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Impurity quadrupole Kondo ground state in a dilute Pr system Y1-xPrxIr2Zn20
Author’s Accepted Manuscript Impurity quadrupole Kondo ground state in a dilute Pr system Y1-xPrxIr2Zn20 Yu Yamane, Takahiro Onimaru, Kazuto Uenishi, Kazuhei Wakiya, Keisuke T. Matsumoto, Kazunori Umeo, Toshiro Takabatake www.elsevier.com/locate/physb
PII: DOI: Reference:
S0921-4526(17)30469-6 http://dx.doi.org/10.1016/j.physb.2017.07.062 PHYSB310129
To appear in: Physica B: Physics of Condensed Matter Received date: 30 June 2017 Accepted date: 27 July 2017 Cite this article as: Yu Yamane, Takahiro Onimaru, Kazuto Uenishi, Kazuhei Wakiya, Keisuke T. Matsumoto, Kazunori Umeo and Toshiro Takabatake, Impurity quadrupole Kondo ground state in a dilute Pr system Y1-xPrxIr2Zn20 Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.07.062 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Impurity quadrupole Kondo ground state in a dilute Pr system Y1-xPrxIr2Zn20 †
Yu Yamane1, Takahiro Onimaru1, Kazuto Uenishi1, Kazuhei Wakiya1 , ‡
Keisuke T. Matsumoto1 , Kazunori Umeo2, and Toshiro Takabatake1 1
AdSM, 2N-BARD, Hiroshima University, Higashi-Hiroshima, JAPAN
Abstract Electrical resistivity and specific heat C of a dilute Pr system Y1-xPrxIr2Zn20 for 0 ≤ x ≤ 0.44 were measured to study the phenomena arising from active quadrupoles of the Pr3+ ion with 4f2 configuration. On cooling, 's of all samples monotonically decrease, while the residual resistivity ratio (300 K)/(3 K) decreases with x. In the whole range x ≤ 0.44, the magnetic contribution of the specific heat divided by temperature Cm/T shows a broad maximum at around 10 K, which can be reproduced by a two-level model with a ground state doublet and a first-excited triplet separated by 30 K. This indicates that the crystalline electric field ground state of the Pr ions remains in the 3 doublet for the cubic Td point group. On cooling, the Cm/T data for x = 0.085 and 0.44 approach constant values at T < 0.3 K as expected from the random two-level model. By contrast, Cm/T for x = 0.044 increases continuously down to 0.08 K, suggesting a non-Fermi liquid state due to the impurity quadrupole Kondo effect. Keywords: quadrupole Kondo effect, non-Fermi liquid, random two-level model Introduction Praseodymium-based intermetallic compounds with Pr3+ ions of the 4f2 configuration have attracted much attention because of a variety of phenomena originating from multipolar degrees of freedom [1-3]. In diluted U4+ or Pr3+ systems with the f 2 configuration, non-Fermi liquid (NFL) behaviors manifest themselves due to interaction of electric quadrupoles of localized f 2 electrons with conduction electrons [4]. The NFL behaviors are understood as an impurity quadrupole Kondo effect in the cubic 3 doublet system, where the quadrupoles of the isolated U4+ or Pr3+ ions could be over-screened by the multichannel conduction bands [4,5]. In this model, the NFL behaviors involve three ingredients, –lnT divergence of specific heat divided by temperature C/T, √ pendence of electrical resistivity , and residual entropy of (1/2)Rln2 at zero temperature [4,5]. In a U diluted system Th1-xUxRu2Si2 for x ≤ 0.07, in fact, both C/T and the magnetic susceptibility show the –lnT dependence and
follows √ where A is a negative coefficient [6]. Similar phenomena were also observed in other uranium-based compounds such as Th1-xUxPd2Si2[7], Y1-xUxPd3[8] and Th1-xUxBe13[9]. In addition, in a dilute Pr system La1-xPrxPb3, C/T shows lnT dependence [10]. However, these observations do not provide firm experimental evidence for the impurity quadrupole Kondo effect, because there are atomic disorder and uncertainty in the crystalline electric-field (CEF) ground states particularly in the U-based systems. Recently, in caged compounds PrT2X20 (T = transition metal, X = Zn, Al, Cd), a variety of phenomena arising from the active quadrupoles such as long-range quadrupole order, superconductivity, NFL behavior, and structural transition have been revealed [11]. They crystallize in the CeCr2Al20-type structure, where the point group of the Pr ion is the cubic Td [12]. In most compounds, the CEF ground state of the Pr ion is the 3 doublet [11], possessing no magnetic dipoles but electric quadrupoles. Thereby, the quadrupole-driven phenomena often manifest themselves in the systems. Notably, PrIr2Zn20 shows an antiferroquadrupole (AFQ) order at TQ = 0.11 K and a superconducting transition at Tc =0.05 K [13,14]. The magnetic entropy at TQ is only 20% of Rln2, suggesting that the quadrupole fluctuations could play a role in the superconducting pair formation. Moreover, at temperatures above TQ, (T) varies with upward convex curvature [15]. This behavior agrees with the theoretical calculation based on the two-channel Anderson lattice model [16]. Thus, formation of quadrupole Kondo lattice was suggested to occur by the interaction between the quadrupoles and the conduction bands [16]. Therefore, we expected dilution of the Pr ions in nonmagnetic host compounds such as YIr2Zn20 may realize the single-site NFL state. Bearing this in mind, we have prepared Y1-xPrxIr2Zn20 (x ≤ 0.44) and measured C and
. We avoided the use of the La counterpart LaIr2Zn20 which undergoes a structural transition at Ts = 200 K [13,17]. In case the Pr ions were sparse enough, impurity effect due to the interaction between the active quadrupoles of the isolated Pr ions and the conduction bands could manifest itself. Experiment Single- and poly-crystalline samples of Y1-xPrxIr2Zn20 for x ≤ 0.44 were prepared by the Zn self-flux method using boron nitride crucibles in purified argon atmosphere [13]. Purity of the used elements was 99.99 % for Y and Pr, 99.9 % for Ir, and 99.9999 % for Zn, respectively. The cubic CeCr2Al20-type structure was confirmed by powder X-ray diffraction measurements at room temperature. The lattice parameters were determined by the analyses of the powder X-ray diffraction patterns. The lattice parameter a
increases linearly from a = 14.1943(1) Å for x = 0 to 14.2215(1) Å for x = 0.44. The atomic compositions were determined by the electron-probe microanalysis (EPMA) with the electron beam of 20 keV using a JEOL JXA-8200 analyzer. Because the resolution of EPMA is not high enough for x < 0.05, we estimated x by comparing the magnetization value at T = 1.8 K with that calculated by the CEF levels of PrIr2Zn20[18]. The electrical resistivity for 3 – 300 K was measured by an AC four-probe method in a laboratory built system. The specific heat data at 0.4 – 300 K and 0.08 – 0.4 K were obtained by the thermal relaxation method using a Quantum Design physical property measurement system (PPMS) and a laboratory built system with a commercial Cambridge mFridge mF-ADR/100s magnetic refrigerator, respectively. Results and Discussion Temperature dependence of the electrical resistivity ρ is plotted in Fig. 1. On cooling, ρ(T) data decrease monotonically and approach residual values at 3 K. The inset of Fig. 1 shows the residual resistivity ratio RRR = (300 K)/ 3 K) with respect to x. It is remarkable that RRR falls from 500 for x = 0 to 50 for x = 0.085 and gradually decreases to 8 for x = 0.44. This change in RRR is attributed to the scattering of the conduction electrons by the disorder between Y and Pr ions. 200
RRR
100
( cm)
100
50
0 0
0.085
0.25
x
x=0 0.024
0.50
0.044
0.23 0.44
Y1-xPrxIr2Zn20 0 0
100
200
300
T (K)
Fig. 1. Temperature variation of the electrical resistivity of Y1-xPrxIr2Zn20 (0 ≤ x ≤ 0.44). The inset shows the x dependence of the residual resistivity ratio RRR = (300 K)/ (3 K).
Temperature dependence of the specific heat C of Y1-xPrxIr2Zn20 for x = 0, 0.044, and 0.44 is shown in Fig. 2. The data of C(T) for x = 0.044 (red closed circles) and 0.44 (blue open squares) coincide with that for x = 0 (black solid line) at T > 50 K. The C(T) data nearly reach 560 J/(K mol) at 300 K which is the Dulong-Petit value expected for a compound with the total atomic number of 23 in the formula unit. This consistency supports that the main phase is Y1-xPrxIr2Zn20 as indicated by the X-ray diffraction analysis and the EPMA. Since the C(T) data show no peak for 3 < T < 300 K, the Pr sites retain cubic symmetry at least above 3 K. Here the phonon contribution Cph was calculated by the following ( ) , where CLa and CY are the specific heat data for equation, nonmagnetic LaIr2Zn20 and YIr2Zn20, respectively. The magnetic specific heat Cm was estimated by subtracting Cph from the measured one. For x =0.044 and 0.44, Cm/ T data at T ≤ 20 K are plotted in the inset of Fig. 2, where Cm/T is normalized by the Pr content. The broad maxima at around 10 K can be reproduced by a two-level model with the ground state doublet and first-excited triplet separated by 30 K as shown with the solid line. The Schottky anomaly is similar to that in PrIr2Zn20. Therefore, the CEF ground state of the Pr ions for x ≤ 0.44 remains in the 3 doublet carrying the active quadrupoles. In fact, Cm/T increases on cooling below 3 K, indicating release of the entropy of the quadrupolar degrees of freedom in the 3 doublet.
Fig. 2. Temperature dependence of the specific heat C of Y1-xPrxIr2Zn20 for x = 0, 0.044, and 0.44. The data for x = 0 are shown by the solid line. The inset displays the magnetic specific heat divided by temperature Cm/T. The solid line shows the calculated values assuming a doublet-triplet two-level model separated by 30 K.
Figure 3 shows the temperature dependence of Cm/T at T < 3 K. On cooling, the values of Cm/T for x = 0.085 and 0.44 approach saturated values at T < 0.3 K. The variations can be reproduced by a random two-level (RTL) model. The specific heat of the RTL model [19, 20] is represented as ( )
∫
( )(
)
,
(
(1)
)
where the density of states of the ground doublet n(E) is assumed to be 1/ in the energy range from zero to . The calculations shown with the solid curves moderately reproduce the data. As shown in the inset of Fig. 3, increases linearly with x, indicating that the 3 doublet splits randomly by the atomic disorder for x ≥ 0.085. On the other hand, Cm/T for x = 0.044 increases continuously on cooling down to the lowest temperature 0.08 K. The data deviate from the curve calculated by the above RTL model, as shown with the (red) solid curve. Because the –lnT dependence in Cm/T appears only for x = 0.044, it may result from the degenerate 3 doublet of the isolated Pr ions. A promising candidate is the impurity quadrupole Kondo effect. In addition to the –lnT dependence of Cm/T, the √ variation of (T) has been observed in the dilute Pr system of Y1-xPrxIr2Zn20 for x < 0.05 (not shown). The anomalous behaviors of Cm and (T) are attributed to the same origin. However, it should be noted that the 30
4
(K)
20
2
2
Cm / T (J / K Pr mol)
x = 0.044
0.085
0 0
0.25
0.3
1
10
x
0.50
0.44
0 0.04
0.1
3
T (K)
Fig. 3. Temperature dependence of the magnetic specific heat divided by temperature Cm/T of Y1-xPrxIr2Zn20 (x = 0, 0.044, and 0.44) for T ≤ 3 K. Solid lines are calculated with a random two-level model [19, 20]. See text for detail. The inset shows the distribution of the energy split between two levels as function of x.
coefficient of √ is positive, in disagreement with the negative sign derived by the impurity quadrupole Kondo model [4,5]. The opposite sign might result from the contribution of magnetic degrees of freedom in the excited triplet. On the other hand, the residual entropy has not been revealed yet. To examine whether the NFL state in the dilute Pr system results from the quadrupole Kondo effect, further systematic studies on the NFL state are indispensable by controlling the parameters such as the Pr concentration, magnetic field, and pressure. Conclusion We have measured the electrical resistivity and the specific heat of the diluted Pr system Y1-xPrxIr2Zn20 for x ≤ 0.44. On cooling, (T) monotonically decreases. A broad maximum at around 10 K in Cm/T can be reproduced by a two-level model with the 3 doublet ground state and the first-excited triplet. The saturated behavior in the Cm/T data for x = 0.085 and 0.44 on cooling for T < 0.3 K can be reproduced by the random two-level (RTL) model. By contrast, the –lnT dependence of Cm/T for x = 0.044 may result from the quadrupolar degrees of freedom in the 3 double ground state in the isolated Pr ions. A promising candidate is the manifestation of the impurity quadrupole Kondo effect due to interaction between the quadrupoles and the multi-channel conduction bands. Acknowledgements We are grateful to Y. Shibata for the electron-probe microanalysis carried out at N-BARD, Hiroshima University. The measurement of the specific heat with the PPMS and the mFridge mF-ADR/100s refrigerator were performed at N-BARD, Hiroshima University. This work was supported by Grants-in-aid from MEXT/JSPS of Japan, Nos. JP26707017, JP15KK0169, JP15H05886, and JP16H01076 (J-Physics). Two of the authors, K. W. and K. T. M., were supported by JSPS Research Fellowships for Young Scientists. †
Present affiliation: Faculty of Engineering, Yokohama National University, Yokohama 240-8501, Japan ‡ Present affiliation: Graduate School of Science and Engineering, Ehime University, Matsuyama 790-8577, Japan
References [1] H. Sato, H. Sugawara, Y. Aoki, and H. Harima, Handb. Magn. Mater. 18, 1 (2009). [2] E. D. Bauer, N. A. Frederick, P.-C. Ho, V. S. Zapf, and M. B. Maple, Phys. Rev. B 65, 100506 (2002). [3] T. Onimaru, T. Sakakibara, N. Aso, H. Yoshizawa, H. S. Suzuki, and T. Takeuchi, Phys. Rev. Lett. 94, 197201 (2005). [4] D. L. Cox and Z. Zawadowski, Adv. Phys. 47, 599 (1998). [5] D. L. Cox, Phys. Rev. Lett. 59, 1240 (1987). [6] H. Amitsuka and T. Sakakibara, J. Phys. Soc. Jpn. 63, 736 (1994). [7] H. Amitsuka, T. Shimamoto, T. Honma, and T. Sakakibara, Physica B 206 & 207, 461 (1995). [8] C. L. Seaman, M. B. Maple, B. W. Lee, S. Ghamaty, M. S. Torikachvili, J.-S. Kang, L. Z. Liu, J. W. Allen, and D. L. Cox, Phys. Rev. Lett. 67, 2882 (1991). [9] F. G. Aliev, H. El Mfarrej, S. Vieira, and R. Villar, Solid State Commun. 91, 775 (1994). [10] T. Kawae, M. Shimogai, M. Mito, K. Takeda, H. Ishii, and T. Kitai, Phys. Rev. B 65, 012409 (2001). [11] T. Onimaru and H. Kusunose, J. Phys. Soc. Jpn. 85, 082002 (2016). [12] T. Nasch, W. Jeitschko, and U. C. Rodewald, Z. Naturforsch. B 52, 1023 (1997). [13] T. Onimaru, K. T. Matsumoto, Y. F. Inoue, K. Umeo, Y. Saiga, Y. Matsushita, R. Tamura, K. Nishimoto, I. Ishii, T. Suzuki, and T. Takabatake, J. Phys. Soc. Jpn. 79, 033704 (2010). [14] T. Onimaru, K. T. Matsumoto, Y. F. Inoue, K. Umeo, T. Sakakibara, Y. Karaki, M. Kubota, and T. Takabatake, Phys. Rev. Lett. 106, 177001 (2011). [15] T. Onimaru, K. Izawa, K. T. Matsumoto, T. Yoshida, Y. Machida, T. Ikeura, K. Wakiya, K. Umeo, S. Kittaka, K. Araki, T. Sakakibara, and T. Takabatake, Phys. Rev. B 94, 075134 (2016). [16] A. Tsuruta and K. Miyake, J. Phys. Soc. Jpn. 84, 114714 (2015). [17] K. Wakiya, T. Onimaru, S. Tsutsui, K. T. Matsumoto, N. Nagasawa, A. Q. R. Baron, T. Hasegawa, N. Ogita, M. Udagawa, and T. Takabatake, Phys. Rev. B 93, 064105 (2016). [18] K. Iwasa, H. Kobayashi, T. Onimaru, K. T. Matsumoto, N. Nagasawa, T. Takabatake, S. Ohira-Kawamura, T. Kikuchi, Y. Inamura, and K. Nakajima, J. Phys. Soc. Jpn. 82, 043707 (2013). [19] P. W. Anderson, B. I. Halperin, and C. M. Varma, Philos. Mag. 25, 1 (1972). [20] W. A. Phillps, J. Low Temp. Phys. 7, 351 (1972).
PII: DOI: Reference:
S0921-4526(17)30469-6 http://dx.doi.org/10.1016/j.physb.2017.07.062 PHYSB310129
To appear in: Physica B: Physics of Condensed Matter Received date: 30 June 2017 Accepted date: 27 July 2017 Cite this article as: Yu Yamane, Takahiro Onimaru, Kazuto Uenishi, Kazuhei Wakiya, Keisuke T. Matsumoto, Kazunori Umeo and Toshiro Takabatake, Impurity quadrupole Kondo ground state in a dilute Pr system Y1-xPrxIr2Zn20 Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.07.062 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Impurity quadrupole Kondo ground state in a dilute Pr system Y1-xPrxIr2Zn20 †
Yu Yamane1, Takahiro Onimaru1, Kazuto Uenishi1, Kazuhei Wakiya1 , ‡
Keisuke T. Matsumoto1 , Kazunori Umeo2, and Toshiro Takabatake1 1
AdSM, 2N-BARD, Hiroshima University, Higashi-Hiroshima, JAPAN
Abstract Electrical resistivity and specific heat C of a dilute Pr system Y1-xPrxIr2Zn20 for 0 ≤ x ≤ 0.44 were measured to study the phenomena arising from active quadrupoles of the Pr3+ ion with 4f2 configuration. On cooling, 's of all samples monotonically decrease, while the residual resistivity ratio (300 K)/(3 K) decreases with x. In the whole range x ≤ 0.44, the magnetic contribution of the specific heat divided by temperature Cm/T shows a broad maximum at around 10 K, which can be reproduced by a two-level model with a ground state doublet and a first-excited triplet separated by 30 K. This indicates that the crystalline electric field ground state of the Pr ions remains in the 3 doublet for the cubic Td point group. On cooling, the Cm/T data for x = 0.085 and 0.44 approach constant values at T < 0.3 K as expected from the random two-level model. By contrast, Cm/T for x = 0.044 increases continuously down to 0.08 K, suggesting a non-Fermi liquid state due to the impurity quadrupole Kondo effect. Keywords: quadrupole Kondo effect, non-Fermi liquid, random two-level model Introduction Praseodymium-based intermetallic compounds with Pr3+ ions of the 4f2 configuration have attracted much attention because of a variety of phenomena originating from multipolar degrees of freedom [1-3]. In diluted U4+ or Pr3+ systems with the f 2 configuration, non-Fermi liquid (NFL) behaviors manifest themselves due to interaction of electric quadrupoles of localized f 2 electrons with conduction electrons [4]. The NFL behaviors are understood as an impurity quadrupole Kondo effect in the cubic 3 doublet system, where the quadrupoles of the isolated U4+ or Pr3+ ions could be over-screened by the multichannel conduction bands [4,5]. In this model, the NFL behaviors involve three ingredients, –lnT divergence of specific heat divided by temperature C/T, √ pendence of electrical resistivity , and residual entropy of (1/2)Rln2 at zero temperature [4,5]. In a U diluted system Th1-xUxRu2Si2 for x ≤ 0.07, in fact, both C/T and the magnetic susceptibility show the –lnT dependence and
follows √ where A is a negative coefficient [6]. Similar phenomena were also observed in other uranium-based compounds such as Th1-xUxPd2Si2[7], Y1-xUxPd3[8] and Th1-xUxBe13[9]. In addition, in a dilute Pr system La1-xPrxPb3, C/T shows lnT dependence [10]. However, these observations do not provide firm experimental evidence for the impurity quadrupole Kondo effect, because there are atomic disorder and uncertainty in the crystalline electric-field (CEF) ground states particularly in the U-based systems. Recently, in caged compounds PrT2X20 (T = transition metal, X = Zn, Al, Cd), a variety of phenomena arising from the active quadrupoles such as long-range quadrupole order, superconductivity, NFL behavior, and structural transition have been revealed [11]. They crystallize in the CeCr2Al20-type structure, where the point group of the Pr ion is the cubic Td [12]. In most compounds, the CEF ground state of the Pr ion is the 3 doublet [11], possessing no magnetic dipoles but electric quadrupoles. Thereby, the quadrupole-driven phenomena often manifest themselves in the systems. Notably, PrIr2Zn20 shows an antiferroquadrupole (AFQ) order at TQ = 0.11 K and a superconducting transition at Tc =0.05 K [13,14]. The magnetic entropy at TQ is only 20% of Rln2, suggesting that the quadrupole fluctuations could play a role in the superconducting pair formation. Moreover, at temperatures above TQ, (T) varies with upward convex curvature [15]. This behavior agrees with the theoretical calculation based on the two-channel Anderson lattice model [16]. Thus, formation of quadrupole Kondo lattice was suggested to occur by the interaction between the quadrupoles and the conduction bands [16]. Therefore, we expected dilution of the Pr ions in nonmagnetic host compounds such as YIr2Zn20 may realize the single-site NFL state. Bearing this in mind, we have prepared Y1-xPrxIr2Zn20 (x ≤ 0.44) and measured C and
. We avoided the use of the La counterpart LaIr2Zn20 which undergoes a structural transition at Ts = 200 K [13,17]. In case the Pr ions were sparse enough, impurity effect due to the interaction between the active quadrupoles of the isolated Pr ions and the conduction bands could manifest itself. Experiment Single- and poly-crystalline samples of Y1-xPrxIr2Zn20 for x ≤ 0.44 were prepared by the Zn self-flux method using boron nitride crucibles in purified argon atmosphere [13]. Purity of the used elements was 99.99 % for Y and Pr, 99.9 % for Ir, and 99.9999 % for Zn, respectively. The cubic CeCr2Al20-type structure was confirmed by powder X-ray diffraction measurements at room temperature. The lattice parameters were determined by the analyses of the powder X-ray diffraction patterns. The lattice parameter a
increases linearly from a = 14.1943(1) Å for x = 0 to 14.2215(1) Å for x = 0.44. The atomic compositions were determined by the electron-probe microanalysis (EPMA) with the electron beam of 20 keV using a JEOL JXA-8200 analyzer. Because the resolution of EPMA is not high enough for x < 0.05, we estimated x by comparing the magnetization value at T = 1.8 K with that calculated by the CEF levels of PrIr2Zn20[18]. The electrical resistivity for 3 – 300 K was measured by an AC four-probe method in a laboratory built system. The specific heat data at 0.4 – 300 K and 0.08 – 0.4 K were obtained by the thermal relaxation method using a Quantum Design physical property measurement system (PPMS) and a laboratory built system with a commercial Cambridge mFridge mF-ADR/100s magnetic refrigerator, respectively. Results and Discussion Temperature dependence of the electrical resistivity ρ is plotted in Fig. 1. On cooling, ρ(T) data decrease monotonically and approach residual values at 3 K. The inset of Fig. 1 shows the residual resistivity ratio RRR = (300 K)/ 3 K) with respect to x. It is remarkable that RRR falls from 500 for x = 0 to 50 for x = 0.085 and gradually decreases to 8 for x = 0.44. This change in RRR is attributed to the scattering of the conduction electrons by the disorder between Y and Pr ions. 200
RRR
100
( cm)
100
50
0 0
0.085
0.25
x
x=0 0.024
0.50
0.044
0.23 0.44
Y1-xPrxIr2Zn20 0 0
100
200
300
T (K)
Fig. 1. Temperature variation of the electrical resistivity of Y1-xPrxIr2Zn20 (0 ≤ x ≤ 0.44). The inset shows the x dependence of the residual resistivity ratio RRR = (300 K)/ (3 K).
Temperature dependence of the specific heat C of Y1-xPrxIr2Zn20 for x = 0, 0.044, and 0.44 is shown in Fig. 2. The data of C(T) for x = 0.044 (red closed circles) and 0.44 (blue open squares) coincide with that for x = 0 (black solid line) at T > 50 K. The C(T) data nearly reach 560 J/(K mol) at 300 K which is the Dulong-Petit value expected for a compound with the total atomic number of 23 in the formula unit. This consistency supports that the main phase is Y1-xPrxIr2Zn20 as indicated by the X-ray diffraction analysis and the EPMA. Since the C(T) data show no peak for 3 < T < 300 K, the Pr sites retain cubic symmetry at least above 3 K. Here the phonon contribution Cph was calculated by the following ( ) , where CLa and CY are the specific heat data for equation, nonmagnetic LaIr2Zn20 and YIr2Zn20, respectively. The magnetic specific heat Cm was estimated by subtracting Cph from the measured one. For x =0.044 and 0.44, Cm/ T data at T ≤ 20 K are plotted in the inset of Fig. 2, where Cm/T is normalized by the Pr content. The broad maxima at around 10 K can be reproduced by a two-level model with the ground state doublet and first-excited triplet separated by 30 K as shown with the solid line. The Schottky anomaly is similar to that in PrIr2Zn20. Therefore, the CEF ground state of the Pr ions for x ≤ 0.44 remains in the 3 doublet carrying the active quadrupoles. In fact, Cm/T increases on cooling below 3 K, indicating release of the entropy of the quadrupolar degrees of freedom in the 3 doublet.
Fig. 2. Temperature dependence of the specific heat C of Y1-xPrxIr2Zn20 for x = 0, 0.044, and 0.44. The data for x = 0 are shown by the solid line. The inset displays the magnetic specific heat divided by temperature Cm/T. The solid line shows the calculated values assuming a doublet-triplet two-level model separated by 30 K.
Figure 3 shows the temperature dependence of Cm/T at T < 3 K. On cooling, the values of Cm/T for x = 0.085 and 0.44 approach saturated values at T < 0.3 K. The variations can be reproduced by a random two-level (RTL) model. The specific heat of the RTL model [19, 20] is represented as ( )
∫
( )(
)
,
(
(1)
)
where the density of states of the ground doublet n(E) is assumed to be 1/ in the energy range from zero to . The calculations shown with the solid curves moderately reproduce the data. As shown in the inset of Fig. 3, increases linearly with x, indicating that the 3 doublet splits randomly by the atomic disorder for x ≥ 0.085. On the other hand, Cm/T for x = 0.044 increases continuously on cooling down to the lowest temperature 0.08 K. The data deviate from the curve calculated by the above RTL model, as shown with the (red) solid curve. Because the –lnT dependence in Cm/T appears only for x = 0.044, it may result from the degenerate 3 doublet of the isolated Pr ions. A promising candidate is the impurity quadrupole Kondo effect. In addition to the –lnT dependence of Cm/T, the √ variation of (T) has been observed in the dilute Pr system of Y1-xPrxIr2Zn20 for x < 0.05 (not shown). The anomalous behaviors of Cm and (T) are attributed to the same origin. However, it should be noted that the 30
4
(K)
20
2
2
Cm / T (J / K Pr mol)
x = 0.044
0.085
0 0
0.25
0.3
1
10
x
0.50
0.44
0 0.04
0.1
3
T (K)
Fig. 3. Temperature dependence of the magnetic specific heat divided by temperature Cm/T of Y1-xPrxIr2Zn20 (x = 0, 0.044, and 0.44) for T ≤ 3 K. Solid lines are calculated with a random two-level model [19, 20]. See text for detail. The inset shows the distribution of the energy split between two levels as function of x.
coefficient of √ is positive, in disagreement with the negative sign derived by the impurity quadrupole Kondo model [4,5]. The opposite sign might result from the contribution of magnetic degrees of freedom in the excited triplet. On the other hand, the residual entropy has not been revealed yet. To examine whether the NFL state in the dilute Pr system results from the quadrupole Kondo effect, further systematic studies on the NFL state are indispensable by controlling the parameters such as the Pr concentration, magnetic field, and pressure. Conclusion We have measured the electrical resistivity and the specific heat of the diluted Pr system Y1-xPrxIr2Zn20 for x ≤ 0.44. On cooling, (T) monotonically decreases. A broad maximum at around 10 K in Cm/T can be reproduced by a two-level model with the 3 doublet ground state and the first-excited triplet. The saturated behavior in the Cm/T data for x = 0.085 and 0.44 on cooling for T < 0.3 K can be reproduced by the random two-level (RTL) model. By contrast, the –lnT dependence of Cm/T for x = 0.044 may result from the quadrupolar degrees of freedom in the 3 double ground state in the isolated Pr ions. A promising candidate is the manifestation of the impurity quadrupole Kondo effect due to interaction between the quadrupoles and the multi-channel conduction bands. Acknowledgements We are grateful to Y. Shibata for the electron-probe microanalysis carried out at N-BARD, Hiroshima University. The measurement of the specific heat with the PPMS and the mFridge mF-ADR/100s refrigerator were performed at N-BARD, Hiroshima University. This work was supported by Grants-in-aid from MEXT/JSPS of Japan, Nos. JP26707017, JP15KK0169, JP15H05886, and JP16H01076 (J-Physics). Two of the authors, K. W. and K. T. M., were supported by JSPS Research Fellowships for Young Scientists. †
Present affiliation: Faculty of Engineering, Yokohama National University, Yokohama 240-8501, Japan ‡ Present affiliation: Graduate School of Science and Engineering, Ehime University, Matsuyama 790-8577, Japan
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