## Abstract

Noble metals that currently dominate the fields of plasmonics and metamaterials suffer from large ohmic losses. Some of the new plasmonic materials, such as doped oxides and nitrides, have smaller material loss, and using them in place of metals carries the promise of reduced-loss plasmonic and metamaterial structures, with sharper resonances and higher field concentrations. This promise is put to a rigorous analytical test in this work, which reveals that having low material loss is not sufficient to have reduced modal loss in plasmonic structures. To reduce the modal loss, it is absolutely necessary for the plasma frequency to be significantly higher than the operational frequency. Using examples of nanoparticle plasmons and gap plasmons one comes to the conclusion that, even in the mid-infrared spectrum, metals continue to hold an advantage over alternative media when it comes to propagation distances and field enhancements. Of course, the new materials still have an application niche where high absorption loss is beneficial, e.g. in medicine and thermal photovoltaics.

This article is part of the themed issue ‘New horizons for nanophotonics’.

## 1. Introduction

Plasmonics and the closely related field of metamaterials have been among the most active and exciting areas of optical sciences over the last decade [1]. The innate ability of plasmonic structures to squeeze optical energy into sub-wavelength volumes and thus achieve a high degree of energy concentration allows one to enhance various linear, and especially nonlinear, optical processes. This ability makes surface plasmon polariton (SPP) modes attractive candidates for use in applications like biosensing and environmental sensing, photovoltaics, nonlinear optics, light sources and all other areas where the extreme localization of electromagnetic fields can improve efficiency, speed or, preferably, both [1–3]. Propagating SPPs in sub-wavelength structures have long been promoted as candidates for future on-chip optical interconnects [4–6]. Metamaterials and metasurfaces relying on plasmonic resonances have been envisioned as the building elements of novel optical systems with widely extended functionality, such as super lenses [7] and optical cloaks and other transformational optics schemes [8].

Alas, the lofty promise of plasmonics and metamaterials has not yet been fulfilled, and the chief culprit behind this disappointing turn of events is the extremely high ohmic loss inherent in all metals, even the noble ones that are currently the mainstay of plasmonics and metamaterials. The salient feature of all metals is the negative real part of the dielectric constant,
1.1
where is the ‘background’ dielectric constant of the bound electrons and ions, is the plasma frequency, *N* is the carrier density, *m* is the effective mass and *γ* is the momentum scattering rate, responsible for the loss. The plasma frequency is typically in the ultraviolet region corresponding to wavelengths of the order of 150 nm, while the momentum relaxation time is on the scale of *γ* ≈ 10^{14} s^{−1}. For frequencies *ω* < *ω*_{p} the real part of the dielectric constant is negative and this fact allows sub-wavelength confinement in the SPP modes of various metal–dielectric structures.

The origin of sub-wavelength confinement in metal-containing structures can be understood from very simple energy conservation considerations [9]. In the dielectric cavity, the energy oscillates between the electric and magnetic fields, just as, in a resonant electric circuit, the energy oscillates between capacitor and inductor. One can also make a comparison with a mechanical oscillator by noting that electric field energy can be thought of as potential energy, while the magnetic field energy plays the role of kinetic energy. Once the dimensions of the resonator become much smaller than the wavelength, the magnetic field decreases and eventually becomes negligibly small—the well-known quasi-static regime. The energy balance between ‘potential’ and ‘kinetic’ energies can no longer be maintained and the mode ceases to exist as whatever energy gets coupled into the cavity is radiated right back. But when free electrons are present, they possess ‘true’ kinetic energy, so now the energy can switch back and forth between the ‘potential’ energy of the electric field and the kinetic energy of free carriers (plus some contribution from magnetic energy). The frequencies at which the energy balance can now be maintained are precisely the frequencies of the sub-wavelength SPP modes.

However, as explained in [9], once the energy is transferred into the form of kinetic motion carriers, it is unavoidably dissipated with the rate *γ*; hence the total energy of the SPP mode also decays at a rate commensurate with *γ* ≈ 10^{14} s^{−1}. Thus, higher confinement always leads to higher loss, at least in the visible and near IR ranges of the optical spectrum.

Therefore, in the last few years there has been a push to develop *alternative plasmonic materials*, including strongly doped ‘conventional’ semiconductors such as InGaAs [10], transparent oxides such as ITO and AlZnO [11], nitrides (TiN, ZrN), and others [12–14]. One of the goals is to find a material with a lower rate of momentum relaxation, and, indeed, this rate is lower at least in some semiconductors than in metals, as shown in table 1, where, for highly doped InGaAs relaxation, the rate is only 10^{13} s^{−1}. The reason for this can be understood by realizing that, with a smaller density of electrons, the Fermi layer in semiconductors lies lower than in metals, and there are fewer states into which the electrons can scatter. Unfortunately, the lower carrier density *N* also means a lower plasma frequency *ω*_{p}. Therefore, all the SPP resonances shift to longer wavelengths. Note that a reduced effective mass in the semiconductor usually does not help much as the materials with small effective mass happen to have narrow bandgaps and correspondingly large background dielectric constants . In general, *γ increases and decreases in lock step with ω*_{p}, because both are related to the density of states in the vicinity of the Fermi level.

So, a reduced *ω*_{p}, in comparison with the noble metals, precludes alternative plasmonic materials from use in the visible and near IR spectral ranges, but, at the same time, it appears to make them adequate, or perhaps even attractive when it comes to longer wavelengths. Indeed, if one considers a spherical ‘metallic’ nanoparticle embedded into a dielectric with a dielectric constant the resonant frequency of the lowest order localized SPP mode is —hence, in the noble metals, it is usually in the blue or green regions of the spectrum, but with a reduced plasma frequency the resonance can be easily shifted to, let us say, the telecom range of 1300–1500 nm or the all-important mid-IR ranges of 3–5 and 8–12 µm. So, assuming that the momentum scattering in the alternative plasmonic materials is reduced, it makes them attractive for applications in mid-IR plasmonics and metamaterials, as has been argued in [10–13].

Alas, at this time, experimental data do not support this optimistic projection as the quality factors of observed resonances in semiconductor nanostructures have been inferior to those seen in metallic designs. While part of the problem may lie in fabrication difficulties, recent numerical analysis [15] has shown that, all said and done, when it comes to the loss, silver is still superior to all new plasmonic materials, including graphene. Yet no clear physical picture behind this ‘superiority’ of Ag has emerged from that work. In this study, we look into the same issue analytically and determine that the main reason for the so-far-disappointing performance of ‘new’ materials is simply *the low value of plasma frequency in them.*

## 2. Kinetic and magnetic inductances

To understand the loss, one shall consider the energy balance in the plasmonic mode [9,16]. Half of the time, all the energy is stored in the form of electrical energy whose density is . The other half of the time the energy is split between the magnetic energy with density and the kinetic energy of the collective motion of the free electrons . Energy conservation, schematically shown in figure 1, requires the total energy to be time independent and therefore , where the time-averaged value of the electrical energy density is with similar expressions for the densities of the kinetic and magnetic energies.

Neglecting the radiation (which can of course be useful as it provides the means for coupling the energy in and out of the mode), neither electric nor magnetic energies are damped. But the kinetic energy of the moving carriers does get damped at twice the rate of the momentum damping, i.e. 2*γ*. It is, therefore, not too difficult to see that the effective rate of energy loss in the mode is
2.1

Now, the magnetic field in the mode is determined by Maxwell's equation . If one assumes that the fields are confined within the characteristic dimension *a*, then, in the absence of a conductivity current, the approximate relation between the magnitudes of two fields can be established as , where is the refractive index of the dielectric, λ is the wavelength in a vacuum and *η*_{0} is the vacuum impedance. Therefore, as we have already noted, as the mode size decreases to less than *λ/n*, the magnetic field *H*_{1} engendered by the time-dependent electric field (displacement current) gets progressively smaller, eventually becoming negligibly small, and so its energy, , is not sufficient to maintain the energy balance.

The field engendered by the conductivity current, *H*_{2} ∼ *I*/*a,* however, does not explicitly depend on the relation between the mode size and the wavelength. This field then can have sufficient magnetic energy to reduce the need for the kinetic energy , where we have introduced *L*_{m} as a ‘conventional’ magnetic inductance and *L*_{k} as the *kinetic inductance*. The kinetic induction is a measure of the inertia of the free electron gas which causes the current to lag in phase behind the voltage.

With these definitions (2.1) the main expression for the effective loss becomes 2.2

This effective loss or, rather, the quality factor determines the maximum field enhancement attainable in the plasmonic structure; hence it is relation (2.2) that we shall now use to compare different plasmonic materials.

## 3. Loss scaling in localized surface plasmon polaritons

Consider now the SPP mode around an elliptical nanoparticle, which we can approximate as a cylinder of length *l* and diameter *a* (figure 2*a*). The length *l* can be adjusted to achieve an SPP resonance at a given wavelength λ no matter what *ω*_{p} is. The electric field, and hence the current, penetrates the metal by the skin depth, roughly λ_{p}/2*π* where is the plasma wavelength. Therefore, the effective cross section in which the current is contained can be evaluated using an ad hoc approximation
3.1

Clearly, if the diameter of the nanoparticle is much smaller than the skin depth, then the field permeates the nanoparticle and , while in large diameter nanoparticles the current is contained in the cylindrical region of skin depth thickness with area .

The total kinetic energy is , while the current is , and using the definition of plasma frequency we obtain 3.2

From this expression, the equation for the kinetic inductance easily follows: 3.3

As one can see, the actual wavelength of light is absent from (3.3) and the kinetic inductance depends only on the ratio between the plasma wavelength and the mode dimension *a*. One can also find the resistance of the nanoparticle as
3.4
which allows us to re-write (2.2) as simply , where , which is exactly what one would obtain from simple resonant electrical circuit consideration.

While kinetic inductance exhibits strong dependence on the transverse dimensions, the magnetic inductance shows only a very weak dependence on the size *a*,
3.5

The expression in the square brackets changes from 1 and 2 as the ratio (*l*/*a*) increases from 1.5 to 5 and thus can be neglected in our order-of-magnitude analysis. Thus we obtain a very simple relation between the two inductances
3.6
and then the value of the effective loss rate can be found as
3.7
where *x* = *πa*/2λ_{p}.

The effective loss and, therefore, the broadening and maximum field enhancement do not depend on the wavelength of light, but only on the plasma wavelength. It is conceivable, therefore, that a metal with larger γ and smaller λ_{p} would have a smaller loss than a doped semiconductor. To illustrate (3.7) we consider an elliptical particle with a diameter *a* = λ/10 made from either gold or InGaAs doped to the degree of 2 × 10^{19} cm^{−3}. As follows from table 1, the scattering rate in the semiconductor is five times smaller than in gold, but the plasma frequency is smaller by a factor of 10. As mentioned above, the resonant frequency of the SPP can be tuned by changing the aspect ratio of the nanoparticle; hence the length *l* of the Au nanoparticle is always longer than that of the semiconductor one. The results are shown in figure 3. For a gold nanoparticle, the SPP resonance can be tuned anywhere in the visible and IR range. In the visible range, the loss rate of the Au SPP is just as high as the scattering rate in gold (), but in the mid-IR range the diameter *a* = λ/10 exceeds λ_{p} and the loss rate decreases to beyond λ = 3 µm, i.e. it is less than the scattering rate in the semiconductor. For the InGaAs nanoparticle the SPP mode exists only at λ ≥ 3 µm, and, since for InGaAs in this range *a* = λ/10 > λ_{p}, the loss of the semiconductor SPP mode always remains higher than that in the Au SPP mode, despite the fact that the scattering rate in the semiconductor is lower. Note that, in the end, at long wavelengths with the effective loss becomes simply , and hence the *proper figure of merit (FOM) for the material* should be *ω*_{p}/*γ* (as first indicated in [15]) and not *ω*/*γ*, which is often used. *According to table 1 the proper FOM is always higher for metals than for any of the semiconducting materials.*

One can also consider a split ring resonator [17] (figure 2*b*) with circumference *l* and wire cross section *πa*^{2}/4. The resonance frequency in it can always be adjusted by properly choosing the size of the gap *d*. The magnetic inductance of the loop once again does not depend on the cross section, , and a result almost identical to (3.7), , can be obtained, indicating the generality of our approach. Using (3.4) one can easily show that the FOM
3.8
is related to the real part of the resistivity of the metal *ρ*_{R} = *mγ*/*Ne*^{2}, as was mentioned in [18]. And this is the main lesson of this work: having a material with low *γ*, i.e. high electron mobility, is not sufficient for low loss sub-wavelength confinement—it is the low resistivity that matters most.

Note that one of the main arguments behind looking for alternative materials is that in order to achieve a high degree of confinement in the plasmonic structure the permittivity of the metal and dielectric should have roughly equal magnitudes and opposite signs, ≈ −, and therefore to achieve good confinement at long wavelength the plasma frequency should be ‘red shifted’ there [10–13]. This argument, however, is totally flawed—no matter how large || is, one can always achieve a high degree of confinement by shifting the resonant SPP frequency via increased capacitance by designing the structure with a sufficiently small gap, changing the eccentricity of the elliptical nanoparticle or using nanoshells and dimers. If anything, reducing the capacitance and shifting the SPP resonance to blue has always presented a problem, but not shifting it to red and beyond!

## 4. Loss scaling in gap surface plasmon polariton waveguides

One can extend this comparison to the case of the gap SPP [19–22] confined within a dielectric of thickness *a* placed between two layers of metal or other plasmonic material as shown in figure 4. If one considers a simple SPP on the metal–dielectric interface, one can achieve high confinement only when ≈ −, and in the IR region one is forced to use a material with a lower electron concentration, such as a doped semiconductor with ensuing losses, or, alternatively, consider a rather complex design of metallic ‘spoof plasmons' [23]. But the gap SPP allows one to achieve sub-wavelength confinement (in a lateral direction) for a very broad range of frequencies no matter how high the absolute value of the dielectric constant of the metal.

The tangential magnetic and electric fields in the gap SPP can be written as
4.1
and
4.2
respectively. The propagation constant *β* and the decay coefficients *q _{d}* and

*q*

_{m}are related as 4.3 and

*k*

_{0}=

*ω*/

*c*, and from (4.3) one obtains 4.4

Applying the boundary condition for the continuity of the tangential component of the electric field immediately yields 4.5

From (4.4) and (4.5) the recursive relation for the decay constant in the dielectric is 4.6

For the gap case of a thin sub-wavelength gap with we have and for the operational wavelength far from the surface plasmon resonance , and one obtains 4.7 Using the Drude formula for the dielectric constant of metal, the decay coefficient in the dielectric medium is 4.8

Substituting this into (4.3), we finally get the dispersion relationship 4.9

After some algebra one can find the dispersive relation for the plasmon under the assumption that the effective propagation constant *β* is not significantly different from the wavevector in dielectric *k*_{d}, which is a typical practical arrangement that still allows sub-wavelength confinement in the lateral plane
4.10

The propagation length can then be found as
4.11
where *n* is the index of the dielectric. Once again, the loss depends only on the ratio *πa*/λ_{p} and the FOM should be the same expression *ω*_{p}/*γ* obtained above for the localized SPP.

For the long-wavelength region where the Drude expression for the dielectric constant (1.1) is valid, once again one can use the expression for the real and imaginary parts of the dielectric constant of the metal to obtain another expression for the FOM,

*It is easy to see (table 1) that the noble metals have higher values of FOM than most if not all ‘new’ plasmonic materials all the way to the far IR range*, just as numerical simulations in [15] have shown.

To illustrate this fact, we perform full modelling of the dispersion and propagation length in gap SPPs for gold and InGaAs waveguides with an SiO_{2} core. The results are shown in figure 5. The propagation constant (effective) index *β* of the gap SPP shown in figure 5*a,c* remains equal to *k*_{d} for as long as and then gradually increases as the field starts penetrating the metal (semiconductor). The propagation length indeed increases proportionally to the gap thickness. For the metal (figure 5*b*) it is almost independent of the wavelength, in full agreement with (4.11), while for InGaAs (figure 5*d*) the propagation length does depend on the wavelength. This can be explained by the fact that, when the wavelength approaches the plasma wavelength, the field penetrates deep inside the semiconductor. At any rate, comparing figure 5*b,d* one can see that the propagation length in metal always exceeds that in semiconductor despite higher material loss in the metal.

## 5. Conclusion

We have compared the performance of alternative plasmonic materials in the IR region with that of metals using a simple fully analytical model. We have shown that the proper FOM, defining loss, broadening and field enhancement, should be the ratio of the plasma frequency *ω*_{p} and the material loss *γ*. In layman's terms, *it indicates that it is always preferable to have many electrons moving relatively slowly rather than a relatively few electrons moving very fast as follows from the quadratic dependence of the kinetic energy on the velocity.* The FOM of all the alternative materials is worse than that of Ag and even Au. Therefore, one cannot expect better performance from the new materials in ‘traditional’ plasmonic applications, where the loss, propagation length or degree of field enhancement is not the most important factor. It is also important to realize that there exist important application niches for alternative plasmonic materials due to their higher melting points [24], their compatibility with existing (Si or III–V) technologies, or simply based on price and availability considerations. These conclusions should not be misunderstood as questioning the usefulness of alternative plasmonic materials as for some applications in medicine [25], photocatalysis [26] and thermal photovoltaics [27]. Both high loss and broad resonances are highly desirable and it is in these applications that the alternative plasmonic materials may find their practical use.

## Competing interests

I declare I have no competing interests.

## Funding

The author acknowledges the generosity of the US Army Research Office (grant no. W911NF-15–1-0629).

## Acknowledgement

The author acknowledges the stimulating discussions with his colleague Prof. P. Noir of Johns Hopkins University.

## Footnotes

One contribution of 15 to a theme issue ‘New horizons for nanophotonics’.

- Accepted October 11, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.