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Elastic properties of valence fluctuating CeRhIn
ARTICLE IN PRESS
Physica B 359–361 (2005) 136–138 www.elsevier.com/locate/physb
Elastic properties of valence fluctuating CeRhIn Haruhiro Higaki, Isao Ishii, Moo-Sung Kim, Daisuke Hirata, Toshiro Takabatake, Takashi Suzuki Department of Quantum Matter, Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima 739-8530, Japan
Abstract We report the temperature dependence of the elastic modulus and the specific heat of CeRhIn. The elastic modulus softens and takes a minimum around 120 K. The magnetic contribution to the specific heat takes a broad maximum around 130 K. These results suggest a renormalized pseudo-gapped state around the Fermi level E F : r 2005 Elsevier B.V. All rights reserved. PACS: 75.30.Mb; 72.55.+s; 71.20.b Keywords: Elastic properties; Ultrasonic investigation; Valence fluctuation; 4f-electron
CeRhIn, which has the ZrNiAl-type hexagonal structure, has been classified as a valence fluctuation system [1]. The unit cell volume of this compound is smaller than that expected for the trivalent lanthanide contraction. The magnetic susceptibility shows a Pauli-paramagnetism-like temperature variation. The magnetic contribution to the electric resistivity has logarithmic temperature dependence between 200 and 300 K. Valence fluctuation systems often show a typical temperature dependence of the elastic moduli due to the electronic Gru¨neisen coupling and/or the deformaCorresponding author. Tel.: +81 82 424 7040;
fax: +81 82 424 7044. E-mail address: [email protected] (T. Suzuki).
tion potential coupling to the quasi-particle band [2]. Recently, Suzuki et al. [3] found that the elastic moduli of isomorphic CeRhSn possessed a crystal electric field (CEF) state in the localized electronic regime, although CeRhSn was categorized as a valence fluctuation compound [4] before their report. We report the temperature dependence of the elastic modulus and the specific heat of singlecrystalline CeRhIn, and propose that a renormalized pseudo-gapped state is formed due to the valence fluctuation nature of this compound. A single crystal of CeRhIn was grown by the Czochralski method from the melt of stoichiometric amounts of the constituent elements in an RF induction furnace. Powder X-ray diffraction
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.014
ARTICLE IN PRESS H. Higaki et al. / Physica B 359– 361 (2005) 136–138
Energy 72
W
C33 [GPa]
EF
∆
71
DOS W
D 70
CeRhIn 69 0
100
200
300
T [K] Fig. 1. T-dependence of elastic modulus of CeRhIn. The solid curve is the fit to experimental data plotted by open circles in a framework of the two-bands model represented in the inset.
8.0
6.0
cm [J/molK]
analysis confirmed that the sample is in single phase with the ZrNiAl-type structure. The temperature T dependence of the elastic stiffness modulus C ii ðTÞ is obtained by using the relation, C ii ðTÞ ¼ rvðTÞ2 , where r is the mass density and v is the ultrasonic velocity. vðTÞ was detected by the phase-comparison type pulse-echo method in the temperature range 3–300 K. Specific heat c as a function of T was measured by the thermal relaxation method from 2 to 300 K using a commercial PPMS (Quantum Design). We also measured c of LaRhIn in order to obtain the magnetic contribution to the specific heat cm for CeRhIn. Fig. 1 shows the temperature dependence of longitudinal C 33 for CeRhIn. This modulus softens as T decreases and takes a minimum around 120 K. Transverse C 44 and C 66 show almost identical T dependence with the minimum around 120 K [5]. It was difficult to reproduce all modes as the strain-multipole response to the CEF state in the localized 4f-electronic regime. They are rather characteristic behavior of the elastic modulus in response to the deformation potential coupling to a pseudo-gapped band structure and thus suggests that 4f-electrons are in the valence fluctuating state. The magnetic specific heat cm is shown in Fig. 2. The data plotted by open circles are
137
4.0
2.0
CeRhIn 0.0
0
100
200
300
T [K] Fig. 2. T-dependence of magnetic specific heat of CeRhIn. The solid curve is the fit with parameters W ¼ 338 K and D ¼ 348 K:
obtained after subtracting the specific heat of LaRhIn from that of CeRhIn. It is noteworthy that the temperature at which cm shows the broad maximum is almost the same T where the elastic modulus shows the minimum. The small peak in cm below 40 K may be due to a fine structure of the electronic density very near E F : In a valence fluctuating state, a quasi-particle band of 4f-electron will appear near the Fermi level E F due to a many-body Kondo effect. We adopt the renormalized two-bands model [6] by assuming identical densities of 4f-electronic states symmetrically lying above and below E F : The model is schematically illustrated in Fig. 1, where W, D and D are the bandwidth, the energy gap and the density of states, respectively. Parameters W ¼ 338 K; D ¼ 348 K and D ¼ 5:00 1024 eV1 cm3 give the best fit to the experimental results of C 33 and cm ; as shown by the solid curves in Figs. 1 and 2, respectively. The elastic modulus has Curie and van Vleck contributions, since it has been defined by the second derivative of the free energy with respect to the elastic strain. The coupling coefficients for the Curie term jgl;G gu;G j2 D and the van Vleck term jhG j2 D which are used in the analysis are listed in Table 1 with the background elastic modulus. We utilized the same definition of coefficients presented in Ref. [6].
ARTICLE IN PRESS H. Higaki et al. / Physica B 359– 361 (2005) 136–138
138
Table 1 Fitting parameters of background stiffness a þ bT; coupling coefficients for the Curie term and the van Vleck term a (GPa)
b (GPa)
jgl;G gu;G j2 D (GPa)
jhG j2 D (GPa)
89.3
0.0116
42.7
9.20
In conclusion, the elastic modulus and the magnetic specific heat of CeRhIn have a minimum and a maximum at almost the same temperature, respectively. The renormalized two-bands model with the pseudo-gap around E F reproduces both elastic and thermal quantities. It is therefore convincing that CeRhIn is in the valence fluctuating state in contrast with the case for CeRhSn. This work was partially supported by the Grantin-Aid for the COE Research (No. 13CE2002)
from the Ministry of Education, Culture, Sports, Science and Technology of Japan and an aid fund from Energia, Inc.
References [1] D.T. Adroja, S.K. Malik, B.D. Padalia, R. Vijayaraghavan, Phys. Rev. B 39 (1989) 4831. [2] M. Yoshizawa, B. Lu¨thi, K.D. Schotte, Z. Phys. B 64 (1986) 169. [3] T. Suzuki, H. Higaki, I. Ishii, M.S. Kim, T. Takabatake, J. Magn. Magn. Mater. 272–276 (2004) e35. [4] M.S. Kim, Y. Echizen, K. Umeo, S. Kobayashi, M. Sera, P.S. Salamakha, O.L. Sologub, T. Takabatake, X. Chen, M.H. Jung, M.B. Maple, Phys. Rev. B 68 (2003) 054416. [5] H. Higaki, I. Ishii, M.S. Kim, D. Hirata, T. Takabatake, T. Suzuki, J. Phys. Soc. Japan, to appear. [6] S. Nakamura, T. Goto, M. Kasaya, S. Kunii, J. Phys. Soc. Japan 60 (1991) 4311.
Physica B 359–361 (2005) 136–138 www.elsevier.com/locate/physb
Elastic properties of valence fluctuating CeRhIn Haruhiro Higaki, Isao Ishii, Moo-Sung Kim, Daisuke Hirata, Toshiro Takabatake, Takashi Suzuki Department of Quantum Matter, Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima 739-8530, Japan
Abstract We report the temperature dependence of the elastic modulus and the specific heat of CeRhIn. The elastic modulus softens and takes a minimum around 120 K. The magnetic contribution to the specific heat takes a broad maximum around 130 K. These results suggest a renormalized pseudo-gapped state around the Fermi level E F : r 2005 Elsevier B.V. All rights reserved. PACS: 75.30.Mb; 72.55.+s; 71.20.b Keywords: Elastic properties; Ultrasonic investigation; Valence fluctuation; 4f-electron
CeRhIn, which has the ZrNiAl-type hexagonal structure, has been classified as a valence fluctuation system [1]. The unit cell volume of this compound is smaller than that expected for the trivalent lanthanide contraction. The magnetic susceptibility shows a Pauli-paramagnetism-like temperature variation. The magnetic contribution to the electric resistivity has logarithmic temperature dependence between 200 and 300 K. Valence fluctuation systems often show a typical temperature dependence of the elastic moduli due to the electronic Gru¨neisen coupling and/or the deformaCorresponding author. Tel.: +81 82 424 7040;
fax: +81 82 424 7044. E-mail address: [email protected] (T. Suzuki).
tion potential coupling to the quasi-particle band [2]. Recently, Suzuki et al. [3] found that the elastic moduli of isomorphic CeRhSn possessed a crystal electric field (CEF) state in the localized electronic regime, although CeRhSn was categorized as a valence fluctuation compound [4] before their report. We report the temperature dependence of the elastic modulus and the specific heat of singlecrystalline CeRhIn, and propose that a renormalized pseudo-gapped state is formed due to the valence fluctuation nature of this compound. A single crystal of CeRhIn was grown by the Czochralski method from the melt of stoichiometric amounts of the constituent elements in an RF induction furnace. Powder X-ray diffraction
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.014
ARTICLE IN PRESS H. Higaki et al. / Physica B 359– 361 (2005) 136–138
Energy 72
W
C33 [GPa]
EF
∆
71
DOS W
D 70
CeRhIn 69 0
100
200
300
T [K] Fig. 1. T-dependence of elastic modulus of CeRhIn. The solid curve is the fit to experimental data plotted by open circles in a framework of the two-bands model represented in the inset.
8.0
6.0
cm [J/molK]
analysis confirmed that the sample is in single phase with the ZrNiAl-type structure. The temperature T dependence of the elastic stiffness modulus C ii ðTÞ is obtained by using the relation, C ii ðTÞ ¼ rvðTÞ2 , where r is the mass density and v is the ultrasonic velocity. vðTÞ was detected by the phase-comparison type pulse-echo method in the temperature range 3–300 K. Specific heat c as a function of T was measured by the thermal relaxation method from 2 to 300 K using a commercial PPMS (Quantum Design). We also measured c of LaRhIn in order to obtain the magnetic contribution to the specific heat cm for CeRhIn. Fig. 1 shows the temperature dependence of longitudinal C 33 for CeRhIn. This modulus softens as T decreases and takes a minimum around 120 K. Transverse C 44 and C 66 show almost identical T dependence with the minimum around 120 K [5]. It was difficult to reproduce all modes as the strain-multipole response to the CEF state in the localized 4f-electronic regime. They are rather characteristic behavior of the elastic modulus in response to the deformation potential coupling to a pseudo-gapped band structure and thus suggests that 4f-electrons are in the valence fluctuating state. The magnetic specific heat cm is shown in Fig. 2. The data plotted by open circles are
137
4.0
2.0
CeRhIn 0.0
0
100
200
300
T [K] Fig. 2. T-dependence of magnetic specific heat of CeRhIn. The solid curve is the fit with parameters W ¼ 338 K and D ¼ 348 K:
obtained after subtracting the specific heat of LaRhIn from that of CeRhIn. It is noteworthy that the temperature at which cm shows the broad maximum is almost the same T where the elastic modulus shows the minimum. The small peak in cm below 40 K may be due to a fine structure of the electronic density very near E F : In a valence fluctuating state, a quasi-particle band of 4f-electron will appear near the Fermi level E F due to a many-body Kondo effect. We adopt the renormalized two-bands model [6] by assuming identical densities of 4f-electronic states symmetrically lying above and below E F : The model is schematically illustrated in Fig. 1, where W, D and D are the bandwidth, the energy gap and the density of states, respectively. Parameters W ¼ 338 K; D ¼ 348 K and D ¼ 5:00 1024 eV1 cm3 give the best fit to the experimental results of C 33 and cm ; as shown by the solid curves in Figs. 1 and 2, respectively. The elastic modulus has Curie and van Vleck contributions, since it has been defined by the second derivative of the free energy with respect to the elastic strain. The coupling coefficients for the Curie term jgl;G gu;G j2 D and the van Vleck term jhG j2 D which are used in the analysis are listed in Table 1 with the background elastic modulus. We utilized the same definition of coefficients presented in Ref. [6].
ARTICLE IN PRESS H. Higaki et al. / Physica B 359– 361 (2005) 136–138
138
Table 1 Fitting parameters of background stiffness a þ bT; coupling coefficients for the Curie term and the van Vleck term a (GPa)
b (GPa)
jgl;G gu;G j2 D (GPa)
jhG j2 D (GPa)
89.3
0.0116
42.7
9.20
In conclusion, the elastic modulus and the magnetic specific heat of CeRhIn have a minimum and a maximum at almost the same temperature, respectively. The renormalized two-bands model with the pseudo-gap around E F reproduces both elastic and thermal quantities. It is therefore convincing that CeRhIn is in the valence fluctuating state in contrast with the case for CeRhSn. This work was partially supported by the Grantin-Aid for the COE Research (No. 13CE2002)
from the Ministry of Education, Culture, Sports, Science and Technology of Japan and an aid fund from Energia, Inc.
References [1] D.T. Adroja, S.K. Malik, B.D. Padalia, R. Vijayaraghavan, Phys. Rev. B 39 (1989) 4831. [2] M. Yoshizawa, B. Lu¨thi, K.D. Schotte, Z. Phys. B 64 (1986) 169. [3] T. Suzuki, H. Higaki, I. Ishii, M.S. Kim, T. Takabatake, J. Magn. Magn. Mater. 272–276 (2004) e35. [4] M.S. Kim, Y. Echizen, K. Umeo, S. Kobayashi, M. Sera, P.S. Salamakha, O.L. Sologub, T. Takabatake, X. Chen, M.H. Jung, M.B. Maple, Phys. Rev. B 68 (2003) 054416. [5] H. Higaki, I. Ishii, M.S. Kim, D. Hirata, T. Takabatake, T. Suzuki, J. Phys. Soc. Japan, to appear. [6] S. Nakamura, T. Goto, M. Kasaya, S. Kunii, J. Phys. Soc. Japan 60 (1991) 4311.