MAGNETOCRYSTALLINE ANISOTROPY AND MAGNETOELASTIC CONSTANTS FOR Nd2Fe14B IN A CRYSTAL FIELD MODEL

(1)

HAL Id: jpa-00224892

https://hal.archives-ouvertes.fr/jpa-00224892

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

MAGNETOCRYSTALLINE ANISOTROPY AND MAGNETOELASTIC CONSTANTS FOR Nd2Fe14B

IN A CRYSTAL FIELD MODEL

H. Szymczak

To cite this version:

H. Szymczak. MAGNETOCRYSTALLINE ANISOTROPY AND MAGNETOELASTIC CON-

STANTS FOR Nd2Fe14B IN A CRYSTAL FIELD MODEL. Journal de Physique Colloques, 1985, 46

(C6), pp.C6-225-C6-228. �10.1051/jphyscol:1985639�. �jpa-00224892�

(2)

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n°9, Tome 46, septembre 1985 page C6-225

MAGNETOCRYSTALLINE ANISOTROPY AND MAGNETOELASTIC CONSTANTS FOR Nd2Fe1 4B IN A CRYSTAL FIELD MODEL

H. Szymczak

Institute of Physios, Polish Academy of Sciences, Warsaw, Poland

Résumé - Le modèle des charges ponctuelles a été Utilisé pour calculer l'ani- sotropie magnétocristalline et les constantes magnétoëlastiques pour Nd.Fe JB.

Il apparaît que la réorientation de l'aimantation observée expérimentalement résulte d'un croisement de niveaux, conduisant à un changement d'état fonda- mental. Le signe et la grandeur de la constante de magnétostriction ont été calculés dans le cadre d'un modèle complet et comparés aux valeurs expérimen- tales .

A b s t r a c t - The point-charge model was used i n order t o c a l c u l a t e the magneto- c r y s t a l l i n e anisotropy and magnetoelastic constants f o r Nd^Fe-.B. I t was shown t h a t the e x p e r i m e n t a l l y observed spin r e o r i e n t a t i o n phenomenon r e s u l t s from the ground s t a t e level c r o s s i n g . The sign and magnitude of the magneto- s t r i c t i o n constant have been c a l c u l a t e d w i t h i n the framework of a developed model and compared w i t h the experimental d a t a .

I - INTRODUCTION

The t e r n a r y rare e a r t h compounds o f the RE,Fe...B - type (RE i s a rare e a r t h o r y t t r i u m ) are w e l l known / 1 , 2 / due t o the extremely high energy product which caused t h e i r a p p l i c a t i o n as e f f e c t i v e permanent magnet m a t e r i a l s . I t was shown,recently /3,k/ t h a t a n i s o t r o p i c magnetic p r o p e r t i e s of the RE-Fe.rB system are associated f i r s t o f a l l w i t h RE i o n s . Such a conclusion i s based mainly on the observation t h a t in the s e r i e s o f RE.Fe.^B compounds there i s a one-to-one correspondence between the d i r e c t i o n o f the easy magnetization and the sign o f the second order Stevens coef- f i c i e n t s f o r RE i o n s . This means, a t the same t i m e , t h a t the m a g n e t o c r y s t a l l i n e a n i - sotropy o f the RE.Fe^B compounds has a predominantly c r y s t a l - f i e l d o r i g i n . The simplest way to c a l c u l a t e the anisotropy c o n s t a n t s , when the c r y s t a l f i e l d i s thought t o be the major c o n t r i b u t i o n to the magnetocrystal1ine energy, is the p o i n t charge model. The p o i n t charge model has severe shortcomings and t h e r e f o r e one can expect t o o b t a i n r e s u l t s o f a s e m i q u a n t i t a t i v e c h a r a c t e r . S t i l l , such r e s u l t s help t o understand deeper the f a c t o r s which determine the o r i g i n o f the magnetocrystal - l i n e a n i s o t r o p y . Support f o r such an approach has been advanced by Cadogan and Coey / $ / who found t h a t the e l e c t r o s t a t i c model is able to e x p l a i n many features o f the magnetic p r o p e r t i e s o f RE.Fe.hB compounds.

In o r d e r t o o b t a i n a more complete p i c t u r e o f the c r y s t a l f i e l d i n RE„Fe.^B compounds, an attempt is made i n t h i s work to c a l c u l a t e , i n the framework of the p o i n t charge model, not only the magnetocrystal1ine constant but also the magnetoelastic c o n s t a n t s . Simultaneously, i t w i l l be shown t h a t the magnetization r e o r i e n t a t i o n observed i n Nd.Fe.jB / 3 , V can be understood, a t l e a s t q u a l i t a t i v e l y , on the basis o f the developed model. D e t a i l e d c r y s t a l f i e l d c a l c u l a t i o n s w i l l be performed f o r Nd2Fejj)B, since f o r t h i s material the exact p o s i t i o n s o f the atoms have r e c e n t l y

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985639

(3)

JOURNAL DE PHYSIQUE

been determined /6,7/.

I I

-

MAGNETOCRYSTALLINE ANISOTROPY

The c r y s t a l f i e l d p o t e n t i a l a t a p o i n t ( ri,Oi,$,) w i t h i n an i o n centered a t t h e o r i - gin, due t o the surrounding i o n s which a r e approximated by an assembly o f p o i n t charges q.(R.,@.,$.) i s g i v e n by t h e formula

J J J J

where t h e summation o v e r a, f o r each n, has terms Zno, Z i l and z:] f o r a l l 1/1 = 1, 2

...

n/. Zna are t h e t e s s e r a l harmonics g i v e n by Hutchings

/8/.

The parameters o f t h e c r y s t a l f i e l d p o t e n t i a l expansion

a r e t h e l a t t i c e sums o v e r t h e p o i n t charges.

F u r t h e r , t o account f o r the s h i e l d i n g o f t h e 4 f e l e c t r o n s from the c r y s t a l l i n e e n v i

-

ronment, a d d i t i o n a l f a c t o r s o f (l-a,) are i n c o r p o r a t e d i n t o e x p r e s s i o n ( 2 ) . Burns/9/

has e s t i m a t e d t h a t u2% 0.5, a4% 0.1 and u6% 0.05.

F i n a l l y , t h e c r y s t a l f i e l d i n t e r a c t i o n has t h e form

where t h e values o f (rn) were taken from / T O / .

N e g l e c t i n g the J-mixing, one can express eq.

( 3 )

i n terms o f t h e Stevens o p e r a t o r e q u i v a l e n t s

/a/:

The l a t t i c e sums were c a l c u l a t e d u s i n g t h e s p h e r i c a l boundary method i n which t h e summation i s c a r r i e d o u t i n s i d e a sphere o f r a d i u s r = 88

8.

F o l l o w i n g /11/ t h e charge o f t h e Hd was taken as

+3

and the c o n t r i b u t i o n s from Fe and B were ignored.

I t was observed t h a t as i n

/S/,

t h e l e a d i n g terms i n HCF a r e t h e second o r d e r terms:

- -

^{6.22 K,}

B20

-

⁼

-

^{3.32 K} ^{f o r} h f s i t e s ,

BZ0 =

-

4.4 K,

BE2 ⁼

-

^{10.58 K} ^{f o r} 49 s i t e s

.

I n o r d e r t o c a l c u l a t e t h e a n i s o t r o p y constants o f t h e system, the f u l l H a m i l t o n i a n

has t o be d i a g o n a l i z e d n u m e r i c a l l y . I n ( S ) ,

-

Hex i s the exchange f i e l d , B

-

t h e Bohr magneton and g i s the g - f a c t o r .

Since, i n o u r case, t h e Zeeman energy i s o f the same o r d e r o f magnitude as the c r y s - t a l f i e l d energy, one can expect t h a t the system w i l l be c h a r a c t e r i z e d by s e v e r a l a n i s o t r o p y constants, even i f t h e c r y s t a l f i e l d i s d e s c r i b e d o n l y by t h e second o r d e r term /12/. Therefore, t h e d i a g o n a l i z a t i o n o f ( 5 ) should be performed f o r seve-

r a l d i r e c t i o n s o f t h e exchangg f i e l d . I n o u r case, we have c a l c u l a t e d t h e energy l e v e l s o f t h e system f o r the He,

11 _[loo], [ l o l l

and [ 0 0 1 l d i r e c t i o n s . Taking 2(g-1) Hex = 250 K / a t T = 0/ t h e f o l l o w i n g values o f t h e second /K2/ and f o u r t h /K4/ o r d e r a n i s o t r o p y constants were o b t a i n e d

(4)

and the r e s u l t i n g a n i s o t r o p y constants f o r T = 0 a r e

-

The a n i s o t r o p y c o n s t a n t s o b t a i n e d a r e o v e r e s t i m a t e d /see /S/ f o r a d i s c u s s i o n / b u t t h e signs o f b o t h the c o n s t a n t s a r e c o r r e c t , i n accordance w i t h t h e canted magnetic s t r u c t u r e observed a t low temperatures /3/.

As the temperature i s going up, t h e magnitude o f the exchange f i e l d i s going down.

By p e r f o r m i n g d i a g o n a l i z a t i o n s f o r d i f f e r e n t values o f Hex, i t was observed t h a t f o r some T = Tcr, namely, those f o r which

Hex ( T c r ) = 0.25 Hex ( 0 )

t h e l e v e l c r o s s i n g e f f e c t occurs. Such an e f f e c t i s o f g r e a t importance f o r t h e mag- n e t i c p r o p e r t i e s o f t h e system. I n p a r t i c u l a r , i t leads t o a change o f the v a l u e o f t h e second o r d e r a n i s o t r o p y from a n e g a t i v e t o a p o s i t i v e one, i n accordance wi t h experimental data. From t h i s p o i n t o f view, t h e cross-over o f t h e ground s t a t e energy l e v e l s i s r e s p o n s i b l e f o r the easy a x i s r e o r i e n t a t i o n observed below T=135 K /3/.

1 1 1

-

MAGNETOELASTIC CONSTANTS

T h e o r e t i c a l c a l c u l a t i o n s o f t h e magnetoel a s t i c c o n s t a n t s a r e based p r i m a r i l y on the s t r a i n dependence o f t h e c r y s t a l l i n e f i e l d s . Therefore, such c a l c u l a t i o n s s h o u l d f o l l o w a r o u t e p a r a l l e l t o t h a t which i s commonly used f o r the case o f the a n i s o t r o - py constants.

I f one expands t h e c r y s t a l f i e l d energy ECF about t h e e q u i l i b r i u m c o n f i g u r a t i o n as a T a y l o r s e r i e s i n the l a t t i c e s t r a i n s ekl

the l i n e a r m a g n e t o e l a s t i c energy i s g i v e n by

Since we a r e concerned o n l y w i t h t h e second o r d e r terms i n the c r y s t a l f i e l d Hami l ton i an

HCF = Di Ji J

i j j

t h e m a g n e t o e l a s t i c Hami l t o n i a n may be w r i t t e n i n t h e form

The f o u r t h - r a n k tensor aD. ./askl i s d i r e c t l y r e l a t e d t o the phenomenological magneto- I J

e l a s t i c t e n s o r Bi jk, d e s c r i b i n g the m a g n e t o e l a s t i c energy:

wtiere a. and a . a r e t h e d i r e c t i o n cosines o f t h e m a g n e t i z a t i o n . J

P o i n t ckarge c a l c u l a t i o n s g i v e t h e f o l l o w i n g values o f B. . / i n

l o 7

erg/cm3/, w r i t t e n

I J i n V o i g t n o t a t i o n :

Bll = 5.4 BI2 =

-

^5.0 ^BI3 ⁼

-

^6.6 ^B44⁼^4.2 ^B66⁼^18.0

.

I t has t o be s t r e s s e d t h a t Md a t t h e 4 f s i t e s g i v e s the major c o n t r i b u t i o n t o mag- n e t o e l a s t i c t e n s o r components.

I n o r d e r t o c a l c u l a t e t h e m a g n e t o s t r i c t i o n c o n s t a n t s from t h e above values o f Bii, the e l a s t i c c o n s t a n t s a r e needed. U n f o r t u n a t e l y , these have been determined o n l y f o r p01 y c r y s t a l l i n e Nd2Fe14~ /13/. For t h i s reason and s i n c e t h e magnetostri c t i o n has

(5)

C6-228

JOURNAL DE PHYSIQUE

a l s o been measured i n p o l y c r y s t a l l i n e samples /13/, we have l i m i t e d o u r s e l v e s t o e s t i m a t i n g o n l y the o r d e r o f magnitude o f X and i t s s i g n . For an i s o t r o p i c medium h =

-

B,+4/3C44. Such an e s t i m a t e g i v e s 1%

-

I O - ~ , i n agreement w i t h t h e experimental data.

I V

-

CONCLUS l ONS

I t was shown t h a t the p o i n t charge model p r o v i d e s a f a i r l y good, s e m i q u a n t i t a t i v e d e s c r i p t i o n o f a n i s o t r o p y and m a g n e t o s t r i c t i o n i n Nd2Fe14B. The s p i n r e o r i e n t a t i o n induced by temperature has been e x p l a i n e d qual i t a t i v e l y, wi t h i n t h e framework o f t h e model, as an e f f e c t o f the l e v e l c r o s s i n g . I t seems t h a t i n o r d e r t o determine t h e a p p l i c a b i l i t y o f t h e e l e c t r o s t a t i c model t o RE2Fe14B, f u r t h e r experiments d e a l i n g w i t h a n i s o t r o p y and m a g n e t o s t r i c t i o n have t o be performed on monocrystal l i n e samples.

Sagawa E l . , Fujimura S., Togawa N., Yamamoto H., Matsuura Y., J. Appl. Phys.

55 / l 984/ 2083.

-

Croat J.J., Herbst J.F., Lee R.W., P i n k e r t o n F.E., J. Appl. Phys.

55

/1984/

2078.

G i v o r d D., L i H.S., P e r r i e r . d e l a B a t h i e R., S o l i d S t . Commun. 50 /1984/ 857.

O e s t e r r e i c h e r H., Spada F., Abache C., Mat. Res. B u l l .

19

/1984711069.

Cadogan J.M., Coey J.M.D., Phys. Rev.

830

/1984/ 7326.

Givord D., L i H.S., Moreau J.M., S o l i d S t . Commun.

50

/1984/ 497.

Shoemaker C.B., Shoemaker D.P., F r u c h a r t R., Acta C r y s t a l l o g r . /1984/ 1665.

Hutchings .M.T., i n : S o l i d S t a t e Physics /Academic Press, New York, 1964/

v o l . 16, pp 227-272.

Burns G., J. Chem. Phys. 42 /1965/ 377.

Freeman A.J., Watson R.E.:P~~S. Rev.

127

/1962/ 2058.

Greedan J.E., Rao V.U.S., J. S o l i d S t . Chem.

6

/1973/ 387.

Szymczak H., Sokolov V.J., Acta Phys. Polonica

-

i n p r e s s .

T u r i l l i G., Szymczak H., Lachowicz H.K., phys. s t a t . s o l i d i

-