3Modes_Writeup_092104
Motivation:
The interconnections on the chip serve as the information link in most cases. The ideal
performance thus is a short circuit. However, the actual behavior of the transmission line
includes delays, losses, cross-talks, dispersion, and substrate noise, etc. All these effects
could degrade the circuit performance.
The microstrip line model (MIS) is one of the basic structures for the on-chip
interconnections. It is worth lots of discussion in order to obtain a detailed understanding
for more complicated interconnect structures, such as coupling effect among the
transmission lines, the 3-D interconnection effects, substrate losses and doping profile
effects for the interconnections.
A typical MIS structure includes a metal ground plane which reduces the coupling noise
between the transmission lines, serving as a medium for the heat radiation. On top of the
ground plane is the Silicon substrate, which varies its thickness and conductivity
(different doping profile) according to the requirement of the components and different
fabrication technologies. An oxide layer is sandwiched between the substrate and the
metal line, which insulates the substrate from the signal (or power) line.
3-modes in the MIS structure
The earliest work was started by Guckel in 1967. In 1971, Hasegawa has first proposed
using the parallel-plate waveguide model to analyze the MIS theoretically and compared
his results with the experiment over a wide range of frequency, substrate resistivity, and
strip width. By solving the attenuation constant  , and the phase constant  , he finally
defined the well-known 3 fundamental modes (namely the Dielectric Quasi-TEM Mode,
the Skin-Effect Mode and the Slow-Wave Mode) for the parallel waveguide structure,
which also exist for the MIS interconnection structure, and it is proved by the
experiments.
We have revisited the analytical method for the parallel waveguide structure, and
regenerated the detailed  ,  map over a variety of resistivity and frequency range for a
certain combination of b1 , b2 ,  1 ,  2 ,  1 ,  2 . The 3-modes and their transition region are
shown in the map, separated by the 3 lines ( 4.0  f  , 0.3  f 0 , and 1.5  f e ), derived in the
literature.
The physical origin of the 3 fundamental modes is as follows (from Hasagawa, Kwon,
etc):
1)
Dielectric Limits: When the product of the frequency and the resistivity of the Si
substrate is large enough to produce a small dielectric loss angle (recall

1
), the Si substrate acts like a dielectric, and the line can be
tan  2  2 
 20 2 f 2
regarded as a microstrip line loaded with a double-layer dielectric consisting of Si and
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3Modes_Writeup_092104
SiO2. As long as the wavelength is much larger than the thickness of the double layer, the
mode is quasi-TEM mode. This is a paraphrase of Dielectric Quasi-TEM mode existing
at high frequencies. Since in reality,  Si b1   SiO2 b2 , almost all the energy is transmitted
through the Si layer, with the velocity almost equal to c /  Si .
2)
Metalic Limits: When the product of the frequency and substrate conductivity
( f 2  f /  2 ) is large enough to yield a small depth of penetration (   1 /  2 f 0 ) into
the silicon, the substrate would behave like a lossy conductor wall, and the line may be
treated as microstrip line on the imperfect ground plane made of silicon. If the depth of
the penetration is much smaller than the thickness of the oxide-layer, we should still do
the fundamental mode analysis. However, this condition rarely holds, because the skin
depth is   1 /  2 f 0 , and under the frequency range of interest,  is varying from
0.5um to 1.6m. The line behaves highly dispersive due to the skin effect.
3)
Semiconductor limit: Frequency is not so high and  2 is moderate, a slow-surface
wave propagates along the line. This mode is generated by the Maxwell-Wagner


mechanism, which says  s 0   SiO2 b1 / b2 . It means a strong interfacial polarization will
occur if b1  b2 , and thus the Maxwell-Wagner permittivity is becoming very large, so
the propagation velocity becomes very slow due to this new effective permittivity.
In the slow-wave mode, almost all the active power is transmitted through the SiO2 layer,
but a large amount of reactive power is exchanged between two layers across the Si
substrate and the Oxide layer with the movement of charge at the interface. The average
amount of magnetic energy stored in the waveguide is equal to that of the electric energy
stored in the SiO2 layer. The magnetic energy is stored mainly within the Si layer because
b2>>b1 (I got this from Hasegawa’s paper, but not fully understand so far). The sum of
these energies is transmitted in the z direction through the thin SiO2 layer. Due to the
energy transfer across the interface, the propagation velocity slows down a lot.
A review of the 3-mode conclusions we got is as follows:
1. When the substrate resistivity is high (greater than approximately 10^4 Homs*cm), at
high frequencies, the substrate acts like an insulator, resulting the dielectric quasiTEM mode;
2. When the substrate resistivity is low (less than approximately 10^-3Ohms*cm), at
high frequency, the substrate acts like an imperfect metal wall having a large skin
effect, and the MIS is in the skin effect mode;
3. The slow-wave mode propagates when the resistivity is neither too low nor too high.
In the low frequency limit ( note that this frequency limit extends into the gigahertz
range at certain substrate resisitivities), the electric field lines do not penetrate into the
semiconductor, whereas the magnetic field lines can fully penetrate into it causing
spatially separated storage of electric and magnetic energies.
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The loss is higher in the transition region between the slow-wave mode and the dielectricquasi TEM mode. The reason is: The dielectric loss in the substrate is due to the
transversal electric field. Also, in the skin effect region, the attenuation constant is larger;
this is due to the longitudinal currents in the substrate.
The Numerical work undergoing and its promising:
So far, we have learned that in order to get the analytical results and avoiding too
complicated derivations, we had neglected the fringing effect, and derived the analytical
result and the 3-mode map based on the parallel-plate waveguide. This approximation is
reasonable since the width of the strip-line is much much larger than the thickness of the
oxide. Other research groups has formed lumped circuit model to analyze the behavior of
the transmission lines. However, in order to get the detailed full wave analysis for the
MIS structures, neither the lumped model, nor the analytical method with several
approximations is good enough. Numerical method is a more powerful way to get the
complicated analysis, though it will cost more time to process.
We have applied the ADI-FDTD method to simulate the wave propagating in the MIS
structure in the time-domain. With the help of Fourier analysis, the frequency response
and the spectrum analysis could be done for this structure. After the numerical result is
obtained, similarly, the attenuation constant  , and the phase constant  could be
extracted from the filed solutions, and a similar map for a MIS structure with different
metal width could be drawn. This is beyond the limit of the pure analytical method.
A further step is to get more accurate analysis of the behavior of more complicated
interconnection structures, such as the coupling effect between the neighboring lines, 3D
interconnections, and the on-chip interconnections with different doping profiles,
geometry, varying with different frequency range. Again, we would be able to extract the
 and  , and draw the map over conductivity and frequencies of those structures. This
map should be similar to the map of the basic parallel-plate waveguide. After the map is
drawn, it could help the designers to make their decisions. For example, if the
interconnection is used as the delay line, we would prefer to use it in the slow-wave mode,
which produces a great delay (about 1/20 of the free space velocity in our example), and
has negligible loss. Or use the line in the quasi-TEM mode, which should produce very
small dispersion. As the frequency getting higher and higher, the structure may not
remain in the slow wave mode as in the traditional IC’s.
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