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Strong anharmonicity of Pr thermal vibration in heavy fermion superconductor PrOs4Sb12
ARTICLE IN PRESS
Physica B 403 (2008) 874–876 www.elsevier.com/locate/physb
Strong anharmonicity of Pr thermal vibration in heavy fermion superconductor PrOs4 Sb12 K. Kanekoa,, N. Metokia,b, H. Kimurac, Y. Nodac, T.D. Matsudaa, M. Kohgid a
Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan b Department of Physics, Tohoku University, Sendai, 980-8578, Japan c Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan d Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
Abstract In order to clarify the detailed structure of PrOs4 Sb12 , single crystal neutron diffraction experiments were carried out and obtained data were analyzed by means of the maximum entropy method (MEM). A space- and time-averaged nuclear density distribution reflecting thermal vibration can be obtained through the MEM analysis. The result unveils a widely spread Pr distribution in the Sb cage at room temperature. The Pr density has a almost flat distribution in the real space centered at the Sb cage, which reaches roughly 0.58 A˚ in the full width at half maximum (FWHM). These features are far from the simple harmonic model, namely, the present result suggests the strong anharmonicity of the Pr potential in the Sb cage. r 2007 Elsevier B.V. All rights reserved. PACS: 61.12.Ld; 63.20.Ry; 74.70.Tx Keywords: PrOs4 Sb12 ; Heavy-fermion superconductivity; Rattling; MEM analysis
Recent discovery on the heavy-fermion superconductivity in PrOs4 Sb12 [1] has attracted considerable interest due to its unconventional superconducting properties such as broken time reversal symmetry in the superconducting state [2]. The nonmagnetic G1 singlet ground state and the existence of a field-induced antiferroquadrupolar (AFQ) order phase suggest an important role of quadrupolar interaction in forming Cooper pairs [3–8]. In addition, the importance of ‘rattling’, large thermal vibration of guest atom in the over-sized host cage, has suggested by recent studies [9]. The strong anharmonicity in the Pr potential and/or the existence of off-center potential minima are discussed in relation with physical properties including an emergence of heavy-fermion superconductivity. Neutron scattering is a powerful tool for the structural study for a disordered system and thermal displacement [10,11]. Recent progress in the structural analysis by using Corresponding author. Tel.: +81 29 282 6830; fax: +81 29 282 5939.
E-mail address: [email protected] (K. Kaneko). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.10.233
a maximum entropy method (MEM) provides an ability to sense the anharmonicity and/or structural disorder without any specific model to represent them. In order to elucidate the detailed structure including thermal motion of PrOs4 Sb12 , single crystal neutron diffraction experiments in combination with the MEM analysis were carried out. A single crystal of PrOs4 Sb12 was grown by the Sb selfflux method. Details of sample preparation is published in Ref. [10]. Single crystal neutron diffraction experiment was carried out on the four circle off-center type diffractometer, FONDER, installed at the research reactor JRR-3 of Japan Atomic Energy Agency. Neutrons with a wavelength of 1.243 A˚ and the maximum scattering angle of 2y156 allow us to measure up to sin y=l0:78. Absorption correction and least-square structural analysis were carried out using the softwares DABEX and RADIEL [12], respectively. The software PRIMA [13] was used for MEM analysis. Three dimensional drawing to display obtained nuclear distributions was made using VESTA [14].
ARTICLE IN PRESS K. Kaneko et al. / Physica B 403 (2008) 874–876
Os
Sb
Pr
40 1.0 30
PrOs4Sb12
1.0
R.T. 0.5 Sb 0
20
-0.4
0
0.4 0.5
10
0
0 -0.4
-0.2
0
0.2
0.4
(x 0 0) (Å) Fig. 2. Cross-sectional drawings of nuclear density distribution of Pr and Sb along the [1 0 0] direction at room temperature. Inset shows those in the same vertical scale.
to that of Sb bSb ¼ 5:57 fm, the height of the density is expected to have the same order if both have the similar distribution. Namely, the nuclear density of Pr widely spreads in the three dimensional space reflecting strong thermal vibration with large amplitude. The amplitude of Pr reaches 0.58 A˚ in the full width at half maximum along the [100] direction which is 3 times larger than that of Sb of 0.16 A˚. Besides, the profile of Pr has a characteristic trapezoidal shape as clearly seen in Fig. 2. In contrast, the distribution of Sb is well represented by Gaussian expected for the harmonic oscillator. The almost flat distribution of Pr in the three-dimensional space is far from that expected for harmonic oscillator, in other words, Pr is under a strong anharmonic potential at room temperature. In order to clarify the relation of this strong anharmonicity with the low temperature physical property, further experiments on temperature dependence is now in progress. We thank Dr. H. Sugawara for giving the useful information on sample preparation. This work was supported by Grants-in-Aid for Scientific Research, Young Scientist (B) (no. 16740212) and in Priority Area ‘‘Skutterudite’’ (nos. 18027013 and 18027015) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Neutron scattering experiments has been carried out under NSPAC 5814 and 6812. References
Fig. 1. Nuclear density distribution of PrOs4 Sb12 at room temperature. Isosurfaces of nuclear density are set at 5 fm=A˚ 3 . Icosahedron cages consisting of Sb are shown for clarity.
dSb (103 fm / Å3)
Pr dPr (fm / Å3)
The total number of 212 independent reflections out of 305 measured data for room temperature were used in the structural analysis. The experimentally determined lattice parameter of 9.303(7) A˚ at room temperature agrees with the reported ones [10,15]. The least-square analysis was ¯ carried out by assuming the reported structure with Im3, where Pr locates at the center of the Sb cage. The anisotropic thermal displacement parameters were applied for Os and Sb, whereas Pr was dealt with isotropic displacement parameter due to the high symmetry of its position. As a result, the number of refined parameters is 11. The final reliable factors of R ¼ 2:06%, wR ¼ 2:06% indicate the goodness of the present analysis. The obtained structural parameters of ySb ¼ 0:3403ð1Þ, zSb ¼ 0:1561ð1Þ and large thermal displacement of Pr U Pr ¼ 0:0431ð36Þ A˚ 2 are consistent with reported ones as well as the lattice constant [10,15]. As a second step, the MEM analysis was carried out based on the result of the least-square analysis. The analysis with 256 256 256 pixels which roughly corre˚ sponds to 0:04 A=pixel gives the better reliable factor of R ¼ 2:05%, wR ¼ 1:52%. Fig. 1 shows the obtained nuclear scattering density distribution (hereafter, nuclear density distribution) at room temperature focused on the Sb icosahedron cage. The isosurface level of nuclear density is set at 5 fm=A˚ 3 . The obtained result well reproduces the structure determined by the least square analysis. Note that the widely spread nuclear density distribution of Pr at the center of the cage is clearly revealed by the present analysis. The cross-sectional drawings of the nuclear density of Pr and Sb along the [1 0 0] direction are shown in Fig. 2. The inset is the result plotted on the same vertical scale. It is clearly seen that the maximum density of Pr at room temperature is quite low, roughly 1/40, as compared to that of Sb. Since the coherent scattering length of Pr, bPr ¼ 4:58 fm is almost comparable
875
[1] [2] [3] [4]
M.B. Maple, et al., J. Phys. Soc. Japan 71 (Suppl.) (2002) 23. Y. Aoki, et al., Phys. Rev. Lett. 91 (2003) 067003. Y. Aoki, et al., J. Phys. Soc. Japan 71 (2002) 2098. M. Kohgi, et al., J. Phys. Soc. Japan 72 (2003) 1002.
ARTICLE IN PRESS 876 [5] [6] [7] [8] [9] [10] [11]
K. Kaneko et al. / Physica B 403 (2008) 874–876 R. Shiina, J. Phys. Soc. Japan 73 (2004) 2257. K. Kuwahara, et al., Phys. Rev. Lett. 95 (2005) 107003. T. Tayama, et al., J. Phys. Soc. Japan 75 (2006) 043707. K. Kaneko, et al., Phys. Rev. B 75 (2007) 094408. T. Goto, et al., Phys. Rev. B 69 (2004) 180511(R). K. Kaneko, et al., J. Phys. Soc. Japan 75 (2006) 034701. M. Takata, et al., Acta Cryst. A 50 (1994) 330.
[12] P. Coppens, et al., Acta Crystallogr. C 44 (1988) 775. [13] F. Izumi, R. A. Dilanian, Recent Research Developments in Physics, Vol. 3, Part II. Transworld Research Network, Trivandrum, 2002, pp. 699–726. [14] K. Momma, F. Izumi, Comm. Crystallogr. Comput. IUCr Newslett. 7 (2006) 106. [15] H. Sugawara, et al., Phys. Rev. B 66 (2002) 220504(R).
Physica B 403 (2008) 874–876 www.elsevier.com/locate/physb
Strong anharmonicity of Pr thermal vibration in heavy fermion superconductor PrOs4 Sb12 K. Kanekoa,, N. Metokia,b, H. Kimurac, Y. Nodac, T.D. Matsudaa, M. Kohgid a
Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan b Department of Physics, Tohoku University, Sendai, 980-8578, Japan c Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan d Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
Abstract In order to clarify the detailed structure of PrOs4 Sb12 , single crystal neutron diffraction experiments were carried out and obtained data were analyzed by means of the maximum entropy method (MEM). A space- and time-averaged nuclear density distribution reflecting thermal vibration can be obtained through the MEM analysis. The result unveils a widely spread Pr distribution in the Sb cage at room temperature. The Pr density has a almost flat distribution in the real space centered at the Sb cage, which reaches roughly 0.58 A˚ in the full width at half maximum (FWHM). These features are far from the simple harmonic model, namely, the present result suggests the strong anharmonicity of the Pr potential in the Sb cage. r 2007 Elsevier B.V. All rights reserved. PACS: 61.12.Ld; 63.20.Ry; 74.70.Tx Keywords: PrOs4 Sb12 ; Heavy-fermion superconductivity; Rattling; MEM analysis
Recent discovery on the heavy-fermion superconductivity in PrOs4 Sb12 [1] has attracted considerable interest due to its unconventional superconducting properties such as broken time reversal symmetry in the superconducting state [2]. The nonmagnetic G1 singlet ground state and the existence of a field-induced antiferroquadrupolar (AFQ) order phase suggest an important role of quadrupolar interaction in forming Cooper pairs [3–8]. In addition, the importance of ‘rattling’, large thermal vibration of guest atom in the over-sized host cage, has suggested by recent studies [9]. The strong anharmonicity in the Pr potential and/or the existence of off-center potential minima are discussed in relation with physical properties including an emergence of heavy-fermion superconductivity. Neutron scattering is a powerful tool for the structural study for a disordered system and thermal displacement [10,11]. Recent progress in the structural analysis by using Corresponding author. Tel.: +81 29 282 6830; fax: +81 29 282 5939.
E-mail address: [email protected] (K. Kaneko). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.10.233
a maximum entropy method (MEM) provides an ability to sense the anharmonicity and/or structural disorder without any specific model to represent them. In order to elucidate the detailed structure including thermal motion of PrOs4 Sb12 , single crystal neutron diffraction experiments in combination with the MEM analysis were carried out. A single crystal of PrOs4 Sb12 was grown by the Sb selfflux method. Details of sample preparation is published in Ref. [10]. Single crystal neutron diffraction experiment was carried out on the four circle off-center type diffractometer, FONDER, installed at the research reactor JRR-3 of Japan Atomic Energy Agency. Neutrons with a wavelength of 1.243 A˚ and the maximum scattering angle of 2y156 allow us to measure up to sin y=l0:78. Absorption correction and least-square structural analysis were carried out using the softwares DABEX and RADIEL [12], respectively. The software PRIMA [13] was used for MEM analysis. Three dimensional drawing to display obtained nuclear distributions was made using VESTA [14].
ARTICLE IN PRESS K. Kaneko et al. / Physica B 403 (2008) 874–876
Os
Sb
Pr
40 1.0 30
PrOs4Sb12
1.0
R.T. 0.5 Sb 0
20
-0.4
0
0.4 0.5
10
0
0 -0.4
-0.2
0
0.2
0.4
(x 0 0) (Å) Fig. 2. Cross-sectional drawings of nuclear density distribution of Pr and Sb along the [1 0 0] direction at room temperature. Inset shows those in the same vertical scale.
to that of Sb bSb ¼ 5:57 fm, the height of the density is expected to have the same order if both have the similar distribution. Namely, the nuclear density of Pr widely spreads in the three dimensional space reflecting strong thermal vibration with large amplitude. The amplitude of Pr reaches 0.58 A˚ in the full width at half maximum along the [100] direction which is 3 times larger than that of Sb of 0.16 A˚. Besides, the profile of Pr has a characteristic trapezoidal shape as clearly seen in Fig. 2. In contrast, the distribution of Sb is well represented by Gaussian expected for the harmonic oscillator. The almost flat distribution of Pr in the three-dimensional space is far from that expected for harmonic oscillator, in other words, Pr is under a strong anharmonic potential at room temperature. In order to clarify the relation of this strong anharmonicity with the low temperature physical property, further experiments on temperature dependence is now in progress. We thank Dr. H. Sugawara for giving the useful information on sample preparation. This work was supported by Grants-in-Aid for Scientific Research, Young Scientist (B) (no. 16740212) and in Priority Area ‘‘Skutterudite’’ (nos. 18027013 and 18027015) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Neutron scattering experiments has been carried out under NSPAC 5814 and 6812. References
Fig. 1. Nuclear density distribution of PrOs4 Sb12 at room temperature. Isosurfaces of nuclear density are set at 5 fm=A˚ 3 . Icosahedron cages consisting of Sb are shown for clarity.
dSb (103 fm / Å3)
Pr dPr (fm / Å3)
The total number of 212 independent reflections out of 305 measured data for room temperature were used in the structural analysis. The experimentally determined lattice parameter of 9.303(7) A˚ at room temperature agrees with the reported ones [10,15]. The least-square analysis was ¯ carried out by assuming the reported structure with Im3, where Pr locates at the center of the Sb cage. The anisotropic thermal displacement parameters were applied for Os and Sb, whereas Pr was dealt with isotropic displacement parameter due to the high symmetry of its position. As a result, the number of refined parameters is 11. The final reliable factors of R ¼ 2:06%, wR ¼ 2:06% indicate the goodness of the present analysis. The obtained structural parameters of ySb ¼ 0:3403ð1Þ, zSb ¼ 0:1561ð1Þ and large thermal displacement of Pr U Pr ¼ 0:0431ð36Þ A˚ 2 are consistent with reported ones as well as the lattice constant [10,15]. As a second step, the MEM analysis was carried out based on the result of the least-square analysis. The analysis with 256 256 256 pixels which roughly corre˚ sponds to 0:04 A=pixel gives the better reliable factor of R ¼ 2:05%, wR ¼ 1:52%. Fig. 1 shows the obtained nuclear scattering density distribution (hereafter, nuclear density distribution) at room temperature focused on the Sb icosahedron cage. The isosurface level of nuclear density is set at 5 fm=A˚ 3 . The obtained result well reproduces the structure determined by the least square analysis. Note that the widely spread nuclear density distribution of Pr at the center of the cage is clearly revealed by the present analysis. The cross-sectional drawings of the nuclear density of Pr and Sb along the [1 0 0] direction are shown in Fig. 2. The inset is the result plotted on the same vertical scale. It is clearly seen that the maximum density of Pr at room temperature is quite low, roughly 1/40, as compared to that of Sb. Since the coherent scattering length of Pr, bPr ¼ 4:58 fm is almost comparable
875
[1] [2] [3] [4]
M.B. Maple, et al., J. Phys. Soc. Japan 71 (Suppl.) (2002) 23. Y. Aoki, et al., Phys. Rev. Lett. 91 (2003) 067003. Y. Aoki, et al., J. Phys. Soc. Japan 71 (2002) 2098. M. Kohgi, et al., J. Phys. Soc. Japan 72 (2003) 1002.
ARTICLE IN PRESS 876 [5] [6] [7] [8] [9] [10] [11]
K. Kaneko et al. / Physica B 403 (2008) 874–876 R. Shiina, J. Phys. Soc. Japan 73 (2004) 2257. K. Kuwahara, et al., Phys. Rev. Lett. 95 (2005) 107003. T. Tayama, et al., J. Phys. Soc. Japan 75 (2006) 043707. K. Kaneko, et al., Phys. Rev. B 75 (2007) 094408. T. Goto, et al., Phys. Rev. B 69 (2004) 180511(R). K. Kaneko, et al., J. Phys. Soc. Japan 75 (2006) 034701. M. Takata, et al., Acta Cryst. A 50 (1994) 330.
[12] P. Coppens, et al., Acta Crystallogr. C 44 (1988) 775. [13] F. Izumi, R. A. Dilanian, Recent Research Developments in Physics, Vol. 3, Part II. Transworld Research Network, Trivandrum, 2002, pp. 699–726. [14] K. Momma, F. Izumi, Comm. Crystallogr. Comput. IUCr Newslett. 7 (2006) 106. [15] H. Sugawara, et al., Phys. Rev. B 66 (2002) 220504(R).