4th Student Conference on Research and Development (SCOReD 2006), Shah Alam, Selangor, MALAYSIA, 27-28 June, 2006

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Nine-Element Lumped Metal Finger Capacitor

Model Extraction Method for RF Applications

Kalavathi Subramaniam1,Student Member,IEEE, Albert Victor Kordesch2, Senior Member,IEEE, Mazlina Esa1, Member, IEEE

1 Dept. Of Radio Communications Engineering,

Faculty of Electrical Engineering, University of Technology, Malaysia (UTM), Skudai, Johor,Malaysia.

2 Silterra Malaysia Sdn. Bhd.,

Kulim, Kedah

Abstract – The need for a good capacitor model increases as the

operating frequency approaches the Gigahertz range. A good

capacitor model should fairly characterize the behavior of the

capacitor to include other passive characteristics such as the

resistance and inductance, not to forget the parasitic elements.

Presented here is a method to extract lumped element models for

metal finger capacitors. The model suggested is a nine element pi

circuit that includes interconnect resistance and inductance as

well as substrate parasitics. This paper identifies the procedures

to extract the nine element lumped metal finger capacitor model

and to optimize the model to fit the measured data up to the self

resonant frequency. A good fit between the modeled and

measured effective capacitance and Q factor have been obtained

in this work . This method can also be applied to MIM

capacitors.

Index terms – lumped elements, capacitance, resistance, inductance,

S-Parameters, Y-Parameters

I. INTRODUCTION The accuracy of passive element circuit simulation

models is essential as the operating frequency approaches the Gigahertz range. This is because at RF and higher frequencies, any passive element does not behave as a pure resistor, capacitor or an inductor. A capacitor that operates at RF frequencies also exhibits some amount of resistance and inductance, and some parasitic capacitance. These elements will be represented by ideal capacitors, resistors and inductors in a lumped element model.

This paper proposes a method to extract the lumped element values for a metal finger capacitor. The method is suggested for a 9 element lumped capacitor model. Figure 1 illustrates the 9 element pi model of a capacitor. This model is made up of three branches. The series branch comprises the nominal capacitance of the dielectric (Cs), the resistance of the interconnects (Rs) and inductive behavior of the interconnects (Ls). Parasitic losses in the capacitor are represented by two parallel branches which are made of the left and right plate oxide capacitance to the substrate (Cox). The frequency

dependent substrate loss parasitic elements (Rsi and Csi) are connected in parallel and they are serially connected to the oxide capacitance. Symmetrical values for the components of the two parallel branches of the pi circuit can be expected if the capacitor is designed to be absolutely symmetrical. However in most cases the values will be different as to correspond to the nonsymmetrical design such as in metal-insulator-metal (MIM) and metal finger capacitors. In MIM capacitors, the top and bottom plates of the capacitor are different metal layers, thus the distance from the plates to ground differ between top and bottom plate resulting in different oxide capacitance between top plate to ground and bottom plate to ground. However in metal finger capacitors, good symmetrical design could be achieved by having the same number of fingers for the left and right plates.

Fig. 1: Nine-Element Lumped Capacitor Model

II. DESIGN AND MEASUREMENTS

Rs Ls Cs

Cox1 Cox2

Rsi1 Csi1 Csi2 Rsi2

In this work, 6 different sizes of metal finger capacitors have been designed. They are 5 layer stacked metal fingers fabricated using Silterra’s industry standard 180nm RF CMOS 6 layer metal process technology. A snapshot of the layout of a 20pF metal finger capacitor with GSG pads is illustrated in Figure 2.

4th Student Conference on Research and Development (SCOReD 2006), Shah Alam, Selangor, MALAYSIA, 27-28 June, 2006

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Two port S-Parameter measurements were done using coplanar ground-signal-ground probes and a vector network analyzer. The test system is usually calibrated using the Impedance-Standard-Substrate that has short, open, thru and load structures on it.

Fig. 2: 20pF Finger Capacitor Layout

Another important area to consider is de-embedding. This can be performed using an on-wafer calibration kit, so that it has similar properties as the device to be measured. SOLT, LTTR, and TRL are to name a few of the on-wafer de-embedding techniques. Reference [6] provides a good insight on the common de-embedding structures. To provide good representation of the device, an appropriate de-embedding technique must be chosen. In this work, open de-embedding has been used.

III. MODEL EXTRACTION Once reliable measurements were obtained, the models were extracted using some simple procedures. The S-parameter measurements were converted to Y parameters. The mathematical equations for this conversion are given in equations 1 to 4 [9]. Y11 = Y0 (1 - S11 ) (1 + S22 ) + S12 S21

(1 + S11 ) (1 + S22 ) - S12 S21 ( 1 ) Y12 = Y0 -2 S12 . (1 + S11 ) (1 + S22 ) - S12 S21 ( 2 ) Y21 = Y0 -2 S21 . (1 + S11 ) (1 + S22 ) - S12 S21 ( 3 ) Y22 = Y0 (1 + S11 ) (1 - S22 ) + S12 S21

(1 + S11 ) (1 + S22 ) - S12 S21 ( 4 ) The model suggested in Figure 1 can be represented by the pi circuit in Figure 3. Y parameter representation of the circuit in Figure3 is given in Equation 5. Equations 6 to 8 were derived from Equation 5.

( 5 ) A = Y11 + (Y12 + Y21 )/2 ( 6 ) B = Y22 + (Y12 + Y21 )/2 ( 7 ) C = (Y12 + Y21 ) ( 8 ) 2

Fig. 3: Equivalent Pi Circuit

The circuit in Figure 1 was made equivalent to the circuit in Figure 3. From Figure 1, A, B and C values can be derived as given by equations 9 to 11. A = [ ( 1/jwCox1 ) + ( 1/Rsi1 + jwCsi1 )

-1 ] –1 ( 9 ) B = [ ( 1/jwCox2 ) + ( 1/Rsi2 + jwCsi2 )

-1 ] –1 ( 10 ) C = [ Rs + j (wLs - 1/wCs) ]

–1 ( 11 ) From the set of data for Y11, Y12, Y21 and Y22, all the elements of capacitor model were extracted. However, the elements were extracted only at certain frequencies. Cs is the nominal capacitance, thus it was extracted from low frequency or dc measurements. From equation 11, it is evident that Rs can be extracted from the real portion of 1/C. This was done using the data at low frequency. Knowing Cs, Ls was extracted from the imaginary portion of 1/C data after resonance at high frequency. The method of extraction of the parasitic elements from the parallel branch is similar for both branches. It is known that the oxide capacitance (Cox) is a frequency independent capacitance. Thus this dominates the parallel branch at low frequencies because as the frequency increases, the frequency dependent substrate loss capacitance (Csi) and resistance (Rsi) have bigger influence on the parallel branch. Hence, Cox1 and Cox2 were extracted from the imaginary portion of 1/A or 1/B respectively at low frequency. Cs however was extracted from the imaginary portion of A (or B) at high frequency. Rs also a frequency dependent element, was extracted at high frequency from the real portion of A (or B). The equations for all these are given below. “A” in the equations should be substituted

4th Student Conference on Research and Development (SCOReD 2006), Shah Alam, Selangor, MALAYSIA, 27-28 June, 2006

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with B when parasitic elements for the second parallel branch are extracted. Rs = real ( 1/C ) ( 12 ) Cox = -1 . ( 13 ) w (im 1/A) Csi = im A ( 14 ) W Rsi = 1 . ( 15 ) Real A

IV. MODEL OPTIMIZATION The extracted values of the elements serve as the initial values for the model. After extracting the elements, manual optimization was done so that the modeled effective capacitance and Q curves fit the measured curves. Optimization could also be done to fit the S-parameter curves. To fit the C curve and SRF, Ls was tuned. It is known that C is extracted from the imaginary portion of Y12. Thus, tuning Ls will help to fit the C curve and also self resonant frequency on C curve. Q was calculated by dividing imaginary Y11 ( or Y22 ) with real Y11 ( or Y22 ). From equation 5, it is known that Y11 is A + C. Thus, to fit Q, Rs was tuned first. As an alternative, Rsi of the appropriate branch could be tuned too. Tuning Csi also provides a reasonable fit for Q. In this work, metal finger capacitors ranging from 2pF to 20pF were modeled using this method and they were able to fit reasonably well to the measured C and Q curves with errors of less than 10%. The table below summarizes the comparison between the initial extraction of the parameters and the values after tuning for a 20pF capacitor that is shown in Figure 2.

TABLE 1 : INITIAL EXTRACTION AND TUNED VALUES COMPARISON

Elements Extracted Value Tuned Value

Ls (H) 5.8786E-11 5.7263E-11

Cs (F) 2.0752E-11 2.0752E-11

Rs (ohm) 2.2347 2.2097E+00

Csi1 (F) 9.7267E-14 9.7267E-14

Csi2 (F) 9.2553E-14 9.2553E-14

Rsi1 (ohm) 6.3100E+02 6.3100E+02

Rsi2 (ohm) 6.9041E+02 6.9041E+02

Cox1 (F) 6.8889E-13 6.8889E-13

Cox2 (F) 6.4468E-13 6.4468E-13

From the table, it can be seen that the initial extraction data has been maintained for all the elements except for Rs and Ls. The difference in extracted and tuned Rs is 1.13 % and difference between extracted and tuned Ls is 2.6%. The tuning range is within 20 % from the extracted value for all the capacitors that have been measured and modeled in this work.

Thus it can be said that the extraction method and results are reliable and did not need much tuning.

V. RESULTS The modeled C and Q curves after parameters tuning match reasonably well with the measured Q and L curves. Figure 4 illustrates the comparison of measured and modeled effective capacitance curve for a 20pF capacitor and the model elements that have been tabulated in Table 1. The error is kept within 6% up to resonance. The average error for effective capacitance over the range from 0 to 20 GHz is about 8%. The comparison curves between measured and modeled Q are plotted in Figure 5. The difference between measured and modeled Q is maintained within 10% up to the resonant frequency. However, the Q turned out to be unexpectedly low for these finger capacitors as finger capacitors generally have high Q. The low Q is caused by the Rs which turned out to be very high which is 2.2097 as in Table 1. A typical value for Rs would be in the range of to 0.4 for this type of designs to produce Q over 100.

Fig. 4: Measured Vs. Modeled Effective Capacitance

Fig. 5: Measured Vs. Modeled Q Factor

Figures 6 and 7 illustrate the measured and modeled S11 and S12 curves for the same capacitor. However, it is seen that the S parameter curves of the modeled capacitor do not match very well with the measured S-parameter curves. The

4th Student Conference on Research and Development (SCOReD 2006), Shah Alam, Selangor, MALAYSIA, 27-28 June, 2006

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differences between the measured and modeled S11 and S12 curves go up to 60%.

Fig. 6: Measured Vs. Modeled S11

Fig. 7: Measured Vs. Modeled S12

VI. CONCLUSION The paper identified a simple method to extract 9 element

lumped metal finger capacitor models for RF applications. The models extracted using this method have fairly good fit to the measured effective capacitance and Q factor curves. The difference is kept within 10 % up to resonance frequency. The extracted element values only serve as the initial value and in most cases optimization is needed to provide better fit to the measured data. However, the difference between the extracted element values and the values after tuning do not vary much (within 20 %). The elements that are supposed to be tuned to fit C and Q curves have also been identified in this work. The data presented indicate the reliability of this simple extraction method.

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

Although in this work, the method has been used to model finger capacitors, it can be utilized to model MIM capacitors

and others which have the similar nine element lumped model pattern, This model can also be modified to fit 11 and 13 element pi models. With further modification, the same method can be utilized 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 reliable measurement data to perform accurate modeling. For this, appropriate de-embedding techniques and measurement tools with very good resolution have to be used.

AknowledgementThe authors are grateful to Silterra Malaysia Sdn.

Bhd. for fabricating the device and Advanced RFIC (S) Pte. Ltd. for providing the measurements.

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