REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 72, NUMBER 1 JANUARY 2001 Spin-resolved off-specular neutron scattering maps from magnetic multilayers using a polarized 3He gas spin filter B. Nickel, A. Ru¨hm,a) W. Donner, J. Major, and H. Dosch Max-Planck-Institut fu¨r Metallforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany and Institut fu¨r Theoretische und Angewandte Physik, Universita¨t Stuttgart, D-70569 Stuttgart, Germany A. Schreyer and H. Zabel Institut fu¨r Experimentalphysik/Festko¨rperphysik, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany H. Humblot Institute Laue­Langevin, 6, rue Jules Horowitz, BP 156, F-38042 Grenoble Cedex 9, France Received 18 April 2000; accepted for publication 3 October 2000 We report on the application of the polarized 3He gas spin filter technique for the measurement of spin-polarized diffuse neutron reflectivity with spin analysis. The spin filter together with a one-dimensional position-sensitive detector enables the simultaneous investigation of sections in reciprocal space while exploiting spin sensitivity. An example of diffuse neutron scattering from a Fe/Cr superlattice demonstrates the potential of the method. In addition we present a first step towards the interpretation of diffuse scattering from magnetic multilayers by exploiting the supermatrix formalism. © 2001 American Institute of Physics. DOI: 10.1063/1.1328403 I. INTRODUCTION which is caused by all local deviations from the average profile. This off-specular scattering is detected as a broad The magnetic structure of thin magnetic superlattices has diffuse background around the forward direction and exhibits attracted much attention in the past few years because of a characteristic angular dependence which eventually reflects their novel and widely tunable interlayer magnetic coupling correlations present between the local imperfections. Thus, phenomena and the associated magneto-transport properties.1 in order to ``reconstruct'' the microscopic details of mag- Various experimental techniques have been devised to un- netic roughness and of magnetic domains, it is mandatory to ravel the magnetic structure of these artificial nanostructures, explore the small diffuse background in the entire reciprocal among them polarized neutron reflectivity which provides space within the first Brillouin zone. Figure 1 b shows sche- detailed information about the laterally averaged magnetiza- matically the diffuse scattering profile associated with the tion profile across the multilayers.2 In the last several years it real structure of Fig. 1 a as a function of the inplane mo- became apparent that the magnetic roughness of the inter- mentum transfer q : The broad featureless component re- faces and the magnetic domains within the ferromagnetic sults from the local uncorrelated steps and kinks at the inter- layers shown schematically in Fig. 1 a play an important faces, and its inverse width is given by the inplane role in understanding of the observed average magnetic in- correlation length R which is assumed here to be micro- terlayer coupling; furthermore, magnetic roughness and mag- scopically small. The other component may be termed small netic domains and in particular the spatial correlations be- angle scattering from magnetic domains of submicrometer tween them seem to be crucial for the performance of such size D . Correlations of roughness or domains between lay- devices.3 The microscopic characterization of such magnetic ers are observed by a characteristic modulation of this dif- imperfections demands techniques which are, on the one fuse scattering as a function of the normal momentum trans- hand, sensitive to local magnetic moments and their spatial fer qz see, for example, Ref. 4 . distribution and, on the other hand, suited to study these real Due to the small amount of local scattering centers in structures in buried layers and interfaces. We will show in highly perfect multilayer systems, the magnetic off-specular this article that spin-resolved off-specular neutron scattering diffuse scattering intensity is usually rather low and, if de- is able to tackle this task, if a proper experimental setup is tectable at all, partially buried under the off-specular scatter- employed. ing contribution from the chemical roughness of the inter- Any diffraction pattern from imperfect magnetic multi- faces. X-ray scattering techniques appear at first to be more layers can be divided into two contributions see Fig. 1 , the favorable than neutron scattering because of the availability specular reflected intensity characterized by inplane mo- of highly brilliant synchrotron sources which provide a large mentum transfer q 0) originating from the laterally aver- x-ray flux at the sample surface. Indeed, several experiments aged density profile and the off-specular diffuse scattering have recently been performed to explore the potential of magnetic off-specular x-ray scattering.5­7 It turns out, how- a ever, that, despite the success of these experiments, magnetic Present address: Advanced Photon Source, IMM-CAT, 9700 South Cass Avenue, Argonne, IL 60439-4832; electronic mail: off-specular x-ray scattering is hampered by the very low adrian@slate.imm.aps.anl.gov magnetic x-ray scattering cross section and the rather com- 0034-6748/2001/72(1)/163/10/$18.00 163 © 2001 American Institute of Physics Downloaded 02 Oct 2001 to 148.6.178.100. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp 164 Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Nickel et al. FIG. 2. Outline of the experimental setup. The neutrons are supplied by a neutron guide NG and diffracted by a monochromator M . After passing a slit, they are polarized by a transmission mirror PM and may be flipped by a Mezei flipper SF1. The sample S is placed in an external field B and can be inclined by an angle i . The reflected or scattered neutrons pass a second flipper SF2 before entering the 3He vessel placed in a homogeneous field B0. PSD acquires the exit angle profiles containing the specular rod SR ( i f) together with the off-specular scattering. can be achieved with 3He filters, they have been proposed as a tool with which to test the electroweak standard model.21 Application in neutron crosscorrelation spectroscopy has also been suggested.22 Further applications of 3He are de- scribed in Ref. 23. Very recently, these filters have been used for polarization analysis in bulk small-angle neutron scattering.24 So far, 3He filters have not yet been utilized for polarized neutron reflectometry or studies of magnetic mul- FIG. 1. a Sketch of an antiferromagnetically ordered magnetic multilayer tilayers like that presented in this article. It is expected, how- exhibiting local imperfections and domains. b Diffraction pattern corre- ever, that the performance25,26 and availability of these de- sponding to a . vices will improve in the near future. As indicated by the results of this article, they could then most effectively be plicated magnetic x-ray interaction potential which seems to used in polarized neutron reflectometry, especially in combi- prevent direct, quantitative access to the local correlations of nation with one- and two-dimensional multidetection the magnetic moments. schemes.24 Off-specular diffuse neutron scattering would in fact In this article we describe the first test of such a neutron avoid all the aforementioned problems associated with x scattering setup realized at the evanescent neutron wave dif- rays, since it is governed by a rather simple scattering cross fractometer EVA which was equipped with a 3He filter and a section with direct access to the spin­spin correlation func- one-dimensional position sensitive detector. We discuss a pi- tion as was demonstrated recently for a single selected off- lot experiment performed on a Fe/Cr multilayer which is specular scan.8 However, it generally suffers from the low known to exhibit a peculiar noncollinear average magnetiza- brilliance of the available neutron sources, thus, the resulting tion profile.27 We further show that the so-called supermatrix very low scattering distribution within the first Brillouin formalism,28 recently developed to analyze polarized neutron zone can only be recorded within reasonable measuring reflectometry data, can be extended to rigorously and effi- times by using multidetection techniques. On the other hand, ciently analyze the spin-resolved diffuse neutron scattering the magnetic diffuse scattering must be separated from the maps obtained in such an experiment. nuclear diffuse neutron scattering arising from chemical roughness of the interfaces by using polarized neutrons and applying a spin analysis of the scattered neutrons. In order to II. EXPERIMENTAL DETAILS combine these two prerequisites, i.e., spin analysis of the Here in Sec. III we describe the essential features of the scattered neutrons and use of multidetector systems, we pro- experimental setup and the sample preparation, thereby fo- pose to use a spin-filtering technique for scattered neutrons cusing on the performance of the 3He filter. by means of a polarized 3He gas which allows one to simul- taneously analyze the polarization of each scattered neutron A. Outline of the off-specular scattering setup recorded in the multidetector system. The off-specular neutron scattering experiments were 3He spin filters have been developed, tested and im- performed at the evanescent neutron wave diffractometer proved since 1991,9­11 and are or will soon be available at EVA at the high flux reactor of the Institute Laue­Langevin several neutron laboratories.12­17 3He filters have already in Grenoble. EVA is designed as a test instrument to explore been successfully applied in experimental studies, e.g., on the potential of surface and interface scattering using tunnel- in-chain spin correlations in YBa2Cu3O6 x Ref. 18 and on ing neutron waves evanescent waves .29,30 A sketch of the neutron optics P-violation effects,19 as well as in paramag- experimental setup is shown in Fig. 2. The instrument is netic scattering experiments on a Fe0.77Ni0.13Mn0.07C0.03 supplied with neutrons of wavelength 5.5 Å by a recently alloy.20 Due to the extremely high degree of polarization that installed new horizontally focusing monochromator, which is Downloaded 02 Oct 2001 to 148.6.178.100. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Spin-resolved neutron scattering 165 made of five pieces of highly oriented pyrolytic graphite TABLE I. Time-dependence of polarization and transmission parameters of HOPG crystals with an individual mosaic spread of 0.3° our 3He gas spin filter, calculated for an initial 3He pressure p 1.45 bar and full width at half maximum FWHM . The focal length is a relaxation time 60 h. conveniently tunable by stepping motor controlled goniom- 0 h 24 h 48 h eter heads, and for reflectivity experiments the focal spot is % % % at the sample position to maximize the flux at the sample P surface. Furthermore, the reflected beam from the monochro- He 55 37 25 T 3.6 1.3 0.6 mator can be tilted vertically for experiments on liquid T 0.0062 0.017 0.0035 surfaces.31 p2 99.7 97.3 89.3 The monochromatic beam is subsequently polarized in the y direction by means of a polarizing transmission mirror ure 3 b shows the magnetic field inside the Braunbek coils PM with a polarizing efficiency of p1 97%. Using down- measured in the presence of the sample magnetization field stream Mezei flipper SF1 flipping efficiency f1 97.5%) the B 400 mT . polarization of the beam can be reversed into the y direc- Prior to the actual spin-resolved off-specular scattering tion. At the sample position a magnetic field B between 50 experiments the performance of the arrangements was tested and 400 mT can be applied parallel to the sample surface (y carefully without the sample and it provided the efficiency direction . The specularly reflected and off-specularly scat- of all components acting on the neutron spin. For each ex- tered neutrons are simultaneously recorded by a linear perimental condition this has been done following the proce- position-sensitive wire counter providing exit-angle scans dures described in Refs. 33 and 34 which allow a clear-cut ( f scans; see Fig. 2 . For analysis of the spin state of each determination of the transmission coefficients T , T of the recorded neutron in the detector a polarized 3He gas neutron setup as well as the aforementioned efficiencies of spin filter is installed between the sample and the detector p1 , p2 , f1, and f2. details of the 3He filter will be given below . In the spin- resolved operation mode, scans of the incidence angle B. Sample preparation i are performed and four f profiles are recorded at each i posi- In order to explore the potential of such an experimental tion, one for each of the four scattering cross sections setup to detect spin-resolved off-specular neutron scattering, , , and , where `` '' refers to the spin upward and `` '' to the spin downward state of the neutron. An f profile is recorded within typically 1000 s. The 3He spin filtering technique is based on the strongly spin-dependent neutron absorption cross section, which is 3000 bn for antiparallel and 5 bn for parallel neu- tron states with respect to the 3He polarization. For a given 3He polarization PHe, the neutron transmission T of the 3He filter can be written as13 T 0.5 exp * 1 PHe , 1 with the dimensionless opacity 0.073 bar 1 cm 1 Å 1 pl , where l is the length of the 3He cell and p the 3He gas pressure. The associated polarization of the neutrons after passing the 3He cell is then given by T p T 2 T . 2 T The 3He spin filter used in this test experiment is a cy- lindrical cell 100 mm in length and 50 mm in outside diam- eter the wall thickness is 5 mm . The cell is fabricated of duran glass with monocrystalline silicon caps of 4 mm thick- ness used for beam entry and exit and filled with polarized 3He gas of pressure p 1.45 bar, resulting in 5.82. After filling the cell with polarized 3He the polarization is 55%, but since it decays with a relaxation time 60 h, it drops from 55% to 33% within a typical measuring time of FIG. 3. a Magnetic field arrangement around the sample position and the 3He filter. S1 and S2 are the slits defining the angular resolution, here 0.5 24 h see Table I . In order to achieve this long relaxation mrad. The Mezei flippers are denoted SF1 and SF2. PM denotes the trans- time, the 3He vessel has to be installed in a homogeneous mission mirror and BC the Braunbeck coils. The sample S , the 3He spin magnetic environment assuring, as a rule of thumb, a small filter (3He , and PSD are also indicated. b Measurements of the homoge- magnetic field gradient (1/B)dB/dx of less than 3 10 4 neity of the transversal fields at the spin filter position as a function of the cm 1.13 In our setup this is realized by a so called Braunbek vertical coordinate parallel to the surface normal of the sample and the distance from the sample. The field at sample position is 56 mT. The fields coil arrangement32 around the 3He vessel see Fig. 3 a . Fig- were measured using a Hall probe. Downloaded 02 Oct 2001 to 148.6.178.100. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp 166 Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Nickel et al. FIG. 4. Schematic view of the sample structure. The terrace structure is due to the growth characteristics of the Nb buffer layer. we have chosen the model system Fe/Cr, which exhibits bi- quadratic coupling between adjacent Fe layers.3,27 A FIG. 5. multilayer sample with 200 periods of 19 Å Fe and 42 Å Cr i - f map generated from 170 i positions. The acquisition time was 650 s. Each point in the intensity map was averaged over 1 mm of the has been grown in the Bochum molecular beam epitaxy PSD 10 channels ( f 0.044°). The intensity ranges from 20 to 20 000 MBE system at 300 °C in ultrahigh vacuum better than counts and the scale is logarithmic. The contour lines are separated by a 10 10 mbar using growth rates of 0.16 Å/s for Cr and 0.2 factor of 2.78. Å/s for Fe on a Nb buffer Fig. 4 on an Al2O3 (11¯02) substrate.35,36 In order to maximize the neutron signal 200 the incidence angle i an f profile is recorded in the posi- Fe/Cr bilayers were grown on a large 5 5 cm2 substrate. tion sensitive detector for all spin combinations ( , For protection against oxidation, the sample was covered , , ). In this way one obtains spin-resolved off- with a 20 Å Cr layer.37 Reflection high energy electron dif- specular neutron scattering maps in the ( i , f) plane. In this fraction performed during growth indicated a smooth growth representation of the data the specularly reflected neutrons front with steps. Energy dispersive x-ray EDX analysis are displayed along the diagonal line corresponding to i gave the relative Fe and Cr concentrations which, combined f see also Fig. 2 . We have performed the experiment in with the measured superlattice period, provided the layer three modes with increasing spin sensitivity: the unpolarized thicknesses. X-ray scattering and EDX spectra taken from mode using an unpolarized neutron beam and no 3He filter the center and near the edges of the samples confirmed the which provides the total off-specular scattering map; the po- lateral homogeneity of the sample, obtained by continuous larized mode using a polarized neutron beam and no 3He rotation of the sample during growth. The chemical homo- filter which provides the ( )-map and the ( )-map; fi- geneity determined with a microprobe was better than 1%. nally, the spin-resolved mode using a polarized neutron The nature of the Nb/Al2O3 (11¯02 epitaxy induces a beam with the 3He installed which provides all four diffuse natural miscut of the Fe/Cr multilayer of 2.3° which in turn maps ( , , , ). In all experiments the incom- gives rise to a pronounced unidirectional step structure with ing neutrons are impinging perpendicular to the miscut- a high step density and a consequently small lateral step­ induced terraces of the sample. step distance of 50 Å.36 Since each Fe/Cr interface may reproduce this step morphology see Fig. 4 , we expect A. Unpolarized maps strong off-specular diffuse scattering which should exhibit a The measured ( i , f) map shown in Fig. 5 discloses a pronounced modulation in the z direction diffuse sheets . specular intensity along qz (the i f line and, superim- The polarized neutron reflectivity of this sample has been posed, strong diffuse scattering which is strongly bunched studied in detail in the past.27 A combined neutron reflectiv- into two diffuse sheets perpendicular to the specular rod. The ity and high angle neutron scattering study implied a com- specular intensity essentially consists of three features: the mensurate antiferromagnetic frustrated spiral structure of the total reflection regime is seen for small angles of incidence, a Cr which causes the noncollinear average coupling between full-order Bragg reflection from the Fe/Cr double-layer unit the Fe layers in this sample.27 For detailed microscopic un- cell occurs at i 2.6°, associated with a perpendicular mo- derstanding of local interlayer coupling the real structure of mentum transfer qz 0.104 Å 1. This periodicity (L 1) is the multilayer is crucial. mediated by the chemical unit cell and eventually by ferro- magnetic coupling present between adjacent Fe layers. The III. EXPERIMENTAL RESULTS additional Bragg reflection observed at the half-order (L 0.5) position is due to doubling of the unit cell caused by In order to record the off-specular scattering from the antiferromagnetic moments between adjacent Fe layers and Fe/Cr multilayer, the sample was magnetized parallel to the is thus purely of magnetic origin. The strongly bunched dif- sample surface with a small external field of B 56 mT in fuse sheets emanating from the full-order and the half-order the easy magnetization direction. This field was maintained Bragg positions are caused by the miscut-induced roughness during the experiment and defined the quantization direction of the Fe/Cr interfaces which is apparently strongly corre- for the impinging neutrons which was, thus, collinear with lated in the growth direction. We note that the diffuse scat- the average magnetization of the sample. For each setting of tering is almost constant in the inplane direction the distor- Downloaded 02 Oct 2001 to 148.6.178.100. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Spin-resolved neutron scattering 167 FIG. 6. Same representation as that in Fig. 5, but using polarized neutrons. The image on the left corresponds to neutrons, and the right image to neutrons. The counting time was 2300 s. Intensities are plotted from 60 to 60 000 counts on a logarithmic scale. The contour lines are separated by a factor of 2.36. tion of the isointensity contours visible at the half-order diffuse sheet is due to Yoneda scattering . This observation can be understood in a straightforward way by the short in- plane length scale of approximately 50 Å which gives rise to this smooth intensity distribution along the half-order and full-order sheet. The noticeable asymmetry in the diffuse in- FIG. 7. Four cross sections as a function of the incident and exit angles. The tensities in the q direction along the sheets is caused by data represent f spectra recorded at 20 different i positions. The record- the unidirectional step structure of the sample38 and by the ing time per spectrum was 2200 s. The contour lines range from 100 to 3000 asymmetric resolution function.39 counts on a logarithmic scale and are separated by a factor of 1.53. B. Polarized maps spin-flip scattering. The spin-flip nature of the diffuse sheet at the half-order peak gives direct evidence that the local The measured polarized contour maps for parallel magnetic roughness at the interfaces which is antiferromag- and antiparallel ( ) incident neutron spin are depicted in netically correlated between adjacent layers has predomi- Fig. 6. On the specular rod, a shift of the critical angle be- nantly perpendicular components with respect to the average tween the and the ( ) states is observed, which is in- magnetization of the multilayer. Interestingly, the spin- dicative of the net magnetization in the sample. In addition, resolved maps disclose that all features of the specular rod, the intensities both specular and off specular around the in particular their spin-flip and nonspin-flip behavior are full-order position (L 1) differ by a factor of 4 between the identically reproduced in the associated diffuse sheets. Since and the ( ) maps. Both observations give unambiguous the spin-flip and nonspin-flip character of the diffuse and the evidence for significant ferromagnetic components in both specular intensity is virtually identical, one is forced to con- the average magnetic structure, determined by the vertical clude that the vertical correlations of the laterally averaged arrangement and the magnetic properties of the Fe layers, Fe moments and of the local Fe moments are strongly re- and the distribution of defects responsible for the diffuse lated. scattering. The simultaneous observation of the antiferro- magnetic component half-order scattering and the ferro- magnetic component implies that the average and local Fe IV. DATA ANALYSIS moments in adjacent layers differ in direction and/or magni- The spin-resolved intensity maps presented in Sec. III tude. contain complete information on spin­spin correlations in the multilayer. If we write C. Spin-resolved maps s The vectorial local spin structures and their spatial cor- i r s j r si s j si r s j r si s j , 3 relation inplane and out of plane are unraveled by the four where si(r) and sj(r ) denote the spin densities at positions spin-resolved off-specular neutron scattering maps obtained r and r within layers i and j, respectively, then the first by use of the 3He filter combined with a multidetection sys- term on the right-hand side is related only to the laterally tem. These maps, which are the primary results of our study, averaged spin structure that determines the specular reflec- are summarized in Fig. 7. The most apparent effect of the tivity. The specular reflectivity of the noncollinear spin struc- spin analysis is that it subdivides the diffuse maps into two ture in our Fe/Cr sample has already been discussed in Ref. regimes indicated by the dashed lines: for angles i and f 27. Here we focus on discussion and theoretical treatment of above the dashed lines the off-specular scattering is essen- the diffuse scattering distribution which arises from the sec- tially of a nonspin-flip character, while for smaller angles ond term on the right-hand side of Eq. 3 . To that end we below these lines the off-specular intensity is dominated by extended the supermatrix formalism recently developed for Downloaded 02 Oct 2001 to 148.6.178.100. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp 168 Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Nickel et al. FIG. 8. Pure specular reflectivity closed circles and line scans parallel to the specular rod extracted from Fig. 5 curve type c for different q offsets. polarized neutron reflectivity28 to off-specular scattering from arbitrary average spin configurations. An outline of the theory is given in the Appendix. FIG. 9. Line scans along the two diffuse sheets shown in Fig. 5 curve type Before we compare the result of our calculations with b at the half-order position top and the full-order position bottom . the two-dimensional experimental intensity maps, we give direct interpretation of line scans extracted from these maps. display virtually the same widths as the corresponding Selected line scans provide interesting insights into specific specular peaks, one is forced to conclude that in particular features of the magnetic roughness. In Fig. 5 three such ge- the vertical correlations of the laterally averaged magnetiza- neric scans are indicated as straight lines, denoted a, b, and c. tion and those of the local scattering centers at the rough 1 Scan a specular scan : This scan measures the lateral interfaces are intimately related. average of the nuclear and magnetic scattering length In Fig. 9 we show diffuse q scans across L 1/2 and 1. density. Within the q range screened in this feasibility study, we 2 Scan b diffuse q observe a smooth diffuse intensity since we pick up only a scan : This scan selects the spin­spin correlations within the layers. Such q small portion of the entire, rather broad diffuse scattering scans have re- cently been presented and qualitatively discussed.8 envelope. Note that a peak width of 0.12 Å 1 in q would be 3 Scan c diffuse q expected for 50 Å correlations which would correspond to z scan : This type of scan gives access to the local spin correlations between the layers. Such the calculated length of the miscut terraces in our sample. In diffuse scans are nowadays standard in x-ray cases,40 but order to determine the width of this broad feature experimen- have so far not been considered in detail in magnetic tally, the accessible q range would need to be properly ex- neutron scattering. tended, e.g., by also examining the transmitted beam.41 A closer inspection of the inplane scans in Fig. 9 reveals a In Fig. 8 we show the measured specular scan and dif- further larger length scale in the diffuse scattering across fuse qz scans for various values of q 0.0003, 0.0006, and the half-order reflection. This additional scattering is caused 0.0009 Å 1. The vertical dashed line separates the spin-flip by magnetic domains with a typical lateral length scale of 2.5 SF and nonspin-flip NSF regimes observed in the spin- m which are antiferromagnetically ordered along the resolved measurements. The apparent similarity of the growth direction of the multilayer. specular and nonspecular peaks is intriguing. Their widths We switch now to discussion of the two-dimensional are indicated in Fig. 8 and allow one to determine the length intensity maps in comparison with model calculations based scales over which the average scattering length density as on the novel supermatrix formalism for off-specular scatter- well as local scattering centers at the interfaces are periodi- ing described in the Appendix. The rigorous theoretical treat- cally correlated: the noticeable width 0.0842 Å 1 of the ment of off-specular diffuse scattering has been worked out specular peaks implies that the vertical correlation length of for x rays in the last decade42 and is nowadays almost rou- the layered structure in our multilayer is limited to about tinely applied to analyze the chemical roughness morphology 2 / 750 Å, i.e., 12 bilayers. This reduced correla- of thin layers and multilayers.38 In the case of magnetic tion length may originate from growth-related layer thick- roughness studied with neutrons this description becomes ness variations or from a diffusion-mediated compositional considerably more complicated, for example, because of the inhomogeneity of the layers due to aging of the sample. Al- vector character of the local magnetic-field distribution though the lateral homogeneity of the vertically averaged which imposes a severe challenge to proper theoretical treat- chemical composition and the overall structure e.g., total ment of the spin-dependent scattering cross section and the thickness of the multilayer was confirmed using EDX and associated spin-flip processes. A recently presented superma- x-ray scattering see Sec. II , local sample characteristics trix approach28 allowed us to readily describe the spin- may vary laterally within each layer or as a laterally inho- resolved neutron reflectivity from arbitrary, e.g., noncol- mogeneous function of depth. Since all nonspecular peaks linear, multilayer structures. The full potential of the spin- Downloaded 02 Oct 2001 to 148.6.178.100. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Spin-resolved neutron scattering 169 resolved off-specular maps presented here, however, can only be exploited when such supermatrix techniques are ex- tended to arbitrary local spin structures. This in fact involves a twofold challenge: 1 the proper theory for local spin­spin correlations asso- ciated with magnetic interfacial roughness and magnetic domains within the multilayer and 2 the development of a convenient spin-dependent off- specular neutron scattering cross section. In the following we demonstrate that the supermatrix approach for specular reflectivity can be extended to off- specular diffuse scattering. An outline of the theory is given in the Appendix and focuses on the case of strongly confor- mal roughness. The calculated intensity maps are based on Eqs. A1 ­ A8 in which the lateral roughness structure fac- tor is modeled through a Gaussian, resulting in S(qx) (2 )3/2 2 2 x exp( 12qx/ x) after integration over the y direc- tion, i.e., the direction transverse to our position sensitive detector see Fig. 2 . According to our line shape analysis the multilayer exhibits coherent growth of approx 12 Fe/Cr bi- layers. For symmetry reasons, we used in our model calcu- lations 16 bilayers of 19 Å Fe and 42 Å Cr, and assumed 90° FIG. 10. Model calculation of the NSF top and SF bottom intensities coupling of two adjacent Fe layers. For simplicity, ideal ex- according to Eq. A7 for a 16 period Fe/Cr multilayer. The data are plotted perimental conditions have been assumed, e.g., an efficiency on a logarithmic scale, and the contour lines are separated by a factor of of 100% for both polarization and spin analysis, as well as 1.53. magnetically saturated Fe layers the presence of cap and buffer layers has been neglected in this first approach . For the combined use of a multidetection system and the polar- the roughness of all interfaces we used a common value of ized 3He gas filtering technique allowed us to obtain spin- 4 Å, close to the value determined in Ref. 36, and did not resolved off-specular neutron scattering maps from magnetic consider eventual systematic variations of the roughness with multilayers which carry detailed information on the spatial depth. The results of the calculations are shown in Fig. 10 distribution of magnetic roughness and magnetic domain after resolution correction taking into account the experimen- structures. Here in Sec. V we will discuss alternatives to our tal parameters of our experimental setup. As mentioned experimental setup and desirable improvements of the theo- above, the resolution and illumination corrections result in a retical treatment. asymmetry of the scattered intensity with respect to the The major disadvantage of the 3He spin analysis tech- specular path. No peculiar correlation function e.g., like that nique is its low detection efficiency of about 3%. With a in Ref. 38 associated with the step distribution was as- conventional setup encompassing a polarizing supermirror, a sumed, which would enhance the asymmetry of the scattered point detector, and variation of the exit angle, a scan of re- intensity with respect to the specular path. The main feature alistically 15­20 exit angles with statistical quality compa- of the experimental result, the spin/flip nature of the half- rable to that of our line scans could be obtained within the order sheet, is reproduced by our calculation. However, no recording time of our PSD spectra. However, even after additional spin/flip due to local spins misaligned with respect combining position sensitive detector PSD channels in to the mean magnetization of each Fe layer was taken into groups of 10, our f profiles still contain about five times account. Such ``loose spins'' and lateral or vertical correla- more data points. The net gain when using a 3He filter in- tions of them would not only lead to additional spin-flip stead of a conventional detection scheme is, thus, a factor of scattering, but also to an interesting magnetic field depen- 5. Such improved performance is especially desirable for dence of the off-specular scattering, which would be differ- studies of complex multilayers in which data points at many ent from the behavior of the specular scattering probing the different momentum transfers perpendicular to the surface average spin directions. Since it was suggested43 that loose are needed to characterize the system. In principle, the con- spins could mediate 90°-type or biquadratic coupling, it ventional setup described above could be improved by using would be interesting in the future to calculate their diffuse the polarizing supermirror in combination with a PSD, rather scattering contributions. than a point detector. Serious complications, however, would arise from the fact that the scattered neutrons leave the V. DISCUSSION sample surface under different exit angles and would, thus, hit the polarizing mirror under different angles. In this work we have focused on experimental aspects In the discussion of the q scans extracted from our data, and the feasability of spin-resolved off-specular neutron scat- it has become evident that for samples with a small lateral tering as well as on theoretical aspects. We demonstrated that coherence length a larger range in q would be desirable. Downloaded 02 Oct 2001 to 148.6.178.100. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp 170 Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Nickel et al. With the experimental setup presented here, this can be eas- volved. In the neutron case, one has to account for birefrin- ily achieved by moving the PSD to negative exit angles, i.e., gence, for both nuclear and a magnetic scattering contrast measuring in transmission geometry. Alternatively, the PSD and the corresponding interface­interface correlations, and, could be rotated by 90° into the y direction see Fig. 2 . This in semiclassical terms, for changes of the orientation of the scheme, which is also frequently used in x-ray neutron magnetic moment as the neutron waves propagate reflectometry,4 would provide spin-resolved qz­qy maps, in- through the multilayer precession , unless the magnetiza- stead of the qz­qx maps our i­ f maps correspond to. tions of all the layers are collinear with the neutron polariza- Note, however, that for laterally anisotropic, e.g. stepped, tion. Note also that a simplified kinematic approach, as is samples like our Fe/Cr multilayer these two maps would not frequently used for x-ray scattering see, e.g., Ref. 38 , be equivalent. While it might again be possible, although less would entirely neglect most of these effects and, thus, inher- effective, to use analyzers other than a 3He filter for such an ently not be able to provide correct information about the experiment, the unrivaled advantage of 3He filters becomes magnetic structure of the sample. Using a numerically more evident when used in combination with two-dimensional de- involved dynamic matrix formalism was therefore funda- tectors, which would allow one to probe both in-plane direc- mental to calculating the spin-resolved off-specular scatter- tions at the same time. This would improve the efficiency of ing from our sample. data collection by another factor of 100 compared to a con- In order to obtain pronounced off-specular diffuse scat- ventional point-detector scheme. Also, once a 3He filter is tering for this feasibility study, we chose a relatively com- filled with polarized helium, it is very convenient to use plicated multilayer sample that exhibited a large number of since it does not require alignment, as opposed to any sophis- Fe/Cr bilayers, a noncollinear spin structure, and a high, pre- ticated and expensive multimirror arrangement that one sumably depth-dependent interface roughness induced by a might consider as an alternative. For a further comparative growth-related miscut. While these sample properties tend to discussion of supermirrors and 3He filters we refer to the complicate data analysis, they in fact provided us with the literature.44,45 desired high diffuse scattering intensities. Note that, although It can be concluded that 3He filters, especially when used the high degree of polarization obtained with our 3He spin in connection with two-dimensional detectors, will provide a filter was associated with a low transmission, it was desirable promising tool for the investigation of advanced magnetic for this first study in order to clearly discriminate between materials. It is expected that such two-dimensional detection nonspin-flip and spin-flip intensities. With the expected im- schemes as those recently used for SANS24 will be available provements of future 3He gas spin filters, similar studies on for polarized neutron reflectometry in the near future. More- simpler samples with a smaller relative amount of diffuse over, an improvement of the 3He polarization from the cur- scattering should become feasible in which further details of rent 55% to 70% is envisaged,25,26 and this would, in our interface morphologies can be revealed. case, improve the degree of polarization by roughly a factor Having understood the gross features of our intensity of 3 and, more important, the transmission from the current maps, we are now working on finalizing the data analysis 3% to as much as 9%­10%. based on the recently developed supermatrix approach. This With regard to the analysis of our spin-resolved diffuse approach provides a convenient and efficient analytical tool intensity maps in light of magnetic roughness, we demon- with which to handle the complexity of such spin-resolved strated that simple assumptions about the average spin struc- diffuse scattering intensities from arbitrary spin structures. ture in the multilayer, about the magnetic scattering potential The first model calculations presented here clearly indicate at the interfaces and about the roughness correlations be- its future potential. We envisage that a supermatrix software tween the interfaces can already explain the gross features of code for analyzing such off-specular intensity maps will be off-specular scattering observed. Theoretical treatment of our available shortly on our homepage.46 data can easily be refined, both in some general and sample- related aspects: The effect of roughness can be included by ACKNOWLEDGMENTS subdividing the interfaces into slabs. Arbitrary roughness The authors thank Thomas Krist, Hahn-Meitner-Institut correlations between the interfaces can be modeled via Knn Berlin, Germany , who kindly provided the polarizing su- in Eq. A2 . An increase of the interface roughness from the permirrors. The support of the management and staff at the substrate to the surface can be introduced, the presence of Intitute Laue­Langevin is gratefully acknowledged. The which was indicated by a comparison of our neutron and work was funded by the Bundesministerium fu¨r Bildung und x-ray reflectivity data. Specific spin configurations and re- Forschung, Berlin, Germany under Contract No. 03- sulting scattering potentials at the interfaces can be tested. DO5MPG. However, due to the arguments given in the following two paragraphs, we deliberately restricted the analysis to the sim- plest treatment possible, on the one hand, in order to keep the APPENDIX: OUTLINE OF THE SUPERMATRIX computation times within a reasonable range and, on the APPROACH TO OFF-SPECULAR SCATTERING other hand, in order not to overinterpret the still relatively The total cross section for neutron reflection and diffuse limited data obtained from our highly complex sample. scattering from a multilayer is42 One has to emphasize that, compared to diffuse x-ray scattering from multilayers, the corresponding computations d tot A¯R ki for the polarized neutron case are considerably more in- d ki ,kf d f , A1 f Downloaded 02 Oct 2001 to 148.6.178.100. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/rsio/rsicr.jsp Rev. Sci. Instrum., Vol. 72, No. 1, January 2001 Spin-resolved neutron scattering 171 where R(k n n 1 n i) is the polarized neutron reflectivity, A ¯ is the netic field. For nqz , nqz 1, with qz as the z compo- cross section of the detected portion of the reflected beam, nent of the momentum transfer within layer n, it can be d /d is the differential cross section for diffuse scattering shown by evaluating the integrals in Eq. A4 that of polarized neutrons, and f is the solid angle subtended f i by the detector. The formalism developed in this Appendix is Cn 2/ 1/2 n n V n n . A6 proposed as a tool to quantitatively interpret off-specular The neutron states i,f i,f(zn) can be calculated scattering intensities from a magnetic multilayer in a spin- n using the supermatrix formalism developed in Ref. 28. If the resolved neutron experiment. Since the neutron optics are polarization states of the incident and scattered beam at the treated according to dynamical scattering theory, the formal- surface, defined by the polarizer and analyzer characteristics, ism is applicable even for highly perfect multilayers. are denoted i,f , and if i,f are the corresponding density We focus in what follows on the differential diffuse scat- matrices, then, after straightforward but somewhat lengthy tering cross section from a multilayer with small roughness. mathematics, one finally arrives at Then d /d can be represented in the form38 N d Ak4 N C 0 * n f 1 R fT, i pf 1 R fT n 1 d K , A2 16 2 nn CnCn n,n 1 where A is the illuminated surface area, k 0 the wave number W 1 R i i , A7 of the incident neutrons, and C ipi 1 R i n are the scattering amplitudes associated with the rough interfaces n 1, . . . ,N. The coef- ficients K or, in terms of the density matrices i,f, nn are determined by the roughness correlations among the interfaces. For simplifying the treatment here, we assume that all interfaces have the same lateral morphology, N 2 described by a structure factor S(q Cn Tr 1 R fT,i pf 1 R fT ) normalized to (2 )2]. n 1 Then Knn nn S(q ) holds for vertically uncorrelated in- terfaces and Knn 1*S(q ) in the case of perfect vertical correlations ideal conformal roughness .38,47 Since our ob- servation indicates a highly correlated roughness morphol- W 1 R i ipi 1 R i 0 1 R i , ipi 1 R i ogy diffuse Bragg sheets , we restrict our considerations here to the latter case, thus obtaining d N W 1 R f* . A8 4 2 i pf 1 R f* d Ak0/16 2 Cn S q . A3 n 1 Here W is Within the distorted wave Born approximation DWBA , the matrix elements Cn are given by47 N W S fT* . . . *S fT 2/ 1/2 n V n 0 S i* ...*S i , 1 n 0 0 n 1 C n 1 n f z V n z i z dz zn A9 zn where S i,f S ( pi,f ) are the transfer supermatrices,28 R i,f f z V n z i z dz, A4 R ( pi,f ) are the reflectance matrices, and pi,f are the vacuum wave numbers perpendicular to the surface. where i,f denote the initial and final neutron spin or In the scalar case x-ray case , and for a simple semi- wave function for the ideal system without roughness, and infinite substrate, the right-hand side of Eq. A8 may be V n(z) is the susceptibility contrast or scattering potential reduced to T fW T i 2 with the transmission coefficients T arising from the root mean square rms roughness n of 1 R . Then, the scattering cross section in Eq. A2 be- interface n, located a distance zn from the surface. As usual, comes we model the scattering potential by the error function in the form d Ak4 2 0 0 T f V 0T i 2S q , A10 1 z z d 2 3 V n n z 2 1 erf V 2 n , A5 which is consistent with the well-known expression for n grazing-angle scattering as given, e.g., in Ref. 47. with V 2 n (4 /k0)(N n 1 N n) as the scattering potential. Following the approach presented in Ref. 28, we define the 1 A. Fert, P. Gru¨nberg, A. Barthelemy, F. Petroff, and W. Zinn, J. Magn. operators N in terms of their eigenvalues and the vector of Magn. Mater. 140, 1 1995 . Pauli matrices and write N nuc mag nuc,mag 2 J. F. Ankner and G. P. Felcher, J. Magn. Magn. Mater. 200, 741 1999 . n Nn Nn bn with Nn 3 denoting the nuclear and magnetic scattering length densities D. T. Pierce, J. Unguris, R. J. Celotta, and M. D. Stiles, J. Magn. Magn. Mater. 200, 290 1999 . in layer n, and b 4 n the unit vector along the respective mag- V. Holy and T. Baumbach, Phys. Rev. B 49, 10668 1994 . 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