Chapter 3
CMOS THz Model i n g
3.1 Introduction
Accurate device models that can take into account the loss from strong
frequency-dependent dispersion and non-quasi-static effects must be c onsid-
ered in CMOS-based THz design. As the most fundamental passive structure,
the accurate modeling of transmission line (T-line) is very important in various
designs [96, 97]. T-line is traditionally char acterized by distributed integer-
order RLGC model as shown in Fig. 3.1(a) [98]. Drude’s classical rela xation-
effect model is deployed for the skin effect w ith R
S
[99, 100]. In addition, the
loss due to diele ctric polarization and dipole rotation can be modeled by a
dielectric-loss of G
D
. However, such an integer-order model is insufficient to
describe the T-line performance a t THz region because the loss term in T-
line is difficult to model the dispersion loss and non-quasi-static effects [101],
which can c ause large deviation at THz freq ue nc y region. Such impact is fur-
ther verified by the measurement r esults and circuit level simulations in this
paper. Moreover, the traditio nal T-line model has a causality issue. Physically,
the real and imaginary parts of both permittivity ε(ω) and permeability µ(ω)
in a propagation medium are not independent of each other, but follow the
Kramers–Kronig relation [102]. As such, the extracted RLC G parameters in
the traditional T-line model may result in a non-causal response in the model
that can induce both accuracy and convergence pr oblems in the time-domain
simulation.
The concept o f fractiona l-order model has been examined to model ca-
pacitor (C) and inductor (L) at high frequenc y region. The I-V relation of a
capacitor is found to follow the fractional-order [103], and the eddy current
and hysteresis effect in inductors are also observed with fractional-order re-
lation [104]. It motivates us to re-examine the RLCG T-line model at THz
39
40 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
C
G
L
R
(a)
C’
G
L’
R
(b)
Figure 3.1: RLGC unit-cell equivalent circuits of T-line: (a) integer-
order model; and (b) fractional-order model.
during device characterization [105]. Note that the fractional-order model has
been deployed to model the surfa ce impedance [106] and describe the abnor-
mal diffusion of voltage and current wave [107]. T he fractional-order-based
impedance-matching network [108, 109] and resonator design [110] have also
been studied. However, no studies investigating the model causality have b een
carried on with measurement verifications at T Hz.
In this chapter, two fractional-order T-line models have been developed for
both CMO S on-chip conventional RLCG T-line modeling a nd metamaterial-
based CRLH T-line modeling a t THz with the following advantages. Firstly,
the fractional-order T-line models can describe dispersion and non-quasi-static
effect in THz. Secondly, by properly deciding the range of fractional-order,
the fractional-order RLC G T-line model does not have the causality issue.
Lastly, the frac tional-order models are still in compact forms that can be
extracted fro m measurement results. The proposed factiona l-order RLCG and
CRLH T-line modes are verified by S-parameter measurement results in 10
MHz ∼ 110 GHz and 220 GHz ∼ 325 GHz, respectively. Compared to the
conventional integer-order models, the proposed fractio nal-order T-line models
demonstrate improved accuracy of characteris tic impedance and propagation
constant, which have significant impacts on THz circuit design.
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