Modorresi, M.

Pahlevane, M.

Roknabadi, M.R.

SocioEconomic Challenges Journal (Sumy State University)

Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   37 
 
A STUDY of INELASTIC ELECTRON-PHONON INTER-
ACTION ON TUNNELING MAGNETORESISTANCE IN 
POLYACETYLENE  
M. R. Roknabadi 1*, M. Pahlevane 2, M. Modarresi 1  
 
1 Department of Physics, Ferdowsi University of Mashhad, Mashhad, Iran. 
2 Department of Physics, faculty of Science University of Mazandaran ,P. O. Box 47415-416, 
Babolsar Iran. 
 
ABSTRACT 
In this research, we study the effect of inelastic electron-phonon interactions on 
magneto-transport properties of a sandwiched Polyacetylene molecule between two 
ferromagnetic electrodes by using a modified Green's function method. The molecule is 
described with the well-known Su–Schrieffer–Heeger Hamiltonian in the tight binding 
form. The self-energy of electrodes is written in the wide-band approximation and spin-
flip is neglected during conduction process. Our calculation results show that with 
increasing of voltage current increases and tunneling magnetoresistance (TMR) is 
decreased. The changes in TMR due to the inelastic interaction is limited in a small bias 
voltage interval and can be neglected in the other bias voltages. These results confirm 
the neglecting of inelastic interactions in previous theoretical works. 
 
Keywords: tunneling magnetoresistance; Green's function; Polyacetylene; 
inelastic electron-phonon interaction. 
 
INTRODUCTION 
Due to its application in electronic devices, transport trough molecule has 
been studied in detail, both theoretically and experimentally. Theoretical and 
experimental works confirm that conducting polymers can act as spin-polarized 
materials [1-4]. We have use the SSH Hamiltonian which a well-known 
Hamiltonian for the calculation of electronic and transport properties of organic 
semiconductors [1,4,5]. In the presence of magnetic field, the magnetic of 
electrodes are oriented in the same direction and the scattering for spin up 
decreases while increases for spin down. By considering a parallel circuit in 
conduction with different spins, the presence of magnetic field can make a short 
circuit and as a result conduction of system decreases. In this work, we study 
spin dependent transport in a Polyacetylene molecule in the presence of 
inelastic electron-phonon interactions. We sandwiched a Polyacetylene 
molecule between two cobalt electrodes as we have schematically shown in 
Figure 1. Because of the weakness in spin-orbit coupling and hyperfine 
                                               
* Cordresponding author: roknabad@um.ac.ir 
38                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
interactions in the organic materials, spin relaxation length is long and spin-flip 
process during conduction can be neglected [2, 6]. The presence of phonon in 
the finite temperature changes the transmission mechanisms which is neglected 
in many previous works. Here, we study the effect of inelastic interactions in 
the central region and neglect the interaction in leads. Walczak developed a 
Green’s function approach to study the electron transport through DNA 
molecule in the presence of electron-phonon interactions [7, 8]. We had used 
the same formalism to study the inelastic electron phonon interaction in organic 
molecules. In this work we will use the same approach for the Polyacetylene 
molecule.  
MODEL AND METHODS 
The electronic part of Hamiltonian can be written as, 
 
 
(1) 
where HL/R denotes Hamiltonian of left/right electrode and HC represents 
molecular Hamiltonian. Other matrix elements show the coupling between 
electrodes and scattering region. Both electrodes are described in the wide-band 
approximation [7,8]. 
 
 
Fig. 1 –Schematic representation of Polyacetylene molecule sandwiched between two 
electrodes 
 
The Polyacetylene  molecule which is considered as a quasi-one-
dimensional chain with N pair of carbon atoms can be described by means of 
modified SSH Hamiltonian as follows [1,9],  
 
                                                                                                            (2)     
 
where 0t  is the hopping integral in a undimerized molecule and and ui is 
the displacement of the ith Carbon atom. The index i runs over the orbitals of 
Carbon chain of molecule. The operator Ci (Ci+) creates/annihilates a ? electron 
with spin ? at the i-th site of Polyacetylene molecule.   
2
0 1 , 1, 1, ,
1
[( ( ))( )]
N
M n n n n n n
n
H t u u d d d d? ? ? ?
?
? ? ?? ? ?
?
? ? ? ? ???
,
, ,
,
0
0
L L C
L C C R C
R C R
H H
H H H H
H H
? ?
? ?
? ?? ? ?
? ?? ?
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   39 
 
The electron-phonon coupling constant is denoted by ?. The electronic 
states in the Polyacetylene chain accompanied with m phonons can be written 
as [7, 8], 
                                                             (3) 
 
 
ai and ai+ are phonon creation and annihilation operators, respectively  and 
| 0?  shows the vacuum state. For the sake of simplicity, we consider only one 
phonon mode with energy w. In this new basis the Hamiltonian of the 
Polyacetylene can be written as, 
 
 (4) 
 
where ? is the energy of electron-phonon interaction, in this work we 
assume that |?|=|?|. In the absence of inelastic electron-phonon interactions 
during conduction process, electron states will be changed from |1, m?  to 
| 4 ,N m? . But in the presence of inelastic interactions, electrons can absorb/emit 
phonon and conduct between two levels with different number of phonons, for 
such a process we can write transmission coefficient as [7],  
  
                                                                                                             (5)  
 
where                                              (?? is the coupling strength between 
molecule and electrodes, ?? is the density of state in the electrodes for spin ?) 
and the Green's function is defined as follows [1, 7, 8], 
 
                                                                              (6) 
 
All the off-diagonal terms in the Hamiltonian which are described the 
interaction between molecule and leads are compressed in the self-energy 
terms. The real part of self-energy is simply a shift in energy axis but the 
imaginary part describes the coupling of molecule with electrodes. In the wide-
band approximation the real part of self-energy is neglected and the imaginary 
part is assumed an energy independent term which is written as 
/ /
i  
2L R L R
?? ?? ? .  The  total  transmission  can  be  defined  as  a  sum  over  all  
incoming/outgoing channels with different number of phonon in the initial and 
final sites. Each transmission mechanism is weighted by an appropriate factor 
Pm, which is the Boltzmann distribution function [7, 8]. 
 
( )| , | 0
!
m
n
an m d
m?
?
?
?? ? ?
' ' '
' '
, , 2
, 1, 1
( ) | |L Rm m m mT G
? ? ? ? ? ?? ? ?? ? ?
, 1( ) [ ]M L RG E H
? ? ? ??? ? ?? ? ?? ??
'
'
'
,
,
, ,
( ) ( )tot m m m
m m
T T? ?
?
? ?? ??
2
/ / /2 | |L R L R L R
? ?? ? ?? ?
, , 0 1 , 1, 1, ,
, ,
[ ( )] [( ( ))( )]M n j j n n n n n n n n
n j
H a a d d t u u d d d d? ? ? ? ? ? ?
?
? ? ?? ? ? ?? ? ?? ? ? ? ? ? ??
40                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
                               (7)  
From the above equation the elastic part of transmission occurs when m 
and m' are equal and can be written     as [7, 8], 
 
 (8) 
 
The total current through junction is derived as follows,     
 
                                                                                                   (9) 
 
 
The elastic part of current can be obtained from the above equation by the 
assumption of m=m' [7,8], 
 
 (10) 
                      
where /
m
L Rf is the Fermi function of the left/right electrode in applied 
voltage V, with chemical potential              ?L/R=?f ± (eV/2) and Fermi energy ?f .  
We calculate transmission coefficient in the presence T??? and absence T?,-
? of magnetic field. Total current and elastic part of current can be calculated 
from Equations (9) and (10). Finally, tunneling magnetoresistance can be 
defined as the change in the current due to the external magnetic field by using 
the following equation [11], 
 
 (11) 
 
where Ip (IAp) is the current in the presence/absence of magnetic field. 
Calculations are performed for the total current and elastic part of current and 
then we have compared the TMR in these two situations. 
RESULTS AND DISCUSSION 
Numerical calculations are carried out for a molecule with 15 carbon pairs 
(30 sites) and the parameters for a Polyacetylene  chain are chosen as (given in 
eV): t0=2.5, ?= 4.1 1/Å and u=0.04 A?. These parameters well describe a 
Polyacetylene chain. We have used the following set of parameters for 
describing Cobalt electrodes (given in 1/eV), ??=0.1367, ??= 0.5772, and we set 
?L=?R=0.2 eV, ?f =-1 eV and w=0.1 eV. By increasing the number of phonons in 
each site the Boltzmann factor tends to zero as a result considering a finite 
number of phonons in each site is a reasonable assumption. Here, Maximum 
number of allowed phonon quanta is chosen to be five and all of the 
calculations are done at room temperature. We also assume that the external 
magnetic field does not change the electronic structure of molecule and the 
',
,
,
( ) ( )ele m m m
m
T T? ?
?
? ?? ??
' ' '
' '
'
,
,
, ,
1 [ (1 ) (1 )]
2
m m m m
tot m L R R Lm m m
m m
I d T f f f f? ?
?
??
??
??
? ? ? ?? ???
',
,
,
1 [ ]
2
m m
ele m m m L R
m
I d T f f? ?
?
??
??
??
? ? ???
( ) ( )
( )
( )
p AP
P
I V I V
TMR V
I V
??
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   41 
 
changes are restricted in the electrodes region. In the presence of magnetic 
field, density of states in electrodes and transmission coefficient through the 
molecule are different for each spin. The presence of inelastic interactions 
emission and absorption of phonons activate new transmission mechanism. 
These new mechanisms are spin independent phenomena and change the 
magneto transport trough the molecule. 
Figure 2 shows the current-voltage characteristic of Polyacetylene 
molecule in the presence of magnetic field. According to current–voltage 
characteristic of Polyacetylene it can be suggested as a switch in electronic 
circuit.    
 
Fig. 2 –The total (T) and elastic (E) part of current as a function of bias voltage for 
electrons with spin up (U) and spin down (D) in the presence of magnetic field 
 
Figure 3(a) presents TMR versus bias voltage in the presence and absence 
of inelastic electron-phonon interactions. The difference between these them is 
plotted in Fig. 3(b). 
 
 
42                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I            
 
Fig. 3 – (a) Elastic part (solid line) and total (dashed line) of TMR (b) difference 
between these situations 
As we expected, due to spin independent nature inelastic electron phonon 
interaction TMR decreases. In the low voltage regime the tunneling probability 
is near zero, current is small and TMR is constant. By increasing voltage 
energy-window opens and resonant tunneling occurs and as a result, current 
increase while TMR decreases. The polaron formation due to the electron 
phonon interaction shifts the current peak and reduces the conductance gap 
[12]. As a result the damping of TMR is occurred in a lower voltage. At higher 
voltages, TMR does not change with the increase of voltage. By increasing the 
applied voltage the energy of electrons is increases and the changes made by 
inelastic interactions are disappeared. A comparison between these results and 
our previous results for Polythiophene [13] confirms the neglecting of inelastic 
electron phonon interaction in the transport calculation of organic molecule, 
even in room temperature.   
CONCLUSIONS 
In summary, we have investigated the effect of inelastic electron-phonon 
interactions  on  the  magneto  transport  in  a  Polyacetylene  molecule.  Our  
calculations indicate that the change of TMR due to inelastic interaction is 
limited in a small bias voltage interval and can be neglected in other bias 
voltages. These results confirm the neglecting of the inelastic interactions in 
previous theoretical works. 
REFERENCES 
[1]  J.  H.  Wei,  X.  J.  Liu,  S.  J.  Xie,  YiJing  Yan,  THE  JOURNAL  OF  CHEMICAL  
PHYSICS 131 (2009) 064906. 
[2] V. Dediu, M. Murgia, F.C. Matacotta, C. Taliani, S. Barbanera, Solid State Commu. 
122 (2002) 181. 
[3] Z. H. Xiong, D. Wu, Z.V. Vardeny, J. Shi, Nature (London) 427 (2004) 821. 
[4] G. Hu, K. He, S. Xie, A. Saxena, THE JOURNAL OF CHEMICAL PHYSICS 129 
(2008) 234708. 
[5] J.  H. Wei,  S.  J.  Xie,  L.  M. Mei,  J.  Berakdar ,  Y.  Yan, New Journal  of Physics 8 
(2006) 82. 
[6] S. Pramanik, C-G Stefanita, S. Bandyopadhyay, N. Harth, K. Garre and M. Cahay, 
Nat. Nanotechnol. 2 (2007) 216. 
[7] K. Walczak, Journal of Mgnetism and Magnetic Materials 305 (2006) 475-482. 
[8] K. Walczak, Physica E 33 (2006) 110–115. 
[9] W. P. Su, J. R. Schrieffer and A. J. Hegeer, Phys. Rev. B 22 (1980) 2099. 
[10] J. Y. Fu, D. S. Liu, S. J. Xie, Physica E 40 (2008) 915–919. 
[11] Bin Wang, Yu Zhu, Wei Ren, Jian Wang and Hong Guo, Phys. Rev. B 75 (2007) 
235415. 
[12] K. Walczak, Central European Journal of Physics 4 (2006) 241-253. 
[13] M. Modarresi, M.R. Roknabadi, N. Shahtahmasbi, D. Vahedi Fakhrabad and H. 
Arabshahi, Physica B 13 (2011) 478-481 


English

A study of inelastic electron-phonon interaction on tunneling magnetoresistance in polyacetylene

Electronic Sumy State University Institutional Repository

Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   37  A STUDY of INELASTIC ELECTRON-PHONON INTER-ACTION ON TUNNELING MAGNETORESISTANCE IN POLYACETYLENE  M. R. Roknabadi 1*, M. Pahlevane 2, M. Modarresi 1   1 Department of Physics, Ferdowsi University of Mashhad, Mashhad, Iran. 2 Department of Physics, faculty of Science University of Mazandaran ,P. O. Box 47415-416, Babolsar Iran.  ABSTRACT In this research, we study the effect of inelastic electron-phonon interactions on magneto-transport properties of a sandwiched Polyacetylene molecule between two ferromagnetic electrodes by using a modified Green's function method. The molecule is described with the well-known Su–Schrieffer–Heeger Hamiltonian in the tight binding form. The self-energy of electrodes is written in the wide-band approximation and spin-flip is neglected during conduction process. Our calculation results show that with increasing of voltage current increases and tunneling magnetoresistance (TMR) is decreased. The changes in TMR due to the inelastic interaction is limited in a small bias voltage interval and can be neglected in the other bias voltages. These results confirm the neglecting of inelastic interactions in previous theoretical works.  Keywords: tunneling magnetoresistance; Green's function; Polyacetylene; inelastic electron-phonon interaction.  INTRODUCTION Due to its application in electronic devices, transport trough molecule has been studied in detail, both theoretically and experimentally. Theoretical and experimental works confirm that conducting polymers can act as spin-polarized materials [1-4]. We have use the SSH Hamiltonian which a well-known Hamiltonian for the calculation of electronic and transport properties of organic semiconductors [1,4,5]. In the presence of magnetic field, the magnetic of electrodes are oriented in the same direction and the scattering for spin up decreases while increases for spin down. By considering a parallel circuit in conduction with different spins, the presence of magnetic field can make a short circuit and as a result conduction of system decreases. In this work, we study spin dependent transport in a Polyacetylene molecule in the presence of inelastic electron-phonon interactions. We sandwiched a Polyacetylene molecule between two cobalt electrodes as we have schematically shown in Figure 1. Because of the weakness in spin-orbit coupling and hyperfine                                                * Cordresponding author: roknabad@um.ac.ir 38                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I             interactions in the organic materials, spin relaxation length is long and spin-flip process during conduction can be neglected [2, 6]. The presence of phonon in the finite temperature changes the transmission mechanisms which is neglected in many previous works. Here, we study the effect of inelastic interactions in the central region and neglect the interaction in leads. Walczak developed a Green’s function approach to study the electron transport through DNA molecule in the presence of electron-phonon interactions [7, 8]. We had used the same formalism to study the inelastic electron phonon interaction in organic molecules. In this work we will use the same approach for the Polyacetylene molecule.  MODEL AND METHODS The electronic part of Hamiltonian can be written as,   (1) where HL/R denotes Hamiltonian of left/right electrode and HC represents molecular Hamiltonian. Other matrix elements show the coupling between electrodes and scattering region. Both electrodes are described in the wide-band approximation [7,8].   Fig. 1 –Schematic representation of Polyacetylene molecule sandwiched between two electrodes  The Polyacetylene  molecule which is considered as a quasi-one-dimensional chain with N pair of carbon atoms can be described by means of modified SSH Hamiltonian as follows [1,9],                                                                                                               (2)      where 0t  is the hopping integral in a undimerized molecule and and ui is the displacement of the ith Carbon atom. The index i runs over the orbitals of Carbon chain of molecule. The operator Ci (Ci+) creates/annihilates a ʌ electron with spin ı at the i-th site of Polyacetylene molecule.   20 1 , 1, 1, ,1[( ( ))( )]NM n n n n n nnH t u u d d d dV V V VVD         ¦¦,, ,,00L L CL C C R CR C RH HH H H HH H § ·¨ ¸ ¨ ¸¨ ¸© ¹Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   39  The electron-phonon coupling constant is denoted by Į. The electronic states in the Polyacetylene chain accompanied with m phonons can be written as [7, 8],                                                              (3)   ai and ai+ are phonon creation and annihilation operators, respectively  and | 0²  shows the vacuum state. For the sake of simplicity, we consider only one phonon mode with energy w. In this new basis the Hamiltonian of the Polyacetylene can be written as,   (4)  where Ȝ is the energy of electron-phonon interaction, in this work we assume that |Ȝ|=|Į|. In the absence of inelastic electron-phonon interactions during conduction process, electron states will be changed from |1, m²  to | 4 ,N m² . But in the presence of inelastic interactions, electrons can absorb/emit phonon and conduct between two levels with different number of phonons, for such a process we can write transmission coefficient as [7],                                                                                                                 (5)   where                                              (ȖĮ is the coupling strength between molecule and electrodes, ȡı is the density of state in the electrodes for spin ı) and the Green's function is defined as follows [1, 7, 8],                                                                                (6)  All the off-diagonal terms in the Hamiltonian which are described the interaction between molecule and leads are compressed in the self-energy terms. The real part of self-energy is simply a shift in energy axis but the imaginary part describes the coupling of molecule with electrodes. In the wide-band approximation the real part of self-energy is neglected and the imaginary part is assumed an energy independent term which is written as / /i  2L R L RȽV V6  .  The  total  transmission  can  be  defined  as  a  sum  over  all  incoming/outgoing channels with different number of phonon in the initial and final sites. Each transmission mechanism is weighted by an appropriate factor Pm, which is the Boltzmann distribution function [7, 8].  ( )| , | 0!mnan m dmVV²  ²' ' '' ', , 2, 1, 1( ) | |L Rm m m mT GV V V V V VH  * *, 1( ) [ ]M L RG E HV V V VHc c  ¦ ¦''',,, ,( ) ( )tot m m mm mT TV VVH H 5¦2/ / /2 | |L R L R L RV VS J U*  , , 0 1 , 1, 1, ,, ,[ ( )] [( ( ))( )]M n j j n n n n n n n nn jH a a d d t u u d d d dV V V V V V VVH O D           ¦40                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                                            (7)  From the above equation the elastic part of transmission occurs when m and m' are equal and can be written     as [7, 8],   (8)  The total current through junction is derived as follows,                                                                                                         (9)   The elastic part of current can be obtained from the above equation by the assumption of m=m' [7,8],   (10)                       where /mL Rf is the Fermi function of the left/right electrode in applied voltage V, with chemical potential              ȝL/R=İf ± (eV/2) and Fermi energy İf .  We calculate transmission coefficient in the presence Tıı and absence Tı,-ı of magnetic field. Total current and elastic part of current can be calculated from Equations (9) and (10). Finally, tunneling magnetoresistance can be defined as the change in the current due to the external magnetic field by using the following equation [11],   (11)  where Ip (IAp) is the current in the presence/absence of magnetic field. Calculations are performed for the total current and elastic part of current and then we have compared the TMR in these two situations. RESULTS AND DISCUSSION Numerical calculations are carried out for a molecule with 15 carbon pairs (30 sites) and the parameters for a Polyacetylene  chain are chosen as (given in eV): t0=2.5, Į= 4.1 1/Å and u=0.04 AÛ. These parameters well describe a Polyacetylene chain. We have used the following set of parameters for describing Cobalt electrodes (given in 1/eV), ȡĹ=0.1367, ȡĻ= 0.5772, and we set ȖL=ȖR=0.2 eV, İf =-1 eV and w=0.1 eV. By increasing the number of phonons in each site the Boltzmann factor tends to zero as a result considering a finite number of phonons in each site is a reasonable assumption. Here, Maximum number of allowed phonon quanta is chosen to be five and all of the calculations are done at room temperature. We also assume that the external magnetic field does not change the electronic structure of molecule and the ',,,( ) ( )ele m m mmT TV VVH H 5¦' ' '' '',,, ,1 [ (1 ) (1 )]2m m m mtot m L R R Lm m mm mI d T f f f fV VVHSff 5  5 ¦³',,,1 [ ]2m mele m m m L RmI d T f fV VVHSff 5 ¦³( ) ( )( )( )p APPI V I VTMR VI V Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I                   41  changes are restricted in the electrodes region. In the presence of magnetic field, density of states in electrodes and transmission coefficient through the molecule are different for each spin. The presence of inelastic interactions emission and absorption of phonons activate new transmission mechanism. These new mechanisms are spin independent phenomena and change the magneto transport trough the molecule. Figure 2 shows the current-voltage characteristic of Polyacetylene molecule in the presence of magnetic field. According to current–voltage characteristic of Polyacetylene it can be suggested as a switch in electronic circuit.     Fig. 2 –The total (T) and elastic (E) part of current as a function of bias voltage for electrons with spin up (U) and spin down (D) in the presence of magnetic field  Figure 3(a) presents TMR versus bias voltage in the presence and absence of inelastic electron-phonon interactions. The difference between these them is plotted in Fig. 3(b).   42                   Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part I             Fig. 3 – (a) Elastic part (solid line) and total (dashed line) of TMR (b) difference between these situations As we expected, due to spin independent nature inelastic electron phonon interaction TMR decreases. In the low voltage regime the tunneling probability is near zero, current is small and TMR is constant. By increasing voltage energy-window opens and resonant tunneling occurs and as a result, current increase while TMR decreases. The polaron formation due to the electron phonon interaction shifts the current peak and reduces the conductance gap [12]. As a result the damping of TMR is occurred in a lower voltage. At higher voltages, TMR does not change with the increase of voltage. By increasing the applied voltage the energy of electrons is increases and the changes made by inelastic interactions are disappeared. A comparison between these results and our previous results for Polythiophene [13] confirms the neglecting of inelastic electron phonon interaction in the transport calculation of organic molecule, even in room temperature.   CONCLUSIONS In summary, we have investigated the effect of inelastic electron-phonon interactions  on  the  magneto  transport  in  a  Polyacetylene  molecule.  Our  calculations indicate that the change of TMR due to inelastic interaction is limited in a small bias voltage interval and can be neglected in other bias voltages. These results confirm the neglecting of the inelastic interactions in previous theoretical works. REFERENCES [1]  J.  H.  Wei,  X.  J.  Liu,  S.  J.  Xie,  YiJing  Yan,  THE  JOURNAL  OF  CHEMICAL  PHYSICS 131 (2009) 064906. [2] V. Dediu, M. Murgia, F.C. Matacotta, C. Taliani, S. Barbanera, Solid State Commu. 122 (2002) 181. [3] Z. H. Xiong, D. Wu, Z.V. Vardeny, J. Shi, Nature (London) 427 (2004) 821. [4] G. Hu, K. He, S. Xie, A. Saxena, THE JOURNAL OF CHEMICAL PHYSICS 129 (2008) 234708. [5] J.  H. Wei,  S.  J.  Xie,  L.  M. Mei,  J.  Berakdar ,  Y.  Yan, New Journal  of Physics 8 (2006) 82. [6] S. Pramanik, C-G Stefanita, S. Bandyopadhyay, N. Harth, K. Garre and M. Cahay, Nat. Nanotechnol. 2 (2007) 216. [7] K. Walczak, Journal of Mgnetism and Magnetic Materials 305 (2006) 475-482. [8] K. Walczak, Physica E 33 (2006) 110–115. [9] W. P. Su, J. R. Schrieffer and A. J. Hegeer, Phys. Rev. B 22 (1980) 2099. [10] J. Y. Fu, D. S. Liu, S. J. Xie, Physica E 40 (2008) 915–919. [11] Bin Wang, Yu Zhu, Wei Ren, Jian Wang and Hong Guo, Phys. Rev. B 75 (2007) 235415. [12] K. Walczak, Central European Journal of Physics 4 (2006) 241-253. [13] M. Modarresi, M.R. Roknabadi, N. Shahtahmasbi, D. Vahedi Fakhrabad and H. Arabshahi, Physica B 13 (2011) 478-481 

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A study of inelastic electron-phonon interaction on tunneling magnetoresistance in polyacetylene

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