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Time-varying transistor-based metamaterial for tunability, mixing, and efficient phaseconjugationAlexander R. Katko, John P. Barrett, and Steven A. Cummer
Citation: Journal of Applied Physics 115, 144501 (2014); doi: 10.1063/1.4871195 View online: http://dx.doi.org/10.1063/1.4871195 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Tunable microwave impedance matching to a high impedance source using a Josephson metamaterial Appl. Phys. Lett. 103, 212601 (2013); 10.1063/1.4832074 Ultra-broadband electromagnetically induced transparency using tunable self-asymmetric planar metamaterials J. Appl. Phys. 114, 163507 (2013); 10.1063/1.4826630 Cryogenic single-shot spectroscopy of a floating poly-silicon gate transistor J. Appl. Phys. 112, 014510 (2012); 10.1063/1.4733944 Induced strain mechanism of current collapse in AlGaN/GaN heterostructure field-effect transistors Appl. Phys. Lett. 79, 2651 (2001); 10.1063/1.1412282 Depletion mode GaAs metal–oxide–semiconductor field effect transistors with Ga 2 O 3 (Gd 2 O 3 ) as the gateoxide J. Vac. Sci. Technol. B 16, 1398 (1998); 10.1116/1.590083
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Time-varying transistor-based metamaterial for tunability, mixing,and efficient phase conjugation
Alexander R. Katko, John P. Barrett, and Steven A. CummerDepartment of Electrical and Computer Engineering, Duke University, 101 Science Drive, Durham,North Carolina 27708-9976, USA
(Received 17 January 2014; accepted 1 April 2014; published online 8 April 2014)
We present a transistor-based microwave metamaterial exhibiting tunability over a wide range of
time scales. By loading a metamaterial with a transistor, we show through theory and simulation
that both the resonant frequency and quality factor of the metamaterial can be dynamically tuned
with a voltage bias. We demonstrate through experiment that such a time-varying transistor-based
metamaterial exhibits this tunability. The tunability is applicable to a wide range of time scales,
from quasi-static effective parameter tuning to parametric pumping for mixing and phase
conjugation. We then apply the metamaterial to a particular application of phase conjugation and
demonstrate through simulation and experiment that a very strong phase conjugated signal is produced.
We experimentally show that the mixing efficiency for a transistor metamaterial is over 30 dB stronger
than that of a varactor-based phase conjugate metamaterial. VC 2014 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4871195]
I. INTRODUCTION
Metamaterials are structures whose bulk properties are
different from the properties of their constituent materials.
Electromagnetic metamaterials have been demonstrated to
yield useful and unusual properties such as a negative index
of refraction,1 enabling the design of structures including an
invisibility cloak2 and superlens.3 While early work focused
on linear and passive metamaterials, recently nonlinear and
active metamaterials have received increasing attention. This
has resulted in nonlinear and active metamaterials based on
embedded devices including varactor diodes,4–7 Schottky
diodes,8 PIN diodes,9 MEMS devices,10 and other methods
of including nonlinearity or active control such as quenching
superconductivity11 or a phase change.12,13 Interesting and
useful effects relying on nonlinearity have also been demon-
strated using metamaterials, including optical bistability,14
optical parametric amplification,15 spatiotemporal solitons,16
tunable Fano resonance,17 and harmonic generation.18–21
Two-terminal devices (such as the aforementioned
diodes and MEMS switches) allow a designer to embed non-
linearity or active (time varying) control within a metamate-
rial unit cell using a simple, easy-to-analyze device. More
degrees of freedom for possible metamaterial designs can be
obtained by considering more complex devices such as tran-
sistors. To date, most transistor-based metamaterial work has
focused on embedding useful circuits into a unit cell, particu-
larly for loss compensation/gain22,23 or non-Foster active
circuits.24 While these types of circuit provide very useful
functionality, design of an effective metamaterial can be
very difficult due to stability concerns. Non-Foster circuits
and gain circuits include effective negative resistors, capaci-
tors, or inductors: these are typically difficult to stabilize for
a range of frequencies or loads.24
In this work, however, we focus on a different and sim-
pler class of transistor-based metamaterials. By focusing on
tunability with a single transistor, we avoid the stability
issues that are typical of gain or non-Foster circuits. We
demonstrate that a simple N-channel MOSFET (NFET) can
be used to dynamically tune the resonant frequency and qual-
ity factor of a metamaterial using a bias voltage. This applies
on a variety of time scales. By using a DC bias, we demon-
strate quasi-static tuning of a metamaterial’s linear proper-
ties. Using a RF bias, on the other hand, allows us to realize
a time-varying mixing metamaterial. We demonstrate how
this time-variance in a metamaterial can be exploited for
phase conjugation (PC). Appropriate selection of a transistor
also allows the mixing efficiency for a process such as phase
conjugation to be significantly larger than that of a varactor-
based metamaterial.
II. DESIGN AND SIMULATION
We begin design of the time-varying transistor metama-
terial (TVTM) by specifying the desired operation of the
transistor. The transistor will be biased such that across the
drain and source terminals it resembles a tunable resistance
RDS. We bias the transistor in its linear range: in this range,
RDS is dependent on the gate-source voltage VGS. The tran-
sistor also presents a non-negligible parasitic capacitance
CDS,eff, so the transistor will act as a tunable resistance in
parallel with a constant capacitance. This will be embedded
within a split-ring resonator (SRR), which can be modeled as
a resonant LC circuit, as shown in Fig. 1(a). A photograph of
a fabricated TVTM is shown in Fig. 1(b). The metamaterial
was designed to resonate near 750 MHz when including the
transistor capacitances.
This inclusion allows us to dynamically tune two essen-
tial parameters for a metamaterial: its resonant frequency f0and the quality of the resonance (the quality factor Q).
Tuning the resonant frequency allows active control of the
metamaterial (e.g., Ref. 5), while tuning the Q determines
the bandwidth of the resonance. This tunability is not a non-
linear effect. Unlike most of the tunable metamaterial
0021-8979/2014/115(14)/144501/6/$30.00 VC 2014 AIP Publishing LLC115, 144501-1
JOURNAL OF APPLIED PHYSICS 115, 144501 (2014)
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literature (e.g., Ref. 5), the tuning element in this work is
operated in the linear regime and nonlinear effects are negli-
gible. Thus, all the tunable parameters and effects we
observe are due to time variance rather than nonlinearity.
There are multiple ways to bias a transistor such that it
presents the desired characteristics. The simplest way is to
apply a bias VGS to the gate of an enhancement-mode N-type
MOSFET transistor operating in the linear regime. VGS mod-
ulates the number of carriers in the conducting channel in
the transistor, thus modulating the resistance.25 Below the
threshold voltage VT the channel does not conduct well,
yielding a large RDS. As VGS is increased to and above VT,
the channel becomes more and more conducting, decreasing
RDS. Thus, by modulating VGS above and below VT, we mod-
ulate RDS. This method also requires very little DC current
from the bias network or supply, allowing low-power RDS
modulation. Using a depletion-mode transistor, such as a
JFET, still yields a tunable RDS. However, in a depletion-
mode transistor, RDS increases as jVGSj increases. The same
tunability can be achieved, allowing us to select appropriate
transistors based on other properties such as the parasitic
capacitances or VT.
For this work, we select an NXP BSS83 N-channel
MOSFET. This transistor has a VT of less than 2.0 V and
relatively small capacitances on the order of � 1 pF. We
measured the current and voltage across the drain and source
terminals, IDS and VDS, as a function of VGS to find the
obtainable RDS values for this particular transistor. As shown
in Fig. 2, RDS ranges from �4000 X below VT to �20 X at
VGS¼ 10 V.
Modeling a SRR as a resonant circuit, as in Fig. 1(b), we
calculate the analytical impedance as a function of frequency
and RDS. We allow RDS to vary over the measured range of
RDS. The normalized analytical impedance is shown in Fig. 3.
When RDS is large, the resonant frequency is determined
by the series combination of CSRR and CDS,eff and the quality
factor Q is determined primarily by RSRR. When RDS is small,
CDS,eff is effectively shorted and the resonant frequency is deter-
mined by CSRR, while the Q is determined by RSRR þ RDS.
To validate this design, we simulate the inclusion of the
selected transistor in a lumped-element SRR model using
LTSpice. We sweep the DC bias of VGS and obtain the effec-
tive impedance of the SRR. A sample of the simulated data
is shown in Fig. 4. We also compute the resonant frequency
and Q as a function of VGS. These are plotted in Figs. 5(a)
and 5(b).
From Fig. 4 in comparison with Fig. 3, we can clearly
see that varying VGS yields a resonant character similar to
(a) (b)
SRR
Equivalent
FET
Equivalent
FET
40 mm
1 mm
E
Hk
FIG. 1. (a) Equivalent circuit for the TVTM with the transistor at its operat-
ing point. (b) Photograph of fabricated TVTM element. The polarization is
indicated.
0 2 4 6 8 1010
1
102
103
104
VGS
[V]
RD
S [Ω
]
FIG. 2. Measured RDS values for the selected transistor in this work as a
function of VGS.
650−25
−20
−15
−10
−5
0
Frequency [MHz]
Ana
lytic
al |Z
| [dB
]
4000 Ω1064 Ω283 Ω75 Ω20 Ω
700 750 800 850
FIG. 3. Normalized analytical impedance of circuit in Fig. 1(b). As RDS
decreases, the resonant frequency and quality factor Q of the circuit are
modulated.
Frequency [MHz]
Sim
ulat
ed |Z
| [dB
]
VGS=10VVGS=5VVGS=3.04VVGS=2.37VVGS=1.7V
−40
−30
−20
−10
650 700 750 800 850
FIG. 4. Spice simulation data for effective impedance of the lumped-
element equivalent TVTM. The f0 and Q shifts are similar to those of the an-
alytical impedance
144501-2 Katko, Barrett, and Cummer J. Appl. Phys. 115, 144501 (2014)
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152.3.216.26 On: Thu, 24 Apr 2014 15:39:38
that of the lumped-element equivalent model. Figure 5
demonstrates that there are several VGS ranges of interest.
The Spice model used for this simulation does not accurately
simulate sub-threshold voltage effects, so for all VGS<VT
¼ 1.7 V the behavior is static. For VGS>VT, both f0 and Qdecrease sharply. Biasing the transistor in this range thus
results in large tunability. If we are interested in mixing
using time variance, for instance, setting the DC voltage
level of VGS near 1.75 V will result in a mixed signal with a
relatively small amplitude for the AC component VGS,AC
because the SRR effective impedance is time-varying. We
demonstrated7 that a PC signal can be generated with a reso-
nant metamaterial by parametrically varying the resonant
frequency. The TVTM allows us to parametrically vary f0 by
varying VGS above VT with a pump signal. For this particular
TVTM, we select the pump frequency fpump¼ 1.275 GHz
with a source signal at fsource¼ 625 MHz in order to generate
a mixed PC signal at fPC¼ 650 MHz. Using Agilent ADS,
we verify that the linear, time-varying mixing products are
generated by sinusoidally varying VGS. The simulated cur-
rent induced in the TVTM is shown in Fig. 6.
As shown in Fig. 6, the desired difference frequency sig-
nal is generated when VGS is sinusoidally varied. We note
that, as the TVTM is operating in the linear regime of the
transistor (with a low VT), the typical nonlinear mixing prod-
ucts at f¼ nf1 þ mf2 for n, m¼ all integers can be neglected.
Thus, the only strong mixed signals are at the sum and
difference frequencies, as expected for a linear, time-varying
material.
As we showed previously,7 the signal at the difference
frequency should be a PC signal. Moreover, by selecting an
appropriate transistor, we can also produce mixed signals
with very high mixing efficiency. JFETs typically have very
small parasitic capacitances, potentially significantly smaller
than a varactor diode such as the one we used in Ref. 7.
They can also realize very large channel conductances and
thus very small RDS. If we are interested in maximizing the
mixing efficiency at a particular mixing product, we can op-
erate the transistor in the nonlinear regime as a nonlinear
time-varying transistor metamaterial (NTVTM). This is eas-
ily accomplished by selecting an appropriate depletion-mode
JFET and allows us to produce very strong mixing products.
The Manley-Rowe equations26 suggest that a purely reactive
nonlinearity is preferable for maximizing mixing efficiency,
but a number of additional considerations lead to a JFET pro-
viding superior performance. First, the Manley-Rowe limits
on mixing efficiency are upper limits. The limit of 100%
conversion efficiency for a reactive nonlinearity and 25% for
a second-order product due to a resistive nonlinearity can,
practically, be very difficult to approach. Moreover, the lim-
its are more restrictive for higher-order products, which we
do not examine here. Second, the Manley-Rowe equations
apply to strictly pure reactive or pure resistive nonlinear ele-
ments. The varactor diode and JFET each possess both non-
linear reactances and resistances: for instance, the varactor is
pumped into forward conduction during part of the RF cycle.
Third, the conversion efficiency depends on the signal power
levels and frequencies: for very high power levels and high
frequencies, the varactor mixing efficiency will likely exceed
that of the NTVTM. At lower power levels, JFET-based
mixers are commonly used in RF applications while
varactor-based mixers require higher power for effective
operation. For these reasons, the strict use of the Manley-
Rowe equations is not applicable for predicting the mixing
performance of the varactor- and JFET-based metamaterials.
We use a CEL NE3509M04 HJ-FET to demonstrate that
very high mixing efficiencies can be realized in a phase con-
jugation metamaterial. The design of the JFET-based
NTVTM is very similar to the MOSFET-based TVTM, so
we only examine the JFET NTVTM for demonstrating very
high mixing efficiency when operated in the nonlinear re-
gime. We proceed to demonstrate the design features of the
TVTM experimentally in the following section.
III. EXPERIMENT
We fabricate the TVTM and test it at microwave fre-
quencies. We use a vector network analyzer to characterize
the resonance of the TVTM when excited by a plane wave,
focusing on the quasi-static tuning regime. The TVTM is
placed in a TEM waveguide to excite its magnetic resonance.
We measure the S-parameters of the TVTM as a function of
VGS, with the results shown in Fig. 7.
The data in Fig. 7 are in good qualitative agreement
with those in Fig. 4. As mentioned previously, the Spice
model used is inaccurate at sub-VT levels of VGS. Similar to
700
f 0 [MH
z]
110
20
30
40
VGS [V]
Q F
acto
r
(a)
(b)
710
720
730
2 3 4 5 6 7 8 9 10
FIG. 5. Spice simulation data for lumped-element equivalent of TVTM. (a)
simulated f0 as a function of VGS and (b) simulated Q as a function of VGS.
550 600 650 700 750−100
0
Frequency [MHz]
Sim
ulat
ed S
igna
l [dB
m]
Source
Signal
Difference
Signal
−80
−60
−40
−20
FIG. 6. ADS simulation data for parametrically varying VGS. The pump sig-
nal is at 1.275 GHz while the source signal is at 625 MHz, generating the
mixed signal at 650 MHz.
144501-3 Katko, Barrett, and Cummer J. Appl. Phys. 115, 144501 (2014)
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the simulation data, we plot f0 and Q as a function of VGS in
Figs. 8(a) and 8(b), respectively, and again observe good
agreement with the simulated data.
The experimental data validate the design and simula-
tion data. By biasing VGS with a DC voltage, we can quasi-
statically vary both f0 and Q for the TVTM. If we vary VGS
near 1.75 V, we obtain a large change in the resonant fre-
quency Df0 for a small change in VGS.
As described in the previous work,7 parametrically vary-
ing f0 should yield mixing products at the sum and difference
frequencies, and the difference frequency signal should be a
PC signal. Similar to our work in Ref. 7, we use a direct
cable connection to deliver a pump signal across the gate
and source terminals of the transistor. Signal generators pro-
vide both the source and pump excitations. The TVTM is
placed over a ground plane, with monopole antennas used to
provide the source signal and measure the generated spec-
trum. The generated spectrum is measured with a spectrum
analyzer. We then excite the TVTM with a pump signal.
As in the simulation, we set the pump frequency fpump to
1.275 GHz and the source frequency fsource to 625 MHz,
yielding a difference signal at fPC¼ 650 MHz. The recorded
data are shown in Fig. 9.
There is a strong difference frequency signal generated
at 650 MHz when the TVTM is present. There is a small
mixed signal when the sources are turned on but the TVTM
is not present: this is due to internal mixing in the
instrumentation. The data verify that the TVTM can be used
for generating a strong mixed signal. We proceed to verify
that the difference frequency signal is in fact a PC signal
using an interferometric process, as detailed in our previous
work.7 We map an interference pattern between two ele-
ments. First, we use two normal, non-PC antennas excited in
phase and then �908 out of phase. Then, with the same
setup, we use two TVTM elements excited in phase and then
�90� out of phase. We expect that the in-phase interference
patterns should be similar. If the TVTM elements are in fact
PC elements, we expect that the interference pattern when
the TVTM elements are �90� out of phase should be similar
to two normal elements being excited þ90� out of phase. In
other words, we expect the TVTM and normal element pat-
terns to be mirror images. For more details, the reader is
referred to Ref. 7. The measured interference patterns are
shown in Fig. 10.
Figure 10 demonstrates that the TVTM is in fact a PC
metamaterial (PCM). Examining the minima, in particular,
we see that the interference pattern for the out-of-phase
TVTM is symmetric with the interference pattern for the
out-of-phase antenna elements. This shows that a �90�
phase shift in the TVTM is equivalent to a þ90� phase shift
650 700 750 800 850
Frequency [MHz]
S21
[dB
]
−2
−1.5
−1
−0.5
00 V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V
FIG. 7. Experimental data for TVTM showing resonance with varying VGS.
As VGS increases f0 decreases.
700
725
750
775
f 0 [MH
z]
0 2 4 6 8 100
Q F
acto
r
VGS
[V]
10
20
30
40
FIG. 8. Experimental tunability data for TVTM. (a) measured f0 as a func-
tion of VGS and (b) measured Q as a function of VGS.
550 600 650 700 750−100
−75
−50
−25
0
Re
cord
ed
Sp
ect
rum
[d
Bm
]
Frequency [MHz]
No Source
NTMNo NTM
Source
PC
Pump
Harmonic
FIG. 9. Measured spectrum analyzer output for 3 cases: No sources turned
on (ambient spectrum), solid black curve; sources turned on but the PC
TVTM not present, dashed red curve; sources turned on with the PC TVTM
present, solid blue. We see a strong difference frequency signal generated at
650 MHz, confirming that the TVTM can be used for mixing.
FIG. 10. Verification that TVTMs can be used as PC elements. The solid
curves are elements excited in phase and dashed curves are elements excited
�90� out of phase. Blue curves are non-PC antenna elements and red curves
are TVTM elements.
144501-4 Katko, Barrett, and Cummer J. Appl. Phys. 115, 144501 (2014)
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152.3.216.26 On: Thu, 24 Apr 2014 15:39:38
in a conventional antenna element. The TVTM can be used
to generate mixed signals including a phase conjugate signal.
We then fabricate a similar NTVTM using the HJ-FET
noted earlier. We wish to characterize the mixing effi-
ciency of the NTVTM for a particular mixing product. In
this case, we examine the first-order difference frequency
xPC ¼ jxpump � xsourcej which we showed above is a PC
signal. Holding the source signal at a constant amplitude,
we measure the amplitude of the PC signal as a function of
the pump power level for both a varactor-based PCM ele-
ment and a NTVTM element. The measured data are
shown in Fig. 11.
The PC mixing product using the NTVTM is �32 dB
larger than that of the varactor PCM over the measured
pump power range. We also note that for pump levels
below approximately �15 dBm, the varactor PCM is virtu-
ally indistinguishable from the noise floor of the measure-
ment equipment. The NTVTM produces an easily
detectable signal even when the varactor PCM signal is
indistinguishable from the noise floor. It is thus well-suited
for mixing applications even with low levels of input
power, providing a significant advantage over varactor-
based nonlinear metamaterials. NTVTMs can be used to
achieve very high mixing-efficiency systems through the
use of time variance and nonlinearity.
IV. CONCLUSIONS
In summary, we demonstrated that tunable time-varying
metamaterials can be made using transistors. By appropri-
ately biasing a transistor on a variety of time scales, we dem-
onstrated how a single metamaterial design can be effective
for multiple applications. With a DC bias, we demonstrated
analytically that a lumped-element model of a properly bi-
ased transistor can be used in a metamaterial to provide a
tunable resonant frequency and quality factor. We verified
this quasi-static tuning using Spice simulations and experi-
ments. We also showed that a RF biased transistor-based
metamaterial can be used for mixing and phase conjugation
in both simulation and experiment. A transistor-based time-
varying metamaterial can yield mixing efficiency signifi-
cantly higher that a similar metamaterial using a varactor
diode. The metamaterials described here demonstrate that
incorporating transistors in metamaterials allows a designer
to obtain properties that can be used for applications such as
frequency agility and high-efficiency phase conjugation.
ACKNOWLEDGMENTS
This work was supported by a Multidisciplinary University
Research Initiative from the Army Research Office (Contract
No. W911NF-09-1-0539).
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−30 −20 −10 0 10 20−100
−80
−60
−40
−20
Pump Power [dBm]
PC
Sig
nal P
ower
[dB
m]
JFET NTVTM
Varactor PCM
FIG. 11. Mixed signal strength comparison for varactor-based PCM and
JFET NTVTM. The JFET NTVTM mixing efficiency is at least 32 dB
greater than the varactor PCM over a wide, realistic range of pump power
levels.
144501-5 Katko, Barrett, and Cummer J. Appl. Phys. 115, 144501 (2014)
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