First-Principles Calculations of x-Dependent Ground Structures and Optical Properties of Ca x La1-x B6.

High transparency in the visible region is desired to manufacture solar control films and glasses for various applications. To improve the visible light transparency of LaB6 nanoparticles which exhibit strong absorption in the near-infrared region, the substitution of La with Ca is investigated using first-principles calculations. Among the numerous atomic replacement configurations in Ca x La1-x B6, all 762 structures existing in the supercells that are up to 8 times the primitive cell are comprehensively evaluated, and the most stable ground structures in Ca x La1-x B6 are deduced. The optical properties of the ground structures are derived by performing high-precision calculations using the HSE06 functional, which reveal that Ca x La1-x B6 with 0 < x < 1/4 is preferred as a solar shielding material with improved visible transparency. This method is effective for the investigation of the effect of substitutional elements in composite compounds on their physical properties.

Introduction

Lanthanoid metal hexaboride, RB6, possesses unique physical properties such as high conductivity, high hardness, and superconductivity.[1] In particular, LaB6 has been studied extensively as a thermionic emission material owing to its low work function and high melting temperature.[2] Recently, it was found that the nanoparticles of LaB6 produce a strong absorption in the near-infrared region coupled with high transparency in the visible light region.[3] This allowed its application in solar control films and glasses, including automotive monolithic, laminated, and polycarbonate glasses, and extended applications in laser welding and cancer therapies employing their efficient photothermal conversion capabilities.[4−13]

The near-infrared absorption of LaB6-dispersed layers occurs with a strong dependence on the particle size and does not occur at a micron scale.[10] Its absorption energy agrees with the plasmon energy (1.3 eV) measured using electron energy-loss spectroscopy.[8] These observations confirm that the near-infrared absorption of LaB6 fine particles is caused by the localized surface plasmon resonance (LSPR) of free electrons, similar to Ag, Au, and ITO (In2O3;Sn) fine particles.

However, the visible light transparency of LaB6 is not as high as the transparent conductive material of ITO and the nearly transparent plasmonic material of Cs0.33WO3, owing to a strong green tint caused by an absorption in the visible region. If the visible light transparency of LaB6 is increased, the tint neutralization and adjustment using color pigments in glass processing can be made easier, and a high near-infrared absorption can be realized at a given visible light transmittance to save energy consumption.

One of the techniques for improving visible light transparency is elemental substitution. A few studies have been reported to control the plasma resonance wavelength of LaB6 by elemental substitution.[14−17] Bao et al. synthesized the alloys of SmLa1–B6 and EuLa1–B6, in which La was replaced by Sm and Eu, respectively, in LaB6, and reported that the plasmon resonance wavelength shifted toward a longer wavelength.[15,16] However, these results indicate that the absorption in the visible region also increases and is not suitable for solar shielding applications because both high visible transparency and strong near-infrared absorption are simultaneously required. Mattox et al. also reported the results for Eu-doped LaEu1–B6 containing B vacancies.[17] The red shift of the plasma resonance wavelength and the decrease in the visible absorption were the same as those reported by Bao et al. and did not improve the solar control properties.

Thus, the previously reported studies were limited to LaB6 with the replacement of La by other lanthanoid elements, where the wavelength control of the plasmon resonance was only toward the long wavelength and the transparency in the visible region was impaired. However, developments that lead to a true improvement in the optical characteristics, that is, a concurrent increase in the visible transparency and near-infrared absorption, has not yet been reported. Therefore, in this study, a detailed investigation of the visible light transparency of LaB6 is conducted using first-principles calculations.

It has been previously reported by our research group that the dielectric functions that determine the optical properties of LaB6 can be predicted with high accuracy by first-principles calculations.[18] The optical properties of nanoparticle dispersions are analyzed by the dielectric functions and Mie scattering theory, but when the particle shape and dielectric functions are modulated during the nanoparticle formation, as in the case of LaB6,[12,19] a new analysis method such as the Mie scattering integration (MSI) method is used to describe the optical response. The MSI method integrates the individual Mie scattering waves from particles with inhomogeneous physical properties and/or shapes in an ensemble.[12,20,21]

The visible transparency of LaB6 was also analyzed recently,[18] and it was found to be essentially derived from the wide-gap band structure and reduced transition probability because of the selection rule that prohibits transitions from the B-p orbital in the valence band to the B-p and La-p orbitals in the conduction band. The spectral blue-side transmission is restricted by the band-edge absorption and the red-side transmission is strongly affected by the LSPR wavelength. Therefore, a wide-gap band structure and low plasma frequency are the requisites for a replacement material for LaB6. In this study, the substitution of La with divalent ions, specifically, alkaline earth metals, is investigated. Alkaline earth metal hexaborides such as CaB6, SrB6, and BaB6 are wide-gap semiconductors/insulators and are isomorphic with LaB6. The electronic structures of CaB6, SrB6, and BaB6[22−24] are reported in prior literature reports, but no detailed studies including the dielectric functions have been published. Therefore, high-accuracy calculations of the optical properties of CaB6, SrB6, and BaB6 were performed to determine the candidate substitution element, and the effects of the elemental substitution in LaB6 at various proportions were examined.

To precisely determine the absorption of the substituted material in the visible region, it is necessary to quantitatively evaluate the effect of interband transitions. However, in a ternary compound in which a portion of La atoms is replaced by another element, the energy and electronic state of the compound strongly depend on the positional relationship between the substituted and matrix elements, and correspondingly different optical characteristics are observed. In the literature, first-principles calculations of several substituted hexaborides have been reported, such as La0.875Sm0.125B6[15] and Ca0.325La0.625B6,[25] but these do not incorporate the various substitutional configurations and calculate only a single type of replacement arrangement. The calculations with only a limited type of arrangement do not afford a satisfactory prediction for the experiments. In the present study, a comprehensive and exhaustive approach is used for the arrangement of elements, and the lowest-energy ground structures of LaB6 with the substitution of La by an alkali earth element at various proportions are identified. Furthermore, the changes in the optical properties with variations in the ground structures are analyzed in detail, and the optical properties of the actual nanoparticle ensembles are inspected.

Results and Discussion

The crystal structure of LaB6 is shown in Figure . LaB6 has a CsCl-type simple cubic crystal structure, where a = b = c and α = β = γ = 90°. The space group belongs to Pm3̅m, consisting of La at the Cs position and B6 octahedral cluster at the Cl position. Similarly, CaB6 has a CsCl-type simple cubic crystal structure in which La is replaced by Ca.

Figure 1

Crystal structure of LaB6.

A previous study reported by our research group shows that the optical characteristics of LaB6 can be reproduced quantitatively by considering the Drude term, which includes the effect of free electrons, and by calculating the effect of interband transitions using a hybrid calculation method that accurately reproduces the band gap.[18] Details are described in the Methods section.

The structures of CaB6, SrB6, and BaB6 are represented in Figure , where La is replaced by Ca, Sr, and Ba. Figure shows the results of the calculation of the dielectric functions of LaB6, CaB6, SrB6, and BaB6. Focusing on the imaginary part of the dielectric function, LaB6 shows a small absorption in the visible region and a large absorption in the low-energy region below 1.8eV, which makes it suitable for application as a solar radiation shielding material.

Figure 2

Dielectric functions of LaB6 and XB6 (X = Ca, Sr, Ba) derived from first-principles calculations using VASP with a HSE06 hybrid functional.

Considering the ε2 values for CaB6, SrB6, and BaB6, the substitution of trivalent La with a divalent alkaline earth metal reduces the free electrons and significantly decreases the absorption in the low-energy side derived from the Drude term. Considering the energy at which ε1(ω) = 0, which determines the screened plasma resonance frequency (Ωsp), the Ωsp values of BaB6, SrB6, and CaB6 are all significantly lower than that of LaB6, that is, BaB6, SrB6, and CaB6 are not expected to absorb near infrared light. In contrast, ε2 value in the visible energy is high for BaB6 and decreases for SrB6 and CaB6 in this order. Therefore, CaB6 was selected as a substitution material for LaB6.

Next, the effect of Ca substitution of La in CaLa1–B6 was examined in detail. Figure shows the convex hull and the mixing energy of each structure of CaLa1–B6 obtained by the first-principles calculations, where the energies of LaB6 and CaB6 are set at 0 eV. For each x of CaLa1–B6, the energies of the possible configurations vary significantly depending on the difference in the substitution sites, that is, the difference in the microscopic crystal structure. The convex hull that traces the envelope of the lowest energies includes all energies of the most stable states of CaLa1–B6.

Figure 3

Results of energy calculations of 762 substitutional configurations showing convex hull in CaLa1–B6.

In addition to LaB6 and CaB6, six compositions are found as the ground states that constitute the most stable state of CaLa1–B6. These include CaLa7B48 (x = 1/8), CaLa5B36 (x = 1/6), Ca2La6B48 (x = 1/4), Ca3La3B36 (x = 1/2), Ca3LaB24 (x = 3/4), and Ca7LaB48 (x = 7/8). The integer corresponds to the number of atoms contained in the calculated unit cell. The most stable state of CaLa1–B6 is composed of these eight types of structures.

In other words, LaB6 and CaLa7B48 coexist when the Ca substitutional fraction x is in the range of 0 < x < 1/8, and CaLa7B48 and CaLa5B36 coexist in the range of 1/8 < x < 1/6. In the range of 1/6 < x < 1/4, CaLa5B36 and Ca3La3B36 coexist, whereas in the range of 1/2 < x < 3/4, Ca3La3B36 and Ca3LaB24 coexist. Furthermore, Ca3LaB24 and Ca7LaB48 coexist in the range of 3/4 < x < 7/8, and in the range of 7/8 < x < 1, a mixed state including Ca7LaB48 and CaB6 can be realized.

At finite temperatures, structures other than the ground-state structures can be realized considering the mixing energies being on the order of room temperature, but it is still important to examine the ground-state structures in detail. The six ground structures excluding the terminal structures are shown in Figure . For simplification, only the atoms for the La site are shown, and the B6 cluster is not drawn. Figure a shows a structure in which 1/8 La atoms are replaced by Ca in a supercell that is 8 times as that of LaB6, and Ca is arranged continuously in the c-axis direction. A structure in which 1/6 La atoms are replaced by Ca is shown in Figure b. Ca is not present at the nearest La site. In the structure in Figure c, 1/4 La atoms are replaced by Ca. Two Ca atoms are connected in the c-axis direction in a manner similar to that shown in Figure a. Figure d shows a structure in which a half of La atoms are replaced by Ca, where La and Ca are clustered in the separated regions. Figure e,f shows structures in which 3/4 and 7/8 La atoms, respectively, are replaced by Ca atoms. These atomic configurations are opposite to those in Figure a,c, where La is placed apart instead of Ca. Thus, the Ca substitution site in the most stable structure varies considerably depending on the fraction of Ca substitution. Figure shows that Ca atoms tend to locate close to each other. This could cause a local bias in charge, but because the large anion of the B62– cluster forms a strong network of sublattice and shields the influence of charge bias, the structure stays stable. The tendency of Ca to cluster is thought to be because of the different electronic behaviors of La and Ca in the B6 sublattice. The shielding of the local charge is confirmed in the small absolute values of mixing energies, as shown in Figure .

Figure 4

Ground structure in CaLa1–B6; (a) CaLa7B48, (b) CaLa5B36, (c) Ca2La6B48, (d) Ca3La3B36, (e) Ca3LaB24, and (f) Ca7LaB48. The boron sublattice is not drawn for clarity.

The changes in the plasma frequencies and dielectric functions of CaB6, LaB6, and the six structures constituting the most stable state were also examined in detail. Table shows the changes in plasma frequencies of these structures. The plasma frequency was found anisotropic; thus, the values in the x, y, and z directions as well as the averages are listed. Because trivalent La is substituted by divalent Ca in CaLa1–B6, it is expected that the number of free electrons and plasma frequency decrease with increasing Ca content in a simple random substitution model. Thus, the plasma frequency of CaB6 is close to 0 eV. However, the data included in Table show that the changes in plasma frequencies upon the variation of x are not uniform. For example, the largest plasma frequency is 4.01 eV for Ca2La6B48. Furthermore, anisotropy is particularly prominent in CaLa7B48 and CaLa5B36. CaLa7B48 includes the substitution of 1/8 La atoms, and the plasma frequencies in the x, y, and z directions are 0.0, 0.0, and 4.3 eV, respectively, that is, the free electrons can move in the z direction but not in the x and y directions.

Table 1

Plasma Frequency of CaLa1–B6

	plasma frequency
composition	x (eV)	y (eV)	z (eV)	average (eV)
LaB6	3.64	3.64	3.64	3.64
CaLa7B48	0.00	0.00	4.31	1.44
CaLa5B36	2.63	2.63	0.00	1.75
Ca2La6B48	3.76	3.57	4.70	4.01
Ca3La3B36	1.49	1.49	1.71	1.56
Ca3LaB24	2.44	2.44	2.44	2.44
Ca7LaB48	2.19	2.58	1.96	2.24
CaB6	0.36	0.36	0.36	0.36

Figure shows the dielectric functions obtained for CaB6, LaB6, and the six ground structures. Here, the averaged values in the x, y, and z directions are shown. LaB6 and CaB6 are represented by blue and gray lines, respectively. In terms of ε2(ω), LaB6 absorbs weakly in the visible light region, and a high absorption is observed in the low-energy side including the near-infrared region. These characteristics are consistent with the measurements[3,8,10] and calculated values[18] reported in previous studies. For applications as a solar shielding material, ε2(ω) should have a large value in the near-infrared region and a small value in the visible region. Replacing a whole La with Ca significantly reduces the free electrons and accordingly decreases the optical absorption in the near-infrared region.

Figure 5

(a) Real and (b) imaginary parts of the calculated dielectric functions of the ground structures in CaLa1–B6. Crystal anisotropy is ignored, and only the averaged profiles are shown.

The dielectric function of CaLa1–B6 does not show a uniform change with variation in the amount of Ca substitution, similar to the trend observed for plasma frequency. Considering ε2(ω), the absorption in the visible region generally decreases with an increase in the amount of Ca substitution, and the transparency in the visible region tends to improve. However, the Ca2La6B48 and Ca3La3B36 compositions with 25 and 50% Ca substitution, respectively, show higher absorptions in the visible region than LaB6, whereas Ca3LaB24 shows a weaker absorption than Ca7LaB48. These phenomena originate from the high anisotropy because of the Ca substitution configurations.

In the real part of the dielectric function, LaB6 has ε1(ω) = 0 on the highest energy side among the other compositions. This shows that Ωsps of CaLa1–B6 shifts toward lower energy in comparison to that of LaB6. However, the change in Ωsp is not uniform with respect to the variation in Ca content, for example, the Ωsps values of Ca3LaB24 and Ca3La3B36 are located on the lower side than that of Ca7LaB48. To obtain sufficient absorption in the near-infrared region, the Ωsp value should be approximately 1.3 eV of CaLa5B36 or greater among those corresponding to the ground-state compositions. That is, the compositions that can be employed as solar radiation shielding materials are CaLa7B48, CaLa5B36, and Ca2La6B48. The range of 0 < x < 1/4 for CaLa1–B6 is preferable because the ε2 values of these compositions are small in the visible range, which is an indication of high transparency in the visible region. This study shows that even when La in LaB6 is substituted with a typical element such as Ca, the change in optical properties is not uniform and varies significantly depending on the substitution fraction and structure.

Next, to determine the characteristics of actual nanoparticle dispersions, the absorption efficiency was calculated using the Mie scattering theory. The optical properties of the nanoparticle dispersions depend on various factors, in particular, the particle shape, dielectric function, and their ensemble inhomogeneities, which are known to have a considerable influence on the optical response.[12,19,20] For LaB6 nanoparticles, the absorption band width has been shown to become 7 times as that of a single spherical particle because of the effect of disk-shaped ensemble inhomogeneity.[12] Lattice defects[19] introduced during the preparation of nanoparticles are analyzed to reduce the relaxation time and increase the base level of ε2, both of which cause their ensemble effects. However, the ensemble inhomogeneity of nanoparticles was ignored in this study as it is outside the scope of this report. An ideal uniform ensemble of spherical particles with an average diameter of 20 nm was assumed, and the absorption efficiency was calculated based on the dielectric function of each structure. For cases where modifications are introduced in the particle shape and/or dielectric function, the corresponding experimental profiles can be predicted by the methods described in refs.[12,20,21]

Figure a shows the absorption efficiencies of LaB6, CaLa7B48, CaLa5B36, and Ca2La6B48. The plasma resonance wavelength shifts toward a high wavelength, in agreement with the corresponding change in ε1(ω). This causes a simultaneous reduction in absorption in the visible region in CaLa7B48 and CaLa5B36. On the other hand, almost no change is introduced in the high-energy side where the band-edge transition is dominant.

Figure 6

(a) Absorption efficiencies and (b) absorption efficiencies normalized by LSPR absorption peak calculated using the Mie scattering theory for LaB6, CaLa7B48, CaLa5B36, and Ca2La6B48 spherical nanoparticles of size 20 nm.

Figure b shows the absorption efficiency normalized by the near-infrared LSPR absorption. The CaLa7B48 and CaLa5B36 nanoparticles show significantly enhanced transparencies at high wavelengths in the visible region and intensified broad-band absorptions in the near-infrared region. In comparison to LaB6, the color tint and transparency significantly improve in addition to an enhanced NIR absorption. The uniform ensemble of the spherical nanoparticles assumed in the present analysis can be produced experimentally by selecting an appropriate synthesis method. For deviations in the nonspherical shape and high relaxation time accompanying ensemble inhomogeneity that are likely introduced in the mechanical production process, the plasmon absorption band should be peaked at the longer-wavelength side with a broader full width at half-maximum, where the composition, Ca2La6B48, could be an attractive choice. For calculations, including the effects of nonspherical particle shape, lattice defects, surface modifications, internal stress, and their ensemble inhomogeneity effects, methods are detailed in the previous researches.[12,19−21]

Conclusions

An alkaline-earth hexaboride was studied as a replacement material to improve the transparency of LaB6, by first-principles calculations, and CaB6 was selected. The most stable structures of CaLa1–B6 among 762 substitutional configurations with different compositions were comprehensively searched, and eight basic ground structures were found. Examinations of the optical characteristics of the ground structures using high-precision calculations revealed that the influence of Ca substitution varied significantly, depending on the substitutional positions and fractions, and the changes in the optical characteristics were not uniform. It was determined that in the range of 0 < x < 1/4 in CaLa1–B6, the transmittance in the visible light region improved while preserving the plasmon absorption in the near-infrared region, whereas for x > 1/4, the plasmon absorption weakened and the relative transmission in the visible region did not improve. Therefore, the compositions with 0 < x < 1/4 in CaLa1–B6 were preferred as solar shielding materials. In general, the physical properties of the substituted composite compounds are effectively analyzed by using this approach, that is, the ground structures are extracted by creating a convex hull through an exhaustive and comprehensive evaluation of low-energy substitution configurations.

Methods

Vienna ab initio simulation package (VASP)[26−29] was used for the first-principles calculations. As the exchange–correlation functional, a hybrid HSE06 functional[30] that reproduced the band gap with high accuracy was used.

The dielectric function ε(ω) that determines the optical characteristics is composed of the real part ε1(ω) and imaginary part ε2(ω), as shown in eq , which are interrelated through the Kramers–Krönig equation (eq ). Thus, if ε2(ω) can be obtained with high accuracy up to a high-energy region, ε(ω) can be obtained by the relationship defined by the Kramers–Krönig equation.ε2(ω) of the dielectric function corresponds to the absorption of photons by electrons and is composed of the Lorentz term, which is the contribution of the interband transition shown in eq , and the Drude term, which is the contribution of free electrons.where V is the unit cell volume; c and v are subscripts indicating the valence band and conduction band, respectively; E is the energy level; is the unit vector; u is the periodic part of the orbit; and w is the weight of the k points.[31]

LaB6 is a conductor whose Fermi energy crosses the conduction band. Therefore, in addition to the Lorentz term, ε2Lorentz(ω), the Drude term is also required. Especially at low energies, the Drude term becomes dominant over the Lorentz term, thereby dictating the low-energy behaviors with the analytical expression of the Drude term. The validity of this method has been confirmed by the excellent agreement with the measured dielectric functions in LaB6, including the low-energy region.[18] The Drude term can be written as eqs and 6 using the plasma frequency ωp, shown in eq , and relaxation time τ = 1/γ.where f is the Fermi–Dirac distribution function, n is the free electron density, e is the charge of the electron, ε0 is the permittivity of the vacuum, and m* is the effective mass of the electron.[32]

The plasma frequency can be obtained from the band structure using eq . The relaxation time τ = 1/γ represents the scattering effect because of the electron–phonon, electron–electron, and electron–lattice defect interactions; however, the influence on the dielectric function because of the fluctuation of γ is not very large,[18] such that we set ℏγ = 0.1 eV in this study.

An infinite number of Ca-substituted configurations exist in the CaLa1–B6 structure. The possible substitutional configurations depend on the size of the supercell. Therefore, in this study, the supercell size was increased stepwise; the LaB6 primitive cell was doubled, tripled, and eventually expanded up to 8 times the primitive cell. First-principles calculations were performed for all 762 structures of CaLa1–B6, which covered all the possible substitutional configurations assumed in all supercells that were up to 8 times the primitive cell and containing 56 atoms. In this step, the generalized gradient approximation-Perdew–Burke-Ernzerhof functional was used as the exchange–correlation functional. The crystal size, shape, and atomic position were fully relaxed. The x-dependence of the most stable energy state of CaLa1–B6, that is, the convex hull, was obtained, and the most stable structure at each substitutional fraction x was identified. Thereafter, for each of the most stable structures obtained, high-precision band structure calculation using the HSE06 functional was performed to derive the optical characteristics. For executing the automatic expansion and replacement operations of the primitive cells, the cluster expansion program ICET[33] was used.