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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