- Email: [email protected]
Role of the incipient Anderson transition in electronic energy transfer in mixed organic crystals
Solid State Communications, Vol. 44, No. 6, pp. 833-836, 1982. Printed in Great Britain.
0038-1098/82/420833--04503.00/0 Pergamon Press Ltd.
ROLE OF THE INCIPIENT ANDERSON TRANSITION IN ELECTRONIC ENERGY TRANSFER IN MIXED ORGANIC CRYSTALS C.M. Soukoulis, J. Klafter and E.N. Economou* Corporate Research Science Laboratories, Exxon Research and Engineering Company, P.O. Box 45, Linden, NJ 07036, U.S.A.
(Received 30 March 1982 by H. Suhl) The role of disorder in impurity bands in mixed organic crystals is examined in terms of the Anderson model. A coherent potential approximation is used to calculate averaged Green's functions; the L (E) criterion is employed to study the transition in the nature of the eigenstates. The transition is found to take place at a concentration ~, which is qualitatively in agreement with experiments. Our interpretation of the experimental results indicate a continuous variation of the transport properties above ~. Electronic energy transfer experiments in mixed organic solids can give a unique opportunity to study basic questions of localization effects. THE NATURE of the electronic eigenstates in disordered systems has been the subject of intensive studies [1 ] since the problem was proposed by Anderson [2]. The basic question that has a direct bearing on physical observables, like the conductivity, is whether the eigenstates are localized or not. Mott [ 1] put forward the concept of minimum metallic conductivity, which was deduced by making certain assumptions about the nature of the eigenstates at the mobility edges [1,3]. Recently Abrahams et al. [4] have developed a scaling argument, following ideas by Thouless et al. [5] which yields predictions for the behavior of the conductivity at T = 0 K in various dimensionalities D. In particular, for D > 2 the conductivity goes continuously to zero at the mobility edges, while for D = 2 there is no metallic conduction at all - no minimum metallic conductivity is expected in either case. For a recent review on localization see [6]. The experimental effort to study these phenomena has always run parallel to the theoretical investigations [7, 8]; however one of the main difficulties is how to distinguish between the disorder induced localization and the correlation (electron-electron) effects [7, 8]. A metal-insulator type transition and the different localization effects can also be observed in experimental studies of electronic energy transfer (EET) in mixed organic crystals [9]. It has recently been proposed [10] that the Anderson transition plays a role in the case of EET in an impurity band of isotopically mixed organic crystals at low temperatures. Experimental studies of * Permanent address: Department of Physics, University of Crete, Iraklion, Crete, Greece. 833
tn'plet EET in isotopically mixed benzene [11 ], naphthalene [9] and phenazine [12] at low temperatures have established the existence of a "critical" concentration ~ of the isotopic impurities below which EET is practically switched off. The EET within the impurity band has been monitored by energy sinks, which were either chemical supertraps [9, 10], or impurity dimers [12] of low concentration n,. Two models have been proposed to explain this phenomenon: (a) The dynamic percolation model by Kopelman et al. [9] which is essentially a kinetic model [10]; (b) The Anderson transition approach. Klafter and Jortner [10] have argued that ~ for triplet EET is a manifestation of the Anderson transition from extended to localized states, due to the presence of diagonal disorder, which originates from inhomogeneous broadening W of the diagonal site excitation energies. The existence of an Anderson transition or of the percolative model in these systems is still controversial. For more arguments on each of the models see [9, 10, 13]. In this letter, we analyze some recent experimental data on triplet EET in isotopically mixed naphthalene [14]. These data give the efficiency of EET as a function of the energy sink concentration n, (chemical supertrap in this case). We show that at least according to our analysis the experimental data may be explained according to the Anderson model approach to the problem as suggested earlier [10], by relaxing the minimum diffusion coefficient [10] condition. We consider some implications of the new results derived by the scaling arguments [4, 6] for understanding the termination of EET in an impurity band, and for the behavior of the
834
INCIPIENT ANDERSON TRANSITION IN MIXED ORGANIC CRYSTALS
wavefunctions above the Anderson transition (D = 3), or above the "quasi transition" (D = 2). Our explanation is based on the recent scaling ideas [4] which do not support the concept of minimum metallic conductivity [ 15 ]. A possible way that Mott's argument for minimum metallic conductivity (or minimum diffusion coefficient) may fail is if the extended eigenfunctions near the mobility edges are very inhomogeneous on a microscopic scale, leaving large portions of the volume inaccessible to the propagating excitation [3 ]. We speculate here that for each energy E in the extended regime (or in the very weakly localized regime) there is a characteristic length ~ [16] such that the wavefunction appears homogeneous on a scale much larger than ~, while on a scale comparable to or smaller than ~, the wavefunction is very inhomogeneous occupying a small fraction of the volume. This speculation of inhomogeneity on a microscopic scale is supported by exact results in D = 1 systems [17], where the length det'med as the union of the regions where the eigenfunction is appreciable is much smaller (for low disorder) than the localization length. Numerical results [18] in D = 2 systems also show that the eigenfunctions are inhomogeneous. We have a gradual increase in the volume (or area) accessible to the propagating excitation as one moves to the extended regime, which is proportional to the participation ratio P. We deffme this participation ratio as P = (1/N) [N i I txi 14] - 1, where a i are the site amplitude of the wavefunction, and N is the total number of sites. According to this definition P = 1 for completely extended states and P = 0 for strongly localized states and large N. In what follows, we will use the numerical results by Yonezawa [18] to evaluate this participation ratio. It should be emphasized that P is a measure for the relative number o f sites over which the eigenfunction is extended, and in this sense, it represents the quantum counterpart of the site percolation probability in the classical theory [ 15 ]. The above picture can reconcile the metal-insulator interpretation and the EET experiments on mixed organic crystals [ 16]. It also provides an alternative check on the basic question of whether or not there exists a minimum metallic conductivity. The supertrap phosphorescence intensity is obviously proportional to the supertrap concentration. It can be argued on the basis of overlap considerations that it is roughly proportional to the participation ratio P [19], that is: I s = ksnsP, where k s is independent of both n s and P. The ground state phosphorescence intensity Ig is clearly independent of n s and P; that is, Is ksnsP Ig+---~s = I g + k s n s P -
asnse I +asnse'
where a s = ks/Ig is independent of n 8 and P, and /tot = Ig + Is" Is[I,o t is the observable in the EET experiments [9, 14] and provide a measure for the EET among the isotopic impurities. In deriving equation (1), no dynamical or matrix element effects were taken into account. Only spatial effects, assuming that the important effect is "geometrical" spread of the wavefunction. Using equation (1) and the experimental data [14] we find the participation ratio P as a function of concentration of impurities e for three different concentrations n s (Fig. 1). We used a s = 3 × 10 4 for the plotting of Fig. 1. We note that there is an almost universal curve for P which starts increasing gradually around c =- 0 . 0 8 - 0 . 1 0 . This smooth increase of the participation ratio as we enter the extended regime is consistent with the idea of a continuous mobility at the mobility edge E e. To describe the impurity band, we employ the Hamiltonian, H = ~
i)ei(i[+
i
~
[i)Vij
(2)
i]
where the diagonal matrix elements ee are random variables characterized by a rectangular probability distribution of width W, W is the inhomogeneous broadening, which is taken to be [20] W ~ 0.1 cm -1 for all concentrations c. Each of the off-diagonal matrix elements depend on the relative i m p u r i t y - i m p u r i t y distance R as, [9-11] Vis(R ) = /3 exp {--ln (AI/3)I(RId) -- 1]},
(3)
where/3 is the near neighbor exchange integral, A is the energy separation between the impurity state and the host exciton band. d is the lattice constant. In our calculations we used/3 = 1 cm -1 and A = 100 cm -1 which are parameters corresponding to triplet excitation of naphthalene in naphthalene-d 8 [9]. In order to determine the dependence of V~S on the impurity concentration, we assume that each impurity molecule interacts with Z neighboring molecules, Z being a fixed parameter. The average value of Vii, V was taken to be Z-1
R/z,
k=O
where Vk = A exp [--In ( A / / 3 ) ] ( ( R ) h / d ) and (R)t~ -
(1)
Vol. 44, No. 6
( 2 k + 1)!! d 2k+lk! X/c"
( R ) k was calculated for D = 2, which represents the case of naphthalene [9], using the probability
(4)
Vol. 44, No. 6
INCIPIENT ANDERSON TRANSITION IN MIXED ORGANIC CRYSTALS
0.9 0.8 0.7
I I I i
nS c D10-4 o iO-3
0.6 0.5
/ °/ /
/7
// //
iO-2
P
• ParticipationRatio
0.4
/
/
0.5
// 0.2
i1 11 o / / o~ cy/D //o / j , /. //
0.1 0
o oz o o4 o oco.o8 o,o 'oiz 'o J4 'o16 o.,8
phonon interactions [16]. Such a behavior was experimentally observed for naphthalene in naphthalene-ds going from 1.8 to 4.2 K [23 ]. We have advanced a possible interpretation of EET experimental results in terms of the gradual growth of the participation ratio above ?. Such a behavior at reflects the same features of the eigenfunctions responsible for the continuous behavior of the mobility at E e. Thus, according to our interpretation, the EET experimental results can be considered as an independent verification of the continuity of the mobility. It is important in our opinion to explore further the theoretical and the experimental foundations of the metal-insulator type transition for EET in impurity bands of organic solids.
c
Fig. 1. The experimentally obtained (D, A, o) participation ratios P for different supertrap concentrations n 8 as a function of the donor concentration c. The solid circles (o) represent P obtained by numerical calculations [18]. The dashed lines are only a guide to the eye and roughly show the size of the errors in the experiments and the numerical calculations. Note that in the experimental points the supertrap concentration is given relative to the concentration c.
REFERENCES 1. 2. 3. 4. 5. 6.
distribution
c 1 [lrcR2]exp(_~rcR2/d2). Pk(R) = -d~2rrg ~.. ~ d2
!
Pk(R) dR is the probability that exactly the k + 1 nearest neighbor is inside a spherical shell of R, R + dR. We now use a CPA calculation and a Bethe Lattice Green's function for the periodic case in order to determine the transition from the L (E) criterion [21 ]. We find that an Anderson transition occurs at a critical concentration ? = 0.09 which qualitatively agrees with experiment [14, 22]. In Fig. 1 we plot also the participation ratio derived from direct diagonalization of large system (100 x 100) by Yonezawa [18], where we have used equation (4) to relate ff to the concentration c. As one can see, the numerically calculated participation ratios are in qualitative agreement with those derived from the experimental data. In passing, we mention that while the localization length increases abruptly at the transition, the participation ratio does not go to its highest value as we enter the extended side, but increases gradually. We add that according to the above arguments, inelastic scattering of the excitations by phonons will help the excitation trapping since it provides an extra channel to bridge the inaccessible regions and make the wavefunction to appear more homogeneous. This corresponds to replacing ~ by a shorter length ~ph determined by the
835
7. 8.
N.F. Mott,Metallnsulator Transitions. Taylor and Francis, London (1974). P.W. Anderson, Phys. Rev. 109, 1492 (1958). N.F. Mott,Phil. Mag. 1144, 265 (1981). E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Ramakrishnan,Phys. Rev. Lett. 42,673 (1979). D.C. Licciardello & D.J. Thouless, Phys. Rev. Lett. 35, 1475 (1975); J. Phys. C l l , 925 (1978). E.N. Economou, Excitations in Disordered Systems (Edited by M. Thorpe) (to be published). N. Giordano,Phys. Rev. B22, 5635 (1980). R.C. Dynes, Invited talk LT-16 (to be published in
Physics B & C). 9.
R. Kopelman, EaM. Monberg & F.W. Ochs, Chem.
Phys. 19,413 (1977). 10. 11. 12. 13. 14. 15. 16. 17. 18.
19.
20.
J. Klafter & J. Jortner, Chem. Phys. Lett. 60, 5 (1978); J. Chem. Phys. 71, 1961 (1979). S.D. Colson, SaM. George, T. Keyes & V. Vaida, J. Chem. Phys. 67, 4941 (1977). D.D. Smith, R.D. Mead & A.H. Zewail, Chem. Phys. Lett. 50,358 (1977). R. Kopelman, Topics in Applied Physics (Edited by F.K. Fong), Vol. 15. Springer, Berlin (1976). D.C. Ahlgren & R. Kopelman, J. Chem. Phys. 70, 3133 (1979). M.H. Cohen & J. Jortner, Phys. Rev. Lett. 30,699 (1973). Y. Imry,Phys. Rev. Lett. 44,469 (1980). C. Papantriantafffllou, E.N. Economou & T.P. Eggarter,Phys. Rev. BI3,910 (1976);Phys. Rev. BI3,920 (1976). S. Yoshino & M. Okazaki, J. Phys. Soc. Japan 43, 415 (1977); P. Prelovsek,Phys. Rev. Lett. 40, 1596 (1978); F. Yonezawa, J. Non-Crystal Solids 35 & 36, 29 (1980). It is worthwhile to point out that in the kinetic approach the supertrap phosphorescence intensity is calculated according to the size of the largest cluster, which is the classical analog of our statement I s ~ P. F. Dupuy, Ph. Pee, R. Lalanne, J.P. Lemaistre, C. Vancamps, H. Port & Ph. Kottis, Mol. Phys. 35, 595 (1978).
836 21. 22.
INCIPIENT ANDERSON TRANSITION IN MIXED ORGANIC CRYSTALS E.N. Economou & M.H. Cohen, Phys. Rev. BS, 2913 (1972). In D = 2, ? marks the transition from strongly localized states to states with long localization
23.
Vol. 44, No. 6
lengths. D.C. Ahlgren & R. Kopelrnan, J. Chem. Phys. 73, 1005 (1980).
0038-1098/82/420833--04503.00/0 Pergamon Press Ltd.
ROLE OF THE INCIPIENT ANDERSON TRANSITION IN ELECTRONIC ENERGY TRANSFER IN MIXED ORGANIC CRYSTALS C.M. Soukoulis, J. Klafter and E.N. Economou* Corporate Research Science Laboratories, Exxon Research and Engineering Company, P.O. Box 45, Linden, NJ 07036, U.S.A.
(Received 30 March 1982 by H. Suhl) The role of disorder in impurity bands in mixed organic crystals is examined in terms of the Anderson model. A coherent potential approximation is used to calculate averaged Green's functions; the L (E) criterion is employed to study the transition in the nature of the eigenstates. The transition is found to take place at a concentration ~, which is qualitatively in agreement with experiments. Our interpretation of the experimental results indicate a continuous variation of the transport properties above ~. Electronic energy transfer experiments in mixed organic solids can give a unique opportunity to study basic questions of localization effects. THE NATURE of the electronic eigenstates in disordered systems has been the subject of intensive studies [1 ] since the problem was proposed by Anderson [2]. The basic question that has a direct bearing on physical observables, like the conductivity, is whether the eigenstates are localized or not. Mott [ 1] put forward the concept of minimum metallic conductivity, which was deduced by making certain assumptions about the nature of the eigenstates at the mobility edges [1,3]. Recently Abrahams et al. [4] have developed a scaling argument, following ideas by Thouless et al. [5] which yields predictions for the behavior of the conductivity at T = 0 K in various dimensionalities D. In particular, for D > 2 the conductivity goes continuously to zero at the mobility edges, while for D = 2 there is no metallic conduction at all - no minimum metallic conductivity is expected in either case. For a recent review on localization see [6]. The experimental effort to study these phenomena has always run parallel to the theoretical investigations [7, 8]; however one of the main difficulties is how to distinguish between the disorder induced localization and the correlation (electron-electron) effects [7, 8]. A metal-insulator type transition and the different localization effects can also be observed in experimental studies of electronic energy transfer (EET) in mixed organic crystals [9]. It has recently been proposed [10] that the Anderson transition plays a role in the case of EET in an impurity band of isotopically mixed organic crystals at low temperatures. Experimental studies of * Permanent address: Department of Physics, University of Crete, Iraklion, Crete, Greece. 833
tn'plet EET in isotopically mixed benzene [11 ], naphthalene [9] and phenazine [12] at low temperatures have established the existence of a "critical" concentration ~ of the isotopic impurities below which EET is practically switched off. The EET within the impurity band has been monitored by energy sinks, which were either chemical supertraps [9, 10], or impurity dimers [12] of low concentration n,. Two models have been proposed to explain this phenomenon: (a) The dynamic percolation model by Kopelman et al. [9] which is essentially a kinetic model [10]; (b) The Anderson transition approach. Klafter and Jortner [10] have argued that ~ for triplet EET is a manifestation of the Anderson transition from extended to localized states, due to the presence of diagonal disorder, which originates from inhomogeneous broadening W of the diagonal site excitation energies. The existence of an Anderson transition or of the percolative model in these systems is still controversial. For more arguments on each of the models see [9, 10, 13]. In this letter, we analyze some recent experimental data on triplet EET in isotopically mixed naphthalene [14]. These data give the efficiency of EET as a function of the energy sink concentration n, (chemical supertrap in this case). We show that at least according to our analysis the experimental data may be explained according to the Anderson model approach to the problem as suggested earlier [10], by relaxing the minimum diffusion coefficient [10] condition. We consider some implications of the new results derived by the scaling arguments [4, 6] for understanding the termination of EET in an impurity band, and for the behavior of the
834
INCIPIENT ANDERSON TRANSITION IN MIXED ORGANIC CRYSTALS
wavefunctions above the Anderson transition (D = 3), or above the "quasi transition" (D = 2). Our explanation is based on the recent scaling ideas [4] which do not support the concept of minimum metallic conductivity [ 15 ]. A possible way that Mott's argument for minimum metallic conductivity (or minimum diffusion coefficient) may fail is if the extended eigenfunctions near the mobility edges are very inhomogeneous on a microscopic scale, leaving large portions of the volume inaccessible to the propagating excitation [3 ]. We speculate here that for each energy E in the extended regime (or in the very weakly localized regime) there is a characteristic length ~ [16] such that the wavefunction appears homogeneous on a scale much larger than ~, while on a scale comparable to or smaller than ~, the wavefunction is very inhomogeneous occupying a small fraction of the volume. This speculation of inhomogeneity on a microscopic scale is supported by exact results in D = 1 systems [17], where the length det'med as the union of the regions where the eigenfunction is appreciable is much smaller (for low disorder) than the localization length. Numerical results [18] in D = 2 systems also show that the eigenfunctions are inhomogeneous. We have a gradual increase in the volume (or area) accessible to the propagating excitation as one moves to the extended regime, which is proportional to the participation ratio P. We deffme this participation ratio as P = (1/N) [N i I txi 14] - 1, where a i are the site amplitude of the wavefunction, and N is the total number of sites. According to this definition P = 1 for completely extended states and P = 0 for strongly localized states and large N. In what follows, we will use the numerical results by Yonezawa [18] to evaluate this participation ratio. It should be emphasized that P is a measure for the relative number o f sites over which the eigenfunction is extended, and in this sense, it represents the quantum counterpart of the site percolation probability in the classical theory [ 15 ]. The above picture can reconcile the metal-insulator interpretation and the EET experiments on mixed organic crystals [ 16]. It also provides an alternative check on the basic question of whether or not there exists a minimum metallic conductivity. The supertrap phosphorescence intensity is obviously proportional to the supertrap concentration. It can be argued on the basis of overlap considerations that it is roughly proportional to the participation ratio P [19], that is: I s = ksnsP, where k s is independent of both n s and P. The ground state phosphorescence intensity Ig is clearly independent of n s and P; that is, Is ksnsP Ig+---~s = I g + k s n s P -
asnse I +asnse'
where a s = ks/Ig is independent of n 8 and P, and /tot = Ig + Is" Is[I,o t is the observable in the EET experiments [9, 14] and provide a measure for the EET among the isotopic impurities. In deriving equation (1), no dynamical or matrix element effects were taken into account. Only spatial effects, assuming that the important effect is "geometrical" spread of the wavefunction. Using equation (1) and the experimental data [14] we find the participation ratio P as a function of concentration of impurities e for three different concentrations n s (Fig. 1). We used a s = 3 × 10 4 for the plotting of Fig. 1. We note that there is an almost universal curve for P which starts increasing gradually around c =- 0 . 0 8 - 0 . 1 0 . This smooth increase of the participation ratio as we enter the extended regime is consistent with the idea of a continuous mobility at the mobility edge E e. To describe the impurity band, we employ the Hamiltonian, H = ~
i)ei(i[+
i
~
[i)Vij
(2)
i]
where the diagonal matrix elements ee are random variables characterized by a rectangular probability distribution of width W, W is the inhomogeneous broadening, which is taken to be [20] W ~ 0.1 cm -1 for all concentrations c. Each of the off-diagonal matrix elements depend on the relative i m p u r i t y - i m p u r i t y distance R as, [9-11] Vis(R ) = /3 exp {--ln (AI/3)I(RId) -- 1]},
(3)
where/3 is the near neighbor exchange integral, A is the energy separation between the impurity state and the host exciton band. d is the lattice constant. In our calculations we used/3 = 1 cm -1 and A = 100 cm -1 which are parameters corresponding to triplet excitation of naphthalene in naphthalene-d 8 [9]. In order to determine the dependence of V~S on the impurity concentration, we assume that each impurity molecule interacts with Z neighboring molecules, Z being a fixed parameter. The average value of Vii, V was taken to be Z-1
R/z,
k=O
where Vk = A exp [--In ( A / / 3 ) ] ( ( R ) h / d ) and (R)t~ -
(1)
Vol. 44, No. 6
( 2 k + 1)!! d 2k+lk! X/c"
( R ) k was calculated for D = 2, which represents the case of naphthalene [9], using the probability
(4)
Vol. 44, No. 6
INCIPIENT ANDERSON TRANSITION IN MIXED ORGANIC CRYSTALS
0.9 0.8 0.7
I I I i
nS c D10-4 o iO-3
0.6 0.5
/ °/ /
/7
// //
iO-2
P
• ParticipationRatio
0.4
/
/
0.5
// 0.2
i1 11 o / / o~ cy/D //o / j , /. //
0.1 0
o oz o o4 o oco.o8 o,o 'oiz 'o J4 'o16 o.,8
phonon interactions [16]. Such a behavior was experimentally observed for naphthalene in naphthalene-ds going from 1.8 to 4.2 K [23 ]. We have advanced a possible interpretation of EET experimental results in terms of the gradual growth of the participation ratio above ?. Such a behavior at reflects the same features of the eigenfunctions responsible for the continuous behavior of the mobility at E e. Thus, according to our interpretation, the EET experimental results can be considered as an independent verification of the continuity of the mobility. It is important in our opinion to explore further the theoretical and the experimental foundations of the metal-insulator type transition for EET in impurity bands of organic solids.
c
Fig. 1. The experimentally obtained (D, A, o) participation ratios P for different supertrap concentrations n 8 as a function of the donor concentration c. The solid circles (o) represent P obtained by numerical calculations [18]. The dashed lines are only a guide to the eye and roughly show the size of the errors in the experiments and the numerical calculations. Note that in the experimental points the supertrap concentration is given relative to the concentration c.
REFERENCES 1. 2. 3. 4. 5. 6.
distribution
c 1 [lrcR2]exp(_~rcR2/d2). Pk(R) = -d~2rrg ~.. ~ d2
!
Pk(R) dR is the probability that exactly the k + 1 nearest neighbor is inside a spherical shell of R, R + dR. We now use a CPA calculation and a Bethe Lattice Green's function for the periodic case in order to determine the transition from the L (E) criterion [21 ]. We find that an Anderson transition occurs at a critical concentration ? = 0.09 which qualitatively agrees with experiment [14, 22]. In Fig. 1 we plot also the participation ratio derived from direct diagonalization of large system (100 x 100) by Yonezawa [18], where we have used equation (4) to relate ff to the concentration c. As one can see, the numerically calculated participation ratios are in qualitative agreement with those derived from the experimental data. In passing, we mention that while the localization length increases abruptly at the transition, the participation ratio does not go to its highest value as we enter the extended side, but increases gradually. We add that according to the above arguments, inelastic scattering of the excitations by phonons will help the excitation trapping since it provides an extra channel to bridge the inaccessible regions and make the wavefunction to appear more homogeneous. This corresponds to replacing ~ by a shorter length ~ph determined by the
835
7. 8.
N.F. Mott,Metallnsulator Transitions. Taylor and Francis, London (1974). P.W. Anderson, Phys. Rev. 109, 1492 (1958). N.F. Mott,Phil. Mag. 1144, 265 (1981). E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Ramakrishnan,Phys. Rev. Lett. 42,673 (1979). D.C. Licciardello & D.J. Thouless, Phys. Rev. Lett. 35, 1475 (1975); J. Phys. C l l , 925 (1978). E.N. Economou, Excitations in Disordered Systems (Edited by M. Thorpe) (to be published). N. Giordano,Phys. Rev. B22, 5635 (1980). R.C. Dynes, Invited talk LT-16 (to be published in
Physics B & C). 9.
R. Kopelman, EaM. Monberg & F.W. Ochs, Chem.
Phys. 19,413 (1977). 10. 11. 12. 13. 14. 15. 16. 17. 18.
19.
20.
J. Klafter & J. Jortner, Chem. Phys. Lett. 60, 5 (1978); J. Chem. Phys. 71, 1961 (1979). S.D. Colson, SaM. George, T. Keyes & V. Vaida, J. Chem. Phys. 67, 4941 (1977). D.D. Smith, R.D. Mead & A.H. Zewail, Chem. Phys. Lett. 50,358 (1977). R. Kopelman, Topics in Applied Physics (Edited by F.K. Fong), Vol. 15. Springer, Berlin (1976). D.C. Ahlgren & R. Kopelman, J. Chem. Phys. 70, 3133 (1979). M.H. Cohen & J. Jortner, Phys. Rev. Lett. 30,699 (1973). Y. Imry,Phys. Rev. Lett. 44,469 (1980). C. Papantriantafffllou, E.N. Economou & T.P. Eggarter,Phys. Rev. BI3,910 (1976);Phys. Rev. BI3,920 (1976). S. Yoshino & M. Okazaki, J. Phys. Soc. Japan 43, 415 (1977); P. Prelovsek,Phys. Rev. Lett. 40, 1596 (1978); F. Yonezawa, J. Non-Crystal Solids 35 & 36, 29 (1980). It is worthwhile to point out that in the kinetic approach the supertrap phosphorescence intensity is calculated according to the size of the largest cluster, which is the classical analog of our statement I s ~ P. F. Dupuy, Ph. Pee, R. Lalanne, J.P. Lemaistre, C. Vancamps, H. Port & Ph. Kottis, Mol. Phys. 35, 595 (1978).
836 21. 22.
INCIPIENT ANDERSON TRANSITION IN MIXED ORGANIC CRYSTALS E.N. Economou & M.H. Cohen, Phys. Rev. BS, 2913 (1972). In D = 2, ? marks the transition from strongly localized states to states with long localization
23.
Vol. 44, No. 6
lengths. D.C. Ahlgren & R. Kopelrnan, J. Chem. Phys. 73, 1005 (1980).