Physica E 9 (2001) 94–98www.elsevier.nl/locate/physe

Excitonic arti�cial atoms in a quantum dot

Pawel Hawrylak ∗

Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Canada K1A OR6

Abstract

We discuss the electronic properties of arti�cial atoms composed of a �xed number of electrons and holes con�ned to aparabolic quantum dot. The calculated emission spectra show distinct features related to the number of excitons. We identifythe relevant “hidden symmetries” and show how they replace Hund’s rules in two-component quantum dots. ? 2001 ElsevierScience B.V. All rights reserved.

Keywords: Quantum dots; Excitons; Arti�cial atoms

1. Introduction

The magnetic �eld and the con�ning potential turnthe continuous spectrum of a quasi-two-dimensionalelectron into a discrete spectrum of two harmonic os-cillators (see for example [1,2]). The most importantaspect of these spectra is the presence of degenera-cies. In a dot with parabolic con�ning potential inzero magnetic �eld dynamical symmetries are respon-sible for the formation of shells of degenerate stateswith di�erent angular momenta [3]. For free elec-trons in a magnetic �eld, the degeneracies of Lan-dau levels have strong implications for the groundstate of interacting electrons. When only electrons arepresent in a partially �lled, spin polarized lowest Lan-dau level, very complicated and not completely un-derstood ground states of the Fractional Quantum Hall

∗ Corresponding author. Fax: +1-613-990-0202.E-mail address: pawel.hawrylak@nrc.ca (P. Hawrylak).

e�ect exist. By contrast, adding to spin polarized elec-trons an equal number of spin polarized particles withopposite charges, i.e. holes, changes the ground statedrastically. Electrons and holes form a condensate ofmagneto-excitons due to “hidden symmetries” [4–7].This many-body ground state is exactly known for anyfractional �lling. The experimental manifestation ofthis condensate is the emission spectrum which doesnot depend on the �lling fraction.We have exploited these QHE analogies in studying

electrons, and electrons and holes in degenerate shellsof a quantum dot [8–14]. For electrons, exact diago-nalization techniques established “generalized Hund’srules” for quantum dots, in analogy to atoms [12].However, we are not aware of the existence in natureof atoms composed of particles of opposite charges,and hence of a guiding principle, such as Hund’srules, determining their electronic properties. We havesearched for such a principle and we describe here re-sults of our calculations [8–11]. These results show

1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(00)00182 -X

P. Hawrylak / Physica E 9 (2001) 94–98 95

that there are rules, “hidden symmetries”, which gov-ern electronic structure of electron–hole complexes inquantum dots. Some of these symmetries are the onesfamiliar from the QHE physics, but also new sym-metries appear related to spin. These calculations arerelevant to a number of “single dot spectroscopies”[13–17]. In particular, Bayer et al. [13] appear to havedemonstrated experimentally the existence of exci-tonic arti�cial atoms and “hidden symmetries”.The starting point of our theory is the single-particle

spectrum. A number of experiments [10,18–20] andnumerical calculations [21] indicate that the e�ectivecon�ning potential in lens shape self-assembled dotsis parabolic. The electron energies Emn = 0(n+ m+1), eigenstates |mn〉 and angular momenta L= m− nare those of two harmonic oscillators with energy 0[1,2]. Strain allows to treat the valence band hole in thee�ective mass approximation as a positively chargedparticle with angular momentum opposite to the elec-tron. Electronic shells are formed by degenerate states.For example, an s-shell consists of a single |0; 0〉 state,p-shell of two |0; 1〉 and |1; 0〉 states, and a d-shell ofthree states |2; 0〉; |1; 1〉; |0; 2〉.The Hamiltonian of the interacting electron–hole

system may be written in second quantization as

H =∑

iEei c

+i ci +

∑

iEhi h

+i hi

−∑

ijkl〈ij|Veh|kl〉c+i h+j hkcl

+12∑

ijkl〈ij|Vee|kl〉c+i c+j ckcl

+12∑

ijkl〈ij|Vhh|kl〉h+i h+j hkhl; (1)

where a composite index j = [m; n; �] has been used.The operators c+i (ci); h

+i (hi) create (annihilate) the

electron or valence band hole in the state |i〉 withthe single-particle energy Ei. The two-body coulombmatrix elements are 〈ij|V |kl〉 for electron–electron(ee), hole–hole (hh) and electron–hole (eh) scatter-ing, respectively [22,23]. We shall study the caseof symmetric e–e, h–h, and e–h interactions. Theimportant property of these interactions is the equiva-lence between electron–hole scattering and electron–electron exchange. To elucidate the physics, weshall often refer to the matrix elements by spec-ifying which type of carrier, which shell, and

whether direct or exchange scattering is involved. Forexample, V pp;xee denotes electron–electron exchangescattering involving two electrons on a p-shell. Theelectron–hole scattering matrix element, e.g. V ppeh ,is equal to e–e exchange matrix elements V pp;xee , as〈10; 10|Veh|01; 01〉= 〈10; 01|Vee|10; 01〉.The interband optical processes in a quantum dot

are described by the set of interband polarization op-erators (P+� ; P

−� ; Pz) which form an algebra of angu-

lar momentum. P+� creates (annihilate) electron–holepairs P+� =

∑i c+i�h

+i;−� (P

−+ =

∑i hi;−�ci�) by anni-

hilating (creating) photons with de�nite circular polar-ization [8,9]. The Pz = 1

2(Ne� + N

h� − Ntot) measures

population inversion, with Ntot the degenracy of theshell. We now need to construct many electron–holestates. The states which are relevant are those whichcan be related to the interband polarization operator.This process is carried forward by exploring the sym-metries of P+� . The symmetries are associated withcommutation properties of P+ with the Hamiltonian[8,9]. This commutator involves a family of two- andfour-particle operators [24]. However, if we neglectthe spin degrees of freedom and consider only statesof a degenerate shell, the degeneracy of single-particlelevels combined with the symmetry of (ee), (hh), and(eh) interactions cause a remarkable cancellation ofthe four particle contribution and lead to a very sim-ple result: [H; P+] = EXP+, where EX is the excitonenergy for a given shell [24]. The commutation re-lation enables the construction of exact eigenstates|N 〉= (P+)N |v〉 of the Hamiltonian by a multiple ap-plication of P+ to vaccum. The energy of these statesdepends linearly on the number of excitons. Hencethe energy of addition=subtraction of excitons fromthese states does not depend on the number of exci-tons N . This is the essence of “hidden symmetry”, aquantum dot analog of the e�ect known in QHE. Themain di�erence between a quantum dot and a QHEsystem is due to spin. Spin starts playing a role whenwe have more than one carrier of the same type. Thishas been anticipated by constructing another relevantoperator, Q+, creating singlet bi-excitons [8,9] Q+ =12

∑i; j (c

+i↓c

+j↑ + c

+j↓c

+i↑)(h

+i↑h

+j↓ + h

+j↑h

+i↓). Rather sur-

prisingly, this bi-exciton operator was shown to satisfya similar commutation relation as the polarization op-erator, [H;Q+] = EXXQ+, with EXX = 2EX. The ap-plication of Q+ to vacuum generates a coherent stateof singlet bi-excitons, and the energy of this bi-exciton

96 P. Hawrylak / Physica E 9 (2001) 94–98

state is twice the energy of a single exciton. Themultiplicative state (P+)2|v〉 and the bi-exciton stateQ+|v〉 are degenerate. However, exact diagonaliza-tion studies show that any small perturbation low-ers the singlet–singlet state [8,9,11]. At this stage wehave constructed a family of relevant operators forthe construction of exact eigenstates of many elec-trons and holes on degenerate shells. We shall nowdiscuss the role of these operators, the energy of gen-erated states, and the excited states by explicit con-struction of many-exciton states and diagonalizationof the Hamiltonian.We start with a single exciton. The single exciton

has been revisited in Ref. [14]. A transformation toJacobi coordinates combined with the identi�cation ofresonant con�gurations yielded a simple understand-ing of the exciton spectrum. It was predicted that thespectrum of a quantum dot with two shells should con-sist of two peaks but a spectrum of a dot with threeshells should consist of �ve peaks: one derived fromthe s-shell, two from the p-shell, and two from thed-shell. These conclusions were nicely veri�ed in re-cent experiments by Bayer et al. [14].Let us now turn to many-exciton states. The �rst

case where electrons and holes are in a degenerateshell is the three exciton complex. The energy of thethree exciton complex, measured from the energyof the occupied s-shell, is simply given by E3X =Ep − V ppeh , where Ep = 2t − (V sp;xee + V sp;xhh )− V pp;dehis the energy of the electron–hole pair on one ofthe p-orbitals (t is the kinetic energy quantizationfor a pair). This energy is lowered by the electron–hole scattering V ppeh . We now consider the �rst casewhere the role of “hidden symmetries” comes toplay, i.e. in the ground state of the four excitoncomplex. We start with the triplet–triplet con�gu-ration, |t〉= (c+10↓h+10↑)(c+01↓h+01↑)|XX〉, where |XX〉is the fully occupied s-shell state. By inspection,the triplet state is a multiplicative state generatedby P+, i.e. |t〉= (P+)2|XX〉. We have written thisstate explicitly as a product of two electron–holepairs, each with energy Ep. Two electrons and twoholes have parallel spins and lower their respec-tive energy by exchange. The ground state energy,measured from the energy of the �lled s-shell, isE4Xt = [2[Ep]− (V pp;xee + V pp;xhh )]. Because exchangeinteraction and electron–hole scattering interactionsare equal and attractive, V pp;xee = V ppeh , the energy of

the two additional excitons in a p-shell of the fourexciton complex is exactly twice the energy of a sin-gle exciton in a three exciton complex, E4Xt = 2E3X.This is so because the lowering of energy due tomixing of electron–hole con�gurations of the threeexciton complex is exactly equal to the exchangeenergy in a single four exciton con�guration.When both electrons and holes are in singlet con-

�gurations, the total number of possible con�gurationsincreases. While there was only one triplet–tripletcon�guration, there are three possible singlet–singletcon�gurations: |a〉= (c+10↑c+10↓)(h+10↓h+10↑)|XX〉, |b〉=1√2(c+10↑c

+01↓+c

+01↑c

+10↓)

1√2(h+10↓h

+01↑+h

+01↓h

+10↑)|XX〉,

|c〉= (c+01↑c+01↓)(h+01↓h+01↑)|XX〉. Electron–hole scat-tering can move an electron–hole pair from con�gu-ration (b) to either (a) or (c) and mix con�gurations.The �nal lowest eigenvalue and eigenstate is [11]E4Xs(1) = 2Ep − 2V ppeh and |1〉= 1√

3(|a〉+ |b〉+ |c〉).

We see that E4Xs(1) = 2(E3X), i.e. the energy of thelowest singlet–singlet state is twice the energy ofthe single exciton in a p-shell. The singlet–singletand triplet–triplet con�gurations are degenerate. Mor-ever, by direct inspection, the state generated by thebi-exciton operator Q+|v〉 is identical to the singlet–singlet ground state |1〉. Hence, we could have writtenthe ground state of the interacting system explicitlywithout resorting to diagonalization of the Hamilto-nian. This illustrates the power of “hidden symme-tries”.Hidden symmetries tell us that the energy to re-

move an exciton from a partially �lled p-shell doesnot depend on the �lling of the shell. Hence, the emis-sion spectrum of the p-shell does not depend on thepopulation of this shell. In order to distinguish spec-tra corresponding to di�erent numbers of excitons weneed to investigate removal of excitons from a �lleds-shell as a function of the �lling of the p-shell. Theseprocesses leave the �nal state exciton droplet in anexcited state, and hidden symmetry no longer applies.The calculated emission spectra [11] for a typical ratioof the Coulomb to kinetic energy V0=t = 0:5 are shownin Fig. 1. The calculated spectra have been Gaussiallybroadened to simulate experiment in which varyingexcitation power increases number of excitons. The1X and 2X recombination spectrum corresponds toa single recombination line, with the 2X emission atslightly lower energy. Let us now discuss emission

P. Hawrylak / Physica E 9 (2001) 94–98 97

Fig. 1. Emission spectra from N exciton ground states to the(N − 1) exciton ground and excited states.

spectra from the ground state of the three exciton com-plex. The �nal 2X states can be either triplet or singletstates. Therefore, one expects to see one emission linefor emission from the p-shell and two emission linesfrom the s-shell. However, only one s-shell emissionline is visible. This is emission to the triplet bi-excitonstate. The emission energy is almost identical to theemission energy of a single exciton. The emission tothe singlet–singlet �nal state is quenched by the quan-tum interference e�ect in a matrix element [11].The emission from the 4X complex to excited states

of the 3X complex consists of two spectra originat-ing from almost degenerate singlet–singlet and triplet–triplet initial 4X states. The recombination spectraconsist of three groups of states: (a) the recombina-tion from the p-shell which is almost degenerate withthe recombination from the 3X complex, (b) the re-combination to the �nal 3X triplet–triplet state, whichis close in energy to the recombination from a single

exciton, (c) the lower energy band of the 3X excitedstates.The emission from the 5X complex to the singlet–

singlet and triplet–triplet 4X �nal states shows a verystrong emission from the p-shell and almost no emis-sion from the s-shell. The weak emission from thes-shell is caused by a similar interference e�ect to thatfor the 3X complex. The recombination to the groundstate is enhanced and to the excited state, with a miss-ing exciton in the s-shell, is reduced. The �nal result,Fig. 1, shows that in the 5X complex the emissionfrom the s-shell is suppressed in comparison with thelower density quantum dot, i.e. with either 4 or 3 ex-citons, a rather counter intuitive result. The emissionfrom the s-shell is recovered when our dot is com-pletely �lled with 6X. The emission from the s-shellcorresponds now to the removal of the s-shell exciton,without mixing with other con�gurations. The energyof this quasi-exciton is renormalized by an exchangeinteraction of the s-shell electron and a hole with elec-trons and holes in a p-shell. This lowers the energy ofthe emission band by the exchange self-energy of theexciton 2(V sp;xee + V sp;xhh ).In summary, we discussed here the electronic prop-

erties of arti�cial atoms composed of a �xed numberof electrons and holes con�ned to a parabolic quan-tum dot. The emission spectra show strong variationwith the number of excitons in the dot. The variationsare due to �lling di�erent electronic shells and for par-tial �lling, due to the condensation of electrons andholes into coherent states. The condensation can beunderstood in terms of relevant symmetries. We iden-tify the relevant “hidden symmetries” and show thatthey replace Hund’s rules in two-component quantumdots. These calculations serve as a �ngerprint of ex-citonic arti�cial atoms observed in “single dot spec-troscopy” experiments. The identi�cation of “hiddensymmetries” opens up possibility of engineering “de-coherence free” Hilbert spaces for quantum informa-tion processing [24]. First step in this direction as beenmade by vertical coupling of pairs of quantum dotswhere entanglement of states on di�erent dots has beendemonstrated experimentally [25].

Acknowledgements

The author acknowledges collaboration with M.Bayer, A. Forchel, S. Fafard, G. Narvaez, K. Hinzer

98 P. Hawrylak / Physica E 9 (2001) 94–98

A. Wojs, M. Korkusinski, and Z. Wasilewski. Partialsupport from the Alexander von Humboldt Stiftungis acknowledged.

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