Machine Learning Model for Predicting the Critical Transition Temperature of Hydride Superconductors
ZHAO Jinbin1,2, WANG Jiantao2,3, HE Dongchang2,3, LI Junlin1, SUN Yan2, CHEN Xing-Qiu2(), LIU Peitao2()
1 School of Materials Science and Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China 2 Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China 3 School of Materials Science and Engineering, University of Science and Technology of China, Shenyang 110016, China
Cite this article:
ZHAO Jinbin, WANG Jiantao, HE Dongchang, LI Junlin, SUN Yan, CHEN Xing-Qiu, LIU Peitao. Machine Learning Model for Predicting the Critical Transition Temperature of Hydride Superconductors. Acta Metall Sin, 2024, 60(10): 1418-1428.
The discovery of hydride superconductors with high critical transition temperature (Tc) under high pressures has received considerable interest in developing superconducting materials that can operate at room temperature and ambient pressure. Although first-principles methods can accurately predict the critical temperature of hydride superconductors, the computational demands are significant because of the expensive calculation of electron-phonon coupling. Hence, constructing an accurate and efficient model for predicting Tc is highly desirable. In this study, a simple and interpretable machine learning (ML) model was developed using the random forest algorithm, which enables the selection of important features based on their importance. Using four physics-based features, namely, the standard deviation of the number of valence electrons, mean covalent radii, range of the Mendeleev number of constituent elements, and hydrogen fraction of the total density of states at the Fermi energy, the optimal ML model achieves high accuracy, with a mean absolute error of 24.3 K and a root-mean-square error of 33.6 K. The ML model developed in this study shows great application potential for high-throughput screening, thereby expediting the discovery of high-Tc superconducting hydrides.
Fund: National Natural Science Foundation of China(52188101,52201030);National Key Research and Development Program of China(2021YFB3501503);Key Research Program of Chinese Academy of Sciences(ZDRW-CN-2021-2-5)
Corresponding Authors:
LIU Peitao, professor, Tel: (024)23971560, E-mail: ptliu@imr.ac.cn; CHEN Xing-Qiu, professor, Tel: (024)23971560, E-mail: xingqiu.chen@imr.ac.cn
Fig.1 Distribution of hydride superconductors (a) critical temperature (Tc) vs pressure (b) number of binary and ternary hydrides in different Tc distribution intervals
Parameter
Meaning of parameter
Value
n-estimators
Number of decision trees
10-30
Max-depth
Maximum depth of decision trees
None, 1, 2, 4
Min-samples-split
Minimum number of samples required to split internal nodes
2, 4, 8
Min-samples-leaf
Minimum number of samples at leaf nodes
1, 2, 4, 8
Table 1 Hyper-parameters used in the sklearn library for the random forest algorithm (For other parameters that are not explicitly specified, default values are used)
Fig.2 Model test errors as a function of number of features for one single training (a) and averaged model test errors of ten independent trainings as a function of training set ratio (b) (The shadow-filled areas show the standard deviation. MAE—mean absolute error, RMSE—root mean square error)
Fig.3 The most important 4 features by recursive selections (a) and Tc predicted by the most accurate machine learning model obtained vs the ground-truth values (b) ([Avg_dev(NValence)]—standard deviation of the number of valence electrons of constituent element, [Mean(CovalentRadius)]—mean covalent radius of constituent element, [Range(Mendeleev Number)]—range of the Mendeleev number of constituent element, HDOS Fraction—hydrogen fraction of the total density of states at the Fermi energy)
Fig.4 Correlation between Tc and four features of Avg_dev(NValence) (a), Mean(CovalentRadius) (b), Range(Mendeleev Number) (c), and HDOS Fraction (d)
Material
Pressure
GPa
HDOS
Range(Mendeleev Number)
Mean(CovalentRadius)
Avg_dev
(NValence)
Tc (RF)
K
Tc (Belli)[59]
K
Tc (Expt.) K
Li2MgH16
250
0.53
91
47.0
0.10
319.3
298.3
473.0[43]
CaHfH12
190
0.28
85
51.6
2.24
324.9
198.3
363.0[60]
CaHfH18
300
0.41
85
45.5
1.62
339.0
332.0
345.0[60]
CaZrH12
300
0.31
85
51.6
0.49
308.2
192.6
343.0[60]
MgH12
500
0.72
24
39.5
0.14
300.5
522.4
340.0[70]
YH10
250
0.41
80
45.6
0.33
312.5
259.6
326.0[89]
MAE
-
-
-
-
-
47.6
125.3
-
RMSE
-
-
-
-
-
68.4
140.3
-
Table 2 Model predictions for the six hydride superconductors with the highest Tc in the dataset[43,59,60,70,89]
Fig.5 Crystal structures and electronic densities of states for the six hydride superconductors with the highest Tc in the dataset (a) Li2MgH16 (250 GPa) (b) CaHfH12 (190 GPa) (c) CaHfH18 (300 GPa) (d) CaZrH12 (300 GPa) (e) MgH12 (500 GPa) (f) YH10 (250 GPa)
1
De Nobel J, Lindenfeld P. The discovery of superconductivity [J]. Phys. Today, 1996, 49: 40
2
Boeri L, Hennig R, Hirschfeld P, et al. The 2021 room-temperature superconductivity roadmap [J]. J. Phys. Condens. Matter, 2022, 34: 183002
3
Kim C J. Superconductor Levitation: Concepts and Experiments [M]. Singapore: Springer, 2019: 1
4
Mangin P, Kahn R. Superconductivity: An introduction [M]. Cham: Springer, 2017: 1
5
Meissner W, Ochsenfeld R. Ein neuer effekt bei eintritt der supraleitfähigkeit [J]. Naturwissenschaften, 1933, 21: 787
6
Hirsch J E, Maple M B, Marsiglio F. Superconducting materials classes: Introduction and overview [J]. Physica, 2015, 514C: 1
7
Bardeen J, Cooper L N, Schrieffer J R. Microscopic theory of superconductivity [J]. Phys. Rev., 1957, 106: 162
8
Bednorz J G, Müller K A. Possible high Tc superconductivity in the Ba-La-Cu-O system [J]. Z. Phys., 1986, 64B: 189
9
Keimer B, Kivelson S A, Norman M R, et al. From quantum matter to high-temperature superconductivity in copper oxides [J]. Nature, 2015, 518: 179
10
Wu M K, Ashburn J R, Torng C J, et al. Superconductivity at 93 K in a new mixed-phase Y-Ba-Cu-O compound system at ambient pressure [J]. Phys. Rev. Lett., 1987, 58: 908
pmid: 10035069
11
Schilling A, Cantoni M, Guo J D, et al. Superconductivity above 130 K in the Hg-Ba-Ca-Cu-O system [J]. Nature, 1993, 363: 56
12
Kamihara Y, Watanabe T, Hirano M, et al. Iron-based layered superconductor La[O1 - x F x ]FeAs (x = 0.05-0.12) with Tc = 26 K [J]. J. Am. Chem. Soc., 2008, 130: 3296
doi: 10.1021/ja800073m
pmid: 18293989
13
Paglione J, Greene R L. High-temperature superconductivity in iron-based materials [J]. Nat. Phys., 2010, 6: 645
14
Wang Q Y, Li Z, Zhang W H, et al. Interface-induced high-temperature superconductivity in single unit-cell FeSe films on SrTiO3 [J]. Chin. Phys. Lett., 2012, 29: 037402
15
Li D F, Lee K, Wang B Y, et al. Superconductivity in an infinite-layer nickelate [J]. Nature, 2019, 572: 624
16
Sun H L, Huo M W, Hu X W, et al. Signatures of superconductivity near 80 K in a nickelate under high pressure [J]. Nature, 2023, 621: 493
17
Li Q, Zhang Y J, Xiang Z N, et al. Signature of superconductivity in pressurized La4Ni3O10 [J]. Chin. Phys. Lett., 2024, 41: 017401
18
Zhang M X, Pei C Y, Du X, et al. Superconductivity in trilayer nickelate La4Ni3O10 under pressure [DB/OL]. arXiv: 2311. 07423, 2023
19
Wang M. Discovery of high-Tc superconductivity in a nickelate [J]. Physics, 2023, 52: 663
王 猛. 液氮温区镍氧化物高温超导体的发现 [J]. 物理, 2023, 52: 663
20
Pickett W E. Colloquium: Room temperature superconductivity: The roles of theory and materials design [J]. Rev. Mod. Phys., 2023, 95: 021001
21
Hu J P. Searching for new unconventional high temperature superconductors [J]. Acta Phys. Sin., 2021, 70(1): 017101
胡江平. 探索非常规高温超导体 [J]. 物理学报, 2021, 70(1): 017101
22
Li J X. Spin fluctuations and uncoventional superconducting pairing [J]. Acta Phys. Sin., 2021, 70(1): 017408
李建新. 自旋涨落与非常规超导配对 [J]. 物理学报, 2021, 70(1): 017408
23
Nagamatsu J, Nakagawa N, Muranaka T, et al. Superconductivity at 39 K in magnesium diboride [J]. Nature, 2001, 410: 63
24
Ashcroft N W. Metallic hydrogen: A high-temperature superconductor? [J]. Phys. Rev. Lett., 1968, 21: 1748
25
Ashcroft N W. Hydrogen dominant metallic alloys: High temperature superconductors? [J]. Phys. Rev. Lett., 2004, 92: 187002
26
Duan D F, Liu Y X, Tian F B, et al. Pressure-induced metallization of dense (H2S)2H2 with high-Tc superconductivity [J]. Sci. Rep., 2014, 4: 6968
27
Gor'kov L P, Kresin V Z. Pressure and high-Tc superconductivity in sulfur hydrides [J]. Sci. Rep., 2016, 6: 25608
doi: 10.1038/srep25608
pmid: 27167334
28
Drozdov A P, Eremets M I, Troyan I A, et al. Conventional superconductivity at 203 Kelvin at high pressures in the sulfur hydride system [J]. Nature, 2015, 525: 73
29
Liu H Y, Naumov I I, Geballe Z M, et al. Dynamics and superconductivity in compressed lanthanum superhydride [J]. Phys. Rev. B, 2018, 98: 100102
30
Somayazulu M, Ahart M, Mishra A K, et al. Evidence for superconductivity above 260 K in lanthanum superhydride at megabar pressures [J]. Phys. Rev. Lett., 2019, 122: 027001
31
Drozdov A P, Kong P P, Minkov V S, et al. Superconductivity at 250 K in lanthanum hydride under high pressures [J]. Nature, 2019, 569: 528
32
Eremets M I, Minkov V S, Drozdov A P, et al. High-temperature superconductivity in hydrides: Experimental evidence and details [J]. J. Supercond. Novel Magn., 2022, 35: 965
33
Zhang S B, Zhang M, Liu H Y. Superconductive hydrogen-rich compounds under high pressure [J]. Appl. Phys., 2021, 127A: 684
34
Troyan I A, Semenok D V, Kvashnin A G, et al. Anomalous high-temperature superconductivity in YH6 [J]. Adv. Mater., 2021, 33: 2006832
35
Snider E, Dasenbrock-Gammon N, McBride R, et al. RETRACTED: Synthesis of yttrium superhydride superconductor with a transition temperature up to 262 K by catalytic hydrogenation at high pressures [J]. Phys. Rev. Lett., 2021, 126: 117003
36
Semenok D V, Kvashnin A G, Ivanova A G, et al. Synthesis of ThH4, ThH6, ThH9 and ThH10: A route to room-temperature superconductivity [DB/OL]. arXiv: 1902. 10206, 2019
37
Semenok D V, Kvashnin A G, Ivanova A G, et al. Superconductivity at 161 K in thorium hydride ThH10: Synthesis and properties [J]. Mater. Today, 2020, 33: 36
38
Li B, Miao Z L, Ti L, et al. Predicted high-temperature superconductivity in cerium hydrides at high pressures [J]. J. Appl. Phys., 2019, 126: 235901
39
Chen W H, Semenok D V, Huang X L, et al. High-temperature superconducting phases in cerium superhydride with a Tc up to 115 K below a pressure of 1 megabar [J]. Phys. Rev. Lett., 2021, 127: 117001
40
Wang H, Tse J S, Tanaka K, et al. Superconductive sodalite-like clathrate calcium hydride at high pressures [J]. Proc. Natl. Acad. Sci. USA, 2012, 109: 6463
doi: 10.1073/pnas.1118168109
pmid: 22492976
41
Li Z W, He X, Zhang C L, et al. Superconductivity above 200 K discovered in superhydrides of calcium [J]. Nat. Commun., 2022, 13: 2863
doi: 10.1038/s41467-022-30454-w
pmid: 35606357
42
Flores-Livas J A, Boeri L, Sanna A, et al. A perspective on conventional high-temperature superconductors at high pressure: Methods and materials [J]. Phys. Rep., 2020, 856: 1
43
Sun Y, Lv J, Xie Y, et al. Route to a superconducting phase above room temperature in electron-doped hydride compounds under high pressure [J]. Phys. Rev. Lett., 2019, 123: 097001
44
Di Cataldo S, Von Der Linden W, Boeri L. Phase diagram and superconductivity of calcium borohyrides at extreme pressures [J]. Phys. Rev., 2020, 102B: 014516
45
Geng N S, Bi T G, Zurek E. Structural diversity and superconductivity in S-P-H ternary hydrides under pressure [J]. J. Phys. Chem., 2022, 126C: 7208
46
Grockowiak A D, Ahart M, Helm T, et al. Hot hydride superconductivity above 550 K [J]. Front. Electron. Mater., 2022, 2: 837651
47
Di Cataldo S, Boeri L. Metal borohydrides as ambient-pressure high-Tc superconductors [J]. Phys. Rev., 2023, 107B: L060501
48
Song P, Hou Z F, Baptista De Castro P, et al. High-pressure Mg-Sc-H phase diagram and its superconductivity from first-principles calculations [J]. J. Phys. Chem., 2022, 126C: 2747
49
Shutov G M, Semenok D V, Kruglov I A, et al. Ternary superconducting hydrides in the La-Mg-H system [J]. Mater. Today Phys., 2024, 40: 101300
50
Chen W H, Huang X L, Semenok D V, et al. Enhancement of superconducting properties in the La-Ce-H system at moderate pressures [J]. Nat. Commun., 2023, 14: 2660
doi: 10.1038/s41467-023-38254-6
pmid: 37160883
51
Zhao W D, Huang X L, Zhang Z H, et al. Superconducting ternary hydrides: Progress and challenges [J]. Natl. Sci. Rev., 2024, 11: nwad307
52
Gao M, Yan X W, Lu Z Y, et al. Phonon-mediated high-temperature superconductivity in the ternary borohydride KB2H8 under pressure near 12 GPa [J]. Phys. Rev., 2021, 104B: L100504
53
Zhang Z H, Cui T, Hutcheon M J, et al. Design principles for high-temperature superconductors with a hydrogen-based alloy backbone at moderate pressure [J]. Phys. Rev. Lett., 2022, 128: 047001
54
Song Y G, Bi J K, Nakamoto Y, et al. Stoichiometric ternary superhydride LaBeH8 as a new template for high-temperature superconductivity at 110 K under 80 GPa [J]. Phys. Rev. Lett., 2023, 130: 266001
55
Zhao W D, Duan D F, Du M Y, et al. Pressure-induced high-Tc superconductivity in the ternary clathrate system Y-Ca-H [J]. Phys. Rev., 2022, 106B: 014521
56
Du M Y, Song H, Zhang Z H, et al. Room-temperature superconductivity in Yb/Lu substituted clathrate hexahydrides under moderate pressure [J]. Research, 2022, 2022: 9784309
57
Sanna A, Cerqueira T F T, Fang Y W, et al. Prediction of ambient pressure conventional superconductivity above 80 K in hydride compounds [J]. npj Comput. Mater., 2024, 10: 44
58
Dolui K, Conway L J, Heil C, et al. Feasible route to high-temperature ambient-pressure hydride superconductivity [J]. Phys. Rev. Lett., 2024, 132: 166001
59
Belli F, Novoa T, Contreras-García J, et al. Strong correlation between electronic bonding network and critical temperature in hydrogen-based superconductors [J]. Nat. Commun., 2021, 12: 5381
doi: 10.1038/s41467-021-25687-0
pmid: 34531389
60
Liu L L, Peng F, Song P, et al. Generic rules for achieving room-temperature superconductivity in ternary hydrides with clathrate structures [J]. Phys. Rev., 2023, 107B: L020504
61
Marsiglio F. Eliashberg theory: A short review [J]. Ann. Phys., 2020, 417: 168102
62
Éliashberg G M. Interactions between electrons and lattice vibrations in a superconductor [J]. Sov. Phys. JETP, 1960, 11: 696
63
Dynes R C. McMillan's equation and the Tc of superconductors [J]. Solid State Commun., 1972, 10: 615
64
Allen P B, Dynes R C. Transition temperature of strong-coupled superconductors reanalyzed [J]. Phys. Rev., 1975, 12B: 905
65
McMillan W L. Transition temperature of strong-coupled superconductors [J]. Phys. Rev., 1968, 167: 331
66
Oliveira L N, Gross E K U, Kohn W. Density-functional theory for superconductors [J]. Phys. Rev. Lett., 1988, 60: 2430
pmid: 10038349
67
Marques M A L, Lüders M, Lathiotakis N N, et al. Ab initio theory of superconductivity. II. Application to elemental metals [J]. Phys. Rev., 2005, 72B: 024546
68
Lüders M, Marques M A L, Lathiotakis N N, et al. Ab initio theory of superconductivity. I. Density functional formalism and approximate functionals [J]. Phys. Rev., 2005, 72B: 024545
69
Choudhary K, Garrity K. Designing high-TC superconductors with BCS-inspired screening, density functional theory, and deep-learning [J]. npj Comput. Mater., 2022, 8: 244
70
Shipley A M, Hutcheon M J, Needs R J, et al. High-throughput discovery of high-temperature conventional superconductors [J]. Phys. Rev., 2021, 104B: 054501
71
Saha S, Di Cataldo S, Giannessi F, et al. Mapping superconductivity in high-pressure hydrides: The Superhydra project [J]. Phys. Rev. Mater., 2023, 7: 054806
72
Sommer T, Willa R, Schmalian J, et al. 3DSC—A dataset of superconductors including crystal structures [J]. Sci. Data, 2023, 10: 816
doi: 10.1038/s41597-023-02721-y
pmid: 37990027
73
Cerqueira T F T, Sanna A, Marques M A L. Sampling the materials space for conventional superconducting compounds [J]. Adv. Mater., 2024, 36: 2307085
74
Stanev V, Oses C, Kusne A G, et al. Machine learning modeling of superconducting critical temperature [J]. npj Comput. Mater., 2018, 4: 29
75
Breiman L. Random forests [J]. Mach. Learn., 2001, 45: 5
Hutcheon M J, Shipley A M, Needs R J. Predicting novel superconducting hydrides using machine learning approaches [J]. Phys. Rev., 2020, 101B: 144505
78
Liu Y, Huang H Y, Yuan J, et al. Upper limit of the transition temperature of superconducting materials [J]. Patterns, 2023, 4: 100841
79
Wines D, Choudhary K. Data-driven design of high pressure hydride superconductors using DFT and deep learning [J]. Mater. Futures, 2024, 3: 025602
80
Ward L, Dunn A, Faghaninia A, et al. Matminer: An open source toolkit for materials data mining [J]. Comput. Mater. Sci., 2018, 152: 60
81
Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set [J]. Phys. Rev., 1996, 54B: 11169
82
Kresse G, Hafner J. Ab initio molecular dynamics for liquid metals [J]. Phys. Rev., 1993, 47B: 558
Wang V, Xu N, Liu J C, et al. VASPKIT: A user-friendly interface facilitating high-throughput computing and analysis using VASP code [J]. Comput. Phys. Commun., 2021, 267: 108033
85
Pedregosa F, Varoquaux G, Gramfort A, et al. Scikit-learn: Machine learning in python [J]. J. Mach. Learn. Res., 2011, 12: 2825
86
Pettifor D G. A chemical scale for crystal-structure maps [J]. Solid State Commun., 1984, 51: 31
87
Pettifor D G. The structures of binary compounds. I. Phenomenological structure maps [J]. J. Phys., 1986, 19C: 285
88
Villars P, Cenzual K, Daams J, et al. Data-driven atomic environment prediction for binaries using the Mendeleev number: Part 1. Composition AB [J]. J. Alloys Compd., 2004, 367: 167
89
Liu H Y, Naumov I I, Hoffmann R, et al. Potential high-Tc superconducting lanthanum and yttrium hydrides at high pressure [J]. Proc. Natl. Acad. Sci. USA, 2017, 114: 6990
