Competing magnetic correlations in the putative ferromagnetic quantum critical system CeTiVGe: V NMR as a local probe
The V nuclear magnetic resonance (NMR), magnetization and specific heat studies on CeTiVGe have been performed to study the evolution from the ferromagnetic Kondo lattice state () to the antiferromagnetic state (), and in particular the emergence of a ”putative” ferromagnetic (FM) quantum critical point (FMQCP) at . From the temperature dependence of the relaxation rate, , and the Knight shift, K, for both the end materials it is concluded that there are considerable competitions between ferro- and antiferro-magnetic correlations. At the critical concentration () quantum critical spin-fluctuations consists of weak but finite antiferromagnetic (AFM) spin-fluctuations admixed with ferromagnetic spin-fluctuations. The putative ferromagnetic quantum critical point in CeTiVGe therefore is not purely ferromagnetic in origin.
pacs:76.60.-k, 75.50.Gg, 75.30.Et, 75.25.Dk
The generic Doniach type of phase diagram based on the interplay of Ruderman, -Kittel, -Kasuya, -Yoshida (RKKY) - and Kondo - interaction between well localized moments is valid strictly only for antiferromagnetic (AFM) isotropic RKKY-type of exchangeDoniach (). For 4 systems this is a prerequisite for the emergence of quantum criticality in general and the formation of a critical point (QCP) in particular. The tuning of AFM correlated 4-systems by an external control parameter (stoichiometry, pressure, magnetic field) towards the quantum critical point QCP is still an active field of correlated matter researchreview (); Gegenwart08 (); Si10 (). Recently some anisotropic Kondo systems gained a lot of interest due to the presence of ferromagnetic correlations in addition to the antiferromagnetic ones and the Kondo interaction. Despite sizable Kondo exchange long range ferromagnetic order is found frequently at low temperatures in theses systems. Therefore one issue in modern correlated matter physics is the search for a ferromagnetic quantum critical point (FMQCP) among 4- and 3- systemsBrando (). Many alloying and pressure studies are conducted so far on various systems but unfortunately due to the fragile interplay between FM and AFM correlations and under an anisotropic crystal electric field these system undergo other novel phase transitions and ”avoid” a FMQCP. For correlated Yb-systems FM order is rare and so far YbNiP is the only FM 4- sytem (with a of 170 mK) which could be tuned towards the FMQCPSteppke13 (). Among the Ce-systems for CeRuPOKrellner07 (), CeRuGeSuellow99 () CeAgSbSidorovat03 () or CeNiSbSidorov05 () FM order was found but upon the application of pressure long range AFM order is induced before approaching the putative FMCP. Furthermore it was shown that in CeFePOLausberg12 () and CePdRhWesterkamp09 () a glass-like state ”Kondo cluster” state forms which also avoids the tuning towards a FMQCP. The local NMR technique was applied successfully on various systems because it probes the local fluctuations and aims in particular to expose the real nature of the magnetic fluctuations (antiferromagnetic - versus ferromagnetic correlations) by temperature- and field- scaling of the spin lattice relaxation rate and the NMR shift . Especially NMR is able to disentangle between finite (AFM) and (FM) excitationsBaenitz13 (); Sarkar13 (). Simultaneously the NMR provides information about the degree of disorder in these alloyed systems which matters at the fragile critical point. For example in the heavy fermion CeFePO where long range order is absent NMR gave clear evidence for FM correlations admixed onto AFM correlations and Kondo interactionBruning08 () whereas in the structural homologue CeRuPO stable long range FM order is established by NMRKrellner07 () measurements. Also, Ru substitution in CeFePO tunes the system towards a critical point and ferromagnetic quantum critical fluctuations have been clearly observed by NMRKitagawa12 (). In contrast to that pressure studies at zero field on the pure CeRuPO reveal a crossover to an AFM ordered state without the formation of a putative FMQCPLengyel15 (). Here, it might encounter a complication by the applied magnetic field in the NMR. This is also supported by pressure experiments under magnetic field where features associated to FM ordering could be resolved. Therefore there is still a ongoing search for tunable new 4-based systems with ferromagnetic correlations as a platform for criticality.
Recently Kittler et al. succeeded to tune the FM ordering temperature of the FM Kondo lattice system CeTiGe ( 14 K) monotonously down to a putative FMQCP by V doping at the Ti site Kittler13 (); Thesis (). The suppression of is continuous with the V concentration, the structure type (hexagonal perovskite P63/ mmc) is preserved and no crossover to other phases is observed. Nonetheless, CeTiGe is a rather anisotropic ferromagnet and upon V-substitution the ratio is reduced and therefore the system is expected to be less anisotropic in it’s magnetic properties (regardless of any possible change of crystal electric field). Furthermore, the end member on the V-rich side CeVGe is known as an antiferromagnetic Kondo system with a of about K and a relatively strong Kondo interaction of K (see figure S5 in the supplemental materialSupplemental ()). It should be noted that in contrast to most of the systems discussed above the V-substitution in CeTiGe is not isoelectric. In Ce(Ti,V)Ge the tetravalent Ti is replaced by Vanadium which leads to a increase in carrier concentration. Therefore, the V-substitution leads to a change in both carrier concentration and the lattice constants that certainly has strong effects on both RKKY- and the Kondo- interactions, as well. Recent pressure studies on CeTiGe suggest an avoided FMQCP but these are isoelectronic studies which could not be completely be compared with V-dopingUdhara18 ().
Ii Experimental details
The powder samples of CeTiVGe ( = 0.113, 0.350, 0.405 and 1.000) used in this experiment were prepared by the same method as reportedKittler13 (). It is well known that these alloys crystallize to hexagonal perovskite (BaNiO-type) structure with P/mmc space group without any structural phase transitions throughout entire concentration. The V NMR has been observed by transient (pulsed) NMR technique with a commercial NMR spectrometer (TecMag Apollo). The MMR line profiles were obtained by integration of the spin-echo magnitude with sweeping external filed at a constant frequency. By simulating the line profile, the isotropic and anisotropic Knight sifts ( and ) have been extracted. The spin-lattice relaxation rate has been measured by the saturation recovery method where the recovery of nuclear magnetization was fitted to a stretched exponential function (see supplemental materialSupplemental ()).
In general, NMR deals with the nuclear spin moment and probes the hypefine field at the nuclear site which has static and dynamic components. The static part of the hypefine field can be measured by the NMR (Knight) shift, , which is related to the electron spin susceptibility, , with , where is the hyperfine coupling constant between the nuclear and the electron spins, and is a temperature independent contribution. In conventional metals is mainly composed by both the orbital interaction associated with non s-electrons and the Fermi contact interaction from -like electrons (). For non-cubic materials like in the present case, the Knight shift exhibits an anisotropic feature reflecting the anisotropy of their susceptibility. In these cases, we often encounter a characteristic NMR spectrum in powder sample owing to the distribution of the Knight shifts, called the powder pattern. For the uniaxial case, we could extract values of ( is parallel to the symmetry axis) and ( perpendicular to the symmetry axis) from the singularities of spectrum. Then isotropic and anisotropic Knight shifts, and , respectively, can be calculated by,
Note that in the present case corresponds to parallel to the c-axis () and to perpendicular to the c-axis ().
The fluctuating part of the hyperfine field is provided via the spin lattice relaxation rate given by the following expression,
where the sum is over the wave vectors within the first Brillouin zone, ) is the imaginary part of the transverse dynamical electron spin susceptibility and is the Larmor frequency for NMR. is the -dependent hyperfine form factor.
Iii Experimental results and discussion
Before discussing the compounds near to the FMQCP ( and 0.405) we present at first the results for mother compounds (AFM, and FM, ).
CeVGe () is the end compound in this series and known to be a heavy fermion itinerant antiferromagnet with 40 K and 6 K. The V NMR have been observed at a fixed frequency of 31.0 MHz (corresponds to the zero shift at = 2.77 T). The V NMR spectra at 8 K, the temperature dependence of and, the relaxation rate divided by temperature, , are depicted in Fig. 1(a), (b) and (d). In order to obtain the isotropic hyperfine coupling constant the observed are plotted as a function of the isotropic magnetic susceptibility setting the temperature as an implicit parameter in Fig. 1(c). From the slope and the y-intercept and were obtained to be +11.16 kOe/Ce- and 1.3%. Here, means that if Ce atom polarized 1 along the external field the V nucleus sees a hyperfine field of 11.16 kOe along the field direction (transferred hyperfine field).
Despite the fact that CeVGe is believed to be an itinerant antiferromagnet with almost quenched magnetic moment on Ce by the Kondo effect, , hence the static susceptibility , has a strong temperature dependence, meaning that has not only a peak at the antiferromagnetic (AF, ) wave vector but also it must have large ferromagnetic (FM) component at . As for the in Fig.1(d), the self-consistent renormalization (SCR) theory for weak itinerant AFM spin correlations predicts type divergence at . Although there is an anomaly at in , the experimental data cannot be fitted to the above formula. Therefore, combining the fact that has the Curie-Weiss (CW) type temperature dependence, we may argue that CeVGe is nominally AF but include a strong contribution of FM correlations, i.e. has rather flat structure extended to . This is considered to be the reason for the depression of the critical divergence at by the external field.
The other end member of the series is the sample, which undergoes FM ordering at about 9 K in zero field (see suppelemental material figure S5Supplemental ()). The V-powder NMR spectra shown in Fig. 2(a), has a large anisotropy ( at 20 K) which is consistent with the anisotropy in magnetization reported for single crystalsInamdar14 (). As is described below, the anisotropy tends to vanish towards the FMQCP but increases slightly again in CeVGe where the anisotropy is reversed as at 8 K.
The which was extracted from and is shown in Fig. 2(b) and in Fig.2(d), as well. One can immediately observed that both and have a characteristic temperature dependence towards the FM ordering temperature. Although we know that scales to (Korringa law) in uncorrelated or weakly correlated metals, in CeTiVGe has a linear dependence on as shown in the insert of Fig. 2(d). From this dependence CeTiVGe can be regarded as a weak itinerant ferromagnet and one can adopt the SCR theory for weak ferromagnets to understand qualitatively its magnetic properties. In the frame work of SCR theory, under an external field may be expressed as,
and well reproduce the experimental shown in Fig. 2(d) for = 6.25 T. Here, is a constant related to the area of the Fermi surface of the magnetic electrons, and is related to the spin-fluctuation parameter, , with the following equation.
The slope of ) vs. plot in the inset of Fig. 2(d) yields the value of to be 21 K. The deviation from 1 of the ratio () gives the degree of itineracy of the 4 electronsMajumder09 (); Majumder10 (); Majumder12 (). indicates that the enhancement of is not confined at but rather extends to a larger values. This situation is similar as in CeVGe, as described above, where we have argued that the dependence of ) has extended towards from the AF wave vector. Therefore, we can conclude that for both the end materials there are considerable competitions between FM and AFM correlations in the present system. This conclusion leads to the following discussion about quantum criticality for the materials in between.
As described above, both ferromagnetic and antiferromagnetic end materials have considerable mixing of ferro- and antiferro-magnetic correlations, meaning has extended to a large range of the -space between and ( being antiferromagnetic wave vector). Now we will discuss the nature of critical behavior in and 0.405 which are close to the putative FMQCP in the phase diagram of CeTiVGe. Here, the V NMR measurements have been performed under different applied magnetic fields. The spectra are less anisotropic compared to and as shown in Fig.3(a, b), and less temperature dependent. The temperature dependence of at 2.76 T and 6.25 T is shown in Fig. 3(c). (T) for both materials fits very well to the constant plus CW terms with the Weiss constant of nearly 18 K. The temperature dependence of the relaxation rate, (T), is shown in Fig.4(a) and (b) for and . As is observed immediately, (T) has a characteristic cross over from low temperature Korringa process () to the temperature independent process above a certain temperature, , which depends on field. The behavior above may be understood by the fact that upon increase of temperature the longitudinal spin density fluctuations saturate around and the transverse fluctuations dominate the spin dynamics. This is just like the case of local moment fluctuations in insulators (so called temperature induced local moment, TILM). In this case, we may evaluate the characteristic correlation time of the spin density fluctuations, , from the following equation analogical to the case for insulating magnets:
where is temperature independent value observed above , is the hyperfine coupling constant and is amplitude of the local spin density. Using the experimental values of , are evaluated and shown in Fig. 4(c) as a function of field. As is seen in Fig. 4(c), for the both samples increases linearly with field, indicating that the characteristic frequency for the spin correlations decreasing with external field, approaching towards a magnetic order. In order to see more details, we plot which is a measure of inverse as a function of temperature in Fig. 5(a) and (b) for and . For both cases (T) above behaves like the CW with the Weiss constant, , which increases as a function of applied magnetic field (figure 5(c)). The enhancement of as a function of field supports the fact that field flavoring magnetic order. for the lowest field for both the samples are very close to zero which indicates quantum critical spin-fluctuations. Note that the field dependent Curie constants correspond to the inverse of .
As is seen in Fig. 4(a) and (b), at temperatures well below the nuclear relaxation process is governed by the Korringa process where interacting electron-hole pair excitations are the primary source of the magnetic excitations. Treating the interaction of the quasiparticles within a random phase approximation, RPA, we have the modified Korringa relation expressed as, ), where is the universal constant with . Here, means the spin-fluctuations are enhanced around , leading to the predominance of FM correlations, and indicates that finite (typically AF) spin-fluctuations are predominantly enhanced. The estimated dependence of at 2 K (a temperature where the modified Korringa law is valid) is shown in the lower panel of Fig.6 which indicates that the dominating ferromagnetic correlations are smoothly reduced with increasing , the effective moment shows the same behavior.
Additional evidence for the decrement of dominant ferromagnetic correlations with V substitution have been further confirmed by Wilson ratio, , where R is the ideal gas constant, C is the Curie constant, is the low temperature susceptibility, and is the electronic specific heat coefficient. The value for and are estimated to be 9.7 and 3.7 at 5 K.
The evidence for the presence of weak AFM spin-fluctuations on the top of dominant FM spin-fluctuations for a system which is close to the FMQCP have also been seen recently in case of Ru doped CeFePO and the appearance of such AFM spin-fluctuations was discussed to be due to the Fermi surface instability which happens in case of local QCP. The presence of considerable AFM correlations even far away from the putative QCP (for sample) indicates the absence of pure QCP in this system. Furthermore, we have seen that the shift anisotropy reduces a lot in the systems towards the QCP and can be considered as a isotropic material which indicates the isotropic nature of local fields at the V site which excludes the possibility of local QCP because for local QCP a 2D XY type of anisotropy is a necessary condition. It is noteworthy that in case of Ca doped SrRuOYoshimura99 () and of MnSiThessieu98 () a continuous change of has been seen like in V doped CeTiGe (present case). A continuous behavior of , , and towards putative QCP found in CeTiVGe have been seen.
It should be mentioned that unusual slow fluctuating glass-like electronic phases near ferromagnetic quantum criticality have been proposed by theory due to the competing interactionsNussinov09 () and in our case competition between dominating FM and weak AFM correlations might induce such states which prevents the formation of a pure FMQCP. But from NMR we did not find any pronounced signature of a glassy-like inhomogeneous state like strong line broadening. It has also been mentioned in theory for an itinerant system that in three dimensional systems the FMQCP is unstable, and the system will undergo a first order phase transition or into a commensurate phaseChubukov04 (). The critical behavior at low temperatures rules out the presence of first order character in the present system but a commensurate state might form near to the FMQCP. It is worth mentioning that the specific heat measurement at zero magnetic field and down to 100 mK of the sample with shows no ordering and follows logarithmic divergences (which we have also seen), but the sample shows a clear signature of bulk ordering (further justifying the absence of strong disorder) at 1 KThesis (). This implies that due to presence of quantum critical FM spin-fluctuations admixed with AFM ones, the system enters to a novel state of matter. To clarify the nature of such state extensively, one needs to do more rigorous NMR study in highly pure single crystals towards very low temperatures and at low fields or to apply other low field techniques like SR.
To conclude, V NMR, magnetization and specific heat measurements have been performed on polycrystalline CeTiVGe systems. NMR as a local probe provides informations about magnetic fluctuations accorss the phase diagram. CeVGe shows a temperature induced local moment behavior above K and a lack of critical fluctuations above . The temperature dependence of of the system with ( at 6.4 T) can be well explained by self consistent renormalization theory for itinerant ferromagnetic systems. At the critical concentration () quantum critical spin-fluctuations consist of weak but finite antiferromagnetic (AFM) spin-fluctuations admixed with ferromagnetic spin-fluctuations. The spin fluctuation parameter and (which probes the strength of FM spin fluctuations) have been estimated for and 0.405 which shows a continuous suppression towards . Surprisingly the critical samples lack the finger print of a pure FMQCP, the NMR power lawMajumder16 () is absent. Instead of that we have strong evidence for the presence of AFM spin fluctuations. From the microscopic NMR study presented here we state that the system is unique and that the putative FMQCP is not a pure one and the correlations involved are rather complex.
We thank M. Brando and C. Geibel for fruitful discussions. Furthermore, we thank H. Rave, C. Klausnitzer and R. Hempel-Weber for technical support.
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