50 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
phase function defined by the following condition [127, 128]:
lim
ω→∞
γ(ω)
jω
⇒ 0. (3.35)
For T-line, the minimum phase function can be calculated by substituting
(3.13) into (3.35) as
lim
ω→∞
γ(ω)
jω
= lim
ω→∞
ω
α
L
+α
C
2
·
√
L
′
C
′
·
sin
(α
L
+α
C
)π
4
− j cos
(α
L
+α
C
)π
4
.
(3.36)
We can observe that (3.3 6) shows very different responses for fractional-
order and integer-order T-line models. For the fractional-order model, (3.36)
equals zero when α
L
+ α
C
< 2. So the causality is always ensur ed as the
minimum phase function condition when (3.35) is satisfied. On the o ther hand,
α
L
and α
C
are both equal to one for the integer-order T-line model, and (3.36)
results in a constant value of
√
L
′
C
′
, where L
′
and C
′
become the normal
inductance and capacitance, respectively. Thus the minimum phase function
condition in (3.3 5) is violated and the model be comes non-causal.
Note that the major reason for the non-causal issue in the traditional inte-
ger T-line model is due to the linear frequency dependence of the propagation
constant γ(ω) when α
L
+ α
C
= 2 in (3.13). This cannot model the disper-
sion loss and non-quasi-static effects in the high-frequency application like
THz. The reality of the integer-order T-line model is lost in the THz re gion,
and so is the causality. In contrast, the non-ideal effects are considered in the
proposed fractional-order T-line model by fractional-order dis persion terms,
which c an largely improve the model r eality. As s uch, both the model accu-
racy and causality are improved. The causality of the fractio nal-order model
can also be verified in numerical calculation by computing the er ror terms in
(3.33), which will be discussed in the following section.
3.4 Prototypin g and Measurement
3.4.1 T-Line Fractional-Order Model Verification
As shown in Fig. 3.5(a), a c oplanar wave guide transmission line (CP W- T L )
testing structure w ith RF-PADs is fabricated with Global Foundry 1P8M
65nm CMOS process, of which the dimensions are given in Fig. 3.5(b). The
CPW-TL is implemented on the top metal layer with thickness of 3.3 µm.
It is measur ed on a CASCADE Microtech Elite-300 probe station by Agilent
PNA-X (N5247A) with frequency sweep up to 110 GHz. The measurement
setup of S- parameters up to 110 GHz is illustrated in Fig. 3.6. The reference
plane of PNA is calibrated to the ends of GSG probes by SOLT method. Note
that both the probes and the impedance standard substrate are provided
by Cascade Microtech. RF-PADs on both sides are de-embedded from the
CMOS THz Modeling 51
5µm
35µm
5µm
5µm
160µm
(a) (b)
Figure 3.5: T-li ne testing structure: (a) die photo, and (b) detailed
dimensions .
measurement results with the “open-short” method. Table 3.1 summa rizes
extracted model parameters of both integer-order and fractional-or de r models
based on measurement results. The parameters of the traditional integer-order
model are ex tracted according to the pro cedure by [121]. The parameters of
fractional-order model are extracted according to Sectio n 3.3.1.
The resulting S-parameters and characteristic impedance (Z
0
) of integer-
order and fractional-order RLGC models are compared in Fig. 3.8. We can
observe that bo th the traditional integer-order mo de l and the proposed
fractional-order model can fit the measurement results in magnitude in Fig.
3.7. Here a relatively large de viation is observed in magnitude of S11 between
the simulation and measurement results. This deviation comes fr om the equip-
ment noise and calibration error, which is unavoidable as the absolute magni-
tude o f S11 is small (−15 ∼ −50 dB). Moreover, the phase delay o f both the
Figure 3.6: Measurement setup of on-wafer S-parameter testing up
to 110 GHz.
52 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Table 3.1: Modeling Parameters of Integer-Order and Fractional-
Order RLGC Model for T-Line
Integer-Order Model Fractio nal-Order Model
Parameter Value Unit Parameter Value Unit
L 247.5 nH/m α
L
0.862 —
C 0.188 nF/m L
′
10022 V s
−α
L
A
−1
/m
R 1200 Ω/m α
C
0.988 —
R
S
12.56 mΩ/m · rad C
′
0.278 As
−α
C
V
−1
/m
G 0.079 S/m R 1200 Ω/m
G
D
19.33 pS/m · rad G 0.079 S/m
Figure 3.7: Verification of fractional-order T-line model with mea-
surement results for magnitude of S11 and S2 1 in dB.
traditional integer-order model and the proposed fractional-order model agree
well with the measurement results as shown in Fig. 3.8. However, it is obser ved
that characteristic impedance Z
0
in the traditional integer-order RLGC model
has de viated from measurement results above 10 GHz, and almost approaches
a constant above 40GHz. On the other hand, the fractional-order RLGC model
closely fits the measured Z
0
up to 1 10 GHz, because it can accurately consider
frequency-dependence loss yet in a compact RLGC form. At 100 GHz, the Z
0
from fractional-o rder and measurement results are 34.3 Ω, which is 3.1 Ω lower
than the one from integer-order model. Note that such difference will keep in-
creasing with frequenc y and largely affects the model accuracy in traditional
integer-order T-line model at THz. Physically, the values of fractional-order
terms (α
L
and α
C
) model the frequency-dependent dispersion loss of the de-
vice at THz. For the T-line fabricated by on-chip CMOS process, large r loss is
CMOS THz Modeling 53
Figure 3.8: Verification of fractional-order T-line model with mea-
surement results for phase delay of S21 and characteristic i mpedance.
observed in meta l layer tha n in dielectric layer. As a result, α
L
has a relatively
large deviation from 1, while α
C
is close to 1. But note that a slight change in
the order -terms (α
L
and α
C
) could bring huge changes in the prefacto rs (L
′
and C
′
) in the THz region as observed in (3.7) and (3.8).
3.4.2 CRLH T-Line Fractional-Order Model
Verification
In order to minimize the charac terization error of each CRLH T-line unit-
cell, one 13-cell CRLH T-line is fabricated with Glo bal Foundry 1 P8M 65nm
CMOS process. As shown in Fig. 3.9, it has a chip size of 145µm × 660µm
excluding the RF Pads. The layout and dimension of each unit ce ll is s hown
in Fig. 3.1 0. The topmost aluminum layer (LB) is exclusively employed as
signal layer for the maximum distance to the bottom ground layer (M1) to
improve the radia tion efficiency. Various components in the CRLH T-line cell
in Fig. 3.2(a) are synthesized by on-chip str uc tures. The L
P
of the CRLH
T-line is synthesized by a microstrip line connected to the ground and C
S
is
implemented with inter- digital capacito r. Both right-handed elements L
S
and
C
P
are contributed by the intrinsic parasitic. Note that a mesh structure is
applied in the ground layer to satisfy the metal density rule.
The 13-c ell CRLH T-line design is verified by circ uit simulation in ADS
from 220 to 325 GHz. From this we get the integer-order and fractional-order
simulation results. The fabricated 13-c ell CRLH T-line structure is measured
on probe station (CASCADE Microtech Elite-30 0) with VNA extender (VDI
WR3.4-VNAX). Two waveguide GSG probes with 50 µm pitch are used for the
S-parameter measurement from 220 to 325 GHz, a s shown in Fig. 3.11. Note
54 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Figure 3.9: Chip mi crograph of fabricated CRLH T-line in 6 5 nm
CMOS proces s.
Figure 3.10: Dimension of each unit-cell and layer configurations of
LWA.
that the testing pads and traces are de-embedded (open, short) from both
sides with recursive modeling technique [129]. We also compare the measure-
ment results of fabr icated CRLH T-line w ith integer-order and fractional-order
circuit simulation as well as EM-simulation by HFSS. The circuit simulations
are conducted with the equivalent circuits of unit-cell shown in Fig. 3.2, and
the values of circuit elements are summarized in Table 3.2, obtained by curve
fitting technique.
As shown in Fig. 3.12 and 3.1 3, the phase and magnitude of S21 are al-
most identical for both fractional-order and integer-order models in the mea-
sured frequency range o f 220–325GHz. But the extracted phase consta nt (β)
of fractio nal-order model is closer to mea surement than that of integer-order
one while considering the dispersion effects. More importantly, the fractional-
order model accurately fits the measurement results at the frequency with
β = 0, which is the boundary b etween left-handed and right-handed regions,
while that from integer-order model is 13 GHz less. Moreover, Fig. 3.14 shows
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