Crystal structure map for materials classification and modeling

Figures & data

Figure 1. Three-step workflow described in the present study. Capital italic name expresses each detailed procedure, CIF2ESC: conversion from a CIF file to electronic-structure-calculation data with output of fingerprint as structure feature for post structure analyses; DISTANCE: evaluation of distances between structure feature vectors; DISTANCE_DISTRIBUTION: calculation of distance distribution; DIAGNOSIS: extraction of inequivalent structures; CMDS: classical multidimensional scaling; PCEV: calculation of principal component eigenvectors; PROJECTION: projection of an arbitrary structure on the dimension-reduced map; ATLS: generation of atom-type list; FINDSYM${ }^{\ast }$: space-group finder ${ }^{\ast }$: Ref [15,16]. Open and solid (and dotted) arrows denote data transfers between the steps and inside each step, respectively. Dotted arrow is applicable if necessary.

Table 1. List of Al${ }_2$O${ }_3$ structures in the order of calculated total energy $\mathit{E}$ (meV/atom) relative to the most stable $\mathit{R}\bar{3}\mathit{c}$ structure, and the energy gap $E_G$ (eV). Figures in parentheses are values given in the materials project site [11]. Color scheme is used for distinguishing different structures in Figures 2 and 4.

ref.	Color	SG	$\mathit{E}$	$E_G$	Remarks
mp-1143	Black	$\mathit{R}\bar{3}\mathit{c}$	0 (0)	5.596 (6.044)	Most stable
mp-7048	Maroon	$\mathit{C}2/\mathit{m}$	+17 (+10)	4.248 (4.605)	Most stable phase of Ga${ }_2$O${ }_3$
mp-2254	Blue	$\mathit{Pna}21$	+18 (+17)	4.592 (4.830)
mp-776,475	Pink	$\mathit{Ia}\bar{3}$	+43 (+30)	4.982 (5.216)	Most stable phase of In${ }_2$O${ }_3$
mp-32,591	Green	$\mathit{Cm}$	+63 (+57)	2.854 (3.075)
mp-754,624	Teal	$\mathit{R}\bar{3}$	+71 (+66)	5.383 (5.797)
mp-754,401	Magenta	$\mathit{Cmc}21$	+74 (+46)	4.022 (4.235)
mp-755,175	Purple	$\mathit{P}\bar{3}1\mathit{c}$	+79 (+72)	5.227 (5.494)
mp-1938	Lime	$\mathit{Pbcn}$	+92 (+94)	5.047 (5.313)
mp-755,483	Olive	$\mathit{R}3\mathit{m}$	+93 (+91)	3.863 (4.098)
mp-754,531	Yellow	$\mathit{P}21/\mathit{c}$	+98 (+77)	4.141 (4.374)
Momida${ }^{\mathrm{\#}}$	Red (thick)	$\mathit{P}1$	+179 (—)	3.725 (—)	Amorphous model
mp-985,587	—	$\mathit{P}321$	+210 (+210)	4.686 (4.807)	Artificial bilayer film
mp-1,245,063	Red (broken)	$\mathit{P}1$	+217 (+204)	3.137 (3.326)	Amorphous model
mp-642,363	Cyan	$\mathit{Cmcm}$	+266 (+281)	4.009 (4.249)	Elastically unstable
mp-638,765	Navy	$\mathit{P}\bar{1}$	+682 (+809)	1.102 (1.136)

^#: Ref [43].

Figure 2. F-Fingerprint of all the Al${ }_2$O${ }_3$ systems listed in Table 1 except for the 2D film model. Radius cutoff $R_{max}=10$ Å is assumed and the fingerprint of dimensions between 0 and 100, 101 and 200, and 201 and 300 corresponds to O-O, O-Al, and Al-Al pairs, respectively. Line colors follow color scheme in Table 1 for distinguishing different structures.

Figure 3. Eigenvalues calculated from the fingerprint of Al${ }_2$O${ }_3$ polymorphs using (a) $R_{\max }=10$ Å and (b) $R_{\max }=5$ Å with Euclidean (blue) and cosine (red) distances.

Figure 4. Map of Al${ }_2$O${ }_3$ structures with the five principal components (PCs) with (a) Euclidean-distance $R_{\max }=10$ Å, (b) cosine-distance $R_{\max }=10$ Å, (c) Euclidean-distance $R_{\max }=5$ Å, and (d) cosine-distance $R_{\max }=5$ Å. Dot colors follow color scheme listed in Table 1. Square dots in each figure denote the three lowest energy structures (SG: $\mathit{R}\bar{3}\mathit{c}$ (black), $\mathit{C}2/\mathit{m}$ (maroon), and $\mathit{Pna}21$ (blue)), and small and large red dots express amorphous model structures (SG: $\mathit{P}1$) by the materials project [11] and momida [43], respectively.

Figure 5. Eigenvector of the five principal components (PCs) with (a) Euclidean-distance $R_{\max }=10$ Å, (b) cosine-distance $R_{\max }=10$ Å, (c) Euclidean-distance $R_{\max }=5$ Å, and (d) cosine-distance $R_{\max }=5$ Å, calculated by using Equation (17)(17) ${\mathbf{u}}_{+}=\left({\mathbf{X}}^T-1{\bar{X}}^{\mathit{T}}\right){\mathbf{Q}}_{+}{\mathbf{\Lambda }}_{+}^{-\mathit{12}}=\left({\mathbf{X}}^T-1{\bar{X}}^{\mathit{T}}\right){\mathbf{X}}_{+}{\mathbf{\Lambda }}_{+}^{-1}$(17) in Al${ }_2$O${ }_3$ polymorphs. Red, lime, blue, maroon, and green lines denote $C_1$, $C_2$, $C_3$, $C_4$, and $C_5$ PC, respectively. Horizontal axis denotes the dimension of the original fingerprint space (see Figure 2). In (a) and (b), the dimensions between 1 and 100, 101 and 200, and 201 and 300 express the O-O, O-Al, and Al-Al pair radius, respectively, and in (c) and (d), the dimensions between 1 and 50, 51 and 100, and 1001 and 150 do the O-O, O-Al, and Al-Al pair radius, respectively.

Figure 6. Regression analysis of the total energy with descriptors up to the second order of the five PCs in Al${ }_2$O${ }_3$ polymorph structures. (a) Euclidean-distance $R_{\max }=10$ Å, (b) cosine-distance $R_{\max }=10$ Å, (c) Euclidean-distance $R_{\max }=5$ Å, and (d) cosine-distance $R_{\max }=5$ Å. Blue, red, and green dots and lines denote the coefficient of determination ($R^2$), its leave-one-out coefficient ($Q^2$), and root mean square error (RMSE), respectively.

Figure 7. Double cross-validation test of the total-energy regression model constructed with the descriptors from the Euclidean distance and $R_{\max }=5$ Å in Al${ }_2$O${ }_3$ polymorph structures. Dot colors follow color scheme listed in Table 1. Red open circle denotes one of the amorphous structure (mp -1,245,063). The model of the number of descriptors of eight shows the smallest standard deviation ($\sim$0.1 eV).

Figure 8. Crystal structure of La${ }_2$Fe${ }_{26-\mathit{y}}$Si${ }_y$. Green, orange, and brown balls denote La, Fe1 and Fe2 atomic sites, respectively. Partial occupation of Si at the Fe2 sites is depicted with blue. The drawing is made with VESTA [9].

Figure 9. Heats of formation $\mathit{H}$ in eV/atom (Equation (22)(22) $\begin{array}{cc}\mathit{H}(\mathit{y};\mathit{i})& =\frac{1}{28}\{\mathit{E}[\mathrm{L}{\mathrm{a}}_2\mathrm{F}{\mathrm{e}}_{26-\mathrm{y}}\mathrm{S}{\mathrm{i}}_{\mathrm{y}};\mathit{i}]-2\mathit{E}[\mathrm{L}\mathrm{a}]\\ \mathit{\hspace{0.17em}}& -(26-\mathit{y})\mathit{E}[\mathrm{F}\mathrm{e}]-\mathrm{yE}[\mathrm{S}\mathrm{i}]\}\end{array}$(22) ) and $\mathrm{\Delta H}$ in meV/atom (Equation (23)(23) $\mathrm{\Delta H}(\mathit{y};\mathit{i})=\mathit{H}(\mathit{y};\mathit{i})-\mathrm{mi}{\mathrm{n}}_j\mathit{H}(\mathit{y};\mathit{j})$(23) ), and spin magnetic moments in ${\mu }_{\mathrm{B}}$ per Fe atom calculated for La${ }_2$Fe${ }_{26-\mathit{y}}$Si${ }_y$. Red dots, green dots, and blue circles denote values for the configurations with Si occupying at only $8\mathit{b}$, at $8\mathit{b}$ and $96\mathit{i}$, and at only $96\mathit{i}$, respectively. Black dots indicate the corresponding values for La${ }_2$Fe${ }_{26}(\mathit{y}=0)$.

Figure 10. (a) $\mathit{F}$-fingerprints and (b) Euclidean eigenvectors of three principal components of Si-Si (0–80) and Fe-Si (80–160) dimensions of La${ }_2$Fe${ }_{24}$Si${ }_2$ (Case 2–3). Line colors used for the $\mathit{F}$-fingerprints in (a) correspond to those of dots in Fig 11. Red, lime, and blue lines in (b) denote the eigenvectors of $C_1$, $C_2$, and $C_3$, respectively.

Figure 11. Structure map of La${ }_2$Fe${ }_{24}$Si${ }_2$ (case 2–3) obtained by dimension reduction of the Euclidean distance. Dot colors denote the different configurations corresponding to the $\mathit{F}$-fingerprint shown in Fig 10(a).

Figure 12. (a) Selected $\mathit{F}$-fingerprints and (b) eigenvectors of three principal components of Si-Si (0–80) and Fe-Si (80–160) dimensions of La${ }_2$Fe${ }_{23}$Si${ }_3$ (case 3–3). Red, lime, and blue lines in (b) denote the eigenvectors of $C_1$, $C_2$, and $C_3$, respectively.

Figure 13. Structure map of La${ }_2$Fe${ }_{23}$Si${ }_3$ (case 3–3) obtained by dimension reduction of the Euclidean distance. Dot colors denote the different configurations corresponding to the $\mathit{F}$-fingerprint shown in Fig. 12(a), where the $\mathit{F}$-fingerprints of the small black dots here are not included.

Stokes HT, Hatch DM. FINDSYM: program for identifying the space-group symmetry of a crystal. J Appl Cryst. 2005;38(1):237–238. doi: 10.1107/S0021889804031528

 Google Scholar

Stokes HT, Hatch DM, Campbell BJ. FINDSYM, ISOTROPY software suite. Available from: https://iso.byu.edu/

 Google Scholar

Momida H, Hamada T, Takagi Y, et al. Theoretical study on dielectric response of amorphous alumina. Phys Rev B. 2006;73(5):054108. doi: 10.1103/PhysRevB.73.054108

 Web of Science ®Google Scholar

The Materials Project. Available from: https://materialsproject.org/

 Google Scholar

Momma K, Izumi F. VESTA: a three-dimensional visualization system for electronic and structural analysis. J Appl Crystallogr. 2008;41:653–658. Available from: https://jp-minerals.org/vesta/en/

 Web of Science ®Google Scholar

Data availability statement

The raw data required to reproduce these findings are available by making an e-mail request to the corresponding author.