Beyond the one-dimensional configuration coordinate model of photoluminescence

Yongchao Jia, Samuel Poncé, Anna Miglio, Masayoshi Mikami, and Xavier Gonze

Phys. Rev. B 100, 155109 – Published 7 October 2019

Abstract

The one-dimensional configuration coordinate model (1D-CCM) is widely used for the analysis of photoluminescence in molecules and doped solids, and relies on a linear combination of the equilibrium nuclear configurations of ground and excited states. It delivers an estimation of the energy barrier at which ground and excited state curves cross, semiclassically linked to nonradiative transition rate and thermal quenching. To assess its predictive power for the latter properties, we propose a new optimized configuration path (OCP) method in which the ground-state and excited-state forces are mixed instead of their configurations. We also define another one-parameter model thanks to a double energy parabola hypothesis (DEPH). We compare the OCP method and the DEPH reference with the 1D-CCM for three paradigmatic $4 f - 5 d$ phosphors $Y_{3} {Al}_{5} O_{12}$ :Ce, ${Lu}_{2} {SiO}_{5}$ :Ce, and ${YAlO}_{3}$ :Ce. We find that the OCP and DEPH methods yield similar results with geometries that have significantly lower ground-state energies than the 1D-CCM for the same $4 f - 5 d$ energy difference. However, the OCP method suffers from the appearance of multiple local minima, rendering the clear determination of the optimal geometry very difficult in practice. Still the OCP method allows one to quantify the deviations from the 1D-CCM, therefore increasing confidence in the lower bound obtained from the DEPH for the $4 f - 5 d$ crossing barrier, and its comparison with the energy of the autoionization thermal quenching mechanism. We expect the OCP approach to be applicable to other luminescent materials or molecules.

Received 9 August 2019
Revised 18 September 2019

DOI:https://doi.org/10.1103/PhysRevB.100.155109

Physics Subject Headings (PhySH)

Electron-phonon coupling Luminescence

Density functional calculations Density functional theory Photoluminescence

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yongchao Jia^1,*, Samuel Poncé², Anna Miglio¹, Masayoshi Mikami ³, and Xavier Gonze ^1,4

¹European Theoretical Spectroscopy Facility, Institute of Condensed Matter and Nanosciences, Université catholique de Louvain, Chemin des étoiles 8, bte L07.03.01, B-1348 Louvain-la-Neuve, Belgium
²Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, United Kingdom
³Functional Materials Design Laboratory, Yokohama R&D Center, 1000, Kamoshida-cho Aoba-ku, Yokohama, 227-8502, Japan
⁴Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Nobel St. 3, Moscow, 143026, Russia

^*yongchao.jia@uclouvain.be

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Vol. 100, Iss. 15 — 15 October 2019

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Images

Figure 1
The one-dimensional configurational coordinate model in ${Ce}^{3 +}$ - or ${Eu}^{2 +}$ -doped phosphors, that share $4 f - 5 d$ excitation. The generalized configuration coordinate $Q$ results from the linear combination of ground-state atomic positions ( $Q_{g}$ ) and excited state ones ( $Q_{e}$ ). Total energies as a function of Q are reported for the electronic ground state (labeled $4 f$ ), and for the electronic excited state (labeled $5 d$ ), the absorption and emission occur between these energies at $E_{g}$ and $E_{e}$ , respectively. The energy lost to the lattice after light absorption defines the Frank-Condon shift $E_{FC, e}$ , and similarly $E_{FC, g}$ is the energy lost to the lattice after light emission. Their sum gives the Stokes shift. One can associate vibrational frequencies to the ground and excited state curves, $Ω_{g}$ and $Ω_{e}$ . They are linked to each spectrum shape through the so-called Huang-Rhys factor $S$ , the ratio between a Frank-Condon shift and the related vibrational frequency. ZPL denotes the zero-phonon line, the direct transition energy when no phonon is involved. $E_{f d}$ is the classical activation barrier at atomic positions $Q_{f d}$ for nonradiative recombination through $4 f - 5 d$ crossing.
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Figure 2
The ground-state and excited-state total energies as a function of $x$ , from the 1D-CCM method and from the DEPH, in three ${Ce}^{3 +}$ -doped phosphors: YAG:Ce, LSO:Ce, and YAP:Ce.
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Figure 3
Electronic band structure of LSO:Ce in the excited state, with different $x$ values for the linear combinations of $Q_{g}$ and $Q_{e}$ geometry from 1D-CCM.
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Figure 4
The ground-state (bottom) and excited-state (top) energy as a function of their difference $Δ E$ , from the 1D-CCM, the OCP method and the DEPH, in three ${Ce}^{3 +}$ -doped phosphors: YAG:Ce (on the left), LSO:Ce (center), and YAP:Ce (on the right). The $4 f - 5 d$ crossing corresponds to $Δ E = 0$ , which is never attained, see text, although clear trends appear for decreasing values of $Δ E$ . The reference DEPH continuous black line corresponds to Eq. (39) with three parameters (per material) determined by first-principles data. The violet full circles are obtained from the one-dimensional configuration coordinate model. The other symbols show computed values from the OCP, including possibly different local minima values for $Δ E$ lower than some threshold: the blue empty triangles are obtained from OCP relaxations starting at the ground-state geometry; the black empty squares are obtained from OCP relaxations starting at the excited-state geometry; the green empty diamond are obtained from test OCP relaxations starting at the OCP geometries with $Λ = 1.4$ , 1.3, and 1.4 for $Y_{3} {Al}_{5} O_{12}$ :Ce, ${Lu}_{2} {SiO}_{5}$ :Ce, and ${YAlO}_{3}$ :Ce, respectively. The arrow indicates the minimum energy point for the ground and excited states.
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Figure 5
Electronic band structure of LSO:Ce in the excited state, for geometries deduced from the OCP method, for four different $Λ$ values ( $Λ = 0, 0.5, 1.0, and 1.5)$ . Using the $Q_{e}$ geometry as input (four leftmost panels), a nearly flat level at 4.8 eV with predominant $5 d$ character can be identified. The $4 f$ state eigenenergies (around 2eV) increase slightly with $Λ$ . If the OCP optimization at $Λ = 1.5$ is started from the $Q_{g}$ geometry, another geometry is found, for which the electronic structure is quite different, see the rightmost panel.
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Figure 6
The ground-state and excited-state total energies as a function of $Λ$ , from the OCP method and from the DEPH, in three ${Ce}^{3 +}$ -doped phosphors: YAG:Ce, LSO:Ce, and YAP:Ce. The DEPH continuous lines correspond to Eqs. (45) and (46) with three parameters (per material) determined by first-principles data (i.e., they are not fitted to the OPC method points).
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