Thermal Expansion in a diluted Ce system La1–x Ce x Cu6

The thermal expansion of a diluted Ce system La1-x Ce x Cu6 for (0.6 ≤ x ≤ 1) has been measured between 10 and 150 K to reveal the change from the coherent heavy Fermion state (0.9 ≤ x ≤ 1) to the incoherent Kondo state (0 < x ≤ 0.73). The large Ce concentration x dependence of the linear thermal expansion coefficient along b-axis αb (T) suggests that the coupling between the 4f 1 electron and the lattice strain is the largest along the b-axis in the three crystallographic axes. The maximum of the magnetic contribution to the volume thermal expansion coefficient β m(T) at T = 50 K is retained in the x range of 0.6 ≤ x ≤ 1, suggesting the crystalline electric field (CEF) level for x = 1 doesn’t change by the substitution. Furthermore, the upturn in β m(T) below 25 K, which should be a precursor of the maximum at T = 2.5 K reported for x = 1, is retained when we decrease x from 1 to 0.6. Because the ground state for x = 0.6 is the incoherent Kondo state, the robustness of the maximum at T = 50 K and upturn in the current x value implies that β m(T) in 10 ≤ T ≤ 150 K is attributed to the CEF and Kondo effects rather than the formation of the heavy Fermion state.


Introduction
Cerium-based intermetallic compounds show a variety of phenomena arising from the strong hybridization between the 4 f 1 electron and the conduction electrons (c − f hybridization), i.e., heavy Fermion states, non-Fermi-liquid behavior, and unconventional superconductivity [1,2,3]. Among them, an orthorhombic CeCu 6 is a well-known heavy Fermion compound exhibiting the large Sommerfeld coefficient γ ∼ 1.5 J/K 2 -mol, the T 2 dependence in the electrical resistivity, and a maximum in the magnetic susceptibility, which is accompanied by the Kondo temperature T K ∼ 3 K [4,5,6,7]. While CeCu 6 crystallizes in the orthorhombic structure with the space group Pnma (#62, D 16 2h ) at room temperature, it shows a structural transition at T s ∼ 220 K, below which the space group changes to the monoclinic P2 1 /c (#14, C 5 2h ) with a small angle modification δβ = 1.36 • . The Au-substituted system CeCu 6−x Au x shows the non-Fermi-liquid behaviors, which have revealed the quantum criticality originating from the competition between the onsite Kondo effect and the intersite Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [8]. The observation of the quantum criticality in CeCu 6−x Au x indicates that the non-substituted CeCu 6 is located near the quantum critical point. On the other hand, the electrical resistivity and magnetic susceptibility measurements on the diluted Ce system La 1−x Ce x Cu 6 have revealed the consecutive change from the coherent heavy Fermion state to the incoherent Kondo state with a crossover concentration x ∼ 0.7 [9].
The thermal expansion of CeCu 6 have been reported in Refs. [10,11]. The volume thermal expansion coefficient β(T ) shows a maximum at T = 2.5 K, which temperature is close to T K ∼ 3 K. Because the maximum is observed in the other heavy Fermion compounds such as CeRu 2 Si 2 [12] and EuNi 2 P 2 [13], it would be attributed to formation of the heavy Fermion state. On the other hand, in our knowledge, the 2 thermal expansion arising from the Kondo effect has not been reported so far. Bearing this in mind, we measured the thermal expansion of the diluted Ce systems La 1−x Ce x Cu 6 (0.6 ≤ x ≤ 1) down to 10 K in order to reveal the change of β(T ) from the heavy Fermion state (0.9 ≤ x ≤ 1) to the incoherent Kondo state (0 < x ≤ 0.73).

Experiments
Single-crystalline samples of La 1−x Ce x Cu 6 were prepared by the Czochralski method with boron nitride or tungsten crucibles as described in the previous papers [9,14]. The thermal expansion was measured at 10-150 K by the active-dummy method with two strain gauges (Kyowa Dengyo, KFLB-12-120-C1-11). For the measurements, the laboratory-made dilatometer was installed on a commercial SQUID magnetometer (Quantum Design, MPMS) and operated with an external device control option.   [11] are plotted with the black solid curve for comparison. All of the α i (T ) data are derived by differentiating the relative length changes dL i (T )/L i with respect to temperature. At first, we discuss the behavior of α i (T ) for the non-substituted system x = 1. The α a (T ) data for x = 1 monotonically decrease on cooling from 150 K to 12 K and the sign changes to negative below 12 K. The α b (T ) data, in contrast, remain constant in the temperature range of 40 ≤ T ≤ 150 K and decrease on further cooling below 40 K. The α c (T ) data show a minimum at T = 25 K, and then an upturn in T < 25 K, . The magnitude and the temperature dependences of α i (T ) are almost in good agreement with those of the previous reports [10,11].

Results and Discussion
Next, we turn to the data of the diluted Ce system La 1−x Ce x Cu 6 for 0.6 ≤ x < 1. The α b (T ) curves mostly decrease with decreasing the Ce concentration x from 1 to 0.6. The α a (T ) and α c (T ) data, in contrast, hardly change in the current x region. Rather, the α a (T ) data for x = 1 are almost identical to those for the nonmagnetic x = 0. The anisotropic change against x would be attributed to the anisotropic coupling between the 4 f 1 electron and lattice strain. In particular, the large change in α b (T ) suggests that the coupling along the b-axis is the largest in the three crystallographic axes.
Here, since the angle modification at T s is small as described in the introduction, we approximate that the system remains in the orthorhombic structure. All the β(T ) data monotonically decrease on cooling from 150 K. For x = 1, the β(T ) data exhibit a minimum at T = 12 K and an upturn on further cooling below 12 K. With decreasing x from 1 to 0, the β(T ) curves gradually decrease, indicating the reduced magnetic contribution. At x = 0 [11], the β(T ) data monotonically decrease down to the lowest temperature of 10 K with no minimum.
In order to discuss the thermal expansion arising from the 4 f 1 electron, we estimated the magnetic contribution to β(T ), β m (T ), by subtracting the data for x = 0 from the β(T ) data for 0.6 ≤ x ≤ 1 as a phonon contribution, i.e., β m (T ) = β x 0 (T ) − β x=0 (T ). Figure 2(b) shows β m (T ) per molar cerium. For x = 1, the β m (T ) data show a maximum at T = 50 K. The maximum was observed in the previous study as well [10], in which it was proposed that it may be due to crystal electric field (CEF) excitations with the level scheme 0-64-128K. In fact, similar maximum has been observed in the magnetic specific heat, which could be reproduced by the Schottky anomaly with the above CEF scheme. In the x range of 0.6 ≤ x ≤ 1, the maximum retains at the same temperature and magnitude, suggesting that the CEF scheme doesn't change by the La-substitution.
Another feature in β m (T ) for x = 1 is the upturn at T < 25 K. Since the upturn should be a precursor of the maximum at T = 2.5 K, we had initially expected that it might be attributed to the formation of the heavy Fermionic state. However, when we decrease the Ce concentration x from 1 to 0.6 in which the ground state changes from the heavy Fermion state to the incoherent Kondo state, the upturn are retained. Therefore, the retaining of the upturn for x = 0.6 implies that it arises from the Kondo effect rather than the formation of the heavy Fermion state. To clarify the origin of the upturn accompanied by the maximum at T = 2.5 K, further low temperature measurements down to 1.8 K are desired. In particular, it is needed to confirm whether the maximum at T = 2.5 K exists or not at x = 0.6.

Summary
To reveal the change of the volume thermal expansion coefficients β(T ) from the coherent heavy Fermion state to the incoherent Kondo state, we performed the thermal expansion measurements on the diluted Ce system La 1−x Ce x Cu 6 for 0.6 ≤ x ≤ 1 in the temperature range of 10 ≤ T ≤ 150 K. The anisotropic Ce concentration x dependence of the linear thermal expansion coefficients α i (T ) (i = a, b, c) suggests that the coupling between the 4 f 1 electron and the lattice strain along the b-axis is the largest in the three crystallographic axes. The maximum in the magnetic contribution to the volume thermal expansion, β m (T ), at T = 50 K is retained in the x range of 0.6 ≤ x ≤ 1, indicating that the crystalline electric field (CEF) levels for x = 1 doesn't change with decreasing x. Moreover, the upturn below 25 K in β m (T ), which should be accompanied by the maximum at T = 2.5 K reported for x = 1, is retained when x is reduced from 1 to 0.6, which implies that β m (T ) in 10 ≤ T ≤ 150 K is attributed to the CEF and incoherent Kondo effects rather than the formation of the heavy Fermion state.