Symmetry Cancellations in the Quadratic Hyperpolarizability of Non

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Symmetry Cancellations in the Quadratic Hyperpolarizability of Non-Centrosymmetric Gold Decahedra Isabelle Russier-Antoine,*,† Julien Duboisset,† Guillaume Bachelier,† Emmanuel Benichou,† Christian Jonin,† Natalia Del Fatti,† Fabrice Vall ee,† Ana S anchez-Iglesias,‡ ‡ ‡ Isabel Pastoriza-Santos, Luis M. Liz-Marzan, and Pierre-Franc- ois Brevet† †

Laboratoire de Spectrom etrie Ionique et Mol eculaire, UMR CNRS 5579, Universit e Claude Bernard Lyon 1,43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France, and ‡Departamento de Quimica Fisica and Unidad Asociada CISC, Universidade de Vigo, 36310 Vigo, Spain

ABSTRACT We report the second harmonic response of non-centrosymmetric gold decahedra with four different sizes, (side length between 17 and 150 nm). We show that, for small sizes, symmetry cancellations occur, leading to quadratic hyperpolarizabilities similar to that of gold spherical nanoparticles despite the non-centrosymmetric shape of the particles. For larger sizes, the quadratic hyperpolarizabilities scale with the volume of the particles, as expected from the multipolar contribution arising from the field retardation effects. SECTION Nanoparticles and Nanostructures

S

the cross-section of the conversion process. Large magnitudes for the quadratic hyperpolarizability tensor of silver and gold spherical nanoparticles with different diameters have therefore been reported by hyper Rayleigh scattering (HRS), also known as incoherent SHG.8-14 The symmetry of the crystalline structure of the material and of the shape of the particles plays an essential role in determining the nature and the intensity of the SH response. No electric dipole SH response can occur from the volume of the particles made from centrosymmetric materials. In the small size regime, their quadratic hyperpolarizability is therefore determined principally by their shape. For small size gold nanospheres, the quadratic hyperpolarizability has been found to arise from the surface of the particle and to scale with the surface area of the particle, the true shape of the particles deviating from that of perfect centrosymmetric spheres. For larger sizes, typically above a diameter of about 40 nm, a volume dependence has been observed due to multipolar effects or field retardation.11,15 A similar behavior for silver nanoprisms has been reported.16 It is the purpose of this paper to further investigate the role of the symmetry of the shape of metallic nanoparticles by determining their SH response. Non-centrosymmetric gold decahedra nanoparticles with different sizes were therefore prepared and investigated by HRS. Our results show that a careful account of the morphology of the particles must be considered. Although the gold decahedra present a non-centrosymmetric shape, their quadratic hyperpolarizabilities obey strong symmetry

mall metallic particles with nanometer dimensions have received a great deal of interest over the past years in many different domains, from chemistry to physics and biology, owing to their peculiar optical and electrical properties.1-4 The synthesis of nanoparticles with a tight control on their size and shape is a requirement for the fabrication of new devices, these often being the main parameters determining the properties of materials at the nanoscale. The unique properties of small coinage metal particles such as gold and silver indeed arise from the collective excitation of the conduction band electrons, which is known as surface plasmon resonance (SPR). This resonance frequency is strongly sensitive to the size and the shape of the particles.4 While spheres, rods, cubes, and flat prisms are the most thoroughly studied shapes, several reports have been published dealing with the synthesis and the detailed characterization of other geometries, such as decahedra, as new targets for applications.5 The properties of these particles are routinely investigated by optical methods, principally linear optical methods such as UV-visible absorption spectroscopy, but more recently nonlinear optical methods have been proven to provide important additional information. Among these, second harmonic generation (SHG), i.e., the conversion of two photons at a fundamental frequency into one photon at the harmonic frequency, has been the focus of many studies. In this case, the SPR excitation can be obtained either at the fundamental or the harmonic frequency, yielding large intensities and greater versatility.6,7 Although the SPR excitation at the fundamental frequency is usually avoided in order to preserve the colloidal sample from degradation, the SPR enhancement at the harmonic frequency provides an important tool to obtain large intensities despite the weak value of

r 2010 American Chemical Society

Received Date: January 7, 2010 Accepted Date: February 11, 2010 Published on Web Date: February 15, 2010

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DOI: 10.1021/jz100018a |J. Phys. Chem. Lett. 2010, 1, 874–880

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150.1 ( 7.6 nm. The size of the decahedra is defined as the side length of the particles (see Figure 1a-d and Figure 1e,f). Figure 1g shows the UV-visible absorbance spectra for the four decahedra colloids in ethanol. Spectra were registered before and after laser exposure to ensure that the solutions were not damaged during the HRS experiments. Enhancement at the harmonic wavelength of 395 nm was expected for the four decahedra sizes through the interband transitions, whereas enhancement at the fundamental wavelength through the SP resonance was expected for the larger particles, 100 and 150 nm principally, and the 50 nm ones to a much lower extent. The band broadening observed at larger sizes arises from the contributions of the retardation of the electromagnetic fields.17 The absolute hyperpolarizability tensor magnitudes for the different size gold decahedra were determined by recording the HRS intensity as a function of their concentration in solution. Indeed, the HRS intensity, corrected for self-absorption at the harmonic and the fundamental frequency, is given by18 IHRS ¼ GÆNs β2s þ Ndeca β2deca æI 2

where I is the fundamental intensity, and NS and Ndeca and βS and βdeca are the number densities and the quadratic hyperpolarizabilities of the solvent molecules (ethanol) and the gold decahedra particles, respectively. G is a general factor containing all geometrical factors as well as absolute constants. In eq 1, the brackets stand for an orientational average owing to the isotropy of the liquid phase. The measurement of the HRS intensity in the absence of particles provided an internal reference for the determination of the absolute values of the quadratic hyperpolarizabilities. The HRS response of neat ethanol was determined by external reference measurements with neat water. The Au nanoparticle concentrations were obtained from the determination of the Au metal concentration from the absorption spectrum at 400 nm. The magnitudes of the quadratic hyperpolarizability tensor βdeca are reported in Table 1. The values obtained for gold spherical metallic particles βsph are also provided for comparison.13 Surprisingly, for all sizes, the measured hyperpolarizability values for the noncentrosymmetric gold decahedra are of the same order of magnitude as those of the corresponding gold spherical particles. For the latter particles, the dependence of the quadratic hyperpolarizability as a function of particle size has been previously studied in detail.13,19 The HRS response is of electric dipolar nature for small particles, arising from the surface and yielding a dependence of the hyperpolarizability with the surface area of the particle. However, for larger particles, retardation effects in the electromagnetic fields must be considered, and a non-negligible quadrupolar contribution is observed. This contribution introduces a volume dependence into the quadratic hyperpolarizability, which dominates for the larger particles.15 For the gold decahedra, because of a genuine non-centrosymmetric shape and therefore an expected strongly allowed electric dipole response, a response dominated by the surface contribution was expected for larger sizes as compared to the case of the pseudospherical particles. The allowed electric dipole contribution should indeed remain a larger contribution

Figure 1. (a-d) TEM images of gold decahedra with 17, 50, 100, and 150 nm side length and (e,f) sketch of a decahedra shape and frame definition. (g) Normalized visible-near-infrared (NIR) spectra of the gold decahedra dispersions in ethanol shown in (a-d).

cancellations, leading to values similar to that recorded for gold spherical particles. The particle characterization was performed with transmission electron microscopy (TEM) imaging and UV-visible absorption spectroscopy. The particle dimensions were obtained by measuring the side length from TEM images. They showed clear evidence that the particles were nearly perfect pentagonal bipyramids with 10 equal triangular faces and somewhat rounded edges (see Figure 1a-d). In all the preparations, regardless the size, a small percentage of triangular nanoparticles were observed. Nevertheless, in all cases, the percentage was lower than 15%. The standard deviations of the corresponding size distributions for the different samples are 17.2 ( 1.5 nm, 54.3 ( 4.2 nm, 99.5 ( 5.5 nm, and


ð1Þ

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Table 1. Quadratic Hyperpolarizabilities of Gold Nanospheres and Decahedraa βsph [10-25 esu]

βsph/atom [10-31 esu]

0.50

16

20

2.7

11

24@PVP 50

3.1 8.0

7.3 2.1

80

22

1.4

100

43

1.4

62

1.8

0.76

0.77

150

109

1.0

129

1.1

0.89

0.62

size (nm) 10

βdeca [10-25 esu]

βdeca/atom [10-31 esu]

1.5

8.6

17

ζVsph

ζVdeca 0.28

0.055 23

4.1

0.24

0.66

0.61

The sizes are given in equivalent diameter D, βsph and βdeca are the quadratic hyperpolarizabilities of the gold nanospheres and decahedra, βsph/ atom and βdeca/atom are the quadratic hyperpolarizabilities calculated per atom for the gold nanospheres and decahedra, and ζVsph and ζVdeca are the dipolar versus quadrupolar weighing parameters for the nanospheres and decahedral particles. 24@PVP stands for the 24 nm diameter poly(vinylpyrrolidone)-coated gold nanospheres. a

Figure 2. Size dependence of the quadratic hyperpolarizability for gold nanospheres and decahedra. For the decahedra, the diameter D corresponds to the equivalent diameter of a sphere having the same number of gold atoms.

as compared to the electric quadrupole contribution at almost all sizes. The latter electric quadrupole response scales with the volume of the particles, and therefore the quadratic hyperpolarizability should scale with the number of atoms in the particles in the domain of rather large sizes, irrespective of the shape of the particle. The quadratic hyperpolarizability values were therefore also calculated per atom and were reported in Table 1 too. For the decahedra, the characteristic size is the side length e, and the decahedra volume V is defined as V = 0.603 e3. The number of atoms per particle is thus Na/deca = 35.6 e3 with e taken in nanometers. For the spherical particles, the characteristic size is simply the diameter D and the volume is V = 0.524 D3 with D taken in nanometers. The number of atoms per particle is in this case Na/sph = 30.9 D3. The quadratic hyperpolarizabilities calculated per atom are reported in Table 1 as βsph/atom and βdeca/atom for the spherical and the decahedral particles and are also plotted in Figure 2. The parameter D displayed on the horizontal axis corresponds to the diameter of the spherical particles and to an equivalent diameter for the decahedral


ones defined as the diameter of the sphere of the same number of atoms. For diameters D larger than 50 nm, a volume-dependent regime clearly appears where the quadratic hyperpolarizability per atom is independent of the shape and the size of the particles (Figure 2). An asymptotic value of about 1 10-31 esu/atom is found. It is pointed here that this value is in principle overestimated for the decahedra owing to the fundamental frequency resonance enhancement occurring for the larger sizes (see Figure 1g). However, this value is similar to that reported for spherical particles and indicates that this overestimation is probably not very large. For the smaller particles, a deviation from this volume dependence appears where the quadratic hyperpolarizability calculated per atom increases greatly as D gets smaller. For spherical particles, it is known that in this regime, the quadratic hyperpolarizability scales with the surface area of the particles.11,15 In this domain of small sizes for the decahedra particles, the quadratic hyperpolarizability value obtained for the particles with 17 nm lateral size is close to, albeit

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slightly weaker than, that observed for the 20 nm spheres. This clearly shows that, although the gold decahedra have a non-centrosymmetric shape, they do not exhibit a strong electric dipole allowed response. However, in this size regime, the details of the chemical synthesis with the incorporation of specific stabilizing agents can severely affect the surface response. Poly(vinylpyrrolidone) (PVP) was used to stabilize the decahedra, whereas the spherical nanoparticles were stabilized with other organic compounds.11 To analyze the influence of the stabilizing layer, spherical particles stabilized with PVP were also synthesized and studied by means of the same HRS method. The quadratic hyperpolarizabilities βsph and βsph/atom obtained for the 24 nm diameter PVP-coated spherical particles are also reported in Table 1 and plotted in Figure 2. These values are very similar to the ones obtained for the 20 nm diameter spherical particles stabilized with different organic compounds. This result indicates that the PVP coating on the particles is not responsible for the similar values of the quadratic hyperpolarizability measured for decahedra and spherical particles. In order to rationalize the above results, a simple approach based on a surface SH response similar to that developed for spherical particles was developed where the gold decahedra are formed as a set of 10 identical facets organized in space to form a pentagonal bipyramid (see Figure 1e,f. The SH response of the 10 facets can then be described within the general framework of the SH response from planar metallic surfaces. In this model, it is customary to define the SH response of flat metallic surfaces with three parameters corresponding to the three origins of the nonlinear response. These three parameters, also called the Rudnick and Stern parameters, correspond to the three nonlinear currents driving the SH fields.20 The first one is a bulk current normal to the interface, and the two others are nonlinear surface currents driven parallel and perpendicular to the interface. The nonvanishing tensor elements of the quadratic susceptibility of a flat metal surface can therefore be recast into these three parameters.21 In the context of small metallic particles, each facet of a non-centrosymmetric particle may be seen as a nanometre scale flat metal surface with a quadratic hyperpolarizability. We can assume furthermore that this quadratic hyperpolarizability is located at the facet center of mass. The quadratic hyperpolarizability of the particle is then calculated from the coherent superposition of the response of the 10 facets. Field retardation has to be taken into account for large nanoparticles, therefore introducing the volume dependence of the quadratic hyperpolarizability at large sizes. In contrast, for sizes much smaller than the wavelength of light, the field retardation can be omitted. Building the coherent superposition of the response of the 10 facets of the decahedra, the orientation in space of the facets must be taken into account. Defining 5 βdeca as the quadratic hyperpolarizability of the decahedra and 5 β0,i as that of the facet i, we write T

βdeca ¼

10 X i ¼1

T 5 Ti ðθ, φ, ψÞβ0, i

decahedra reference frame. This tensor depends on the standard three Euler angles (θ,φ,ψ), although only two are necessary here: 2 3 cos θi sin θi cos φi sin θi sin φi 6 7 5 Ti ðθi , φj Þ ¼ 4 -sin θi cos θi cos φi cos θi sin φi 5 ð3Þ 0 -sin φi cos φi where θi ∈ {0, 2π/5, 4π/5, 6π/5, 8π/5} depending on the facet considered, and φi = φ0 if the facet belongs to the upper pentagonal pyramid and φi = π - φ0 if it belongs to the lower one. The decahedra are not single crystals but 5-fold-twinned particles. They therefore exhibit single-crystal structures for each facet. For a 10 facet pentagonal bipyramid, the coherent superposition leads to a vanishing quadratic hyperpolarizability 5 βdeca, similarly to the case of a perfect sphere. This cancellation can be readily inferred from the decahedra general shape symmetry since it proceeds from tensorial considerations for quadratic nonlinear optical phenomena.22 This cancellation is more obvious if eq 2 is used. Although further considerations on the in-plane symmetry of the quadratic hyperpolarizability of each facet is not necessary,5 the problem can be considerably simplified if it is assumed that the nonlinearity is dominant along the direction normal to the facet surface.23 In this case, the quadratic hyperpolarizability tensor of each facet reduces to a single nonvanishing tensor element, namely β0,i,^^^, where the symbol ^ stands for the direction normal to the surface of the facet. Using eq 2, we obtain for the elements βdeca,ZZZ and βdeca,XXX: βdeca;ZZZ ¼ 5½cos3 φ0 þ cos3 ðπ -φ0 Þβ0;^^^ ¼ 0 ð4aÞ βdeca;XXX ¼ ½sin3 φ0 þ sin3 ðπ -φ0 Þ½

i ¼1

sin3 θi β0;^^^ ¼ 0 ð4bÞ

and, similarly, a vanishing value for all other elements. By centrosymmetry along the bipyramid axis, the corresponding components vanish. In the plane perpendicular to the bipyramid axis, a residual 5-fold symmetry is observed for the two upper and lower pentagonal pyramids. The 5-fold symmetry leads to vanishing resulting tensor elements, as seen in eqs4 for two of them. Hence, despite the non-centrosymmetric shape of the decahedra, a symmetry cancellation of the quadratic hyperpolarizability occurs, similarly to the case of perfect spheres. Deviation from the perfect shape, in particular on the edges and the corners of the facets, still breaks the perfect cancellation, similarly to the case of the spherical particles. As a result, the quadratic hyperpolarizabilities of gold decahedra and gold spherical particles are similar in the surface dominated regime where defects-induced symmetry breaking plays the determining role. To confirm the origin of the HRS response of the gold decahedra, the linear polarization angle γ of the incident light was varied and the HRS intensity was collected with a vertical polarization. The polar plots obtained for gold decahedra are shown in Figure 3. The HRS intensity can be rewritten as a function of the angle γ as11 V IHRS ¼ aV cos4 γ þ bV cos2 γ sin2 γ þ cV sin4 γ ð5Þ

ð2Þ

with the introduction of 5 Ti(θ,φ,ψ), the frame transformation tensor of the ith facet from the frame of the facet itself to the


5 X

where the different parameters aV, bV, and cV can be related to the initial parameters of eq 1.

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Figure 3. Polar plot of the HRS intensity as a function of the incoming fundamental beam polarization angle: experimental points (empty circles) and fit (solid line) to the experimental data points using eq 5 for 17 nm gold decahedra (a) and 100 nm gold decahedra (b).

It is then also possible to introduce the following weighting parameter:19 ! bV -ðaV þ cV Þ V ð6Þ ζ ¼ bV to define a quantitative value of the ratio of the electric dipole to the electric quadrupole contributions. A value of ζV = 0 corresponds to a pure electric dipole response prevailing in the small size surface dependent regime and ζV = 1 to a pure electric quadrupole response prevailing in the limit of the large size pure volume dependent regime. The values of the ζV parameter for the four decahedra colloids are reported in Table 1. The ζV values confirm that, for all sizes larger than 50 nm, the signal is dominated by the electric quadrupolar response, whereas for the smaller sizes it is dominated by the electric dipolar response. Even for the smaller size decahedra, a non-negligible quadrupolar component is observed with ζVdeca = 0.28. These results further support the observation of two regimes with a transition at sizes around 50 nm in terms of equivalent diameters. Figure 4 displays the ζV parameter for both the decahedra and the spheres.13 The ζV parameter is larger at small diameters for the decahedra than for the spherical particles, and the reverse is true at larger diameters. This trend is not fully understood yet, but we speculate that the shape of the gold decahedra can be closer to that of the perfect pentagonal bipyramid shape, i.e., with less surface defects, at small diameters. On the contrary, for larger sizes, the shape of the spherical particles approaches more closely that of the perfect shape. In conclusion, a detailed analysis of the SH response of particles with centrosymmetric and non-centrosymmetric shape is presented. Gold nanospheres, representative of the class of centrosymmetric particles, and gold decahedra, representative of the class of non-centrosymmetric particles, are discussed. The size dependence of their quadratic hyperpolarizability clearly exhibits two domains. At large sizes, the quadratic hyperpolarizability is shown to scale with the volume of the particles, irrespective of their shape, and as a result, the quadratic hyperpolarizability per atom is found to


Figure 4. Size dependence of the ζV parameter. For the decahedra, the diameter D corresponds to the equivalent diameter of the sphere having the same number of gold atoms.

be constant and equal to about 1 10-31 esu/atom. For smaller sizes, i.e., average equivalent diameter of less than about 50 nm, the quadratic hyperpolarizability per atom is shown to be much larger . This domain of size dependence is dominated by a scaling of the quadratic hyperpolarizability with the surface area of the particles due to breaking of the perfect symmetry by shape defects. For gold nanospheres, the defects induce a shape which deviates from that of the perfectly centrosymmetric shape. For gold decahedra, we found that the quadratic hyperpolarizability is not larger than that measured for the spherical particles, despite the noncentrosymmetric shape. This is ascribed to the symmetry cancellations in the quadratic hyperpolarizability of the pentagonal bipyramid shape.

EXPERIMENTAL SECTION Tetrachloroauric acid (HAuCl4 3H2O) and sodium borohydride (NaBH4) were purchased from Aldrich.

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Poly(vinylpyrrolidone) (PVP, MW = 10 000-40 000) and N, N-dimethylformamide (DMF) were supplied by Fluka. All chemicals were used as received. Pure grade ethanol and Milli-Q grade water were used to make up all solutions. Gold decahedra with different average side lengths of 17, 50, 100, and 150 nm were synthesized as follows:5 for small decahedra (17 and 50 nm side length), 0.1 or 1.1 mL (for 17 and 50 nm particles, respectively) of 0.114 M HAuCl4 aqueous solution was added to 15 mL of 2.5 mM PVP (MW = 40 000) solution in DMF, in a 50 mL vessel, and the mixture was irradiated with ultrasound until complete disappearance of the Au3þ charge transfer-to-solvent absorption band at 325 nm. Then 0.7 mL of a preformed seed solution of Au nanoparticles ([Au] = 5 10-5 M) with 2-3 nm diameter was added and further sonicated for complete reduction, as indicated by a constant plasmon band position. For large decahedra (100 and 150 nm side length), the same procedure was used, but using 50 nm (side length) decahedra as seeds (0.806 mL, 0.115 M). Prior to seed addition, 0.452 or 0.904 mL of a 0.1037 M HAuCl4 aqueous solution was added to the PVP solution in DMF to obtain the 100 or 150 nm Au particles, respectively. Samples were centrifuged at 2000 rpm (17 and 50 nm) or 1500 rpm (100 and 150 nm) to remove excess PVP and enable the subsequent transfer into ethanol. Ultrasonic irradiation was performed with a Sonopuls HD2200 ultrasonic horn (6.35 mm tip radius) homogenizer operating at a frequency of 20 kHz and at 30% of the maximum power (200 W). TEM images were obtained with a JEOL JEM 1010 transmission electron microscope operating at an acceleration voltage of 100 kV. UV-visible-NIR spectra were measured with a Cary 5000 UV-vis-NIR spectrophotometer (Figure 1g). The light source for the HRS experiments consisted of a mode-locked femtosecond Ti:sapphire laser delivering at the fundamental wavelength of 790 nm pulses with a duration of about 180 fs at a repetition rate of 76 MHz. After passing through a low-pass filter to remove any unwanted harmonic light generated prior to reaching the cell, the fundamental beam of about 900 mW was focused by a microscope objective into a 1 cm 1 cm spectrophotometric cell containing the metal particles. The HRS light was collected at an angle of 90 from the incident direction by a 2.5 cm focal length lens. The second harmonic light was separated from its linear counterpart by a high-pass filter and a monochromator positioned at the second harmonic wavelength. The HRS light was then detected with a cooled photomultiplier tube, and the pulses produced were counted with a photon counter. The fundamental beam was chopped at about 130 Hz to enable a gated photon counting mode, allowing automatic subtraction of the noise level. The fundamental input beam was linearly polarized, and the input angle of polarization γ was selected with a monitored rotating half-wave plate.

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Corresponding Author: *Corresponding author. Tel: þ33 472 431 914. Fax: þ33 472 445 871. E-mail: [email protected].


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Symmetry Cancellations in the Quadratic Hyperpolarizability of Non

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