Divergent Thermal Expansion and Grüneisen Ratio in a Quadrupolar Kondo Metal

In solid state physics, non-Fermi liquid phases describe unconventional metallic states of matter. Such exotic phases have been extensively studied in the framework of heavy fermion (HF) quantum criticality [1]. A magnetic quantum critical point typically forms in Ce- and Yb-based intermetallic systems that are located in between a Kondo screened Fermi-liquid and a long range antiferromagnetically ordered state. The divergence of the Grüneisen parameter, defined as the ratio of volume thermal expansion to specific heat, is a universal signature of pressure sensitive quantum critical points [2, 3].

In addition, there has been a particular focus on exploring non-Fermi liquid states that are related to electric quadrupole moments and on identifying potential links between these states and magnetic HF quantum criticality [4, 5, 6, 7]. Quadrupolar ground states form, for instance, in non-Kramers Pr- and U-based intermetallics, given that certain symmetry constraints are fulfilled. Possible cause for unconventional metallic behavior in these materials is the quadrupolar Kondo effect, which was postulated by Cox in 1987 [8]. Here, the simultaneous overscreening of a localized quadrupole moment by two-channels of conduction electrons, which are related to their spin degree of freedom, leads to non-Fermi liquid behavior in specific heat ($C/T\propto\log 1/T$), electrical resisitivity ($\rho/\rho_{0}\propto 1+A\sqrt{T}$) and quadrupole susceptibility ($\chi_{\mathrm{Q}}\propto\log{1/T}$) as well as an unconventional residual entropy of $S=R\log\sqrt{2}$ [9].

Cubic Pr-based 1-2-20 systems are prototypical to study such novel quadrupolar  [10, 11] as well as higher multipolar related correlation effects [12, 13, 14]. PrIr₂Zn₂₀, for instance, has a well defined quadrupolar non-Kramers $\Gamma_{3}$ ground state doublet (point group $T_{d}$) [15] and displays clear signatures of the quadrupolar Kondo lattice effect [4], which are cut off by antiferroquadrupolar order at 0.11 K [16]. Recent studies on highly diluted Y_1-xPr_xIr₂Zn₂₀ provided direct evidence of the single-impurity quadrupolar Kondo effect, based on characteristic non-Fermi liquid behaviors found in the specific heat, electrical resistivity and elastic constant [5, 17]. Nevertheless, evidence for the residual entropy $S=R\log\sqrt{2}$ remains elusive so far [5].

The concept of the Grüneisen parameter can be generalized to all irreducible representations of elastic strains. Such a generalized Grüneisen parameter is expected to diverge close to a quantum critical point provided that the associated stress couples to a relevant operator of the critical fixed-point [2, 18]. For the single-impurity quadrupolar Kondo fixed-point, this is the case for the quadrupolar stress $\sigma_{\mathrm{u}}$, that breaks the cubic symmetry, and, as a consequence, destabilizes the non-Fermi liquid physics and quenches the residual entropy. This implies that the quadrupolar Grüneisen ratio $\Gamma_{\mathrm{u}}=(\partial_{T}\varepsilon_{\mathrm{u}})/C$ diverges in a characteristic manner. Here, $\varepsilon_{\mathrm{u}}=(2\varepsilon_{zz}-\varepsilon_{xx}-\varepsilon_{yy})/\sqrt{3}$ is the strain component of the $\Gamma_{3}$ doublet that is conjugate to $\sigma_{\mathrm{u}}$ with the corresponding thermal expansion $\alpha_{\mathrm{u}}=\partial_{T}\varepsilon_{\mathrm{u}}$ and $C=C_{\mathrm{m}}/V_{\mathrm{m}}$ the specific heat with the molar volume $V_{\mathrm{m}}$ and the molar specific heat $C_{\mathrm{m}}$. Using a thermodynamic identity, $\Gamma_{\mathrm{u}}=1/T(\mathrm{d}T/\mathrm{d}\sigma_{\mathrm{u}})_{S}$ also quantifies the adiabatic change of temperature upon the variation of $\sigma_{\mathrm{u}}$ and thus describes an elastocaloric effect [19].

The application of a magnetic field $\boldsymbol{H}$ will split the non-Kramers $\Gamma_{3}$ doublet resulting in a finite quadrupole moment $\langle Q\rangle\sim\chi_{\mathrm{Q}}H^{2}$ of each Pr³⁺ ion, that is proportional to $H^{2}$ for small fields due to time-reversal symmetry. This induces a local quadrupolar strain via the elastic coupling $g_{\Gamma_{3}}$, and after averaging over Pr disorder results in a homogeneous strain $\varepsilon_{\mathrm{u}}\sim g_{\Gamma_{3}}n_{\mathrm{Pr}}\chi_{\mathrm{Q}}H^{2}/c_{\mathrm{u}}$ where $n_{\mathrm{Pr}}$ is the Pr³⁺ density and $c_{\mathrm{u}}=(c_{11}-c_{12})/2$ the corresponding elastic constant. The quadrupolar Kondo effect predicts $\chi_{\mathrm{Q}}\sim\log 1/T$ resulting in a singular temperature dependence of the quadrupolar strain $\varepsilon_{\mathrm{u}}\sim n_{\mathrm{Pr}}H^{2}\log 1/T$. Finally, using that the specific heat at low temperatures is dominated by the contribution of the Pr³⁺ ions, $C_{\mathrm{m}}\sim n_{\mathrm{Pr}}T\log 1/T$ for small magnetic fields [5], one expects for the quadrupolar Grüneisen parameter $\Gamma_{\mathrm{u}}\sim H^{2}/(T^{2}\log{1/T})$. Consequently, thermal expansion and magnetostriction experiments are ideally suited for the investigation of the non-Fermi liquid behavior associated with the quadrupolar Kondo effect.

In this work, we perform such measurements on highly diluted single crystalline Y_1-xPr_xIr₂Zn₂₀. As our key finding, we confirm experimentally that the temperature dependence for the quadrupolar strain $\varepsilon_{\mathrm{u}}$ and the associated quadrupolar Grüneisen parameter $\Gamma_{\mathrm{u}}$ is indeed consistent with the suggested quadrupolar Kondo scenario [5, 17]. Remarkably, we also find that the volume strain $\varepsilon_{\mathrm{B}}$ shows a similar singular behavior in zero magnetic field although isotropic stress does not directly couple to a relevant operator of the quadrupolar Kondo fixed-point. We speculate that an indirect coupling might be generated either by elastic anharmonicities or by static strain disorder accounting for the experimentally observed divergence in the volume thermal expansion.

Ultrahigh-resolution thermal expansion and magnetostriction measurements were carried out in a dilution refrigerator using a miniaturized capacitive dilatometer [20]. Central to this study is a highly diluted Y_1-xPr_xIr₂Zn₂₀ single crystal with $x=0.036$, for which relative length changes were measured along a cubic $\langle 100\rangle$ direction, with magnetic fields applied either parallel or perpendicular to the measurement direction. The magnetic field direction is defined as [001] in the following, so that $\varepsilon_{zz}$ and $\varepsilon_{xx}$ correspond to longitudinal $\varepsilon_{\parallel}$ and transverse strain $\varepsilon_{\perp}$, respectively. As the applied field does not break the symmetry within the $(x,y)$ plane, we can assume that $\varepsilon_{xx}=\varepsilon_{yy}$. The measurement of longitudinal and transverse strain in magnetic field then allows to infer the values of the isotropic strain $\varepsilon_{\mathrm{B}}=\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz}$ and the quadrupolar strain $\varepsilon_{\mathrm{u}}$, as detailed in the Supplemental Material (SM) [21]. An unavoidable side effect of the experimental technique is a small force of approximately $4$ N [20] acting on the sample along the measurement direction, which corresponds to a tiny unixial stress of a few MPa. To clarify whether this effect has an impact on the deduced relative length changes, we performed complementary measurements on a $[111]$ oriented single crystal with a comparable Pr-concentration of $x=0.033$. For further characterization, a special uniaxial stress capacitive dilatometer [27], that exerts a roughly fifteen times larger force on the sample than the miniaturized dilatometer, was employed. For details on the single crystalline samples examined in this study, see SM [21].

First, we discuss the quadrupolar thermal expansion coefficient $\alpha_{\mathrm{u}}=\partial_{T}\varepsilon_{\mathrm{u}}$ of Y_1-xPr_xIr₂Zn₂₀ with $x=0.036$. The respective data is derived from the measurement of the longitudinal $\alpha_{\parallel}$ and the volume thermal expansion $\beta$ for $\boldsymbol{H}\parallel[001]$ via the the relation $\alpha_{\mathrm{u}}=\sqrt{3}(\alpha_{\parallel}-\beta/3)$. A detailed derivation of this relation and an overview of the data of $\alpha_{\parallel}$ and $\beta$ is provided in the SM [21].

The quadrupolar thermal expansion coefficient $\alpha_{\mathrm{u}}$ is shown in Fig. 1(a) on a logarithmic temperature scale ranging from 0.05 K to 4 K for various magnetic fields up to 10 T. For small fields, $\alpha_{\mathrm{u}}$ increases down to lowest temperatures. At around 1.5 T a maximum develops whose position shifts to higher temperatures as a function of increasing field. At the same time, the height of the maximum continuously decreases with $H$. For $\mu_{0}H\gtrsim 8$ T, this peak is quantitatively captured by crystal electric field (CEF) calculations [21] as indicated by the dashed lines.

The ratio $\alpha_{\mathrm{u}}/(\mu_{0}H)^{2}$ is shown in the inset of Fig. 1(b). For temperatures $k_{\mathrm{B}}T\gg\mu_{\mathrm{B}}\mu_{0}H$, the data collapses onto a single curve that exhibits a crossover from a $\alpha_{\mathrm{u}}/H^{2}\sim\partial_{T}\chi_{\mathrm{Q}}\sim 1/T$ behavior at low temperature, consistent with the quadrupolar Kondo effect, to a $1/T^{2}$ dependence at high temperature as expected for a Curie susceptibility $\chi_{\mathrm{Q}}\sim 1/T$ of a fully localized $\Gamma_{3}$ doublet[21]. The crossover temperature of $\sim 0.6$ K is consistent with previous studies [5, 17]. By using the molar 4f specific heat $C_{\mathrm{m}}$ [21], measured on a single crystal from the same batch with a comparable Pr-concentration of $x=0.044$, we evaluate the quadrupolar Grüneisen parameter that follows the expected singular behavior $\Gamma_{\mathrm{u}}\sim H^{2}/(T^{2}\log 1/T)$ for low magnetic fields as shown by the black solid line in Fig. 1(b). The crossover field of approximately 2 T is comparable to the crossover field as suggested by the measurement of $\alpha_{\mathrm{u}}$.

Within the framework of the quadrupolar Kondo model, the deviation from the characteristic quadrupole Kondo behavior for larger magnetic fields can be attributed to both a channel- and a quadrupolar-asymmetry, induced in linear and quadratic order in $H$, respectively. At small fields, the first effect is expected to dominate and results in a crossover temperature $T^{*}\sim H^{2}$, see Ref. [9], separating non-Fermi and Fermi-liquid behavior. In order to specify the origin of the magnetic field induced crossover in highly diluted Y_1-xPr_xIr₂Zn₂₀, Fig. 2 displays the data of $\alpha_{\mathrm{u}}/(\mu_{0}H)^{2}$ on a logarithmic temperature scale at small magnetic fields up to 2.5 T. The crossover temperature $T^{*}$, which is determined as the temperature at which deviations from the universal single-impurity quadrupole Kondo behavior arise, is indicated by an arrow for each magnetic field. The inset displays the characteristic temperature $T^{*}$ as a function of magnetic field, whereby the red solid line indicates a quadratic magnetic field dependence. Indeed, the characteristic temperature $T^{*}$ estimated from the quadrupolar thermal expansion data can be well scaled with $T^{*}\sim H^{2}$. This indicates that the field induced channel-asymmetry is the dominating perturbation at low magnetic field leading to the experimentally found deviations from the quadrupole Kondo behavior at low temperature, which is in very good agreement with the theoretical expectation.

The results on the quadrupolar thermal expansion coefficient $\alpha_{\mathrm{u}}$ and the quadrupolar Grüneisen parameter $\Gamma_{\mathrm{u}}$, which are indicative of the single-impurity quadrupole Kondo effect, are further corroborated by the quadrupolar magnetostriction $\varepsilon_{\mathrm{u}}$ displayed in Fig. 3. The quadrupolar magnetostriction coefficient $\varepsilon_{\mathrm{u}}$ is derived from the measurement of the longitudinal $\varepsilon_{\parallel}$ and the volume magnetostriction coefficient $\varepsilon_{\mathrm{B}}$ by using the relation $\varepsilon_{\mathrm{u}}=\sqrt{3}(\varepsilon_{\parallel}-\varepsilon_{\mathrm{B}}/3)$ for $\boldsymbol{H}\parallel{[001]}$. The data of $\varepsilon_{\parallel}$ and $\varepsilon_{\mathrm{B}}$ used for the calculation of $\varepsilon_{\mathrm{u}}$ is provided in the SM [21]. By analyzing the initial quadratic field dependence of $\varepsilon_{\mathrm{u}}\sim n_{\rm Pr}\chi_{\mathrm{Q}}H^{2}$ [21], we extract the quadrupolar susceptibility $\chi_{\mathrm{Q}}$ that is shown for various Pr doping $x$ in the inset of Fig. 3. The CEF calculation, that predicts a Curie-like $1/T$ temperature dependence on top of a constant Van Vleck contribution, can only capture the behavior of $\chi_{\mathrm{Q}}$ at elevated temperature. At low $T$ and for the lowest doping concentration $x=0.036$ our results are consistent with a logarithmic temperature dependence of $\chi_{\mathrm{Q}}$. For higher doping, the susceptibility is suppressed at low temperatures indicating a quenching of the quadrupolar Kondo effect, likely induced by interactions between Pr³⁺ ions.

We now turn to the discussion of the volume thermal expansion coefficient $\beta=\partial_{T}\varepsilon_{\mathrm{B}}$. In Fig. 4(a), $\beta$ is shown for the highly diluted single crystal with $x=0.036$. Remarkably, the thermal expansion coefficient at zero field increases down to lowest temperatures. This continuous increase is quenched by the application of a field and, for $\mu_{0}H\gtrsim 6$ T, the volume thermal expansion practically vanishes. Similarly, $\beta$ is also suppressed by larger doping as shown in the inset for zero field.

The continuous increase of $\beta$ on cooling at zero field and low doping is unexpected at first sight as the influence of the quadrupolar moments should in lowest order average out in the bulk thermal expansion. A parasitic signal could potentially arise from the presence of a small uniaxial stress from the sample holder that is on the order of a few MPa in case of the miniaturized dilatometer and breaks the isotropy of the environment. In order to clarify this influence, we compare in Fig. 4(b) the uniaxial thermal expansion along crystallographic $[001]$ and $[111]$ directions measured on two single crystals with almost the same doping level. While a uniaxial stress along $[111]$ only activates the bulk strain $\varepsilon_{\mathrm{B}}$, a uniaxial stress along $[001]$ induces both bulk and quadrupolar strains where the latter couples linearly to the $\Gamma_{3}$ ground state doublet of the Pr³⁺ ions. At very low temperature, we find indeed a difference between $\alpha_{[001]}$ and $\alpha_{[111]}$ that we attribute to the presence of a small uniaxial stress exerted by the dilatometer.

To demonstrate the tremendous effect of uniaxial stress along [001], we performed a complementary measurement using a uniaxial stress dilatometer with an applied uniaxial stress approximately fifteen times larger than in the miniaturized dilatometer, see Fig. 4(c). Under higher uniaxial stress, the divergence of $\alpha_{[001]}$ is suppressed and a maximum forms at around 0.11 K, suggesting the breakdown of the two-channel Kondo effect due to the linear in strain splitting of the $\Gamma_{3}$ ground state doublet.

Figure 4(d) displays the temperature dependence of $\alpha_{[111]}$ measured with both the miniaturized and the uniaxial stress dilatometer at zero magnetic field. As there is no notable difference between the two data sets, we conclude that the divergent temperature dependence of $\alpha_{[111]}$ is in fact an intrinsic volume effect and not artificially induced by the uniaxial stress applied externally along the measurement direction. This result is striking as it is markedly different from the molar specific heat of the 4f electrons $C_{\mathrm{m}}$ as shown by the red symbols. The thermal expansion is well described by a fit $\alpha_{\mathrm{fit}}=\gamma C_{\mathrm{m}}+b/T$ (blue line) that assumes two contributions. The first obeys Grüneisen scaling with a constant $\gamma$ and the second contribution diverges like $1/T$, which is reminiscent of the behavior of $\alpha_{\mathrm{u}}/H^{2}$ shown in the inset of Fig. 1(b). Consequently, the bulk Grüneisen ratio $\Gamma_{\mathrm{B}}=V_{\mathrm{m}}3\alpha_{[111]}/C_{\mathrm{m}}\approx 3V_{\mathrm{m}}\gamma+3V_{\mathrm{m}}b/(TC_{\mathrm{m}})$ exhibits a divergence as a function of temperature similar to $\Gamma_{\mathrm{u}}/H^{2}$ shown in Fig. 1(b).

In the following, we speculate on the origin of this surprising finding of an intrinsic bulk thermal expansion that increases down to lowest temperatures. The elastic coupling $g_{\Gamma_{3}}$ to the non-Kramers $\Gamma_{3}$ doublet leads in second order perturbation theory to the following correction to the elastic Hamiltonian

Here, $\varepsilon_{\mathrm{u/v}}(\vec{r}_{\alpha})$ are the two components of the local quadrupolar strain doublet at the position $\vec{r}_{\alpha}$ of the Pr³⁺ ion with index $\alpha$. After averaging over Pr disorder this Hamiltonian recovers the renormalization of the quadrupolar elastic constant $\delta c_{\mathrm{u}}=-n_{\mathrm{Pr}}g_{\Gamma_{3}}^{2}\chi_{\mathrm{Q}}$ whose temperature dependence was confirmed by elastic constant measurements [17]. Treating the Hamiltonian of Eq. (1) in perturbation theory, one obtains a correction to the free energy density of the form $\delta f\sim\frac{\delta c_{u}}{2}\langle\varepsilon_{\mathrm{u/v}}^{2}\rangle$. Here, a finite expectation value $\langle\varepsilon_{\mathrm{u/v}}^{2}\rangle$ might arise either from dynamic zero-point fluctuations of the quadrupolar strain or from static strain disorder [21]. This results in a contribution to the bulk thermal expansion $\delta\beta=\partial_{p}\partial_{T}\delta f=-\frac{1}{2}(\partial_{p}n_{\mathrm{Pr}}g_{\Gamma_{3}}^{2}\langle\varepsilon_{\mathrm{u/v}}^{2}\rangle)\partial_{T}\chi_{\mathrm{Q}}$. The divergence $\delta\beta\sim 1/T$ originating from $\chi_{Q}\sim\log 1/T$ is consistent with the observations in Fig. 4(d).

Hence, dynamic as well as static strain fluctuations can in principle explain the observed singular behavior in $\beta$. However, both scenarios also imply a contribution to the specific heat $\delta C_{\mathrm{m}}\sim 1/T$, which so far has not been successfully identified. A possible reason is a small prefactor of $\delta C_{\mathrm{m}}$, or, equivalently, a large effective Grüneisen parameter $\delta\Gamma_{\mathrm{B}}=\delta\beta/\delta C_{\mathrm{m}}$ characterizing the hydrostatic pressure dependence of $n_{\mathrm{Pr}}g_{\Gamma_{3}}^{2}\langle\varepsilon_{\mathrm{u/v}}^{2}\rangle$. This dependence can arise from elastic anharmonicities, e.g., via the elastic coupling $g_{\Gamma_{3}}(p)$ or, in case of strain disorder, from a response of $n_{\mathrm{Pr}}\langle\varepsilon_{\mathrm{u/v}}^{2}\rangle$ to pressure changes. From Fig. 4(d) we can estimate a lower bound $\Gamma_{\mathrm{B}}>2.15$  GPa^-1, and with the isothermal compressibility $\kappa_{T}\approx 0.01$ GPa^-1, estimated from elastic constant measurements on PrIr₂Zn₂₀ [28], this implies a dimensionless Grüneisen parameter $\delta\Gamma_{\mathrm{B}}\kappa_{T}$ at least of the order of 200 to be compared with a typical value of 1 for metals [29]. As elastic anharmonicities are in general unlikely to account for such large values, this suggests static strain disorder to be at its origin. Hydrostatic pressure will influence the local strain distribution and might thus affect sensitively the quadrupolar environment of each Pr ions resulting in a large response. Local strain fields were also invoked to explain the temperature and field dependence of the specific heat [5] and the elastic constant [17], respectively. Future nuclear quadrupolar resonance (NQR) measurements might help clarifying this issue.

In this work, we established the symmetrized quadrupolar expansivity, i.e., thermal expansion and magnetostriction, as well as the quadrupolar Grüneisen parameter as a highly sensitive probe to uncover quadrupolar fluctuations. For the example of highly diluted Y_1-xPr_xIr₂Zn₂₀ we demonstrated that it gives access to the quadrupolar susceptibility. Our experimental results are in excellent agreement with expectations for the single-impurity quadrupolar Kondo effect and corroborate its emergence in this material. On a more general note, we anticipate the quadrupolar expansivity and quadrupolar Grüneisen parameter to become an important tool also for the investigation of other materials with pronounced quadrupolar correlations, in particular, for materials close to a nematic quantum critical point as discussed for certain cuprate and iron pnictide superconductors [30, 31].