EPJ Appl. Metamat. 5, 9 (2018) © S. Hrabar, published by EDP Sciences, 2018 https://doi.org/10.1051/epjam/2018005 Available online at: Metamaterials’2017 epjam.edp-open.org Metamaterials and Novel Wave Phenomena: Theory, Design and Applications REVIEW First ten years of active metamaterial structures with “negative” elements Silvio Hrabar* University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, Zagreb, HR-10000, Croatia Received: 18 September 2017 / Accepted: 27 June 2018 Abstract. Almost ten years have passed since the first experimental attempts of enhancing functionality of radiofrequency metamaterials by embedding active circuits that mimic behavior of hypothetical negative capacitors, negative inductors and negative resistors. While negative capacitors and negative inductors can compensate for dispersive behavior of ordinary passive metamaterials and provide wide operational bandwidth, negative resistors can compensate for inherent losses. Here, the evolution of aforementioned research field, together with the most important theoretical and experimental results, is reviewed. In addition, some very recent efforts that go beyond idealistic impedance negation and make use of inherent non-ideality, instability, and non-linearity of realistic devices are highlighted. Finally, a very fundamental, but still unsolved issue of common theoretical framework that connects causality, stability, and non-linearity of networks with negative elements is stressed. Keywords: Active metamaterial / Non-Foster / Causality / Stability / Non-linearity 1 Introduction dispersion becomes significant. The same behavior applies for metamaterials, in which above effects occur to the All materials (except vacuum) are dispersive [1,2] due to resonant energy redistribution in some kind of an inevitable resonant behavior of electric/magnetic polari- electromagnetic structure [2]. Thus, every passive material zation. The simplest physical picture deals with mechanical (or metamaterial) with negative or less-than-unity effec- inertia of polarized particles, caused by their finite mass. tive parameters is inevitably narrowband. This fact is Due to inertia, the particles cannot respond instantaneous- mathematically expressed in the form of well-known ly to an external electromagnetic field. Thus, there will be a energy-dispersion constraints [1,2]: phase lag/lead between particle movement and applied ∂½eðvÞ ∂½mvðÞ oscillating field. Clearly, this effect causes a change of > 0; > 0: ð1Þ relative permittivity and permeability with frequency (i.e. ∂v ∂v the dispersion). An electrical engineering model of this Very similar equations apply for any reactive element in phenomenon comprises an RLC resonant circuit (Lorentz circuit-theory (Foster’s reactance theorem) [3,4]: dispersion model with “parallel” (fp) and “series” (fs) resonant frequency [1,2], Fig. 1). ∂½XðvÞ ∂½BðÞv > 0; > 0: ð2Þ Here, the resonant energy re-distribution between ∂v ∂v electric and magnetic fields describes the mechanical inertia. Inspection of the Lorentz curve (Fig. 1) shows Here e and m stand for permittivity and permeability that the dispersion is negligible if a material is used far while X and B are reactance and susceptance, respectively, below its parallel resonant frequency (fp). Actually, this is symbol v represents angular frequency. the case of most of ordinary dielectrics in radiofrequency Obviously, ordinary reactive circuit elements (capaci- and microwave regime. tors and inductors) obey Foster’s theorem. However, one However, if one wants to make use of phase lead effect might (hypothetically) define new “negative” (or “non- (i.e. of the occurrence of negative or less-than-unity Foster”) elements: negative capacitor (Cn < 0) and negative effective constitutive parameters: EpsilonNeGative, inductor (Ln < 0) (Fig. 2). MuNeGative, EpsilonNearZero, and MuNearZero), the Due to negative capacitance/inductance, the terminal current of these elements is negative (it flows outward from * e-mail: [email protected] the positive terminal). Thus, a negative capacitor and a This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 S. Hrabar: EPJ Appl. Metamat. 5, 9 (2018) Fig. 1. Lorentz dispersion model. Solid: real part of relative Fig. 2. Solid: reactance of positive (Foster) reactive elements, permittivity/permeability, dashed: imaginary part of relative dashed: reactance of negative (non-Foster) reactive elements [20]. permittivity/permeability. Fig. 3. Linear dispersion-free models of ideal negative elements. (a) Negative capacitor, (b) negative inductor, and (c) negative resistor. positive feedback is needed in other to assure impedance sign flipping in the case of NICs (Zin ∼ ÀZload) or impedance sign flipping accompanied with the impedance inversion in the case of NIIs (Zin ∼ À1/Zload), Zin and Zload being the input and load impedance, respectively. In order to get acquainted with the aforementioned principle, it is instructive to briefly explain the simplest voltage-conversion NIC (Fig. 4). The (grounded) load impedance that should be inverted (Z)is shown in the left part of Figure 4. The load can be a lumped element of any type (a capacitor, an inductor, or a resistor). In order to invert the load impedance, one swaps the sign of Fig. 4. Simplified explanation of the generation of negative the input current (I). This is done by inserting an additional impedance by a voltage-converting NIC [2]. “dependent” voltage source of amplitude 2V in series with impedance Z (the middle part of Fig. 4). This dependent source is actually a voltage amplifier with gain equal to +2 negative inductor behave as (reactive) sources, i.e. they are (the right part of Fig. 4). The voltage drop across the inevitably active devices [2]. impedance is now equal to ÀV (the left electrode has It is important to notice that energy-dispersion potential of V and the right one has potential of 2 V). This constraints (and Foster’s theorem) have been derived causes the inversion of the impedance Zin (as seen from the assuming that there are no losses present [1,3–5]. So, input terminals): strictly speaking, hypothetical negative resistor (an active V V V element that delivers real power [4]) should not be termed Z ¼ ¼ ¼ ¼Z: ð3Þ “ ” in I V À2V V as a non-Foster element . However, one might say that a Z À Z negative capacitor, a negative inductor, together with a negative resistor form a complete group of (hypothetical) A more general equation that allows use of complex gain “negative circuit elements”. These ideal elements are function of realistic active element can be found in postulated to be linear and dispersionless (Fig. 3). [2,6,7,10]. In practice, “negative” elements are mimicked by There are also some electronic elements (such as tunnel appropriate electronic circuitry, so called “Negative Imped- diodes and Gunn diodes [4]) that show negative resistance ance Converters”,[6,7] and “Negative Impedance Inverters”, effects (but not negative capacitance or negative induc- [8,9], proposed more than 60 years ago. In essence, these are tance!) per se. They again contain some kind of positive specially designed amplifiers with positive feedback. A feedback, which occurs at a microscopic level due to S. Hrabar: EPJ Appl. Metamat. 5, 9 (2018) 3 Fig. 5. Transmission-line models of different metamaterials (a) passive metamaterial, (b) active non-Foster-based metamaterial, and (c) active gain-based metamaterial. background semiconductor physics. Traditionally, these by the bandwidth of the realized non-Foster element itself. elements have been used for construction of amplifiers and Thus, the inherent energy-dispersion constraints (1) would oscillators in the microwave community [4]. be bypassed. The first potential application of negative capacitors Theoretical analysis from [15] initiated a new way of and inductors (that has been investigating for many thinking and encouraged researchers all around the world years and it is still active nowadays) is broadband to try to bring this interesting idea to reality. The field of matching of small antennas [10]. Briefly, this matching active “negative” metamaterials has been growing rapidly. relies on the compensation of frequency dispersion of an Nowadays it goes beyond broadband (almost dispersion- ordinary reactive network with the “inverse” dispersion of less) operation, and includes power-flow reversal, non- a “negative” non-Foster network. This compensation linear, and oscillatory behavior. This paper points out some yields (theoretically) infinite bandwidth. Apart from of the most important (in Author’s personal opinion) matching, several (mostly theoretical) studies, that theoretical and experimental results obtained so far. On the investigated the use of active elements in periodic other hand, it also tries to stress the “bottlenecks” of this structures for various microwave applications such as technology, related to some very fundamental but still manipulation of bianisotropy [11]andemulationof unclear issues. perfectly matched layers (ideal absorbers) [12,13], were published in the past. Particularly interesting is an idea of “artificial atoms” 2 Evolvement of the research field and that comprise circuits based on OPerational AMPlifiers current state of the art (even including NIC circuits) [14]. Those “atoms” were proposed for construction