## Abstract

Dielectric and ohmic losses in metamaterials are known to limit their practical use. In this paper, an all-electronic approach for loss compensation in metamaterials is presented. Each unit cell of the meta-material is embedded with a cross-coupled transistor pair based negative differential resistance circuit to cancel these losses. Design, simulation and experimental results for Split Ring Resonator (SRR) metamaterials with and without loss compensation are presented. Results indicate that the quality factor (Q) of the SRR improves by over 400% at 1.6GHz, showing the effectiveness of the approach. The proposed technique is scalable over a broad frequency range and is limited only by the maximum operating frequency of transistors, which is reaching terahertz in today’s semiconductor technologies.

© 2012 Optical Society of America

## 1. Introduction

Metamaterials derive their exotic electromagnetic (EM) properties from their engineered structure made of metal and dielectric material instead of constituent properties of atoms and molecules [1–3]. However, metamaterials suffer from metallic and dielectric losses that eventually limit their overall performance. This issue is exacerbated in 3D metamaterials where there is more interaction between the electromagnetic waves and the bulk metamaterial [4]. Loss mitigation approaches are needed for metamaterial devices to be practical for real applications. One approach for loss compensation is to introduce a gain medium into metamaterials [4, 5]. However, it was noted in [6] that this optical pumping based method for loss compensation may be highly impractical and unscalable. Instead, an electrical approach for loss compensation that works by injecting electrical current from a semiconductor into a chunk of metamaterial may be more suitable [6]. One example of an electronic method for loss compensation, recently shown for Split Ring Resonators (SRRs), is to insert additional smaller SRRs with an amplifier signal chain to sense, amplify and inject amplified signal into the metamaterial to compensate for loss [7, 8]. However, additional antenna elements affect the original metamaterial response and add design complexity especially at shorter wavelengths with reduced dimensions. Another Field-Effect Transistor (FET) based solution was proposed for a single ring resonator at 100MHz [9] using a single-ended circuit network with complex biasing and tuning circuit, with mediocre improvement in reflection coefficient |*S*_{11}| of just −1.55dB. An operational amplifier based solution with no experimental results was also proposed in [10]; this approach has been used for broadening metamaterial response [11, 12] and not for loss compensation. Another method for loss compensation utilized negative resistance region in non-linear diodes [13] without practical implementation. The paper [13] also acknowledged that ”such diodes bring with them the risk of instability arising from parasitic oscillations”. In this paper, we present a practical realization of a loss compensation circuitry using cross-coupled transistor pair. The circuit is fully symmetrical and provides a tunable bias control to adjust impedance and guarantee stability.

## 2. Negative differential resistance circuit topology

In this paper, we present an active-transistor embedding approach for loss compensation. The embedded circuitry provides a Negative Differential Resistance (NDR) to compensate for the loss in each unit cell. The concept of NDR is explained as follows. By definition, NDR circuit exhibits a negative value for differential impedance
$\frac{\mathrm{\Delta}V}{\mathrm{\Delta}I}$, while keeping a positive value for impedance
$\frac{V}{I}$, where *V* and Δ*V* are the voltage phasor and its variation across the network, and *I* and Δ*I* represent the corresponding current phasor and its variation flowing into the NDR network respectively (|Δ*V*| < |*V*|, |Δ*I*| < |*I*|). Δ*V* and Δ*I* represent the voltage and current induced by the incident EM radiation and
$\mathrm{\Delta}I=\frac{\mathrm{\Delta}V}{R}$ where resistance *R* models the undesirable ohmic and dielectric loss in the equivalent RLC representation of the metamaterial as shown in Fig. 1(a). At resonant frequency of the metamaterial,
$\omega =\frac{1}{\sqrt{LC}}$, the strength of the resonance is affected by its loss *R*. This also affects the (ε,μ) parameters of effective medium. Connecting a NDR circuitry in parallel with the metamaterial RLC circuit will inject a current
$\mathrm{\Delta}I=-\left|\frac{\mathrm{\Delta}V}{R}\right|$ that essentially compensates for current through the lossy resistor *R*. Thus, NDR circuit provides an effective NDR of value −*R* at the resonant frequency. While this concept of NDR has been applied for the first time in metamaterials, it has been widely used in circuits community. For example, oscillations in Voltage Controlled Oscillators (VCO) are prevented from decaying out using NDR circuits [14, 15]. NDR circuits can be implemented using very few active and passive components that can all be potentially integrated into a monolithic chip with size negligible compared to the metamaterial unit cell and and with low power dissipation. Aggressive scaling of the semiconductor technology to nanometer dimensions where transistors are reaching maximum oscillation frequencies *f _{max}* and transit-time frequencies

*f*of terahertz values [16] allows for such a prospect. A terahertz modulator based on embedding of GaAs High Electron Mobility Transistors (HEMT) in metamaterial was recently demonstrated [17].

_{T}In this paper, we demonstrate this approach of loss compensation for a class of planar meta-materials based on single-split SRR. The SRR structure couples to both electric and magnetic field [18] and have been a very popular choice with practical demonstrations of negative permeability [19]. Their performance is limited by the ohmic and dielectric losses.

For this work, we implemented SRR with dimensions shown in Fig. 1(b). The metallic split ring is made of 1.4 mil thick copper (1 mil=0.00254 cm). The dielectric substrate is a 62 mil FR4 substrate. The real part of its permittivity is 4.8 and the loss tangent is 0.027 at 2GHz. The equivalent circuit model of SRR can be characterized at the gap of the SRR by measuring the reflection coefficient. The SRR was simulated using Microwave Studio by CST. For experimental verification, SRR was excited by the loop antenna [20], which is discussed later.

The implementation of the NDR circuit and its connection to the SRR is shown in Fig. 1(b). It consists of just 3 transistors, 2 resistors and 2 inductors. In principle, this circuit is a cross-coupled transistor pair consisting of Q1 and Q2, Lc is a high Q inductor and Rc is a resistor to bias the transistors Q1 and Q2. Q3 is a current sink whose current can be tuned by its gate voltage *V _{bias}*. In our implementation, Q1, Q2 and Q3 were pseudomorphic HEMT (pHEMT) and could also have been Metal Oxide Semiconductor Field Effect Transistors (MOSFET) or Bipolar Junction Transistors (BJT). The active NDR circuit can be regarded as a one-port network looking into the drains of transistor Q1 and Q2 and connected across the gap of the SRR. When appropriately biased, the circuitry imparts NDR to small signal variations across the gap, equal and opposite in polarity to the loss

*R*. The negative impedance imparted by this cross-coupled NDR circuitry can be approximated by $-\frac{2}{{g}_{m}}$, where

*g*is the small-signal transconductance of Q1 or Q2 and is related to their aspect ratio and bias current [21]. We can tune the NDR by changing the aspect ratio of transistors during the design phase or by tuning the bias voltage of current sink Q3. The active NDR circuit also provides a reactive component for the SRR, whose value can be tuned by changing Lc in Fig. 1(b). By running an EM-circuit co-simulation in Agilent ADS design environment and microwave CST studio environment, we obtain the reflection |

_{m}*S*

_{11}| of SRR as shown in Fig. 2(a), which is −2dB for the SRR without active circuit at the resonant frequency of 1.68GHz and −18dB for the SRR with active circuit at the slightly shifted resonant frequency of 1.7GHz. The slight frequency shift is due to process variations and parasitics of Q1 and Q2, which can be incorporated by pre-embedding the parasitics during the design phase. This issue may be exacerbated at higher frequencies where more than one iteration may be needed to correct for parasitics.

One of the important design considerations is the stability of the closed loop system formed by the SRR metamaterial and the active NDR circuit. Based on the well-known Barkhausen’s stability criteria for oscillations in linear electronic circuits, the proposed network with SRR and NDR circuit will not oscillate, i.e. it is stable, as long as the net resistance across the SRR remains positive. However, since we also expect the net resistance to be close to zero for perfect loss compensation, one needs to provide enough margin to account for component tolerances and temperature variations. The proposed circuit provides a convenient mechanism to adjust the negative resistance across the SRR given by
$-\frac{2}{{g}_{m}}$ and also the margin for stability, by tuning the current *I* with the cross coupled transistor pair, through *V _{bias}* control of Q3 in Fig. 1(b); the

*g*is a function of operating point of a transistor (

_{m}*I*,

*V*), where

_{g}*I*is the current and

*V*is the gate voltage [21].

_{g}## 3. Results and discussion

Next we provide measurement results for loss compensation. For the SRR shown in Fig. 1(b), the resonance frequency is measured around 1.7 GHz. The circuit topology of the loss compensation circuit is shown in Fig. 1(b). Q1, Q2 and Q3 are implemented using Avago ATF-54143 low noise enhancement mode pHEMT transistors in a surface mount plastic package. Lc is a high-Q 1nH surface mount inductors. Rc is a surface mount 100Ω resistor. We intentionally added more loss by introducing 2 surface mount 1Ω resistors in series at two ends of the SRR.

Excitation was provided using a loop antenna as proposed in [20]. We made the loop antenna with tinned copper wire, slightly larger than the SRR and placed at a distance of 2 mm over the SRR. Measurements were taken with vector network analyzer Agilent HP8510C. The loop antenna has a self-resonant frequency at 3.9 GHz, much higher than the expected metamaterial resonance. The SRR with loss compensation circuitry is mounted underneath the loop antenna and an expected resonance at around 1.7GHz in |*S*_{11}| is measured. The bias voltage *V _{bias}* of the current sink Q3 of the active NDR circuit is set to two values, which are 0V (no loss compensation) and 0.350V ( maximum loss compensation). The current in the active circuit is measured to be just 3.6mA when

*V*is set to 0.350V. The measurement setup is shown in Fig. 2(b). The measurement results in Fig. 3(a) show that resonance strength has been improved from −8dB to −32dB, which also agrees well with simulation in Fig. 2(a). The resonant frequency was shifted slightly because of the parasitics of the active and passive components. Experiments confirm that the network becomes unstable if

_{bias}*V*is increased further; increasing

_{bias}*g*increases the negative conductance and causes self-oscillations. One could easily tune

_{m}*V*real-time to ensure stability.

_{bias}We also retrieved the effective permeability of the SRR simulated with plain wave excitation in CST using the electromagnetic parameter retrieval method proposed in [22]. The Eq. used to retrieve the effective permeability [22] is

*d*is the thickness of the slab (18 mm),

*k*is the wave number of the incident wave,

*S*

_{11}is the reflection coefficient and

*S*

_{21}is related to the transmission coefficient

*T*by

*S*

_{21}=

*T*exp(

*ikd*) [23]. The extracted permeability of original SRR and SRR embedded with loss compensation circuit are shown in Fig. 3(b). We used the Lorentz oscillator model, which are shown in dashed lines in Fig. 3(b), to fit the curve of extracted μ. The Eq. of Lorentz oscillator model is $\mu =1+\frac{{\omega}_{p}^{2}}{{\omega}_{0}^{2}-{\omega}^{2}-i\gamma \omega}$, in which ω

*is the plasma frequency, ω*

_{p}_{0}is the resonance frequency and γ is the damping parameter. The quality factor

*Q*is defined as $Q=\frac{{\omega}_{0}}{\gamma}$. We calculated the

*Q*of the SRR based on the Lorentz model. The

*Q*of the SRR is increased by 401.3% when the loss compensation circuit is embedded into SRR. Metamaterial arrays designed using this unit cell with NDR circuitry will also exhibit loss compensated response. We have simulated the effect of neighbor interactions and found that results are unaffected [24].

## 4. Conclusion

In conclusion, this paper provides a method of loss compensation in metamaterials through embedding of active transistor based NDR circuit directly into each metamaterial unit cell. This approach is valid for any metamaterial construct that are based on patterned metallic inclusions in the dielectric medium. Because of the negligible size of the embedded circuit, it does not interfere the EM response of the metamaterial. It was also shown that the NDR value can be tuned off-chip and this will allow one to guarantee stability of the circuitry and also accommodate for any process and environmental variations. The resonance frequency of metamaterial can also be tuned through additional on chip varactors in active NDR circuit. As the semiconductor technology scales to nanometer dimensions, proposed electronic method for loss compensation can be utilized in metamaterials from RF to terahertz frequencies. Since continued transistor scaling also comes with reduced power consumption, the overall power consumption for loss compensation circuit will be amenable for practical implementations. This approach can be combined with current approaches based on gain medium that have only been applied at infrared and visible frequencies to achieve loss compensation in metamaterials over a broad frequency range. The proposed loss compensation can be used in bulk metamaterials to make ideal absorbers, cloaks, perfect lens, modulators, phase shifters, etc. with applications in radar imaging, cellular mobile communications, satellite communications, antennas and security screening.

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