Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
1
NEW ANALYTIC FORMULAS FOR SERIES
MUTUAL IMPEDANCE COMPONENT
CALCULATIONS OF COUPLED INTERCONNECTS
ON LOSSY SILICON SUBSTRATE
Hasan Ymeri, Bart Nauwelaers, Karen Maex, and David De Roest
Abstract
A new analytic model for series mutual impedance of coupled interconnects
on lossy silicon substrate is presented. The model includes the frequency-dependent
distribution of the current on the silicon substrate (the substrate skin effect). From
this model easy formulas for the accurate calculation of the frequency dependent
distributed mutual inductance and the associated series mutual resistance of coupled
interconnects on silicon substrate are derived. The validity of the proposed formulas
has been checked by a comparison with the equivalent-circuit model data and
corresponding full wave solutions. Through this work, it is found that the effect of
the semiconducting substrate return path on the transmission behaviour of the
interconnects must to be well modeled for the accurate prediction of the resistance
and inductance over the whole frequency range.
Index terms - Interconnects, silicon substrate, on-chip interconnects, skin
effect, mutual impedance.
I - INTRODUCTION
Accurate modeling and calculation of electrical properties of interconnects is essential and crutial
for the design of high-speed integrated VLSI circuits. Particularly, as the sub-nano second
transition time of a signal becomes comparable to its propagation delay on interconnect lines, the
transmission line effects on the IC interconnects become extremely important. [1-3]. The
transmission behaviour of interconnect lines on silicon oxide-silicon semiconducting substrate has
already been characterized by experiment [4-5], by full wave or quasi-static analytical and
numerical procedures [6-12], and recently in terms of analytical closed-form models or CADoriented equivalent-circuit approach [13,14], respectively. The effect of a silicon semiconducting
substrate, which is negligible at low frequency, has a strong effect on the transmission line
characteristics of interconnection lines and making the parameters frequency dependent. The
conducting silicon substrate cause the capacitive and inductive coupling effects in the structure. If
the conductivity of silicon substrate is low, the capacitive coupling effects dominates the overall
coupling behaviour. If the conductivity of silicon substrate is higher the inductive coupling starts
increasing (capacitive coupling effects starts to decrease) and to be more important for on-chip
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
2
interconnect. In some previous references [4,6,8,10], it has been shown that the exact current
distribution induced by a strip interconnect conductors on a silicon semiconducting substrate may
be predicted by means of a integral equations approach (spectral domain approach) or Fourier
series technique, respectively. Such a current distribution is of paramount importance for the
determination of transmission line parameters per unit length of IC interconnects.
In this paper, starting from several reasonable approximations, closed-form expression for
the mutual impedance per unit length of coupled IC interconnects with silicon substrate is
proposed, valid in a wide frequency range. Then, formulas for the direct evaluation of the mutual
inductance and resistance per unit length are derived and compared with those existing in
literature, which are approximate, analyticaly or numericaly calculated and represent a highfrequency limit. The comparisons show that the new expressions are more general accounting for
finite conductivity and permitivity (conductive and displacement currents) of the silicon substrate,
are more accurate in the low-frequency range, allow the direct evaluation of the IC interconnect
and silicon substrate impedances and yield to the same results of existing formulations at high
frequencies. Finally, discussions and a conclusion are followed.
II - ANALYSIS
To develope the expressions for series mutual impedance per unit length of coupled interconnects
Zm the given structure (Fig. 1a) can be regarded as a system of inductively coupled transmission
lines with silicon semiconducting substrate acting as a return path as shown in Fig. 1b. Results
obtained from full-wave analysis [2,10] have shown that the influence of the finite substrate
thickness d can be neglected for practical dimensions (i.e., d >> wi, d >> Ti and d >> tox). The
silicon substrate is therefore assumed to be infinitely thick in the following derivation (d → ∞).
The following assumptions are used in this deduction:
• As a first step, the cross-section of the lines are assumed rounded, which is realistic for most
IC interconnect micrometre and sub-half micrometre designs. To model actual rectangular
conductors, we define an equivalent diamater 2rieq (i = 1,2) as the mean of the diameter of the
circles inscribed in the conductors (2rieq = (wi + Ti)/2). The other geometrical dimensions H, h
and s are consequently redefined as Heq = H + (T1-w1)/4, heq = h + (T1-w1)/4 + (T2-w2)/4 and
seq = s + (w1-T1)/4 + (w2-T2)/4 (see Fig. 1b).
• Skin effect in the interconnect conductors will be neglected and the proximity effect of the
silicon substrate return current on the conductor currents will be neglected, too.
• Current density Jsi = σEsi + εsi∂Esi/∂t and the electric field intensity Esi in silicon
semiconducting substrate are parallel to the axial currents in the signal interconnect
conductors. Hence the components in the x and y direction are Ex = Ey = Jx = Jy = 0.
• Current in silicon substrate, and hence the electric field intensity, does not change in the
direction of propagation, i.e., ∂Ez/∂z = ∂Jz/∂z = 0.
• The excitation source is sinusoidally time varying (exp(jωt)).
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
w1
3
w2
s
T1
T2
tox
SiO2
silicon
d
Fig. 1a Coupled interconnects on general lossy silicon substrate.
The mutual impedance per unit length of the coupled conductors above the
semiconducting half-space may be derived in several ways. For this reason a straight filament
paralel to a semiconducting half-space as shown in Fig. 2 will be first regarded. The conductivity,
permittivity and permeability of silicon substrate are designated as σ, ε, µ, respectively.
w2
2r 1 eq
s
T2
h
s eq
heq
w1
2r 2 eq
Region 1
T1
H eq
H
Region 2
Silicon substrate
µ ε σ
Fig. 1b. Parameter definition of the round-sectioned lines.
According to Maxwell’s equations, calculation expressions of the electric and magnetic
field expressed by the magnetic vector potential A are given by
E = − jω A +
Copyright SBMO
jω
∇(∇ ⋅ A)
k2
(1)
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
H=
4
1
∇×A
µ
(2)
where k2 = ω2εµ is the propagation constant of the medium. The current filament, parallel to the
z-axis and displaced at x = 0, y = b, radiates in the presence of the silicon semiconducting halfspace. It is well known that the electromagnetic field due to a straight current filament in the free
space is [16]
E0 = −
π
ζ 0k

 kr  
I (ω ) − j γ + ln  1 z
2π
 2  

2
(3a)
I (ω )
1ϕ
2πr
H0 =
(3b)
where r = ((x2 + (y - b)2)1/2, ζ0 is the free-space wave impedance, the Fourier transform of the
current I(ω) = ℑ[i(t)] has been introduced and γ is the Euler’s constant. Due to the impressed field
(3), an unknown current density J is induced in the silicon semiconducting half-space: because of
the particular geometry, this current has only the component along the z-axis (see assumptions),
which is a function of x and y, namely J = J(x,y)1z. Since most of the induced current is confined
in a limited zone of the semiconducting substrate just beneath the source, useful hints and
unexpectedly accurate results are obtained assuming the width of the silicon substrate to be
infinite.
y
I
σ=0
ε0
µ = µ0
b
Region 1
z
Region 2
σ
ε
µ = µ0
x
Fig. 2. Straight filament parallel to a semiconducting half-space.
Thus, in the two regions depicted in Fig. 2, the magnetic vector potential A1(x,y) above the
silicon semiconducting substrate verify the equation
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
5
∇ 2 A1 ( x, y ) = 0, − ∞ < x < ∞,
y≥0
(4)
and the vector potential A2(x,y) in the silicon substrate will verify the equation:
∇ 2 A2 ( x, y) − jωµ (σ + jωε ) A2 ( x, y) = 0, − ∞ < x < ∞,
y≥0
.(5)
A general solution of eqs. (4) and (5), respectively, may be looked for in the form [17]
∞
A1 ( x, y ) = ∫ [C11 (λ )e λy + C12 (λ )e −λy ] cos(λx)dλ ,
for
0
∞
A2 ( x, y ) = ∫ [C 2 (λ )e my cos(λx)dλ ,
0
for
y≥0
(6a)
y≤0
(6b)
where m = (λ2 + jωµ(σ + jωε))1/2 and having considered that A2 must vanish at large distance
from the current filament, when y tends to -∞. In order to determine the unknown coefficients C11,
C12, and C2, we have to take into account, that when the silicon semiconducting half-space is
absent, the magnetic flux density B0 of a single current filament can be expressed in the form [17]
µ0 I
2π
∫
µ0 I
2π
∫
B0 x = ±
B0 y =
∞
0
∞
0
e
e
− b− y λ
− b− y λ
cos(λx)dλ
(7a)
sin(λx)dλ .
(7b)
Using these expressions and imposing at the boundary surface y = 0, the continuity of the
tangential components of the magnetic field and of the normal component of the magnetic flux
density, the following expressions were obtained:
x 2 + ( y − b) 2
µ0 I
+
ln
2π
x 2 + ( y + b) 2
A1 ( x, y ) =
µI
π
∫
∞
0
A2 ( x, y ) =
e
−(b + y )λ
µ r λ + λ + jωµ (σ + jωε )
2
µI
π
∫
∞
0
e −bλ e y
(8)
cos(λx)dλ
λ2 + jωµ (σ + jωε )
µ r λ + λ2 + jωµ (σ + jωε )
cos(λx) dλ .
(9)
III - MUTUAL SERIES IMPEDANCE PER UNIT LENGTH
The application of the Ampere’s law for the derivation of the inductances from the magnetic flux
linkages is impractical for the system of interconnect conductors on lossy silicon substrate.
In Sect. 2 the magnetic vector potential is introduced in Maxwell’s equations for the
horizontal magnetic field intensity above the lossy silicon semiconducting substrate. This leads to
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
6
the quasi-static voltage drop ∂V/∂z in the z-direction that also appears in the classical line
equations,
∂V
= −[ Z s ]I .
∂z
(10)
Hence, the total “corrected” series impedance per unit length Zs of the interconnect line
can be derived. This method provides the best insight into the relation between the purely
mathematical equations and the physical reality of the problem.
The magnetic vector potential is used in order to find the quasi-static potential drop ∂V/∂z,
at any point (x,y) in space along the line parallel to the z direction. this allows the total series and
mutual impedances per unit length, eq. (10), to be evaluated.
The axial electric field intensity along the lossy silicon substrate is
∂V ( x, y = 0)
,
∂z
E zs ( x, y = 0) = − jωA1 ( x, y = 0) −
(11)
and at any point (x,y) above the lossy silicon substrate
E za ( x, y ) = − jωA1 ( x, y ) −
∂V ( x, y )
.
∂z
(12)
Subtracting eq. (11) from eq. (12), we get
E za ( x, y ) = E zs ( x, y = 0) − jω [ A1 ( x, y ) − A1 ( x, y = 0)]
−
∂
[V ( x, y ) − V ( x, y = 0)].
∂z
(13)
The last term in eq. (13) represents the total scalar voltage drop, in the axial z-direction, of
the distributed parameter circuit consisting of interconect conductor and lossy silicon substrate,
equation (10).
From the definition of the magnetic vector potential
y
A1 ( x, y ) − A1 ( x, y = 0) = µ 0 ∫ H 1x ( x, y )dy
0
(14)
where the x-component of the magnetic field intensity above the lossy silicon substrate is given
by
H 1 x ( x, y ) =
1 ∂A1 ( x, y )
.
µ0
∂y
(15)
The final simplified equation for electric field component in z-direction above the lossy
silicon substrate is
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
E za ( x, y) = −
− jω
µI
π
7
µ I
x 2 + (b + y ) 2
∂V ( x, y) ∂V ( x, y = 0)
− jω 0 ln 2
+
∂z
2π
∂z
x + (b − y ) 2
∫
∞
0
e −(b+ y )λ
µ r λ + λ2 + jωµ (σ + jωε )
cos(λx)dλ .
(16)
In order to develop expression for mutual impedance per unit length of coupled
interconnect conductors above the lossy silicon substrate a current I flowing in the round
interconnect conductors as depicted in Fig. 1b is now regarded. It can be assumed that the current
density is constant over the conductor’s cross-section because the skin-effect in the interconnect
conductors can be neglected due to their geometrical dimensions.
Substituting the coordinates for the round interconnect conductors, that are (x = 0, y = hk)
and (x = xn, y = hn), respectively, in general expression (16), simplifying the resulting equation
and rearranging the terms, finally we get
(hk + hn ) 2 + Dkn2
µ0
Z m = jω
ln
+
2π
(hk − hn ) 2 + Dkn2
jω
µ ∞
e −(h +h )λ
cos( Dkn λ )dλ
π ∫0 µ r λ + λ2 + jωµ (σ + jωε )
k
n
(17)
where
• hk = Heq + heq +2r2eq + r1eq and hn = Heq + r2eq are the hieght of the k-th and n-th conductors
above a lossy silicon substrate, respectively
• Dkn = seq + r1eq + r2eq is the horizontal distance between the k-th and n-th interconnect
conductors
according to Fig. 1b.
IV - CLOSED-FORM EXPRESSION FOR SERIES MUTUAL IMPEDANCE
It is known that the presence of a silicon semiconducting substrate in a transmission system make
the propagation parameters analysis very complex. The calculation of series mutual impedance of
coupled interconnects according to eq. (17) is based on relation which contain infinite integral
with complex arguments. The calculations using numerical techniques are tedious, heavy and
time consuming, even with various simplifications and efficient mathematical transformations
(spectral domain approach, for example) [2,4,11,12,15]. While these numerical tools are very
accurate they are valid for a limited range of frequencies only, and medium frequencies are not
covered in some cases (slow wave mode of propagation).
In this section based on the thoretical analysis in Sects. 2 and 3, it will be demonstrated
and analytically proved that simple and sufficiently accurate expression for series mutual
impedance can be derived which is valid for the whole range of frequencies (slow wave mode,
quasi-TEM mode, and skin-effect mode). They are especially designed for micron and submicron
VLSI technoligies.
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
8
Derivation of new closed-form formula for mutual impedance of coupled interconnects is
based on the assumption that the silicon substrate return currents are concentrated in a fictious flat
plane p as complex penetration depth
1
p=
jωµ 0 (σ + jωε )
(18)
which is related to the penetration depth
2
ωµ 0 (σ + jωε )
δ =
(19)
by
1
1
= (1 + j )
p
δ
(20)
where for silicon substrate µr = 1.
Let us now consider two interconnect lines running parallel to the same silicon
semiconducting substrate as shown in Fig. 1b. Let us consider the general expression for mutual
impedance per unit lenght given in (17) and let p, defined in eq. (18), be introduced in it. Then the
integral in eq. (17) will have the following form:
I2 = ∫
∞
0
Define
e − ( hk + hn ) λ
1
λ+ λ + 2
p
cos( Dkn λ )dλ .
(21)
2
k = pλ
β=
Dkn
hk + hn
q=
hk + hn
.
2p
Inserting the above relations in eq. (21), we obtain:
I2 = ∫
(L)
e −2 qk
k + k 2 +1
cos(2qβk )dk
where (L) is the path of integration shown in Fig. 3.
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
9
Im(k)
L’
Re(k)
C→∞
L
Fig. 3 The complex plane of the corresponding path of integration.
Regarding the integral I2 as a function of q, and applying the definition of the complex
cosine function, it can be expressed in the form:
I 2 (q ) = ∫
e −2 (1+ jβ ) qk + e −2(1− jβ ) qk
k + k 2 +1
( L)
dk
(22)
Let us take into consideration now only the case of β < 1, in which the integrandus is
regular and is equal to zero, if k → ∞, in the half plane under the dotted line, so instead of the path
(L) the positive part of the real axis can be used, that is the integration can be carried out
according to the real variable k.
Considering the derivative of eq. (22) with respect to q, we get:
∂I 2 (q )
1 ∞ (1 + jβ ) 2ke −2 (1+ jβ ) qk + (1 − jβ )2ke −2 (1− jβ ) qk .
=− ∫
dk
2 0
∂q
k + k 2 +1
(23)
(It can be proved that the sequence of differentiation and integration can be interchanged.)
Integral expression (23) contain the term of the form
2k
k + k 2 +1
.
The presence of this term in the integrandus of (23) prevents a closed-form evaluation of
the integral. Now, we propose the following approximation for this term
1
≈ 1 − e − 2k − k 3 e −2k .
3
k + k 2 +1
2k
(24)
The curves corresponding to the left and right sides are plotted in Fig.4a, and the relative
error of their differences is shown in Fig. 4b. Eq. (24) means, that in expression (22) the following
substitution can be done:
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
10
1 − e −2 k − (k 3 e −2 k ) / 3
.
≈
2k
k + k 2 +1
1
(25)
Introducing the approximation (25) into the relation (23), it yields
∂I 2 ( q )
k 3 e −2 k
1 ∞
= − ∫ 1 − e − 2 k −
2 0
3
∂q

 (1 + jβ )e −2 (1+ jβ ) qk + (1 − jβ )e − 2(1− jβ ) qk dk .

(
)
(26)
Carrying out the integration, the following simple result is obtained:
(a)
(b)
Fig. 4. The function in integrand of (23) and its approximation by (24).
(a) The solid line represents the approximation function and the dotted line is integrand term in (23).
(b) Relative error between the functions in eq. (24).
∂I 2 (q )
1
1 + jβ
1 − jβ
=−
+
+
∂q
2q 4[1 + (1 + jβ )q ] 4[1 + (1 − jβ )q ]
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
+
11
1
1 + jβ
1
1 − jβ
+ K.
+
4
16 [1 + (1 + jβ )q ] 16 [1 + (1 − jβ )q ] 4
(27)
The integration according to the variable q yields:
1
1
1
I 2 ( q) = − ln q + ln[1 + (1 + jβ ) q] + ln[1 + (1 − jβ ) q]
2
4
4
−
1
1
1
1
−
+ K.
3
48 [1 + (1 + jβ )q ]
48 [1 + (1 − jβ )q ]3
If q → ∞, then I2(q) → 0, so constant of integration is K = -(1/4)ln(1+β2). Thus
2


p 
 + β 2 
 1 +
Hm 
1 

I 2 (q ) = ln  

4
1+ β 2










1 
1
1

− 
+
3
3
48  H m

 Hm



(1 + jβ ) + 1
(1 − jβ ) + 1 

  p

 p
 
(28)
where
Hm =
hk + hn
.
2
With substitution of eq. (28) in (17), we get the following expression for mutual
impedance in closed-form:
Z m = jω
(hk + hn ) 2 + Dkn2
µ0
ln
2π
(hk − hn ) 2 + Dkn2
2




p 


 + β 2 
 1 +
1  Hm 


−
ln
2 

2

+
β
1







µ0 µr 



+ jω

.
2π  


 
1
1

1 
+
3
3 
 24   H
 H

   m (1 + jβ ) + 1  m (1 − jβ ) + 1  
   p
  p
  
Copyright SBMO
(29)
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
12
V - DISCUSSION OF THE RESULTS
In order to prove and validity of the given approach the frequency-dependent mutual inductance
and resistance per unit length [Zm = Rm + jωLm] calculated using the closed-form expression (29)
are compared with the CAD-oriented equivalent-circuit model procedure and full-wave solutions
using the spectral domain approach [15,18].
However, generaly, the interconnect resistance (R) includes the resistance of the signal
lines Rs (called signal line resistance or internal conductor line resistance) and the serial substrate
resistance Rsub (called internal silicon substrate resistance). Similarly, the total inductance
associated with interconnect line over silicon substrate L includes the external inductance Lex
(result from the magnetic field distribution outside the signal line and silicon substrate), the self
(internal) inductance of the signal line Ls (result from the magnetic field penetration into the
signal conductor), and the self (internal) inductance of the silicon substrate Lsub (result from the
magnetic field penetration into the substrate). The internal inductance and resistance of the signal
conductor and of the silicon substrate, respectively, are governed by the skin effect (i.e., the skin
depth associated with the signal conductor and silicon substrate). As a result, it is known
qualitatively that at high frequencies, the total inductance is dominated by the external inductance
of IC interconnects and the internal inductance of substrate can be neglected. However, for a
given frequency there has not been a way of quantitatively determining how large or small the
contribution of the internal substrate inductance is compared to the external or total inductance of
the structure. With the analytic model introduced in our paper, we can quantitatively determine
what percentage of the total inductance is associated with the internal silicon substrate inductance
for a given frequency. For a special case of perfectly ground plane (no silicon present) the
resistance of the signal line and the total resistance of the structure naturally are the same. In our
model skin and proximity effects in signal conductors are in general negligible because conductor
cross-sections are small compared to skin depth. This means that, in general, the resistance and
inductance of our structures is clearly frequency-dependent. This leads to the conclusion that the
calculated frequency dependence must be attributed to the skin effect in the silicon substrate.
Knowing the fact that the skin depth at our highest calculation frequency of 10 Ghz is greater than
thickness of signal conductors, the skin effect in the interconnect conductors can be neglected.
In order to prove the validity of the given approach mutual per unit length series
impedance parameters (resistance and inductance per unit length) calculated using our analytic
model are compared with the results of the full-wave analysis (spectral domain approach) and
with those obtained by using equivalent circuit modeling technique [15,18]. In Fig. 1a, an
asymmetric coupled interconnect structure is depicted. For this interconnect structure two sets of
geometries used in [18] and [15] are analyzed.
The first geometry have the following electrical and geometrical parameters [18]:
d = tsi = 500 µm, tox = 2 µm, w1 = 4 µm, w2 = 1 µm, T1 = T2 = 1 µm, εsi = 11.8, ρsi = 0.01 Ωcm,
εox = 3.9 and s = 4 µm.
Fig. 5a,b gives another examples for the good egreement between results of our model and
those obtained in [18]. A strong increase of the resistance with increasing a frequency can be
notized. Especially for wide lines the substrate resistance and the skin-effect in silicon substrate
play an important role. For higher frequencies, the inductance of the silicon substrate is negligible,
but becomes important at lower frequencies (slow wave mode).
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
13
(a)
(b)
Fig. 5. Mutual series impedance components as a function of frequency (a) resistance per unit length and (b)
inductance per unit length. The solid lines indicate results obtained by using proposed analytic model, and the dashed
lines correspond to calculations obtained by using spectral-domain approach and equivalent-circuit model of [18].
The second geometry have the following electrical and geometrical parameters:
• d = tsi = 300 µm, tox = 3 µm, w1 = 2 µm, w2 = 1 µm, T1 = T2 = 1 µm, εsi = 11.8, ρsi = 0.01
Ωcm, εox = 3.9 and s = 2 µm.
Fig. 6a shows the results of the mutual series resistance per unit length as a function of
frequency. Similarly, in Fig. 6b, the results for mutual series inductance per unit length are plotted
as a function of frequency. These results illustrate how the mutual series impedance per unit
length (resistance and inductance) changes as the magnetic fields penetrate into the silicon
substrate.
Numerical results presented in the paper show that the skin effect in the highly conductive
silicon substrate leads to a strong frequency dependence of the tramsmission line matrices [R] and
[L].
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
14
(a)
(b)
Fig. 6. Mutual series impedance components as a function of frequency (a) resistance per unit length and (b)
inductance per unit length. The solid lines indicate results obtained by using proposed analytic model, and the dashed
lines correspond to calculations obtained by using spectral-domain approach and equivalent-circuit model of [15].
A comparison of the results from the analytic model with equivalent-circuit model data
and full-wave simulations (Figs. 5 and 6) shows that analytic model yields very good results,
without having to apply heavy and time consuming full-wave field solvers.
VI - CONCLUSION
The determination of the electrical properties of the interconnects represents a critical design and
analysis problem in the high-speed IC circuits in order to minimize signal distortion. Lossy silicon
semiconducting substrate leads to a significant frequency dependency of the transmission line
parameter matrices [R], [L], [G], and [C] characterizing uniform interconnect systems in IC or
VLSI/WSI circuits on a voltage-current level of description. In contrast. skin and proximity
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
15
effects in the signal lines are in general negligible because conductor cross-sections are small
compared to skin depth.
In this paper, for the first time, we have introduced the closed form expressions for mutual
resistance and inductance per unit length of the coupled interconnects on lossy silicon substrate.
These expressions can be used to determine resistance and inductance (impedance) for any given
frequency, resistivity and permittivity of silicon semiconducting substrate. The results presented
here illustrate that the resistivity and the geometry of the signal line as well as the frequency play
a strong role in determining the relative importance of the resistance and inductance of the general
lossy multiconductor interconnects.
The effect of the magnetic field penetration inside the lossy (CMOS) silicon substrate on
the mutual impedance per unit length was investigated. It was shown that depending on the
resistivity and dimensions of the interconnect lines, large changes in the resistance and inductance
can occur at low and medium frequencies due to this effect (frequency range of the slow-wave
mode). Due to the strong skin effect in the silicon substrate, the substrate resistance increases
rapidly with a frequency. The inductances which consider the silicon substrate effect show much
larger values than those without the effect because of the magnetic flux penetrating into the
silicon.
It was demonstrated in the paper that the inductive and resistive effects due to the silicon
substrate can’t be neglected any more in today’s high speed IC circuit designs. For the accurate
estimation of the transmission line parameter matrices a silicon substrate effects has to be taken
into account.
Originally derived for high-speed IC interconnect and VLSI technologies, these analytical
formulas apply in many practical cases with MCM or MMIC circuits, provided that no important
skin effect occurs in the line conductors. An analysis of how resistance and inductance change
due to the signal conductor is more involved and will be investigated in the forthcoming papers.
REFERENCES
[1] D.C. Edelstein, G. A. Sai-Halasz, and Y.-J. Mii, “VLSI on-chip interconnection performance simulations and
measurements”, IBM J. Res. Develop., vol. 39, pp. 383 - 401, July 1995.
[2] E. Groteluschen, L. S. Dutta, and S. Zaage, “Quasi-analytical analysis of the broad-band properties of
multiconductor transmission lines on semiconducting substrates", IEEE Trans. Comp., Packag., Manufact.
Technol., B, vol. 17, pp. 376 - 382, Aug. 1994.
[3] A. Deutsch, G. V. Kopcsay, P. J. Restle, H. H. Smith, G. Katopis, W. D. Becker, P. W. Coteus, C. W. Surovic,
B. J. Rubin, R. P. Dunne, Jr., T. Gallo, K. A. Jenkins, L. M. Terman, R. H. Dennard, G. A. Sai-Halasz, B. L.
Krauter, and D. R. Knebel, “When are transmission-line effects important for on-chip interconnects ?”, IEEE
Trans. Microwave Theory & Tech., vol. 45, pp. 1836 - 1844, Oct. 1997.
[4] S. Zaage and E. Groteluschen, “Characterization of the broadband transmission behaviour of interconnections
on silicon substrate”, IEEE Trans. Components, Hybrids Manuf. Technol., vol. 16, pp. 686 - 691, Nov. 1993.
[5] V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of
broad-band transmission for coplanar waveguides on CMOS silicon substrates”, IEEE Trans. Microwave
Theory & Tech., vol. 46, pp. 632 - 640, May 1998.
[6] T. Shibata and E.Sano, “Characterization of MIS structure coplanar transmission lines for investigation of signal
propagation in integrated circuits”, IEEE Trans. Microwave Theory & Tech., vol. 38, pp. 881 - 890, July 1990.
[7] H. Ymeri, B. Nauwelaers, and K. Maex, “Computation of capacitance matrix for integrated circuit interconnects
using semi-analytic Green’s function method”, Integration, the VLSI J., vol. 30, pp. 55 - 63, Dec. 2000.
Copyright SBMO
ISSN 1516-7399
Journal of Microwaves and Optoelectronics, Vol. 2, N.o 3, July 2001.
16
[8] H. Grabinski, B. Konrad, and P. Nordholz, “Simple formulas to calculate the line parameters of interconnects on
conducting substrates”, in Proc. IEEE 7th Topical Meeting Electrical Performance of electronic Packaging
(EPEP 98), west Point, NY, Oct. 26 - 28, 1998, pp. 223 - 226.
[9] H. Ymeri, B. Nauwelaers, K. Maex, and D. De Roest, “A new approach for the calculation of line capacitance of
two-layer IC interconnects”, Microwave Opt. Technol. Lett., vol. 27, pp. 297 - 302, Dec. 2000.
[10] H. Ymeri, B. Nauwelaers, and K. Maex, “On the modelling of multiconductor multilayer systems for
interconnect applications”, Microelectronics Journal, vol. 32, pp. 351 - 355, April 2001.
[11] H. Hasegawa, M. Furukawa, and H. Yanai, “Properties of microstrip line on Si-SiO2 system”, IEEE Trans.
Microwave Theory & Tech., vol. 19, pp. 869 - 881, Oct. 1971.
[12] A. Tripathi, Y. C. Hahm, A. Weisshaar, and V. K. Tripathi, “Quasi-TEM spectral domain approach for
calculating distributed inductance and resistance of microstrip on Si-SiO2 substrate”, Electron. Lett., vol. 34, pp.
1330 - 1331, June 1998.
[13] J.-K. Wee, Y.-J. Park, H.-S. Min, D.-H. Cho, M.-H. Seung, and H.-S. Park, “Modeling the substrate effect in
interconnect line characteristics of high-speed VLSI circuits”, IEEE Trans. Microwave Theory & Tech., vol. 46,
pp. 1436 - 11443, Oct. 1998.
[14] D. F. Williams, “Metal-insulator-semiconductor transmission lines”, IEEE Trans. Microwave Theory & Tech.,
vol. 47, pp. 176 - 181, Feb. 1999.
[15] J. Zheng, Y.-C. Hahm, V. K. Tripathi, and A. Weisshaar, “CAD-oriented equivalent-circuit modeling of on-chip
interconnects on lossy silicon substrate”, IEEE Trans. Microwave Theory & Tech., vol. 48, pp. 1443 - 1451,
Sept. 2000.
[16] R. F. Harrington, Time Harmonic Electromagnetic Fields. London: McGraw-Hill, 1961.
[17] J. A. Tegopoulos and E. E. Kriezis, Eddy Current in Linear Conducting Media. Amsterdam: Elsevier, 1985.
[18] J. Zheng, Y.-C. Hahm, A. Weisshaar, and V. K. Tripathi, “CAD-oriented equivalent circuit modeling of on-chip
interconnects for RF integrated circuits in CMOS technology”, 1999 International Microwave Symposium,
Paper MO1D-3, Anaheim, CA, June 1999.
[19] Y. Eo and W. R. Eisenstadt, “High-speed VLSI interconnect modeling based on S-parameter measurements”,
IEEE Trans. Components, Hybrids Manuf. Technol., vol. 16, pp. 555 - 562, Aug. 1993.
Copyright SBMO
ISSN 1516-7399