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Low-temperature magnetism of Nd 2 CuO 4 : An ultrasonic investigation P. Richard, 1,2, * M. Poirier, 1 and S. Jandl 1 1 Regroupement Québécois sur les Matériaux de Pointe, Département de Physique, Université de Sherbrooke, Sherbrooke, Canada, J1K 2R1 2 Department of Physics, Boston College, Chestnut Hill, Massachusetts 02134, USA sReceived 9 August 2004; revised manuscript received 26 January 2005; published 29 April 2005d We have performed an ultrasonic study of Nd 2 CuO 4 in order to investigate the low temperature magnetism due to the Nd 3+ moments. In addition to small anomalies corresponding to the Cu 2+ spin reorientations observed at 33 and 67 K, a large magnetic field and frequency dependent anomaly is detected around 4 K in the C 66 elastic constant and the corresponding attenuation variations. The frequency dependence of this anomaly, which disappears under a magnetic field of 2 T applied along the in-plane Cu-O bonds, is related to a resonant phenomenon rather than to a phase transition as previously reported. The ultrasonic measurements suggest the development of local domains, due to a frustration of the magnetic structure caused by the competition between the Nd 3+ -Cu 2+ and the Nd 3+ -Nd 3+ interactions at low temperature. In contrast to the common belief, the C 66 elastic constant and attenuation anomalies above 2 T characterize a field-induced transition of the Nd 3+ spins which occurs at critical field lower than the one associated with the Cu 2+ spins. DOI: 10.1103/PhysRevB.71.144425 PACS numberssd: 75.25.1z, 74.25.Ld, 74.72.Jt I. INTRODUCTION In order to clarify the role played by magnetism in the high-temperature superconductors, much effort has been de- voted in the past years to the understanding of the 2-1-4 electron-doped superconductors sRE 2-x Ce x CuO 4 , RE=Pr, Nd, Smd magnetic properties. In contrast to the hole-doped materials, for which a low concentration of holes suppresses the Cu 2+ antiferromagnetic sAFd order, the AF order persists in Nd 2-x Ce x CuO 4 for dopings as high as the optimal cerium concentration x = 0.15. 1 Furthermore, recent experiments have established the AF order as a competing ground state in the electron-doped superconductors. 2 Among the 2-1-4 electron-doped superconductors, Nd 2 CuO 4 has aroused a lot of interest owing to the strongly coupled Cu 2+ and Nd 3+ magnetic subsystems, as illustrated by the spin reorientations found around 33 and 70 K, well below the Cu 2+ Néel tem- perature s,250 Kd. 3–6 Several experimental techniques have been used for their study, such as crystal-field infrared transmission, 7 Raman scattering, 8 muon spin relaxation, 4 and neutron measurements. 5,9,10 As the temperature is decreased, Nd 2 CuO 4 exhibits at least three magnetic phases called phase I s70 K & T & 250 Kd, phase II s33 K & T & 70 Kd, and phase III sT & 33 Kd; the magnetic structure being the same in phases I and III. In between two consecutive phases, each three-layer- spin unit, formed by one Cu 2+ layer sandwiched between two Nd 3+ layers, rotates as a whole by 90° in the ab plane, with the adjacent unit rotating in opposite direction. While the magnetic structure of Nd 2 CuO 4 is noncollinear in zero ap- plied field, 11–13 the magnetic structure becomes collinear un- der a magnetic field of 4.4 and 0.75 T applied along the f100g and the f110g directions, respectively. 11 As shown in a ther- mal conductivity study of Nd 2 CuO 4 , these field-induced tran- sitions are accompanied by the closure of a ,0.3 meV gap in an acoustic magnon branch at k =0, leading to magnon heat transport at temperatures as high as 18 K. 14 Even though Nd 3+ spins are involved in these magnetic transitions, a spontaneous ordering of the Nd 3+ subsystem at low temperature, due to a Nd 3+ -Nd 3+ interaction, remains controversial. Hence, while x-ray magnetic scattering data indicate that Nd 3+ ions are polarized at 37 K, 15 the removal of the Kramers doublet degeneracies, as observed by crystal- field infrared transmission, indicates that these ions are al- ready polarized by the Cu 2+ subsystem at a temperature as high as 140 K. 7 An enhancement of neutron scattering mag- netic peak intensities around 3 K has been interpreted by Matsuda et al. as a direct Nd 3+ -Nd 3+ interaction, 5 while Lynn et al. has estimated the Nd 3+ ordering temperature around 1.5 K. 6 The ultrasonic technique is known to be very sensitive to the magnetic structure of crystals. Hence, in a previous ul- trasonic study of Nd 2 CuO 4 at a frequency of 54.3 MHz, anomalies found around 33.5 K on the C 66 and the C 11 -C 12 elastic moduli have been associated with the phase IIphase III magnetic transition. 16 Moreover, this study has revealed a strong anomaly s,5% d around 5 K in the C 66 elastic con- stant, accompanied by a peak in the attenuation. While no frequency dependence of this anomaly has been reported, a shift towards the low temperatures, as well as a decrease in its amplitude up to the maximum magnetic field used s3Td, have been observed under a magnetic field applied along the f100g direction. 16 Weaker anomalies s,0.1% d have also been measured at the same temperature in the C 11 and C 44 elastic moduli. These anomalies around 5 K were attributed to a spontaneous ferroelastic phase transition, whose origin was later associated with a loss of spin orthogonality in the magnetic structure. 17 According to free energy calculations, opposite signs between the Nd 3+ -Nd 3+ and the Cu 2+ -Cu 2+ interactions would be responsible for this symmetry lowering. 18 More recently, acoustic measurements at lower frequencies performed on polycrystalline NCCO bars re- vealed a softening of the Young’s modulus below 20 K. 19 Because Ce doping shifts this anomaly to lower tempera- PHYSICAL REVIEW B 71, 144425 s2005d 1098-0121/2005/71s14d/144425s7d/$23.00 ©2005 The American Physical Society 144425-1

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Page 1: : An ultrasonic investigation

Low-temperature magnetism of Nd2CuO4: An ultrasonic investigation

P. Richard,1,2,* M. Poirier,1 and S. Jandl1

1Regroupement Québécois sur les Matériaux de Pointe, Département de Physique, Université de Sherbrooke,Sherbrooke, Canada, J1K 2R1

2Department of Physics, Boston College, Chestnut Hill, Massachusetts 02134, USAsReceived 9 August 2004; revised manuscript received 26 January 2005; published 29 April 2005d

We have performed an ultrasonic study of Nd2CuO4 in order to investigate the low temperature magnetismdue to the Nd3+ moments. In addition to small anomalies corresponding to the Cu2+ spin reorientationsobserved at 33 and 67 K, a large magnetic field and frequency dependent anomaly is detected around 4 K inthe C66 elastic constant and the corresponding attenuation variations. The frequency dependence of thisanomaly, which disappears under a magnetic field of 2 T applied along the in-plane Cu-O bonds, is related toa resonant phenomenon rather than to a phase transition as previously reported. The ultrasonic measurementssuggest the development of local domains, due to a frustration of the magnetic structure caused by thecompetition between the Nd3+-Cu2+ and the Nd3+-Nd3+ interactions at low temperature. In contrast to thecommon belief, theC66 elastic constant and attenuation anomalies above 2 T characterize a field-inducedtransition of the Nd3+ spins which occurs at critical field lower than the one associated with the Cu2+ spins.

DOI: 10.1103/PhysRevB.71.144425 PACS numberssd: 75.25.1z, 74.25.Ld, 74.72.Jt

I. INTRODUCTION

In order to clarify the role played by magnetism in thehigh-temperature superconductors, much effort has been de-voted in the past years to the understanding of the 2-1-4electron-doped superconductorssRE2−xCexCuO4, RE=Pr,Nd, Smd magnetic properties. In contrast to the hole-dopedmaterials, for which a low concentration of holes suppressesthe Cu2+ antiferromagneticsAFd order, the AF order persistsin Nd2−xCexCuO4 for dopings as high as the optimal ceriumconcentration x=0.15.1 Furthermore, recent experimentshave established the AF order as a competing ground state inthe electron-doped superconductors.2 Among the 2-1-4electron-doped superconductors, Nd2CuO4 has aroused a lotof interest owing to the strongly coupled Cu2+ and Nd3+

magnetic subsystems, as illustrated by the spin reorientationsfound around 33 and 70 K, well below the Cu2+ Néel tem-peratures,250 Kd.3–6 Several experimental techniques havebeen used for their study, such as crystal-field infraredtransmission,7 Raman scattering,8 muon spin relaxation,4 andneutron measurements.5,9,10

As the temperature is decreased, Nd2CuO4 exhibits atleast three magnetic phases called phase Is70 K&T&250 Kd, phase II s33 K&T&70 Kd, and phase IIIsT&33 Kd; the magnetic structure being the same in phases Iand III. In between two consecutive phases, each three-layer-spin unit, formed by one Cu2+ layer sandwiched between twoNd3+ layers, rotates as a whole by 90° in theab plane, withthe adjacent unit rotating in opposite direction. While themagnetic structure of Nd2CuO4 is noncollinear in zero ap-plied field,11–13 the magnetic structure becomes collinear un-der a magnetic field of 4.4 and 0.75 T applied along thef100gand thef110g directions, respectively.11 As shown in a ther-mal conductivity study of Nd2CuO4, these field-induced tran-sitions are accompanied by the closure of a,0.3 meV gap inan acoustic magnon branch atk =0, leading to magnon heattransport at temperatures as high as 18 K.14

Even though Nd3+ spins are involved in these magnetictransitions, a spontaneous ordering of the Nd3+ subsystem atlow temperature, due to a Nd3+-Nd3+ interaction, remainscontroversial. Hence, while x-ray magnetic scattering dataindicate that Nd3+ ions are polarized at 37 K,15 the removalof the Kramers doublet degeneracies, as observed by crystal-field infrared transmission, indicates that these ions are al-ready polarized by the Cu2+ subsystem at a temperature ashigh as 140 K.7 An enhancement of neutron scattering mag-netic peak intensities around 3 K has been interpreted byMatsudaet al.as a direct Nd3+-Nd3+ interaction,5 while Lynnet al.has estimated the Nd3+ ordering temperature around 1.5K.6

The ultrasonic technique is known to be very sensitive tothe magnetic structure of crystals. Hence, in a previous ul-trasonic study of Nd2CuO4 at a frequency of 54.3 MHz,anomalies found around 33.5 K on theC66 and theC11-C12elastic moduli have been associated with the phase II→phaseIII magnetic transition.16 Moreover, this study has revealed astrong anomalys,5%d around 5 K in the C66 elastic con-stant, accompanied by a peak in the attenuation. While nofrequency dependence of this anomaly has been reported, ashift towards the low temperatures, as well as a decrease inits amplitude up to the maximum magnetic field useds3 Td,have been observed under a magnetic field applied along thef100g direction.16 Weaker anomaliess,0.1%d have alsobeen measured at the same temperature in theC11 and C44elastic moduli. These anomalies around 5 K were attributedto a spontaneous ferroelastic phase transition, whose originwas later associated with a loss of spin orthogonality in themagnetic structure.17 According to free energy calculations,opposite signs between the Nd3+-Nd3+ and the Cu2+-Cu2+

interactions would be responsible for this symmetrylowering.18 More recently, acoustic measurements at lowerfrequencies performed on polycrystalline NCCO bars re-vealed a softening of the Young’s modulus below 20 K.19

Because Ce doping shifts this anomaly to lower tempera-

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tures, the authors concluded that it could not be related to theminimum observed inC66sTd in the ultrasonic experimentsperformed on single crystals.16,17 The authors interpretedrather this anomaly in terms of a paraelastic contributionfrom the relaxation of Nd3+ 4f electrons between the levelsof the ground state doublet split by the interaction with theantiferromagnetically ordered Cu2+ lattice; the value of thesplitting was then found to be in agreement with inelasticneutron scattering,20 infrared transmission,21 and specificheat experiments.22

In this paper, we further investigate the magnetic proper-ties of Nd2CuO4 at low temperature with the ultrasonic tech-nique. Beyond the confirmation of previous data, we havecharacterized more thoroughly the low temperature anoma-lies observed around 5 K by performing experiments as afunction of frequency and magnetic field. We focused ourstudy on theC66 elastic constant, which exhibits the stron-gest anomalies at low temperatures, and on the correspond-ing attenuation variations. Neither a spontaneous ferrroelas-tic transition16,17 nor a paraelastic contribution19 can explainour results; we rather propose to relate these anomalies togrowing magnetic domains due to the frustration of the Nd3+

magnetic subsystem arising from the competition betweenNd3+-Cu2+ and Nd3+-Nd3+ interactions.

II. EXPERIMENT

Preliminary ultrasonic measurements on our largeNd2CuO4 samples have indicated that the strong ultrasonicattenuation below 10 K, which increases and distorts the sig-nal as the frequency is enhanced, does not allow a detailedstudy as a function of frequency. In order to reduce the at-tenuation significantly and avoid such complications, wechose a Nd2CuO4 single crystalsla=0.54 mm;lb=2.37 mm;lc=0.47 mmd grown by the flux method having a shortpropagation lengthla. Usually, samples are measured usingan ultrasonic interferometer working either in the transmis-sion or the reflection modes. However, samples with a shortpropagation length cannot be studied with the standard mul-tiple echo technique since the acoustic echoes overlap andcannot be separated from the electric pulse. For this reason, aCaF2 delay line is introduced between the sample and thereceiving transducer to enable measurements of smallsamples in the transmission mode, for the first transmittedpulse only.23,24 Using a frequency feedback, the phase shiftof the first transmitted pulse is maintained constant with re-spect to a reference signal. Both the amplitude and the fre-quency variations of that pulse are measured. This modifiedtechnique generally yield highly reliable results for the ve-locity variations since it is based on a null signal detection.The amplitude of the transmitted echo can, however, beweakly polluted by interference effects originating from mul-tiple reflections inside the crystal, which overlap with vari-ous relative phases. The use of LiNbO3 transducers allows usto generate and detect, at 24 MHz and corresponding oddovertones, transverse waves propagating along thea axis andpolarized along theb axis sC66d. While the absolute attenu-ation and elastic constant values are not accessible, the varia-tions of attenuationasTd−as40 Kd and the relative varia-

tions of the elastic constantfC66sTd-C66s40 Kdg /C66s40 Kdare obtained with high accuracy. A magnetic fields0–10 Tdhas been applied along various orientations. Both the varia-tion of the sound attenuation and the relative variation of theelastic modulusC66 were obtained in the 1.7–80 K tempera-ture range.

III. RESULTS AND DISCUSSION

Figure 1 shows the amplitude of the first transmitted pulseat 208 MHz, in the 2–80 K temperature range. As the tem-perature is decreased from 80 K, three dips are observedaround 67, 33, and 4 K, respectively. The dip intensities areenhanced as the temperature is decreased similarly to theNd3+ magnetic susceptibility,14,25 suggesting a magnetoelas-tic coupling involving directly the Nd3+ magnetic momentsrather than the Cu2+ ones. Furthermore, the Cu2+ Néel phasetransition has been detected neither in our measurements norin previous ultrasonic studies,16,17indicating a weak couplingof the Cu2+ sublattice with the acoustic phonons. The firstdip observed at 67 K, which is revealed using the ultrasonictechnique, is associated with phase I→phase II transition.According to the enhancement of the attenuation with fre-quency, the corresponding anomaly could not be detectedbelow 208 MHz. Also, no anomaly could be identified out ofthe noise on the relative variations of the elastic constantC66around 67 K. The second dip is associated with the phaseII→phase III magnetic transition, as reported elsewhere.16

The anomaly around 4 K, which is much stronger than theothers, is the main subject of this paper. It is also worthnoting that no hysteresis associated with the measurementshas been observed.

The attenuation variationsDasTd and the relative varia-tions of the elastic constantC66 due to the phase II→phaseIII magnetic transition, are given in Fig. 2. For sake of clar-ity, the DC66sTd /C66 curves obtained at various frequencies

FIG. 1. Amplitude of the first transmitted pulse at 208 MHz.Arrows correspond to Cu2+ spin reorientations.

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have been separated upwards from each other. BothDasTdandDC66sTd /C66 exhibit a symmetric anomaly around 33 K.The shape of the anomalies supports the hypothesis of a spinreorientation. While a peak of attenuation is observed duringthe rotation of the Nd3+ and the Cu2+ spins, a small softeningoccurs in DC66sTd /C66. Furthermore, the variations in theattenuation and in the elastic constantC66 are the same belowand above the transition. This indicates that the magneticstructure has not been significantly modified by the spin re-orientation. WhileDC66sTd /C66 is not affected by the usednfrequency, the amplitude of theDasTd anomaly varies with an2 frequency dependence. Moreover, the temperature atwhich the anomaly occurs is not frequency dependent, as it isexpected at a phase transition.

The situation is more complicated in the case of theanomaly around 4 K, as illustrated in Fig. 3, which showsDC66/C66 and Da at various frequencies. Both theDa andthe DC66/C66 anomalies around 4 K are stronger than thosedetected at 33 K by two orders of magnitude, due to a muchstronger elastic coupling compatible with the Nd3+ magneticmoments reinforcement at low temperature. As for the tran-sition at 33 K, a peak of attenuation is accompanied by asoftening of theC66 elastic constant. However, the anomaliesaround 4 K differ from those at 33 K in their amplitude,shape and frequency dependence.

Contrary to the phase II→phase III transition, theDa andDC66/C66 anomalies are not symmetric and their temperatureprofiles differ below and above the transition. The smallwiggles observed on the attenuation data are extrinsic effectslikely originating from overlapping echoes: they are not ob-served systematically at all frequencies and they have nocounterparts on the variations of the elastic constant. Thesewiggles are observed only in the temperature range wherethe variations of the elastic constantC66sTd vary rapidly,

leading to strong modulations of the relative phases betweenthe overlapping echoes and consequently to modulations ofthe measured attenuation amplitude. Interestingly, the tem-perature for which the attenuation is maximum does not cor-respond to the minimum ofDC66/C66, but is rather close tothe maximum of its temperature derivative, indicated in Fig.3 by upward arrows. Also noticeable is the frequency depen-dence of the 4 K anomalies. As shown in Fig. 3, theDC66/C66 and theDa anomalies are shifted towards lowertemperatures as the frequency is enhanced. This shift is ac-companied respectively by a decrease and an increase in theamplitude of theDC66/C66 and theDa anomalies. The tem-perature profiles of the anomalies sharpen as frequency in-creases and the reinforcement of theDa anomaly amplitudefollows a n dependence rather than an2 dependence. Such afrequency dependence ofDa andDC66/C66 cannot be attrib-uted to a phase transition. This situation suggests that theanomalies are related either to a resonance or a relaxationphenomenon characterized by coefficients varying with tem-perature. However, the relaxation model would not produce apeak inDC66/C66 unless the relaxation timetsTd does notvary monotonically with temperature. For this reason, weargue that the anomalies shown in Fig. 3 are more compat-ible with a resonance phenomenon. For example, such amodel has already been introduced in order to describe thedamping of ultrasonic waves due to dislocations.26 Hence,Da and DC66/C66 could be related, respectively, to the dis-sipative and the elastic terms of a forced harmonic oscillatorresponse to a fixed excitation frequency, when the param-eters which characterize the resonance phenomenon varywith temperature.

Since the 4 K anomalies are observed only inNd2−xCexCuO4, they should be related to the Nd3+ ion mo-

FIG. 2. Relative variation of the elastic constantC66 stop paneldand variation of attenuationsbottom paneld due to the phaseII→phase III magnetic transition. TheDC66/C66 curves have beenshifted for sake of clarity.

FIG. 3. Relative variation of the elastic constantC66 stop paneldand variation of the attenuationsbottom paneld at various frequen-cies, measured using a transverse wave propagating along thea axisand polarized along theb axis. The arrows correspond to the criticaltemperatureTa ssee the textd.

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ments. In order to confirm that these anomalies are the con-sequences of their magnetic properties, measurements havebeen performed under magnetic field. The low temperaturemagnetic field dependences ofDC66/C66 and Da at 148MHz, which are illustrated respectively in the top and bot-tom panels of Fig. 4 for a field applied along thea axisspropagation directiond, confirm that the anomalies have amagnetic origin. The influence of the magnetic field on the 4K anomalies is similar to the one produced by a frequencyincrease: while these anomalies are shifted towards lowertemperatures, they get sharper as the magnetic field is en-hanced. However, both amplitudes of theDC66/C66 and theDa anomalies decrease under magnetic field and no anomalyis observed above 4 T in our temperature range. Furthermore,the temperature profiles ofDC66/C66 become very differentabove and below the minimum of theDC66/C66 anomaly.While the profile ofDC66/C66 above that minimum is veryabrupt above 2 T, it is not significantly affected by the ap-plied field below 2 T. Moreover, all theDC66/C66 curvescorresponding toH,2 T reach the same curve below theminimum of the anomaly. Above that field, the low tempera-ture values ofDC66/C66 are no longer in coincidence.

As mentioned earlier, theDC66/C66 and theDa anomaliesaround 4 K can be interpreted as the elastic and the dissipa-tive terms of a forced harmonic oscillator response, respec-tively. The two parameters which characterize the resonancephenomenon are the center temperatureTa sdefined as themaximum temperature derivative ofDC66/C66d and the fullwidth DTa. On the different plots shown in Figs. 3 and 4,only Ta values could be measured with high precision atdifferent frequencies and magnetic field values. TheseTadata are reported in Fig. 5 as a function of magnetic field at24 and 148 MHz frequencies. Similar data at other frequen-cies are not included for the sake of clarity.

Even though thea and the b axes are magneticallyequivalent, we have included in Fig. 5 data obtained with thefield applied along these two axes. We recall that we usetransverse waves propagating along thea axis and polarizedalong theb axis to obtainC66; different field orientationsimply different orientations of the Nd3+ moments relative tothe polarization vectore. For H ' e sH ia axis, black andwhite upward trianglesd, the characteristic temperatureTa de-creases with increasing field. If the values ofTa are quitedifferent in zero field for the two frequencies, they merge asa single line above 2 T: this behavior has been verified at allfrequencies. ForH i e, sH ib axis, black and white circlesd, Tarather increases with field forH&1.5 T; it then decreasesrapidly for higher field values before reaching the universalH-T line observed for the other field orientation. These mea-surements confirm that the low temperatureDC66/C66 andDa anomalies cannot be related to a magnetic phase transi-tion, in contradiction with previous studies using only onefrequency.16–18

While the Nd3+ magnetic subsystem is involved in theresonance phenomenon that occurs around 4 K, a strongNd3+-Cu2+ coupling remains, as indicated by data obtainedwith various magnetic field orientations. The results at 148MHz are reported in the inset of Fig. 5. Similarly to theanomalies at 33 and 67 K, which are related to the Cu2+ spinscoupled to the Nd3+ moments, the low temperature anoma-lies are not significantly affected by a magnetic field as highas 7.25 T applied along thec axis. Only a small shift ofTafor H ic axis is observed, possibly due to a small canting ofthe Nd3+ moments out of theab planes. While the 4 Kanomaly disappears above a field of 4 T applied along theaandb axes, a more drastic effect is observed when the field isoriented along thef110g direction. In such a configuration,Ta

FIG. 4. Relative variation of the elastic constantC66 stop paneldand variation of the attenuationsbottom paneld at 148 MHz undermagnetic field. The field is applied along thea axis spropagationdirectiond.

FIG. 5. Magnetic field dependence of the low temperatureanomaly critical temperatureTa, at 24 and 148 MHz, for field ap-plied along thea axis and along theb axis. The magnetic fielddependence ofTa at 148 MHz for various field orientations areshown in the inset. The gray symbols refer to magnetization datafrom Chernyet al. ssee Ref. 11d. Lines are guides for the eye.

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decreases rapidly above 0.5 T, and persists up to 0.75 T.These results are consistent with magnetization data onNd2CuO4, which are most sensitive to the Nd3+ magneticmoments. Considering that the Nd3+ spin orientation isdriven by the Cu2+ magnetic sublattice, a magnetizationstudy has reported critical fields ofHc1=4.4 T and Hc2=0.75 T associated with the transition from a noncollinear toa collinear magnetic structure at 0.7 K, for magnetic fieldsapplied along the f100g and the f110g directions,respectively.11 However, these transitions are detected only atlow temperature and are not observed above 2.2 K. The val-ues ofTa obtained from magnetization have been included inFig. 5 sgray trianglesd and we observe that our universalH-T line extrapolates towards these magnetization data.

Now we may ask the question: what is the origin of the 4K anomaly observed in Nd2−xCexCuO4 crystals? On the onehand, a paraelastic contribution due to the splitting of theNd3+ Kramers ground state doublet as suggested by Corderoet al.19 cannot explain the frequency and magnetic field de-pendences observed in our ultrasonic experiment. Indeed, aparaelastic contribution should decrease in amplitude withincreasing frequency without any shift in temperature; more-over, since the ground state splitting should increase withmagnetic field, the anomalies would then move to highertemperatures, as the Schottky anomaly in the specific heatmeasurements.22 The frequency and magnetic field trends re-ported here are opposite to these observations. If a paraelas-tic contribution is still possible in this system, it is not de-tected in the megahertz frequency range. On the other hand,although our data are fully consistent with previous ultra-sonic studies, they cannot be explained by a spontaneousferroelastic phase transition.16–18The frequency and the mag-netic field orientationsa axis versusb axisd dependences ofthese anomalies obtained in the present study invalidate thishypothesis since the energy of the probe used is much lowerthan the typical energies involved. These dependences arealso incompatible with a possible resonance of the ultrasonicwave with the gap observed below 4.5 T in an acoustic mag-non branch:14 the gap energy being two to three orders ofmagnitude larger than the energies associated with the usedultrasonic frequenciess1 meV<2.43105 MHzd. Besides,thermal conductivity measurements show similar results inNd2CuO4 and Pr2CuO4

14 while the ultrasonic anomalyaround 4 K is observed only in Nd2−xCexCuO4 crystals.

Below 2 T, the experimental data of Fig. 5 rather suggestthe presence of domains, which resonate with the ultrasonicwaves. Hence, these domains should involve either a charac-teristic length or a characteristic time. Using the absolutesound velocity of a transverse wave propagating along theaaxis and polarized along theb axis, which is4.223103m/s,27 the corresponding length or correlationlength that would be probed in our experiment ranges be-tween 15 and 175mm. On the other hand, the correspondingtime or correlation time that would be probed ranges be-tween 4 and 42 ps. These domains could also be related to aNd3+ spin glass phase.

A likely candidate to explain the development of thesemagnetic domains is the competition between theNd3+-Cu2+ exchange interaction, which is responsible for thepolarization of the Nd3+ sublattice at temperatures as high as

140 K,7 and the Nd3+-Nd3+ interaction which is enhancedsignificantly at low temperature. In Fig. 5, the discrepancyobserved between the frequency dependent regime data andthe extrapolation to lower field of theH-T universal lineobtained above 2 T suggests that the magnetic structure isfrustrated. Hence, as the temperature is lowered, the mag-netic moments of the Nd3+ Kramers ions increase rapidly, asindicated by the in-plane Nd2CuO4 susceptibility.13,14,25Con-sequently, the frustrated magnetic structure of the Nd3+ sub-system gets unstable. Knigavkoet al.have suggested that theNd3+-Nd3+ and the Cu2+-Cu2+ interactions have oppositesigns.18 This idea has also been pointed out in an inelasticneutron scattering study.28 Consequently, the orientation ofthe Nd3+ ions, as imposed by the Cu2+ subsystemvia theNd3+-Cu2+ anisotropic exchange interaction is not the oneadopted in the absence of the Cu2+ subsystem.

The frustration of the Nd3+ magnetic subsystem shouldlead to the formation of local magnetic domains where theNd3+ and Cu2+ magnetic moments are not parallel. Hence,there are some indications that the magnetic structure ofNd2CuO4 at low temperature has a lower symmetry than theusually reported one. While the supposedmmmsymmetry ofthe Nd2CuO4 magnetic structure does not allow any magne-toelectric effect, a non-negligible electric polarization alongthe f100g direction is observed at 4.2 K as an external mag-netic field is applied along thef010g direction.29 Moreover, asplitting along the AM direction of the Nd3+ spin wavedispersion30 has been attributed to a tilting of the Nd3+ spinswith respect to the Cu2+ sublattice, which is estimated to beless than 1°.31

TheH-T line above a magnetic field of 2 T applied alongthe Cu-O bonds could be associated with a field-inducedtransition of the Nd3+ spins from a noncollinear to a collinearmagnetic structure. While the field-induced transition of theCu2+ subsystem is observed up to 18 K in thermal conduc-tivity without significant variation of the critical fieldHc1=4.5 T,14 the magnetization of Nd2CuO4, directly sensitiveto the Nd3+ magnetic moments, shows small variations ofthat critical field which extrapolate to theH-T line measuredwith the ultrasonic technique. Nevertheless, these magnetiza-tion data have been unambiguously attributed to a transitionfrom a noncollinear to a collinear magnetic structure.11

Hence, the noncollinear magnetic structure of the Nd3+ sub-system at low temperature transits to a collinear configura-tion at a critical field lower than the corresponding field-induced transition of the Cu2+ spins, in contrast to thecommon belief.11 On the other hand, the magnetic anisotropybetween thea and b axes, as well as the frequency depen-dence of theH-T line below 2 T, suggest that the Nd3+

noncollinear→collinear magnetic transition is not completein this range, but is rather characterized by magnetic do-mains which indicate a beginning of Nd3+ ordering. Hence,no long range self-ordering of the Nd3+ spins is observedbelow 2 T.

Even though the characterization of the resonance phe-nomenon due to the presence of these magnetic domainswould need a better understanding of their magnetic struc-ture, which cannot be determined with only ultrasonic mea-surements, we can adequately assume that the damping co-efficient and the resonance frequency of the resonance

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phenomenon are somehow related to the Nd3+ uniform mag-netic susceptibilityxsq=0,Td, which increases rapidly atlow temperature. In the relaxation model, frequently used formagnetic domains,32,33 the damping coefficient is supposedto be inversely proportional toxsq=0,Td and such a depen-dence will also apply here. As for the resonance frequency,we expect that it will increase with the domains magnetiza-tion, and thus be proportional toxsq=0,Td. Such tempera-ture dependent parameters are sufficient to reproduce quali-tatively the attenuation and the elastic constantC66sTdobserved experimentally. According to Ref. 26,DC66sTd andDasTd can be described, in the case of a resonant phenom-enon, by

DC66sTd = An2 − n0

2sTdfn2 − n0

2sTdg2 + G2sTdn2 , s1d

DasTd = BnGsTdn

fn2 − n02sTdg2 + G2sTdn2 , s2d

whereA andB are constants. Using the strongly temperaturedependent parametersGsTd~1/xsq=0,Td and n0~xsq=0,Td, where the uniform paramagnetic susceptibilityxsq=0,Td can be obtained from the the crystal-field parametersgiven in Ref. 34, simulations ofDC66sTd andDasTd at vari-ous frequencies have been performed and reported in Fig. 6.Even though the Nd3+-Nd3+ and Nd3+-Cu2+ interactions have

not been included in the calculation ofxsq=0,Td, thesesimulations reproduce rather very well both the shapes andthe frequency dependence of the experimental data shown inFig. 3, especially the displacement with frequency, confirm-ing that the ultrasonic anomalies found around 4 K onsinglecrystals are described by a resonant phenomenon involvingstrongly temperature dependent parameters, rather than by arelaxation process. A quantitative comparison will be onlypossible using an adequate model including the Nd3+-Nd3+

and Nd3+-Cu2+ interactions, which is outside the scope of thepresent study.

IV. CONCLUSION

In this study, we have shown that the ultrasonic techniqueis a powerful tool to characterize the magnetic structure ofNd2CuO4. In addition to an anomaly related to the phaseII→phase III transition, an anomaly related to the phaseI→phase II transition is observed using this technique. More-over, the use of small samples has allowed a frequency studyof the ultrasonic anomalies previously reported around 4 Kon both the ultrasonic attenuation and theC66 elastic modu-lus. These anomalies have been found to be field orientationand frequency dependent, preventing their association with aphase transition, in contrast to previous studies using onlyone ultrasonic frequency. This behavior is also incompatiblewith the resonance of an ultrasonic wave with an energy gapin an acoustic magnon branch.14 Owing to the anomaly pro-files, which cannot be reproduced by any relaxation modelusing a relaxation time varying monotonically with tempera-ture, we propose that these anomalies are associated with aresonance phenomenon which is somehow related to localmagnetic domains due to the frustration of the Nd3+ mag-netic subsystem arising from the competition between theNd3+-Cu2+ and the Nd3+-Nd3+ interactions. The latter in-crease as the Nd3+ low temperature magnetic moments. Fi-nally, we have shown that the Nd3+ magnetic moments can-not order by themselves without a magnetic field of 2 Tapplied along the Cu-O bonds. Above that critical field, theH-T line obtained is associated with a transition of the Nd3+

spins from a noncollinear to a collinear magnetic structure,which occurs at a lower critical field than the one requiredfor the Cu2+ spins.

ACKNOWLEDGMENTS

We thank M. Castonguay for technical assistance, as wellas S. N. Barilo and D. I. Zhigunov for the Nd2CuO4 sample.We also thank P. Fournier and A.-M. Tremblay for their care-ful reading of the manuscript, and C. Bourbonnais for thediscussions we had about our experimental results. Finally,we acknowledge support from the National Sciences and En-gineering Research Council of CanadasNSERCd and leFonds Québécois de la Recherche sur la Nature et les Tech-nologies du Gouvernement du Québec.

FIG. 6. sColor onlined. Simulated relative variation of the elasticconstantC66 stop paneld and variation of the attenuationsbottompaneld obtained at various frequencies using the resonance modeldescribed in the text.

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