Semiconductor Nano RingsSemiconductorNanorings

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Eur. Phys. J. B 19, 599606 (2001) THE EUROPEANPHYSICALJOURNAL B

cEDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2001

Coherent dynamics of magnetoexcitons in semiconductornanorings

K. Maschke1,a, T. Meier2, P. Thomas2, and S.W. Koch2

1 Institut de Physique Appliquee, Ecole Polytechnique Federale, 1015 Lausanne, Switzerland2 Department of Physics and Material Sciences Center, Philipps University, Renthof 5, 35032 Marburg, Germany

Received 27 September 2000

Abstract. The coherent dynamics of magnetoexcitons in semiconductor nanorings following pulsed opticalexcitation is studied. The calculated temporal evolution of the excitonic dipole moment may be understood

as a superposition of the relative motion of electrons and holes and a global circular motion associatedwith the magnetic-field splitting of these states. This dynamics of the electron-hole pairs can be generatedeither by local optical excitation of an ordered ring or, alternatively, by homogeneous excitation of ringswith broken rotational symmetry due to disorder or band tilting.

PACS. 71.35.Cc Intrinsic properties of excitons; optical absorption spectra 71.70.Ej Spin-orbit coupling,Zeeman and Stark splitting, Jahn-Teller effect 78.66.-w Optical properties of specific thin films 42.50.MdOptical transient phenomena: quantum beats, photon echo, free-induction decay, dephasings and revivals,optical nutation, and self-induced transparency

1 Introduction

Most recently it became possible to fabricate ring-like

semiconductor nanostructures and to measure some oftheir optical properties [13]. Theoretical studies of ex-citons and biexcitons in semiconductor nanorings pene-trated by a magnetic field have predicted Aharonov-Bohmoscillations [4] in the respective binding energies and oscil-lator strengths [57]. It has been found that for magneticfluxes= (n+1/2)0, wherenis an integer and 0is theflux quantum 0 = hc/e both, binding energy and oscil-lator strength pass through a maximum for excitons, andthrough a minimum for biexcitons. They can in principlebe studied using conventional linear and nonlinear opticalexperiments.

In general, a magnetic flux will lead to a circular mo-

tion of both electrons and holes. For optical excitationclose to the gap, electrons and holes move in opposite di-rection, at least in a situation where the Coulomb interac-tion can be neglected. The corresponding partial electronor hole currents are largest for fluxes = (n+ 1/4)0 or= (n+ 3/4)0. For spatially homogeneous optical exci-tation, the photo-generated carriers follow the well knownpersistent current scenario [8]. In this situation, the opti-cal excitation would just lead to a transient change of themagnetic moment of the ring.

In the present work we concentrate on the underlyingelectron-hole dynamics, which can in principle be revealedby local optical pulse excitation. We show that the latter

will give rise to a rotating electric dipole moment and,a e-mail: [email protected]

accordingly, to a circularly polarized radiation in the GHzto THz frequency range that is determined by the effectivemasses of the electrons and holes, the diameter of the ring

and the magnetic field strength. The circular motion ofthe excitonic dipole moment is found to survive in thepresence of the Coulomb interaction, provided that therings are sufficiently small such that their circumferenceremains of the order of the extension of the relative motionof the correlated electron-hole pair.

We study the dynamics of the dipole moment afteroptical excitation in the excitonic energy range of a semi-conductor nanoring. The model system and the numericalapproach are presented in Sections 2 and 3. Results forthe noninteracting electron-hole system are given in Sec-tion 4 in order to introduce the fundamental physics andnotions used in this report. The case of excitonic excita-tions is discussed in Section 5. In both Sections 4 and 5,we consider local excitation to generate the initial wavepackets. This restriction is lifted in Section 6, where wetreat the case of homogeneous optical excitation over theentire ring. In this case, the symmetry breaking necessaryto form spatially localised wave packets required to re-veal the dynamical behaviour, is obtained by introducingspatially inhomogeneous one-particle potentials within thering. Both, tilting of the bands or energetic site disorderare considered.

It is shown that the dynamical behaviour of the photo-generated excitonic dipole can be described as a super-position of the intra-excitonic wave packet dynamics [9],which is characterized by frequencies in the THz-range,and the dynamics resulting from the magnetic-field split-ting of one of the contributing excitonic relative-motion


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600 The European Physical Journal B

states, corresponding to frequencies in the GHz- to THz-range. This excitonic doublet corresponds to an opticallyforbidden dipole transition in a totally homogeneous sys-tem that becomes, however, allowed when the rotational

symmetry is broken either by inhomogeneous excitationor by single-particle potentials inherent to the system it-self. The splitting of this state, which is twofold degen-erate in the absence of the magnetic field, is periodic inthe magnetic flux and vanishes whenever is a multipleof half the flux quantum 0. This effect will be termed asquantum-confined Zeeman effect. Possible experimentalrealizations are discussed in the concluding Section 7.

2 The model system

In order to reduce the numerical effort, we restrict ourcalculations to a two-band model with spinless particles.At first sight, this may seem to be a rather severe approx-imation. It is, however, directly suitable for the descrip-tion of some interesting physical situations. In real nanos-tructures fabricated from III-V-material, the valence-bandstructure close to the center of the Brillouin zone, andthus the optical resonances near the fundamental edge,consist of energetically well separated levels related tothe heavy hole (hh), the light hole (lh) and the split-off(so) valence band. We may then study the situation nearthe hh-transitions. The latter are still twofold degener-ate, but choosing circularly polarized light pulses, only onespecies of transitions is excited due to the relevant selec-tion rules [10]. This situation can be adequately described

within a two-band model. Assuming spinless particles, wedisregard spin-flip processes, which would essentially leadto a depopulation of the excited levels and thus to a damp-ing of the signals. In nanostructures, these processes are,however, expected to be slow in comparison with the timescale relevant for the dynamics considered here [11], andthus they may safely be neglected.

We consider a ring consisting ofNsites equally spacedwith nearest neighbour separationd and with radius R =Nd/(2). The sites at positions n carry two states eachwith energiescn and

vn, which give the diagonal elements

Tcnn and Tvnn of the Hamiltonian matrix T in real-space

representation. The nondiagonal matrix elements of the

Hamiltonian define the coupling of the local basis statesto its neighbours. In the presence of a magnetic field andfor nearest-neighbour interactions the nonvanishing ma-trix elements are given by

Tcn,n+1= Jcexp[i2(/N)/0] ,

Tcn+1,n = Jcexp[i2(/N)/0] (1)for the conduction band. Accordingly, the holes in the va-lence band are described by

Tvn,n+1= Jvexp[i2(/N)/0] ,Tvn+1,n = Jvexp[i2(/N)/0] . (2)

Compared to the case without magnetic field, the tight-binding coupling parameters Jc and Jv are thus multi-

plied by a phase factor depending on the magnetic flux through the ring [7,12,13].0= hc/eis the flux quantum.

In the absence of Coulomb interaction the systemHamiltonian reads

H0=nm

Tcnmc+n cm+nm

Tvnmd+n dm. (3)

Here,c+n (cm) creates (destroys) an electron at site n (m)in the conduction band and d+n (dm) creates (destroys) ahole at siten (m) in the valence band. The eigenvalues ofthis Hamiltonian form two bands (E(qi), =c, v)

E(qi) = 2Jcos(qid) (4)defined for the wave vectors qi, i= 1...N, where

qi= ki+ 2

Nd0, (5)

and whereki are the wave vectors without magnetic field

ki = d

+2

d

i

N (6)

The effective masses at the extrema of the valence andconduction band are then given by

m=

2

2Jd2 , (7)

or equivalently

mR

2J

N2 =

2

2(2)2 = const. (8)

The physically relevant parameters are the effectivemassesmand the radius R of the ring. The parametersJandNare chosen in such a way that i) the dispersionsclose to the band extrema are representative for realis-tic semiconductor nanostructures, and ii) that the numer-ics remains manageable. The calculations discussed belowwere performed for a ring with radiusR = 11 nm contain-ing N= 20 sites, which corresponds to a site separationd= 34A. The chosen coupling parameters Jc = 50 meVand Jv = 10 meV correspond to effective masses close toGaAs values mc = 0.064m0 and mv = 0.32m0, where m0

is the free electron mass. In Section 4 we will also usevalues Jc = Jv = 50 meV for tutorial reasons. A mag-netic flux of0/4 through the ring would require a mag-netic field of about 12 T. As shown in the next section,the relevant time scale for the description of the electron-hole dynamics is of the order of 1 to 100 ps, which iswell within the coherent excitonic regime of a semicon-ductor nanostructure. Larger rings would require smallermagnetic fields, but the relevant time scales also becomelarger, as is shown in Section 5, so that phase-breakinginteractions will eventually destroy the coherent dynam-ics one intents to observe. We therefore believe that thechosen parameters are a good compromise.

Perturbation potentials destroying the rotational sym-metry of the ring can easily be introduced in our descrip-tion by taking site dependent electron and hole energies


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K. Maschke et al.: Coherent dynamics of magnetoexcitons in semiconductor nanorings 601

cn and vn. Disorder in the conduction band is modeled

by choosing cn randomly from a Gaussian distributionfunction exp((cn)2/22c ), where c characterizes the en-ergetic scale of the disorder. Eventually, we will consider

also a tilting of the conduction band described bycn= tsin[2(n 1)/N], (9)

which corresponds to an additional external potentialvarying linearly with the y -coordinate in the plane of thering, given site n= 1 lies on the positive x-axis. We willfurther assume that the disorder potentials for electronsand holes or the respective tilting potentials are corre-lated, as is often the case in semiconductor nanostructures.We takevn= (Jv/Jc)

cn= (mc/mv)

cn,i.e., we weight the

potential perturbations inversely with the masses.The Coulomb interaction is included using the

monopole-monopole form

HC = 1

2

nm

(c+n cn d+n dn)Vnm(c+mcm d+mdm), (10)

where

Vnm= U0d

Nd

sin( N

|n m|) +a0 (11)

In principle, HC includes the repulsion between electronsand between holes, as well as the electron-hole attraction.In the low-excitation regime considered here, only the lat-ter is relevant. The Coulomb-interaction potential, equa-tion (11), is regularized to avoid the divergence of the

exciton binding energy in one dimension. The parametersU0and a0characterize the strength of the interaction andits spatial variation, respectively. The sin(x) function inthe denominator of equation (11) is needed to describeproperly the spatial distance between sites n and m onthe ring.

The electron-hole dynamics is of central importance forthe here discussed physical phenomena. It is thus essen-tial to treat the relative motion of the Coulomb-correlatedelectron-hole pair in a proper way. This requirement is sat-isfied by our present model. It describes correctly the ef-fects of the electron-hole correlation on the optical absorp-tion spectrum where it yields characteristic resonances

near the absorption edges, which in a bulk situation wouldbe termed excitonic bound states (see also [15]). It isthus able to account for the reduced relative motion of theelectron-hole pair on the length scale of the Bohr radiusand of the diameter of the nanoring. The present modelhas previously been used to study the center-of-mass andthe relative motion of a Coulomb-correlated electron-holepair in a disordered one-dimensional semiconductor [16].

The coupling of the electronic system to a classicaloptical electric field is given by

HI= E(t)P = E(t)n

(ndncn+nc+n d

+n ), (12)

whereP is the total optical interband polarization, whichis obtained by summing over all microscopic polarizations

ndncn, and n are the on-site dipole matrix elementsallowing the optical transitions between valence and con-duction bands. Homogeneous excitation is obtained withn= , inhomogeneous excitation at site 1 corresponds to

the choicen= 1n. The electric field pulse is chosen as

E(t) =E01

tLexp(iLt)exp

t

2

t2L

(13)

The time-evolution of the system is then determined bythe total Hamiltonian

H=H0+HC+HI. (14)

3 The numerical approach

The time evolution of the interband coherenceplk =

dlck

is obtained from the Heisenberg equation together withthe total Hamiltonian H. In this work we adopt linearresponse theory, i.e., we solve the system of differentialequations

itplk =n

Tcknplnm

Tvmlpmk+Vlkplk+E(t)l lk.

(15)

The electron and hole densitiesnclland nvll can be obtained

from the conservation laws [14] valid in the linear responseregime

nc

ll =k

pklp

kl (16)

nvll =k

plkplk.

The electron and hole dipole moments dc and dv definedwith respect to the center of the ring are given by

dx = el

nllR cos l (17)

dy = el

nllR sin l,

wherel = 2(l1)/Ndefines the angular position of sitel. The signs depend on the charge of the particle in band, negative signs correspond to electrons, positive signs toholes. The total electron-hole dipole moment d is

d= dc + dv. (18)

The optical 2-spectra are calculated by Fourier trans-forming the optical interband polarization plk(t) corre-sponding to a short pulse (tL = 0.1 ps) excitation withcentral frequency L close to the lowest transition. Thespectra are calculated assuming a polarization decay timeT2 = 50 ps, while we use T2 for the calculation ofthe coherent dynamics.

The dynamical motion of the dipole may be consideredas a source of electromagnetic radiation. The associated


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/d 0 /dwave vector

energy

Fig. 1. Band structure of a ring composed from N= 6 sitesfor equal electron and hole masses. The vertical dashed linesindicate direct transitions at wave vectorskm, the vertical solidlines those at wave vectors qm= km+

1

4

2Nd

shifted by the mag-netic flux = 0/4. Dotted lines indicate the lowest indirecttransitions for inhomogeneous excitation.

electric field is obtained from the second time derivativeof the dipole moment, which constitutes the source termin Maxwells equations. The corresponding polarizationangle is determined by

= arctand2dy/dt

2

d2dx/dt2 (19)

4 Local excitation of the noninteractingsystem

In order to elucidate the underlying basic physical mech-anism, we consider first the case U0 = 0 a nd Jc =Jv = 50 meV. Figure 1 shows the valence and conduc-tion bands as well as the relevant optical transitions forN = 6. The wave vectors km for the magnetic-field-freecase = 0 are denoted by vertical dashed lines, those for= 0/4 by solid lines. For = 0 there are two twofolddegenerate optical dipole transitions (at k =/3d andk= 2/3d), and two nondegenerate transitions (atk = 0and k = /d). For = 0/4 the degeneracies are lifted

and the spectrum consists of 6 lines. It can easily be seenthat one would obtain three twofold degenerate transitionsfor = 0/2.

The optical2-spectrum, which corresponds to the lin-ear absorption spectrum, is shown in Figure 2 for N= 20.The lines pointing downward show the 11 transitions for = 0, the upward pointing lines show the 20 nondegen-erate transitions for = 0/4. Note that the amplitudesare determined by the degeneracies of the correspondingtransitions.

We concentrate on the transitions close to the funda-mental gap. As can be seen in Figure 2, the two lowesttransitions in the spectrum for = 0/4 are strongly de-pendent on the magnetic flux. They merge for

0/2.

In the following we assume inhomogeneous excitation,where the optical excitation takes place at a single site

0 100 200 300 400energy/ meV

0

absorption(arb.units)

Fig. 2. Optical spectrum due to direct transitions for a ho-mogeneous ring withN = 20 sites, equal electron and holemasses, Jc = Jv = 50 meV, and without Coulomb interac-tion,U0 = 0. Pointing downwards: no magnetic flux. Pointingupwards: = 0/4.

0 1 2 3 4energy/ meV

absorption(a

rb.units)

(a)

0 1 2 3 4energy/ meV

abso

rption(arb.units)

(b)

Fig. 3. Optical spectrum for the homogeneous ring withN = 20 close to the fundamental gap, no Coulomb interac-tion, U0 = 0, and local excitation at site n = 1. The mag-netic flux is = 0/4. (a) Equal electron and hole massesJc = Jv = 50 meV. The higher energy peak is due tothe twofold degenerate indirect transitions indicated by dot-ted lines in Figure 1. (b) Different electron and hole massesJc= 50 meV, Jv = 10 meV. The degeneracy is lifted.

only (n= 1, on the positive x-axis). In this situation ad-ditional transitions become allowed which are not shownin Figure 2. In order to resolve the spectrum for this situ-ation, we present an enlarged view of the low-energy tran-sitions in Figures 3 and 4. Here and in the following theenergy zero corresponds to the lowest transition energy ofthe perfect noninteracting system for = 0. In the spec-trum for= 0/4, Figure 3a, we find two absorption linesspaced by about 2.5 meV. The lower peak corresponds tothe shifted k = 0 transition, the higher one is associatedwith the first indirect transitions, which are twofold de-generate in the case of equal electron and hole masses(see dotted lines in Fig. 1).

The optical pulse has an average excitation energyL close to the fundamental gap and a temporal width


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K. Maschke et al.: Coherent dynamics of magnetoexcitons in semiconductor nanorings 603

2(arb.units)

(a)

2(arb.units)

(b)

2(arb.units) (c)

-11 -10 -9 -8energy/ meV

2(arb.units)

(d)

Fig. 4. Optical spectrum close to the fundamental gap forthe ring with N = 20 sites for different electron and holemasses, Jc= 50 meV, Jv = 10 meV, with Coulomb interactionU0 = 16.5 meV. (a) Homogeneous ring, local excitation at siten = 1 and magnetic flux = 0/4. (b) Homogeneous ring,homogeneous excitation, magnetic flux = 0. The peak is thelowest excitonic resonance. (c) Tilted bandstructure, t = 0.2meV, homogeneous excitation, magnetic flux = 0. In addi-tion to the lowest excitonic resonance the next higher excitonicstates become optically allowed. (d) Same as c), but with mag-netic flux = 0/4. The second resonance is splitted.

0 10 20time (ps)

-1

0

1

totaldip

olemoment(arb.units) (a)

dy

dx

0 20 40time (ps)

(b)

dy

dx

Fig. 5.Total dipole momentd(t) = (dx(t), dy(t)) as a functionof time. (a) Equal electron and hole masses, cf.Figure 3a. (b)Different masses, cf. Figure 3b.

oftL = 1 ps. The resulting total dipole moment dof thephoto-generated electron-hole pair is shown in Figure 5a,where the x-componentdx(t) and the y -componentdy(t)are shown as a function of time. Electron and hole startat siten = 1 and move into opposite directions with equalvelocities, since their masses are equal. Thusdx(t) remainszero whiledy(t) oscillates. The tracedy(dx) shows the lin-

-1 0 11

0

1

(a)

-1 0 1-1

0

1

(b)

-1 0 11

0

1

(c)

-1 0 1-1

0

1

(d)

-1 0 11

0

1

(e)

-1 0 1-1

0

1

(f)

-1 0 11

0

1

(g)

-1 0 1-1

0

1

(h)

Fig. 6. Trace of the total dipole moment d(t) = (dx(t), dy(t))(a g) and of the second time derivative of the total dipole mo-ment (h) for = 0/4. Vertical axes: y-direction, horizontalaxes: x-direction. (a) Equal electron and hole masses, inho-mogeneous excitation at site n = 1 of the homogeneous ring,U0 = 0, cf. Figure 3a. The polarization is linear. (b) Differentmasses, inhomogeneous excitation at siten = 1 of the homoge-neous ring,U0 = 0,cf.Figure 3b. The polarization is circular.(c) Inhomogeneous excitation at site n = 1 of the homoge-neous ring, U0 = 16.5 meV, cf. Figure 4a. (d) Same as (c),but homogeneous excitation of a ring with tilted band struc-

ture, t = 0.1 meV, cf . Figure 4d. Note the qualitative simi-larity with Figure 6c. (e) Homogeneous excitation of the ringwith tilted bandstructure and disorder, c = t = 0.1 meV,U0 = 16.5 meV. (f) Homogeneous excitation of the ring withdisorder and without band tilting, c = 0.1 meV, t = 0,U0 = 16.5 meV. (g) Same as (f), but without magnetic flux,= 0. (h) Second time derivative of the total dipole momentfor the situation of Figure 6d.

ear polarization of this motion (see Fig. 6a). The oscil-lation frequency of dy(t) is determined by the energeticseparation of the involved transitions. For small energeticpulse widths like in our calculation only the two lowesttransitions contribute and the motion is purely sinusoidal.

In realistic semiconductors electron and hole masseswill be different. In this case the situation may change


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considerably. Figure 3b shows the spectrum close to thegap for a ring with Jc = 50 meV andJv = 10 meV for ex-citation at siten = 1 and = 0/4. The peak with lowestenergy corresponds again to the direct transition between

the lowest electron state and the highest hole state, whichwould correspond to thek = 0 transition in the absence ofthe magnetic flux. The second peak, however, is now dueto the indirect transition between the same electron stateand the next hole state. Both transitions are nondegen-erate because of the different masses. As for energeticallynarrow pulses only a single electron k-state contributes,the electron density is homogeneously distributed over thering after the light pulse, yielding dc = 0. On the otherhand, the two contributing hole states form a wave packetthat moves around the ring with a velocity given by theenergetic phase factors of the two states involved. Thisleads to a circularly polarized motion of the total dipolemoment d(t) as seen in Figures 5b and 6b, where both

dx(t) and dy(t) as well as the trace dy(dx) are shown.

5 Local excitation of the interacting system

Coulomb interaction between electron and hole is ex-pected to be important for the wave packet dynamics.In the following we assume U0 = 16.5 meV, which corre-sponds to a binding energy of about 10 meV [15].

Magnetic field effects in the low-energy excitations canonly be expected if the corresponding electron-hole pairsare not too tightly bound, or in other words if theexcitonic Bohr radius remains comparable with the ring

circumference. As already mentioned above, small ringsare required in any case, since the coherent dynamics canonly take place on a time scale smaller than the dephasingtime. For given effective masses, the differences in energyleading to the circular motion and thus the time scale ofthe circular dynamics, depend solely on the ring radius Rand increase as 1/R2 with decreasingR. In particular, theenergetic distances do not depend on the number of sitesNnor on the site separation d or the coupling parame-tersJfor fixed masses and radius.

Similar to the previous section, we first consider a ho-mogeneous ring and local excitation at siten= 1. The low-est transitions are shown in Figure 4a. We see a single peak

near 10 meV and a split peak near9.4 meV. Compar-ison with the absorption spectrum for homogeneous exci-tation (Fig. 4bd) shows that this doublet originates froma higher twofold degenerate (for = 0) excitonic boundstate with odd parity, which is optically forbidden for ho-mogeneous excitation. This transition becomes, however,allowed if the symmetry is broken as e.g. under local ex-citation conditions, where the circular symmetry of thesystem is perturbed by the local excitation itself. Alterna-tively, the symmetry may be broken by a perturbing one-particle potential. This is also evident from Figures 4c and4d, which show the absorption spectra for homogeneousexcitation for = 0 and for = 0/4, but where now ad-ditional band-tilting potentialscn = tsin[2(n

1)/N]

and vn = (Jc/Jv)tsin[2(n1)/N] are added to theconduction and valence band site energies, respectively. In

this case, the symmetry is broken by the additional poten-tials, and the optical transition into these states becomesallowed even under homogeneous excitation conditions.

The doublet splitting in Figure 4a is due to the mag-

netic flux, as can also been seen in Figure 4d. This splittingis periodic with period 0/2. This effect could be calledquantum confined Zeeman effect. The period halfing isdue to the fact that the two peaks cross at = 0/2.

Similar to the situation treated in the previous sec-tion, the optical excitation of the doublet structure leadsto electron and hole wave packets moving in opposite di-rections, but now with different velocities. The time scaleof the rotation is determined by the small splitting ofthe doublet. The rotation frequency is thus considerablysmaller than the superimposed linear oscillation which isdue to the contribution of the lowest excitonic transition,and which scales with the energetic distance between thefirst absorption peak and the doublet. This explains therosette-like trace of the total dipole moment shown in Fig-ure 6c. With our parameters, the circular component hasa period of 100 ps, while that of the linear component hasa period of the order of 5 ps.

6 Homogeneous excitationof the inhomogeneous system

6.1 Band tilting

As already mentioned in the previous Section 5, spectraand dynamical behaviour similar to Figures 4a and 6ccan be obtained also under homogeneous optical excita-tion conditions, provided that the rotational symmetry isbroken by an additional one-particle potential. This situ-ation will be investigated in the present section. We con-sider the sinusoidal perturbation potential equation (9),cn = tsin[2(n1)/N] and vn = (Jc/Jv)cn, appliedto the conduction and valence band site energies, respec-tively.

Figures 4b, 4c and 4d show the spectra close to theabsorption edge for = 0 and t = 0, = 0 a n dt = 0.2 meV, as well as for = 0/4 andt= 0.2 meV,

respectively. The dynamics of the total dipole moment isshown in Figure 6d fort= 0.1 meV. It closely resemblesthat of Figure 6c for the perfect ring with single-site exci-tation. The additional stationary dipole moment seen inFigure 6d is caused by the tilting of the valence and con-duction bands. The stationary dipole moment increaseswith the amplitude of the sinusoidal potential, whereasthe amplitude of the dynamic motion decreases. The latterfeature is due to the fact that the excitonic wave functionbecomes more confined for larger amplitudes t and fi-nally localizes in the region where the differences betweenthe site energies are smallest.

The source term in Maxwells equations, which is givenby the second time derivative of the dipole moment, isshown in Figure 6h. It essentially reflects the dynamicalpart of the total dipole moment shown in Figure 6d.


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6. R.A. Romer, M.E. Raikh, Phys. Rev. B 62, 7045 (2000);Phys. Stat. Sol. (b) 221, 535 (2000).

7. T. Meier, P. Thomas, S.W. Koch, Eur. Phys. J. B, sub-mitted.

8. M. Buttiker, Y. Imry, R. Landauer, Phys. Lett. A 96, 365(1983).

9. J. Feldmann, T. Meier, G. von Plessen, M. Koch, E.O.Gobel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer,W. Schafer, H. Nickel, Phys. Rev. Lett. B 70, 3027 (1993);F. Jahnke, M. Koch, T. Meier, J. Feldmann, W. Schafer,P. Thomas, S.W. Koch, E.O. Gobel, H. Nickel, Phys. Rev.B 50, 8114 (1994).

10. C. Sieh, T. Meier, F. Jahnke, A. Knorr, S.W. Koch, P.Brick, M. Hubner, C. Ell, J. Prineas, G. Khitrova, H.M.Gibbs, Phys. Rev. Lett. 82, 3112 (1999); T. Meier, S.W.Koch, M. Phillips, H. Wang, Phys. Rev. B 62, 12605(2000).

11. T. Armand, X. Marie, P. Le Jeune, M. Brousseau, D.Robart, J. Barrau, Phys. Rev. Lett. 78, 1355 (1997).

12. N. Byers, C.N. Yang, Phys. Rev. Lett. 7, 46 (1961).13. F. Bloch, Phys. Rev. B 2, 109 (1970).14. K. Victor, V.M. Axt, A. Stahl, Phys. Rev. B 51, 14164

(1995).

15. The terminology may be somewhat misleading. The notionof binding energy, exciton and Bohr radius shouldbe used with some care. In a ring all states are discrete, i.e.they are all bound states with a finite extension. In addi-tion, the form of the Coulomb interaction matrix elementin a nanoring takes care of the ring geometry, and it differstherefore from that in a one-dimensional chain. Neverthe-less we will use these notions, since similar to the bulksituation the interaction lowers the transition energiesat the absorption edge with respect to the noninteractingcase. In the here adopted terminology, binding energy

just means the energetic difference between the lowest ab-sorption peaks calculated with and without interaction.

16. D. Brinkmann, J.E. Golup, S.W. Koch, P. Thomas, K.Maschke, I. Varga, Eur. Phys. J. B 10, 145 (1999).

17. J. Feldmann, K. Leo, J. Shah, D.A.B. Miller, J.E.Cunningham, T. Meier, G. von Plessen, A. Schulze, P.Thomas, S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992);T. Meier, G. von Plessen, P. Thomas, S.W. Koch, Phys.Rev. Letters 73, 902 (1994); T. Meier, G. von Plessen, P.Thomas, S.W. Koch, Phys. Rev. B 51, 14490 (1995); V.G.Lyssenko, G. Valusis, F. Loser, T. Hasche, K. Leo, M.M.Dignam, K. Kohler, Phys. Rev. Lett. 79, 301 (1997).

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