PHYSICAL REVIEW B VOLUME 55, NUMBER 22 1 JUNE 1997-II Morphology and indirect exchange coupling in Fe/Cr/Fe 110... trilayers Jo¨rg Schwabenhausen, Tobias Du¨rkop, and Hans-Joachim Elmers Physikalisches Institut, Technische Universita¨t Clausthal, D 38678 Clausthal-Zellerfeld, Germany Received 11 November 1996; revised manuscript received 18 February 1997 We investigated the indirect exchange coupling in W 110 /Fe/Cr/Fe/Cr films. Magnetic interface anisotro- pies of the W/Fe and Fe/Cr interface differ in sign and thus allow the preparation of Fe films with orthogonal adjustable uniaxial anisotropies. The bilinear (J1)and biquadratic (J2) exchange coupling constants were determined from magnetization curves independent of magnitude and sign. We determined J1 and J2 as a function of Cr interlayer thickness (tCr 0 ­ 4 nm and temperature (T 100­ 300 K . Moreover, we varied the morphology of the interlayer as determined by high-resolution low-energy electron diffraction using different substrate temperatures (Tp,Cr 100­ 500 K during evaporation of the Cr spacer. Results for the exchange coupling versus spacer thickness are related to the particular lateral thickness fluctuation of the spacer. Our results are in good agreement with the bilinear coupling predicted by ab initio theories. Extrinsic biquadratic coupling models explain J2 observed experimentally. S0163-1829 97 03922-2 I. INTRODUCTION face an interface diffusion involving two monolayers ML's may occur even at room-temperature deposition. The layer Magnetic exchange interactions between ferromagnetic by layer growth mode leads to the observation of a short films separated by nonmagnetic spacer layers1 have been in- oscillation period 2 ML of the bilinear coupling term27,23 in vestigated to a large extent because of both possible applica- addition to the long oscillation period 9 ML which still can tions exploiting the giant magnetoresistance effect2 and the be observed for a worse growth. The same long oscillation expectation of a new fundamental understanding of the na- period was also found for Fe/Cr 211 Ref. 28 and Fe/ ture of magnetism.3 The exchange interaction is usually Cr 110 superlattices29 in agreement with the theoretically dominated by a qualitatively well-understood4­6 Heisenberg- predicted period.30,5 Biquadratic coupling constants as re- type interaction bilinear in the films magnetizations and os- ported for Fe/Cr 100 trilayers or superlattices by several cillating in sign with increasing spacer thickness. An addi- authors23,10,11,7 considerably vary in strength. This can be tional biquadratic term supporting a 90° alignment of the understood from the fact that the structure of the spacer plays magnetization vectors was observed initially in Fe 100 / a major role or from the fact that for the 100 symmetry a Cr/Fe Ref. 7 and in Co 100 /Cu/Co.8 Since then it has been separation of the biquadratic from the bilinear term requires detected in several systems for a review see Ref. 9 and a careful analysis of magnetization data.23 recently has attracted great attention10­12 because the origin We present coupling data on the Fe 110 /Cr/Fe trilayer of the biquadratic coupling term is still an open question. with uniaxial anisotropies in the magnetic Fe 110 films Several models for its explanation considering intrinsic13­16 aligned orthogonally. Recently, it was shown that in a and extrinsic mechanisms17­20 have been proposed. Intrinsic trilayer with orthogonal uniaxial anisotropies bilinear and bi- models usually predict biquadratic coefficients which are 2 quadratic coefficients can be determined in a straightforward orders of magnitude smaller than bilinear coefficients and manner independently from each other from magnetization thus cannot explain the large experimentally values. Extrin- curves.12 In the present study we intentionally change the sic models take into account the nonideal structure of the morphology of the Cr 110 spacer layer by varying the sub- spacer layer, spatially varying thickness of the spacer,17 strate temperature during Cr deposition on the first flat loose spins in the spacer material,19 and even pinholes Fe 110 surface. The thickness distribution of the spacer through the spacer layer.20 layer was determined by high-resolution low-energy electron In many cases the interfaces in the layered structures de- diffraction HRLEED . We measured the exchange coupling viate far from the ideal interfaces. It has been shown that for different thickness distributions and as a function of tem- both bilinear and biquadratic coupling strongly depend on perature in order to assess the role of loose spins and pin- the interface structure21 and alloying.23 Obviously, a conclu- holes in our samples. For spacers deposited below room tem- sive experimental study of the coupling must include a de- perature the growth mode switches to a one-dimensional step termination of the spacer structure. structure thus allowing a quantitative test of extrinsic theo- One of the most frequently tested systems is the Fe/Cr ries of the biquadratic coupling. system. Similar lattice constants aFe 0.2866 nm and aCr 0.2886 nm and surface energies Fe 2.9 J m 2 and II. EXPERIMENT Cr 2.0 J m 2 promise a layer by layer growth of Cr on Fe close to systems which can be handled by theory. This The samples examined in this paper were deposited onto a simple growth mode was indeed observed for Cr films on W 110 surface by molecular-beam epitaxy. The base pres- perfect Fe 100 whisker surfaces22 although recent sure during deposition of the films was below 1 10 10 investigations23­26 show that at the open 100 Fe/Cr inter- mbar. Thicknesses of the films were determined by a 10 0163-1829/97/55 22 /15119 11 /$10.00 55 15 119 © 1997 The American Physical Society 15 120 SCHWABENHAUSEN, DU¨RKOP, AND ELMERS 55 MHz quartz monitor, which was calibrated by Auger elec- tron spectroscopy AES . The absolute error of the layer thicknesses given below is approximately 5%. The first Fe layer with fixed thickness of 18 atomic layers was deposited onto W 110 in an optimized mode described previously,31 starting at a substrate temperature Ts 300 K in order to avoid Stranski-Krastanov islanding, and raising the tempera- ture to 600 K with increasing thickness. This procedure leads to a smooth surface with an average terrace width of 50 nm.32 The wedge-shaped Cr spacer layer was grown at vari- ous substrate temperatures Tp,Cr 100­570 K. Then, the sec- ond Fe layer of equal thickness as the first Fe layer was grown at Ts 300 K, avoiding any interdiffusion. At Fe 110 /Cr interfaces no interdiffusion occurs at room temperature,33 in contrast to the observations for 100 interfaces.23­26 The second Fe layer was covered by additional 5 ML Cr for reasons of increased anisotropy. The coverage also serves as a protection against residual gas adsorption. All Fe and Cr layers grow pseudomorphically onto the first Fe layer pre- serving the 110 orientation, as was shown by LEED. Magnetization loops were measured in situ using the lon- gitudinal magneto optical Kerr effect with the external field in the plane of incidence. The angle of incidence was 15°. It was shown in Ref. 12 that the Kerr signal for this type of samples is proportional to m(H) m1(H) m2(H), mi be- ing the magnetic moment component parallel to the external field H. This is certainly a special property of our samples and geometry. The growth mode of the Cr interlayer as depending on the substrate temperature was determined by HRLEED in a sec- ond apparatus. The W 110 /Fe 110 base layer and the Cr layers were grown under similar conditions except that the lowest available substrate temperature for the HRLEED in- vestigations was 230 K. III. Cr INTERLAYER GROWTH For characterizing the growth of the Cr interlayer, we measured LEED spot profiles of the specular beam with nearly normal incidence 3.75° of the primary electrons. Parameters of the profiles were the temperature of the sub- strate during deposition of the Cr spacer layer Tp,Cr , the average thickness DCr in units of monolayers , DCr tCr /dz , with thickness tCr and layer distance dz aCr / 2, and the incident energy E of electrons, which in our case is equivalent to the normal component Kz 2( 2mE/ )cos of the scattering vector. For fixed pa- rameters, the reflected intensity was measured as a function of in-plane components of the scattering vector. In the fol- lowing we use a Cartesian system with the x and y axes FIG. 1. Contour lines of equal intensities in the specular LEED along the 001 and 11¯0 directions. We present the re- spot for samples W 110 /18Fe/3Cr with the Cr film deposited at flected intensity as a function of K sample temperatures T x and Ky in units of p,Cr 230, 350, and 450 K as indicated in the K figure. Kinetic energy of the electron beam near normal incidence is 1 1 ¯ 0 K1 1¯ 2 2 /aCr 31 nm 1. Contourlines of equidistant intensity values are shown in E 120 eV. Parallel components Kx and Kyof the scattering vector Fig. 1 for 3 ML Cr deposited on a smooth W 110 /Fe 110 are normalized to K11¯ K1 1¯0 . substrate at Tp,Cr 230, 350, and 450 K. The incident energy E 120 eV was adjusted close to an out-of-phase value, For preparation at Tp,Cr 230 K we observe a splitting of the Kzdz/2 3.5 (E 111 eV , where wavelets from neighbor- specular beam in two shoulders, spread along the 11¯0 di- ing atomic levels are out of phase. This diffraction condition rection. This splitting indicates a one-dimensional step struc- is well suited for a determination of the lateral step structure. ture in real space with step edges along the 001 direction. 55 MORPHOLOGY AND INDIRECT EXCHANGE COUPLING . . . 15 121 FIG. 3. Positions Ky of the shoulders in the specular spot pro- files see Fig. 2 , measured along the direction of K11¯vs the normal component Kz of the scattering vector, for the sample W 110 /18Fe/ 3Cr with the Cr film deposited at Tp,Cr 230 K. The mean number N of atomic rows in the steps resulting from the maximum separa- tion in the out-of-phase scattering condition is N 14. similar to those observed for homoepitaxial Fe 110 layers on Fe 110 see Ref. 32 and we will follow the guidelines FIG. 2. Spot profiles of the specular beam along the direction of given in this reference to extract quantitative information on K 1 1¯0 left panel and K 001 right panel at incident kinetic ener- the structure in real space. gies E as indicated in the figure, for the sample W 110 /18Fe/3Cr The positions K with the Cr film deposited at T y,0 of the shoulders are shown in Fig. 3 as p,Cr 230 K. Parallel components K a function of Kz . The distance between the shoulders is pro- x and Ky of the scattering vector are normalized to K1 1 ¯ K11¯0 . portional to the deviation of Kz from the in-phase value, For higher deposition temperatures Tp,Cr 350 and 450 K the except for Kz very close to the in-phase scattering vector, reflected intensity switches to a two-dimensional distribu- where Ky,0 shows a finite value Ky,0 0.007K11¯ independent tion, indicating a two-dimensional step structure with step of Kz . This behavior implicates the facet structure shown in edges along several directions. A characteristic change of the Fig. 4. The period G of the ridge structure causes the intensity distribution occurs between 350 and 450 K. For 350 Kz-independent shoulders appearing close to the in-phase K deposition the contour lines are formed hexagonally condition34 and can be estimated from the position G whereas for 450 K deposition they show a nearly fourfold (0.007) 1 143 (Gdy 29 nm . As was shown in Ref. 32 symmetry. From the width of the intensity profile at half the maximum separation between the shoulders at an out-of- maximum we roughly estimated mean values for the terrace phase condition, Ky is related to the number N of atomic widths N ¯(Tp,Cr) in units of dy 0.204 nm and found in- rows per terrace, N K11¯/ Ky for the regular step array creasing terrace widths with increasing deposition tempera- shown in Fig. 4. For Tp,Cr 230 K we result in N 14. Con- tures, N ¯(230 K 9, N¯ 300 K 10, N¯ 350 K 30, N¯ 400 sidering a statistical distribution of terrace widths, N denotes K 60, N ¯ 450 K 60, and N¯ 500 K 100, respectively. the maximum of the distribution function, N ¯ the mean value We also determined the terrace width for fixed T of terrace widths. p,Cr as a function of the Cr thickness D The roughness w h(r) h 2 , given by the mean Cr and found decreasing values for N ¯ with increasing D quadratic deviation of the height h(r) at site r from the mean Cr . For the case of Tp,Cr 230 K we will discuss the one- dimensional distribution of terrace widths in detail. Intensity profiles I(E) along the directions of K10 and K11¯ are shown in Fig. 2. Along the 11¯0 direction we observe a pair of symmetric shoulders, which can be separated from the cen- tral spike. As Kz approaches an out-of-phase energy 111 or 182 eV where the wavelets of neighboring levels are in antiphase, the shoulders take all of the reflected intensity and the distance of their maxima from the central peak increases with increasing deviation from the in-phase energy. Along the 001 direction the profiles only show the central peak superimposed by a homogeneous background independent of Kx . This feature again confirms that nearly all steps are along the 001 direction. The profiles for this sample are FIG. 4. Periodic triangular staircase model. 15 122 SCHWABENHAUSEN, DU¨RKOP, AND ELMERS 55 In the following we again discuss the results for Tp,Cr 230 K in detail, where the intensity profiles can be explained completely by the facet model. Using a model with statistically distributed angles of the facets kinematic scattering theory results in32 I00 1 sin2 Kzdz H 1 /2 I , 3.2 tot H 1 2 sin2 Kzdz/2 with H denoting the height of the facet structure as indicated in Fig. 4. The numerical result for Hobtained by a fit of this equation to the experimental data shown in Fig. 5 for Tp,Cr 230 K is H 3.3. Of course the roughness W is related to H. Assuming the facet structure of Fig. 4 the exact relation38 is W2 H2/12 H/6, resulting in W 1.2 in very good agreement with the independently determined value W 1.3. Combining the information of terrace width and height of the periodic structure we determine the period G see Fig. 4 of the ridge structure, G 2(H 1)N. For Tp,Cr 230 K we result in G 120, again in agreement with the magnitude G 143 as determined from the spot profile I(K1 1¯) . For in- creasing thickness DCr we observe an increasing height of the ridge structure. Because the terrace width decreased with increasing thickness, the period G shows to be nearly inde- pendent of DCr . The formation of the periodic step structure can be understood following the model of kinetic roughen- FIG. 5. Intensity I ing. A barrier prevents atoms to jump between different lev- 00 of the central spike normalized to the to- tally reflected intensity Itot . Itot by integration over the K interval els, however the atoms diffuse freely on a particular atomic Kx , Ky 0.1K1 1¯, for samples W 110 /18Fe/3Cr with the Cr film level until they stick at a step edge. In this layer-restricted deposited at Tp,Cr 230, 300, 350, 400, 450, 500 K as indicated in diffusion model the period G is given approximately32 by the figure. Full lines are fits of type I00 /Ttot exp (Kz G 2N 2 D, D representing the mean number of atomic Kz,n)Wdz 2 , resulting in roughness W as indicated in the figure. layers. G 122 for DCr 3 is in agreement with the ex- perimentally determined G thus providing a strong support height h¯, can be determined from the intensity profile near for this growth mode, which was also observed for Fe on the in-phase scattering condition, Kz,i 2 i/dz . For Kz Fe 110 . Kz,i the intensity of the central spike I00(Kz) normalized to For Tp,Cr 300 K this analysis of the intensity profiles the total reflected intensity Itot I(Kz)dK can be approxi- results in a period G 300. At higher deposition tempera- mated by the relation35 tures the analysis based on a periodic model is certainly no longer appropriate. The increase both of the terrace width I00 /Itot exp Kz Kz,i 2w2 . 3.1 and the roughness with increasing Tp,Cr might indicate a mo- notonously increasing period G with increasing Tp,Cr . We determined I00 /Itotfor the samples W 110 /18 Fe/3 Cr for varying deposition temperatures of the Cr film as shown in Fig. 5. For I IV. MAGNETIC DATA ANALYSIS 00 we inserted the intensity I(0) measured for K 0. Itotwas determined from the integration of I over the For the data analysis we assume a homogeneous in-plane K interval Kx , Ky 0.1K11¯. The roughness W w/dz re- magnetization in each of both layers pointing at angles i sults from the fit of Eq. 3.1 to the experimental data and is with respect to the external field, which was applied along indicated in the figure. We observed a strong increase of W the 001 axis of our samples see Fig. 6 . The idea of our for Tp,Cr increasing from 230 to 300 K. This is very surpris- method is that in Fe layer 1, in the following named the ing because the usual behavior is just the other way around, driver layer, the magnetization is fixed along the 001 direc- a smoother surface with increasing temperature. A related tion by a very strong uniaxial anisotropy. In Fe layer 2, phenomenon might be the reentrant layer-by-layer growth named sensor layer, the easy axis of the magnetization is observed for Pt on Pt 111 at low temperatures.36,37 A second oriented at 90° along the 11¯0 direction. However, the temperature region for a smoother surface, although pro- uniaxial anisotropy of the sensor is weak. Then the direction nounced more weakly, is near Tp,Cr 400 K. Above this tem- of the magnetization in the sensor layer rotates from its equi- perature I00 /Itot cannot be explained by a model with single librium position under the combined action of the indirect steps. A new maximum occurs for Kzdz/2 3.8 indicating a coupling and of the external field H. Knowing the value of severe structural change. This observation coincides with a the weak anisotropy, the sensor magnetization thus probes decrease of the Cr Auger signal. Presumably an interdiffu- the indirect coupling energy. sion of Cr and Fe starts at this temperature. In order to formulate the idea quantitatively we write 55 MORPHOLOGY AND INDIRECT EXCHANGE COUPLING . . . 15 123 TABLE I. In-plane anisotropy constants for driver and sensor layer as explained in the text. Effective second-order anisotropy constants K (1) (1) p (i) are K p (1) K p K4xy for the driver layer ( 1 0) and K (2) (2) p (2) K p K4xy for the sensor layer ( 2 /2). The thickness of the layers is t 3.6 nm. Constant Unit Driver (i 1) Sensor (i 2) K(i) s,p mJ/m2 0.42 4 a 0.54 3 b K(i) s,4xy mJ/m2 0.14 5 a 0.02 2 b K(i) v,p 105 J/m3 0.78 10 b 0.78 10 b K(i) v,4xy 105 J/m3 0.32 5 b 0.32 5 b tK(i) p mJ/m2 0.70 0.26 kK(i) 4xy mJ/m2 0.02 0.09 tKp (i) mJ/m2 0.68 0.35 aFrom Ref. 39. bThis work. 2) is crucial for the determination of coupling constants. Therefore we repeated the study of Ref. 39, resulting in FIG. 6. a Schematic cross section of samples W 110 / D2Fe/ slightly deviating values. Moreover, we determined K(2) p and DCrCr/D1Fe/5Cr. Fe layers of thickness ti Didz , number of layers K(2) 4xy explicitly for every sample and at every temperature of Di with distance dz 0.203 nm, are separated by a Cr spacer of measurement. For this purpose a part of every sample was thickness tCr DCrdz . Easy axis e.a. of each Fe layer is indicated not covered with the driver layer. in the figure. Directions of magnetic moments mi are defined by As long as K(i) (i) p /K4xy 1, the direction of the easy axis is angles i , assuming homogeneous magnetization in each magnetic either 001 or 11¯0 and determined by the sign of the layer. b Schematic cross section of a typical wedge sample of second-order constant, K(i) linearly varying spacer thickness. p . For the driver layer the volume type and both interface anisotropies have the same sign and (i) down the free enthalpy per area g( add up to the desired strong easy axis along 001 , K 0. 1 , 2) of the coupled p system: For the sensor layer the strong interface anisotropy of the W/Fe interface supports an easy axis along 11¯0 whereas the g 1 , 2 J1cos 1 2 J2cos2 1 2 volume type anisotropy has an easy axis along 001 . By varying the thickness of the sensor layer we can thus control JstH cos 1 cos 2 fk,1 1 fk,2 2 . 4.1 the anisotropy in the sensor layer in a wide range. For the The indirect exchange coupling between the two magnetic thickness t 3.6 nm of both Fe layers, which was fixed in Fe layers is represented by a bilinear and biquadratic cou- this study, values are summarized in Table I. pling term with constants J Qualitative insight is obtained by the following rigid 1 and J2 .The third term is the Zeeman energy per area of the magnetization in an external driver approximation. A weak-coupling energy J1 , J2 (1) field H equal film thickness t) of both Fe layers. Using the Kp with respect to the driver layer anisotropy leaves the notation of Ref. 39, the anisotropy energy per area for layer magnetization of the driver layer nearly locked for decreas- i is given by ing field values along the direction of the external field, i.e., 1 0. For a small deviation of 2 from equilibrium posi- f i i tion k,i tK p cos2 i tK4xysin2 icos2 i , 4.2 2 /2 we can describe the sensor anisotropy by an effective anisotropy term Kp cos2 2 of second order only K(i) (i) p and K4xy denoting the second- and fourth-order in-plane with K anisotropy constants. The anisotropy is a sum of volume type p K p K4xy . Equation 4.2 then simplifies to the rigid driver approximation: index v) and surface type index s) contributions: g 2 tK K i i i p J2 cos2 2 J1 JstH cos 2 . 4.5 p Kv,p 1/t Ks,p , 4.3 Minimization of g results in cos 2 (J1 JstH)/2(tKp J2). K i i i From cos 4xy Kv,4xy 1/t Ks,4xy . 4.4 2(HA) 0 we determine the bilinear constant, J1 JstHA and from the initial increase of cos 2(H) the bi- For the driver layer we take the surface anisotropy con- quadratic constant J2 . Hence, J1and J2 can be determined stants for Cr/Fe/Cr films as determined previously,39 see from the sensor magnetization curve independently from Table I. Volume anisotropy constants for the driver layer each other. Certainly, the rigid driver model is valid only for were assumed to be the same as for the sensor layer, because the approximation of weak-coupling constants. In general, the pseudomorphic growth of the whole trilayer implicates magnetization curves generated by minimization of the exact the same residual strain in driver and sensor layer and hence free enthalpy Eq. 4.2 are fitted to the experimental curves the same magnetoelastic contribution to the volume type with J1 and J2 as parameters. As long as J1 and J2 are small anisotropies. The anisotropy of the sensor layer W/Fe/Cr (i with respect to the anisotropy constant Kp both parameters 15 124 SCHWABENHAUSEN, DU¨RKOP, AND ELMERS 55 FIG. 7. Magnetization loops represented by normalized Kerr rotations Kerr / Kerr,max m/msat vs external field H for samples W 110 / 18Fe/DCrCr/18Fe/5Cr, consisting of 2 Fe layers of 18 atomic layers each, separated by a Cr spacer consisting of DCr atomic layers, and cov- ered by a cap layer of 5 atomic layers of Cr. D 18 of the Fe layers and spacer thickness DCr as included in the figure. Data are measured for de- creasing field, only full circles . Full lines are fitting curves from minimization of g, resulting in the coupling parameters J1 , J2 in mJ/m2) as indicated in the figure. result independently from each other from the fit. For large For the strongest indirect coupling and small field, the interlayer coupling, J1 and J2 are related to some extent, thus magnetization directions are aligned antiparallel to each resulting in an increasing error as discussed below. other. The critical field Hc needed to get the onset of non- collinear configuration can be estimated by the following V. RESULTS consideration. At Hc both magnetization directions start to deviate from Figure 7 shows experimental magnetization curves from a 1 0 and 2 . We assume small devia- tions angles xi, 1, i.e., sample with wedgelike varying spacer thickness D 1 2 and 2 2 . At Cr . We some critical field the gain of free enthalpy g g(0, ) measure the Kerr rotation Kerr , normalized to the satura- g( 2 , 2 ) becomes positive and the magnetization tion value, Kerr,max , which was assumed to be proportional directions will deviate from the antiparallel alignment. In a to the magnetization component parallel to the external field. quadratic approximation this condition results in the critical Simulated magnetization curves using the free enthalpy field H model given by Eq. 4.2 were fitted to the experimental data c with parameters J1 and J2 . For the fit we only use magneti- zation data measured at decreasing absolute field values, in 2 2 2 H order to avoid metastable magnetization states. When the c HA 2 2 Hk,1 2 2 Hk,2 2 2 , 5.1 interlayer coupling is weak, as in the case of DCr 12.4 ML, we observe a magnetization curve composed of the sum of with the effective fields HA ( J1 2J2)/Jst and Hk,i the constant easy axis signal of the driver layer and the hard 2(K(i) (i) p K4xy)/Js . The effective anisotropy field of the axis curve of the sensor layer. At H 0 we only see the driver layer is Hk,1 1.1 kOe and of the sensor layer Hk,2 signal of the driver layer. The sensor layer signal increases 0.3 kOe. Therefore the deviation of the driver magneti- with H, the nonlinear jump to the saturation state at zation is smaller than the deviation of the sensor layer . In H 0.6 kOe is caused by the fourth-order anisotropy term. In a rough approximation the ratio is given by the saturated state the signal is twice as large as the remanent Hk,1 /Hk,2 3.5 . With these values we result in signal, thus confirming that magnetization components of H 0.6 kOe both for the bottom left case (DCr 3.2 ML and sensor and driver layer contribute equally to the total signal. for the right center row (DCr 3.6 ML of Fig. 7 in good For decreasing DCr the sensor curve is shifted to positive agreement with the exact calculation. fields indicating an increasing antiferromagnetic coupling Results for the bilinear coupling J1 as a function of inter- J1 0. For DCr 5 ML the slope of the sensor curve at layer thickness are summarized in Fig. 8 a . For the case of Kerr / Kerr,max 0.5 is decreased and prevents the signal weak-coupling constants J1 ,J2 0.1 mJ/m2 we avoided the from saturation. This observation indicates a strong 90° cou- elaborate fit procedure and present the data resulting from pling, J2 0. the rigid driver model instead. At some points we checked 55 MORPHOLOGY AND INDIRECT EXCHANGE COUPLING . . . 15 125 FIG. 8. Coupling parameters J1 a and J2 b vs DCr for samples W 110 / DFe/DCrCr/DFe/5Cr. The parameter Tp,Cr indicates the sample temperature during deposition of the Cr spacer layer. Full circles result from numerical fits of simulated magnetization curves resulting from minimization of g, open circles from rigid driver approximations. Full lines are guide-to-the-eye curves. that for such small coupling constants values resulting from towards J2 0 at DCr 5 ­ 7 ML. For DCr 2.5 ML the error both evaluation procedures are equal. Independent of Tp,Cr , both for J J 1 and J2 increases considerably due to the strong 1 shows a sharp transition from a ferromagnetic coupling coupling between the two parameters. Therefore coupling for DCr D0 (2.5 0.1) ML to an antiferromagnetic cou- data are not available for DCr D0 .The largest 90° coupling pling at DCr D0 , followed by a minimum value at DCr magnitudes were observed for the interlayer deposited at D1 3.4 0.1 ML. For the lowest deposition tempera- room temperature see Ref. 12 . We did not find positive ture, Tp,Cr 100 K, J1 remains negative up to the thickest Cr values for J2 although our model would have been able to thickness used in this study, DCr 15 ML. In this case no detect them. indication of an oscillation of J1 was observed. For higher The antiferromagnetic extremum J1(D1) of the bilinear deposition temperatures J1 changes its sign a second time at coupling constant is shown in Fig. 9 as a function of the DCr D2 7 ML. D2 decreases with increasing deposition interlayer deposition temperature. We found a surprising temperature. large antiferromagnetic coupling for Tp,Cr below room tem- The biquadratic coupling J2 shown in Fig. 8 b shows perature. J1(D1) increased by a factor of 3 when Tp,Cr is negative values 90° coupling at DCr 2.5 ML decreasing decreased below room temperature. Deposition above room 15 126 SCHWABENHAUSEN, DU¨RKOP, AND ELMERS 55 ridge structure Gdy 29 nm is still of the order of magnitude of the exchange length. In this case the magnetization vector oscillates around a mean direction given by 2 , with a small amplitude, thus satisfying the local bilinear coupling to some extent. As discussed by Slonczewski17 the latter effect gives rise to an effective 90° coupling term (J2) but the homoge- neous magnetization model as declared in Eq. 4.2 is still valid. Moreover, for strong antiferromagnetic coupling J1 tKp and vanishing external field one expects in the case of homogeneous magnetization a compensation of the driver layer and sensor layer signal. This was observed in the ex- periment see Fig. 7 . Contrarily, in the case of strongly in- homogeneous magnetization, one would have expected a su- perposition of magnetization curves from independent antiferromagnetic and ferromagnetic regions. Then the ferro- magnetic coupled regions showing easy axis hysteresis loops would have resulted in a considerable remanent signal. This argument again supports our model of virtually homoge- neous magnetization. There has been an attempt to explain the biquadratic cou- FIG. 9. Maximum antiferromagnetic coupling constant J1(D1) pling component by an intrinsic electronic mechanism. How- observed at a spacer thickness D1 3.4 ML and biquadratic cou- ever, the values predicted by these models are 1­2 orders of pling constant J2(D0) at spacer thickness D0 2.5 ML where the magnitude too small to explain our experimental data. Ex- bilinear coupling changes sign vs deposition temperature Tp,Cr . trinsic models that we take into account are the loose spin Full lines are guide-to-the-eye curves. mechanism, pinholes through the spacer layer and Sloncze- wski's mechanism of interfacial roughness. The loose spin temperature results in gradually decreasing values of mechanism would result in a strong decrease of J J 2 with 1(D1) with increasing T p,Cr . We show data for the biqua- increasing temperature. Because in our case J dratic coupling constant J 2 does not vary 2(D0) observed for an interlayer with the temperature this mechanism can be excluded. Fer- thickness D0 2.5 ML where J1 changes its sign for the first romagnetic pinholes in the spacer with a size larger than the time. J2 could still be determined with satisfying accuracy exchange length would have caused a superposition of easy for this interlayer thickness. The 90° coupling J2 slightly axis loops which was not observed. The effect of pinholes increases with increasing Tp,Cr from 100 to 310 K. This in- which are small enough to allow for a homogeneous magne- crease is not significant with respect to the error bars. For tization can in principle not be distinguished from the effect further increased Tp,Cr 400 K, however, J2 is strongly re- of interfacial roughness, which shall be discussed below in duced. detail. Slonczewski17 showed how spatial fluctuations of the lo- VI. DISCUSSION cal bilinear coupling J1(x) caused by spatial fluctuations of the spacer layer thickness account for an effective biqua- The main purpose of this paper is the study of the influ- dratic coupling. In the simplest version of the model J1 is a ence of the morphology of the interlayer on the indirect cou- one-dimensional periodic step function J pling. Because the Cr interlayer was grown on a smooth 1 J ¯1 J Fe 110 surface, one of the interlayer interfaces is flat. The 1sgn sin( x/L) with the period L and amplitude J structure of the second interface is known from the HRLEED 1 . This assumption results in a biquadratic coupling con- stant investigation. Consequently, we know the lateral distribution of the interlayer thickness DCr . Unfortunately, we could not 4 J1 2L reach layer by layer growth of the interlayer. Therefore, we J2 3A coth t/L . 6.1 could not determine directly the bilinear coupling constant J1 as a function of a virtually constant interlayer thickness. Slonczewski considered a thickness fluctuation of only 1 Because of the inhomogeneous interlayer thickness and atomic layer of a Cr 100 interlayer. For the Cr 100 inter- the corresponding inhomogeneous interlayer coupling we layer the bilinear coupling J have to consider inhomogeneous magnetization directions 1 shows opposite signs for odd and even numbers of atomic layers. These short period os- within the Fe layers. For this purpose the length scale of the cillations cannot be expected for Cr 110 layers because the lateral DCr distribution has to be compared with the magnetic magnetic moments of antiferromagnetic Cr cancel in the exchange length 2l in the sensor layer. The relevant ex- 110 plane. However, the one-dimensional ridge structure of change length 2l 4A/Kp 29 nm in our samples can be the 110 spacer layer deposited at Tp,Cr 230 K causes a estimated, using the bulk value for the exchange constant fluctuating bilinear coupling similar to the model structure A 2 10 11 J/m and the effective anisotropy constant tKp described above. When the average thickness of the spacer is 0.35 mJ/m2 of the sensor layer. For deposition tempera- thin, the roof tops of the ridge structure provide regions with tures Tp,Cr 300 K the terrace width Ndy 2 nm of the Cr antiferromagnetic coupling whereas in the valleys the cou- interlayer is much smaller than 2l. Even the period of the pling is ferromagnetic due to the long period oscillation of 55 MORPHOLOGY AND INDIRECT EXCHANGE COUPLING . . . 15 127 netic calculations taking into account the ridge structure of the spacer are highly desirable. The weak increase of J2(D0) with Tp,Cr increasing from J2 1.2 mJ/m2 at 100 K to J2 1.8 mJ/m2 at 310 K might be a result of the increasing period of the ridge struc- ture with increasing deposition temperature. An increase of the period directly results in an increase of bilinear coupling Eq. 6.1 . However, the experimentally observed increase is much lower than expected from Eq. 6.1 . For Tp,Cr 310 K we observed a structural change of the spacer from the one-dimensional ridge structure to a two- dimensional pattern. A spatially fluctuating J1 in two dimen- sions will also result in a biquadratic coupling but with a reduced magnitude. This reduction compensates the increase caused by the period to some extent. For higher deposition temperatures Tp,Cr 400 K the biquadratic coupling becomes rather weak although the period G increases monotonously. However, Eq. 6.1 is not valid when the period exceeds the exchange length. This is very likely the case for high depo- sition temperatures. We now discuss the thickness dependence of J2 . From Fig. 8 b we see that the biquadratic coupling constant J2 shows only negative values as expected from Eq. 6.1 . J2 is largest for small Cr thicknesses. As explained above the experimental error of J2 increases drastically for Cr thicknesses below DCr D0 .Therefore no unequivocal maxi- mum of J2 could be detected, except in the case of Tp,Cr 310 K. Our model of fluctuating bilinear coupling at ridges and valleys would result in an extremum for J2 at the spacer thickness D0 where J1 changes sign: An increasing spacer thickness reduces the ferromagnetic coupling in the valleys whereas the average antiferromagnetic coupling at the ridges is nearly constant or even reduced. Thus, the am- plitude J1 rapidly decreases with increasing DCr . Conse- quently, J2 will decrease as observed in the experiment. A quantitative analysis of the thickness dependence requires a micromagnetic analysis. A similar monotonously decreasing J2 with increasing DCr was observed in the case of Fe/Cr/Fe in the 100 orientation.11 In this reference the authors described the thickness dependence of J 1.4 2 by a power law J2 DCr . Our results for J2(DCr) cannot be described with such a unique FIG. 10. Coupling parameters J power law. 1 a and J2 b vs DCr for samples W 110 /18Fe/ D Our experimental data for the bilinear coupling J CrCr/18Fe/5Cr with the spacer deposited at 1 can be T compared with theoretical considerations. Stiles4 recently p,Cr 400 K, measured at temperatures T 100, 200, and 300 K. Symbols and full lines as in Fig. 8. calculated the strength and period of oscillatory exchange coupling for Fe/Cr layers by first-principle methods. For the bilinear coupling. Hence, the coupling constant J1 can be 110 orientation he found for the strongest long period os- described by the same function as in Ref. 17 to a very good cillation a coupling strength t2CrJ1 /(1.0) nm2 approximation. Now, the meaning of L in Eq. 6.1 is not the 3.2 mJ/m2. For the case of the first antiferromagnetic terrace width but the period of the ridge structure G. We maximum at tCr 0.7 nm the theoretical value is J1 estimate the amplitude J1 3 mJ/m2 at the spacer thickness 6.5 mJ/m2. Previously measured coupling strengths for D0 where J1 changes it sign from the maximum value ob- Fe/Cr were much smaller than the calculated values. Our served for J1 . Inserting the period of the ridge structure (L measured maximum of antiferromagnetic coupling for low G 29 nm for Tp,Cr 230 K and L G 61 nm for temperature deposition of the Cr is exceptionally large and Tp,Cr 300 K and the Fe layer thickness t 3.6 nm, Eq. 6.1 reaches nearly half of the theoretical value. This is very sur- results in J2 4.5 mJ/m2 (Tp,Cr 230 K and J2 prising taking into account the fluctuating spacer thickness 9.5 mJ/m2 (Tp,Cr 300 K . These values are in sufficient still observed for the low temperature deposition. At DCr agreement with the experimentally observed magnitude J2 3 ML D1 the height H 3 of the ridge structure is con- 1.6 mJ/m2 to support the model. Detailed micromag- siderable. Following Ref. 17 the measured value of J1 is the 15 128 SCHWABENHAUSEN, DU¨RKOP, AND ELMERS 55 mean value of the contributions from different interlayer For higher deposition temperatures Tp,Cr 500 K we ob- thicknesses, J1(DCr) nJ1(n)p(n,DCr), where the contri- served a nonvanishing ferromagnetic coupling J1 0 even bution from the coupling constant J1(n) at the interlayer for interlayer thicknesses DCr 12 ML see Fig. 8 a . This thickness n is weighted with the probability p(n,DCr) of the effect is presumably caused by the onset of interdiffusion. It occurrence of interlayer thickness n at a total deposited is quite plausible that Fe-rich Fe/Cr alloy regions produced thickness DCr . The function p(n,DCr) is not known exactly by the interdiffusion enhance the ferromagnetic coupling. in our case and it is impossible to extract J1(n) from As expected, we did not observe any significant tempera- J1(DCr). However, the maximum antiferromagnetic cou- ture dependence of J1 see Fig. 10 a . From theoretical con- pling constant J1(D1) observed in our experiment for low- siderations the temperature dependence of J1 should follow temperature deposition is in good agreement with the theo- the temperature dependence of the magnetization Js of the Fe retically predicted value. layers.5 Since the Curie temperature of the Fe layers is far The sharp decrease of J1(D1) for deposition tempera- above room temperature, Js is expected to be nearly constant tures Tp,Cr 310 K coincides with the increase of the rough- for the range of temperatures applied in our experiment. ness W. An increase of W will certainly result in a broader distribution p(n,DCr) and consequently in a decrease of VII. SUMMARY maximum values. Especially, contributions from thin inter- layer thicknesses n 0 and 1 with presumably very strong We determined bilinear and biquadratic coupling con- ferromagnetic coupling will rapidly decrease the antiferro- stants in Fe/Cr/Fe 110 trilayers for varying spacer mor- magnetic coupling. This may explain the nonlinear decrease phologies. The large coupling constants occurring for spacer of J layer deposition below room temperature exceed previously 1(D1) near T p,Cr 310 K although the increase of the roughness is rather small. reported values. The bilinear coupling constant does not de- The origin of the ferromagnetic maximum near D pend significantly on the temperature of measurement in the Cr 8 ML appearing for T interval T 100­300 K. The maximum value of the antifer- p,Cr 310 K apparently is an oscillating bilinear coupling J romagnetic coupling is in agreement with theoretical calcu- 1(n), significantly damped by the fluctu- ating spacer thickness. The oscillation period 2 nm esti- lations using ab initio methods. The biquadratic coupling can mated from the half period observed in our experiment co- be explained from the fluctuating bilinear coupling caused by incides with the long wavelength period as determined both the rough interface. We compared experimental data with for the 100 and for the 211 orientation.28 This surprising Slonczewski's micromagnetic model. independence of the long-wavelength period on the orienta- tion has been attributed to either a lens structure near the ACKNOWLEDGMENTS center of the Cr Fermi surface40 or to the N-centered The authors are indebted to U. 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