CEF~Excitation ~I
0 200 400 600 B00 Fpequency Shift (cm-i)
Fig. 14. U n p o l a r i z e d R a m a n spectra of different R B 6 ( R = La, Ce, Nd). Magnetic excitations are o b s e r v e d in CeB 6 n e a r 372 cm -1 and in N d B 6 n e a r 95 cm -1. In all t h r e e spectra o n e observes t h e R a m a n - a c t i v e T2g p h o n o n n e a r 680 cm 1 and a p h o n o n density of states n e a r 180 cm 1.
In the case of cubic crystal symmetry the F1, F 3 and F 5 symmetry components are R a m a n allowed. In fig. 15 the well known vibrations of the B 6 octahedra have been included to demonstrate the corresponding symmetry. T h e peak near 95 cm-1 appears only in F 5 symmetry with t h e / ' 1 and F 3 components being zero.
T h e symmetry analysis is consistent with the identification of the 95 cm-1 line as due to a crystal-field excitation, but does not allow a separation of the two transitions.
Figure 16 shows the t e m p e r a t u r e d e p e n d e n c e of the magnetic excitation of NdB 6 near 95 cm -1 together with the p h o n o n density of states near 170cm -1 (Pofahl et al. 1986). A t 300 K the peak at 95 cm -~ is the center of two transitions
F (2)---->
F~ 1) and the /.~1)__+/~6" T h eexcitation /~2)._._). &
( 2 4 m e V = 190cm -1) is neither observable in R a m a n scattering nor in n e u t r o n scattering due to the small transition matrix element ( L o e w e n h a u p t and Prager 1986). By cooling from 300 K down to 7 K, Pofahl and co-workers find a shift of the center of the excitation from 95 cm -~ to 92 cm -1 and a decrease in the linewidth of the peak from 36 cm -1 to 16 c m - 1 ( f i g . 16).Measuring with a high resolution of 2 c m - 1 and by extending the m e a s u r e m e n t time by a factor of about ten, Pofahl and co-workers could resolve at 300 K, as
L I G H T S C A T T E R I N G IN I N T E R M E T A L L I C C O M P O U N D S 183
g
°I
°I I/1 ¢-
ol t -
I:1 ¢.) t O
i , 22, i 22, ' I '
I I ' ,,3
I ~. vibrotions ..
I • of lhe
[ unpolorized .---B6-oclohedron '.
i ~ ~ ~ \..
I T= 77K
T2g (F~)-Symmetry I
I . :..
:~..:~: • ..~, ,...~.~;,,,,,,; ~;..-.,::~'!': ",.< , .,...: ,.. y.'.:...,~.~,~'2t, . . . =--: - .
I
i
Alg+4Eg (q +4F3)-Symmetry
I i-
100 200 550 700 1000 1200 1400 F r e q u e n c y S h i f t (cM 1}
Fig, 15. S y m m e t r y analysis of t h e C E F tran- sitions and the R a m a n - a c t i v e p h o n o n s of NdB 6 for t h e (110) face at 7 7 K . T h e un- polarized s p e c t r u m is s h o w n at t h e top.
g
0
!
¢-
I:I ¢ J
tO
0
I I I I I
C E F : 9 2 c m -1
i P h o n o n
~
.... 7K7 7 K
• ;" . . . . " ~ "ii
I I I i
50 100 150 200 250
F r e q u e n c y S h i f t (¢m -1)
Fig. 16. R a m a n scattering intensities of N d B 6 at different temperatures• T h e p e a k at 170 cm -1 corresponds to t h e p h o n o n density of states and decreases u p o n cooling due to the Bose factor, By cooling down f r o m 300 K to 7 K the center of t h e C E F transitions shifts f r o m 95 cm -1 to 92 cm ~.
184 E. Z I R N G I E B L and G. GI~INTHERODT
¢..
3
. Q
¢.. I11
E
r -
tt) u
I I I
92 98 CI~ 1
I I
... • ':.' - 3 0 0 K ... ;:..".". ".~">...~.•.: ..
92 98 crff 1
I I
:: ."
,: : :"::~:~, 77 K
.(..
.,:..',....:~./.:. ;.:'
50
I
75 100 125
F r e q u e n c y S h i f t (cm -1)
Fig. 17. R a m a n spectra of the CEF transitions of NdB 6 at 300 K and 7 7 K measured with a high instrumental resolution of 2 cm -1 (=0.25 meV).
well as 77 K, two peaks at 92cm -1 and at 98 cm -1 as shown in fig. 17. A t room temperature both peaks have the same intensity• At 77 K the intensity of the excitation at 98 cm -a has decreased compared to that of the peak at 92 cm -a.
These experimental results can only be explained by a C E F level scheme with the /,~1) state 92 cm -~ above the F~ 2> ground state and the F 6 state 98 cm -a above the F~ I) state• At 300K both F 8 states are nearly equally populated, yielding comparable intensities of the two peaks at 92 cm-1 and 98 cm-1 in fig. 17. A t low temperatures (77 K), the/,~1) state is less populated than the F~ 2) state, yielding an increased intensity of the F~2)1---~ F~ ~) excitation at 92 cm -1 compared to the F(aa)---~ F 6 excitation near 98 c m - . Taking into account the observation of two peaks of equal intensity at high temperatures and the non-observance of a third one due to its weak intensity, one can find only one point in the L e a - L e a s k - W o l f scheme of Nd 3÷ for cubic crystal fields that fits all the results: hence Pofahl and co-workers obtained x = - 0 . 8 2 and W = - 2 . 7 6 c m -1. The resulting levels, to- gether with the transition probabilities between the levels, are indicated in fig. 18.
~(1}
G(2).
273K (190 cm -11
I
9.5 132K (92crn -11I
9.5 0.3 OK (0crn -1)Fig. 18. CEF level scheme of NdB 6 with the values of the corresponding transition matrix ele- ments indicated.
LIGHT SCATTERING IN INTERMETALLIC COMPOUNDS 185 2.6. Light scattering from CEF excitations in CeCu2Si 2
A great deal of attention has been recently afforded to "heavy-fermion"
systems, a group of intermetallic compounds which behave like localized f- moment systems at high temperatures, but yet display many features of a simple Fermi-liquid at low temperatures [for a review of the activities on these materials, see Stewart (1984), and Grewe and Steglich (chapter X in this handbook)]. Of special interest are three such materials, CeCu2Si 2 (Steglich et al. 1979b), UBe13 (Ott et al. 1983) and UPt 3 (Stewart et al. 1984), which have been shown to possess a superconducting ground state in which the heavy f electrons are thought to participate in spite of their room temperature predisposition towards localized magnetism. Given its evident importance, the 4f-electronic excitation spectrum of CeCu2Si 2 has been widely investigated by neutron scattering to characterize the magnetic fluctuations of the 4f electrons and the effects of the CEF on the Ce multiplet. CEF excitations were first observed with levels reported by Horn et al.
(1981b) at 140 K and 364 K (100cm -1 and 260cm -1, respectively), whereas subsequent neutron scattering studies, while clearly observing the peak at higher energy, have been unable to confirm the lower energy transition (Stassis et al.
1986, Johnson et al. 1985).
Electronic Raman scattering experiments on oriented single-crystal samples of CeCu2Si 2 were performed by Cooper et al. (1986) using a polarized 4880 or 5145 A line of an argon laser as an excitation source.
In tetragonal surrounding the Ce ~+ (J-- ~-) multiplet is expected to split into the three doublets with two of F 7 symmetry and one of F 6 symmetry. Electronic transitions between these levels should manifest the symmetries allowed by the direct products of these states:
Cooper et al. (1986) reported on the observation of crystal-field excitations of CeCu2Si 2 as being a broad hump centered roughly at 290 cm -1 in the Raman spectra of CeCu2Si 2 as shown in fig. 19. For comparison the spectrum of the isostructural d-band metal LaCu2Si 2 is also shown, exhibiting no Raman signal around 290 cm -1, thus strongly confirming the interpretation of the 290 cm -1 excitation in CeCu2Si 2 as being due to 4f electrons. This identification is further supported by the temperature dependence of the h2g + Big spectrum (Cooper et al. 1986). As expected of electronic transitions, the crystal-field peak at 290 cm -1 narrows and becomes stronger as the temperature is lowered, mimicking the sharpening Fermi factor. The appearance of the crystal-field peak in the (A2g q- Big )- as well as the (A2g + B2g )- symmetry-type Raman spectra confirms that it has the symmetry of the purely antisymmetric representation of the CeCu2Si 2 space group, A2g , characteristic of a F7-F 7 transition. No evidence of a FT-F 6 transition is seen according to Cooper and co-workers (Cooper et al. 1986).
186 E. ZIRNGIEBL and G. G/.)NTHERODT 4 0 0
.-'= 500 ~ c "
>._ 200- k- o9 7- LH I - IO0- z
A2g+ B2g 3 0 K
LaCu2Si2~ ~
Fig. 19. Comparison of A2g +Bzg spectrum of Ce- Cu2Si 2 (upper) with that of
0 , LaCu2Si 2 (lower) at 30K.
0 160 260 500 460 5(30 6 ( 3 0 760 800 Resolution, 10cm -1. After
ENERGY SHIFT (cm -I) Cooper et al. (1986).
The temperature dependence of the crystal-field linewidth [full width at half maximum (FWHM)] observed by Cooper et al. (1986) is shown in fig. 20. This temperature dependence has been calculated only for cubic Kondo systems (Becker et al. 1977, Lopes and Coqblin 1986), but the results adequately describe the general features of the observed linewidth in anisotropic CeCu2Si 2. In these models, the dominant damping mechanism at high temperatures (A < T, with A the crystal-field splitting) results from elastic scattering (i.e., the creation of electron-hole pairs) within each of the crystal-field levels, giving a linear depen- dence of linewidth on temperature (Lopes and Coqblin 1986):
F = 4~-[n(EF)]2(IJ7712 +
214812)T,
a < Z.Here, J77 and J88 are exchange integrals between electrons within the F 7 and F 8 crystal-field levels, respectively, while n(EF) is the conduction-band density of states at the Fermi energy.
At low temperatures (T < A), damping chiefly results from transitions between the crystal-field levels, promoted by the exchange interaction between the conduc- tion electrons and the Ce 3÷ 4f electron. This leads to a saturation of the linewidth at sufficiently low temperatures, as described by Lopes and Coqblin (1986)
F = 4~'[n(EF)I 2
(IJT,12A
1 + 2e-a/r~ j , T < a .k
The crystal-field splitting is described by zl in this relation, while J78 is the exchange term between the F 7 and F 8 levels.
An informal application of these results to CeCu2Si 2 (A = 406 K), presuming equal exchange terms and a weak splitting of the F 8 level, indicates that these two mechanisms should be of roughly equal importance down to about 160 K. Below this temperature, the inelastic damping term quickly begins to predominate. This
170 160- I E o 150-
v
T
~n 140- LLt 7 130 l
120- ~ J
II0
LIGHT SCATTERING IN INTERMETALLIC COMPOUNDS 187
o sb ,oo ,6o z6o zso soo sso 400
T E M P E R A T U R E (K)
Fig. 20. Observed crystal-field linewidth (FWHM) vs. tem- perature for the
Azg
crystal- field peak in CeCu2Si 2. Res- olution: 10cm -1. Line drawn is a guide to the eye. After Cooper et al. (1986).behavior is reflected in the linewidth, observed by Cooper et al. (1986), wherein one notes a linear dependence above 200 K, with saturation occurring at lower temperatures (see fig. 20).
3. Phonon Raman scattering 3.1. Introduction
The primary interest in investigating mixed-valence materials using Raman scattering arose from the estimate (Varma 1976) that the inverse timescale of the 4f charge fluctuations inducing volume changes of up to 15%, may be of the order of the phonon frequencies. Hence, early Raman scattering experiments in inter- mediate-valence materials by G/intherodt et al. (1977a, 1978, 1981b, c), Treindl and Wachter (1979, 19~0) and Stiisser et al. (1981, 1982) were concerned with the investigation of phonon anomalies and their relation to the electron-phonon interaction. In particular, polarized Raman scattering (G/intherodt et al. 1981b, c, Kress et al. 1981) has provided an experimental test of the relative importance of the different charge deformabilities introduced in the lattice dynamical model calculations (G/intherodt et ah 1981b, Kress et al. 1981, Bilz et al. 1979).
The interaction of phonons with the 4f electrons in IV compounds has been studied theoretically in different and complementary terms by a large number of authors (Ghatak and Bennemann 1978, Grewe and Entel 1979, Bennemann and Avignon 1980, Entel and Sietz 1981, Miura and Bilz 1986, Wakabayashi 1980, Matsuara et ah 1980). On the other hand, the experimental evidence for phonon frequency renormalizations due to valence fluctuations is on the whole rather limited. Phonon anomalies have been observed only for some IV compounds of Sm (Mook et al. 1978a, 1982, Hillebrands and G/intherodt 1983, Giintherodt et al. 1978), of Tm (Mook and Holtzberg 1981, Treindl and Wachter 1979), of Yb
188 E. Z I R N G I E B L and G. G O N T H E R O D T
(Giintherodt et al. 1983, 1985a) and only in two cases of Ce (Giintherodt et al.
1983, 1985b, Blumenr6der et al. 1985). Moreover, no consistent interpretation of these observed anomalies has been given so far based on microscopic concepts.
Therefore, we shall review the accumulated data only in the framework of some phenomenological ideas first introduced by Zirngiebl and co-workers (Zirngiebl et al. 1986b, Mock et al. 1986). We shall discuss the behavior of acoustic and optical phonon mode frequencies hw in IV compounds according to the following classification regimes: h~o >> F c and hco -< Fc, where F~ denotes the charge fluctua- tion rate. For hw >> F c the phonon "sees" a static mixture of divalent ions. The mode frequency behavior in alloys of stable divalent and trivalent R ions as found, e.g., in Gd~_xEUxB 6 by Ishii et al. (1976a) is shown in fig. 21 and varies nearly linearly as a function of the averaged valence. On the basis of this result one expects the phonon mode frequencies of the IV compound to be intermediate between those of the divalent and trivalent reference compounds according to the valence mixing ratio. In the phonon frequency range characterized by ho)-< Fc, however, the phonon should soften compared to the stable valence reference compounds, because the charge fluctuation rate can easily follow the movement of the ions and thereby soften the lattice.
It has been emphasized (Mock et al. 1986) that besides a systematic under- standing of the occurrence or absence of elastic and phonon anomalies in different IV compounds the concept introduced above allows for a first experimental estimate of charge fluctuation rates. The direct experimental investigation of these has not been feasible, unlike the magnetic relaxation rates, which have been investigated intensively by quasielastic neutron scattering (Holland-Moritz et al.
LATTICE CONSTANT (~) 4.110 4.118 4.127 4.]32 4.138 4.157 4.178 1300'
•EE
O 1200,z n l
1100
ILl ee
ii
":'--,~. Alg
o ~
~ o