Designing Pr-based advanced photoluminescent materials using machine learning and density functional theory

Abstract

This work presents a machine learning approach to predict novel perovskite oxide materials in the Pr-Al-O and Pr-Sc-O compound families with the potential for photoluminescence applications. The predicted materials exhibit a large bandgap and high Debye temperature, and have remained unexplored thus far. The predicted compounds (Pr$_3$AlO$_6$, Pr$_4$Al$_2$O$_9$, Pr$_3$ScO$_6$ and Pr$_3$Sc$_5$O$_{12}$) are screened using machine learning approach, which are then confirmed by density functional theory calculations. The study includes the calculation of the bandgap and density of states to determine electronic properties, and the optical absorption and emission spectra to determine optical properties. Mechanical stability of the predicted compounds, as demonstrated by satisfying the Born-Huang criterion. By combining machine learning and density functional theory, this work offers a more efficient and comprehensive approach to materials discovery and design.

Graphical Abstract

Appendix

Mechanical properties

The elastic tensor matrices for Pr$_3$AlO$_6$ has form:

$$\begin{aligned} \text{C}_{ij} (\text{GPa})= \begin{bmatrix} 204.85 &{} 67.26 &{} 84.16 &{} 0 &{} 0 &{} 0 \\ 67.26 &{} 185.02 &{} 85.07 &{} 0 &{} 0 &{} 0 \\ 84.16 &{} 85.07 &{} 237.75 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 52.50 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 45.87 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 38.66 \end{bmatrix} \end{aligned}$$

(5)

In the case of Pr$_3$AlO$_6$, which is part of the orthorhombic crystal system, the essential conditions are as follows:

• 1. $\text{C}_{11} > 0$

• 2. $\text{C}_{11}\times \text{C}_{22} > \text{C}_{12}^2$

• 3. $\text{C}_{11}\times \text{C}_{22}\times \text{C}_{33} \ + \ 2\text{C}_{12}\times \text{C}_{13}\times \text{C}_{23} \ - \ \text{C}_{11}\times \text{C}_{23}^2 \ - \ \text{C}_{22}\times \text{C}_{13}^2 \ - \text{C}_{33}\times \text{C}_{12}^2 > 0$

• 4. $\text{C}_{44} > 0$

• 5. $\text{C}_{55} > 0$

• 6. $\text{C}_{66} > 0$

The elastic tensor matrices for Pr$_4$Al$_2$O$_9$ has form:

$$\begin{aligned} \text{C}_{ij} (\text{GPa})= \begin{bmatrix} 205.77 &{} 83.24 &{} 82.13 &{} 2.68 &{} 0 &{} 0 \\ 83.24 &{} 195.80 &{} 81.60 &{} -2.54 &{} 0 &{} 0 \\ 82.13 &{} 81.60 &{} 169.47 &{} -1.73 &{} 0 &{} 0 \\ 2.68 &{} -2.54 &{} -1.73 &{} 54.80 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 61.25 &{} -1.83 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1.83 &{} 61.43 \end{bmatrix} \end{aligned}$$

(6)

For Pr$_4$Al$_2$O$_9$ which belongs to the monoclinic crystal system, the necessary criteria are given as:

• 1. C$_{11}>$ 0

• 2. C$_{22}>$ 0

• 3. C$_{33}>$ 0

• 4. C$_{44}>$ 0

• 5. C$_{55}>$

• 6. C$_{66}>$ 0

• 7. $\big [$C$_{11}$ + C$_{22}$ + C$_{33}$ + 2$\times$(C$_{12}$ + C$_{13}$ + C$_{23}$)$\big ]>$ 0

• 8. C$_{33}\times$C$_{55}$ - C$_{35}^2>$ 0

• 9. C$_{44}\times$C$_{66}$ - C$_{46}^2>$ 0 (x) C$_{22}$ + C$_{33}$ - 2$\times$C$_{23}$ > 0

• 10. C$_{22}\times$(C$_{33}\times$C$_{55}$ - C$_{35}^2$) + 2$\times$C$_{23}\times$C$_{25}\times$C$_{35}$ - (C$_{23}$)$^2\times$C$_{55}$ - (C$_{25}$)$^2\times$C$_{33}$ > 0

• 11. 2$\times$[C$_{15}\times$C$_{25}\times$(C$_{33}\times$C$_{12}$ - C$_{13}\times$C$_{23}$) + C$_{15}\times$C$_{35}\times$(C$_{22}\times$C$_{13}$ - C$_{12}\times$C$_{23}$) + C$_{25}\times$C$_{35}\times$(C$_{11}\times$C$_{23}$ - C$_{12}\times$C$_{13}$)] - [C$_{15}\times$C$_{15}\times$(C$_{22}\times$C$_{33}$ - C$_{23}^2$) + C$_{25}\times$C$_{25}\times$(C$_{11}\times$C$_{33}$ - C$_{13}^2$) + C$_{35}\times$C$_{35}\times$(C$_{11}\times$C$_{22}$ - C$_{12}^2$)] + C$_{55}\times$g > 0 where, g = [C$_{11}\times$C$_{22}\times$C$_{33}$ - C$_{11}\times$C$_{23}\times$C$_{23}$ - C$_{22}\times$C$_{13}\times$C$_{13}$ - C$_{33}\times$C$_{12}\times$C$_{12}$ + 2$\times$C$_{12}\times$C$_{13}\times$C$_{23}$ ].

The elastic tensor matrices for Pr$_3$ScO$_6$ has form:

$$\begin{aligned} \text{C}_{ij} (\text{GPa})= \begin{bmatrix} 185.28&{} 92.99 &{} 98.50 &{} -4.21 &{} -14.20 &{} 6.47 \\ 92.99 &{} 214.09 &{} 93.92 &{} -4.96 &{} 7.98 &{} -5.98 \\ 98.50 &{} 93.92 &{} 195.43 &{} 8.53 &{} -11.13 &{} -0.95 \\ -4.21 &{} -4.96 &{} 8.53 &{} 60.74 &{} -0.29 &{} -8.01 \\ -14.20 &{} 7.98 &{} -11.13 &{} -0.29 &{} 45.79 &{} -4.37 \\ 6.47 &{} -5.98 &{} -0.95 &{} -8.01 &{} -4.37 &{} 50.59 \end{bmatrix} \end{aligned}$$

(7)

For Pr$_3$ScO$_6$ which belongs to the rhombohedral-2 crystal system, the necessary criteria are given as;

• 1. C$_{11}$ - C$_{12}$ > 0

• 2. C$_{13}^2 < (1/2)\times$C$_{33}$(C$_{11}$ + C$_{12}$)

• 3. C$_{14}^2$ + C$_{15}^2 < (1/2)\times$C$_{44}\times$(C$_{11}$-C$_{12}$) = C$_{44}\times$C$_{66}$

• 4. C$_{44}$ > 0

The elastic tensor matrices for Pr$_3$Sc$_5$O$_{12}$ has form:

$$\begin{aligned} \text{C}_{ij} (\text{GPa})= \begin{bmatrix} 222.24 &{} 82.08 &{} 82.08 &{} 0 &{} 0 &{} 0 \\ 82.081 &{} 222.24 &{} 82.081 &{} 0 &{} 0 &{} 0 \\ 82.081 &{} 82.081 &{} 222.24 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 56.54 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 56.54 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 56.54 \end{bmatrix} \end{aligned}$$

(8)

For Pr$_3$Sc$_5$O$_{12}$ which belongs to the cubic crystal system, the necessary criteria are given as;

• 1. C$_{11}$ - C$_{12}$ > 0

• 2. C$_{11}$ + 2C$_{12}$ > 0

• 3. C$_{44}$ > 0