Journal of Low Temperature Physics

, Volume 175, Issue 1–2, pp 498–507 | Cite as

Non-Fermi Liquid Behaviour in the Heavy-Fermion Kondo Lattice Ce2Rh3Al9

  • M. Falkowski
  • A. M. Strydom


In the heavy fermion class of strongly correlated electron systems, the Landau Fermi liquid description of metals has become a rather fragile basis on which to formulate an understanding of their ground state. The proximity to cooperative phenomena such as magnetic order and superconductivity and the amenability of Ce- and Yb-based compounds to be tuned into quantum criticality have been found to have extraordinary effects on the T→0 thermal scaling of electronic and magnetic properties. A collection of non-Fermi liquid scaling relations have thus far been proposed in the search for universality. Here we report on the physical properties of the heavy fermion Kondo lattice Ce2Rh3Al9. The low-temperature specific heat and electrical resistivity are best described by power laws in their temperature dependence, and we model these according to the expectation for a system close to a magnetic phase transition. We demonstrate how applied magnetic fields drive the transition from the Kondo coherent state, through a cross-over phase, and into Fermi-liquid behaviour at high fields and low temperatures.


Heavy fermions Kondo lattice Non-Fermi liquid 

1 Introduction

Highly correlated electron systems, and in particular the heavy-fermion class of materials, have proven over the last 3 decades to be exceptionally rich in physics. New phenomena emerging from magnetic instabilities in metals have been an enduring and fertile research field in condensed matter physics. Two topics have remained at the forefront of investigations into highly correlated electrons systems: the coexistence of superconductivity and magnetic order, and unconventional effects at the proximity of cooperative electron behaviour [1, 2]. When a phase transition is either suppressed to zero using a suitable tuning parameter such as magnetic field or pressure [3], or alternatively when a magnetic phase transition happens to be situated arbitrarily close to zero [4, 5] under ambient conditions, quantum criticality arises. This condition governs the scaling of thermal electronic properties, for which a quantitative description of scaling relations and the notion of universality in the presence of quantum criticality remains elusive. The standard model of metals invokes the well-established Fermi-liquid paradigm. Here, the point of departure is that electronic properties of a metal in its ground state can be described by assuming that excitations within the sea of fermions may be renormalized out of the energy spectrum. In an interacting system, and with the extreme example of highly correlated electron systems when electron-electron interactions are turned on adiabatically from the ground state, single-particle excitations of the Fermi liquid evolve into quasiparticle excitations of the composite electron fluid [6]. Thus it is that even in heavy-fermion systems where Coulomb energy may exceed the electron kinetic energy many times over, ground states that are not close to cooperative behaviour may still be adequately described by Fermi-liquid theory.

On the other hand, non-Fermi liquid behaviour describes an anomalous state in a metal when electronic properties are found not to scale according to the simple, non-interacting condition, and this is the subject of this paper. A small but growing number of correlated systems are now realized to behave in a manner that is not reconcilable with Fermi-liquid behaviour, yet without a magnetic phase transition that may otherwise have the physics in such systems governed by the existence of a quantum critical point. We focus in this work on selected compounds in the R2T3X9 series, where R is a rare-earth element, T stands for a d-block element, and X is a p-electron element such as Al or Ga.

The existence and exploratory physical properties of compounds in the series Ce2T3X9 (T=Rh, Ru, Ir; X=Al, Ga) were first announced by Buschinger et al. [7]. Ce2Rh3Al9 is classified as a non-magnetically ordered (reportedly [8] down to 1.4 K) moderate heavy-fermion system. The nature of the ground state in this compound remains controversial. The magnetic susceptibility at intermediate temperatures produces intermediate valent type scaling, which is thought [9] to be understandable in terms of an energy scale T χ ≃150 K much higher than the Kondo-lattice characteristic energy T K∼20 K found in the electrical resistivity [9]. One plausible description of the physics in Ce2Rh3Al9 is therefore that there are two energy scales at work, separated by an order of magnitude in temperature. An inelastic neutron study [10] attributed a localized picture to the magnetic density in Ce2Rh3Al9. A more recent x-ray photoluminescent spectroscopy plus density of state calculation study [11] however supported a Ce mixed-valent state in Ce2Rh3Al9. Our study of Ce2Rh3Al9 with results reported here is intended to explore the low-temperature region in search of the origin of non-Fermi liquid scaling in the electrical resistivity and specific heat.

2 Experimental Detail

Polycrystalline samples of the two compounds Ce2Rh3Al9 and La2Rh3Al9 were synthesized by direct argon arc-melting of stoichiometric quantities of the elements in ultra-high purity argon gas with further in-situ purification of argon. The starting elements were all of purity 99.99 wt.% or better. The samples were remelted several times to promote homogeneity, and mass losses were confined to 0.6 % or less. The sample ingots were then given prolonged heat treatment in order to improve phase formation and purity. Powder x-ray diffraction surveys confirmed the single-phase nature of the synthesized sample material in the expected orthorhombic Y2Co3Ga9 structure type with space group Cmcm. The lattice parameter values obtained by Rietveld refinement (see Fig. 1) for respectively Ce2Rh3Al9 and La2Rh3Al9 are a=13.1414 Å and 13.3932 Å; b=7.6883 Å and 7.7315 Å; c=9.5586 Å and 9.54 Å. The determined values are in close agreement with formerly reported values [7].
Fig. 1

Powder x-ray diffractogram obtained for Ce2Rh3Al9, together with a full profile Rietveld refinement (solid line), Bragg position markers (vertical bars), and difference plot. The sample specimen is single-phase to within the limits of detection. (Color figure online)

Physical properties were studied using a PPMS-9T system from Quantum Design, San Diego, equipped with low temperature facilities including a 3He recirculating insert and a 3He−4He dilution refrigerator. Magnetic properties were studied in a squid-type MPMS magnetometer also from Quantum Design, in conjunction with a 3He insert from iQuantum Corporation in Tsukuba, Japan.

3 Results

The temperature variation of the molar magnetic susceptibility χ(T) and its inverse χ −1(T) for Ce2Rh3Al9 up to 400 K recorded in a magnetic field B=0.1 T are shown in Fig. 2 (main panel). The high temperature region of χ(T) does not show typical Curie-Weiss behavior as would otherwise be expected for a well-localized magnetic moment. We find instead a broad and flat plateau in the region 100–300 K. This feature in χ(T) is characteristic of fluctuating valence in Ce-compounds, examples being CeRhIn [12], CeRhSb [13], and Ce2Co3Ge5 [14]. A similar behavior in χ(T) was previously reported also for Ce2T3X9 (T=Rh, Ru, Ir; X=Al and Ga) [7, 9, 15]. We have applied the interconfiguration fluctuation (ICF) model proposed by Sales and Wohlleben [16]. This model bases the progression of χ(T) upon a characteristic temperature that is related to fluctuations between two states of integral valence. χ(T) is described by the fractional occupation of the two states, which for Ce are the nonmagnetic 4f 0 and magnetic 4f 1 states. Using the ICF model for Ce2Rh3Al9 we can determine how much of the χ(T) can be considered as intrinsic to the valence stability. The least-squares fit presented in Fig. 2 accurately describes our χ(T) data in the range between 10 K and 400 K according to the expression [14, 16]:
$$\begin{aligned} \chi({T})=\frac{N_{\mathrm{Av}}\mu_{\mathrm{eff}}\left[1-\nu\left(T\right)\right ]}{3k_{\mathrm{B}}\left(T+T_{\mathrm{sf}}\right)}+\chi_{\mathrm{imp}}({T})+\chi_{\mathrm{0}}({T}), \end{aligned}$$
where ν(T)={1+6exp[−E ex/k B(T+T sf)]}−1 is the fractional occupation of the non-magnetic ground state, χ imp(T)=C imp/(T+T imp) is a Curie-Weiss like impurity contribution that scales with C imp=6.2×10−2 emu K/mol Ce and T imp=−5.4 K, while χ 0=4.4×10−4 emu/mol Ce collects contributions of the conduction electron paramagnetism, core-electron diamagnetism, and Van Vleck paramagnetism. The effective moment in the full 4f 1 state on Ce ions is taken as μ eff=2.54 μ B, T sf=128.7 K measures the spin fluctuation temperature and E ex=530.8 K denotes an excitation energy between the ground and the excited state. Assuming that the impurity contribution originates from unquenched Ce+3 ions, their concentration \(n = C_{\mathrm{imp}}/C_{\mathrm{Ce}}^{3+}\) takes the value of ≃5.8 at.%. In relation to the Coqblin-Schrieffer model [17], solved by Rajan [18] for unquenched Kondo spins such as Ce (J=5/2), we use for T 0=T sf with [18, 19] T 0=(2J+1)T L/2π and T K=0.6745T L, we find T K=91 K for the Kondo temperature of Ce2Rh3Al9. The magnetization of this compound measured at T=0.47 K (inset of Fig. 2) confirms the very small residual magnetic moment that effectively remains at low temperature as a result of intermediate valence.
Fig. 2

Overall magnetic susceptibility (left) and inverse susceptibility (right) of Ce2Rh3Al9. The fit on the χ(T) plot represents intermediate valent behaviour as discussed in the text. Inset: low-temperature magnetization of Ce2Rh3Al9. (Color figure online)

Below 50 K, χ(T) is found to increase strongly as T→0. Down to 0.45 K no anomaly indicative of magnetic ordering or a phase transition is observed in χ(T) in our lowest measuring field of 0.01 T. Figure 3 illustrates the susceptibility of Ce2Rh3Al9 at low temperatures and in various fields, normalized to the 1.9 K values to facilitate comparison. Here we note a remarkable change of curvature in progressively higher fields. χ(T) in the lowest field of 0.01 T tends to saturation as T→0. With progressively higher fields however an inflection point emerges out of very low temperatures. The inflection temperature shifts towards higher values until a weak plateau appears in a field of 4 T that becomes yet more clearly expressed in a field of 7 T. The dashed line in Fig. 3 marks the inflection points on the 3 high-field χ(T) curves (B=2 T, 4 T, and 7 T). We return to the significance of this point in the discussion further below.
Fig. 3

Semi-log plot of the low-temperature susceptibility of Ce2Rh3Al9 normalized to 1.9 K, in various fields. The green line on the 7 T data illustrates \(\chi({T})\sim\sqrt{T}\) behaviour that is significant for the proximity to a quantum critical point [22]. The dashed line traversing the three highest field plots connects inflection points found on these curves. (Color figure online)

One approach to understand the low-temperature χ(T) data of Ce2Rh3Al9, is shown by the line superimposed onto the 7 T data in Fig. 3. This line illustrates a χ(T)=χ 0+A χ T n (n=1.6) temperature dependence, which is most pronounced for the 7 T data but evidently so also for the 4 T and 2 T data sets, albeit with the plateau in χ(T) shifted to lower temperatures as the field value is decreased. This particular temperature dependence of χ(T) was predicted by Moriya and Takimoto [22] for weakly interacting spin fluctuations in the proximity of a quantum phase transition. In Ce2Rh3Al9 it was established [23] that spin fluctuations due to Rh 4d electrons are well separated from the 4f-electron dynamics which are responsible for the non-Fermi liquid behaviour that we are concerned with here.

The electrical resistivities of Ce2Rh3Al9 and La2Rh3Al9 are shown in Fig. 4. Our data shown here on ρ(T) of Ce2Rh3Al9 extend the temperature range in former studies [8] down to 0.07 K. La2Rh3Al9 shows a monotonous and normal metallic behaviour in its resistivity, but for Ce2Rh3Al9 the overall ρ(T) dependence is strongly temperature dependent over practically the entire temperature range. We assess the 4f-electron derived magnetic resistivity, ρ 4f (T) by means of subtracting the temperature dependent part of ρ(T) for La2Rh3Al9 from that of Ce2Rh3Al9 (blue symbols in Fig. 4). Close to and below room temperature, ρ 4f (T) of Ce2Rh3Al9 increases with decreasing temperature and, more particularly, a ρ(T)∼−logT type scattering is evident which is attributable to the incoherent Kondo effect. The resistivity shown for Ce2Rh3Al9 reveals two distinct features, both consistent with previous findings [7, 9]. The first is a maximum in ρ 4f (T) close to 100 K produced by the onset of phase-translational coherence towards low temperature. At still lower temperature, a clear deflection occurs in the range 10<T<20 K that indicates a change in electron scattering dynamics. A description of the ground state of Ce2Rh3Al9 evidently requires deeper analysis at considerably lower temperatures.
Fig. 4

Electrical resistivity of La2Rh3Al9 (black triangles), Ce2Rh3Al9 (black circles) and the magnetic part (blue circles) attributable to the 4f electrons in Ce2Rh3Al9. The solid line at the high-temperature region of ρ 4f (T) illustrates incoherent Kondo scattering in Ce2Rh3Al9. (Color figure online)

Figure 5(a) illustrates how the resistivity of Ce2Rh3Al9 in constant applied fields up to 7 T decreases in a quasi-linear manner below 5 K before further curvature develops below 1 K. By comparison with ρ(T) of Ce2Rh3Al9, we note in Fig. 4 from ρ(T) of La2Rh3Al9 that the total conduction-electron scattering below ≃10 K hardly involves any electron-phonon scattering at all, and phonons are evidently being excited to any significant extent only at higher temperatures. We thus attribute the temperature dependent part of ρ for Ce2Rh3Al9 in the region below ∼10 K entirely to magnetic 4f-electron scattering. In Fig. 5(b) we plot the low-temperature iso-field ρ(T) data of Ce2Rh3Al9 for various applied fields. As can be seen in the inset, a double-log plot proves power-law behaviour unambiguously. The 4f-electron resistivity of Ce2Rh3Al9 is governed by consistent power-law behaviour of the form ρ(T)=ρ 0+A ρ T n over more than one decade in temperature, and across our range of applied fields. It is important to note furthermore that the power-law behaviour in resistivity of Ce2Rh3Al9 is an attribute of the very low limit of temperature, i.e. below ∼0.8 K. We collect in Fig. 6 values of the field evolution of the power-law coefficient n. While Fermi-liquid like values of n≈2 adequately describe the temperature dependence of ρ(T) in zero field, and again in fields upward of B≃3 T, there is a dip in the values of this coefficient at intermediate fields, reaching down to n≈1 in B=1 T. This coefficient for the temperature dependence of electrical resistivity is a hallmark of non-Fermi liquid behaviour, and it is found in quantum critical systems such as CeCu5.9Au0.1 [24], YbRh2(Si0.95Ge0.05) [25], and Sr3Ru2O7 [26].
Fig. 5

(a) Electrical resistivity of Ce2Rh3Al9 displays quasi-linear behaviour, and a negative magnetoresistance developing below 5 K. (b) Semi-log plot of the low-temperature resistivity in applied fields. Following the demonstrable power-law behaviour over more than one decade in temperature (see inset), power-law fits ρ(T)=ρ 0+A ρ T n are used to describe the data below 0.8 K. Inset: log-log plot of \(\Delta\rho\left(T\right)/\Delta\rho\left(0.8~K\right)\), where Δρρ(T,B)−ρ(0,B). (Color figure online)

Fig. 6

Field dependence of the resistivity power-law coefficient n, obtained from the ρ(T)∝A ρ T n quadratic term (see Fig. 5(b)). (Color figure online)

The low-temperature specific heat of Ce2Rh3Al9 in the form C P(T)/T is displayed in Fig. 7. We have subtracted the lattice heat capacity, obtained from the non-magnetic analogue La2Rh3Al9 to obtain the magnetic 4f-electron specific heat C 4f that is plotted in Fig. 7. The two elements Rh and Al both have nuclear spins and thus a nuclear contribution may be expected as part of the specific heat measured on Ce2Rh3Al9. At the lowest temperature (0.2 K) of data displayed in Fig. 7, Al contributes a nuclear specific heat value [20] of at most C N/T=0.0035 J/mol K2. For the case of Rh, the nuclear specific heat [21] at 0.2 K amounts to C N/T=0.002 J/mol K2. The sum of these values can thus be expected to contribute ≃0.0055 J/mol K2 or less at 0.2 K, which is equal to less than 2 % of the value of heat capacity assigned to the 4f-electrons of Ce2Rh3Al9 at this temperature. While it appears tempting at first sight to assign a C 4f (T)/T∼−logT temperature dependence to the specific heat of Ce2Rh3Al9 below ∼5 K, we find that a much more plausible description of the zero-field data is according to the prediction of \(C_{\mathrm{P}}({T})/T=\gamma_{0} - A_{C}\sqrt{T}\) due to Moriya and Takimoto [22]. The adherence of the low-temperature specific heat of Ce2Rh3Al9 to this model can be assessed by the dashed line, superimposed onto the zero-field data in Fig. 7. By contrast, the specific heat in applied fields show progressive deviation from the illustrated T→0 power-law divergence. Instead, a near-constant C 4f /T dependence develops, and the plateau of constant C 4f /T becomes wider with higher fields. As is the case with the electrical resistivity in comparing with the quasi-linear behaviour in ρ(T) for B=0, we find that there is an intermediate region of non-Fermi liquid behaviour in the specific heat (C 4f (T)/T∼−logT) from 8 K to 0.8 K, and which is in agreement with earlier work [7]. However, in our extended low-temperature measurements it is found that in zero field C 4f (T)/T then turns moderately downward below 0.3 K and settles into a weakly increasing trend with further lowering of temperature that is characteristic of the non-ordered, heavy fermion Kondo ground state. As is the case with magnetic susceptibility and electrical resistivity in the above, the Fermi-liquid state is recovered by applying appropriate fields, but this state evidently displaces the Kondo effect in the ground state of Ce2Rh3Al9.
Fig. 7

4f-electron derived magnetic specific heat of Ce2Rh3Al9 in zero field (squares), and progressively higher applied fields. The dashed line proposes a \(C_{\mathrm{4}f}({T})/T=\gamma_{0} -A_{C}\sqrt{T}\) dependence in B=0, according to the model of Moriya and Takimoto [22]. (Color figure online)

Finally, we collect in Fig. 8 on a B vs. T phase diagram a number of scaling points obtained from the physical properties studied in this work. We denote by the dark grey shaded area the B vs. T region in which distinct non-Fermi liquid behaviour is found in the low-temperature limit. The symbols associated with χ(T,B) data denote the inflection point in χ(T) (see Fig. 3), and those with C 4f (T,B) connect the temperature below which C 4f (T)/Tconstant develops (see Fig. 7). The point denoted by ρ indicates the upper temperature and field within which non-Fermi liquid behaviour is found in the electrical resistivity.
Fig. 8

Field-temperature phase diagram of Ce2Rh3Al9. The shaded region near B=0, T=0 is the quantum critical region where the resistivity follows ρ(T)∼T behaviour. The points at higher fields and temperatures are associated with change-over scales, from the Kondo coherent state at low fields, to the Landau Fermi liquid state that is achieved at high fields and very low temperatures. (Color figure online)

4 Conclusions

An important point of departure in the study of quantum criticality and the non-Fermi liquid effects that arise in metals as a result, is the distinction between quantum criticality in itinerant electron materials, and that found in local-moment systems. 4f-electron compounds based on Ce or Yb dominate the latter class of systems. A burning issue in creating an understanding of the so-called unconventional or local-moment quantum criticality in heavy fermion systems, is whether the heavy electrons form a spin-density wave composed of heavy, itinerant quasiparticles, or whether the Kondo screening prevalent in heavy fermion systems cause the heavy quasiparticles to decompose during the onset of quantum criticality [1]. Aside from comprehensive evidence for the destruction of the Kondo effect occurring in YbRh2Si2, the cubic system Ce3Pd20Si6 has recently been found [27] to be a supreme example of the phenomenon of Kondo breakdown, and moreover in the anomalous situation of cubic crystal symmetry of the magnetic ions. In this work we have forwarded evidence that the heavy-fermion Kondo lattice Ce2Rh3Al9 displays non-Fermi liquid behaviour at ambient conditions at low temperature. Our studies of specific heat and electrical resistivity at the lowest temperatures demonstrate that magnetic order seems to be avoided in this compound, an observation that may have its roots in the extremely small magnetic moment that resides on the Ce+3 ions as a result of severe Kondo screening. The electrical resistivity however demonstrates how the electron scattering rate becomes decidedly non-Fermi liquid like in applied magnetic fields of 0.5–1.5 T, and for higher fields Fermi-liquid scaling is recovered. Our results presented here open up the question of whether Kondo breakdown associated with quantum criticality is in fact contingent on magnetic ordering among the Kondo spins, or whether the Kondo effect itself in heavy fermions may be intrinsically instable in the presence of quantum criticality.



A.M.S. gratefully thanks the URC of UJ, and the SA-NRF (78832) for financial assistance. M.F. acknowledges support from the UJ Faculty of Science and URC for Postdoctoral Fellowship. Douglas Britz is thanked for experimental assistance.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of JohannesburgAuckland ParkSouth Africa

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