J. SINGII: Complex Charge Carrier in Molecular Crystals 423

phys. stat. sol. (b) 103, 423 (1981)

Subject classification: 13 and 13.5.1 ; 22.9

Research School of Chemistry, The Australian A'ational University, Canberra1)

Formation of a Complex Charge Carrier Due to the Interaction between a Free Frenkel Exciton and Excess Electron in Molecular Crystals BY J. SINGIS

Microscopic theory of a complex charge carrier formed by the binding of a free exciton and a n excess electron is briefly described. The potential of interaction between the exciton and excess electron is shown to be attractive as a function of the dipole moments induced by the excess electron on the molecule that carries the exciton. The complex charge carrier, like exciton, is shown to move throughout the molecular crystal.

Es wird die mikroskopische Theorie eines komplexen Ladungstragers, der durch die Bindung eines freien Exzitons und eines ~berschuBelektrons gebildet wird, kurz dargelegt. Es wird gezeigt, daB das Wechselwirkungspotential zwischen dem Exziton und dem uberschussigen Elektron anziehend ist in Abhangigkeit von den Dipolmomenten, die durch die uberschussigen Elektronen auf dem Molekiil, das das Exziton tragt, induziert werden. Es mird gezeigt, daB der komplexe Ladungs- trager, wie das Exziton, sich durch den gesamten Molekiilkristall bewegt.

1. Introduction It has been observed t,hat the presence of charge carriers in molecular crystals decreases the prompt [l, 21 and delayed [3] fluorescence as well as causes external photoemission [4]. Interaction of an excess charge carrier wit,h an exciton can decrease the fluores- cence in two possible ways: (i) The exciton may dissociat'e and the excess energy is either given to the free charge carrier to excite it' to some other higher conducting st.ates or t o a trapped charge carrier to detrap i t and then to excite i t to some other conducting &ate. Either the whole of the excess energy is given in the form of addi- t.ional kinetic energy to the charge carrier and thus to increase t,he carrier mobility in t,he crystal, or a part of the energy is given to excite t'he free charge carrier t o another excited &ate and the rest is to increase its kinetic energy 1151. (ii) The exciton and excess charge carrier may form a bound state if the net potential of interaction between them is at,tractive [F].

In the former process the fluorescence or phosphorescence is decreased and t,he photo- conductivity of the crystal is enhanced, whereas in the latter only the fluorescence is decreased. Agranovich and Zakhidov [C;] have recently demonstrated that the electro- static potential of interaction between bound exciton and an electron is attractive and a bound state can be formed. A proper microscopic theory of interaction bet'ween excitons and excess charge carriers is, however, not yet developed.

The purpose of this paper is, therefore, to present the microscopic view of the inter- action between a free (moving) pair of Frenkel exciton and excess electron in a mole- cular crystal. Hence the problem as well as the approach both, in this paper, are dif- ferent from those of Agranovich and Zakhidov who have estimated (not through microscopic theory) the potential of interaction between a localized pair of Frenkel exciton and excess electron. Although here the t'heory is presented for a pair of exciton and electron only the approach can easily be extended to st.udg the interactions be-

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l) Box 4, P.O., Canberra A.C.T. 2600, Australia.

tween a free pair of exciton and excess hole or trapped pair of exciton and excess charge carriers or pair of trapped exciton and free excess charge carriers or vice versa. The theory is worked out for crystals with one molecule per unit cell. It is derived that t he exciton-excess electron potential of interaction is attractive and can give rise to an excited bound state of a complex charge carrier (exciton + electron) that moves throughout the crystal. Here we have a free electron that induces an electric field on the neighbouring molecules which may or may not be excited. Our interest here is, however, to study the potential of interaction between the induced electric field and an excited molecule. As discussed by Hochstrasser [7] the lattice vibrations have little influence on such potentials in molecular crystals. We have, therefore, neglected the exciton-lattice and electron-lattice interactions for the present problem. Spin-orbit coupling is also neglected, which is a well accepted approximation for molecular crys- tals. More details of the theory including other possible combinations of excitons and charge carriers and extending the treatment to crystals with more than one molecule per unit cell shall be published elsewhere.

u$o) and alj(o) are, respectively, the creation and annihilation operators of an electron with spin G in the j-th molecular orbital of the molecule a t 1. The ket ( 1 , j , o) = = I?+vl,j,Jr - 1 ) ) represents the product of the molecular orbital wave function asso- ciated to (1, j ) and the eigenfunction associated with the electronic spin u,

y& - 1) = yl,# - 1 ) 01 or = yl,j(p - 1 ) B . (5) a and or -$. We will denote by gi ( j = g i ; gt = 1, 2, ... , g) to the occupied molecular orbitals up to the highest occupied one (9) (inclusive) of a molecule in its ground state, and by f to the empty ones or the higher energy MO’s. The ground state of a molecule is defined when all the gi-MO’s are fully occupied and all the f-MO’s are completely empty.

For describing the formation of bound state of the complex charge carrier we proceed as follows: We assume that there is one free Frenkel exciton with wavevector k and spin S (8 = 0 for singlet, and S = 1 for triplet) and a free excess electron with wave aector K - 12 and spin G present in the crystal. We will investigate the possibility that a complex charge carrier with wave vector K can be formed due to the Coulomb inter- action between the exciton and excess electron. We describe the state vector of the crystal with one such complex charge carrier as

are the one electron spin eigenfunctions corresponding to o =

IK, S, f , f ’ ) = Cff,(k, K - k; n) &ids) aj’~--lo(u) (0) , (6) k, a

Formation of a Complex Charge Carrier in Molecular Crystals 425

where f and f ’ , respectively, denote the excited states of the molecules by exciting an electron to the f-th MO, and by creating an excess electron in f’-th MO. C,--(k, K - k ; n) is the expansion coefficient and n denotes an additional quantum number associated with the internal structure of the complex charge carrier. We have considered only a particular pair of excited states due to the occupation of f and f‘ MO of the niole- cules. S represents the exciton’s spin state; S = 0 for singlet and 8 = 1 for triplet.

BTk(S) and UJK-k((T) are the creation operators of exciton and electron, respectively, defined by.

BJk(X) N-112 C eik’t Bt (@ lo> 9 (7 a) 2

N is t,he total number of unit cells in the crystal and B$(S) the creation operator of an excitation on molecule a t I, and given by

Using (8) one can easily verify that B$(S) and a$(o), in (6), commute. We now seek the solution of the following Schrodinger equation:

IK, 8, f , f’) = wfr(K) IK x, f , f’> . (9) Using (6) to (S), operating by a typical term as <Ol a~/k,,(u) Bfk,(S) of (6) from the left- hand side in (9), and then neglecting the overlap of the intermolecular wave functions the following equation is derived for any particular pair of molecules a t I and 1‘ carrying the exciton and excess electron, respectively, a t any instant of time (there will be N 2 number of such pairs of molecules in the crystal), we get

Ewo + Ef(k’ ) + Er(k”) - wfp(K)I Cfr(k’, k” ; n) +

x t(l, f ; l’, f‘l u ll’, f ‘ ; I , f> - (I, g ; l’, f ‘ l u 1 2 ’ 3 f’; 2 , q>l = 0 ,

+ Cf- , (k ,K - k; n) exp { i [ ( k - k’) . 2 + ( K - k - k”) . Z’]} x k

(10) where Ef(k’ ) and E#’) represent the energy of Frenkel exciton with wave vector k’ in the f-th MO and that of the free excess electron with wave vector k“ in the f‘-th MO, respectively. These energies can be expressed in terms of the effective mass approxima- tion as m E,(k’) = E,(O) + 7 = E,(O) - --,

hZk‘2

2M 2M* h27/’2 h2@ Er(k”) x Ey(0) + - - - Ef.(O) - - 2mz 2rn: ’

where E f ( 0 ) and Er(0) are the energy of exciton and the excess electron a t the centre of the Brillouin zone, respectively. M * and rnz are, respectively, t,he effective masses

426 J. SIXGH

of the exciton and excess electron. vlorlT represents the gradient corresponding to the moving carries coordinates in the crystal and does not refer to a change in the posi- tion of the lattice sites. The microscopic expressions for E,(k’) and EY(k”) , including all the configuration interactions, are not required here and therefore shall not be given. The last two matrix elements of (10) are

e2 ( I , f ; 1‘, f ’ l U Il’, f ’ ; 1, f ) = y&(rl - 1 ) y,?“(rz - 1 ’ ) x ss lr, - r 2 l

x yl,,7(r2 - 1’) y l ,g(r l - 1 ) d3r1 d3r2 . (14)

While (13) represents the Coulomb interaction between an excess electron in MO f’ at 1’ and another excited electron in MO f a t 1, (14) represents that between the same excess electron in MO f‘ and I’ and an excited hole in MO g at 1. I n other words, this can easily be derived, (13) represents the Coulomb interaction between a negative molecular ion in the crystal at 1’ and an excited niolecule at 1, and (14) represents that between the same niolecular ion at I’ and the molecule at 1 in its ground state. Here onwards, therefore, we will mean that g denotes the ground state and f an excited state of a molecule.

Let us transform the carrier coordinates in (13) and (14) as

ifi = r, - I , & = r2 - 1 , (15)

and then expand

by assuming that I & - [,I < I I - 1’1 and retaining the terms only of the first order in (& - Ez), ((16) is not applicable when 1 = Z’). With the help of (11) to (16) we can bring (10) into the following solvable forin:

where

Equation (19) represents total dipole moment induced by an excess electron (at Z’) on the niolecule a t Z in its j-th (ground g or excited f ) electronic state. Usually the induced dipole moment on an excited molecule is greater than that on a molecule in its ground state, Ipf{,’l > Thus the potential U ( 11 - Z’ l ) , which is the interaction potential between two moving particles with masses N * and m; (see (17)) , is an attractive one, and conclusively the particles can form a bound excited state in

Formation of a Complex Charge Carrier in Molecular Crystals 427

the crystal. Theenergy of the crystal with such a bound st.ate can be calculated by applying another transformation of coordinates as

N * l + mzl’ M * + rn: 1 - 1 ’ r R and - = L , K = Ic’ + k“ .

Use of (21) in (17) yields the energy eigenvalue of the crystal with one complex charge carrier as

where

1 1 1 Mt = M * +rnz and -=-

/Ah rn:+M*.

Corresponding to (22) the wavefunction of the complex charge carrier is written as

GffT(R, L) = eiK.L yif‘(R) . (25)

The coefficient of expansion Cff , (k , K - k ; n) can be determined from (20) and ( 2 5 ) in the following form:

( 2 6 ) For an infinite crystal t.he sum in (26) can be converted into an integration.

3. Discussion

The formation of the bound state of a complex charge carrier depends on the potential of interaction, U ( R ) , between the excess electron and exciton. Let us consider a model crystal in which the niolecular dipole nioments of (18) can be considered as

(p;/,’)& = (pJ f ’ , 0, 0) -2s , = (0, pjf’, 0) 2g , (p;/,’)z = (0 , 0 , p q 22 . (27)

We can expand pjf’ in terms of the permanent dipole moments and polarizability [lo]. We will, however, retain here the terms up to the linear polarization only,

@(R) -- $(O) + aJf’ E ( R ) , (28) where p’(0) represents the permanent dipole moment of the molecule in its electronic state j and in absence of the excess electron a t a distance R. a%*’ and E(R) are the second-rank tensor of static polarizability of molecule, and the electrostatic field induced by the excess electron on the molecule in its electronic state j . Using (28) wc can write the potential of interaction as

(29)

where E ( R ) = eR/&R3; E is the dielectric constant of the crystal. Consider a crystal of centrosymmetric molecules where pg(0) = 0, and if the molecule remains centro- symmetric in the excited state too, pf(0) = 0. I n such crystals the second term of the potential (28) is responsible for creating the complex charge carrier. The depth of potential of interaction between a stationary exciton and electron is estimated by

eR U ( R ) = - R3 * [(pf(O) - pU(0)) + (orff’ - o r q E(R) ] ,

428 J. SINGH: Complex Charge Carrier in Molecular Crystals

Agranovich and Zakhidov in centrosynimetric crystals. If, however, the molecule gets distorted in its excited state thenpf(0) +O. I n such crystals depending on the separa- tion between exciton and electron the potential of interact?on may depend on both or one of the terms of (28). Likewise the crystals of non-centrosymmetric molecules with non-zero $(O) andpf(O), andyO(0) +pf(O) the potential of interaction will be as given in (29). It is to note here that the measurements of the dipole moments and polariza- bilities are done at 4.2 K [11] where the influence of phonons cannot be dominant. This further supports the approximation of neglecting the exciton-phonon and elec- tron-phonon interactions.

We have so far considered the case of a complex charge carrier where its constituents, the exciton and electron, are not carried on by the same molecule, i.e. 1 + 1’ a t the same time. I n case the exciton and electron both occupy the same molecule a t the same time the potential energy of interaction is given by the difference between (13) and (14) for I = 1‘. Thus the relative magnitude of ( I , f ; I, f ’ l 17 IZ, f ’ ; I , f ) and (I, 8 ; I , f ‘ l U 11, f‘; I , g) will decide whether the interaction energy is positive (repulsive) or negative (attractive). When the potential energy is positive the bound state will not be possible until they are carried away on different molecules during which the exciton may or may not dissociate depending on the relative magnitude of the ex- citons own binding energy. However, if the energy of interaction between the exciton and electron is negative, a bound complex charge carrier state is possible with no relative motion between exciton and electron because V R = R = 0 (in (23)).

As derived in (10) the binding energy of the complex charge carrier does not depend on the exciton spin within the approximations used here. Thus with a singlet exciton a doublet bound state and with a triplet exciton a quartet bound state can be formed; this would of course have to take place within the lifetime of the corresponding exciton states.

It is rather easy to show that such a bound complex charge carrier state can also be formed by more than one exciton and one excess charge carrier in the crystal. Here, however, the other competitive process of exciton-exciton collision with a rate as fast as = lo9 ~ m - ~ s-l [9] would also play a role against the formation of a complex bound state with many excitons.

Acknowledgement

I am very thankful to Prof. Y. P. Craig, FRS, for going through the manuscript and useful comments.

References [l] 11. POPE, J. BURGESS, and N. WOTHERSPOON, Chem. Phys. Letters 12, 140 (1971). [2] P. SCHLOTTER and H. BAESSLER, Chem. Phys. 19,353 (1977). [3] L. PETER and G. VAUBEL, phys. stat. sol. (b) 48, 587 (1971).

[4] D. HAARER and G. CASTRO, Chem. Phys. Letters 12, 277 (1971). [5 ] J. SINGH, Chem. Phys., to be published. [6] V. M. AGRANOVICH and A. A. ZAKHIDOV, Chem. Phys. Letters, to be published. [7] R. M. HOCHSTRASSER, Accounts chem. Res. 6, 263 (1973). [8] H. HAKEN, Quantum Field Theory of Solids, North-Holland, Publ. Co., 1976. [9] J. SINGH, J. Phye. C 13, 3639 (1980).

A. MANY, J. LEVINSON, and I. TEUCHNER, Mol. Cryst. liquid Cryst. 6, 121 (1968).

[lo] A. D. BUCKINGHAM, Intermolecular Forces in Advances in Chemical Physics, Vol. 12, Ed.

[ll] R. M. HOCHSTRASSER and D. A. WIERSMA, J. chem. Phys. 66,528 (1972). J. 0. HIRSCHFELDER, Interscience Publ., London, New York, Sydney 1967 (p. 107).

(Received June 11, 1980)