close

Вход

Забыли?

вход по аккаунту

?

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. Broken or indistinct print, colored or poor quality illustrations and
photographs, print bleed-through, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
UMI
UMI Microform 3241904
Copyright 2007 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Ml 48106-1346
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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. Further reproduction prohibited without permission.
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. Inter­
national Conference on Computer Aided-Design, 2004, pp. 887-891.
[4] Z. Zhu and J. K. W hite, “Fastsies: a fast stochastic integral equation solver for
modeling the rough surface effect,” in Proc. IE E E /A C M International Confer­
ence on Computer-aided Design, San Jose, CA, 2005, pp. 675-682.
[5] C. S. Chang and A. P. Agrawal, “Fine line thin dielectric circuit board char­
acterization,” IE E E Trans. Components, Packaging, and Manufacturing Tech.,
vol. 18, pp. 842-850, Dec. 1995.
[6 ] G. Brist, S. Hall, S. Clouser, and T. Liang, “Non-classical conductor losses due
to copper foil roughness and treatm en t,” presented at th e Electronic Circuits
World Convention (ECWC) 10, Anaheim, CA, Feb. 22-24, 2005.
[7] H. Tanaka and F. Okada, “Precise measurements of dissipation factor in mi­
crowave printed circuit boards,” IE E E Trans, on Instruments and Measurements,
vol. 38, pp. 509-514, Apr. 1989.
[8 ] J. R. Brews, “Transmission line models for lossy waveguide interconnections in
vlsi,” IE E E Trans. Electron Devices, vol. 33, no. 15, pp. 1356-1365, 1986.
[9] K. C. G upta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines.
MA: Artech House, Sep. 1979.
Norwood,
[10] S. P. Morgan, “Effects of surface roughness on eddy current losses at microwave
frequencies,” J. Appl. Phys., vol. 20, pp. 352-362, Apr. 1949.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
[11] E. O. Ham m erstad and F. Bekkadal, “M icrostrip handbook,” University of
Trondheim, Norway, ELAB Report pp. 86-194, 1987.
[12] Z. Wu and L. E. Davis, “Surface roughness effect on surface im pedance of super­
conductors,” J. Appl. Phys., vol. 76, pp. 3669-3672, Sep. 1994.
[13] L. Rayleigh, The Theory of Sound.
New York: Macmillan, 1929.
[14] S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,”
Commun. Pure Appl. Math., vol. 4, pp. 351-378, 1951.
[15] A. E. Sanderson, Effect of Surface Roughness on Propagation of the T E M Mode.
N.Y.: Academic Press, 1971, vol. 7, pp. 1-57.
[16] S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radio­
physics, Vol. f: Wave Propagation Through Random Media. Berlin: SpringerVerlag, 1989, ch. 5.
[17] L. P roekt and A. C. Cangellaris, “Investigation of the im pact of conductor surface
roughness on interconnect frequency-dependent ohmic loss,” in Proc. Electronic
Components and Technology Conf. (ECTC), New Orleans, LA, May 27-30, 2003,
pp. 1004-1010.
[18] R. Garcia-Molina, A. A. M aradudin, and T. A. Leskova, “The impedance bound­
ary condition for a curved surface,” Phys. Reps., vol. 194, pp. 351-359, 1990.
[19] A. A. M aradudin, “The impedance boundary condition for a one-dimensional,
curved, m etal surface,” Opt. Commun., vol. 103, pp. 227-234, 1993.
[20] ------ , “The impedance boundary condition a t a two-dimensional rough metal
surface,” Opt. Commun., vol. 116, pp. 452-467, 1995.
[21] C. L. Holloway and E. F. Kuester, “Power loss associated w ith conducting and
superconducting rough interfaces,” IE E E Trans. Microwave Theory and Tech.,
vol. 48, pp. 1601-1610, Oct. 2000.
[22] ------ , “Impedance-type boundary conditions for a periodic interface between a
dielectric and a highly conducting medium,” IE E E Trans. Antennas and Propa­
gation, vol. 48, pp. 1660-1672, Oct. 2000.
[23] V. Twersky, “On the non-spectular direction of electromagnetic waves,” J. Appl.
Phys., vol. 22, pp. 825-835, Jun. 1951.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
[24] M. A. Biot, “Some new aspects of the reflection of electrom agnetic waves on a
rough surface,” J. Appl. Phys., vol. 28, pp. 1455-1463, Dec. 1957.
[25] ------ , “Generalised boundary conditions for multiple scatter in acoustic reflec­
tion,” J. Ac. Soc. Am ., vol. 44, pp. 1616-1622, 1968.
[26] J. R. W ait, “Guiding of electromagnetic waves by uniformly rough surfaces, part
i,” IR E Trans. Antennas and Propagation, vol. 7, pp. S154-S162, Dec. 1959.
[27] L. Tsang and J. A. Kong, Scattering o f Electromagnetic Waves, Vol. 3: Advanced
Topics. New York: Wiley Interscience, 2001, ch. 1, pp. 1-57.
[28] J. T. Johnson, L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and
Y. Kuga, “Backscattering enhancement of electromagnetic waves from twodimensional perfectly conducting random rough surfaces: Comparison of Monte
Carlo simulations w ith experimental d ata,” IE E E Trans. A ntennas and Propa­
gation, vol. 44, pp. 748-756, May 1996.
[29] Q. Li, C. H. Chan, and L. Tsang, “Monte Carlo simulations of wave scattering
from lossy dielectric random rough surfaces using the physics based two grid
m ethod and the canonical grid m ethod,” IE E E Trans. A ntennas and Propaga­
tion, vol. 44, pp. 752-763, Apr. 1999.
[30] L. Zhou, L. Tsang, V. Jandhyala, Q. Li, and C. H. Chan, “Emissivity simulations
in passive microwave remote sensing w ith 3-dimensional numerical solutions of
Maxwell equations,” IE E E Trans. Geoscience and Remote Sensing, vol. 42, pp.
1739-1748, 2004.
[31] P. Xu and L. Tsang, “Scattering by rough surface using a hybrid technique com­
bining the multilevel UV m ethod w ith the sparse-m atrix canonical grid m ethod,”
Radio Science, vol. 40, p. RS4012, 2005.
[32] A. A. M aradudin, Ed., Light Scattering and Nanoscale Surface Roughness.
Springer Verlag, Nov. 2006.
[33] A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume II:
multiple scattering, turbulence, rough surfaces and remote sensing. New York:
Academic Press, 1977.
[34] L. Tsang, J. A. Kong, and K. H. Ding, Scattering o f Electromagnetic Waves,
Vol. 1: Theory and Applications. New York: Wiley Interscience, 2000.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
[35] L. Tsang, J. A. Kong, K. H. Ding, and C. 0 . Ao, Scattering o f Electromagnetic
Waves, Vol. 2: Numerical Simulations. New York: Wiley Interscience, 2001.
[36] L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves, Vol. 3: Advanced
Topics. New York: Wiley Interscience, 2001.
[37] V. Celli, A. A. M aradudin, A. M. Marvin, and A. R. M cGurn, “Some aspects
of light scattering from a randomly rough m etal surface,” J. Opt. Soc. Am . A,
vol. 2, pp. 2225-2239, Dec. 1985.
[38] I. Simonsen, T. A. Leskova, and A. A. M aradudin, “Light scattering from an
amplifying medium bounded by a randomly rough surface: A numerical study,”
Physical Review B, vol. 64, Jul. 2001.
[39] A. Ishimaru, J. D. Rockway, and Y. Kuga, “Rough surface green’s function based
on the first-order modified perturbation and smoothed diagram m ethods,” Waves
in Random Media, vol. 10, pp. 17-31, Jan. 2000.
[40] R. Lim, K. L. Williams, and E. I. Thorsos, “Acoustic scattering by a threedimensional elastic object near a rough surface,” J. Ac. Soc. Am ., vol. 107, pp.
1246-1262, Mar. 2000.
[41] E. I. Thorsos, D. R. Jackson, and K. L. Williams, “Modeling of subcritical pen­
etration into sediments due to interface roughness,” J. Ac. Soc. Am ., vol. 107,
pp. 263-277, Jan. 2000.
[42] V. Twersky, “M ultiple scattering of radiation by an arbitrary configuration of
parallel cylinders,” J. Ac. Soc. Am ., vol. 24, pp. 42-46, Jan. 1952.
[43] ------ , “Multiple scattering of waves and optical phenom ena,” J. Opt. Soc. Am .,
vol. 52, pp. 145-171, Feb. 1962.
[44] ------ , “M ultiple scattering of electromagnetic waves by arbitrary configurations,”
Journal of Mathematical Physics, vol. 8 , pp. 589-610, Mar. 1967.
[45] L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering o f Electromagnetic
Waves, Vol. 2: Numerical Simulations. New York: Wiley Interscience, 2001,
ch. 4, pp. 124-151.
[46] 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. 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.
Документ
Категория
Без категории
Просмотров
0
Размер файла
2 562 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа