Numerical conversion efficiency of thermally isolated Seebeck nanoantennas

The approach of using resonant nanoantennas for harvesting applications has gained considerable attention over the last decade since they introduce a novel and efficient mechanism to confine light into small volumes, and thus, provide an enabling technology for energy gathering in the visible or infrared region.^1–3 Nanoantennas are metallic nanostructures that resonate to the free-propagating electromagnetic waves by inducing a high frequency AC resonant current along their volume.^4,5 This resonant current is subsequently exploited either to sense or recover the electromagnetic energy.^6,7 Different harvesting mechanisms such as the use of fast rectifying diodes coupled to the antennas have been extensively investigated in order to recover this confined energy in an efficient way,^8–11 thereby opening, a new route for the advanced design of harvesting devices.

In this regard, Seebeck nanoantennas are devices that have recently reappeared as an alternative to harvest the electrical energy confined by such nanoantennas.^12–15 These devices are simple nanometer-sized thermocouples which act as nanoantennas and generate DC power by the Seebeck effect when they operate at resonance. Furthermore, Seebeck nanoantennas exploit the temperature gradients occurring along their structure which are induced by the resistive heating generated by the resonant current. The thermal gradients along the nanometer-sized thermocouples in turn generate a DC Seebeck voltage V_OC that can be sensed at the open edges of the nanoantennas;^16,17 hence, defining a transduction mechanism to harvest energy.

Nanoantennas exhibit some advantages as energy harvesters when the thermoelectric Seebeck effect is employed as the recovering mechanism. For example: (i) they can be tuned to resonate at a wide range of frequencies, (ii) they can generate DC power from any high-frequency electromagnetic wave because their speed of operation is not limited by an external load element, such as rectifying diodes, (iii) their architecture reduces the technological process required for their manufacture, implying only two-steps of high resolution lithography and metal thin-film deposition, and (iv) numerical results show they can exhibit higher conversion efficiencies $ηe$ than the currently fabricated rectennas.¹⁸ 

Even though the Seebeck nanoantennas present some advantages for energy harvesting applications, their conversion efficiency $ηe$ falls far below from the efficiency $ηth$ that standard metallic thermocouples exhibit.^14,19 The main reason is that the thermal gradients induced inside the antennas are undoubtedly poor because the heat they generate dissipates in a uniform way towards the surroundings. In order to maximize the thermal gradients along the nanoantennas, the outer ends of their arms must be kept as cool as possible and their center as hot as possible.

In this letter, we evaluate the thermal isolation as a strategy to increase the performance of the Seebeck nanoantennas and discuss their suitability and scope for harvesting applications (advantages or disadvantages). This type of optimization is achieved by employing the so-called air-bridge technology,^20–22 technique which allows the suspension of the devices in the air above the substrate. Moreover, this strategy technology removes the major thermal conduction pathway out of the harvesters (towards the substrate), confining heat at the center of the structures and also enhances the lateral dissipation. The air-bridge technology has been successfully employed to increase the performance of antenna-coupled microbolometers,²⁰ and in this proposal, its extended use in Seebeck nanoantennas is also considered.²³ 

The performance of the isolated nanoantennas is determined by considering a multi-physics problem in which the heat-diffusion and electromagnetic equations are solved. Numerical simulations are performed with COMSOL Multi-physics (ver5.2) commercial software package based on the finite element method that solves the equations required for this analysis.

A related point to consider is that this thermal analysis is presented as an aid in the design of Seebeck nanoantennas and the main goal is to unveil the limits of thermal isolation as a strategy of optimization. Furthermore, the efficiency of the Seebeck nanoantennas is expected to be further increased by tuning their geometrical parameters by proposing different geometries or employing optical concentrators such as Fresnel zone plates. The effect of all these parameters on the conversion efficiency of the Seebeck nanoantennas is currently under study and is within the scope of future contributions.

The goal of this contribution is to establish the dependence of the efficiency of the Seebeck nanoantennas with their thermal optimization. For this purpose, we evaluate and compare the response of the devices as shown in Fig. 1.

We consider two types of architectures: (a) the case of nanoantennas laying on the top of a half-space SiO₂ substrate (Fig. 1(a)), and (b) the case of nanoantennas laying on top of a suspended SiO₂ membrane (200nm-thick) with two lateral windows which define a narrow bridge of length L (Fig. 1(b)). In the first type of architecture, the thermal isolation of the nanoantennas is not optimized; they are employed as reference devices whose response is affected by the heat dissipation towards the substrate. In the second type of architecture the conduction pathways of the nanoantennas are optimized by means of a narrow bridge on the SiO₂ membrane (which confines the heat along the axis of the antennas). The bridge is supported by (horizontal) silicon pillars which, in turn, substantially enhance the lateral dissipation of heat and will cool the ends of the nanoantennas due to its high thermal conductivity (⁠$κSi∼153$ W/mK).¹⁹ 

Because the focus of this study is on the thermal isolation of the devices, their effects on conversion efficiency and their limits for harvesting applications, we select to arbitrarily analyze the simple case of dipole nanoantennas, even though higher conversion efficiencies are exhibited by different dipole geometries such as the tapered or the diabolo structures.^3,24 It is worth highlighting that the efficiency of the Seebeck nanoantennas is expected to increase further by proposing different geometries than the dipoles, but here, we evaluate the limits of thermal isolation and thermo-electric properties of metals. The proposed nanoantennas were tuned to recover the thermal wavelengths around $10.6μm$ (28.3 THz). This task was achieved by adjusting their length to $2.5μm$⁠, a resonant length that has been reported for gold dipoles.²⁵ 

On the other hand, the arms of the nanoantennas are made of different metals, nickel and titanium (denoted by different colors in the figure), and thus, they form thermocouples whose bimetallic junction or hot-point is located at the center of the nanoantennas, where the Joule heating is higher. The DC open voltage V_OC that these devices could generate by Seebeck effect (when they operate at resonance) is given by the relationship:^16,17

where $ΔT$ denotes the temperature difference between the center (hot-point) and the extremes of the antennas, and S_A and S_B denote the Seebeck coefficients of the metals. Nickel and titanium are used as building materials due to the considerable difference in their Seebeck coefficients (S_Ni= $−19.5μV/K$ and S_Ti =7.19 $μV/K$⁠)²⁶ and their low thermal conductivities (⁠$κNi=90$ W/mK and $κTi=21.8$ W/mK),¹⁶ which enhance the performance of the devices.

The analysis is performed by exciting the devices with a monochromatic plane wave whose polarization matches that of the nanoantennas. The irradiance of the plane wave S, is arbitrarily set to 117 W/cm² (E₀=29700 V/m) and is kept fixed for each single frequency of excitation. This irradiance intensity is commonly employed for the experimental characterization of infrared antennas.²⁷ The numerical model is built by using the refractive index of materials in the wavelength range of interest.²⁸ 

Once the Seebeck nanoantennas are designed, we introduce a simplified method to evaluate the figures-of-merit of the harvesting devices, such as the conversion efficiency. This method consists in evaluating the temperature distribution that the incident light induces on the devices. From these calculations, the parameter $ΔT$ can be extracted and the Seebeck voltage V_OC can be easily derived from imaginary electrodes by using the relationship (1). Moreover, the Seebeck voltage can be used to find other parameters such as the responsivity or the conversion efficiency of the devices.

The temperature distribution of the nanoantennas is evaluated by considering the nanoantennas as the only source of heat. The heating of the nanoantennas is considered as full optical problem since heat is generated by the ohmic losses associated to the resonant current. The density of heat q_th [W/m³] is found by using the Joule expression q_th=J$⋅$E, where J denotes the current density generated by the electric-field distribution E inside the nanoantennas; field-distribution automatically computed by COMSOL. The density of heat q_th is then employed to numerically solve the heat-conduction equation inside and outside the nanoantennas and find the temperature distribution of the whole devices (by imposing that the bottom of the silicon pillars is thermalized at room temperature).

The conversion efficiency of the devices can be evaluated, from a theoretical perspective, by assuming that devices are incorporated into an acquisition system, and thus, inducing the electrical power exchange between both elements.^16,29–31 In this case, the nanoantennas can supply a maximum amount of electrical DC power P_DC to the system given by:²⁹ 

where R_int refers to the internal resistance of the thermocouples, which is considered as the effective ohmic resistance of the materials.^16,19 The proposed devices present an ohmic resistance of $4.34Ω$ and $25.8Ω$⁠, which are calculated by using the resistivity values reported for nickel and titanium (⁠$6.9μΩ·cm$ and $41.6μΩ·cm$⁠, respectively). This power can only be supplied if the impedance of the acquisition system matches the internal resistance of the nanoantennas R_int.^19,29

On the other hand, the power received by the antennas P_rec can be evaluated by using the relationship:

where P_rad and P_loss are the power re-emitted by the antenna and the power dissipated by the ohmic losses, respectively.

The overall energy conversion efficiency $ηe$ can thus be evaluated as the ratio of the power generated by the thermocouples P_DC to the power received by antennas P_rec, expressed as:

Fig. 2,(a) shows the temperature map that both types of dipole architectures exhibit for a normal incident plane wave whose wavelength is $10.6μm$ (28.3THz). Numerical results show that the architecture based on the silicon bridge increases its temperature by more than three degrees above room-temperature. As mentioned above, this air-bridge architecture confines the resistive heat all along the axe of the antenna but also cools their ends due the low-thermal conductivity of the silicon pillars supporting the bridge (⁠$κSi∼153$ W/mK). For instance, the temperature difference between the hot spot and the open ends $ΔT$ reached by the bridge structure is around 920 mK, whereas, the half-space structure reaches a $ΔT$ around 210 mK. Those thermal gradients will induce on the nanoantennas Seebeck voltages around $22μV$ and $5μV$⁠, respectively.

The conversion efficiency of the antennas $ηe(%)$ as a function of frequency is determined by using the relationship (4) shown in Fig. 3(c). Numerical results show that the dipole antennas on the bridge structures (open squares plot) exhibit conversion efficiencies around 10^-4 %, 2 orders of magnitude higher than the previously reported Seebeck nanoantennas¹⁴ (from 10^-6 to 10^-4%) and 5 orders of magnitude higher than rectennas.¹⁸ On the other hand, the conversion efficiency values are close to the theoretical conversion values of standard thermocouples based on metals, reaching the limits of this thermoelectric technology.^16,29

It is worth mentioning that even if the enhancement in the temperature gradient is not a surprise, by performing this analysis, we were able to determine the scope of the thermal isolation on the response of the antennas. On the other hand, the presence of thermal isolation makes it more difficult to manufacture an actual device. For this reason, in order to make the Seebeck nanoantennas useful for harvesting applications, different strategies should be considered such as use of different geometries or materials. For instance, materials with higher Seebeck coefficients could be employed. Attractive candidates could be the antennas based on semiconductor materials such as the photoconductive antennas,³² which exhibit Seebeck coefficients thousands of times larger than the Seebeck coefficients of metals, promising open voltages in millivolts.

Finally, we proceeded to evaluate the relevance of the geometrical parameters of the bridge, such as the width W and the length L, on the efficiency of the nanoantennas (Fig. 1). In order to determine this dependence, the next step is to perform a series of simulations in which the length L of the bridge was varied (from $0.4μm$ to $2.6μm$⁠) while the width W is kept fixed. This procedure is repeated for different width values (from $0.4μm$ to $1.2μm$⁠). The numerical results are shown in Fig. 4(a). Higher efficiencies can be obtained by reducing the width of the bridge or by increasing its length, as could be expected. These type of geometrical variations in turn reduce the contact area between the dipole and the bridge, and this indeed, represents a better thermal isolation of the resonant structure, and as a result, higher thermal gradients.

A similar study is performed in order to find the relevance of the efficiency on the substrate thickness. To perform this analysis, the width and the length of the bridge are kept fixed to $0.6μm$ and $2.4μm$⁠, respectively. Results show that the efficiency of the Seebeck nanonatennas decreases by increasing the thickness of the bridge (Fig. 4(b)). This result shows that in thicker substrates, the heat flow through the silicon pillars is blocked, which translates in the suppression of the cooling process at the ends of the antennas.

In summary, we have evaluated by numerical simulations a new strategy to increase the conversion efficiency of the Seebeck nanoantennas, a strategy which is based on the thermal optimization of the device. Numerical results show that by using a 200nm-thin SiO₂ membrane (air-bridge technology), which suspends the antennas in the air above the substrate, higher thermal gradients and higher conversion efficiencies can be obtained. These type of air-bridge architectures allow efficiencies to reach around 10^-4 %, 2 orders of magnitude higher than the previously reported Seebeck antennas and 5 orders of magnitude higher than rectennas. Due to the difficulties of achieving the experimental realization of air-bridge architectures, some other strategies should be employed in order to make the Seeebeck nanoantennas useful for harvesting applications.

The corresponding author gratefully acknowledges the support from The National Research System (SNI) of Mexico under the research grant 56874.