Truly bulk-sensitive spectroscopic measurements of valence in heavy fermion materials†

Intermediate valence is one of the typical phenomena of systems with strong electronic correlation. The Anderson impurity model predicts a scaling of the valence with the reduced temperature T/T_K, which is difficult to observe by traditional surface-sensitive electronic spectroscopies. This paper presents results obtained by resonant inelastic X-ray scattering (RIXS), a bulk-sensitive configuration- and chemical-specific technique. The temperature dependence of the valence of YbInCu₄ and YbAgCu₄ was measured by tuning the incident energy to the resonance of the Yb²⁺ spectral component. In the case of YbInCu₄ a sharp valence transition, as known from thermodynamical measurements, has been found. The valence of YbAgCu₄ reveals a smooth dependence consistent with a Kondo temperature T_K =  70 K. These findings establish RIXS as a powerful tool for measuring bulk electronic properties of solids.

1. Introduction

Mixed valence arises in strongly correlated solids from the conflict between the natural tendency of electrons to form extended bands and the opposite tendency to occupy atomic-like localized states. This leads to spectacular physical properties including Kondo and heavy fermion behaviour as well as unconventional magnetism and superconductivity (Lee et al., 1986).

Valence fluctuations can be revealed by various transport, thermo­dynamic and spectroscopic measurements. The latter probe, in the most direct way, the nature of the electronic ground state but typically only sample a thin layer (∼10–20 Å) below the surface. Recent experiments at higher photon energy (Sekiyama et al., 1997; Braicovich et al., 1997) with larger probing ranges (∼20–50 Å) highlight the limitations of conventional measurements. This dis­advantage is overcome by photon in–photon out spectroscopies like resonant inelastic X-ray scattering (RIXS), which probes the electronic states of a much thicker (∼10 µm in our experiment) bulk-like layer. RIXS is also element-specific and can distinguish between different electronic configurations. These characteristics make RIXS an ideal tool to obtain bulk information about intermediate-valence systems, as we prove in the present study of valence in the two Yb-based Kondo systems YbInCu₄ and YbAgCu₄. The valence is predicted by the Anderson impurity model (AIM) to scale with the reduced temperature T/T_K, where T_K is the Kondo temperature (Bickers et al., 1987). YbInCu₄, a moderately heavy fermion, was chosen as a model compound. It exhibits a first-order transition at T_V  =  42 K, where the valence suddenly decreases from 2.96 to 2.83, and T_K changes from ∼20 K to ∼400 K. Such a small valence difference gives rise to large changes in all the physical properties. For YbAgCu₄, a smooth transition with temperature is expected, consistent with T_K ≃ 60–100 K. The mentioned valence changes have been hinted at by a wealth of non-spectroscopic measurements. Spectroscopic results in these and in related Kondo systems are controversial, due to possible contributions from perturbed surface layers (Malterre et al., 1997). The present findings unambiguously demonstrate the validity of the AIM predictions providing results that are related to truly bulk states.

2. Experimental

The experiment was performed on the ESRF beamline ID16. The incident radiation was monochromated by an Si(111) monochromator. X-rays scattered at 90° by the sample were energy-analyzed with a Rowland circle-based spectrometer equipped with a silicon crystal cut along the (620) direction. We measured Lα₁ (3d–2p) RIXS spectra of ytterbium at selected excitation energies across the L₃ (2p_3/2) threshold, and L₃ absorption spectra (XAS) by recording the intensity of the Lα₁ fluorescence while scanning the incident photon energy. These partial fluorescence yield (PFY) spectra are free from the lifetime broadening of the deep core-hole (Hämäläinen et al., 1991). The overall resolution was 1.3 eV. The samples were polished single crystals mounted on an He closed-cycle refrigerator allowing accurate temperature control between 15 K and 300 K.

3. Results and discussion

The ground state of ytterbium in YbInCu₄ and YbAgCu₄ is a coherent superposition of Yb²⁺ (4f¹⁴) and Yb³⁺(4f¹³) states. Fig. 1 shows the electronic configuration of ytterbium in the ground state and in the intermediate and final states of the RIXS process, together with the main transitions connecting the different configurations. The fingerprints of both Yb²⁺ (4f¹⁴) and Yb³⁺(4f¹³) are visible in the Yb L₃ PFY spectra of YbInCu₄, recorded below and above the transition temperature (Fig. 2). The intensity of the Yb²⁺ component drops at T_V, reflecting the sudden change of valence. For comparison we show the same absorption spectra measured in the traditional total fluorescence yield mode. These results are consistent with previous data (Cornelius et al., 1997) but the use of the PFY detection results in superior resolution and much better defined spectral features.

Figure 1 Total energy level scheme for a Yb ion in mixed-valent ytterbium compounds. The arrows indicate the relevant XAS and RIXS transitions. The XAS final states are the intermediate states of the RIXS process.

Figure 2 Ytterbium L3 absorption spectrum in YbInCu4 measured on the same sample in partial fluorescence yield (PFY) mode (top) and in conventional total fluorescence yield (TFY) mode (bottom). In the PFY mode only the emitted radiation corresponding to 3d5/2–2p3/2 decay (hνout =  7415 eV) was recorded. The spectra exhibit features corresponding to the 2+ and 3+ configurations in the initial state. Their intensity change across the transition temperature (TV = 42 K) reflects the valence change.

The quantitative determination of the Yb²⁺ weight in the ground state is necessary to evaluate the deviation from the integer 3+ valence. The corresponding XAS signal is small. However, the sensitivity to the divalent configuration can be greatly enhanced by tuning the excitation energy to the peak of the Yb²⁺ XAS signal and recording the corresponding RIXS spectrum. This is clearly illustrated by Fig. 3, showing RIXS spectra collected at various energies around the Yb²⁺ resonance. At the peak of the resonance the Yb²⁺ signal dominates the spectrum for  T  =  15 K (below the transition). Above the transition temperature ytterbium is closer to trivalent, and the divalent contribution is less intense.

Figure 3 RIXS spectra of ytterbium in YbInCu4. The incident energy was scanned in the region of the Yb^2+ signal (thick part of absorption curve in the inset). At its resonance the divalent peak is the most prominent spectral feature below TV (T = 15 K). Above TV (T  =  295 K) the Yb valence is closer to 3 and the divalent peak is less intense. At this temperature the weak contribution from the intermediate state following 2p–4f quadrupolar excitation is observed (see Fig. 6 for more details).

The temperature dependence of the divalent signal is most clearly observed at resonance. Fig. 4 shows the change of the spectral shape for both compounds. The small valence changes yield large changes in the weight of the divalent component, which is proportional to (1 − n_h), where n_h is the number of f-holes (n_h = 0 for Yb²⁺, n_h = 1 for Yb³⁺). The spectra of YbInCu₄ exhibit a variation only across T_V. For YbAgCu₄ the line shape changes continuously between the two extreme cases of Fig. 4. The temperature dependence was followed continuously by monitoring the divalent peak intensity (shaded region in Fig. 4). The measured intensity was corrected to account for the overlapping tail of the trivalent signal, determined from the fluorescence spectrum excited 30 eV above the absorption threshold (dashed line).

Figure 4 Yb Lα1 RIXS spectra in YbInCu4 and YbAgCu4 excited at the maximum of the Yb^2+ resonance. The small change of valence with temperature (Δnf = 0.13 for YbInCu4 and 0.065 for YbAgCu4) results in a large change of the divalent peak intensity. The shaded area indicates the spectral region whose intensity was monitored versus temperature. The dashed line is the fluorescent contribution that was subtracted from the RIXS spectra to estimate the intensity of the divalent component (shown in Fig. 5).

The temperature evolution of the divalent component is presented in Fig. 5. The measured intensities have been converted into absolute (1 − n_h) values by using the low-temperature n_h values of Lawrence et al. (1994) (0.83 for YbInCu₄ and 0.87 for YbAgCu₄). A very different behaviour with temperature is identified: in YbInCu₄ the edge located around T_V  =  42 K reveals the sudden valence change at the transition temperature. In YbAgCu₄ the evolution is smooth and exhibits a concavity change around 70 K. The dashed curve is the prediction of the AIM, calculated within the non-crossing approximation (NCA), assuming a Kondo temperature T_K  =  70 K.

Figure 5 The continuous intensity evolution of the divalent RIXS signal has been converted into 1  –   nh values (nh  = number of f-holes) by adopting the XAS value nh = 0.83 for YbInCu4 and 0.87 for YbAgCu4. YbInCu4 shows a sharp transition. The evolution of YbAgCu4 agrees with the NCA calculation for TK  =  70 K (dashed curve).

RIXS data at the L edges of rare earths contain a further independent indicator of valence. In fact, in addition to the dipolar 2p–5d transition, a 2p–4f quadrupolar excitation is also possible in the absorption step of the scattering process. The quadrupolar transition is much weaker than the dipolar one and is detected only for excitation energies well below the 2p–5d absorption threshold (Dallera et al., 2000). Nonetheless, this channel is very interesting because it directly involves the 4f states. This transition is therefore a very powerful indicator of rare-earth valence changes driven by external temperature and pressure conditions, especially in Yb, where the 4f shell is close to being filled. The quadrupolar transition cannot occur in divalent (4f¹⁴) metallic ytterbium. In compounds, following hybridization with the 5d orbitals, n_h holes are available for the transition. The quadrupolar feature is indicated by the arrows in Fig. 6, displaying spectra excited at the resonance of the 2p–4f transition. It appears at a lower energy transfer because the addition of an electron into the 4f states leaves the system in a less excited state. The peak is evident at high temperature when ytterbium is in an almost trivalent state. At low temperature it appears only as a shoulder. Both the dipolar and quadrupolar features carry in principle the same information on the Yb valence state. However, since the dipolar channel involves extended conduction states, a quantitative comparison of their relative intensities is not straightforward and should explicitly take into account the material-dependent 5d density of states near E_F, and subtle details of the resonance profile.

Figure 6 When the excitation energy is tuned below the 2p–5d dipolar excitation threshold the intensity of the quadrupolar 2p–4f transition is enhanced. The arrows point to the RIXS peak following quadrupolar excitation (hνin = 8940 eV). This feature is well seen at high temperature, where both compounds are close to the trivalent configuration. At low temperature the quadrupolar feature is seen as a weak shoulder.

The present results confirm that the Anderson model provides a sound theoretical framework also for the spectral properties of Kondo systems. Previous inconsistencies are most likely due to the short probing depth of electron spectroscopies, and to perturbed surface layers. They demonstrate that RIXS is an extremely interesting bulk probe of intermediate valence thanks to the possibility of resonantly enhancing the contribution of a specific electronic configuration of interest. Extensions to other strongly correlated systems like the transition metal oxides, and to experiments under pressure or in external fields, which are precluded to electron spectroscopies, are being considered.

Footnotes

†Presented at the `International Workshop on High-Resolution Photoemission Spectroscopy of Correlated Electron Systems' held at Osaka, Japan, in January 2002.