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PHYSICAL REVIEW B 1 JULY 1999-IVOLUME 60, NUMBER 1

Delocalization of tightly bound excitons in disordered systems

Richard BerkovitsThe Minerva Center for the Physics of Mesoscopics, Fractals and Neural Networks, Department of Physics, Bar-Ilan Univers

Ramat-Gan 52900, Israel~Received 28 December 1998; revised manuscript received 19 February 1999!

The localization length of a low-energy tightly bound electron-hole pair~excitons! is calculated by exactdiagonalization for small interacting disordered systems. The exciton localization length~which corresponds tothe thermal electronic conductance! is strongly enhanced by electron-electron interactions, while the localiza-tion length~pertaining to the charge conductance! is only slightly enhanced. This shows that the two-particledelocalization mechanism widely discussed for the electron pair case is more efficient close to the Fermienergy for an electron-hole pair. The relevance to experiment is also discussed.@S0163-1829~99!08125-4#

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Interacting electrons in random potentials are in the ceof recent studies in mesoscopic systems because of theievance to many different phenomena such as the Coulgap,1 persistent currents,2 the two-dimensional~2D! metal-insulator transition,3 and delocalization of two-particlestates.2,4–6

Although the subject of two-particle state delocalizatidue to electron-electron~e-e! interaction has been the centof many recent studies,2 most of those studies concentratethe situation of two particles in an empty band of a ondimensional ~1D! system. This is an idealized situatiowhich is not very relevant to experimental systems. A mrelevant situation, i.e., two excited electrons above the Fesea, was considered in Refs. 6,7. In Ref. 7 it was showntwo electrons close to the Fermi energy show no enhanment in 1D systems. Nevertheless, it has been emphasthat the situation may be more favorable for highdimensions,6 for which, up to now, only the empty band cawas studied.8 Recently it has been pointed out9 that when theeffect of the Coulomb gap is taken into account the telectron delocalization scenario becomes less likely.

In this paper we consider a different type of two-particdelocalization, namely the delocalization of a tightly bouelectron-hole pair~strongly bound excitons!. This exciton isanalogous to the classical compact electron-hole pair extions discussed in Ref. 1, which can transfer energy acthe system but not charge. This is different from most preous discussions of two-particle delocalization which weconcerned with the charge conductance of the system. Tdelocalization of the electron-hole will manifest itself in thenhancement of the heat conductance of the disorderedtem, an effect which can be measured in the laboratory.deed, the relevance of the two-particle delocalization snario to electron-hole pairs has been noted by Imry.6 In Ref.9 it was suggested that not only is the two-particle localition length enhancement relevant to an electron-hole pais also much more effective than for the two-electron cabecause of the Coulomb enhancement of the exciton denThis scenario will be verified in this paper by explicit calclation of the localization length. Moreover, the existencedelocalized excitons may have interesting consequencelow-temperature hopping conductance, which will be d

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cussed in more detail after presenting the evidence forexciton localization length enhancement.

Our study is based on the following interacting manparticle tight-binding Hamiltonian:

H5(k, j

ek, jak, j† ak, j2V(

k, j~ak, j 11

† ak, j1ak11,j† ak, j !1H.c.

1H int , ~1!

where ek, j is the energy of a site (k, j ), chosen randomlybetween2W/2 andW/2 with uniform probability, andV is aconstant hopping matrix element. The interaction Hamtonian is given by

H int5U (k, j . l ,p

ak, j† ak, jal ,p

† al ,p

urWk, j2rW l ,pu/b, ~2!

whereU5e2/b andb is the lattice unit.We considerLx3Ly samples of size 332, 333, 334,

and 335 with M56,9,12,15 sites andN52,3,4,5 electronscorrespondingly~i.e., 1/3 filling!. The calculation of the electron transmission required also the calculation of theN11electron Hamiltonian for each case. Them3m Hamiltonian@wherem5(N

M)] matrix is numerically diagonalized and athe eigenvectorsuC l

N& and eigenvaluesElN are obtained. The

strengthU of the interaction is varied between 0230V andthe disorder strength is chosen to beW530V. The results areaveraged over 6000, 1000, 200, 25 different realizationsdisorder for theM56,9,12,15 samples.

The zero-temperature tunneling density of state~TDOS!for the N11 electron is defined in the usual way10

n~«!5(l

U K C lN11U(

k, jak, j

† UC0NL U2

d„«2~ElN112E0

N!…,

~3!

which for the noninteracting case will reproduce the singparticle density of states above the Fermi energy. The a

aged TDOS ^n(«)&5(0.4D8)21*«20.2D8«10.2D8^n(«8)&d«8 ~here

^ . . . & stands for an average over realizations of disorder! fordifferent values of interaction is presented in Fig. 1~a!. Sincethe distance to the many-particle band edgeB increases as a

26 ©1999 The American Physical Society

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PRB 60 27BRIEF REPORTS

function of the interaction strengthU, we rescale the singleparticle level spacingD to offset this trend. The rescale‘‘single-particle’’ level spacing is defined asD85B/N(M2N), which is equal toD in the noninteracting case. Folarge valuesD8 there is little influence of the interactiostrength onn(«), while for small values ofD8 there is achange. While forU50 the TDOS is rather monotonouexcept for a remanent of the level repulsion atD850, thereis a signature of the Coulomb gap1 at U530V. Thus, theTDOS shows the expected behavior as function of the inaction strength.

We shall define a tightly bound exciton as the movemof an electron from its ground-state position to a neighborsite, which may be expressed by the application of a creaoperator Jk, j

† 5Jk, j2x†1Jk, j

1x†1Jk, j2y†1Jk, j

1y† ~where Jk, j6x†

5ak, j 61† ak, j and Jk, j

6y†5ak61,j† ak, j ) on the ground-state ei

genvector. The overlap of an excited state with a statetained out of the ground state by the creation of an excitosite (k, j ) is then given byu^C l

N11uJk, j† uC0

N11&u2. One mayalso define the exciton density of state~EDOS!

nE~«!5(l

U K C lN11U(

k, jJk, j

† UC0N11L U2

3d„«2~ElN112E0

N11!… ~4!

FIG. 1. The average transport density of state~a! and averageexciton density of state~b! as function of the rescaled singleelectron level spacing of a 333 system for different values of interaction strength.

r-

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b-at

in an analogous way to the TDOS. The EDOS then descrthe weight of a single tightly bound exciton at a given eergy. The average EDOSnE(«)& ~calculated in the sameway as the average TDOS! for different values ofU is plot-ted in Fig. 1~b!. It is clear that the interaction strength influences the EDOS in a different way than the TDOS. Therno sign of of the Coulomb gap, instead there is a mononous enhancement of the EDOS, as one may expect fromclassical theory of tightly bound excitons which predictsconstant EDOS as function of«.1

The localization properties of a system manifest theselves in the transport behavior of the system. For a sinparticle system one may connect the localization lengthj tothe transmissiont(rW,rW 8,«)5^rWu(«2H1 ih)21urW 8& via Ref.11 j(«)215^ lnut(rW,rW 8,«)u&/urW2rW 8u. In Ref. 12 a many-particleLandauer formula connecting transmission and conductafor an interacting system coupled to external leads wasveloped. For the many-particle case the transmission ininteracting region is given by

FIG. 2. The average log of the charge~regular symbols! andheat transmission~heavy symbols! as function of system length fodifferent values of interaction strength.~a! «5D8 ~b! «52D8.

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28 PRB 60BRIEF REPORTS

t~rW,rW 8,«!5(l

^C lN11uarW

†uC0N&^C0

NuarW 8uC lN11&

«2~ElN112E0

N!1 ih, ~5!

whereh→0. In the numerical calculation we assumedrW andrW 8 to correspond to the opposite corners of the sample,rW5(1,1) andrW 85(1,Ly).

The heat transport~namely the exciton transport! can beprobed, for example, by attaching leads which excitetightly bound exciton at one point of the sample and absit on another point. In this case the heat transmission ismulated as

th~rW,rW 8,«!5(l

^C lN11uJrW

†uC0N11&^C0

N11uJrW 8uC lN11&

«2~ElN112E0

N11!1 ih.

~6!

The localization lengths of the system are extracted frthe numerical data in the following way: We calculate taverage log of the~heat! transmission between the two coners of the sample for the low-lying excitations^ lnut(h)(«)u&5^ln„(2D8)21u*«2D8

«1D8t (h)(rW,rW 8,«8)d«8u…&. The results arepresented in Figs. 2~a! for ^ lnut(h)(«5D8)u& and in Fig. 2~b! for@^ lnut(h)(«52D8)u&#. From the slope of ln t& vs Ly one candeduce the localization length according toj}Ly /^ ln t&.11

For the charge transmission we obtain the following locization length j50.85b, 0.86b, 0.70b, 0.71b at U50, 10V, 20V, 30V for «5D8 and j51.01b, 1.20b, 1.08b, 0.95b at U50, 10V, 20V, 30Vfor «52D8. Thus, the charge transmission is somewhathanced for weak interactions, but is suppressed for stroninteractions. This is in line with our expectations on thefluence of the interaction on the charge conductance.13,14 Onthe other hand, for the heat transmission the exciton loization length jh50.33b, 0.47b, 0.59b, 0.59b corre-sponding to U50, 10V, 20V, 30V for «5D8 and jh50.37b, 0.55b, 0.69b, 0.72b corresponding to U50, 10V, 20V, 30V for «52D8. Thus, the heat transmission and the exciton localization length are significantly ehanced by the interactions.

Thus, low-energy excitons show a significant enhanment of their localization length, while electrons in that rgion show only a moderate enhancement. This can be unstood as the result of the fact that for electrons interacting

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the Coulomb interaction the joint density of states of twelectrons drastically decreases at small energies due toCoulomb gap in the one-electron density of states. This leto a suppression of the enhancement of the localizalength which strongly depends on the two-particle densitystates. However, for excitons, Coulomb effects increasedensity of states making these arguments more plausThis can be clearly seen for the stronger interaction stren(U520V,30V) for which jh are strongly enhanced, whilejis suppressed compared to the noninteracting value. Afrom a direct calculation of the two-electron transmissiothe suppression of the two electron localization length clto the Fermi energy is evident.

An enhancement ofjh will of course manifest itself inelectronic thermal conduction measurements of disordesystems. For a case in whichjh.j, which has not beenachieved for the parameters we have chosen in our mobut cannot be ruled out for larger systems or at a differparameter range, interesting behavior may emerge. Forample, it is quite possible that forj@b the exciton localiza-tion length may become infinite so that the electronic coductivity is exponentially small while the electronic thermconductivity changes as a power of temperature.15,9 Anotherinteresting effect in such a regime is the possibility thatlow temperatures the delocalized excitons may assist etron hopping much more effectively than phonons.9,16,17Thiswill result in a prefactor of the variable-range hopping prportional or even equal to a universal valuee2/h. Such pref-actors were observed in the QHE regime~see references ananalysis in Refs. 16,17! and recently at zero magnetic field.18

In conclusion, we have shown that the two-particle decalization mechanism close to the Fermi energy is mumore efficient for an electron-hole pair than for two electro~or holes!. This is a result of the enhancement of the excitdensity of states near the Fermi energy by Coulomb intetion opposed to the suppression of the electronic densitystates. This results in a large enhancement of the electrthermal conductance while the conductance is only modately enhanced.

I would like to thank B. I. Shklovskii for initiating thisstudy and for many important discussions and suggestiThis research was supported by The Israel Science Foutions Centers of Excellence Program.

1B. I. Shklovskii and A. L. Efros,Electronic Properties of DopedSemiconductors~Springer, Heidelberg, 1984!.

2For a recent review which covers persistent currents and tparticle delocalization see T. Guhr, A. Mu¨ller-Groeling, and H.A. Weidenmuller, Phys. Rep.299, 191 ~1998!, and referencetherein.

3S. V. Kravchenkoet al. Phys. Rev. Lett.77, 4938 ~1996!; D.Popovic, A. B. Folwer, and S. Washburn,ibid. 79, 1543~1997!.

4O. N. Dorokhov, Zh. Eksp.Teor. Fiz.98, 646 ~1990! @Sov. Phys.JETP71, 360 ~1990!#.

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o-

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1469 ~1993!.12Y. Meir and N. Wingreen, Phys. Rev. Lett.68, 2512~1992!.13R. Berkovits and Y. Avishai, Phys. Rev. Lett.76, 291 ~1996!.14T. Vojta, F. Epperlein, and M. Schreiber, Phys. Rev. Lett.81,

4212 ~1998!.

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