Analysis and circuit model of a multilayer semiconductor slow-wave microstrip line A.K. Verma, Nasimuddin and E.K. Sharma Abstract: An analytical single-layer reduction quasi-static formulation to accurately compute all the line parameters of metal insulator semiconductor (MIS) and Schottky contact multilayer slow- wave microstrip lines is presented. It is valid for a wide range of parameters and its validity is compared with the full-wave spectral domain analysis technique. We also obtain a circuit model, which is able to accurately explain the experimental results, including dispersion at the lower end of the frequency range, for both the MIS and Schottky contact microstrip lines. Useful data to design passive components based on these lines are also presented. 1 Introduction Both metal insulator semiconductor (MIS) and Schottky contact microstrip lines are used with the production of electronics components such as variable phase shifters, voltage-tunable filters and tunable microstrip antennas. MIS lines are also used to form interconnects between active devices in MMIC and VLSI technologies. The propagation characteristics of MIS lines are controlled by the resistivity of the substrate and the operating frequency [1–24] . These line structures have significantly different propagation characteristics as compared to those of a standard microstrip line on a dielectric substrate. The effective dielectric constant of a standard microstrip line on a dielectric substrate is always less than the relative permittivity of the substrate and it increases with increase in the frequency. Such a microstrip line only works in the dielectric mode. However, a microstrip line in either a MIS or a Schottky structure works in three different modes. In the so-called slow-wave mode at low frequencies its effective relative permittivity is much larger than the relative permittivity of the substrate. This is due to Maxwell- Wagner interfacial polarisation. Unlike a standard micro- strip, the effective relative permittivity of the MIS and Schottky microstrip lines decreases with an increase in frequency. This is due to the disappearance of the interfacial polarisation. In this process the microstrip line moves from the slow-wave mode to the dielectric mode. At the lower end of the frequency range, the MIS and Schottky microstrip lines show a significant increase in effective relative permittivity with a decrease in the frequency. This is due to the finite conductivity of the strip and the ground plane. Unlike a standard microstrip, a slow-wave microstrip line with a conductor loss also shows an increase in the real characteristic impedance with a decrease in the frequency. For a lossy slow-wave microstrip line, the imaginary characteristic impedance becomes negative. The properties of MIS and Schottky microstrip lines have been experi- mentally investigated in [2–6]. These structures have been analysed using several methods, including the parallel plate waveguide model [1, 3, 6] , the mode matching method [16, 22], full-wave spectral domain analysis (SDA) [13, 17, 18] , finite-difference time- domain [19] and the device-level transport analysis method [15]. In the Fourier domain, quasi-static methods have also been used [20–22]. The parallel plate waveguide model is not a realistic description of a narrow width microstrip line. Most of the full-wave methods and device-level transport methods do not take into account the conductor loss. These methods are also computationally slow and are not suitable for interactive computer-aided design (CAD). They are also not suitable for the synthesis of lines and components. Normally a full-wave method, including the SDA, does not account for the effect of the conductor loss on the propagation characteristics. However, Liou and Lau [13] have incorporated finite conductivity in their SDA for- mulation for the coplanar MIS structure. A circuit model is also available that accounts for the finite conductivity of the strip conductor [23] . No such effort has been expanded for the MIS and Schottky contact slow-wave microstrip lines. Hasegawa et al. [3] reported a set of circuit models that are valid only for very wide microstrip lines without any conductor loss. Their model could not fully explain their own measured slow-wave factor, an increase in the real characteristic impedance and the appearance of a negative imaginary characteristic impedance. Likewise, the circuit model of Jager [6] is also only valid for a very wide microstrip line working in the slow-wave mode. However, it does take into account the conductor loss and it is able to explain the observed experimental features of the Schottky contact microstrip line at low frequencies. The present work is the outcome of our desire to develop a fast and accurate model suitable for CAD implementation that takes into account all three modes of propagation on a slow-wave miscrostrip line with conductor losses. Our formulation is valid for a slow-wave microstrip line in a multilayer environment, which occurs in the CMOS [24] and HEMT [13] technologies. A suspended slow-wave structure can also be formed in a multilayer case [7, 8]. The formulation is based on the so-called single-layer reduction (SLR) process. The SLR formulation is only valid in the A.K. Verma and E.K. Sharma are with the Department of Electronic Science, University of Delhi South Campus, New Delhi- 110021, India Nasimuddin is with the Department of Engineering, Macquarie University, NSW 2109, Australia r IEE, 2004 IEE Proceedings online no. 20040775 doi:10.1049/ip-map:20040775 Paper received 18th November 2003 IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 5, October 2004 441
Transcript
Analysis and circuit model of a multilayersemiconductor slow-wave microstrip line
A.K. Verma, Nasimuddin and E.K. Sharma
Abstract: An analytical single-layer reduction quasi-static formulation to accurately compute allthe line parameters of metal insulator semiconductor (MIS) and Schottky contact multilayer slow-wave microstrip lines is presented. It is valid for a wide range of parameters and its validity iscompared with the full-wave spectral domain analysis technique. We also obtain a circuit model,which is able to accurately explain the experimental results, including dispersion at the lower end ofthe frequency range, for both the MIS and Schottky contact microstrip lines. Useful data to designpassive components based on these lines are also presented.
1 Introduction
Both metal insulator semiconductor (MIS) and Schottkycontact microstrip lines are used with the production ofelectronics components such as variable phase shifters,voltage-tunable filters and tunable microstrip antennas.MIS lines are also used to form interconnects betweenactive devices in MMIC and VLSI technologies. Thepropagation characteristics of MIS lines are controlled bythe resistivity of the substrate and the operating frequency[1–24]. These line structures have significantly differentpropagation characteristics as compared to those of astandard microstrip line on a dielectric substrate. Theeffective dielectric constant of a standard microstrip line ona dielectric substrate is always less than the relativepermittivity of the substrate and it increases with increasein the frequency. Such a microstrip line only works in thedielectric mode. However, a microstrip line in either a MISor a Schottky structure works in three different modes. Inthe so-called slow-wave mode at low frequencies its effectiverelative permittivity is much larger than the relativepermittivity of the substrate. This is due to Maxwell-Wagner interfacial polarisation. Unlike a standard micro-strip, the effective relative permittivity of the MIS andSchottky microstrip lines decreases with an increase infrequency. This is due to the disappearance of the interfacialpolarisation. In this process the microstrip line moves fromthe slow-wave mode to the dielectric mode. At the lowerend of the frequency range, the MIS and Schottkymicrostrip lines show a significant increase in effectiverelative permittivity with a decrease in the frequency. This isdue to the finite conductivity of the strip and the groundplane. Unlike a standard microstrip, a slow-wave microstripline with a conductor loss also shows an increase in the realcharacteristic impedance with a decrease in the frequency.For a lossy slow-wave microstrip line, the imaginary
characteristic impedance becomes negative. The propertiesof MIS and Schottky microstrip lines have been experi-mentally investigated in [2–6].
These structures have been analysed using severalmethods, including the parallel plate waveguide model [1,3, 6], the mode matching method [16, 22], full-wave spectraldomain analysis (SDA) [13, 17, 18], finite-difference time-domain [19] and the device-level transport analysis method[15]. In the Fourier domain, quasi-static methods have alsobeen used [20–22]. The parallel plate waveguide model is nota realistic description of a narrow width microstrip line.Most of the full-wave methods and device-level transportmethods do not take into account the conductor loss. Thesemethods are also computationally slow and are not suitablefor interactive computer-aided design (CAD). They are alsonot suitable for the synthesis of lines and components.Normally a full-wave method, including the SDA, does notaccount for the effect of the conductor loss on thepropagation characteristics. However, Liou and Lau [13]have incorporated finite conductivity in their SDA for-mulation for the coplanar MIS structure. A circuit model isalso available that accounts for the finite conductivity of thestrip conductor [23]. No such effort has been expanded forthe MIS and Schottky contact slow-wave microstrip lines.
Hasegawa et al. [3] reported a set of circuit models thatare valid only for very wide microstrip lines without anyconductor loss. Their model could not fully explain theirown measured slow-wave factor, an increase in the realcharacteristic impedance and the appearance of a negativeimaginary characteristic impedance. Likewise, the circuitmodel of Jager [6] is also only valid for a very widemicrostrip line working in the slow-wave mode. However, itdoes take into account the conductor loss and it is able toexplain the observed experimental features of the Schottkycontact microstrip line at low frequencies.
The present work is the outcome of our desire to developa fast and accurate model suitable for CAD implementationthat takes into account all three modes of propagation on aslow-wave miscrostrip line with conductor losses. Ourformulation is valid for a slow-wave microstrip line in amultilayer environment, which occurs in the CMOS [24]and HEMT [13] technologies. A suspended slow-wavestructure can also be formed in a multilayer case [7, 8]. Theformulation is based on the so-called single-layer reduction(SLR) process. The SLR formulation is only valid in the
A.K. Verma and E.K. Sharma are with the Department of Electronic Science,University of Delhi South Campus, New Delhi- 110021, India
Nasimuddin is with the Department of Engineering, Macquarie University,NSW 2109, Australia
dielectric and the slow-wave modes. We will present animproved SLR, called SLR (I) that is valid for all threemodes of propagation. We will also present more design-oriented information suitable for MMIC designers. How-ever, even SLR (I) does not explain the dispersionbehaviour of a slow-wave microstrip at lower frequencies.Finally, following Jager [6] a circuit model will be developedto explain the deserved experimental features for slow-wavemicrostrip lines. The results obtained using the SLR andcircuit models will be compared against available experi-mental results obtained on the MIS and Schottky contactmicrostrip lines.
2 SLR formulation for a multilayer slow-wavemicrostrip line
A six-layer slow-wave microstrip line structure is shown inFig. 1. It can be easily reduced to many useful structures suchas the MIS and Schottky contact microstrip lines [2, 5, 6],suspended MIS and Schottky microstrip lines [7] and amicrostrip line in HEMT [13] or in CMOS structures [24].The structure can be analysed in the Fourier domain byusing the SLR formulation that is based on a variationalmethod. The SLR model is able to handle the multilayersubstrates, the multilayer superstrate and the effect of the topconductor shield on the propagation characteristics. In orderto compute the dielectric and conductor losses of themultilayer microstrip line, the SLR converts a multilayermicrostrip structure to a single layer microstrip structure. Theequivalent single-layer substrate has an equivalent relativepermittivity (er,eq) and an equivalent loss tangent (tan deq).The thickness of the equivalent single-layer substrate is thetotal thickness between the ground-plane and the stripconductor of the original structure. The width of the centralstrip is kept unchanged. Figure 1 shows the SLR scheme toreduce the six-layer microstrip structure into an equivalentsingle-layer microstrip structure. The scheme is valid for agreater number of layers. The scheme works in two steps.
Step 1 is called the Wheeler’s transformation and itconverts the multilayer inhomogeneous medium of themicrostrip line into a homogenous medium for the micro-strip line. The homogenous medium has a complex relativepermittivity ereff , which is the complex effective relativepermittivity of the original structure. Although, ereff iscomputed using a variational method, we still call thetechnique a Wheeler’s transformation, because he intro-duced the concept of the filling-factor that transforms themicrostrip in the inhomogeneous medium to the microstripin the homogeneous medium [25].
Step 2 is called the inverse Wheeler’s transformation andit converts the microstrip in the homogeneous medium intoa standard microstrip in the inhomogeneous medium. In theprocess, we obtain ereq and tan deq of the equivalent single-layer microstrip line. Thus, the complex equivalentpermittivity, ereq contains information about the relative
permittivity, the loss tangent (or resistivity of the substrate)and the thickness of each layer of the original multilayerstructure. We will now summarise all the requiredexpressions to compute the propagation characteristics ofa multilayer microstrip line.
The ereff of step 1 is computed in the Fourier domainfrom the complex line capacitance that is obtained by avariational method:
1
C¼ 1
p
Z10
~ff bð ÞQ
2
1
bY db ð1Þ
where b is the Fourier variable. The total charge, Q on thestrip, is obtained from an assumed cubic charge distribution[26]. The Galerkin method can also be used in the Fourierdomain to obtain a slightly improved result [20] but at thecost of a considerably longer computation. The admittanceparameter Y* is related to the Green’s function for themultilayer structure. The parameter Y* for a six-layerstructure is obtained by the transverse transmission linemethod [27]:
Each substrate layer is characterised by a complex relativepermittivity given by:
eri ¼ e0ri j1
orie0or e0ri je00ri tan di
where, e0ri is the real relative permittivity, ri is the substrateresistivity and tan di is the substrate loss tangent of the ithlayer of the dielectric medium. Also, eo is the permittivity offree space.
The complex effective dielectric constant of a multi-layered slow-wave microstrip is obtained from the followingexpression:
where, C0 is the line capacitance of the structure when allthe dielectric layers are replaced by air.
Step 2 computes the ereq of the equivalent single layer
microstrip line:
ereq ¼ereff 1
qþ 1 ¼ e0req je00req ð4Þ
The equivalent relative permittivity and the equivalent losstangent of the equivalent single-layer microstrip line aregiven by:
e0req ¼ Re ereq
ð5Þ
tan deq ¼e00reqe0req¼ e00reff
e0reff þ q 1ð6Þ
where the permittivity independent filling-factor q isobtained from the Wheeler’s formula summarised in [28].The slow-wave factor (b/b0) of the multilayer microstrip lineis computed by:
bbo¼
ffiffiffiffiffiffiffie0reff
qð7Þ
Its complex characteristic impedance is obtained from:
Z ¼ Z0ffiffiffiffiffiffiffiereff
p ð8Þ
where Z0 is the characteristic impedance of the microstripline on an air substrate. The characteristic impedance of theopen microstrip line on an air substrate is obtained from theclosed-form expressions of Hammersted and Jensen [29].For a shielded microstrip line, it can be obtained by usingexpressions given in [30].
Once the equivalent relative permittivity e0req and loss
tangent, tandeq for the equivalent single-layer microstripline are obtained, the dielectric loss of the multilayer slow-wave microstrip line in units of decibels per unit length canbe computed from [31]:
ad ¼27:3
lo
e0req e0reff 1
ffiffiffiffiffiffiffie0reff
pe0req 1h i tan deq ð9Þ
where, lo is the free-space wavelength. We can adoptWheeler’s incremental inductance rule [32] in the SLRformulation in order to compute the conductor loss of themultilayer slow-wave microstrip line [33]. Wheeler’s methodis usually suitable for conductors with a thickness that is afew multiples of the skin depth. However, its presentformulation in terms of the fractional characteristicimpedance works satisfactorily for a conductor thicknessgreater than 1.1ds and a strip width that is a few multiples ofthe skin depth. Unlike the perturbation method, Wheeler’smethod does not require information about the chargedistribution on the strip. We have tested the accuracy of ourformulation of Wheeler’s incremental inductance ruleagainst the experimental results of Goldfarb and Platzkar[34] in [33]. Goldfarb and Platzkar used microstrip linewidths of between 10 and 350mm on a 100mm GaAssubstrate with a 200nm silicon nitride passivation layer. Theattenuation was measured up to 40GHz. Wheeler’s methodhas an average accuracy of about 6% against theexperimental results. It is expected that the method willalso provide sufficiently accurate results in the slow-wavecase. Thus, the conductor loss in decibels is:
where e0req of the equivalent single-layer substrate is obtainedfrom (5) and h¼ h1+h2+h3 is the total substrate thickness.The difference in the characteristic impedance of themicrostrip line is given by:
DZ ¼ Z er1 ¼ er2 ¼ . . . ¼ 1; W ds; hþ ds; t dsð Þ Z er1 ¼ er2 ¼ . . . ¼ 1;W ; h; tð Þ
ð12Þ
where, ds is the skin depth of the strip conductor. Thecharacteristic impedance of a microstrip line on an airsubstrate is obtained using the closed-form expressions ofHammerstad and Jensen [29] modified for the finitethickness of the strip conductor [7]. The total loss in thestructure is:
aT ¼ ad þ ac in units of decibels per unit length ð13Þ
We have tested the validity of the SLR formulation bycomparing computed propagation characteristics of theMIS microstrip line against results obtained using the SDAmodel at 1GHz [20]. Figures 2a–2d show computed resultsfor the slow-wave factor, attenuation constant, realcharacteristic impedance and imaginary characteristic im-pedance with respect to the conductivity of the dopedsilicon layer of a MIS microstrip line.
The silicon substrate of the MIS structure has a thicknessh1¼ 0.25mm, er1¼ 12, and the loss-less SiO2 passivationlayer has a thickness h2¼ 0.001mm, er2¼ 4 and s2¼ 0. Thestrip width is 0.16mm (W/h¼ 0.637) and the operatingfrequency is 1 GHz. In the dielectric and slow-wave modesboth methods show almost identical results. However,beyond s1¼ 1000S/m, the slow-wave mode moves to theskin-effect mode and the slow-wave factor reduces. InFig. 2a the SDA shows such a mode transition. However,the SLR does not show such a behaviour. Likewise, in theslow-wave region, the attenuation constant reduces to aminimum value and it again increases for s14400S/m.Figure 2b shows the results obtained using SDA. TheSLR does not exhibit this characteristic. However in Fig. 2c,the real characteristic impedance computed by the SLRmethod at 1GHz shows an excellent agreement with resultsof SDA up to s1o4000S/m. There is a difference inthe nature of the real characteristic impedance beyond thisvalue of s1. Figure 2d shows further agreement betweenthe two methods for results on the imaginary characteristicimpedance up to s1o2000S/m. For s142000S/m,SDA shows a negative imaginary characteristic impedance;however, the SLR method does not produce a negativevalue. Therefore, we can say that the SLR method resultsshow an accurate behaviour only in the dielectric andslow-wave regions and not in the transition region betweenthe slow-wave mode and the skin-effect mode. In thisregion the TEM nature of the mode propagation breaksdown and the TM nature of mode becomes important.Therefore, some modification is needed to the SLRformulation to improve the result so that it is valid in allthree regions.
We have seen that the current SLR formulation does notgive correct results for the line characteristics in thetransition region from the slow-wave mode to the skin-effect mode. We can improve the SLR model for a highconductivity semiconductor substrate by assuming that thesemiconductor substrate acts as a lossy conductor. Thestudy of Gilb and Balanis [17] shows that two maximumattenuation peaks occur in the two transition regions, i.e.transition from the dielectric mode to the slow-wave modeand the transition from the slow-wave mode to the skin-effect mode. They studied the attenuation behaviour at1GHz with respect to the variation of the conductivity of adoped silicon substrate for zero losses up to s very highvalues approaching to the level of a conductor. Thedielectric loss reaches a minimum level at smin in theslow-wave region. At smax the second maximum peakoccurs in the transition region between the slow-wave modeand the skin-effect mode. In the skin-effect mode, the highlydoped silicon layer behaves more like a conductor. Thus, tomodel the propagation behaviour, we can divide theconductivity (s) of a silicon-substrate into three regions:
1. The s¼ 0 to s¼ smin region: The SLR technique in itsoriginal form is valid in this range of conductivity for thecomputation of both the slow-wave factor and the dielectricattenuation. The value of smin can be estimated by thefollowing expression obtained from the parallel-platewaveguide model [3]:
For the MIS structure corresponding to Figs. 2a–2d, thesmin is 581S/m.
2. The s¼ smin to s¼smax region: At s¼ smax, themaximum attenuation peak occurs in the transition regionfrom the slow-wave mode to the skin-effect mode. We canassume that the skin-effect mode region starts at s¼ smax.At this stage, the skin-depth in the silicon substrate becomesequal to the thickness of the silicon substrate, h1. Thus, atthe operating frequency f, the smax can be obtained in termsof the thickness of the silicon substrate by:
smax ¼1
pfmoh21
ð15Þ
The smax in the transition region can also be estimated byone of three expressions provided by Gilb and Balanis [17].The SLR can again compute the slow-wave factor as usual.However, the dielectric loss for the range sminososmax iscomputed using the normalised conductivity or the normal-ised loss-tangent of the silicon substrate. In the transitionregion, the mode is changing from TEM to TM. Weassume that the changes leading to mode transition can beexplained by the normalised conductivity of the substratesilicon but stress that the mode remains as TEM. Thenormalised conductivity is given by:
sn ¼s
smax; tan dn ¼
tan dtan dmax
ð16Þ
The complex dielectric constant of a doped semiconductorsubstrate in this region can be written as:
er1 ¼ e0r1 jsn
oe0or e0r1 je0r1 tan dn
Now the SLR method is applied to compute attenuationdue to the dielectric loss using (9).
3. The smax to very high s region. y the region proper:This is the proper skin-effect mode. In this region the
1
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19
22
100 101
101
100
10−1
10−2
102 103 104
SLR(I)
SLR
SDA [20]
SLR(I)
SLR
SDA [20]
SLR(I)
SLR
SDA [20]
atte
nuat
ion
cons
tant
, dB
/mm
slow
-wav
e fa
ctor
, /
o
conductivity of the silicon substrate, S/m
100 101 102 103 104
conductivity of the silicon substrate, S/m
a
b
100 101 102 103 104
conductivity of the silicon substrate, S/m
c
100 101 102 103 104
conductivity of the silicon substrate, S/m
d
5
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45
−5
0
5
10
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imag
inar
y ch
arac
teris
tic im
peda
nce,
Ωre
al c
hara
cter
istic
impe
danc
e, Ω
SLR(I)
SLR
SDA [20]
Fig. 2 Various characteristics of MIS microstrip line againstsilicon substrate conductivitya Slow-wave factorb Attenuation constantc Real characteristic impedanced Imaginary characteristic impedance
electric field does not fully penetrate the doped siliconsubstrate. Therefore, the bottom portion of the siliconsubstrate can be treated as a conductor whereas theremaining part of the silicon substrate can be treated as adoped substrate. In this process the effective thickness of thesubstrate is reduced. It is given by:
h1eff ¼ dss ð17Þwhere, dss is the skin depth in the silicon substrate. In thisregion, the SLR model computes the slow-wave factor onreplacing h1 by h1eff. The conductivity of the siliconsubstrate is taken to be s itself. We note that with a changein frequency or conductivity of the silicon substrate, h1effchanges as dss changes. Thus, the effective thickness of thesubstrate is frequency and doping-level dependent. Thechanging thickness of the substrate compensates for thetreatment of the TM mode by the TEM mode. The presenttreatment of the TM mode by using the TEMmode is onlyphenomenological.
We take a slightly different approach for the computationof the attenuation constant in the skin-effect region. Weassume that the silicon substrate acts as a poor conductorground plane and the dielectric loss is really a conductorloss in the silicon substrate. This assumption effectivelymeans that we treat a two-layer microstrip line as a single-layer microstrip line with the SiO2 layer as the substrate andthe highly doped silicon layer of thickness dss as a variablethickness lossy ground plane. Wheeler’s inductance rule,discussed in Section 2 can be used to compute the conductorloss, which is mainly the loss in the silicon substrate. Thecharacteristic impedance difference used in the Wheeler’sinductance rule is given by:
DZ ¼Z Er1 ¼ Er2 ¼ 1; W dst; hðþððdss=2Þ þ ðdst=2ÞÞ; t dstÞ Z Er1 ¼ Er2 ¼ 1;W ; h; tð Þ
ð18Þ
where, h is the total thickness of the substrate. The dst is theskin depth in the strip conductor.
In Figs. 2a–2d we have presented results obtained usingboth the improved SLR model, i.e. SLR (I), and theoriginal SLR model. They clearly show the improvementachieved by the SLR (I). Similar to the SDA formulation,the SLR formulation is valid in all three modes ofpropagation. In Fig. 2d the results obtained using SDAshow a negative imaginary impedance in the highconductivity region of the semiconductor substrate. How-ever, the SLR formulations, including the SLR (I) modeldoes not show this behaviour. The circuit model, presentedin the following Section does show such results.
Figures 3a and 3b show variations in the effective relativepermittivity and attenuation constant for the MIS micro-strip structure with respect to the loss tangent of the dopedsilicon substrate. The structural parameters are:W¼ 0.6mm, h1¼ 0.5mm, h2¼ 0.135mm, (w/h¼ 0.94)er1¼ 9.7, er2¼ 9.7, tan d2¼ 0, t¼ 0.003mm and f¼ 1GHz.The figure also includes results obtained using SDA asadapted by Gilb and Balanis [17]. The results obtainedusing the improved SLR (I) model closely follow the resultsof SDA whereas the original SLR formulation is valid onlyup to the slow-wave region, tan d1otan d1min. We note thatin Fig. 2a the results obtained using the SLR model deviatesignificantly in the conductivity range 2–40S/m from theresults of the SDA formulation of Mesa et al. [20]. Thisrange of conductivity for the silicon substrate is compatablewith the dielectric mode and the transition region from thedielectric mode to the slow-wave mode. The differencebetween the results obtained using the SLR formulation
and SDA could be due to the choice of the basis functionand the accuracy of the numerical computation of the SDA.However, Fig. 3a shows a good agreement between theresults obtained using the SLR model and the SDA of Gilband Balanis [17] under similar conditions.
The improved SLR formulation is valid in both thedielectric and the skin-effect mode regions. Therefore, wecan use the SLR formulation to investigate the behaviour ofa slow-wave microstrip line in a CMOS configuration, i.e. amicrostrip line on a Si-SiO2 composite substrate with a SiO2
overlay [24]. For such a study, we have taken the Si-SiO2
MIS structure from Hasegawa et al. [3], without theSiO2 overlay. We note that a SiO2 overlay with a thicknessof h3¼ 1, 2 and 5mm does not have any noticeable influenceon the propagation characteristics of a slow-wave micro-strip line in a CMOS structure. Likewise, the SLR methodcan be used to determine the propagation characteristics ofa multilayer slow-wave microstrip line on a HEMTstructure [13].
The SLR method can be used to study the effect ofthe top shield on the slow-wave propagation characteristics.Figures 4a and 4b show such a study on the slow-wavefactor and real characteristic impedance for a microstripline on the Si-SiO2 MIS structure, taken from Hasegawaet al. [3]. Only a very closely placed top shield, i.e. at 1mm, isable to decrease the slow-wave factor and even thenthe effect is minimal. However, there is a small increasein the characteristic impedance. The total loss is notinfluenced by the presence of the top shield. This is due to
0
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effe
ctiv
e re
lativ
e pe
rmitt
ivity
loss tangent of the lower sunstrate tan 1
10−2 100 102 104 106
SLR(I)
SLR
SDA [17]
SLR(I)
SLR
SDA [17]
a
loss tangent of the lower sunstrate tan 1
10−2 100 102 104
103
102
101
10−1
10−2
10−3
100
106
b
atte
nuat
ion
cons
tant
, dB
/m
Fig. 3 Real value of effective relative permittivity and attenuationconstant of MIS microstrip line against loss tangent of lowersubstratea Real value of effective relative permitivityb Attenuation constant
near-total confinement of the electric field in the compositesubstrate.
A slow-wave microstrip line usually has a low character-istic impedance. However, our studies show that changingthe strip width from 0.01mm to 1.0mm can change thecharacteristic impedance from 50 to 3 O. Such a largevariation in the characteristic impedance is not obtainednormally even for a microstrip line on either plastic oralumina substrates. There is also a significant change in theslow-wave factor for a change in the width of the strip from0.01mm to 1mm. Figures 5a–5d show the changes in theline parameters of the MIS slow-wave microstrip line whenthe width of the strip conductor is varied. These results areobtained using the SLR formulation for a silicon substratehaving resistivity values of r1¼ 0.1, 1.0 and 10 O cm. Thestructural details are shown in Fig. 5a. This information isuseful in the development of passive components and thedesign of matching networks for MIS and Schottky contactmicrostrip line environments.
4 Circuit model for a slow-wave microstrip line
Experimental studies on the dispersion behaviour at lowfrequencies have been performed for the MIS [2] andSchottky contact microstrip lines [6]. There is an increase inthe slow-wave factor and real characteristic impedance witha decrease in the frequency. The imaginary characteristicimpedance also becomes negative. Finally, in the slow-waveregion the decrease in the loss with a decrease in frequency
saturates to a fixed level. These specific characteristics aredue to a finite conductivity and a finite width of the stripand ground conductors. However, the SLR formulation isnot able to explain these results.
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-wav
e fa
ctor
expt. [3] SLR
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frequency, GHz
10−2 10−1 100 102
frequency, GHz
10−2 10−1 100 102
h3/h = 2
h3/h = 5
h3/h = 1
expt. [3] SLR
h3/h = 2
h3/h = 5
h3/h = 1
real
cha
ract
eris
tic im
peda
nce,
Ω
a
b
Fig. 4 Slow-wave factor and real characteristics impedance ofshielded MIS microstrip line against frequencya Slow-wave factorb Real characteristic impedance
0
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0.01 0.03 0.05 0.07 0.10 0.20 0.30 0.40 0.50 1.00
0.01 0.03 0.05 0.07 0.10 0.20 0.30 0.40 0.50 1.00
1=0.1Ω cm1=1 Ω cm1 =10 Ω cm
1=0.1 Ω cm1=1 Ω cm1 =10 Ω cm
loss
, dB
/mm
d c T
c
slow
-wav
e fa
ctor
, /
o
width of strip conductor W, mm
width of strip conductor W, mm
a
b
0.01 0.03 0.05 0.07 0.10 0.20 0.30 0.40 0.50 1.00width of strip conductor W, mm
c
0.01 0.03 0.05 0.07 0.10 0.20 0.30 0.40 0.50 1.00width of strip conductor W, mm
d
1=0.1Ω cm1=1 Ω cm1 =10 Ω cm
1=0.1Ω cm1=1 Ω cm1 =10 Ω cm
102
101
10−1
10−2
10−3
100
102
101
10−1
10−2
100
0
5
10
15
20
25
30
35
40
45
50
real
impe
danc
e, Ω
imag
inar
y ch
arac
teris
tic im
peda
nce,
Ω
SLRSLRSLR
Fig. 5 Various characteristics of MIS microstrip line against widthof strip conductora Slow-wave factorb Dielectric, conductor and total lossc Real characteristic impedanced Imaginary characteristic impedance
We have seen that a SLR-based computation gives thepropagation constant, i.e. slow-wave factor, on theattenuation constant and the characteristic impedance fora multilayer slow-wave microstrip line as separate para-meters. However, in reality these parameters influence eachother. The interaction between the parameters can beobtained through a circuit model. Our circuit is based onthe model of Jager [6] and is shown in Fig. 6.
The circuit model shown in Fig. 6 is based on atransmission line and is valid for a multilayer slow-wave
microstrip line. The resistance Rs accounts for any loss inthe semiconductor due to a longitudinal current [23]. In thedielectric and slow-wave modes, Rs is large and it can beneglected. However, Rs improves results on the propagationcharacteristics especially in the transition region. In the skin-effect mode region, a lossy silicon substrate is treated as aground plane, therefore Rs could be neglected as thestructure becomes a single-layer microstrip line. Its presencemay overcompensate the loss. In order to check thefull-wave results of Mesa et al. [20] and Gill and Balanis[17] using our circuit model, we have to disregard Rs.Since [17] and [20] used perfect strip and groundconductors, there is no division of current between the stripand ground plane.
The SLR and SLR (I) formulations are used to obtainthe primary line constants, i.e. R, L, C and G of a circuitmodel. To check the validity of the circuit model, Hasegawaet al. [3] also compared their experimental results forpropagation parameters and their results obtained using acircuit model in R, L, C and G format for the slow-wavemode. However, they did not validate their model againsttheir own experimental results in the skin-effect moderegion. They modelled each layer of the substrate separatelyin a shunt arm in the circuit model. Our transmission-line-based circuit model takes care of all layers at the same time.More circuit elements can be added to incorporateadditional physical phenomena, such as losses due to thesurface-wave, leaky wave mode and radiation.
The primary line constants of a multilayer slow-wavemicrostrip line are computed from the conductor loss,dielectric loss and real part of effective relative permittivity:
R ¼ 2Zac resistance per unit length ð19Þ
G ¼ 2ad
Zconductance per unit length ð20Þ
C ¼ffiffiffiffiffiffiffie0reff
pcZ
capacitance per unit length ð21Þ
L ¼Z
ffiffiffiffiffiffiffie0reff
pc
inductance per unit length ð22Þ
where, Z is the characteristic impedance of the multilayerslow-wave microstrip line and c is the free-space velocityof a EM wave. We have already discussed the computationof e0reff , Z, ad and ac by using the SLR formulationin Section 2 and Section 3. The attenuation constant ac iscomputed using Wheeler’s inductance rule for a stripthickness of t41.11dss. For tr1.11dss, we assume auniform current distribution in the strip conductor andthe circuit resistance is:
R ¼ 1
smtW
where sm is the conductivity of the strip conductor andW is its width. The series impedance of the circuit model isgiven by:
Z 0 ¼ RsZRs þ Z
ð23Þ
where, Z¼R+joL. The resistance:
Rs ¼1
s1h1W
is due to the longitudinal current in the semiconductorsubstrate for dSSZh1. The resistance is:
Rs ¼1
s1dssW
for dss/h1 where, dss ¼ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipfm0s1
pis the skin depth in the
semiconductor substrate. We are computing R and Rs intwo regions based on the skin depth. This gives adiscontinuity in the results.
The circuit model shown in Fig. 6 can calculate thecomplex characteristic impedance, phase constant and totalloss of a structure. The complex characteristic impedance Zois given by:
Zo ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z 0
Gþ joCð Þ
sð24Þ
The complex propagation constant g* is given by
g ¼ aþ jb ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ 0 Gþ joCð Þ
pð25Þ
The total loss is aT¼ real(g*) in units of neper per unitlength, the phase constant is b¼ imag(g*) in units of radiansper unit length and the slow-wave factor is b/b0, where b0 isthe free-space phase constant.
5 Validation of the circuit model
We have verified the accuracy of our SLR formulationand circuit model by comparing our calculated slow-wavefactor, attenuation and complex characteristic impedance ofthe MIS microstrip line with the classical experimentalresults of Hasegawa et al. [2, 3] and the recent experimentalresults of Wang et al. [15]. Hasegawa et al. [2, 3]used experiment at conditions of W/h¼ 0.64–6.4 and aresistivity of the n-type silicon- substrate in the range 0.001-85 Ocm in the frequency range 30–4000MHz.Their experimental method is not as accurate as ameasurement technique using a probe station with vectornetwork analyses (VNA). Their measurements on theimaginary characteristic impedance tend to have a greatererror. Moreover, they do not provide information on theconductivity and thickness of the strip conductor andground plane. For our calculations we have assumedconductivity of 5.8 107S/m and a thickness of 0.005mmfor all strip samples. Hasegawa et al. [2, 3] mentioned thatboth the conductivity and thickness were different forvarious samples. The lack of this information has influenced
L
GC
Rs
R
Fig. 6 Circuit model of multilayer slow-wave microstrip line
the validation of our model for the loss and the imaginarycharacteristic impedance. Over the frequency range of 30–4000MHz and for a resistivity of r140.15O cm themicrostrip line works in the slow-wave to dielectric modeand for a resistivity of r1o0.15O cm, it works in the slow-wave to skin-effect mode. We have compared both ourcircuit model and SLR (I) with the experimental results ofHasegawa et al. [3, 4] for both cases mentioned above.However, for the sake of brevity detailed comparisons arenot presented.
We find that the circuit model is essential to account forthe effect of a finite conductivity and a finite thickness of thestrip and ground conductors on the propagation character-istics mentioned above. The circuit model also providesbetter results as compared to the results obtained solelyusing SLR (I) in the transition region. In this region theresults obtained using the SLR (I) formulation show abetter agreement with the experimental results, especially forthe imaginary characteristic impedance as compared to theresults of the circuit model. This may be due to possibleerrors in the experimental results, as recognised byHasegawa et al. [3] and may also be due to our assumedconductivity and thickness of the strip and groundconductors.
Our circuit model also shows a better agreement with theexperimental results of Wang and Dutton [15] than theirdevice-level-based EM–model. We also present a compar-ison of the results obtained using the SLR and circuitmodels for a Schottky contact microstrip line with theexperimental results of [4–6] for W/h¼ 0.3–9.1 andr1¼ 0.1–12O cm. In [4–6] the experimental results werenot compared with results obtained by any theoreticalmethod. Even although we have successfully validated ourmodels against each case, in order to avoid duplication weare not presenting detailed information.
For a Schottky contact slow-wave microstrip line,changing the applied reverse DC bias voltage controls thepropagation characteristics of the line structure. The siliconsubstrate is suitably doped to achieve the desired resistivityof the substrate. The total substrate thickness is h1+h2,where h2 is thickness of the depletion layer formed by theSchottky contact. The depletion depth is controlled byapplying a variable reverse bias voltage and is given by:
h2 ¼2 V þ vdð Þere0
ne
12
ð26Þ
where V is the applied reverse DC bias voltage, vd isdiffusion potential of the Schottky barrier, e is charge on anelectron (1.60219 1019 C), er is the relative permittivity ofthe substrate (for silicon er¼ 11.7), e0 is permittivity of freespace (8.854 1014F/cm), and n is the doping concentra-tion of the carrier per cubic centimeters. Normally theresistivity of a doped substrate is known and we can obtainthe doping concentration for a n-type silicon substrate fromthis information [35]. For any applied voltage, we obtain thedepletion layer thickness by using (26).
For the Schottky contact microstrip line, Jager et al. [5]used values of W¼ 0.17mm, h (¼ h1+h2)¼ 0.04mm, i.e.,W/h¼ 4.25 and r1¼ 10O cm. They also used an aluminiumconductor; however, they do not provide the conductorthickness. We have taken t¼ 0.003mm in our calculations.This information is taken from the set of experimental datagiven by Jager [6]. Figures 7a–7d show the experimentalresults on the slow-wave factor, attenuation constant, realcharacteristic impedance and imaginary characteristic im-pedance respectively. The figures also show the resultsobtained by using the SLR and circuit models. The figures
0
5
10
15
20
25
30
35
40
45
50
SLR
SLR
CM expt. [5]
CM expt. [5]
V = 0 V
V = 12 V
slow
-wav
e fa
ctor
, /
oSLR
SLR
CM expt. [5]
CM expt. [5]
V = 0 V
V = 12 V
SLR
SLR
CM expt. [5]
CM expt. [5]
V = 0 V
V = 12 V
10110−110−2 100
frequency, GHz
a
10110−110−2 100
frequency, GHz
b
10110−110−2 100
frequency, GHz
10110−110−2 100
frequency, GHz
c
d
atte
nuat
ion
cons
tant
, dB
/mm
SLR
SLR
CM expt. [5]
CM expt. [5]
V = 0 V
V = 12 V
0
5
10
15
20
25
Rea
l cha
ract
eris
tic im
peda
nce,
Ω
101
10−1
10−2
10−3
100
−20
−15
−10
−5
0
5
10
15
imag
inar
y ch
arac
teris
tic im
peda
nce,
Ω
Fig. 7 Various characteristics of Schottky contact microstrip linea Slow-wave factorb Attenuation constantc Real characteristic impedanced Imaginary characteristic impedance
clearly show that the circuit model (CM) is able to explainthe dispersion in the propagation constant and character-istic impedance both at the lower end and upper end of thefrequency range. At the lower end of the frequency rangethis is due to conductor loss, and at the upper end of thefrequency range it is due to the loss produced by Maxwell-Wagner interfacial polarisation.
6 Conclusions
The slow-wave MIS and Schottky microstrip lines form abasic component in MMIC and VLSI technologies. Wehave presented a comprehensive quasi-static SLR formula-tion to accurately compute all the line parameters over awide range of structural parameters. The SLR method isvalid for the three propagation modes supported by suchstructures. It has the accuracy of a full-wave methodwithout its large computational and analytical efforts. Themethod conveniently works for a multilayer case. It issuitable for the development of a synthesis algorithm todesign components and matching networks using the slow-wave microstrip line. However, it is not able to explaindispersion at the low frequencies. We have developed a CMto explain this behaviour of the MIS and Schottky contactmicrostrip lines.
7 Acknowledgments
We express our gratitude to the unknown reviewers forcritical comments that significantly improved an earlierversion of this paper. We express our gratitude to MIT, theGovernment of India for the award of a grant to studyoptically controlled devices. The first author expresses hisgratitude to the University of Delhi, India for granting hissabbatical leave and to the University of Magdeburg,Germany for the award of a fellowship. The second authorexpresses his gratitude to ARC, the Government ofAustralian for the award of a PDF.
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