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Anisotropy of the neodymium-copper exchange interaction in Nd2CuO4
Journal of Magnetism and Magnetic Materials 226}230 (2001) 973}975
Anisotropy of the neodymium-copper exchange interaction in Nd CuO A.A. Nugroho *, V. Nekvasil, I. VeltruskyH , S. Jandl, P. Richard, A.A. Menovsky , F.R. de Boer , J.J.M. Franse
Van der Waals - Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, Netherlands Jurusan Fisika Institut Teknologi Bandung, jl. Ganesha 10, Bandung 40132, Indonesia Institute of Physics, Czech Academy of Sciences, Cukrovarnicka& 10, 162 53 Prague 6, Czech Republic De& partement de Physique, Universite& de Sherbrooke, Sherbrooke, Que& bec, Canada J1 K 2R1
Abstract A recent infrared transmission study of the I , J"9/2, 11/2, 13/2 and 15/2 multiplets of Nd> in the antiferromagnetic ( Nd CuO single crystals has revealed a splitting of the Kramers doublets of the order of a few cm\ due to the exchange interaction between Nd and Cu. The present paper shows that these splittings can be described by an e!ective anisotropic exchange Hamiltonian for the Nd> ion expressed in terms of single-electron spherical tensor operators up to the sixth order, applicable to all levels of the 4f> con"guration. A least-squares "t of the eight observed splittings using six parameters provides a good description of the splittings. The contributions of the second- and fourth-order terms to the splittings are two orders of magnitude larger than that of the isotropic term in the exchange operator. This "nding is compatible with symmetry considerations indicating that the isotropic term in the Hamiltonian vanishes in the case when only the nearest Cu neighbors of Nd are considered. 2001 Elsevier Science B.V. All rights reserved. Keywords: Exchange coupling*anisotropic; Superconductivity*high T
A magnetic coupling between Nd and Cu in Nd CuO , a perovskite-like antiferromagnetic insulator which becomes superconducting below ¹ &24 K when doped with Ce or Th, has been the subject of intense experimental and theoretical studies because of its contribution to intricate magnetic properties of this compound including several spin reorientation transitions (see Ref. [1] and references therein). In particular, several experimental studies have revealed that the magnetic exchange interaction between Nd and Cu splits the ground state Kramers doublet of Nd> in Nd CuO by &4$1 cm\ [2}4]. The infrared transmission experiments up to 7000 cm\ yielded a splitting of the order of
* Corresponding author: Tel.: #31-020-5255718; fax: #31020-5255788. E-mail address: [email protected] (A.A. Nugroho).
a few cm\ also for Kramers doublets within remaining J multiplets of the I term [4]. Kinks in the temperature dependence of these doublet splittings, appearing at temperatures in which the individual spin reorientation transitions in the Cu magnetic sublattice occur, provide another manifestation of the Nd}Cu coupling [4]. The spin reorientation transitions have been interpreted using a mean "eld model within which an isotropic part of the exchange term vanishes for symmetry reasons and the anisotropic interaction arising from the orbital degeneracy of the Nd> ions is represented by a pseudo-dipolar term [1]. An earlier analysis of the above-mentioned doublet splittings revealed, in qualitative agreement with Ref. [1], that the dominant contribution to the Nd}Cu coupling is provided by anisotropic terms in the exchange Hamiltonian [4]. In this paper we present an analysis of the doublet splittings [4] based on the complete Hamiltonian within the molecular "eld approximation [5,6] describing the exchange interaction of the 4f electrons of a Nd ion in a surrounding cluster of the
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 6 3 - X
974
A.A. Nugroho et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 973}975
Table 1 Splitting of the Nd> Kramers doublets in Nd CuO as measured by infrared transmission [4] and calculated using Eq. (3) in cm\ J multiplet
CF level symmetry
9/2 11/2
Observed doublets
1988 1991 1993.5 1997 2001 2004.5 2007 2010.5 2014 2018 2076.5 2078.5 2410 2415.5 2418.5 3948.5 3951.5 3954 5864.5 5872
13/2
15/2
Observed splittings
Calculated splittings
5.5 3.5
5.65 3.67
3.5
3.51
4.0
3.74
2.0
2.12
3.0
2.90
2.5
2.66
2.0
1.32
Transitions involving the 5.5 cm\ ground state level.
Table 2 The best-"t values of the exchange parameters in cm\ Exchange parameter IO
!7.03 !490.85 !177.37 !256.98 14.66 46.84
S-state Cu ions, magnetically polarized by a coupling within the Cu sublattice which is much stronger than the Nd}Cu studied interaction. The considered perturbation single-ion Hamiltonian H includes the crystal "eld (CF) term H and the ex!$ change term H : H"H #H , (1) !$ where H " B C . (2) !$ IO IO I O C and B are, respectively, the single-electron irreduIO IO cible tensor operators and the crystal "eld parameters;
H is written in the form [5,6] H "!2 ¹ I (i)s(i)n, (3) IO O I O G where are the exchange parameters, ¹ I (i) are the unit IO O irreducible tensor operators which act on the orbital part of the Nd wave function, s(i) is the spin of a Nd electron and n is the unit vector directed along the crystallographic a-axis in our case. The Nd site symmetry C implies that there are "ve and six independent para meters in H and H , respectively We note that Eq. !$
(3) includes the isotropic term, k"q"0. A least-squares procedure has been used to "t the splitting of the eight observed Kramers doublet splittings using Eqs. (1)}(3). The crystal "eld parameters were taken from an earlier infrared absorption study [7]. Attempts to "t these splittings with a single exchange parameter entirely failed [4] as expected from the above-men tioned symmetry considerations [1]. The inclusion of leads to only a partial improvement of the "t; which indicates that the higher-order terms in Eq. (3), neglected in Ref. [1], are important. The results of the "nal "t considering all six parameters as unknowns are sumIO marized in Table 1; the corresponding best-"t values of are given in Table 2. We note that the quality of the "t IO as well as the best-"t values of the second- and fourthorder parameters do not change signi"cantly when the isotropic and the sixth order terms are neglected. En-
A.A. Nugroho et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 973}975
visaged infrared measurements in applied magnetic "elds should enable to further test the reliability of the best-"t values of . Their independent test is also provided by IO the Kramers doublet at &6585 cm\: a very small calculated splitting of 0.11 cm\ compares well with an experimental "nding that this splitting is zero within the experimental accuracy [4]. We thank M.O. Tjia for a careful reading as well as useful comments of the manuscript. V.N. and I.V. gratefully acknowledge the Grant Agency of the Czech Republic for its Grant No. 202/00/1602. This work was supported by Van der Waals}Zeeman Instituut.
975
References [1] R. Sachidanandam, T. Yildirim, A.B. Harris, A. Aharony, O. Entin-Wohlman, Phys. Rev. B 56 (1997) 260. [2] P. Dufour, S. Jandl, C. Thomsen, M. Cardona, B.M. Wanklyn, C. Changkang, Phys. Rev. B 51 (1995) 1053. [3] U. Staub, P. Allenspach, A. Furrer, H.R. Ott, S.W. Cheong, Z. Fisk, Solid State Commun. 75 (1990) 431. [4] S. Jandl, P. Richard, V. Nekvasil, D.I. Zhigunov, S.N. Barilo, S.V. Shiryaev, Physica C 314 (1999) 189. [5] P.M. Levy, Phys. Rev. 135 (1964) 155. [6] W.P. Wolf, J. Phys. (Paris) 32 (1971) C1}26. [7] S. Jandl, P. Dufour, P. Richard, V. Nekvasil, D.I. Zhigunov, S.N. Barilo, S.V. Shiryaev, J. Lumin 78 (1998) 197.
Anisotropy of the neodymium-copper exchange interaction in Nd CuO A.A. Nugroho *, V. Nekvasil, I. VeltruskyH , S. Jandl, P. Richard, A.A. Menovsky , F.R. de Boer , J.J.M. Franse
Van der Waals - Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, Netherlands Jurusan Fisika Institut Teknologi Bandung, jl. Ganesha 10, Bandung 40132, Indonesia Institute of Physics, Czech Academy of Sciences, Cukrovarnicka& 10, 162 53 Prague 6, Czech Republic De& partement de Physique, Universite& de Sherbrooke, Sherbrooke, Que& bec, Canada J1 K 2R1
Abstract A recent infrared transmission study of the I , J"9/2, 11/2, 13/2 and 15/2 multiplets of Nd> in the antiferromagnetic ( Nd CuO single crystals has revealed a splitting of the Kramers doublets of the order of a few cm\ due to the exchange interaction between Nd and Cu. The present paper shows that these splittings can be described by an e!ective anisotropic exchange Hamiltonian for the Nd> ion expressed in terms of single-electron spherical tensor operators up to the sixth order, applicable to all levels of the 4f> con"guration. A least-squares "t of the eight observed splittings using six parameters provides a good description of the splittings. The contributions of the second- and fourth-order terms to the splittings are two orders of magnitude larger than that of the isotropic term in the exchange operator. This "nding is compatible with symmetry considerations indicating that the isotropic term in the Hamiltonian vanishes in the case when only the nearest Cu neighbors of Nd are considered. 2001 Elsevier Science B.V. All rights reserved. Keywords: Exchange coupling*anisotropic; Superconductivity*high T
A magnetic coupling between Nd and Cu in Nd CuO , a perovskite-like antiferromagnetic insulator which becomes superconducting below ¹ &24 K when doped with Ce or Th, has been the subject of intense experimental and theoretical studies because of its contribution to intricate magnetic properties of this compound including several spin reorientation transitions (see Ref. [1] and references therein). In particular, several experimental studies have revealed that the magnetic exchange interaction between Nd and Cu splits the ground state Kramers doublet of Nd> in Nd CuO by &4$1 cm\ [2}4]. The infrared transmission experiments up to 7000 cm\ yielded a splitting of the order of
* Corresponding author: Tel.: #31-020-5255718; fax: #31020-5255788. E-mail address: [email protected] (A.A. Nugroho).
a few cm\ also for Kramers doublets within remaining J multiplets of the I term [4]. Kinks in the temperature dependence of these doublet splittings, appearing at temperatures in which the individual spin reorientation transitions in the Cu magnetic sublattice occur, provide another manifestation of the Nd}Cu coupling [4]. The spin reorientation transitions have been interpreted using a mean "eld model within which an isotropic part of the exchange term vanishes for symmetry reasons and the anisotropic interaction arising from the orbital degeneracy of the Nd> ions is represented by a pseudo-dipolar term [1]. An earlier analysis of the above-mentioned doublet splittings revealed, in qualitative agreement with Ref. [1], that the dominant contribution to the Nd}Cu coupling is provided by anisotropic terms in the exchange Hamiltonian [4]. In this paper we present an analysis of the doublet splittings [4] based on the complete Hamiltonian within the molecular "eld approximation [5,6] describing the exchange interaction of the 4f electrons of a Nd ion in a surrounding cluster of the
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 1 6 3 - X
974
A.A. Nugroho et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 973}975
Table 1 Splitting of the Nd> Kramers doublets in Nd CuO as measured by infrared transmission [4] and calculated using Eq. (3) in cm\ J multiplet
CF level symmetry
9/2 11/2
Observed doublets
1988 1991 1993.5 1997 2001 2004.5 2007 2010.5 2014 2018 2076.5 2078.5 2410 2415.5 2418.5 3948.5 3951.5 3954 5864.5 5872
13/2
15/2
Observed splittings
Calculated splittings
5.5 3.5
5.65 3.67
3.5
3.51
4.0
3.74
2.0
2.12
3.0
2.90
2.5
2.66
2.0
1.32
Transitions involving the 5.5 cm\ ground state level.
Table 2 The best-"t values of the exchange parameters in cm\ Exchange parameter IO
!7.03 !490.85 !177.37 !256.98 14.66 46.84
S-state Cu ions, magnetically polarized by a coupling within the Cu sublattice which is much stronger than the Nd}Cu studied interaction. The considered perturbation single-ion Hamiltonian H includes the crystal "eld (CF) term H and the ex!$ change term H : H"H #H , (1) !$ where H " B C . (2) !$ IO IO I O C and B are, respectively, the single-electron irreduIO IO cible tensor operators and the crystal "eld parameters;
H is written in the form [5,6] H "!2 ¹ I (i)s(i)n, (3) IO O I O G where are the exchange parameters, ¹ I (i) are the unit IO O irreducible tensor operators which act on the orbital part of the Nd wave function, s(i) is the spin of a Nd electron and n is the unit vector directed along the crystallographic a-axis in our case. The Nd site symmetry C implies that there are "ve and six independent para meters in H and H , respectively We note that Eq. !$
(3) includes the isotropic term, k"q"0. A least-squares procedure has been used to "t the splitting of the eight observed Kramers doublet splittings using Eqs. (1)}(3). The crystal "eld parameters were taken from an earlier infrared absorption study [7]. Attempts to "t these splittings with a single exchange parameter entirely failed [4] as expected from the above-men tioned symmetry considerations [1]. The inclusion of leads to only a partial improvement of the "t; which indicates that the higher-order terms in Eq. (3), neglected in Ref. [1], are important. The results of the "nal "t considering all six parameters as unknowns are sumIO marized in Table 1; the corresponding best-"t values of are given in Table 2. We note that the quality of the "t IO as well as the best-"t values of the second- and fourthorder parameters do not change signi"cantly when the isotropic and the sixth order terms are neglected. En-
A.A. Nugroho et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 973}975
visaged infrared measurements in applied magnetic "elds should enable to further test the reliability of the best-"t values of . Their independent test is also provided by IO the Kramers doublet at &6585 cm\: a very small calculated splitting of 0.11 cm\ compares well with an experimental "nding that this splitting is zero within the experimental accuracy [4]. We thank M.O. Tjia for a careful reading as well as useful comments of the manuscript. V.N. and I.V. gratefully acknowledge the Grant Agency of the Czech Republic for its Grant No. 202/00/1602. This work was supported by Van der Waals}Zeeman Instituut.
975
References [1] R. Sachidanandam, T. Yildirim, A.B. Harris, A. Aharony, O. Entin-Wohlman, Phys. Rev. B 56 (1997) 260. [2] P. Dufour, S. Jandl, C. Thomsen, M. Cardona, B.M. Wanklyn, C. Changkang, Phys. Rev. B 51 (1995) 1053. [3] U. Staub, P. Allenspach, A. Furrer, H.R. Ott, S.W. Cheong, Z. Fisk, Solid State Commun. 75 (1990) 431. [4] S. Jandl, P. Richard, V. Nekvasil, D.I. Zhigunov, S.N. Barilo, S.V. Shiryaev, Physica C 314 (1999) 189. [5] P.M. Levy, Phys. Rev. 135 (1964) 155. [6] W.P. Wolf, J. Phys. (Paris) 32 (1971) C1}26. [7] S. Jandl, P. Dufour, P. Richard, V. Nekvasil, D.I. Zhigunov, S.N. Barilo, S.V. Shiryaev, J. Lumin 78 (1998) 197.