Modeling of High Speed
MetalInsulatorSemiconductor Interconnections:
The Effect of ILD on SlowWave Attenuation
By
Liyong Wang
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF SCIENCE
Major Subject: Electrical Engineering
Approved:
John F. McDonald
Thesis Advisor
Rensselaer Polytechnic Institute
Troy, New York
April, 1998
(For Graduation May, 1998)
Table of Contents
List of Tables *
List of Figures *
Abstract *
Acknowledgements *
Chapter 1 Introduction *
Chapter 2 Basic Physics Concepts *
2.1 Transmission Line and Microstrip *
2.2 Transmission Line Theory *
2.2.1 Lossless Line *
2.2.2 Low Loss Line *
2.3 Conformal Transformations *
2.4 Effective Dielectric Constant *
2.5 SkinEffect *
2.6 Silicon SlowWave *
2.7 Electrical Properties of Materials under Low Temperature *
2.7.1 Silicon *
2.7.2 Metals *
Chapter 3 Basic Modeling *
3.1 Validity of QuasiTEM assumption *
3.2 Computing Output Waveform *
3.3 Characteristic Impedance and Effective Dielectric Constant *
3.4 Attenuation Factor *
3.4.1 Conductor Loss *
3.4.2 Substrate loss *
3.5 Total attenuation and propagation factor *
Chapter 4 Extended Modeling *
4.1 Equivalent Circuit Model *
4.2 Computing k *
4.3 Effective Dielectric Constant *
4.4 Result and discussion *
4.4.1 Pulse propagation along microstrip line *
4.4.2 Conductor and Substrate Attenuation Constants a_{c} and a_{d} *
4.4.3 Total Attenuation Constant a *
4.4.4 Factors that Affect Pulse Propagation *
4.4.4.1 Pulse Width *
4.4.4.2 Insulator Layer Thickness *
4.4.4.3 Insulator Dielectric Constant *
4.4.4.4 Substrate Resistivities *
4.4.4.5 Conductor Materials *
4.4.4.6 Cryogenic Temperature *
Chapter 5 Conclusions and Future Works *
References *
Table 11 Typical interconnections and maximum usable length [3] *
Table 21 Common line structures for wave propagation [19] *
Table 31 Calculation of electrical parameters *
Figure 21 A typical microstrip structure *
Figure 22 Distributed circuit model for transmission line *
Figure 23 Illustration of current distribution in a microstrip vs. wire dimension and frequency. Current density increases with darkness. *
Figure 24 Resistance of 1 cm microstrip vs. frequency at different dimensions *
Figure 25 Electric and magnetic field within a microstrip structure *
Figure 26 Resistivity of borondoped silicon vs. temperature *
Figure 27 Resistivity of arsenicdoped silicon vs. temperature *
Figure 28 Resistivities of several metals vs. temperature *
Figure 31 Simplified microstrip line structure *
Figure 32 typical electric field pattern of a microstrip *
Figure 33 Characteristic Impedance vs. w/h at zero frequency *
Figure 34 Effective dielectric constants vs. frequency *
Figure 35 Characteristic impedance vs. frequency *
Figure 36 Current distribution profiles in a microstrip *
Figure 37 Surface recessions for calculating skineffect resistance *
Figure 38 Equivalent circuit model *
Figure 41 Microstrip line structure of extended model *
Figure 42 Equivalent circuit of extended model *
Figure 43 First order approximation of electric field in a MIS microstrip structure *
Figure 44 Actual electric field line in a MIS microstrip structure *
Figure 45 Actual electric field lines in MIS microstrip structure with thick insulator layer *
Figure 46 Comparison of k value between actual and simplified result. *
Figure 47 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm, including 1mm SiO_{2} layer. *
Figure 48 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm, ignoring the SiO_{2} layer. *
Figure 49 Conductor attenuation factors vs. frequency under different conductor resistivities *
Figure 410 Substrate attenuation factor vs. frequency under different substrate resistivities *
Figure 411 Total attenuation factor a versus frequency for Cu, Al, W, and polySi microstrips. *
Figure 412 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm vs. different pulse width: 5ns, 500ps, 50ps, and 25ps *
Figure 413 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm vs. different insulator layer thickness: 0m m, 1m m, 4m m, and 13m m. *
Figure 414 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm vs. different insulating dielectric constants: 4, 2.4, 1.6 and 1. *
Figure 415 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm vs. different substrate resistivities: 13W cm, 50W cm, 100W cm, and Infinity. *
Figure 416 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm vs. different conductor materials with resistivities of: 0 m W cm, 2.7 m W cm (Al), 10 m W cm (W), and 100m W cm (polySi). *
Figure 417 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm under temperatures of: 300K and 76K *
Figure 51 Electric field line in even and odd mode *
Because of rapid progress made in VLSI industry, the circuit delay due to interconnects becomes the dominate factor that limits the speed of VLSI chips because of shrinkage of device gate length, increasing of clock speed and larger die dimension. A theoretical model for a picosecond metalinsulatorsemiconductor (MIS) microstrip interconnection is presented in this thesis using a quasiTEM approximation. We have studied, in particular, the influence of an underlying SiO_{2} insulator layer as well as several other physical effects: slowwave substrate coupling, conductor resistance loss, skineffect degradation, distributed RC propagation, and cryogenic cooling. Although metalinsulatorsemiconductor transmission lines have been studied extensively by several previous authors, the influence of an underlying insulator has been usually ignored. In some cases, however, the insulator has been included by means of fullwave, frequencydomain approach. This fullwave approach is usually complicated and requires timeconsuming computer solution. A relatively simple model for the metalinsulatorsemiconductor transmission line is developed by extending a previous work [6]. The SiO_{2} layer simply decouples the interconnect conductor from the substrate. The extension in this thesis mathematically takes this into account. It also incorporates skineffect energy losses in metal conductors, and the slowwave losses due to finite substrate resistivity.
First, Several basic physics concepts that help to understand the modeling of microstrip structure are reviewed. The author then studies the previous model on which the new model bases. After that, the new model is presented and investigated thoroughly. The author explores several physical factors that affect the signal propagation in microstrip lines: pulse width, insulator layer thickness, insulator dielectric constant, substrate resistivities, conductor materials, and cryogenic temperature. Finally, further multiconductor and multilayer modeling and test structure are suggested by author as future works.
The excellent results demonstrated in this thesis show that the model is suitable for VLSI interconnection design, scaling and implementation in the manufacturing process.
This research is supported by the SRC Center of Advanced Interconnect Science and Technology, Rensselaer, contract ID: 448.024. This work would not have been possibly done without this funding.
I would sincerely like to thank my advisor, Prof. John F. McDonald, for his technical guidance and financial support in developing the models. I am also very grateful to Prof. Yannick L. Le Coz, for his kind and patient assistance in paper publishing. I greatly appreciate the helps from our group, Steven Nicholas, Samuel Steidl, Steven R. Carlough, Matthew Ernest, Xin Ma and Thomas Krawczyk. Their helps and friendship are of immense worth.
I would also like to give my special heartfelt thanks to my fiancée. Her endless love is the priceless treasure to give me light to overcome the darkest time. Finally, thankyou to my mom, dad, and sister in my far homeland  China, for their love and support.
Development of very highspeed integrated circuits is currently of great technological interest. The wiring cross sections are reducing and the lines are packed closer together, while at the same time the propagated signals switch with faster rise time. As processor cycle times become shorter and chips become larger and more complex, the performance of onchip interconnections becomes more important. Onchip interconnection delay and crosstalk are expected to become important problems. The average interconnection length L per gate on VLSI chips in known to increase with the gate count G, as in the following empirical formula:
(11)
where K and a are empirical constants, and A is the logic cell layout area. This equation can also be derived theoretically [1] on the basis of the Rent’s rule [2] concerning the pin versus gate count relationship. Table 11 show several typical interconnections used in today’s industry and the maximum interconnection lengths.
Interconnection type 



Onchip 
1–2 
130–260 
0.7–1.4 
Thinfilm carriers 
10–25 
1.25–4 
20–45 
Ceramic 
75–100 
0.4–0.7 
20–50 
Printedcircuit boards 
60–100 
0.06–0.08 
40–70 
Table 11 Typical interconnections and maximum usable length [3]
When the wavelengths of pulses are comparable with wire dimension, the wires act like transmission lines and microwave theory must be applied to solve these kinds of problems. In these cases, study of propagation of ultrafast transients becomes important. The challenges in VLSI techniques are bringing into focus the need of understanding pulse distortions caused by loss mechanisms in microwave region such as skineffect, dielectric dispersion and silicon slowwave. Onchip interconnects have unique characteristics, namely very high capacitive and inductive coupling and resistive losses, and very nonuniform transmissionline structures. Most of their parameters are frequencydependent. These characteristics make major limitation of further improvement of onchip delays. For example, reducing wiring dimensions results in appreciably resistive lines, even when the best conductive metals, such as copper, are used. Also the semiconducing substrates used in MMIC’s and VLSI interconnects cause dielectric losses to become a major contributor to signal attenuation. In addition, high dielectric loses, such as slowwave loss, can cause a significant change in the propagation constant, and a severe pulse dispersion.
As one of the most widely used and most important microwave interconnections, microstrip line gains a lot of attentions for past 40 years. The conventional lumpedcapacitance model assumes that signal propagation delays are much shorter than pulse rise times, but this assumption is no longer valid for highspeed VLSI circuits. Hasegawa and Seki [4] have shown that the traditional lumpedcircuit RC representation is no longer adequate when the switching speeds are faster than 100 ps. Because the propagation time of an ideal Transverse Electromagnetic Mode (TEM) wave over 1mm distance is typically 8–10 ps on a semiconductor substrate (e _{r} = 12–13), validity of such an approach becomes very doubtful in highspeed VLSI’s.
To date, a number of approximation techniques have been used for analyses of microstrip interconnections. Hasegawa, et al., [5] have thoroughly investigated the MetalInsulatorSemiconductor (MIS) microstrip structures. They introduced skineffect mode, slowwave mode, and dielectric quasiTEM mode propagating in microstrips under different frequencies and substrate resistivity conditions. Goossen and Hammond collect many theoretical formulas from previous works and use quasiTEM assumption to analysis a typical MIS microstrip line [6]. Y. R. Kwon, et al., also use quasiTEM assumption but to analysis Coplanar Waveguide (CPW) instead of MIS structure [7].
All of works above are analytic approaches. When encounter with multiconductor and multilayer problems, these analytic approaches are no long practical because of the unwieldy mathematical complexity. In these cases, a fullwave approach has its advantages to give accurate characterization of dielectric losses in structures with semiconducting substrates.
Spectral Domain Approach (SDA) has been discussed extensively in previous works. The SDA is simpler to formulate for multilayer, multiconductor structures and also in taking metal loss into consideration. It gives quick and accurate results, over a wide range of conductivities and frequencies. A very good explanation of the method can be found in [8] and an excellent list of references is given in [9]. A lossy multilayer, multiconductor microstrip structure is analyzed in [10] using SDA method. The paper gives attenuation and propagation constants over a wide range of substrate parameters and frequencies, covering all 3 propagation mode regions: low loss, slow wave, and skin effect. Spectral domain approach (SDA) with complex permittivity for the dielectric layers is also used to analyze microstrip structures.
There are several other approaches published in past decade, including modematching approach [11] [12] [13], Fouriertransformed domain analysis [14], finiteelement method [15], integral equation method [16], point matching method, finite difference method, boundary element method, step current density approximation, as well as other unique ways of solving this problem [17] [18]. Among them, modematching approach is another popular method that has also been applied to give accurate results for a limited range of substrate parameters.
Although these fullwave approaches mentioned above can give rigorous simulation results, their electromagnetic boundaryvalue problems are usually too laborious and timeconsuming for direct application in CAD programs. They require complicated computer programming. Also, they usually do not provide a clear analysis review of the effect of geometrical dimensions and technological line constants on the electrical parameters of microstrip. At this point, analytic approaches have their advantages of simplicity and ease of implementing into circuit simulators.
The purpose of this thesis is to investigate the onchip MIS microstrip line delay in very highspeed integrated circuits by analytic approaches, using quasiTEM assumptions. We extended a previous work [6] to develop our new theoretical model for a MIS microstrip interconnection. We have included several highfrequency physical effects: slowwave substrate coupling, conductor resistance loss, skineffect degradation, distributed RC propagation. As extensions, we have studied, in particular, the influences of an underlying insulator layer and cryogenic cooling effects. Our research results in this thesis show that this insulator layer is very important for signal attenuation and can not be simply ignored. This insulator layer decouples the interconnect conductor from substrate. The simple approximate formulas to express the attenuation and dispersion property in this paper are suitable for the purpose of desk calculations or the computer aided design of microwave integrated circuits.
Several basic physics concepts are needed for modeling the microstrip interconnections. Simplified microstrip line structure has to be identified to represent major characters of actual onchip interconnections. Transmission line theory is a necessity of understanding the propagation of signals along this kind of lines because the signal transmission is under microwave region at very high switchspeeds such as picosecond or faster. Conformal transformation is a very powerful mathematical way to calculate electrical parameter such as capacitance, effective dielectric constant, and characteristic impedance. It transforms coordinates system to another geometry in which the Laplace’s equation could be possibly solved while conserves certain values. Skineffect is obvious at high frequencies because the current is more packed at the region near the surface. This causes additional frequencydependent resistance and internal inductance. Slowwave loss is another very important effects on silicon substrate due to the finite resistivity of substrate layer. Unlike electric field, the magnetic field can feel free to penetrate the silicon substrate and this induces the separation of the electric field and magnetic field and finally causes slowwave mode propagation. The temperature dependence of electrical properties of materials such as metals and semiconductor are also studied since people are looking for the possibility of improving chip performance by chilling the chips under liquid nitrogen temperature. The resistivity of metals decreases dramatically. But semiconductors show their unique trend under lower temperature. The resistivity has a peak at around 100K and drops on both side of the peak. In the rest of the chapter, these physics effects are investigated more deeply and should be very helpful for understanding the calculation shown later in this thesis.
2.1 Transmission Line and MicrostripA "transmission line" is defined in [19] as any structure that guides a propagating electromagnetic wave from one point to the other. However, the common use of this term is far more restrictive. It is usually required that the electrical length of the line be at least several percent of a wavelength at the highest frequency of interest.
The transmission mode in a transmission line can be divided into two categories, TEM mode and nonTEM mode. If the electric and magnetic fields, which surround the transmission line, are normal both to each other and to the direction of energy propagation, the signal is transmitting in Transverse Electromagnetic mode (TEM). Two features can be used to distinguish a TEM mode.
Not all the transmission lines support the TEM mode propagation. For a transmission line that is surrounded by an inhomogeneous dielectric, the TEM mode can be supported at only dc frequency. This is because the electric field lines can not satisfy all the necessary boundary conditions at the dielectric interfaces and conductor surfaces without inclusion of a longitudinal component of the electric field at the dielectric interface. Several commonly used transmission lines are listed in Table 21. The propagation modes in these lines are also specified.
Symbol 
Common Name 
TEM Type 
Transmission Line 
Coaxial cable 
Yes 
Yes 

Stripline 
Yes 
Yes 

Balanced twowire line 
Yes 
Yes 

Microstrip 
No 
Yes 

Slotline 
No 
Yes 

Rectangular wave guide 
No 
No 
Table 21 Common line structures for wave propagation [19]
Among the transmission line above, the most important practical one for VLSI onchip interconnection is microstrip principally because it is easily made in the photolithographic fabrication technology predominant today. It is defined as a transmission line consisting of a strip conductor and a ground plane separated by a dielectric medium. However, since the dielectric layer is usually semiconductor such as silicon, an insulator layer is frequently inserted between the conductor and dielectric substrate. This structure is often call MIS structure although the conductor is not necessary to be a metal (e.g. polysilicon). Fig. 21 illustrates a typical microstrip line structure.
Figure 21 A typical microstrip structure
Circuits built with microstrip transmission lines have several important advantages such as:
It is important to remember that the microstrip is intrinsically dispersive (Table 21). It is incapable of supporting a pure TEM wave. This nonTEM property of microstrip is caused by the existence of three different dielectric constants in the line cross section. The inhomogeneous dielectric experienced by the fringing field of the microstrip leads to a discontinuity of the field at the interfaces and ultimately, the presence of contributions from longitudinal components.
2.2 Transmission Line Theory
Figure 22 Distributed circuit model for transmission line
For a uniform transmission line, the differential equations for the line voltage V and current I can be expressed in frequency domain as [20]:
(21)
, (22)
where impedance and admittance . R, L, C, and G are the line resistance, inductance, capacitance, and conductance per unit length and are in general frequencydependent.
It is worth of mentioning that quasiTEM behavior is assumed for typical interconnection geometry for the transmission line cross section is a small fraction of the wavelength in the frequency range of interest.
The general solution to Equations (21) and (224) can be expressed as [20]
(23)
where the complex propagation factor is defined as
. (24)
The characteristic impedance is in the form of
. (25)
V_{a}, V_{b}, I_{a}, and I_{b} in equation (23) are constants that can be determined by boundary conditions at the two ends of the transmission line. The complex propagation factor g can also be written into a form of real part and imagine part as , where is the attenuation constant and is the propagation constant.
2.2.1 Lossless LineIn the case of lossless line, R = G = 0, the propagation delay per unit length is
. (26)
And the propagation velocity is of course
. (27)
Please note that for TEM mode propagation, the condition below
(28)
is held. m is permeability and is equal to m_{0}m_{r}, e is dielectric constant and is equal to e_{0}e_{r}.
The characteristic impedance is defined as
. (29)
2.2.2 Low Loss LineIn the case when losses are small but not negligible, i.e.
and ,
the attenuation and propagation constant per unit length according to [20], can be approximated by
, (210)
. (211)
If the series resistive loss is small but finite, some pulse distortion will be encountered because the different frequency components attenuate the same amount but are shifted in phase differently. The rise time is degraded by dispersion. However, for interconnections with maximum length l_{max}, if Rl_{max} << Z_{0}, dispersion is usually negligible, and the total signal line resistance results in a dc drop.
If the medium in which the conductors are located has finite resistance, then the dielectric e must be replaced by , where is the loss tangent of the lossy material. The effective conductivity of the material is s , where , and the shunt conductance G is then given by . The complex propagation factor g can then be expressed as
. (212)
2.3 Conformal TransformationsConformal transformation is a mathematical technique that allows particular transmission line geometry to be transformed into a new geometry with a second coordinate system without changing certain values. If the second coordinate system is judiciously chosen, the new geometry may be more amenable to being solved by Laplace’s equation than be the original geometry.
Consider a function , which satisfies the Laplace’s equation
.
We can define a new coordinate system, in which the problem is more easily solvable. In general, the coordinates in new system has follow relationship with the old one:
(213)
.
If the transformation or mapping satisfies the CauchyRiemann conditions, which is
(214)
,
then the new function can be proved that it still satisfies the Laplace’s equation,
There are some values that are not changed after the mapping. For example, let us consider the energy stored in electric field in both coordinate systems. In the (x, y) system, this is simple
. (215)
With the conformal transformation, we can prove the following result is correct [19]:
. (216)
Since the energy stored in electric field is directly related with capacitance of a structure, this equation shows that the capacitance of this structure in the (u, v) system is identical to the one in the original (x, y) system. This is the reason that the conformal transformation is a very useful method to calculate capacitance and effective dielectric constant.
2.4 Effective Dielectric ConstantThe concept of an effective dielectric constant is introduced to describe the permittivity encountered by a wave on a transmission line structure that contains the multidielectric interface, for example, a airsubstrate interface. It describes the interaction of the field with a dielectric, which are both the substrate and the open air.
Once the correct values of L and C of a guild line structure have been found, the e _{eff} can be solved by equation (28). This e _{eff} has a value lying between e _{re }0 and e _{0}. Also, owing to the nowTEM nature of this transmission line, it is dependent on the frequency of a signal. Qualitatively, more field lines are in the substrate dielectric at high frequencies than at low frequencies, leading to an effective permittivity that increases with frequency. Once the formula of effective dielectric constant is derived, e_{eff} can be used by substituting e_{r} which is located anywhere.
The value of e_{eff} is stable at both very low or very high frequencies. But there is a certain frequency region where e_{eff} varies significantly. If a pulse contains frequency components in this region, dramatic changes in the shape of a waveform will result from the action of this function.
2.5 SkinEffectAt high frequencies, current tends to crowd on the surface of the conductor. The crowding decreases the effective flow area and increases its effective resistance due to the simple resistance formula
. (217)
where r
is resistivity, l is the length of a line, S is the flow area.
Also the conductor exhibits an internal inductance, L_{int}, due to magnetic flux penetration, in addition to the external inductance. A reduction in phase velocity and an increase in series impedance and attenuation are caused by the penetration of the electric field into the conductor as it propagates along the surface. For an arbitrary solid conductor, the skin effect results in a surface resistance, R_{s}, and reactance , which are both frequencydependent, and, in general, not equal. However, at sufficiently high frequency, the skin depth is much smaller than the conductor crosssectional dimensions, Then . The additional series impedance due to skin effect will be proportional to the square root of the frequency. The skin depth d is defined as the penetration distance at which the current density is attenuated by 1 neper (1/e = –8.7 decibels) and is equal to [21]:
(218)
where f, r
, and m
are the frequency, conductor resistivity, and permeability of the medium, respectively. At low frequency or very small wire dimension, the skin depth d
is much larger than the conductor cross section. The current approaches uniform distribution, and the resistance is approximately equal to R_{dc}. On the other hand, at high frequency and relatively large dimension, the d
becomes smaller than the cross section, and both R and L are frequencydependent.
Figure 23 Illustration of current distribution in a microstrip vs. wire dimension and frequency.
Current density increases with darkness.
Figure 24 Resistance of 1 cm microstrip vs. frequency at different dimensions
Weeks et al. developed a numerical technique [22] that is very efficient for rectangular conductors. Chari and Silveter, on the other hand, reported to determine and using finiteelement method [23]. However, these two methods require excessive computer time because of the large number of elements needed to achieve accuracy at high frequencies. Wheeler [21] developed a very simple rule of calculating the skin effect resistance when d < wire dimension. The rule states that the effective resistance in a circuit is equal to the change of reactance caused by the penetration of magnetic flux into the conductor. While the internal inductance is due to the flux penetration of d /2. The and can be calculated as follows [21]:
(219)
, (220)
where n is the direction normal to the surface and L_{ext} is the external circuit inductance, which is associated with the magnetic field surrounding the conductors. Using equations (212), (219), and (220), a more approximate formula for resistance can be obtained as:
. (221)
A reasonable results can also be obtained by using the approximations in [3]. For high frequency, we have
and . (222)
Please note that for highly resistive interconnections, conductor loss dominates. Therefore, the dielectric loss can be ignored for most practical digital circuitswitching speeds. In the case of wiring with small R, the dielectric loss should be included in the analysis.
2.6 Silicon SlowWaveAlthough the resistivity of GaAs wafer could be high enough to form a fairly low loss dielectric for a microstrip line, the resistivity of the silicon wafer used in semiconductor device manufacture varies with the needs of the semiconductor device designer and usually do not have very high values. In this case, this silicon substrate can not be considered to be a good insulator.
Because of the SiSiO_{2} double layer dielectric, there exists a slowwave loss in the silicon substrate. The electric field is confined in the SiO_{2} layer while the magnetic field feels free to penetrate the silicon substrate. This separation of electric and magnetic field causes the slowwave mode propagation. Fig. 25 illustrates the distribution of electric field and magnetic field.
Figure 25 Electric and magnetic field within a microstrip structure
Although the compound dielectric is made up of materials with an e _{r} = 12 (Si) and e _{r} = 4 (SiO_{2}), the slow wave mode may propagate as slowly as if the dielectric had an e _{r} of 1600. At certain circumstances, the silicon substrate slowwave loss could be very severe and dominant. The silicon dioxide layer grown on the surface of the silicon can help to increase the resistance to ground seen by the upper metal and thus reduce the attenuation due to the substrate slowwave.
2.7 Electrical Properties of Materials under Low TemperaturePeople are thinking about improving the performance of chips simply by putting the chips into liquid nitrogen. Although high cost of external equipment may prevent industry to manufacture them commercially, low metal resistance and low noise still make this idea a kind of newsworthy way of promoting chip speed.
2.7.1 SiliconThe temperature dependence of conductivity, drift mobility and Hall mobility of n and ptype silicon has been investigated extensively [25]. The formula of the conductivity in semiconductor is given by:
(223)
where, q is the electron charge; n and p are the electron and hole concentrations respectively. m_{n} and m_{p} are the mobility of the electron and hole in semiconductors. The temperature dependence of intrinsic carrier concentrations in Si is given by: [24]
. (224)
The temperature dependence of the carrier mobilities are given as [26]
(225)
.
From these formula, one can see that the carrier concentrations grow with the increasing temperature but the carrier mobility goes down. This controversy effect results that the conductivity reaches its lowest value at around 100K. The conductivity goes down on the both side of the peak. The data measured in [26] is shown below:
Figure 26 Resistivity of borondoped silicon vs. temperature
Figure 27 Resistivity of arsenicdoped silicon vs. temperature
2.7.2 MetalsFigure 28 Resistivities of several metals vs. temperature
The temperature dependent of resistivities of several metals is shown in Fig. 28. The resistivities of metals, unlike semiconductors, drop monotonically and dramatically from room temperature down to the liquid nitrogen temperature. For example, Al has resistivity of 2.74 mW cm at 300K but only has 0.21 mW cm at 80K, which is only 7.66% of that in room temperature.
The simplified microstrip transmission line shown in Fig. 31 is studied extensively by Goossen and Hammond [6]. The insulator layer is omitted in their analysis. Several physics effects are included such as inhomogeneous dielectric layer, conductor resistive loss, skineffect loss and substrate slowwave dielectric loss. Most of the effects are frequencydependent.
Figure 31 Simplified microstrip line structure
The dispersive wire is caused by several factors. The conductor is resistive because of the small dimension and nonzero resistivity. The frequencydependent skineffect causes the different attenuation in the different frequencies. The effective dielectric constant, which causes the geometric dispersion, is used in the calculation due to the inhomogeneous dielectrics. As mentioned in section 2.4, severe signal dispersion occurs at the region where effective permittivity transits. The substrate slowwave loss turns very severe when the substrate resistivity gets smaller. All these effects are taken into account in the Goossen’s analysis.
3.1 Validity of QuasiTEM assumptionQuasiTEM mode propagation is assumed throughout the analysis. The analyses are still valid for frequencies up to the highest frequency where TEM modes can propagate in the microstrip interconnections. A useful reference quantity, which identifies the approximate transition center of the effective permittivity function, is the cutoff frequency for the surfacewave mode [27] [28]. The formula is given by
. (31)
In general, a microstrip can be expected to preserve the rise time of a temporal pulse if the rising edge contains only spectral components at frequencies less than one decade below the cutoff frequency.
This cutoff frequency is related to the substrate thickness inversely and is about 50 GHz for a silicon wafer thickness of 450 m m, which corresponds to switching speeds of 7 ps. Conductor resistivity also introduces limitations to the validity of the quasiTEM approximation. These limitations can be examined by finding the ratio of the longitudinal and tangential electric fields of the mode. If the ratio is much smaller than one, the quasiTEM approximation then is justified. Because only the order of magnitude of this ratio is interested, the ratio is calculated using a parallelplate model. [6]
t >>d (32)
t << d
where s is the conductivity of the conductor, d is the skindepth in the conductor, Equ. (218). t is the thickness of the conductor (Fig. 31). This ratio is much smaller than one for all our examples. For the largest e _{r} = 12, smallest s » 1.5 mW cm, at 10 GHz frequency, the skindepth d is 0.616 mm. In the case of t >> d , the ratio is 4.58´ 10^{6} and in the case of t << d , the ratio is 1.41´ 10^{6}. Both of them is much smaller then 1.
To the first order approximation, the effects of dielectric layer are ignored in Goossen’s work. This layer is usually relatively thin. At 100 MHz, the capacitance introduced by the insulator is a negligible as long as t_{ox} << h (Fig. 31). Because the sample interval in Fast Fourier Transformation is about 100 MHz, this dielectric layer is treated as an open circuit at zero frequency and as a short circuit at all other frequencies. This means that when doing FFT, the first element of array of attenuation factor a will be set to 0 manually.
3.2 Computing Output WaveformThe voltage waveform V is a function of position z and time t. For a given input waveform , what we are interested is to find at certain z. This is accomplished by Fourier transformations:
(33)
where F and F^{1} denote the forward and inverse Fourier transforms respectively. a
and b
are the attenuation and propagation constants respectively. Because we have already derived the relationship between input and output signal in frequency domain, the forward and inverse Fourier transforms is the key to transfer signals from frequency domain to time domain. So problem now focuses on finding formulas of a
and b
.
Please note that there are different kinds of loss mechanism lying in a microstrip line. We will analysis them one by one and finally add them together.
3.3 Characteristic Impedance and Effective Dielectric ConstantAssuming QuasiTEM mode is necessary to calculate electrical parameters. We have approved that this assumption is justified under certain conditions. Calculating characteristic impedance and effective dielectric constant begins with computing line inductance L and capacitance C [29]. The basic idea is that the inductance L does not change much with changing the filling dielectrics if the magnetic permeability is all the same among the dielectrics. Usually The relative magnetic permeability of the substrate material is m _{r} = 1. On the other words, the inductance per unit length depends only upon the conductor geometry and is absolutely independent of the geometry and the dielectric properties of the supporting structure. If the relative dielectric constant e_{r} of the dielectric layer is 1, the microstrip shown in picture 32 would be a TEM line and the relation between the inductance and capacitance, L and C, would be according the property of the TEM line. For the case of , because L is not a function of e _{r} (same permeability), it is still the same as the one when e _{r} = 1. So the L could be calculated analytically or numerically by temporarily setting e _{r} = 1 and calculating the capacitance. The L then can be derived from the C by:
. (34)
Then the characteristic impedance Z_{0} could be derived from L and C.
Once the electrical parameters of airfilled line are calculated, the parameters of other similar structure can then be derived by the relationship shown in Table 31.





Impedance Z_{0} 

Wavelength l _{0} 
l _{0} 

Attenuation a _{0} 
Table 31 Calculation of electrical parameters
where and .
If we assume that the line parameters of the air line in Table 31 (a) are impedance Z_{0}, wavelength l _{0} and attenuation a _{0} per unit length. Then if the structure dimensions remain same but the wire is fully embedded in a dielectric medium with a relative dielectric constant e _{r}, Table 31 (b), one obtains the new line parameter. However, if the wire is partially filled with dielectric support material with a relative dielectric constant e _{r}, Table 31 (c), one obtains basically the similar line parameter formulas but using e _{eff} instead of e _{r}. Table 31 (d) gives a structure with reduced dimensions. This insures that the electrical dimension of the two basic line parameters in the same as (c). Table 31 (e) shows a structure with reduced the ground plane spacing h_{0} to h_{2} such that the characteristic impedance of the line is the same as (c). Please note that the electrical parameters of any microstrip can be computed by the formula given in Table 31, if the characteristic impedance Z_{0} and the dielectric constant e _{eff} are known.
The structure of our interest is shown in Fig. 31 (c). A typical electric field pattern in this kind of structure appears in Fig. 32.
Figure 32 typical electric field pattern of a microstrip
By using conformal mapping method, we could obtain exact analytic solution for characteristic impedance Z_{0}:
, (35)
where K and are complete elliptic integrals of the first kind with modulus m. By noticing that
,
we get
.
For very narrow strips or very wide strips, one obtains the simple expressions:
W w << h (36)
W w >> h
The rigorous solution for computing Z_{0} can be expressed in terms of rational functions or series expansions as follows:
W , ; (37)
W , .
The following formulas approximate Equ. (37) by rational function approximation and give an accuracy of ± 0.25% for 0 £ w/h £ 10 which is the range of importance for most engineering applications.
W (38)
W .
The w/h dependence of characteristic impedance is shown in Fig. 33.
Figure 33 Characteristic Impedance vs. w/h at zero frequency
As shown in Table 31, effective dielectric constant is needed in order to calculate electrical parameters in the inhomogeneous dielectrics media layer. There are many methods to compute the effective dielectric constant. In [30], the authors used a method based on variational calculus. While the authors in [31] start from the numerical determination of Green’s function. These methods are numerical analyses and have some disadvantage as stated in Chapter 1. Analytically, the effective dielectric constant can be calculated by conformal transformation similar as computing the characteristic impedance. If the capacitance of microstrip structure is C_{0} without the dielectric and C with partial dielectric filling, the effective dielectric constant at dc can be written as:
. (39)
The derivative steps are omitted here. The approximation function of the final rigorous results can be expressed as
, (310)
where .
One class of functions that fulfills this requirement is the class of irrational functions.
(311)
The final approximation with a rational function is
(312)
(313)
The line capacitance is based on the ratio of microstrip width and substrate thickness w/h for the same reason of parallel plate capacitance. However, as the thickness of the upper conductor increases, electric field lines from the ground plane will reorient and terminate along the vertical edges of the metal. This reorientation causes the capacitance of the microstrip system to increase with the increasing upper conductor thickness. We can solve this kind of problem by using the concept of Effective Width. The formulas has been given [19] as:
, for ; (314)
, for .
The effective dielectric constant is frequencydependent because more field lines are in the substrate dielectric at high frequencies than at low frequencies. Yamashita, Kazuhiko, and Ueda reported an empirical formula that describes the frequency dependence of e _{eff} [32]. The formula is obtained by fitting the curves calculated by Integral Equation Method. The final result is:
(315)
where .
Here, f is signal frequency and c is velocity of light in vacuum.
Recently the Kirschning and Jansen dispersion mode [33] has been found to be the most accurate compared against the measured value [34]. Getsinger’s formula generally describing microstrip dispersion as [35]
. (316)
Kirschning and Jansen modeled the formula of P(f ) as follow:
(317)
,
,
,
,
where fh represents the normalized frequency in unit of (GHz cm). The accuracy of this expression is better that 0.6% in the range 0.1 £ w/h £ 100, 1 £ e_{r} £ 20 and 0 £ h/l_{0} £ 0.13.
Once the characteristic impedance without dielectrics and effective dielectric constant are computed, the characteristic impedance with dielectrics can be computed by the formula in Table 31:
. (318)
Figure 34 and figure 35 shows the frequency dependence of effective dielectric constant and characteristic impedance respectively.
From the picture one can see that there is a transition area in frequency domain. As a consequence, spectral components below the transition propagate with one velocity, while the higher frequencies travel at another, lower velocity. The signal will have severe dispersion if most of its frequencies are within this area.
Figure 34 Effective dielectric constants vs. frequency
Figure 35 Characteristic impedance vs. frequency
3.4 Attenuation Factor
The attenuation along the microstrip is basically divided into two different parts: the ohmic skin loss in conductor and the loss due to the dielectric substrate. Both of them are due to the finite resistivity of the materials.
3.4.1 Conductor LossAt low frequency, the conductor loss factor a _{c} is computed from the microwave theory (218) and given by:
(319)
where r _{c} is the resistivity of the metal.
At high frequency, however, the crosssection current distribution is not even due to the skin effect. Pucel et al. [36] derived the formulas for the conductor loss considered with both finite resistivity and skin effect. If the current distribution were known, one could compute the ohmic attenuation factor directly using the expression shown in [37]:
. (320)
Here, and are the surface skin resistivity in W /square for the microstrip and ground plane, respectively. J_{c} and J_{g} are the corresponding surface current densities. I is the magnitude of the total current density. The quantity m _{c} and m _{g} are permeability of the strip line and ground. And the r _{c} and r _{g} are the bulk resistivity of them respectively. The current distribution on microstrip structure is shown in the Fig. 36 [38] for a strip with nonzero thickness. The strip conductor contributes the major part of the skin loss.
Figure 36 Current distribution profiles in a microstrip
The exact expression is never been derived because the unknow current distribution and the mathematical complex. Wheeler developed a technique based on the "incremental inductance rule" [21]. The technique expresses the series skin resistance R_{c} per unit length in terms of that part of the total inductance per unit length which is attributable to the skin effect, that is to the inductance produced by the magnetic field within the conductors. From Wheeler, one has
, (321)
where is the surface skin resistivity of wall j. denotes the derivative of L with respect to the incremental recession of wall j. n_{j} is the vector normal to this wall. To apply equation (321) to the microstrip line, we assume that the stored magnetic energy per unit length is not affected by the presence of the dielectric. The Fig. 37 shows the surface recessions to be considered.
Figure 37 Surface recessions for calculating skineffect resistance
From Fig. 37, one obtains that
. (322)
From [39][40], we have approximate results of L as:
, ; (323)
, .
where is the effective conductor width, which is given by
, ; (324)
, .
With assuming R_{sc} = R_{sc} = R, one finally has following result after lengthy but straightforward derivation:
, ; (325)
, ;
,
where is the surface skin resistance and m is the permeability of the conductor.
3.4.2 Substrate lossAn expression of dielectric slowwave loss a _{d} due to the nonzero conductivity in the substrate has been derived by Welch and Pratt [41]. The dielectric attenuation constant for a TEM wave in a homogeneous medium is given by
(326)
And the final result with inhomogeneous medium is given by
(327)
Total attenuation and propagation factorFigure 38 Equivalent circuit model
Fig. 38 shows the microwave equivalent circuit for Wheeler’s model. The circuit is the same as the one shown in Fig. 21. Only some notations are changed. From the derivation of microwave circuit in Chapter 2, we have
(24)
and . (328)
To find the expression of a and b , the equations (24) and (328) above has to be solved. By expanding both equations, we have
and
. (329)
Solving this equation, we have
(330)
,
where
(331)
By analyzing the microwave circuit shown in Fig 38 in low loss condition, we have the relationship that
, (332)
,
,
where c is the velocity of light in vacuum. Substitute (332) into (331), we finally get
(333)
.
At low loss conditions, i.e. b _{0} >> a _{c}, a _{d}, the equations (330) are simplified as
(334)
.
One thing may be worthy to be noticed. Although the model is derived under low loss condition, It actually fits into all the frequencies. According to [6], at sufficiently high frequencies, (~ 10s GHz), even the most lossy line is in the lowloss regime.
Although Wheeler’s work gives us a relatively simpler approach to predict the microstrip losses, simply ignoring the effects of insulator layer is not accurate and appropriate. They overlook a key mitigating parameter for controlling attenuation and dispersion. The function of the insulator layer is decoupling interacts between the conductor and substrate semiconductor. This effect of insulator layer is not negligible when the dielectric thickness is thick enough. Our calculation shows that even with 1 mm SiO_{2} layer, the changing of the attenuation is evident. The extended model takes this insulator layer into account. Fig. 41 shows a typical microstrip structure. We consider in our model several physical effects: insulator layer, slowwave substrate coupling, conductor resistance, skineffect degradation, distributed RC propagation, and signal dispersion.
Figure 41 Microstrip line structure of extended model
4.1 Equivalent Circuit ModelIn our model, the effects of the insulator are taken into account. The thickness of the insulator layer is t_{i}, Fig. 41. The equivalent circuit model for the MIS structure is shown below:
Figure 42 Equivalent circuit of extended model
A capacitance C_{i} is inserted to express the effects of the insulator layer because we assume this insulator layer is lossless.
By investigating this equivalent circuit, the relations between voltage and current are changed to
(41)
.
From the equations above, we have
(42)
.
Let
, (43)
then the equations (42) are simplified as
(44)
.
Similar as the derivation in the last chapter , then to find out the a and b, we set
. (45)
Here, we introducing a new parameter k, which is the ratio of the substrate capacitance and insulator capacitance.
. (46)
The k demonstrates how big the decoupling effects of the insulator layer. If k = 0, C_{i} is equal to infinity, which means the insulator is completely ignored. If the k has a nonzero value, the effects of the insulator are taken into account. The larger the k value is, the more evident of the decoupling effects. Solving this equation, and again applying the low loss condition
, (332)
,
and .
We get the attenuation and propagation factor again as
, (330)
,
but the f_{1} and f_{2} are changed to
, (47)
.
Please note that if k is equal to 0, the equations (47) are exactly the same as the equation (333) shown in the Chapter 3.
4.2 Computing kTo the first order approximation, if we assume the electric field was like below, Fig. 43.
Figure 43 First order approximation of electric field in a MIS microstrip structure
Then substrate and insulator basically form a parallel plate capacitor and the formulas of the capacitance are given by:
, (48)
,
,
where e_{s} and e_{i} are the dielectric constant of substrate and insulator respectively.
However, this approximation is not quite accurate because the actual electric field should be like below.
Figure 44 Actual electric field line in a MIS microstrip structure
The electric field does not only concentrate below the conductor, but spreads out beyond the shadow of conductor. As the result, both insulator capacitor and substrate capacitor increase. Especially the substrate capacitor increases a lot comparing with the insulator capacitor. When the thickness of the insulator layer increase and the conductor is pushed away from the substrate dielectric, the electric field spreads out further and the value of k increase dramatically, Fig. 45.
Figure 45 Actual electric field lines in MIS microstrip structure with thick insulator layer
Figure 46 Comparison of k value between actual and simplified result.
The k value is determined numerically using a field solver program QuickCAP^{TM} [42]. Fig. 46 shows the k value with respect to thickness of the insulator layer. One can see the big difference shown in the figure that is based on the two assumptions of electric field contours.
4.3 Effective Dielectric ConstantSince we inserted an insulator layer between the conductor and the substrate, the structure of the microstrip system is changed, Fig. 41. The formula of effective dielectric constant has to be recalculated. Jiri Svacina [43] gave a pretty accurate formula of effective dielectric constant for double layer dielectrics by using conformal mapping method. The formulas are:
, (49)
where
,
.
For narrow microstrip w £ h,
,
,
for wide microstrip w ³ h. The effective line width for dielectric constant w_{eff} is
. (410)
The accuracy of these formulas is within 2% as acclaimed.
4.4 Result and discussionBy finding pulse propagation along microstrip in time domain, several physics effects are investigated with the basic and extended model mentioned before such as frequency dependency of attenuation and propagation constants, signal dispersion and attenuation, conductor resistance, skineffect degradation, distributed RC propagation. We have also examined, in particular, the influence of an underlying insulator layer, as the basic model has not included such a layer. Many new results are discovered. We have found that the attenuation factor is strongly dependent on the dielectric thickness and dielectric constant of the insulator layer. Another interesting phenomena is the microstrip behavior under cryogenic temperature. We observe a small amount of attenuation under very low substrate resistivity, which is unexpected. We also try to separate the attenuation factors due to the conductor loss and due to substrate slowwave. We found that at some high pulse frequencies, substrate loss becomes a dominant factor of pulse attenuation. At this frequency region, insulator layer plays a very important role for decoupling the interaction between conductor and substrate, and thus relieves the slowwave loss due to the finite substrate resistivity.
4.4.1 Pulse propagation along microstrip lineFig. 47 and 48 show the square pulse propagation along a microstrip line. The dimensions of the microstrip structure are shown in the legends of these figures. Silicon substrate with resistivity of 13W cm is chosen because it is compatible for most of the Si techniques. the relative dielectric constant of Si is 11.8. The insulator layer is made of SiO_{2} since it has been extensively used in Si IC industry. The relative dielectric constant of SiO_{2} is 4. Copper is selected as the conductor metal with resistivity of 1.72 mW cm, which is getting popular in today’s industry for its low resistivity and good electron migration properties. The pulse width is 50 ps with 12 ps rise and fall time. This pulse is suitable for about 8 GHz clock cycle.
Figure 47 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm, including 1mm SiO_{2} layer. 
Figure 48 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm, ignoring the SiO_{2} layer. 
These simulations tell us that there may exists a relatively high attenuation with pulse propagation if the pulse frequency is high enough. In the microstrip structure shown in Fig. 47, the pulse remains only lower than half after 9mm propagation. Also severe pulse dispersion is observed. The edge of pulses is degraded and pulses have long tails. After long distance transmission, the pulses seem to have a tendency of being split into two pieces. This is probably because of the existence of two major values of effective dielectric constant and characteristic impedance.
Compare with Fig. 47 and 48, the pulse attenuation is 40% reduced with only 1 mm SiO_{2} layer. This confirms that ignoring the effects of insulator thickness, which is expressed in the basic modeling (Chapter 3), is not proper. The function of this insulator layer is not only cutting off the dc current penetration from conductor to substrate, but also decoupling the interaction between conductor and substrate at high frequencies and such reducing the slowwave loss. However, the dispersion is not improved with the insulator layer.
4.4.2 Conductor and Substrate Attenuation Constants a_{c} and a_{d}Figure 49 and 410 shows the frequencydependent of conductor and substrate attenuation factor. The conductor attenuation factor a_{c} is strongly frequencydependent. However, the substrate attenuation factor is more stable throughout frequencies. At low frequencies, substrate attenuation is the dominant attenuation factor. E.g. for copper microstrip, the attenuation factor at 1 GHz is only 2.32, while the substrate attenuation factor is all beyond the 20 at the same frequency. However, at very high frequency, the skineffect becomes the major loss comparing with substrate slowwave loss. The attenuation factor due to it could reach as high as a thousand.
Figure 49 Conductor attenuation factors vs. frequency under different conductor resistivities 
Figure 410 Substrate attenuation factor vs. frequency under different substrate resistivities 
(i) 
(ii) 
(iii) 
(iv) 
Figure 411 Total attenuation factor a versus frequency for Cu, Al, W, and polySi microstrips.
a – Line cross section = 1´0.6 mm, Substrate resistivity = 13 W cm
b – Line cross section = 1´0.6 mm, Substrate resistivity = 100 W cm
c – Line cross section = 10´2 mm, Substrate resistivity = 13 W cm
d – Line cross section = 10´2 mm, Substrate resistivity = 100 W cm
Figure 411 shows a set of curves of total attenuation factor a. Several popularly used conductor materials, Cu, Al, W, and polySi are investigated with variant silicon substrate resistivities.
As expected, the total loss monotonically increases with the increasing conductor resistivity and the decreasing substrate resistivity. At relatively low line resistivity and low substrate resistivity, the conductor loss and substrate slowwave loss are comparable, as shown in (i – c) and (ii – c). We can see that the conductor loss becomes dominant at several hundred GHz. However, in other circumstances, the conductor loss is the major loss factor and the curve is very similar to the conductor loss factor a_{c} as shown before (Fig. 49). At very high line resistivity, such as polySi, the conductor loss become so high that changing the substrate resistivity does not help much (iv – a, b).
4.4.4 Factors that Affect Pulse Propagation 4.4.4.1 Pulse WidthThe propagation of pulses with different pulse width is simulated in Figure 412. The clock pulses frequency in picture (a), (b), (c), and (d) corresponds with 32 GHz, 16 GHz, 1.6 GHz, and 160 MHz. From the picture we can see that higher frequency pulses have higher amount of attenuation, as we expected, because they contain more high frequency component and attenuation factors increase at higher frequency. In the picture (a), which has 160 MHz pulses, we can hardly see any attenuation and dispersion in signals. This is why today’s industries do not pay much attention about the clock pulse attenuation. However, at high frequency, which may be not too far in the future, the attenuation and dispersion begin to show and finally become very severe, Fig. 412 (a).
(a) 
(b) 
(c) 
(d) 
Figure 412 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm
vs. different pulse width: 5ns, 500ps, 50ps, and 25ps
4.4.4.2 Insulator Layer Thickness
The main effect of insulator layer is decoupling interaction between conductor and substrate in a microstrip and then reducing the slowwave losses in resistive substrate. Fig 413 shows a set of signals with different SiO_{2} layer thickness. We choose copper as the conductor and large wire dimensions (4´ 1m m) to minimize the influence from conductor loss.
Fig. 413 (a) is the case in which the SiO_{2} layer is simply ignored, which is modeled by Wheeler [6]. Significant losses and dispersion is observed for 50 ps pulse width. With an only 1 m m SiO_{2} layer thickness, (b) shows waveform attenuation is reduced by approximately 40%. At 4 m m oxide layer thickness, (c), attenuation within the 3 mm distance is only 10%. With 13 m m oxide layer thickness, the pulses can run about 6 mm with <10% losses. However, the dispersion still can not be eliminated.
(a) 
(b) 
(c) 
(d) 
Figure 413 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm
vs. different insulator layer thickness: 0m
m, 1m
m, 4m
m, and 13m
m.
4.4.4.3 Insulator Dielectric Constant
Choosing insulator materials with dielectric constant can also help to reduce the substrate losses because of the same reason described in last section, i.e. reduce the interaction between conductor and substrate. Now a day, lots of materials with low dielectric constant is of researcher’s interest. Again, we choose same conductor material and dimensions for minimizing the conductor loss.
(a) 
(b) 
(c) 
(d) 
Figure 414 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm
vs. different insulating dielectric constants: 4, 2.4, 1.6 and 1.
Fig. 414 above shows the effects of insulator dielectric constant. Picture (a) is the case in which normal SiO_{2} is used. However, Picture (b) shows pulse propagation results when the SiO_{2} layer is replaced with lowdielectric constant paryleneF, whose relative dielectric constant is 2.4. Attenuation is reduces 20% compared with using SiO_{2}. Picture (c) gives the results with aerogel or parylenF copolymer network dielectric. In this case, an additional 15% attenuation reduction is achieved. Picture (d) shows the best case in which the insulator layer is nothing but air, whose relative dielectric constant is 1. With changing dielectric constant in insulator layer, one can not have better results than (d).
4.4.4.4 Substrate ResistivitiesAt relatively high frequency, substrate slowwave loss is dominant loss factor. Increasing substrate resistivity is the best way to reduce the resistive loss in this layer. Fig. 415 demonstrates the control of substrate losses. The conductor is chosen by the same reasons described in section 4.4.4.2.
(a) 
(b) 
(c) 
(d) 
Figure 415 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm
vs. different substrate resistivities: 13W
cm, 50W
cm, 100W
cm, and Infinity.
By increasing substrate resistivity from 13 W cm to 50 W cm, the loss is lessened 50%. Further 40% reduction can be attained by increasing the substrate resistivity to 100 W cm. However, a silicon wafer with 100 W cm resistivity is very hard to get from material suppliers. Other substrate materials, such as GaAs, SiC, may be used to achieve this reduction. If the substrate resistivity is an infinity, the signal will propagate almost without any attenuation. This case is suitable for SOI technology, in which an insulator is acted as a substrate. However, the dispersion seems getting worse at higher substrate resistivities.
4.4.4.5 Conductor Materials
(a) 
(b) 
(c) 
(d) 
Figure 416 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm
vs. different conductor materials with resistivities of:
0 m
W
cm, 2.7 m
W
cm (Al), 10 m
W
cm (W), and 100m
W
cm (polySi).
Although substrate loss is the dominant loss factor in pulse propagation, the conductor losses is still not a totally negligible lossy factor. Especially at very high frequencies, the conductor will finally become a controlling attenuation factor (Fig. 411). Fig. 416 shows the pulse propagation within different conductor materials. Figure (a) shows the condition with zero conductor resistivity, which occurs with superconductors. There is still a lot of attenuation in this situation because of the silicon slowwave losses. The degradation of using aluminum is not very significant but visible comparing between (a) and (b). Noticeable changes can also be seen in (c) and (d), in which the aluminum conductor is changed with tungsten and polySi. Especially in the polySi case, lots of losses are observed. Please notice the dispersion in these cases is not like the one in previous sections. The edges of signals are extremely softened. Whereas, for instance, the pulse seems to be split in Fig. 415 (d). This is because the conductor loss goes up monotonically with frequency as shown in Fig 49, so the high frequency components have higher attenuation.
4.4.4.6 Cryogenic TemperaturePulse propagation at cryogenic temperature presents a very interesting result. The effects of low temperature mainly locate on changing the resistivity of materials. According the material property described in Chapter 2, conductor resistivity goes down at low temperature monotonically. However, the resistivity of semiconductor has a peak at about 100K and goes down on both sides. At liquid nitrogen temperature, 76K, the conductor resistivity is only about 1/5 ~ 1/10 of the resistivity in room temperature. The resistivity of substrate also decreases a lot. Fig 417 compares pulse propagation at room temperature, 300K (a) and at liquid nitrogen temperature, 76K (b).
(a) 
(b) 
(c) 
(d) 
Figure 417 Square pulse propagation along a microstrip at length intervals 0, 1, 2, 3, 6, and 9mm
under temperatures of: 300K and 76K
The interest thing is that, instead of expected greater pulse attenuation, we observe a small amount of attenuation. Further investigation of attenuation factor shows that the factor a has its maximum value at certain substrate resistivity. This unique effect can not be obtained if the oxide layer is ignored. With the same microstrip structure except the insulator layer is ignored, (d), the attenuation is so significant that (c) the signal totally dies after 6 mm propagation. We think this phenomenon can be explained, as the electric field in substrate is push back toward conductor because the substrate resistivity is so low that the electric field can no longer penetrate this layer, hence the substrate loss is decreased. In spite of low attenuation, the edge is softened so much that the square waveform finally turns out like a sine function. The high frequency component seems to be totally wiped out. The actual physical interpretation is still under further investigation.
We have analyzed several physical factors that affect the pulse propagation along a microstrip line. The attenuation of pulses is smaller at wider pulse width, thicker insulator layer, lower insulator relative dielectric constant, higher substrate resistivities, and lower conductor resistivities. We also observed an interesting phenomenon of pulse propagation at cryogenic temperature. The attenuation is unexpectedly low although the substrate resistivity is very small. These results can be used to choose proper microstrip structure parameters at high frequency interconnection design.
5. Conclusions and Future Works
With the shrinkage of device gate length, increasing of clock speed and larger die dimension, the circuit delay due to interconnects becomes the dominate factor that limits the speed of VLSI chips. Accurate and efficient circuit models for these interconnections are necessary and helpful for the circuit designers.
A new theoretical model for a metalinsulatorsemiconductor (MIS) microstrip interconnection is presented in this paper using a quasiTEM approximation. We have studied, in particular, the influence of an underlying insulator layer as well as several other physical effects: slowwave substrate coupling, conductor resistance loss, skineffect degradation, distributed RC propagation, and cryogenic cooling effects. Our new theoretical model is developed by extending a previous work [6]. The circuit model in this paper ignored the insulator layer by simply treated it as an open circuit in DC and as a short circuit in other frequencies. The research results in this thesis show that simply ignoring this insulator layer is not accurate and proper. This insulator layer decouples the interconnect conductor from the substrate. Our extension mathematically takes this into account. It also incorporates skineffect energy losses in metal conductors, and the slowwave losses due to finite substrate resistivity.
Our investigation of microstrip interconnection shows that the slowwave loss is a very important loss factor and deserves the most effort to be improved. Both conductor attenuation factor and substrate attenuation factor increase with increasing frequency. The signals that contain higher frequencies have greater attenuation. Lowing conductor resistivity or enlarging the conductor dimension can reduce the attenuation because both of them cut some conductor resistance off. Generally, the improvement is not very evident. However, further reduction of attenuation could be reached by increasing the substrate resistivity. The insulator layer is another very important factor that influences the line attenuation. Both increasing the layer thickness and choosing low dielectric constant materials as insulator help to reduce the losses.
Still, the results of this model are only compared with the data published in previous paper. More experiment works need to be done to justify the correctness and accuracy of this model. Active test structures for MIS microstrip line are suitable structures for this purpose. We will basically put bunch of microstrips with different wire dimensions at different height in some insulator layer, such as SiO_{2} or BCD. Very wide bandwidth distributed amplifiers are needed to be implemented into the test structure to drive the microstrip line actively. High frequency pulse (picosecond and subpicosecond) generators and detectors are also necessary for recording and characterization signals. Fortunately, the problems associated with this creation, detection, and characterization this kind of signals have been in large part solved [44] [45].
Because the trend of higher interconnect density, MultiConductor, Multilayer models are to be developed since they are more realistic than the single conductor and single layer models. Although lots of research has been done by previous works, all of them are numerical analyses which are timeconsuming and not suitable for CAD tools. We have notice that by putting two wires close enough and driving them in odd mode, more electric field lines are attracted around wires’ vicinity and stay in the insulator layer instead of going through the substrate, Fig. 51.
Figure 51 Electric field line in even and odd mode
The substrate losses may be reduced because more electric field lines stay in insulator layer, which is a lossless media. The field lines in substrate are also reduced.
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