- Email: [email protected]
Multipolar interactions in the Anderson lattice
ARTICLE IN PRESS
Physica B 359–361 (2005) 720–722 www.elsevier.com/locate/physb
Multipolar interactions in the Anderson lattice Gen’ya Sakurai, Yoshio Kuramoto Department of Physics, Tohoku University, Aoba Aza Aramaki Sendai, Miyagi 980 8578, Japan
Abstract The wave number dependence of RKKY multipolar interactions is derived for the Anderson-type model with orbital degeneracy in the simple cubic lattice. With the spherical Fermi surface and one conduction electron per cell, the Kohn anomaly arises in the magnetic interaction near the G point of the Brillouin zone. This result may be relevant to the incommensurate magnetic structure observed in the quasi-cubic compound CeB2 C2 : r 2005 Elsevier B.V. All rights reserved. PACS: 71.20.Eh; 71.27.+a; 75.30.Et Keywords: Incommensurate magnetic structure; Kohn anomaly; RKKY interaction
The RB2 C2 compounds (R ¼ rare earths) crystallize in the tetragonal LaB2 C2 -type structure(P4/ mbm), and some of them exhibit incommensurate magnetic structures [1]. For example, CeB2 C2 undergoes a magnetic ordering at T N ¼ 7:3 K, and a successive transition at T t ¼ 6:5 K. Below T t ; an incommensurate magnetic structure is found. The propagation vector is q ¼ ðd; d; d0 Þ; with d ¼ 0:16 and d0 ¼ 0:10: Other RB2 C2 systems often exhibit similar incommensurate magnetic structures [1]. Thus, propagation vectors should be determined mostly by conduction bands near Fermi surfaces. CeB2 C2 can be considered as Corresponding author. Tel.: +22 221 0713.
E-mail address: [email protected] (G. Sakurai).
pffiffiffi quasi-cubic because the distance a= 2 between R ions in the tetragonal plane is almost the same as the distance c along the c-axis [2], where a and c are lattice constants for the base-centered tetragonal structure. The Fermi surfaces of RB2 C2 compounds involve an ellipsoidal piece around the Z point in the tetragonal Brillouin zone (BZ), according to the de Haas–van Alphen (dHvA) experiment on the isostructural compound LaB2 C2 [3,4]. The RKKY interaction should largely be determined by hybridization in the case of R ¼ Ce. We consider the following form of hybridization: rffiffiffiffiffiffi i 4p X h ^ ikRi f y ðiÞcks þ h:c: ; V k Y n3m ðkÞe (1) ms N ikms
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.205
ARTICLE IN PRESS G. Sakurai, Y. Kuramoto / Physica B 359– 361 (2005) 720–722 Γ4u2, Γ5u
Γ2u
0.01
0.00
~ Du(q)
where V k is the strength of hybridization, N is the number of lattice, and Y lm ðOÞ is the spherical harmonics. A 4f electron is created by f yms ðiÞ at site Ri with the z-component m of the orbital angular momentum l ¼ 3 and spin s: A conduction electron with its wave number k and spin s is annihilated by cks : We simplify the Fermi surface as a spherical one in the simple cubic pffiffiffi BZ (SCBZ) with the lattice constant aq a= 2: Then the result can be mapped into the the quasi-cubic BZ (QBZ) by taking a tetragonal lattice with lattice constant a and c ¼ aq : In the previous paper [5], we derived the RKKY interaction for the nearest and the next-nearest neighbors for the Anderson lattice model with orbital degeneracy. In the incommensurate structure, however, interactions at large distance may also play an important role. In this paper, we compute the intersite interaction by taking account of 643 lattice spacings, and make a Fourier transform to the reciprocal space. The dimensionless Fermi wave number kF aq is taken as ð3p2 Þ1=3
3:09; with one conduction electron per Ce ion. The multipolar interactions are projected into the lowlying CEF eigenstates for which we take G8 quartet of j ¼ 52: This quartet includes the m ¼ 12 doublet, which is likely to be the ground CEF state as suggested from the magnetic susceptibility. We take X a ðiÞ as a multipole operator in lowlying crystal electric field (CEF) states at site Ri : In this paper, only the f 0 intermediate state at each site is considered. The multipolar coupling between X a ðiÞ and X b ðjÞ is given by Dab ðRi Rj Þ: We first calculate DðRi Rj Þ in the real space and use ~ the fast Fourier transform to derive DðqÞ with ~ wave number q: The matrix DðqÞ is diagonalized for each q: With the time-reversal symmetry, it is ~ possible to decompose DðqÞ into dipole and ~ u ðqÞ and the quadrupole part octupole parts D ~ g ðqÞ for general q: D ~ u ðqÞ in the SCBZ. Fig. 1 shows eigenvalues of D The minimum gives the most stable propagation vector and symmetry. Although not shown in Fig. 1, we have checked that quadrupoles are less likely to order than magnetic multipoles under the present set of parameters. At high symmetry points such as G and R, multipolar couplings are characterized by irreducible representations
721
–0.01
Γ4u1 –0.02
M
X
Γ
R
X
Γ
M
R
Wavenumber q Fig. 1. Wave number dependence of the dimensionless ~ u ðqÞ: magnetic multipolar couplings, D
G2u ; G4u ; and G5u : The minimum occurs near the G point at the wave number ðd00 ; d00 ; d00 Þ; with d00 ’ 0:03 in the SCBZ. The symmetry is close to G4u1 : Since we have 2kF ’ 0:98ð2p=aq Þ; the Kohn anomaly can arise near G point in the reduced zone scheme. It is well known that the Lindhard function does not show a peak at 2kF ; but only its derivative diverges. With orbital angular momenta, generalized polarization functions are given by X
dk k0 ;q
kk0
/
X
f ðk Þ f ðk0 Þ ^ l m ðk^ 0 Þ Y l 1 m1 ðkÞY 2 2 k k0
ð2Þ
l
i l 1 þl 2 þl 3 Y l 3 m3 ð^qÞwl 31 l 2 ðjqj=kF Þ
l 3 m3
Z
dOY nl 3 m3 ðOÞY l 1 m1 ðOÞY l 2 m2 ðOÞ;
where f ðÞ is the Fermi distribution function, k is the energy dispersion of free electron, and l 1 ; l 2 are even integers with 0pl 1 ; l 2 p6: Some of these functions have a hump or a dip at jqj ’ 2kF as shown in Fig. 2. They accumulate to give a sharp extremum near jqj ¼ 2kF : A rough measure of the wave number for the Kohn anomaly is given by dF 1 2kF =ð2p=aq Þ in units of the cubic reciprocal lattice. Actually, dF differs to some extent from d00 obtained numeril cally. This is because extrema in wl 31 l 2 ðxÞ are off from x ¼ 2 in general. Taking the information from the dHvA experiment of LaB2 C2 [4], we now take 2kF ¼ 0:86ð2p=aq Þ; which gives dF ¼ 0:14:
ARTICLE IN PRESS G. Sakurai, Y. Kuramoto / Physica B 359– 361 (2005) 720–722
722
relevant CEF states, we have checked that the anomaly is not prominent. In conclusion, our simplified spherical model gives an incommensurate magnetic order as the most stable multipole order, provided the quasi-quartet is the CEF ground state. This result should be relevant in understanding the incommensurate magnetic structures observed in RB2 C2 : The authors are grateful to Dr. H. Kusunose, Dr. K. Kubo, T. Onimaru and H.N. Kono for helpful discussions.
0 6
χ24
χ22 0
4
χ22 χ02
2
0.5
2
χ22
–4χ
l3 (χ)/π l1l2
1.0
χ00
0.0
χ26 0
1
χ
2
8 3
Fig. 2. Radial parts of generalized polarization functions, l wl 31 l 2 ðxÞ with x ¼ jqj=kF :
Numerically, the anomalies arise at ðd00 ; d00 ; 0Þ and at ðd00 ; d00 ; d00 Þ with d00 ’ 0:17 in the QBZ, which has a good correspondence with experimental value of d ¼ 0:16: However, our isotropic model is too crude to explain the experimental anisotropy with d ¼ 0:10: We note that a similar anomaly near the G point also occurs with m ¼ 12 doublet as the low-lying CEF eigenstates. With the G7 doublet as the
References [1] K. Ohoyama, K. Kaneko, T. Onimaru, A. Tobo, K. Ishimoto, H. Onodera, Y. Yamaguchi, J. Phys. Soc. Japan 72 (2003) 3303. [2] K. Ohoyama, K. Kaneko, K. Indoh, H. Yamauchi, A. Tobo, H. Onodera, Y. Yamaguchi, J. Phys. Soc. Japan 70 (2001) 3291. [3] H. Harima, Newsletter of Scientific Research on Priority Areas (B) ‘‘Orbital Ordering and Fluctuations’’, vol. 2, No. 1, 2001, p. 7 (in Japanese). [4] R. Watanuki, T. Terashima, K. Suzuki, J. Phys. Soc. Japan 71 (2002) 693. [5] G. Sakurai, Y. Kuramoto, J. Phys. Soc. Japan 73 (2004) 225.
Physica B 359–361 (2005) 720–722 www.elsevier.com/locate/physb
Multipolar interactions in the Anderson lattice Gen’ya Sakurai, Yoshio Kuramoto Department of Physics, Tohoku University, Aoba Aza Aramaki Sendai, Miyagi 980 8578, Japan
Abstract The wave number dependence of RKKY multipolar interactions is derived for the Anderson-type model with orbital degeneracy in the simple cubic lattice. With the spherical Fermi surface and one conduction electron per cell, the Kohn anomaly arises in the magnetic interaction near the G point of the Brillouin zone. This result may be relevant to the incommensurate magnetic structure observed in the quasi-cubic compound CeB2 C2 : r 2005 Elsevier B.V. All rights reserved. PACS: 71.20.Eh; 71.27.+a; 75.30.Et Keywords: Incommensurate magnetic structure; Kohn anomaly; RKKY interaction
The RB2 C2 compounds (R ¼ rare earths) crystallize in the tetragonal LaB2 C2 -type structure(P4/ mbm), and some of them exhibit incommensurate magnetic structures [1]. For example, CeB2 C2 undergoes a magnetic ordering at T N ¼ 7:3 K, and a successive transition at T t ¼ 6:5 K. Below T t ; an incommensurate magnetic structure is found. The propagation vector is q ¼ ðd; d; d0 Þ; with d ¼ 0:16 and d0 ¼ 0:10: Other RB2 C2 systems often exhibit similar incommensurate magnetic structures [1]. Thus, propagation vectors should be determined mostly by conduction bands near Fermi surfaces. CeB2 C2 can be considered as Corresponding author. Tel.: +22 221 0713.
E-mail address: [email protected] (G. Sakurai).
pffiffiffi quasi-cubic because the distance a= 2 between R ions in the tetragonal plane is almost the same as the distance c along the c-axis [2], where a and c are lattice constants for the base-centered tetragonal structure. The Fermi surfaces of RB2 C2 compounds involve an ellipsoidal piece around the Z point in the tetragonal Brillouin zone (BZ), according to the de Haas–van Alphen (dHvA) experiment on the isostructural compound LaB2 C2 [3,4]. The RKKY interaction should largely be determined by hybridization in the case of R ¼ Ce. We consider the following form of hybridization: rffiffiffiffiffiffi i 4p X h ^ ikRi f y ðiÞcks þ h:c: ; V k Y n3m ðkÞe (1) ms N ikms
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.205
ARTICLE IN PRESS G. Sakurai, Y. Kuramoto / Physica B 359– 361 (2005) 720–722 Γ4u2, Γ5u
Γ2u
0.01
0.00
~ Du(q)
where V k is the strength of hybridization, N is the number of lattice, and Y lm ðOÞ is the spherical harmonics. A 4f electron is created by f yms ðiÞ at site Ri with the z-component m of the orbital angular momentum l ¼ 3 and spin s: A conduction electron with its wave number k and spin s is annihilated by cks : We simplify the Fermi surface as a spherical one in the simple cubic pffiffiffi BZ (SCBZ) with the lattice constant aq a= 2: Then the result can be mapped into the the quasi-cubic BZ (QBZ) by taking a tetragonal lattice with lattice constant a and c ¼ aq : In the previous paper [5], we derived the RKKY interaction for the nearest and the next-nearest neighbors for the Anderson lattice model with orbital degeneracy. In the incommensurate structure, however, interactions at large distance may also play an important role. In this paper, we compute the intersite interaction by taking account of 643 lattice spacings, and make a Fourier transform to the reciprocal space. The dimensionless Fermi wave number kF aq is taken as ð3p2 Þ1=3
3:09; with one conduction electron per Ce ion. The multipolar interactions are projected into the lowlying CEF eigenstates for which we take G8 quartet of j ¼ 52: This quartet includes the m ¼ 12 doublet, which is likely to be the ground CEF state as suggested from the magnetic susceptibility. We take X a ðiÞ as a multipole operator in lowlying crystal electric field (CEF) states at site Ri : In this paper, only the f 0 intermediate state at each site is considered. The multipolar coupling between X a ðiÞ and X b ðjÞ is given by Dab ðRi Rj Þ: We first calculate DðRi Rj Þ in the real space and use ~ the fast Fourier transform to derive DðqÞ with ~ wave number q: The matrix DðqÞ is diagonalized for each q: With the time-reversal symmetry, it is ~ possible to decompose DðqÞ into dipole and ~ u ðqÞ and the quadrupole part octupole parts D ~ g ðqÞ for general q: D ~ u ðqÞ in the SCBZ. Fig. 1 shows eigenvalues of D The minimum gives the most stable propagation vector and symmetry. Although not shown in Fig. 1, we have checked that quadrupoles are less likely to order than magnetic multipoles under the present set of parameters. At high symmetry points such as G and R, multipolar couplings are characterized by irreducible representations
721
–0.01
Γ4u1 –0.02
M
X
Γ
R
X
Γ
M
R
Wavenumber q Fig. 1. Wave number dependence of the dimensionless ~ u ðqÞ: magnetic multipolar couplings, D
G2u ; G4u ; and G5u : The minimum occurs near the G point at the wave number ðd00 ; d00 ; d00 Þ; with d00 ’ 0:03 in the SCBZ. The symmetry is close to G4u1 : Since we have 2kF ’ 0:98ð2p=aq Þ; the Kohn anomaly can arise near G point in the reduced zone scheme. It is well known that the Lindhard function does not show a peak at 2kF ; but only its derivative diverges. With orbital angular momenta, generalized polarization functions are given by X
dk k0 ;q
kk0
/
X
f ðk Þ f ðk0 Þ ^ l m ðk^ 0 Þ Y l 1 m1 ðkÞY 2 2 k k0
ð2Þ
l
i l 1 þl 2 þl 3 Y l 3 m3 ð^qÞwl 31 l 2 ðjqj=kF Þ
l 3 m3
Z
dOY nl 3 m3 ðOÞY l 1 m1 ðOÞY l 2 m2 ðOÞ;
where f ðÞ is the Fermi distribution function, k is the energy dispersion of free electron, and l 1 ; l 2 are even integers with 0pl 1 ; l 2 p6: Some of these functions have a hump or a dip at jqj ’ 2kF as shown in Fig. 2. They accumulate to give a sharp extremum near jqj ¼ 2kF : A rough measure of the wave number for the Kohn anomaly is given by dF 1 2kF =ð2p=aq Þ in units of the cubic reciprocal lattice. Actually, dF differs to some extent from d00 obtained numeril cally. This is because extrema in wl 31 l 2 ðxÞ are off from x ¼ 2 in general. Taking the information from the dHvA experiment of LaB2 C2 [4], we now take 2kF ¼ 0:86ð2p=aq Þ; which gives dF ¼ 0:14:
ARTICLE IN PRESS G. Sakurai, Y. Kuramoto / Physica B 359– 361 (2005) 720–722
722
relevant CEF states, we have checked that the anomaly is not prominent. In conclusion, our simplified spherical model gives an incommensurate magnetic order as the most stable multipole order, provided the quasi-quartet is the CEF ground state. This result should be relevant in understanding the incommensurate magnetic structures observed in RB2 C2 : The authors are grateful to Dr. H. Kusunose, Dr. K. Kubo, T. Onimaru and H.N. Kono for helpful discussions.
0 6
χ24
χ22 0
4
χ22 χ02
2
0.5
2
χ22
–4χ
l3 (χ)/π l1l2
1.0
χ00
0.0
χ26 0
1
χ
2
8 3
Fig. 2. Radial parts of generalized polarization functions, l wl 31 l 2 ðxÞ with x ¼ jqj=kF :
Numerically, the anomalies arise at ðd00 ; d00 ; 0Þ and at ðd00 ; d00 ; d00 Þ with d00 ’ 0:17 in the QBZ, which has a good correspondence with experimental value of d ¼ 0:16: However, our isotropic model is too crude to explain the experimental anisotropy with d ¼ 0:10: We note that a similar anomaly near the G point also occurs with m ¼ 12 doublet as the low-lying CEF eigenstates. With the G7 doublet as the
References [1] K. Ohoyama, K. Kaneko, T. Onimaru, A. Tobo, K. Ishimoto, H. Onodera, Y. Yamaguchi, J. Phys. Soc. Japan 72 (2003) 3303. [2] K. Ohoyama, K. Kaneko, K. Indoh, H. Yamauchi, A. Tobo, H. Onodera, Y. Yamaguchi, J. Phys. Soc. Japan 70 (2001) 3291. [3] H. Harima, Newsletter of Scientific Research on Priority Areas (B) ‘‘Orbital Ordering and Fluctuations’’, vol. 2, No. 1, 2001, p. 7 (in Japanese). [4] R. Watanuki, T. Terashima, K. Suzuki, J. Phys. Soc. Japan 71 (2002) 693. [5] G. Sakurai, Y. Kuramoto, J. Phys. Soc. Japan 73 (2004) 225.