Absorption peak decomposition of an inhomogeneous nanoparticle ensemble of hexagonal tungsten bronzes using the reduced Mie scattering integration method

Recent optical analyses of cesium-doped hexagonal tungsten bronze have accurately replicated the absorption peak and identified both plasmonic and polaronic absorptions in the near-infrared region, which have been exploited in various technological applications. However, the absorption peaks of tungsten oxides and bronzes have not generally been reproduced well, including those of the homologous potassium- and rubidium-doped hexagonal tungsten bronzes that lacked evidence of polaronic subpeaks. The present study reports a modified and simplified Mie scattering integration method which incorporates the ensemble inhomogeneity effect and allows precise peak decomposition and determination of the physical parameters of nanoparticles. The decomposed peaks were interpreted in terms of electronic structures, screening effect, and modified dielectric functions. The analysis revealed that the plasma frequencies, polaron energies, and the number of oxygen vacancies decrease in the dopant order Cs → Rb → K. The coexistence of plasmonic and polaronic excitations was confirmed for all the alkali-doped hexagonal tungsten bronzes.


Materials and methods
K x WO 3 , Rb x WO 3 , and Cs x WO 3 powders (where x = 0.20, 0.25, 0.30, and 0.33 refers to the initial M/W weight ratio; M = K, Rb, Cs) were prepared by mixing the raw materials of alkali carbonates and tungstic acid and heating at 550 °C for 1-3 h in a 1% H 2 -N 2 atmosphere, followed by homogenization heating at 800 °C for 1 h in a N 2 atmosphere.The resulting deep blue powders are known to be highly reduced tungsten oxides containing many V O s 33 .Thus, the true chemical formulas would be K x WO 3-y , Rb x WO 3-y , and Cs x WO 3-y , although y is generally ignored in the following contents.
The obtained powders were pulverized and mixed with the dispersant in methyl isobutyl ketone (MIBK) solvent.The NP-dispersion liquid was prepared in a bead mill, diluted to a fixed concentration with MIBK, and transferred to a glass holder for transmittance profile measurements in the visible and infrared wavelengths using a U-4100 spectrophotometer (Hitachi High Tech Corp., Tokyo, Japan).Powder XRD measurements were conducted using an X'Pert-PRO/MPD diffractometer (Spectris Co., Ltd., Tokyo, Japan) with Cu-Kα radiation (λ = 1.54 Å).

Absorption strengths of the HTB nanoparticles
Structural characterization of the alkali HTB powders prepared was done by powder XRD and the Rietveld method.The main phase in all HTBs was hexagonal of space group P6 3 /mcm 34 .Trace amounts of WO 3 and WO 2.92 were contained in K 0.20 WO 3 , Rb 0.20 WO 3 and Cs 0.20 WO 3 , which were found as optically negligible.All the other compositions were single phased.Further structural details including small unsymmetric strains due to pseudo Jahn-Teller distortion are reported elsewhere 35 .
The transmittance profiles of HTB NPs with nominal alkali ratios of 0.20, 0.25, and 0.30 are shown in Fig. 2a-c.All profiles exhibit a large transmission peak in the visible spectrum and a large absorption region at NIR wavelengths.However, only Cs-HTB presents a distinct dip around 850 nm.This dip was attributed to polaronic transition of localized electrons provided by V O s 4,23 , whereas the main broadband absorption centered around 1500 nm comprised the LSPR of free electrons activated in directions perpendicular (⊥) and parallel (ǁ) to the c-axis 4,20 .
Figure 1.Schematic showing the ensemble inhomogeneity effect: absorption profiles of a single particle (blue) and a particle ensemble (orange).
The relative absorption strengths of the dispersed Cs-, Rb-, and K-HTB NPs were determined in terms of the molar absorption coefficient χ, which is defined in Lambert Beer's law log 10 (I/I 0 ) = − χ l c, where I 0 and I denote the intensities of the incident and transmitted light, respectively, l is the optical length (cm), and c is the molar concentration (mol/L).The absorption coefficient of the bulk is inapplicable to a NP dispersion, whereas the commonly used absorbance or transmittance is inaccurate as a measure because the filler concentration is unknown.For these reasons, we selected χ as a measure of the absorption strength 6 .The χ of a solution containing 1 mol of NP filler must be measured under the same Mie scattering conditions, i.e., the same particle shape and size, same degree of dispersion (particle distribution), and same refractive constant of the medium.These constraints are crucial because the solute in the present case is not a homogeneous dye but discrete NPs that undergo Mie scattering and LSPR.
Figure 2d-f presents the χs corresponding to Fig. 2a-c.The χs were measured under the same Mie scattering condition with the particle size within 25-30 nm.All χ profiles exhibit an absorption peak between 0.5 and 2.0 eV.The peak sizes considerably decreased as Cs → Rb → K, which cannot be deduced from Fig. 2a-c unless the filler amounts are specified.The absorptions of all HTB NPs in Fig. 2d-f decreased as the doping amount decreased from M/W = 0.30-0.20,although the small decrement suggests that the dopant should not be the only source of the absorption.Moreover, the magnitude of the decrement reduced from Cs-HTB to K-HTB (where K 0.25 WO 3 and K 0.30 WO 3 were reversed either by error or by an unclarified reason.See ref. 35 ).If the alkali dopant alone caused the NIR absorption [36][37][38] , such changes are not expected.Therefore, both the dopant-donated and V O -donated electrons must be involved in the low-energy absorption.
The absorption peak of Cs-HTB in Fig. 2d comprises the main peak at 0.8 eV and a subpeak at 1.4 eV, whereas the subpeaks of Rb-and K-HTB are not clearly defined 6,[24][25][26] .The absorption peaks were decomposed using Mie theory with consideration of the ensemble inhomogeneity.In previous work 29 , we examined the effects of particle size distribution and particle shape on absorption peaks of Cs-HTB dispersions using TEM and small-angle x-ray scattering and concluded that their effects were negligible as compared to those due to the variations in dielectric functions.So, the optical profiles of Cs x WO 3-y (0.20 ≤ x ≤ 0.33, 0.17 ≤ y ≤ 0.46) NPs were decomposed by integrating 10,000 Mie scattered waves from randomly computer-generated particles, each having a different plasma frequency and polaron energy 4,29 .To avoid large-scale calculations, we here assume a normal distribution of parameters defining the absorptions.In addition, we replace the complex arithmetic of Mie scattering by a simple formula under the quasistatic approximation 39 .These simplified procedures are described in the next section.

Simplified procedure of the Mie scattering integration method
NIR absorption from a single NP of an alkali HTB is contributed by LSPR⊥ and LSPRǁ of free electrons and polaronic transition of trapped electrons 2,4,35 .The optical behavior of free electrons is described by the Drude oscillator in the free electron model, whereas the interband transitions of trapped electrons is described by the Lorentz oscillator.Thus, we assume the Drude and Lorentz terms in the dielectric function (in cgs unit) as follows: p ω 2 +iωγ , denote the high-frequency permittivity, plasma frequency, and relaxation constant, respectively, and ω T , Ω E 2 , and Γ in the Lorentz term denote the peak frequency, peak area, and peak width, respectively.The plasma frequency is expressed by Eq. ( 2), where N and m* denote the carrier density and effective mass, respectively.
The dielectric function is assumed to reduce to the Drude term at low energies 1 .Under the quasistatic approximation, the Mie extinction cross-section is given by 31 where λ is the wavelength, ε m is the permittivity of the surrounding medium, and A is a constant.Im(z) indicates the imaginary part of the complex number z.The quasistatic approximation is valid when 2πε m d/λ << 1 (d = particle size) 31 , thus applicable to a present particle of d approximately less than 30 nm in MIBK of ε m = 2.25.Substituting the Drude term of Eq. (1) into Eq.( 3), we can express the LSPR absorption in terms of Drude parameters alone as follows: where B is a constant.
The absorption due to polaronic excitation is incorporated via the Lorentz term in Eq. ( 1).Alternately, it may be directly expressed by the extinction cross section, quoting from Austin and Mott 22,40 , where A p is a temperature-dependent constant, ħ is reduced Planck's constant, E B is the binding energy, k is the Boltzmann constant, and τ is the absolute temperature (K).Equation ( 5) was derived for small polarons by contributions of Reik and Heese 41 , Bogomolov et al. 42 and others, and reviewed by Austin and Mott 40 .Small polarons were suggested in Cs-HTB, as the Voronoi local charge calculation revealed the extra charges on W ions adjacent to V O 5 .As both expressions of the polaronic term involve three parameters, i.e., ω T , Ω E , and Γ for the Lorentz expression and A p , E B , and τ for the Austin-Mott expression, nine parameters must be described for each particle.Below, we illustrate the MSI procedure with the Austin-Mott expression.
The total extinction cross-section of an N-particle ensemble is given by: where X is a general term for the parameters involved in the Drude, Lorentz, or Austin-Mott terms (each particle has nine independent Xs).Correlations between Xs of different particles can be neglected in the present dilute system, as previously shown by measuring σ ext for dispersions of different inter-particle distances 29 .We also assume no correlations between Xs inside a particle.An actual single NP has both aspects of LSPR⊥ and LSPRǁ, which we treat mathematically as 2/3 particles in the ensemble oriented such as to activate only LSPR⊥ and 1/3 particles only LSPRǁ (Thus, coefficient B of σ LSPR⊥ ext is set around twice that of σ

LSPR|| ext
).This treatment assumes no correlation, but explains the experimental curves very well, as shown later.
Let us introduce a uniform variation of X given by X M ± 3.0s X , where X M and s X denote the mean (central) value and standard deviation of X, respectively.The total number of values in X can be an arbitrary odd number greater than (say) 10 and less than 100, depending on the nature of the ensemble.If the number is too small, calculated σ ext profiles do not fit to experimental values.As the number increases, they converge to a fixed fitted curve.Here, we illustrate with 61 particles labeled No. 0 to No. 60.The value of X of particle No. i, X i , is then given by: Here, the parameters of particles No. 0, 30, and 60 are given by X 0 = X M − 3.0s x , X 30 = X M and X 60 = X M + 3.0s x , respectively.That is, every parameter takes a value within ± 3.0s X from the mean value with a uniform interval.The value of s X is restricted to be within 1/3 (33%) because the parameter values must be positive ( X 0 = X M − 3.0s x > 0).
The extinction profiles of the 61 particles calculated by Eq. ( 6) are shown in Fig. 3a.The experimental molar absorption coefficient is plotted in the same chart (note that the extinction reduces to absorption in our particle sizes of concern, as the scattering becomes negligible).After summing these 61 extinction profiles, we expect a large discrepancy from the experimental profile.In the actual distribution, however, the probability of occurrence decreases with increasing deviation of the parameter from its mean value.Therefore, we assign a normally distributed weighting multiplier p i to particle No. i (Fig. 3b): (2) The σ ext of absorption factor Y (Y = LSPR⊥, LSPRǁ, or polaron) is assumed to depend only on X.It is calculated as Among the nine parameters, we varied the most significant and decisive parameters (namely, p⊥ , p and E B 29 ) and the other parameters were determined as the best fit values through a trial and error.The total extinc- tion cross-section is generally expressed as ( 8) The summations over i, j, and k are independent and can be simplified to a summation over i.The total extinction cross-section from the 61 NPs, which will be fitted to the experimental curves, is then given by: On executing the analysis, the successful range of s X is 1/10-1/3, depending on the nature of the ensemble and the extent of parameter inhomogeneities.In the present ensemble, s x of p⊥ , p and E B was taken to be 1/3.Note also that ε ∞ must be greater than unity.In the course of our trials, peak decompositions using Eq. ( 11) generally gave unique results within a narrow range without imposing any a priori conditions.In our previous analysis 4, 29 , the MSI fitting was set under two constraints, fixing the p⊥ / p|| ratio to 5.11/3.09and taking the same bandwidth for LSPR⊥ and LSPRǁ.However, these constraints were found not just necessary but could possibly lead to slight deviation from the true value if strong screening effect (see below) occurred between peaks.Notably, this MSI method requires no information on the filler quantity throughout the analysis; the useful physical parameters of the NPs are obtained simply from the transmittance or absorbance data of the NP dispersions.

Analysis of the absorption peaks of HTB NPs
The Mie scattered waves were integrated such that the total σ ext of the ensemble given by Eq. ( 11) fitted the experimental absorption profiles.The shape of calculated profiles gradually changed and converged to a fixed one as the number of virtual particles increased, as shown in Fig. 3c.It suggests that as low as 11 particles may be sufficient to calculate for this ensemble, though below we present the result with 61 particles.As shown in Fig. 4, the molar absorption coefficients were successfully decomposed into their three main components, yielding numerical values of Ω p , γ, ε ∞ , E B , A p , Ω E , Γ, ω T and P (see Table 1).A small discrepancy seen at energies greater than 2.0 eV arises from the band-edge absorption around 3.5 eV and slight interband excitations due to the insufficient bandgap between 2.0 and 3.5 eV.The results of the simplified method almost matched those of the original MSI method, as shown for Cs 0.32 WO 3 in parentheses in Table 1.
The important trend in Table 1 is the shift in Ω p , P, E B , Ω E , and P pol toward lower energy in the order of Cs → Rb → K, reflecting a consistent decrease in free and localized electrons.This indicates that the amount of V O should decrease in the same dopant order because the alkali content was fixed at x = 0.33.Chemical and XRD Rietveld analyses of sintered bulk HTBs yielded the same results of the decreasing V O as Cs → Rb → K 35 .
To understand the triplet peaks of LSPR⊥, LSPRǁ and polarons in Fig. 4a-c, the electronic structures of HTB are quoted 43 in Fig. 5.According to first-principles calculations 5,43 , the alkali-donated electrons occupy the W-d yz and d zx orbitals (Fig. 5a) while the V O -donated electrons occupy the W − d xy , d x 2 −y 2 orbitals in addition to the (11)     polaron peak is derived entirely from the V O electrons because it did not emerge without the presence of V O even if the dopants existed 4,5 .Hence, the intensities of the LSPR⊥ and polaron peaks decreased in the dopant order of Cs → Rb → K, as expected (Fig. 4d).On the other hand, the LSPRǁ peak remained nearly unchanged, even though it is contributed by both the Cs and V O electrons.
To discuss the peak intensities, one needs additionally to inspect the electron screening effect for contiguous excitations 4,44 .A typical behavior of absorption peaks due to this effect is simulated as shown in Fig. 6.For a given set of anisotropic LSPR peaks, the interband transition (polaron) peak was postulated to approach from the high energy side by varying ħω T .Parameter values used are ε ∞⊥ = 2.1, ħΩ p⊥ = 1.69 eV and ħγ ⊥ = 0.29 eV for LSPR⊥, ε ∞ǁ = 6.0, ħΩ pǁ = 3.00 eV and ħγ ǁ = 0.30 eV for LSPRǁ, and ħΩ E = 0.85 eV and ħΓ = 0.40 eV for the polaron peak, respectively.Here, the extinction cross section, σ ext , was calculated under quasistatic approximation 39 using Eq. ( 3).The anisotropic σ ⊥ and σ ǁ were separately calculated and averaged with weights by σ = (2σ ⊥ + σ � )/3 .When the polaron peak approaches the LSPR⊥ and LSPRǁ peaks from the high energy side, both the LSPR peaks decrease their intensities and are pushed toward the low energy side (screening).When ħω T falls behind the LSPRǁ peak energy at around 1.1 eV, the relative locations are interchanged and the LSPRǁ peak is intensified (anti-screening).Therefore, the screening effect of K-HTB with less V O and weaker polaron peak increases the relative peak intensity of LSPR⊥ and LSPRǁ, which explains the behavior of LSPRǁ peaks in Fig. 4d.It should be noted that the screening effect of nearby peaks can be automatically incorporated in σ, as done in Fig. 6, if one includes the polaron term in Eq. ( 6) not as σ polaron ext but in the expressions of σ LSPR⊥ ext and σ LSPR ext by using anisotropic ε ⊥ and ε ǁ that include the polaron term.This procedure was adopted in our previous work 4 .
Returning to the triplet peaks in Fig. 4, it has been clarified 29,32 that, for both plasmonic and polaronic absorptions, the peak gets weakened and peak position redshifted when V O is decreased.Therefore, it should be the relative redshift and diminution of the polaron peak with respect to the decreased plasmon peak, which caused the less prominent polaronic shoulder in Rb-and K-HTBs, in contrast to the distinct shoulder observed in Cs-HTB.The large difference in χ of M-HTBs as shown in Fig. 2d-f is now clearly resolved as attributed to the difference in the V O amount involved.
Table 1 also reveals changes in the physical properties of NPs relative to the bulk properties.For instance, Ω p is notably smaller in the NPs than in bulk crystals (bulk data given in 35 ). Figure 7 compares the dielectric functions of a single NP derived from the data in Table 1 with those in bulk crystals.The screened plasma frequencies at ε 1 = − 4.5 in the ε 1 profiles of the NPs dispersed in MIBK (ε m = 2.25) are considerably decreased from those of bulk HTBs, from 0.96 to 0.61 eV in Cs-HTB NPs, from 0.97 to 0.62 eV in Rb-HTB NPs, and from 0.83 to 0.57 eV in K-HTBs NPs.The polaron peak in the ε 2 profile is also substantially diminished and redshifted in the NPs relative to bulk.These deteriorations of 31-36% in the dielectric properties are naturally attributed to the decrease in alkali ions on the particle surfaces 7,45 and the decrease in interior V O s 29 during the milling process.These clarifications show that MSI analysis for peak decomposition is highly useful in analyzing the physical properties of a NP in the ensemble without incurring a heavy calculation load.The present method should also be applicable to the optical analyses of various ensembles including noble metal colloids, chalcogenide and perovskite quantum dots, and metal oxide nanoparticle dispersions, if their functional model is known.

Figure 3 .
Figure 3. (a) Extinction profiles (dotted lines) of 61 virtual particles with scattered dielectric parameters.The experimental profile (dashed line) is also shown for comparison.(b) Assumed occurrence-ratio distribution of the extinction profiles.(c) Extinction profiles calculated for 1, 5, 11, and 61 virtual particles.

Figure 5 .
Figure 5. densities of states in the W-d orbitals of (a) V O -free Cs 0.33 WO 3 and (b) V Oincorporated Cs 0.33 WO 2.83 [quoted from Ref. 43 ]; (c) schematic of W-d and O-p hybridized orbitals in a WO 6 octahedron projected onto the x-z plane; (d) molecular orbital diagrams of HTB 43 showing the W-d orbitals accommodating the V O -and Cs-donated electrons.

Table 1 .
Plasmon and polaron parameter values obtained by decomposing the optical absorption profiles of the NPs.The polaron component was calculated using both the Austin-Mott and Lorentz expressions.Values of Cs-HTB in parentheses are those using the original MSI method 4 .