Raman study of spin excitations in the tunable quantum spin ladder Cu(Qnx)(Cl 1−x Br x ) 2
G. Simutis, S. Gvasaliya, F. Xiao, C. P. Landee, A. Zheludev
RRaman study of spin excitations in the tunable quantum spin ladderCu(Qnx)(Cl − x Br x ) G. Simutis, ∗ S. Gvasaliya, F.Xiao, C. P. Landee, and A. Zheludev Neutron Scattering and Magnetism, Laboratory for Solid State Physics, ETH Z¨urich, Z¨urich, Switzerland Department of Physics, Durham University, Durham, United Kingdom Department of Physics, Clark University, Worcester, MA 01610, USA (Dated: October 22, 2018)Raman spectroscopy is used to study magnetic excitations in the quasi one dimensional S = 1 / − x Br x ) . The low energy spectrum is found to be dominatedby a two-magnon continuum as expected from the numerical calculations for the Heisenberg spinladder model. The continuum shifts to higher energies as more Br is introduced. The cutoff of thescattering increases faster than the onset indicating that the increase of exchange constant alongthe leg is the main effect on the magnetic properties. The upper and lower continuum thresholdsare measured as a function of Br content across the entire range and compared to estimates basedon previous bulk studies. We observe small systematic deviations that are discussed. I. INTRODUCTION
Heisenberg S = 1 / The limiting cases of non-interactingdimers and non-interacting spin chains have exact solu-tions. The ground state and excitations change dramat-ically with a changing ratio of exchange constants alongthe ladder leg (J || ) and rung (J ⊥ ), respectively. On theexperimental side, recent progress in organic quantummaterials led to the discovery of several excellent spin-ladder prototype compounds. Perhaps the “cleanest” re-alizations of the model are DIMPY - (C H N) CuBr and BPCB - (C H N) CuBr for the strong-leg andstrong-rung case, respectively. The recently character-ized family of materials Cu(Qnx)(Cl − x Br x ) , whereQnx stands for quinoxaline (C H N ), opened a new op-portunity of continuously varying the leg to rung ex-change ratio, albeit in a rather narrow range. Mostprevious studies of these materials concentrated on thebulk measurements and provided rough estimates ofthese parameters: J || = 1 .
61 meV, J ⊥ = 2 .
95 meV( J ⊥ /J || = 1 .
83) and J || = 1 .
99 meV, J ⊥ = 3 .
26 meV( J ⊥ /J || = 1 .
64) for x = 0 and x = 1, respectively. Todate, the only direct study of the excitation spectrumare neutron experiments on powder samples of the Br-rich material . Unfortunately, more detailed neutronstudies are hindered by the challenges of growing suit-able single crystals and by the need to fully deuteratethe organic ligand.An alternative approach to studying magnetic exci-tations in quantum spin systems is provided by lightspectroscopy. This technique has been applied to a va-riety of materials, including dimer compounds and spinchains. Raman spectroscopy has also proven useful inprobing quantum criticality.
For spin ladders, Ramantechniques are particularly appealing. The simplest caseof 2-magnon scattering essentially probes energy-energycorrelations between exchange bonds. In the generalcase, the measured signal is a combination of specific (Qnx) Cl/BrCuCH NJ ll J ⊥ bac FIG. 1: Sketch of the Cu(Qnx)(Cl − x Br x ) ladder with thickarrows showing the principle exchange paths. The interactionalong the leg of the ladder proceeds through quinoxaline(Qnx)molecules. Part of H and C ions that are further away fromthe magnetic ions are not shown. The exchange along therung is due to halogen ions. spin spin correlation functions with unknown coefficientsthat depend on the electronic structure. For the sim-ple Heisenberg spin ladder model though, contributionsof interactions on the rungs and legs are proportionate,and therefore the spectrum can be calculated to withina single scale factor. Several spin ladder materialshave been studied with Raman scattering to date, in-cluding BiCu PO , Sr − x − y Ca x Y y Cu O andSrCu O . However, none of those actually correspondto the simple Heisenberg ladder model, involving nextnearest neighbor, inter-lader or 4-spin “cyclic” exchangeinteractions. While some Raman data do exist for thealmost perfect BPCB ladder system , the limited signalstrength precludes quantitative analysis.In the present work we use Raman spectroscopy toinvestigate magnetic excitations in Cu(Qnx)(Cl − x Br x ) for the entire range of Br concentrations. We are ableto measure the evolutions of both the gap energy andthe magnon bandwidth, and study these parameters as a a r X i v : . [ c ond - m a t . s t r- e l ] M a r I n t en s i t y ( c oun t s / s ) -1 Raman shift (cm ) I n c r ea s i ng T Cu(Qnx)Br (bb) FIG. 2: Temperature dependence of the spectrum for the Br-end compound. A broad continuum appearing at low tem-peratures is interpreted as two-magnon scattering. The insetshows the integrated intensity of the scattering continuum asa function of temperature. function of Br concentration. Comparing our results toestimates based on previous bulk measurements, we notesystematic discrepancies. These findings are discussed inthe context of existing numerical calculations of Ramanspectra for the ideal Heisenberg spin ladder model.
II. EXPERIMENT
The Cu(Qnx)(Cl − x Br x ) crystals used in this studywere grown by slow diffusion in methanol solution .They crystalize in a monoclinic C 2/m space group,with lattice parameters a=13.237 , b=6.935 ˚A, c=9.775˚A, β =107.88 ◦ for pure Cu(Qnx)Cl and a=13.175˚A, b=6.929 ˚A, c=10.356 ˚A, β =107.70 ◦ for pureCu(Qnx)Br . Antiferromagnetic chains of S = 1 / ions bridged by Qnx molecules run along the crys-talographic b axis. As shown in Fig. 1, these chains arecoupled into ladders by rung superexchange paths viapairs of halogen ions. As Cl is gradually replaced byBr, the most significant crystalographic change is the in-crease of the lattice constant c , which affects the Cu-Cl/Br-Cu bond angle, and thereby results in a changeof the rung exchange constant. Interestingly, the mag-netic properties are modified more strongly along theleg. This is because of a complex orbital overlap throughQnx molecules. A subtle crystalographic change alongthe leg due to chemical pressure leads to a considerablechange in the magnetic properties. As-grown crystals ofa typical size 2x1x1 mm were aligned using an X-Raydiffractometer. In all cases, the surfaces studied con-tained a well-defined b axis, which coincides with the legof the ladder. -1 Raman shift (cm ) I n t en s i t y ( a r b . un i t s ) Cu(Qnx)Br T = 4 K
FIG. 3: Spectra of the Br-end compound in different polar-ization settings. The strongest magnetic signal is observed inthe configuration where polarisation is parallel to the leg ofthe ladder as expected from theoretical prediction . I n t en s i t y ( a r b . un i t s ) -1 Raman shift (cm ) C u ( Q n x ) C l C u ( Q n x ) B r T = 4 K(bb)
FIG. 4: The spectra from the two end compounds. The sharpfeatures are due to lattice vibrations. The broad continuumis due to two-magnon scattering. The most intense phononsare shaded.
Raman spectroscopy measurements were carried outin a backscattering geometry, using a λ = 532 nm solidstate laser. A low power of 0.5 mW was used in order tolimit sample heating. The spectra were obtained a usingTrivista tripple spectrometer and a liquid nitrogen cooledCCD detector with reduced etaloning. The samples werecooled down using a helium flow optical cryostat fromCryovac. Most measurements were performed at a tem-perature of 4 K. In order to obtain good quality spectraat base temperature, measurements were taken in sev-eral acquisitions which allowed increasing signal to noiseratio and removing detector response to cosmic muons.The total counting time was of the order of 10 hours persample for the base temperature measurements. Shorterscans were performed when studying temperature andpolarisation dependencies.The polarisation of incoming and detected light wasselected by using a λ /2 plate and a polariser, respec-tively. Additionally, a λ /4 plate was introduced afterthe polariser to correct for the efficiency of the gratingas function of the polarization of the light. The spectrawere normalised to a standard light source, in order toremove the nonuniform response of the spectrometer. Inthe text below we define the polarizations of incomingand analyzed light in terms of the crystal axes. In thisnotation, the polarization of both incoming and analyzedlight along the ladder direction (crystallographic b axis)is denoted as “ ( bb )”. Due to the way the crystals grew,we could not perfectly control the geometry of the exper-iment in the transverse direction, therefore we use “(xx)”to indicate that both polarizations are transverse to thedirection of the leg. III. EXPERIMENTAL RESULTS
At high temperatures, the obtained spectra are domi-nated by phonons. As the temperature is lowered downto about 60 K, a broad continuum of excitations developsat low energies, as shown in Fig. 2. From its temperaturebehavior, comparison with neutron results and theoret-ical expectations , we interpret this broad low-energyfeature as coming from two-magnon magnetic scatter-ing. The inset shows the integrated intensity of the two-magnon scattering as a function of temperature, whichis similar to one observed in a similar strong-rung ladderBPCB. The shape of this signal remains unchanged indifferent polarization configurations that we could access,but its intensity is strongest in ( bb ) geometry as shownin Fig. 3. Due to the shape of the crystals we could notaccess ( yy ) geometry in a controlled manner. From nowon all the data shown and discussed here were taken in( bb ) configuration. The low-energy spectra for the twoend compounds in the series are shown in greater detailin Fig. 4. The sharp peak in the middle of the magneticcontinuum is a phonon related to the halogen ion move-ment, as it shifts significantly as a function of Br content.In order to extract the information about the spin ex-citation gap and the magnon bandwidth, we studied thespectral region close to the onset E − (Fig. 5) and up-per bound E + (Fig. 6) of the continuum. The corre-sponding threshold energies were determined empiricallyby linear extrapolation, as shown. We empirically fit-ted two straight lines below and above the thresholds,and defined the intersection of the two as the thresholdvalue. The linear fit covered an energy range of ± − around E − , excluding the data in its immediate vicinity(0.6 cm − on either side). A similar analysis was per-formed on the upper edge of the spectrum. In that case I n t en s i t y ( a r b . un i t s ) -1 Raman shift (cm ) T = 4 K x = 0 0.1 0.150.20.60.80.9 (bb)
FIG. 5: The onset region for selected different concentra-tions.The spectra are offset for a clearer presentation of data.The fitted straight lines are presented as dotted lines showingthe estimates of the onset. -1 Raman shift (cm )T = 4 K x =00.10.150.20.60.80.9 I n t en s i t y ( a r b . un i t s ) (bb) FIG. 6: The cutoff region for selected different concentrations.The spectra are offset for a clearer presentation of data. Thefitted straight lines are presented as dotted lines showing theestimates of the cutoff. The sharp phonons for the x = 0compound are truncated for the visual clarity of the data. the linear fitting range was 15 cm − below E + and up to30 cm − above it, excluding 1.2 cm − in the immediatevicinity. The obtained values for the upper and lowercontinuum thresholds are plotted in Figures 7a and 7b,respectively, in solid triangles. IV. DISCUSSION
The overall shape of the observed scattering is quali-tatively in very good agreement with theoretical predic-tions for a simple Heisenberg spin ladder with J ⊥ /J || ∼ .
7. The two-magnon continuum is rather symmetric asexpexted for present ratio of exchange constants. As Clis replaced by Br, both the onset and the cutoff of thescattering increases steadily, suggesting that the energyscale involved in the exchange is increasing. More im-portantly, the cutoff increases at a greater rate than theonset. This indicates that upon increase of Br concentra-tion, the exchange along the leg of the ladder is increas-ing faster than the exchange along the rung pushing thesystem towards the isotropic spin ladder case.For a quantitative analysis we consider the extractedvalues for the onset and cutoff of the scattering contin-uum. It has been shown in that the onset of the scat-tering corresponds to the value of twice the gap size ∆.We make use of these considerations for a comparisonwith expectations and related earlier studies. In Fig. 7b,2∆ estimated from bulk magnetic measurements andneutron scattering experiments is plotted in circles andopen triangles respectively for a direct comparison withour result. Through the entire concentration range, theRaman data reproduces the trends observed by othertechniques rather well. However, the measured thresholdis consistently below estimates based on previous studies,by roughly 10 %.Since Cu(Qnx)(Cl − x Br x ) materials are strong-rungladders, they are expected to have well-defined single-magnon excitations across the entire Brillouin zone (see,for example, ). Assuming non-interacting magnons, wecan relate E + to the upper bound of a q = 0 2-magnoncontinuum computed using the single-magnon dispersionrelation. Computational studies have indeed shown thisto be the appropriate description In this fashion wefirst estimated E + /J || for two specific ladder species with J ⊥ /J || = 2 and J ⊥ /J || = 1, for which the magnon disper-sion relation has been previously derived using DMRGtechniques. For the two cases, we obtain E + /J || = 6 . E + /J || = 4 .
08, respectively. We then used a linearinterpolation of E + /J || vs. J ⊥ /J || to obtain E + for theactual exchange constants in Cu(Qnx)(Cl − x Br x ) , as es-timated by bulk magnetometry. The resulting estimatefor E + is plotted in semi-filled circles in Fig. 7a. Onceagain, the experimental trend is well reproduced, but theobserved upper Raman continuum threshold is typically10% below the estimate.Bulk magnetic measurements may not provide a par-ticualrly reliable estimate for J ⊥ /J || , but actually area very robust way to determine the spin gap ∆. Atleast for the lower continuum threshold, the observed10% discrepancy therefore requires explanation. It seemsthat the most likely cause are terms in the spin Hamilto-nian of CQX unaccounted for by the simple Heisenbergladder model. In particular, the culprit could be weakinter-ladder coupling. If such interactions produced a Br concentration E ne r g y ( m e V ) a)b) FIG. 7: Values of the onset(b) and cutoff(a) of magnetic scat-tering for different concentrations obtained in present study(black triangles). The estimates using the results from pre-vious studies are also plotted. Red circles represent valuesestimated by using results of Povarov et al. and extrapolat-ing DMRG calculations of Schmidiger et al. as described inthe text. The blue triangles are taken from inelastic neutronstudy of Br-end compound by Hong et al. The dashed linesare guides to the eye. On the otherhand, the gap extracted from magnetometry assuminga purely one-dimensional model would not be correct.Compared to the true spin gap (the global minimum inthe 3-dimensional magnon dispersion), it would be toolarge by about half the transverse magnon bandwidth, or0.15 meV. That would account for the observed discrep-ancies on the lower threshold of the Raman continuum.The failings of an analysis that assumes a purely one-dimensional model would then propagate to the estimateof J ⊥ /J || and the upper continuum threshold. V. CONCLUSION
In summary, we have obtained high quality magneticraman scattering data on a family Cu(Qnx)(Cl − x Br x ) which are close to being ideal Heisenberg systems. Theobserved scattering from two-magnon continuum hasbeen found to shift to higher energies as more Br is in-troduced. The faster increase of the scattering sutoffis consistent with the system approaching the isotropicladder. While the trend and observations are generallyconsistent with previous bulk measurements, some de-viations persist. We hope that this work will stimulatenumerical calculations of the exact shape of the Ramancontinuum for the partucular values of J ⊥ /J || found inCu(Qnx)(Cl − x Br x ) for a direct comparison with ex- periment. Such a comparison will also clarify if the in-consistencies between gap energies deduced from Raman,neutron and bulk experiments are indeed due to inter-ladder coupling. Alternatively, the discrepancy may bedue to an intrinsic feature of the Heisenberg ladder, suchas magnon-magnon interactions. ∗ Electronic address: [email protected] T. M. Rice, S. Gopalan, and M. Sigrist, EPL (EurophysicsLetters) , 445 (1993), URL http://stacks.iop.org/0295-5075/23/i=6/a=011 . E. Dagotto and T. M. Rice, Science . T. Giamarchi,
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