Evolutionary search for cobalt-rich compounds in the yttrium-cobalt-boron system
Takahiro Ishikawa, Taro Fukazawa, Guangzong Xing, Terumasa Tadano, Takashi Miyake
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Evolutionary search for cobalt-rich compounds in the yttrium-cobalt-boron system
Takahiro Ishikawa, ∗ Taro Fukazawa,
2, 1
Guangzong Xing, Terumasa Tadano,
3, 1 and Takashi Miyake
2, 1 ESICMM, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan CD-FMat, National Institute of Advanced Industrial Science and Technology,1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan CMSM, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan (Dated: February 4, 2021)Modern high-performance permanent magnets are made from alloys of rare earth and transitionmetal elements, and large magnetization is achieved in the alloys with high concentration of tran-sition metals. We applied evolutionary search scheme based on first-principles calculations to theY-Co-B system and predicted 37 cobalt-rich compounds with high probability of being stable. Fo-cusing on remarkably cobalt-rich compounds, YCo and YCo , we found that, although they aremetastable phases, the phase stability is increased with increase of temperature due to the contri-bution of vibrational entropy. The magnetization and Curie temperature are higher by 0.22 T and204 K in YCo and by 0.29 T and 204 K in YCo than those of Y Co which has been well studiedas strong magnetic compounds. PACS numbers: 61.50.Ah, 75.50.Bb, 75.50.Cc, 75.50.Ww
I. INTRODUCTION
Rare-earth magnets are strong permanent magnets,which mainly consist of rare-earth elements and 3d tran-sition metals (Fe and/or Co). High Fe/Co concentrationgives rise to high magnetization and rare earths are asource of high magnetocrystalline anisotropy which is es-sential for high coercivity. Rare-earth magnets have beendeveloped since the discovery of large magnetocrystallineanisotropy in an alloy of yttrium and cobalt, YCo , andneodymium magnets are the strongest type of perma-nent magnet commercially available, and its main phaseis formed by Nd Fe B compound , which has the satura-tion magnetization of 1.85 T at 4.2 K, magnetocrystallineanisotropy field of 5.3 MA/m at room temperature, andCurie temperature of 586 K .The magnetization is expected to be further increasedusing iron-richer compounds than Nd Fe B, and RT ( R = rare earth; T = Fe, Co) systems have attracted con-siderable attention as potential candidates for strongerpermanent magnets than Nd Fe B . It has long beenknown that RT compounds are thermodynamically un-stable in a bulk form and are stabilized by partial substi-tution of the third element for T , i.e. R ( T x X x ) ( X =Al, Si, Ti, V, Cr, Nb, Mo, W) . However, the magne-tization decreases with the increase of x , and a search iscurrently underway for the best third elements, in otherwords, the elements maximizing the stabilization andminimizing the decrease of the magnetization. On theother hands, thin films of NdFe N x and Sm(Fe x Co x ) have been fabricated by the epitaxial growth on W- andV-buffered MgO(001) substrates. The films have highermagnetization, Curie temperature, and anisotropy fieldthan Nd Fe B . In addition, YFe with the ThMn structure is experimentally obtained in multi-phases byrapid quenching method . Recently, first-principles cal-culations predict that, in Y Fe and Y(Fe x Co x ) with x of 0–0.7, the magnetization and Curie temperature are enhanced by the transformation from ThMn into mon-oclinic C /m structures .In the present study, we search for novel stable com-pounds with novel Fe/Co-rich rare-earth compounds us-ing composition and crystal structure prediction schemebased on first-principles calculations and evolutionary al-gorithm. Here, we focused on Y-Co-B system for thefollowing reasons: (i) Y is favorable for theoretical treat-ment because it has no f electron in its ground-state elec-tronic configuration, (ii) Co has a hexagonal close-packed(hcp) structure in the simple substance and is expectedto be compatible with Y having hcp compared with Fehaving a body-centered cubic structure, which makes awide variety of compositions stable, and (iii) B can playa role for stabilizing Y-Co compounds and forming novelcrystal structures, similarly to the case of Nd Fe B. Asa result, we found 37 cobalt-rich compounds includingremarkably Co-rich YCo and YCo . II. COMPUTATIONAL DETAILS
We used the evolutionary construction scheme of aformation-energy convex hull to search for stable com-pounds in the Y-Co-B system. First, we created an ini-tial set of Y-Co-B compounds using the structure data ofexperimentally reported YFe , Y Fe , Y Fe , NdFe ,NdFe , Nd Fe , Sm Fe , YB , YB , CoB, YCo B ,Y CoB , Nd Fe B, Sm Fe N , and SmCo B , whichare included in the Materials Project database and theSpringerMaterials database . For the pure elements, weused the hcp structures for Co and Y, and a rhombo-hedral R ¯3 m structure for B. Next, we constructed thepreliminary convex hull of Y-Co-B system by perform-ing the structural optimizations for the compounds inthe initial set. Then, by applying evolutionary opera-tors, “mating”, “mutation”, and “adaptive mutation” to target compounds selected from the compounds whosedistance to the convex hull is small (0–4.4 mRy/atom),new compositions and structures with high probabilityof being stable were created. Repeatedly performing thecreation of compounds and the update of the convex hull,we searched for stable compounds.We combined our evolutionary construction code withthe Quantum ESPRESSO (QE) code to perform theoptimizations of the structures created by the opera-tors. We used the generalized gradient approximation byPerdew, Burke and Ernzerhof (PBE) for the exchange-correlation functional in the framework of the projectoraugmented wave (PAW) method . We got the PAW po-tentials from the QE web site . The energy cutoff wasset at 100 Ry for the wave function and 800 Ry for thecharge density. The maximum number of atoms in cal-culation cell is 84, and the k -space integration over theBrillouin zone (BZ) was carried out on 12 × ×
12, 8 × ×
8, 6 × ×
6, and 4 × × ab initio simulation package (VASP) and thePHONOPY code . Second-order interatomic force con-stants were computed by the finite-displacement methodbased on harmonic approximation, as implemented inPHONOPY. Total number of atoms in each supercell is ∼
100 or larger, which was sufficient to reach the conver-gence of the vibrational free energy. The energy cutofffor the wave function was set at 400 eV, and the k -pointmesh was generated automatically in such a way thatthe mesh density in the reciprocal space is larger than450 ˚A − . The convergence criteria of energy and forceminimization were set to 10 − and 10 − eV, respectively.For the stable compounds, we calculated the intersitemagnetic couplings using the Liechtenstein’s method .For this purpose, we used AkaiKKR , a first-principlesprogram of Korringa-Kohn-Rostoker (KKR) Green’sfunction method, within the local density approximation.The Curie temperature T C is evaluated from a classicalspin model within the mean-field approximation. Othercomputational details are the same to the settings inRef. 28. III. RESULTS
In this study, we searched for stable and metastablecompounds with the convex-hull distance ∆ E less than4.4 mRy/atom (59.8 meV/atom). This tolerance is asso-ciated with the approximations and the omission of tem-perature effects in first-principles calculations , andthe possibility of the stabilization by inclusion and/orsubstitution of other elements. We created 4120 struc-tures up to the 11th generation by applying the evo-lutionary construction technique to the Y-Co-B system(Y − x − y Co x B y , 0 ≤ x ≤
1, 0 ≤ y ≤
1) and pre-
FIG. 1: Projection of the formation-energy convex hull ofY − x − y Co x B y on the xy plane. The solid lines are the edgesof the convex hull, and the dots show the compounds with theconvex-hull distance less than 4.4 mRy/atom. dicted Y Co, YCo, YCo , YCo , Y Co , Y Co , YB ,YB , CoB, YCo B , YCo B , and Y Co B as stablecompounds on the convex hull (See Fig. S1 in Supple-mental Material [SM] ). Figure 1 shows the closeup ofthe convex hull in the Co-rich region of 0 . ≤ x ≤ and YCo emerge as Co-richer compounds thanYCo . The ∆ E values are 2.72 mRy/atom for YCo and 3.92 mRy/atom for YCo . In addition, for themetastable YCo phase, we obtained an orthorhombic Imma structure, which is more stable by 0.27 mRy/atomthan the CaCu -type structure used to construct the pre-liminary convex hull (See Fig. S8 and Table S7 in SM forthe details of the structure ). Another important pointis that Y Co B transforms into YCo with an orderedtetragonal structure P /mnm , going through the smallenergy region of ∆ E less than 0.82 mRy/atom. Y Co Btakes the Nd Fe B-type structure with P /mnm in-cluding four formula units in the unit cell. The low-energy path connecting between Y Co B and YCo isachieved by step-by-step eliminating the B atoms fromthe unit cell (See Fig. S30 in SM ). See Fig. S2 in SMfor the details of other metastable compounds .Hereafter, we focus on the novel Co-rich compounds,YCo and YCo . The structures are assigned as tri-clinic P ¯1 for YCo and monoclinic C /m for YCo (See Tables S27 and S28 and Fig. S28 and S29 in SM forthe details of the structures ). Figure 2 shows I /mmm (ThMn -type) YCo viewed along the b axis, P ¯1 YCo viewed along the [1¯10] direction, and C /m YCo viewed along the b axis. P ¯1 YCo and C /m YCo are achieved by adding the Co atoms into ThMn -type FIG. 2: Projections of (a) YCo with a tetragonal I /mmm (ThMn -type) structure along the b axis, (b) YCo with atriclinic P ¯1 structure along the [1¯10] direction, and (c) YCo with a monoclinic C /m structure along the b axis. Framesshow the unit cells, and large and small balls represent theY and Co atoms, respectively. The solid (broken) arrowsshow the areas where additional Co atoms are inserted bythe transformation from YCo (YCo ) into YCo (YCo ).Crystal structures were drawn with VESTA . YCo . YCo (YCo ) is obtained by the insertion ofthe four Co atoms per formula unit into the area shownby the solid (broken) arrows of YCo (YCo ), parallelto the (101) plane of ThMn -type YCo .We investigated the dynamical and thermodynamicalstability of YCo and YCo by performing phononcalculations. Figure 3 shows the phonon dispersioncurves of P ¯1 YCo and C /m YCo . No imaginaryphonon modes were detected in the dispersion curves,which indicates that the two structures are dynami-cally stable at 0 K. We investigated the variations ofthe convex-hull distance for YCo and YCo with in-crease of temperature by considering the entropy con-tribution, including electronic and vibrational free ener-gies (Fig. 4). We compared the static formation ener-gies of I /mmm YCo , P ¯1 YCo , and C /m YCo FIG. 3: Phonon dispersions of (a) YCo with a triclinic P -1structure and (b) YCo with a monoclinic C /m structure.FIG. 4: Phase stability of YCo , YCo , and YCo againstthe decomposition into Y Co and Co. between QE and VASP and confirmed the errors are0.03 mRy/atom, 0.12 mRy/atom, and 0.32 mRy/atom,respectively. Since these results are reasonably consis-tent with each other, we discuss the finite-temperaturethermodynamic stability of YCo and YCo basedon the VASP results. With increasing the tempera-ture up to 1200 K, the convex-hull distances decrease to0.25 mRy/atom for YCo , 1.48 mRy/atom for YCo ,and 2.72 mRy/atom for YCo . Regarding the struc- TABLE I: Comparison of volume per atom V , magnetic mo-ment per atom m , magnetization M , and Curie temperature T C among Y Co , YCo , YCo , YCo , Y Fe , YFe ,YFe , and YFe .structure V m M T C (˚A /atom) ( µ B /atom) (T) (K)Y Co R ¯3 m I /mmm P ¯1 11.90 1.505 1.474 1378YCo C /m Fe R ¯3 m I /mmm P ¯1 12.65 2.093 1.928 719YFe C /m ture of Y Co , a rhombohedral R ¯3 m (Th Zn -type)structure is stable in the low-temperature region, whilethe entropy contributions make the hexagonal P /mmc (Th Ni -type) more stable at temperatures above 780 K.This temperature-induced phase transition was properlyconsidered in the results shown in Fig. 4.Next, we investigated the magnetic properties of P ¯1YCo and C /m YCo . Table I lists volume per atom V , magnetic moment per atom m , total magnetization M , and Curie temperature T C for Y Co YCo , YCo ,and YCo . The V and m values decrease and increasewith the increase of the Co concentration, respectively.Consequently, the M value increases to 1.474 T in YCo and 1.539 T in YCo , which are larger than those inY Co and YCo . Furthermore, we obtained thatYCo and YCo show the T C value of 1378 K, whichis higher by 204 K and 98 K than those of Y Co andYCo , respectively. Although the mean-field approxi-mation tends to overestimate T C , the differences of the-oretical T C values among magnet compounds have beenfound to be qualitatively consistent with those in exper-iments . Therefore, we expect the trend of the en-hancement in T C is realistic.We also calculated the V , m , M , and T C values for Fe-based compounds. Although the m value increases withincrease of the Fe concentration from Y Fe throughYFe , it turns to the decrease in YFe due to the de-crease of the magnetic moment of Fe at the 4 e site. Asa result, the M value shows the increase to 1.928 T inYFe , whereas there is no further increase in YFe . Incontrast to the case of the Co-based compounds, the T C value decreases to 719 K in YFe , which is almost equal to that of Y Fe , and is largely decreased in YFe . IV. CONCLUSION
In conclusion, we searched for stable compounds in theY-Co-B system, Y − x − y Co x B y , using the evolution-ary construction technique of a formation-energy con-vex hull. Focusing on Co-rich (0 . ≤ x ≤
1) and lowenergy (∆ H ≤ . and YCo . In addition, we obtained a new stable struc-ture of YCo , and a low energy path connecting betweenY Co B and YCo . Phonon calculations predict thatYCo and YCo are dynamically stable and the hulldistance ∆ H is decreased to 1.48 mRy/atom for YCo and 2.72 mRy/atom for YCo with increase of tempera-ture to 1200 K due to the contribution of the vibrationalfree energy. The calculated M and T C values are 1.474 Tand 1378 K in YCo and 1.539 T and 1378 K in YCo ,which are larger than those in Y Co and YCo . Weperformed the same calculations for YFe and YFe ,and obtained the results that, although the T C valuesare decreased in YFe and YFe , the M values are in-creased.This study provides novel Co-rich compounds, YCo and YCo , with high magnetization and high Curietemperature, whereas further studies are required to getthe knowledge about their magnetocrystalline anisotropyand coercivity, which is crucial for the application of RT and RT systems to high-performance permanentmagnet. For example, it is important to accumulatethe data about the variation of the magnetocrystallineanisotropy and coercivity by systematically replacing Ywith the other R elements. Acknowledgments
This work was supported by the Ministry of Educa-tion, Culture, Sports, Science and Technology (MEXT)as “The Elements Strategy Initiative Center for Mag-netic Materials (ESICMM)” (JPMXP0112101004) and“Program for Promoting Researches on the Supercom-puter Fugaku” (DPMSD). The computation was partlyconducted using the facilities of the Supercomputer Cen-ter, the Institute for Solid State Physics, the Universityof Tokyo, the supercomputer of ACCMS, Kyoto Univer-sity, and the HPCI System Research project (ProjectID:hp200125). ∗ Electronic address: [email protected] G. Hoffer and K. Strnat, IEEE Trans. Magn. , 487 (1966). M. Sagawa, S. Fujimura, H. Yamamoto, and Y. Matsuura,IEEE Trans. Magn. , 1584 (1984). S. Hirosawa, Y. Matsuura, H. Yamamoto, S. Fujimura,M. Sagawa, and H. Yamauchi, J. Appl. Phys. , 873(1986). K. Ohashi, Y. Tawara, R. Osugi, and M. Shimao, J. Appl.
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