Optical localized heating in the nanoscale has recently attracted great attention due to its unique small hot spot size with high energy. However, the hot spot size is conventionally constrained by the diffraction limit. Plasmonic localized heating can provide solutions to this limitation in nanoscale patterning, cancer treatment, and data storage. Plasmonic approaches to overcome the diffraction limit in hot spot size have mainly utilized the excitation of surface plasmon or localized surface plasmon resonance. However, achieving plasmonic localized heating by the excitation of magnetic polariton has not been researched extensively yet. In this work, we numerically investigated the optical response of a nanoscale metamaterial composed of a gold nanowire array and a gold film separated by an ultrathin polymer spacer using ANSYS High Frequency Structural Simulator. A strong absorption peak at the wavelength of 760 nm was exhibited, and the underlying physical mechanism for the strong absorption was verified via the local electromagnetic field distribution to be magnetic resonance excitation. An inductor-capacitor circuit model was used to predict the magnetic resonance wavelength and compare with the numerical results for varied geometrical parameters. Volume loss density due to the strong local optical energy confinement was transferred as heat generation to an ANSYS thermal solver to obtain the local temperature profile. The steady state temperature profile shows an average temperature of 145 °C confined in a local area as small as 33 nm within the spacer, with a full-width at half-maximum of 50 nm along the x-direction. Moreover, the temperature rise from ambient drops to half its maximum value at a distance of 5 nm from the top of the spacer along the z-direction. This clearly demonstrates plasmonic localized heating beyond the diffraction limit via magnetic polariton excitation. Furthermore, the transient temperature profile shows that the system reached steady state within ∼0.36 *μ*s.

## I. INTRODUCTION

The spot size of an optically focused light is constrained by the Abbe diffraction limit, which in turn limits the hot spot size that can be generated by the focused light or laser.^{1} This hot spot is too large for some applications such as nanoscale patterning,^{2} photothermal therapy,^{3–5} and super high areal density data recording.^{6,7} However, through the excitation of plasmonic resonances in certain metamaterials, the diffraction limit can be overcome to obtain hot spots on the order of tens of nanometers. Cao *et al.* utilized the local temperature rise that results from plasmonic modes in metallic nanostructures to grow semiconductor nanowires and carbon nanotubes.^{8} Sotiriou *et al.* used the localized thermal heating effect induced by the plasmonic coupling of silica coated gold and Fe_{2}O_{3} nanoparticles to kill cancer cells.^{9} The localized heating effect in the previous two applications was due to the excitation of surface plasmon resonance. Challener *et al.* proposed a method for localized heating in a heat assisted magnetic data recording system.^{10} The method utilized localized surface plasmons and it efficiently enhanced the data storage density. Apart from surface plasmon resonance and localized surface plasmon resonance, applying magnetic polariton (MP) to achieve plasmonic localized heating is not well understood yet.

The unique behavior of MP excitation has enabled it to be used in some applications such as energy harvesting^{11,12} and sensing.^{13} Several metamaterial structures were employed to excite MP. These structures include periodic strips coupled to a metallic film,^{14} slit arrays,^{15} and deep gratings.^{16} Moreover, MP excitation has also been found and numerically investigated in film-coupled nanoparticles.^{17}

Exciting plasmonic resonance modes through utilizing nanowires, which can be easily applied to films, is a fast and inexpensive way of achieving spectrally tunable localized heating. A layer of monodispersed nanoparticles, through self-assembly, can be applied to a film by several methods such as utilizing long-range attractive interactions or capillary forces between the nanoparticles.^{18–21} However, neither the excitation of MP between film-coupled horizontally aligned nanowire arrays nor the plasmonic localized heating through this MP excitation has been investigated extensively yet.

In our previous work,^{22} we investigated the optical properties of a film-coupled horizontally aligned gold nanowire (AuNW) array metamaterial, and selective absorption due to MP was observed. In this work, we further analyze the optical and thermal behaviors of the structure. An inductor-capacitor (*LC*) circuit model will be used to predict the MP resonance wavelength and compare with the numerical results for varied geometrical parameters. Moreover, a detailed study of the local steady state and transient temperature profiles due to the resonant light absorption of MP will be presented to demonstrate plasmonic localized heating at the nanometer scale.

The structure of interest is shown in Fig. 1. The film-coupled nanowire metamaterial structure consists of an array of horizontally aligned gold nanowires (AuNWs) that are periodically dispersed on a gold film (Au film) with an ultrathin polymer spacer in between. This structure rests on a silicon substrate which does not affect the optical properties due to the optically opaque Au film. The AuNWs have a diameter *d* and periodicity *p*, and the thickness of the gold film and the polymer spacer are *h* and *t*, respectively. The structure is assumed to be uniform and infinitely long along the axial direction of the AuNWs.

## II. RESULTS AND DISCUSSION

A continuous broadband electromagnetic plane wave was incident on the structure with a spectral range from 500 nm to 1000 nm. The plane of incidence is the plane formed by the surface normal and the incident wavevector *K*_{inc} and depending on the orientation of the electric and magnetic fields with respect to the plane of incidence, the wave is either called a transverse electric (TE) wave or a transverse magnetic (TM) wave. Simulation of the optical response of the structure was performed through the use of the numerical finite element based solver, High Frequency Structural Simulator (HFSS). The optical behavior of the structure under TE and TM waves was previously studied,^{22} and an absorptance peak was found for TM normal incidence at a wavelength of 760 nm. This absorptance peak is due to the excitation of MP. The optical constants for gold and silicon were taken from Refs. 23 and 24, respectively. For the lossless polymer spacer, the refractive index n was taken as 1.5.^{25} Refer to Ref. 22 for a detailed modeling setup of the optical simulation.

To explain the underlying mechanism for the excitation of MP, the electromagnetic field distribution is presented in Fig. 2(a) at the resonance wavelength of 760 nm. The plot is a cross-sectional view of the metamaterial structure, and the axes represent the dimensions of the structure. The color of the contour indicates the strength of the normalized magnetic field $log\u2009|H/H0|$ inside the structure, and the arrows represent the strength and direction of the electric field *E*. A red ellipse is drawn around the spacer with the arrows showing the direction of current density *J* inside the polymer spacer between the AuNW and Au film. It can be seen from Fig. 2(a) that the magnetic field is strongly enhanced inside the current loop, and a confinement of the electromagnetic energy can also be observed. This resonance behavior only occurs in the case of TM incidence since the magnetic field must be parallel to the AuNWs in order to excite an in-plane resonant current loop within the structure. This type of magnetic response has been found and studied in other grating structures where it was determined as MP.^{14,16,26} However, MP resonance has not been studied in film-coupled nanowires yet.

Figure 2(b) shows the electrical field along the top surface of the Au film. This will be used to determine parameters in the inductor–capacitor (*LC)* circuit model that will be discussed next. Since a resonance current loop is excited within the structure at MP resonance, an *LC* circuit model is used to analogize the structure to an electric circuit, as well as predict the resonance wavelength by zeroing the total impedance of the electric circuit. This model was previously used to successfully predict the MP resonance wavelength in film coupled structures consisting of 1D gratings^{26} and 2D patches.^{27} These structures have one feature in common: the top gratings are basically a rectangular or parallelogram shape in contrast to the nanowire structure proposed in this paper. We modified the *LC* circuit model in order to account for this difference in structure. Through this modification, the film-coupled AuNW structure can be treated as an effective film-coupled grating structure based on the electromagnetic field distribution shown in Fig. 2. A simplified schematic view is plotted in Fig. 3(a) to help explain the *LC* circuit model which is shown in Fig. 3(b). The dimensions of the metamaterial structure are as follows: the AuNW radius *R* is 125 nm, the polymer spacer thickness *t* is 10 nm, and the period *p* is 400 nm. Based on the electric field distribution as well as the material properties, the effective inductor and capacitor components in the circuit can be defined. This was accomplished through approximating the AuNW as a plate with an effective thickness $heff$ and a gap distance $dgap$ away from the polymer spacer. The mutual inductances between the effective AuNW and Au film can be calculated from $Lm,NW=Lm,film=\mu 0w(t+dgap\u22120.5heff)/(2l)$, where $\mu 0$ is the permeability of free space, $w$ is the width of the effective plate, and $l$ is the length of the structure in the y-direction which can be cancelled out through the calculations. Note that the effect of drifting electrons is not neglected as it contributes to the kinetic inductance for both the AuNW and the Au film, which are defined as $Lk,NW$ and $Lk,film$, respectively. The kinetic inductances can be calculated from $Lk,NW=Lk,film=\u2212w/(\omega 2heffl\epsilon 0\epsilon \u2032m)$, where $heff$ is the height of the effective plate, $\omega $ is the angular frequency of the incident light, and $\epsilon \u2032m$ is the real part of the dielectric function of gold. Moreover, the effective plate and the Au film separated by the polymer spacer can be treated as a parallel plate capacitor with capacitance $Cspacer=c1\epsilon d,spacer\epsilon 0wl/t$, where $c1$ is a coefficient that accounts for non-uniform charge distribution at the metal surface (taken as 0.2 for our calculations^{28}), and $\epsilon d,spacer$ is the dielectric function of the polymer spacer. Furthermore, capacitance due to the air gap should also be considered as $Cgap=c1\epsilon d,air\epsilon 0wl/(dgap\u22120.5heff)$, where $\epsilon d,air$ is the dielectric function of air.

Determining the geometric parameters of the effective Au plate ($w$, $dgap$, and $heff$) will now be discussed. The excited current loop at MP resonance is simplified as an ellipse. The long axis of the ellipse is $a=aiR$, where $ai$ is the long axis coefficient and $R$ is the radius of the AuNW. The short axis of the ellipse is $b=(t/2)+biR$, where $bi$ is the short axis coefficient. The long and short axis coefficients are based on the distribution of the electric field in the structure shown in Fig. 2(b). The long axis coefficient $ai$ is determined as 188/250 = 0.75, as the electric field decayed to 0 at *x* = 94 nm, which is the edge of the current loop. The short axis coefficient $bi$ is obtained based on the approximation that the current penetrates into the Au film by two penetration depths. The penetration depth $\delta =\lambda /(4\pi \kappa )$ is found to be 13.28 nm, where $\kappa $ is the extinction coefficient.^{23} Therefore, $bi$ is equal to 26.56/125 = 0.212. The effective width is determined as the distance between the two intersection points between the ellipse and circle. Next, the distance between the plate and the polymer spacer $dgap$ is found through $w=2R2\u2212(R\u2212dgap)2$, which is based on the geometric relations as seen from Fig. 3(a). Furthermore, the height of the effective plate is then found through $heff=0.5(biR)$, which is equal to one penetration depth $\delta $. By collecting the previous parameters together, the total impedance can be expressed by Equation (1), where $Ctot$ is the total capacitance of $Cspacer$ and $Cgap$, calculated via Equation (2)

where

MP resonance occurs when the total impedance $Ztot$ is zero. Therefore, the MP resonance wavelength $\lambda 0$ can be calculated by Equation (3), where $c0$ is the speed of light

The predicted MP resonance wavelengths for different geometries are now compared in Fig. 4 with the ones obtained from HFSS. This is done to verify that the *LC* model follows the same trends as the HFSS results. Figures 4(a) and 4(b) show the effects on the absorptance of the metamaterial structure by varying the AuNW diameters and spacer thicknesses, respectively. Fig. 4(a) shows that the absorption peak redshifts with increased nanowire diameter. On the other hand, it is observed from Fig. 4(b) that as the spacer thickness increases, the spectral absorptance blueshifts. Figs. 4(c) and 4(d) show the absorption peak (MP resonance) wavelengths as a function of AuNW diameter and spacer thickness, respectively. The non-varying dimensions in both plots are the basic dimensions used throughout the paper. Both plots compare the MP resonance wavelength predicted by the *LC* circuit model with that obtained from the HFSS simulations. It can be seen from the two comparison plots that the *LC* circuit model predictions agree well with the HFSS results with a maximum relative error of 5%. Fig. 4 demonstrates the spectral tunability of the absorptance peak of the proposed film-coupled AuNW array metamaterial structure, which allows for a wider field of applications for localized heating.

The geometric dependence of the MP resonance wavelength observed in Fig. 4 can be understood by the *LC* circuit model. When the AuNWs' diameter increases, the MP resonance peaks redshift because of increasing $Lm$, $Lk$, and $Ctot$ which leads to a larger resonance wavelength based on Equation (3). However, increasing the spacer thickness leads to a smaller spacer capacitance $Cspacer$, resulting in a smaller MP resonance wavelength or blueshift in Fig. 4.

In order to study the thermal response of the metamaterial structure, the solution of the HFSS simulation under MP excitation was transferred into an ANSYS thermal solver. The thermal conductivities of the materials used in the thermal simulation were taken as constant values: 315 W/(m·K), 148 W/(m·K),^{29} 0.026 W/(m·K),^{30} and 0.215 W/(m·K),^{31} for gold, silicon, air, and the polymer spacer, respectively. Refer to Ref. 22 for a detailed modeling setup of the thermal simulation. Fig. 5(a) shows the contour of the internal heat generation. The axes represent the physical dimensions of the structure, and the contour represents the amount of heat generation. It can be observed that the heat generation is greatly enhanced in the bottom of the AuNW and the top of the Au film. The absorbed energy can now dissipate to the rest of the structure through conduction and convection. Moreover, this heat generation confinement matches the confinement that is observed in the electromagnetic field distribution in Fig. 2(a) but with only the metallic material absorbing the energy. To demonstrate localized heating, the steady state temperature distribution was obtained and is shown in Fig. 5(b). The axes represent the physical dimensions of the structure, and the contour represents the steady state temperature. For an incident power of 0.2 mW over a 400 nm $\xd7$ 400 nm domain, the steady state temperature of the hot spot in the polymer spacer increased to an average of 145 °C at the top of the polymer spacer where it coincides with AuNW. This enhancement is confined in a region that is approximately 33 nm wide as can be seen from the zoomed-in steady state temperature in Fig. 5(b). This verifies, numerically, that this structure can be used to overcome the diffraction limit of light which is on the scale of a few hundred nanometers. The total heat flux contour throughout the metamaterial structure was also obtained to help understand the heat transport process as can be seen from Fig. 5(c). The axes represent the physical dimensions of the structure, and the contour represents the total heat flux. It can be observed that the majority of the heat flux is directed from the heat generation regions towards the bottom of the metamaterial structure. This means that conduction heat transfer is dominating rather than convection, which will be verified next.

To check for the convection heat transfer effect, multiple convection heat transfer coefficients, $hconv$ (5 W/(m^{2}·K)– 50 W/(m^{2}·K)), were used for the surfaces in contact with air, namely, the outer surface of the nanowires and the top surface of the polymer spacer. Fig. 6 shows the steady state temperature rise (ΔT) at three temperature probes in the polymer spacer. The x-axis represents the convection heat transfer coefficient, and the y-axis represents ΔT from the ambient temperature (22 °C). The inset of Fig. 6 shows the location of the probes that were used for the plot. The top and bottom probes are 0.5 nm deep into the polymer spacer from its top and bottom surfaces, respectively. Meanwhile, the center path is located halfway into the polymer spacer. Moreover, the three probes were placed at the center of the structure with respect to the x-direction. It can be seen from Fig. 6 that convection heat transfer did not have a significant effect on the steady state temperature. To understand why convection heat transfer was not significant, a comparison was made between the thermal resistance for convection, $\u2009Rconv=1/(hconvA)$, and conduction across the polymer spacer, $Rcond=t/(kA)$, where $hconv$ is the convection heat transfer coefficient, $A$ is the cross sectional area that heat crosses, $t$ is the thickness of the polymer spacer, and $k$ is the conductive heat transfer coefficient. It was found that the convective thermal resistance was six orders of magnitude higher than the conductive thermal resistance. This means that heat would transfer through conduction much more readily than through convection. To check for the radiation heat transfer effect, an emissivity of 1 was used for the radiating bodies. Radiation heat transfer was found to be insignificant as the steady state temperature did not change when applying radiation boundary conditions for a black body, which represents the maximum radiated power. To verify this, first, the radiation thermal resistances for the AuNW and the polymer spacer were calculated by $\u2009Rrad=1/(hradA)$. The temperatures of the AuNW and the polymer spacer were taken as uniform with a value equal to the maximum temperature of the structure. This resulted in the minimum radiation thermal resistance and was used in order to be able to make a comparison with the conduction thermal resistance across the polymer spacer. It was found that radiation thermal resistance from the AuNW was five orders of magnitude larger than the conductive thermal resistance, and for the polymer spacer, it was six orders of magnitude larger. This indicates that conduction is the dominant mode of heat transfer in the structure, and radiation heat transfer is insignificant in this case.

Fig. 7(a) shows the ΔT profile along three paths that cross the polymer spacer at different depths. The y-axis represents ΔT from the ambient temperature (22 °C). The inset of Fig. 7(a) shows the three paths that were used for the plot. The top and bottom paths are 0.5 nm deep into the polymer spacer from its top and bottom surfaces, respectively. Meanwhile, the center path is located halfway into the polymer spacer. Fig. 7(a) provides us with a more accurate way of knowing the width of the localized hot spot, which is determined to be ∼33 nm with an average ΔT of 123 °C. Note that the full-width at half-maximum (FWHM) in the polymer spacer is 50 nm along the x-direction. The FWHM is defined as the width (in the polymer spacer, along the x-direction) at which ΔT is half its maximum value. Moreover, to determine the amount of incident power required to heat up the structure to various temperatures, the effect of the incident power on the steady state temperature is also studied and is shown in Fig. 7(b). The plot shows the steady state temperature as a function of incident power. The temperature probes used for this plot are the same ones used for Fig. 6. Moreover, the depth of the structure in the y-direction was fixed at 400 nm to determine the amount of incident power in milliwatts, which in the simulations was varied from 0.02 mW to 0.2 mW. Note that the temperature of the top of the polymer spacer is significantly affected by the incident power, while the temperature of the bottom of the polymer spacer does not change significantly when increasing the incident power. This is because the Au film is highly conductive, which leads to the temperature of the bottom of the polymer spacer to remain at the same temperature as the Au film. On the other hand, the temperature of the top of the polymer spacer increases because a larger heat flux induces a larger temperature difference between the top and bottom surfaces of the polymer spacer.

The transient temperature results for the continuous light source are shown in Fig. 8. Note that the ANSYS thermal solver uses adaptive time-stepping and the chosen initial time step was 3 ns, the minimum time step was 0.3 ns, and the maximum time step was 30 ns. To study how ΔT and the width of the hot spot develop with increasing time, ΔT along the center path of the polymer spacer is plotted in Fig. 8(a) at different time steps. The width of the localized hot spot along the center path is determined to be ∼16 nm with a ΔT of ∼66.5 °C after ∼0.6 *μ*s. Moreover, a vertical path across the Au film, the polymer spacer, and the AuNW is used in Fig. 8(b) to show ΔT change across different components of the structure, with increasing time. It can be seen from Fig. 8(b) that ΔT across the Au film remains constant with increasing time. In the polymer spacer region, a sharp increase in ΔT is observed and this increase becomes even sharper with increasing time. However, the temperature across the AuNW is also relatively constant, but this time, it changes with time. The cause of the constant temperature in the components made from gold is that gold has a high thermal conductivity which leads to a more uniform temperature profile. Note that, within the polymer spacer along the z-direction, the steady state ΔT reduces to half its maximum value at a distance of 5 nm from the top of the polymer spacer. Furthermore, to determine when the structure reaches steady state, the transient temperature response for three temperature probes was studied and the results are presented in Fig. 8(c). The temperature probes used here are the same ones used in Fig. 6. Moreover, sufficient time was given for the temperature to stabilize and the system to reach steady state. The time scale used for the transient response was 0.6 *μ*s. The x-axis in the plot represents the time in *μ*s, while the y-axis represents ΔT at the probes. It can be seen from Fig. 8(c) that the temperatures of the three probes stabilize at around ∼0.36 *μ*s which is when the relative change in ΔT slows down to below 1%. This means that the system reaches steady state within ∼0.36 *μ*s.

## III. CONCLUSION

Plasmonic localized heating at the nanoscale was demonstrated through numerically investigating a film-coupled AuNW array metamaterial. HFSS simulations showed that the structure exhibits selective absorption at MP resonance, which, with the chosen structure dimensions, was at a wavelength of 760 nm. Moreover, the ANSYS thermal solver simulations indicated a steady state average temperature of 145 °C in a hot spot of ∼33 nm, where the electromagnetic energy was confined, demonstrating nanoscale plasmonic localized heating. In the polymer spacer, the FWHM was 50 nm along the x-direction, and ΔT dropped to half its maximum value at a distance of 5 nm from the top of the spacer along the z-direction. Furthermore, the transient temperature profiles showed that the system reaches steady state within ∼0.36 *μ*s. The findings of this work may facilitate applications in the fields of energy harvesting (by using multi-sized nanowires to enable broadband absorption), data storage (high speed, high density storage), and biotherapy (killing cancer cells without harming nearby healthy cells).

## ACKNOWLEDGMENTS

This work was supported by the National Science Foundation under CBET-1454698. H.A. would like to thank King Saud University and the Saudi Arabian Cultural Mission (SACM) for their sponsorship for his Ph.D. study at ASU.