Heavy fermion materials exhibit a rich variety of exotic correlated states, such as non-Fermi liquid and unconventional superconductivity, which often occur in the vicinity of a magnetic quantum critical point.1,2,3 The special non-Fermi liquid behavior may be compared to experiment to reveal the effective low-energy physics of the correlated electrons.4, 5 Taking the two-dimensional (2D) antiferromagnetism as an example, the conventional spin density wave (SDW) scenario based on the fluctuations of the magnetic order parameter has predicted as a function of temperature the resistivity, ρ~T, and the specific heat, C/T~−ln T, in the quantum critical regime.6 While these have been confirmed in some heavy fermion compounds such as CeCu2Si2, disparate temperature dependences of the resistivity and the specific heat have been observed in both stoichiometric and slightly Ge-doped YbRh2Si2 as the systems approach the quantum critical point.7 This indicates the failure of the conventional SDW scenario in describing the low-energy physics and points to possible breakup of the composite (heavy) quasiparticles near the antiferromagnetic quantum critical point in the YbRh2Si2 family.

Such unconventional quantum criticality in heavy fermion systems defy a satisfactory theoretical understanding because of the non-perturbative nature of the Kondo lattice physics despite of many experimental and theoretical efforts in recent years. The well-known mean-field approximation only captures the hybridization physics qualitatively, but fails to provide more insight into the detailed nature of the quantum criticality.8 Exact numerical methods such as the density matrix renormalization group approach9 are typically limited to one-dimensional (1D) systems, while most heavy fermion materials exhibit strong quasi-2D or three-dimensional features.10,11,12 Dimensionality is a key ingredient in determining the non-Fermi liquid behavior. The study of quasi-1D systems may be useful for our understanding of the heavy fermion physics and the associated unconventional quantum critical behavior.

In this work, we report experimental studies on the single crystal CeCo2Ga8 and show that it is a quasi-1D Kondo lattice system with a quantum critical point near ambient pressure. After the successful synthesis of the single crystals, we noticed that a polycrystalline CeCo2Ga8 has been reported previously,13 in which electrical resistivity and thermoelectric power have been measured but shown no interesting properties. In this work, we carried out more detailed investigations of the single crystal CeCo2Ga8. A picture of the single crystal is shown in Fig. 1a. The compound adopts the YbCo2Al8-type orthorhombic structure (space group Pbam, No. 55) with the lattice parameters a = 12.3792(7) Å, b = 14.3053(9) Å, and c = 4.0492(3) Å. The whole structure can be viewed as built of fused polyhedral of Ga that are interstitially stabilized by Co and Ce. The Co atoms are situated in tricapped trigonal prisms formed by nine Ga atoms. The Ce atoms form a chain along the c-axis located in the center of the pentagon formed by five CoGa9 cages in the ab plane. The Ce–Ce distances between neighboring chains are about 6.5 and 7.5 Å, much longer than the intra-chain distance of 4.05 Å, indicating that it might be a quasi-1D system. As a result, the single crystal prefers to grow up along the c-axis as shown in Fig. 1.

Fig. 1
figure 1

Crystalline structure of CeCo2Ga8. a Picture of a single crystal of CeCo2Ga8 of about 4 mm in length along the c-axis. b Side view of the YbCo2Al8-type orthorhombic structure with face-shared CoGa9 cages along the c-axis. c Stereoscopic view of the crystal structure showing the quasi-1D chains of Ce atoms along the c-axis. Each unit cell contains four Ce atoms and the interchain Ce–Ce distances are 6.5 and 7.5 Å, respectively

Our measurements reveal characteristic non-Fermi liquid behavior in the normal state, where the resistivity exhibits linear temperature dependence between 0.1 and 2 K and the specific heat grows logarithmically with lowering temperature between 2 and 10 K. These non-Fermi liquid behaviors are similar to those observed in YbRh2Si2 despite of the different dimensionality of the two compounds, i.e., quasi-1D vs. quasi-2D, and may therefore be of potential interest for future investigations. The resistivity shows a coherence peak at T * ≈ 20 K. Above T *, the Hall coefficient obeys the skew scattering theory as seen in most heavy fermion compounds.14 The magnetic susceptibility deviates from the usual Curie–Weiss law below about 150 K but could be well fitted with the typical 1D Bonner-Fisher formula15 from 300 K down to T *, implying the quasi-one dimensionality of the underlying Kondo lattice. Our first-principles calculations also yield several flat Fermi sheets originating from the quasi-1D f-electron bands along the c-axis. Combining these experimental and theoretical results, it suggests the quasi-1D nature of the heavy fermion compound CeCo2Ga8, which provides an interesting basis for future investigation of the Kondo lattice physics in 1D. Resistivity measurements under pressure confirm the quantum critical behavior in a large temperature–pressure range.


Resistivity and Seebeck coefficient

Figure 2 presents the temperature dependence of the resistivity ρ and the Seebeck coefficient S along the c-axis. Both curves show similar logarithmic temperature dependence between about 20 and 90 K originating from the incoherent Kondo scattering16 of the conduction electrons by the localized f-moments. A broad peak appears at T * ≈ 20 K, which marks a crossover from the insulating-like behavior to the metallic behavior at lower temperatures. Above 90 K, the resistivity also exhibits metallic behavior where the Kondo scattering is suppressed and the transport property is governed by the electron–phonon scattering. These features are common in Ce-based heavy fermion materials. Importantly, as shown in the inset of Fig. 2a, the resistivity exhibits T-linear behavior over a decade in temperature from 2 to 0.1 K, in resemblance of those found in CeCoIn5 and YbRh2Si2.17, 18 This indicates that the single crystal CeCo2Ga8 locates near a quantum critical point.4, 5, 7 However, we observe no sign of superconductivity down to 0.1 K.

Fig. 2
figure 2

Temperature dependence of the electrical resistivity and Seebeck coefficient. a Electrical resistivity ρ of CeCo2Ga8 single crystal along the c-axis at zero field and H = 7 T. The two sets of data are almost superimposable, showing small magnetoresistance. The inset plots the resistivity below 2 K measured in a dilution refrigerator, indicating linear-in-T (non-Fermi liquid) behavior over a decade in temperature from 2 K down to 0.1 K. No sign of superconductivity is observed down to 0.1 K. b Temperature dependence of the Seebeck coefficient at zero field, showing similar behavior as in the resistivity

We note that our sample exhibits a large residual resistivity, ρ 0 ≈ 90 μΩ cm, and a small residual resistivity ratio, RRR ≈ 1.2. To improve the quality of the single crystals, we have grown many samples but found similar value for the RRR while ρ 0 could be reduced by a factor of 2 (see, e.g., Fig. 3a, where ρ 0 ~60 μΩ cm and RRR ~1.1 at ambient pressure). This seems to suggest that the small RRR might be an intrinsic property of this compound at ambient pressure. Closeness to a quantum critical point and small values of T * may lead to a large ρ 0. However, this contribution typically decreases rapidly with pressure, in contrast to the weak pressure dependence of our measured ρ 0. We speculate that the large ρ 0 might be partly due to the impurity (Ga) scattering enhanced by the quasi-1D nature of the charge transport in CeCo2Ga8. However, our single crystal X-ray diffraction data show no significant site/chemical disorder. More work are needed to further improve the sample quality and solve this issue.

Fig. 3
figure 3

Pressure dependence of the c-axis resistivity as a function of temperature. a Temperature dependence of the resistivity under pressure. b Temperature-pressure phase diagram derived from the resistivity data, where the coherence temperature, T *, and the Fermi liquid temperature, T FL, are both plotted for comparison. Details on the determination of the Fermi liquid regime (n = 2) can be found in the Supplementary Fig. S1. Also shown is the pressure dependence of the resistivity coefficient, A, defined as ρ − ρ 0 = AT 2 in the Fermi liquid regime

The possible existence of impurities raises the question concerning the origin of the power-law behavior in the resistivity. To explore this, we further performed pressure measurements of the resistivity (on a different sample). Figure 3a plots the pressure dependence of the c-axis resistivity as a function of temperature. As shown in the Supplementary Fig. S1, our detailed analysis on the power-law behavior of the resistivity reveals continuous variation of the upper and lower boundaries of the non-Fermi liquid regime (n = 1, where n is the resistivity exponent defined as ρ~Tn). The large temperature range of the non-Fermi liquid regime is not unusual. In CeCoIn5, the upper boundary goes above 10 K with T * ~50 K at ambient pressure,17 while in CeRhIn5 it follows roughly T */2 at high pressures.19 The pressure variation of the coherence temperature, T *, estimated from the resistivity peak and the Fermi liquid temperature, T FL , obtained from the upper boundary of the Fermi liquid regime (n = 2) are plotted in Fig. 3b. We see that both T * and T FL increase with increasing pressure, consistent with the usual expectation for the Ce-based heavy fermion compounds. The fact that T FL approaches zero below 2 GPa indicates that CeCo2Ga8 at ambient pressure indeed locates near a quantum critical point. In particular, the resistivity coefficient, A, defined as ρ − ρ 0 = AT 2 in the Fermi liquid regime, increases by almost two orders of magnitude from 12 to 2 GPa and tends to diverge approaching the ambient pressure. Since the effective mass of heavy quasiparticles follows m * A 1/2. This indicates that the heavy electrons also have an enhanced effective mass, as has been observed in CeRhIn5 near the pressure-induced quantum critical point. These results are typical for Ce-based heavy fermion compounds.

Heat capacity

Figure 4a plots the measured specific heat C p as a function of temperature at zero field. At T = 300 K, C p is about 260.8 J/mol K, close to the Dulong–Petit limit,20 3nR = 274.2 J/mol K, where n = 11 is the total number of atoms per formula unit and R is the gas constant. The high-temperature specific heat above 30 K can be well fitted taking into account the contributions of conduction electrons and phonons,21, 22 which yields a specific heat coefficient of the conduction electrons, γ 0 = 6.3 mJ/mol K2. We note that this fit is just a theoretical approximation. The background electronic and phonon contributions should be better compared with that of a non-magnetic reference compound. Unfortunately, we have failed to grow the single crystal LaCo2Ga8. Yet our first-principles calculations for the hypothetical LaCo2Ga8 crystal with the same lattice structure yield \(\gamma _0^{th} = 8.4\) mJ/mol K2, in rough agreement with the above fitting result. The inset of Fig. 4a shows the magnetic entropy S m after subtracting the above electronic and phonon contributions. We see S m becomes saturated and reaches R ln 2 at about 20 K, close to T * determined from the resistivity peak.23 This implies a doublet ground state for the Ce f-electrons. Figure 4b plots C p/T as a function of the temperature on a semilogarithmic scale. We see a minimum at about 10 K that separates clearly the low-temperature behavior from the high-temperature phonon contributions. Below 10 K, C p/T grows logarithmically with lowering temperature down to 2 K, a characteristic signature of quantum criticality. An extrapolation of the logarithmic behavior to high temperatures also gives an onset temperature of about 20 K, consistent with the previously determined T * from the resistivity peak. This indicates that T * indeed marks the onset of lattice coherence. At 1 K, the specific heat coefficient reaches ~800 mJ/mol K2, a hundred times of that of the background conduction electrons but comparable to those of typical heavy fermion compounds such as CeCoIn5.24 Applying a magnetic field of 9 T suppresses the divergence at low temperatures and gives rise to a Fermi liquid state with roughly temperature-independent C p/T, as shown in the inset of Fig. 4b.

Fig. 4
figure 4

Temperature dependence of the specific heat. a The zero-field specific heat C p as a function of temperature on a linear scale. The solid line represents the theoretical fit between 30 and 300 K using a Debye–Einstein model for the phonon contribution. The inset shows the magnetic entropy calculated from the specific heat data after subtracting the phonon contribution. b Temperature dependence of C p/T on a semilogarithmic scale, showing a logarithmic divergence between 2 and 10 K. The inset plots the field variation of C p/T at low temperatures for H = 0, 5, 9 T in perpendicular to the c-axis

Magnetic susceptibility and Hall coefficient

Figure 5 gives the measured magnetic susceptibility and the Hall coefficient. The field-cooling (FC) and zero-field-cooling (ZFC) data are essentially superimposable. The susceptibilities increase monotonically with decreasing temperature, showing no sign of magnetic transitions. However, the magnetization for H || c is nearly three times as large as that of H c at T = 2 K, as shown in the Fig. 5a. This indicates a strong anisotropy in CeCo2Ga8 and the c-axis is the easy axis. Above 150 K, a Curie–Weiss fit using χ(T) = C/(T − θ), where C is the Curie constant and θ is the Weiss temperature, yields the moment μ eff = 2.74 μ B for H c and μ eff = 2.43 μ B for H || c, close to the free-ion moment of Ce3+, 2.54 μ B. However, the fit fails below T = 150 K, well above the coherence temperature, T * ≈ 20 K. While this has been observed in many heavy fermion materials and often ascribed to the crystal field effect, we find here that an alternative formula for 1D spin chain,15 χ(T) = C/Te η|J|/T where J is the exchange coupling and η accounts for the anisotropy, could yield an excellent fit for χ|| from 300 K down to T *, giving an effective antiferromagnetic coupling η|J| = 6.9 K and an effective moment μ eff = 2.74 μ B, in agreement with the Curie–Weiss fit at high temperatures. This suggests that the f-electrons in CeCo2Ga8 are well localized and form a quasi-1D spin chain at high temperatures, consistent with the quasi-1D crystal structure shown in Fig. 1. The c-axis Hall coefficient is also measured and presented in Fig. 5b. We find R H is proportional to ρχ above T * and governed by the incoherent skew scattering following the typical behavior in most other heavy fermion systems.14 The deviation below T * signals the onset of the f-electron coherence.

Fig. 5
figure 5

Temperature dependence of the susceptibility and Hall coefficient. a The FC (red and black) and ZFC (green) susceptibility of CeCo2Ga8 along (χ||) and perpendicular (χ) to the c-axis. The dashed line is the fit to χ|| using the formula of 1D spin chain. The inset shows the the Curie–Weiss fit (dashed lines) to the inverse susceptibility, yielding μ eff = 2.74 μ B for H c and μ eff = 2.43 μ B for H || c at high temperatures. b The Hall coefficient (H c, j || c) as a function of temperature. The inset compares the Hall coefficient with the prediction (dashed line) of the skew scattering theory

First-principles calculations

The quasi-1D nature of the underlying Kondo lattice is further supported by the first-principles density functional theory (DFT) calculations using the full-potential linearized augmented-plane-wave method as implemented in the WIEN2K package.25 Although DFT cannot treat correctly the strong electronic correlations, it usually yields qualitatively good predictions on the Fermi surface topology of the f-electron bands and has therefore been widely used as a starting point for understanding the electronic properties of many heavy fermion compounds. Calculations using strongly correlated methods such as the dynamical mean-field theory are too time-consuming for this compound, as its unit cell contains 4 formula units with a total number of 44 atoms including 4 Ce-ions, as shown in the inset of Fig. 6a. Our DFT calculations treat the f-electrons as fully itinerant and take into account the spin-orbit coupling. We use the refined lattice parameters as listed in the Supplementary Tables S1 and S2. The GGA-PBE functional26 is adopted for the exchange correlation energy with R MT  × K max = 8.0 and 1000 k-point meshes over the Brillouin zone. The obtained band structures and Fermi surfaces are presented in Fig. 6. We see that the Ce f-bands split into J = 5/2 and J = 7/2 multiplets with an energy difference of about 0.3 eV. The J = 5/2 multiplet have the lower energy and locate slightly above the Fermi energy. Importantly, we see that the itinerant f-bands are only dispersive along the Γ − Z path in the Brillouin zone (namely, the c-axis in real space), but remain flat in the ab plane, indicating the quasi-1D property of the itinerant f-electrons. The Fermi surfaces show two different types of topologies, including four flat sheets originating from the quasi-1D f-electron bands of the four Ce ions in each unit cell and quasi-2D cylindrical Fermi surfaces from the conduction electrons owing to the layered structure perpendicular to the b-axis. As may have been clearly seen in Fig. 1, the heavy electron physics originates from the hybridization between the Ce f-chains and the conduction electrons from the surrounding CoGa9-cages. Our calculations yield a specific heat coefficient of about 22.9 mJ/mol K2, much smaller than the experimentally observed value of ~800 mJ/mol K2, indicating the great enhancement due to quantum criticality.

Fig. 6
figure 6

First-principles calculations for the band structures and Fermi surfaces. a The band structures, showing flat f-electron bands within the ab plane and a spin-orbit splitting of about 0.3 eV. The inset shows the unit cell containing 4 formula units and a total number of 44 atoms. b The Fermi surfaces showing four flat sheets from the four Ce f-chains along the c-axis and several cylindrical sheets from the Co/Ga-layered structure perpendicular to the b-axis. The colors have no special meaning and are generated automatically from the WIEN2K code to distinguish different Fermi sheets

Discussion and conclusion

We have synthesized the single crystals of CeCo2Ga8 and carried out systematic investigations of its magnetic, thermodynamic, and transport properties. We found that CeCo2Ga8 behaves like a quasi-1D Kondo lattice and locates in the close vicinity of a magnetic quantum critical point. It exhibits non-Fermi liquid behavior with a T-linear resistivity below 2 K and a logarithmically divergent-specific heat between 2 and 10 K at ambient pressure. In the conventional Hertz–Millis theory of quantum criticality, these scaling behaviors are predicted for the 2D antiferromagnetic quantum critical point but not expected in a quasi-1D Kondo lattice system. It will be crucial to examine the dimensionality of the critical spin fluctuations. On the other hand, the observed disparate temperature dependences in the two quantities below 2 K are very similar to experimental observations in YbRh2Si2.18 These suggest a possibly unconventional quantum critical point whose nature is yet to be explored.7 We note that there are also other Ce-128 compounds including CeCo2Al8, CeFe2Ga8, and CeFe2Al8,13, 27,28,29 none of which has been well studied. Synthesis of good single crystals seems more challenging in this family of Ce-128 compounds compared to the famous Ce-115 family, possibly due to their structural differences. Nevertheless, a systematic exploration of the whole family is worthwhile for future investigations. The effect of chemical tuning may help us to achieve a better understanding of the Kondo lattice physics and, in particular, the role of dimensionality on heavy fermion quantum criticality.


Sample preparation and characterization

Single crystals of CeCo2Ga8 were grown using a Ga self-flux method in alumina crucible, which was sealed in a fully evacuated quartz tube. The crucible was heated to 1100 °C for 10 h, then cooled slowly to 630 °C at which point the Ga flux was spun off in a centrifuge, and subsequently quenched in cold water. Rodlike single crystals were yielded with the length of ~4 mm. Elemental analysis was conducted via energy dispersive X-ray (EDX) spectroscopy using a Hitachi S-4800 scanning electron microscope at an accelerating voltage of 15 kV, with an accumulation time of 90 s. The result of EDX indicated the composition of CeCo2Ga8 was stoichiometric. Single crystal X-ray diffraction was carried out on a RIGAKU Saturn CCD Diffractometer with a VariMax confocal optical system at 213(2) K using Mo K α radiation (λ = 0.71073 Å). The crystal structure was refined by full-matrix least-squares fitting on F 2 using the SHELXL-2014/7 program.

Transport, heat capacity, and magnetic measurements

The magnetic susceptibility (χ) was measured in a Quantum Design Magnetic Property Measurement System from 2 to 300 K under various applied magnetic fields up to 50 kOe in FC and ZFC modes. A well-crystallized sample was picked out for the study of magnetic anisotropy with the field perpendicular to and along the c-axis, respectively. The specific heat was measured in a Physical Property Measurement System (PPMS) with He-3 option. The electrical resistivity (ρ) along the c-axis was measured in PPMS upon cooling from 300 to 2 K and in a top-loading dilution refrigerator using the standard low-frequency lock-in technique below 2 K. High-pressure resistivity measurements up to 12 GPa were performed with a standard four-probe method in a palm cubic anvil cell apparatus using glycerol as pressure transmitting medium in order to maintain a good hydrostatic pressure condition.30

Data availability

The authors declare that all source data supporting the findings of this study are available within the article and the file.