Magnetic field effects on the specific heat of a diluted Pr system Y_1-xPr_xIr₂Zn₂₀

A class of compounds containing Ce, Pr, Yb, and U ions exhibit non-Fermi liquid (NFL) behaviors in physical properties at low temperatures.¹ The NFL behaviors have been explained in terms of two concepts. One is a magnetic instability in the vicinity of the quantum critical point, where the quantum fluctuations are divergently enhanced by the competition between the on-site single-channel Kondo effect and the inter-site Ruderman-Kittel-Kasuya-Yosida interaction.² This competition have been explained by the Doniach phase diagram.³ The other is a single-site two-channel Kondo effect, where an impurity spin is over-screened by two-channel conduction bands.⁴ When an electric quadrupole of the impurity is regarded as the local spin, the two-channel Kondo effect is translated to the quadrupole Kondo effect.⁵ The theory of the quadrupole Kondo model predicted novel properties, e.g., the entropy of (1/2)Rln2 remaining at T = 0 and single-site NFL behaviors such as specific heat divided by temperature C/T $∝$ –lnT, quadrupole susceptibility χ_Q $∝$ –lnT, and electrical resistivity ρ/ρ₀ $∝$ 1 + A$T$⁠, where ρ₀ is the residual resistivity and A is the coefficient.⁵ Despite much experimental effort on diluted Pr and U alloys,^6–8 the single-site quadrupole Kondo effect has not been established yet. It is mostly because a superconducting impurity phase and atomic disorder affect the properties of Heusler-type Pr compounds and the crystalline electric field (CEF) ground state in diluted U systems is not well defined.

The Pr-based compounds PrT₂X₂₀ (T = transition metal, X = Zn, Al, and Cd) crystallizing in a cubic structure have attracted much attention.⁹ The CEF ground state of the Pr³⁺ ion with 4f ² configuration is the non-Kramers Γ₃ doublet with active quadrupoles. The quadrupole degrees of freedom give rise to a variety of phenomena such as long-range quadrupole order, unconventional superconductivity, and NFL behaviors.⁹ Notably, PrIr₂Zn₂₀ shows an antiferroquadrupolar (AFQ) order at T_Q = 0.11 K below which a superconducting transition occurs at T_c = 0.05 K.¹⁰ Moreover, the magnetic specific heat divided by temperature, C_m/T, shows –lnT dependence and ρ(T) exhibits an upward curvature in a moderately wide temperature range above T_Q. It suggests the formation of a quadrupole Kondo lattice.¹¹ Recently, we have studied its diluted Pr system Y_1-xPr_xIr₂Zn₂₀ for x ≤ 0.44. The CEF ground state retains the non-Kramers Γ₃ doublet with active quadrupoles.^12,13 For x ≤ 0.44, C_m/T and the differential electrical resistivity Δρ are scaled well by using a characteristic temperature T₀ defined as the temperature where the magnetic entropy reaches 0.75Rln2.¹³ Here, T₀ may indicate an energy scale for the NFL behaviors due to the multi-channel Kondo model.^5,14 Furthermore, for x < 0.05, the NFL behaviors of C_m/T $∝$ –lnT and Δρ $∝$ 1 + A$T$ were observed at T < T₀.¹³ The anomalous temperature dependences are explained by the calculation with the single-site quadrupole Kondo model.

In the present work, we report the specific heat of Y_1-xPr_xIr₂Zn₂₀ for x = 0.044 in magnetic fields of B $≤$ 12 T. To obtain distinct NFL signals from the Pr ions, we chose a sample with x = 0.044, in which C_m/T shows the –lnT behavior down to 0.08 K at B = 0.¹³ Since the Pr ions are well isolated in this system, neither magnetic nor quadrupole order was observed. Magnetic fields were applied along the [100] direction whereby the Zeeman effect is expected to split the Γ₃ doublet much wider than those in B || [110] and [111].

A single crystal was grown from a starting composition of x = 0.05 in Y_1-xPr_xIr₂Zn₂₀ by the procedure reported in the previous papers.^12,13 The Pr composition of the crystal could not be reliably determined by the wavelength dispersive electron-probe microanalysis with 1% resolution. Therefore, we determined the value of x as 0.044 by comparing the measured magnetization value at B = 1 T and T = 1.8 K with that calculated by using the CEF parameters for PrIr₂Zn₂₀.¹⁵ The measurements of C were done by a thermal relaxation method using a Quantum Design physical property measurement system (PPMS) from 0.4 to 300 K in B $≤$ 12 T. For 0.08 $≤$T $≤$ 0.5 K, C was measured in B $≤$ 6 T by using a laboratory-built system installed in a commercial Cambridge mFridge mF-ADR/100s adiabatic demagnetization refrigerator.

Figure 1 shows the temperature dependences of C for Y_1-xPr_xIr₂Zn₂₀ with x = 0.044 in B ≤ 12 T applied along the [100] direction. In B = 0, C(T) is increased on cooling below 1 K and saturated to a constant value below 0.3 K. The data for T < 0.3 K is largely enhanced when B is increased to 6 T, which can be explained by considering nuclear contribution as described below. The Hamiltonian of a ¹⁴¹Pr nucleus with the spin I = 5/2 in a magnetic field B is represented by

where A_hf = 0.052 K is a hyperfine coupling of a Pr ion,¹⁶ J is the total angular momentum of 4f electrons, g_N = 1.72 the nuclear g factor, and μ_N the nuclear magneton. By taking the isothermal magnetization M along B into consideration, the term of $J⋅B$ is equivalent to $(M/gJμB)⋅B$⁠, where g_J = 4/5 is the Lande’s g factor for J = 4 and $μB$ the Bohr magneton. Here, we used the values of M(B) measured for PrIr₂Zn₂₀ (Ref. 11) as shown with the opened circles in the inset of Fig. 1. The nuclear specific heat C_N(T) calculated for B = 2, 4, and 6 T are shown in Fig. 1 with the colored solid curves. By increasing B, C_N is enhanced and the calculated value reproduces the upturn of the experimental data for T < 0.3 K in B = 6 T. In B = 4 T, C(T) shows a broad maximum at around 0.4 K, which shifts to higher temperatures with increasing B and becomes a shoulder at 3 K in B = 12 T. The evolution of the Schottky specific heat results from the splitting of the CEF ground state doublet by the Zeeman effect via the excited triplet as described later.

To estimate the 4f-electron contribution, C_4f, the calculated C_N and the phonon contribution C_ph (black solid curve in Fig. 1) were subtracted from the measured C data. Here, C_ph was calculated by the equation, C_ph = xC_La+(1 − x)C_Y, where C_La and C_Y are the C data for nonmagnetic compounds LaIr₂Zn₂₀ and YIr₂Zn₂₀, respectively. The temperature variation of C_4f is plotted in the inset of Fig. 2. The data of C_4f in B = 4 T exhibit a maximum at around 0.4 K, which shifts to high temperatures as B is increased. The magnitude of the maximum does not change in B up to 12 T, which is a characteristic of the Schottky specific heat. This indicates that the ground state doublet is split by the magnetic fields for B ≥ 4 T. Thereby, the split state loses the quadrupole degrees of freedom.

The main panel of Fig. 2 shows the temperature dependences of C_4f/T. The –lnT dependence appears at T < 0.5 K in B = 0 and 2 T, which agrees with the prediction of the single-site quadrupole Kondo model.⁵ For B $≥$ 4 T, on the other hand, C_4f/T data deviate from the –lnT variation at low temperatures and exhibit a maximum which shifts to higher temperatures.

By integrating the C_4f/T data, we estimated the 4f-electron contribution to the entropy, S_4f, which is plotted as function of temperature in Fig. 3. Here, the measured S_4f is vertically offset by 1.40 J/K Pr-mol for B = 0 and 0.48 J/K Pr-mol for B = 2 T, respectively, so that they agree with the data of B = 4 T in the temperature range of T > 4 K. The S_4f curve moves to high temperatures with increasing B, which is the consequence of the shift of the maximum temperature of C_4f(T) as shown in the inset of Fig. 2. To estimate S_4f(T) in B, the CEF effect and the Zeeman effect are considered. The CEF Hamiltonian is represented with the CEF parameters W and X, and the Stevens operators $Onm$ as follows,

The colored solid curves in Fig. 3 are calculated by considering the CEF and Zeeman effect. Since the ionic radius of the Y³⁺ ion is smaller than that of the Pr³⁺ ion, we modified the CEF parameters for PrIr₂Zn₂₀, which were determined by the inelastic neutron scattering experiments,¹⁵ to be more relevant for the diluted Y(Pr)Ir₂Zn₂₀ system. Here, we increased the absolute value of W to adjust the energy scale of the CEF. The entropy was calculated with the modified CEF parameters of W = −1.50 K and X = 0.537 as shown with the solid curves. For B$≥$ 4 T, the experimental data moderately agree with the solid curve. This suggests that the entropy of the Γ₃ doublet at B = 0 is released at higher temperatures when the magnetic fields are increased above 4 T. For B = 2 T, however, the calculated curve is larger than the experimental data in the temperature range of T < 0.9 K. Moreover, in B = 0, the S_4f data are about 0.25Rln2 at the lowest temperature of 0.08 K, although the calculation remains at a constant value of Rln2 for T < 2 K. This discrepancy indicates that the magnetic entropy must be released by another mechanism except the CEF and Zeeman effects. In fact, C_4f/T for B = 0 and 2 T follows –lnT in the temperature range where the value of S_4f is less than Rln2. It suggests that the residual entropy and the –lnT dependence result from the active quadrupoles in the doublet. Furthermore, the anomalous behaviors agree with the prediction of the single-site quadrupole Kondo model.⁵ Therefore, the single-site quadrupole Kondo effect may arise from the on-site hybridization between the local quadrupoles and the two-channel conduction bands. To confirm that the quadrupole Kondo effect causes the NFL behaviors of C_4f and the residual entropy in S_4f, measurements of specific heat and electrical resistivity in magnetic fields are currently under investigation using different Pr concentrations.

In conclusion, we have reported the specific heat of the Pr diluted system Y_1-xPr_xIr₂Zn₂₀ with x = 0.044 in magnetic fields of B ≤ 12 T applied along the [100] direction. We obtained the 4f-contribution to the specific heat C_4f and estimated the entropy S_4f. In B = 0 and 2 T, C_4f/T shows the –lnT dependence in the temperature range 0.08 < T < 0.5 K. In B ≥ 4 T, however, C_4f shows a broad maximum which shifts to higher temperatures with increasing B. The S_4f (T) data in B $≥$ 4 T can be reproduced by considering the splitting of the Γ₃ doublet due to the Zeeman effect via the first excited Γ₄ triplet. In magnetic fields below 2 T, however, the calculated entropy is much larger than that evaluated from the experimental data. Therefore, the –lnT behaviors of C_4f/T in B = 0 and 2 T may arise from the single-site quadrupole Kondo effect, where the active quadrupoles of the Γ₃ doublet hybridize with the two-channel conduction bands.

We are grateful to S. Hoshino, J. Otsuki, H. Kusunose, A. Tsuruta, K. Miyake, R. J. Yamada, G. B. Park, K.-I. Imura, and A. Tanaka for valuable discussions. The specific heat measurements with the PPMS and the mFridge mF-ADR 100/s refrigerator were performed at N-BARD, Hiroshima University. This work was financially supported by Grants-in-Aid from MEXT/JSPS of Japan, Nos., JP15KK0169, JP15H05886, JP16H01076 (J-Physics), and JP18H01182.