- Email: [email protected]
Elastic property of TbRu4P12 under pressure
ARTICLE IN PRESS Physica B 404 (2009) 3271–3274
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Elastic property of TbRu4 P12 under pressure Y. Nakanishi a,, K. Ito a, T. Kamiyama a, M. Nakamura a, M. Yoshizawa a, M. Ohashi b, G. Oomi c, M. Kosaka c, C. Sekine d, I. Shirotani c a
Graduate School of Engineering, Iwate University, Morioka 020-8551, Japan Department of Physics, Kyushu University, Fukuoka 810-8560, Japan c Department of Physics, Saitama University, Saitama 338-8570, Japan d Faculty of Engineering, Muroran Institute of Technology, Muroran 050-8585, Japan b
a r t i c l e in fo
PACS: 71.20.Eh 71.27.þa 71.70.Ch 75.20.Hr Keywords: Skutterudite Ultrasonic measurement Multipolar moment High pressure
abstract We investigated elastic properties of the filled skutterudite compound with heavy lanthanide TbRu4P12 by means of ultrasonic measurements under pressure for the first time. TbRu4P12 undergoes a two successive phase transition from a paramagnetic to an antiferromagnetically ordered phase at TN 20 K, then to another phase transition at T1 10 K. We found that a clear elastic anomaly was observed at the two successive phase transition. Furthermore, it is found that they both show a significant pressure dependence. A steep decrease closely associated with TN is suppressed significantly, but it hardly shifts by applying the pressure. On the other hand, a slight anomaly associated with T1 is gradually suppressed and shifts to lower temperatures by applying the pressure. We argue the elastic behavior and the possible interpretation in each ordering phase. & 2009 Elsevier B.V. All rights reserved.
1. Introduction There has been much interest in the remarkable electronic behavior of the filled skutterudite compounds with general formula RTr4 X12 , where R ¼ rare earth alkali or actinide metal, Tr ¼ Fe, Ru, Os and X ¼ P, As, Sb [1]. They exhibit a wide variety of physical properties including superconductivity, metal–insulator (MI) transition, magnetic ordering, heavy fermions (HFS) and multipolar ordering. Since compounds with a light lanthanide element have been studied much more frequently than compounds with a heavy lanthanide element, the investigation of the filled skutterudite compounds with the heavy rare earth element has made insufficient progress until now. The smaller ionic radius due to the lanthanide contraction prevents the crystal structure from forming stably. The system with a heavy lanthanide element is expected to exhibit new physical phenomena based on the orbital degrees of freedom due to its more highly degenerated 4flevel schemes as compared with that of a light lanthanide one. TbRu4P12, in fact exhibits the complicated magnetic phase diagram where the orbital degrees of freedom of Tb ion possibly plays an important role [2–4]. Besides, a possibly nested Fermi surface property of RRu4 P12 with the nesting vector q ¼ ð1; 0; 0Þ is
Corresponding author. Tel./fax: +8119 6216356.
E-mail address: [email protected] (Y. Nakanishi). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.07.098
essentially important to understand the complicated physical properties [5]. The interest in TbRu4P12 is due to a successive two antiferromagnetic phase transition at TN ¼ 20 K and T1 ¼ 10 K in zero field, as evidenced by the magnetic susceptibility and specific heat measurements [2,3]. Furthermore, the field-induced magnetic phase is observed below TN , suggesting strongly great importance of the orbital degrees of freedom and highly degenerated lowlying 4f levels of Tb ion in TbRu4P12. A remarkable elastic anomaly at the successive two antiferromagnetic transition has been found in the elastic constants as a function of temperature. We proposed the 4f-ground state split by CEF effect to be G4 or G5 triplet in our previous paper [4]. To extend the last investigation and understand further the low temperature properties including the complicated magnetic phase diagram we performed the ultrasonic measurements on TbRu4P12 under hydrostatic pressures. In general, ultrasonic measurement under hydrostatic pressures is rarely performed, partly due to its difficulty. We succeeded, however, to do the measurements on TbRu4P12 for the first time. Application of hydrostatic pressure is expected to induce stronger quadrupole strain and intersite quadrupole interactions, which then affected elastic properties. This prompted us to explore the elastic properties under hydrostatic pressures related to the complicated magnetic phase diagram. In this paper, we give results of the elastic as well as magnetic properties of polycrystal TbRu4P12 and discuss the possible low
ARTICLE IN PRESS 3272
Y. Nakanishi et al. / Physica B 404 (2009) 3271–3274
temperature properties governed by 4f-electronic state of Tb ion as a function of temperature and hydrostatic pressures.
1.2 GPa
1.01
0.8 GPa
2. Experiment
1.00 0.5 GPa
CL (a.u)
Single-phase cylindrical polycrystalline sample of TbRu4P12 was synthesized at high pressure and high temperature using a wedge-type cubic-anvil high-pressure apparatus. The stoichiometric amounts of each metal and red phosphorus powders were reacted at around 1100 3 C and 4 GPa. The samples were characterized by powder X-ray diffraction using CuKa1 radiation and silicon as a standard. TbRu4 P12 has the cubic structure with a lattice constant of 8:0338 A at 300 K. The sound wave velocity v ˚ was detected by an ultrasonic apparatus based on the phasecomparison method. The absolute value of the elastic constant could not be estimated explicitly because the thickness of the sample was not large enough to do that. Hydrostatic pressure up to 1.2 GPa was generated for ultrasonic measurements by means of a Cu–Be piston-cylinder pressure cell and a Teflon capsule inside it. The cell has two layers, i.e., an outer cylinder and an inner one. The inner diameter and the length of the inner cylinder are 8 and 40 mm, respectively. The sample with two ultrasonic transducers bonded to it, together with a tin manometer, was placed in the Teflon capsule filled with a pressure medium (a 1:1 mixture of Fluorinert FC70 and FC77). Details are described elsewhere [6]. The measurements were carried out down to the temperature of liquid helium, 4.2 K, or down to 1:5 K by pumping. The effective hydrostatic pressure inside the Teflon capsule was calibrated by measuring the superconducting transition temperature of the Sn manometer. Additionally, detailed magnetization measurements were performed with a superconducting quantum interference device (SQUID) magnetometer up to 0.6 GPa. Hydrostatic pressure was applied by utilizing a zirconia piston and a Cu–Be piston-cylinder cell with a pressure medium (a 1:1 mixture of Fluorinert FC70 and FC77). To calibrate the pressure at the sample position at low temperatures, the shift in Tc of the Sn metal was monitored by an inductance measurement.
0.25 GPa 0.99
ambient
0.98
0.97
0
10 20 Temperature (K)
Fig. 1. Temperature dependence in zero field of the longitudinal elastic constant CL ðTÞ at selected pressures. The arrows indicate the phase transition point, determined by the dCL =dT vs. T curves.
0.025
3. Results
TbRu4P12
0.020
P = 0.6 GPa M (emu)
Fig. 1 depicts the temperature dependence in zero field of the longitudinal elastic constant CL ðTÞ at selected pressures. Each of the curves are shifted by an arbitrary vertical offset so as to avoid overlap and for clarity. At ambient pressure, a clear anomaly of softening and cusp were observed in the elastic constant at antiferromagnetic phase transition temperature of TN 20 K. Furthermore, a clear cusp was also recognized at T1 10 K. An application of hydrostatic pressure, however, change the overall temperature dependence of the elastic constant below the transition temperature. The remarkable elastic anomaly associated with the AF magnetic ordering is suppressed significantly by applying pressure, while TN shifts hardly. The elastic anomaly at T1 , however, shifts slightly to a lower temperature with an increase in the pressure. The anomaly is also suppressed by applying pressure, and almost undetectable at a pressure of 0.5 GPa. It is noted that an elastic softening toward the antiferromagnetic transition temperature is rapidly suppressed by the application of hydrostatic pressure. Since it is most likely to be attributed to the ground state property of Tb ions in TbRu4P12 formed by the CEF effect as reported in our previous article, this fact suggests a large change of the ground state of the 3þ Tb ion split by CEF effect. It will be discussed in detail later.
TbRu4P12
0.015
P = 0.4 GPa 0.010
P = 0.2 GPa 0.005 P = ambient 0.000
0
10
20
30 40 50 Temperature (K)
60
70
80
Fig. 2. Temperature dependence of the magnetic susceptibility at selected pressures. The dotted vertical lines denote the phase transition temperatures T1 and TN .
ARTICLE IN PRESS Y. Nakanishi et al. / Physica B 404 (2009) 3271–3274
3273
30 TN ΔC 0.01
15 T1
10 5 0
CL (a.u.)
20
ΔCL (a.u.)
Temperature (K)
25
0.5
1.0 1.5 Pressure (GPa)
ΔC
P = 0.25 GPa
TbRu4P12 0.0
1.00
0.99
2.0
Fig. 3. Pressure dependence of the phase transition temperatures T1 and TN . The solid lines are a guide for the eyes.
4. Discussion First, we would like to discuss the complicated magnetic phase diagram for TbRu4 P12. The phase transition at 20 K, which is strongly believed to be an antiferromagnetic ordering, shows the pronounced pressure dependence. This phase is, however, somewhat different from a conventional antiferromagnetic ordering due to the following reasons. This phase transition is accompanied with a large jump in the temperature dependence of the electrical resistivity rðTÞ [2]. Effects of Fermi surface nesting play an important role to cause the jump as mentioned before. The most prominent feature of the ordered phase is the phase boundary. The boundary seems to disappear gradually with increasing magnetic fields. That is to say, there is a critical point where the transition terminates in the magnetic phase diagram. The obtained magnetic phase diagram of TbRu4P12 is analogous to that of TmS where antiferromagnetic ordering and ferroquadrupolar ordering would occur simultaneously at 6.5 K [7]. The phase boundary also gradually disappears with increasing magnetic
10 T (K)
20
0.00 1.0 @ TN @ T1 ξ (a.u.)
Fig. 2 depicts the temperature dependence of the magnetic susceptibility wðTÞ for TbRu4P12 at selected pressures. A clear cusp was observed at the transition temperatures TN at ambient pressure. There is also a small shoulder located at T1 . These data are in good agreement with the previous one [2]. The cusp at TN shifts slightly by applying pressure. An application of hydrostatic pressure, however, does not change the shape of the cusp at TN . This is a prominent feature in wðTÞ, markedly in contrast to the strong pressure dependence observed in the elastic constant CL ðTÞ. The shoulder at T1 is slightly suppressed by applying pressure. An upturn appearing below around 6 K might be related to the presence of magnetic impurity. Fig. 3 depicts the pressure dependence of the transition temperatures TN and T1 for TbRu4P12. The TN exhibits a monotonic increase up to 0.5 GPa, while a slight decrease upon further increasing the pressure. On the other hand, T1 is nearly constant or slightly increase with increasing the pressure. Fig. 4(a) depicts the pressure dependence of a relative change of elastic constant determined by the extrapolation of the linear fit to below and above TN as shown in the inset of Fig. 4(a). Unfortunately, the same estimation with respect to T1 could not be done in the present study since the data are not enough. The relative change of the CL ðTÞ associated with TN is roughly followed by the fitting curve as expected by DCL ¼ 0:016 expð2:28 PÞ.
0
0.5
0.0
0.0
0.5
1.0 Pressure (GPa)
1.5
Fig. 4. Pressure dependence of a relative change of the elastic constant CL below the phase transition temperature in zero field (a) and the estimated coupling constant deduced by the theoretical formula in terms of Eq. (3) in text. Inset of (a) shows the behavior of CL around the phase transition temperature with an expanded scale and the definition of DC.
fields. Furthermore, from the behavior let us conjecture the pressure–temperature phase diagram of a liquid–gas transition. As pointed out in our previous article, the behavior of the elastic constant CL ðTÞ around TN is also reminiscent of that around the metal–insulator transition in SmRu4P12 at which an octupolar ordering possibly occurs [8]. There is no clear evidence of which kind of an order parameter causes this phase transition at present. It is noted that again, this phase transition should be distinguished from a conventional magnetic phase transition derived from a well-localized 4f-electronic state. Second, we would like to discuss the pressure evolution of the elastic constant CL ðTÞ. We consider the free energy F expanded around the second-order phase transition by the power series of the order parameter QG and the elastic strain eG based on Landau phenomenological theory so as to describe the lattice instability due to the phase transition below TN as follows [9]: 4
F ¼ 12 aQ 2 þ 14bQ zQ 2 eG þ xQ 2 e2G þ :
ð1Þ
From the second derivative of the free energy F with respect to the elastic strain eG , one can obtain the temperature dependence
ARTICLE IN PRESS 3274
Y. Nakanishi et al. / Physica B 404 (2009) 3271–3274
at 0.8 and 1.2 GPa, the bend located at 0.5 GPa in the pressure dependence of TN is not sure in this stage.
of the elastic constants as follows [10]: 2
@2 F 2z 2Q 2 x; CG ðTÞ ¼ 2 ¼ CGð0Þ ðTÞ b @e
ð2Þ
where CGð0Þ ðTÞ is the lattice part and originates mainly from anharmonic effects of the crystal. a and b denote the second- and fourth-order coefficients of free-energy. z and x denote the fitting parameters related to the elastic softening and hardening, respectively. As readily recognized in Fig. 1, the elastic softening below TN characterized by z is negligibly small. The significant decrease of the elastic constant in the ordered phase is most likely to be related to the decrease of x under pressure. Since the x denotes strength of a coupling between the elastic strain and the order parameter, the result indicates that the coupling constant would be weaker rapidly with increasing pressure. The change of carrier number under pressure or the change of the expectation value /Q S might cause the present results under pressure. If the latter case is true, the disappearance of a characteristic softening toward TN can be reasonably explained by an effective reduction of a Curie term in the quadrupolar susceptibility for TbRu4P12. Such a significant change caused by hydrostatic pressure lets us imagine that the primary order parameter Q in the ordered state would not only be a magnetic moment but a multipolar moment or a combination of them, although it is an open question to be solved. We speculate that the magnetic ordering and the other ordering, which is not known, may occur simultaneously at the same point as observed in TmS [7]. In order to check our conjecture, microscopic measurements under hydrostatic pressure are highly required. ¨ Finally, we would like to comment on the estimated Gruneissen parameter O ¼ 23:2 for TN and T1 . Although T1 is still a controversial issue whether to be intrinsic or not, an anomaly due to the T1 observed in the present elastic constants and the mSR measurements, performed recently give a strong piece of evidence that T1 is not ascribable to a second phase such as TbP, but being a intrinsic phase transition [11]. Since the absolute value of the sound velocity could not be estimated in the present measurement on TbRu4P12 and the sample used is polycrystalline, we use the absolute values of the elastic constants CB ¼ 193:5 GPa at 4.2 K observed in the isostructural compound PrRu4P12. By using the following formula, where T is TN or T1,
O¼
@lnT CB @T ¼ @lnV T @P
ð3Þ
we can thus obtain the Gruneissen parameter O ¼ 23:2 and 32.3 for TN and T1 , respectively. Here, we used the data up to 0.5 GPa to determine @T =@P for TN . Since the elastic anomaly is so small up
5. Summary To summarize, we measured the longitudinal elastic constant CL ðTÞ and the magnetic susceptibility wL ðTÞ for polycrystalline TbRu4P12 under hydrostatic pressure. Significant elastic suppression accompanied by TN 20 K is induced by an application of pressure, while little change is observed in the magnetic susceptibility wL ðTÞ. The results give evidence that the ordered state below 20 K, believed to be an antiferromagnetic ordering until now, is not a conventional magnetic ordering. We anticipate that it would be a multipolar ordering or a combination of a quadrupolar ordering and magnetic one. The strange phase boundary due to TN may be related to the large Gruneissen parameter O ¼ 38:7 and 23.2 at T1 and TN , respectively. The same measurements in magnetic fields simultaneously are currently in progress, so as to establish a detailed P–H–T phase diagram.
Acknowledgments The technical support and help with low-temperature measurements by T. Fujino, D. Inoue and R. Ishihara is gratefully acknowledged. The work was performed at the Cryogenic Laboratory, Center for Instrumental Analysis, Iwate University. This work was supported by a Grant-in-Aid for Science Research Priority Area ‘‘Skutterudite’’ (no. 15072202) of the Ministry of Education, Culture, Sports, Science and Technology of Japan and in part supported financially by the discretionary expenditure of the president of Iwate University. References For example, refer to this URL /http://www.org.kobe-u.ac.jp/skut/S. C. Sekine, et al., Phys. Rev. B. 62 (2000) 11581. C. Sekine, et al., Phys. B 359–361 (2005) 306. T. Fujino, et al., J. Phys. Soc. Japan Suppl. A 77 (2008) 856. H. Harima, et al., J. Phys. Soc. Japan Suppl. A 77 (2008) 114. P. Sun, et al., Phys. Rev. B 75 (2007) 054114. Y. Nakanishi, et al., Phys. Rev. B 64 (2001) 184434. M. Yoshizawa, et al., J. Phys. Soc. Japan 74 (2005) 2141. For example, L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, vol. 5, Pergamon, Oxford, 1980, p. 446. [10] A. Imaduddin, et al., J. Phys. Soc. Japan 71 (2002) 1965. [11] W. Higemoto, private communication. [1] [2] [3] [4] [5] [6] [7] [8] [9]
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Elastic property of TbRu4 P12 under pressure Y. Nakanishi a,, K. Ito a, T. Kamiyama a, M. Nakamura a, M. Yoshizawa a, M. Ohashi b, G. Oomi c, M. Kosaka c, C. Sekine d, I. Shirotani c a
Graduate School of Engineering, Iwate University, Morioka 020-8551, Japan Department of Physics, Kyushu University, Fukuoka 810-8560, Japan c Department of Physics, Saitama University, Saitama 338-8570, Japan d Faculty of Engineering, Muroran Institute of Technology, Muroran 050-8585, Japan b
a r t i c l e in fo
PACS: 71.20.Eh 71.27.þa 71.70.Ch 75.20.Hr Keywords: Skutterudite Ultrasonic measurement Multipolar moment High pressure
abstract We investigated elastic properties of the filled skutterudite compound with heavy lanthanide TbRu4P12 by means of ultrasonic measurements under pressure for the first time. TbRu4P12 undergoes a two successive phase transition from a paramagnetic to an antiferromagnetically ordered phase at TN 20 K, then to another phase transition at T1 10 K. We found that a clear elastic anomaly was observed at the two successive phase transition. Furthermore, it is found that they both show a significant pressure dependence. A steep decrease closely associated with TN is suppressed significantly, but it hardly shifts by applying the pressure. On the other hand, a slight anomaly associated with T1 is gradually suppressed and shifts to lower temperatures by applying the pressure. We argue the elastic behavior and the possible interpretation in each ordering phase. & 2009 Elsevier B.V. All rights reserved.
1. Introduction There has been much interest in the remarkable electronic behavior of the filled skutterudite compounds with general formula RTr4 X12 , where R ¼ rare earth alkali or actinide metal, Tr ¼ Fe, Ru, Os and X ¼ P, As, Sb [1]. They exhibit a wide variety of physical properties including superconductivity, metal–insulator (MI) transition, magnetic ordering, heavy fermions (HFS) and multipolar ordering. Since compounds with a light lanthanide element have been studied much more frequently than compounds with a heavy lanthanide element, the investigation of the filled skutterudite compounds with the heavy rare earth element has made insufficient progress until now. The smaller ionic radius due to the lanthanide contraction prevents the crystal structure from forming stably. The system with a heavy lanthanide element is expected to exhibit new physical phenomena based on the orbital degrees of freedom due to its more highly degenerated 4flevel schemes as compared with that of a light lanthanide one. TbRu4P12, in fact exhibits the complicated magnetic phase diagram where the orbital degrees of freedom of Tb ion possibly plays an important role [2–4]. Besides, a possibly nested Fermi surface property of RRu4 P12 with the nesting vector q ¼ ð1; 0; 0Þ is
Corresponding author. Tel./fax: +8119 6216356.
E-mail address: [email protected] (Y. Nakanishi). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.07.098
essentially important to understand the complicated physical properties [5]. The interest in TbRu4P12 is due to a successive two antiferromagnetic phase transition at TN ¼ 20 K and T1 ¼ 10 K in zero field, as evidenced by the magnetic susceptibility and specific heat measurements [2,3]. Furthermore, the field-induced magnetic phase is observed below TN , suggesting strongly great importance of the orbital degrees of freedom and highly degenerated lowlying 4f levels of Tb ion in TbRu4P12. A remarkable elastic anomaly at the successive two antiferromagnetic transition has been found in the elastic constants as a function of temperature. We proposed the 4f-ground state split by CEF effect to be G4 or G5 triplet in our previous paper [4]. To extend the last investigation and understand further the low temperature properties including the complicated magnetic phase diagram we performed the ultrasonic measurements on TbRu4P12 under hydrostatic pressures. In general, ultrasonic measurement under hydrostatic pressures is rarely performed, partly due to its difficulty. We succeeded, however, to do the measurements on TbRu4P12 for the first time. Application of hydrostatic pressure is expected to induce stronger quadrupole strain and intersite quadrupole interactions, which then affected elastic properties. This prompted us to explore the elastic properties under hydrostatic pressures related to the complicated magnetic phase diagram. In this paper, we give results of the elastic as well as magnetic properties of polycrystal TbRu4P12 and discuss the possible low
ARTICLE IN PRESS 3272
Y. Nakanishi et al. / Physica B 404 (2009) 3271–3274
temperature properties governed by 4f-electronic state of Tb ion as a function of temperature and hydrostatic pressures.
1.2 GPa
1.01
0.8 GPa
2. Experiment
1.00 0.5 GPa
CL (a.u)
Single-phase cylindrical polycrystalline sample of TbRu4P12 was synthesized at high pressure and high temperature using a wedge-type cubic-anvil high-pressure apparatus. The stoichiometric amounts of each metal and red phosphorus powders were reacted at around 1100 3 C and 4 GPa. The samples were characterized by powder X-ray diffraction using CuKa1 radiation and silicon as a standard. TbRu4 P12 has the cubic structure with a lattice constant of 8:0338 A at 300 K. The sound wave velocity v ˚ was detected by an ultrasonic apparatus based on the phasecomparison method. The absolute value of the elastic constant could not be estimated explicitly because the thickness of the sample was not large enough to do that. Hydrostatic pressure up to 1.2 GPa was generated for ultrasonic measurements by means of a Cu–Be piston-cylinder pressure cell and a Teflon capsule inside it. The cell has two layers, i.e., an outer cylinder and an inner one. The inner diameter and the length of the inner cylinder are 8 and 40 mm, respectively. The sample with two ultrasonic transducers bonded to it, together with a tin manometer, was placed in the Teflon capsule filled with a pressure medium (a 1:1 mixture of Fluorinert FC70 and FC77). Details are described elsewhere [6]. The measurements were carried out down to the temperature of liquid helium, 4.2 K, or down to 1:5 K by pumping. The effective hydrostatic pressure inside the Teflon capsule was calibrated by measuring the superconducting transition temperature of the Sn manometer. Additionally, detailed magnetization measurements were performed with a superconducting quantum interference device (SQUID) magnetometer up to 0.6 GPa. Hydrostatic pressure was applied by utilizing a zirconia piston and a Cu–Be piston-cylinder cell with a pressure medium (a 1:1 mixture of Fluorinert FC70 and FC77). To calibrate the pressure at the sample position at low temperatures, the shift in Tc of the Sn metal was monitored by an inductance measurement.
0.25 GPa 0.99
ambient
0.98
0.97
0
10 20 Temperature (K)
Fig. 1. Temperature dependence in zero field of the longitudinal elastic constant CL ðTÞ at selected pressures. The arrows indicate the phase transition point, determined by the dCL =dT vs. T curves.
0.025
3. Results
TbRu4P12
0.020
P = 0.6 GPa M (emu)
Fig. 1 depicts the temperature dependence in zero field of the longitudinal elastic constant CL ðTÞ at selected pressures. Each of the curves are shifted by an arbitrary vertical offset so as to avoid overlap and for clarity. At ambient pressure, a clear anomaly of softening and cusp were observed in the elastic constant at antiferromagnetic phase transition temperature of TN 20 K. Furthermore, a clear cusp was also recognized at T1 10 K. An application of hydrostatic pressure, however, change the overall temperature dependence of the elastic constant below the transition temperature. The remarkable elastic anomaly associated with the AF magnetic ordering is suppressed significantly by applying pressure, while TN shifts hardly. The elastic anomaly at T1 , however, shifts slightly to a lower temperature with an increase in the pressure. The anomaly is also suppressed by applying pressure, and almost undetectable at a pressure of 0.5 GPa. It is noted that an elastic softening toward the antiferromagnetic transition temperature is rapidly suppressed by the application of hydrostatic pressure. Since it is most likely to be attributed to the ground state property of Tb ions in TbRu4P12 formed by the CEF effect as reported in our previous article, this fact suggests a large change of the ground state of the 3þ Tb ion split by CEF effect. It will be discussed in detail later.
TbRu4P12
0.015
P = 0.4 GPa 0.010
P = 0.2 GPa 0.005 P = ambient 0.000
0
10
20
30 40 50 Temperature (K)
60
70
80
Fig. 2. Temperature dependence of the magnetic susceptibility at selected pressures. The dotted vertical lines denote the phase transition temperatures T1 and TN .
ARTICLE IN PRESS Y. Nakanishi et al. / Physica B 404 (2009) 3271–3274
3273
30 TN ΔC 0.01
15 T1
10 5 0
CL (a.u.)
20
ΔCL (a.u.)
Temperature (K)
25
0.5
1.0 1.5 Pressure (GPa)
ΔC
P = 0.25 GPa
TbRu4P12 0.0
1.00
0.99
2.0
Fig. 3. Pressure dependence of the phase transition temperatures T1 and TN . The solid lines are a guide for the eyes.
4. Discussion First, we would like to discuss the complicated magnetic phase diagram for TbRu4 P12. The phase transition at 20 K, which is strongly believed to be an antiferromagnetic ordering, shows the pronounced pressure dependence. This phase is, however, somewhat different from a conventional antiferromagnetic ordering due to the following reasons. This phase transition is accompanied with a large jump in the temperature dependence of the electrical resistivity rðTÞ [2]. Effects of Fermi surface nesting play an important role to cause the jump as mentioned before. The most prominent feature of the ordered phase is the phase boundary. The boundary seems to disappear gradually with increasing magnetic fields. That is to say, there is a critical point where the transition terminates in the magnetic phase diagram. The obtained magnetic phase diagram of TbRu4P12 is analogous to that of TmS where antiferromagnetic ordering and ferroquadrupolar ordering would occur simultaneously at 6.5 K [7]. The phase boundary also gradually disappears with increasing magnetic
10 T (K)
20
0.00 1.0 @ TN @ T1 ξ (a.u.)
Fig. 2 depicts the temperature dependence of the magnetic susceptibility wðTÞ for TbRu4P12 at selected pressures. A clear cusp was observed at the transition temperatures TN at ambient pressure. There is also a small shoulder located at T1 . These data are in good agreement with the previous one [2]. The cusp at TN shifts slightly by applying pressure. An application of hydrostatic pressure, however, does not change the shape of the cusp at TN . This is a prominent feature in wðTÞ, markedly in contrast to the strong pressure dependence observed in the elastic constant CL ðTÞ. The shoulder at T1 is slightly suppressed by applying pressure. An upturn appearing below around 6 K might be related to the presence of magnetic impurity. Fig. 3 depicts the pressure dependence of the transition temperatures TN and T1 for TbRu4P12. The TN exhibits a monotonic increase up to 0.5 GPa, while a slight decrease upon further increasing the pressure. On the other hand, T1 is nearly constant or slightly increase with increasing the pressure. Fig. 4(a) depicts the pressure dependence of a relative change of elastic constant determined by the extrapolation of the linear fit to below and above TN as shown in the inset of Fig. 4(a). Unfortunately, the same estimation with respect to T1 could not be done in the present study since the data are not enough. The relative change of the CL ðTÞ associated with TN is roughly followed by the fitting curve as expected by DCL ¼ 0:016 expð2:28 PÞ.
0
0.5
0.0
0.0
0.5
1.0 Pressure (GPa)
1.5
Fig. 4. Pressure dependence of a relative change of the elastic constant CL below the phase transition temperature in zero field (a) and the estimated coupling constant deduced by the theoretical formula in terms of Eq. (3) in text. Inset of (a) shows the behavior of CL around the phase transition temperature with an expanded scale and the definition of DC.
fields. Furthermore, from the behavior let us conjecture the pressure–temperature phase diagram of a liquid–gas transition. As pointed out in our previous article, the behavior of the elastic constant CL ðTÞ around TN is also reminiscent of that around the metal–insulator transition in SmRu4P12 at which an octupolar ordering possibly occurs [8]. There is no clear evidence of which kind of an order parameter causes this phase transition at present. It is noted that again, this phase transition should be distinguished from a conventional magnetic phase transition derived from a well-localized 4f-electronic state. Second, we would like to discuss the pressure evolution of the elastic constant CL ðTÞ. We consider the free energy F expanded around the second-order phase transition by the power series of the order parameter QG and the elastic strain eG based on Landau phenomenological theory so as to describe the lattice instability due to the phase transition below TN as follows [9]: 4
F ¼ 12 aQ 2 þ 14bQ zQ 2 eG þ xQ 2 e2G þ :
ð1Þ
From the second derivative of the free energy F with respect to the elastic strain eG , one can obtain the temperature dependence
ARTICLE IN PRESS 3274
Y. Nakanishi et al. / Physica B 404 (2009) 3271–3274
at 0.8 and 1.2 GPa, the bend located at 0.5 GPa in the pressure dependence of TN is not sure in this stage.
of the elastic constants as follows [10]: 2
@2 F 2z 2Q 2 x; CG ðTÞ ¼ 2 ¼ CGð0Þ ðTÞ b @e
ð2Þ
where CGð0Þ ðTÞ is the lattice part and originates mainly from anharmonic effects of the crystal. a and b denote the second- and fourth-order coefficients of free-energy. z and x denote the fitting parameters related to the elastic softening and hardening, respectively. As readily recognized in Fig. 1, the elastic softening below TN characterized by z is negligibly small. The significant decrease of the elastic constant in the ordered phase is most likely to be related to the decrease of x under pressure. Since the x denotes strength of a coupling between the elastic strain and the order parameter, the result indicates that the coupling constant would be weaker rapidly with increasing pressure. The change of carrier number under pressure or the change of the expectation value /Q S might cause the present results under pressure. If the latter case is true, the disappearance of a characteristic softening toward TN can be reasonably explained by an effective reduction of a Curie term in the quadrupolar susceptibility for TbRu4P12. Such a significant change caused by hydrostatic pressure lets us imagine that the primary order parameter Q in the ordered state would not only be a magnetic moment but a multipolar moment or a combination of them, although it is an open question to be solved. We speculate that the magnetic ordering and the other ordering, which is not known, may occur simultaneously at the same point as observed in TmS [7]. In order to check our conjecture, microscopic measurements under hydrostatic pressure are highly required. ¨ Finally, we would like to comment on the estimated Gruneissen parameter O ¼ 23:2 for TN and T1 . Although T1 is still a controversial issue whether to be intrinsic or not, an anomaly due to the T1 observed in the present elastic constants and the mSR measurements, performed recently give a strong piece of evidence that T1 is not ascribable to a second phase such as TbP, but being a intrinsic phase transition [11]. Since the absolute value of the sound velocity could not be estimated in the present measurement on TbRu4P12 and the sample used is polycrystalline, we use the absolute values of the elastic constants CB ¼ 193:5 GPa at 4.2 K observed in the isostructural compound PrRu4P12. By using the following formula, where T is TN or T1,
O¼
@lnT CB @T ¼ @lnV T @P
ð3Þ
we can thus obtain the Gruneissen parameter O ¼ 23:2 and 32.3 for TN and T1 , respectively. Here, we used the data up to 0.5 GPa to determine @T =@P for TN . Since the elastic anomaly is so small up
5. Summary To summarize, we measured the longitudinal elastic constant CL ðTÞ and the magnetic susceptibility wL ðTÞ for polycrystalline TbRu4P12 under hydrostatic pressure. Significant elastic suppression accompanied by TN 20 K is induced by an application of pressure, while little change is observed in the magnetic susceptibility wL ðTÞ. The results give evidence that the ordered state below 20 K, believed to be an antiferromagnetic ordering until now, is not a conventional magnetic ordering. We anticipate that it would be a multipolar ordering or a combination of a quadrupolar ordering and magnetic one. The strange phase boundary due to TN may be related to the large Gruneissen parameter O ¼ 38:7 and 23.2 at T1 and TN , respectively. The same measurements in magnetic fields simultaneously are currently in progress, so as to establish a detailed P–H–T phase diagram.
Acknowledgments The technical support and help with low-temperature measurements by T. Fujino, D. Inoue and R. Ishihara is gratefully acknowledged. The work was performed at the Cryogenic Laboratory, Center for Instrumental Analysis, Iwate University. This work was supported by a Grant-in-Aid for Science Research Priority Area ‘‘Skutterudite’’ (no. 15072202) of the Ministry of Education, Culture, Sports, Science and Technology of Japan and in part supported financially by the discretionary expenditure of the president of Iwate University. References For example, refer to this URL /http://www.org.kobe-u.ac.jp/skut/S. C. Sekine, et al., Phys. Rev. B. 62 (2000) 11581. C. Sekine, et al., Phys. B 359–361 (2005) 306. T. Fujino, et al., J. Phys. Soc. Japan Suppl. A 77 (2008) 856. H. Harima, et al., J. Phys. Soc. Japan Suppl. A 77 (2008) 114. P. Sun, et al., Phys. Rev. B 75 (2007) 054114. Y. Nakanishi, et al., Phys. Rev. B 64 (2001) 184434. M. Yoshizawa, et al., J. Phys. Soc. Japan 74 (2005) 2141. For example, L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, vol. 5, Pergamon, Oxford, 1980, p. 446. [10] A. Imaduddin, et al., J. Phys. Soc. Japan 71 (2002) 1965. [11] W. Higemoto, private communication. [1] [2] [3] [4] [5] [6] [7] [8] [9]