1. science
2. physics

International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
Volume 2 Issue 1 (January 2015)
www.ijirae.com
The phenomenology of manganites as spin glass
Young In Lee*
Department of Electrical and Biological Physics, Kwangwoon University
Abstract— In this study, we investigate the relationship between ferromagnetism / charge-ordering and spin glasses in
manganese oxides. The phase of colossal ferromagnetism in manganites may be considered as the ferromagnetic
ordering between block spins consisting of many random spins with respective majority spin directions. Magnetization
and susceptibility are obtained at lower and higher temperature approximations of the Brillouin function. In addition,
the resistivity is obtained from the electric susceptibility. The magnetic-field dependence on the Brillouin function can
clarify of resistivity and magnetization characteristics. The optical conductivities of the colossal ferromagnetism and
the charge-ordered states are calculated. The former is also calculated in terms of pinning charge density waves as
observed in the manganites.
Keywords— Manganites, Colossal Magnetoresistance, Block Spin, Spin Glass, Ferromagnetism
I. INTRODUCTION
Manganese oxides with the generic formula of Ln1-xAxMnO3 (Ln=La, Pr, Nd; A= Ca, Sr, Ba, Pb) have attracted much
attention because colossal magnetoresistance (CMR) [1-6] has been observed in these materials. Crystalline manganese
oxides form perovskite structure in which the Mn-O layer plays an important role. In nearly all manganese oxides with a
composition rate of x>0.5, paramagnetic or ferromagnetic (FM) phases occur above the antiferromagnetic (AF)
temperature. Half-doped manganites, with x=1/2, are particular. These systems form FM zigzag chains that show coupled
AF at low temperatures magnetically, in what is known as the magnetic CE phase [7]. Moreover, the ground state is an
orbital-ordered and charge-ordered insulator. This behaviour is generic and is observed experimentally in Nd1/2Sr1/2MnO3
[8,9], Pr1/2Sr1/2MnO3 [10], Pr1/2Ca1/2MnO3 [11], La1/2Ca1/2MnO3 [12,13], Nd1/2Ca1/2MnO3 [14], and in the half-doped
layered manganite La1/2Sr3/2MnO4 [15]. The insulating charge-ordered state can be transformed into a metallic FM state
with the application of an external magnetic field, a transition that is accompanied by a change in the resistivity of
several orders of magnitude [8,16]. What is unique in the compound Nd1/2Sr1/2MnO3 [8,9] is that another distinct phase
transition occurs at TCO=158 K, where TCO represents the critical temperature for the charge-ordered transition. When
this phase transition occurs at TCO, the resistivity jumps by more than two orders of magnitude from the typically metallic
value and the ferromagnetic magnetization disappears, which indicates a simultaneous ferromagnetic-toantiferromagnetic transition. It is known that the band of Mn3+ is split into eg and t2g by the exchange interaction and the
crystal field. When an electron and a hole coexist in the eg- band, the band is split into the conduction band of a hole and
the localized band of an electron by Jahn-Teller distortion [17,18]. Charge stripe patterns regarded as charge density
waves (CDWs), in earlier work, were also observed [1]. The double-exchange model suggested by Zener [1] represents
the mainstream theoretical explanation thus far.
In this study, we use renormalization group theory [19] in relation to the finite sized block-spin concept, in which each
block spin is sized so as to have a finite nonzero total spin. We can calculate various quantities of the resultant system
relative to these block spins using Curie’s law [20].
Since the use of the Curie-Weiss model is inevitable for deducing our results, we must now refer to our previous work
[21,22].
II. THE PHENOMENOLOGY OF MANGANITES AS SPIN GLASS
We treat colossal magnetoresistive manganites as spin glasses. We postulate that spin glasses are composed of spin
clusters spin clusters are regarded as finite-sized block spins [23-27]. If magnetic field H is applied along the z-direction,
the Hamiltonian for a spin glass consisting of random block spins is given by [20],
 
H
g  B H  Si

i 1
 g B H
S
iz
,
i 1
(1)

where g is the Landé g-factor for a block spin,  B is the Bohr magneton, and Si is the spin operator for a block
spin.
The magnetization is then given by
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International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
Volume 2 Issue 1 (January 2015)
www.ijirae.com
 M z   g  B
 S
iz

i
  NB g B  Sz 
S

 M z   N B g  B {
S z exp[ g  B H S z / (k BT )]} / Z
S z  S
S

Z
exp[ g  B H S z / (k BT )]
(2)
S z  S
1
S   ( )N ,
2
where k B is the Boltzmann’s constant, N is the number of random spins in a block spin, N B is the number of block
spins, and   0 represents an infinitesimal value.
The resultant magnetization and the freezing temperature T f are given as
 M z  N B g  B SBS [ g B HS / (k BT )]
2S  1
2S  1
1
1
coth(
x) 
coth(
x)
2S
2S
2S
2S
S 1
BS ( x  0) 
x
3S
BS [ x ] 
(3)
x
BS ( x  1)  1 
1 S
e
S
S 1
1 
g  B HS / (k BT f )  1  e
3S
S
 M z |T   0,
g  B HS /( k BT f )
S
1 1 
 a0   e
S S
g  B HS /( k B T f )
S
where the Brillouin function is approximated at two asymptotic limits and a0 is a constant for compensation.
The real and imaginary susceptibilities are then calculated via

 Re( M z ) 
H

 '' 
 Im(M z ) 
H
   ' i  ''
'
(4)
Im  M z  N B g  B SBS [ g  B HS / (k BT )]
1
1
2S  1
2S  1
B S [ x ] 
tanh(
x) 
tanh(
x),
2S
2S
2S
2S
where the imaginary part is f (

  ) for the real part of f ( ) with the phase angle of  .
2
Using the Curie-Weiss theory [20], the ferromagnetic temperature between finite block spins is expressed as
TFM 
2z
S ( S  1) J ,
3k B
(5)
where the magnetic susceptibility is
 H 
c
T  TFM
(6)
2
c  N ( g  B ) S (S  1) / 3k B ,
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International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
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where z is the number of nearest block spins and J is the exchange integral between the nearest block spins.
The pseudo-magnetization is then given by
 M zE   g eL  Siz 
i
  N B g eL  S z 
S

 M zE   N B g eL{
S z exp[ g ( B H 
S z  S
S
Z

exp[ g (  B H 
S z  S
E
eEL) S z / (k BT )]} / Z
H
E
eEL) S z / (k BT )]
H
(7)
1
S   '( ) N ,
2
where the effective magnetic field becomes
H (
E

eEL   B H    ) /  B ,
H
H
(8)
and E is the electric field, L is the size of the sample,   h / 2 is the Planck’s constant divided by 2 , and  is the
external angular frequency.
The average number density of an electron in the absence of H is given by


0




d
d
f ( )d  

 F
   B H  eEL     F
0 1  exp
0 1  exp
kBT
k BTeff
 k BT ln[1  exp
(9)
F
(  B H  eEL     F )
]  k BTeff ln[1  exp
],
kBT
k BTeff
where f ( ) is the Fermi-Dirac distribution, N ( ) is the density of states, the i variables are positive constant
parameters, and  F denotes the Fermi energy.
Here the effective temperature is given by
k BTeff  k BT   H  B H   E eEL   .
(10)
The resulting pseudo-magnetization and freezing temperature T f are related according to
E
eEL)S / (k BT )]
H
2S  1
2S  1
1
1
BS [ x ] 
coth(
x) 
coth(
x)
2S
2S
2S
2S
S 1
BS ( x  0) 
x
3S
 M zE  N B g eLSBS [ g ( B H 
x
BS ( x  1)  1 
1 S
e
S
(11)
g ( B H 

S 1
1 
g ( B H  E eEL) S / (k BT f )  1  e
3S
H
S
E
eEL) S /( k BT f )
H
S
g ( B H 

1 1 
 e
S S
 a0
E
eEL ) S /( k BT f )
H
S
.
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International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
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The resultant real and imaginary electric susceptibilities are then calculated as follows

 Re(M zE ) 
E

 E '' 
 Im( M zE ) 
E
 E   E ' i  E ''
E ' 
(12)
E
eEL)S / (k B T )]
H
1
1
2S  1
2S  1
BS [ x ] 
tanh(
x) 
tanh(
x),
2S
2S
2S
2S
Im  M zE  N B g eLSB S [ g ( B H 
where the imaginary part is f (

  ) for the real part of f ( ) with a phase angle of  .
2
Let us consider the exchange integral between the nearest block spins.
Assuming that all states are possible and that the state is governed by the Fermi-Dirac distributions in a block spin, it is
given by
 M z   Ng  B  S z 

 g  B SH
 M z   Ng  B

 
 g
Sz
d (  g B S z H )
1  exp(  g  B S z H )

B SH

 g  B SH
1
d (  g B S z H )
1  exp(  g  B S z H )


  g  B SH
N 

[ln
 H

 g  B SH

 
 g

B SH
(13)
1
d ( g  B S z H )]
1  exp(  g  B S z H )
1
1
1 g B H
1
1 g B H
If S  ,  S z  tanh[
], and S block spin  S  N tanh[
],
2
2
4 k BT
2
4 k BT
and it then becomes
S
g B H
1
N tanh
2
4 k BT
Ji   2
( J i  J Dirac or J )
(14)
2
J  J Dirac
g B H
S
 J Dirac N 2 tanh 2
,
1 2
4k BT
( )
2
where  denotes each spin wavefunction and J Dirac is the Dirac type exchange integral between two nearest single
spins.
From the following Maxwell equation

  D 
 H 
 J,
t

where k is the wavevector of the magnetic field H and  is the wavenumber of the electric field E.

The displacement vector D is given by
D  ( 0 ' i 0 '') E ,
(15)
(16)
where  ' (  '' ) is the real (imaginary) dielectric constant and  0 is the electric permittivity in a vacuum.
This, then becomes
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International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
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 
 
ik  H  i ( 0 ' i 0 '') E  J
(17)



Re( J )   0 '' E   E.
1.00
Theory
data
R(H)/R(H=0)
0.99
0.98
0.97
0.96
0
1
2
3
4
5
Magnetic Field [T]
Fig.1 The magnetic field dependence on the resistivity is shown for a La2/3Sr1/3MnO3 single crystal at T=280 K [4].
<Mz>(H)/<Mz>(H=5T)
1.00
Theory
data
0.99
0.98
0.97
0.96
0.95
0
1
2
3
4
5
Magnetic Field [T]
Fig.2 The magnetic field dependence on the magnetization is shown for a La 2/3Sr1/3MnO3 single crystal at T=280 K [4].
The resistivity of spin glasses is given by
R  1/  
1 1 1
,
 0   E ''
(18)
where the lower temperature limit of T  0  x   is used and the higher temperature limit of T    x  0 is used
in Eq. (12). As shown in Fig. 1 and 2, the magnetic field dependence on the resistivity and magnetization in comparison
with the experimental data is plotted within two temperature approximations.
The resultant DC-type current in ferromagnetic states is expressed as
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International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
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Resistivity [Ohm cm]
0.1
Theory
Theory
Data
0.01
1E-3
0
2
4
6
8
10
Magnetic Field [T]
Fig. 3 The magnetic field dependence on the resistivity for charge-ordered states at 141 K for Nd0.5Sr0.5MnO3 [5].

t
t
J  J 0 tan( (eV  H  B ( H   H )  E g ))  J 0 tan( Eg )

E


t
t
 {J 0 tan( (eV  H  B H  E g ))  J 0 tan( E g )}

E

  CO E   CO
(19)
V
,
L
where  CO is the conductivity in charge-ordered states, E g is the energy gap,  H is the part from spontaneous
magnetization, the Matsubara relationship between the temperature and the imaginary time
t


i , and  is a
 kBT
constant [20].
The resistivity of the charge-ordered states is given as follows
RCO  1/  CO  1/
 1/ (

( LJ ) |V 0
V

e

1
LJ 0 ) / [{1  tanh 2 (
(
k BT  H  B ( H   H )  E g ))}
k BT
k BT  E
E
 {1  tanh 2 (
(20)


1
(
k BT  H  B H  Eg ))}].
k BT  E
E
As shown in Fig. 3, the resistivity of charge-ordered states in comparison with the experimental data is plotted within
two temperature approximations.
From Eq. (19), the magnetization of the charge-ordered states is given as
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International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
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5
Arbitrary Unit]
4
3
High Temperature Limit
Low Temperature Limit
2
1
0
0
2
4
6
8
10
Arbitrary Unit]
Fig. 4 The optical conductivity is shown using two limits.
 M z   B (n  n )   B
 B A

e
J
 Adt

 J  Adt
e


t
ln(cos( (eV  H  B ( H   H )  E g )))
H

E
(eV 
B ( H   H )  Eg )
E
 A


t
 B
ln(cos( (eV  H B H  Eg )))
H
e

E
(eV 
B H  Eg )
E
B A




ln(cosh(
(eV  H  B ( H   H )  E g )))

e
k BT
E
(eV  H  B ( H   H )  E g )
E
 A



 B
ln(cosh(
(eV  H  B H  E g ))),

e
k BT
E
(eV  H  B H  E g )
E
where the Matsubara relationship between the temperature and time is given as [20]
(21)
t

i
and A represents the

kBT
surface.
Next, we consider the optical conductivities.
The optical conductivity of spin-glass-like manganites is expressed as
 ( )   0 E ''(  B H   B H 

 ),
H
(22)
as shown in Fig. 4.
The optical conductivity for charge-ordered states is given as follows
 CO ( )
(


e

1
LJ 0 )[{1  tanh 2 (
(
k BT  H  B ( H   H )     E g ))}
k BT
k BT  E
E
E
{1  tanh 2 (
(23)



1
(
k BT     H  B H  Eg ))}].
k BT  E
E
E
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Let us investigate the optical conductivity,  ( ) in manganese oxides from the viewpoint of CDWs.
Fig. 5 The charge density wave is confined in a standard potential well where we assume a standard well amongst many different types in the presence
of pinned CDWs.
Assuming the CMR manganites are governed by charge density waves (CDWs) [1], the conductivity of pinned CDW
is calculated under the conjecture that CDWs are confined within a quantum well in the presence of pinning. Amongst
many different potential wells in a material with a CDW, we choose as the most typical one, postulating an average and
standard potential well, to accommodate a commensurate pinning of a CDW as illustrated in Fig. 5.
We can then calculate the current density in the presence of the applied voltage V to be
J  nh ev  ne ev  J '  J 0 tanh(
e(V  VT )
)  J ',
2kBT
(24)
where n0 is the total number density of the electrons and holes, ne is the electron number density, nh is the hole number
density, the average velocity of an electron is assumed to be identical that of a hole,
ne  n0
1
1
, nh  n0
, T is the temperature, VT is the height of the potential well,
e(V  VT )
e(V  VT )
1  exp(
)
1  exp(
)
k BT
k BT
J 0 is a normalization constant, J ' is the current density from other mechanism, and J (V  0)  0 .
Transport in normal states is explained in the form of a quantum well originating from the pinning of the CDW as
shown in Fig. 5.
The current density is rewritten as
J  J 0 {tanh(
eV  eVT
eV
eV  eVT
eV
)  tanh( T )}  n0 evF {tanh(
)  tanh( T )}
2k BT
2k BT
2k BT
2k BT
E

V
,
L
(25)
where vF is the Fermi velocity,  is the conductivity, and the effective voltage is changed into
(eV 

1
k BT    ) / e as shown in Fig. 5.
E
E
The optical conductivity in paramagnetic states is given by
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2
 ( ) 
n0 e vF L
[tanh(

1
(eV 
k B T    )
E
E
eV 

1
k BT     eVT
E
E
eV
)  tanh( T )]   0 ,
2kBT
2k BT
(26)
as shown in Fig. 6 and  0 is a constant from different mechanisms.
180
Experiment (T=290 K)
Theory
160
120
-1
[ cm ]
140
-1
100
80
60
40
20
0
0.0
0.5
1.0
1.5
2.0
heV]
Fig. 6 The optical conductivity in normal states is plotted where the given parameters are the effective temperature, k BT eff  1.2 eV at

1 ,
E
eVT  2.25 eV and the experimental data [6].
450
400
Experiment (T=9 K)
Theory
300
-1
 cm ]
350

250
200
150
100
50
0
0.0
0.5
1.0
1.5
2.0
h[eV]
Fig. 7 The optical conductivity in ferromagnetic states is plotted where the given parameters are the effective temperature, k BT eff  1.2 eV at

1 ,
E
eVT  2.25 eV ,  B H  5.1 eV, and the experimental data [6].
The optical conductivity in ferromagnetic phases is given by
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 ( ) 
n0 e2 vF L

1
(eV 
k B T    )
E
E
eV 
{[tanh(
eV 
 [tanh(


1
k BT     H  B ( H   H )  eVT
E
E
E
eV
)  tanh( T )]
2k BT
2k B T
(27)


1
k BT     H  B H  eVT
E
E
E
eV
)  tanh( T )]}
2k B T
2k BT
  0 ',
where  H is from spontaneous magnetization, this conductivity is plotted in Fig. 7, and  0 ' is a constant contributed by
different mechanisms.
III. CONCLUSIONS
In conclusion, the renormalization group theory [19] can be extended to a finite-sized block spin system to elucidate
the phenomenology of manganites as spin glasses. Colossal ferromagnetism in manganites can be explained by Diractype exchange integral between block spins. The magnetic field dependence on the resistivity and magnetization
originates from the variation of the Brillouin functions. The optical conductivity of the CMR and charge-ordered states
are calculated. The transport of CMR manganites can be also explained by pinned CDWs, that is, moving CDWs
confined in a quantum well amongst many different types whose potential barrier is induced by impurity pinning or
commensurate pinning. As shown in Fig. 4-7, the optical conductivity levels in CMR manganites can be determined not
only in terms of finite block spin phenomenology but also by CDW pinning.
ACKNOWLEDGMENT
The present Research has been conducted by the Research Grant of Kwangwoon University in
2014.
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