Collective Modes in the Luminescent Response of Si Nanodisk Chains with Embedded GeSi Quantum Dots

: In this paper, we study the effects of GeSi quantum dot emission coupling with the collective modes in the linear chains of Si disk resonators positioned on an SiO 2 layer. The emission spectra as a function of the chain period and disk radius were investigated using micro-photoluminescence (micro-PL) spectroscopy. At optimal parameters of the disk chains, two narrow PL peaks, with quality factors of around 190 and 340, were observed in the range of the quantum dot emission. A numerical analysis of the mode composition allowed us to associate the observed peaks with two collective modes with different electric ﬁeld polarization relative to the chain line. The theoretical study demonstrates the change of the far-ﬁeld radiation pattern with increasing length of the disk chain. The intensive out-of-plane emission was explained by the appearance of the dipole mode contribution. The obtained results can be used for the development of Si-based near-infrared light sources.


Introduction
Many results have now been obtained indicating that the modification of the optical properties of materials by using various types of micro-and nano-resonators can be successfully applied to create efficient photon radiation sources, lasers with low threshold powers and narrow generation lines, optical filters with controlled bandwidth, and others [1][2][3][4][5][6][7][8][9][10][11].This approach is already being used for solving the urgent problem of creating compact, efficient light sources compatible with Si technology.The solution of this problem can lead to great progress in the development of silicon photonics.One of the promising variants is tunable structures with dielectric Mie resonators, allowing control of the light-matter interaction [6][7][8][9][10][11][12].Emission enhancement can be achieved due to an increase in the probability of optical transitions, when some kinds of emitters, which can be quantum dots, are placed at the maximum of the electric field inside the Mie resonator (Purcell effect [12,13]).In the frame of Si-based technology, GeSi quantum dots (QDs) can be considered a suitable variant of radiation sources with emission bandwidth coinciding with the highly demanded near-infrared range [14][15][16][17][18].The choice of GeSi QDs as an emitting medium is determined by several factors.The localization of the charge 2 of 14 carriers on GeSi QDs weakens the influence of surface defects on the emitting properties of the structures.The possibility of the growth of structures with GeSi quantum dots on silicon-on-insulator (SOI) substrates allows limiting the propagation of light in one of the directions and simplifies the preparation of dielectric resonators.The luminescence signal of GeSi QDs is characterized by a large width, which allows us to use it for studies of the interaction with different modes of resonators.In addition, for this type of heterostructure, there are well-developed methods of spatial ordering of QDs [19][20][21][22][23][24][25][26][27], which can be used for the development of devices with high efficiency of interaction of a small number of emitters (down to single emitter [1][2][3]) with the modes of micro-and nano-resonators.GeSi QDs formed on silicon emit at room temperature in the wavelength range of 1.3-1.55µm, which makes them promising for the creation of light sources for silicon photonics.The best results in QD emission enhancement were achieved by creating structures supporting high quality states [28][29][30][31][32][33][34][35][36].The high quality factor is an inherent property of a special type of states, bound states in continuum (BIC) [37].There are two types of BICs: the first type is the symmetry-protected BIC arising inthe center of the Brillouin zone (in the Γ-point) of some periodical photonic structure and completely decoupled from the radiation continuum due to the high symmetry [38][39][40]; the second type is off-Γ BIC, which can be obtained by tuning the parameters of the photonic system [41][42][43][44].This latter BIC is also called the accidental BIC or Friedrich-Wintgen state (FW-BIC) [45].The most intensive studies of BICs have been conducted for PhC structures [18,42,46]; however, there are many works dealing with dielectric Mie resonators.A large amount of theoretical and experimental material has been accumulated in this direction, both on single resonators [8,9,47] and on chains of resonators [5,7,10,11,44,[48][49][50][51].The possibility of creating experimental systems on the basis of disk resonator chains supporting high quality states, such as symmetry-protected and accidental BICs, was demonstrated [44,50].Recently, a new approach for achieving the giant Q-factors in finite-length periodic arrays of subwavelength optical Si resonators was proposed [7].It is based on the interference between the band-edge mode and a standing mode in the resonator chain, which leads to the suppression of radiative losses and the formation of high Q localized state.The collective band-edge modes were studied also in linear chains of Si nanopillars with embedded GeSi quantum dots [10], where the effective coupling of GeSi QDs to the Mie resonance modes of the nanopillars was demonstrated.The 10-fold enhancement of the luminescent signal due to the excitation of resonant antisymmetric magnetic and electric dipole modes was obtained for the nanopillar trimer.In the next paper by the same authors [11], a Q-factor of around 500 was experimentally measured for an array of 11 pillars, which demonstrates the potential of engineering optical nanopillar cavities for active photonic devices on an Si platform.The high quality factor was provided by the formation of the collective dipolar mode in a nanopillar array.
Until now, as far as we know, experimental structures with chains of Si disk resonators supporting BIC states (symmetry-protected or accidental) have not yet been realized and investigated.There are papers with impressive results on anapole-based energy transfer along Si disk chains [52,53], but unfortunately no BIC states were detected there [53].The main results on the realization of BIC states in disk resonator chains are obtained on systems with ceramic microwave resonators [44,50], which cannot be directly incorporated into the existing Si technology.In the present paper, we have investigated structures with linear chains of Si disk resonators on a SiO 2 layer, depending on the period and radius of disks in order to verify the possibility of the formation of high quality states in this system.At optimal parameters, we have demonstrated the enhancement of the emission of GeSi quantum dots.The micro-PL study demonstrates the appearance of two narrow PL peaks even for short chains with the number of disks N ≥ 2. By analysis of the mode composition, the correspondence between the observed PL peaks and two high quality collective modes were found.The change of radiation pattern from in-plane to out-of-plane was detected at the transition from the case of a single disk resonator to a linear chain of disks.The obtained results can be used for the development of near-infrared light sources compatible with Si-based technology.

Materials and Methods
To fabricate the linear chains of disk resonators with embedded GeSi QDs, a siliconon-insulator (SOI) substrate with a 180 nm-thick top Si layer and a 3 µm-thick SiO 2 layer was used.At the first step, a 110 nm-thick Si buffer layer was grown by molecular beam epitaxy at 500 • C ata rate of 0.6 Å/s.At the next step, ten layers of GeSi QDs separated by 15 nm-thick silicon spacer layers were grown at 600 • C. Each QD layer was formed by the deposition of 7 monolayers of Ge at a growing rate of 0.05 Å/s.At the final stage, the structure was covered with a 15 nm-thick Si layer.Si spacers and cap-layer were grown at a rate of 1 Å/s.The total thickness of the multilayer structure was 440 nm.Then, using e-beam lithography and plasma-etching, linear chains of disk resonators with different diameters and periods were formed on the surface of the SiO 2 layer.In the first step, by means of e-beam lithography, the chains of holes were created in positive resist PMMA-950K with a thickness of 200 nm.The radius of the holes in the resist layer varied from 330 ± 10 nm to 385 ± 10 nm with a step of 15 nm.The distance between the linear chains d was chosen to avoid the coupling of different chain resonators.It was varied depending on the period of the chain a.In our experimental implementation, for a = 1 µm, d = 3 µm; for a = 2 µm, d = 4 µm; for a = 3 µm, d = 6 µm; for a = 10 µm, d = 10 µm; for a = 15 µm, d = 15 µm were chosen.Then, a thin layer of chromium (20 nm) was deposited by magnetron sputtering on top of the resist layer with holes.After lift-off lithography, the periodic arrays of Cr disks were obtained, which served as a mask for the fabrication of the resonators.In the next step, anisotropic plasma chemical etching of the top Si layer with GeSi QDs was carried out in SF 6 /CHF 3 gas mixture through a Cr mask down to the SiO 2 layer.The etching rate was ~10 nm/s.According to scanning electron microscopy (SEM) measurements, the linear chains of disk resonators with radii 330 ± 10 nm, 345 ± 10 nm, 365 ± 10 nm, 385 ± 10 nm were obtained for each value of period.A schematic representation and scanning electron microscopy image of the obtained 1D arrays of disks with embedded GeSi QDs are shown in Figure 1.As well as the full-length chains, we investigated chains with different lengths.Such structures were fabricated by ultrasonic treatment, which resulted in removing some parts of the chains and the formation of chains with different lengths.

Materials and Methods
To fabricate the linear chains of disk resonators with embedded GeSi QDs, a con-on-insulator (SOI) substrate with a 180 nm-thick top Si layer and a 3 μm-thick layer was used.At the first step, a 110 nm-thick Si buffer layer was grown by molec beam epitaxy at 500 °C ata rate of 0.6 Å/s.At the next step, ten layers of GeSi QDs s rated by 15 nm-thick silicon spacer layers were grown at 600 °С.Each QD layer formed by the deposition of 7 monolayers of Ge at a growing rate of 0.05 Å/s.At the stage, the structure was covered with a 15 nm-thick Si layer.Si spacers and cap-layer grown at a rate of 1 Å/s.The total thickness of the multilayer structure was 440 nm.T using e-beam lithography and plasma-etching, linear chains of disk resonators with ferent diameters and periods were formed on the surface of the SiO2 layer.In the step, by means ofe-beam lithography, the chains of holes were created in positive r PMMA-950K with a thickness of 200 nm.The radius of the holes in the resist layer va from 330 ± 10 nm to 385 ± 10 nm with a step of 15 nm.The distance between the li chains d was chosen to avoid the coupling of different chain resonators.It was va depending on the period of the chain a.In our experimental implementation, for a = 1 d = 3 μm; for a = 2 μm, d = 4 μm; for a = 3 μm, d = 6 μm; for a = 10 μm, d = 10 μm; for a μm, d = 15 μm were chosen.Then, a thin layer of chromium (20 nm) was deposite magnetron sputtering on top of the resist layer with holes.After lift-off lithography periodic arrays of Cr disks were obtained, which served as a mask for the fabricatio the resonators.In the next step, anisotropic plasma chemical etching of the top Si with GeSi QDs was carried out in SF6/CHF3 gas mixture through a Cr mask down to SiO2 layer.The etching rate was ~10 nm/s.According to scanning electron micros (SEM) measurements, the linear chains of disk resonators with radii 330 ± 10 nm, 345 nm, 365 ± 10 nm, 385 ± 10 nm were obtained for each value of period.A schematic resentation and scanning electron microscopy image of the obtained 1D arrays of d with embedded GeSi QDs are shown in Figure 1.As well as the full-length chains investigated chains with different lengths.Such structures were fabricated by ultras treatment, which resulted in removing some parts of the chains and the formatio chains with different lengths.The luminescent properties of the structures were studied by the cro-photoluminescence (micro-PL) method with high spatial and spectral resolut The luminescent properties of the structures were studied by the micro-photoluminescence (micro-PL) method with high spatial and spectral resolutions.The micro-PL signal was excited by a continuous laser at the wavelength of 532 nm.The laser radiation was focused in a spot of ~2 µm with a Mitutoyo M Plan APO 50× objective (numerical aperture NA = 0.42).The excitation power was 5 mW.Measurements were carried out in the normal incidence geometry of the exciting and detected beams.The PL signal collection angle was 2ϑ = 48 • .The micro-PL signal was detected by a cooled Ge detector using a Bruker IFS 125HR high-resolution Fourier spectrometer.The spectra were recorded with resolution 4 cm −1 .The measurements were carried out at a temperature of 77 K. To analyze the obtained experimental results, we have simulated the emission spectra of the studied structures with resonator chains.The near-field components distributions and dispersion dependences of the eigenmodes are numerically calculated using COMSOL Multiphysics software.

Results and Discussion
The micro-PL study shows that the formation of linear chains of Si disks with embedded GeSi QDs led to PL spectra modification as compared with the original, unprocessed area (Figure 2).The most significant changes were observed for chains with period 1 µm (Figure 3).In contrast, at periods ≥ 2 µm, the spectra are very similar to the spectrum of a single disk and represent broad PL peaks with a maximum position depending on the disk radius (Figure 2b).At period 1 µm and disk radius ≈ 385 nm, two narrow peaks at E 1 ≈ 827 meV and E 2 ≈ 849 meV appear in the PL spectra (Figure 2a, red curve, the data are given for maximum chain length N = 51).These peaks are present in the spectra of even very short chains, for example, when the number of resonators in the chain N = 2.With increasing N, these peaks became narrower and more intensive.At N = 51, the quality factors of the peaks were 190 (peak at 827 meV) and 340 (peak at 849 meV).These values were obtained on the basis of PL spectrum fitting by a set of Lorentz functions (Figure S1 in Supplementary Materials).
carried out in the normal incidence geometry of the exciting and detected beams.The PL signal collection angle was 2ϑ = 48°.The micro-PL signal was detected by a cooled Ge detector using a Bruker IFS 125HR high-resolution Fourier spectrometer.The spectra were recorded with resolution 4 cm −1 .The measurements were carried out at a tempera ture of 77 K. To analyze the obtained experimental results, we have simulated the emis sion spectra of the studied structures with resonator chains.The near-field components distributions and dispersion dependences of the eigenmodes are numerically calculated using COMSOL Multiphysics software.

Results and Discussion
The micro-PL study shows that the formation of linear chains of Si disks with em bedded GeSi QDs led to PL spectra modification as compared with the original, unpro cessed area (Figure 2).The most significant changes were observed for chains with period 1 μm (Figure 3).In contrast, at periods ≥ 2 μm, the spectra are very similar to the spec trum of a single disk and represent broad PL peaks with a maximum position depending on the disk radius (Figure 2b).At period 1 μm and disk radius ≈ 385 nm, two narrow peaks at E1 ≈ 827 meV and E2 ≈ 849 meV appear in the PL spectra (Figure 2a, red curve, the data are given for maximum chain length N = 51).These peaks are present in the spectra of even very short chains, for example, when the number of resonators in the chain N = 2 With increasing N, these peaks became narrower and more intensive.At N = 51, the quality factors of the peaks were 190 (peak at 827 meV) and 340 (peak at 849 meV).These values were obtained on the basis of PL spectrum fitting by a set of Lorentz functions (Figure S1 in Supplementary Materials).(a) Micro-PL spectra measured at 77 K on the samples with linear chains of Si disks with embedded GeSi QDs (red curve, disk radius ≈ 385 nm and period ≈ 1 μm, the number of disks in the chain N = 51).For comparison, the PL spectra of the non-processed area (gray spectrum) and the single disk (dark cyan spectrum) are presented.The laser radiation was focused in a spot of ~2 μm In the case of the non-processed area, all QDs under the laser beam were excited, while in the case of a disk chain, only QDs in two disks in the central part of chain were excited.Therefore, the number of excited QDs in the disk chain is about 3 times less than in the non-processed area.Taking this fact into account, it was possible to estimate the true ratio (~6 times for the low-energy narrow peak) of PL signal intensities of the disk chain (red spectrum) and the non-processed area (gray spectrum).(b) The PL spectra in dependence on the disk radiusat fixed period of a chain, a = 2 μm (at this period the disks can be considered as isolated).
The gap between the narrow peaks depended on the number of disks in the chain, i increased with N (Figure 4).At N = 2, the gap between peaks is smallest and consists o ≈17 meV (E1 ≈ 831 meV and E2 ≈ 848 meV).By increasing the number of disks, the nar (a) Micro-PL spectra measured at 77 K on the samples with linear chains of Si disks with embedded GeSi QDs (red curve, disk radius ≈ 385 nm and period ≈ 1 µm, the number of disks in the chain N = 51).For comparison, the PL spectra of the non-processed area (gray spectrum) and the single disk (dark cyan spectrum) are presented.The laser radiation was focused in a spot of ~2 µm.In the case of the non-processed area, all QDs under the laser beam were excited, while in the case of a disk chain, only QDs in two disks in the central part of chain were excited.Therefore, the number of excited QDs in the disk chain is about 3 times less than in the non-processed area.Taking this fact into account, it was possible to estimate the true ratio (~6 times for the low-energy narrow peak) of PL signal intensities of the disk chain (red spectrum) and the non-processed area (gray spectrum).(b) The PL spectra in dependence on the disk radiusat fixed period of a chain, a = 2 µm (at this period the disks can be considered as isolated).
The gap between the narrow peaks depended on the number of disks in the chain, it increased with N (Figure 4).At N = 2, the gap between peaks is smallest and consists of ≈17 meV (E 1 ≈ 831 meV and E 2 ≈ 848 meV).By increasing the number of disks, the narrower peak was practically unchanged in its position, while the broader peak slightly shifted (by ~4 meV) to longer wavelengths.Also, the dependence of the PL peak position on the radius of the disks was observed.The PL spectra at the period of the chain a = 1 µm (Figure 3a) showed the blueshift of narrow peaks with decreasing radius.
rower peak was practically unchanged in its position, while the broader peak slightly shifted (by ~4 meV) to longer wavelengths.Also, the dependence of the PL peak position on the radius of the disks was observed.The PL spectra at the period of the chain a = 1 μm (Figure 3a) showed the blueshift of narrow peaks with decreasing radius.To understand the nature of the observed narrow peaks, we performed simulations for model structures that represented the linear chains of disk resonators with parameters corresponding to experimental ones.First, we studied the eigenmodes of the single disk resonator with radius 385 nm and height 440 nm.In the range 790-870 meV, we have found many eigenmodes (~60) and made a selection by the value of the quality factor.We have chosen a two-fold degenerate mode with energy of 831 meV and a quality factor of 97, an order of magnitude higher than the others.The near-field configuration shown in Figure 5a indicates that this is the Mie mode with orbital angular momentum m = 3 rower peak was practically unchanged in its position, while the broader peak slightly shifted (by ~4 meV) to longer wavelengths.Also, the dependence of the PL peak position on the radius of the disks was observed.The PL spectra at the period of the chain a = 1 μm (Figure 3a) showed the blueshift of narrow peaks with decreasing radius.To understand the nature of the observed narrow peaks, we performed simulations for model structures that represented the linear chains of disk resonators with parameters corresponding to experimental ones.First, we studied the eigenmodes of the single disk resonator with radius 385 nm and height 440 nm.In the range 790-870 meV, we have found many eigenmodes (~60) and made a selection by the value of the quality factor.We have chosen a two-fold degenerate mode with energy of 831 meV and a quality factor of 97, an order of magnitude higher than the others.The near-field configuration shown in Figure 5a indicates that this is the Mie mode with orbital angular momentum m = 3 To understand the nature of the observed narrow peaks, we performed simulations for model structures that represented the linear chains of disk resonators with parameters corresponding to experimental ones.First, we studied the eigenmodes of the single disk resonator with radius 385 nm and height 440 nm.In the range 790-870 meV, we have found many eigenmodes (~60) and made a selection by the value of the quality factor.We have chosen a two-fold degenerate mode with energy of 831 meV and a quality factor of 97, an order of magnitude higher than the others.The near-field configuration shown in Figure 5a indicates that this is the Mie mode with orbital angular momentum m = 3 (magnetic octupole, MO mode).The far-field pattern demonstrated the predominantly in-plane emission (Figure 5b).
(magnetic octupole, MO mode).The far-field pattern demonstrated the predominantly in-plane emission (Figure 5b).At the second step, we studied the modes of linear chains with different numbers of resonators.Special attention is paid to the evolution of the MO mode.Already at N = 2 the mode splitting occurs and two collective modes were formed (Figure 6).One of the modes, with larger electric field amplitude in the gap between disks, shifted downward in energy scale, while the other shifted upward.In the following, we will refer to the former as the "longitudinal" mode (with a spread electric field along the chain) and the latter as the "transverse" mode.By increasing N, the splitting between modes increased and after N = 10 practically did not change.According to the calculation results, with enough long chains (N ≥ 10) the "longitudinal" mode had energy ≈ 822 meV, while the "transverse" mode achieved energy ≈ 841 meV, which is in good agreement with the experiment.The "transverse" mode is characterized by a higher field concentration inside disk than the "longitudinal" one.This corresponds well to the experimentally revealed difference in the quality factor of low-and high-energy peaks (Figure 2a).We would like to point out that the near field configurations of the considered modes (see the wave field pattern in Figure S3) are very similar to configurations of the true BICs found for infinitely long rods and cylinders [56][57][58], namely, the field pattern of our "transverse" mode is similar to one of the symmetry-protected BIC, while the field pattern of the "longitudinal" mode is close to one of the accidental BIC studied in these works.At the second step, we studied the modes of linear chains with different numbers of resonators.Special attention is paid to the evolution of the MO mode.Already at N = 2 the mode splitting occurs and two collective modes were formed (Figure 6).One of the modes, with larger electric field amplitude in the gap between disks, shifted downward in energy scale, while the other shifted upward.In the following, we will refer to the former as the "longitudinal" mode (with a spread electric field along the chain) and the latter as the "transverse" mode.By increasing N, the splitting between modes increased and after N = 10 practically did not change.According to the calculation results, with enough long chains (N ≥ 10) the "longitudinal" mode had energy ≈ 822 meV, while the "transverse" mode achieved energy ≈ 841 meV, which is in good agreement with the experiment.The "transverse" mode is characterized by a higher field concentration inside disk than the "longitudinal" one.This corresponds well to the experimentally revealed difference in the quality factor of low-and high-energy peaks (Figure 2a).We would like to point out that the near field configurations of the considered modes (see the wave field pattern in Figure S3) are very similar to configurations of the true BICs found for infinitely long rods and cylinders [56][57][58], namely, the field pattern of our "transverse" mode is similar to one of the symmetry-protected BIC, while the field pattern of the "longitudinal" mode is close to one of the accidental BIC studied in these works.
Photonics 2023, 10, x FOR PEER REVIEW 6 of 14 (magnetic octupole, MO mode).The far-field pattern demonstrated the predominantly in-plane emission (Figure 5b).At the second step, we studied the modes of linear chains with different numbers of resonators.Special attention is paid to the evolution of the MO mode.Already at N = 2 the mode splitting occurs and two collective modes were formed (Figure 6).One of the modes, with larger electric field amplitude in the gap between disks, shifted downward in energy scale, while the other shifted upward.In the following, we will refer to the former as the "longitudinal" mode (with a spread electric field along the chain) and the latter as the "transverse" mode.By increasing N, the splitting between modes increased and after N = 10 practically did not change.According to the calculation results, with enough long chains (N ≥ 10) the "longitudinal" mode had energy ≈ 822 meV, while the "transverse" mode achieved energy ≈ 841 meV, which is in good agreement with the experiment.The "transverse" mode is characterized by a higher field concentration inside disk than the "longitudinal" one.This corresponds well to the experimentally revealed difference in the quality factor of low-and high-energy peaks (Figure 2a).We would like to point out that the near field configurations of the considered modes (see the wave field pattern in Figure S3) are very similar to configurations of the true BICs found for infinitely long rods and cylinders [56][57][58], namely, the field pattern of our "transverse" mode is similar to one of the symmetry-protected BIC, while the field pattern of the "longitudinal" mode is close to one of the accidental BIC studied in these works.Figure 6.The left panel shows the calculated spectral positions of "longitudinal" (red symbols) and "transverse"(blue symbols) modes depending on the number of Si disks in the chain positioned on SiO 2 layer.The right panels demonstrate the near-field distributions for "transverse" (top panels) and "longitudinal" (bottom panels) modes in the chains of 2 and 5 disks.We show |E| as in Figure 5.The radius of the disks is 385 nm, and the period is 1 µm.The refractive index and extinction coefficient of silicon and SiO 2 were taken according to [54,55].
Figure 7 demonstrates the results of calculations of quality factors for "transverse" and "longitudinal" modes depending on the chain period and on the number of disks in the chain.Figure 6.The left panel shows the calculated spectral positions of "longitudinal" (red symbols) and "transverse"(blue symbols) modes depending on the number of Si disks in the chain positioned on SiO2 layer.The right panels demonstrate the near-field distributions for "transverse" (top panels) and "longitudinal" (bottom panels) modes in the chains of 2 and 5 disks.We show |E| as in Figure 5.The radius of the disks is 385 nm, and the period is 1 μm.The refractive index and extinction coefficient of silicon and SiO2 were taken according to [54,55].
Figure 7 demonstrates the results of calculations of quality factors for "transverse" and "longitudinal" modes depending on the chain period and on the number of disks in the chain.The dependence of the quality factor on the chain period a (calculated at N = 11) showed that for the "transverse" mode the optimal period is slightly smaller than experimental value (a = 1 μm) and consisted of 950 nm, while for the "longitudinal" mode it is equal nm.So, the chosen experimental period is very close to optimal one.The Q dependence on the period calculated for infinite chain is presented in Supplementary Materials (Figure S4).This dependence demonstrates that the quality factor can reach the value Q = 7500 at a = 1400 μm, if we turn off the material losses in the SiO2 substrate.
The Q dependence on N (calculated at a = 1 μm) shows that the quality factor of the "transverse" mode increased with chain length, while the quality factor of the "longitudinal" mode tended to stabilize at 200 after an initial growth.For the "transverse" mode, the quality factor reached the value of ~1100 at N = 11, demonstrating an almost linear dependence.Such dependence is preserved up to N = 20, and then the increase in the quality factor slows down, reaching saturation (Figure S5) near the value ~3000.The existing studies of the dependence Q(N) carried out on chains of dielectric resonators [44,50] show the dependence Q~N 2 for BIC at Γ-point and Q~N for accidental BIC.However, it is also shown there that the presence of radiation losses can lead to a weaker dependence on N [50].
The theoretical consideration of infinite chains can give more detailed physical insights into the nature of the observed modes, so we have calculated the dependence of Q on the wave vector kx (along the chain line) and dispersion dependence of radiation losses γ = Im[ω(kx)] for the "transverse" and the "longitudinal" modes in the infinite chain of Si disks (Figure 8a,b).These dependences demonstrated the growth of Q near the Γ-point for the "transverse" mode and the presence of Q maximum out of the Γ-point (at kx =0.36π/a) for the "longitudinal" mode.This behavior indicated that these modes can be considered as candidates for BICs, but the non-zero radiation losses at maximum Q points do not allow them to be so classified.The dependence of the quality factor on the chain period a (calculated at N = 11) showed that for the "transverse" mode the optimal period is slightly smaller than experimental value (a = 1 µm) and consisted of 950 nm, while for the "longitudinal" mode it is equal to 1020 nm.So, the chosen experimental period is very close to optimal one.The Q dependence on the period calculated for infinite chain is presented in Supplementary Materials (Figure S4).This dependence demonstrates that the quality factor can reach the value Q = 7500 at a = 1400 µm, if we turn off the material losses in the SiO 2 substrate.
The Q dependence on N (calculated at a = 1 µm) shows that the quality factor of the "transverse" mode increased with chain length, while the quality factor of the "longitudinal" mode tended to stabilize at 200 after an initial growth.For the "transverse" mode, the quality factor reached the value of ~1100 at N = 11, demonstrating an almost linear dependence.Such dependence is preserved up to N = 20, and then the increase in the quality factor slows down, reaching saturation (Figure S5) near the value ~3000.The existing studies of the dependence Q(N) carried out on chains of dielectric resonators [44,50] show the dependence Q~N 2 for BIC at Γ-point and Q~N for accidental BIC.However, it is also shown there that the presence of radiation losses can lead to a weaker dependence on N [50].
The theoretical consideration of infinite chains can give more detailed physical insights into the nature of the observed modes, so we have calculated the dependence of Q on the wave vector k x (along the chain line) and dispersion dependence of radiation losses γ = Im[ω(k x )] for the "transverse" and the "longitudinal" modes in the infinite chain of Si disks (Figure 8a,b).These dependences demonstrated the growth of Q near the Γ-point for the "transverse" mode and the presence of Q maximum out of the Γ-point (at k x =0.36π/a) for the "longitudinal" mode.This behavior indicated that these modes can be considered as candidates for BICs, but the non-zero radiation losses at maximum Q points do not allow them to be so classified.
We think that the main reason of finite Q values is that the dispersion branches are located in regions with one and two open diffraction channels (Figure 9).The leakage into the 2nd diffraction continuum can leads to a significant decrease of Q factors [50,59].
In the experiments we have obtained a lower value Q = 340 than predicted theoretically, which could be for several reasons.The first possible reason was of some imperfections in the experimental structures, which could have led to additional radiation losses.However, we think that this is not the main reason.The second, main reason in our opinion, is the absorption on free charge carriers excited by laser pumping during the micro-PL measurements.At given sizes of disks the excitation, already at low pump powers, could have led to a significant concentration of charge carriers in the disk, since the "diffusive escaping" is problematic.In general, this could even have caused the overheating of the sample.In our experiments, we controlled the power of pumping to avoid the displacement of PL peaks to the long-wavelength region due to overheating of the sample.The and extinction coefficient of silicon were taken according to [54].The results of calculations performed for the disk chains positioned on SiO2 substrate are presented in the Supplementary Materials (Figure S6).
We think that the main reason of finite Q values is that the dispersion branches are located in regions with one and two open diffraction channels (Figure 9).The leakage into the 2nd diffraction continuum can leads to a significant decrease of Q factors [50,59].Blue and red curves show the dispersion dependences for the "transverse" and the "longitudinal" modes, respectively, in infinite chains of Si disks.The radius of disks is 385 nm, and the period is 1 μm.The calculations were done without substrate, the refractive index of surroundings was taken as n = 1.The refractive index and extinction coefficient of silicon are taken according to [54].
In the experiments we have obtained a lower value Q = 340 than predicted theoretically, which could be for several reasons.The first possible reason was of some imperfections in the experimental structures, which could have led to additional radiation losses.However, we think that this is not the main reason.The second, main reason in our opinion, is the absorption on free charge carriers excited by laser pumping during the micro-PL measurements.At given sizes of disks the excitation, already at low pump Calculations were done without substrate, and the refractive index of the surroundings was taken as n = 1.The refractive index and extinction coefficient of silicon were taken according to [54].The results of calculations performed for the disk chains positioned on SiO 2 substrate are presented in the Supplementary Materials (Figure S6).The refractive index and extinction coefficient of silicon were taken according to [54].The results of calculations performed for the disk chains positioned on SiO2 substrate are presented in the Supplementary Materials (Figure S6).
We think that the main reason of finite Q values is that the dispersion branches are located in regions with one and two open diffraction channels (Figure 9).The leakage into the 2nd diffraction continuum can leads to a significant decrease of Q factors [50,59].Blue and red curves show the dispersion dependences for the "transverse" and the "longitudinal" modes, respectively, in infinite chains of Si disks.The radius of disks is 385 nm, and the period is 1 μm.The calculations were done without substrate, the refractive index of surroundings was taken as n = 1.The refractive index and extinction coefficient of silicon are taken according to [54].
In the experiments we have obtained a lower value Q = 340 than predicted theoretically, which could be for several reasons.The first possible reason was of some imperfections in the experimental structures, which could have led to additional radiation losses.However, we think that this is not the main reason.The second, main reason in our opinion, is the absorption on free charge carriers excited by laser pumping during the micro-PL measurements.At given sizes of disks the excitation, already at low pump Blue and red curves show the dispersion dependences for the "transverse" and the "longitudinal" modes, respectively, in infinite chains of Si disks.The radius of disks is 385 nm, and the period is 1 µm.The calculations were done without substrate, the refractive index of surroundings was taken as n = 1.The refractive index and extinction coefficient of silicon are taken according to [54].
The most interesting result of our theoretical study is the change of the far-field pattern with increasing length of the disk chain.The typical far-field distribution for magnetic octupole mode is shown in Figure 5b.Such a far-field pattern is very suitable for use in planar device structures, but for up-emission light sources it is major disadvantage.However, in our structures, as the number of disks in the chain increased, the far-field distribution changed (Figure 10).For both modes, it is clearly seen that the probability of up-emission increased even for a two-disk chain (Figure 10b,d).For the "transverse" mode, at N = 3, the angle distribution of emission is broader than that for the "longitudinal" mode (Figure 10f).avoid the displacement of PL peaks to the long-wavelength region due to overheating of the sample.
The most interesting result of our theoretical study is the change of the far-field pattern with increasing length of the disk chain.The typical far-field distribution for magnetic octupole mode is shown in Figure 5b.Such a far-field pattern is very suitable for use in planar device structures, but for up-emission light sources it is major disadvantage.However, in our structures, as the number of disks in the chain increased, the far-field distribution changed (Figure 10).For both modes, it is clearly seen that the probability of up-emission increased even for a two-disk chain (Figure 10 b,d).For the "transverse" mode, at N = 3, the angle distribution of emission is broader than that for the "longitudinal" mode (Figure 10f).,d,f,h) show the corresponding cross-sections of the far-field radiation pattern, calculated without SiO 2 layer, in the homogeneous medium approximation.The radius of the disks is 385 nm, and the period of the chains is 1 µm.The refractive index and extinction coefficient of silicon and SiO 2 were taken according to [54,55].
To explain the obtained result, one can assume that at the formation of the collective state another mode with high out-of-plane emission was mixed with the main octupole mode.This assumption can be verified by analysis of the mode composition for the case of the infinite disk chain.To make the calculations as close as possible to our experiment, we placed inside each disk a point dipole, and considered all possible orientations of this dipole (the X-dipole is oriented along the chain, the Y-dipole is transverse to the chain line orientation, and the Z-dipole is oriented along the vertical axis of the disk).
It turned out that Z-dipole predominantly excited the "longitudinal" collective mode in the chain.The up-emission intensity spectrum of Z-dipole is shown in Figure S7.There is only one intensive peak at λ ≈ 1.506 µm (E ≈ 823 meV), which corresponds to our "longitudinal" mode.In Figure 11, the near-field distribution for this mode is shown.We have plotted separately the E x , E y , E z components distribution in XY, XZ, YZ-sections.It is clearly seen that the collective "longitudinal" mode is a superposition of the TM mode (magnetic octupole) and the two TE modes (dipole and quadrupole modes).
mode.This assumption can be verified by analysis of the mode composition for the case of the infinite disk chain.To make the calculations as close as possible to our experiment, we placed inside each disk a point dipole, and considered all possible orientations of this dipole (the X-dipole is oriented along the chain, the Y-dipole is transverse to the chain line orientation, and the Z-dipole is oriented along the vertical axis of the disk).
It turned out that Z-dipole predominantly excited the "longitudinal" collective mode in the chain.The up-emission intensity spectrum of Z-dipole is shown in Figure S7.There is only one intensive peak at λ ≈ 1.506 μm (E ≈ 823 meV), which corresponds to our "longitudinal" mode.In Figure 11, the near-field distribution for this mode is shown.We have plotted separately the Ex, Ey, Ez components distribution in XY, XZ, YZ-sections.It is clearly seen that the collective "longitudinal" mode is a superposition of the TM mode (magnetic octupole) and the two TE modes (dipole and quadrupole modes).The X-dipole predominantly excited the "transverse" collective mode in the chain.The up-emission spectra of the X-dipole is shown in Figure S67.These spectra were richer in peaks than the Z-dipole spectra, but there was only one intensive peak in the range of interest to us, at λ ≈ 1.488 µm (E ≈ 833 meV), which corresponded to our "transverse" mode.In Figure 12, the near-field distribution for this mode is shown.It is clearly seen that the collective "transverse" mode is also a superposition of TM mode (magnetic octupole) and two TE modes (dipole and quadrupole modes).The principal difference of the "transverse" collective mode case is another orientation of admixed dipole mode relative to the chain line.For the "transverse" collective mode, the admixed dipole mode was oriented along Y-axis, while for "longitudinal" collective mode the dipole mode was oriented along the chain line.that the collective "transverse" mode is also a superposition of TM mode (magnetic octupole) and two TE modes (dipole and quadrupole modes).The principal difference of the "transverse" collective mode case is another orientation of admixed dipole mode relative to the chain line.For the "transverse" collective mode, the admixed dipole mode was oriented along Y-axis, while for "longitudinal" collective mode the dipole mode was oriented along the chain line.It should be noted that the analysis of the mode composition in the case of a single disk resonator demonstrates the presence of an admixture of the quadrupole mode (Figure S8).But the dipole mode is added only in the case of the linear chain of disks.Thus, one can conclude that the intensive up-emission is provided by an appearance of the dipole mode admixture.It should be noted that the analysis of the mode composition in the case of a single disk resonator demonstrates the presence of an admixture of the quadrupole mode (Figure S8).But the dipole mode is added only in the case of the linear chain of disks.Thus, one can conclude that the intensive up-emission is provided by an appearance of the dipole mode admixture.

Conclusions
As a result of this study, it was obtained that for structures with linear chains of Si disks on a SiO 2 substrate it is possible to select such parameters (disk radius, period, and length of the chain), which ensure the existence of high-quality collective modes.These modes are distant relatives of the BIC states found for infinitely long rods in [56][57][58].Embedding quantum dots in such disk chains led to multiple amplification of the QD emission due to interaction with these collective modes.The change of the far-field radiation pattern from in-plane to out-of-plane emission is found with increasing length of the disk chain.The result is explained by the appearance of the dipole mode admixture.infinite chain; Figure S5: The dependences of quality factors on the number of disks calculated for "transverse" and "longitudinal" modes in the chains of Si disks positioned on SiO 2 .Figure S6: The dependences of losses γ = Im[ω(k x )] on the wave vector k x (along the chain line) calculated for "transverse" and "longitudinal" modes in the infinite chains of Si disks.Figure S7: The calculated up−emission intensity spectra for infinite linear chains of silicon disk resonators with embedded point dipoles.Figure S8: The near field distributions for eigenmode at λ = 1.493 µm (≈830 meV) for an isolated Si disk.

Figure 1 .
Figure 1.Left-a schematic view of the structure under study.Right-a scanning electron croscopy image of the sample with linear chains of disk resonators.The disk diameter is 770 the period is 1 μm, and the distance d between chains is 3 μm.

Figure 1 .
Figure 1.Left-a schematic view of the structure under study.Right-a scanning electron microscopy image of the sample with linear chains of disk resonators.The disk diameter is 770 nm, the period is 1 µm, and the distance d between chains is 3 µm.

Figure 2 .
Figure 2.(a) Micro-PL spectra measured at 77 K on the samples with linear chains of Si disks with embedded GeSi QDs (red curve, disk radius ≈ 385 nm and period ≈ 1 μm, the number of disks in the chain N = 51).For comparison, the PL spectra of the non-processed area (gray spectrum) and the single disk (dark cyan spectrum) are presented.The laser radiation was focused in a spot of ~2 μm In the case of the non-processed area, all QDs under the laser beam were excited, while in the case of a disk chain, only QDs in two disks in the central part of chain were excited.Therefore, the number of excited QDs in the disk chain is about 3 times less than in the non-processed area.Taking this fact into account, it was possible to estimate the true ratio (~6 times for the low-energy narrow peak) of PL signal intensities of the disk chain (red spectrum) and the non-processed area (gray spectrum).(b) The PL spectra in dependence on the disk radiusat fixed period of a chain, a = 2 μm (at this period the disks can be considered as isolated).

Figure 2 .
Figure 2.(a) Micro-PL spectra measured at 77 K on the samples with linear chains of Si disks with embedded GeSi QDs (red curve, disk radius ≈ 385 nm and period ≈ 1 µm, the number of disks in the chain N = 51).For comparison, the PL spectra of the non-processed area (gray spectrum) and the single disk (dark cyan spectrum) are presented.The laser radiation was focused in a spot of ~2 µm.In the case of the non-processed area, all QDs under the laser beam were excited, while in the case of a disk chain, only QDs in two disks in the central part of chain were excited.Therefore, the number of excited QDs in the disk chain is about 3 times less than in the non-processed area.Taking this fact into account, it was possible to estimate the true ratio (~6 times for the low-energy narrow peak) of PL signal intensities of the disk chain (red spectrum) and the non-processed area (gray spectrum).(b) The PL spectra in dependence on the disk radiusat fixed period of a chain, a = 2 µm (at this period the disks can be considered as isolated).

Figure 3 .
Figure 3. (a) Micro-PL spectra in dependence on the disk radius R at fixed period of a chain, a = 1 μm.The arrows indicate the positions of narrow peaks.(b) Micro-PL spectra depending on the period of the chain at fixed radius R = 385 nm.The number of disks in the chain N = 51.

Figure 4 .
Figure 4. (a) The dependence of the micro-PL spectra on the number of the disks in the chain N at fixed period and radius, a = 1 μm and R = 385 nm.(b)The dependence of the gap between the narrow PL peaks on the number of disks in the chain at fixed radius R = 385 nm.The red symbols correspond to the positions of the low-energy peak, and the blue symbols correspond to the positions of the high-energy peak.The dashed lines are guides to the eye.We have processed the spectra and plotted the dependence of peak intensity on the number of resonators in the chain (the results are presented in Supplementary Materials, FigureS2).

Figure 3 .
Figure 3. (a) Micro-PL spectra in dependence on the disk radius R at fixed period of a chain, a = 1 µm.The arrows indicate the positions of narrow peaks.(b) Micro-PL spectra depending on the period of the chain at fixed radius R = 385 nm.The number of disks in the chain N = 51.

Figure 3 .
Figure 3. (a) Micro-PL spectra in dependence on the disk radius R at fixed period of a chain, a = 1 μm.The arrows indicate the positions of narrow peaks.(b) Micro-PL spectra depending on the period of the chain at fixed radius R = 385 nm.The number of disks in the chain N = 51.

Figure 4 .
Figure 4. (a) The dependence of the micro-PL spectra on the number of the disks in the chain N at fixed period and radius, a = 1 μm and R = 385 nm.(b)The dependence of the gap between the narrow PL peaks on the number of disks in the chain at fixed radius R = 385 nm.The red symbols correspond to the positions of the low-energy peak, and the blue symbols correspond to the positions of the high-energy peak.The dashed lines are guides to the eye.We have processed the spectra and plotted the dependence of peak intensity on the number of resonators in the chain (the results are presented in Supplementary Materials, FigureS2).

Figure 4 .
Figure 4. (a) The dependence of the micro-PL spectra on the number of the disks in the chain N at fixed period and radius, a = 1 µm and R = 385 nm.(b)The dependence of the gap between the narrow PL peaks on the number of disks in the chain at fixed radius R = 385 nm.The red symbols correspond to the positions of the low-energy peak, and the blue symbols correspond to the positions of the high-energy peak.The dashed lines are guides to the eye.We have processed the spectra and plotted the dependence of peak intensity on the number of resonators in the chain (the results are presented in Supplementary Materials, FigureS2).

Figure 5 .
Figure 5. (a) The near field distribution, calculated for the Mie mode with an orbital angular momentum m = 3 in a single Si resonator positioned on the SiO2 layer.(b)The cross-section of the far field radiation pattern for this mode, calculated without SiO2 layer, in the homogeneous medium approximation.The radius of disks was 385 nm.The refractive index and the extinction coefficient of silicon and SiO2 were taken according to[54,55].

Figure 5 .
Figure 5. (a) The near field distribution, calculated for the Mie mode with an orbital angular momentum m = 3 in a single Si resonator positioned on the SiO 2 layer.(b)The cross-section of the far field radiation pattern for this mode, calculated without SiO 2 layer, in the homogeneous medium approximation.The radius of disks was 385 nm.The refractive index and the extinction coefficient of silicon and SiO 2 were taken according to[54,55].

Figure 5 .
Figure 5. (a) The near field distribution, calculated for the Mie mode with an orbital angular momentum m = 3 in a single Si resonator positioned on the SiO2 layer.(b)The cross-section of the far field radiation pattern for this mode, calculated without SiO2 layer, in the homogeneous medium approximation.The radius of disks was 385 nm.The refractive index and the extinction coefficient of silicon and SiO2 were taken according to[54,55].

Figure 7 .
Figure 7.The dependences of quality factors of "transverse" and "longitudinal" modes (a) on the chain period and (b) on the number of Si disks in the chain with period 1 μm.The dependence on the chain period is calculated for N = 11.The radius of disks is 385 nm.The refractive index and extinction coefficient of silicon and SiO2 were taken according to[54,55].

Figure 7 .
Figure 7.The dependences of quality factors of "transverse" and "longitudinal" modes (a) on the chain period and (b) on the number of Si disks in the chain with period 1 µm.The dependence on the chain period is calculated for N = 11.The radius of disks is 385 nm.The refractive index and extinction coefficient of silicon and SiO 2 were taken according to[54,55].

Figure 9 .
Figure 9. Diagram showing the number of open diffraction channels, Λ, depending on the frequency ω and the wave vector kx.Blue and red curves show the dispersion dependences for the "transverse" and the "longitudinal" modes, respectively, in infinite chains of Si disks.The radius of disks is 385 nm, and the period is 1 μm.The calculations were done without substrate, the refractive index of surroundings was taken as n = 1.The refractive index and extinction coefficient of silicon are taken according to[54].

Figure 8 .
Figure 8.The dependences of quality factors (a) and radiation losses γ = Im[ω(k x )] (b) on the wave vector k x (along the chain line) calculated for the "transverse" and the "longitudinal" modes in the infinite chains of Si disks.The radius of the disks was 385 nm, and the period was 1 µm.Calculations were done without substrate, and the refractive index of the surroundings was taken as n = 1.The refractive index and extinction coefficient of silicon were taken according to[54].The results of calculations performed for the disk chains positioned on SiO 2 substrate are presented in the Supplementary Materials (FigureS6).

Figure 9 .
Figure 9. Diagram showing the number of open diffraction channels, Λ, depending on the frequency ω and the wave vector kx.Blue and red curves show the dispersion dependences for the "transverse" and the "longitudinal" modes, respectively, in infinite chains of Si disks.The radius of disks is 385 nm, and the period is 1 μm.The calculations were done without substrate, the refractive index of surroundings was taken as n = 1.The refractive index and extinction coefficient of silicon are taken according to[54].

Figure 9 .
Figure 9. Diagram showing the number of open diffraction channels, Λ, depending on the frequency ω and the wave vector k x .Blue and red curves show the dispersion dependences for the "transverse" and the "longitudinal" modes, respectively, in infinite chains of Si disks.The radius of disks is 385 nm, and the period is 1 µm.The calculations were done without substrate, the refractive index of surroundings was taken as n = 1.The refractive index and extinction coefficient of silicon are taken according to[54].

Figure 10 .
Figure 10.The left panels (a,c,e,g) show the near-field distributions calculated for the "longitudinal" (a,e)" and transverse" (c,g) modes in the model structures corresponding to the experimental chains of resonators with N = 2, 3 positioned on SiO 2 layer.The right panels (b,d,f,h) show the corresponding cross-sections of the far-field radiation pattern, calculated without SiO 2 layer, in the homogeneous medium approximation.The radius of the disks is 385 nm, and the period of the chains is 1 µm.The refractive index and extinction coefficient of silicon and SiO 2 were taken according to[54,55].

Figure 11 .Figure 11 .
Figure 11.The near-field distributions calculated at λ = 1.506 (E ≈ 823 meV) μm for the infinite chain of resonators with embedded Z-dipoles.The radius of disks was 385 nm, the period of the chain is 1 μm, and the height H = 440 nm.The Z-dipole was placed at a depth of 150 nm from the top plane of the disk.In XY-plane, this dipole was positioned on the Y-axis at a distance 2R/3 from the symmetry axis of the disk.The dashed lines show the positions of the XY-and XZ-sections.The panels (a,d,g) are related to the Ex component.The panels (b,e,h)-Ey component.The panels (c,f,i)-the Ez Figure 11.The near-field distributions calculated at λ = 1.506 (E ≈ 823 meV) µm for the infinite chain of resonators with embedded Z-dipoles.The radius of disks was 385 nm, the period of the chain is 1 µm, and the height H = 440 nm.The Z-dipole was placed at a depth of 150 nm from the top plane of the disk.In XY-plane, this dipole was positioned on the Y-axis at a distance 2R/3 from the symmetry axis of the disk.The dashed lines show the positions of the XYand XZ-sections.The panels (a,d,g) are related to the E x component.The panels (b,e,h)-E y component.The panels (c,f,i)-the E z component.The refractive index and extinction coefficient of silicon and SiO 2 were taken according to [54,55].

Figure 12 .
Figure 12.The near-field distributions calculated at λ = 1.488 μm (E ≈ 833 meV) for the infinite chain of resonators with embedded X-dipoles.The radius of the disks is 385 nm, the period of the chain is 1 μm, and the height H = 440 nm.The X-dipole is placed at a depth of 150 nm from the top plane of the disk.In XY-plane, this dipole is positioned on the Y-axis at a distance of 2R/3 from the symmetry axis of the disk.The dashed lines show the positions of the XY-and XZ-sections.The panels (a,d,g) are related to the Ex component.The panels (b,e,h)-Ey component.The panels (c,f,i)-Ez component.

Figure 12 .
Figure 12.The near-field distributions calculated at λ = 1.488 µm (E ≈ 833 meV) for the infinite chain of resonators with embedded X-dipoles.The radius of the disks is 385 nm, the period of the chain is 1 µm, and the height H = 440 nm.The X-dipole is placed at a depth of 150 nm from the top plane of the disk.In XY-plane, this dipole is positioned on the Y-axis at a distance of 2R/3 from the symmetry axis of the disk.The dashed lines show the positions of the XYand XZ-sections.The panels (a,d,g) are related to the E x component.The panels (b,e,h)-E y component.The panels (c,f,i)-E z component.