Multiple scattering camouflaged as magnetic stripes in single crystals of superconducting (La,Sr)_2CuO_4
A.-E. Ţuţueanu, T.B. Tejsner, M.E. Lǎcǎtuşu, H.W. Hansen, K.L. Eliasen, M. Boehm, P. Steffens, C. Niedermayer, K. Lefmann
MM ULTIPLE SCATTERING CAMOUFLAGED AS MAGNETICSTRIPES IN SINGLE CRYSTALS OF SUPERCONDUCTING (L A ,S R ) C U O A.-E. T¸ ut¸ueanu , T.B. Tejsner , M.E. Lˇacˇatus , u , H.W. Hansen , K.L. Eliasen , M. Boehm , P.Steffens , C. Niedermayer , K. Lefmann Institute Max von Laue Paul Langevin, 38042 Grenoble, France Nanoscience Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, Denmark Department of Energy Conversion and Storage, Technical University of Denmark, 2800 Kgs. Lyngby,Denmark Glass and Time, IMFUFA, Department of Science and Environment, Roskilde University, 4000 Roskilde,Denmark Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, 5232 Villigen, Switzerland * email: [email protected] A BSTRACT
Neutron diffraction has been a very prominent tool to investigate high-temperature su-perconductors, in particular through the discovery of an incommensurate magnetic signalknown as stripes. We here report the findings of a neutron diffraction experiment on thesuperconductor (La,Sr) CuO , where a spurious signal appeared to be magnetic stripes.The signal strength was found to be strongly dependent on the neutron energy, peaking at E = Since their discovery more than 30 years ago [1], considerable scientific effort has been put into the quest ofunderstanding the emergence of superconductivity in ceramic cuprates. During this formidable amount ofresearch, much evidence points to the interplay between magnetism and superconductivity as playing animportant role for the mechanisms that form the superconducting Cooper pairs [2].Lanthanum-based cuprate superconductors possess a very complex electronic phase diagram with inter-twined orders ranging from an antiferromagnetic Mott insulator to a metal. The long-range antiferromagneticorder in these compounds is destroyed by as little as 2% strontium doping which introduces one electronhole per substituted atom [3]. Upon hole doping, these materials are known to form so called magneticstripes, also known as incommensurate antiferromagnetic (IC AFM) order, depicted as insulating domainsof antiferromagnetically arranged spins separated by one dimensional ”rivers of charge” within the CuO layers. Spin stripes were first proposed by Tranquada et al. in La − x Nd Sr x CuO [4] and later confirmedin La − x Sr x CuO (LSCO), as summarized by Yamada et al. [5]. In neutron scattering experiments, spinstripes are observed as a quartet of incommensurate peaks around the AFM reflection with a wavevectortransfer of q IC = ( ± δ , 1 ± δ , 0 ) , or crystallographical equivalent positions, such as q IC = ( ± δ , ± δ , 0 ) . a r X i v : . [ c ond - m a t . s up r- c on ] J un ultiple scattering camouflaged as magnetic stripes in single crystals of superconducting(La,Sr) CuO Since their initial discovery, numerous neutron scattering experiments have focused on characterizingthe static and dynamic spin correlations under different conditions in the attempt of determining theirrelationship with superconductivity. The evidence so far points, in broad terms, to a competition betweensuperconductivity and the static magnetic order, as the second was seen to vanish at optimal (and higher)doping [6] and is enhanced by an applied magnetic field, which is known to suppress superconductivity[7, 8].From the early magnetic scattering studies performed on cuprates, multiple scattering was acknowledged asa possible contaminant of the stripe signal [4]. Here we present the results of a systematic neutron studyaimed at determining the magnitude and origin of the double scattering events in a highly underdopedsuperconducting La Sr CuO single crystal. The findings presented here can be taken into account asguidelines in the planning phase of future experiments (i.e. by fine tuning the neutron beam energy) in orderto obtain optimal measurements of magnetic stripes. The sample used throughout this study is a highly underdoped La − x Sr x CuO crystal with nominal doping x = I4/mmm space groupnumber 139) to the low temperature othorombic one (
Bmab space group number 64). The measurementsrevealed a transition temperature T s = ± x = ± T onsetc = ± layers in the scattering plane, making it possible to access wavevectors in the ( h , k , 0 ) - plane.Throughout this paper the orthorhombic notation is used, meaning that a commensurate antiferromagneticreflection would be observed at q = (
0, 1, 0 ) , and crystallographically equivalent positions, and the stripesignal, due to its incommensurate magnetic modulation, should be visible at ( ± δ , 1 ± δ , 0 ) , and equivalentpositions, where δ ∼ x in this low-doping regime [5]. In this notation, the 10 K lattice parameters of a x = a o = b o = c o = Initial neutron diffraction measurements were performed with incident and outgoing energies of 4.6 meV,corresponding to k i = k f = − , where k i and k f are the wave vectors of the initial and final neutronbeams, respectively. This is a standard configuration of cold triple-axis instruments that allows for an optimalfiltering of higher order scattering by cooled polycrystalline beryllium. The measurements revealed thepresence of a peak at the antiferromagnetic position, which has previously not, to our knowledge, been CuO reported in homogeneously doped systems of x > ( ± δ obs , 1 ± δ obs , 0 ) were also measured, as depicted in Figure1a, with incommensurability δ obs = δ x ∼ x ∼ a) Figure 1: a) Elastic scattering grid scan around the (
0, 1, 0 ) reflection taken at 2.5 K on LSCO with x = T c [15]. Secondly, a decrease in incommensurability with increasing temperature wasmeasured (Figures 1c,d), whereas the spin modulation in stripe signals was earlier found to be invariable totemperature variations. These two temperature effects were invariant to sample annealing (Figure 1c).Because the temperature dependence of the peaks resembles that of the tetragonal-to-orthorhombic phasetransition around 360 K, multiple scattering involving reflections allowed only in the orthorhombic phasewas soon proposed as the origin of the signal. Following the textbook by Shirane, Shapiro, and Tranquada[16], the strict condition for multiple scattering to occur is that more than one reciprocal lattice point lieson the surface of the Ewald sphere. Its presence can be tested by either rotating the crystal around thecontaminated scattering vector or by changing the neutron energy of the diffraction experiment ( i.e. changingthe radius of the Ewald sphere). It should be mentioned that in a real experiment, where the incident beamis not perfectly monochromatized, the Ewald sphere should be seen rather as a spherical shell, the thicknessof which corresponds to the spread in wave vectors of the neutron beam.The method of changing wavelength of the diffraction experiment was employed in our experiments atRITA-II and the data is shown in Figure 2. Our findings clearly support the hypothesis of multiple scattering,since the incommensurate side peaks were only found at certain energies (4.5 meV to 4.7 meV) and theintensity of the central (
0, 1, 0 ) peak greatly decreased when the initial and final energies were modified. Themore than one order of magnitude difference in intensity can be visualised in Figure 3 where 1 dimensionalcuts thought the grids are plotted for two different energies.Figure 2 also shows the result of a virtual experiment by the McStas ray-tracing simulation package [17, 18]of a hypothetical crystal showing single scattering at the (0 1 0) reflection, in effect providing the resolutionfunction of the instrument. Through the same McStas simulation it was found that the width of incomingenergy, E i , on the sample was 0.13 meV (FWHM). This value is the relevant number to determine the effective CuO thickness of the Ewald sphere and is not equal to the energy resolution of the spectrometer (which is 0.20 meVFWHM), because the energy resolution of the secondary spectrometer is here not taken into account. f) g) h) c) d) e) a) b) Figure 2:
Diffraction measurements performed around the (
0, 1, 0 ) reflection at 125 K. The caption in eachfigure indicates the value of the initial and final energy of the neutron beam. Sub-figure a) shows in red thedirection of the two diagonal scans plotted in 2D in Figure 3. Panel h) shows the simulated resolution of theinstrument as presented in the text. -0.05 0 0.05 012 b) -1 -0.05 0 0.05012 10 -2 a) Figure 3:
Elastic diagonal scans taken at different values of the initial and final energy, noted in the caption,in the direction exemplified in Figure 2a. The solid line represents a gaussian fit to the data.
Due to the fact that the signal appears in a very narrow energy range and because we know the space groupsof the crystallographic phases allowed in this sample, we were able to determine the reflections involvedin a potential double scattering event. Following the treatment by Shirane, Shapiro, and Tranquada [16],we construct the Ewald sphere corresponding to our experiment as shown in Figure 4a. In our particularcase, the condition for double scattering is fulfilled if another allowed reflection of the low temperatureorthorombic (LTO) phase lies on the surface of the sphere. This poses two requirements: (1) the existence ofa secondary reflection wavevector k (cid:48) f of the same length as k i ; (2) the two scattering wavevectors q (cid:48) and q (cid:48)(cid:48) should correspond to allowed reflections in the LTO phase (space group no. 64 in the unconventional Bmab setting [19]). k (cid:48) f is geometrically defined as: CuO k (cid:48) f = k i − q (cid:48) (1)Using the scattering angle θ = arcsin ( q /2 k i ) to calculate the angle ϕ = − θ and knowing the latticeparameters corresponding to the LTO phase we are able to rewrite the components of k i and q (cid:48) as: q (cid:48) x = h (cid:48) · π a o q (cid:48) y = k (cid:48) · π b o q (cid:48) z = l (cid:48) · π c o (2) k i x = k i sin ϕ k i y = k i cos ϕ k i z = h (cid:48) , k (cid:48) and l (cid:48) are the Miller indices of the allowed reflections in the LTO phase. We have investigatedindices with values between − k (cid:48) f in theenergy range 4.5 − k (cid:48) f = (cid:113) ( k i x − q (cid:48) x ) + ( k i y − q (cid:48) y ) + ( k i z − q (cid:48) z ) (4)Checking the above mentioned condition (1), meaning comparing the values of k i and k (cid:48) f , two pair ofreflections appear to lie on the surface of the Ewald sphere (
0, 2, − ) (equivalent to (
0, 2, 2 ) ) and (
1, 1, − ) (equivalent to (
1, 1, 3 ) ). However, condition (2) is only fulfilled for the first pair since both (
1, 1, − ) and (
1, 1, 3 ) require the disallowed q (cid:48)(cid:48) reflections ( ( −
1, 0, 3 ) and ( −
1, 0, − ) respectively) in order to produceintensity at q = q (cid:48) + q (cid:48)(cid:48) = (
0, 1, 0 ) . In contrast, (
0, 2, − ) and (
0, 2, 2 ) reflections pair with q (cid:48)(cid:48) = ( −
1, 2 ) and ( − − ) respectively, which are only allowed in the orthorombic phase and not in the tetragonalone. This explanation is thus in direct agreement with the specific temperature dependence of the signal aspresented in Figure 1.Figure 4b shows the excellent correlation between the fulfillment of condition (1) for the (
0, 2, − ) reflectionand the peak in integrated intensity of the measured data. The red symbols close to y = k (cid:48) f (cid:39) k i for the (
0, 2, − ) which takes place inthe same energy range as the observed increase in scattered intensity. The (
0, 2, 2 ) reflection behaves in anequivalent manner. We attribute the complex pattern of multiple peaks around the (
0, 1, 0 ) reflection, as it can be observedin Figure 1a, to a combination of the double scattering mechanism, presented above, and twinning of thereflections. It is well known that LSCO crystals in the low temperature orthorombic phase are composed oftwo sets of two twin orientations sharing the (
1, 1, 0 ) or ( −
1, 0 ) planes [20]. This co-existence of a 4-phasestructure with equal lattice parameters but different orientations of the ( a , b ) planes is observed in neutronscattering experiments as a peak splitting of the Bragg reflections along the h and k directions (as exemplifiedin Figure 1 from Ref. [20]).Thus, if we consider all possible combinations of q = q (cid:48) + q (cid:48)(cid:48) , where q (cid:48) and q (cid:48)(cid:48) are the wavevectors ofthe 4 satellite peaks around (
0, 2, 2 ) and ( − − ) respectively (i.e. ( ± t , 2 ± t , 2 ) and ( ± t , − ± t , − ) ),we obtain a total of 16 peaks appearing around the (
0, 1, 0 ) reflection (as depicted by the red circles inFigure 5). Note that the incommensurability ( t ) of the satellite peaks is related to the twin splitting ( ∆ = ◦ − ( b / a ) [20]) as t = ∆ /2, in the small angle approximation, and it does not affect the l valuesince there is no splitting along the c-axis. In Figure 5b we plot the calculated pattern obtained as gaussiandistributions centered at the calculated peak positions and with tuned amplitudes as to match the measureddata shown in Figure 5a. The twin splitting used in the calculations corresponds to a lattice parametersratio b / a = b o / a o = CuO Figure 4: (a) Graphical representation of an instance of double scattering within the Ewald sphere. k i and k f define the incoming and outgoing neutron beam, q (cid:48) and q (cid:48)(cid:48) denote the tested reflections allowed in theLTO phase and k (cid:48) f is the secondary reflection wavevector. (b) Blue symbols show the integrated intensityof the measured signal (see Figure 2) and follow the left side y-axis. The red symbols follow the right sidey-axis and show the calculated difference between the magnitude of the two scattering wavevectors k i and k (cid:48) f for the case q (cid:48) = (
0, 2, − ) and q (cid:48)(cid:48) = ( −
1, 2 ) . The x-axis shows the incoming and outgoing energy ofthe neutron beam ( E i = E f ).fractions of the 4 twin phases. However, we have chosen not to address this involved optimisation scheme,since we do not think is would add to the general understanding. The good agreement between the measureddata and the calculated reconstruction implies that the observed multiple peaks pattern can be regarded as asuperposition of the twinning patterns of all the reflections that simultaneously intersect the surface of theEwald sphere. a) b ) a) Figure 5: (a) Elastic scattering grid scan around the (
0, 1, 0 ) reflection taken at 2.5 K on LSCO with x = Although numerous studies from the literature follow the behaviour of static spin stripes in LSCO supercon-ductors as a function of doping, temperature and applied magnetic field, this spurious process has not beenaddressed in any of the publications to our knowledge. There are two main ways in which the additionaldouble scattering signal went unreported: a) at higher doping ( x ∼ q values compared to the spurious one, thus, depending on the scanning direction, one is able tomeasure the stripe signal without detecting the spurion (for example by performing scans of the sampleorientation, A3). Secondly, as we have argued throughout this paper, by tuning the incoming and outgoingneutron beam energy the contamination can easily be avoided. Indeed, many of the experiments presentedin the literature have been carried out at energies outside the 4.5 − CuO That multiple scattering can make forbidden peaks appear was already mentioned by the first discovery ofstripes [4], where Tranquada et al. reported a neutron energy (13.9 meV) of being almost free of spuriousscattering at (0,1,0). In this respect, not all findings in the present article are new as such. However, the effectof multiple scattering, and in particular the effect that twinning causes IC peaks to appear, is in our workthoroughly documented, as a warning and explanation to future experimentalists.
Our neutron diffraction measurements and calculations support the hypothesis of a double scattering event,originating from the (
0, 2, − ) - and ( −
1, 2 ) -type reflections, corroborated with twinning of the crystal, asthe cause for the observed varying intensity of the (
0, 1, 0 ) reflection and the appearance of incommensuratepeaks around it.Because the intensity of the static magnetic stripe signal is very low in cuprates in general and in LSCO inparticular [24], the contribution of multiple scattering should always be carefully considered and the neutronenergy should be tuned as to minimize this intrinsic effect, which could easily lead to detrimental pollutionof the experimental data. In particular for cold-neutron experiment on the LSCO-family of compounds,energies around 4.6 meV should be avoided. As alternatives, energies of 3.7 meV (the BeO filter edge) or 5.0meV (the Be-filter edge) can be strongly recommended. Our findings explain why, historically, best difractionresults on stripes in cuprate superconductors was found for exactly these energies. Although the multiplescattering explanation of spurious signals was noted early by Tranquada et al. , we have here presented acomprehensive investigation of this effect that we now believe to be fully understood.We emphasize the need for applying these considerations for the planning of neutron diffraction experimentson similar systems, in order to minimize contamination by spurious signals. Acknowledgements
We thank Jean-Claude Grivel and Maria Retuerto for valuable assistance with the preparation of the singlecrystal samples.We are grateful to Philippe Bourges for assisting us with a related test experiment at the LLB and for bringingthe idea of multiple scattering to our attention.We thank the students from the University of Copenhagen Neutron Scattering Course as well as Linda Udby,Ursula B. Hansen, and Sonja Holm-Dahlin for participating in the initial measurements.We thank Anne Bartholdy and Henrik Jacobsen for assisting in the neutron scattering characterization of thestructural phase transition of the crystals.This work was supported by the Danish Agency of Science and Technology through DANSCATT. Ana ElenaT , ut , ueanu was supported by a Ph.D. grant from the Institute Laue-Langevin.The neutron scattering data were obtained at the SINQ neutron source, Paul Scherrer Institute, Switzerland. References [1] J.G. Bednorz and K.A. M ¨uller, Possible high T c superconductivity in the Ba-La-Cu-O system, in: TenYears of Superconductivity: 1980–1990 , Springer, 1986, pp. 267–271.[2] E. Fradkin and S.A.K. andJ. M. Tranquada, Colloquium: Theory of intertwined orders in high tempera-ture superconductors,
Reviews of Modern Physics (2015), 457.[3] E. Dagotto, Correlated electrons in high-temperature superconductors, Reviews of Modern Physics (3)(1994), 763.[4] J. Tranquada, B. Sternlieb, J. Axe, Y. Nakamura and S. Uchida, Evidence for stripe correlations of spinsand holes in copper oxide superconductors, Nature (6532) (1995), 561.[5] K. Yamada, C. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya,G. Shirane et al., Doping dependence of the spatially modulated dynamical spin correlations and thesuperconducting-transition temperature in La − x Sr x CuO , Physical Review B (10) (1998), 6165. CuO [6] M.-H. Julien, Magnetic order and superconductivity in La − x Sr x CuO : a review, Physica B: CondensedMatter (2003), 693–696.[7] S. Katano, M. Sato, K. Yamada, T. Suzuki and T. Fukase, Enhancement of static antiferromagneticcorrelations by magnetic field in a superconductor La − x Sr x CuO with x = Physical Review B (22) (2000), R14677.[8] B. Lake, H. Rønnow, N. Christensen, G. Aeppli, K. Lefmann, D. McMorrow, P. Vorderwisch, P. Smeibidl,N. Mangkorntong, T. Sasagawa et al., Antiferromagnetic order induced by an applied magnetic field ina high-temperature superconductor, Nature (6869) (2002), 299–302.[9] M. Fujita, K. Yamada, H. Hiraka, P. Gehring, S. Lee, S. Wakimoto and G. Shirane, Static magneticcorrelations near the insulating-superconducting phase boundary in La − x Sr x CuO , Physical Review B (6) (2002), 064505.[10] S. Wakimoto, S. Lee, P. M. Gehring, R. J. Birgeneau and G. Shirane, Neutron scattering study of softphonons and diffuse scattering in insulating La Sr CuO , Journal of the Physical Society of Japan (12) (2004), 3413–3417.[11] M. Kofu, S.-H. Lee, M. Fujita, H.-J. Kang, H. Eisaki and K. Yamada, Hidden Quantum Spin-Gap Statein the Static Stripe Phase of High-Temperature La − x Sr x CuO Superconductors,
Physical review letters (4) (2009), 047001.[12] K. Lefmann, C. Niedemayer, A. Abrahamsen, C. Bahl, N. Christensen, H. Jacobsen, T. Larsen, P. Hafliger,U. Filges and H. Ronnow, Realizing the full potential of a RITA spectrometer,
Physica B (2006),1083–1085.[13] P. Radaelli, D. Hinks, A. Mitchell, B. Hunter, J. Wagner, B. Dabrowski, K. Vandervoort, H. Viswanathanand J. Jorgensen, Structural and superconducting properties ofLa − x Sr x CuO as a function of Sr content, Physical Review B (6) (1994), 4163.[14] C. Bahl, K. Lefmann, A. Abrahamsen, H. Ronnow, F. Saxild, T. Jensen, L. Udby, N. Andersen, N. Chris-tensen, H. Jakobsen, T. Larsen, P. Hafliger, S. Streule and C. Niedemayer, Inelastic neutron scatteringexperiments with the monochromatic imaging mode of the RITA-II spectrometer, Nuclear Instrumentsand Methods B (2) (2006), 452–462.[15] T. Croft, C. Lester, M. Senn, A. Bombardi and S. Hayden, Charge density wave fluctuations inLa − x Sr x CuO and their competition with superconductivity, Physical Review B (22) (2014), 224513.[16] G. Shirane, S.M. Shapiro and J.M. Tranquada, Neutron scattering with a triple-axis spectrometer: basictechniques , Cambridge University Press, 2002.[17] K. Lefmann, P.K. Willendrup, L. Udby, B. Lebech, K. Mortensen, J.O. Birk, K. Klenø, E. Knudsen,P. Christiansen, J. Saroun, J. Kulda, U. Filges, M. Konnecke, P. Tregenna-Piggott, J. Peters, K. Lieutenant,G. Zsigmond, P. Bentley and E. Farhi, Virtual experiments: the ultimate aim of neutron ray-tracingsimulations,
J. Neutron Research (2008), 97.[18] P.K. Willendrup and K. Lefmann, McStas (i): Introduction, use, and basic principles for ray-tracingsimulations, J. Neutron Research (accepted) (2019).[19] J.K. Cockcroft, Crystallographic Space Group Diagrams and Tables.[20] M. Braden, G. Heger, P. Schweiss, Z. Fisk, K. Gamayunov, I. Tanaka and H. Kojima, Characteriza-tion and structural analysis of twinned La − x Sr x CuO ± δ crystals by neutron diffraction, Physica C:Superconductivity (3–4) (1992), 455–468.[21] A.T. Rømer, J. Chang, N.B. Christensen, B. Andersen, K. Lefmann, L. M¨ahler, J. Gavilano, R. Gilardi,C. Niedermayer, H.M. Rønnow et al., Glassy low-energy spin fluctuations and anisotropy gap inLa Sr CuO , Physical Review B (14) (2013), 144513.[22] J. Chang, C. Niedermayer, R. Gilardi, N. Christensen, H. Rønnow, D. McMorrow, M. Ay, J. Stahn,O. Sobolev, A. Hiess et al., Tuning competing orders in La − x Sr x CuO cuprate superconductors by theapplication of an external magnetic field, Physical Review B (10) (2008), 104525.[23] B. Khaykovich, S. Wakimoto, R. Birgeneau, M. Kastner, Y. Lee, P. Smeibidl, P. Vorderwisch and K. Ya-mada, Field-induced transition between magnetically disordered and ordered phases in underdopedLa − x Sr x CuO , Physical Review B (22) (2005), 220508.[24] A.E. Tutueanu and et al, Magnetic stripes and fluctuations in strongly underdoped (La,Sr) CuO , inpreparation (2019).(2019).