scalable interconnect model by erik harker a senior …

SCALABLE INTERCONNECT MODEL

FOR TOWER JAZZ SBC18

Erik Harker

A senior thesis submitted to the faculty of

Brigham Young University - Idaho

in partial fulfillment of the requirements for the degree of

Bachelor of Science

Department of Physics

Brigham Young University - Idaho

April 2014

BRIGHAM YOUNG UNIVERSITY - IDAHO

DEPARTMENT APPROVAL

of a senior thesis submitted by

Erik Harker

This thesis has been reviewed by the research committee, senior thesis coor-dinator, and department chair and has been found to be satisfactory.

Date Evan Hansen, Advisor

Date David Oliphant, Senior Thesis Coordinator

Date Richard Hatt, Committee Member

Date Stephen McNeil, Commitee Member

ABSTRACT

SCALABLE INTERCONNECT MODEL

FOR TOWER JAZZ SBC18

Erik Harker

Department of Physics

Bachelor of Science

A scalable interconnect model of transmission lines specific to the SBC18

Tower Jazz process was created with the intent to be used at Maxim Inte-

grated Products. The model has the ability to predict the series inductance

and resistance of the line and the shunt capacitance and conductance due to

the substrate. Both single ended and coupled lines were modeled. The model

allows one to select one of the 6 metal layers, set the line width and the line

separation in the case of the coupled lines. Simulations were done in Maxwell

Q2D to have a standard to compare the model to. The skin effect was taken

into account in order to improve reliability at higher frequencies. The single

ended line correctly prediced a rise in resistance, and conductance as with fre-

quency, and a slight decrease in capacitance and inductance. The coupled line

model also accurately predicted the capacitance, resistance and conductance.

The model for the series and mutual inductance did not follow the predicted

frequency dependence, but the values given were within the accepted error.

ACKNOWLEDGMENTS

I would like to thank my mentor, Dr. Garth Sundberg, and those at Maxim

Integrated in Beaverton, OR, for this project and guidance and experience

I gained while working there. I also thank the faculty at Brigham Young

University-Idaho for their helpful feedback throughout the semester. And of

course I want to thank my wife, Ashley for all that she does for me.

Contents

Table of Contents vii

List of Figures ix

1 Introduction 11.1 Project Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Transmission Line Uses . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Why This Is Important . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 History 52.1 History of Maxim Integrated Products . . . . . . . . . . . . . . . . . 52.2 History of Semiconductor Devices . . . . . . . . . . . . . . . . . . . . 62.3 Transmission Line Theory . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Process 93.1 Maxwell Q2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Results 154.1 Single Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Coupled Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Conclusion 215.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Bibliography 23

A Matlab Code 25

List of Figures

3.1 Single Ended Cross Section . . . . . . . . . . . . . . . . . . . . . . . 103.2 Coupled Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Single Ended Line Model . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Coupled Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1 Single 10µm Wide Metal Layer 6 Resistance . . . . . . . . . . . . . . 164.2 Single 10µm Wide Metal Layer 6 Inductance . . . . . . . . . . . . . . 164.3 Single 10µm Wide Metal Layer 6 Conductance . . . . . . . . . . . . . 174.4 Single 10µm Wide Metal Layer 6 Capacitance . . . . . . . . . . . . . 174.5 Coupled 10µm Wide with 10µm Metal Layer 6 Resistance . . . . . . 184.6 Coupled 10µm Wide with 10µm Metal Layer 6 Inductance . . . . . . 184.7 Coupled 10µm Wide with 10µm Metal Layer 6 Conductance . . . . . 194.8 Coupled 10µm Wide with 10µm Metal Layer 6 Capacitance . . . . . 194.9 Coupled 10µm Wide with 10µm Metal Layer 6 Mutual Inductance . . 204.10 Coupled 10µm Wide with 10µm Metal Layer 6 Mutual Capacitance . 20

Chapter 1

Introduction

1.1 Project Background

While working for Maxim Integrated Products in Beaverton Oregon, a project arose,

to work on modeling transmission lines for the Tower Jazz SBC18 process. The

SBC18 process refers to a specific process that is used to make devices on silicon

wafers. One diffecence about this process was the specific dimensions that each layer’s

interconnects had.

The development of each new process costs the company millions of dollars, and

to make a test wafer just for transmission lines would be very expensive. To help

reduce costs the designers rely on computer simulations to tell them how a layout

will react before they are ever produced. The scalable model that was created would

help them know how the transmission lines would act under given frequencies.

2 Chapter 1 Introduction

1.2 Transmission Line Uses

In wafers that have devices built into the silicon, these devices are connected with

transmission lines in the interlayer dielectric (ILD). These lines are made of copper

because of its high conductivity. At low frequencies these transmission lines would

appear to have little or no reactance, but at the higher frequencies that are needed

for modern computing even the connecting transmission lines must be modeled. It

was important to know how these lines react because at high frequency the reactence

of a line could cause data to be lost.

The project was to create a scalable model for both single ended and coupled

transmission lines. The model would predict the conductance, resistance, capacitance

and, inductance. It was important that they be modeled up to 10GHz, not that these

lines would ever be used at those speeds but to make sure the model was accurate.

1.3 Why This Is Important

The model was scalable in that the designer would be able to enter the desired line

width, depth, length, and for the coupled lines, the separation. Extreme limits were

used to make sure all feasible options were available to the designer. We limited the

length of the line to 1mm, the width of the line to 50µm, and the separation of the

lines to 50µm. The depths of the transmission lines are set at specific heights that

are predetermined by the process specifications.

A project like this had been previously done to model another process’s transmis-

sion lines and the designers found the convenience of it enough that they requested

another model be made. Just as before, one needed to compare the predicted model

against something, but the whole point of this is to save money so having test wafer

made just for this would be too expensive. Instead it was decided that the model

1.3 Why This Is Important 3

would be compared to a trusted electromagnetic simulator made by Ansys called

Maxwell Q2D. It had previously been tested to be accurate within 10%, and ignoring

the ends of the lines had not been a problem before at Maxim. These simulations

would help build a Matlab code that would predict the behavior of the interconnect.

The final Matlab code was put into SPICE to be used by the engineers.

4 Chapter 1 Introduction

Chapter 2

History

2.1 History of Maxim Integrated Products

Jack Gifford and others founded Maxim Integrated Products in April 1983 in Sun-

nyvale, California. The founders felt that there was a market for a supplier of high-

quality analog and mixed-signal integrated circuits. By 1987 they were already turn-

ing a profit. Maxim now boasts annual revenues of more than 2 billion dollars [1],

and has locations throughout the world, with their headquarters located in San Jose,

California. [2]

Maxims products have been designed for use in a variety of products for companies

such as Apple, Samsung, and Cisco. A large portion of the company’s business in

the years leading up to 2013 came from Samsung, with 50% of the companys revenue

coming from that company alone. Maxim designed the no-touch swipe-to-answer

system that was used on the Samsung, Galaxy S4 cell phone.

6 Chapter 2 History

2.2 History of Semiconductor Devices

Michael Faraday was the first man known to see the effects of semiconductors. In 1833

he noticed that the resistance of silver sulfide decreased with temperature, which was

contrary to the reaction of metals. In 1878 Edwin Herbert Hall discovered the Hall

effect, in that he discovered that charge carriers in solids are deflected in magnetic

fields. And, in 1899 Eduard Riecke theorized that there were both positive and

negative charge carriers with different mobilities. These were important discoveries

that were necessary for the understanding of semiconductor devices. [3]

The most essential of all semiconductor devices are field effect transistors (FET).

A FET is built using doped semiconductors and uses electric fields to move charges

and build paths for current to flow. FETs are the basic building block of modern

computing, being needed for logic gates, as well for digital memory. All semiconduc-

tors must be connected to build a circuit, and they are connected using transmission

lines.

2.3 Transmission Line Theory

There are two types of transmission line that are used in modern in-wafer: single

ended lines, and coupled lines. Coupled lines are favored in many instances because

having two lines is a way to assure correct data transfer.

The capacitance per length of a single transmission line is:

L= εrε0KCL

h, (2.1)

where εr is the relative permittivity of the material, ε0 is the relativity of free space,

KC1 is the capacitive fringing factor, w is the width of the transmission line and h is

the height between the bottom of the transmission line and the ground plane below.

2.3 Transmission Line Theory 7

The mutual capacitance per length of coupled transmission line is:

L= εrε0KCLKL1

d2, (2.2)

where KL1 is the inductive fringe factor and d is the distance between the two lines. [5]

The impedance, Z0, of the of the line was calculated based on thickness of oxide

layer, oxide material, and line width assuming substrate is grounded. The equation

C, (2.3)

The resistatance of in the shunt term, RSi is calculated using the conductivity of

the silicon,the relativity of free space and, relative permittivity of the material:

RSi =ρεrε0CSi

, (2.4)

The series inductance, L, is found using the Greenhouse equation [4]:

L =N2µrµ0A

l, (2.5)

With N being the number of turns (which will be a scaling factor), µr is the

relative permeability, µ0 is the permeability of free space, l being the length of the

coil, and A is the area.

8 Chapter 2 History

Chapter 3

Process

3.1 Maxwell Q2D

The modeling process started in Ansys Maxwell Q2D, a software that was made for

the purpose of modeling transmission lines. The process of creating a two-dimensional

model was fairly straightforward. Going into the pull down menus in the software,

one enters the oxide material, which will give the resistivity in Ωµm. The ILD is then

set in the same way. The dielectric constant for silicon, silicon oxide, silicon nitride,

and plastic need to be set or confirmed they are the correct values.

The next step is to set up the specific layout of the transmission lines to be

simlated. The first part was to create a box of silicon. It had to be 300µm in height

to meet specifics for the SBC18 process, and the width was set to be infinite. This

was done so we dont have complications caused by edge effects. The ILD was set

on top of the silicon and was set to be 11.51µm thick. Next, the nitride layer was

created on top of the ILD and was set to be 0.6µm thick. A copper layer was added

to the bottom to supply a constant ground reference. The thickness was set at 150µm

but the exact thickness of copper was inconsequential. The last thing that will be

10 Chapter 3 Process

Figure 3.1 A cross section of a metal layer 6 transmission line. This wasnot drawn to scale.

constant for all the simulations was a cap layer of plastic over the top of the nitride;

the thickness of this is also unimportant.

The metal layers were then added. For the SBC18 process there are 6 different

layer options for our transmission lines, and each of these has a specific unchanging

height associated with it. The specific heights have been withheld for copyright, but

suffice it to say, metal layers 5 and 6 are thicker than 1 through 4. Metal layer 6 is

unique in that the ILD, nitride and plastic above had to be raised in order for it to

fit around it as seen in Figures 3.1 and 2. This is approximated by building a box of

each with the walls of the box being the same thickness as the layer is deep.

The simulations were run many times per metal layer, cycling through predeter-

mined widths, never being thinner than the metal layer was tall. The range in widths

is far beyond what one would expect the designers to ever use, but it was done for the

sake of thoroughness. For the coupled transmission lines the spacing between then

lines was also scaled along with the line widths. After the tests were run, the data

were saved and then imported into Matlab to begin the fitting.

3.2 Circuit Analysis 11

Figure 3.2 A cross section of metal layer 6 coupled transmission lines. Thiswas not drawn to scale.

3.2 Circuit Analysis

But before the data in Matlab could be used, there had to be a circuit design that

would properly represent the circuit. A simple transmission line has series inductance

and series resistance with two capacitance terms. The transmission line model that

was used was more accurate, in order to take into account the addition of the substrate

between our transmission line and the copper ground. Being that this transmission

line will be used for high frequencies, terms that take skin effect into account must

be present.

The terms for skin effect were simplified into only having one additional set of

inductance and resistance terms. In order to make the model more accurate for very

high frequencies, more would need to be added in parallel. It has been shown and

approved by the engineers that this additional complication was not be needed.

Figure 3.3 The chosen layout for the single ended line.

3.3 Matlab

In Matlab, the constants were set first, the same as in Maxwell Q2D. We then set

up a selection of if-then statements that would change the height, widths, resistivity,

and separation (for the coupled lines). The first of the calculations was calculating

the series resistance and series inductance terms. As stated our model will take

into account skin effect, so those inductance and resistance terms were next to be

calculated. Our shunt terms were: the capacitance caused by the ILD, the capacitance

caused by the silicon, and the shunt resistance caused by the silicon. Figure 3.4 shows

this layout.

The coupled lines were a bit more complicated. The whole circuit was doubled

to represent each line and would have the addition of coupling, which brought linked

inductance and mutual capacitance between the lines. This was was put into Y-

parameters. The Y-parameters are then changed into Z-parameters, which will be

needed for translating this into SPICE, the desired language for the final product.

3.3 Matlab 13

Figure 3.4 The chosen layout for the coupled line.

Once the basic equation were made, the scaling terms in our equations were then

manually changed to best fit the imported data from Maxwell Q2D. There were two

scaling factor used: one that would be combined with our mutual capacitance and

another to help fit the DC resistance. Simple guess and check was used to fit these

scaling factors.

Chapter 4

Results

4.1 Single Lines

The model suggested the resistance of our single ended transmission line increased

almost to linearly with frequency for most metal layers and, inductance decreased as

frequency rose. Conductance also rose with frequency and capacitance tended to fall

as frequency increased.

These predictions stick close to the values that were accepted as reliable from

Maxwell Q2D. The model stayed consistently within 10% of the predicted values. In

Figures 4.1-4 one can see the results for the 10µm wide metal layer 6 single transmis-

sion lines:

4.2 Coupled Lines

The model of the coupled lines accurately predicted that the resistance would increase

linearly just as the single ended lines did. The modeling of the shunt capacitance cor-

rectly showed a decrease as frequency rose. Modeling the series inductance correctly

16 Chapter 4 Results

predicted the initial values within error, but the frequency dependence was unreliable.

The mutual inductance between the two lines were also off, the results from Maxwell

Q2D were approximately flat, but the model showed a rise as frequency increased.

The conductance was more accurate and showed an increase with frequency. The

mutual capacitance was also accurate, showing an increase with frequency.

Figure 4.1 Single 10µm wide 1000µm long metal layer 6, Resistance vs.Frequencey

Figure 4.2 Single 10µm wide 1000µm long metal layer 6, Inductance vs.Frequencey

4.3 Limitations 17

Figure 4.3 Single 10µm wide 1000µm long metal layer 6, Conductance vs.Frequencey

Figure 4.4 Single 10µm wide 1000µm long metal layer 6, Capacitance vs.Frequencey

4.3 Limitations

The limitations of the model for single and the coupled lines are; that the results

could be more accurate if the skin effect portion of the topology was improved. This

could be achieved by putting an inductor and resistor in parallel with the skin effect

resistor, R2 in Figure 3.4. This was not done in this model because at the time of

the experiment, the results would not need any more accuracy than what was already

achieved.

Figure 4.5 Coupled 10µm wide 1000µm long with 10µm separation metallayer 6, Resistance vs. Frequencey

Figure 4.6 Coupled 10µm wide 1000µm long with 10µm separation metallayer 6, Inductance vs. Frequencey

The values given by the model were within 10% of the simulated values, but with

many of the coupled lines the frequency dependence was lost. This was brought to

the attention of the engineers that would be implementing the model and it was said

that further work to improve the performance was unnecessary. With that the model

was called complete.

4.3 Limitations 19

Figure 4.7 Coupled 10µm wide 1000µm long with 10µm separation metallayer 6, Conductance vs. Frequencey

Figure 4.8 Coupled 10µm wide 1000µm long with 10µm separation metallayer 6, Capacitance vs. Frequencey

Figure 4.9 Coupled 10µm wide 1000µm long with 10µm separation metallayer 6, Mutual Inductance vs. Frequencey

Figure 4.10 Coupled 10µm wide 1000µm long with 10µm separation metallayer 6, Mutual Inductance vs. Frequencey

Chapter 5

Conclusion

5.1 Summary

While on an internship with Maxim Integrated Products, a multi-billion dollar in-

tegrated circuit designing company, the author constructed a scalable interconnect

model of the transmission lines for the Tower Jazz SBC18 process. This code con-

tained models of single and coupled transmission lines. The model would be scalable

over width, depth and length, as well as separation for the coupled lines.

The process started on Ansys Maxwell Q2D, running hundreds of simulation,

systematically changing the width/depth/separation to have reliable data for com-

parison. A topology was then selected that would be an accurate approximation

for the transmission lines. Fitting the data was done by calculating the values for

the inductance, capacitance, resistance and conductance. Scaling factors were also

needed.

The model was deemed acceptable with the values being within 10% of the data

from Q2D. The frequency dependence was very close for single ended lines and coupled

lines except for the inductance on the couple transmission lines, where only the general

22 Chapter 5 Conclusion

values were close.

5.2 Future Work

There is a need for more scalable interconnect models for Maxim and other integrated

circuit companies. This was the second time that Maxim had a transmission line

model created for one of their processes and it stands to reason that they will continue

to need more. As speeds in computing increase the need for an understanding of how

each transmission line will react at high frequencies becomes more important.

As stated before the short-comings of the model made for the SBC18 process were

accepted, but as speeds increase the model will have to be refined to better match

the frequency dependence. Another problem that may arise is that there was no

comparing the model to a physical wafer. The whole point of the experiment was to

save money and creating a wafer of just transmission lines would not be cost effective.

But as time goes on it will be important to recalibrate the values given by Maxwell

Bibliography

[1] History of Maxim http://www.maximintegrated.com/company/profile.cfm

[2] History of Maxim http://web.archive.org/web/20080720003507/http://www.

sanjosemagazine.com/main/?p=450

[3] Lidia ukasiak and Andrzej Jakubowski. “History of Semiconductors,” Journal of

Telecommunications and Information Technology. (2010)

[4] Inductance equation picture http://www.allaboutcircuits.com/vol 5/chpt 1/6.

[5] Charles S Walker. “Capacitanc, Inductance and Crosstalk Analysis,”(Norwood,

MA 1990), pg 57.

24 BIBLIOGRAPHY

Appendix A

Matlab Code

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Transmission lines for Tower Jazz SBC18 %

% June 6, 2011 %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all; %clear all variables

close all; %close all figures

length=1000; %Lenght of line in umeters

% user is to activate user interactive mode, set to 1 or 0

user =1;

%checkspek is used when checking SPECTRE values, set to 1 or 0

checkspek=0;

done=0;

if (user ==1)

26 Chapter A Matlab Code

P1orP2=input(’Ports 1&3 or 2&4 for Ls Rs Cs Gs, 1&2 for CM, 1&2 for LM [1 2 3 4]: )’);

layer=input(’Enter Layer Number from 1 to 6: ’);

if (layer==1)

width=input(’Choose Width [.52 .8 1 2 5 8 10 20 50]: ’);

seperation=input(’Choose Seperation [.52 .8 1 2 5 8 10 20 50]: ’);

if (layer==2)

if (layer==3)

if (layer==4)

if (layer==5)

width=input(’Choose Width [1.59 2 5 8 10 20 50]: ’);

seperation=input(’Choose Seperation [1.59 2 5 8 10 20 50]: ’);

if (layer==6)

width=input(’Choose Width [2.81 5 8 10 20 50]: ’);

seperation=input(’Choose Seperation [2.81 5 8 10 20 50]: ’);

if (user==0)

layer=1;

width=5;

seperation=.52;

P1orP2=3;

% Enter constants here

c0=3e14; %Speed of light in um/s

u0=4*pi*1e-13; %permeability in H/um

e0=8.854e-18; %permittivity in F/um

ICEo=e0;

ros_Al=2.67e1; %conductivity of Al in S/um

ros_Cu=5.01e1; %conductivity of Cu in S/um

er_Si=11.9; %Dielectic constant of Silicon

ICEsi=er_Si;

ros_Si=12.5; %conductivity of Silicon

er_Si02=4; %Dielectic constant of Silicon Oxid

er_Si3N4=7; %Dielectic constant of Silicon Nitr

er_Pst=4.2; %Dielectic constant of plastic

ICEr=3.9; % Rel permittivity of thermal SiO2

t_substrate=300;%Thickness of substrate

p_sub=80000; %substrate resistivity in ohm*um

%skewing parameters

rwaferx=1;

cwaferx=1;

%end skewing parameters

% Set case to account for up or down. Set up_or_down to -1,0, or +1.

%up_or_down=-1;

up_or_down=0;

%up_or_down=1;

% Sheet Resistivities in Ohm/sqr

if (layer ==1)

ps_m = 0.082;

if (layer ==2)

ps_m = 0.082;

if (layer ==3)

ps_m = 0.082;

if (layer ==4)

ps_m = 0.066;

if (layer ==5)

ps_m = 0.018;

if (layer ==6)

ps_m = 0.0105;

%Thinkness of Metal Layers in umeters

if (layer ==1)

t_m = ###;

if (layer ==2)

t_m = ###;

if (layer ==3)

t_m = ###;

if (layer ==4)

t_m = ###;

if (layer ==5)

t_m = ###;

if (layer ==6)

t_m = ###;

% Min seperation of the two lines

s_min=t_m;

% Thinkness variance (up down)

if (layer ==1)

t_ud_m = 0.08 * up_or_down;

if (layer ==2)

if (layer ==3)

if (layer ==4)

if (layer ==5)

if (layer ==6)

. (Code left out to save space)

freq=frequency*1e-9;

w=2*pi*frequency;

%Convert S-parameters to Y-Parameters

Y=StoY(S);

Z=StoZ(S);

%Extract Simulated L, R, C, G

Rsim=real(-1./Y(:,3));

Lsim=imag(-1./Y(:,3))./(2*pi*freq1);

Gsim=2*real(Y(:,1)+Y(:,3));

Csim=2*imag(Y(:,1)+Y(:,3))./(2*pi*freq1);

Lmsim=imag(Z(:,3))./(2*pi*freq1);

Cmsim=imag(-Y(:,3))./(2*pi*freq1);

%metal drawn-to-finished feature size change

dm =t_ud_m;

lf_m=length+2*dm;

wf_m=width+2*dm;

% ILD thickness in um;

if (layer == 1)

tox= 1.14;

if (layer == 2)

tox= 2.46;

if (layer == 3)

tox= 3.78;

if (layer == 4)

tox= 5.1;

if (layer == 5)

tox= 7.72;

if (layer == 6)

tox= 11.31;

%use er=4.2 for plastic, er=2.85 for bcb, er_eff=2.5 for air since BCB is still

%5um thick over lines on unpackaged dice

if (layer == 1)

erair=3.9;

if (layer == 2)

erair=3.9;

if (layer == 3)

erair=3.9;

if (layer == 4)

erair=3.9;

if (layer == 5)

erair=3.9;

if (layer == 6)

erair=4.2;

%calculate p.u.l. C for ILD_global

er_eff_ILD = (ICEr+erair)/ 2 + (ICEr-erair) / 2 * (1/ sqrt(1 + 12 *tox/ wf_m));

Z0_ILD=120*pi/ (sqrt(er_eff_ILD) *(wf_m/ tox+ 1.393+ 0.667* log(wf_m/ tox + 1.444)));

CILD = sqrt(er_eff_ILD)/(c0*Z0_ILD);

%calculate p.u.l. C for Si

w_eff_Si = CILD * tox/(ICEr*ICEo);

er_eff_Si = (ICEsi + ICEr) / 2 + (ICEsi - ICEr)/2 * (1 /sqrt(1 + 12 *

t_substrate/w_eff_Si));

Z0_Si = 120 * pi/ (sqrt(er_eff_Si) * (w_eff_Si/ t_substrate + 1.393 + 0.667 *

log(w_eff_Si/t_substrate + 1.444)));

CSi = cwaferx * sqrt(er_eff_Si) / (c0 * Z0_Si);

GSi = (1/rwaferx*1) * (1/p_sub) * CSi / (ICEsi*ICEo);

if (layer ==1)

CM_scale=0.773.*seperation.^0.214;

if (layer ==2)

if (layer ==3)

if (layer ==4)

if (layer ==5)

CM_scale=0.955.*seperation.^-0.10;

if (layer ==6)

CM_scale=1.466.*seperation.^-0.16;

% modified based on TX09Z measurement data

xw=0.85*(0.95/(wf_m^(0.22)));

CM=CM_scale*(((2/t_m^0.43)*(135.7/((4*seperation/s_min)^xw)+yw)*

1e-18*(1.0+0.04)) *(lf_m/2)); %used to have "pack" in it. Combined Cm and CM into one formula

%calculate shunt equivalent circuit parameters for substrate

f0pt25=.2;

f0pt2=.2;

f0pt4=.5;

CILD1=CILD*(lf_m/2)*(1-(1-(f0pt25*wf_m^(f0pt2)+f0pt4))*(s_min/seperation));

CSi1=CSi*(lf_m/2)*(1-(1-(f0pt25*wf_m^(f0pt2)+f0pt4))*(s_min/seperation));

RSi1=1/(GSi*(lf_m/2)*(1-(1-(f0pt25*wf_m^(f0pt2)+f0pt4))*(s_min/seperation)));

%calculate series inductance terms (internal and external)

eps = -0.008044;

x0 = -0.196224;

aa = 7.534187;

B1 = log(t_substrate/(1));

C1 = 1-(B1/aa-x0);

tscale=1-0.5*(C1+sqrt(C1*C1+4*eps*eps))+.051;

%tscale=1;

Ldc =( 2e-13 * lf_m * (log(2 * lf_m/ (wf_m +t_m)) +0.5005 +((wf_m+t_m)

/(3*lf_m))))*tscale;

Ldc500um_norm= (2e-13* 500* (log(2* 500 /(wf_m +t_m)) +0.5005 +

((wf_m+t_m)/ (3*500))))*tscale;

if (layer ==1)

Li = 0.01 *Ldc500um_norm;

if (layer ==2)

if (layer ==3)

if (layer ==4)

if (layer ==5)

if (layer ==6)

Li = 0.13* Ldc500um_norm;

Le = Ldc - Li;

GMD=wf_m+seperation;

Lm=2e-13*lf_m*(log((lf_m/GMD)+(1+(lf_m^2)/(GMD^2))^(0.5))-

(1+(GMD^2)/(lf_m^2))^(0.5)+(GMD/lf_m));

Km=Lm/Le;

%DC resistance

Rdc = ps_m * (lf_m/ wf_m);

if (layer == 1)

n0=100;

if (layer == 2)

n0=100;

if (layer == 3)

n0=100;

if (layer == 4)

n0=100;

if (layer == 5)

n0 = (1/((wf_m)^2.25)) + 5.2;

if (layer == 6)

n0 = (1/((wf_m)^2.25)) + 5.0633;

%calculate R1 and R2 for series branch

R2 = (n0+1) / n0*Rdc;

R1 = Rdc * R2 / (R2-Rdc);

% Calculations for Y21

Zcm=1./(j*w.*CM); %Same as Zb0

Zsh=(1./(j*w.*CILD1))+(RSi1.*(1./(j*w.*CSi1)))./(RSi1+(1./(j*w.*CSi1))); %Same as Zb2

Zs1=(j*w.*Le)+(((j*w.*Li+R2).*(R1))./((j*w.*Li+R2)+(R1))); %Same as Zb1

Zb3=((Zsh.*(1./(j*w.*CM)))./(Zsh+(1./(j*w.*CM))))+Zs1; %Check

Zb4=((Zsh.*Zb3)./(Zsh+Zb3))+(1./(j*w.*CM)); %Check

Zb5=(Zs1.*Zb4)./(Zs1+Zb4);

Zetotal=(Zb5.*Zsh)./(Zb5+Zsh);

%for Cases 1 and 2

y11=1./Zetotal;

y21=((-Zcm./Zs1)-(2.*Zsh./Zs1)-(2.*Zcm.*Zsh)./(Zs1.*Zs1)-(Zcm./Zs1)+

(y11.*Zcm.*Zsh./Zs1))./(Zcm+Zsh+(Zcm.*Zsh./Zs1));

%for case 3 (Compare CM with ports of P1&P2 open P3&P4)

ZCM1=((Zs1).*(Zsh))./((Zs1)+(Zsh))+Zcm;

ZCM2=((ZCM1.*Zsh)./(ZCM1+Zsh))+Zs1;

ZCM3=(ZCM2.*Zcm)./(ZCM2+Zcm);

ZCM4=(ZCM3.*Zsh)./(ZCM3+Zsh);

YCM11=1./ZCM4;

YCM21=-((2.*Zs1./Zsh)+2+(2.*Zs1./Zcm)-(YCM11.*Zs1))./(Zcm+(Zs1.*Zcm./Zsh)+Zs1);

%for case 4 (Compare LM with ports on P1&P2 short P3&P4)

ZLm=j*w.*Lm;

ZLe=j*w.*Le;

Zs0=(((j*w.*Li+R2).*(R1))./((j*w.*Li+R2)+(R1)));

YLM11=((1./Zcm)+(1./Zsh)+((Zs0+ZLe)./((Zs0+ZLe-ZLm).*(Zs0+ZLe+ZLm)))); %4 port Y11

YLM21=((Zs0+ZLe)./((Zs0+ZLe-ZLm).*(Zs0+ZLe+ZLm)))-(1./Zcm)-(1./(Zs0+ZLe-ZLm)); %4 port Y21

ZLM21=(-YLM21./(YLM11.*YLM11-YLM21.*YLM21)); %4 port Z21

Y31=-YLM11+(1./Zcm)+(1./Zsh); %4 port Y31

Y41=-YLM21-(1./Zcm); %4 port Y41

Zs = (((w.* j* Li+ R2)* R1)./(R1 + R2 + j* w.* Li)) +(j* w.* Le);%Total Z for seires

Ys = 1 ./ Zs;

Zp = ((-j./(w.*2*CSi1))*(RSi1/2))./((RSi1/2)+j*(-1./(w.*2*CSi1)))+(-j./(w.*2*CILD1));

%Total Z for shunt

Yp = 1 ./ Zp;

Z011maxwell=sqrt((Rpul+j*w.*Lpul)./(Gpul+j*w.*Cpul));

Z012maxwell=sqrt((Rmpul+j*w.*Lmpul)./(Gmpul+j*w.*Cmpul));

gammamaxwell=sqrt((Rpul+j*w.*Lpul).*(Gpul+j*w.*Cpul));

Z0DDmaxwell=2*(Z011maxwell-Z012maxwell);

Z0CCmaxwell=(1/2)*(Z011maxwell+Z012maxwell);

%Zmodel0DD11=sqrt(ZLM11(:,1)./(YLM11(:,1)));

%Z0DD11=sqrt(Z(:,1)./(Y(:,1)));

%%%%%%%%%%%%%%%%

% Plot results %

%%%%%%%%%%%%%%%%

fig=1;

if (P1orP2==1 || P1orP2==2)

figure(fig);fig=fig+1;

plot(1e-9*frequency,real(-1./y21));hold on

plot(1e-9*freq1,Rsim,’g-o’);

%axis([0 10 3 6])

plot(1e-9*frequency,length*1e-6*Rpul,’r--’);hold off;grid on

legend(’Model’,’SPECTRE’,’Q2D’);

xlabel(’Frequency (GHz)’)

ylabel(’Resistance (\Omega)’)

title([’R vs. f for two ’,num2str(width),’\mum wide by 1000 \mum long M’

,num2str(layer), ’ lines with a ’ ,num2str(seperation), ’\mum seperation’])

boldify

plot(1e-9*frequency,1e9*imag(-1./y21)./w);hold on

plot(1e-9*freq1,1e9*Lsim,’g-o’);

%axis([0 10 1.15 1.45])

plot(1e-9*frequency,1e9*length*1e-6*Lpul,’r--’);hold off;grid on

ylabel(’Inductance (nH)’)

title([’L vs. f for two ’,num2str(width),’\mum wide by 1000 \mum long M’

boldify

plot(1e-9*frequency,2e12.*((imag(y11)+imag(y21))./w));hold on

plot(1e-9*freq1,1e12*Csim,’g-o’);

%axis([0 10 .0 .5])

plot(1e-9*frequency,1e12*length*1e-6*(Cpul),’r--’);hold off;grid on

ylabel(’Capacitance (pF)’)

title([’C vs. f for two ’,num2str(width),’\mum wide by 1000 \mum long M’

boldify

plot(1e-9*frequency,2*(real(y11)+real(y21)));hold on

plot(1e-9*freq1,Gsim,’g-o’);

%axis([0 10 0 1e-3])

plot(1e-9*frequency,length*1e-6*Gpul,’r--’);hold off;grid on

ylabel(’Conductance (S)’)

title([’G vs. f for two ’,num2str(width),’\mum wide by 1000 \mum long M’ ,

num2str(layer), ’ lines with a ’ ,num2str(seperation), ’\mum seperation’])

boldify

% figure(fig);fig=fig+1;

% plot(1e-9*frequency,2e12.*imag(1./Zsh)./w);hold on

% plot(1e-9*frequency,1e12*length*1e-6*(Cpul-Cmpul),’g-o’);hold off; grid on

% legend(’Model’,’Q2D’);

% xlabel(’Frequency (GHz)’)

% ylabel(’Capacitance (pF)’)

% boldify

if (P1orP2==3)

%plot(1e-9*frequency,1e12*CM*2*ones(size(frequency,1),1));hold on

plot(1e-9*frequency,1e12*imag(-YCM21)./(2*pi*frequency));hold on

plot(1e-9*freq1,1e12*Cmsim,’g-o’);

axis([0 10 0 .2])

plot(1e-9*frequency,1e12*length*1e-6*(Cmpul),’r--’);hold off;grid on

ylabel(’Capacitance (pF)’)

title([’Mutual C vs. f for two ’,num2str(width),’\mum wide by 1000 \mum long

M’ ,num2str(layer), ’ lines with a ’ ,num2str(seperation), ’\mum seperation’])

boldify

if (P1orP2==4)

%plot(1e-9*frequency,1e9*Lm*ones(size(frequency,1),1));hold on

plot(1e-9*frequency,1e9*imag(ZLM21)./(2*pi*frequency));hold on

plot(1e-9*freq1,1e9*Lmsim,’g-o’);

axis([0 10 0.8 1.4])

plot(1e-9*frequency,1e9*length*1e-6*Lmpul,’r--’);hold off;grid on

ylabel(’Inductance (nH)’)

title([’Mutual L vs. f for two ’,num2str(width),’\mum wide by 1000 \mum long M’

boldify

if (checkspek==1)

close all;

Lsim(1)

scalable interconnect model by erik harker a senior …

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