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Ab initio calculation of the magnetism in GdFe12
imlt Journal of Magnetism and Magnetic Materials 104-107 (1992) 1447-1448 North-Holland
Ab initio calculation of the magnetism in GdFe12 Joakim Trygg, B6rje Johansson and M.S.S. Brooks Condensed Matter Theory Group, Department of Physics, UniL,ersityof Uppsala, Box 530, S-751 21 Uppsala, Sweden Electronic structure calculations by means of the LMTO-ASA method have been performed for the hypothetical rare earth-transition metal compound GdFe12 with the ThMn~z structure. The R-4f magnetic moments were obtained from the standard Russel-Saunders scheme but the radial 4f spin density was otherwise part of the self-consistent band calculation. The influence of localized 4f magnetism upon the conduction band magnetism is found to give noticeable changes in the local moments of the iron. The presence of the 4f spin moment is found to induce a redistribution of the conduction electron spin moment between the rare earth and iron sites while the total conduction moment remains practically constant. Powerful m a g n e t materials are b a s e d on c o m p o u n d s such as SmCo 5 a n d NdzFe14 B [1,2]. This has lead to a r e n e w e d interest in the basic physics of 4 f - 3 d intermetallic c o m p o u n d s . Recently a new class of iron rich ternary c o m p o u n d s have b e e n p r o p o s e d as promising c a n d i d a t e s for new p e r m a n e n t m a g n e t materials. T h e s e c o m p o u n d s have the general formula composition RFe12_~M x a n d crystallize in the body c e n t e r e d tetragonal ThMn12 structure. T h e binary c o m p o u n d s RFe~2 do not exist but the s t r u c t u r e can be stabilized by i n t r o d u c i n g a small fraction of Ti, W, V, Cr or M o instead of iron [3,4]. F r o m the theoretical point of view these c o m p o u n d s are just as interesting as R 2 F e j 4 B . In o r d e r to u n d e r s t a n d the transition m e t a l c o n t r i b u t i o n to the m a g n e t i c p r o p e r t i e s electronic s t r u c t u r e calculations have already b e e n p e r f o r m e d for materials of this type [5,6]. To get f u r t h e r insight into the m a g n e t i c p r o p e r t i e s in these c o m p o u n d s , o n e has to take into account the influence of the localized 4f m a g n e t i s m of the rare e a r t h atom u p o n the c o n d u c t i o n b a n d magnetism basically originating from the 3d electrons of the transition m e t a l atoms. I n ' this p a p e r we p r e s e n t electronic structure calculations for GdFe~2 which give us the o p p o r t u n i t y to study the interplay b e t w e e n the localized 4f m a g n e t i s m a n d the c o n d u c t i o n b a n d magnetism. W e treat the G d 4f states within the local spin density a p p r o x i m a t i o n ( L S D A ) a n d c o m b i n e this with the L M T O calculation of the c o n d u c t i o n b a n d electronic structure. In o r d e r to describe the c o m b i n a t i o n of localized 4f m a g n e t i s m a n d i t i n e r a n t m a g n e t i s m of the spd conduction electrons we used the same a p p r o a c h as Brooks et al. [7]. W e t r e a t the localized 4f states as o u t e r core states in the sense t h a t they are not allowed to hybridize with the c o n d u c t i o n bands. To do this we fix the total 4f spin m o m e n t by applying the s t a n d a r d R u s s e l - S a u n d e r s (RS) coupling s c h e m e to the 4f shell a n d we also fix the total n u m b e r of 4f electrons. Subject to this constraint, the 4f spin densities are calculated self-consistently. T h e 4f spin densities cont r i b u t e to the total spin densities, a n d h e n c e to the spin
up and spin down potentials. In the absence of hybridization of the 4f states the i n t e r a c t i o n b e t w e e n the 4f electrons and c o n d u c t i o n electrons is electrostatic plus local exchange. In this m a n n e r the influence of 4f m o m e n t s u p o n the m a g n e t i s m is calculated ab initio a l t h o u g h the choice of 4f occupation n u m b e r and mom e n t is not. T h e ThMn~2 structure is body c e n t r e d tetragonal with 13 a t o m s p e r unit cell a n d t h e r e are t h r e e n o n e q u i v a l e n t iron sites 8(f), 8(i) a n d 8(j) a n d o n e G d site 2(a). T h e calculations were p e r f o r m e d using the L M T O t e c h n i q u e in the atomic s p h e r e approximation ( A S A ) [8]. Since G d F e l 2 does not exist as a stable c o m p o u n d we are lead to e s t i m a t e the lattice constants from YFe12. T h e W i g n e r - S e i t z r a d i u s was chosen to be S = 1.853 A for G d a n d S = 1.418 A for Fe a n d for the axis ratio we used c / a = 0.5545 a n d the lattice c o n s t a n t a was set equal to 8.495 A,. T h e frozen core a p p r o x i m a t i o n was used a n d the valence charge density was iterated until it was self-consistent with a m a x i m u m e r r o r in the valence charge density of the o r d e r of 10 -3 . F o r the local exchange and correlation part of the ( L S D A ) p o t e n t i a l the p a r a m e t r i z a t i o n of V o n B a r t h - H e d i n [9] was used. A n g u l a r m o m e n t a up to two were included on the G d a n d Fe sites in the electronic structure calculation, which gives a matrix d i m e n s i o n of 117 × 117 for the L M T O - m a t r i x . Selfconsistency was o b t a i n e d with 30 k-points in the irreducible part of the Brillouin zone. It is well known that the total 4f m o m e n t of the G d atom couples antiparallel to the spin m o m e n t of Fe atoms. T h e r e a s o n for this is that the 4f spin couples to the c o n d u c t i o n b a n d m a g n e t i z a t i o n by the local exc h a n g e - c o r r e l a t i o n potential. T h e limited extension of the 4f wave-function m e a n s t h a t this coupling will essentially take place within the rare e a r t h a t o m a n d t h e r e f o r e mainly with the G d 5d electrons a n d will p r o d u c e a parallel spin alignment. It is t h e n easy to see t h a t a large 4f spin m o m e n t will e n h a n c e the local 5d spin m o m e n t quite significantly, since the polarization from the 4f m o m e n t d e p e n d s u p o n the n u m b e r of u n p a i r e d 4f spins. T h e n due to hybridization b e t w e e n
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
1448
J. Trygg et al. / Ab initio calculation for GdFel2
Table 1 Local magnetic d-moment on the various crystallographic sites. Calc I and Calc II refer to the calculation discussed in the text, i.e. calculations with and without a 4f spin moment respectively. All moments are in units of bohr magnetons (uB) Site
Calc I
Calc II
Difference
Gd (a) Fe (f) Fe (i) Fe (j)
- 0.402 1.684 2.531 2.316
- 0.255 1.658 2.523 2.293
- 0.147 0.026 0.008 0.023
5d states a n d the 3d states the 4f spin m o m e n t couples indirectly to the 3d spin m o m e n t . Since t h e 3 d - 5 d hybridization is less strong for the majority spins t h a n for minority spins it will lead to a d e c r e a s e in the 5d c o n t e n t in the majority b a n d and the 5d spin up occupation b e c o m e s less t h a n the 5d spin down occupation. T h u s the total 5d m o m e n t is opposite to the 3d m o m e n t . T h e r e f o r e o n e has to e n s u r e that the spin m o m e n t of the G d 4f states and the Fe 3d states are antiparallel, otherwise the calculation may converge wrongly. W e have p e r f o r m e d two different calculations for GdFe12, o n e with the 4f spin m o m e n t in accord a n c e with RS coupling (Calc I) a n d o n e w h e r e the 4f spin m o m e n t is set equal to zero (Calc II). In table 1 we give the results for the calculated local d m o m e n t s a n d in table 2 the calculated partial occupation numbers for the majority a n d minority spins for the d states of G d a n d Fe. F r o m the results we can see t h a t the spin m o m e n t on the iron site i is nearly saturated, i.e. the spin-up b a n d is almost completely filled a n d t h e r e fore the inclusion of the 4f m o m e n t in the calculation will not induce any c h a n g e in the occupation n u m b e r . T h e inclusion of the 4f m o m e n t only gives noticeable changes in the local m o m e n t s for the iron atoms on
Table 2 Calculated partial d occupation numbers for various crystallographic sites. Calc 1 and Calc II refer to the calculation discussed in the text, i.e. calculations with and without a 4f spin moment, respectively Site Gd (a) Fe (f) Fe (i) Fe (j)
Calculation I
Calculation II
spin up
spin down
spin up
spin down
0.669 4.091 4.515 4.390
1.071 2.407 1.985 2.074
0.742 4.079 4.510 4.377
0.997 2.421 1.986 2.084
sites f a n d j. T h e s e two sites arc b o t h s u r r o u n d e d by two G d atoms. T h e fact that site f is more influenced by the 4f spin m o m e n t t h a n site j can be u n d e r s t o o d from the fact that the f site m o m e n t is less s a t u r a t e d t h a n the site j m o m e n t . T h e different sizes of the iron m o m e n t can be u n d e r s t o o d from the distribution of the n e a r e s t n e i g h b o u r iron atoms [10]. Site i has only one iron atom at a relatively close distance a n d t h e n 4 iron atoms at a considerably larger distance. Due to this relative lack of close iron atoms the associated magnetic m o m e n t b e c o m e s particularly large. Iron site j has 4 n e i g h b o u r i n g iron atoms at an i n t e r m e d i a t e dislance. T h e r e f o r e also the site j m o m e n t b e c o m e s large, but still s o m e w h a t less t h a n for the i site. Also the f site has 4 n e i g h b o u r s at the same distance as the j site, but in addition t h e r e are two Fe atoms at a very close distance. This is the r e a s o n why the m o m e n t on site f is the lowest one of the t h r e e iron m o m e n t s . T h e total c o n d u c t i o n m a g n e t i c m o m e n t is calculated to be 24.61 # B / f . u . w h e n we include the 4f m o m e n t and 24.62 /x~/f.u. w h e n the 4f m o m e n t is set equal to zero. T h u s the p r e s e n c e of the 4f spin m o m e n t induces a redistribution of the spin m o m e n t b e t w e e n the rare e a r t h a n d iron sites while the total c o n d u c t i o n electron m o m e n t r e m a i n s constant. B. J o h a n s s o n is grateful to the Swedish N a t u r a l R e s e a r c h Council for financial support.
References [1] K.H.J. Buschow, in: Ferromagnetic Materials, vol. 4, eds. E.P. Wohlfarth and K.H.J. Buschow (North-Holland, Amsterdam, 1988). [2] K.H.J. Buschow, in: Supermagnets, Hard Magnetic Materials, Lecture Notes Nato-ASI, eds. G.J. Long and F. Grandjean (Kluwer, Dordrecht, 1991). [3] D.B. de Mooij and K.H.J. Buschow, J. Less-Common Met. 136 (1988) 207. [4] K. Ohashi, Y. Tawara, R. Osugi, J. Sakurai and Y. Komura, J. Less-Common Met. 139 (1988) L1. [5] R. Coehoorn, Phys. Rev. B 41 (1990) 11790. [6] R. Coehoorn, in: Supermagnets, Hard Magnetic Materials, Lecture Notes Nato-ASl, eds. G.J. Long and F. Grandjean (Kluwer, Dordrecht, 1991). [7] M.S.S. Brooks, L. Nordstr6m and B. Johansson, J. Phys.: Condens. Matter 3 (1991) 2357. [8] O.K. Andersen, Phys. Rev. B 12 (1975) 3060, H.L. Skriver, The LMTO Method (Springer, Berlin, 1984). [9] U. von Barth and L. Hedin, J. Phys. C 5 (1972) 1629. [10] P. Villars and L.D. Calvert, Person's Handbook of Crystallographic Data for Intermetallic Phases (American Society for Metals, Ohio, 1986).
Ab initio calculation of the magnetism in GdFe12 Joakim Trygg, B6rje Johansson and M.S.S. Brooks Condensed Matter Theory Group, Department of Physics, UniL,ersityof Uppsala, Box 530, S-751 21 Uppsala, Sweden Electronic structure calculations by means of the LMTO-ASA method have been performed for the hypothetical rare earth-transition metal compound GdFe12 with the ThMn~z structure. The R-4f magnetic moments were obtained from the standard Russel-Saunders scheme but the radial 4f spin density was otherwise part of the self-consistent band calculation. The influence of localized 4f magnetism upon the conduction band magnetism is found to give noticeable changes in the local moments of the iron. The presence of the 4f spin moment is found to induce a redistribution of the conduction electron spin moment between the rare earth and iron sites while the total conduction moment remains practically constant. Powerful m a g n e t materials are b a s e d on c o m p o u n d s such as SmCo 5 a n d NdzFe14 B [1,2]. This has lead to a r e n e w e d interest in the basic physics of 4 f - 3 d intermetallic c o m p o u n d s . Recently a new class of iron rich ternary c o m p o u n d s have b e e n p r o p o s e d as promising c a n d i d a t e s for new p e r m a n e n t m a g n e t materials. T h e s e c o m p o u n d s have the general formula composition RFe12_~M x a n d crystallize in the body c e n t e r e d tetragonal ThMn12 structure. T h e binary c o m p o u n d s RFe~2 do not exist but the s t r u c t u r e can be stabilized by i n t r o d u c i n g a small fraction of Ti, W, V, Cr or M o instead of iron [3,4]. F r o m the theoretical point of view these c o m p o u n d s are just as interesting as R 2 F e j 4 B . In o r d e r to u n d e r s t a n d the transition m e t a l c o n t r i b u t i o n to the m a g n e t i c p r o p e r t i e s electronic s t r u c t u r e calculations have already b e e n p e r f o r m e d for materials of this type [5,6]. To get f u r t h e r insight into the m a g n e t i c p r o p e r t i e s in these c o m p o u n d s , o n e has to take into account the influence of the localized 4f m a g n e t i s m of the rare e a r t h atom u p o n the c o n d u c t i o n b a n d magnetism basically originating from the 3d electrons of the transition m e t a l atoms. I n ' this p a p e r we p r e s e n t electronic structure calculations for GdFe~2 which give us the o p p o r t u n i t y to study the interplay b e t w e e n the localized 4f m a g n e t i s m a n d the c o n d u c t i o n b a n d magnetism. W e treat the G d 4f states within the local spin density a p p r o x i m a t i o n ( L S D A ) a n d c o m b i n e this with the L M T O calculation of the c o n d u c t i o n b a n d electronic structure. In o r d e r to describe the c o m b i n a t i o n of localized 4f m a g n e t i s m a n d i t i n e r a n t m a g n e t i s m of the spd conduction electrons we used the same a p p r o a c h as Brooks et al. [7]. W e t r e a t the localized 4f states as o u t e r core states in the sense t h a t they are not allowed to hybridize with the c o n d u c t i o n bands. To do this we fix the total 4f spin m o m e n t by applying the s t a n d a r d R u s s e l - S a u n d e r s (RS) coupling s c h e m e to the 4f shell a n d we also fix the total n u m b e r of 4f electrons. Subject to this constraint, the 4f spin densities are calculated self-consistently. T h e 4f spin densities cont r i b u t e to the total spin densities, a n d h e n c e to the spin
up and spin down potentials. In the absence of hybridization of the 4f states the i n t e r a c t i o n b e t w e e n the 4f electrons and c o n d u c t i o n electrons is electrostatic plus local exchange. In this m a n n e r the influence of 4f m o m e n t s u p o n the m a g n e t i s m is calculated ab initio a l t h o u g h the choice of 4f occupation n u m b e r and mom e n t is not. T h e ThMn~2 structure is body c e n t r e d tetragonal with 13 a t o m s p e r unit cell a n d t h e r e are t h r e e n o n e q u i v a l e n t iron sites 8(f), 8(i) a n d 8(j) a n d o n e G d site 2(a). T h e calculations were p e r f o r m e d using the L M T O t e c h n i q u e in the atomic s p h e r e approximation ( A S A ) [8]. Since G d F e l 2 does not exist as a stable c o m p o u n d we are lead to e s t i m a t e the lattice constants from YFe12. T h e W i g n e r - S e i t z r a d i u s was chosen to be S = 1.853 A for G d a n d S = 1.418 A for Fe a n d for the axis ratio we used c / a = 0.5545 a n d the lattice c o n s t a n t a was set equal to 8.495 A,. T h e frozen core a p p r o x i m a t i o n was used a n d the valence charge density was iterated until it was self-consistent with a m a x i m u m e r r o r in the valence charge density of the o r d e r of 10 -3 . F o r the local exchange and correlation part of the ( L S D A ) p o t e n t i a l the p a r a m e t r i z a t i o n of V o n B a r t h - H e d i n [9] was used. A n g u l a r m o m e n t a up to two were included on the G d a n d Fe sites in the electronic structure calculation, which gives a matrix d i m e n s i o n of 117 × 117 for the L M T O - m a t r i x . Selfconsistency was o b t a i n e d with 30 k-points in the irreducible part of the Brillouin zone. It is well known that the total 4f m o m e n t of the G d atom couples antiparallel to the spin m o m e n t of Fe atoms. T h e r e a s o n for this is that the 4f spin couples to the c o n d u c t i o n b a n d m a g n e t i z a t i o n by the local exc h a n g e - c o r r e l a t i o n potential. T h e limited extension of the 4f wave-function m e a n s t h a t this coupling will essentially take place within the rare e a r t h a t o m a n d t h e r e f o r e mainly with the G d 5d electrons a n d will p r o d u c e a parallel spin alignment. It is t h e n easy to see t h a t a large 4f spin m o m e n t will e n h a n c e the local 5d spin m o m e n t quite significantly, since the polarization from the 4f m o m e n t d e p e n d s u p o n the n u m b e r of u n p a i r e d 4f spins. T h e n due to hybridization b e t w e e n
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
1448
J. Trygg et al. / Ab initio calculation for GdFel2
Table 1 Local magnetic d-moment on the various crystallographic sites. Calc I and Calc II refer to the calculation discussed in the text, i.e. calculations with and without a 4f spin moment respectively. All moments are in units of bohr magnetons (uB) Site
Calc I
Calc II
Difference
Gd (a) Fe (f) Fe (i) Fe (j)
- 0.402 1.684 2.531 2.316
- 0.255 1.658 2.523 2.293
- 0.147 0.026 0.008 0.023
5d states a n d the 3d states the 4f spin m o m e n t couples indirectly to the 3d spin m o m e n t . Since t h e 3 d - 5 d hybridization is less strong for the majority spins t h a n for minority spins it will lead to a d e c r e a s e in the 5d c o n t e n t in the majority b a n d and the 5d spin up occupation b e c o m e s less t h a n the 5d spin down occupation. T h u s the total 5d m o m e n t is opposite to the 3d m o m e n t . T h e r e f o r e o n e has to e n s u r e that the spin m o m e n t of the G d 4f states and the Fe 3d states are antiparallel, otherwise the calculation may converge wrongly. W e have p e r f o r m e d two different calculations for GdFe12, o n e with the 4f spin m o m e n t in accord a n c e with RS coupling (Calc I) a n d o n e w h e r e the 4f spin m o m e n t is set equal to zero (Calc II). In table 1 we give the results for the calculated local d m o m e n t s a n d in table 2 the calculated partial occupation numbers for the majority a n d minority spins for the d states of G d a n d Fe. F r o m the results we can see t h a t the spin m o m e n t on the iron site i is nearly saturated, i.e. the spin-up b a n d is almost completely filled a n d t h e r e fore the inclusion of the 4f m o m e n t in the calculation will not induce any c h a n g e in the occupation n u m b e r . T h e inclusion of the 4f m o m e n t only gives noticeable changes in the local m o m e n t s for the iron atoms on
Table 2 Calculated partial d occupation numbers for various crystallographic sites. Calc 1 and Calc II refer to the calculation discussed in the text, i.e. calculations with and without a 4f spin moment, respectively Site Gd (a) Fe (f) Fe (i) Fe (j)
Calculation I
Calculation II
spin up
spin down
spin up
spin down
0.669 4.091 4.515 4.390
1.071 2.407 1.985 2.074
0.742 4.079 4.510 4.377
0.997 2.421 1.986 2.084
sites f a n d j. T h e s e two sites arc b o t h s u r r o u n d e d by two G d atoms. T h e fact that site f is more influenced by the 4f spin m o m e n t t h a n site j can be u n d e r s t o o d from the fact that the f site m o m e n t is less s a t u r a t e d t h a n the site j m o m e n t . T h e different sizes of the iron m o m e n t can be u n d e r s t o o d from the distribution of the n e a r e s t n e i g h b o u r iron atoms [10]. Site i has only one iron atom at a relatively close distance a n d t h e n 4 iron atoms at a considerably larger distance. Due to this relative lack of close iron atoms the associated magnetic m o m e n t b e c o m e s particularly large. Iron site j has 4 n e i g h b o u r i n g iron atoms at an i n t e r m e d i a t e dislance. T h e r e f o r e also the site j m o m e n t b e c o m e s large, but still s o m e w h a t less t h a n for the i site. Also the f site has 4 n e i g h b o u r s at the same distance as the j site, but in addition t h e r e are two Fe atoms at a very close distance. This is the r e a s o n why the m o m e n t on site f is the lowest one of the t h r e e iron m o m e n t s . T h e total c o n d u c t i o n m a g n e t i c m o m e n t is calculated to be 24.61 # B / f . u . w h e n we include the 4f m o m e n t and 24.62 /x~/f.u. w h e n the 4f m o m e n t is set equal to zero. T h u s the p r e s e n c e of the 4f spin m o m e n t induces a redistribution of the spin m o m e n t b e t w e e n the rare e a r t h a n d iron sites while the total c o n d u c t i o n electron m o m e n t r e m a i n s constant. B. J o h a n s s o n is grateful to the Swedish N a t u r a l R e s e a r c h Council for financial support.
References [1] K.H.J. Buschow, in: Ferromagnetic Materials, vol. 4, eds. E.P. Wohlfarth and K.H.J. Buschow (North-Holland, Amsterdam, 1988). [2] K.H.J. Buschow, in: Supermagnets, Hard Magnetic Materials, Lecture Notes Nato-ASI, eds. G.J. Long and F. Grandjean (Kluwer, Dordrecht, 1991). [3] D.B. de Mooij and K.H.J. Buschow, J. Less-Common Met. 136 (1988) 207. [4] K. Ohashi, Y. Tawara, R. Osugi, J. Sakurai and Y. Komura, J. Less-Common Met. 139 (1988) L1. [5] R. Coehoorn, Phys. Rev. B 41 (1990) 11790. [6] R. Coehoorn, in: Supermagnets, Hard Magnetic Materials, Lecture Notes Nato-ASl, eds. G.J. Long and F. Grandjean (Kluwer, Dordrecht, 1991). [7] M.S.S. Brooks, L. Nordstr6m and B. Johansson, J. Phys.: Condens. Matter 3 (1991) 2357. [8] O.K. Andersen, Phys. Rev. B 12 (1975) 3060, H.L. Skriver, The LMTO Method (Springer, Berlin, 1984). [9] U. von Barth and L. Hedin, J. Phys. C 5 (1972) 1629. [10] P. Villars and L.D. Calvert, Person's Handbook of Crystallographic Data for Intermetallic Phases (American Society for Metals, Ohio, 1986).