Magnetic excitations in SrCu2O3: a Raman scattering study
A. Gössling, U. Kuhlmann, C. Thomsen, A. Löffert, C. Gross, W. Assmus
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Magnetic excitations in SrCu O : a Raman scattering study A. G¨oßling, U. Kuhlmann, and C. Thomsen
Institut f¨ur Festk¨orperphysik, Technische Universit¨at Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
A. L¨offert, C. Gross, and W. Assmus
Physikalisches Institut, J.W. Goethe-Universit¨at, Robert-Mayer-Str. 2-4, D-60054 Frankfurt a. M., Germany (Dated: 6 February 2003)We investigated temperature dependent Raman spectra of the one-dimensional spin-ladder com-pound SrCu O . At low temperatures a two-magnon peak is identified at 3160 ±
10 cm − andits temperature dependence analyzed in terms of a thermal expansion model. We find that thetwo-magnon peak position must include a cyclic ring exchange of J cycl /J ⊥ = 0 . − .
25 with acoupling constant along the rungs of J ⊥ ≈ − (1750 K) in order to be consistent with otherexperiments and theoretical results. PACS numbers: 78.30.-j, 75.50.Ee
Antiferromagnetic copper oxide spin-ladders have beeninvestigated intensively from a theoretical and exper-imental point of view. Superconducting under highpressure they form a bridge between 1D Heisenbergchains and a 2D Heisenberg square-lattice, which is alsoa model for high- T C superconductors. The magneticground state of a spin ladder is surprisingly not a longranged N´eel state but a short ranged resonating valencebond state. The first excited state is separated fromthe ground state by a finite energy ∆. The spin gapwas first predicted theoretically and later confirmedexperimentally. Typical realizations of a two-leg spin-ladder are thecompounds SrCu O (Ref. 7) and (Sr,Ca,La) Cu O . The first compound is a prototype of weakly coupledCu O spin-ladders while the latter consists of a lad-der and a Cu O edge sharing chain part. In contrastto SrCu O , (Sr,Ca,La) Cu O is intrinsically dopedwith holes. A schematic view of the compound SrCu O is shown in Fig. 1. Cu atoms, represented by d -orbitals,are antiferromagnetically coupled via an intermediateoxygen p -orbital by superexchange . The Sr atoms arelocated in between the planes containing the Cu O atoms. The coupling constants along the legs and therungs are denoted with J k and J ⊥ . The interladder cou-pling is negligible to first approximation because the su-perexchange via a Cu-O-Cu path with a 90 ◦ bond an-gle has a smaller orbital overlap than with a bond an-gle of 180 ◦ . Thus, a ladder can be considered an iso-lated quasi one-dimensional object with a HeisenbergHamiltonian H = J ⊥ P rung S i · S j + J k P leg S i · S j . Inthe literature coupling ratios J k /J ⊥ ∼ . − . taking the Hamiltonian mentionedabove into account for analyzing the data. On the otherhand with almost identical Cu-Cu distances in leg andrung direction one would expect an isotropic ra-tio J k /J ⊥ ∼
1. This picture was also confirmed byan analysis of optical conductivity data. Recently theinclusion of a cyclic ring exchange J cycl was suggestedin order to resolve the discrepancy between the geo- metrical considerations and J k /J ⊥ ∼ . − . This ring exchange can be understood as a superpo-sition of clockwise and counter clockwise permutationsof four spins around a plaquette (positions ABCD inFig. 1). A term H cycl = J cycl P i K iABCD with K ABCD =( S A S B )( S C S D ) + ( S A S D )( S B S C ) − ( S A S C )( S B S D ) hasto be added to the conventional Heisenberg Hamilto-nian H . In order to achieve the isotropic limit of J k /J ⊥ ∼ J cycl /J ⊥ ∼ . − . In this paper we studied the Raman spectra of the twoleg S = spin-ladder compound SrCu O . In additionto phonons we investigated the two-magnon peak in thiscompound at temperatures between 25 K and 300 K.We find that the inclusion of a ring exchange is neces-sary for the understanding of the magnetic properties ofSrCu O . O (p x ) Cu (d x²-y² ) J (cid:1) J (cid:0) J cycl AB CD Sr ba FIG. 1: Schematic view of the two-leg ladder SrCu O pro-jected on the ab -plane. The Sr atoms are located in betweenthe planes containing the Cu O atoms. The magnetic cou-pling constants along the rungs and the legs are indicated by J ⊥ and J k and the cyclic ring exchange by J cycl . The cou-pling between two Cu d -orbitals is caused by superexchangeinteraction via an O p -orbital. Polycrystalline SrCu O was grown under high pres-sure as described by L¨offert et al. The crystallographicstructure was verified by x-ray diffraction. In addition toSrCu O small amounts of Sr Cu O , CuO and Cu Owere detected. Measurements were performed using aLabRam spectrometer (Dilor) including a grating with600 and 1800 grooves/mm in backscattering geometryand a multichannel CCD detector. An Ar + laser spot(488 nm) was focused by a 80 × microscope objective onsmall individual crystallites on the sample surface with adiameter of 1-2 µ m. The spectra in Fig. 2 were mea-sured at room temperature. The crystallite was cho-sen by using a polarization microscope : The samplesurface was illuminated normally with linear polarizedwhite light. The reflected light was detected throughan analyzer crossed to the polarizer with a CCD cam-era. Being an anisotropic material the reflected light isin general elliptically polarized. As for the ladder com-pound LaCuO . , we assume the direction along theladder ( a -axis) to be the one with the largest anisotropyin SrCu O , whereas the b and c axes are approximatelyoptically isotropic. Thus, there should be a difference inbrightness along and perpendicular to the a -axis whenrotating the sample. If the incident wave vector k i isparallel to the a -axis the image stays dark while rotat-ing the sample. For k i ⊥ a the crystallite appears darkand bright for four times when turning the sample 360 ◦ .The spectra in Fig. 2 were measured on a crystallite for k i ⊥ a . For our measurements down to 25 K (spectra inFig. 3) the sample was mounted on the cold finger of anOxford microcryostat.The Raman spectra of SrCu O can be divided intoa low (100-1700 cm − ) and a high-energy (2000-4600cm − ) region as shown in Figs. 2 and 3. We verifiedthe chemical composition of our polycrystalline sampleby an analysis of the phonons in the low energy part ofthe spectra. We start with a factor group analysis (FGA)of the vibrational modes according to Rousseau et al. The unit cell of SrCu O consists of two formula units.Structure analysis attributes SrCu O to the spacegroup Cmmm ( D h ). The FGA results inΓSrCu O = 2 A g +4 B u +2 B g +4 B u +4 B u +2 B g (1)Having a center of inversion only even modes show Ra-man activity, resulting in six Raman active phonons. Weexpect the two A g modes in the spectra measured in par-allel polarization while the four even B modes should beobserved in crossed polarization.Figure 2 shows four spectra measured at T =295 K twoin parallel and two in crossed polarization. The axeswere identified using the described analysis with polar-ized light; comparing Fig. 2 to the spectra of the very sim-ilar ladder compound Sr Cu O , we identifythe a (leg) and b (rung) axes as indicated in the figure. In ( bb ) polarization only a single phonon at 550 cm − is observed, while in ( aa ) two phonons at 310, 550 cm − ,and in addition several broad peaks around 1150 cm − are visible. We assign these two phonons as the two al- FIG. 2: Raman spectra in parallel and crossed polarizationmeasured at T =295 K. The insets show the low energy partsof the spectra measured in parallel polarization. lowed A g modes. In Sr Cu O modes at 316 cm − and 549 cm − were observed. The latter is believed tobe the breathing mode of the O-ladder atoms located onthe ladder legs , which is in accordance with our as-signment. The structure at about 1150 cm − was alsoobserved in Sr Cu O , its origin attributed to two-phonon processes. Knowing only few phonons at theΓ point the details of the broad structure around 1150cm − in SrCu O remain unclear. In summary, we iden-tified the two allowed A g ladder phonons at 310 and 550cm − in SrCu O at room temperature, the B g modeswere too weak to be observed, and the B g not allowedfor our geometry.In Fig. 2 in addition to the phonons a peak around3000 cm − was observed in ( aa ) and ( bb ) polarizationswhile it was absent in the crossed polarizations ( ab ) and( ba ). It can be shown from the two-magnon RamanHamiltonian that for a two-leg Heisenberg ladder thetwo-magnon peak is forbidden in all crossed polariza-tions. The observed spectra follow the selection rulesof a two-magnon peak.In Fig. 3 the Raman spectra of the high-energy re-gion are presented for different temperatures. The spec-tra were measured on another crystallite between 25 Kand 300 K. Comparing them to the aligned spectra inFig. 2 we assign the upper spectrum as ( aa ) and thelower one as ( bb ) polarized. In ( aa ) polarization a sharppeak is visible which broadens and shifts to lower energies FIG. 3: Raman spectra measured with the same polarizationas in Fig. 2 with λ ex = 488 nm at 25 K (solid), 150 K (dashed)and 250 K (dotted). Insets: peak positions (top) and edgeposition (bottom) of the two-magnon peak as a function oftemperature. with increasing temperature. In ( bb ) polarization a muchbroader peak is measured; it remains broad also down tothe lowest temperatures. We identify these features tobe the two-magnon peaks which were also observed inSr Cu O at a lower energy (maximum for T =8-20K at 2900-3000 cm − ). The energy of the peakmaxima as a function of temperature is shown in theupper inset of Fig. 3. In ( bb ) polarization the left edgeposition is plotted as a function of temperature in thelower insets of Fig. 3 due to difficulties in determiningthe precise peak maxima. The two-magnon peak shiftsfor T >
75 K almost linearly with temperature and satu-rates at low temperatures at 3160 cm − (peak position)and 3130 cm − (edge position).Using a simple model of thermal expansion we areable to understand the temperature dependence of thesepeaks. The starting point is a system containing two cop-per d -orbitals and one intermediate oxygen p -orbital witha 180 ◦ bond angle as found in flat ladder compounds,hence the model is not only applicable to SrCu O butalso, e.g., to Sr Cu O . The two copper atoms arecoupled by superexchange with the coupling constant J . We assume that the two-magnon energy scales lin-early with J . The average distance d between two cop-per atoms is a function of temperature. Due to an an-harmonic interatomic potential the average distance in-creases in first order as d = d + Aω/ (exp( ω/T ) − d is the distance at T =0 K, A is a constant, rep- resenting the strength of the expansion, and ω is an aver-aged phonon energy (Einstein model). Within the three-band Hubbard-model J equals 4 t pd /ε pd ( U d − + ε − pd ) with t pd being the overlap integral between Cu-O sites, U theCoulomb repulsion, and ε pd the charge transfer energy. The overlap integral t pd ∝ d − and the Coulomb repul-sion U ∝ d − are a functions of d as shown by Harrison. (p. 643) Using the parameters C , , J can be written as J = C d − + C d − (2)The temperature dependence of the two-magnon energy,which is proportional to J , arises from a Taylor expansionaround d . E − mag = E − Bω exp( ω/T ) − E is the two-magnon energy at T =0 K, B is a dimen-sionless constant proportional to A , and ω is the aver-aged phonon frequency. Other power-law exponents inEq. 2 have been used in similar systems in the litera-ture (e.g., Kawada et al. ); the further analysis doesnot depend much on this choice as long B is simply a pa-rameter. The temperature dependence of the peak/edgemaxima of both spectra in the insets of Fig. 3 were fit-ted simultaneously using Eq. (3) with the parameters B = 0 . ± . ω = 160 ±
10 cm − , E ( aa ) = 3160 ± − and E ( bb ) = 3130 ±
10 cm − . The fit shows excel-lent agreement with the experimental data. The averagedphonon frequency ω is comparable to those of high- T C cuprate materials. The different values for E in the( aa ) and ( bb ) spectra in Fig. 3 are caused by the differ-ence between peak edge and peak maximum. For thefurther analysis we take only the energy E ( aa )=3160cm − into account. The observed temperature depen-dence is in accordance with the assumed magnetic originof the peak. The common picture of superexchange isconfirmed by our measurements.We now show that in order to compare the energy E with existing experiments and theoretical results ina consistent way the cyclic ring exchange must be in-cluded. Note, that it is not possible to determine themagnetic constants and the spin gap directly from thetwo-magnon peak position. In order to obtain these val-ues from our spectra we have to make several assump-tions based on calculations: (i) Ab initio calculations yielded J k /J ⊥ = 1 . J cycl /J ⊥ = 0 . − .
25. Cyclic ring ex-change values on the same order ( J cycl /J ⊥ = 0 . − . Cu O (Refs. 16,17).(iii) Schmidt et al. have shown recently in a theoreti-cal Raman study for a Heisenberg two-leg ladder with J k /J ⊥ = 1 . J cycl /J ⊥ = 0 . E ≈ . · J ⊥ . Be-cause the ratio J k /J ⊥ = 1 . O we take E /J ⊥ = 2 . ± . ≈ . · J ⊥ − . · J cycl which hasbeen derived by Nunner et al. for a two-leg Heisenbergladder with ring exchange having adjusted for the dif-ferent definitions of J cycl . The expression for ∆ is validapproximately in the range J k /J ⊥ = 1 . − .
3. As a re-sult we obtain from our experiment J ⊥ = 1170 − − (1680 − J ⊥ = 1215 cm − results in ∆ = 260 −
470 cm − (380 −
680 K) for the spin gap. The latter value is in goodagreement with spin gap values measured with suscepti-bility (420 K), NMR (680 K) and neutron scatter-ing (380 K) on Sr Cu O . In contrast, excluding thespin exchange coupling would yield a spin gap value ofapproximately 840 K, which is out of scale by a factor of1 . − . J cycl /J ⊥ = 0 . − .
25. Thelower limit of J cycl /J ⊥ so obtained corresponds to ∆ from NMR, the upper one to ∆ from neutron scattering.In conclusion we found the two allowed A g phononsat 310 cm − and 550 cm − in SrCu O and studiedthe magnetic excitations in this compound. A peakwith an energy of 3160 ±
10 cm − at low temperaturesstarts to shift to lower energies with increasing temper-ature. We identified this peak as the two-magnon peakand confirmed this by its temperature dependence. Wederived a simple expression which describes excellentlythe peak energy as a function of temperature. For lowtemperatures we are able to explain the two-magnonpeak energy E =3160 cm − in accordance with exist-ing theoretical and experimental results by including acyclic ring exchange using the values: J k /J ⊥ = 1 . J cycl /J ⊥ = 0 . − . J ⊥ ≈ − (1750 K) and∆=260-470 cm − (380 −
680 K).We thank K. P. Schmidt and G. S. Uhrig for helpfuland valuable discussions. This work was supported bythe Deutsche Forschungsgemeinschaft, SPP 1073. E. Dagotto and T. M. Rice, Science , 618 (1996). M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mˆori,and K. Kinoshita, J. Phys. Soc. J. , 2764 (1996). P. W. Anderson, Nature , 1196 (1987). E. Dagotto, J. Riera, and D. Scalapino, Phys. Rev. B ,5744 (1992). T. M. Rice, S. Gopalan, and M. Sigrist, Europhys. Lett. , 445 (1993). M. Azuma, Z. Hiroi, M. Takano, K. Ishida, and Y. Kitaoka,Phys. Rev. Lett. , 3463 (1994). Z. Hiroi, M. Azuma, M. Takano, and Y. Bando, J. SolidState Chem. , 230 (1991). E. M. McCarron, M. A. Subramanian, J. C. Calabrese, andR. L. Harlow, Mat. Res. Bull. , 1355 (1988). P. W. Anderson, Phys. Rev. , 350 (1950). C. de Graaf, I. de P. R. Moreira, F. Illas, and R. L. Martin,Phys. Rev. B , 3457 (1999). R. S. Eccleston, M. Uehara, J. Akimitsu, H. Eisaki, N. Mo-toyama, and S. I. Uchida, Phys. Rev. Lett. , 1702 (1998). N. Ogita, Y. Fujita, Y. Sakaguchi, Y. Fujino, T. Nagata,J. Akimitsu, and M. Udagawa, J. Phys. Soc. Jpn. , 2684(2000). S. M. Kazakov, S. Pachot, E. M. K. S. N. Putelin, E. V.Antipov, C. Chaillout, J. Capponi, P. G. Radaelli, andM. Marezio, Physica C , 139 (1997). M. Windt, M. Gr¨uninger, T. Nunner, C. Knetter, K. P.Schmidt, G. S. Uhrig, T. Kopp, A. Freimuth, U. Ammer-ahl, B. B¨uchner, et al., Phys. Rev. Lett. , 127002 (2001). S. Brehmer, H.-J. Mikeska, M. M¨uller, N. Nagaosa, andS. Uchida, Phy. Rev. B , 329 (1999). M. Matsuda, K. Katsumata, R. S. Eccleston, S. Brehmer,and H.-J. Mikeska, Phys. Rev. B , 8903 (2000). T. S. Nunner, P. Brune, T. Kopp, M. Windt, andM. Gr¨uninger, Phys. Rev. B , 180404(R) (2002). K. P. Schmidt, C. Knetter, M. Gr¨uninger, and G. S. Uhrig,Phys. Rev. Lett, , 167201 (2003). A. L¨offert, C. Gross, and W. Assmus, J. Cryst. Growth , 796 (2002). S. Sugai, T. Shinoda, N. Kobayashi, Z. Hiroi, and M.Takano, Phys. Rev. B , R6969 (1999). D. L. Rousseau, R. P. Bauman, and S. P. S. Porto, J.Raman Spectrosc. , 253 (1981). D. C. Johnston, M. Troyer, S. Miyahara, D. Lidsky,K. Ueda, M. Azuma, Z. Hiroi, M. Takano, M. Isobe,Y. Ueda, M. A. Korotin, V. I. Anisimov, A. V. Mahajan,and L. L. Miller, arXiv:cond-mat/0001147 (unpublished). Z. V. Popovi´c, M. J. Konstantinovi´c, V. A. Ivanov, O. P.Khuong, R. Gaji´c, A. Vietkin, and V. V. Moshchalkov,Phys. Rev. B , 4963 (2000). A. Gozar, G. Blumberg, B. S. Dennis, B. S. Shastry, N. Mo-toyama, H. Eisaki, and S. Uchida, Phys. Rev. Lett. ,197202 (2001). S. Sugai, and M. Suzuki, Phys. Stat. Sol. B , 653(1999). Note that the crystallographic directions of the rungs andlegs are different in other ladder compounds. M. V. Abrashev, C. Thomsen, and M. Surtchev, PhysicaC , 297 (1997). P. A. Fleury, and R. Loudon, Phys. Rev. , 514 (1968). E. M¨uller-Hartmann, and A. Reischl, Eur. Phys. J. B ,173 (2002). T. Kawada, and S. Sugai, J. Phys. Soc. Jpn. , 3897(1998). W. A. Harrison,
Elementary electronic structure (WorldScientific Publishing Co. Pte. Ltd., 1999). P. Knoll, C. Thomsen, M. Cardona, and P. Murugaraj,Phys. Rev. B , 4842 (1990). K. Ishida, Y. Kitaoka, K. Asayama, M. Azuma, Z. Hiroi,and M. Takano, J. Phys. Soc. Jpn. , 3222 (1994). K. Ishida, Y. Kitaoka, Y. Tokunaga, S. Matsumoto,K. Asayama, M. Azuma, Z. Hiroi, and M. Takano, Phys.Rev. B53