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Spatial distribution of Gaussian fluctuations of the molecular field in Ising ultra-thin films
Journal of Magnetism and Magnetic Materials 99 (1991) 253-260 North-Holland
Spatial distribution of Gaussian fluctuations of the molecular field in Ising ultra-thin films Z. Onyszkiewicz Instituteof Physics, A. Mickiewicz
University, ul. Grunwaldrka
6, 60-780 Poznari, Poland
and A. Wierzbicki Department
of Physics, Dalhousie University
Halifax-B3
H3 J5, Canada
Received 16 January 1991
The Gaussian fluctuation approximation is used to obtain the spatial distribution both of the mean fluctuation of the molecular field and the magnetization in the king model of ferromagnetic ultra-thin films.
1. Introduction For investigation of ultra-thin films containing R d 10 monatomic layers parallel to the film surface the exact calculation of the spatial distribution of magnetization a, = a(r), where r is the number of a monatomic layer, r = 1, 2, 3,. . . , R, is very important because the dependence u(r) essentially affects the spectrum of elementary excitations in the film [l]. For obvious reasons at higher temperatures and especially near the Curie point T,, it is equally important to find the spatial distribution of the mean magnetization fluctuations in a fihn or, of an equivalent quantity, the spatial distribution of the mean fluctuations of the molecular field gy, = gy(r). This paper aims at presenting possibly the most accurate calculations of 6y, for various temperatures T, for the simplest model of an ultra-thin film. It presents the calculation of molecular field fluctuations performed within the Gaussian fluctuation approximation (GFA), which is a modified version of the high density expansion method as has been proposed in refs. [2,3]. GFA is an improvement over the molecular field approximation (MFA) due to the self-consistent inclusion of Gaussian fluctuations of this field. The essential new element of GFA is the summing up of the partial sums of Feynman diagrams of the same structure of recurrent formulae at each stage of the calculations. Owing to this procedure the theory becomes internally consistent and does not lead to unphysical results such as, for example, a complex Curie temperature (see ref. [4]). We consider a thin film with ferromagnetic order of spins. According to the Valenta [5] model the film is divided into R monatomic layers parallel with the surfaces of the film. The position of each atom is given by the number of the layer r = 1, 2,. . . , R and the bidimensional vector f. In each plane r the periodicity conditions are assumed. The specific properties of thin films are a consequence of the physical fact of lack of neighbours of atoms in the boundary layers. Our considerations are restricted to the nearest 0304-8853/91/$03.50
0 1991 - Elsevier Science Publishers B.V. All rights reserved
254
Z. Onyszkiewicz,
A. Wierzbicki
/ Molecular field in Ising ultra-thin films
neighbour interactions only. Finally, let us assume that the magnetic properties satisfactorily described by a simple cubic Ising model (s = i)
of the thin film are
2. Gaussian fluctuation of the molecular field As a starting point to GFA we choose the following decomposition
of the Hamiltonian
H=(H-H,)+H,=H,+H,,
(1) (2)
where the perturbative part H, is defined by the transformation H+H,=H(S’+W),
(3)
Wr;=
(4)
where sr;-
(S,z>
is the fluctuation operator of the z-component of the spin and (St)
= Tr[ S,?, esCF-“)],
p = (k,T)-’
(5)
and F is the free energy. According to the rules of the thermodynamical (SF) = ($7
perturbation
e-gn,)o/(e-B”l)O,
expansion we can write
(6)
where ( . . . ). = Tr[ . . . po] and po= (Tr[exp(-PH,)])-i
exp(-PH,).
The right-hand side of eq. (6) can be expanded into a series with respect to the perturbing term HI. From such an infinite series we now choose a certain partial sum which can be represented graphically as
where m
=I r
and symbols -
+ Fl r
+V]+... r
r
(8)
denote the renormalized interaction line, (9)
(10)
Z. Onyszkiewicz,
A. Wierzbicki
/ Molecular field in Ising ultra-thin fihns
6Yr
R=8
0.4
l\. l.
t
\-
I
e ./*
Y’
“._-.__._-.”
d /*
-1’ ‘..-_.-
l\
b
I
.‘.___/
. . .__._-._A
‘\
0.2 -
,
/’
‘\
l\
‘..
255
-.-
\
, .
-.--.-
C
-6 I*
b
* .-
-.--.
r*
0.
I
6Y,
R=7
0.4
0.
-\
,
/’
‘\\ \ ‘L- _._ _J’ , ,I’ I \ \
‘\ .__.__.’
\
..
/ -. .
._-.-_.’
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l.
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r
,*
--.__.-.-
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9
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e
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-\\
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a
-._-•--.--.-
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.
b
a r
1
R=IO 0.4
t
‘b.
<.’
l\ .
‘.---.__.‘\
. l\
e
/’
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/’
‘.-_.__._-.-__-0’
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_._-_. _-.’
d /*
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-~_--.--~__.-_.--_.-
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--.--
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.--.
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d
_-.-_.--.C
a
R=9
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l. . ‘\. ‘\ \
,* . .
., 0.2 -
\\
k’ ‘L_.__.__._-.’
e
/ ,
.--.--.-_.__.’
/’ _,
d ,
,* C
‘*__.-_.--.--4’
l\
/
\ -.
0.
0
I Fig. 1. Spatial distribution I = 0.5, (e)
of the mean fluctuations
t = 0.6
and different
V- -.-_.
3
4
b
__._-.x./’
----._-.-_.-~--~_-2
7
5
of the molecular
6
7
8
a 9
IO
r
field 6y, for temperatures:
(a)
1= 0.2,
(h) t - 0.3, (c) I = 0.4, (d)
thickness R of the film. The solid line represents the values (I3f_$y,)/R.
Z. Onyszkiewicz,
256
A. Wierzbicki
/ Molecular field in Ising ultra-thin film
rK,W~
(11)
L(ryr) =hl 2 cash ;y,,
(12) (13)
....., =
Yr= PI [4($)
+ 0 - ~r,,K%>
+ (1 - &dw+l>l~
I,.,,(k) is the planar Fourier transform of the bilinear coupling parameter ZrrrfrT= I. As a result of calculating the infinite sums (8) and (9) we obtain: (SF) =
L/,, 26
--m
e-ix2 th(+y, + xSy,)
dx,
04)
where [4(1+O - &,,I(1 is the mean Gaussian surfaces of the film.
th2(+y, + xay,))
- th2(-ly,+l + x~Y,+I))]
fluctuation
(1 - %,,)(l- th2(:yr-l+ xQ-I))
+
(15)
dx)‘/l
of the molecular field y, in the r th monoatomic
layer parallel to the
3. Results Eqs. (14) and (15) are solved numerically and the results are presented graphically. Fig. 1 presents the spatial distribution of the mean fluctuations of the molecular field 6y, for different temperatures t = 2k,T/3Z and different thicknesses, R, of the film. The same spatial distribution 6y, near
\ ‘\
\
\\
\
0.30
:
R=7
-
.__ /
.--*_-
l
t, = 0.8296 I
I
2
3
4
5
6
7
Fig. 2. Spatial distribution of the mean fluctuations of the molecular field 6y, near the critical (Curie) point line represents the values @_ ,6y,)/R.
tC for R = 7. The solid
2. Onyszkiewicz,
A. Wierzbicki
/ Molecular field in Ising ultra-thin film
-- __ l’
=I
I
.--.--* \
I
.’
I
i
?’ I
;
+
.--. \ : \ \
-*l..
l,’ . ‘\. 1’
d
-a a b
‘\ \ \ \
0.5
0’
__
\
*,.--.,
-.
$0
c
,’ :
l
\ \ \ :
,.--.,
d
‘.
I
\ \ \
:
\ \ \.
\
I
- 0.5
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:
\ \ \
R=4
-9
I
./’ /
I
9
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P--*--*--~
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I
0.:
-_
0,
-
\
;
251
‘b
!
R=5
,
,.--.,
R=6
\
e P. ,*’
‘*
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e
tI
-
2
r
3
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0
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e
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=r‘q
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I
I
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./
9’
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__
I
\
i
:
: .’
/*--*--*--01\
I
d
0.5
R=7
C \ \ \ l
. \
\ \\ :
,I :
\‘. “b
\
:
\ \ \ \ \ .
r
..--
l t
__a__
C_.--.--.--._-*
./’ w-•--*--*---t
.
.’
I23456
r
.;
d
,,*--*\
1
/’
t
I
e
1’
r’
‘0
\
\\ .\
e \
\ \
\. R=8
0 I234567
r
I2345678
r
Fig. 3. Spatial distribution of the magnetization 0, = 2(Sf) for different temperatures; (a) t = 0.2, (b) f = 0.3, (c) t = 0.4, (d) 1= 0.5, (e) f = 0.6. The solid line represents the mean value of magnetization CJ= (Ef_ p,)/R.
258
2. Onyszkiewicz,
__ -- __ -- -- __ 0;
l’
__.--.--.--.--.__
‘*
A. Wierzbicki
‘a
/ Molecular field in Ising ultra-thin film
=r
\\
. /’ ._r .--. --.---.--.,
- - -- __ _____- .- l_-.--.--.__.__.__.__e
1
.’
‘*
II i
,.
_A--.--.. -
.’
\\
:
i
__.--.--.-_.__*
I
,/
.
/
I’
d’
1
d
‘\
I
\ \
/
\ \ .
:
0.5 A--., ,*-
‘\. \ \
f I
’
-*\
‘0 \
\
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:
i
, \ \\
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1’
\
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R=9
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.--.
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a--.--
lr
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--a \
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R=lO_
r
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Fig. 3 (continued).
r
Fig. 4. Temperature dependence of the mean fluctuations the molecular field Sy, for different monatomic layers.
Fig.
I
5. Temperature dependence of the fluctuation 8y = Zf_116y,)/R for different thicknesses R of the film.
259
Z. Onyszkiewicz, A. Wierzbicki / Molecular field in Ising ultra-thin fZms
0.6
0
I
\\,\\I
I
0.5
I
-
t
Fig. 6. Magnetization a = (Er_,q)/R versus temperature different thicknesses R of the film.
1 I
for
: 0.5 I
I
I
I
I
80
I
I
I
I
I23456789101112
Fig. 7. The Curie temperature
I
I
I
R
tC versus the film thickness R.
the critical point (the Curie point tc) for the film with R = 7 is shown in fig. 2. Spatial distributions of the magnetization known as magnetization profiles a, = 2( S,Z) for different temperatures t and different thicknesses of the film R, are shown in fig. 3. Fig. 4 illustrates the temperature dependence of the mean fluctuations of the molecular field in different monatomic layers of the film. The temperature dependencies of the mean value of the molecular field fluctuations in the whole film determined as
(16) for ultra-thin films of different thickness, are shown in fig. 5. Then in fig. 6 we show the temperature dependence of the mean value of the magnetization (17) for ultra-thin films of different thickness temperature on the thin film thickness R.
R. Finally,
fig. 7 presents
the dependence
of the Curie
4. Conclusions Basic parameters characterizing the magnetic properties of an Ising ultra-thin ferromagnetic film have been calculated applying the GFA method. The fluctuations of the molecular field 6y, were found to be greatest in the vicinity of the film surface (see fig. 1). This result confirms ones intuitions, as the spins lying
Z. Onyszkiewicz,
260
A. Wierzbicki
/ Molecular field in Ising ultra-thin f&s
the surface have fewer neighbours than the others in the film. A deviation from the rule is observed in the vicinity of the Curie point (see fig. 2) which is a consequence of the molecular field value tending to zero as the film temperature approaches t,. Nevertheless, the ratio 6y,/y, is, for t x t,, again greater near the film surface than inside the film. An interesting result is the temperature dependence of the molecular field fluctuations obtained for individual monatomic layers, presented in fig. 5. A maximum in 6y, for any r is observed to occur at t,. The results obtained seem valuable to us as they can be experimentally verified by the measurements of fluctuations of the molecular field 6y, in individual monatomic layers applying the technique based on the Mossbauer effect. Moreover, the results obtained on the spatial distribution of magnetization a, and the Curie temperature dependence on the film thickness are qualitatively consistent with the results obtained by other methods [l&8]. In conclusion, we can say that GFA can be successfully applied in investigation of magnetic thin film systems. In particular, taking into regard the relative simplicity of GFA, it can be applied in the investigation of even more complex thin-film systems. on
References [l] [2] [3] [4] [5] [6] [7] [8]
Z. Z. Z. Z. L. L. U. G.
Onyszkiewicz, Phys. Stat. Sol. 41 (1970) 161. Onyszkiewicz, Phys. Lett. A 76 (1980) 411. Onyszkiewicz, Physica A 103 (1980) 257. Onyszkiewicz and A. Wierzbicki, Phys. Lett. A 116 (1986) 335. Valenta, Czech. J. Phys. 7 (1957) 127. Wojtczak, Czech. J. Phys. B 20 (1970) 247. Kokowska, A.R. Ferchmin and H. Cofta, Acta Phys. Polon. A 45 (1974) 443. Wiatrowski, T. Balcerzak, L. Wojtczak and J. Mielnicki, Phys. Stat. Sol. (b) 138 (1986) 189.
Spatial distribution of Gaussian fluctuations of the molecular field in Ising ultra-thin films Z. Onyszkiewicz Instituteof Physics, A. Mickiewicz
University, ul. Grunwaldrka
6, 60-780 Poznari, Poland
and A. Wierzbicki Department
of Physics, Dalhousie University
Halifax-B3
H3 J5, Canada
Received 16 January 1991
The Gaussian fluctuation approximation is used to obtain the spatial distribution both of the mean fluctuation of the molecular field and the magnetization in the king model of ferromagnetic ultra-thin films.
1. Introduction For investigation of ultra-thin films containing R d 10 monatomic layers parallel to the film surface the exact calculation of the spatial distribution of magnetization a, = a(r), where r is the number of a monatomic layer, r = 1, 2, 3,. . . , R, is very important because the dependence u(r) essentially affects the spectrum of elementary excitations in the film [l]. For obvious reasons at higher temperatures and especially near the Curie point T,, it is equally important to find the spatial distribution of the mean magnetization fluctuations in a fihn or, of an equivalent quantity, the spatial distribution of the mean fluctuations of the molecular field gy, = gy(r). This paper aims at presenting possibly the most accurate calculations of 6y, for various temperatures T, for the simplest model of an ultra-thin film. It presents the calculation of molecular field fluctuations performed within the Gaussian fluctuation approximation (GFA), which is a modified version of the high density expansion method as has been proposed in refs. [2,3]. GFA is an improvement over the molecular field approximation (MFA) due to the self-consistent inclusion of Gaussian fluctuations of this field. The essential new element of GFA is the summing up of the partial sums of Feynman diagrams of the same structure of recurrent formulae at each stage of the calculations. Owing to this procedure the theory becomes internally consistent and does not lead to unphysical results such as, for example, a complex Curie temperature (see ref. [4]). We consider a thin film with ferromagnetic order of spins. According to the Valenta [5] model the film is divided into R monatomic layers parallel with the surfaces of the film. The position of each atom is given by the number of the layer r = 1, 2,. . . , R and the bidimensional vector f. In each plane r the periodicity conditions are assumed. The specific properties of thin films are a consequence of the physical fact of lack of neighbours of atoms in the boundary layers. Our considerations are restricted to the nearest 0304-8853/91/$03.50
0 1991 - Elsevier Science Publishers B.V. All rights reserved
254
Z. Onyszkiewicz,
A. Wierzbicki
/ Molecular field in Ising ultra-thin films
neighbour interactions only. Finally, let us assume that the magnetic properties satisfactorily described by a simple cubic Ising model (s = i)
of the thin film are
2. Gaussian fluctuation of the molecular field As a starting point to GFA we choose the following decomposition
of the Hamiltonian
H=(H-H,)+H,=H,+H,,
(1) (2)
where the perturbative part H, is defined by the transformation H+H,=H(S’+W),
(3)
Wr;=
(4)
where sr;-
(S,z>
is the fluctuation operator of the z-component of the spin and (St)
= Tr[ S,?, esCF-“)],
p = (k,T)-’
(5)
and F is the free energy. According to the rules of the thermodynamical (SF) = ($7
perturbation
e-gn,)o/(e-B”l)O,
expansion we can write
(6)
where ( . . . ). = Tr[ . . . po] and po= (Tr[exp(-PH,)])-i
exp(-PH,).
The right-hand side of eq. (6) can be expanded into a series with respect to the perturbing term HI. From such an infinite series we now choose a certain partial sum which can be represented graphically as
where m
=I r
and symbols -
+ Fl r
+V]+... r
r
(8)
denote the renormalized interaction line, (9)
(10)
Z. Onyszkiewicz,
A. Wierzbicki
/ Molecular field in Ising ultra-thin fihns
6Yr
R=8
0.4
l\. l.
t
\-
I
e ./*
Y’
“._-.__._-.”
d /*
-1’ ‘..-_.-
l\
b
I
.‘.___/
. . .__._-._A
‘\
0.2 -
,
/’
‘\
l\
‘..
255
-.-
\
, .
-.--.-
C
-6 I*
b
* .-
-.--.
r*
0.
I
6Y,
R=7
0.4
0.
-\
,
/’
‘\\ \ ‘L- _._ _J’ , ,I’ I \ \
‘\ .__.__.’
\
..
/ -. .
._-.-_.’
/’
l.
C
3,4r
123456
12345678
r
,*
--.__.-.-
d
9
-..--._-..~
0
e
f
c
-\\
I2
a
-._-•--.--.-
0
.
b
a r
1
R=IO 0.4
t
‘b.
<.’
l\ .
‘.---.__.‘\
. l\
e
/’
‘\ 0.2
/’
‘.-_.__._-.-__-0’
\
_._-_. _-.’
d /*
,’
-~_--.--~__.-_.--_.-
C P
_I’
‘\_
--.--
-.
.--.
,’
-~-_.--._-.__.-_.-
0 6Yr
d
_-.-_.--.C
a
R=9
0.4 -
l. . ‘\. ‘\ \
,* . .
., 0.2 -
\\
k’ ‘L_.__.__._-.’
e
/ ,
.--.--.-_.__.’
/’ _,
d ,
,* C
‘*__.-_.--.--4’
l\
/
\ -.
0.
0
I Fig. 1. Spatial distribution I = 0.5, (e)
of the mean fluctuations
t = 0.6
and different
V- -.-_.
3
4
b
__._-.x./’
----._-.-_.-~--~_-2
7
5
of the molecular
6
7
8
a 9
IO
r
field 6y, for temperatures:
(a)
1= 0.2,
(h) t - 0.3, (c) I = 0.4, (d)
thickness R of the film. The solid line represents the values (I3f_$y,)/R.
Z. Onyszkiewicz,
256
A. Wierzbicki
/ Molecular field in Ising ultra-thin film
rK,W~
(11)
L(ryr) =hl 2 cash ;y,,
(12) (13)
....., =
Yr= PI [4($)
+ 0 - ~r,,K%>
+ (1 - &dw+l>l~
I,.,,(k) is the planar Fourier transform of the bilinear coupling parameter ZrrrfrT= I. As a result of calculating the infinite sums (8) and (9) we obtain: (SF) =
L/,, 26
--m
e-ix2 th(+y, + xSy,)
dx,
04)
where [4(1+O - &,,I(1 is the mean Gaussian surfaces of the film.
th2(+y, + xay,))
- th2(-ly,+l + x~Y,+I))]
fluctuation
(1 - %,,)(l- th2(:yr-l+ xQ-I))
+
(15)
dx)‘/l
of the molecular field y, in the r th monoatomic
layer parallel to the
3. Results Eqs. (14) and (15) are solved numerically and the results are presented graphically. Fig. 1 presents the spatial distribution of the mean fluctuations of the molecular field 6y, for different temperatures t = 2k,T/3Z and different thicknesses, R, of the film. The same spatial distribution 6y, near
\ ‘\
\
\\
\
0.30
:
R=7
-
.__ /
.--*_-
l
t, = 0.8296 I
I
2
3
4
5
6
7
Fig. 2. Spatial distribution of the mean fluctuations of the molecular field 6y, near the critical (Curie) point line represents the values @_ ,6y,)/R.
tC for R = 7. The solid
2. Onyszkiewicz,
A. Wierzbicki
/ Molecular field in Ising ultra-thin film
-- __ l’
=I
I
.--.--* \
I
.’
I
i
?’ I
;
+
.--. \ : \ \
-*l..
l,’ . ‘\. 1’
d
-a a b
‘\ \ \ \
0.5
0’
__
\
*,.--.,
-.
$0
c
,’ :
l
\ \ \ :
,.--.,
d
‘.
I
\ \ \
:
\ \ \.
\
I
- 0.5
\
\
:
\ \ \
R=4
-9
I
./’ /
I
9
__
P--*--*--~
.’
\ :
I
0.:
-_
0,
-
\
;
251
‘b
!
R=5
,
,.--.,
R=6
\
e P. ,*’
‘*
\
\
.’ 0
e
tI
-
2
r
3
---&I 1234
0
0
r
e
\
/
.
---lo 12345
I
__ _
=r‘q
a, .’
Y--
,I 1’
c
--.--.,
\ ‘0
_-.-_ /* t,,
\
I
I
/ : 0.5 -
./
9’
\\ \\ \
__
I
\
i
:
: .’
/*--*--*--01\
I
d
0.5
R=7
C \ \ \ l
. \
\ \\ :
,I :
\‘. “b
\
:
\ \ \ \ \ .
r
..--
l t
__a__
C_.--.--.--._-*
./’ w-•--*--*---t
.
.’
I23456
r
.;
d
,,*--*\
1
/’
t
I
e
1’
r’
‘0
\
\\ .\
e \
\ \
\. R=8
0 I234567
r
I2345678
r
Fig. 3. Spatial distribution of the magnetization 0, = 2(Sf) for different temperatures; (a) t = 0.2, (b) f = 0.3, (c) t = 0.4, (d) 1= 0.5, (e) f = 0.6. The solid line represents the mean value of magnetization CJ= (Ef_ p,)/R.
258
2. Onyszkiewicz,
__ -- __ -- -- __ 0;
l’
__.--.--.--.--.__
‘*
A. Wierzbicki
‘a
/ Molecular field in Ising ultra-thin film
=r
\\
. /’ ._r .--. --.---.--.,
- - -- __ _____- .- l_-.--.--.__.__.__.__e
1
.’
‘*
II i
,.
_A--.--.. -
.’
\\
:
i
__.--.--.-_.__*
I
,/
.
/
I’
d’
1
d
‘\
I
\ \
/
\ \ .
:
0.5 A--., ,*-
‘\. \ \
f I
’
-*\
‘0 \
\
\
:
i
, \ \\
.
\
\
1’
\
I
\
,
:
e
\
I
.
:
R=9
123456789
\\ :
\
,._-.-_* l’
\\
I
\\ \.
b
C \
*/ /*
cl
1,
-.
I
\
\
\
I’
\
!
__.__*,
I
‘.
I
.--.
I
:
l\
a--.--
lr
‘? \
--a \
/
R=lO_
r
12345678910
r
I
Fig. 3 (continued).
r
Fig. 4. Temperature dependence of the mean fluctuations the molecular field Sy, for different monatomic layers.
Fig.
I
5. Temperature dependence of the fluctuation 8y = Zf_116y,)/R for different thicknesses R of the film.
259
Z. Onyszkiewicz, A. Wierzbicki / Molecular field in Ising ultra-thin fZms
0.6
0
I
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I
0.5
I
-
t
Fig. 6. Magnetization a = (Er_,q)/R versus temperature different thicknesses R of the film.
1 I
for
: 0.5 I
I
I
I
I
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I
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Fig. 7. The Curie temperature
I
I
I
R
tC versus the film thickness R.
the critical point (the Curie point tc) for the film with R = 7 is shown in fig. 2. Spatial distributions of the magnetization known as magnetization profiles a, = 2( S,Z) for different temperatures t and different thicknesses of the film R, are shown in fig. 3. Fig. 4 illustrates the temperature dependence of the mean fluctuations of the molecular field in different monatomic layers of the film. The temperature dependencies of the mean value of the molecular field fluctuations in the whole film determined as
(16) for ultra-thin films of different thickness, are shown in fig. 5. Then in fig. 6 we show the temperature dependence of the mean value of the magnetization (17) for ultra-thin films of different thickness temperature on the thin film thickness R.
R. Finally,
fig. 7 presents
the dependence
of the Curie
4. Conclusions Basic parameters characterizing the magnetic properties of an Ising ultra-thin ferromagnetic film have been calculated applying the GFA method. The fluctuations of the molecular field 6y, were found to be greatest in the vicinity of the film surface (see fig. 1). This result confirms ones intuitions, as the spins lying
Z. Onyszkiewicz,
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/ Molecular field in Ising ultra-thin f&s
the surface have fewer neighbours than the others in the film. A deviation from the rule is observed in the vicinity of the Curie point (see fig. 2) which is a consequence of the molecular field value tending to zero as the film temperature approaches t,. Nevertheless, the ratio 6y,/y, is, for t x t,, again greater near the film surface than inside the film. An interesting result is the temperature dependence of the molecular field fluctuations obtained for individual monatomic layers, presented in fig. 5. A maximum in 6y, for any r is observed to occur at t,. The results obtained seem valuable to us as they can be experimentally verified by the measurements of fluctuations of the molecular field 6y, in individual monatomic layers applying the technique based on the Mossbauer effect. Moreover, the results obtained on the spatial distribution of magnetization a, and the Curie temperature dependence on the film thickness are qualitatively consistent with the results obtained by other methods [l&8]. In conclusion, we can say that GFA can be successfully applied in investigation of magnetic thin film systems. In particular, taking into regard the relative simplicity of GFA, it can be applied in the investigation of even more complex thin-film systems. on
References [l] [2] [3] [4] [5] [6] [7] [8]
Z. Z. Z. Z. L. L. U. G.
Onyszkiewicz, Phys. Stat. Sol. 41 (1970) 161. Onyszkiewicz, Phys. Lett. A 76 (1980) 411. Onyszkiewicz, Physica A 103 (1980) 257. Onyszkiewicz and A. Wierzbicki, Phys. Lett. A 116 (1986) 335. Valenta, Czech. J. Phys. 7 (1957) 127. Wojtczak, Czech. J. Phys. B 20 (1970) 247. Kokowska, A.R. Ferchmin and H. Cofta, Acta Phys. Polon. A 45 (1974) 443. Wiatrowski, T. Balcerzak, L. Wojtczak and J. Mielnicki, Phys. Stat. Sol. (b) 138 (1986) 189.