Introduction

While most of the rare earth (RE) elements show a very stable trivalent state in solids, some of them, especially Ce, Yb, and Eu, may display different valence states depending on the chemical composition, the temperature, and the applied pressure or magnetic field1. This valence instability, which is caused by the f shell being almost empty (Ce), almost full (Yb) or almost half-filled (Eu), is connected with a hybridization between localized 4f and delocalized valence states2. Weak hybridization results in a stable trivalent RE state with large local magnetic moments which order at low T due to RKKY exchange. An increasing hybridization first results in an effective antiferromagnetic exchange interaction between the conduction electrons and the local moments, called Kondo interaction3, 4. As a result the local moments gets screened by the conduction electrons, the magnetic order disappears, leaving a paramagnetic ground state with strongly renormalized conduction electrons, referred to as Kondo lattices and heavy fermion systems. In these systems the occupation of the f shell, i.e. the number of 4f electrons n f for Ce or 4f holes nh for Yb, shows a weak decrease with increasing hybridization and the valence v starts to deviate from the trivalent state nf,h = 1. Models based on the Kondo interaction have proven to be very appropriate to describe this regime5, 6. In the limit of strong hybridization, one enters the intermediate valence regime where a strong quantum mechanical mixing of different valence state occurs and real charge fluctuations are dominant1, 2. Theoretically this regime can be described with the Anderson model, which takes both the large Coulomb repulsion between f electrons and the hybridization between 4f and conduction electrons explicitly into account and allows for both spin and charge fluctuations. In this case, n f and the valence can considerably deviate from 1+ and 3+, respectively. Since the competition between the RKKY interaction on one hand and the Kondo interaction and valence fluctuations on the other hand leads to very unusual states and properties, such systems have attracted a lot of attention7,8,9,10. Particularly fascinating are phase transitions at absolute zero temperature, called quantum phase transitions, which are often connected to unconventional superconductivity. Knowledge of the RE valence and of the occupation of the f level is crucial for an understanding of the exotic properties of such systems.

The simplicity of the X-ray absorption/emission measurements at the L2,3 edges of RE elements and the large X-ray penetration depth providing bulk specific results made this technique a standard for the study of the valence in Kondo lattice and intermediate valent systems as a function of composition and temperature. Hence a large number of studies have been performed to determine n f as well as the Kondo or valence fluctuating energy scale using the T dependence of n f 11,12,13,14,15,16,17,18,19. However, the accuracy to which n f could be determined with standard XAS or photoemission measurements was limited to maybe a few percent11. Since in Kondo lattices the deviation from the trivalent state is small and typical changes in n f between 3 and 300 K are only of the order of a few percent, these studies of the T-dependent changes concentrated on systems with a relatively high TK ≥ 100 K. However, the most interesting cases are those where the deviation from trivalent is small, since this is the regime where the transition from a magnetic ordered to a paramagnetic ground state occurs3, 20. The strong development of resonant X-ray emission spectroscopy (RXES) has opened new possibilities for studying even minute valence changes in Kondo lattices and to determine n f with very high accuracy11, 21. The Yb case is especially suited for RXES because the weak signal of the “minority” Yb2+ state is on the low energy side of the absorption edge, i.e. in a region with very low background by other emission lines. RXES allows to selectively enhance this Yb2+ signal by at least one order of magnitude21.

The extended and detailed knowledge accumulated on the Kondo alloy systems Yb(Rh1−xCo x )2Si2 and Yb(Rh1−xIr x )2Si2 makes them perfect candidates for a systematic investigation of the relation between n f and the Kondo scale20, 22,23,24,25. With increasing Co doping the Kondo scale is gradually reduced by almost two orders of magnitude from about 25 K for YbRh2Si2 to less than 1 K for YbCo2Si222. In contrast substituting Ir for Rh increases TK by about a factor of two up to 45 K26, 27. YbNi4P2 is a Kondo lattice system just at the verge of the transition from a ferromagnetic to a paramagnetic ground state, with a Kondo temperature of 8 K which increases upon substituting As for P28, 29.

We have used the high accuracy offered by this method for a high precision study of the T dependence of n f and its relation to bulk properties in Yb-based Kondo lattices. Our experimental data reveal an almost perfect scaling of the low temperature valence with TK, in agreement with theoretical predictions. In contrast, the temperature scale on which the valence changes occur is almost the same in all compounds despite the fact that the Kondo temperatures differ by several orders of magnitude. This unexpected finding is in stark contradiction to predictions of the Kondo model. Comparing with the data on other Yb Kondo lattices that are available in the literature, we find that this behavior is observed in all of them and even seems to extend into the mixed-valent regime. We discuss the influence of excited crystal field levels and lattice vibrations as possible origins of this unexpected observation, but substantial theoretical work seems necessary in order to explain our findings.

Results

Resonant X-ray emission spectroscopy

In Fig. 1 the obtained RXES spectra as a function of temperature are shown for four selected compounds. The data have been normalized to the peak intensity of the Yb3+ emission line. For YbIr2Si2 and YbRh2Si2 with a relatively high Kondo temperature the temperature effects are sizeable and easily seen by eye. In YbCo2Si2 where the Kondo scale is small the deviation from integer valent Yb3+ is very small and consequently the intensity of the Yb2+ line is very weak. However, even in this case the variation of the Yb2+ intensity with temperature can be measured and quantified with extremely high accuracy thanks to the excellent performance of the spectrometer yielding very high quality data. This is shown in Fig. 2 for the low temperature spectra of YbIr2Si2 and YbCo2Si2 and explained in detail in Supplementary Notes 1 and 2.

Fig. 1
figure 1

Yb L3-Lα1 RXES spectra as a function of temperature. Data are shown for four Yb Kondo lattice systems with different TK from 45 K (YbIr2Si2) down to well below 1 K (YbCo2Si2). The incident photon energy was set to the maximum of the 2+ resonance at the Yb L3 absorption edge. All spectra are normalized to the intensity of the Yb3+ emission line. The inset shows a zoom-in on the Yb2+ region of the YbCo2Si2 spectra. Even though TK is well below 1 K in this compound, small but systematic changes of the Yb2+ intensity all the way up to room temperature are observed

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Fig. 2
figure 2

Quantitative analysis of the RXES spectra. The spectra are fitted using three components corresponding to the Yb3+ (blue filling) and Yb2+ (green) emission lines and the weak quadrupole transitions E2 (white) in the pre-edge region, respectively21. The line shape of the Yb3+ and Yb2+ lines is not symmetric and has been determined from the experimental data by looking at the difference spectrum between high and low T data (see Supplementary Note 1). The Yb valence is then determined from the ratio of the Yb2+ and Yb3+ integrated intensities knowing that the Yb3+ emission line is reduced to 13% of its maximum intensity when the incident photon energy is set to the Yb2+ resonance of the L3 absorption edge (see ref. 21 and Supplementary Note 2). The same lineshapes are used to fit the Yb2+ and Yb3+ components of the RXES spectra for all samples and at all temperatures

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Temperature dependence of the 4f occupancy

The experimentally determined deviation from the trivalent (one hole) Yb state Δnh(T) = 1 − nh(T) as a function of temperature is plotted in Fig. 3a for all studied Yb Kondo lattices. The absolute changes in the 4f occupancy and, in particular, the low temperature values Δnh(T → 0) =: Δnh(0) are consistent with the expectation that Δnh(0) increases as TK increases. Surprisingly however, the temperature scale Tv on which the valence changes occur seems to be very similar in all measured compounds despite the very different Kondo scales, TK. Here we identify Tv with the temperature at which Δnh(T) = 1/2Δnh(0) following ref. 30. The dashed lines are fits to the data using the empirical form Δnh(T) = Δnh(0)/[(1 + (21/s − 1))(T/Tv)2]s, with Δnh(0) and Tv free parameters and s = 0.2131, which approximates the observed T dependencies well. Note that other ways of assigning a scale Tv to each curve in Fig. 3 can be used. This will change the absolute numbers for Tv but it will not affect our finding that Tv is very similar among all studied compounds (see Supplementary Note 6).

Fig. 3
figure 3

Evolution of the 4f occupancy with temperature. Temperature dependence of Δnh(T), the deviation of the Yb valence from 3+, in absolute values (a) and normalized to its low-temperature value (b). The error bars show the 2σ confidence interval calculated from the covariance matrix of the fits of the 2+ and 3+ amplitudes in the experimental spectra. The absolute error for Δnh(T) in a is comparable for all compounds and within the symbol size. The relative error in b increases with smaller values of Δnh(T) but stays within ±5% even for the smallest values of Δnh(T). The dashed lines are fits to the data points with an analytic expression from which the temperature scale of the valence fluctuations, Tv, and the low temperature value, Δnh(0), are obtained (see text for details). Previously reported results on other Yb Kondo lattices13,14,15,16, 19 are shown in faded colors in b for comparison, together with the experimental technique that was used to obtain the data

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In order to be able to better compare the temperature dependence for the different samples we show the changes in Δnh(T) normalized to their low-temperature values in Fig. 3b. Apparently, there is no significant difference in the characteristic temperature scale Tv of the valence changes neither within the Co doping series Yb(Rh1−xCo x )2Si2 nor with respect to the other measured Yb Kondo lattices. It should be noted that the temperature dependencies obtained here also resemble those reported in the literature for other Yb Kondo lattices, some with a TK as high as several hundred Kelvin13,14,15,16, 19. Also in these cases a clear T dependence in the Yb valence between 3 and 300 K, the range most easily accessible to X-ray spectroscopies, was observed. For comparison we show those data in Fig. 3b together with our results. The plot clearly demonstrates that even though the Kondo scale differs by up to four orders of magnitude between the different compounds, the Yb valence shows a very similar T response regardless of TK. In contrast, the measured low temperature values of the Yb valence correlate very well with the Kondo scale determined in other experiments.

It should be noted that Yb-systems which are in the strong intermediate valent regime, i.e. with valence well below 3+, do not show a significant change in the valence in the T range below 300 K. As an example we show in Supplementary Fig. 4 results for YbAl2. This system presents a valence of about 2.2 and a characteristic valence fluctuation energy of the order of 2500 K32. The experimental RXES spectra at 3 and 300 K lie on top of each other. Thus any change in the valence is smaller than the experimental resolution. A further intermediate valent system for which precise RXES data exist, YbAlB4, with a valence of the order of 2.75, shows a T dependence of the valence below 300 K (Supplementary Fig. 4)18, but the change is much weaker than in the systems shown above. This suggest that within the strong intermediate valent regime both the magnitude of the T-induced change in the valence, as well as the T scale on which it occurs do change with the strength of charge fluctuations. This temperature behavior is much more in line with theoretical predictions according to which the characteristic temperature scale for valence changes in the mixed valent regime should increase only linearly with the bare hybridization strength Γ, and not exponentially as in the Kondo regime (c.f. Supplementary Note 3). However, our data on the YbT2Si2 series demonstrate that once the system enters from the strongly mixed-valent into the Kondo regime, Tv stops to decrease and stays within a factor of two at a universal value, even when TK drops by four orders of magnitude. We note that this effect is also clearly visible on recent X-ray absorption spectroscopy (XAS) data on YbNi3Ga9 under pressure33. When the pressure is increased from p = 0 to p = 16.1 GPa, the valence at low T < 10 K increases continuously from v = 2.6 to v = 2.88, but the T scale on which v is increasing towards 3+ does not change significantly (Fig. 2 in ref.33).

Our findings are summarized in Fig. 4 where we plot for each Kondo lattice compound the experimentally measured low temperature deviation from the trivalent Yb state, i.e. Δnh(0), against Tv, the temperature scale of the valence changes, and TK, the Kondo energy scale inferred from magnetic susceptibility, entropy or Mössbauer measurements28, 34, 35. For comparison, we again included the results reported in the literature13,14,15,16, 19, 35 in panel (b) and also (a) if Tv has been determined from the X-ray spectroscopy data. While there is a clear correlation between Δnh(0) and TK, Tv seems to be independent of either of these two.

Fig. 4
figure 4

Dependence of the low temperature valence on characteristic temperature scales. The deviation of the Yb valence from 3+ at low temperatures, Δnh(0), is plotted against a the characteristic temperature scale of the valence changes, Tv, extracted from the data shown in Fig. 3, and b the Kondo energy scale, TK, obtained from thermodynamic measurements. Data points taken from previous reports in the literature13,14,15,16, 19 are shown in shaded colors. The dashed lines are guides to the eye. The data show that Δnh(0) scales with TK as expected, over several orders of magnitude, while Tv does not scale with Δnh(0) or TK and is very similar in all investigated Yb Kondo lattices

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Discussion

Within the usually used approximations the Kondo model predicts that both the low temperature valence, 3 − Δnh(0), and the temperature scale, Tv, on which the valence changes occur will increase with increasing Kondo scale TK (see e.g. ref. 30, n.b. Supplementary Note 5). Figure 4b clearly shows that there is indeed a very nice correlation between Δnh(0) measured with RXES and the thermodynamically determined TK. The linear relation between log Δnh(0) and log TK suggests a power law dependence between both properties, i.e. \({\mathrm{\Delta }}n_{\mathrm{h}}(0) \approx aT_{\mathrm{K}}^n\). The dashed line in Fig. 4b corresponds to n = 2/3 and a = 1/200. Within the Kondo model both Δnh(0) and TK depend on several parameters. Therefore the exact relation between both properties is a complex one. However, in a simple approximation, Δnh(0) is proportional to TK/Γ, where Γ stands for the bare hybridization width of the ground state doublet30, 36. TK itself depends exponentially on Γ which results in a weak logarithmic dependence of Γ on TK (see Supplementary Note 3). In total that should lead to a sublinear dependence of Δnh(0) on TK. Therefore, given the approximations made, the n = 2/3 power law we observed is in reasonable agreement with theoretical expectations.

The observed perfect scaling between Δnh(0) and TK strongly indicates that the Kondo effect is the dominant interaction responsible for Δnh. It is therefore a big surprise that the T-induced increase of the Yb valence towards 3+ occurs on almost the same temperature scale Tv for all compounds (Fig. 4a), independently of Δnh(0) and TK. The Kondo model has proven to be very successful in explaining even fine details of the properties of Yb-Kondo lattices5, 6, but it predicts the T scale of the recovery of the 3+ state to be proportional to TK, an intuitively trivial relation (n.b. Supplementary Note 5). One therefore needs to look for possible mechanisms that could modify the T dependence expected within the simple approximation usually used for Kondo systems.

A first candidate is the crystal electric field (CEF), which is not taken into account in those simple approximations, but which is known to be important in RE compounds. CEF leads to a splitting of the J = 7/2 multiplet of Yb3+ into four doublets (or one quartet and two doublets in case of a cubic local Yb symmetry). Usually for Kondo physics only the lowest doublet is considered but at higher temperatures, when T becomes comparable to the CEF splitting, excited CEF levels get thermally populated and become relevant. However, even for a single Kondo ion, an exact calculation of nh including CEF is extremely demanding, and therefore to the best of our knowledge has only been done for one very specific case36, 37. Hence the effect of CEF on the T dependence of nh is not very clear at the moment. Naively, because the Kondo scale depends exponentially on the degeneracy N of the effective local moment, involvement of excited CEF states corresponds to an increase in the effective degeneracy which should result in an increase of the effective Kondo scale with increasing T. Because of the exponential dependence of TK on N, this effect would become stronger upon decreasing TK of the ground state. Thus it would provide a way for making Tv independent of the ground state TK (for a more detailed argument see Supplementary Note 4). A similar effect would also be expected if the hybridization of an excited CEF level is stronger than the hybridization strength of the ground state doublet. Different hybridization strengths for different CEF levels have rarely been considered so far. But recent theoretical and experimental results indicate that the hybridization depends on the symmetry of the CEF level38, 39, and that this might result in unexpected phenomena40, 41. In YbRh2Si2, where the hybridization of the CEF ground state seems to be relatively weak, such an effect might become important. An STM study of YbRh2Si2 indeed provided evidence that in this compound the evolution of the Kondo lattice at higher temperatures is strongly influenced by excited CEF levels42. Different compounds might have very different CEF schemes, but the overall CEF splitting, i.e. the energy difference between the CEF ground state and the highest excited CEF level do not differ much and are typically in the range 200 to 400 K. Thus both the absolute value and the range of the overall CEF splitting are compatible with the value and range of Tv.

Another factor that could be relevant for nh(T) is the lattice. It seems that thermal expansion does not have a significant effect on nh(T). In Yb Kondo lattices increasing the volume results in a shift of the valence towards the 2+ state, because of its larger atomic volume25. Therefore normal thermal expansion should lead to an increase of Δnh(T) with increasing T, opposite to the experimental observation. On the other hand, Tv is about 300−400 K, which is similar to the Debye temperatures ΘD of these compounds. ΘD does not show large variations between the studied Yb compounds, typically differing by not more than ±30%. These differences would be even smaller if one would consider a ΘD only corresponding to the vibration of Yb atoms. Thus it is tempting to discuss the universal Tv scale in terms of a phonon-induced mechanism. The renormalization of the quasiparticles at the Fermi level E F due to electron−phonon coupling is indeed expected to vanish on the scale of ΘD which would lead to a decrease of the renormalized electronic density of states at E F on the same T scale. As the Kondo energy is proportional to the renormalized electronic density of states, one would therefore expect TK to decrease with increasing temperature on a T scale related to the Debye temperature. But at the same time, the exchange coupling itself should also be affected by thermally excited phonons and a preliminary analysis indicates that it might increase with increasing T, thus compensating the effect of the change in renormalization. Therefore the effect of enhanced thermal vibration is not yet clear and requires more theoretical and experimental studies. On a general level, in strongly correlated electron systems, the effect of the lattice on the electronic properties is usually just ignored. However, recent studies indicate that taking the lattice into account might severely change the expected electronic properties43. Very recently, a study of the effect of zero point and thermal ionic motion on the superexchange J in cuprates indicates a strong motional broadening of J giving rise to substantial effects on some electronic and magnetic properties44.

Our results imply that a determination of the Kondo scale based on the T dependence of the valence determined with X-ray spectroscopy, as frequently done, is at least questionable. Instead this T dependence might not necessarily be related to the Kondo scale. RXES is intimately related to standard L2,3 edge XAS and other X-ray and electron spectroscopic techniques. Previous studies comparing the different techniques showed that hard and soft X-ray photoemission yield the same low T valences and T dependences as X-ray absorption/emission techniques for Yb compounds19, 21. Part of the previously reported results on other Yb compounds which we included in Figs. 3 and 4 have been obtained with X-ray spectroscopies other than RXES and show the same T dependence. Thus the surprising T-dependent behavior reported here is seen independent of the employed spectroscopic technique and seems to be universal among the investigated Yb Kondo lattice systems. It would be interesting to see whether the valence changes in Ce Kondo lattices show a similar temperature behavior independent of the Kondo scale TK. Such a result might provide some further insight into the origin of the unexpected temperature dependence of Δnh(T).

The high sensitivity and accuracy in determining Δnh with RXES and the nice relation observed between TK and Δnh(0) suggest some immediate application in the field of Kondo lattices. In Kondo lattices where TK is of the order of or smaller than the magnetic ordering temperature, thermodynamic, transport and magnetic data cannot reliably determine TK since all these properties are then controlled by the RKKY energy scale. In such systems, a RXES measurement of Δnh(0) can provide a much more reliable estimation of TK using the power law scaling between TK and Δnh(0). This can be easily understood since Δnh(0), in contrast to many other properties, is not affected by the RKKY exchange.

The high accuracy of the present RXES data also allows settling a controversy in the field of Kondo lattice systems. YbRh2Si2 is very close to the quantum critical point separating the antiferromagnetically ordered ground state from the paramagnetic ground state45. This results in very unusual thermodynamic, transport and magnetic properties, e.g. in a divergence of the effective mass of the conduction electrons45. Several models have been proposed to explain these unusual properties, but there is no consensus yet46,47,48. The topic is subject of an intense debate. One of the proposed models relies on the presence of a valence quantum critical point, i.e. a valence transition which is suppressed to T = 0 by pressure or chemical tuning48. However the present data and analysis demonstrate that throughout the alloy series Yb(Rh1−xCox)2Si2 and including YbIr2Si2, the T dependence of the valence is the same, and that the valence stays constant from lowest T up to 20 K within a (relative) accuracy of Δv ~ 5×10−4. This excludes the presence of a valence transition, and thus the presence of a valence quantum critical point. The valence quantum critical point scenario of ref. 48 does not seem to be relevant for YbRh2Si2.

In summary, we used RXES to study the evolution of the valence in the Kondo alloy Yb(Rh1−xCo x )2Si2 and a few further Yb-based Kondo lattices as a function of composition and temperature. The Kondo temperature TK in the studied compounds cover the range from very small values TK ≈ 0.25 K up to quite large values TK = 40 K. With present RXES techniques the Yb valence in such systems can be determined with very high accuracy. This allows us to study the deviation from a strict trivalent state, i.e. the deviation in the number of holes nh in the 4f shell from unity, Δnh = 3 − v, as a function of composition and temperature with a precision of the order of 0.1%. Combining the present results and literature data, we reveal an almost perfect linear correlation between log Δnh(T = 0) determined from RXES at T < 10 K and log TK determined from thermodynamic properties. This linear correlations extends over three orders of magnitude, from TK ≈ 0.25 K and Δnh(0) = 0.004 up to TK ≈ 500 K and Δnh(0) = 0.4. Upon increasing T, all systems evolve towards a pure trivalent state, but surprisingly the T scale on which Δnh decreases with increasing T is almost the same for all compounds, and thus does not scale at all with TK. Looking for the origin of this unexpected universal T scale we identify two possible candidates, excited CEF levels and lattice vibrations. However the effect of both properties on nh(T) has not been studied yet and is thus presently not clear. Therefore our results urge for more experimental and theoretical studies on how excited CEF levels and lattice vibrations affect the T dependence of the valence. Without respective information available it is not clear whether excited CEF levels or lattice vibrations can explain our observations, or whether another mechanism is active. The very similar temperature scale Tv observed in all compounds in this study gives, at least, a good idea of the energy scale of the relevant mechanism.

Methods

Sample details

Single crystals of Yb(Rh1−xCo x )2Si2 (x = 0.00, 0.07, 0.12, 0.38, 0.68, 1.00), YbIr2Si2, and YbNi4P2 have been grown and characterized as described previously49, 50. The sharpness of the Laue diffraction pattern taken from the samples suggests high crystalline quality which for YbRh2Si2 and YbCo2Si2 is also confirmed by excellent ARPES and STM data6, 42. The physical properties of the samples have been extensively studied with e.g. specific heat, resistivity and magnetic susceptibility measurements. From these data a consistent picture of the Kondo scale in the Yb(Rh1−xCo x )2Si2 has emerged showing a gradual decrease of the single ion Kondo temperature TK from about 25 K for x = 0 to well below 1 K for x = 122, 25. The onset of coherence effects in the resistivity which is typical for Kondo lattices scales similarly from about 120 K for x = 0 to ~ 1.5 K for x = 1.

Resonant X-ray emission spectroscopy

The RXES data have been taken with the X-ray emission spectrometer of the former ESRF beamline ID16, now upgraded and reopened as ID20, looking at the Yb L3-Lα1 emission line. Measuring with energy resolution in the scattered beam significantly reduces the life-time broadening of the spectra and furthermore can be used to enhance the weak Yb2+ contributions in the Kondo lattices11. For each sample we have taken the Yb L3 XAS spectrum, then fixed the incident photon energy to the maximum of the Yb2+ resonance and measured the Yb Lα1 RXES spectrum. The experimental procedure and the data analysis are described in detail in ref. 21. The data have been taken twice for each sample during two different experiments and using two different sample growth batches. The results of the first and the second run of experiments are exactly the same showing perfect reproducibility of our experimental findings. As a hard X-ray photon-in photon-out technique, Yb L3 edge RXES is a bulk sensitive technique free of surface effects. The available temperature range was 3−300 K and the data were taken starting from low temperature.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.