) Js where f a (F) = Ho on the surface z — f (x) The plane-wave representation of 2-D G reen’s function g\ (F, r') for z' > f (x ') is given by 9i (r, r') = gx (x, z\ x', z') - j f Jf e x p ( - j k x (x; - x ) ) e x p ( - j k l z ( z ' - z)) = - J d K ------------------------. (2.25) We first define unknown fields a (F) and b (f) on the surface as follows: a (F) = 6 (F) = a (x, / (x)) = a (x) = f a (f) 6 (x, f (x )) = b(x) = ^ J l + (2.26) n •V fa (r). (2.27) Similar to 2-D MoM formulation, we apply the periodic boundary condition with the period L such th a t a (f) and 6 (r) on the surface can be represented using Fourier series expansion: a ( x 0) = ' Y ^ a mle x p { - j k xmlXo) (2.28) y ^ P m ' exp { - j k xm>xQ) (2.29) m' b ( x 0) = where kxm = Coefficients a m/ can be determ ined by specifying the constant magnetic field H 0 as the boundary condition: Ho = y 2 am' exP (- jkx m' Xo) • Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.30) 16 Next coefficients ,6m>can be solved from th e following m atrix equation ^ m’ (2.31) ' f i m ' B mrni mf where the m atrix elements are given by [ dxo exp (j k xmx 0 + j k \ zmf (a;0)) exp { - j k xm'X0) ( _ \ J So [ dxo exp (j (kxm - kxm/) x 0) exp ( j k lzmf (x0)) ( J So Bmm' = I dxo exp ( j k x m Xo + j k \ z m f (x0)) exp ( — j k So = - 1 J (2.32) \ C t X rCi z m / / x m 'X o ) 1 J \ k lzm dxo exp ( j ( k xrn - k x m>) x 0) exp (j k l z m f (x0)) ( JSo ). (2.33) \klzm j Note th a t So is the 0-th patch of the periodic infinite surface and fo = ( xq, f (®o)) is a point on th e 0 -th patch. After the m atrix equation is solved, the power absorption is calculated by p * a , rough j ..,* . /1 +, f d f ' J/ ”dx'ipl^j _ = 1 W 77^ 2n - ’V'ipi (2.34) N Ss bid* A x j. i —1 where N s is the number of discretization on the surface. Remember the power dissipation on the sm ooth surface is given by w L \ h£ -‘ a,sm ooth — 2&S ’ y^-OOJ Hence, we get the ratio of power dissipation as follows: Ns -Pa,rough Pa, sm ooth & L \H 0 R e^ka -A xi. i=1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.36) 17 2.6 N u m erica l R e su lts 2.6.1 2-D SPM2 Results Figures 2.2-2.4 illustrate the 2-D SPM2 results of power absorption ratio. In Figure 2.2, the results are for a Gaussian correlation function with I = h. The RMS height h varies from 0.2n m to 1.6fim. We note th a t the absorption ratio increases with frequency. It also increases with RMS height. In Figure 2.3, th e results are repeated for the case of I = 1.5h. The absorption ratios are smaller th a n those of Figure 1 because a larger correlation length gives a smoother surface. The results show th a t for the Gaussian correlation function, saturation is consistent w ith M organ’s findings. In Figure 2.4, the results are illustrated for surfaces w ith exponential correlation func tions exhibiting larger absorption th an surfaces w ith Gaussian correlation function. Also the absorption ratio results do not saturate which is distinctly different from the results given by Morgan and the Ham m erstad and Bekkadal formula. The results of Figures 2.2-2.4 show th a t the absorption depends on all three of the roughness characteristics, viz., RMS height, correlation length and correlation function. 2.6.2 2-D SPM2, Mo M and T-Matrix Comparison The 2-D small perturbation m ethod only requires com putation of one integral and thus it is convenient for evaluating power absorption for different random rough sur faces. However, 2-D SPM2 involves a second-order approxim ation and its accuracy needs to be verified. Figure 2.5 compares 2-D SPM2 results w ith MoM results. The modeled surface profiles are Gaussian w ith h = 1.2p m and (I = 2h). In case of a smoother surface (I = 2h), the 2-D SPM2 and MoM results are in good agreement. The results also show saturation close to 1.2 for the absorption ratio. In th e case of a rougher surface (I = h), the absorption ratio for SPM2 is larger th a n for MoM. The difference is about 17% at 10 GHz. On the other hand, both numerical results agree w ith each other well. Figure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 2 -*• 1.8 jZ ■s - q ^ y m - H - 1.0 urn -© - 1 .6 um 1.6 JZ Cl 3o 4 1.2 Figure 2.2: Power absorption ratio as a function of frequency from 2-D SPM2: Gaus sian correlation function (I = h) w ith varying RMS height. 2.6 compares 2-D MoM with 2-D T-m atrix results for the sm oother surface case (h = 1.2//m and I — 2h). The 2-D T -m atrix curve is less sm ooth th a n th e 2-D MoM curve because of different numbers of realization (100 for T -m atrix and 600 for MoM). Figure 2.7 shows the simulated power absorption using MoM. The modeled surface profile param eters are h = 2/iin and I = 0.7h (I = h ) with an exponential correlation function. In this case, the roughness of th e conductor is quite significant so th a t the power absorption ratio goes up to around 3 at 30 GHz. In contrast, the classical Hamm erstad and Bekkadal formula would approach an asym ptotic value of only 2. 2.6.3 Conclusion The effects of a random rough surface on the absorption by a metallic surface at microwave frequencies are analyzed by using the analytic small perturbation method, the numerical m ethod of moments m ethod and the numerical T -m atrix method. The rough surfaces in this chapter are considered as a two-dimensional problem where the surface height only varies in one horizontal direction. A constant magnetic field is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Figure 2.3: Power absorption ratio from 2-D SPM2: Gaussian surface (I = 1.5h) enforced as the boundary condition. The results show significant difference between absorption of a rough surface and th a t of a sm ooth surface. The absorption depends on the root mean square height, correlation length, and correlation function of the ran dom rough surface. The similarities w ith and differences from Morgans classical result and the H am m erstad and Bekkadal formula are discussed. It is shown th a t for m ulti scale rough surfaces such as surface w ith exponential correlation function,saturation of absorption does not occur, or occurs at much higher frequencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 Figure 2.4: Power absorption ratio from 2-D SPM2: Exponential surface (I = 1.5h). jZ ■6 o E w H 3 - SPM MoM - e - SPM 0 MoM ~ . « (l=2h) (l=2h) (l=h) (l=h) M l ............... ; : .^0 — JZ Zi o CL (GHz) 10 Figure 2.5: 2-D SPM2 vs. 2-D MoM: Gaussian surface (h = 1.2/um, I = 2h). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 % o 1.15 E CL 1.05 MoM T-m atrix 2 0 4 6 8 10 Frequency (GHz) Figure 2.6: 2-D MoM vs. 2-D T-m atrix (h — 1.2/xm, I = 2h) 3.5 |“ |^| -B - 1 = 0 .7 h (GHz) Figure 2.7: Power absorption ratio from 2-D MoM: Exponential surface (I — 2jim). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 Chapter 3 MODELING ABSORPTION OF RANDOM ROUGH INTERFACE BETWEEN CONDUCTOR AND DIELECTRIC MEDIUM IN TWO-DIMENSIONAL PROBLEM 3.1 In tro d u c tio n The underlying physics of modeling power absorption due to surface roughness is focused on the analysis of the interactions of electromagnetic waves w ith th e rough surface. Research by Sanderson [15] shows th a t the Rayleigh-Rice perturbation tech nique [13], [14] gives good results for periodic surface roughness when slopes are small to m oderate. Rytov [16] and Tsang et al. [52] further speculate th a t this may also be true for random or non-periodic roughness. In C hapter 2, we have ap plied Rayleigh-Rice perturbation technique of second order to a random rough surface in two-dimensional problem and derived a closed-form formula of power absorption enhancement factor [53]. Similar to M organ’s assumption, we enforced a constant magnetic field as the boundary condition and only solved th e fields in th e conductor as a one-medium problem. In this chapter, we extend the SPM2 approach to two-media studies including both the dielectric region and conductor region. The rough interface is modeled by the characteristics of RMS height, correlation length, and correlation function. The absorption is calculated by two methods: the analytic small p erturbation m ethod to second order (SPM2) and the numerical system transfer operator m atrix (T-m atrix) method. The results of absorption based on SPM2 are in term s of th e spectral density of the random rough surface. Furtherm ore, instead of assuming constant magnetic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 Z X Figure 3.1: A plane wave impinging on a rough surface w ith incident angle fields on th e rough interface, the new methods in this paper take into account both dielectric region and conductor region by considering a plane wave impinging on the interface. 3.2 T w o -M ed ia A n a ly tic S m all P e r tu r b a tio n M e th o d 3.2.1 Two-media Formulation Consider a random rough surface profile z = f (x), as shown in Figure 3.1. In a 2-D transverse magnetic (TM) problem, the magnetic field in the y direction is denoted as ip. Let ip and tp\ be the magnetic fields in the upper dielectric and lower conductor region, respectively. A plane wave w ith incident angle 6i is expressed as follows: Vw (r) = exp (ikixx - ikizz ) (3.1) where kiX — ksindi, kiZ — kcosdi, and k is the wave number in th e dielectric. Since the tangential electric and magnetic fields are continuous at th e boundary, namely, ip (f) = ip\ (r) and n-Vtp (r) = j-n-Vipi (r) where e and e-, are th e perm ittivity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 of the dielectric and conductor, we can define surface field unknowns as follows: a(x) = 'ip1 ( x ‘, fJ (Vx )/ )/ T 1 V \ji + ( = (3.2) £ ) ( f)w w (3-3) • Applying the extinction theorem to both dielectric and conductor regions [45] gives (?) + [ dx a ( x ) \j 1 + ( Js = n ■V g (r, r') - g (r , r') j ^ K x ) 0 for z ’ < J dx a(x)^J 1 + n ■Wgi (r, r') - gi (r, r') b(x) = for z! > (3.4) f{x') 0 (3.5) f(x') where g (F, r') and gi (r, f') are the plane wave representation of th e G reen’s function in the two regions: 9( r, r' ) = i i r°° ^~ dkx— exp (ikx( x ' - x) + ikz \ z ' - z\) 47T J —oo -oo POO 1 — / d k x - — exp ( i k x ( x f — x) + i k \ z \z' — z\ % 9i (r, r') I (3.6) — 47t J _ oc (3.7) k lz here kz — \ J k 2 — k 2, k\z = \ J k \ — k2, ki — 5 is the skin depth (5 = y j a is the conductivity of the conductor, g, is its magnetic permeability, and u = 2 irf is the angular frequency. We use a second order perturbation method, letting A (kx) and B (kx) be the Fourier transform s of the field unknowns a(x) and b(x). 3.2.2 Zeroth-order, First-order, and Second-order Solution Balancing (3.4) and (3.5) to second order after substituting (3.6) and (3.7) gives the solutions of zeroth, first and second order as follows. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 The zeroth-order solution is: A ^ ( k x) = A 06 ( k x - k ix) (3.8) B ® ( k x) = B 0S (kx - kix) (3.9) where a = ,kizE 2kT \ 1 T skizi B0 = - --------- kiz^i (3-10> '* T ( 3 .1 1 ) &k\zi and kiZi = \ / k \ - k 2x and 5 (kx) is the Dirac delta function. The first-order solution is: ^4(1) (kx) = M (kx) F (kx - kix) ( 3 .1 2 ) -B(1) (kx) = B i (kx) F (kx - kj^) ( 3 .1 3 ) where F (kx) is the Fourier transform of / (x) and A\ (kx) and B\ (kx) obey the fol lowing two simultaneous equations: - ikzA x (kx) + — B x (kx) = - k z2A 0 - kx (kx - kix) A 0 - — ikzB 0 £l ( 3 .1 4 ) £i ik\z M (kx) + B x (kx) = - k \ zA 0 - kx (kx - kix) A 0 + ik lzB 0. (3 .1 5 ) For the second-order solution, we only need to calculate th e ensemble average of A & and B (2>: ( A {2) (kx)) = A 2S(kx - kix) ( 3 .1 6 ) (B<V(kx)) = B 25(kx — kix) ( 3 .1 7 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 where A 2 and B 2 obey two simultaneous equations: -ikiZA 2 H (3.18) B2 — £1 7,3 _ /-OO -i^A 0 / dkxW ( k x - k ix) -OO f*QO kixkizA 0 I dhxVV (kx ^ix) ^ (kx k{x) J —00 oo / ^ dfcxw (fee-fete)-Ai (kx) •00 oo / _ dkx kx) (kx kix) A \ (kx) •00 k 2 ~ Z*00 + -^ fS 0 / f s . r°° ikiz I dkxW (kx £1 J —00 kix) B \ (kx) (3.19) ik\ZiA<i + B 2 = p _ foo i^fA o / dkxW (kx - kix) " J — OO / / b.2 / *1 zi OO d k x ^V (kx ' kix) ^ ( ^ r kix) •00 oo (kx kix) Ax (fcx) •00 -00 oo ^ roo f° + ' ^ B 0 dkx (kix dkxW kx) (kx — k(kx ^ ) kix) A \ (kx) J —00 •c ^ / 5.2.5 oo dkxW {kx ^i:r) -^1 (^x) • •00 Coherent and Incoherent Scattered Field The to tal scattered fields ips can be determined using the extinction theorem [54]: dx 1ps ( ? ) = f < a ( x ) \l 1 + ( df \ _ , x _, x e J n • V g (r,F) —g (r , r') ~ b ( x ) for z ' > f(x'). Js (3.20) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 To calculate absorption, we need to sum the contribution from coherent and in coherent scattered fields respectively. In the case of 2-D rough surface, th e coherent scattering occurs at zeroth order and second order while incoherent scattering occurs at first order. Using the aforementioned solution of surface fields gives the coherent solution to second order. The zeroth-order coherent scattered field is ^ 0) (kx) = (kx - kix) (3.21) where T(o) = k i z£i - e h z i ' s kiZe i + £kiZi The second-order coherent scattered field is < ^ f ) (kx) > = ^ 5 ( k x - k ix) (3.23) where $ 2) (3.24) k lh 2 ^o) 2 i f 7. 00 /•O O 1 ~ - k , — I dkxA i (kx) W (kix J —oo 1 f°° ~ I dkxA i (kx) W (kix Kiz J —oo 1~ -A 2 22 1£ 2 f°° £i J kx) kx) i (k%x kx) dkxB i (kx) W ( k ^ ~ kx) £i 2 The solution of first-order incoherent scattered field is ^ (kx) = (kx) F ( k x - kix) (3.25) where kzAo kx(kx kiX)Ao ikzA \ (kx) T ikz Bq El B \ (kx) £i c Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .26) 28 3.2.4 Power Absorption due to Coherent and Incoherent Scattering In the 2-D problem, power scattered per unit area in the z direction can be calculated by using Poynting’s theorem S s ■z = Re 2 cue — V (3.27) dz Since the to tal incident power per unit area in the —z direction is S inc • (—z) = | cos 6i, the emissivity of the lossy conductive medium is given by ' Z)coh _ ' z ) incoh ‘S'inc ( £■) S[nc ■( z') _ 2Re 2)* (3.28) } ~ k ^ 9 - f kk d k x W ] where the second and the th ird term represent coherent reflectivity and the last term is incoherent reflectivity. Note th a t as a result of th e analytic derivation, the limits of integration in (3.24) are from —oo to oo, whereas in the last term of (3.28) they are from —k to k. On the other hand, for a surface w ith length —^ to ^ in the x direction, the average power absorption can be calculated directly by using th e fields on the surface Pn = J d S ( S - h) -§ R e (3.29) dk^ [ IUEX J ^ , , (k'x - k x) 2 (A* (kx) B ( k ' x) ) \ . P u ttin g the solutions of the surface fields into (3.29) and dividing by the incident power cos 9i gives the absorptivity of th e lossy conductive medium: av 1 hiz£ i &k\zi (3.30) k iZ£ i + z k iz i i r°° —/ dkxA \ (kx) B x (kx) W (kx - kix) tUJSiT]] (cos 9i J ^ —Re —Re ( ( A ^ B q + A qB% { iu i£ iT ] COS 9 i \ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 Here rj is the wave impedance in the dielectric and the limits of integration are from —oo to oo. We will show ev and av in the following numerical results, proving th a t SPM2 obeys reciprocity and energy conservation. 3.2.5 Emissivity of Metal Surface Next we consider a special case where the lower medium is a highly conductive metal such as copper. On such a m etal surface, the RMS height h and correlation length I are in th e order of micrometers. The wave length k in the dielectric is in the order of cm - 1 at microwave frequencies. The incoherent reflectivity is negligible because it is in the order of O (k 3h2l) . Also note th a t k j |fci| is of order 10- 4 and e / |ei| is of order 10-8 . We then calculate the solution of surface fields to th e first order of k since k is much smaller th an k\. The zeroth-order solution remains the same as in (3.10)—(3.11). For first-order solution, approxim ating (3.14)-(3.15) by assuming kz = ^Jkf — kf « i |A;X| and kjiX <C \kx \ gives (3.31) B x (kx) = 2ki (-& i + k lz) (3.32) A pproximating (3.18)—(3.19) by using (3.31)-(3.32) and assuming W (k{X —kx) & W (kx) gives the second-order solution (3.33) (3.34) P u ttin g (3.31)-(3.34) into (3.24) gives the second-order scattered field Note th a t the first term in is purely imaginary. For th e integral f dkx, the integral limit of kx is of order l / l and is much larger th an k. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Next we calculate the emissivity to the first order of k. A pproxim ating (3.22) by assuming |fci| k sin 9t gives +> = k\ cos 6i —k k\ cos di + k (3.36) Approximating term s in (3.28) to the first order of k by assuming th a t cos 9i is not small gives kd cos 9i (3.37) R e ^ 2). (3.38) 2 2 R e { $ ° ) * $ 2>} = P u ttin g (3.37)-(3.38) into (3.28) gives 2 k 28 2R eipi2) (3.39) where 2kh2 , ■Re/ui A Re-0 ^ COS &i 2k c o s 9t / dkxW (kx) Refci*. (3.40) Here the integral in (3.40) is convergent because W (kx) Rek \z asym ptotically ap proaches as kx becomes large. So we can extend the limits of integration from —oo to oo. The emissivity of the m etal surface becomes 4k dkxW (kx) Refcix. cos 9i J,, — _O cQ F 2kd Akh2 ~ + cos 9i 6 cos 9i (3.41) Dividing (3.41) by (3.37) gives the power absorption ratio between rough surface and sm ooth surface, leading to {Pa, rough) P a,sm ooth __ 2 kS/ = 1+ 2 cos 9i h2 ^2 (3.42) 2 ~S J dkxW (kx) Refcx^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 N u m erica l A p p ro a ch U sing T -m a trix M e th o d To validate the SPM2 results, we use a numerical T -m atrix m ethod to com pute power absorption as well as magnetic field on the rough surface. We apply th e periodic boundary condition w ith the period L. This is a valid approxim ation in random rough surface scattering provided th a t the period contains many peaks and valleys and many correlation lengths, i.e., L 3> I [45]. Using the T -m atrix m ethod on (3.4) and (3.5) to formulate m atrix equations gives Aa + Bf3 = A\a = V (3.43) Bi/3. (3.44) In the above two equations, the m atrix elements are as follows, assuming So is the rough surface w ith length L in the x direction and x 0 is a point on So'. (3.45) kXm df 'xm ’x m ' (3.46) -5(m +iV m + l) ( m / +jVm + l ) /. dxo Gxp [ i (kxm kxm' kix) So ^kzmf ( x o)] , £ I /C; (3.47) 1 (m + iVm+ 1) (m f+ N m +1) kxm df 'xm 'x m 1 (3.48) 1(m+Nm+ 1)(m'+Nm+ 1) 'x m 'xm ' °iVm x l V= 2L Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.49) 32 where kxm — kiX ^ , kXm' — k;x j , kzm \ / kf kf^m, k\zm \/~k~i kxm, and m = —N m, —N m + 1 , . . . , N m - 1, N m, m ' = -iV m, -iV m + 1 , . . . , N m - 1, lVm. Surface fields can be further obtained after solving the unknowns a and /3: Nm a ( x 0) b( x 0) = ex p ( i k ixx 0) ^ a m exp (ikxmx 0) m=—ATm Nm = exp (ikiXXo) exp (zfcxmx0) . m=—Nm E (3.50) (3.51) Then we can compute the power absorption by the conductor, for a given width w in the y direction and length L in the x direction. Pa,rough = ^R e J ^ dxipl^J 1 + j n ■VV’i Ns = ^ R e S " b ia*Ax 2a L ^ Z= 1 where (3.52) (3.53) is the number of discretization elements on the surface. In the numerical implementation, we take L = 20/ , N m = 30, and the surface discretization is chosen as A x = min { ^ , R , ^ } . To calculate the average power absorption, we use a M onte-Carlo simulation approach. We generate a large number of realizations of rough profiles. Solving the T -m atrix equations we then calculate the absorption ratio for every realization and the average absorption is computed. For the simulation results shown in the next section, 600 realizations are used. 3.4 3-4-1 R e s u lts an d D iscussion Absorption by Copper Surface using SPM2 In the following examples, we assume a conductor with th e conductivity of pure copper (a = 5.8 x 107 S/m ) and a dielectric w ith a relative perm ittivity of 4.0. The SPM2 results are based on equations in Sections 3.3.2-3.3.4. They are independent of angle of incidence, as explained in Section 3.3.5; an arbitrary 6i < 90° can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 1.25 § 1.15 o e CL CL 1.05 Frequency (GHz) Figure 3.2: Power absorption ratio as a function of frequency from SPM2: Gaussian correlation function (h = lp m ) with varying correlation length. chosen. Figures 3.2-3.4 illustrate the results of power absorption ratio between rough surface and sm ooth surface. In Figure 3.2, the results are for a Gaussian correlation function w ith h = 1 pm. The correlation length I varies from 2.0 pm to 3.0 pm. We note th a t the absorption ratio increases w ith frequency. It also increases when the correlation length gets smaller. In Figure 3.3, the results are repeated for the case of h = 0.75 pm. The absorption ratios are smaller th an those of Figure 3.2 because a smaller RMS height gives a smoother surface. In Figure 3.4, th e results are illustrated for surfaces w ith exponential correlation functions exhibiting larger absorption than surfaces with Gaussian correlation function. The two-media results do not assume any artificial boundary condition and also dem onstrate th a t the absorption depends on all three of the roughness characteristics: RMS height, correlation length and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 1.15 J= ■os E aw 1.05 Frequency (G H z) Figure 3.3: Power absorption ratio: surface w ith Gaussian correlation function (h = 0.75/xm). correlation function. Note th a t th e power due to incoherent scattering vanishes in this case because the wave number k\ of the lower conductor is much greater than the wave number k of the upper dielectric. 3-4-2 SPM2 and T-matrix Comparison Figure 3.5 compares 2-D SPM2 results w ith T-m atrix results. The modeled surface profiles are Gaussian w ith h = 0.48 ^m and correlation length I = 1.5 pm, I = 2.0 pm, and I = 2.5 pm. respectively. The numerical T -m atrix results are in good agreement with the analytic SPM2 results for rough surfaces of small slope. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 0 2 3 4 5 6 7 8 9 10 Frequency (G H z) Figure 3.4: Power absorption ratio: surface w ith exponential correlation function (h = ljum). 3-4-3 Surface Fields by T-matrix The magnetic fields on the conductor surface are calculated numerically by the Tm atrix method, for each given rough surface realization. Figure 3.6 illustrates the magnitude of the magnetic surface field based on one rough surface profile. As nor malized by the incident field, the to tal magnetic field on th e surface is close to twice the value of the incident field due to scattering from a well conducting surface. The variation of the surface magnetic field is very small, which is consistent w ith M organ’s assumption th a t the surface field is constant. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 “ € h T-matrix: K 2 .5 |im SPM2: l=2.5|j,m -EH T-matrix: l=2.0|j.m -&■ SPM2: 1=2.O^m 1.08 “V * T-matrix: l=1.5n.m SPM2: 1=1 .Stun so rSnT--3^' 1.06 E CL » C 3l c. O CL 1.04 t— 1.02 Frequency (GHz) Figure 3.5: SPM2 versus T-m atrix: surface w ith Gaussian correlation function (h = 0.48pm). 3-4-4 SPM2 Two-media and One-medium Comparison In C hapter 2, we followed M organ’s assum ption by enforcing constant magnetic fields on the rough interface and applied SPM2 only in the conductor region. The power absorption ratio is given by the same closed-form formula as (3.42). Figure 3.7. compares the power absorption ratio using two-media SPM2 and th e formula based on M organ’s boundary condition. The absorption are illustrated for surfaces with Gaussian correlation function and exponential correlation function (h — 1 /mi and I = 2 pm). The results are in good agreement between the two methods. 3-4-5 Emissivity of Soil Surface using SPM2 Besides calculating power absorption ratio due to the rough interface between a di electric and a good conductor such as copper, the SPM2 can also be used to solve the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 E -3CD -0.5 ■5 -1 O CD M -20 -15 -10 X (fim) E 2.002 .998 .996 -20 -15 -10 X (^m) Figure 3.6: A surface profile and m agnitude of surface magnetic fields. emissivity of a less conductive medium, such as soil, in applications of remote sensing. Next we use a common soil profile w ith Gaussian correlation function (h = 2.4 cm and I — 12.0 cm). The frequency of the incident plane wave is 5.0 GHz. The relative perm ittivity of the soil is 15.57 + 3.71* and the upper region is air. The emissivity of the soil surface at different incident angles is illustrated in Figure 3.8. As expected, the emissivity gives the same results as when using the absorptivity formula. Unlike for th e copper region, th e wave number of the soil medium is com parable to th a t of the air medium. As a result, the emissivity has contribution from bo th coherent and incoherent fields scattered by the soil surface. Also note th a t the emissivity of the soil surface for the exponential correlation function does not exist using SPM2 because of divergent integrals. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 1.B 1.7 - /V Gaussian (Two-media) Gaussian (Morgan's boundary condition) Exponential (Two-media) Exponential (Morgan's boundary condition) 1.6 1.5 1.4 1.3 1.2 0 2 3 4 5 6 7 8 9 10 Frequency (G H z) Figure 3.7: Two-media SPM2 vs. one-medium SPM2: h = 1/xm and I = 2/j,m. 3.5 C onclusion We have applied the 2-D analytic small perturbation m ethod to second order and numerical T-m atrix m ethod to study the effects of a random rough surface on power absorption at microwave frequencies. The new methods take into account both di electric and lossy conductive media. The results show th e absorption depends on the characteristics of rough surfaces: RMS height, correlation length, and correlation function. Surface magnetic fields are also obtained numerically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 0.8 5 □ — Emissivity Absorptivity CO 0.75 0.7 0 .6 ' LU 0.6 Incident Angle (Degree) Figure 3.8: Emissivity and absorptivity versus incident angle: surface w ith Gaussian correlation function (h = 2.4cm, I = 12.0cm), / = 5.0GHz, and £i/eo = 15.57 + 3.71*. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Chapter 4 POW ER ABSORPTION OF RANDOM ROUGH SURFACE OF CONDUCTOR WITH CONSTANT MAGNETIC-FIELD BOUNDARY CONDITION IN THREE-DIMENSIONAL PROBLEM 4.1 In tro d u c tio n In C hapter 2 and C hapter 3, we have applied Rayleigh-Rice perturbation technique to study the rough surface effect on conductor loss in two-dimensional (2-D) problems where the surface height only varies in one horizontal direction. The practical topo logical features of the conductor surface on interconnect and microelectronic package, after artificially roughened, are most likely three-dimensional (3-D), i.e., th e surface height actually varies in both horizontal directions. In this chapter, we extend the SPM2 approach from a two-dimensional problem to a three-dimensional problem. Random rough surface is characterized by RMS height, correlation length, and correlation function. To simplify the formulation of 3-D SPM2, we start with M organ’s assumption th a t the magnetic field on th e surface is constant. Following this specific boundary condition, we derive analytically th e electric fields in the conductive medium up to the second order and calculate th e additional power absorption due to the surface roughness. The result of absorption based on SPM2 is in term s of the spectral density of the random rough surface and skin depth in the conductor at microwave frequencies. Results are compared w ith previous results of 2-D problem and the difference are discussed. It is also shown th a t enforcing constant magnetic field as the boundary condition is valid for 2-D problem, b u t not for 3-D problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 (a) (b) Figure 4.1: Visualization of rough surfaces in three-dimensional problem: (a) Gaus sian correlation function; (b) Exponential correlation function. 4.2 R a n d o m R ou gh Surface in 3-D P ro b le m For a 3-D problem of random rough surface, the height function / (x, y ) is treated as a stationary Gaussian random process. The two point ensemble average of the random process is ( / (®i, Vi) f (^ 2 , 2/2 )) = h2C ( |s i - x 2\ , \yi - 2/2 1) (4.1) where h2C (x , y) is the correlation function. Two common correlation functions are the Gaussian correlation function with C (x, y) = exp [ - (x 2 + y 2) /I 2] (4.2) and exponential correlation function with C (x, y) = exp a / z 2 + y2/ l } (4.3) where I is the correlation length. Similar to the 2-D case, th e 3-D exponential cor relation profile appears significantly rougher th an th a t for th e Gaussian correlation function as shown in Figure 4.1. In generating th e roughness profiles, we use the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 spectral density function W (kx , k v ) which is the Fourier transform of th e correlation function. The spectral density of the Gaussian correlation function is given by h 2l W ( kx , ky ) = — exp {kl + kl) l2 ( 4 .4 ) 47T and th a t of the exponential correlation function by h 2l W ( k X, k y ) ( 4 .5 ) [(kl + kl) E + l f 2' 4.3 3-D A n a ly tic S m all P e r tu r b a tio n M e th o d 4-3.1 Governing Equations The governing equations for z > f (x, y ) are J = dS' jjw /iiG i (f, f') - [ fi x Hi (r')] - y x Gi (f, f') - [ f i x E i (r')] } 0 ( 4 .6 ) where h is the normal unit vector pointing outward th e conductor surface and rji is the wave impedance inside the conductor. The dyadic G reen’s function for z > f (x, y ) is G i (r,r') JJ = d k xd k y { e x p ( - j k xx - j k yy - j k l z z) ( 4 .7 ) exp ( j k xx' + j k vy ’) exp ( j k u f (x', y ')) & ( k i z ) & ( k \ z ) H- ^ 1 (& 12:) h \ ( k \ z ) } and V x G i ( r , r ') J = d k xd k y { e x p ( - j k xx - j k yy - j k l z z) exp { j k xx' + j k vy') exp ( j k l z f (x', y' )) h k 1z - h ( k u ) ei (k iz) + ei (k iz) hi (k iz) } Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 4 .8 ) 43 where the TM polarization vector is e ( k u ) = e \ ( k u ) kxz ~t \ kxz and th e T E polarization vector is h i ( k u ) = e ( k i z ) x k. We further define the unknowns a (x', y') and b (xf, y') as tangential magnetic and electric fields on the surface dx'dy'a(x',y') = dS'ifch x H i(r ') (4.9) dx’dy'b (x', y') = dS' h x E \ ( f' ) . (4.10) P u ttin g (4.7)-(4.10) into (4.6) yields _ g^ 2 JJ dkxdky{exp ( - j k xx - j k vy - j k lzz) ■ exp ( j k xx' + j k yy') exp (j k lzf (x', y')) ■ [ Js' d x 'd y ' { j u — ^ — \je ( k l z ) e ( k u ) + j h i ( k l z ) hi ( k u ) } • a (x', y') k 1z = Vi k i z L -I - h i ( k u ) ei ( k u ) + &i ( k u ) hi ( k u ) • b (xf, ?/)}} (4.11) 0 where z > f (x,y). 4-3.2 Perturbation to Second Order Applying small perturbation theory to second order exp ( j k u f ( x \ y')) = 1 + j k u f (x1, y') - ^ gives ^ ’V ^ (4.12) d (x', y') = a (0) (x', y') + a (1) (x', y') + o(2) (x', y') (4.13) b ( x ', y ’) = b(0) (x',y') + 6 (1) (x',y') + 6 (2) ( x ' , y ' ) . (4.14) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 Also we represent surface height function f (x,y) and unknown functions a (a ■',y') and b (x', y') in spectrum domain as below F (kx, ky) = F {2)(kx, k y) = - j k xF ( k x, k y) = I d x ' d y ' e x p ( j k xx ' + j k yy') f (x',y') — !-2 (2 7 t) (4.15) J s’ [ dx'dy'exp ( jk xx ' + j k yy') f 2 (x',y') — ^ -2 (2ir) Js' — ^ -2 (2 7 t) J s’ [ d x 'd y ' e x p ( j k xx ' + j k yy ' ) d ^ X dx' (4.16) (4.17) (4.18) A a (kx, k y) = — I dx'dy'exp (jkxx ' + j k vy ' ) a x (x',y') (27r) J s ' A y ( k x,ky ) = — !-2 ds'dy'exp^s'+j^y')®!/^!^) (27tJ J s ' f (4.19) A z (kx, k y) = —^ 2 (27 t ) [ dx'dy'exp (jkxx ' + j k yy ' ) a z (x',y') (4.20) JS ' and B r (kx, k y) = t ~ ~ 2 / dz'dy'exp(jA ;x£' + B„ —i = B z (kx, ky) = (27 t) -2 (4.21) Js" [ dx'dy'exp (j k xx ' + j k yy')by (x’,y') (4.22) [ dx'dy'exp (j k xx ' + j k yy')bz ( x ' , y ' ) . (4.23) (2ir) J s' — !-2 (2 tt) J s' P u ttin g (4.12)-(4.23) into the governing equation (4.11), we obtain e (klz) e (klz) + hi (kiz) hi (kXz) ■ A (kx, ky) ~f“ j k lz f f dk'xdk'yF (kx - k'x, ky - k'y) A (k'x, k'y) dk'xdk'yF W (kx - K , ky - k'y) A (k'x, k'y) hi (Aq2) &i (kiA + §i (k\z) hi (ku) B (kx, ky) + j k U f f dk ' jF y F (kX ~ Fx, ky - k'y) B (k'x, Fy) ~ F2lz / f d k ' j k ' y F ^ ( ^ - Fx , ky ~ k'^ B ( K , k^) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.24) 45 Here A ( kx . ky) and B (kx , ky) can also be expanded to the second order: A ( k x ,ky) = A ^ ( k x ,k y) + A ^ ( k x ,ky) + A ^ ( k x , k y) (4.25) B ( k x, k y) = B {0) ( k x , k y ) + B ^ ( k x , k y ) + B ^ ( k x , k y ) . (4.26) Note th a t once A (kx , ky) is determined by specifying boundary condition, B (kx , ky) can be solved by balancing term s to different orders in (4.24). 4-3.3 Balancing to Second Order In this 3-D problem, we follow Dirichlet-type boundary condition as in 2-D study, which specifies H\ = yHo on the surface. Then a (x y r) and A (kx , k y) can be deter mined as d ^ { x ’,y') = - VlH 0x (4.27) a (1) (a/,y') = -rji^H o Z (4.28) a W ( x ’,y') = 0 (4.29) and H(0) (kx, ky) = -rjxHoS (kx) 5 (ky) x (4.30) A {1)(kx, k y) = rjiH0j k xF (kx, k y) z (4.31) A ^ ( k x, k y) = 0. (4.32) Because n • b(x', y') = 0, we have one m = 0,1,2 more set of relationship for B ^ (k±) when : B^(k±) = 0 B i m) { k ± ) = - j (4.33) J dk'± { k± - k'±) • F ( k ± - k'± ) B r{ l) (fc±) • (4-34) To solve for B ^ (kx, k y), B ^ (kx, k y), and B ^ (kx, k y), it is more convenient to define new coordinate system (p , q , z ), where p and q are the unit vectors th a t are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 parallel to the projections of e and h on the horizontal plane: q = p e (klz) = x k y - ykx (4.35) kp x k x + yk y z x q— kn = (4.36) Balancing (4.24) to zeroth order, we obtain D(o )/,. - k i k yr ) i H 08 ( k x ) Dv [Kx,Ky) — d (o) 1 °q A/1znp _ ;„ \ \Kx, Ky) ~ 8 (ky) kukxTjiHoS (kx) 8 (ky) ^ (4.38) (4.39) = 0. B f \ k x ,ky) (4.37) Balancing (4.24) to first order, we obtain B ' P ( K , k y) = k\ k u kp h •®2 ^ (^X) k y ) 1 k lz 1 kx - j kyPi HoF ( k:r. ky) k Xz j kxPi HoF ( k x , ky) kx (4.40) (4.41) (4.42) j k y T j x H o F ( k X 1k y ) • Balancing (4.24) to second order, we obtain B W ( k x,ky) = (4.43) — -rjxHoGx (kx, ky) klz ■kxr)xH0G 3 (kx, k y) kykxz Ck x - k l z) F ( k x, ky) 2 kn (4.44) ■ ^ V i H o G x (kx, k y) Bq2^ (kx, ky) = /Cl +kxzpxH0G2 (kx, ky) ( k\z kx kx V 2 B ^ ( k x, k v) = kp kx (4.45) pxH qG 4 (kx, k y) where Gx (kx, ky) = I J d k 'x d k 'y F ( k x - k'x , k v - k'y ) k'x F (k 'x , k'y ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.46) 47 JJ (kx-)ky) ( (4.47) ! ^ x \ kyk>x kxk'y l J + fcW j J dk'xdk'yF (k, k f hy k t^x) + ¥ t ( K lz + (t« > ( K - K ) ( * t - ki fc'fcl U + (\ K ([llz ( ki - K ) F kyk'y+kxk'x KKr. V fci G$ {k>X) ky) k'x i ky dk'xdk'yF (kx ^ (4.48) K ) f (K ,K ) -fcu ^1^p^p (kxky h . \ , V $ l\ - k u K 0 + k’l z ) k^kpk’p kykx) (kxkx + kyky) +k'y ^k\ Jj G 4 (kx 5 ky dk'Jk'yF ( k , - k ' „ l : „ - k's) F (k',, (4.49) (k, - K ) ■ | ( t + !& « » -* ■ )) +■ | ( I - *i) + ^ ) . + (fcy —fcy) • -<4 (t ? + 1c 'K « » - *o) Ky 4-3.4 *i, J Power Dissipation To Second Order After solving A (0) (kx, ky), A (1) (kx, ky), A ^ ( k x, k y), 5 (0) (kx, ky), 5 (1) (kx, ky), and 5 (2) (kx, k y), we can calculate power dissipation on the conductor to second order using Poynting’s theorem: Pn >/ dS- 1 1 + (g r - w a {/ / [ m ( K , ^y ) + flW {K, k'y) + ^ d k 'J k y exp ( ~ j k ' xx - jk'yy ) ■ ( K , k'y)]}- J / / d^dfc^exp (jk'xx + j^ y ) • [AW* ( K , K ) + A ^ * (k'x, k'y) + A<2)* (k'x , k'y)] x n}. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.50) 48 Taking the ensemble average and using the property of (F{k\x, k \y )F (/u2xj ^ 2jy)) — S {h\x k2xjkiy ^2y) ^ly)i (4-51) we then obtain the power absorption by the rough surface after simplifying the results (4.52) Notice th a t the first term of (-Pa)roUgh) corresponds to the power dissipation on the smooth surface (Pa,sm ooth)- Thus the absorption enhancement factor is obtained by taking the ratio of ( P a ,r0Ugh) and (Pa,smoo th ): (4.53) a.sm ooth 4.4 R e s u lts an d D iscu ssion In the discussion below, we refer to the 3-D SPM2 problem w ith a constant surface magnetic field as a one-medium 3-D SPM2 problem. Figure 4.2 illustrates the onemedium 3-D SPM2 results of power absorption ratio for surfaces w ith Gaussian and exponential correlation function (h = 0.5fim, I = 2h and h = 1 /im. I = h). Same as seen in the previous 2-D SPM2 results in C hapter 2 and C hapter 3, power ab sorption is higher for surface with exponential correlation function th a n th a t with Gaussian correlation function because the exponential correlation function appears much rougher th an the counterpart. Figures 4.3 and 4.4 illustrate the comparison between 2-D SPM2 and 3-D SPM2 for surface profiles with Gaussian and exponential correlation function (h = 0.5jim, I = 2h and h = 1/rni, I = h). Comparing 2-D and 3-D results, we can find the following noticeable differences: one-medium 3-D SPM2 results show less power absorption ratio th an the 2-D SPM2. For surface w ith expo nential correlation function, the absorption given by 3-D SPM2 is slightly less than the 2-D SPM2 results. For Gaussian correlation function, such difference becomes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 ~©~ Gaussian h=0.5um 1=1 um 2.6 2.4 - 0 “ Exponential h=0.5um 1=1 um - E h Gaussian h=1um 1=1 um - A - Exponential h=1um 1=1 um 2.2 J= ■o5 £ CL CL Frequency (G H z) Figure 4.2: 3-D SPM2: Gaussian correlation function versus exponential correlation function. more significant. The other main difference between 2-D and 3-D results is for rough surface w ith Gaussian correlation function, the absorption ratio reaches a maximum value at certain frequency and decreases slowly beyond th a t frequency. However, such differences between one-medium 3-D SPM2 results and 2-D SPM2 results hardly fit into any reasonable explanation. Physically, the surface area in th e 3-D problem is larger th an in the 2-D problem and thus the power absorption is expected to be greater for 3-D case as well. Furtherm ore, as the microwave frequency moves higher, the surface current stays closer to the surface. For Gaussian correlation function, the surface area is finite (no fine multiscale features) and the absorption is expected to increase slowly w ith frequency till saturation at some point instead of exhibiting the slow decay after reaching the maximum value as shown in Figures 4.3 and 4.4. The contradiction between physics and one-medium 3-D SPM2 results indicates the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 1.9 1.8 - a - 2D SPM 2 . - B - 2D SPM 2 3D SPM 2 - © - 3D SPM 2 h=0.5um 1=1 um h=1um 1=1 um h=0.5um 1=1 um h=1um 1=1 um ............. -0 -3 " ; j i : : i i i .a --0 1.7 1. 6 J* _C ■5 ex' 1 a. w 1.5 f o 50*r / 1.4 j g I 1.3 _____A if T :S :? :7 € L « -i W ' _________ . 1.2 / 1.1 1 0 ------ 6 8 . r-i n r i d — Fl !_ © — 0 , o 10 12 14 o <fa i I i 16 18 20 Frequency (GHz) Figure 4.3: 3-D SPM2 versus 2-D SPM2 for Gaussian correlation function. original boundary condition may not be valid for th e 3-D problem. In other words, the constant magnetic field boundary condition fails in the 3-D problem, although it is found to work well in the 2-D problem. We will show in th e next chapter th a t the correct formulation to tackle the 3-D problem is to take into account both dielectric and conductive media. Only after applying 3-D SPM2 to two-media case can we get results which agree with the physics of th e problem. 4.5 C onclusion We have extended the analytic small perturbation m ethod to second order in a threedimensional problem where the surface height varies in both horizontal directions. To simplify the formulation, we follow M organ’s assum ption and apply a constant magnetic field as the boundary condition such th a t we only need to focus on the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 - B - 2 D S P M 2 h=0.5um 1=1 um 2.6 - B - 2 D S P M 2 h=1um 1=1 um - e - 3 D S P M 2 h=0.5um 1=1 um ...... 3 D S P M 2 h=1um 1=1 um 2.4 .* 3 P " I 2.2 .C ■ 5 o E « sz cn 3O CL CL JET Frequency (GHz) Figure 4.4: 3-D SPM2 versus 2-D SPM2 for exponential correlation function. conductor region. The results of power absorption enhancement factor are similar to the two-dimensional problem, b u t the differences are inconsistent w ith th e physics which indicates M organ’s assumption fails in three-dimensional problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 Chapter 5 MODELING ABSORPTION OF RANDOM ROUGH INTERFACE BETWEEN DIELECTRIC AND CONDUCTIVE MEDIUM IN THREE-DIMENSIONAL PROBLEM 5.1 In tro d u c tio n The Rayleigh-Rice technique [13], [14] is a well-known approach to study wave scatter ing by rough surfaces. This m ethod is based on perturbation theories and it assumes th a t th e height of the surface roughness is small compared to a wavelength and also th a t th e slope of the roughness is small. In C hapter 2 and C hapter 3, we have applied Rayleigh-Rice small perturbation m ethod of second order (SPM2) to two-dimensional problems and derived a closed-form formula of power absorption enhancement factor, also found in [53], [55], in which the 2-D SPM2 formula utilizes a model of random rough surfaces. All the previous results in [53] and [55] were based on 2-D problems with the assumption th a t the surface height varies only in one horizontal direction. In real ity typical surface roughness is most likely to have three-dimensional (3-D) random profiles. In C hapter 4, we made an initial attem p t to solve a 3-D problem using Mor gan’s constant magnetic field boundary condition. In this chapter, we include both dielectric and conductive medium in the formulation and develop a model of power absorption for a three-dimensional problem where the surface height varies in both horizontal directions. A random rough surface model is used to characterize the phys ical interface between the two media. The characteristics include root mean square (RMS) height, correlation length, and correlation function. The effects of surface Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 roughness on power absorption are then analyzed by considering incident plane wave impinging on the interface. The absorption is calculated by two methods: the ana lytic small perturbation m ethod to second order and the numerical system transfer operator m atrix (T-m atrix) method. The results of absorption based on 3-D SPM2 are in term s of the spectral density of the random rough surface. We further compare the three-dimensional results to the previous two-dimensional results and show sig nificant difference. The power absorption enhancement factor exhibits saturation for the Gaussian correlation function, b u t not for the exponential correlation function. 5.2 D eriv a tio n o f 3-D S P M 2 Form ula In this section, we derive a closed-form 3-D SPM2 formula of power absorption en hancement factor due to rough interface between a dielectric and a highly conductive medium. This work is an extension of previous work, in which we studied wave scat tering at the interface between two dielectric regions for rem ote sensing applications [27]. The simplification here takes advantage of an assum ption th a t th e conductivity of lower conducting medium is much greater th an th a t of th e dielectric medium. In typical dielectric-metal layers occurring on interconnect and package, th e magnitude of wave number in the conductor is a few thousand times larger th a n th a t in the dielectric medium at microwave frequencies. 5.2.1 Emissivity and Absorptivity of a Dielectric-Conductor Interface In a two-media problem as shown in Figure 5.1, let £ and £\ denote the perm ittivity of th e upper and lower half space, respectively. Consider a plane electromagnetic wave E i = e.Lexp ( i klxx + i k iyy —i ki Zz ) incident upon the interface w ith an incident altitudinal angle 6i and azim uthal angle (\>t. Here we use exp ( —iu>t) as the phasor notation. Note th a t kiX = k sin Q%cos </>*, kiy — k sin 0,; sin 0 ,, k, z = k cos (9*, kpi — k sin $i and k lzt = y j k f — k?pi where k and k \ are the wave numbers in the upper and lower media. C refers to the unit polarization vector of the incident wave. The rough Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 4 '*** k" Figure 5.1: A three-dimensional random rough interface between dielectric and con ductor. interface is characterized by a random height function z = / (x,y), where f (x, y) is a random function w ith zero mean ( f (x,y)) — 0. Define wavevectors k t± = klxx + kiyy, k± — kxx + kyy, dk± = dkxdky and let kp = ^ k 2. + k 2, kz = ^Jk2 —k 2, k\z = ^Jk2 — k 2. The emissivity of a vertically polarized (TM) wave and a horizontally polarized (TE) wave, respectively, in the direction of (0*, 7r + A,) is given by applying small perturbation m ethod to second order [27]: \Rvo\ —2 Re \ K v OJh h /*7T/2 I rn/2 — / ddk sin 0k / lS0i Jo Jo 1 - |i4 o | 2 l cos 6 i d(f)kk2 cos2 6k f S (*x) ‘ + ff (tit) (*x) w ( U - fex) eh = (5.1) j 2 Re { i W e (e2)*} /»7t/2 /*27T r*l2 / d0k sin 0k / Id(f>kk cos 9k J o Jo Jo •W (k± - h i ) I/ i ! ( u ) f + £ '( i x ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.2) 55 where the second and third term s on the right hand side represent the reflectivity of power due to coherentscattering and th e fourth of power due term represents th e reflectivity to incoherent scattering. Here W (k±) is the spectral density function of the rough surface, R vo and Rho are the Fresnel reflection coefficients of the TM and f j ^ are the first-order and second-order and T E wave, scattering coefficients: £\kiz &k\zi . R vO — — ;-------- ;— j— £\kiz + sk.1 zi rn%z rT rn*lzi1 = i) (T \ feh i k±) f (1 ) ( k s ) V -L f Jhh = k\ k fc j - k 2 I 1.21, = b.2b. t v \tv z { (5-4) 2 k k iz k \ zi +fci k2u , + \ n -lz rv h ,\z l T k ft,z ' . Sm k2k . r v \l^ iz , . - <f>i) 2 k 2k iz p i,. l z 'V \" JIZ . (5 -3 ) , 4_ r . 2 I . I ^ .i (5‘5) . V ° '° ; . •vl z i k2 - k \ z k \ z i COS ( jp k / / 4*i) OO (fc± - fci±) k l k k l •OO OO dfcxfE (fcx - fcix) •O O ■[-kkLi^ + ^ " sin 2 ( 0 fc - <fc) + Pfc Tfc~ fcT ^ A> 2 A '1 2 (^Ay ~ ^ - 2 k p i k pk k l ( k u + k z ) k izi cos ((j>k - <£*) - k k z k Xz ( k \ - k 2) k j zi cos2 (0fc - <&))]} fee = f a + \ \ k . 2+ l . C0S ^ 1 2 ^ ^ 2 / ^22 d ^122 ^(i) _ (fci ~~ k^) k\ z k fh e ~ T 2k iz - &) . . I Ap, k/ i l 2 fht ,.j 2 4 . AK/ l 2 l. S l n ( ^ fc (5-8) . ^*) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. , . 56 /i? (M = ? / (5-10) OO d k ± W (k± - ki±) •OO / OO d k x W (kL - k x ) •OO •[sin2 (0 * - fa) k 2 k k^ 2 k A/l hjZ ~r A/ + cog2 (& - &) fci I p }”' 1,2 I z Next we consider the case of our interest where the lower medium is a highly conduc tive m etal such as copper. On such a m etal surface, th e RMS height h and correlation length I are in the order of micrometers. The wave number k in th e dielectric is typi cally in the order of cm-1 . Also note th a t k / \ki | is of order 10-4 , and e / |ei| is of order 10-8 . The power absorption due to incoherent and coherent scattering is analyzed in the following subsections. 5.2.2 Reflection due to Incoherent Scattering Let R^ncoh and i?[|lcoh denote the fourth term s on right hand side of (5.1) and (5.2), representing the respective incoherent reflectivity of the TM and T E wave. Note th a t kx = k sin 6k cos (pk. ky = k: sin 0k sin (f)k and kz = k cos dk- Because of th e integration limits of d6k d(pk) kx and ky are in the same order of k, therefore (kx —kix) I 1 and (ky — k y) ( < 1. Consider a random rough surface w ith Gaussian correlation function for the term W (kx — k x ) in R[[lcoh and Rfncoh- Then th e spectral density can be approxim ated as r , hH2 I - [(h - fe )2 + (t, - h , f ] p \ W ( t 1 - f c u.) = _ e Xp | - l ---------------- j ---------------- h*P (5'n ) The following approximations are also valid for incoherent scattering, kx <C k , ky <S k , k z — y j k \ — k — « k , kzi ~ k - Here we take k / k , ( k / k ) 2 as first-order and second-order smallness, respectively. Approximating Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / “ (*x) 57 2 fhh (^-l) 2 ) fhe (*x) 2 an<^> fee (^ -0 to first order, we obtain: k2 2k cos 0i . . k\ + k cos 9k k + ki cos 6i S k f (i) J eh . 1 (5.12) fc?______ 2 k cos Oj »k\V!cos w Ok+k K| >v1fe+fei 1 | rvj. m cos( uo<J% M) Jhh m 2kcos$i fc? f(l) J he . .. 1--------a— + T Tk i k ----------a , s in fci cos9k cos9i +■ fci 2 (5.13) ■[ - cos (<f>k — (pi) + sin 6k sin 9t ^ .. _ & ) 2k cos 9i cos ((f>k - fa) (ki + k cos 9k k cos 0 * + kiL (5.14) (5.15) 27r d<pk yields P u ttin g (5.11)—(5.15) into i?”ncoh and i?||lcoh and integrating over JQ 'incoh k4 fi2£ 2 2 h cos 9j cos 9i 4 A; + fci cos 0* />tt/2 d9k sin 9k cos2 $*,{ to + r>h ■^incoh h ki cos 9k + k k Ah2e COS 9 i ’ k\ cos 9i k cos 9i + k\ 2 h ki + k cos 9k In (5.16) and (5.17), note th a t [l + 2ki cos 0i k + k 1 cos 0i + 2 2 2 (5.16) k\ —:------ ww + k cos 9k) 71— sin 2 9k sin 2 9f\ }. />7r/ 2 L d9k sin 9k cos2 9k ■ h k\ cos 9k + k k\ COS $i k cos Qi+k 1 (5.17) 2 and integral J ^ 2 d9k are all in the order of 1. Therefore, _R"ncoh and 7?fncoh are of order O (k4fi2£2) and is negligible since h and I are in the order of micrometers whereas k is in th e order of cm-1 . For a random rough surface with exponential correlation function, th e spectral density can be approxim ated as h2l2/ (2ir) similar to (5.6). Repeating the steps above also leads to the incoherent reflectivity of order O (k4h2£2). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 5.2.3 Absorption due to Coherent Scattering Because the incoherent reflectivity is negligible in case th a t k h , k l -C 1, the power absorption of the rough interface is solely determ ined by coherent scattering waves. Unlike incoherent scattering, coherent scattering covers evanescent waves. Therefore, the scattering coefficients and in (5.7) and (5.10) have the limits of integration from —oo to oo for kx and ky. To evaluate th e integration of fj jj and assume th a t kp = 0 ( 1 / 1 ) , we k, kz = y j k 2 —k 2p « ikp, k\Zi = y j k \ — k^ « k\ and k\z = y j k j - k p2. Taking k / k \ , (k/k{) as first-order and second-order smallness, we obtain, to first order, fcicos 6 j - k k\ cos 6i + k v0 k cos Ot —k\ - /tco sft + fc, POO n r , ® = -T^fT _ (k± ) [h - k u / C ( J b Oi J _ (X) +ikp(sin 2 (4>k - / (5' 19) 0 (5.20) j) - sin 2 0 *)]} OO d k ± { W (k± ) [h + kiz (5.21) •OO +ikp cos2 (<j)k - (()i)]}. Further, to first order of k w ith (5.18) and (5.19), we obtain 1 - \Rvo\2 = COS Vi l - \ R ho\2 = 2kdcos0i. Note th a t ki = (1 + 8 —y j 1/ (itfnocr). i) (5.22) (5.23) /8 where 8 denotes th e skin depth in the conductor with Here / isthe frequency, and /i 0 and a are th e perm eability and conductivity of the conductive medium, respectively. Combining (5.18) and (5.20), (5.19) and (5.21), approxim ating them to first order and removing im aginary terms, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 we obtain (5.24) 2 R e { ^ 0 f f l* } 4k - r dk A w {k±) cos 6 ii J -o o I Re (klz) -2Re {R h o fi? * } = 4k cos Bi Os f (5.25) dkx (k±) Re (klz) Substituting (5.22) and (5.24) into (5.1), (5.23) and (5.25) into (5.2), and making use of the property h 2 — f™ d k± W ( k x ) , we then obtain the absorptivity of the TM and TE wave due to coherent scattering: 2k5 4k + cos 9i cos 9i OO h2 r°° d k x W (k±) Re(fci*)] ■f— 15 rough / •O O 2k5 cos 9i + 4 k cos 0, = rough h2 r— 1 5.2.4 (5.26) (5.27) d k x W (k±) Re(fcu)]. <5 3-D Power Absorption Enhancement Factor The power absorption enhancement factor is a ratio of the power loss dissipated in a conductor with a rough surface compared to th a t dissipated in th e same conductor with a sm ooth boundary. For a smooth surface, th e power absorptivity of the TM and T E wave are given by (5.22) and (5.23). Thus, the enhancement factor can be obtained by taking th e ratio of a'('otlgh and 2k 5 / cos 9t, or a('ough and 2k5 cos 9p. { P a , rough) rough rough 2k5/ cos 9i p ,a,sm ooth = h2 82 2 2 1 2 ~ kS cos 9i f°° [° *u (5.28) 2i dkxdky <( W (kx, ky) Re \ j — - k 2x - k% > . o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 The 3-D formula of absorption enhancement factor in (5.28) has th e same form for the TM and T E wave and is independent of angle of incidence; an arbitrary Oi < 7t / 2 can be chosen. Please note th a t, intuitively, the application to guided waves on quasi-TEM interconnect structures and loss prediction in planar circuits corresponds to the TM case as 6 , —> ir/2. Similar to the 2-D closed-form formula of power absorption enhancement factor th a t we derived in [53] and [55] based on small perturbation method, (5.28) also depends on the RMS height and spectral density of th e rough surface, as well as the skin depth in the conductor. T he integrand of (5.28) asym ptotically approaches PE (kx, k y) / (52y j k l + k%) as kx and ky become large. Thus, the integral is convergent for both Gaussian and exponential correlation function. If we make the rough surface profile uniform in th e y direction to reduce the three-dimensional problem to a two-dimensional problem, then th e spectral density W (k±) becomes PE (kx) 5 ( k y), where 5 (ky) is a Dirac delta function. Substituting PE (kx) S (ky) into (5.28) leads back to th e 2-D formula (2.13). Also notice unlike two-dimensional surface, M organ’s assum ption of a constant surface magnetic field is no longer valid for three-dimensional rough interface. It is necessary to take into account both dielectric and conductor media to analyze the field scattering on the interface. In C hapter 4, we show th a t applying 3-D small perturbation m ethod of second order by forcing constant magnetic field Hoy on the interface leads to an erroneous form of enhancement factor where Re {k\z} in (5.28) is replaced by Re [ k \z + ky/ k i z }. 5.3 N u m erica l A p p ro a ch U sing T -m a tr ix M e th o d In this section we apply numerical T -m atrix m ethod to solve for the power absorption of the rough interface. Note th a t we now use exp (juit) as phasor notation. Let S denote th e rough interface, y and rji are the wave impedance of dielectric and conductor, respectively. Consider a normal incident plane wave E inc — yHoX exp(jkz) and E i, H i are the electric and magnetic field in conductor. Next we define surface Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 field unknowns as follows: dr±a(r±) = dSrjin x H \ ( f ) (5.29) dr±b (Fj_) = d S n x E \ (r) (5.30) where F = (x, y, f (x, y)), r± = (x,y) and n is the unit normal vector pointing into conductor. To apply the numerical T -m atrix method, we use the periodic boundary condition with the period L x and L y in the x and y direction. This is a valid approximation in random rough surface scattering provided th a t th e period contains a number of correlation lengths. Define S mn as the (m, n )-th patch of the infinite periodic surface, where m, n are integers from —oo to oo. Let F0o = (xo, yo, f (x0, yo)) denote a point a and b on 5oo-The periodic boundary condition indicates th a t th e field unknowns are also periodic and thus can be represented by the following Fourier series: a(x,y,z) (^ 0 ) Vo) (5.31) OO ^ ^ ^(x,y,z)m'n' OXp ( j k xrn'Xo m',nf=—oo jkyn'Uo) b(x,y,z) (xo,yo) (5.32) oo ^ ^ P(x^y,z)m'n' 6 Xp ( j k xm'Xo m' ,n'=—oc where kxm 2e7trm / L x , kyn — I‘ ttti/ L y, kpmn — \J^xm "F jk y nryo) &ftd kzrnn — k ^ k^mn. Q-xm'n' 5 ^ym'n'i &zm'n1 nnd $xm,n,'j ftym'n'i Pzm'n' nre Fourier series coefficients of un known surface fields a and b. The subscript (x , ?/, z) refers to th e three dimensional components of a, b and a , (3. To solve for these unknown coefficients, we need six m atrix equations in total. From Huygen’s principle and the extinction theorem for electrical fields in the two media [27], [54], we firstobtain four m atrix equations as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 follows: k •^yn \ ^ / i ^ v ^ Ot-xm'n' -f*-mnm!n' '~ x iiti t ■ *^ i n i i i i i n b , ^a;m , , r x ” , m ^ ^ ~t" * ™pmn r , OO b .v “ m 'r>/— , /? i * (5.33) H y m 'n ’-^-n , 77i ,77 = — o o J^ L b b b ^ 1 ^ z m n rhp m n r O X E OO . fcpm n \T XT-"1 / v K>zmn , , m ,7i ——oo ^n ~A by~ l 'bvpmn ^ b b ^A_ 2 _ A_ 2 ^ 7A 7^ 7 _ ^ / b, b7 b7 *4 rhz m n ™p v ^ H ym 'n '-^r] /n ' (5.35) , , m ,77 ——oo A *i zm.n.iviin. k '*'1 ! ^lzmn^yn X“ bn A ^pmn ^ D ^ / v Q y m 'n ' -Brnnm 'n' 777, ,71, ——oo iT~L • ^ l^ p m n r OO J kyn / 7 v / f /O D rH' xxim n 'n n ’ -^11111111 J->m n m fnii ' v / D Qtzm'n' -Bp m ' , n ——oo oo ~t~ i 7, pTYlTl / v , , ^1 , ^xm 771 ,71 = — OO ^ T J V- r 4 00 ™ 1 zm.n,™: ^lzm n^iCTTi p77X7X b _ Pxm'n'■£*-mnm,n ^ I^XIIl II lllllllt ' , , m ' n ——oc n (5.34) Q;a:m/n/ -^-77i.7i77i/7i/ , 7 ‘ ljy T l b r. V / 7t, 771 ,7 7 — — o o V -^ 4 / v Q ym 'n'■ ^■ m nm 'n' ™pTnn , , r m ,n ——oo A 7 jl ™r 077777™! ,,77 ,— — o o I T 7 b ^ ^Lym,n , -^r\ oo ™1 A ^ ^ brvzm.rt.r^it b 3:77171 -l , 2^7-ffo^mO^nO-^^-^T/ t , . 1Hill H x m 'nAt' -^ ■1m n mL'nII' .' m ',n — — o o k^pmn k\ r^ a E m 'r>'=— rv~i ^pmn^l / ™pmn , , r m ,n ——oo ™pmn , , r m ,n ——oo ^ jz m n ryjx b m rX \ ™A ”T ' T 7 r / § /O ^ D i-'ym 'n ' ^ m n m 'r / 771 ,71 — — OO and ^ 7 /7 1 ^ ZD 1 // v^ ^C^xm'n' x n i ii ■ ^mnm'n' ^■‘i niiiii II fcpmn , , r m , n ' — — OO V -^ k lz m n k x m 1 7 fclfcpmn ^ E / ^r ,, 777 , 77 — — 0 0 r ZD (5.36) 73 ^m rP- x 'xm n v'n 11' ■ ^ 'i n i i im i i i'n i 11' - l k^ \l zzrn m nn k^ yy nn ,7 ,7 i\ K'lfcpmn r V / ^ , 777 ,77 — — OO /? r H ym 'n* ■ Dn °° b mn rx’p k\ ^ XTYl T7 / v Qtym’n' -£*77 ftpmn ,~, r 7 7 7 , TV — — o o ^ ^ $ z m fn f B p m ' ,n '= —oo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 where / k \zm n \ 1 . ^p m n 5 ^TtiO 1 (m = 0) \ and ■A-rnnm'n' j B m n m 'n ' are given by 0 (m ^ 0) (5.37) A m n m 'n ' dxodyo exp [j (kxm - kxm>) x 0 ^S qq ~^~3 {J^yn kyn'^j Hq j k zmn f ( xo, yo) \ and (5.38) ^ Soo d x dy exp[j (kxm 0 0 ~^~3 ij^yn kxm>) x 0 kyn' ) yo + j h zmn f ( x 0, y 0)]- From (5.29) and (5.30), we have n • a (r±) = n - b (r ± ) = 0 which is equivalent to a z (r±) = bz (r.l) = j9 / ( F L) , X- & r (fj_) + y - dy 9 J (F±) + g d f (r± ) xdx " dy a± (r ± ) (5.39) b± (F j .). (5.40) Taking Fourier series expansion on both sides of (5.39) and (5.40) gives the last two equations (5.41) CXz L XLy m —m n —77= —oo ^ Q-ym'n' ^ ( j k x(rn—777' ) ) 777 — 777, 77'—77= —OO (5.42) & L XL y Pxm'n’ ( j k X( 777—m' ) ) F(m—777/ ) (77 — 77/ ) { 777 — 777, 7 7 '— 7 7 = — OO + 777E777 Pym'n' ( j kX(777—m' ))F{m—m')(n~n')} — , 77'—77=—00 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 where Fmn is the Fourier series coefficients of the surface height function defined by Fmn exp ( - j k xmx - j k yny ) Thus we have six m atrix equations (5.33)—(5.36), (5.41) and (5.42) to solve for six unknowns a x, a y, a z, i3x, (3y, and [3Z, which in tu rn gives th e surface fields a and b from (5.31) and (5.32). Then we can compute the power absorption by the conductor for a rough surface with length of L x and L y in the x and y direction (5.44) In the numerical implementation, we take L x = L y = 51 and \ m \ , \n\ , \m' \ , \n'\ < 3 in the six m atrix equations. The surface discretization is chosen as A x = A y = 0.055/ such th a t \m \ , |n| < 45 in (5.43). To calculate the average power absorption, we use a M onte-Carlo simulation approach. We generate a large num ber of realizations of three-dimensional profiles. Solving the T-m atrix equations we then calculate the absorption ratio for every realization and the average absorption is computed. For the simulation results shown in the next section, 500 realizations are used. 5.4 5-4-1 R e s u lts an d D iscussion 3-D SPM2 Results In the following examples, we assume a conductor with the conductivity of pure copper (ct = 5.8 x 107 S/m ) and a dielectric with a relative perm ittivity of 4.0. Figures 5.35.4 illustrate the results of power absorption ratio between rough surface and smooth surface based on the 3-D SPM2 formula. In Figure 5.2 the results are for a Gaussian correlation function with RMS height h = 1.0 /mi. The correlation length I varies from 2.0 /im to 3.5 /mi. We note th a t the absorption ratio increases w ith frequency. It Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 1.5 1.4 JZ 1.3 -5 o E JZ Cl 1.2 3O (GHz) 0 2 4 6 8 10 Figure 5.2: Power absorption ratio: surface w ith Gaussian corrrelation function (h = 1 / i m ) with varying correlation length. also increases when the correlation length gets smaller. In Figure 5.3, th e results are repeated for the case of h = 0.75 n m. The absorption ratios are smaller th an those of Figure 5.2 because a smaller RMS height gives a smoother surface. In Figure 5.4, the results are illustrated for surfaces w ith exponential correlation functions exhibiting larger absorption th an surfaces w ith Gaussian correlation function. The results of Fig. 5.2-5.4 show th a t the absorption ratios satu rate for Gaussian correlation func tion because of the finite rough surface area. However, the absorption ratios do not saturate for exponential correlation function because the surface contains multiscale roughness since th e spectral density of the exponential correlation function decays slowly with kx and ky. The results also dem onstrate th a t th e absorption depends on all three of the roughness characteristics: RMS height, correlation length and corre lation function. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 1.25 ------------ 1— 2.0|i m - B- 2. 5^m 1.2 3.0n.m 3.5|r.m 1 J — ^ 1 * 11 1 11 t _i m n T 11 1 n ri-n □ □ rjl 21.15 v ~y 0 ! 0 0 0 -'cT 0 0 0 (GHz) 10 Figure 5.3: Power absorption ratio: surface w ith Gaussian correlation function (h 0.75 lira) w ith varying correlation length. 5.4.2 3-D SPM2 and 2-D SPM2 Comparison In [53], we followed M organ’s assum ption by enforcing constant magnetic fields on a two-dimensional rough interface and applied SPM2 only in the conductor region. The power absorption ratio is obtained by the following closed-form formula: ( P a , rough/ 1) a,sm ooth = 1+ 2 JP 52 f J 0 dkxW (kx) Re 82 kl (5.45) Figure 5.5 and Figure 5.6 compare the power absorption ratio using 3-D SPM2 and 2D SPM2 based on M organ’s boundary condition. The absorption ratios are illustrated for surfaces with Gaussian correlation function and exponential correlation function (h = 1 /rm and I = 2 p m ). The results show more significant power absorption by rough surface w ith three-dimensional configuration th a n those w ith two-dimensional configuration. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Figure 5.4: Power absorption ratio: surface w ith exponential correlation function (h — 1 //m) with varying correlation length. 5-4-3 3-D SPM2 and T-matrix Comparison Fig. 5.7 and Fig. 5.8 compare 3-D analytic results based on small perturbation m ethod of second order w ith numerical T-m atrix results. Note th a t th e validity of the Tm atrix m ethod requires the slopes of the surface profile to be much smaller than unity. Therefore, the correlation length I has to be much larger th a n th e RMS height h. The modeled surface profiles are for Gaussian correlation function with RMS height h = 0.5 /mi, h = 0.75 p m and correlation length I = 3.0 /jm, I — 3.5 pm. The numerical T-m atrix results are in good agreement with the analytic 3-D results of small perturbation m ethod for rough surfaces with small slope. 5.5 C onclusion In this chapter, we use a random surface model w ith correlation functions and spec tral densities to characterize different roughness profiles in three-dimensional problem. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 1.5 - B - 2-D SPM2 -© - 3-D SPM2 1.4 1.3 (GHz) 0 2 4 6 8 10 Figure 5.5: 3-D SPM2 versus 2-D SPM2: surface w ith Gaussian correlation function (h = 1 jum, I = 2 /im). The key result is the closed-form 3-D SPM2 formula of power absorption enhancement factor, which can be used in conjunction w ith an interconnect model w ith a perfectly smooth conductor surface to quantify the im pact of surface roughness on conductor loss. The spectral densities in the 3-D SPM2 model can be further obtained from measured height profiles. The extraction of surface spectral densities and correla tion between loss measurements and the theoretical model are discussed in the next chapter. The small perturbation m ethod and T-m atrix m ethod are generally valid for rough surfaces w ith small RMS height and small slope. For larger RMS heights and slopes, numerical methods of exact solutions can be obtained by using th e m ethod of moments with RWG basis functions. These have been done for microwave scattering from soil surfaces [28], [29], [30] and [31]. These numerical methods can be used in th e future to study random rough surfaces of interconnects. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 2 -B - 2-D SPM2 - e - 3-D SPM2 1.2 (GHz) Figure 5.6: 3-D SPM2 versus 2-D SPM2: surface w ith exponential correlation function (h = 1 fim, I — 2 /mn). 1.14 iii i 1.12 1. 1 1.08 cn 3O 1.06 / / S/ 1.04 1.02 1 0 if _ -J ; fl fi f| II I n n - - "O'“- -<■>---- ^ --- ■© ii i i i Ir-JFr-r^n-r¥n-|0TT-fl-lTfi -©- 3-D SPM2 (h=0.75pm) -i T-matrix (h=0.75pm) 1 -B- 3-D SPM2 (h=0.5|im) -j -El- T-matrix (h=0.5pm) j (GHz) 10 Figure 5.7: 3-D SPM2 versus T-m atrix: surface w ith Gaussian correlation function (I = 3.0 n m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.8: 3-D SPM2 versus T-m atrix: surface w ith Gaussian correlation function (I = 3.5 /im). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 Chapter 6 ESTIMATION OF ROUGHNESS-INDUCED POWER ABSORPTION FROM MEASURED SURFACE PROFILE DATA 6.1 In tro d u c tio n For interconnect waveguide structures on high-speed microelectronic package sub strates and boards, the surface roughness between dielectric and conductor layers may introduce significant additional power loss th a t can be detrim ental for insertion loss limited designs. In C hapters 2-5, we have applied analytic small perturbation m ethod of second order (SPM2), numerical m ethod of moments (MoM) and T-m atrix m ethod to quantify the roughness effect on power absorption for one-dimensional ( 1 D) and two-dimensional (2-D) surfaces [53], [55], [56]. The SPM2 formula of absorp tion enhancement factor for 2-D metallic rough surface in a three-dimensional (3-D) problem is given by (5.28) where 5 is the skin depth at microwave frequencies, h is the root-m ean-square (RMS) height of surface and W (kx , k y) is th e surface power spectral density (PSD) in 2-D form. The ultim ate purpose of this paper is to ex tract the PSD from real interconnect surfaces and use it to estim ate corresponding roughness-induced power loss. The 2-D PSD is designated as th e preferred quantity for specifying surface rough ness [57]. The conceptual approach to obtain 2-D PSD from measured surface height profiles is to take the m agnitude squared of the 2-D Fourier transform of the d ata record, known as the periodogram. An ensemble average of PSD estim ates needs to be taken afterwards because the periodogram estim ate has large variance and makes it a noisy estim ator. A simplification can be made for an isotropic surface, i.e., if Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 the topology has the same statistics regardless of direction. One can obtain 1-D PSD from averaged periodogram estimates [58], [59] and convert it into th e 2-D form [60]. However, in practice the conversion is difficult to perform numerically due to the presence of the derivative of the inherently noisy 1-D PSD. Our approach discussed in Section 6.2 takes an indirect route to obtain the 2-D PSD from an isotropic surface. Instead of taking the 2-D Fourier transform of the sur face height d ata directly, we first extract the 1-D PSD from the averaged periodogram estim ate as in [59] and then compute the correlation function by taking an inverse Fourier transform. Next we take a fast Fourier-Bessel transform [61], also referred to as Hankel transform , of the correlation function to get the 2-D PSD. T he procedure is valid if we assume the correlation function for a statistically isotropic surface has the same form for both 1-D and 2-D cases. Finally we use the extracted 2-D PSD to evaluate the power absorption enhancement factor based on (5.28). In Section 6.3, this four-step process is validated by synthetic d ata with given correlation function and PSD. Next we compute the corresponding power absorption enhancement factor. Results show th a t the extracted PSD yields accurate roughness-induced power loss in the SPM2 model despite the bandw idth lim itation due to space resolution and finite surface size. We further apply the procedures to analyze a real m etal surface of an interconnect structure and dem onstrate good correlation between th e estim ated loss and the measured loss up to 20 GHz. 6.2 E stim a tin g P o w er A b so r p tio n w ith P S D E x tra c tio n We consider a 2-D profile of a rough surface described by a topographic height function / (x, y ). The expression of 2-D PSD of th e rough surface W 2d {kx , ky) is defined by W 2 D (kX,ky) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.1) 73 where L is the surface length in both directions and (•) denotes th e operation of en semble average. In practice, L is finite and the 2-D periodogram is com puted by 2-D fast Fourier transform . To obtain an acceptably sm ooth PSD for typical 2-D surface profiles th a t contain 512 x 512 d ata points, normally at least a few hundred of mea sured 2-D profiles are required for averaging. Thus, applying (6.1) directly to extract the 2-D PSD can be infeasible due to the time-consuming process in measurement and d ata processing. Our approach of extracting the 2-D PSD is formulated by considering a 2-D profile of a statistically isotropic rough surface. The expression of th e 1-D PSD of the rough surface, W\ n (k), can be obtained by taking sweeps in any direction of a 2-D profile, e.g., in the x direction, and averaging the 1-D periodogram: 1 1 J 2\ d x f (x) exp {—j k x ) (6 .2) To get a large enough number of sweeps for averaging, we may either subdivide a long sweep into small segments as in [59] or take small segments from th e 2-D profile horizontally and vertically, as long as the correlation length of th e rough surface is considerably smaller th an the segment size. The extracted W \ d (k ) has a bounded shape in the spatial frequency domain, where the lower bound is due to the finite surface length L and the upper bound is due to th e smallest resolution A x . Next we take the inverse Fourier transform of W \ d (k) to get th e correlation func tion of the rough surface C (p): / OO d k W m (k ) exp (jkp) (6.3) •OO where p = \ J x 2 + y 2 is the polar radius. Note th a t the correlation function C (p) does not depend on polar direction since the surface is isotropic. The bandw idth lim itation of W m (k) may introduce ringing artifacts in the extracted C (p) due to the Gibbs phenomenon. This problem can be minimized by choosing a large L and a small A x which leads to a wide-band W \d (k). Some approxim ations could be further Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 made to cut off the oscillation as some researchers have pointed out th a t th e surface correlation functions in general should not be regarded as reliable for distances greater th an about 1/10 of the surface extent [62], [63]. The th ird step is generally taking the 2-D Fourier transform of th e correlation function to get the 2-D PSD W 2d (kx , ky): W 2D (kx , ky) ^ = 4^2 J (6.4) poo 7*00 J dxdyC (x, y) exp {—j k xx —j k yy ) . For an isotropic surface where C (x,y) only depends on p, the 2-D Fourier transform can be simplified as Fourier-Bessel transform , also referred to as Hankel transform: 1 W 2D (kp) = — y where C (p )J o (kpp) pdp (6.5) kp= \ f k 2 + k 2 and Jo (kpp) is the Bessel function of zeroth order. We use a fast algorithm [61] for the numerical im plem entation of the transform . The last step is substituting the extracted 2-D PSD W 2d (kp) into the formula of power absorption enhancement factor in (5.28). For an isotropic surface, (5.28) can be simplified as ( Ja ,ro u g h ) (6 .6 ) Ja ,s m o o th = 2h2 A t: f ° ° ( 1 + - 82 T T - —8 J J o — dkp Py \ k- ppW 2D (kp) ’' ™ v-pj I Re yi ' s2J — -pi where the integrand asymptotically approaches W 2D (kp) / 8 2 as kp becomes large. In the next section we select some sample results for synthetic and real rough surfaces and dem onstrate how the above four-step procedure works. 6.3 R e s u lts a n d D iscussion 6.3.1 Estimating Loss from Synthetic Surfaces To validate the procedure for PSD extraction, we first generate a 2-D profile of syn thetic random rough surface based on the Gaussian correlation function h 2 exp {—p 2 / I 2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 ip 0"4oft -20 .-25 .-30 Extracted Given 10 5 Figure 6.1: One-dimensional power spectral density W io(k): E xtracted versus syn thetic model. with RMS height h = 1 pm and correlation length I = 2 pm. The surface size is 150 pm by 150 pm and the number of sample points is 1024 x 1024. Figure 6.1 illus trates the extracted W m (k) compared w ith the given 1-D spectral density exp . Note th a t the extracted W \o (k) is obtained by averaging periodograms from all hor izontal and vertical sweeps of the given surface profile. Figure 6.2 illustrates the extracted C (p) compared w ith the given Gaussian correlation function. Some slight ringing is noticeable due to the limited bandw idth of W \ d (k). Figure 6.3 illustrates the extracted W 2o (kp) compared w ith the given 2-D spectral density 1,2/2 ( k 2l2 \ exp ( — j. Figure 6.4 illustrates the absorption enhancement factor com puted by the extracted W 2d (kp) and by the given 2-D PSD for two different synthetic Gaussian surfaces. The results show th a t the procedures are acceptably accurate and can be combined with the SPM2 formula to estim ate the roughness-induced conductor loss. 6 .3.2 Estimating Loss from Real Measured Surfaces Here we study the metal surface of a real interconnect structure after removing di electric materials by reactive ion etching. The 2-D profile of th e surface is obtained Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 Extracted Given x' pGjm) 2 4 6 8 10 Figure 6.2: Correlation function C(p): E xtracted versus synthetic model. by scanning w ith an atomic-force microscope (AFM). The surface size is 50 pm. by 50 p m and the number of sample points is 1024 x 1024. The RMS height of the surface profile is 0.85 pm. Figures 5-7 illustrate the extracted C (p) and W 2d (kp), as well as th e absorption enhancement factor based on the extracted W 2d (kp). In addition, we find it possible to choose a random rough surface model to approxi m ate the measured surface profile to a large extent. Figures 6 .5-6.7 include such comparison w ith a differentiable-exponential correlation function model [47] given where h = 0.85 pm , l\ = 1.4 pm and l2 = 0.53 pm. Figure 6 .8 illustrates the visualization of the m etal surface from AFM d ata and the simulated rough surface based on the differentiable-exponential correla tion function model. One can notice th a t the overall roughness scale matches between the two images. Next we correlate measured loss w ith the estim ated loss, using th e AFM measured surface profile data. The measured loss in term s of attenuation constant is obtained by measuring the S param eters of two microstrip lines w ith different length b u t the same m etal surface characteristics. Estim ating the loss, on the other hand, requires modeling of the smooth surface case, in which the attenuation constant can be split Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 -24 Extracted — Given 10 4 10 5 Figure 6.3: Two-dimensional power spectral density W 2 D(kp): E xtracted versus syn thetic model. into conductor and dielectric loss by setting the dielectric loss tangent to zero. In our simulation, we use a commercial 2-D field solver (Ansoft 2D) and use th e following geometric param eters from cross-sectioning of samples: trace w idth = 65.5 pm; trace thickness = 15.3 jum; dielectric thickness = 30.6 pm. The relative dielectric con stant and dielectric loss tangent are 3.4 and 0.017, respectively. The medium above the dielectric is air. The conductivity of m etal is 4.5 x 107 S/m . The final form of the estim ated loss is obtained by multiplying the attenuation constant of smoothconductor loss w ith the extracted absorption enhancement factor in Figure 6.7 and then adding the simulated attenuation constant of dielectric loss. Figure 6.9 illustrates the measured and estim ated attenuation constants. The blue curves (dash-square and solid-diamond) are the respective attenuation constants of conductor loss and total loss for the smooth case, com puted by th e Ansoft 2D simulator. The black solid curve refers to the attenuation constant extracted directly from measurement. The red solid-circle curve refers to the roughness-corrected attenuation constant by taking into account the absorption enhancement factor in Figure 6.7. The pink dash curve refers to the roughness-corrected attenuation constant by applying classic Hammer- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 1.5 1.4 W ~ e - Extracted (h=1gm, l=2pm) Given (h=1gm, l=2gm) -e- Extracted (h=0.5pm, l=2pm) — Given (h=0.5pm, l=2pm) ■£).! 1 0 2 4 6 8 . _4> j(GHz) 10 Figure 6.4: Absorption enhancement factor: Extracted versus synthetic model. stad and Bekkadal formula (2.1) in [11] and [64], The plot dem onstrates excellent correlation between the measured loss and the estim ated loss by 3-D SPM2 formula. The results of H am m erstad and Bekkadal formula also agree w ith th e measured loss. However, it happens coincidentally in this case as the formula was developed from a two-dimensional problem with periodic ridged surface and it only depends on the RMS height and skin depth. In contrast, th e 3-D SPM2 formula is based on a physical three-dimensional problem w ith random rough surface and th e formula includes the RMS height, skin depth and spectral density to b etter estim ate th e roughness-induced absorption at any given microwave frequency. 6.4 C onclusion In this chapter, we present a methodology for extracting th e two-dimensional power spectral density of a statistically isotropic random rough surface from height measure ments by utilizing fast Fourier-Bessel transform. We compute the additional propa gation loss due to surface roughness by integrating the extracted spectral density via the formula of absorption enhancement factor. Results for a m icrostrip dem onstrate good correlation between measured and estim ated loss up to 20 GHz. It is also pos- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 x 10 —e - Extracted Modeled Figure 6.5: Correlation function C(p): Extracted versus synthetic model. sible to choose a random rough surface model for th e measured surface and use it to predict the roughness effect on power loss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 ►- Extracted - Modeled i EEEEIEEEc EEIe Ee Ec EIEi : jiiilSISB: HHHHHi 1====;==!!==;==i=|E;=i io 7kp(m'1) Figure 6 .6 : Two-dimensional power spectral density W 2D(&/>): E xtracted versus syn thetic model. CL CL 1.2 Extracted M odeled J (GHz) Figure 6.7: Absorption enhancement factor: E xtracted versus synthetic model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6 .8 : Surface visualization: AFM measured surface (left) versus synthetic model (right). R o u g h /M easu red Sm ooth / Ansoft 2D (tan 6=0.017) Sm ooth / Ansoft 2D (tan 5=0) Rough / Ansoft 2D + Ham-Bek Rough /A nsoft 2D + 3DSPM2 rc.-.e(GHz) Figure 6.9: A ttenuation constant: measured loss versus estim ated loss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 BIBLIOGRAPHY [1] A. Deutsch, C. Surovic, R. K rabbenhoft, G. V. Kopcsay, and B. Chamberlin, “Losses caused by roughness of metallization in printed-circuit boards,” in Proc. IE E E 1 4 th Topical Meeting on Electrical Performance of Electronic Packaging (EPEP), Oct. 24-26, 2005, pp. 39-42. [2] R. Patrikar, C. Dong, and W. Zhuang, “Modelling interconnects w ith surface roughness,” Microelectronics Journal, vol. 33, no. 11, pp. 929-934, 1983. [3] Z. Zhu, A. Demir, and J. K. W hite, “Stochastic integral equation m ethod for modeling the rough surface effect on interconnect capacitance,” in Proc. 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New York: Wiley Interscience, 2000, ch. 9, pp. 389-407. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [47] A. K. Fung, Microwave scattering and emission models and their applications. Boston and London: Artech House, 1994, ch. 2, pp. 119-121. [48] P. J. Chandley, “Surface roughness measurements from coherent light scatter ing,” Optics, and Quantum Elec., vol. 8 , pp. 323-327, 1976. [49] G. Rasigni, M. Rasigni, J. P. Palm ari, C. Dussert, F. Varnier, and A. Lebaria, “Statistical param eters for random and pseudorandom rough surfaces,” J. Opt. Soc. Am ., vol. 5, pp. 99-103, 1988. [50] G. Rasigni, F. Varnier, M. Rasigni, and J. P. Palm ari, “Spectral-density function of the surface roughness for polished optical surfaces,” J. Opt. Soc. Am ., vol. 73, p p . 1235-1239, 1983. [51] M. L. Boyd and R. L. Deavenport, “Forward and specular scattering from a rough surface,” J. Ac. Soc. Am ., vol. 53, pp. 791-801, 1973. [52] L. Tsang, J. A. Kong, and R. Shin, Theory o f Microwave Rem ote Sensing. York: Wiley Interscience, 1985. New [53] L. Tsang, X. Gu, and H. Braunisch, “Effects of random rough surface on absorp tion by conductors at microwave frequencies,” IE E E Microwave and Wireless Components Letters, vol. 16, pp. 221-223, Apr. 2006. [54] W. C. Chew, Waves and Fields in Inhomogeneous Media. Press, 1995, pp. 430-433. New York: IEEE [55] X. Gu, L. Tsang, H. Braunisch, and P. Xu, “Modeling absorption of rough inter face between dielectric and conductive medium,” Microwave and Optical Tech nology Letters, vol. 49, pp. 7-13, Jan. 2007. [56] X. Gu, L. Tsang, and H. Braunisch, “Modeling effects of random rough in terface on power absorption between dielectric and conductive medium in threedimensional problem,” IE E E Trans. Microwave Theory and Tech., in press, 2006. [57] J. C. Stover, Optical Scattering: Measurement and Analysis. McGraw-Hill, 1990, ch. 2, pp. 32-44. New York: [58] H. Stark and J. W. Woods, Probability, Random Process, and Estim ation Theo ries fo r Engineers, 2nd ed. Englewood Cliffs, New Jersey: Prentice Hall, 1994, ch. 10, pp. 492-511. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 [59] J. M. Elson and J. M. Bennett, “Calculation of th e power spectral density from surface profile d ata,” Applied Optics, vol. 34, no. 1, pp. 201-208, Jan. 1995. [60] E. L. Church and P. Z. Takacs, “The optim al estim ation of finish param eters,” in Optical Scatter: Applications, Measurement, and Theory, vol. 1530, 1991, pp. 71-78. [61] M. Guizar-Sicairos and J. C. Gutierrez-Vega, “Com putation of quasi-discrete Hankel transform s of integer order for propagating optical wave fields,” J. Opt. Soc. Am ., vol. 2 1 , no. 1, pp. 53-58, Jan. 2004. [62] J. M. Bendat and A. G. Piersol, Measurement and Analysis o f Random Data. New York: Wiley, 1966. [63] M. S. Shunmugam and V. Radhakrishnan, “Selection and fitting of reference lines for surface profiles,” in Proc. Instn. Mech. Engrs., no. 190, 1976, pp. 193-201. [64] T. C. Edwards and M. B. Steer, Foundation o f interconnects and microstrip design, 3rd ed. Chichester: John Wiley and Sons Ltd., 2000, ch. 5, pp. 148-150. [65] V. Twersky, “Reflection and scattering of sound by correlated rough surfaces,” J. Ac. Soc. Am ., vol. 73, pp. 68-84, 1983. [6 6 ] L. Tsang, J. A. Kong, and K. H. Ding, Scattering o f Electromagnetic Waves, Vol. 1: Theory and Applications. New York: Wiley Interscience, 2000, ch. 2, pp. 97-98. [67] ------ , Scattering of Electromagnetic Waves, Vol. 1: Theory and Applications. New York: Wiley Interscience, 2000, ch. 1, pp. 24-30. [6 8 ] L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol. 2: Numerical Simulations. New York: Wiley Interscience, 2001, pp. 533-543. [69] R. Barakat and E. Cohen, “Numerical results for scattering by a hemisphere on a plane,” J. Ac. Soc. Am ., vol. 39, no. 4, pp. 753-755, 1966. [70] J. A. Ogilvy, Theory of wave scattering from random rough surfaces. Bristol and Philadelphia and New York: Adam Hilger, 1991, ch. 6 , pp. 151-161. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [71] A. K. Hamid, I. R. Ciric, and M. Hamid, “Electrom agnetic scattering by hemi spherical bosses on a infinite plane surface,” in Proc. IE E E A ntennas and Prop agation Society International Symposium (AP-S), Jun. 27-Jul. 0 2 , 1993, pp. 8285. [72] P. G. Huray, “Signal scattering from im purities in PCB s,” in Proc. IT E SO -Intel International Workshop on Signal Integrity, G uadalajara, Mexico, Apr. 7, 2005, pp. 135-178. [73] ------ , “3-D model for surface roughness losses,” presented a t the IEEE C PM T/C A S Oregon Joint C hapter Workshop on Microprocessor and Commu nication Platform Design Technology, Beaverton, Oregon, Jun. 1, 2006. [74] E. Sijercic, O. Oluwafemi, S. Hall, et a l, “Surface roughness simulation m ethod ology for the copper interconnect,” presented at the M editerranean Microwave Symposium, Sep. 2006. [75] E. Sijercic, S. Hall, O. Oluwafemi, P. G. Huray, A. M oonshiram, and G. Brist, “Modeling surface roughness for 30 gigabit per second chip-to-chip signals,” in Proc. Intel Design and Test Technology Conf. (D TTC ), San Jose, CA, Aug. 2831, 2006, Intel Internal. [76] E. Sijercic, private communication, Feb. 12, 2006. [77] P. Ye, “Analysis of microstrip transm ission lines with a spherical air inclusion in the substrate,” M aster thesis, University of South Carolina, May 2005. [78] L. Tsang, C. E. M andt, and K. H. Ding, “Monte Carlo simulations of the ex tinction rate of dense media w ith random ly distributed dielectric spheres based on solution of maxwell’s equations,” Optics Letters, vol. 17, pp. 314-316, Mar. 1992. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 A p p e n d ix A SCATTERING AND ABSORPTION OF ELECTROMAGNETIC WAVES ON A PLANE WITH HEMISPHERICAL BOSSES A .l In tro d u c tio n For interconnect waveguide structures in high-speed microelectronic packages, the surface roughness between dielectric and conductor layers may cause significant ad ditional power loss at microwave frequencies as the current travels along an increased surface area compared to on a smooth surface. In Chapters 2-6, we have charac terized the random rough surface by a random height function w ith spectral density and characterization function. Solution methods in C hapters 2-6 include the small perturbation m ethod to second order, T -m atrix m ethod and th e m ethod of moments. We have also extracted the spectral density from real life surfaces. C haracterization by stochastic random height function and the solution of the wave scattering equation for such characterization is the main them e of this dissertation. In this Appendix, we digress to study the hemispherical-boss characterization of rough surface and the m ethod of solution. The small irregularities are modeled by controlling the size of the bosses as well as their distribution on the plane. The hemispherical boss approach was originally used by Twersky [23] and has been ex tensively studied by him [65] and other researchers such as Biot [24], [25], W ait [26] and Barakat [69] (also refer to the review by Ogilvy [70]). However, those literatures either were focused on finding appropriate effective boundary conditions for the total field on the embossed plane or only took into account the dipole fields regardless of the location of bosses. Hamid [71] further solved a backscattering problem for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 different embossed surfaces including multiple scattering of a plane electromagnetic wave by an array of hemispherical bosses. However, his results cannot be applied to the interconnect rough surface because the boss radius is assumed to be comparable to the wave length in the dielectric and the separation between successive bosses is a few times larger th an the boss radius. In [21] and [22], Holloway explicitly dis cussed hemispherical-boss modeling as an alternative for conductor loss prediction in planar circuits. Recent applications to interconnect rough surface modeling include the works of Sijerciic [75], [74], [76], Huray [72], [73] and Ye [77] et ah, in which a hemispherical boss model was used to estim ate th e surface roughness induced losses and current flow on a transm ission line. In this appendix, we take into account randomly distributed hemispherical bosses over a conducting plane based on multiple scattering theory of Foldy-Lax equations. Numerical solutions of such equations were com puted to study th e effects of densely packed spheres. Up to 4000 spheres were used in such simulations [78], [35]. To resemble the real rough surface of interconnect structures, the distribution of such bosses is mostly like to be very dense so th a t the separation between two bosses is much smaller th an the boss radius. In such a case, the dipole approxim ation is not accurate enough to calculate the close range interaction of scattered electromagnetic field. To solve this problem, we apply Foldy-Lax multiple scattering equations [6 8 ] to an embossed surface, on which a vertically polarized incident plane wave propagates parallel to the flat plane. To simplify the analysis, we assume infinite conductivity for the plane and bosses. We then make low frequency approxim ation and derive a multipole solution of scattering field in the order up to three. The results show sig nificant improvement of accuracy for closely located bosses by increasing the order of multipole solution. We further illustrate the distribution of surface currents for many bosses with a random distribution. The power absorption on a lossy embossed surface can also be obtained from the field solution for the perfectly conducting surface by assuming th a t the skin depth in the lossy conductor is much less th an th e radii of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 Reference plane: z=0 Incident plane wave x Figure A .l: A flat plane w ith a random distribution of hemispherical bosses. bosses at given microwave frequency. We compute th e absorption and enhancement factors for different embossed rough surfaces and show the absorption is dependent on the actual distribution of the bosses. A .2 A .2.1 M u ltip o le S olu tion o f H em isp h erica l B o ss S c a tte rin g Multiple Scattering Equations Consider an incident plane electromagnetic wave propagating in a dielectric region on top of a perfectly conducting plane which has a distribution of hemispherical bosses, as shown in Figure A .I. We first remove th e flat plane based on Image Theory and replace the hemispherical bosses w ith spheres centered at th e same locations. Using exp (—icot) as phaser notation, the vertically polarized (TM) plane wave is given by H inc(f) = y e x p (ik x ) (A .l) E inc(f) = - z r j exp (ikx) (A.2) where k and rj are the respective wave number and wave impedance in dielectric. To apply multiple scattering equations, we write th e incident plane wave in term s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 of spherical waves in the spherical coordinates (r, #,</>): Einc (r) = ^ 2 [amn)R 9Mmn ( k r , 8 , 4>) + a ^ R g N mn ( k r , 8, 0 )] . m,n (A.3) where 8 and <f> are the respective altitudinal and azim uthal angles w ith respect to the origin, a!££?, dmn are the vector spherical wave coefficients and R g M mn, R g N mn are vector spherical wave functions of Bessel form [67]. ^ denotes E l= -n > m,n i.e., to, n go through the vector spherical wave coefficients as n — 1,2, ••• , N max. m = 0, ±1, ± 2, • • • , ± n . The final field exciting the I-th boss is: (T) = E m,n ( ^ ) + ( t r n) ] (A.4) where f f f is defined as a vector pointing from F; to r. W m n ^ and Wmn ® are the unknown exciting field coefficients of bosses due to magnetic and electric multipoles respectively. Next we substitute (A.3) and (A.4) into Foldy-Lax multiple scattering equations [6 8 ] which states th a t the field exciting boss I is th e sum of the incident field and scattered field from all bosses j except boss I. The exact equations are given by w. = exp (A.5) Nb + £ A „ { k r W ] ) T i M\ ^ M^ Nb + J2 B mnhlv{k7W ])TlN\H N 2 U) and w. = exp ( ik i- r i) a ,W Nb + £ B „ (k W ])T ^ v ^ ) Nb + ^2 Amn^u{krWj)TlN\,W(N)U) 3=hi¥=i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.6 ) 93 where Nb is the number of bosses, ki = kx, n — 1,2, • • ■ , N max and v = 0, ± 1, ± 2, • • • , ± /n are T-m atrix elements of spheres [6 6 ]. A mntiu (krjrj) and B mnilv {krjr]) and are defined as translational operators for electric and magnetic multipoles in addition theorem [6 8 ]. Once exciting field coefficients Wmn and are solved, th e scattered field are Wmn given by Nb Nb E, « = E £ +E E Z=1 m ,n (k m) TiM>wZw> (A.7) 1=1 m ,n and Nb H s (r) = ir\ Nb E E »™ ( k m ) t ( n )w W {1) + (.M){1) E E * - mn (=1 m ,n _ 1=1 m ,n (A.8 ) where M mn, N mn are vector spherical wave functions of Hankel form [67]. A .2.2 Low Frequency Approximation For rough surfaces on interconnect structures, th e roughness scale is in the order of micrometers. So th e radius of the boss a is of order 10~6. The wave number k in dielectric at microwave frequencies, on the other hand, is in the order of cm -1 . We define a small quantity £ = ka and use it to normalize the scattering equations in the following steps. Also note th a t in the proxim ity of given boss I, th e phase change exp (iki ■f i ) is negligible. Simplifying (A.6 ) w ith the assumptions above, we obtain Nb ] C Amn^ where w£n){l) = f ^ N) = TvN)™fv){j) and A mnii„ (krfTj) = (A ’9) £ n + " + 1 A mntlv (krjr,])■ Bmn/iu {krjr]) term s vanish because of the decoupling between the electric parts and the magnetic parts of multipoles when ka and k r are small (of order 1 CT4). Conse quently, only the electric parts of electric multipole interact in (A.9). For the elements Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 in A mniiv (krirj), taking the largest order p = n + v for h ^ 1 (k r ^ j ) as p varies from \n — u\ to n + u, we obtain A.mnfiiy (krirj) Ijiv Iran ( — 1 )m a (A.10) (p, u\ — m , n \n + u) a (u, n, n + u) n+i/ + 1 Y n + ™ ( ^ n r j 5 <finrj) ( 2 n + 2 v - 1]]{ w f \ ) i1 ' 3 ' 5 where (2 n + 1 ) (n —m )\ 4 -7rn (n 4-1) (n + m)\ 'Im n a( p, u\ — m ,n \n + u) — (—l)~ m+M ( 2 n + (A .ll) 2 z/+ 1 ) (A.12) (n —m )\ (u + p)\ (n 4- u + m — p)\ 1A (n + m)! (y — p) \ {n + u — m + p)\ u n p —m n +u I ( v n n+ u m — p, 0 0 0 ;.2n Here | a (v , n, n + u) = ^ n + J " (^nrj> 0 rff7) = n p, —m n + z^ \ m — pi J 2 r> (u -[2n (n + 1) —2n (u + n + 1) ( 2 n + 1)(|A.13) + 1) Pn+™ ( C0S ^ O rj) eXP [* (P' ~ m ) < ?W I ■ ( u n and I \0 0 n+u \ 0 (A -14) are the W igner 3j symbols [6 8 ] and J Pn+.™ (cos 0) is the associate Legendre polynomial. Similarly, only the magnetic parts of magnetic multipole interact and (A.4) can be simplified as Nb w, C 1<4$ + (M)(j) (A.15) The expressions of vector spherical wave coefficients a i ^ \ a+n and T -m atrix coeffi- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 cients T^M\ are « = (n) (_1) r ( M) " “ = 1 (2n + 1) -i \m+i ( aJ fW 'AA6> T ^ iifn T l) Jn {ka) ft!,1’ (te ) ” Je '*-5 (A'17) 2n_i ^ _______ - i _______ ~ ( 2 „ + l)H(2„-l)M _r 2 « - l i o M ? +1 dP ~m (cos 0) _ l W > (* r)] ~ (A18) + 1) ( AI ON - ( 2 n + l)! ! ( 2 n - l ) ! ! ( } Solving th e multiple scattering equations (A.9) and (A. 15), we obtain th e unknown and Wmn® for all bosses I = 1,2, ••• ,Nb- field coefficients Note th a t the and W mn^ is Nb (N ^ ax + size of the system m atrix for solving 2 Amax) by Nb (Nfn.lx + 2Armax). The to tal scattered field due to th e multiple scattering from all bosses can be com puted as follows: E.if) = Nb E E " ™ l~ 1 m,n Nb H s (f) = (m )T W v > W W — - IT] (A.20) (A-21) .1=1 m,n where N m n(krri) = £,n+2N m n(krri). Finally summing up the scattered field and incident field, we obtain the solution of to tal electromagnetic field. A .2.3 Power Absorption on a Lossy Embossed Surface Here we compute the power absorption by a lossy conducting surface. Let H\\ (x, y) denote the magnetic field tangential to the surface at a point (x , y. f (x , y)). The elec tric field E (x, y) can also be decomposed into E ± (x , y ) and E\\ (x , y) which refer to the respective normal and tangential components of th e electric field on the surface. For a lossy conductor with high conductivity a such as copper, H\\ (x, y) can be ap proxim ated by the magnetic field which we are able to solve for a perfectly conducting Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 surface with identical boss distribution. Furtherm ore, at high frequencies when the skin depth S in the conductor is much less th an the radii of the bosses, E ± (x , y) is approxim ated as —^ (1 —i) H\\ (x, y). Applying Poynting’s theorem w ith H\\ (x, y) and E ± (x,y), we calculate the power absorption on th e lossy surface as follows: ^O,rough = e l f dSn- — E ± (x, y) x i/Jj (x, y) j (A.22) “ /_ /. "T +(S +(f) 1 1 where L x and L y are the surface length in the x and y direction. Treating power absorption on a flat sm ooth lossy surface as a special case where no boss is present, we obtain / kx 2 frkjL 2 1 1 2 I l ~2(j() I■^r*nc y) I ' (A.23) ~2~ In the next section, we first illustrate some sample results for th e to tal electric and l magnetic field on perfectly conducting embossed surfaces using the multipole solution for n = 1, 2, and 3. We further compute the power absorption enhancement factor, defined as p“,r°"Rh , for different lossy embossed surfaces. ■* a ,s m o o th A .3 A.3.1 R e s u lts an d D iscu ssion Scattering by Two Nearby Bosses In the following examples, we assume a perfectly conducting surface and a dielectric with a relative perm ittivity of 4.0. The radius of hemispherical bosses is 2 microme ters. The frequency of incident plane wave is 10 GHz. We have to consider multiple scattering between bosses if the number of bosses is greater th an one. For a typical hemispherical boss model to resemble the interconnect rough surface, th e spacing be tween bosses are usually smaller th an the boss radius. We thus need a higher order multipole solution to capture the multiple scattering effect accurately. Figures A.2 and A.3 show the boundary conditions n x E and n ■H for two nearby bosses whose Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 centers are located at (—| , 0 , 0) and ( |,0 , 0 ) . Here the distance between two boss centers is s — 2.1 a and a is the boss radius. The results are plotted in the same scale for multipole order n = 1,2 and 3. It can be noticed th a t th e dipole approxim ation (n = 1) brings large errors for the field distributed in the region between two closely placed bosses. Such errors can be significantly suppressed by increasing th e order of multipole solution to n = 2 and 3. Same behaviors are shown in Figures A.4 and A.5 for two bosses placed at (0 , —§, 0 ) and ( 0 , f , 0 ) w ith s = 2 . 1 a. The distribution of surface current J s, defined as h x 77, are shown in Figures A . 6 and A.7 based on multipole order n = 3. The difference is th a t in th e close interaction region between two bosses, the surface current density is the highest when two bosses are located parallel to the direction of the incident field (i.e., x direction), while the current density is the lowest when two bosses are located in th e perpendicular direction of the incident field (i.e., y direction). A .3.2 Scattering by a Dense Distribution o f Bosses Figure A . 8 illustrates the surface current distribution for five closely located bosses based on multipole order n = 3. The center boss is located a t th e origin and the rest four bosses are placed at (s, s, 0 ), ( s , —s, 0 ), (—s , s , 0 ) and (—s, —s, 0 ) respectively where s = 1.5556a. It can be seen th a t th e surface current w ith high density flows between the corner boss and center boss in the close interaction region through the multiple scattering mechanism. We can further take into account a random distribution of hemispherical bosses in our model. Figure A.9 illustrates the surface current distribution for 200 randomly distributed bosses based on multipole order n = 3. It is noticeable th a t the overall surface current distribution is also random b u t following th e distribution of boss positions. For local nearby bosses, the current distribution on th e surface exhibit similar behaviors as in the two-boss and five-boss results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 A .3.3 Power Absorption and Enhancement Factor Here we com pute the power absorption and enhancement factor for different lossy embossed surfaces. Table A .l lists the absorption and enhancement factor for above 2-boss (in X and Y axis), and 5-boss cases. In addition, we add a 4-boss case where the bosses are located at (s, s, 0), (s, —s, 0), (—s, s, 0) and (—s, —s, 0) w ith s = 1.05a, as well as a 9-boss case where the bosses are located at (—s, s,0 ), (0 ,s,0 ), (s, s, 0), (—s, 0,0), (0,0,0), (s, 0,0), (—s, —s, 0), (0, —s, 0), and (s, —s, 0) w ith s = 2.1a. The boss radius a is 2ym . The surface dimension in the x and y direction is L x — L y = 12.6y m . The conductivity of the lossy conductor is assumed as 5.8 x 107 S/m . The absorption is com puted based on (A.22) and (A.23) at the frequency of 10 GHz. It can be seen th a t the embossed rough surfaces enhance th e power absorption compared to the flat sm ooth surface. The enhancement factor depends on the size and distribution of the bosses as the those factors determine the scattering of electromagnetic field and the surface current. For example, the 9-boss case has th e highest density of boss distribution which causes the surface current to flow along a rougher p ath compared with the rest cases. Consequently, the surface absorbs the most power in the lossy conductor, leading to the highest enhancement factor. A .4 C onclusion In this appendix, we model the random rough surface by considering randomly dis tributed hemispherical bosses on a sm ooth conducting plane surface. Compared with the random rough surface model in Chapters 2-6, the hemispherical boss model is a gross assumption since the real interconnect rough surfaces do not consist of an embossed plane. From a given surface profile, it is difficult to choose param eters such as the number and size of bosses, as well as their relative positions. Because of these limitations, there have not been many applications of such a hemispherical boss model to analyze random rough surfaces since the original idea was proposed in 1950’s. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 Table A .l: Absorption and enhancement factor for different embossed surfaces N u m b e r o f b o sse s smooth (X10- 12W) fa,rough (X 10- 12W) fa,rough/fa,smooth 2 (X-axis) 2.098 2.682 1.277 2 (Y-axis) 2.098 2.553 1 .2 2 1 4 2.098 3.080 1.467 5 2.098 3.375 1.608 9 2.098 4.394 2.093 Nevertheless, the hemispherical boss model reproduces some im portant features of wave scattering and sheds some light on the current distributions. We apply multiple scattering equations to study the scattering of electromagnetic waves from a perfectly conducting plane surface w ith a random dense distribution of hemispherical bosses. We derive a multipole solution up to th ird order to analyze close range interactions between nearby bosses. Results show significant improvement of accuracy compared with the traditional dipole approxim ation solution. Absorption on a lossy embossed surface is obtained from the field solution of perfectly conducting surface. The surface current and absorption enhancement factor are further computed numerically. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 n=1 Figure A.2: Boundary condition of electric field (|n x E |) for 2 bosses in X axis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 n=1 Figure A.3: Boundary condition of magnetic field (|n ■H |) for 2 bosses in X axis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 Figure A.4: Boundary condition of electric field (|n x E |) for 2 bosses in Y axis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 n=1 Figure A.5: Boundary condition of magnetic field (|ra • H |) for 2 bosses in Y axis. Reproduced with permission of the copyright owner. Furiher reproduction prohibited without permission. 104 U„l-|l*H| X (|itn) Figure A. 6 : D istribution of surface current on 2 bosses in X axis. |Jal=|iixH| X (|im ) Figure A.7: D istribution of surface current on 2 bosses in Y axis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 w. hiwhi X (|j.m) Figure A. 8 : D istribution of surface current on 5-boss surface. IJ MiixHI -80 -60 -40 -20 0 20 40 X (|im) Figure A.9: D istribution of surface current on 200-boss surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 VITA Xiaoxiong Gu was born in Shanghai, China, 1978. He received th e B.S. degree in electronic engineering from Tsinghua University, Beijing, China in 2000, th e M.S. degree in electrical engineering from the University of Missouri, Rolla in 2002, and the Ph.D. degree in electrical engineering from the University of W ashington, Seattle in 2006, respectively. Mr. Gu worked as a research intern at Intel Corporation in th e summers of 2003 and 2005, and he was with IBM T. J. W atson Research Center in th e summer of 2004. He now works as a research staff member at IBM Research, Yorktown Heights, New York. His research interests include characterization of high-speed interconnect and microelectronic packaging, signal and power integrity, and com putational electromag netics. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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