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Nine-Element Lumped Metal-Insulator-Metal (MIM) Capacitor Modelfor RF Applications.

Kalavathi Subramaniam 3,Student Member,IEEE, Albert Victor Kordesch2, Senior Member,IEEE,Mazlina Esa', Member, IEEE

'Dept. OfRadio Communications Engineering,Faculty ofElectrical Engineering,

University of Technology, Malaysia (UTM),Skudai, Johor,Malaysia.

2 Silterra Malaysia Sdn. Bhd.,Kulim, Kedah

3 kala subramaniam(,silterraxcom

Abstract- In this paper we suggest using a nine

element model to represent metal-insulator-metal(MIM) capacitors up to 20GHz. The model suggested isa nine element pi circuit that includes interconnectresistance and inductance as well as substrate parasitics.This paper also identifies the procedures to extract thenine element lumped MIM capacitor model and tooptimize the model to fit the measured data up to theself resonant frequency. A good fit between themodeled and measured effective capacitance and Q

factor have been obtained in this work.

1. Introduction

The accuracy of passive element circuit simulationmodels becomes very essential as the operatingfrequency approaches the 20 Gigahertz range. This isbecause at RF and higher frequencies, any passiveelement does not behave as a pure resistor, capacitor or

an inductor. A capacitor that operates at RF frequenciesalso exhibits some amount of resistance and inductance,and some parasitic capacitance. These elements will berepresented by ideal capacitors, resistors and inductors ina lumped element model.

This paper proposes a method to extract thelumped element values for a MIM capacitor. Themethod is suggested for a 9 element lumped capacitormodel.

Seven element lumped models for MIM capacitorsat radio frequencies have been reported in the past [1].Figure l(a) illustrates the model reported in [1]. Thereported model comprises the nominal capacitor,interconnect resistance and inductance on the mainseries branch. The two parallel branches consist ofparasitic capacitance and resistance between the top andbottom plates to the substrate respectively. These are

given by Ra, Rb, Ca and Cb. The difference betweenthe model in Figure l(a) and the model suggested inthis paper is that the model suggested here includes the

frequency dependent resistance and capacitance in thesubstrate. Also, it excludes the top and bottom plateresistance to substrate as the MIM capacitor isseparated by the silicon dioxide from substrate andthus, only capacitance is observed there.

Figure 1 (b) illustrates the 9 element pi model of a

capacitor that is suggested in this paper. This model ismade up of three branches. The series branch comprisesthe nominal capacitance of the dielectric (Cs), theresistance of the interconnects (Rs) and inductivebehavior of the interconnects (Ls). Parasitic losses inthe capacitor are represented by two parallel brancheswhich are made of the top and bottom plate oxidecapacitance to the substrate (Cox). The bottom plate hasa much larger parasitic capacitance compare to the top.The frequency dependent substrate loss parasiticelements (Rsi and Csi) are connected in parallel to thesubstrate . In general, symmetrical values for thecomponents of the two parallel branches of the pi

circuit could only be expected if the capacitor isdesigned to be absolutely symmetrical. However inmost cases the values will be different as to correspondto the nonsymmetrical design such as in metal-insulator-metal (MIM) capacitors. In MIM capacitors,the top and bottom plates of the capacitor are differentmetal layers, thus the distance from the plates to groundare different resulting in different oxide capacitancebetween top plate to ground and bottom plate to ground.This is different from metal finger capacitors, where a

symmetrical design can be achieved by having the samenumber of fingers for the left and right plates.

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Csr mwr capaciEr

*n.rcihncu, the me~tats indits dbe to

XC psreti: between th.opttm. pite npdteand

(a)Ls Cs

IS

Cox2-

Csi2 I Rsi2

.t.

(b)Figure I (a) :Seven-Element Lumped MIM Capacitor Model [3]

(b): Nine-Element Lumped MIM Capacitor Model

2. Design

In this work, different sizes of MIM capacitors havebeen designed. They are MIM 5.5 which means theinsulator layer is in between Metal 5 and Metal 6.These have been fabricated using Silterra's industrystandard 1 80nm RF CMOS 6 layer metal processtechnology. Figure 2 illustrates the top view layout andthe cross section of a 0.9pF MIM capacitor. Thecapacitor has been constructed under the top metalbecause this will provide greater distance from groundand thus results in lower parasitic capacitance toground.

t.FtIIRI

.4.nk'C6 ..1.

j:TQt P:.ti

.. t.GA~~~~~~~~~~~~~~~~~~~~~~~~~~:ii

(b)

Figure 2: 0.9 pF MIM Capacitor (a)Layout(b) Cross Section [3]

The MIM capacitors were designed to beconnected to Ground-Signal-Ground (GSG) pads.Figure 3 is an illustration of a 1.8 pF MIM capacitor ina GSG pad configuration. The signal pads areconnected to the device using stacks of metal 1 to metal6 and vias interconnecting them. Metals I to 6 were

stacked to reduce resistance in the interconnects as theresistance in the metal lines will be parallel. Thesemetal interconnects are connected using vias to avoidany potential difference in between them. This willeradicate the existence of overlap capacitance inbetween the metal lines of different layers. The groundpads are connected also using stacks of metal lines as inthe signal interconnects. However, the ground pads are

also connected to the substrate of the wafer.Another important area to consider is de-

embedding. This can be performed using severalmethods. Reference [2] provides a good insight on thecommon de-embedding structures. To provide goodrepresentation of the device, an appropriate de-embedding technique must be chosen. In this work,open-short de-embedding is used. Open and shortstructures were also fabricated on the same chip as thedevice. For open structures, they are only replicas of thestructure in Figure 3 without the device under test(DUT) MIM Capacitor in the middle. The shortstructure is designed by also removing the MIMcapacitor from Figure 3 and then connecting the signaland ground probes together.

(a)

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| + ~~~E caputor

.g.~~~~~~~~~~..

Figure 3: 1.8 pF MIN Capacitor connected to GSG pads

3e Measurements

Two port S-Parameter measurements were made usingInfinity ground-signal-ground probes and a E8364B50GHz vector network analyzer. Before initializingmeasurements, the planarization of the infinity probes isperformed. This is to ensure that the Ground, Signaland Ground tips of the probe have the same amount ofcontact with the measurement device. After this, the testsystem is usually calibrated using the Impedance-Standard-Substrate (ISS) that has short, open, thru andload structures on it.

4. Model Extraction

Once reliable measurements were obtained, the modelswere extracted using some simple procedures. The S-parameter measurements were first converted to Yparameters. The mathematical equations for thisconversion are given in equations 1 to 4 [4].

y _o (1 IO .O+ u 2

(1+ SD)(l + S2)-S12 S21

g 4~~Y(-2S12)y_2=

(1 + SX)(1 + S22) - S21

y21 = Y0(-2S21)(1 + SI Xl + S22) SU21

me2 Y(1 + SI)- 12S21

(1 + S11)(1 + S22) - SU2S1

[1]

[2]

[4]

[4]

The model suggested in Figure 1 (b) can be representedby the pi circuit in Figure 4. Y parameter representationof the circuit in Figure 3 is given in Equation 5.Equations 6 to 8 were derived from Equation 5.

YpA+Cypi =A+1_n

-CD)

-B+C/A =Yll + (Y12 + Y21)12

- + (Y12 +Y21)

2

7)

U )

The circuit in Figure 1 was made equivalent to thecircuit in Figure 3. From Figure 1, A, B and C valuescan be derived as given by equations 9 to 11.

A~~~~~~~~~

- - (9)

[(1/jaCtl) + (If R2B + Ja72) ]

wCox2fi + (1 R5i2 + wC32Z -

[R5 + J(a-Z, - l l WCO)

From the set of data for YlI, Y12, Y21 and Y22, allthe elements of the MIM capacitor model were

extracted. However, the elements were extracted only atcertain frequencies.

Cs is the nominal capacitance, thus it was extractedfrom low frequency or dc measurements. Fromequation 11, it is evident that Rs can be extracted fromthe real portion of 1/C. This was done using the data atlow frequency. Knowing Cs, Ls was extracted from theimaginary portion of 1/C data after resonance at highfrequency.

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The method of extraction of the parasitic elementsfrom the parallel branch is similar for both branches. Itis known that the oxide capacitance (Cox) is frequencyindependent. This dominates the parallel branch at lowfrequencies but as the frequency increases, thefrequency dependent substrate loss capacitance (Csi)and resistance (Rsi) have bigger influence on theparallel branch. Hence, Coxl and Cox2 were extractedfrom the imaginary portion of 1/A or 1/B respectivelyat low frequency. Cs however was extracted from theimaginary portion of A (or B) at high frequency. Rsalso a frequency dependent element, and was extractedat high frequency from the real portion of A (or B). Theequations for all these are given below. "A" in theequations should be substituted with B when parasiticelements for the second parallel branch are extracted.

R= real-

L =imag-sC

-1c(imag AX

imagACSi =

( 12)

( 13 )

( 14)

( 15 )

R = 1 (16)Si realA

5. Model Optimization

The extracted values of the elements serve as the initialvalues for the model. After extracting the elements,manual optimization was done so that the modeledeffective capacitance (C) and quality factor (Q) curves

fit the measured curves. Optimization could also bedone to fit the S-parameter curves. The effectivecapacitance and quality factor for the measured datawas calculated using equations 17 andl8.

C -imagY12(17)

2af

imagYl1 (18)

realYl 1

To fit the C curve and SRF, Ls was tuned. It isknown that C is extracted from the imaginary portion ofY12. Thus, tuning Ls will help to fit the C curve and

also self resonant frequency on C curve. Q wascalculated by dividing imaginary Y,, ( or Y22 ) with realY (or Y22 ). From equation 5, it is known that Y,1 is A+ C. Thus, to fit Q, Rs was tuned first. As analternative, Rsi of the appropriate branch could betuned too. Tuning Csi also provides a reasonable fit forQ.

In this work, MIM capacitors ranging from 1.7 pFto 12.7 pF were modeled using this method and theywere able to fit reasonably well to the measured Ccurve with errors less than 5 % and Q curves witherrors of less than 10%. Table 1 summarizes thecomparison between the initial extraction of theparameters and the values after tuning for a 6.8 pFMIM capacitor.

Table I: Initial Extraction and Tuned Values Comparison

Elements Extracted Value Tuned ValueLs (H) 326 E-11 Ole-11Cs (F) 6.8266 E-12 6.8266 E-12Rs (ohm) 1.2499 1.1562Csil (F) 5.4584 E-15 5.4584 E-15Csi2 (F) 1.9931 E-14 1.9931 E-14Rsil (ohm) 6.8456 E+02 6.8456 E+02Rsi2 (ohm) 3.0609 E+02 3.0609 E+02Coxl (F) 2.9001 E-14 2.9001 E-14Cox2 (F) P.7081 E-14 8.7081 E-14

From the table, it can be seen that the initialextraction data has been maintained for all the elementsexcept for Rs and Ls. The difference in extracted andtuned Rs is 9.2 % and difference between extracted andtuned Ls is 8.3 %. The tuning range is within 20 % fromthe extracted value for all the capacitors that have beenmeasured and modeled in this work. Thus it can be saidthat the extraction method and results are reliable anddid not need much tuning.

6. Results

The modeled C and Q curves after parametertuning match reasonably well with the measured C andQ curves. Figure 5 illustrates the comparison ofmeasured and modeled effective capacitance curve (C)for a 6.8 pF capacitor and the model elements that havebeen tabulated in Table 1. For all the capacitors thatwere modeled using this method, the difference betweenthe measured C and Q curves with the modeled C and Qcurves is kept within 6 % up to resonance. The averageerror for effective capacitance over the range from 0 to20 GHz is about 5 %. The comparison curves betweenmeasured and modeled Q are plotted in Figure 6. Thedifference between measured and modeled Q ismaintained within 10% up to the resonant frequency.

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The average difference between the measured andmodeled Q is 6.2 %.

753-12

6.5E-1 2

45E-120

3.5E-1 2

2.5E-1 2

1 5&-12

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12 -

0o8 _ -312 REAL(Model)S-12 IM(Model)

0.6 S2Real(Meas); s ~~~~~~~~~S12Im (M^eas)

0.4

02

0

0 5E+09 1E+10 15E+10 2E+1O 2 5E+10Frequency (Hz)

Figure 5: Measured Vs. Modeled Effective Capacitance

1000

100

1

0.10.0E'

Q Meas Vs. Model

-Q Modeled. Q Measured

_ " -T ----- -- l--- -1---t

+00 5.0E+09 1.OE+10 1.5E+10 2.OE+10 2.5E+'

Frequency (Hz)Figure 6: Measured Vs. Modeled Q Factor

Figures 7 to 10 illustrate the measured and modeledSI 1, S 12, S21 and S22 curves for the same capacitor.The difference between the measured and modeledparameters is kept within 20 %. The measured andmodeled effective capacitance curves for the capacitorsin this work are given in Figure 11.

-S3 1 REAL (mod)-311 IM (Mod)

. S11 Real (Meas).k . S311 Im (Meas)

-.00 - . ,- -1 . __ .~ _. ....................M..*,"

N\ /

. . . .10 5 10 15 20

Freq (GHz)Figure 7: Measured Vs. Modeled S 1I

25

1.2

0.8

tr06

N

0.2

0

-0.2

1

0.6-0.4

00.2

-0.2-

-0A -

5 10Freq (GHz)15 20 25Figure 8: Measured Vs. Modeled S1,!

-S21 RE(Model)S-21 lM(Model)

S21 Real (Meas)4 _ + ~~~~~~~~~S21Imr (Meas8)

0 5 'req (GHZ 1 20 25

Figure 9: Measured Vs. Modeled S2 1

-S22 RE(Model)S-22 IM(Model)

S22 Real (Meas)3S22 Im (Meas)

0 5 10 15 2ii) 25Freq (GHz)

Figure 10: Measured Vs. Modeled SII

5.OOE-11 -

4.00E-11 -

3.00-E1 1 -

2.00E-11 -_- 1 OOE-1I1

O.OE+00-1 .00E-1 1 -

-2.OOE-t 1 -_

-3.OOE-I 1 --

-4.00E- 11

03OE+00 50OE+09 1E.0E+10 15E+10 2.03E10 - 25E+10Frequency (Hz)

Figure 11: Measured Vs. Modeled Effective Capacitance for DifferentSizes

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7. Conclusions

The paper identified a simple method to extract 9element lumped MIM capacitor models for RFapplications. The models extracted using this methodhave fairly good fit to the measured effectivecapacitance and Q factor curves. The difference is keptwithin 10 % up to resonance frequency. The extractedelement values only serve as the initial value and inmost cases optimization is needed to provide better fitto the measured data. However, the difference betweenthe extracted element values and the values after tuningdid not vary much (within 20 %). The elements thatneed to be tuned to fit C and Q curves have also beenidentified in this work. The data presented indicate thereliability of this simple extraction method.

It has been observed in this work that the modelfits the measured data better when the size of thecapacitor is small. For increased size of the capacitor,we observe increased error percentage between themeasured and modeled C, Q and S-parameters data.

Although in this work, the method has been used tomodel MIM capacitors, it can be utilized to modelmetal finger capacitors and others which have thesimilar nine element lumped model pattem, This modelcan also be modified to fit 11 and 13 element pi models.With further modification, the same method can beutilized for t-models. T-models however will use Z-parameters instead of Y-parameters as in this case.

Finally, it is also important to have reliablemeasurement data to perform accurate modeling. Forthis, appropriate de-embedding techniques andmeasurement tools with very good resolution have to beused.

Acknowledgments

The authors are grateful to Silterra Malaysia Sdn. Bhd.for fabricating the device and providing facilities tomeasure them.

3) Goh M.W.C., Lim Q., Keating R.A., KordeschA.V., Yusman M. Y "Design of Radio FrequencyMetal-Insulator-Metal (MIM) Capacitors",Proceedings 7h International Conference onSolid-State and Integrated CircuitsTechnology,2004. Vol.1, 18-21 Oct.2004, pp 209-212

4) Appendix B: Scattering Parameters Relationshipat http://www.sss-mag.com/pdf/hpan95-1.pdf.

5) Aparicio R. and Hajimiri A., "Capacity Limitsand Matching Properties of IntegratedCapacitors", IEEE Journal of Solid-State Circuit,vol. 37, pp 384-393, March 2002.

6) See G.H., Mazlina Esa, and KordeschA.V.,"Spiral Inductor Macro Model Extractionand Optimization", Proceeding of IEEE NationalSymposium on Microelectronics 2003,2003. pp22-25.

7) Ng C.H., Ho C.S., Toledo and Chu S.F."Characterization and Comparison of Single andStacked MIMC in Copper Interconnect Processfor Mixed-Mode and RF Applications", ElectronDevice Letters, IEEE Vol.25, Issue 7, July 2004.pp 489-491.

8) Blood W.,Ling F., Kamgaing T., Myres T., andPetras M. " Simulation, Modeling and Testing ofEmbedded RF Capacitors in Low TemperatureCofired Ceramic",51st Electronic Components andTechnology Conference 2001. 29 May-I June2001, pp852-857.

9) Kolding T. E. "On-Wafer CalibrationTechniques for Giga-Hertz CMOSMeasurements", Proceedings of the 1999International Conference on MicroelectronicTest Structure, 1999.15-18 March 1999, pp105-110.

References

1) Chunqi G., Anh D.M., Zheng Z, and BoylandF.,"A Scalable RF Model of the Metal-Oxide-Metal (MOM) Capacitor", TechnicalProceedings of the 2001 International Conferenceon Modeling and Simulation ofMicrosystems,2001, 482-485.

2) Kolding T. E. "A Four-Step Method for De-Embedding Gigahertz On-Wafer CMOSMeasurements ",IEEE Transactions of ElectronDevices,Vol.47, Issue 4, April 2000, pp 734-740.

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