JOURNAL OF APPLIED PHYSICS VOLUME 88, NUMBER 2 15 JULY 2000 Magnetic phase diagrams of the trilayers with the noncollinear coupling in the form of the proximity magnetism model Shi-shen Yana) and P. Grušnberg Institut fušr Festkošrperforschung, Forschungszentrum Jušlich GmbH, D-52425 Jušlich, Germany Liang-mo Mei Physics Department, Shandong University, Jinan, Shandong 250100, People's Republic of China Received 23 December 1999; accepted for publication 6 April 2000 The magnetic phase diagrams of Fe/Mn/Fe trilayers with the noncollinear interlayer coupling in the form of the proximity magnetism model were theoretically studied. The C ­ C phase diagram in the remanent magnetization state predicts very rich spin configurations. The H ­ C and H ­ C phase diagrams show that the spin configurations of Fe/Mn/Fe trilayers depend strongly on the external magnetic field, the anisotropy of Fe layers, and the coupling coefficients C and C . Our experimental results of noncollinear spin configurations of Fe/Mn/Fe trilayers strongly support the magnetic phase diagrams based on the proximity magnetism model. © 2000 American Institute of Physics. S0021-8979 00 00914-2 I. INTRODUCTION where C 0, C 0, and 0 . Here C and C are the coupling coefficients, and as above is the angle be- The magnetic phase diagrams of the layered magnetic tween the magnetization vectors of the two FM layers. The structures have been greatly enriched owing to the antiferro- origin of the simple square dependence is based on the as- magnetic AF interlayer coupling and noncollinear cou- sumption that the same tilt angle between the magnetiza- pling, which were found in many magnetic layered struc- tures, such as Fe/Cr/Fe,1­3 Fe/ Al,Au /Fe,4 Fe/ Cu,Ag /Fe,5 tion vectors of the nearest neighbor atomic planes of the Fe/Mn/Fe6,7 trilayers and Fe/Cr multilayers.8 Theoretically interlayer away from the antialignment is only small and so the magnetic phase diagrams strongly depend on the energy higher terms in the series expansion of the coupling energy expression of the interlayer coupling. For the layered mag- as a function of can be neglected. This translates to the netic structures with nonmagnetic metallic interlayers, the same dependence on the angle between the magnetization interlayer coupling can be characterized by the following vectors of the two FM layers as shown in Eq. 2 . If neither phenomenological energy expression: C nor C vanishes, the mutual equilibrium orientation of the two magnetizations in the FM layers is not collinear. But E if one of the two coefficients C and C vanishes for the c j1 cos j2 cos2 , 1 perfect spacers no thickness fluctuation , the equilibrium of where is the angle between the magnetization vectors of the two magnetizations is collinear AF or FM coupling . the two ferromagnetic FM layers. Here the first term with Comparable mixtures of even and odd monolayers of the the parameter j spacers may give C 1 represents the bilinear coupling which C , which constitutes an orthogonal aligns the magnetic moments parallel if j coupling. 1 0 and antiparal- lel if j Up to now, based on the interlayer coupling energy in 1 0. The second term with j2 describes the 90° cou- pling which creates a perpendicular alignment of the mag- the form of Eq. 1 several authors have analyzed the spin netic moments for j configurations in the coupled multilayers10­15 but few au- 2 0. However, strictly speaking, Cr and Mn are not ``non- thors have done the work based on Eq. 2 , whereas the magnetic'' metal spacers as often tacitly assumed. So an- proximity magnetism model has been successfully used to other phenomenological model, i.e., the proximity magne- describe the very strong 90° coupling in CoFe/Mn/CoFe tism model9 was suggested by Slonczewski, which is based trilayers,6 the 50° coupling in Fe/Cr multilayers,8 and the on the helicoidal quasi-AF ordering of the interlayers such trivial noncollinear coupling in Fe/Mn/Fe trilayers.7 Experi- as Cr and Mn in conjunction with the long-range lateral mentally the difference between Eqs. 1 and 2 shows up in thickness fluctuation due to interfacial roughness. In this a subtle difference concerning saturation after remagnetiza- model the exchange coupling energy per unit area can be tion. Though Eq. 1 implies full saturation of the M ­ H written as: curve at a finite critical external field, Eq. 2 implies an asymptotic approach toward saturation. Theoretically we can E expect that the magnetic phase diagrams will be different c C 2 C 2, 2 owing to the different energy expressions of the interlayer a coupling. Author to whom correspondence should be addressed; present address: Center for Materials for Information Technology, The University of Ala- On the other hand, both Eqs. 1 and 2 predict very rich bama, Tuscaloosa, AL 35487-0209; electronic mail: syan@mint.ua.edu spin configurations in the remanent magnetization states, but 0021-8979/2000/88(2)/983/5/$17.00 983 © 2000 American Institute of Physics Downloaded 10 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html 984 J. Appl. Phys., Vol. 88, No. 2, 15 July 2000 Yan, Grušnberg, and Mei only a few spin configurations have been experimentally experimental values of coupling coefficients C and C , found, such as FM coupling, AF coupling,1 90° coupling,2­6 and the coupling angle 1 2 at any given external 135° coupling in FeNi/Ag multilayers,16 and 50° coupling in field. Fe/Cr multilayers.8 Only recently, trivial noncollinear cou- For simplicity and in order not to lose universality, we pling states were found in Fe/Mn/Fe Ref. 7 and Fe/Cr/Fe allow 0 1 2 in Eq. 3 . The minimum total energy trilayers3 and they seem to be a common phenomenon for should satisfy the equations E/ i 0 (i 1,2), i.e., samples with interfacial roughness. So it is necessary to give more experimental evidence for the noncollinear spin con- E 2C figurations to prove the theoretical predictions in the mag- 1 2 2C 1 2 1 netic phase diagrams. In this article, we first report the the- Kt oretical magnetic phase diagrams of the magnetic trilayers with the noncollinear interlayer coupling in the form of the 2 sin 4 1 HMt sin 1 0, 4a proximity magnetism model, and then we present some ex- E perimental results of noncollinear spin configurations of Fe/ 2C 1 2 2C 1 2 Mn/Fe trilayers to support our calculations based on the 2 proximity magnetism model. Kt 2 sin 4 2 HMt sin 2 0, 4b II. THEORETICAL MAGNETIC PHASE DIAGRAMS with the stability condition that all the eigenvalues of the As an example we calculate the magnetic phase dia- matrix M defined by M grams of a real system like Fe/Mn/Fe or Fe/Cr/Fe trilayers i j 2E/ i j should be positive, i.e., with the following assumptions. In Fe/Mn/Fe trilayers the spins of Fe are assumed to lie parallel to the film plane, and 2E 2E 2 so there is not a static demagnetizing field. We also assume 2 2 2E that the spins within an individual Fe layer remain parallel to 1 2 1 2 one another because of a strong intralayer-exchange cou- 2C 2C 2Kt cos 4 1 HMt cos 1 pling. The details of the actual magnetic structures of the Mn or Cr are not considered, but its contribution to the total 2C 2C 2Kt cos 4 2 HMt cos 2 energy is represented by the exchange energy in the form of 2C 2C 2 0. 5 Eq. 2 . In our case, the sample plane is parallel to 001 crystallographic plane and the external field is along the in- It is easy to find that there are two kinds of solutions plane easy axis 100 direction or equivalent . Taking into which satisfy Eqs. 4 and 5 : the symmetrical solution 1 account the cubic anisotropy energy of Fe, Zeeman energy, 2 symmetrical phase, i.e., the two magnetizations are and interlayer coupling energy in the form of Eq. 2 , we symmetrical with respect to the direction of the applied field write the total energy E per unit area in the following form: and the asymmetrical solution 1 2 asymmetrical phase, i.e., the two magnetizations are asymmetrical with E Ea Eh Ec , respect to the direction of the applied field . But it is difficult E to give the analytical expressions of ( a Kt sin 2 1 2 sin 2 2 2 /4, 1 , 2). So the bound- 3 aries of the phases and the magnetization curves are, in gen- Eh HMt cos 1 cos 2 , eral, obtained numerically. In fact, Eqs. 4 and 5 some- times have more than one set of solutions of ( E 1 , 2) for a c C 1 2 2 C 1 2 2, given field H owing to different local positions of the local where Ea is the anisotropy energy, Eh is the Zeeman energy, minimum energy. In this case we should choose the set of and Ec is the interlayer coupling energy of the proximity solution of ( 1 , 2) which corresponds to the global mini- magnetism model.9 Here t, M, K, and H are, respectively, the mum energy. Therefore, we directly compare the different thickness of Fe layers, the saturation magnetization of Fe values of E( 1 , 2) at a given field H for all sets of layers, the first-order cubic crystal anisotropy of Fe layers, ( 1 , 2) to find the global minimum, and then get the solu- and the external field; 1 or 2 is the angle between the tion ( 1 , 2) to construct the phase diagram and the mag- magnetization vector of the first or second Fe layer and the netization curve. field direction; and C and C are the coupling coefficients However, for zero external field we can give the analyti- which are the only two adjustable independent constants in cal solutions of the boundaries of the phase diagram accord- our calculation. 1 2 (0 ) is the angle be- ing to Eqs. 3 ­ 5 . The C ­ C phase diagram at zero field tween the two magnetization vectors of the Fe layers at a is shown in Fig. 1. We have known that there are two phases: given external field we call it coupling angle . the symmetrical and the asymmetrical. At zero field the The theoretical magnetization curves and magnetic boundaries of the two phases are two straight lines in phase diagrams are obtained by minimizing the total energy C ­ C phase diagram, i.e., C ­ 3C 0 and 3C ­ C of Eq. 3 with respect to 1 and 2 at a given external field 0. The symmetrical phase can be divided into two sub- H for the appropriate C and C . By fitting in this way the phases S1 and S2 , as shown in Fig. 1. In the S1 region theoretical magnetization curves to the experimentally mea- (C ­ 3C 0) the spin configuration of the two magnetiza- sured hysteresis loops, one can quantitatively determine the tions is symmetrical with respect to the easy axes, and the Downloaded 10 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html J. Appl. Phys., Vol. 88, No. 2, 15 July 2000 Yan, Grušnberg, and Mei 985 region 3C C 0 and C C 0 the spin configura- tion is asymmetrical but symmetrical with respect to the hard axes , and the angle is in the range of /4 /2. From the above we can see that the C ­ C phase dia- gram at zero field is quite different from the j1­ j2 phase diagram see Fig. 1 in Ref. 13 . In j1­ j2 phase diagram there are wide regions of ( j1 ,j2) where the two magnetizations are collinear FM coupling phase I and AF coupling phase II in Fig. 1 of Ref. 13 . In Figs. 2 a ­2 d we show the magnetic phase diagrams for different cases, where the external field is along the 100 direction of the easy axis. In our calculations we use the following typical experimental values of Fe layers: M 1707 G s, K 4.76 105 erg/cm3, and t 5 10 7 cm. In Fig. 2 a (C 0), the two magnetizations are always anti- parallely aligned at zero field AF coupling . For small but FIG. 1. The C ­ C phase diagram at zero field. S and AS represent, respectively, the symmetrical and the asymmetrical configurations. The finite values of C , when the field is increased the system boundaries of the two phases solid lines are two straight lines, i.e., C goes from the symmetrical configuration to the asymmetrical 3C 0 and 3C C 0. The dotted line is C C 0. The insetted and then to the symmetrical again. For big but finite values symbols show schematically the nonequal spin configurations of the two magnetizations in the Fe layers. of C the system is always symmetrical with respect to the applied field the 100 direction of the easy axis . In Fig. 2 b (C C ) the two magnetizations are al- angle between the two magnetizations is in the range of ways perpendicularly aligned at zero field 90° coupling . 3 /4 . Only when C For any given C 0 and C 0, can the two C 0, the system goes from asym- magnetizations be aligned antiparallel AF coupling . In the metrical to symmetrical when the field is increased. S2 region (3C C 0) the spin configuration is sym- Figures 2 c and 2 d are two more universal cases. In metrical, and the angle is in the range of 0 /4. Only Fig. 2 c we show a H ­ C phase diagram for fixed C when C 0 and C 0, can the two magnetizations be (C 2.0 Kt/4 0.1190 erg/cm2 . For small but finite values aligned parallel FM coupling . of C , the system goes from the symmetrical configuration The asymmetrical phase can also be divided into two to the asymmetrical and then to the symmetrical again when subphases AS1 and AS2 , as shown in Fig. 1. In AS1 region the field is increased. For middle values of C , the spin (C 3C 0 and C C 0 the spin configuration is configuration is asymmetrical at low field and become sym- asymmetrical with respect to the easy axes but symmetrical metrical at big field. For big but finite values of C the with respect to the hard axes , and the angle is in the range system is always symmetrical. of /2 3 /4. Only when C C 0, can the two mag- In Fig. 2 d we show the H ­ C phase diagram for fixed netizations align perpendicularly 90° coupling . In the AS2 C C 2.0 Kt/4 0.1190 erg/cm2 . For small or big given FIG. 2. The magnetic phase diagrams for different cases: a H ­ C phase diagram for fixed C 0; b H ­ C C phase diagram; c H ­ C phase diagram for fixed C 2 Kt/4 0.119 erg/cm2; and d H ­ C phase diagram for fixed C 2.0 Kt/4. The external field is along the 100 direction of the easy axis. Downloaded 10 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html 986 J. Appl. Phys., Vol. 88, No. 2, 15 July 2000 Yan, Grušnberg, and Mei FIG. 3. The hysteresis loop measured by MOKE at room temperature marked by circles and the theoretical fitting by Eq. 3 solid line . The Mn layer in this sample was deposited at 50 °C. The inset shows the details of the hysteresis loop in the high field range. The thickness of Mn layer D FIG. 4. Some typical hysteresis loops measured by MOKE at room tem- Mn nm), the coupling coefficients C and C (erg/cm2), and the cou- pling angle deg in remanence are, respectively, D perature marked by various symbols and the theoretical fittings by Eq. 3 Mn 1.66 nm, C solid lines . For these samples the Mn layers were deposited at 150 °C. C 0.118 erg/cm2, and 90°. The thickness of Mn layer DMn nm), the coupling coefficients C and C (erg/cm2), and the coupling angle deg in remanence, are respec- tively, up triangles DMn 0.52, C 0.27, C 0.19, 78; rectangles values of C , the system is always symmetrical. But for DMn 0.63, C 0.13, C 0.52, 148.5; down triangles DMn 0.66, middle values of C , the field dependence of spin configu- C 0.133, C 0.405, 141.6; and circles DMn 0.77, C 0.16, ration is complicated. If (2/3)(Kt/4) C C 6(Kt/4), the 0.215, 100. spin configuration goes from asymmetrical to symmetrical when the field is increased. If 6(Kt/4) C 6.15(Kt/4), the spin configuration will experience symmetrical, asym- at room temperature and the theoretical fittings solid lines metrical, and symmetrical for a given C when the field is only by adjusting C and C in Eq. 3 for different increased. samples. In Fig. 3 the remanent magnetization of about half The phase transition induced by the external field in Fig. of the saturation value and the big jump at near zero field 2 is always of the first order. indicate that the individual magnetizations switch between By the way, our calculations above are not limited to the different easy axes but the angle between them remains 90°. Fe/Mn/Fe or Fe/Cr/Fe trilayers. They are also appropriate for We can see that the theoretical fittings are in good agreement Fe/Mn or Fe/Cr multilayers with infinite bilayers when the with the experimental hysteresis loops obtained by MOKE. coefficients C and C in Eq. 3 are, respectively, replaced Especially, the following characters of the hysteresis loops by the coefficients 2C m and 2C m . Here C m and C m are well described by the proximity magnetism model: first, are the coupling coefficients in the mutilayers. all the hysteresis loops show an asymptotic approach toward saturation; and second, there are some noncollinear coupling III. COMPARING WITH THE EXPERIMENTAL RESULTS states as shown in Fig. 4 except for the well known 90° coupling as shown in Fig. 3. Our theoretical fittings to the In this section, we present our experimental results of experimental hysteresis loops indicate that the coupling noncollinear spin configurations of Fe/Mn/Fe trilayers to angles in the remanent magnetization states are, respectively, support our calculations based on the proximity magnetism 78°, 148°, 141.6°, and 100° when the Mn layer thicknesses model. The Fe 5 nm/Mn 0­4 nm/Fe 5 nm trilayers were are, respectively, 0.52, 0.63, 0.66, and 0.77 nm. Further de- deposited in UHV by the MBE method on GaAs/Fe 1 nm/Ag tails will be reported. 150 nm substrate­buffer system and covered by 50 nm ZnS Figures 5 a and 5 b show the dependence of spin con- layer. The bottom 5 nm Fe layer was grown on Ag buffer at figurations on the external field. In Fig. 5 a the magnetiza- room temperature for the first 4 ML and at 200 °C for the tion hysteresis loop marked by down-triangles has been rest. We prepared the Mn layer with the growth rate of about shown in Fig. 4 , the angle between the two magnetizations 0.9 nm/min at different temperatures from 150 to 200 °C. is 141.6° at zero field. When the external field is less than The top 5 nm Fe layer was deposited at 200 °C. We mea- 373 Oe, the two magnetizations are symmetrical. When the sured hysteresis loops and examined the coupling by longi- field is in the range from 373 to 676 Oe, the two magnetiza- tudinal magneto-optic Kerr effect MOKE . The external tions are asymmetrical. When the field is bigger than 676 Oe, field was applied along the 100 direction of the easy axis, the two magnetization become symmetrical again and and parallel to both the sample plane and the incidence plane asymptotically approach the direction of the external field. of the laser. Other details of the experiments are the same as In Fig. 5 b the magnetization hysteresis loop has been described elsewhere.2,7,17,18 shown in Fig. 3 , the two magnetizations are perpendicular to One direct and simple way to examine the theoretical each other at zero field. When the external field is less than magnetic phase diagrams is to compare the experimental 355 Oe the switching field HSW in Fig. 3 , the two magne- hysteresis loops with the theoretical fittings. In Figs. 3 and 4 tizations are asymmetrical. When the field is bigger than 355 we show some typical hysteresis loops measured by MOKE Oe, the two magnetizations are symmetrical and asymptoti- Downloaded 10 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html J. Appl. Phys., Vol. 88, No. 2, 15 July 2000 Yan, Grušnberg, and Mei 987 anisotropy of Fe layers, and the coupling coefficients C and C . The above calculations based on the proximity magne- tism model are strongly supported by our experimental re- sults of noncollinear coupling in Fe/Mn/Fe trilayers. From the viewpoint of theory, the good agreement be- tween the experimental hysteresis loops of Fe/Mn/Fe trilay- ers and the theoretical calculations based on the proximity magnetism model suggests that the Fe/Cr/Fe system should be reexamined by the same idea since both Mn and Cr are antiferromagnetic materials. From the viewpoint of applica- tions, the Fe/Mn/Fe trilayers supply a new artificial spin sys- tem where an arbitrary coupling angle in the ground state is available by modifying the interlayer thickness and growth conditions of the films, and where the switching of the mag- netization vectors in the two Fe layers is easy to control by the external field. Since the giant magnetoresistance depends on the relative orientation of the magnetization vectors in the magnetic layers, the findings in this article may be useful in the design of magnetoresistive sensors. ACKNOWLEDGMENTS The authors thank R. Schreiber, F. Voges, D. Olligs, and FIG. 5. The dependence of spin configurations on the external field. The line P. Rošttlašnd for their help. S.-s. Y. is pleased to thank the with circles or triangles represents angle 1 or 2 between the magne- Alexander von Humboldt Foundation for support. tization of the first or second Fe layer and the external field; the solid line represents the angle between the two magnetizations: a DMn 0.66 nm 1 P. Grušnberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sowers, Phys. and b DMn 1.66 nm. Rev. Lett. 57, 2442 1986 . 2 M. Rušhrig, R. Schašfer, A. Hubert, R. Mosler, J. A. Wolf, S. Demokritov, and P. Grušnberg, Phys. Status Solidi A 125, 635 1991 . cally approach the direction of the external field. 3 D. T. Pierce, J. Unguris, R. J. Celotta, and M. D. Stiles, J. Magn. Magn. From the above experimental hysteresis loops of Fe/ Mater. 200, 290 1999 . Mn/Fe trilayers, we can see that although the spin configu- 4 A. Fuss, S. Demokritov, P. Grušnberg, and W. Zinn, J. Magn. Magn. rations and phase transitions induced by the external field Mater. 103, L221 1992 . 5 Z. Celinski, B. Heinrich, and J. F. Cochran, J. Magn. Magn. Mater. 145, seem to be complicated, they are quantitatively described by L1 1995 . the proximity magnetism model. Physically, all possible spin 6 M. E. Filipkowski, J. J. Krebs, G. A. Prinz, and C. J. Gutierrez, Phys. Rev. configurations can be explained in accordance with the prin- Lett. 75, 1847 1995 . 7 ciple of minimum total energy in the Fe/Mn/Fe system see S.-s. Yan, R. Schreiber, F. Voges, C. Osthošver, and P. Grušnberg, Phys. Rev. B 59, R11641 1999 . Eq. 3 . Approximately but simply speaking, in order to 8 A. Schreyer, J. F. Ankner, Th. Zeidler, H. Zabel, M. Schašfer, J. A. Wolf, reduce the anisotropy energy and as a result reduce the total P. Grušnberg, and C. F. Majkrzak, Phys. Rev. B 52, 16066 1995 . energy to the minimum, phase transitions occur. In fact, if 9 J. C. Slonczewski, J. Magn. Magn. Mater. 150, 13 1995 . 10 there is no anisotropy, the spin configurations will always be B. Dieny, J. P. Gavigan, and J. P. Rebouillat, J. Phys.: Condens. Matter 2, 159 1990 . symmetrical about the external field. 11 R. W. Wang, D. L. Mills, Eric E. Fullerton, J. E. Mattson, and S. D. Bader, Phys. Rev. Lett. 72, 920 1994 . 12 IV. CONCLUSIONS N. S. Almeida and D. L. Mills, Phys. Rev. B 52, 13504 1995 . 13 V. V. Kostyuchenko and A. K. Zvezdin, Phys. Rev. B 57, 5951 1998 . In conclusion, the magnetic phase diagrams of Fe/Mn/Fe 14 F. C. Nošrtemann, R. L. Stamps, A. S. Carrico, and R. E. Camley, Phys. trilayers with noncollinear interlayer coupling in the form of Rev. B 46, 10847 1992 . 15 T. L. Fonseca and N. S. Almeida, Phys. Rev. B 57, 76 1998 . the proximity magnetism model were theoretically studied. 16 B. Rodmacq, K. Dumesnil, P. Mangin, and M. Hennion, Phys. Rev. B 48, The C ­ C phase diagram in remnant magnetization states 3556 1993 . predicts very rich magnetic phases. The H ­ C 17 J. A. Wolf, Q. Leng, R. Schreiber, P. Grušnberg, and W. Zinn, J. Magn. and H ­ C phase diagrams show that the spin configurations of Fe/ Magn. Mater. 121, 253 1993 . 18 M. Schašfer, S. Demokritov, S. Mušller-Pfeiffer, R. Schašfer, M. Schneider, Mn/Fe trilayers depend strongly on the external field, the and P. Grušnberg, J. Appl. Phys. 77, 6432 1995 . Downloaded 10 Mar 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html