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Spectroscopic evidence of a Kondo scale in cerium compounds
PHYSICA[
Physica B 186-188 (1993) 38-43 North-Holland
Spectroscopic evidence of a Kondo scale in cerium compounds M. Grioni, D. Malterre, P. Weibel, B. Dardel and Y. Baer lnstitut de Physique, Universit~ de Neuchgttel, Switzerland
We present a high-resolution photoemission (PES) and temperature-dependent inverse photoemission (BIS) investigation of Ce compounds, and in particular, of the paradigmatic system CeSix. The PES intensity near the Fermi level follows, as in the calculated spectral function of the Anderson Hamiltonian, the evolution of the Kondo temperature TK. In BIS, the predicted collapse of the Kondo resonance reflects the breakdown of the singlet ground state above TK. These results provide an important confirmation of the validity of the Kondo model of cerium compounds, and contradict recent claims of inconsistencies with spectroscopic data.
1. Introduction
Strong correlation effects among the shallow 4f electrons are responsible for the unusual properties of cerium compounds. The 4f states exhibit a partially localized character, that reflects the competition between a strong intra-atomic Coulomb interaction and a weak hybridization with the conduction states [1-3]. It is now widely accepted that the essential physics of these materials, neglecting coherence effects that may develop at very low temperature, is well described by the degenerate Anderson impurity model [4-6]. This model explicitly contains high-energy scales (el, U~f) associated with local charge fluctuations (fo~_fl, f~ ~_f2) and generates a small energy scale 6 = kBTK, which depends exponentially on the hybridization strength (A). 6 is the energy separation between the hybridized singlet ground state and a manifold of degenerate magnetic states with essentially f~ character. This small energy scale, which is the characteristic energy of 4f spin fluctuations, controls all the lowtemperature physical properties of the impurity system. For T >> T K the 4f electrons behave as localized and carry a finite magnetic moment; below T K the moment is quenched, and for T ~ T K a description in terms of a local Fermi liquid is appropriate. Values of the static susceptibility X(0) and the low-T specific heat, that are usually much larger than in normal metals, indicate that the interaction with the local impurity renormalizes the quasiparticle density of states at E L by a factor of order 1/T K. High-energy spectroscopies have played an imCorrespondence to: M. Grioni, Institut de Physique, Universit6 de Neuch~tel, CH-2000 Neuch~ttel, Switzerland.
portant role in establishing this picture by allowing direct estimates of the parameters of the Anderson model [7-9]. The observation of two structures with 4f character in valence band spectra [10[, and of satellites in the core spectra [11] was crucial in demonstrating the inconsistency of the early models of intermediate valence in Ce. A wealth of spectroscopic data collected over the past decade indicate that all cerium compounds belong to the so-called Kondo limit (defined by lefl>>A) of the Anderson Hamiltonian, so that the 4f occupation nf is always close to unity. Photoemission and inverse photoemission stand out among other spectroscopic techniques for their fundamental character. Within the sudden approximation, and with the caveats discussed, e.g., in ref [7], PES and BIS spectra are proportional to the negative and, respectively, the positive frequency part of the 4f spectral density function pf (the imaginary part of the one-particle Green's function) [4]. This quantity, essential to describe the properties of the many-body system, exhibits all the characteristic energy scales of the Anderson Hamiltonian. Structures at - e ~ (PES) and - e l + U , (BIS) reflect fl ~._~fo and f~ ~_~f2 charge fluctuations, while the low-energy scale is revealed by a sharp peak (the Kondo resonance) immediately above E F, accounting for the low-lying excitations responsible for the large values of X(0) and 7. The Kondo resonance is also predicted to exhibit sidebands, corresponding to spin-orbit and crystal field excited states, both in the PES and the BIS parts [6]. The direct observation of these sidebands in highresolution PES spectra has indeed provided a strong confirmation of the consistency of the model [12]. Our general understanding of the physics of a Ce impurity seems therefore satisfactory. The Anderson
0921-4526/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
M. Grioni et al. / Spectroscopic evidence of
model provides a unified picture of the thermodynamic properties and of the results of different spectroscopies, all described by a common set of parameters [7,13]. A recent investigation of the et-~ transition in metallic Ce, that pushes to the limit the analysis of the experimental data, suggests that this unified description even holds quantitatively [14]. This model also predicts that the characteristic temperature dependence of the physical properties, reflecting the low-energy scale kBTK, should be observed in the spectroscopic measurements. A high-resolution PES study has actually revealed a subtle temperature effect in the 4f spectrum of CeSi2 [15]. However, this result has recently been questioned [16] and it has been claimed that PES spectra do not exhibit the temperature and hybridization dependence expected from the Kondo model. Our purpose is to present spectroscopic data that, on the contrary, clearly reveal the existence of a Kondo scale, and therefore cannot be explained without the Kondo model. We shall also discuss why possible ambiguities that may exist in the interpretation of temperature-dependent PES spectra are absent in BIS, where the effects are more characteristic, since they occur over a scale of several eV's.
2. High-resolution PES results PES spectra of Ce and its compounds reveal a peak (the 'ionization peak') with mostly fo character at - 2 eV, and, n e a r EF, the tail and the sidebands of the Kondo resonance, reflecting final states with a 4f occupation close to that of the ground state. At the lowest order of a 1/Nf expansion the integrated PES intensity from the Kondo resonance is equal to nf(1 nf) [4], which is vanishing small in the atomic limit A ~-~0 (nf ~ 1). The calculated spectral function shows a transfer of intensity between the ionization peak and the shallow structure as a function of A, so that the intensity n e a r E F is largest in strongly hybridized materials. This general trend has been experimentally verified over a wide range of values of T K [7,12]. Nonetheless, the difficulty of isolating in a reliable way the 4f contribution to the measured spectra represents an obstacle in an accurate quantitative analysis. It is possible to take advantage of the photon energy dependence of the 4f cross section, but this process is not completely unambiguous; especially the fo contribution is often masked by band states. CeAg represents a very favorable exception, because the Cederived signal is superposed to a flat Ag sp density of states. The 4f spectrum of fig. 1, obtained after subtraction of the spectrum of LaAg (a compound with a similar band structure, but no occupied f states) is dominated by the ionization peak at --2.5eV. The curve of fig. l(b) is the 4f spectral function calculated
TK
39
in Ce compounds l
i
i
i
(c)
,.....-
)
-2 Energy (eV)
0=E F
Fig. 1. Ce 4f spectrum of CeAg, obtained from a difference of CeAg and LaAg spectra, from ref. [17]. (b) 4f spectral function calculated within the NCA of the Anderson model; el=-2.2eV, A=0.05eV. (c) Same as (b) but e l = - 1 . 6 e V and A=0.12eV to reproduce the 4f occupation number nf = 0.94 of the LDA calculation of ref. [18].
in the noncrossing approximation (NCA) [6] of the Anderson model with the following parameters: ef = - 2 . 2 eV, Uff = 0% A = 0.05 eV, spin-orbit and crystalfield splittings Aso = 0.28 eV and ACEF = 0.028 eV, and degeneracy Nf = 4 (ref. [17]). This set of parameters yields a 4f occupation nf = 0.995 and a Kondo temperature of about 3 K, which is consistent with the observation of ferromagnetic order below T c = 5.6 K. The predictions of the model are therefore adequate, even if a finite value of U , , and therefore a n f2 contribution to the ground state, could slightly modify the calculated T~. It is interesting to consider, in the same figure, the result of another NCA calculation where the hybridization has been increased to A = 0.12eV in order to reproduce the value n f = 0 . 9 4 obtained in a state-of-the-art density functional calculation performed in the local-density approximation (LDA) [18]. Obviously neither the shape of the spectrum nor the calculated T r of 60 K are compatible with experiment. Besides demonstrating the existence of still unsolved problems in the L D A treatment of correlated systems, this comparison illustrates a rather general point: in the Kondo regime, proper to Ce compounds, small variations of nf lead to important changes in the physical and spectroscopic properties. A more appropriate parameter to describe the importance of hybridization in these materials is the weight of the fo configuration in the ground state, ( 1 - nf). Small variations of nf correspond, in this limit, to large variations of ( 1 - nf). In the case of CeAg, for instance, the best fit of fig. l(b) yields (1 - nf) = 0.005, a value one order of magnitude smaller than the L D A estimate of 0.06. The strength of the hybridization affects not only the intensity ratio of the ,fl, and ,f0, peaks, but also
40
M . G r i o n i et al. / Spectroscopic evidence o f T r in Ce c o m p o u n d s
the relative weight of the Kondo resonance and its sidebands. When A is reduced, the other parameters being kept constant, the Kondo resonance becomes sharper and its intensity decreases. For small values of hybridization the spectral weight near E v is concentrated in the crystal field, and eventually only in the spin-orbit satellites, and the Kondo resonance is too weak to be observed [12]. This is consistent with high-resolution PES spectra of heavy fermions, like Cefu6, which exhibit a single peak at -0.28 eV [19]. CeSi x is an ideal case to test this prediction, because, by varying the concentration of Si vacancies, it is possible to tune the value of the Kondo temperature. The stoichiometric compound CeSi 2 has a nonmagnetic ground state, with a Kondo temperature estimated, by different measurements, between 40 and 100K [20,21]. The ground state of CeSix remains nonmagnetic in the range 1.85 ~
i
At 300 K the first peak is somewhat broader and weaker and the second one has almost disappeared. The calculation, performed with e t = - 1 . 5 eV and A = 0.088 eV and degeneracy N t = 2, yields a 4f occupation nf = 0.96 and a Kondo temperature T K = 50 K. The sharp peak in the T = 15 K spectrum mostly reflects excited crystal field levels, while the weak tail of the Kondo resonance is washed out by the finite resolution. The agreement with experiment is satisfactory, and the value of T K is consistent with the estimates of ref. [20]. The calculation also accounts for the observed temperature dependence of the shape and the negligible variation of the integrated intensity. It has been claimed [16] that similar changes could formally result from more conventional broadening mechanisms and that the spectral weight of the features near E v does not scale with T K. We shall see, however, that in the CeSi x system the respective weight of the two structures near E F actually follows the modifications of T K and that the alternative interpretation proposed in ref. [16] is inconsistent with the temperature evolution of the BIS spectra. The 4f spectra of CeSi 2 and CeSil.6, normalized to the same intensity at - 0 . 6 eV, are compared in fig. 3. A detailed analysis of these spectra will be presented elsewhere. Here we just wish to underline the dramatic intensity variation of the structure at - E v. As expected, the 4f spectrum of CeSi]. 6 is actually quite similar to those of CeAg, C e C H 6 o r CeAI:, materials with Kondo temperatures lower than 10 K. The weak signal at E v reflects the low T K and the fact that the measurement has been carried out at T ~> T K. The remarkable evolution of the spectral function observed in fig. 3 is therefore fully consistent with the predictions of the Anderson model. Possible sources
i
j I.•
CeSi2 •~.~..~...~...
i
s.~,~..~
T (K)
o
I
i
• CeSi 2 o CeSil.6
,". ~,
t.~,~,¢~/~'''~
o
% ~ o
. °
O •, ....
-500
?..oo
-250 01=EF Energy (meV)
Fig. 2. Experimental 4f spectra of CeSi2 (circles) obtained as a difference of HeII and HeI spectra, and calculated NCA 4f spectral functions (lines), with the following parameters: e f = - l . 5 e V , zl=0.088eV, Aso=0.3eV, ACEF=0.035eV (Nf- 2) (from ref. [22]).
0
..~ o
-500
-250 O=EF Energy (meV)
Fig. 3. Experimental 4f spectra of CeSi2 and CeSi~.o at T = 15 K. The different intensities at E v reflect the variation of the estimated Kondo temperatures (~50 K for CeSi2 and
41
M. Grioni et al. / Spectroscopic evidence of T K in Ce compounds i
of error, like the existence of distinct surface and bulk contributions [23] or the effect of finite U , [14,24], as well as the details of the normalization procedure, could somewhat modify the actual values of T K extracted from the spectra, but by no means could call into question the validity of this approach.
i
CeSi 2
i
?~.~
/,.,.
,~**-~
..;.~.
•
T= 15 K
3, Temperature dependence of the BIS spectral function I
In the preceding section we have mentioned that the Kondo model interpretation of the temperature dependence of PES data is at present controversial. BIS measurements can solve that controversy, because the approach of ref. [16], that does not contain the thermodynamic scale kBTK, is incompatible with the large energy scale over which temperature effects are observed in BIS. The different influence of temperature on the PES and BIS spectral functions can be qualitatively understood with the help of a very simple model. Figure 4 shows the relevant energy levels of a Ce impurity system in the initial state (N electrons) and in the PES ( N - 1) and BIS (N + 1) final states. We consider, for clarity, the limit of zero bandwidth (the levels are discrete) and large U; we also neglect spin-orbit and crystal-field splittings. The singlet ground state (n e < 1) is separated by an energy 8 from the magnetic (nf = 1) states, and at T ~ T K only the ground state is populated. As temperature increases, the low-lying excited states are progressively populated so that the average 4f occupation (n¢) increases, according to a scaling law in T / T K [6]. As discussed elsewhere [25], the removal of one f electron in photoemission couples in the same way the singlet and magnetic initial states to the two accessible final states, so that the PES spectral function has a very weak
I Efl
~t Energy (eV)
8
Fig. 5. BIS spectra of CeSi2 measured below (T = 15 K, full circles) and above (T = 300 K, open circles) the Kondo temperature TK~ 50 K. The solid and dashed lines are spectra calculated according to the simplified approach described in the text (from ref. [22]). temperature dependence. In BIS the situation is completely different, since the singlet ground state couples to the complete set of final states containing f~ or f2 components, whereas the magnetic states give access only to the f2 final state. When the magnetic states become thermally populated, spectral weight is transferred from the Kondo resonance (the two low-lying final states in fig. 4) to the f2 structure several eV's above it, following the evolution of (1 - (nf)). The inclusion of spin-orbit splitting does not modify this conclusion, because both the Kondo resonance and its satellite at ~ + Aso have the same temperature dependence. A similar argument holds if crystal field splitting is considered. The BIS spectra of CeSi 2 of fig. 5 confirm this picture. Due to the poor energy resolution of BIS ( - 0 . 6 e V ) the Kondo resonance and its sidebands appear as a single, 0.8 eV wide, peak; the f2 structure,
/
U+ ef
I
EF=0
If2> //
Lf0> Ill>
kBTr
alf0> + I]lfl> N-I
N
N+I
UPS BIS Fig. 4. Schematic total-energy levels (zero-bandwidth, large U, degeneracy Nf = 14) of the neutral impurity system (N electrons) and of the PES ( N - 1) and BIS (N + 1) final states. The lowest-lying level is a singlet state with mixed fl and fo character, while the first excited states are in an almost pure fl configuration. Solid and dashed lines indicate allowed transitions from the ground state and the thermally excited magnetic states, and highlight the different temperature dependence of the PES and BIS spectra.
42
M. Grioni et al. / Spectroscopic evidence of Tr in Ce compounds
broadened by multiplet effects [8], is centered at - 5 eV. A direct comparison of the two spectra already reveals a transfer of spectral weight between the two structures. A full calculation of the BIS spectral function of the Anderson model including at the same time the effects of finite temperature and of finite U (to treat the f2 peak) is beyond our present capabilities. To analyse our results we must therefore resort to a simplified approach. We assume that the BIS spectrum at T > 0 is a superposition of spectra corresponding to the ground and excited states, with the statistical weights of the respective populations in the initial state: A(¢o, T) = a( T)AGs(Oa) + b( T ) A ~ ( w ) .
(1)
AGs(~O) is the spectrum, calculated at T = 0 K in the Gunnarsson-Sch6nhammer model [4], that gives the best fit to the low-temperature experimental spectrum. A ex(w) is the spectrum associated with the excited magnetic states. As these states have a pure 4f' character, Aex(~0) presents only the f2 structure. The coefficients a(T) and b(T) can be obtained from an NCA calculation of (ne) [22,25]: for CeSi 2 and the set of parameters used to fit the PES spectra of fig. 2, (nf) is found to increase from 0.965 at T = 15 K to 0.977 at 300 K, corresponding to a substantial 35% change in (1 - ( n f)). To account for the experimental energy resolution the calculated spectra have been convoluted with a gaussian (0.6 eV FWHM), and the 4f 2 structure has further been broadened to simulate the experimental lineshape. Finally, the 4f spectra have been superposed to a background which simulates non-f states and inelastic contributions. This background is admittedly somewhat arbitrary, but it does not markedly influence our analysis, since it is temperature independent. The agreement between calculated and experimental spectra is very satisfactory. We have observed a similarly large temperature dependence in CePd 3 [25], a compound with a Kondo temperature of - 2 4 0 K. Figure 6 shows the 4f BIS contribution extracted from experimental spectra measured at 15 K and 300 K, and the results of a calculation similar to that described above. Once again the agreement is very good. On the other hand we anticipate very small temperature effects in more strongly hybridized (large TK) materials like CeRh 3 or CeNis, because the variation of ( h i ) is very small in the accessible temperature range. As predicted by the Anderson model the combined effect of selection rules and the different symmetry of the ground and excited states produce a transfer of intensity over several eV's. This mechanism is intrinsically different from the usual temperature broadening of the Fermi function, normally observed in metals. The Fermi function in fact affects the shape of the
0"961 092
~1.0
"~
.-x"'" `
,
0.0
Log(T/TK)
CePd 3 ,
," -,
a) EF=0
2 4 6 8 Energy (eV) Fig. 6. (a) Experimental 4f BIS spectral function of CdPd 3 at T = 15 K (full circles) and at T = 300 K (open circles), after subtraction of the non-f contribution, as in ref. [25] (the lines are guides to the eye). (b) Spectral functions calculated for the same temperatures according to the simplified approach described in the text. Inset: temperature-dependent NCA calculation of the mean 4f occupation number. The black squares correspond to the nf values at T = 15 K and 300 K. spectra on a moderate energy range (about 100 meV at room temperature), but not the integrated spectral weight and couldn't possibly explain the experimental data. Therefore, despite the poor resolution, BIS is sensitive to subtle changes occurring on a scale set by T K. This sensitivity is not unique to BIS, and we have observed similar effects in the Ce 3d core spectra. In the limit nf ~-~1, however, the magnitude of the effect is expected to be largest in BIS [4].
4. Conclusion We have shown that the spectroscopic properties of CeSi x and other Ce-based materials are well described by the Anderson model. The intensity distribution between the Kondo resonance and its sidebands, as measured by photoemission with high resolution, emerges as an important indicator of hybridization in Ce compounds. The observation of large temperature effects proves that, despite its poor energy resolution, BIS is sensitive to the low-energy excitations and to the breakdown of the singlet ground state, with the additional advantage of a weak surface sensitivity. In conclusion, both the PES and BIS spectral functions bear the unmistakable mark of the underlying Kondo energy scale.
Acknowledgements This work has been supported by the Fonds National Suisse de la Recherche Scientifique. Two of the authors (M.G. and D.M.) are indebted to E. Bovary
M. Grioni et al. / Spectroscopic evidence of T K in Ce compounds
for disclosing some results on arsenides prior to publication.
References [1] P. Fulde, Electron Correlations in Molecules and Solids, Springer Series in Solid State Science No. 100 (SpringerVerlag, Berlin, 1991). [2] J.M. Lawrence, ES. Riseborough and R.D. Parks, Rep. Prog. Phys. 44 (198l) 1. [3] P.A. Lee, T.M. Rice, J.W. Serene, L.J. Sham and J.W. Wilkins, Comm. Condens. Matter Phys. 12 (1986) 99. [4] O. Gunnarsson and K. Sch6nhammer, Phys. Rev. B 28 (1983) 4315. [5] P. Coleman, Phys. Rev. B 29 (1984) 3035. [6] N.E. Bickers, D.L. Cox and J.W. Wilkins, Phys. Rev. B 36 (1987) 2036. [7] J.W. Allen, S.J. Oh, O. Gunnarsson, K. Sch6nhammer, M.B. Maple, M.S. Torikachvili and I. Lindau, Adv. Phys. 35 (1986) 275. [8] Y. Baer and W. Schneider, in: Handbook on the Physics and Chemistry of Rare-Earths, Vol. 10, eds. K.A. Gschneidner Jr., L. Eyring and S. H/ifner (NorthHolland, Amsterdam, 1987) p. 1. [9] J.C. Fuggle, F.U. Hillebrecht, Z. Zolnierek, R. Lfisser, Ch. Freiburg, O. Gunnarsson and K. Sch6nhammer, Phys. Rev. B 27 (1983) 7330. [10] L.I. Johansson, J.W. Allen, T. Gustafsson, I. Lindau and S.B.M. Hagstr6m, Solid State Commun. 28 (1978) 53.
43
[11] J.C. Fuggle, F.U. Hillebrecht, Z. Zolnierek, R. Lfisser and A. Platau, Phys. Rev. Lett. 45 (1980) 1597. [12] F. Patthey, B. Delley, W.-D. Schneider and Y. Baer, Phys. Rev. Lett. 55 (1985) 1518. [13] F. Patthey, J.-M. Imer, W.-D. Schneider, H. Beck, Y. Baer and B. Delley, Phys. Rev. B 42 (1990) 8864. [14] L.Z. Liu, J.W. Allen, O. Gunnarsson, N.E. Christensen and O.K. Andersen, Phys. Rev. B 45 (1992) 8934. [15] F. Patthey, W.-D. Schneider, Y. Baer and B. Delley, Phys. Rev. Lett. 58 (1987) 2810. [16] J.J. Joyce, A.J. Arko, J. Lawrence, P.C. Canfield, Z. Fisk, R.J. Bartlett and J.D. Thompson, Phys. Rev. Lett. 68 (1992) 236. [17] P. Weibel, D. Malterre, M. Grioni, B. Dardel and Y. Baer, Z. Phys. B 87 (1992) 165. [18] P. Monachesi, S. Fraizzoli and E.G. Moroni, J. Magn. Magn. Mater. 98 (1991) 130. [19] F. Patthey, W.-D. Schneider, Y. Baer and B. Delley, Phys. Rev. B 34 (1986) 2967. [20] H. Yashima, H. Mori, T. Satoh and K. Kohn, Solid State Commun. 43 (1982) 193. [21] R.M. Galera, A.P. Murani and J. Pierre, Physica B 156-157 (1989) 801. [22] D. Malterre, M. Grioni, P. Weibel, B. Dardel and Y. Baer, Europhys. Lett. 20 (1992) 445. [23] C. Laubschat, E. Weschke, C. Holtz, M. Domke, O. Strebel and G. Kaindl, Phys. Rev. Lett. 65 (1990) 1639. [24] O. Gunnarsson and K. Sch6nhammer, Phys. Rev. B 31 (1985) 4815. [25] D. Malterre, M. Grioni, P. Weibel, B. Dardel and Y. Baer, Phys. Rev. Lett. 68 (1992) 2656.
Physica B 186-188 (1993) 38-43 North-Holland
Spectroscopic evidence of a Kondo scale in cerium compounds M. Grioni, D. Malterre, P. Weibel, B. Dardel and Y. Baer lnstitut de Physique, Universit~ de Neuchgttel, Switzerland
We present a high-resolution photoemission (PES) and temperature-dependent inverse photoemission (BIS) investigation of Ce compounds, and in particular, of the paradigmatic system CeSix. The PES intensity near the Fermi level follows, as in the calculated spectral function of the Anderson Hamiltonian, the evolution of the Kondo temperature TK. In BIS, the predicted collapse of the Kondo resonance reflects the breakdown of the singlet ground state above TK. These results provide an important confirmation of the validity of the Kondo model of cerium compounds, and contradict recent claims of inconsistencies with spectroscopic data.
1. Introduction
Strong correlation effects among the shallow 4f electrons are responsible for the unusual properties of cerium compounds. The 4f states exhibit a partially localized character, that reflects the competition between a strong intra-atomic Coulomb interaction and a weak hybridization with the conduction states [1-3]. It is now widely accepted that the essential physics of these materials, neglecting coherence effects that may develop at very low temperature, is well described by the degenerate Anderson impurity model [4-6]. This model explicitly contains high-energy scales (el, U~f) associated with local charge fluctuations (fo~_fl, f~ ~_f2) and generates a small energy scale 6 = kBTK, which depends exponentially on the hybridization strength (A). 6 is the energy separation between the hybridized singlet ground state and a manifold of degenerate magnetic states with essentially f~ character. This small energy scale, which is the characteristic energy of 4f spin fluctuations, controls all the lowtemperature physical properties of the impurity system. For T >> T K the 4f electrons behave as localized and carry a finite magnetic moment; below T K the moment is quenched, and for T ~ T K a description in terms of a local Fermi liquid is appropriate. Values of the static susceptibility X(0) and the low-T specific heat, that are usually much larger than in normal metals, indicate that the interaction with the local impurity renormalizes the quasiparticle density of states at E L by a factor of order 1/T K. High-energy spectroscopies have played an imCorrespondence to: M. Grioni, Institut de Physique, Universit6 de Neuch~tel, CH-2000 Neuch~ttel, Switzerland.
portant role in establishing this picture by allowing direct estimates of the parameters of the Anderson model [7-9]. The observation of two structures with 4f character in valence band spectra [10[, and of satellites in the core spectra [11] was crucial in demonstrating the inconsistency of the early models of intermediate valence in Ce. A wealth of spectroscopic data collected over the past decade indicate that all cerium compounds belong to the so-called Kondo limit (defined by lefl>>A) of the Anderson Hamiltonian, so that the 4f occupation nf is always close to unity. Photoemission and inverse photoemission stand out among other spectroscopic techniques for their fundamental character. Within the sudden approximation, and with the caveats discussed, e.g., in ref [7], PES and BIS spectra are proportional to the negative and, respectively, the positive frequency part of the 4f spectral density function pf (the imaginary part of the one-particle Green's function) [4]. This quantity, essential to describe the properties of the many-body system, exhibits all the characteristic energy scales of the Anderson Hamiltonian. Structures at - e ~ (PES) and - e l + U , (BIS) reflect fl ~._~fo and f~ ~_~f2 charge fluctuations, while the low-energy scale is revealed by a sharp peak (the Kondo resonance) immediately above E F, accounting for the low-lying excitations responsible for the large values of X(0) and 7. The Kondo resonance is also predicted to exhibit sidebands, corresponding to spin-orbit and crystal field excited states, both in the PES and the BIS parts [6]. The direct observation of these sidebands in highresolution PES spectra has indeed provided a strong confirmation of the consistency of the model [12]. Our general understanding of the physics of a Ce impurity seems therefore satisfactory. The Anderson
0921-4526/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
M. Grioni et al. / Spectroscopic evidence of
model provides a unified picture of the thermodynamic properties and of the results of different spectroscopies, all described by a common set of parameters [7,13]. A recent investigation of the et-~ transition in metallic Ce, that pushes to the limit the analysis of the experimental data, suggests that this unified description even holds quantitatively [14]. This model also predicts that the characteristic temperature dependence of the physical properties, reflecting the low-energy scale kBTK, should be observed in the spectroscopic measurements. A high-resolution PES study has actually revealed a subtle temperature effect in the 4f spectrum of CeSi2 [15]. However, this result has recently been questioned [16] and it has been claimed that PES spectra do not exhibit the temperature and hybridization dependence expected from the Kondo model. Our purpose is to present spectroscopic data that, on the contrary, clearly reveal the existence of a Kondo scale, and therefore cannot be explained without the Kondo model. We shall also discuss why possible ambiguities that may exist in the interpretation of temperature-dependent PES spectra are absent in BIS, where the effects are more characteristic, since they occur over a scale of several eV's.
2. High-resolution PES results PES spectra of Ce and its compounds reveal a peak (the 'ionization peak') with mostly fo character at - 2 eV, and, n e a r EF, the tail and the sidebands of the Kondo resonance, reflecting final states with a 4f occupation close to that of the ground state. At the lowest order of a 1/Nf expansion the integrated PES intensity from the Kondo resonance is equal to nf(1 nf) [4], which is vanishing small in the atomic limit A ~-~0 (nf ~ 1). The calculated spectral function shows a transfer of intensity between the ionization peak and the shallow structure as a function of A, so that the intensity n e a r E F is largest in strongly hybridized materials. This general trend has been experimentally verified over a wide range of values of T K [7,12]. Nonetheless, the difficulty of isolating in a reliable way the 4f contribution to the measured spectra represents an obstacle in an accurate quantitative analysis. It is possible to take advantage of the photon energy dependence of the 4f cross section, but this process is not completely unambiguous; especially the fo contribution is often masked by band states. CeAg represents a very favorable exception, because the Cederived signal is superposed to a flat Ag sp density of states. The 4f spectrum of fig. 1, obtained after subtraction of the spectrum of LaAg (a compound with a similar band structure, but no occupied f states) is dominated by the ionization peak at --2.5eV. The curve of fig. l(b) is the 4f spectral function calculated
TK
39
in Ce compounds l
i
i
i
(c)
,.....-
)
-2 Energy (eV)
0=E F
Fig. 1. Ce 4f spectrum of CeAg, obtained from a difference of CeAg and LaAg spectra, from ref. [17]. (b) 4f spectral function calculated within the NCA of the Anderson model; el=-2.2eV, A=0.05eV. (c) Same as (b) but e l = - 1 . 6 e V and A=0.12eV to reproduce the 4f occupation number nf = 0.94 of the LDA calculation of ref. [18].
in the noncrossing approximation (NCA) [6] of the Anderson model with the following parameters: ef = - 2 . 2 eV, Uff = 0% A = 0.05 eV, spin-orbit and crystalfield splittings Aso = 0.28 eV and ACEF = 0.028 eV, and degeneracy Nf = 4 (ref. [17]). This set of parameters yields a 4f occupation nf = 0.995 and a Kondo temperature of about 3 K, which is consistent with the observation of ferromagnetic order below T c = 5.6 K. The predictions of the model are therefore adequate, even if a finite value of U , , and therefore a n f2 contribution to the ground state, could slightly modify the calculated T~. It is interesting to consider, in the same figure, the result of another NCA calculation where the hybridization has been increased to A = 0.12eV in order to reproduce the value n f = 0 . 9 4 obtained in a state-of-the-art density functional calculation performed in the local-density approximation (LDA) [18]. Obviously neither the shape of the spectrum nor the calculated T r of 60 K are compatible with experiment. Besides demonstrating the existence of still unsolved problems in the L D A treatment of correlated systems, this comparison illustrates a rather general point: in the Kondo regime, proper to Ce compounds, small variations of nf lead to important changes in the physical and spectroscopic properties. A more appropriate parameter to describe the importance of hybridization in these materials is the weight of the fo configuration in the ground state, ( 1 - nf). Small variations of nf correspond, in this limit, to large variations of ( 1 - nf). In the case of CeAg, for instance, the best fit of fig. l(b) yields (1 - nf) = 0.005, a value one order of magnitude smaller than the L D A estimate of 0.06. The strength of the hybridization affects not only the intensity ratio of the ,fl, and ,f0, peaks, but also
40
M . G r i o n i et al. / Spectroscopic evidence o f T r in Ce c o m p o u n d s
the relative weight of the Kondo resonance and its sidebands. When A is reduced, the other parameters being kept constant, the Kondo resonance becomes sharper and its intensity decreases. For small values of hybridization the spectral weight near E v is concentrated in the crystal field, and eventually only in the spin-orbit satellites, and the Kondo resonance is too weak to be observed [12]. This is consistent with high-resolution PES spectra of heavy fermions, like Cefu6, which exhibit a single peak at -0.28 eV [19]. CeSi x is an ideal case to test this prediction, because, by varying the concentration of Si vacancies, it is possible to tune the value of the Kondo temperature. The stoichiometric compound CeSi 2 has a nonmagnetic ground state, with a Kondo temperature estimated, by different measurements, between 40 and 100K [20,21]. The ground state of CeSix remains nonmagnetic in the range 1.85 ~
i
At 300 K the first peak is somewhat broader and weaker and the second one has almost disappeared. The calculation, performed with e t = - 1 . 5 eV and A = 0.088 eV and degeneracy N t = 2, yields a 4f occupation nf = 0.96 and a Kondo temperature T K = 50 K. The sharp peak in the T = 15 K spectrum mostly reflects excited crystal field levels, while the weak tail of the Kondo resonance is washed out by the finite resolution. The agreement with experiment is satisfactory, and the value of T K is consistent with the estimates of ref. [20]. The calculation also accounts for the observed temperature dependence of the shape and the negligible variation of the integrated intensity. It has been claimed [16] that similar changes could formally result from more conventional broadening mechanisms and that the spectral weight of the features near E v does not scale with T K. We shall see, however, that in the CeSi x system the respective weight of the two structures near E F actually follows the modifications of T K and that the alternative interpretation proposed in ref. [16] is inconsistent with the temperature evolution of the BIS spectra. The 4f spectra of CeSi 2 and CeSil.6, normalized to the same intensity at - 0 . 6 eV, are compared in fig. 3. A detailed analysis of these spectra will be presented elsewhere. Here we just wish to underline the dramatic intensity variation of the structure at - E v. As expected, the 4f spectrum of CeSi]. 6 is actually quite similar to those of CeAg, C e C H 6 o r CeAI:, materials with Kondo temperatures lower than 10 K. The weak signal at E v reflects the low T K and the fact that the measurement has been carried out at T ~> T K. The remarkable evolution of the spectral function observed in fig. 3 is therefore fully consistent with the predictions of the Anderson model. Possible sources
i
j I.•
CeSi2 •~.~..~...~...
i
s.~,~..~
T (K)
o
I
i
• CeSi 2 o CeSil.6
,". ~,
t.~,~,¢~/~'''~
o
% ~ o
. °
O •, ....
-500
?..oo
-250 01=EF Energy (meV)
Fig. 2. Experimental 4f spectra of CeSi2 (circles) obtained as a difference of HeII and HeI spectra, and calculated NCA 4f spectral functions (lines), with the following parameters: e f = - l . 5 e V , zl=0.088eV, Aso=0.3eV, ACEF=0.035eV (Nf- 2) (from ref. [22]).
0
..~ o
-500
-250 O=EF Energy (meV)
Fig. 3. Experimental 4f spectra of CeSi2 and CeSi~.o at T = 15 K. The different intensities at E v reflect the variation of the estimated Kondo temperatures (~50 K for CeSi2 and
41
M. Grioni et al. / Spectroscopic evidence of T K in Ce compounds i
of error, like the existence of distinct surface and bulk contributions [23] or the effect of finite U , [14,24], as well as the details of the normalization procedure, could somewhat modify the actual values of T K extracted from the spectra, but by no means could call into question the validity of this approach.
i
CeSi 2
i
?~.~
/,.,.
,~**-~
..;.~.
•
T= 15 K
3, Temperature dependence of the BIS spectral function I
In the preceding section we have mentioned that the Kondo model interpretation of the temperature dependence of PES data is at present controversial. BIS measurements can solve that controversy, because the approach of ref. [16], that does not contain the thermodynamic scale kBTK, is incompatible with the large energy scale over which temperature effects are observed in BIS. The different influence of temperature on the PES and BIS spectral functions can be qualitatively understood with the help of a very simple model. Figure 4 shows the relevant energy levels of a Ce impurity system in the initial state (N electrons) and in the PES ( N - 1) and BIS (N + 1) final states. We consider, for clarity, the limit of zero bandwidth (the levels are discrete) and large U; we also neglect spin-orbit and crystal-field splittings. The singlet ground state (n e < 1) is separated by an energy 8 from the magnetic (nf = 1) states, and at T ~ T K only the ground state is populated. As temperature increases, the low-lying excited states are progressively populated so that the average 4f occupation (n¢) increases, according to a scaling law in T / T K [6]. As discussed elsewhere [25], the removal of one f electron in photoemission couples in the same way the singlet and magnetic initial states to the two accessible final states, so that the PES spectral function has a very weak
I Efl
~t Energy (eV)
8
Fig. 5. BIS spectra of CeSi2 measured below (T = 15 K, full circles) and above (T = 300 K, open circles) the Kondo temperature TK~ 50 K. The solid and dashed lines are spectra calculated according to the simplified approach described in the text (from ref. [22]). temperature dependence. In BIS the situation is completely different, since the singlet ground state couples to the complete set of final states containing f~ or f2 components, whereas the magnetic states give access only to the f2 final state. When the magnetic states become thermally populated, spectral weight is transferred from the Kondo resonance (the two low-lying final states in fig. 4) to the f2 structure several eV's above it, following the evolution of (1 - (nf)). The inclusion of spin-orbit splitting does not modify this conclusion, because both the Kondo resonance and its satellite at ~ + Aso have the same temperature dependence. A similar argument holds if crystal field splitting is considered. The BIS spectra of CeSi 2 of fig. 5 confirm this picture. Due to the poor energy resolution of BIS ( - 0 . 6 e V ) the Kondo resonance and its sidebands appear as a single, 0.8 eV wide, peak; the f2 structure,
/
U+ ef
I
EF=0
If2> //
Lf0> Ill>
kBTr
alf0> + I]lfl> N-I
N
N+I
UPS BIS Fig. 4. Schematic total-energy levels (zero-bandwidth, large U, degeneracy Nf = 14) of the neutral impurity system (N electrons) and of the PES ( N - 1) and BIS (N + 1) final states. The lowest-lying level is a singlet state with mixed fl and fo character, while the first excited states are in an almost pure fl configuration. Solid and dashed lines indicate allowed transitions from the ground state and the thermally excited magnetic states, and highlight the different temperature dependence of the PES and BIS spectra.
42
M. Grioni et al. / Spectroscopic evidence of Tr in Ce compounds
broadened by multiplet effects [8], is centered at - 5 eV. A direct comparison of the two spectra already reveals a transfer of spectral weight between the two structures. A full calculation of the BIS spectral function of the Anderson model including at the same time the effects of finite temperature and of finite U (to treat the f2 peak) is beyond our present capabilities. To analyse our results we must therefore resort to a simplified approach. We assume that the BIS spectrum at T > 0 is a superposition of spectra corresponding to the ground and excited states, with the statistical weights of the respective populations in the initial state: A(¢o, T) = a( T)AGs(Oa) + b( T ) A ~ ( w ) .
(1)
AGs(~O) is the spectrum, calculated at T = 0 K in the Gunnarsson-Sch6nhammer model [4], that gives the best fit to the low-temperature experimental spectrum. A ex(w) is the spectrum associated with the excited magnetic states. As these states have a pure 4f' character, Aex(~0) presents only the f2 structure. The coefficients a(T) and b(T) can be obtained from an NCA calculation of (ne) [22,25]: for CeSi 2 and the set of parameters used to fit the PES spectra of fig. 2, (nf) is found to increase from 0.965 at T = 15 K to 0.977 at 300 K, corresponding to a substantial 35% change in (1 - ( n f)). To account for the experimental energy resolution the calculated spectra have been convoluted with a gaussian (0.6 eV FWHM), and the 4f 2 structure has further been broadened to simulate the experimental lineshape. Finally, the 4f spectra have been superposed to a background which simulates non-f states and inelastic contributions. This background is admittedly somewhat arbitrary, but it does not markedly influence our analysis, since it is temperature independent. The agreement between calculated and experimental spectra is very satisfactory. We have observed a similarly large temperature dependence in CePd 3 [25], a compound with a Kondo temperature of - 2 4 0 K. Figure 6 shows the 4f BIS contribution extracted from experimental spectra measured at 15 K and 300 K, and the results of a calculation similar to that described above. Once again the agreement is very good. On the other hand we anticipate very small temperature effects in more strongly hybridized (large TK) materials like CeRh 3 or CeNis, because the variation of ( h i ) is very small in the accessible temperature range. As predicted by the Anderson model the combined effect of selection rules and the different symmetry of the ground and excited states produce a transfer of intensity over several eV's. This mechanism is intrinsically different from the usual temperature broadening of the Fermi function, normally observed in metals. The Fermi function in fact affects the shape of the
0"961 092
~1.0
"~
.-x"'" `
,
0.0
Log(T/TK)
CePd 3 ,
," -,
a) EF=0
2 4 6 8 Energy (eV) Fig. 6. (a) Experimental 4f BIS spectral function of CdPd 3 at T = 15 K (full circles) and at T = 300 K (open circles), after subtraction of the non-f contribution, as in ref. [25] (the lines are guides to the eye). (b) Spectral functions calculated for the same temperatures according to the simplified approach described in the text. Inset: temperature-dependent NCA calculation of the mean 4f occupation number. The black squares correspond to the nf values at T = 15 K and 300 K. spectra on a moderate energy range (about 100 meV at room temperature), but not the integrated spectral weight and couldn't possibly explain the experimental data. Therefore, despite the poor resolution, BIS is sensitive to subtle changes occurring on a scale set by T K. This sensitivity is not unique to BIS, and we have observed similar effects in the Ce 3d core spectra. In the limit nf ~-~1, however, the magnitude of the effect is expected to be largest in BIS [4].
4. Conclusion We have shown that the spectroscopic properties of CeSi x and other Ce-based materials are well described by the Anderson model. The intensity distribution between the Kondo resonance and its sidebands, as measured by photoemission with high resolution, emerges as an important indicator of hybridization in Ce compounds. The observation of large temperature effects proves that, despite its poor energy resolution, BIS is sensitive to the low-energy excitations and to the breakdown of the singlet ground state, with the additional advantage of a weak surface sensitivity. In conclusion, both the PES and BIS spectral functions bear the unmistakable mark of the underlying Kondo energy scale.
Acknowledgements This work has been supported by the Fonds National Suisse de la Recherche Scientifique. Two of the authors (M.G. and D.M.) are indebted to E. Bovary
M. Grioni et al. / Spectroscopic evidence of T K in Ce compounds
for disclosing some results on arsenides prior to publication.
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