ELSEVIER Physica B 230-232 (1997) 176-178

Giant lattice softening in the normal state of CeRu2 Takashi Suzuki a' *, Hiroshi Goshima a, Shigenobu Sakita a, Toshizo Fujita a,

Masato Hedo b, Yoshihiko Inada b, Etsuji Yamamoto c, Yoshinori Haga c, Yoshitika Onuki b "Department of Physics, Hiroshima University, Hi9ashi-Hiroshima 739, Japan

bDepartment of Physics, Osaka University, Toyonaka 560, Japan CAdvanced Science Research Center, JAERI, Tokai-mura 319-11, Japan

Abstract

Elastic moduli of high-quality single-crystalline CeRu 2 have been measured between 2 and 300 K by an ultrasonic technique. Both shear moduli of(C11 - C12)/2 and C44 show a giant softening without structural transition. In contrast to the shear moduli, the bulk modulus CB indicates a normal temperature dependence without the anomalous softening. These results reveal that there exists a remarkable structural fluctuation corresponding to shearing strain in the normal state of CeRu 2. The analysis taking account of the deformation potential coupling between the strain and an electronic energy band demonstrates that degenerate and peaked density of states is responsible for the softening.

Keywords: CeRu2; Lattice softening; Ultrasonic measurement; Electronic band structure

I. Introduction 2. Experimental

CeRu2 with C15-type cubic structure has attrac- ted much interest because a spatially modulated superconducting order parameter is suggested in the mixed state [1,2]. The direct evidence for the modulated order parameter, however, has not been observed. On the other hand, the electronic struc- ture in the normal conducting state is well de- scribed by an itinerant-4f band model, which has been confirmed both experimentally I-3] and theo- retically [4]. In the present work, we have found a giant lattice softening in shear moduli in the normal conducting state of CeRu2 by means of ultrasonic measurements. The origin of the lattice softening is discussed in terms of an electronic de- formation coupling [5] between strain and the elec- tronic band.

*Corresponding author.

A high-quality single crystal with the residual resistivity ratio of ,--270 has been obtained by Czochralski-pulling in a tetra-arc furnace followed by purification with a solid state electrotransport method. Using a phase comparison method, we measured changes in sound velocity v correspond- ing to transverse elastic stiffness ½(Cll - C12) and C44, and longitudinal Cll . Both transverse elastic moduli have pure symmetry because ½(Cll - C 1 2 )

is a response to strain exx - eyy with Eg representa- tion in a cubic structure and C44 is a response to exy with Tzg representation. However, longitudinal C11 is a mixed symmetric mode because of C11 = CB -}- 2 ( C l l -- C12 ) where CB = ½(Cll + 2C12)is the

bulk modulus in response to non-propagative strain ex~ + eyy + ezz with Alg. We can obtain the bulk modulus using the above relation with experi- mental results of Cl l and ½ ( C l l - C 1 2 ) . All moduli C are calculated from sound velocity by the

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T. Suzuki et al. /Physica B 230-232 (1997) 176-178 177

relation C = p v 2 where p = 10.65 g/cm 3 is the mass density of CeRu2 at room temperature.

3. Results and discussion

Fig. 1 shows the temperature (IT) dependence of pure symmetric elastic moduli ½(Cl1-C12), C44 and Cs. Giant lattice softening is found in ½(Cll - C12) and C44. Especially, the magnitude of ½ ( C l l - Cla) at 18.2K is about 55% of that at room temperature. Both shear modes shows a gradual hardening after marking a minimum with decreasing T. There is no divergence at the min- imum point, which suggests that no structural transition undergoes in CeRu2. This has been con- firmed by our preliminary low-temperature X-ray diffraction measurement. At the superconducting transition temperature Tc =6.2K, all elastic moduli start to soften again corresponding to the growth of superconducting order parameters. In magnetic fields above 5 T where the superconduc- tivity is destroyed, the softening below Tc disap-

71 CeRu 2 . . . .

J I ' I ' I

~ " 2 : ............

r~ 22

~ 21

15/'es- ' ' ' ' ' '

156 ~

154

152

0 ' 160 ' 260 ' 300 T ( K )

Fig. 1. 1 C12), Temperature dependence of (a) transverse 2(Cll - (b) transverse C44 and (c) bulk modulus CB. Dashed curves are theoretical fits.

pears and the gradual hardening looks to persist down to OK [6]. In contrast to the significant softening of shear moduli, the total-symmetric bulk modulus shows normal temperature depend- ence as is seen in usual solids without structural instability. Therefore, the data shown in Fig. 1 demonstrate that a structural fluctuation, which is aiming to reduce the crystal symmetry from cubic to lower symmetry, persists to very low temper- atures in CeRu 2 and it is not removed by a phase transition. The analysis by means of the deforma- tion potential coupling between strain and elec- tronic bands has succeeded in explaining such an unusual lattice-softening in several metallic com- pounds [7, 8].

The Fermi surfaces of CeRu2 determined from de Haas van Alphen effect [3] are explained by a 4f- itinerant band model [4]. Little temperature de- pendence of magnetic susceptibility displayed in Fig. 2, which has been measured in the field of B = 0.1 T along the [100] direction by a SQUID magnetometer, supports that the 4f electrons in CeRu2 are itinerant. The band theory is considered to well describe the electronic structure around the Fermi level EF.

If a band constructing the Fermi surface is de- generated and it splits into two subbands with the band energy E = Eo +_ drer , in response to the shearing strain er, the deviation of the elastic modulus Cr(T) from the background Cr BG is cal- culated by the formula [5] within a rigid band approximation;

C r ( T ) = CBr ~ -- 2d 2 N ( E ) dE, (1) - o o

3.0

2.5

2.0

1.5

10 0.5

0.0

' i • i , i • i

I

" ~ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1

CeRu 2

B/ / [ 100] B=0 .1T

, 510 , I , 100 I , I , i ,

150 200 250

r(K) 300

Fig. 2. Temperature dependence of magnetic susceptibility.

178 T. Suzuki et al. / Physica B 230-232 (1997) 176-178

where Eo, dr, F, N(E) and f are the band energy without strain, a deformation potential coupling coefficient, a symmetry representation, DOS and the Fermi distribution function, respectively. If the band is sufficiently narrow and d2N(EF) is sufficiently large, a significant lattice-softening is expected from the formula. In case that the magni- tude of the second term of Eq. (1) exceeds the back- ground, a structural transformation takes place. As the band theory for CeRu2 [4] gives a sharp peaked density of states (DOS)just above EF, we have tried a fit to data assuming a simplified DOS which is isosceles-triangular shaped with bandwidth 2W and the center of which shifts by S from E F. In the fitting, T dependence of Cr B6 is omitted because it is less than one-tenth of the softening found in CeRu2, and DOS at EF is kept as N(EF) = 160 states/Ryd Ce that was given by the theory [4]. The result of fit is shown by dashed curves in Fig. l(a) and (b). The best fit is obtained for ½(Cxl -C12) . The fitting parameters are listed in Table 1. The T-dependence of½(C1 a - C12) is well reproduced except the grad- ual hardening below the minimum point at 18.2 K. This discrepancy is due to the simplification of the DOS shape. We have preliminarily confirmed that not only the significant softening but also the grad- ual hardening is well reproduced by using the DOS that is given by recent band calculation [9]. For C,4, we made a fit using the same values of S and 2W obtained for ½(Cll - C12). As seen in Table 1, the smaller softening of C44 than that of ½(C1~ - C12) leads to the small value of dr Finally, the present analysis evidently suggests that there exists an electronic band with a structural instabil- ity giving rise to the giant softening in shear moduli but the magnitude of dZrN(Ev) is not sufficient for completing a substantial structural transition.

Table 1 The list of the fitting parameters

Mode ½(Cll C12) C44

2d2r N(EF) 6.818 GPa 4.774 GPa Cr BG 9.941 GPa 25.28 GPa 2 W 1000 K 1000 K S 76.66 K 76.66 K F E~ T2g

4. Conclusion

The ultrasonic measurements have been carried out on CeRu2. Giant lattice-softening without structural transition in shear moduli is found in the normal conducting state. In contrast to the shear moduli, the bulk modulus has no anomalous sof- tening. These are the evidence for the presence of structural instability aiming to reduce the crystal symmetry. The analysis by means of the deforma- tion potential coupling reveals that a degenerate band which splits in response to shearing strain gives rise to the giant softening. However, the mag- nitude of the deformation potential coupling co- efficient and DOS is insufficient for occurrence of the substantial structural transition.

Acknowledgements

The authors thank Prof. H. Har ima for showing the recent band calculation before publication and Profs. B. Liithi and M. Yoshizawa for fruitful dis- cussion. One of the authors (HG) is grateful to JSPS Research Fellowships for Young Scientists. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan.

References

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[2] R. Modler, P. Gegenwart, M. Lang, M. Deppe, M. Weiden, T. Lfihmann, C. Geibel, F. Steglich, C. Paulsen, J. L. Tholence, N. Sato, T. Komatsubara, Y. Onuki, M. Tachiki and S. Takahashi, Phys. Rev. Lett. 76 (1996) 1292.

[3] M. Hedo, Y. Inada, T. Ishida, E. Yamamoto, Y. Haga, Y Onuki, M. Higuchi and A. Hasegawa, J. Phys. Soc. Japan 64 (1995) 4535.

[4] M. Higuchi and A. Hasegawa, J. Phys. Soc. Japan 65 (1996) 1302.

[-5] B. Liithi, J. Magn. Magn. Mater 52 (1985) 70. [6] T. Suzuki, H. Goshima, S. Sakita, T. Fujita, M. Hedo, Y.

Inada, E. Yamamoto, Y. Haga and Y. Onuki, J. Phys. Soc. Japan 65 (1996) 2753.

[-7] M. Niksch, B. L/ithi and J. Kfibler, Z. Phys. B 68 (1987) 291, and references therein.

[-8] M. Nohara, T. Suzuki, Y. Maeno, T. Fujita, I. Tanaka and H. Kojima, Phys. Rev. B. 52 (1995) 570.

[9] H. Harima, private communication.