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The dielectric-metal transition mechanism for the thread-like structure polymers
SYnTI,IE==TIIE: I IITALS ELSEVIER
Synthetic Metals 68 (1995) 99-102
The dielectric-metal transition mechanism for the thread-like structure polymers O.A. Ponomarev, E.S. Shikhovtseva Department of Physics, Ufa Scientific Center of the Russian Academy of Sciences, Ufa, Russian Federation Received 8 April 1993; revised 22 July 1994
Abstract
The paper presents a model of the polymer film dielectric-metal transition under a small external influence. The conductive pulse propagation along the polymer thread has a soliton nature. The polymer film current-voltage characteristic is obtained on the basis of the described model. Keywords: Film; Conductivity; Dielectric-metal transition
1. I n t r o d u c t i o n
l The existence of anomalously high spontaneous conductivity discovered in some polymers with thread-like structure (for example, polyarylenephthalides) was reported recently [1-5]. The polyarylenephthalide experimental results [5-8] show the switch effect: from the normal state of 10-14 (~'~ cm)-1 conductivity to the conductive state of 109 (12 cm) -1. This conductivity jump is obtained in strong electric fields (more than 105 V/cm) and thin films (less than 3 #m) when the external influence is small enough (the transition threshold pressure is less than 102 Pa). A dielectric-metal transition model based on the chemical structure of thread-like polymers is given below. 2. The m o d e l
Obviously, the observed huge conductivity jump of polymer films under the influence of negligible external pressure is not caused by a change of the volume structure of the film. The reason for this phenomenon is the chemical structure of the polymer chain. The bridges between the side chains are known to be able to go from the dielectric state into the metal one and back under an external influence or under the influence of neighbouring bonds of the polymer spine. The central carbon atom in a polyarylenephthalide fragment has an sp 3 hybridization in a dielectric state of the bridge (Fig. l(a)) and the benzene ring plane
0379-6779/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 3 7 9 - 6 7 7 9 ( 9 4 ) 0 2 2 8 3 - 5
"~'C
(a)
FI Fig. 1. Polyarylene m o n o m e r in dielectric (a) and conductive (b) state.
of the side chain is perpendicular to the spine (let us choose an angle 4)= 0 ° for this state). In the conductive bridge (Fig. l(b)) the central carbon atom has an sp 2 hybridization and the benzene ring plane becomes parallel to the spine (q5=90°). The bridge-bridge interaction energy depends on 4). Let us consider the mechanism of such a dielectric-metal transition for a polymer based on butadiene. The nth fragment structure of the dielectric state polymer (4)= 0 °) is shown in Fig. 2(a). The destruction of the C 1 = O bond and the creation of the C]=C2 bond are the beginning of the transition (Fig. 2(b)). Then the transformations of double bonds to single ones and vice versa in the spine from C1 to the next central
100
0,4. Ponornarev, E.S. Shikhovtseva / Synthetic Metals 68 (1995) 99-102
Thus, we have presented the mechanism of the persistent conductive state propagation by the transformation of central carbon atoms of sp 3 hybridization to those of sp 2 hybridization. This mechanism is based on the soliton transport of the side-chain benzene-ring rotation along the spine. In the created conductive state the polymer becomes polyaeetylene-like and its conductivity may be described by the soliton mechanism similar to the SSH model [9]. But in our paper we do not deal with this problem. The Hamiltonian in the representation of second quantization has the form:
"\C"CH =CH --CH~--------CH-HC/ t., HC (a) \
\\C ~CH--CH = C H ~ /
t
I He,,. ¢ C H HC
, ~ = ~C.na+,~ai,,o.+ ~ J
c o s ( A q S , / 2 ) a+oain+go _
+ EJiia i+~.ay,¢+ ( 1 / 2 ) ~ V0,ma,+~a,,oaf~ ajm~
(b)
i ~*j
+ (1/2).~,/32//+ (/3/2)£[(+, + 1 - ~b,)z
CH==CH---
+ ( & _ 1 - (~.)21 + (U/2)E(1 - cos(24,.)) HH~cJ ~ H (c)
n-~l
[ \,C~ CI...I==CI| // ~
CH==CH---
2
I (d) r
\
"\. ~CI-I==C~H ==CI-I~
H?
(1)
Here a~:(r is a creation operator for an electron of E~ energy on the ith thread and nth fragment, J cos(A~b~/2) is an exchange integral between atoms situated in nth and (n + 1)th knots, ¢~ is a side-chain rotation operator J/j is an exchange integral between the polymer threads, Vijnm is Coulomb interaction of the electrons, /~ is a side-chain angular-momentum operator, I is the moment of inertia of the side chain with regards to rotation of angle ~b,, /3 is an indirect exchange interaction parameter for neighbouring side chains, U is the anharmonicity parameter, and A=4 for the polymers under consideration.
3. The side-chain dynamics
?
{Hea\ CH [(e)
n+'l
Fig. 2. Dielectric-metal evolution of the polymer based on butadiene.
carbon atom and in the side chain from O to C1 occur. Such pulse propagation along the spine causes destruction of the C1-O bond in the next fragment (Fig. 2(c)) and the pulse propagation along the side chain causes the inverse transition CI=Cz to C1-C2 in the nth monomer. Thus, we have a conductive state (Fig. 2(d)) in the first fragment (~b=90 °) and conductive pulse movement (Fig. 2(e)) along the spine. In other words, the transformation of the dielectric fragments
The side-chain dynamics is determined by angle Sn along the polymer spine. The side-chain energy E obtained from Eq. (1) yields the system of equations for the qSn coordinates: d~b,, dt
aE 0P,
P,, I (2)
dPn dt
~E a$~
/3(2~b,- ~. +a - ~.-1) - U sin(AqS.)
Here, in the classic limit, Sn and Pn are the side-chain rotation angle and side-chain angular momentum, respectively. Then in the continuum approximation one can write:
/
with a central carbon atom - ~ to the conductive I ones with a carbon atom /\ C - -- propagates along the polymer thread. A similar process takes place in polyarylenephthalide films.
I O2~(x' t) _ ~ ( x , t) at z K ~-------T- U sin[A~b(x, t)]
(3)
where K=/3L 2, Lc is the monomer size along the polymer spine, x = Lcn.
OM. Ponomarev, E.S. Shikhovtseva / Synthetic Metals 68 (1995) 99-102
By substitutions h4)(x, t) =y(x, t), t'= (UA/I)mt and x' = (UA/K)lr2x the Eq. (3) can be reduced to an ordinary
101
As a result the number of polymer threads turned into the conductive state is
sine-Gordon equation: O2y(x,t) at,2
n = n o ~ exp{
aZy(x,t) ~x,2 + sin y(x, t) = 0
Ek~-~iP } cosh(biE/kT)
(4) where no is the number of side chains near the electrode which can cause soliton generation in their polymer threads under external influence, and T is the film temperature. As one can see that n increases step by step as the inequality:
Under the boundary conditions:
y(t', - oo ) =y(t', + oo ) = 0 the solution of Eq. (4) gives us: 4)(x, t) = (4/A) arctan{exp[ + y((UA/K)l/2(X-Xo)
oi
-
v(u~/o"at)]}
(5)
H e r e ~/= ( 1 - v 2 ) -1/2 where v is a dimensionless soliton velocity. The initial conditions determine the type of solution. It may be the soliton solution with ' + ' or ' - - ' sign (the two choices in sign in Eq. (5) represent solutions of Eq. (4) with appropriate boundary conditions: 4) = 0 ° or 4)= -+ 90 ° at x = _+ oo) and oscillated near 4)= 0 ° or 4)=90 ° solutions. The latter ones are not considered below. However, they are also experimentally observed [7] in the polymer films. The kink is generated by the influence on the side chain nearest to the electrode. The interaction energy of the side chain and electrode is Z2e2/2Xo, where Z is the side-chain charge, Xo is the equilibrium distance between charge Z and electrode. So, E ° = E c - Z 2 e 2 / 2 x o is the activation energy of the bond destruction when there are no external influences on the side chain. H e r e Ec is the active bond energy. If the external field E is applied to the film, then there are two activation energies of bond destruction for one bond: E 2
Z2e 2
[E 2
2(
Z2e'2"]] 1/2
Ea - a i P +_biE <~kT
has been satisfied.
4. Discussion
Initially we consider the influence of external conditions on the film properties in the case of the film to be made into conductive form. The side chain is shifted from the equilibrium position Xo with energy - Uo to some xl with energy - U1 (/do> U1) by external pressure. Such a side-chain transition is equivalent to a bond C - O destruction and it causes the polymer chain transformation into the conductive state. Although the kink generation is improbable and there are not many conducting polymer spines in the film, the film conductivity is determined by them. The inverse metal-dielectric transition is associated with antikink generation and takes place when the external pressure stops its action. The transition of initially conductive film to the dielectric state may be caused by mechanical influence (shaking, shift and so on) on the polymer-electrode contact.
Here w is the oscillation frequency of the side chain relative to its rotational axis. If then in addition to the field E, an external pressure P is applied to the film, the activation energy of bond destruction is
Ea~ = E ° - a P ++_bE
(6)
where
a = (Z2e2/2xo)o~b = (2E°)~/2/o~
) 4
E
where a is the linear compression coefficient. The terms proportional to E z are neglected in Eq. (6). Under real physical conditions we have a set of activation energies Eic (this fact is labelled by index i) and, thus, ET. Then instead of Eq. (6) the equation: Eia+ __ - E , oi - a i P
takes place.
+biE Fig. 3. Current-voltage characteristic of the initially dielectric polymer film.
102
0 . 4 . Ponomarev, E.S. Shikhovtseva / Synthetic Metals 68 (1995) 99-102
2
7
E
Fig. 4. Current-voltage characteristic of the initially conductive polymer film.
The energy barrier between - U o and -U1 is decreased by the external electric field and so the transition may be induced by only electric field without the pressure or the electric field and pressure together. One carbon obtain a current-voltage characteristic (Fig. 3) for an initially dielectric polymer film. The line 0-1 shows the dielectric state of the film. Then, the electric field achieves a value sufficient for the side chain to overcome the barrier between - U o and -U1 (the point 1) and the film goes to the conductive state via 1-2. As the activation energy of transition - U I ~ - U o is less than that of - U o ~ -U1 the con-
ductive state of a film does not revert to point 0 but only to point 3. The points 5 and 9 (and similar ones) correspond to the coming into operation of the next side chains with greater activation energies. The line 13-14 describes the improbable process of the decrease in activation energy with increase of electric field. The initially conducting film has a current-voltage characteristic (Fig. 4) with consequent side chains switching off. The real experimental data may give a mixture of Figs. 3 and 4.
References [1] O.V. Demicheva, E.I. Shkliarova, A.V. Volkova, S.G. Smirnova, V.M. Andrejev, D.N. Rogachev and L.N. Grigorov, Vysok. Soedin., Ser. B, 32 (1990) 659. [2] L.N. Grigorov, Macromol. Chem. Symp., 37 (1990) 159. [3] I.S. Jenikolopian, L.N. Grigorov and S.G. Smirnova, Pis'ma Zh. Teor. Fiz., 49 (1989)( 326. [4] V.M. Arhangorodskii, E.G. Gook, A.M. Jeljashevich, A.N. Ionov, V.M. Tuchkevich and I.G. Shliman, Dokl. Acad. N a u k SSSt~ 309 (1989) 603. [5] A.N. Lachinov, A. Yu. Zherebov and V.M. Kornilov, Pis'ma Zh. Teor. Fiz., 52 (1990) 742. [6] V.M. Komilov and A.N. Lachinov, Synth. Met., 53 (1992) 71. [7] A. Yu. Zherebov and A.N. Lachinov, Synth. Met., 46 (1992) 181. [8] A.N. Lachinov, A. Yu. Zherebov and V.M. Kornilov, Zh. Eksp. Teor. Phys., 102 (1992) 187. [9] W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett., 42 (1979) 1698; W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. B, 22 (1980) 2099; 28 (1983) 1138 (E).
Synthetic Metals 68 (1995) 99-102
The dielectric-metal transition mechanism for the thread-like structure polymers O.A. Ponomarev, E.S. Shikhovtseva Department of Physics, Ufa Scientific Center of the Russian Academy of Sciences, Ufa, Russian Federation Received 8 April 1993; revised 22 July 1994
Abstract
The paper presents a model of the polymer film dielectric-metal transition under a small external influence. The conductive pulse propagation along the polymer thread has a soliton nature. The polymer film current-voltage characteristic is obtained on the basis of the described model. Keywords: Film; Conductivity; Dielectric-metal transition
1. I n t r o d u c t i o n
l The existence of anomalously high spontaneous conductivity discovered in some polymers with thread-like structure (for example, polyarylenephthalides) was reported recently [1-5]. The polyarylenephthalide experimental results [5-8] show the switch effect: from the normal state of 10-14 (~'~ cm)-1 conductivity to the conductive state of 109 (12 cm) -1. This conductivity jump is obtained in strong electric fields (more than 105 V/cm) and thin films (less than 3 #m) when the external influence is small enough (the transition threshold pressure is less than 102 Pa). A dielectric-metal transition model based on the chemical structure of thread-like polymers is given below. 2. The m o d e l
Obviously, the observed huge conductivity jump of polymer films under the influence of negligible external pressure is not caused by a change of the volume structure of the film. The reason for this phenomenon is the chemical structure of the polymer chain. The bridges between the side chains are known to be able to go from the dielectric state into the metal one and back under an external influence or under the influence of neighbouring bonds of the polymer spine. The central carbon atom in a polyarylenephthalide fragment has an sp 3 hybridization in a dielectric state of the bridge (Fig. l(a)) and the benzene ring plane
0379-6779/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 3 7 9 - 6 7 7 9 ( 9 4 ) 0 2 2 8 3 - 5
"~'C
(a)
FI Fig. 1. Polyarylene m o n o m e r in dielectric (a) and conductive (b) state.
of the side chain is perpendicular to the spine (let us choose an angle 4)= 0 ° for this state). In the conductive bridge (Fig. l(b)) the central carbon atom has an sp 2 hybridization and the benzene ring plane becomes parallel to the spine (q5=90°). The bridge-bridge interaction energy depends on 4). Let us consider the mechanism of such a dielectric-metal transition for a polymer based on butadiene. The nth fragment structure of the dielectric state polymer (4)= 0 °) is shown in Fig. 2(a). The destruction of the C 1 = O bond and the creation of the C]=C2 bond are the beginning of the transition (Fig. 2(b)). Then the transformations of double bonds to single ones and vice versa in the spine from C1 to the next central
100
0,4. Ponornarev, E.S. Shikhovtseva / Synthetic Metals 68 (1995) 99-102
Thus, we have presented the mechanism of the persistent conductive state propagation by the transformation of central carbon atoms of sp 3 hybridization to those of sp 2 hybridization. This mechanism is based on the soliton transport of the side-chain benzene-ring rotation along the spine. In the created conductive state the polymer becomes polyaeetylene-like and its conductivity may be described by the soliton mechanism similar to the SSH model [9]. But in our paper we do not deal with this problem. The Hamiltonian in the representation of second quantization has the form:
"\C"CH =CH --CH~--------CH-HC/ t., HC (a) \
\\C ~CH--CH = C H ~ /
t
I He,,. ¢ C H HC
, ~ = ~C.na+,~ai,,o.+ ~ J
c o s ( A q S , / 2 ) a+oain+go _
+ EJiia i+~.ay,¢+ ( 1 / 2 ) ~ V0,ma,+~a,,oaf~ ajm~
(b)
i ~*j
+ (1/2).~,/32//+ (/3/2)£[(+, + 1 - ~b,)z
CH==CH---
+ ( & _ 1 - (~.)21 + (U/2)E(1 - cos(24,.)) HH~cJ ~ H (c)
n-~l
[ \,C~ CI...I==CI| // ~
CH==CH---
2
I (d) r
\
"\. ~CI-I==C~H ==CI-I~
H?
(1)
Here a~:(r is a creation operator for an electron of E~ energy on the ith thread and nth fragment, J cos(A~b~/2) is an exchange integral between atoms situated in nth and (n + 1)th knots, ¢~ is a side-chain rotation operator J/j is an exchange integral between the polymer threads, Vijnm is Coulomb interaction of the electrons, /~ is a side-chain angular-momentum operator, I is the moment of inertia of the side chain with regards to rotation of angle ~b,, /3 is an indirect exchange interaction parameter for neighbouring side chains, U is the anharmonicity parameter, and A=4 for the polymers under consideration.
3. The side-chain dynamics
?
{Hea\ CH [(e)
n+'l
Fig. 2. Dielectric-metal evolution of the polymer based on butadiene.
carbon atom and in the side chain from O to C1 occur. Such pulse propagation along the spine causes destruction of the C1-O bond in the next fragment (Fig. 2(c)) and the pulse propagation along the side chain causes the inverse transition CI=Cz to C1-C2 in the nth monomer. Thus, we have a conductive state (Fig. 2(d)) in the first fragment (~b=90 °) and conductive pulse movement (Fig. 2(e)) along the spine. In other words, the transformation of the dielectric fragments
The side-chain dynamics is determined by angle Sn along the polymer spine. The side-chain energy E obtained from Eq. (1) yields the system of equations for the qSn coordinates: d~b,, dt
aE 0P,
P,, I (2)
dPn dt
~E a$~
/3(2~b,- ~. +a - ~.-1) - U sin(AqS.)
Here, in the classic limit, Sn and Pn are the side-chain rotation angle and side-chain angular momentum, respectively. Then in the continuum approximation one can write:
/
with a central carbon atom - ~ to the conductive I ones with a carbon atom /\ C - -- propagates along the polymer thread. A similar process takes place in polyarylenephthalide films.
I O2~(x' t) _ ~ ( x , t) at z K ~-------T- U sin[A~b(x, t)]
(3)
where K=/3L 2, Lc is the monomer size along the polymer spine, x = Lcn.
OM. Ponomarev, E.S. Shikhovtseva / Synthetic Metals 68 (1995) 99-102
By substitutions h4)(x, t) =y(x, t), t'= (UA/I)mt and x' = (UA/K)lr2x the Eq. (3) can be reduced to an ordinary
101
As a result the number of polymer threads turned into the conductive state is
sine-Gordon equation: O2y(x,t) at,2
n = n o ~ exp{
aZy(x,t) ~x,2 + sin y(x, t) = 0
Ek~-~iP } cosh(biE/kT)
(4) where no is the number of side chains near the electrode which can cause soliton generation in their polymer threads under external influence, and T is the film temperature. As one can see that n increases step by step as the inequality:
Under the boundary conditions:
y(t', - oo ) =y(t', + oo ) = 0 the solution of Eq. (4) gives us: 4)(x, t) = (4/A) arctan{exp[ + y((UA/K)l/2(X-Xo)
oi
-
v(u~/o"at)]}
(5)
H e r e ~/= ( 1 - v 2 ) -1/2 where v is a dimensionless soliton velocity. The initial conditions determine the type of solution. It may be the soliton solution with ' + ' or ' - - ' sign (the two choices in sign in Eq. (5) represent solutions of Eq. (4) with appropriate boundary conditions: 4) = 0 ° or 4)= -+ 90 ° at x = _+ oo) and oscillated near 4)= 0 ° or 4)=90 ° solutions. The latter ones are not considered below. However, they are also experimentally observed [7] in the polymer films. The kink is generated by the influence on the side chain nearest to the electrode. The interaction energy of the side chain and electrode is Z2e2/2Xo, where Z is the side-chain charge, Xo is the equilibrium distance between charge Z and electrode. So, E ° = E c - Z 2 e 2 / 2 x o is the activation energy of the bond destruction when there are no external influences on the side chain. H e r e Ec is the active bond energy. If the external field E is applied to the film, then there are two activation energies of bond destruction for one bond: E 2
Z2e 2
[E 2
2(
Z2e'2"]] 1/2
Ea - a i P +_biE <~kT
has been satisfied.
4. Discussion
Initially we consider the influence of external conditions on the film properties in the case of the film to be made into conductive form. The side chain is shifted from the equilibrium position Xo with energy - Uo to some xl with energy - U1 (/do> U1) by external pressure. Such a side-chain transition is equivalent to a bond C - O destruction and it causes the polymer chain transformation into the conductive state. Although the kink generation is improbable and there are not many conducting polymer spines in the film, the film conductivity is determined by them. The inverse metal-dielectric transition is associated with antikink generation and takes place when the external pressure stops its action. The transition of initially conductive film to the dielectric state may be caused by mechanical influence (shaking, shift and so on) on the polymer-electrode contact.
Here w is the oscillation frequency of the side chain relative to its rotational axis. If then in addition to the field E, an external pressure P is applied to the film, the activation energy of bond destruction is
Ea~ = E ° - a P ++_bE
(6)
where
a = (Z2e2/2xo)o~b = (2E°)~/2/o~
) 4
E
where a is the linear compression coefficient. The terms proportional to E z are neglected in Eq. (6). Under real physical conditions we have a set of activation energies Eic (this fact is labelled by index i) and, thus, ET. Then instead of Eq. (6) the equation: Eia+ __ - E , oi - a i P
takes place.
+biE Fig. 3. Current-voltage characteristic of the initially dielectric polymer film.
102
0 . 4 . Ponomarev, E.S. Shikhovtseva / Synthetic Metals 68 (1995) 99-102
2
7
E
Fig. 4. Current-voltage characteristic of the initially conductive polymer film.
The energy barrier between - U o and -U1 is decreased by the external electric field and so the transition may be induced by only electric field without the pressure or the electric field and pressure together. One carbon obtain a current-voltage characteristic (Fig. 3) for an initially dielectric polymer film. The line 0-1 shows the dielectric state of the film. Then, the electric field achieves a value sufficient for the side chain to overcome the barrier between - U o and -U1 (the point 1) and the film goes to the conductive state via 1-2. As the activation energy of transition - U I ~ - U o is less than that of - U o ~ -U1 the con-
ductive state of a film does not revert to point 0 but only to point 3. The points 5 and 9 (and similar ones) correspond to the coming into operation of the next side chains with greater activation energies. The line 13-14 describes the improbable process of the decrease in activation energy with increase of electric field. The initially conducting film has a current-voltage characteristic (Fig. 4) with consequent side chains switching off. The real experimental data may give a mixture of Figs. 3 and 4.
References [1] O.V. Demicheva, E.I. Shkliarova, A.V. Volkova, S.G. Smirnova, V.M. Andrejev, D.N. Rogachev and L.N. Grigorov, Vysok. Soedin., Ser. B, 32 (1990) 659. [2] L.N. Grigorov, Macromol. Chem. Symp., 37 (1990) 159. [3] I.S. Jenikolopian, L.N. Grigorov and S.G. Smirnova, Pis'ma Zh. Teor. Fiz., 49 (1989)( 326. [4] V.M. Arhangorodskii, E.G. Gook, A.M. Jeljashevich, A.N. Ionov, V.M. Tuchkevich and I.G. Shliman, Dokl. Acad. N a u k SSSt~ 309 (1989) 603. [5] A.N. Lachinov, A. Yu. Zherebov and V.M. Kornilov, Pis'ma Zh. Teor. Fiz., 52 (1990) 742. [6] V.M. Komilov and A.N. Lachinov, Synth. Met., 53 (1992) 71. [7] A. Yu. Zherebov and A.N. Lachinov, Synth. Met., 46 (1992) 181. [8] A.N. Lachinov, A. Yu. Zherebov and V.M. Kornilov, Zh. Eksp. Teor. Phys., 102 (1992) 187. [9] W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett., 42 (1979) 1698; W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. B, 22 (1980) 2099; 28 (1983) 1138 (E).