Chapter 3 Broadband and Scalable On-chip Inductor
3.1 Broadband accuracy for on-chip inductors
3.4.4 Varying substrate resistivity effect
Regarding substrate resistivity effect on inductors’ frequency response and performance, EM simulation using ADS Momentum with extensive calibration is conducted to accurately predict the broadband characteristics under wide range of substrate resistivities (ρsi = 0.05 ~ 1KΩ −cm). Three operation models such as TEM,
slow-wave, and eddy current are presented. The improved T-model parameters manifest themselves physics-base through relevant correlation with ρsi over three operation mdoes. In this section, we will use ADS Momentum simulation with extensive calibration conducted to predict the broadband characteristics under varying ρsi. Figure 3.47 indicate good math between ADS Momentum, measurement, and T-model in terms of S11, S21, L(ω), Re(Zin(ω)), and Q(ω) fro inductors on standard substrate of ρsi = 10Ω −cm. We find one reference to discuss above mention. It divide to three operation modes, such as TEM, slow wave, and eddy current corresponding to wide range of ρsi (0.05 ~ 1KΩ −cm).
Figure 3.47 Comparison between ADS momentum simulation, measurement, and improved T-model for on-chip inductor (a) S11 (mag, phase) (b) S21 (mag, phase) (c) L(ω), Re(Zin(ω)) (d) Q(ω)
Figure 3.48 (a)-(d) show Qm, fm, fLmax, and fSR as function of ρsi. Qm is the maximum Q and fm is the frequency responsible for Qm. fLmax is the frequency corresponding to maximum L. Interesting result is identified in region of ρ =0.05 ~ 10
Ω −cm. where fSR drops monotonically with reducing ρsi while Qm reveals a hump due to initial increase and then drop with further reduction of ρsi. The drop of fSR and increase of Qm suggest that the spiral coil is getting into resonator mode, i.e.
slow-wave model. As fro high resistivity region of ρsi > 10 Ω −cm, fSR saturates at maximum while Qm increases continuously with ρsi This region is so called TEM model or inductor model, which favors inductor operation with high Q attributed to suppressed resonance in substrate of dielectric property. Regarding the very low resistivity of ρsi < 0.5 Ω −cm, fSR saturates at minimum and Qm drops drastically. The spiral coil is driven into eddy current model or skin effect mode where ρsi is so small that the skin depth is thinner than the substrate thickness and becomes the limiting factor. In the following, improved T-model parameters are extracted from the simulated S-parameters under various ρsi to verify if the model parameters can reflect the physical properties responsible for the three modes of operation.
10m 100m 1 10 100 1k
by ADS Momentum simulation
Figure 3.49 indicate how the resistive elements (Rp , Rsub, Rloss, Rloss1, Rloss2) and inductive elements (Lsub, Lsub1, Lsub2) vary with varying ρsi. Quite interestingly, Rp just follows exactly the same trend as that of Qm vs. ρsi with a hump in slow-wave model while the others show monotonic increase with ρsi in slow-wave and TEM modes and near saturation in eddy current mode. Regarding the capacitive elements (Cp, Csub, Cox1, Cox2) in figure 3.48 (c)-(d), all four capacitances demonstrate monotonic increase with reduction of ρsi in slow-wave mode, saturation in TEM mode while different behaviors in eddy current mode. The larger capacitances associated with lower ρsi.
Concerning the RLC model parameter effect in determining Qm and fSR over the wide range of ρsi, figure 3.50 reveal a monotonic increase of Qm due to increase of Rp while hump appears for other resistive or inductive elements such as Rsub, Rloss and Lsub
corresponding to the region of slow-wave mode. The result supports an important point that Rp, a new element introduced in our T-model is the key parameter to explicitly guide substrate engineering for on-Si-chip inductor to achieve maximum Qm. Figure 3.51 present the capacitances’ (Cp, Csub, Cox1, Cox2) effect on fSR where monotonic increase of fSR with lowering capacitance is demonstrated for all four capacitances.
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Figure 3.50 Qm vs. Improved T-model parameters under varying ρsi (a) Rp (b) Rsub, (c)
100 150 200 250
12
3.4.5 Broadband accuracy
Mag
(
S21)
(dB)Mag(
S21)
(dB)mea_4.5 T_model_4.5 mea_2.5
T_model_2.5 mea_3.5
T_model_3.5
Figure 3.52 Comparison of improved T-model and measured S11, S21 (mag, phase) for inductors. Coil numbers (a) N=2.5 (b) N=3.5 (c) N=4.5 (d) N5.5
The improved T-model has been verified by comparison with measurement in terms of S-parameters (S11, S21), L(ω), Re(Zin(ω)), and Q(ω) over up to 20GHz. Figure 3.52 (a) ~ (d) indicate the comparison for magnitude and phase of S11 and S21
between the model and measurement. According to figure 3.52, match is achieved for all coil numbers even beyond resonance, which happened at fSR << 20GHz for larger coil number (N=3.5, 4.5, 5.5). It is an obvious improvement over the original T-model and even better match is achieved as compared to EM simulation (figure 3.47). All three parameters are frequency dependent that is critically related to the spiral conductor loss and Si substrate loss. Figure 3.53 (a) ~ (d) can accurately fit to the
measured L(ω), and Re(Zin(ω)) by the improved T-model for all inductors operating up to 20 GHz. Besides, the model can exactly capture the full band behavior of Re(Zin(ω)) even beyond resonance such as the dramatic increase prior to resonance, peak at resonance, and the curve drop after the peak. And Q(ω) is the most important parameter governing RF IC performance. Figure 3.53 (d) shows the excellent match with the measured Q(ω) over broadband of 20 GHz.
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Figure 3.53 Comparison of improved T-model and measured L(ω), Re(Zin(ω)) for inductors. Coil numbers (a) N=2.5 (b) N=3.5 (c) N=4.5 (d) N=5.5
3.4.6 Scalability
Another important feature is the good scalability w.r.t dimension for all model parameters. Figure 3.54 (a) ~ (d) illustrates present good match with a linear function of coil numbers for each model parameter. These parameters represent the spiral coil’s RLC network. Figure 3.55 (a) ~ (d) illustrates also present good match with a linear function of coil numbers that have been involved model parameters, Csub, 1/Rsub, Lsub1, Lsub2, Rloss, Rloss1, Rloss2. These parameters have proven scalability and these also suggests that T-model can be used for pre-layout simulation and optimization.
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Cp =-0.48905+4.7087N Cp T-model
Rp =1953.09815+103.8559 N Cox1 =0.10465+17.1179N
Cox2 =17.70615+15.9049N
Rp
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Lsub, Lsub1,2 (nH)
Coil Number, N Lsub2 =-0.357+0.294 N
Lsub1 =0.0111+0.1306N Rloss1=12.7896+18.3736N Rloss2=17.287+36.63 N