Lightning-Rod Effect of Plasmonic Field Enhancement on Hydrogen-Absorbing Transition Metals

The plasmonic enhancement of electromagnetic field energy density at the sharp tips of nanoparticles or nanoscale surface roughnesses of hydrogen-absorbing transition metals, Pd, Ti, and Ni, is quantitatively investigated. A large degree of energy focusing is observed for these transition metals in the microwave region, even surpassing the enhancement for noble metals according to the conditions. Pd, for instance, exhibits peak field enhancement factors of 6000 and 2 × 108 in air for morphological aspect ratios of 10 and 100, respectively. Metal surfaces possibly contain such degrees of nano- or micro-scale native random roughnesses, and, therefore, the field enhancement effect may have been unknowingly produced in existing electrical and optical systems. In addition, for future devices under development, particularly in hydrogen-related applications, it is desirable to design and optimize the systems, including the choice of materials, structures, and operating conditions, by accounting for the plasmonic local energy enhancement effect around the metal surfaces.


Theory and Calculation Methods
We calculate the field enhancement factors, which represent the intensity ratios for fields around the object (metals in this case) to those in the absence of the object, or the original incident fields, for prolate-spheroidal metal nanoparticles in air, H 2 , or vacuum, and H 2 O. We specifically calculate the field enhancement factors at the tips of the prolate-spheroidal metal nanoparticles, to represent ere e is the eccentricity of the particle shape: (3 field enhancement factor is then calculated as (4 te that this field enhancement factor is defined as the ratio of field intensities and not fie gnitudes. Incidentally, for the spherical case, which provides L1 = 1/3, η reduces to the equation field enhancement factor derived in Reference [7]. The empirical complex dielectric functions tals and of the surroundings on the frequencies listed in References [7] and [26] are used for t putations in this study. We assume that ε′ = 1 and ε″ = 0 throughout the entire frequencies for a and vacuum. The electrostatic calculations carried out in this study are valid for particle sizes range of 10-100 nm, for which the phase retardation is negligible throughout the particle, the fie ancement will be largest, and metal nanoparticles and nanoscale roughnesses will, therefo ome most applicable, as discussed in the Results and Discussion section.

Results and Discussion
Firstly, as a reference, we present in Figure 2 the calculated electromagnetic field enhancement factors for the simple, spherical nanoparticle case, which corresponds to the spheroid's aspect ratio, a/b, of one. The peaks seen in these spectra are associated with the resonance or surface mode, characterized by internal electric fields with no radial nodes. A local energy enhancement around 10 times is decently observed for the hydrogen-absorbing transition metals, Pd, Ni, and Ti, in this spherical-shape case. Incidentally, the results in Figure 2 for shorter wavelengths are consistent with those reported in Reference [7]. It should be noted that the field enhancement factors, for our calculations, are independent of the particle size under the quasistatic approximation, and are valid for particle diameters around the range of 10-100 nm [7]. Figure 3 shows the field enhancement factors for spheroidal metal nanoparticles with an aspect ratio of three. It is observed that even for such a relatively small aspect ratio or morphological surface sharpness, enhancement factors over 100 are attainable for Pd, Ni, and Ti for a wide range of frequencies, through visible to infrared. Such nanoparticles or nanoscale surface roughness thus concentrate electromagnetic or optical energy in their vicinity like antennae. The artifact discontinuities for the curves for Cu and Ti around 1 and 4 µm, respectively, in Figures 3 and 4 are incidentally because of the discontinuities in the source empirical data of the dielectric functions. Remarkably, electromagnetic field enhancement factors of several thousand are observed for Pd, Ni, and Ti for the aspect ratio of 10 ( Figure 4). The resonant peak enhancement factors for Au, Ag, and Cu are observed to be even larger, on the order of 10 5 . Among the whole metal elements, Al and the noble metals Ag, Au, and Cu are known to exhibit distinctively higher field enhancement factors than other metals because of their high electrical conductivities [7,46]. Therefore, the combination of such noble-metal spheroidal nanoparticles and hydrogen-absorbed transition metals, available, for instance, by coating bulk metal surfaces by colloidal metal nanoparticles, may be another strategy for applications to harvest the photonic or electromagnetic energy focusing effect. Strikingly, as observed in Figure 5, the field enhancement factor of Pd for the sharp particle or surface morphology case of an aspect ratio of 100 reaches the order of 10 8 in the infrared region, even exceeding those for the noble metals. This consequence is consistent with the results reported in Reference [26] that the plasmonic field enhancement factors of the hydrogen-absorbing transition metals become higher than those of noble metals for planar metal surfaces. Incidentally, the peak or resonant wavelength of Ti may unfortunately locate outside of the range of frequencies handled in this study. Figures 6 and 7 summarize the dependence of the peak field enhancement factors and wavelengths, respectively, on the aspect ratio. It is observed that the electromagnetic field enhancement factors dramatically increase with the aspect ratio, namely, with the sharpness of the metal surfaces. As the sharpness increases, the resonant peak wavelength red-shifts. Large-aspect-ratio metal particles or high-curvature edges of surface irregularities exhibit high polarizabilities and, thus, large dipole moments, particularly at the resonance, to produce strong local field enhancement in the vicinity of such edges [44,47]. To discuss the electromagnetic similarity between isolated metal spheroidal particles and rough surfaces, it is worth mentioning that the detailed numerical calculation results reported in Reference [40], where rough metal surfaces were modeled as prolate hemispheroids protruding from a grounded flat plane, are quantitatively similar to our results in Figures 6 and 7, for the peak field enhancement factors and wavelengths.
The large field-enhancement effect on the hydrogen-absorbing transition metals, Pd, Ni, and Ti, observed in the series of calculations in this study can be used for various hydrogen-energy applications. As discussed in Reference [26], potential applications include hydrogen storage, sensing [48,49], laser fusion [46], and condensed-matter fusion. In addition, for the reported experiments so far, for instance, in the condensed-matter nuclear fusion field, it is highly possible that the deuterium-absorbed Pd, Ni, and Ti surfaces contained certain degrees of nano-or micro-scale native random roughnesses [39,40,50,51] corresponding to such morphological aspect ratios as those studied in this article. Therefore, some of the experimental material systems may have unknowingly benefited from the plasmonic field enhancement effect. The electrostatic calculation results shown in this paper are valid for particle sizes smaller than the fields' wavelengths at which the phase retardation is negligible throughout the particle object. In addition, the dielectric functions of materials used for our calculations are empirical values for bulk materials, whose validity is debatable when the particle sizes become smaller than 10 nm, because of the electron mean free path limitation or scattering of conduction electrons off particle surfaces [44,47,52,53]. The calculation results for optical wavelengths under the quasistatic approximation are, therefore, valid for metal particles with diameters in the range of 10-100 nm. Metal particles with sizes smaller and larger than these limits both exhibit broader plasmon resonances and smaller field enhancements, because of the surface scattering losses and the radiative losses or electrodynamic damping, respectively [47,52,53]. Therefore, the choice of particle sizes, 10-100 nm, for our calculations is most suitable for plasmon-enhanced electromagnetic and optical applications, because of the largest field enhancements. This size aspect should, therefore, also be accounted for in the optimized design of the material structures in potential applications. In addition, surface plasmons located in between multiple metallic objects with nanoscale separation distances, or so-called "gap plasmons" [54][55][56], would also provide large field enhancements on the conditions. The gap-plasmon effect for hydrogen-energy applications is important partially because gap plasmons are also commonly observed in real structures such as rough metal surfaces, and will be discussed in future work.        The large field-enhancement effect on the hydrogen-absorbing transition metals, Pd, Ni, and Ti, observed in the series of calculations in this study can be used for various hydrogen-energy applications. As discussed in Reference [26], potential applications include hydrogen storage, sensing [48,49], laser fusion [46], and condensed-matter fusion. In addition, for the reported experiments so far, for instance, in the condensed-matter nuclear fusion field, it is highly possible that the deuteriumabsorbed Pd, Ni, and Ti surfaces contained certain degrees of nano-or micro-scale native random roughnesses [39,40,50,51] corresponding to such morphological aspect ratios as those studied in this article. Therefore, some of the experimental material systems may have unknowingly benefited from the plasmonic field enhancement effect. The electrostatic calculation results shown in this paper are valid for particle sizes smaller than the fields' wavelengths at which the phase retardation is negligible throughout the particle object. In addition, the dielectric functions of materials used for our calculations are empirical values for bulk materials, whose validity is debatable when the particle sizes become smaller than 10 nm, because of the electron mean free path limitation or scattering of conduction electrons off particle surfaces [44,47,52,53]. The calculation results for optical wavelengths under the quasistatic approximation are, therefore, valid for metal particles with diameters in the range of 10-100 nm. Metal particles with sizes smaller and larger than these limits both exhibit broader plasmon resonances and smaller field enhancements, because of the surface scattering losses and the radiative losses or electrodynamic damping, respectively [47,52,53]. Therefore, the choice of particle sizes, 10-100 nm, for our calculations is most suitable for plasmon-enhanced electromagnetic and optical applications, because of the largest field enhancements. This size aspect should, therefore, also be accounted for in the optimized design of the material structures in potential applications. In addition, surface plasmons located in between multiple metallic objects with nanoscale separation distances, or so-called "gap plasmons" [54][55][56], would also provide large field enhancements on the

Conclusions
In this work, we numerically investigated the lightning-rod effect of plasmonic field enhancement on hydrogen-absorbing transition metals. A large degree of energy focusing was observed for these transition metals in the microwave region, even surpassing the enhancement for noble metals according to the conditions. Pd, for instance, exhibited peak field enhancement factors of 6000 and 2 × 10 8 in air for morphological aspect ratios of 10 and 100, respectively. The metal surfaces possibly contained such degrees of nano-or micro-scale native random roughnesses, and, therefore, the field enhancement effect may have been unknowingly produced in existing electrical and optical systems. Active utilization of the plasmonic local energy enhancement effect around the metal surfaces by proper material and structure choices, such as the introduction of sharp nanoparticles or sharply roughened surfaces, can potentially improve hydrogen-related device performance.
Author Contributions: K.T. conceived the idea for the study. N.F. carried out the calculations and analyzed the data. Both authors contributed to discussion of the results and composed the manuscript.
Funding: This research was funded by the Thermal & Electric Energy Technology Foundation and the Research Foundation for Opto-Science and Technology.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Conclusions
In this work, we numerically investigated the lightning-rod effect of plasmonic field enhancement on hydrogen-absorbing transition metals. A large degree of energy focusing was observed for these transition metals in the microwave region, even surpassing the enhancement for noble metals according to the conditions. Pd, for instance, exhibited peak field enhancement factors of 6000 and 2 × 10 8 in air for morphological aspect ratios of 10 and 100, respectively. The metal surfaces possibly contained such degrees of nano-or micro-scale native random roughnesses, and, therefore, the field enhancement effect may have been unknowingly produced in existing electrical and optical systems. Active utilization of the plasmonic local energy enhancement effect around the metal surfaces by proper material and structure choices, such as the introduction of sharp nanoparticles or sharply roughened surfaces, can potentially improve hydrogen-related device performance.