Anisotropic magnetic-field response of quantum critical fluctuations
in Ni-doped CeCoIn
This paper demonstrates the anisotropic response of quantum critical fluctuations with respect to the direction of the magnetic field in Ni-doped CeCoIn by measuring the magnetization and specific heat . The results show that at for both the tetragonal and directions exhibits dependencies, and that at follows a function, which are the characteristics of non-Fermi-liquid (NFL) behaviors. For , both the and dependencies change into nearly temperature-constant behaviors by increasing , indicating a crossover from the NFL state to the Fermi-liquid state. For , however, the NFL behavior in persists up to , whereas exhibits temperature-independent behavior for . These contrasting characteristics in and reflect the anisotropic nature of quantum critical fluctuations; the -axis spin component significantly contributes to the quantum critical fluctuations. We compare this anisotropic behavior of the spin fluctuations to superconducting properties in pure CeCoIn, especially to the anisotropy in the upper critical field and the Ising-like characteristics in the spin resonance excitation, and suggest a close relationship between them.
The role of spin fluctuations in unconventional superconductivity is a long-standing subject in the physics of strongly correlated electron systems. The unconventional superconducting (SC) phase commonly emerges in the vicinity of magnetic orders in many strongly correlated electron systems, such as high- cuprates, FeAs-based alloys, and heavy fermion compounds. In particular, the heavy fermion compounds often exhibit SC order proximity to a magnetic quantum critical point (QCP), corresponding to a magnetic phase transition at zero temperature. Hence, quantum critical fluctuations that are enhanced around the QCP are expected to play a critical role in the SC order of the heavy fermion compounds.
Among the heavy fermion superconductors, CeCoIn has attracted continuous interest because of its anomalous SC properties coupled with magnetic correlations rf:Petrovic2001 (). This compound has a HoCoGa-type tetragonal structure [Fig. 1(b), inset] and exhibits a SC order below . The magnetically mediated pairing mechanism of the SC order is inferred from the -wave () symmetry of the SC gap rf:Izawa2001 (); rf:An2010 (); rf:Park2008 (). The inelastic neutron scattering experiments have revealed that a resonance excitation involving the tetragonal -axis spin component develops in the SC state rf:Stock2008 (); rf:Raymond2015 (); rf:Song2016 (); rf:Mazzone2017 (); rf:Stock2018 (); rf:Eremin2008 (); rf:Chubukov2008 (); rf:Michal2011 (). Furthermore, applying the magnetic field yields another SC phase that coexists with an incommensurate antiferromagnetic (AFM) modulation (the so-called phase) at very low temperatures below 0.3 K and at high fields just below for rf:Bianchi2003-1 (); rf:Kakuyanagi2005 (); rf:Young2007 (); rf:Kenzelmann2008 (); rf:Aperis2008 (); rf:Agterberg2009 (); rf:Yanase2008 (); rf:Yanase2009 (). All of these features indicate a close coupling between the anisotropic spin correlations and the SC state, but the nature of the spin correlations with respect to CeCoIn has not yet been fully uncovered.
A key to clarifying the relationship between the spin correlations and the anomalous SC properties is expected to be found in the field-induced non-Fermi-liquid (NFL) behaviors observed under when applied along the axis. At , the specific heat divided temperature exhibits dependence, and both electrical resistivity and magnetization follow nearly -linear functions down to very low temperatures rf:Bianchi2003-2 (); rf:Paglione2003 (); rf:Tayama2002 (). It is widely believed that spin fluctuations enhanced near an AFM QCP are responsible for these NFL behaviors rf:Paglione2003 (); rf:Bianchi2003-2 (); rf:Tokiwa2013 (). In fact, substituting the ions for elements in CeCoIn, such as Nd for Ce rf:Hu2008 (); rf:Raymond2014 (), Rh for Co rf:Zapf2001 (); rf:Yoko2006 (); rf:Yoko2008 (); rf:Ohira-Kawamura2007 (), and Cd, Hg, and Zn for In rf:Pham2006 (); rf:Nicklas2007 (); rf:Yoko2014 (); rf:Yoko2015 (), can induce long-range AFM orders. Moreover, possible field-induced AFM ordering at extremely low temperatures () has been proposed by a recent quantum oscillation measurement for pure CeCoIn rf:Shishido2018 ().
In contrast, the substitutions of Sn for In rf:Bauer2005 (); rf:Bauer2006 (); rf:Ramos2010 () and Ni for Co rf:Otaka2016 () do not induce the AFM phase, but simply yield paramagnetic ground states through the suppression of the SC phase. In a recent study, we have revealed that in the mixed compound CeCoNiIn, the SC transition temperature monotonically decreases from 2.3 () to 0.8 K () with increasing ; subsequently, the SC order disappears above the critical Ni concentration rf:Otaka2016 (). At this concentration, the NFL behaviors are realized around the zero field, characterized by the dependence in the specific heat divided by the temperature, the weak diverging behavior in the magnetization, and the nearly -linear behavior of the electrical resistivity rf:Otaka2016 (). These NFL features are quite similar to those seen in pure CeCoIn, strongly suggesting that the NFL anomaly observed in Ni-doped CeCoIn also originates from the AFM quantum critical fluctuations. Furthermore, the effective magnetic moment for , estimated from the Curie-Weiss law at high temperatures, is nearly independent of and coincides well with that calculated from the multiplet in the Ce ion rf:Otaka2016 (), suggesting that the Ce 4 electrons are mainly responsible for the magnetic properties in pure and Ni-doped CeCoIn.
The observation of the NFL behavior at the zero field in Ni-doped CeCoIn provides an opportunity to investigate the magnetic anisotropy of the quantum critical fluctuations. In pure CeCoIn, in contrast, it is difficult to perform such an investigation with typical macroscopic measurements, because the quantum critical behavior is suppressed (or masked) by the SC phase at low magnetic fields and is visible only at very low temperatures above ( for and for ) rf:Ronning2005 (); rf:Hu2012 (). Consequently, the magnetic anisotropy of the quantum critical fluctuations remains unclear. In this paper we demonstrate the anisotropic changes of the NFL behaviors in the magnetization and specific heat between and in CeCoNiIn, and we discuss the relationship between the anisotropic spin fluctuations and the SC properties in pure and Ni-doped CeCoIn.
Ii Experiment Details
A single crystal of CeCoNiIn with was grown using an Indium flux technique, the details of which are described elsewhere rf:Otaka2016 (). The energy dispersive x-ray spectroscopy (EDS) and the inductively coupled plasma mass spectrometry (ICP-MS) measurements for the sample indicated that the actual Ni concentration approximately coincided with the starting (nominal) value within the deviation of , including the experimental error. Furthermore, through the EDS measurements, we confirmed the homogeneous distributions of the elements in the single crystal prepared for the experiments. The magnetization along the and axis was measured in temperatures as low as 0.1 K and in the magnetic field () at up to 8 T with a capacitively detected Faraday force magnetometer rf:Sakakibara94 (). A commercial SQUID magnetometer (MPMS, Quantum Design) was used for the magnetization measurements in the temperature range of 2.0–300 K and the magnetic field at up to 5 T. The specific heat was measured in the temperature range of 0.31–4 K and in the field range of 0–7 T with a conventional quasiadiabatic technique.
Figures 1(a) and 1(b) show the temperature dependencies of the - and -axis magnetization divided by the magnetic field respectively. Note that the data are plotted with logarithmic scales for both the vertical and horizontal axes. for both directions showed qualitatively similar features. Namely, at the lowest field () exhibited diverging behavior with a function () as the temperature decreased. The dependence in was realized in a very wide temperature range of 0.1–10 K for both directions. It is natural to conclude that this NFL behavior originates from the quantum critical fluctuations, because similar NFL behaviors are also found in various macroscopic quantities in pure CeCoIn rf:Bianchi2003-2 (); rf:Paglione2003 (); rf:Tokiwa2013 () and its doped alloys rf:Bauer2005 (); rf:Bauer2006 (); rf:Yoko2017 (). In both the - and -axis magnetization, the diverging feature was reduced by further applying , and the -constant behavior was then realized at low temperatures.
From a quantitative viewpoint, however, a significant anisotropy was found in the NFL region between the -axis and -axis magnetization and , respectively. The exponent of at in  was larger than the value  in , as the details of those evaluation procedures are described later. Furthermore, the magnitude of at and was twice that of . The diverging behavior in the temperature variation of was thus stronger than that of . Indeed, this feature can be verified by considering the variation of as a function of temperature (Fig. 2). exhibited a peak with a magnitude of at . The peak structure was also observed in the other physical quantities, such as the electrical resistivity rf:Petrovic2001 (), and its origin is considered a development of a coherent heavy-fermion state below this temperature. The value was reduced to with decreasing temperatures, down to . However, the spin fluctuations, associated with the NFL behavior, enhanced the value at low temperatures again; for increased with a decrease in temperature below and then reached 1.96(10) at 0.11 K. In a high magnetic-field region, in contrast, for and 8 T exhibited a saturation to the values of 1.55(3) and 1.46(2) at low temperatures, respectively, remaining with magnitudes comparable to those in high temperatures. These experimental results suggest that the NFL anomaly involved mainly the -axis spin component.
Figures 3(a) and 3(b) show the specific heat divided by the temperature obtained under various fields along the and axis, respectively. For , was markedly enhanced below for , although its temperature dependence became weak with increasing at high temperatures. This enhancement is considered to be caused by the Zeeman splitting of the nuclear spins. Such an effect should also be included in for . To eliminate this contribution, we estimated the nuclear Schottky anomaly by performing a calculation based on the natural abundance of the nuclear spins in the sample [Fig. 3(b), inset]. At and 0.4 K, the fraction of in was estimated to be 14% for and 12% for . Note that the contribution of Ni and Co nuclear spins was only 10% in ; therefore, the ambiguity of the Ni/Co concentration () in the sample is negligible in the estimation of .
Figures 4(a) and 4(b) display the specific heat data obtained by subtracting the nuclear spin contribution for and , respectively. for increased with decreasing temperature, with a nearly dependence at temperatures as low as 0.31 K. As displayed in Fig. 4(a), this feature was markedly suppressed by applying along the axis, and eventually became nearly independent of temperature at . The feature of suppression in coincides fairly well with that observed in [Fig. 1(a)]; hence, these behaviors are attributed to a crossover from the NFL to Fermi-liquid (FL) states.
However, it was found that the diverging behavior in was not suppressed as much by for . At 0.4 K, the reduction of the specific heat at , , was estimated to be 5% for , whereas it was 14% for . In addition, at continued to increase with the decreasing temperature for , whereas it was nearly independent of temperature for . Similar weak dependence of for was also found at a very high region above in pure CeCoIn rf:Ronning2005 (). This weak dependence in for is in stark contrast to the rapid reduction of with for the same direction; was markedly suppressed by applying and then became constant at low temperatures for [Fig. 1(b)]. These contrasting features in and for strongly suggest that the fluctuating spin component is perpendicular to the applied direction; that is, the -axis spin component significantly contributes to the quantum critical fluctuations in CeCoNiIn. This situation is similar to that expected in the Ising model with a transverse magnetic field, in which the transverse magnetic field does not align the spins but yields a quantum paramagnetic state with short-range spin correlations rf:Sachdev99 (). However, it should be remembered that the anisotropy of magnetic moments in the present system was not so strong that it can be regarded as simply the Ising-like anisotropy. In addition, the spins of the itinerant heavy quasiparticles, rather than the completely localized spins, were likely responsible for the quantum critical fluctuations. Hence, the deviation from the Ising-like characteristics of the magnetic moments would lead to a suppression of the quantum critical fluctuations and would then stabilize the FL state at a high region above , even for .
In Figs. 5(a) and 5(b) we summarize the exponent of at low temperatures for and , respectively. In these plots, was estimated using a simple relation: . The effect of the Van Vleck susceptibility may be included in the estimation of using an alternative formula: . However, we confirmed that the trends seen in Figs. 5(a) and 5(b) did not depend on the finite value, at least up to /T Ce, which is about half of the magnitude of the experimentally observed magnetic susceptibility at 300 K rf:Otaka2016 ().
As displayed in Fig. 5(a), the NFL state with governed the low region in the plane for , and the suppression of the NFL state at the high region was realized as a reduction of toward . A similar gradual suppression of the NFL behavior with was also observed in . In fact, when the characteristic temperature below which deviates from the function [see Fig. 4(a)] was plotted onto the image map of in Fig. 5(a), we found that the curve traced the contour of well. This consistency between and reflected the occurrence of the NFL-to-FL crossover for .
However, the situation for was very different, as shown in Fig. 5(b). The finite value for was rapidly suppressed by , and a large region was then distributed in the plane. In contrast, the curve entered deeply into the region. Note that for , did not exhibit the -constant behavior ascribed to the FL state, even below , although could be defined in the data. It is likely that the -axis spin component of the quantum critical fluctuations, which was not significantly influenced by for , led to this inconsistency between and for , as argued previously.
The present investigation of CeCoNiIn revealed the clear anisotropic response of the NFL behaviors in and with respect to the direction; the crossover from the NFL state to the FL state occurred for , whereas the NFL behavior persisted at least up to for . In this section, we compare this anisotropic NFL behavior to the SC properties in pure CeCoIn.
First, we find that the anisotropy concerning the stability of the SC phase in pure CeCoIn qualitatively coincides with that of the quantum critical fluctuations in Ni25%-doped CeCoIn. The SC order parameter in the pure compound has a characteristic temperature scale of , corresponding to . In addition, this SC state is broken at for , although it persists up to for rf:Ikeda2001 (). In the Ni25%-doped alloy, at takes on the NFL characteristics up to the same ranges as in the pure compound. This coincidence of the stability of the NFL and SC states concerning implies that the spin correlations, similar to those yielding the NFL behavior in the Ni25%-doped alloy, play a critical role in the occurrence of the SC order in the pure compound. If this is the case, such spin correlations should be concerned with the determination of through both the SC condensation energy and the paramagnetic spin susceptibility, yielding a Pauli paramagnetic effect rf:Yanase2008 (), because of CeCoIn is considered Pauli limited rf:Izawa2001 (); rf:Ikeda2001 ().
Second, it is remarkable that the -axis spin component is primarily responsible for the quantum critical fluctuations in Ni25%-doped CeCoIn. Indeed, such anisotropic spin fluctuations and excitations are also observed in the SC phase of pure CeCoIn. The recent inelastic neutron scattering experiments for pure and Nd-doped CeCoIn have revealed that the spin resonance excitation emerging in the SC phase has a nearly Ising nature along the axis rf:Raymond2015 (); rf:Mazzone2017 (). This similarity in the spin polarization suggests that the spin resonance excitation in pure CeCoIn and the NFL behavior in Ni-doped CeCoIn have similar origins. The spin fluctuations in Ni25%-doped CeCoIn may have an energy distribution centered at , because the NFL behavior at in and persists down to very low temperatures. However, once the SC order occurs, as in pure CeCoIn, the spin fluctuations may have gapped energy due to the SC condensation, detected as the spin resonance excitation in the inelastic neutron scattering measurements. In this situation, the coherency and Ising-like polarization of the spin fluctuations may be somewhat enhanced along with the variation of the ground state from the paramagnetic NFL state to the SC ordered phase. In fact, it has been demonstrated that the spin resonance excitation may condensate into AFM ordering rf:Song2016 (); rf:Stock2018 (); rf:Michal2011 (), supporting the above suggestion that the quantum critical fluctuations and the spin resonance excitation have similar origins because the quantum critical fluctuations likely originate from the AFM instabilities in pure CeCoIn and its doped alloys rf:Paglione2003 (); rf:Pham2006 (); rf:Yoko2017 ().
Despite the aforementioned considerations, the microscopic nature of the quantum critical fluctuations has not yet been uncovered. We believe that the relationship of the spin fluctuations between pure and Ni25%-doped CeCoIn would be clarified by comprehensive investigations using the inelastic neutron scattering technique on CeCoNiIn with a wide range. Such investigations could provide a key to understanding the anomalous SC properties coupled with the magnetic correlations in CeCoIn.
Our magnetization and specific heat measurements for CeCoNiIn revealed anisotropic NFL behavior, depending on the direction. For , the diverging behaviors in the temperature variations of and changed into nearly -constant behaviors, reflecting the NFL-to-FL crossover with increasing . For , however, the NFL behavior in persisted up to , although was sufficiently reduced with for . These anisotropic responses in and indicate that the quantum critical fluctuations are suppressed by the -axis magnetic field more effectively than by the -axis field because they are composed mainly of the -axis spin component. We compared this feature to the SC properties in pure CeCoIn, especially to the anisotropy in the upper critical field and the Ising-like characteristics in the spin resonance excitation, and suggested a close coupling between them.
Acknowledgements.We are grateful to Y. Oshima, I. Kawasaki, R. Otaka, and Rahmanto for their experimental support, and to D. Ueta and T. Masuda for their assistance with the specific-heat measurements prior to this study. M.Y. expresses gratitude to Y. Yanase for fruitful discussions. This study was supported in part by JSPS KAKENHI Grant Number 17K05529, and the research completed at ISSP was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “J-Physics” (15H05883) from JSPS.
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