Electronic energy transfer in an impurity band

..

Chemical Physics 38 (1979) 227-237 .O North-HollandPublishing Company

..

_., :

ELECTRONIC ENERGY TRANSFER IN AN IMI’URITY BAND

-.

,:

_..

l&us GODZIK g Department of Chemical Physics, 171e Weizmann Institute of Science, Rehovbt, Israel

and Joshua JORTNER Department of Chemisny, Tel-Aviv University, Tel-Aviv. Israel

Received 28 August 1978

In this paper we explore the dynamics of triplet electronic energy transfer in an impurity band of a substitutionally-- ‘... ‘. disordered material. The general characteristics of the solutions of the master equation in the strong scattering regime have 1 been studied, leading to a generalized diffusion equation that contains a memory term, which results in a timedependent diffusion coefficient. Approximate results for the average density of excitSon were derived within the framework of the ... pair-approximation, resulting in explicit expressions for the time-resolved spectral diffusion, the mean-sqntie displacement;,. as well as the time-dependent diffusion coefficient for electronic transfer processes induced by exchange interactions..

::

1sIntroduction The problem of electronic energy transfer (EET) in impurity bands of substitutionally-disordered molecuIar and ionic crystals is of considerable experimental ‘and theoretical interest. From the experimental point of view, recent information concerning the migration of electronic excitation between randomly substituted molecules stems from the following sources: (1) Luminescence quenching of the emission from the impurity band. EET in an impurity band of an isotopic or a chemical substituent was monitored by interrogating the emission yield for an energy-acceptor, such as a chemical supertrap or.an impurity dimer [l-4]. . (2) Spectral diffusion studies probed by timeresolved-fluorescence-line-narrowing experiments [5, 61. A small fraction of the impurities within the inhomogeneously broadened absorption lineshape was excited by a narrow laser pulse, resulting inEET from $ On leave from: Lehrstuhl fti TheoretischeChemie, TechnischeUniversEit Miinchen, Lichtenbergstrak4, 8046 Garching,Germany.

_. the initially excited states to other im&riti& whi& -’ I-,_ are located at different microscopio host environ- .. .. : merits. (3) Spatial diffusion stud&s, which monitk the .-_ rakge of the spread of the electromc exiitatiori ‘.. c .’--:: energy in an impurity band was conducted by the -. ; transient picosecond grating method [7]. From the point of view of general methodology. two physically distinct mechanisms~of EET irki im- . : purity barid may be realized. When the mean free-_ .-:-. path of the electronic excitatiotikonsiderably exceedsthe impurity-impurity separation, the EET process :.I’-. is coherent. Such a situation was theoretically explored 1 for low-temperature, neat molecular cr)istal& &here :_ -in the absence of effective exciton-phonon scattering processes the square root of the mean-square _displacement is proportional to the tinie ,[8] _To the-best of % our knowledge no experimental evidence is y&avail; .‘., able for coherent HET m an impurity band, and we:.. .. have to consider the second ekreme limit if EET.irr.1 such a system. This involves the ikcoberent; strong-. ‘! scattering, situation where allbhase relatiobs betskeen ,. different impurity sites are eroded. Consequently; the-. tipurity molecules have lost all me_mOy rega@iAg L. : . . : ..

228

K. Godzik, J. Jortntir /Electron?

previous energy migration events. This state of affairs can be specified In terms of a master equation for energy hopping in the impurity band, which is characterized in terms of probabilities of EET between pairs of impurities. Some of the available experimental -evidence regarding luminescence quenching, spectral diffusion and spatial diffusion studies in impurity bands is consistent with the strong scattering picture. At this point one should inquire whether the strong scattering picture for EET in an impurity band implies that the energy migration process is amenable to a theoretical description in terms of an ordinarydiffusion process. Does there exist a simple diffusion equation for EET in an impurity band, as is the case for an ordered system, and does it make sense to define a diffusion coefficient, which determines the-meansquare displacement of the electronic energy, by a relation similar.to the Einstein formula [9]? Sakun [IO] and Haan and Zwanzig [l l] addressed themselves to this problem demonstrating that one can derive a generalized diffusion equation for the average density of excitation (ADE), where the diffusion coefficient is characterized by a dispersive behavior. The pioneering studies of Sakun IlO] and of Haan and Zwanzig [_Il] confined their studies to the case of EET induced by the Forster-Dexter dipoledipole coupling mechanism [ 12,131. Recent experimental studies of triplet EET in impurity bands of isotopically-mixed molecular crystals resulted in some interesting observations of a “critical concentration” for EET [l-4]. A coherent interpretation of these experiments requires the extension of the studies of incoherent strong scattering EET, induced by shortrange exchange interactions,-which is the subject matter of the present study. In this note we shall discuss the approximate solutions for the master equation for migration of electronic excitations in a substitutionally-disordered system, deriving an explicit expression for the time-resolved spectral diffusion, for the mean-square displacement of the excitation, as well as for the time-dependent diffusion coefficient when the EET process is induced by exchange interactions.

2. Microscopic model We consider a substitutionally-disordered twocomponent system, which consists of N molecules of

energy transfer in an impurity band

an inert, optically-transparent host andNr impurity molecules, which are randomly distributed at the lattice sites:Confining ourselves to flash excitation of the impurity the initial conditions at time t = 0 imply that the probability of finding the moleculei, located at fj in the excited state, will be independent of the configurations of the other (Nr - 1) molecules

pj(R,t=O)=F(fj);

R=(r,,r*

,._.,mArI).

(1)

In what follows we will assume that the total impurity number density, c =Nr/N, is large relative to the density C* of optically-excited molecules C*
G-9

that the probability of exciton-excitori interaction processes is negligible. The microscopic time evolution of the excitation probability pj of an individual impurity molecule j at so

time I will be described by a master equation of the gain-loss(birth-death) type: (3) where pi =4

exp(t/r) _

(34

The sum over n is taken over all the impurities and %vjk are the rates for EET between the molecules k andi, and we assert that Wnj = 0 when n =i_ Finally, T is the lifetime of the excited state due to radiative and nonradiative decay. Thus the additional loss term due to the ftite Iifetime of the excited state has been omitted in eq. (3) since it was easily incorporated by the transformation (3a), and the effects of radiative decay can be included in the foal result. The master equation (3) can easily be rewritten in matrix form: dp(tlldt = WP@) >

(4)

with

(5) This transfer matrix W is real, symmetric and singular. The latter feature expresses the conservation of the number of excited impurities. The formal solution of this fust order differential equation (4) is given by p(R,t)=exp(tW)*p(R,O).

(6)

229

K. Godzik, I. Jortner f Etectrchic energy transfer in an impurity band

Before proceeding to.the treatment of eq. (6) for an impurity band, it will be useful to consider two cases, which allow an exact solution of the master equation (3) (A) The regular krttice. Let us assume that the Nr Impurities occupy a regular Iattice or a superlattice. The problem then is characterized by translational symmetry and can readily be solved by taking the Iattice Fourier transform of the site excitation probability.

G exp(* - m)~k(t),

p,(t) =f

(7)

I

where k is the lattice wavevector. For a singIe excited impurity in the system we utilize the initial condition

to that of case (A). The diffusion coefficient DlcZ~for a dilute random system evaluated in the ,mean field approximation is therefore identical with (11) up to a factor of impurity density: D

03)

In the next section we will see, that the treatment of EET in a random system by the mean field approximation is inadequate, as substitutional disorder results in some interesting memory effects.

3. Equations of motion for the averaged density of excitation

@I

PJLO)= &I,, > to get

p,(t) = exp -t (

C w,, I

)

X&T

exp ik*rno+t [1

1 _I

m

c

~Vn, co@ - ~,*m) .

(9) This equation results in an explicit expression for the diffusion coefficient by expanding co@ - mm) to the lowest order in k and averaging over the angular dependence of k: r,,. Eq. (9) reduces then to: p,(t) =

kc

exp(ik

l

rno f k*Dt) ,

m

I

with D being the diffusion constant for the regular lattice

The treatment of a random system without invoking the mean field approximation is very difficult due to the loss of transIational symmetry. A general solution over the whole time s&e and the entire concentration range was not yet derived. It is therefore sometimes advantageous to reformulate the problem to gain further insight into the general characteristics of the solution. This will be accomplished in the first part of this chapter, resulting in an alternative formal relation. Subsequently, we shall be concerned with approximations which allow for the solution of the average density of excitation (ADE) in a certain time- and concentration range. The quantity we are interested in is the average density of excitation @DE), P(r, t),

PC;,t>

=($hj- r)PjR f)),

(14)

where D=$&,w,,;

mm =r,-r,

_

(11)

m

(B) The mean-field result. For a dilute random system it is tempting to average the master equation (3) directly by repiacing the restricted sums over the impurity sites by

&CC, n

02)

denotes the usual configurational average [14]. Here. V is the volume of the system. Since the probability pj(R, 0) is taken to be independent of the configuration of the other (N - 1) molecules the initial condition for the ADE can be shown to be

n

P(r, t = 0) = pF(r) i.

where now the summation is taken over all the lattice sites. Within this mean-field approximation translational symmetry is restored and the solution reduces

(16)

where p =Nr/Vis the impuiity number density_ We now proceed to the derivation of an integro-dif-

K. Godzik, J. Joortner / Electronic energy transfer in nn impuri~

230

ferentid equation for the ADE. In what follows it will turn out to be convenient to introduce the Green’s function representation for the solution of eq. (6) i; r)P(r’, 0) ,

P(r, r) =Jd3r’G(r,

where the configurationally-averaged is defined as G(r,r’; f) =p-1 i?ic

i

(17) Green’s function


r’)) , (18)

k

w

[exP(~w)l jk .

=

A. By inspection of the diagrammatic series it can be seen that one can separate out the sum of all irreducible graphs, go and rewrite eq. (25) in the form of a Dyson equation 1151 <=!kr+goD~,

(26)

where the matrix element uj, = rvj& )

with Ajk

band

Performing now the &place transform of eqs. (18) and (19)

(27)

is sometimes referred to as the “interactor” between the sites j and k 1161. Performing now the configurational average we obtain the corresponding Dyson equation for the configurational&averaged Green’s function, eq. (1 S), G(r, r’; s) =@s)-r

g(r, r’; s) ,

(28)

with A@,$‘= J

dt eeS’A(r, t) ,

(20)

0

E=a+aUG

(29)

and

we obtain
U(r,f’;S)=z G(r,r’;s)

=p -r CC(s(r,-f)gjrP(Tk-_r3), i k

The Laplace transform of eq. (17) now takes the form

where glk is the Green’s function for the probability of excitation before averaging over different cotifigurations: gj&= [(s - W)--‘I,

.

(221

Applying successively the operator identity _

1

=Q-!-

s-A

s

ss-A

(23)

Multiplication of (31) by ?(r”, I-, s) p followed by integration with respect to r results in

(24)

sP(r, s) - P(r, 0) = s sd%‘@(r, r’, s) P(r’, s) ,

=6(f

if’)

+pw(f

(32)

(33)

m

The first few B, coefficients, as well as their appropriate diagrammatic expansion, are given in appendix

-

pu-l(r,r’,s).

(34)

+

(25)

-f’)/s

Bearranging (34) we get: s(r, s) - P(r, 0) =j@(r

result will be expressed in the coincise form

&kTnzO&(jk)s-‘. n I

(r”, r; s) p(r, s) = P(r”, 0) ,

which can be rewritten by means of eq. (28) as:

@(f,fr,s)

gjk = 6jk + ST’ Wjk + s -* C WjnW& + _e-. n

(31)

where we have introduced the auxiliary quantity:

a

where

ms

sP(r, s) = p” sd3r’c”(r, r’; s) P(r’, 0) _

ps Jd”r”‘-’

to eq. (22) reWhS in the following identity for gik gjk=s-%jk

(30)

(21)

.I-

- r’, O)P(r’, s) d3i

d3r’[&(f - r’, s) - $(r - r’, 0)] P(r’, s) ,

(35)

with #(r - r’, t) being the Laplace transform of @(r r’, s)_ Applying the inverse Laplace transform to (35) we finally obtain an intergo-differential equation for

K. Go&k. J. Joher /Electronic energy transferin an impurityband

_’

the ADE

;P(r,

231

ible graphs, but rather to exploit a special property of the transfer matrix W. W, eq. (5), can be segregated into a diagonal and a nondiagonal contribution:

t) =sd3r’Q(r - r’, O)P(;‘, r)

(40) Defining now the bare diagonal Green’s function ge as Eq. (36) is similar to the matrix equation given by Sakun [lo]. To proceed, we would like to establish the relation between eq. (36) and the mean-field result. Exploiting the relation 4(r - r’, 0) = lim s@(f - r’, s) , S’W

the following Dyson equation is obtained for g, eq. (22) 9 = 90

- r”)] . . (38)

Inserting eq. (38) into eq. (36) we get the final result:

+

s

d3r’

r’) [P(r’,

t dr a@ - r’, 7) s 0

a7

P(r’,

t) -

P(r,

t -

7)

f)]

.

(3%

The frst term on the rhs of eq. (39) is cert&ly identical to the mean-field expression for the averaged form of the master equation (3). The second contribution to the rhs of eq. (39) constitutes a memory term which introduces retardation effects due to the distribution’of the transfer rates in random systems. This memory term results in a time-dependent diffusion coefficient. This dispersive behavior of the diifusion coefficient is well known in the theory of eiectronic conduction in amorphous solids [17-191, while it was introduced for the case of electronic excitations of impurities in matrices by Sakun [lo] and by &an and Zwanzig [l l] . Eq. (39) can be solved by a density expansion summing up all the diagrams of the corresponding order in the density. This approach is quite tedious and it turns out that for actual calculations it is more convenient not to partition the Green’s functions, eqs. (21) and (22), with respect to the sum of irreduc-

+ %wridg

(42)

-

Iteration of eq. (42) be means of the operator identity (23) results in the expansion 9 = i!O

@(r - i, 0) = p [w(r - r’) - 6 (r - r’) sd?‘w(r

r) = p rd”$w (r -

(41)

(37)

it turns out that one can derive an exact expression for $(r - r’, 0) by taking into account only diagrams of the order s-r (appendix B):

$P(r,

,

90 = [s - WP

+ gOWndg0

+ ~O%d~Ow,d~O

+ .-. 1

(43)

Again this perturbation series cannot be solved exactly. The simplest approximation one can perform will be the pair-approximation, where in the case of the nondiagonal elements of g we restrict the intermediate sites in eq. (43) to be either the initial or the final site, and in the case of the diagonal elements only repeated transfer between two sites is allowed for. Bearing in mind that go is diagonal, i.e. cso)ii = g$$ eq. (43) can be recast in the form

(44) For the diagonal elements of g eq. (44) simplifies to (45)

(46) Eq. (45) can now be summed up immediately, and one obtains: 1 L$C

gii’# (

Wjm&Wmj

m

-I*

(47)

1

Inserting eqs. (5) and (41) into eq. (47), followed by the application of the pair-approximation, results in

K. Godzik, J_ Jorfner /Electronic

232

(

St

-1

Cswim . m S+Wjm

)

(48)

Eq. (48) cannot be inverted generally. Invoking the short time or the Iow concentration limit we obtain the following relation between s and the transition m&ix elements wjk . SB\Vjk

_

m

COSh(lVj~t),

9

.

(53)

Substituting eqs. (41) into (53) and confining ourselves to the lowest order result in the short time approximation, one obtains for the inverse Laplace transform of gjk(s): Ajk(t)

z t?Xp(-Wj$)

sinh(wjkr) .

X [d3r’CS (rj - r) A&t) S (rk 7 r’)) 6 (r’) ,

a+

{eLwcrm)’

C

m

t] -

1)

1-

Eq. (59) is the low density approximation equation

(52)

- g~wjkg~wkj)-’

(57)

P”&, G =iw&5 - 11hJl

Xcosh[iv(r,)

Eq. (51) is again sununed up to give gjk =gfwj&i(I

G(r)(G(r_)A&)) ,

(58)

P&r, t) =6(r)

(51)

*

= &I/P)

m9

with

7jj =g~Wj*&lVkj

r)Ajj(f))

(4%

where the index m refers to impurity sites. For the nondiagonal elements of g, eq. (44), reduces within the pair-approximation to gjk = { 1 t 5 + 7i’f+ a--) $\vjkgf

X
where the site indices j’ and k label a fixed impurity pair. Now performing the configurational averaging in eqs. (57) and (58), we obtain for the diagonal contribution

The inverse Laplace transform is then simply: Aij(t) = I-I’ exp(-wi,t)

t) = f&/p) $d3r’6 (r - r’) 6 (r’>

P&r,

the following explicit expression for gji &j(S)=

energy fransfer in an impurity i?and

(54)

We now proceed to the evaluation of the ADE utilizing eqs. (17), (18), (50) and (54). In the evaluation of the.Green’s function (18) it is convenient to separate the diagonal and the nondiagonal contribu-

pd(r, f) = 6 (r) exp c h-t{I + c [e-w(rm) ’ Lm Xcosh[w(r,)

t] -

1)

1,

W) of the

(60)

which was derived by Huber et al. [21], who have advanced a heuristic approximation to account for back transfer to the initially excited impurity. The present treatment provides within the framework of the pair approximation, a proper derivation of the probability to find the initially-excited impurity at time t still in its excited state. Our treatment results in a theoretical expression for the interpretation of spectral diffusion experiments; Next, we turn to the off-diagonal contribution to the ADE. The configurational averaging procedure of eq. (58) results in P,&r,

t) = p

exp[-w(r)

t] sinh[w(r) t] .

(60

4. Mean-square displacement and diffusion coefficient

The ADE takes the form

The quantities of physical interest for the hopping process of electronic excitation in the impurity band can now be derived by means of eqs. (59) and (61). For the mean square displacement we obtain (r? = Jd3n2P(r,

t) .

(62)

,233.

K. Godzik.J. Jortner / EIectronic.energytrarisferin’animpuriy. band

rate constant for EET induced by exchange interac-

As is evident from eq. (59) only the nondiagonal contribution to the ADE survives and we get = ipJd3rr2(l

f]} .

-exp[-2w(r)

tions (63)

W(T)= (2rrjIz) K*F exp(-2r/L) where K iS aconstant

The time dependent diffusion coefficient is defined as: D(t) +;.@,

=;sd

3w2w(r) exp[-2w(r)

t] , (64)

for any form of the transition probability W(T);however, it should be borne in mind that these expressions are. vahd only within the framework of the pair approximation, i.e., for short times or/and for low impurity concentrations. To the best of our knowIedge, the time dependence of the diffusion coefficient has been handled only for the case of dipole-dipole multipolar interactions [ IO,1 11. For the sake of completeness we shah quote the general results for the case of EET induced by multipoIar interactions. The interaction’is taken to be of the conventional form .w(r) = T-‘&/r)”

,

our analysis it is convenient to rewrite the rate con; stant in the form. w(r)=r-l

exp[y(l -r/&J]

,

_

(70)

where y is now related to the critical radius I& for exchange EET and to the excitation lifetime 7 by the .’ expressions Y = &IL

,

T-I exp 7 = (&r//r) K*F .

(71) :

From eqs. (66), (70) and (63), we’obtain for the mean-square displacement

(65)

where r is the radiative lifetime of the excitation and transfer probabilities are equal, while the impurity concentration is expressed now in tern& of the reduced quantity p =@rR~)-l.

= 5Cy+sRf J

(72)

~. (73)

and T = t/r is the reduced time. Integration by parts of eq. (72) yields

(67)

and for the diffusion coefficient

= K1 7 0

o(T) = $(nr)-2 2-(“-5)/nCR;

with .

eWX)],

3C= 2Texp r’,

One then gets for the mean-square displacement - 2 “CR,,l?[(n -_ 5)/n] T””

dxx4[1 - exp(&

0

where

mi)

x r[(n - 5)/n] z-+yn

with the .dirnension’of energy,

L is a characteristic-length of the order of a lattice constant, F = _f d&f&), F*(E) is the spectral overlap’ integral with f&F?) being the donor emission specttim and F,(E) is the acceptor absorption spectrum. For

Ro the critical radius for EET where the decay and,

(&+-b-s)/

~.. :(69)

,

(68)

Haan and Zwanzig [I I ] advanced an elegant scaling argument showing that eqs, (67) and (68) are the first order terms in a power series expansion of and ofD in the density c Their argument holds for any EET process induced by the multipolar interac1 tion [eq. (65)]. From recent. experimental studies_on electronic triplet EET in isotopicahy mixed organic crystals [ 1, 41 it would be worthwhile to investigate the diffusion process in the case of short-range exchange forces. Dexter ]13] derived an expression for the

d.vx5 eBx exp(-JC e-7

Kl~$&-5R&7C,

(74)

(75)

which can easily be reduced .- to the fmal form
$.X-%z, 1

I,

Jc)

u=l’

(76)

~(a, ac>being the incomplete gamma function_ For srt~@values of Jc (
(77)

K. Godzik, J. Jortner / Electronic energy transfer in a?zinzpunS~band

234.

end with eqs. (75) and (76) we obtain the following result for the mean-square displacement n+l (p) =

36.y-5@

c (-l)n Jc n=lJ (1 +n)en!

Zc
(78)

'

and the diffusion coefficient (7%

m9=moa9, with D(0) = 12y-‘i?R~r-’

er ,

$Q) = 1 + ;r

3Pj(l + n>s n !] -

[(-I)”

(80)

For large values of Jc (>IO), eq. (74) can be rewritten in an alternative way


J dy(ln Jc - my)5 eCy 0

I$ view of the pair approximation inherent in the : derivation of eq. (64), we shall be interested essentially in the short time EET dynamics, expressed in terms’of eqs. (79), (80) and (85). From these results several conclusions emerge conceming the short-time behavior.of the apparent diffusion coefficient for triplet EET. First; regarding concentration dependence it is apparent from eq. (64) that D(T) a cover the entire time regime where the pair approximation holds. The same density dependence of II is exhibited for EET induced by multipolar coupling. Second, the short-time t dependence ofD(T) is D(n a r?&“), where G(T) represents a power series in T, as is evident from eq. (79). This behavior differs from the short-time dependence of D(T) in the case of EET induced by multipolar coupling which, according to eq. (68), is given by a single. term L)(T) . . a a-(“-“)!“.

5. Numerical

3f>lO

031)

.k=O

with

resuks for triplet EET

We have performed some model calculations for EET in an impurity band which is induced by shortrange exchange interactions. For this purpose we have to Drodde a rouah estimate of the critical radius I

Ilk@) = (-l)k

dv hrk~ e-J’ _

(i) f

(Q

0

Approximating the integral in eq. (82) for large values of 3Cby the Laplace transform of lr?@), which can be shown to be

[

y dy In”(~) eey = $ 0

1,

I”(a) cI=l

(83)

one obtains another pair of equations for
2

(-l)JQPJ(l)

k=O

Xh+@f)

3c > 10

;

(84)

and

Dfl= &#@I$0

(_l)k(;)rtk)(l)

X [(S - k)/JC] lnGk@)

;

K>lO.

(85)

R. = +L ln(2nh-*rK*fl ,

036)

where K = ez Y”*/~ao, where a0 is the inter-impurity spacing, Y is a dimensionless quantity smaller than unity and E is the dielectric constant. Reasonable guesses are Y/e = 10m2, a0 = 7 A, whereupon K = lo-* eV. Taking F= 0.5-l eV-‘, and the triplet lifetime 1 s < r < 10m6 s, we estimate Ro to be in the range 5L to 14L. Reasonable values of 7, eq. (71), arethusy= 10-25. In view of the limited validity range of the pair approximation we shall limit the discussion to the short-time behavior of the.excitation dynamics. Fig. 1 exhibits the time dependence of the diffusion coefficient for exchange coupling with 7 = 10; as well as for high-order multipolar interactions with n = 8 and n = 10. Several features of these results should be noted. First, for the case of exchange interaction the diffusion coefficient D(0) is ftite at t = 0 in contrast to the divergence of D(0) for any case of multipolar interaction. Second; it is interesting to note that the zero-time diffusion coefficient D(0) = &p_fd3rrz w(r)

K. Gddzik, J. Jortner /

: Fig. 1. The apparent time-dependent diffusion coefficient D(T) pIotted as a function of the reduced time T = fir; Solid curve: exchange interaction, broken atid dotted curves:multipolar interaction with n = 8 and n - 10.

obtained from eq. (64) within the framework of the pair approximation is identical with the mean. field diffusion coefficient derived by adopting a con---. tinuum description to eq. (13). This observation can b&zrationalized by noting that the time derivation of the ADE, eqs. (59) and (61), at 1= 0 is given by ..

I t=o

‘. (87)

m

... reduced time T.,Solid_cwe:

+ 24 exp(-2&p-u3/L)

tie +ajo; operational conch&n

exp(A2p-“e/L) t 24 c~p(-2&-rl~L)

+ ...I , where we have restricted the l&ice sums over nearest neighbors and next nearest neighbors_ ThusDcL: exhibits a rather complicated density dependence .. which differs from.D(O), as well as from the mek field result. To obtain further informatidn on the dyn.. amiss of EET in an impurity band &display in fig. 2. the time dependence of the mean-square displacement, the deviation of this curve.from a linear I dependence, refl&~g &e break&~ of the &onventio& diff& sion model:We have also included iri Gg. 2 some results for.@(t)) induced by high-order tiultipolar .

broken

of

so that back transfer does not play any role at the initial stages of the process. Third, the linear density dependence of D(0) differs from the density dependence of the (time-independent) diffusion coefficient in a regularly ordered lattice. When the EET process is enhanced by exchange interactions, eq. (11) results for a cubic superlattice of impurities DCL = [exp(y)jdr] p-q3[6

exchange jn&&tion,

: and dotted curve?.:multipolarinteraction 9th n = 8 and II 7 : .., -‘ i 10. :’ :: ;... ..coupli&From the& rest& it .&p&rent that the ‘, grossfeatures of EET induced by.exchange intcractions]are qualitatively sin&r to energy migration .: induced by high-order ‘multipo& mteractions. In spite :’ this ‘qualitative fanri&resembi&&between the : 1resultS of excha&e~cou$ir@ and m&~multi~olar’inter~ actionsit should bc noted that these results differin res&tto the functional t&i~d$&n&?nce of .the ;’ :, .’ apparetit &ffusion coefiiidnt;-, :_:...: ,,,, ::.,.,:.

.,: -:

eme&g

from:. ,,:‘,’

236

K_ Godzik, J. Jortner / Electronic energy transfer in an impurity band

:in a perturbative series in the transfer matrix W [eq. (5)I

gjk's -l~~-n(,!fn)jk=i-l%

s-"&(jk), n=o

(A-2)

description, one has to sum over all intermediate path points and aU field points. Eqs. (A-3)-(AS) are repre‘. sented by the following graphs: ) BoQkI:

:--*----J

k

Bl(jW;

i

k

Bz(jkb

_

I

where the first expansion coefficients are given by Bo( jI+

fijk y

(A-3)

Bl(jk)= wik - 6$ c

tvil ,

I

Bz($C)=

C

WinWnk

-

Wfk

(A-4)

C

(Wj1-b

+

-

i#j

Wkl)

1.

__-->__i

+

k

I#k

1

n

j.!-

+s,

~3C174=

P

,- 5;’

c

11.12

‘~jl,~Vjl~2

.k+j

j

c Wjn%mbvmk n,m

‘I Wjnh’nk c

+ wjk c (1vjl11vj12 +Wjl,WkI,t 'lJ2

c

wjl,wj12wjZ3*

11J2J3

~vkll’vkl2)

64.6)

J

I,=12

’

k

---_,__ j ,k

f

Irreducible graphs are defmed as diagrams which cannot be simplified by a partioning across a wide line into lower order diagrams. Due.to this rule we obtain for the sum of irreducible graphs, Z ’ Y;--* -- ;

::__.f J

+

,,

,2

___*!!I k

---J.l=_l,. k

j

k

j

k

{*

{j-k+

+ *

+--}+

jek+.---}

+----

k

(4) Interactions which-stem from the diagonal part of W, -SwlWi,, are represented by a fme line: .--

+

.

(3) Each actual hop between two sitesj and k, s-l Wjk,i.e. contributions from the nondiagonal part of W, is represented by a wide line: -. j

k

'2

_-__,_-__ .TiL=tl)

A diagrammatic representation of eqs. (A.3)-(A.6) can now be easily introduced by the following prescription: (I) Each diagram represents exactly one path. (2) Sjk is represented by a broken line: y--*-,--

+

(N’jJJ + WnJ, + WkJJ/)

i

J

k

i

(A.51

11

- 8ik

1

k_

I

Appendix B: Expression fo&r

- r’; 0)

’

where 1is a dummy index, representing a field point in contrast td the path points, e.g., j and k which arc connected by wide lines. (5) Going.from the diagrammatic to the algebraic

Considering only irreducible graphs of the order s-l we get: r ,, fl’-f’,S)Z:;--,--;

r

+. ;-_,__

f r’

’

K. Godzik,L Jortner / Electronic energy transferin an impun@ band

which correspond to the following algebraic expressions: o(r - r’, s) qE

(6(Ii-_1)[6jk- Sjk qSmlWjl

=S(r-r’)

XT

1

6(fk - r’))

C(6@j-~))-S-1C(6(Tk_T) [I.

w$]=;6(r

1 r’)[l-

,-$d3rr’w(r

-

r”)] .

(B-1) ‘The inverse of u can easily be obtained by means of the relation s

_~(r - r’s) u-l (r’ - r”, s) d3r’ = 6 (r - r”)

03.2)

and we get a-+

, References [l] -R. Kopelman, E.&i. Monberg and F.W. Ochs, Chem. Phys. 19 (1977) 413; 21 (1977) 373. [2] K-E. Mauser, H. Port and H.C. Wolf, Chem. Phys. 1 (1973) 74. [ 31 H. Port, D. Vogel and H.C. Wolf, Chem. Phys. Letters 34 (1975) 23. [4] D.D. Smith, R.D. Mead and A.H. Zewail, Chem. Phys. Lztters 50 (1977) 358. [S] R. Flach, D.S. Hamilton, P.M. Seizer and W.M. Yen, Pbys. Rev. B15 (1977) 1248. [6] D.S. Hamilton, P.M. Seizer and W.M. Yen, Phys. Rev. B16 (1977) 1858. [7] J.R. Sakedo, A.E. Siegman, D.D. Dlott and M.D. Foyer, Phys. Rev. Letters 41(1978) 131. [8] M. Grover and R. Silbey, 3. Chem. Phys. 54 (1971) 4843. [9 ] A. Einstein, Investigation oxi the theory of the brownian movement (Dover, New York, 1956). [lo] V.P. Sakun, Soviet Phys. Solid State 14 (1973) 1906. [ll] S-W. Haan and R. Zwanzig, J. Chem. Phys. 68 (1978) 1879. [12] T. Fiirster, 2. Naturforsch. 4a (1949) 321. 1131 D.L. Dexter, J. Chem. Phys. 21(1953) 836. (141 S.F. Edwards, Phil. Msg. 3 (1958) 1020. [15] F.J. Dyson, Phys. Rev. 92 (1953) 1331. (161 J.M. Ziman, J. Phys. C2 (1969) 1230. [17] N.F. Mott and E.&Davis, Electronic processes in noncrystalline materials (Clarendon Press, Oxford, 1971). [18] P.N. Butcher, J. Phys. C: Solid State Phys. 5 (1972) 1817; 7 (1974) 879. [19] M. Poilak and T.H. Geballe, Phys. Rev. 122 (1961) 1742. [20] M. Abmmowitz and I.A. Steg? eds., Handbook of mathematical functions (Dover, New York, 1968). [21] D.L. Huber, D.S. Hamilton and B. Bamett, Phys. Rev. 816 (1977) 4642.

X[1-p~-'~d3~~'~(,p") ' 1I' - r’, s) =p-16(r

- r’)

P-3)

The auxiliary quantity @(r, r’, s) [eq. (34)] has now the explicit form @(r - r’, s) = 6 (r - r’) + ps-lw(r - r’) - 6 (r - r’) 1 - ps-’ sd”r”w(r

- r”)

[

1

-’ .

Q-4)

Its La&ace transform at t = 0 can be obtained due to the simple relation $(r - r’, 0) = lim s@(r - r’, s) s+” = pw(r - r’) - p6 (r - r’) Jd3r”w(r

- r”) .

(B.5)

23-i