Paramagnetic spin correlations in CaFe2As2 single crystals

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Magnetic correlations in the paramagnetic phase of CaFe2As2 (T_N=172 K) have been examined by means of inelastic neutron scattering from 180 K (~ 1.05 T_N) up to 300 K (~1.8 T_N). Despite the first-order nature of the magnetic ordering, strong but
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    a  r   X   i  v  :   1   0   0   1 .   2   8   0   4  v   2   [  c  o  n   d  -  m  a   t .  s  u  p  r  -  c  o  n   ]   7   J  u   l   2   0   1   0 Paramagnetic Spin Correlations in CaFe 2 As 2  Single Crystals S.O. Diallo, 1 D.K. Pratt, 1 R.M. Fernandes, 1 W. Tian, 1 J.L. Zarestky, 1 M. Lumsden, 2 T.G. Perring, 3 C.L. Broholm, 4 N. Ni, 1 S.L. Bud’ko, 1 P.C. Canfield, 1 H.-F. Li, 1 D. Vaknin, 1 A. Kreyssig, 1 A.I. Goldman, 1 and R.J. McQueeney 1 1 Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011 USA 2  Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA 3  ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX, United Kingdom  4 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 USA (Dated: July 8, 2010)Magnetic correlations in the paramagnetic phase of CaFe 2 As 2  ( T  N   = 172 K) have been examinedby means of inelastic neutron scattering from 180 K ( ∼ 1 . 05 T  N  ) up to 300 K (1 . 8 T  N  ). Despite thefirst-order nature of the magnetic ordering, strong but short-ranged antiferromagnetic (AFM) cor-relations are clearly observed. These correlations, which consist of quasi-elastic scattering centeredat the wavevector  Q AFM  of the low-temperature AFM structure, are observed up to the highestmeasured temperature of 300 K and at high energy transfer (¯ hω >  60 meV). The  L  dependenceof the scattering implies rather weak interlayer coupling in the tetragonal  c -direction correspondingto nearly two-dimensional fluctuations in the ( ab ) plane. The spin correlation lengths within theFe layer are found to be anisotropic, consistent with underlying fluctuations of the AFM stripestructure. Similar to the cobalt doped superconducting BaFe 2 As 2  compounds, these experimen-tal features can be adequately reproduced by a scattering model that describes short-ranged andanisotropic spin correlations with overdamped dynamics. PACS numbers: 74.70.-b,75.30.Et,78.70.Nx I. INTRODUCTION The appearance of superconductivity (SC) in doped A Fe 2 As 2  materials ( A  = Ca,Sr,Ba) is linked to the sup-pression of antiferromagnetic (AFM) ordering found inthe parent compounds. 1–3 Other unconventional super-conductors share a similar phase diagram, suggestingthat the AFM spin fluctuations may be responsible forpairing of electrons in the SC state. Certainly, theAFM fluctuation spectrum itself is directly influencedby superconductivity. 4–6 The appearance of a gap andresonance-like feature in the paramagnetic spectrum of these compounds below the superconducting tempera-ture  T  C   also closely resembles 7 other unconventional su-perconductors and highlights the coupling of the ironspins with electronic charge carriers. Recent neutronscattering results elegantly show that the magnetic reso-nance can infer details about the symmetry of the super-conducting gap. 4 In order to understand the nature of the superconduct-ing pairing, details of the normal state spin fluctuationsmust be first understood. These spin fluctuations areexpected to be unusual 8–11 since magnetic frustration inthe tetragonal paramagnetic phase leads to an additionalnematic 12 degree-of-freedom due to the weak net mag-netic coupling of nearest-neighbor Fe sublattices. TheAFM ordering in parent  A Fe 2 As 2  compounds below  T  N  is observed to occur either simultaneously with or aftera structural transition from tetragonal to orthorhombicat  T  S  .The magnetic excitations in  A Fe 2 As 2  in stripeAFM ordered parent compounds have been extensivelystudied 13–17 and indicate itinerant spin waves whose ex-citation spectrum can be adequately described by largenearest ( J  1 ) and next-nearest-neighbor ( J  2 ) in-plane ex-change constants, a substantial interlayer coupling ( J  c ),and strong Landau damping  γ  .The primary role of doping is to suppress both theAFM and structural transitions, resulting in strong AFMspin fluctuations. In the context of establishing the con-nection between spin fluctuations and superconductivity,it becomes imperative to examine the evolution of thesespin excitations across the entire phase diagram of doped A Fe 2 As 2 . There have been several theoretical and ex-perimental studies on the effects of charge doping on thespin excitations in the superconducting compounds. 18–23 In general, these studies show that the normal state spinfluctuations are quasi-two-dimensional (2D) and stronglydamped while remaining peaked at  Q AFM , the magneticordering wavevector of the stripe AFM state. It is un-clear if the quasi-2D nature and strong damping are fea-tures that appear only with sufficient doping, or whetherthese are common features of the spin fluctuations in thetetragonal phase.In order to address these questions, we examine thespin correlations that occur in the parent compoundCaFe 2 As 2  above the AFM ordering temperature  T  N  .Inelastic neutron scattering data on CaFe 2 As 2  single-crystals show that above the simultaneous first ordertransitions at  T  N   =  T  S   = 172 K, 24 the spin gap col-lapses and spin wave scattering centered at the stripeordering wavevector  Q AFM  is replaced by short-rangedand quasi-elastic AFM correlations that extend up to atleast 60 meV. Just above  T  N   (at  T   = 180 K) the low en-ergy magnetic response is quasi-elastic with anisotropicin-plane correlations. We find the in-plane correlationlength to be  ξ  T  +  ≃  8 ˚A along the orthorhombic  a -axisand  ξ  T  −  ≃ 6 ˚A along  b . Weak modulations of the scat-  2 FIG. 1:  Summary of scans performed at  T   = 10 K, 140 K (emptysymbols) and 180 K (filled symbols) on HB3 ( E  f   = 41.2 meV) andHB-1A with spectrometer configurations described in the text. (a)shows ( hhL ) plane in reciprocal space where the scans at HB3 wereperformed. (b)-(f) shows the various cuts investigated, as indicatedin (a). (g) and (h) show the scans performed in the ( h 0 L ) planeusing HB1A. No diffuse magnetic scattering was observed in the( h 0 L ) plane. The magnetic signal is only observed centered atwavevectors  Q = Q AFM . tered intensity are also observed along the  c -axis indi-cating a two-dimensional character to the paramagneticfluctuations. In general, spin correlations weaken andbroaden further in momentum and energy with increas-ing temperature, but are still observed up to the highestmeasured temperature of 300 K. These observations canbe explained in the context of spin dynamics overdampedby particle-hole excitations. In particular, we use a phe-nomenological theoretical model with in-plane and inter-plane magnetic anisotropy to consistently fit our data forall temperatures, obtaining the ratios  J  1 /J  2  ≃  0 . 55 and J  c /J  2  ≃  0 . 1. We find that the spin fluctuations in theparamagnetic phase of the parent compound bear a closeresemblance to the paramagnetic fluctuations in the su-perconducting compositions.This article is laid out as follows. In section II below,the experimental conditions under which the experiment FIG. 2:  Temperature evolution of the neutron scattering signalmeasured on HB3 at ¯ hω  = 10 meV. (a) Background estimatemeasured away from  Q AFM  at (002) (solid symbols) and (003)(open symbols) showing no anomaly at  T  N  . Solid lines are lin-ear fits to the temperature dependent intensity. (b) Intensity at Q AFM  = (1 / 2 ,  1 / 2 ,  1) and (1 / 2 ,  1 / 2 ,  0). The solid line is thenon-magnetic background estimate obtained from averaging thefits at (002) and (003), shown in panel (a). were performed and the sample details are presented.The data analysis and results are presented in sectionIII. Finally, a discussion and a summary are given in sec-tion IV. II. EXPERIMENTAL PROCEDURES Inelastic neutron scattering measurements were per-formed on a single crystal mosaic ( ∼ 400 small single-crystal samples) of CaFe 2 As 2  with a total mass of  ∼  2 grams that are co-aligned to within 1.5 degreesfull-width-at-half-maximum (FWHM). The preparationmethods of the single-crystals have been describedelsewhere. 24 Data were collected using the HB3 andHB1A triple-axis spectrometers at the High Flux Iso-tope Reactor at Oak Ridge National Laboratory andthe MAPS chopper spectrometer at the ISIS facility atRutherford Appleton Laboratory. HB3 was operated inrelaxed resolution for measurement of the diffuse scat-tering signals in the paramagnetic phase, with fixed finalenergy ( E  f  ) configurations,  E  f   = 14.7 meV and 41.2meV, and 48’-60’-80’-120’ collimation. The sample wasmounted in a closed-cycle refrigerator and oriented forscattering in the tetragonal ( hhL ) plane. HB1A was op-erated with fixed incident neutron energy of 14.7 meVand 48’-40’-40’-136’ collimation and the sample mountedin the ( h 0 L ) plane. The MAPS experiment was per-formed at  T   =180 K, with an incident energy of 100 meVusing the same sample aligned with the  c -axis along theincident beam direction.To avoid confusion, the data is exclusively presented intetragonal units and we define  Q  =  2 πa  ( h i +  k  j ) +  2 πc  L k as the momentum transfer indexed according to the I  4 /mmm  tetragonal cell with lattice parameters a  = 3.88˚A and  c  = 11.74 ˚A at 300 K. The vectors i ,  j  and k  are thefundamental translation unit vectors in real space. Forcomparison with the AFM low temperature orthorhom-  3                                                                                                                                                                                                                                                                   FIG. 3:  (Color online).  L  and  h -dependence of the scatter-ing at ¯ hω  = 10 meV measured on the HB3 instrument with E  f   =41.2 meV. The scans are performed along the (1 / 2 ,  1 / 2 , L )and ( h, h,  3) directions for temperatures  T   =300 K [(a) and (b)]and  T   =180 K [(c) and (d)]. The solid lines in (a) and (c) corre-spond to fits to the dynamical susceptibility described in Eq. (5).The solid lines in (b) and (d) are guide to the eye, and based onLorentzian fits to the data. The dashed line is an estimate of back-ground scattering. Panels (c) and (d) show the sharper magneticscattering at 180 K. bic ( o ) structure, we note the following relations betweenthe Miller indices of the two phases,  h =( H  o  +  K  o ) / 2, k =( H  o − K  o ) / 2, and  L = L o . For convenience, we some-times use the reduced momentum transfer  q = Q − Q AFM in reciprocal lattice units (rlu) where  Q  is the momen-tum transfer to the sample and  Q AFM =( h 0 ,k 0 ,l 0 ) thereciprocal lattice vector which defines the AFM low tem-perature zone center. Typical AFM wave vectors studiedare (1/2, 1/2,  L ) with  L  = odd.After suitable subtractions of the non-magnetic (back-ground) scattering, the observed magnetic inelastic neu-tron scattering data is cast in terms of the dynamicalstructure factor  S  ( Q ,ω ), which is related to the imagi-nary part of the dynamic spin susceptibility  χ ′′ ( Q ,ω ) viathe fluctuation-dissipation theorem, S  ( Q ,ω ) = 2( r 0 ) 2 F  2 ( Q )4 πµ 2 B χ ′′ ( Q ,ω )1 − e − ¯ hω/kT   (1)where ( r 0 ) 2 = 290 . 6 mbarns Sr − 1 is a conversion fac-tor to bring the intensity into absolute units of mbarnsmeV − 1 Sr − 1 f.u. − 1 (Sr=Steradian, f.u.=Formula Unit)and  F  ( Q ) is the magnetic form factor for the Fe 2+ ion. III. ANALYSIS AND RESULTSA. Survey of reciprocal space Fig. 1 shows the ( hhL ) plane in reciprocal space wherethe low energy measurements were performed. It alsoshows several  Q -scans taken at temperatures below (10K and 140 K) and above  T  N   (180 K) the concomitantstructural and N´eel ordering temperature  T  N   =  T  S   = 172K. Below  T  N  , magnetic Bragg peaks appear at  Q AFM  =(1 / 2 ,  1 / 2 , L ) positions with  L  = odd that describe theordered AFM stripe structure. Figs. 1(b)-(f) show var-ious cuts, as indicated in Fig. 1(a), through the ( hhL )scattering plane at a finite energy transfer of 10 meV and E  f   = 41.2 meV. Similar scans performed with  E  f   = 14 . 7meV show qualitatively the same results. Below  T  N  ,sharp excitations are observed at  Q AFM  that srcinatefrom very steep spin wave excitations in the orderedstate. 13,15 The difference in sharpness of the spin wavepeaks in the [ h,h, 0] and [0 , 0 ,L ] directions is due tothe anisotropy in the spin wave velocity, as discussedin Ref.[15]. When the sample is warmed up above  T  N  ,strong intensity remains at  Q AFM  position with muchbroader lineshapes indicating short-ranged AFM spincorrelations. The scans shown in Fig. 1 are stronglyinfluenced at higher angles by the presence of aluminumphonon scattering from the sample holder and low anglescattering from the direct beam. Both contributions leadto very high background levels and limit the range of the Q -scans.Since the stripe AFM ordering may be frustrated in thetetragonal structure, we searched for evidence of spin cor-relations at other wavevectors in addition to the strongcomponents of the diffuse excitations near  Q AFM . Nomagnetic diffuse scattering was observed along varioussymmetry directions in the ( h 0 L ) plane. In particular, noevidence of magnetic scattering was seen at the wavevec-tor (1,0, L  = even) corresponding to N´eel (C-type) AFMfluctuations (see Figs. 1(g) and (h)). In the ( hhL ) plane,there are indications of weak peaks in the extended  Q -scans at wavevectors other than  Q AFM  which may arisefrom additional magnetic modulations in the paramag-netic phase. For example, very weak peaks can be ob-served at (002) and (003) (see Figs. 1(b), (d), and (e))which would correspond to the presence of ferromagneticcorrelations and A-type magnetic correlations (ferromag-netic within the layer, AFM between layers), respectively.In order to understand the development of correlationsat  Q AFM  and address the potential existence of addi-tional magnetic modulations (002) and (003), the tem-perature dependence was measured at various points inthe ( hhL ) plane at an energy transfer of 10 meV. Fig.2(a) shows the temperature evolution of the scatteredintensities at (002) and (003). The intensities show noanomaly at  T  N  , but rather the intensity increases linearlywith temperature as expected for a phonon background.This background is observed throughout the scatteringplane and partly arises from aluminum phonon scatter-  4 FIG. 4:  (Color online) The dynamic magnetic susceptibility of CaFe 2 As 2  as a function of energy at  Q AFM  = (1 / 2 ,  1 / 2 ,  3) for T   = 140 K (blue solid squares),  T   = 180 K (grey solid circles) and T   = 220 K (red solid diamonds). Above the magneto-structuraltransition  T  N   =  T  S  = 172 K, a broad magnetic spectrum is ob-served as quasi-elastic response near Q AFM . Data taken at  T   = 180K were fit to a Lorentzian form given in Eq. (2) convoluted withthe instrumental resolution (black solid line). The Lorentzian half-width Γ T   at  T   = 180 K is 10 meV. At  T   = 220 K, we estimate theenergy linewidth to be ∼ 13 meV using the expression Γ T   =  γ  (  aξ T  ) 2 and the fitted value of   γ   (temperature independent Landau damp-ing defined in the text) and that of the correlation length  ξ T   at 220K. The calculated Lorentzian scattering at  T   = 220 K is shown asa red solid line. In contrast, sharp spin waves having an energy gap∆ of   ∼ 7 meV are observed in the ordered phase at  T   = 140 K. ing from the sample holder, with little or no magneticcontribution. We can use the intensity at (002) and (003)as an estimate for this phonon background. In Fig. 2(b),the temperature dependence at  Q AFM  = (1 / 2 ,  1 / 2 ,  1)and also (1 / 2 ,  1 / 2 ,  0) is shown. The intensity at  Q AFM increases from low temperatures as expected for the in-creasing Bose population factor of the low lying spin wavemodes. At  T  N  , there is a sharp drop in the intensity con-sistent with the first-order transition to the paramagneticstate. The paramagnetic intensity decreases above  T  N  and is nearly at background by 300 K. At (1 / 2 ,  1 / 2 ,  0)the intensity is nearly at background level below  T  N  , in-creases sharply at the transition, and decreases slowly athigher temperatures.The  L − dependence of the paramagnetic scattering isconsistent with weak antiferromagnetic correlations be-tween layers atop a constant magnetic background, asillustrated in Fig. 2(b). Figs. 3(a) and (c) show the  L -dependence of the scattering at ¯ hω  = 10 meV and alongthe (1 / 2 ,  1 / 2 , L ) direction for  T   = 300 K and 180 K,respectively. The  L -dependence displays sinusoidal vari-ation with maxima at odd values of   L . The lines showncorrespond to fits to the dynamic susceptibility and willbe described in detail below. The intensity drops by afactor of 2 between 180 K and 300 K indicating the grad-ual evolution of the system to less correlated quasi-2Dspin fluctuations similar to the reduction in intensity for FIG. 5:  (Color online). Magnetic excitations in CaFe 2 As 2  mea-sured on the MAPS spectrometer with an incident energy of  E  i  = 100 meV at  T   = 10 K (left panel) and 180 K (right panel).The data shows magnetic intensity as a function of the [ h, h ] di-rection and the energy transfer after averaging over the transverse[ h, − h ] direction in the range 0.4 < h < 0.6. Given the fixed crystalorientation with incident beam along  L , the  L  component of thewave vector varies with the energy transfer as indicated. Excita-tions below  T  N   are consistent with steep spin waves and diffusemagnetic excitations are observed above  T  N   at  T   = 180 K. cuts along the [ h,h, 3] direction shown in Figs. 3 (b)and (d). The temperature dependence of Al phonons isprimarily responsible for the increase in background be-tween 180 K and 300 K. However, the constant magneticbackground itself is also weakly temperature dependentas inferred from Fig. 2 (a). We return to this in sectionIIIC.In all, the surveys of magnetic scattering intensitiesabove  T  N   in the ( hhL ) and ( h 0 L ) planes indicate thatthe AFM spin correlations are restricted to the vicinity of  Q AFM , the wavevector of the stripe ordered phase. Oneessential difference in the paramagnetic phase is an in-crease in the  c -axis anisotropy and tendency towards 2Dspin fluctuations, as indicated by the weakly modulatedrod of scattering along  L . This is entirely analogous tothe behavior of AFM spin fluctuations in the optimallydoped superconductors, where interlayer correlations arevery weak 5,23 . B. Energy dependence Fig. 4 depicts the dynamical structure factor χ ′′ ( Q AFM ,ω ) / ¯ hω  at  Q AFM  = (1 / 2 ,  1 / 2 ,  3) and ¯ hω <  22meV for temperatures above and below  T  N  . In order toobtain  χ ′′  from the raw data, a non-magnetic backgroundwas estimated by measurements at  Q  = (0 . 35 ,  0 . 35 ,  3)and subtracted. Below  T  N  , the low energy magneticspectrum consists of spin waves with a sizeable spin gapof 7 meV (see Ref. 15). Just above  T  N  , the gap collapsesand the spin wave scattering is replaced by gapless, dif-fusive excitations. At 180 K, the diffusive excitations canbe fit to a quasi-elastic Lorentzian form, χ ′′ ( Q AFM ,ω )¯ hω  =  A (¯ hω ) 2 + Γ 2 T  (2)with an energy linewidth of Γ T   = 10  ±  1 meV. Theparameter  A  is an arbitrary intensity scale factor. As  5 FIG. 6:  (Color online). Constant energy slices (∆ E   =  ± 5 meV)through the excitation spectrum of CaFe 2 As 2  in the ( h,k ) planeat  T   = 10 K and 180 K, as observed on MAPS. Energy slicesare chosen to correspond to odd values of   L . Intensity shown isin absolute units (mbarn Sr − 1 meV − 1 per formula unit). Below T  N  , well defined spin waves are observed around  Q AFM  (see alsoRef. [16]). Above  T  N  , strong but short-range magnetic correlationsremain around Q AFM  and extend up to at least 60 meV. Solid linesin panel (c) show the directions along which the cuts in Fig. 7 aretaken, with (+) designating longitudinal cuts and (-) transversecuts. the temperature is increased, the Lorentzian half-widthgrows rapidly. At  T   = 220 K, the spectrum weakensconsiderably with temperature and appears flat withinthe energy range measured, thus the Lorentzian widthsbecome large and poorly defined. This rapid increase inthe quasi-elastic linewidth with temperature is explainedbelow. C. Spectrum of paramagnetic spin fluctuationsnear Q AFM At temperatures just above  T  N   ( T   = 180 K,  T/T  N   =1.05) we used the MAPS spectrometer to perform a de-tailed survey of the spin fluctuations in the paramagneticphase in the vicinity of  Q AFM . For detailed modeling, theMAPS measurements were normalized in absolute scat-tering units of mb Sr − 1 meV − 1 f.u. − 1 by comparison toa vanadium standard. Data were collected at 180 K andalso in the AFM ordered phase at 10 K (with an inci-dent energy of 100 meV). As in previous work, 16 we use   100200300200300400    I  n   t  e  n  s   i   t  y   (   A .   U .   ) 0.250.50.75 h in [hhL]100200300 -0.4-0.200.20.4 [1/2+  ε,  1/2-  ε,  L] 60 meV12 meV39 meV T=180 K L=1L=3L=5(+)(-) FIG. 7:  Longitudinal (+) and transverse (-) constant-energy cutsmeasured around  Q = Q AFM =(1 / 2 ,  1 / 2 , L ) on the MAPS spec-trometer. The cuts are taken at ¯ hω  =12, 39 and 60 meV, whichcorrespond to  L  = 1 , 3 and 5, respectively. Solid lines are best fitsof Eq. (6) to the data. Dotted lines indicate the fitted background,and the dashed lines show the instrumental resolution function. the MSLICE program 25 to visualize the data and to takeone and two dimensional cuts through main crystallo-graphic symmetry directions for subsequent data analysiswith the TOBYFIT suite of analysis programs describedbelow. 26 Where possible, symmetry equivalent cuts andslices were added to improve statistics. Fig. 5 showsslices of the neutron intensity along the [ h, h ] directionas a function of energy transfer after averaging over the[ h, − h ] direction. Below  T  N  , the data in the left panel of Fig. 5 show a steep plume of intensity centered at  Q AFM arising from AFM spin waves. The sizeable exchangecoupling along  c  leads to variations in the structure factoralong  L , observed as energy-dependent intensity oscilla-tions that are peaked at the AFM zone centers; ¯ hω  = 12meV ( L =1), 39 meV ( L =3), and 60 meV ( L =5). Analy-sis of the AFM spin wave spectra is described in detail inRef. [16]. Above  T  N  , the right panel of Fig. 5 indicatesthat the magnetic spectrum is much broader in  Q , andenergy dependent oscillations are much less pronounced,confirming short-ranged spin correlations within the Felayer and a weakening of interlayer correlations.Fig. 6 shows the neutron intensity for several constantenergy slices at  T   = 10 K and 180 K. The energies arechosen such that  L  = 1, 3, and 5, in order to measure thespin fluctuations in ( h, k )-planes containing  Q AFM  (i.e.measuring the spin correlationswithin the Fe layers). Be-low  T  N  , Figs. 6 (a)-(c) show three constant energy slices
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