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On magnetocrystalline anisotropy of Co2+ ions substituted in RFeO3 orthoferrites
Volume 47A, number 2
PHYSICS LETTERS
11 March 1974
ON MAGNETOCRYSTALLINE ANISOTROPY OF Co2~IONS SUBSTITUTED IN RFeO 3 ORTHOFERRITES~ Cz. RUDOWICZ Institute of Theoretical Physics, A UM, 60— 769 Poznan, Poland Received 7 December 1973 2 + ions in orthoferrites is proposed. One-ion anisotropy constants K A mechanism of magnetocrystalline anisotropy of Co 1 and K2 at 00 K are derived and discussed.
In orthoferrites RFeO3, where R is Y, La or Lu, substitution of small amounts ofCo ions causes a spin-reorientation, not exhibited by the pure cornpounds Co However, ions in RFeO [1, 2]. Recently, the valency 2+ [3,of4]. 3 was clarified as being Co none of the known magnetocrystalline anisotropy models of Co2~ions [5—7] is directly applicable to the present case. In RFeO3, the n.n.n. ions to an iron ion form a slightly deformed octahedron. The dominant crystal field (C.F.) component ~ on an Fe-site in RFeO3 should be cubic only. The remaining non-cubic C.F. component ~~flcshould be of the order of several cm 1~The principal axes i~, ~ of ~ as determined by EPR in Fe3~: YAIO 3 [8] are distorted from those in an ideal perovskite structure axes ~‘]. 2~[the in RFeO The Hamiltonian for a Co 3 is: ~,
~‘
~‘
c +~(so +~C’~ ex +~(nc +
ex (1) 2~is taken The isotropic exchange Hamiltonian for Co in the molecular field approximation: 2JCo—Fe .Z(SFe)SC0 (2) z z _=Hz ~Co z 2~concentrations, there are six At low Co to each Co2+(Zr6). The direction of 3~ions n.n. Fe the z-axis is assumed along the antiferromagnetic vecex
=
tor m. The value Of ~Co—Fe for Co: RFeO3 15 unknown. Neglecting structural differences between KCoF3 and RFeO3, we tentatively resort to the rela\
In this paper we shall neglect the anisotropic exchange ~C~(Co—Fe)in eq. (1) [101. After ref. [51,the effect of spin-orbit interaction 4T 2~can be ~ inside the lowest cubic term 1 of Co described by an effective ~ We use a general form of ~ derived by tensor methods,J9]. The dominant scalar part of ~C 50is XiS ~ The non-scalar part of ~Ksoexpressed by effective orbital 1= 1 and spin S tensor operators is: ~
so
o
(3)
~ ~(2)~(2)~ (pq) ~TJ q
o
where the summation (p, q) goes over (±2,be±2)exand 2)(S) should and O~, pressed in cubic axes i~, ~. Since 77, ~ diverge but slightly from the idealized axes 77’, we shall use the latter [91. For estimating the value of b’ the matrix of cornplete and F-~so inside the basis 4T of twelve states F6, r8, F8’ 7 arising from 1 was evaluated [9]. On diagon~ization its eigenvalues E(F6 F,) (functions of X, b, c) were fitted by computer to the experimental energies 300, 780, 860 cm~for i = 8, 8’, 7 respectively for Co2~:MgO (for [71,KCoF where the Dq = 930 cm I We obtained~=202 c = + 12, whence b’ = b c = —29 33~=211), cm~.Inb ortho= —17, ferritesDq = 1220 cm~but the values of E(F 1) remain unchanged in this range of Dq [7]. Thus, the in RFeO sign and2~order of magnitude of b’ should be the same for Co 3.
(~2, ±2). The ~(2)(7’)
~,
~,
~‘,
~‘,
—
—
1/2
1, ~Fe—Fe = 25 [9]. cm~ [9J, we tion ~Co—Fe = ~~Co—Co ~Fe—Fe) Taking obtain ~Co—Fe ~Co—Co = 10 cm16 cm~.This yields H~= 480 cm’ at 0°K. ~ Supported in part by the Institute of Physics, Polish Academy of Science, Warsaw.
We write the quadratic component of ~nc as: = + F(O(~)+ The principal axes of ~
are the (1
,
2,
(4)
3) axes of 169
Volume 47A, number 2
ref. [8]. The
~,
PHYSiCS LETTERS
77, ~ and 1,2,3 axes of ref. [8] corre-
spond to an Fe-site number (4). Reflection of 34 in the c -- ~‘ plane gIves ~2 The axes for site 3 (1) are ohtamable by rotation of those, for site 4(2) about the c-axis by [~1 2~:YAIQ EPR data on Co 3 are not available as ~ 2~:ZnMoO In various structures. e.g.. CoO. KCoF3. Co 4, Co2+: ZnWO 4 a negative ~ was found. whereas ~ and F ranged from 10 over 100 cm . respectively [~I This paper contains results on the one-ion magnetocrystalline anisotropy constants K1 and K2 at 0~K 2+ in RFcO for Co 3 accounted for by the Hamiltonian (1). From the above discussion it is seen that we have to diagonalise ~CYand K~simultaneously as whereas ~ and 7f~~ can be treated as a nerturbation. With H > 0, the lowest eigenstate of 7f is: 0) all, ~ + hO.1 ---i) cl--l.-~-),For X are 2]l found cni as: . the+ coefficients a, b.c and H, =--0.3038 480 cm and 0.0723. The functions rn 0.9450. 1, rn,.) are quantized along the z1-axis for each of the lour inequivalent sites (1). To calculate the matrix elements of ~ and ~ between twelve 1k) states of 7f , the s and s have to he transformed to coordinate with the z-axis taken along 2+. a given z,•-axis.the Onone-ion calculatwe derive ing the freeenergy energyfan for(Ia, Co anisotropy i). Assuming equal distribu~O.
tion of Co2~ions on the inequivalent sites (i), we average fan~I. i) over three possible orientations of the cubic ~‘-axis (1) and over (/) vve outain the magnetic anisotropy energy per Coion in the form/an = Ki cos2 0 + K 4 0. With accos curacy to linear terms in h’. ~. F at7 0°K:
170
K1
=
--
II March 1974
0b’
~+\/~F
COS
2e)t
K~ =
~P b’
0
0
where P0, t~. b, c) are found as. 0.1 22 and 0.36]. Taking tentatively h = --30, ~ = --60. F = - 5 cm 1 and the angle (the 1. 3-axis, the c-axis) = 52g. we get K1 = + 19 cm The value K1(exp) +65 cm ~/ion found in [31 should he divided by a factor of 2 because of a presumable misinterpretation of K0 in ref. [11.Direct insertion of 0.5- 1 0~Oe/ion 121 into the formula as done in ref. [31yields K1 36 cm~/ion instead of ~ Considering the approximations made in deriving eq. (5). our theoretical model can he said to account well for the anisotropy of Co+ ions in RFeO3 predieting the correct sign and order of magnitude of K - The author indebtedreading to Dr L.the Kowalewski discussions andiscritically manuscript.for
References Il R.L. White, J. Appi. Phys. 40 (1969)1061. 121 LHo~e~~~?. Uitert Proc. and RIntermag. Hecker, Conf. J. AppI1972. 3] 1j.Makjno and Y.~ Hidaka, Kyoto. 4] K.P. Below et a!., Fiz. Tver. Tela 15 (1973) 2244.
15] J. Kanamori, Prog. Theor. Phys. 17 (1957) 177. 16]
J.C.
Slonczewski, Phvs. Rev. 110(1958)1341.
171 M. Tachiki, Prog. Theor. Phys. 23(1960)1055.
18] RI. White et al., Phys. Rev. 136 (1964) A231 9] For detailed references, see C.Z; Rudowicz, Ph.!). Thesjs. in preparation. 110] M.O. Sturge ci al., Phys. Rev. 180 (1969) 413.
PHYSICS LETTERS
11 March 1974
ON MAGNETOCRYSTALLINE ANISOTROPY OF Co2~IONS SUBSTITUTED IN RFeO 3 ORTHOFERRITES~ Cz. RUDOWICZ Institute of Theoretical Physics, A UM, 60— 769 Poznan, Poland Received 7 December 1973 2 + ions in orthoferrites is proposed. One-ion anisotropy constants K A mechanism of magnetocrystalline anisotropy of Co 1 and K2 at 00 K are derived and discussed.
In orthoferrites RFeO3, where R is Y, La or Lu, substitution of small amounts ofCo ions causes a spin-reorientation, not exhibited by the pure cornpounds Co However, ions in RFeO [1, 2]. Recently, the valency 2+ [3,of4]. 3 was clarified as being Co none of the known magnetocrystalline anisotropy models of Co2~ions [5—7] is directly applicable to the present case. In RFeO3, the n.n.n. ions to an iron ion form a slightly deformed octahedron. The dominant crystal field (C.F.) component ~ on an Fe-site in RFeO3 should be cubic only. The remaining non-cubic C.F. component ~~flcshould be of the order of several cm 1~The principal axes i~, ~ of ~ as determined by EPR in Fe3~: YAIO 3 [8] are distorted from those in an ideal perovskite structure axes ~‘]. 2~[the in RFeO The Hamiltonian for a Co 3 is: ~,
~‘
~‘
c +~(so +~C’~ ex +~(nc +
ex (1) 2~is taken The isotropic exchange Hamiltonian for Co in the molecular field approximation: 2JCo—Fe .Z(SFe)SC0 (2) z z _=Hz ~Co z 2~concentrations, there are six At low Co to each Co2+(Zr6). The direction of 3~ions n.n. Fe the z-axis is assumed along the antiferromagnetic vecex
=
tor m. The value Of ~Co—Fe for Co: RFeO3 15 unknown. Neglecting structural differences between KCoF3 and RFeO3, we tentatively resort to the rela\
In this paper we shall neglect the anisotropic exchange ~C~(Co—Fe)in eq. (1) [101. After ref. [51,the effect of spin-orbit interaction 4T 2~can be ~ inside the lowest cubic term 1 of Co described by an effective ~ We use a general form of ~ derived by tensor methods,J9]. The dominant scalar part of ~C 50is XiS ~ The non-scalar part of ~Ksoexpressed by effective orbital 1= 1 and spin S tensor operators is: ~
so
o
(3)
~ ~(2)~(2)~ (pq) ~TJ q
o
where the summation (p, q) goes over (±2,be±2)exand 2)(S) should and O~, pressed in cubic axes i~, ~. Since 77, ~ diverge but slightly from the idealized axes 77’, we shall use the latter [91. For estimating the value of b’ the matrix of cornplete and F-~so inside the basis 4T of twelve states F6, r8, F8’ 7 arising from 1 was evaluated [9]. On diagon~ization its eigenvalues E(F6 F,) (functions of X, b, c) were fitted by computer to the experimental energies 300, 780, 860 cm~for i = 8, 8’, 7 respectively for Co2~:MgO (for [71,KCoF where the Dq = 930 cm I We obtained~=202 c = + 12, whence b’ = b c = —29 33~=211), cm~.Inb ortho= —17, ferritesDq = 1220 cm~but the values of E(F 1) remain unchanged in this range of Dq [7]. Thus, the in RFeO sign and2~order of magnitude of b’ should be the same for Co 3.
(~2, ±2). The ~(2)(7’)
~,
~,
~‘,
~‘,
—
—
1/2
1, ~Fe—Fe = 25 [9]. cm~ [9J, we tion ~Co—Fe = ~~Co—Co ~Fe—Fe) Taking obtain ~Co—Fe ~Co—Co = 10 cm16 cm~.This yields H~= 480 cm’ at 0°K. ~ Supported in part by the Institute of Physics, Polish Academy of Science, Warsaw.
We write the quadratic component of ~nc as: = + F(O(~)+ The principal axes of ~
are the (1
,
2,
(4)
3) axes of 169
Volume 47A, number 2
ref. [8]. The
~,
PHYSiCS LETTERS
77, ~ and 1,2,3 axes of ref. [8] corre-
spond to an Fe-site number (4). Reflection of 34 in the c -- ~‘ plane gIves ~2 The axes for site 3 (1) are ohtamable by rotation of those, for site 4(2) about the c-axis by [~1 2~:YAIQ EPR data on Co 3 are not available as ~ 2~:ZnMoO In various structures. e.g.. CoO. KCoF3. Co 4, Co2+: ZnWO 4 a negative ~ was found. whereas ~ and F ranged from 10 over 100 cm . respectively [~I This paper contains results on the one-ion magnetocrystalline anisotropy constants K1 and K2 at 0~K 2+ in RFcO for Co 3 accounted for by the Hamiltonian (1). From the above discussion it is seen that we have to diagonalise ~CYand K~simultaneously as whereas ~ and 7f~~ can be treated as a nerturbation. With H > 0, the lowest eigenstate of 7f is: 0) all, ~ + hO.1 ---i) cl--l.-~-),For X are 2]l found cni as: . the+ coefficients a, b.c and H, =--0.3038 480 cm and 0.0723. The functions rn 0.9450. 1, rn,.) are quantized along the z1-axis for each of the lour inequivalent sites (1). To calculate the matrix elements of ~ and ~ between twelve 1k) states of 7f , the s and s have to he transformed to coordinate with the z-axis taken along 2+. a given z,•-axis.the Onone-ion calculatwe derive ing the freeenergy energyfan for(Ia, Co anisotropy i). Assuming equal distribu~O.
tion of Co2~ions on the inequivalent sites (i), we average fan~I. i) over three possible orientations of the cubic ~‘-axis (1) and over (/) vve outain the magnetic anisotropy energy per Coion in the form/an = Ki cos2 0 + K 4 0. With accos curacy to linear terms in h’. ~. F at7 0°K:
170
K1
=
--
II March 1974
0b’
~+\/~F
COS
2e)t
K~ =
~P b’
0
0
where P0, t~. b, c) are found as. 0.1 22 and 0.36]. Taking tentatively h = --30, ~ = --60. F = - 5 cm 1 and the angle (the 1. 3-axis, the c-axis) = 52g. we get K1 = + 19 cm The value K1(exp) +65 cm ~/ion found in [31 should he divided by a factor of 2 because of a presumable misinterpretation of K0 in ref. [11.Direct insertion of 0.5- 1 0~Oe/ion 121 into the formula as done in ref. [31yields K1 36 cm~/ion instead of ~ Considering the approximations made in deriving eq. (5). our theoretical model can he said to account well for the anisotropy of Co+ ions in RFeO3 predieting the correct sign and order of magnitude of K - The author indebtedreading to Dr L.the Kowalewski discussions andiscritically manuscript.for
References Il R.L. White, J. Appi. Phys. 40 (1969)1061. 121 LHo~e~~~?. Uitert Proc. and RIntermag. Hecker, Conf. J. AppI1972. 3] 1j.Makjno and Y.~ Hidaka, Kyoto. 4] K.P. Below et a!., Fiz. Tver. Tela 15 (1973) 2244.
15] J. Kanamori, Prog. Theor. Phys. 17 (1957) 177. 16]
J.C.
Slonczewski, Phvs. Rev. 110(1958)1341.
171 M. Tachiki, Prog. Theor. Phys. 23(1960)1055.
18] RI. White et al., Phys. Rev. 136 (1964) A231 9] For detailed references, see C.Z; Rudowicz, Ph.!). Thesjs. in preparation. 110] M.O. Sturge ci al., Phys. Rev. 180 (1969) 413.