Supporting Information: Efficient Second-Harmonic Generation in. Nanocrystalline Silicon Nanoparticles

Transcription

1 Supporting Information: Efficient Second-Harmonic Generation in Nanocrystalline Silicon Nanoparticles Sergey V. Makarov,, Mihail I. Petrov, Urs Zywietz, Valentin Milichko, Dmitry Zuev, Natalia Lopanitsyna,, Alexey Kuksin,, Ivan Mukhin, George Zograf, Evgeniy Ubyivovk, Daria Smirnova, Sergey Starikov,, Boris N. Chichkov, and Yuri S. Kivshar, Department of Nanophotonics and Metamaterials, ITMO University, St. Petersburg , Russia Nanotechnology Department, Laser Zentrum Hannover e.v., Hannover D-30419, Germany Laboratory of Chemical Thermodynamics, Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow , Russia Moscow Institute of Physics and Technology, Moscow Russia Interdisciplinary Resource Centre for Nanotechnology, St. Petersburg State University, St. Petersburg , Russia Nonlinear Physics Centre, Australian National University, Canberra ACT 2601, Australia 1

2 Contents I. Calculation of dark-field spectra II. Novel interatomic potential for silicon III. Evaluation of the cooling rate IV. Setup for nonlinear measurements V. Numerical calculations of SHG from Si nanoparticle with bulk and surface nonlinear optical responses 2

3 I. Calculation of dark-field spectra In order to calculate dark-field spectra properly, we carry out full-wave simulations in COM- SOL, taking into account presence of a silica substrate, finite angle of signal collecting, and realistic material dispersions S1. This approach reveals disappearing of some modes in the dark-field scheme of optical measurements. For instance, magnetic quadrupole is almost not visible due to specific structure of its power pattern (Fig. S1). On the other hand, an electric quadrupole (EQ) is clearly seen in the dark-field scheme and in experimental spectra (see Fig.1d of the main text). Therefore, we use the positions of electric (ED) and magnetic (MD) dipoles as well as EQ to determine realistic diameter of the nanoparticles. This method is more precise for ablative nanoparticles as compared with measurements by scanning electron microscope, because the nanoparticles are always covered by an oxide layer with thickness 2 3 nm. II. Novel interatomic potential for silicon In this work we developed new interatomic potential for atomistic simulation of crystallization. The potential has the form of angular-dependent potential (ADP) S2. For ADP the total potential energy U is given by the following formula: U = i>j ϕ(r ij ) + F (ρ i ) + 1 (µ k i ) i i,k + 1 (λ kl i ) 2 1 νi 2, 2 6 i,k,l i (S1) where 3

4 EQ 0.14 Glass EQ 0.12 Scattering cross section, µm nm 350nm 300nm 250nm nm Wavelength, nm Wavelength, nm Figure S1: Numerically calculated scattering spectra of spherical silicon nanoparticles with different diameters and different scattered energy collection: in all angles (left) and in a segment of 49 o (right) as shown by the insets. Red arrows indicate positions of electric quadrupole (EQ). 4

5 ρ i = j i µ k i = j i ρ(r ij ), u(r ij )r k ij, λ kl i = j i w(r ij )r k ijr l ij, (S2) ν i = k λ kk i. Here indices i and j enumerate atoms and superscripts k, l = 1, 2, 3 refer to the Cartesian components of vectors and tensors. The first term in (S1) represents interactions between atoms with a pair potential ϕ. The summation is over all j-th neighbors of i-th atom within the cutoff distance r cut = 6.2 Å. F is the embedding energy which is a function of the total electron density ρ.the first and the second terms in (S1) give principal contribution to the system energy. The µ and λ terms introduce non-central interactions through the dipole vectors and quadrupole tensors. They are aimed to penalize deviations of local environment from the cubic symmetry. The force-matching method S3 is used for the development of the potential, as implemented in the Potfit code S4. This method provides a way to construct physically justified interatomic potentials from the fitting database which does not contain experimental data. The idea is to adjust the interatomic potential functions to optimally reproduce per-atom forces (together with total energies and stresses) computed at the ab initio level for a finetuned set of reference structures. The reference data are calculated using the DFT code VASP 5.2 S5. The reference structures contain approximately 200 atoms in a simulation box with periodic boundary conditions. The exact number of atoms depends on the phase structure, its density and state. The Brillouin zone is sampled with the Monkhorst-Pack k-point mesh. The cut-off energy of a plane-wave basis set is 250 ev. We use projector augmented wave pseudopotentials in- 5

6 cluded in VASP package and the exchange-correlation functional within generalized-gradient approximation (GGA). Four electrons 3s 2 3p 2 are taken into account. For the construction of the ADP we use 14 configurations containing 2560 atoms altogether. These configurations represent solid and liquid states. All these atomic configurations have been taken from classical atomistic simulations at different temperatures and densities (for this purpose the MEAM potential from S6 was applied at the initial step). The fitting procedure consists of the following steps: (1) fitting the parameters of a new potential to the reference ab initio database; (2) testing the calculated potential with respect to certain properties (lattice parameters, melting temperature, etc.); (3) recalculating the initial set of configurations with the fitted potential. This procedure is performed in an iterative manner in order to improve the description of the desired properties. The target function for minimization is given by the following sum: Z = Z f + Z C, (S3) Z f = N f i=1 α=x,y,z (f ADP iα f DFT iα ) 2 (f DFT iα ) 2, (S4) Z C = N C j=1 (A ADP j A DFT j ) 2 (A DFT j ) 2. (S5) The reference data are represented by per-atom forces f and integral characteristics A (one value of energy and six components of the stress tensor for each configuration). The Z f and Z C are two parts of target function for forces and integral characteristics, respectively. N f is the total number of atoms. N C is the total number of configurations. The index DFT 6

7 denotes the reference values; ADP denotes the values computed with the fitted potential. The basic properties of Si calculated with a new potential are summarized in Table S1 and compared with the experimental data. Table S1: Results calculated with ADP in comparison with the existing experimental data S7,S8 : the cohesive energy E c, the equilibrium lattice parameter a, the elastic constants C ij, melting temperature T m, surface tension γ at T = 1550K. experiment simulation with ADP E c (ev) a (Å) C 11 (GPa) C 12 (GPa) T m (K) γ (N/m) The used interatomic ADP potential is specified by the spline nodes coordinates. In the tables below all potential functions are listed with the appropriate accuracy that allows the potential reproduction. The final potential functions are shown in figure S2. Table S2: conditions. The pair and density potential functions and the derivatives for the boundary r (Å) ϕ (ev) ρ ϕ (ev/å) ρ (Å 1 )

8 Table S3: The u and w potential functions and the derivatives for the boundary conditions (in ( ev/å 2 ) and ( ev/å 3 )). r (Å) u ( ev/å) w ( ev/å 2 ) u w Table S4: Embedded functions F and the derivatives for the boundary conditions. ρ F (ev) F (ev)

9 Figure S2: (color online). Potential functions of the developed ADP. Potential functions u(r) and w(r) are given in ev/å and ev/å 2, respectively. III. Evaluation of the cooling rate We need to estimate the cooling rate at laser printing for comparison of simulation results with experimental data. There are two different regime of cooling: at moving in air from the irradiated film to the receiving substrate; at cooling by direct contact with receiving substrate. K may be estimated with use the kinetic theory of collisions with gas molecules at moving of the particle in air. For such approach, the time derivative of internal energy Ė n 3k K E ν, where n is total number of Si atoms in the particle, k is Boltzmann constant, E is the change in the internal energy at single collision, ν is collision frequency. This equations are given with taking into account that atomic heat capacity is equal 3k. We use simple estimation ν V πr 2 ρ, where V is velocity of the particle relative to air molecules, r is the radius of the particle and ρ is atomic density of air. For order of magnitude estimation purposes, we take E as k (T T 0 ), where T 0 = 300 K. Final formula for estimation of K may be written as: K (T T 0)V πr 2 ρ n 3 (S6) 9

10 This formula gives K r 1 because n r 3. We estimate value of V about 10 3 m/s and K about 0.1 K/ps for the particle with r = 40 nm. At direct contact with receiving substrate, we use Fourier expression for the heat flux q at the estimation of K: q = χ T T 0 λ q πr 2 Ė χ T T 0 πr 2 n 3k K λ K χπr2 n 3k T T 0 λ (S7) where χ is parameter of thermal conductivity, λ is length of thermal disturbance in the substrate. For λ = 10 nm and χ = 10 W m 1 s 1, we estimate that K is more than 50 K/ps for the particle with r = 40 nm. Though these estimates are rough, such consideration allows us to understand the crystallization kinetics. The more detailed studies of the crystallization process are scope of our future work. 10

11 IV. Setup for nonlinear measurements Figure S3 shows the detailed configuration of the sample-scanning nonlinear optical microscope we constructed for the single nanoparticle SHG measurements. Yb:laser 1050 nm / 150 fs Pockels cell Attenuator spectrometer autocorrelator Pin-hole Filters power meter Spectrometer CCD Objective x50 Objective x10 Figure S3: Schematic of setup for nonlinear confocal optical micro-spectroscopy. 11

12 V. Numerical calculations of SHG from Si nanoparticle with bulk and surface nonlinear optical responses In order to gain a deeper insight into the origin of the second-harmonic generation in polycrystalline silicon nanoparticles, we have made numerical modeling, comparing different mechanisms of nonlinear signal generation. The second-order nonlinear polarization in nanoparticles made of materials with central symmetry is described by the expression (1) in the main text. We are guided by the results of the experimental measurements, shown in Fig. 4 in the main text, to identify the physical mechanism responsible for nonlinear signal generation. We start from modeling of linear optical response from Si nanoparticles of different diameters at the fundamental and harmonic wavelengths: 1050 nm (fundamental), and 525 nm (SH). The spectral dependence of the scattering cross section, normalized to geometrical cross section, is shown in Fig. S4a. One can see that at 1050 nm there is a magnetic dipole resonance (A) in the considered size range, the corresponding field distribution is shown in the inset. For larger particles, the electric dipole mode (B) is excited. We reveal a more complicated structure of the scattering spectra for wavelength 525 nm. To identify these modes, we plot the spectrum of the total energy of the electric field inside a nanoparticle in Fig. S4b. One can see two sharp peaks (C and D), which correspond to electrical octupole (C) and magnetic quadrupole mode (D), according to the multipolar decomposition made analytically by using Mie theory. The respective field distributions are also shown in the insets of Fig. S4b. Next, we model the size dependence of SHG signal. We employ the nonlinear polarization expressed in Eq. (1) in the main text. Our initial hypothesis is that the main source of the SH signal comes from multiple boundaries of crystallites inside the fabricated nanoparticle. In order to account for a specific polycrystalline structure of our nanoparticles and emulate such sources distribution, we use a coarser mesh grid, with element size comparing to the sizes of crystallites grains ( 10 nm). The mesh grains allow us to model the mesoscopic effects 12

13 related to the polycrystalline structure of nanoparticles. The simulated results describe well the experimentally measured dependence of SH signal shown in Fig. 4 of the main text. One can see that the resonant modes influence the SHG signal spectra as the enhancement of SH emission corresponds well to the resonances supported by nanoparticles at 525 nm. To verify our approach further, we make a comparative analysis of SHG from a homogeneous Si nanoparticle without the inner multigrain crystallinity. As the first step, we simulate SHG from such nanoparticle governed by the bulk nonlinear sources coming solely from the second and third terms in the expression Eq. 1(c), setting the parameters β = 0,γ = δ 0. The calculated results are shown with orange line in Fig. S4c. One can conclude that (i) the simulated curve significantly differs from the experimentally measured dependence, and (ii) the modes at 525 nm do not influence the SH spectrum much. Such volume nonlinear source occurs due to inhomogeneity of the electric field inside the nanoparticles and is not related to the grain structure, which can be seen from the field distribution picture (see Fig. S4c inset E). Moreover, the polarization of the emitted SH signal is rotated for almost 90 degree for these bulk contributions relatively to the experimental one (see Fig. S5), while simulating interfacial nonlinearities gives the proper result describing the experiment (see Fig. 4). Finally, we address the contribution of the outer nanoparticle surface into the total SHG signal. The surface of any isotropic nanoparticles is a defect, which locally lifts the inversion symmetry. The nonlinear surface polarization P (2ω) surf is related to the second-order nonlinear susceptibility tensor, which is nonzero at the surface of nanoparticle. It can be expressed in the form of (1b) with three nonzero components χ, χ, χ. We performed the simulations of the SHG determined by each,, and surface polarization components. In our simulations we used the following parameters of the surface tensor χ χ, and χ 20χ according to Ref. S9. In Fig. S6 we plot the SHG intensity as a function of nanoparticle diameter. One can see that only the χ component is defined by the mode structure at the second harmonic, and is similar to what was measured experimentally. On the other hand, the χ signal is 10 times weaker than χ, and 13

14 (a) Scattering cross section A 12 A B 10 B nm 525nm (b) D Energy of electric field in nanoparticle C C D 1050 nm 525nm 0 (c) Intensity of SH signal 1 E E Bulk component Grains component F 0.5 F Nanoparticle diameter (nm) Figure S4: (a) Modeled scattering cross section, normalized to geometrical cross section, depending on nanoparticle size, at the fundamental and harmonic wavelengths: 1050 nm (red dashed), and 525 nm (green solid). The field distributions are shown in the inset for magnetic dipole (A) and electric dipole (B) resonances. (b) The normalized energy of electric field trapped inside the nanoparticles of different sizes plotted at two wavelengths: 1050 nm (red dashed), and 525 nm (green solid). The field distribution is shown in the inset for electrical octupole (C) and magnetic quadrupole (D). (c) The calculated SH signal from the grained (solid purple line) and bulk (orange line) nonlinear sources in arb. un. (not to scale). The field distribution of the generated SH field at the resonances is shown in insets E and F. 14

15 Grains component Figure S5: Calculated dependence of the polarization of the SHG signal from a Si nanoparticle with grained (purple) and bulk (orange) nonlinear sources. 15

16 5000 weaker than the χ contribution, which dominates the resultant SHG signal and does not coincide with the experimental measurements. These results exclude the surface of the studied nanoparticle as the leading source of the measured nonlinear signal, leaving the nonlinear sources induced at the grains surfaces to be the main mechanism responsible for SHG. SHG signal, a.u x10 x Diameter, nm Figure S6: Calculated dependence of the second harmonic signal related to the different sources of surface nonlinearity. Different components are plotted with blue (χ ), red (χ ), and green (χ ) lines. 16

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