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Abstract
A coupled silicon nanocuboid dimer is employed to improve the magnetic dipole (MD) emission. Finite difference time domain (FDTD) simulations reveal that the nanocuboid dimer supports coupled magnetic resonance and confines a large magnetic field within the gap. The coupling and magnetic-field-enhancing capability are stronger than that of the nanosphere dimer due to the nonspherical symmetry of the nanocuboid. The MD resonance and the magnetic quadrupole (MQ) of the nanocuboid dimer are used to improve the emission of the MD emitter positioned within the gap. It is revealed that the MD resonance can lead to a large enhancement factor of 262 for the radiative decay rate at the wavelength of 653 nm. The enhancement factor is about 3.8 times that caused by the nanosphere dimer. The peak of the radiative decay rate can be tuned in a broad wavelength range by tailoring the structure parameters of the nanocuboid and an enhancement factor of over 230 is achieved at the long wavelength range. For a nanocuboid dimer with a larger size, the MQ resonance can lead to a higher enhancement factor of about 368 at the wavelength of 819 nm, which is nearly 1.5 times that caused by the nanosphere dimer.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Magnetic effects are usually neglected in light-matter interactions because the magnetic dipole (MD) transition is about five orders of magnitude lower than the electric dipole (ED) transitions [1,2]. Since the observation of the stronger MD transitions in the lanthanide ions, however, much attraction is drawn to MD transition and enhancing MD emission becomes an important topic in various researches [3–10]. For example, many efforts are devoted to developing approaches based on plasmonic nanoantennas such as split-ring resonator and diabolo nanoantennas to improve the MD emission [11–14]. By carefully designing the plasmonic structures, MD emission can be strongly modified and the enhancement factor of the radiative decay rate reaches a high value of over 200 [14]. Despite the large MD emission enhancement, the intrinsic absorption loss in plasmonic structures results in quenching of the MD emission and then lowers the quantum efficiency.
To avoid this problem, all-dielectric nanoantenna with high refractive index is used to enhance the MD emission since these types of nanoantenna can support Mie resonance mode, which results in magnetic hotspots and then enhances the MD emission with minimal absorption losses [15–22]. A drawback of this approach is that the magnetic hotspots of the Mie modes are confined within the dielectric nanoantenna, where it is difficult to position a MD emitter. To make MD emitter spatially overlapped by the magnetic hotspots, approaches based on nano-doughnut-shaped silicon disk and silicon nanocavity are proposed to enhance the MD emission and the enhancement factor of the radiative decay rate achieves a high value of more than 350 [19–21]. Positioning MD emitters in a dimer of silicon particles is another approach to enhance the MD emission. By positioning an MD emitter within the gap of the silicon nanospheres dimer, a relatively high enhancement factor of over 250 is achieved for the radiative decay rate since the magnetic quadrupolar (MQ) resonance can confine relatively large magnetic field in the gap [22]. In this case, however, the strongest magnetic field is not confined in the gap of the dimer but inside the silicon nanoparticles, which limits the MD emission to be further improved. On the other hand, it is demonstrated that nanocuboid dimer can enhance large magnetic field in the gap in both visible and gigahertz spectral region [23]. But little is known about the effects of the nanocuboid dimer on the MD emission in the visible wavelength range.
In this paper, silicon nanocuboid dimer is proposed to improve the MD emission by positioning the MD emitter within the silicon nanocuboid dimer. Finite difference time domain (FDTD) simulations [24] are first used to calculate the scattering cross section spectra of the silicon nanocuboid dimer to reveal the magnetic resonances of the silicon nanocuboid dimer and provide information on its interactions with MD emission enhancement. It is revealed that the coupling and magnetic-field-enhancing capability is stronger than that of the nanosphere dimer due to the nonspherical symmetry of the nanocuboid. As a result of the stronger magnetic-field-enhancing capability, the MD resonance of the nanocuboid dimer lead to an enhancement factor of 262 for the radiative decay rate of the MD emitter, which is positioned in the silicon nanocuboid dimer. This enhancement factor is nearly 3.8 times that of the MD emitter positioned within the nanosphere dimer. The effects of the gap distance, the position of the MD and the structure parameters on the MD emission are also investigated. Finally, the radiative decay rate enhancement of a nanocuboid dimer with larger size is also investigated. It is found that the MQ mode of the nanocuboid dimer can lead to a higher enhancement factor of 368.
2. Structure and simulating method
The structure of the coupled silicon nanocuboid dimer consists of two nanocuboids with a gap distance of d, as is shown in Fig. 1. The length, width and height of the silicon nanocuboid in the dimer is represented by l, w, h. FDTD simulation is employed to investigate the scattering property of the dimer and the effects of the dimer on the MD emission. The simulation is performed in a domain spanning ± 500 nm, ± 500 nm, ± 1000 nm in x, y, and z direction with the nanocuboid dimer positioned at the center. All six boundaries of the computational domain are set as perfectly matched layers. Nonuniform grid meshing is used in the simulation with size set as 15 nm at the periphery of the simulating domain and 1 nm in the immediate vicinity of the nanoantennas. In the simulation, the refractive index of the silicon is adopted from Ref [25]. and the dispersion curve is shown in Fig. 1(b).
Fig. 1 (a) Schematic of the coupled silicon nanocuboid dimer; (b) The refractive index and extinction coefficient of silicon.
In order to investigate the scattering properties of the nanocuboid dimer, total-field scattered-field (TFSF) source is used in the simulation. The TFSF source is included in a monitor box which is used to calculate the total scattered power. The scattering cross section is then calculated as:
where Pscat is the total scattered power [W] and Iinc is the incident source intensity [W/m2]. An xy plane monitor is positioned at z equal to 0 nm to obtain the magnetic field intensity profile. To investigate the effects of the dimer on the MD emission, we calculate the radiative decay rate enhancement of a MD emitter positioned at the gap of the dimer. In the calculation, a MD source with magnetic field parallel to the y-axis is used and the radiative decay rate enhancement is calculated as the ratio of far-field radiative power of the MD emitter with the nanoantenna to that without the nanoantenna.3. Results and discussion
3.1 Scattering properties of the silicon nanocuboid dimer
The scattering properties of the nanocuboid dimer are first investigated by employing FDTD to calculate the scattering cross section spectrum of the nanocuboid dimer. In the calculation, the silicon nanocuboid has a length l of 140 nm, a width w of 60 nm and a height h of 150 nm. The distance between the two nanocuboid is set as 10 nm. A plane wave with x-polarization is illuminated on the silicon nanocuboid along negative z direction. For comparison, the scattering cross section spectrum of the silicon nanocuboid monomer is also calculated and the calculated results are shown in Fig. 2(a). It is observed that the scattering cross section spectrum of the nanocuboid monomer peaks at the wavelength of 498 nm and 585 nm. These two peaks are attributed to the ED resonance and the MD resonance, respectively [17]. When two nanocuboids are positioned together, coupling occurs between the MD resonance mode of the two nanocuboids and redshifts the MD resonance from 585 nm to 650 nm. The coupling can enhance the magnetic field intensity within the gap. This is supported by Fig. 3(a) and Fig. 3(b), where it is observed that the magnetic field intensity enhancement (|H|2/|H0|2) at the center of the gap reaches a high value of over 260 at 650 nm.
Fig. 2 Scattering cross section spectra. (a) nanocuboid monomer and nanocuboid dimer. (b) nanosphere monomer and nanosphere dimer.
Fig. 3 Normalized magnetic field intensity profile. (a) nanocuboid monomer (at 585 nm). (b) nanocuboid dimer (at 650 nm). (c) nanosphere monomer (at 630 nm). (d) nanocuboid dimer (at 650 nm). H denotes the magnetic field and H0 denotes the magnetic field of the incident light. White arrows show the direction of magnetic field and the length of the arrow denotes the magnetic field intensity.
Further investigation is performed to compare the scattering cross section spectra of the nanocuboid dimer and the nanosphere dimer. The radius of the nanosphere is set as 78 nm to ensure that the nanosphere dimer has the same coupled MD resonance mode at about 650 nm as the nanocuboid dimer (see Fig. 2(b)). It is observed that the redshift (~65 nm) of the MD resonance mode for the nanocuboid dimer exceeds three times that (~20 nm) of the nanosphere dimer, indicating that the coupling is stronger in the nanocuboid dimer than in the nanosphere dimer. In addition, magnetic field intensity in the center of the gap of the nanocuboid dimer is about four times that of the nanosphere dimer. In other words, the nanocuboid dimer is more efficient to improve the magnetic field in the gap.
The higher magnetic-field-enhancing capability should be attributed to the nonspherical symmetry of the nanocuboid. For both the nanocuboid and the nanosphere monomer, MD resonance results in quasi-spherical symmetric magnetic field distribution with intensity decreasing quickly from the center to the edge. In the nanocuboid, XZ surface is closer to the center due to its nonspherical symmetry and larger magnetic field is leaked out (see Fig. 3(a) and 3(c)). It in turn strengthens the coupling and increases the magnetic field within the gap when two nanocuboids are placed close to construct a dimer, as is shown in Fig. 3(b) and 3(d). This also indicates that the nanocuboid dimer can lead to a higher emission enhancement for the MD emitter positioned within the gap than the nanospherical dimer. The judgment is examined and investigated in the following section.
3.2 MD emission properties of the silicon nanocuboid dimer
3.2.1 Improving MD emission by the MD resonance of the silicon nanocuboid dimer
The MD emission properties of the silicon nanocuboid dimer are first investigated by calculating the MD emission enhancement caused by the MD resonance mode. For this purpose, we calculate and compare the emission spectra of the MD positioned at the center of the gap of the nanocuboid and the nanosphere dimer. In the calculation, the structure parameters of the nanocuboid dimer and the nanosphere dimer are the same as those used in Section 3.1 and the MD is positioned at the center of the gap with dipole moment paralleling to the y axis. The calculated results are shown in Fig. 4, from which two main results can be observed. First, a large radiative decay rate enhancement of 262 is achieved for the nanocuboid dimer at the wavelength of 653 nm. The enhancement should be attributed to the MD resonance since the peak wavelength of the decay rate enhancement spectra matches that of the scattering cross section spectrum (see Fig. 2(a)). In addition, the enhancement factor of 262 is 3.8 times that (about 69) of MD positioned in the nanosphere dimer, which is consistent with the enhancement factor of the magnetic field intensity. This finding confirms the judgment that the nanocuboid dimer is more efficient than the nanosphere dimer to improve the MD emission.
Fig. 4 Radiative decay rate enhancement spectra. (a) nanocuboid monomer and dimer. (b) nanosphere monomer and dimer.
The calculation above demonstrates that MD resonance of the silicon nanocuboid dimer can result in large enhancement for MD emission. In the following, we investigate the effects of the gap distance d of the dimer, the position of the MD and the structure parameters (l, w, h) of nanocuboid on the MD emission.
First, we investigate the effects of the gap distance by calculating radiative decay rate enhancement of the MD in silicon nanocuboid dimer with different distance gap. In the calculation, l, w and h of the silicon nanocuboid are set as 140 nm, 60 nm and 150 nm, respectively. The gap distance d is changed from 4 nm to 50 nm and the MD emitter is assumed to be positioned at the center of the gap. It is found that the peak value of the radiative decay rate enhancement spectra is increased as d decreases and the enhancement factor reaches the maximum value of nearly 350 for the dimer with 4 nm-gap. In addition, the enhancement spectra are redshifted, as is shown in Fig. 5(a). The redshift happens because the coupled MD resonance mode is shifted to the longer wavelength for the dimer with smaller gap. In Fig. 5(b), the decrease in gap distance redshifts the peak of the scattering cross section spectra related to MD resonance mode. This trend is the same with that of the nanosphere dimer and nanodisk dimer, which is reported in Ref [22,26].
Fig. 5 Effects of gap distance on (a) radiative decay rate enhancement spectra and (b) scattering cross section spectra.
Second, we investigate the effects of the position of the MD by calculating the radiative decay rate enhancement of the MD positioned in different places in the vertical z-axis. In the calculation, l, w and h of the silicon nanocuboid are respectively set as 140 nm, 60 nm and 150 nm. The calculated results are shown in Fig. 6(a). It is observed that the radiative decay rate enhancement is maximized at the center of the gap (Z = 0) and decreases as the MD is positioned far away from the center of the gap. The trend of the variation matches that of magnetic field intensity (see Fig. 6(b)). These findings indicate that the MD resonance should be adjusted to overlap with the MD emitter so as to maximize the MD emission.
Fig. 6 (a) Radiative decay rate enhancement spectra as a function of Z position. (b) Normalized magnetic field intensity. H denotes the magnetic field and H0 denotes the magnetic field of the incident light.
Third, we investigate the effects of the structure parameters by calculating the radiative decay rate enhancement of the MD in silicon nanocuboid dimer with varied structure parameters (l, w, h). In the calculation, when one parameter is varied, others keep constant. The calculated results are shown in Fig. 7, from which two results can be observed. First, the peak values of the decay rate enhancement spectra remain larger than 250 when the parameters (l, w, h) are varied separately, and a maximum enhancement of 262 is achieved with l, w and h set as 140 nm, 60 nm and 150 nm, respectively. Second, in Fig. 7(a)-7(c), the peak of the decay rate enhancement spectra shifts to the long wavelength when l, w and h of the nanocuboid increases separately. The redshift happens because the MD resonance of the nanocuboid dimer is shifted to the longer wavelength when these parameters increase, as is shown in Fig. 7(d)-7(f). These figures also demonstrate that the peak of the decay rate enhancement spectrum (the black squares in Fig. 7(d)-7(f)) roughly fits with the MD resonance (the red spots in Fig. 7(d)-7(f)) with only several nanometers offset, which confirms the conclusion that the peaks of radiative decay rate enhancement result from the MD resonance of the nanocuboid dimer.
Fig. 7 (a-c):Radiative decay rate enhancement spectra. (d-f): Peak wavelengths of the radiative decay rate enhancement and the scattering cross section spectra (related to the MD resonance). (a), (d): varied l with w and h kept as constants of 60 nm and 150 nm; (b), (e): varied w with l and h kept as constants of 140 nm and 150 nm; (c), (f): varied h with l and w kept as constants of 140 nm and 60 nm.
It is observed from the calculation above that the peak of the radiative decay rate enhancement can be tuned by tailoring the structure parameters of the nanocuboid. Actually, the peak of radiative decay rate enhancement can be tuned to any randomly given wavelength by properly designing the structure parameters of the nanocuboid. For this purpose, the radiative decay rate enhancement at a randomly given wavelength (for example, 400 nm, 500 nm, 600 nm, 700 nm and 800 nm) is maximized by optimizing the structure parameters (l, w, h) of the nanocuboid dimer. In the optimizing process, a computational method based on particle swarm optimization [27] is used and the MD emitter is assumed to be positioned at the center of the gap. The results are shown in Fig. 8. The peak of the radiative decay rate enhancement spectrum occurs at the wavelength of 400 nm, when l, w, and h are set as 73 nm, 17 nm, and 73 nm, respectively. By adjusting these parameters to 171 nm, 78 nm, and 196 nm, the peak of radiative decay rate enhancement spectrum is redshifted to 800 nm. For the structure with the radiative decay rate enhancement spectra peaking at 600 nm,700 nm and 800 nm, the peak enhancement factor is larger than 230. For the structure with the radiative decay rate enhancement spectra peaking at 400 nm and 500 nm, the enhancement factor is less than 200 since lots of energy can be absorbed by the silicon nanocuboids due to the large extinction coefficient within this wavelength range (see Fig. 1(b)).
Fig. 8 Radiative decay rate enhancement spectra peaking at 400 nm, 500 nm, 600 nm, 700 nm and 800 nm. Black: l = 73 nm, w = 17 nm, h = 73 nm; red: l = 90 nm, w = 50 nm, h = 105 nm; blue: l = 140 nm, w = 56 nm, h = 120 nm; Green: l = 160 nm, w = 75 nm, h = 145 nm; Purple: l = 171 nm, w = 78 nm, h = 196 nm.
3.2.2 Improving MD emission by the MQ resonance of the silicon nanocuboid dimer
In addition to investigating the MD emission enhancement caused by the MD resonance mode, the enhancement factor caused by the MQ resonance mode is also calculated since the latter can also lead to MD emission enhancement [22]. For this purpose, we calculate the radiative decay rate enhancement of a MD emitter positioned within the gap of nanocuboid, nanodisk, and nanosphere dimer with larger size. In the calculation, l, w and h of the nanocuboid are respectively assumed to be 218 nm, 275 nm, and 200 nm. The radius and thickness of the nanodisk is assumed to be 148 nm and 200 nm. The radius of the nanosphere is set as 150 nm. The gap distance of all the nanoantennas is assumed to be 10 nm and the MD is assumed to be positioned at the center of the gap. The calculated results are shown in Fig. 9, where two main findings can be observed. First, the radiative decay rate enhancement spectrum caused by the nanocuboid dimer shows a sharp peak at the wavelength of 819 nm, and a broad peak at the wavelength of 1047 nm. These two peaks respectively result from the MQ resonance and MD resonance of the nanocuboid dimer [22]. Second, the MQ resonance of the nanocuboid dimer can lead to higher radiative decay rate enhancement than that of the nanodisk dimer and nanosphere dimer. More specifically, the radiative decay rate enhancement caused by the MQ resonance of the nanocuboid dimer reaches a high value of 368. This value is about 1.5 times that (249) caused by the nanosphere dimer and 1.1 times that (335) caused by the nanodisk dimer. The MD emission enhancement factor of 368 is also higher than the values caused by the nanocavity and nano-doughnut-shaped silicon disk, which are presented in previous studies [19,21]. These results demonstrate that the nanocuboid dimer is an efficient approach to improve MD emission.
Fig. 9 The radiative decay rate enhancement spectra of the silicon nanocuboid dimer, nanodisk dimer and nanosphere dimer with (a) and without (b) the SiO2 substrate.
For a realistic system, a substrate is usually needed to support the nanocuboid dimer. This can be fabricated by the following process, which is presented in Fig. 10. A silicon film with suitable thickness is first deposited on the silicon dioxide (SiO2) substrate by using plasma enhanced chemical vapor deposition, and a layer of electron beam resist is then coated on the silicon film. Electron beam lithography is employed to write pattern (the designed nanocuboid dimer) in the electron beam resist. With the resist pattern as mask, reactive-ion etching is used to etch the silicon layer and the nanocuboid dimer structure is obtained after removing the resist. To examine whether the physical laws achieved in this paper can be extended to the realistic system, we calculate the radiative decay rate enhancement of a MD emitter positioned at the center of the gap of the nanocuboid dimer supported by the SiO2 substrate. For comparison, we also calculate the radiative decay rate enhancement of a MD emitter positioned in the gap of a nanodisk dimer and a nanosphere dimer supported by SiO2 substrate. In the calculation, the parameters of the nanocuboid, the nanodisk and the nanosphere dimer are the same to that of the structures without the substrate, which are simulated and discussed in the paragraphs above. The calculated results are shown in Fig. 9b. It is observed that the nanocuboid dimer supported by the SiO2 substrate has similar magnetic emission properties to that without the substrate except that the radiative decay rate enhancement is relatively lower. Moreover, the radiative decay rate enhancement caused by both the MD resonance and the MQ resonance of the nanocuboid dimer is higher than that of the nanodisk dimer and the nanosphere dimer. These findings indicate that the physical laws achieved in this paper can be extended to the realistic system (i.e., nanocuboid dimer with the substrate). For practical fabrication of the nanocuboid dimer with the substrate, the structure parameters shall be further optimized to maximize radiative decay rate enhancement.
4. Conclusion
In summary, coupled silicon nanocuboid dimer is proposed to improve the MD emission by positioning the MD emitter within the silicon nanocuboid dimer. FDTD simulations demonstrate coupling occurs between the silicon nanocuboid dimer and both the MD resonance and MQ resonance of the dimer can significantly improve the radiative decay rate of the MD emitter positioned in the gap of the dimer. The MD resonance can lead to an enhancement factor of 262 for the radiative decay rate at the wavelength of 653 nm, which is nearly 3.8 times that of the MD emitter positioned within the nanosphere dimer. By tailoring the structure parameters of the nanocuboid, an enhancement factor of over 230 is achieved for the radiative decay rate at the long wavelengths of 600 nm, 700 nm and 800 nm. At the short wavelength range, however, the enhancement factor is relatively low due to the large energy absorbed by the silicon nanocuboids. Finally, it is revealed that the MQ resonance of the nanocuboid dimer with larger size can lead to a higher radiative decay rate enhancement of ~368 at the wavelength 819 nm, which is larger than that of the nanodisk dimer and nanosphere dimer.
Funding
Guangdong Natural Science Funds for Distinguished Young Scholar (2014A030306005); National Natural Science Foundation of China (11774099); Natural Science Foundation of Guangdong Province, China (2016A03031339); Foundation for High-level Talents in Higher Education of Guangdong Province, China (Yue Cai-Jiao [2013]246, Jiang Cai-Jiao [2014]10); Science and Technology Program of Guangzhou (201607010176).
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