Introduction

Tungsten bronzes possess moderately large bandgaps and abundant conduction electrons, enabling both high visible transparency and intense near-infrared (NIR) absorption1,2,3,4,5. Nanoparticles (NPs) of tungsten bronzes, in which this property is enhanced, have been employed in solar control windows6,7,8,9 and related applications. A growing number of studies10,11,12,13,14,15,16 are devoted to alkali tungsten bronze NPs and their applications, but the absorption mechanism and related electronic structures, including intensive optical-peak analyses, have not been sufficiently reported4,17,18.

Low-energy optical absorption peaks of tungsten oxides and tungsten bronzes have not been accurately decomposed thus far, although their origins are variously discussed and accepted19,20,21,22. Recently, however, the absorption peak profile of Cs-doped hexagonal tungsten bronze (Cs-HTB) NPs was reproduced precisely4 to reveal dual components involving localized surface plasmon resonance (LSPR) of free electrons and polaronic transitions of trapped electrons3,4,20,23. This drew attention to the isostructural alkali HTBs doped with K and Rb, as they produced no observable polaronic subpeak6,24,25,26.

To correctly decompose the optical peak of K- and Rb-doped HTB, one must consider the effect of the ensemble inhomogeneity27,28,29 caused by slight random variations in the particle shapes and physical properties of the individual NPs in the ensemble. Such inhomogeneity is evidenced in the large scatter in the plasmon peak energy, as directly observed in electron energy-loss spectroscopy30. According to optical analysis, the scatter in the plasma frequency of a NP ensemble of Cs-HTB reaches 30%4. Ensemble inhomogeneity is prominently observed in mechanically processed NPs, in which the particle shape, surface composition, internal oxygen vacancy (VO) population, and other features are modified during processing. Being composed of numerous Mie scattered waves, the absorption profile is especially sensitive to variations in particle shape and plasma frequency, which cause peak broadening and redshift as schematized in Fig. 1. In most studies, the observed profiles are described by a single Mie scatterer31,32, whereas an inhomogeneous ensemble actually comprises a large number of slightly different Mie scatterers. To properly treat an inhomogeneous ensemble, we previously proposed a Mie scattering integration (MSI) method with computer-generated random numbers representing spontaneous inhomogeneities29. This method well reproduced the complex optical profiles of plasmonic and polaronic excitations in Cs-HTB NPs and gave a unique route to determine the physical parameters of NPs in the ensemble4,29. However, to reproduce the experimental profiles, the original method required a special program that simulates the actual physical process involving a large number of particles (at least 1000–10,000). The present paper develops a modified and simplified MSI method that lowers the calculation load by greatly reducing the number of virtual particles. When applied to K- and Rb-HTB NPs, the new method not only provided an excellent simulation result that facilitated clarification of the absorption peak origins but also disclosed the features that distinguish K- and Rb-HTBs from Cs-HTB.

Figure 1
figure 1

Schematic showing the ensemble inhomogeneity effect: absorption profiles of a single particle (blue) and a particle ensemble (orange).

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Materials and methods

KxWO3, RbxWO3, and CsxWO3 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% H2–N2 atmosphere, followed by homogenization heating at 800 °C for 1 h in a N2 atmosphere. The resulting deep blue powders are known to be highly reduced tungsten oxides containing many VOs33. Thus, the true chemical formulas would be KxWO3-y, RbxWO3-y, and CsxWO3-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 P63/mcm34. Trace amounts of WO3 and WO2.92 were contained in K0.20WO3, Rb0.20WO3 and Cs0.20WO3, 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 elsewhere35.

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 VOs4,23, 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-axis4,20.

Figure 2
figure 2

Transmission profiles (a–c) and molar absorption coefficients (d–f) of CsxWO3, RbxWO3 and KxWO3 (x = 0.20, 0.25, 0.30) NP dispersions with different alkali content.

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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 log10(I/I0) =  − χ l c, where I0 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 strength6. 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 K0.25WO3 and K0.30WO3 were reversed either by error or by an unclarified reason. See ref.35). If the alkali dopant alone caused the NIR absorption36,37,38, such changes are not expected. Therefore, both the dopant-donated and VO-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 defined6,24,25,26. The absorption peaks were decomposed using Mie theory with consideration of the ensemble inhomogeneity. In previous work29, 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 CsxWO3-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 energy4,29. 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 approximation39. 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 electrons2,4,35. 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:

$${\upvarepsilon } = {\upvarepsilon }_{\infty } - \frac{{{\Omega }_{{\text{p}}}^{2} }}{{{\upomega }^{2} + {\text{i}}\upomega \upgamma }} + \frac{{{\Omega }_{{\text{E}}}^{2} }}{{{\upomega }_{{\text{T}}}^{2} - {\upomega }^{2} - {\text{i}}{\upomega \Gamma }}},$$
(1)
$${\Omega }_{{\text{p}}}^{2} = \frac{{4\pi Ne^{2} }}{{m^{*} }},$$
(2)

where ε∞, Ωp, and γ in the Drude term, \({\varepsilon }_{\infty }- \frac{{{\Omega }_{{\text{p}}}^{2} }}{{{\upomega }^{2} + {\text{i}}\upomega \upgamma }}\), denote the high-frequency permittivity, plasma frequency, and relaxation constant, respectively, and ωT, ΩE2, 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 energies1. Under the quasistatic approximation, the Mie extinction cross-section is given by31

$$\begin{array}{*{20}c} {\sigma_{ext} = A\frac{1}{\lambda }\text{Im}\left( {\frac{{\varepsilon - \varepsilon_{m} }}{{\varepsilon + 2\varepsilon_{m} }}} \right),} \\ \end{array}$$
(3)

where λ is the wavelength, εm 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 2πεmd/λ << 1 (d = particle size)31, thus applicable to a present particle of d approximately less than 30 nm in MIBK of εm = 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:

$$\sigma_{ext}^{LSPR} = \frac{{B\gamma \Omega_{p}^{2} \omega^{2} }}{{\left( {\varepsilon_{\infty } + 2\varepsilon_{m} } \right)^{2} \omega^{4} + \left( {\varepsilon_{\infty } + 2\varepsilon_{m} } \right)\left[ {\left( {\varepsilon_{\infty } + 2\varepsilon_{m} } \right)\gamma^{2} - 2\Omega_{p}^{2} } \right]\omega^{2} + \Omega_{p}^{4} }},$$
(4)

where B is a constant.

Reflecting the crystalline anisotropy in HTB, the Drude parameters have components perpendicular (⊥) and parallel (ǁ) to the c-axis2. Therefore, the Drude factor involves six parameters: \(\varepsilon_{\infty \bot }\), \({\Omega }_{{{\text{p}} \bot }}\), \(\gamma_{ \bot }\), \(\varepsilon_{\infty \parallel }\), \({\Omega }_{{{\text{p}}\parallel }}\) and \(\gamma_{\parallel }\).

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 Mott22,40,

$$\begin{array}{*{20}c} {\sigma_{ext}^{polaron} = \frac{{A_{p} }}{\hbar \omega }\exp \left[ { - \frac{{\left( {\hbar \omega - 2E_{B} } \right)^{2} }}{{8E_{B} k\tau }}} \right],} \\ \end{array}$$
(5)

where Ap is a temperature-dependent constant, ħ is reduced Planck’s constant, EB 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 Heese41, Bogomolov et al.42 and others, and reviewed by Austin and Mott40. Small polarons were suggested in Cs-HTB, as the Voronoi local charge calculation revealed the extra charges on W ions adjacent to VO5. As both expressions of the polaronic term involve three parameters, i.e., ωT, ΩE, and Γ for the Lorentz expression and Ap, EB, 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:

$$\begin{array}{*{20}c} {\sigma_{ext} = \mathop \sum \limits_{X} \left\{ {\sigma_{ext}^{LSPR \bot } + \sigma_{ext}^{LSPR\parallel } + \sigma_{ext}^{polaron} } \right\},} \\ \end{array}$$
(6)

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 σext for dispersions of different inter-particle distances29. 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 \(\sigma_{ext}^{LSPR \bot }\) is set around twice that of \(\sigma_{ext}^{LSPR||}\)). 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 XM ± 3.0sX, where XM and sX 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 σext 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, Xi, is then given by:

$$X_{i} = X_{M} + \left( {0.1i - 3.0} \right)s_{x} .$$
(7)

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 ± 3.0sX from the mean value with a uniform interval. The value of sX 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):

$$p_{i} = \frac{1}{{\sqrt {2\pi } }}\exp \left[ { - \frac{{\left( {0.1i - 3.0} \right)^{2} }}{2}} \right] .$$
(8)
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.

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The σext of absorption factor Y (Y = LSPR⊥, LSPRǁ, or polaron) is assumed to depend only on X. It is calculated as

$$\begin{array}{*{20}c} {\sigma_{ext}^{Y} = \mathop \sum \limits_{i = 0}^{60} p_{i} \sigma_{ext,i}^{Y} \left( {X_{i} } \right).} \\ \end{array}$$
(9)

Among the nine parameters, we varied the most significant and decisive parameters (namely, \({\Omega }_{{{\text{p}} \bot }}\), \({\Omega }_{{{\text{p}}\parallel }}\) and \(E_{B}\)29) and the other parameters were determined as the best fit values through a trial and error. The total extinction cross-section is generally expressed as

$$\sigma_{ext} = \mathop \sum \limits_{i} p_{i} \sigma_{ext,i}^{LSPR \bot } \left( {{\Omega }_{p \bot ,i} } \right) + \mathop \sum \limits_{j} q_{j} \sigma_{ext,j}^{LSPR\parallel } \left( {{\Omega }_{p\parallel ,j} } \right) + \mathop \sum \limits_{k} r_{k} \sigma_{ext,k}^{polaron} \left( {E_{B,k} } \right).$$
(10)

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:

$$\sigma_{ext} = \mathop \sum \limits_{i = 0}^{60} p_{i} \sigma_{ext,i}^{LSPR \bot } \left( {{\Omega }_{p \bot ,i} } \right) + \mathop \sum \limits_{i = 0}^{60} p_{i} \sigma_{ext,i}^{LSPR\parallel } \left( {{\Omega }_{p\parallel ,i} } \right) + \mathop \sum \limits_{i = 0}^{60} p_{i} \sigma_{ext,i}^{polaron} \left( {E_{B,i} } \right).$$
(11)

On executing the analysis, the successful range of sX 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 \({\Omega }_{{{\text{p}} \bot }}\), \({\Omega }_{{{\text{p}}\parallel }}\) 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 analysis4,29, the MSI fitting was set under two constraints, fixing the \({\Omega }_{p \bot }\)/\({\Omega }_{p||}\) ratio to 5.11/3.09 and 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 σext 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 Ωp, γ, ε∞, EB, Ap, ΩE, Γ, ωT 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 Cs0.32WO3 in parentheses in Table 1.

Figure 4
figure 4

Molar absorption coefficients of (a) Cs0.33WO3, (b) Rb0.33WO3 and (c) K0.33WO3 NPs decomposed into the LSPR⊥c, LSPRǁc, and polaron components and (d) their peak area fractions.

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Table 1 Plasmon and polaron parameter values obtained by decomposing the optical absorption profiles of the NPs.
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The important trend in Table 1 is the shift in Ωp, P, EB, ΩE, and Ppol 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 VO 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 VO as Cs → Rb → K35.

To understand the triplet peaks of LSPR⊥, LSPRǁ and polarons in Fig. 4a–c, the electronic structures of HTB are quoted43 in Fig. 5. According to first-principles calculations5,43, the alkali-donated electrons occupy the W‒dyz and dzx orbitals (Fig. 5a) while the VO-donated electrons occupy the W − \({d}_{xy}\), \({d}_{{x}^{2}-{y}^{2}}\) orbitals in addition to the W‒dyz and dzx orbitals (Fig. 5b). These W‒d electrons are hybridized with the O‒p orbital electrons (Fig. 5c) to construct t2g* and eg* states in the π and σ bonds, respectively (see the molecular orbital diagrams in Fig. 5d). Herein, the geometrical electronic distributions across the overlapped orbitals suggest that the W − \({d}_{xy}\), \({d}_{{x}^{2}-{y}^{2}}\) electrons in eg* are related only to LSPR⊥, whereas the W − dyz and dzx electrons in t2g* are related to both LSPR⊥ and LSPRǁ. Thus, the LSPR⊥ peak should be contributed by both the VO and Cs electrons in W − \({d}_{xy}\), \({d}_{{x}^{2}-{y}^{2}}\), dyz and dzx, whereas the LSPRǁ peak should be contributed by the Cs and VO electrons in W − dyz and dzx. The polaron peak is derived entirely from the VO electrons because it did not emerge without the presence of VO even if the dopants existed4,5. 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 VO electrons.

Figure 5
figure 5

Calculated densities of states in the W‒d orbitals of (a) VO-free Cs0.33WO3 and (b) VO-incorporated Cs0.33WO2.83 [quoted from Ref.43]; (c) schematic of W‒d and O‒p hybridized orbitals in a WO6 octahedron projected onto the x‒z plane; (d) molecular orbital diagrams of HTB43 showing the W‒d orbitals accommodating the VO- and Cs-donated electrons.

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To discuss the peak intensities, one needs additionally to inspect the electron screening effect for contiguous excitations4,44. 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 ħωT. Parameter values used are ε∞⊥ = 2.1, ħΩp⊥ = 1.69 eV and ħγ⊥ = 0.29 eV for LSPR⊥, ε∞ǁ = 6.0, ħΩpǁ = 3.00 eV and ħγǁ = 0.30 eV for LSPRǁ, and ħΩE = 0.85 eV and ħΓ = 0.40 eV for the polaron peak, respectively. Here, the extinction cross section, σext, was calculated under quasistatic approximation39 using Eq. (3). The anisotropic σ⊥ and σǁ were separately calculated and averaged with weights by \(\sigma =(2{\sigma }_{\perp }+{\sigma }_{\parallel })/3\). 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 ħωT 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 VO 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 \({\sigma }_{ext}^{polaron}\) but in the expressions of \({\sigma }_{ext}^{LSPR\perp }\) and \({\sigma }_{ext}^{LSPR\parallel }\) by using anisotropic ε⊥ and εǁ that include the polaron term. This procedure was adopted in our previous work4.

Figure 6
figure 6

The electron screening and anti-screening effects of contiguous excitations calculated for the different locations of polaron peak with respect to LSPR⊥ and LSPRǁ peaks. Parameter values used are ε∞⊥ = 2.1 eV, Ωp⊥ = 1.69 eV, γ⊥ = 0.29 eV, ε∞ǁ = 6.0 eV, Ωpǁ = 3.00 eV, γǁ = 0.30 eV, ΩE = 0.85 eV, Γ = 0.40 eV, and ωT = varied.

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Returning to the triplet peaks in Fig. 4, it has been clarified29,32 that, for both plasmonic and polaronic absorptions, the peak gets weakened and peak position redshifted when VO 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 VO amount involved.

Table 1 also reveals changes in the physical properties of NPs relative to the bulk properties. For instance, Ωp is notably smaller in the NPs than in bulk crystals (bulk data given in35). 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 ε1 =  − 4.5 in the ε1 profiles of the NPs dispersed in MIBK (εm = 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 ε2 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 surfaces7,45 and the decrease in interior VOs29 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 7
figure 7

Effective dielectric functions of (a) Cs0.33WO3, (b) Rb0.33WO3 and (c) K0.33WO3 NPs at low energies, showing significant modifications from their respective bulk dielectric functions. Red arrows indicate the screened plasma frequencies.

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Conclusions

The optical absorption profiles of reduced K-, Rb-, and Cs-HTB NPs were decomposed using the modified MSI method and analyzed in terms of the dopant and VO electrons. The number of virtual particles to be integrated was considerably lowered by approximating the parameter variations as normal distributions. The new calculation scheme incorporated only 61 particles with varied parameters and its results minutely differed from those of the original method, which was rigorously executed with 10,000 particles. It was suggested that number of virtual particles may be minimized to 11 in the present ensemble.

The new method decomposes the absorption peaks into two LSPR and one polaronic excitation, as previously analyzed for Cs-HTB4. In NPs of all the alkali HTB, free electrons coexisted with trapped electrons having similar energies. The plasma frequencies and polaron energies of NPs decreased in the dopant order Cs → Rb → K, consistent with the VO decrease observed in sintered bulk plates35. It was found that the LSPR⊥ peak is derived from Vo electrons in W−\({d}_{xy}\) and W−\({d}_{{x}^{2}-{y}^{2}}\) orbitals and Cs electrons in W‒dyz and W‒dzx, whereas the LSPRǁ peak is derived from Cs and VO electrons in W‒dyz and W‒dzx orbitals. The decomposed peak location and intensity were consistently interpreted in terms of the electronic structures, screening effect, and modified dielectric functions. The decreased and redshifted LSPR⊥ and polaron peaks, combined with the relatively stable LSPRǁ peak, caused the shoulder-less appearance of the absorption peaks of the K- and Rb-HTBs NPs. The decomposition of the absorption peaks also enabled to determine the effective dielectric functions and physical parameters of an average NP in the ensemble. VOs largely influenced the optical properties of the alkali HTBs, and their effect increased in the dopant order K → Rb → Cs. The validity of the present MSI method is not limited to the present HTB materials but may be extended to other plasmonic and polaronic systems, including noble metal, metal oxide, chalcogenide, and perovskite NPs.