J. SINGH and H. BAESSLER: Exciton Dissociation in Molecular Crystals 147

phys. stat. sol. (b) 62, 147 (1974)

Subject classification: 13.5.1; 22.9

Fachbereieh Physikalische Chemie der Universitat MarburglLahn

Theory of Exciton Dissociation in Molecular Crystals at the Interface of a Metal

BY J. SINGH and H. BAESSLER

The phenomenon of dissociation of anthracene singlet excitons a t the interface of the crystal and an aluminium metal layer has been treated quantitatively applying the con- figuration interaction scheme. HMOs are used as the wave functions of anthracene mole- cule. Calculated values of the rate constants are two orders of magnitude lower than the experimental ones. This discrepancy has mainly been attributed to the use of HMOs in the calculations. It has been suggested, on the basis of the results obtained for a different set of MOs, that the agreement between theory and experiment would improve by using proper molecular wave functions.

Mit dem Konfigurationswechselwirkungschema wird das Dissoziationsphanomen von Anthrazen-Singulett-Exzitonen an der Grenzflache des Kristalls und einer Aluminium- metallschicht quantitativ behandelt. Als Wellenfunktionen des Anthrazenmolekiils wurden HMOs benutzt. Die berechneten Werte der Ratenkonstanten sind zwei GroOenordnungen niedriger als die experimentellen Werte. Diese Diskrepanz wird hauptsachlich der Be- natzung von HMOs in den Berechnungen zugeschrieben. Es wird auf der Grundlage der fur verschiedene Satze von 310s erhaltenen Ergebnisse angenommen, da13 die Uberein- stimmung zwischen Theorie und Experiment durch Benutzung geeigneterer Molekiilfunktio- nen verbessert werden kann.

1. Introduction

It has recently been shown [l] that non-radiative decay of a Frenkel exciton striking the int'erface of a molecular cryst'al and a metal is determined by two competitive processes : energy transfer to metal electrons and dissociation. Dis- sochtion is the result, of a charge transfer reaction following the scheme

&I* + metal + M+ + e- (metal); (oxidative dissociation) , M* +- metal --f M- + e+ (metal) ; (reductive dissociation) ,

where M* denotes the excited molecule a t the interface. This means that an excited molecule can either transfer an electron or a hole to the metal thus generating a hole or an electron respect'ively inside the crystal. These charge carriers can be detected in course of a photoconductivity experiment. From their studies Killesreiter and Baessler [l] concluded that dissociation is a very efficient channel for non-radiative exciton decay occurring a t a maximum rat'e constant of the order of l O I 3 s-l.

These authors also suggested that the process of dissociation of excitons a t a metal contact can be interpreted with the help of the configuration inter- action model. Following these ideas the present paper f u r the first time reports on a quantitative determination of the rate constants of dissociation of anthracene singlet excitons a t an aluminium contact. 10'

148 J. SINGH and H. BAESSLER

Fig. 1. The energetic aspect of exciton dissociation a t the interface of an anthracene crystal and a n aluminium contact. A, is electron affinity, I, is the ionization potential and E,, is the exciton energy. E, = A , + E,, and E, = I , - E,,. E, and E, are the energetic positions of the metal state which a hole or an electron, respectively, occupies after dis-

sociation

CB

/eve/

VB

2. Theory

The rate a t which an exciton decays non-radiatively to form a pair of charge carriers is determined by the interaction between exciton state and ionized state. Since in an anthracene crystal the motion of singlet excitons can be de- scribed by an incoherent hopping process, the exciton state will be identified with a localized excited state of a single molecule embedded in the crystal lattice. According to Fano [2] the wave function of the mixed state is composed of the wave function y E of the bound molecular state, i.e. the exciton state, and the wave function of the ionized states yj$ and characterized by a hole inside the crystal and an electron inside the metal or vice versa:

where a, b , and c are the binding coefficients. In Fano's treatment y& are energy states in continuum. Therefore. the second and the third terms of (1) are express- ed as integrals over the energy range E', covered by the continuum of ionized states. This is necessary if decay of the bound molecular state leads to genera- tion of carriers in a manifold of delocalized crystal states. In the present treat- ment y& is expressed as the product of the wave function y , of the carrier inside the crystal and the wave function ym of the countercharge inside the metal:

Y' = q m v c * (2) Since, as Munn and Siebrand [3] have shown, charge transport along the c'-direc- tion of an anthracene crystal can best be described by a hopping process, it is reasonable to assume that immediately after t h e dissociation everit the charge carrier created inside the crystal is located at that particular molecule which formerly was excited. Therefore yc is to be identified with the wave function of an anthracene ion embedded in the crystal lattice. Allowance for carrier motion is made by assuming that the energetic width of this state equals the width AE of the valence or conduction band of the crystal which is of the order of k T at room temperature.

Because of energy conservation during exciton dissociation the countercharge must be in an energy interval AE. In the case of exciton oxidation this zone of accessible states is a t an energy E,,, above the crystal valence band, and in the case of excilon reduction it is loeated a t E,,, below the conduction band. E,,, de- notes the exciton energy (see Fig. 1 ). The overlap between neutral and ionized states is therefore

<cPElY$> = BEE' * (3)

E denotes the energy of the charge carrier in the crystal and E' is that of its image carrier in the metal. The transition rate (KcT) of carriers between a bound

Exciton Dissociation in Molecular Crystals a t the Interface 149

state (E) and a continuum state ( E ' ) can be given by the golden rule [4]

where @ ( E l ) is the density of states in the continuum (metal) within an energy range AE. V& arc the coupling matrix elements of the two configurat#ions as

and

where H is the one-electron Hamiltonian operator of a neutral molecule in a crystal in contact with a metal surface. n is the number of molecules in one ab plane of the crystal lattice. The quantity ( l / n 2 V $ } gives the average value of coupling per molecule in an ab plane. The wave function qE of the excited state which is required t o evaluate the matrix elements (equation (4)) can be constructed from Huckel molecular orbi- tals (HMOs) [5] of an anthracene neutral molecule:

n

q = 2 caxa > z

where xz is the SCF-Zp, atomic orbital of carbon as calculated by Clementi et a]. [el. The highest occupied molecuiar orbital (HOMO) is taken to represent the wave function of a hole, and the lowest empty molecular orbital (LEMO) as that of an electron, forming the exciton localized on a molecule. It is further assumed that the wave functions of a neutral molecule corresponding to HOMO and LEMO, do not change by putting an excess charge onto the molecule. In other words, the wavefunction of an excess electron in a LEMO is assumed to be identical with that of LEMO of a neutral molecule.

Since we have considered the exciton dissociation process along c'directioii of anthracene for which the experimental data are avGilable, the metal wave functions ym are taken to be one-dimensional Bloch waves with a wave vector perpendicular t o the ab crystal plane. They enter the forbidden gap of the adjacent crystal as damped waves :

z q m ( z ) = ym(z = 0) exp

where K(z ) is the damping constant along c'-direction. Because qm(- z ) = = A exp (--iK lz l ) , (6) becomes

For a rectangular potential barrier of the thickness z = d equation (7) reduces to

It should be noted that the iinal results will be independent of A = 1-112, where 1 is a characteristic length associated with the free electron state in the metal. Inserting of (2) and (8) into (5) yields

150 J. SINGH and H. BAESSLER

This agrees with the experimental observation that the coupling between exciton state and ionized state decreases exponentially with the interaction distance.

The damping constants can be expressed as 1 h

and K f = - 1 [2m;(E, - E , , o ) ] ~ / ~ . (10) h K- = - [2~i:(E, - Ec,o)1112

E,,o and E , locate the energetic positions of conduction and valence band of the crystal and E , and E, indicate the energy of the free electron and hole state in the metal, respectively. rn: and m; are the effective masses of electron and hole, respectively. K * values calculated from (10) are by a factor of two lower than the experimental data. This is due to an overestimate of the potential barrier by assuming rectangular shape. Where Kf enters the present calcula- tions the experimental value Kf = 0.23 A-l is used.

3. Evaluation of the Integrals

The Hamiltonian H can be written as

Vn is the neutral molecular potential given by [7 ]

zeff is the effective charge on the carbon atom. J,, and Kni are the usual Coulomb and exchange operators. The third term of (11) is due to the image potential between the charge carrier localized on the molecule and the corresponding countercharge in the metal. D is the dielectric constant of the crystal. The molecular integral can be written as

where ( q E 1 II 1~:) == 8; + 8: 2- EZ 4- I*, (13)

p and n denote the location of the molecules in the crystal lattice. Interaction with 39 molecules has been considered (see Fig. 2 ) in the calculation. Equation (14) can be easily evaluated in the way it has been done for the energy band structure calculation of anthracene and other molecular crystals [TI, except that here carriers are assumed to be localized a t a particular molecule.

In the evaluation of the hybrid integrals involved in the Coulomb and ex- change terms, only two centre integrals have been taken into account. Their calculation is based on the Miilliken's approximation. The values of the various integrals (equation (14)) are not reported here. They are, however, available on request.

Exciton Dissociation in Molecular Crystals a t the Interface 151

Fig. 2. Number of anthracene: molecules considered for the calculations

third loyer seco!d loyer first, loyer ,

The program was run on TR 4 elect,ronic computer. The density of metal dates for holes (g+(E') ) and electrons (e - (E' ) ) calculated

on the basis of free electron model

2n h3

e ( E ) =: - Vm(2mE)'/' AE

is 0.50 and 0.36, respectively. It gives the number of states for a charge carrier of mass m in a volume V = 2 a x 2 b x l and in an cnergy range AE = k T , which is taken as the width of the energy bands in the anthracene crystals. a and b are the lattice parameters of the unit cell of anthracene crystal (a = = 8.66 A and b = 6.038 A).

4. Results and Discussion

Ths numerical results of the rate constant for oxidation of an anthracene singlet state located in the first molecular layer adjacent to the aluminium ifiter- face is kh) = 1011 s-l. For exciton reduction k a ) = 1O1O s-l is obtained. These figures can be compared with the data derived from experiments yielding the rate constant for exciton dissociation as a function of the spacing 'd' between the crystal surface and the metal contact, the spacing being produced by means of a molecular fatty acid film. For hole production by singlet excitons kc^ = = kcT,O exp (-did,) was found with kAT,O = 3 x l O I 3 s-l. Since the limit d + 0 corresponds to the dissociation of an exriton located a t a molerule adjacent to the metal contact, k$i,o and k y ) should be identical.

Theoretical and experimental values for k: differ by more than two orders of magnitude. This is most likely due to uncertainties in both the values.

The experimental value kcT.o was obtained by extrapolation. The curve was fitted to experimental data obtained for reaction distances 19 A 5 d 5 30 A . Moreover, the quantum yield for the photoelectric carrier production which enters in the determination of kcT,O was not precisely known, because of diffi- culties in accounting for geminate recombination. This will cause an uncer- tainty of a t least one order of magnitude in kCT,O. Owing to several approxima- tions in the theoretical treatment as mentioned below the coupling matrix ele- ment is also associated with an uncertainty which enters as its square in the calculated rate constant.

152 J. SINGH and H. BAESSLER: Exciton Dissociation in Molecular Crystals

For example, it has been assumed that the wave function corresponding to various energy levels of a neutral molecule is same as that of the energy levels corresponding to an ion and also in the evaluation of the hybrid integrals only two centre integrals were included.

The use of HMOs as the molecular wave functions may lead to a significant underestimate of the valnes of the molecular integrals. In order to get an idea how the rate constants depend on the choice of the molecular wave functions we calculated them with the help a recent set of anthracene MOs reported by Mathur and Singh [S] iiicluding all overlaps in the molecule. We ob- tained kh) x 1015 s-1 and ka , x 1012 s-l. The latter value of kh) is about two orders of magnitude higher than the experimental one. Moreover, as compared to former values obtained from HMOs, k;) is increased by four orders of magni- tude but k 6 ) only by two orders of magnitude. At present it is difficult to dis- cuss this discrepancy since the latter MOs of anthracene have not been tested so far.

The increase obtained in the values of rate constants by using a different set of wave functions, indicates how drastically kCT,O is affected by the choice of the wave function. A more appropriate set of MOs will probably improve the agreement between theory and experiment. Nevertheless the present treat- ment shows that dissociation is a very efficient channel for non-radiative elec- tronic decay for an exciton in the neighbourhood of a metal. The rate constants of dissociation obtained for the second layer of the crystal are k&) =z 2 x lo9 s-1 and k& ;= 2 x lo8 s-l using experimental values for the damping constants K'. It should be noted that even in the second molecular layer of an aluminium coated anthracene crystal, these rate constants are still high enough to ensure that exciton decay proceeding directly from this layer effectively competes with exciton diffusion to the first molecular layer followed by exciton decay a t a rate constant k&,O.

Acknowledgements

Our thanks are due to Prof. P. 0. Lowdin for a valuable discussion on this paper during the International Summer Institute on Quantum Chemistry, 1973. The authors would like to thank the authorities of the Zentrale RechenanIage for providing the computer facilities. Financial support from Deutsche For- schungsgemeinschaft is gratefully nclinowledged.

References [l] H. KILLESREITER and H. BAESSLER, phys. stat. sol. (b) 51, 657 (1972); Chem. Phys.

121 U. FAKO, Phys. Rev. 124, 1866 (1961). [3] R. W. MUNN and W. SIEBRAND, J. chem. Phys. 52, 6391 (1970); Disc. Faraday Soc.

51, 17 (1971). [4] R. P. FEYNMANN and A. R. HIBBS, Quantum Mechanics and Path Integrals, McGraw-

Hill Publ. Co., 1965 (pp. 147 to 152). [51 J. N. MURREL, Mol. Phys. 4, 205 (1961). [6] E. CLEMENTI, C. C. J. ROOTHAAN, and Y. YOSHIMINE, Phys. Rev. 127, 161 (1962). [7] See e.g. S. C. MATHUR, J. SINCA, and D. C. SINGH, J. Phys. C4, 3122 (1971).

[S] S. C. MATIIUR and D. C. SINGH, Chem. Phys. Letters 5, 249 (1970).

Letters 11, 411 (1971).

R. S ~ B E Y , J. JORTNER. S. A. RICE, and M. T. VALA, JR., J. chem. Phys. 42,733 (1965).

(Received December 21, 1973)