Quasiparticle Self-Consistent GW Analysis of Excited States and Model Construction of 3d2 Luminescent Centers in α-Aluminum Oxide

Harutaka Saito; Katsuhiro Suzuki; Kazunori Sato; Takao Kotani

doi:10.2320/matertrans.MT-MG2022024

Abstract

Cr⁴⁺-doped crystals are promising materials for infrared solid laser applications. In order to choose proper host materials of Cr⁴⁺ from the wide variety of crystals, multiplet energy levels of Cr⁴⁺ should be evaluated without experimental information. A newly developed first-principles method for determining the mean-field Hamiltonian of the localized luminescent center based on quasiparticle self-consistent GW was applied to 3d² transition metal ions (Cr⁴⁺ and V³⁺) in α-Al₂O₃. Our method gave good agreement with the effective Coulomb parameters and crystal-field parameters estimated from experimental data. Multiplet energy structures were therefore well reproduced by our method. These results show the possibility of designing new laser host materials within the framework of the first-principles calculations.

1. Introduction

The first discovery of forsterite (Mg₂SiO₄):Cr⁴⁺, which shows a broad tunable laser operation in near infrared region (1.15–1.35 µm)¹⁾ accelerated the investigation of Cr⁴⁺-doped crystals for the applications to femtosecond lasers²^,³⁾ and tunable solid-state lasers.⁴^–⁶⁾ The advantages of using Cr⁴⁺ are as follows: a simple multiplet energy structure, a broad absorption band overlapping with the working wavelengths of commercial pump lasers;⁷⁾ Moreover, as a laser device, a broad tuning range, the possibility for direct pumping, the ability of producing femtosecond pulses, and safe emission wavelength range for our eyes.⁸⁾ Therefore, laser emissions from Cr⁴⁺ have been extensively used in medicine,⁹⁾ spectroscopy,¹⁰⁾ and telecommunications systems.¹¹⁾

However, appropriate host materials for Cr⁴⁺ have been demanded even now. There are two reasons for the lack of promising host materials. First, the fluorescent quantum yield of Cr⁴⁺-doped crystals is quite low due to the nonradiative transition of Cr⁴⁺. Even for practically used crystals such as Y₃Al₅O₁₂ and Mg₂SiO₄ show low quantum efficiencies about 14–22% and 9%, respectively.¹²⁾ Second, not all Cr cations exist as Cr⁴⁺: There also exist Cr³⁺ cations.¹³⁾ These disadvantages limit the possibilities of Cr⁴⁺-doped systems.

Computational material design (CMD) is a key concept to investigate various combinations of materials with high efficiency and design optimized materials for desired properties. In order to perform CMD for Cr⁴⁺-doped crystals, a first-principles method of emission properties of localized transition metal (TM) luminescent centers is necessary.

So far, some configuration interaction (CI) methods in quantum chemistry have been proposed for the calculations of multiplet energy levels and emission spectra.¹⁴⁾ However, CI assumes a cluster model of each system. This is not appropriate for the system with an impurity embedded in a solid crystalline material. Moreover, CI cannot be easily applied to systematic calculations of various systems, which is not desired for CMD application.

Density functional theory (DFT) has been widely used as a first-principles calculations method of ground state properties of materials. DFT based methods are thus appropriate for systematic calculations. However, usual approximation in DFT, such as local density approximation (LDA) or generalized gradient approximation (GGA) cannot reproduce the electronic structures of semiconductors due to insufficient consideration of electron correlations.¹⁵⁾ Furthermore, DFT based methods have difficulties in predicting the excited state properties.¹⁶⁾ These problems have prevented the development of estimating luminescent properties based on DFT.

In order to go beyond the usual DFT approaches, we employed quasiparticle self-consistent GW (QSGW) method¹⁷^–¹⁹⁾ implemented in ecalj package developed by Kotani et al. In fact, Suzuki et al. developed a new calculation method of multiplet energy levels of rare earth (RE) ions and RE-doped materials based on QSGW method.²⁰⁾ We apply this method to TM-doped crystals. For our purpose, we focus on 3d² systems in the present study. As a first trial, we calculated fundamental systems, Cr⁴⁺ and V³⁺ in α-Al₂O₃, and compare the mutiplet energy levels of these systems.

2. Computational Method

Computational procedures are essentially the same as the method explained in Refs. 20, 21). In this section, outline of the computational methods is given.

2.1 QSGW calculations of Cr⁴⁺ and V³⁺ in α-Al₂O₃

According to an experimental study,²²⁾ Cr⁴⁺ stably exist in a center of a distorted octahedron in α-Al₂O₃. They observed Cr⁴⁺ signals in electron spin resonance and optical absorption spectra, which are different from those of Cr³⁺. V³⁺ is also known to substitute octahedral Al³⁺ sites in α-Al₂O₃. These are confirmed by the facts that experimental spectra of Cr⁴⁺ and V³⁺ in α-Al₂O₃ are well described by assuming these cations to occupy the distorted octahedral symmetry sites.²³^–²⁵⁾ Since TM ion substitutes Al site, the structure thus should be relaxed to be more stable. We prepare a 2 × 2 × 2 rhombohedral supercell (80 atoms/cell) including one TM per super cell. Structure optimizations of both unit cell volume and atomic positions are performed by projected augmented wave method implemented in Quantum ESPRESSO package.²⁵^,²⁶⁾ The cut off energy is 60 Ry, and k-point mesh is 6 × 6 × 6. We extracted 1 × 1 × 1 local structure around TM ion from the optimized supercell for QSGW electronic calculations due to its high computational cost.

For the QSGW calculations, we employ ecalj package developed by Kotani. Technically, we adopted QSGW80 method, which is a hybrid method of QSGW 80% plus LDA 20% for exchange correlation functionals.²⁷⁾ We set k-point mesh 8 × 8 × 8 for the LDA+U as a preprocessing of QSGW80 calculation, 4 × 4 × 4 for the QSGW80 calculation, and 10 × 10 × 10 to construct the maximally localized Wannier functions (MLWFs)²⁸⁾ of 3d orbitals. In order to preserve charge neutrality after substituting Al³⁺ with Cr⁴⁺ in α-Al₂O₃, we impose negative background charge to the unit cell.

2.2 N-body atomic Hamiltonian of transition metal luminescent center

We assume the model Hamiltonian of 3d orbitals of TM luminescent center can be mainly described by 2 terms as

\begin{equation*} H=H_{\text{CF}}+H_{\text{C}}, \end{equation*}

where H_CF stands for crystal-field (CF) term and H_C the effective Coulomb interaction term. For CF term, we assume C_3v symmetry. Therefore, we have three parameters Dq, Dσ, and Dτ for CF term. The cubic CF parameter Dq describes the e_g-t_2g gap of 3d orbitals, and trigonal CF parameters Dσ and Dτ explains small splitting of t_2g states into e_g and a_1g.²⁹⁾

For the effective Coulomb term, we have three modified Slater-Condon parameters F₀, F₂, and F₄.³⁰⁾ F₀ describes spherical component, and F₂, F₄ are anisotropic components of the effective Coulomb interaction.

Since H is a ₁₀C₂ × ₁₀C₂ matrix, we cannot directly compare H with TM 3d states of QSGW80 band structure. We thus apply Hartree-Fock approximation to H, and obtain Hartree-Fock model Hamiltonian (HFMH). We determined these six parameters of HFMH so as to minimize the difference between eigenvalues of HFMH and ones of onsite 3d Hamiltonian extracted from TM 3d states in QSGW80 band structure by means of MLWFs. After determination of six parameters, we can calculate the multiplet structure of TM ion by the exact diagonalization of H.

3. Results and Discussions

In this section, we will discuss the estimation of HFMH parameters based on QSGW80 electronic structures in Sec. 3.1, and multiplet energy structures in Sec. 3.2.

3.1 Electronic structure and HFMH parameters of α-Al₂O₃:Cr⁴⁺ and V³⁺

Figure 1 shows the band structures and the density of states (DOS) of (a) α-Al₂O₃:Cr⁴⁺ and (b) α-Al₂O₃:V³⁺ calculated by QSWG80. QSGW80 predicted the band gap energy of α-Al₂O₃ as 9.2 eV, which is only 0.5 eV higher than experimental value 8.7 eV.³⁰⁾ The eigenvalues of onsite 3d Hamiltonian constructed by MLWFs basis, and of HFMH with optimized parameters are also illustrated next to the electronic structures in Fig. 1.

Fig. 1

The electronic structures of Cr⁴⁺- and V³⁺-doped α-Al₂O₃ calculated by QSGW80, eigenvalues of onsite 3d atomic Hamiltonian of TM ion, and ones of HFMH for (a) α-Al₂O₃:Cr⁴⁺, (b) α-Al₂O₃:V³⁺. Red lines in band structures and red regions in DOS indicates TM 3d states.

For the electronic structure, we can see some similarities and differences between (a) and (b). The red flat lines in Fig. 1 correspond to the energy levels of TM 3d states. Since both Cr⁴⁺ and V³⁺ have 3d² electron configurations, the splitting of 3d orbitals is almost the same. However, the following differences of splitting affect the estimation of HFMH parameters: The magnitude of gap between lower two states and higher two states, and the relative position of the third highest 3d energy level around 3 eV of majority spin states. The position of the energy levels of TMs relative to the host band structure of α-Al₂O₃ is also different: The energy levels of V³⁺ are shifted to higher energy about 3 eV compared to ones of Cr⁴⁺. However, constant shifts of energy levels do not make any differences of HFMH parameters.

We determined the HFMH parameters so as to reproduce the onsite 3d Hamiltonian derived from the QSGW80 calculations. We can confirm reasonable agreement between eigenvalues of onsite 3d Hamiltonian and of HFMH for both Cr⁴⁺ and V³⁺ cases as shown in Fig. 1. The determined HFMH parameters are listed in Table 1. They are consistent with experimental analysis. For the modified Slater-Condon parameters, F₂ value estimated from QSGW80 (0.119 eV for Cr⁴⁺/0.117 eV for V³⁺) is about 15% smaller than experimental analysis (0.139 eV/0.129 eV) while F₄ values agree well for both systems. For 10Dq, agreement is within 10%. Moreover, the trend of larger interaction in Cr⁴⁺ than in V³⁺ was reproduced.

Table 1 HFMH parameters of Cr⁴⁺ and V³⁺ in α-Al₂O₃. We show three modified Slater-Condon parameters F₀, F₂, F₄, and three CF parameters 10Dq, Dσ, Dτ.

3.2 Multiplet energy structure of α-Al₂O₃:Cr⁴⁺ and V³⁺

Multiplet energy structures of Cr⁴⁺ and V³⁺ are calculated using determined HFMH parameters. Figure 2 and Fig. 3 show 10Dq dependence of multiplet energy level splitting of V³⁺ and Cr⁴⁺, respectively. Such a diagram is called a Tanabe-Sugano (TS) diagram.

Fig. 2

Tanabe-Sugano diagram of α-Al₂O₃:Cr⁴⁺. (a) Calculated by the parameters from QSGW80. (b) Calculated by the parameters extracted from experimental optical spectrum.²³⁾ See text for detailed explanations.

Fig. 3

Tanabe-Sugano diagram of α-Al₂O₃:V³⁺. (a) Calculated by the parameters from QSGW80. (b) Calculated by the parameters extracted from experimental optical spectrum.²⁴⁾ Notations are the same as Fig. 2.

In Fig. 2, (a) represents the TS diagram predicted by our method, and (b) by the experimental analysis of the observed spectrum. 10Dq = 0 eV line, roughly corresponds to free ion state. The symbols stand for ones of irreducible representations under spherical symmetry. Red symbols correspond to spin singlet states, and green ones triplet states. We can see the larger splitting between the singlet and triplet states for larger 10Dq. The red broken line 10Dq = 10Dq_QSGW80 corresponds to the CF strength in α-Al₂O₃, thus the intersection points with that line and black solid curves show multiplet energy structure of α-Al₂O₃:Cr⁴⁺, extracted next to the TS diagram. Multiplet energy structure calculated using parameters from experimental analysis is illustrated in Fig. 2(b). In the middle of (a) and (b), corresponding multiplet energy levels are connected by dotted lines. The symbols indicate irreducible representations under octahedral symmetry. The same illustrations and notations are employed in Fig. 3.

The multiplet structure of α-Al₂O₃:Cr⁴⁺ and V³⁺ predicted by our method qualitatively reproduced the sequences of excited states obtained from experimental analysis as one can observe in Fig. 2 and Fig. 3. Moreover, our method gave good agreement within 20% with experimental analysis for both systems. In Fig. 2, our method systematically underestimated the energy levels about 10%, which reflects the error in F₂ value of Cr⁴⁺ 0.119 eV (our method)/0.139 eV (experimental analysis). We see larger splitting of, e.g., ³T_2g(³F) for Cr⁴⁺ in Fig. 2(b) than in (a), because of larger values of Dσ and Dτ. On the other hand, in Fig. 3, most of the excited states above 2 eV were overestimated by our method though lower excited states ¹E_g and ¹T_2g were underestimated. This can be attributed to the overestimation of 10Dq value in V³⁺.

Let us compare important multiplet energy levels ³T_2g(³F), ³T_1g(³P), and ³A_2g(³F) clearly observed in the experimental optical spectra, which correspond to energy levels described by green lines in Fig. 2 and Fig. 3. The predicted/experimental values are 2.43 eV/2.57 eV for ³T_2g(³F), 3.21 eV/3.64 eV for ³T_1g(³P), and 4.99 eV/5.25 eV for ³A_2g(³F) in the case of α-Al₂O₃:Cr⁴⁺. In the same way, 2.31 eV/2.04 eV for ³T_2g(³F), 3.17 eV/3.11 eV for ³T_1g(³P), and 4.77 eV/4.25 eV for ³A_2g(³F) in the case of α-Al₂O₃:V³⁺ are observed. Note that the mean values of small splitting due to trigonal symmetry are shown here. From these comparisons, we see our method underestimates the differences between Cr⁴⁺ and V³⁺, however qualitatively reproduces the trend that Cr⁴⁺ has larger energy levels than V³⁺.

4. Conclusion

We constructed the model Hamiltonian of 3d² TM (Cr⁴⁺ and V³⁺) doped in α-Al₂O₃ based on the QSGW80 electronic structure calculated by the QSGW80 method. We found that the QSGW80 method reproduced the band gap energy of α-Al₂O₃ well. Our method gave reasonable HFMH parameters, and thus yielded multiplet energy structures which are consistent with experimental analysis.

Our method will contribute to the investigation of Cr⁴⁺-doped solid state laser materials within the first-principles framework. In order to obtain infrared light emission from Cr⁴⁺, the first excited state of Cr⁴⁺ will be essential. Though our method cannot give quantitative prediction of multiplet energy levels, we can apply our method to predict the qualitative information of multiplet energy levels. The most important shortcoming of Cr⁴⁺ will be the presence of Cr³⁺. By combining formation energy analysis of Cr³⁺ and Cr⁴⁺, we will be also able to estimate the stability of Cr⁴⁺.

Acknowledgments

This work is partly supported by JSPS KAKENHI (Grant No. 20K05303, 18H05212, 22K04909), and JST CREST (Grant No. JP-MJCR18I2).

REFERENCES

Corresponding author

Register with J-STAGE for free!