Exciton—Exciton Interactions and Photoconductivity in Crystalline AnthraceneSangil Choi and Stuart A. Rice

Citation: The Journal of Chemical Physics 38, 366 (1963); doi: 10.1063/1.1733666 View online: http://dx.doi.org/10.1063/1.1733666 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/38/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Exciton–charge carrier interactions in the electroluminescence of crystalline anthracene J. Chem. Phys. 63, 4187 (1975); 10.1063/1.431177 CarrierExciton Interactions in Crystalline Anthracene J. Chem. Phys. 57, 1770 (1972); 10.1063/1.1678469 Exciton—Exciton Collision Ionization in Anthracene Crystal J. Chem. Phys. 46, 3475 (1967); 10.1063/1.1841241 Photoconduction in Anthracene Induced by Triplet Excitons J. Chem. Phys. 41, 3657 (1964); 10.1063/1.1725792 Exciton—Exciton Interaction and Photoconductivity in Anthracene J. Chem. Phys. 40, 1173 (1964); 10.1063/1.1725280

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 38, NUMBER 2 15 JANUARY 1963

Exciton-Exciton Interactions and Photoconductivity in Crystalline Anthracene

SANG-IL CHOI AND STUART A. RICE*

Department of Chemistry and Institute for the Study of Metals, University of Chicago, Chicago 37, Illinois

(Received 6 September 1962)

In this paper we consider the mechanism of photoconductivity in crystalline anthracene. It is shown that two excitons may interact to form a pair of charge carriers and an unexcited molecule. The computed rate of generation of charge carriers is 3.7X 108 cm-a sect, in satisfactory agreement with the (approximate) experimental value of 7.2X 1()8 cm-a seC' when the exciton concentration is 1.2X 10'0 cm-a. Other qualita­tive features of the proposed mechanism are in agreement with observation if electron-hole recombination is accounted for. Recent experiments by Silver demonstrating a photocurrent proportional to the square of the light intensity, and by McGlynn demonstrating the necessity for singlet states as the kinetic intermediate in charge-carrier generation are in agreement with the model proposed.

I. INTRODUCTION

FROM the many studies of the electrical properties of molecular crystals made in recent years,! the

following generalizations may be drawn:

(a) The dark current in an organic crystal is very small. The temperature dependence of the conductivity corresponds to an activation energy very much smaller in magnitude than the energy required for the ioniza­tion of a molecule.

(b) A large photocurrent can be generated with spectral response identical to the absorption spectrum of the crystal,2 The frequency threshold for photo­conductivity is (approximately) equivalent to the first excitation frequency of the isolated molecule.

(c) There is a large class of space charge and trap­ping effects which determines the conductivity of a real, imperfect, and probably impure crystal,3 It is clear that these three broad deductions from experi­ment are insufficient to determine the mechanism responsible for the observed conductivity. In par­ticular, the observations codified under (a) and (b) raise important questions about the energy balance in the excitation process. It is the purpose of this paper to suggest a mechanism for the creation of a photo current in a molecular crystal. Our considerations are guided by the two observations cited in (b) and the principle of conservation of energy.

We consider in the following text only the mecha­nism of charge-carrier production following photon absorption. It is well known that the absorption of light by a molecule in a molecular crystal leads to a state in which the excitation can migrate from molecule

* Alfred P. Sloan Fellow. 'C. G. B. Garrett, Semiconductors, edited by H. B. Hannay

(Reinhold Publishing Corporation, New York, 1959), p. 634; J. Kommendeur, J. Phys. Chern. Solids 22, 339 (1961).

2 R. G. Kepler, Phys. Rev. 119, 1226 (1960); A. Bree and L. E. Lyons, J. Chern. Phys. 22, 1630 (1954).

a H. Kallmann and M. Pope, Symposium on Electrical Con­ductivity in Organic Solids (Interscience Publishers, Inc., New York, 1961) p. 1.

to molecule.4 There is, therefore, an "exciton band' analogous to the electron bands familiar from the theory of metals. The reason for the migration is, of course, that a state in which one of N crystallographi­cally identical molecules is excited cannot be a sta­tionary state of the crystal. The rate at which the excitation migrates depends on the interaction between the molecules of the crystal. Two obvious limiting cases may be distinguished:

(1) The excitation transfers from molecule to molecule in a time short compared to the time required for the displacement of the equilibrium positions of neighboring molecules. This transfer therefore occurs without local lattice deformation and is characterized as the free-exciton case.

(2) The excitation transfers from molecule to molecule in a time long compared to the time required for the displacement of neighborin'g molecules. Transfer of excitation in this case causes a local deformation to travel through the crystal, and is characterized as the localized exciton case.

The low-temperature (200 K) spectrum of anthracene4

shows spectral splittings due to impurities and free excitons. We may therefore safely assume that in the vicinity of 3000 K only the free excitons need be con­sidered; in the following analysis we always consider the motion of the excitation through the undistorted anthracene lattice.

Granted that the absorption of light by a crystal leads to exciton states, what is the relationship of these states to the generation of photoelectrons? The most widely held view is that an exciton may interact with an impurity or defect or may diffuse to the surface of the crystal.5,6 In each case it is assumed that the result of the interaction is the dissociation of the

4 A. S. Davydov, Theory oj Molecular Excitons, translated by M. Kasha and M. Oppenheimer, Jr. (McGraw-Hill Book Com­pany, Inc., 1962).

5 L. E. Lyons, J. Chern. Phys. 23, 220 (1955). 6 H. P. Kallman and M. Pope, J. Chern. Phys. 36,2482 (1962).

366

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PHOTOCONDUCTIVITY IN CRYSTALLINE ANTHRACENE 367

exciton into a hole and an electron. The authors know of no quantitative (or even qualitative) estimates of the transition probability for the processes envisaged. Moreover, it has never been made clear how the requisite energy for dissociation is obtained. In our opinion this is a serious drawback to the mechanism(s) mentioned; we do not mean that they are impossible, but only that quantitative considerations are necessary to make them credible.7

A different mechanism was proposed by Northrop and Simpson.8 These investigators concluded, from studies of photoconductance and fluorescence quench­ing in pure and doped anthracene, that exciton­exciton interaction was responsible for the generation of photoelectrons. The proposal has often been dis­missed on the grounds that it requires the photocurrent to be quadratic in the incident light intensity, a conse­quence which disagrees with experiment. This objection is, however, not valid if recombination is taken into account. The time rate of change of the concentration of electrons ne is describable by

(1)

where n is the concentration of excitons (proportional to the incident light intensity) and k~nep is the assumed bimolecular recombination rate of holes (concentra­tion p) and electrons. The current is often measured under steady-state conditions, wherein (dne/ dt) = O. Since the exciton concentration is proportional to the light intensity, and the electric current i is proportional to the electron concentration, we find from Eq. (1) and the conditions, ne=P and (dne/dt) =0

io:: no:: I, (2)

where I is the incident light intensity. The exciton­exciton interaction mechanism is, therefore, com­patible with the observed photocurrent-light-intensity relationship.9 Moreover, in most cases the first elec­tronic excitation of the isolated molecule is more than

7 It has been suggested that the energy of the lowest conducting state may lie lower than the lowest singlet exciton state. Accord­ing to the classical calculations of Lyons and Mackie (Proc. Chern. Soc. 1962, 71) the energy of this conducting state should lie about 2.6 eV above the ground state, or about 0.5 eV below the lowest singlet exciton state. If true, conservation of energy in the carrier generation process is accounted for. Despite the attractive­ness of this conclusion we do not believe that the calculations by Lyons and Mackie settle the problem, for, these investigators assume that the charge is sufficiently localized that a point charge formalism is valid. Even if the carrier electron is trapped on a molecule forming a negative ion, the delocalization of the 71' elec­tron will decrease the polarization of the surroundings greatly. A better estimate of the polarization energy would start from a charge distribution on the negative ion, as required by the mole~u­lar wavefunction, approximately uniformly spread on the nng carbons. The greatly diminished charge density then leads to smaller polarization energies than those estimated by Lyons and Mackie.

8 D. C. Northrop and O. Simpson, Proc. Roy. Soc. (London) A244, 377 (1958).

9 See Sec. III for the case of pulsed-light experiments.

b 6.036

C 11.163

~ a,c 124.7°

halfway to the ionization potential in the crystal.lO Thus, two excitons provide sufficient energy for an ionization.

As a result of recent work, it appears likely that the original experiments of Northrop and Simpson were conducted under conditions for which surface generation of charge carriers predominates. Surface generation is dominant when the incident light is all absorbed within a short distance of the (imperfect) surface. Under these conditions the photocurrent is sensitive to surface structure, to impurities, etc. Only when there is bulk absorption of light is the homo­geneous exciton-exciton mechanism for charge-carrier generation likely to be dominant.

To demonstrate the plausibility of the exciton­exciton mechanism, it must be shown that the transition probability for excitation of an electron into a con­ducting state when two excitons "collide" is com­patible with the observed rate of generation of elec­trons. In a previous note Choi and Ricell considered the case of a tetragonal lattice of hydrogen atoms, and showed that the exciton-exciton mechanism was com­patible with the measurements of Northrop and Simpson. In this paper we consider the same problem for the anthracene crystal. It is shown that the exciton­exciton mechanism is able to account for the observed rate of generation of electrons. The authors regard this result as strong evidence in favor of the exciton­exciton mechanism for the generation of photoelec­trons.

Note added in proof. Recent experiments by M. Silver, D. Olness, M. Swicord, and R. C. Jarnagin (private communication) with weakly absorbed light show a photocurrent proportional to f2 and a rate of carrier generation in agreement with that computed in this paper.

II. DESCRIPTION OF THE CALCULATION

In the following analysis we neglect the effects of lattice vibrations and internal molecular vibrations on the electronic states of the crystal; the crystal is

10 L. E. Lyons and J. C. Mackie, Proc. Chern. Soc. 1962, 71. 11 S. 1. Choi and S. A. Rice, Phys. Rev. Letters 8, 410 (1962).

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368 S. CHOI AND S. A. RICE

treated as if all the molecules are rigidly fixed at their equilibrium positions.

The anthracene crystal belongs to the C2hs space group of monoclinic crystals with two molecules in a unit cell.12 One molecule is located at a corner of the cell and the second molecule is situated in the center of the a- b face of the cell (see Fig. 1). The orientation of the second molecule can be obtained by reflection of the first molecule in the a-c plane. The unit cell is char­acterized by the lattice constantsl3

a=8.561A,

b=6.036A,

c= 11.163A,

fJ= 1240 42'.

The free single exciton states of a crystal with two translationally nonequivalent molecules in a unit cell were first discussed by Davydov4 in a treatment of some characteristics of the electronic absorption spectrum of the anthracene crystal. A detailed group theoretical discussion of excitons in such crystals was given by H. Winston in 1951.14 The starting point of this analysis is the crystal Hamiltonian for our model which may be written

X= LXlm+t L Vii,lm, (3) I,m (ijr' 1m)

where Xlm is the Hamiltonian for the lth molecule in the mth unit cell, when the molecule is isolated. Vii,lm is the interaction potential between the mole­cules ij and lm.

For purposes a simple treatment of the electronic states of the isolated molecule is sufficient. It is well known that the lower excited states of aromatic mole­cules may be described in terms of the transitions of the 7r electrons from bonding to antibonding 7r-electron orbitals.ls We consider each molecule to have a frame­work of (J' bonds on which the 7r electrons are confined. Then X lm is a self-consistent field Hamiltonian for the isolated molecule, while Vii,lm is the sum of the frame­frame, frame-electron, and electron-electron inter­actions.

We confine attention to the lowest singlet exciton state. To write the wavefunction of this exciton state we need to know the ground state and the first singlet excited state of the isolated molecule. Denote the ground state and the first singlet excited state of the molecule (lm) by the normalized functions y"lm(r) and y,,'lm(r), respectively. Then, the zeroth approxi­mations to the ground-state wavefunction of the

12 A. 1. Kitaigorodskii, Organic Chemical Crystallography trans­lated from Russian by Consultant Bureau (Consultant Bureau, New York, 1961), p. 420.

13 D. V. J. Cruickshank, Act. Cryst. 9, 915 (1956). 14 H. Winston, J. Chern. Phys. 19, 156 (1951). 16 R. Daudel, R. Lefebvre, and C. Moser, Quantum Chemistry

(Interscience Publishers, Inc., New York, 1959),

electrons in the crystal can be expressed in the form

(4) am

where a= 1,2, and m=l, 2, ••• , N with N the number of unit cells in the crystal. For the first singlet exciton states we write, corresponding to Eq. (4), the wave­functionl6

1 N q,'±(K)=-( )lL exp(iK·R".)[ll,m)± 12,m)], (5)

2N "....1

where I a, m)=y,,'am(r) II y"pj(r). (6)

(/Ji"'am)

Implicit in the assumed product form of the above state functions is the neglect of the overlap of the electronic wavefunctions between different molecules. In other words, the state functions are not antisym­metrized with respect to the exchange of electrons between molecules. As usual, the values of the mo­mentum vector K are limited to the first Brillouin zone. In this zone, K takes on the N discrete values

K= L27r a;-Ip . i N

i" ~,

(7)

where the Pi are whole numbers within the range de­noted by

i= 1,2,3, (8)

and the ai-I are vectors characterizing the unit cell in the reciprocal lattice. In the direct lattice, the basic vectors are denoted a, h, c. Note that the state repre­sented by Eq. (5) describes mobile excitation energy with momentum vector hK in the crystal. Equation (5) may therefore be considered to be the momentum eigenfunction of the "exciton"4 in the crystal.

By use of Eq. (5) the electronic states of a crystal which has absorbed radiation of a given frequency may be described in terms of a dilute gas of "excitons" which obey Bose-Einstein statistics. The excitons can be destroyed through interaction with impurities, defects or surfaces or by the emission of radiation (unimolecular reaction). Also, it is possible to produce new species in the excitation gas through interaction between excitons. The simplest assumption is that the exciton-exciton interaction is a bimolecular reaction leading, possibly, to a free electron, and a hole (positive ion). In the present work we identify the reaction rate constant (aside from constant factors) with the transi­tion probability per unit time for the transition between the "double" exciton state and the state which repre­sents the crystal with one ionized molecule and a free

16 The function (5) is not a stationary state of the tight binding Hamiltonian, but this does not influence our calculation because we do not calculate the energy of the single-exciton state. Our interest is only in the transition between th(double exciton_state and the ionized state.

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PHOTOCONDUCTIVITY IN CRYSTALLINE ANTHRACENE 369

electron. Consider now the double exciton state repre­sented by the state function

2

'l'k,k,(a) (r) = I: Bap<a)if;k,k,(a,p) (r); a= 1,2,3,4, (9) aJl=-l

where

if;k,k' (a ,P) (r) = [2N(2N -1)]-1

xI: exp(iK·Rn+iK'·Rn,) I an, (3n')~(n, n'), (10) R.n'

~(n, n') = 1 if arf={3,

TABLE 1.

~ 1 2 3 4

Bn(a) ! ! ! -1 B ,2(a) 1 .1 -1 1 • B

21(a) 1 1 ! .1 -;l •

B22

(a) ! -1 -1 -l

task is to calculate the elements of the energy matrices appearing in Eq. (14).

(K, K', a I X I k, p, ±) = 1 if a= (3, nrf=n',

=0 if a=fj, n=n',

I an, (3n')=if;'"",(r)if;'pn,(r) II if;ri(r). <ri"""n¢pn')

(11) (1 )iI: I: exp[ip.Rm-iK·Rn-iK'.Rn,] 2N 2N-1 n,n' m

The argument r indicates that the state vector is a function of the electron coordinates. The coefficients Bafj(a) are determined by the condition that the Hamil­tonian matrix be diagonal. We do not obtain the exact values of the BafJ(a) in this paper, but rather will use a set of approxmate values of accuracy suitable for our calculation. The last state of the crystal which we need consider is one in which only one molecule is ionized and all other molecules are in the ground state. This state may be represented by the function

'l'(±)k per) =_1_I: exp(ip·Rm)[[ 1m, k)± 12m, k)] , (2N)1 m

(12) where

I am, k)=ua.m(k) II'if;fJn, (13) fJn¢am

and uam(k) describes the state of an isolated singly charged positive ion and a free electron with wave vector k.

First-order time-dependent perturbation theory gives the transition probability between the states described by Eq. (9) and the ionized states described by Eq. (8) in the form

Ja= ~~ f··· f {I (K, K', a I X I k, p, +) 12

+ I (K, K', a I X I k, p, -) 12}O(EJ-Ei)dk, (14)

where

1 K, K', a)='l'(a)K,K',

I k, p, + ) ='l'(+)kP,

1 k, p, - ) ='l'(-\p, (15)

and Ej, Ei are the energies of the final and the initial states, as described by Eqs. (9) and (12). Our main

2

X { I: Bap<a)~(n, n') (an, (3n' I X

X[llm, k)± 12m, k)]}, (16) where

X= I:Xam+t I: Vam,fJn. (17) am afl'l,¢{3n

The interaction potential between two molecules con­sists of three parts,

where the first term is the interaction energy between the (J' frames of the anthracene molecules and the second term is the interaction between one (J' frame and the 1r electrons on the other molecule. The third term is the Coulombic interaction energy between the 1r elec­trons on one molecule and those on the other molecule. Since each term in the interaction energy contains coordinates pertaining to only two molecules the only values of m which make nontrivial contributions are nand n',

(K, K', a I X I k, p, ±)

1 2N(2N-1)iE, exp( -iK·Rn-iK'·Rn,)

2

X It I: BafJ(a)~(n, n') [exp(ip· Rn) a,fJ=l

x (an, (3n' I Van ,fJn,(lln, k)± 12n, k»

+ exp(ip·Rn,)(an,{3n'l V an,fJn,(11n' k)± 12n', k»]}.

(19)

Each matrix element on the right side of Eq. (19) depends only on a, (3, and Rn-Rn" Define the relative coordinates and the coordinates of the center of mass K.,., rn and Kc and Pn by the relations

Kr=HK-K'), rn=Rn-Rn"

Kc=K+K', (20)

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370 S. CHOI AND S. A. RICE

Then Eq. (19) can be reduced to

(K, K', a 1 X 1 k, p, ±)

o(Kc-p) 2 (2N -1) t {~ exp( -iKT • rn+i!p' rn)

2

xC! L:Ball(a).6.(n,O) (an,{30 lVan,lio(11n,k)± i2n,k»)] a.Il~1

2

+ L: exp( -iKT • rn-i!p· r n)[! L: Batl(a).6.(n, 0) n a,~1

x (an, (30 1 Van,llo(llO, k)± 120, k») ]}. (21)

The matrix elements (an, (30 1 Van,lio 1 an, k) can be further simplified to read

(an, (30 1 Van,liO 1 an, k)

(22)

with the specified integration over the coordinates of all 11" electrons belonging to molecules an and {30. As shown in Eq. (18), Van,IlO may be split into three parts. The first two terms contain either no electron coordi­nates or the electron coordinates of only one molecule. Therefore these two terms do not contribute to the matrix element displayed in Eq. (22). The only inter­action potential which need be considered is the coulombic interaction between the 11" electrons in one molecule and those in the other molecule. Thus

(an, (30 1 Van,liO 1 an, k)

= f··· f 1f;/*an1f;/*llo(f: :")Uan(k)1f;llodTI4andTI41l0' (23) ~IJ tJ

where rij is the distance between the ith electron on the molecule an and the jth electron on the molecule (30.

where the arguments of the molecular orbitals are the coordinates of an electron in each molecule with origins at the centers of each molecule, and R is the distance between two electrons.

In order to choose the proper representation of Uk (r) we consider the three important properties of such continuum state functions; (a) they behave as plane waves far away from the molecule. (b) In the region occupied by the molecule the wavefunction oscillates rapidly because of the large negative potential energy. (c) The continuum wavefunction is orthogonal to all filled molecular orbitals. We choose, as a basic function which takes into account all of the above properties, an orthogonalized plane wave17,18

8

Uk(r) = (211" )-l{ exp(ik· r) - L:(CPa, exp(ik· r»CPa}. a=l

(25)

The second term in Eq. (25) is added to orthogonalize uk(r) to all of the occupied molecular orbitals of the excited molecule. This function is approximately normalized to a delta function of the k vector. Equa­tion (24) in terms of atomic orbitals and Huckel coefficients now becomes

(an, (30 1 Van,tlO 1 an, k)

=V2 (211")-!{" f IEC(8) j exp(ik· ran)Vj(r"n)

14

X {L:C(7) iC(8)jVi(rtlO) vj(rllo) }dTandTIlO, id

(26)

To proceed further we need explicit expressions for where the 1I"-electron states. Accurate wavefunctions are not available and we therefore use Slater determinants of Huckel molecular orbitals.15 In this approximation to the molecular electronic structure, the ground state corresponds to that electronic configuration in which all fourteen 11" electrons are in the lowest seven molecular orbitals CPi while the lowest singlet excited state has the configuration with one electron elevated to the lowest antibonding orbital from the highest bonding orbital. uan(k) will be represented by a configuration with one electron from the highest bonding orbital elevated to a continuum level uk(r), which will be defined later. In terms of molecular orbitals, Eq. (23) may be written

(an, (30 1 Van,liO 1 an, k)

(24)

and Vj(ran) is the atomic orbital of thejth carbon atom with its coordinate originating in the center of molecule.

In Eq. (26), the following integrals appear:

A (jj if) =. { .. fVj(r) exp(ik· r)~vi(r/)vf(r')drdr/,

(28)

B(ilj if) =. f··· fVi(r)vl(r)~vi(r')vtCr')drdr/. (29)

We approximate these integrals by the following ex-

17 M. H. Cohen and V. Heine, Phys. Rev. 122, 1821 (1961). 18 J. C. Phillips and L. Kleinman, Phys. Rev. 118, 1153 (1960).

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PHOTOCONDUCTIVITY IN CRYSTALLINE ANTHRACENE 371

pressions:

A(j;iJ)= :Jr··/Vj(r) eXP(ik.r)dr}

x{r·· /vi(r')v/(r')dr'}' (30)

B(il; if) = ;:Jr·· /vi(r)vl(r)dr}

X {r·· / v;(r')v/(r')dr'}, (31)

where Ro and R'o are the distances between the centers of the charge distributions represented by the inte­grands of the integrals on the right side of the above two equations. We neglect all A's and B's which include overlap integrals between atomic orbitals of different atoms. This approximation is consistent with the use of Huckel orbitals.15 When the approximations cited are incoorporated in Eq. (26) we are led to the relation

14 14

(an, (30 I JC I an, k)=v:2(21r)-!:EC(7)[C(8)[ :Elj(k) [=1 i=1

X{C(8)~- tc(a)jtC(8)fc(a)f~}' RjI a=1 f=1 RfI

(32)

where C(a) i denotes the coefficient of the atomic orbital of the ith atom in the ath molecular orbital, and Rij

represents the distance between the ith atom of the molecule an and the jth atom of the molecule (30. The integration over r in Eq. (27) can be carried out easily and leads to

lj(k) = /vj(r) exp(ik·r)dr=cos8!(k) exp(ik·qj),

(33)

where qj is the position vector of the jth atom and 8 is the angle between the vector k and the normal to the molecular plane. It is convenient to use as a representa­tion for Vj( r) a Slater 2pz orbital,

Vj( r) = (a5/1r) ir; COsOj exp( -ar;) , (34)

where rj is measured from the center of the jth atom and a=3.08X108 cm-1•19 In terms of a and k= I k I,

f(k) = -21r(a5/1r)!(16ak/[i(a2+k2)3JI· (35)

As defined by Eq. (21), the transition probability amplitude is a linear combination of matrix elements of the type given in Eq. (32) with values of an and (30 which correspond to all molecules in the crystal. To simplify the calculation we make the following as­sumptions: (a) only nearest-neighbor interactions are

19 ]. C. Slater, Phys. Rev. 36, 57 (1930).

important, (b) the contribution from the interaction between molecules on different a-b planes can be neglected. With these approximations the sum over n in Eq. (21) can be carried out easily since only the terms with rn=a, -a, b, -b, 0, +(a+b), and - (a+b) contribute to the result.

To complete the calculation of the transition proba­bility, Eq. (14), we need the energy of the initial state and an explicit expression for the energy of the final state as a function of p and k. For our purposes it is sufficient to take the initial energy Ei to be equal to the sum of the ground state energy of the crystal and the two singlet excitons

Ei= Eo+2D+2€1+€(K) +€(K'). (36)

In Eq. (36) Eo is the ground-state energy of the crystal, D is the difference between the interaction energy of an excited molecule with neighboring unexcited molecules and the interaction energy of an unexcited molecule with the same neighboring unexcited molecules. €l is the excitation energy of an isolated molecule, and €(K) is the usual "resonance energy" which appears due to the migration of the excitation energy. For the final energy Ef we neglect possible anisotropy with respect to the direction of k and also possible p de­pendence and assume the simple form

(37)

where I is the ionization energy of a molecule in the crystal, and m is the electronic mass. For anthracene the value for I is taken to be 5.6 eVlO and the singlet exciton energy is taken to be 3.1 eV. From these data, the value of k which satisfies the equality E f = Ei is calculated to be 3.96X107/cm.

To obtain the four sets of B(a) a{3 which must be deter­mined from the solutions of a secular equation, we make a simple approximation. It is assumed that the four double excitons denoted by ~K,K,(a,II), Eq. (10), are equivalent. Then four linearly independent, ortho­gonal, normalized sets of double excitons are obtained with the use of the four sets of coefficients B a{3(a) given in Table I.

The preceding description gives all the information necessary to evaluate the transition probability from which the rate of ionization in a unit volume is deter­mined as the product of the transition probability with the number of pairs of excitons in a unit volume. If there are n excitons per unit volume, then there are t[n(n-l)] possible pairs in the unit volume. We assume these pairs are distributed uniformly over the four double exciton states. Then the rate of ionization per second per cm3 is given by

CR=Hn2) {::h+:J2+:J3+:J4 1. (38)

We have computed the rate of ionization assuming the steady-state concentration of excitons,8 n, to be 3.SX

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372 S. CHot AND S. A. RtCE

1012 cm-3 in the limit of K·ai=O, K'·ai=O. Such conditions are satisfied for excitons created by the absorption of ultraviolet light. It is found that

<R= 3.23X 10l3/sec-cm3

for the anthracene crystal.

III. DISCUSSION

(39)

Note added in proof. Silver et at. (private communica­tion) have observed a photocurrent in anthracene which increases with the square of the light intensity and also increases with temperature in the superlinear region. Because of the observed temperature dependence the possibility of a spurious /2 dependence due to the action of different recombination centers may be discarded. In Silver's experiments only long-wavelength (4150-4550 A) weakly absorbed light was used. The results clearly show that:

(a) With strongly absorbed light, carriers are gen­erated at or near the illuminated surface,

(b) with high-intensity weakly absorbed light, carriers are generated approximately uniformly through­out the bulk of the crystal. The bimolecular rate con­stant is found to be 5 X 10-12 cm3 secI, whereas from Eqs. (38) and (39) the theory presented herein pre­dicts the value 2.6X 10-12 cm3 seci . The agreement between theory and experiment must be regarded as remarkably good. Moreover, the recent experiments of M. Kleinerman, L. Azarraga, and S. P. McGlynn D. Chem. Phys. 37, 1825 (1962) ] unambiguously establish that the first excited singlet state is the kinetic precursor to free-charge carriers and that the carrier generation act requires the participation of two or more molecules. Both of these observations concur with the requirements of the proposed exciton-exciton mechanism for charge carrier generation. To consider further the exciton-exciton mechanism discussed in this paper, two questions must be examined: the nature of the approximations in the theoretical analysis and the relevance of the model in view of the available experimental data.

The approximations made in the analysis presented in Sec. II are basically two in number: approximate wavefunctions have been employed in the calculation of transition matrix elements and the range of interac­tion between molecules has been restricted to nearest neighbors.

Consider first the anthracene molecular wavefunction for which we have adopted the Huckel approximation. The Huckel theory of aromatic hydrocarbons is based on the assumption that the one-electron Hamiltonian is of the form T+ V, where V is a potential which is the same for all the electrons under consideration. With the auxiliary assumptions that each molecular orbital can be represented as a linear combination of atomic orbitals, overlap integrals may be neglected, that only

nearest-neighbor atomic interactions need be con­sidered and that the total energy of the electrons is additive in the individual electron energies, the theory accounts rather well for the major features of the spectra of benzenoid hydrocarbons.I5 The phrase "rather well" must be understood to mean semiempirical under­standing and the prediction of correlations, but not accurate quantitative calculations. For example, the transition from the highest occupied to the lowest unoccupied molecular orbital is predicted, correctly, to be short-axis polarized. If overlap is included, the Huckel approximation can also consistently account for excitation from the ground state to all lower singlet states.15 However, the Huckel approximation cannot account for the observed depression of the lowest triplet state below the lowest singlet state. This splitting arises from the differences in electron repulsion in the two excited states. In our calculations only the proper­ties of the first excited state of the isolated molecule are used. To the extent that the Huckel theory ade­quately accounts for the transition to that state, we may consider the corresponding wavefunction a reason­able approximation. It should be noted that our calculations do not depend on the use of Huckel or­bitals, but other more accurate molecular wavefunc­tions are not available at present. One of the likely principal errors involved in the use of Huckel orbitals is avoided in our calculation in that we have neglected overlap between adjacent molecules in the crystal. Overlap integrals are particularly sensitive to details of the shape of the molecular wavefunctions and the usual Huckel orbitals based on Slater functions have inaccurate tails.

From the approximate molecular orbitals we have constructed wavefunctions for the single exciton and double exciton states, always neglecting overlap be­tween adjacent molecules. This is likely to be an excellent approximation since the binding energy of the molecular crystal is small. The fact that the first excited state may be considered an internal excitation and is far removed from a Rydberg state suggests that overlap of an excited molecule with its neighbors will also not significantly alter our results.

Finally, we have assumed a form for the wave­function of the free electron which accounts for the Pauli principle. Since the major contribution to the transition rate comes from the region where the electron wave function and the molecular wavefunction are simultaneously largest, it is of utmost importance to orthogonalize the electron wave function to the occupied core states. Indeed, it has been shown by Phillips and Kleinman18 and Cohen and Heine17 that the effect of the orthogonalization required by the Pauli principle is so large that deep attractive potentials are replaced by repulsive pseudopotentials. Without ortho­gonalization as described, the transition matrix ele­ments could easily be one or two orders of magnitude in error. With orthogonalization, it is possible to make

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PHOTOCONDUCTIVITY IN CRYSTALLINE ANTHRACENE 373

quantitative calculations of the electronic structure of solids and we presume a similar accuracy attends the use of orthogonalization in the present case.

The reader will note the further approximation made in computing the coefficients B(a)a(J' Since the splitting between the four double exciton states is very small compared to the energy difference between the ground state and the first excited singlet state, the error made in assuming the four states to be equivalent is likely to be small.

It is difficult to assess the error involved in allowing only nearest neighbor interactions. From the results of band calculations and overlap integral calculations20 we conclude that the neglect of interactions between molecules in different a-b planes is justified. The restriction to nearest-neighbor interactions in the other directions is not likely to be in error by more than 20%-30% if experience obtained in calculations on the properties of simple liquids can be applied to this case.21

Consideration of all the preceding approximations leads us to believe that the calculation presented is probably accurate to within an order of magnitude. Great care must be exercised in interpreting effects which depend on differences between experiment and theory of less than an order of magnitude.

We turn now to the more interesting question of the relationship between the model mechanism discussed and experimental fact. It is important to note that in Kepler's2 elegant mobility experiments on anthracene, no electrodes were in contact with the crystal. Although hole or electron injection into a crystal may occur when electrodes are in direct contact with the crystal,3 Kepler's observations clearly show that carrier injec­tion is not necessary for photoconductivity to be ex­hibited. The arguments advanced by Kallmann, there­fore, need not be considered herein. In view of the recent experiments by Silver, we conclude that: (a) Exciton-exciton interaction in bulk leads to carrier generation. The theoretical and experimental genera­tion rates are in agreement. (b) When a crystal is illuminated with strongly absorbed light, surface generation of carriers is the dominant mechanism.

The argument considered in this paper permits a temperature variation of the photocurrent through only two mechanisms: thermal expansion of the solid and temperature dependent scattering of the electron. Kepler has convincingly demonstrated that the mobility decreases as the temperature is increased, an observa­tion in agreement with a free electron-isolated scatterer model. Expansion of the crystal lattice will cause a very small decrease in the rate of generation of carriers: The effect of expansion is small because we have neglected any overlap between adjacent molecules. Since we have

20 J. L. Katz and S. A. Rice (to be published). 21 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular

Theory of Gases and Liquids (John Wiley & Sons, Inc., New York, 1954), Chap. 4.

not treated the electronic-vibration interactions of the molecule we cannot decide whether the photocon­ductivity should or should not exhibit an activation energy related to particular vibrational modes. More­over, the observed activation energies for photo­conduction are so small ("'-'0.1-0.3 eV) that it is pos­sible that photoconduction at constant volume would show no such activation. Until this question is settled experimentally, further theoretical speculation is diffi­cult.

Note added in proof. Following the discovery by Silver et al. of an exciton-exciton induced photocurrent with weakly absorbed light we must add a third mecha­nism contributing to the temperature dependence of the photocur~~nt. For the case of weakly absorbed light, the tranSItIOns to the upper electronic state arise mainly from excited vibrational levels of the ground electronic state. An increase of temperature raises the population of the upper vibrational levels and therefore should lead to an increased photocurrent. Indeed, this effect has been observed by Silver.

Two comments are in order at this time. Even though we have not considered interactions with vibrations it is easily seen that if the double exciton lacks suffici~nt energy to ionize by only a small amount, then the contribution of one or two vibrational quanta would make the entire difference between vanishing and non­vanishing of the photocurrent. The basic mechanism remains as described and the vibrational interaction can be considered a perturbation. Also, in a sense interaction with vibrational modes determines th; ratio of free to localized excitons. This ratio may be expected to be temperature dependent and would lead to an apparent activation energy for photoconduction (since the concentration of free excitons increases as T increases). The importance of the two effects men­tioned can only be assessed by detailed calculations which we have not attempted in this paper.

In conclusion, we have shown that the photocon­ductivity of anthracene crystals can be accounted for if it is assumed that the charge carriers are generated by an exciton-exciton interaction. The phenomenology of this scheme is in agreement with the available data if hole-electron recombination is accounted for. While the proposed mechanism may not be the only source of photo carriers, the very good agreement between experi­ment and theory strongly supports the suggestion that exciton-exciton interactions are important.

ACKNOWLEDGMENTS

We wish to thank the U. S. Air Force Office of Scien­tific Research, the U. S. Public Health Service, and the National Science Foundation for fmancial support. We are also indebted to Joseph L. Katz for assistance in the calculations. It is a great pleasure to thank Pro­fes~or M .. Silver and co-workers for communicating theIr e:l"'penmental results before pUblication.

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