Supplementary Figure S1 SEM and optical images of Si 0.6 H 0.4 colloids. a, SEM image of Si 0.6 H 0.4 colloids. b, The size distribution of Si 0.

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1 Supplementary Figure S1 SEM and optical images of Si 0.6 H 0.4 colloids. a, SEM image of Si 0.6 H 0.4 colloids. b, The size distribution of Si 0.6 H 0.4 colloids. The standard derivation is 4.4 %.

2 Supplementary Figure S2 The optical properties of individual Si 0.75 H 0.25 silicon colloid before and after annealing. a, Transmission spectra of different colloids before (black line) and after (red line) annealing at the indicated temperature. The optical images of the corresponding colloid before (black framed image) and after (red framed image) the annealing process are shown in the upper framed inset of figure. The SEM image of the corresponding annealed colloid is also shown in the lower inset of figure. b and c, The same as a for different colloids. In all cases the scale bar of SEM images is 200 nm.

3 Supplementary Figure S3 The optical properties of individual Si 0.6 H 0.4 silicon colloid before and after annealing. a, Transmission spectra of different colloids before (black line) and after (red line) annealing at the indicated temperature. Optical images of the corresponding colloid before (black framed image) and after (red framed image) annealing are shown in the upper framed inset of figure. SEM images of the corresponding annealed colloid are also shown in the lower insets of the figures. b and c, The same as a for different colloids. In all cases the scale bar of SEM images is 200 nm.

4 Supplementary Figure S4 The influence of the annealing process on the particle size and refractive index of silicon colloids. a, The diameter (black line) and the refractive index (pink line) of the fabricated silicon colloids as a function of annealing temperature for the Si 0.75 H 0.25 colloids case. b, The same as a, for the Si 0.6 H 0.4 colloids case. The diameters of colloids are obtained from the SEM images of the Supplementary Fig. S2 and S3. The refractive index is obtained by fitting the Mie theory calculation to the experimental data, as shown in the Supplementary Fig. S5.

5 Supplementary Figure S5 The Mie theory fitting to the experiments. a, Transmission spectra of three different Si 0.75 H 0.25 colloids (yellow, green and red) after annealing at the temperature shown in the inset. The calculated extinction (black line) and scattering (orange line) efficiencies of single annealed Si 0.75 H 0.25 colloids are shown. The size and refractive index values of colloids are shown in the supplementary Fig. S4. The contribution of the electric (blue dash line) and the magnetic (blue solid line) resonances are also shown. The Mie resonances are highlighted with orange (magnetic dipole), grey (electric dipole) and blue (magnetic quadrupole) arrows. b, The same as a, but for Si 0.6 H 0.4 colloids.

Supplementary Figure S6 The optical properties of Si 0.75 H 0.25 colloids suspension without annealing process. a, Transmission spectrum of a suspension with a concentration of 0.005% v/v.

6 Supplementary Figure S6 The optical properties of Si 0.75 H 0.25 colloids suspension without annealing process. a, Transmission spectrum of a suspension with a concentration of 0.005% v/v. For comparison, the calculated extinction (black) and scattering (orange) efficiency of single Si 0.75 H 0.25 colloid immersed in toluene are also shown. A particle size of 430 nm with a refractive index value 70% of that of bulk silicon has been assumed (see the Supplementary Fig. S4). The contribution of electric like (blue dash line) and magnetic like (blue solid line) Mie resonances are also shown. b, The transmission spectrum of the suspension with a concentration value of 0.01% v/v. c, The transmission spectrum of the suspension with a concentration value of 0.02% v/v. The photos in the right panels show the corresponding samples that were measured. Because of the low refractive index values of original Si 0.75 H 0.25 colloids, the magnetic like Mie resonance dramatically weakens when dispersed in toluene

7 Supplementary Figure S7 The physical origin of the magnetic response of silicon colloids in the optical range. a, Schematic view of the light impinging a metallic split ring metamaterial building block (upper panel), corresponding extinction (black line) and scattering (red line) efficiency spectra with the magnetic resonance peak highlighted by an orange arrow (middle panel), and current intensity & magnetic field map plots at the magnetic resonance wavelength (lower panel). The field plot map corresponds to the xy plane inside the split ring structure. b, the same as a but for the silicon nanocavity. The cut plane of the displacement current density and the magnetic field distribution correspond to the cross section of the structure in the xz plane. The inset figure in the middle panel of b shows the extinction and scattering efficiency of a single polystyrene (PS) colloid with the same size of the silicon colloid.

8 Supplementary Figure S8 The magnetic and electric polarizabilities, as well as the scattering pattern obtained from Mie theory. a, The magnetic (red) and electric (blue) polarizabilities as a function of the light wavelength. The solid and dash lines correspond to the real and the imaginary parts of the polarizabilities. The upper panel corresponds to the silicon colloids case and the lower panel corresponds to the PS colloids case. The size of the silicon and PS colloids is the same as that of the Supplementary Fig. S7. b, Scattering angular patterns of silicon (upper panel) and PS (lower panel) colloids under magnetic dipole resonances (the corresponding wavelength is indicated by a dashed line in a. The incident light polarization is shown in the Supplementary Fig. S7b.

9 Supplementary Note 1 The optical properties of individual fabricated silicon colloids before and after annealing. The results presented in the Supplementary Fig. S2 and Fig. S3 are obtained from 24 silicon colloids. The results presented in the Supplementary Fig. S2 are obtained from the optical spectra of 12 different Si 0.75 H 0.25 silicon colloids. All colloids show similar optical features that are equally red-shifted upon the annealing process. From the comparison to the Mie theory we can estimate the refractive index of the annealed samples. Since all different optical resonances appear at the same wavelength values, we can conclude silicon colloids are monodisperse and the 600 ºC annealed colloids have a refractive index value of 90% of that of bulk silicon (see the Supplementary Figs. S4, S5 and Note 2 below). The Supplementary Fig. S3 shows optical experiments for the case of 12 different Si 0.6 H 0.4 colloids. In this case, the annealed colloids achieve a refractive index value of 75% of that of bulk silicon. Supplementary Note 2 The influence of annealing on the size and refractive index of silicon colloids. From SEM images of colloids (Figure 1, the Supplementary Fig. S2 and S3) we can obtain their diameter and from the fit of optical experiments to the Mie theory we achieved their refractive index values. In the calculation of Mie theory, the refractive index of pure silicon, n Si is taken from Ref. [42]. We assumed the refractive index of hydrogenated silicon colloids, n a Si: H p. nsi, where p is the ratio between the refractive index values of hydrogenated and the bulk silicon colloids. The fitting results for both Si 0.75 H 0.25 and Si 0.6 H 0.4 samples after different annealing temperature are shown in the Supplementary Fig. S4. As expected, by increasing the annealing temperature, the refractive index increases. For the 300 o C annealing case, the refractive index of Si 0.75 H 0.25 samples is 70% of that of bulk silicon. However, after submitting the samples to 600 o C annealing process the refractive index of colloids outstandingly increase to 90% of that of bulk silicon. With this high refractive index, a-si:h colloids are not only able to have good optical performance in air but also in a liquid environment case such as water [26] or toluene. For Si 0.6 H 0.4 particles, because of the high hydrogen content, the refractive index is much lower than that of Si 0.75 H 0.25 samples at any annealing temperature. Even after annealing at 600 o C, the refractive index is 74% of that of bulk silicon, equivalent to a refractive index value of 2.6, which is not enough for sustaining a huge magnetic response in a liquid media. The Supplementary Fig. S5 shows the Mie theory calculation results.

10 Supplementary Discussion The physical origin of the magnetic response of silicon colloids in the optical range. Silicon is not a magnetic material. However, due to both, the huge refractive index value, and the small particle size, silicon nanocavities show strong magnetic resonances at optical frequencies. The underlying physics is similar to that of noble metal based metamaterials. In the Supplementary Fig. S7, we compare the properties of silicon colloids with a typical metamaterial building block reported in the literature [5-7, 44-46]. The upper panel of the Supplementary Fig. S7a, shows a schematic view of the well known split ring resonator made of gold. This structure has extensively been reported because of its strong magnetic response in the optical frequency region. The LC oscillator has widely been used to mimic the behavior of the split ring structure [5-7, 44-46]. It is well known that the LC oscillator circuit has a well-defined resonance frequency, which scales inversely with the size of the nanostructure. When the frequency of the incident light equals to the oscillator resonance frequency and the electric component of the incident light is parallel to the direction of the capacitor (corresponding to the metallic gap region of the split ring structure), an oscillating current is induced. Then, the induced current produces an oscillating magnetic dipole perpendicular to the split ring plane, along the z direction. The middle panel of the Supplementary Fig. S7a shows both the scattering and extinction efficiency of a single gold split ring. Although a detailed discussion can be found in Ref. [45], we will summarize the optical properties of the split ring resonator. Two strong scattering and extinction peaks appear in the wavelength region of interest. The longer wavelength peak corresponds to the magnetic dipole resonance and the shorter wavelength one corresponds to the electric dipole resonance. The origin of the electric dipole resonance is the same as that appearing for metallic nanoparticles [45]. For the magnetic dipole resonance, the lower panel of the Supplementary Fig. S7a shows the induced oscillating current, J (left), and the magnetic field (right) distribution at the resonant wavelength (1339 nm). Just as what we should expect, a strong oscillating current loop is observed at the split ring, this inducing an oscillating magnetic dipole along the z direction and a local magnetic field enhancement observed in the center part of the split ring structure. It is important pointing out the extinction coefficient is much larger than the scattering coefficient this being a test of the high optical absorption of the gold nanostructures in the spectral zone of interest. After understanding the split ring case we can discuss the magnetic response of the silicon colloids. Similarly to the induced current appearing for conductive materials, displacement currents appear for the case of dielectric systems. Based on the classical electromagnetic field theory, the displacement current, J D, can be written as, J D = D t = iωd = ( iωε)e, which is proportional to the dielectric permittivity, ε, of silicon. If the permittivity (or refractive index) is big enough, the displacement

11 current increases, this producing a strong magnetic dipole inside the dielectric colloid (see lower panel of the Supplementary Fig. S7b) [15, 16, 22, 23, 26, 27, 30-32, 47-49]. The loop shape of the displacement current is crucial for the magnetic response of dielectric materials and is analogous to the loop current in the split ring resonator. The size of the dielectric structure limits the resonance frequency of the magnetic response [22, 23, 26, 27, 30-32]. Actually, this strong magnetic response already was predicted by the Mie scattering theory. The magnetic dipole resonance above explained corresponds to the lowest order Mie resonance of a high refractive index spherical nanocavity [15, 22, 23, 26, 27, 30-32, 49]. It means that the scattered light near this magnetic dipole resonance follows the magnetic dipole radiation pattern (details discussed in the Supplementary Fig. S8b). We have used the Mie scattering theory to show the magnetic response and also to explain the optical features of silicon colloids. More detailed discussions of the magnetic response of high refractive index structures, and its connection to the Mie scattering, have extensively been reported [15, 16, 22, 23, 26, 27, 30-32, 47-49]. In the Supplementary Fig. S7b we show the calculated scattering and the extinction spectra of a single silicon colloid, similar to those used in the experiments (see Figure 1e of the main body of the paper). In the Inset of the Supplementary Fig. S7b, we also show the scattering and extinction spectra of a polystyrene (PS) colloid with the same size as that of the silicon colloid. The PS particle does not show Mie resonances in the wavelength region of interest due to the small refractive index of the particle. Therefore we can conclude that high refractive index nanocavities show strong magnetic resonances similar to the resonant modes of gold metamaterial building blocks. However, there are some differences between gold and silicon nanostructures we describe next. Firstly, at variance to metallic nanostructures, we are dealing with dielectric nanoparticles with no free electrons and therefore no plasmons, in which the magnetic response comes from the optical resonance of the spherical nanocavity. Also, the silicon colloid shows several scattering peaks corresponding to higher order Mie resonances (see the Supplementary Fig. S7b), for example electric dipole, magnetic quadropole resonances etc. Secondly, the optical losses of silicon in the wavelength region of interest (see the Supplementary Fig. S7) are much smaller than those appearing for metallic nanostructures. The absorption coefficient of bulk gold is four to five orders of magnitude larger than that for bulk silicon in the spectral region of interest. This is pointed out in the difference between the extinction and the scattering coefficients. By subtracting the scattering spectrum from the extinction one we can derive the absorption spectrum. In the case of the silicon colloid, there are no obvious differences between both spectra near the magnetic resonance (see middle panel of the Supplementary Fig. S7b), and therefore the absorption of silicon colloids, as expected, is very small. However this is not the case for the metallic nanostructure as it can be seen from the middle panel of the Supplementary Fig. S7a. Finally, to better describe the magnetic response of silicon colloids, the magnetic and electric dipole polarizabilities α m and α e of a single fabricated silicon colloid are calculated by using the Mie theory [22, 25, 27]. The magnetic and electric dipole polarizabilities can be written as = and =, where P is the electric

12 dipole moment and M is the magnetic dipole moment. In the Supplementary Fig. S8a, the red and blue lines correspond to the magnetic and electric dipole polarizabilities respectively. Also, the solid and the dash lines represent the real and imaginary part of the polarizabilities respectively. For low refractive index materials like PS colloids (lower panel), the magnetic dipole polarizability, α m, shows pretty low values, these being smaller than those of α e, as it can be expected for standard dielectric materials. However for the Si colloid case (upper panel), α m shows large values, and for certain wavelength values corresponding to the magnetic resonance around 1256 nm, α m is larger than α e (see vertical dashed line in the Supplementary Fig. S8a). The Supplementary Fig. S8b compares the scattering pattern in the x-y plane of the silicon colloid here reported (upper panel) and a PS colloid of the same size (lower panel) at 1256 nm (black dash vertical line in the Supplementary Fig. S8a). The electric and the magnetic field components of the incident light are along x and y directions respectively (as shown in the Supplementary Fig. S7b). A huge difference between the scattering pattern of Si and PS colloids can be seen. The scattered light along the y direction is minimum for the Si colloid but maximum for the PS colloid. As radiating dipoles emits in the perpendicular direction to the dipole direction [50], we can see that the scattering pattern of the PS particle corresponds to an electric dipole, but it turns out to be a magnetic one for the case of the silicon colloid (see the Supplementary Fig. S8b). All the above results demonstrate that the silicon colloid has a magnetic response in the optical region of interest, and it can be a building block for metamaterials. Supplementary References [44] Yen T. J., Padilla W. J., Fang N., Vier D. C., Smith D. R., Pendry J. B., Basov D. N. and Zhang X. Terahertz magnetic response from artificial materials, Science 303, 1494 (2004). [45] Enkrich C., Wegener M., Linden S., Burger S., Zschiedrich L., Schimidt F., Zhou J. F., Koschny Th. and Soukoulis C. M. Magnetic metamaterials at telecommunication and visible frequencies, Phy. Rev. Lett. 95, (2005). [46] Boudarham G., Feth N., Myroshnychenko V., Linden S., de Abajo J. G., Wegener M. and Kociak M. Spectral imaging of individual split-ring resonators, Phys. Rev. Lett. 105, (2010). [47] Schuller J. A., Zia R., Taubner T. and Brongersma M. L. Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles. Phys. Rev. Lett. 99, (2007). [48] Popa B. I. and Cummer S. A. Compact dielectric particles as a building block for low loss magnetic metamaterials, Phys. Rev. Lett. 100, (2008). [49] Ginn J. C., Brener I., Peters D. W., Wendt J. R., Stevens J. O., Hines P. F., Basilio L. I., Warne L. K., Ihlefeld J. F., Clem P. G. and Sinclair M. B. Realizing optical magnetism from dielectric metamaterials, Phys. Rev. Lett. 108, (2012). [50] Jackson J. D. Classical Electrodynamics (John Wiley & Sons, Inc, 1962).

Negative refractive index response of weakly and strongly coupled optical metamaterials.

$Negative refractive index response of weakly and strongly coupled optical metamaterials.$ Negative refractive index response of weakly and strongly coupled optical metamaterials. Jiangfeng Zhou, 1 Thomas Koschny, 1, Maria Kafesaki, and Costas M. Soukoulis 1, 1 Ames Laboratory and Department