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Reorientation transition of the magnetization in thin ferromagnetic films
s u r f a c e science Surface Science307-309 (1994) 1109-1113
ELSEVIER
Reorientation transition of the magnetization in thin ferromagnetic films D.K. Morr, P.J. Jensen
*, K . - H . B e n n e m a n n
Institute for Theoretical Physics, Freie Universitiit Berlin, Arnimallee 14, D-14 195 Berlin, Germany
(Received 20 August 1993)
Abstract
The orientation of the magnetization M in thin ferromagnetic films is studied by use of a Greens function type theory within RPA approximation. For the temperature dependence of the effective anisotropy a phenomenological approach is applied. Due to the competing lattice anisotropy and the dipole interactions the magnetization turns from a perpendicular to an in-plane orientation with increasing temperatures. A vanishing overall magnetization may result in a temperature range around the reorientation temperature TR, before an in-plane magnetization appears.
This transition metal a n d rare earth metal films with thicknesses of a few monolayers are known to be ferromagnetically ordered [1]. A two-dimensional (2D) Heisenberg magnet with only isotropic and short range interactions should not exhibit any ordered state at finite temperatures [2]. However, a global magnetization in these systems is induced by additional (weak) interactions like lattice anisotropies a n d / o r the dipole coupling (shape anisotropy), and one obtains a Curie temperature Tc of the order of the exchange coupling [3], as observed experimentally. For certain thin film systems like Fe/Cu(100) [4-7], Fe/Ag(100) [5,8], and C o / A u ( l l l ) [9] a perpendicular magnetization has been observed at low temperatures. With increasing temperatures (and also with increasing film thicknesses) the magnetization turns into the film plane at a
* Corresponding author.
reorientation temperature T R. In a temperature range around T R all components of the magnetization might vanish simultaneously [4,8]. Recent experiments with an improved spatial resolution observed a multi-domain state at T R [6]. This also leads to an overall vanishing or at least strongly reduced magnetization. Furthermore, Brillouin light scattering experiments on a magnetic field induced reorientation of the thin film magnetization report a strong increase of the scattering intensity around T R [10], which might indicate an occurrence of critical fluctuations. Therefore, the vanishing magnetization at T R may result from either a multi-domain state or a 2D isotropic Heisenberg system without long range order, due to the mutual cancellation of anisotropies. The reorientation of the thin film magnetization was attempted to be explained by the rotational entropy, but the strong fluctuations in the 2D system were neglected [11]. In Ref. [12] these fluctuations were considered by use of a renor-
0039-6028/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0039-6028(93)E0825-F
1110
D.K. Morret al./ SurfaceScience307-309 (1994)1109-1113
malization procedure, yielding a temperature dependent effective anisotropy Keff composed of the uniaxial lattice anisotropy and the dipole coupling. According to these calculations Keff changes sign at the reorientation temperature, since the two parts of the effective anisotropy renormalize differently. However, other calculations obtain quite different results [13,14]. Though it is accepted that the reorientation of M is caused by the change of sign of K~ff, a reliable calculation of the temperature dependence of the lattice anisotropy and the dipole coupling is currently not available. In this paper we use a spin wave theory for the investigation of the magnetic properties around TR. For the calculation of M(T) we make an ansatz for the temperature dependence of the effective anisotropy K~ff(T), in order to simulate the experimental observations. We start with the following Hamiltonian [15]
~=
1j E SiSj- ~K 1 E S{Sj (i,j) (i,j)
1 2
Mz'x
1 [~ [,~ 2M~, x ~--£ Jo dkxJo dkr exp( E~,x/O) - 1' (2)
a 0 = 1, and with the spin wave dispersion relations 2K
D
E z = 4 - 2y k + J
A
j Yk - ~- (~a(kx, ky)
+40)]Mz,
[
K
E x = 4 - 2y k - ~--~Tk+ ~-~(4 - 7~) A
+7(2ga(k.,k,) + 2 S 2 - 3Eb(k~, kr))]Mx .
2 (S •Sj + s sf) (i,j)
q_lAi~ [ sisj (SiFij)(Sjrij) ) i,j I r3 -- 3 r5" ,
components of the average magnetization perpendicular (M~) and parallel (M~) to the film plane, assuming a uniform phase:
(3)
The following denotations have been used: "~k = COS(kx) + c o s ( k y ) ,
(1)
where the z-axis is chosen to be along the film normal. E(i,j> denotes the sum over nearest neighbors and S~ is a 1/2 Heisenberg spin at lattice site i. J is the exchange coupling between the spins and D the quartic in-plane anisotropy. K is the uniaxial anisotropy, confined mainly to the surface/interface layers and preferring for K > 0 a perpendicular magnetization. The last term in (1) represents the magnetic dipole interaction, with A = izo(glzB)2/a 3 and rii is the distance between lattice sites i and j in units of the lattice constant a 0. For simplicity we assume a magnetic monolayer (square 2D lattice) first. Our treatment can be straightforwardly extended to the case of several atomic layers. We will explain later how our result will change in this case. The Hamiltonian Eq. (1) is solved by using a Greens function theory within the RPA approximation [15]. We obtain a self-consistent equation for the
a=Ea = l
1
~
1
+ a,b=l E
( a 2 + b 2)
3/2,
cos( ak x) + cos(ak r )
Ea(kx, k y ) =
E
a3
a=l
+2
Z
cos(ak x) cos(bky),
a,b= a
( a2 + b2) 3/2
cos(aky) a3
F~a(kx, k y ) = • a=l
+ Z
a,b=l
2b: cos(ak~) cos(bky) (a 2 + b 2 ) 5/2
(4) O=kBT/J
is the relative temperature, k B is Boltzmann's constant, and krl=(kx, ky) is the 2D wave vector parallel to the film plane. The spin wave energy gaps Ez°,x= Ez,x(k x = O, k r = O)
D.K. Morr et al. / Surface Science 307-309 (1994) 1109-1113
> 0, Eq. (3), cause a thin film magnetization and are given by [K D E° = 2 7 - 7
[
A ]
2~'c7 ] M== 2
Kef f
l
U
t
~
o.zt
~
01
I
Keel
and
Ex°= - - 7 - -
Mx'
with c = 1.078 numerically calculated. In order to observe a magnetization perpendicular to the film ( E ° > 0) the uniaxial anisotropy K has to overcome the magnetic dipole interaction ( ~ 27rA). Such strong K for the surface/interface layers have been obtained in recent calculations [17]. With temperature independent coefficients, as given in the Hamiltonian (1), one always obtains, depending on the sign of Ez° and Ex°, either a perpendicular or an in-plane magnetization up to the Curie temperature Tc, respectively. Thus, a reorientation of the magnetization can only be achieved by a change of sign of the effective a n i s o t r o p y Kef f with increasing temperatures. Due to the lack of a reliable theory for the temperature dependence of Kefe around TR, we introduce a temperature function f(T) and put Keff(T)=Kefr(O)'f(T). This results in temperature dependent spin wave energy gaps, Eq. (3). According to experimental observations, f(T) should satisfy the following requirements: (1) f(T) should be continuous and r f(T) l <_1; (2) T R is given by f ( T n) = 0; (3) f(T) > 0 for T < T R and f(T)
[(TR-T)/AT, f(T) = [ s i g n ( r l ~ - T),
ITR-TIAT.
(5)
The parameters T R and AT determine f(T)= 0 and the slope of f(T) at T R. In Fig. 1 we present the resulting magnetization for TR/T c = 0.78 and for different parameters of AT. A gap of vanishing magnetization I M(T)I around T R is obtained, in addition a steeper slope of f(T= T R) a n d / o r a smaller TR/T c lead to a narrower gap. M z vanishes slightly below TI~ and Mx appears above T R. In a finite temperature range around T R the effective
K/J=0.02
0.4
~
J M~
1111
0
~
M
\
~
.m,m ~^.
0.6
A/g=0.001
D/a=0.o01 %/%=0.78 "'."~/ ~ ' " ~
0.8
1
T e m p e r a t u r e T/T c
Fig. 1. Perpendicular (M z) and in-plane (M x) monolayer magnetization for the parameter A T / T c =0.05 (full line), 0.10 (dashed line), and 0.20 (dotted line) of the temperature function Eq. (5). We have chosen T R / T c = 0.78 and the coupling constants in units of the exchange coupling J as given in the figure.
anisotropy is, though not zero, too weak to induce a global ordering in the magnetic plane. Thus, in this region the film corresponds to an almost isotropic 2D Heisenberg model. Different choices of the temperature function f(T) result in a quite similar behavior of the magnetization. For comparison with experimental data our formalism can be extended to the case of thin films with d atomic layers. The strong uniaxial anisotropy K is (mainly) confined to the surface and interface layers, thus is nearly independent of d, whereas the dipole interaction is almost proportional to d for thin films. Then a magnetization reorientation is induced by an increasing film thickness. The reorientation thickness d R is given by K = 2rrcAd R and is expected to be d R --4 - 7 atomic layers [4-8]. Also the parameters T R and AT of Eq. (5) should depend on d, resulting for increasing d in a smaller TR, for example. For the same TR/T c and AT/T c the temperature region of vanishing magnetization decreases for increasing d. However, the main features of M(T) for thin films are the same as obtained for a monolayer, as confirmed by recent calculations [191. In Figs. 2 and 3 we compare the results of our model with experiments [4,8]. Note that for small
1112
~Z
D.K. Morr et aL / Surface Science 307-309 (1994) 1109-1113
0.4
the presence of magnetic domains as observed recently [6]. Since our calculations only consider a homogeneous magnetization, domain structures have not been taken into account in this work. This would require a calculation assuming a spatial variation of the magnetization M ( r ) . Actually one expects that for K > 2~-cA a domain structure may be more favorable than a homogeneous magnetization, as shown previously for T = 0 in particular for a striped domain pattern [18]. If domain averaging yields M = 0, then the magnetization should vanish for a larger temperature range around T R. It is interesting to note that a domain structure due to competing interactions in Eq. (1) affects the perpendicular orientation (Mz), but not the in-plane orientation ( M x) of the magnetization. This may be the reason for the asymmetric magnetic behavior around T R as observed experimentally [8]. Therefore, a comparison of our results with experiments requires a domain averaging. In Fig. 3 we indicate how our results, referring to a homogeneous magnetic system, might change if such a domain averaging is performed. Note that [ M [ > 0 [6] may actually occur in the reorientation region in thin films due to small external magnetic fields or, in the case of a wedge shaped specimen, magnetic stray fields from neighboring ordered regions.
K/J=O.04
A/J=0.001 D/J=O.O01
TR/Tc=0.83
0.3
AT/Tc=O.11 o N
0.2
O) 0.1
0
I
P 0.8
0.6
1
Temperature T/T c
Fig. 2. Comparison of the perpendicular ( M z) and in-plane ( M x) monolayer magnetization with experimental results for fcc Fe/Cu(100) from Pappas et al. [4]. The calculations are performed for Tc = 360 K, T R / T c = 0.83, and A T / T c = 0.11. The other coefficients are chosen as indicated in the figure. The square and triangle symbols refer to the measured M z and M , magnetization, respectively. Although these results are obtained for a monolayer, the qualitative features of our model remain unchanged in the case of several atomic layers.
T R / T c ~ 0.5 we obtain with our model a much narrower region of diminished magnetization than measured experimentally. This might be due to
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AT/Tc=0'175
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Temperature T/T c Fig. 3. Comparison of the perpendicular ( M z) and in-plane ( M x) monolayer magnetization with experimental results for bcc Fe/Ag(100) from Qiu et al. [8] assuming T R / T c = 0.48 and A T / T c = 0.175. Tc of the thin film with six atomic layers of bcc Fe is estimated to be ~ 800 K. Note that here T R / T c is markedly smaller than T R / T c used in Fig. 2, resulting in a much narrower range of vanishing magnetization. Our results refer to a homogeneous system (single domain). If domain averaging yields Edom,i,s M = 0, results indicated by the dashed curves would be obtained. Thus, the magnetization is diminished in a wider temperature range in better agreement with experiments [6,8].
D.K. Morr et al. / Surface Science 307-309 (1994) 1109-1113
A different magnetic behavior is obtained if we use different temperature functions fz(T) and fx(T) for the calculation of the perpendicular and in-plane components of M. If fx(T) changes sign at a lower temperature than the respective function f~(T), both M~ and M x are simultaneously nonzero, corresponding to a rotation of M(T). In this work we have investigated the reorientation of the thin film magnetization from a perpendicular to an in-plane orientation. Using an ansatz for the temperature dependence of the effective anisotropy Keff(T) we obtain a region of vanishing magnetization around the reorientation temperature TR. Concerning a comparison with experiments, one should note that we have considered a perfect film morphology, but not the effects due to lattice strain and inhomogenities, which result in a thickness dependent Tc and film magnetization. Also a domain structure should be taken into account for the magnetization reorientation. We thank D.L. Mills for useful discussions. This work has been partly supported by the Deutsche Forschungsgemeinschaft (DFG).
1. References [1] For a recent review on magnetism of thin films, see: J. Magn. Magn. Mater. 100 (1991). [2] N.M. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. [3] V.L. Pokrovsky, Adv. Phys. 28 (1979) 595, and references therein. [4] D.P. Pappas, K.P. K[imper and H. Hopster, Phys. Rev. Lett. 64 (1990) 3179.
1113
[5] D.P. Pappas, C.R. Brundle and H. Hopster, Phys. Rev. B 45 (1992) 8169. [6] R. Allenspach and A. Bischof, Phys. Rev. Lett. 69 (1992) 3385. [7] J. Thomassen, F. May, B. Feldmann, M. Wuttig and H. Ibach, Phys. Rev. Lett. 69 (1992) 3831. [8] Z.Q. Qiu, J. Pearson and S.D. Bader, Phys. Rev. Lett. 70 (1993) 1006. [9] R. Allenspach, M. Stampanoni and A. Bischof, Phys. Rev. Lett. 65 (1990) 3344. [10] J.R. Dutcher, J.F. Cochran, I. Jacob and W.F. Egelhoff, Jr., Phys. Rev. B 39 (1989) 10430. [11] P.J. Jensen and K.H. Bennemann, Phys. Rev. B 42 (1990) 849; Solid State Commun. 83 (1992) 1057. [12] D. Pescia and V.L. Pokrovsky, Phys. Rev. Lett. 65 (1990) 2599; M.G. Pini, A. Rettori, D. Pescia, N. Majlis and S. Seizer, Phys. Rev. B 45 (1992) 5037. [13] R.P. Erickson and D.L. Mills, Phys. Rev. B 46 (1992) 861. [14] P. Politi, A. Rettori and M.G. Pini, Phys. Rev. Lett. 70 (1992) 1183; A.P. Levanyuk and N. Garcia, Phys. Rev. Lett. 70 (1992) 1184. [15] In this work we have assumed a static lattice. The temperature dependence of the effective anisotropy is caused by thermal fluctuations. A possible influence on the anisotropy by structural changes, for example due to thermal lattice expansion or a structural transition, as observed in Ref. [7], is not considered here. At present, it remains unclear how this will affect the lattice anisotropy. [16] R.A. Tahir-Kheli and D. ter Haar, Phys. Rev. 127 (1962) 88; D.T. Hung, J.C.S. Levy and O. Nagai, Phys. Status Solidi (b)93 (1979) 351. [17] J.P. Gay and R. Richter, Phys. Rev. Lett. 56 (1986) 2728; D.S. Wang, R. Wu and A.J. Freeman, Phys. Rev. Lett. 70 (1993) 869; Phys. Rev. B 47, (1993) 14932. [18] Y. Yafet and E.M. Gyorgy, Phys. Rev. 38 (1988) 9145; A. Kashuba and V.L. Pokrovsky, Phys. Rev. Lett. 70 (1993) 3155. [19] D.K. Morr, P.J. Jensen and K.-H. Bennemann, to be published.
ELSEVIER
Reorientation transition of the magnetization in thin ferromagnetic films D.K. Morr, P.J. Jensen
*, K . - H . B e n n e m a n n
Institute for Theoretical Physics, Freie Universitiit Berlin, Arnimallee 14, D-14 195 Berlin, Germany
(Received 20 August 1993)
Abstract
The orientation of the magnetization M in thin ferromagnetic films is studied by use of a Greens function type theory within RPA approximation. For the temperature dependence of the effective anisotropy a phenomenological approach is applied. Due to the competing lattice anisotropy and the dipole interactions the magnetization turns from a perpendicular to an in-plane orientation with increasing temperatures. A vanishing overall magnetization may result in a temperature range around the reorientation temperature TR, before an in-plane magnetization appears.
This transition metal a n d rare earth metal films with thicknesses of a few monolayers are known to be ferromagnetically ordered [1]. A two-dimensional (2D) Heisenberg magnet with only isotropic and short range interactions should not exhibit any ordered state at finite temperatures [2]. However, a global magnetization in these systems is induced by additional (weak) interactions like lattice anisotropies a n d / o r the dipole coupling (shape anisotropy), and one obtains a Curie temperature Tc of the order of the exchange coupling [3], as observed experimentally. For certain thin film systems like Fe/Cu(100) [4-7], Fe/Ag(100) [5,8], and C o / A u ( l l l ) [9] a perpendicular magnetization has been observed at low temperatures. With increasing temperatures (and also with increasing film thicknesses) the magnetization turns into the film plane at a
* Corresponding author.
reorientation temperature T R. In a temperature range around T R all components of the magnetization might vanish simultaneously [4,8]. Recent experiments with an improved spatial resolution observed a multi-domain state at T R [6]. This also leads to an overall vanishing or at least strongly reduced magnetization. Furthermore, Brillouin light scattering experiments on a magnetic field induced reorientation of the thin film magnetization report a strong increase of the scattering intensity around T R [10], which might indicate an occurrence of critical fluctuations. Therefore, the vanishing magnetization at T R may result from either a multi-domain state or a 2D isotropic Heisenberg system without long range order, due to the mutual cancellation of anisotropies. The reorientation of the thin film magnetization was attempted to be explained by the rotational entropy, but the strong fluctuations in the 2D system were neglected [11]. In Ref. [12] these fluctuations were considered by use of a renor-
0039-6028/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0039-6028(93)E0825-F
1110
D.K. Morret al./ SurfaceScience307-309 (1994)1109-1113
malization procedure, yielding a temperature dependent effective anisotropy Keff composed of the uniaxial lattice anisotropy and the dipole coupling. According to these calculations Keff changes sign at the reorientation temperature, since the two parts of the effective anisotropy renormalize differently. However, other calculations obtain quite different results [13,14]. Though it is accepted that the reorientation of M is caused by the change of sign of K~ff, a reliable calculation of the temperature dependence of the lattice anisotropy and the dipole coupling is currently not available. In this paper we use a spin wave theory for the investigation of the magnetic properties around TR. For the calculation of M(T) we make an ansatz for the temperature dependence of the effective anisotropy K~ff(T), in order to simulate the experimental observations. We start with the following Hamiltonian [15]
~=
1j E SiSj- ~K 1 E S{Sj (i,j) (i,j)
1 2
Mz'x
1 [~ [,~ 2M~, x ~--£ Jo dkxJo dkr exp( E~,x/O) - 1' (2)
a 0 = 1, and with the spin wave dispersion relations 2K
D
E z = 4 - 2y k + J
A
j Yk - ~- (~a(kx, ky)
+40)]Mz,
[
K
E x = 4 - 2y k - ~--~Tk+ ~-~(4 - 7~) A
+7(2ga(k.,k,) + 2 S 2 - 3Eb(k~, kr))]Mx .
2 (S •Sj + s sf) (i,j)
q_lAi~ [ sisj (SiFij)(Sjrij) ) i,j I r3 -- 3 r5" ,
components of the average magnetization perpendicular (M~) and parallel (M~) to the film plane, assuming a uniform phase:
(3)
The following denotations have been used: "~k = COS(kx) + c o s ( k y ) ,
(1)
where the z-axis is chosen to be along the film normal. E(i,j> denotes the sum over nearest neighbors and S~ is a 1/2 Heisenberg spin at lattice site i. J is the exchange coupling between the spins and D the quartic in-plane anisotropy. K is the uniaxial anisotropy, confined mainly to the surface/interface layers and preferring for K > 0 a perpendicular magnetization. The last term in (1) represents the magnetic dipole interaction, with A = izo(glzB)2/a 3 and rii is the distance between lattice sites i and j in units of the lattice constant a 0. For simplicity we assume a magnetic monolayer (square 2D lattice) first. Our treatment can be straightforwardly extended to the case of several atomic layers. We will explain later how our result will change in this case. The Hamiltonian Eq. (1) is solved by using a Greens function theory within the RPA approximation [15]. We obtain a self-consistent equation for the
a=Ea = l
1
~
1
+ a,b=l E
( a 2 + b 2)
3/2,
cos( ak x) + cos(ak r )
Ea(kx, k y ) =
E
a3
a=l
+2
Z
cos(ak x) cos(bky),
a,b= a
( a2 + b2) 3/2
cos(aky) a3
F~a(kx, k y ) = • a=l
+ Z
a,b=l
2b: cos(ak~) cos(bky) (a 2 + b 2 ) 5/2
(4) O=kBT/J
is the relative temperature, k B is Boltzmann's constant, and krl=(kx, ky) is the 2D wave vector parallel to the film plane. The spin wave energy gaps Ez°,x= Ez,x(k x = O, k r = O)
D.K. Morr et al. / Surface Science 307-309 (1994) 1109-1113
> 0, Eq. (3), cause a thin film magnetization and are given by [K D E° = 2 7 - 7
[
A ]
2~'c7 ] M== 2
Kef f
l
U
t
~
o.zt
~
01
I
Keel
and
Ex°= - - 7 - -
Mx'
with c = 1.078 numerically calculated. In order to observe a magnetization perpendicular to the film ( E ° > 0) the uniaxial anisotropy K has to overcome the magnetic dipole interaction ( ~ 27rA). Such strong K for the surface/interface layers have been obtained in recent calculations [17]. With temperature independent coefficients, as given in the Hamiltonian (1), one always obtains, depending on the sign of Ez° and Ex°, either a perpendicular or an in-plane magnetization up to the Curie temperature Tc, respectively. Thus, a reorientation of the magnetization can only be achieved by a change of sign of the effective a n i s o t r o p y Kef f with increasing temperatures. Due to the lack of a reliable theory for the temperature dependence of Kefe around TR, we introduce a temperature function f(T) and put Keff(T)=Kefr(O)'f(T). This results in temperature dependent spin wave energy gaps, Eq. (3). According to experimental observations, f(T) should satisfy the following requirements: (1) f(T) should be continuous and r f(T) l <_1; (2) T R is given by f ( T n) = 0; (3) f(T) > 0 for T < T R and f(T)
[(TR-T)/AT, f(T) = [ s i g n ( r l ~ - T),
ITR-TI
(5)
The parameters T R and AT determine f(T)= 0 and the slope of f(T) at T R. In Fig. 1 we present the resulting magnetization for TR/T c = 0.78 and for different parameters of AT. A gap of vanishing magnetization I M(T)I around T R is obtained, in addition a steeper slope of f(T= T R) a n d / o r a smaller TR/T c lead to a narrower gap. M z vanishes slightly below TI~ and Mx appears above T R. In a finite temperature range around T R the effective
K/J=0.02
0.4
~
J M~
1111
0
~
M
\
~
.m,m ~^.
0.6
A/g=0.001
D/a=0.o01 %/%=0.78 "'."~/ ~ ' " ~
0.8
1
T e m p e r a t u r e T/T c
Fig. 1. Perpendicular (M z) and in-plane (M x) monolayer magnetization for the parameter A T / T c =0.05 (full line), 0.10 (dashed line), and 0.20 (dotted line) of the temperature function Eq. (5). We have chosen T R / T c = 0.78 and the coupling constants in units of the exchange coupling J as given in the figure.
anisotropy is, though not zero, too weak to induce a global ordering in the magnetic plane. Thus, in this region the film corresponds to an almost isotropic 2D Heisenberg model. Different choices of the temperature function f(T) result in a quite similar behavior of the magnetization. For comparison with experimental data our formalism can be extended to the case of thin films with d atomic layers. The strong uniaxial anisotropy K is (mainly) confined to the surface and interface layers, thus is nearly independent of d, whereas the dipole interaction is almost proportional to d for thin films. Then a magnetization reorientation is induced by an increasing film thickness. The reorientation thickness d R is given by K = 2rrcAd R and is expected to be d R --4 - 7 atomic layers [4-8]. Also the parameters T R and AT of Eq. (5) should depend on d, resulting for increasing d in a smaller TR, for example. For the same TR/T c and AT/T c the temperature region of vanishing magnetization decreases for increasing d. However, the main features of M(T) for thin films are the same as obtained for a monolayer, as confirmed by recent calculations [191. In Figs. 2 and 3 we compare the results of our model with experiments [4,8]. Note that for small
1112
~Z
D.K. Morr et aL / Surface Science 307-309 (1994) 1109-1113
0.4
the presence of magnetic domains as observed recently [6]. Since our calculations only consider a homogeneous magnetization, domain structures have not been taken into account in this work. This would require a calculation assuming a spatial variation of the magnetization M ( r ) . Actually one expects that for K > 2~-cA a domain structure may be more favorable than a homogeneous magnetization, as shown previously for T = 0 in particular for a striped domain pattern [18]. If domain averaging yields M = 0, then the magnetization should vanish for a larger temperature range around T R. It is interesting to note that a domain structure due to competing interactions in Eq. (1) affects the perpendicular orientation (Mz), but not the in-plane orientation ( M x) of the magnetization. This may be the reason for the asymmetric magnetic behavior around T R as observed experimentally [8]. Therefore, a comparison of our results with experiments requires a domain averaging. In Fig. 3 we indicate how our results, referring to a homogeneous magnetic system, might change if such a domain averaging is performed. Note that [ M [ > 0 [6] may actually occur in the reorientation region in thin films due to small external magnetic fields or, in the case of a wedge shaped specimen, magnetic stray fields from neighboring ordered regions.
K/J=O.04
A/J=0.001 D/J=O.O01
TR/Tc=0.83
0.3
AT/Tc=O.11 o N
0.2
O) 0.1
0
I
P 0.8
0.6
1
Temperature T/T c
Fig. 2. Comparison of the perpendicular ( M z) and in-plane ( M x) monolayer magnetization with experimental results for fcc Fe/Cu(100) from Pappas et al. [4]. The calculations are performed for Tc = 360 K, T R / T c = 0.83, and A T / T c = 0.11. The other coefficients are chosen as indicated in the figure. The square and triangle symbols refer to the measured M z and M , magnetization, respectively. Although these results are obtained for a monolayer, the qualitative features of our model remain unchanged in the case of several atomic layers.
T R / T c ~ 0.5 we obtain with our model a much narrower region of diminished magnetization than measured experimentally. This might be due to
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.
.
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,
0.6
Temperature T/T c Fig. 3. Comparison of the perpendicular ( M z) and in-plane ( M x) monolayer magnetization with experimental results for bcc Fe/Ag(100) from Qiu et al. [8] assuming T R / T c = 0.48 and A T / T c = 0.175. Tc of the thin film with six atomic layers of bcc Fe is estimated to be ~ 800 K. Note that here T R / T c is markedly smaller than T R / T c used in Fig. 2, resulting in a much narrower range of vanishing magnetization. Our results refer to a homogeneous system (single domain). If domain averaging yields Edom,i,s M = 0, results indicated by the dashed curves would be obtained. Thus, the magnetization is diminished in a wider temperature range in better agreement with experiments [6,8].
D.K. Morr et al. / Surface Science 307-309 (1994) 1109-1113
A different magnetic behavior is obtained if we use different temperature functions fz(T) and fx(T) for the calculation of the perpendicular and in-plane components of M. If fx(T) changes sign at a lower temperature than the respective function f~(T), both M~ and M x are simultaneously nonzero, corresponding to a rotation of M(T). In this work we have investigated the reorientation of the thin film magnetization from a perpendicular to an in-plane orientation. Using an ansatz for the temperature dependence of the effective anisotropy Keff(T) we obtain a region of vanishing magnetization around the reorientation temperature TR. Concerning a comparison with experiments, one should note that we have considered a perfect film morphology, but not the effects due to lattice strain and inhomogenities, which result in a thickness dependent Tc and film magnetization. Also a domain structure should be taken into account for the magnetization reorientation. We thank D.L. Mills for useful discussions. This work has been partly supported by the Deutsche Forschungsgemeinschaft (DFG).
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