Nonlinear Transient Dynamics of Photoexcited Resonant Silicon ...
Supporting Information: Nonlinear Transient Dynamics of Photoexcited Resonant Silicon Nanostructures Denis G. Baranov,∗,† Sergey V. Makarov,‡ Valentin A. Milichko,‡ Sergey I. Kudryashov,‡ Alexander E. Krasnok,‡ and Pavel A. Belov‡ †Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia ‡ITMO University, St. Petersburg 197101, Russia E-mail: [email protected]
S1: Calculation of volume-averaged fields Calculation of the one- and two-photon absorption rates W1,2 requires knowing of the electric ˜ 2 ˜ 4 field value averaged over the nanoparticle volume, E and E . We should relate these values to the instantaneous amplitudes of electric and magnetic dipole moments, whose dynamics is governed by Eq. (1) in the main text. To do that, we assume that at each moment electric field inside the particle can be represented as a sum of electric dipole and magnetic dipole modes: √ E(r) = EMD (r) + EED (r) = AMD j1 ( εkr) [eθ cos ϕ − eϕ sin ϕ cos θ] + " # √ √ √ 0 j1 ( εkr) ( εkr · j1 ( εkr)) √ AED 2 √ er cos ϕ sin θ + (eθ cos θ cos ϕ − eϕ sin ϕ) εkr εkr
(S1)
The values AMD and AED are found by integrating the near-field current of the field 1
distribution E and equating the resulting dipole moments to known values of p ˜ and m: ˜ 1 m ˜ = 2c
Z r × (−iω) V
Z p ˜= V
ε−1 EMD d3 r 4π
ε−1 EED d3 r 4π
(S2)
(S3)
Now, volume-averaged electric field can be directly calculated by integrating expression (S1) over the nanoparticle volume. This procedure is repeated at each step of numerical simulations.
S2: Competition of one- and two-photon absorption
Figure S1: The rates of one- and two-hoton absorption as a function of light intensity. Figure S1 shows the absorbed power density due to one- and two-photon absorption as a function of light intensity inside silicon at wavelength of 650 nm. It indicates that for intensities above 5 GW/cm2 TPA quickly becomes the dominating absorption mechanism.
2
S3: Permittivity of excited silicon This is a well-know result and can be found, e.g., in Refs.: 1,2 2 ωpl (ρeh ) ρeh i ε(ω, ρeh ) = εIB (ω ) 1 − − 2 1− , ρbf ω + 1/(τe2 (ρeh )) ωτe (ρeh ) ∗
(S4)
where the above mentioned ρeh -dependent bandgap shrinkage effect on interband transitions is accounted by introducing effective photon frequency ω ∗ = ω + Θρeh /ρbgr . Here, the characteristic renormalization EHP density ρbgr ≈ 1 × 1022 cm−3 , the factor Θ is typically about 5% of the total valence electron density (≈ 2×1023 cm−3 in Si) to provide the ultimate 50% electronic direct bandgap renormalization, i.e., h ¯ Θ ≈ 1.7 eV of the effective minimal gap ≈ 3.4 eV in silicon, and ρbf is the characteristic band capacity of the specific photo-excited regions of the first Brillouine zone in the k-space (e.g., ρbf (L) ≈ 4 × 1021 cm−3 for L-valleys and ρbf (X) ≈ 4.5 × 1022 cm−3 for X-valleys in Si), affecting interband transitions via the band-filling effect. 1 The bulk EHP frequency ωpl is defined as 2 ωpl (ρeh ) =
4πρeh e2 , εhf (ρeh )m∗opt (ρeh )
(S5)
where the averaged over L- and X- valleys effective optical (e-h pair) mass m∗opt ≈ 0.18me (Ref. 1 ) is a ρeh -dependent quantity, varying versus transient band filling due to the band dispersion and versus bandgap renormalization. The high-frequency electronic dielectric constant εhf was modeled in the form εhf (ρeh ) = 1 + εhf (0) × exp(−ρeh /ρscr ), where the screening density ρscr ≈ 1 × 1021 cm−3 was chosen to provide εhf → 1 in dense EHP. The electronic damping time τe in the regime of dense EHP at the probe frequency ωpr was taken, similarly to metals, in the random phase approximation as proportional to the inverse −1 bulk EHP frequency ωpl . Here, τe is evaluated for h ¯ ω > kB Te in the form τe (ρeh ) ≈ 3 ×
102 /(ωpl (ρeh )), accounting multiple carrier scattering paths for the three top valence subbands, and multiple X-valleys in the lowest conduction band of silicon.
3
S4: Optical characterization The broadband spectral measurements of the reflected signal from a single nanoparticle (Rs ) were carried out by means of strong focusing and collection of light (λ=400–900 nm) to a spectrometer (Horiba LabRam HR) through an achromatic objective with a numerical aperture NA=0.95. Raman scattering data were measured by a micro-Raman apparatus (Raman spectrometer HORIBA LabRam HR, AIST SmartSPM system). As a source 632.8-nm HeNe laser were used. The Raman spectra were recorded through the 100× microscope objective (NA=0.9) and projected onto a thermoelectrically cooled charge-coupled device (CCD, Andor DU 420A-OE 325) with a 600-g/mm diffraction grating. Individual nanoparticle spectrum was recorded by a commercial spectrometer (Horiba LabRam HR) when the nanoparticle was precisely places (accuracy about 100 nm) in the center of the laser beam (0.86-µm diameter) focused on the substrate.
DM 515 nm 220 fs
laser
BS
P λ 4
PD
O
F DM
delay
sample
Figure S2: Experimental setup for the pump-probe transmittance measurements, where BS is beam splitter, DM is dielectric mirror, O - objective, P - polarizer, PD - photodetector.
4
relaxation time (ps)
10000
1000
100
10
1 -3
carrier density (cm ) Figure S3: Relaxation time of EHP as a function of EHP density. Data from 13 (red square), 12 (red triangle), 8 (green squares), 11 (black stars), 10 (black triangles), 9 (black squares), and calculated Auger relaxation for c-Si (black dotted line).
S5: Time-domain transmittance measurements We exploit a laser pulses at 515 nm wavelength and 1 kHz repetition rate, which are split by two pulses with orthogonal linear polarizations and have 10-times difference between their intensities. Maximum energy of the pump pulse in non-damage regime is about 10 nJ. The pulses are focused by a NA=0.25 objective onto the nanoparticles array, providing focused laser spot size of about r1/e ≈ 2.5 µm. The energy of the transmitted probe pulse is measured by a GaP photodetector, whereas the pump pulse is filtered by a polarizer.
S6: Recombination For silicon, the carrier trapping (ΓTR ) on point defects 3 and self-trapping 4 is well known to be the dominating mechanism at low EHP densities (< 1020 cm−3 ) and low crystallinity. So-called bi-molecular (ΓBM ) non-radiative and radiative recombination (both resonant and non-resonant) processes govern relaxation dynamics mostly in amorphous silicon. 5,6 For crys-
5
Figure S4: Highest value of EHP density inside a resonant nanoparticle during the pulse action as a function of radius and incident intensity. talline silicon in a broad range of EHP densities, the relaxation proceeds via Auger recombination (ΓA ). 7 However, different methods of silicon fabrication yield the materials with different defects types and concentrations, resulting in different coefficients for ΓTR , ΓBM , and ΓA . 5,6,8 Fig. S3 represents some experimental data from previous works on EHP relaxation in different silicon-based materials and nanostructures 8–13 and their comparison with Auger recombination in c-Si given as τ = (4· 10−31 ρ2eh )−1 s, revealing much faster relaxation dynamics for amorphous and nanocrystalline states as compared to pure crystalline silicon.
S7: Generated EHP density In Fig. S4 we show the highest value of EHP density attainable during the pulse action assuming that nanoparticle radius obeys MD resonance condition.
S8: Estimation of switching intensity Developed theory allows us to obtain an evaluation of the threshold pulse intensity Iswitch necessary for transition to the unidirectional scattering regime. Assume that pulse wavelength λ is tuned to the magnetic dipole resonance of unexcited particle. The first Kerker 6
√ condition, when αe = αm , is satisfied at λKerker ≈ 2.2R ε0 . 14 Given the constant excitation wavelength, we conclude that refractive index of the photoexcited silicon should be decreased by 10% to achieve the Kerker’s condition: δn ≈ −0.1n, δε = δ (n2 ) ≈ −0.2ε0 . The main contribution to modified permittivity at optical and IR frequencies is provided by the free 2 2 carriers term in Eq. (S4) and it can be roughly estimated as δε ≈ δ(−ωpl ω ), where ωpl is 2 the bulk EHP frequency defined as ωpl = 4πρ0 e2 / εhf m∗opt . Here, the effective optical (e-h pair) mass m∗opt ≈ 0.18me and high-frequency electronic dielectric constant εhf → 1 in dense EHP. 2 To estimate peak EHP density ρ0 created by the incident pulse, we note that during a typical 100 − 200 fs pulse duration EHP relaxation can be neglected as it occurs on a ps scale. Further, we notice that two-photon absorption is more efficient at characteristic intensities required for activation of Huygens regime. Therefore we integrate the EHP rate equation (Eq. (2) of the main text) and estimate peak EHP density as ρ0 ≈ Finally, recalling that at magnetic dipole resonance αm =
3i , 2k3
τ 16π¯ h
Im χ(3) |Ein |4 .
and that electric field inside
the particle is enhanced by a factor of F ∼ 6αm /(kR4 ε), we substitute this formula into expression for δε and arrive at the desired evaluation of threshold intensity: 2
Iswitch =
2
c π 4λ Re(εSi )
2 s
h ¯ Re(εSi )m∗ · e2 τ Im χ(3)
(S6)
S9: FBR relaxation The FBR dynamics of a nanoantenna upon incidence of 10 GW/cm2 pulse is shown in Fig. S5 on a ps time scale. The FBR dynamics reveals that the reverse switching from Huygens regime with FBR ≈ 4 to the unexcited regime occurs during approximately 4 ps.
7
Figure S5: Front-to-back ratio of a MD resonant silicon nanoantenna for a 200 fs pulse (shown by the shaded area) on picosecond time scale.
References 1. Sokolowski-Tinten, K.; von der Linde, D. Generation of dense electron-hole plasmas in silicon. Phys. Rev. B 2000, 61, 2643–2650. 2. Makarov, S.; Kudryashov, S.; Mukhin, I.; Mozharov, A.; Milichko, V.; Krasnok, A.; Belov, P. Tuning of Magnetic Optical Response in a Dielectric Nanoparticle by Ultrafast Photoexcitation of Dense Electron-Hole Plasma. Nano Lett. 2015, 15, 6187–6192. 3. Jepsen, P.; Schairer, W.; Libon, I. H.; Lemen, U.; Hecker, N.; Birkholz, M.; Lips, K.; Schall, M. . Appl. Phys. Lett. 2001, 79, 1291. 4. Mao, S.; Quere, F.; Guizard, S.; Mao, X.; Russo, R. Dynamics of femtosecond laser interactions with dielectrics. Appl. Phys. A 2004, 79, 1695. 5. Esser, A.; Seibert, K.; Kurz, H.; Parsons, G.; Wang, C.; Davidson, B.; Lucovsky, G.; Nemanich, R. J. Ultrafast recombination and trapping in amorphous silicon. Phys. Rev. B 1990, 41, 2879. 6. Fauchet, P.; Hulin, D.; Migus, A.; Antonetti, A.; Kolodzey, J.; Wagner, S. Initial stages of trapping in a-Si: H observed by femtosecond spectroscopy. Phys. Rev. Lett. 1986, 57, 2438. 8
7. Shank, C. R.; Yen, R.; Hirlimann, C. Time-Resolved Reflectivity Measurements of Femtosecond-Optical-Pulse-Induced Phase Transitions in Silicon. Phys. Rev. Lett. 1986, 50, 454. 8. He, W.; Zakar, A.; Roger, T.; Yurkevich, I. V.; Kaplan, A. Determination of recombination coefficients for nanocrystalline silicon embedded in hydrogenated amorphous silicon. Opt. Lett. 2015, 40, 3889–3892. 9. Fauchet, P.; Hulin, D.; Vanderhaghen, R.; Mourchid, A.; Nighan, W. The properties of free carriers in amorphous silicon. Journal of Non-Crystalline Solids 1992, 141, 76–87. 10. Esser, A.; Heesel, H.; Kurz, H.; Wang, C.; Parsons, G. N.; Lucovsky, G. Femtosecond spectroscopic study of ultrafast carrier relaxation in hydrogenated amorphous silicon a-Si:H. J. Appl. Phys. 1993, 73, 1235. 11. Shkrob, I.; Crowell, R. Ultrafast charge recombination in undoped amorphous hydrogenated silicon. Phys. Rev. B 1998, 57, 12207–12218. 12. Yang, Y.; Wang, W.; Boulesbaa, A.; Kravchenko, I. I.; Briggs, D. P.; Puretzky, A.; Geohegan, D.; Valentine, J. Nonlinear Fano-Resonant Dielectric Metasurfaces. Nano Lett. 2015, 15, 7388–7393. 13. Shcherbakov, M. R.; Vabishchevich, P. P.; Shorokhov, A. S.; Chong, K. E.; Choi, D.-Y.; Staude, I.; Miroshnichenko, A. E.; Neshev, D. N.; Fedyanin, A. A.; Kivshar, Y. S. Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures. Nano Lett. 2015, 15, 6985. 14. Luk‘yanchuk, B. S.; Voschinnikov, N. V.; Paniagua-Dominguez, R.; Kuznetsov, A. I. Optimum Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Index. ACS Photon. 2015, 2, 993.
9
S1: Calculation of volume-averaged fields Calculation of the one- and two-photon absorption rates W1,2 requires knowing of the electric ˜ 2 ˜ 4 field value averaged over the nanoparticle volume, E and E . We should relate these values to the instantaneous amplitudes of electric and magnetic dipole moments, whose dynamics is governed by Eq. (1) in the main text. To do that, we assume that at each moment electric field inside the particle can be represented as a sum of electric dipole and magnetic dipole modes: √ E(r) = EMD (r) + EED (r) = AMD j1 ( εkr) [eθ cos ϕ − eϕ sin ϕ cos θ] + " # √ √ √ 0 j1 ( εkr) ( εkr · j1 ( εkr)) √ AED 2 √ er cos ϕ sin θ + (eθ cos θ cos ϕ − eϕ sin ϕ) εkr εkr
(S1)
The values AMD and AED are found by integrating the near-field current of the field 1
distribution E and equating the resulting dipole moments to known values of p ˜ and m: ˜ 1 m ˜ = 2c
Z r × (−iω) V
Z p ˜= V
ε−1 EMD d3 r 4π
ε−1 EED d3 r 4π
(S2)
(S3)
Now, volume-averaged electric field can be directly calculated by integrating expression (S1) over the nanoparticle volume. This procedure is repeated at each step of numerical simulations.
S2: Competition of one- and two-photon absorption
Figure S1: The rates of one- and two-hoton absorption as a function of light intensity. Figure S1 shows the absorbed power density due to one- and two-photon absorption as a function of light intensity inside silicon at wavelength of 650 nm. It indicates that for intensities above 5 GW/cm2 TPA quickly becomes the dominating absorption mechanism.
2
S3: Permittivity of excited silicon This is a well-know result and can be found, e.g., in Refs.: 1,2 2 ωpl (ρeh ) ρeh i ε(ω, ρeh ) = εIB (ω ) 1 − − 2 1− , ρbf ω + 1/(τe2 (ρeh )) ωτe (ρeh ) ∗
(S4)
where the above mentioned ρeh -dependent bandgap shrinkage effect on interband transitions is accounted by introducing effective photon frequency ω ∗ = ω + Θρeh /ρbgr . Here, the characteristic renormalization EHP density ρbgr ≈ 1 × 1022 cm−3 , the factor Θ is typically about 5% of the total valence electron density (≈ 2×1023 cm−3 in Si) to provide the ultimate 50% electronic direct bandgap renormalization, i.e., h ¯ Θ ≈ 1.7 eV of the effective minimal gap ≈ 3.4 eV in silicon, and ρbf is the characteristic band capacity of the specific photo-excited regions of the first Brillouine zone in the k-space (e.g., ρbf (L) ≈ 4 × 1021 cm−3 for L-valleys and ρbf (X) ≈ 4.5 × 1022 cm−3 for X-valleys in Si), affecting interband transitions via the band-filling effect. 1 The bulk EHP frequency ωpl is defined as 2 ωpl (ρeh ) =
4πρeh e2 , εhf (ρeh )m∗opt (ρeh )
(S5)
where the averaged over L- and X- valleys effective optical (e-h pair) mass m∗opt ≈ 0.18me (Ref. 1 ) is a ρeh -dependent quantity, varying versus transient band filling due to the band dispersion and versus bandgap renormalization. The high-frequency electronic dielectric constant εhf was modeled in the form εhf (ρeh ) = 1 + εhf (0) × exp(−ρeh /ρscr ), where the screening density ρscr ≈ 1 × 1021 cm−3 was chosen to provide εhf → 1 in dense EHP. The electronic damping time τe in the regime of dense EHP at the probe frequency ωpr was taken, similarly to metals, in the random phase approximation as proportional to the inverse −1 bulk EHP frequency ωpl . Here, τe is evaluated for h ¯ ω > kB Te in the form τe (ρeh ) ≈ 3 ×
102 /(ωpl (ρeh )), accounting multiple carrier scattering paths for the three top valence subbands, and multiple X-valleys in the lowest conduction band of silicon.
3
S4: Optical characterization The broadband spectral measurements of the reflected signal from a single nanoparticle (Rs ) were carried out by means of strong focusing and collection of light (λ=400–900 nm) to a spectrometer (Horiba LabRam HR) through an achromatic objective with a numerical aperture NA=0.95. Raman scattering data were measured by a micro-Raman apparatus (Raman spectrometer HORIBA LabRam HR, AIST SmartSPM system). As a source 632.8-nm HeNe laser were used. The Raman spectra were recorded through the 100× microscope objective (NA=0.9) and projected onto a thermoelectrically cooled charge-coupled device (CCD, Andor DU 420A-OE 325) with a 600-g/mm diffraction grating. Individual nanoparticle spectrum was recorded by a commercial spectrometer (Horiba LabRam HR) when the nanoparticle was precisely places (accuracy about 100 nm) in the center of the laser beam (0.86-µm diameter) focused on the substrate.
DM 515 nm 220 fs
laser
BS
P λ 4
PD
O
F DM
delay
sample
Figure S2: Experimental setup for the pump-probe transmittance measurements, where BS is beam splitter, DM is dielectric mirror, O - objective, P - polarizer, PD - photodetector.
4
relaxation time (ps)
10000
1000
100
10
1 -3
carrier density (cm ) Figure S3: Relaxation time of EHP as a function of EHP density. Data from 13 (red square), 12 (red triangle), 8 (green squares), 11 (black stars), 10 (black triangles), 9 (black squares), and calculated Auger relaxation for c-Si (black dotted line).
S5: Time-domain transmittance measurements We exploit a laser pulses at 515 nm wavelength and 1 kHz repetition rate, which are split by two pulses with orthogonal linear polarizations and have 10-times difference between their intensities. Maximum energy of the pump pulse in non-damage regime is about 10 nJ. The pulses are focused by a NA=0.25 objective onto the nanoparticles array, providing focused laser spot size of about r1/e ≈ 2.5 µm. The energy of the transmitted probe pulse is measured by a GaP photodetector, whereas the pump pulse is filtered by a polarizer.
S6: Recombination For silicon, the carrier trapping (ΓTR ) on point defects 3 and self-trapping 4 is well known to be the dominating mechanism at low EHP densities (< 1020 cm−3 ) and low crystallinity. So-called bi-molecular (ΓBM ) non-radiative and radiative recombination (both resonant and non-resonant) processes govern relaxation dynamics mostly in amorphous silicon. 5,6 For crys-
5
Figure S4: Highest value of EHP density inside a resonant nanoparticle during the pulse action as a function of radius and incident intensity. talline silicon in a broad range of EHP densities, the relaxation proceeds via Auger recombination (ΓA ). 7 However, different methods of silicon fabrication yield the materials with different defects types and concentrations, resulting in different coefficients for ΓTR , ΓBM , and ΓA . 5,6,8 Fig. S3 represents some experimental data from previous works on EHP relaxation in different silicon-based materials and nanostructures 8–13 and their comparison with Auger recombination in c-Si given as τ = (4· 10−31 ρ2eh )−1 s, revealing much faster relaxation dynamics for amorphous and nanocrystalline states as compared to pure crystalline silicon.
S7: Generated EHP density In Fig. S4 we show the highest value of EHP density attainable during the pulse action assuming that nanoparticle radius obeys MD resonance condition.
S8: Estimation of switching intensity Developed theory allows us to obtain an evaluation of the threshold pulse intensity Iswitch necessary for transition to the unidirectional scattering regime. Assume that pulse wavelength λ is tuned to the magnetic dipole resonance of unexcited particle. The first Kerker 6
√ condition, when αe = αm , is satisfied at λKerker ≈ 2.2R ε0 . 14 Given the constant excitation wavelength, we conclude that refractive index of the photoexcited silicon should be decreased by 10% to achieve the Kerker’s condition: δn ≈ −0.1n, δε = δ (n2 ) ≈ −0.2ε0 . The main contribution to modified permittivity at optical and IR frequencies is provided by the free 2 2 carriers term in Eq. (S4) and it can be roughly estimated as δε ≈ δ(−ωpl ω ), where ωpl is 2 the bulk EHP frequency defined as ωpl = 4πρ0 e2 / εhf m∗opt . Here, the effective optical (e-h pair) mass m∗opt ≈ 0.18me and high-frequency electronic dielectric constant εhf → 1 in dense EHP. 2 To estimate peak EHP density ρ0 created by the incident pulse, we note that during a typical 100 − 200 fs pulse duration EHP relaxation can be neglected as it occurs on a ps scale. Further, we notice that two-photon absorption is more efficient at characteristic intensities required for activation of Huygens regime. Therefore we integrate the EHP rate equation (Eq. (2) of the main text) and estimate peak EHP density as ρ0 ≈ Finally, recalling that at magnetic dipole resonance αm =
3i , 2k3
τ 16π¯ h
Im χ(3) |Ein |4 .
and that electric field inside
the particle is enhanced by a factor of F ∼ 6αm /(kR4 ε), we substitute this formula into expression for δε and arrive at the desired evaluation of threshold intensity: 2
Iswitch =
2
c π 4λ Re(εSi )
2 s
h ¯ Re(εSi )m∗ · e2 τ Im χ(3)
(S6)
S9: FBR relaxation The FBR dynamics of a nanoantenna upon incidence of 10 GW/cm2 pulse is shown in Fig. S5 on a ps time scale. The FBR dynamics reveals that the reverse switching from Huygens regime with FBR ≈ 4 to the unexcited regime occurs during approximately 4 ps.
7
Figure S5: Front-to-back ratio of a MD resonant silicon nanoantenna for a 200 fs pulse (shown by the shaded area) on picosecond time scale.
References 1. Sokolowski-Tinten, K.; von der Linde, D. Generation of dense electron-hole plasmas in silicon. Phys. Rev. B 2000, 61, 2643–2650. 2. Makarov, S.; Kudryashov, S.; Mukhin, I.; Mozharov, A.; Milichko, V.; Krasnok, A.; Belov, P. Tuning of Magnetic Optical Response in a Dielectric Nanoparticle by Ultrafast Photoexcitation of Dense Electron-Hole Plasma. Nano Lett. 2015, 15, 6187–6192. 3. Jepsen, P.; Schairer, W.; Libon, I. H.; Lemen, U.; Hecker, N.; Birkholz, M.; Lips, K.; Schall, M. . Appl. Phys. Lett. 2001, 79, 1291. 4. Mao, S.; Quere, F.; Guizard, S.; Mao, X.; Russo, R. Dynamics of femtosecond laser interactions with dielectrics. Appl. Phys. A 2004, 79, 1695. 5. Esser, A.; Seibert, K.; Kurz, H.; Parsons, G.; Wang, C.; Davidson, B.; Lucovsky, G.; Nemanich, R. J. Ultrafast recombination and trapping in amorphous silicon. Phys. Rev. B 1990, 41, 2879. 6. Fauchet, P.; Hulin, D.; Migus, A.; Antonetti, A.; Kolodzey, J.; Wagner, S. Initial stages of trapping in a-Si: H observed by femtosecond spectroscopy. Phys. Rev. Lett. 1986, 57, 2438. 8
7. Shank, C. R.; Yen, R.; Hirlimann, C. Time-Resolved Reflectivity Measurements of Femtosecond-Optical-Pulse-Induced Phase Transitions in Silicon. Phys. Rev. Lett. 1986, 50, 454. 8. He, W.; Zakar, A.; Roger, T.; Yurkevich, I. V.; Kaplan, A. Determination of recombination coefficients for nanocrystalline silicon embedded in hydrogenated amorphous silicon. Opt. Lett. 2015, 40, 3889–3892. 9. Fauchet, P.; Hulin, D.; Vanderhaghen, R.; Mourchid, A.; Nighan, W. The properties of free carriers in amorphous silicon. Journal of Non-Crystalline Solids 1992, 141, 76–87. 10. Esser, A.; Heesel, H.; Kurz, H.; Wang, C.; Parsons, G. N.; Lucovsky, G. Femtosecond spectroscopic study of ultrafast carrier relaxation in hydrogenated amorphous silicon a-Si:H. J. Appl. Phys. 1993, 73, 1235. 11. Shkrob, I.; Crowell, R. Ultrafast charge recombination in undoped amorphous hydrogenated silicon. Phys. Rev. B 1998, 57, 12207–12218. 12. Yang, Y.; Wang, W.; Boulesbaa, A.; Kravchenko, I. I.; Briggs, D. P.; Puretzky, A.; Geohegan, D.; Valentine, J. Nonlinear Fano-Resonant Dielectric Metasurfaces. Nano Lett. 2015, 15, 7388–7393. 13. Shcherbakov, M. R.; Vabishchevich, P. P.; Shorokhov, A. S.; Chong, K. E.; Choi, D.-Y.; Staude, I.; Miroshnichenko, A. E.; Neshev, D. N.; Fedyanin, A. A.; Kivshar, Y. S. Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures. Nano Lett. 2015, 15, 6985. 14. Luk‘yanchuk, B. S.; Voschinnikov, N. V.; Paniagua-Dominguez, R.; Kuznetsov, A. I. Optimum Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Index. ACS Photon. 2015, 2, 993.
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