Microsoft Word - rjs-vol-ii-p.samarasekara-final1.0.doc.doc


© 2007 Faculty of Science 
University of Ruhuna 

RUHUNA JOURNAL OF SCIENCE 
Vol. 2, September 2007, pp. 1-9 
http://www.ruh.ac.lk/rjs/rjs.html 
ISSN 1800-279X 
 
  

1 

Abstract. The energy of ultra-thin sc(001) ferromagnetic film with three layers will be 
investigated using classical Heisenberg Hamiltonian. Energy curves show several 
minimums indicating that the film can be easily oriented in these directions under the 
influence of certain values of demagnetization factor. When the demagnetization factor is 

given by 
ωµ 0
dN =8, the angle corresponding to first minimum is 0.6 radians for sc(001) 

ferromagnetic lattice. Under the influence of demagnetization factor of 
ωµ 0
dN = 7.5, energy 

minimum can be observed at 0.6 radians for bcc(001) lattice. The energy curve of bcc(001) 
ferromagnetic lattice is smoother compared with that of sc(001) lattice.  

 

Keywords: materials, thin films, Heisenberg Hamiltonian, demagnetization factor 

1. Introduction:  
      The properties of ferromagnetic ultra-thin films have been investigated using 
classical model of Heisenberg Hamiltonian with limited number of terms (Usadel 
and Hutch 2002).  Because of the difficulties of understanding the behavior of 
exchange anisotropy and its applications in magnetic sensors and media technology, 
exchange anisotropy has been extensively investigated in recent past (David et al. 
2004). Ferromagnetic films are thoroughly studied nowadays, due to their potential 
applications in magnetic memory devices and microwave devices. Bloch spin wave 
theory has been used to investigate magnetic properties of ferromagnetic thin films 
(Martin and Robert 1951). Although the magnetization of some thin films is 
oriented in the plane of the film due to dipole interaction, the out of plane 
orientation is preferred at the surface due to the broken symmetry of uniaxial 
anisotropy energy. Previously two dimensional Heisenberg model has been used to 
explain the magnetic anisotropy in the presence of dipole interaction (Dantinzger et 
al. 2002). Ising model has been used to study magnetic properties of ferromagnetic 
thin films with alternating super layers (Bentaleb et al. 2002).      

Effect of demagnetization factor on total energy of 
ultra-thin ferromagnetic films with three layers 

P. Samarasekara 
Department of Physics, University of Ruhuna, Matara, Sri Lanka 

Correpondence: pubudus@phy.ruh.ac.lk 



2                                  Samarasekara:  Effect of demagnetization…
                         R u h u n a  J o u r n a l  o f  S c i e n c e  2 ,  p p . 1 - 9  ( 2 0 0 7 )  
 

          For the very first time, the variation of energy of ferromagnetic ultra-thin 
films with demagnetization factor will be described in this report. The energy of 
non-oriented ultra-thin ferromagnetic films with two and three layers has been 
calculated using Heisenberg Hamiltonian with second order perturbation, under the 
effect of limited number of energy parameters (Samarasekara 2006). The properties 
of perfectly oriented thick ferromagnetic films have been investigated by classical 
Heisenberg model (Samarasekara 2006). The variation of energy with angle and 
number of layers has been studied for thick films up to 10000 layers. The total 
magnetic energy has been calculated using two different methods depending on 
discrete and continuous variation of thickness. For bcc(001) lattice, the easy and 
hard directions calculated using both methods were exactly same.  

2. Model and discussion: 
The classical model of Heisenberg Hamiltonian is given as following (Samarasekara 
2006), 

∑ ∑ ∑∑ −−−+=
≠nm m m

z
m

z
m

nm mn

nmnmnm

mn

nm
nm SDSDr

SrrS
r

SS
SS

J
H

mm
,

4)4(2)2(
53 )()()

).)(.(3.
(

2
.

2 λλ
ω

rrrrrrrr

 
             ∑ ∑−−−

nm m
msmnd SinKSSNH

,
0 2)]./([ θµ

rrr
        

For the Heisenberg Hamiltonian given in above equation, total energy can be 
obtained as following (Samarasekara 2006). 

E(θ)=E0+ εα
rr

. + εε
rr

..
2
1

C =E0 αα
rr

..
2
1 +− C                   

The matrix elements of above matrix C are given by 

0

2
2cos

4
3

)
4

(
µ

θ
ωω d

nmnmnmmn
N

JZC +Φ−Φ−−= −−−  

          ∑
=

−− −−+Φ−+
N

mmmmn DJZ
1

)2(22 )cos(sin2)]2cos
4

3
4

([{
λ

λλ θθθ
ωω

δ  

         

}2sin4
4

cossin)sin3(coscos4
0

)4(222 θ
µ

θθθθθ s
d

outinm K
N

HHD +−++−+

 
θθεα 2sin)()( B

rr
=  are the terms of matrices with 

θ
ω

θ λλλλ
2)4(

)2(

1
cos2

4
3

)( DDB
N

m
m ++Φ−= ∑

=
−                     (1) 

Here (Samarasekara 2006) 



Samarasekara:  Effect of demagnetization…                              3   
R u h u n a  J o u r n a l  o f  S c i e n c e  2 ,  p p . 1 - 9  ( 2 0 0 7 )  
            

])1(2[
2 100

ZNNZ
J

E −+−= 10 )1(2{ Φ−+Φ+ NN )2cos8
3

8
}( θ

ωω
+  

      )2sincossincos(cos
0

)4(4)2(2 θ
µ

θθθθ s
d

outinmm K
N

HHDDN +−+++−  

E0 is the energy of the oriented thin ferromagnetic film. Here J, nmZ − , ω, nm−Φ , θ, 

,,,,,, )4()2( sdoutinmm KNHHDD  m, n and N are  spin exchange interaction, 
number of nearest spin neighbors, strength of long range dipole interaction, 
constants for partial summation of dipole interaction, azimuthal angle of spin, 
second and fourth order anisotropy constants, in plane and out of plane applied 
magnetic fields, demagnetization factor, stress induced anisotropy constant, spin 
plane indices and total number of layers in film, respectively. When the stress 
applies normal to the film plane, the angle between mth spin and the stress is θm.  
Matrix elements for a film with three layers (N=3) can be given as following 
(Samarasekara 2006), 

0
1132232112

2
)2cos31(

4 µ
θ

ω dNJZCCCC +−Φ+−====  

0
223113

2
)2cos31(

4 µ
θ

ω dNJZCC +−Φ+−==  

)2(

0
21213311 )2cos2(

2
)2cos31)((

4
)( m

d D
N

ZZJCC θ
µ

θ
ω

+−+Φ+Φ−+==  

                     
θθθθθθ 2sin4cossin)sin3(coscos4 )4(222 soutinm KHHD +++−+  

)2(

0
1122 )2cos2(

2
)2cos31(

2
2 m

d D
N

JZC θ
µ

θ
ω

+−+Φ−=  

           θθθθθθ 2sin4cossin)sin3(coscos4 )4(222 soutinm KHHD +++−+      
If the second or fourth order anisotropy constants are invariants inside an ultra thin 
film, then D1(2)=D2(2)=D3(2) and D1(4)=D2(4)=D3(4). Under some special conditions 
(Samarasekara 2006), C+ is the standard inverse of a matrix, given by matrix 

element
C

cofactorC
C nmmn

det
=+ . For the convenience, the matrix elements C+mn will 

be given in terms of C11, C22, C32, and C31 only.  

33

1131
2

32
2

31221111

2
322211

11
)(2)(

++ =
−+−

−
= C

CCCCCCC
CCC

C  

322321

1131
2

32
2

31221111

11323132
12

)(2)(
++++ ===

−+−

−
= CCC

CCCCCCC
CCCC

C  



4                                  Samarasekara:  Effect of demagnetization…
                         R u h u n a  J o u r n a l  o f  S c i e n c e  2 ,  p p . 1 - 9  ( 2 0 0 7 )  
 

31

1131
2

32
2

31221111

3122
2

32
13

)(2)(
++ =

−+−

−
= C

CCCCCCC
CCC

C  

)(2)( 1131
2

32
2

31221111

2
31

2
11

22
CCCCCCC

CC
C

−+−

−
=+                (2)         

Matrices C and C+ are highly symmetric, and total energy can be given as 
(Samarasekara 2006),  
E(θ)=E0-0.5[C+11(α12+α32)+C+32(2α1α2+2α2α3)+C+31(2α1α3)+α22C+22]           
From equation 1,  

θ
ω

θθ λλ
2)4()2(

21031 cos2)(4
3

)()( DDBB ++Φ+Φ+Φ−==  

θ
ω

θ λλ
2)4()2(

102 cos2)2(4
3

)( DDB ++Φ+Φ−=  

Because in this case, α1=α3 
E(θ)=E0-0.5[2C+11α12+4C+32α1α2+2C+31α12+α22C+22]             (3)           
First simulation will be carried out for 

510
)4()2(

======
ωωωωωω
msoutinm Dand

KHHDJ
 

For sc(001) lattice, Z0=4, Z1=1, Z2=0, Φ0=9.0336, Φ1= -0.3275 and Φ2=0 (Usadel 
and Hucht 2002),  

ωµ
θ

ωωωω 0
32232112 22cos2456.008.10 d

NCCCC
++−====  

ωµωω 0
3113 2 dNCC ==  

θ
ωω

2cos2456.2008.103311 +==
CC

)sin3(coscos20
2 222

0

θθθ
ωµ

−+− d
N

 

                     θθθ 2sin40cos10sin10 +++  

 θ
ω

2cos49.20164.2022 +=
C

)sin3(coscos20
2 222

0

θθθ
ωµ

−+− d
N

 

                θθθ 2sin40cos10sin10 +++  

 θθ
ω
α

ω
α

2sin)cos1047.3( 231 +==  

θθ
ω
α

2sin)cos10716.3( 22 +=  



Samarasekara:  Effect of demagnetization…                              5   
R u h u n a  J o u r n a l  o f  S c i e n c e  2 ,  p p . 1 - 9  ( 2 0 0 7 )  
            

78.40 −=
ω
E

θ2cos67.9+

)2sin10cos10sin10cos5cos10(3
0

42 θ
ωµ

θθθθ +−+++− d
N

 

For sc(001), 3-D plot of energy versus angle and 
ωµ 0
dN  is given in figure 1. The 

graph indicates several energy minimums at different values of angle and 
demagnetization factor. Under the influence of these demagnetization factors, the 
sample can be easily oriented in certain directions corresponding to energy 

minimums. For example, one minimum can be observed at 
ωµ 0
dN =8, and the angle 

corresponding to this demagnetization factor can be obtained from figure 2. Angle 
corresponding to first minimum is 0.6 radians.     
 

 

Figure 1: 3-D plot of energy versus angle and 
ωµ 0
dN

 for sc(001) ferromagnetic lattice 



6                                  Samarasekara:  Effect of demagnetization…
                         R u h u n a  J o u r n a l  o f  S c i e n c e  2 ,  p p . 1 - 9  ( 2 0 0 7 )  
 

 

Figure 2: Energy versus angle at 
ωµ 0
dN

=8 for sc (001) lattice 

For bcc(001) lattice, Z0=0, Z1=4, Z2=0, Φ0=5.8675 and Φ1=2.7126 (Usadel and 
Hucht 2002), 

ωµ
θ

ωωωω 0
32232112 22cos03.232.39 d

NCCCC
+−−====  

ωµωω 0
3113 2 dNCC ==  

θ
ωω

2cos03.232.393311 −==
CC

)sin3(coscos20
2 222

0

θθθ
ωµ

−+− d
N

 

                     θθθ 2sin40cos10sin10 +++  

 θ
ω

2cos07.464.7822 −=
C

)sin3(coscos20
2 222

0

θθθ
ωµ

−+− d
N

 

                θθθ 2sin40cos10sin10 +++  

 θθ
ω
α

ω
α

2sin)cos10565.3( 231 +==  

θθ
ω
α

2sin)cos1053.1( 22 +=  



Samarasekara:  Effect of demagnetization…                              7   
R u h u n a  J o u r n a l  o f  S c i e n c e  2 ,  p p . 1 - 9  ( 2 0 0 7 )  
            

44.760 −=
ω
E

θ2cos67.10+   

              )2sin10cos10sin10cos5cos10(3
0

42 θ
ωµ

θθθθ +−+++− d
N

 

3-D plot of energy versus angle and 
ωµ 0
dN  for bcc(001) ferromagnetic lattice is 

given in figure 3. This graph also indicates several energy minimums. Energy is 

minimum at
ωµ 0
dN = 7.5. The figure 4 has been drawn to find the easy directions 

corresponding to this energy minimum at 
ωµ 0
dN = 7.5. Energy minimum can be 

observed at 0.6 radians, and this angle gives the easy direction at 
ωµ 0
dN = 7.5. This 

graph is smoother compared with the energy curve given in figure 2.   
 

 
 

Figure 3: 3-D plot of energy versus angle and 
ωµ 0
dN

 for bcc(001) ferromagnetic lattice 



8                                  Samarasekara:  Effect of demagnetization…
                         R u h u n a  J o u r n a l  o f  S c i e n c e  2 ,  p p . 1 - 9  ( 2 0 0 7 )  
 

 
 

Figure 4: Energy versus angle at 
ωµ 0
dN

=7.5 for bcc (001) lattice 

3. Conclusion: 
          3-D plots and 2-D plots of energy indicate several minimums implying that 
the film can be easily oriented in these directions given by angles corresponding to 
minimums under the influence of certain values of demagnetization factor. At 

ωµ 0
dN =8, the angle corresponding to first minimum is 0.6 radians for sc(001) 

ferromagnetic lattice. Under the influence of demagnetization factor given by 

ωµ 0
dN = 7.5, energy minimum can be observed at 0.6 radians for bcc(001) lattice. 

This simulation can be carried out for any other values of these energy parameters 
as well.    
 



Samarasekara:  Effect of demagnetization…                              9   
R u h u n a  J o u r n a l  o f  S c i e n c e  2 ,  p p . 1 - 9  ( 2 0 0 7 )  
            

References: 
Bentaleb M., El Aouad N. and Saber M 2002. Magnetic properties of the spin -1/2 Ising 

Ferromagnetic thin films with alternating superlattice configuration, Chinese J. Phys., 
40(3): 307- 314 

Dantziger M., Glinsmann B., Scheffler S., Zimmermann B. and Jensen P.J 2002. In-plane 
dipole coupling anisotropy of a square ferromagnetic Heisenberg monolayer, Phys. Rev., 
B(66): 094416 1-6  

Lederman David, Ricardo Ramirez and Miguel Kiwi 2004. Monte Carlo simulations of 
exchange bias of ferromagnetic thin films on FeF2(110), Phys. Rev. B70: 184422 1-7  

Martin J. Klein and Robert S. Smith 1951. Thin ferromagnetic films, Phys. Rev. 81: 378-
380  

Samarasekara P 2006. Second order perturbation of Heisenberg Hamiltonian for non-
oriented ultra-thin ferromagnetic films, Elec. J. Theo. Phys., 3(11): 71-83  

Samarasekara P 2006. A solution of the Heisenberg Hamiltonian for oriented thick 
ferromagnetic films, Chinese J. Phys., 44(5): 377-386  

Usadel KD and Hucht A 2002. Anisotropy of ultrathin ferromagnetic films and the spin 
reorientation transition, Phys. Rev. B 66: 024419 1-6