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Abstract
We present a theoretical study on the plasmonic response of borophene, a monolayer 2D material that is predicted to exhibit metallic response and anisotropic plasmonic behavior in visible wavelengths. We investigate plasmonic properties of borophene thin films as well as borophene nanoribbons and nanopatches where polarization-sensitive absorption values in the order of 50% is obtained with monolayer borophene. It is demonstrated that by adding a metal layer, this absorption can be enhanced to 100%. We also examine giant dichroism in monolayer borophene which can be tuned passively (patterning) and actively (electrostatic gating) and our simulations yield 20% reflected light with significant polarization rotation. These findings reveal the potential of borophene in the manipulation of phase, amplitude and polarization of light at the extreme subwavelength scales.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recent advances in 2D materials signal a bright future for the next generation, compact electronic and photonic devices [1]. With reduced dimensionality and material thicknesses reaching down to atomic and molecular levels, 2D materials provide access to unique electrical, optical and mechanical properties that can enable compact, sub-diffraction limit and yet efficient devices. Another landmark phenomenon that plays a crucial role in the trend of smaller devices is plasmonics. Plasmons are collective oscillation of free electrons at the interface of metals and dielectrics [2–5]. Conventional plasmonic materials are usually noble metals like silver and gold, and literature around these plasmonic materials is vast. Plasmonic materials are also utilized in designing metamaterials and metasurfaces that could modify the amplitude, phase and polarization of light [6]. Metasurfaces and metamaterials find applications in wide range of areas including absorbers [7,8], photodetectors [9,10], solar cells [11], retarders [12], waveguides [13] and lenses [14], and negative refraction and cloaking [15]. However, ohmic losses in metals restrict the optical applications of plasmonics that cannot tolerate such optical losses [16]. Graphene was the first 2D material to be discovered and investigated in detail. Despite its monolayer thickness, theoretical and experimental studies in the recent years have unearthed the capability of graphene in supporting plasmons and interacting strongly with light. Initial demonstrations included, but were not limited to, nanostructured graphene (i.e. nanoribbons) which supports localized and propagating surface plasmon polaritons (LSPP and PSPP) with an exceptional tunability of plasmonic response with different doping methods (i.e. electrical, chemical and optical) [5,17]. The literature abounds with several options in the family of 2D van der Waals materials with astonishing properties; hyperbolic phonon polaritons in hBN [18], valley-dependent physics in transitional metal dichalcogenides for spintronics [19] and anisotropic plasmons in black phosphorus (BP) [20] are only a few examples out of many.
Despite having access to 2D semi-metal (e.g. Graphene), insulators (e.g. hBN) and semiconductors (e.g. MoS2), a 2D metal has been elusive. Recently, borophene was introduced as a 2D material that supports plasmons in the visible and C-band ranges of electromagnetic spectrum with high electron density and hence is called a 2D metal even though in bulk form it is a semiconductor [21,22]. After initial demonstration of experimental growth of borophene on Ag (111) substrate [23,24], optical and electronic properties of borophene deserves more in-depth investigation. The high electron density and strong anisotropy of borophene has garnered attention as an outstanding 2D material that has potential to excite PSPPs and LSPPs [25,26]. Moreover, owing to its highly dispersive optical response in different in-plane crystallographic directions, it can give rise to several interesting physical phenomena. Similar to BP [27], borophene has innate hyperbolic response and can support anisotropic plasmons and canalization by itself which obviates the need to be integrated with other materials through laborious nanofabrication techniques [28]. Additionally, borophene surmounts the shortcoming of BP due to the fact that unlike BP which supports LSPPs in mid-IR, borophene supports plasmons in the visible range thanks to its high carrier density compared to BP [20,22].
2. Results and discussion
2.1. Optical model of borophene
In the 2D limit, boron atoms form bonds in a hybrid lattice of hexagonal and triangle distributions. Theoretical calculations predict several polymorphs of borophene; however, only three phases α, β12, and χ3 have attracted scrutiny. Borophene allotropes are unstable in free-standing form and a distribution of vacancies is required to make it experimentally feasible. Growth of borophene using molecular beam epitaxy (MBE) on Ag (111) surface alleviates the stability issue since Ag passivates the sp2 hybridization. Therefore, α phase which lacks vacancies (like zigzag and armchair directions of BP) is not experimentally stable. Of the mentioned three phases α and χ3 have large anisotropy in their crystal directions which yield anisotropy in optical response of borophene [21,23]. α phase is schematically illustrated in Fig. 1(a) with marked critical directions. Since we aim to theoretically investigate the anisotropic nature of plasmons here, we chose α phase which possesses the strongest anisotropy and highest density of electrons [22]. The dispersion relation of α phase is calculated, and the χ3 counterpart is provided in Appendix 4.1. All the upcoming discussions are applicable for χ3 as well and hereafter, α phase is used in this study.
Fig. 1. (a) Schematic of α phase borophene monolayer under impinging light, (b) dispersion relation of α phase free-standing borophene in x and y directions (dotted lines) and their respective 2DEG model for comparison. Effect of periodicity (p) on absorption (A) versus wavelength (λ) for a free-standing monolayer borophene nanoribbon with w = 50 nm in (c) x direction and (d) y direction. The arrows in (b) represent the resonances in the large periodicity limit of panels (c) and (d) and the discussed parameters for x direction are illustrated in the inset of (c).
Plasmon excitation occurs at the interface of dielectric and metal, where the permittivity of the metal is negative at the wavelength of interest, driven by negative charge oscillation [5]. Since the carrier density is quite large, this phenomenon is modeled macroscopically by semi-classical Drude model which treats electrons as free carriers [20]. Initial theoretical studies on borophene suggest that it has a very high-density of free electrons (i.e. Drude weight) compared to other 2D materials [22]. Recently, monolayer borophene is verified to have metallic properties and host Dirac cones [21,26]. Therefore, borophene is a viable candidate to enable plasmonic characteristics at visible frequencies. Figure 1(b) demonstrates the dispersion relation for α phase borophene (dotted lines) in two different crystallographic directions, x and y, where x direction is chosen to be the one with higher conductivity with Drude parameters taken from those reported by Huang et al. and Mannix et al. [22,23]. Moreover, the 2DEG counterpart which is estimated in the work of Huang et al. is also provided for comparison. The dispersion relation infers that our calculated dispersion is almost identical to the one reported by Huang et al.; as a result, our model successfully mimics the optical characteristics of first-principle calculations of borophene [22]. It is also evident from Fig. 1(b) that the accessible wavelengths for LSPP resonance of borophene reach the visible range.
Given that borophene is a metal, it enables intraband transitions which are commonly modeled by Drude equation. Since plasmonic response usually relies on intraband transitions (i.e. free carrier response), it is highly tunable and covers broader response bands. Interband transitions, on the other hand, are modeled by phenomenologically adopted Kramers-Kronig relation [29]. Based on this discussion, the conductivity of borophene can be modeled by Drude model and is given by
In order to calculate the dispersion relation, a free-standing borophene sheet was used in simulations where a plane wave is impinging onto the ribbon arrays with polarization perpendicular to the ribbons for each direction (x and y) separately. Two examples for each of the two anisotropic directions for w = 50 nm are plotted in Figs. 1(c) and 1(d). In these plots, the resonance frequency remains unaltered due to the suppression of coupling between neighboring ribbons when the periodicity (p) is approximately an order of magnitude larger than w. This behavior agrees with plasmonic response of 2D materials. In order to access the wave-vector values (q) through the selected w values, Fabry-Perot analysis is used. When the ribbon length is finite, electrons (i.e. excited plasmons) oscillate and back-scatter from the edges of the ribbon. Assuming that the ribbon is homogeneous, standing wave patterns will form that correspond to LSPPs, which are described by 2qw + 2ϕR = 2nπ. In this equation, ϕR is the phase picked up through reflection from the edges and n is the mode number. Following the discussion by Nikitin et al. for graphene an anomalous phase of ϕR = -0.75π is chosen [31], which yields q = 0.75π/w. The dispersion (dotted lines) is illustrated in Fig. 1(b). The calculated dispersion is also compared to the 2DEG model (solid lines) defined as ω = (σe2q/2mj ɛ0)0.5 which is used to fit to ab-initio calculated dispersion in [22]. Each two sets of dispersion curves are almost equivalent except for large wavevector values (i.e. small ribbon width) which is attributed to inhomogeneous doping in the edges of the ribbon and strong interaction of neighboring ribbons [31]. It is also evident from Fig. 1(b) that there exists strong anisotropy in the plasmonic response of borophene. Moreover, Fig. 1(c) infers that as the periodicity increases, the interaction between the ribbons gets weaker, the absorption peak blueshifts and the peak value of absorption decreases. This reduction in absorption is due to smaller surface coverage of borophene. This discussion is also valid for y direction illustrated in Fig. 1(d). In this figure, the narrow line at 0.8 µm is the higher order mode LSPP resonance (more information in Fig. 7 of Appendix).
2.2. Anisotropic plasmons
In order to investigate LSPP resonances in borophene further, the effect of ribbon width on absorption is studied where borophene ribbon is placed on an SiO2 substrate. Figure 2(a) [2(b)] illustrates reflectance versus the wavelength and ribbon width in x (y) direction when the periodicity is kept constant at p = 100 nm. As the width of the ribbon (w) increases, the supported LSPPs are realized in longer wavelengths which is expected, and the reflectance peak broadens as a result of the increased optical losses at longer wavelengths [20]. Like Fig. 1(d), the higher order plasmonic modes are discernable in Figs. 2(a) and 2(b). The spatial electric field distribution data are summarized in Figs. 2(c)–2(e). Figure 2(c) [2(d)] illustrates the real part of electric field, Ex (Ez) in a xz cross section for a ribbon with w = 50 nm and p = 100 nm at its corresponding LSPP resonance wavelength, 970 nm. An important implication from the electric field pattern of Fig. 2(c) is that only even modes are accessible for LSPP resonances [31]. This figure also vindicates high field enhancement and localization. Besides, Fig. 1(e) depicts the total electric field intensity in xz cross section which shows a field intensity enhancement of approximately four orders of magnitude.
Fig. 2. Reflection versus width and wavelength for a ribbon with periodicity p = 100 nm patterned in (a) x-direction and (b) y-direction on a SiO2 substrate. Real part of (c) Ex and (d) Ez in xz cross section for a ribbon with w = 50 nm and p = 100 nm at 970 nm resonance. (e) Total electric field intensity in xz cross section (log scale) at 970 nm resonance.
In order to shed light on anisotropic LSPPs in borophene, we elaborate on direction-dependent extinction spectra. We calculate absorption of a 2D square patch pattern of borophene on a SiO2 substrate [Fig. 3(a)] while the polarization angle of the impinging light (θ) is chosen to be 0, 45 and 90°. The polarization angle is defined with respect to x axis. The geometric values are set as w = 30 nm and p = 60 nm. Figure 3(b) shows the absorption in a borophene square patch for the mentioned three polarization values. For θ = 0°, There is 30% absorption around 1.75 µm which is attributed to the LSPP resonance in the higher conducting x direction. When the polarization is set to θ = 45°, the peak intensity at 1.75 µm drops and a second peak emerges at 2.88 µm which is likewise associated with LSPP resonance in y direction. Finally, when the polarization is (θ = 90°), the first peak disappears and the LSPP resonance in y direction at 2.88 µm becomes dominant. Such a strong dichroism is in line with expectations since borophene is patterned along its principal axes and the diagonal terms of conductivity (i.e. σxy and σyx) are zero in the absence of magnetic field. As a result, as long as the polarization of the incident light is aligned with in-plane crystal axes, only one of the x or y LSPP resonances can be excited which will be the one parallel to the polarization of the impinging field. These resonances can be tuned significantly by varying geometric or carrier density parameters w, p and n, increase of which would respectively result in blue-shift, red-shift and blue-shift of the resonances corresponding to x and y resonances. These trends are implied in Figs. 1(b)–1(d), Figs. 2(a)–2(b) and Figs. 7(b) and 7s(g). On the other hand, the two resonances can be excited simultaneously, although with lower intensity, when the polarization of light is not parallel to the optical axes.
Fig. 3. Demonstration of dichroism caused by anisotropic LSPPs. (a) Schematic of patterned borophene in square patches of 30 nm side on SiO2 substrate with a periodicity value of 60 nm, (b) absorption versus wavelength for three different polarization values θ = 0, 45 and 90°. Field intensity values for θ = 45° on the surface of borophene square patches (xy cross section) for (c) Ex at the first (x) resonance λ = 1.75 µm and (d) Ey at y resonance λ = 2.88 µm.
Figure 3(c) shows the intensity of the x component of the electric field at its corresponding LSPP resonance λ = 1.75 µm when the polarization is set to 45°. This panel suggests that there is almost two orders of magnitude field enhancement and the signature dipolar response along x direction is also recognizable. The y counterpart is also shown in Fig. 3(d) where all the values are the same except for the sampled wavelength which is the LSPP resonance in y direction, λ = 2.88 µm. The absorption peak in Fig. 3(b) hits 0.4 in borophene. The polarization-sensitive nature of borophene provides ample opportunities to apply this feature to birefringence applications in visible range, such as biological tissue imaging [32], wave retarders [33], nonlinear optics [34] and polarization sensing. We also note that the intensities of absorption values in both directions can be boosted to near unity values which would render the absorption for θ = 45° to more than 0.5, by adding a metallic back reflector to form a vertical plasmonic metal-insulator-borophene cavity [20]. The results for this configuration are not provided here for the sake of brevity and are briefly discussed in Appendix 4.3.
2.3. Anisotropic plasmon enhanced birefringence
The large dichroism that stems from anisotropic LSPP resonances has significant implications in photonics. As mentioned in the introduction, metal-based metasurfaces used to dominate birefringence effects that lead to phenomena such as anomalous reflection and refraction [14]. It was the advent of graphene that changed this perspective; many 2D van der Waals materials even in monolayer thickness have strong anisotropy in their plasmonic response that allows hyperbolic and elliptic plasmons to exist. This fact implies a significant advantage in that the need for complicated metasurface cell fabrication is eradicated [35]. Not only do these materials exhibit inherent linear birefringence effect, but also, they enable active metasurfaces meaning that their response is reconfigurable in real time by changing their free electron concentrations [36,37]. Even though isotropic graphene can demonstrate giant birefringence if patterned, natural anisotropic 2D materials provide even stronger birefringence when patterned in nanoribbon structure. As a result, phase, amplitude and polarization state of light can be manipulated and exploited. In this section, we will briefly analyze a simple configuration using borophene that can rotate the polarization of light in reflection and transmission mode from which manifests the potential of borophene in photonics applications.
The conductance tensor of a continuous anisotropic 2D material like borophene can be described by
Fig. 4. Conductivity versus wavelength for (a) continuous borophene, (b) patterned borophene in form of nanoribbons in x direction with w = 34 nm and p = 68 nm. Simulated far-field (c) polarization ellipse major angle, and (d) electric field ratio between Ey and Ex versus wavelength, when the applied field is linearly polarized with θ = 45°, in transmission and reflection mode for continuous and patterned (ribbon) borophene. Polarization ellipse of continuous and patterned borophene at 1550 nm in (e) reflection and (f) transmission mode.
The critical point in Fig. 1(b) at 1500 nm marks the distinction between two different regions of operation in the patterned borophene. The real parts of the conductivities in both directions are positive for all of the wavelength values. Our results show two orders of magnitude larger conductivity for x direction compared to y in the vicinity of resonance, 1550 nm. Below λ = 1500 nm, the imaginary parts of the two conductivity values have similar signs (Im(σx).Im(σy) > 0) which represents elliptic region. In the elliptic region, anisotropy exists, and plasmons can evolve in elliptic manner in both directions and wavefronts would form elliptic propagation pattern. Contrarily, when λ > 1500 nm the imaginary parts of the conductivities have opposite signs (Im(σx).Im(σy) < 0) which marks the hyperbolic region [28].
One of the implications of hyperbolic region is that it can support hyperbolic plasmons in the case of a patterned structure which can give rise to polarization ellipse rotation. In order to delve into this phenomenon better, simulations have been carried out for the structure whose effective conductivity is shown in Fig. 4(b). Figures 4(c) and 4(d) summarize the polarization ellipse major angle and magnitude ratio of electric field in two different in-plane directions (x and y). These results are recorded for normal incidence with polarization set to 45° in the far-field limit. The monitor is also normal to the borophene plane. It is evident from these two panels that at 1550 nm (C-band), the polarization of the reflected and transmitted light is the strongest for the patterned film, reaching 25°. We recognize that the continuous borophene film also demonstrates polarization ellipse rotation in reflectance as much as 15°, however, the reflected power is less than 8% which demonstrates poor efficiency (Fig. 10 of Appendix 4.4). In contrast, in the patterned case, the reflection is enhanced above 20% and the polarization rotation is also increased as seen in illustrated in Fig. 4(e) which demonstrates the reflected polarization ellipse at 1550 nm, for continuous and patterned borophene layers on SiO2 substrate. It is worth pointing out that the EMT successfully models the response where the peak in conductivity matches the resonance of the FDTD simulated transmittance and reflectance result of Fig. 9. The effect of patterning on polarization rotation is more pronounced in the transmittance mode; the continuous borophene film results in negligible polarization rotation (Fig. 9) unlike the patterned configuration, where the rotation reaches 17°. Moreover, the hyperbolic region can be passively and actively tuned respectively with geometry parameters such as w and p and electrostatic gating to modify carrier density. Therefore, this simple structure can significantly rotate the polarization state of the incoming linearly polarized light in one atomic layer thickness which reveals the strength of borophene in dynamically tunable manipulation of phase, amplitude and polarization of impinging light as a whole which is an essential issue in photonics. The metallic 2D borophene in this study can be tailored to the existing wealth of electro-optical tuning mechanisms and quantum well configurations established in the work of Sherrott et al. with BP and can lead to next generation of 2D heterostructure photonics [37].
3. Conclusion and outlook
In conclusion, we propose Drude model to discuss the optical characteristics of borophene. LSPP resonances and the corresponding field enhancement and localization in patterned borophene monolayers is studied. Our numerical simulations suggest orders of magnitude enhancement in electric field intensity in both principal axis directions. In particular, we investigate the linear birefringence behavior of LSPPs in borophene using 2D patterned patches. A functional structure is proposed which can rotate the polarization state of impinging light in the C-band. Having access to the rich family of existing 2D materials from insulators to semi-metals and semiconductors, the addition of borophene to the family of 2D materials as a 2D metal can open new paths in improving the functionality of future photonics as well as creating new applications. As the experimental realization of borophene keeps maturing [40], we envisage that borophene metal can be used to improve electronic junction characteristics with other 2D materials as well. Specifically, since boron has a stable 2D form of insulator, hexagonal boron nitride (hBN), it can create an opportunity for miniature all-boron electronics and photonics [21].
4. Appendix
4.1. Anisotropic plasmon enhanced birefringence
The Drude model values used in this study (α) as well as the ones for χ3 phase are summarized in Table 1.
For all the simulations using Lumerical FDTD solutions, the mesh sizes along the in-plane and vertical directions are respectively set to 1 nm and 0.03 nm. Table 2 summarizes the dispersion relation data points for α phase discussed in the main body as well as χ3 phase discussed here in appendix.
Table 2. Summary of dispersion relation values for α and χ3 phases of borophene.
Figure 5(a) depicts schematically the χ3 phase borophene which fits to the 2DEG model proposed in the work of Huang et al. [22]. Using the Drude model Eqs. (1) and (2) in the manuscript, the permittivity values for α phase of borophene is illustrated in Fig. 6.
Fig. 5. (a) Schematically represented χ3 borophene and (b) dispersion relation of χ3 phase free-standing borophene in x and y directions (dotted lines) and their respective 2DEG model for comparison.
Fig. 6. (a) Real and (b) imaginary parts of permittivity for a monolayer α phase borophene in x and y directions.
Fig. 7. Real part of Ex at (a) 1470 nm and (b) 790 nm. Real part of Ez at (c) 1470 nm and (d) 790 nm. The field profiles are in yz cross-section and where w = 50 nm and p = 100 nm. The two left panels are the fundamental LSPP mode and the two right panels are the higher order LSPP mode. The wavelength values are read from Fig. 1(d) at p = 100 nm.
4.2. Analysis of Drude parameters
In order to provide further insight into the variables in this equation, effect of each one is investigated in this section briefly. This study is carried out on y direction LSPP of α borophene. The width of the nanoribbon in the mentioned crystal direction is 50 nm and the periodicity is set to 100 nm. The mesh in the out-of-plane direction is 0.03 nm as stated earlier and the polarization of the impinging light is along y direction. The effect of significant Drude model parameters, ɛr, m* (mj = my), n, τ and d on absorption is shed light on in Fig. 8. Panels (a) through (e) in the left side of Fig. A3 illustrate the dependence of LSPP resonance on different Drude model parameters. The panels (f) through (j) on the right-hand side illustrate spectral absorption of points picked from their left panel counterpart which are colored as red squares. In all these simulation results, the default value for parameters are taken from Table 1 in the main manuscript for the y direction and only one parameter is changed at a time so that the effect of varying parameters can be visualized independently.
Fig. 8. Dependence of resonance wavelength on different Drude parameters (a) normalized effective mass (m*/m0), (b) free carrier density (n), (c) monolayer thickness (d) and (d) relative DC permittivity (ɛr). (e) Dependence of absorption strength (A) and full width at half maximum on carrier lifetime (τ). (f) – (j) Absorption versus wavelength counterparts for the red-colored dots in (a) – (d).
Taking Fig. 8(a) into consideration, it is obvious that the resonance wavelength increases for increased effective mass. This is not surprising since according to the 2DEG dispersion model (which fits to all our material models as discussed in Fig. 1(b) and Fig. 5(b), there is a square root dependence of resonant wavelength on effective mass. Writing this equation in terms of resonance wavelength instead of resonant angular frequency, we get
Fitting a square root function to the plot in Fig. 8(a), we get a function of the form λfit=0.6412(m*)0.5+0.0031 which proves the mentioned λ∝ (m*)0.5 dependence. On this figure, three different spots are selected, and their respective spectral absorption is provided in Fig. 8(f). Due to the curves in this figure, increasing the effective mass results in red-shifted resonance with higher absorption and narrower full width at half maximum (FWHM). This behavior can be deduced from Eq. (6) in that the higher the effective mass, the higher the real and imaginary parts of permittivity. Besides, the sharp and narrow resonance in the vicinity of 0.8 µm in Fig. 8(f) for the case of m* = 6m0 is the higher order mode LSPP resonance for the main resonance sitting at 1.6 µm. The discussion about square root dependence of resonance on effective mass can also be applied to the results of Figs. 8(b) and 8(c) and is avoided here for the sake of brevity.
Figure 8(d) illustrates the weak dependence of resonance wavelength on DC permittivity of borophene. This weak dependence is predictable since in the wavelength range of interest, the DC permittivity value is much smaller than the second term in Eq. (6). Moreover, since the resonance wavelength is almost unaltered with varying the carrier life time, it is avoided and the absorption intensity and FWHM are provided in Fig. 8(e) instead. Together with Fig. 8(j), it illustrates that the intensity (FWHM) of spectral absorption is increased (decreased) as τ is chosen to be larger which is a natural result of a dielectric function of this form.
4.3. Boosting absorption in vertical plasmonic cavity
In this section, we aim to elaborate briefly on the ramifications of constructing a vertical plasmonic cavity by incorporating layers from bottom to top as a thick metal, SiO2 dielectric with thickness t and patterned borophene patch (patterned in both x and y directions) on top. It is a well-established practice to form such cavities to increase field confinement and enhancement and as a result, the absorption. This scheme indeed increases absorption to over 0.5 in both directions in borophene. In order to look at this effect closely, we have simulated a 50 nm by 50 nm patch of α phase borophene on SiO2 and Au as illustrated schematically in Fig. 9(a). The polarization of the impinging light is set to θ = 45° so that both resonances in the fundamental crystal directions are accessible. The plot in Fig. 8(b) shows the absorption in the structure for t = 400 nm. It is obvious that adding a back reflector does indeed increase the absorption peak in both x (1.6µm) and y (3.5µm) LSPP resonances. These resonances are the fundamental LSPP modes in the mentioned directions. It is worth pointing out that in the results of Fig. 3 of the main text, the maximum absorption for θ = 45° polarization is only 0.2. The absorption in Fig. 9(b) for the same polarization reaches 0.5 in both directions which emphasizes the fact that a vertical cavity boosts absorption in borophene and intensifies all the discussed phenomena such as electric field magniude.
Fig. 9. (a) Schematic of the square patch borophene nanopattern forming a vertical cavity together with SiO2 and Au with all of the simulation parameter values illustrated, (b) total absorption for the structure in (a) where the insulator thickness t is chosen to be 400 nm and (c) total absorption (A) versus insulator thickness t and wavelength (λ).
Figure 9(c) illustrates the dependence of spectral absorption on the thickness of the SiO2 insulator (t). This illustration captures most of the critical information about resonances in a cavity. The dashed oblique lines are the cavity resonance modes that link the insulator thickness to the wavelength, linearly. The lowermost oblique line is the fundamental cavity mode and the other ones represent higher order modes and this behavior is a signature of interference and cavities that interact with LSPP modes and render them split. There are two modes (dark and bright) for x direction LSPPs and one for y direction at the higher wavelengths.
4.4. Transmittance and reflectance in borophene film versus ribbons
Funding
Office of Naval Research (N00014-17-1-2425).
Acknowledgments
K.A. acknowledges support from the Office of Naval Research Young Investigator Program (ONR-YIP) Award (N00014-17-1-2425). The program manager is Brian Bennett.
Disclosures
The authors declare no conflicts of interest.
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