CCS International Symposium 2018, Univ. Tsukuba, 16 Oct. 2018

Computational magnetic materials discovery
Takashi Miyake
CD-FMat, AIST, Tsukuba
Collaborators
Tien-Lam Pham, Hieu-Chi Dam (JAIST)
Hiori Kino, Kiyoyuki Terakura (NIMS)

Materials discovery
ü Superconductor having highest Tc?
ü Harder material than diamond ?
ü Strong magnet ?
…

Atomic species + Composition + Structure

2

History of permanent magnet

Nd2Fe14B

ü High saturation magnetization
ü High coercivity → High magnetocrystalline anisotropy
ü High Curie temperatureü Phase stability3

Magnet compound vs Magnet materialIntrinsic properties

Sample dependent(microstructure, interfaces, …)

Easy axis

Nd-Fe-B magnet

Hardaxisü Saturation magnetization
ü Magnetocrystalline anisotropyü Curie temperature

K. Hono (NIMS)ü Coercivity
ü Microstructure4Magnet compounds
Magnetic properties• Saturation magnetization• Magnetocrystalline anisotropy• Curie temperature
Structural properties• Phase stability

5

Fe, Co, Mn, …
Nd, Sm, …
B, C, N, …
Other elements ???

Slater-Pauling curve
Calculation by H. Akai

Hard-magnetic compounds

2:14:1 type
Nd2Fe14B
1:5 type (m=1, n=0)
SmCo5, YCo5
2:172:17:3 type (m=3, n=1)
Sm2Co17, Sm2Fe17N3
1:12 type (m=2, n=1)
SmFe11Ti, NdFe11TiN

Hirayama, Miyake and Hono, JOM 67, 1344 (2015)

Rm-nT5m+2n6

NdFe12N film
Magnetic anisotropy

Magnetization

Y. Hirayama et al., Scripta Materialia 95, 70 (2015)7

Chemical substitution

Ce, Pr, Nd, Sm, …

8

B, C, NR (Fe1-a Ta )12 X
Fe(8j)
(Ti, Cr, V, Mn, Co, Ni, …
N••••

Magnetization
Magnetocrystalline anisotropy
Curie temperature
Formation energyNdFe(8i)
Fe(8f)

High-throughput computational screening
Direct screening of 1,280 compounds by first-principles calculation

Körner, Krugel and Elsässer, Sci. Rep. 6, 24686 (2016)9

Topological materials

10T. Zhang et al., arXiv: 1807.08756
c.f. F. Tang et al., arXiv:1807.09774
M.G. Vergniory et al., arXiv:1807.10271

Structure datafrom database

Electronic structure
by first-principles calc.

39,519 materials from ICSD

8,056 are topological
5,005 topological semimetals
1,814 topological insulators
1,237 topological crystalline insulators

Exhaustive search ??
Material datafrom database{ xi } material

Candidate compounds
by chemical substitution{ x´i } material

Evaluate properties by
first-principles calculation{ y´i } property

Recommend new compounds

11Virtual screening
Material data

{ xi } material{ yi } property

12

Machine-learning modely = f (x)Candidate compounds
by chemical substitution{ x´i } material
Validation by
first-principles calculation

Recommend new compounds{ y´i } property

Virtual screening

Material data

R-T-X
Candidate compounds
by chemical substitution

Nd-Fe-BValidation by
first-principles calculation

Recommend new compounds

13Machine-learning modely = f (x)③{ y´i } formation energy

① Computational database14

RTX materials
T4d-5d = ["Y", "Zr", "Nb","Mo", "Tc", “Ru", "Rh", "Pd","Ag", "Cd", "Hf", "Ta", "W","Re", "Os", "Ir", "Pt", "Au",“Hg"]
T3d = ["Ti", "V", "Cr", "Mn","Fe", "Co", "Ni", "Cu", “Zn"]ü 9485 Fe compounds
ü 9541 Co compounds
ü 9932 Ni compounds

http://oqmd.org

R = ["La", "Ce", "Pr", "Nd","Pm", "Sm", "Eu", "Gd","Tb", "Dy", "Ho", "Er", "Tm",“Yb", “Lu"]X = ["B", "C", "N", "O"]

Optimized structure
Total energy
Magnetic momentetc.

② Machine-learning model
Kernel regression

f (⃗x) =N!

αik(⃗x(i), ⃗x)i=1 !
Nf (⃗x) =αik(⃗x(i), ⃗x)N!
i=1(i)f(⃗x)=
αk(⃗x,⃗⃗x′ ) 2i′⃗i=1k(⃗x, x ) = exp(−β||⃗x − x || 2)⃗′) = exp(−β||⃗x − x⃗′|| )k(⃗x,xGaussian kernel′ ) = exp(−β||⃗′ ||2 )⃗⃗k(⃗x,xx−xNN1!
!
2(j)(i) (j)2 )}2′′⃗⃗ασ ={y−k(⃗x,⃗xik(⃗x, x ) = C exp(−β||⃗x − x || )i=1
N j=1 ⃗′k(⃗x, x ) = C exp(−β||⃗x − x⃗′||2)NN
1 kernel!
!
Laplacian2(j)(i) (j) 2ασ={y−k(⃗x, ⃗x )}iN
N1!
!
2(j)N ⃗(i) (j) 2⃗′T
j=1i=1′σ ={y k(⃗x−, x ) =αiCk(⃗
x , ⃗x x )}Kαexp(−β||⃗− x+λα||)N j=1 1 N i=1 N!
2(j) !
(i) (j) 2T15

16② Descriptor
Orbital Field Matrix (OFM)

Pham et al., STAM 18, 756 (2017)

Sci. Technol. Adv. Mater. 18 (2017) 759

T. LAM PHAM et al.

T. LAM PHAM et al.

T. LAMs of all surrounding atoms in the local structurepied
f orbitals of thetations of all surrounding atoms in the local⃗
A central
Oatom and the coordination number.
⃗es in determining theusing Oatom and the coordination number. Aattransition-metalsite p in a localstructure
be represented
atompcanat sitep in a local structure can be rep
sites.
whicharedefined
OFM with theelements
X ,theusing
OFMwiththe elements X p , which aretheinterpretabilityof ourows:
as follows:
structuraland physical

1. OFM representation for an Na atom in a regular octahedral site surrounded by six Cl atoms: atomic one-hot vector for Na
idgeFigure
regression(KRR)(left), representation for the
six Cl atoms surrounding the Na atom (middle), and representation for the Na atom surrounded by sixnnppCl atoms (right).
!
!
ues and similarity
meaTTpppp⃗⃗⃗ k × wk ,⃗X
=O×OX=O×O×w,kking prediction models
Nprojectors [36,37], k=1
and employed pymatgen 4.3.0 [27]!
Generalization
pk=1
l magnetic moments
Fij =Xof(3)
to prepare the VASPninput files with 0.1eV Gaussianij ,p mesh density
pnp ppsmearing of MITRelaxSet
and a k-point!
θ! p θoys, formationenergies
p isk 520 keV. The VASPof 150 Å− 3 .pThe energy cutoﬀpkkX=o,where F is the OFM representing the entire molecule.
p by default.
ij scalar relativistic
i ojeﬀects
PAW includes
Wep′② Orbital Field Matrix (OFM)

Pham et al., STAM 18, 756 (2017)

Formation energy

T. LAM PHAM et al.

• Data: 4,220 R-T-X compounds• DFT: VASP+GGA• Machine learning: KRR + OFM

Predicted energy (eV/atom)hnol. Adv. Mater. 18 (2017) 763

RMSE = 0.190 eV/atom
MAE = 0.112 eV/atom
R2 = 0.98

DFT energy (eV/atom)

3. Comparison of formation energies calculated using DFT and those predicted through machine learning (ML-predicted),
OFM.

a)

17(b)

② OFM, OFM118

Pham et al., STAM 18, 756 (2017)

Formation energy

T. LAM PHAM et al.

• Data: 4,220 R-T-X compounds• DFT: VASP+GGA• Machine learning: KRR + OFM

Predicted energy (eV/atom)hnol. Adv. Mater. 18 (2017) 763

RMSE = 0.190 eV/atom
MAE = 0.112 eV/atom
R2 = 0.98

DFT energy (eV/atom)

3. Comparison of formation energies calculated using DFT and those predicted through machine learning (ML-predicted),
OFM.

a)

(b)

OFM1 + Manhattan distance + Laplace kernel regressor
RMSE = 0.110 eV/atom
MAE = 0.067 eV/atom

Pham et al., JCP 148, 204106 (2018)

R2 = 0.994

③ Phase stability

19EfDEXEH

DE = Ef - EH
DE < 0 : Stable
https://materialsproject.org/wiki/images/6/6f/Convexhull.png

Curie temperature of R-T bimetals

20Collect experimental TC’s of 101 R-T alloys from database (AtomWork)
H. Kino (NIMS)

Descriptors
ü Represent a compound by a vector x
ü 27 primary descriptors characterizing crystal constituent elements andJ. Phys. Soc. Jpn.
structureCategory
Atomic properties oftransition metals (T)Atomic properties ofrare-earth metals (R)Structural information(S)

Descriptors
ZT , rT , rTcv , IPT , χT , S 3d , L3d , J3d
ZR , rR , rRcv , IPR , χR , S 4 f , L4 f , J4 f , g J , J4 f g J ,J4 f (1 − g J )CT , CR , dT −T , dT −R , dR−R , NT −R , NR−R ,NR−T

Table I. Transition metal, rare-earth and structural descriptors. See also thesupporting information.

x = (ZT, rT, … , ZR, rR, ..., CT, CR, ...)

21

Machine learning22

ü Select a subset of 27 primary descriptors that maximizes prediction
accuracy (Exhaustive search for 227-1 combinations)ü Gaussian kernel-ridge regression
ü 10-fold cross validation

original prescription, we add another dedescriptors, which all
we must define. We
bgroups here and make use of them in theleave-C
We utilize hierarchal
clustering
R-out analysis,
etween descriptors is one minus absolute
orrelation coeﬃcient. We can define thethat are made of tree nodes containing, of the clustering. For example, we cand = 0.5. Two of them have the same dehe T and R categories, while we have twol S category. (We call the original cluster
ster by the hierarchical analysis group.)a group, while the other S category de-LETTERS

Table II. The best R2 score and descriptors as a function of thdescriptors n.
n2
34
5
678

score
0.870153
0.942222
0.953386
0.954294
0.954391
0.954452
0.954448

descriptor(s)CR ,ZT
CR ,ZR ,ZT
J3d ,CR ,ZR ,ZT
L3d ,J3d ,CR ,ZR ,ZT
L3d ,J3d ,χT ,CR ,ZR ,ZTL3d ,J3d ,χT ,CR ,ZR ,ZT ,rTcv
L3d ,J3d ,χT ,IPT ,CR ,ZR ,ZT ,rTcv

Damet al.,J. Phys.
Jpn.J3d(accepted)
tors
among
them,a set ofSoc.
ZT andis the best (0TC of R-T bimetals

Prediction

Figure 6: experimental TC versus CV predicted TC after fixing errors.

Understanding?

23

Subgroup relevance analysis

24

Dam et al., J. Phys.
LETTERSSoc. Jpn. (accepted)

J. Phys. Soc. Jpn.
0.875866

0.954451

0.588678

Etot0.506824⎞0.954451

0.9544510.930175

0.779022

0.880249

0.948763

0.948466
0.651662

0.946715

0.953017

si = s(xs−xi)s(x − xi)
i =0.948598
0.894670

0.888323

0.953824

0.954451

⎛⎛

⎞
2
2
1
ϵ1
ϵ⎜
⎜
⎟
i ⎟ ⎜− i ⎟⎟! ⎜exp⎠P (ϵi) = P! (ϵi)2=exp
−⎝⎠ ⎝
2
22
2πs2s
2si
2πsi
i i

0.911396

0.932963

0.932963

0.873747

0.906548

1 - |r |Etot

⎛⎞

⎛

⎞

0.917286

0.817842

0.925426

0.946501

0.924265

0.909167

0.890967

0.949931

0.885542

0.851642

0.932963

0.898889

22N1 Nϵϵ(⎜1⎟ ⎜(ii ⎟⎟ ⎜⎟P (ϵ1, ϵ2,P· (ϵ· ·1,,ϵϵN2), ·=· ·C, ϵN )!= C 2 exp! ⎜⎝− 2 exp⎠ ⎝−⎠22
i=1 2πsii=1 2πs2si2sii

R2 scores of the subgroup relevance analysis on the hierarchical clustering of the descriptors. TheFig. 2.

cov(X, Ycov(X,)Y)ρX,Y = ρX,Y =σX σY σX σYgroup R (green) starts from L4 f to rRcv . The group T (red) starts from IPT to rT . The group S (cyan) is starts from

Pearson’s
d to C correlation. The group d is made of the descriptor d . Horizontal values are strong relevance values and tiltedvalues are weak relevance values. The values of the vertical axis are the one minus the absolute values of thecoefficient
Pearson’s correlation coeﬃcient. Paths of the highest value (0.954451) are colored in yellow dashed lines. SeeTTTTRTR

details in the main body also.

covariancecov(X, Ycov(X,) = ⟨XY −⟨X⟩⟩⟨Y⟨Y ⟩⟩ − ⟨Y ⟩⟩)=⟨X − −⟨X⟩⟩⟨Yrelevance. We evaluate score of a subgroup belonging to the group under the condition that
we must use at least one descriptor in the subgroup and we can add any descriptors belonging

Subgroup relevance analysis

Dam et al., J. Phys.
LETTERSSoc. Jpn. (accepted)

0.917286

0.817842

0.954451
0.930175

0.925426

0.779022

0.948763
0.946501

0.880249

0.954451
0.953017
0.909167

0.946715
0.890967

0.949931

0.888323

0.948598
0.894670

0.953824

0.954451
0.911396
0.885542

0.932963
0.932963

0.851642

0.906548

0.932963
0.898889

0.873747

0.506824

0.588678

0.954451

0.924265

0.875866

0.948466
0.651662

J. Phys. Soc. Jpn.

1 - |r |25LETTERS

Table II. The best R2 score and descriptors as a function of the number ofiption, we add another de- descriptors n.
which we must define. We
Fig. 2. R2 scores of the subgroupon the hierarchical clustering of the descriptors. Then relevance
scoreanalysis descriptor(s)group R (green) starts from L4 f to rRcv . The group T (red) starts from IPT to rT . The group S (cyan) is starts fromnd make use of them in the2 0.870153 CR ,ZTdT T to CT . The group dT R is made of the descriptor dT R . Horizontal values are strong relevance values and tilted
archal clustering analysis,
3 0.942222 CR ,ZR ,ZT
values are weak relevance values. The values of the vertical axis are the one minus the absolute values of the4 0.953386
J3d(0.954451),CR ,ZR ,ZareT colored in yellow dashed lines. Seeptors is one minus absolute
Pearson’s correlation coeﬃcient. Paths of the highest value
5 0.954294
L3d ,J3d ,CR ,ZR ,ZTﬃcient. We can define thedetails in the main body also.
6 0.954391
L3d ,J3d ,χT ,CR ,ZR ,ZT
e of tree nodes containing
Result by
7 0.954452
L3d ,J3d ,χT ,CR ,ZR ,ZT ,rTcv
ring. For example, we canexhaustive search
8 0.954448
L3d ,J3d ,χT ,IPT ,CR ,ZR ,ZT ,rTcv
relevance. We evaluate score of a subgroup belonging
to the group under the condition thatof them have the same dewe must use at least one descriptor in the subgroup and we can add any descriptors belonging

Summaryn High-throughput screening by a combined
computational and machine-learning techniquesn Virtual screening of Nd-Fe-B compounds using
kernel-ridge regression and orbital-field-matrix
n Kernel-ridge regression of Curie temperature:important descriptors and descriptor groups by
subgroup relevance analysis

26