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On the path length of an excess electron interacted with optical phonons in a molecular chain
Physics Letters A 372 (2008) 5725–5726
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Physics Letters A www.elsevier.com/locate/pla
On the path length of an excess electron interacted with optical phonons in a molecular chain V.D. Lakhno Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region 142290, Russia
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 May 2008 Received in revised form 1 July 2008 Accepted 4 July 2008 Available online 10 July 2008 Communicated by V.M. Agranovich
We show that in a molecular chain with dispersionless phonons at zero temperature, a “quasistationary” moving soliton state of an excess electron is possible. As the soliton velocity vanishes, the path length of the excess electron exponentially tends to infinity. It is demonstrated that in the presence of dispersion, when the soliton initial velocity exceeds the maximum group velocity of the chain, the soliton slows down until it reaches the maximum group velocity and then moves stationarily at this maximum group velocity. A conclusion is made of the fallacy of some works were the existence of moving polarons in a dispersionless medium is considered infeasible. © 2008 Elsevier B.V. All rights reserved.
PACS: 63.20.kd Keywords: Phonon emission Soliton Mobility Dispersion
The authors of [1] reason that in a molecular chain without dispersion, a soliton (polaron) formed by an excess electron cannot move stationary. The arguments are based on the fact that the phonons obeying the dispersion law Ω 2 = Ω02 + V 02 K 2 have a zero group velocity at V 0 = 0 and, therefore the phonon environment cannot follow the soliton motion. The main aim of this work is to show that a soliton can move. Of interest here is to find the length of a path that a soliton having at the initial moment the velocity V travels until it finally stops. Following [1], we can write Hamiltonian H which describes the electron motion along a molecular chain in a continuum approximation: H =−
1
a
+
M 2a
∗
Ψ (x, t )
h¯ 2 ∂ 2 2m ∂ x2
∂ u (x, t ) ∂t
Ψ (x, t ) dx +
2
χ a
Ψ (x, t )2 u (x, t ) dx
+ Ω02 u 2 (x, t ) + V 02
∂ u (x, t ) ∂x
2 dx. (1)
In expression (1) energy is reckoned from the bottom of the conductivity band, Ψ (x, t ) is a normalized electron wave function, m is the electron effective mass, χ is a constant for the electron interaction with the chain displacement u (x, t ), M is the reduced mass of the elementary cell, a is the lattice constant.
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.07.009
For steady solutions of the form u (x, t ) = u (ξ ), Ψ (x, t ) =
ϕ (ξ ) exp(i [mV x − ( W + mV 2 /2)t ]/¯h), ξ = (x − V t )/a expression (1) implies the following motion equations:
−ε Ω02 −
d2 u dξ 2
+ Ω02 u +
h¯ 2
d2 ϕ
2ma2
dξ 2
where
χ M
|ϕ |2 = 0,
(2)
+ χ uϕ = W ϕ ,
(3)
ε = ( V 02 − V 2 )/a2 Ω02 . The solution of Eq. (2) is written as:
u (ξ ) = −
χ M Ω02
2
dξ ω(ξ − ξ )ϕ (ξ ) ,
(4)
where:
Θ(ξ ) ξ sin √ , ε < 0, |ε | |ε | |ξ | 1 , ε > 0, ω(ξ ) = √ exp − √ 2 |ε | |ε |
ω(ξ ) = √
(5) (6)
where Θ(ξ ) = 1 for ξ > 0 and Θ(ξ ) = 0 for ξ < 0. For ε = 0, Eq. (3) is reduced to a stationary nonlinear Schrödinger equation, whose normalized solution has the form: 1
ϕ (ξ ) = √
2r
ch−1 (ξ/r ),
r = 4M (¯hΩ0 )2 /mχ 2 a2 .
(7)
In the absence of dispersion (V 0 = 0) and small soliton velocities, when |ε | 1, from (4), (5) we get:
5726
V.D. Lakhno / Physics Letters A 372 (2008) 5725–5726
u (ξ ) = c sin ξ/ |ε | , c = −π r χ
/2M Ω02 a|
ξ < c1r ,
ε| sh π r /2 |ε| ,
(8)
(10)
whole length of the chain decaying exponentially on each side of the soliton center. In this situation the quantity E ph has a finite value even providing infinite limits on integral (11) and corresponds to the energy of the phonon environment following the charge along the chain. In this case the path length is infinite. As ε < 0 (i.e. when V > V 0 ) emission of phonons becomes possible which leads to slowing down of the soliton. At any initial velocity the slowing down will take place until the soliton reaches the velocity V 0 , after which it will move stationarily at this velocity. The path length of the soliton until it reaches the velocity V 0 for ( V − V 0 )/ V 0 1 is given by the expression:
(11)
L /a ∼ exp
(9)
where c 1 is a constant of the order of 1. From (8) follows that for ξ = x − V t, a soliton moving in positive direction leaves behind a “tail” of the chain oscillations loosing on the way its kinetic energy E kin = m∗∗ V 2 /2 (where m∗∗ is the soliton effective mass) until it finally stops. The distance L which a soliton will travel is found from the condition: E ph = E kin , E ph =
M 2
c1 r − L /a
( V 2 + V 02 ) du dξ a2
2 + Ω02 u 2
dξ.
Substituting expression (9) into (11) and using (10) with small we express the soliton path length as: L /a ∼ V 6 exp(π raΩ0 / V ).
ε
(12)
So, in the absence of dispersion, as V → 0, the soliton path length L tends to infinity. We emphasize that in the limit ε → 0 expression (12) gives an exact asymptotic estimation of minimal soliton path length. This, in turn, means that at any initial velocity of the soliton, its path length is equal to infinity. The reason is that expending its kinetic energy on the formation of the phonon tail, the soliton is slowing down, i.e. its velocity is decreasing to such value that formula (12) becomes applicable. This fact justifies the idea of the possibility of “quasisteady” states of solitons and polarons at small V , for which all the calculations of the effective mass were made. Taking account of dispersion (V 0 = 0) changes qualitatively this picture. At arbitrarily small values of V 0 and V < V 0 , when ε > 0, Eq. (2), according to (6), has some localized solutions. Since for ε > 0 the soliton velocity is less than the maximum group velocity V 0 , in the steady case u (ξ ) is different from zero over the
π raΩ0
(13)
V 2 − V 02
which transforms into (12) as V 0 → 0. According to (13), the soliton path length tends to infinity as V → V 0 , accordingly, infinite time is required for a soliton to reach the velocity V 0 . Therefore the statements of [2] that polarons with V > V 0 cannot move are erroneous, as are the conclusions drawn on the basis of these statements. Acknowledgement The work was done with the support from the RFBR, Project N 07-07-00313. References [1] A.S. A.S. A.S. [2] A.E. A.E. A.E. A.E.
Davydov, V.Z. Enol’skii, Zh. Eksp. Teor. Fiz. 79 (1980) 1888; Davydov, V.Z. Enol’skii, Phys. Status Solidi B 143 (1987) 167; Davydov, V.Z. Enol’skii, Zh. Eksp. Teor. Fiz. 94 (1988) 177. Myasnikova, E.N. Myasnikov, Zh. Eksp. Teor. Fiz. 115 (1999) 180; Myasnikova, E.N. Myasnikov, Zh. Eksp. Teor. Fiz. 116 (1999) 1386; Myasnikova, E.N. Myasnikov, Phys. Rev. B 52 (1995) 10457; Myasnikova, E.N. Myasnikov, Phys. Lett. A 286 (2001) 210.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On the path length of an excess electron interacted with optical phonons in a molecular chain V.D. Lakhno Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region 142290, Russia
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 May 2008 Received in revised form 1 July 2008 Accepted 4 July 2008 Available online 10 July 2008 Communicated by V.M. Agranovich
We show that in a molecular chain with dispersionless phonons at zero temperature, a “quasistationary” moving soliton state of an excess electron is possible. As the soliton velocity vanishes, the path length of the excess electron exponentially tends to infinity. It is demonstrated that in the presence of dispersion, when the soliton initial velocity exceeds the maximum group velocity of the chain, the soliton slows down until it reaches the maximum group velocity and then moves stationarily at this maximum group velocity. A conclusion is made of the fallacy of some works were the existence of moving polarons in a dispersionless medium is considered infeasible. © 2008 Elsevier B.V. All rights reserved.
PACS: 63.20.kd Keywords: Phonon emission Soliton Mobility Dispersion
The authors of [1] reason that in a molecular chain without dispersion, a soliton (polaron) formed by an excess electron cannot move stationary. The arguments are based on the fact that the phonons obeying the dispersion law Ω 2 = Ω02 + V 02 K 2 have a zero group velocity at V 0 = 0 and, therefore the phonon environment cannot follow the soliton motion. The main aim of this work is to show that a soliton can move. Of interest here is to find the length of a path that a soliton having at the initial moment the velocity V travels until it finally stops. Following [1], we can write Hamiltonian H which describes the electron motion along a molecular chain in a continuum approximation: H =−
1
a
+
M 2a
∗
Ψ (x, t )
h¯ 2 ∂ 2 2m ∂ x2
∂ u (x, t ) ∂t
Ψ (x, t ) dx +
2
χ a
Ψ (x, t )2 u (x, t ) dx
+ Ω02 u 2 (x, t ) + V 02
∂ u (x, t ) ∂x
2 dx. (1)
In expression (1) energy is reckoned from the bottom of the conductivity band, Ψ (x, t ) is a normalized electron wave function, m is the electron effective mass, χ is a constant for the electron interaction with the chain displacement u (x, t ), M is the reduced mass of the elementary cell, a is the lattice constant.
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.07.009
For steady solutions of the form u (x, t ) = u (ξ ), Ψ (x, t ) =
ϕ (ξ ) exp(i [mV x − ( W + mV 2 /2)t ]/¯h), ξ = (x − V t )/a expression (1) implies the following motion equations:
−ε Ω02 −
d2 u dξ 2
+ Ω02 u +
h¯ 2
d2 ϕ
2ma2
dξ 2
where
χ M
|ϕ |2 = 0,
(2)
+ χ uϕ = W ϕ ,
(3)
ε = ( V 02 − V 2 )/a2 Ω02 . The solution of Eq. (2) is written as:
u (ξ ) = −
χ M Ω02
2
dξ ω(ξ − ξ )ϕ (ξ ) ,
(4)
where:
Θ(ξ ) ξ sin √ , ε < 0, |ε | |ε | |ξ | 1 , ε > 0, ω(ξ ) = √ exp − √ 2 |ε | |ε |
ω(ξ ) = √
(5) (6)
where Θ(ξ ) = 1 for ξ > 0 and Θ(ξ ) = 0 for ξ < 0. For ε = 0, Eq. (3) is reduced to a stationary nonlinear Schrödinger equation, whose normalized solution has the form: 1
ϕ (ξ ) = √
2r
ch−1 (ξ/r ),
r = 4M (¯hΩ0 )2 /mχ 2 a2 .
(7)
In the absence of dispersion (V 0 = 0) and small soliton velocities, when |ε | 1, from (4), (5) we get:
5726
V.D. Lakhno / Physics Letters A 372 (2008) 5725–5726
u (ξ ) = c sin ξ/ |ε | , c = −π r χ
/2M Ω02 a|
ξ < c1r ,
ε| sh π r /2 |ε| ,
(8)
(10)
whole length of the chain decaying exponentially on each side of the soliton center. In this situation the quantity E ph has a finite value even providing infinite limits on integral (11) and corresponds to the energy of the phonon environment following the charge along the chain. In this case the path length is infinite. As ε < 0 (i.e. when V > V 0 ) emission of phonons becomes possible which leads to slowing down of the soliton. At any initial velocity the slowing down will take place until the soliton reaches the velocity V 0 , after which it will move stationarily at this velocity. The path length of the soliton until it reaches the velocity V 0 for ( V − V 0 )/ V 0 1 is given by the expression:
(11)
L /a ∼ exp
(9)
where c 1 is a constant of the order of 1. From (8) follows that for ξ = x − V t, a soliton moving in positive direction leaves behind a “tail” of the chain oscillations loosing on the way its kinetic energy E kin = m∗∗ V 2 /2 (where m∗∗ is the soliton effective mass) until it finally stops. The distance L which a soliton will travel is found from the condition: E ph = E kin , E ph =
M 2
c1 r − L /a
( V 2 + V 02 ) du dξ a2
2 + Ω02 u 2
dξ.
Substituting expression (9) into (11) and using (10) with small we express the soliton path length as: L /a ∼ V 6 exp(π raΩ0 / V ).
ε
(12)
So, in the absence of dispersion, as V → 0, the soliton path length L tends to infinity. We emphasize that in the limit ε → 0 expression (12) gives an exact asymptotic estimation of minimal soliton path length. This, in turn, means that at any initial velocity of the soliton, its path length is equal to infinity. The reason is that expending its kinetic energy on the formation of the phonon tail, the soliton is slowing down, i.e. its velocity is decreasing to such value that formula (12) becomes applicable. This fact justifies the idea of the possibility of “quasisteady” states of solitons and polarons at small V , for which all the calculations of the effective mass were made. Taking account of dispersion (V 0 = 0) changes qualitatively this picture. At arbitrarily small values of V 0 and V < V 0 , when ε > 0, Eq. (2), according to (6), has some localized solutions. Since for ε > 0 the soliton velocity is less than the maximum group velocity V 0 , in the steady case u (ξ ) is different from zero over the
π raΩ0
(13)
V 2 − V 02
which transforms into (12) as V 0 → 0. According to (13), the soliton path length tends to infinity as V → V 0 , accordingly, infinite time is required for a soliton to reach the velocity V 0 . Therefore the statements of [2] that polarons with V > V 0 cannot move are erroneous, as are the conclusions drawn on the basis of these statements. Acknowledgement The work was done with the support from the RFBR, Project N 07-07-00313. References [1] A.S. A.S. A.S. [2] A.E. A.E. A.E. A.E.
Davydov, V.Z. Enol’skii, Zh. Eksp. Teor. Fiz. 79 (1980) 1888; Davydov, V.Z. Enol’skii, Phys. Status Solidi B 143 (1987) 167; Davydov, V.Z. Enol’skii, Zh. Eksp. Teor. Fiz. 94 (1988) 177. Myasnikova, E.N. Myasnikov, Zh. Eksp. Teor. Fiz. 115 (1999) 180; Myasnikova, E.N. Myasnikov, Zh. Eksp. Teor. Fiz. 116 (1999) 1386; Myasnikova, E.N. Myasnikov, Phys. Rev. B 52 (1995) 10457; Myasnikova, E.N. Myasnikov, Phys. Lett. A 286 (2001) 210.