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Magnetic instabilities in CeRu2Si2 compounds
Physica B 259 —261 (1999) 48—53
Magnetic instabilities in CeRu Si compounds S. Raymond *, D. Raoelison , S. Kambe , L.P. Regnault , B. Fa k , R. Calemczuk , J. Flouquet , P. Haen, P. Lejay CEA-Grenoble, De& partement de Recherche Fondamentale sur la Matie% re Condense& e, SPSMS, 17 rue des Martyrs, 38054 Grenoble Cedex, France Centre de Recherche sur les Tre% s Basses Tempe& ratures, CNRS, BP 166, 38042 Grenoble Cedex, France
Abstract Neutron scattering experiments were recently performed in compounds of the CeRu Si family. The topic addressed is the proximity of magnetic instabilities. The first one is the proximity of a ¹"0 K quantum phase transition experimentally achieved with lanthanum doping. The second kind of instability is linked to the competition between antiferromagnetic and ferromagnetic interactions in these materials. The pseudo-metamagnetic transition of CeRu Si and the occurrence of an antiferromagnetic phase in the ferromagnetic compound CeRu Ge were studied. Comparisons with macroscopic measurements are made using a simple analysis of the data. 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic instability; Metamagnetism; CeRu Si ; Neutron scattering
1. Introduction Among heavy-fermion (HF) compounds, CeRu Si is a well-known enhanced Pauli paramagnet with a value of the linear coefficient c of the specific heat (C"c¹ at low temperatures) c"350 mJmol/K. This compound undergoes a pseudo-metamagnetic transition for a field H "7.8 T. Various anomalies observed at H in
macroscopic measurements (specific heat, susceptibility, resistivity, dilatometry) are indicative of a strong interplay between the magnetism and electronic and lattice properties [1]. In this paper we report recent experiments performed on CeRu Si and its alloys using inelastic neutron scattering (INS). Experiments performed on large single crystals using a triple axis spectrometer allow to measure point by point the dynamical spin susceptibil-
* Corresponding author. Tel.: 33-4-76-883738; fax: 33-4-7688-51-09; e-mail: [email protected].
ity s (Q, u) of the 4f electrons (For a review, see for example Ref. [2]). The spin dynamics of CeRu Si is characterized by short-range magnetic correlations of inelastic nature [3]. The spectrum is characteristic of spin fluctuations: the signal, broad in both Q and u space, is peaked around two instability vectors: k "(0.31, 0, 0) and k "(0.31, 0.31, 0). Recently, the existence of a third vector k "(0, 0, 0.35), which was not clearly observed in the sample of the Grenoble group, was observed by the Osaka group [4]. The application of a magnetic field clearly shows the occurrence of two components participating to the dynamical response by suppressing one of them [5,6]. The first one is quasielastic with a characteristic energy width (half-width at half-maximum of a Lorentzian line shape) C "2 meV independent of the 11 wavevector and is thus ascribed to single site effects. The second one is inelastic in nature with an inelasticity u "1.2 meV and a width C "0.75 meV for the vector '1 Q"k . It is characteristic of intersite effects with a cor relation length of the order of 3 unit cells. This latter contribution is suppressed by a magnetic field of 10 T while the former is independent of the magnetic field up
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 7 7 2 - 8
S. Raymond et al. / Physica B 259–261 (1999) 48—53
to 10 T. Alloying the pure compound CeRu Si induces long-range antiferromagnetic order for a concentration of 8% of lanthanum [7] and 5% of germanium, respectively [8]. More surprisingly, in the CeRu (Si Ge ) \V V compounds, a ferromagnetic ground state occurs for x'0.8 preceded by an antiferromagnetic phase [9]. In this context we report experiments performed on three systems: (i) Ce La Ru Si in order to compare the spin \V V dynamics for compounds with various x in relation with the renewed interest in ¹"0 K quantum phase transitions [10], (ii) CeRu Si under high magnetic field in order to study the competition between antiferromagnetic and ferromagnetic interactions [11], and (iii) CeRu Ge , a ferromagnet with ¹ +8 K and m +2l , in order to address the existence of the antiferromagnetic phase antecedent to the ferromagnetic one.
2. Analysis of the data Concerning the INS experiments, we will first comment on the phenomenological description used in these studies. Knowing that the scattering cross section consists of two components, a description of the experimental data must include two terms corresponding to intersite, s (Q, u), and single site, s (u), fluctuations, '1 11 respectively. In the past decade, the dynamical susceptibility was written as a sum of these two contributions [5,6] s(Q, u)"s (u)#s (Q, u). 11 '1
(1)
This formulation allows to describe the magnetic excitation spectrum over the whole Brillouin zone. Nevertheless, the parameters extracted from Eq. (1), which is purely phenomenological, are different from the one often found in theoretical work [12,13], where the random phase approximation (RPA) is used to describe the susceptibility: s (u) 11 s(Q, u)" . 1!J(Q)s (u) 11
49
with C "C (1!J(Q)s ) and s "s /(1!J(Q)s ). / 11 11 / 11 11 The intensity measured by INS is directly proportional to the scattering function which is equal to the imaginary part of the susceptibility multiplied by the Bose factor. A recent microscopic model that treats the RKKY and the Kondo effect on an equal footing derives a dynamical susceptibility composed of two terms [14], which are related to intraband and interband transitions in a hybridization gap picture. This calculation bears striking similarities to our INS results and a detailed comparison is promising.
3. Magnetic excitation spectrum near the magnetic instability A crystal of Ce La Ru Si with x"0.075( \V V x "0.08 was grown using the Czochralsky method. We have already reported elastic neutron scattering measurements [15] in this compound that show the occurrence of a small moment with a magnetic ordering for Q"k below ¹"1.8 K, with a correlation length of the order of 200 As and a value of the moment of 0.02 l at ¹"50 mK. Similar small moment magnetic order has also been observed in UPt and URu Si , but this may be related to a slight concentration disorder for Ce La Ru Si . The dynamics was studied [10] at the \V V wavevector Q"k , which does not exhibit magnetic ordering. The magnetic excitation spectrum measured around the vector k is shown in Fig. 1 for various wave vectors. The signal broadens with increasing q. For larger q values, the signal corresponds to local fluctuations as shown in the scan at Q"(0.44, 1, 0) in Fig. 1. For the wave vector k , the energy width extracted using the analysis described in Section 2 is found to be
(2)
Around the instability wavevector k , we use J(Q)" J(k )#Aq where Q"k #q. Far from k , the suscepti bility is simply s (u) considering that J(Q)"0. The 11 drawback of this analysis is that J(Q) has no simple expression over the whole Brillouin zone. The single site contribution is determined by an amplitude s and 11 a single energy scale C corresponding to the Kondo 11 temperature, i.e. the mechanism of the relaxation of the 4f moments. The resulting expression for the imaginary part of the total susceptibility is s uC s (Q, u)" / / u#C /
(3)
Fig. 1. Energy scans performed at Q"(0.69#q, 0.69!q, 0) and Q"(0.44, 1, 0) at ¹"1.5 K on 4F2 (LLB) in Ce La Ru Si .
50
S. Raymond et al. / Physica B 259–261 (1999) 48—53 Table 1 Parameters of the SCR spin fluctuation theory extracted from INS at 1.5 K
¹ (K) ¹ (K) J(k) (meV) y
x"0
x"0.075
11.3 14.2 3.3 0.23
11.0 16.5 2.5 0.07
Fig. 2. Wave vector dependence of the quantity J(k#q)s in 11 Ce La Ru Si with x"0, 0.075 at 1.5 K. \V V
C(Q"k )"0.3 meV. This value is much smaller than that of the pure compound where C(Q"k )"0.8 meV, thus stressing the decrease of the energy scale as the magnetic instability is approached. The enhancement factor J(Q)s can be extracted from the energy width of 11 these spectra as shown in Fig. 2. The larger value found in the doped sample is in agreement with the decreasing energy scale. In addition, its stronger wave vector dependence shows an increase of the correlation length. To go one step beyond a qualitative analysis of the data, the self-consistent renormalised (SCR) spin fluctuation theory [13] was used in order to link our measurements to the macroscopic ones, and in particular to the specific heat. In the phenomenological SCR theory, the starting point is an expression of the dynamical susceptibility using three parameters: ¹ describes the extension of the spin fluctuations in q space (¹ "Aq/2, where q is the zone boundary), ¹ describes the extension of the spin fluctuations in u space (¹ "¹ C s /n), and y de 11 11 scribes the proximity of the instability, y"1/(2¹ s ) /I (y"y at ¹"0 K and y P0 at the instability). These quantities can be directly extracted from our neutron scattering data. The results are given in Table 1. On the other hand, these parameters can be extracted from a fit of the specific heat at low temperatures [16] (Fig. 3). The corresponding parameters are given in Table 2. For each composition, a Fermi liquid (FL) behavior is always recovered at low enough temperatures estimated roughly to y ¹ . A general agreement is found between the para meters determined from those two different experimental techniques, except for J(k) which is significantly different. The model is in fact an oversimplification of the spin dynamics where anisotropy and degenerescence of the instability vectors are neglected. In Ref. [16], the agreement was not so good because the early analysis of the
Fig. 3. Temperature dependence of the magnetic specific heat of Ce La Ru Si for x"0, 0.05, 0.075. Solid lines are calcu\V V lations using the parameters given in Table 2.
Table 2 Parameters of the SCR spin fluctuation theory extracted from specific heat at low temperatures
¹ (K) ¹ (K) J(k) (meV) y
x"0
x"0.075
16 14.1 11 0.31
11 14.2 3.5 0.05
neutron scattering data was made using Eq. (1) rather than Eq. (2), the latter being closer to the expression of Moriya.
4. Magnetic excitation spectrum across the pseudo-metamagnetic transition We have reinvestigated the spin dynamics of CeRu Si on both sides of its pseudo-metamagnetic transition [11]. The antiferromagnetic correlations disappear in high magnetic fields: Figs. 4 and 5 show cuts of the response
S. Raymond et al. / Physica B 259–261 (1999) 48—53
51
Fig. 4. Energy scans performed at Q"(0.69, 0.69, 0) at ¹" 2.5 K for various magnetic fields on IN8 (ILL) in CeRu Si .
Fig. 6. (a) Bulk, staggered, and local susceptibility versus magnetic field of CeRu Si . (b) Field variation of the ferromagnetic and antiferromagnetic couplings.
Fig. 5. Q scans performed at constant u"1.6 meV at ¹" 2.5 K for various magnetic fields on IN8 (ILL) in CeRu Si .
function at constant Q"k and constant energy u" 1.6 meV, respectively. When the field increases the response broadens in energy as well as in wave vector. The instability wave vector does not change with field within the accuracy of our data. Attempts to measure the dynamical response characteristic of the ferromagnetic (H'H ) phase were unsuccessful. Only a long range
and static response was found in this phase, corresponding to the development of sharp ferromagnetic Bragg peaks. The change in the nature of the spin response function from short-range and inelastic at low fields to
long-range and elastic at high fields was described using a simple model based on the analysis of the data presented in Section 2. Knowing that the local susceptibility decreases in accordance with a single-impurity Kondo model while the uniform susceptibility increases nonlinearly and the staggered susceptibility also decreases as shown in Fig. 6a, we infer that the ferromagnetic exchange coupling must increase dramatically above H .
The variation of the ferromagnetic and antiferromagnetic couplings are shown in Fig. 6b. Both quantities cross near H with the scaling J(Q"0, H"H )+J(Q"k ,
H"H )+kH +C /2. The factor 2 between the ex
* change parameters and the Kondo-like energy scale C may be intrinsic (various measurements estimate * ¹ in the range 10—30 K) or may be an artefact of our ) simple model.
5. Competition between antiferromagnetism and ferromagnetism in CeRu2Ge2 Early measurements on this compound revealed the occurrence of a more or less pronounced antiferromagnetic phase antecedent to the ferromagnetic ground state
52
S. Raymond et al. / Physica B 259–261 (1999) 48—53
[17,18], raising the question of the intrinsic behavior of CeRu Ge . We performed elastic neutron scattering ex periments in order to measure the temperature dependence of the ferromagnetic and antiferromagnetic order parameters. Sample C1 was grown in Grenoble and exhibits a sharp transition at ¹ "7.7 K and a hump at ! ¹ "8.2 K in specific heat measurements and in the , derivative of the electrical resistivity as shown in Fig. 7. The linear coefficient of the specific heat is found to be c"22 mJ/molK and the data below ¹ are consistent ! with the occurrence of a gap in the spin wave spectrum of the order of 9 K. Sample C2 was grown in Toyama and shows only one anomaly in the derivative of the resistivity and in the specific heat at ¹ . Both samples exhibit ! long-range order with the incommensurate propagation vector k "(0.3, 0, 0) but with quite a different behavior related to their different macroscopic properties. In sample C1, the antiferromagnetic phase has a domain of existence of the order of 0.7 K with a Ne´el temperature ¹ "8.2 K and a moment of m(¹ )"0.5 l , where ,
¹ is the temperature at which the antiferromagnetic
signal is maximum. The neutron intensity at this point is shown in Fig. 8 together with the measurements performed at the ferromagnetic point. The ferromagnetic order parameter gives a value of ¹ +8 K. Sample C2 is characterized by a distribution of Ne´el temperatures around a nominal value estimated at 9 K and a moment m(¹ )"0.1 l . The temperature variation of the anti ferromagnetic intensity is shown in the inset of Fig. 8 for comparison with sample C1. The weakness of the magnetic moment explains why no anomaly is observed in the specific heat measurements for sample C2. Prelimi-
Fig. 8. Ferromagnetic and antiferromagnetic order parameters of CeRu Ge measured on IN12 (ILL). The inset shows the antiferromagnetic order parameter of sample C2.
nary measurements of the critical scattering was performed in sample C1 as well as inelastic neutron scattering measurements. Quasistatic ferromagnetic fluctuations appear below 30 K. As the temperature is lowered, antiferromagnetic fluctuations develop and transform into long range order at ¹ . INS measurements revealed , a magnetic excitation spectrum composed of two components as in CeRu Si . The energy scale is considerably lower than that in CeRu Si . A single site signal already observed in powder measurements [19] gives an estimate of the Kondo temperature of the order of 1 K. This single site signal of width C "0.15 meV at ¹"12.5 K is 11 enhanced at the ferromagnetic point (C "0.02 meV at '1 ¹"12.5 K). In the paramagnetic state, the compound is thus already near a ferromagnetic instability. More experiments are needed to study the spin dynamics and its temperature dependence in the paramagnetic phase.
6. Discussion
Fig. 7. Specific heat of CeRu Ge . The inset shows the deriva tive of the electrical resistivity.
Concerning the study of the magnetic instability, a similar scenario occurs in CeCu Au compounds, \V V which orders for x'0.1. This latter concentration was studied using INS technique by the Karlsruhe group. Qualitatively, the behavior is similar to the one we observed in our samples, with a reduction of the energy scale as the instability is approached [20]. A link to the non Fermi liquid properties observed in macroscopic measurements [21] in this compound was tentatively made, invoking a two-dimensional spin fluctuation spectrum [22,23]. This surprising possibility is still under investigation regarding new results from very recent
S. Raymond et al. / Physica B 259–261 (1999) 48—53
experiments performed with improved resolution [24]. The spin dynamics in this family of alloys is more complicated since the instability vector changes with the concentration. Also, the intrinsic behavior is restricted to very low temperatures due to low characteristic energy scales. When an analysis is made in the SCR spin fluctuation framework [16], a FL behavior can be expected to be restored below ¹"20 mK. Our observation of the change of nature of the spin dynamics of CeRu Si from short range and inelastic to long range and elastic across the pseudo-metamagnetic transition are in agreement with recent NMR measurements performed on both Si and Ru sites. The nuclear spin-lattice relaxation rate 1/¹ follows a Korringa law for fields up to 15 T, although its enhancement is strongly reduced above H . The field dependence of the spin-echo
decay rate 1/¹ is well explained by the contribution of the nuclear spins only. As in our studies, no ferromagnetic fluctuations are found above H [25,26]. A recent
theoretical model also describes the metamagnetic transition of CeRu Si with a magnetic coupling that depends on the field [27]. More generally, we can ascribe the competition between antiferromagnetism and ferromagnetism to exchange couplings varying with a configuration parameter, for example the volume of the unit cell as in the exchange inversion model [28]. This can be drawn from the large magnetoelastic effects observed in HF compounds. Within this picture, the Fermi surfaces of CeRu Ge in the ferromagnetic state and CeRu Si above H are rather similar [29—31]. The exchange coup lings (of RKKY nature) are directly linked to the susceptibility of the conduction electrons and hence to the topology of the Fermi surface. In the future more studies are needed to characterize the intrinsic behavior of CeRu Ge and understand the relation between the topology of the Fermi surface and the dominant magnetic interactions. Acknowledgements The measurements were performed on various triple axis spectrometers at Institut Laue Langevin (I.L.L.), Grenoble and Laboratoire Le´on Brillouin (L.L.B.), Saclay. We thank J.M. Mignot for useful discussions and
53
help during our stay at LLB and T. Fukhuara for providing us with a CeRu Ge sample (C2). References [1] J. Flouquet et al., Physica B 215 (1995) 77, and references therein. [2] G. Aeppli, C. Broholm, in: K.A. Gschneidner et al. (Eds.), Handbook on the physics and chemistry of rare earths, Vol. 19, Elsevier, Amsterdam, 1994, p. 123. [3] L.P. Regnault et al., Phys. Rev. B 38 (1988) 4481. [4] M. Sato et al., ISSP Report, Tokyo, 1997. [5] J.L. Jacoud et al., Physica B 156&157 (1989) 818. [6] J. Rossat-Mignod et al., J. Magn. Magn. Mat. 76&77 (1988) 376. [7] S. Que´zel et al., J. Magn. Magn. Mat. 76&77 (1988) 403. [8] S. Dakin et al., J. Magn. Magn. Mat. 108 (1992) 117. [9] P. Haen et al., J. Phys. Soc. Jpn. 65 (1996) Suppl. B 16, and references therein. [10] S. Raymond et al., J. Low Temp. Phys. 109 (1997) 205. [11] S. Raymond et al., J. Phys. Condens. Matter 10 (1998) 2363. [12] Y. Kuramoto, Physica B 156&157 (1989) 789. [13] T. Moriya, T. Takimoto, J. Phys. Soc. Jpn. 64 (1995) 960. [14] C. Pe´pin, M. Lavagna, in these Proceedings (SCES ’98), Physica B 259 —261 (1999). [15] S. Kambe et al., Physica B 223&224 (1996) 135. [16] S. Kambe et al., J. Phys. Soc. Jpn. 65 (1996) 3294. [17] M.J. Besnus et al., Physica B 171 (1991) 350. [18] J.D. Thompson et al., Physica B 199&200 (1994) 589. [19] B.D. Rainford et al., J. Magn. Magn. Mat. 108 (1992) 119. [20] A. Schro¨der et al., Physica B 241-243 (1998) 868. [21] H.v. Lo¨hneysen et al., Phys. Rev. Lett. 72 (1994) 3262. [22] A. Rosch et al., Phys. Rev. Lett. 79 (1997) 159. [23] A. Schro¨der et al., Phys. Rev. Lett. 80 (1998) 5623. [24] O. Stockert et al., Phys. Rev. Lett. 80 (1998) 5627. [25] K. Ishida et al., Phys. Rev. B 57 (1998) 11054. [26] T. Matsuda et al., in these Proceedings (SCES ’98), Physica B 259—261 (1999). [27] H. Sato, F.J. Okhawa, Phys. Rev. B 57 (1998) 5891. [28] C. Kittel, Phys. Rev. 120 (1960) 335. [29] C.A. King, G.G. Lonzarich, Physica B 171 (1991) 161. [30] H. Aoki et al., Physica B 206&207 (1995) 26. [31] F.S. Tautz et al., Physica B 206&207 (1995) 29.
Magnetic instabilities in CeRu Si compounds S. Raymond *, D. Raoelison , S. Kambe , L.P. Regnault , B. Fa k , R. Calemczuk , J. Flouquet , P. Haen, P. Lejay CEA-Grenoble, De& partement de Recherche Fondamentale sur la Matie% re Condense& e, SPSMS, 17 rue des Martyrs, 38054 Grenoble Cedex, France Centre de Recherche sur les Tre% s Basses Tempe& ratures, CNRS, BP 166, 38042 Grenoble Cedex, France
Abstract Neutron scattering experiments were recently performed in compounds of the CeRu Si family. The topic addressed is the proximity of magnetic instabilities. The first one is the proximity of a ¹"0 K quantum phase transition experimentally achieved with lanthanum doping. The second kind of instability is linked to the competition between antiferromagnetic and ferromagnetic interactions in these materials. The pseudo-metamagnetic transition of CeRu Si and the occurrence of an antiferromagnetic phase in the ferromagnetic compound CeRu Ge were studied. Comparisons with macroscopic measurements are made using a simple analysis of the data. 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic instability; Metamagnetism; CeRu Si ; Neutron scattering
1. Introduction Among heavy-fermion (HF) compounds, CeRu Si is a well-known enhanced Pauli paramagnet with a value of the linear coefficient c of the specific heat (C"c¹ at low temperatures) c"350 mJmol/K. This compound undergoes a pseudo-metamagnetic transition for a field H "7.8 T. Various anomalies observed at H in
macroscopic measurements (specific heat, susceptibility, resistivity, dilatometry) are indicative of a strong interplay between the magnetism and electronic and lattice properties [1]. In this paper we report recent experiments performed on CeRu Si and its alloys using inelastic neutron scattering (INS). Experiments performed on large single crystals using a triple axis spectrometer allow to measure point by point the dynamical spin susceptibil-
* Corresponding author. Tel.: 33-4-76-883738; fax: 33-4-7688-51-09; e-mail: [email protected].
ity s (Q, u) of the 4f electrons (For a review, see for example Ref. [2]). The spin dynamics of CeRu Si is characterized by short-range magnetic correlations of inelastic nature [3]. The spectrum is characteristic of spin fluctuations: the signal, broad in both Q and u space, is peaked around two instability vectors: k "(0.31, 0, 0) and k "(0.31, 0.31, 0). Recently, the existence of a third vector k "(0, 0, 0.35), which was not clearly observed in the sample of the Grenoble group, was observed by the Osaka group [4]. The application of a magnetic field clearly shows the occurrence of two components participating to the dynamical response by suppressing one of them [5,6]. The first one is quasielastic with a characteristic energy width (half-width at half-maximum of a Lorentzian line shape) C "2 meV independent of the 11 wavevector and is thus ascribed to single site effects. The second one is inelastic in nature with an inelasticity u "1.2 meV and a width C "0.75 meV for the vector '1 Q"k . It is characteristic of intersite effects with a cor relation length of the order of 3 unit cells. This latter contribution is suppressed by a magnetic field of 10 T while the former is independent of the magnetic field up
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 7 7 2 - 8
S. Raymond et al. / Physica B 259–261 (1999) 48—53
to 10 T. Alloying the pure compound CeRu Si induces long-range antiferromagnetic order for a concentration of 8% of lanthanum [7] and 5% of germanium, respectively [8]. More surprisingly, in the CeRu (Si Ge ) \V V compounds, a ferromagnetic ground state occurs for x'0.8 preceded by an antiferromagnetic phase [9]. In this context we report experiments performed on three systems: (i) Ce La Ru Si in order to compare the spin \V V dynamics for compounds with various x in relation with the renewed interest in ¹"0 K quantum phase transitions [10], (ii) CeRu Si under high magnetic field in order to study the competition between antiferromagnetic and ferromagnetic interactions [11], and (iii) CeRu Ge , a ferromagnet with ¹ +8 K and m +2l , in order to address the existence of the antiferromagnetic phase antecedent to the ferromagnetic one.
2. Analysis of the data Concerning the INS experiments, we will first comment on the phenomenological description used in these studies. Knowing that the scattering cross section consists of two components, a description of the experimental data must include two terms corresponding to intersite, s (Q, u), and single site, s (u), fluctuations, '1 11 respectively. In the past decade, the dynamical susceptibility was written as a sum of these two contributions [5,6] s(Q, u)"s (u)#s (Q, u). 11 '1
(1)
This formulation allows to describe the magnetic excitation spectrum over the whole Brillouin zone. Nevertheless, the parameters extracted from Eq. (1), which is purely phenomenological, are different from the one often found in theoretical work [12,13], where the random phase approximation (RPA) is used to describe the susceptibility: s (u) 11 s(Q, u)" . 1!J(Q)s (u) 11
49
with C "C (1!J(Q)s ) and s "s /(1!J(Q)s ). / 11 11 / 11 11 The intensity measured by INS is directly proportional to the scattering function which is equal to the imaginary part of the susceptibility multiplied by the Bose factor. A recent microscopic model that treats the RKKY and the Kondo effect on an equal footing derives a dynamical susceptibility composed of two terms [14], which are related to intraband and interband transitions in a hybridization gap picture. This calculation bears striking similarities to our INS results and a detailed comparison is promising.
3. Magnetic excitation spectrum near the magnetic instability A crystal of Ce La Ru Si with x"0.075( \V V x "0.08 was grown using the Czochralsky method. We have already reported elastic neutron scattering measurements [15] in this compound that show the occurrence of a small moment with a magnetic ordering for Q"k below ¹"1.8 K, with a correlation length of the order of 200 As and a value of the moment of 0.02 l at ¹"50 mK. Similar small moment magnetic order has also been observed in UPt and URu Si , but this may be related to a slight concentration disorder for Ce La Ru Si . The dynamics was studied [10] at the \V V wavevector Q"k , which does not exhibit magnetic ordering. The magnetic excitation spectrum measured around the vector k is shown in Fig. 1 for various wave vectors. The signal broadens with increasing q. For larger q values, the signal corresponds to local fluctuations as shown in the scan at Q"(0.44, 1, 0) in Fig. 1. For the wave vector k , the energy width extracted using the analysis described in Section 2 is found to be
(2)
Around the instability wavevector k , we use J(Q)" J(k )#Aq where Q"k #q. Far from k , the suscepti bility is simply s (u) considering that J(Q)"0. The 11 drawback of this analysis is that J(Q) has no simple expression over the whole Brillouin zone. The single site contribution is determined by an amplitude s and 11 a single energy scale C corresponding to the Kondo 11 temperature, i.e. the mechanism of the relaxation of the 4f moments. The resulting expression for the imaginary part of the total susceptibility is s uC s (Q, u)" / / u#C /
(3)
Fig. 1. Energy scans performed at Q"(0.69#q, 0.69!q, 0) and Q"(0.44, 1, 0) at ¹"1.5 K on 4F2 (LLB) in Ce La Ru Si .
50
S. Raymond et al. / Physica B 259–261 (1999) 48—53 Table 1 Parameters of the SCR spin fluctuation theory extracted from INS at 1.5 K
¹ (K) ¹ (K) J(k) (meV) y
x"0
x"0.075
11.3 14.2 3.3 0.23
11.0 16.5 2.5 0.07
Fig. 2. Wave vector dependence of the quantity J(k#q)s in 11 Ce La Ru Si with x"0, 0.075 at 1.5 K. \V V
C(Q"k )"0.3 meV. This value is much smaller than that of the pure compound where C(Q"k )"0.8 meV, thus stressing the decrease of the energy scale as the magnetic instability is approached. The enhancement factor J(Q)s can be extracted from the energy width of 11 these spectra as shown in Fig. 2. The larger value found in the doped sample is in agreement with the decreasing energy scale. In addition, its stronger wave vector dependence shows an increase of the correlation length. To go one step beyond a qualitative analysis of the data, the self-consistent renormalised (SCR) spin fluctuation theory [13] was used in order to link our measurements to the macroscopic ones, and in particular to the specific heat. In the phenomenological SCR theory, the starting point is an expression of the dynamical susceptibility using three parameters: ¹ describes the extension of the spin fluctuations in q space (¹ "Aq/2, where q is the zone boundary), ¹ describes the extension of the spin fluctuations in u space (¹ "¹ C s /n), and y de 11 11 scribes the proximity of the instability, y"1/(2¹ s ) /I (y"y at ¹"0 K and y P0 at the instability). These quantities can be directly extracted from our neutron scattering data. The results are given in Table 1. On the other hand, these parameters can be extracted from a fit of the specific heat at low temperatures [16] (Fig. 3). The corresponding parameters are given in Table 2. For each composition, a Fermi liquid (FL) behavior is always recovered at low enough temperatures estimated roughly to y ¹ . A general agreement is found between the para meters determined from those two different experimental techniques, except for J(k) which is significantly different. The model is in fact an oversimplification of the spin dynamics where anisotropy and degenerescence of the instability vectors are neglected. In Ref. [16], the agreement was not so good because the early analysis of the
Fig. 3. Temperature dependence of the magnetic specific heat of Ce La Ru Si for x"0, 0.05, 0.075. Solid lines are calcu\V V lations using the parameters given in Table 2.
Table 2 Parameters of the SCR spin fluctuation theory extracted from specific heat at low temperatures
¹ (K) ¹ (K) J(k) (meV) y
x"0
x"0.075
16 14.1 11 0.31
11 14.2 3.5 0.05
neutron scattering data was made using Eq. (1) rather than Eq. (2), the latter being closer to the expression of Moriya.
4. Magnetic excitation spectrum across the pseudo-metamagnetic transition We have reinvestigated the spin dynamics of CeRu Si on both sides of its pseudo-metamagnetic transition [11]. The antiferromagnetic correlations disappear in high magnetic fields: Figs. 4 and 5 show cuts of the response
S. Raymond et al. / Physica B 259–261 (1999) 48—53
51
Fig. 4. Energy scans performed at Q"(0.69, 0.69, 0) at ¹" 2.5 K for various magnetic fields on IN8 (ILL) in CeRu Si .
Fig. 6. (a) Bulk, staggered, and local susceptibility versus magnetic field of CeRu Si . (b) Field variation of the ferromagnetic and antiferromagnetic couplings.
Fig. 5. Q scans performed at constant u"1.6 meV at ¹" 2.5 K for various magnetic fields on IN8 (ILL) in CeRu Si .
function at constant Q"k and constant energy u" 1.6 meV, respectively. When the field increases the response broadens in energy as well as in wave vector. The instability wave vector does not change with field within the accuracy of our data. Attempts to measure the dynamical response characteristic of the ferromagnetic (H'H ) phase were unsuccessful. Only a long range
and static response was found in this phase, corresponding to the development of sharp ferromagnetic Bragg peaks. The change in the nature of the spin response function from short-range and inelastic at low fields to
long-range and elastic at high fields was described using a simple model based on the analysis of the data presented in Section 2. Knowing that the local susceptibility decreases in accordance with a single-impurity Kondo model while the uniform susceptibility increases nonlinearly and the staggered susceptibility also decreases as shown in Fig. 6a, we infer that the ferromagnetic exchange coupling must increase dramatically above H .
The variation of the ferromagnetic and antiferromagnetic couplings are shown in Fig. 6b. Both quantities cross near H with the scaling J(Q"0, H"H )+J(Q"k ,
H"H )+kH +C /2. The factor 2 between the ex
* change parameters and the Kondo-like energy scale C may be intrinsic (various measurements estimate * ¹ in the range 10—30 K) or may be an artefact of our ) simple model.
5. Competition between antiferromagnetism and ferromagnetism in CeRu2Ge2 Early measurements on this compound revealed the occurrence of a more or less pronounced antiferromagnetic phase antecedent to the ferromagnetic ground state
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S. Raymond et al. / Physica B 259–261 (1999) 48—53
[17,18], raising the question of the intrinsic behavior of CeRu Ge . We performed elastic neutron scattering ex periments in order to measure the temperature dependence of the ferromagnetic and antiferromagnetic order parameters. Sample C1 was grown in Grenoble and exhibits a sharp transition at ¹ "7.7 K and a hump at ! ¹ "8.2 K in specific heat measurements and in the , derivative of the electrical resistivity as shown in Fig. 7. The linear coefficient of the specific heat is found to be c"22 mJ/molK and the data below ¹ are consistent ! with the occurrence of a gap in the spin wave spectrum of the order of 9 K. Sample C2 was grown in Toyama and shows only one anomaly in the derivative of the resistivity and in the specific heat at ¹ . Both samples exhibit ! long-range order with the incommensurate propagation vector k "(0.3, 0, 0) but with quite a different behavior related to their different macroscopic properties. In sample C1, the antiferromagnetic phase has a domain of existence of the order of 0.7 K with a Ne´el temperature ¹ "8.2 K and a moment of m(¹ )"0.5 l , where ,
¹ is the temperature at which the antiferromagnetic
signal is maximum. The neutron intensity at this point is shown in Fig. 8 together with the measurements performed at the ferromagnetic point. The ferromagnetic order parameter gives a value of ¹ +8 K. Sample C2 is characterized by a distribution of Ne´el temperatures around a nominal value estimated at 9 K and a moment m(¹ )"0.1 l . The temperature variation of the anti ferromagnetic intensity is shown in the inset of Fig. 8 for comparison with sample C1. The weakness of the magnetic moment explains why no anomaly is observed in the specific heat measurements for sample C2. Prelimi-
Fig. 8. Ferromagnetic and antiferromagnetic order parameters of CeRu Ge measured on IN12 (ILL). The inset shows the antiferromagnetic order parameter of sample C2.
nary measurements of the critical scattering was performed in sample C1 as well as inelastic neutron scattering measurements. Quasistatic ferromagnetic fluctuations appear below 30 K. As the temperature is lowered, antiferromagnetic fluctuations develop and transform into long range order at ¹ . INS measurements revealed , a magnetic excitation spectrum composed of two components as in CeRu Si . The energy scale is considerably lower than that in CeRu Si . A single site signal already observed in powder measurements [19] gives an estimate of the Kondo temperature of the order of 1 K. This single site signal of width C "0.15 meV at ¹"12.5 K is 11 enhanced at the ferromagnetic point (C "0.02 meV at '1 ¹"12.5 K). In the paramagnetic state, the compound is thus already near a ferromagnetic instability. More experiments are needed to study the spin dynamics and its temperature dependence in the paramagnetic phase.
6. Discussion
Fig. 7. Specific heat of CeRu Ge . The inset shows the deriva tive of the electrical resistivity.
Concerning the study of the magnetic instability, a similar scenario occurs in CeCu Au compounds, \V V which orders for x'0.1. This latter concentration was studied using INS technique by the Karlsruhe group. Qualitatively, the behavior is similar to the one we observed in our samples, with a reduction of the energy scale as the instability is approached [20]. A link to the non Fermi liquid properties observed in macroscopic measurements [21] in this compound was tentatively made, invoking a two-dimensional spin fluctuation spectrum [22,23]. This surprising possibility is still under investigation regarding new results from very recent
S. Raymond et al. / Physica B 259–261 (1999) 48—53
experiments performed with improved resolution [24]. The spin dynamics in this family of alloys is more complicated since the instability vector changes with the concentration. Also, the intrinsic behavior is restricted to very low temperatures due to low characteristic energy scales. When an analysis is made in the SCR spin fluctuation framework [16], a FL behavior can be expected to be restored below ¹"20 mK. Our observation of the change of nature of the spin dynamics of CeRu Si from short range and inelastic to long range and elastic across the pseudo-metamagnetic transition are in agreement with recent NMR measurements performed on both Si and Ru sites. The nuclear spin-lattice relaxation rate 1/¹ follows a Korringa law for fields up to 15 T, although its enhancement is strongly reduced above H . The field dependence of the spin-echo
decay rate 1/¹ is well explained by the contribution of the nuclear spins only. As in our studies, no ferromagnetic fluctuations are found above H [25,26]. A recent
theoretical model also describes the metamagnetic transition of CeRu Si with a magnetic coupling that depends on the field [27]. More generally, we can ascribe the competition between antiferromagnetism and ferromagnetism to exchange couplings varying with a configuration parameter, for example the volume of the unit cell as in the exchange inversion model [28]. This can be drawn from the large magnetoelastic effects observed in HF compounds. Within this picture, the Fermi surfaces of CeRu Ge in the ferromagnetic state and CeRu Si above H are rather similar [29—31]. The exchange coup lings (of RKKY nature) are directly linked to the susceptibility of the conduction electrons and hence to the topology of the Fermi surface. In the future more studies are needed to characterize the intrinsic behavior of CeRu Ge and understand the relation between the topology of the Fermi surface and the dominant magnetic interactions. Acknowledgements The measurements were performed on various triple axis spectrometers at Institut Laue Langevin (I.L.L.), Grenoble and Laboratoire Le´on Brillouin (L.L.B.), Saclay. We thank J.M. Mignot for useful discussions and
53
help during our stay at LLB and T. Fukhuara for providing us with a CeRu Ge sample (C2). References [1] J. Flouquet et al., Physica B 215 (1995) 77, and references therein. [2] G. Aeppli, C. Broholm, in: K.A. Gschneidner et al. (Eds.), Handbook on the physics and chemistry of rare earths, Vol. 19, Elsevier, Amsterdam, 1994, p. 123. [3] L.P. Regnault et al., Phys. Rev. B 38 (1988) 4481. [4] M. Sato et al., ISSP Report, Tokyo, 1997. [5] J.L. Jacoud et al., Physica B 156&157 (1989) 818. [6] J. Rossat-Mignod et al., J. Magn. Magn. Mat. 76&77 (1988) 376. [7] S. Que´zel et al., J. Magn. Magn. Mat. 76&77 (1988) 403. [8] S. Dakin et al., J. Magn. Magn. Mat. 108 (1992) 117. [9] P. Haen et al., J. Phys. Soc. Jpn. 65 (1996) Suppl. B 16, and references therein. [10] S. Raymond et al., J. Low Temp. Phys. 109 (1997) 205. [11] S. Raymond et al., J. Phys. Condens. Matter 10 (1998) 2363. [12] Y. Kuramoto, Physica B 156&157 (1989) 789. [13] T. Moriya, T. Takimoto, J. Phys. Soc. Jpn. 64 (1995) 960. [14] C. Pe´pin, M. Lavagna, in these Proceedings (SCES ’98), Physica B 259 —261 (1999). [15] S. Kambe et al., Physica B 223&224 (1996) 135. [16] S. Kambe et al., J. Phys. Soc. Jpn. 65 (1996) 3294. [17] M.J. Besnus et al., Physica B 171 (1991) 350. [18] J.D. Thompson et al., Physica B 199&200 (1994) 589. [19] B.D. Rainford et al., J. Magn. Magn. Mat. 108 (1992) 119. [20] A. Schro¨der et al., Physica B 241-243 (1998) 868. [21] H.v. Lo¨hneysen et al., Phys. Rev. Lett. 72 (1994) 3262. [22] A. Rosch et al., Phys. Rev. Lett. 79 (1997) 159. [23] A. Schro¨der et al., Phys. Rev. Lett. 80 (1998) 5623. [24] O. Stockert et al., Phys. Rev. Lett. 80 (1998) 5627. [25] K. Ishida et al., Phys. Rev. B 57 (1998) 11054. [26] T. Matsuda et al., in these Proceedings (SCES ’98), Physica B 259—261 (1999). [27] H. Sato, F.J. Okhawa, Phys. Rev. B 57 (1998) 5891. [28] C. Kittel, Phys. Rev. 120 (1960) 335. [29] C.A. King, G.G. Lonzarich, Physica B 171 (1991) 161. [30] H. Aoki et al., Physica B 206&207 (1995) 26. [31] F.S. Tautz et al., Physica B 206&207 (1995) 29.