Realization of Kondo chain in CeCo$_2$Ga$_8$

TL;DR

This study demonstrates that CeCo$_2$Ga$_8$ is a quasi-one-dimensional heavy-fermion material exhibiting Kondo coherence primarily along its c-axis, providing a concrete example of a Kondo chain with significant magnetic anisotropy.

Contribution

The paper provides experimental evidence that CeCo$_2$Ga$_8$ is a Kondo chain with strong magnetic anisotropy, highlighting its quasi-one-dimensional heavy-fermion behavior.

Findings

Resistivity shows Kondo coherence below 17 K along c-axis.

Magnetic anisotropy ratio of exchange interactions is 4-5.

CeCo$_2$Ga$_8$ is a quasi-one-dimensional heavy-fermion compound.

Abstract

We revisited the anisotropy of the heavy-fermion material CeCo$_2$Ga$_8$ by measuring the electrical resistivity and magnetic susceptibility along all the principal $\mathbf{a}$-, $\mathbf{b}$- and $\mathbf{c}$-axes. Resistivity along $\mathbf{c}$-axis ($\rho_c$) shows clear Kondo coherence below about 17 K, while both $\rho_{a}$ and $\rho_{b}$ remain incoherent down to 2 K. The magnetic anisotropy is well understood within the theoretical frame of crystalline electric field effect in combination with magnetic exchange interactions. We found the anisotropy ratio of these magnetic exchange interactions, $|J_{ex}^c/J_{ex}^{a,b}|$, reaches a large value of 4-5. We, therefore, firmly demonstrate that CeCo$_2$Ga$_8$ is a quasi-one-dimensional heavy-fermion compound both electrically and magnetically, and thus provide a realistic example of \textit{Kondo chain}.

Tables

Table 1. Table 1: CEF parameters, energy levels and wave functions in CeCo 2 Ga 8 at zero magnetic field.

CEF parameters
	$B_2^0$=$-$22.0(5) K,	$B_2^2$=0.8(1) K,	$B_4^0$=0.6(1) K,	$B_4^2$=0.45(5) K,	$B_4^4$=1.4(1) K
Energy levels and Eigenstates
$E$(K)	$\mid \frac{5}{2},+\frac{5}{2}⟩$	$\mid \frac{5}{2},+\frac{3}{2}⟩$	$\mid \frac{5}{2},+\frac{1}{2}⟩$	$\mid \frac{5}{2},-\frac{1}{2}⟩$	$\mid \frac{5}{2},-\frac{3}{2}⟩$	$\mid \frac{5}{2},-\frac{5}{2}⟩$
0	0	$-$0.2763	0	$-$0.0371	0	0.9604
0	0.9604	0	$-$0.0371	0	$-$0.2763	0
145(5)	0	0.9609	0	0.0055	0	0.2767
145(5)	0.2767	0	0.0055	0	0.9609	0
445(5)	0.0341	0	0.9993	0	$-$0.0156	0
445(5)	0	$-$0.0156	0	0.9993	0	0.0341

Equations

$${\hat{H}}_{CEF}=B_2^0{\hat{O}}_2^0+B_2^2{\hat{O}}_2^2+B_4^0{\hat{O}}_4^0+B_4^2{\hat{O}}_4^2+B_4^4{\hat{O}}_4^4,$$

$${\hat{H}}_{Zee}=-g{\mu }_B\hat{\text{J}}\cdot \text{B},$$

$${\hat{H}}_{ex}=-\sum _{<i,j>}(J_{ex}^a{S}_i^x{S}_j^x+J_{ex}^b{S}_i^y{S}_j^y+J_{ex}^c{S}_i^z{S}_j^z),$$

$$\begin{array}{rl}\frac{1}{{\chi }_{\alpha }}& =\frac{1}{C}[T-({\theta }_{CEF}^{\alpha }+{\theta }_{ex}^{\alpha })]\\ & =\frac{1}{{\chi }_{\alpha }^{CEF}}-\frac{{\theta }_{ex}^{\alpha }}{C},(\alpha =a,b,c)\end{array}$$

Full text

A lattice in Condensed Matter Physics typically means a periodic repetition and extension in a multi-dimensional space. An example is crystalline lattice, in which the atoms are connected by chemical bondings and construct a stereo networkAshcroft and Mermin (1976). The concept also applies to more virtual systems, like vortex latticeAbrikosov (1957), skyrmion latticeMühlbauer et al. (2009), Kondo lattice et al. A Kondo singlet, the “unit cell” of Kondo lattice, comes into being when a localized magnetic moment immersed into fermi sea of a metalColeman (2012) is quenched by entangling with conduction-electron spin to form a magnetic singlet [Fig. 1(a)]. For a dense lattice of local moments [Fig. 1(b)], if the Kondo coupling is sufficiently strong, the Kondo singlets communicate to each other and develop a coherent narrow band of which the low-energy quasiparticles are of greatly enhanced effective mass [Fig. 1(c)]. The situation may change if the dense lattice reduces to one dimension (1D) or quasi-1D, say, the percolation path of Kondo coherence can be realized only in one direction but fails in other directions [Fig. 1(d)]. Systems like this may be termed as a Kondo chain.

“Dimensions are critical”, as Coleman declared in Ref. Coleman (2012). This is because lower dimension means more phase space for long-wavelength fluctuations and a larger magnetic frustration parameterCusters et al. (2012), the latter of which dictates the way that the system undergoes from a quantum ordered state to a disordered state: a conventional spin-density-wave (SDW) type quantum critical point (QCP)Hertz (1976); Millis (1993) or an unconventional Kondo-destruction type QCPSi et al. (2001); Coleman and Schofield (2005); Gegenwart et al. (2008). Kondo destruction generically requires large spin fluctuations and thus favors lower dimension. So far, most known examples of Kondo-destruction QCP were observed in materials between 2D and 3DCusters et al. (2003); Park et al. (2011); Custers et al. (2012); Shishido et al. (2010); Kim and Aronson (2011); Luo et al. (2014, 2015), while examples of 1D or quasi-1D have been rareKrellner et al. (2011). The situation in the 1D limit seems elusive: on one hand, long-range magnetic order is hard to stabilize in 1D materials, but on the other hand, a Fermi-surface reconstruction due to $4f$ localized-delocalized transition is still possible according to the predicted global phase diagram for heavy-fermion compoundsCusters et al. (2012). Extensive material bases are required to elucidate these issues.

Recently, Wang et al reported the synthesis and physical properties of CeCo2Ga8Wang et al. (2017). This compound crystalizes in the YbCo2Al8-type orthorhombic structureKoterlin et al. (1989), in the space group Pbam (No. 55). The crystalline structure of CeCo2Ga8 can be viewed as individual Ce chains along $\mathbf{c}$ axis, and each chain is surrounded by five polyhedral CoGa9 cages in the $\mathbf{ab}$ plane. Since the inter-chain Ce-Ce distances (6.5 and 7.5 Å) are much longer than the intra-chain distance (4.05 Å), the compound is considered as a candidate of quasi-1D Kondo latticeWang et al. (2017). Electrical resistivity, magnetic susceptibility and specific heat measurements manifested that it does not exhibit long-range ordering down to 0.1 K, but show non-Fermi-liquid behavior with a large electronic Sommerfeld coefficient \gamma_{0}$$\sim800 mJ/(mol$\cdot$K2). First-principles calculations revealed flat Fermi sheets arising from itinerant $4f$ band, characteristic of quasi-1D nature. All these suggest that CeCo2Ga8 sits nearby a quantum critical point. However, direct evidence for its quasi-1D nature is still lacking, or equivalently the question is, whether it realizes a Kondo chain at low temperature.

Experimentally, one may expect some physical properties for a Kondo chain compound. Electrically, coherent Kondo scattering would occur only in one direction, or the difference in Kondo-coherence temperature for various directions is huge; and magnetically, before Kondo effect sets in, the magnetic properties and exchange interactions exhibit a great anisotropy. In this paper, we performed careful measurements of resistivity and susceptibility on single crystalline CeCo2Ga8 along all the principal $\mathbf{a}$-, $\mathbf{b}$- and $\mathbf{c}$-axes. We observe all these supposed features for a Kondo chain. Theoretical calculations based on crystalline electric field (CEF) effect were carried out to understand these anisotropies.

Single crystalline CeCo2Ga8 was grown by a Ga self-flux method as described previouslyWang et al. (2017). The as-grown samples mostly are needle-like with typical length $\sim$3 mm along $\mathbf{c}$-axis, and about 1$\times$1 mm2 in cross-section. We carefully polished the crystals to make the long-side along $\mathbf{a}$-, $\mathbf{b}$- and $\mathbf{c}$-axes, respectively, and the orientations are verified by X-ray diffraction. These polished single crystals are used for anisotropic electric resistivity and magnetic susceptibility measurements, for which Physical Property Measurement System (PPMS, Quantum Design) and Magnetic Property Measurement System (MPMS, Quantum Design) were employed.

We start with electrical resistivity, as shown in Fig. 2. Here the resistivities were measured for electrical currents along all $\mathbf{a}$-, $\mathbf{b}$- and $\mathbf{c}$-axes, and results are denoted as $\rho_{a}$, $\rho_{b}$ and $\rho_{c}$, respectively. At room temperature, the resistivity values respectively are 0.372, 0.257 and 0.106 m$\Omega\cdot$cm. Upon cooling, both $\rho_{b}$ and $\rho_{c}$ decrease, while $\rho_{a}$ slightly increases. A tiny bump is visible in both $\rho_{a}(T)$ and $\rho_{b}(T)$ for temperatures near 170 K, which is probably due to spin-flip scattering by the excitations between CEF levels. We will discuss the CEF effect later on in detail. A common feature for lower temperature is that below $\sim$50 K all $\rho(T)$ curves turns up logarithmically, reminiscent of incoherent Kondo scatteringKondo (1964). In the low temperature limit, $\rho_{c}(T)$ turns down, manifesting the establishment of Kondo coherence. The coherence temperature can be defined as T^{c}_{coh}$$\approx17 K. Our previous work has systematically studied $\rho_{c}$ under high pressureWang et al. (2017). The sub-Kelvin measurements there revealed non-Fermi-liquid behavior (\rho_{c}$$\propto$$T) at low temperature, and a recovery of Fermi liquid (\rho_{c}$$\propto$$T^{2}) under pressure, implying that the compound sits in the vicinity of a quantum critical regime. In contrast, both $\rho_{a}$ and $\rho_{b}$ remain incoherent down to 2 K. It should be noted that $\rho_{b}(T)$ tends to level off below 3 K. It is possible that Kondo coherence may occur below 2 K. Any way, $\rho_{a}$ and $\rho_{b}$ require further investigations in the future. Noteworthy that the previously known quasi-1D Kondo-lattice candidate YbNi4P2 actually displays coherent Kondo scatterings for currents along both inter-chain and intra-chainKrellner and Geibel (2012). This places CeCo2Ga8 in a regime with even lower dimension than YbNi4P2.

Turning now to magnetic susceptibilities ($\chi$) as shown in Fig. 3. The magnetic anisotropy for field parallel and perpendicular to $\mathbf{c}$ has been reported in our earlier paperWang et al. (2017), while the in-plane anisotropy remains unknown. This is now complemented in Fig. 3. We find generically \chi_{c}$$>$$\chi_{b}$$\gtrsim$$\chi_{a} over the full temperature range, only except for below 5 K where $\chi_{a}(T)$ and $\chi_{b}(T)$ slightly cross. The anisotropic ratio $\chi_{c}/\chi_{a,b}$ reaches 3.5 at 2 K. We calculated the powder-average susceptibility $\chi_{avg}$=(\chi_{a}$$+$$\chi_{b}$$+$$\chi_{c})/3, the inverse of which has been displayed in the inset to Fig. 3(a). By fitting $\chi_{avg}(T)$ to a standard Curie-Weiss law, we derive the effective moment \mu_{eff}$$=2.62 $\mu_{B}$/f.u. This value is relatively larger than that of a free Ce3+ (2.54 $\mu_{B}$), suggesting that the Co ions also carry some magnetic moment. It is very common that Co ions in Ce-contained compounds are magnetic, see for e.g. CeCoAsOSarkar et al. (2010) and CeCo2As2Thompson et al. (2014).

It is well known that the CEF effect plays a key role in the anisotropic magnetism of rare earth ions. The CEF Hamiltonian for a $D_{2h}$ point group contains five termsBau ,

[TABLE]

where $B_{l}^{m}$ ($l$=2,4; $m$=0,2,4) are the CEF parameters, and $\hat{O}_{l}^{m}$ are Steven s operatorsStevens (1952); Hut . Note that $\hat{O}_{2}^{2}$ and $\hat{O}_{4}^{2}$ turn on the in-plane anisotropy. In addition, the Zeeman effect and exchange interaction should also be taken into account,

[TABLE]

where $\hat{\mathbf{J}}$=($\hat{J}_{x}$,$\hat{J}_{y}$,$\hat{J}_{z}$) is the total angular momentum operator, $g$=6/7 is the Landé factor of Ce3+ ions, $\mu_{B}$ is Bohr magneton, $J_{ex}^{a,b,c}$ are the components of the nearest-neighbor exchange interaction with Ce3+ moment along $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$, respectively.

An exact calculation of the paramagnetic susceptibility in the presence of CEF and magnetic interactions have been performed by P. BoutronBoutron (1973). At high temperature, a series expansion of $\chi$ in $1/T$ leads to the expressions of $1/\chi(T)$ as following,

[TABLE]

in which $C$ is the Curie constant, and $\theta_{ex}^{\alpha}$=$\frac{j(j+1)}{3}J_{ex}^{\alpha}$ with $j$=5/2 for Ce3+. This approximation turns out to be rather good for the temperature region T$$\gg$$|\theta_{ex}^{\alpha}|Cho et al. (1996); Gingras et al. (2000); Luo et al. (2012). Eq. (4) tells us that the total Weiss temperature is a sum of both $\theta_{CEF}$ and $\theta_{ex}$. The way to get the intrinsic magnetic correlation is to separate out $\theta_{ex}$, as $\theta_{CEF}$ usually is more dominant. We numerically calculated $\chi^{CEF}$, and fitted to our experimental data. The results are shown in $1/\chi$ in Fig. 3(b). The best fitting parameters include $B_{l}^{m}$ that have been summarized in Table 1. The calculation also results in the exchange interactions J_{ex}^{a}$$\approx2.5 K, J_{ex}^{b}$$\approx3 K, J_{ex}^{c}$$\approx$$-12 K. Note that the signs of $J_{ex}^{c}$ and $J_{ex}^{a,b}$ are opposite. This indicates that the magnetic correlations are probably antiferromagnetic for intra-chain, but are ferromagnetic for inter-chain. The anisotropy ratio of magnetic interactions can thus be estimated |J_{ex}^{c}/J_{ex}^{a,b}|$$\sim4-5. It, therefore, is reasonable to draw a conclusion that CeCo2Ga8 is magnetically quasi-1D. It should be emphasized that the strongest exchange interaction, $J_{ex}^{c}$, is comparable to but relatively smaller than $T_{coh}^{c}$, which is consistent with the fact that the compound resides on the edge of a magnetic instabilityWang et al. (2017).

We should admit that at low temperature, the fittings are not excellent. There are several reasons for such deviation. (i) At low temperatures, the condition of T$$\gg$$\theta_{ex}^{\alpha} is not satisfied, especially for $\mathbf{c}$-axis, therefore the approximation of Eq. (4) is no longer valid. (ii) Kondo effect becomes more and more important at low temperature. This is particularly the case for $\mathbf{c}$-axis, as Kondo coherence is seen in $\rho_{c}$ below 17 K. (iii) As the 1D spin-chain nature becomes more prominent, $\chi(T)$ no longer simply obeys the Curie-Weiss lawBonner and Fisher (1964); Wang et al. (2017), which is not considered in the CEF theory, either. (iv) The CEF parameters $B_{l}^{m}$ can be temperature dependent, but is beyond the scope of the present work. Despite of these shortcomings, the agreement between calculations and experiments is decent.

A schematic of the CEF splitting is presented in Fig. 4(a). The $j$=5/2 multiplet splits into three Kramers doublets, with the first and second excited doublets sitting at $\sim$145 K and 445 K above the ground states. The ground state can be expressed in term of $|\Psi_{0}\rangle$=\alpha|\pm 5/2\rangle$$+$$\sqrt{1-\alpha^{2}-\beta^{2}}|\mp 3/2\rangle$$+$$\beta|\pm 1/2\rangle, with $\alpha$=0.9604 and $\beta$=$-$0.2763. The large $\alpha^{2}$($\gg$1/6) makes the orbital severely oblateWillers et al. (2015), see the calculated contour of the $4f$ charge distribution in Fig. 4(a) Freeman and Watson (1962); Bau . For Ce3+, an oblate orbital typically renders a magnetic easy-axis. This can be seen from the simulated isothermal magnetizations [Fig. 4(b)]. Since in this simulation magnetic exchange interactions and Kondo effect are not considered, a direct comparison with experimental results is not perfect, but $\mathbf{c}$ axis being the magnetic easy-axis is inescapable. Intuitively, this should be an important premise to form a quasi-1D magnetic system.

Going a little further, we also calculated the CEF contribution to electrical resistivity ($\rho_{CEF}$) and specific heat ($C_{4f}$), the results of which are displayed in Fig. 4(c). $\rho_{CEF}$, as we mentioned before, arising from spin-flip scattering by CEF excitations, shows an obvious hump near 150 K, which is in agreement with the experimental $\rho_{a}$ and $\rho_{b}$ (Fig. 2) (Note not ). This provides additional evidence for the validity of our analysis. A broad peak near 75 K is expected in $C_{4f}(T)$, i.e. Schottky anomaly. A direct comparison to experiment is not available now, because all our attempts to synthesize the non-magnetic counter-part LaCo2Ga8 failed, and we are not able to extract $C_{4f}(T)$ from the total specific heat of CeCo2Ga8.

To draw a conclusion, we carefully measured the electrical resistivity and magnetic susceptibility of the heavy-fermion compound CeCo2Ga8 along all three principal $\mathbf{a}$-, $\mathbf{b}$- and $\mathbf{c}$-axes. Anisotropies can be seen in both resistivity and susceptibility. While $\rho_{c}$ displays coherent Kondo scattering below 17 K, the Kondo scattering in $\mathbf{a}$ and $\mathbf{b}$ remain incoherent down to 2 K. The magnetic anisotropy can be well understood by theoretical calculations based on CEF theory. The calculations confirm $\mathbf{c}$-axis as the easy axis, and also derives an anisotropy ratio of magnetic exchange interactions |J_{ex}^{c}/J_{ex}^{a,b}|$$\sim4-5. We, therefore, firmly demonstrate that CeCo2Ga8 is a quasi-one-dimensional heavy-fermion compound both electrically and magnetically, and thus provide a realistic example of Kondo chain. This finding has an immediate ramification for the global phase diagram of heavy-fermion materialsCusters et al. (2012). Our previous specific heat measurements suggested unconventional quantum critical behavior in CeCo2Ga8Wang et al. (2017). Altogether, a Kondo-destruction QCP seems possible in the quasi-1D limit. This, of course, requires more evidence to clarify. One should also be noted that the YbCo2Al8 family has many other Ce128 counterpartsKoterlin et al. (1989) whose physical properties remain unclear. They open an important means to explore heavy-fermion materials for studies on quantum criticality and dimensionality.

Finally, it is worthwhile to mention that, although the system on the whole behaves quasi-1D electrical and magnetic properties, the inter-chain coupling along $\mathbf{b}$ seems relatively stronger than along $\mathbf{a}$. This can be seen from many aspects of view. The magnitude of $\rho_{b}$ is smaller than that of $\rho_{a}$, and moreover, coherence-like Kondo scattering seems to be on the verge in the former. Magnetically, $\chi_{b}$ is a little larger than $\chi_{a}$, and most importantly, the estimated exchange interaction is also slightly stronger in $\mathbf{b}$. A weak in-plane anisotropy is likely.

We thank Hua Chen for helpful discussion. Y. Luo is supported by the 1000 Youth Talents Plan of China. The authors acknowledge National Natural Science Foundation of China (Grant No. 11674115, 11522435, 11774399 and 11474330) and the National Key R&D Program of China (Grant No. 2017YFA0303103, 2016YFA0401704, 2017YFA0302901 and 2016YFA0300604).