Letter pubs.acs.org/journal/apchd5
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
‡
S Supporting Information *
ABSTRACT: Optically generated electron−hole plasma in high-index dielectric nanostructures was demonstrated as a means of tuning their optical properties. However, until now an ultrafast operation regime of such plasma-driven nanostructures has not been attained. Here, we perform pump−probe experiments with resonant silicon nanoparticles and report on dense optical plasma generation near the magnetic dipole resonance with an ultrafast (about 2.5 ps) relaxation rate. On the basis of experimental results, we develop an analytical model describing the transient response of a nanocrystalline silicon nanoparticle to an intense laser pulse and show theoretically that plasma-induced optical nonlinearity leads to ultrafast reconfiguration of the scattering power pattern. We demonstrate 100 fs switching to a unidirectional scattering regime upon irradiation of the nanoparticle by an intense femtosecond pulse. Our work lays the foundation for developing ultracompact and ultrafast all-optical signal processing devices. KEYWORDS: nanophotonics, silicon nanoparticles, magnetic Mie resonance, dense electron−hole plasma, Huygens source
S
low-intensity photoexcitation is relatively slow (tens of hundreds of picoseconds), being comparable with the fastest electronic devices. Such slow EHP relaxation in silicon-based nanoantennas was demonstrated in previous studies,17,18 where ultrafast (subpicosecond) reconfiguration of resonant silicon (Si) nanoparticles was achieved only via relatively weak Kerrtype optical nonlinearity. Therefore, for achieving effective alloptical signal processing by means of a single dielectric nanoparticle, two basic problems should be solved: (i) achieving of a few picosecond time of EHP relaxation and (ii) optimization of the modulation depth and time of switching. In this paper, we report on the experimental observation of a ∼2.5 ps operation regime of a nonlinear all-dielectric nanoantenna, which is 1 order of magnitude faster than previously reported for resonant Si nanoparticles. On the basis of the pump−probe experiments, we build an analytical model describing the transient dynamics of optical scattering on a dielectric nanoparticle with magnetic and electric optical responses. The developed model allows us to analyze the excitation and radiation of the electric and magnetic dipole modes in the strongly nonlinear regime of EHP generation and ultrafast relaxation. In particular, we describe reconfiguration of
ilicon presents a versatile and low-cost platform for data processing at speeds of 100 Gbit/s and beyond.1 This is possible thanks to a broad range of inherent optical nonlinearities such as stimulated Raman scattering,2 Kerr effect,3 two-photon absorption (TPA),4 thermo-optical effect,5 and electron−hole plasma (EHP) generation.6 However, the relatively large size of such devices and their high power consumption have remained an obstacle for the realization of ultracompact optical structures. Miniaturization of an optical device is possible with use of high-index dielectric (including silicon) nanoparticles, which due to their Mie resonances enhance light−matter interaction at the nanoscale.7−13 Recently, such all-dielectric nanostructures have been studied in the context of huge enhancement of optical nonlinearities.14−18 The ability to control the scattering behavior of nanoparticles via nonlinear effects may open unique opportunities for effective light manipulation by using only a single dielectric nanoparticle. The choice of optical nonlinearity for data processing is an important factor, which determines the ultimate performance of a photonic device. Refractive index change induced by EHP nonlinearity can be significantly larger than that of Kerr nonlinearity,1 avoiding the necessity to use high-Q large resonators and photonic crystals. The generation of EHP was employed for tuning of silicon nanoantenna optical properties in the IR and visible regions.15,16 However, in contrast to the instantaneous Kerr mechanism, the EHP relaxation process at © 2016 American Chemical Society
Received: May 25, 2016 Published: July 27, 2016 1546
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of a single nanoparticle. The existence of the MD response is confirmed by numerical modeling in CST Microwave Studio, revealing a prominent ring-like electric field distribution within the nanoparticle, which is a signature of magnetic dipoole resonance (see inset in Figure 2a). The nanoparticles were fabricated by means of plasmaenhanced chemical vapor deposition in a SH3 atmosphere, which usually results in the formation of completely amorphous Si:H material with a typical broad maximum at 480 cm−1 in the Raman spectrum (Figure 2b, black curve). However, at the final stage of the lithographical procedure the resulting nanoparticles represent a partially recrystallized state with the narrow Raman peak at 501 cm−1 with an asymmetric shape caused by the influence of the remaining amorphous component19 (Figure 2b, red curve). According to our Raman microspectroscopy measurements (for details see the Supporting Information), the amount of crystalline fracture Ic/(Ic + Ia) is about 0.45, where Ic and Ia are the integrated intensities of the crystalline and amorphous phases. The partial recrystallization of the initially amorphous Si:H film is supposed to be caused by an electronbeam processing step with 25 kV acceleration voltage.20 The resulting partially recrystallized material has an average grain size of about d ≈ 2 nm, according to the expression d = 2π B /Δω , where Δω is the peak shift for the microcrystalline material as compared with that of the c-Si and B = 2 cm−1 nm2.19 Such material properties are preferable for enhancement of radiative recombination of generated EHP carriers, reducing thermal recombination.21 The ultrafast operation of the conical nanoantennas was further confirmed in a series of pump−probe experiments. The results of measurements are presented in Figure 3a, where the relative transmittance change ΔT/T0 of the probe beam through the array is shown as a function of the delay between two pulses Δτ with respect to the pump beam. The laser source with a 515 nm wavelength was used, which is close to the MD resonance of the conical nanoparticles. The duration of the pump pulse is 220 fs, the intensity is 200 ± 100 GW/cm2, and the repetition rate is 1 kHz. This measurement at the highest possible intensity in the nondamage regime provides information on the EHP relaxation time. Fitting the transmission transient in Figure 3a results preferably in a decay time of ∼2.5 ± 0.5 ps. Relatively small values of modulation depth are described by slight mismatching of pump and probe beams on a 15 × 15 μm2 nanoparticle array. The observed value of 2.5 ps is the smallest observed for EHP decay in silicon-based resonant nanoparticles, compared to ∼30 ps17 and ∼25 ps18 (Figure 3b). The spectral positions and quality factors of electric and magnetic Mie resonances depend strongly on the shape of the particle. Although these results were obtained for conical particles, our conclusion applies to other geometries involvivng spherical particles, since the EHP relaxation rate is mostly dictated by the material structure and instantaneous EHP density, rather than the geometric shape of the nanoparticle.22 The comparison of the results for the nanoparticles with those for bulk amorphous and nanocrystalline silicon phases is given in the Supporting Information.
the radiation pattern during the pulse action from dipole-like to a unidirectional regime mediated by the generation of EHP in the nanoparticle, schematically shown in Figure 1.
Figure 1. Schematic illustration of transient dynamics of silicon nanoparticle scattering properties during its photoexcitation by a femtosecond laser pulse with duration τ. Optical absorption causes electrons to fill the conduction band of silicon, therefore modifying its permittivity and nanoantenna radiation pattern.
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PUMP−PROBE EXPERIMENT In order to demonstrate the possibility of ultrafast silicon nanoantenna operation, we have designed the pump−probe experiment with an array of silicon nanoparticles supporting a magnetic optical response. The cylindrical nanoparticles have bottom/top diameters of 140 nm/40 nm, a height of 220 nm, and a period of 600 nm. Results of linear optical measurements (see Figure 2a) show a pronounced resonance near 500 nm, which is attributed to the magnetic dipole (MD) Mie resonance
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Figure 2. (a) Reflection spectrum of cone silicon nanoparticles with a height of 220 nm and bottom/top diameters of 140 nm/40 nm. Filled area below the curve represents the spectrum of laser pulses in the pump−probe experiments. (b) Raman spectra of initial a-Si:H film (black curve) and nanoparticles fabricated from the film (red curve).
ANALYTICAL THEORY Now we derive the master equations describing the temporal response of a Si nanoparticle exposed to a short optical pulse. We model a spherical nanoparticle as a combination of electric 1547
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The above equations fully describe dynamics of the electric and magnetic dipole moments provided that EHP density, which determines dipole polarizabilities αe,m, is known at all times. However, in our case the field intensity itself describes EHP generation, which we address below. In order to describe the dynamics of laser-induced EHP, we employ the rate equation that can be found in the literature.25−27 For the sake of simplicity, we will describe volume averaged concentration ρeh(t) neglecting its spatial distribution and therefore diffusion of carriers across the particle volume. Indeed, at ρeh > 1020 cm−3 the thermal velocity of hot free electrons is about ve ≈ 5 × 105 cm/s,28 whereas the corresponding electron−electron scattering time is about ∼100 fs. Taking into account the ring-like MD mode structure, the characteristic EHP homogenization time governed by a ballistic motion of electrons can be estimated as τhom ≈ R/2ve ≈ 100 fs, where R is the nanoparticles radius. We also ignore thermal effects, because of their negligible influence on permittivity at the picosecond time scale under nondamage irradiation conditions. The rate equation for EHP therefore takes the form dρeh dt
= −Γρeh +
W1 W2 + ℏω 2ℏω
(2)
Here, W1,2 stand for the volume-averaged rates of EHP generation via one- and two-photon absorption, and Γ is the phenomenological EHP relaxation rate constant, which is dependent on EHP density.22 The absorption rates are written ω ̃ |2 ⟩Im(ε) a n d i n t h e u s u a l f o r m a s W1 = 8π ⟨|E in
Figure 3. (a) Relative transmittance change as a function of probe delay. Inset: Colorized SEM image of the array of Si nanoparticles. (b) Electron−hole plasma relaxation times for resonant nanoparticles made of a-Si:H at a pump wavelength of 800 nm (orange circle17), for resonant nanoparticles made of pc-Si at 1350 nm (blue square18), for nonresonant nanocrystals inside an a-Si:H bulk matrix at 800 nm (red squares23), and relaxation time measured in this work at 515 nm for resonant nanoparticles made of nc-Si/a-Si:H (green square). Green dotted line corresponds to our model for the nc-Si/a-Si:H material.
ω
̃ |4 ⟩Im χ (3), where ⟨...⟩ denotes averaging over the W2 = 8π ⟨|E in nanoparticle volume. Nonlinear susceptibility Im χ(3) may be expressed through experimentally accessible TPA coefficient β εc 2
via Im χ (3) = 8πω β , where β is a function of a pump laser wavelength.29 Noting different functional dependence of oneand two-photon absorption on incident light intensity, we may expect that TPA will dominate over a one-photon process beyond a certain level of intensity. Simple calculations show that it occurs for intensities above 5 GW/cm2 within silicon (see Supporting Information). The averaged electric fields should be related to the instantaneous values of electric and magnetic dipole moments. This is done by integrating the total field of the two spherical harmonics corresponding to the given values of p̃ and m̃ (see Supporting Information). Generally, the total EHP relaxation rate (Γ) can be represented as a polynomial function of EHP density: Γ = ΓTR(ρ0eh) + ΓBM(ρ1eh) + ΓA(ρ2eh), where each term corresponds to trapping (ΓTR), bimolecular (ΓBM), and Auger (ΓA) mechanisms. Our experimental results allow us to take realistic parameters for our model. We describe the observed ultrafast recombination based on the measurements of nc-Si from ref 23 including the Auger recombination mechanism, because of mixed a-Si/nc-Si composition of our material.30 Therefore, the resulting relaxation rate is represented via the following terms: ΓTR = 1.5 × 1011 s−1 (ref 23), ΓBM = 1.1 × 10−10 ρeh cm3 s−1 (ref 23), and ΓA = 4 × 10−31 ρeh2 s−1 (ref 30). The resulting dependence of relaxation time on EHP density is shown in Figure 3c, demonstrating good agreement with the experimental values. The last essential element of our dynamical model is the expression relating permittivity of excited silicon ε to EHP
(p) and magnetic (m) dipoles, which are related to incident monochromatic electromagnetic fields via p = αeE and m = αmH. The corresponding dipole polarizabilities are expressed in 3i terms of the Mie coefficients a1 and b1 as αe = 3 a1 and 2k
3i
αm = 3 b1, respectively,24 where k = ω/c is the free-space wave 2k vector. The above expressions relate incident field and induced dipole moments in the stationary regime. In order to obtain dynamical equations for dipole moments, it is convenient to rewrite these expressions as αe−1p = E and αm−1m = H. Assuming that the spectrum of the incident pulse is centered at the frequency ω0, we further represent incident waves as E(t) = Ẽ (t) exp(−iω0t) and H(t) = H̃ (t) exp (−iω0t), where Ẽ (t) and H̃ (t) denote slowly varying amplitudes of electric and magnetic fields, respectively. Correspondingly, nanoparticle dipole moments are p(t) = p̃(t) exp(−iω0t) and m(t) = m̃ (t) exp (−iω0t). Expanding inverse polarizabilities ∂α
−1
into Taylor series αe,m−1 = αe,m−1(ω0) + ∂e,mω (ω − ω0) and performing a Fourier transform of expressions relating induced dipole moments to the incident fields, we obtain the differential equations that dictate the optical dynamics: i i
∂αe−1 dp̃ + αe−1p̃ = Ẽ (t ) ∂ω dt
∂αm−1 dm̃ + αm−1m̃ = H̃ (t ) ∂ω dt
(1) 1548
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Figure 4. Optical dynamics of a Si nanoparticle upon EHP generation. (a) Temporal dynamics of photoexcited Si permittivity for a particle irradiated by a resonant Gaussian pulse of different intensities. Shaded area represents the envelope of the pulse intensity. (b) Dynamics of ED and MD phases excited in the particle during the pulse action for the largest pulse intensity. Inset: Corresponding amplitudes of ED and MD. (c) Dynamical reconfiguration of nanoantenna directivity: FBR of a nanoparticle during the pulse action. Scattering diagrams of the incident beam at the largest intensity are shown in the two insets.
Figure 5. (a) Map of switching time τswitch as a function of incident pulse wavelength and peak intensity. (b) Wavelength dependence of the switching intensity (for 200 fs pulse) calculated for a series of FBR values.
density ρeh. Generally, this dependence is represented as the following expression:16,26 ε(ω , ρeh ) = εSi + Δεbgr + Δεbf + ΔεD
intensities I0 determined as I0 = (c/8π)|E|2. Pulse duration is taken as τ = 200 fs; pulse center position is t0 = 400 fs. Figure 4a shows the time-dependent real part of Si permittivity for various I0. It shows a pronounced dip near the pulse center around t = 500 ps, where EHP density is maximal. Electric dipole (ED) and magnetic dipole (MD) phases for the largest peak intensity (I0 = 40 GW/cm2) are shown in Figure 4b. The phase of the MD abruptly jumps around the pulse center in such a way that ED and MD oscillate almost in-phase. The corresponding MD and ED amplitudes are shown in the inset in Figure 4b. While the ED amplitude smoothly follows the incident pulse shape, the MD amplitude also experiences a sharp change at the pulse maximum. At this moment, due to the large EHP density, the particle is detuned from the MD resonance condition, resulting in an abrupt decrease of the MD amplitude. Therefore, transition to unidirectional scattering during the rest of the pulse may be expected. In order to address radiation directivity evolution, we calculate the angular emitted power distribution created by perpendicular electric and magnetic dipoles in the E- and Hplane as8
(3)
where εSi is the permittivity of nonexcited material, whereas Δεbgr, Δεbf, and ΔεD are the contributions from band gap renormalization, band filling, and the Drude term. The Drude contribution strongly dominates the others in the IR range and at relatively low intensities.1 Band filling is important when EHP density becomes comparable with the capacity of the conduction band (ρeh > 1020 cm−3). Band gap renormalization plays significant role at wavelengths where permittivity dispersion dε/dω is considerable ( 15 GW/cm2 is the most optimal for the ultrafast reconfiguration of the nc-Si nanoantenna. In practice, achieving an FBR = ∞ is not necessary, whereas FBR = 3−5 is sufficient for significant change of the transmitted optical signal through the nanoswitch, yielding a relatively low range of incident intensities, I0 = 15−60 GW/cm2, at λ = 500− 1500 nm (see Figure 5b). Particularly, at I0 = 15 GW/cm2, λ = 550−700 nm, τ = 100 fs, and realistic focal spot area ≈ 1 μm2, a 15 pJ pulse is sufficient for considerable switching (FRB ≈ 3) at the ultrafast time. Such energy and intensity levels are far below the damage threshold of Si (Idamage ≈ 102−103 GW/cm2, refs 16−18) for this type of nanoparticle, which is significantly larger than the typical thresholds (101−102 GW/cm2) for plasmonic nanostructures.32−34 Remarkably, our simple expression for Iswitch gives the same ∼τ−1/2 scaling as the damage
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Author Contributions
§ D.G.B. and S.V.M. contributed equally to this work. S.V.M., V.A.M., and S.I.K. conducted the experiments. D.G.B., S.V.M., and A.E.K. developed the analytical model. A.E.K. and P.A.B. managed the work. All authors participated in discussions and contributed to the manuscript.
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Notes
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The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We are thankful to Andrey Ionin and Pavel Damilov for assistance with pump−probe measurements and to Yuri Kivshar, Arseniy Kuznetsov, and Maxim Shcherbakov for fruitful discussions. The experimental and theoretical parts of the work were financially supported by grants of the Russian Science Foundation, Nos. 15-19-00172 and 15-19-30023, respectively. D.G.B. acknowledges support from the Russian Foundation for Basic Research (project No. 16-32-00444).
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