Abstract
A new method for highresolution quantitative measurement of the dielectric function by using scattering scanning nearfield optical microscopy (sSNOM) is presented. The method is based on a calibration procedure that uses the sSNOM oscillating dipole model of the probesample interaction and quantitative sSNOM measurements. The nanoscale capabilities of the method have the potential to enable novel applications in various fields such as nanoelectronics, nanophotonics, biology or medicine.
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Introduction
Scattering scanning nearfield optical microscopy (sSNOM)^{1,2,3,4,5} has attracted massive interest in the past couple of decades because of its capabilities for probing the optical properties of unlabeled samples at subdiffraction resolutions. To date, sSNOM has been successfully employed for multiple applications such as nanoimaging^{3,6,7}, characterization of plasmonic structures^{8,9,10,11}, nearfield spectroscopy^{4,5,12,13,14}, nanochemical characterization^{15,16}, or for the measurement of the dielectric function in the infrared domain^{17,18}. In the present work, we propose a method based on sSNOM that allows the measurement of the dielectric function of a material with nanoscale lateral resolution. Associated with database matching^{19,20} this method can conduct to the identification of investigated materials.
Many different techniques – like Quantitative Phase Microscopy^{21}, TransIllumination Microscopy^{22}, Optical Coherence Tomography^{23,24}, Multiphoton Microscopy^{25}, or Refracted NearField Technique^{26} – had been reported to date to be capable of measuring, mapping or profiling the dielectric function (the refractive index). Compared to these, the method that we propose provides three massive advantages: (a) subwavelength resolution; (b) the possibility to perform measurements on any solid material sample, irrespectively of its transparency and (c) the method is suitable not only for dielectrics, but also for metals and semiconductors.
Essentially different from a Plasmonic Force Microscope^{27} used for local dielectric response mapping (where measuring extremely low forces may be an issue), sSNOM is typically built as an upgrade^{28} to an Atomic Force Microscope (AFM) allowing for simultaneous sSNOM/AFM imaging. In this configuration, the metalcoated tip of a nanoprobe is brought into the proximity of a sample and is driven into sinusoidal oscillations above the sample’s surface, with the frequency f_{o}. An external laser source laterally illuminates the tip, while the sample is moved pointbypoint in a raster scan process. One of the key problems related to sSNOM imaging is background signal, which affects the detection of the nearfield scattered light. Two combined methods are typically used to diminish the background light: higher harmonic demodulation (HHD)^{29} and pseudoheterodyne detection (PD)^{30}. HHD takes advantage of the nonlinear dependence of the nearfield scattered light intensity on the tip – sample distance. As the tip is oscillating with the frequency f_{o} above the sample, demodulation on a higher harmonic nf_{o} assures an important suppression of the background^{29}. PD refers to an interferometric detection method in which the nearfield scattered light interferes with a reference beam (which has the same wavelength). The phase of the reference beam is modulated by means of a vibrating mirror (with frequency M and amplitude A), which causes the appearance of two sidebands around each harmonic component of the HHD signal. It has been proven that the background light does not affect these sidebands^{30}, which have components located at nf_{o} ± mM (with ). The amplitude of a spectral component u_{n,m} which is located at nf_{o} + mM is given by^{30}:
where c_{n} represents the Fourier spectral components of the nearfield scattered light intensity (σ) and ρ_{m} are the spectral components of the reference beam intensity (E_{R}). Symbols and stand for the real and the imaginary parts, respectively.
The mathematical models that can accurately describe the physical phenomenon of the interaction between the incident beam, the probe (usually an AFM tip) and the sample have been thoroughly discussed to date^{11,17,18,31,32,33,34}. The most popular models among these are the Oscillating Point Dipole Model (OPDM)^{31,32,33}, which is based on the approximation of the probe with a sphere and the Finite Dipole Model (FDM)^{34} in which the tip is treated as a conductive spheroid with physical characteristics of the probe. FDM has been successfully employed to date in the frame of various experiments based on infrared illumination for quantitative measurement of the dielectric function and the local absorption^{17,18}; the results were obtained by employing Fourier Transform Infrared Spectroscopy (FTIR) and thus the applicability was limited to the infrared domain. The method that we propose combines the OPDM and sSNOM imaging conducted in the visible domain in the purpose of quantitatively determining the dielectric function of an investigated material. While both FDM and OPDM based methods are capable of subwavelength resolution, the key advantage of using one over the other consists in reduced mathematical complexity for OPDM, as well as in the reduced number of parameters that are involved in this model. Probing the dielectric function at nanoscale holds massive potential for applications in various fields such as materials science^{15,16,35}, nanoelectronics^{12,36,37}, biology^{23,25,38,39}, or medicine^{24,40,41,42}.
Results
The proposed method for measuring the dielectric function is based on calculating a calibration factor between an experimental image and the OPDMbased simulated signal in the case of an investigated material of wellknown dielectric function, under a particular sSNOM imaging configuration. Once this calibration factor is known, it is further on used for determining the dielectric function of a second material present on the investigated sample, which is initially unknown. This can be achieved using the experimental sSNOM image generated by the unknown material together with the calibration factor and running the OPDM backwards. Calculating the dielectric function of the second material allows for its exact identification via database matching^{19,20}.
Before going further with the description of the implemented algorithm, two observations need to be emphasized.
O1: an important step of the algorithm consists in determining the value of a particular Fourier spectral component c_{n} corresponding to the investigated material with unknown dielectric function. The components c_{n} are functions of the local dielectric function of the sample, c_{n} = f (ε_{s}) and are complex numbers; to determine both real and imaginary parts, one will need a set of two equations. This is the reason for which two images detected from two successive spectral components (located at nf_{o} + mM and nf_{o} + (m + 1)M) need to be used. All our calculations and experiments were done for n = 2 and m = 1.
O2: the pixel value of an sSNOM image I_{n,m} (achieved for the frequency nf_{o} + mM) is proportional with the amplitude of the harmonic component u_{n,m} on which the detection is employed^{17,30}:
The parameter C_{n,m} plays the role of a calibration factor and it is specific to particular setup configurations consisting in detector sensitivity, the output scale of the lockin amplifier, its amplifier factor, offsets, etc.
Algorithm for determination of the dielectric function of an unknown material
Supposing a sample containing two different materials from which one of them is a material with known dielectric function, the algorithm for measuring the dielectric function of the other material is described in the following:

1
The first step consists in collecting two images, I_{2,1} and I_{2,2} detected at the frequencies 2f_{o} + M and 2f_{o} + 2M, respectively, while recording with sufficient precision the values of the configuration parameters involved (introduced in the Methods section). The image areas of the wellknown material will be annotated as the subimages and . Similarly, the image areas of the unknown material will be annotated as the subimages and .

2
The second step is to use the experimental values of the configuration parameters and the equations in the OPDM mathematical model to calculate the corresponding and amplitudes for the material of known dielectric function.

3
The third step is to calculate the calibration factors from equation (2) and to determine the corresponding and amplitudes for the material with unknown dielectric function.

4
The fourth step is to resolve the following set of equations based on the equation (1) in order to determine the real and imaginary parts of the spectral component of the nearfield scattered light signal for the unknown material:

5
The fifth step consists in resolving the equation , thus obtaining the dielectric function of the unknown material. Knowing the dielectric function of a particular material allows for its facile identification via database matching^{19,20}.
It can be observed that the entire algorithm is based on the proportional relationship between the results generated by the mathematical model and the experimental results. It is important to note that this relationship is valid as long as the values of the sSNOM configuration parameters are well known, as these are needed for calculating the dielectric function based on the mathematical model that we introduced.
Demonstration of the method. For demonstrating the proposed method we use two samples, one that contains Si and SiO_{2} regions and one that contains Pt and Al_{2}O_{3} regions. Both samples are introduced in the Methods section. For each of these two samples, we run the experiment in two scenarios: in the first scenario we consider the first material as known and the second material as the material of supposedly unknown dielectric function and in the second scenario we switch roles. Results for the Si/SiO_{2} sample are relevant for samples containing semiconductor/dielectric materials, while the results for the Pt/Al_{2}O_{3} sample are relevant for metallic/dielectric materials.
Application of the algorithm on the Si/SiO2 sample
In the first experiment, SiO_{2} will be considered the unknown material whose dielectric function we want to determine for the used wavelength of 638 nm, while Si will be regarded as the material of known dielectric function. For sample areas containing both Si and SiO_{2} regions, two images I_{2,1} and I_{2,2} are collected at the frequencies 2f_{o} + M and 2f_{o} + 2M, respectively. In the AFM image illustrated as Fig. 1a, we mark nine Si regions. To demonstrate the proposed method, for each Si region the dielectric function of the corresponding SiO_{2} surrounding region is determined for the wavelength of 638 nm. In Fig. 1b,c the corresponding areas of the two materials are graphically delimited for the central region (region 5 in Fig. 1a). Average pixel values are calculated for each area to obtain , , and (as shown in Fig. 1). The area of each circle that delimits a Si region is chosen by considering all the pixels with values higher than 10% of the local intensity peak; the area of the second circle used for delimiting the SiO_{2} surrounding region is chosen so that the ratio between the Si and SiO_{2} areas equals the overall ratio between the areas of Si and SiO_{2} of the whole image.
Using the algorithm described in the previous subsection, we calculate nine values of the dielectric function, one for each of the nine regions of supposedly unknown dielectric function (illustrated in Fig. 1). We calculated the average between the nine measured values and the mean absolute deviation (theoretical value: ε_{SiO2} = 2.379 for 638 nm):
The second experiment ran on the Si/SiO_{2} sample demonstrates the capability of the method to determine the complex dielectric function of a semiconductor. In this second experiment the Si areas are considered of supposedly unknown dielectric function, while considering SiO_{2} as the known material. In this case, the average value, together with the mean absolute deviation for Si will be:
The theoretical value: ε_{Si} = 14.9960.144j for 638 nm.
Application of the algorithm on the Pt/Al2O3 sample
The method for dielectric function determination was applied on a second sample containing a Pt/Al_{2}O_{3} boundary area (see Methods section and Fig. 2). Because of significant differences in the shape of the features that are present on the surface of different samples, the regions corresponding to certain materials cannot be defined always by specific geometric shapes as in the previous experiment. Thus, for this sample, instead of using circular and donut areas (as for the first sample), the measurements are done along horizontal scanning lines of the images, in the purpose of emphasizing the versatility of the presented method. For a single scanning line, the pixel value for Pt was obtained as an average of the pixels contained in the Pt part of the scanning line and the pixel value for Al_{2}O_{3} was obtained as an average of the pixels contained in the Al_{2}O_{3} part of the scanning line.
The determinations were done in a similar manner with the ones performed for the Si/SiO_{2} sample. In a first scenario, Al_{2}O_{3} was considered as a known material and the dielectric function for Pt was determined at the wavelength of 638 nm and in a second scenario Pt was considered as the known material and the dielectric function for Al_{2}O_{3} was determined at the same wavelength. Ten different scanning lines were used to calculate the average and the mean absolute deviation. For this sample, the method returned the following results for the dielectric function of Pt and Al_{2}O_{3}, respectively:
Theoretical values are ε_{Pt} = –11.834–19.773j and ε_{Al2O3} = 3.118 (for 638 nm).
Discussion
The obtained results demonstrate that the dielectric function of a material can be measured with good precision using the proposed method that combines sSNOM imaging and the OPDM. The key requirement of this method is that in the sSNOM image used as support for measuring the dielectric function of one or more materials, sSNOM data collected on one or more materials of known dielectric function needs to be included as well. Using the sSNOM data collected in the regions corresponding to the material of known dielectric function, a calibration factor is calculated and used for determining the dielectric function of other materials contained in the same sSNOM image.
Since the resolution of an sSNOM setup depends on the radius of curvature of the metallic probe used for scanning the sample, typically lying in the range of 1040 nm^{27,43,44,45}, this method allows for determining the dielectric function of nanoscale sample components, which enables new perspectives for novel characterization methods of high potential usefulness for fields such as material science, nanoelectronics, biology, medicine or others.
The proposed method exhibits versatility with respect to the approaches used for defining the regions corresponding to different materials. Thus, among the most important factors is the exact localization of the known material. Afterwards, one can use the pixels contained in areas of different shapes (as in the first experiment), or even the pixels contained along a scanning line of the images (like in the second experiment).
The performed experiments demonstrate that the proposed method for highresolution dielectric function measurement is highly effective for different material classes such as dielectrics, semiconductors and metals. Comparing the averaged measured values (see equations (4, 5, 6, 7)) with the actual values of the dielectric functions for the four materials involved in the experiments (see Methods), small discrepancies can be observed and they are mainly connected to measurement errors.
As the resolution of sSNOM images is not limited by optical diffraction, a high impact of these quantitative measurements is expected – especially in the field of electronic nanochips industry or in the constantlygrowing field of photonic integrated circuits. For example, in the actual requirements in MOSFETs industry, the optical constant measurement of SiO_{2} thin films stands particularly important^{36}. On the other hand, the AFM has become indispensable in the semiconductor industry for dimension metrology^{46}.
The main limitation of this method is given by the dielectric function dependency on the light wavelength, as our method offers the possibility to measure the corresponding dielectric function for a single wavelength at one time.
In summary, a new method for quantitative highresolution measurement of the dielectric function was introduced. The performed experiments demonstrate high measurement precision and enforce the idea that a combined sSNOM/AFM system can be regarded as a powerful tool for simultaneous metrology and optical properties measurements. Such a tool has the potential to enable novel applications in the fields of nanoelectronics, nanophotonics, material science, biology or medicine.
Methods
Materials
The semiconductor/dielectric sample that we have used consists in a SiO_{2} thin layer (26.6 nm thick) deposited on a Si substrate. The sample contains periodic circular holes, with a diameter of 500 nm, that penetrate the SiO_{2} layer reaching the Si substrate. The values for the dielectric functions of the two materials are as follows: ε_{Si} = 14.996–0.144j for Si and ε_{SiO2} = 2.379 for SiO_{2} (at wavelength 638 nm)^{19,20}.
The metal/dielectric sample was a 10 nm thick rectangular domain of Pt deposited on an Al_{2}O_{3} substrate and the surface boundary between the two materials was investigated. The values of the dielectric functions of the two materials are as follows: ε_{Pt} = −11.834–19.773j for Pt and ε_{Al2O3} = 3.118 for Al_{2}O_{3} (at wavelength 638 nm)^{19,20}.
The values of the dielectric functions at 638 nm were obtained by employing the least squares method to the data provided by the available databases^{19,20}.
Mathematical model and calculations
Previous studies have already shown that the intensity of the nearfield scattered light σ is proportional with the amplitude of the incident light phasor E_{o} and the effective polarizability, α_{eff} , where the effective polarizability has the form^{31,32,33}:
In equation (8), α stands for the polarizability of the tip, with its formula:
where a is the tip diameter and ε_{t} is the electric permittivity of the tip. The reflection coefficient β(ε_{s}) is a parameter that depends on the local dielectric function of the sample ε_{s} by the relation:
is the instantaneous distance from the tip of the probe to the sample’s surface and it can be defined as:
Here, d_{o} stands for the minimum separation distance between the tip and the sample during the probe oscillation above the sample, z_{o} is the oscillation amplitude of the probe, f_{o} is the oscillation frequency of the probe and t is time.
Based on the equations presented until now, the intensity of the nearfield scattered light (as a function of β and time) can be rewritten in the following form:
Starting from this point, one can spectrally analyze the function given by equation (12) using the exponential Fourier transformation method. Using the variablechanging u(t) = 2πf_{o}·t, the Fourier coefficients c_{n} will be given by:
In a brief evaluation of equation (13), it can be observed that the only variable implied is β, which is linked to the sample’s electric permittivity ε_{s} by equation (10) thus, the notation c_{n} = f(ε_{s}) is justified. The other parameters are usually known because they characterize the system setup. Fig. 3a illustrates the schematic frequency spectrum of the nearfield scattered light, σ.
The integral in equation (13) is not a common one and its calculation requires special mathematical algorithms; calculating its expression for a general variable n can be regarded as difficult. However, for a given value for n, the integration complexity decreases, allowing for calculation via software computational engines.
In the pseudoheterodyne scheme, the nearfield scattered light interferes with the reference beam E_{R}, which can be mathematically written as:
where ρ is the amplitude of the reference beam phasor, A is the oscillation amplitude of the reference mirror, λ is the wavelength of the beam, M is the oscillation frequency of the reference mirror, Ψ_{R} is the mean phase difference between the two interferometric pathways and t is time.
The mathematical function in equation (14) can be expanded in a Fourier series^{47}:
where the coefficients ρ_{m} are the Fourier coefficients given by:
In equation (16), J_{m} stands for the Bessel function of order m.
The interference signal U between the nearfield scattered light σ and the reference beam E_{R} will have the spectral components u_{n,m}, introduced by equation (1).
In Fig. 3b we represent the frequency spectrum of the interference signal U with the sidebands around each harmonic component of the cantilever oscillation frequency^{30}.
Software calculations and simulations
Calculations and simulations have been performed using the WOLFRAMAlpha online platform and the MATLAB software platform. More precisely, the WOLFRAMAlpha online platform was used for calculating the integral functions required by the spectral components analysis of the sSNOM signal, while the MATLAB software platform was used for sSNOM signal simulation, image analysis and calculation of the calibration factors.
Experimental data acquisition
Experimental data was collected by using a homemade pseudoheterodyne sSNOM setup upgrading an AFM Quesant 350^{28}. The sSNOM configuration parameters during data acquisition were set to the following values: beam wavelength, λ = 638 nm; oscillation frequency of the probe, f_{o} = 60 kHz; oscillation amplitude of the probe, z_{o} = 50 nm; oscillation frequency of the reference mirror, M = 1000 Hz; oscillation amplitude of the reference mirror, A = 267 nm; mean phase difference between the two interferometric pathways (in the pseudoheterodyne scheme), Ψ_{R} = π; ratio between the reference and the incident beam intensities (ρ_{o} and E_{o}, respectively) is 1. The investigating Ptcoated nanoprobe has a tip radius of curvature less than 35 nm. The values of the dielectric function of Platinum is ε_{Pt} = −11.834–19.773j at the wavelength of 638 nm^{19,20}.
Additional Information
How to cite this article: Tranca, D. E. et al. Highresolution quantitative determination of dielectric function by using scattering scanning nearfield optical microscopy. Sci. Rep. 5, 11876; doi: 10.1038/srep11876 (2015).
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Acknowledgements
The LANIR research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/20122015) under grant agreement n°280804. This communication reflects the views only of the authors and the Commission cannot be held responsible for any use which may be made of the information contained therein. The presented work has been supported as well by the research grants PNIIPTPCCA20113.21162. The work of D.E. Tranca is supported by the Sectoral Operational Programme Human Resources Development (SOP HRD), financed from the European Social Fund and the Romanian Government under the contract number POSDRU/159/1.5/S/137390/.
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D.E.T. performed the experimental investigation, the mathematical calculations and the analysis of data. C.S. and G.A.S. contributed at various stages of the experiment and participated in the interpretation of the data. D.E.T. and S.G.S. analyzed the images and wrote the paper, which all authors read and commented on. R.H. and S.A.M.T. contributed to results interpretation and to the manuscript editing. G.A.S. coordinated the experiment and supervised the project.
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Tranca, D., Stanciu, S., Hristu, R. et al. Highresolution quantitative determination of dielectric function by using scattering scanning nearfield optical microscopy. Sci Rep 5, 11876 (2015). https://doi.org/10.1038/srep11876
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DOI: https://doi.org/10.1038/srep11876
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