Mapping the local dielectric constant of a bio-nanostructured system

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Introduction
The dielectric constant, or relative permittivity, is a fundamental physical property which is crucial for describing various phenomena, either biological, chemical, or physical. It is a material property associated to the decrease of the electric force among two-point charges due to the medium. Therefore, it modulates the interaction between charged particles within materials and also the interaction of electromagnetic radiation with matter. Accordingly, it plays a fundamental role in fields as diverse as, for example, the full understanding of proteins [1,2] or in the development of solar cells [3].
Natural photonic crystals are exciting nanostructured systems in which the dielectric properties play a fundamental role [4]. Many of them are biological systems where the richness of colors, produced by different strategies found by Nature, is astonishing [5,6]. Studies of the origin of physical colors in insects are numerous in the literature and the most used tools are non-local optical reflectance, electron microscopies, and scanning probe microscopies, which give support to theoretical models aiming to describe the measured optical properties [7]. However, all these techniques directly reveal only the structure with nanometric resolution, the local dielectric response is indirectly inferred from the model [8][9][10]. Despite the large number of studies, the local dielectric properties of natural photonic crystals remain essentially undetermined due to the great difficulties in measuring the dielectric response at the nanometric scale [11]. The nanometric local relative permittivity of a natural photonic crystal have not been directly measured yet.
Fumagalli et al. [12][13][14][15], and Riedel et al. [16] developed several techniques of Electrostatic Force Microscopy (EFM) to extract the relative permittivity at the nanoscale, allowing new fields to be explored. Here we use EFM to map the relative 4 permittivity of nanostructures within the wings of the Chalcopteryx rutilans damselfly [17][18][19]; nanostructures which make it a natural photonic crystal. We obtain quantitative information about the wing structure and its local relative permittivity properties. We also simulate the optical reflectance using the extracted spatial profile of the relative permittivity and compare it with the measured reflectance in the visible range, obtaining a good correlation. In this way, we can provide a full description of the origin of the shimmering colors of the Chalcopteryx rutilans damselfly male posterior wings. This technique should be useful in the study of similar systems and could be included in their advanced characterization toolkit, enhancing, thus, the investigation possibilities of natural photonic crystals.

Results and Discussion
In damselflies, color has many functions, the most important being sex recognition, courtship and territory defense behaviors [19]. In Chalcopteryx rutilansa damselfly found in the Amazonian rain forestthose functions are performed by the male by displaying its strongly iridescent hind wings. This phenomenon of iridescence results from physical effects such as diffraction and interference, and all observed colors result from a multilayer structure, i.e., these wings are natural one-dimensional photonic crystals [7][8][9][10]. 6 clearly visible. Detailed electron microscopies and mass spectrometry studies of this natural nanostructured system can be found in Valeriano [17] and Carr et al. [18].

Relative permittivity determination via EFM
The EFM measurements were performed in the conventional double pass mode, which means that the probe executes two scansthe first scan measures the sample topography in tapping mode and the second scan mimics that profile at a defined lift height Hlift applying a voltage Vdc between the tip and the conductive substrate [20].
The tip is forced to oscillate, during the second pass, at the cantilever's resonance frequency, 0 . Variations in the local relative permittivity properties of the sample will lead to different tip-sample force gradients, which promote a shift f0 in the tip oscillation frequency [20,21] which is, approximately, where dF/dz is the tip-sample force gradient and K is the spring constant of the cantilever.
The tip-sample-substrate system configures a capacitor with the sample (wing) as part of the relative permittivity region, so the force among tip and substrate can be modeled where C is the system capacitance and Vdc is the applied tip-sample bias.
From relations (1) and (2) we have: This equation relates the frequency shift Δf0 with the applied bias voltage Vdc. The biasindependent term in equation (3) is defined as ∆ 0  given by Since f0, K and VDC are well determined, local variations of the measured frequency shift Δf0 are associated with changes in the capacitance second derivative in equations (3) and (4). The capacitance depends both on the geometry and on the relative permittivity of the medium, so we only need to use a suitable capacitive model to be able to determine the local relative permittivity of the sample from the EFM data.
The capacitor model applied here considers a conical tip and an infinite flat surface with relative permittivity r and thickness h, which has a capacitance expressed as Inserting equation (5) into equation (4) leads to an expression from which the relative permittivity r can be extracted: where R is the tip apex radius, is the tip conical angle, z is the tip-sample distance, and h is the sample thickness. Figure 3 shows a scheme of the model, which is commonly applied for such configuration [12][13][14][15][16].
This capacitor model is valid within the following conditions: i) tip-sample distance z between 0 and 100 nm; ii) maximum sample thickness h around 100 nm; iii) sample relative permittivity r smaller than ~ 100; iv) tip nominal radius R between 30 and 200 nm; v) tip cone angle  between 10° and 45° [12].

Determination of the coefficient ∆
In the EFM mode, the microscope measures the frequency shift for each bias voltage in each position on the sample.
We varied the bias voltage from -10 V to +10 V, in steps of 1 V. Plotting the frequency shift as a function of bias voltage, we find a parabolical behavior, as can be seen in where is the tip-sample surface potential difference due to their different work functions [20], we obtain the coefficient ∆ 0 .

Construction of the relative permittivity map
From the AFM topography image, the thickness h is determined for each pixel, as shown in the Experimental section.
EFM measurements were carried out on the same sample region varying the bias voltage from -10 V to +10 V, resulting in 21 images, where each one is an array of frequency shift values. For the same pixel element in the sample image, we have 21 pairs of values: a frequency shift and its respective bias voltage. Through equation (7) we obtain an ∆ 0 coefficient for each pixel element in the sample.
The parameters in equation (6), less the ∆ 0 coefficient and the thickness h, are the same for all pixel elements.

Chalcopterix rutilans damselfly wings
Using the protocol described above, we constructed relative permittivity maps of the three colored regions: Red region (a), Blue region (b), and Yellow/Green region (c), as seen in Figure 5.
The parameters used for the Red region are 0 = 61.106 kHz, k = 1.19 nN/nm and R = (34.8 ± 0.2) nm; for the Blue region they are 0 = 62.111 kHz, k = 1.87 nN/nm and As done for the Al2O3 case (see Experimental Section), the substrate region was set with r = 1since in this region there is only air between the probe and the gold substrate [15].  During the preparation of the wing samples, there is a variation of about 10 nanometers in the thicknesses of the slices. This variation has not impacted the results, as can be seen in Figure 6, which demonstrates the reliability of this technique for thin samples.

13
The relative permittivity of the layers ranges from 6 ± 1 to 8 ± 1. The main differences between the colored regions lie in the thickness and number of layers. Each multilayered structure is wrapped by a wax layer which is identified as the irregular region at the boundary of the wings in the maps of Figure 5, and as a valley in the relative permittivity profile with ( ) ~ 4 in Figure 6.
Using Time-of-Flight Secondary Ion Mass Spectrometry (TOF-SIMS), Carr et al. [18] concluded that the wing layer composition consists of mostly chitin with an alternating content of melanin, where chitin builds up the structure and melanin modulates the relative permittivity along the cross-section. From the results shown in Figure 6, we can see that, in addition, the number of layers and their thicknesses varies from one color region to the other. Comparing the composition of the layers measured in the TOF-SIMS study [18] with the relative permittivity maps of this work, we can say that melanin-rich layers have the relative permittivity around 8 ± 1, while low melanin concentration layers present a value around 6 ± 1.

The structural color
As shown in Figure 1, the posterior wings of the male Chalcopteryx rutilans has a wide range of structural colors that covers almost the entire visible spectrum. We now seek to correlate the relative permittivity information obtained via EFM with the photonic behavior of the wing.
The wing has several layers with thicknesses comparable to the wavelength of light in the visible range. Through refraction and diffraction, which depend on the thicknesses and refractive indexes of the layers, the light wavelengths that will be reflected with higher intensities are selected, generating the displayed colors. As shown above (see Lyddane-Sachs-Teller relation has been extended to other crystalline systems and disordered materials [23][24][25] but its application for the present case, chitin with a varying concentration of melanin, is not straightforward. In any case, there is a direct relation between the refractive indexes and the measured relative permittivity.
Therefore, to simulate the optical reflectance of the damselfly wing, we consider that the refractive index varies along the cross section of each region of the wing following the continuous profiles shown in Figure 6, treating the minimum and maximum values of each profile as fitting parameters. The continuous profiles are discretized in layers thin enough to give a smooth as possible variation and then the transfer matrix technique is used to simulate the reflectance of the structure. This is actually a similar process as that used by Vukusic and Stavenga [6] and Stavenga et al. [7,8] except that in those works the spatial profile of the refractive index was taken to be proportional to the gray scale in TEM images, that is, it was assumed that the optical density is directly proportional to the electronic density. We consider our method to be more reliable since there is a direct relation between the relative permittivity and the refractive index.

Sample preparation
Damselfly wing samples were produced by ultramicrotomy. Fragments of the chosen colored region were embedded, with no previous cleaning procedure, in epoxy resin.
The polymerized resin block was trimmed in a wedge shape, with the wing fragment section perpendicular to the wedge axis. Sections 40 nm thick of the apex wedge were cut using a diamond knife and placed on 10 mm x 10 mm Au/Cr coated silicon wafer pieces with 60 nm/20 nm film thickness, respectively. A conductive substrate surface is necessary for the proposed εr determination method. The sample also needs to be less than 100 nm thick to eliminate the influence of the stray capacitance [12].
Therefore, samples of three different color regions of the wing were prepared, namely Blue, Red and Green/Yellow ones.
An essential characteristic of the experiment is that both the sample and the conductive substrate need to be present within the AFM image. This is a key condition since the conductive substrate establishes a reference level in the analysis. Having both in the imaged region guarantees that the cantilever amplitude and, consequently, the effective radius of the tip will be the same for different materials, which is critical for the applied capacitor model [27]. Also, having the conductive surface and the sample in the same AFM image guarantees the precision in the determination of the thickness of the sample.

Determination of the parameters:
The sample thickness h is directly determined via AFM imaging. The microscope control software also determines the resonance frequency 0 and the cantilever elastic constant K, using the Thermal tune method [28,29].
A critical parameter is the tip-sample distance z, that consists of the height plus the cantilever amplitude . The value of is adjusted and presented by the microscope with remarkably high accuracy and precision. The value of Zamp is obtained by the standard amplitude-distance curve method.
Another critical parameter is the effective radius R. To obtain it, we carry out EFM measurements on the gold surface and determine the coefficient ∆ 0 . Over the gold surface, the effective thickness of the sample h goes to zero, thus ℎ = 0 [13,30], and equation (6) can be seen as a function of R.
The coefficient ∆ 0 depends on z and to obtain the correct R, it is necessary to perform the EFM measurements of the sample and the substrate with the same height reference. EFM measurements with both the substrate and sample in the same image solve this issue.
The conical angle is = 0.261 rad, as informed by the tip producer. We used the AC240TM-R3 probe model by Oxford Instruments Asylum Research in all measurements.

Al2O3 reference sample
We made reference samples of a material with a well-known relative permittivity. In this way, applying the method to this reference sample, we validated the technique presented in this paper to measure the relative permittivity of the nanostructures in the damselfly wings.
Our reference samples were photolithographically defined 5 µm radius disks of Al2O3 films deposited by ALD on Au/Cr coated silicon wafer pieces with 60 nm/20 nm thicknesses, respectively.
The topographic image of an Al2O3 disk by AFM is shown in Figure 8a. The sample thickness was (21.0 ± 0.2) nm, referred to the gold surface.
In the EFM mode, the microscope measures the frequency shift for each bias voltage in each position on the sample. We varied the bias voltage from -10 V to +10 V, in steps of 1 V. Plotting the frequency shift versus bias voltage, we obtain a parabolical behavior. Fitting the data with equation (7) we obtain the ∆ 0 coefficient. Doing this operation for each pixel we build the image of alpha coefficients as we can see in Figure 8b.
The gold surface has a higher ∆ 0 coefficient than the Al2O3 disk, Figure 8b. This means that the frequency shifts on gold are larger for all bias voltages, so the tipsubstrate interaction on the gold surface is stronger than on the alumina disk.
The tip radius was obtained using the ∆ 0 coefficient for measurements on the gold surface for h = 0, and was R = (36.7 ± 0.2) nm, Figure 8b. The tip-sample distance z is 44 nm, determined by the sum of 40 nm of the setup lift height with the 4 nm of the cantilever oscillation amplitude [16]. We choose a small cantilever oscillation amplitude 22 during the EFM measurements in order to keep the tip not far from the sample surface, in a range where this method is valid [12]. c) 23 The free oscillating frequency 0 = 73.403 kHz and the cantilever elastic constant k = 2,24 nN/nm, were calculated using the Thermal tune.
The height h, Figure 8a, and the ∆ 0 coefficient, Figure 8b, are different for each pixel in the sample, the other parameters mentioned above and seen in equation 6 are the same for all pixels. Solving equation (6) for we constructed the map of the dielectric constant as shown in Figure 8c. In the reconstructed dielectric image, the region corresponding to the substrate was set to = 1 since this region corresponds to the air relativity permittivity [15].

Reflectance measurements
For measurements of the reflectance spectra, a halogen lamp with a color temperature of 3200 K (OLS1 FIBER ILLUMINATOR, Thorlabs) was used, which allows reflection measurements in the spectral range from 350 nm to 1100nm. A set of biconvex lenses focused the source light on the desired colored region and another set of biconvex lenses delivered the reflected light to an optical fiber with 0.22 numerical aperture. The light was delivered through this optical fiber to the spectrometer (USB4000, Ocean Optics). Spectral data were acquired with the Spectra Suite software (Ocean Optics).
Spectral measurements were made in three colored regions of male damselfly hind wings, namely Yellow/Green, Red, and Blue region. In this work we study the light captured at 60ᵒ in relation to the normal of the wing.