Wideband Lumped Element Model for On-Chip Interconnects on Lossy Silicon Substrate Sheng Sun1, Rakesh Kumar1, Subhash C. Rustagi1, Koen Mouthaan2 and T. K. S. Wong3 1

2

Institute of Microelectronics, Singapore Department of Electrical & Computer Engineering, National University of Singapore, Singapore 3 School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore

Abstract — This work presents the fully lumped element model for wideband on-chip interconnects, with large scalability of line lengths up to 8000 Pm, and widths down to 100 nm. Both the series and the shunt lumped elements of the model are determined based on the frequency asymptotic technique without any optimization. The equivalent lumped circuit is derived and verified to accurately recover the frequency-dependent parameters up to 40 GHz, where all the parasitic effects are accounted for based on the EM simulation and measurement. The convergence of the cascaded lumped element model is analyzed for the minimal requirement of desired wideband and distributed performance. The frequency responses of the proposed model agree well with those of the distributed transmission line model and measurements. Finally, the model is validated by comparing simulated and measured Scatteringparameters. Index Terms — Interconnect, lumped element model, frequency-dependent, transmission line, silicon substrate, wideband.

I. INTRODUCTION Over the past decade, silicon-based CMOS technology has proven to be the most cost-effective process for the development of advanced radio frequency (RF) and mixed-signal integrated circuits. As the operating frequency increases, an accurate and scalable model of on-chip interconnects becomes a fundamental and dominant requirement in different applications over a very wide frequency range, such as 3.1–10.6 GHz ultrawideband (UWB) systems, 60 GHz RF systems, and 77 GHz radar systems. Due to the lossy nature of the low-resistivity silicon substrate, the wideband behavior of on-chip interconnects becomes very lossy and frequency-dependent. The onedimensional metal-insulator-semiconductor (MIS) transmission line can be characterized by rigorous analytical methods and treated by different equivalentcircuit descriptions [1]–[3]. Up to 100 GHz, the closedform expressions are developed in [4] for the equivalentcircuit parameters of the dominant TM0 mode, and the full-wave analysis, finite difference time-domain (FDTD)

(a)

(b)

Fig.1. Lumped element model for on-chip interconnects. (a) Type-I. (b) Type-II.

method [5] and spectral domain approach (SDA) [6], are extended to characterize the MIS coplanar transmission line, where the metal conductor loss is also included. The transmission line can be characterized by the generalized Telegrapher’s equations with a series impedance R (Z )  jZ L (Z ) and a shunt admittance G (Z )  jZC (Z ) , where R (Z ) , L (Z ) , G (Z ) and C (Z ) are the frequency-dependent distributed per-unit-length (p.u.l.) resistance, inductance, conductance and capacitance, respectively. Unfortunately, it is difficult to be directly applied to the circuit-level simulators, such as SPICE and ADS, particularly while carrying out the time domain simulation, due to the distributed effects of the transmission line at high frequencies. Therefore, the equivalent circuits of interconnects consisting of only ideal lumped elements are generally used in the circuits design [7], [8]. Based on the EM simulation data, the equivalent-circuit parameters are extracted up to 10 GHz, including the frequency-dependent shunt admittance by additional dielectric time constant relationship [7] and series impedance with effective substrate current loops by approximating rational function [8]. On the other hand, the complex image approach is employed [9] to derive the approximate closed-form expressions for the series impedance parameters R (Z ) and L (Z ) . Meanwhile, the shunt admittance parameters G (Z ) and C (Z ) are derived in terms of high- and low-frequency asymptotic solutions [10]. So far, all of the above mentioned lumped element models are limited to about 10 GHz. It is imperative and absolutely necessary however to develop fully lumped

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element models for a variety of wideband applications. In this paper, two simple circuit topologies, as shown in Fig. 1, are investigated with different series impedance configurations to account for metal as well as substrate skin effects. The circuit topology with a pair of additional series resistance (R2) and inductance (L2) is connected in parallel with the resistance (R1) shown in Fig. 1 (a), and in parallel with the inductance (L1) shown in Fig. 1 (b). Each lumped element of series impedance and shunt admittance is extracted based on the frequency asymptotic technique, and the determined frequency-dependent parameters have good agreements with those of directly received full-wave EM simulation data by SONNET. Finally, the model is validated from the simulated and experimental results up to 40 GHz with large scalability of lengths up to 8000 Pm, and widths down to 100 nm.

(a)

II. WIDEBAND LUMPED ELEMENT MODEL

A. Series Lumped Elements The frequency dependent series impedance of a single interconnect on lossy silicon substrate can be represented in terms of the equivalent circuits shown in Fig. 1. Both configurations in Fig. 1 have the same merging results and the only difference is the value of each individual element. The p.u.l. parameters are first extracted from the Sparameters obtained by full-wave EM simulator SONNET. Without any optimization, each element can be directly obtained as follows, Type-I: R1 lim R (Z ) and L1 lim L (Z ) (1a) Z of

Z of

(b)

lim R (Z ) lim R (Z ) R2

Z o0

Z of

lim R (Z )  lim R (Z ) Z of

Z o0

 lim L(Z )  lim L(Z )  lim R(Z )  lim R(Z )

lim R (Z ) L2

Z of

2

Z o0

lim L (Z )

Z o0

Z o0

(1b)

Z o0

lim R (Z ) and L1 2

Z of

R1 R2 R1  R2  Z 2 R1 L2 R1  R2 2  Z 2 L2 2

Lc(Z )

R1 L2 L R1  R2 2  Z 2 L2 2 1 2

Type-II: R c(Z )

Z o0

Z of

lim L (Z )  lim L (Z ) Z o0

R1 

Z of

lim L (Z ) lim L (Z ) L2

Z R2 L1

Lc(Z )

Z of

Subsequently, the merging impedance Rc(Z )  jZ Lc(Z ) in the simple equivalent extraction are then obtained as,

(2a)

2

2

2

2

R2  Z ( L1  L2 )

2

(2c)

2

R2 L1  Z L1 L2 ( L1  L2 ) 2

(2b)

2

R2  Z ( L1  L2 )

2

(1d)

2

2

Z o0

2

Z o0

Type-I: Rc(Z )

(1c)

Z o0

 lim R(Z )  lim R(Z ) and  lim L(Z )  lim L(Z )

lim L (Z ) R2

Z of

2

Z of

Type-II: R1

Fig.2. Comparison between the original simulated and recovered frequency-dependent p.u.l. parameters of the wideband on-chip interconnect. SiO2 substrate: Hr=4.1, Si substrate: Hr=11.9 and U=10 :-cm.

and

2

(2d)

Fig. 2(a) shows the comparison between the original simulated values of R (Z ) , L (Z ) and the recovered values of R c(Z ) , Lc(Z ) . The maximum relative error is smaller than 2.9 % for the series resistance and 1.2% for

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the series inductance over the frequency range 50 MHz to 40 GHz. B. Shunt Lumped Elements Similarly, the shunt lumped elements, including oxide capacitance C1, silicon capacitance C2 and silicon conductance G2 are derived as, (3a) C1 lim C (Z ) Z o0

lim C (Z ) lim C (Z ) C2

Z o0

Z of

(3b)

lim C (Z )  lim C (Z ) Z o0

Z of



lim G (Z ) lim C (Z )  lim C (Z ) G2

Z of

Z o0

Z of

lim C (Z )

2



2

(3c)

(a)

Z o0

Then, the merging admittance G c(Z )  jZC c(Z ) is given by [10], 2

G c(Z )

Z G2 C1 2

2

2

G2  Z (C1  C2 )

(4a)

2

2

C c(Z )

2

Z C1C2 (C1  C2 )  G2 C1 2

2

G2  Z (C1  C2 )

2

(4b)

As shown in Fig. 2(b), the maximum relative error is smaller than 4.7% for the shunt conductance and 2.0% for the shunt capacitance over the frequency range 50 MHz to 40 GHz. III. SIMULATED AND EXPERIMENTAL VALIDATION In this section, the modeling technique described above is validated in terms of the scattering parameters by cascading several basic cells with length selection in [11]. For the short length of 500 Pm, the lumped element model with 4 cells is found to have a good agreement with the transmission line model, as well as the direct EM simulation and measurement over frequency range of interest, as shown in Fig. 3 (a). Fig. 3(b) indicates a good accuracy of lumped element model for predicting the long length interconnects. For the long length case with distributed effects, the cell number of complete equivalent circuit is increased and it is found that 32 cells are enough for the 8000 Pm case, as shown in Fig. 3(b). On the other hand, the model also can be directly applied on the experimental data. As shown in Fig. 4, the model validation is implemented based on the measured S-parameters. The p.u.l. parameters are firstly extracted from measured S-parameters, and lumped element values are then derived by Eq. 1 and 3. The recovered results are plotted together with those from transmission line model and measurement. Up to 40 GHz, the model provides a

(b) Fig.3. Model validation based on the EM simulated results. (a) Short line length: 500 µm. (b) Long line length: 8000 µm.

very good and scalable agreement with selected line widths of 8 Pm, 0.5 Pm, and 100 nm. Interestingly, the behavior of nano-scale transmission line with width of 100 nm, becomes very lossy and weak changing with frequency, while that of wider lines strongly depends on frequency. VI. CONCLUSION In this paper, a wideband lumped element model is presented and validated up to 40 GHz. All the lumped elements including the series impedance components and shunt admittance components are directly extracted without any optimization or high-order approximation. The obtained ideal lumped element model can be directly used in the SPICE-compatible simulation for the wideband frequency applications in the design of RF integrated circuits.

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ACKNOWLEDGEMENT The authors wish to acknowledge the assistance and support of the device modeling & CAD group of Institute of Microelectronics, Singapore. This work was carried out for the A*STAR project No. 0421140045. REFERENCES

(a)

(b)

(c) Fig. 4. Model validation based on the measured results with short line length of 500µm. (a) Wide line width: 8µm. (b) Narrow line width: 0.5µm. (c) Nano line width: 100 nm.

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