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Numerical studies of the effect of initial conditions on the quantum yield in sensitized luminescence
Volume 114A, number 7
PHYSICS LETTERS
10 Match 1986
NUMERICAL STUDIES OF THE EFFECT OF INITIAL CONDITIONS O N T H E Q U A N T U M Y I E L D IN S E N S I T I Z E D L U M I N E S C E N C E :~ L. S K A L A I and V.M. K E N K R E Department of Physics and Astronomy, University of New Mexico. Albuquerque, NM 87131, USA Received 1 August 1985: revised manuscript received 12 September 1985: accepted for publication 23 December 1985
The effect of initial conditions involved in the creation of the exciton through illumination on the quantum yield of sensitized luminescence in molecular crystal is investigated numerically. The dependence of the effect on the degree of the coherence of exciton motion and on the size of the crystal is discussed. And, the initial spatial distribution of the exciton corresponding to the maximum effect is analyzed.
The generalized master equation [1] (GME)
Im(t ) = -i.
(2)
t
dPm(t)/dt =
f0 dt' ~n
[Wran(t - t')Pn(t' )
- Wnm(t - t')Pm(t')] + Ira(t),
(I)
which is an exact consequence of the LiouviUe-von Neumann equation for the density matrix, has been used a great deal in recent times for the investigation of the motion of various quasiparticles in solids, particularly Frenkel excitons in molecular crystals [2,3]. The structure of this evolution equation for the probability Pro(t) of finding the exciton at site m - and its memory functions Wmn(t ) - enables one to address and describe successfully a number of important features of quasiparticle transport, particularly coherence. However, the usual form of the'GME which is employed for these investigations is the one obtained by dropping a term Im(t ) called the initial term from the evolution [ 1,2]. This initial or inhomogeneous term depends on the initial values of the off-diagonal elements of the density matrix [2] O(t),
~:: Work supported in part by the National Science Foundation under grant nos. DMR-8513058 and DMR-8111434. On leave of absence from Faculty of Mathematics and Physics, Charles University, 121 16 Prague 2, Czechoslovakia. 0.375-9601/86]$ 03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
Here, L is the Liouville operator, P is the projection operator and Ira> denotes a localized basis state. The term vanishes identically for several important conditions: initial occupation of a single lattice site or of arbitrary sites but with completely random phases [ 1] ; or of a single Bloch state of the crystal [4] ; or of a thermalized combination of Bloch states as would be appropriate to a system in equilibrium [2]. It has also been shown how the initial term may be calculated [4] and then used as driving term in the GME to calculate required quantities. However, the fact that detailed effects of the initial term are still unknown has prompted us to undertake a numerical investigation o f its effects which we report below. It is hoped that this note will lead to further work of a more analytical nature into this subject. The observable we study is the quantum yield of luminescence and the system we consider is a molecular crystal in which an exciton propagates and is trapped by guest molecules, The exciton has a finite lifetime r in both the host crystal and the guest molecules. The ratio of the photons collected from the guest molecules to the photons put into the host initially is the observable guest quantum yield ¢~G- It has been analyzed thoroughly elsewhere [2,3] and it can be written as 395
Volume 114A, number 7
PHYSICS LETTERS
10 March 1986
Rmax
~)0 = 2 Pn(O)~nr(1/r) .
~/~ + .~r(1/.~ ) ,
n
1/c+ @rr(1/r)
(4) .
where tildes denote Laplace transforms. In eqs. (3)(5) the initial probability distribution of the exciton over host sites n is given by Pn(O), the @ are probability propagators, ff/rnn(t) being the probability that the ruth site is occupied at time t by the exciton given that the nth site was occupied at time 0, r is the trap site, and c is the rate o f capture o f the exciton by the guest molecule once it is within its influence. For simplicity, we have taken a single guest molecule in the above expressions and in the numerical study below. Factors of 1IN, where N is the total number o f host molecules in the crystal studied, are to be replaced by the relative guest concentration as explained elsewhere [2]. We carried out numerical calculations for onedimensional linear chains with periodic boundary conditions with the nearest neighbour interaction V and the number o f molecules N. Values r = 10 - 8 s and V = 1012 S-1 , which are typical of singlet excitons in molecular crystals [5] were assumed in the calculations. The scattering o f excitons at rate a leading [2,6] to the e x p ( - ~ t ) damping o f the memory functions Wmn~) and initial term Im(t ) was assumed. The initial term I n and propagators ~5nr corresponding to a = 0 may be found in ref. [4]. Modification o f these results for c~> 0 is straightforward and will not be given here. We focused our attention on the ratio R = ~G/~G,I 0 which describes the contribution of the initial term to the quantum yield q~G relative to that of the GME without the term, thus eliminating consideration of the capture rate c. The results are presented in figs. 1 - 4 below. The first numerical investigation was aimed at finding the maximum possible effect of the initial term on the quantum yield. We considered a linear segment of 5 molecules on which the exciton could move via nearest-neighbour interactions V and be scattered at rate c< For every value o f a, we generated 1000 sets of arbitrary (random) numbers for the real and imaginary parts o f the exciton amplitude at the initial time and from them the initial density matrix. We calculated 396
J i~1
05
,
104
lO s
I06
707
~0B
.tO9
1040
Fig. 1. The maximum value of O ~ / ~ plotted as a function of the scattering rate a for a finite periodic linear chain showing the effect of the degree of transport coherence on the relative contribution of the initial term to the guest quantum yield.
the yield ratio R and found Rma x, i.e. the maximum value o f R . The variation o f R m a x with the scattering rate a is plotted in fig. 1. We see that Rma x changes monotonously from 1.1 for highly coherent conditions to zero for incoherent ones. We make two observations: (i) the effect of the initial term on the quantum yield is most significant in the highly coherent case (small c0 as expected, and (ii) it can be substantial, i.e. c)lG can be of the order of ~ 0 . There are two reasons why the effect is small in the region ~ 1/z ~ V. One is the vanishing of the Irn(t ) term due to damping (which also happens for large a >> l / r ) . The other is that for c~ ~ 1/r ~ V, the propagators themselves are largely site independent which makes ~1 essentially proportional to the sum over all sites of In(I/r), and the sum vanishes as a result of the obvious summation rule EnIn(t ) = O. The second numerical investigation was concerned with the variation of the maximum yield ratio Rma x with the size of the crystal, i.e. with the number N of molecules. We restricted our calculations to the coherent case (a = 0) because the effect is largest in that limit, as fig. 1 shows. The results are plotted in fig. 2. The value o f R m a x is very low for the dimer ( N = 2) which appears to be a special case in which initial el2 fects are negligible. F o r N = 3 , R m a x ~ 1.1 so that initial effects can certainly be important. F o r N 2> 3, the value o f R m a x decreases slowly and non-monotonically with N as shown and appears to approach a non-zero limiting value.
Volume 114A, number 7
PHYSICS LETTERS
10 Match 1986
Rmax
-0.4
O.5"
-0.2
0
=
5
'/0
N
I
fE
2
3
4.
5
6
7
Fig. 2. The maximum value of ~ / q ~ plotted as a function o f the number of molecules in the linear chain showing the effect of the system size.
Fig. 4. The ratio ~ / ~ for an initial gaussian exciton packet plotted as a function of the width o of the gaussian showing that the initial condition effect vanishes for both small and large packet width.
The third investigation we undertook was into the spatial distribution of the exciton which would result in the maximum initial condition effect. As it is known [2,4] that the initial effect is negligible both in the fully localized and the fully delocalized limit, we expect R to equal its maximum value when the spatial distribution is intermediate. To examine this distribution we determined the initial amplitudes and from them the initial probabilities Pro(O)of Finding the exciton at sites m = 1,2, ..., N, corresponding to R = Rma x. The results for two different segments, one o f N = 11, and the other o f N = 18, are plotted in fig. 3a and fig. 3b, respectively. The probabilities seem to be roughly oscillatory in space, i.e. to pattern of alternatingly large and small density.
Fig. 3 suggest that the situation "intermediate" between the fully localized and fully delocalized limits which would give the largest initial condition effects is a spatially oscillatory probability rather than one representing a packet o f finite width. To investigate the point further, we consider a segment with N = 11 and took gaussian initial amplitudes rather than arbitrary (i.e. random) ones as in the work corresponding to figs. 1 - 3 . The yield ratio R calculated as a function o f the width a o f the gaussian is plotted in fig. 4. In agreement with the results of refs. [1,2,4] initial condition effects are seem to vanish for small and large o respectively and reach the target value for intermediate o. We point out that (i) the initial effects are much less pronounced for these packet interme-
Pm(OJ
P=(o)
0.4
0.3 0.1 0.2 a
0.05
0.1
0
=, 0
2
4
6
8
10
m
0
i
0
5
10
L
m
15
Fig. 3. The initial spatial distribution o f the exciton leading to the maximum initial condition effect for two system sizes: (a) N =lland(b) N =18.
397
Volume 114A, number 7
PHYSICS LETTERS
diate conditions than for the oscillatory intermediate ones o f figs. 3, and (ii) for the gaussians o f fig. 4, we found R to be negative, i.e. the yield to be reduced from the I(t) = 0 case. In summary, we see numerically that the initial condition effects ( I ) are largest for the coherent limit and reduce monotonically as the scattering rate is increased, (2) decrease non-monotonically as the size o f the crystal increases, (3) seem to be largest for initial exciton occupation which is spatially oscillatory rather than gaussian-like. Most existing sensitized luminescence observations [ 7 - 9 ] entail states which are essentially at the delocalized limit [4] and therefore involve negligible I(t) effects. However, the numerical investigations presented in this note show explicitly how substanting they can be for other (intermediate) illumination conditions, and present intriguing features which await a thorough theoretical understanding. We hope
398
10 March 1986
that efforts in this direction will be undertaken because we believe that initial state effects are both important and poorly understood in this field.
References [1 ] R.W. Zwanzig, in: Lectures in theoretical physics, Vol. 3, eds. W. Downs and J. Downs (Boulder, 1961). [2] V.M. Kenkre, in: Exciton dynamics in molecular crystals and aggregates, ed. G. HiShler (Springer, Berlin, 1982). [3] V.M. Kenkre, in: Energy transfer processes in condensed matter, ed. B. Di Bartolo (Plenum, New York, 1984). [4] V.M. Kenkre, J. Star. Phys. 19 (1978) 333. [5] M. Pope and C.E. Swenberg, Electronic processes in organic solids (Clarendon, Oxford, 1982). [6] P. Reineker, in: Exciton dynamics in molecular crystals and aggregates, ed. G. Hbhler (Springer, Berlin, 1982). [7] H.C. Wolf, Advan. At. Mol. Phys. 3 (1967) 119. [8] R.C. PoweU and Z. Soos, J. Lumin. 11 (1975) 1. [9] D. Schmied, in: Organic molecular crystals, eds. P. Reineker, H. Haken and H.C. Wolf (Springer, Berlin, 1983).
PHYSICS LETTERS
10 Match 1986
NUMERICAL STUDIES OF THE EFFECT OF INITIAL CONDITIONS O N T H E Q U A N T U M Y I E L D IN S E N S I T I Z E D L U M I N E S C E N C E :~ L. S K A L A I and V.M. K E N K R E Department of Physics and Astronomy, University of New Mexico. Albuquerque, NM 87131, USA Received 1 August 1985: revised manuscript received 12 September 1985: accepted for publication 23 December 1985
The effect of initial conditions involved in the creation of the exciton through illumination on the quantum yield of sensitized luminescence in molecular crystal is investigated numerically. The dependence of the effect on the degree of the coherence of exciton motion and on the size of the crystal is discussed. And, the initial spatial distribution of the exciton corresponding to the maximum effect is analyzed.
The generalized master equation [1] (GME)
Im(t ) = -i
(2)
t
dPm(t)/dt =
f0 dt' ~n
[Wran(t - t')Pn(t' )
- Wnm(t - t')Pm(t')] + Ira(t),
(I)
which is an exact consequence of the LiouviUe-von Neumann equation for the density matrix, has been used a great deal in recent times for the investigation of the motion of various quasiparticles in solids, particularly Frenkel excitons in molecular crystals [2,3]. The structure of this evolution equation for the probability Pro(t) of finding the exciton at site m - and its memory functions Wmn(t ) - enables one to address and describe successfully a number of important features of quasiparticle transport, particularly coherence. However, the usual form of the'GME which is employed for these investigations is the one obtained by dropping a term Im(t ) called the initial term from the evolution [ 1,2]. This initial or inhomogeneous term depends on the initial values of the off-diagonal elements of the density matrix [2] O(t),
~:: Work supported in part by the National Science Foundation under grant nos. DMR-8513058 and DMR-8111434. On leave of absence from Faculty of Mathematics and Physics, Charles University, 121 16 Prague 2, Czechoslovakia. 0.375-9601/86]$ 03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
Here, L is the Liouville operator, P is the projection operator and Ira> denotes a localized basis state. The term vanishes identically for several important conditions: initial occupation of a single lattice site or of arbitrary sites but with completely random phases [ 1] ; or of a single Bloch state of the crystal [4] ; or of a thermalized combination of Bloch states as would be appropriate to a system in equilibrium [2]. It has also been shown how the initial term may be calculated [4] and then used as driving term in the GME to calculate required quantities. However, the fact that detailed effects of the initial term are still unknown has prompted us to undertake a numerical investigation o f its effects which we report below. It is hoped that this note will lead to further work of a more analytical nature into this subject. The observable we study is the quantum yield of luminescence and the system we consider is a molecular crystal in which an exciton propagates and is trapped by guest molecules, The exciton has a finite lifetime r in both the host crystal and the guest molecules. The ratio of the photons collected from the guest molecules to the photons put into the host initially is the observable guest quantum yield ¢~G- It has been analyzed thoroughly elsewhere [2,3] and it can be written as 395
Volume 114A, number 7
PHYSICS LETTERS
10 March 1986
Rmax
~)0 = 2 Pn(O)~nr(1/r) .
~/~ + .~r(1/.~ ) ,
n
1/c+ @rr(1/r)
(4) .
where tildes denote Laplace transforms. In eqs. (3)(5) the initial probability distribution of the exciton over host sites n is given by Pn(O), the @ are probability propagators, ff/rnn(t) being the probability that the ruth site is occupied at time t by the exciton given that the nth site was occupied at time 0, r is the trap site, and c is the rate o f capture o f the exciton by the guest molecule once it is within its influence. For simplicity, we have taken a single guest molecule in the above expressions and in the numerical study below. Factors of 1IN, where N is the total number o f host molecules in the crystal studied, are to be replaced by the relative guest concentration as explained elsewhere [2]. We carried out numerical calculations for onedimensional linear chains with periodic boundary conditions with the nearest neighbour interaction V and the number o f molecules N. Values r = 10 - 8 s and V = 1012 S-1 , which are typical of singlet excitons in molecular crystals [5] were assumed in the calculations. The scattering o f excitons at rate a leading [2,6] to the e x p ( - ~ t ) damping o f the memory functions Wmn~) and initial term Im(t ) was assumed. The initial term I n and propagators ~5nr corresponding to a = 0 may be found in ref. [4]. Modification o f these results for c~> 0 is straightforward and will not be given here. We focused our attention on the ratio R = ~G/~G,I 0 which describes the contribution of the initial term to the quantum yield q~G relative to that of the GME without the term, thus eliminating consideration of the capture rate c. The results are presented in figs. 1 - 4 below. The first numerical investigation was aimed at finding the maximum possible effect of the initial term on the quantum yield. We considered a linear segment of 5 molecules on which the exciton could move via nearest-neighbour interactions V and be scattered at rate c< For every value o f a, we generated 1000 sets of arbitrary (random) numbers for the real and imaginary parts o f the exciton amplitude at the initial time and from them the initial density matrix. We calculated 396
J i~1
05
,
104
lO s
I06
707
~0B
.tO9
1040
Fig. 1. The maximum value of O ~ / ~ plotted as a function of the scattering rate a for a finite periodic linear chain showing the effect of the degree of transport coherence on the relative contribution of the initial term to the guest quantum yield.
the yield ratio R and found Rma x, i.e. the maximum value o f R . The variation o f R m a x with the scattering rate a is plotted in fig. 1. We see that Rma x changes monotonously from 1.1 for highly coherent conditions to zero for incoherent ones. We make two observations: (i) the effect of the initial term on the quantum yield is most significant in the highly coherent case (small c0 as expected, and (ii) it can be substantial, i.e. c)lG can be of the order of ~ 0 . There are two reasons why the effect is small in the region ~ 1/z ~ V. One is the vanishing of the Irn(t ) term due to damping (which also happens for large a >> l / r ) . The other is that for c~ ~ 1/r ~ V, the propagators themselves are largely site independent which makes ~1 essentially proportional to the sum over all sites of In(I/r), and the sum vanishes as a result of the obvious summation rule EnIn(t ) = O. The second numerical investigation was concerned with the variation of the maximum yield ratio Rma x with the size of the crystal, i.e. with the number N of molecules. We restricted our calculations to the coherent case (a = 0) because the effect is largest in that limit, as fig. 1 shows. The results are plotted in fig. 2. The value o f R m a x is very low for the dimer ( N = 2) which appears to be a special case in which initial el2 fects are negligible. F o r N = 3 , R m a x ~ 1.1 so that initial effects can certainly be important. F o r N 2> 3, the value o f R m a x decreases slowly and non-monotonically with N as shown and appears to approach a non-zero limiting value.
Volume 114A, number 7
PHYSICS LETTERS
10 Match 1986
Rmax
-0.4
O.5"
-0.2
0
=
5
'/0
N
I
fE
2
3
4.
5
6
7
Fig. 2. The maximum value of ~ / q ~ plotted as a function o f the number of molecules in the linear chain showing the effect of the system size.
Fig. 4. The ratio ~ / ~ for an initial gaussian exciton packet plotted as a function of the width o of the gaussian showing that the initial condition effect vanishes for both small and large packet width.
The third investigation we undertook was into the spatial distribution of the exciton which would result in the maximum initial condition effect. As it is known [2,4] that the initial effect is negligible both in the fully localized and the fully delocalized limit, we expect R to equal its maximum value when the spatial distribution is intermediate. To examine this distribution we determined the initial amplitudes and from them the initial probabilities Pro(O)of Finding the exciton at sites m = 1,2, ..., N, corresponding to R = Rma x. The results for two different segments, one o f N = 11, and the other o f N = 18, are plotted in fig. 3a and fig. 3b, respectively. The probabilities seem to be roughly oscillatory in space, i.e. to pattern of alternatingly large and small density.
Fig. 3 suggest that the situation "intermediate" between the fully localized and fully delocalized limits which would give the largest initial condition effects is a spatially oscillatory probability rather than one representing a packet o f finite width. To investigate the point further, we consider a segment with N = 11 and took gaussian initial amplitudes rather than arbitrary (i.e. random) ones as in the work corresponding to figs. 1 - 3 . The yield ratio R calculated as a function o f the width a o f the gaussian is plotted in fig. 4. In agreement with the results of refs. [1,2,4] initial condition effects are seem to vanish for small and large o respectively and reach the target value for intermediate o. We point out that (i) the initial effects are much less pronounced for these packet interme-
Pm(OJ
P=(o)
0.4
0.3 0.1 0.2 a
0.05
0.1
0
=, 0
2
4
6
8
10
m
0
i
0
5
10
L
m
15
Fig. 3. The initial spatial distribution o f the exciton leading to the maximum initial condition effect for two system sizes: (a) N =lland(b) N =18.
397
Volume 114A, number 7
PHYSICS LETTERS
diate conditions than for the oscillatory intermediate ones o f figs. 3, and (ii) for the gaussians o f fig. 4, we found R to be negative, i.e. the yield to be reduced from the I(t) = 0 case. In summary, we see numerically that the initial condition effects ( I ) are largest for the coherent limit and reduce monotonically as the scattering rate is increased, (2) decrease non-monotonically as the size o f the crystal increases, (3) seem to be largest for initial exciton occupation which is spatially oscillatory rather than gaussian-like. Most existing sensitized luminescence observations [ 7 - 9 ] entail states which are essentially at the delocalized limit [4] and therefore involve negligible I(t) effects. However, the numerical investigations presented in this note show explicitly how substanting they can be for other (intermediate) illumination conditions, and present intriguing features which await a thorough theoretical understanding. We hope
398
10 March 1986
that efforts in this direction will be undertaken because we believe that initial state effects are both important and poorly understood in this field.
References [1 ] R.W. Zwanzig, in: Lectures in theoretical physics, Vol. 3, eds. W. Downs and J. Downs (Boulder, 1961). [2] V.M. Kenkre, in: Exciton dynamics in molecular crystals and aggregates, ed. G. HiShler (Springer, Berlin, 1982). [3] V.M. Kenkre, in: Energy transfer processes in condensed matter, ed. B. Di Bartolo (Plenum, New York, 1984). [4] V.M. Kenkre, J. Star. Phys. 19 (1978) 333. [5] M. Pope and C.E. Swenberg, Electronic processes in organic solids (Clarendon, Oxford, 1982). [6] P. Reineker, in: Exciton dynamics in molecular crystals and aggregates, ed. G. Hbhler (Springer, Berlin, 1982). [7] H.C. Wolf, Advan. At. Mol. Phys. 3 (1967) 119. [8] R.C. PoweU and Z. Soos, J. Lumin. 11 (1975) 1. [9] D. Schmied, in: Organic molecular crystals, eds. P. Reineker, H. Haken and H.C. Wolf (Springer, Berlin, 1983).