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Numerical studies of the trapping of Frenkel excitons in one-dimensional systems
Chemical Physics 146 ( 1990) 409-4 13 North-Holland
Numerical studies of the trapping of Frenkel excitons in one-dimensional systems D.L. Huber Department of Physics, Universityof Wisconsin-Madison, Madison, WI 53706, USA
and W.Y. cbing Department of Physics, University of Mssouri-Kansas,
City Kansas City, MO 64110, USA
Received 15 January 1990
Numerical studies of the trapping of Frenkel excitons on a onedimensional lattice are reported. The effects of trapping are introduced through a non-Hermitian decay term in the Hamiltonian. Each of the traps, which are assumed to occupy interstitial positions, can receive excitation from a single center. The equations of motion for the exciton correlation functions of a finite array of centers are integrated to obtain the decay of the k=O mode and the probability of finding an exciton in any mode, following excitation of the k=O mode. Results for the latter are compared with the predictions of calculations by Hemenger and Pearlstein.
1. Introduction
The dynamics of Frenkel excitons in disordered systems continues to be a rich source of problems, both experimental and theoretical. Although, initially, most of the interest focused on systems with a random distribution of transition frequencies and/ or interactions, recently, attention has been drawn to models with a random distribution of trapping centers [ l-41. The simplest example of this class of problems is one where the system, apart from traps, has translational symmetry; the only disorder is that associated with the traps. In this paper we analyze trapping in one dimension. We investigate a onedimensional model where the traps are randomly distributed in interstitial sites that are associated with a periodic array of optically active centers. A new numerical technique is introduced which allows a direct calculation of the decay probabilities of a finite array of centers for arbitrary trap concentration and strength, this making it possible to evaluate approximate theories for the trapping under controlled conditions.
The remainder of the paper is divided into three sections. The model Hamiltonian and attendant numerical techniques are discussed in section 2. Results specific to one dimension are presented in section 3. Section 4 is devoted to a discussion of the results, with particular emphasis on the limiting behavior at long times and low trap concentrations. Here contact is made with earlier analytic studies of the trapping problem in one dimension [ 5-7 1.
2. Model Hamiltonian Our treatment of the trapping problem is based on a model Hamiltonian where the presence of the traps is represented by a non-Hermitian decay term [ 3-61. The full Hamiltonian takes the form
Here ai and a T are the exciton annihilation and cre-
0301-0104/90/S 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
410
D.L. Huber, W.Y. Ching / Trappingof Frenkei excitor
ation operators in the site representation; Vi is the single-center transition frequency, and U, is the interaction between centers on sites i andj. The third term in eq. ( 1) models the trapping, which is assumed to involve interstitial traps, each of which couples to only one active center. The parameter fi, which is zero if there is no trap associated with the ith site, is identified with the decay rate of the excited state of the ith center in the absence of coupling between centers, i.e. the probability for finding the ith center in the excited state under these conditions varies as exp( - 2rJ) (radiative decay is assumed to be unimportant ) . In our calculations, we will assume an ideal onedimensional system with nearest-neighbor interactions. In this case, the Hamiltonian reduces to X=VC
U:ai+UC i
i$Gi(r)=
C W,/Gj(t),
(5)
i
where Wii=S,i(vi-ir;)+(l-S,)V,,
(6)
with the initial conditions Gi(O)=Ai
e
(7)
The simulation studies that have been carried out involve the integration of the equations of motion for a chain of N centers and NC traps, the latter distributed at random. The Ai were set equal to N -‘12, corresponding to the initial state being a zone-center (k=O) exciton, as might be created through optical absorption. From the set of Gi, two functions were calculated
(Uj+Ui+l+a:+iai) i
p(t)=
-i C riU:Ui,
(2)
I
i$, G,(t) 2/N,
(8)
ii, IGi(f) 1’.
(9)
I
i
where r,=r or 0, depending on whether site i has a trap associated with it. We further assume a fraction, c, of the sites with traps, and that the traps are distributed at random. The numerical calculations involve a set of correlation functions of the form #’ Gi(t)=
C A,(Olai(t)a,’ j
IO> 7
(3)
where 10) refers to the exciton vacuum state, Aj is a number, and ai(t)=exp(i.%?)aiexp( -iPt). The function Gi( t) has a simple interpretation which becomes evident when it is written in the form Gi(t)=(OIuie-i”Y’CAi~f j
IO),
and
Q(t)=
The function P(t) is the square of the modulus of the overlap between the state vector at time t, I v,) = exp( -i&?) I yo) and the initial state vector ( vo) =N-‘/2Cuf IO). As such it is identified with the probability of finding an exciton in the k= 0 mode at time t. The function Q(t) has a slightly different interpretation. Q(t) can be written as
Q(t)=
7 <~tlaT lo>
.
(10)
(4)
having made use of the property (0 I exp (i%?) =O. Eq. (4) shows that Gi.(t) is the overlap between the state vector of the system at time t that has evolved from the initial state &$a,? IO) and the state vector a: IO) corresponding to the ith center being in the excited state and all other centers in the ground state. The function Gi( t) obeys the equation of motion (fi=l) xl For a related application of correlation (Green’s) function techniques to the optical lineshape associated with Frenkel excitons in disordered systems we ref. [ 8 1.
In light of the completeness relation for one-exciton states
where the I v) are eigenstates of the full Hamiltonian, Q(t) can be expressed as
Q(t)=
“$, I12,
(12)
and thus is the probability of finding an exciton in any mode at time t. Q(t) is equal to unity in the ab-
D.L. Huber, W. Y. Ching /Trapping of Frenkel excitons
411
sence of traps for arbitrary transition frequencies Vi and interaction parameters U,. In contrast, P(t) is equal to unity only when there are no traps and the Hamiltonian has translational symmetry so that the initial state v. is an eigenstate. Of the two functions, only P(t) is directly accessible through optical studies. P(t) can be inferred from the decay of the total fluorescence (i.e. integrated over the emission line) that is emitted after pulsed, broadband excitation at t = 0.
3. Numerical results Extensive numerical calculations of P( t) and Q(t) were carried out for a chain of 2000 centers (N=2000)for0.l~c~l.0,usingeqs.(5)-(9),with initial conditions G,(O) = 2000-“2. The results, which are independent of the transition frequency, V, are shown in a semilog plot in fig. 1 (P(t) ) and fig. 2 ( Q( t ) ) . They were calculated with r= 2 U. Time is in units of Ut z T. Although not always evident from the figure, the decay is non-exponential, except in the limit c=l, where P(t)=Q(t)=exp(-2rt). (Note that the slight ripple seen in some curves at long times is an indication of the onset of numerical instabilities in the integration of the differential equations. ) The non-exponential behavior is shown in greater detail in fig. 3, which is a semilog plot of P(t) and Q(t)forr=U,c=O.landO,
-74
\
1
Fig. 1. In P( T) versus T, T= Ut. r= 2 V, N= 2000. In descending order,thecurvesarecalculatedwithc=O.1,0.2,0.3,0.4,0.5,0.6, 0.7,0.8,0.9 and 1.0.
Fig. 2. In Q( T) versus T, T= Ut. r= 2 U, N= 2000. In descending order,thecurvesarecalculatedwithc=O.l,O.2,0.3,0.4,0.5,0.6, 0.7,0.8,0.9 and 1.0.
Fig. 3. In P( T) (broken curve) and In Q(T) (solid curve) versus T; ~0.1; r= CJ.The integration of the equations of motion becomes unstable for Tz 15. The points are values of In Q(t) obtained from eqs. (14) and (IS) with T=Ut, c=O.i, andf(r/ U) z2.5.
being determined by the stability limit of the integration. In order to extract information about the time dependence, we have plotted In ( -In P( t ) ) and ln( -In Q(t)) versus In T with results shown in fig. 4. The straight-line behavior is indicative of the fact that both P(t) and Q(t) vary as exp( -aTX) with x=2/3 (P(t)) and x=1/2 (Q(t)), over the indicated interval. We return to this point in section 4.
D.L. Huber, W. Y. Ching / Trapping of Frenkel excitons
412
0.3. 0.2
-0.5.
2.0
2.1
22
2.3
2..
a.3
2.6
2.7
2.8
2.9
3.0
LW) Fig. 4. In( -In P(T)) (broken curve) and In( -1n Q(t)) (solid curve) versus In T. c= 0.1 and r= U. In the linear region, marked by the straight line, In P(t) -t*” and In Q(t) - t ‘I’.
The focus of this paper has been on the numerical simulation of the trapping of Frenkel excitons, using a model where the trapping is introduced by adding a non-Hermitian term to the Hamiltonian. We have treated a one-dimensional lattice with nearest-neighbor interactions in which the traps occupied interstitial positions. Calculations have been carried out for the probability of finding the system in the state initially excited, P(t), and for the probability of there being an exciton present in any mode, Q( t ) . As is evident from figs. 1 and 2, P(t) decays much more rapidly than Q(t). Such a result is not surprising. The random distribution of traps breaks the translational symmetry of the Hamiltonian. Under these conditions the k=O mode is no longer an eigenstate of the Hamiltonian. The trapping term not only removes zone center excitons from the system, but also causes them to decay into other exciton modes. When c= 1, there is a periodic array of traps; only k=O excitons are present, and both P(t) and Q(t) decay as exp( - 2Z7). It is worth noting that although P(t) d Q(t), they both have the same initial slope: dt
dQ(t=O+ 1 = =
Q(t) =c2 $,
N( 1 -c)“-‘av(t)
,
(14)
where qNiS giVen by
4. Discussion
Wt=O+ 1
As mentioned above, the calculations reported here pertain to a system with interstitial traps. Analytical studies of trapping in one dimension for a model with substitutional traps have been reported by Hemenger and Pearlstein [ 61. Although in one dimension the substitutional model is fundamentally different from the interstitial model in that transport along the chain is disrupted by the presence of the traps, they were able to establish a connection between the two models in the low concentration, strong trapping limit, CK 1 and r/Ul=* 1. They derive an expression for Q(t), which in the notation of this paper takes the form
dt
_2cr
9
(13)
as can be verified by direct calculation from eqs. ( 5 )(9). Moreover, from figs. 1 and 2, it is apparent that Q(t) and P(t) are approximately equal for short times, T= Ut 6 0.5.
2 qN(t)= N(N+ 1) k=r : cot2(2&J (k&d)
X ex
f
- &f(r/U)Utsin’
&( N+l
>I ’
(15)
with f (I'/ U) equal to 8 U/lY We have fit the data shown in fig. 3 (~~0.1, r= U) using eqs. (14) and ( 15) and treating the scale factor f as an adjustable parameter. As shown by the points in fig. 3, good agreement with the solid curve is obtained forfz2.5. We have also repeated the calculation for r= 2 U and 0.5 Uand obtained similar agreement with f= 1.4 and 2.0, respectively. In the case of r= 2 U, the fit was over the interval 0~ TG 4, whereas for r=O.5U, we had O
D.L. Huber, W.Y. Ching /Trapping of Frenkel excitons
As a final note, we mention that although the calculations have been done for onedimensional systems, the numerical approach can be applied to systerns of arbitrary dimension. Results for trapping in three-dimensional systems will be presented elsewhere [ 9 1, along with a comparison with the predictions of the average T-matrix approximation [ 21. Acknowledgement
Computer time was provided by the Office of Basic Energy Sciences, Department of Energy. We would like to thank the authors of refs. [ 3,4] for providing us with preprints of their work.
413
References [ 1] V.M. Kenkre, Phys. Stat. Sol. 89b (1978) 651. (21 D.L. Huber, Phys. Rev. B 22 (1980) 1714; 24 (1981) 1083. [ 31 P.E. Pan-is, Phys. Rev. Letters 62 (1989) 1392; Phys. Rev. B 40 (1989) 4928. [4] J.W. Edwards and P.E. Parris, Phys. Rev. 40 (1989) 8045. [5] R.M. Pearlstein, J. Chem. Phys. 56 (1972) 2431. [6] R.P. Hemenger and R.M. Pearlstein, Chem. Phys. 2 ( 1973) 424. [ 7 ] R.P. Hemenger, K. Lakatos-Lindenberg and R.M. Pearlstein, J. Chem. Phys. 60 ( 1974) 327 1. [8] D.L. Huber and W.Y. Chin& Phys. Rev. B 39 (1989) 8652. [ 91 D.L. Huber and W.Y. Chin& Phys. Rev. B, to be published.
Numerical studies of the trapping of Frenkel excitons in one-dimensional systems D.L. Huber Department of Physics, Universityof Wisconsin-Madison, Madison, WI 53706, USA
and W.Y. cbing Department of Physics, University of Mssouri-Kansas,
City Kansas City, MO 64110, USA
Received 15 January 1990
Numerical studies of the trapping of Frenkel excitons on a onedimensional lattice are reported. The effects of trapping are introduced through a non-Hermitian decay term in the Hamiltonian. Each of the traps, which are assumed to occupy interstitial positions, can receive excitation from a single center. The equations of motion for the exciton correlation functions of a finite array of centers are integrated to obtain the decay of the k=O mode and the probability of finding an exciton in any mode, following excitation of the k=O mode. Results for the latter are compared with the predictions of calculations by Hemenger and Pearlstein.
1. Introduction
The dynamics of Frenkel excitons in disordered systems continues to be a rich source of problems, both experimental and theoretical. Although, initially, most of the interest focused on systems with a random distribution of transition frequencies and/ or interactions, recently, attention has been drawn to models with a random distribution of trapping centers [ l-41. The simplest example of this class of problems is one where the system, apart from traps, has translational symmetry; the only disorder is that associated with the traps. In this paper we analyze trapping in one dimension. We investigate a onedimensional model where the traps are randomly distributed in interstitial sites that are associated with a periodic array of optically active centers. A new numerical technique is introduced which allows a direct calculation of the decay probabilities of a finite array of centers for arbitrary trap concentration and strength, this making it possible to evaluate approximate theories for the trapping under controlled conditions.
The remainder of the paper is divided into three sections. The model Hamiltonian and attendant numerical techniques are discussed in section 2. Results specific to one dimension are presented in section 3. Section 4 is devoted to a discussion of the results, with particular emphasis on the limiting behavior at long times and low trap concentrations. Here contact is made with earlier analytic studies of the trapping problem in one dimension [ 5-7 1.
2. Model Hamiltonian Our treatment of the trapping problem is based on a model Hamiltonian where the presence of the traps is represented by a non-Hermitian decay term [ 3-61. The full Hamiltonian takes the form
Here ai and a T are the exciton annihilation and cre-
0301-0104/90/S 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
410
D.L. Huber, W.Y. Ching / Trappingof Frenkei excitor
ation operators in the site representation; Vi is the single-center transition frequency, and U, is the interaction between centers on sites i andj. The third term in eq. ( 1) models the trapping, which is assumed to involve interstitial traps, each of which couples to only one active center. The parameter fi, which is zero if there is no trap associated with the ith site, is identified with the decay rate of the excited state of the ith center in the absence of coupling between centers, i.e. the probability for finding the ith center in the excited state under these conditions varies as exp( - 2rJ) (radiative decay is assumed to be unimportant ) . In our calculations, we will assume an ideal onedimensional system with nearest-neighbor interactions. In this case, the Hamiltonian reduces to X=VC
U:ai+UC i
i$Gi(r)=
C W,/Gj(t),
(5)
i
where Wii=S,i(vi-ir;)+(l-S,)V,,
(6)
with the initial conditions Gi(O)=Ai
e
(7)
The simulation studies that have been carried out involve the integration of the equations of motion for a chain of N centers and NC traps, the latter distributed at random. The Ai were set equal to N -‘12, corresponding to the initial state being a zone-center (k=O) exciton, as might be created through optical absorption. From the set of Gi, two functions were calculated
(Uj+Ui+l+a:+iai) i
p(t)=
-i C riU:Ui,
(2)
I
i$, G,(t) 2/N,
(8)
ii, IGi(f) 1’.
(9)
I
i
where r,=r or 0, depending on whether site i has a trap associated with it. We further assume a fraction, c, of the sites with traps, and that the traps are distributed at random. The numerical calculations involve a set of correlation functions of the form #’ Gi(t)=
C A,(Olai(t)a,’ j
IO> 7
(3)
where 10) refers to the exciton vacuum state, Aj is a number, and ai(t)=exp(i.%?)aiexp( -iPt). The function Gi( t) has a simple interpretation which becomes evident when it is written in the form Gi(t)=(OIuie-i”Y’CAi~f j
IO),
and
Q(t)=
The function P(t) is the square of the modulus of the overlap between the state vector at time t, I v,) = exp( -i&?) I yo) and the initial state vector ( vo) =N-‘/2Cuf IO). As such it is identified with the probability of finding an exciton in the k= 0 mode at time t. The function Q(t) has a slightly different interpretation. Q(t) can be written as
Q(t)=
7 <~tlaT lo>
.
(10)
(4)
having made use of the property (0 I exp (i%?) =O. Eq. (4) shows that Gi.(t) is the overlap between the state vector of the system at time t that has evolved from the initial state &$a,? IO) and the state vector a: IO) corresponding to the ith center being in the excited state and all other centers in the ground state. The function Gi( t) obeys the equation of motion (fi=l) xl For a related application of correlation (Green’s) function techniques to the optical lineshape associated with Frenkel excitons in disordered systems we ref. [ 8 1.
In light of the completeness relation for one-exciton states
where the I v) are eigenstates of the full Hamiltonian, Q(t) can be expressed as
Q(t)=
“$, I
(12)
and thus is the probability of finding an exciton in any mode at time t. Q(t) is equal to unity in the ab-
D.L. Huber, W. Y. Ching /Trapping of Frenkel excitons
411
sence of traps for arbitrary transition frequencies Vi and interaction parameters U,. In contrast, P(t) is equal to unity only when there are no traps and the Hamiltonian has translational symmetry so that the initial state v. is an eigenstate. Of the two functions, only P(t) is directly accessible through optical studies. P(t) can be inferred from the decay of the total fluorescence (i.e. integrated over the emission line) that is emitted after pulsed, broadband excitation at t = 0.
3. Numerical results Extensive numerical calculations of P( t) and Q(t) were carried out for a chain of 2000 centers (N=2000)for0.l~c~l.0,usingeqs.(5)-(9),with initial conditions G,(O) = 2000-“2. The results, which are independent of the transition frequency, V, are shown in a semilog plot in fig. 1 (P(t) ) and fig. 2 ( Q( t ) ) . They were calculated with r= 2 U. Time is in units of Ut z T. Although not always evident from the figure, the decay is non-exponential, except in the limit c=l, where P(t)=Q(t)=exp(-2rt). (Note that the slight ripple seen in some curves at long times is an indication of the onset of numerical instabilities in the integration of the differential equations. ) The non-exponential behavior is shown in greater detail in fig. 3, which is a semilog plot of P(t) and Q(t)forr=U,c=O.landO,
-74
\
1
Fig. 1. In P( T) versus T, T= Ut. r= 2 V, N= 2000. In descending order,thecurvesarecalculatedwithc=O.1,0.2,0.3,0.4,0.5,0.6, 0.7,0.8,0.9 and 1.0.
Fig. 2. In Q( T) versus T, T= Ut. r= 2 U, N= 2000. In descending order,thecurvesarecalculatedwithc=O.l,O.2,0.3,0.4,0.5,0.6, 0.7,0.8,0.9 and 1.0.
Fig. 3. In P( T) (broken curve) and In Q(T) (solid curve) versus T; ~0.1; r= CJ.The integration of the equations of motion becomes unstable for Tz 15. The points are values of In Q(t) obtained from eqs. (14) and (IS) with T=Ut, c=O.i, andf(r/ U) z2.5.
being determined by the stability limit of the integration. In order to extract information about the time dependence, we have plotted In ( -In P( t ) ) and ln( -In Q(t)) versus In T with results shown in fig. 4. The straight-line behavior is indicative of the fact that both P(t) and Q(t) vary as exp( -aTX) with x=2/3 (P(t)) and x=1/2 (Q(t)), over the indicated interval. We return to this point in section 4.
D.L. Huber, W. Y. Ching / Trapping of Frenkel excitons
412
0.3. 0.2
-0.5.
2.0
2.1
22
2.3
2..
a.3
2.6
2.7
2.8
2.9
3.0
LW) Fig. 4. In( -In P(T)) (broken curve) and In( -1n Q(t)) (solid curve) versus In T. c= 0.1 and r= U. In the linear region, marked by the straight line, In P(t) -t*” and In Q(t) - t ‘I’.
The focus of this paper has been on the numerical simulation of the trapping of Frenkel excitons, using a model where the trapping is introduced by adding a non-Hermitian term to the Hamiltonian. We have treated a one-dimensional lattice with nearest-neighbor interactions in which the traps occupied interstitial positions. Calculations have been carried out for the probability of finding the system in the state initially excited, P(t), and for the probability of there being an exciton present in any mode, Q( t ) . As is evident from figs. 1 and 2, P(t) decays much more rapidly than Q(t). Such a result is not surprising. The random distribution of traps breaks the translational symmetry of the Hamiltonian. Under these conditions the k=O mode is no longer an eigenstate of the Hamiltonian. The trapping term not only removes zone center excitons from the system, but also causes them to decay into other exciton modes. When c= 1, there is a periodic array of traps; only k=O excitons are present, and both P(t) and Q(t) decay as exp( - 2Z7). It is worth noting that although P(t) d Q(t), they both have the same initial slope: dt
dQ(t=O+ 1 = =
Q(t) =c2 $,
N( 1 -c)“-‘av(t)
,
(14)
where qNiS giVen by
4. Discussion
Wt=O+ 1
As mentioned above, the calculations reported here pertain to a system with interstitial traps. Analytical studies of trapping in one dimension for a model with substitutional traps have been reported by Hemenger and Pearlstein [ 61. Although in one dimension the substitutional model is fundamentally different from the interstitial model in that transport along the chain is disrupted by the presence of the traps, they were able to establish a connection between the two models in the low concentration, strong trapping limit, CK 1 and r/Ul=* 1. They derive an expression for Q(t), which in the notation of this paper takes the form
dt
_2cr
9
(13)
as can be verified by direct calculation from eqs. ( 5 )(9). Moreover, from figs. 1 and 2, it is apparent that Q(t) and P(t) are approximately equal for short times, T= Ut 6 0.5.
2 qN(t)= N(N+ 1) k=r : cot2(2&J (k&d)
X ex
f
- &f(r/U)Utsin’
&( N+l
>I ’
(15)
with f (I'/ U) equal to 8 U/lY We have fit the data shown in fig. 3 (~~0.1, r= U) using eqs. (14) and ( 15) and treating the scale factor f as an adjustable parameter. As shown by the points in fig. 3, good agreement with the solid curve is obtained forfz2.5. We have also repeated the calculation for r= 2 U and 0.5 Uand obtained similar agreement with f= 1.4 and 2.0, respectively. In the case of r= 2 U, the fit was over the interval 0~ TG 4, whereas for r=O.5U, we had O
D.L. Huber, W.Y. Ching /Trapping of Frenkel excitons
As a final note, we mention that although the calculations have been done for onedimensional systems, the numerical approach can be applied to systerns of arbitrary dimension. Results for trapping in three-dimensional systems will be presented elsewhere [ 9 1, along with a comparison with the predictions of the average T-matrix approximation [ 21. Acknowledgement
Computer time was provided by the Office of Basic Energy Sciences, Department of Energy. We would like to thank the authors of refs. [ 3,4] for providing us with preprints of their work.
413
References [ 1] V.M. Kenkre, Phys. Stat. Sol. 89b (1978) 651. (21 D.L. Huber, Phys. Rev. B 22 (1980) 1714; 24 (1981) 1083. [ 31 P.E. Pan-is, Phys. Rev. Letters 62 (1989) 1392; Phys. Rev. B 40 (1989) 4928. [4] J.W. Edwards and P.E. Parris, Phys. Rev. 40 (1989) 8045. [5] R.M. Pearlstein, J. Chem. Phys. 56 (1972) 2431. [6] R.P. Hemenger and R.M. Pearlstein, Chem. Phys. 2 ( 1973) 424. [ 7 ] R.P. Hemenger, K. Lakatos-Lindenberg and R.M. Pearlstein, J. Chem. Phys. 60 ( 1974) 327 1. [8] D.L. Huber and W.Y. Chin& Phys. Rev. B 39 (1989) 8652. [ 91 D.L. Huber and W.Y. Chin& Phys. Rev. B, to be published.