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Magnetic anisotropy of Er3Fe5O12
Journal of Magnetism and Magnetic Materials 157/158 (1996) 523-524
journal of magnetism ~ H and magnetic ~ H materials
N
ELSEVIER
Magnetic anisotropy of Er3FesO12 M. Savosta *, V. Doroshev, V. Borodin Donetsk Physical and Technical Institute, R. Luxemburg str. 72, 340114 Donetsk, Ukraine Abstract
The low-temperature magnetic properties of erbium iron garnet have been calculated as a function of temperature and the orientation of the magnetization. The calculation is based on an effective spin Hamiltonian approach. The results are in agreement with available experimental data. An anomaly in the magnetic anisotropy near 6 K has been predicted and observed in this work. Keywords: Erbium iron garnet; Exchange interaction; Magnetic anisotropy; Magnetic structure; Spin Hamiltonian
The calculation of the magnetic anisotropy of rare-earth iron garnets is a nontrivial problem due to the low symmetry (D z) of the rare-earth sites. The corresponding Hamiltonians describing the crystal field and anisotropic R E - F e exchange contain nine and ten independent parameters, respectively [1]. To simplify the task, the approximation of isotropic E r - F e exchange has been used in the case of Er3Fe5012 (ErlG) [2]. The aim of the present work is to show that the low-temperature magnetic properties of EflG can be qualitatively explained using an effective spin Hamiltonian including a term describing the anisotropy of exchange. The effective spin Hamiltonian used is [3]:
used available values of Er 3+ exchange splittings in two inequivalent sites deduced from the far- infrared spectra [5], and the anisotropy energy A ( H a ~ 0) = F[111] - F[001] = 0.76 cm - I per Er 3+ ion we obtained by torque studies of the garnet EroA2Y2.saF%O12 at T = 4.2 K. However, since the exchange splittings and the anisotropy energy are invariant to the permutation of the indices x ~ y as well as to the change of sign of the components Gi, ambiguity remains. Starting with the neutron data for the low-temperature noncollinear magnetic smmture of the Er 3+ sublattice in ErlG (magnetization along the (001) direction) [6], we arrived at the following values:
H=tzB
G x = +30.3 cm -1,
~
(HEGiSI + H a g i S I - H a c ~ i H a
i=x,y,z
G z = + 18.2
- H . c~;HE),
(1)
where S' = 1 / 2 is the effective spin, H E is the unit vector along the magnetization of the RE sublattice, H a is the applied field, g and G are the magnetic and exchange tensors for the ground state Kramers doublet, respectively, a n d a i and ce~ are the second-order terms which formally take into account the influence of higher-lying energy levels of the Er 3+ ion. For the magnetic g-tensor we used the values determined by Orlich et al. [4] from the optical spectra of Er 3+ in Er3GasO12; this structure is similar to ErlG: gx = 4.3,
gy
s o m e
=
--
4.3,
gz = 11.3.
(2)
The signs were attributed in accordance with Ref. [2]. To determine the components of the exchange G-tensor we
* Corresponding author. Fax: + 38-0622-550-127; email: doroshev @host.dipt.donetsk.ua.
a 'x = + 2 . 6 ,
Gy
=
+4.1 cm -1,
cm-1;
(3)
a'y= +2.7,
ce'z = + 1 . 0 .
(4)
Using the values in Eqs. (2)-(4) we calculated the low-temperature magnetic properties of ErlG: magnetic smacture, magnetization and anisotropy. The calculated magnetic umbrella structure for M along the (111) direction at T = 4.2 K, with /x=5.0/xB, ~=-1°;
/Z=5.9/XB, ~O'= + 4 0 °,
where /z and /x' are the magnetic moments of Er 3+ ions in two inequivalent sites, and ~O and ~b' are the angles between the moments in a plane (110) with the [111] direction and are positive for a deviation towards the (001) axis, was found to be close to the NMR data [7]: / x = 5 . 8 ( 3 ) / x B, ~p= +12(-T-10)°; / Z = 5.45(75)/XB, ~0'= +48(-T-9) °. Our numerical analysis predicted a jump in the magnetization curve for the case where both the exchange and applied fields are parallel to the [110] direction at T ~ 0 K
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 1 0 0 3 - 3
524
M. Savosta et al. / Journal of Magnetism and Magnetic Materials 157/158 (1996) 523-524
(see inset of Fig. 1). Contributing to this feature are those Er 3+ ions which have the local y-axis directions parallel to the exchange field. The magnetic moment in this case is given by
10 E o
¢
i
,
i
Er3F%0~2 ~ ~
-
.-/t--"
•
experimental predicted
o
m = my = l t z B gy sign( Gy + H a g y ) t h ( IXBGy/2kT ) -~- ~B(
O/ty -1- 2HaO/y)
"
(5)
o
x_
,¢,
The above-mentioned jump is related to the change of sign of the first term in Eq. (5) at Ha ay//gy. Unfortunately, the predicted jump falls into the region of unsaturated magnetization which makes a direct observation of the effect difficult. However, inspecting the experimental results of Ref. [8] we have found an anomaly in the angular dependence of the spontaneous magnetization Ms of ErIG at T = 4.2 K, determined by an extrapolation to Ha ~ 0 from the magnetization curves in low magnetic fields ( H a < 3 T). To demonstrate that this anomaly can be caused by the predicted jump in the magnetization curve, we plotted in Fig. 1 the calculated angular dependence of magnetization M s ( T ~ 0 K) extrapolated from H, = 2.3 T together with the data from Ref. [8] as well as the calculated angular dependence of magnetization at Ha = 0. Assuming that the anisotropy of the Fe 3+ ions in ErIG is similar to that in YIG, we used Hamiltonian (1) to calculate the temperature of the spin reorientation transition [001] ~ [111], Tt = 76 K, and found it to be in surprisingly good agreement with recent experimental data [9]. W e also predicted an anomaly of the magnetic anisotropy in ErIG at low temperatures caused by a strong anisotropy of the exchange G-tensor for the Er 3+ ions. In accordance with Eq. (3) the exchange splitting of the ground state doublet for the case when magnetization lies along the local y-axis (i.e. for M I][ll0]) is small compared with the splitting for other directions of M. This
I
10
=
[001]
"7" --
20
[111]
-m
J'----'-.
=2 .m~
• 15
[1101
\u/
, .
kl&/
)M tzs mole 1
• ~Ii]/
j
L. . . . f ........ i
=
°.
~'=~ 1o
I
E 0
'
M//[11°1t H T/
/ i001
UL'
,
1
30
3
~
I6
q
i
60
I
90
orientation in (110) plane Fig. 1. Calculated (lines) and measured (points) angular dependencies of low-temperature magnetization for ErIG. The solid line corresponds to spontaneous magnetization (H a = 0) and the dashed line to magnetization extrapolated from H~ = 2.3 T: The experimental points are from Ref. [8]. The inset shows the predicted jump on the magnetization curve.
20
30
40
temperature K Fig. 2. Comparison of the predicted and experimental temperature dependencies of anisotropy constant K 1 for ErIG. leads to the abrupt temperature dependence of the free energy near direction [110] and, for example, to the appearance of a maximum on the temperature dependence of the anisotropy constant K 1 defined as K 1 = 1 8 ( F u 2 f 0 0 0 - 9 ( F r o - f 0 0 0 - 2(F110 - Fo0m) at T ~ 6 K.
To verify this prediction we carried out torque studies of ErIG magnetocrystalline anisotropy in the temperature region 4 . 2 - 3 4 K. The measurements were performed in a magnetic field H, = 4.7 T. As can be seen from Fig. 2, the theory predicts the temperature of the effect correctly, but the calculated magnitude of the anisotropy constant K~ differs substantially from the experimental results. It is clear that the approach we used may only give a qualitative description of the magnetic anisotropy. Concluding, we wish to stress that the simple spin Hamiltonian approach used accounts for some low temperature properties of ErIG related to the splitting of the lowest Kramers doublet of Er 3÷ ions by an anisotropic exchange interaction with the iron sublattice. Acknowledgement: The research described in this publication was made possible in part by Grant N K55100 from the Joint Fund of the Government of Ukraine and the International Science Foundation. W e would like to thank Dr. V. Nekvasil for many helpful discussions. References [1] V. Nekvasil and I. Veltrusky, J. Magn. Magn. Mater. 86 (1990) 315. [2] I. Veltrusky, Czech. J. Phys. B 37 (1987) 30. [3] W.P. Wolf, Proc. Int. Conf. on Magnetism, Nottingham, 1964 (Inst. Phys./Phys. Soc., London, 1965) p. 555. [4] E. Orlich, S. Hiifner and P. Grtinberg, Z. Phys. 231 (1970) 144. [5] A.J. Sievers and M. Tinkhan, Phys. Rev. 129 (1963) 1195. [6] R. Hock, H. Fness, T. Vogt and M. Bonnet, Z. Phys. B 82 (1991) 283. [7] V.A. Borodin, V.D. Doroshev, T.N. Tarasenko, M.M. Savosta and P. Novfik, J. Phys.: Condens. Matter 3 (1991) 5881. [8] M. Guillot, A. Marchand, F. TchGou, P. Feldmann and H. Le Gall, Z. Phys. B 44 (1981) 41. [9] R. Hock, J. Baruchel, H. Fuess, B. Antonini and P. Paroli, J. Magn. Magn. Mater. 104-107 (1992) 453.
journal of magnetism ~ H and magnetic ~ H materials
N
ELSEVIER
Magnetic anisotropy of Er3FesO12 M. Savosta *, V. Doroshev, V. Borodin Donetsk Physical and Technical Institute, R. Luxemburg str. 72, 340114 Donetsk, Ukraine Abstract
The low-temperature magnetic properties of erbium iron garnet have been calculated as a function of temperature and the orientation of the magnetization. The calculation is based on an effective spin Hamiltonian approach. The results are in agreement with available experimental data. An anomaly in the magnetic anisotropy near 6 K has been predicted and observed in this work. Keywords: Erbium iron garnet; Exchange interaction; Magnetic anisotropy; Magnetic structure; Spin Hamiltonian
The calculation of the magnetic anisotropy of rare-earth iron garnets is a nontrivial problem due to the low symmetry (D z) of the rare-earth sites. The corresponding Hamiltonians describing the crystal field and anisotropic R E - F e exchange contain nine and ten independent parameters, respectively [1]. To simplify the task, the approximation of isotropic E r - F e exchange has been used in the case of Er3Fe5012 (ErlG) [2]. The aim of the present work is to show that the low-temperature magnetic properties of EflG can be qualitatively explained using an effective spin Hamiltonian including a term describing the anisotropy of exchange. The effective spin Hamiltonian used is [3]:
used available values of Er 3+ exchange splittings in two inequivalent sites deduced from the far- infrared spectra [5], and the anisotropy energy A ( H a ~ 0) = F[111] - F[001] = 0.76 cm - I per Er 3+ ion we obtained by torque studies of the garnet EroA2Y2.saF%O12 at T = 4.2 K. However, since the exchange splittings and the anisotropy energy are invariant to the permutation of the indices x ~ y as well as to the change of sign of the components Gi, ambiguity remains. Starting with the neutron data for the low-temperature noncollinear magnetic smmture of the Er 3+ sublattice in ErlG (magnetization along the (001) direction) [6], we arrived at the following values:
H=tzB
G x = +30.3 cm -1,
~
(HEGiSI + H a g i S I - H a c ~ i H a
i=x,y,z
G z = + 18.2
- H . c~;HE),
(1)
where S' = 1 / 2 is the effective spin, H E is the unit vector along the magnetization of the RE sublattice, H a is the applied field, g and G are the magnetic and exchange tensors for the ground state Kramers doublet, respectively, a n d a i and ce~ are the second-order terms which formally take into account the influence of higher-lying energy levels of the Er 3+ ion. For the magnetic g-tensor we used the values determined by Orlich et al. [4] from the optical spectra of Er 3+ in Er3GasO12; this structure is similar to ErlG: gx = 4.3,
gy
s o m e
=
--
4.3,
gz = 11.3.
(2)
The signs were attributed in accordance with Ref. [2]. To determine the components of the exchange G-tensor we
* Corresponding author. Fax: + 38-0622-550-127; email: doroshev @host.dipt.donetsk.ua.
a 'x = + 2 . 6 ,
Gy
=
+4.1 cm -1,
cm-1;
(3)
a'y= +2.7,
ce'z = + 1 . 0 .
(4)
Using the values in Eqs. (2)-(4) we calculated the low-temperature magnetic properties of ErlG: magnetic smacture, magnetization and anisotropy. The calculated magnetic umbrella structure for M along the (111) direction at T = 4.2 K, with /x=5.0/xB, ~=-1°;
/Z=5.9/XB, ~O'= + 4 0 °,
where /z and /x' are the magnetic moments of Er 3+ ions in two inequivalent sites, and ~O and ~b' are the angles between the moments in a plane (110) with the [111] direction and are positive for a deviation towards the (001) axis, was found to be close to the NMR data [7]: / x = 5 . 8 ( 3 ) / x B, ~p= +12(-T-10)°; / Z = 5.45(75)/XB, ~0'= +48(-T-9) °. Our numerical analysis predicted a jump in the magnetization curve for the case where both the exchange and applied fields are parallel to the [110] direction at T ~ 0 K
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 1 0 0 3 - 3
524
M. Savosta et al. / Journal of Magnetism and Magnetic Materials 157/158 (1996) 523-524
(see inset of Fig. 1). Contributing to this feature are those Er 3+ ions which have the local y-axis directions parallel to the exchange field. The magnetic moment in this case is given by
10 E o
¢
i
,
i
Er3F%0~2 ~ ~
-
.-/t--"
•
experimental predicted
o
m = my = l t z B gy sign( Gy + H a g y ) t h ( IXBGy/2kT ) -~- ~B(
O/ty -1- 2HaO/y)
"
(5)
o
x_
,¢,
The above-mentioned jump is related to the change of sign of the first term in Eq. (5) at Ha ay//gy. Unfortunately, the predicted jump falls into the region of unsaturated magnetization which makes a direct observation of the effect difficult. However, inspecting the experimental results of Ref. [8] we have found an anomaly in the angular dependence of the spontaneous magnetization Ms of ErIG at T = 4.2 K, determined by an extrapolation to Ha ~ 0 from the magnetization curves in low magnetic fields ( H a < 3 T). To demonstrate that this anomaly can be caused by the predicted jump in the magnetization curve, we plotted in Fig. 1 the calculated angular dependence of magnetization M s ( T ~ 0 K) extrapolated from H, = 2.3 T together with the data from Ref. [8] as well as the calculated angular dependence of magnetization at Ha = 0. Assuming that the anisotropy of the Fe 3+ ions in ErIG is similar to that in YIG, we used Hamiltonian (1) to calculate the temperature of the spin reorientation transition [001] ~ [111], Tt = 76 K, and found it to be in surprisingly good agreement with recent experimental data [9]. W e also predicted an anomaly of the magnetic anisotropy in ErIG at low temperatures caused by a strong anisotropy of the exchange G-tensor for the Er 3+ ions. In accordance with Eq. (3) the exchange splitting of the ground state doublet for the case when magnetization lies along the local y-axis (i.e. for M I][ll0]) is small compared with the splitting for other directions of M. This
I
10
=
[001]
"7" --
20
[111]
-m
J'----'-.
=2 .m~
• 15
[1101
\u/
, .
kl&/
)M tzs mole 1
• ~Ii]/
j
L. . . . f ........ i
=
°.
~'=~ 1o
I
E 0
'
M//[11°1t H T/
/ i001
UL'
,
1
30
3
~
I6
q
i
60
I
90
orientation in (110) plane Fig. 1. Calculated (lines) and measured (points) angular dependencies of low-temperature magnetization for ErIG. The solid line corresponds to spontaneous magnetization (H a = 0) and the dashed line to magnetization extrapolated from H~ = 2.3 T: The experimental points are from Ref. [8]. The inset shows the predicted jump on the magnetization curve.
20
30
40
temperature K Fig. 2. Comparison of the predicted and experimental temperature dependencies of anisotropy constant K 1 for ErIG. leads to the abrupt temperature dependence of the free energy near direction [110] and, for example, to the appearance of a maximum on the temperature dependence of the anisotropy constant K 1 defined as K 1 = 1 8 ( F u 2 f 0 0 0 - 9 ( F r o - f 0 0 0 - 2(F110 - Fo0m) at T ~ 6 K.
To verify this prediction we carried out torque studies of ErIG magnetocrystalline anisotropy in the temperature region 4 . 2 - 3 4 K. The measurements were performed in a magnetic field H, = 4.7 T. As can be seen from Fig. 2, the theory predicts the temperature of the effect correctly, but the calculated magnitude of the anisotropy constant K~ differs substantially from the experimental results. It is clear that the approach we used may only give a qualitative description of the magnetic anisotropy. Concluding, we wish to stress that the simple spin Hamiltonian approach used accounts for some low temperature properties of ErIG related to the splitting of the lowest Kramers doublet of Er 3÷ ions by an anisotropic exchange interaction with the iron sublattice. Acknowledgement: The research described in this publication was made possible in part by Grant N K55100 from the Joint Fund of the Government of Ukraine and the International Science Foundation. W e would like to thank Dr. V. Nekvasil for many helpful discussions. References [1] V. Nekvasil and I. Veltrusky, J. Magn. Magn. Mater. 86 (1990) 315. [2] I. Veltrusky, Czech. J. Phys. B 37 (1987) 30. [3] W.P. Wolf, Proc. Int. Conf. on Magnetism, Nottingham, 1964 (Inst. Phys./Phys. Soc., London, 1965) p. 555. [4] E. Orlich, S. Hiifner and P. Grtinberg, Z. Phys. 231 (1970) 144. [5] A.J. Sievers and M. Tinkhan, Phys. Rev. 129 (1963) 1195. [6] R. Hock, H. Fness, T. Vogt and M. Bonnet, Z. Phys. B 82 (1991) 283. [7] V.A. Borodin, V.D. Doroshev, T.N. Tarasenko, M.M. Savosta and P. Novfik, J. Phys.: Condens. Matter 3 (1991) 5881. [8] M. Guillot, A. Marchand, F. TchGou, P. Feldmann and H. Le Gall, Z. Phys. B 44 (1981) 41. [9] R. Hock, J. Baruchel, H. Fuess, B. Antonini and P. Paroli, J. Magn. Magn. Mater. 104-107 (1992) 453.