Extended states and dynamical localization in the random-dimer model

a r X i v :c o n d -m a t /9505120v 1 25 M a y 1995

Extended states and dynamical localization in the random-dimer model

F.Dom′?nguez-Adame ?

Departamento de F′?sica de Materiales,Facultad de F′?sicas,Universidad Complutense,E-28040Madrid,Spain

Angel S′a nchez ?and Enrique Diez ?

Departamento de Matem′a ticas,Escuela Polit′e cnica Superior,Universidad Carlos III,E-28911Legan′e s,Madrid,Spain (February 1,2008)

We study quantum di?usion of wavepackets in one-dimensional random binary subject to an applied electric ?eld.We consider three di?erent cases:Periodic,random,and random dimer (paired)lattices.We analyze the spatial extent of electronic wavepackets by means of the time-dependent inverse participatio ratio.We show that the delocalized states recently found in random dimer lattices become spatially localized under the action of the applied ?eld (dynamical localization)but wavepackets are much less localized than in purely random lattices.We conclude that the resonant tunneling e?ects causing delocalization play an important role even in the presence of the electric ?eld.

PACS number(s):71.50.+t,72.15.Rn,73.20.Dx

I.INTRODUCTION

The existence of one-dimensional (1D)disordered lat-tices with a number of extended electronic states large enough to contribute to transport properties in a rel-evant fashion has been undoubtedly established during this decade.The interest on this problem arose from the pioneering works of Flores [1]and Dunlap,Wu and Philips [2],which stimulated a considerable e?ort devoted to understand these delocalization phenomena [3–9].The common feature of the models studied so far is that they consist of a host (discrete or continuum)system where defects are placed randomly although their distribution exhibit spatial correlation.This spatial correlation is usually introduced by imposing that impurities appear always in pairs (dimers)or in more complicated group-ing schemes.Speci?cally,results for 1D random models with paired disorder,i.e.,with defects forming dimers,which exhibit delocalization have been put on solid theo-retical grounds [10].Interestingly,spatial correlations in 1D random systems lead to new and unexpected phenom-ena not only in electronic systems but also in the case of quantum ferromagnets [11],Frenkel excitons [12],classi-cal vibrations [13]and excitations [14].It thus becomes clear that the study of random systems with correlated disorder is of interest in a wide range of physical prob-lems,and that,ultimately,such a line of research may lead to the development of a variety of new devices and applications.

In Ref.[10],it was shown that delocalized electronic states arise in spite of the inherent disorder due to reso-nant phenomena taking place at dimers which,in turn,lead to a transmission coe?cient of di?erent segments forming the lattice close to unity,no matter what the length of the segment is.The transmission coe?cient is exactly unity for the resonant energy at a single dimer defect and,most important,is very close to unity for

electron energies near the resonant one,as demonstrated by perturbative and numerical calculations.Once the ex-istence of these bands of extended states is put forward and the reasons for its appearance are understood in the isolated,non-interacting model,an interesting question immediately arises,namely what is the e?ect of external perturbations on the delocalized states?In particular,since we are going to concern ourselves with electronic states,the ?rst perturbation that has to be studied is an applied electric ?eld;since applied electric ?elds lead to localization even in periodic lattices,one should ex-pect that delocalized states in random correlated systems might also be spatially localized.But the key question is to elucidate whether the physical mechanisms giving rise to delocalization in unperturbed systems are of relevance in the presence of the electric ?eld or,on the contrary,they are immaterial at all.The answer to this question is not trivial:Competition between quantum coherence due to correlated disorder and the loss of quantum coher-ence due to the misalignement of local electronic levels under the action of the ?eld will be the main mechanism governing this system,and the prevailing factor among these two is di?cult to foresee.

In this letter we present a ?rst study in the above di-rection.We consider the problem of quantum di?usion of wavepackets initially localized in random-dimer mod-els (RDMs),as introduced in Ref.[2],under the action of a uniform electric ?eld.The way we carry out such an analysis is by comparing electronic amplitudes in three di?erent binary systems,namely periodic,unpaired dis-ordered lattices and paired disordered ones.The study of periodic systems will allow us to establish the main features of dynamical localization in periodic binary sys-tems.This is required for a better understanding of wavepacket dynamics when an amount of randomness is introduced in the system.To get an estimation of the spreading of the wavepacket as a function of time we will

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use the time-dependent inverse participatio ratio(IPR). By means of this quantity,we will be able to show below that,although all states in random dimer models become localized under the action of the electric?eld,they ac-quire an spatial structure much more extended that their counterparts in the purely random lattice.

II.MODEL

The unperturbed random-dimer model is described by the following1D tight-binding Hamiltonian[2]

H0= n E n c?n c n+V n(c?n+1c n+c?n c n+1).(1) Here c n and c?n are electron annihilation and creation operators in the site representation.The hopping matrix element V is assumed to be constant over the whole lat-tice,whereas on-site energies E n can only take on two values E A and E B,with the additional constraint that E B are assigned at random to pairs of lattices sites.In Ref.[2],it was found that for|E A?E B|<2|V|,an ini-tially localized wavepacket becomes delocalized and its mean-square displacement grows in time as t3/2(super-di?usion).For|E A?E B|=2|V|the mean-square dis-placement behaves asymptotically as t(di?usion).Oth-erwise the wavepacket remains localized.

As we mentioned above,we are interested in quantum di?usion of wavepackets under an applied electric?eld. In particular,we investigate the time behaviour for dif-ferent sets of constituent parameter E A,E B,and V.The perturbed Hamiltonian is written as follows

H=H0?F n nc?n c n,(2) where F is the electric?eld(we use units such that e=ˉh=1in the rest of the paper).In order to solve the corresponding Sch¨o dinger equation we express the wave function in terms of localized Wannier states.In doing so,it can be seen that the time-dependent amplitudes ψn(t)satisfy the following equation

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Results

corresponding to both kinds of disordered lat-tices are shown in Fig.2,for E A =0,V =?1,F =0.02,and three di?erent values of the on-site energy E B .The concentration of B-sites is c =0.2in all cases.We can ob-serve that Bloch oscillations are completely absent in ran-dom lattices.This fact can be explained by the absence of translational invariance and,consequently,by scattering of electrons with the lattice,which destroys the quantum coherence required to observe such phenomenon.In both kind of random lattices the IPR presents strong ?uctua-tions at small time scales,but it can be observed that its average value over larger times is constant.Such small ?uctuations depend on the particular realization of the disorder and on the initial position of the wavepacket.However,for a given concentration c ,the mean value de-pends only on the electric ?eld and on the hopping ma-trix element (the larger the electric ?eld or the hopping matrix element,the closer to unity the IPR).

This far,we have summarized the common features of states of both random lattices.It is now the moment to comment the main di?erences between paired and un-paired lattices.When |E A ?E B |<2|V |,i.e.whenever the defect energy lies within the allowed band [Fig.2(a)],the mean value of the IPR is smaller for paired lattices,meaning that the wavepacket spreads over larger portions of the system.From this plot,it can be appreciated that the di?erence between both IPR values is about an order of magnitude,and hence the spatial extent of wavefunc-tions in paired and unpaired lattices will largely di?er.Thus,when the unperturbed (F =0)paired lattice sup-ports extended states,the resulting dynamical localiza-tion under the action of the electric ?eld is much less e?ective than in unpaired lattices.It is therefore rea-sonable to expect that the transport properties of the two systems will also exhibit speci?c features:For in-stance,the hopping conductivity has to be much larger in the dimer lattice than in the random lattice,due to the increased tunneling probability between neighboring localized states.A smaller degree of localization in the dimer lattice is also observed,although to a somewhat lesser extent,in the critical situation |E A ?E B |=2|V |[Fig.2(b)].On the contrary,the dynamics in both lat-tices is almost identical whenever the unperturbed lattice only supports localized states [Fig.2(c)].

A better understanding of this result is achieved if one considers that the (initially strongly localized)wavepacket is the combination of plane waves in a con-tinuous band [17].Since the energy spectrum of the paired disordered lattices presents a band of extended states,the lattice behaves as a selective electronic ?lter,and those components whose wavenumber belongs to this band can propagate over larger distances,producing a larger spreading of the resulting wavepacket.The obser-vation of this behaviour,as we have reported,is therefore a clear consequence of the fact that the unperturbed lat-tice supports extended states.Finally,the absence of Bloch oscillations in paired disordered lattices indicates that their extended states are no longer Bloch states.

Bloch states are characterized by a complete quantum coherence with a perfectly de?ned phase.This is not the case in the RDM,where electronic states increment its phase by a factor of πwhenever they pass over a dimer defect [2,4],and the position of each dimer defect is in any case a random variable.

FIG.2.Inverse participatio ratio as function of time in a binary random lattices with E A =0,and V =?1,for F =0.02with (a)E B =1,(b)2,and (c)3.Solid and dotted lines correspond to paired and unpaired disordered lattices,respectively.

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IV.CONCLUSION

We have studied quantum di?usion of wavepackets driven by an applied external?eld in periodic and ran-dom(unpaired and paired)binary lattices.The spatial degree of localization of wavepackets initially localized at a single lattice site has been properly described by means of the time-dependent IPR.In binary periodic lat-tices we have con?rmed the dynamical localization under electric?elds as well as Bloch oscillations,for which the wavepacket oscillates in time with a well-de?ned period proportional to the inverse of the electric?eld.Quantum dynamics in disordered lattices also exhibits dynamical localization although it turns out to be much more in-tricated:In particular,no evidence of Bloch oscillations (regular behavior)is observed.What is most important for the purposes of the present work,we have determined that dynamical localization is less e?ective in paired dis-ordered lattices than in unpaired ones,provided that the energy of defects lies within the band of extended states.Thus,it can be concluded that extended states can spread over larger segments of the lattice,giving rise to a smaller localization of the wavepacket in the presence of the electric?eld.Therefore,the resonant tunneling ef-fects causing delocalization plays an important role,even in the presence of the applied?eld.

The results we have reported in this letter provide an-other piece of evidence supporting the true extended na-ture of states near the resonant energy in the random dimer model.From plots in Figs.1and2(a),one can observe that the value of the IPR for the random dimer model is about the minimum.of the Bloch oscillations of the periodic lattice.This is to be compared with the ramdom case,whose IPR is close to half of the IPR of the periodic lattice.It is then clear that electric-?eld-localized states in the random dimer model are much closer to those of the periodic lattice than to the purely random system.In view of this,we envisage that the transport properties of random dimer lattices under elec-tric?elds will also be close to those of periodic lattices, this being an experimentally veri?able,qualitative pre-diction.To conclude,we mention that another question stemming from this work is whether the same behavior will be found in more realistic models such as the contin-uum random dimer model[4]or the square well model [12].These models have already given rise to quantita-tive predictions of e?ects that should be observed in real devices and therefore they are very appealing in order to?nd physically relevant consequences of dynamical lo-calization in dimer models.Work in this direction is in progress.

ACKNOWLEDGMENTS

This work has been supported by CICYT(Spain)un-der project MAT95-0325.