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Abstract
We present numerical simulations of scattering-type scanning near-field optical microscopy (s-SNOM) of 1D plasmonic graphene junctions. A comprehensive analysis of simulated s-SNOM spectra is performed for three types of junctions. We find conditions when the conventional interpretation of the plasmon reflection coefficients from s-SNOM measurements does not apply. Our approach can be used for other conducting 2D materials to provide a comprehensive understanding of the s-SNOM techniques for probing the local transport properties of 2D materials.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Scattering type Scanning Near-Field Microscopy (s-SNOM) is a powerful tool for studying the local electronic properties of surfaces and 2D materials in a few tens of nanometer size domains [1–3]. Simulation of the s-SNOM signal obtained in experiments is an essential part of a technique for retrieving materials’ physical properties. The available mathematical modelings of s-SNOM experiments are developed mainly for homogeneous surfaces [4–10]. Similar calculations for the case of a composite wafer of planar layers covered by graphene [11–13] have also been reported. Besides, one can find reports of simulated s-SNOM of 1D graphene plasmon junctions [14,15].
The s-SNOM images give a good qualitative understanding of a sensed object on a length scale much smaller than the wavelength of light [16]. In particular, s-SNOM is a powerful tool for visualizing the electric properties of surfaces and atomically thin 2D materials [17]. However, retrieving accurate quantitative information on the length scale beyond the plasmon wavelengths is still a challenge. Machine learning techniques offer some progress in this direction [18]. The plasmon’s reflectance is one of the most valuable physical quantities of interest [1,14]. In graphene, plasmon reflectance was reported both analytically and numerically for various types of 1D junctions: discontinuously changed doping in graphene supported by a homogeneous dielectric substrate [19,20], homogeneously doped graphene sheet supported by a wafer with discontinuously changed dielectric permittivity [21], scattering regions, such as a gap in graphene [22,23], Gaussian-profile spatial distortion of graphene doping [22], and 1D corrugations of graphene sheet [15]. In all the mentioned works, reflection coefficients are calculated for almost ideal conditions when the plasmon was generated very far from the junction. However, in the case of s-SNOM scanning of a junction, the distance from the spot under the s-SNOM tip sourcing plasmons is comparable with the tip’s diameter and height above the sample. Therefore, direct measurements of the reflected plasmon intensity become complicated because of the tip’s indispensable presence and its electric field. Here, we demonstrate that reconstruction of the plasmon reflection coefficient from s-SNOM measurements is non-trivial, especially when the field reflection coefficient is larger than $10$%.
2. Formulation of the problem
In this work, we develop a model for numerical simulation of the s-SNOM signal of a 2D conducting material accounting for the electric field modulation from the oscillating tip, see Fig. 1(a) and Sec. 1 of Supplement 1. Following experimental setup [1–3], we use TM (transverse magnetic) polarization shown in Fig. 1(a), such that the dipole moment lies in the $xz$-plane. The numerical results presented here are calculated for $\theta _1=\theta _2=0^\circ$ (unless otherwise stated), while the formalism is available for arbitrary angles (see Supplement 1). The conductivity $\gamma _{\omega }$ [24,25] characterizes a 2D material. In the case of graphene, we assume the Drude model [26]:
where $\mathcal {E}_{\mathrm {F}}$ is the Fermi energy and $\nu$ is the electron scattering rate. Following Refs. knoll2000basic and hillenbrand2001basic, our simulations employ a model of s-SNOM of a bare plane homogeneous dielectric surface. That model was extended to enable simulations of the s-SNOM of the layered planar structure of homogeneous materials covered by graphene (or other 2D conducting material) [11–13]. We consider a cylindrical tip parallel to the sample surface with a diameter smaller than a characteristic size of the $E$-field inhomogeneity.
Fig. 1. (a) Schematics of the s-SNOM experimental setup to probe a flat dielectric covered by a homogeneous conducting 2D material. The three plasmon junctions geometries are shown in (b) discontinuous doping, (c) gap in graphene, and (d) discontinuous wafer, where a cylindrical tip is sketched on top of the plane in yellow. (e) shows results of the simulated $S_2$ scans for the incident photon energy $\hbar \omega =120$ meV on two graphene junctions with discontinuous doping, shown in panel (b). The chosen doping level contrasts in two cases give the same plasmon reflection coefficient of 14%. The positions of the relaxed signal values far from the junction (Rx) and (Rx,2), brightest (BL) and darkest (DL) lines are labeled correspondingly. Dielectric permittivity of the wafer supporting the junctions is $\varepsilon _{1}=3.9$, and the Fermi energy of graphene on the left side is $\mathcal {E}_{\mathrm {F}1}=300$ meV, which corresponds to the plasmon wavelength of $\lambda _{p1}=154$ nm. The Fermi levels on the right side are given in the legend for both cases. We use an electron scattering rate of $\hbar \nu =10$ meV. The minimum and maximum distances from the center of the tip’s cylinder to the sample are $h_{\mathrm {min}}-a=5$ nm and $h_{\mathrm {max}}-a=50$ nm, correspondingly, where $a=30$ nm is the tip’s radius. The $S_2$ signal is normalized to the absolute value of the relaxed signal on the left side far away from the junction.
We consider three common plasmonic junctions, shown in Fig. 1(b)-(d), which we refer to as discontinuous conductivity, a gap in graphene, and a discontinuous wafer, correspondingly. The complete self-consistent solutions with a spheroid-like or small-ball-like tip shape require substantial efforts (see, for instance, supporting information for Ref. [14]). We consider a cylindrical tip model to reduce the initial 3D problem to a one-dimensional one and enable a solution in the Fourier domain. We take into account the sample’s back-action on the tip’s polarization, the tip’s oscillation, and the signal’s demodulation, see Sec. 1 and 2 of Supplement 1. We have achieved an excellent run-time for the s-SNOM scans simulations for all the three types of junctions owing to the high parallelizability of the resulting numerical schemes, which enables us to simulate numerous scans for different parameters and verify the technique used in the experiments to retrieve plasmon reflection coefficients [1]. The main goal of the present research is to determine under which conditions simple assumptions about the relationship between the s-SNOM $S_2$ signal and the plasmon reflection coefficient are valid. Note that reflection coefficients for the junctions shown in Fig. 1(b)-d can also be obtained from the numerical eigenmode solutions [27] and compared with the plasmon reflection coefficients extracted from the s-SNOM scans. The latter are found from the contrasts of the brightest (or the darkest) line formed by the standing plasmon waves, as demonstrated in Fig. 1(e). While we find that this procedure is reliable in many instances, there are situations when those dependencies are much more complex. The non-unique relationships between the s-SNOM signal contrasts and reflection coefficients are explained below by certain arbitrariness in post-processing the s-SNOM scans.
3. Simulation results and discussion
To simulate the s-SNOM response of graphene junctions, we employ a Fourier transform method for periodically repeated junctions, as shown in Sec. 3 and 4 of Supplement 1. Technically, the Poisson equation solutions in these geometries are the same as in the case of the diffraction problem of a far-field incidence on the sample [28–30]. The only difference is that instead of the far-field, harmonics of the near-field generated by the tip’s dipole are “scattered” of the sample (see Sec. 3 and 4 of Supplement 1and Ref. 31 for details). To eliminate spurious size effects, we have chosen the maximal height of the tip’s dipole above the sensed surface $h_{\mathrm {max}}$ to be much smaller than the system’s period $d=W+L$, and the plasmon wavelengths in the regions between the junctions to be much smaller than $W$ and $L$ defined in Supplement 1.
During the s-SNOM measurements, the tip senses the standing wave, resulting from the interference between the tip and the edge-reflected plasmon. The interference pattern disappears at distances larger than the plasmon propagation length determined by parameter $\nu$ in Eq. (1). As a result, the $S_{2}$ signal changes from a constant, which we call a relaxed signal intensity, to an oscillating signal with a growing amplitude as the tip approaches the junction. This signal oscillation amplitude is usually used in experiments to determine the reflection coefficient from the plasmon junctions [1]. It is natural to assume that the envelope of the signal oscillations, normalized to its relaxed value, depends only on the junction’s reflection coefficient independently of the junction type. This assumption enables one to calibrate experimental setups using the s-SNOM signal from a junction with the known reflection. For instance, a graphene edge or a gap in graphene of length larger than a quarter of the plasmon wavelength almost entirely reflects plasmons [22,32]. We will show below that such assumptions are not always applicable.
In Fig. 2(a), we demonstrate typical $S_{2}$ signals for three types of junctions shown in Fig. 1(b)-d. The Fermi energies are the same on the left side of each junction, while on the right side, the Fermi energies are chosen to result in the same plasmon reflection coefficient [19,27]. One of the most straightforward estimations of the $S_2$ signal variation (or signal contrast) is based on fitting numerical results to the expression: $\pm ae^{b\left (x-x_{0}\right )}$, where $x_{0}$ is the junction’s position. However, our results show that this fitting procedure does not always work when processing a wide variety of simulated scans. It is more convenient to use the most straightforward approach for defining $S_2$ contrast, namely, the absolute difference between the brightest (or the darkest) line and the relaxed signal. The exact choice depends on the relation between the relaxed signals on two sides of the junction:
One should note that Eq. (2) is not the result of some self-consistent mathematical derivation. We use it because of similar constructions of $S_{2}^{\mathrm {BL}}$ and $S_{2}^{\mathrm {Rx}}$ have already been used in the experiments [1]. We define $S_2$ contrast depending on the relation between the relaxed signals on both sides of the junction; see Table 1. To show the conventionality of Eq. (2), we developed a classification of the brightest and darkest lines on the left side of a junction. The main idea of the classification is to show the results calculated by both expressions in Eq. (2), taking into account the fact that plasmon reflection in some cases may be dependent on the evanescent plasmons, which are non-symmetrical on the left and right sides of a junction, specifically, when the plasmon wavelengths $\lambda _{\mathrm {p}1}$ and $\lambda _{\mathrm {p}2}$ are different semenenko2020reflection). According to Eq. (2) and Table 1, in Fig. 3, we plot the $S_2$ signal contrast versus the absolute values of the reflection from junctions for three different types and various dielectric permittivities of the wafers supporting the junctions’ left sides. Note that the plasmon reflections can be calculated independently without involving the s-SNOM tip simulations [19,21,27]. We assign each of the brightest and darkest lines a marker corresponding to the values of the conditional expressions. Thus, the triangle (and its variations) is used when the plasmon wavelength on the right side is smaller than on the left side. Star (and its variations) is used in the opposite case, when $\lambda _{\mathrm {p}2}>\lambda _{\mathrm {p}1}$. In the case of the gap in graphene junction, we assume $\lambda _{\mathrm {p}2}=0$. Straight oriented markers ($▴$, ${\triangle}$, ${\bigstar}$ and ${\star}$) are plotted for the cases when Eq. (2) uses $S_{2}^{\mathrm {BL}}$. Their overturned versions ($▾$, $▿$, ${\bigstar}$, and ${\star}$ ) are plotted when $S_{2}^{\mathrm {DL}}$ are used. A solid marker designates a line for which the condition in Eq. 2 is true; otherwise, a marker is open and oppositely oriented. The mentioned rules are summarized in Tab. 1 and demonstrated on examples of s-SNOM scans in Fig. 2.
Fig. 2. Examples of $S_2$ scans from three types of junctions depicted in Figs. 1(b)-d. Parameters of the junctions are the following: (discontinuous doping) $\mathcal {E}_{\mathrm {F}2}=210$ meV, (discontinuous wafer) $\varepsilon _{2}=3.9$, (gap in graphene) $L=10$ nm. The simulations are done for an incident photon energy $\hbar \omega =120$ meV, Fermi energy of graphene on the left side of a junction $\mathcal {E}_{\mathrm {F}1}=300$ and the dielectric permittivity of a wafer on the left side is $\varepsilon _{1}=7.0$, which corresponds to the plasmon wavelength of $\lambda _{p1}=94$ nm. The electron scattering rates on both sides of the junction are the same $\hbar \nu =10$ meV. The minimum and maximum spans between the tip and the sample are $h_{\mathrm {min}}-a=5$ nm, $h_{\mathrm {max}}-a=50$ nm, where $a=30$ nm is the tip’s radius. The scans are normalized to the absolute value of the relaxed signal on the junction’s left side. A vertical dashed line shows the junction’s position. Markers illustrate the classification rules for the brightest and darkest lines used to calculate reflection coefficients according to Eq. (2) and Table 1.
Fig. 3. $\left |S_{2}\right |$ signal contrast on the left side of a junction, i.e., $S_2$ maximum normalized to its relaxed value $S_{2}^{\mathrm {Rx}}$, versus modulus of the plasmon reflection coefficient for different dielectric permittivities of the medium below graphene. On the left side of a junction: (a) $\varepsilon _{1}=1.0$, (b) $3.9$, (c) $7.0$ and (d) $10.0$. These simulations are done at fixed $h_{\mathrm {min}}=35$ nm and $h_{\mathrm {max}}=80$ nm. All the simulations use incident photon energy $\hbar \omega =120$ meV, graphene Fermi energy $\mathcal {E}_\mathrm {F}=300$ meV (or $\mathcal {E}_{\mathrm {F}1}=300$ meV on the left side of the junction for the case of discontinuous doping junction), electron scattering rate $\hbar \nu =10$ meV, and the tip’s radius $a=30$ nm. The markers have the same meanings as in Fig. 2. The black dashed lines in figure panels correspond to the simple assumption about the proportionality between the signal contrast and the plasmon reflection.
Table 1. Brightest and darkest lines (BL and DL) classification rules used for building Fig. 3. Superscripts (Rx) and (Rx,2) denote relaxed values of s-SNOM signal on the left and the right sides of the junction, correspondingly.
It can be concluded from Figs. 3(a-d), that the assumption about the proportionality of the $S_2$ signal contrast (the black dashed lines) with the reflection coefficient calculated independently according to Ref. [19–23,27] works reasonably well for $\varepsilon _{1}=1$ and $\varepsilon _{1}=3.9$. However, for the larger values of $\varepsilon _{1}$, multiple families of curves appear using all alternatives in Eq. (2). Moreover, even a single branch does not follow a straight line (see the filled markers). The deviations in Figs. 3(c) and (d) are most prominent when the plasmon wavelength on the left side of a junction is smaller than the one on the right side (see star-markers). We relate this to the electric field generated by the long-plasmon-wavelength edge forming the junction. The short-plasmon-wavelength edge does not affect its counterpart because the edge’s field penetration length into another side is about half of the plasmon wavelength [27]. Therefore, we find that the assumption about the proportionality between the signal contrast and the plasmon reflection breaks down the cases of high-k dielectric substrates on the left side of the junction. Finally, to demonstrate that our conclusions are not dependent on the incident angle, we repeat the calculations for $\theta _1=\theta _2=30^\circ$ in Fig. 4 using all other parameters the same as in Fig. 3.
Fig. 4. $\left |S_{2}\right |$ signal contrast on the left side of a junction versus modulus of the plasmon reflection coefficient for different dielectric permittivities of the medium below graphene using the same parameters as in Fig. 3.
4. Conclusion
We provide an analytical model of s-SNOM of a conducting 2D material supported by a dielectric wafer that can be used to choose a proper tip oscillation regime. We applied a model for simulating the s-SNOM signal for three types of 1D junctions. The model considers the near-field signal’s tip oscillation, modulation, and demodulation. This approach covers a broad range of experimentally relevant parameters. Our numerical approach is time efficient and allows us to explore a wide range of parameters to optimize plasmonic circuits and waveguides [33–35].
We have analyzed many simulated scans from different types of junctions and found that shapes of the s-SNOM signal profiles depend on multiple parameters, such that no simple fitting expression can cover all cases. We have found that the assumption about the proportionality of the $S_2$ signal contrast and the reflection coefficient works well if the plasmonic junction is deposited on a low-k SiO$_2$ wafer, but it fails in cases of high-k substrates. In addition, the assumption breaks down in most cases when the plasmon wavelength at the side of the junction, where the contrast is measured, is smaller than that at the other side.
Our findings demonstrate that within a particular parameter space, the s-SNOM response can be used to extract material’s properties at the side of a junction, which is non-accessible by the s-SNOM tip. Our computationally efficient approach can generate many training data sets for machine learning approaches for processing s-SNOM scans [36].
Funding
Office of the Vice President for Research and Economic Development, University at Buffalo (100013666); State University of New York Research Seed Grant Program (RSG201091.1); National Science Foundation (DMR - 2045425); Solar System Exploration Research Virtual Institute (80NSSC19MO2015).
Acknowledgments
We acknowledge support from the Vice President for Research and Economic Development (VPRED), SUNY Research Seed Grant Program, and the Center for Computational Research at the University at Buffalo [37]. This work was partly supported by the RISE2 node of NASA’s Solar System Exploration Research Virtual Institute under NASA Cooperative Agreement 80NSSC19MO2015. M. L. acknowledges support from the NSF Faculty Early Career Development Program under Grant No. DMR - 2045425.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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