CMOS Metamaterial Devices 31
2.3 Resonant-Type Metamaterial
Metamaterial-based resonators have been explor ed recently for CMOS MMIC
applications. The planar SRR structure can be considere d as a magnetic dipole
excited by the magnetic field (H-Field) along the ring axis a s shown in Figure
2.12(a). Figure 2.12(c) shows the equivalent circuit of SRR unit cell, in which
the equivalent inductance is coupled to the external applied magnetic flux. As
the dual counterpart of SRR, CSRR shown in Figure 2.12(b) was proposed
by [78] based on the well-known complementary theory. CSRR shows the
metamaterial property of negative permittivity (ε) at resonance freq ue nc y,
and can be considered as an electric dipole excited by the electric field (E-
Field) a long the ring axis. Figure 2.12(d) shows the equivalent circuit of CSRR
unit cell, in which the equivalent LC resonator is driven by the external applied
electric field.
In the recent years, there are several works proposed for the oscillator
design with high-Q metamaterial resonators. SRR or CSRR-based oscillator
design is explored in PCB scale at 5.5∼5.8GHz [79, 80, 81]. TL-SRRs have
also been studied on PCB substrate with operating frequencies below 10 GHz
[82, 83]. A single-ended T-line loaded with SRRs (STL-SRRs) was designed
with silicon substrate for 60 GHz MMIC applications [84]. This structure with
multiple SRRs occupies a large silicon area and has weak EM coupling between
T-line and SRR load, both of which will contribute to more energy loss. A
24 GHz CMOS oscillator based on open-lo op multiple-SRR is presented as
another kind of metamaterial resonator in [85].
In the regime from millimeter-wave to THz, the challenge is how to design
Figure 2.12: Layout topologies of SRR in (a), CSRR in (b), equivalent
circuits of SRR in (c), CSRR in (d).
32 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
a low-energy-loss, strong EM coupling and high-area-efficiency metamaterial
resonator for the MMIC applications. As explored in the mm-wave r egion,
both SRR and CSRR can be applied as the load for a T-line to build compact
on-chip metamateria l-based resonators with high-Q factor. When coupling
either SRR or CSRR as load to a host T-line, a plasmonic medium with
single negative ε or µ could be formed, called electric or magnetic plasmonic
medium. A s harp band-gap or stop-band is formed in such a medium at the
resonant frequency such that the EM-wave ca n be perfectly reflected back
into the host T-line to form a stable standing-wave. Ideally, the EM-energy is
stored in the compac t SRR or CSRR str uc ture, where the energy density can
be significantly increased with a high-Q factor.
2.3.1 T-Line Loade d with Split Ring Resonator
As shown in Figure 2.13(a), T-line loaded with SRR (TL-SRR) can b e im-
plemented on chip by the topmost metal layer. By e xciting SRR with the
magnetic flux generated from the differential current in the host T-line, a
magnetic plasmonic medium is formed by TL-SRR in the vicinity of SRR
resonance frequency. SRRs are excited by the magnetic flux g enerated by the
differential current flowing in the host T-line. The equiva lent circuit of TL-
SRR unit-cell is de pic ted in Figure 2.13(b). L and C are the intrinsic series
inductance and shunt capacitance of T-line, while L
s
and C
s
are the equiva-
lent inductance and capacitance of SRR. M is the mutual inductance between
SRR and T-line.
Figure 2 .13: (a) Standing-wave formed by perfect reflection at DTL-
SRR; (b) equivalent circuit of DTL-SRR with condition to form per-
fect reflection.
CMOS Metamaterial Devices 33
The metamaterial property for TL-SRR can be analyzed by T -line model
[86]. Recall that T-line is usually modeled by distributed series impedance
(Z) and shunt admittance (Y ) with determined ε and µ, respectively. It can
be shown that ε > 0 and µ < 0 co nditio n can be satisfied by T L -SRR in the
frequency range
s
1
L
′
s
C
′
s
< ω <
s
L + L
′
s
LL
′
s
C
′
s
(2.20)
where L
′
s
= C
s
M
2
ω
2
and C
′
s
= L
s
/(M
2
ω
2
) are the equivalent series induc-
tance and capacitance of SRR. Note tha t M needs to be sufficiently high for
a negative µ. As such, differential host T-line is deployed in the design with
SRRs placed in b etween as clos e as possible.
2.3.2 T-Line Loade d with Complementary Split Ring
Resonator
What is more, as shown in Figure 2.14(a), T-line loaded with CSRR (TL-
CSRR) can be implemented on chip by eng raving CSRRs on the T-line with
the use of the topmos t meta l layer. By exciting CSRR with the E-Field in
the ho st T-line, an electric plasmonic medium is for med by T L -CSRR in the
vicinity of CSRR resonance frequency. The equivalent circuit of TL-CSRR
unit-cell is depicted in Figure 2.14 (b). L
C
and C
C
are the equivalent induc-
tance and capa citance of the CSRR resonator. By comparing the equivalent
circuit of TL-CSRR unit-cell with the T-line unit-cell, it can be observed that
Figure 2 .14: (a) Standing-wave formed by perfect reflection at DTL-
CSRR; (b) equivalent circuit of DTL-CSRR with condition to form
perfect reflection.
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