3Modes_Writeup_092104
Motivation:
The interconnections on integrated circuits form a complex metallic network which
contains hundreds of transmission lines ranging from millimeters to centimeter in length.
These interconnect networks are key to transporting electromagnetic energy throughout
the chip that is obtained from external microwave sources. The Metal Insulator
Semiconductor (MIS) microstrip line model is one of the basic structures for the on-chip
interconnections. A typical MIS structure includes a metal ground plane. 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
We are extending previous work on MIS structures (Hasegawa, 1971) to include more
complex physics and the effects of external EM radiation. We analyze the MIS over a
wide range of frequency, semiconductor substrate doping (resistivity), and strip width. By
solving for the attenuation factor  , and the phase factor  , three fundamental modes
are defined (namely the Dielectric Quasi-TEM Mode, the Skin-Effect Mode and the
Slow-Wave Mode).
We have generated a detailed  ,  map over a large resistivity and frequency range.
The physical origin of the 3 fundamental modes is as follows
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
SiO2. As long as the wavelength is much larger than the thickness of the double layer, the
mode is quasi-TEM mode. This analogous to the 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 interconnect
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 perform 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.
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3Modes_Writeup_092104
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. 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 reduces
considerably.
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.
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:
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.
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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,
fwhich 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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