188 Lumped Elements for RF and Microwave Circuits Table 5.5 ABCD-, S-, Y-, and Z-Matrices for Ideal Lumped Capacitors ABCD Matrix S-Parameter Matrix Y-Matrix Z-Matrix ͫ j ␻ C −j ␻ C −j ␻ Cj ␻ C ͬ ΄ 1 −j ␻ C 01 ΅ 1 −j ␻ C + 2Z 0 ΄ −j ␻ C 2Z 0 2Z 0 −j ␻ C ΅ ͫ 10 j ␻ C 1 ͬ 1 Z 0 − j2 ␻ C ΄ −Z 0 −j2 ␻ C −j ␻ C −Z 0 ΅΄ −j ␻ C −j ␻ C −j ␻ C −j ␻ C ΅ References [1] Ballou, G., Capacitors and Inductors in Electrical Engineering Handbook, R. C. Dorf, (Ed.), Boca Raton, FL: CRC Press, 1997. [2] Walker, C. S., Capacitance, Inductance and Crosstalk Analysis, Norwood, MA: Artech House, 1990. [3] Durney, C. H., and C. C. Johnson, Introduction to Modern Electronic Genetics, New York: McGraw-Hill, 1969. [4] Zahn, M., Electromagnetic Field Theory, New York: John Wiley, 1979. [5] Ramo, S., J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 2nd ed., New York: John Wiley, 1984. [6] Abrie, P. D., Design of RF and Microwave Amplifiers and Oscillators, Norwood, MA: Artech House, 1999, Chap. 7. [7] Weber, R. J., Introduction to Microwave Circuits, New York: IEEE Press, 2001. [8] Weber, R. J., Introduction to Microwave Circuits, New York: IEEE Press, 2001. [9] American Technical Ceramics, Huntington Station, NY. [10] Dielectric Lab, New York. [11] AVX Corporation, Myrtle Beach, SC. [12] Ingalls, M., and G. Kent, ‘‘Monolithic Capacitors as Transmission Lines,’’ IEEE Trans. Microwave Theory Tech., Vol. MTT-35, November 1987, pp. 964–970. [13] de Vreede, L. C. N., et al., ‘‘A High Frequency Model Based on the Physical Structure of the Ceramic Multilayer Capacitor,’’ IEEE Trans Microwave Theory Tech., Vol. 40, July 1992, pp. 1584–1587. 189 Capacitors [14] Sakabe, Y., et al., ‘‘High Frequency Measurement of Multilayer Ceramic Capacitors,’’ IEEE Trans. Components, Packaging Manufacturing Tech.—Part B, Vol. 19, February 1996, pp. 7–12. [15] Murphy, A. T., and F. J. Young, ‘‘High Frequency Performance of Multilayer Capacitors,’’ IEEE Trans. Microwave Theory Tech., Vol. 43, September 1995, pp. 2007–2015. [16] Goetz, M. P., ‘‘Time and Frequency Domain Analysis of Integral Decoupling Capacitors,’’ IEEE Trans. Components, Packaging Manufacturing Tech.—Part B, Vol. 19, August 1996, pp. 518–522. [17] Fiore, R., ‘‘RF Ceramic Chip Capacitors in High RF Power Applications,’’ Microwave J., Vol. 43, April 2000, pp. 96–109. [18] Lakshminarayanan, B., H. C. Gordon, and T. M. Weller, ‘‘A Substrate-Dependent CAD Model for Ceramic Multilayer Capacitors,’’ IEEE Trans. Microwave Theory Tech., Vol. 48, October 2000, pp. 1687–1693. [19] Semouchkina, E., et al., ‘‘Numerical Modeling and Experimental Investigation of Reso- nance Properties of Microwave Capacitors,’’ Microwave Optical Tech. Lett., Vol. 29, April 2001, pp. 54–60. [20] Fiore, R., ‘‘Capacitors in Broadband Applications,’’ Applied Microwave and Wireless, May 2001, pp. 40–54. 6 Monolithic Capacitors Monolithic or integrated capacitors (Figure 6.1) are classified into three catego- ries: microstrip, interdigital, and metal–insulator–metal (MIM). A small length of an open-circuited microstrip section can be used as a lumped capacitor with a low capacitance value per unit area due to thick substrates. The interdigital geometry has applications where one needs moderate capacitance values. Both microstrip and interdigital configurations are fabricated using conventional MIC techniques. MIM capacitors are fabricated using a multilevel process and provide the largest capacitance value per unit area because of a very thin dielectric layer sandwiched between two electrodes. Microstrip capacitors are discussed briefly below. The interdigital capacitors are the topic of the next chapter and MIM capacitors are treated in this chapter. All metals printed on a GaAs substrate will establish a shunt capacitance to the back side ground plane, C, given by C = C p + C e (6.1) where C p is the parallel plate capacitance and C e is the capacitance due to edge effects. The parallel plate capacitance to the backside metal may be expressed as C p = A 152 × 10 − 8 pF/ ␮ m 2 (75- ␮ m substrate) (6.2) = A 91 × 10 − 8 pF/ ␮ m 2 (125- ␮ m substrate) where A is the top plate area in square microns. As an approximation, C e can be taken as [1] 191 192 Lumped Elements for RF and Microwave Circuits C e = P 3.5 × 10 − 5 pF/ ␮ m (75- ␮ m substrate) (6.3) = P 5 × 10 − 5 pF/ ␮ m (125- ␮ m substrate) where P is the perimeter of the capacitor in microns. An accurate printed capacitor model must treat the capacitor as a microstrip section with appropriate end discontinuities as discussed in Chapter 14. Monolithic MIM capacitors are integrated components of any MMIC process. Generally, larger value capacitors are used for RF bypassing, dc blocking, and reactive termination applications, whereas smaller value capacitors find usage as tuning components in matching networks. They are also used to realize compact filters, dividers/combiners, couplers, baluns, and transformers. MIM capacitors are constructed using a thin layer of a low-loss dielectric between two metals. The bottom plate of the capacitor uses first metal, a thin unplated metal, and typically the dielectric material is silicon nitride (Si 3 N 4 ) for ICs on GaAs and SiO 2 for ICs on Si. The top plate uses a thick plated conductor to reduce the loss in the capacitor. The bottom plate and the top plate have typical sheet resistances of 0.06 and 0.007 ⍀/square, respectively, and a typical dielectric thickness is 0.2 ␮ m. The dielectric constant of silicon nitride is about 6.8, which yields a capacitance of about 300 pF/mm 2 . The top plate is generally connected to other circuitry by using an airbridge or dielectric crossover, which provides higher breakdown voltages. Typical process variations for microstrip and MIM capacitors are compared in Table 6.1. Normally MIM capacitors have two plates, however, three plates and two- layer dielectric capacitors have also been developed. 6.1 MIM Capacitor Models Several models for MIM capacitors on GaAs substrate have been described in the literature [2–5]. These include both EC and distributed models, which are discussed next. Figure 6.1 Monolithic capacitor configurations: (a) microstrip, (b) interdigital, and (c) MIM. 193 Monolithic Capacitors Table 6.1 Capacitance Variations of Microstrip and MIM Capacitors on GaAs Substrate Capacitor Range Design Uncertainty Process Variation Microstrip (shunt only) 0.0–0.1 pF ±2% ±2% MIM 1.0–30.0 pF ±5% ±10% MIM 0.1–1.0 pF ±5% ±20% MIM 0.05–0.1 pF ±5% ±30% 6.1.1 Simple Lumped Equivalent Circuit When the largest dimension of the MMIC capacitor is less than ␭ /10, in the dielectric film at the operating frequency, the capacitor can be represented by an equivalent circuit, as shown in Figure 6.2, where B and T depict the bottom and top plate, respectively. The model parameter values can be calculated from the following relations: C = ⑀ 0 ⑀ rd Wᐉ d = ⑀ rd 10 − 15 36 ␲ Wᐉ d (F) (6.4a) R = 2 3 R s W ᐉ (6.4b) G = ␻ C tan ␦ = 1 18 ⑀ rd f Wᐉ d × 10 − 6 tan ␦ (mho) (6.4c) where ⑀ rd and tan ␦ are the dielectric constant and loss tangent of the dielectric film, respectively; R s is the surface resistance of the bottom plate expressed in ohms per square; and W, ᐉ, and d are in microns, and f is in gigahertz. The value of L can be obtained from (2.13a) in Chapter 2. The conductor (Q c ) and dielectric (Q d ) quality factors can be expressed as Figure 6.2 EC model of a MIM capacitor. 194 Lumped Elements for RF and Microwave Circuits Q c = 1 ␻ RC = 3W 2 ␲ f 2R s ᐉ C = 27 × 10 6 d fR s ᐉ 2 ⑀ r (6.5a) Q d = 1 tan ␦ (6.5b) where f is in gigahertz and ᐉand d are in microns. The total quality factor Q T is given by Q T = ͫ 1 Q c + 1 Q d ͬ − 1 (6.6) Figure 6.3 shows another simple lumped EC. Model parameter values for MIM capacitors on a 125- ␮ m-thick GaAs substrate are given in Table 6.2. The model parameters were extracted from measured S-parameter data as discussed in Chapter 2. Empirically fit closed-form values for such capacitors were obtained as follows: L (nH) = 0.02249 × log (10 × C ) + 0.01 (6.7a) C 1 (pF) = 0.029286 × C + 0.007 (6.7b) C 2 (pF) = 0.00136 × C + 0.004 (6.7c) where C is capacitor value in picofarads, the substrate thickness is 125 ␮ m, capacitor range is 1 to 30 pF, and the frequency range is dc to 19 GHz. 6.1.2 Coupled Microstrip-Based Distributed Model Mondal [2] described a distributed lumped-element MIM capacitor model based on coupled microstrip lines. The model parameter values can be either extracted Figure 6.3 MIM capacitor and its EC model. 195 Monolithic Capacitors Table 6.2 Typical Model Parameter Values for MIM Capacitors C (pF)* W = ᐉ ( ␮ m) L (nH) C 1 (pF) C 2 (pF) Q** at 10 GHz 1.0 58 0.0325 0.001 0.0054 120.0 2.0 82 0.0393 0.0129 0.0067 60.0 5.0 130 0.0482 0.0219 0.0108 24.0 10.0 182 0.055 0.0363 0.0176 12.0 15.0 223 0.0589 0.0509 0.0244 8.0 20.0 258 0.0618 0.0656 0.0312 5.0 *Based on 300 pF/mm 2 MIM capacitance. **Q = 1/ ␻ CR. from the measured two-port S -parameter data or approximately calculated as discussed later. The cross-sectional view and the distributed model based on coupled transmission lines of a MIM capacitor are shown in Figure 6.4. The model parameters are defined as follows: L 11 = inductance/unit length of the top plate; L 22 = inductance/unit length of the bottom plate; L 12 = mutual inductance between the plates/unit length of the capacitor; R 1 = loss resistance/unit length of the top plate; R 2 = loss resistance/unit length of the bottom plate; G = loss conductance of the dielectric/unit length of the capacitor; C 12 = capacitance/unit length of the capacitor; C 10 = capacitance with respect to ground/unit length of the top plate; C 20 = capacitance with respect to the ground/unit length of the bottom plate. C 10 and C 20 are due to substrate effects. The voltage and current equations, relating the model parameters, based on coupled-mode transmission lines can be written as follows: − ΄ ∂v 1 ∂x ∂v 2 ∂x ΅ = ͫ R 1 + j ␻ L 11 j ␻ L 12 j ␻ L 12 R 2 + j ␻ L 12 ͬͫ i 1 −i 2 ͬ (6.8) 196 Lumped Elements for RF and Microwave Circuits Figure 6.4 MIM capacitor: (a) cross-sectional view and (b) distributed model. − ΄ ∂i 1 ∂x −∂i 2 ∂x ΅ = ͫ G + j ␻ (C 10 + C 12 ) −(G + j ␻ C 12 ) −(G + j ␻ C 12 ) G + j ␻ (C 20 + C 12 ) ͬͫ v 1 v 2 ͬ (6.9) where ␻ is the operating angular frequency. Equations (6.8) and (6.9) are solved for the Z-matrix [6] by applying the boundary conditions i 1 (x = ᐉ) and i 2 (x = 0) = 0, and the values of the LE parameters are obtained by comparing the measured two-port S-parameters for the capacitor and converting it into the Z-matrix. The LE parameter values can also be calculated by using analytical equations as described here: C 12 = ⑀ 0 ⑀ rd W /d (6.10) or C 12 = W × capacitance per unit area and 197 Monolithic Capacitors C 20 = C p + (C 2 − C p ) и ⑀ re ⑀ r (6.11) C 10 = C T − C 20 (6.12) where C T = ͩ Z 0m c √ ⑀ re ͪ − 1 (6.13) C p = ⑀ 0 ⑀ r W h (6.14) C 2 = 1 2 и ͩ Z os c √ ⑀ r ͪ − 1 (6.15) and c is the velocity of light, Z om and ⑀ re are the characteristic impedance and effective dielectric constant of the microstrip (Figure 6.5) of capacitor width and GaAs as the substrate, respectively, and Z os is the characteristic impedance of the stripline of width W. Terms ⑀ r and ⑀ rd are the dielectric constants for the substrate and capacitor film, respectively. The various inductance values are calculated using the following relations: Figure 6.5 (a) Stripline used for calculating C 2 capacitance and (b) microstrip used for calculating C T capacitance. [...]... Tangent 12.4 13.0 14.0 15. 0 16.0 17.0 18.0 1,288 1,3 45 1,266 1,222 1,333 1,306 1,229 356 351 3 35 3 35 336 349 358 0.1 15 0.099 0.062 0.082 0.082 0. 057 0.044 0.014 0.030 0.0 25 0.021 0.012 0.008 0.014 Table 6.9 Dielectric Constant and Loss Tangent of Single-Crystal SrTiO3 Material at Various Temperatures Temperature (؇C) ⑀ rd tan ␦ × 10−4 50 − 25 0 + 25 438 380 350 320 4.1 4 .5 4.8 5. 2 In BSTO or simply BST... of several MIM capacitors Figure 6.6 Microstrip distributed EC model of MIM capacitor 200 Lumped Elements for RF and Microwave Circuits Figure 6.7 Modeled and measured S 21 for 2- and 10-pF MIM capacitors on 1 25- ␮ m-thick GaAs substrate Figure 6.8 Modeled and measured S 11 for 2- and 10-pF MIM capacitors on 1 25- ␮ m-thick GaAs substrate When the MIM capacitor value is small, on the order of, say, 0.2... Figure 6. 15 shows 206 Lumped Elements for RF and Microwave Circuits Table 6 .5 EM Simulated Phase Angle of S 11 for Various Feed Configurations Connected to a 10-pF MIM Capacitor, with GaAs Substrate Thickness of 75 ␮ m Frequency (GHz) a Feed Configuration b c d 2 4 6 8 10 12 14 16 18 20 −168.8 −174.8 180.0 176.4 173.3 170.6 168.1 1 65. 6 163.2 160.8 −163 .5 −174.9 179.9 176.4 173.4 170.7 168.1 1 65. 7 163.3... (pF) (for h = 1 25 ␮ m) 0 .5 2.0 5. 0 10.0 15. 0 20.0 42 82 130 182 223 258 42 82 130 182 223 258 7.68 8.08 8.44 8. 75 8. 95 9.13 67. 05 52.3 42.08 35. 51 31.68 28.63 0.0014 0.0029 0.0049 0.0068 0.0084 0.0101 relation was derived that relates the designed capacitor value C to the measured value C m by the following equation: C m = C (1 + 0.012/C ) (6.21) where units are in picofarads 6.1.4 EC Model for MIM... 40V for 104-␮ m2 capacitor area and about 30V for a 1 05- ␮ m2 capacitor area Table 6.6 compares EC model parameters for MIM and MIMIM capacitors with W = 50 ␮ m and ᐉ = 25 ␮ m Figure 6.18 compares the series inductance of these capacitors when the capacitors aspect ratio ᐉ /W = 0 .5 In this configuration the series inductance is lower because of the wider and shorter conductor used compared 210 Lumped Elements. .. compared 210 Lumped Elements for RF and Microwave Circuits Figure 6.17 (a) Cross-sectional view of a MIMIM capacitor on Si substrate and (b) EC model for MIM and MIMIM capacitors Table 6.6 EC Model Parameters for MIM and MIMIC Capacitors with 25 × 50 ␮ m2 Size on Si Substrate C (pF) MIM MIMIM L s (nH) R s (⍀) C ox (fF) R sub (k⍀) 1.02 1.91 0.128 0.124 0.61 0.36 7.01 8.49 4 .50 4.03 Figure 6.18 Series... thinner than at other places and this effect is more pronounced for smaller capacitor areas Based on measured data an empirical Monolithic Capacitors 201 Figure 6.9 Q -factor versus frequency for various MIM capacitors on 1 25- ␮ m-thick GaAs substrate Figure 6.10 Series SRF of various MIM capacitors on 1 25- ␮ m-thick GaAs substrate 202 Lumped Elements for RF and Microwave Circuits Table 6.4 Distributed... commonly used silicon nitride (Si3N4 ) material Such circuits are connected with external capacitors on a carrier or in a package, resulting in a higher parts count and an increase in the package size and assembly costs To overcome these drawbacks and minimize the wire bond effects at higher 208 Lumped Elements for RF and Microwave Circuits Figure 6. 15 (a–d) Three-port connections of MIM capacitors frequencies,... 6.11 (a) MIM capacitor structure on Si substrate and (b) EC model (From: [9] © 2002 John Wiley Reprinted with permission.) ͩ ͩ L 1 = 2 × 10−4l ln 2l W + t1 + 0 .5 + W + t1 3l L 2 = 2 × 10−4l ln 2l W + t2 + 0 .5 + W + t2 3l g1 = 1 ␻ C tan ␦ ͪ ͪ (6.22f) (6.22g) (6.22h) 204 Lumped Elements for RF and Microwave Circuits C ox = 0 .5 × 103Wl⑀ rox /(36 × 3.14 159 d ox ) 1 ␻ C ox tan ␦ ox (6.22j) 1 и l и W и ⑀... 0. 05 dB Another SrTiO3 film capacitor with a dielectric constant of 180 and having a frequency of operation up to 20 GHz was reported by Nishimura et al [22] 216 Lumped Elements for RF and Microwave Circuits Figure 6.22 BST capacitor: (a) capacitance versus frequency and (b) capacitance versus temperature (From: [21] © 19 95 IEEE Reprinted with permission.) Figure 6.24 shows the I-V characteristics for . ( ␮ m) ⑀ re Z om (for h = 1 25 ␮ m) 0 .5 42 42 7.68 67. 05 0.0014 2.0 82 82 8.08 52 .3 0.0029 5. 0 130 130 8.44 42.08 0.0049 10.0 182 182 8. 75 35. 51 0.0068 15. 0 223 223 8. 95 31.68 0.0084 20.0 258 258 9.13 28.63. 188 Lumped Elements for RF and Microwave Circuits Table 5. 5 ABCD-, S-, Y-, and Z-Matrices for Ideal Lumped Capacitors ABCD Matrix S-Parameter Matrix. relations: Figure 6 .5 (a) Stripline used for calculating C 2 capacitance and (b) microstrip used for calculating C T capacitance. 198 Lumped Elements for RF and Microwave Circuits [L] = 1 c 2 ͫ C 10 +