Muon spin rotation and relaxation in PrNdOsSb:
Magnetic and superconducting ground states
Muon spin rotation and relaxation (SR) experiments have been carried out to characterize magnetic and superconducting ground states in the PrNdOsSb alloy series. In the ferromagnetic end compound NdOsSb the spontaneous local field at positive-muon () sites below the ordering temperature is greater than expected from dipolar coupling to ferromagnetically aligned Nd moments, indicating an additional indirect RKKY-like transferred hyperfine mechanism. For , spin relaxation rates in zero and weak longitudinal applied fields indicate that static fields at sites below are reduced and strongly disordered. We argue this is unlikely to be due to reduction of Nd moments, and speculate that the Nd- interaction is suppressed and disordered by Pr doping. In an sample, which is superconducting below K, there is no sign of “spin freezing” (static Nd magnetism), ordered or disordered, down to 25 mK. Dynamic spin relaxation is strong, indicating significant Nd-moment fluctuations. The diamagnetic frequency shift and spin relaxation in the superconducting vortex-lattice phase decrease slowly below , suggesting pair breaking and/or possible modification of Fermi-liquid renormalization by Nd spin fluctuations. For , the SR data provide evidence against phase separation; superconductivity and Nd magnetism coexist on the atomic scale.
pacs:74.70.Tx, 75.30.Mb, 75.40.-s, 76.75.+i
Rare-earth-based materials are in many ways ideal for studies of the interaction between superconductivity and magnetism in metals. An example is the filled skutterudite family of isostructural lanthanide intermetallics MBHH09 (), where the unconventional heavy-fermion superconductor PrOsSb and its alloys have been the subject of considerable interest BFHZ02 (); MFHY06 (). The isomorph NdOsSb is a ferromagnet with Curie temperature K SSNS03 (); HYBF05 (); MHYH07 (). In order to elucidate the interplay between the superconductivity of PrOsSb and the ferromagnetism of NdOsSb, the alloy system PrNdOsSb has been investigated.
Figure 1 gives the phase diagram obtained from thermodynamic and transport measurements HYYD11 (), together with transition temperatures from our muon spin rotation and relaxation (SR) data as discussed in Sec. III.
Superconductivity persists up to , and Nd “spin freezing” (static magnetism, with or without long-range order) appears above . This is evidence for competition between superconductivity and magnetism for the ground state of the system, as well as the possibility of a ferromagnetic quantum critical point near –0.5. The rate of decrease of the superconducting transition temperature with Nd concentration is nearly the same in the (PrNd)OsSb and Pr(OsRu)Sb alloy systems HYBY08 (), contrary to the behavior of “conventional” superconductors where magnetic impurities are much more effective than nonmagnetic ones in suppressing . There is evidence that Nd substitution does not affect the Pr CEF splitting nearly as strongly as Ru doping MHYH07 ().
A number of aspects of superconductivity and magnetism on the atomic scale are readily accessible to the SR technique. Early SR studies of PrOsSb revealed two important phenomena: the absence of nodes in the superconducting energy gap MSHB02 (), and the onset of a spontaneous internal magnetic field below the superconducting transition temperature ATKS03 () that indicated broken time-reversal symmetry in the superconducting state. Later SR experiments found no change of the muon Knight shift below HSKO07 (), suggesting -wave pairing, resolved a discrepancy between inductive and SR penetration-depth measurements SMBH09 (), ruled out a second phase transition in the superconducting state MHMS10c (), and studied the effects of La and Ru doping AHSO05 (); ATSK07 (); Shu07d (); SHAH11 ().
This article reports SR experiments in the PrNdOsSb alloy series, undertaken with the goal of providing microscopic characterization of the ground states of these systems. Their paramagnetic states have also been investigated via SR Knight shift measurements, the results of which will be reported in a companion article 111P.-C. Ho et al. (unpublished).. Three major regions of Nd concentration have been found in experiments to date HYYD11 (): (local-moment ferromagnetism, disordered in the alloys), (coexistence of superconductivity and ferromagnetism?), and (coexistence of superconductivity with Nd local moments). After a brief description of the SR experiments in Sec. II, Sec. III reports and discusses the results from alloys in each of these regions. Conclusions are given in Sec. IV. The principal results of this work are (1) the discovery of reduced and strongly disordered static magnetic fields below at stopping sites in the alloys, (2) the absence of any static Nd magnetism, ordered or disordered, for above 25 mK, and (3) microscopic-scale coexistence of superconductivity and magnetism for .
SR experiments were carried out at the M15 beam line of the TRIUMF accelerator facility, Vancouver, Canada, using a dilution refrigerator to obtain temperatures in the range 25 mK–3 K. Millimeter-sized PrNdOsSb single crystals, , 0.45, 0.50, 0.55, 0.75, and 1.00, were grown using the self-flux technique BSFS01 (). Solid solubility is good across the alloy series HYBY08 (); HYYD11 (). The samples were characterized by x-ray diffraction and magnetic susceptibility measurements. Each SR sample consisted of a mosaic of crystals glued to a mm 6N Ag plate using GE 7031 varnish, which was attached to the cold finger of the cryostat with a thin layer of Apiezon grease. The crystals were partially oriented, as their (100) faces tended to be parallel to the mounting plate, but the cubic symmetry renders most averages over sites (e.g., dipolar field averages) independent of orientation. To ensure isothermal conditions an Ag foil was wrapped around the sample and firmly attached to the cold finger. Magnetic fields in the range 0–200 Oe were applied.
In time-differential SR experiments spin-polarized positive muons () are implanted in a sample, precess in the local fields at the interstitial sites, and decay with a mean lifetime s via the reaction . The decay positrons are emitted preferentially in the direction of the spin, and are detected using scintillation counters. This asymmetry yields the dependence of the spin polarization on time between implantation and muon decay. Typically events are obtained for a single measurement of .
In transverse-field SR (TF-SR) the applied field , which is usually the dominant field at sites, is perpendicular to the initial spin orientation. The spins precess, and the positron count rate for a given direction oscillates with time at a frequency near 222Frequency shifts from this value (e.g., the Knight shift in metals) are due to internal fields in the sample.; here kHz/G is the gyromagnetic ratio. In a disordered (inhomogeneous) static field the precession frequencies are distributed and the oscillations are damped, or do not appear at all if the disorder is sufficiently strong.
In longitudinal-field SR (LF-SR) the spins are initially oriented parallel to the applied field , and hence precess only in local fields generated by the sample such that the resultant field is not parallel to . Zero-field SR (ZF-SR) can be considered a limiting case of LF-SR. Here again, any distribution of local-field magnitudes results in damping of the oscillation expected from a unique field, although functions for LF- and TF-SR are quite different.
In both cases, thermal fluctuations of the local fields give rise to dynamic relaxation (the spin-lattice relaxation of NMR). This is the only relaxation mechanism for the components parallel to local static fields observed in ZF- and LF-SR.
For a given positron counter, the count rate is related to by
(ignoring uncorrelated background counts), where is the initial count rate, is the lifetime, is the initial count-rate asymmetry (spectrometer-dependent but typically 0.2), and the polarization function is the projection of the time-dependent ensemble spin polarization (normalized to 1 at ) on the direction to the counter. Various experimental configurations (applied field direction, counter orientations, etc.) yield specific forms for , from which information on static and dynamic magnetic properties of the sample can be extracted. Such configurations, and the information they reveal in the case of PrNdOsSb alloys, are discussed below in Sec. III. More details of the SR technique can be found in a number of monographs and review articles Sche85 (); Brew03 (); Blun99 (); LKC99 (); YaDdR11 ().
In a ZF-SR experiment in a multidomain or polycrystalline magnet with a sufficient number of randomly oriented domains or crystallites, the static local fields are oriented randomly with respect to the initial direction of the spin polarization . At each site the spin has a nonprecessing “longitudinal” component parallel to of magnitude , where is the angle between and . The projection of this longitudinal component back along the initial spin direction (the axis of the spectrometer counter system) is therefore . For randomly oriented the angular average , leading to a “1/3” longitudinal contribution to the spin polarization in zero applied field KuTo67 (); *HUIN79. The “2/3” transverse contribution oscillates if is reasonably constant. Any distribution of field magnitudes (i.e., of precession frequencies) leads to damping of the oscillation. This leaves only the 1/3 component at late times, which is time independent if there are no other relaxation mechanisms; otherwise it relaxes dynamically. Hence dynamic and static relaxation can be separated in ZF-SR (and weak-LF-SR) experiments if the former is sufficiently slow compared to the latter.
In nonzero longitudinal field the resultant field is no longer randomly oriented. Then is greater than 1/3, and approaches 1 asymptotically as ; the spins are “decoupled” KuTo67 (); *HUIN79 from their local fields. In the absence of dynamic relaxation, one expects a polarization function of the form
where describes the distribution of precession frequencies, and , the fraction of longitudinal component, is a monotonically increasing function of Prat07 ().
Since we will be dealing with disordered spin systems in the following sections, we consider polarization functions appropriate to such cases. In zero applied field the static Gaussian Kubo-Toyabe ( KT) function KuTo67 (); *HUIN79
describes the polarization when the distribution of each Cartesian component of is Gaussian with zero mean and rms width . The Gaussian KT function models relaxation in dipolar fields from a densely populated lattice of randomly oriented moments, electronic or nuclear, when the moment magnitudes are fixed but their orientations are random on the atomic scale. The Gaussian distributions result from the central limit theorem of statistics, since a given muon is coupled to many lattice moments. For randomly located sites in a lattice with a low concentration of moments, however, the wings of the distribution are dominated by single nearby moments, and the distribution of field components becomes nearly Lorentzian WaWa74 (); UYHS85 (). This leads to the “Lorentzian KT” ZF-SR polarization function , where is the half-width of the field distribution Kubo81 (); UYHS85 (). Clearly the form of the polarization function changes appreciably between the dilute and concentrated limits.
Noakes and Kalvius NoKa97 () have approached the problem of relaxation in the intermediate-dilution regime by generalizing the KT result phenomenologically, assuming that in a moderately diluted lattice the KT distribution width is itself distributed. The physical meaning of this approach is discussed further in Sec. IV. A Gaussian distribution of with mean and rms width results in the “Gaussian-broadened Gaussian” (GbG) KT ZF-SR polarization function NoKa97 ()
where is the effective spin relaxation rate and is the ratio of the distribution widths.
GbG KT polarization functions are shown in Fig. 2 together with the Lorentzian KT function.
The ratio parameterizes the departure from single-Gaussian KT behavior NoKa97 (). For , and with increasing the minimum near decreases in depth. For does not approach the Lorentzian function, so that is not an interpolation function between the Gaussian and Lorentzian limits. Instead, becomes monotonic and nearly independent of ; for this reason the condition was imposed in fits of Eq. (4) to the data. For small , independent of for all .
Often the experimental SR asymmetry signal has a contribution from muons that miss the sample and stop in the mounting plate or cryostat cold finger. This plate is usually silver, for good thermal contact and also because Ag nuclear moments are small and spin relaxation in Ag is negligible. In the following the plots of ZF- and LF-SR data give the polarization function for the sample rather than the total asymmetry . The relation between these quantities is
where is the total initial asymmetry and is the fraction of muons that stop in the silver.
Iii Results and Discussion
In the end compound NdOsSb the onset of ferromagnetism below the Curie temperature K gives rise to a spontaneous internal field at sites, and therefore to oscillations in ZF-SR and weak-LF-SR. Figure 3 shows early-time data taken in a longitudinal field Oe,
which decouples the moments from nuclear dipolar fields above (cf. Sec. II), but has little effect in the ferromagnetic state; the observed internal fields are two orders of magnitude greater than except near . The data are well fit by the damped Bessel function
where is the zeroth-order cylindrical Bessel function, is the (dynamic) longitudinal spin relaxation rate, is the fraction of longitudinal signal component, describes the damping of the oscillation, is the initial phase, and is the dominant value of the local-field magnitude distribution. A damped cosine function does not provide as good a fit.
The choice of the Bessel function is motivated by the Fourier-transform relation between the polarization function and the distribution of precession frequencies. In an incommensurate single- sinusoidal spin-density wave, the precession frequency distribution is , the Fourier transform of which is the Bessel function AFGS95 (). The distribution is characterized by a singularity at and a broad distribution of lower frequencies. Thus represents the polarization when the experimental frequency distribution has this character (inset of Fig. 3). An actual incommensurate spin-density wave in NdOsSb is ruled out by the considerable evidence for a ferromagnetic ground state HYBF05 (). Broadening of the singularity is accounted for by damping the Bessel function [Eq. (III.1)].
spin polarizations in NdOsSb at various temperatures below are shown in Fig. 4, together with damped Bessel-function fits.
Figure 5 gives the temperature dependencies of , , and .
The dominant frequeny exhibits an order-parameter-like increase below . At 0.80 K is small but finite (cf. Fig. 4), indicating that is slightly higher than this. The damping rate , being considerably larger than , is mainly due to static disorder. It is much smaller than until approaches from below; here it increases slightly, suggesting a spread of transition temperatures [See; e.g.; ]MVBdR94. The dynamic rate increases as from above due to critical slowing down of Nd spin fluctuations 333In the motionally narrowed limit the rate is proportional to the correlation time, followed by a decrease below as the Nd moments freeze. The behavior of all these quantities is that of a conventional ordered magnet.
The value of at low temperatures is G. Assuming the saturation moment found from magnetization isotherms HYBF05 (), the experimental -Nd coupling constant is . For comparison we have calculated the dipolar coupling tensor from ferromagnetically aligned Nd moments , assuming muons stop at the probable site as found in PrOsSb ATKS03 (). The field at this site is given by . The principal axes of are parallel to the crystal axes; the principal-axis values and the corresponding values of for are given in Table 1.
|Crystal axis||site coordinate|
In general three frequencies are expected from the three inequivalent sites in the cubic structure for nonero .
All the are smaller in magnitude than the observed field, so that a significant RKKY-like transferred hyperfine interaction between Nd spins and spins is necessary to account for the difference. The interaction strength required to do this depends on the orientation of , which is not known at present. Broadened unresolved resonances corresponding to the 370-G spread in principal-axis dipolar fields ( spread in angular precession frequencies) might contribute to the spectral weight below the peak in (inset of Fig. 3).
In PrNdOsSb alloys the Pr ions are in nonmagnetic crystal-field ground states, resulting in a substitutionally diluted lattice of Nd ions 444But see the discussion in Sec. IV.. For Pr doping has reduced the magnetic transition temperature from 0.8 K to 0.55 K HYYD11 (). Although there has been no direct confirmation of ferromagnetic order in the diluted alloys, the Nd concentration dependence of the “Curie” temperature and paramagnetic-state properties in the alloys tend smoothly towards their values in NdOsSb as HYYD11 (). In PrNdOsSb the paramagnetic Curie-Weiss temperature is positive and .
Figure 6 shows early-time weak-LF-SR spin polarization data from PrNdOsSb at mK and Oe.
Compared to data from NdOsSb at 25 mK (Fig. 4) the oscillation is almost completely damped, indicating a broad distribution of local fields. The deep minimum of the Gaussian KT function (Fig. 2) is not observed, and we have therefore fit the GbG KT polarization function [Eq. (4)] to the data. As in Sec. III.1, we take dynamic relaxation into account via an exponential damping factor:
The fit of Eq. (7) to the data at 25 mK, the early-time portion of which is shown in Fig. 6, is tolerable but not perfect; it is, however, considerably better (reduced ) than that of a number of other candidate functions as follows.
The damped cosine and damped Bessel functions discussed in Sec. III.1 ( and 1.39, respectively).
The “-function/Gaussian” function LFGK00 ()
The “power KT” function CrCy97 ()
The “Voigtian KT” function [See; e.g.; ]MSKG10
The functions in (1) and (2) above oscillate with defined nonzero frequencies, whereas the last two are phenomenological interpolations between the Gaussian and Lorentzian KT functions. The poor fits to the oscillating functions are clear evidence that for the spread in is considerably greater than the average.
Below increases in an order-parameter-like fashion with decreasing temperature, to at 25 mK. This is 20% of in NdOsSb at the same temperature, to be compared with the much smaller decrease in transition temperature; . The change in polarization behavior between and 1 is drastic, and the average frequency for should not be compared in detail to the spread in frequencies for . Nevertheless, the difference for this relatively light Pr doping is quite striking. In the neighborhood of varies rather smoothly without an abrupt transition, suggesting an effect of nonzero and/or an inhomogeneous spread of transition temperatures. The inset of Fig. 7 shows that is nearly constant (0.6) at low temperatures, jumping suddenly to 1 near .
The longitudinal rate behaves similarly in the and samples, exhibiting a cusp near the magnetic transition. For the sample, however, the cusp occurs at a temperature well below , where has increased to more than 60% of its low-temperature value. This behavior may also be due to a distribution of transition temperatures. It should be noted, though, that Eq. (7) characterizes the entire polarization function and hence the entire sample volume; the good fits to this function are thus evidence against macroscopic inhomogeneity or phase separation.
iii.3 PrNdOsSb, , 0.50, and 0.55
In the concentration range the transition from static magnetism to superconductivity is occurring, perhaps with coexistence of the two phases on the microscopic scale and with the possibility of one or more quantum critical points near . Figure 8 shows early-time spin polarization data for alloys with , 0.50, and 0.55.
The data for and 1.00, discussed above, are repeated for comparison. As is the case for PrNdOsSb (Sec. III.2), the data are fit best by the damped GbG KT function [Eq. (7)]. The two-component structure associated with a distribution of quasistatic local fields is present but in attenuated form, since the damping rate is large in these alloys.
Figure 9 gives the temperature dependencies of , , and for , 0.50, and 0.55.
Both rates increase significantly below 0.3–0.5 K. The increase of indicates the onset of a distribution of quasistatic local fields in this temperature range. The transitions occur close to temperatures determined from ac susceptibility measurements HYYD11 ().
In this Nd concentration range is suppressed significantly compared to in NdOsSb, continuing the trend found in PrNdOsSb (Sec. III.2). It can be seen in Fig. 9 that never exceeds at low temperatures. This corresponds to a spread of 65 Oe in fields at sites, about 11% of the average field in NdOsSb. A decrease in is expected due to the dilution of the Nd moment concentration Noak91 () but not to this extent, as discussed in Sec. IV. The low-temperature values of do not vary monotonically with , but exhibit a marked minimum for . The width ratios [inset of Fig. 9(a)] behave remarkably similarly for the , 0.50, and 0.55 alloys: at low temperatures , and then increases toward 1 at 0.25 K more continuously than for (inset of Fig. 7). The dynamic relaxation rates [Fig. 9(b)] differ considerably from the corresponding data for and 1 in that there is no sign of a peak at or near and the rates remain large down to 25 mK.
Figure 10 shows LF-SR spin polarization in PrNdOsSb at mK for in the range 15–821 Oe.
This alloy is superconducting below a transition temperature K from ac susceptibility measurements, compared to K for the end compound PrOsSb HYBY08 (); HYYD11 (). The suppression of superconductivity by Nd doping has been discussed HYYD11 () in two alternative scenarios: two-band superconductivity, as found in PrOsSb MBFS04a (); SBMF05 (), and the Fulde-Maki multiple pair-breaking theory FuMa66 (). The main issues addressed by SR experiments are therefore the magnetism associated with Nd moments, and its effect on the superconductivity of this alloy.
iii.4.1 Transverse Field
TF-SR experiments were carried out in PrNdOsSb in an applied field . Damped oscillations were observed both above and below . The asymmetry data [ in Eq. (1)] were fit to a cosine polarization function with combined Gaussian and exponential damping:
together with an undamped oscillation from muons that stopped in the silver plate and cold finger. Combined Gaussian and exponential damping is necessary to fit TF-SR in PrOsSb MSHB02 ().
Figure 11 shows weak-TF-SR asymmetry data obtained from PrNdOsSb for Oe, at 1.604 K (above ) and 25 mK (well below ).
The signal from muons that did not stop in the sample has been subtracted. It can be seen that at 25 mK the precession frequency decreases and the damping rate increases markedly compared to 1.604 K.
The value of for the surrounding silver serves as a reference, and is also plotted in Fig. 12(a). The decrease in and increase in in the superconducting state of the sample are expected from the diamagnetic response and the field distribution in the vortex lattice, respectively. But normally these changes begin much more abruptly just below , as is observed in PrOsSb MSHB02 (); MHMS10c ().
The curve in Fig. 12(b) is a fit of the relation
where the temperature dependence of the superconducting-state Gaussian rate is modeled by
and is the normal-state Gaussian rate due to nuclear dipolar fields. The two contributions are added in quadrature because the nuclear dipolar fields that give rise to are randomly oriented and uncorrelated with the vortex-lattice field. The power law used to model [Eq. (10)] is merely to indicate the form through the exponent , and has little physical significance, although is found in an early two-fluid phenomenology. The fits yield , much smaller than the usual values (3–4) in conventional superconductors. The exponential rate , which is quite significant, is not approximately constant, as in PrOsSb SMAT07 (), but increases below and exhibits an inflection point at 0.5 K.
There is no evidence for “freezing” of the Nd moments. Static magnetism of full Nd moments would affect both the muon precession frequency and the damping much more strongly than observed. Further evidence for the lack of spin freezing is discussed below.
iii.4.2 Zero Field
In most conventional superconductors the only ZF-SR relaxation mechanism is provided by nuclear dipolar fields at sites. These are not affected by superconductivity, so that the relaxation rate is constant through the transition. The spin relaxation is then well fit by the KT function [Eq. (3)]. In PrOsSb, however, it was necessary SMAT07 () to use the exponentially damped Gaussian KT function
The exponential damping was attributed to dynamic fluctuations of hyperfine-enhanced Pr nuclear moments SMAT07 (). Equation (11) models the case where the local fields fluctuate around static averages rather than around zero 555This relaxation is longitudinal, since it characterizes the relaxation of components of spins parallel to their time-averaged local fields.. Alternatively, the “dynamic KT function” KuTo67 (); *HUIN79, appropriate when the local fields fluctuate as a whole around zero with a fluctuation rate , might be considered.
Figure 13 shows the ZF spin polarization in PrNdOsSb at a number of temperatures in the range 25 mK–2.50 K.
As in PrOsSb, the data can be well fit with Eq. (11) at all temperatures, with a significant contribution by the exponential damping. The dynamic KT function scenario seems unlikely, assuming that decreases with decreasing temperature. If the overall relaxation of the dynamic KT function would decrease monotonically with decreasing KuTo67 (); *HUIN79, contrary to observation (Fig. 13). If the relaxation is in the motionally narrowed limit and the rate increases with decreasing , but then the relaxation would be exponential at all temperatures KuTo67 (); *HUIN79, again contrary to observation. We conclude that the damped static KT function is the better choice.
The temperature dependencies of and are shown in Fig. 14.
In contrast to the transverse-field Gaussian rate [Fig. 12(b)], the zero-field Gaussian rate shows almost no temperature dependence through the superconducting transition down to 25 mK. The normal-state values of and are similar and essentially the same as in PrOsSb, consistent with their attribution to Sb nuclear dipolar fields ATKS03 (); SMAT07 (). The dynamic rate in the normal state is also essentially the same as in PrOsSb, but increases dramatically below 0.4 K, indicating the onset of a new relaxation mechanism at this temperature.
We note that the observed constant here and in PrNdOsSb are in contrast to the increase of observed in PrOsSb below and attributed to broken time-reversal symmetry in the superconducting state ATKS03 (); SHAH11 (). Nd doping appears to have restored time-reversal symmetry in the superconductivity of the alloys. This seems somewhat paradoxical, since in the BCS theory spin scattering of conduction electrons breaks time-reversed Cooper pairs.
In Fig. 15 the temperature dependencies of for and for Oe are compared.
Although both relaxation rates show upturns in the region 0.4–0.5 K, they are clearly different: is larger than , and exhibits a jump at 666 is a purely dynamic rate that describes “lifetime broadening,” and thus is always a lower bound on .. These features are discussed in more detail in Sec. IV.
Further characterization of the unusual spin relaxation behavior observed in ZF-SR was obtained from LF-SR experiments in PrNdOsSb at and 200 Oe, mK. These fields are an order of magnitude larger than that needed to decouple the field distribution width of 1 Oe determined from the ZF relaxation rates KuTo67 (); *HUIN79, and hence would completely suppress the relaxation if it were due solely to static fields. But Fig. 16 shows that spin relaxation in these fields is considerable, even though reduced by the field.
The LF polarization function is subexponential, i.e., exhibits more upward curvature than an exponential function, for Oe. This signals an inhomogeneous distribution of dynamic relaxation rates, with the initial slope of giving the average rate, and slowly relaxing regions dominating at late times after the rapidly relaxing regions have lost their spin polarization. The power exponential function
fits the LF data well (curves in Fig. 16 for and 200 Oe), yielding a so-called stretched exponential John06 () (, inset of Fig. 16). Although this form is phenomenological and has no theoretical significance, it often fits subexponential data well. For consistency, power-exponential damping of the KT function
was used to fit the ZF data. The resulting value of for is close to 1 (inset of Fig. 16), justifying our previous use of simple exponential damping in this case, but decreases with increasing field. The value of in the stretched exponential is not the average rate but a rough characterization of the relaxation; is the time at which has decreased to of its initial value. Different values of should not be compared if is also varying, since then the shape of the polarization function is changing.
Magnetic transition temperatures obtained from weak-LF-SR in PrNdOsSb, and the superconducting transition from TF-SR in the sample are in good agreement with previous results HYYD11 () (Fig. 1). In the latter sample, however, there is no sign of a transition in other data at 0.5 K, where a marked increase in ZF-SR dynamic relaxation is seen (Fig. 14). Specific-heat and other measurements on a sample in this temperature range would be desirable.
For all Nd concentrations, the SR spectra (after subtraction of the background Ag signal) exhibit either a single component (TF-SR) or the two-component structure that is intrinsic to LF-SR. Furthermore, the total sample asymmetry is observed not to change within 5% at the transitions, magnetic or superconducting; there is no “lost asymmetry.” Thus there is no evidence in our data for phase separation anywhere in the phase diagram.
Magnetism in NdOsSb.
The weak-LF-SR data from NdOsSb reveal a fairly standard ferromagnet, with evidence of disorder from the damped Bessel-function form of the polarization function below . A distribution of magnitudes is necessary for this damping; a random distribution of field directions with fixed field magnitude results in an undamped oscillation in the polarization function. The most likely origin of this distribution is disorder in Nd orientations in the ferromagnetic state. We note, however, that as discussed below much broader spreads of are observed in Pr-diluted alloys, the mechanism for which might play a role in NdOsSb.
The observation of a peak at in the dynamic relaxation rate indicates critical slowing down of Nd spin fluctuations as the transition is approached from above, and thus is evidence that the transition is second order. Critical slowing down is somewhat in disagreement with the conclusion that the transition is mean-field-like HYBF05 (), however, since in that case the critical region around would be expected to be quite small.
Magnetism in PrNdOsSb: reduced moments or reduced coupling strengths?
The markedly reduced value of (and hence ) and the striking difference in polarization function compared to NdOsSb (Sec. III.2), are the salient results of SR experiments in PrNdOsSb. Noakes Noak91 (); Noak99 () has reported Monte Carlo calculations of distributions under various conditions of random site dilution, moment direction, and moment magnitude. For fixed moment magnitudes and random moment orientations he found that with increased dilution the spin polarization function retains the Gaussian KT form with its deep minimum (Fig. 2) down to moment concentration 0.5. This is clearly not observed in PrNdOsSb (Fig. 6). Noakes showed Noak99 () that an inhomogeneous distribution of moment magnitudes can give rise to the shallower minimum exhibited by the GbG KT function [Eq. (4), ; cf. Fig. 2] NoKa97 ().
The usual mechanism for local-moment suppression in metals is the Kondo effect, which for ions is normally observed only in Ce-, Yb-, and (very occasionally) Pr-based materials. This restriction to the ends of the lanthanide series is well understood [See; e.g.; ]Hews93: the energy difference between the level and the Fermi energy increases with increasing atomic number. This decreases the effective exchange interaction due to -conduction electron hybridization at the Fermi surface, to which the Kondo temperature is exponentially sensitive, and thus quenches the Kondo effect 777A similar argument involving holes applies at the high- end of the lanthanide series.. An inhomogeneous Kondo effect may be involved in Pt- and Cu-doped CeNiSn, where the GbG polarization function also fits the ZF-SR data well KFTK97 (). To the authors’ knowledge, however, Kondo screening has never been observed in Nd compounds.
As noted above, the Pr ions are in nonmagnetic crystal-field ground states, but it should not be assumed that they are magnetically inert. Exchange interactions, mediated by a RKKY-like indirect mechanism, might admix magnetic excited CEF states into the nonmagnetic Pr ground state so that they contribute to . It is hard to see how would be reduced by this effect, however.
is a sum of terms, each of which is proportional to the squared product of the static Nd moment magnitude and the Nd- coupling strength Abra61 (). An alternative mechanism for variation and reduction of might invoke a negative contribution of admixed Pr states to the indirect Nd- coupling. Variation of moment magnitudes and coupling strengths seem indistinguishable because of the product form, and the latter has the advantage of avoiding a mysterious reduction of Nd moments. As far as we know, such a mechanism has not been addressed theoretically.
Magnetism in PrNdOsSb, .
The SR experiments in this concentration range are dominated by effects of Nd magnetism; there is no sign in the data of a superconducting contribution to . The superconducting state with broken time-reversal symmetry found in PrOsSb is not found in these alloys.
The pattern of good fits to the GbG polarization function with reduced values of is continued. Figure 17 shows the dependence of on Nd concentration for the alloys (including , where vanishes, and ),
together with the experimental precession frequency and the maximum calculated precession frequency assuming dipolar interactions only (Sec. III.1) in NdOsSb.
In general the initial curvature of is related to the precession frequency distribution: , where is the Van Vleck second moment of the high-field resonance line HUIN79 (); Abra61 (). This result is independent of the functional form of the static field distribution, as long as the second moment is defined 888The second moment diverges for the Lorentzian distribution.. For the GbG KT function NoKa97 () (Sec. II), so that . Furthermore, in a diluted lattice of randomly oriented moments the second moment is proportional to the moment concentration 999Ref. Abra61 (), Chap. IV.: . A lattice-sum calculation of for dipolar coupling of 1.73 Nd moments to spins at sites in NdOsSb yields . The curve in Fig. 17 gives (), a lower limit for the expected relaxation rate since an additional RKKY-based -Nd transferred hyperfine interaction is present (Sec. III.1). The observed values of fall significantly below this curve.
At 25 mK depends significantly on , exhibiting a marked minimum at (Fig. 17). This behavior is also hard to understand: in general would be expected to track , i.e., should decrease monotonically with decreasing in the neighborhood of (Fig. 1). The minimum suggests that the moment suppression is related to an approach to quantum criticality at .
The behavior of the dynamic relaxation at low temperatures differs significantly between the , 0.50, and 0.55 alloys and the higher-concentration materials. In the former the rate remains large as [Fig. 9(b)] rather than decreasing with decreasing temperature below as in the alloy and NdOsSb (Figs. 7 and 5, respectively). This behavior signals the persistence of spin fluctuations to low temperatures. Normally the amplitude of thermal fluctuations decreases at low temperatures with the decreasing population of low-lying excitations such as spin waves. Persistent spin dynamics (PSD) such as seen here are often found in geometrically frustrated spin liquids [See; e.g.; ]LMM11p79, but only occasionally in systems with long-range order [E.g.; ]PZZA12. PSD may be related to the absence of spin freezing in the alloy, discussed below. They might also be considered as a source of the reduced static fields (Fig. 17), except that this is also seen in the alloy where PSD are absent (Fig. 7).
PrNdOsSb: superconductivity and no spin freezing.
As noted above, the observation that the TF-SR asymmetry does not change in the superconducting state (Fig. 11) is strong evidence that the latter occupies essentially all of the sample; there is no evidence for separation of superconducting and magnetic phases.
From Fig. 14 the ZF-SR Gaussian KT rate does not deviate from its average by more than , i.e., the field does not change by more than 0.8 G over the entire temperature range. This is even more compelling evidence against static Nd magnetism than the TF-SR data (Fig. 12), since frozen Nd moments would produce local fields much greater than 0.8 G. An estimate of such a field can be obtained from the calculated rms dipolar relaxation rate (curve in Fig. 17), which is G for . Since this calculation uses the measured saturation moment of /Nd ion in NdOsSb, the data suggest an upper limit on any static moment of in PrNdOsSb.
SR is sensitive to the onset of static magnetism independent of the degree of order; the technique is as applicable to spin glasses as to ordered magnets. Sche85 (); Brew03 (); Blun99 (); LKC99 (); YaDdR11 (). The absence of evidence for static Nd magnetism, ordered or disordered, down to 25 mK in PrNdOsSb is perhaps the most surprising result of this study. If were to scale as the Nd concentration the freezing temperature would be 0.2 K; its suppression by at least an order of magnitude is extraordinary.
In PrNdOsSb the values at of the diamagnetic shift and Gaussian relaxation rate are of the same order of magnitude as in the end compound PrOsSb for comparable . In the London limit (penetration depth length ) the rms width of the vortex-lattice field distribution in a conventional superconductor is related to the London penetration depth by , where is the flux quantum Bran88 (). Assuming that is approximated by the TF-SR Gaussian relaxation rate , at least near 101010This approximation is well obeyed in PrOsSb (Ref. SMBH09 ())., the zero-temperature fit value of yields Å, compared to 3610 Å in PrOsSb.
The increase below of the diamagnetic shift and are much faster in superconducting PrOsSb MHMS10c () than for (Fig. 12). One possible mechanism for this behavior is strong pair-breaking by Nd spins. A more speculative possibility is that the mass renormalization characteristic of heavy-fermion superconductivity Varm85 (), which affects the temperature dependence of the penetration depth VMS-R86 (), is strongly modified by interaction with fluctuating Nd moments.
The increases of both and below (Fig. 15) indicate a marked effect of superconductivity on the Nd spin dynamics. As noted in Sec. III.1, an increase in relaxation rate with decreasing temperature signals slowing down of spin fluctuations. The observation that is not surprising, since a contribution to relaxation from a static field distribution is possible in TF-SR but not for the 1/3 component of ZF-SR. The extra transverse relaxation may reflect inhomogeneity in the vortex lattice. The increases of both and below 0.4–0.5 K may be due to a further reduction of the Nd spin fluctuation rate at a crossover or transition, although there is no anomaly in in PrNdOsSb at this temperature HYYD11 ().
SR data from the PrNdOsSb alloy series reveal a disordered reduction by Pr doping of the spontaneous static local field due to static Nd-ion magnetism that is is well beyond that expected from dilution (Fig. 17). Kondo-like reduction of Nd moments is highly unlikely, suggesting by default an effect involving the Nd- coupling strength. The absence of static moments (upper limit ) in a sample down to 25 mK may be due to a related suppression of indirect Nd-Nd exchange. The origins of these phenomena remain unclear, and future work is called for to elucidate their mechanisms.
Acknowledgements.We are grateful to the Centre for Material and Molecular Sciences, TRIUMF, for facility support during these experiments. Thanks to J. M. Mackie and B. Samsonuk for assistance with data taking and analysis, and to Y. Aoki and W. Higemoto for useful discussions. This research was supported by the U.S. National Science Foundation, Grants No. 0422674 and No. 0801407 (UC Riverside), No. 0802478 and No. 1206553 (UC San Diego), No. 1104544 (CSU Fresno), and 1105380 (CSU Los Angeles), by the U.S. Department of Energy, Grant No. DE-FG02-04ER46105 (UC San Diego), by the National Natural Science Foundation of China (11204041), the Natural Science Foundation of Shanghai, China (12ZR1401200), and the Research Fund for the Doctoral Program of Higher Education of China (2012007112003) (Shanghai), and by the Japanese MEXT (Hokkaido).
- (1) M. B. Maple, R. E. Baumbach, J. J. Hamlin, P.-C. Ho, L. Shu, D. E. MacLaughlin, Z. Henkie, R. Wawryk, T. Cichorek, and A. Pietraszko, in Properties and Applications of Thermoelectric Materials, NATO Science for Peace and Security Series B: Physics and Biophysics, edited by V. Zlatić and A. C. Hewson (Springer Netherlands, 2009) pp. 1–18
- (2) E. D. Bauer, N. A. Frederick, P.-C. Ho, V. S. Zapf, and M. B. Maple, Phys. Rev. B 65, 100506(R) (2002)
- (3) M. Maple, N. Frederick, P.-C. Ho, W. Yuhasz, and T. Yanagisawa, J. Supercond. Novel Magn. 19, 299 (2006)
- (4) H. Sato, H. Sugawara, T. Namiki, S. R. Saha, S. Osaki, T. D. Matsuda, Y. Aoki, Y. Inada, H. Shishido, R. Settai, and Y. Onuki, J. Phys.: Condens. Matter 15, S2063 (2003)
- (5) P.-C. Ho, W. M. Yuhasz, N. P. Butch, N. A. Frederick, T. A. Sayles, J. R. Jeffries, M. B. Maple, J. B. Betts, A. H. Lacerda, P. Rogl, and G. Giester, Phys. Rev. B 72, 094410 (2005)
- (6) M. B. Maple, Z. Henkie, W. M. Yuhasz, P.-C. Ho, T. Yanagisawa, T. A. Sayles, N. P. Butch, J. R. Jeffries, and A. Pietraszko, J. Magn. Magn. Mater. 310, 182 (2007)
- (7) P.-C. Ho, T. Yanagisawa, W. M. Yuhasz, A. A. Dooraghi, C. C. Robinson, N. P. Butch, R. E. Baumbach, and M. B. Maple, Phys. Rev. B 83, 024511 (2011)
- (8) P.-C. Ho, T. Yanagisawa, N. P. Butch, W. M. Yuhasz, C. C. Robinson, A. A. Dooraghi, and M. B. Maple, Physica B 403, 1038 (2008)
- (9) D. E. MacLaughlin, J. E. Sonier, R. H. Heffner, O. O. Bernal, B.-L. Young, M. S. Rose, G. D. Morris, E. D. Bauer, T. D. Do, and M. B. Maple, Phys. Rev. Lett. 89, 157001 (2002)
- (10) Y. Aoki, A. Tsuchiya, T. Kanayama, S. R. Saha, H. Sugawara, H. Sato, W. Higemoto, A. Koda, K. Ohishi, K. Nishiyama, and R. Kadono, Phys. Rev. Lett. 91, 067003 (2003)
- (11) W. Higemoto, S. R. Saha, A. Koda, K. Ohishi, R. Kadono, Y. Aoki, H. Sugawara, and H. Sato, Phys. Rev. B 75, 020510 (2007)
- (12) L. Shu, D. E. MacLaughlin, W. P. Beyermann, R. H. Heffner, G. D. Morris, O. O. Bernal, F. D. Callaghan, J. E. Sonier, W. M. Yuhasz, N. A. Frederick, and M. B. Maple, Phys. Rev. B 79, 174511 (2009)
- (13) D. E. MacLaughlin, A. D. Hillier, J. M. Mackie, L. Shu, Y. Aoki, D. Kikuchi, H. Sato, Y. Tunashima, and H. Sugawara, Phys. Rev. Lett. 105, 019701 (2010)
- (14) Y. Aoki, W. Higemoto, S. Sanada, K. Ohishi, S. R. Saha, A. Koda, K. Nishiyama, R. Kadono, H. Sugawara, and H. Sato, Physica B 359-361, 895 (2005)
- (15) Y. Aoki, T. Tayama, T. Sakakibara, K. Kuwahara, K. Iwasa, M. Kohgi, W. Higemoto, D. E. MacLaughlin, H. Sugawara, and H. Sato, J. Phys. Soc. Jpn. 76, 051006 (2007)
- (16) L. Shu, Ph.D. dissertation, University of California, Riverside (2007)
- (17) L. Shu, W. Higemoto, Y. Aoki, A. D. Hillier, K. Ohishi, K. Ishida, R. Kadono, A. Koda, O. O. Bernal, D. E. MacLaughlin, Y. Tunashima, Y. Yonezawa, S. Sanada, D. Kikuchi, H. Sato, H. Sugawara, T. U. Ito, and M. B. Maple, Phys. Rev. B 83, 100504 (2011)
- (18) P.-C. Ho et al. (unpublished).
- (19) E. D. Bauer, A. Ślebarski, E. J. Freeman, C. Sirvent, and M. B. Maple, J. Phys.: Condens. Matter 13, 4495 (2001)
- (20) Frequency shifts from this value (e.g., the Knight shift in metals) are due to internal fields in the sample.
- (21) A. Schenck, Muon Spin Rotation Spectroscopy: Principles and Applications in Solid State Physics (A. Hilger, Bristol & Boston, 1985)
- (22) J. H. Brewer, “Muon spin rotation/relaxation/resonance,” in digital Encyclopedia of Applied Physics, edited by G. L. Trigg, E. S. Vera, and W. Greulich (WILEY-VCH Verlag GmbH & Co KGaA, 2003)
- (23) S. J. Blundell, Contemp. Phys. 40, 175 (1999)
- (24) S. L. Lee, S. H. Kilcoyne, and R. Cywinski, eds., Muon Science: Muons in Physics, Chemistry and Materials, Scottish Universities Summer School in Physics No. 51 (Institute of Physics Publishing, Bristol & Philadelphia, 1999)
- (25) A. Yaouanc and P. Dalmas de Réotier, Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter, International series of monographs on physics (Oxford University Press, New York, 2011)
- (26) R. Kubo and T. Toyabe, in Magnetic Resonance and Relaxation, edited by R. Blinc (North-Holland, Amsterdam, 1967) pp. 810–823
- (27) R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida, T. Yamazaki, and R. Kubo, Phys. Rev. B 20, 850 (1979)
- (28) F. L. Pratt, J. Phys.: Condens. Matter 19, 456207 (2007)
- (29) R. E. Walstedt and L. R. Walker, Phys. Rev. B 9, 4857 (1974)
- (30) Y. J. Uemura, T. Yamazaki, D. R. Harshman, M. Senba, and E. J. Ansaldo, Phys. Rev. B 31, 546 (1985)
- (31) R. Kubo, Hyperfine Interact. 8, 731 (1981)
- (32) D. R. Noakes and G. M. Kalvius, Phys. Rev. B 56, 2352 (1997)
- (33) A. Amato, R. Feyerherm, F. N. Gygax, A. Schenck, H. v. Löhneysen, and H. G. Schlager, Phys. Rev. B 52, 54 (1995)
- (34) D. E. MacLaughlin, J. P. Vithayathil, H. B. Brom, J. C. J. M. de Rooy, P. C. Hammel, P. C. Canfield, A. P. Reyes, Z. Fisk, J. D. Thompson, and S.-W. Cheong, Phys. Rev. Lett. 72, 760 (1994)
- (35) In the motionally narrowed limit the rate is proportional to the correlation time
- (36) But see the discussion in Sec. IV.
- (37) M. Larkin, Y. Fudamoto, I. Gat, A. Kinkhabwala, K. Kojima, G. Luke, J. Merrin, B. Nachumi, Y. Uemura, M. Azuma, T. Saito, and M. Takano, Physica B 289-290, 153 (2000)
- (38) M. R. Crook and R. Cywinski, J. Phys.: Condens. Matter 9, 1149 (1997)
- (39) A. Maisuradze, W. Schnelle, R. Khasanov, R. Gumeniuk, M. Nicklas, H. Rosner, A. Leithe-Jasper, Y. Grin, A. Amato, and P. Thalmeier, Phys. Rev. B 82, 024524 (2010)
- (40) D. R. Noakes, Phys. Rev. B 44, 5064 (1991)
- (41) M.-A. Méasson, D. Braithwaite, J. Flouquet, G. Seyfarth, J. P. Brison, E. Lhotel, C. Paulsen, H. Sugawara, and H. Sato, Phys. Rev. B 70, 064516 (2004)
- (42) G. Seyfarth, J. P. Brison, M.-A. Méasson, J. Flouquet, K. Izawa, Y. Matsuda, H. Sugawara, and H. Sato, Phys. Rev. Lett. 95, 107004 (2005)
- (43) P. Fulde and K. Maki, Phys. Rev. 141, 275 (1966)
- (44) L. Shu, D. E. MacLaughlin, Y. Aoki, Y. Tunashima, Y. Yonezawa, S. Sanada, D. Kikuchi, H. Sato, R. H. Heffner, W. Higemoto, K. Ohishi, T. U. Ito, O. O. Bernal, A. D. Hillier, R. Kadono, A. Koda, K. Ishida, H. Sugawara, N. A. Frederick, W. M. Yuhasz, T. A. Sayles, T. Yanagisawa, and M. B. Maple, Phys. Rev. B 76, 014527 (2007)
- (45) This relaxation is longitudinal, since it characterizes the relaxation of components of spins parallel to their time-averaged local fields.
- (46) is a purely dynamic rate that describes “lifetime broadening,” and thus is always a lower bound on .
- (47) D. C. Johnston, Phys. Rev. B 74, 184430 (2006)
- (48) D. R. Noakes, J. Phys.: Condens. Matter 11, 1589 (1999)
- (49) A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, 1993)
- (50) A similar argument involving holes applies at the high- end of the lanthanide series.
- (51) G. Kalvius, S. Flaschin, T. Takabatake, A. Kratzer, R. Wäppling, D. Noakes, F. Burghart, A. Brückl, K. Neumaier, K. Andres, R. Kadono, I. Watanabe, K. Kobayashi, G. Nakamoto, and H. Fujii, Physica B 230-232, 655 (1997)
- (52) A. Abragam, The Principles of Nuclear Magnetism (Oxford University Press, Oxford, 1961)
- (53) The second moment diverges for the Lorentzian distribution.
- (54) Ref. Abra61 (), Chap. IV.
- (55) P. Carretta and A. Keren, “NMR and SR in highly frustrated magnets,” in Introduction to Frustrated Magnetism: Materials, Experiments, Theory, Springer Series in Solid-State Sciences, Vol. 164, edited by C. Lacroix, P. Mendels, and F. Mila (Springer, Heidelberg Dordrecht London New York, 2011) p. 79
- (56) M. Pregelj, A. Zorko, O. Zaharko, D. Arčon, M. Komelj, A. D. Hillier, and H. Berger, Phys. Rev. Lett. 109, 227202 (2012)
- (57) E. H. Brandt, Phys. Rev. B 37, 2349 (1988)
- (58) This approximation is well obeyed in PrOsSb (Ref. SMBH09 ()).
- (59) C. M. Varma, Phys. Rev. Lett. 55, 2723 (1985)
- (60) C. M. Varma, K. Miyake, and S. Schmitt-Rink, Phys. Rev. Lett. 57, 626 (1986)