1. Introduction

Advances in nanotechnology have made it possible to structure metal films on length scales down to a few nanometers. As was recognized, this allows for the efficient manipulation of light at the nanoscale, and the scientific field of plasmonics emerged [1–3].

One of the main driving forces of this field of research is the application of plasmonic sensing in ultra-small volumes [4–7]. In the early days, this was led by the promise of obtaining spectroscopic fingerprints of only few molecules in surface enhanced Raman scattering (SERS) [8].

More recently, the development of coupled bright and dark plasmonic structures allowed the production of more sensitive geometries that even allow for the sensing of minute refractive index changes in very small volumes [9]. In both cases, the large field-enhancements obtained near the surface of resonant metallic structures are the enabling effects.

The scientific advances of recent years have shown that these metallic structures can be regarded as optical nanoantennas [10,11]. In this picture, the resonant metallic structures are considered as optical antennas and the sensed material as the equivalent to an antenna-driven circuit in electronics. This view gives rise to the recognition that sensing cannot only be carried out on dielectric substances but also on non-resonant metallic structures. This is especially useful in cases where particles are too small to have a significant scattering cross-section [12,13] or where the inherent damping of the metal prevents strong plasmonic resonances. The latter is the case in the optical detection of hydrogen in palladium [14].

The optical detection of hydrogen is of particular interest for two applications: the detection of explosive mixtures of hydrogen with air at concentrations ranging from 4 to 75%, and the investigation of loading/unloading kinetics of hydrogen in nanostructured materials, e.g., for fuel cell applications. Many examples of plasmonic hydrogen sensing have already been proposed [15–20]. Palladium (Pd) serves as an ideal material for the achievement of these two goals since it is one of the simplest model systems for the incorporation of hydrogen in metal lattices [21,22].

In order to investigate palladium kinetics on minute length scales and for very low concentrations, a move from conventional designs relying on extended films [23] or arrays of plasmonic elements [16] towards single structures is desirable [24]. However, the use of, e.g., simple single Pd nanodisks is precluded by the large intrinsic damping and the resulting broad resonance profile as well as low scattering efficiency. Antenna-enhanced geometries can overcome these problems by coupling a highly resonant antenna to the system under investigation which can greatly enhance nanoscale optical effects [12,25–27].

The application of this approach to hydrogen sensing has been demonstrated experimentally by Liu et al. through the coupling of a single gold bowtie antenna to a Pd nanodisk [13]. Unfortunately, in their paper no theoretical simulations of optical spectra were included. However, numerical modeling is crucial for the efficient design and optimization of an optical sensor geometry. Furthermore, simply using the dielectric functions for Pd and hydrided Pd given by Vargas et al. [14] in simulations will result in a spectral blueshift upon hydrogen exposure. Contrary to this, Liu et al. experimentally observed a redshift.

Here, we use numerical simulations to understand the underlying physics responsible for the observed spectral shifts and resolve the apparent discrepancy. By not only including the index of refraction change but also the volume expansion of Pd, for the first time, we are able to model the sensing behavior of the experimentally investigated Pd disk and gold antenna system. In addition, we show how adding a second antenna arm can substantially improve the sensing performance.

2. System under consideration

The antenna-enhanced system under consideration is shown in Fig. 1 and consists of a single gold bowtie antenna (side length a = 108 nm, height 40 nm) separated by a gap g from a Pd disk (diameter d = 60 nm, height 40 nm) on a SiO₂ substrate.

 

Fig. 1 Schematic view of the system under consideration. Pd disk with diameter d = 60 nm and height 40 nm situated next to a single gold bowtie antenna with side length a = 108 nm and height 40 nm.

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The system is similar to the one in [13] where the authors manufacture this structure using electron-beam lithography and record dark-field scattering spectra of single antenna-enhanced sensor elements. In the absence of hydrogen they find a strong scattering maximum associated with the antenna-resonance at a wavelength of 638 nm. When exposed to a concentration of 1% hydrogen in nitrogen, the resonance undergoes a redshift of 5 nm as well as a strong decrease of the scattering amplitude and significant broadening. An increase of the concentration to 2% in nitrogen yields a total redshift of 9 nm as well as further broadening. The observed effect is mostly reversible, showing some hysteresis due to the hydrogen loading/unloading.

When Pd is exposed to hydrogen gas, the molecules are split at the surface and atomic hydrogen enters the Pd lattice. At low concentrations the hydrogen forms a solid solution in the host lattice, the so-called α-phase. For higher concentrations a less mobile palladium-hydride (PdH) β-phase with increased lattice parameter is formed. Starting from impurities or lattice defects the β-phase forms domains and for very high concentrations most of the system is converted, leading to a pronounced increase of the lattice constant by as much as 4% [21]. In the following PdH will always refer to the hydride system with a H/Pd atomic ratio of 0.82 as given in [14].

Concurrent to the hydride-formation the dielectric function of Pd is also modified. This is caused by an electron transfer from the hydrogen atoms to the electronic system of Pd, changing the density of states at the Fermi-surface [19]. This change substantially modifies both the real and imaginary parts of the dielectric function of Pd [14].

A qualitative explanation for the effect of antenna enhanced hydrogen sensing is that the hydrogen-induced changes of the dielectric function and the lattice constant of Pd are enhanced by the antenna resonance, leading to a redshift. In order to obtain a more quantitative understanding and enable optimization of this sensor structure, detailed numerical analysis is desirable.

The numerical analysis has been performed using the HFSS [28], a commercial finite element (FEM) electromagnetic field solver. Pd/PdH and gold are described using the refractive indices given in [14] and [29], respectively. The permittivity of the SiO₂ substrate is taken as 2.25. The footprint of the single gold bowtie antenna is a regular triangle with side length l = 116 nm where the corners are rounded with 5 nm radius, resulting in an effective bowtie side length of a = 108 nm.

The top edge of the bowtie as well as of the cylinder is rounded with 2.5 nm radius. The whole system is surrounded by a perfectly matched layer (PML) where the substrate extends into the PML. The detector was placed parallel to the substrate surface at 1500 nm distance and spans a solid angle _{$\Omega =\{(\theta ,\varphi )|\theta =[0,7.5°],\varphi =[0,360°]\}$}. The integrated differential scattering cross-section σ_s used in the following discussion is given by

3. Simulation results

The main advantage that simulations offer compared to experiments is the possibility to study different effects related to material property changes independently from each other. We start with the change of dielectric function for a gap size of g = 5 nm in the top row of Fig. 2 . The first surprising result is that a change of the dielectric function of the disk from Pd to PdH results in the expected amplitude decrease and spectral broadening but combined with a blueshift of 3 nm instead of a redshift.

 

Fig. 2 Calculated scattering spectra for two values of the dielectric function of the Pd disk and scaling factor s. The gap size g before scaling is fixed at 5 nm. A change of the dielectric function from Pd to PdH blueshifts the antenna resonance by 3 nm whereas scaling the Pd disk with a factor of s = 1.04 causes a redshift of 13 nm. The superposition of these two effects yields a total redshift of 10 nm.

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What looks like a contradiction to the experimental results of [13] can be resolved by considering the hydrogen-induced lattice expansion in the Pd disk. It can be modeled in a simple way by multiplying both the diameter and the height of the disk with a scaling factor s. Due to the large field-enhancement in the area between the antenna and the disk (Fig. 3a ), the system should be extremely sensitive to the decreasing gap size associated with the scaling.

 

Fig. 3 (a) Calculated enhancement of the electric field on a plane parallel to the substrate through the center of the bowtie. (b) Calculated single particle scattering spectra for different disk diameters and an initial gap size of g = 7 nm. Note the linear redshift of the antenna resonance with increasing scaling factor s.

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Figure 3b shows the influence of scaling on the antenna resonance for an initial gap size of 7 nm. We observe a redshift linear in scaling that reaches 7 nm for a value of s = 1.04 while the scattering amplitude and linewidth remain mostly constant. The scaling parameter ranges from 1.00 to 1.05, chosen in accordance with PdH values of 1.038 given in literature [21].

The two competing effects of dielectric function mediated blueshift and lattice expansion induced redshift can be clearly identified in Fig. 2. Whereas the dielectric change causes a blueshift of 3 nm and a strong broadening of the resonance, the scaling by a factor of s = 1.04 induces a redshift of 13 nm. The superposition of these contributions results in a total redshift of 10 nm with a simultaneously broadened lineshape. This behavior agrees very well with the experimental data in [13].

To examine the influence of gap size on the magnitude of these two effects we calculated scattering spectra for different gap sizes and the parameter cases described in Fig. 2. In Fig. 4 the markedly different scaling behavior of the two competing effects can be observed. Whereas the blueshift induced by the dielectric change in the Pd remains mostly constant at 3 nm, the redshift associated with the disk's scaling can be tuned by changing the gap size. This scaling behavior enables the tailoring of the total hydrogen-induced redshift.

 

Fig. 4 Extracted spectral shifts due to the Pd/PdH transition and the growing of the Pd disk for different initial gap sizes g. Whereas the Pd/PdH-induced blueshift remains mostly constant at 3 nm, the magnitude of the scaling-induced redshift can be tuned by varying the gap size.

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Although we are able to numerically reproduce the direction of hydrogen-induced spectral shifts and the concurrent resonance broadening, there are still differences in magnitude between our simulated results and the experimental data reported in [13]. These differences are most probably due to deviations in gap size between the antenna and the disk as well as strain effects between the structures and the substrate which are neglected in our numerical calculations.

Still, our approach allows us to numerically explore optimized nanoantenna-enhanced hydrogen sensor geometries. A straightforward way to increase the sensitivity is to add a second antenna arm to the current design. To demonstrate this, we simulate a Pd nanodisk in the center of a full bowtie antenna.

Figure 5b shows that the total hydrogen-induced redshift is more than doubled by adding a second bowtie antenna opposite the first. However, this addition also redshifts the resonance position by 52 nm with regard to the single bowtie and thus requires a different figure of merit (FOM) to accurately compare the two designs. Since the full width at half maximum (FWHM) of our simulated spectra is difficult to determine we calculate and compare the _{${\text{FOM}}_{\lambda }=(\Delta \lambda /\Delta n)/{\lambda }_0$} in analogy to the _{${\text{FOM}}^{\text{*}}=(\Delta I/\Delta n)/I_0$} defined by J. Becker et al. [30]. Here _{${\lambda }_0$} is the resonance position and the refractive index change can safely be set to _{$\Delta n=1$} for the purpose of simply comparing different hydrogen sensing geometries.

 

Fig. 5 (a) Calculated enhancement of the electric field on a plane parallel to the substrate through the center of the double bowtie system. (b) Calculated scattering spectra for a double bowtie geometry with center Pd disk and a gap size of g = 5 nm. A significant enhancement of the total redshift can be observed.

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We obtain values of _{${\text{FOM}}_{\lambda }=1.4\cdot {10}^{-2}$} for the single bowtie and _{${\text{FOM}}_{\lambda }=3.5\cdot {10}^{-2}$} for the double bowtie geometry with an enhancement factor of 2.5. This strong enhancement can be understood in terms of the field enhancement in both geometries (Figs. 3a and 5a). Although the maximum enhancement is similar for both systems, the double bowtie geometry includes two hot spots of the electric field where the system is highly sensitive to the scaling of the center disk. This field concentration in the feed gap of an antenna pair is responsible for the strongly increased redshift that we observe.

Interestingly, this enhancement of the spectral shift should be even more pronounced in antenna geometries that do not exhibit field focusing behavior on their own.

To explore this, we consider a gold cut-wire (cuboid) antenna (width 60 nm, length 97 nm, height 40 nm) separated by a gap of g = 5 nm from the palladium disk. The length of the cut-wire was chosen to match its resonance position to the single bowtie from Fig. 2.

Since the field hot spots of a single cut-wire antenna are located at its corners away from the Pd disk [13], we expect a decreased spectral shift compared to the single bowtie geometry. Figure 6a shows that the cut-wire system indeed exhibits a much lower redshift of 5 nm when moving from Pd to PdH and scaling the disk by a factor of s = 1.04. This is in excellent agreement with the experimental results.

 

Fig. 6 (a) Calculated scattering spectra for a cut-wire antenna situated next to a Pd disk with a gap size of g = 5 nm. (b) Calculated scattering spectra for a double cut-wire geometry with center Pd disk and a gap size of g = 5 nm. A significant enhancement of the total redshift comparable to the double bowtie geometry can be observed.

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However, when considering a double cut-wire geometry by adding a second antenna arm to the system, we observe a large redshift of 28 nm (Fig. 6b), comparable to the double bowtie geometry. This shows that in contrast to single antennas, spectral shifts in double antenna geometries are governed less by the shape of the antenna and more by the field enhancement created in the feed gap between the two antenna arms.

Although the positioning of a Pd disk into the feed gap of two antenna arms adds some fabrication complexity compared to positioning it next to a single antenna, we believe that this further enhancement of the sensor response will soon be verified experimentally.

4. Conclusions and outlook

In conclusion, we have numerically investigated hydrogen-induced spectral shifts in antenna-enhanced hydrogen sensing geometries and found two competing effects: a blueshift attributed to the change of the dielectric function in Pd that is mostly unaffected by changes in gap size and a lattice-expansion induced redshift that can be tailored by changing the gap size. The insight we gained into the behavior of this system enables us to accurately model antenna-enhanced hydrogen sensors. The numerical characterization of structures with an added second antenna arm shows that this is a straightforward way to improve the sensing performance. Our modeling allows further optimization beyond simple antenna geometries and thus paves the way towards the realization of extremely sensitive plasmonic hydrogen sensors.