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Magnetic anisotropy in R2Fe14B compounds
Journal of Magnetism and Magnetic Materials 109 (1992) Li51-LI52 North-Holla.ad
Letter to the Editor
Magnetic anisotropy in R 2Fe 14B compounds S.S. Jaswal a n d A.A. Kusov Behlen Laboratoo, of Physics and Center ]i~r Materials Research and Analysis, Universit3,of Nebraska, LincobL NE 68588-0111, USA Received 25 October 1991
The sums of the magnetic dipole-dipole interactions are carried out to find their contribution to intrinsic anisotropy in R.~Fet4B compounds. The dipolar contributions in R:Fe~4B (R = Y, Gd, Lu) along with the experimental data are used to deduce the anisotropy due to the non-4f spin-orbit interactions common to all the R2Fet4B compounds. Then the dipolar and non-4f spin-orbit contributions together with ex0erimentai data lead to an estimate of the single-ion anisotropy due to 4f electrons in R2Fet4B compounds with non-zero orbital angular momentum. Thus we are able to separate the various contributions to the intrinsic anisotropy in R 2Fe~4B compounds.
A permanent-magnet material must have reasonable uniaxial anisotropy for high coercivity. The anisotropy arises from magnetostatic dipole-dipole and spin-orbit interactions. It is normally assumed that the dipolar contribution to the intrinsic anisotropy (K u) is zero as given by macroscopic theory for uniform magnetization. This is true only for simple highly s, mmetric lattices (cubic, hcp) where the local ficlA correction is isotropic [1]. For complex low-symmctr3. lattices of permanent-magnet materials, one mu',t sum the dipole-dipole interactions to find their contribution to K u. We report here the first calculation of the dipolar contributior~s to K~ in R2Fe~4B compounds and use these results mong with the experimental data to deduce, the contributions due to 4f and non-4f spin-orbit interactions. The R2Fet4B compounds are ietragonal in structure [2] with the intrinsic uniaxiai anisotropy ener~, density .K~, defined as K u = U(!O0)-
U(001), where U(100) and U(001) are the energy densities with the magnetization in the [100] and [001] directions respectively. The dipolar densities are calculated for a cell in the middle of a 251 × 251 × 5 slab of primitive cells by numerically summing the dipole-dipole interactions
m , . . m ~ - 3(m,'r,~)(m)'r,.ij/r,'- ~ u,, --
.3
(1)
ltj
between the dipoles rn~ and rn~. separated by a distance rij. This size of the slab was chosen to ensure proper convergence of the sum. The dipole moments used in these calculations are those obtained from band-~t~ acture calculations [3] and the experimental va!ue~ [4] (table 1). The dipolar contribution to the intrinsic anisotropy K '~ was obtained by adding 2rrM, 2 to the calculated result to account for. the shape anisotropy, where M~ is the saturation magnetization. U
Table I Magnetic moments # of various atoms in R 2 Fe~4B compounds studied Atom
Y
Nd
Gd
Tb
Dy
Fe(4c)
Fe(4e)•
Fe(8j~)
Fe(Sj 2)
Fe(16K~)
Fe(16.K2)
B
~t [/.tu]
-0.55 a)
2.75 a)
-6.4t't
- 8 . 7 t,)
- 9 . 7 t'~
2.59;°
2.13 ~'~
2.12 ")
2.74 ~')
2.12 ;')
2.18 '')
-I.).2 ;')
;o Ref. [3]. ~'~ Ref. [4]. 0304-8853/92/$05.00 ~ 1992 - Elsevier Science Publishers B.V. All rights reserved
LETTER
T . Q T H E E D I T O R~ r
i
S.S. JaswaL A.A. Kusot' / Magnetic anisotropy in R 2 Fe t4 B compounds
L152
Table 2 Calculated and experimental anisotropy energy densities in 107 crg/cm 3 Kud YzFet4B Gd.,Fe~4B Lu:,Fet4B Nd:,Fet4B Tb:,Fe~4B Dy, Fe ~4B
--u/(expt
-0.13 -0.56 -0.13 0.0 -0.76 - 0.87
1.14 0.70 1.17 12.5 7.9 3.86
,o (4 K)
K~°
K~i
1.27 1.26 1.30 1.3 1.3 1.3
11.2 7.4 3.43
~ Ref. [51.
The results for several R2Fe~4B (R = Y, Gd, Lu, Nd, Tb and Dy) are listed in table 2. It is interesting to note that K~~ is zero for NdzFe~4B which is a ferromagnet with the moments of the magnetic atoms being of the same order of magnitude. Thus the macroscopic theory holds for such a system. This is no longer true for the other systems which are ferrimagnetic in nature. K~d due to the local field anisotropy ranges from - 0 . 1 5 for Y2Fet4 B to - 6 . 2 for Dy2Fe~4B in units of 2arM~. Thus, as the moment of a rareearth atom increases, so does the magnitude of K~d, for a ferrimagnctic system. K~,! is always negative, i.e., the dipolar contribution favors the (001) plane for the easy axis in fcrrimagnetic systems. Assuming that spin-orbit effect can be separated into non-4f and 4f contributions, enc can write K~ as K,, = K~ + __,K~"+K~i,
(2)
where -K ~° is the anisotropy e n e r ~ density due - U to the spin-orbit interactions of non-4f electrons and K u~ is the single-ion anisotropy due to the spin-orbit interactions of 4f electrons. Now K~~ is zero in Y, Gd and Lu compounds. Thus, from K~" for these compounds and the results are u shown in table 9 ~x~e find that K~"= 1.3 >,: 10 7 c r g / c m ~ for all three of these compounds supporting the assumption about the spin-orbit interactions made in eq. (2). Because of the simiJarity of the non-4f electronic structure of R~Fe~4B, it is quite reasonable to approximate K~" by 1.3 x 107 erg/cm 3 for all the rare-earth compounds favoring uniaxiai anisotropy. ~.
,~
. . ~
From ~x "~'' the calculated values of K~ and I,! ' the experimental values [5] of K,, one can deduce K~~, the single-ion anisotropy due to the 4f electrons. The results for Nd, Tb and Dy compounds are shown in table 2. From table 2 we see that K~d and --uK~°tend to cancel each other in the ferrimagnetic rare-earth compounds. Also -K - ~ ~" is an order of magnitude smaller than K~]i in compounds with large uniaxial anisotropy such as Nd2Fe~4B. As a result, the uniaxial anisotropy in these compounds is primarily determined by the 4f single-ion contribution which is in agreement with the experimental data [6]. In conclusion, we have calculated dipolar contributions to the intrinsic anisotropy in Rz FelaB compounds which favor easy axis in the (001) plane for ferrimagnetic systems. Their values for Y, Gd and Lu compounds are used to find the universal value of non-4f spin-orbit contributions to the anisotropy for the whole rare-earth series. Finally, our calculations, which isolate the dominant single-ion contribution due to 4f spin-orbit interactions, will be important for first-principles calculations of these contributions to the anisotropy of R 2Fe 14 compounds.
Acknowledgements We are indebted for financial support of this research to the U.S. Department of Energy (Grant No. DE-FG02-86ER45262), the Nebraska Energy Office and the Cornell National Supercomputing Facility.
References [1] H.J.G. Draaisma and W.J.M. de Jonge, Jo Appl. Phys. 64 (1988) 3610. [2] J.F. Herbsl, J.J. Croat, F.E. Pinkerton and W.B. Yelon, Plays. Rcv. B 29 (1984) 4171,. [3] S.S. Jaswal, Phys. Rev. B 41 ! 199[)) 9697. [4] K.II.J. Buschow, Met. Sci. Rep. ! (1986) 1. [5] J.F. Herbst, Rev. Mod. Phys. (in press). [6] D.J. Seilmyer, Z.S. Shah and S.S. Jaswal, Mater. Sci. Eng. B 6 (1990) 137.
Letter to the Editor
Magnetic anisotropy in R 2Fe 14B compounds S.S. Jaswal a n d A.A. Kusov Behlen Laboratoo, of Physics and Center ]i~r Materials Research and Analysis, Universit3,of Nebraska, LincobL NE 68588-0111, USA Received 25 October 1991
The sums of the magnetic dipole-dipole interactions are carried out to find their contribution to intrinsic anisotropy in R.~Fet4B compounds. The dipolar contributions in R:Fe~4B (R = Y, Gd, Lu) along with the experimental data are used to deduce the anisotropy due to the non-4f spin-orbit interactions common to all the R2Fet4B compounds. Then the dipolar and non-4f spin-orbit contributions together with ex0erimentai data lead to an estimate of the single-ion anisotropy due to 4f electrons in R2Fet4B compounds with non-zero orbital angular momentum. Thus we are able to separate the various contributions to the intrinsic anisotropy in R 2Fe~4B compounds.
A permanent-magnet material must have reasonable uniaxial anisotropy for high coercivity. The anisotropy arises from magnetostatic dipole-dipole and spin-orbit interactions. It is normally assumed that the dipolar contribution to the intrinsic anisotropy (K u) is zero as given by macroscopic theory for uniform magnetization. This is true only for simple highly s, mmetric lattices (cubic, hcp) where the local ficlA correction is isotropic [1]. For complex low-symmctr3. lattices of permanent-magnet materials, one mu',t sum the dipole-dipole interactions to find their contribution to K u. We report here the first calculation of the dipolar contributior~s to K~ in R2Fe~4B compounds and use these results mong with the experimental data to deduce, the contributions due to 4f and non-4f spin-orbit interactions. The R2Fet4B compounds are ietragonal in structure [2] with the intrinsic uniaxiai anisotropy ener~, density .K~, defined as K u = U(!O0)-
U(001), where U(100) and U(001) are the energy densities with the magnetization in the [100] and [001] directions respectively. The dipolar densities are calculated for a cell in the middle of a 251 × 251 × 5 slab of primitive cells by numerically summing the dipole-dipole interactions
m , . . m ~ - 3(m,'r,~)(m)'r,.ij/r,'- ~ u,, --
.3
(1)
ltj
between the dipoles rn~ and rn~. separated by a distance rij. This size of the slab was chosen to ensure proper convergence of the sum. The dipole moments used in these calculations are those obtained from band-~t~ acture calculations [3] and the experimental va!ue~ [4] (table 1). The dipolar contribution to the intrinsic anisotropy K '~ was obtained by adding 2rrM, 2 to the calculated result to account for. the shape anisotropy, where M~ is the saturation magnetization. U
Table I Magnetic moments # of various atoms in R 2 Fe~4B compounds studied Atom
Y
Nd
Gd
Tb
Dy
Fe(4c)
Fe(4e)•
Fe(8j~)
Fe(Sj 2)
Fe(16K~)
Fe(16.K2)
B
~t [/.tu]
-0.55 a)
2.75 a)
-6.4t't
- 8 . 7 t,)
- 9 . 7 t'~
2.59;°
2.13 ~'~
2.12 ")
2.74 ~')
2.12 ;')
2.18 '')
-I.).2 ;')
;o Ref. [3]. ~'~ Ref. [4]. 0304-8853/92/$05.00 ~ 1992 - Elsevier Science Publishers B.V. All rights reserved
LETTER
T . Q T H E E D I T O R~ r
i
S.S. JaswaL A.A. Kusot' / Magnetic anisotropy in R 2 Fe t4 B compounds
L152
Table 2 Calculated and experimental anisotropy energy densities in 107 crg/cm 3 Kud YzFet4B Gd.,Fe~4B Lu:,Fet4B Nd:,Fet4B Tb:,Fe~4B Dy, Fe ~4B
--u/(expt
-0.13 -0.56 -0.13 0.0 -0.76 - 0.87
1.14 0.70 1.17 12.5 7.9 3.86
,o (4 K)
K~°
K~i
1.27 1.26 1.30 1.3 1.3 1.3
11.2 7.4 3.43
~ Ref. [51.
The results for several R2Fe~4B (R = Y, Gd, Lu, Nd, Tb and Dy) are listed in table 2. It is interesting to note that K~~ is zero for NdzFe~4B which is a ferromagnet with the moments of the magnetic atoms being of the same order of magnitude. Thus the macroscopic theory holds for such a system. This is no longer true for the other systems which are ferrimagnetic in nature. K~d due to the local field anisotropy ranges from - 0 . 1 5 for Y2Fet4 B to - 6 . 2 for Dy2Fe~4B in units of 2arM~. Thus, as the moment of a rareearth atom increases, so does the magnitude of K~d, for a ferrimagnctic system. K~,! is always negative, i.e., the dipolar contribution favors the (001) plane for the easy axis in fcrrimagnetic systems. Assuming that spin-orbit effect can be separated into non-4f and 4f contributions, enc can write K~ as K,, = K~ + __,K~"+K~i,
(2)
where -K ~° is the anisotropy e n e r ~ density due - U to the spin-orbit interactions of non-4f electrons and K u~ is the single-ion anisotropy due to the spin-orbit interactions of 4f electrons. Now K~~ is zero in Y, Gd and Lu compounds. Thus, from K~" for these compounds and the results are u shown in table 9 ~x~e find that K~"= 1.3 >,: 10 7 c r g / c m ~ for all three of these compounds supporting the assumption about the spin-orbit interactions made in eq. (2). Because of the simiJarity of the non-4f electronic structure of R~Fe~4B, it is quite reasonable to approximate K~" by 1.3 x 107 erg/cm 3 for all the rare-earth compounds favoring uniaxiai anisotropy. ~.
,~
. . ~
From ~x "~'' the calculated values of K~ and I,! ' the experimental values [5] of K,, one can deduce K~~, the single-ion anisotropy due to the 4f electrons. The results for Nd, Tb and Dy compounds are shown in table 2. From table 2 we see that K~d and --uK~°tend to cancel each other in the ferrimagnetic rare-earth compounds. Also -K - ~ ~" is an order of magnitude smaller than K~]i in compounds with large uniaxial anisotropy such as Nd2Fe~4B. As a result, the uniaxial anisotropy in these compounds is primarily determined by the 4f single-ion contribution which is in agreement with the experimental data [6]. In conclusion, we have calculated dipolar contributions to the intrinsic anisotropy in Rz FelaB compounds which favor easy axis in the (001) plane for ferrimagnetic systems. Their values for Y, Gd and Lu compounds are used to find the universal value of non-4f spin-orbit contributions to the anisotropy for the whole rare-earth series. Finally, our calculations, which isolate the dominant single-ion contribution due to 4f spin-orbit interactions, will be important for first-principles calculations of these contributions to the anisotropy of R 2Fe 14 compounds.
Acknowledgements We are indebted for financial support of this research to the U.S. Department of Energy (Grant No. DE-FG02-86ER45262), the Nebraska Energy Office and the Cornell National Supercomputing Facility.
References [1] H.J.G. Draaisma and W.J.M. de Jonge, Jo Appl. Phys. 64 (1988) 3610. [2] J.F. Herbsl, J.J. Croat, F.E. Pinkerton and W.B. Yelon, Plays. Rcv. B 29 (1984) 4171,. [3] S.S. Jaswal, Phys. Rev. B 41 ! 199[)) 9697. [4] K.II.J. Buschow, Met. Sci. Rep. ! (1986) 1. [5] J.F. Herbst, Rev. Mod. Phys. (in press). [6] D.J. Seilmyer, Z.S. Shah and S.S. Jaswal, Mater. Sci. Eng. B 6 (1990) 137.