1. Introduction

While metals are widely used for electrodes in many optoelectronic devices, the device efficiency is limited by optical loss in metals [1, 2]. In particular, when electrodes are nanostructured to reduce the shading loss of bulk metals, surface plasmons can be excited and the metal absorption can be substantial [3–8]. The problem of the parasitic absorption becomes critical when a high device efficiency is desired. Moreover, if the device operates over a broad band as in many photovoltaic devices, metal absorption needs to be controlled over the whole spectrum of interest and this poses a significant scientific and engineering challenge.

The problem of metal absorption persists even to the mid-infrared (IR) range where photodetectors are important for night vision surveillance, fire fighting, and missile approach warning [9]. In thin IR detectors, a popular approach to concentrate light in thin active layers is to use surface plasmons and these waves involve significant parasitic loss [10–12]. For example, at an interface between InSb and Ag, surface plasmons are dissipated more than 5 times faster in Ag than in InSb at the light wavelength of 5 μm. Thus, for reduced metal loss in the detectors, it is desirable to avoid surface plasmons and other resonances, sacrificing light concentration near metals. Moreover, the loss problem in the detectors is not always solved by simply avoiding resonances and further loss reduction is needed.

Past effort to reduce metal loss focused mostly on applications in plasmonics and metamaterials [13]. To reduce loss, alternative materials have been explored such as graphene [14] and transparent conducting oxides [6]. These materials may also be used to avoid plasmonic excitations and further reduce loss in IR applications. However, for electrodes, it is difficult to obtain electrical properties as good as noble metals with the alternative materials. A different approach for loss reduction is to compensate loss by incorporating an active gain medium near metal structures [15]. In this case, however, the realization of low loss is fundamentally limited to very narrow frequency bands [16]. Compared to these material approaches, geometric manipulation of metallic structures to suppress loss is advantageous in maintaining good electrical properties and may potentially reduce loss over a broad band. For loss reduction by geometric manipulation, it was suggested that sharp edges in metallic structures should be eliminated to provide smooth current flow [17]. However, as a variety of optical effects are possible by geometric manipulation, it is valuable to quest for other superior possibilities for geometry-induced loss reduction.

It is known from the microwave circuit theory that an inductor subjected to an oscillating field suppresses Joule heat dissipation at high frequencies. This suggests that the metal loss in IR applications may be suppressed by increasing the Faraday inductance of the metal nanostructures. Moreover, metals can be nanostructured such that electric current can flow in tortuous paths. In this case, compared to straight electron paths at the same metal filling fraction, the effective conductivity would be smaller so that the structured metal behaves like a low-loss dielectric material [18, 19]. Thus, it is important to determine if these two loss suppression effects due to Faraday inductance and electron path length would be significant enough to be practically useful in IR applications.

Here, we investigate the effects of IR loss suppression in non-plasmonic metal nanocoil structures by controlling Faraday inductance and electron path length. Surprisingly, we find that the nanocoiled metals can behave almost like a vacuum with a negligible optical loss in the IR region. Our detailed analysis confirms that the suppression of optical loss is due to both a large Faraday inductance and a long electron path. For an example nanocoil structure, the loss is reduced in comparison to flat films by more than an order of magnitude over most of the very broad spectrum between short and long wavelength IR. To investigate the usefulness of this effect in IR detectors, we include a photoactive material in the coiled metal structures. In this case, the fraction of absorption in the active material increases by two orders of magnitude compared to non-coiled structures. Moreover, we demonstrate that the coiling of the IR detector nanostrips increases the photoresponsivity by 15 times without additional electrical loss.

A few remarks would be useful before we begin our detailed discussion. First, it has been reported that increasing the Faraday inductance in metal structures can reduce the plasma frequency [20–22]. This property is due to Lenz’s law according to which the increased Faraday inductance of the structures hinders the change in the movement of conduction electrons in the metal. We point out that the Faraday inductance bears important implications for loss suppression in nanostructured metals. For Drude metals, the hindered acceleration or deceleration of the electrons increases the electron collision time and the metal loss is suppressed. Second, our approach cannot be used to reduce loss in plasmonics. For surface plasmons, the Faraday inductance effect would be insignificant because of relatively small magnetic fields. Third, our approach is to be used when the metal structures support direct electric current. When the structures are electrically disconnected, the off-resonance metal loss is in general unimportant but they cannot be used as electrodes as electric current is not supported. Fourth, we use the term Faraday inductance in distinction from the kinetic inductance [23]. Kinetic inductance is negligible in the structures in our study.

2. Metal nanocoil array

To investigate the loss suppression effect, we consider a monolayer array of metal nanocoils shown in Fig. 1. We choose this simple structure because it allows straightforward optical modeling from which the underlying physics of loss suppression can be revealed. The nanowire is helically wound with an outer radius R and a pitch p with a winding angle θ. It has a rectangular cross-section with a width l cosθ and a thickness δ. The nanocoils form a periodic monolayer array with the distance between the neighboring nanocoil axes equal to a and light polarized in the $\widehat{\mathbf{z}}$ direction is incident in the direction normal to the monolayer surface. We focus on this polarization because the loss part of the effective dielectric function for metal nanostructures is in general maximized when light moves the electrons in the same direction as the polarization at long wavelengths of light.

We begin by finding expressions for useful geometric quantities. Referring to Fig. 1, we see that the electron path length in a nanocoil per unit length in the $\widehat{\mathbf{z}}$ direction is longer than a straight nanowire aligned along the same direction by a factor of

 

Fig. 1 Illustration of a nanocoil array. Top: nanocoil array with the direction of light incidence (k) and polarization (E₀). Bottom: a single nanocoil with its geometrical parameters, cylindrical coordinates, tangential and normal vectors (ŝ) and $(\widehat{\mathbf{n}})$, and electric fields.

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The Faraday inductance per unit z-direction length of the nanocoil is given by

We describe the optical properties of the nanocoil array in terms of the three geometric parameters in Eqs. (1)–(3). When the structure is much smaller than the light wavelength, the optical properties can be described by a lumped circuit model using quasi-static approximation [13, 18, 19, 23, 24]. The model predicts that, when θ and f are small, the absorptance A of the nanocoil array is given by [25]

The absorption decrease is directly related to the effective dielectric function ε_eff of the nanocoil array. By definition,

In Eq. (6), we see that σ_eff is inversely proportional to η². Thus, by elongating the electron path at a fixed metal fraction, we can decrease the effective conductivity and hence the metal loss. Further, Eq. (7) reveals that the electron relaxation time increases as the Faraday inductance increases. This increased electron collision time results in the reduction of absorption in the metal.

Figure 2 shows comparison of the absorptance A and the effective dielectric function ε_eff for an aluminum nanocoil array between the numerical solutions and the model predictions based on Eqs. (4) and (5). The reflectance of the nanocoil array is negligible compared to the absorptance. The dielectric function of aluminum was modeled by $\overline{h}/\tau =0.05307eV$ and $\overline{h}{\omega }_p=12eV$ where ω_p is the plasma frequency. For accurate calculations, we used the finite element method with a nonuniform mesh concentrated in the metal. The effective dielectric function was extracted from the numerical calculations using the method in Ref. [26]. The model predictions agree well with numerical solutions for both A and ε_eff. The effective dielectric function exhibits the typical behavior of a Drude metal with a low effective plasma frequency ω_p,eff corresponding to 6 meV in photon energy. When the energy is above ∼ 0.1 eV, the A approaches zero and the ε_eff is close to the dielectric function of vacuum. This low loss of the structure at high frequencies is the consequence of the combined effect of a large Faraday inductance and a long electron path as Eqs. (4)–(7) demonstrate. Namely, the loss of the metallic structure becomes negligible above the plasma frequency ω_p,eff.

 

Fig. 2 Absorptance and effective dielectric function of a nanocoil array. (a) Numerical solutions (solid circles) and model predictions (solid line) of absorptance and (b) numerical solutions (open square for the real part and solid square for the imaginary part of ε_eff) and model predictions (solid line) of effective dielectric function for the nanocoil array as a function of photon energy. The aluminum wire of cross-section 20 nm × 20 nm is wound into a coil with outer radius R = 500 nm and pitch p = 100 nm. The coils form a monolayer array with the center-to-center distance a = 1.5 μm and light polarized parallel to the coil axes is incident normally on the monolayer surface.

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The degree of loss suppression in the nanocoil array can be assessed by comparing the loss of the structure with a flat film. The film has the same mass as the nanocoil array per unit surface area. The absorptance ratio of the film to the nanocoil array is shown in Fig. 3. Overall, the absorptance ratio is of the order of ten, meaning that absorption is reduced by the nanocoil structure by this factor. This effect is valid over a very broad spectrum between short (∼ 1.2 μm) and long wavelength (∼ 20 μm) infrared. The two dips at 0.63 eV and 0.84 eV correspond to the small absorption peaks in Fig. 2(a) which are due to resonances in the nanocoil array. We note that the suppression of metal absorption translates into the reduction in thermal radiation by Kirchhoff’s law. Thus, the loss reduction would be useful for solar selective surfaces that should absorb sun light strongly and suppress the heat loss that occurs through thermal radiation [28]. Because the thermal radiation from the selective surfaces is typically in the broad IR range, the nanocoil array suppresses the radiation loss significantly. As the frequency becomes higher (> 1 eV) and approaches the diffraction limit i.e. ω ∼ 2πc/a, the absorptance of the nanocoil structure becomes significant so that sunlight absorption is strong.

 

Fig. 3 Absorptance ratio of a flat film to a nanocoil array. The two structures has the same mass of aluminum per unit surface area so that the film thickness is fa = 8.2 nm. The structural parameters of the nanocoil array are in Fig. 2. Experimentally determined dielectric function was used for calculations [27]. The incident light polarization is parallel to the coil axes.

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In addition to the Faraday inductance effect, decreasing the metal filling fraction f reduces the loss as Eq. (4) shows. However, the filling fraction effect alone cannot achieve the dramatic change in the optical properties of structured metals. For example, Fig. 4 compares the transmittance of the nanocoil and the nanotube monolayer array at the same metal filling fraction. While the nanocoil array is highly transparent, the transmittance of the nanotube array is very small due to strong reflection. In fact, the optical properties of the nanotube array are what one could expect based on the Maxwell-Garnett theory [29]. Contrary to the nanotube array, the reflectance of the nanocoil array is negligible. Calculations showed that even the small deviation from perfect transmission in the nanocoil array is mostly due to absorption rather than reflection. Even though the two structures are similar, the Faraday inductance and the electron path length are much larger for the nanocoil array and this is the reason for the dramatic difference in the optical response of the two structures. The relative contribution of the two effects to loss reduction in the nanocoil array can be seen in Fig. 4. The long electron path, which is independent of frequency, contributes to its high transmittance over the whole spectrum in Fig. 4 and the large Faraday inductance further increases transmittance at high energies above $\overline{h}{\omega }_{p,eff}=6meV$.

 

Fig. 4 Numerical solutions of transmittance between the nanocoil and the nanotube aluminum monolayer array. The filling fraction, the center-to-center distance, and the outer diameter are the same for the two structures. The structural parameters of the nanocoil array is in Fig. 2. The incident light polarization is parallel to the axes of the nanocoils and the nanotubes.

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The loss suppression of the nanocoil array is remarkably insensitive to the incidence angle. In Fig. 5(a), the polar and azimuthal angles θ_inc and ϕ_inc for the direction of incident light are defined and light is polarized in the yz plane. When ϕ_inc = 0°, the light polarization is in the z direction. In this case, we can expect from our modeling that absorption is strongly suppressed well above the effective plasma frequency, ω_p,eff, even for off-normal incidence. Indeed, Fig. 5(b) shows that absorption is negligible for any θ_inc above ∼ 0.1 eV. However, as ϕ_inc deviates from 0, the E field is not parallel to the z direction and the absorption spectrum can change appreciably. When θ_inc is kept constant, the maximum deviation of the incident E field from the z direction happens at ϕ_inc = ±90°. However, even in this case, we find a spectral window between 0.05 and 0.1 eV where metal loss is efficiently suppressed as shown in Fig. 5(c). The size of the window becomes smaller than the ϕ_inc = 0° case at off-normal incidences because of the high energy resonance at ∼ 0.6 eV. This resonance is highly sensitive to θ_inc near the normal direction. Namely, the resonance is negligible at θ_inc = 0° but becomes strong at θ_inc = 30°. Based on the observations in Fig. 5, it can be said that the strong loss suppression effect of the nanocoil array persists from θ_inc = 0° to 60°.

 

Fig. 5 Angular dependence of metal loss in nanocoil array. (a) Definition of incidence angles for nanocoil arrays. (b and c) Angular dependence of metal absorption in a nanocoil array when (b) ϕ_inc = 0° and (c) ϕ_inc = 90°. Light is polarized in the yz plane. The structural parameters of the nanocoil array are the same as in Fig. 2.

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3. Nanocoil IR detectors

The effect of loss suppression in metals is useful in IR detectors for which metal electrodes are widely used. We consider an example photodetector structure where metal-semiconductor-metal nanostrips are curled into nanocoils as shown in Fig. 6(a). The two metal nanowires on both sides of the semiconductor nanostrip are electrodes and the semiconductor is a photoactive material. For the coiled nanostrip monolayer array, we calculate the partial absorptance [30] in semiconductor (InSb) and metal (Ag) when θ = 2.5° and show, in Fig. 6(b), the resulting spectra above the semiconductor band gap which corresponds to the free photon wavelength of 7.3 μm. The spectra show that the absorption enhancement in semiconductor is over a broad band without any resonant features. The excitation of surface plasmons is negligible and the optical diffraction is almost absent. A useful parameter to characterize the effect of metal loss suppression in IR detectors is the fraction of absorption in semiconductor, ρ_s, defined by

 

Fig. 6 Metal loss suppression in IR detectors. (a) Schematic of a coiled metal-semiconductor-metal strip. The width of the metal (Ag) and the semiconductor (InSb) strips is w_m = 10 nm and w_s = 40 nm, respectively, and both strips have a thickness 10 nm. (b) Spectrum of partial absorptance in semiconductor and metal for a monolayer array of the composite nanocoils in (a) with a periodicity of a = 750 nm and a coil winding angle θ = 2.5°. Light polarized parallel to the coil axes is incident normally on the monolayer surface. (c) Fraction of absorption in semiconductor, ρ_s, as a function of θ for the mono-layer array of the composite nanocoils. The ρ_s for an array of flat composite strips with the same dimensions and periodicity is shown in dashed line. (d and e) The ρ_s as a function of θ_inc when (d) ϕ_inc = 0° and (e) ϕ_inc = 90°. The definition of the angles is given in Fig. 5. The ρ_s for an array of flat composite strips with the same dimensions and periodicity for θ_inc = 60° is shown in blue line. (f) Angular dependence of partial absorptance in metal, A_m, for the θ = 2.5° structure at ϕ_inc = 90°.

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When the incident E field deviates the farthest from the z direction by increasing ϕ_inc to 90°, the polar angle dependence of the loss suppression becomes appreciable as shown in Fig. 6(e). As θ_inc increases, the loss suppression effect becomes smaller. However, even when θ_inc = 60°, the ρ_s increases by 18 times when the flat nanostrips are coiled at θ = 2.5°. The absorption spectra of the nanocoil arrays reveal that the decrease in ρ_s for increasing θ_inc at ϕ_inc = 90° is due to a high energy resonance. For example, Fig. 6(f) compares the partial absorptance in Ag between θ_inc = 0° and 30° at ϕ_inc = 90°. The θ = 2.5° structure exhibits a resonance at λ = 2.1 μm at θ_inc = 30°. The resonance tail extends into our spectral range of interest 3.5 μm < λ < 7.3 μm. This is the reason for the strong θ_inc dependence of ρ_s in Fig. 6(e). However, the loss suppression effect is still strong for longer wavelengths. For example, the θ_inc dependence of absorption is negligible over a very broad spectrum of 5 μm ≤ λ ≤ 20 μm in Fig. 6(f) for θ_inc = 30°. If we chose a material with a band gap lower than InSb such as HgCdTe and InAsSb, the spectral range of the IR detector shifts to longer wavelengths where the resonance tail is negligible. Therefore, in this case, the IR detector would exhibit the loss suppression effect that is almost independent of the incident polar angle irrespective of the azimuthal angle.

The loss suppression effect can be probed optically for metal nanocoil arrays in Fig. 2. For fabrication of the structures, various techniques can be used including direct laser writing [31], interference lithography [32], glancing angle deposition [33], and zinc oxide growth [34]. Here, to see if the loss suppression effect is realizable in IR detectors, we consider photoconductance measurement on the coiled Ag-InSb-Ag composite nanostrip arrays. Figure 7(ab) shows a schematic image of coiled nanostructures and electrical connections for the measurement. The structures are fabricated by the recently developed methods that use the inherent stress gradient over the nanostrip thickness [35,36]. The flat strained nanostrips are self-coiled as released from the mesa lines by chemical etching. To control the coiling properties of the Ag-InSb-Ag nanostrips, thin IR transparent nanostrips can be deposited on them before coiling [35]. The Ag nanowires on the right and left side of the nanostrips are connected to two Ag plates separately and the plates are electrically biased. Light is incident on the structure from the top and the conductance change by irradiation is probed. Figure 7(c) shows the calculated change in conductance G per unit incident power P as a function of the coil winding angle. The ΔG/P, which is the photoresponsivity per unit applied voltage, increases as θ decreases and is enhanced by 15 times at θ = 2.5° compared to flat strips. Therefore, not only can the loss suppression effect be probed in photoconductivity experiments but also will it be useful in IR detectors. The coiling-induced enhancement factor of ΔG/P is smaller than that of ρ_s because ρ_s is affected by both A_s and A_m whereas the ΔG/P is determined by A_s only. Note that, even though the electron path length per unit length of the coil axis is a strong function of θ, the total electron path length is independent of θ in the considered experiment because the strip length is fixed. Thus the resistance of the Ag nanowires, which accounts for electrical loss, is the same for all the structures while the optical loss suppression depends strongly on θ. Detailed analysis (not shown) suggests that the experiment should be performed at low temperatures and light intensity should not be too high because, otherwise, the InSb conductivity becomes large and many electrons in the Ag nanowire pass to InSb before reaching to the nanowire end. Because the InSb layer width is only 40 nm, the typical carrier transit time between the electrodes would be much shorter than the carrier lifetime. Thus, the nanostructured detectors would exhibit higher photoconductivity gain in comparison to macroscopic devices [37].

 

Fig. 7 Photoresponsivity enhancement in IR detectors by metal loss suppression. (a) Schematic of a coiled metal-semiconductor-metal nanostrip array fabricated by releasing strained strips from mesa lines. (b) A close-up view of the array showing electrical connections to measure photoconductance. (c) Calculated change in conductance per unit incident power as a function of the coil winding angle at a temperature T = 77 K. The light source is a black body at T = 500 K that is frequency filtered within 3.5 μm < λ < 7.3 μm. The incident power density is 0.2 W/m². Nanostrips with a length L = 200 μm are coiled. The material and structural parameters are the same as in Fig. 6.

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While the metal loss suppression enhances absorption in the photoactive material, this effect is to be distinguished from conventional light trapping [38–41]. As can be inferred from the absence of resonance features in Fig. 6(b), this structure does not concentrate light in the semiconductor through surface plasmons [39] or optical diffraction [40]. Thus, the strategy of metal loss suppression can be combined with other light-trapping techniques. For example, we can enhance absorption in the semiconductor while keeping the metal loss minimal by embedding the coiled nanostrip arrays in diffracting dielectric structures with a periodicity comparable to the light wavelength.

4. Conclusion

In conclusion, we have demonstrated that proper design of metal nanostructures can considerably suppress optical absorption in metals over a broad IR spectrum by increasing the Faraday inductance and the electron path length of the structures. Our findings will be useful in various optical applications. First, the high transparency of metal nanocoil arrays can be useful for IR transparent windows in cold weather. In this case, the metal nanostructures are embedded in the IR glass windows and the loss suppression effects result in high optical transparency. When these windows are frosted in cold weather, the metal structure can be resistively heated to remove the frost [42–44]. Second, the IR loss suppression will enable efficient solar selective surfaces. The extremely low IR loss of the metal structures translates to a highly efficient heat retention of the structures in vacuum because thermal radiation loss is suppressed. At the same time, the strong resonances in the visible wavelengths will absorb sunlight strongly. Third, the loss suppression effect will be useful in increasing the efficiency of optoelectronic devices which consist of metals and other photoactive materials by selectively reducing loss in metals. We have demonstrated that the loss suppression effect could be experimentally realized in photoconductive IR detector structures. Importantly, our method to suppress optical loss does not increase electrical loss in the IR detectors. More generally, our findings could be useful in other metallic metamaterials that require low loss. Although the loss suppression effect was investigated for coiled structures in the present study for analytical simplicity, the application of the physical principles can be extended to a variety of other geometries and further work is required to find the full potential of this effect in diverse applications.