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On direction of the magnetization of thin films and sandwiches as a function of temperature-II
Solid State Communications, Printed in Great Britain.
ON DIRECTION
Vol. 83, No. 12, pp. 1057-1059,
1992.
0038-1098/92 $5.00 + .OO Pergamon Press Ltd
OF THE MAGNETIZATION OF THIN FILMS AND SANDWICHES AS A FUNCTION OF TEMPERATURE-II P.J. Jensen and K.H. Bennemann
Freie Universitit
Berlin, Institut fiir theoretische
Physik, Arnimallee 14, D-1000 Berlin 33, F.R.G.
(Received 28 February 1992 by P.H. Dederichs) The perpendicular to in-plane transition of thin ferromagnetic films is discussed. A simple theory by P.J. Jensen & K.H. Benemann, Phys. Rev. B42,849 (1990) is compared with a renormalization treatment of the problem by D. Pescia t V.L. Pokrovsky, Phys.Rev.Lett. 65,2599 (1990). Furthermore, extensions of these theories are studied.
IN A RECENT PAPER [1] we studied the direction of magnetization M, of thin ferromagnetic films. A number of systems, e.g., Fe/Cu( 100) [2], Fe/ Ag(lO0) [3], and Co/Au(l 1 1) [4], are magnetized perpendicularly at T = 0 due to the strong surface anisotropy. For increasing temperature and also for increasing tihu thickness the magnetization turns usually into the film plane, before the system reaches its Curie temperature T,. This rotation was observed to be smooth and reversible, but is accompanied by a loss of long range magnetic order near the reorientational temperature TR [2], at which the perpendicular components of M, vanish. In Ref. [l] we tried to explain this effect by the larger in-plane directional entropy. However, then TR is of the order of the lattice anisotropy energy &, which is usually much smaller than the Curie temperature of the film, even if this energy may be strongly enhanced for 3dtransition metal films [S]. Shortly after the simple theory by Jensen et al. appeared another study of the same problem was published by Pescia and Pokrovsky [6]. Using a renormalization procedure they obtained a much higher reorientational temperature T, of the order of T,. At T, an abrupt jump of M, from perpendicular to in-plane orientation is predicted by this theory. In this paper we would like to point out that our theory may also yield TR w T, as obtained by the renormalization treatment, if the strong correlations of the 2D Heisenberg system are properly included by us. In addition, we show that around TR the directional entropy, as considered by us earlier [l], may cause a significant contribution to the orientational behaviour of the film magnetization and explain the observed continuous transition of M,. First we sketch the essential steps of the theoretical approaches used by us [l] and by Pescia
and Pokrovsky (61. The main contribution to the anisotropic free energy F.&, of N spins are the (competing) uniaxial lattice anisotropy Kz and magnetic dipolar interaction a, which in general depend on the temperature T. Hence, Fmi,(TV 0) = N( Q!- K2) COS2 B N -ln[l + (27r - l)sine], - TN(T)
(1)
where 8 is the angle between M, and the film normal. A strong K2 > 0 may overcome the dipolar energy (which always favours an in-plane magnetization), leading to a perpendicular orientation of M, for thin films at T = 0 [5]. M( 7’) denotes the number of spins which move coherently. N/N(T) amounts to the number of ‘giant spins” or spin blocks which may gain directional entropy by turning from perpendicular to in-plane orientation. The entropy expression yields correctly the limiting cases for 0 = 0 and 8 = x/2. If X( 7) is large, the directional entropy can be neglected. Previously (11, we used K(T) = 1, thus neglecting spin correlations in the 2D Heisenberg magnet. In consequence, we obtained TR < T,, in contrast to experimental observations [2-4], indicating the poor quality of the approximation ./v( 7) fi: 1. As shown by Pescia and Pokrovsky [6] by use of a renormalization procedure the anisotropic free energy can be put into the form F,i,(T,l?)
= N[G(T) -k2(T)]cosZe+..m,
(2)
with effective parameters & = ctZ2(T) and & = K2Z3( T) dependent on temperature. In principle they should also depend on the film thickness d. The renormalization procedure [6] yields
1057
Z(T) = 1 - $nf, 2
1058
MAGNETIZATION
OF THIN
and a reorientational temperature TR of the order of T, or the exchange energy J. At TR the dipolar and the anisotropy energies mutually cancel: (I = KzZ(TR). This causes a discontinuous transition of the magnetization direction [7]. In the following we show that using an appropriate choice of N(T) we are able to obtain from equation (1) also the high TR m T, and furthermore a continuous transition of Ms. Note that in Ref. [2] a continuous reorientational transition of M,(T) was observed, accompanied with an almost vanishing magnetization in a temperature range of 20-30K around TR. Two possible explanations for this were proposed. On the one hand, the long range ordered magnetic film may fragment into many static magnetic domains, yielding a vanishing overall magnetization M,. Note that magnetic domain fragmentation has been observed for very thin Co films on Au(1 1 1) [4]. However, since the domain wall width for an in-plane N&e1 wall is given approximately by W(7) a ,/J(T)/[&(T) - a(T)] [8], W(T) becomes very large at T = TR. Hence, the use of domain fragmentation for explaining IM,I+ 0 becomes questionable if the wall width is comparable to the domain size. A second explanation would be that the long range magnetic order disappears, because at T GZTR anisotropy and dipolar interactions cancel and the efictive anisotropy becomes too weak to sustain the long range order. Thus the film is expected to become almost an isotropic 2D Heisenberg system, breaking up into spin blocks with an appropriate size N(T) = exp(4rJ/T) [9]. Then, at larger temperatures the long range order is again established by in-plane anisotropies and the dipolar interaction. Note that a transition into a state with vanishing magnetization at T fir TR is supported by inelastic light scattering experiments observing a magnet field induced reorientational transition [lo]. Around T u TR a strong increase of absorption is observed, indicating possibly the existence of critical fluctuations. This is not expected for a static domain state. In view of this we extend our previous theory, equation (I), as follows [l]. We assume that at TM TR the thin film magnetization breaks up into spin blocks of size M(T). Note that in equation (1) the coefficients & and a are replaced by &(T) and G(T), which are the renormalized parameters [6]. The X(7) spins move coherently, maintaining an almost ferromagnetic order due to the strong exchange coupling. The number of spins N(T) gain an entropy of the order ln2x when turning into a film plane, behaving like a corresponding “giant spin”. In particular, for a magnetic film with long
FILMS AND SANDWICHES
Vol. 83, No. 12
range ordering the number of spins N(T) should approximately be of the order of the total number of spins in the film. Then the directional entropy is negligible, see equation (1). One obtains now from equation (1) with Kz + i&(T) and a + ii(T) the equilibrium angle eM by determining aF,,,/&?, = 0. The result is sine&T)
= &
({S-
l)?
(3) with c = 2x - I. If the number M(T) of spins in a spin block is large, one obtains almost the same value for the reorientational temperature TR as previously obtained by Pescia et al. [6]. For large M(T) the directional entropy is only a small correction to the anisotropic free energy. However, inclusion of this entropy causes as observed a continuous transition, and not a discontinuous one [l 11. In conclusion we have discussed how to obtain from our previous theory [l] TR = T,, in agreement with Pescia and Pokrovsky [6]. The number X(T) of coherently moving spins reflecting the strong correlations of the 2D Heisenberg model controls TR. Only if M(T) is relatively small, one finds TX < T,. Furthermore, cancellation of anisotropy and dipolar coupling at TM TR may cause a destruction of the long range magnetic order into spin blocks. In this case we obtain a continuous rotation of M, around TR.
Acknowledgements - We thank V. Pokrovsky useful and very illuminating discussions regarding problem of the 2D Heisenberg model and for critical discussion of our earlier estimate of
for the the TR.
REFERENCES 1. 2. 3.
4.
5. 6.
P.J. Jensen & K.H. Bennemann, Phys. Rev. B42, 849 (1990). D.P. Pappas, K.P. Krimer & H. Hopster, Phys. Rev. L&t. 64, 3179 (1990). C. Liu, E.R. Moog & S.D. Bader, Phys. Rev. L&t. 60, 2422 (1988); M. Stampanoni, Appl. Phys. A49, 449 (1989); B. Heinrich, J.F. Cochran, A.S. Arrot, S.T. Purcell, K.B. Urquhart, J.R. Dutcher 8c W.F. Egelhoff, Appl. Phys. A49,473 (1989). C. Chappert & P. Bruno, J. Appl. Phys. 64, 5736 (1988); P. Bruno & J.P. Renard, Appl. Phys. A49, 499 (1989); R Allenspach, M. Stampanoni & A; Bischof, Phys. Rev. Lett. 65, 3344 (1990). J.G. Gav & R. Richter. Phvs. Rev. L.&t. 56, 2728 (1686). D. Pescia & V.L. Pokrovsky, Phys. Rev. Lett. I
w
Vol. 83, No. 12 7.
‘.
MAGNETIZATION
OF THIN FILMS AND SANDWICHES
65, 2599 (1990). Two remarks should be added: First the renormalization procedure as used in Ref. [6] is valid for a magnetic monolayer (2D Heisenberg system). However, spin fluctuations are expected to be less important for thicker magnetic films. Hence the renormalization calculation should be performed also as a function of the film thickness d. For thick 6lms the magnetization should always be inplane. Secondly, in the case of in-plane magnetization the magnetic order is no longer induced by the uniaxial anisotropy Kz as assumed in Ref. [a]. Rather for T > TR the magnetic properties are characterized by the action of in-plane anisotropies and the dipolar
1059
interaction on a 2D planar (XY-) magnet. In view of this the previous studies should be re-examined. 8. A. Hubert, Theorie &r Dorncinenwhitde in geordneten Medien, Springer-Verlag, Berlin (1974). 9. J.M. Kosterlitz & D.J. Thouless, in Progess in Low Temperature Physics (Edited by D.F. Brewer), Vol. VIIB, p. 394. North Holland 1978 ; V.L. Pokrovsky, Adv. Phys. 243, 595 1979 [ 1i 10. J.R. Dutcher, J.F. Cochran, I. Jacob & W.F. Egelhoff, Jr., Phys. Rev. B39, 10430 (1989). 11. As noted already by Pescia and Pokrovsky [6], consideration of higher order anisotropy terms may also cause a continuous rotation of M,.
ON DIRECTION
Vol. 83, No. 12, pp. 1057-1059,
1992.
0038-1098/92 $5.00 + .OO Pergamon Press Ltd
OF THE MAGNETIZATION OF THIN FILMS AND SANDWICHES AS A FUNCTION OF TEMPERATURE-II P.J. Jensen and K.H. Bennemann
Freie Universitit
Berlin, Institut fiir theoretische
Physik, Arnimallee 14, D-1000 Berlin 33, F.R.G.
(Received 28 February 1992 by P.H. Dederichs) The perpendicular to in-plane transition of thin ferromagnetic films is discussed. A simple theory by P.J. Jensen & K.H. Benemann, Phys. Rev. B42,849 (1990) is compared with a renormalization treatment of the problem by D. Pescia t V.L. Pokrovsky, Phys.Rev.Lett. 65,2599 (1990). Furthermore, extensions of these theories are studied.
IN A RECENT PAPER [1] we studied the direction of magnetization M, of thin ferromagnetic films. A number of systems, e.g., Fe/Cu( 100) [2], Fe/ Ag(lO0) [3], and Co/Au(l 1 1) [4], are magnetized perpendicularly at T = 0 due to the strong surface anisotropy. For increasing temperature and also for increasing tihu thickness the magnetization turns usually into the film plane, before the system reaches its Curie temperature T,. This rotation was observed to be smooth and reversible, but is accompanied by a loss of long range magnetic order near the reorientational temperature TR [2], at which the perpendicular components of M, vanish. In Ref. [l] we tried to explain this effect by the larger in-plane directional entropy. However, then TR is of the order of the lattice anisotropy energy &, which is usually much smaller than the Curie temperature of the film, even if this energy may be strongly enhanced for 3dtransition metal films [S]. Shortly after the simple theory by Jensen et al. appeared another study of the same problem was published by Pescia and Pokrovsky [6]. Using a renormalization procedure they obtained a much higher reorientational temperature T, of the order of T,. At T, an abrupt jump of M, from perpendicular to in-plane orientation is predicted by this theory. In this paper we would like to point out that our theory may also yield TR w T, as obtained by the renormalization treatment, if the strong correlations of the 2D Heisenberg system are properly included by us. In addition, we show that around TR the directional entropy, as considered by us earlier [l], may cause a significant contribution to the orientational behaviour of the film magnetization and explain the observed continuous transition of M,. First we sketch the essential steps of the theoretical approaches used by us [l] and by Pescia
and Pokrovsky (61. The main contribution to the anisotropic free energy F.&, of N spins are the (competing) uniaxial lattice anisotropy Kz and magnetic dipolar interaction a, which in general depend on the temperature T. Hence, Fmi,(TV 0) = N( Q!- K2) COS2 B N -ln[l + (27r - l)sine], - TN(T)
(1)
where 8 is the angle between M, and the film normal. A strong K2 > 0 may overcome the dipolar energy (which always favours an in-plane magnetization), leading to a perpendicular orientation of M, for thin films at T = 0 [5]. M( 7’) denotes the number of spins which move coherently. N/N(T) amounts to the number of ‘giant spins” or spin blocks which may gain directional entropy by turning from perpendicular to in-plane orientation. The entropy expression yields correctly the limiting cases for 0 = 0 and 8 = x/2. If X( 7) is large, the directional entropy can be neglected. Previously (11, we used K(T) = 1, thus neglecting spin correlations in the 2D Heisenberg magnet. In consequence, we obtained TR < T,, in contrast to experimental observations [2-4], indicating the poor quality of the approximation ./v( 7) fi: 1. As shown by Pescia and Pokrovsky [6] by use of a renormalization procedure the anisotropic free energy can be put into the form F,i,(T,l?)
= N[G(T) -k2(T)]cosZe+..m,
(2)
with effective parameters & = ctZ2(T) and & = K2Z3( T) dependent on temperature. In principle they should also depend on the film thickness d. The renormalization procedure [6] yields
1057
Z(T) = 1 - $nf, 2
1058
MAGNETIZATION
OF THIN
and a reorientational temperature TR of the order of T, or the exchange energy J. At TR the dipolar and the anisotropy energies mutually cancel: (I = KzZ(TR). This causes a discontinuous transition of the magnetization direction [7]. In the following we show that using an appropriate choice of N(T) we are able to obtain from equation (1) also the high TR m T, and furthermore a continuous transition of Ms. Note that in Ref. [2] a continuous reorientational transition of M,(T) was observed, accompanied with an almost vanishing magnetization in a temperature range of 20-30K around TR. Two possible explanations for this were proposed. On the one hand, the long range ordered magnetic film may fragment into many static magnetic domains, yielding a vanishing overall magnetization M,. Note that magnetic domain fragmentation has been observed for very thin Co films on Au(1 1 1) [4]. However, since the domain wall width for an in-plane N&e1 wall is given approximately by W(7) a ,/J(T)/[&(T) - a(T)] [8], W(T) becomes very large at T = TR. Hence, the use of domain fragmentation for explaining IM,I+ 0 becomes questionable if the wall width is comparable to the domain size. A second explanation would be that the long range magnetic order disappears, because at T GZTR anisotropy and dipolar interactions cancel and the efictive anisotropy becomes too weak to sustain the long range order. Thus the film is expected to become almost an isotropic 2D Heisenberg system, breaking up into spin blocks with an appropriate size N(T) = exp(4rJ/T) [9]. Then, at larger temperatures the long range order is again established by in-plane anisotropies and the dipolar interaction. Note that a transition into a state with vanishing magnetization at T fir TR is supported by inelastic light scattering experiments observing a magnet field induced reorientational transition [lo]. Around T u TR a strong increase of absorption is observed, indicating possibly the existence of critical fluctuations. This is not expected for a static domain state. In view of this we extend our previous theory, equation (I), as follows [l]. We assume that at TM TR the thin film magnetization breaks up into spin blocks of size M(T). Note that in equation (1) the coefficients & and a are replaced by &(T) and G(T), which are the renormalized parameters [6]. The X(7) spins move coherently, maintaining an almost ferromagnetic order due to the strong exchange coupling. The number of spins N(T) gain an entropy of the order ln2x when turning into a film plane, behaving like a corresponding “giant spin”. In particular, for a magnetic film with long
FILMS AND SANDWICHES
Vol. 83, No. 12
range ordering the number of spins N(T) should approximately be of the order of the total number of spins in the film. Then the directional entropy is negligible, see equation (1). One obtains now from equation (1) with Kz + i&(T) and a + ii(T) the equilibrium angle eM by determining aF,,,/&?, = 0. The result is sine&T)
= &
({S-
l)?
(3) with c = 2x - I. If the number M(T) of spins in a spin block is large, one obtains almost the same value for the reorientational temperature TR as previously obtained by Pescia et al. [6]. For large M(T) the directional entropy is only a small correction to the anisotropic free energy. However, inclusion of this entropy causes as observed a continuous transition, and not a discontinuous one [l 11. In conclusion we have discussed how to obtain from our previous theory [l] TR = T,, in agreement with Pescia and Pokrovsky [6]. The number X(T) of coherently moving spins reflecting the strong correlations of the 2D Heisenberg model controls TR. Only if M(T) is relatively small, one finds TX < T,. Furthermore, cancellation of anisotropy and dipolar coupling at TM TR may cause a destruction of the long range magnetic order into spin blocks. In this case we obtain a continuous rotation of M, around TR.
Acknowledgements - We thank V. Pokrovsky useful and very illuminating discussions regarding problem of the 2D Heisenberg model and for critical discussion of our earlier estimate of
for the the TR.
REFERENCES 1. 2. 3.
4.
5. 6.
P.J. Jensen & K.H. Bennemann, Phys. Rev. B42, 849 (1990). D.P. Pappas, K.P. Krimer & H. Hopster, Phys. Rev. L&t. 64, 3179 (1990). C. Liu, E.R. Moog & S.D. Bader, Phys. Rev. L&t. 60, 2422 (1988); M. Stampanoni, Appl. Phys. A49, 449 (1989); B. Heinrich, J.F. Cochran, A.S. Arrot, S.T. Purcell, K.B. Urquhart, J.R. Dutcher 8c W.F. Egelhoff, Appl. Phys. A49,473 (1989). C. Chappert & P. Bruno, J. Appl. Phys. 64, 5736 (1988); P. Bruno & J.P. Renard, Appl. Phys. A49, 499 (1989); R Allenspach, M. Stampanoni & A; Bischof, Phys. Rev. Lett. 65, 3344 (1990). J.G. Gav & R. Richter. Phvs. Rev. L.&t. 56, 2728 (1686). D. Pescia & V.L. Pokrovsky, Phys. Rev. Lett. I
w
Vol. 83, No. 12 7.
‘.
MAGNETIZATION
OF THIN FILMS AND SANDWICHES
65, 2599 (1990). Two remarks should be added: First the renormalization procedure as used in Ref. [6] is valid for a magnetic monolayer (2D Heisenberg system). However, spin fluctuations are expected to be less important for thicker magnetic films. Hence the renormalization calculation should be performed also as a function of the film thickness d. For thick 6lms the magnetization should always be inplane. Secondly, in the case of in-plane magnetization the magnetic order is no longer induced by the uniaxial anisotropy Kz as assumed in Ref. [a]. Rather for T > TR the magnetic properties are characterized by the action of in-plane anisotropies and the dipolar
1059
interaction on a 2D planar (XY-) magnet. In view of this the previous studies should be re-examined. 8. A. Hubert, Theorie &r Dorncinenwhitde in geordneten Medien, Springer-Verlag, Berlin (1974). 9. J.M. Kosterlitz & D.J. Thouless, in Progess in Low Temperature Physics (Edited by D.F. Brewer), Vol. VIIB, p. 394. North Holland 1978 ; V.L. Pokrovsky, Adv. Phys. 243, 595 1979 [ 1i 10. J.R. Dutcher, J.F. Cochran, I. Jacob & W.F. Egelhoff, Jr., Phys. Rev. B39, 10430 (1989). 11. As noted already by Pescia and Pokrovsky [6], consideration of higher order anisotropy terms may also cause a continuous rotation of M,.