- Email: [email protected]
Polarizability of polaron in conjugated polymer
18 February 2002
Physics Letters A 294 (2002) 113–116 www.elsevier.com/locate/pla
Polarizability of polaron in conjugated polymer K. Chen a,b,∗ , X. Sun a , L.H. Tang c a Research Center for Theoretical Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China b National Laboratory of Infrared Physics, Shanghai Institute of Technical Physics, Shanghai 200083, China c Department of Physics, Hong Kong Baptist University, Hong Kong, China
Received 7 April 2001; received in revised form 29 November 2001; accepted 3 December 2001 Communicated by A.R. Bishop
Abstract For conjugated polymers with non-degenerate ground state, the excited state of a polaron is stable, and its static polarizability is negative. Then the photoinduced polarization reversion (PPR) can occur in a polymeric molecule with a polaron induced by doping. Here the PPR produced by a polaron needs only one photoexcitation; but that produced by a biexciton needs to be photoexcited twice. Therefore this Letter provides a easier way to observe the PPR in the polymers. 2002 Elsevier Science B.V. All rights reserved. PACS: 33.90.+h; 36.20.-r; 32.80.Wr
It has been proved that a polymeric molecule with a biexciton [1] possesses negative static polarizability, meanwhile the static polarizability of the same molecule with an exciton is positive. Then a new photoinduced phenomenon—photoinduced polarization reversion (PPR)—can occur in a polymeric molecule with an exciton: the electric dipole of the molecule with an exciton is reversed by absorbing one photon [2]. Many groups are trying to observe the PPR in experiments. Since the PPR produced by the biexciton is two-photon process: first the molecule is photoexcited to have an exciton, then it is reexcited from the exciton to biexciton, its efficiency is quite low. If we can find a way, which is one-photon process, to get the
* Corresponding author.
E-mail address: [email protected] (K. Chen).
PPR, it will be much easier to demonstrate the PPR in polymers. This Letter shows us that the polaron in the polymeric molecule can provide a new PPR, which needs only one photoexcitation. The key point to get the negative static polarizability is to produce two electronic states (up-one ψu and down-one ψd ), which are close to each other but far away from the higher excited states, then ψu possesses negative static polarizability. The self-trapped exciton in polymers produces two localized electronic states in gap. Besides, the polaron induced by doping in polymers can also produce two gap states [3]. If the polarizability of the up-gap-state ψu of the polaron is negative, the PPR can be realized in a polymeric molecule with a polaron by exciting the electron from ψd to ψu , and this kind of PPR will be one-photon process. This Letter gives an analytic calculation to prove that the up-gap-state of the polaron does possess negative polarizability.
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 8 1 4 - 3
114
K. Chen et al. / Physics Letters A 294 (2002) 113–116
It is well known that the polaron in the polymer can be analytically described by the TLM model [4,5]: ∂ψ2s ∂ψ1s + + − ψ1s ivf ψ2s (y) (y) HTLM = dy ∂y ∂y s + + ˜ + ∆(y) ψ1s (y)ψ2s (y) + ψ2s (y)ψ1s (y) 1 + 2 ∆2 (y) . (1) 4g vf In this Letter, the Fermi velocity vf = 7 Å eV and the coupling constant g 2 = 0.37. For the polaron, its order parameter is [6–11] ˜ ∆(y) = ∆˜ 0 − K0 vf tanh K0 (y + y0 ) (2) − tanh K0 (y − y0 ) , K0 vf , tanh(2K0 y0 ) = (3) ∆˜ 0
−
∗ v+0 (y) = −v−0 (y).
(b) Valance band, εkv = − k 2 vf2 + ∆˜ 20 , u− (k, y) = Nk eiky εk + ∆˜ 0 − kvf
(6)
(7)
+ y0 )
ε0 tanh K0 (y + y0 ) 2vf − tanh K0 (y − y0 ) ε0 tanh K0 (y + y0 ) − n− 2vf − tanh K0 (y − y0 ) ,
(8)
+ γ ∗ (1 + i) tanh K0 (y − y0 ) . (9)
3π = 0, 4
(12)
where θ = arcsin K0 vf /∆˜ 0 , γ = (ln ∆˜ 0 /∆0 )−1 . For the polymers with non-degenerate ground state, when γ = 0.1, θ = 1.45971, y0 = 1.45371
− γ (1 − i) tanh K0 (y + y0 )
(11)
where n+ , n− are the populations in the up-gap-state and down-gap-state, respectively. For photoexcited electron-polaron, n+ = 2, n− = 1, Eq. (11) becomes θ + γ tan θ −
+ γ ∗ (1 − i) tanh K0 (y − y0 ) ,
v− (k, y) = −Nk e
ε02 K0 vf tanh K0 (y 2 2 2 ˜ ε(k vf + ∆0 )
− tanh K0 (y − y0 )
(5)
− γ (1 + i) tanh K0 (y + y0 )
εk + ∆˜ 0 + kvf
(10)
+ n+
For up-gap-state,
∗ v+0 (y) = −v−0 (y).
For hole-polaron there is one electron in the downgap-state, the up-gap-state is empty. For electronpolaron, there are two electrons in the down-gap-state, one in the up-gap-state. The photoexcitation of the polaron is that one electron is excited from the downgap-state to the up-gap-state by absorbing one photon. In photoexcited hole-polaron, there is one electron in the up-gap-state, no electron in down-one. In photoexcited electron-polaron, there are two electrons in the up-gap-state and one electron in down-one. The order parameter of the photoexcited polaron is [6]
kf ˜ ∆(y) 1 2 dk ∆ex (y) = −2g vf − π ε
(4)
For down-gap-state, 1 u−0 (y) = K0 (1 + i) sech K0 (y + y0 ) 4 + (1 − i) sech K0 (y − y0 ) , 1 v−0 (y) = − K0 (1 − i) sech K0 (y + y0 ) 4 + (1 + i) sech K0 (y − y0 ) .
iky
u+0 (y) = u∗−0 (y),
−kf
and electron’s wave functions are: (a) Gap states, εg = ±ε0 , ε0 = ∆˜ 20 − K02 vf2 .
u+0 (y) = u∗−0 (y),
(c) Conduction band, εc = −εv ,
and ε0 =
K0 = 0.993836 vf , ∆˜ 0
∆˜ 20 − K02 vf2 = 0.070 eV.
∆˜ 0 , vf
K. Chen et al. / Physics Letters A 294 (2002) 113–116
When γ = 0.08, θ = 1.47977, y0 = 1.55095
K0 = 0.99586
115
(a) Down-gap-state, 2 χ−0 (ω) = +0|p|−0 1 1 − × ω+0 − ω−0 − ω ω−0 − ω+0 − ω ci|p| − 0 2 +
∆˜ 0 , vf
vf , ∆˜ 0
and
ci
ε0 = 0.056 eV.
×
When γ = 0.04, θ = 1.52283, y0 = 1.86728
K0 = 0.99885
(15)
∆˜ 0 , vf
(b) Up-gap-state, 2 χ+0 (ω) = −0|p|+0 1 1 − × ω−0 − ω+0 − ω ω+0 − ω−0 − ω ci|p| + 0 2 +
vf , ∆˜ 0
and ε0 = 0.028 eV. We can see that ε0 is getting smaller when γ decreases, it indicates that the polarizability of the up-gap-state can become negative when γ is small enough. For the polymers with degenerate ground state (γ = 0), √ √ 2 2 vf −1 tanh , y0 → 2 2 ∆˜ 0 it means that the photoexcited polaron cannot be dissociated into two separated solitons. So the photoexcited polaron is still stable for the polymers with degenerate ground state. The dynamical polarizability can be calculated by means of linear response function [12], for the eigenstate m, m|p|n 2 χm (ω) = n
×
1 1 − ωn − ωm − ω ωm − ωn − ω
1 1 − . ωi1 − ω−0 − ω ω−0 − ωi1 − ω
, (13)
where p is the electric dipole: +∞ + + ey ψ1s (y)ψ1s (y) + ψ2s (y)ψ2s (y) dy. (14) p= −∞
By using above wave functions, we can calculate the polarizabilities of different eigenstates.
ci
1 1 − × . ωi1 − ω+0 − ω ω+0 − ωi1 − ω (16) (c) Valence band, 2 χvi (ω) = −0|p|vi 1 1 − × ω−0 − ωvi − ω ωvi − ω−0 − ω +0|p|vi 2 + vi
1 1 − × ωi1 − ω+0 − ω ω+0 − ωi1 − ω cj |p|vi 2 +
cj
1 1 − , × ωcj − ωvi − ω ωvi − ωcj − ω
(17) where |ci and |vi are the states in conduction band and valence band. The total polarizability of the polymeric molecule with a polaron is χ(ω) = n−0 χ−0 (ω) + n+0 χ+0 (ω) + nvi χvi (ω), vi
(18)
116
K. Chen et al. / Physics Letters A 294 (2002) 113–116
When γ = 0.1, the static polarizabilities of both photoexcited hole-polaron and electron-polaron become positive: αhole = 1.0 × 104(ea)2 /eV, αelectron = 4.4 × 103 (ea)2/eV. Conclusion:
Fig. 1. The dynamical polarizability of the excited electron-polaron.
where n+0 , n−0 and nvi are the populations in the down-gap-state, up-gap-state and valence band, respectively. For the photoexcited hole-polaron, χ(ω) = χ+0 (ω) + 2 (19) χvi (ω). vi
For the photoexcited electron-polaron, χvi (ω). χ(ω) = χ−0 (ω) + 2χ+0 + 2
1. The excited state of the polaron is stable in the polymer. 2. The static polarizability of the photoexcited polaron can be negative. 3. The PPR can occur in a polymeric molecule with a polaron, and it is an one-photon process.
Acknowledgements This work was supported by the Grant FRG 9900/II-53 from the Hong Kong Baptist University, the NSF of China by the Grants 20074007, 90103034, 10104006, and Shanghai Education Commission Grant 973-G1999075203.
(20)
vi
When γ = 0.04, the dynamical polarizability of the photoexcited electron-polaron is shown in Fig. 1. The static polarizability is α = χ(ω = 0). When γ = 0.04, the static polarizabilities of both photoexcited hole-polaron and electron-polaron are negative: αhole = −1.2 × 104 (ea)2/eV, αelectron = −3.9 × 104(ea)2 /eV. When γ = 0.08, αhole = 7.4 × 103 (ea)2/eV, αelectron = −1.4 × 103(ea)2 /eV. Here, αhole becomes positive, but αelectron remains negative.
References [1] V. Klimov, D.W. McBranch, N. Barashkov, J. Ferraris, Phys. Rev. B 58 (1998) 7654. [2] X. Sun, R.L. Fu, K. Yonemitsu, K. Nasu, Phys. Rev. Lett. 84 (2000) 2830. [3] A.J. Heeger, S. Kivelson, J.R. Schreiffer, W.P. Su, Rev. Mod. Phys. 60 (1988) 781. [4] H. Takayama, Y.R. Lin-Lin, K. Maki, Phys. Rev. B 21 (1980) 2388. [5] S.A. Brazovskii, N.N. Kirova, Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 6. [6] D.K. Campbell, A.R. Bishop, Nucl. Phys. B 200 (1982) 297. [7] A. Scott, F. Chu, D. Mclaughlin, Proc. IEEE 61 (1973) 1443. [8] W.A. Bardeen, M.S. Chanovitz, S.D. Drell, M. Weinstein, T.M. Yan, Phys. Rev. D 11 (1975) 1094. [9] S.-S. Shei, Phys. Rev. D 14 (1976) 535. [10] R.F. Dashen, D. Hasslacher, A. Neveu, Phys. Rev. D 12 (1975) 2443. [11] D.K. Campell, Y.-T. Liao, Phys. Rev. D 14 (1976) 2093. [12] P.M. Chaikin et al., Principle of Condensed Matter Physics, Cambridge University Press, 1995, Chapter 7.
Physics Letters A 294 (2002) 113–116 www.elsevier.com/locate/pla
Polarizability of polaron in conjugated polymer K. Chen a,b,∗ , X. Sun a , L.H. Tang c a Research Center for Theoretical Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China b National Laboratory of Infrared Physics, Shanghai Institute of Technical Physics, Shanghai 200083, China c Department of Physics, Hong Kong Baptist University, Hong Kong, China
Received 7 April 2001; received in revised form 29 November 2001; accepted 3 December 2001 Communicated by A.R. Bishop
Abstract For conjugated polymers with non-degenerate ground state, the excited state of a polaron is stable, and its static polarizability is negative. Then the photoinduced polarization reversion (PPR) can occur in a polymeric molecule with a polaron induced by doping. Here the PPR produced by a polaron needs only one photoexcitation; but that produced by a biexciton needs to be photoexcited twice. Therefore this Letter provides a easier way to observe the PPR in the polymers. 2002 Elsevier Science B.V. All rights reserved. PACS: 33.90.+h; 36.20.-r; 32.80.Wr
It has been proved that a polymeric molecule with a biexciton [1] possesses negative static polarizability, meanwhile the static polarizability of the same molecule with an exciton is positive. Then a new photoinduced phenomenon—photoinduced polarization reversion (PPR)—can occur in a polymeric molecule with an exciton: the electric dipole of the molecule with an exciton is reversed by absorbing one photon [2]. Many groups are trying to observe the PPR in experiments. Since the PPR produced by the biexciton is two-photon process: first the molecule is photoexcited to have an exciton, then it is reexcited from the exciton to biexciton, its efficiency is quite low. If we can find a way, which is one-photon process, to get the
* Corresponding author.
E-mail address: [email protected] (K. Chen).
PPR, it will be much easier to demonstrate the PPR in polymers. This Letter shows us that the polaron in the polymeric molecule can provide a new PPR, which needs only one photoexcitation. The key point to get the negative static polarizability is to produce two electronic states (up-one ψu and down-one ψd ), which are close to each other but far away from the higher excited states, then ψu possesses negative static polarizability. The self-trapped exciton in polymers produces two localized electronic states in gap. Besides, the polaron induced by doping in polymers can also produce two gap states [3]. If the polarizability of the up-gap-state ψu of the polaron is negative, the PPR can be realized in a polymeric molecule with a polaron by exciting the electron from ψd to ψu , and this kind of PPR will be one-photon process. This Letter gives an analytic calculation to prove that the up-gap-state of the polaron does possess negative polarizability.
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 8 1 4 - 3
114
K. Chen et al. / Physics Letters A 294 (2002) 113–116
It is well known that the polaron in the polymer can be analytically described by the TLM model [4,5]: ∂ψ2s ∂ψ1s + + − ψ1s ivf ψ2s (y) (y) HTLM = dy ∂y ∂y s + + ˜ + ∆(y) ψ1s (y)ψ2s (y) + ψ2s (y)ψ1s (y) 1 + 2 ∆2 (y) . (1) 4g vf In this Letter, the Fermi velocity vf = 7 Å eV and the coupling constant g 2 = 0.37. For the polaron, its order parameter is [6–11] ˜ ∆(y) = ∆˜ 0 − K0 vf tanh K0 (y + y0 ) (2) − tanh K0 (y − y0 ) , K0 vf , tanh(2K0 y0 ) = (3) ∆˜ 0
−
∗ v+0 (y) = −v−0 (y).
(b) Valance band, εkv = − k 2 vf2 + ∆˜ 20 , u− (k, y) = Nk eiky εk + ∆˜ 0 − kvf
(6)
(7)
+ y0 )
ε0 tanh K0 (y + y0 ) 2vf − tanh K0 (y − y0 ) ε0 tanh K0 (y + y0 ) − n− 2vf − tanh K0 (y − y0 ) ,
(8)
+ γ ∗ (1 + i) tanh K0 (y − y0 ) . (9)
3π = 0, 4
(12)
where θ = arcsin K0 vf /∆˜ 0 , γ = (ln ∆˜ 0 /∆0 )−1 . For the polymers with non-degenerate ground state, when γ = 0.1, θ = 1.45971, y0 = 1.45371
− γ (1 − i) tanh K0 (y + y0 )
(11)
where n+ , n− are the populations in the up-gap-state and down-gap-state, respectively. For photoexcited electron-polaron, n+ = 2, n− = 1, Eq. (11) becomes θ + γ tan θ −
+ γ ∗ (1 − i) tanh K0 (y − y0 ) ,
v− (k, y) = −Nk e
ε02 K0 vf tanh K0 (y 2 2 2 ˜ ε(k vf + ∆0 )
− tanh K0 (y − y0 )
(5)
− γ (1 + i) tanh K0 (y + y0 )
εk + ∆˜ 0 + kvf
(10)
+ n+
For up-gap-state,
∗ v+0 (y) = −v−0 (y).
For hole-polaron there is one electron in the downgap-state, the up-gap-state is empty. For electronpolaron, there are two electrons in the down-gap-state, one in the up-gap-state. The photoexcitation of the polaron is that one electron is excited from the downgap-state to the up-gap-state by absorbing one photon. In photoexcited hole-polaron, there is one electron in the up-gap-state, no electron in down-one. In photoexcited electron-polaron, there are two electrons in the up-gap-state and one electron in down-one. The order parameter of the photoexcited polaron is [6]
kf ˜ ∆(y) 1 2 dk ∆ex (y) = −2g vf − π ε
(4)
For down-gap-state, 1 u−0 (y) = K0 (1 + i) sech K0 (y + y0 ) 4 + (1 − i) sech K0 (y − y0 ) , 1 v−0 (y) = − K0 (1 − i) sech K0 (y + y0 ) 4 + (1 + i) sech K0 (y − y0 ) .
iky
u+0 (y) = u∗−0 (y),
−kf
and electron’s wave functions are: (a) Gap states, εg = ±ε0 , ε0 = ∆˜ 20 − K02 vf2 .
u+0 (y) = u∗−0 (y),
(c) Conduction band, εc = −εv ,
and ε0 =
K0 = 0.993836 vf , ∆˜ 0
∆˜ 20 − K02 vf2 = 0.070 eV.
∆˜ 0 , vf
K. Chen et al. / Physics Letters A 294 (2002) 113–116
When γ = 0.08, θ = 1.47977, y0 = 1.55095
K0 = 0.99586
115
(a) Down-gap-state, 2 χ−0 (ω) = +0|p|−0 1 1 − × ω+0 − ω−0 − ω ω−0 − ω+0 − ω ci|p| − 0 2 +
∆˜ 0 , vf
vf , ∆˜ 0
and
ci
ε0 = 0.056 eV.
×
When γ = 0.04, θ = 1.52283, y0 = 1.86728
K0 = 0.99885
(15)
∆˜ 0 , vf
(b) Up-gap-state, 2 χ+0 (ω) = −0|p|+0 1 1 − × ω−0 − ω+0 − ω ω+0 − ω−0 − ω ci|p| + 0 2 +
vf , ∆˜ 0
and ε0 = 0.028 eV. We can see that ε0 is getting smaller when γ decreases, it indicates that the polarizability of the up-gap-state can become negative when γ is small enough. For the polymers with degenerate ground state (γ = 0), √ √ 2 2 vf −1 tanh , y0 → 2 2 ∆˜ 0 it means that the photoexcited polaron cannot be dissociated into two separated solitons. So the photoexcited polaron is still stable for the polymers with degenerate ground state. The dynamical polarizability can be calculated by means of linear response function [12], for the eigenstate m, m|p|n 2 χm (ω) = n
×
1 1 − ωn − ωm − ω ωm − ωn − ω
1 1 − . ωi1 − ω−0 − ω ω−0 − ωi1 − ω
, (13)
where p is the electric dipole: +∞ + + ey ψ1s (y)ψ1s (y) + ψ2s (y)ψ2s (y) dy. (14) p= −∞
By using above wave functions, we can calculate the polarizabilities of different eigenstates.
ci
1 1 − × . ωi1 − ω+0 − ω ω+0 − ωi1 − ω (16) (c) Valence band, 2 χvi (ω) = −0|p|vi 1 1 − × ω−0 − ωvi − ω ωvi − ω−0 − ω +0|p|vi 2 + vi
1 1 − × ωi1 − ω+0 − ω ω+0 − ωi1 − ω cj |p|vi 2 +
cj
1 1 − , × ωcj − ωvi − ω ωvi − ωcj − ω
(17) where |ci and |vi are the states in conduction band and valence band. The total polarizability of the polymeric molecule with a polaron is χ(ω) = n−0 χ−0 (ω) + n+0 χ+0 (ω) + nvi χvi (ω), vi
(18)
116
K. Chen et al. / Physics Letters A 294 (2002) 113–116
When γ = 0.1, the static polarizabilities of both photoexcited hole-polaron and electron-polaron become positive: αhole = 1.0 × 104(ea)2 /eV, αelectron = 4.4 × 103 (ea)2/eV. Conclusion:
Fig. 1. The dynamical polarizability of the excited electron-polaron.
where n+0 , n−0 and nvi are the populations in the down-gap-state, up-gap-state and valence band, respectively. For the photoexcited hole-polaron, χ(ω) = χ+0 (ω) + 2 (19) χvi (ω). vi
For the photoexcited electron-polaron, χvi (ω). χ(ω) = χ−0 (ω) + 2χ+0 + 2
1. The excited state of the polaron is stable in the polymer. 2. The static polarizability of the photoexcited polaron can be negative. 3. The PPR can occur in a polymeric molecule with a polaron, and it is an one-photon process.
Acknowledgements This work was supported by the Grant FRG 9900/II-53 from the Hong Kong Baptist University, the NSF of China by the Grants 20074007, 90103034, 10104006, and Shanghai Education Commission Grant 973-G1999075203.
(20)
vi
When γ = 0.04, the dynamical polarizability of the photoexcited electron-polaron is shown in Fig. 1. The static polarizability is α = χ(ω = 0). When γ = 0.04, the static polarizabilities of both photoexcited hole-polaron and electron-polaron are negative: αhole = −1.2 × 104 (ea)2/eV, αelectron = −3.9 × 104(ea)2 /eV. When γ = 0.08, αhole = 7.4 × 103 (ea)2/eV, αelectron = −1.4 × 103(ea)2 /eV. Here, αhole becomes positive, but αelectron remains negative.
References [1] V. Klimov, D.W. McBranch, N. Barashkov, J. Ferraris, Phys. Rev. B 58 (1998) 7654. [2] X. Sun, R.L. Fu, K. Yonemitsu, K. Nasu, Phys. Rev. Lett. 84 (2000) 2830. [3] A.J. Heeger, S. Kivelson, J.R. Schreiffer, W.P. Su, Rev. Mod. Phys. 60 (1988) 781. [4] H. Takayama, Y.R. Lin-Lin, K. Maki, Phys. Rev. B 21 (1980) 2388. [5] S.A. Brazovskii, N.N. Kirova, Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 6. [6] D.K. Campbell, A.R. Bishop, Nucl. Phys. B 200 (1982) 297. [7] A. Scott, F. Chu, D. Mclaughlin, Proc. IEEE 61 (1973) 1443. [8] W.A. Bardeen, M.S. Chanovitz, S.D. Drell, M. Weinstein, T.M. Yan, Phys. Rev. D 11 (1975) 1094. [9] S.-S. Shei, Phys. Rev. D 14 (1976) 535. [10] R.F. Dashen, D. Hasslacher, A. Neveu, Phys. Rev. D 12 (1975) 2443. [11] D.K. Campell, Y.-T. Liao, Phys. Rev. D 14 (1976) 2093. [12] P.M. Chaikin et al., Principle of Condensed Matter Physics, Cambridge University Press, 1995, Chapter 7.