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Khoptyar et al. Vol. 24, No. 7 /July 2007 /J. Opt. Soc. Am. B 1527

Migration-assisted nonlinear quenchingin random media

Dmitry Khoptyar,1,* Johannes Gierschner,1,2 and Hans-Joachim Egelhaaf1,3

1Institute of Physical and Theoretical Chemistry, Eberhard-Karls-Universität Tübingen, Auf der Morgenstelle 8,D-72076 Tübingen, Germany

2Laboratory for Chemistry of Novel Materials, University of Mons-Hainaut, Place du Parc 20, 7000 Mons, Belgium3Christian-Doppler-Labor für Oberflächenoptische Methoden, Johannes-Kepler-Universität Linz,

Altenbergerstrasse 69, A-4040 Linz, Austria*Corresponding author: dmitry.khoptyar@uni-tuebingen.de

Received January 8, 2007; accepted March 26, 2007;posted April 16, 2007 (Doc. ID 78655); published June 15, 2007

A simple and highly accurate model is presented for the decay of the excited state density of randomly distrib-uted luminescent centers (e.g., rare-earth ions or fluorescent dyes) affected by migration-assisted nonlinearquenching (e.g., upconversion or singlet–singlet annihilation). The model relates quenching efficiency, inter-preted in terms of the time-dependent quenching coefficient, to parameters of underlying Förster energy trans-fer phenomena and to center concentration. The accuracy of the model is verified by comparison with MonteCarlo simulations. The model sets up a rigorous basis for the characterization of nonlinear quenching in Er-doped glasses and disordered organic optoelectronic materials. © 2007 Optical Society of America

OCIS codes: 060.2410, 130.3130, 140.4480, 160.4890, 160.5690, 260.2160.

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. INTRODUCTIONonlinear quenching [1] is a special type of Förster donoruorescence quenching by acceptors [2,3]. In contrast to

inear donor–acceptor quenching, where the excitationnergy of initially excited donors is dissipated via irre-ersible transfer to acceptors, in the nonlinear quenchingase, the donor excitation energy is transferred to otherxcited donors, which then act simultaneously as both en-rgy donors and acceptors. This so-called upconversionrocess (or singlet–singlet annihilation) leads to shorteronor luminescence (fluorescence) lifetimes and de-reased donor luminescence (fluorescence) quantumields. In many cases, nonlinear quenching is intensifiedue to the presence of energy migration, i.e., Förster ra-iationless energy transfer (homo-transfer) between ex-ited and unexcited donors. The overall phenomenon iseferred to here as migration-assisted nonlinear quench-ng (MANLQ).

MANLQ is encountered in various active laser materi-ls and typically is a deleterious process that causes lossf excitation (pumping) energy. Due to the strong distanceependence of Förster energy transfer probabilities un-erlying MANLQ on the microscopic scale, the lossesrow nonlinearly with donor concentration increase. Inhe case of erbium (Er)-doped silica glass, where the phe-omenon is typically referred to as (migration-assisted)pconversion, it sets a fundamental limit to the maxi-um useful Er-doping level. The corresponding phenom-

non in condensed organic materials is referred to asinglet–singlet annihilation, which is frequently observedn pump–probe experiments at high excitation densities4].

0740-3224/07/071527-8/$15.00 © 2

Since the losses caused by MANLQ are proportional tohe square of excited donor concentration, modeling meth-ds based on Green function formalism, which were ear-ier developed [2,3] to describe donor–acceptor quenching,re not readily applicable in the case of MANLQ. For theame reason the steady-state case of cw pumping and theynamic case of excited states density decay are treatedn a different way. A comprehensive model for MANLQ inhe steady-state case has recently been presented [5]. Dy-amic MANLQ models available to date [1,6–8] are appli-able only for a limited range of donor concentrations.

The phenomenology of MANLQ is well understood onhe basis of extensive Monte Carlo simulation (MCS)tudies [8–12]. It has been shown that macroscopicANLQ effects on the decay kinetics can be accounted for

y the term n2�t�kNL�t�, where n�t� is the donor excitationrobability and kNL�t� is the time-dependent nonlinearuenching coefficient. For high donor concentrationsnd/or fast migration, kNL�t� may be approximated by aime-independent (upconversion) constant CNL, whichonstitutes the so-called quadratic model. CNL is given byhe so-called kinetic limit [13], i.e., the maximum possibleuenching rate ensured by very efficient migration. Thepplicability range for the quadratic model typically cor-esponds to very strong losses that heavily deteriorate theerformance of an active material. In contrast, for theractically important case when losses are low or moder-te, kNL�t� is essentially time dependent. In fact, in thebsence of migration and/or for low donor concentrations,hich corresponds to so-called static quenching limit,

NL�t� is proportional [14] to 1/�t. This dependenceradually changes to the constant CNL when the center

007 Optical Society of America

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1528 J. Opt. Soc. Am. B/Vol. 24, No. 7 /July 2007 Khoptyar et al.

oncentration increases. Accurate analytical modeling ofNL�t� for practically important donor concentrations,hich would allow us to avoid highly time-consumingCS, is therefore important for the characterization and

ptimization of the materials affected by MANLQ.In this paper we present a rigorous model for MANLQ

n the dynamic (decay) regime that is widely used for ma-erial characterization. We confirm the accuracy of theodel by comparing its results to MCSs.

. MODELe consider optically active centers, which are randomly

nd uniformly distributed in the host material with anverage number concentration c and assume that theinimum separation between the centers may not be

loser than some minimum distance R0 that is given byhe structural properties of the host material. The centersre susceptible to two kinds of Förster resonant energyransfer (FRET) phenomena, namely, energy transfer up-onversion (ETU) and migration. Their interplay causesANLQ on the microscopic scale. We assume that tran-

ition dipole moments for the centers have spherical sym-etry on the time scale of energy transfer. The centers

an be, for example, Er ions in silica-germanium glass orye molecules in isotropic matrices. Finally, we also as-ume that all the centers have the same radiative lifetimeand that MANLQ is the only quenching mechanism.ETU is FRET from one excited center (energy donor) to

nother excited center (energy acceptor). As a result, theonor is deactivated to the ground state while the accep-or is first promoted to a higher excited state and thenromptly nonradiatively relaxes back to the initial excitedtate. The net result is the annihilation of one excitedtate, which can be illustrated by the following energyransfer sequence:

D* + D* ——→ETU

D + D** ——→nonradiatve relaxation

D + D*.

ince the lifetime of higher excited states is typically veryhort, in the following we will limit our consideration to awo-level approximation, i.e., consider only transitions be-ween the ground state and the first excited state.

Migration is nonradiative FRET from an excited centero an unexcited one, i.e., D*+D→D+D*. Since excitationnergy is conserved, migration itself does not causeosses. Migration transports excitation energy over theenters as a series of successive hops from center to centerntil two neighboring centers are excited and one of them

s deactivated due to ETU. In this way migration intensi-es MANLQ, in analogy to donor–acceptor quenching.The probabilities (rates) for ETU �Pu�r�� and migration

Pm�r�� are strongly dependent on the distance �r�, be-ween the interacting centers. In the point-dipole approxi-ation, i.e., at distances much larger than the lengths of

he transition dipole moments, they are given by Försterheory as

Pu�r� =1

�

Ru6

r6 ,

Pm�r� =1

�

Rm6

r6 , �1�

here Ru and Rm are the Förster critical radii for ETUnd migration, respectively.On a macroscopic scale, the decay of center excitation

robability �n�t�� affected by MANLQ is modeled by theate equation

d

dtn�t� = −

n�t�

�− n2�t�kNL�t�,

n�0� = n0, �2�

hich yields deactivation kinetics in the form

n�t� =n0 exp�− t/��

1 + n0�0

t

kNL�t��exp�− t�/��dt�

. �3�

here n0 is an initial excitation probability obtained aftershort excitation pulse at t=0. The time-dependent non-

inear quenching coefficient kNL�t� relates macroscopicANLQ efficiency to center concentration, minimum cen-

er separation, and Förster critical radii for migration andTU.For kNL�t� we adopt the expression given by Philipsen

t al. [10]:

kNL�t� =1

2

4�

cn2�t��0

�

Pu�r�f**�r,t�r2dr, �4�

here f**�r , t� is the probability distribution (at time t) perunit volume�2 of finding two excited donors at distance r.**�r , t� accounts for the temporal evolution of excited cen-er distribution due to the interplay of ETU, migration,nd spontaneous relaxation. By solving the rate equa-ions for a pair of two centers affected by ETU and migra-ion (cf. Appendix A) we show that f**�r , t� is equal to

f**�r,t� = c2n2�t�H�r − R0��R2�t�exp�− Pu�r�t�

+

�0

t

exp�− Puc�r�t��R2�t��dt�

�0

�

R2�t��dt� � . �5�

ere H�r−R0� is the Heaviside step function, and R2�t� ishe deactivation kinetics for a pair of two excited centersue to energy migration:

R2�t� = exp�− 2t/�M�, �6�

here 1/�M is a Markovian (i.e., time-independent) mi-ration rate for one center in a pair. We calculate an ap-roximate value for � in Appendix B as

M

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Qrt

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Ia

a(cttrnR

TItB(p

A

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Khoptyar et al. Vol. 24, No. 7 /July 2007 /J. Opt. Soc. Am. B 1529

�M = 2��DD−2 , �7�

here �DD is the critical concentration for migration,hich we define by

�DD =4���

3c

Rm6

�2Rm6 + 4Rm

3 Ru3

. �8�

ubstituting Eq. (5) into Eq. (4) and integrating, we ar-ive at the following expression for the nonlinear quench-ng coefficient:

kNL�t� = R2�t�Q�t� +

�0

t

Q�t��R2�t��dt�

�0

�

R2�t��dt�

, �9�

here Q�t� is the static (i.e., not migration assisted) non-inear quenching rate [1,8,14]:

Q�t� =�DA

4�t�erf2���KL

�DA

�t/� . �10�

�t� is related to center concentration, c, and ETU criticaladii, Ru, via the critical concentration for ETU ��DA� andhe MANLQ kinetic limit [13] ��KL�:

�DA = �4/3����Ru3c, �11�

�KL =2

3�

Ru6

R03 c. �12�

To present our results in more computation-friendlyorm, we note that Eq. (9) can be approximated by a sumf stationary and nonstationary terms, in direct analogyo the recipe suggested by Gösele [15]:

kNL�t� = �KL/� − kST

�KL/� Q�t� + kST, �13�

kST =

�0

�

Q�t�R2�t�dt

�0

�

R2�t�dt

. �14�

t can be shown by direct computation that the discrep-ncy between Eqs. (9) and (13) is negligible.It is worth noting at this point that Eqs. (9) and (13)

re similar to what was established [2,3,16,17] for thelinear) quenching of donors by a constant number of ac-eptors. Here we have shown that the formulas hold inhe nonlinear case of MANLQ. A notable difference lies inhe meaning of R2�t�. For donor–acceptor quenching R2�t�epresents the deactivation kinetics of a single excited do-or center due to migration, whereas in the present case2�t� is the deactivation kinetics of an excited pair.The evaluation of Eq. (14) results in

kST =�DA

�2�����M

arctan�2��KL��M

�DA�� . �15�

his result is obtained in the Markovian approximation.t can be generalized to the non-Markovian case using theime-dependent migration rate kM

p �t� found in Appendix. The generalization is done by replacing R2�x� in Eq.

14) with non-Markovian deactivation kinetics [2] for aair in the form of

exp− 2�0

t

kMp �t��dt� . �16�

fter integration we arrive at the result

kST =�KL

� 1 − F �DA�DD

2���KL , �17�

F�x� = 1 − ��x exp�x2�erfc�x�. �18�

he difference in kST computed by using either Eq. (15) orqs. (17) and (18) is negligible; however, the latter formu-

as are easier evaluated by the standard mathematicalackages (e.g., MATHCAD). We therefore recommend usingqs. (10)–(13) in combination with Eqs. (17) and (18) for

omputing the nonlinear quenching coefficient kNL�t�. Theoefficient kNL�t� thus obtained is then used in Eq. (3) forodeling experimental relaxation kinetics affected byANLQ.

. DISCUSSION AND COMPARISON TOONTE CARLO SIMULATIONS

n the present model the contribution of direct (static)TU transfer to MANLQ is given by the nonstationary

first) term of the quenching coefficient kNL�t� [cf. (13)],hich is proportional to Q�t�. The energy migration con-

ribution to MANLQ is accounted for by the stationaryerm kST. Q�t� depends on center concentration and ETUadius as cRu

3 whereas the dependence of kST on centeroncentration and energy transfer radii is more complex.n the following, we explore the effects of the critical radiind center concentration on kST and then proceed with annalysis of kNL�t� given by linear combination of Q�t� andST.In Fig. 1, kST is plotted as a function of the critical ra-

ius of migration �Rm�. We observe that kST grows at leasts Rm

3 for sufficiently low Rm. By making use of thesymptotic expression F�x��1−��x, �x�1�, and usingqs. (17) and (18), we find that

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1530 J. Opt. Soc. Am. B/Vol. 24, No. 7 /July 2007 Khoptyar et al.

kST ��DA�DD

2�, �19�

hich holds if kST��KL /�. Using Eqs. (8) and (11), thisan be further simplified as follows:

kST �8�3Ru

3Rm6 c2�−1

9�2Rm6 + 4Rm

3 Ru3

��19.5�−1c2Ru3Rm

3 , Rm � Ru

13.8�−1c2Ru3/2Rm

9/2, Rm � Ru.

�20�

or larger values of Rm, kST saturates at the maximumossible quenching rate, �KL /� (the kinetic limit), which isiven by the particular values of c and Ru but is indepen-ent of Rm [cf. Eq. (12)].To compare our model to those presented earlier, Fig. 1

lso shows the curves for kST, which follow from the mod-ls derived in [1,6,7] and the Gösele diffusion model15,18] (see also Appendix C). Comparing our results withhose given in [6,7] we took into account that their Ru de-ermines the ETU probability for a single center in a pairnd thus is �62 smaller than the present Ru, which corre-ponds to the total ETU probability in a pair. In the hop-ing quenching regime �Rm�Ru�, the difference betweenur model and those described in [1,6,7] lies only in theumerical factor [cf. Eq. (20)], which is 19.5 for the formernd 9.7 for the latter. The notable advantage of theresent model is that the transition to the kinetic limit isaturally included in the equations for kST, which was nothe case in the earlier models. In the diffusion regime, i.e.,Rm�Ru�, kST is proportional to Ru

3/2Rm9/2, which is consis-

ent with the qualitative predictions of the Gösele diffu-ion model and gives better correspondence to MCSs thanhe models of [1,6,7]. This justifies the definition of �DDiven by Eq. (8) and the empirical correction factor intro-uced in Appendix B.In Fig. 2 we compare our calculations with the results

ig. 1. Comparison of the contribution of migration, kST (nor-alized to the center lifetime, �) to the nonlinear quenching co-

fficient kNL�t�, following from different models. Solid line: kSTersus Rm in the present model, calculated using Eq. (17), as-mptotically approaches the kinetic limit (12) (dotted line) forarge Rm. Line with triangles: kST calculated according to [1].here the kinetic limit is given as upper bound for kST withoutmooth transition as in the present model. Line with squares: kSTomputed according to [6] in the limit of high center concentra-ions. Curve with circles: kST

G following from the diffusion model,alculated according to Eqs. (C1)–(C4). Parameters for all com-utations: c=5�1025 m−3, R0=0.35 nm, Ru=1 nm. Approximateoundaries for diffusion regime, hopping regime, and the kineticimit are depicted by thin vertical dashed lines.

f MCS. kNL�t� is plotted as a function of time for a seriesf different values of Rm, using the same values of centeroncentration �c=5�1025 m−3� and ETU critical radiusRu=1 nm� in all cases. The MCSs were performed simi-ar to [5,8].

When Rm�Ru (i.e., migration is negligible), kNL�t� isiven by Eq. (10), which corresponds to the situationhen excitations relax on initially excited centers [i.e.,

tatic quenching,Q�t�]. When Rm�Ru, kST dominates over�t�, except for the beginning of the decay, and gives ma-

or contribution to kNL�t�. Even if the contribution of Q�t�s pronounced only at the beginning of the decay, it mayot be safely disregarded since it corresponds to the stron-est quenching at highest excitation densities after aumping pulse. The fast variation of kNL�t� in the begin-ing of the decay follows from our model under the as-umption of uniform center distribution and is not relatedo center aggregation (e.g., so-called clustering).

To illustrate the dependence of kNL�t� on center concen-ration, in Fig. 3 we plot the curves of kNL�t� calculatedccording to Eqs. (10), (13), and (15), for a set of meaning-ul Er-doping levels in aluminum-silica-germanium glass

ig. 2. Nonlinear quenching coefficient kNL�t� (normalized toenter lifetime, �) versus time, calculated according to Eq. (13)nd using Eqs. (8), (10)–(12), (17), and (18) (solid curves). Modelredictions are compared to Monte Carlo simulations (dots). Thearameters for computations and simulations are c=51025 m−3, R0=0.35 nm, Ru=1 nm. The maximum value of kNL�t�

or these parameters, corresponding to Rm→�, is given by the ki-etic limit (dotted line). Note that all curves start in the kinetic

imit. The curve for Rm=0 gives Q�t�.

ig. 3. Nonlinear quenching coefficient kNL�t� (normalized toenter lifetime, �) versus time, calculated as in Fig. 2 for differententer concentrations, c (solid curves). The model predictions cor-espond well to Monte Carlo simulations (dots). Parameters foralculations and simulations: Rm=2 nm, Ru=1 nm, R0=0.35 nm.he static quenching rate, Q�t�, given by Eq. (10), is plotted forhe smallest (c=5�1025 m−3, dashed curve) and the largest cen-er concentrations (c=5�1026 m−3, dashed–dotted curve).

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Khoptyar et al. Vol. 24, No. 7 /July 2007 /J. Opt. Soc. Am. B 1531

solid curves) together with the corresponding MCS re-ults (dots). To illustrate the importance of migration ef-ects, the static quenching rate Q�t�, given by Eq. (10), islotted for the smallest �c=1�1025 m−3� and the largestc=5�1026 m−3� center concentrations. The parametersor calculations and modeling are given in the figure cap-ion.

For low center concentration, the contribution of migra-ion to kNL�t� is small, and kNL�t� only weakly deviatesrom Q�t� (cf. Fig. 3. dashed curve). This is in spite of theact that for the present choice of parameters, the transferate of ETU is �Ru /Rm�6=1/64 times smaller than theate of migration. This can be understood by saying thatigration intensifies MANLQ due to excitation energy

ransfer over large distances via successive hops fromenter to center. The efficiency of this process thus scaless c2 [cf. Eqs. (8), (11), and (19)]. The contribution of staticuenching, accounted for by Q�t�, is proportional to c andt low concentrations dominates over the contribution ofigration, given by kST. When the center concentration

ncreases, kST grows faster than Q�t� (cf. Fig. 3. dashed–otted curve) and becomes the major contribution toNL�t� at sufficiently high center concentrations. For evenigher center concentration, kST becomes comparableith the maximum quenching rate given by the kinetic

imit in Eq. (12). In accordance with Eqs. (13) and (18),he overall kNL�t� saturates at �KL /�. Physically this cor-esponds to the situation when energy transfer via migra-ion is so efficient that the distribution of excitation en-rgy over the centers is uniform (i.e., all the centers havequal excitation probability), therefore there is no furtherncrease of quenching rate kNL�t� due to migration.

To illustrate the effect of MANLQ on the deactivationinetics, we plot in Fig. 4 the quenching efficiency (QE) ofANLQ versus Ru and Rm for a center concentration of

=1026 m−3 and different initial excitation probabilitiesgiven in the figure). QE is defined by

ig. 4. Quenching efficiency calculated according to Eq. (21) asfunction of Ru and Rm for different initial excitation levels, n0

given in the figures). Center concentration c=1026 m−3.

QE = 1 −1

n0��

0

�

n�t��dt�. �21�

rom an inspection of Fig. 4 it becomes clear that with in-reasing excitation density �n0� QE grows due to the de-reasing mean distance between singly excited states. En-rgy migration, i.e., increasing Rm, leads to enhanced QE,he fastest change being observed for Rm�Ru. The en-ancing effect of energy migration on QE is more pro-ounced for small values of n0 (and for large values of c).or the center concentration of c=1026 m−3 used in thisxample, QE at Ru=Rm=1.5 nm is approximately 1% for0=0.01. This value increases to more than 70% for a cen-er concentration of c=3�1027 m−3, as it is typicallyound in organic crystals (assuming that the point dipolepproximation still holds).In summary, QE strongly grows with increasing center

oncentration, excitation density, and both critical radii.or possible applications this means that the perfor-ance of the active medium may be significantly deterio-

ated by MANLQ unless one or several of these four pa-ameters are optimized in order to keep losses on aeasonably low level.

It should be noted that the QE, as defined in Eq. (21),haracterizes losses during the decay when the center ex-itation probability decreases from n0 to 0. The rigorousreatment of MANLQ in the steady state, when the exci-ation probability, n, is a constant, is presented in [5],hich together with the present paper provides the com-lete modeling toolbox for characterization and analysisf MANLQ in disordered materials.

. CONCLUSIONSe present an analytical model for migration-assisted

onlinear quenching in a sample of randomly and uni-ormly distributed luminescence centers. The model inter-rets MANLQ efficiently in terms of a time-dependentuenching rate kNL�t� and describes explicitly the depen-ence of kNL�t� on center concentration, minimum centereparation, and FRET parameters (critical radii) for non-inear energy transfer and energy migration. The modelredictions coincide well with Monte Carlo simulation re-ults. The model calculations are easily implemented inny standard mathematical package and allow us tovoid highly time-consuming Monte Carlo simulations forhe accurate interpretation of experimental decay kinet-cs affected by MANLQ.

PPENDIX Ao find the excitation distribution function, we considerhe rate equations for a pair of centers separated by a dis-ance r (cf. Fig. 5):

�f00�r,t�

�t= − 2Wexf

00�r,t� + 2Wdef0*�r,t�,

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Fmstsets

wg

Teb

Tios

Iaf

Nb

S=

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ADptiadsMb

Ttaa

Fscestbt

1532 J. Opt. Soc. Am. B/Vol. 24, No. 7 /July 2007 Khoptyar et al.

�f0*�r,t�

�t= Wexf

00�r,t� − �Wex + Wde�f0*�r,t�

+ �Wde + �1/2�Pu�r��f**�r,t�,

�f**�r,t�

�t= 2Wexf

0*�r,t� − �2Wde + Pu�r��f**�r,t�, �A1�

here the probability that both centers in the pair are si-ultaneously excited is designated as f**�r , t�, whereas

*0�r , t� and f00�r , t� are the probabilities that only one orone of the centers are excited, respectively. The prob-bilities are related to one another and to the spatial cor-elation function for the centers fD�r� as

f**�r,t� + 2f*0�r,t� + f00�r,t� = c2fD2 �r�. �A2�

or randomly and uniformly distributed centers withinimal separation R0, fD�r� is given by the Heaviside

tep function H�r−R0�. In Eq. (A1) Wex and Wde are theotal excitation and de-excitation rates for the centers, re-pectively. In the present case when the centers can relaxither due to spontaneous emission or MANLQ and canransfer and receive energy due to migration to/from theurrounding centers the rates are

Wex = n�t�/�M, �A3�

Wdex = 1/� + �1 − n�t��/�M + kNL, �A4�

here 1/�M is an average probability (rate) for energy mi-ration. The equations (A1) obey the initial conditions

limt→0

f**�r,t� = c2n02H�r − R0�,

limt→0

f*0�r,t� = c2n0�1 − n0�H�r − R0�,

limt→0

f00�r,t� = c2�1 − n0�2H�r − R0�. �A5�

his means that at t=0, the distribution of excitation en-rgy is uniform. Additionally, since there is no interactionetween the centers at sufficiently large distance:

ig. 5. Illustration of transitions inside a pair of two centersusceptible to ETU. In the pair either (a) none, (b) one, or (c) bothenters can be excited. Each of the centers can be excited or de-xcited independently with the transition rates Wex and Wde, re-pectively; the rates account for spontaneous relaxation, migra-ion, and MANLQ with surrounding centers; additionally, whenoth centers are excited (c), each of them can be de-excited owingo ETU inside the pair with the probability Pu�r� /2.

limr→�

f**�r,t� = c2n2�t�,

limr→�

f*0�r,t� = c2n�t��1 − n�t��,

limr→�

f00�r,t� = c2�1 − n�t��2. �A6�

his system of equations is exactly the same as presentedn [10] where it was solved numerically in the frameworkf the continuum model. Here we present an analyticalolution to these equations.

We first exclude f00�r , t� from Eq. (A1) by using Eq. (A2).n the resulting equations, which relate f**�r , t� to f*0�r , t�nd fD

2 �r�, we substitute the first two by the auxiliaryunctions g**�r , t� and g*0�r , t�, defined as

f**�r,t� = c2n2�t�g**�r,t�H�r − R0�,

f*0�r,t� = c2n�t��1 − n�t��g*0�r,t�H�r − R0�.

�A7�

eglecting small terms proportional to n2�t�, after simpleut tedious computations, we arrive at

�

�tg**�r,t� = − �2/�M + Pu�r��g**�r,t� + �2/�M�g*0�r,t�,

�

�tg*0�r,t� = − �1/�M�g*0�r,t� + 1/�M. �A8�

olving this system with the initial conditions g**�r ,0�1 and g*0�r ,0�=1 yields

g**�r,t� =2/�M + Pu�r�exp�− �2/�M + Pu�r��t�

2/�M + Pu�r�,

g*0�r,t� = 1. �A9�

quation (5) readily follows from Eqs. (A7) and (A9) usinghe definition for R2�t� given by Eq. (6).

PPENDIX Buring the derivation of the expression for excitationrobability distribution in Appendix A it was assumedhat the migration rate for a single center �1/�M� is timendependent. This constitutes the so-called Markovianpproximation for migration, when the generally time-ependent (non-Markovian) migration rate in disorderedystems is approximated by an average value. The non-arkovian migration rate for a single isolated center can

e found as [2]

kMs �t� = 4�c�

0

�

Pm�r�exp�− Pm�r�t�r2dr. �B1�

here the exponent term has the meaning of a spatial dis-ribution of surrounding centers available for migrationt time t. In the Markovian approximation, �M is defineds

Ttw

wrpalwHrdptti

w

BE

ATsbmqchsmitveoti

w(

a

Hm

aaMsTr�a

adcsbtdtfm

AWtP2Tc

R

Khoptyar et al. Vol. 24, No. 7 /July 2007 /J. Opt. Soc. Am. B 1533

�M = ��0

�

exp−�0

t

kMs �t��dt�dt�−1

. �B2�

o obtain the migration rate for one center in a pair whenhe environments for both centers are strongly correlatede modify (B1) as follows:

kMp �t� = 4�c�

0

�

Pm�r�exp�− �2Pm�r� + 4�Pu�r�Pm�r��t�r2dr,

�B3�

here the factor of 2 in front of Pm�r� accounts for the cor-elation of the environments of two excited centers in aair, which compete for the neighboring centers that acts acceptors for deactivation via migration. If this corre-ation were neglected, the deactivation rate of a pairould be twice the deactivation rate of an isolated center.owever, if it is taken into account, the total deactivation

ate for the excited pair is just �2 times larger than theeactivation rate of an isolated center. The additional em-irically chosen factor 4�Pu�r�Pm�r� in the argument ofhe exponent in Eq. (B3) makes our model consistent withhe Gösele diffusion model [15,18] (see also the discussionn Sec. 3 and Appendix C).

Evaluating Eq. (B3) and using �DD defined by Eq. (8)e arrive at

kMp �t� =

1

2

�DD

�t�. �B4�

y replacing kMs in the definition for the Markovian rate

q. (B2) with kMp we obtain Eq. (7).

PPENDIX Che present model as well as those of [1,6,7] belong to theo-called hopping quenching models [2,3], which areased on the continuous-time random walk (CTRW)ethod. From investigations of linear donor–acceptor

uenching it is known that hopping models are quite ac-urate for high migration efficiency, i.e., in the so-calledopping regime �Rm�Ru�. In the opposite case, i.e., in theo-called diffusion regime �Rm�Ru�, an alternativeodel, the so-called diffusion approach by Gösele [15,18]

s preferable. To the best of our knowledge, the validity ofhe diffusion model in the case of MANLQ has never beenerified. To test the diffusion model we modify the originalquations [15,18], replacing Ru by 2−1/6Ru, which is donen the grounds discussed in Section 3. This yields the sta-ionary migration-related term of the Gösele model, kST

G ,n the form

kSTG = 4�cDF, �C1�

here D is the diffusion coefficient given by the GAFGachanour, Anderson, Fayer) method [19] as

D = 0.434�

3RDD

3 c4/3RDD2

�, �C2�

nd is the collision radius defined by

F

F =RDA

2�4 2

�3/4�

�5/4�RDA2

D�1/4 I3/4�z0�

I−3/4�,z0�, �C3�

z0 =1

2�2RDA

R02RDA

2

D�1/2

. �C4�

ere �x� is a gamma function, and Im�x� is a mth orderodified Bessel function of the second kind.In agreement with what was established for donor–

cceptor quenching, in the diffusion regime kSTG calculated

ccording to Eqs. (C1)–(C4) indeed better corresponds toCS than calculated according to [1,6,7] whereas it

trongly overestimates quenching in the hopping regime.he present model follows the kST

G curve in the diffusionegime and the curves for the hopping models for �RmRu� [cf. Eq. (20)]. This makes it uniformly accurate for

rbitrary values of Rm and Ru.It is noteworthy that Eqs. (C1)–(C4) are based on the

ssumption that energy diffusion occurs only among theonors. In the MANLQ case, both acceptor and donor ex-itations diffuse through the system with identical diffu-ion coefficients, D. We attempted to account for this facty substituting D→2D in Eqs. (C1)–(C4), i.e., by takinghe effective diffusion coefficient as the sum of individualonor and acceptor diffusion coefficients. Unfortunatelyhis deteriorates the agreement with MCS and shows thaturther efforts are needed in order to adapt the Göseleodel to the MANLQ case.

CKNOWLEDGMENTSe acknowledge the financial support of this research by

he European Commission through the Human Potentialrogramme (RTN “NANOCHANNEL,” contract HPRN-CT-002-00323). We thank Alfred Meixner (University ofübingen) for indispensable support and encouraging dis-ussions while working on the paper.

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