Supporting Information: Resonant non-plasmonic nanoparticles for. efficient temperature-feedback optical heating

Transcription

1 Supporting Information: Resonant non-plasmonic nanoparticles for efficient temperature-feedback optical heating George P. Zograf, Mihail I. Petrov,,, Dmitry A. Zuev, Pavel A. Dmitriev, Valentin A. Milichko, Sergey V. Makarov,, and Pavel A. Belov Department of Nanophotonics and Metamaterials, ITMO University, St. Petersburg 97, Russia University of Eastern Finland, Department of Physics and Mathematics, Yliopistokatu 7, 8, Joensuu, Finland

2 Contents I. Model of optical heating of spherical nanoparticles II. Heating of spherical particles of different permittivities III. Heating in medium with higher refractive index and thermal conductivity (water and ) IV. Mode decomposition V. Raman shift VI. Radiation pattern at magnetic quadrupole resonance 2

3 I. Model of optical heating of spherical nanoparticles We consider a thermal diffusion equation in a steady-state regime inside a nanoparticle T (r): T (r) = q(r) κ, (S) where q(r) is the heating power density. Owing to high thermal conductivity κ of the nanoparticle material, the distribution of the heating power inside the nanoparticle is highly homogeneous and can be set as a constant q(r) = q. This simplifies the equation S, where temperature inside the nanoparticle is only a function of radius (r): d dt r 2 r2 dr dr = q. κ (S2) To distinguish temperatures inside (T ) and outside (T 2 ) the nanoparticle (NP): T r 2 r r2 r = q(r) inside NP, (S3) κ T r 2 r r2 2 = outside NP. (S4) r After some straightforward calculations we derive : T = qr2 6κ C r + C inside NP, (S5) T 2 = C 2 r + C 2 outside NP. (S6) We need to apply boundary conditions to the general solutions. In order to provide correct formulation of the optical heating problem, we employ boundary conditions for a perfect thermal contact between the the nanoparticle and surrounding medium: 3

4 T r=r = T 2 r=r, κ T r r=r = κ 2 T 2 r r=r. (S7) (S8) Also, we use the following limit cases: T 2 r T C 2 = T T r= < C = (S9) (S) The solution of the equation S2 with these boundary conditions gives an expression for the temperature inside the nanoparticle with radius R: T = P ) ( r2 + P 8πκ R R 2 4πκ 2 R + T (S) where P = 4/3πR 3 q is the total absorbed power, and T is the temperature of the surrounding medium. As the thermal conductivity of nanoparticle is much larger than that of surrounding medium (κ 2 ), i.e. κ κ 2, the temperature inside the nanoparticle is homogeneous, which simplifies the expression for the temperature increase inside the nanoparticle: δt NP = T T = P 4πκ 2 R. (S2) From this expression it follows that the nanoparticle temperature is determined by the total absorbed power P, size of the nanoparticle R and the thermal conductivity κ 2 of the surrounding medium. The power absorbed by the nanoparticle can be calculated via its 4

5 absorption cross-section C abs and incident light intensity I: P = C abs I, (S3) where the absorption cross-section C abs can be calculated analytically from Mie theory or with the use of commercial software Comsol Multiphysics in case of nanoparticle on a substrate. Absorption C abs cross-section for spherical nanoparticles is linked with extinction and scattering cross-sections C abs = C ext C sca, where C ext and C sca are following: C sca = W sca I = 2π k 2 (2n + )( a n 2 + b n 2 ), (S4) n= C ext = W ext I = 2π k 2 (2n + )Re(a n + b n ), (S5) n= where W sca, W ext are the scattered and extinct energies by nanosphere, respectively, k = 2πn r /λ, n r is the refractive index, I is the incident intensity, a n and b n are the scattering coefficients that describe interaction of the nanoparticle with a plane wave. Magnetic permeabilities of the surrounding media and nanoparticles are equal to one in the optical range. These assumptions lead to simplified form of the scattering coefficients: a n = mψ n(mx)ψ n(x) ψ n (x)ψ n(mx) mψ n (mx)ξ n(x) ξ n (x)ψ n(mx), (S6) b n = ψ n(mx)ψ n(x) mψ n (x)ψ n(mx) ψ n (mx)ξ n(x) mξ n (x)ψ n(mx), (S7) with x = ka, and the Riccati - Bessel functions : ψ n (ρ) = ρj n (ρ), ξ n (ρ) = ρh () n (ρ), where j n and h () n are spherical Bessel and Hankel functions, respectively. 5

6 II. Heating of spherical particles of different permittivities In Fig.2 in the main text, 2D heating maps for fixed λ/d ratios and various imaginary and real parts of nanospheres permittivity are presented. According to Fig.2a, it is evident that for small particles (λ/d = ) the temperature maximum lies in the region of negative values real part of permittivty, i.e. in the region of metals, whereas in the area of positive values of real parts, no resonances can be observed. Indeed, low-loss dielectric nanoparticles do not support effective heat absorption regime until they become big enough (relatively to the wavelength) to possess Mie resonances. For example, in Fig.2 we consider the wavelengthto-diameter values: λ/d = 3.5 and λ/d = 2.8, where dipolar and quadrupolar modes can be excited. In Fig.S, we represent dependencies of light-to-heat conversion on imaginary part of dielectric permittivity for the metallic (Fig.Sa) and dielectric (Fig.Sb) regions. All curves have maximum where the optimal ratio γ rad γ Ohmic is fulfilled, whereas the T for two limit cases: Im(ɛ) and Im(ɛ). a b 8 ΔT (K) Im( ε) Im( ε) Figure S: Theoretical dependence of the temperature increase on imaginary part of permittivity. The values of real part are fixed at resonant positions according to Fig.2 in the main text. In metallic region : (a) yellow - λ/d =, Re(ɛ) = -2.25; green - λ/d = 3.5, Re(ɛ) = -.9; blue λ/d = 2.8, Re(ɛ) = In dielectric region (b) red λ/d = 3.5, solid: Re(ɛ) = 24.25; dashed: Re(ɛ) =.; black λ/d = 2.8, solid: Re(ɛ) = 5.25; dashed: Re(ɛ) = 7.. Spherical nanoparticles in air are considered. On one hand, the smallest metallic nanoparticles give the highest values for optical heating (see yellow curve in Fig. Sa). On the other hand, these values can be achieved only at too low values of imaginary part of dielectric permittivity (Im(ɛ) <.5), which is not 6

7 Temperature increase achievable for realistic metals. III. Heating in medium with higher refractive index and thermal conductivity (water and ) a 5 4 metals.45 dielectrics λ/d = b metals dielectrics.45.4 λ/d = 3.5 Ge c metals dielectrics.7.45 λ/d = Tmax T Ge max Im( ε ) 3 2 Au.6.6 Au Fe 2 O 3 Ag Ag.4.4 Ag SiO.4 SiO Re( ε) Re( ε) Re( ε) Si.6 Au Fe 2 O 3 Si.6T.4T.2T max max max Figure S2: Optical heating a spherical nanoparticle in water. Theoretically calculated (Mie theory) heating maps for spherical nanoparticles with fixed wavelength(λ)/diameter(d) ratio for different real and imaginary parts of permittivity in homogeneous medium (water): (a) λ/d = ; (b) λ/d = 3.5; (c) λ/d = 2.8. Green lines with arrows depict values of Re(ɛ) and Im(ɛ) for different materials (dispersion). The orientation of arrows corresponds to increase of wavelength from minimum to maximum indicated by numbers in microns. In the main text, we discuss that the silicon nanospheres can provide more efficient light-to-heat conversion as compared to golden ones.the high efficiencies are based on high quality factor of non-plasmonic nanoresonators. However, Mie-resonances are dependent on the refractive index of a host medium. Indeed, the closer values of the refractive indices of medium and resonator, the worse is the Q-factors of Mie resonances. In Fig. S2, we consider the same dependencies as for Fig.3 in the main text, but for homogeneous or water media. The calculations show that silicon remains wider wavelength range for effective optical heating than gold. The highest temperature values both for gold and silicon are almost the same in water. 7

8 a c diameter (nm) diameter (nm) water b d ΔT (K) ΔT (K) Figure S3: 2D heating maps for gold and silicon spherical nanoparticles surrounded by and water medium. Light intensity is I =. mw/µm 2, i.e. similar to that in Fig.3. (a) Silicon in water, (b) gold in water, (c) silicon in, and (d) gold in IV. Mode decomposition Optical modes characterization is crucial for nanoparticles to define their heating properties. For instance, in Fig.4(b) in the main text, we use experimental and theoretical dark-field spectra for a silicon nanosphere to determine the diameter of the nanosphere as 26 nm. However, some modes are hardly visible in our optical scheme, e.g. a magnetic quadrupolar mode. In order to know all supported modes in the nanoresonator, we carry out an additional theoretical modes decomposition for this sphere by analizing the scattering coefficients got 5 2 from the Mie theory. The results are shown in Fig. S4, where the magnetic quadrupole is clearly visible near λ 7 nm. This mode, providing strong optical heating in our experiments, remains even for nanoparticle inside (Fig. S4b) or water(fig. S4c). 8

9 a b c scattering (arb. u.)..8.6 a.4 2 b 2 a.2 air b 2 b b a 2 a scattering (arb. u.) b 2 a b 3 3 a 2 b a scattering (arb. u.) a 3 a 2 b 2 a water b Figure S4: Scattering cross-section modes decomposition for spherical silicon nanoparticle with diameter 26 nm in air (a), (b), and water (c). a - electric dipolar mode, b - magnetic dipolar mode, a 2 - electric quadrupolar mode, b 2 - magnetic quadrupolar mode, a, a 2, b, b 2 - higher orders of magnetic and electric dipolar and quadrupolar modes V. Raman shift According to the Balkanski s theory [5], a Raman signal (spectral position and intensity) is temperature dependent. In Fig. S5 we show the dependence of thermal shift of Raman peak for silicon calculated by using Eq.(4) in the main text ΔΩ (cm -¹) ΔT (K) Figure S5: Theoretical dependence of Raman shift on temperature increase for silicon. 9

10 VI. Radiation pattern at magnetic quadrupole resonance a scattering (arb. u.) Dark-Field apperture 49⁰ ED MQ c NA =.42 k ED (785 nm) MQ (75 nm) E b scattering (arb. u.) Upper halfspace π ED MQ air Figure S6: Scattering of a single Si sphere on a substrate. Numerically calculated scattering spectra for (a) dark-field scheme (collection angle around 49 o ) and (b) for collection angle 8 o in upper half-space. (c) Numerically calculated radiation patterns at MQ (λ=75 nm) and ED (λ=785 nm) for a spherical silicon nanoparticle with diameter D=26 nm on a substrate In the main text of the manuscript, we show that the magnetic quadrupole (MQ) mode gives strong contribution into the absorption cross section. However, MQ is hardly visible in the scattering spectra measured in the dark-field experiment. This effect is related to the radiation pattern of silicon nanoparticle at MQ resonance. In Figs. S6a,b the simulations of the scattering spectra collected in the 49 degree aperture ( dark-field scheme ) and the upper half-space (8 degree), respectively, are presented. The magnetic quadrupole contribution almost vanishes in the spectrum, when the scattered light energy is collected in the aperture. This effect has simple explanation if one looks at the radiation pattern shown in Fig. S6c. At the incline incidence, the bigger part of the energy is not collected into the numerical aperture NA=.42. The redistribution of the magnetic (MD) and electric (ED) dipole resonances relative intensities in Figs. S6a,b is also related to radiation pattern of MD mode.