Spin correlation time in thin film near the Curie point

Volume 92A, number 7

PHYSICS LETTERS

29 November 1982

SPIN CORRELATION TIME IN THIN FILM NEAR THE CURIE POINT W. KORNETA Department of Physics, Technical University, Malczewskiego 29, Radom, Poland

and Z. PYTEL Institute of Physics of the Polish Academy of Sciences, Michatowska 4, Radom, Poland Received 12 August 1982

The spatial distribution and the temperature dependence of the spin correlation time in a thin ferromagnetic film near the Curie temperature are considered.

1. Deduction ofan expression for the spin correlation time. A thin ferromagnetic film in the critical region near the Curie temperature Tc is considered. A simple cubic structure with spacing a and (1, 0, 0) surface orientation of the film is assumed. The film consists of n layers indexed by v parallel to its surfaces. The number of atoms in a layer is N and the position of an atom in a layer is described by the vector j. We denote by s~1(t)the temporal deviation of one component of the magnetization density from thermal equilibrium value in a position (z.j). For temperatures near the Curie temperature s~1(t)satisfies a diffusion equation with a damping term, which can be written in the following form:

as~(t)= ~(5(t) ~t

T~

+~

B~.(T,n)spj~(t)),

(1)

where the summation runs over nearest neighbours of the position (vi), T~denotes the spin lattice relaxa. tion time in depending the vth layer ) are certain coefficients on and v, s/,B,~.(T, n and nthe temperature T. In surface layers there is a smaller number of nearest neighbours included in the summation. Such an equation describing the decay of fluctuations of the magnetization density can be obtained for example by assuming that the thermal equilibrium value of the magnetization density in any position of the vth layer is determined by the following equation: 0 03l-9163/82/0000—0000/$02.75 © 1982 North-Holland

(Sr)

F

(~

~ (2) kBT where In,. denotes the nearest neighbour exchange integral between spins situated in layers v and ~/,F(X) is a function and kB is the Boltzmann constant. Follow. ing the procedure described by Oguchi [1] we obtain as~,(t) 1 s~,(t)+ G~(T,n) =

,

~‘

(

—

~

k~T ~ I~.s~.1~(t)) , (3)

—

p

where G~(T,n) is determined by G~(T,n) =

ax~

for

X1)

=

~

f-~T(Sp)

.

(4)

Because of translational symmetry in the plane parallel to film surfaces we can assume the solution of eq. (1) to have the following form: n

2h C(rh)s~h(v)eth1e_D(nh1~ , (5) d where C(rh),D(rh) and ~~h(~) are certain functions. The two-dimensional vector h corresponds to the component of the wave vector which is parallel to the film and it changes over the first Brillouin zone. The mode number r corresponds to the component of the wave vector which is perpendicular to the film and it assumes discrete values from 1 to n. Substituting (5) Spj(t)=

~I fh

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Volume 92A, number 7

PHYSICS LETTERS

29 November 1982

into (1) we obtain for given r and h the equation in matrix form (6) D(rh)s~h= RA(h2)s~h

Using eqs. (6), (9) and (5) it can be shown that

where R and A(h2) are n X n matrices. The matrix A has elements (vv’) determined as ~ and the matrix A(h2) has diagonal elements equal to 1 B~,(T,n)(4 + h2) and off.diagonal elements equal to 6~ iB~+ 1(T, n). Bys~hwe denoted the2) right nwith dimensional eigenvector matrix RA(h left eigenelements equal to ~~h(~)’ of Wethe can also define vectors S~hof this matrix with elements denoted by ~~h(~)• It is known that

where R~)and A~2(h2)denote the diagonal elements of the inverse matrix R1 and the square of the inverse matrix A—1(h2). Performing the integral in the last equation, leaving only the dominant term for h2

,

—

~

=

R~)fh d2h A~2(h2),

(12)

= 0 as the most one and introducing the evident form of important R~1we obtain =

T~A~](0),

(13)

where A~J’(0)denotes the diagonal element of the inverse matrix A~(h2)with h2 = 0. This formula al-

n ~

~=

~rh@’)~rh(~’) =

i

(7)

.

From eq. (5) for t = 0, taking into consideration eq. (7), we obtain C(rh) =

~ sPf(0)eth1s~h(v). (8) ~ We can write the equation for the general Fourier transform of the magnetic susceptibility ~(rh) in a thin film [2] in a form similar to eq. (6): —~

N

= A(h2)~h (9) where a~hdenotes the right eigenvector of the matrix A(h2), and its elements are O~.h(P).We can also determine the left eigenvector O~hof this matrix, The spin correlation time p~in the vth layer parallel to the surface of the film is defined as

p~=

and I/2kBT near can be expressed as (1 + 2 cos[ir/(n + 1)]} where e equal to TITC

f dt~ ~

— —

1 is

(s~j(t)s~ 1(0)),

(10)

and it can be measured near T~using Mossbauer effect techniques [3,4] placing Mössbauer atoms in different(10), positions theaccount film. Substituting (8) into takingininto eq. (7) and(5) theand fluctuation—dissipation theorem, we obtain n

2h ~ p~,= fh d

lows to obtain spin correlation every layer one of the film forthe temperatures abovetime and for below T~if the decay of fluctuations of the magnetization density is described by eq. (1). We apply eq. (13) to the Valenta model [5] of a thin ferromagnetic film where the decay of fluctuations of the magnetization density near was described by Duszewski [2]. In this case the thermal equilibrium value of the magnetization density determines eq. (2) withIn,’ equal to I for any pair of spins and F(X) denoting 0.5 tanh(X). The decay of fluctuations of the magnetization density is described by eq. (3) where the coefficient G~(T,n) equals 4(S~)2and the spin lattice relaxation time T~is the same in all layers. The Curie temperature of such a film is given by [2] kBTc = ~I{4 ÷ 2 cos[ir/(n + 1)]} , (14)

v’l

(~

the reduced temperature. Eq. (13) in this model assumes the form 4 + 2 cos[ir/(n + 1)] D~_1D~~ p~ (15) 2) D~ e)(1 v4(S~) where(1theD~for from 1 ton satisfy the following recursion formula —

/4 + 2 cos[7r/(n + 1)] 4)D~ 1—D~2, (16) D~ ~~‘- e)(1 — 4
n

—

D’(rh)s~h(v)s~h(v’)

r1

with D 0 equal to unity and D_1 equal to zero. ApplyX

I~X(r’h)c’~’h(v’)a~’h(v)).

(11)

ing formulae (15) and (16) we have determined the spatial distribution and the temperature dependence

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PHYSICS LETTERS

of the spin correlation time in a thin film with one hundred layers. We have also calculated the critical index w(p~)of the spin correlation time p~,in some layers. This index is defined as the derivative of in p~, with respect to in e. Because the film is symmetric only the first fifty layers are shown in all figures. In the discussion of the results we take into account that in the model considered the average correlation length in one direction equals a/~/~[2].

29 November 1982

8

10

12

14

16

18 I

—0.2 —0.4 —0.6 —0.8 —1.0

w(p~)

2. Spin correlation time above Tc. The spontaneous magnetization (S1) above Tc vanishes and in this case analytical expressions for D~and p~,can be obtained [6]. Spatial distributions of the spin correlation time for reduced temperatures 8 X l0~ and 5 X i~—~ are shown in fig. 1. It can be noticed that the spin correlation time increases when the reduced temperature decreases. For all temperatures p~decreases near the surfaces of the film over a distance approximately equal to the correlation length Critical indices w(p~) in the range of reduced temperatures between 2~ and 2—19 in layers 1, 10, 20 and 50 are shown in fig. 2. All the curves in this figure have one inflection point in the temperature in which the correlation length is three times smaller than the thickness of the film. The ~.

12

b

Fig. 2. Spin correlation time critical indices W(pp) above T~ in layers 1, 10, 20 and 50 in the range of reduced temperatures between 2~ and 2—’~.The critical index of the spin correlation time averaged over ali layers is shown by the dashed line. The reduced temperature is 2~where 1 is on the horizontal axis.

critical index of the spin correlation time averaged over all layers of the film is also shown in this figure by the dashed line. The value of the spin correlation time critical index for any temperature and in any layer of the film is determined by the relation between the correlation length and the distance of the layer considered to both surfaces of the film. In the vth layer the value of w(p~)equals —0.5 as in the infinite threedimensional ferromagnet [6] for temperatures where the distances of this layer to both surfaces of the film are bigger than 2~or it equals —1 as in the infinite two-dimensional ferromagnet [6] for temperatures where these distances are smaller than 2~.For temperatures where the distance of the wth layer to one surface of the film is smaller than 2~and the distance to the other surface is bigger than 2~the value of w(p~) tends to zero.

10

8

6

4 a

2

10

20

30

40

50

v

3. Spin correlation time below Tc. The spontaneous magnetization in the Valenta model [5] of a thin ferromagnetic film is given by the following expression L7]: 1/2kK(k) sn(2K(k)v/(n + 1)) , (17) (Sr) = (6e*/rr) where e~equals ir2/6(n + 1)2,K(k) is an elliptic integral of the first kind and k is determined by the condition e/e*

Fig. 1. Spatial distributions of the spin correlation time p~ above Tc for reduced temperatures 8 x i03 (curve a) and 5 X 10~ (curve b).

=

[s/i +k2(2/ir)K(k)]2

—

1

.

(18)

The spatial distributions of the spin correlation time for reduced temperatures 4 X 10~and 2 X i05 347

Volume 92A, number 7

PHYSICS LETTERS

29 November 1982 8

10

12

14

16

1

pp —02 -0.4 —06

w(p~) —1 0

1

10

20

30

40

50

v

Fig. 3. Spatial distributions of the spin correlation time ~ below Tc for reduced temperatures 4 X i~—~ (curve a) and 2 X i0~ (curve b). Values given by the curve b should be multiplied by a factor 10.

are shown in fig. 3. In any layer the spontaneous magnetization decreases with decreasing reduced temperature, whereas the correlation time increases. For the reduced temperature 4 X l0—~the spin correlation time is constant in the middle of the film and it increases near the surfaces where the spontaneous magnetization decreases. At a distance from the surface of the film that is approximately equal to the correlation length the spin correlation time decreases. Critical indices w(p~)in the range of reduced temperatures between 2~ and 2—15 in the layers 1, 7, 30 and 50 are shown in fig. 4. The value of w(p~)depends, as in section 2, on the relation between the correlation length and the distances of the layer considered to both surfaces of the film. All obtained results are easily generalized to the Valenta model of a semiinfinite ferromagnet. We conclude that the values of the spin correlation time and its critical index ob-

348

Fig. 4. Spin correlation time critical indices w(pp) below Tc in layers 1,7, 30 and 50 in the range of reduced temperatures between 2—~and

2i ‘~. The reduced temperature is 2—1 where 1 is on the horizontal axis.

tamed in an experiment depend on the range of investigated temperatures and the distances from the surfaces of the medium where the measurements are made. The authors would like to thank Professor L. Wojtczak for the assistance and guidance throughout the course of this work. The work is sponsored by the Institute of Physics of the Polish Academy of Sciences. References [1] T. Oguchi, Seminar on Phase transitions (Cetniewo, Poland, 1973). [2] J. Duszewski, Phys. Stat. Sol. (b) 103 (1981) 167. [3] H.Wegener,Z.Phys. 186 (1965) 498. [4] M.A. Kobeissi and C. Hohenemser, Hyp. mt. 4(1978) 480.

[5] L. Valenta, Czech. J.

Phys. 7 (1957) 136. [6] W. Korneta and Z. Pytel, Phys. Lett. 89A (1982) 238. [7] W. Korneta and Z. Pytel, Czech. J. Phys., to be published.