Tunable Mie resonance in complex-shaped gadolinium niobate

Nanoscale particles described by Mie resonance in the UV–vis–NIR region are in high demand for optical applications. Controlling the shape and size of these particles is essential, as it results in the ability to control the wavelength of the Mie resonance peak. In this work, we study the extensive scattering properties of gadolinium niobate particles with complex bar- and cube-like shapes in the UV–vis–NIR region. We perform our experimental analysis by characterizing the morphology and extinction spectra, and our theoretical study by implementing a Mie scattering model for a distribution of spherical particles. We can accurately model the size distribution and extinction spectra of complex shaped particles and isolate the contribution of aggregates to the extinction spectra. We can separate the contributions of dipoles, quadrupoles, and octupoles to the Mie resonances for their respective electric and magnetic parts. Our results show that we can tune the broad Mie resonance peak in the extinction spectra by the nanoscale properties of our system. This behavior can aid in the design of lasing and luminescence-enhanced systems. These dielectric gadolinium niobate submicron particles are excellent candidates for light manipulation on the nanoscale.


Introduction
Submicron particles that support Mie resonance in the ultraviolet-visible (UV-vis) and near-infrared (NIR) regions offer a range of applications such as lasing [1], enhanced photoluminescence [2], low-loss meta-optics [3], coloring [4], enhanced Raman scattering [5], among others.The most studied materials for these applications are spherical crystalline silicon [4,6], perovskites [1,2,7], and gold and silver nanoparticles (NPs) [8].The Mie resonance is not restricted to spherical particles and has been observed in cubic halide perovskites (CsPbBr 3 ) [1], randomly-shaped MAPbI 3 [2], and fullerene-like MoS 2 NPs [9].Among the studied materials, all-dielectrics with high refractive indices, such as silicon [5] and germanium [10], offer a great alternative to the plasmonic gold and silver NPs [11].Replacing the plasmonic NPs with all-dielectrics overcomes limitations such as energy Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.dissipation and absorption.The properties of all-dielectrics make them attractive for applications in quantum and thermal photonics [11].
The orientation of the applied electric field must be considered in light scattering from cubes, rectangular particles, cylinders, and spheroids (oblate, prolate).For example, the cubic symmetry possesses ten different symmetry characters, resulting in twenty subdeterminants by the leading multipoles [20].When considering randomly orientated complex-shaped particles, the cubic model of Mie scattering would have to include several terms in the expansion in spherical harmonics.For example, in [19] for a simpler geometry concludes that at least 25 terms are needed.For that many terms, it is more convenient to work with a size distribution of spheres [21,22].Although it is possible to extend the theoretical approach to fit the experimental results, it is often difficult to obtain a good fit between theory and experiment for non-spherical particles.This fit is mainly achieved by numerical solutions using finite-element methods, such as FullWave (Synopsys, Inc., Mountain View, USA) or Lumerical (Ansys Ltd, Canada) software.
It is possible to compare theoretical calculations with farfield measurements such as absolute absorbance (derived from diffuse reflectance) [23], UV-vis extinction spectra [24], and angle-resolved and total scattering [25].These measurements are proportional to the absorption (Q abs ) and extinction (Q ext ) efficiencies, where the scattering efficiency Recently, the synthesis of nanoscale gadolinium niobate (GdNb 2 O 6 ) with cube-and bar-like morphologies was reported [27].This compound exhibits photoluminescence emission before and after surface water removal [27,28].In addition, GdNb 2 O 6 is an efficient host matrix for Tm 3+ and Nd 3+ ions with intense emission in the blue and NIR regions.This material has an indirect band gap of 3.52-3.57eV and 3.48-3.55eV for the as-prepared and dehydrated powders, respectively [27].The GdNb 2 O 6 compound is isostructural to RETiNbO 6 (rare earth (RE) = La, Ce, Pr, Nd, Eu), with an aeschynite orthorhombic crystalline phase and a Pnma space group [29,30].The high dielectric constant RETiNbO 6 compounds are known to be good dielectric resonators in the microwave region [31,32].
This study shows that the cube-and bar-like morphologies of GdNb 2 O 6 exhibit Mie resonance in the UV-vis-NIR region.A simple model of spherical particles can describe this Mie resonance phenomenon by considering a particle-size distribution.Although the analysed particles are not spherical and their refractive index function (with respect to wavelength) is approximated, we obtain an excellent fit between the calculated and measured extinction spectra using a size distribution of spheres.To do this, we use two Gaussian functions to fit the size distribution.One Gaussian describes the size distribution of small and the second one the size distribution of large spherical particles.This size distribution describes the different dimensions of the faceted GdNb 2 O 6 particles and their aggregates with high accuracy.Hence, our method replaces the time-consuming procedure of measuring each particle size individually with a simple calculation.We see a shift in the central wavelength of the Mie resonance by changing the particle size of our systems and studying them using Mie theory.We calculate the Mie resonant modes for the single and multi-particle cases and separate the electric and magnetic contributions for dipole, quadrupole, and octupole.We show that our theoretical calculations fully support our experimental results.

Characterization
To measure the particle size by scanning electron microscopy (SEM), they are first dispersed in ethanol by sonication for 5 min and then dropped-cast onto a grid.A JEOL JSM-7401F (secondary electron (SEI) detector) SEM is used to take the images.We use an accelerated voltage of 4.0-4.5 kV and a working distance of 3 mm for the images.We use ImageJ software [33] to measure the particle dimensions.
We use an integrated sphere implemented in the Edinburgh Instruments FLS980 spectrometer to measure the absolute reflectance of the powder samples.The data acquisition settings are excitation and emission beam slits of 10 and 0.4, 0.1 nm step, and 0.25 s dwell time.Each spectrum is the average of four measurements.
To obtain the extinction spectra (UV-vis-NIR), we use the PerkinElmer Lambda 12 spectrometer in ABS mode.The settings for the measurements are 250-1100 nm range and a 1 nm step interval.To analyse the effect of the refractive index on the position of the Mie resonance peak, we disperse the powders in water, tetraethyl orthosilicate (TEOS), chloroform, and 1,2-dichlorobenzene, with refractive indices of 1.3330, 1.3830, 1.4458, and 1.5514, respectively.
The dynamic light scattering (DLS) particle-size measurements are performed using the Malvern Panalytical Zetasizer Nano-ZS90.For the measurements, the particles are dispersed in aqueous media.Dispersion of the particle powders for all measurements are done by sonication for 5 min.
X-ray diffraction (XRD) patterns were measured using a D8 ADVANCE Eco, BRUKER diffractometer equipped with a 1D LYNXEYE detector.The Cu anode (CuKα, 1.5418 Å wavelength) is operated at 40 kV and 25 mA.The 2-Theta scans are performed in the Bragg-Brentano mode with a step size of 0.015˚and scan speed of 8 s step −1 .

Numerical calculations
We use Wolfram Mathematica 12.1.1 to compute the absolute absorbance and extinction spectrum in terms of wavelength using Mie theory for a homogeneous sphere of a given diameter.As a starting point, we use the code provided by Sarid and Challener in chapter 9 of [26].Bohren and Huffman originally developed this code in appendix A of [34], (see also [35] for a similar code in Python.)We approximate the real part of the refractive index using known data for a similar compound, potassium niobate (KNbO 3 ), available from the refractiveindex.infodatabase [36].The imaginary part of the refractive index (extinction coefficient k) is obtained from Expression 1 [37] using the absolute absorbance (Q abs ) measured for powder samples in the air for wavelengths in the 250-800 nm range: Expression 1 where α is the absorption coefficient, OD is the optical density, and x is the sample thickness (0.2 cm in our case).
We then fit the experimentally measured absolute absorbance (Q abs ) with one Gaussian function and use it to calculate the extinction spectra (Q ext ).We calculate Q ext in two ways: (1) using Mie theory for spherical particles with a weighted average from the particle-size distribution (∼300 particles) measured from SEM images (Mie-measured size) and (2) by fitting the experimentally measured Q ext (250-1100 nm range) using Mie theory with a spherical particle-size distribution of two Gaussian functions (Mie-2-Gaussian) (figure S1, all S figures are in the Supporting Information).
For the measured particle-size distribution, we approximate the size of the bar-like particles as a cylinder with two equally weighted dimensions called length and width (figure S2(a)).
For the cube-like morphology, we approximate the size as a rectangular prism with three equally weighted dimensions: length, width, and diagonal (figure S2(b)).For the fit (Mie-2-Gaussian Size), we first compute the Mie extinction spectra for 490 spheres with diameters ranging from 20 to 1000 nm.
We then perform the least squares fit of the experimentally measured Q ext using a weighted two-Gaussian size distribution to find five parameters: spherical particle diameters D 1 , D 2 (Gaussian peaks), standard deviations σ 1 , σ 2 of the two Gaussians, and their relative weights w 1 , 1-w 1 (table 1).
To describe the particle-size distribution (PSD) in terms of the diameter d, we use a superposition of two Gaussian functions (figure S1) given by the Expression 2: We fit Q ext for particles dispersed in water (refractive index of 1.3330), assuming a random orientation of the particles in the solution.Therefore, we need to account all these dimensions in the measured size distribution.The final Q ext is given by the weighted average using the PSD as: Expression We calculate the optical band gap (E g ) from the diffuse reflectance F(R) using the Kubelka-Munk (K-M) method [38].The function F(R) is proportional to the attenuation coefficient (α), which allows a modification [39] of the Tauc equation [40,41]: where h is Planck's constant (J•s), B is the absorption constant, and υ is the photon frequency (Hz).We have shown in a previous study [27] that GdNb 2 O 6 is an indirect band gap material.Hence, we plot (F(R)•hυ) 0.5 versus hυ (photon energy in eV) to obtain the band gap value.We use the relationship proposed by Herve and Vandamme to calculate the refractive index [42,43]: , where E g is the material band gap, A = 13.6 eV is the hydrogen ionization energy, and B = 3.4 eV.

Results and discussion
We synthesize the bar-and cube-like morphologies of GdNb 2 O 6 using a simple, inexpensive hydrothermal method.We control the size and morphology of these submicron particles solely by the pH and temperature of the precursor solution [27], where seeds are formed.Particles' growth occurs from a supersaturated solution.Higher pH values within a narrow temperature range result in smaller particles [27].As with zeolites [44] and silicalites [45], the aspect ratio of the particles (length/width) is controlled by alkalinity, which increases with KOH concentration.We present the morphology and size distribution of the analysed GdNb 2 O 6 submicron particles in figures 1(a1), (a2)-(d1), (d2).We study four systems with different size distributions, designated as S1-S4.The crystalline structure of GdNb 2 O 6 is confirmed by XRD analysis (figure 1(e)).All systems (S1-S4) show a similar XRD pattern without any contamination peaks.
To describe the particle-size distribution, we use a superposition of two Gaussian functions (figure S1) described by five parameters: D 1 , D 2 , σ 1 , σ 2 , and w 1 (table 1).We obtain the values of these parameters by calculating through Mie scattering theory the corresponding Q ext that fits the experimental results.The average particle size is largest for S1 and smallest for S4 (S1 > S2 > S3 > S4), as shown in table 1. System 1 consists of bar-like particles (figures 1(a1), (a2)), where the percentage of smaller particles (∼295 nm) is about 35% and the percentage of larger particles (∼500 nm) is about 65% (table 1).System 2 contains mainly cube-like particles and some bar-like particles (figures 1(b1), (b2)), with the percentage of smaller particles (∼330 nm) about 60% and the percentage of larger particles (∼580 nm) about 40%.Systems S3 and S4 have a more homogeneous cube-like particle distribution.
Figures 2(a)-(d) show the experimental extinction and absolute absorbance spectra of the four systems (S1-S4).All Q ext spectra contain a broad peak followed by a dip.The peak is redshifted with a larger average particle size as predicted by Mie theory [34].The peak position is 483, 488, 438, and 371 nm for systems S1 to S4, respectively.In addition, the dip in the UV region is more pronounced for system S3, where the smaller particle-size distribution is narrowest (figure 3(c), table 1).The absolute absorbance spectra (figures 2(a)-(d)) lack the broad peak component.This behaviour indicates that the broad peak is scattering-related [9,46].These results highlight the importance of particle size, as a change in size shifts the Mie resonance to a different wavelength.As a result, the different sizes can be excited at different wavelengths to obtain lasing or enhanced photoluminescence [1,2].In addition, the S1 and S2 particles, although different in shape, have similar extinction spectrum characteristics (figure 2(a)).As mentioned in the experimental section, the difference in shape is averaged out by considering all possible orientations that affect the extinction spectra.
Figures 3(a   (Measured) closely match the size distributions calculated using two Gaussian functions (2-Gaussian).On the other hand, for S3 and S4, particles with sizes around 600 nm are practically absent from the measured size distribution.We have not observed particle sizes larger than 600 nm by SEM.However, we believe that this region of the particle-size distribution (second Gaussian peak, figures 3(c) and (d)) results from small aggregates of two or three particles (figure S3).In support of this claim, we observe that the center of the second Gaussian peak in the calculated size distribution is approximately twice the size of the first peak (figures 3(c) and (d), table 1).In general, the maximum particle size observed in the UV-vis-NIR region of the extinction spectra (figure 2(a)) for these GdNb 2 O 6 particles is ∼1000 nm, as larger particles contribute to longer wavelengths in the NIR.We also performed DLS measurements for comparison.The obtained size distribution (figure 3) was used to calculate the extinction spectra (figure S4).We can see that the obtained spectra with both intensity and volume size distributions describe the experimental Q ext poorly, especially for the larger and elongated particles as in S1 and S2.For these two systems, the size is concentrated in a single value (figure 3), and the Q ext spectra behave as if it were produced by a single particle (figure S4).
In figure 4 we compare the experimental and calculated extinction spectra.The calculated Q ext of a single particle shows a ripple structure with its Mie resonance centred at the peak of the experimental extinction spectra.This ripple structure averages out considering the Mie resonance fit with the particle size distribution of two Gaussian functions (Mie-2-Gaussian size, figure 4).For this fit, the experimental and calculated extinction spectra are almost identical, showing that our calculated two-Gaussian size distribution does indeed describe all our systems.We also obtain Q ext with a weight function directly from the measured size distribution (Miemeasured size) without a fit, where the experimental and calculated spectra shift slightly.For S1, the peak of the measured size distribution is slightly shifted with respect to the first Gaussian peak (centred at 295 nm with standard deviation of 60 nm, table 1) of the 2-Gaussian size distribution.This shift is manifested in the Mie-Measured Size Q ext (figure 4(a)) as a narrowing of the spectrum compared to the experimental and Mie-2-Gaussian size Q ext , accordingly.For S2, where the measured and the two Gaussian size distributions mostly agree, the dip and the peak of the Mie-2-Gaussian size and Mie-measured size Q ext spectra are at the same location (figure 4 (b)).For S3 and S4, where sizes larger than 600 nm are missing, the spectra calculated with measured data only partially agree with the Mie-2-Gaussian size extinction spectra (figures 4(c), (d)).The match is lacking for longer wavelengths (>450 nm) affected by the larger particle sizes (second gaussian peak, figures 3(c), (d), table 1).As expected, the calculated Q ext with the two-Gaussian size distribution is in best agreement with the experimental one in all cases.However, the Q ext obtained with the measured size distribution (Mie-measured size) agrees well with the experimental Q ext spectra without the need for fitting.The comparison between Mie-2-Gaussian size and Mie-measured size is meaningful and allows us to isolate the contribution of particle aggregates to the Q ext spectra.That is, the poor agreement in the 450-1100 nm wavelength range for the Mie-measured size spectra (figures 4(c), (d)) is due to the exclusion of the sizes of the aggregated particles in the measured distribution.
We measure the extinction spectra of all systems (S1-S4) in different solvents and observe a redshift of the peak with decreasing refractive index of the media, as predicted by Mie theory (figure 5) [34].Reducing the difference between the refractive index of the particles and the medium reduces the scattering efficiency.These results provide additional evidence to support our conclusion that the broad peak in the extinction spectra is due to a Mie resonance.This behaviour can be useful in the design of lasing and luminescence-enhanced systems [1,2].It is possible to control the wavelength of the Mie-resonance by tuning the refractive index of the solvent.For example, for S3 the position of the Mie resonance peak (figure 5(c)) is shifted  from 375 to 442 nm by changing the refractive index of the medium from 1.5514 to 1.3330.
Figure 6 shows the extinction spectrum of a single 300 nm particle and its electric and magnetic dipole (ED, MD), quadrupole (EQ, MQ), and octupole (EO, MO) contributions.The extinction spectrum (Mie-1-particle) contains multiple peaks, the so-called ripple structure [34].The dipole, quadrupole, and octupole contributions are the sum of the magnetic and electric parts.Each mode sum (ED + MD, EQ + MQ, and EO + MO) fits a peak in the extinction spectrum of a single particle (figure 6(a)).We can see that the electric and magnetic contributions are different for a single particle (figure 6(b)).Also, the magnetic modes are more pronounced than the electric modes in the extinction spectrum.Hence, the peaks of the magnetic modes correspond to the peaks observed in the extinction spectra.
We show the extinction spectrum calculated with the 2-Gaussian size distribution for S3 and the contribution of the ED, MD, EQ, MQ, EO, and MO modes in figure 6(c).The extinction spectrum (Mie-2-Gaussian size, figure 6(c)) does not contain multiple peaks (up to ∼300 nm) in contrast to the case of a single particle (figure 6(a)), but the sum peaks follow a similar behaviour for the dipole, quadrupole, and octupole.Figure 6(d) shows that the electric and magnetic parts for the multi-particle case are similar for each of the dipole, quadrupole, and octupole contributions.Since, Mie-2-Gaussian size fit is a weighted superposition of modes from particles of different sizes.We obtained similar results for the remaining systems (S1, S2, and S4, figures S5-7).In the 240-300 nm wavelength range, we can observe the ripple structure (figure 6(c)) due to the contribution of small particles with a narrow size distribution (figure 3(c)).
Figure 6(e) shows the dependence of light scattering (Q sct ) on particle size.The dependence is practically linear, the highest scattering intensity (Mie resonance) almost coinciding with the particle size.For particle sizes of 300, 500, and 700 nm (figure 6(b) and S8), the highest scattering efficiency is around 400, 628, and 862 nm, respectively.These calculations agree with the experimental results in figure 2, where the Mie resonance is redshifted with increasing average particle size.Also, the 400, 628, and 862 nm wavelengths coincide with the MO peak for the single particle of size 300, 500, and 700 nm, respectively (figures 6(b) and S8).Even though the MO peak is not the strongest, several parts (most notably EO, MQ, EQ) contribute at this wavelength, making MO the highest peak.
Figure 7 shows the total and x, y, and z components of the electric near-field distribution amplitude for a 300 nm particle.We plot the field for three cases corresponding to the peak of ED (510 nm), EQ (440 nm), and EO (375 nm).For a wavelength of 510 nm, the ED and EQ largely overlap (figure 6(b)).At 440 nm, the EQ still overlaps significantly with the ED and slightly with the EO, and at 375 nm, the EO contribution dominates with only a small overlap with the ED and EQ (figure 6(b)).The total electric field (|E T |) for a wavelength of 375 nm is mainly concentrated around the particle surface (figure 7).At 440 nm, the field is intense at the centre of the particle.At 510 nm, the electric field is strong over a large part of the particle volume.

Conclusion
This work shows that submicron bar-and cube-like GdNb 2 O 6 particles support a Mie resonance in the UV-vis-NIR region.We apply a simple Mie model for a two Gaussian size distribution of spherical particles to support these experimental results.Using this model, we approximate the size distribution of the GdNb 2 O 6 particles by fitting the measured extinction spectra and obtaining an excellent agreement.As a result, even though our particles have a complex shape and we approximate their refractive index function, the model agrees with our measurements without ad-hoc approximations.We can also isolate the contribution of aggregates to the extinction spectra.We can tune the Mie resonance peak by varying the particle size and the refractive index of the surrounding media.We calculate the Mie resonance modes for single-and multi-particle cases.The electric and magnetic contributions are different for a single particle.The location of the Mie scattering peak redshifts linearly with increasing particle size, and the maximum intensity is at the peak of the magnetic octupole.For the multi-particle systems, the electric and magnetic dipole, quadrupole, and octupole contributions are a superposition of different modes of all particle sizes.In this case, the weighted average of the particle sizes washes out most of the differences between the electric and magnetic parts.This superposition manifested itself as a softening of the extinction spectra.
Our theoretical and experimental results conclude that Mie scattering of a distribution of spherical particles models the studied dielectric GdNb 2 O 6 submicron particles with complex bar-and cube-like shapes.These submicron particles are excellent candidates for applications in quantum and thermal photonics.
) and (b) show that the particle-size distributions of S1 and S2 measured from the SEM images

Figure 6 .
Figure 6.Calculated extinction spectra for (a) a single particle of 300 nm size and (c) S3 (Mie-2-Gaussian size) with a sum of an electric and magnetic dipole, quadrupole, and octupole parts.(b), (d) The same case but now with separate ED, EQ, EO (solid line) and MD, MQ, MO (dashed line) contributions.(e) Calculated spectral map of scattering efficiency from Mie theory (Q sct ) for spherical particles of 20-1000 nm in size versus wavelength.