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Exciton states in quantum dot solids: excitation transfer and dynamic decorrelation Garnett W. Bryant* Atomic Physics Division, National Institute of Standards and Technology, 100 Bureau Dr., Stop 8423, Gaithersburg, MD 20899-8423, USA

Abstract Coherent exciton transport in quantum dot solids is determined by electron and hole interdot tunneling and dipole– dipole interdot excitation transfer. We present a tight-binding theory of coupled dots to understand the interdot coupling and hybridization of states in dot solids. Results show that signiﬁcant coupling is possible. Exciton dynamics in a dot solid is simulated by use of a Hubbard model that includes interdot carrier tunneling, electron–hole attraction and Forster exciton excitation transfer. Dynamic exciton decorrelation is driven by carrier tunneling. Dynamic exciton dephasing is driven by excitation transfer. Published by Elsevier Science B.V. Keywords: Electronic structure; Excitons; Nanocrystals; Quantum computing; Quantum dots

1. Introduction Ensembles of quantum dots (QD) and nanocrystals ordered in one, two, and three dimensions are being studied intensively [1–5]. Interdot exciton transfer mediated by dipole–dipole interaction, lasing, enhanced nonlinear optical effects, controlled tuning of radiative lifetimes and metal– insulator transitions have been observed. As fabrication processes improve, control of dot size and lattice ordering should be enhanced, allowing dot solids to support coherent transport across the entire ensemble. This would make QD solids ideal for use as nanoarchitectures for nanoscale computing and quantum information processing. Quantum information processing requires qubits that can be manipulated and entangled. In QD *Tel.: +1-301-975-2595; fax: +1-301-990-1350. E-mail address: [email protected] (G.W. Bryant). 0921-4526/02/$ - see front matter Published by Elsevier Science B.V. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 4 5 7 - 0

solids, excitons could serve as the qubits, qubit manipulation and entanglement could be achieved by exciton–exciton interaction, and the dot array would provide the ensemble of qubits needed for useful quantum computing. Coherent exciton dynamics in QD solids should be determined by interdot carrier tunneling and dipole–dipole interdot excitation transfer. We present a tight-binding theory of coupled dots to understand the coupling and hybridization of states in dot solids. Results show that signiﬁcant coupling is possible, depending on the spatial distribution and symmetry of the states of the dots, indicating that carrier tunneling could be important in exciton dynamics. We simulate exciton dynamics in a dot solid by use of a Hubbard model that includes interdot carrier tunneling, electron–hole attraction and exciton excitation transfer to identify the effects of each on exciton dynamics in QD solids.

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2. Theory Electron and hole states in coupled QDs are calculated by use of the empirical tight-binding (ETB) method [6–10]. The ETB approach is an atomistic approach well suited for calculations of electronic states of coupled QDs with atomic-scale interfaces and variations in composition and shape. In this paper, we consider dot molecules with two coupled dots. We assume that atoms in the individual dots occupy the sites of an FCC lattice. We can model spherical, hemispherical, tetrahedral or pyramidal QDs. Here, we study, as one example, coupled spherical nanocrystals. We assume that the dots are connected epitaxially at a common interface with the same FCC lattice in each dot. This assumption greatly simpliﬁes the model for these initial calculations. In our ETB model, each atom is described by its outer valence s orbital, 3 outer p orbitals and a ﬁctitious excited s* orbital, included to mimic the effects of higher lying states [11]. We include on-site and nearestneighbor coupling between orbitals. Spin–orbit coupling has also been included in the theory but will not be considered for the results presented here. This simpliﬁcation allows us to focus on how the spatial distribution and atomic orbital of dot states determines interdot coupling and hybridization. We explicitly exclude surface states in our calculations by passivating surface dangling bonds. As an example, we consider dot molecules made from CdS/ZnS core/shell nanocrystals. Empirical tight-binding parameters are taken from Lippens and Lannoo [8]. The dynamics of excitons in QD solids is studied by use of a Hubbard model for excitons [12]. The model for a square array of dots is shown schematically in Fig. 1. One electron (e) level and one hole (h) level describe each QD in the array. This gross simpliﬁcation for the electronic levels of a QD allows us to focus on the dynamical effects of interdot carrier tunneling and excitation transfer. Any size, dimension or array geometry can be considered. The key parameters of the model are: Vc ; and Vc;nn the electron–hole intradot and nearest-neighbor interdot attraction; Te and Th ; the electron and hole interdot hopping; and Vf ; the long-range dipole–dipole Forster excitation

Fig. 1. Schematic for the Hubbard model used to describe exciton dynamics in a quantum dot solid. Model parameters are discussed in the text.

transfer [3], here restricted to be exciton hopping between nearest neighbors. Exciton dynamics is studied by simulating the time evolution of exciton states that either are localized initially at a chosen dot or are initially distributed throughout the array in the ground state for an exciton with negligible carrier tunneling. The exciton oscillator strength is monitored in the time evolution. In this model, we assume that the electron–hole pair has an intradot transition dipole, normalized to give unit oscillator strength for an electron–hole pair localized to a single dot. The total exciton oscillator strength, which is the coherent sum of its intradot dipole moments on each dot, can be greater or less than one depending on whether the intradot moments contribute in phase or not.

3. Results We ﬁrst discuss interdot carrier tunneling by considering dot molecules. For a single dot, the symmetry of electron levels is determined by the symmetry of the spatial envelope function because

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the electron levels are made from atomic s orbitals. For spherical dots, the electron envelope functions are 1S, 1P, 1D, etc. In the dot molecule, electron states form symmetric/antisymmetric pairs with approximately symmetric splitting about the uncoupled states. The coupling lifts the degeneracy of P and D states. There is a close analogy between the hybridization of electron states in dot molecules and molecular hybridization because the symmetry of the coupled electron states is determined by the symmetry on the spatial envelope functions. Px states lying along the molecular axis strongly couple to form s bonds and Py;z states weakly couple to form p bonds. The strength of the interdot coupling, as illustrated by the level splitting DE between symmetric/antisymmetric pairs of states, is shown in Fig. 2 for dot molecules made from two CdS/ZnS core/shell nanocrystals. The CdS radius is R ¼ 4a; where a is the CdS lattice constant. Results are shown as a function of

Fig. 2. Symmetric/antisymmetric level splitting DE for states in dot molecules made from CdS/ZnS core/shell nanocrystals. Results are shown as a function of ZnS shell thickness and center-to-center separation S:

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the ZnS shell thickness and the dot-to-dot centerto-center separation S: The dots are just touching and the coupling is through just a few atoms when S ¼ 2ðR þ TÞ where T is the shell thickness. When S is smaller, the dots connect through a common interface, DE can be substantial (10–90 meV) and DE depends strongly on the thickness of the barrier provided by the two shells but only weakly on the width of the common interface. The spatial distribution of the state is a critical factor in determining DE: S and Px states, with a large distribution along the molecular axis, have larger DE: The analogy with molecular hybridization is more complex for hole states because the symmetry of hole states is determined by both the envelope function symmetry and the atomic orbital symmetry. Nonetheless, large DE are possible for hole states. Thus, interdot carrier tunneling can be signiﬁcant for both electrons and holes. Detailed results will be presented elsewhere. In this paper, we discuss the dynamics of excitons in 1D arrays. Results for 2D arrays are similar. The time evolution for an exciton initially localized at site 12 in a 1D array with 24 dots is shown in Fig. 3. This array is long enough that end effects are minimized for short times. The largest energy scale typically will be the Coulomb energy Vc ; so we scale all other parameters and time t by this energy. Time markers for 1 ps are shown on the ﬁgure for Vc ¼ 10 and 100 meV. For comparison, expected coherence times can be on the order of 50 ps [13]. The time period in Fig. 3 is reasonable for coherent time evolution. The dependence on electron tunneling Te is shown. The oscillator strength is 1 at t ¼ 0; but drops rapidly when the time evolution begins and the electron hops away from the hole. The initial decrease in the oscillator strength is greater for larger Te : The most rapid oscillations are due to beating between components of the exciton state with the electron/hole pair correlated and uncorrelated. At longer times (Vc t=2pX20), the oscillator strength decreases almost to 0 and then is greater than 1. This variation results from the spreading of the localized exciton due to the excitation transfer which mixes states with the exciton localized at other dots and with different phases. At the

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short time scales and dephasing from excitation transfer at longer times.

4. Conclusions

Fig. 3. Time evolution of the oscillator strength of an exciton initially localized at dot 12 in a 24 dot chain. The dependence on electron tunneling Te is shown. The parameters used are Th ¼ 0; Vc;nn ¼ 0:1Vc and Vf ¼ 0:04Vc : Vertical dotted lines are 1 ps time markers for Vc ¼ 10 and 100 meV.

longest times, the time dependence becomes increasingly complicated as the exciton spreads out to and is scattered back from the array edge. Dephasing due to excitation transfer takes longer to develop for larger Te ; because excitation transfer is inhibited when the electron hops away from the hole. The time evolution is qualitatively similar, except for the change in the overall time scale, for all Te provided that the hopping bandwidth 4 Te is less than Vc : In this case, the electron–hole pair retains its excitonic character. For larger Te ; strong mixing of correlated and uncorrelated states occurs and the pair loses its excitonic character. Calculations for different Vf and for different chain lengths conﬁrm that the time scale for the dephasing at longer times depends on the rate of excitation transfer while the rapid initial decrease in oscillator strength depends on the decorrelation from carrier tunneling. Excitons initially distributed globally throughout the array also show rapid decorrelation on

Coherent exciton dynamics in quantum dot solids is determined by electron–hole pair attraction, electron and hole interdot tunneling and interdot excitation transfer due to exciton dipole– dipole interactions. Tight-binding calculations show that interdot tunneling can be signiﬁcant for electon and hole states that have signiﬁcant amplitude at interdot connections. Interdot tunneling will be important for exciton dynamics involving these states. Simulations of exciton dynamics show that an exciton localized to a single dot in a dot array can rapidly decorrelate. On longer time scales, the exciton dephases due to the excitation transfer. Both effects could be critical for determining how to use excitons in quantum dot arrays as qubits for quantum computation. Dynamic decorrelation will reduce the exciton character of the electron–hole states and blur the assignment of qubits to localized excitons. Dynamic dephasing due to excitation transfer will induce additional phase change of the qubits that will have to be accounted for in any coherent manipulation and entanglement of these excitons.

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