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tion ofulated byn return

Physics Letters A 347 (2005) 255–261

www.elsevier.com/locate/pl

Dynamical localization of a two-band system in real-space

Xian-Ke Peng∗, Jian-Li Shao, Duan Suqing, Xian-Geng Zhao

Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China

Received 16 May 2005; accepted 29 July 2005

Available online 15 August 2005

Communicated by J. Flouquet

Abstract

With the method of the long-time averaged occupation probability (LAOP), we investigate the conditions of localizaan electron in the coupling two-band system under the action of ac and dc–ac fields. The values of LAOP can be calcFloquet theorem. We find with varying the electric field, the values of LAOP show some peaks, at which the electron carepeatedly to the initial site or stay in the initial site forever. 2005 Elsevier B.V. All rights reserved.

PACS: 73.20.Dx; 72.15.Rn; 63.20.Kr

tohet ofsly,onse ofnyd/orma-fer-c-

fineding-onary

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1. Introduction

The response of semiconductor superlatticesexternal fields is a subject of interest both for tfundamental physics study and the developmenthe high-quality semiconductor devices. Previouthe transport and the dynamic properties of electrin the semiconductor superlattices in the presencelectric fields have been studied extensively. Maphenomena have been predicted theoretically anobserved in experiments because of the availableture experimental technology, such as negative difential conductivity[1,2], absolute negative condu

* Corresponding author.E-mail address: pengxk_1980@163.com(X.-K. Peng).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2005.07.087

tance[3], inverse Bloch oscillation[4], band collapse[5–7], dynamical localization[8–13], and fractionalWannier–Stark ladders[14]. Especially for the onset othe dynamical localization, there has been a sustaresearch interest. Within a single-band tight-bindmodel, Dunlap and Kenkre[8] have obtained the analytical solutions of a charged particle hoppinga discrete lattice under the influence of an arbitrtime-dependent electric field. Then Zhao[10] givesthe exact dynamic localization conditions of a chargparticle in a dc–ac electric field, and Duan[13] dis-cusses the case of a dc-bichromatic electric fieldpoints out that when the dynamical localization hapens, it just corresponds to the collapse points ofquasi-energy band. In a word, they find that antially localized particle remains localized if the ratof the field magnitude to the frequency is a root of

.

256 X.-K. Peng et al. / Physics Letters A 347 (2005) 255–261

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lat-

emnen

s aer,nsg

ged

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tialthendm-

enandonenndsest-en

ofeld

r

wongnttate

-

er-

of a

as ofuet

x

i-

lfn

roth order Bessel function, which can be accounfor the band collapse of the quasi-energy miniband[5]and has been verified experimentally in an opticaltice [15].

However, can the electron of the two-band systlocalize in the initial lattice point under the actioof the external electric fields? What happens whthe ratio of the field magnitude to the frequency iroot of zeroth order Bessel function? In this Lettwe investigate the dynamical localization conditioof a single electron within a two-band tight-bindinmodel through a method called the long-time averaoccupation probability (LAOP)[16]. Compared withthe methods of the previous work, LAOP can quatitatively describe the localization effect. Using thmethod, we find the electron can return to the inisite repeatedly in the pure ac field signified bypeaks of the LAOP. As the dc field is introduced aits amplitude is suitably chosen, the electron can copletely localize in the initial well and oscillate betwethe bands under the combined action of the dcac fields. In addition, we also study the localizatieffect by changing the coupling coefficient betwethe Bloch bands and the site energy of the two barespectively in the presence of the dc–ac field. Interingly, the electron can perform Rabi oscillations whwe change the site energy.

2. Theoretical model

We consider the standard tight-binding modela two-band system in a time-dependent electric fiE(t):

H(t) = ∆a

∑m

|a,m〉〈m,a| + ∆b

∑m

|b,m〉〈m,b|

− Ra

∑m

(|a,m〉〈m + 1, a| + h.c.)

+ Rb

∑m

(|b,m〉〈m + 1, b| + h.c.)

+ edE(t)∑m

m(|a,m〉〈m,a| + |b,m〉〈m,b|)

(1)+ µE(t)∑m

(|a,m〉〈m,b| + h.c.).

In this expression, the integerm (−N � m � N) is thelattice site and|a,m〉 (|b,m〉) represents the Wannie

state localized on sitem referring to the banda (b);∆a and∆b are the site energies belonging to the tbands;Ra and Rb are the nearest-neighbor hoppimatrix elements;e andd are, respectively, the electrocharge and the lattice constant;µ is the dipole momenof the interband coupling. Expressing the particle s|ψ(t)〉 as a linear combination of Wannier states|i,m〉(i = a, b),

(2)∣∣ψ(t)

⟩ = ∑i,m

Cim(t)|i,m〉.

Cim(t) are the time-dependent amplitudes〈m, i|ψ(t)〉.

We obtain from Eq.(1) the following evolution equation for the amplitudes:

(3)

i(∂/∂t)Cam

= (∆a + edE(t)m

)Ca

m + µE(t)Cbm

− Ra

(Ca

m+1 + Cam−1

),

i(∂/∂t)Cbm

= (∆b + edE(t)m

)Cb

m + µE(t)Cam

+ Rb

(Cb

m+1 + Cbm−1

).

We puth̄ = 1 throughout this Letter.In fact, the lattice number of semiconductor sup

lattices is finite. Physically speaking, solving Eq.(3)with N = 5 andN = 10,20, . . . will give nearly in-distinguishable solution, so we just study the casesystem of 11 lattice points.

The Hamiltonian is periodic in time, so there iscomplete set of Floquet wave functions as solutionthe Schrödinger equation in the framework of Floqtheory:

(4)∣∣ψα(t)

⟩ = exp(−iεαt)∣∣uα(t)

⟩,

with quasi-energy εα and T -periodic function|uα(t)〉 = |uα(t + T )〉, whereT is the periodicity ofthe external fields.

We introduce the time-evolution operatorU(t,0),which satisfies the following equation:

(5)i∂

∂tU(t,0) = H(t)U(t,0),

where the HamiltonH(t) can be written as a matrilabelled by Wannier sites by using Eq.(3). We canintegrate Eq.(5) numerically over one period, and dagonalizeU(T ,0) to get the quasi-energyεα and theinitial Floquet state|uα(0)〉. We suppose the initiastate of the system is|I,K〉, which is an element othe Wannier states|i,m〉, then the quantum evolutio

X.-K. Peng et al. / Physics Letters A 347 (2005) 255–261 257

Flo-

a

ibed

y

ofenin

ndoree-

,ni-

ediale

of

tita-dver

on

1re-pt

process of the system can be described in terms ofquet states as follows:

(6)∣∣ψ(t)

⟩ = ∑α

exp(−iεαt)∣∣uα(t)

⟩⟨uα(0)

∣∣I,K⟩.

According to Eq.(6), we can get

(7)U(t,0) =∑α

exp(−iεαt)∣∣uα(t)

⟩⟨uα(0)

∣∣.Then |uα(t)〉 can be obtained at arbitrary time incycle of the external fields by using Eq.(7) with thenumerical results forU(T ,0) from integrating Eq.(5).

The Floquet states in real-space can be descras:

(8)∣∣uα(t)

⟩ = ∑i,m

|i,m〉�=|I,K〉

ciα,m|i,m〉 + cI

α,K |I,K〉.

Upon substituting Eq.(8) into Eq.(6), we get∣∣ψ(t)⟩ = ∑

α

cI†α,K(0)cI

α,K(t)exp(−iεαt)|I,K〉

+∑α

∑i,m

|i,m〉�=|I,K〉

cI†α,m(0)ci

α,m(t)

(9)× exp(−iεαt)|i,m〉.Then we defineP I

K(t) as the occupation probabilitof the component of state|I,K〉 in the state|i,m〉, soP I

K(t) can be written as:

P IK(t) =

∑α

∣∣cIα,K(0)

∣∣2∣∣cIα,K(t)

∣∣2

+ Re∑α

α �=β

cIα,K(0)c

I†β,K(0)c

I†α,K(t)cI

β,K(t)

(10)× exp[−i(εβ − εα)t

].

To investigate the localization effect, we introduceP̄ IK

to express the long-time averaged value ofP IK(t). By

setting 0= t0 < t1 < · · · < tQ−1 < tQ = T and λ =max�tj = max(tj − tj−1) (1� j � Q), we can arriveat the following equation:

P̄ IK = lim

τ→∞1

τ

τ∫0

P IK(t) dt

= limM→∞

1

MT

M∑n=1

T∫P I

K

(t + (n − 1)T

)dt

0

= limλ→0

1

T

Q∑j=1

∑α

∣∣dIα,K(0)

∣∣2∣∣dIα,K(ξj )

∣∣2�tj

+ Re limλ→0

1

T

Q∑j=1

∑α,βα �=β

δ(εβ − εα)dIα,K(0)

(11)× dI†β,K(0)d

I†α,K(ξj )d

Iβ,K(ξj )�tj ,

whereξj ∈ [tj−1, tj ]. Thus the analytical expressionthe long-time averaged occupation probability is givby Eq. (11). However, it is not convenient to usepractice. With definition of�tj = T/Q andξj = tj−1,Eq.(11)can be cast in a more explicit form:

P̄ IK = lim

Q→∞1

Q

Q−1∑q=0

∑α

∣∣dIα,K(0)

∣∣2∣∣∣∣dIα,K

(qT

Q

)∣∣∣∣2

+ Re limQ→∞

1

Q

Q−1∑q=0

∑α,βα �=β

δ(εβ − εα)dIα,K(0)

(12)× dI†β,K(0)d

I†α,K

(qT

Q

)dIβ,K

(qT

Q

).

In a concrete calculation we can adjustQ, which is fi-nite, to obtain the desired precision. However, we fithat all Q � 10 generate the same solution; therefwe setQ = 10 in this Letter. If we choose 11 latticpoints of a two-band model,N equals 5 in the practical calculation.

As a convention,Pa(t) andPb(t) are, respectivelythe occupation probability of the electron on an itial site referring to banda andb at arbitrary timet ;while PaL andPbL represent the long-time averagoccupation probability of the electron on the initsite in the bandsa andb, respectively. Thus we havP(t) = Pa(t) + Pb(t) andPL = PaL + PbL to expressrespectively the occupation probability and LAOPthe electron on the initial site. Then the value ofPL

can be used to describe the localization effect quantively. If PL ≈ 1, it indicates that an initially localizeelectron remains localized on the original site foreand could not travel to other sites. If we drawPL asa function of external field, the dynamic localizatihappens at the peaks ofPL, though the value ofPL

is small. In this case,P(t) oscillates between 0 andperiodically, meaning that the electron can returnpeatedly to the initially occupied lattice point. Excefor the above two cases, delocalization happens.

258 X.-K. Peng et al. / Physics Letters A 347 (2005) 255–261

s

ct

i-henc-oerveted

Afeen,thep-

We set the frequency of ac fieldω as the unit ofenergy and take the electron to be initially in banda onthe middle lattice point (I = a, K = 0, soPa(t) = P a

0andPaL = P̄ a

0 ) throughout this Letter.

Fig. 1. The LAOPPaL (PbL) versusedE1/ω, where the parameterare(∆b − ∆a)/ω = 1.25, µ/ed = 0.12 and(Ra + Rb)/ω = 0.24.The solid and dotted lines correspond toPaL andPbL, respectively.

3. Numerical results and discussion

First, we study the case of a pure ac fieldE(t) =E1 cos(ωt). In order to see the localization effeclearly, we drawPaL andPbL as functions ofedE1/ω,as shown inFig. 1, where the parameters are(∆b −∆a)/ω = 1.25,µ/ed = 0.12 and(Ra +Rb)/ω = 0.24.The solid line and the dotted line correspond toPaL

andPbL, respectively. Fortuitously, there is an obvous peak accompanied by several small peaks wedE1/ω is near the root of zeroth order Bessel funtion, but it shifts to left due to the interaction of the twbands. Subsequently, we select some points to obsthe evolution of the occupation probability, as depicin Fig. 2(a)–(d).

Fig. 2(a) shows the case of the first main peakwhereedE1/ω = 2.304, which is near the first root othe zeroth order Bessel function. Recurrences are simplying that the electron can return repeatedly toinitial site. That is to say, dynamical localization hapens. A similar case can be seen fromFig. 2(b), which

Fig. 2. The time evolution of the occupationP(t). (a)edE1/ω = 2.304, (b)edE1/ω = 5.403, (c)edE1/ω = 1.343, (d)edE1/ω = 4.0.

X.-K. Peng et al. / Physics Letters A 347 (2005) 255–261 259

derC

veron-

by

nionatini-s

eld–orcanan

a-

e

ld,oone

ll.ichcil-the

asehen,u-ag-n ofperbe

hen

s

thehee-

end

ak.ise,Intial

in-rse-theami-tly

nthesi-o-

shows that of the second sharp peak B whenedE1/ω

is near the second root of the Bessel function of orzero.Fig. 2(c) shows the case of the first small peakof PaL whenedE1/ω = 1.343. The oscillation ofP(t)

is irregular and sometimes it is beyond 0.5, but nereally close to unity, which reveals that the electrcan still be localized.Fig. 2(d) shows the time evolution of P(t) with edE1/ω = 4.0, wherePaL andPbL

do not form peaks and vary very slowly as labelledpoint D in Fig. 1. Compared withFig. 2(c), P(t) isless than 0.5 all the time indicating that the electrotravels among all the lattice sites, and delocalizathappens. FromFig. 2(c) and (d), we can conclude ththe electron has a great opportunity to stay in thetial site wherePaL (PbL) form peaks, even the peakare small.

Now we consider the case of a dc–ac electric fiE(t) = E0 + E1 cos(ωt). Under the action of the dcac field, the electron could stay in the initial well fever. This is very interesting because the electronbe completely restricted to the original site. We csee this case fromFig. 3.

Fig. 3 showsPaL (PbL) as a function ofedE1/ω

with different magnitude of the dc field, and other prameters are(∆b − ∆a)/ω = 1.25, µ/ed = 0.12 and(Ra +Rb)/ω = 0.24. The solid line and the dotted lincorrespond to, respectively,PaL andPbL for both pan-els. Generally speaking,PaL is larger thanPbL. Butwith the increasing of the magnitude of the ac fiePbL may exceedPaL due to the interaction of the twbands. When the amplitude of the ac field is small,can see there is a zone wherePL is very near 1, indi-cating that the electron is restricted to the initial weFollowing the plateau are some small peaks, at whthe electron can escape from the initial site and oslate around the initial site. This can be seen fromfollowing observation of the time evolution ofP(t).We find the stronger the dc field is, the better the cof the localization is (we have studied the case wedE0/ω < 1, i.e.,edE0/ω is not an integer. We findwhen the strength of the dc field is properly modlated, the electron can also be localized), for the mnitude of the dc field decreases, the phenomenolocalization vanishes quickly, as shown in the uppanel ofFig. 3. For a weak dc field, the electron cancompletely confined only whenedE1/ω is very small.

For the sake of clarity, we next take a look at ttime evolution ofP(t). Then we select a localizatio

Fig. 3. The LAOPPaL (PbL) as a function ofedE1/ω withedE0/ω = 4.0 in the upper panel andedE0/ω = 10.0 in thelower panel, where the other parameters are(∆b − ∆a)/ω = 1.25,µ/ed = 0.12 and(Ra + Rb)/ω = 0.24. The solid and dotted linecorrespond toPaL andPbL, respectively.

state in the lower panel ofFig. 3 and examine thedynamical evolution of the electron, as the lineα il-lustrates inFig. 4(a). As we can see,P(t) oscillatesaround 0.999, meaning that the electron stays ininitial site and the localization happens perfectly. Tline β in Fig. 4(a) corresponds to the case of a dlocalized stateedE1/ω = 6.97 wherePaL and PbL

form valley.P(t) decreases rapidly, implying that thelectron escapes from the initially occupied lattice amoves to other sites.

Fig. 4(b) shows the time evolution ofP(t) withedE1/ω = 4.8, which corresponds to a smooth pein the lower panel ofFig. 3. P(t) decreases slowlyCompared withFig. 4(a), one can see the electronlocalized on the initial site in an earlier length of timand it could travel to other sites in the late time.fact, the electron can repeatedly return to the inisite at the peaks, but need a longer time due to thefluence of the dc field. However, for the time is ordeof magnitude higher than the driving period, this bhavior can be observed in the experiment. Fromabove discussed cases, one can know that the dyncal localization of the electron can be judged perfecby the peaks ofPL.

Today, it is possible for us to fabricate differekinds of high-quality semiconductor devices with trapid progress of technology. Variations in compotion of heterostructures are used to control the m

260 X.-K. Peng et al. / Physics Letters A 347 (2005) 255–261

(a) (b)

Fig. 4. The time evolution of the occupationP(t). The parameters are those of the lower panel ofFig. 3. (a) The lineα corresponds toedE1/ω = 1.0, which is a localized point, while the lineβ corresponds toedE1/ω = 6.97, which is a delocalized point. (b)edE1/ω = 4.8,which corresponds to a small peak.

to

ing.nd

D)ca-ng-

u-

d

el oly

,of-

udey of

theds.

Fig. 5. The LAOPPaL (PbL) as a function ofedE1/ω with differ-ent µ/ed . In the upper panelµ/ed = 0.01, whileµ/ed = 0.05 inthe lower panel. In both cases,edE0/ω = 10,(∆b − ∆a)/ω = 1.25and(Ra + Rb)/ω = 0.24. The solid and dotted lines correspondPaL andPbL, respectively.

tion of electrons and holes through band engineerThe techniques of molecular-beam epitaxy (MBE) ametal-organic chemical vapor deposition (MOVCmay produce the best material for electronic applitions. So we give a brief numerical analysis of chaing the internal parameters of the system.

Taking the dipole moment of the interband copling, for example, we drawPaL andPbL as functionsof edE1/ω with µ/ed = 0.05 in the lower panel anµ/ed = 0.01 in the upper panel inFig. 5, where theother parameters are the same as the lower panFig. 3. The solid line and the dotted line respective

f

Fig. 6. Graph of the LAOPPaL (PbL), as functions of(∆b − ∆a)/ω, where the parameters areedE0/ω = 10,edE1/ω = 1.5, µ/ed = 0.08 and(Ra + Rb)/ω = 0.24. The solidand dotted lines correspond toPaL andPbL, respectively.

correspond toPaL andPbL for both panels. Obviouslythere is still a localization zone in a small rangeedE1/ω. Compared withFig. 3, one can find the probability of the electron moving to bandb on the initialsite increases with the enlargement of the magnitof µ/ed , and decreases otherwise, but the propertdynamical localization does not change at all.

We are also interested in the case of changingdifference of the site energy between the two banThen we drawPaL and PbL as functions of(∆b −∆a)/ω in Fig. 6, with edE0/ω = 10, edE1/ω = 1.5,µ/ed = 0.08 and(Ra +Rb)/ω = 0.24. Also, the solid

X.-K. Peng et al. / Physics Letters A 347 (2005) 255–261 261

tnelsvi-

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singks.af-

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nd

alantof

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.

Fig. 7. The time evolution of the occupation probabilityP(t) with(∆b −∆a)/ω = 1.176. The other parameters are the same asFig. 6.

line and the dotted line correspond toPaL andPbL,respectively. We findPL is always around 1 excep(∆b − ∆a)/ω = 4.735,5.781. . . , hence the electrostill maintains completely localized and never travto other wells. We can notice that there are two obous resonant peaks as plotted inFig. 6. Interestingly,we can see fromFig. 7 thatPa(t) andPb(t) oscillatebetween 1 and 0 alternately andP(t) remains 1 at theresonant peak, which reveals that the electron exhthe behavior of Rabi oscillations on the initial site.

In addition, it is worth to mention thatPaL andPbL

periodically show some sharp peaks as the increaof (∆b − ∆a)/ω, though there are no resonant peaIn a word, the behavior of the electron is greatlyfected by changing(∆b − ∆a)/ω.

4. Conclusions

In summary, we have quantitatively studied tproblem of the dynamical localization of the electrmoving in the two-band system driven by an ac adc–ac fields with the method of LAOP. We find thelectron can return repeatedly to the initially occup

lattice or completely stay in the initial site under apropriate conditions, indicating that one can confithe electron in a well by manipulating the field asystem parameters.

Acknowledgements

This work is supported in part by the NationNatural Science Foundations of China under GrNo. 10274007 and a grant of the China AcademyEngineering and Physics.

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