First-Principles Calculations of Structural, Elastic, Electronic, and Optical Properties of Pure and Tm²⁺-Doped Alkali‒Earth Chlorides MCl₂ (M = Ca, Sr, and Ba)

Abstract

The structural, elastic, and electronic properties of pure and Tm²⁺-doped alkali–earth chlorides are studied in detail using the hybrid density-functional theory as implemented in the CRYSTAL09 code. The calculated local geometrical structures of the dopant site, Mulliken populations, electronic-band structures, and density of states for the pure and doped crystals are analyzed and compared to reveal the changes induced by the impurity ion. Additionally, the electronic and optical properties of the Tm²⁺-ions doped into alkali–earth chlorides are modeled by employing the exchange-charge model of the semi-empirical crystal-field theory with the help of the optimized local coordination structures around impurities. The successful simulation of the 4f-5d transition spectrum of Tm²⁺ ions in SrCl₂ is demonstrated as an example to show the validity of the combined theoretical scheme of the first-principles and crystal-field models.

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5.1 Introduction

Rare-earth ions (or lanthanides, chemical elements from La [No. 57] to Lu [No. 71]) have been actively used for more than several decades in numerous optical applications such as phosphor materials for displays and lighting, solid-state lasers operating in various spectral regions from infrared to ultraviolet, infrared to visible up-conversion materials, etc. [1 and references therein]. The physical reason behind such a wide range of applications is that all these elements have an unfilled 4f electron shell, which is shielded from the crystalline environment by the outer completely filled 5s and 5p electron shells. A large number of energy levels, which arises from various interelectron interactions within the 4f shell, can be additionally split by a crystal field. The electronic transitions between all those energy levels cover a wide energy range and can be employed for various applications.

In a vast majority of these applications, these ions are introduced into crystalline solids in their stable trivalent state. However, there are a few lanthanide ions whose divalent state is also stable, e.g., Eu²⁺, Tm²⁺. One of the main differences between the trivalent and divalent lanthanides (apart from the electrical charge) is that the charge-transfer transitions and the parity allowed 5d-4f transitions of the divalent ions are located at lower energies than those of the trivalent ions; because of this they can more easily studied experimentally. Regarding Tm²⁺ ions, it should be kept in mind that the 4f¹³ electron shell has only two energy levels: the ground state ²F_7/2 and the excited state ²F_5/2 located at approximately 9500 cm⁻¹ [2]. As a result, the 4f-4f and 4f-5d transitions of the Tm²⁺ ions do not overlap and can be easily studied separately.

The MCl₂ (M = Ca, Sr, and Ba) crystals are ideal candidates for studies of the spectroscopic properties of Tm²⁺ ions. First, no charge-compensation defects are required after thulium ions are introduced into these hosts. Second, there is reliable experimental information about the absorption/emission spectra of these materials [2–8]. Moreover, the earth-alkaline halides are also important from an application point of view because the laser action has been achieved in CaF₂:Tm²⁺ [9]. The spectra of Tm²⁺ were modeled in our earlier publication [10].

Theoretical interpretation of peculiar features of impurity-ion spectra in crystals can be advanced by the combination of the semi-empirical crystal-field (CF) theory and the modern density-functional theory (DFT)–based computational methods. Such a combination highlights the strong features of both computational approaches and allows for creating a complementary picture of the electronic properties of doped optical materials. The short comparative list of the advantages and disadvantages of both methods is as follows: The CF theory can handle the multiplet structure of the impurity ion energy levels including low-symmetry splitting of the J manifolds and admixture of the electronic configurations of the opposite parity. These questions are beyond the current opportunities of currently used DFT-based computational packages because they are essentially based on the one-electron approximation. At the same time, the electronic-band structure of a solid simply cannot be obtained within the framework of a CF theory, whereas such calculations are already among the standard ones in the DFT-based methods. However, from calculations of the density of states of a doped material, it is possible to locate the ground state of an impurity ion in the band gap. Determination of the ground-state position in the host band gap is a crucial issue and has decisive importance for potential applications of a given crystal with intentionally introduced impurities. After that, the energy levels of an impurity ion, calculated with the help of the CF theory, can be superimposed onto the host’s band structure starting from the ground state. Such an approach was used earlier for a number of crystals doped with transition-metal and rare-earth ions [11–13].

Recently we presented an example of such consistent joint application of these two different approaches to the description of the optical properties of Tm²⁺ions in SrCl₂ [10], and the success of such an approach was demonstrated and confirmed by good agreement between the experimental and calculated optical properties of SrCl₂:Tm²⁺. This system has not been thoroughly studied so far: To illustrate such a statement, we also mention that earlier the crystal-field parameters for the 4f electrons of the Tm²⁺ ions in similar compounds of CaF₂, SrF₂, and BaF₂ were analyzed in Refs. [14, 15], but they were not analyzed in CaCl₂, SrCl₂ and BaCl₂.

Based on the above-given consideration and examples, we decided to apply the DFT-based calculations to three alkali metal halide crystals, namely, CaCl₂, SrCl₂, and BaCl₂ doped with Tm²⁺ ions. The structural, electronic, optical, and elastic properties of these compounds were determined using first-principles calculation: The location of the 4f and 5d energy levels of the Tm²⁺ ions in the band gaps was identified. All obtained results are compared with the experimental data (if available), and agreement between the theory and the results of the experiment is discussed.

5.2 Crystal Structure

All three compounds studied here crystallize in different space groups. The corresponding structural data are summarized in Table 5.1, whereas Fig. 5.1 depicts unit cells for each of the studied crystals as well as the local coordination structures of alkaline-earth cations.

Table 5.1 Calculated and experimental structural data for MCl₂ (M = Ca, Sr and Ba) crystals

Fig. 5.1

Schematic representations for the unit cell of pure MCl₂ crystal and the local coordination structure around M cation (M = Ca [a], Sr [b], and Ba [c]). Drawn with VESTA [19]

The Ca atoms in CaCl₂ are sixfold coordinated by the chlorine ions, and the Ba and Sr atoms in BaCl₂ and SrCl₂ are eightfold and ninefold coordinated by the chlorine atoms, respectively.

As a common feature of all these crystals, we note here that the metal‒chlorine distance is rather large. In CaCl₂, there are two Ca–Cl distances of 2.704 Å and four distances of 2.765 Å. In SrCl₂, all eight Sr‒Cl bonds are equal to 3.019 Å. Finally, Ba–Cl distances in BaCl₂ vary in a wide range from 2.864 to 3.579 Å.

5.3 Methods of Calculations

5.3.1 Ab Initio Calculations

The first-principles calculations of the structural, elastic, and electronic properties of pure and Tm²⁺-doped hosts were performed using the periodic ab initio CRYSTAL09code [20] based on the linear combination of the atomic-orbitals method. Closed-shell and spin-polarized calculation forms were adopted for the pure and doped systems, respectively. To achieve better agreement between the calculated and experimental properties of the considered materials and improve the description of the strongly correlated lanthanide 4f orbitals [21], we employed the hybrid exchange‒correlation (EXC) functional WC1PBE consisting of a PBE correlation part and a Wu-Cohen exchange part with a fractional mixing (16 %) of the nonlocal Hartree‒Fock exchange [22]. The positive performance of such a hybrid EXC functional was confirmed by our recent works [23–25]. The local Gaussian-type basis sets (BS) were chosen as follows: the all-electron BSs in the form of 86-311G and 86-511d21G were used for the Cl [26] and Ca [27] atoms, respectively; the full-relativistic effective core pseudo-potentials (ECP), i.e., ECP28MDF and ECP46MDF—together with their corresponding valence BSs from the Stuttgart/Cologne group—were applied to Sr and Ba atoms [28], respectively; for Tm atom, the inner electrons [Ar]3d¹⁰ were described by the scalar-relativistic ECP developed by Dolg et al. [29]; the other electrons 4s²4p⁶4d¹⁰5s²5p⁶4f¹³6s²5d¹ were explicitly treated by the segmented-contraction type of the valence BS (14s13p10d8f)/[10s8p5d4f] [30] related to the ECP we used; and all the primitive functions with exponents less than 0.1 bohr⁻² were excluded because it is well-known that in the LCAO calculations for crystals, the BS of a free atom must be modified because the diffuse functions may cause numerical problems due to the large overlap with the core functions of the neighboring atoms in a densely packed crystal [31].

The Monkhorst‒Pack schemes for the 8 × 8 × 8 and 4 × 4 × 4 k‒point meshes in the Brillouin zone were applied to the pure and doped cases, respectively. The truncation criteria for bielectronic integrals (Coulomb and HF exchange series) were set to 8, 8, 8, 8, and 16, and a predefined “extra extra large” pruned DFT integration grid was adopted together with much higher DFT density and grid-weight tolerances (values 8 and 16). The tolerance of the energy convergence on the self-consistent field iterations was set to 10⁻⁸ Hartree. The Anderson scheme with 60 % mixing was used to achieve the converged solutions for all the calculation cases, and the total spin of the doped systems was locked to a value of 1/2 in the first 15 cycles due to the single positive hole occupation in the Tm²⁺4f orbitals. In the geometry optimization with full structural relaxation, the convergence criteria of the root mean square of the gradient and the nuclear displacement were set to 0.00006 Hartree/bohr and 0.00012 bohr, respectively. For the elastic constant calculation, the default parameter values recommended in Ref. [32] were used.

5.3.2 Crystal-Field Calculations and Exchange-Charge Model

The parameterized effective Hamiltonians for the 4f¹³ and 4f¹²5d configurations of Tm²⁺ ions can be respectively written as:

$$ \varvec{H}( 4 {\text{f}}^{ 1 3} ) = \varvec{E}_{\text{avg}} + \varsigma_{\text{f}} A_{so} ({\text{f}}) + \varvec{H}_{{\varvec{CF}}} ( 4 {\text{f}}) $$

(1)

$$ \begin{aligned} \varvec{H}( 4 {\text{f}}^{ 1 2} 5 {\text{d}}) & = \Delta_{E} ({\text{fd}}) + \sum\limits_{k = 2,4} {F^{k} ({\text{fd}})f_{k} + \sum\limits_{j = 1,3,5} {G^{j} ({\text{fd}})g_{j} } } \\ & \quad + \sum\limits_{k = 2,4,6} {F^{k} ({\text{ff}})f_{k} } + \varsigma_{\text{f}} ({\text{ff}})A_{so} ( {\text{ff)}} + \alpha ({\text{ff}})L(L + 1) + \beta ({\text{ff}})G(G_{2} ) \\ & \quad + \gamma ({\text{ff}})G(R_{7} ) + \sum\limits_{h = 0,2,4} {M^{h} ({\text{ff}})m_{h} + \sum\limits_{k = 2,4,6} {P^{k} ({\text{ff}})p_{k} } } \\ & \quad + \varsigma_{\text{d}} ({\text{d}})A_{so} ( {\text{d)}} + \varvec{H}_{{\varvec{CF}}} ( 4 {\text{f}}) + \varvec{H}_{{\varvec{CF}}} ( 5 {\text{d}}) \\ \end{aligned} $$

(2)

where all notations and terms/operators/parameters are defined according to the standard practice [33]. The sum of all the interactions, except for the terms denoted as H _CF (nl)(nl = 4f or 5d), describes a quasi‒free ion configuration Hamiltonian, the parameters of which can be taken as the free-ion ones with a reasonable reduction, caused by the nephelauxetic effect. Shortly, where E _avg represents the energy barycenter of the 4f¹³ configuration. The term denoted asΔ _E (fd) is the (barycenter-to-barycenter) energy difference between both the 4f¹³ and the 4f¹²5d configurations; the ζ _f, ζ _f(ff), and ζ _d(d) entries are the spin-orbit coupling parameters for 4f electrons in the 4f¹³ configuration, 4f¹² core and for the 5d single electron, correspondingly. The Slater radial integrals for the radial part of the electrostatic interaction among the 4f electrons or between the 4f and 5d electrons are denoted as F ^k(ff)(k = 2,4,6), F ^k(fd)(k = 2,4), and G ^j(fd)(j = 1,3,5). The two-body corrections terms describing the configuration interaction are denoted by α(ff), β(ff), and γ(ff). Finally, the M ^h(ff)(h = 0,2,4) and P ^k(ff)(k = 2,4,6) are related to the spin‒spin and spin‒other orbit relativistic interactions and the electrostatically correlated spin‒orbit interactions between 4f elections, respectively (usually the ratios of M ⁰:M ²:M ⁴ and those of P ²:P ⁴:P ⁶ are kept fixed as described and explained in Ref. [34]). The last term, H _CF (nl), is the CF interaction produced by the crystal lattice ions, which is felt by the 4f or 5d electrons of rare-earth ions and leads to the splitting of the J manifolds of an impurity ion. It is a standard practice to expand this term as follows:

$$ \varvec{H}_{{\textbf{CF}}} (nl) = \sum\limits_{k,q = - k, \cdots k} {B_{q}^{k} (nl)\varvec{C}_{\varvec{q}}^{{{\mathbf{(}}\varvec{k}{\mathbf{)}}}} } $$

(3)

where C ^k q are the spherical functions with rank k and order q defined by Wybourne [35]; and B ^k q(nl) are the crystal-field parameters (CFP). The former act on the angular parts of an impurity ion’s wave functions, whereas the latter reflect the local symmetry of an impurity ion’s site. The CFP values can be calculated using the crystal-structure data and the exchange-charge model (ECM) as shown below. More details related to this method of calculations can be found in our recent work [36].

The choice of the quasi‒free -ion parameter values for the 4f¹³ and 4f¹²5d configurations of Tm²⁺ ions was performed according to data in the earlier literature. Thus, Pan et al. [37] reported the quasi‒free ion parameter values for the 4f⁶5d configuration of the Eu²⁺ ions in SrCl₂. From their data, it is possible to obtain the reduction ratio between the free ion’s values and those in the crystalline environment. The free ion‒parameter values for the 4f^N−15d configurations of Eu²⁺(N = 6) and Tm²⁺(N = 12) ions were calculated by the standard atomic-physics codes of Cowan [38, 39], and then the same reduction ratio as for the Eu²⁺ ions was directly applied to the Tm²⁺ ions. The reduced quasi‒free ion parameter values for the 4f¹²5d configuration of Tm²⁺ ions (using the standard notations in Ref. [33]) were obtained as follows: F ²(ff) = 84176, F ⁴(ff) = 59189, F ⁶(ff) = 45569, ζ _f(ff) = 2483, α(ff) = 17.26, β(ff) = ‒624.5, γ(ff) = 1820, M ⁰(ff) = 3.81, P ²(ff) = 695, ζ _d(d) = 1053, F ²(fd) = 11611, F ⁴(fd) = 5354, G ¹(fd) = 5198, G ³(fd) = 4051, and G ⁵(fd) = 3056 (all in cm⁻¹). For the 4f¹³ configuration of the Tm²⁺ ions, the parameter ζ _f value can be approximately taken from that of ζ _f(ff). The parameters Δ _E (fd) and E _avg are respectively adjusted to 34756 and 3912 cm⁻¹ to obtain the best agreement between the experimental and calculated energy levels.

The basic ideas of the exchange-charge model of the crystal field as outlined in Ref. [40] are as follows: The CFPs of any impurity ion, i.e., B ^k q(nl), are represented as a sum of two different contributions that arise from the point charges of all crystal lattice ions and introduced fictitious exchange charges (placed in the space between an impurity ion and ligands) due to the spatial overlap of the impurity-ion and ligand wave functions:

$$ \begin{aligned} B_{q}^{k} (nl) = & - e^{2} \left\langle {nl} \right.\left| {r^{k} } \right|\left. {nl} \right\rangle \sum\limits_{i} {q_{i} } \beta_{k} \cdot ( - 1)^{k} C_{ - q}^{k} (\theta_{i} ,\varphi_{i} )/R_{i}^{k + 1} \\ & \quad + e^{2} \frac{2(2k + 1)}{2l + 1}\sum\limits_{L} {S_{k}^{nl} (R_{L} )} \cdot ( - 1)^{k} C_{ - q}^{k} (\theta_{L} ,\varphi_{L} )/R_{L} \\ \end{aligned} $$

(4)

where the indexes i and L are, respectively, used to enumerate all the crystal lattice ions and the ligand ions in the nearest-neighbor coordination shell; (R _i , θ _i , φ _i ) and (R _L , θ _L , φ _L ) are the spherical coordinates of the ith crystal lattice ions and the Lth ligand ions in the reference system centered at the Tm²⁺ ions; and the ion charges q _i for Sr and Cl atoms can be taken as +2 and ‒1 as the first approximation. However, in our calculations we went further and took the values of these charges from the Mulliken population analysis as obtained by the performed ab initio calculations. The reduction factor β _k (k = 2, 4, …2 l) is defined as 1 − σ _k (the so-called shielding constant [41]) for the 4f electrons, which accounts for the screening effects produced by the outer filled 5s and 5p electron shells. However, for the 5d electrons, it is set to 1 (no screening because this is an outer orbital). The <nl|r ^k|nl> entry is the radial integral of r ^k (electron radial coordinate) between nl orbitals of the Tm²⁺ ions. The total overlap integral $S_{k}^{nl} \left( {R_{L} } \right)$ can be further expressed as:

$$ S_{k}^{nl} (R_{L} ) = G_{s}^{nl} S_{s}^{nl} (R_{L} )^{2} + G_{\sigma }^{nl} S_{\sigma }^{nl} (R_{L} )^{2} + \gamma_{{_{k} }}^{nl} G_{\pi }^{nl} S_{\pi }^{nl} (R_{L} )^{2} $$

(5)

where $\gamma_{2}^{{4{\text{f}}}} = 3/ 2,\,\gamma_{4}^{{4{\text{f}}}} = 1/ 3,\,\gamma_{6}^{{4{\text{f}}}} = - 3/ 2,\gamma_{2}^{{5{\text{d}}}} = 1,$ and $\gamma_{4}^{{5{\text{d}}}} = { - 4/3}$ [40]; $S_{s}^{nl} \left( {R_{L} } \right)$ = <nl0|3s0> , $S_{\sigma }^{nl} \left( {R_{L} } \right)$ = <nl0|3p0> , and $S_{\pi }^{nl} \left( {R_{L} } \right)$ = <nl±1|3p±1> ; and $G_{s}^{nl} ,\,G_{\sigma }^{nl}$ and $G_{\pi }^{nl}$ entries are the dimensionless adjustable parameters. They are determined by matching the calculated energy levels of an impurity ion to the experimental positions of its three lowest in energy absorption bands. For the sake of simplicity (and this turns out to be a very good approximation indeed in most cases), these three parameters can be approximated to a single value G(nl). In such a case, the value of this only parameter of the model can be easily found by matching the calculated position of the first excited state to the lowest absorption band in the experimental absorption/excitation spectrum. The moment of the nl-electron’s radial coordinate r <nl|r ^k|nl> , as well as the overlap integrals between the nl orbitals of the Tm²⁺ ions and the outer 3s and 3p orbitals of Cl⁻ ions for various interionic distances, can be calculated using the corresponding numerical radial-wave functions provided by the Cowan code [38].

After all CFPs are determined, the “f-shell” program code written by M.F. Reid [33] can be employed to calculate the 4f¹³ and 4f¹²5d CF energy levels and simulate the 4f-5d transition spectrum of Tm²⁺ ions in all considered cases. The results of these calculations are described below after the description of properties of pure CaCl₂, SrCl₂, and BaCl₂.

5.4 Ab Initio Calculations for Pure CaCl₂, SrCl₂, and BaCl₂ Crystals

The calculated structural parameters of pure MCl₂ (M = Ca, Sr, and Ba) crystals are listed in Table 5.1. As seen from that table, agreement between the experimental and calculated structural characteristics is very good, especially for SrCl₂ and BaCl₂, including not only the crystal lattice constants but also the fractional coordinates of all ions in a crystal lattice. The difference between the calculated and experimental structural parameters is somewhat greater in the case of CaCl₂, but this can be due to the fact that the corresponding experimental data are rather old [16], and the accuracy of those measurements might not be too high (as is evidenced by the number of digits after the decimal point in the corresponding structural data).

Good agreement between the optimized and experimental structural parameters allows calculations of the electronic, optical, and elastic properties of these compounds.

Figure 5.2 shows the calculated band structures of all three studied chlorides. The common feature is that for all compounds the valence band is rather narrow (3–4 eV), whereas the conduction band is rather wide. The upper states in the valence bands are very flat, which suggests a weak dispersion of holes in the reciprocal space. The lowest states of the conduction bands are also rather flat (low mobility of electrons), except for the vicinity of the Brillouin zone center. The band gaps are direct for CaCl₂ and BaCl₂ and is indirect for SrCl₂. The calculated band gaps are 7.28 eV (CaCl₂), 7.42 eV (SrCl₂) and 6.58 eV (BaCl₂), which is in good agreement with the experimental data of 6.9 eV (CaCl₂), 7.5 eV (SrCl₂), and 7 eV (BaCl₂), with all experimental data taken from Ref. [42].

Fig. 5.2

Calculated band-structure diagrams for pure MCl₂ (M = Ca [a], Sr [b], and Ba [c]) crystals. In the cases of (a) and (c), the letters Γ, Z, T, Y, S, X, U, and R stand for the high-symmetry k points (0,0,0), (0,0,1/2), (0,1/2,1/2), (0,1/2,0), (1/2,1/2,0), (1/2,0,0), (1/2,0,1/2) and (1/2,1/2,1/2), respectively, whereas the high-symmetry k points (1/2,0,1/2), (1/2,1/4,3/4), (1/2,1/2,1/2), (0,0,0) and (3/4,3/8,3/8) corresponding to the letters X, W, L,Γ and K, respectively, are chosen in the case of (b). b Reproduced from Ref. [10] by permission of John Wiley & Sons Ltd

Composition of the calculated electronic bands can be analyzed with the help of the density-of-states (DOS) and partial-density-of-states (PDOS) diagrams shown in Fig. 5.3. The valence band in all cases is formed by the 3p states of Cl ions, whereas the conduction band is basically composed of the d states of metals (3d states of Ca, 4d states of Sr, and 5d states of Ba). These are the most important states, which determine the optical properties of these crystals. The remaining DOS peaks are located deeply in energy; they come from the completely filled 3s states of Cl and p states of metal ions and do not manifest themselves in the optical spectra.

Fig. 5.3

Calculated PDOS/DOS diagrams for pure MCl₂ [M = Ca (a), Sr (b) and Ba (c)] crystals. b Reproduced from Ref. [10] by permission of John Wiley & Sons Ltd

Table 5.2 lists the calculated elastic constants for all materials. All non-zero values of the elastic constants C _ij (the number of these non-zero components is determined by the crystal-lattice symmetry), bulk moduli, and Poisson’s ratio—along with the sound velocities corresponding to different directions in the crystal lattices are listed in the table. Low values of the bulk moduli suggest that all three crystals are very soft substances.

Table 5.2 Calculated elastic constants C _ij (GPa), bulk moduli B (GPa), Poisson’s ratios ν, and longitudinal and transverse sound velocities v _l,t (km/s) for MCl₂ (M = Ca, Sr and Ba) crystals

Comparison of our calculated results with those available in the literature [43–45] yields good agreement between corresponding sets of data. We also went further with an analysis of the elastic anisotropy of these compounds, which can be visualized by plotting three-dimensional surfaces of the Young moduli along various directions in the crystal lattice. In the case of an an orthorhombic crystal (such as CaCl₂ and BaCl₂), such a three-dimensional surface is described by the following expression [46]:

$$ \frac{ 1}{E} = l_{1}^{4} S_{11} + 2l_{1}^{2} l_{2}^{2} S_{12} + 2l_{1}^{2} l_{3}^{2} S_{13} + l_{2}^{4} S_{22} + 2l_{2}^{2} l_{3}^{2} S_{23} + l_{3}^{4} S_{33} + l_{2}^{2} l_{3}^{2} S_{44} + l_{1}^{2} l_{3}^{2} S_{55} + l_{1}^{2} l_{2}^{2} S_{66} , $$

(6)

where E is the value of the Young’s modulus in the direction determined by the direction cosines $l_{1} ,\,l_{2} ,\,l_{3}$ ; and S _ij are the elastic compliance constants, which form the matrix inverse to the matrix of the elastic constants C _ij . The same surface in the case of a cubic crystal (SrCl₂) is given as

$$ \frac{ 1}{E} = S_{11} - 2\left( {S_{11} - S_{12} - \frac{1}{2}S_{44} } \right)\left( {l_{1}^{2} l_{2}^{2} + l_{2}^{2} l_{3}^{2} + l_{1}^{2} l_{3}^{2} } \right). $$

(7)

The distance from the center of such a surface to a point at its surface is equal to the Young modulus in a given direction determined by the direction cosines $l_{1} ,\,l_{2} ,\,l_{3}$. Figure 5.4 shows these surfaces plotted for the studied systems, and as seen from the surfaces, the anisotropy of the elastic properties is well-pronounced, especially for CaCl₂ and BaCl₂.

Fig. 5.4

Directional dependences of the calculated Young moduli for the MCl₂ (M = Ca, Sr, and Ba, from left to right) crystals (the axes units are GPa)

5.5 Ab Initio Calculations for Tm²⁺-doped CaCl₂, SrCl₂, and BaCl₂ Crystals

Introduction of an impurity ion into a crystal can considerably change its optical properties. Even the color of a crystal can be changed because the purity-ion’s energy levels, which appear in the band gap, can cause absorption in the visible range.

To model the geometrical structures of the Tm²⁺-doped MCl₂ (M = Ca, Sr, and Ba) crystals, we constructed three super-cells from the optimized structural data of the pure host’s primitive cells by respectively using the symmetrical super-cell transformation matrices (1 − 1 0; 1 1 0; 0 0 2), (2 0 0; 0 2 0; 0 0 2) and (1 − 1 0; 1 1 0; 0 0 1) [31]. Each studied super-cell contains eight MCl₂ chemical formula units, and one M ²⁺cation was replaced by the Tm²⁺ ion. In other words, the geometry optimizations were performed on the form of M ₇TmCl₁₆ for all cases. At first, due to the differences in the ionic radii of the Tm²⁺ and substituted metal ions (Ca²⁺, Sr²⁺, and Ba²⁺), the lattice constants and interionic distances in the doped crystal will be somewhat different from those ones for the pure host. Table 5.3 illustrates this structural difference between the pure and thulium-bearing crystals.

Table 5.3 Calculated lattice parameters and volumes of the pure and Tm²⁺-doped MCl₂ (M = Ca, Sr and Ba) supercells

As seen from Table 5.3, the volume of a unit cell of CaCl₂ doped with Tm²⁺ is slightly increased (by approximately 0.68 %), whereas the unit cells of SrCl₂ and BaCl₂ undergo a compression by 1.21 and 2.41 %, respectively. Such behavior can be explained by a simple comparison of the ionic radii of all ions involved. However, inspection of the database of the ionic radii given by Shannon [47] shows that the ionic radii of Tm²⁺ ions with eightfold and ninefold coordination numbers (CN) are absent. Due to the consideration of the chemical similarity across the whole lanthanide series, we can propose a linear ionic-radius dependence of divalent lanthanide ions on their atomic number Z and coordination number N to bypass the problem we encountered above as follows:

$$ R(Ln^{2 + } ) = 0.0521N - 0.0176Z + 1.9382 $$

(8)

where the values of the two slopes and the intercept were obtained by fitting the experimental data set collected in Table 5.4 [47]. The calculated ionic radius of Tm²⁺using Eq. (8) is slightly greater than that of Ca²⁺, but slightly smaller than those of Sr²⁺ and Ba²⁺, as shown in Table 5.4. This straightforward equation reveals the change reason for the super-cell volume with respect to the pure-host case after Tm²⁺-doping.

Table 5.4 Effective ionic radii (in Å) of divalent lanthanides and M ²⁺(M = Ca, Sr and Ba) cations with different CNs

Another geometrical manifestation of doping effects is the change of characteristic interionic distance. Comparison of the M-Cl (M = Ca, Sr, and Ba) and Tm–Cl distances in the pure and doped crystals is given by Table 5.5. Again, in accordance with the general trends in the change of the unit cell volumes from Table 5.3, the Tm–Cl distances are slightly increased, if the Tm²⁺ ions enter CaCl₂, and decreased when the Tm²⁺ ions occupy the Sr and Ba sites in SrCl₂ and BaCl₂, respectively.

Table 5.5 Calculated distances (in Å) from the dopant site to the chlorine ligands in the first coordination shell before and after Tm²⁺-doping in CaCl₂, SrCl₂ and BaCl₂ crystals

Introduction of an impurity ion also produces local changes of the electron-density distribution, which can be evaluated by comparison of the calculated effective Mulliken charges of the crystal-lattice ions before and after doping (Table 5.6).

Table 5.6 Effective Mulliken charge Q(e) of individual ion and “metal-chlorine” bond population P (milli e) in the local cluster composed of the dopant site and its nearest chlorine ligands before and after Tm²⁺-doping in CaCl₂, SrCl₂ and BaCl₂ crystals

The Mulliken charge calculations indicate an increasing degree of covalency for MCl₂ crystals when M changes along the Ca‒Sr‒Ba series due to the decreasing effective charge of M cation. The Tm²⁺ effective charge is in all three hosts slightly greater than the charge of the substituted ion, which causes then a slight increase of the charge of the surrounding chlorine ions to maintain the local charge balance. It can be also noticed that the effective charges of all ions are different from their formal charges, which indicates formation of covalent bonds between the neighboring ions. Such a covalent chemical bonding behavior has been confirmed by the analysis of the bond populations between the dopant site and its nearest chlorine ligands before and after Tm²⁺-doping. In addition, the electron-density distributions around the Ba²⁺ and Tm²⁺ ions occupying the same site in BaCl₂ are not similar but rather very different: The electrons lost by Ba atom trend to go to those chlorine ligands far away from the cation site, whereas most of the transferred electrons from the Tm atom stop in the coordination environment composed of the first seven chlorine ligands as indicated by the opposite dependences of the “metal‒chlorine” bond populations on the chemical bond length for the pure and doped cases.

Finally, the changes of the electronic band structure that take place on doping are shown in Fig. 5.5.

Fig. 5.5

Calculated band-structure diagrams for Tm²⁺-doped MCl₂ [M = Ca (a and b), Sr (c and d), and Ba (e and f)] crystals. The red two-way arrows indicate the energy separation between the host valence and conduction bands. ALPHA and BETA represent the electronic pictures with the up and down spins, respectively. c, d Adapted from Ref. [10] by permission of John Wiley & Sons Ltd

The 4f states of the Tm²⁺ ions are clearly seen in the band gap. Because Tm²⁺ ions have 13 4f electrons, spin-polarized calculations were performed to distinguish between the spin-up and spin-down states. The band-structure diagrams from Fig. 5.5 allow for the following estimation of the position of the Tm²⁺ 4f ground state in the band gap above the top of the host’s valence band: 1.05 and 2.91 eV (spin-up and spin-down states in CaCl₂, respectively), 1.37 and 3.20 eV (spin-up and spin-down states in SrCl₂, respectively), and 1.39 and 3.12 eV (spin-up and spin-down states in BaCl₂, respectively).

Certain asymmetry of the spin-up and spin-down state distributions is also clearly seen in Fig. 5.6, which presents the DOS/PDOS diagrams for the Tm²⁺-doped crystals. In the case of SrCl₂ (cubic crystal and O_h local symmetry around the Tm²⁺ site), it was also possible to determine the symmetry properties of the split electron states and assign the O_h group irreducible representations as shown in Fig. 5.6.

Fig. 5.6

Calculated PDOS/DOS diagrams for Tm²⁺-doped MCl₂ [M = Ca (a), Sr (b), and Ba (c)] crystals. b Adapted from Ref. [10] by permission of John Wiley & Sons Ltd

5.6 Crystal-Field Modeling of the Tm²⁺ Spectra in SrCl₂ Crystal

As described in Sect. 5.3.2, the overlap integrals between an impurity ion and the ligands are needed to apply the ECM. These integrals, in the case of the Tm²⁺ and Cl⁻ ions, were calculated numerically for various Tm²⁺‒Cl⁻ distances. They vary smoothly with the interionic separation as shown in Figs. 5.7 and 5.8.

Fig. 5.7

Calculated dependence of overlap integrals on the distance between Tm²⁺ and Cl⁻ ions (<4f0|3s0> [solid line], <4f0|3p0> [dashed line], and <4f1|3p1> [dash-dotted line]). Reproduced from Ref. [10] by permission of John Wiley & Sons Ltd

Fig. 5.8

Calculated dependence of overlap integrals on the distance between Tm²⁺ and Cl⁻ ions (<5d0|3s0> [solid line], <5d0|3p0> [dashed line], and <5d1|3p1> [dash-dotted line]). Reproduced from Ref. [10] by permission of John Wiley & Sons Ltd

Based on the structure of the Tm²⁺-doped SrCl₂ optimized by the first-principles calculation, the non-zero 4f and 5d CFPs of Tm²⁺ ions in SrCl₂ can be estimated as follows—$B_{0}^{4}$(5d) = ‒18944, $B_{4}^{4}$(5d) = $B_{- 4}^{4}$(5d) = ‒11321, $B_{0}^{4}$(4f) = ‒567, $B_{4}^{4}$(4f) = $B_{- 4}^{4}$(4f) = ‒339, $B_{0}^{6}$(4f) 340, and $B_{4}^{6}$(4f) = $B_{- 4}^{6}$(4f) = ‒636 (all in cm⁻¹)—where the G(4f) and G(5d) parameter values can be fitted from the experimental 4f‒4f and 4f‒5d spectra [2] and are equal to 7.3 and 1.1, respectively.

With all of these parameters, it was possible to calculate the energy levels of the 4f¹³and 4f¹²5d electron configurations of the Tm²⁺ ions in SrCl₂. In addition, probabilities of the electric dipole transitions between all possible states were also calculated. Finally, the simulated and measured 4f‒5d spectra of SrCl₂:Tm²⁺ are shown together shown in Fig. 5.9 where the pure 4f‒5d transition lines are represented by the vertical bars, and the broad bands are reproduced by using the Gaussian-shaped curves with a full width at half maximum E _width = 400 cm⁻¹, which are displaced from the zero-phonon lines by E _shift = 200 cm⁻¹. It can be seen from Fig. 5.9 that the simulated spectrum is in good agreement with the experimental one including the positions of the absorption maxima and relative intensity of the absorption bands.

Fig. 5.9

Experimental absorption spectrum (top) and simulated 4f¹³‒4f¹²5d transition spectrum (below) for SrCl₂:Tm²⁺. The vertical bars show the calculated locations and relative intensities of pure electronic transitions. Reproduced from Ref. [10] by permission of John Wiley & Sons Ltd

5.7 Summary

Detailed analysis of the structural and electronic properties of pure and Tm²⁺-doped CaCl₂, SrCl₂, and BaCl₂ is presented in this chapter. First-principles calculations of the structural, electronic, and elastic properties were performed. Special emphasis was placed on variation of the structural and electronic properties on doping with Tm²⁺ ions. In particular, the position of the ground 4f level of thulium ions in the band gap of each considered material was determined. The calculated local structure of the impurity Tm²⁺ ions were applied to the understanding of the 4f‒5d absorption spectrum of Tm²⁺ ions in SrCl₂ by employing semi-empirical CF theory. The successful theoretical simulation reveals that such a hybrid method is a good tool for describing the structural, electronic, and related optical properties of lanthanide defects in solids.