Volume 48A, number 6 PHYSICS LE’JTERS 29 July 1974

ENERGY TRANSFER AND ANDERSON LOCALIZATION

R. ORBACH* Physics Department University of Tel Aviv, Ramat Aviv (Tel Aviv), Israel

Received 11 June 1974

The temporal decay characteristics of the donor fluorescence for inhomogeneously broadened optical lines is shown to be a direct determination of excitation localization, in the Anderson sense.

The question of spatial localization of excitations for an inhomogeneously broadened transition was first examined in detail by Anderson [ 11. He showed that, for short range transfer interactions (falling off faster than l/R;, where Rii is the distance between active sites) excitation localization would occur below a criti- cal concentration of active centers. His results were ap- plied to donor-donor (Cr3’) transfer in ruby by Lyo [2], who showed that localization took place for Cr concentrations less than - 0.3 - 0.4 at.%. It has proven remarkably difficult to verify this result experimentally, though a preliminary observation has been reported [3]. In this letter, we propose a simple and direct method for determination of donor-donor localization on the basis of the time dependence of the donor flu- orescence decay.

It has been realized for some time [4] that the do- nor fluorescence decay is exponential only if there exists substantial donor-donor energy transfer. In such a limit, excitation of acceptor sites is ‘shared’ amongst the donor sites, and one can think of two systems weak- ly coupled to one another, but the donor system al- ways in strong internal dynamic equilibrium (the ac- ceptor system assumed to be in thermodynamic equi- librium with the bath). When, however, the donor- do- nor transfer is inhibited (by localization in the Ander- son sense, for example) Inokuti and Hirayama [5] showed that donor fluorescence is not exponential, but rather has the time dependence (for power law

* Supported in part by the National Science Foundation and the U.S. Office of Naval Research. Support is also gratefully acknowledged from the John Simon Guggenheim Memorial Foundation in the form of a one year fellowship grant.

Permanent address: Department of Physics, University of California, Los Angeles, California 90024, USA.

donor-acceptor coupling)

exp [ -@l-3/s)E ; 3’s ) 0 1 (1)

where C is the acceptor concentration, Co a ‘critical transfer coefficient (defined in ref. [S]), f. the donor decay constant in the absence of acceptors, and s = 6, 8, 10 corresponding to dipole-dipole, dipole-quadrupole, quadrupole-quadrupole donor-acceptor transfer inter- actions (tie. the range of the interaction goes as Rf2).

It is seen from (1) that the donor fluorescence de- cay is most definitely not exponential. One cannot sim- ply average over the individual donor acceptor energy transfer rates, but instead each process must be consid- ered separately, leading to the complicated form of

(1). Our proposal is as follows. Anderson [ 11, Lyo [2],

and more recent work [6] has shown that a critical concentration for energy transfer between active sites is present if the inhomogeneous broadening of the op- tical transition line exceeds the strength of the energy transfer (donor-donor) Hamiltonian, and if the interac- tion is of sufficiently short range (falling off faster than dipolar). It is now believed that in many transi- tion metal and rare earth doped hosts, the energy trans- fer Hamiltonian between donor sites is quadrupolar or higher in character, or exchange [7]. These interac- tions all have range dependences sufficiently short that the Anderson criterion is relevant, and a donor concentration exists below which donor-donor energy transfer cannot occur. Thus, the donor fluorescence should exhibit a time dependent decay according to (1) at low donor concentrations. As one increases the donor concentration in excess of the critical concentra- tion, the presumably abrupt transition to a state of finite donor-donor excitation conductivity will take

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Volume48A, number6 PHYSICSLETTERS 29 July 1974

placeat the criticalvalue. Thisconcentrationwill de- seeif suchbehavioroccurs.Onecanthen usethe tech-pendon themagnitudeof the inhomogeneousbroaden- mqueto probethe dependenceof the critical energybig of the donorfluorescenceline, andthe strength~ transferconcentrationon inhomogeneouswidth, spe-andrangedependenceof the donor-donortransfer cific energytransfermechanisms,andhost topology,Hamiltonian (seeref. [2] foran exampleof a detailed by examininga variety of donorsandhosts.evaluation).Abovethis critical concentration,the do-

The authorwishesto thank ProfessorR. Reisfeldnor fluorescencedecayshouldbe exponential.Weem- for themanyhelpful discussionswhich led to therea-phasizethat the transitionbetweenthe two regimes lization thatdonor fluorescentdecaycan serveas a(asa functionof concentration)shouldbe abrupt. tool for investigatingtheAndersonlocalization mech-Thereis , or thereis not,donor-donortransferaccord-

anism.ing to the Andersoncriterion. Thisresultcanbesome-what softenedif thereexistsresidualdipole-dipolecoupling,butall estimatesof this ratefor intra-config- Referencesuration transitionsappeartobe small.

In summary,theconceptof Andersonlocalization [1] P.W. Anderson,Phys.Rev. 109 (1958)1492.canbe directly testedin optical transitionswith large [2] S.K. Lyo,Phys.Rev.B3 (1973)3331.inhomogeneousbroadening.Donor-donorenergy [3] J.C.Koo, J.L. Davis andS. Geschwind,Bull. Am. Phys.transferis requiredif the donorfluorescencedecayis Soc. 18 (1973)472.[4] For confirmationexperiments,seeW.B. Gandrudandto beexponential.Forconcentrationsbelowthe criti- H.W. Moos,J.Chem.Phys.49 (1968)2170.

cal value,no donor-donorenergytransfertakesplace [5] M. Inokuti andF. Hirayama,J.Chem.Phys.43 (1965)andthe donorfluorescentdecayceasesto beexponen- 1978.tial, taking the form(1). Thetransitionbetweenthese [6] J.M. Ziman,J.Phys.C2 (1962)1230;

R. Abou-Chaira,P.W. AndersonandD.J.Thouless,J.Phys.distinctbehaviorsis abrupt,andoccursat accessible C6 (1973) 1734.donor concentrations(within the rangeof 0.3 at.%for M.J. Weber,Phys.Rev. B4 (1971)2933.ruby,for example).It wouldbe of greatinterestto [7] R.J. Birgeneau,J.’Chem.Phys.50 (1969)4282.

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