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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152.3.216.26 On: Thu, 24 Apr 2014 15:39:38

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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