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AB-initio calculation of indirect anisotropic bilinear interactions in DyZn
Journal
of Magnetism
AB-INITIO DyZn
and Magnetic
Materials
CALCULATION
54-57
(1986) 461-462
OF INDIRECT
461
ANISOTROPIC
BILINEAR
INTERACTIONS
IN
D. SCHMITT Laboratoire
Louis N&l, CNRS,
166X. 38042 Grenoble
cedex, France
In metallic rare-earth systems, conduction electrons having a non-spherical orbital character may produce an anisotropic bilinear exchange coupling between 4f ions in addition to the usual isotropic Heisenberg term. An ab-initio calculation of such a coupling has been performed for the cubic compound DyZn, using APW band wavefunctions. The magnitude of the tetragonal-type coupling Jy(q) is far from negligible and arises mainly from d-eg conduction electrons.
The indirect exchange coupling between 4f ions via conduction electrons has been widely invoked in rareearth metals and intermetallic compounds [l]. In addition to the leading term, i.e. the isotropic Heisenbergtype interaction, anisotropic bilinear coupling may occur due to orbital effects within the 4f shell as well as in the conduction band [2,3]. Such a coupling has been recently invoked to explain the raising of degeneracy of the paramagnetic crystal field levels in PrSb [4] and TbP
151. The influence of conduction electrons has been extensively investigated with regard to the isotropic coupling [6-S], but less attention has centered on the evaluation of the anisotropic term from a “realistic” band structure. An ab-initio calculation of such a coupling is presented here for the cubic CsCl-type compound DyZn, using APW band wavefunctions and a formalism previously described for the calculation of the isotropic term [S]. Phenomenologically, bilinear interactions may be written in terms of irreducible representations of the cubic symmetry group [9]: X=
- C If/
C ~~~,(ij)J;Jb.
p,p
=
X,
y,
z
PP’
=sPa+Gw+Xc,
(1)
these interactions,
i.e:
J’.‘(q) = 2J,,(q) -L(q) =
-J,,.,(q)
31J,,(q) -J*(q)].
J’.‘(q) = J,x(q) -J,v(q). The isotropic term, J”(q). has been extensively discussed in a previous paper [8]. Regarding the anisotropic coupling, Jy.‘(q), it is more convenient to consider instead a single term, namely J,,(q), since all the others in eq. (4) may be easily derived by appropriate symmetries. In addition. by connection with paramagnetic Curie temperature [9], the quantity:
w(q)
= +J( J + 1)5,;(q)
(5)
will be discussed here, and drawn in the simple cubic Brillouin zone (fig. l), with labels associated with the tetragonal symmetry involved in OA(q )( X = (i,O,O), X(0.0, f), M=(i. i.O)and M’=(i,O. {)in units of 2n/a). The calculated values are consistent with the tetragonal symmetry of the coupling, with a raising of degeneracy at the points X and M in comparison with the isotropic interaction [8], and no raising of degeneracy
where Xa=
- c J”(ij)J’.JJ IfI is the usual isotropic bilinear
.X”=
- c +J’.‘(ij)[2J/J; t+i
(2) Hamiltonian, -<;A,‘-
+ +~y.*(.ij)[ J:J; - J;J;]
and
J;J;]
(3)
is the “ tetragonal-type“ anisotropic bilinear coupling involving only diagonal terms. .%” is the anisotropic coupling which includes only off-diagonal terms; it is not considered in the following, because it was found to be negligible compared to GX? [4,5]. The calculation provided the Fourier transform of
0304-8853/86/$03.50
0 Elsevier
Science Publishers
-4
X
I-
M
R
X
M
rR
Fig. 1. Fourier transform O:(q) for DyZn. Points are calculated, lines are least-square fits with coefficients (see text): 0.*.(100)=41 K. 0;(001,=52 K, 0;(110)=1.3 K. O;(lOl) = 4.5, O,t.(lll) = 2.1 K. 0;z(200) = - 32.4 K. 0;(002) = -20.5 K.
B.V.
462
D. Schmitt / Indirect amsotropic bihnenr rnteracrions rn DvZn
along the line TR. The resulting splitting is the strongest for the X-point, where it reaches 65 K (= 12% of the overall dispersion), a value far from negligible compared to the static (q = 0) bilinear exchange parameter O* = O;(q = 0) = 153 K. This splitting leads to anisotropic interionic coupling parameters OA(lmn), between ions distant from each other of Rn, = la + mb + nc.
(6)
These parameters may differ by = 25% for the different neighbours equivalent by cubic symmetry but not equivalent relative to the z-axis (e.g. (100) and (001) see caption of fig. 1). This anisotropy appears to be weaker than in PrSb or TbP [4,5], and indeed it has never been observed in inelastic neutron scattering experiments performed on isomorphous compounds, for example HoZn [lo] or TmCu [ll], the order of magnitude of the present calculated splitting remaining smaller than the experimental resolution. This might be the signature of anisotropic bilinear coupling intrinsicnlly larger in NaCl-type compounds than in CsCl-type ones. Conduction electrons with s character do not contribute to Jy(q) due to their spherical distribution; p electrons account for = 1% of P(q). Thus the d electrons make the dominant contribution to the anisotropic bilinear coupling, in particular those of es-type symmetry (= 75% of Jy(q)). The remaining contribution arises from mixed e,-t,, terms. This result is readily explained by the predominance of d electrons in the vicinity of the Fermi level and above it [8]. On the other hand the contributions from ea- and t,,-electrons oppose one another. the raising of degeneracy of @f(q) favouring the points X and M in the former case ( Jy,‘( X, M) > 0), the points X’ and M’ in the latter one ( Jy.‘( X, M) i 0). Focusing on the origin of J?(q) within the band structure, it turns out that all energy bands must be considered, including k-states well above the Fermi level. It was already the case for the isotropic coupling J”(q) [8]. The bands close to the Fermi energy favour rather the points X and M for @A(q), i.e. Jy,‘( X, M) > 0. while the interband contributions involving the highest energy bands favour the points X’ and M’, i.e. JY,‘(X, M) < 0. Shifting the Fermi energy to value 0.42 or 0.432 Ry corresponding to 4.8 or 5.2 conduction electrons per unit cell (respectively, 0.426 Ry and 5 electrons for DyZn) does not change drastically the overall features of @:(q). The splitting at the point M increases with
the Fermi energy while the one at X is largest for E, = 0.426 Ry. However, a consequence of this splitting is to alter the domains of stability of different possible magnetic structures as a function of the number of conduction electrons in a rigid-band model (see fig. 13 of ref. [S]). In particular the antiferromagnetic structures with propagation vector Q = X or M are more stable than without anisotropic coupling, and their domain of stability always grows compared with ferromagnetism or structure with Q = R. In addition the point which is favoured by this splitting, i.e. having the highest O:(Q), fixes the relative direction of the magnetic moments and of the propagation vector in the magnetic cell, below the ordering temperature. This can be compared with the experimental situation in light rare-earth-zinc compounds, which are antiferromagnets with Q = X’, i.e. with magnetic moments parallel to the propagation vector [12]. This result is in disagreement with the present calculation in DyZn, for both the type of structure and the sign of the splitting at Q = X. This seems to indicate a noticeable modification of the band structure throughout the RZn series, and reveals the limits of any ab-initio quantitative estimation within the rigid band model. In conclusion, this ab-initio calculation shows that indirect anisotropic bilinear interactions are always present in rare-earth intermetallic compounds, as soon as the conduction electrons have a noticeable non-s character, as is the case for DyZn where d-electrons are predominant around and far above the Fermi level. See for example A.J. Freeman, in: Magnetic Properties of Rare-Earth Metals. ed. R.J. Elliott (Plenum, New York, 1972) chap. 6. PI T.A. Kaplan and D.H. Lyons, Phys. Rev. 129 (1963) 2072. 131 N.L. Huang Liu. K.J. Ling and R. Orbach, Phys. Rev. B 14 (1976) 4087. [41 C. Vettier. D.B. McWhan, E.I. Blount and G. Shirane, Phys. Rev. Lett. 39 (1977) 1028. 151 A. Loidl. K. Knorr, J.K. Kjems and B. Liithi. 2. Phys. B 35 (1979)253. The Electronic Structure of Rare-Earth 161 See B. Coqblinin: Metals and Alloys (Academic Press, London, 1977) chap. 6. Phys. [71 P.A. Lindg&rd, B.N. Harmon and A.J. Freeman. Rev. Lett. 35 (1975) 383. WI D. Schmitt and P.M. Levy, J. Magn. Magn. Mat. 49 (1985) 15. [91 D. Schmitt, J. Magn. Magn. Mat. 44 (1984) 109. PO1 B. Hennion and J. Pierre, J. Phys. F 8 (1978) 2617. 1111 P. Morin, D. Schmitt and C. Vettier, J. Magn. Magn. Mat. 40 (1984) 287. [121 P. Morin and J. Pierre, Phys. Stat. Sol. (a) 30 (1975) 549.
of Magnetism
AB-INITIO DyZn
and Magnetic
Materials
CALCULATION
54-57
(1986) 461-462
OF INDIRECT
461
ANISOTROPIC
BILINEAR
INTERACTIONS
IN
D. SCHMITT Laboratoire
Louis N&l, CNRS,
166X. 38042 Grenoble
cedex, France
In metallic rare-earth systems, conduction electrons having a non-spherical orbital character may produce an anisotropic bilinear exchange coupling between 4f ions in addition to the usual isotropic Heisenberg term. An ab-initio calculation of such a coupling has been performed for the cubic compound DyZn, using APW band wavefunctions. The magnitude of the tetragonal-type coupling Jy(q) is far from negligible and arises mainly from d-eg conduction electrons.
The indirect exchange coupling between 4f ions via conduction electrons has been widely invoked in rareearth metals and intermetallic compounds [l]. In addition to the leading term, i.e. the isotropic Heisenbergtype interaction, anisotropic bilinear coupling may occur due to orbital effects within the 4f shell as well as in the conduction band [2,3]. Such a coupling has been recently invoked to explain the raising of degeneracy of the paramagnetic crystal field levels in PrSb [4] and TbP
151. The influence of conduction electrons has been extensively investigated with regard to the isotropic coupling [6-S], but less attention has centered on the evaluation of the anisotropic term from a “realistic” band structure. An ab-initio calculation of such a coupling is presented here for the cubic CsCl-type compound DyZn, using APW band wavefunctions and a formalism previously described for the calculation of the isotropic term [S]. Phenomenologically, bilinear interactions may be written in terms of irreducible representations of the cubic symmetry group [9]: X=
- C If/
C ~~~,(ij)J;Jb.
p,p
=
X,
y,
z
PP’
=sPa+Gw+Xc,
(1)
these interactions,
i.e:
J’.‘(q) = 2J,,(q) -L(q) =
-J,,.,(q)
31J,,(q) -J*(q)].
J’.‘(q) = J,x(q) -J,v(q). The isotropic term, J”(q). has been extensively discussed in a previous paper [8]. Regarding the anisotropic coupling, Jy.‘(q), it is more convenient to consider instead a single term, namely J,,(q), since all the others in eq. (4) may be easily derived by appropriate symmetries. In addition. by connection with paramagnetic Curie temperature [9], the quantity:
w(q)
= +J( J + 1)5,;(q)
(5)
will be discussed here, and drawn in the simple cubic Brillouin zone (fig. l), with labels associated with the tetragonal symmetry involved in OA(q )( X = (i,O,O), X(0.0, f), M=(i. i.O)and M’=(i,O. {)in units of 2n/a). The calculated values are consistent with the tetragonal symmetry of the coupling, with a raising of degeneracy at the points X and M in comparison with the isotropic interaction [8], and no raising of degeneracy
where Xa=
- c J”(ij)J’.JJ IfI is the usual isotropic bilinear
.X”=
- c +J’.‘(ij)[2J/J; t+i
(2) Hamiltonian, -<;A,‘-
+ +~y.*(.ij)[ J:J; - J;J;]
and
J;J;]
(3)
is the “ tetragonal-type“ anisotropic bilinear coupling involving only diagonal terms. .%” is the anisotropic coupling which includes only off-diagonal terms; it is not considered in the following, because it was found to be negligible compared to GX? [4,5]. The calculation provided the Fourier transform of
0304-8853/86/$03.50
0 Elsevier
Science Publishers
-4
X
I-
M
R
X
M
rR
Fig. 1. Fourier transform O:(q) for DyZn. Points are calculated, lines are least-square fits with coefficients (see text): 0.*.(100)=41 K. 0;(001,=52 K, 0;(110)=1.3 K. O;(lOl) = 4.5, O,t.(lll) = 2.1 K. 0;z(200) = - 32.4 K. 0;(002) = -20.5 K.
B.V.
462
D. Schmitt / Indirect amsotropic bihnenr rnteracrions rn DvZn
along the line TR. The resulting splitting is the strongest for the X-point, where it reaches 65 K (= 12% of the overall dispersion), a value far from negligible compared to the static (q = 0) bilinear exchange parameter O* = O;(q = 0) = 153 K. This splitting leads to anisotropic interionic coupling parameters OA(lmn), between ions distant from each other of Rn, = la + mb + nc.
(6)
These parameters may differ by = 25% for the different neighbours equivalent by cubic symmetry but not equivalent relative to the z-axis (e.g. (100) and (001) see caption of fig. 1). This anisotropy appears to be weaker than in PrSb or TbP [4,5], and indeed it has never been observed in inelastic neutron scattering experiments performed on isomorphous compounds, for example HoZn [lo] or TmCu [ll], the order of magnitude of the present calculated splitting remaining smaller than the experimental resolution. This might be the signature of anisotropic bilinear coupling intrinsicnlly larger in NaCl-type compounds than in CsCl-type ones. Conduction electrons with s character do not contribute to Jy(q) due to their spherical distribution; p electrons account for = 1% of P(q). Thus the d electrons make the dominant contribution to the anisotropic bilinear coupling, in particular those of es-type symmetry (= 75% of Jy(q)). The remaining contribution arises from mixed e,-t,, terms. This result is readily explained by the predominance of d electrons in the vicinity of the Fermi level and above it [8]. On the other hand the contributions from ea- and t,,-electrons oppose one another. the raising of degeneracy of @f(q) favouring the points X and M in the former case ( Jy,‘( X, M) > 0), the points X’ and M’ in the latter one ( Jy.‘( X, M) i 0). Focusing on the origin of J?(q) within the band structure, it turns out that all energy bands must be considered, including k-states well above the Fermi level. It was already the case for the isotropic coupling J”(q) [8]. The bands close to the Fermi energy favour rather the points X and M for @A(q), i.e. Jy,‘( X, M) > 0. while the interband contributions involving the highest energy bands favour the points X’ and M’, i.e. JY,‘(X, M) < 0. Shifting the Fermi energy to value 0.42 or 0.432 Ry corresponding to 4.8 or 5.2 conduction electrons per unit cell (respectively, 0.426 Ry and 5 electrons for DyZn) does not change drastically the overall features of @:(q). The splitting at the point M increases with
the Fermi energy while the one at X is largest for E, = 0.426 Ry. However, a consequence of this splitting is to alter the domains of stability of different possible magnetic structures as a function of the number of conduction electrons in a rigid-band model (see fig. 13 of ref. [S]). In particular the antiferromagnetic structures with propagation vector Q = X or M are more stable than without anisotropic coupling, and their domain of stability always grows compared with ferromagnetism or structure with Q = R. In addition the point which is favoured by this splitting, i.e. having the highest O:(Q), fixes the relative direction of the magnetic moments and of the propagation vector in the magnetic cell, below the ordering temperature. This can be compared with the experimental situation in light rare-earth-zinc compounds, which are antiferromagnets with Q = X’, i.e. with magnetic moments parallel to the propagation vector [12]. This result is in disagreement with the present calculation in DyZn, for both the type of structure and the sign of the splitting at Q = X. This seems to indicate a noticeable modification of the band structure throughout the RZn series, and reveals the limits of any ab-initio quantitative estimation within the rigid band model. In conclusion, this ab-initio calculation shows that indirect anisotropic bilinear interactions are always present in rare-earth intermetallic compounds, as soon as the conduction electrons have a noticeable non-s character, as is the case for DyZn where d-electrons are predominant around and far above the Fermi level. See for example A.J. Freeman, in: Magnetic Properties of Rare-Earth Metals. ed. R.J. Elliott (Plenum, New York, 1972) chap. 6. PI T.A. Kaplan and D.H. Lyons, Phys. Rev. 129 (1963) 2072. 131 N.L. Huang Liu. K.J. Ling and R. Orbach, Phys. Rev. B 14 (1976) 4087. [41 C. Vettier. D.B. McWhan, E.I. Blount and G. Shirane, Phys. Rev. Lett. 39 (1977) 1028. 151 A. Loidl. K. Knorr, J.K. Kjems and B. Liithi. 2. Phys. B 35 (1979)253. The Electronic Structure of Rare-Earth 161 See B. Coqblinin: Metals and Alloys (Academic Press, London, 1977) chap. 6. Phys. [71 P.A. Lindg&rd, B.N. Harmon and A.J. Freeman. Rev. Lett. 35 (1975) 383. WI D. Schmitt and P.M. Levy, J. Magn. Magn. Mat. 49 (1985) 15. [91 D. Schmitt, J. Magn. Magn. Mat. 44 (1984) 109. PO1 B. Hennion and J. Pierre, J. Phys. F 8 (1978) 2617. 1111 P. Morin, D. Schmitt and C. Vettier, J. Magn. Magn. Mat. 40 (1984) 287. [121 P. Morin and J. Pierre, Phys. Stat. Sol. (a) 30 (1975) 549.