1. Introduction

With the rapid development of nanotechnology and optoelectronic materials, a series of new systems and devices have been developed at an increasing rate during the past decades. Nano-rectenna as one of the new nano devices that have been used as infrared (IR) detectors [1], novel solar cells [2], optoelectronic devices [3], near-field imaging [4], and biochemical sensors [5] has attracted great attention of scientists. A nano-rectenna is comprised of an optical nanoantenna and a metal-insulator-metal (MIM) diode. Under illumination of visible and IR light, a tremendous electric field (hot spot) can be generated at the feed point of the metallic nanoantenna due to the surface plasmon resonance [6,7], meanwhile an alternating current (AC) will be induced on the antenna surface. Moreover, the MIM diode embedded in the nanoantenna is used to rectify the AC oscillations to DC power based on the asymmetric of the tunnel junction [8,9]. Recently, many efforts have been engaged in the study of the nano-rectenna devices, including achieving large local field by designing and optimizing the geometry structure of the antenna [10,11], and high rectification efficiency by selecting appropriate materials and setting specific geometric parameters of the diode [12,13]. However, the photoelectric response and power conversion efficiency of the nano-rectenna devices are very low due to the difficulty in manufacturing and the impedance mismatch between the antenna and diode.

In addition, the mature theory and calculation formulae for designing and analyzing the nano-rectenna are still in great demand. Therefore, the theoretical study may provide useful information in designing and simulating nano-rectenna devices. So far the theoretical study is mainly focused on characterizing the optical performance of the nanoantennas with different simulation methods [14–17], such as the dipole [18], bowtie [19], spiral [20] and log-periodic [21]. Among them, the spiral nanoantenna is good at collecting electromagnetic energy [22–25] owing to its broad bandwidth, strong resonance in the long wavelength, high local field enhancement and linearly (as well as circularly) polarized wave. When the spiral nanoantenna is integrated with a rectifier or a load element to constitute a nano-rectenna [26,27], DC signals can be generated. However, usually the photoelectric conversion efficiency is very low, which may be caused by poor coupling performance of the nanoantenna, poor rectifying performance of the diodes, and/or the antenna-diode impedance mismatch.

To improve the optical and electrical properties of the spiral nano-rectenna, i.e. enhance the collection and utilization of IR radiation energy, some scientific issues still need to be studied. First, the influence of changes in the geometric parameters of the spiral nanoantenna and in the material and structure of the rectifier on the optoelectronic properties of nano-rectenna is still unclear. Second, besides the near field, far-field radiation characteristics of the spiral nano-rectenna should be also considered because the optical antenna is capable of functioning as both receiver and emitter. The far-field simulation can modify and control the properties of photon emission in terms of the directivity, polarization and emission intensity [28]. Third, the mechanism about the excitation mode of resonance spectrum and the relationship of electric field and output current needs further investigation, which is in favor of improving photoelectric conversion efficiency. So it is extremely important to design a spiral nanoantenna integrated with a MIM diode so as to simulate the actual devices both optically and electrically, and then optimize various parameters of the spiral nanoantenna and rectifier.

In this work we proposed and designed an IR-response spiral nano-rectenna comprised of a receiving spiral nanoantenna and a MIM diode. The response wavelength range is 5~30 μm, as approximately 80% of the solar radiation is absorbed by the Earth's surface and atmosphere, which will be reemitted as IR radiation mainly in this wave band with a peak wavelength of 10.6 μm [22]. The shape of the spiral is square rather than circular since the square spiral antennas have advantages in size over the circular ones. When the same antenna gain is obtained, the width of the square spiral is about 75% of the diameter of the circular one [29]. Here Au-TiO_x-Ti diode is used as the rectifier due to the large difference in the work function between Au and Ti, which is in favor of a high rectifying effect. In addition, Au and Ti arms may exhibit strong plasmon effect owing to their metallic property. The model construction and simulated calculation are carried out by using Ansoft HFSS, which cannot only calculate the local field intensity of a dielectric layer, but also define some electrical parameters via field calculation like output current, induced voltage and output power.

2. Modeling of the spiral nano-rectenna and calculation methods

When the IR radiation is shed on the nanoantenna surface, an alternating voltage V_IRcos(ωt) is generated between the two metal electrodes in the diode (Fig. 1(a)). Owing to the nonlinear tunneling behavior originated from the asymmetry in the diode, the MIM junction can act as a rectifier that leads to a net current flow in one direction [14]. For efficient IR radiation collecting, the MIM diode should have a fast response time, which is determined by tunneling speed of electrons. In the equivalent circuit of the nano-rectenna (Fig. 1(b)), the diode consists of a junction capacitor C_D and a nonlinear voltage-dependent resistor R_D(V), and is connected with a resistor R_A of the spiral nanoantenna. The cutoff frequency is defined as Eq. (1).

 

Fig. 1 Design of the square spiral nano-rectenna. (a) MIM diode band diagram under IR illumination, (b) equivalent circuit of the nano-rectenna, and (c) structure of the nano-rectenna.

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To minimize the response time of the diode and achieve a high cutoff frequency, the diode capacitance has to be very small according to Eq. (1). The diode capacitance C_D can be calculated by Eq. (2), where ε_r is the relative permittivity of the oxide in the MIM diode, ε₀ is the permittivity of free space, A is the junction area, and d is the thickness of the oxide. Moreover, the free-electron Drude mode [30] is applied to describe the dispersion behavior of the metals, where the change of the metal permittivity versus frequency in the IR range can be precisely simulated. The metal complex relative permittivity function ε(ω) can be described by Eq. (3), where ω is angular frequency, 1/γ_D is collision frequency, and ω_p is plasma frequency. The ε₁(ω) and ε₂(ω) are the real and imaginary part, respectively.

The respective plasma frequency of Au (ω_p(Au)) and Ti (ω_p(Ti)) is 1.37 × 10¹⁶ s^‒1 and 3.83 × 10¹⁵ s^‒1, while respective collision frequency of Au (γ_D(Au)) and Ti (γ_{D (Ti)}) is 4.08 × 10¹³ s^‒1 and 7.20 × 10¹³ s^‒1 [31,32]. Thus the dielectric constant can be obtained at different frequencies. For example, the respective dielectric constant of Au (ε_(Au)) and Ti (ε_(Ti)) is (‒5045 + 1077i) and (‒3602 + 138i) at the frequency of 30 THz. The results agree with the reported values [23,33].

The structure of a square spiral nano-rectenna is comprised of a receiving antenna and an Au-TiO_x-Ti diode. All the geometric parameters are shown in Fig. 1(c). The two arms of the antenna are Au and Ti. The initial size of the antenna is a = 300 nm, b = 1 μm, h = 100 nm, n = 1, which represents respectively the width of the spiral arms, spacing of the spiral arms, thickness of the spiral antenna, and spiral turns based on the Au arm. The dielectric layer is TiO_x and the initial parameters are d = 5 nm and A = 100 × 100 nm², where d is thickness of the dielectric layer and A is contact area of the tunnel junction. It is assumed that the nano-rectenna is present in the vacuum. The overall structure is excited by an incident plane wave propagating along z-axis with electric field intensity of 1 V/m and a linear polarization along y-axis. The linearly polarized wave is superior to a circularly polarized wave for the square spiral nano-rectenna since it is convenient to study the accumulated surface charges in a single spiral arm when the nano-rectenna is coupled to the incident light, though the latter has a higher IR collection efficiency [23,34,35].

The electromagnetic wave incident on the spiral nano-rectenna can stimulate the surface plasmon oscillations due to interactions between the incident light and free electrons in the metals, resulting in a huge local field across the dielectric layer of the rectenna. The local field enhancement factor K is defined as $K=|E(\omega )|/\left|E_0\right|$, where E(ω) is frequency-dependent electric field induced via coupling of nanoantenna with the incident light. Observation point of the K is at the center of the dielectric layer. As the frequency changes, the K can reach a maximum value caused by surface plasmon resonance, which is expressed as K_max. The resonance wavelength is λ_rs. Similarly, based on the Maxwell's equations in three-dimensional electromagnetic theory, output current of the nano-rectenna is $\nabla \times H=-i\omega \epsilon (\omega )E+\sigma E$, and the induced voltage between the two metal electrodes is expressed as $\nabla \times E=i\omega \mu H$.

All the calculations are performed using HFSS. The meshes were refined until the convergence and the simulations run long enough so as to reduce the solution error, i.e. the maximum number of passes is 20 and the maximum delta energy per pass is 0.01. In addition, it is simulated with radiation condition, the solution frequency is 28.3 THz and the sweep frequency range is from 10 THz to 60 THz (5~30 µm).

3. Results

3.1 E-field and phase distribution

With the aforementioned model, the first step is to explore coupling effect of the spiral nano-rectenna to the IR radiation. The intensity and phase of electric field distribution of the square spiral nano-rectenna are simulated upon excitation of incident light. The intensity distribution in xy plane is shown in Fig. 2(a) when the intensity of incident electric field ($\overrightarrow{E_y}$) is 1 V/m. A weak electric field exists on the antenna surface, while a greatly enhanced electric field is across the dielectric layer (Fig. 2(b)). The phase distribution (Fig. 2(c)) in the xy plane represents the phase changes significantly at the interface between the vacuum and metal as shown in orange/red and blue regions. It indicates that such change is very fast along the y-polarization direction at the interface of vacuum medium and metal, leading to the coupling between the nano-rectenna and incident IR radiation. The fact is that the IR radiation coupled with the antenna is transmitted to the dielectric layer, resulting in the tremendous local field due to the surface plasmon resonance. As shown in Fig. 2(d), the electric field vector lies in the xy plane. The electric field vector outside the nano-rectenna is along the polarization direction of the y axis, while the one inside is scattered. Since the induced charge in the inner arm is positive and the outer is negative, the electric field vector in an individual arm points to the outward from the inner and the one between the two arms points inward, as the mutual attraction between the positive and negative charges results from Coulomb interaction [36].

 

Fig. 2 The electric field and phase distribution of square spiral nano-antenna upon excitation of the incident light. (a) electric field intensity distribution in xy plane when the incident electric field intensity is 1 V/m, (b) zoom-in view of the middle in Fig. 2(a), i.e. the electric field intensity distribution over tunnel junction, (c) electric field phase distribution in xy plane, and (d) electric field vector distribution in xy plane.

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The intensity and phase of electric field distributions confirm that the IR radiation energy is mainly concentrated in the dielectric layer due to the coupling effect of nano-rectenna. Then the major issue of the target device is whether or not the rectification effect can be observed. Based on the above theoretical model, the square spiral nano-rectenna has been fabricated experimentally and the I-V curve has been measured using a semiconductor analyzer (Fig. 3(a)). The nonlinearity and asymmetric effect are observed, indicating the rectification effect [10]. The asymmetric factor of the MIM diode is determined to be about 2.0 by absolute value of the ratio from forward current to reverse current under the same numerical voltage, which is expressed as $f=\left|I_F/I_R\right|$. Another characteristic of a rectifier is its sensitivity [10], which is given by the ratio of second derivative and first derivative of the I-V characteristics (Fig. 3(b-d)), i.e. $S=\left|I^{\text{'}\text{'}}(V)/I^{\text{'}}(V)\right|$, where $I^{\text{'}}(V)=dI/dV$ and $I^{\text{'}\text{'}}(V)=d^{2}I/d{V}^2$. The maximum sensitivity is determined to be 0.44 at a DC bias point where maximum curvature of the I-V curve occurs. It is noted the values for both asymmetric factor and maximum sensitivity are quite low, which may be mainly because in our experiments it is very difficult to prepare the dielectric layer with the expected property, neither the accurate fabrication of the device. If the proper devices could be fabricated in future, a much higher asymmetric factor would be achieved. Nevertheless, the rectification effect can be obtained in the proposed device with an MIM diode and the conduction is due to charge tunneling through a thin insulating layer.

 

Fig. 3 Nano-rectenna device performance. (a) I-V characteristic curve of the MIM diode, (b) the first derivative (dI/dV), (c) second derivative (d²I/dV²), and (d) sensitivity (I^”/I^’) vs. bias voltage for the MIM diode.

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3.2 Optimization of the nanoantenna structure

According to the aforementioned results, therefore, the photoelectric properties of the proposed device can be analyzed and optimized via adjusting different parameters of the square spiral nanoantenna and the MIM rectifier, respectively. The nanoantenna optimization includes the width of the spiral arms, spacing of the spiral arms and the antenna thickness; while the geometry and composition (the thickness, permittivity, conductivity of the dielectric layer and the contact area of the tunnel junction) will be considered for the MIM rectifier. Moreover, the far-field radiation behavior of the nano-rectenna has also an important impact on the photoelectric properties. Thus, it is possible to maximize the utilization of light energy and photoelectric conversion efficiency if all the parameters can be modulated accurately.

The resonance spectra of local field and output current for the spiral nanoantenna are calculated as shown in Fig. 4, i.e. the width of the spiral arms, spacing of the spiral arms and the antenna thickness. It indicates that each resonance spectrum has three major resonance peaks in the wavelength range of 5~30 μm and the peak electric field intensity is enhanced from short wavelength to long wavelength. The trends of the three resonance peaks are consistent with the variation of each parameter. However, the changes of the local field and the output current are not the same as those of the different parameters. The calculation results (Fig. 4(a-c)) indicate that the three resonance peaks exhibit significant redshift and increased peak electric field as the width of the spiral arms increases, and show a larger redshift and increased peak electric field as the spacing of the spiral arms increases, but have almost no shift and a weakened peak electric field as the antenna thickness increases. These are because, as the width and spacing of the spiral arms increase, the enlarged distance between positive and negative charges can lead to weakened restoring force (Fig. 4(a-b)) and reduced oscillation frequency [37]. Moreover, increased dipole moment results in the redshift of the resonance wavelength and a larger local electric field. The variation of the antenna thickness does not change such features. So the resonance spectra are almost no shift. In addition, the increase in the antenna thickness can suppress the electric field intensity because the reduced thickness has a small cross section, making the interface of vacuum and antenna close to the observation point and the lightning rod effect lead to a strong local field [38].

 

Fig. 4 Theoretical calculation of the local field enhancement factor K vs wavelength at (a) different width of spiral arms, (b) different spacing of spiral arms, and (c) thickness of the spiral antenna; and output current vs wavelength at (d) different arm width, (e) different spacing of spiral arms, and (f) thickness of the spiral antenna.

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In addition, the spacing of the spiral arms is usually larger than the arm width in the same spiral antenna, i.e. the decline rate of the restoring force is more significant. So the redshift in the resonance wavelength caused by the spacing change is much larger than that caused by the arm width. The resonance wavelength of the second and third-order peaks varies with the spacing of the spiral arms (Fig. 4(b)), indicating a direct proportion relationship between them for both peaks. The second-order resonance wavelength can be expressed as λ_rs = 8.4b + 4.85 (μm) and the third-order one is λ_rs = 14.6b + 7.04 (μm). When the arm width is constant, the length of the spiral antenna is proportional to the spacing of the spiral arms. The fact that the resonance wavelength is proportional to the spiral length is similar to the standing wave model of dipole antenna λ_eff × j/2 = L, which can be used to select the experimental response wavelength in terms of the spacing size.

For the output current, i.e. the rectified DC current, it is noted that only the second-order and third-order resonance peaks will be considered for the output current and discussions on other parameters thereafter, since the first-order resonance peak intensity is much weaker than the other two peaks. The results of Fig. 4(d-e) show that the output current increases first and then starts to decrease if the arm width is larger than 500 nm or the spacing is larger than 1.2 μm, which indicates the enhancement of local field can facilitate the tunneling probability when the excited electrons pass through the dielectric layer, giving rise to a large output current. However, if the arm width exceeds a certain value, there will be a big drop of the rectified current in the antenna arms [10], i.e. an incredibly wide antenna arm may lead to the loss of output current when the electrons transfer along the antenna. Similarly, when the spacing exceeds a certain value, the increased region of the vacuum background may result in a large energy loss, and the long transmission distance of electrons from the positive (negative) electrode to the external load can increase the resistance. Instead, the output current of Fig. 4(f) enlarges with the increase of the antenna thickness, although the local field reduces. This is because a large thickness can lower the antenna impedance, causing a large current, which is similar to the bow-tie rectenna [19].

3.3 Optimization of the rectification effect

On the basis of the optimal results of the spiral nanoantenna, therefore, appropriately increasing the width or spacing of the spiral arms and reducing the antenna thickness are in favor of improving the utilization of light energy and generation of large current. Then the next step is aimed at the optimization and analysis of the MIM diode so as to investigate the influence of the rectifier on the device performance, mainly including theoretical calculation of the geometry and composition.

Here the geometric parameters of the dielectric layer refer to the thickness of the layer and contact area of the tunnel junction. The calculated local field enhancement factor varies with the IR wavelength for the dielectric layer with different thicknesses (Fig. 5(a)). The three resonance wavelengths exhibit almost no shift and the local field intensity has little decrease as the thickness of the dielectric layer increases. Thus, the changes in the thickness of the dielectric layer have little impact on the strength of the surface plasmon resonance. The same is true for the resonance wavelength. However, the output current of the nano-rectenna decreases as the thickness increases, and the current decay is more significant at a large thickness (Fig. 5(b)). The peak current of the third-order resonance is slightly stronger than the second-order resonance, while both of them change from about 24 nA to 12 nA when the thickness varies from 3 to 7 nm. This is because the increasing thickness of the dielectric layer leads to an enlarged potential barrier, which greatly reduces the probability of electron tunneling and, thereby, the exponential reduction in the output current. In addition, the zero-bias resistance R₀ of the diode is mainly determined by the thickness d of the dielectric layer and the height $\varphi$ of the potential barrier [39], i.e. $R_0\propto \frac{d}{\sqrt{\varphi }}{e}^{d\sqrt{\varphi }}$. The R₀ increases sharply when the d increases, causing a rapid drop in the current.

 

Fig. 5 (a) Theoretical calculation of local field enhancement factor vs wavelength at different thickness of the dielectric layer, (b) output peak current vs dielectric layer thickness for the second and third-order resonance, (c) different contact area between metal and semiconductor of the tunnel junction, i.e. 50 × 50, 100 × 100, 150 × 150, 200 × 200 nm², respectively, (d) local field enhancement factor vs wavelength at different contact area of the tunnel junction, and (e) output peak current vs contact area for the second-order and third-order resonance.

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Figure 5(c) shows the tunnel junction with different contact area of 50 × 50, 100 × 100, 150 × 150, 200 × 200 nm², respectively. The calculated local field intensity varies with the IR wavelength for the tunnel junction with different contact area (Fig. 5(d)). For the devices with different contact area, it is found that the peak position of the local field enhancement factor is at 5.7, 13 and 21 μm, respectively. So the resonance wavelength is independent of the contact area. In addition, the increasing contact area induces significant decay of the local field intensity, and the evanescent field almost disappears when the contact area is 200 × 200 nm². The output peak current of the second and third-order resonance declines gradually as the contact area increases (Fig. 5(e)), for which the decay rate becomes fast at a large contact area. Both two peaks are about from 24 nA to 4 nA when the contact area is from 50 × 50 to 200 × 200 nm². The increase in the contact area results in an increase in the capacitance of a diode. Accordingly the cutoff frequency of the diode decreases, resulting in more distorted current signals. Moreover, the diode with an oversized contact area has a low resistance and thus exhibits very low rectification capability. Therefore, the obtained results confirm the reason that in current experiments one always tries to reduce the thickness of the dielectric layer and the contact area of the tunnel junction [13].

Besides the discussion about the geometry, the composition of the dielectric layer is also calculated and analyzed in order to further optimize the rectifier. For an MIM (Au-TiO_x-Ti) diode rectenna, the increasing oxidation degree of the dielectric layer improves the dielectric property of the diode, i.e. the relative permittivity of TiO_x increases when x changes from 0 to 2. As the average permittivity of TiO₂ is about 100, the relative permittivity for TiO_x can be assumed to range from 2 to 100. Furthermore, the conductivity of the dielectric layer can have an impact on the rectifying characteristics of the nano-rectenna, which increases gradually as x decreases, corresponding to the change from a semiconductor to a metal for the TiO_x. Here the conductivity of TiO_x is set as 10⁻⁸, 1800, 3600, 10000 and 50000 S/m for TiO₂, Ti₂O₃, TiO, Ti₂O and Ti_nO, respectively. The Ti₂O and Ti_nO have a high conductivity with the metallic characteristic. The calculated resonance spectra of the local field for TiO_x with different relative permittivity are shown in Fig. 6(a).

 

Fig. 6 Theoretical calculation of local field enhancement factor K vs wavelength (a) at different relative permittivity of TiO_x and (d) at different conductivity of TiO_x, and peak field intensity vs different relative permittivity (b) at the second-order resonance and (c) at the third-order resonance.

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The maximum field enhancement factor of the second and third-order resonance varies with the relative permittivity is plotted in Fig. 6(b) and Fig. 6(c), respectively. With the increase of the relative permittivity for TiO_x, the three resonance peaks exhibit a slight redshift and the peak intensity of local field in the dielectric layer significantly decreases, i.e. both the second and third peaks decay exponentially. The slight redshift can be explained by the resonant cavity perturbation theory [40], which is expressed in Eq. (4), where ω₀ and ω is the angular frequency of the pre-perturbation and post-perturbation, respectively. The increase of the permeability (Δμ) is almost 0 in the cavity. The relative permittivity of TiO_x becomes larger with the increase of the oxidation degree, i.e. Δε > 0. Thus, a reduced resonance frequency leads to redshift. Meanwhile, the exponential decay of the local field originates from the surface plasmon oscillation. The decay factor is ${\kappa }^{-1}=\frac{\lambda }{2\pi }\left(\frac{{\epsilon }_m^{\text{'}}+{\epsilon }_d}{{\epsilon }_d^2}\right)$, where $\lambda$ is the incident light wavelength, ${\epsilon }_m^{\text{'}}$ is the real part of metal permittivity, and ${\epsilon }_d$ is the permittivity of the dielectric material. It can be seen that the decay factor is dependent on the relative permittivity for the dielectric material, i.e. the increasing relative permittivity results in a significant reduction of the plasmonic coupling strength and, thereby, a rapid decrease in the peak electric field. The local field enhancement factor changes with different conductivity of TiO_x at the IR wavelength (Fig. 6(d)). The three resonance peaks exhibit blueshift to some degree and the local field intensity decreases as the conductivity of TiO_x increases. This is because the increasing conductivity σ ( = neμ) associates with significantly increase of the carrier concentration n, which makes major contribution to high frequency photons, i.e. the high frequency vibrations of TiO_x dielectric layer result in the blueshift [41]. In addition, the decrease in the insulation of dielectric layer will lead to weakened blocking of the tunneling electrons and, thus, the declined strength of the local field. The above optical phenomena are similar to those for the bow-tie nano-rectenna [19], so does the influence on the electrical properties.

3.4 Far field radiation characteristics

All of the above calculations are focused on the near field for the collection and utilization of infrared energy, from which one can know how to design the rectifier antenna accurately. However, the far field emitting characteristics of the device also have a significant impact on the rectifier antenna, especially the optical properties. According to the reciprocity theorem [42], an antenna is not only a receiver to generate strong local field, but also a transmitter to tailor the properties of far-field radiation in terms of the emission intensity, directivity and polarization. It is reported that the characteristics of far-field radiation are largely affected by the number of spiral turns [28,43]. In this work, four different spiral turns are used to calculate the directivity, far and near-field intensity of the square spiral nano-rectenna (Fig. (7)), i.e. 1, 1.5, 2 and 2.5 turns, with respect to the Au arm. The directivity D of an antenna is a measure of the ability for an antenna to concentrate radiated power into a certain direction [44], which is defined as the maximum of directive gain, i.e. $D(\theta ,\phi )=4\pi \cdot p(\theta ,\phi )/P_{rad}$, where θ and φ represent the spherical coordinates with respect to the antenna orientation.

 

Fig. 7 Far-field radiation characteristics of the square spiral nano-rectenna at different spiral turns. (a) Simulation model for 1, 1.5, 2 and 2.5 turns, respectively, with respect to Au arm; (b) three-dimensional directivity of the nano-rectenna for the far-field radiation at different spiral turns; (c) two-dimensional directivity of the nano-rectenna for the far-field radiation at different spiral turns, and the internal brown line represents E plane while the outside red line is H plane; (d) and (e) electric field amplitude of far and near-field radiation at different spiral turns, respectively.

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The three and two-dimensional directivity of the far-field radiation for the nano-rectenna with different spiral turns are shown in Fig. 7(b) and 7(c), respectively. It is found that the two spherical lobes are perpendicular to the antenna surface, i.e. the polarization direction of the antenna emitting is along the z-axis, and the directivity increases with the increase of the number of turns (D = 2.49, 3.15, 4.06 and 4.18 dB for the number of turns n = 1, 1.5, 2 and 2.5, respectively). For one turn of the spiral antenna, the length of the rectenna is 5.5 μm, which is about half of the 10.6 μm wavelength of the incident light. The directivity of D_{n = 1} = 2.49 dB is only slight larger than D_λ/2 = 2.15 dB of an ideal thin-wire half-wave dipole [44]. The length of a spiral antenna changes from a half wavelength to full wavelength as n increases, and eventually exceeds the length of the full wavelength. The IR light with any wavelength may be excited owing to the asymmetry between the nano-rectenna structure and the incident light. In addition, it can be seen from the two-dimensional directivity diagram that the FWHM_(far) (full-width-half-maximum) of the emission pattern in the far field increases first and then decreases, which reaches the maximum with the number of 2 turns.

The three-dimensional far-field radiation and the near-field distribution are shown in Fig. 7(d) and 7(e), respectively. Both the far and near-field intensity increases first and then decreases as the number of turns increases, which reaches the maximum with n = 2 (E_far = ‒118.10, ‒104.63, ‒98.36 and ‒100.50 dB, and E_near = ‒23.43, ‒9.98, ‒4.07 and ‒6.76 dB for the number of turns n = 1, 1.5, 2 and 2.5, respectively). The reason may be that the coupling of the incident light increases with the increase of the number of turns, while the energy dissipation loss may arise and the far-field emission properties along a certain direction can be weakened. The figures also indicate that the far-field distribution is continuous and exhibits sphere-like shape, while the near-field distribution is discontinuous. So the change in the near-field intensity is very dramatic. Moreover, the intensity of the far-field radiation is almost three orders of magnitude lower than that of near-field one. The monitor range for the near field is within 20 μm. So there may be an optimal number of the turns (n = 2) to ensure high performance of the antenna radiation and obtain large intensity for both near and far fields.

4. Discussions

Based on the aforementioned optimal antenna and rectifier, a nano-rectenna with high performance can be obtained, from which the mechanisms can be analyzed by tuning various parameters. Furthermore, in order to comprehensively evaluate the design significance of the nano-rectenna from theory to application, three parts need much more in-depth discussion, i.e. excitation mode of resonance spectra, relationship between electric field and output current, and photoelectric conversion efficiency of the nano-rectenna.

4.1 Odd-order mode resonance

It is known that the resonance spectra of the square spiral nano-rectenna exhibit three peaks upon IR radiation. The local field intensity increases from the first to the second, and then to the third-order resonance. Since the proposed nano-rectenna is not perfectly symmetrical to the polarization direction of the incident radiation, both modes of the odd and even-order resonance can be excited. As plotted in Fig. 2(d), the Coulomb interaction is not only between the positive and negative charges in the two ends of one arm, but also between the two adjacent arms, resulting in the formation of three different dipole moments. Thus, the optical antenna structure can be approximated described using a dipole-dipole model [45]. The three different dipole-dipole Coulomb interactions can lead to different hybrid peaks, corresponding to the three types of resonance. The different strength of the dipole moment represents three peaks with different intensity. In addition, the quality factor (Q factor) is defined as the ratio of λ_rs /Δf [46], where λ_rs and Δf is the resonance wavelength and FWHM_near of the resonance in the near field curves, respectively. The respective Q factor of the first, the second and the third-order peak is around 1.8, 2.0, and 4.3, which further indicates that the third-order peak has a sharper shape and a better detectability.

4.2 Relationship between electric field and output current

When the nano-rectenna is illuminated with IR radiation, the photoemission current density J across the metal-semiconductor interface is assumed to be proportional to the square of the electric field in terms of the theory of electron photoemission [47], i.e. $J=C_{em}(\omega )\cdot {\left|E_n\right|}^2$, where proportionality coefficient C_em(ω) is dependent on the wavelength of the incident light, work function of the metal and Fermi level of the dielectric material. Thus the photocurrent (i.e. the output current of the nano-rectenna) is $I(\omega )=C_{em}(\omega ){{\displaystyle {\oint }_S\left|E_n\right|}}^{2}dS$, where S is the cross-section area that electrons pass. As a result, a high local field is in favor of improving the output current of the nano-rectenna. Such relationship may explain the phenomena of changes in some parameters of the antenna structure, such as the width and spacing of the spiral arms. Owing to the influence of other factors like impedance, rectification and propagation loss, however, this relationship may be invalid for interpreting the impact of dielectric layer on the optoelectronic properties of a rectenna. In addition, a spiral arm with an oversized width and spacing may cause increasing propagation loss. Thus, the output current is not positively correlated with the electric field. Moreover, the output current of the device is achieved with a rectifier, which agrees with the experimental results [48]. In addition, the nano-rectenna may also exhibit thermoelectric effect (i.e. the Seebeck effect) to some degree [27], as the power loss is inevitable when the current flows.

4.3 Photoelectric conversion efficiency

For a square spiral nano-rectenna, conversion efficiency is an important parameter to evaluate its performance. A time-dependent alternating current is generated upon the absorption of IR radiation by the antenna and transfers to the tunnel junction so as to be converted to direct current via the diode rectification, and then transports to an external load. As reported previously [26,49], the photoelectric conversion efficiency of a nano-rectenna is defined as $f=P_{out}/P_{in}$, where P_in is the input power of the IR radiation, P_out is the output power of the nano-rectenna. The input power $P_{in}=I_0\cdot S_{in}$, where S_in is the area of the spiral antenna that receives the radiation and $S_{in}\approx 29a\cdot b$ for the number of turns n = 1, and I₀ is the incident light intensity, i.e. $I_0=\frac{1}{2}\sqrt{\frac{\epsilon }{\mu }}\cdot E\cdot E^{*}=\frac{1}{2\eta }{E}_0^2$, η is wave impedance in a medium, and η = η₀ when the environment is vacuum, i.e. η₀ = 377 Ω. The output power $P_{out}=\frac{1}{2}\mathrm{Re}\left({\displaystyle {∯}_{s}SdS}\right)$, where S is the Poynting vector, S = E × H. The physical meaning of P_out is equivalent to the DC Voltage multiplied by DC current [26,27]. A large output power in a small receiving area should be achieved so as to acquire high conversion efficiency. Here the initial parameters are the aforementioned optimal ones, i.e. a = 300 nm, b = 1 μm, n = 1, S_in ≈8.7 μm², I₀ = 1.33 × 10⁻³ W/m² when the incident electric field intensity is 1 V/m. So the P_in is calculated to be 1.16 × 10⁻¹⁴ W. The output power of the second and third-order peak is 2.04 × 10⁻¹⁶ W and 2.60 × 10⁻¹⁶ W, respectively, corresponding to the respective conversion efficiency of f_second = 1.76% and f_third = 2.24%. Similarly, if keeping all the above parameters constant, while n = 2, then, S_in ≈17.1 μm². The output power of the second and third-order peak becomes 2.36 × 10⁻¹⁶ W and 3.51 × 10⁻¹⁶ W, respectively, corresponding to the respective conversion efficiency of f_second = 1.04% and f_third = 1.54%. Such lower values may be due to a larger receiving area and impedance mismatch for the device with n = 2, although it is the optimized value considering far and near-field intensity. It is noted that the output power of the spiral rectenna is higher than that of the bow-tie nano-rectenna [19], while its receiving area is larger than that of the bow-tie at the respective optimal conditions, resulting in a slightly lower efficiency for the spiral rectenna than the bow-tie one [19]. It is believed that the impedance mismatch between the antenna and diode also plays an important role here.

5. Conclusion

In conclusion, the square spiral nano-rectenna coupling the mid and far IR radiation has been systematically analyzed by using Ansoft HFSS method. The intensity and phase of electric field distributions of theoretical simulation and the I-V characteristic curve of experimental test validate the rationality of the designed device. The results of theoretical calculation show that the variation of each parameter has a great impact on the optical and electrical properties of the spiral nano-rectenna. Three types of resonance occur in the IR radiation of 5~30 μm due to the dipole-dipole Coulomb interactions. The resonance wavelength exhibits redshift with the increasing width and spacing of the spiral arms and relative permittivity of TiO_x, while slight blueshift happens with the increasing conductivity of TiO_x. The peak intensity of the local field increases with the increasing width and spacing of the spiral arms; while decreases with the increasing arm thickness, relative permittivity and conductivity of TiOx, thickness of the dielectric layer and contact area of the tunnel junction. Moreover, the output current rectified by the MIM diode is about tens of nA at 1 V/m of the incident electric field intensity, and the induced voltage of the nano-rectenna is approximately tens of nV. It is known that the correlation between the local field and output current is not always positive if other factors dominate the optoelectronic properties like impedance, rectification and propagation loss. Furthermore, the directivity of the spiral nano-rectenna increases with increasing number of turns. The far and near-field intensity decreases after the initial increase and reaches the maximum value with n = 2. In addition, considering the optoelectronic characteristics and cost of a real device, an optimized structure can be determined as a = 300 nm, b = 1 μm, h = 100 nm, n = 1 turn, d = 5 nm, A = 100 × 100 nm². The photoelectric conversion efficiency of the optimized square spiral nano-rectenna is about f_second = 1.76% and f_third = 2.24%. The efficiency can be improved via further optimizing the structure of the spiral nano-rectenna. Thus, the proposed nano-rectenna in this work may provide theoretical support for the fabrication of real optoelectronic devices that can efficiently collect and convert IR energy.