báo cáo khoa học hệ spin

…

1

aν z

rν j

Spin wave in ultrathin magnetic film
Le Ngan, Bach Thanh Cong

Computing materials Science Laboratory, Faculty of Physics, VNU University of Science,
Rj
334 Nguyen Trai, Hanoi,
ν Viet nam
Abstract
In this work the temperature dependent dispersive law for spin wave in ultrathin magnetic films
consisting from several atomic spin layers is calculated using double time Green function technique
and anisotropic exchange Heisenberg model. The chain of Green functions is decoupled within
Tyablikov-Bogoliubov approximation. Dependence of spin wave spectra on the number of layers and
temperature is analyzed.

Keywords: Magnetic thin film, Heisenberg model, Double times Green function
1. Introduction

At present time magnetism of low dimensional spin systems involves much interest of
researchers. In [1] the ground magnetic state of perovskitenanoclusters is studied using
density functional theory (DFT) method. Thermodynamic properties at finite temperature of
ultra- thin magnetic films (quasi- two dimensional case)is proposed to calculate by the
functional integral method [2]. The spin waves theory in magnetic single layer with dipole
and isotropic

exchangeinteractions is developed
in [3].The aim of this research is to investigate the spin wave in the case of anisotropic
exchange interaction in untrathin magnetic films using the popular double – time Green’s
function method [4]. We will restrict our treatment to ferromagnets.
2. Model and formalism

We consider a magnetic thin film having
cubic structure and consisting of n atomic
spin layers.Each atom has a magnetic

ma = − g µ B S

Fig. 1: Arrangement of thin film atomic
spin lattice relative to coordinate system

moment
. The Oz axis of
coordinates system is chosen perpendicular to the surfaces of the thin film and atomic spin
planes are parallel to xOyplane (see figure 1).There is a translational symmetry of spin
arrangement in xOy plane andthe number of spin in every plane N is to be very large (N ~ ∞).

rν j
is the position vector of spin

Sν j ( rν j = R j + aν zˆ )

. Here a is the lattice constant,

ν

is an

Rj
index of layer and

aν zˆ

is the two-component vector describing position of the spin on the xOy

plane,
is the component of the position vector on the Oz axis.Heisenberg
Hamiltoniandescribed theinteracting spin system in the thin film is written as:

ν=


2

H =−

1
z
∑ Jν j ,ν ' j ' Sν j Sν ' j ' − g µB B0 ∑ Sν j
νj
2 ν j ,ν ' j '
(1)

rν j
The first term in (1) is an exchange interaction between two spins at sites

and

Jν j ,ν ' j ' = Jνν ' ( R j − R j ' )

depends only on the distance,

rν ' j '
, which

. This exchange integral is a periodical

Rj − Rj '

function of two dimensionallattice vector
. The second term in (1) is the energy of the
spin system in the external field orientated along the Oz axis. Normally, we use the up, down

Sν±j = Sνx j ± iSνyj , Sνz j
and Z components of spin operator,

. And Hamiltonian (1) is rewritten in

Sν±j , Sνzj

terms of

H =−

as

1
+ −
z
z
z
∑ Jν ,ν ' ( R j − R j ' ) Sν j Sν ' j ' + Sν j Sν ' j ' − g µ B B0 ∑ Sν j
νj
2 ν j ,ν ' j '

(

)

(2)
In order to study the kinetics of the system at finite temperature, we introduce the following
retarded double times Green function:

Gν j ,ν ' j ' ( t , t ') =

Sν+j ( t ) ; Sν−' j ' ( t ' )

= −iθ ( t − t ' )  Sν+j ( t ) , Sν−' j ' ( t ' ) 
−
(3)

An equation of motion for Green function (3) in the energy representation has been obtained
as:

r
r

EGν jν ' j ' ( E ) = 2 Sνz j δνν 'δ jj ' − ∑ Jν1ν R j1 − R j
ν1 j1

− Sνz1 j1 Sν+j ; Sν−' j '

E

+ g µ B B0

(

Sν+j ; Sν−' j '

E

}

){

Sνz j Sν+1 j1 ; Sν−' j '

E

(4)

The lower symbol E in (4) denotes the energy representation. In what follows we drop this
symbol for clarity. The chain for Green function (4) can be decoupled using Bogolyubov Tyablikov’s procedure, where:

Sνz j Sν+1 j1 ; Sν−' j '

→ Sνz j

Sν+1 j1 ; Sν−' j '

Sνz1 j1 Sν+j ; Sν−' j '

→ Sνz1 j1

Sν+j ; Sν−' j '
(5)


3

Because of translational symmetry in the thin film plane, the average value of the projection
Sνz j = Sνz
of moment spin onto the z-axis does not depend on j and j 1, i.e.
.The formula (4)
now becomes:

r
r

E
−
g
µ
B
−
J

R
−
R
∑

B 0
ν1ν
j1
j
ν1 j1


(

)


Sνz1  Gν jν ' j ' ( E ) = 2 Sνz δνν 'δ jj '


(

− ∑ Jν1ν R j1 − R j
ν1 j1

)

Sνz Gν1 j1ν ' j ' ( E )
(6)

Due to the translational symmetry of the spin distribution in the thin film planes, Green
k
functioncan be expanded in the Fourier series of wave vector space

Gν jν ' j ' ( E ) =

ik ( R j − R j ' )
1
∑ Gνν ' ( k , E )e
N k

(7)

One has from (6)


z 
z
z
%
 E − g µ B B0 − ∑ J%
ν1ν (0) Sν1  Gνν ' ( k , E ) = 2 Sν δνν ' − ∑ Jν 2ν ( k ) Sν Gν 2ν ' ( k , E )
ν1
ν2


(8)
ikR j
J%
νν ' ( k ) = ∑ Jνν ' ( R j ) e

j

Where

(9)

We use the nearest neighbor approximation (n.n.) then Fourier image of exchange integral (9)
becomes:


 2 J s ( cos k x a + cos k y a ) when ν ' = ν
J%
νν ' ( k ) = 
when ν ' = ν ± 1

J p

Js J p
(

(10)

) denotes the in-plane (next plane) n.n. exchange.

3. Spin wave spectra in single and double layers thin films
3.1 Single layer film

For single layer film with isotropic exchangewe obtain the following retarded Green function
from (8):

2 Sz
2 Sz
G11 ( E , k ) =
=
z
z
E − E( k)
E − g µ B B0 − J%
+ J%
11 ( 0 ) S
11 ( k ) S
(11)


4

The spin wave spectrum as a pole of the Green function is derived as:

E (k , T ) = g µ B B0 +  4 − 2 ( cos k x a + cos k y a )  mJ s

(12)

m = Sz / S

In (12),
is an average magnetization per sites which is temperature dependent
and satisfies following equation:

m = 1−

1
∑
NS k

1
 ε ( k ,τ ) 
exp
 −1
 τ 

(13a)

Here the dimensionless energy and temperature are

ε (k ,τ ) =

k T
E (k , T )
τ= B
Js
Js

(13b)

;

The isotropic exchange case is not interest because of absence of magnetization in monolayer
spin film with isotropic Heisenberg exchange (Mermin-Wagner theorem) and it is not proper
described in Tiablikov-Bogolivbov approximation. Then we study the more interesting case

J s = J s21 + J s22

J s1 J s 2

of single layer film with anisotropic exchange:
(
(
) is n.n. exchange
along Ox (Oy) direction).We obtain the following spin wave spectrum formonolayer film:

ε k,a =

{

[

] }

1
gµ B B0 + 2 J s1 1 + J s 2 − cos k x a − ρ cos k y a mS
J s1

ρ = J s2 / J s1

(14)

τ = k BT / J s1

characterizes anisotropy and
is dimensionless temperature.

Figure2 and 3 show the temperature dependence of the magnetization of the monolayer thin
film with anisotropic exchange. Figure show 4 and 5 show the spin wave spectra for the cases
where magnetizations are described in figures 2, and 3. One sees from figures 4, 5 that
intensity of spin wave reduces with increasing temperature
enhancement of spin wave in single layer film.

τ

and the anisotropy gives to


5

1

1

0 .9

0 .8
0 .8

0 .6
m

m

0 .7
0 .6

0 .4
T
0 .2

0

0 .0 0 5

0 .0 1

0 .0 1 5 τ 0 .0 2
α

0 .0 2 5

0 .5
T

c

0 .0 3

0 .4

0 .0 3 5

Fig. 2:Temperature dependence of magnetization
of single layer film with S=1 on temperature for

ρ = 1.7

0

τ

0 .0 4

0 .0 6

α

0 .0 8

c

0 .1

Fig. 3: Temperature dependence of magnetization of
single layer film with S=2 on temperature for

ρ = 1.7

10

25
E ( τ= 0.01)

E (τ= 0.01)

k

E ( τ= 0.02)

8

k

20

k

E ( τ= 0.05)

E (τ= 0.03)

k

k

15

E (τ= 0.096)
k

E

E

k

k

6
4

10

2

5

0

0 .0 2

0

0 .5

1

1 .5

2

2 .5

3

3 .5

4

4 .5

k

Fig. 4: Spin wave energy spectra in the first
Brillouin zone of single layer cubic spin film at
different temperatures for S=1, anisotropic

ρ = 1.7

exchange parameter

0

0

0 .5

1

1 .5

2

k

2 .5

3

3 .5

4

4 .5

Fig. 5: Spin wave energy spectra in the first
Brillouin zone of single layer cubic spin filmat
different temperatures for S=2, anisotropic exchange

ρ = 1.7

parameter

3.2 Double layers thin film

Similarly, by symmetry we obtain the expression for two Green functions in case of two
layers film:



1
1
÷
G22 ( k , E ) = G11 ( k , E ) = m ( E − Ek ,m ) 
−
 E − ( Ek , m + J p m ) E − ( E k , m − J p m ) ÷




(15)


1
1
G12 ( k , E ) = G21 ( k , E ) = J p m 2 
−
 E − ( Ek , m + J p m ) E − ( E k , m − J p m )



÷
÷


(16)


6

Ek±,m
We obtain the two branches for energy spectrum of spin waves

ξ k+,m =

Ek+,m
Js

=

{

1
g µ B B0 +  4 − 2 ( cos k x a + cos k y a )  J s mS
Js

}
(17)

ξ

−
k ,m

Ek−,m 1
=
=
g µ B B0 +  4 − 2 ( cos k x a + cos k y a ) + 2η  J s mS
Js
Js

{

}

Parameter η =Jp/Js characterizes an anisotropy behavior of exchanges and magnetization obeys the
following equation:

m=

Sz
S

≅ 1−

1
η m2
− +
S
S
=
1
−
2S 2
2N



1

÷
α

÷
exp
ξ
τ

−
1
k ,α =± 
k ,m


∑

(

)

Figure 6, and 8 show the temperature dependence of the magnetization of the double layers
thin film with anisotropic exchange between layers. Figure show 7 and 9 show the spin wave
spectra for the cases where magnetizations are described in figures 2, and 3.One sees from
figures 7, and 9 that intensity of spin wave reduces (increases) with increasing (reducing)
temperature (anisotropy) like in monolayer case. But differing with single layer case, there is
an energy gap between two spin waves branches and the gap is wider when temperature is
lower. Crossing between branches at different temperatures in the Brillouin zone is also
possible.
1

10

(E )- (0 .01)
k

0 .9

8

0 .8

(E )+(0.01)

gap
τ = 0 .0 1

k

-

6

0 .6

(E ) (0 .09)
k

(E )+(0.09)
k

E

m

k

0 .7

4

0 .5

gap
τ = 0 .0 9

2

0 .4
T
0

0 .0 1

0 .0 2

0 .0 3

0 .0 4

τ

0 .0 5

0 .0 6

c

0 .0 7

0 .0 8

0 .0 9

Fig. 6: The dependence of magnetization on

η = 1 .2

temperature for

, S = 1.

0
0

0.5

1

1.5

2 k 2.5

3

3.5

4

4.5

Fig. 7: Dependence of the spin wave energy spectrum

k

on the wave vector
0.01 and τ = 0.09 for

at different temperatures τ =

η = 1 .2

,S=1.


7

1

10

0.9

8

(E )- (0.01)
k

(E )+(0.01)

gap τ = 0 .0 1

k

6

(E ) (0.1)

4

(E )+(0.1)

-

k

E

m

k

0.8
0.7

τ = 0 .1

2

0.6
0.5
0

gap

k

Tc
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

τ

Fig. 8: Temperature dependence of magnetization for
double layer film for S = 1,and anisotropic exchange

η = 0.2

0
0

0.5

1

1.5

2

k

2.5

3

3.5

4

4.5

Fig. 9: Energy spectrum of spin waves in the first Brillouin
zone of double layers film at different temperatures τ=0.01;
τ=0.1 for S=1,and anisotropic exchange parameter between

η = 0.2

parameter between layers
layers

Acknowledgements
Authors thank the NAFOSTED grant 103.02.2012.37 for support.
References
[1]Nguyen Thuy Trang, Bach Thanh Cong, Pham Huong Thao, Pham The Tan, Nguyen Duc
Tho,Hoang Nam Nhat, Physica B 406 (2011) 3613.
[2] Bach Thanh Cong, Pham Huong Thao, Physica B 426 (2013)144–149.
[3] E. Meloche, J. I. Mercer, J. P. Whitehead, T. M. Nguyen, and M.L. Plumer, Phys. Rev. B 83
(2011) 174425.
[4] S.V. Tyablikov, Methods in the quantum theory of magnetism, Plennum press, New York, 1967.