© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Phys. Status Solidi B 246, No. 7, 1642–1645 (2009) / DOI 10.1002/pssb.200844388 p s sbasic solid state physics

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Structure and transition of charged excitons

Li Ge1, Thomas F. George2, and Xin Sun*, 1

1 Research Center for Quantum Manipulation and Department of Physics, Fudan University, Shanghai 200433, P.R. China 2 Office of the Chancellor and Center for Nanoscience, Departments of Chemistry & Biochemistry and Physics & Astronomy,

University of Missouri–St. Louis, St. Louis, Missouri 63121, USA

Received 26 September 2008, revised 31 December 2008, accepted 6 March 2009

Published online 12 May 2009

PACS 71.20.Rv, 71.35.Pq, 78.66.Qn

* Corresponding author: e-mail xin_sun@fudan.edu.cn

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Different from a metal or semicon-ductor where carriers are fermions (electrons or holes, with spin s = 1/2 and charge q = ±e), in a conducting polymer, carriers can be either fermions or bosons. The former are polarons, and the latter are charged solitons (in degenerate polymers such as trans-(CH)

x) or bipolarons (in nondegen-

erate polymers such as cis-(CH)x) [1, 2]. The bipolaron is a

bound state of two electrons or holes (with opposite spins) via the electron–lattice interaction. Actually, B is a local-ized Cooper pair. Since a polymer chain is very flexible, the bond structure of the lattice can easily be distorted, and two electrons can be self-trapped under the electron–lattice interaction to form a bound state B. This means that the electron–lattice interaction produces an indirect attrac-tion between electrons, which overcomes the direct Cou-lomb repulsion. Although the formation probability of a bipolaron is lower than that of a polaron, B plays important roles in electrical and optical processes of polymers [3–12]. For example, the slow formation of a hole bipolaron produces a bias-stress effect in polymeric thin-film transistors [3]. It is also proposed the participation of bipolarons can open one more channel of luminescence [13]. Electroluminescence is produced by the recombination of excitons. When a positive polaron meets a negative po-laron, they form a neutral exciton with spin 0s = (singlet)

or 1s = (triplet). Since the triplet is non-emissive, the elec-troluminescence efficiency is limited to 25%. However, in nondegenerate polymers, both the polaron and bipolaron are carriers. When a positive (negative) bipolaron meets a negative (positive) polaron, they also form a bound state, but it is no longer a neutral exciton with 0s = or 1. Rather, it is a charged exciton, and its spin is 1/2. This means there is no triplet in the charged exciton, which has only two com-ponents with s = ±1/2, and this fact favors luminescence. This paper provides the bond structure and charge dis-tribution of the charged exciton. The transition probabili-ties of this kind of exciton are also presented. 2 Structure of a charged exciton In polymer the-ory, the bipolaron and exciton are commonly described by the SSH (Su–Schrieffer–Heeger) model [1] including the extended Hubbard terms:

e a,H H H= +

e 0 1 e 1, ,

,

( 1) ( ) ( 1) ( H.c.)l l

l l l s l s

l s

H t t a aα φ φ +

+ += - + - + + - +È ˘Î ˚Â

, , 1

,

1 1( 1) ( 1) ,

2 2 2 2l s l s l l

l s l

U Vn n n n

- +

Ê ˆ Ê ˆ+ - - + - -Ë ¯ Ë ¯Â Â

2

2

a 1

1 d( ) .

2 d

l

l l

l

H k Mt

φφ φ

+

È ˘Ê ˆ= + +Í ˙Ë ¯Î ˚ (1)

In a polymer, a positive polaron p+ attracts a negative polaron

p– to form an exciton, which is neutral and has spin s = 0

(singlet, emissive) or s = 1 (triplet, non-emissive). However,

in non-degenerate polymers, bipolarons (B) which are doubly

charged and spinless are also carriers, such that B2+ (B2–) at-

tracts p– (p+) to form a new kind of exciton – the charged ex-

citon. Since the charged exciton has spin s = 1/2, the triplet

state does not appear and is emissive. The structure and tran-

sition probability of the charged exciton are presented in this

paper.

Phys. Status Solidi B 246, No. 7 (2009) 1643

www.pss-b.com © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

Here, H consists of the electron Hamiltonian e

H and lattice energy

aH .

eH describes the electron–lattice coupling and

electron interaction, in which ,l s

a+

,

( )l sa is the creation (an-

nihilation) operator of a π electron with spin s at the l th lat-tice site,

, , ,l s l s l sn a a

+

= is the electron number, 0t is the trans-

fer integral between the nearest-neighbor sites, α is the electron–phonon coupling, ( 1)l

l luφ = - , where

lu is the

displacement of the l th lattice site from the equilibrium po-sition,

et is a confinement parameter, which characterizes

the non-degeneracy of the conjugated polymer, and U and V are the on-site and nearest-neighbor Coulomb repulsion strengths. In

aH , k is the spring constant, and M is the

mass of a lattice unit. In conjugated polymers, the width of the energy band is W > 10 eV, which is much larger than U and V (in this paper U = 3 eV and V = 0.5 eV), so that the electron–electron interaction can be treated by the Hartree–Fock approximation. Here it should be mentioned that while in Ref. [13] the collision of a positive bipolaron 2

B+ and a

negative polaron p- in an external electric field was stud-ied, the electron–electron interaction was not considered, so that Ref. [13] does not give information about the bound state of 2

B+ and p- . This present paper includes the elec-

tron–electron interaction and provides the structure and transitions of the charged exciton. Because the mass of an electron is much smaller than that of a lattice unit, the electronic motion is an adiabatic process. At any instant, electrons are in the eigenstates of

eH :

{ }( )e

.

lH

µ µ µψ ε φ ψ= (2)

The total energy of the polymer chain in any configuration

{ }l

φ is

{ }( ) { }( ) 21

tot 12

occ

( ) .l l l l

l

E kµ

φ ε φ φ φ+

= + +Â Â (3)

Then the force exerting on the l th lattice unit is

tot,

l

l

EF

φ

∂= -

∂ (4)

Y 1

Y 4

Y 3

Y 2

Figure 1 Electronic energy spectrum of the charged exciton.

0 50 100 150 200−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Site

F/0.1nm

Figure 2 (online colour at: www.pss-b.com) Lattice configura-

tion of the charged exciton.

and the motion of the lattice unit is determined by

d,

d

l

l

vM F

t=

d.

d

l

lv

t

φ= (5)

Equations (2) and (5) should be solved self-consistently. In a numerical calculation, the time step tD must be much smaller than the period of an atomic atom oscillation, T. In our case, T ~ 100 fs and Δt = 0.5 fs. When a positive bipolaron meets a negative polaron, they merge into a charged exciton. The electronic energy spectrum and lattice configuration of the charged exciton can be obtained by solving Eqs. (2)–(5), where the results are shown in Figs. 1 and 2. In the spectrum of the charged exciton (Fig. 1), there are four gap states:

1ψ is occupied by

one electron, 4

ψ is occupied by two electrons, and 2

ψ and

3ψ are empty (these four wave functions are shown in Fig. 3). In the lattice configuration (Fig. 2), there is a bump at the bottom of the valley, which is a signature of the merger of the polaron and bipolaron. From the obtained wave functions of the electrons, Eq. (2), the charge distribution of the charged exciton can be determined by

2

occ

( )n

nµ

ρ ψ=Â (6)

and the result is shown in Fig. 4. 3 Transitions of a charged exciton In the energy spectrum of the charged exciton (Fig. 1), the parities of the electronic states are changed alternatively. In particular,

1ψ

and 3

ψ are antisymmetric, 2

ψ and 4

ψ are symmetric. Therefore, the transition from

1ψ to

2ψ is dipole allowed,

and that from 1

ψ to 3

ψ is forbidden. For the charged exci-ton, there is one electron staying in

1,ψ no electron in

2ψ

and 3

ψ , and 4

ψ and the valence band are fully filled, so that the electron in

1ψ can only transit to

2.ψ The matrix ele-

ment of the dipole transition 12r and the lifetime

1τ staying

1644 Li Ge et al.: Structure and transition of charged excitons

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

ph

ysic

ap s sstat

us

solid

i b

0 50 100 150 200−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Site

Y1

0 50 100 150 200−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Y2

Site

0 50 100 150 200−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Site

Y3

0 50 100 150 200−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25Y

4

Site

Figure 3 (online colour at: www.pss-b.com) Wave functions of gap states. in

1ψ can be calculated from the wave functions

1ψ and

2ψ

as solutions of Eq. (2):

3

8

1 23 2

12 12

35 10 s ,

4

c

e r

τ

ω

�-

= = ¥ (7)

where 12

ω� is the energy difference between the states 1

ψ and

2ψ (0.5–1.0 eV, where the exact value depends on the

characteristics of the polymer). The result indicates that the

0 50 100 150 200−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Figure 4 (online colour at: www.pss-b.com) Charge distribution

of the charged exciton.

electron in 1

ψ transits to 2

ψ very quickly, after which the lattice configuration and energy spectrum are changed. Figure 5 is the resultant energy spectrum, which is just an excited polaronp*. The relaxation process of the lattice configuration from the charged exciton to the excited pola-ron is shown in Fig. 6. For the excited polaron p*, there are two gap states,

2ψ

and 3

ψ (Fig. 5), where 2

ψ is occupied by one electron and

3ψ is empty. Since they have opposite parities, the electron in

2ψ will transit to

3ψ by emitting one photon, and the ex-

cited polaron returns to a normal polaron. From 2

ψ and 3

ψ , the matrix element and the transition probability

23A of the

Y 1

Y 4

Y 3

Y 2

Figure 5 Electronic energy spectrum of the excited polaron.

Phys. Status Solidi B 246, No. 7 (2009) 1645

www.pss-b.com © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

050

100150

200

0

5

10

15

20

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

t/0.5fsSite

F/0.1nm

Figure 6 (online colour at: www.pss-b.com) Time evolution of

the lattice configuration from the charged exciton to the excited

polaron.

excited polaron can be obtained as

23 2

823 23

23 3

42 10 /s ,

3

e rA

c

ω

�= = ¥ (8)

where 23

ω� is the energy difference between the states 2

ψ and

3,ψ and its value is 1.0–2.0 eV. Therefore, the transi-

tion of a charged exciton involves a two-step emission, where the first step is from a charged exciton to an excited polaron, and the second is from the excited polaron to a normal polaron. These results tell us that, as well as the singlet neutral exciton, the charged exciton is emissive. Recently, some experiments have found that the electrolu-minescence efficiency η ranges from 22% to 63% [14–18], indicating that η can exceed 25%. The charged exciton proposed here may provide some contribution to this en-hancement. 4 Conclusion In conjugated polymers, when a posi-tive (negative) bipolaron meets a negative (positive) pola-ron, they form a charged exciton which is different from the neutral exciton. The neutral exciton has spin 0 (singlet) or 1 (triplet), where the latter is non-emissive and heavily suppresses the electroluminescence efficiency. However, the charged exciton has spin s = 1/2 and no triplet, being emissive.

Acknowledgments This work was supported by the Na-

tional Natural Science Foundation of China under Grant

20674010 and the Doctoral Foundation Program of the Chinese

Ministry of Education.

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