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Electronic structure and magnetostriction in bulk gadolinium
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) e249–e250
Electronic structure and magnetostriction in bulk gadolinium Ulrike Nitzschea,*, Manuel Richtera, Igor Chaplyginb, Klaus Koepernika, Ingo Opahlea, Helmut Eschriga a
IFW Dresden, Department of Theoretical Solid State Physics, P.O. Box 27 01 16, Dresden D-01171, Germany b Institut fur . Physik, TU Chemnitz, Chemnitz D-09107, Germany
Abstract We have investigated the spontaneous anisotropic magnetostriction of element hcp Gd metal in two different approximations to density functional theory, the local spin density approximation (LSDA) and the so-called LSDA+U. As a rough model for the paramagnetic state with large persisting 4f-moments several static AFM arrangements were used. The obtained value of spontaneous volume magnetostriction is reasonable, but the sign of the anisotropic magnetostriction is different in LSDA and LSDA+U. r 2004 Published by Elsevier B.V. PACS: 75.80.+q; 71.20.Eh; 71.15.Nc Keywords: Magnetostriction; Gadolinium; Density functional calculation; LSDA+U
Gadolinium is a lanthanide with half-filled f-shell showing the electronic structure of a trivalent 5dtransition metal with superposed inert f-states. Below a quite high Curie temperature TC of 294 K its magnetic ground state is ferromagnetic (FM). The magnetic moment amounts to 7.55 mB [1], consisting of a local 4f-moment (7 mB) and the itinerant electron contribution. Both gadolinium and its compounds exhibit large and anisotropic spontaneous magnetoelastic effects. The change of lattice parameters due to magnetic order in Gd is comparable in magnitude with related single-ion contributions in other rare-earth systems. The experimental value of the volume magnetostriction is 0.5%, the changes of lattice constants are larger in c-direction (Dc/cE 0.3%) than in a-direction (Da/aE 0.1%) [2]. At temperatures above TC, Gd is paramagnetic (PM) with fluctuating f-moments. Such a state cannot be described in density functional theory without resorting to model assumptions. It is well accepted that the persistent magnetic moments fluctuate at a time-scale
*Corresponding author. Tel.: +49 4659 463; fax: 49 4659 490. E-mail address: [email protected] (U. Nitzsche). 0304-8853/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.jmmm.2003.12.399
which is much slower than the time-scale of electron motion [3]. This fact allows an adiabatic approximation that, for any given moment, results in a specific disordered spin configuration. The configurations are quasi-static at the time-scale of electron motion but exhaust a huge configuration space on the long run. Four particular collinear antiferromagnetic (AFM) representatives of this space have been choosen to models the system in the PM ensemble state:
AFM0 and AFM1 are structures with AFM stacking in c-direction and a period of one and two layers, respectively, whereas AFM2 and AFM3 show AFM order in a-direction and mainly FM (AFM2) or AFM (AFM3) arrangement in c-direction. To estimate volume and anisotropic magnetostriction. Total energy calculations for different magnetic states were performed within the frame-work of density functional theory. Volume and c/a ratio were relaxed
ARTICLE IN PRESS e250
U. Nitzsche et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) e249–e250
and compared with the related values of the FM reference state. The calculations were carried out with the full-potential local-orbital minimum-basis bandstructure scheme (FPLO) [4] employing a minimum basis of 4f, 5spd and 6sp states and about 3000 points for k-space integration in the irreducible part of the Brillouin zone. Using the local spin-density approximation in Perdew-Wang92 parameterization [5] a volume magnetostriction between 0.5% and 1.5% and an anisotropic magnetostriction between 0.2% and 1.1% were found. Besides a conspicuous dependence of magnetostriction in a-direction on the specific AFM configuration, the magnetostriction is larger in c-direction (0.6–1.1.%) than in a-direction ( 0.1–0.4%) for all choosen magnetic states, in agreement with the experiment. This satisfactory result is in contrast to the wrong magnetic ground state: all AFM-states are lower in energy than the FM state, a known LSDA problem [6,7]. The splitting of the f-states is too small. Minority f-states can hybridize with the 5d-states close to the Fermi energy Ef, leading to a stabilization of AFM. This problem is avoided in the LSDA+U model. Within this model, the strong intraatomic correlations of the f-localized states are explicitly taken into account. One effect of this model is to remove the unphysical hybridization with occupied d-states. The FM state is now lower in energy than any of the AFM states, as in experiment. We used the LSDA+U (in the so-called atomic limit) with a parameter U = 6 eV and J = 0. Variation of U in the range from 3 to 7 eV gives marginal changes in total energy and structural parameters. The result for the anisotropic magnetostriction is now opposite to the experiment: The striction in a-direction is much larger than in c-direction leading to a wrong sign in anisotropy independent of the chosen AFM order (Fig. 1). On the other hand, the volume magnetostriction is comparable with both experiment and LSDA results. Further investigations on other compounds are required to gain more insight into the origin of anisotropic magnetostriction in Gd system.
Fig. 1. Relative changes between the FM reference state and several AFM structures compared with experiment, Note: in Ref. [7] the authors used an AFM0 structure.
The authors thank Martin Rotter for drawing their attention to this topic and Helge Rosner and Ulrich . RoXler for interesting and clarifying discussions. The work was supported by SFB 463, TP B11 and RTN ck F-electron.
References [1] H.E. Nigh, S. Legvold, F.H. Spedding, Phys. Rev. 132 (1963) 1092. [2] A. Lindbaum, M. Rotter in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, Vol. 14, Elsevier Science B.V., Amsterdam, 2002, p. 307. [3] B.L. Gyorffy, A.J. Pindor, J.B. Staunton, G.M. Stocks, H. Winter, J. Phys. F 15 (1985) 1337. [4] K. Koepernik, H. Eschrig, Phys. Rev. B 59 (1999) 1743; H. Eschrig, K. Koepernik, I. Chaplygin, cond-mat/0301558. [5] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244–13249. [6] A.B. Shick, W.E. Pickett, C.S. Fadley, Phys. Rev. B 60 (1999) 10763. [7] Ph. Kurz, G. Bihlmayer, S. Blugel, . J. Phys.: Condens. Matter 14 (2002) 6353.
Journal of Magnetism and Magnetic Materials 272–276 (2004) e249–e250
Electronic structure and magnetostriction in bulk gadolinium Ulrike Nitzschea,*, Manuel Richtera, Igor Chaplyginb, Klaus Koepernika, Ingo Opahlea, Helmut Eschriga a
IFW Dresden, Department of Theoretical Solid State Physics, P.O. Box 27 01 16, Dresden D-01171, Germany b Institut fur . Physik, TU Chemnitz, Chemnitz D-09107, Germany
Abstract We have investigated the spontaneous anisotropic magnetostriction of element hcp Gd metal in two different approximations to density functional theory, the local spin density approximation (LSDA) and the so-called LSDA+U. As a rough model for the paramagnetic state with large persisting 4f-moments several static AFM arrangements were used. The obtained value of spontaneous volume magnetostriction is reasonable, but the sign of the anisotropic magnetostriction is different in LSDA and LSDA+U. r 2004 Published by Elsevier B.V. PACS: 75.80.+q; 71.20.Eh; 71.15.Nc Keywords: Magnetostriction; Gadolinium; Density functional calculation; LSDA+U
Gadolinium is a lanthanide with half-filled f-shell showing the electronic structure of a trivalent 5dtransition metal with superposed inert f-states. Below a quite high Curie temperature TC of 294 K its magnetic ground state is ferromagnetic (FM). The magnetic moment amounts to 7.55 mB [1], consisting of a local 4f-moment (7 mB) and the itinerant electron contribution. Both gadolinium and its compounds exhibit large and anisotropic spontaneous magnetoelastic effects. The change of lattice parameters due to magnetic order in Gd is comparable in magnitude with related single-ion contributions in other rare-earth systems. The experimental value of the volume magnetostriction is 0.5%, the changes of lattice constants are larger in c-direction (Dc/cE 0.3%) than in a-direction (Da/aE 0.1%) [2]. At temperatures above TC, Gd is paramagnetic (PM) with fluctuating f-moments. Such a state cannot be described in density functional theory without resorting to model assumptions. It is well accepted that the persistent magnetic moments fluctuate at a time-scale
*Corresponding author. Tel.: +49 4659 463; fax: 49 4659 490. E-mail address: [email protected] (U. Nitzsche). 0304-8853/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.jmmm.2003.12.399
which is much slower than the time-scale of electron motion [3]. This fact allows an adiabatic approximation that, for any given moment, results in a specific disordered spin configuration. The configurations are quasi-static at the time-scale of electron motion but exhaust a huge configuration space on the long run. Four particular collinear antiferromagnetic (AFM) representatives of this space have been choosen to models the system in the PM ensemble state:
AFM0 and AFM1 are structures with AFM stacking in c-direction and a period of one and two layers, respectively, whereas AFM2 and AFM3 show AFM order in a-direction and mainly FM (AFM2) or AFM (AFM3) arrangement in c-direction. To estimate volume and anisotropic magnetostriction. Total energy calculations for different magnetic states were performed within the frame-work of density functional theory. Volume and c/a ratio were relaxed
ARTICLE IN PRESS e250
U. Nitzsche et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) e249–e250
and compared with the related values of the FM reference state. The calculations were carried out with the full-potential local-orbital minimum-basis bandstructure scheme (FPLO) [4] employing a minimum basis of 4f, 5spd and 6sp states and about 3000 points for k-space integration in the irreducible part of the Brillouin zone. Using the local spin-density approximation in Perdew-Wang92 parameterization [5] a volume magnetostriction between 0.5% and 1.5% and an anisotropic magnetostriction between 0.2% and 1.1% were found. Besides a conspicuous dependence of magnetostriction in a-direction on the specific AFM configuration, the magnetostriction is larger in c-direction (0.6–1.1.%) than in a-direction ( 0.1–0.4%) for all choosen magnetic states, in agreement with the experiment. This satisfactory result is in contrast to the wrong magnetic ground state: all AFM-states are lower in energy than the FM state, a known LSDA problem [6,7]. The splitting of the f-states is too small. Minority f-states can hybridize with the 5d-states close to the Fermi energy Ef, leading to a stabilization of AFM. This problem is avoided in the LSDA+U model. Within this model, the strong intraatomic correlations of the f-localized states are explicitly taken into account. One effect of this model is to remove the unphysical hybridization with occupied d-states. The FM state is now lower in energy than any of the AFM states, as in experiment. We used the LSDA+U (in the so-called atomic limit) with a parameter U = 6 eV and J = 0. Variation of U in the range from 3 to 7 eV gives marginal changes in total energy and structural parameters. The result for the anisotropic magnetostriction is now opposite to the experiment: The striction in a-direction is much larger than in c-direction leading to a wrong sign in anisotropy independent of the chosen AFM order (Fig. 1). On the other hand, the volume magnetostriction is comparable with both experiment and LSDA results. Further investigations on other compounds are required to gain more insight into the origin of anisotropic magnetostriction in Gd system.
Fig. 1. Relative changes between the FM reference state and several AFM structures compared with experiment, Note: in Ref. [7] the authors used an AFM0 structure.
The authors thank Martin Rotter for drawing their attention to this topic and Helge Rosner and Ulrich . RoXler for interesting and clarifying discussions. The work was supported by SFB 463, TP B11 and RTN ck F-electron.
References [1] H.E. Nigh, S. Legvold, F.H. Spedding, Phys. Rev. 132 (1963) 1092. [2] A. Lindbaum, M. Rotter in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, Vol. 14, Elsevier Science B.V., Amsterdam, 2002, p. 307. [3] B.L. Gyorffy, A.J. Pindor, J.B. Staunton, G.M. Stocks, H. Winter, J. Phys. F 15 (1985) 1337. [4] K. Koepernik, H. Eschrig, Phys. Rev. B 59 (1999) 1743; H. Eschrig, K. Koepernik, I. Chaplygin, cond-mat/0301558. [5] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244–13249. [6] A.B. Shick, W.E. Pickett, C.S. Fadley, Phys. Rev. B 60 (1999) 10763. [7] Ph. Kurz, G. Bihlmayer, S. Blugel, . J. Phys.: Condens. Matter 14 (2002) 6353.