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High-field ultrasonic study on PrPtBi with non-Kramers doublet ground state
ARTICLE IN PRESS
Physica B 346–347 (2004) 416–420
High-field ultrasonic study on PrPtBi with non-Kramers doublet ground state O. Suzuki*, H. Suzuki, T. Takamasu, H. Kitazawa, G. Kido Nanomaterials Laboratory, National Institute for Materials Science, 3-13 Sakura, Tsukuba 305-0003, Japan
Abstract We have measured the elastic constants of PrPtBi single crystal with the cubic MgAgAs-type structure as functions of temperatures and magnetic fields. A remarkable softening of 10% in ðc11 c12 Þ=2 was observed below 20 K with decreasing temperatures, while c44 shows a monotonous increase. These behaviors are attributed to the crystal-field ground state non-Kramers doublet G3 : The enormous softening in ðc11 c12 Þ=2 prefers a ferro-quadrupolar order of O02 or O22 ; which brings about a structural phase transition of cubic–tetragonal or cubic-orthorhombic. The magnetic field dependences of c11 and ðc11 c12 Þ=2 up to 25 T were explained qualitatively in terms of a quadrupolar susceptibility assuming the crystal-field level proposed by inelastic neutron scattering. Above 1:8 K the field dependence shows no clear indication of phase transition, which mean paramagnetic phase is stable above 1:8 K up to 25 T: r 2004 Elsevier B.V. All rights reserved. PACS: 43.35.A,B,C; 63.20; 71.20.E; 62.65 Keywords: Elastic constant; Ultrasonic measurement; Non-Kramers doublet; Quadrupolar susceptibility; PrPtBi
1. Introduction In prospect of the exotic magnetism many experimental and theoretical studies are carried out on the systems with the orbital degrees of freedom. Much interest is paid to the compounds, which shows non-Kramers doublet G3 ground state with the quadrupolar degrees of freedom without spin degrees of freedom, for a review see Ref. [1]. For example, PrPb3 shows a quadrupolar order in the ground G3 doublet and forms an anomalous phase diagram, in which the quadrupolar order is stabilized by magnetic fields [2,3]. *Corresponding author. Tel.: +81-29-859-5018; fax: +8129-859-5010. E-mail address: [email protected] (O. Suzuki).
Kondo-like heavy-Fermi liquid behaviors were reported in PrInAg2 ; PrFe4 P12 and PrOs4 Sb12 in spite of non-magnetic ground states [4–6]. PrPtBi with the cubic MgAgAs-type structure has been found as a semiconductor with nonKramers doublet G3 ground state [7]. The specific heat of polycrystalline sample shows a sharp peak at 1:35 K accompanied with a long-range ordering. Magnetic susceptibility of PrPtBi follows Curie–Weiss law above 50 K and shows temperature-independent behavior below 50 K: This result indicates that the ground state has a non-magnetic character. In the cubic crystal-electric-field (CEF) Hund’s ground multiplet J ¼ 4 of Pr3þ ion splits into the singlet G1 ; doublet G3 ; two triplets G4 and G5 : The magnetic entropy estimated by specific heat measurements reaches R ln 2 below 10 K;
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.01.118
ARTICLE IN PRESS O. Suzuki et al. / Physica B 346–347 (2004) 416–420
where R is a gas constant. The low-temperature behavior of the magnetic susceptibility and the magnetic entropy indicate that the ground state is the non-Kramers doublet G3 : The possibility of ferromagnetic or antiferro-magnetic order at 1:35 K is excluded because the magnetic susceptibility shows no anomaly. The most characteristic feature in PrPtBi is the elastic property at low temperatures. The temperature dependence of the transverse elastic constant in polycrystalline sample shows an enormous softening of about 20% below 10 K and shows a temperature-independent behavior below around 1 K: From the results of the magnetic and the elastic and the thermal measurements, the anomalies at 1:35 K can be interpreted as a quadrupolar ordering. Until now, no experimental study on a PrPtBi single crystal has been reported because of the difficulty to prepare the single crystalline sample. Recently, we have succeeded in growing single crystals by the Bridgman method. By means of ultrasound technique, we measured the elastic constants of a PrPtBi single crystal in order to investigate the anisotropy of electronic properties originated from non-Kramers doublet G3 : Symmetric elastic constants can give rich information about CEF state and quadrupolar order parameter.
417
lar-strain and intersite quadrupolar interaction, respectively. gG is the coupling constant for the quadrupole-strain and g0G for the intersite quadrupole interaction. OG and Om l mean the equivalent operator of the quadrupolar- and multipolar moments, respectively. We used the following form of the temperature- and the magnetic-field dependence of elastic constants [8]: 0 QS cG ¼ c0G Ng2G wQS G =ð1 gG wG Þ:
ð3Þ
Here wQS G is the quadrupolar-strain susceptibility. c0G and N are the background part of elastic constants and the number of rare-earth ions in unit volume, respectively. We assumed that the background parts of elastic constants is proportional to temperature as c0G ¼ aT þ b; here a and b are constants. The elastic constant ðc11 c12 Þ=2 shows a divergent softening proportional to the reciprocal temperatures in the non-Kramers doublet G3 ground state, while c44 shows a Van-Vleck-type temperature-independent behavior at low temperature. The bulk modulus cB ¼ ðc11 þ 2c12 Þ=3 shows a monotonous increase in the localized-4f systems such as PrPtBi, while valence fluctuation compound such as SmB6 shows a softening [9].
3. Experimental details 2. Quadrupolar-strain susceptibility The temperature- and the magnetic-field dependence of elastic constants give information about CEF state regarding to the quadrupolar degrees of freedom. The ultrasound-induced strain interacts to 4f-electronic quadrupolar moments via the CEF potential. The quadrupolar total Hamiltonian is written as follows: H ¼ HCEF gJ mB JH gG OG eG g0G oOG > OG ; HCEF ¼ B4 ðO04 þ 5O44 Þ þ B6 ðO06 21O46 Þ:
ð1Þ ð2Þ
Here HCEF is the CEF Hamiltonian for the cubic crystals. The second term on the right-hand side in Eq. (1) is Zeeman term. The third and fourth term in the right-hand side of Eq. (1) mean quadrupo-
Single crystals of PrPtBi were prepared by the Bridgman method using Mo crucibles. The typical dimension of the samples are 2 mm 3 mm 3 mm with rectangular shape. LiNbO3 transducers were bonded on a pair of parallel faces of a sample by RTV silicone rubber (Shin-Etsu chemical, Co., Ltd.). Sound velocities were measured by our ultrasound apparatus based on the phase comparison method. Elastic constants c is estimated by the formula c ¼ rv2 ; here r is the mass density 11:56 g=cm3 of PrPtBi and v sound velocity. Ultrasonic measurements in high magnetic fields were performed using a water-cooled magnet at Tsukuba Magnet Laboratory, NIMS. A 4 He cryostat and a 3 He evaporation refrigerator were used for the low-temperature ultrasonic measurements down to 1:8 K and down to 0:6 K; respectively.
ARTICLE IN PRESS O. Suzuki et al. / Physica B 346–347 (2004) 416–420
418
4. Results and discussion Fig. 1 shows the temperature dependence of elastic constants of PrPtBi in zero magnetic field. The characteristic softening is observed in c11 and ðc11 c12 Þ=2: These modes are described by the quadrupolar susceptibility of the G3 quadrupole moments O02 or O22 : c11 is the mixed mode of G1 and G3 : The temperature dependence of the bulk modulus cB ¼ ðc11 þ 2c12 Þ=3 increase monotonously in higher temperature above 20 K; which indicates that localized 4f-electron is stable. The pure G3 mode ðc11 c12 Þ=2 shows a slow increase from the higher temperature side and begin to soften below about 20 K: The softening of ðc11 c12 Þ=2 becomes steep at around 4 K and continues down to 0:6 K: The magnitude of the softening is 2% in c11 and 10% in ðc11 c12 Þ=2: This quite
140 C11
Elastic constant (GPa)
130 75
CB = (C11+2C12)/3
70
PrPtBi
(C11-C12)/2
50
40
0
C44
20
40
60
80
100
Temperature (K) Fig. 1. The elastic constants in PrPtBi as functions of temperatures in zero magnetic field. Solid lines and broken lines for ðc11 c12 Þ=2 are calculation fit with the quadrupolarstrain interaction jgG3 j ¼ 35 K and background part of elastic constants, respectively. The bulk modulus cB ¼ ðc11 þ 2c12 Þ=3 is calculated from the results of c11 and ðc11 c12 Þ=2:
large softening in ðc11 c12 Þ=2 clearly manifest the G3 non-Kramers doublet ground state, which brings the Curie-type softening originated from the quadrupolar degeneracy of O02 and O22 : The observed magnitude of softening in ðc11 c12 Þ=2 in the present experiments is smaller than the softening in the transverse elastic constant in polycrystalline PrPtBi reported by Kasaya et al. In our polycrystalline measurements, we observed similar softening of 25% in transverse mode and upturn at around 1 K (not shown). In the single crystal case, the strength of the softening in ðc11 c12 Þ=2 strongly depends on the sample quality. The higher quality sample, which shows clear VanVleck-type temperature-independent behavior in the magnetic susceptibility (not shown), softening in ðc11 c12 Þ=2 becomes larger. Other elastic modes in single crystals show quite small dependence on the sample quality. In the present experiments we used the single-crystalline sample with the largest softening of ðc11 c12 Þ=2 among our samples. The detailed sample preparation and sample dependence in other measurements will be discussed elsewhere [10]. In the present experiment remarkable elastic softening in ðc11 c12 Þ=2 is observed, however, the existence of the quadrupolar ordering temperature TQ is not clear. It is contrast to the clear anomalies in polycrystalline sample at TQ ¼ 1:35 K: The phase transition temperature is considered to be lower in our single crystalline sample than in polycrystalline sample. In order to determine the nature of quadrupolar interaction we calculated the quadrupolar susceptibility for ðc11 c12 Þ=2: The solid line for ðc11 c12 Þ=2 in Fig. 1 are calculation fit using Eq. (3). We assumed the CEF level scheme obtained by inelastic neutron scattering by Kasaya et al. [11]. The obtained quadrupolar-strain interaction is jgG3 j ¼ 35 K: The intersite quadrupolar interaction is too small to determine the magnitude. The large softening of 10% in ðc11 c12 Þ=2 in the present experiment prefer the ferro-quadrupolar order in the G3 ground state. The large softening in ðc11 c12 Þ=2 has been also observed in the cubic compound Ce3 Pd20 Ge6 which shows the cubic– tetragonal structural phase transition accompanied with ferro-quadrupolar order of O02 [12]. Recent X-ray powder diffraction of PrPtBi by
ARTICLE IN PRESS O. Suzuki et al. / Physica B 346–347 (2004) 416–420
2.5 PrPtBi C11 k//u//H//[001]
∆C/C0 (%)
2.0
2.5 PrPtBi (C11-C12)/2 k//H//[110] u//[110]
2.0
1.8 K
1.5
0
∆C/C (%)
Naher et al. [13] detected an abrupt increase of nuclear Bragg reflection below TQ ¼ 1:35 K: This result can be interpreted as an indication of the lattice distortion resulted from a ferro-quadrupolar order. From the results of our elastic data, the expected structural change is cubic-tetragonal for O02 order or cubic–orthorhombic for O22 : In order to observe macroscopic structural change, thermal expansion measurements are further needed. In order to investigate elastic property in magnetic fields we have measured field dependence of elastic constants. Figs. 2 and 3 displays relative changes of elastic constant c11 and ðc11 c12 Þ=2 in PrPtBi as functions of magnetic fields at various temperatures. The field dependence of c11 and ðc11 c12 Þ=2 at 1.8 and 4:2 K shows a monotonous increase up to 25 T: No clear indication of phase transition was observed. From the results of temperature dependence measurements PrPtBi is in paramagnetic phase above 1:8 K up to 25 T in fields along the ½0 0 1 and ½1 1 0 axes. To investigate the nature of quadrupolar phase in magnetic fields, it is needed to measure at lower temperature. The solid lines in Figs. 2 and 3 show the calculation fit for c11 and ðc11 c12 Þ=2: We used quadrupolar-strain interaction jgG3 j ¼ 35 K
419
1.0 0.5 4.2 K 0.0 0
5
10
15
20
25
H (T) Fig. 3. The relative changes of transverse elastic mode ðc11 c12 Þ=2 in PrPtBi as functions of magnetic fields along the [1–10] axis at various temperatures. The data is scaled by the value at H ¼ 0: Solid lines are calculation fit with the quadrupolar interaction of jgG3 j ¼ 35 K:
obtained from the temperature dependence of ðc11 c12 Þ=2 in Fig. 1. The magnetic field dependence in ðc11 c12 Þ=2 can be explained qualitatively by assuming the CEF level scheme obtained by inelastic neutron scattering [11]. The field dependence of c11 and ðc11 c12 Þ=2 at 1:8 K shows a tendency of saturation above 12 T; although the origin is not clear. Further investigation is needed to elucidate the quadrupolar effects in magnetic fields.
1.8 K
1.5
Summary 1.0 4.2 K 0.5 0.0 0
5
10
15
20
25
H (T) Fig. 2. The relative changes of longitudinal elastic mode c11 in PrPtBi as functions of magnetic fields along the ½0 0 1 axis at various temperatures. The data is scaled by the value at H ¼ 0: Solid lines are calculation fit with the quadrupolar interaction of jgG3 j ¼ 35 K:
The temperature- and the magnetic-field dependence of elastic constants of PrPtBi single crystal have been measured by ultrasound technique. In the absence of magnetic fields, temperature dependence of c11 and ðc11 c12 Þ=2 shows remarkable softening at low temperature, while c44 shows monotonous increase without softening. These softenings are attributed to the CEF ground state non-Kramers doublet G3 with the degeneracy of quadrupolar moments O02 and O22 : From the results of temperature dependence of ðc11 c12 Þ=2 quadrupolar-strain interaction was estimated to be
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O. Suzuki et al. / Physica B 346–347 (2004) 416–420
jgG3 j ¼ 35 K: The enormous softening of 10% in ðc11 c12 Þ=2 prefer a ferro-quadrupolar order of O02 or O22 ; which is accompanied with a structural phase transition of cubic–tetragonal or cubic– orthorhombic. The elastic constants as functions of magnetic fields show monotonous increase up to 25 T at 1.8 and 4:2 K: No clear indication of phase transition was observed up to 25 T: These results imply paramagnetic phase is stable up to 25 T above 1:8 K:
Acknowledgements The authors thank the staff members of the Tsukuba Magnet Laboratory, NIMS for operation of the magnets.
References [1] D.L. Cox, A. Zawadowski, Adv. Phys. 47 (1998) 599. [2] M. Niksch, W. Assmus, B. Luthi, . H.R. Ott, J.K. Kjems, Helv. Phys. Acta. 55 (1982) 688.
[3] T. Tayama, T. Sakakibara, K. Kitami, M. Yokoyama, % K. Tenya, H. Amitsuka, D. Aoki, Y. Onuki, Z. Kletowski, J. Phys. Soc. Jpn. 70 (2001) 248. [4] A. Yatskar, W.P. Beyermann, R. Movshovich, P.C. Canfield, Phys. Rev. Lett. 77 (1996) 3637. [5] E.D. Bauer, N.A. Frederick, P.-C. Ho, V.S. Zapf, M.B. Maple, Phys. Rev. B 65 (2002) 100506. [6] Y. Aoki, T. Namiki, T.D. Matsuda, K. Abe, H. Sugawara, H. Sato, Phys. Rev. B 65 (2002) 064446 and references there in. [7] H. Suzuki, M. Kasaya, T. Miyazaki, Y. Nemoto, T. Goto, J. Phys. Soc. Japan 66 (1997) 2566. [8] P. Thalmeier, B. Luthi, . in: K.A. Gschneider Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, Vol. 14, Chapter 96, Elsevier, Amsterdam, 1991. [9] S. Nakamura, T. Goto, M. Kasaya, S. Kunii, J. Phys. Soc. Jpn. 60 (1991) 4311. [10] H. Suzuki, unpublished. [11] M. Kasaya, H. Suzuki, D. Tazawa, M. Shirakawa, A. Sawada, T. Osakabe, Physica B 281–282 (2000) 579. . [12] O. Suzuki, T. Horino, Y. Nemoto, T. Goto, A.Donni, T. Komatsubara, M. Ishikawa, Physica B 261 (1999) 334; . T. Goto, T. Horino, Y. Nemoto, T. Yamaguchi, A.Donni, T. Komatsubara, Physica B 312 (2002) 492; Y. Nemoto, T. Yamaguchi, T. Horino, M. Akatsu, . T. Yanagisawa, T. Goto, O. Suzuki, A. Donni, T. Komatsubara, Phys. Rev. B 68 (2003) 184109. [13] S. Naher, H. Suzuki, M. Mizuno, Y. Xue, M. Kasaya, S. Kunii, T. Kitai, Physica B 329–333 (2003) 631.
Physica B 346–347 (2004) 416–420
High-field ultrasonic study on PrPtBi with non-Kramers doublet ground state O. Suzuki*, H. Suzuki, T. Takamasu, H. Kitazawa, G. Kido Nanomaterials Laboratory, National Institute for Materials Science, 3-13 Sakura, Tsukuba 305-0003, Japan
Abstract We have measured the elastic constants of PrPtBi single crystal with the cubic MgAgAs-type structure as functions of temperatures and magnetic fields. A remarkable softening of 10% in ðc11 c12 Þ=2 was observed below 20 K with decreasing temperatures, while c44 shows a monotonous increase. These behaviors are attributed to the crystal-field ground state non-Kramers doublet G3 : The enormous softening in ðc11 c12 Þ=2 prefers a ferro-quadrupolar order of O02 or O22 ; which brings about a structural phase transition of cubic–tetragonal or cubic-orthorhombic. The magnetic field dependences of c11 and ðc11 c12 Þ=2 up to 25 T were explained qualitatively in terms of a quadrupolar susceptibility assuming the crystal-field level proposed by inelastic neutron scattering. Above 1:8 K the field dependence shows no clear indication of phase transition, which mean paramagnetic phase is stable above 1:8 K up to 25 T: r 2004 Elsevier B.V. All rights reserved. PACS: 43.35.A,B,C; 63.20; 71.20.E; 62.65 Keywords: Elastic constant; Ultrasonic measurement; Non-Kramers doublet; Quadrupolar susceptibility; PrPtBi
1. Introduction In prospect of the exotic magnetism many experimental and theoretical studies are carried out on the systems with the orbital degrees of freedom. Much interest is paid to the compounds, which shows non-Kramers doublet G3 ground state with the quadrupolar degrees of freedom without spin degrees of freedom, for a review see Ref. [1]. For example, PrPb3 shows a quadrupolar order in the ground G3 doublet and forms an anomalous phase diagram, in which the quadrupolar order is stabilized by magnetic fields [2,3]. *Corresponding author. Tel.: +81-29-859-5018; fax: +8129-859-5010. E-mail address: [email protected] (O. Suzuki).
Kondo-like heavy-Fermi liquid behaviors were reported in PrInAg2 ; PrFe4 P12 and PrOs4 Sb12 in spite of non-magnetic ground states [4–6]. PrPtBi with the cubic MgAgAs-type structure has been found as a semiconductor with nonKramers doublet G3 ground state [7]. The specific heat of polycrystalline sample shows a sharp peak at 1:35 K accompanied with a long-range ordering. Magnetic susceptibility of PrPtBi follows Curie–Weiss law above 50 K and shows temperature-independent behavior below 50 K: This result indicates that the ground state has a non-magnetic character. In the cubic crystal-electric-field (CEF) Hund’s ground multiplet J ¼ 4 of Pr3þ ion splits into the singlet G1 ; doublet G3 ; two triplets G4 and G5 : The magnetic entropy estimated by specific heat measurements reaches R ln 2 below 10 K;
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.01.118
ARTICLE IN PRESS O. Suzuki et al. / Physica B 346–347 (2004) 416–420
where R is a gas constant. The low-temperature behavior of the magnetic susceptibility and the magnetic entropy indicate that the ground state is the non-Kramers doublet G3 : The possibility of ferromagnetic or antiferro-magnetic order at 1:35 K is excluded because the magnetic susceptibility shows no anomaly. The most characteristic feature in PrPtBi is the elastic property at low temperatures. The temperature dependence of the transverse elastic constant in polycrystalline sample shows an enormous softening of about 20% below 10 K and shows a temperature-independent behavior below around 1 K: From the results of the magnetic and the elastic and the thermal measurements, the anomalies at 1:35 K can be interpreted as a quadrupolar ordering. Until now, no experimental study on a PrPtBi single crystal has been reported because of the difficulty to prepare the single crystalline sample. Recently, we have succeeded in growing single crystals by the Bridgman method. By means of ultrasound technique, we measured the elastic constants of a PrPtBi single crystal in order to investigate the anisotropy of electronic properties originated from non-Kramers doublet G3 : Symmetric elastic constants can give rich information about CEF state and quadrupolar order parameter.
417
lar-strain and intersite quadrupolar interaction, respectively. gG is the coupling constant for the quadrupole-strain and g0G for the intersite quadrupole interaction. OG and Om l mean the equivalent operator of the quadrupolar- and multipolar moments, respectively. We used the following form of the temperature- and the magnetic-field dependence of elastic constants [8]: 0 QS cG ¼ c0G Ng2G wQS G =ð1 gG wG Þ:
ð3Þ
Here wQS G is the quadrupolar-strain susceptibility. c0G and N are the background part of elastic constants and the number of rare-earth ions in unit volume, respectively. We assumed that the background parts of elastic constants is proportional to temperature as c0G ¼ aT þ b; here a and b are constants. The elastic constant ðc11 c12 Þ=2 shows a divergent softening proportional to the reciprocal temperatures in the non-Kramers doublet G3 ground state, while c44 shows a Van-Vleck-type temperature-independent behavior at low temperature. The bulk modulus cB ¼ ðc11 þ 2c12 Þ=3 shows a monotonous increase in the localized-4f systems such as PrPtBi, while valence fluctuation compound such as SmB6 shows a softening [9].
3. Experimental details 2. Quadrupolar-strain susceptibility The temperature- and the magnetic-field dependence of elastic constants give information about CEF state regarding to the quadrupolar degrees of freedom. The ultrasound-induced strain interacts to 4f-electronic quadrupolar moments via the CEF potential. The quadrupolar total Hamiltonian is written as follows: H ¼ HCEF gJ mB JH gG OG eG g0G oOG > OG ; HCEF ¼ B4 ðO04 þ 5O44 Þ þ B6 ðO06 21O46 Þ:
ð1Þ ð2Þ
Here HCEF is the CEF Hamiltonian for the cubic crystals. The second term on the right-hand side in Eq. (1) is Zeeman term. The third and fourth term in the right-hand side of Eq. (1) mean quadrupo-
Single crystals of PrPtBi were prepared by the Bridgman method using Mo crucibles. The typical dimension of the samples are 2 mm 3 mm 3 mm with rectangular shape. LiNbO3 transducers were bonded on a pair of parallel faces of a sample by RTV silicone rubber (Shin-Etsu chemical, Co., Ltd.). Sound velocities were measured by our ultrasound apparatus based on the phase comparison method. Elastic constants c is estimated by the formula c ¼ rv2 ; here r is the mass density 11:56 g=cm3 of PrPtBi and v sound velocity. Ultrasonic measurements in high magnetic fields were performed using a water-cooled magnet at Tsukuba Magnet Laboratory, NIMS. A 4 He cryostat and a 3 He evaporation refrigerator were used for the low-temperature ultrasonic measurements down to 1:8 K and down to 0:6 K; respectively.
ARTICLE IN PRESS O. Suzuki et al. / Physica B 346–347 (2004) 416–420
418
4. Results and discussion Fig. 1 shows the temperature dependence of elastic constants of PrPtBi in zero magnetic field. The characteristic softening is observed in c11 and ðc11 c12 Þ=2: These modes are described by the quadrupolar susceptibility of the G3 quadrupole moments O02 or O22 : c11 is the mixed mode of G1 and G3 : The temperature dependence of the bulk modulus cB ¼ ðc11 þ 2c12 Þ=3 increase monotonously in higher temperature above 20 K; which indicates that localized 4f-electron is stable. The pure G3 mode ðc11 c12 Þ=2 shows a slow increase from the higher temperature side and begin to soften below about 20 K: The softening of ðc11 c12 Þ=2 becomes steep at around 4 K and continues down to 0:6 K: The magnitude of the softening is 2% in c11 and 10% in ðc11 c12 Þ=2: This quite
140 C11
Elastic constant (GPa)
130 75
CB = (C11+2C12)/3
70
PrPtBi
(C11-C12)/2
50
40
0
C44
20
40
60
80
100
Temperature (K) Fig. 1. The elastic constants in PrPtBi as functions of temperatures in zero magnetic field. Solid lines and broken lines for ðc11 c12 Þ=2 are calculation fit with the quadrupolarstrain interaction jgG3 j ¼ 35 K and background part of elastic constants, respectively. The bulk modulus cB ¼ ðc11 þ 2c12 Þ=3 is calculated from the results of c11 and ðc11 c12 Þ=2:
large softening in ðc11 c12 Þ=2 clearly manifest the G3 non-Kramers doublet ground state, which brings the Curie-type softening originated from the quadrupolar degeneracy of O02 and O22 : The observed magnitude of softening in ðc11 c12 Þ=2 in the present experiments is smaller than the softening in the transverse elastic constant in polycrystalline PrPtBi reported by Kasaya et al. In our polycrystalline measurements, we observed similar softening of 25% in transverse mode and upturn at around 1 K (not shown). In the single crystal case, the strength of the softening in ðc11 c12 Þ=2 strongly depends on the sample quality. The higher quality sample, which shows clear VanVleck-type temperature-independent behavior in the magnetic susceptibility (not shown), softening in ðc11 c12 Þ=2 becomes larger. Other elastic modes in single crystals show quite small dependence on the sample quality. In the present experiments we used the single-crystalline sample with the largest softening of ðc11 c12 Þ=2 among our samples. The detailed sample preparation and sample dependence in other measurements will be discussed elsewhere [10]. In the present experiment remarkable elastic softening in ðc11 c12 Þ=2 is observed, however, the existence of the quadrupolar ordering temperature TQ is not clear. It is contrast to the clear anomalies in polycrystalline sample at TQ ¼ 1:35 K: The phase transition temperature is considered to be lower in our single crystalline sample than in polycrystalline sample. In order to determine the nature of quadrupolar interaction we calculated the quadrupolar susceptibility for ðc11 c12 Þ=2: The solid line for ðc11 c12 Þ=2 in Fig. 1 are calculation fit using Eq. (3). We assumed the CEF level scheme obtained by inelastic neutron scattering by Kasaya et al. [11]. The obtained quadrupolar-strain interaction is jgG3 j ¼ 35 K: The intersite quadrupolar interaction is too small to determine the magnitude. The large softening of 10% in ðc11 c12 Þ=2 in the present experiment prefer the ferro-quadrupolar order in the G3 ground state. The large softening in ðc11 c12 Þ=2 has been also observed in the cubic compound Ce3 Pd20 Ge6 which shows the cubic– tetragonal structural phase transition accompanied with ferro-quadrupolar order of O02 [12]. Recent X-ray powder diffraction of PrPtBi by
ARTICLE IN PRESS O. Suzuki et al. / Physica B 346–347 (2004) 416–420
2.5 PrPtBi C11 k//u//H//[001]
∆C/C0 (%)
2.0
2.5 PrPtBi (C11-C12)/2 k//H//[110] u//[110]
2.0
1.8 K
1.5
0
∆C/C (%)
Naher et al. [13] detected an abrupt increase of nuclear Bragg reflection below TQ ¼ 1:35 K: This result can be interpreted as an indication of the lattice distortion resulted from a ferro-quadrupolar order. From the results of our elastic data, the expected structural change is cubic-tetragonal for O02 order or cubic–orthorhombic for O22 : In order to observe macroscopic structural change, thermal expansion measurements are further needed. In order to investigate elastic property in magnetic fields we have measured field dependence of elastic constants. Figs. 2 and 3 displays relative changes of elastic constant c11 and ðc11 c12 Þ=2 in PrPtBi as functions of magnetic fields at various temperatures. The field dependence of c11 and ðc11 c12 Þ=2 at 1.8 and 4:2 K shows a monotonous increase up to 25 T: No clear indication of phase transition was observed. From the results of temperature dependence measurements PrPtBi is in paramagnetic phase above 1:8 K up to 25 T in fields along the ½0 0 1 and ½1 1 0 axes. To investigate the nature of quadrupolar phase in magnetic fields, it is needed to measure at lower temperature. The solid lines in Figs. 2 and 3 show the calculation fit for c11 and ðc11 c12 Þ=2: We used quadrupolar-strain interaction jgG3 j ¼ 35 K
419
1.0 0.5 4.2 K 0.0 0
5
10
15
20
25
H (T) Fig. 3. The relative changes of transverse elastic mode ðc11 c12 Þ=2 in PrPtBi as functions of magnetic fields along the [1–10] axis at various temperatures. The data is scaled by the value at H ¼ 0: Solid lines are calculation fit with the quadrupolar interaction of jgG3 j ¼ 35 K:
obtained from the temperature dependence of ðc11 c12 Þ=2 in Fig. 1. The magnetic field dependence in ðc11 c12 Þ=2 can be explained qualitatively by assuming the CEF level scheme obtained by inelastic neutron scattering [11]. The field dependence of c11 and ðc11 c12 Þ=2 at 1:8 K shows a tendency of saturation above 12 T; although the origin is not clear. Further investigation is needed to elucidate the quadrupolar effects in magnetic fields.
1.8 K
1.5
Summary 1.0 4.2 K 0.5 0.0 0
5
10
15
20
25
H (T) Fig. 2. The relative changes of longitudinal elastic mode c11 in PrPtBi as functions of magnetic fields along the ½0 0 1 axis at various temperatures. The data is scaled by the value at H ¼ 0: Solid lines are calculation fit with the quadrupolar interaction of jgG3 j ¼ 35 K:
The temperature- and the magnetic-field dependence of elastic constants of PrPtBi single crystal have been measured by ultrasound technique. In the absence of magnetic fields, temperature dependence of c11 and ðc11 c12 Þ=2 shows remarkable softening at low temperature, while c44 shows monotonous increase without softening. These softenings are attributed to the CEF ground state non-Kramers doublet G3 with the degeneracy of quadrupolar moments O02 and O22 : From the results of temperature dependence of ðc11 c12 Þ=2 quadrupolar-strain interaction was estimated to be
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jgG3 j ¼ 35 K: The enormous softening of 10% in ðc11 c12 Þ=2 prefer a ferro-quadrupolar order of O02 or O22 ; which is accompanied with a structural phase transition of cubic–tetragonal or cubic– orthorhombic. The elastic constants as functions of magnetic fields show monotonous increase up to 25 T at 1.8 and 4:2 K: No clear indication of phase transition was observed up to 25 T: These results imply paramagnetic phase is stable up to 25 T above 1:8 K:
Acknowledgements The authors thank the staff members of the Tsukuba Magnet Laboratory, NIMS for operation of the magnets.
References [1] D.L. Cox, A. Zawadowski, Adv. Phys. 47 (1998) 599. [2] M. Niksch, W. Assmus, B. Luthi, . H.R. Ott, J.K. Kjems, Helv. Phys. Acta. 55 (1982) 688.
[3] T. Tayama, T. Sakakibara, K. Kitami, M. Yokoyama, % K. Tenya, H. Amitsuka, D. Aoki, Y. Onuki, Z. Kletowski, J. Phys. Soc. Jpn. 70 (2001) 248. [4] A. Yatskar, W.P. Beyermann, R. Movshovich, P.C. Canfield, Phys. Rev. Lett. 77 (1996) 3637. [5] E.D. Bauer, N.A. Frederick, P.-C. Ho, V.S. Zapf, M.B. Maple, Phys. Rev. B 65 (2002) 100506. [6] Y. Aoki, T. Namiki, T.D. Matsuda, K. Abe, H. Sugawara, H. Sato, Phys. Rev. B 65 (2002) 064446 and references there in. [7] H. Suzuki, M. Kasaya, T. Miyazaki, Y. Nemoto, T. Goto, J. Phys. Soc. Japan 66 (1997) 2566. [8] P. Thalmeier, B. Luthi, . in: K.A. Gschneider Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, Vol. 14, Chapter 96, Elsevier, Amsterdam, 1991. [9] S. Nakamura, T. Goto, M. Kasaya, S. Kunii, J. Phys. Soc. Jpn. 60 (1991) 4311. [10] H. Suzuki, unpublished. [11] M. Kasaya, H. Suzuki, D. Tazawa, M. Shirakawa, A. Sawada, T. Osakabe, Physica B 281–282 (2000) 579. . [12] O. Suzuki, T. Horino, Y. Nemoto, T. Goto, A.Donni, T. Komatsubara, M. Ishikawa, Physica B 261 (1999) 334; . T. Goto, T. Horino, Y. Nemoto, T. Yamaguchi, A.Donni, T. Komatsubara, Physica B 312 (2002) 492; Y. Nemoto, T. Yamaguchi, T. Horino, M. Akatsu, . T. Yanagisawa, T. Goto, O. Suzuki, A. Donni, T. Komatsubara, Phys. Rev. B 68 (2003) 184109. [13] S. Naher, H. Suzuki, M. Mizuno, Y. Xue, M. Kasaya, S. Kunii, T. Kitai, Physica B 329–333 (2003) 631.