Chapter 2. Inductor Design for RFIC Applications

Transcription

1 Chapter 2 Inductor Design for RFIC Applications 2.1 Introduction A current carrying conductor generates magnetic field and a changing current generates changing magnetic field. According to Faraday s laws of induction changing magnetic field induces electromotive force. The property of a conductor by which a change in current induces an electromotive force (EMF) is called inductance (L). If EMF is induced due to current in the same conductor then it is called self inductance and if it is due to the current in nearby conductors then it is known as mutual inductance. Lenz law states that EMF thus generated, opposes the change in the current. EMF = L di dt Mutual Inductance between a filamentary circuit m and another filamentary circuit n could be calculated from Neumann s formula given in Eq L m,n = µ 4π C m (2.1) C n ds m.ds n r mn (2.2) where C m and C n are the contours of the filamentary circuits, ds m and ds n are the circuit elements, and r mn is the distance between the circuit elements ds m and ds n. Inductors are passive two terminal devices which oppose the change in current. Any simple straight filament of a conductor has inductance associated with it and would act like an inductor. To improve the value of inductance, a straight filament could be wound as shown in Fig By winding, the magnetic flux linkage is improved with increase in number of turns and the area of the coil. Inductance of a typical solenoid coil is given by Eq L = µn 2 A (2.3) l As can be observed from this equation, value of inductance can be varied by varying geometry parameters: area of the coil (A), number of turns (N), length of the coil (l), and by varying the surrounding material of the inductor to increase the permeability (µ). Rapid developments in integrated circuit (IC) technology happened because of digital applications, where there was no necessity to incorporate inductors explicitly (In an IC, any conductor line would 6

2 Figure 2.1 A Solenoid Coil Inductor Figure 2.2 Planar Inductors: a) Square Inductor b) Octagonal Inductor c) Hexagonal Inductor d) Circular Inductor have inductance associated with it). Inductors have evolved from discrete components on printed circuit boards to integrated devices on chip [6, 36, 37]. In RFICs, inductors are essential for applications like matching, oscillators, DC-DC converters, filters and so on. To obtain required values of inductance in range of 1 to 100 nh, an approximate two dimensional projection of solenoid like a planar rectangular, polygonal or circular spiral is utilized as shown in Fig Generally on-chip inductors are fabricated in top metal layers which are incorporated on top of or embedded within dielectrics, with a substrate beneath. In CMOS RFICs on-chip spiral inductors are fabricated on top of silicon dioxide with silicon substrate, as shown in Fig Electromagnetic flux path of the inductors is through the silicon dioxide layer and the silicon substrate beneath. For the frequencies at which the inductors are employed, there would be significant eddy current effects in the conductive portions of the inductor and the substrate. Eddy current losses in substrate, attributed to its conductivity, causes parasitic losses. To connect one end of the spiral to the output an underpass in the 7

3 Figure 2.3 Cross section of an inductor on top of the silicon dioxide with silicon substrate. Top layer is aluminum, middle layer is silicon dioxide and the bottom layer is silicon substrate. Underpass through silicon dioxide connecting both ends of inductor in the top layer, could also be seen lower metal layer or overpass in higher metal layer is essential. As inductors are fabricated in the top most metal layer, an underpass is preferred. 2.2 Modeling on-chip inductor To enable the utilization of inductors in high frequency circuits, a model which accounts for all the high frequency parasitic effects must be available. In earlier analog IC design, inductors were not considered as a standard passive component like resistors and capacitors. With the advent of RFICs, a model for inductor became inevitable. To account for all the parasitics and to extract the exact behavior of inductors incorporated in CMOS RFICs, several approaches for modeling inductors on silicon were reported with numerical modeling, curve fitting, and empirical formulae [38 42]. All these models were limited in their application. An accurate model which accounts for the physical effects was first proposed in [43]. A physical model for planar spiral inductors on silicon which accounts for eddy current effects in the conductor, crossover capacitance between the spiral and center-tap, capacitance between the spiral and substrate, substrate ohmic loss, and substrate capacitance, shown in Fig. 2.4, was proposed in [43]. The equivalent lumped model of planar inductor with SiO 2 beneath and a silicon substrate consists of series inductance and resistance, oxide capacitance, substrate resistance and substrate capacitance as shown in Fig Series Inductance: The conductive portion of inductor has self inductance associated with each straight conductor in a single turn and mutual inductance between each of the other conductors. Unlike a solenoid, inductance of these spiral structures is complex. It is extracted from self inductance of each segment of the conductive portion and the mutual inductance between each of the segments of the conductive portion. Considering a planar rectangular spiral, which has a rectangular cross section, self inductance of a wire is given by Eq. 2.4 and mutual inductance is given by Eq. 2.5 [44] ( L self = 2l ln 2l w + t w + t ) 3l (2.4) 8

4 Figure 2.4 Physical Model of Inductor Figure 2.5 Lumped Physical Model of Inductor 9

5 where L self is self inductance in nh; l is the wire length in cm; w is width in cm and t is thickness in cm. M = 2lQ (2.5) where M is the mutual inductance in nh, l is the wire length in cm and Q is the parameter for mutual inductance which is given by the Eq Q = ln l ( ) l 2 ( ) GMD GMD GMD GMD l l where GMD is the geometric mean distance between two parallel wires. Pitch of the wires is approximately equal to GMD. Greenhouse developed a formula for calculation of inductance of planar rectangular inductors based on the Grover s formulae [45]. Total inductance L T is given by Eq. 2.7 (2.6) L T = (Self Inductance of all segments) + (P ositive Mutual Inductances) (2.7) (Negative Mutual Inductances) Applying Greenhouse method, series inductance (L s ) can be calculated. Various geometric parameters like spacing between conductors, width of the conductors, number of turns, internal radius, thickness of conductors, and so on affect the value of inductance. To have rough estimate of the inductance of rectangular spiral inductors Eq. 2.8 can be used. L µ 0 n 2 r (2.8) Inductance of on-chip inductors is a function of frequency. Self inductance is sum of the internal inductance, due to the magnetic field that is inside the conductor and the external inductance, due to the magnetic field outside the conductor [46]. As frequency increases, due to the skin effect, magnetic field inside the conductor decreases, and the value of internal inductance decreases, thus decreasing the total inductance at higher frequencies. But the increase in total inductance of onchip inductor is attributed to the capacitive coupling with oxide and substrate capacitance. This LC tank resonates causing increase in the total inductance, till the self resonant frequency. In this thesis the effective inductance which is function of frequency is referred to as inductance. Series Resistance: The conductive portions of inductor contribute to series resistance. At higher frequencies eddy current effects become significant and the series resistance increases. Owing to the induced eddy currents, current density in a conductor becomes non-uniform. Current density 10

6 in a conductor confines itself to the surface or the skin of conductor which is known as skin effect. Depth of penetration of the current, also known as skin depth is given by the formula Eq ρ δ = πµf where ρ is the resistivity of conductor, µ is the permeability and f is the frequency of operation. In case, eddy currents are caused in a conductor by an adjacent one, it is called proximity effect. It has been shown that proximity effect is lesser when conductors are placed side by side compared with conductors on top of another with dielectric separating them [43]. Considering the skin effect, the effective resistance of the conductor could be calculated from Eq R = (2.9) ρ.l w.t eff (2.10) where ρ is the resistivity of the conductor, l is the total length, w is the width and t eff is the effective thickness which is given by Eq ( ) t eff = δ. 1 e t δ (2.11) where t is the thickness of conductor and δ is the skin depth at the frequency of operation obtained from Eq In this thesis Aluminum is used for the conductor lines. Considering the resistivity of the Aluminum to be ohm-meter the skin depth at 20 GHz would be approximately 580 nm. Sputtered Aluminum has higher resistivity compared to the bulk Aluminum target. In this work Aluminum has a maximum thickness of 300 nm owing to fabrication related constraints. Hence skin effect can be safely neglected. Coupling Capacitance with Underpass: Spiral structure has a capacitance (C s ) with the underpass which connects one port of the inductor to the other. This capacitance is significant compared to the inter-turn capacitance. Inter-turn capacitance is generally neglected because the adjacent conductors are treated to be at same potential. The distance between underpass and the spiral, and quality of dielectric separating them determines the coupling capacitance. It may allow the direct flow of signal between the two ports bypassing the inductor. This series capacitance, C s is obtained from Eq. 2.12, where n is the number of overlap turns of conductor with the underpass, w is the width of the conductor and t oxspiral Underpass is the thickness of oxide between the spiral and underpass. ɛ ox C s = n.w 2. (2.12) t oxspiral Underpass Fabricated inductors do not have a underpass and hence the coupling capacitance with underpass is ignored in further modeling and calculations. 11

7 Oxide Capacitance: Conducting portion of the inductor has a capacitance (C ox ) with the substrate, and the oxide layer in between acts as dielectric. For ease of computations this capacitance is distributed equally to the two ports of the inductors. This capacitance (Eq. 2.13) depends on, the quality of oxide layer which is indicated by the dielectric constant (ɛ ox ), thickness of oxide (t ox ), length (l) and width (w) of the conductor. C ox = 1 2.l.w.ɛ ox t ox (2.13) Substrate Capacitance and Resistance: In Complementary Metal Oxide Semiconductor (CMOS) RFICs, substrate has inherent MOS capacitance and conductance associated with it due to their large conductivity. Capacitance (Eq. 2.14) and resistance (Eq. 2.15) associated with silicon substrate depend on the length (l) and width of the conductor (w), capacitance of the substrate per unit area (C sub ), and conductance of the substrate per unit area (G sub ). C sub and G sub are functions of the substrate doping concentration. C Si = 1 2.l.w.C sub (2.14) R Si = 2 l.w.g sub (2.15) For ease of computation both the substrate capacitance and resistance are distributed equally between the two ports. Both electric and magnetic flux of inductor complete their path through the substrate. Due to the significant conductivity of the substrates, eddy currents are created causing significant losses and reduction in inductance. Substrate losses contribute more compared to other parasitics Other Models In addition to the model described which computes the different lumped elements using a static field analysis, a frequency-dependent compact model for inductors in high ohmic substrates, which is based on an energy point-of-view, is developed [47]. In another work a frequency independent model which is fully compatible with both ac and transient analysis is presented [48]. This is a wide-band physical and scalable 2π equivalent circuit model for on-chip spiral inductors based on physical derivation and circuit theory. Closed-form formulas were generated to calculate the RLC circuit elements directly from the inductor layout. The 2π model accurately captures r(f) and l(f) characteristics beyond the self-resonant frequency. Electromagnetic simulators could always be used to understand the effects of parasitics, which have drastic impact on passive devices. Nevertheless frequency independent and scalable models will enable the designer to optimize the layout of the inductor for desired inductance value, maximizing the value of quality factor, under minimal time. Model proposed in [43] with required modifications is used in this thesis. 12

8 2.3 Design parameters of inductor and their effects on performance A field solver could be employed to investigate the effects of various geometry parameters on the performance of inductor. Simulations are performed by varying a certain geometry parameter with others held constant, and its effect on the inductance (L) and quality factor (Q) are analyzed. The effect of geometry parameters on the performance, using field solvers, is a widely analyzed topic [49, 50]. Here the effect of geometry parameters on the performance factors are reiterated. During this analysis the focus is also on the effect on self resonance frequency so as to increase the bandwidth of inductor. Electromagnetic field solvers are effective tools in determining the influence of various geometry parameters of spiral inductor on its performance. Various simulations are performed using Ansoft High Frequency Structural Simulator (HFSS) by varying the design parameters of spiral inductors such as: a) thickness of the metal, b) width of the metal, c) spacing between metal lines, d) internal radius of spiral structure, e) number of turns, and so on. A three and half turn spiral inductor with silicon dioxide thickness of 2.2 µm, and aluminum as spiral and underpass metal, is used for most of the investigations (Fig. 2.9), unless otherwise stated. In order to establish the accuracy of simulation results the simulated S-parameters are compared with the hfss simulations and measurement results of fabricated inductors by Feng Ling et al [49]. Similar trends in the S-parameters have been obtained as shown in Fig. 2.6 and Fig Figure 2.6 S-parameters of HFSS and measured results obtained in the work by Ling, Feng, et al. Systematic analysis of inductors on silicon using EM simulations. Electronic Components and Technology Conference, Proceedings. 52nd. IEEE, To further verify the simulations results, S-parameters have been obtained using the physical model described above for the following values: three and half turns, 10 µm wide metal lines and inner radius of 90 µm. The S-parameters extracted from model are compared with the S-parameters obtained from 13

9 Figure 2.7 S parameters obtained from HFSS simulations of typical 3.5 turn inductor. HFSS simulator. The results are shown in Fig Again similar trends in the S-parameters could be observed. Also dc inductance value is calculated using modified wheeler formula [51] given by Eqn and compared with inductance obtained from HFSS simulations at frequencies less than 1 GHz for a 3.5 turn inductor with a spacing of 5 µm, line width of 10 µm and outer diameter of 180 µm. The value of dc inductance calculated is 2.4 nh and inductance at frequencies less than 1 GHz obtained from HFSS is 2.36 nh. Both the values are in good agreement. L = K 1 µ 0 n 2 d avg 1 + K 2 ρ (2.16) where K 1 and K 2 are layout dependent factors and are equal to 2.34 and 2.75 respectively for a square spiral. n is number of turns which is 3.5 for the simulated structure. d avg and ρ are given by below equations: and d avg = (d out + d in ) 2 (2.17) ρ = d out + d in d out d in (2.18) 14

10 Figure 2.8 Comparison of S-parameters obtained from physical model and HFSS. Figure 2.9 A spiral inductor with 3.5 turns. Two open ends of the highlighted portion indicate the two terminals of the inductor. 15

11 and d out and d in are outer and inner diameters of the spiral inductor respectively. Having established the HFSS simulation capability, physical parameters of interest are varied to analyze the parameter sensitivity. All the simulation results obtained are in good agreement with the measured results of fabricated inductors in [52] which are summarized in the Fig Figure 2.10 Measured frequency at Q max and self-resonant frequency for various fabricated inductor configurations, where N is number of turns, A is inner opening diameters, W is coil conductor width,s is conductor inter-turn space and D is substrate contact guard ring distances. Image courtesy: Chao, Chuan-Jane, Characterization and modeling of on-chip spiral inductors for Si RFICs,Semiconductor Manufacturing, IEEE Transactions on 15.1 (2002): Effect of thickness of metal Thickness of aluminum, both underpass and spiral is varied from 0.2 µm to 1 µm. As thicknesses of spiral and underpass increase, inductance and quality factor of spiral inductor increase. But there is insignificant shift in the self resonant frequency and the value of inductance. Fig and Fig show the effect of increasing thickness of aluminum on inductance and quality factor respectively. At lower frequencies increasing the metal thickness decreases the series resistance as can be observed from the Eq. 2.10, due to increase in cross section of the metal. This results in higher quality factor with increasing thickness. As the frequency increases skin and proximity effects cause an increase in series resistance Effect of width and spacing of metal lines This investigation is done in two parts: (1) increasing width of the metal turns with constant metal to metal spacing and (2) increasing spacing between metal turns with constant width. Though these investigations are done on a wide range of data, only two cases are shown: (a) width varied from 1 µm to 17 µm with a constant metal to metal spacing of 7 µm (Fig and 2.14) and (b) metal to metal spacing of spiral inductor varied from 1 µm to 9 µm with a constant width of 5 µm (Fig and 2.16). For a constant spacing between metal to metal of spiral, as the width of the aluminum is increased, it is observed that the self resonant frequency of inductor decreases. Wider metal inductor has the highest inductance till its resonant frequency compared to others. Similarly, for a constant width of aluminum, 16

12 Figure 2.11 Variation of inductance with metal thickness Figure 2.12 Variation of quality factor with metal thickness 17

13 Figure 2.13 Variation of inductance with spacing of 7 µm with various width Figure 2.14 Variation of quality factor with spacing of 7 µm with various width 18

14 if the spacing between metal turns is increased then the shift in self resonant frequency towards lower frequencies is observed. Higher inductance and quality factor till self resonant frequency are achieved at an optimal value of pitch between the metal lines. Aluminum metal thickness of 500 nm is used for these investigations. At lower frequencies lesser spacing between metal lines aids in maximum magnetic flux coupling increasing the inductance and quality factors. Due to the proximity effects there is a non uniform current distribution at higher frequencies which in turn effects the magnetic flux distribution. Hence at higher frequencies an optimal value of spacing and width would be desired for higher quality factor at the required value of inductance. Subtle changes in width and spacing of the metal lines cannot be accurately described by the physical model and hence field solver is suitable for such situations. Figure 2.15 Variation of inductance with metal width of 5 µm with various spacing between metals Effect of internal radius Internal radius of a 3.5 turn spiral inductor has been varied from 1 to 60 µm. As the internal radius of spiral inductor is increased, self resonant frequency of inductor decreases significantly. Fig and Fig show the results of spiral inductor with varied internal radius. A 500 nm thick aluminum is used for these simulations. As the internal radius is increased, both the inductance and quality factor improve and also the area of the inductor increases significantly. As internal radius decreases the mutual inductance between the conductors carrying currents in opposite directions increases therefore inner turns of inductor contribute lesser towards the total inductance. 19

15 Figure 2.16 Variation of quality factor with metal width of 5 µm with various spacing between metals Figure 2.17 Variation of L with internal radius 20

16 Figure 2.18 Variation of Q with internal radius Hence hollow spiral inductors are preferred. Eq indicates that there is an increase in inductance value with increase in internal radius Effect of number of turns As the number of turns are increased from 1.5 to 5.5 with same internal radius, self resonant frequency of inductor decreases significantly. Fig and Fig show the results of spiral inductor with increased number of turns. Inductance and quality factor increase with number of turns. Figure 2.19 Variation of L with number of turns Eq indicates an increase in inductance value with number of turns. This is because of the increase in total self inductance due to additional metal lines. Mutual inductance between the adjacent lines also increases due to additional number of turns. As the number of turns increase the area occupied by the inductor increases. Thus the capacitive coupling with the substrate and oxide increases, as can be inferred from Eq Thus as both the inductance value and the capacitances values increase with increase in number of turns, self resonant frequency decreases. 21

17 Figure 2.20 Quality factor variation with increase in number of turns 2.4 Magnetic field distribution in on-chip inductors Distribution of magnetic field in the vicinity of inductor is also simulated and is shown in Fig It is essential to understand the magnetic field distribution for proper placement of magnetic materials. It could be observed that the significant amount of magnetic field surrounds the metal lines. This has direct effect on the placement of magnetic materials as discussed in latter chapter. 2.5 Summary Based on the above simulation results following conclusions could be drawn: Decreasing internal radius and number of turns contribute significantly towards increasing the self resonant frequency of the spiral inductor. As the self resonant frequency increases, the feasibility of spiral inductor at higher frequencies and the bandwidth increases. Optimization of geometry parameters is essential to obtain high performance inductor which occupies minimal area. But the improvement in inductance and quality factor that could be obtained with optimization in geometry parameters is insufficient for designers. Employment of magnetic material could help in avoiding this shortcoming. The H-field distribution shows that magnetic field is confined along the metal lines. It is important to consider proper placement of magnetic materials for effective increase in inductance and quality factor. 22

18 Figure 2.21 Magnetic field distribution in an inductor HFSS simulations have also been performed on the fabricated inductor and a good agreement is there between the measured and simulated values of inductance. These results are discussed in chapter 4 and chapter 5 respectively. Electromagnetic field solvers do not take into account the domain and domain dynamic behavior involved in the magnetic material employed with inductors. Therefore the exact behavior of the magnetic material incorporated with inductors and its effects at high frequencies are unpredictable with field solvers. Micromagnetic solvers are utilized separately to determine the domain properties of magnetic material. 23