# Photo-Thermoelectric Conversion of Plasmonic Nanohole Array
## 3 Results and Discussion
### Quantification of Plasmonic Local Heating  
As a proof of concept, we estimated the amount of plasmonic local heating generated at the Ag nanohole arrays, experimentally and numerically. The Ag nanohole arrays with a periodicity of 300 nm produced a maximum electric current of 173 nA at a wavelength of 690 nm. The conductivity of the 40 nm-thick Ag film, as measured via a four-probe method, was \(1.23\times 10^{7}\) S/m. In addition, we measured the conductivity of the nanohole arrays by preparing a sample consisting of a film entirely covered in Ag nanohole arrays. The measured conductivity of these nanohole arrays was \(0.60\times 10^{7}\) S/cm, which was nearly half of the value measured for the Ag film. The thermal conductivities of the Ag film and nanoholes were estimated by using the Wiedemann-Frantz law, which states that the ratio of the thermal and electrical conductivities of a metal is proportional to the temperature [20]. The Lorenz number for Ag, \(2.42\times 10^{-8}\) W \(\Omega\)/K\({}^{2}\), was used for this estimation [21,22]. The estimated thermal conductivities of the Ag film and nanoholes, being 88.1 and 58.0 W/m\(\cdot\)K, respectively, are much greater than those of the resistant polymer and air [23]. We used these conductivity values for the nanohole area and Ag film in a thermoelectric simulation model, simulating the thermoelectric performance of the device by using a thermoelectric module within the COMSOL Multiphysics package. We determined that the temperature of the heat source on the device generates the same value of current as that obtained experimentally. Consequently, the thermoelectric simulation revealed that plasmonic local heating by 4.61 K, at the Ag nanoholes, generated a photocurrent of 173 nA, the measured current. We also calculated the plasmonic local heating generated at the Ag nanoholes by using the COMSOL software package, including its heat transfer module. Supplementary Materials Figure S4 illustrates the model used for this calculation. Briefly, the model involves a single nanohole in a unit that is surrounded by heat flux layers to permit heat flow from the inside to the outside of the unit. Perfect matching layers were put on the top and bottom of the unit. Experimental values were used for morphological parameters and for the conductivity of the Ag film. In the simulation model, illumination was by a continuous wave light source, with a single wavelength, and an optical power of \(2.6\times 10^{2}\) W/cm\({}^{2}\). Since plasmonic local heating occurs as a result of the Joule effect, the heat power density \(q\) can be calculated using Equation (1) [24]. \[q(\mathbf{r})=\frac{\omega}{2}\;\text{Im}(\varepsilon(\omega))\varepsilon_{0} \big{|}\mathbf{E}(\mathbf{r})\big{|}^{2} \tag{1}\]  
where \(\varepsilon_{0}\) is the vacuum permittivity, \(\varepsilon(\omega)\) is the relative permittivity of the material, and \(\big{|}\mathbf{E}(\mathbf{r})\big{|}^{2}\) is the intensity of the electric field. Figure 3 shows a cross-sectional SEM image of the Ag nanohole alongside the modeled electric field distribution for the same view of the Ag nanohole. The electric field distribution was calculated for a wavelength of 690 nm. Strong electric field confinement at the edge of the Ag film, inside the nanohole, was observed, indicating that the edges of the film are the main sites of plasmonic local heating. The edges of the Ag disk at the bottom of the nanohole were also expected to be heat sources; however, we concluded that the Ag disks did not contribute directly to thermoelectric conversion, owing to the electrical discontinuity. Heat transfer in a system can be described by a conventional heat transfer expression [25],  
\[\rho(\mathbf{r})c(\mathbf{r})\frac{\delta T(\mathbf{r})}{\delta t}=\
abla\kappa( \mathbf{r})\
abla T(\mathbf{r})+q(\mathbf{r}) \tag{2}\]  
where \(T(\mathbf{r})\) is the temperature, \(\rho(\mathbf{r})\) is the mass density, \(c(\mathbf{r})\) is the specific heat, and \(\kappa(\mathbf{r})\) is the thermal conductivity. In fact, [PERSON] et al.