Langmuir 1996, 12, 45-50
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Surface and Thin Film Magnetization of Transition Metals† P. J. Jensen* and K. H. Bennemann Institut fu¨ r Theoretische Physik, Freie Universita¨ t Berlin, Arnimallee 14, D-14 195 Berlin, Germany Received November 14, 1994. In Final Form: February 6, 1995X The dependence of the Curie temperature and the direction of magnetization on the reduced dimensionality is investigated. The diminished coordination number can lead to competing effects on the magnetic ordering. For certain cases the Curie temperature for surfaces or thin films is larger than for the respective bulk system. A strong magnetic surface anisotropy causes possibly a canted magnetization of the surface. In thin films a reorientation of the thin film magnetization is observed in some systems with increasing temperature and film thickness, accompanied by a range of strongly diminished magnetization. This is due to the fragmentation of the uniform thin film magnetization with perpendicular orientation into a periodic magnetic domain structure. The domain phase is stabilized by the long range magnetic dipole interaction. We show that a small applied magnetic field is able to restore the uniform phase.
1. Introduction The magnetic properties of surfaces and thin film systems have been studied intensively in the last year both on fundamental grounds as well as for technological applications.1,2 The development of new methods for preparation and characterization allows a controlled growth of very thin films in the monolayer range. Thus also comparison with theoretical results is possible which always assume a definite atomic structure. We investigate here the effects of reduced dimensionality on the magnetic ordering, the Curie temperature Tc, and the direction of magnetization. Especially for 3d ferromagnetic transition metals the reduced coordination number leads to competing influences on Tc. The weakened magnetic order due to the lowered number of nearest neighbors may be masked by a larger surface magnetic moment. For certain systems the critical temperature can be even larger on surfaces or in thin films than in the bulk system.3-7 The direction of the magnetization is determined by anisotropic interactions. For (quasi-) two-dimensional (2D) systems these interactions are very important since they induce a long range magnetic order in thin films.8 The magnetization direction depends also on temperature and film thickness. A rotation from a perpendicular to an in-plane orientation with increasing temperature and/or thickness was observed for a number of thin film systems, e.g., Fe/Cu(100),5,9,10 Fe/Ag(100),11 Co/Au(111),12 and others.13 This †
Presented at the symposium on Advances in the Measurement and Modeling of Surface Phenomena, San Luis, Argentina, August 24-30, 1994. X Abstract published in Advance ACS Abstracts, January 1, 1996. (1) For a recent review see J. Magn. Magn. Mater. 1991, 100. (2) Falicov, L. M.; et al. J. Mater. Res. 1990, 5, 1299. (3) Kaneyoshi, T. J. Phys.: Condens. Matter 1991, 3, 1299, and references therein. (4) Stampanoni, M. Appl. Phys. A 1989, 49, 449. (5) Thomassen, J.; May, F.; Feldmann, B.; Wuttig, M.; Ibach, H. Phys. Rev. Lett. 1992, 69, 3831. (6) Weller, D.; Alvarado, S. F.; Gudat, W.; Schro¨der, K.; Campagna, M. Phys. Rev. Lett. 1985, 54, 1555. Mulhollan, G. A.; Garrison, K.; Erskine, J. L. Phys. Rev. Lett. 1992, 59, 3240. Tang, H.; Weller, D.; Walker, T. G.; Scott, J. C.; Chappert, C.; Hopster, H.; Pang, A. W.; Dessau, D. S.; Pappas, D. P. Phys. Rev. Lett. 1993, 71, 444. (7) Jensen, P. J.; Dreysse´, H.; Bennemann, K. H. Europhys. Lett. 1992, 18, 463; Surf. Sci. 1992, 269/270, 627. (8) Maleev, S. V. Sov. Phys. JETP 1976, 43, 1240. Pokrovsky, V. L.; Feigel’man M. V. Sov. Phys. JETP 1977, 45, 291. (9) Pappas, D. P.; Ka¨mper, K. P.; Hopster, H. Phys. Rev. Lett. 1990, 64, 3189. (10) Allenspach, R.; Bischof, A. Phys. Rev. Lett. 1992, 69, 3385. (11) Qiu, Z. Q.; Pearson, J.; Bader, S. D. Phys. Rev. Lett. 1993, 70, 1006.
0743-7463/96/2412-0045$12.00/0
reorientation is accompanied by a region of strongly reduced or even vanishing global magnetization. Two explanations for this phenomenon were proposed. First is the existence of an isotropic 2D Heisenberg magnet without any long range remanent magnetic order due to mutually cancelling anisotropies.14 This is supported by the inelastic scattering of light which is strongly enhanced around the reorientation, possibly indicating the occurrence of critical fluctuations.15 Second the appearance of a magnetic domain structure as observed experimentally.10 Oppositionally oriented magnetic domains lead to a vanishing total thin film magnetization. We show here that the first mechanism may be significant only if the reorientation temperature TR is close to Tc. In addition we investigate the action of a small applied magnetic field on the domain structure. In the following section the dependence of the Curie temperature for surfaces and thin films on the reduced dimensionality is reviewed. In section 3 the direction of magnetization on surfaces and thin films determined by anisotropic interactions is discussed, including the magnetic reorientation in thin films. In section 4 we present new results for domain phases in presence of the uniaxial anisotropy, the magnetic dipole coupling, and an external magnetic field. A conclusion is given in section 5. 2. Magnetization and Curie Temperature The ferromagnetic state of a system may be described by a simple Heisenberg model
1 Hex ) - I 2
∑bµibµj
(1)
〈ij〉
with I > 0 the interatomic Heisenberg exchange integral, b µi the localized magnetic moment on lattice site i with magnitude µi, and 〈ij〉 denotes the sum over nearest neighbor pairs. The ferromagnetic order parameter or magnetization m(T) is given by the average value of the component of b µi along the quantization axis. A mean field (12) Allenspach, R.; Stampanoni, M.; Bischof, A. Phys. Rev. Lett. 1990, 65, 3344. (13) Fritzsche, H.; Kohlhepp, J.; Elmers, H. J.; Gradmann, U. Preprint, 1994. (14) Morr, D. K.; Jensen, P. J.; Bennemann, K. H. Surf. Sci. 1994, 307-309, 1109. (15) Dutcher, J. R.; Cochran, J. F.; Jacob, I.; Egelhoff, W. F., Jr. Phys. Rev. B 1989, 39, 10430.
© 1996 American Chemical Society
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Jensen and Bennemann
estimate of the critical temperature yields
Tc ∝ qIµ2
(2)
with q the coordination number. For nonuniform systems the quantities q, I, and µ2 have to be replaced by averaged values. The reduced coordination number at the surface leads to a diminished surface magnetization which is stabilized by the semi-infinite bulk system. The different behavior of the surface and the bulk magnetization is readily observed in the critical region T j Tc, where the magnetization can be written as a power law
mbulk ∝ (Tc - T)β
(3)
msurf ∝ (Tc - T)β1
(4) 1/ 2
The mean field values of the critical exponents are β ) and β1 ) 1.16 A more exact renormalization group calculation for three-component (Heisenberg) spins yields β ≈ 0.37 and β1 ≈ 0.84.17 Experimental observations for Ni(100) and Ni(110) surfaces obtain β1 ≈ 0.8 ( 0.2,18 and for an EuS(111) surface β1 ≈ 0.72 ( 0.03.19 On the other hand for the magnetic 3d transition metals (Cr-Ni), which are known to exhibit an itinerant electronic structure, the reduced number of nearest neighbors causes a narrowing of the 3d bandwidth W ∝ q1/2. In general this feature leads to an enhanced density of states n(F) at the Fermi energy and thus to an enhanced surface magnetic moment µsurf > µbulk. Electronic band structure calculations yield a 35% increase of the surface atomic magnetic moment µsurf ) 2.98µB for a Fe(100) surface20 (µbulk ) 2.21µB). Experimental investigations obtain for the surface layer of a Fe(100) thin film µsurf ∼ 2.4 - 2.6µB.21 For magnetic 4f (rare earth) metals the atomic magnetic moment depends only slightly on the environment and stays almost constant on surfaces. Also the interatomic exchange interaction I may change its value at surfaces, in special in the case of surface relaxations and reconstructions. However, this quantity is hard to calculate, thus in the following we will assume I to be the same for all nearest neighbor pairs. We merely like to vary the atomic magnetic moment µ, for which calculated and also measured values are accessible for a large number of systems.20 To compare the following calculations with former works one may put Jij ) Iµiµj. As can be seen from eq 2 the effect of reduced dimensionality leads to competing influences on the Curie temperature. The reduced coordination number q may be counterbalanced by an enhanced surface magnetic moment µsurf. A large value of µsurf (or the surface exchange coupling Jsurf) may even overcome the reduced coordination number, leading to the possibility of a surface critical temperature Tc,surf > Tc,bulk. In this case a ferromagnetically ordered surface may exist on top of a disordered bulk for Tc,bulk < T < Tc,surf,3 thus representing a ferromagnetic thin film. However, the film thickness is not well defined since the strongly magnetized surface induces a weak magnetic ordering in the subsurface layers which is exponentially decaying with increasing distance from the surface, as sketched in Figure 1. Also the critical (16) Mills, D. L. Phys. Rev. B 1971, 3, 3887. (17) Fisher, M. E. J. Vac. Sci. Technol. 1973, 10, 685. (18) Alvarado, S. F.; Campagna, M.; Ciccacci, F.; Hopster, H. J. Appl. Phys. 1982, 53, 7920. (19) Dauth, B. H.; Alvarado, S. F.; Campagna, M. Phys. Rev. Lett. 1987, 58, 2118. (20) Freeman, A. J.; Fu, R. J. Magn. Magn. Mater. 1991, 100, 497. (21) Bateson, R. D.; Ford, G. W.; Bland, J. A. C.; Lauter, H. J.; Heinrich, B.; Celinski, Z. J. Magn. Magn. Mater. 1993, 121, 189.
Figure 1. Magnetization m(z) of layers with distance z from the surface of a fcc(100) face. A strong surface magnetic moment µsurf ) 1.5µbulk causes a larger surface Curie temperature Tc,surf compared to the bulk critical temperature Tc,bulk. The magnetization is calculated within a mean field approximation for two different temperatures T ) 0.958Tc,bulk and T ) 1.083Tc,bulk. The latter case corresponds to a magnetically ordered surface on top of a disordered bulk. The dashed curves refer to exponentially decaying m(z) fitted to the mean field results.
Figure 2. Mean field calculation of the Curie temperature of a semi-infinite magnetic system dependent on the ratio of the surface magnetic moment µsurf to the bulk moment µbulk. For µsurf/µbulk > 1.22 a critical temperature Tc,surf > Tc,bulk is obtained.
behavior will change considerably. In Figure 2 a mean field calculation is presented for Tc,bulk and Tc,surf as a function of µsurf/µbulk. Experimentally for a Gd surface Tc,surf - Tc,bulk j 60 K was observed,6 probably caused by an enhanced surface exchange coupling. For thin films the influence of reduced dimensionality on the magnetic properties is even more pronounced than for surfaces of a semi-infinite bulk. For a number of systems the Curie temperature as a function of the film thickness d was measured1,4,22 which do not show the universal behavior as predicted by the simple scaling law Tc,bulk - Tc(d) ∝ d-λ, with λ ) 1/ν the “shift” exponent, and ν the critical exponent of the bulk correlation length.23 By use of a generalized molecular field theory we were able to simulate some of the different experimental findings by assuming different magnetic moments for the surface and/or interface layers,7 Figure 3. In addition it was shown that especially for very thin films in the monolayer range (22) Yi Li; Baberschke, K. Phys. Rev. Lett. 1992, 68, 1208. (23) Bergman, D. J.; Imry, Y.; Deutscher, G.; Alexander, S. J. Vac. Sci. Technol. 1973, 10, 674.
Surface and Thin Film Magnetization
Langmuir, Vol. 12, No. 1, 1996 47
Figure 4. Sketch of the deviation of the direction of the surface magnetization from the in-plane orientation (θ ) π/2) for the first three surface layers i ) 1, 2, 3 dependent on the surface anisotropy K2 according to ref 32. Above a certain threshold K2,c a canted surface magnetization emerges. Figure 3. Thin film Curie temperature as a function of the film thickness d for a fcc(100) film obtained by using a generalized mean field approximation.7 The ratio of the surface magnetic moment µsurf to the bulk moment µbulk on both surfaces is chosen as indicated. For µsurf/µbulk > 1.3 a maximum of Tc(d) emerges.
the actual atomic geometry has a distinct influence on the existence of an ordered magnetic state and the magnitude of the Curie temperature. In the case of an island type growth leading to spatially limited magnetic systems, a sharp phase transition is no longer present.24 Nevertheless one can define, e.g., the maximum of the specific heat as the ferromagnetic ordering temperature. For isolated magnetic islands or free clusters, this temperature increases as the average coordination number increases.25 Thus the order temperature of a film consisting of bulky islands may be larger than a flat film despite the same amount of deposed magnetic material.26 A very intriguing consequence of the reduced dimensionality is the possible existence of an atomic magnetic moment for surface layers or in thin films which vanishes in bulk. An important quantity is the (paramagnetic) density of states at the Fermi energy n(F) which must exceed a certain value in order to generate a (ferro-)magnetic state (Stoner criterion). As mentioned earlier a surface or, more pronounced, for thin films n(F) is enhanced; thus the possibility arises that a metal which barely fails to fulfill the Stoner criterion may be magnetic on surfaces or in thin films. Consequently a large number of 4d and 5d transition metal monolayers were calculated to be ferromagnetic.20,27 However, experimental results are not conclusive yet. On the other hand a definite ferromagnetic state was found for small Rh clusters.28 3. Anisotropy and the Direction of Magnetization A second important magnetic quantity is the direction of magnetization, which at surfaces and thin films also exhibits unusual features. The exchange coupling, eq 1, produces a parallel mutual alignment of the spins but (24) Binder, K.; Rauch, H.; Wildpaner, V. J. Phys. Chem. Solids 1970, 31, 391. Merikoski, J.; Timonen, J.; Manninen, M.; Jena, P. Phys. Rev. Lett. 1991, 66, 938. (25) Jensen, P. J.; Dreysse´, H.; Bennemann, K. H. Nanostruct. Mater. 1994, 3, 365. (26) Farle, M.; Baberschke, K. Phys. Rev. Lett. 1987, 58, 511. Farle, M.; Baberschke, K.; Stetter, U.; Aspelmeier, A.; Gerhardter, F. Phys. Rev. B 1993, 47, 11571. (27) Blu¨gel, S. Phys. Rev. Lett. 1992, 68, 851. (28) Cox, A. J.; Louderback, J. G.; Bloomfield, L. A. Phys. Rev. Lett. 1993, 71, 923.
does not favor a certain direction of the magnetization relative to the lattice. The magnetic orientation at surfaces and in thin films is mainly determined by two different anisotropic contributions which for 3d metals are much weaker than the exchange coupling. First the lattice anisotropy originating from the relativistic spinorbit interaction is present due to the broken inversion symmetry. It was calculated29 and also measured30 that this anisotropy may be 10-100 times stronger than in bulk but is confined only to the surface/interface layers. Second, unlike in cubic bulk systems the long range magnetic dipolar coupling does not vanish at surfaces and in thin films. This interaction prefers always a magnetization direction parallel to the surface or film plane (polar angle θ ) π/2) and is also denoted by shape anisotropy or demagnetizing field. In addition for thin film systems anisotropic interactions cause a remanent magnetization. An isotropic 2D Heisenberg magnet should not exhibit any long range order at finite temperatures (MerminWagner theorem31). However, due to anisotropic interactions, which are always present in magnetic systems, an ordered state is induced with a Curie temperature of the order of the exchange coupling.8 Consequently, ferromagnetic thin films with a few atomic layers are observed at room temperatures. The dipole coupling and the second order uniaxial lattice anisotropy are given by the following expression
∑i
Hani ) -K2
z
2
(µi /µi) +
µB2
∑ 2 ij
(
b µ ib µj
3
rij3
)
(µ bib r ij)(µ bjb r ij) rij5 (5)
rij connects the with µB the Bohr magneton. The vector b lattice sites i and j. We assume for the anisotropy constant K2 > 0; i.e., the lattice anisotropy prefers a magnetization perpendicular to the surface or film plane and thus competes with the dipolar coupling. Together with the exchange coupling, eq 1, this expression is the Hamiltonian considered in the following calculations. First we review briefly the behavior of the direction of the surface magnetization. For small values of K2 the magnetization is always oriented in-plane (θ ) π/2). If K2 exceeds a certain value K2,c, the magnetization of the surface layer(s) may be canted (θi < π/2), as sketched in Figure 4; i.e., the surface magnetization is no longer (29) Gay, J. P.; Richter, R. Phys. Rev. Lett. 1986, 56, 2728. Wang, D. S.; Wu, R.; Freeman, A. J. Phys. Rev. Lett. 1993, 70, 869; Phys. Rev. B 1993, 47, 14932. (30) Gradmann, U. J. Magn. Magn. Mater. 1986, 54-57, 733; 1991, 100, 481. (31) Mermin, N. M.; Wagner, H. Phys. Rev. Lett. 1966, 17, 1133.
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parallel to the bulk magnetization.32 The canting angles θi(K2) behave similar to a continuous (second order) phase transition near the threshold value K2,c and depend on temperature. A canted surface magnetization is possibly observed for Gd.6 For certain thin films a rotation of the magnetization from a perpendicular to an in-plane orientation with increasing film thickness and temperature was observed. In this system the lattice anisotropy is strong enough to overcome the dipolar coupling and cause a perpendicular magnetization.5,9-12 Since the dipole energy increases with film thickness whereas the strong lattice anisotropy is confined only to the surface layers, the magnetization turns into the film plane for thicker films. The temperature behavior is harder to explain. In general the total entropy is expected to be larger for an in-plane direction of the magnetization, which becomes more important for larger temperatures.33 Because the lattice anisotropy and the dipolar coupling are different interactions, the action of temperature on these two couplings is also different.34,35 The ability of the lattice anisotropy to align the magnetization along a certain direction seems to weaken more strongly than for the dipolar coupling, thus above a certain temperature the magnetization is directed in-plane. The temperature dependence of the anisotropic couplings is estimated within a renormalization treatment35 or simulated phenomenologically.14 The magnetic reorientation is accompanied by a region of strongly reduced or even vanishing global thin film magnetization.9,10,11 Since at the reorientation the leading anisotropic interactions are expected to cancel mutually, a 2D Heisenberg system with almost isotropic interactions given by eq 1 may be realized, i.e., all components of the magnetic moments b µi feel the same coupling. If no other anisotropic couplings are present, this should result in a lack of any long range magnetic order in a finite temperature range. In the following we estimate this range of vanishing magnetization. We assume that a temperature-dependent effective anisotropy Keff(T) (the sum of lattice anisotropy and dipole coupling) is present and changes its sign at TR (0 < TR < Tc). Furthermore, a perpendicular magnetization for T < TR is assumed. The general idea is that when Keff(T) drops below a small but nonzero value at a temperature Tz j TR, the effective anisotropy is too weak to maintain a magnetically ordered state, as depicted in Figure 5. To estimate the difference Tz - TR we apply the following simple model. The Curie temperature Tc of a thin film with a constant anisotropy K is approximately given by34
Tc ≈ πIµ2/ln
( ) π2Iµ2 K
(6)
In the case of a temperature-dependent anisotropy Keff(T) the magnetization vanishes at a temperature Tz < TR given by
Tz ≈ πIµ2/ln
(
π2Iµ2 Keff(Tz)
)
(7)
Replacing Tz by TR on the left side of eq 7, one obtains the following scaling equation for Keff(Tz): (32) Mills, D. L. Phys. Rev. B 1989, 39, 12 306. Endo, Y. Phys. Rev. B 1992, 46, 11129. (33) Jensen, P. J.; Bennemann, K. H. Phys. Rev. B 1990, 42, 849; Solid State Commun. 1992, 83, 1057. (34) Mills, D. L. J. Magn. Magn. Mater. 1991, 100, 515. (35) Pescia, D.; Pokrovsky, V. L. Phys. Rev. Lett. 1990, 65, 2599. Erickson, R. P.; Mills, D. L. Phys. Rev. B 1992, 46, 861.
( )
Keff(Tz) ) K
K π Iµ2
(Tc-TR)/TR
2
(8)
This means that the magnetization vanishes in a finite temperature range Tz < T < TR for effective anisotropies 0 < Keff(T) < Keff(Tz), Figure 5. A similar treatment is applied to the in-plane direction leading to a temperature Tx > TR above which the in-plane magnetization appears. The values of Tz and Tx are determined by the temperature dependence of Keff(T) around TR. One can see immediately from eq 8 that Keff(Tz) is the smaller and thus Tz the closer to TR the larger the temperature difference Tc - TR. This possible mechanism for the vanishing magnetization was studied by us in greater detail.14 The uniform magnetization was calculated within a spin wave model by use of a Greens function theory with the Hamiltonian given by eqs 1 and 5. The change of sign of the effective anisotropy at TR was introduced phenomenologically. The resulting range of vanishing magnetization by assuming this uniform 2D Heisenberg model can be compared to the measured range only if the reorientation temperature TR is close to Tc. Thus to explain the experimental observations in general this model has to be improved, in special by consideration of a magnetic domain structure.36 Also higher order anisotropies should be taken into account.13 4. Domain Structure In thin films the long range property of the magnetic dipole interaction has a very important consequence for the magnetic phases. It was shown at T ) 0 for Heisenberg type as well as for Ising type magnetic moments that the long range dipolar interaction leads to an instability of the uniform magnetic phase with perpendicular orientation.37,38 The uniform phase breaks up into a periodic system of stripe-shaped regions with uniform magnetic order and alternating orientation along the film normal (stripe domains). In the case of strong exchange coupling and comparable weak anisotropy as encountered in 3d metals, these stripe domains are separated by Bloch type domain walls.38,39 Also experiments with an improved spatial resolution observed a magnetic domain structure.10 Thus the region of diminished magnetization around the magnetic reorientation may be caused by the occurrence of magnetic domains, which exhibit a remanent magnetic order but cancel mutually. We have calculated the stripe domain structure at T ) 0 by use of the Hamiltonian equation (1) together with eq 5. For simplicity we assume a single magnetic layer first. Specific care should be regarded to the dipole sums, especially to oscillating ones which appear in the case of modulated (domain) magnetic phases. Since the energies of the different phases have to be calculated with high accuracy, the slowly converging sums are replaced by rapid converging ones in the spirit of the Ewald summation.40 For the stripe domain phase we have assumed the following model.38 The direction of magnetization varies periodically with periodicity L along the y-direction, whereas is uniform along the x-direction. Inside the domains the magnetization is directed along the (zdirection. The domains are separated by 180° Bloch type (36) Kashuba, A.; Pokrovsky, V. L. Phys. Rev. Lett. 1993, 70, 3155. (37) Garel, T.; Doniach, S. Phys. Rev. B 1982, 26, 325. Gabay, M.; Garel, T. J. Phys. (Paris) 1985, 46, 5. Czech, R.; Villain, J. J. Phys.: Condens. Matter 1988, 1, 619. Taylor, M. B.; Gyorffy, B. L. J. Phys.: Condens. Matter 1993, 5, 4527. (38) Yafet, Y.; Gyorgy, E. M. Phys. Rev. B 1988, 38, 9145. (39) Hubert, A. Theorie der Doma¨ nenwa¨ nde in geordneten Medien; Springer Verlag: Berlin, 1974. (40) Benson, H.; Mills, D. L. Phys. Rev. 1969, 178, 839.
Surface and Thin Film Magnetization
Figure 5. Sketch of the temperature induced magnetic reorientation in a thin film. The effective anisotropy Keff(T) changes its sign at the reorientation temperature TR. Between Keff(Tz) > Keff(T) > Keff(Tx) the anisotropy is too weak to induce a magnetization leading to a vanishing long range ordered phase in a temperature range Tz < T < Tx near the reorientation between the perpendicular (mz) and the in-plane (mx) direction of magnetization.
Figure 6. Periodicity L and wall width b in multiples of the lattice constant a0 of the stripe domain phase dependent on the uniaxial lattice anisotropy K2. A cosine profile of the domain wall is chosen (see text), E0 is the demagnetizing energy, J the exchange coupling, and µ the magnetic moment. Note the logarithmic scale for L and b.
walls; i.e., the magnetization inside the walls has a constant magnitude but exhibits no component along the wall normal (the y-direction). The domain wall is simply simulated by a cosine function with wall width b. Energy minimization according to the two parameters L and b leads to the most stable domain phase. The involved energies are given in units of the demagnetizing energy E0 ) 2πc(µµB)2/a03, i.e., the magnetic dipole energy difference between uniform perpendicular and in-plane magnetization, with a0 the lattice constant, and c ) 1.078 for a square lattice. Then for K2/E0 > 1 the uniform perpendicular phase is more stable than the in-plane one. The resulting solution of L and b as a function of the lattice anisotropy K2 is shown in Figure 6, the respective energy difference ∆E between domain and uniform phases in Figure 7. The periodicity L raises almost exponentially, whereas the wall width b reaches a constant value with increasing K2. For Iµ2/E0 ) 25 and K2/E0 > 0.997 the domain phase is more stable than the uniform phases.38 Note that ∆E is quite small. Around the magnetic reorientation the binding energy is largest; consequently the observation of a domain structure should be most
Langmuir, Vol. 12, No. 1, 1996 49
Figure 7. Energy difference between the stripe domain and the uniform phases as a function of K2. For the other denotations see Figure 6. For K2/E0 > 1 the uniform perpendicular magnetization is more stable than the in-plane one. However, for K2/E0 > 0.997 the domain phase energy is always lower than the energy of the uniform magnetization.
Figure 8. Thin film magnetization in the presence of an applied magnetic field H0 with perpendicular orientation as a function of K2. The different curves refer to different ratios H0/E0 as indicated in units of 10-4. For the other denotations, see Figure 6. To compare with experimental observations the ratio K2/E0 has to be viewed as a slowly varying function of temperature.
probable near the reorientation. In addition we have considered a different domain profile which resembles the shape of a single Bloch wall in bulk (tanh profile).41 This domain profile leads to a lower domain phase energy and to a similar behavior of L and b. The consideration of other domain walls like the Ne´el wall type39 is always unfavorable. Why can nevertheless a perpendicular uniform magnetization be observed? First, the uniform phase may be in a metastable state and reside there for a long time because of the small energy difference ∆E. Second, a small external magnetic field may be present which stabilizes the uniform magnetization. To calculate the latter µi is included in the possibility, a Zeeman term -H B ∑ib Hamiltonian with a perpendicularly oriented magnetic field H B ) (0,0,H0). We use a similar domain profile as before but with different extensions of the domains pointing in the +z- and -z-direction, resulting in a domain wall motion and in a nonzero perpendicular magnetization. When a magnetic field is applied, the periodicity L increases markedly, whereas the wall width b stays almost constant. The magnetization increases linearly with H0. (41) Jensen, P. J. To be published.
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Figure 9. Phase diagram of the thin film magnetic phases in the K2-H0 plane for T ) 0. The dashed line refers to the phase boundary between uniform phases with perpendicular and inplane orientations. A stripe domain phase is stable in the dashed region. For the other denotations see Figure 6.
Above a certain field strength H0,z the uniform phase becomes more favorable, leading to a jump in the magnetization. This threshold value decreases rapidly with increasing K2, reflecting the decreasing binding energy of the domain phase. Thus already very small magnetic fields are sufficient to stabilize the uniform magnetization. Only near the magnetic reorientation the threshold magnetic field H0,z reaches considerable values. For Fe we estimate H0,z ≈ 18 G for K2/E0 ) 1. In Figure 8 we show the magnetization as a function of K2 for certain values of H0/E0. The resulting phase diagram in the K2H0 plane is sketched in Figure 9. With the assumption of a weak perpendicularly oriented magnetic field, one can qualitatively simulate the observed asymmetric magnetization profile around the reorientation.11 For this the lattice anisotropy and the dipole coupling are assumed to be temperature dependent; thus the ratio K2/E0 is a slowly varying function of the temperature. 5. Conclusion In this work we have investigated the dependence of the Curie temperature and the direction of magnetization on reduced dimensionality. The diminished coordination number always leads to a weaker magnetic order. On the other hand especially for magnetic 3d transition metals
Jensen and Bennemann
this can be counterbalanced by enhanced magnetic moments or exchange interactions for the surface layers. In extreme cases the Curie temperature for surfaces or thin films may be larger than for the respective bulk system. Also the magnetic anisotropy is strongly enhanced for surface layers, leading possibly to a canted magnetization at surfaces. For thin films anisotropic interactions are even more important, since they suppress the strong thermal fluctuations in 2D system and induce a long range magnetic order. In some cases a reorientation of the thin film magnetization is observed with increasing temperature and film thickness. The long range property of the magnetic dipole interaction leads to an instability of the uniform thin film magnetization with perpendicular orientation. In this case the ferromagnetic phase breaks up into a magnetic domain pattern, resulting in a diminished or vanishing total magnetization especially around the magnetic reorientation. However, a small applied magnetic field is able to restore the uniform phase. The possible realization of a 2D Heisenberg magnet with isotropic interactions due to mutual cancellation of anisotropies around the reorientation does not seem to cause the vanishing magnetization in general. The question arises whether the magnetic reorientation is a phase transition or not.34 Around a phase transition the free energy exhibits a nonanalytical behavior leading to strong fluctuations of the order parameter. In a magnetic field induced reorientation a strong increase of inelastically scattered light was observed,15 possibly due to a magnetic domain structure and/or the occurrence of critical phenomena. Calculations obtain a strong increase of transversal fluctuations, leading to a diminished magnetization.34 However, we like to emphasize that the consideration of additional phases like the domain structure may alter the nature of the reorientation transition drastically, for example by inducing a pure rotation of the magnetization without any critical fluctuations. This demands the calculation of the uniform and the domain phases at finite temperatures.41 Also higher order lattice anisotropies should be considered,13 as well as thin films with a few atomic layers. However, we expect that the general features of the magnetic reorientation emerge already for a single magnetic layer. Acknowledgment. This work was supported by the Deutsche Forschungsgeimeinschaft (DFG). LA940906G
