Anomalous frequency and intensity scaling of collective and local modes in a coupled spin tetrahedron system
Kwang-Yong Choi, Hiroyuki Nojiri, Naresh S. Dalal, Helmuth Berger, Wolfram Brenig, Peter Lemmens
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Anomalous frequency and intensity scaling of collective and local modes in a coupledspin tetrahedron system
Kwang-Yong Choi
Department of Physics, Chung-Ang University, 221 Huksuk-Dong, Dongjak-Gu, Seoul 156-756, Republic of Korea
Hiroyuki Nojiri
Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577, Japan
Naresh S. Dalal
Department of Chemistry and Biochemistry, Florida State University andNational High Magnetic Field Laboratory, Tallahassee, Florida 32306-4390, USA
Helmuth Berger
Institute de Physique de la Matiere Complexe, EPFL, CH-1015 Lausanne, Switzerland
Wolfram Brenig
Institute for Theoretical Physics, TU Braunschweig, D-38106 Braunschweig, Germany
Peter Lemmens
Institute for Condensed Matter Physics, TU Braunschweig, D-38106 Braunschweig, Germany (Dated: November 29, 2018)We report on the magnetic excitation spectrum of the coupled spin tetrahedral systemCu Te O Cl using Raman scattering on single crystals. The transition to an ordered state atT ClN =18.2 K evidenced from thermodynamic data leads to the evolution of distinct low-energy mag-netic excitations superimposed by a broad maximum. These modes are ascribed to magnons withdifferent degree of localization and a two-magnon continuum. Two of the modes develop a substan-tial energy shift with decreasing temperature similar to the order parameter of other Neel orderedsystems. The other two modes show only a negligible temperature dependence and dissolve abovethe ordering temperature in a continuum of excitations at finite energies. These observations pointto a delicate interplay of magnetic inter- and intra-tetrahedra degrees of freedom and an importanceof singlet fluctuations in describing a spin dynamics.
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I. INTRODUCTION
Frustration and competition of magnetic interactions isone of the central concepts in condensed matter physics. It is related to unusual ground states and (quantum)-criticality as well as exotic low-lying excitations. Thelatter may reach a fascinating complexity due to the di-chotomy of local singlet versus collective magnon states.A prominent example is found in the 3D pyrochlore lat-tice antiferromagnets (AF), consisting of corner-sharingtetrahedra. Such systems have a macroscopic number ofclassical ground states. Weak residual interactions of lat-tice and orbital origin lift these degeneracies in the limitto low temperatures. Furthermore, order-by-disorder ef-fects may be observed.A weakly coupled counterpart of the pyrochlore sys-tem with S=1/2 is realized in the oxohalide Cu Te O X (X=Br,Cl) and Cu Te O Cl compounds. FourCu clusters form a distorted tetrahedron, which alignsin chains along the c axis. The tetrahedra are separatedby lone-pair ions within the ab plane that allow an easymodification of the important in-plane exchange paths along oxygen and halogenoid ions using substitutions, chemical modifications and pressure. Unlike the py-rochlore system, each tetrahedron is isolated while stillbeing coupled by inter-tetrahedral couplings. From spintopology point of view the arrangement of Cu realizesall prerequisites for quantum criticality.Cu Te O X shows an incommensurate magnetic or-dering at T BrN =11.4 K in X=Br and T
ClN =18.2 Kin X=Cl.
The observed ordered magnetic moment0.395(5) µ B of Cu is strongly reduced for X=Br, com-pared to the classical value of ∼ µ B . In contrast, forX=Cl the moment of 0.88(1) µ B is closer to the saturatedone. Since the unit cell volume decreases by 7% as Bris substituted by Cl, the ratio between intra-tetrahedraand inter-tetrahedra couplings seems to be a crucial fac-tor for understanding the respective magnetic behavior.Indeed, the ab initio calculation shows that exchangepaths vary with composition because the Cl 3p orbital atthe Fermi level is more strongly distorted towards the Cu3d orbital than the Br 4p orbital. Moreover, for X=Cl in-plane inter-tetrahedral diagonal interaction is estimatedto be nearly as strong as the intra-tetrahedral interac-tion. With applied pressure T BrN =11.4 K is systemati-cally reduced, implying a decrease of the magnetic inter-tetrahedra coupling strengths and an enhanced degreeof frustration. The effect of symmetry can be studiedcomparing Cu Te O Cl with Cu Te O Cl as in thelatter system the spin tetrahedra have a larger separationwithin the ab plane including an inversion center. Asa result it shows a more mean-field like character of themagnetic properties.In spite of the similar magnetic ordering structure ofCu Te O X with X=Br,Cl, the detailed magnetic prop-erties differ from each other. First, the effect of anexternal field and pressure on T N is opposite. ForX=Br, T N decreases with increasing external field orpressure while for X=Cl T N increases with increasingfield or pressure. Second, thermal conductivity differsfrom each other. The bromide shows a round maxi-mum at low temperature while the chloride displays alevelling-off followed by a steep increase for temperaturebelow 15 K. Third, inelastic neutron scattering (INS)measurements uncovered that the two compounds showa marked difference in the temperature dependence ofmagnetic excitations. For X=Br, upon heating the in-tensity of magnetic excitations decreases monotonicallywhile undergoing no change in lineshape. For X=Cl,however, the magnetic continuum shifts to lower energyand then evolves to a quasielastic diffusive response aboveT
ClN . This suggests that the nature of spin dynamics ofboth compounds is unlike.In previous Raman investigations ofCu Te O Br various magnetic excitations havebeen observed in the spin singlet channel that providedevidence for the presence of a longitudinal magnon inthis system. The latter is an important prerequisitefor a proximity to quantum criticality. In contrast,the magnetic excitations of the X=Cl system have notbeen fully addressed due to the lack of sizable singlecrystals. To enhance our understanding of this weaklyinteracting tetrahedron system and to differentiate thespin dynamics of X=Br and Cl a thorough Ramanspectroscopy investigation of X=Cl is indispensable.In this paper, we report dc magnetic susceptibility,high-field magnetization, and Raman scattering mea-surements of large single crystals of Cu Te O Cl . Theanisotropic magnetization suggests that the ground stateis given by a long-range ordered state. However, we ob-serve an intriguing richness of the magnetic Raman spec-trum that consists of four sharp peaks as well as of aweaker, broad continuum. The former are interpretedin terms of magnon excitation. The latter is due totwo-magnon scattering, whose temperature dependenceis indicative of a minor contribution from localized fluc-tuations. The scaling of the modes points to a sizablecontribution of singlet fluctuations to a spin dynamics. II. EXPERIMENTAL SETUP
Single crystals of Cu Te O Cl were prepared by thehalogen vapor transport technique, using TeCl and Cl as transport agents. Magnetic susceptibility was mea-sured by a SQUID magnetometer (MPMS, Quantum De-sign). High-field magnetization measurements were car-ried out by means of a standard inductive method. Afast sweeping pulsed field was generated by a capacitorbank of 90 kJ. The sample is directly immersed in liquid He to maintain a temperature of 0.4 K. Raman scatter-ing experiments were performed using the excitation line λ = 514 . + laser in a quasi-backscatteringgeometry. A comparably small laser power of 0.1 mWwas focused to a 0.1 mm diameter spot on the surface ofthe single crystal in contact gas. The scattered spectrawere collected by a DILOR-XY triple spectrometer anda nitrogen cooled charge-coupled device detector.
III. EXPERIMENTAL RESULTSA. Magnetic susceptibility and magnetization
Figure 1 displays the temperature dependence of themagnetic susceptibility χ ( T ) in a field H = 0 . || c and H ⊥ c axis, respectively. Our results confirm earlierdata. With decreasing temperature χ ( T ) showsa broad maximum around T max = 24 K. This is asso-ciated with the onset of short-range magnetic ordering.Upon further cooling, χ ( T ) exhibits a kink around 18.2 Kand then drops to a finite residual value as T →
0. Thekink is identified with a transition to a long-rang orderedstate at T
ClN =18.2 K as evidenced by the λ -like anomalyof dχ/dT (see the inset of Fig. 1). H c ( e m u / m o l ) Temperature (K)
H ll c Cu Te O Cl H=0.1 T
Temperature (K) d / d T ( e m u / m o l / K ) T N FIG. 1: Temperature dependence of magnetic susceptibility ofCu Te O Cl , χ ( T ), in a an applied field of H = 0 . || c and H ⊥ c axis, respectively. Inset: derivative of magneticsusceptibility, dχ/dT . The kink at T ClN =18.2 K correspondsto long-range magnetic ordering.
In Ref. 3 χ ( T ) was approximated in terms of anisolated tetrahedral model with a spin-gapped state of∆ ≈ J = J ∼ . J and J are twointra-tetrahedral exchange interactions. However, itis difficult to extract accurately the exchange interac-tions and spin gap from an analysis of χ ( T ) since sig-nificant inter-tetrahedral couplings smear out the spingap features. Actually, χ ( T ) becomes slightlyanisotropic for temperatures below T max due to the onsetof long-range correlations. In addition, χ ( T ) approachesa finite value as T → M agne t i z a t i on ( B / C u ) H (T) Cu Te O Cl H ll cH cT=0.4 K
FIG. 2: High-field magnetization of Cu Te O Cl for H || cand H ⊥ c axis at T=0.4 K, respectively, measured using apulsed magnetic field. Shown in Fig. 2 is the high-field magnetization. Weobserve an anisotropic magnetization behavior. For fieldsapplied perpendicular to the c axis, the magnetizationdisplays a linear field dependence, i.e. the susceptibilityis field independent. In contrast, for field applied alongthe c axis the magnetization is reduced with a concavecurvature in the studied field interval up to 30 T. Werecall that a magnetization plateau at half the saturationvalue has been predicted for a linear chain of spin tetra-hedra in a spin gapped ground state. The absence of ahalf magnetization plateau together with the anisotropysuggests that the ground state is governed by a classi-cally ordered state rather than by a spin singlet state.On a qualitative level, the anisotropic magnetizationbehavior is compatible with helical magnetic ordering.The linear field dependence for H ⊥ c is associatedwith an easy-plane type magnetization. The change ofthe magnetization slope for H k might be related to aspin-flop transition with an incommensurate wave vector. B. Raman scattering
The low energy Raman spectra of Cu Te O Cl aredisplayed in Fig. 3 for (cc), (aa), (ca), and (ab) polar-izations at 3 K. We do not find any distinct temperaturedependence of the optical phonon modes in the frequencyregime 80 −
700 cm − . Similar observations have beenmade for the other tetrahedra based compounds.
23 4939 * (ab)(aa)(ca) * Cu Te O Cl at 3 K (cc) Raman shift (cm -1 ) I n t en s i t y ( a r b . un i t s ) FIG. 3: (Online color) Low-frequency Raman spectra ofCu Te O Cl for (cc), (aa), (ca), and (ab) polarizations atT=3 K. The dashed lines denote the position of four mag-netic signals. The numbers give their respective energy inthe unit cm − . The asterisks denote low-frequency phononmodes that superimpose the magnetic signals. Hereafter, we will focus on the magnetic excitationswhich differ from the phonons by their characteristic en-ergy scale and the variation of both intensity and energywith temperature. These excitations are composed offour peaks at 23 (P1), 39 (P2), 49 (P3), and 67 cm − (P4) as well as of a weak broad continuum (2M) extend-ing from 30 to 120 cm − . With increasing temperaturesan additional quasielastic signal (QC) is observed. Themodes are observed for all polarizations, indicating thatspin tetrahedra are networked in all three dimensions.However, also antisymmetric Dzyaloshinsky-Moriya in-teractions may release the Raman scattering selectionrules. In Fig. 4 the detailed temperature dependence of Ra-man scattering data in ( ab ) polarization of Cu Te O Cl is displayed. For a quantitative analysis we fit them toGaussian profiles after subtracting peaks with phononorigin. The resulting peak frequencies and intensities aredepicted in Fig. 5 on a logarithmic temperature scale and
20 40 60 80 100 120 20 40 60 80 100 120 * ***
21 K17 K13 K11 K7 K R a m an i n t en s i t y ( a r b . un i t s ) Raman shift (cm -1 ) * (ab) Cu Te O Cl * Raman shift (cm -1 ) *
30 K
FIG. 4: (Online color) Temperature dependence of low-frequency Raman spectra in (ab) polarization. The aster-isks denote low-frequency phonon modes. Pi (i=1-4) and 2Mcorrespond to the four sharp magnetic signals and the two-magnon continuum, respectively. in Fig. 6 on a log-log plot as a function of the reducedtemperature, t=1-T/T N .The intensity as well as the energy of modes P1-P4 arerenormalized to a different extent with increasing tem-peratures. This evolution can be used to characterize themodes in addition to their absolute frequencies. Whilethe intensities of P1 and P2 are comparably small anddrop too fast to allow a detailed analysis the modes P3and P4 show a moderate decrease of intensity following amore rapid drop in the proximity of T N . In contrast, thetwo-magnon scattering intensity increases and even formsa maximum at 25 K, i.e. very close to the maximum inthe magnetic susceptibility. This increase of the scat-tering continuum resembles observations in the stronglyfrustrated, 2D Shastry-Sutherland system SrCu (BO ) .In the latter system it is due to the localization of tripletexcitations on a strongly frustrated lattice with a tem-perature independent spin gap. It is noteworthy thatin the latter system the intensity drops to zero for smalltemperatures while in the spin tetrahedron case finite in-tensity remains.Dividing the intensity of magnetic quasielastic scatter-ing (∆ ω ≈
0) by T leads to a measure of the fluctu-ations of the magnetic energy density. This quantity isproportional to the specific heat of a quantum spin sys-tem as also observed in SrCu (BO ) . In the lowerpanel of Fig. 5 this renormalized intensity is plotted forCu Te O Cl with a sharp maximum at T N which is sim-ilar to the earlier reported specific heat data. With respect to energy the peak frequencies of themodes P3 and P4 behave different from P1 and P2. Therenormalization for T
For X=Br there ex-ists also a high energy mode with very weak temperaturedependence. This triangular, broadened mode might beunderstood as composed of several modes, i.e. an overlapof P3 and P4 of Cu Te O Cl . Although the higher en-ergy modes in the two systems show several similaritiesthere is one major difference in the behavior at highertemperatures, T > T N . While the broad triangular modefor X=Br survives several times T N , the sharp modesfor X=Cl rapidly disappear and dissolve in the broader2M signal. The interpretation of this difference is notstraightforward and could be due to a weakly first ordercontribution to the phase transition in Cu Te O Cl .Anticipating the later detailed discussion, we summa-rize that our study unveils a survival of zero-dimensionalquantum fluctuations attributed to individual spin tetra-hedra even though at lower temperatures long range or-dering takes place due to the coupling of the spin entities. C. A comparison of neutron and Raman scattering
In the following we will compare Raman scattering[Fig. 4 of Ref. 8] with neutron scattering [Fig. 7(a) ofRef. 14] addressing first the results for X=Br. Two com-ponents of the magnetic excitation spectrum have beenobserved in neutron scattering; (i) a flat, constant energycomponent and (ii) a dispersive excitation. In a coupledtetrahedra system localized, dispersionless excitationsare expected to occur due to intra-tetrahedral interac-tions and dispersive excitations due to inter-tetrahedralcoupling. In this light, the former is related to a spin gapfeature while the latter to an incommensurate magneticordering. The simultaneous observation of two compo-nents points to the coexistence of long range order witha spin-gapped ground state. As discussed before, in Ra-man scattering a strongly temperature dependent and aweakly temperature dependent feature exist. The smallshift of the latter signal is due to the damping of a Gold-stone mode. Since the softened spectral weight is small,we conclude that the ground state and spin dynamicsof X=Br is governed by spin singlet fluctuations. Thisis supported by the strongly reduced magnetic moment0.395(5) µ B of Cu . For temperatures above T
BrN theintensity of the continuum is monotonically suppressedwithout any change in lineshape.The similarity of the spectral response in INS andRaman scattering is a striking feature considering dif-ferent mechanisms for the scattering processes. Ramanspectroscopy probes simultaneous two-spin flip processesleading to a two-magnon continuum. Thus, the magneticcontinuum is proportional to twice the magnon densityof states. In contrast, INS corresponds to a spin-spin cor-relation function in momentum space. As the availableINS experiments have been performed on polycrystallinesamples the close correspondence between the two spec-troscopic results might be based on the averaging of theINS intensity over momentum space. This intensity isroughly given by the one-magnon density of state.Next, we turn to the discussion of the chloride. INSshows a magnetic excitation spectrum that again consistsof two components, that is, a flat, dispersionless band at6 meV (48 cm − ) and a dispersive lower energy com-ponent. Its energy scale is smaller, 3 meV (24 cm − )and has a gap of 2 meV (16 cm − ). INS on a poly-crystalline sample shows a progressive shift of spectralweight to lower energy with increasing temperature andits transfer to a QC diffusive response in the paramag-netic state [compare Fig. 4 and Fig. 7(b) of Ref. 14].More recently it has been shown that the dispersive modepartially softens and remains gapped while being furtherbroadened. The softening of the magnetic excitationimplies the damping of short-range magnetic fluctuationsby thermal fluctuations. Therefore, we conclude thatthe spin dynamics of X=Cl is dominated by long-rangeordering in contrast to the case with X=Br. This is con-sistent with the larger ordered moment of 0.88(1) µ B inX=Cl. D. Analysis of the magnetic Raman scattering
Here we will discuss several options for the potentialorigin of the four sharp peaks at 23 (P1), 39 (P2), 49(P3), and 67 cm − (P4). As shown in Fig. 5, the peaksshift to lower frequency with increasing temperature andvanish below the magnetic ordering temperature. Thus,they might originate from a transverse magnon excitationat q = 0.Another possible interpretation is in terms of a longi-tudinal magnon. Such excitations have been observed inRaman scattering experiments of the sister compoundof X=Br. To be more specific we start by consider- ing the eigenstates of the isolated tetrahedron at site r with Hamiltonian H ( r ) = J [( S r + S r ) · ( S r + S r )]+ J ( S r · S r + S r · S r ). These consist of two singlets s , , at least one of which is the ground state, threetriplets t α , , , and one quintuplet q α . At J = J thetwo singlets form a degenerate ground state. While J ≈ J applies to X=Cl, we assume J < J for definit-ness and s to be the ground state. This is similarto X=Br. The actual structure of the inter-tetrahedralcoupling in the tellurates is an open issue, but to afirst approximation may be modeled by a molecular field H MF = P r l M r l · S r l with some incommensurate or-der parameter M r l for T < T N . As suggested by theanalysis in Refs. 17 and 18, H MF will mix the groundstate singlet with the triplet components along the localquantization axis of the molecular field. A priori , suchmixing does not have to be restricted to only one of thetriplets, as in Ref. 17, but may involve all three. Dis-carding the high-energy quintuplet this would imply fourlow-energy excited states directly observable at zero mo-mentum transfer by Raman scattering, i.e. one singlet s at energy 2 J − J and three longitudinal magnonswhich vanish above T N . The 3 × T N -as for the remaining one- andtwo-magnon contributions from the quintuplet.From this one should conclude first, that one of thefour modes P1-P4 corresponds to the excited singlet. Incomparison to the 23.2 cm − mode for X=Br, P1 wouldbe a likely candidate for this, leaving P2-P4 for the longi-tudinal magnons. Second, the singlet mode among P1-P4should be affected only weakly by an external magneticfield, while the remaining three should be field-dependentthrough their triplet admixture. This should be investi-gated by future Raman studies in finite magnetic fields.Finally, for a second order transition the longitudinalmagnon energies should vanish as T → T N for T < T N .However, while some softening is observable in all of themodes in Fig. 5, their behavior would be more indicativeof a weakly first order transition. Indeed and dependingon the molecular field, first order transitions may occurfor coupled tetrahedra - as already noted in the commentunder Ref. 12 in Ref. 17. IV. CONCLUSIONS
To conclude, we have presented a magnetic sus-ceptibility, high-field magnetization, and Raman scat-tering study of the coupled spin tetrahedral systemCu Te O Cl . Several distinct magnetic excitations areobserved as one-magnon modes in addition to a two-magnon continuum. The exceptionally rich magnetic ex-citation spectrum evidences the significance of a localizedspin singlet dynamics arising from zero-dimensional spintetrahedra topology even though the static ordered mo-ment has nearly a classical value. Acknowledgements
This work was supported by the German Science Foun-dation and the ESF program
Highly Frustrated Mag- netism . Work at the EPFL was supported by the SwissNSF and by the NCCR MaNEP. Work at FSU was sup-ported by NSF (DMR-0506946). We acknowledge impor-tant discussions with R. Valent´ı. See, for example,
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