The transition of energy from the 4f to the 5d state is a fundamental element driving various applications, such as phosphors and optoelectronic devices. The positioning of the 4f ground states and the 5d excited states significantly influences this energy shift. In our research, we delve into the placement of these states utilizing a hybrid density functional theory (DFT) combined with spin–orbit coupling (SOC) via the supercell method. Additionally, we scrutinize the transition energy, applying the constrained density functional theory (cDFT) approach in conjunction with the ΔSCF method. Our study illustrates that the synergy of cDFT and SOC generates a discrepancy of about 2% for Ce1 and 4% for Ce2 when comparing the calculated results to experimental data. Moreover, We have determined the positions of the 4f ground states to be 2.73 eV above the Valence Band Maximum (VBM) for Ce1 and 2.70 eV for Ce2. We also note a tight correlation between the 5d levels identified in the experimental data and the theoretical outcomes derived from wave function calculations at the CASPT2 accuracy level.

The performance of optoelectronic devices and phosphors hinges on the optical and electronic attributes of both host materials and activators. Numerous applications, including displays,1 white light-emitting diode (LEDs),2 depend on the effectiveness of these phosphors and activators. Lanthanides (Ln) stand out as preferred activators, owing to their distinctive electronic structure and the transition from 4f to 5d levels. This specific transition generates a notable yet broad peak in both absorption and emission spectra. The form and intensity of these peaks heavily rely on the host lattice, affecting the energy difference between the Ln dopants’ lowest 4fn level and the first 4fn−15d level, as reported by Qin et al.3 Consequently, the precise positioning of the Ln 4f and 5d levels is crucial, influencing intensity, energy transitions, Ln valence, luminescence, and charge carrier trapping properties.4,5 This underlines the importance of accurately understanding the 4f and 5d levels’ locations in Ln ions for the creation of efficient optoelectronic devices and phosphors.

A prominent optical material is Y2SiO5 (YSO) doped with Rare-Earth (RE) ions, particularly Ce-doped YSO, recognized for its scintillation properties. Its luminescence has been thoroughly probed, both experimentally6,7 and theoretically.8 However, previous theoretical investigations into Ce-doped YSO using the ab initio model potentials (AIMPs) method encounter limitations due to their dependence on empirical data and the accompanying computational expenses.9,10 This reliance restricts the AIMP model’s adaptability to various host materials and RE ions. Given YSO doped with RE’s significant potential as an optical material, it is essential to delve into fresh theoretical avenues. Such strategies must efficiently and precisely discern luminescent characteristics, ensuring their broad material applicability without compromise.

RE ions in inorganic compounds are typically excited from 4f → 5d using either quantum chemistry methods or solid-state methods. Quantum chemical techniques, such as coupled-cluster (CC)11 and complete active space self-consistent field (CASSCF),12 are highly accurate but are often used for finite-sized solids and molecules due to their computational cost. Time-dependent density functional theory (TDDFT13), despite its success in closed-shell molecules, is still not well-tested for larger molecules and solids, especially those with open shells.14 Furthermore, TDDFT has only been applied to systems with defects using clusters to approximate infinite solids.14 Many-body perturbation theory (MBPT) based on GW approximation is another solid-state technique that corrects the Kohn-Sham eigenvalues of a traditional DFT with quasi-particle corrections.15 GW + BSE correction of MBPT is currently the best approach for studying the optical properties of solids.16 However, its application is limited to small molecules and high-symmetry solids due to its computational cost. Therefore, running calculations for defect studies using supercell sizes is not feasible.17 Additionally, it is not possible to calculate forces on atoms in solids using this approach.

To address the limitations of DFT for calculating excitation processes and to overcome the limitations that are offered by the aforementioned excitation approaches, we employ the constrained DFT (cDFT).18,19 This method is an adaptation of generic DFT that partially mitigates its shortcomings, enabling the calculation of excitation processes.18 Although cDFT does not offer excitonic effects by itself, it has proven to be an effective approach for simulating neutral excitation in molecular and solid systems.17,20 It has also been used in high-throughput studies of Ln scintillators, as demonstrated in the works of Canning et al.19 and Jia et al.16,17

In this paper, we aim to determine the energy transitions of the Ce3+ ion doped in YSO host through a complete ab initio technique. To enhance the accuracy of our calculations, we include the Heyd–Scuseria–Ernzerhof (HSE21) functional with the Spin-Orbit Coupling (SOC) effect. It has been demonstrated that electronic property predictions with the screened hybrid functional can be comparable to GW0@LDA + U calculations.22 Additionally, we construct a supercell of 1 × 2 × 1 relative to the host unit cell to minimize impurity-impurity interactions. The relative energy levels of the 4f and 5d states of Ce3+ are identified through a combination of cDFT and hybrid functional calculations. Once the energy levels are determined, we employ the ΔSCF method to obtain the 4f → 5d transition in the Ce3+ ion.

The initial atomic positions used in this study were obtained from our previous work,23 in which we characterized the two distinct spectroscopic sites present in YSO. These sites, known as site1 (Y/Ce1) and site2 (Y/Ce2), are each surrounded by 7 and 6 oxygen atoms (Fig. 1), respectively. These sites with the aforementioned coordination number (CN) belong to X2-YSO crystallographic form, the other form being X1-YSO has CN numbers of 9 and 7.23 The X2-YSO has a stronger luminescent intensity in comparison with X1-phase, and thus it is more crucial for practical application and luminescent efficiency.24 The calculations are carried out by employing pseudopotential plane-wave DFT methodology as implemented in the Vienna Ab initio Software Package (VASP).25 The Ce: 5d14f1, Y: 4 s24p64d15 s2, Si: 3 s23p2, and O:2 s22p4 electrons are treated as valence electrons, whereas the core electrons and electron-ion interactions are determined using the projected augmented wave (PAW) method.26 The contribution due to exchange and correlation is expressed by the generalized gradient approximation (GGA) theory as described by Perdew-Burke-Ernzerhof (PBE),27 and the partial occupancy of orbitals was determined using the tetrahedron method with Blöchl corrections.28 The cut-off energy was set to, 520 eV, while the k-point set for the unit cell was selected to be 4 × 8 × 6. The convergence threshold was set to 1 × 10−8 eV, and the force criterion for geometry optimization was 0.000 01 eV/Å.

FIG. 1.

Schematic view of impurity sites in X2-YSO. Site1 with coordination number 7, and site2 with coordination number 6.

FIG. 1.

Schematic view of impurity sites in X2-YSO. Site1 with coordination number 7, and site2 with coordination number 6.

Close modal
In order to tune the HSE parameters a series of Gamma-point only calculations were performed. These calculations are based on the PBE pre-optimized unit cell of the pure system. First, the percentage of exact exchange α, according to Eq. (1), was varied to reach the optimal level, which in this case means obtaining a band gap value similar to the experimental band gap. Once, the approximate percentage of exact exchange is set, the screening parameter ω is tuned. In the end, an HSE calculation with sufficiently dense k-mesh was chosen to validate the band gap result and tuned parameters. The subsequent HSE calculations are conducted based on the tuned parameters.
ExcHSE=αExHF,SR(ω)+(1α)ExPBE,SR(ω)+ExPBE,LR(ω)+EcPBE
(1)
As Fig. 2 shows the SOC plays an important role in the identification of the energy levels especially if we are dealing with heavy elements such as Ln, due to this fact we have included the SOC effect. The SOC effect is documented here using the implemented method in VASP. The SOC is most effective in the vicinity of the core region, so the implementation of SOC in VASP is only considered within the PAW spheres.30 So, the contribution of SOC is included using the following perturbed Hamiltonian (zeroth-order-regular approximation):
H̃SO=ij|piϕi|HSO|ϕjpj|,
(2)
and the HSO is given below according to the zeroth-order-regular approximation:
HSOαβ=2(2mec)2K(r)rdV(r)drσαβL.
(3)
The α and β denote spin up and spin down, and the angular momentum operators L are defined as L=r×p. The V(r) is the spherical part of the effective all electronic potential within the PAW sphere, and its relation with K(r) is
K(r)=1V(r)2mec22,
(4)
where me is the effective mass of the particle and c is the speed of light. Besides, in VASP inclusion of SOC effects coincides with the inclusion of relativistic effects. The core electrons are treated fully relativistic and the effect for valence electrons is considered through scalar relativistic approximation.31 
FIG. 2.

The source and magnitude of the energy level splittings in 4fn → 5d configuration. The figure is adapted from Ref. 29.

FIG. 2.

The source and magnitude of the energy level splittings in 4fn → 5d configuration. The figure is adapted from Ref. 29.

Close modal

To simulate a dopant in a periodic lattice, we use the supercell approach with periodic boundary conditions. We construct a 1 × 2 × 1 supercell based on the pre-optimized unit cell of the host crystal and then replace one of the Y3+ sites by a Ce3+ ion. Such supercell contains 128 atoms, and a Ce3+ doping concentration of 3.125%. This doping concentration of Ce3+ is usually higher than the experiments. However, the distance between these Ce3+ ions in the periodic cells are greater than 12 Å, which is sufficiently large to reduce the possibility of interaction between impurities. We then relax the atomic positions while keeping the cell dimensions fixed. Once the supercell is relaxed, we perform a ground-state calculation to determine the position of the Ce-4f level relative to the valence band maximum (VBM) of the host material. The occupied 4f level is typically very localized and atomic in nature and has almost no bandwidth, so the VBM-4f gap is well-defined. Nevertheless, a modified HSE calculation was performed to improve the positioning of levels. The cut-off energy is kept at 520 eV, and the conjugate gradient algorithm is used to relax the system. The sampling of the Brillouin zone was done by implementing the Γ-point only kpoint grid. A convergence threshold of 1 × 10−6 eV was used to reach self-consistency.

Two distinct methods are employed for calculating excited states. The first is the constrained occupancy approach. This method begins by calculating the ground states. The energy eigenvalues are then analyzed to identify levels with 4f and 5d character. To mimic the effect of a photon removing an electron from valence 4f and transferring it to 5d, the 4f levels are manually emptied, and the lowest-lying 5d-like band is filled. Kohn-Sham (KS) eigenvalues and total energies for ground and excited states are subsequently determined. Since KS eigenvalues cannot replace optical transition levels,32,33 only the difference between total energies (ΔSCF) of these two states is considered for energy transition. Absorption energy is obtained using ΔSCF, represented as Eabs = e0*g0 in Fig. 3. Later, the constraint on occupancy is lifted, allowing the perturbed lattice to relax due to a change in its electronic configuration. This relaxation pertains only to atomic positions, leaving the lattice vectors unaltered. This state, depicted in Fig. 3, is denoted as g1, and the emission energy is calculated as Eems = e1*g1. The Stokes shift is determined by subtracting the absorption energy from the emission energy.

FIG. 3.

Configuration coordinate diagram.

FIG. 3.

Configuration coordinate diagram.

Close modal

In the latter approach, a relaxed Ce-doped 1 × 2 × 1 supercell, acquired during ground state calculations, undergoes two separate HSE calculations for each Ce site. Initially, a neutral HSE calculation is conducted to determine the ground state location of 4f states when a 4f level is occupied. Additionally, a modified HSE functional relaxes a mono-positively charged supercell structure of Ce-doped YSO. This process, which adheres to the Frank-Condon principle,34 reveals the electronic structure of excited states at the conduction level and ascertains the location of low-lying 5d states. The next step involves obtaining the 4f → 5d transition energies by analyzing the 4f eigenvalues from a neutral calculation and subtracting them from the valence band edge of the corresponding host cell, expressed as a = 4f − VBMhost. Additionally, the 5d eigenvalues of the Ce impurity are identified by evaluating the mono-positively charged calculation. The crystal field effect eliminates the degeneracy of the 5d levels, as depicted in Figs. 7(b) and 7(d), allowing the determination 4f to 5di where i is the indices for different 5d orbital orientations.

Before determining the transition energies, the 5di eigenvalues are subtracted from the corresponding host conduction band minimum (CBM), expressed as b = CBMhost − 5di. Finally, the transition energies between 4f → 5di are computed by subtracting a and b from the experimental or modified band gap energy of the host in its primitive cell, expressed as 4f → 5di = Egab. This entire process is visually outlined in Fig. 4.

FIG. 4.

The transition energy: The host valence band maximum of the corresponding cell is subtracted from occupied 4f eigenvalue (a = 4fVBMhost). Then 5d eigenvalues that are obtained from +1-charged supercell calculation are subtracted from the host conduction band minimum of the corresponding cell (b = CBMhost − 5d). Lastly, the two values are subtracted from the calculated host band gap, Eg = 7.4013 eV. 4f → 5d = Egab.

FIG. 4.

The transition energy: The host valence band maximum of the corresponding cell is subtracted from occupied 4f eigenvalue (a = 4fVBMhost). Then 5d eigenvalues that are obtained from +1-charged supercell calculation are subtracted from the host conduction band minimum of the corresponding cell (b = CBMhost − 5d). Lastly, the two values are subtracted from the calculated host band gap, Eg = 7.4013 eV. 4f → 5d = Egab.

Close modal

As we mentioned earlier the KS eigenvalues are not considered true states of the system, however, since we only consider their energy difference and we subtract these energy differences from the KS band gap, one should end up with a true value that can be considered as true transition energy between 4f and 5d states.

In the initial stage of our study, we diligently fine-tuned the parameters of the HSE functional to match the experimental band gap, which are tabulated in Table I. Our parameter optimization process is presented in Fig. 5, which reveals that the optimal value for the fraction of HF exact exchange α is 0.40, while the screening value μ is 0.19. Notably, these values were derived from a Gamma-point only HSE calculation, and we subsequently validated them using a kpoint mesh of 1 × 3 × 2.

TABLE I.

Band gap of C2/c Y2SiO5. This study: *.

MaterialMethodBand gap (eV)
Y2SiO5 Expt. 7.435  
Y2SiO5 Expt. 7.436  
Y2SiO5 PBE 4.8210  
Y2SiO5 PBE* 4.819 
Y2SiO5 mBJ 6.1410  
Y2SiO5 HSE* 7.40 
MaterialMethodBand gap (eV)
Y2SiO5 Expt. 7.435  
Y2SiO5 Expt. 7.436  
Y2SiO5 PBE 4.8210  
Y2SiO5 PBE* 4.819 
Y2SiO5 mBJ 6.1410  
Y2SiO5 HSE* 7.40 
FIG. 5.

Tuning of HSE parameters based on the experimental band gap. The data points are extracted from Gamma-point only calculations. As can be seen in the figure, the experimental band gap gets reproduced at α = 40% exact exchange and ω = 0.19 screening.

FIG. 5.

Tuning of HSE parameters based on the experimental band gap. The data points are extracted from Gamma-point only calculations. As can be seen in the figure, the experimental band gap gets reproduced at α = 40% exact exchange and ω = 0.19 screening.

Close modal

The information provided in Table II illustrates the variation in bond lengths at site1 and site2. This change in bond length during excitation can be understood through the Franck-Condon Principle,37,38 which explains that during an electronic transition such as ionization, the nuclear arrangement of the molecule initially remains largely unchanged due to the nuclei’s significantly greater mass compared to electrons. This results in the electronic transition happening quicker than the nuclei’s response. However, post-transition, the nucleus has to realign itself to the new electronic configuration, a process that involves vibration, and a rhythmic motion of the atoms in the molecule relative to each other. This motion can bring about a change in the bond length. Moreover, when a transition from the 4f to the 5d states occurs, electrons are moved from the inner 4f shell to the more external 5d shell in an atom’s electronic configuration. In such a scenario, bond length generally decreases. The underpinning reason for this decrease is the alterations in the orbital overlap’s nature and magnitude, along with the differences in shielding and penetration abilities between 4f and 5d electrons.39 This explanation supports the data in Table II, where the bond length has almost decreased in all cases in both sites except for the Ce–O5 at the site1.

TABLE II.

Bond lengths (in Å) of Ce3+ at site1 and site2 in ground and excited states.

Bond lengthsite1 - gr.site1-exc.site2-gr.site2-exc.
Ce–O1 2.32 2.29 2.297 2.219 
Ce–O2 2.49 2.46 2.411 2.368 
Ce–O3 2.71 2.70 2.373 2.307 
Ce–O4 2.38 2.32 2.390 2.335 
Ce–O5 2.49 2.50 2.385 2.326 
Ce–O6 2.44 2.40 2.391 2.337 
Ce–O7 2.45 2.28   
Bond lengthsite1 - gr.site1-exc.site2-gr.site2-exc.
Ce–O1 2.32 2.29 2.297 2.219 
Ce–O2 2.49 2.46 2.411 2.368 
Ce–O3 2.71 2.70 2.373 2.307 
Ce–O4 2.38 2.32 2.390 2.335 
Ce–O5 2.49 2.50 2.385 2.326 
Ce–O6 2.44 2.40 2.391 2.337 
Ce–O7 2.45 2.28   

Figure 6 displays the projected density states of pure YSO using the modified HSE functional, which results in a band gap value that is identical to experimental measurements. The dominant states at the VBM are Op orbitals, whereas Yd orbitals are the main constituents of the CBM.

FIG. 6.

PDOS of pure YSO with and without SOC.

FIG. 6.

PDOS of pure YSO with and without SOC.

Close modal

In the lower section of Fig. 6, we observe the effects of SOC on the electronic structure of YSO. The band gap calculated without SOC yields a value of 7.401 30, while the inclusion of the SOC effect slightly enlarges the band gap to 7.421 30. This difference corresponds to an increase of 20 meV. Notwithstanding the small alteration in the band gap energy, the constitution of the VBM and CBM remains essentially invariant, with a minor shift ascribed to SOC effects. This limited shift is predominantly attributable to the constituent elements of the system. Neither silicon nor oxygen, being lightweight elements, produce a significant SOC effect. Concurrently, the heaviest atom in the system, Yttrium (Y), doesn’t have a noticeable impact on the SOC effect since it is still categorized as a light element according to Russell-Saunders coupling theory, where it states elements with Z < 40 is considered as light.40 Consequently, the cumulative effect of SOC on the physical properties of pure YSO is deemed insignificant.

Table III presents the calculated absorption and emission energies, as well as the Stokes shift. The calculations were performed on a YSO standard unit cell with 64 atoms, as it has been demonstrated by Jia et al.41 that the energy difference between cells with 64 atoms and 128 atoms is negligible. Even though the compound studied by Jia et al. is Lu2SiO5 (LSO), the structure of the cell is identical (C2/c). Thus, we opted to use the 64 atoms cell for the cDFT approach as it is a more cost-effective option.

TABLE III.

Calculated total energies, absorption, and emission energies of Ce: Y2SiO5 based on cDFT approach.

Site1 (eV)Site2 (eV)
Eg0 −746.058 −745.744 
Ee0* −741.929 −741.868 
Eg1 −746.058 −745.744 
Ee1* −742.121 −742.346 
Eabs 4.130 3.876 
Eabs6  3.41 3.30 
Eems 3.938 3.398 
Eems6  3.12 2.73 
Stokes shift 0.191 0.478 
Site1 (eV)Site2 (eV)
Eg0 −746.058 −745.744 
Ee0* −741.929 −741.868 
Eg1 −746.058 −745.744 
Ee1* −742.121 −742.346 
Eabs 4.130 3.876 
Eabs6  3.41 3.30 
Eems 3.938 3.398 
Eems6  3.12 2.73 
Stokes shift 0.191 0.478 

The results show larger calculated values for absorption and emission at both sites, compared to the experimental values. The discrepancy between the calculated and experimental values is ∼0.7 eV for both energies at Ce1, while at Ce2, the difference is slightly smaller, around 0.5 eV. A possible reason for this mismatch is the lack of consideration of SOC in the calculations. This connection can be inferred easier by referring to Fig. 2, which shows that the magnitude of the SOC effect is on the order of 1000 cm−1 (0.12 eV), comparable to the observed discrepancy.

It is worth noting that SOC is a relativistic correction that affects heavier atoms more significantly, and since Ce has unpaired electrons, the SOC effect on this atom is likely to be more pronounced. Hence, it is reasonable to focus on the SOC correction that corresponds to Ce, represented by Esoc.

In VASP, Esoc is defined as the energy difference between two collinear spin configurations, where the spins are oriented parallel and anti-parallel to the quantization axis. The quantization axis is defined by the direction of the external magnetic field or the preferred orientation of the magnetic moments in the system. Thus, by calculating Esoc, we can account for the effect of SOC on the electronic and magnetic properties of the system, and it is defined as follows:
Esoc=ijEsocij,
(5)
Esocij can be expanded as,
Esocij=δRiRjδliljnkωkfnkαβψ̃nkα|pĩ×ϕi|Hsocαβ|ϕjpj̃|ψ̃nkβ,
(6)
where, ωk is k-point weights, fnk is the Fermi weights, ϕi are the partial waves of an atom centered at Ri, ψ̃nkα is the α Spinor-component of the pseudo-orbital where its band-index is n and Bloch-vector is k.30 

We observed two distinct values of Esoc for the absorption and emission processes. So, we applied them separately to account for the differences. The calculated SOC effects for Ce1 during absorption and emission processes are Esoc = −0.65396 and Esoc = −0.77685 eV, respectively. Similarly, the Esoc for Ce2 during absorption and emission processes are Esoc = −0.65397 and Esoc = −0.77850 eV, respectively. The SOC-corrected values of absorption and emission energy are listed in Table IV, and they are obtained by adding the corresponding Esoc energy to the absorption and emission energies presented in Table III.

TABLE IV.

The SOC-corrected values of absorption and emission energies. The updated values are achieved by adding Esoc to the corresponding energies in Table III. This study: *, Exp.6 

Site1 (eV)Site2 (eV)
Eabs3.47 3.22 
Eabs6  3.41 3.30 
Eems3.16 2.62 
Eems6  3.12 2.73 
Stokes shift 0.314 0.602 
Site1 (eV)Site2 (eV)
Eabs3.47 3.22 
Eabs6  3.41 3.30 
Eems3.16 2.62 
Eems6  3.12 2.73 
Stokes shift 0.314 0.602 

The corrected values exhibit an excellent agreement between experimental and calculated results, particularly at Ce1. For Ce1, the discrepancy for both absorption and emission is around 0.05 eV higher than the experiment, whereas for Ce2, we observe a larger difference at the order of 0.10 eV lower than the experiment. It is worth mentioning that these comparisons are made with one experiment only, and if we compare the calculated absorption energy with the average of experiments listed in Table V for Ce1, Eg = 3.48 eV, then the discrepancy becomes almost zero. However, this is not the case for Ce2, as the average absorption energy for experiments at this site is 3.31 eV, while the calculated value is 3.22 eV, indicating a difference of 0.09 eV.

TABLE V.

Transition energies, 4f → 5di. The procedure for the calculated value in this table is depicted in Fig. 4. This study: *.

4f → 5di = 1–5 (eV)
IonSiteMethod12345
Ce3+ Ce1 Expt.6  3.41 4.03 4.54 ⋯ ⋯ 
Ce3+ Ce1 Expt.46  3.55 4.19 4.68 ⋯ ⋯ 
Ce3+ Ce1 Expt.47  3.48 4.13 4.72 ⋯ ⋯ 
Ce3+ Ce1 Expavg 3.48 4.11 4.65 ⋯ ⋯ 
Ce3+ Ce1 CASPT28  3.41 4.03 4.46 5.75 6.35 
Ce3+ Ce1 HSE* 3.511 4.45 4.64 4.92 5.21 
Ce3+ Ce1 HSE + SOC* 3.27 4.21 4.53 4.65 4.97 
Ce3+ Ce1 CASPT2+SOC8  3.53 4.16 4.59 5.89 6.49 
Ce3+ Ce2 Expt.6  3.30 3.80 ⋯ ⋯ ⋯ 
Ce3+ Ce2 Expt.46  3.32 3.78 4.79 ⋯ ⋯ 
Ce3+ Ce2 Expt.47  3.31 3.79 4.78 ⋯ ⋯ 
Ce3+ Ce2 Expavg 3.31 3.79 4.78 ⋯ ⋯ 
Ce3+ Ce2 CASPT28  3.44 3.64 3.91 6.29 7.09 
Ce3+ Ce2 HSE* 3.44 3.50 4.22 4.85 5.21 
Ce3+ Ce2 HSE + SOC* 3.30 3.43 4.15 4.79 5.09 
Ce3+ Ce2 CASPT2 + SOC8  3.52 3.76 4.04 6.42 7.21 
4f → 5di = 1–5 (eV)
IonSiteMethod12345
Ce3+ Ce1 Expt.6  3.41 4.03 4.54 ⋯ ⋯ 
Ce3+ Ce1 Expt.46  3.55 4.19 4.68 ⋯ ⋯ 
Ce3+ Ce1 Expt.47  3.48 4.13 4.72 ⋯ ⋯ 
Ce3+ Ce1 Expavg 3.48 4.11 4.65 ⋯ ⋯ 
Ce3+ Ce1 CASPT28  3.41 4.03 4.46 5.75 6.35 
Ce3+ Ce1 HSE* 3.511 4.45 4.64 4.92 5.21 
Ce3+ Ce1 HSE + SOC* 3.27 4.21 4.53 4.65 4.97 
Ce3+ Ce1 CASPT2+SOC8  3.53 4.16 4.59 5.89 6.49 
Ce3+ Ce2 Expt.6  3.30 3.80 ⋯ ⋯ ⋯ 
Ce3+ Ce2 Expt.46  3.32 3.78 4.79 ⋯ ⋯ 
Ce3+ Ce2 Expt.47  3.31 3.79 4.78 ⋯ ⋯ 
Ce3+ Ce2 Expavg 3.31 3.79 4.78 ⋯ ⋯ 
Ce3+ Ce2 CASPT28  3.44 3.64 3.91 6.29 7.09 
Ce3+ Ce2 HSE* 3.44 3.50 4.22 4.85 5.21 
Ce3+ Ce2 HSE + SOC* 3.30 3.43 4.15 4.79 5.09 
Ce3+ Ce2 CASPT2 + SOC8  3.52 3.76 4.04 6.42 7.21 

Figure 7, shows the Density of States (DOS) for the Ce-doped YSO. The main composition of the VBM and the CBM is constituted of Op and Yd orbitals, respectively, a configuration similar to that of pure YSO. The Partial DOS (PDOS) at ground state for Ce-doped YSO is depicted in subfigures (a) and (b) of Fig. 7.

FIG. 7.

Orbital projected DOS of Ce-doped YSO, at Ce1 and Ce2 in a 1 × 2 × 1 supercell. The blue-filled states indicate the 4f orbitals of the Ce3+ while they are occupied: (a) and (b). When 4f states are unoccupied, we can observe the 5d states in the band gap that are filled in red. The black dotted lines show the position of the Fermi level: (c) and (d).

FIG. 7.

Orbital projected DOS of Ce-doped YSO, at Ce1 and Ce2 in a 1 × 2 × 1 supercell. The blue-filled states indicate the 4f orbitals of the Ce3+ while they are occupied: (a) and (b). When 4f states are unoccupied, we can observe the 5d states in the band gap that are filled in red. The black dotted lines show the position of the Fermi level: (c) and (d).

Close modal

The location of the 4f states within the band gap, as indicated by Zhou et al.,10 contrasts with our observations. Zhou et al. reported the 4f states to be 4.78 eV away from the VBM, a different finding compared to our analysis. However, the current results align closely with the findings of Ning et al.,42 who utilized the AIMPs approach on Ce: LSO. Their research identified the 4f - VBM gap as 2.81 and 3.07 for Ce1 and Ce2, respectively. In a related study, Vedda et al.43 conducted experiments on Ce: LSO, ascertaining the 4f - VBM gap to be within the range of 2.6–2.9 eV. Despite the difference in the chemical compound of the host, the host still shares similar crystal (C2/c) and site (C1) symmetries for the Ce dopant. These consistent findings across different studies suggest that the VBM-4f gap in Ce: YSO aligns well with previous studies.

For more comprehensive comparison, the Dorenbos model can be employed.4,44 The Dorenbos relation is an empirical model that is widely used to estimate the energy of 4f of trivalent and divalent Ln ions in a given host lattice based on the peak position of the lowest 4f → 5d level of at least one of the Ln ions in that host. Based on this empirical approach and with the use of experimental data for Ce, and Pr-doped YSO and LSO, Kolk together with Dorenbos et al. have produced the probable location of 4f and the lowest lying 5d levels inside the band gap for all the Ln in YSO.45 The estimated value that they have obtained for the 4f - VBM gap is around 3.53 eV. It should be added that they have not identified this value to a specific site, Ce1 or Ce2, as they mention their data for the photocurrent experiment do not show any doublet structure.45 Lastly, the inclusion of SOC has shifted the 4f ground states by 0.20 eV at both sites. This has reduced the 4f-VBM gap down at Ce1 to 2.73 eV, and to 2.70 eV at Ce2. With this shift, the agreement with the experimental work of Vedda et al. has improved.43 

Figures 7(c) and 7(d) illustrates the mono-positively charged state of Ce-doped YSO. At this phase, all atomic positions are held consistent with the neutral Ce-doped YSO to generate an excitonic effect. This effect occurs between the hole at the VBM and an electron in one of Ce’s lower 5d states, adhering to the Franck-Condon principle. These low 5d states, close to the CBM, display notable splitting due to the crystal field (CF), marked in red. A comparison of the red peaks at Ce1 and Ce2 highlights more prominent peaks or splittings at Ce2 in both SOC and non-SOC scenarios. This observation is attributed to the smaller size of Ce2 compared to Ce1. Therefore, the CF splitting is stronger at Ce2 than Ce1, which agrees with previous studies conducted using DFT + CASPT2.8,10

The significant impact of SOC is evident in Figs. 7(c) and 7(d), particularly in the lower section where the 5di levels undergo a downward shift towards the VBM at both sites. This shift measures 0.38 eV at Ce1 and 0.04 eV at Ce2. Interestingly, the direction of the shift for the remaining levels is not uniform at both sites. This shift, combined with the presence of SOC, results in four out of the five lowest peaks falling within the band gap. A clearer illustration of the gap between these peaks and their shifts due to SOC incorporation is provided in Fig. 8. Here, the 5d2−i levels are marked based on their relative positions to 5d1.

FIG. 8.

Location of 5di levels with respect to lowest lying 5d1 of the Ce atom at Ce1 and Ce2 with and without SOC.

FIG. 8.

Location of 5di levels with respect to lowest lying 5d1 of the Ce atom at Ce1 and Ce2 with and without SOC.

Close modal

Table V outlines the transition energies from 4f to 5d in two unique sites of the YSO. Details about the calculation and methodology used for determining these energy transitions are elaborated in Sec. II.

The table’s sixth row reveals the computed value of the non-SOC but HSE functional of the supercell structure. The initial transition value at Ce1 closely aligns with the experimental value, with calculated values showing a minor difference from the experimental average for the lowest lying 5d: 0.03 at Ce1 and 0.13 at Ce2. This indicates a more precise definition of the lowest lying 5d level in both experiment and AIMPs calculation at both impurity sites.

However, the disparity between the experimental and calculated values grows for the other 5di levels, notably for the last two, 5d4 and 5d5. Here, the absence of clear, sharp peaks and the low intensity, as observed in Fig. 8, complicate the measurement process. This difficulty likely accounts for the data deficiency for these two levels, precluding any comparison.

Incorporating SOC has uniformly reduced the energy of 5di levels at both sites. At Ce1, the energy shift is ∼1600 cm−1 (0.2 eV), and at Ce2, it is about 1000 cm−1 (0.1 eV). These observations align with Wen et al.’s findings for Ce: YSO, where the shift was around 0.12 and 0.1 eV for Ce1 and Ce2, respectively.8 Moreover, Ning et al. reported a similar 1000 cm−1 shift for both Ce: LSO sites.42 However, a distinctive difference is noted in this study: while previous works reported an increase in 5d levels with SOC, a decline is observed here.

Overall, it is hard to state the SOC has uniformly improved all the levels as we observed in the result of cDFT approach. However, it has certainly improved the agreement with the experimental average at 5d2 Ce1, and at 5d1, 5d3 Ce2.

In this study, we investigated the electronic and optical properties of Ce-doped YSO, with a focus on the 4f → 5d transition. We used a band gap-modified HSE functional with Gamma-point only and found that the results were in good agreement with experimental data, particularly when the SOC effect was included. Although the effect of SOC was not consistent for all 5di levels, it improved the transition energies overall, especially when using the cDFT approach. We observed a consistent overshoot in the cDFT results compared to experimental data, but this deviation was reduced with the inclusion of SOC. Specifically, the deviations between calculated values and experiments for absorption and emission were 1.75% and 1.28% at Ce1, and 2.42% and 4.02% at Ce2, respectively.

Computational time often hinders SOC + HSE calculations. Despite this, our research reveals that using the Gamma-point calculations yield results accurate enough for comparison with experimental data. Moreover, our study was conducted on a sufficiently large supercell, making it useful for large materials without compromising the environmental effect surrounding the impurity as is the case with the AIMPs approach. This approach does not require any assumptions to be made, thereby reducing one source of uncertainty.

Another constraint is the need for data to fine-tune parameters. Fore example, the DFT + U approach, which requires tuning the U parameter for each lanthanide to accurately simulate the location of 4f electrons in the band gap. Instead, we employed the HSE + SOC method combined with a supercell approach. This approach helped us pinpoint the location of the 4f ground states within the band gap, taking into account only the associated band gap of the host material. Consequently, we were able to simulate the doped system entirely using first principles, eliminating the need for any pre-existing data. Our findings were closely aligned with prior calculations and experimental data for Ce-doped LSO, despite the comparatively larger deviation (around 0.7 eV) from the empirical model developed through the Dorenbos model for Ce-doped YSO and LSO. Notably, our methodology showed the most consistent agreement with two distinct studies on Ce-doped LSO, both theoretically and experimentally. This outcome implies that our approach may be applicable to other lanthanides and host materials, even in the absence of significant prior knowledge.

In conclusion, our simpler approach, compared to wave function methods like AIMPs, could be considered as a reliable alternative for evaluating phosphors and identifying optical properties in laser materials, even without prior data. It holds promise for high-throughput phosphors applications, though its applicability to other Ln groups and host materials still awaits confirmation.

This work is supported by the Knut and Alice Wallenberg Foundation through Grant No. KAW-2016.0081. The simulations were performed using computational resources provided by the Swedish National Infrastructure at the NSC, Linköping University, and at the PDC, Royal Institute of Technology.

The authors have no conflicts to disclose.

Amin Mirzai: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Aylin Ahadi: Funding acquisition (equal); Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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