Comments on the applicability of the Anderson localization model for electronic energy transfer in mixed organic solids

Volume 60, number 1

CHEMICAL

COMMENTS ON THE APPLICABILITY FOR ELECTRONIC Joseph KLAFTER Department

ENERGY

PHYSICS

LETTERS

OF THE ANDERSON

TRANSFER

15 December 1978

LOCALIZATION

MODEL

IN MIXED ORGANIC SOLIDS

and Joshua JORTNER

of Chemistry,

Tel-Aviv

University,

Tel-Aviv, Israel

Received 31 August 1978

We discuss the conditions for Anderson localization of an electronic excitation in an impurity band of an isotopically mixed omit sohd and dwell on the experimental impkations of this phenomenon.

1. Prologue

2. Effects of disorder

Recent experimental studies of triplet electronic energy transfer (EET) in isotopically mixed benzene [l] , naphthalene [Z-5] and phenazine [6] at low temperatures have established the existence of a “critical” concentration Z of the isotopic impurity below which EET in the impurity bad, as interrogated by trapping by energy sinks (chemical supertraps [l-S] or impurity dimers [6]), is abruptly switched off. We have proposed [7,8] that ?? for triplet EET is a manifestation of the Anderson transition (AT) from extended to localized states [9] *, occurring as a consequence of diagonal disorder, which originates from +&e inhomogeneous broadening (IB) of the site-excitation energies. This proposal was not received with overwhelming enthusiasm by Monberg and Kopelman (MK) [1 I] who prefer their “dynamic percolation” kinetic model [2-51. As organic solids are expected to provide extremely useful model systems for the understanding of electronic structure and dynamics in disordered materials, it is important to explore further the theoretical foundation and the experimental implications of the AT model for EET in isotopic impurity bands of organic solids.

Three general types of disorder will be considered, together with their effects on the electronic structure and dynamics in an impurity band: (A) Substitutional disorder in a random binary alloy. While early work [12] proposed that substitutional disorder results in localization. it is now agreed 1133 that this type of disorder does not result in localized states in the monomer impurity band_ (B) Off-diagonal disorder arising from the interimpurity excitation-exchange integrals J,due to the random distribution of impurities, and which does not result in localization [14]. (C) Diagonal structural disorder, originating from the IB, W, of the diagonal site-excitation energies, which is expected to result in the AT [9,10]. In a binary alloy sources (A) and (B) of disorder always prevail and do not lead to localization_ One should now inqu:ze what is the evidence for IB of isotopic impurity states? MK have proposed [1 l] that W is negligibly small, in contrast to our original conjecture [7] that W = 0.1-5 cm-l. This dispute has to be settled on the basis of experimental data. Chemical substitution in organic solids results in well documented IB, which was probed by studies of hole-burning and of coherent optical effects 11%171. A more subtle question is what is the magnitude of IB originating from random strains as well as small local perturbation induced by the isotopic impurity_ The recent experimental observations of substantial, W = 4 cm-l, inhomogeneous broadening of dilute isotopic

* Lye [lo] applied the Andersot EET in ruby_

model to localization of

5

Vdune

60. mmber

1

CI-EMICAL

PHYSICS

impurity state5 of phenazine in pheMZineiig [63 and of W =: 0.1 cm-1 for the monomer band of naphthalene in naphthalene-dg 1181 provide support to our proposal [?I that localization cf electronic excitations in isotopically-mixed crystals can occur as th;. consequence of diagonal disorder.

3. The AT

32,

(1)

where 2 is the coordination number. The Iocalization in the centre of the band will then occur provided that [9,10,19]
(21


4. Energetics of the impurity band We have now and of u on the gether with eqs. -aate of c. It is 6

15 December 1978

tegral between a pair of impurities separated by n host molecules is determined by the superexchange interaction J = L@/AY

to determine the dependence of .T impurity concentration C, which to(1) and (2) will result in the &iigenerally agreed that the transfer in-

(3)

,

where 0 is the nearest-neighbour exchange and A represents the trapdepth. We have used [7,8,19] a continuum approximation recastingJ in the form J = B exp [-@r/d

We have now to consider the interplay between the effects of diagonal disorder (IB) and of offdiagonal disorder (distribution of J terms). UtiQing the localization function method [I43 we have demonstrated 1191 that the simultaneous effects of IB, expressed in terms of W, and of the spread o in the J transfer integrals results in the shift of c to lower values than those predicted in the basis of the original AT model when o = 0. In the limit of large o/W, z * 0 and no AT is exhibited. For an exactly soluble model which involves Iorentzian distributions of the site excitation energies and of the transfer integrals the original AT model applies when [19] WfaZ

LETTERS

-

1)1,

o = InCAi@ ,

(3a)

where r is the interimpurity separation and d the lattice constant_ Three cdculations of (J) were recently performed by us. (A) As a preliminary estimate we took (J) 0: exp(-c&j) where Q = r(3/2) (&J-I/2 d corresponds to the Wigner-Seitz radius in a 2-D system, which is of interest to us for the study of the triplet impurity band in naphthalene and in phenazinc. (B) Next, a proper configurational averaging of eq. (3a) was employed setting J = Jdi PfJy where P(J) is the distribution function of J. For a 2-D system we got 1191
n

Fig. 1. Nmnericalsimulattoasof the energeticsof the &xtp*urityband in a 2-D 128 X 128 square kttice cont.&big a random distribution of isotopic impurities. kq~urity conceutration.~ C = O.Oi, 0.03, 0.05, 0.07 and 0.10. P(n) represents the distributionof the number of host molecute~separating an impurity-impurity pair. The resultsfor c&i C were obtained from averqin~ over S-10 separate&milatior?s.FOX C 2 0.05 the fluctuationsof P(n) beiweesl diffezent experiments were nq.&ibly small. {a) Exbiiits the concentration dependence of the confiiratiomdiy averaged energy exchange term W. The circles denote
proximation
pendence of the varknce of the &txibution off. The circles denote o = [Z$=, P(r~)Jcn)~ - (Zg& P(n)J(n)}* 1‘i* where P(n) iS obtained Fromihe numeriml simuhtions (methoO f(J)). The squaresrepresenttie resultsof the contixmum%Pproximation (method CB_O.

impurity monomer band is roughly obtained by setting pt Q: Ct) ia. eq. (3), which is akin to our procedure (A). To provide further theoretical evidence for the appl.icabiIity of our averaging procedure we have utilized [22] the results of a soluble model of a random binary tioy On az~infinite Cayley tree (Bethe Iattice), Heinricbs [23j has shown that for

sufficiently high v&es of K the width of the monomer band in the absence of diagonal disorder is B = ~K@~JLI)C~~~_ W result is in order-of-magnitude agreement with our averaging procedures, being higher by five order of magnitude than the P&K estimate [I I]. Vie pus conclude that’ the MK procedure for the evaluation of ,_O_l CI-IX-~ the ordinary AT will be exl$?ited and eq. (2) is appi&able for the estimate of C. As evident from fig. la, W/v = 5 X 10-3 results in ?? w 0.1 for A/j3 = 100, which is quite a reasonable estimate. We thus assert that AT in the isotopic impurity baad at sufficiently low temperatures will be exhibited for W 2 0-I cm-l , a value of fB compatible with the ava& able spectroscopic data [ 181. Finally, we would like to point out that we did not assert [7] that W varies with 5 [ 111, as the accuracy of the avaiiable experimental F data [i--6] is insumcient to establish any such correlation.

Consider exciton dynamics in the impurity band when tie temperature is sufficiently low to preclude appreciable tbermtiy activated hopping betwem loc&&d Gates, however kT > FVso r.hat all the states in the narrow impuriry band are therm&y accessible. The macroscopic diffusion coefficient is D = 0 for C2;.The mi.&~um diffusion coefficient was evaluated using the random phase model i24-273, a simple rough estimate for a 2-D system being 181 D, = Z&3,/ hp h$(A/R), so that for A//3 = 100 and 9 = I cm-l, cm3- s--1 [S]. Ear C > si we ex% = 1O-4--1O-s pect that on the tim.e scale of the lifetime, 7? of the electronic excitation, a diffusion process of the excitation among the trap molecules prevails -uM,l an energy sink is reached. The branching ratio, f, for emission from the energy sink is @en by (41

CxEMrcAL PHYSICS LEXTF.Rs

Voiume 60, number 1 6, = Iltr[$W(D(c)r)-1/2]

f 0.577)-1

(for 2-D) , (5)

k, = 47r~(C)n,U?J

(for 3-D) , (6)

where k, is the rate for trapping of the excitation by the energy sink, rY* and ns represent the number density per unit area and the number density of the ener,gy sink, respectively while (R) is the reaction radiUS. The AT model predicts a sharp rise in flC) for C > z where exciton dynamics can be described in terms of strong scattering, temperature independent diffusive motion. The AT within the impurity band is amenable to experimental observation provided that fiC=c>=

1,

i.e.&r*

1 atC=Z?.

(7a)

This condition will be satisfied for electronic excitations characterized by Iong lifetimes. On the other hand, when _JC=i?)QI,

i.e_ ktr 4 1 at C = c ,

(7b)

the competition between excitation trapping and decay will be exhibited at C > ??_ Such a state of affairs will be realized by impurity excitations characterized by short lifetimes. Thus, condition (7a) implies a critical behavior where the threshold at C = cis independent of the concentration of the energy sink and of the Lifetime r_ On the other hand, for a system satisfying condition (7b) c cannot be observed, but rather a gradual change off willbe revealed at C > ?? and a kinetic behavior will be exhibited. These considerations elucidate features of the concentration dependence of the branching ratio f, eq. (4) for triplet and for singIet isotopic impurity states in crystalline benzene and naphthalene. TripIet impurity excitons [l-6] are characterized by to case (7aj, a lifetime of 5 =r 1 s correspond@ where a genuine critical concentration for EET is observed. On the other hand, for singIet rmpurity excitons [l-S], where 7 = 10-7 s, condition (7b) holds whereupon the kinetic threshoid observed which is located at C > ?_ In this case, the concentration dependence off will be determined by the concentration of the energy sink as well as by T.

6. Erosion of the concentration

15 December 1978 threshold for the AT

This carr originate from the following causes: (A) Nonstatistical distribution of impurities. For a microscopically inhomogeneous material transport can be well described in terms of classical percolation theory. Isotopic impurities do not, however, show any tendency for nonrandom clustering. (B) Trapping from localized states. At C < F and at low T EET from localized states to energy sinks can occur when the spatial extent of the localized states is comparable to the average separation between the enerOv sinks. Thus the supertrap concentration n, has to be kept sufficiently low to reduce this effect. Model calculations [S] for the localization length 1281 in the impurity band indicate that ‘z’is amenable to experimental observation over a broad range of ns_ (C) Thermally activated hopping [ 11,29]_ Phonon assisted EET alIows for energy propagation at C < ??, at higher temperatures_

f. Some experimental implications of the AT model

(A) The critical concentration ?? is independent of the concentration ns of the enerOT sink and of the excitation lifetime T. The critical concentration c is amenable to experimental observation iu a system where xc) = 1 and where n, is still sufficiently low to minimize effect (6.B). In a system where f(i?) 4 1, r will not be observed, but rather a kinetic threshold will be exhibited at C > z_ (D) The kinetic concentration threshold wi!l depend on both ns and 7 according to eq. (4). We believe that for low-temperature triplet EET in benzene [l], naphthalene [2] and phenazine [6] the AT is exhibited and prediction (B) holds_ For singlet EET in benzene [l] and naphthalene [3] the kinetic threshold is revealed following predictions (C) and (II). Prediction (D) regarding the n, dependence of the kinetic threshold for singlet excitations in benzene is borne out by experimentaI facts [I ,3]_ Finally, at higher temperatures the AT should be eroded by thermahy assisted hopping, as indeed exhibited in phenazine [6J_ Thus oniy triplet EET is of inherent interest for the study of localization in disordered mate&Is.

Volume 60, number 1 8. The Monberg-Kopelman

CEIEMIC;u. PHYSICS LETTERS criticism

MK have argued a,gainst an interpretation of F as marking the AT for triplet EET in benzene and naphthalene, dwelling on “‘effects of guest clusterization, exciton lifetime, sensor (energy-sink) concentration and exciton-phonon coupling” [ 111. Regarding the effects of guest clusterization we have considered the monomer band providing in section 4 a proper consistent procedure for the evaluation of the configurationally averages band width. We predict that AT wil: be exhibited in tripIet impurity bands for W 2 0.05 cm-l, which is compatible with available spectroscopic data. Concerning the effects of n, and of 7 on the “critical” concentration for the termination of EET we were able to provide in sections 5 and 7 a coherent physical picture which accounts for the different behavior of triplet and of singlet excitations. Finally, we would like to make two comments on the effects of exciton-phonon coupling (EPC). First, to the best of our knowledge there is no complete theoretical treatment of the erosion of the AT by phonons. In this context we should probably d&nguish between the effects of “strong” EPC, i.e. thernial diagonal interaction, which may enhance iocahzation 130,311 and the effects of “weak” EFC, i.e. thermal modulation of the transfer integrals which wilI lead to deIocalization_ From the operational point of view one can consider at present only the effects of thermally activated hopping [29]_ We agree with MK [lI] that effect (SC) will erode the AT at sufficiently high temperatures, however, the cardinal question is whether this effect was important in the low-temperature experiments [l---6] reported up to date? The experiments of Smhh et al_ [6] on phenazinc provide a clear demonstration for the erosion of the AT with increasing temperature from I.3 K to 3 IL Kopeiman’s current approach to triplet EET 12,111 in low-temperature impurity bands invoIves essentially a kinetic model where D(C) is finite at all C (which he believes is due to negligibly small. IB, i.e. I#/(& + 0 and/or thermal hopping effects). Kopelman thus envisions that ?? -+ 0 when 7 + m. On the other had, our AT model predicts a genuine “critical” behavior for triplet EET, involving a djscontinuity in the low-temperature macroscopic diffusion coefficient [7,8]. The ditochomy between Kopehnan’s

15 December 1978

“dynamic percolation” mode? [2-&l 11 and the AT model has not yet been resolved experimentally. Some experimental evidence for AT [9,lO] in impurity bands of transition-metal and of rare-earth ions in ionic crystals was reported [32,33], however, also that area is stiil under active study. The AT in ionic impurity bands bears a close analogy to the AT of small polarons in transition-metal oxides which was documented [34]. Regarding EET in isotopic impurity bands we have presented a set predictions which should be confronted with experiment.

9. EpiIogue The possibility of observing an AT in an impurity band of a mixed organic solid is not restricted, of course, to isotopic impurities. It will be extremely interesting to explore exciton dynamics in chernically-substituted organic solids provided that the transfer integrals fall off faster than the dipole-dipole coupling, an essential condition for the realization of the AT [9.10]. From the theoretical point of view several interesting problems require further study. First, the basic problem of the AT in an impurity band has to be explored further. AII the legitimate theoretical treatments of the AT probIem addressed themselves to a one-component solid subjected to the effects of diagonal and off-diagonal disorder. In that context the concepts of a coordination number of the connectivity are loosely defined for an impurity band. Second, the problem of cross-relaxation within the impurity band deserves further investigation. To onterrogate the iocalized states in the middle of the band at C > 3 one requires efficient cross-relaxation. & kT S W thisis not difficult. However, if this is the case one cannot extract information from energy-resolved studies, as reported by Koo et al. for ruby l32,33] and which we have recommended f7] for isotopic impurity bands. Third, it wii be important to elucidate the mechanism of thermally activated hopping which erodes the AT transition at higher temperatures. It is an intriguing question whether it proceeds via a small polaron mechanism [30,31], by one-phonon, or Raman type, or multiphonon processes? For impurities consisting of large molecules the small polaron contributions may be important. This treatment 9

Volume 60, nurr~ber1

cEIEMIcAL

PHYSICS LJC-ITERS

will result in the noncoherent pair EET probabilities y,,. Fourth, it will be inteiesting to provide a coherent physical.picture for EFX in the high-tempemture thermal hopping regime. We believe that this will be best accomplished by using modern methods [35,36] for the solution of the Pauli master equation for incoherent EET, dp(t)/dr = yfir) where p(r) is the vector of the site population probabilities at y is the transfer matrix [37]. Such a treatment will result in dispersivetransfer, characterizedby a timedependent diffusion coefficient, for EET in impurity bands. Acknowledgercent

We are grateful to Professor Raoul Kopelman for providing ‘us with preprints of his recent work [l l] well ahead of publication, for conununicating to us his yet unpubiished results and for stimulating discussions_ References ill VI [31 141 t51 161 171 181

[91 WI r111

D-S_ Colson, SM. George, T_ Keyes and V. Vaida, J. Chem. Phyn 67 (1977) 4941. R Kopehnan, EM_ Monber~ and F_W_ O&i Chem. Fhyr 19 (1977) 413_ R. Kopelman, EM. Monberg and F.W. Ochs, Chem. Phys 21 (1977) 373. R Kopehnan, EM. Monberv, F-W_ Ochs uld P.N. Prasad, J. Chem Phyys 62 (1975) 292. R KopeIman, Topics AppL Phys. 15 (1976) 298. D-D. Smith. R-D_ Mead and A_& Zewail, Chem. Phys. Letters 50 (1977) 358. J. Klafter and 3. Jortner, Chem. Phys. Letters 49 (1977) 410_ J. Klafter and J. Jortner. Erectronic Enew Transfer in Impurity Bands of Mixed Organic Soli& to be pub lished P-W_ Ande_mn, Phys. Rev. 109 (1958) 1492. SK_ Lye, Phys. Rev_ B3 (1973) 3331, EM_ Yonber~ and R Kopehnan, Chem Phys Letters 58 (1978) 497.

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15 December 1978

1121 EN. Economou, M-H. Cohen, K.F. Freed and S.

Kirkpatrick, in: Amorphous and liquid semiconductors, ed_ J. Taut
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WI Ml WI 1231 1241

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1321 r331 [341 1351 1361

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