Kondo-lattice ferromagnets and their peculiar order along the magnetically hard axis

TL;DR

This paper reveals that Ce- and Yb-based Kondo-lattice ferromagnets predominantly order along their magnetically hard axis, a common but previously underestimated phenomenon, largely unaffected by structural or magnetic variations.

Contribution

It demonstrates that ordering along the hard magnetic axis is the typical behavior in these systems, challenging prior beliefs and suggesting a universal aspect of their magnetic properties.

Findings

Order along the hard axis is common in Ce- and Yb-based Kondo ferromagnets.

Kondo coherence maximum occurs just above the Curie temperature.

Transition shows non-mean-field behavior influenced by anisotropy.

Abstract

We show that Ce- and Yb-based Kondo-lattice ferromagnets order mainly along the magnetically hard direction of the ground state Kramers doublet determined by crystalline electric field (CEF). Here we argue that this peculiar phenomenon, that was believed to be rare, is instead the standard case. Moreover, it seems to be independent on the Curie temperature $T_\mathrm{C}$, crystalline structure, size of the ordered moment and type of ground state wave function. On the other hand, all these systems show the Kondo coherence maximum in the temperature dependence of the resistivity just above $T_\mathrm{C}$ which indicates a Kondo temperature of a few Kelvin. An important role of fluctuations is indicated by the non-mean-field like transition in specific heat measurements as well as by the suppression of this effect by a strong Ising-like anisotropy. We discuss possible theoretical scenarios.

Tables

Table 1. Table 1: Ce- and Yb-based Kondo-lattice ferromagnets. YRCS = Yb(Rh 0.73 Co 0.27 ) 2 Si 2 , T C subscript 𝑇 C T_{\textnormal{C}} = Curie temperature, T K subscript 𝑇 K T_{\textnormal{K}} = Kondo temperature, μ 𝜇 \mu = magnetic moment, CEF = crystalline electric field, ESR = electron spin resonance (9.4 GHz, T > 3 𝑇 3 T>3  K, main crystalline directions), MFT = mean-field transition, GS = ground state wave function, n.a. = not available, ∗ = moments order along the CEF magnetically hard axis.

System	$T_{\text{C}}$ (K)	$T_{\text{K}}$ (K) a	crys.	ordered	CEF	easy	order	ESR	MFT	coherence
			struc.	$\mu$ (${\mu }_{\text{B}}$) b	anisotropy	axis	axis	signal		max. in $\rho \left(T\right)$
∗CeAgSb2 Myers_et_al_1999 ; Andre_et_al_2000 ; Sidorov_et_al_2003 ; Araki_et_al_2003 ; Takeuchi_et_al_2003 ; Jobiliong_et_al_2005	9.6	16	tetra.	0.41	3	$ab$-plane	$c$-axis	not found	weak	yes
∗CeRuPO Krellner_et_al_2007 ; Krellner_Geibel_2008 ; Krellner_et_al_2008	15	7	tetra.	0.3	3	$ab$-plane	$c$-axis	found	no	yes
∗CeFeAs0.7P0.3O Jesche_et_al_2012	7.5	5	tetra.	0.3	3	$ab$-plane	$c$-axis	found	no	yes
∗YRCS Klingner_et_al_2011 ; Lausberg_et_al_2013 ; Gruner_et_al_2012	1.3	7.5	tetra.	0.1	6	$ab$-plane	$c$-axis	found	no	yes
∗YbNi4P2 Krellner_et_al_2011 ; Krellner_Geibel_2012 ; Steppke_et_al_2013 ; Gegenwart_et_al_2015  b	0.15	8	tetra.	0.05	5	$c$-axis	$ab$-plane	not found	no	yes
∗YbIr3Ge7 Rai_et_al_2019	2.6	16	rhomb.	0.05	4	$c$-axis	$ab$-plane	not found	no	yes
∗YbNiSn Bonville_et_al_1992 ; Kasaya_et_al_1991 ; Generalov_et_al_2017	5.6	10	orth.	0.85	1.8	$a$-axis	$c$-axis	not found	no	yes
∗YbPtGe Katoh_et_al_2009	5.4	9.4	orth.	1	2	$a$-axis	$c$-axis	n.a.	no	yes
∗YbRhSb Muro_et_al_2004 ; Umeo_et_al_2012  c	4.3	30	orth.	0.4	2	$a$-axis	$c$-axis	n.a.	no	yes
∗YbPdSi Tsujii_et_al_2016  d	8	13	orth.	0.26	n.a.	$b$-axis	$c$-axis	n.a.	no	yes
∗$\beta -$CeNiSb3 Thomas_et_al_2007  e	6	10	orth.	0.9	1.5	$b$-axis	$c$-axis	n.a.	no	yes
∗CeIrGe3 Muro_et_al_1998 ; Kawai_et_al_2008 ; Anand_et_al_2018  f	4.8	12	tetra.	0.14	1.3	$a$-axis	$c$-axis	n.a.	1st-order	yes
CeTiGe3 Fritsch_et_al_2015	14	30	hexag.	1.5	10	$c$-axis	$c$-axis	n.a.	yes	yes
CeRu2Al2B Matsuoka_et_al_2012 ; Baumbach_et_al_2012 ; Bhattacharyya_et_al_2016   g	13	23	tetra.	1	$>40$	$c$-axis	$c$-axis	n.a.	n.a.	yes
a Reported or estimated from entropy. b In-plane anisotropy: orthorhombic point symmetry site for Yb.
c Under pressure of about 2 GPa. At zero pressure the order is canted AFM with a very small ordered moment (0.003 ${\mu }_{\text{B}}$
along the $b$-axis). $T_{\text{K}}$ is the value at zero pressure.
d Largest moment in a complex structure with 3 Yb sites and 3 different moment sizes. e Two Ce sites.
f Transition into a canted AFM at 8.7 K, which is probably 1st-order. Recent neutron experiments suggest the magnetic
structure is more complex than a collinear FM Anand_et_al_2018
g AFM transition at 14.3 K.

Full text

Kondo-lattice (KL) systems are typically intermetallic compounds based on trivalent Ce or Yb atoms and are characterized by the Kondo effect at low temperatures and subsequent Kondo coherence at even lower temperatures. The degenerate ground state multiplet ($J=5/2$ for Ce and $J=7/2$ for Yb) is split by the crystalline electric field (CEF), making Kramers doublets the prevalent ground state. Only in cubic structures the ground state can be a quartet, which is prone to multipolar order Shiina_et_al_1997 . The first excited state is usually located at several tens of Kelvins above the ground state and does not participate in the magnetic ordering. In fact, depending on the strength of the Kondo and Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions, transition temperatures are usually in the order of a few Kelvin, often between 2 and 12 K, or below 1 K in systems with a very large distance ($>6$ Å) between the Ce atoms, like in Ce4Pt12Sn25 Kurita2010 , or strong Kondo effect, like in YbRh2Si2 Trovarelli_et_al_2000 .

It is well established that among all KL systems, there are more than two hundred that show antiferromagnetic (AFM) order at low temperature, while only very few show ferromagnetic (FM) order Stewart_2001 ; Brando_et_al_2016 . The reason for this difference is still unclear. It has recently been proposed that it could result from a $p$-type form factor of the Kondo coupling Ahamed_et_al_2018 . To our knowledge, the first FM KL system was discovered by Sato et al. in 1988 Sato_et_al_1988 . In the following 30 years an increasing number of such compounds have been discovered and studied. The main interest and hope was to find exotic heavy-fermion (HF) superconductivity near a FM quantum critical point (QCP) as it has often been found in the HF AFM systems Steglich_et_al_1979 ; Mathur_et_al_1998 . But up to now, apart from some U-based ferromagnets Aoki_et_al_2011 , no Ce- or Yb-based FM KL superconductor has been found. One of the reasons for the absence of superconductivity in quantum critical metallic ferromagnets has been suggested to be the absence of quantum critical fluctuations due to the intrinsic first order phase transition at the FM quantum phase transition Belitz_et_al_1999 ; Brando_et_al_2016 .

The discovery of YbNi4P2 Krellner_et_al_2011 with the lowest Curie temperature among pure compounds ever observed ($T_{\textnormal{C}}$ = 0.15 K), and the subsequent observation of a FM QCP Steppke_et_al_2013 , reopened the discussion about the existence of FM QCPs Steppke_et_al_2013 ; Gegenwart_et_al_2015 . Along with this unexpected observation, another peculiar feature in this system caught our attention: The magnetic moments in the FM ordered state point along the magnetically CEF hard axis and not, as naively expected, along the easy axis Comment_01 . In fact, ordering along the easy direction (with a larger moment) is expected because the gain in energy in the ordered state is proportional to the square of the size of the ordered moment. Such a behavior was first observed by Bonville et al. in YbNiSn with a $T_{\textnormal{C}}$ of 5.6 K Bonville_et_al_1992 and it was surprisingly also found in Yb(Rh0.73Co0.27)2Si2, a tetragonal system with a large CEF magnetic anisotropy of about 6 Lausberg_et_al_2013 ; Hamann_et_al_2018 . There is no clear explanation for this counter-intuitive phenomenon, but at least two theoretical approaches were recently proposed: i) A Monte-Carlo calculation based on the Heisenberg model with competing FM and AFM ordering combined with competing anisotropies in exchange interactions and $g$-factors, which could reproduce the experimental results for Yb(Rh0.73Co0.27)2Si2 very well Andrade_et_al_2014 . This model, however, seems not to work for large CEF anisotropies Hamann_et_al_2018 and does not provide any explanation for the particular choice of exchange couplings in the system, and it does not work for quantum critical systems like YbNi4P2. ii) Another and more general approach is the one proposed by Krüger et al.: Large fluctuations in an easy basal plane favor ordering along the transversal hard axis Kruger2014 . The idea of having a fluctuations induced transition would work for any classical or quantum ferromagnet, provided that the anisotropy is not too large.

In the course of our studies on other KL ferromagnets like CeRuPO Krellner_Geibel_2008 and YbIr3Ge7 Rai_et_al_2019 , we have realized that even more of these systems show this peculiar behavior. Here, we present part of these studies and compile a list of FM KL systems to show that this phenomenon, rather than a rare occurrence, is instead the general case.

We start showing the main properties of these FM KL systems that order along the hard axis by taking a look at the temperature dependence of the magnetic ac-susceptibility $\chi^{\prime}(T)$ of YbNi4P2, Yb(Rh0.73Co0.27)2Si2, CeRuPO, and YbIr3Ge7 plotted in Fig. 1. $\chi^{\prime}(T)$ was measured with modulated field along the two principal directions of the tetragonal crystalline structure, $H_{ac}\parallel c$ and $H_{ac}\perp c$. At high temperatures, both susceptibilities follow the same $T$-dependence, because of the dominant Curie-Weiss contribution of the full moment of trivalent Ce and Yb. However, their absolute values differ significantly due to the magnetocrystalline anisotropy caused by the CEF of the tetragonal structure. At low temperatures, just above $T_{\textnormal{C}}$, these susceptibilities cross each other at a temperature T_{\textnormal{0}}\ (marked by an arrow in the first panel of Fig. 1), which inevitably indicates that the magnetic moments order along the magnetically hard direction of the CEF. This has also been confirmed by magnetization measurements at $T<T_{\textnormal{C}}$ Steppke_et_al_2013 ; Lausberg_et_al_2013 ; Krellner_Geibel_2008 ; Rai_et_al_2019 . Below $T_{\textnormal{C}}$, the measured $\chi^{\prime}(T)$ perpendicular to the ordered moments remains constant, while the behavior of $\chi^{\prime}(T)$ parallel to the moments depends on the ratio between the coercive field and the modulated field used in the measurements: For instance, in YbNi4P2 $\chi^{\prime}(T)$ stays constant below $T_{\textnormal{C}}$ whereas in Yb(Rh0.73Co0.27)2Si2 it decreases steeply. The fact that $T_{\textnormal{0}}$ is just above $T_{\textnormal{C}}$ implies that there is no correlation between this behavior and the CEF first excited state which is located at much higher temperatures in these systems. There are indeed FM systems in which the susceptibilities cross each other at a high $T_{\textnormal{0}}$ because of the CEF excited states like CeSix Sato_et_al_1988 ; Pierre_et_al_1990 ($T_{\textnormal{0}}$ $\approx 70$ K) or CeCu0.18Al0.24Si1.58 Maurya_et_al_2017 ($T_{\textnormal{0}}$ $\approx 40$ K).

To demonstrate that this peculiar behavior, that was believed to be rare, is instead the general rule, we present a list of all KL ferromagnets known to date in table 1. There are twelve systems that order along the hard axis and only two exceptions, CeTiGe3 Fritsch_et_al_2015 and CeRu2Al2B Matsuoka_et_al_2012 ; Baumbach_et_al_2012 . We also present in the following other measurements performed on the same four systems of Fig. 1 to emphasize some characteristic properties that are common to all KL ferromagnets, which will help us to derive some conclusions about the origin of this behavior. We should also mention that our analysis is valid for systems with a single site for Ce or Yb. Systems with more sites for Ce or Yb are obviously more complex and might deviate from the general trend, like in the case of YbPdGe Enoki_et_al_2012 ; Tsujii_et_al_2015 . However, YbPdSi Tsujii_et_al_2016 and $\beta-$CeNiSb3 Thomas_et_al_2007 have more than one site per magnetic atom, but order along the hard axis.

A common property of KL ferromagnets is the presence of the Kondo effect with a Kondo temperature $T_{K}$ of a few Kelvins, often close to $T_{\textnormal{C}}$. $T_{K}$ is listed in table 1 and, if not reported, it was estimated by us from the magnetic entropy $S_{\textnormal{m}}$ extracted from specific heat measurements with $S_{\textnormal{m}}(\frac{1}{2}T_{\textnormal{K}})=\frac{1}{2}R\ln 2$. The presence of the Kondo effect can also be seen in the temperature dependence of the resistivity, exemplarily plotted in Fig. 2 for the same four systems of Fig. 1. The maximum in $\rho(T)$ at $T_{K}^{*}>T_{\textnormal{C}}$ indicates the Kondo coherence temperature Krellner_et_al_2011 ; Krellner_et_al_2007 ; Klingner_et_al_2011 ; Rai_et_al_2019 . This behavior is similar in all ferromagnets showing order along the hard axis. In contrast to this, materials in which the Kondo effect is absent or $T_{K}$ is small show ordering along the easy axis, like CeNiSb2 Thamizhavel_et_al_2003 . Although CeTiGe3 Fritsch_et_al_2015 and CeRu2Al2B Matsuoka_et_al_2012 ; Baumbach_et_al_2012 show a sizeable $T_{K}^{*}$, they order along the easy axis. A hint why these two systems do not follow the common rule can be found in their huge Ising-type anisotropy, which limits fluctuations to longitudinal ones along the c-axis. This would strongly reduce the possibility for fluctuation induced order. This assumption is also supported by the shapes of the second order phase transition at $T_{\textnormal{C}}$ measured in specific heat. While for CeTiGe3 the transition is mean-field-like as expected for an Ising-system, for the other compounds ordering along the hard axis, the transistion is $\lambda$-shaped, signifying that strong fluctuations are present around $T_{\textnormal{C}}$. This is shown in Fig. 3 for the same four compounds of Figs. 1 and 2.

Regarding theoretical proposals, the evidence for fluctuations in the temperature dependence of the specific heat near $T_{\textnormal{C}}$ also strongly favors the theory by Krüger et al. Kruger2014 based on strong transversal fluctuations over a purely MFT as the one based on competing anisotropies by Andrade et al. Andrade_et_al_2014 . In addition, it has recently been shown that in Yb(Rh1-xCox)2Si2 a model with competing anisotropy of the exchange interaction which is supposed to overcome the CEF anisotropy is unlikely, because the huge CEF anisotropy ($>10$) for small $x$ would require a huge inverse anisotropy in the exchange interactions Hamann_et_al_2018 . A direct approach to get information on exchange interactions is to measure the dispersion relation of magnetic excitation, e.g. magnons, using inelastic neutron scattering (INS). Therefore one might expect that the anisotropy of the exchange interactions in the FM systems is a problem which can unambiguously be settled using this approach. For one of the compounds discussed here, CeAgSb2, such a detailed INS study has been performed Araki_et_al_2003 . This study indeed concluded that all experimental results, including the FM ordering along the hard axis, can be fully explained by a huge anisotropy of the exchange interactions Araki_et_al_2003 . As we show in detail in the supplemental material, however, there are problems in their analysis: In contrast to the view suggested in Araki_et_al_2003 , the magnon dispersion relations typically determined by INS do not provide a unique answer concerning exchange interactions on their own. Specifically in the case of CeAgSb2, we also demonstrate that the huge anisotropy of the exchange parameters deduced from INS in Araki_et_al_2003 are in clear contradiction to the anisotropy of the susceptibility observed at high temperatures. This contradiction points on its own to a yet unidentified phenomena which promotes the hard axis ordering at low temperature. A similar discrepancy between a huge anisotropy of exchange interactions deduced from INS experiments and a much weaker anisotropy of the susceptibility at high temperature is also observed in the system CeRu2Al10 with AFM order along the hard axis Robert_et_al_2012 ; Strigari_et_al_2012 ; Dean_et_al_2018 ; Adroja_2013 . The authors of Robert_et_al_2012 conducted this type of analysis and concluded that this leads to an ”unrealistically large” value for the hard axis exchange interaction. The similarity between these cases suggests the problem addressed in our paper is not only relevant to FM, but also for AFM Kondo systems.

Table 1 also shows what these KL ferromagnets do not have in common: For instance, the crystalline structure, the size of the ordered moment or $T_{\textnormal{C}}$. The ground state wave functions are also very different. While it has been proposed that FM correlations are essential for the observability of an ESR signal in a KL system Abrahams_2008 ; Kochelaev_2009 , the situation still seems unclear. Inspired by a work done on some KL systems a few years ago Krellner_et_al_2008 , in which the detection of the ESR signal at 9.4 GHz was attributed to the presence of FM correlations, we investigated the ESR response of some of these systems and included whether such a signal has been found or not in Tab. 1. Unfortunately, there does not seem to be any systematic relation and although the ESR signal is undoubtedly connected to ferromagnetism, the latter is not the only deciding factor for the occurrence of an ESR signal.

Having ruled out the role of CEF excited states and a model with competing exchange interactions, and considering that the only common features between the systems in Tab. 1 are the presence of the Kondo effect, fluctuations at $T_{\textnormal{C}}$ and the possibility of sizeable (transversal) fluctuations perpendicular to the hard directions, it seems that the most possible scenario is that in which fluctuations are the driving force, a sort of order-by-disorder mechanism like the one proposed by Krüger et al. Kruger2014 . However, while there is a qualitative match for the susceptibility curves between theory and experiment, there are still other details that do not match, e.g. the proposed first order transition versus the observed second order transition. Further comparisons require detailed measurements of the direction dependence of fluctuations, which are possible by neutron scattering or NMR experiments. More systems and information might also be needed to finally unravel the origin of this mysterious behavior. In fact, there are some FM systems for which only polycrystals are available, like CePd Kappler_et_al_1991 or CePdIn2 daSilva_et_al_2009 ; Carleschi_et_al_2015 ; or systems which show FM ordering only at very high pressure, like YbCu2Si2 Tateiwa_et_al_2014 ; Dung_et_al_2009 (at about 11.5 GPa), for which not much information about the CEF anisotropy at high-$p$ is available. We would also like to mention, that this phenomenon has also been observed in cerium and actinide intermetallics Cooper_Siemann_1979 and some AFM KL systems, i.e. CeRu2Al10 Kondo_et_al_2013 , CeOs2Al10 Khalyavin_et_al_2013 and CeRhIn5 Takeuchi_2001 . In these systems the change from hard-axis to the easy-axis ordering has been attributed to the weakening of the Kondo hybridization. But this seems to be in contrast to what has been observed, e.g., in Yb(Rh1-xCox)2Si2, in which increasing the Kondo hybridization drives the moments into the easy plane Klingner_et_al_2011 ; Hamann_et_al_2018 .

In conclusion, we observe that Ce- and Yb-based Kondo-lattice ferromagnets order mainly along the magnetically CEF hard direction. This behavior is independent on $T_{\textnormal{C}}$, crystalline structure, size of the ordered moment and type of ground state wave function. On the other hand, all these systems show Kondo temperatures of a few Kelvin, often close to $T_{\textnormal{C}}$, and they have in common a relatively small CEF anisotropy. CEF excited states are too high in energy to be responsible for this behavior. Specific heat measurements indicate that the second order phase transition is not mean-field like, pointing to an important role of fluctuations, which might induce such an order along the hard axis. However, the intrinsic mechanism leading to this kind of order in all KL ferromagnets remains unknown. We further note that a huge Ising-type anisotropy prevents this unexpected type of ordering and leads to conventional order along the easy axis.