Supplementary Submission to Electronic Physics Auxillary Publication Services (E-PAPS)
For Paper entitled “Observation of Magnetic-field Induced Phase Transition in BiFeO
3
by Highfield
Electron Spin Resonance: Cycloidal to Homogeneous Spin Order”
In this supplimentary submission to E-PAPS, a Landau-Ginzburg (LG) formalism for the free energy will be considered. Accordingly, a theory for the field-induced phase transition of
BiFeO
3
and changes in the ESR spectra will be developed. In this section, we will present the solution to the Euler-Lagrange equation for minimization of the free energy, which was arrived at using the thermodynamic theory of perturbation [1]. We will also consider the question of the accuracy of such an approximation.
Theory of the Field-induced Transition of BiFeO
3
The BiFeO
3
spin structure is characterized by the unit antiferromagnetic vector l

that in spherical coordinates can be represent as l


 sin
 cos

, sin
 sin

, cos
 
, where

is the polar angle of the l

vector,

is the azimuthal angle of l

, and c
ˆ is the polar axis. The Landau-
Ginzburg energy [2] of the spin structure is the sum
F

F
L

F exch

F an

F m
. (1)
The first term F
L
in (1) is the magneto-electric coupling that is linear in gradient (i.e.,
Lifshitz invariant), given as
F
L
   
P z l
 x
 x l z
 l y
 y l z

   
P z sin
2


 x
 cos
   y
 sin


; (2a) where P z
is spontaneous polarization, and

is the inhomogeneous relativistic exchange constant
(inhomogeneous magneto-electric constant). The Lifshitz invariant is responsible for the creation
Benjamin Ruette, et. al. 1

of the spatially modulated spin structure in BiFeO
3
, as will be shown below. The second term
F exch
in (1) is the inhomogeneous exchange energy, given as
F exch

A i


, x , y z
  i
2 
A

 
2  sin
2
  
2

; (2b) where A is a stiffness constant. The third term F an
in (1) is the anisotropy energy, given as
F an
 
K u l z
2 
K
2 l z
4  
K u


K u

2 K
2
 sin
2
 
K
2 sin
4

; (2c) where K u
is the uniaxial magnetic anisotropy constant, and K
2
is the second order anisotropy.
The fourth term F m
in (1) is the magnetic energy, given as
F m
  


H

2
 total ,
2
; (2d) where

 is the magnetic susceptibility of the media in the direction perpendicular to the antiferromagnetic vector l

, and

H total ,

is the component of the total magnetic field perpendicular to the antiferromagnetic vector l

.
The total field consists of the sum of the externally applied field internal field

H , and an effective


D that originates from the magneto-electric-like Dzyaloshinsky-Moria DM interaction, given as

H total


H
 l
 
D ; (3) where

D


0 , 0 ,
 
P z

, where

is the homogeneous magneto-electric constant. The
Dzyaloshinsky-Moria interaction W
DM
is the antisymmetric exchange, given as
W
DM

D z

 m y l x
 m x l y

, (4) where m is the unit vector of magnetization.
Benjamin Ruette, et. al. 2

Taking into account that H total ,


H total
 l
  
H total
 l
 
, we can find the magnetic energy to be
F m
 


2

H
2 
  z
2 sin
2
 
2
  z sin

( H x sin
 
H y cos

)


2
. (5)
The Dzyaloshinsky-Moria term in this equation can be attributed to the effective anisotropy constant
K’ u
K u

  

  
P z

2
2

K u

2 K
2
. (6)
Working within the framework of the thermodynamic theory of perturbation [1], we treat the last two terms in (1) as perturbations. The structure of the spin cycloid will be found by the minimization of F
0
. The total free energy is then treated as a sum of F
0 and
W

1
V

V d
3 x

( F an
(

0
( x ),

0
( x ))

F m
(

0
( x ),

0
( x ))) , where the last two terms of (1) are averaged over the spin cycloid and where a zero-order approximation is used for the functional forms of

0
( x ),

0
( x ) .
The Lagrange equations d
 d r






F
0
 
 r







F
0
 

0 ; d
 d r






F
0
 

 r






F
0
 

0 for the minimization problem are
2 A
     
P z sin 2


 cos

 
 y
 sin

 
 x



A

 
2 sin
 

0
,
2 A
 sin
2      sin 2
 
      
  
P z sin
2 
 cos
  y
  sin
  x



0 .
(7a)
(7b)
The solution of these equations are [see also 2]

0
 arctg ( q y q x
) , (8a)
Benjamin Ruette, et. al. 3


0
 q x x
 q y y (8b)
Equation (8a) requires that the spins lie in the plane of the wave-vector. This requirement is in agreement with the experimentally observed cycloidal spin structure [3].
Substituting the solution (8) into equation (1) for the volume-averaged free energy under a magnetic field parallel to the [001] c
-direction of H


H x
, 0 , H z

, the following expression can be derived
F cycloid

F
0
 
W
  

H 2
2

 
P z
2 q

Aq
2 
K u

2

3
8
K
2
 

H x
2 cos
2

4
 

H z
2
4
. (9)
Minimization of (9) occurs for q

 
P z
4 A
and
 

2
(the plane of the spiral is perpendicular to the in-plane component of the field). Substituting these values in to equation (9) and taking into account equation (6) gives
F cycloid
  

H 2
2

1
16 A
  
P z

2 
K u
2

5
8
K
2
 

  
P z

2
4
 

H
2 z
4
. (10)
A phase transition to the homogeneous state will occur at critical field H c
, when the energy of the homogeneous state is equal to that one of the cycloidal one. The energy of the homogeneous state,
   

2
 const , under an external field of H
 
H x
, 0 , H z

is
F
Homogeniou s
 



H
2
2

K u

K
2
 

  
P z

2
2
 

  
P z

H x
. (11)
Taking into account that the field was directed along the rhombohedral axis,

H


H

2 / 3 , 0 , H / 3

, the following quadratic equation for H c
can be obtained
H с
2 
4 6
  
P z

H с

6

K

 u

9 K
2
2



3
  
P z

2 
4


3

A
  
P z

2 
0 . (12)
Benjamin Ruette, et. al. 4

To estimate the accuracy of the thermodynamic theory of perturbation, we compare our results with the rigorous solution for the problem in the case of a magnetic fied applied along the c-axis [4]. With increasing field H and anisotropy K u
, the period of the spin cycloid increases and the spin structure given by eq. (8) becomes increasingly anharmonic. In a first order approximation that includes anharmonicity, the solution for

( x ),

( x ) is an elliptic function
[2,4]. The deviation from a sinusoidal profile is proportional to


H
H c


2
, and thus increases with increasing field. In the phase transition region, although the spin structure differs noticeably from a sinusoidal profile, the deviation of the value for the critical field as estimated by the thermodynamic perturbation theory of eq. (12) from the experimental one is ≤10%. We consider this acceptable for our purposes.
Theory of the ESR Signal at Fields of H>H c
To consider the dynamic properties of the cycloidal modulation of BiFeO
3
that are conditioned by oscillation of the l
 vector, we use a Lagrangian, given as (i.e., see [5] and reference cited there within)
L

2



2 l
 2


2



H total

 l l
 


F ; (13) where

1.73x10
7 cm / g is the gyromagnetic ratio, F is the free energy given in (1), and H total is total field given in (3).
In the high field regime ( H>H c
), the cycloidal spin structure is destroyed and a homogenously magnetized state formed. Accordingly, we can omit all terms in the Lagrangian that have spatial gradients. As in the experiment H y
=0, the Lagrangian is then simplified to
Benjamin Ruette, et. al. 5

L

1
2



2 d
 dt
2
 d
 dt
2 sin
2
 

2



H total

 l l


K u
' sin
2
 
K
2 sin
4



2


H
2 
2
  
P z
 sin
 
H x sin
 

H x
 sin
 cos
 
H z cos
 
2

. (14)
Lets consider a small deviation of the ferromagnetic vector from the equilibrium state:

 x , y , t

  o
 
 
 x , y , t

and 
 x , y , t

  o
 
 
 x , y , t

; where

and

are small; and the angles corresponding to the equilibrium state are
 o
=
 o
=90 o
. The Lagrange equations
 d dt




L
 




L
 

0 ;
 d dt




L
 




L
 

0 in the linear approximation of

and
 can be written as




2 d
2 dt
2

2 K u
  

  
P z

2
  

H z
2
  

  
P z

H x
  

H x
H z
 
0
,




2 d 2 dt 2
 

H x
2
  

  
P z

H x
  

H x
H z
 
0
.
(15a)
(15b)
Considering only small deviations from the equilibrium sate, in the form of harmonic oscillations

 x , y , t

   e i
  t and

 x , y , t

   e i
  t , the following matrix representation can be obtained for equations (15)





2

2

2 K u



  
P z

2 
H 2 z

  
P z

H x

H x
H z


2
2

H x
H z

H 2 x

  
P z

H x












0 .
(16)
By setting the determinate of the matrix equal to zero, the ESR resonance frequency can be obtained as
 

2



2

 b

2 b
2 
4 c
(17a)
Benjamin Ruette, et. al. 6

where, b

2 K u



H 2 z

  
P z

2 
2
  
P z

H x

H 2 x
; c



2 K

 u
   
P z

2 
H 2 z
   
P z

H x




H 2 x
   
P z

H x

 
H x
H z

2
. (17b)
[1] L.D. Landau, E.M. Lifshitz, Statistical Physics. Course of Theoretical Physics, Vol.
V, Pergamon, Oxford, 1997.
[2] I. Sosnowska and A. Zvezdin, J. Magnetism and Magnetic Materials 140-144, 167
(1995).
[3] I. Sosnowska, T. Peterlin-Neumaier, and E. Steichele, J. Phys. C 15, 4835 (1982).
[4] M.M.Tehranchi, N.F.Kubrakov, and A.K.Zvezdin Ferroelectrics, 1997, v.204, pp.181-188.
[5] A.K. Zvezdin and A. Mukhim, JETP V. 102 (vol. 8), 577 (1992). (In Russian)
Benjamin Ruette, et. al. 7