Electronic-Structure Theory of Semiconductor Quantum Dots
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flect quantum size, quantum shape, interfacial strain, and surface effects and (b) the nature of "many-particle" interactions such as electron-hole exchange (underlying the "red shift"), electronhole Coulomb effects (underlying excitonic transitions), and electron-electron Coulomb (underlying Coulomb-blockade effects). Interestingly, while the electronicstructure theory of periodic solids has been characterized since its inception by a diversity of approaches (all-electron versus pseudopotentials; Hartree Fock versus density-functional; computational schemes creating a rich "alphabetic soup," such as APW, LAPW, LMTO, KKR, OPW, LCAO, LCGO, plane waves, MRS BULLETIN/FEBRUARY 1998
ASW, etc.), the theory of quantum nanostructures has been dominated mainly by a single approach so widely used that I refer to it as the "Standard Model": the effective-mass approximation (EMA) and its extension to the "k • p"5~7 (where k is the wave vector and p is the momemtum). In fact, speakers at nanostructure conferences often refer to it as "theory" without having to specify what is being done. The audience knows. The essential idea5"7 is sweeping in its simplicity: The single-particle wave functions i//,(r) of three-dimensional (3D)-periodic bulk, two-dimensional (2D)-periodic film/well, one-dimensional (lD)-periodic wires, or zerodimensional (OD)-periodic dot are expanded by a handful of 3D-periodic Bloch orbitals taken from the Brillouin zone center (F point) of the underlying bulk solid. The physical accuracy of this representation is naturally highest for systems closest to the reference from which the basis functions are drawn (F point of the 3D bulk). It decreases as one wanders away from the Brillouin zone center and as dimensionality (D) is reduced in the sequence 3D^> 2D^> ID—> OD. For example,8 reproducing the energy of the X u 3D bulk state in GaAs within 1 meV requires Nb = 150 (F-like) Bloch bands. However if only Nb = 10 (Flike) bands are used, the error increases to 300 meV. If Nb is further reduced to eight bands, the error in X k goes up to 20 eV and the curvatures (hence, effective masses) of the bulk valence bands develop an unphysical negative sign.8 Application of direct diagonalization and "first-principles k • p" to the 2D GaAs/ AlAs superlattices9 showed that the k • p errors versus Nb parallel those in bulk GaAs. The severity of such errors is miti-
gated by the central feature of the Standard Model: Instead of computing the parameters of the Hamiltonian from the basis set used, they are fit to the observed band properties of the bulk5"7 (bandgaps, masses) or the nanostructures themselves.10'1 The Standard Model has been eminently successful in describing the spectroscopy and transport of highdimensional nanostructures such as 2D quantum-well structures 7 and laser devices.12 Its success in describing lower dimensional nanostructures such a 0D quantum dots, while very impressive (e.g., see the good agreement with the excited-state energies of the CdSe dot10'11), is sometimes clouded by the fact that the par
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