Introduction

An enhanced effective electron mass m* is considered as a hallmark of a large class of strongly correlated metals. In such heavy-electron systems, many exotic quantum phenomena, including unconventional superconductivity, non-Fermi-liquid behavior, quantum criticality, and even topological semimetals1,2,3,4,5,6,7 occur. Among the materials with the largest Sommerfeld coefficients γ ∝ m* are the rare-earth-based heavy fermion systems YbPtBi8,9, YbT2Zn20 (T= Fe, Co)10 and PrAg2In11, also dubbed as ‘super’-heavy electron systems12,13. In these cubic systems, γ reaches record values as high as 10 Jmol−1 K−2. The Yb-variants have in common that they are characterized by small characteristic energy scales10,14 of the order of kB ⋅ 1…10 K, including the Kondo scale kBTK as well as excited crystal electric field (CEF) levels at kBTCEF. Correspondingly, it has been suggested that the hybridization of the conduction electrons1 with a large number of degenerate CEF states is the source of the high electronic mass10,15,16. However, given the multitude of small energy scales, a definite determination of the energy scales and thus a quantitative description of the unusually high γ has proven difficult14,17,18.

For YbPtBi, specifically, the following characteristic temperature scales8,9,14,17,19,20 have been found so far (see Fig. 1a, c, d). Since the Yb3+ Kramers ion resides on a site with cubic symmetry, the CEF is expected to split the J = 7/2 multiplet into two doublets of type Γ6 and Γ7 and a Γ8 quartet21. The analyses of several experiments consistently find that the highest CEF excited state is a doublet (likely Γ6) with TCEF,2 ~ 60…100 K. Even though various studies20 favor Γ7 to be the ground state and Γ8 to be the first excited state with TCEF,1 ~ 1…10 K, the reverse assignment with a Γ8 ground state was also found to be compatible with a number of experimental results20. In addition, symmetry-breaking distortions at low temperature, giving rise to additional level splittings, could not be conclusively ruled out so far17,20,22,23. These uncertainties have not only hampered estimates for the absolute value of TCEF,1, but also of the second important temperature scale TK14 which is believed to be in the same energy range. Finally, for very low temperatures below TN ≈ 400 mK, YbPtBi orders antiferromagnetically. This order is fragile12,24,25,26 as it can be suppressed towards a quantum critical point by a small external magnetic field, and non-Fermi-liquid behavior was discovered in the quantum critical region24.

Fig. 1: Ambient-pressure thermodynamics of YbPtBi.
figure 1

a Schematic representation of proposed characteristic energy scales of YbPtBi. TN indicates the temperature of magnetic ordering, TK the possible temperature range of the Kondo crossover and TCEF,1 and TCEF,2 the possible temperature range of the first and second excited crystal-electric field levels. b Schematic of the setup to measure the elastocaloric effect28. A sample with attached thermocouple is placed across a gap on the sample carrier, which is screwed into the pressure cell (not shown). Different compressive uniaxial pressures can be exerted when the mechanical contact is closed by a uniaxial pressure cell. The direction of applied force is indicated by the big black arrows. The strain-induced temperature changes, ΔT, are recorded with the thermocouple. c–e Ambient-pressure properties of YbPtBi: total specific heat, Cp, and magnetic specific heat, Cm24, (c) thermal expansion24, α, (d) and elastocaloric temperature amplitude, ΔT (e), from this work. The highlighted temperature range indicates the temperature range of the Kondo/crystal-electric field scales, and the dotted line indicates TN, as also shown in (a).

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In this paper, we clarify the hierarchy of energy scales in YbPtBi, something only made possible through conducting thermodynamic measurements of the elastocaloric effect under well-controlled, symmetry-breaking, uniaxial pressure p. In contrast to magnetic field, which breaks time-reversal symmetry and thus affects all Kramers-degenerate states, the lattice strain ϵ associated with uniaxial pressure can only lift degeneracies stabilized by crystallographic symmetries. As we will show below, it is through this elastic level splitting that the application of uniaxial pressure can be used to sensitively probe the single-ion physics27 associated with the Γ8 state. As a result, we successfully disentangle the thermodynamic features resulting from CEF excitations and other energy scales which can be attributed to the Kondo scale in YbPtBi. Overall, this analysis, in line with literature results, places the ‘super’-heavy YbPtBi in the limit of TK > TCEF,1 and provides support for the notion that the extremely high γ value results from the hybridization of conduction electrons with a quasi-sextet CEF ground state.

Results

Experimental elastocaloric effect data

The elastocaloric effect ΔT/Δϵ describes a temperature change ΔT that is induced by varying the strain by Δϵ. Thermodynamically it is given by

$${\left.\frac{{{\Delta }}T}{{{\Delta }}\epsilon }\right\vert }_{S}=-\frac{{\left.\partial S/\partial \epsilon \right\vert }_{T}}{{\left.\partial S/\partial T\right\vert }_{\epsilon }}=-\frac{T}{{C}_{V}}{\left.\frac{\partial S}{\partial \epsilon }\right\vert }_{T},$$
(1)

with S being the entropy and CV the heat capacity at constant volume V.

As recently established in ref. 28, ΔT/Δϵ can be determined with high precision in an a.c. version of the technique by applying an oscillation with amplitude Δϵ in piezo-driven uniaxial pressure cells29,30 and measuring the resulting ΔT using a thermocouple (see Fig. 1b). Δϵ is determined through a capacitive displacement sensor (not shown). A constant, finite strain ϵ can be superimposed, so that ΔT/Δϵ can be mapped out as a function of ϵ. In this work, we follow this experimental method to determine ΔT/Δϵ for YbPtBi with uniaxial pressure p applied along the crystallographic [1 0 0] direction, resulting in a finite ϵ. We implemented following important modifications to the technique: first, we use a uniaxial pressure cell that also incorporates a force sensor29 since the applied force/pressure p is a better control parameter than the conjugated strain ϵ in these type of devices. Second, we mount the sample free-standing in a sample carrier which is designed in such a way that only compression (denoted by a negative sign of p) can be applied when the mechanical contact in the carrier is closed (see Fig. 1b, ref. 31). This allows us to determine precisely the neutral point p = 0 (and ϵ = 0) at any given temperature which is important for the symmetry arguments presented below.

To illustrate the fingerprint of relevant energy scales in YbPtBi at ambient pressure in our data, we first compare on the right of Fig. 1 the elastocaloric temperature amplitude ΔT(T) for T ≳ 1.2 K (panel (e)) at ambient pressure pa ≈ 0 and Δϵ ≈ const. with literature data24 on the molar specific heat \({C}_{p}(T)\propto T{\left.\partial S/\partial T\right\vert }_{{p}_{a}}\) (panel (c)) and the thermal expansion \(\alpha (T)\propto {\left.\partial S/\partial p\right\vert }_{T}\propto {\left.\partial S/\partial \epsilon \right\vert }_{T}\) (panel (d)) (see Supplementary Note 2 for a discussion of the equivalence of different thermodynamic quantities at ambient pressure). Upon cooling, ΔT shows a clear feature around T* ≈ 7.6 K (see Supplementary Note 1 for criterion) with a concomitant sign change of ΔT. α(T) also exhibits a similar feature as ΔT(T), including a sign change, at a slightly lower temperature. Simultaneously, the magnetic contribution Cm to the specific heat Cp, obtained after subtracting the specific heat of the non-moment bearing Lu analogue24, shows a clear peak at T*. Previously, these prominent features in α and Cm were interpreted either to be solely due to CEF effects or to combined CEF/Kondo effects. Here, we will provide an alternative interpretation, and we will show that the first excited CEF level is in fact located much lower in energy. Upon further cooling down to ≈1.2 K, the lowest temperature of our experiment, ΔT(T) remains featureless and small, similar to α(T). At even lower temperatures Cp(T) and α(T) show clear features associated with a magnetic phase transition at TN ≈ 0.4 K.

Now we turn to the behavior of ΔT under finite, symmetry-breaking uniaxial pressure p, shown in Fig. 2. In our experiments, we were able to apply compression as high as ≈−1.8 GPa, which is a large value for uniaxial compression in these devices31. Clearly, the data sets as a function of T (Fig. 2a) and p (Fig. 2b) reveal significant changes of ∣ΔT∣. Key observations can be summarized as follows: first, as ∣p∣ is increased, a low-temperature extremum emerges at Text ≈ 2.7 K for p ≈ −0.3 GPa, which is increased to 4.0 K by p ≈ −1.8 GPa. Second, the feature at T* ≈ 7.6 K remains visible for all pressures and its position is only barely affected by p (see inset of Fig. 2a). Third, ΔT increases strongly, from almost zero, in a monotonic and, to first approximation, in a near linear manner with ∣p∣ for any given T. Only for larger ∣p∣ ≳ 0.5 GPa are some deviations from this p-linear behavior, in particular at lowest T, observed.

Fig. 2: Elastocaloric effect of YbPtBi under finite, symmetry-breaking uniaxial pressure.
figure 2

Elastocaloric temperature amplitude, ΔT, induced by a small oscillating uniaxial strain with amplitude \({{\Delta }}\epsilon \approx {{{\rm{const}}}}. < 0\) at constant offset uniaxial pressures, p, as a function of temperature, T, (a) and as a function of p at constant T (b). Note that the p ≈ 0 data is also shown in Fig. 1 on a logarithmic T scale. The data spacing in (a) is ~0.1 GPa and in (b) 1 K (for T < 10 K) and 2 K (for T > 10 K). The arrows in panel (a) indicate the position of the characteristic temperatures Text and T⋆. The inset to (a) shows the evolution of Text and T⋆ with p (for criteria, see Supplementary Note 1).

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In general, this elastocaloric effect data contains contributions from all relevant energy scales, in particular the CEF and Kondo energy scales. In the following we will use generic qualitative arguments and explicit modeling of the elastocaloric effect of single-ion CEF states to disentangle these contributions. In particular, we will demonstrate that the strong change of temperature ΔT with uniaxial pressure p results from the response of the first excited quartet CEF level to symmetry breaking, whereas the behavior of ΔT at p = 0 (including the anomaly at T⋆) most likely originates from the formation of the coherent Kondo state.

To facilitate the discussion of the elastocaloric effect of CEF levels, we will from now on focus on the notion of strain ϵ, since Young’s modulus Y100 ≔ ∂p/∂ϵ∣T ≈ const. (see Supplementary Note 5), and assume temperatures \(T={{{\mathcal{O}}}}({{{\Delta }}}_{{{{\rm{CEF}}}}})/{k}_{{{{\rm{B}}}}}\) where ΔCEF is the energy difference between ground state and first excited CEF energy level. The two dominant effects of finite ϵ are expected to be (i) shifting the CEF levels and (ii) a possible lifting of degenerate CEF levels due to lowering the crystal symmetry. The latter scenario will most likely only be relevant when the degenerate state is not the ground state, since the degeneracy is otherwise prone to be lifted already at zero strain through a spontaneous Jahn-Teller distortion or multipolar order32,33. The elastocaloric effect ΔT/Δϵ is expected to be significantly distinct in these two cases.

The two scenarios are visualized separately in Fig. 3a: In case (i), left sketch of the figure, the CEF energy level is unambiguously associated with ϵ, necessary for the applicability of Grüneisen scaling ∂p/∂T∣V ∝ CV/V (see Supplementary Note 6). We obtain ∂S/∂ϵ∣T ∝ T∂S/∂T∣ϵ, therefore ΔT/Δϵ ≈ const. as a function of strain, see Eq. (1). Thus, in this case, we expect a large intercept of ΔT/Δϵ at ϵ = 0 and no significant change with ϵ.

Fig. 3: Modeling of the elastocaloric effect of YbPtBi.
figure 3

a Generic considerations of the impact of a strain ϵ on the CEF energy level difference ΔCEF and the resulting behavior of ΔT/Δϵ at constant temperature \(T={{{\mathcal{O}}}}({{{\Delta }}}_{{{{\rm{CEF}}}}})/{k}_{{{{\rm{B}}}}}\). Left: scenario (i), induced shift of the first excited level. Right: scenario (ii), strain-induced splitting of the first excited level. b Left: model specific heat CV(T) at ϵ = 0. Right: strain-induced splitting of ΔCEF as a function of ϵ used in model 1 and model 2, both within scenario (ii) (see text for details). c Comparison of experimental data of the elastocaloric effect and the results of the two model calculations. For the experimental data, Y100 ≈ 120 GPa was used47 to express p in terms of ϵ (see text and SI). The data spacing for both experimental data and model calculations in the top panel is ~0.08% and in the bottom panel 1 K.

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In case (ii), the strain-induced symmetry lowering leads to a splitting of the first excited CEF energy level for both compressive and tensile strains. Hence, at ϵ = 0 we must have ∂S/∂ϵ∣T = 0, correspondingly ΔT/Δϵ = 0 and Grüneisen scaling is no longer applicable. We also must have ∂S/∂ϵ∣T ∝ ϵ. Therefore, in scenario (ii), we expect that the magnitude of ΔT/Δϵ is expected to increase rapidly from a small value at ϵ = 0 with increasing tension or compression (see right sketch of Fig. 3a).

Our data (see Figs. 2 and 3c) is characterized by a large change of ΔT/Δϵ with ϵ and only a small finite intercept at ϵ = 0. We can therefore conclude that the elastocaloric effect under finite strains is dominated by a strain-induced splitting of a first excited CEF level. This CEF level has to be the Γ8 quartet, since the Yb3+ Kramers doublets are protected by time-reversal symmetry and cannot be split by the application of strain.

We note that these considerations also imply that measurements of the thermal expansion α at ambient pressure will only display anomalies of excited CEF levels when these levels are doublets that only shift with strain. In contrast, the excited CEF Γ8 level will leave almost no fingerprint in \(\alpha (p=0)\propto {\left.\partial S/\partial \epsilon \right\vert }_{T}(\epsilon =0)\approx 0\) when the level splitting is the dominant effect. Therefore, a finite α seems unlikely to be related to the physics of the Γ8 level, contrary to what has been discussed in previous studies on YbPtBi17 under the assumption of validity of Grüneisen scaling (see Supplementary Note 6).

Modeling of the elastocaloric effect

To extract quantitative information on the CEF states from the elastocaloric data, in particular on the estimate of energy of the Γ8 level in YbPtBi, we performed model calculations of ΔT/Δϵ using a simplified model with a minimal number of free parameters. Our model is based on a Schottky-type specific heat Cp of a two-level system with four-fold degenerate first excited state at energy ΔCEF. We do not include a possible hybridization-induced broadening of the CEF levels, because (i) Kondo hybridization does not significantly change the energy levels of the f electrons and (ii) the applied stress yields an approximately volume conserving strain which only has a minor effect on the Kondo hybridization with the CEF levels as such. We note that only the occupation of the f electron energy levels, which is by definition only weakly affected by the Kondo coupling, is measured in our elastocaloric experiments. This is in contrast to spectroscopic experiments, where the single-particle spectral function is measured and thus the quasielastic linewidth is strongly broadened, which renders effects from the CEF level splitting unobservable. In addition, since our measured signal is ΔT ∝ 1/Cp, we added an electronic contribution of the form γT to the total model specific heat Cp. To reduce the number of parameters, we omitted phononic contributions because they are small below ~ 10 K (see Fig. 1c). The full model specific heat at ϵ = 0 is shown in the left panel of Fig. 3b. Using a value of ΔCEF/kB = 1.6 K, this model reproduces the broad hump in the literature Cp(T) data8,24 around T ≈ 800 mK.

To parameterize the response to strain, we use two approximations, which we call model 1 and model 2. Model 1 comprises a linear splitting by strain via \({{{\Delta }}}_{{{{\rm{CEF}}}}}(\epsilon )={{{\Delta }}}_{{{{\rm{CEF}}}}}(0)\left(1\pm {\beta }_{1}| \epsilon | \right)\). The choice for this model is motivated by considering the effect of a tetragonal distortion on the CEF-Hamiltonian21\({{{{\mathcal{H}}}}}_{{{{\rm{CEF}}}}}^{{{{\rm{cubic}}}}}\to {{{{\mathcal{H}}}}}_{{{{\rm{CEF}}}}}^{{{{\rm{cubic}}}}}+{g}_{zz}\epsilon {O}_{2}^{0}\) perturbatively with an elastic constant gzz that characterizes the distortion and \({O}_{2}^{0}\) the Stevens operator that emerges in tetragonal symmetry (see Supplementary Note 8). The energy of the Γ8 state then changes as \({E}_{{{{\Gamma }}}_{8}}\to {E}_{{{{\Gamma }}}_{8}}\pm 6{g}_{zz}\epsilon\) for small ϵ.

Naturally deviations from the ϵ-linear behavior of the CEF excitation energy will arise for larger ∣ϵ∣. In an attempt to better describe the magnitude of our experimental data, we include a second-order term of the expansion in ϵ in model 2. The corresponding energy splitting we use then reads \({{{\Delta }}}_{{{{\rm{CEF}}}}}(\epsilon )={{{\Delta }}}_{{{{\rm{CEF}}}}}(0)\left(1\pm {\beta }_{1}| \epsilon | +{\beta }_{2}| \epsilon {| }^{2}\right)\).

Figure 3c shows a comparison of the experimental data for ΔT/Δϵ (left column) to the calculations for model 1 (middle column) and model 2 (right column). For both models we use ΔCEF(0)/kB = 1.6 K and β1 = 70. Since the lowest temperature of our experiments is \({T}_{\min }\approx 1.2\,{{{\rm{K}}}}\), we restrict the calculations to this temperature range. Therefore, the model results reflect the effects associated with the higher-energy branch (blue dotted lines in the right panel of Fig. 3b).

Clearly, the results for model 1 already capture many of the essential observations of the experiment on a qualitative level. It reproduces the low-temperature minimum of ΔT/Δϵ as a function of T around T ≈ 3 K under finite ϵ, as well as the approximately linear change of ΔT/Δϵ with ϵ.

However, model 1 is not sufficient to account for the data also on a quantitative level. Repeating our calculations within model 2 and setting β2 = 1750, we obtain the results shown in the right column of Fig. 3c). Now much of the experimental data can be very well reproduced over the full strain range.

Importantly, the good agreement between experiment and model calculations is only achieved when Γ8 is the first excited state (scenario (ii)). In the Supplementary Note 7, we also show model calculations for the reverse scenario (we call it scenario (iii)) assuming Γ8 is the ground state and the first excited state is a doublet, even if this scenario is unlikely due to the inherent instability of a symmetry-protected degeneracy of the ground state towards Jahn-Teller distortions. We find that scenario (ii) and (iii) can be clearly distinguished based on our experimental data by considering the value of Text for ϵ → 0. Specifically, we find that the finite value of Text for ϵ → 0 (see inset of Fig. 2a) is only compatible with scenario (ii). In fact, the value of Text for ϵ → 0 restricts the possible values of ΔCEF(0), for which a reasonably good description of the experimental data within model 2 can be achieved. This leads us to estimate that ΔCEF(0)/kB = (1.6 ± 0.3) K in YbPtBi.

Discussion

Overall, our analysis establishes the Γ8 quartet to be the first excited state with an energy difference to the doublet ground state of ΔCEF ≈ kB ⋅ 1.6 K. However, the physics of single-ion CEF levels does not capture the feature at T⋆ ~ 7.5 K in ΔT/Δϵ which persists for all uniaxial pressures (see Fig. 2a and Fig. 3c). The model would also predict that ΔT/Δϵ → 0 for ϵ → 0, required by restoring the cubic crystal symmetry. Instead we observe a small but finite ΔT/Δϵ, which we attribute to contributions from other energy scales than the Γ8 one.

Since the remaining doublet CEF level excitation is located at much higher temperatures around 60…100 K, it can most likely be excluded as the source for the anomaly at T⋆. The remaining temperature scale which is known to be relevant in YbPtBi is set by the Kondo temperature TK. Remarkably, the thermal expansion data24 shown in Fig. 1d reveals the onset of negative α below T ≈ 20 K that persists down to T ≈ T⋆. In general, a negative thermal expansion in cubic systems is exceptional34. To rationalize this, we note that the volume of Yb3+ is smaller than the one of Yb2+. Therefore, even tiny hybridization-induced changes35,36 of the strictly trivalent state of Yb can be the origin of a negative α. Thus, all experimental data are consistent with the expectations for the formation of the Kondo system with TK ≈ T⋆ ≈ 10 K. However, solely based on the present elastocaloric data, we cannot entirely rule out the presence of yet another, so far undiscovered, characteristic energy scale in YbPtBi around kBT⋆.

From the analysis above, the hierarchy of temperature scales in YbPtBi can now be assigned to TCEF,2 > T⋆ > TCEF,1. Therefore, this opens the possibility that the conduction electrons do not only hybridize with the Yb3+ CEF Γ7 doublet ground state, but also with the first excited Γ8 quartet state. Effectively, hybridization then takes place with an Yb quasi-sextet (N = 6). This approximately matches the analysis15 of the generalized Kadowaki-Woods ratio A/γ2, with A being the Fermi-liquid coefficient of the resistivity, shows that ρ(T) for YbPtBi8,10 falls between the values of A expected for N = 6 and N = 8. This is nonetheless remarkable for two reasons. First, for T ≪ TCEF,1 the assumption of a quasi-sextet might be no longer valid. Second, the hybridization strength can also depend on the symmetry of the underlying CEF level37,38,39,40, an aspect which has so far been rarely considered. Our study shows that YbPtBi with two low-lying CEF levels of different symmetry below T⋆ ≈ TK might be an interesting reference system to quantify the relevance of symmetry-dependent hybridization strength. Based on the present results, an explicit modeling of the elastocaloric effect including all relevant degrees of freedom might be a promising pathway towards an improved quantitative description of the many-body physics in heavy fermions.

In summary, through measurements and analyses of the elastocaloric effect, we have firmly established the crystal-electric field levels and therefore the hierarchy of energy scales in the ‘super’-heavy fermion material YbPtBi. We find that the energy difference ΔCEF between the ground state and the first excited quartet CEF level with ΔCEF ≈ kB ⋅ 1.6 K is significantly smaller than the proposed Kondo scale kBTK ≈ kB ⋅ 10 K, putting both the Γ7 ground state doublet and the Γ8 quartet below kBTK. This allows for the possibility that conduction electrons hybridize with a quasi-sextet (N = 6) Yb3+ ground state, providing support for theoretical models that assign the anomalously large electron mass to an enhanced degeneracy of the CEF levels.

At a more general level, our work demonstrates that measurements of the elastocaloric effect under finite pressures41,42,43,44 enable us to control and quantify strain-induced changes of the crystal-electric field schemes and disentangle relevant low-energy scales in correlated electron systems. This approach will also be particularly relevant for the field of quantum magnets, in which the unambiguous determination of single-ion CEF states is essential for a microscopic description of their unusual magnetic properties.

Methods

Crystal growth

Single crystals of YbPtBi were grown from ternary melts rich in Bi45. High purity, elemental Yb, Pt, and Bi were combined in an alumina crucible in a 10(Yb):10(Pt):80(Bi) atomic ratio, sealed in an amorphous silica tube under 1/6 atmosphere of Ar and heated to 1100 ∘C over 4 h. After dwelling at 1100 ∘C for 3 h, the ampule was cooled to 900 ∘C over 2 h and then slowly cooled to 600 ∘C over 231 h. After dwelling at 600 ∘C for a few hours the ampule was taken out of the furnace, inverted into a metal centrifuge cup and rotated a several thousand rpm for times less than 15 s45 so as to separate the remaining, very Bi rich, remaining liquid from the grown YbPtBi crystals. The resultant crystals were well faceted and could be as large as 5–10 mm on a side. Given that YbPtBi is air sensitive, crystals were sealed in evacuated silica tubes for long-term storage.

Elastocaloric effect measurements

The samples were polished for measurements under finite uniaxial pressures29 into a bar with dimensions of 100 μm × 140 μm × 1000 μm, with the long axis being the strain axis. For measurements of the a.c. elastocaloric measurements, the d.c. voltages on the piezoelectric actuators were modulated by a small a.c. voltage with frequency f = 56.13 Hz (see Supplementary Note 4) on the tension stack. For the measurements of the induced temperature change ΔT, an chromel-AuFe0.07% thermocouple46 was fixed to the sample with a tiny amount of Stycast 1266 (see Supplementary Note 3). The thermocouple was anchored on the cell body. The voltage on the thermocouple was amplified by a low-temperature transformer mounted on the low-temperature stage and subsequently read out by a Lock-In amplifier. Further description on the uniaxial pressure cell and modeling of the elastocaloric effect are included in the main text (see Fig. 1).