Electron spin resonance of the itinerant ferromagnets LaCrGe₃, CeCrGe₃ and PrCrGe₃

The interplay between local 4f magnetism and itinerant 3d ferromagnetism is an important topic in heavy fermion systems and superconducting pnictide systems [1, 2]. This is also relevant for systems such as CeCrGe₃ which show 4f Kondo lattice and heavy fermion behavior in the presence of ferromagnetic order of itinerant 3d moments [3, 4]. A comparison of CeCrGe₃ with LaCrGe₃ revealed a remarkably large Sommerfeld coefficient [3] which characterizes the specific heat in the magnetically ordered state. The roles of spin fluctuations and quantum criticality in itinerant ferromagnetic systems were investigated in the systems LaV_x Cr_1−x Ge₃ and CeCr_1−x Ti_x Ge₃ where a strong suppression of ferromagnetic order was observed [5, 6]. Recently, LaCrGe₃ has drawn considerable attention in the search for a pressure driven ferromagnetic quantum critical point [7–9]. For the investigation of the spin dynamics of metals with strong ferromagnetic correlations the spectroscopic method of electron spin resonance (ESR) provided important results in 3d metals like TiBe₂ [10], ZrZn₂ [11, 12], NbFe₂ [13] as well as 4f heavy fermion metals like YbRh₂Si₂ [14] and CeRuPO [15, 16]. Moreover, ESR is probing the magnetic ion locally and directly in contrast to magnetization measurements which probe the bulk of the sample. Here we report the Cr³⁺ ESR in LaCrGe₃, CeCrGe₃, and PrCrGe₃ in which the magnetism is based on different interactions between 3d and 4f electrons, namely pure itinerant 3d magnetism (LaCrGe₃, 4f⁰), heavy fermion ferromagnetism with 3d–4f hybridization (CeCrGe₃, 4f¹), and itinerant 3d/local 4f magnetism (PrCrGe₃, 4f²). Distinct differences in the ESR properties of these compounds are mainly found in the ferromagnetically ordered state.

Polycrystalline samples of RECrGe₃ with RE = La, Ce and Pr were prepared and characterized as described previously [3]. All magnetic properties were measured using a quantum design superconducting quantum interference device—vibrating-sample magnetometer (QD SQUID VSM). Specific heat and electrical four-point resistivity were measured in a commercial QD physical property measurement system (PPMS) equipped with a ³He option. The ESR experiments were carried out using a standard continuous wave spectrometer at X-band frequencies (ν = 9.4 GHz). The temperature was varied between 4 and 295 K with a He-flow cryostat.

ESR probes the absorbed power P of a transversal magnetic microwave field as a function of a static and external magnetic field μ₀ H [17]. To improve the signal-to-noise ratio, we used a lock-in technique by modulating the static field, which yields the derivative of the resonance signal dP/dH. The measured ESR spectra were fitted with a Lorentzian function including the influence of the counter-rotating component of the linearly polarized microwave field [13]. From the fit, we obtained the resonance field H_res (which determines the ESR g-factor g = hν/μ_B H_res), and the linewidth ΔH (half-width at half maximum). In metals, ΔH is a direct measure of the spin lattice relaxation time 1/T₁ and its temperature and frequency dependences reveal the nature of the participating relaxation mechanisms.

3.1. Magnetic, transport, and thermal properties

Measurements of the electrical transport and thermodynamic properties were carried out for PrCrGe₃ as was done previously for LaCrGe₃ and CeCrGe₃ in reference [3]. Furthermore, we performed DC-susceptibility measurements on all RECrGe₃ (see figure 1(a)) in a field of 0.32 T to get the magnetic response close to the ESR resonance field H_res (see figure 5). The magnetic properties of LaCrGe₃ and CeCrGe₃ are consistent with earlier reports [3, 18, 19].

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Interestingly, PrCrGe₃ exhibits two anomalies, one at ∼102 K due to the ordering of itinerant Cr-moments, and another at ∼16 K suggesting a low-temperature transition due to the ordering of Pr³⁺ moments. From linear fitting the inverse susceptibility data with a Curie–Weiss law (χ⁻¹ = (T − Θ)/C_CW) for the temperature range between 200 K and 400 K which is well above the ordering temperature T_C (defined as the inflection point of χ(T)) we obtained the effective moments (μ_eff) and Weiss temperatures (Θ), see summarized values in table 1. The Cr contribution in the effective moment μ_eff (Cr) for CeCrGe₃ and PrCrGe₃ can be estimated assuming stable 3+ valence states for the rare earth ions. A similar approach was discussed previously [3, 6]. The resulting moments μ_eff (Cr) summarized in table 1 are less than expected for free Cr³⁺ ions (3.8 μ_B). These reduced moments hint to an itinerant character of the Cr magnetism in all measured RECrGe₃ compounds.

Table 1. Summary of magnetic susceptibility data analysis of RECrGe₃.

RE	μeff, total(μB)	μeff(RE)(μB)	μeff(Cr) (μB)	${T}_{\text{C}}^{\enspace \chi }(\mathrm{C}\mathrm{r})$ (K)	${T}_{\text{C}}^{\enspace \chi }(4f)$ (K)	Θ (K)
La	2.44	0	2.44	90	—	104
Ce	3.38	2.54	2.2	70	—	72
Pr	4.10	3.58	2.0	102	16	105

Positive and large values of Θ for all cases are indicative of a strong ferromagnetic exchange interaction in these systems. The electrical resistivity data, ρ(T), in figure 1(b) exhibit a sharp drop below the magnetic transition temperatures due to the reduction of spin disorder resistivity. Furthermore, while LaCrGe₃ and PrCrGe₃ show metallic behavior, ρ(T) for CeCrGe₃ is similar to that of heavy fermion materials [3]. The resistivity anomaly at 102 K in PrCrGe₃ is due to magnetic order of Cr-moments and the anomaly near 16 K is likely due to the ordering of Pr³⁺ moments. The kink temperatures in ρ(T) are consistent with the T_C values determined from χ(T) (see figures 1(a) and (b)).

Specific heat C(T) data are presented in figures 1(c) and 2. The bulk nature of the magnetic transitions are confirmed by pronounced anomalies in C(T) near T_C. CeCrGe₃ exhibits a significant enhancement of the Sommerfeld coefficient γ = 130 mJ mol⁻¹ K⁻² (renormalization of ∼45) at low temperatures, making it a moderate heavy fermion system [3, 6]. PrCrGe₃, on the other hand, exhibits a smaller γ of about 66 mJ mol⁻¹ K⁻² but shows a large nuclear specific heat contribution (α_n) at low temperatures. As shown in figure 2, below 1 K, C(T)/T for PrCrGe₃ increases dramatically due to the nuclear Schottky effect which is caused by the strong interaction between the nuclear magnetic moments with the strong magnetic field produced by the 4f electrons at the nuclear site resulting in the splitting of the nuclear hyperfine levels [20–22]. Low temperature C(T < 2 K) data of PrCrGe₃ were fitted with C/T = γ + α_n T⁻³. Remarkably, α_n amounts to ∼454 mJ K mol⁻¹ which is significantly higher than α_n of pure Pr metal (∼56 mJ K mol⁻¹) at zero applied field [23] suggesting the presence of a strong hyperfine interaction in PrCrGe₃.

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3.2. Electron spin resonance

Typical ESR spectra at room temperature, in the paramagnetic state of LaCrGe₃, CeCrGe₃, and PrCrGe₃ are shown in figure 3. For all compounds the spectra could be well described by symmetric Lorentzian line shapes, indicating sample grain sizes being smaller than the microwave penetration depth. This resulted in the powder-averaged parameters g = 2.10 ± 0.02 and μ₀ΔH = 55 ± 2 mT. The deviations from the Lorentzian shapes are due to an anisotropy of the g factor which, however, could not be reliably resolved by fitting with powder-averaged Lorentzians due to broad lines and broad background structures.

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The obtained g factor describes the S = 3/2 spins of Cr³⁺ ions which are centered in octahedra which form face sharing chains in the hexagonal (P6₃/mmc space group) crystal structure. However, for an insulating environment and in an octahedral field, one expects for local Cr³⁺ (Hund's rule ${}^{4}F_{3/2}$ state) a g value slightly below 2 for the lowest lying level which is effectively an S state because of the quenched orbital moment [17]. In order to explain the deviation from the observed g value, Δg, one needs to take into account that the Cr 3d states are delocalized in narrow d-bands, leading to the band ferromagnetism in all three compounds [24]. Therefore, the resonance may be interpreted as a conduction ESR of conduction spins in a narrow band which is based almost entirely on Cr d-states. Then Δg is an effect of spin–orbit coupling of the conduction electrons and depends on the band structure [25].

Figure 4 illustrates the temperature dependence of the derivative microwave absorption for the three compounds. For temperatures above the ordering temperatures the wavy features correspond to the paramagnetic resonance signal. Interestingly, in LaCrGe₃ this feature is also visible in the ordered temperature region indicating a ferromagnetic resonance signal. Close to the magnetic phase transitions the microwave absorption is dominated at low fields (<0.1 T) by pronounced non-resonant structures reflecting a strongly field dependent magnetization. These structures have been disregarded when fitting the derivative microwave absorption with a Lorentzian shape.

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The results of a Lorentzian line fitting are displayed in figure 5. The effect of ferromagnetic ordering is to shift the line toward lower resonance fields which is nicely seen for LaCrGe₃. For CeCrGe₃ and PrCrGe₃ the line becomes undetectable due to extreme broadening and shift. The reason for this quite different behavior to LaCrGe₃ may be the much stronger magnetic anisotropy in CeCrGe₃ and PrCrGe₃ which arises from the 4f states. In the ordered state of LaCrGe₃ demagnetization fields could be a relevant contribution to the observed resonance field. Assuming for the investigated polycrystalline powder spherically shaped samples with a demagnetizing factor of $\mathcal{N}=1/3$ we obtain demagnetization $\mathcal{N}M\left(T,{\mu }_{0}H=320\right.$ mT) values below 0.1 mT in the entire temperature range. Thus, this 'sample-shape anisotropy' has a tiny effect, leaving anisotropy fields as a major source for the observed line shift.

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The results of the ESR study on LaCrGe₃, CeCrGe₃ and PrCrGe₃ are similar in the paramagnetic region but strongly depend on the presence of 4f electrons for temperatures close to and below the ferromagnetic ordering temperature T_C, see figure 5. The coupling between 4f and 3d magnetism obviously influences the spin resonance of Cr 3d electrons. Therefore, the effect of 4f magnetism seems to be relevant at relatively high temperatures although 4f magnetic ordering occurs well below the ordering of the 3d moments. For example, in PrCrGe₃ the Pr ordering is reflected in an additional increase of the magnetic susceptibility below 30 K (see figure 1(a)) whereas 3d magnetic ordering is at T_C = 102 K. Approaching T_C in the paramagnetic region, the linewidth increases due to critical fluctuations and a reduction of exchange narrowing processes [26] whereas the resonance field decreases because internal fields are built up toward T_C. Below T_C, in PrCrGe₃ and CeCrGe₃, the anisotropy inherent in the 4f magnetism leads to additional broadening such that the 3d resonance is unobservable below T_C (see figures 5(b), (c), (e) and (f)).

In case of LaCrGe₃ (4f⁰), the pure 3d magnetism determines the resonance properties. Besides the clear signatures at ferromagnetic ordering there are also anomalies around 35 K and 125 K in the linewidth and resonance field (see figures 5(a) and (d), respectively). Around 35 K, measurements of the thermoelectric power show a broad hump which was related to the ordering of Cr moments [3]. The small maximum in the linewidth data around 125 K has no correspondence to other quantities and is the result of a broad, unresolved background line superimposed on the main line. Upon decreasing the temperature below 125 K the main line starts to dominate and the fitted ESR parameters approach the main line properties.

Interestingly, there is no clear difference in the temperature dependence of the ESR parameters between PrCrGe₃ and CeCrGe₃ compounds. However, one would expect a difference since both compounds show a different coupling among their 4f–3d electron systems. PrCrGe₃ (4f²) has a stable, classic 4f moment [a non-magnetic singlet is possible which, however, is not indicated in the magnetic susceptibility (see upturn in χ(T) below 30 K, figure 1(a))]. CeCrGe₃ (4f¹) is a heavy fermion ferromagnet where 4f and conduction electron (3d) states are hybridized leading to Kondo lattice behavior of the electrical resistivity and an enhanced Sommerfeld coefficient of the specific heat [3]. The Kondo effect was shown to be essential for the observability of a Yb³⁺-based heavy ESR in YbRh₂Si₂ [27, 28]. In this respect, for temperatures above T_C, it seems remarkable that the Cr 3d resonance is very similar in PrCrGe₃ and CeCrGe₃, regardless of the presence or absence of 4f conduction-electron hybridization.

We acknowledge valuable discussions with Christoph Geibel. ZH would like to acknowledge the Polish National Agency for Academic Exchange (NAWA) for ULAM fellowship.

All data that support the findings of this study are included within the article (and any supplementary files).