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We theoretically study the metal-insulator-metal (MIM) structure based ultrathin broadband optical absorber which consists of a metallic substrate, a dielectric middle layer, and a nanostructured metallic top layer. It is found that, there exists an effective permittivity, εnull, for the top nanostructured metallic layer which leads to unit-absorption (zero-reflection) of the MIM structure. Importantly, this εnull exhibits abnormal dispersion behaviors. Both its real and imaginary parts increase monotonically with the wavelength. To obtain such naturally non-existing permittivity, we investigate the optical properties of two typical types of metal-dielectric nanocomposites, namely, thoroughly mingled composites using Bruggeman’s effective medium theory, and more realistic Au nanosphere-in-dielectric structures using numerical permittivity retrieval techniques. We demonstrate that the εnull-type dispersions, and consequently, perfect absorption can be obtained over a broad spectral range when the filling factor of the metal component is close to the percolation threshold. The result not only explains the recently reported broadband absorbers made of randomly deposited Au nanoparticles [M. K. Hedayati, et al, Adv. Mater. 23, 5410 (2011)], but also provides theoretical guidelines for designing ultrathin broadband plasmonic absorbers for a wealthy of important applications.
© 2015 Optical Society of America
1. Introduction
Ultrathin broadband optical absorbers have been of great interest in many important fields, including photovoltaics [1–3], photothermovoltaics [4,5], and nanostructure-assisted ablation and ionizations [6,7], to name a few. To date, various materials have been implemented to reduce the reflection and to increase the absorption, such as black paints, nanowires [8–10], carbon nanotube forests [11], and black silicons [12]. Those absorbers are broadband, but their thicknesses are large (>wavelength).
Recently, it was found that the thickness of an absorber can be significantly decreased using plasmonic nanostructures, e.g., nanovoids [13,14], nanogrooves [15,16], and metal-insulator-metal (MIM) structures [17]. Among those ultrathin plasmonic absorbers, MIM structures are particularly interesting due to their unique design. In general, a MIM structure consists of a nanostructured metallic thin layer on the top and an opaque metallic substrate at the bottom spaced by a dielectric thin layer in the middle [18]. The two metallic parts not only form an optical cavity which can have strong resonant absorption of light, but also can serve as electrodes, making MIMs an appealing design in optoelectronics [19]. On the other hand, because of the resonance nature, MIM absorbers are intrinsically narrow-band. To address this issue, in the last few years, many efforts have been made to broaden the resonance using specially designed multi-resonance plasmonic “atoms”, such as crossed trapezoid [20], metallic split ring [21], nano-cross [4], and metallic sawtooth [22]. With those multi-resonance designs, improved bandwidths have been achieved, however, often at the expense of absorption efficiency. Meanwhile, Elbahri group [23,24], and Qiu group [25,26] reported that, instead of using specially designed nanostructures, high efficient broadband absorption can be realized by simply using a nanometer-thin random Au nanostructure as the top layer of MIM. The absorption spectra of this type of random MIM structures are flat, very different from the conventional resonance-induced absorption spectra which have resonant peaks, and cannot be explained by the commonly used near-field coupling picture. We, hereby, in this work focus on the mechanism and the design of this type of broadband MIM absorbers.
In parallel with the development of plasmonic thin film absorber, Capasso and his associates recently introduced the concept of metasurface, and pointed out that a perfect absorption (zero-reflection) can be realized by simply using a thin layer of lossy material on a lossy substrate [27]. Different from conventional planar antireflection thin films which are quarter wavelength thick and narrowband, the thickness of lossy thin-film based absorbers can be much smaller than the working wavelength because the phase change at an interface is no longer 0 or π. Based on this mechanism, Gan and his associates showed that near-unit absorption can even be achieved by a 1.5 nm Ge film [28].
Inspired by the idea of metasurface, in this work, we study the MIM absorbers with randomly structured metallic top layer from the perspective of thin-film interferences. Instead of focusing on the resonance behavior of MIM, we describe the top structured metallic layer as a thin layer of homogeneous lossy material, and calculate (1) the permittivity of the top layer needed for perfect absorption, and (2) the effective permittivity of metallic nanostructures (i.e., metal-dielectric composite (MDC) material) with the aim of revealing the mechanism of the reported broadband absorption of MIM and further giving guidelines for designing broadband MIM absorbers in different scenarios.
The manuscript is organized as following. First, we numerically search for the permittivity of the top metallic layer, εnull(λ), required for broadband perfect absorption (i.e., zero-reflection). Second, the effective permittivity of ideal MDCs is investigated using Bruggeman’s effective medium theory. Finally, we study the optical properties of a realistic MDC structure, the Au-Nanosphere-In-Dielectric (ANID) material, using 3D full-wave simulations, and discuss how to build ultrathin broadband absorbers using MDC materials.
2. Searching for the zero-reflection effective permittivity, εnull, for nanostructured top layer of MIM absorber
Figure 1(a) depicts a model MIM structure, which consists of a thin layer of metallic nanostructures (i.e., lossy medium with effective permittivity εtop and dtop in thickness) on the top, a thin dielectric layer in the middle with permittivity εmid (dmid in thickness), and an opaque metallic substrate at the bottom with permittivity εsub. The absorbance, A, and reflectivity R = 1-A, are decided by the geometrical parameters, dtop, and dmid, the optical parameters, εtop, εmid and εsub, and the angle and polarization of the incident light. Since here we focus on the case of ultrathin absorbers (i.e., dtop, dmid << λ) in which the phase changes are small (<<π), the absorption, as well as the reflection, is not sensitive to incident angle. Hence, for simplicity, we only discuss the case of normal incident in this work.
Fig. 1 (a) Sketch of a model MIM structure. (b) Reflection, R, as a function of Re(εtop) and Im(εtop) at 500 nm. Here, dtop = dmid = 20 nm, and εmid = 2.25. (c) Reflection map of the MIM structure at different wavelengths (from 400 nm to 800 nm). (d) Zero-reflection permittivity, εnull, as a function of wavelength. Here, “×” labels the zero-reflection points εnull, and “+” marks the center of the reflectivity isolines, εc(R) at R = 0.1, 0.2, 0.3, and 0.4.
To understand how εtop influences R, we first take a typical MIM structure with dtop = dmid = 20 nm and εmid = 2.25 as an example, sweep both Re(εtop) and Im(εtop), and calculate R using transfer matrix, as shown in Fig. 1(b). Zero-reflection (i.e., unit absorption) point, εnull, is found in the upper half of the complex εtop plane.
Interestingly, function R(εtop) has a very low changing rate dR/dεtop in the vicinity of εnull, and as a result, R stays low in a considerable large area in the complex εtop plane. For instance, the enclosed area by the isoline A = 10%, reaches 29.2 at 500 nm. It is also observed that the isolines in the R map are circular. Define εc(R) the center of the isoline of reflectivity R. We find that its imaginary part, Im(εc(R)), increases monotonically with R, while the real part, Re(εc(R)), stays constant. In other words, when Im(εtop) is slightly larger than Im(εnull), the reflection, R, will stay low even if there is a considerably large deviation of Re(εtop) from Re(εnull). Therefore, to achieve a low reflectivity (high absorbance), the key is to have a Im(εtop) slightly larger than Im(εnull).
We furthermore calculate R(εtop) maps at different wavelengths, and found that εnull is wavelength-dependent, as shown in Fig. 1(c). Both the real part, Re(εnull), and imaginary parts, Im(εnull), exhibit abnormal dispersion behavior, and increase monotonically with wavelength with large increasing rates, dRe(εnull)/dλ, and dIm(εnull)/dλ (Fig. 1(d)). As a result, to realize broadband absorption, Re(εtop) and Im(εtop) need to vary over a large range monotonically. For example, in Fig. 1(d), dRe(εnull)/dλ is approximately 0.019, and therefore, the value of Re(ε top) needs to change from 3.25 at 400 nm to 8.95 at 700 nm in order to have a 300 nm absorption band. In nature, no such material exists. This abnormal dispersion property of εnull is the reason why, since long, it has been challenging to build ultrathin broadband absorbers.
Zero-reflection permittivity, εnull, is dependent on the geometrical parameter, dtop, and dmid, as well as the optical property of dielectric middle layer, εmid. To illustrate the influences of those parameters, first, we fix dmid at 20 nm, vary dtop from 10 nm to 50 nm, and calculate εnull(λ), as shown in Fig. 2(a-b). It is found that a larger dtop leads to a smaller Re(εnull) and a smaller Im(εnull). Then, we fix dtop and change dmid from 0 nm to 60 nm. The calculated εnull (Fig. 2c-d) shows that R(εnull) is reversely related to dmid while Im(εnull) is insensitive to dmid. Finally, we investigate the role of εmid by varying it from 2 to 10. As shown in Fig. 2(e-f), when εmid increases Re(εnull) decreases but Im(εnull) only has minor changes. The above results indicate that one can optimize the absorption of a MIM structure with a given top material by tuning parameter dtop, dmid, and εmid.
Fig. 2 (a) and (b) shows the spectra of Re(εnull) and Im(εnull) with varied dtop (10 nm, 20 nm, 30 nm, 40 nm and 50 nm), respectively. Here, dmid = 20 nm, εmid = 2.25. (c) and (d) depicts the spectra of Re(εnull) and Im (εnull) with different dmid (0 nm, 15 nm, 30 nm, 45 nm, and 60 nm),respectively. Here, dtop = 20 nm, εmid = 2.25. (e) and (f) shows the spectra of Re(εnull) and Im(εnull) with varied εmid = 2, 4, 6, 8, and 10, respectively. Here, dtop = 20 nm, dmid = 20 nm.
From Fig. 2, it is evidence that thinner is the top layer, higher will Re(εnull) and Im (εnull) be. This is the reason why the reported ultrathin absorber is made high refractive index material Ge [28]. From this perspective, we also learn that, to have broadband absorption, the top layer needs to be ‘thick’. Otherwise, if dtop < 10 nm, both the real and imaginary part of εmid will need to be larger than 10, a value hard to achieve in the optical spectral regime.
It is worth emphasizing that, although εnull(λ) is dependent on dtop, dmid, and εmid., it always exhibits abnormal dispersion behaviors for ultrathin absorber. In fact, a similar phenomenon is also found in the radio wave regime [29].
3. Dispersion engineering using Bruggeman’s metal-dielectric nanocomposites
In the previous section, we learn that to build an ultrathin broadband absorber, materials with εnul–type abnormal dispersion over a broad spectral range are needed. However, no such material exists in nature. It is known that the optical response of solids can commonly be described by Lorentzian model:
And, εnull-type of abnormal dispersion only occurs on the blue side of resonance (Fig. 3(a)) in a very narrow spectral range. Metamaterials are therefore needed in order to achieve such an abnormal dispersion property over a broad spectral range, and in this work, we focus on the simplest case, the metal-dielectric-composites.Fig. 3 (a) Permittivity spectra by Lorentzian model. (b,c) Calculated effective permittivity spectrum maps of metal-dielectric nanocomposite at different filling factors by effective medium theory. (d) Real (solid line) and imaginary (dash line) parts of effective permittivity as a function of wavelength. The green, blue and red color denotes filling factors of 0.21, 0.33, and 0.50 (i.e., larger than, equal to, and smaller than the percolation threshold), respectively. (e) Calculated absorbance spectrum map at different filling factors. Here, dtop = 50 nm, dmid = 30 nm, εmid = 2.25. (f) Calculated absorbance spectra of a MIM structure different filling factors (0.10, 0.33, and 0.70).
One of the most commonly used models for MDCs is effective medium theory (EMT) by Bruggeman, which assumes that the components in the medium are thoroughly mingled with each other [31]. In EMT, the effective permittivity of MDCs can be calculated by an explicit formula:
Here fm, and fd is the filling factor of metal and dielectric, respectively. Using Eq. (2), we calculate the effective permittivity of a typical MDC made of Au and SiO2 at different wavelengths and different filling factors, as shown in Fig. 3(b) and 3(c). Two distinguished regimes are observed, namely, the low filling factor regime and the high filling factor regime. In the low filling factor regime, there exists a resonance which is originated from the metal component. When the filling factor increases this resonance redshifts and broadens. Once entering the high filling factor regime, i.e., filling factor > = 1/3 (a value known as the percolation threshold of Bruggerman’s composite, fperc) [30], the resonance will disappear and the permittivity will become metal-like.We are particularly interested in the case of fm ≈fperc, where Im(εeff) exhibits an abnormal dispersion property over a broad spectral range. As aforementioned, this specific type of dispersion allows us to construct an ultrathin broadband MIM absorber. To demonstrate it, we take a MIM structure with dtop = 50 nm, dmid = 30 nm, εmid = 2.25 as an example, and calculate the absorbance of the structure using transfer matrix, as shown in Fig. 3(e). As predicted, broadband absorption appears when the filling factor is close to the percolation threshold of MDC material. When fm = fperc, the absorbance is larger than 90% from 400 nm to 700 nm; on the contrary, when fm is far away from fperc, the absorbance becomes considerably lower (<80%). It is noticed that at the percolation threshold, there is still considerable discrepancy between Re(εeff) and Re(εnull). Nevertheless, since Im(εeff) follows the εnull-type of dispersion and is larger than Im(εnull), the absorber can tolerate a large deviation of Re(εeff), and A can still stay above 90% as discussed in previous section.
Note that the optical property of MDC is also a function of the permittivity of the dielectric component, εdie, and this provides an additional dimension for tailoring the dispersion of MDC materials and building broadband ultrathin absorbers with given geometries. Figure 4 shows the calculated effective permittivity of MDC with different εdie. It is evident that higher εdie leads to larger Re(εeff) and larger Im(εeff). Compared with εnull calculated in previous section (Fig. 1, and Fig. 2), we learn that a higher εdie will allow the construction of a thinner absorber.
Fig. 4 (a) Real (solid line) and imaginary (dash line) parts of effective permittivity spectra with εdie = 2, 5, and 8 and fm = 0.33.(b) Calculated absorbance of a MIM structure with different εdie (2, 5, and 8). Here, dtop = 50 nm, dmid = 30 nm.
4. Effective permittivity of Au nanosphere dispensed in dielectic matrix
Bruggeman’s model assumes that the components in MDCs are thoroughly mingled with each other and does not consider the effects of microscopic features of the components. However, in reality, microscopic features of the metal component in MDC often have strong influences on the effective permittivity, and MDCs made of the same materials with the same filling factor but different topological features may exhibit very different optical responses [31,32]. Due to the complexity, numerical simulations are needed to retrieve the optical properties of each specific MDC.
In the following, we will numerically study the optical properties of a realistic model MDC structure, the ANID, which is widely used in the field of optical metamaterials for achieving naturally non-existing optical properties [33,34].
4.1 ANID with periodic Au nanospheres
For simplicity, ANIDs with periodic Au nanospheres are first investigated. Figure 5(a) illustrates the structure which is composed with a 3D array of 5 nm Au nanospheres in cubic lattice embedded in SiO2 matrix (ε = 2.25). Here, the filling factor is decided by the periodicity of the nanospheres. We numerically simulated the reflectivity and transmittance spectra of a planar thin film of the ANID using a commercial finite-different time-domain (FDTD) solver (Lumerical Inc.) and then, retrieved the effective permittivity using the method developed by Smith et al. [35] In the FDTD simulation, mesh size of 0.25 nm, normal plane wave excitation, and periodic boundary condition are used.
Fig. 5 (a) Sketch of periodic ANID consisting of a 3D array of Au nanospheres (D = 5 nm) embedded in SiO2 matrix (n = 1.5). (b) Effective permittivity of ANIDs with fm = 0.23, 0.52, and 0.70. (c-d) Retrieved effective permittivity of ANIDs material at different wavelengths and different filling factors. (e) Absorbance of the MIM structure constructed with ANID, and dtop = dmid = 20 nm, εmid = 2.25. (f) Absorbance spectra of the discrepancy structure with different filling factors (fm = 0.23, 0.52, and 0.70).
Figure 5(c) and 5(d) illustrate the numerically retrieved effective permittivity map of the ANID material. Similar to the results of Bruggeman’s MDCs. it exhibits two distinct regimes separated by the percolation threshold (filling factor ~52%). Figure 5(b) shows the typical permittivity spectra of ANID at different filling factors. In the low filling factor regime, there is a strong resonance which redshifts monotonically with the filling factor, fm, of the metal part. When fm crosses the percolation threshold and enters the high filling factor regime, the resonance disappears and the material becomes metal-like. At the percolation threshold, the width of resonance peak of ANID reaches its maximum, making the construction of broadband absorbers possible.
Using the retrieved effective permittivity, we calculate the absorbance of the MIM structure using transfer matrix. Figure 5(e) shows the absorbance map of a typical MIM structure with dtop = dmid = 20 nm, and εmid = 2.25. The absorbance reaches 90% in the whole visible spectral range (400 nm – 700 nm) when the filling factor is 52% (approximately at the percolation threshold).
We also calculated the absorbance spectra of the same structure by directly simulating the whole MIM structure using FDTD. The results agree well with those calculated by transfer matrix. It verifies the permittivity retrieval method in this work. In addition, it also implies that near-field couplings between ANID and the Au substrate are weak and can be ignored in this case despite the dielectric spacer is only 20 nm. Therefore, the mechanism of this type of MIM absorber is fundamentally different from the conventional MIM absorbers in which the near-field couplings between the two metal layers cause the strong light absorption [17].
We plotted the effective permittivity of ANID at filling factor of 52% as shown in Fig. 5(b), and find that there is a considerable deviation of εeff from εnull, while the absorbance is kept high. This phenomenon can be understood by the fact that MIM absorber has large tolerance for εtop, as illustrated in Fig. 1(b) and 1(c).
Despite the similarity discussed above, εeff of ANID materials has many significant differences from the case of EMT, including the different percolation threshold (52% for ANID, but 33.3% for Bruggeman material), narrower resonances in the low filling factor regime, and more abrupt behavior change of optical responses at the percolation threshold. All those differences are rooted in the geometrical differences at the microscopic scale, which on one hand brings complexities, and on the other hand, provides additional flexibilities for engineering the permittivity of MDCs. In the following part, the detailed geometrical effects will be discussed.
4.2 Size effect of Au nanosphere on the effective permittivity of ANIDs
The most important geometrical parameter of ANID is the diameter of the nanospheres. To investigate the effects, we change the diameter of the nanospheres from 5 nm to 10 nm, and calculate the effective permittivity. Two differences are observed, as shown in Fig. 6. First, the resonance peak redshifts, and second, the resonance becomes stronger. These phenomena can be understood by considering the stronger couplings induced by larger particle sizes, which have also been observed in other plasmonic systems.
Fig. 6 (a) Real and (b) imaginary part of the ANID material with different nanosphere diameters, (i.e., 5 nm and 10 nm) and same filling factor fm = 52%
4.3 ANID with nonperiodic Au nanoparticles
In addition to the particle size, the spatial arrangement of the Au nanospheres also has strong influences on εeff of an ANID material because the interparticle coupling strengths, and consequently εeff are sensitive to the particle-particle distance. To illustrate this phenomenon, in the numerical model, we shift each of the Au NPs from its original position in cubic lattice by a randomly generated small distance (< = 1 nm) in a randomly chosen direction, and then retrieve the effective permittivity using FDTD simulation, as shown in Fig. 7. The abrupt behavior change of εeff observed in the periodic ANID structure at the percolation threshold disappears, and instead, εeff varies smoothly with fm in a way similar to the case of EMT. Meanwhile, we notice that the permittivity of this ANID material still shares some features with the periodic ANID material, e.g., the narrow resonance in the low filling factor regime, a phenomenon which stems from the single-particle resonance.
Fig. 7 (a, b) Numerically retrieved real and imaginary part of the effective permittivity of an ANID material made of randomly arranged Au NPs. (c,d) Numerically calculated effective permittivity at percolation threshold of ideal Bruggeman’s MDC, periodic ANID, and random ANID.
From above discussions, we learn that the EMT can be used to make a qualitative prediction of the effective permittivity of random MDC materials, while fullwave simulations are needed for quantitative analysis.
5. Discussions
5.1 Role of the metallic component
In the previous sections, Au is used as the metallic component. However, the method of using MDC material to construct ultrathin broadband absorber is universal and can be applied to different metals. To demonstrate this point, we calculate the εeff of composite materials (random ANID) made of SiO2 and different metals, namely Au, Ag and Cu at the percolation threshold, as shown in Fig. 8. Similar to the case of Au, all these materials exhibit εnull-type of abnormal dispersion over a broad spectral range at the low frequency regime (i.e., below their bulk plasma frequency), and therefore, allow us to construct ultrathin broadband MIM absorbers.
Fig. 8 (a) Real and (b) imaginary part of effective permittivity of MDC with different metallic materials (i.e., Au, Ag and Cu) calculated using the EMT.
5.2 Fundamental limit for ultrathin broadband absorber
Finally, it is worth mentioning that the real part and imaginary part of the permittivity of a material are linked via the Kronig-Krammer relation. Because of this constriction, no material can own the εnull-type of abnormal dispersion over an infinitively wide spectral range, and this yields a fundamental limit (i.e., an upper bound) for the bandwidth of ultrathin absorbers. Indeed, Rozanov has investigated the case of thin-film absorbers with a perfect electrical conductor substrate, and gave the upper bound to its bandwidth using rigid mathematics [29,36].
6. Summary
In this work, we theoretically studied the design of MIM-based ultrathin broadband absorbers. With the top nanostructured metallic layer treated as a homogeneous material, we find that there exists an effective permittivity of the top layer, εnull, which yields unit-absorption (i.e., zero-reflection). Calculations also show that both Re(εnull) and Im(εnull) increase monotonically with the wavelength, and no material with such an abnormal dispersion property exists in nature. To realize such a naturally non-existing dispersion, we investigated the optical properties of two different types of metamaterials, the ideal Bruggeman-type MDCs and a more realistic MDC structure, the ANID. Numerical investigations show that, for both structures, when the filling factor of the metal component is near the percolation threshold, εnull-type abnormal dispersion can indeed be obtained, and consequently near-unit absorption (A > 90%) can be achieved over a broad spectral range (400 nm – 700 nm). Meanwhile, numerical results also revealed sophisticated dependences the optical constants of ANIDs on their local geometrical features, namely, the size and the arrangement of the nanospheres. It provides us additional degrees of freedom for tailoring the optical properties of ANIDs for various applications. This work not only explains the recently reported broadband absorption observed on randomly structured Au films from the perspective of metasurface, but also lays out the guidelines for the design and optimization of broadband ultrathin MIM absorbers, which own a broad spectrum of applications.
Acknowledgments
We gratefully acknowledge funding from the National Natural Science Foundation of China (No. 11321063, 11374152, and 61306123), the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions, and the 1000 Young Talent Program of China.
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