1. Introduction

Many branches of optics require miniaturization of optical devices. For example: optoelectronics and telecommunications would benefit from smaller and faster light modulators [1]; optical biosensing would profit from smaller and smarter biosensors that can be incorporated even in mobile phones [2] and optical computing would experience a strong boost from smaller optical elements and scalable circuits [3]. To achieve miniaturization, new optical materials are required. Recently, a whole set of new 2D atomic materials has been experimentally obtained [4] which allowed one to significantly reduce the footprint of optical devices. These new materials showed a great promise in optics: graphene (semimetal) [5], was used in photonics [6], optoelectronics [7], and plasmonics [8] in a wide wavelength range from visible to far-infrared light; hexagonal boron nitride (dielectric) demonstrated interesting phonon-polaritons [9] in infrared light; molybdenum disulphide (semiconductor) was used as a broadband saturable absorber [10]. Just a few years ago, ultrathin metal films were added to this family [11–13]. They demonstrate tuneable plasmons [11], electron confinement [13,14] and strong optical anisotropy [14]. There exist many works devoted to quantum size effects [15] in ultrathin metal nanostructures and their non-local behaviour [16]. Various miniature optical devices were fabricated with the help of atomic materials and their heterostructures. At the same time, there is still a huge demand for ultrathin materials with designed/controlled optical characteristics.

To achieve a control over optical properties of ultrathin materials - nanostructuring can be used. Indeed, optical properties of a nanostructured metal can be very different from the properties of a host material due to the presence of localized plasmon resonances [17]. Here, we discuss highly oriented single-crystalline gold quantum-dot gold films as a new prospective material for photonics. This novel class of materials consists of densely packed single-crystal Au nano-dots of regular ultrathin height fabricated on a simple substrate with the help of ultrahigh-vacuum molecular-beam epitaxy pulsed-laser-deposition [18]. We show that these materials of thickness smaller than 10 nm behave as a good dielectric with reasonably high refractive index (n≈2.75) and very low absorption (k≈0.005) at telecom wavelengths. This compares favourably with the artificial metamaterials fabricated from graphene oxide in microfluidic channels [19]. In addition, the studied metasurfaces demonstrate elements of topological photonics. Namely, we observe topological darkness [20] and associated phase singularities [21,22] in such materials for the case where thickness of the material was just 7.8 nm. To the best of our knowledge, this is the first observation of topological darkness in a random extremely thin metasurface placed on a simple substrate.

2. Results and discussion

Gold films of highly oriented single-crystalline gold quantum-dots (HOSG-QDs) studied in this work were produced by a unique ultrahigh-vacuum molecular-beam-epitaxy pulsed-laser-deposition system. The details of fabrications are provided in Ref. 16 and Methods. Post-fabrication, the crystallinity of the HOSG-QDs is verified by in-situ reflection high-energy electron diffraction (RHEED), X-ray diffraction (XRD), and atomic force microscopy (AFM). An AFM image and a schematic diagram of a typical studied sample are shown in Fig. 1. The employed fabrication method differs significantly from the established dewetting process [23] which is normally employed to produce gold nano-island films. In our case, HOSG-QDs were grown directly in the first deposition step removing the necessity of the second annealing step. This yields Au films that are crystalline and more densely packed than those produced by dewetting.

 

Fig. 1. A typical HOSG-QD sample morphology. a) An AFM image of a sample. b) A schematic diagram of the same HOSG-QD sample on a MgO substrate.

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Optical properties of the fabricated samples were measured by spectroscopic variable angle ellipsometry that allows one to extract optical constants of heterostructures with the help of the Fresnel theory. Spectroscopic ellipsometry routinely measures the spectroscopic parameters Ψ and Δ. These are connected to the ratio of the complex amplitudes of sample reflections r_s and r_p for s and p-polarizations, respectively, as: $\tan (\Psi )\textrm{exp} (i\Delta ) = \frac{{{r_p}}}{{{r_s}}}$. It is worth noting that zero reflection in p-polarization corresponds to Ψ = 0°, while zero reflection in s-polarization corresponds to Ψ = 90°. More details are given in Methods.

The first unexpected result of our study was the observation of topological darkness (TD) in reflection in the HOSG-QD samples as thin as 7.8 nm. TD is a phenomenon that ensures topologically protected zero reflection from a layered sample [20,24,25] and yields associated phase singularities [20,21,26]. It gained large interest due to the possibility to be used in label-free optical sensing [2,27–30]. To predict and confirm the presence of TD in an optical heterostructure we make use of the theory described in our previous work [31].

(In parenthesis we note that topological darkness can be observed in both reflection and transmission. The case of 100% light absorption in a heterostructure is more complicated and requires some additional constraints imposed on a system, see [28,32].)

Figure 2(a) shows the ellipsometric spectra of sample A measured at different angles of incidence. By fitting the measured data with the Fresnel theory, we extracted the refractive index, n, and the absorption coefficient, k, of a HOSG-QDs film plotted in Fig. 2(b). The extracted parameters (n,k) were the same for all angle of incidences and light polarizations. One can clearly see that the optical response of the HOSG-QD films is governed by the strong localized surface plasmon resonances located at λ ≈ 600 nm. One of the important consequences of this fact is the disappearance of absorption in infrared part of the spectrum (k goes to zero when λ becomes larger). This point will be discussed below.

 

Fig. 2. Topological darkness in HOSG-QD for a sample A. a) Ellipsometry measurements made at different angles of incidence fitted using Fresnel theory. The inset shows the model for fitting. EML stands for effective medium layer. b) Refractive index of the HOSG-QD layer extracted from the fit. c) A zero-reflection surface (ZRS) calculated for the MgO substrate and HOSG-QD layer plotted alongside the extracted complex refractive index of the HOSG-QDs (blue line). The intersection point (the red dot) corresponds to the point of topological darkness for the sample. d) The TD point is experimentally verified using variable angle ellipsometry at θ = 72.1°.

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For the sample geometry shown in the inset of Fig. 2(a), there exists zero reflection surfaces (ZRS) in (n, k, λ) co-ordinates where the reflection from the sample is exactly zero under some angle of incidence and some wavelength [20,28,31]. (In short, in order to find angles and wavelengths for which the reflection coefficient of an incident light is zero, we enforce the condition r_p,s(n, k, θ, λ) = 0, where p and s represents p- and s-polarizations, respectively. As r is a complex number, this places two constraints on the parameter space. These constraints imply the potential existence of values of n and k that result in zero reflection for each pair of θ and λ. This generates the zero-reflection surface(s) that depend on thickness of the top film.) Fig. 2(c) plots ZRS for p-polarized light calculated for sample A as a yellow surface. We see that the dispersion curve of the HOSG-QD films extracted with the help of Fresnel fitting (Fig. 2(b)) has branches from both sides of ZRS. Through the application of the Jordan-Brouwer theory [33–35] we can state that the dispersion curve will inevitably intersect ZRS at some point (the red dot in Fig. 2(c)) and hence our sample should demonstrate exactly zero reflection at some wavelength and angle of incidence. This zero reflection would be topologically protected against surface roughness and thickness variations due to the topological nature of Jordan-Brouwer theorem.

The red dot on the ZRS of Fig. 2(c) corresponds to the angle of incidence θ = 72.1° and the wavelength of λ ≈ 646 nm. We have measured the ellipsometric spectra of Ψ for sample A at θ = 72.1° and found that indeed sample A shows exactly zero p-polarized reflection at λ ≈ 646 nm, see Fig. 2(d). (In parenthesis, we remind that Ψ = 0 corresponds to r_p= 0). Figure 2(d) also shows the result of Fresnel modelling of the spectra in which the extracted dispersion curve plotted in Fig. 2(b) was used. One can see an excellent agreement between the experimental results and the modelling data, suggesting that this extremely thin (with a thickness of 7.8 nm as measured by AFM and confirmed by Fresnel modelling) nanostructured gold sample can be described as an effective medium layer. In addition, the inset to Fig. 2(d) shows the Heaviside π-jump in the ellipsometric phase Δ measured at θ = 72.1° which confirms the presence of TD in our sample [20,31].

In proving the presence of TD in the studied gold nanodot samples, we have followed an algorithm which was fully explained and applied in our previous work [31]. A slightly abridged explanation is given below, and the four basic steps are demonstrated in their application to the HOSG-QD sample A in Fig. 2. First, spectroscopic ellipsometry is used to measure spectra of Ψ and Δ at various incidence angles (Fig. 2(a)). A transfer matrix model is then constructed from the known thicknesses of each layer and the complex refractive indices of the substrate layer(s). Second, the complex refractive index of the effective medium layer (EML) is extracted by fitting the transfer model to the measured ellipsometry data. Third, the zero-reflection surface is constructed by calculating the frequencies and incidence angles that produce zero reflection for each potential value of n and k. This surface is then plotted alongside the extracted refractive index and the intersection with the ZRS is found (Fig. 2(c)). Finally, the TD point predicted by the transfer matrix model is experimentally verified using additional reflection measurements (Fig. 2(d)) to validate the model. Although it would be possible to find this minimum in reflected p-polarised light via a trial-and-error method without the TD analysis, the ZRS intersection assures us that the reflectance at the TD point is exactly zero. Additionally, it shows that the existence of zero reflection minima is resilient to sample imperfection due to topological protection.

This algorithm was applied to other HOSG-QD samples (the samples B and C) and a comparison between the samples A, B, and C is detailed in Fig. 3. All three samples were shown to exhibit TD at different wavelengths and incidence angles. This demonstrates a degree of tunability in the location of the TD point based on different chosen growth parameters. Such tunability is connected to homogeneity of the samples or, more specifically, to size distribution of gold nano-islands (nano-dots) in the samples. Indeed, from Fig. 3(a) and (b), we see that the localised plasmon resonances of HOSG-QD films show a wide degree of variability.

 

Fig. 3. Optical properties of three direct growth HOSG-QD samples. a, b) Refractive indices of the three GNI samples. c, d) Confirmation of TD in samples B (left) and C (right) using the TD theory.

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For homogeneous samples (such as sample A), the gold nano-islands have homogeneous size distribution which leads to a well-defined localised plasmon resonances at ∼600 nm (Fig. 3(a) and (b)). For less homogenous samples, the broad size distribution of nano-islands leads to broadening of localised plasmon resonances as it was observed for samples B and C (Fig. 3(a) and (b)). In addition, for samples B and C we observe non-vanishing values of k in the infrared part of the spectrum (k ∼ 0.5 at telecom wavelengths) which is probably connected with residual conductivity of the samples. Despite this, we still were able to predict topological darkness for all three samples through the intersection of the dispersion curves with the ZRSs (Fig. 2(c), Fig. 3(c) and Fig. 3(d)). The predicted TD points were subsequently observed for all samples. These results are summarised in Table 1.

Table 1. Properties of topological darkness in the studied samples. 

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The demonstration of TD in these samples presents an advancement in the application of the TD theory as, to the best of own knowledge, there has not been an example of TD arising from localised surface plasmon resonances of ultrathin inhomogeneous films in the literature. It is remarkable that such a disordered system may exhibit exact zero reflection behaviour; it is also difficult to imagine how zero reflection may be proven in a disordered sample without TD theory. Additionally, the studied HOSG-QD films are particularly well positioned for applications in optical biosensing [2] as techniques for functionalising gold surfaces are both well established and an area of active research due to gold’s relatively strong adhesion to biological molecules and its chemical stability. The large degree of tunability of dispersion relations of HOSG-QD films is based on varying the substrate crystal orientation and growth parameters. This could be advantageous for future optical sensor designs as it has been recently shown that the angle of intersection of the dispersion curve and ZRS for a TD point significantly affects the sensor’s sensitivity due to the ‘scissor effect’ [28]. With this recent addition to the TD theory, it is now possible to optimise the HOSG-QD growth parameters for biosensor sensitivity, implying even greater potential for biosensor designs of this type.

In addition to their use as optical label-free biosensors, direct growth HOSG-QD films may also be used in place of de-wetted gold nano-island films (GNIFs) in their established applications. Previous work has already demonstrated that the direct growth HOSG-QD films have improved performance in surface enhanced Raman scattering (SERS) applications over de-wetted GNIFs [36,37]. In order to evaluate their potential for other dewetting GNIFs applications we may compare the morphology and refractive index of the resulting samples from each technique. Carr et al. carried out spectroscopic ellipsometric measurements of various de-wetted GNIFs that were annealed at a variety of temperatures [38]. They found that their GNIFs refractive indices had a very similar structure to the ones extracted from the direct growth HOSG-QD films with the significant deviation of the peak refractive index of the direct growth GNIFs being approximately twice that of the de-wetted samples. This suggests that the localised plasmon resonances exhibited in HOSG-QD films are much greater in magnitude than those for the de-wetted samples which explains there improved SERS performance and implies that HOSG-QD films may outcompete the de-wetting samples in their other applications, such as Surface enhanced infrared absorption spectroscopy (SEIRA). This is likely due to the greater substrate coverage reached by the direct growth technique over the de-wetting process [39,40].

Reducing inter-island distance has been a focus for improvements in de-wetted gold films and other structures for the purposes of SEIRA sensing techniques where the advantage of lower lattice spacing is clear [41–43]. There have been recent attempts to push the de-wetting techniques further with the addition of extra (and sometimes simultaneous) de-wetting and growth steps in order to increase the performance of the GNIFs [44–46]. There are also emerging techniques that start from gold nano particles instead of thin films [47,48]. Though these techniques do allow for greater coverage than simple de-wetting, they are notably more complex and less successful at reducing inter-island distance than the direct growth techniques described here.

Finally, we address interesting near infrared/infrared optical properties of HOSG-QD films. Figure 2(b) suggests that HOSG-QD films could behave as a dielectric with large refractive index (n ≈ 2.75) and reasonably small absorption (k ∼ 0.005) at telecom wavelengths. Such behaviour was observed only in the samples with small inhomogeneity (sample A). The larger the wavelength of the light, the better dielectric HOSG-QD films will become. We plotted the extracted refractive index of sample A alongside glass BK7 and Si for comparison in Fig. 4(a), demonstrating relatively large refractive index and small extinction coefficient in near infrared/infrared ranges. Figure 4(b) compares n/k for HOSG-QD, glass BK7 and Si in a wide infrared range. Considering the ultrathin HOSG-QD films’ thicknesses, mechanical properties and ease of fabrication alongside these dielectric properties, we conclude that these direct growth HOSG-QD films may be useful for future infrared optoelectronic devices and could prove to be competitive in telecommunication applications.

 

Fig. 4. Optical properties of sample A versus silicon and BK7 glass a) Refractive index of sample A, silicon and BK7 glass b) The ratio of the refractive index over the extinction coefficient for the same materials.

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3. Conclusions

We studied properties of optical metasurfaces produced by direct grown of nanostructured gold films with the help of ultrahigh-vacuum molecular-beam-epitaxy pulsed-laser-deposition. We showed that these ultrathin (<10 nm) crystalline nanostructured gold films placed on a simple MgO substrate demonstrate topological darkness – guaranteed exactly zero p-polarized reflection – and phase singularities that can be useful for optical sensing applications. Therefore, the studied metasurface can be employed as elements of topological photonics. The optical properties of the studied films were governed by strong localized plasmon resonance in the red part of spectrum. This led to dielectric behaviour of the ultrathin nanostructured gold in near and far infrared parts of the spectrum providing a metasurface with large refractive index (n ≈ 2.75) and small absorption (k ∼ 0.005) at telecom wavelengths. The quality of dielectric behaviour of the studied ultrathin nanostructured gold films becomes better in far infrared where one can anticipate possible applications.

4. Methods

Fabrication and characterization: Each highly oriented single-crystalline gold quantum-dots (HOSG-QDs) were prepared on a 1 cm ${\times} $ 1 cm MgO(001) substrate using a unique ultra-high vacuum molecular beam epitaxy pulsed laser deposition (UHV MBE PLD system) equipped with a solid-state Nd:YAG laser (266 nm laser output wavelength), and in-situ reflection high-energy electron diffraction (RHEED) for growth monitoring [18]. The MgO(001) substrate is first loaded as received into the UHV-MBE-PLD system with a base pressure less than 6 ${\times} $ 10⁻⁹Torr. The substrate is subsequently outgassed at 310°C for 45 mins before annealing at 900°C for another 30 mins to obtain a clean MgO(001) surface. Prior to gold deposition, the substrate is reduced to the growth temperature of 350°C. The Au target is pre-laser-ablated using 2000 pulses at 10 Hz frequency. Au is subsequently pulsed laser deposited onto clean MgO(001) at 1 Hz. The laser energy is fixed at about 3.25 Jcm⁻² for all depositions. The samples are annealed for a further 10 mins at 350°C before cooling down to room temperature. All temperature ramps are fixed at 25°Cmin⁻¹.

Ellipsometer measurement: Two types of ellipsometers were used. First, a J. A. Woollam M-2000F variable angle spectroscopic ellipsometer with a 245-1690 nm spectral range and 1nm-1.6 nm resolution was used. The light source was a 75 W Xe lamp producing a reasonably smooth ultraviolet-visible-IR continuum spectrum. Second, a customer made a J.A. Woollam V-VASE ellipsometer was used to obtain room-temperature spectroscopic ellipsometric measurements from 0.6 eV to 6.2 eV in 0.02 eV steps. Spectroscopic ellipsometry (SE) measurements are conducted at incident angles of 50°, 60° and 70° in the reflectance mode. Each data point consists of 100–300 polarizer rotation with an auto-adjustable retarder (compensator) to ensure high accuracy in the whole range. The ellipsometry data was fitted using a transfer matrix model performed with the help of the J. A. Woollam software package WVASE32. The model was used for extraction of the HOSG-QD layer refractive indices and Fresnel modelling of the optical spectra. The HOSG-QD layer thickness was fitted with the initial value taken from AFM measurements. The refractive index of the MgO substrate was taken from Palik’s handbook of optical constants of solids [49].

TD model: The Fresnel models used for construction of ZRS were written in Mathematica. The thicknesses extracted from optical fitting were used; substrates were modelled using the Palik data [49]. A set of Fresnel transfer matrices were calculated for each sample and combination of θ, λ, n and k at resolutions of 0.1 RIU, 0.1 nm and 0.1°. The smallest value of r_p for each value of θ and λ is then found to construct the zero-reflection surface. All the calculated r_p values in the ZRS were less than 0.005.