- Email: [email protected]
Exciton polarons in molecular aggregates: a dynamical coherent potential approximation approach
16 April 1999
Chemical Physics Letters 303 Ž1999. 649–656
Exciton polarons in molecular aggregates: a dynamical coherent potential approximation approach Tsuyoshi Kato ) , Fumio Sasaki, Shunsuke Kobayashi Electrotechnical Laboratory, 1-1-4, Umezono, Tsukuba 305-8568, Japan Received 22 October 1998; in final form 15 February 1999
Abstract A dynamical coherent potential approximation ŽDCPA. is formulated within the Hartree approximation for an exciton interacting with phonons. Two-exciton polaron spectra can be calculated by the present method. For the one-exciton interacting with Einstein phonons by a linear short-range coupling, it is shown that the present formulation is equivalent to the well-known single-site approximation DCPA. For a nonlocal exciton–phonon coupling, the result is compared with a variational band theory and they show good agreement. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction There has recently been interest in optical properties of molecular aggregates because of their possible applications as optical devices. For example, the J-aggregate of pseudo-isocyanine bromide ŽPIC–Br. is one such aggregate that has been intensively studied w1x since its discovery. Spectroscopic studies of J-aggregates including the absorption and nonlinear spectroscopies, such as pump–probe experiments, have provided interesting results w2–5x. To interpret those results, many theoretical models have been proposed based on the Frenkel exciton model w6–11x. Concerning the vibronic structure of one-exciton states, i.e., exciton polarons, detailed studies have clarified the various types of exciton polarons appearing in the absorption spectra w12,13x. However, the vibronic structure of multi-exciton states, multi-exciton polarons, has not received much attention, although the formation of the multi-exciton polaron affects nonlinear optical responses through peak shifts and intensity redistributions. There remain a number of questions concerning the multi-exciton polaron such as its dispersion relation. From a fundamental point of view, knowledge of the multi-exciton polaron is also important. It may shed light on the relation between the molecular electronic coupling and the effective inter-excitation force modulated by the exciton–phonon coupling, as well as providing the first step for analyzing the dynamics in the multi-exciton manifold. In this Letter, we report a new formulation of the dynamical coherent potential approximation based on the Hartree approximation ŽHA–DCPA. for the one- and two-exciton polarons. Various types of exciton–phonon
)
Corresponding author: Fax: q81 298 54 5459; e-mail: [email protected]
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 2 3 4 - 1
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
650
interactions Ži.e., short- or long-range. and the phonon dispersion can be treated by the present theory. It is shown that the present formulation is equivalent to the single-site approximation ŽSSA. DCPA w12x when we consider the one-exciton interacting with Einstein phonons by a linear short-range coupling. For the nonlocal exciton–phonon coupling, our result is compared with a variational band theory and they show good agreement. The performance of our approach is shown by calculating the two-exciton polaron dispersion of the PIC–Br J-aggregate.
2. Theoretical 2.1. The model We start with the molecular crystal Hamiltonian for an aggregate composed of N two-level molecules interacting with phonons, Hˆ s Hˆex q Hˆph q Hˆex – ph ,
Ž 1.
where Hˆex , Hˆph and Hˆex – ph denote the exciton Hamiltonian, the phonon Hamiltonian and the linear exciton–phonon interaction, respectively. Hˆex s ´ Ý a†n a n q
Ý
n
n, msn"1
t n m a†n a m ,
Ž 2a .
Hˆph s Ý " Va ba† ba ,
Ž 2b .
a
and Hˆex – ph s Ý HˆaX .
Ž 2c .
a
In Eq. Ž2a., a†n Ž a n . creates Žannihilates. the molecular excitation at site n. The molecular transition energy and the intermolecular excitation transfer energy are denoted by ´ and t n m , respectively. Under the cyclic boundary condition, Hˆex is diagonalized by Frenkel exciton operators, e.g., a†k s Ž1r 'N . Ý n expŽi kn. a†n for the creation of one-exciton states and a†K k s Ž2rN .Ý n ) m exp i K Ž n q m.4 sin k Ž n y m.4 a†n a†m for two-exciton states w14x. k s 2 jprN Ž j s 0," 1," 2, . . . . for one-exciton states, and K s hprN Ž h s 0," 1," 2, . . . . and k s lprN Ž l s 1,2, . . . , N y 1. for two-exciton states where the parity of h and l are opposite. j and h are defined in wyŽ N y 1.r2, Ž N y 1..r2x for odd N and in wyŽ N y 2.r2, Nr2x for even N. In Eqs. Ž2b. and Ž2c., a specifies the phonon mode, e.g., momentum or site, and Va 4 represents its dispersion. ba 4 and bb† 4 satisfy the Bose commutation relation w ba , bb† x s da , b . The linear exciton–phonon interaction can be written in the form of HˆaX s
Ý ½ f qŽqa . ba q f˜qŽqa . ba† 5 a†q a q X
q, q
X
,
X
Ž 3.
X
with coupling constants f qŽ ,aq.X 4 and f˜qŽ ,aq.X . We use the suffix q as the exciton quantum number for both the one- and two-exciton manifolds, since the Hamiltonian Ž1. conserves the number of excitationŽs., thus the oneand two-exciton subspaces are decoupled.
½
5
2.2. The effectiÕe Hamiltonian and the condition for the coherent potential To calculate the exciton self-energy Žcoherent potential., we define the effective Hamiltonian by using the undetermined self-energy matrix, w s Ž z .x q, qX s s Ž z .q, qX , as w15x Hˆeff Ž z . s
Ý ´ q dq , q q s Ž z . q , q 4 a†q a q q Hˆph , X
q, q
X
X
X
Ž 4.
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
651
where the complex energy z is composed of the exciton energy E and the exciton damping constant g ) 0, z s E q ig . ´ q are the exciton eigenvalues. Eq. Ž1. is rewritten by using Eqs. Ž2c. and Ž4. as Hˆ s Hˆeff zX y Hˆph q Ý D HˆaX Ž zX .
ž
/
Ž 5a .
a
with D HˆaX Ž zX . s
Ý ½ f qŽqa . ba q f˜qŽqa . ba† y s Ža . ž zX y Hˆph / q , q 5 a†q a q X
q, q
X
X
X
,
Ž 5b .
X
where we use an arbitrary notation for the self-energy matrix as Žsee Eq. Ž12..
s Ž z . s Ý s Ža .Ž z . .
Ž 5c .
a
Note that RŽ zX . represents the total Žexciton q phonon. energy in Eqs. Ž5a. and Ž5b.. According to the ordinary CPA w16,17x, the coherent potential is chosen so that the thermal averaged t-matrix vanishes when we consider Hˆeff as an unperturbed Hamiltonian and Ý a D HˆaX as a perturbation in the right-hand side of Eq. Ž5a.. For such a partition, the t-matrix is given as Tˆ Ž zX . s Ý tˆ Ž a . Ž zX . q Ý tˆ Ž a . Ž zX . Gˆeff Ž zX . a
a
Ý tˆ Ž b . Ž zX . q . . . ,
Ž 6.
b Ž/ a .
where Gˆeff is the retarded Green’s operator for the unperturbed system, Gˆeff Ž zX . s 1rw zX y Hˆeff Ž zX .x, and tˆŽ a . Ž zX . is the t-matrix defined by the unperturbed Hamiltonian Hˆeff and a perturbation D HˆaX . The CPA condition mentioned above is expressed as ²Tˆ Ž zX . : s 0
Ž 7.
where ² . . . : stands for the phonon thermal average Žthroughout this Letter, we do not distinguish exciton operators and their matrix representations.. Here, we adopt the Hartree approximation, neglecting the phonon mode correlation. Eq. Ž7. is approximated by ² tˆ Ž a . Ž zX . : ' t Ž a . Ž z . s 0 ;a .
Ž 8.
The approximated CPA condition for the thermal averaged t-matrix t Ž a . Ž z . is expressed in terms of Green’s functions, Geff Ž z . and g Ž a . Ž z .4 , as Geff Ž z . s g Ž a . Ž z .
;a .
Ž 9.
In Eq. Ž9., Geff Ž z . is given as Geff Ž z .
q,q
X
¦
s vac a q
1 zy
Ý ´ p d p , p q s Ž z . p , p 4 a†p a p . X
X
p, p
X
;
a†qX vac ,
Ž 10 .
X
where vac : is the exciton vacuum. The Green’s function g Ž a . Ž z . is given by taking thermal average of the Green’s operator defined by gˆ Ž a . Ž zX . s
1 z y hˆ X
½ ž z y Hˆ / q Hˆ q Hˆ 5 Ža .
X
X a
ph
,
Ž 11a.
ph
with hˆ Ž a . Ž z . s
Ý ´ q dq , q q s˜ Ž a . Ž z . q , q 4 a†q a q X
q, q
X
X
X
,
Ž 11b.
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
652
where a new notation s˜ Ž a . Ž z . s s Ž z . y s Ž a . Ž z . is used. Geff Ž z . describes the exciton in the coherent potential s , and g Ž a . Ž z . ' ² gˆ Ž a . Ž zX .: represents the exciton in the auxiliary potential s˜ Ž a . Ž z . that is induced by the coupling with all the phonon modes b 4 other than the relevant coupling mode a . From Eqs. Ž9. to Ž11a. we can see that our approximation is nothing but finding wavevector-dependent mean fields for the exciton. The calculation of the matrix g Ž a . Ž z . can be carried out by the continued fraction expansion method originally developed in the SSA–DCPA formulation w12,18,19x. Combined with the expression of Geff Ž z . and using the relation of Eq. Ž9., for instance, we get the closed expression for the self-energy matrix at absolute zero temperature as 1
s Ž z . s Ý FŽa . P a
g 0Ž a .
Ž z y " Va .
y1
yF
Ža .
P
P F˜ Ž a . ,
2 g 0Ž a . Ž z y 2 " Va .
y1
y PPP
Ž 12 .
P F˜ Ž a .
with the coupling matrices w F Ž a . x q, qX s f q,Ž aqX. and w F˜ Ž a . x q, qX s f˜q,Ž aqX.. The Green’s function g 0Ž a . Ž z . is defined by g 0Ž a . Ž z .
q,q
X
¦
s vac a q
1 z y hˆ
Ža .
Ž z.
;
a†qX vac .
Ž 13 .
Eq. Ž12. is a self-consistent one for the coherent potential. Since s Ž z . Žor s˜ Ž a . Ž z .. is determined by that are expressed in terms of s˜ Ž a . Ž zX .Ž RŽ zX . - RŽ z .. as seen in Eqs. Ž11b. and Ž13.. At a finite temperature, the expression of g Ž a . Ž z . includes the phonon absorption term, therefore the coherent potential at energy RŽ z . must be determined consistently with the coherent potentials at energy EX ) RŽ z . and EXX - RŽ z .. An iterative procedure is required in this case w20x. Once the self-energy matrix is obtained, the partial density of qth exciton state is calculated by g 0Ž a . Ž zX .4
rq Ž E . s y
1 p
¦
I vac a q
1 zy
Ý ´ p d p , p q s Ž z . p , p 4 a†p a p . X
X
p, p
X
;
a†q vac .
Ž 14 .
X
Consider one-exciton states interacting with Einstein phonons by a linear short-range coupling expressed as HˆnX s 'S " V Ž bn q bn† . a†n a n ,
Ž 15a.
s Ž 'S rN . " V Ý exp yi Ž k y kX . n4 Ž bn q bn† . a†k a k X , k,k
Ž 15b.
X
where S is the Huang–Rhys factor Ž a specifies the site n of local dispersionless phonons, Va 4 s V in Eq. Ž2b... It is apparent that the single-site approximation applied in Ref. w12x for the one-exciton polaron can be regarded as a Hartree approximation with respect to site-localized phonons. Therefore, the SSA–DCPA and the HA–DCPA result in the same coherent potential. It is easier to calculate the coherent potential by the SSA–DCPA than by the HA–DCPA in this case. However, our approach can treat more general exciton–phonon couplings ŽEinstein and dispersive phonons, local and nonlocal coupling. and be applied to two-exciton polaron problems from a unified viewpoint. We point out that if translational symmetry is assumed, the coherent potential has following properties under the Hartree approximation, with respect to site localized phonons w20x:
s˜ Ž n . Ž z . s Ž z.
X
k,k
X
s eyiŽ kyk . s˜ Ž ny1. Ž z .
k,k
X
s d k , kX s Ž z .
k,k
k,k
X
,
5
Ž 16 .
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
653
for one-exciton states, and
s˜ Ž n . Ž z . s Ž z.
X
K k , K ,k
X
s ey2 iŽ KyK
X
K k , K ,k
X
X
.
s˜ Ž ny1. Ž z .
s dK , K X s Ž z .
K k , K ,k
X
X
K k , K ,k
X
,
5
Ž 17 .
for two-exciton states. These properties reduce the costs of the coherent potential calculations.
3. Numerical calculations The effects of the simultaneous presence of the local and nonlocal exciton–phonon couplings on the formation of excited bound states has been investigated by using the variational band theory w21,22x. It was found that for certain values of the coupling parameter the lowest one-exciton polaron band shows a nontrivial shape, double minimum with respect to the momentum coordinate. Here, we consider the same exciton–phonon interaction studied in Ref. w21x, HˆnX s " V Ž bn q bn† .
'S a†n a n q U a†nq1 a n q a†n a nq1 y Ž a†ny1 a n q a†n a ny1 . 4
.
Ž 18 .
A new coupling parameter U comes from a dependence of the exciton transfer energies on lattice coordinates, such as librations that promote the exciton transfers between neighboring molecules. The one-exciton eigenenergy ´ k s ´ y B cos k is assumed; B s 2 t n, n " 1 ) 0 is the half-bandwidth of one-dimensional J-aggregates. We consider two correlative phonon pictures in the HA–DCPA calculations: Ž1. site-localized phonons and Ž2.
Fig. 1. The lowest one-exciton polaron dispersion for Ž Br " V ,S,U . s Ž1.0,1.0,0.5., g r " V s 0.01 and N s 25. Large circles, triangles, small circles and squares show the results obtained by the HA–DCPA with localized phonons, the HA–DCPA with delocalized phonons, the variational principle and the bare exciton dispersion, respectively.
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
654
delocalized phonons whose creation operator is given by b†k s Ž1r 'N . Ý n expŽi kn. bn†. The difference between these two pictures does not affect the results obtained from the variational approach, since the two phonon pictures are related by a unitary transformation. The trial wavefunction for the lowest one-exciton polaron with the crystal momentum k is represented by < k:s
Ý exp Ž i kn . a nk yn a†n exp X
X
n, n
X
½y Ý Ž b m
k † k) myn bm y b myn bm
.
5 <0: ,
Ž 19 .
where a nk 4 and bnk 4 are the variational parameters and <0: is the vacuum for the exciton and phonons w21,23x. On the other hand, as is apparent from Eq. Ž8., i.e., each phonon mode is treated independently, the two pictures yield different results within the HA–DCPA. Examples of calculated lowest one-exciton polaron dispersion for a parameter set Ž Br" V ,S,U . s Ž1.0,1.0,0.5 . are shown in Fig. 1. We see that the variational approach and the HA–DCPA with respect to local phonons show good agreement. It might be said that our approximation leads to a proper exciton polaron dispersion, since almost the same results are obtained by two very different methods. The delocalized phonon-based HA–DCPA does not give a similar result. According to a general variational principle, the existence of two solutions for the band shape, i.e., two sets of the lowest exciton polaron eigenenergies, implies that the spatial spread of the phonon can be regarded as a variational parameter. It is expected that an introduction of such a parameter into the present method will remove the ambiguity in choosing the phonon basis, and improve the exciton self-energy. In other words, starting from the local phonon basis, i.e., a single-site approximation, a proper extension of the spatial spread of phonons is expected to take into account the correlated exciton scattering among different sites.
Fig. 2. The two-exciton polaron dispersion for Ž Br " V ,S,U . s Ž0.9,0.5,0., g r " V s 0.05 and N s14. The closed and open circles represent the lowest polarons for each quantum number Ž K, k . and higher energy Žquasi-continuum nŽ / 0.-phonon. polaron states, respectively. Note that the abscissa represents the crystal momentum 2 K conjugate to the center of mass for a pair of excitations.
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
655
Fig. 2 shows the lowest two-exciton polaron dispersion calculated by the local phonon-based HA–DCPA with the on-site exciton–phonon interaction expressed as Eq. Ž15a.. A parameter set appropriate for the PIC–Br J-aggregate, Ž Br" V ,S,U . s Ž0.9,0.5,0. w24x, and the two-exciton eigenenergy, ´ K , k s ´ Kq k q ´ Ky k w14x, are assumed. We can see that the bandwidth is renormalized about the half of the bare two-excitons Ž, 4 B . by the polaron effect. A qualitative difference from the lowest one-exciton polaron dispersion is found. Although the lowest one-exciton polaron band is completely separated from its one-phonon continuum above it, only about half of the lowest polaron states are located below the one-phonon continuum in two-exciton polaron manifold w24x.
4. Summary We have developed a new theoretical scheme to obtain the exciton Green’s function based on the Hartree approximation ŽHA–DCPA. for an exciton interacting with phonons. Our method is capable of treating general linear exciton–phonon coupling in one- and two-exciton manifolds from a unified viewpoint. The local phonon-based HA–DCPA is shown to account for the lowest one-exciton polaron band shape for the nonlocal exciton–phonon coupling reported by Zhao et al. w21x. The calculation of optical spectra within the present scheme is straightforward. For example, the two-photon absorption spectrum is calculated by introducing appropriate vertex parts for the three-particle Green’s function composed of the one- and two-exciton polaron Green’s functions w20x. Since our approach is a mean field approximation, the correlated exciton scattering is treated incorrectly. To improve this, a formulation that takes into account the spatial spread of phonons is suggested.
Acknowledgements One of the authors ŽTK. would like to thank Dr. Shuji Abe of the Electrotechnical Laboratory for his helpful suggestions. The support of the Japan Science and Technology Corporation is gratefully acknowledged.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x
T. Kobayashi ŽEd.., World Scientific, Singapore, 1996, and references therein. B. Kopainsky, J.K. Hallermeier, W. Kaiser, Chem. Phys. Lett. 83 Ž1981. 498. H. Fidder, D.A. Wiersma, Phys. Stat. Sol. Žb. 188 Ž1995. 285. F. Sasaki, S. Kobayashi, Appl. Phys. Lett. 63 Ž1993. 2887. K. Misawa, S. Machida, K. Horie, T. Kobayashi, J. Luminesc. 64 Ž1995. 239. P.O.J. Scherer, S.F. Fischer, Chem. Phys. 86 Ž1984. 269. H. Fidder, J. Knoester, D.A. Wiersma, J. Chem. Phys. 95 Ž1991. 7880. E.W. Knapp, Chem. Phys. 85 Ž1984. 73. J. Knoester, J. Chem. Phys. 99 Ž1993. 8466. F.C. Spano, Chem. Phys. Lett. 234 Ž1995. 29. F.C. Spano, J.R. Kuklinski, S. Mukamel, Phys. Rev. Lett. 65 Ž1990. 211. H. Sumi, J. Phys. Soc. Jpn. 36 Ž1974. 770. V.L. Broude, E.I. Rashba, E.F. Sheka, in: Spectroscopy of Molecular Excitons, Springer Series in Chemical Physics, vol. 16, Springer, Berlin, 1985. H. Ishihara, K. Cho, Phys. Rev. B 42 Ž1990. 1724. S. Abe, J. Phys. Soc. Jpn. 57 Ž1988. 4029. B. Velicky, S. Kirkpatrick, H. Ehrenreich, Phys. Rev. 175 Ž1968. 747. E.N. Economou, in: Green’s Function in Quantum Physics, Springer Series in Solid-State Sciences, vol. 7, Springer, Berlin, 1983. R.E. Lagos, R.A. Friesner, J. Chem. Phys. 81 Ž1984. 5899.
656 w19x w20x w21x w22x w23x w24x
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656 Y. Won, R.E. Lagos, R.A. Friesner, J. Chem. Phys. 84 Ž1986. 6567. T. Kato, in preparation. Y. Zhao, D.W. Brown, K. Lindenberg, J. Chem. Phys. 106 Ž1997. 2728. Y. Zhao, D.W. Brown, K. Lindenberg, J. Chem. Phys. 100 Ž1994. 2335. Y. Toyozawa, Prog. Theor. Phys. 26 Ž1961. 29. T. Kato, F. Sasaki, S. Abe, S. Kobayashi, Chem. Phys. 230 Ž1998. 209.
Chemical Physics Letters 303 Ž1999. 649–656
Exciton polarons in molecular aggregates: a dynamical coherent potential approximation approach Tsuyoshi Kato ) , Fumio Sasaki, Shunsuke Kobayashi Electrotechnical Laboratory, 1-1-4, Umezono, Tsukuba 305-8568, Japan Received 22 October 1998; in final form 15 February 1999
Abstract A dynamical coherent potential approximation ŽDCPA. is formulated within the Hartree approximation for an exciton interacting with phonons. Two-exciton polaron spectra can be calculated by the present method. For the one-exciton interacting with Einstein phonons by a linear short-range coupling, it is shown that the present formulation is equivalent to the well-known single-site approximation DCPA. For a nonlocal exciton–phonon coupling, the result is compared with a variational band theory and they show good agreement. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction There has recently been interest in optical properties of molecular aggregates because of their possible applications as optical devices. For example, the J-aggregate of pseudo-isocyanine bromide ŽPIC–Br. is one such aggregate that has been intensively studied w1x since its discovery. Spectroscopic studies of J-aggregates including the absorption and nonlinear spectroscopies, such as pump–probe experiments, have provided interesting results w2–5x. To interpret those results, many theoretical models have been proposed based on the Frenkel exciton model w6–11x. Concerning the vibronic structure of one-exciton states, i.e., exciton polarons, detailed studies have clarified the various types of exciton polarons appearing in the absorption spectra w12,13x. However, the vibronic structure of multi-exciton states, multi-exciton polarons, has not received much attention, although the formation of the multi-exciton polaron affects nonlinear optical responses through peak shifts and intensity redistributions. There remain a number of questions concerning the multi-exciton polaron such as its dispersion relation. From a fundamental point of view, knowledge of the multi-exciton polaron is also important. It may shed light on the relation between the molecular electronic coupling and the effective inter-excitation force modulated by the exciton–phonon coupling, as well as providing the first step for analyzing the dynamics in the multi-exciton manifold. In this Letter, we report a new formulation of the dynamical coherent potential approximation based on the Hartree approximation ŽHA–DCPA. for the one- and two-exciton polarons. Various types of exciton–phonon
)
Corresponding author: Fax: q81 298 54 5459; e-mail: [email protected]
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 2 3 4 - 1
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
650
interactions Ži.e., short- or long-range. and the phonon dispersion can be treated by the present theory. It is shown that the present formulation is equivalent to the single-site approximation ŽSSA. DCPA w12x when we consider the one-exciton interacting with Einstein phonons by a linear short-range coupling. For the nonlocal exciton–phonon coupling, our result is compared with a variational band theory and they show good agreement. The performance of our approach is shown by calculating the two-exciton polaron dispersion of the PIC–Br J-aggregate.
2. Theoretical 2.1. The model We start with the molecular crystal Hamiltonian for an aggregate composed of N two-level molecules interacting with phonons, Hˆ s Hˆex q Hˆph q Hˆex – ph ,
Ž 1.
where Hˆex , Hˆph and Hˆex – ph denote the exciton Hamiltonian, the phonon Hamiltonian and the linear exciton–phonon interaction, respectively. Hˆex s ´ Ý a†n a n q
Ý
n
n, msn"1
t n m a†n a m ,
Ž 2a .
Hˆph s Ý " Va ba† ba ,
Ž 2b .
a
and Hˆex – ph s Ý HˆaX .
Ž 2c .
a
In Eq. Ž2a., a†n Ž a n . creates Žannihilates. the molecular excitation at site n. The molecular transition energy and the intermolecular excitation transfer energy are denoted by ´ and t n m , respectively. Under the cyclic boundary condition, Hˆex is diagonalized by Frenkel exciton operators, e.g., a†k s Ž1r 'N . Ý n expŽi kn. a†n for the creation of one-exciton states and a†K k s Ž2rN .Ý n ) m exp i K Ž n q m.4 sin k Ž n y m.4 a†n a†m for two-exciton states w14x. k s 2 jprN Ž j s 0," 1," 2, . . . . for one-exciton states, and K s hprN Ž h s 0," 1," 2, . . . . and k s lprN Ž l s 1,2, . . . , N y 1. for two-exciton states where the parity of h and l are opposite. j and h are defined in wyŽ N y 1.r2, Ž N y 1..r2x for odd N and in wyŽ N y 2.r2, Nr2x for even N. In Eqs. Ž2b. and Ž2c., a specifies the phonon mode, e.g., momentum or site, and Va 4 represents its dispersion. ba 4 and bb† 4 satisfy the Bose commutation relation w ba , bb† x s da , b . The linear exciton–phonon interaction can be written in the form of HˆaX s
Ý ½ f qŽqa . ba q f˜qŽqa . ba† 5 a†q a q X
q, q
X
,
X
Ž 3.
X
with coupling constants f qŽ ,aq.X 4 and f˜qŽ ,aq.X . We use the suffix q as the exciton quantum number for both the one- and two-exciton manifolds, since the Hamiltonian Ž1. conserves the number of excitationŽs., thus the oneand two-exciton subspaces are decoupled.
½
5
2.2. The effectiÕe Hamiltonian and the condition for the coherent potential To calculate the exciton self-energy Žcoherent potential., we define the effective Hamiltonian by using the undetermined self-energy matrix, w s Ž z .x q, qX s s Ž z .q, qX , as w15x Hˆeff Ž z . s
Ý ´ q dq , q q s Ž z . q , q 4 a†q a q q Hˆph , X
q, q
X
X
X
Ž 4.
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
651
where the complex energy z is composed of the exciton energy E and the exciton damping constant g ) 0, z s E q ig . ´ q are the exciton eigenvalues. Eq. Ž1. is rewritten by using Eqs. Ž2c. and Ž4. as Hˆ s Hˆeff zX y Hˆph q Ý D HˆaX Ž zX .
ž
/
Ž 5a .
a
with D HˆaX Ž zX . s
Ý ½ f qŽqa . ba q f˜qŽqa . ba† y s Ža . ž zX y Hˆph / q , q 5 a†q a q X
q, q
X
X
X
,
Ž 5b .
X
where we use an arbitrary notation for the self-energy matrix as Žsee Eq. Ž12..
s Ž z . s Ý s Ža .Ž z . .
Ž 5c .
a
Note that RŽ zX . represents the total Žexciton q phonon. energy in Eqs. Ž5a. and Ž5b.. According to the ordinary CPA w16,17x, the coherent potential is chosen so that the thermal averaged t-matrix vanishes when we consider Hˆeff as an unperturbed Hamiltonian and Ý a D HˆaX as a perturbation in the right-hand side of Eq. Ž5a.. For such a partition, the t-matrix is given as Tˆ Ž zX . s Ý tˆ Ž a . Ž zX . q Ý tˆ Ž a . Ž zX . Gˆeff Ž zX . a
a
Ý tˆ Ž b . Ž zX . q . . . ,
Ž 6.
b Ž/ a .
where Gˆeff is the retarded Green’s operator for the unperturbed system, Gˆeff Ž zX . s 1rw zX y Hˆeff Ž zX .x, and tˆŽ a . Ž zX . is the t-matrix defined by the unperturbed Hamiltonian Hˆeff and a perturbation D HˆaX . The CPA condition mentioned above is expressed as ²Tˆ Ž zX . : s 0
Ž 7.
where ² . . . : stands for the phonon thermal average Žthroughout this Letter, we do not distinguish exciton operators and their matrix representations.. Here, we adopt the Hartree approximation, neglecting the phonon mode correlation. Eq. Ž7. is approximated by ² tˆ Ž a . Ž zX . : ' t Ž a . Ž z . s 0 ;a .
Ž 8.
The approximated CPA condition for the thermal averaged t-matrix t Ž a . Ž z . is expressed in terms of Green’s functions, Geff Ž z . and g Ž a . Ž z .4 , as Geff Ž z . s g Ž a . Ž z .
;a .
Ž 9.
In Eq. Ž9., Geff Ž z . is given as Geff Ž z .
q,q
X
¦
s vac a q
1 zy
Ý ´ p d p , p q s Ž z . p , p 4 a†p a p . X
X
p, p
X
;
a†qX vac ,
Ž 10 .
X
where vac : is the exciton vacuum. The Green’s function g Ž a . Ž z . is given by taking thermal average of the Green’s operator defined by gˆ Ž a . Ž zX . s
1 z y hˆ X
½ ž z y Hˆ / q Hˆ q Hˆ 5 Ža .
X
X a
ph
,
Ž 11a.
ph
with hˆ Ž a . Ž z . s
Ý ´ q dq , q q s˜ Ž a . Ž z . q , q 4 a†q a q X
q, q
X
X
X
,
Ž 11b.
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
652
where a new notation s˜ Ž a . Ž z . s s Ž z . y s Ž a . Ž z . is used. Geff Ž z . describes the exciton in the coherent potential s , and g Ž a . Ž z . ' ² gˆ Ž a . Ž zX .: represents the exciton in the auxiliary potential s˜ Ž a . Ž z . that is induced by the coupling with all the phonon modes b 4 other than the relevant coupling mode a . From Eqs. Ž9. to Ž11a. we can see that our approximation is nothing but finding wavevector-dependent mean fields for the exciton. The calculation of the matrix g Ž a . Ž z . can be carried out by the continued fraction expansion method originally developed in the SSA–DCPA formulation w12,18,19x. Combined with the expression of Geff Ž z . and using the relation of Eq. Ž9., for instance, we get the closed expression for the self-energy matrix at absolute zero temperature as 1
s Ž z . s Ý FŽa . P a
g 0Ž a .
Ž z y " Va .
y1
yF
Ža .
P
P F˜ Ž a . ,
2 g 0Ž a . Ž z y 2 " Va .
y1
y PPP
Ž 12 .
P F˜ Ž a .
with the coupling matrices w F Ž a . x q, qX s f q,Ž aqX. and w F˜ Ž a . x q, qX s f˜q,Ž aqX.. The Green’s function g 0Ž a . Ž z . is defined by g 0Ž a . Ž z .
q,q
X
¦
s vac a q
1 z y hˆ
Ža .
Ž z.
;
a†qX vac .
Ž 13 .
Eq. Ž12. is a self-consistent one for the coherent potential. Since s Ž z . Žor s˜ Ž a . Ž z .. is determined by that are expressed in terms of s˜ Ž a . Ž zX .Ž RŽ zX . - RŽ z .. as seen in Eqs. Ž11b. and Ž13.. At a finite temperature, the expression of g Ž a . Ž z . includes the phonon absorption term, therefore the coherent potential at energy RŽ z . must be determined consistently with the coherent potentials at energy EX ) RŽ z . and EXX - RŽ z .. An iterative procedure is required in this case w20x. Once the self-energy matrix is obtained, the partial density of qth exciton state is calculated by g 0Ž a . Ž zX .4
rq Ž E . s y
1 p
¦
I vac a q
1 zy
Ý ´ p d p , p q s Ž z . p , p 4 a†p a p . X
X
p, p
X
;
a†q vac .
Ž 14 .
X
Consider one-exciton states interacting with Einstein phonons by a linear short-range coupling expressed as HˆnX s 'S " V Ž bn q bn† . a†n a n ,
Ž 15a.
s Ž 'S rN . " V Ý exp yi Ž k y kX . n4 Ž bn q bn† . a†k a k X , k,k
Ž 15b.
X
where S is the Huang–Rhys factor Ž a specifies the site n of local dispersionless phonons, Va 4 s V in Eq. Ž2b... It is apparent that the single-site approximation applied in Ref. w12x for the one-exciton polaron can be regarded as a Hartree approximation with respect to site-localized phonons. Therefore, the SSA–DCPA and the HA–DCPA result in the same coherent potential. It is easier to calculate the coherent potential by the SSA–DCPA than by the HA–DCPA in this case. However, our approach can treat more general exciton–phonon couplings ŽEinstein and dispersive phonons, local and nonlocal coupling. and be applied to two-exciton polaron problems from a unified viewpoint. We point out that if translational symmetry is assumed, the coherent potential has following properties under the Hartree approximation, with respect to site localized phonons w20x:
s˜ Ž n . Ž z . s Ž z.
X
k,k
X
s eyiŽ kyk . s˜ Ž ny1. Ž z .
k,k
X
s d k , kX s Ž z .
k,k
k,k
X
,
5
Ž 16 .
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
653
for one-exciton states, and
s˜ Ž n . Ž z . s Ž z.
X
K k , K ,k
X
s ey2 iŽ KyK
X
K k , K ,k
X
X
.
s˜ Ž ny1. Ž z .
s dK , K X s Ž z .
K k , K ,k
X
X
K k , K ,k
X
,
5
Ž 17 .
for two-exciton states. These properties reduce the costs of the coherent potential calculations.
3. Numerical calculations The effects of the simultaneous presence of the local and nonlocal exciton–phonon couplings on the formation of excited bound states has been investigated by using the variational band theory w21,22x. It was found that for certain values of the coupling parameter the lowest one-exciton polaron band shows a nontrivial shape, double minimum with respect to the momentum coordinate. Here, we consider the same exciton–phonon interaction studied in Ref. w21x, HˆnX s " V Ž bn q bn† .
'S a†n a n q U a†nq1 a n q a†n a nq1 y Ž a†ny1 a n q a†n a ny1 . 4
.
Ž 18 .
A new coupling parameter U comes from a dependence of the exciton transfer energies on lattice coordinates, such as librations that promote the exciton transfers between neighboring molecules. The one-exciton eigenenergy ´ k s ´ y B cos k is assumed; B s 2 t n, n " 1 ) 0 is the half-bandwidth of one-dimensional J-aggregates. We consider two correlative phonon pictures in the HA–DCPA calculations: Ž1. site-localized phonons and Ž2.
Fig. 1. The lowest one-exciton polaron dispersion for Ž Br " V ,S,U . s Ž1.0,1.0,0.5., g r " V s 0.01 and N s 25. Large circles, triangles, small circles and squares show the results obtained by the HA–DCPA with localized phonons, the HA–DCPA with delocalized phonons, the variational principle and the bare exciton dispersion, respectively.
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
654
delocalized phonons whose creation operator is given by b†k s Ž1r 'N . Ý n expŽi kn. bn†. The difference between these two pictures does not affect the results obtained from the variational approach, since the two phonon pictures are related by a unitary transformation. The trial wavefunction for the lowest one-exciton polaron with the crystal momentum k is represented by < k:s
Ý exp Ž i kn . a nk yn a†n exp X
X
n, n
X
½y Ý Ž b m
k † k) myn bm y b myn bm
.
5 <0: ,
Ž 19 .
where a nk 4 and bnk 4 are the variational parameters and <0: is the vacuum for the exciton and phonons w21,23x. On the other hand, as is apparent from Eq. Ž8., i.e., each phonon mode is treated independently, the two pictures yield different results within the HA–DCPA. Examples of calculated lowest one-exciton polaron dispersion for a parameter set Ž Br" V ,S,U . s Ž1.0,1.0,0.5 . are shown in Fig. 1. We see that the variational approach and the HA–DCPA with respect to local phonons show good agreement. It might be said that our approximation leads to a proper exciton polaron dispersion, since almost the same results are obtained by two very different methods. The delocalized phonon-based HA–DCPA does not give a similar result. According to a general variational principle, the existence of two solutions for the band shape, i.e., two sets of the lowest exciton polaron eigenenergies, implies that the spatial spread of the phonon can be regarded as a variational parameter. It is expected that an introduction of such a parameter into the present method will remove the ambiguity in choosing the phonon basis, and improve the exciton self-energy. In other words, starting from the local phonon basis, i.e., a single-site approximation, a proper extension of the spatial spread of phonons is expected to take into account the correlated exciton scattering among different sites.
Fig. 2. The two-exciton polaron dispersion for Ž Br " V ,S,U . s Ž0.9,0.5,0., g r " V s 0.05 and N s14. The closed and open circles represent the lowest polarons for each quantum number Ž K, k . and higher energy Žquasi-continuum nŽ / 0.-phonon. polaron states, respectively. Note that the abscissa represents the crystal momentum 2 K conjugate to the center of mass for a pair of excitations.
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656
655
Fig. 2 shows the lowest two-exciton polaron dispersion calculated by the local phonon-based HA–DCPA with the on-site exciton–phonon interaction expressed as Eq. Ž15a.. A parameter set appropriate for the PIC–Br J-aggregate, Ž Br" V ,S,U . s Ž0.9,0.5,0. w24x, and the two-exciton eigenenergy, ´ K , k s ´ Kq k q ´ Ky k w14x, are assumed. We can see that the bandwidth is renormalized about the half of the bare two-excitons Ž, 4 B . by the polaron effect. A qualitative difference from the lowest one-exciton polaron dispersion is found. Although the lowest one-exciton polaron band is completely separated from its one-phonon continuum above it, only about half of the lowest polaron states are located below the one-phonon continuum in two-exciton polaron manifold w24x.
4. Summary We have developed a new theoretical scheme to obtain the exciton Green’s function based on the Hartree approximation ŽHA–DCPA. for an exciton interacting with phonons. Our method is capable of treating general linear exciton–phonon coupling in one- and two-exciton manifolds from a unified viewpoint. The local phonon-based HA–DCPA is shown to account for the lowest one-exciton polaron band shape for the nonlocal exciton–phonon coupling reported by Zhao et al. w21x. The calculation of optical spectra within the present scheme is straightforward. For example, the two-photon absorption spectrum is calculated by introducing appropriate vertex parts for the three-particle Green’s function composed of the one- and two-exciton polaron Green’s functions w20x. Since our approach is a mean field approximation, the correlated exciton scattering is treated incorrectly. To improve this, a formulation that takes into account the spatial spread of phonons is suggested.
Acknowledgements One of the authors ŽTK. would like to thank Dr. Shuji Abe of the Electrotechnical Laboratory for his helpful suggestions. The support of the Japan Science and Technology Corporation is gratefully acknowledged.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x
T. Kobayashi ŽEd.., World Scientific, Singapore, 1996, and references therein. B. Kopainsky, J.K. Hallermeier, W. Kaiser, Chem. Phys. Lett. 83 Ž1981. 498. H. Fidder, D.A. Wiersma, Phys. Stat. Sol. Žb. 188 Ž1995. 285. F. Sasaki, S. Kobayashi, Appl. Phys. Lett. 63 Ž1993. 2887. K. Misawa, S. Machida, K. Horie, T. Kobayashi, J. Luminesc. 64 Ž1995. 239. P.O.J. Scherer, S.F. Fischer, Chem. Phys. 86 Ž1984. 269. H. Fidder, J. Knoester, D.A. Wiersma, J. Chem. Phys. 95 Ž1991. 7880. E.W. Knapp, Chem. Phys. 85 Ž1984. 73. J. Knoester, J. Chem. Phys. 99 Ž1993. 8466. F.C. Spano, Chem. Phys. Lett. 234 Ž1995. 29. F.C. Spano, J.R. Kuklinski, S. Mukamel, Phys. Rev. Lett. 65 Ž1990. 211. H. Sumi, J. Phys. Soc. Jpn. 36 Ž1974. 770. V.L. Broude, E.I. Rashba, E.F. Sheka, in: Spectroscopy of Molecular Excitons, Springer Series in Chemical Physics, vol. 16, Springer, Berlin, 1985. H. Ishihara, K. Cho, Phys. Rev. B 42 Ž1990. 1724. S. Abe, J. Phys. Soc. Jpn. 57 Ž1988. 4029. B. Velicky, S. Kirkpatrick, H. Ehrenreich, Phys. Rev. 175 Ž1968. 747. E.N. Economou, in: Green’s Function in Quantum Physics, Springer Series in Solid-State Sciences, vol. 7, Springer, Berlin, 1983. R.E. Lagos, R.A. Friesner, J. Chem. Phys. 81 Ž1984. 5899.
656 w19x w20x w21x w22x w23x w24x
T. Kato et al.r Chemical Physics Letters 303 (1999) 649–656 Y. Won, R.E. Lagos, R.A. Friesner, J. Chem. Phys. 84 Ž1986. 6567. T. Kato, in preparation. Y. Zhao, D.W. Brown, K. Lindenberg, J. Chem. Phys. 106 Ž1997. 2728. Y. Zhao, D.W. Brown, K. Lindenberg, J. Chem. Phys. 100 Ž1994. 2335. Y. Toyozawa, Prog. Theor. Phys. 26 Ž1961. 29. T. Kato, F. Sasaki, S. Abe, S. Kobayashi, Chem. Phys. 230 Ž1998. 209.