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What is a good conductor for metamaterials or plasmonics

Costas M. Soukoulis
  • Corresponding author
  • Ames Laboratory–US DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
  • Institute of Electronic Structure and Lasers (IESL), FORTH, 71110 Heraklion, Crete, Greece
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Thomas Koschny
  • Ames Laboratory–US DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Philippe Tassin / Nian-Hai Shen
  • Ames Laboratory–US DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Babak Dastmalchi
  • Ames Laboratory–US DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
  • Institute of Electronic Structure and Lasers (IESL), FORTH, 71110 Heraklion, Crete, Greece
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-04-13 | DOI: https://doi.org/10.1515/nanoph-2014-0013

Abstract

We review conducting materials like metals, conducting oxides and graphene for nanophotonic applications. We emphasize that metamaterials and plasmonic systems benefit from different conducting materials. Resonant metamaterials need conductors with small resistivity, since dissipative loss in resonant metamaterials is proportional to the real part of the resistivity of the conducting medium it contains. For plasmonic systems, one must determine the propagation length at a desired level of confinement to estimate the dissipative loss.

Keywords: optics; metamaterials; plasmonics; conductors

1 Introduction

Artificial nanophotonic media – photonic crystals (PCs), negative index materials (NIMs), metamaterials (MMs), and plasmonic structures – enable the realization of novel electromagnetic properties unattainable in naturally occurring materials. There has been a truly amazing amount of innovation during the last few years [1–5] and more is yet to come. Spurred by new opportunities, scientists have produced exotic concepts that exploit these new materials: (i) we can now specify how to make a lens whose resolution is limited not by the wavelength of light, but only by our ability to build a material to the necessary specifications [6]; (ii) we can guide radiation along an arbitrary trajectory, for instance to hide an object from sight [7, 8]; and (iii) we can design and manufacture materials with magnetic response in the terahertz and optical domains. Clearly, nanophotonics can develop mold-breaking technologies for a plethora of applications, where control over light (or more generally electromagnetic radiation) is a prominent ingredient – among them telecommunications, solar energy harvesting, biological and terahertz imaging and sensing, optical isolators, nano-lasers, quantum emitters, sensors, polarizers, and medical diagnostics, to name just a few.

However, many serious obstacles must be overcome before the impressive possibilities of MMs and plasmonics, especially in the optical regime, will lead to applications. One of these obstacles is dissipative loss, which needs to be overcome to advance nanophotonic structures towards real-life applications. At this point, we already want to emphasize that a good conducting material for plasmonics is not necessarily the same as a good material for metamaterials. For resonant metamaterials, we need conductors with small resistivity, whereas for plasmonics we need to have large propagation length and high confinement for the surface plasmon polaritons [9, 10].

2 Resonant metamaterials with low loss

Metamaterials are tailored, man-made materials composed of subwavelength building blocks (“photonic atoms”), densely packed into an effective medium [1–4]. In this fashion, optical properties that simply do not exist in either naturally occurring or conventional artificial materials become reality. A particularly important example of such a photonic atom is the split-ring resonator (SRR) [11] – essentially a tiny electromagnet – which allows for artificial magnetism at elevated frequencies, enabling the formerly missing control of the magnetic component of electromagnetic light waves. Negative magnetic response (i.e., μ<0) above the SRR eigenfrequency, combined with the more usual negative electric response from metal wires (i.e., ε<0), can lead to a negative index of refraction. Following the original theoretical proposal in 1999 [11], MMs were realized at microwave frequencies in 2000 [12] and entered the optical domain (from infrared to the visible) during 2004–2011 [4, 13]. In 2007, negative-index MMs reached the red end of the visible spectrum [4, 14] (see Figure 1 in ref. [4]) and, in 2011, a NIM operating at a free-space wavelength of 660 nm was realized [15].

The field of metamaterials (MMs) has seen spectacular experimental progress in recent years [1–4]. However, large intrinsic losses in metal-based structures have become the major obstacle towards real-world applications, especially at optical wavelengths. Most MMs to date are made with metallic constituents, resulting in significant dissipative loss. These losses originate in the Joule heating caused by the large electric currents in meta-atoms and the poor conductivity of metals and other available conductors at optical frequencies. One promising way of overcoming dissipative loss is based on introducing gain materials in metamaterials [16]. Therefore, it is of vital importance to understand the mechanism of the coupling between a meta-atom and the gain medium [17, 18]. Counter-intuitively, pump-probe experiments of split-ring resonators on top of a gain substrate have revealed that the transmission of a metamaterial may be reduced when gain is added to a metamaterial [19]. Computer simulations have confirmed this effect and attributed it to the characteristic impedance mismatch created by the meta-atom-gain coupling [17]. In addition, these ideas can be used to incorporate gain to obtain new nanoplasmonic lasers [16, 20].

Whether or not a gain medium is added to a metamaterial structure, we want to have strong electric and magnetic resonances to achieve low dissipative loss. We have recently addressed the question of what makes a good conductor for MMs and we have developed a fairly general model of MMs with a single resonance in the magnetic dipole response [9]. It turns out that the fraction of power dissipated in the material can be expressed through a dimensionless figure-of-merit called the dissipation factor, ζ, which is proportional to the real part of the resistivity Re(ρ). This analysis is valid for all metamaterials with a resonant subwavelength constituent.

Since the figure-of-merit for conductors in resonant MMs comes down to the real part of the (high-frequency) resistivity, we need to identify new materials with smaller resistivity. Finding materials with smaller resistivity would have an important impact on the field of metamaterials [9]. It must be noted here that the imaginary part of the permittivity of different conductors (metals, conducting oxides) may not correctly characterize the corresponding intrinsic losses, but we should adopt the real part of resistivity [Re(ρ)] for the dissipative loss evaluation [9].

In the scope of materials whose response can be satisfactorily described by a Drude model, Re(ρ) is essentially determined by γ/ωp2, where γ is the collision frequency and ωp=2πfp represents the plasma frequency. This establishes a good figure-of-merit for conducting materials (noble metals (Ag, Au, Cu), Al, alkali-noble intermetallic alloys (KAu, LiAg), and nitrides of transient metals (ZrN, TiN)) in resonant MMs. For instance, from the fitted Drude model for AZO within 100–300 THz (see Table 1), the characteristic loss term γ/fp2 equals 1×10-3 [THz-1], while in contrast, for gold within almost the same frequency range (up to 460 THz), the loss term has a much smaller value, 2.45×10-5 THz-1. Figure 1 shows the real (left column) and the imaginary (right column) part of permittivity of the conducting oxide AZO (data reproduced from Refs. [21, 24, 25]) and the Drude model fittings [21, 24]. Our Drude fit presented in Table 1, with ε=3.48, fp=366.2 THz and γ=135.1 THz, is shown together with the models listed in Refs. [21] and [24] with and without a 2π factor taken into account for the corresponding collision frequency γ. It is found from Figure 1 that, by missing the 2π factor for γ, the Drude model renders an unrealistically low imaginary part of permittivity values (see curves with blue squares and circles). In addition, the experimental Johnson and Christy data [22] for gold are also presented in Figure 1 with our fitted Drude model, for an intuitive comparison to AZO data. However, the imaginary part of permittivity Im(ε) is not a correct figure-of-merit for characterizing conducting materials.

Table 1

Fitted parameters of Drude model and value of γ/fp2 (characterizing the ohmic loss) for different materials, i.e., AZO, Au, ZrN and Cu.

Comparison of data for gold and AZO between 100 and 300 THz: (A) Re(ε)
						and (B) Im(ε). For AZO, our fitted Drude model is listed together
						with the models in Refs. [21] and [24] w/ and w/o a 2π factor taken
						into account in the corresponding collision frequency. For gold, our Drude
						fitted model is shown consistent with experimental data by Johnson and
						Christy [22].
Figure 1

Comparison of data for gold and AZO between 100 and 300 THz: (A) Re(ε) and (B) Im(ε). For AZO, our fitted Drude model is listed together with the models in Refs. [21] and [24] w/ and w/o a 2π factor taken into account in the corresponding collision frequency. For gold, our Drude fitted model is shown consistent with experimental data by Johnson and Christy [22].

Finally, we illustrate our comparison of conducting materials with the fishnet structure, a typical resonant metamaterial at optical wavelengths. In Figure 2, we show the retrieved effective material parameters – i.e., real part of the refractive index [Re(n)], the permittivity [Re(ε)], and the permeability [Re(μ)] – for an AZO- and a Au-based fishnet metamaterial. The geometry of the fishnet is schematically presented in the inset of Figure 2 (the parameters are given in the figure caption). According to Figure 2, we find that the fishnet made from Au-MgO-Au possesses a negative index with simultaneously negative ε and μ within some frequency band, but that the AZO-based fishnet does not show any interesting feature of magnetic resonance. This is due to the fairly high intrinsic loss of AZO (see last column of Table 1), dampening the fishnet resonance enough to preclude negative permeability. Based on the data for ZrN within the 200–600 THz frequency band, the loss characterization term γ/fp2 equals 1.74×10-4, which is about an order of magnitude larger than that of gold, making the achievement of negative n or μ far from feasible.

Retrieved real part of effective refractive index [Re(n)], permittivity
						[Re(ε)], and permeability [Re(μ)] for fishnet structure made
						by AZO-MgO-AZO (A) and Au-MgO-Au (B), respectively. The fishnet structure
						[schematically shown as inset of Figure
							2(B)] has dimensions
							ax=500 nm,
							ay=600 nm,
							wx=200 nm,
							wy=350 nm,
							tm=30 nm, and
							td=40 nm.
Figure 2

Retrieved real part of effective refractive index [Re(n)], permittivity [Re(ε)], and permeability [Re(μ)] for fishnet structure made by AZO-MgO-AZO (A) and Au-MgO-Au (B), respectively. The fishnet structure [schematically shown as inset of Figure 2(B)] has dimensions ax=500 nm, ay=600 nm, wx=200 nm, wy=350 nm, tm=30 nm, and td=40 nm.

3 Surface plasmons with large propagation length

Dissipative loss in plasmonic systems is manifested by the decay of surface plasmon polariton (SPP) excitations (see Figure 3). Unfortunately, there is no simple criterion to estimate the propagation length from a single constitutive parameter. Only if Im(ε)<<∣Re(ε)∣, a simplified expression can be found, but for many conducting materials, it is not applicable. In addition, there is an intrinsic trade-off between the degree of confinement and the propagation length of a SPP that must be considered. In most cases, therefore, it is necessary to calculate the propagation length from the dispersion relation for a given level of confinement.

TM surface modes of (A) a single interface structure, (B) a two-dimensional
						sheet material (e.g., graphene).
Figure 3

TM surface modes of (A) a single interface structure, (B) a two-dimensional sheet material (e.g., graphene).

Figure 4 provides an overview of the propagation and confinement figures of merit for SPPs at a single material/air interface [10, 26]. The propagation length of an SPP, propagating at the interface in the z-direction (∼exp[i(βz-ωt)]), can be obtained from Lp=1/∣Im(β)∣. The SPP wavelength is λSPP=2π/∣Re(β)∣, and the lateral decay length is δ=1/Re[β2-(ω/c)2]. We can then define the two figure-of-merits for a plasmonic system: for the propagation length we have FOMProp=LP/λSPP, and for the degree of confinement we have FOMConf=λSPP/δ. Three different degrees of confinement (FOMConf=2, 4 and 8) are presented for the comparison in Figure 4.

Propagation length of SPPs figure-of-merit (FOMProp) vs. frequency
						for different materials at three different FOMconf.
Figure 4

Propagation length of SPPs figure-of-merit (FOMProp) vs. frequency for different materials at three different FOMconf.

At optical frequencies, silver has the longest propagation length of the listed materials (100 SPP wavelengths) at a low degree of confinement (FOMConf=2), but at a higher degree of confinement (FOMConf=8), it exhibits the shortest propagation length equal to one SPP wavelength. If strong confinement is desired, Al is a better choice for the conducting medium. At around 100 THz, transparent conducting oxides (AZO, ITO GZO) have FOMProp=10 for weak confinement and FOMProp=2∼3 for medium confinement, respectively. At 20–30 THz, SiC with weak confinement (FOMConf=2) has a surprisingly large propagation length: its FOMProp is nearly 60. Graphene is another conducting medium that sustains SPP modes [5, 27]. SPPs on graphene are extremely well confined (FOMConf=264), but they have rather short propagation lengths (FOMProp≈1) [9]. Recent graphene plasmonics experiments [28] have indeed demonstrated propagation lengths of about one SPP:

Lp/λSPP=12πIm(β)/Re(β)=12πγp=12π0.1351.18 with the plasmon damping rate γp≈0.135 from Ref. [28]. Nevertheless, graphene remains a fascinating material for terahertz applications [29], because of its atomic thickness, its easy tunability and its extreme subwavelength lateral confinement of surface plasmons. Figure 5 plots the ratio of the surface plasmon wavelength to the free-space wavelength versus frequency in different materials.

Ratio of the surface plasmon wavelength (λspp) over the
						free-space wavelength (λ0) at three different
							FOMConf.
Figure 5

Ratio of the surface plasmon wavelength (λspp) over the free-space wavelength (λ0) at three different FOMConf.

4 Conclusions

The lowest dissipative loss in resonant metamaterials on the one hand and plasmonics systems on the other hand is not achieved by the same conductors. For use in resonant metamaterials, the real part of the frequency-dependent resistivity is the correct quantity to judge the merits of the conducting material. For plasmonics applications, the propagation length (dissipative loss) and the degree of confinement of the SPP modes must be determined. We have compared a number of commonly used plasmonic materials while considering the inherent trade-off between propagation length and confinement. We believe it is very worthwhile continuing the research effort to develop better conducting materials, because of the considerable improvement such materials would bring to the nanophotonics field.

Acknowledgments

Work at Ames Laboratory was partially supported by the US Department of Energy, Office of Basic Energy Science, Division of Materials Science and Engineering (Ames Laboratory is operated for the US Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358), and by the US Office of Naval Research, Award No. N00014-14-1-0474. Work at FORTH was supported by the European Research Council under the ERC Advanced Grant No. 320081 (PHOTOMETA).

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About the article

Corresponding author: Costas M. Soukoulis, Ames Laboratory–US DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA; and Institute of Electronic Structure and Lasers (IESL), FORTH, 71110 Heraklion, Crete, Greece, e-mail:


Received: 2014-06-04

Accepted: 2014-09-04

Published Online: 2015-04-13


Citation Information: Nanophotonics, ISSN (Online) 2192-8614, ISSN (Print) 2192-8606, DOI: https://doi.org/10.1515/nanoph-2014-0013.

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© 2015 Costas M. Soukoulis et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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