Indication of Ferromagnetic Quantum Critical Point in Kondo Lattice CeRh$_6$Ge$_4$

TL;DR

This study presents resistivity measurements under pressure on CeRh$_6$Ge$_4$, revealing a continuous ferromagnetic quantum critical point, which is rare among clean metallic ferromagnets, and suggests its significance in understanding quantum criticality.

Contribution

It provides evidence of a ferromagnetic quantum critical point in CeRh$_6$Ge$_4$, a rare case where the transition remains continuous under pressure, challenging typical behavior in similar materials.

Findings

Continuous decrease of Curie temperature with pressure

Observation of non-Fermi liquid behavior near the transition

Band structure suggests unusual electronic state

Abstract

We report resistivity measurements under pressure for Kondo-lattice ferromagnet CeRh$_6$Ge$_4$, and present that a quantum ferromagnetic (FM) phase transition is easily achieved. In most clean metallic ferromagnets, a quantum critical point (QCP) at zero field is avoided by changing the FM transition to a discontinuous transition or to an antiferromagnetic transition. In CeRh$_6$Ge$_4$, to the contrary, the Curie temperature of 2.5 K decreases continuously as increasing pressure without any clear signature that the transition changes to first order. The obvious non Fermi liquid behavior is observed in the vicinity of the quantum FM phase transition. The experimental data do not contradict a picture in which CeRh$_6$Ge$_4$ shows the FM QCP at zero field. Band structure calculation suggests the unusual electronic state of CeRh$_6$Ge$_4$ among Ce-based Kondo lattices. CeRh$_6$Ge$_4$ deserves further investigations and will be a key material to understand the matter of the FM QCP.

Full text

A quantum critical point (QCP) is a phase transition at zero temperature accompanied by a continuous change of the order parameter. A feature of a QCP in metal is characterized by an anomalous electronic state affected by quantum fluctuations. In the metallic antiferromagnetic (AFM) case, a large number of materials show the AFM QCP by changing control parameters such as pressure, chemical substitution, and magnetic field. In specific materials, for instance heavy fermion systems or $d$-electron systems such as Fe-pnictides, they can offer a stage to induce exotic superconductivity, and have been extensively studied. In the metallic ferromagnetic (FM) case, however, a QCP at zero field as shown in Fig. 1(a) is hardly achieved.Brando A route to avoid a FM QCP is an appearance of the AFM state, as shown in Fig. 1(b). Actually, many ferromagnets show the alternation of the ground state before the termination of the FM state.Sidorov ; Kotegawa2 ; Lengyel ; Valentin_LaCrGe3 ; Kaluarachchi ; Kaluarachchi2 More crucial route to prevent a QCP is an appearance of a tricritical point (TCP), above which the second order transition changes to the first order transition, as expressed in Fig. 1(c). In this case, a metamagnetic transition from a polarized paramagnetic (PM) state to a FM state is induced by applying magnetic field along the easy axis, and the wing structure of the first-order planes is formed. A QCP emerges at a finite magnetic field, where the first-order metamagnetic transition terminates, but it is not at zero field. This kind of phase diagram has been investigated theoretically, Yamada ; Belitz ; Millis ; Binz ; Yamaji ; Imada and qualitatively reproduced by experiments in many systems such as MnSi, ZrZn2, UGe2, UCoAl, URhAl and U3P4.Pfleiderer_MnSi ; Uhlarz ; Kabeya ; Valentin ; Kotegawa1 ; Aoki ; Kimura ; Shimizu ; Araki Sr3Ru2O7 is also included in this category.Wu Originally, the change of the order in the FM transition and the emergence of the metamagnetic transition have been discussed to originate in the band structure in the itinerant system, as discussed in UGe2,Sandeman but a crucial point is a theoretical suggestion that the FM quantum phase transition is generically first order by considering soft particle-hole excitations about the Fermi surface, which are always present in metals.Belitz ; Belitz2 If this is correct, a FM QCP is impossible to be realized at zero field. It is an intriguing challenge to find a FM QCP in actual materials. As another important theoretical suggestion, it has been proposed that the wing structure is lost by disorder.Sang The QCP at zero field can be obtained by disorder effect, but it is not an intrinsic form of ferromagnets. An investigation in a clean system is essential.

A good example being close to a FM QCP is YbNi4P2, which shows extremely low $T_{\rm Curie}=170$ mK.Krellner The FM transition is second order in spite of the low $T_{\rm Curie}$. Substitution of As for P expands the lattice and suppresses the FM state. The 10% doping can induce the FM QCP,Steppke but a control by pressure is more desirable to prevent from inducing disorder. Another good example is FM superconductor UCoGe with $T_{\rm Curie}=2.5$ K. The FM state is easily suppressed by pressure, but the order of the FM transition has been controversial. The bulk measurements have shown the feature of the continuous transition under pressure in UCoGe,Elena ; Slooten ; Gael while the NQR measurement suggests that the FM transition is discontinuous, that is, first order.Ohta ; Manago It is an intriguing issue whether this discrepancy is specific to UCoGe or common behavior in metallic ferromagnets, but other examples to be investigated are very few.

Recently, CeRh6Ge4 was presented as a novel example, which would be in the vicinity of a FM QCP.Matsuoka It is a stoichiometric Kondo lattice with weak ferromagnetism below $T_{\rm Curie}=2.5$ K. CeRh6Ge4 crystalizes in non-centrosymmetric hexagonal structure with the space group of $P$6̄$m$2 (No. 187, $D_{3h}^{1}$). Voswinkel It is similar to UCoAl with $P$6̄2$m$ (No. 189, $D_{3h}^{3}$), which follows the phase diagram shown in Fig.1(c).Aoki ; Kimura In CeRh6Ge4, the translational symmetry is realized for all the Ce ions. It is composed of Ce-Rh1-Ge1 layer and Rh2-Ge2 layer, as shown in Fig. 2. The nearest Ce-Ce distance is 3.855Å along the $c$ axis, and the Ce-Ge2 distance (3.125 Å) and Ce-Rh2 distance (3.147 Å) are shorter. The easy axis is perpendicular to the $c$ axis, and the spontaneous magnetization of $\sim 0.16$ $\mu_{B}$/Ce is also induced perpendicular to the $c$ axis below $T_{\rm Curie}$.Matsuoka2 The reduced ordered moment and the small entropy of $\sim 20$ % of $R\ln 2$ just above $T_{\rm Curie}$ indicate the strong Kondo effect of this material.Matsuoka The specific heat suggests that the FM transition is of second order; therefore, CeRh6Ge4 is most likely to be located in the vicinity of a FM QCP. Another notable point is that the residual resistivity ratio (RRR) is more than 30,Matsuoka ; Matsuoka2 ensuring high quality of the crystal. In this letter, we show the pressure effect of this clean material to reveal how the FM state is suppressed, and discuss the unusual aspect in the band structure.

The single crystal of CeRh6Ge4 was grown by a Bi-flux method.Voswinkel It is needle-shape grown along the $c$ axis, and the typical thickness is about 50 $\mu$m. The electrical resistivity was measured using a four-probe method, in which the current direction was the $c$ axis. The electrical contacts of wire were made by a spod-weld method. The high pressure was applied by utilizing an indenter-type pressure cell and Daphne7474 as a pressure-transmitting medium.Indenter ; Murata The low temperature was achieved using a dilution refrigerator or a 3He cryostat. Band structure calculation was obtained through a full-potential LAPW (linear augmented plane wave) calculation within the LDA (local density approximation).

Figure 3(a) shows the temperature dependence of the electrical resistivity of CeRh6Ge4 measured at ambient pressure and at several magnetic fields along the easy axis. The FM transition at $T_{\rm Curie}=2.5$ K is clearly seen at zero field, while it easily changes to the crossover under weak magnetic fields. This is consistent with the interpretation that the ground state of CeRh6Ge4 is the FM state, although it has not been confirmed experimentally whether the FM state is completely collinear or not. Figures 3(b-d) show the temperature dependence of the electrical resistivity of CeRh6Ge4 measured at several pressures up to 2.17 GPa. The low residual resistivity $\rho_{0}$ of 1.35 $\mu\Omega$cm ensures the high quality of the present single crystal. The resistivity possesses the shoulder at around 60 K, which is an indication of the Kondo effect, and it moves to higher temperatures as increasing pressure as well as most Ce-based materials. Figure 3(c) shows the resistivity at low temperatures, and the experimental data are shifted by adding the expedient factors in Fig. 3(d) to avoid the overlap of the data. The FM transition is clearly observed at $\sim 2.5$ K at ambient pressure, and it decreases sensitively and gradually as increasing pressure. At 0.70 GPa, the transition is still clearly seen at $\sim 0.6$ K, and the small anomaly still remains at around $0.2-0.3$ K at 0.80 GPa, as indicated by the arrows. The anomaly disappears at around 0.99 GPa, where temperature dependence of the resistivity is linear, as indicated by the red line. This suggests a realization of the non-Fermi liquid (NFL) state in the vicinity of the quantum phase transition between the FM and PM states. The $T$-linear resistivity differs from a prediction by a SCR theory; $T^{5/3}$ for three-dimensional FM QCP,Moriya but is consistent with those observed in YbNi4P2 and UCoGe.Krellner ; Steppke ; Elena ; Slooten ; Gael At 1.23 GPa, the feature of the NFL is weakened, and the ground state changes over to the FL state at higher pressures. The change from the NFL to the FL states can be checked in Fig. 3(e), where the resistivity is plotted as a function of $T^{2}$. Linearity, which corresponds to the FL state, is hardly obtained at 0.99 GPa and even at 1.23 GPa, while it comes up above 1.67 GPa at the low-temperature regions. The slope at low temperatures gives $A$ coefficient in $\rho(T)=\rho_{0}+AT^{2}$, and the $A$ drastically decreases under higher pressures.

The pressure-temperature phase diagram of CeRh6Ge4 is shown in Fig. 4. The $T_{\rm Curie}$ continuously decreases and seems to reach 0 K at $P_{C}\sim 0.85$ GPa. The trial fitting gives $T_{\rm Curie}\propto(P_{C}-P)^{\beta}$ with $\beta\sim\frac{3}{5}$ (not shown), which is somehow smaller than $\beta=\frac{3}{4}$ predicted for that a three-dimensional clean ferromagnet.Millis2 In the experimental resolution, there is no clear signature that the transition changes to first order. The observation of the NFL state is a strong indication of the critical fluctuations at low temperatures. The top panel shows the pressure dependence of the $A$ coefficient. Here, the FL state is not realized at 0.99 and 1.23 GPa but they are determined by a tentative fitting at low temperatures as indicated in Fig. 3(d). The $A$ shows the divergent behavior, which is reminiscent of the presence of a QCP.

Figures 5(a-c) show the effects of the magnetic field in the vicinity of $P_{C}$. At 0.80 GPa, where $T_{\rm Curie}$ is expected to be close to $\sim 0.25$ K, the resistivity at zero field shows the $T$-linear behavior above 0.4 K. The applied field of 1.5 T suppresses the scattering especially below $\sim 2$ K, indicating that the broad crossover occurs at around 2 K. The NFL behavior at low temperatures is suppressed as increasing magnetic field. In the field dependence measured at 0.5 K shown in Figs. 5(b) and (c), the resistivity slightly decreases at lower fields due to the suppression of the scattering, and turns to increase towards higher fields due to the occurrence of the transverse magnetoresistance. There is no clear signature of the field-induced phase transition corresponding to the case shown in Fig. 1(c). If a system has a metamagnetic transition, which is relevant with a presence of a TCP, the corresponding anomaly is often detected in resistivity,Kotegawa1 ; Aoki ; Kimura but they are negligible in CeRh6Ge4 within the experimental resolution. This suggests that a TCP in CeRh6Ge4 is absent or located at least below $\sim 0.5$ K even if it exists. In most of the clean ferromagnets, the TCP appears above 5 K;Brando therefore, the electronic state in CeRh6Ge4 is quite unusual.

In YbNi4P2, one-dimensionality of the Fermi surface has been proposed to be a key factor to avoid a first-order phase transition,Krellner ; Steppke and it still opens to argument. To examine the electronic state of CeRh6Ge4, we performed the band structure calculation utilizing the lattice parameters obtained in Ref. 35. Figure 6(a) and (b) show the energy dispersion and the density of states (DOS) near the Fermi level for the PM state. The partial DOS originating in Ce-$4f$ orbits forms the two-peak structure above $E_{F}$. The $E_{F}$ is located at the tail of the peak structure of Ce-$4f$ orbits, or the local minima in the total DOS, giving the relatively low and gently-sloping DOS at $E_{F}$. This is in contrast to the typical LDA calculation for Ce-based heavy fermion compounds, where $E_{F}$ is located at the shoulder of the peak structure,Sheikin ; Ishikawa even in a nonmagnetic system.Suzuki It should be noted that the valence of Ce in CeRh6Ge4 is suggested to be close to 3+, because of the sufficiently occupied electron number in the tail. The relatively low DOS must be relevant with the weak magnetism in CeRh6Ge4, and the absence of the steep structure in the DOS in the vicinity of $E_{F}$ can be an important factor to avoid a clear discontinues FM transition. The two peaks originating in Ce-$4f$ orbits are separated well by the atomic spin-orbit coupling, indicating weak hybridization effect, whereas the significant broadened tail is brought by the hybridization. These two features suggest that only specific orbital in Ce-$4f$ has the strong hybridization with the conduction electrons, probably giving a key ingredient to induce the unusual electronic state in CeRh6Ge4. The calculated Fermi surfaces are shown in Fig, 6(c), where two sets of the Fermi surfaces appear because of lack of the space inversion symmetry. Two-dimensional feature is seen at the 159 and 160 electron sheets, and a similarity to the Fermi surface of YbNi4P2 is not clear.Krellner Experimental determination of the Fermi surfaces is an important issue to clarify the origin inducing the unusual electronic state in CeRh6Ge4.

In conclusion, we established the $P$-$T$ phase diagram of the novel weak ferromagnet CeRh6Ge4. The $T_{\rm Curie}$ of 2.5 K is easily suppressed by a low pressure of $P_{C}\sim 0.85$ GPa. The continuous disappearance of the FM state and the obvious NFL behavior suggest the presence of the FM QCP at zero field or at least in the immediate vicinity of zero field. CeRh6Ge4 behaves as a material showing a FM QCP. In UCoGe, however, the FM transition has been suggested to be microscopically discontinuous to the contrary to the continuous behavior in other bulk measurements. It is interesting to check the electronic state in the vicinity of $P_{C}$ for CeRh6Ge4 from the microscopic point of view. This would be a crucial issue to clarify what happens at the limit where a TCP goes to zero. Unfortunately, superconductivity did not appear in CeRh6Ge4, but this may be a benefit for the research of the pure quantum FM transition. Another notable point is the interplay between the matter of a quantum FM transition and the band structure. The band structure calculation suggests the peculiar electronic state in CeRh6Ge4. CeRh6Ge4 is a key material to know whether a FM QCP can be realized in actual material, and also how to achieve it.

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP15H05882, JP15H05885, JP18H04320, and 18H04321 (J-Physics), 15H03689.and 15H05745.