Field-dependent collective ESR mode in YbRh 2 Si 2

Electron spin resonance experiments in YbRh 2 Si 2 Kondo lattice ð T K C 25K Þ at different ﬁeld/ frequencies ð 4 : 1 r n r 34 : 4GHz Þ and H ? c revealed: (i) a strong ﬁeld dependent Yb 3 þ spin–lattice relaxation, (ii) a weak ﬁeld and T -dependent effective g -value, (iii) a suppression of the ESR intensity beyond 15% of Lu-doping, and (iv) a strong sample and Lu-doping ð r 15 % Þ dependence of the ESR data. These results suggest that the ESR signal in YbRh 2 Si 2 may be due to a coupled Yb 3 þ -conduction electron resonant collective mode with a subtle ﬁeld-dependent spins dynamic.


Introduction
The heavy-fermion (HF) Kondo lattice YbRh 2 Si 2 ðT K C25 KÞ is an antiferrromagnetic (AF, T N ¼ 70 mK) tetragonal (I4/mmm) intermetallic compound. At low-T ðTtT K Þ the magnetic susceptibility exhibits a HF behavior and at high-T ðT\200 KÞ an anisotropic Curie-Weiss with a full Yb 3þ magnetic moment ðm eff C4:5m B Þ is observed [1][2][3]. The AF ordering of YbRh 2 Si 2 may be driven to T N % 0 by fields of H ?c $650 Oe and H Jc $7 kOe [4]. At these fields a quantum critical point (QCP) is observed with non-Fermi-liquid (NFL) behavior [1,2]. Therefore, among other systems [5,6], YbRh 2 Si 2 is particularly an interesting system to study quantum criticality and NFL behavior. Electron spin resonance (ESR) experiments at low-T ðTt20 KÞ in YbRh 2 Si 2 by Sichelschmidt et al. [7] have reported on a narrow ð1002200 OeÞ single dysonian resonance with no hyperfine components, T-dependence of the linewidth, DH, and a g-value anisotropy consistent with Yb 3þ in a metallic host of tetragonal symmetry. As in the early work of Tien et al. [8] the observation of a narrow Yb 3þ ESR in the intermediate valence compound of YbCuAl and, recently, in a dense Kondo system below T K were unexpected results [7]. Nevertheless, various reports were already published on the ESR of Yb 3þ in stoichiometric YbRh 2 Si 2 [9], YbIr 2 Si 2 [10], and YbRh 2 Si 2 doped with non-magnetic impurities as Ge [11] and La [12]. Also, the ESR of Ce 3þ in dense Kondo systems was communicated [13]. However, it is not clear yet what mechanism allows the observation of the Yb 3þ ESR line with local magnetic moment features in these highly correlated electron systems.
The main purpose of this work is to investigate the H-dependent ESR data in the NFL phase of YbRh 2 Si 2 (4:2tTt10 K; 0oHt10 kOe). We found an unexpected H-dependence behavior of the Yb 3þ ESR data in YbRh 2 Si 2 that, we hope, will contribute to the understanding of the observed ESR signal in this system.

Experiment
Single crystals of Yb 1Àx Lu x Rh 2 Si 2 (0rxt1:00) were grown from In and Zn-fluxes as reported elsewhere [14][15][16]. The structure and phase purity were checked by X-ray powder diffraction. The high quality of our undoped crystals was confirmed by X-rays rocking curves which revealed a mosaic structure of maximum c-axis angular spread of t0:015 3 . The electrical residual resistivity ratio, r 300 K =r 1:9 K , for the In and Zn-flux grown crystals were 35 and 10, respectively [14][15][16]. For the ESR spectra $2 Â 2 Â 0:5 mm 3 single crystals were used. The ESR experiments were carried out in a Bruker S, X, and Q-bands (4.1, 9.5 and 33:8 GHz) spectrometer using appropriated resonators and T-controller systems. A single anisotropic resonance, with no hyperfine components, from the Kramer doublet ground state was observed at all bands. The dysonian lineshape ðA=B % 2:5Þ corresponds to a microwave skin depth smaller than the size of the crystals [17]. Fig. 1 shows the YbRh 2 Si 2 ESR X-band spectra at 4.2 K and H ?c for single crystals grown in In and Zn-fluxes. From the anisotropy of the field for resonance, H r ðyÞ (not shown), one can obtain the angular-dependence of the effective g-value which is given by hn=m B H r ðyÞ ¼ gðyÞ ¼ ½g 2

Results and discussions
?c cos 2 y þ g 2 Jc sin 2 y 1=2 . From the fitting of the experimental H r ðyÞ we obtain g Jc t0:6ð4Þ and g ?c ¼ 3:60ð7Þ. The inset of Fig. 1 displays, for H ?c , the Korringa-like [18,19] linear thermal broadening of the linewidth, DHðTÞ ¼ a þ bT, and the fitting parameters for the crystal grown in Zn-flux at X and Q-bands. As it will be shown below, the large values measured for a and b indicates that Zn impurities were incorporated in this crystal. Fig. 2 presents the relative normalized X-band integrated ESR intensities for Yb 1Àx Lu x Rh 2 Si 2 at 4.2 K and H ?c as a function of x, I 4:2 ðxÞ=I 4:2 ð0Þ. The ESR intensities were determined taking into consideration the crystal exposed area, skin depth and spectrometer conditions. These results show that, while for xr0:15 the Yb 3þ ESR intensity is nearly constant, for 0:15oxt1:00 the intensity vanish completely and no ESR could be detected. It is worth mention that for x40:15 and T\200 K, w ?c ðTÞ follows an Curie-Weiss law with a full Yb 3þ magnetic moment and that for xr0:15 there is no appreciable changes in the thermodynamic properties of these compounds [14,15]. The absence of resonance for x40:15 strongly suggests that the observed ESR for xo0:15 cannot be associated to a single Yb 3þ ion resonance but rather to a resonant collective mode of exchange coupled Yb 3þ -ce (conduction electrons) magnetic moments. We argue that a strong Yb 3þ -ce exchange coupling may broadens and shifts the ce resonance toward the Yb 3þ resonance allowing their overlap and building up a Yb 3þ -ce coupled mode with possibly bottleneck/dynamic-like features. Evidences for bottleneck/ dynamic-like features in the Lu-doped crystals will be published elsewhere. An internal field caused by the Yb 3þ local moments may be responsible for the shift of the ce resonance [21]. Moreover, the Lu-doping may disrupt the collective mode coherence and, probably, may also opens the bottleneck/ dynamic regime [22].    measured at S, X and Q-bands for H ?c . In this T-interval, and within the error bars, it is also found that DH ¼ a þ bT for the three bands. This suggests a Korringa-type of mechanism for the Yb 3þ spin-lattice relaxation (SLR), i.e. the Yb 3þ local moment is exchange coupled to the conduction electrons [18]. The residual linewidth, a, and relaxation-rate, b ¼ DH=DT, are given in Fig. 3a.
Notice that the minimum relaxation rate, b, is found at X-band, HC1900 Oe. The actual determination of the residual linewidth, a ¼ DHðT ¼ 0Þ, would require measurements at lower-T, therefore, the obtained values should be consider just as fitting parameters.
A H-dependent SLR-rate b is not expected for a normal local magnetic moment-ce exchange coupled system, where the Korringa-rate is frequency/field independent [19]. However, since the Yb 3þ and ce magnetic moments carry unlike spins and different g-values, the H-dependence of b may be an anomalous manifestation of a bottleneck-like behavior. Fig. 3b shows that the T-dependence of the effective g-values are slightly different in the three bands, with minimum effective g 4:2 -values also at the X-band. The effective g-value accuracy is much higher than that obtained from hn=m B H r ðyÞ ¼ gðyÞ ¼ ½g 2 ?c cos 2 y þ g 2 Jc sin 2 y 1=2 because proper experimental conditions were chosen for these H ?c measurements. Fig. 4 displays the H-dependence of b and g 4:2 -values. Notice that both parameters have minimum values at the X-band field, HC1900 Oe. Fig. 5 shows the X-band DHðTÞ for 4:2rTr21 K and H ?c . The data were fitted to DHðTÞ ¼ a þ bT þ cd=½expðd=TÞ À 1 taking into consideration all the contributions to DH in a metallic host. The 1st and 2nd terms are the same as above. The 3rd is the relaxation, also via an exchange interaction with the ce, of a thermally populated Yb 3þ excited crystal field state at d K above the ground state [23]. The fitting parameters are in the inset of Fig. 5. This analysis does not consider any direct Yb 3þ spin-phonon contribution [23]. The S and Q-band DHðTÞ data for 7tTt20 K also show exponential behaviors with c % 200ð70Þ Oe=K and d % 75ð20Þ K.
Within a molecular field approximation the effective gðTÞ may be written as g eff ¼ gð1 þ lw ?c ðTÞÞ. Fig. 6 presents a plot of Dg=g ¼ ðg eff ðTÞ À gð15ÞÞ=gð15Þplw ?c ðTÞ for Tr15 K and X-band for our x ¼ 0 crystals [15] and that from Refs. [7,24]. A linear correlation is obtained with l values in the interval of À2 kOe=m B 4l4 À 3 kOe=m B . In the Appendix, it is shown that the shift that gives the temperature dependence of g eff arises from anisotropic exchange interactions between the Yb 3þ ions. Fig. 7 shows the comparison between theory and experiment. Therefore, these results definitely indicate that the T-dependence of the effective gðx; TÞ is nothing but a consequence of the shift of the    field for resonance toward higher fields due to an AF internal molecular field and has nothing to do with a Dg ¼ c=lnðT K =TÞ divergence [7]. Moreover, the expected g-shift caused by the exchange interaction between the Yb 3þ and ce local moments, J fce , can be estimated from the largest Korringa-rate value measured at Q-band in our In-flux crystal, b ffi 45 Oe=K. Within a single band approximation [23] and absence of q-dependence of the Yb 3þ -ce exchange interaction, J fce ðqÞ ffi J fce ð0Þ, [26] one can write where in a normal metal the SLR-rate, b, and g-value depend on the competition between the Korringa/Overhauser relaxation and the ce SLR [18,29,19,28]. Then, the increase of b by the addition of non-magnetic impurities to YbRh 2 Si 2 (Lu, Zn in Fig. 1 and La in Ref. [12]), may be associated to ''opening'' the bottleneck regime due to the increase in the ce spin-flip scattering [19,28]. The results and discussion about the non-magnetic Lu impurities effects and bottleneck behavior will be the subject of a forthcoming publication. Another striking result reported in Fig. 4 is the non-monotonic H-dependence of b and effective g-value of the Yb 3þ -ce resonant collective mode. Admixtures via Van Vleck terms [30] may be disregarded because this contribution should scale with H. Therefore, we believe that the low H-tunability [1][2][3] of the ESR parameters in YbRh 2 Si 2 is an ''intrinsic'' property of the NFL state near a QCP, where the strength of the Yb 3þ -ce magnetic coupling may be subtly tuned and allows the formation of the resonant collective mode. We attribute the absence of low H-dependent ESR results in previous reports [7] to the presence of ''extrinsic'' impurities and/or Rh/Si defects [15,31] that increase the SLR, b, and residual linewidth, a. Thus, as for our Zn-flux crystals, hiding the low H-dependence of the ESR parameters in the NFL phase.
In the bottleneck scenario, the Yb 3þ -ce resonant collective mode presents the strongest bottleneck regime (smallest b) at H % 1900 Oe. However, due to the subtle details of the coupling between the Kondo ions and the ce in a Kondo lattice and to strong impurity effects, these resonant collective modes may not be always observable, unless extreme bottleneck regime is achieved. The proximity to a QCP and/or the presence of enhanced spin susceptibility may favor this condition [13,32]. The bottleneck scenario for the Yb 3þ -ce resonant collective mode may also explain the absence of Yb 3þ hyperfine ESR structure [33].
Recent calculations by Abrahams and Wolfle have suggested that the ESR linewidth may be strongly reduced by a factor involving the heavy fermion mass and quasiparticle ferromagnetic (FM) exchange interactions ðm=m Ã Þð1 ÀŨ~w þÀ ff ;H ð0ÞÞ [34]. These results indicate that the estimation of the linewidth from the Kondo temperature, T K ðDH ¼ k B T K =gm B Þ is an over estimation. However, these calculations may not be contemplating all the possibilities and have to be taken with care when applied to the dynamic of the ESR of YbRh 2 Si 2 compound because, (i) it presents an AF Yb 3þ 2Yb 3þ exchange interaction (although other works in literatures have claimed in favor of the existence of FM fluctuation in YbRh 2 Si 2 [35,36] and (ii) samples with the same thermodynamic properties present quit different linewidths (see Fig. 1). Furthermore, the anisotropy in the ESR in YbRh 2 Si 2 reflects both single-ion crystal field effects and the Yb 3þ -Yb 3þ and Yb 3þ -ce interactions. In principle, the analysis of crystal field effects is straightforward although somewhat hindered by the inability to detect a signal when the field is along the c-axis. The anisotropy of the Yb 3þ -Yb 3þ and Yb 3þ -ce is more difficult to determine and in the latter case more critical. The application of the resonant collective mode model is based on the assumption that the Yb 3þ -ce coupling is dominated by a scalar interaction between the Yb 3þ ground state doublet pseudo-spins S ps , and the spins of the conduction electrons, s, with the consequence that the total spin s þ S ps is (approximately) a constant of the motion [37,38]. In the presence of uniaxial anisotropy, only the component of the total spin along the symmetry axis is a constant of the motion. How the lower symmetry affects the formation of the collective mode is an unsolved problem requiring further study [34]. Finally, we hope that our results motivate new theoretical approaches to understand the dynamics of strong exchange coupled magnetic moments of unlike spins and g-values, as Yb 3þ and ce, and explore the general existence of a resonant collective mode with a bottleneck/dynamic-like behavior.

Summary
In summary, this work reports low H-dependent ESR, below T K C25 K, in the NFL phase of YbRh 2 Si 2 ðTt10 KÞ. It is suggested that the observed ESR in YbRh 2 Si 2 corresponds to a Yb 3þ -ce resonant collective mode in a strong bottleneck-like regime, which is highly affected by the presence of impurities and/or defects. The analysis of our data allowed us to give estimations for the Yb 3þ 2Yb 3þ exchange parameter, J ff , and a lower limit for the Yb 3þ -ce exchange parameter, jJ fce j.
In applying this equation to the resonance in YbRh 2 Si 2 it must be kept in mind that the g-factors and susceptibilities refer to the ground state doublet. We assume that the relevant doublet susceptibilities take the form C ?;J =ðT þ y ?;J Þ where C ? and C J are constants and fit the experimental data for w ?c to the form w 0 þ C ?c =ðT þ y ?c Þ in the temperature range of the resonance experiment (see inset of Fig. 7) with the result y ?c ¼ 1:48 K. We chose y Jc and the overall amplitude g 0 ?c as adjustable parameters. The measured values g ?c ðTÞ were then fit to the function: g ?c ðTÞ ¼ g 0 ?c 1 À y ?c À y Jc T þ y ?c 1=2 ð3Þ % g 0 ?c 1 À 0:5 y ?c À y Jc T þ y ?c with the best fit given by g 0 ?c ¼ 3:66 and y Jc ¼ 1:09 K (see Fig. 7).
The result shown in Fig. 6 is in accord with Eq. (4) and Fig. 7 since the T-dependent part of w ?c ðTÞ is proportional to ðT þ y ?c Þ À1 , the same factor that is present in Eq. (4). The results outlined in the preceding paragraph show that the T-dependence of g ?c ðTÞ is associated with the difference in the longitudinal and transverse exchange interaction which is reflected in the difference between y ?c and y Jc . Although it has not been possible to observe the resonance with the static field along the c-axis, there is a corresponding shift there as well, which takes the form: Note that the shift in g Jc ðTÞ is in the opposite direction from the shift in g ?c ðTÞ.