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Exciton trapping dynamics at very low temperatures
Journal of Luminescence 45(1990) 279—281 North-Holland
279
EXCITON TRAPPING DYNAMICS AT VERY LOW TEMPERATURES P.E. PARRIS and John Wm. EDWARDS
*
Department of Physics, University of Missouri — Rolla, Rolla, Missouri 6540], USA
We consider the decay of particles or excitations undergoing quantum mechanical transport in a medium containing randomly-placed, irreversible trapping centers. We find, as observed earlier, that the trapping efficiency can be reduced for coherent excitations relative to the case where particle motion is diffusive. Indeed, our calculations show that in d-dimensions the untrapped fraction of coherent excitations should decay at very long times as a stretched exponential, P(t) — 3)], which is asymptotically slower than that associated with diffusive transport. In strongly disordered media exp[— Atd/(ti± the decay can be substantially slower than this, exhibiting, e.g., a power law decay, with a power which depends upon the localization length of localized eigenstates near the band edge.
1. Introduction The dynamical behavior of a particle or excitation moving through a medium containing randomly placed traps or reaction centers is of central importance to numerous applications in chemistry and condensedmatter physics [1—4].It is, e.g., of particular relevance to sensitized luminescence studies. Strong theoretical results have been obtained regarding this behavior when particle motion is diffusive [4]. Specifically, it has been shown for diffusing particles that the asymptotic decay of the survival probability P(t) has a stretched-exponential form, P( t) — exp[ — At~”~2)], resulting from anomalously long-lived particles that find themselves in large, rarely-occurring, trap-free regions of the medium. Such regions lead to asymptotic tails in the distribution of trapping times for the eigenmodes of the random system, and the stretched exponential decay that results is sometimes referred to as ‘Lifshitz-tail’ behavior due to the similarity, in both functional form and origin, to the tails (first derived by Lifshitz [5]) that appear in the density-of-states of energetically disordered quantummechanical systems. At very low temperatures, however, the mean-freepath for phonon scattering becomes large, and the survival probability associated with quasiparticle trapping in low-temperature condensed phases may be expected to exhibit deviations from diffusive behavior [1,3]. It is this low-temperature, coherent limit of the trapping problem that we address here. Specifically, we consider the asymptotic survival probability for a single quantum-mechanical particle moving at zero temperature in a d-dsmenssonal medtum containsng randomly
*
Permanent address: Dept. of Physics, University of Nevada, Las Vegas, NV 89154, USA.
0022-2313/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
placed irreversible traps in fixed concentration q. In a recent paper [3] we were able to show for a tractable one-dimensional model that the asymptotic decay at zero-temperature is slower than when transport between the traps is diffusive, reinforcing earlier conclusions of Pearlstein, and coworkers [1]. In this paper we discuss the extension of these results to higher dimensions and to energetically disordered systems. Space constraints allow us to give only a brief outline of our analysis; details will be presented elsewhere [6]. A single particle moving in an isotropic solid contaming a fractional concentration q of interstitial trapping impurities can be described by the Liouville—von Neumann equation of motion [1,3] d ‘d — — F H 1 — ‘1 Pi — i ‘ f) ~ P for the single-particle density matrix p ( t). In (1), H represents a single-band Hamiltonian of the usual form. The kernel .1, defined through its action on the density matrix (which we express in the site or Wannier basis) [~1)]m,n
=
yLPm.n(8mn
+
o~,,),
(2)
describes irreversible dephasing and loss of amplitude from sites { n ) which are connected (through phonon operators) to states of lower energy associated with the trapping impurities. The evolution of the system described by (1) can also be described in terms of an effective, but non-Hermitian, Hamiltonian j~0, That is, we can write d p/d t = — i[ .t. p], in which the effective Hamiltonian
n.>iy
~V’=H—
(3)
I
contains terms which assign to the trap-coupled sites an imaginary component of the energy, giving them a finite lifetime i- = (2y)~ arising from the coupling to the
280
P.E. Parris andJ. Wm. Edwards
trap molecules [3]. The evolution is now compactly expressable in terms of the eigenstates and complex eigenvalues of this effective Hamiltonian, which resembles that of a disordered binary alloy having imaginary site energies for one species. In particular, the imaginary part of the eigenvalue for a given eigenvector determines the rate at which a particle in that particular state decays to the traps. The asymptotic decay of the survival probability will be determined by the distribution of the most slowly decaying eigenstates of the effective Hamiltonian. In analogy to the diffusion prob-
/
Exciton trapping dynamics
mean intertrap spacing, and hence will experience the imaginary potential of a significant number of sites. We consider, then, the lowest mode of the continuum Hamiltonian H= ~ 2/2 m + V( r) (4) associated with a particle in a trap-free void of radius R centered at the origin, where V(r)= —iqy0(r— R), and J( X) is the Heaviside step function. The lowest eigenstate of (4) is a rotationally symmetric solution to the ‘s-wave’ equation
lem [2,4], we expect the most important class of these
1
states to be those which are centered on asymptotically
-:i r
large trap-free voids surrounded by regions of more typical trap density. We distinguish between two cases: (i) the ordered case, in which the transport Hamiltonian H contains no energetic disorder, and (ii) the disordered case in which H contains a random site-diagonal term of the Anderson form.
d
—
1I
d~p(r)
~r’~
dr
+
—
K(r)]~p(r)
=
0,
(5)
where c = 2mE and ~(r) = 2mV(r). Solving this simpie square-well problem for the most slowly-decaying mode, we obtain a quasiparticie wavefunction which decays with a rate given asymptotically in R by 2
—
1/2
~
FR= —2 Im[E
2. Ordered solids When the region within a trap-free void is ordered, a long-wavelength state with an energy near the band edge will be strongly scattered by the disorder outside
the void, and will vanish exponentially into the region populated by traps. (Note that it is the traps themselves which are the source of the localizing disorder.) The larger the trap-free region associated with an eigenstate of this type, the smaller the fraction of the amplitude which will have any significant overlap with the traps, and the smaller will be the decay amplitude associated with that state. As a consequence, the asymptotic distribution of small decay amplitudes will be determined by the distribution of large, compact (i.e., nearly spherical) trap-free voids. Non-compact voids of the same volume (hence same statistical weight) exhibit a faster decay
than non-spherical ones, due to the larger perimeter
0]=x01(4mq-y) R , (6) where x01 is the first root of the lowest normalizable eigenstate of (5) which is regular at the origin. It follows, that a particle created in a trap-free region of volume V(R)
= CdR’~,will have a decay which is asymptotically bounded by 1 2 — 1/2 31 PR(t)—f(R)expl—x01t(4mqy) R I. (7)
The function f( R) may depend upon the exact initial conditions, but is expected to vary, at most, algebraically with R, and is thus slowly varying compared to the statistical weight associated with the distribution of trap-free voids. Averaging (7) over the probability to find the particle created in a trap-free region of given volume, PR X exp( — XVR), where X = — ln(I — q), we obtain by saddle-point analysis a stretched-ex-
ponential decay [6] _~~i ‘8 ~ 1 exp~ which is slower than that associated with diffusive mo-
p’t~
which is in contact with traps [2]. It suffices, therefore,
tion [2,4]. In writing (8) we have neglected any algebraic
to calculate the decay amplitude associated with a rotationally-symmetric void of radius R (centered at the origin for convenience) surrounded by a region of more
prefactors.
typical trap density. To this end, we use the fact that we are interested in asymptotically long wavelengths and work in the continuum. In the Bloch states I k) (the
3. Disordered solids
discrete Fourier transforms of the site states) the transport part of the Hamiltonian 2/2m, can associated be written, withforasmall free k, in the H = mass k particle of form effective m. We now represent the region outside the void with a constant effective potential equal to the average value V= —iqy of the absorptive part of the potential existing in that region. This representation is based upon our expectation that the
When the solid contains intrinsic energetic disorder aside from that associated with the irreversible traps, some fraction trap-free void will of the be exponentially states centeredlocalized, inside awith given a distribution of localization lengths characteristic of the disorder. Delocalized states will decay quickly, having substantial overlap with traps, and so P(t) will be asymptotically limited by exponentially-localized states centered in large, statistically-rare trap-free voids. There
tails of the wavefunction in the absorptive region, while
will, in general, be a minimum localization length
small compared to R, will be long compared to the
which for sufficiently strong disorder will be of the
~,
FE. Parris andJ. Wm. Edwards order of the lattice spacing a. The decay of such a state
centered in a trap free void of radius R will then be determined by the overlap with traps at the boundary of the region. It will, therefore, be an exponential function of the void radius, i.e., FR — 2A exp( — R/~).By averaging the decay associated with this rate over the volume of the trap-free void, as in sect. 2, we obtain a very slow, generalized power law decay (9) 1. Statistical fluctuawith v(t) = X~[2~ln(At/qd)]~ tions will be much greater in this case, and it is not as clear that compact voids will play as important a role as they do in the ordered case. One can argue, e.g., that at a given time the dominant contribution to the decay will come from localized states centered at a given
optimal distance from the perimeter of the trap-free void in which they are contained. Thus, contributions from trap-free regions of very different shape, but the same volume, can have comparable decay times. This effect is not expected to qualitatively change the form of the decay. (Indeed, if anything this would seem to suggest a slower decay than that which we have ob-
/
Exciton trapping dynamics
281
tamed.) It is to be hoped that sufficently low temperatures can be obtained experimentally, which would allow the observation of this anomalous behavior. Investigallons are currently under way to extend the zero-
temperature analysis given above to finite temperatures, where quasiparticle transport is only partially coherent [1,6].
References [1] R.M. Pearlstein, J. Chem. Phys. 56 (1972) 2431; R.P. Hemenger, K. Lakatos-Lindenberg and R.M. Pearlstein, J. Chem. Phys. 60 (1974) 3271. [2] J. Klafter, G. Zumofen and A. Blumen, J. de Phys. Lett. 45 (1984) L49.
[3] P.E. Parris, Phys. Rev. Lett. 62 (1989) 1392. [4] B.Ya Balagurov and V.G. Vaks, Zh. Exp. Teor. Fiz. 65 (1973) 1939 [Englishtranslation: Soy. Phys. JETP 38 (1974) 9681; M.D. Donsker and S.R.S. Varadhan, Commun. Pure AppI. Math. 28 (1975) 525. [5] I.M. Lifshitz, Adv. Phys. 13 (1964) 483. [6] J.Wm. Edwards and P.E. Paths, Phys. Rev. B40 (1989) 8045, and unpublished results.
279
EXCITON TRAPPING DYNAMICS AT VERY LOW TEMPERATURES P.E. PARRIS and John Wm. EDWARDS
*
Department of Physics, University of Missouri — Rolla, Rolla, Missouri 6540], USA
We consider the decay of particles or excitations undergoing quantum mechanical transport in a medium containing randomly-placed, irreversible trapping centers. We find, as observed earlier, that the trapping efficiency can be reduced for coherent excitations relative to the case where particle motion is diffusive. Indeed, our calculations show that in d-dimensions the untrapped fraction of coherent excitations should decay at very long times as a stretched exponential, P(t) — 3)], which is asymptotically slower than that associated with diffusive transport. In strongly disordered media exp[— Atd/(ti± the decay can be substantially slower than this, exhibiting, e.g., a power law decay, with a power which depends upon the localization length of localized eigenstates near the band edge.
1. Introduction The dynamical behavior of a particle or excitation moving through a medium containing randomly placed traps or reaction centers is of central importance to numerous applications in chemistry and condensedmatter physics [1—4].It is, e.g., of particular relevance to sensitized luminescence studies. Strong theoretical results have been obtained regarding this behavior when particle motion is diffusive [4]. Specifically, it has been shown for diffusing particles that the asymptotic decay of the survival probability P(t) has a stretched-exponential form, P( t) — exp[ — At~”~2)], resulting from anomalously long-lived particles that find themselves in large, rarely-occurring, trap-free regions of the medium. Such regions lead to asymptotic tails in the distribution of trapping times for the eigenmodes of the random system, and the stretched exponential decay that results is sometimes referred to as ‘Lifshitz-tail’ behavior due to the similarity, in both functional form and origin, to the tails (first derived by Lifshitz [5]) that appear in the density-of-states of energetically disordered quantummechanical systems. At very low temperatures, however, the mean-freepath for phonon scattering becomes large, and the survival probability associated with quasiparticle trapping in low-temperature condensed phases may be expected to exhibit deviations from diffusive behavior [1,3]. It is this low-temperature, coherent limit of the trapping problem that we address here. Specifically, we consider the asymptotic survival probability for a single quantum-mechanical particle moving at zero temperature in a d-dsmenssonal medtum containsng randomly
*
Permanent address: Dept. of Physics, University of Nevada, Las Vegas, NV 89154, USA.
0022-2313/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
placed irreversible traps in fixed concentration q. In a recent paper [3] we were able to show for a tractable one-dimensional model that the asymptotic decay at zero-temperature is slower than when transport between the traps is diffusive, reinforcing earlier conclusions of Pearlstein, and coworkers [1]. In this paper we discuss the extension of these results to higher dimensions and to energetically disordered systems. Space constraints allow us to give only a brief outline of our analysis; details will be presented elsewhere [6]. A single particle moving in an isotropic solid contaming a fractional concentration q of interstitial trapping impurities can be described by the Liouville—von Neumann equation of motion [1,3] d ‘d — — F H 1 — ‘1 Pi — i ‘ f) ~ P for the single-particle density matrix p ( t). In (1), H represents a single-band Hamiltonian of the usual form. The kernel .1, defined through its action on the density matrix (which we express in the site or Wannier basis) [~1)]m,n
=
yLPm.n(8mn
+
o~,,),
(2)
describes irreversible dephasing and loss of amplitude from sites { n ) which are connected (through phonon operators) to states of lower energy associated with the trapping impurities. The evolution of the system described by (1) can also be described in terms of an effective, but non-Hermitian, Hamiltonian j~0, That is, we can write d p/d t = — i[ .t. p], in which the effective Hamiltonian
n.>iy
~V’=H—
(3)
I
contains terms which assign to the trap-coupled sites an imaginary component of the energy, giving them a finite lifetime i- = (2y)~ arising from the coupling to the
280
P.E. Parris andJ. Wm. Edwards
trap molecules [3]. The evolution is now compactly expressable in terms of the eigenstates and complex eigenvalues of this effective Hamiltonian, which resembles that of a disordered binary alloy having imaginary site energies for one species. In particular, the imaginary part of the eigenvalue for a given eigenvector determines the rate at which a particle in that particular state decays to the traps. The asymptotic decay of the survival probability will be determined by the distribution of the most slowly decaying eigenstates of the effective Hamiltonian. In analogy to the diffusion prob-
/
Exciton trapping dynamics
mean intertrap spacing, and hence will experience the imaginary potential of a significant number of sites. We consider, then, the lowest mode of the continuum Hamiltonian H= ~ 2/2 m + V( r) (4) associated with a particle in a trap-free void of radius R centered at the origin, where V(r)= —iqy0(r— R), and J( X) is the Heaviside step function. The lowest eigenstate of (4) is a rotationally symmetric solution to the ‘s-wave’ equation
lem [2,4], we expect the most important class of these
1
states to be those which are centered on asymptotically
-:i r
large trap-free voids surrounded by regions of more typical trap density. We distinguish between two cases: (i) the ordered case, in which the transport Hamiltonian H contains no energetic disorder, and (ii) the disordered case in which H contains a random site-diagonal term of the Anderson form.
d
—
1I
d~p(r)
~r’~
dr
+
—
K(r)]~p(r)
=
0,
(5)
where c = 2mE and ~(r) = 2mV(r). Solving this simpie square-well problem for the most slowly-decaying mode, we obtain a quasiparticie wavefunction which decays with a rate given asymptotically in R by 2
—
1/2
~
FR= —2 Im[E
2. Ordered solids When the region within a trap-free void is ordered, a long-wavelength state with an energy near the band edge will be strongly scattered by the disorder outside
the void, and will vanish exponentially into the region populated by traps. (Note that it is the traps themselves which are the source of the localizing disorder.) The larger the trap-free region associated with an eigenstate of this type, the smaller the fraction of the amplitude which will have any significant overlap with the traps, and the smaller will be the decay amplitude associated with that state. As a consequence, the asymptotic distribution of small decay amplitudes will be determined by the distribution of large, compact (i.e., nearly spherical) trap-free voids. Non-compact voids of the same volume (hence same statistical weight) exhibit a faster decay
than non-spherical ones, due to the larger perimeter
0]=x01(4mq-y) R , (6) where x01 is the first root of the lowest normalizable eigenstate of (5) which is regular at the origin. It follows, that a particle created in a trap-free region of volume V(R)
= CdR’~,will have a decay which is asymptotically bounded by 1 2 — 1/2 31 PR(t)—f(R)expl—x01t(4mqy) R I. (7)
The function f( R) may depend upon the exact initial conditions, but is expected to vary, at most, algebraically with R, and is thus slowly varying compared to the statistical weight associated with the distribution of trap-free voids. Averaging (7) over the probability to find the particle created in a trap-free region of given volume, PR X exp( — XVR), where X = — ln(I — q), we obtain by saddle-point analysis a stretched-ex-
ponential decay [6] _~~i ‘8 ~ 1 exp~ which is slower than that associated with diffusive mo-
p’t~
which is in contact with traps [2]. It suffices, therefore,
tion [2,4]. In writing (8) we have neglected any algebraic
to calculate the decay amplitude associated with a rotationally-symmetric void of radius R (centered at the origin for convenience) surrounded by a region of more
prefactors.
typical trap density. To this end, we use the fact that we are interested in asymptotically long wavelengths and work in the continuum. In the Bloch states I k) (the
3. Disordered solids
discrete Fourier transforms of the site states) the transport part of the Hamiltonian 2/2m, can associated be written, withforasmall free k, in the H = mass k particle of form effective m. We now represent the region outside the void with a constant effective potential equal to the average value V= —iqy of the absorptive part of the potential existing in that region. This representation is based upon our expectation that the
When the solid contains intrinsic energetic disorder aside from that associated with the irreversible traps, some fraction trap-free void will of the be exponentially states centeredlocalized, inside awith given a distribution of localization lengths characteristic of the disorder. Delocalized states will decay quickly, having substantial overlap with traps, and so P(t) will be asymptotically limited by exponentially-localized states centered in large, statistically-rare trap-free voids. There
tails of the wavefunction in the absorptive region, while
will, in general, be a minimum localization length
small compared to R, will be long compared to the
which for sufficiently strong disorder will be of the
~,
FE. Parris andJ. Wm. Edwards order of the lattice spacing a. The decay of such a state
centered in a trap free void of radius R will then be determined by the overlap with traps at the boundary of the region. It will, therefore, be an exponential function of the void radius, i.e., FR — 2A exp( — R/~).By averaging the decay associated with this rate over the volume of the trap-free void, as in sect. 2, we obtain a very slow, generalized power law decay (9) 1. Statistical fluctuawith v(t) = X~[2~ln(At/qd)]~ tions will be much greater in this case, and it is not as clear that compact voids will play as important a role as they do in the ordered case. One can argue, e.g., that at a given time the dominant contribution to the decay will come from localized states centered at a given
optimal distance from the perimeter of the trap-free void in which they are contained. Thus, contributions from trap-free regions of very different shape, but the same volume, can have comparable decay times. This effect is not expected to qualitatively change the form of the decay. (Indeed, if anything this would seem to suggest a slower decay than that which we have ob-
/
Exciton trapping dynamics
281
tamed.) It is to be hoped that sufficently low temperatures can be obtained experimentally, which would allow the observation of this anomalous behavior. Investigallons are currently under way to extend the zero-
temperature analysis given above to finite temperatures, where quasiparticle transport is only partially coherent [1,6].
References [1] R.M. Pearlstein, J. Chem. Phys. 56 (1972) 2431; R.P. Hemenger, K. Lakatos-Lindenberg and R.M. Pearlstein, J. Chem. Phys. 60 (1974) 3271. [2] J. Klafter, G. Zumofen and A. Blumen, J. de Phys. Lett. 45 (1984) L49.
[3] P.E. Parris, Phys. Rev. Lett. 62 (1989) 1392. [4] B.Ya Balagurov and V.G. Vaks, Zh. Exp. Teor. Fiz. 65 (1973) 1939 [Englishtranslation: Soy. Phys. JETP 38 (1974) 9681; M.D. Donsker and S.R.S. Varadhan, Commun. Pure AppI. Math. 28 (1975) 525. [5] I.M. Lifshitz, Adv. Phys. 13 (1964) 483. [6] J.Wm. Edwards and P.E. Paths, Phys. Rev. B40 (1989) 8045, and unpublished results.