Strong magnetic response of submicron Silicon particles in the infrared

A. García-Etxarri; R. Gómez-Medina; L. S. Froufe-Pérez; C. López; L. Chantada; F. Scheffold; J. Aizpurua; M. Nieto-Vesperinas; J. J. Sáenz

doi:10.1364/OE.19.004815

1. Introduction

Electromagnetic scattering from nanometer-scale objects has long been a topic of great interest and relevance to fields from astrophysics or meteorology to biophysics and material science [1–6]. During the last decade nano-optics has developed itself as a very active field within the nanotechnology community. Much of it has to do with plasmon (propagating) based subwavelength optics and applications [7], the synthesis of negative-index optical metamaterials [8–10] or the design of optical antennas [11–13].

Magnetic effects, a key ingredient of relevant microwave applications, cannot be easily exploited in the optical range (visible or infrared) due to intrinsic natural limitations of optical materials. The quest for magnetic plasmons and magnetic resonant structures at optical frequencies [14] has been mainly focused on metallic structures with the unavoidable problems of losses and saturation effects inherent to metals in the optical range. Moreover, their size is often required to be comparable to, or a fraction of, the operating wavelength in order to provide a non-negligible response, and their magnetic response is strongly affected and influenced by an intrinsic quadrupolar contribution, which introduces additional radiation losses and impurity of radiation and polarization [15].

As an alternative approach, there is a growing interest in the theoretical and experimental study of high-permittivity dielectric objects as constitutive elements of new metamaterials [16–25], negatively refractive photonic crystals [26, 27] and antennas based on dielectric resonators [28–30]. Large field resonant enhancements in the gap between two dielectric silicon microdisks have been predicted [31] to be comparable to those founded between metal nanoparticles. Interesting magnetodielectric properties are usually associated to low loss and large dielectric constants which, being accessible at microwave and terahertz frequencies, are still a challenging issue in the infrared (IR) and visible frequency ranges [32]. However, it has recently been shown [25] that silicon rod arrays could exhibit a true metamaterial left handed dispersion branch in the visible to mid-IR despite the moderate value of the refractive index (m ∼ 3.5).

Motivated by these results, we have analyzed in detail the magnetodielectric properties of loss-less dielectric spheres made of moderate permittivity materials in the infrared regime. The scattering resonances of homogeneous spheres have been analyzed in the past [1–3]. Both electric and magnetic Mie resonances were studied for very large dielectric particles [33] and in smaller particles of very large refractive index, and at wavelengths being 100 times their radius [34]. We concentrate here on the relevance of the first magnetic dipolar resonance in non-absorbing dielectric nano-spheres (e.g. silicon spheres in the mid-IR). As we will see, submicron Silicon spheres (with radius a ≈ 200nm) can present a strong magnetic resonance response at telecom and infrared frequencies, (namely, wavelengths λ ≈ 2ma ≈ 1.5μm). Moreover, in this frequency range, and based on standard Mie theory [1], we have found that the scattering of these Si nanoparticles can be perfectly described by dipolar electric and magnetic fields; quadrupolar and higher order contributions being negligible. Interestingly, the magnetic resonance line remains well defined and resolved even for lower index materials, as Rutile-like TiO₂ with an average index m ≈ 2.45 in the near-IR, even though in this latter case the electric dipole resonance line strongly overlaps with the magnetic dipole one, as well as with those of higher order multipoles. Subsequently, as an illustration we estimate the consequence of this magnetic resonance to build composites of Si spheres with a substantial effective magnetic permeability. Thus such particles constituting important candidates to be considered as metamaterial elements.

2. Extinction resonances of a dielectric sphere. The Mie theory revisited

Consider a non-absorbing dielectric sphere of radius a, index of refraction m_p and dielectric permittivity $ɛ_{p} = m_{p}^{2}$ in an otherwise uniform medium with real relative permittivity ɛ_h, permeability μ_h = 1, and refractive index $m_{h} = \sqrt{ɛ_{h}}$ . The magnetic permittivity of the sphere and the surrounding medium is assumed to be 1. Under plane wave illumination, and assuming linearly polarized light, the incident wave is described by

E = E_{0} u_{Z} e^{i k X} e^{- i ω t}, B = B_{0} u_{Y} e^{i k X} e^{- i ω t}

where k = m_hω/c = m_h2π/λ, λ being the wavelength in vacuum and B₀ = μ₀H₀ = −(m_h/c)E₀ (see Fig. 1). The field scattered by the sphere can be decomposed into a multipole series, (the so-called Mie’s expansion), characterized by the electric and magnetic Mie coefficients {a_n} and {b_n}, respectively; (a₁ and b₁ being proportional to the electric and magnetic dipoles, a₂ and b₂ to the quadrupoles, and so on). We shall find useful to write the Mie coefficients in terms of the scattering phase-shifts α_n and β_n [1]

a_{n} = \frac{1}{2} (1 - e^{- 2 i α_{n}}) = i sin α_{n} e^{- i α_{n}}

b_{n} = \frac{1}{2} (1 - e^{- 2 i β_{n}}) = i sin β_{n} e^{- i β_{n}}

where

tan α_{n} = \frac{m^{2} j_{n} (y) {[x j_{n} (x)]}^{'} - j_{n} (x) {[y j_{n} (y)]}^{'}}{m^{2} j_{n} (y) {[x y_{n} (x)]}^{'} - y_{n} (x) {[y j_{n} (y)]}^{'}},

tan β_{n} = \frac{j_{n} (y) {[x j_{n} (x)]}^{'} - j_{n} (x) {[y j_{n} (y)]}^{'}}{j_{n} (y) {[x y_{n} (x)]}^{'} - y_{n} (x) {[y j_{n} (y)]}^{'}},

m = m_p/m_h being the relative refractive index, x = ka the size parameter and y = mx. j_n(x) and y_n(x) stands for the spherical Bessel and Neumann functions, respectively, and the primes indicate differentiation with respect to the argument. In absence of absorption, i.e. for real m, the phase angles α_n and β_n are real, then the extinction and scattering cross sections, σ_ext and σ_S, respectively, have the common value [1]

σ_{S} = σ_{ext} = \frac{2 π}{k^{2}} \sum_{n = 1}^{\infty} (2 n + 1) {{sin}^{2} α_{n} + {sin}^{2} β_{n}} = \sum_{n = 1}^{\infty} {σ_{E, n} + σ_{M, n}}

Fig. 1 Incident field vector.

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In the small particle limit (x ≪ 1) and large particle permittivities (m ≫ 1) the extinction cross section presents characteristic sharp resonance peaks. The values of y = mx at which the angles α_n or β_n are π/2, 3π/2, . . ., etc, define the resonance points. At each resonance, the extinction cross section is of the order of λ² and it is independent of either the particle size or the refractive index [1]

σ_{S}^{res} = {σ_{ext}^{res} |}_{{x ≪ 1; m ≫ 1}} \approx \frac{2 π}{k^{2}} (2 n + 1)

Asymptotically, the first resonance peak occurs at y = π (i.e. λ = m2a) corresponding to the magnetic dipole term of coefficient b₁.

Interesting consequences and applications of well defined Mie resonance lines, associated to low loss and large dielectric constants, are accessible for different materials at microwave and terahertz frequencies. However, as m decreases there is an increasing overlap between the wavelength dependent cross-section peaks, and the sphere resonant response smears out. Since usually non-absorbing materials present low refractive index in the infrared (IR) and visible frequency ranges, Mie resonances of small particles in these regimes have not been considered in detail. However, the above analysis on the cross-section, carried out for Si nanoparticles in the IR, indicate that the aforementioned asymptotic behavior can be extended to a relative refractive index as low as m ≈ 3.

In Fig. 2 we show the scattering cross section map of a non-absorbing Mie sphere as a function of the refractive index m and the y parameter, y = m(2πa/λ). Brightness in the color-map is proportional to σ_ext while color code represents the contribution of electric and magnetic dipoles to the total cross section. The RGB (Red, Green, Blue) code is formed by taking RGB = σ_E,1R + σ_M,1G + σ_resB with $σ_{res} \equiv \sqrt{σ_{ext}^{2} - σ_{E, 1}^{2} - σ_{M, 1}^{2}}$ . The whole map is normalized to avoid over-saturation. Hence, green areas correspond to parameter ranges where the magnetic dipole contribution dominates the total scattering cross section, while red areas represent regions where the electric dipole contribution is dominating. The remaining blue-saturated areas are dominated by higher order multipoles.

Fig. 2 Scattering cross section map of a non-absorbing Mie sphere as a function of the refractive index m and the y parameter, y = mka = m(2πa/λ). Green areas correspond to parameter ranges where the magnetic dipole contribution dominates the total scattering cross section, while red areas represent regions where the electric dipole contribution is dominating. The remaining blue-saturated areas are dominated by higher order multipoles. Brightness in the color-map is proportional to the total cross section. White horizontal lines represent the y-range covered by figures 3 (m = 3.5) and 5 (m = 2.45).

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In the micrometer wavelength regime, within the transparent region of silicon (λ ≳ 1100 nm), the index of refraction can well be approximated by a real constant $m_{p} \approx \sqrt{12} \sim 3.5$ (see for example [35]). The scattering (or extinction) cross section of a Si sphere of radius a = 230nm in vacuum (m_h = 1) is plotted in Fig. 3. Although there is a partial overlap between the first dipolar peaks, the magnetic line of the first resonant peak (at λ ≈ 2ma) is still very well resolved. Interestingly, for wavelengths larger than λ ≈ 1200nm, the cross section is completely determined by the first b₁ and a₁ coefficients. In other words, in this regime Si particles can be treated as non-Rayleigh dipolar particles.

Fig. 3 Scattering cross-section σ_S versus the wavelength λ for a 230nm Si sphere (the refraction index m = 3.5 is constant and real in this wavelength range). The contribution of each term in the Mie expansion is also shown. The green line corresponds to the magnetic dipole contribution.

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3. Magnetic and electric resonances of dipolar particles

Let us consider further in some detail the scattering properties of small dielectric particles (x ≪ 1). In the x ≪ 1 limit, the sphere is sufficiently small so that only the dipole scattered fields are excited. The induced dipole moments, proportional to to the external (polarizing) fields, E and B, are usually written in terms of the particle electric and magnetic polarizabilities α_E and α_M, respectively :

p = ɛ_{0} ɛ_{h} α_{E} E, m = \frac{1}{μ_{0} μ_{h}} α_{M} B;

where

α_{E} = i {(\frac{k^{3}}{6 π})}^{- 1} a_{1}, α_{M} = i {(\frac{k^{3}}{6 π})}^{- 1} b_{1} .

By using the definition of the phase angles α₁ and β₁, [cf. Eqs. (2) – (5)], we can rewrite the polarizabilities as

α_{E} = \frac{α_{E}^{(0)}}{1 - i \frac{k^{3}}{6 π} α_{E}^{(0)}}, α_{M} = \frac{α_{M}^{(0)}}{1 - i \frac{k^{3}}{6 π} α_{M}^{(0)}},

where

α_{E}^{(0)} = - \frac{6 π}{k^{3}} tan α_{1}, α_{M}^{(0)} = - \frac{6 π}{k^{3}} tan β_{1} .

In terms of these magnitudes, the extinction and scattering cross sections then read

σ_{ext} = k Im {α_{E} + α_{M}}

σ_{S} = \frac{k^{4}}{6 π} {{| α_{E} |}^{2} + {| α_{M} |}^{2}}

Of course, in absence of absorption,

α_{E}^{(0)}

and

α_{M}^{(0)}

are real quantities and we recover the well known optical theorem result σ_ext = σ_S.

In particular, in the Rayleigh limit, when y = 2πma/λ ≪ 1, $α_{E}^{(0)}$ and $α_{M}^{(0)}$ approach the quasi-static form,

{α_{E}^{(0)} |}_{y ≪ 1} \approx 4 π a^{3} \frac{m^{2} - 1}{m^{2} + 2}, {α_{M}^{(0)} |}_{y ≪ 1} \approx 4 π a^{3} (m^{2} - 1) \frac{k^{2} a^{3}}{30}

From Eq. (14), together with Eq. (10), we recover the well known expression for the polarizability of a Rayleigh particle including radiative corrections [6, 36, 37]. Notice that in this limit the magnetic polarizability is negligible. Very small particles always behave as point electric dipoles.

However, for particle sizes that are not extremely subwavelength, the quasi-static approach fails to describe the resonant behavior. As y increases, (i.e. as λ decreases), there is a crossover from electric to magnetic behavior as shown by Fig. 3 for a Si particle. The y values at which the quasi-static polarizability $α_{E}^{(0)}$ ( $α_{M}^{(0)}$ ) diverges define the electric (magnetic) dipolar resonances. Near the first b₁-resonance, the particle essentially behaves like a magnetic dipole (cf. Fig. 2). If λ decreases further, a₁ peak dominates and the sphere becomes again an electric dipole. Notice however that, due to the overlap between the electric and magnetic responses, the radiation field near the resonances does not correspond to a fully pure electric or a fully pure magnetic dipolar excitation. Yet, as seen in Fig.2 for the aforementioned Si sphere, the magnetic dipole contribution is, at its peak, about five times larger than the electric dipole one.

As a further example, we have plotted in Fig. 4 (left and central panels) both the near field amplitude maps in the XZ plane crossing the origin of coordinates (y = 0nm) and far field radiation patterns at the resonant peaks of a₁ and b₁, (cf. Fig. 3), numerically calculated from the full Mie solution (and not only from the dipolar approximation). The magnetic and electric nature of both resonances is clearly supported by these near and far field maps. For λ = 1250nm (left column), the electric field map shows the typical two-lobe distribution corresponding to an electric dipole oriented along the z axis. The far-field radiation pattern also presents the characteristic donut-like shape of a dipole oriented along the z-axis with a slight distortion originated by the weaker contribution of the magnetic dipole. For λ = 1680nm, the circular distribution of the electric field enhancement inside the sphere (mid-top panel) is a clear signature of the magnetic dipole oriented along the direction normal to the plane (y direction). The corresponding far field radiation pattern (mid-bottom panel) is also consistent. Cross sections of the near fields on the XZ and XY planes through the origin of coordinates are included as supplementary material ( Media 1) and ( Media 2).

Fig. 4 Maps for the modulus of the total electric an magnetic vectors normalized to the incoming electric and magnetic field respectively (E_tot/E_inc and H_tot/H_inc), for a Si nanoparticle of radius a = 230nm under plane wave illumination, (cf. Fig. 1). XZ planes crossing y = 0 are displayed. The left and central panels correspond to λ = 1250nm and λ = 1680nm of the electric and magnetic resonance peaks of Fig. 3, respectively. The right panel corresponds to the situation where the electric and magnetic resonances contribute equally to the scattering cross section (λ = 1525nm). The corresponding far field scattering radiation patterns for the three wavelengths are shown in the bottom row. The corresponding maps for YZ and XZ planes can be seen in the suplementary figure files ( Media 1) and ( Media 2), respectively.

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It is interesting to notice that there is no restriction for the magnetic field enhancement of an electric dipole to be substantially large, mainly inside the particle (mid-left plot in Fig. 4). Equally, the electric field enhancement in and around the magnetic dipole can present large values (top-center plot in Fig. 4).

For a general wavelength, different modes contribute to the optical response of the system. We show this effect if the right panel of Fig. 4 for the special case where both the electric and magnetic resonances contribute equally to the scattering cross section of the Si nanosphere (λ = 1525nm, cf. Fig. 3). The near field patterns present mixed features of both electric and magnetic dipoles, and so does the far field with an interesting omnidirectional emission pattern as a result of the sum of both coherent contributions.

Silicon nanoparticles with radius of the order of 200–250nm are then fully characterized by electric and magnetic polarizabilities as given by Eqs. 10 in the mid and near-IR region. By tuning the appropriate resonance wavelength, Si particles exhibit a strong magnetic response, (only slightly perturbed by their electric dipole counterpart).

It should be remarked that if the relative refractive index is smaller, the overlap between the Mie resonance lines increases and, as it can be seen in Fig. 2, the first electric resonance smeared out for a refraction index lower than m ≲2.5. Interestingly, the magnetic resonance line remains well defined and resolved even for lower index materials. As an example, in Fig. 5 we represent the scattering cross section of particle with $m = \sqrt{6} \approx 2.45$ which would correspond to the averaged refractive index of TiO₂ (rutile) in the near IR [35]. For comparison purposes, in order to cover the same y-range (y = mka) as in Fig. 3, we consider a larger sphere radius of a = 330nm. As seen in this figure, beyond 1400nm the scattering is still described by the sum of the electric and magnetic dipoles. While the magnetic resonance is still visible, the peak corresponding to the electric dipole, a₁, now disappears due to the strong overlap with the magnetic dipole one, as well as with those of higher order multipoles.

Fig. 5 Scattering cross-section σ_S versus the wavelength λ for a sphere of radius a = 330nm (its relative refractive index is $m = \sqrt{6} \approx 2.45$ , constant and real in this wavelength range). The contribution of each coefficient in the Mie expansion is also shown. The green peak at longer wavelengths corresponds to the magnetic dipole contribution.

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4. Composite of magnetodielectric Si particles

Among other possible applications, the strong dipolar magnetic resonance of Silicon submicron spheres endows them the potential of being exploited as elements of negative index materials, or in general of magnetodielectric metamaterials. Most of the current published work about magnetic resonances and dielectric metamaterials concentrate on dielectric spheres with very large relative permittivities (larger than ≈ 100) [17–21]. In the infrared spectral range, such high permittivities can be achieved by using materials with infrared polariton resonances which however present a relatively large absorption. Therefore, the non-absorbing dipolar response, both electric and magnetic of Si submicron spheres is of interest.

In practice, determining the effective constitutive parameters of such systems is a difficult problem, that can be solved only under (sometimes severe) approximations [38–40]. Lorentz-Lorenz theory is usually used to describe the effective response of composites made of an arbitrary arrangement of dipolar particles. Within this approach, if the composite is made of ρ spherical particles per unit volume, the effective permittivity ɛ_eff and permeability μ_eff are simply given by the Clausius-Mosotti equation [18, 21, 38]

\frac{ɛ_{eff} - 1}{ɛ_{eff} + 2} = f \frac{α_{E}}{4 π a^{3}}; \frac{μ_{eff} - 1}{μ_{eff} + 2} = f \frac{α_{M}}{4 π a^{3}}

where f = 4πa³ρ/3 is the filling fraction. This approach neglects positional correlations between particles and multiple scattering effects that could be relevant at relatively large filling fractions [40]. However, Eq. 15 would give a first guess on the relevance of the observed magnetic resonances in possible physical applications.

In order to obtain a preliminary estimation of the role that the Si spheres, addressed in Fig. 3, may have as building blocks of a continuum medium, or metamaterial, we plot in Fig. 6 the resulting values of the effective permittivity and permeability of an arbitrary arrangement of Si spheres for filling fractions f = 0.25 (a) and f = 0.5 (b) respectively. The free space wavelength at the magnetic resonance is about 4 times the diameter of the spheres. As it can be seen, the effective medium presents a remarkable dispersion in the effective permeability which is pivotal for most of the applications. For instance, one can play with these values of the effective constitutive parameters at chosen wavelengths to obtain real and imaginary parts of the product ɛ_effμ_eff that are adequately situated in the fourth quadrant of the complex plane to yield, in the selected branch, a convenient effective refractive index $m_{eff} = \sqrt{ɛ_{eff} μ_{eff}}$ , either with negative real part or not [19].

Fig. 6 Effective real and imaginary permittivities and permeabilities for an arbitrary arrangement of Si spheres in an otherwise homogeneous medium with ɛ_h = μ_h = 1 for two different filling factors f = 0.25 (a) and f = 0.5 (b).

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We should stress the absence of absorption (i.e. heating) in the system. Notice, however, that the effective constants are complex even in the case of a non-absorbing medium. Their imaginary parts being associated with the extinction of the propagating beam due to scattering (see, for example, Sec. 4.3 in Ref. [1] and [19, 41]).

5. Conclusion

We have analyzed the dipolar electric and magnetic response of lossless dielectric spheres made of both low and moderate permittivity materials. Based on the Mie expansion, we have derived general expressions for the electric and magnetic polarizabilities of dielectric spheres. We found that submicron Silicon particles present a strong magnetic resonant response in the mid-near infrared (wavelengths ≈ 1.2 – 2μm). Interestingly, the light scattered by these Si nanoparticles of appropriate size is perfectly described by dipolar electric and magnetic fields, being quadrupolar and higher order contributions negligible in this frequency range. A preliminary estimation of the effective optical constants of an arbitrary arrangement of Si spheres suggest a remarkable dispersion in the effective permeability which is pivotal for most of the applications. These results can play an important role, not only in the field of metamaterials or optical antennas, but also in tailoring the light transport through complex dielectric media like photonic glasses [42, 43] with intriguing magnetic properties. Fabrication of new materials made of highly monodisperse subwavelength silicon spheres [44] may then lead to a new generation of magnetodielectric optical materials. At the magnetic or electric resonance wavelengths the extinction cross section is of the order of λ² reaching its maximum theoretical limit (independent of the particle size or refractive index) [1] (see also [30]). The large dipolar cross-section of magnetodielectric spheres near a magnetic resonance would also imply strong radiation pressure magnetic forces [37, 45, 46] leading to new concepts related to the optical forces on magnetic particles.

Acknowledgments

We appreciate interesting discussions with S. Albaladejo, B. Garcia-Camara, F. González, L. Inclán, F. Moreno, J.M. Saiz, I. Suárez and K. Vynck. This work has been supported by the EU NMP3-SL-2008-214107-Nanomagma and NoE Nanophotonics4Energy 248855, the Spanish MICINN Consolider NanoLight (CSD2007-00046), MAT2009-07841 GLUSFA and FIS2009-13430-C01 and C02, as well as by the Comunidad de Madrid Microseres-CM (S2009/TIC-1476) and PHAMA-CM (S2009/MAT-1756). F.S. and L.C. acknowledge financial support from the Swiss National Science Foundation, Projects No. 126772 and 117762. Work by R.G.-M. and L.S.F.-P. was supported by the MICINN “Juan de la Cierva” Fellowship.

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Strong magnetic response of submicron Silicon particles in the infrared

Abstract

1. Introduction

2. Extinction resonances of a dielectric sphere. The Mie theory revisited

3. Magnetic and electric resonances of dipolar particles

4. Composite of magnetodielectric Si particles

5. Conclusion

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (6)

Equations (15)

Optics Express