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abstract: 'Starting with the heavy fermion compound CeNi$_9$Ge$_4$, the substitution of nickel by copper leads to a dominance of the RKKY interaction in competition with the Kondo and crystal field interaction. Consequently, this results in an antiferromagnetic phase transition in CeNi$_{9-x}$Cu$_x$Ge$_4$ for $x>0.4$, which is, however, not fully completed up to a Cu-concentration of $x=1$. To study the influence of single-ion effects on the AFM ordering by shielding the $4f$-moments, we analyzed the spin diluted substitution series La$_{0.5}$Ce$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$ by magnetic susceptibility $\chi$ and specific heat $C$ measurements. For small Cu-amounts $x\leq 0.4$ the data reveal single-ion scaling with regard to the Ce-concentration, while for larger Cu-concentrations the AFM transition (encountered in the CeNi$_{9-x}$Cu$_x$Ge$_4$ series) is found to be completely depressed. Calculation of the entropy reveal that the Kondo-effect still shields the 4$f$-moments of the Ce$^{3+}$-ions in CeNi$_8$CuGe$_4$.'
address:
- '$^1$ CPM, Institut für Physik, Universität Augsburg, 86159 Augsburg, Germany'
- '$^2$ Institut für Festkörperphysik, Technische Universität Wien, 1040 Wien, Austria'
author:
- 'L Peyker$^1$, C Gold$^1$, E-W Scheidt$^1$, H Michor$^2$ and W Scherer$^1$'
title: 'Evolution of single-ion crystal field and Kondo features in Ce$_{0.5}$La$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$'
---

Introduction
============

One of the most outstanding Fermi-liquid systems is the heavy fermion compound CeNi$_9$Ge$_4$, which turns out to display the largest ever recorded value of the electronic specific heat $\Delta
C/T\approx5.5$Jmol$^{-1}$K$^{-2}$ without showing any magnetic order [@Michor2004]. The dilution of the magnetic Cerium 4$f$-moments in Ce$_{1-x}$La$_x$Ni$_9$Ge$_4$ reveals single-ion scaling with regard to the Ce-concentration [@Killer2004]. Therefore, the unique behavior of CeNi$_9$Ge$_4$ could be mainly attributed to a single-ion effect. Gradually replacing Ni by Cu changes both, the 3$d$ electron number and the lattice parameters. This substitution influences the crystal field and leads to a formation of long range antiferromagnetic order in CeNi$_{9-x}$Cu$_x$Ge$_4$ for $x>0.4$ which culminates in a transition temperature of $T_{\rm{N}}=175$mK for $x=1$ [@Peyker2009]. Even though the maximum of the magnetic specific heat $\Delta C(T_{\rm{N}})$ of CeNi$_{8}$CuGe$_4$ reaches less than 15% of the theoretical expected value, the transition was discussed in terms of a reduced long range antiferromagnetic order due to the presence of the Kondo-effect [@Peyker2009]. At a suitable concentration of $x \simeq 0.4$, where a crossover between single-ion and magnetic ordered behavior occurs, the system exhibits a quantum critical phase transition (QCP)[@Peyker2009].\
In the present work we studied the influence of Kondo-shielding on the antiferromagnetic ordering and how far single-ion effects are still present when crossing the phase transition from a Kondo-state ($x \leq 0.4$) to an antiferromagnetic coherent state ($x \geq 0.4.$). Therefore we performed magnetic susceptibility $\chi$ and specific heat $C$ measurements of the spin diluted substitution series La$_{0.5}$Ce$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$ and compared them to the pure CeNi$_{9-x}$Cu$_x$Ge$_4$ series.

Experimental Details
====================

Polycrystalline samples of

![\[fig1\](Color online) The magnetic susceptibility $\chi$ of La$_{0.5}$Ce$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$ normalized per Ce-mol. The filled symbols represent the La-diluted samples, while the open symbols represent the pure Ce-compounds CeNi$_{9-x}$Cu$_x$Ge$_4$ for identical Cu-contents.](fig1.eps){width="22pc"}

La$_{0.5}$Ce$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$ were prepared by arc melting the pure elements under argon atmosphere. Subsequently the samples were annealed at 950$^{\circ}$C for two weeks in evacuated quartz tubes. Less than 0.8% weight loss occurred during the melting process. X-ray powder diffraction, optical emission spectroscopy in an indicatively coupled plasma (ICP-OES) and energy dispersive X-ray spectroscopy (EDX) indicated that the samples display a single phase character. The system crystalizes in the tetragonal LaFe$_9$Si$_4$-type structure (space group *I*4/$mcm$). For details of the preparation and the measurements on the pure Ce-compounds CeNi$_{9-x}$Cu$_x$Ge$_4$ see [@Peyker2009].\
Figure\[fig1\] shows the magnetic susceptibility $\chi(T)$ in the temperature range 2K$<$$T$$<$400K of La$_{0.5}$Ce$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$ normalized per Ce-mol (filled symbols) in comparison with the pure Ce-alloys (open symbols) taken from [@Peyker2009]. For a direct comparison of the magnetically dilute solid solution $\mathrm{La}_{0.5}\mathrm{Ce}_{0.5}\mathrm{Ni}_{9-x}\mathrm{Cu}_x\mathrm{Ge}_4$ with the corresponding solid solution $\mathrm{CeNi}_{9-x}\mathrm{Cu}_x\mathrm{Ge}_4$ with undiluted Ce-sublattice, the specific heat and magnetic susceptibility data are normalized to the Ce-content. In case of the Ce-normalized magnetic susceptibility ($\chi_{\mathrm{Ce-mol}}$), we first subtracted the nonparamagnetic La-contribution and then scaled the data with the Ce-concentration, using the following equation: $$\label{eq.norm}
\chi_{\mathrm{Ce-mol}}=2\cdot\biggl(\chi(\mathrm{La}_{0.5}\mathrm{Ce}_{0.5}\mathrm{Ni}_{9-x}\mathrm{Cu}_x\mathrm{Ge}_4)
-0.5\cdot\chi(\mathrm{La}\mathrm{Ni}_{9}\mathrm{Ge}_4)\biggr)$$. For $T>80$K, $\chi(T)$ follows a modified Curie-Weiss law, $\chi(T)=C/(T-\Theta)+\chi_0$, yielding an effective magnetic moment of $\mu_\mathrm{eff}\approx2.5\mu_\mathrm{B}$, as theoretically expected for a Ce$^{3+}$-lattice. In the low temperature range ($T<80$K) the data scales for $x\leq0.4$ with the Ce-concentration indicating single-ion behavior. For $x>0.4$ the single-ion character vanishes and the temperature dependence of the La-substituted samples deviate from the behavior of the pure Ce-compounds, which follow a Curie-Weiss-law only down to 100K, due to the formation of an antiferromagnetic transition at lower temperatures [@Peyker2009]. The different temperature dependence of $\chi
(T)$ is due to the absence of antiferromagnetic correlations in the La-substituted system and results from the dilution of the magnetic moments.\
The specific heat $C$ normalized per Ce-mol and divided by temperature $T$ is displayed in Fig.\[fig2\] for La$_{0.5}$Ce$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$ in the temperature range between 0.05K and $T$$<$300K.

![\[fig2\](Color online) The specific heat $C$ divided by temperature $T$ of La$_{0.5}$Ce$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$ normalized per Ce-mol (filled symbols) and of the undiluted Ce-compounds CeNi$_{9-x}$Cu$_x$Ge$_4$ (open symbols).](fig2.eps){width="22pc"}

As already known from literature [@Killer2004] a normalization to CeNi$_9$Ge$_4$ for $x=0$ is not possible due to the coherence of the Kondo-lattice. This is, however, not true for the diluted sample, where a logarithmic increase of $C/T$ below 1.5K is observed, indicating non-Fermi-liquid-behavior. For $0<x\leq0.4$ the data scale with the undiluted compounds which is in agreement with single-ion effects as already observed in the magnetic susceptibility. The absence of the antiferromagnetic transition for $x>0.4$ in the Ce-diluted compounds is also in line with their susceptibility behavior. The stronger increase of $C/T$ of the pure Ce-alloys compared to the diluted compounds is due to the additional entropy required for the antiferromagnetic ordering.\

Discussion and conclusions
==========================

In order to study the antiferromagnetic transition of CeNi$_8$CuGe$_4$ in a more quantitative manner, we take a closer look at the $C/T$-difference of the diluted and undiluted systems. Due to the fact, that only the pure Ce-compounds order antiferromagnetically, the difference in the specific heat $\Delta
C/T$ provides an entropy, which only belongs to the formation of long range magnetic order. The left picture in Fig.\[fig3\] displays $\Delta C/T$, showing the contribution of the antiferromagnetic ordering, and the associated entropy $S$ in CeNi$_8$CuGe$_4$. The estimated value of the entropy $S=1.2$J/molK is about 20% of the theoretical expected value of $R\ln2\approx 5.8$J/molK. This is in line with the presence of a partially Kondo-screened long range antiferromagnetic order which has been analyzed in terms of the resonant-level model model by Schotte and Schotte in Ref. [@Peyker2009]. A model calculation with a RKKY coupling parameter $J=2.3$K and a Kondo temperature $T_K=1.3$K approximately reproduces the reduced magnitude of the AF specific heat anomaly and the enhanced electronic specific heat anomaly of CeNi$_8$CuGe$_4$. This calculation implies Kondo-screening of an ordered Ce-moment along the $c$-axis which reduces the Ce moment to 36% of its CF ground state value of $\mu_c$. Considering a reduction of the local symmetry at the Ce-sites due to substitutional disorder present in CeNi$_8$CuGe$_4$ we expect some reduction of $\mu_c$ as compared to CeNi$_9$Ge$_4$ with $\mu_c=2\mu_{\rm B}$ [@Michor2006] for the CF ground state. The Kondo-screened ordered moments of CeNi$_8$CuGe$_4$ are thus expected to range between 0.5–0.7$\mu_{\rm B}$.\
Further details can be drawn from the calculation of

![\[fig3\](Color online) The differences of $C/T$ (left panel) and $C$ (right panel) of CeNi$_8$CuGe$_4$ and La$_{0.5}$Ce$_{0.5}$Ni$_8$CuGe$_4$ and the resulting entropy $S$ and enthalpy $H$, respectively.](fig3.eps){width="30pc"}

the enthalpy $H$. Therefore the difference of the specific heat $\Delta C$ was integrated as displayed in the right panel in Fig.\[fig3\]. From the estimated value of $H=0.35$J/mol an internal magnetic field of $B=0.13$T is determined, using the relation $H=N_\mathrm{A}0.5\mu_\mathrm{B}B$ with the reduced magnetic moment of 0.5$\mu_\mathrm{B}$ discussed above. This means that an external magnetic field of about 0.13T should lead to a suppression of the longrange antiferromagnetic order. With the knowledge of this critical magnetic field an estimation of the Néel-temperature $T_\mathrm{N} = 87$mK can be made which is by a factor of two smaller than the observed Néel-temperature $T_\mathrm{N}\approx175$mK of CeNi$_8$CuGe$_4$ [@Peyker2009]. From our thermodynamic considerations, taking into account the experimental Néel-temperature, a reduced Ce magnetic moment of 0.24$\mu_\mathrm{B}$ would be expected.

Summary
=======

Comparative studies of the specific heat and the magnetic susceptibility on the diluted system La$_{0.5}$Ce$_{0.5}$Ni$_{9-x}$Cu$_x$Ge$_4$ verify that the behavior of the pure Ce system CeNi$_{9-x}$Cu$_x$Ge$_4$ in the none ordered magnetic region ($x\leq0.4$) is driven by a single-ion Kondo-effect. In the magnetic ordered phase ($x > 0.4$) the Kondo-effect still influences the magnetic ordering, leading to a reduction of the magnetic moments and therefore to a reduced antiferromagnetic contribution in the specific heat at $T_\mathrm{N}$, as it is also predicted in [@Peyker2009], where a resonant-level model in combination with a molecular field approach is used. Thermodynamic calculations support these results.\

Acknowledgments
===============

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Contract No. SCHE487/7-1 and by the COST P16 ECOM project of the European Union.

[1]{} url \#1[[\#1]{}]{}urlprefix\[2\]\[\][[\#2](#2)]{} Michor H, Bauer E, Dusek C, Hilscher G, Rogl P, Chevalier B, Etourneau J, Giester G, Killer U and Scheidt E-W [2004]{} [*[J. Magn. Magn. Mater.]{}*]{} [**[272]{}**]{} [227–228]{} ISSN [0304-8853]{}

Killer U, Scheidt E-W, Eickerling G, Michor H, Sereni J, Pruschke T and Kehrein S [2004]{} [*[Phys. Rev. Lett.]{}*]{} [**[93]{}**]{} ISSN [0031-9007]{}

Peyker L, Gold C, Scheidt E-W, Scherer W, Donath J G, Gegenwart P, Mayr F, Eyert V, Bauer E and Michor H [2009]{} [*[J. Phys.: Condens. Matter]{}*]{} [ **[21]{}**]{} [235604]{}

Michor H, Adroja D T, Bauer E, Bewley R, Dobozanov D, Hillier A D, Hilscher G, Killer U, Koza M, Manalo S, Manuel P, Reissner M, Rogl P, Rotter M and Scheidt E-W [2006]{} [*[Physica]{}*]{} [B]{} [**[378-80]{}**]{} [640–643]{} ISSN [0921-4526]{}
