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Inelastic neutron scattering study of magnetic excitations in Sr2RuO40206304

Inelastic neutron scattering study of magnetic excitations in Sr2RuO40206304
Inelastic neutron scattering study of magnetic excitations in Sr2RuO40206304

a r X i v :c o n d -m a t /0206304 v 1 17 J u n 2002

Inelastic neutron scattering study of magnetic excitations in Sr 2RuO 4.

M.Braden a,b,c,?,Y.Sidis b ,,P.Bourges b ,P.Pfeuty b ,J.Kulda d ,Z.Mao e ,Y.Maeno e,f

a

Forschungszentrum Karlsruhe,IFP,Postfach 3640,D-76021Karlsruhe,Germany b

Laboratoire L′e on Brillouin,C.E.A./C.N.R.S.,F-91191-Gif-sur-Yvette CEDEX,France c

II.Phys.Inst.,Univ.zu K¨o ln,Z¨u lpicher Str.77,D-50937K¨o ln,Germany d

Institut Laue-Langevin,Boite Postale 156,38042Grenoble Cedex 9,France e

Department of Physics,Kyoto University,Kyoto 606-8502,Japan f

International Innovation Center (IIC),Kyoto 606-8501,Japan

(Dated:June 18,2002)

Magnetic excitations in Sr 2RuO 4have been studied by inelastic neutron scattering.The magnetic ?uctuations are dominated by incommensurate peaks related to the Fermi surface nesting of the quasi-one-dimensional d xz -and d yz -bands.The shape of the incommensurate signal agrees well with RPA calculations.At the incommensurate Q -positions the energy spectrum considerably softens upon cooling pointing to a close magnetic instability :Sr 2RuO 4does not exhibit quantum criticality but is very close to it.ω/T -scaling may be ?tted to the data for temperatures above 30K.Below the superconducting transition,the magnetic response at the nesting signal is not found to change in the energy range down to 0.4meV.

PACS numbers:PACS numbers:78.70.Nx,75.40.Gb,74.70.-b

I.INTRODUCTION

Sr 2RuO 4is still the only superconducting layered per-ovskite besides the cuprates[1];however,in contrast to the cuprate high temperature superconductors (HTSC),superconductivity in Sr 2RuO 4develops in a well de?ned Fermi-liquid state [1,2].Nevertheless the superconduct-ing state and the pairing mechanism in Sr 2RuO 4are highly unconventional.The present interest in this com-pound goes far beyond the simple comparison with the cuprate high temperature superconductors.

The extreme sensitivity of the superconducting tran-sition temperature on non-magnetic impurities suggests a non-phonon mechanism [3].It is further established that superconductivity in Sr 2RuO 4is made of spin-triplet Cooper pairs and breaks time-reversal symmetry [2,4].The strongest experimental argument for that can be found in the uniform susceptibility measured either by the NMR-Knight-shift or polarized neutrons experi-ments [5,6]and in the appearance of spontaneous ?elds in the superconducting state reported by μSR [7].A spin-triplet p-wave order parameter had been proposed before these experiments [8],in the idea that supercon-ductivity arrises from a dominant interaction with ferro-magnetic ?uctuations in analogy to super?uid 3He.Rice and Sigrist stressed the comparison with the perovskites SrRuO 3and CaRuO 3which order ferromagnetically or are nearly ferromagnetic respectively [9].Evidence for ferromagnetic ?uctuations in Sr 2RuO 4was inferred from NMR-experiments :Imai et al.found that 1

2 gies is indeed dominated by the incommensurate?uctu-

ations occurring very close to the calculated position at

q i=(0.30,0.30,0).Furthermore,we found a pronounced

temperature dependence which can explain the temper-

ature dependence of both Ru-and O-1

d?dω=

k f

|Q|2)Sαβ(Q,ω)(1)

where k i and k f are the incident and?nal neutron wave vectors,r20=0.292barn,F(Q)is the magnetic form fac-tor.The scattering-vector Q can be split into Q=q+G, where q lies in the?rst Brillouin-zone and G is a zone-center.All reciprocal space coordinates(Q x,Q y,Q z)are given in reduced lattice units of2π/a or2π/c. According to the?uctuation-dissipation theorem,the spin correlation function is related to the imaginary part of the dynamical magnetic susceptibility times the by one enhanced Bose-factor:

Sαβ(Q,ω)=1

1?exp(? ω/k B T)

(2)

In case of weak anisotropy,which is usually observed in a paramagnetic state,χαβ”(Q,ω)reduces toχ”(Q,ω)δα,β. Note that for itinerant magnets,anisotropy of the suscep-tibility can occur due to spin-orbit coupling.F(Q)can

3

be described by the Ru +magnetic form factor in ?rst ap-proximation.Once determined,the magnetic response is converted to the dynamical susceptibility and calibrated in absolute units by comparison with

acoustic phonons,

using a standard procedure depicted in [20].

III.RESULTS AND DISCUSSION

A.

RPA analysis of the magnetic excitations

At low temperature Sr 2RuO 4exhibits well de?ned Fermi-liquid properties;therefore,it seems appropriate to assess its magnetic excitations within a RPA approach basing on the band structure.Density functional calcula-tions in LDA were performed by several groups and yield good agreement with the Fermi-surface determined in de-Haas van Alphen experiments.The bare non-interacting susceptibility,χ0(q ),can be obtained by summing the matrix elements for an electron hole excitation [28]:χ0(q ,ω)=?

k ,i,j

M k i,k +q j

f (εk +q ,j )?f (εk ,i )

1?I (q )χ0(q )

(4)

with the q -dependent interaction I (q ).For the nesting positions Mazin and Singh get I (q )χ0(q )=1.02,which already implies a diverging susceptibility and a magnetic instability.

In Fig.1we show a scheme of the (hk0)-plane in re-ciprocal space.Due to the body centering in space-group I4/mmm any (hkl)-Bragg-points have to ful?ll the condi-tion (h+k+l)even;(100)for instance is not a zone-center but a Z-point.The boundaries of the Brillouin-zones are included in the ?https://www.sodocs.net/doc/5a17666355.html,rge ?lled circles indicate the position of the incommensurate peaks as predicted by the band-structure and as observed in our previous work.The thick lines connecting four of them correspond to the ridges of enhanced susceptibility also suggested in refer-ence [17].

wave?vector (red. lattice units)

FIG.1:Upper part :Scheme of the (hk0)-plane in reciprocal space.Thin lines show the boundaries of the Brillouin-zones and small ?lled and open circles the zone-centers and the Z-points of the body centered https://www.sodocs.net/doc/5a17666355.html,rge ?lled circles indi-cate the position of the incommensurate peaks and thick lines connecting four of them correspond to the walls of enhanced susceptibility also suggested in reference [17].The dashed double arrows show the paths of the constant energy scans frequently used in this work:along [010]-direction –y -scan,transverse to the Q -vector,r -scan,and along the diagonals in [110]-direction,d -scans.Lower part :Imaginary part of the generalized susceptibility calculated by RPA along the three paths indicated in the upper part as dotted lines.

In the meanwhile several groups have performed simi-lar calculations which all agree on the dominant incom-mensurate ?uctuations [23,29,30,31,32].However there are serious discrepancies concerning the detailed struc-ture of the spin susceptibility away from q i .These dis-crepancies mainly rely on the parameters used to describe the electronic band structure,on the choice of I(q )and on the inclusion of more subtle e?ects such as spin-orbit or Hund couplings.

In order to compare directly to the INS experiments,see equations (1)and (2),it is necessary to perform the RPA analysis by taking into account both real and imag-inary part of the susceptibility.Morr et al.[32]report such calculations obtained by ?tting the band structure to the ARPES results [33].They ?nd in addition to the peak at q i quite strong intensity near P i =(0.3,0.5,0),(the middle of the walls,see Fig.1),which is even com-

parable to that at q i in the bare susceptibility.Simi-lar results were obtained in references[23,29].Ng and Sigrist[30]?nd much less spectral weight in the ridges at(0.3,q y)for0.3

We have performed the full RPA analysis basing on the LDA band structure reported in reference[17,19] in order to accompany our experimental investigations. We?rst calculate the bare electron hole susceptibility χ0from the usual expression,equation(3).For the band energies (k x,k y,k z)we use the expressions of Mazin and Singh[19]for the three mutually non hybridizing tight-binding bands in the vicinity of the Fermi level:

xy(k x,k y,k z)=400mev(?1+2(cos(k x)+cos(k y))

?1.2cos(k x)cos(k y))(5) xz(k x,k y,k z)=400mev(?.75+1.25cos(k x)?

.5cos(k x/2)cos(k y/2)cos(k z/2))(6) yz(k x,k y,k z)=400mev(?.75+1.25cos(k y)?

.5cos(k x/2)cos(k y/2)cos(k z/2))(7) We also used the crude approximation that matrix-elements for transitions between bands of the same char-acter are equal to one and others zero.The q-dependent ”ferromagnetic”interaction(Stoner factor)I(q)is taken to be equal to(following reference[17]):I(q)=320mev

FIG.3:Height of the magnetic signal at the incommensurate position as function of Q l;the dashed line shows the depen-dence expected from the Ru1+-form factor and the solid line that assuming the spin-density distribution observed recently in Ca1.5Sr0.5RuO4[39],see text.

inelastic signal.The SDW propagation vector?nally fa-vored in the static ordering corresponds to the out-of-phase coupling between neighboring layers.This static interlayer-coupling might be even due to the CDW always coupled to a SDW,which,however,has not yet been dis-covered so far.One may add that the stripe ordering in La1.67Sr0.33NiO4,which di?ers from the Sr2Ru1?x Ti x O4-case in many aspects but nevertheless may be consid-ered as a mixed SDW-CDW-ordering,occurs at the same propagation vector[36].

The wider Q l-dependence of the incommensurate sig-nal shown in Fig.3can give information about the anisotropy of the excitation,since INS measures only the spin component perpendicular to Q.We consider the diagonal susceptibilityχ”αβwith tetragonal symmetry χ±:=χ”xx=χ”yy=χ”zz.The measured intensity is then given by:

d2σ

Q2l

|Q|2)χ”zz+(1+

6

in

of not

a of

a the the cir-

an-four In

the to to

6d) due

7

to a smaller scattering angle).The energy and temper-ature dependence of the additional contribution corre-sponds to that predicted by the full RPA analysis for the γ-band magnetic contribution shown in Fig.6b).The spectral weight at Q=(0.15,0.15,0)relative to that of the nesting feature increases upon increasing temperature or energy as it is expected for a signal directly related to the van Hove singularity.Due to the agreement between the INS results and the RPA calculations,we suggest to interpret the additional broad scattering as being mag-netic in origin;however,a polarized neutron study would be highly desirable.This scattering further might be rel-evant for a quantitative explanation of the NMR-data [10,20],in particular its temperature independent part. Sr2RuO4is not close to ferromagnetic order but sub-stitution of Sr through Ca yields such order for the con-centration Ca1.5Sr0.5RuO4[15].This doping e?ect was explained in a band structure calculation[40]as arising from a down shift in energy of theγ-band pushing the van Hove-singularity closer to the Fermi-level.In such samples theγ-band magnetic scattering should there-fore become strongly enhanced.Indeed?rst INS studies on these compounds reveal broad signals similar to the additional scattering described above,but much stronger [41].This strongly supports a magnetic interpretation of the scattering in?gures5and6.

https://www.sodocs.net/doc/5a17666355.html,bined temperature and energy dependence

of the incommensurate signal

The energy and temperature dependence has been studied in more detail by performing r-scans across the incommensurate position,since in this mode the back-ground is almost?at.However,due to phonon contribu-tions we could not extend the measurements to energies higher than12meV,which is slightly below the lowest phonon frequency observed at q i[42].In particular the large scans required to cover the broad magnetic signal prevent any analysis within the phonon band frequency range on a non-polarized thermal triple axis spectrome-ter.

The results of the scans are given in Fig.7.At low temperature we?nd an energy spectrum in good agree-ment to that published earlier[20].In the range up to 12meV we observe at all temperatures an energy inde-pendent peak width,which,however,increases upon in-crease of temperature.For temperatures much higher than160K the background considerably increases,see Fig.2,and prevents a detailed analysis within reason-able beam-time.In Fig.7b)we show the temperature dependence of the peak width averaged over the di?erent energies which agrees well to the results obtained from the single scans with less statistics reported in reference [20].Even at the lowest temperature the width of the signal remains?nite.

The spectral functions have been?tted by a single

re-FIG.7:Observed imaginary part of the generalized suscep-tibility as function of energy and temperature;lines are?ts with a single relaxor a).Temperature dependence of the av-eraged full width at half maximum(FWHM)corrected for the experimental resolution and temperature dependence of the square of the FWHM b).Temperature dependence of the amplitude and its inverse c)and of the characteristic energy d)in the relaxor behavior?tted to part a).Lines in b-d)are guides to the eye.

laxor behavior[43]:

χ”(q i,ω)=χ (q i,0)

Γω

8 Our?nding that the characteristic energy in the range

6-9meV is well de?ned only at low temperatures can be

related with the far-infrared c-axis re?ectance study by

Hildebrand et al.[44],since the optical spectrum shows

a resonance in this energy range at low temperatures.

In reference[20]we have compared the temperature de-

pendence of the incommensurate signal with that of the

spin-lattice relaxation rate T1measured by both17O and

101Ru NMR experiments[10].These NMR-techniques

probe the low energy spin?uctuations(ω→0with re-

spect to INS measurements);furthermore,they integrate

the?uctuations in q-space.(1/T1T)is related to the

generalized susceptibility and the INS results by[45]:

(1/T1T) k Bγ2n

ω ω→0(10)

with|A(q)|the hyper?ne?elds.(1/T1T)corresponds hence to the slope of the spectral function in Fig.7a) times the extension of the signal in Q-space.The new data perfectly agrees to the former result,the loss of the incommensurate signal upon heating may explain al-most entirely the temperature dependent contribution to (1/T1T)[10].

The temperature dependence of the magnetic excita-tion spectrum at the incommensurate position may be analyzed within the results of the self consistent renor-malization theory described in reference[43].In an anti-ferromagnetic metal the transition is governed by a sin-gle parameter related to the Stoner-enhancement at the ordering wave-vector,δ=1?I(q i)χ0(q i).The charac-teristic entities of the magnetic excitations are then given by:

κ2∝δ(11)

Γ∝δ(12)

1

T

-scaling,as it has been claimed for the high temperature superconductor La2?x Sr x CuO4[46,47,48]and for CeCu5.9Au0.1[49]. One would expect that the susceptibility is given by:

χ”(q i,ω,T)∝T?αg(

ω

T

forα=0.75and1.0.Only the data at higher temperatures agree with the scaling concept,demonstrat-ing that Sr2RuO4is not a quantum critical point.The schematic inset may illustrate the phase diagram,where the magnetic transition is determined by some parame-ter r(external pressure or composition).At the critical transition one would observe quantum criticality in the entire temperature range,whereas for r-values where the transition is suppressed quantum criticality is observed only at higher temperatures.One then may expect a cross-over temperature T?where the system transforms from an unconventional metal at high temperatures to-wards a Fermi-liquid at low temperatures[50].Only in the temperature range above T?the magnetic excitations should exhibit the related

ω

FIG.8:

χ”(q i ,ω,T )multiplied by ωαas a function of

ω

There is actually little known about the value of the su-

perconducting gap in https://www.sodocs.net/doc/5a17666355.html,ube et al.[57]have

reported an Andreev re?ection study where the open-

ing of the gap is clearly visible in an astonishingly large energy range.The quantitative analysis of the spectra

is quite involved;assuming a p-wave order parameter

Laube et al.obtain2?=2.2meV which may be com-

pared to the value expected within BCS-theory2?=3.55 k B T c=4.97K=0.43meV.Our data shows that there

is no change in the excitation spectrum for energies well

below the reported value of2?,but it has not been pos-

sible to investigate the lowest energies due to the strong elastic incoherent signal.With further increased resolu-

tion(k f=1.3?A?1)we have scanned the energy range 0.3–0.7meV at0.80K again without evidence for a reso-

nant feature.Furthermore,the comparison of two scans at constant Q=(1,0,0)did not yield any di?erence below

and above T c meaning no ferromagnetic spin susceptibil-

ity enhancement.

The theory presented by Morr et al.[32]should be considered as being quite reliable in the case of Sr2RuO4, since the RPA approach to the magnetic excitations is so successful in the normal state,and since Sr2RuO4ex-hibits well de?ned Fermi-liquid properties at tempera-tures below25K.Therefore,the data in Fig.9gives strong evidence against a simple p-wave order parame-ter in Sr2RuO4with a maximum value of the gap of the order of the reported value[57].However,in the meanwhile there are several indications that the order parameter is more complex.The recent speci?c heat data on the highest quality single crystals[24]points to the existence of line nodes in the gap function which were then shown to be aligned parallel to the a,b-plane(hor-izontal line nodes).Such line nodes were explained by Zhitomirsky and Rice through a proximity e?ect between the activeγ-band and the more passive one-dimensional bands[27].A modulation of the gap function along the c-direction will wipe out the resonance predicted for the non-modulated p-wave gap,since for the Q-position an-alyzed,(0.70.30),the electron hole excitation involves parts of the Fermi surface which are fully,partially or not gapped at all.In this sense the absence of any tempera-ture dependence in the magnetic excitation spectrum is consistent with the presently most accepted shape of the gap function.Further theoretical as well as experimental studies are required to clarify the possibility of a resonant feature at other positions in(Q,ω)-space.

IV.CONCLUSIONS

Using assemblies of several crystals of Sr2RuO4we have analyzed the magnetic excitations by INS.The in-commensurate signal arising from the nesting between the one-dimensional bands shows an asymmetry which is well explained by the full RPA analysis as a contribution mainly from theγ-band.

The energy dependence of the incommensurate signal varies with temperature and exhibits a general softening of the spectrum upon cooling.This behavior indicates that Sr2RuO4is approaching the corresponding SDW in-stability at low temperature even though this compound is not at a quantum critical composition.This inter-pretation is con?rmed by the fact that the generalized susceptibility exhibits someω/T-scaling only above~30 K,i.e.in the temperature range where also the transport properties indicate non-Fermi-liquid behavior.

The analysis of magnetic excitations besides the nest-ing ones shows only minor contributions.There is some evidence for additional magnetic scattering closer to the zone-center but still not peaking at the zone-center.This interpretation gets support from the fact that similar scattering is observed in nearly ferromag-netic Ca1.5Sr0.5RuO4and from various RPA calculations which?nd excitations mainly related to theγ-band in this q-range.

The magnetic excitations in Sr2RuO4may be com-pared to the distinct types of magnetic order which have been induced by substitution.The dominant excitations re?ect the SDW reported to occur in Sr2Ru1?x Ti x O4at small Ti concentrations.The less strong excitations sit-uated more closely to the zone-center and most likely related to theγ-band become enhanced through Ca-substitution which drives the system towards ferromag-netism,but only for rather high Ca-concentration. Upon cooling through the superconducting transition we do not observe any change in the magnetic excita-tion spectra,which-combined with recent calculations [32]-indicates that the order-parameter in Sr2RuO4does not possess simple p-wave symmetry.These experimen-tal?ndings are still in agreement with a p-wave order parameter modulated by horizontal line nodes. Acknowledgements.We would like to thank O. Friedt,H.Y.Kee,D.Morr and R.Werner for stimulat-ing discussions and P.Boutrouille(LLB)and S.Pujol (ILL)for technical assistance.Work at Cologne Univer-sity was supported by the Deutsche Forschungsgemein-schaft through the Sonderforschungsbereich608.Work at at Kyoto was supported by a grant from CREST, Japan Science and Technology Corporation.

*electronic mail:braden@ph2.uni-koeln.de

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学习。起初,我很不情愿。渐渐地,我开始全身心地投入到学习中,并且觉得我不能让父亲失望。在那段时间里,我还遇到了一个小学同学,她学习成绩很好。我记得父亲说过的话,并问她如何学习。她坦率地告诉了我。这样,我在班上得了第十名,数学得了第一名。 在回家的路上,风吹在树叶上,好像在为我鼓掌。在我父亲知道这件事之后,他嘴角露出一丝微笑,让我继续努力。 人们说,“没有压力,就没有动力。”在生活中,如果没有逆耳忠言,一个人似乎就失去了通往成功之路的风向标。因此,让我们接受那些不在我们心中的建议,驶向成功!

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