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Replacing Noble Metals with Alternative Materials in Plasmonics and Metamaterials: how good an idea?

Noble metals that currently dominate the fields of plasmonics and metamaterials suffer from large ohmic losses. New plasmonic materials, such as doped oxides and nitrides, have smaller material loss, and, using them in place of metals carries promise of reduced-loss plasmonic and metamaterial structures, with sharper resonances and higher field concentration. This promise is put to a rigorous analytical test in this work which reveals that having low material loss is not sufficient to have a 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 advantage over the alternative media. The new materials may still find application niche where the high absorption loss is beneficial, e.g. in medicine and thermal photovoltaics.

Alas, the lofty promise of plasmonics and metamaterials has not been yet fulfilled and the chief culprit behind this disappointing turn of event is extremely high ohmic loss inherent to all metals, even the noble ones that are currently the mainstay of plasmonics and metamaterials. The salient feature of all the metals is the negative real part of the dielectric constant, where ε ∞ is the "background" dielectric constant of the bound electrons and ions, 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 the capacitor and inductor. One can also make a comparison with the mechanical oscillator by noting that electric field energy can be thought of as a potential energy, while the magnetic field energy plays the role of kinetic energy. Once the dimensions of the resonator become much smaller than wavelength, the magnetic field decreases and eventually becomes negligently small -well known quasi-static regime. The energy balance between the "potential" and "kinetic" energies can no longer be maintained and the mode ceases to exist as whatever the energy gets coupled into the cavity is radiated right back. But when the free electrons are present they possess the "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 the rate commensurate with 3 Therefore in the last few years there has been a push to develop alternative plasmonic materials, such as strongly doped "conventional" semiconductors such as InGaAs [10] transparent oxides such as ITO, AlZnO [11] as well as nitrides (TiN, ZrN) and others [12][13][14].
One of the goals is to find a material with lower rate of the momentum relaxation, and indeed, this rate is lower at least in some semiconductors than in metals, a as shown in Table I, where for highly doped InGaAs relaxation rate is only 10 13 s -1 . The reason for it can be understood by realizing that with 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 lower plasma frequency and it causes that all the SPP resonances shift to longer wavelengths. Note that reduced effective mass in the semiconductor usually does not help much as the materials with small effective mass happen have the narrow bandgaps and correspondingly large background dielectric constant ε ∞ . In general, γ increases and decreases in lock step with ωp. since both are related to the density of states in the vicinity of the Fermi level.
So, reduced, in compassion to the noble metals, p ω precludes using alternative plasmonic materials from use in 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 d ε the resonant frequency of the lowest order localized SPP mode is ( ) in the noble metals it is usually in the blue or green regions of the spectrum, but with reduced plasma frequency the resonance can be easily shifted to let us say telecom range of 1300-1500nm or all-important mid-IR ranges of 3-5 and 8-12 micrometers. So, assuming that the momentum scattering in the alternative plasmonic materials is reduced, it makes them attractive for applications in mid-infrared plasmonics and metamaterials, as has been argued in [10][11][12][13].
Alas, at this time, experimental data do not support this optimistic projection as quality factors of the 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 so-far-disappointing performance of "new" materials is simply the low value of plasma frequency in them.

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 the electrical energy whose density is The other half time the energy is split between the magnetic energy with It is therefore not too difficult to see that the effective rate of energy loss in the mode is Now, magnetic field in the mode is determined by the Maxwell equation one assumes that the fields are confined within characteristic dimension a, then in the absence of conductivity current the approximate relation between the magnitudes of two fields can be established as 1 λ is the wavelength in vacuum, and 0 η is the vacuum impedance. Therefore, as we have already noted, as the mode size decreases to less than / n λ the magnetic field engendered by the timedependent electric field (displacement current), 1 H , gets progressively smaller eventually becoming negligibly small and so is its energy, is the plasma wavelength. Therefore the effective cross-section in which the current is contained can be approximated as Clearly, if the diameter of the nanoparticle is much smaller than the skin depth, then the field permeates the nanoparticle and The total kinetic energy is From this expression the for the kinetic inductance easily follows.
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 and then the value of effective loss rate can be found as ( )  Table I, the scattering rate in the semiconductor is five time 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 semiconductor one. The results are shown in Fig.3. For gold nanoparticle the SPP resonance can be tuned anywhere in the visible and IR range. In the visible range the loss in of the Au SPP is just as high as the scattering rate in gold ( Note that one of the main arguments behind looking for alternative materials is that in order to achieve high degree of confinement in the plasmonic structure the permittivity of metal and dielectric should have roughly equal magnitudes and opposite signs, m d ε ε ≈ − , and therefore to achieve good confinement at long wavelength the plasma frequency should be "red shifted" there [10][11][12][13]. This argument, however, is totally flawed -no matter how large is m ε one can always achieve high degree of confinement by shifting the resonant SPP frequency via increased capacitance by designing the structure with a sufficiently small gap, changing eccentricity of the elliptical nanoparticle or using nanoshells and dimers. If anything, reducing the capacitance and shifting the SPP resonance to the blue has always presented the problem, but not shifting it to the red and beyond!

Loss Scaling in Gap SPP waveguides.
One can extend this comparison to the case of the gap SPP [18][19][20][21] confined within the dielectric of thickness a placed between the two layers of metal or other plasmonic material as shown in Fig.1 Applying the boundary condition for the continuity of the tangential component of electric field immediately yields cosh( From (14) and (15) Substituting this into (13) we finally get the dispersion relationship ( ) After some algebra one can find the dispersive relation for the plasmon under assumption that effective propagation constant β is not significantly different from the wavevector in dielectric kd which is a typical practical arrangement.
( ) The propagation length then can be found as 1 2 prop p ca L n π β λ γ = = (21) where n is the index of dielectric. Once again, the loss depends only on the ratio for the metal (Fig.5b) it is almost independent on the wavelength in full agreement with (21) while for InGaAs (Fig.5d) the propagation length does depend on the wavelength. This can be explained by the fact that when wavelength approaches the plasma wavelength the field penetrates deep inside the semiconductor. At any rate, comparing Fig.5b and 5d one can see that propagation length in metal always exceeds that in semiconductor despite higher material loss in the metal.

Conclusions
We have compared performance of the alternative plasmonic materials in the IR region with that of the metals using a simple fully analytical model. We have shown that the proper figure of merit, defining loss, broadening and field enhancement should be the ratio of plasma frequency ωp and material loss γ. In laymen terms it indicates that it is always preferable to have many electrons moving relatively slow rather than a relatively few electrons moving very fast. The figure of merit 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 the "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 the alternative plasmonic materials due to them having a higher melting point [22], their compatibility with existing (Si or III-V) technologies, or simply based on the price and availability considerations. Furthermore, some applications in medicine [23], photocatalysis [24], and thermal photovoltaics [25] actually require high loss and broad resonances and it is in these applications that the alternative plasmonic materials may find their practical use.