Designing quantum dots and quantum-dot solids

Designing quantum dots and quantum-dot solids

Physica E 11 (2001) 72–77 www.elsevier.com/locate/physe Designing quantum dots and quantum-dot solids Garnett W. Bryanta; ∗ , W. Jask(olskib a Atomi...

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Physica E 11 (2001) 72–77

www.elsevier.com/locate/physe

Designing quantum dots and quantum-dot solids Garnett W. Bryanta; ∗ , W. Jask(olskib a Atomic

Physics Division, National Institute of Standards and Technology, Building 221, A267, Gaithersburg, MD 20899-8423, USA b Instytut Fizyki, UMK, Grudzia . dzka 5, Toru0n, Poland

Abstract To design nanocrystal nanosystems, a theory for nanocrystals is needed which can be applied to nanocrystals with atomic-scale variations in composition, structure and shape. We present an empirical tight-binding theory for nanocrystal heteronanostructures. Electronic structure and optical absorption spectra are obtained for CdS=HgS=CdS and ZnS=CdS=ZnS quantum-dot quantum-well barrier=well=barrier nanocrystals. Comparison with experiment shows that tight-binding theory provides a good description of nanosystems with monolayer variations in composition. Well=barrier and well=barrier=well nanocrystal heteronanostructures also are modeled to illustrate state and energy-level engineering in nanocrystal nanosystems. c 2001 Elsevier Science B.V. All rights reserved.  PACS: 71.24.+q; 71.15.Fv; 71.35.Cc; 73.61.Tm Keywords: Quantum dots; Tight binding theory; Heteronanostructures

1. Introduction The development of nanocrystal nanosystems requires uniform, monodisperse nanocrystals with precisely determined geometry, nanocrystals tailored with internal structure for enhanced functionality (quantum-dot quantum-wells), and arrays of close-packed nanocrystals (quantum-dot solids). To provide an understanding of these nanosystems, models of nanocrystals must describe variations in ∗ Corresponding author. Tel.: +1-301-975-2595; fax: +1-301975-3038. E-mail addresses: [email protected] (G.W. Bryant), [email protected] (W. Jask(olski).

composition on the monolayer scale, realistic shapes and faceting, and coupling between closely spaced nanocrystals. A theory on the atomic scale is needed. In this paper, we present an empirical tight-binding theory of nanocrystal nanosystems. We apply the theory to the individual nanocrystals that will serve as the building blocks for nanocrystal architectures and show that complex nanocrystals with atomic-scale structure can be modeled. Recently, several groups have shown that nanostructures can be fabricated inside nanostructures [1– 6]. These quantum-dot quantum-well (QDQW) heteronanostructures are multishell nanocrystallites. Fabrication of a QDQW, for example a ZnS=CdS

c 2001 Elsevier Science B.V. All rights reserved. 1386-9477/01/$ - see front matter  PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 1 7 9 - 5

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QDQW with a ZnS core, a CdS shell and a ZnS cladding shell [1], produces a barrier=well=barrier nanocrystal a radial band-proEle with an internal quantum well (the CdS shell) inside the (ZnS) quantum dot. The QDQW is formed via a layer-by-layer chemical growth process in a colloidal solution. Single-band [7] and multi-band [8] eFective mass theories have been developed for QDQWs but these eFective mass theories cannot give a complete description of QDQWs. An atomic model is essential for QDQWs since the internal wells are no more than a few monolayers thick. In this paper, tight-binding theory is used to obtain the electronic structure and optical absorption spectra for CdS=HgS=CdS and ZnS=CdS=ZnS QDQW nanocrystals. Comparison with experiment shows that tight-binding theory provides a good description of nanosystems with monolayer variations in composition. Issues for designing other multilayer nanocrystal heteronanostructures are discussed. Well=barrier and well=barrier=well nanocrystal heteronanostructures are modeled to illustrate state and energy-level engineering in these nanocrystal nanosystems. 2. Theory Electron and hole states in these nanoparticle nanosystems are calculated by use of the empirical tight-binding (ETB) method [9 –11]. Effective mass kp models can describe InAs=GaAs self-assembled quantum dots with abrupt interfaces and nanometer-scale dimensions [12]. The ETB approach is an atomistic approach better suited for calculating the electronic states of QDQWs, with internal interfaces and monolayer-wide, atomic-scale shells, than eFective mass or kp models. Our ETB theory can be used to model single nanocrystals or coupled nanocrystal systems. In this paper, we focus on single nanoparticles. We assume that atoms in these nanoparticles occupy the sites of a regular fcc lattice. We can model spherical, hemispherical, tetrahedral or pyramidal nanoparticles. The eFect of the surrounding colloidal solution is not included. Each atom is described by its outer valence s orbital, the 3 outer p orbitals and a Ectitious excited s∗ orbital, that is included to mimic the eFects of higher lying states [13]. The empirical Hamiltonians are determined by

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adjusting the matrix elements to reproduce known band gaps and eFective masses of the bulk band structures. We include on-site and nearest-neighbor coupling between orbitals. Spin–orbit coupling has also been included in the theory but will not be considered for the results presented here because the empirical spin–orbit coupling is not known for HgS. In this paper, we present results for spherical nanoparticles. The electron and hole energies and eigenstates close to the band edges are found by diagonalizing the Hamiltonians with an iterative eigenvalue solver. The QDQWs have the order of 5000 –25,000 atoms. Typically, we End several hundred states closest to the band edge. We explicitly exclude the eFects of surface states in our calculations by passivating the surface dangling bonds. The passivation is modeled by shifting the energy of the dangling bonds above the conduction band so that the dangling bonds do not modify states near the band gap. No lattice relaxation is included. In self-assembled quantum dots such as InAs=GaAs, strain eFects due to the large (7%) lattice mismatch can be important [12]. In CdS=HgS QDQWs, the lattice mismatch is only 0.5% and strain eFects should be small. In ZnS=CdS QDQWs, the lattice mismatch is 7% and could be important. For ZnS=CdS, we compare our calculations with experiments done on samples with large inhomogeneous broadening. A theory including strain eFects is not needed to describe these experiments. Strain effects are well deEned for self-assembled InAs=GaAs QDs because the strained InAs QD is surrounded by bulk GaAs. In QDQWs, the internal well is only a couple of monolayers thick and should relax to the lattice constant of the core and cladding. Any additional strain eFects due to lattice relaxation near the surface are not as well deEned for QDQWs as for self-assembled QDs because the QDQW is in solution rather than in a crystalline bulk matrix. We estimate the energies for diFerent transitions by Erst taking the diFerences between the energies for the electron and hole single-particle states. Then we make a simple estimate for the electron–hole Hartree energy and include the estimate for the exciton binding energy. We should include the eFects of Coulomb mixing [7]. For small QDQWs, the electron states are widely separated in energy and the Coulomb mixing of electron–hole pair states with diFerent electron states can be ignored. Hole states are more closely spaced

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Fig. 1. Transition density of states for a CdS=HgS=CdS QDQW. The dimensions are indicated. No Coulomb eFects are included. The nanocrystal is centered about a bond midpoint.

and the Coulomb mixing of pair states with diFerent hole states could be important. These eFects will be included in future work. Spectra are calculated by evaluating the dipole matrix elements using the electron and hole eigenstates found in the ETB calculations [10,11]. In the ETB approach, dipole matrix elements are not needed to deEne the Hamiltonian. We End the dipole matrix elements between orbitals on the same atom by use of calculated atomic dipole matrix elements [14]. The dipole matrix elements between bonding orbitals on nearest-neighbor sites are chosen by reasonable guesses. The qualitative structure in the calculated spectra is insensitive to variations in the dipole matrix elements for bond orbitals for choices of these matrix elements less than the bond length. However, the absolute magnitude of the spectra is sensitive to the choice of these parameters. For our calculations we are interested in the relative strengths of diFerent peaks, but not in the absolute magnitudes. 3. Results The calculated transition density of states (electron– hole transition energy) for the CdS=HgS=CdS QDQW studied by Mews with size selective spectroscopy [2] is shown in Fig. 1. The corresponding calculated optical spectrum is shown in Fig. 2. HgS acts as a

Fig. 2. Spectrum for a CdS=HgS=CdS QDQW. The dimensions are the same as in Fig. 1. No Coulomb eFects are included. The nanocrystal is centered about a bond midpoint. A level scheme for the lowest transitions is shown.

quantum well for both electrons and holes in this QDQW. For this structure, the lowest two electron states (the 1S-like and 1P-like states) lie below the CdS bulk band edge and are localized near the HgS shell. Hole states are more easily trapped in the HgS well so multiple hole states lie below the CdS bulk valence band edge. A comparison of the transition density of states and the optical spectrum show that many possible transitions are optically weak. The ground-state transition (labeled e in Fig. 2) is weak and will appear only in emission. The lowest transition seen in absorption (labeled la in Fig. 2) and the Erst upper transition seen in absorption (ua) are excited state transitions involving an excited hole state and the 1P-electron state, respectively, as shown in the level scheme. The splitting between the Erst two hole states (20 meV) determines the Stokes shift between the ground state emission and the Erst absorption peak. The measured Stokes shift is 19 meV [2]. The measured splitting between the lowest and upper absorption peaks of 60 meV [2] is also consistent with the energy splittings shown in the calculated level scheme. When a reasonable estimate for the exciton binding energy (100 meV, see [7]) is subtracted from the electron–hole ground state transition energy, the predicted emission

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Fig. 3. Experimental absorbance for ZnS=CdS=ZnS nanocrystal heteronanostructures [1] and the predicted lowest absorption band, indicated by the horizontal arrows, for each structure. The diameter of each structure is indicated.

peak at 1.87 eV lies very close to the emission peak observed at 1.86 eV. Calculations for other HgS-well thicknesses and for other locations of the HgS well inside the CdS nanocrystal show that the energy level splittings are not too sensitive to these geometrical parameters. The good agreement between the calculated level spacings and the observed spectra does not depend on precise choices for the geometrical parameters. Measured absorbance spectra for ZnS=CdS=ZnS QDQWs [1] are shown in Fig. 3. In this nanosystem, CdS acts as the quantum well. For structures with a ZnS cladding, the cladding is one monolayer wide. The shell nanostructures have either one or two monolayers of CdS, as indicated. The electronic structure for ZnS=CdS=ZnS QDQWs is similar to the level structure for CdS=HgS=CdS QDQWs. The lowest electron and hole states are localized near the CdS well. The ground state transition is weak but there are two dominant excited state transitions near threshold. The horizontal arrows in Fig. 3 indicate the energy

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range spanned by these two strong transitions. In each case, the calculated electron–hole pair transition has been shifted to higher wavelength by 15 nm as a simple estimate for the exciton binding. For the 2 nm ZnS nanocrystal, this shift corresponds to a 270 meV binding energy. For the 3 nm CdS nanocrystal, the shift corresponds to a 140 meV binding energy. These are reasonable estimates for the pair binding in these particles. The calculated positions for strong absorption agree well with the measured absorbance thresholds. Well=barrier and well=barrier=well nanocrystal heteronanostructures could also be used to tailor states and energy levels in nanocrystals. As examples, we consider CdS=ZnS and CdS=ZnS=CdS nanocrystals. CdS=ZnS is typical of nanocrystals grown with a high-barrier capping material to passivate the surface of the core dot. CdS=ZnS=CdS structures have not been grown yet but could be grown from the CdS=ZnS structure. The hole charge density in a 2-nm radius CdS nanocrystal is shown in Fig. 4. The ground state is P-like, is pushed out toward the surface, and is even. The Erst excited state is pulled toward the center, has an S part, and is odd. This level ordering produces an optically weak ground state transition. This ordering is typical for the QDQWs and all the CdS dot sizes studied. For this CdS dot, the two hole levels are split by 30 meV. When the dot is capped with a ZnS barrier, the more extended states, like the ground state of the CdS core, are more sensitive to the barrier potential. As the barrier thickness increases, the level ordering of the Erst two hole states switches (as shown in Fig. 4), the S-like odd state becomes the ground state and the ground state becomes optically active. The splitting between the hole levels with the reversed ordering is as big as the splitting between the hole levels in the CdS nanocrystal. These results show that the capping layer is not just a passivant but can signiEcantly alter the states in the nanocrystal. We have considered here the case where the barrier material is a semiconductor similar to the core. We expect a similar sensitivity to the capping layer when the cap is a molecular surfactant used to functionalize the core dot, passivate the core surface, or act as a glue to hold dots together in a quantum-dot solid. Well=barrier=well structures could be used, in analogy with coupled 2D quantum wells, to tailor the states by exploiting the interwell tunneling to modify the states trapped in the core and cladding wells. As an

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Fig. 5. Dependence of the emission and absorption thresholds of a CdS=ZnS=CdS nanocrystal on clad well thickness. The core radius and barrier thickness are indicated.

Fig. 4. Charge density for the hole ground state (solid curve) and Erst excited state (dashed curve) in a 2-nm radius CdS nanocrystal (upper panel) and in a 2-nm radius CdS=ZnS nanocrystal (lower panel). The state symmetries are shown.

example, a CdS=ZnS=CdS well=barrier=well nanocrystal heteronanostructure with a 1 nm core well and a 1 nm barrier is considered. The dependence of the ground state transition energy on cladding well thickness is shown in Fig. 5. For thin clad wells, the ground state remains optically active, as it was for the well=barrier structure. For thicker clad wells, the order of hole states switches back to the ordering in homogeneous nanocrystals with a dark ground state and an optically active excited state. Resonant coupling between electron states in the core and clad wells is shown in Fig. 6 by the dependence of electron states on clad-well thickness. Resonant coupling occurs between states with the same spatial angular momentum. The 1S and 2S states couple resonantly through the ZnS barrier deEned by the ZnS band edge at EZnS . The 1P and 2P states also couple. In this case, they couple above the ZnS barrier. After the resonant crossover, the 2S state is the only state trapped in

Fig. 6. Dependence of the electron energy levels of a CdS=ZnS=CdS nanocrystal on clad well thickness. The core radius and barrier thickness, the state symmetries, and the ZnS conduction band edge are indicated.

the core well below the ZnS band edge. All of the other trapped states are in the clad well. The resonant coupling between hole states is more complicated due to the mixed symmetry of the hole states. The jump in absorption threshold in Fig. 5 is due to the level repulsion between resonantly coupled hole states.

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4. Conclusion

References

A comparison of the observed and calculated level structure for CdS=HgS=CdS and ZnS=CdS=ZnS nanocrystal heteronanostructures shows that the tight-binding approach provides a good description for nanosystems with monolayer variations in composition. Calculations conErm that the shell can act as an electron and hole trap. However, care must be taken in describing these trapped states as well states. Depending on the shell thickness and band oFset, these trapped states can have a large interface character. In principle, the ordering of barrier and well materials could be interchanged as in, for example, CdS=ZnS=CdS. Resonant coupling can occur between states trapped in the core and in the clad of such well=barrier=well nanocrystal nanostructures. For electron states, which are formed from cation s orbitals, resonant coupling occurs between states with the same energy and same spatial angular momentum. For hole states, resonant coupling is more complicated. Even for the simpler well=barrier nanocrystal, the single well=barrier heterostructure can signiEcantly alter the states of the core dot. All of these issues must be modeled carefully to design nanocrystal nanosystems.

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