Development of an eight-band theory for quantum dot heterostructures

Abstract

We derive a nonsymmetrized 8-band effective-mass Hamiltonian for quantumdot heterostructures (QDHs) in Burt's envelope-function representation. The 8×8 radial Hamiltonian and the boundary conditions for the Schrödinger equation are obtained for spherical QDHs. Boundary conditions for symmetrized and nonsymmetrized radial Hamiltonians are compared with each other and with connection rules that are commonly used to match the wave functions found from the bulk k ·p Hamiltonians of two adjacent materials. Electron and hole energy spectra in three spherical QDHs: HgS/CdS, InAs/GaAs, and GaAs/AlAs are calculated as a function of the quantum dot radius within the approximate symmetrized and exact nonsymmetrized 8×8 models. The parameters of dissymmetry are shown to influence the energy levels and the wave functions of an electron and a hole and, consequently, the energies of both intraband and interband transitions.PACS numbers: 73.20. Dx, 73.40.Kp, 73.40.Lq This model explicitly includes eight bands around the Γ point of the Brillouin zone, namely, electron, heavy-, light-, and spin-orbit split-off hole bands (each of them is twice-degenerate due to the spin), and treats all other bands as remote. Along with more simple models, the 8×8 k ·p Hamiltonian has been used to investigate different QDs (see, e. g. .Recently, one has begun to apply multiband effective-mass Hamiltonians to investigate elastic, electronic, and optical properties of multilayer nanostructures such as quantumdot heterostructures (QDHs): CdS/HgS 11 , InAs/GaAs 12,13 , GaAs/Al x Ga 1−x As 14,15 , and CdS/HgS/CdS/H 2 O 16,17 . However, it should be emphasized, that multiband k ·p Hamiltonians are derived for homogeneous bulk materials, i.e. under the assumption that all effective-mass parameters are constant. This is important, because at a certain step of the derivation, wavenumbers k are declared as operatorsp/h that do not commute with the functions of coordinates. But, at the heterointerfaces of the multilayer nanostructures, there occurs an abrupt change of effective-mass parameters from their values in one material to those in the adjacent material. Inside a thin transitional layer that contains the heterointerface, the ordering of the differential operators and coordinate-dependent effective-mass parameters in the multiband Hamiltonian becomes crucial. In QDs with an infinitely high confining potential for electrons and holes, all components of the wave function vanish at the heterointerface, and there remains a possibility of applying the bulk multiband k ·p Hamiltonian straightforwardly. [3][4][5][6][8][9][10][11] There are two ways to proceed from QDs to QDHs.(i) The first way is to use an appropriate bulk multiband Hamiltonian for each constituent material separately, and then to match the obtained homogeneous solutions at the abrupt heterojunctions applying the connection rules (CRs) that are usually obtained by imposing the continuity of the wave function envelopes and of the normal to the heterointerface component of the velocity. 11,16...

Optical properties and optimization of electromagnetically induced transparency in strained InAs/GaAs quantum dot structures

“…The asymmetry parameter present in the Burt-Foreman formalism but not in the Luttinger-Kohn formalism is shown to lead to changes in approximately Ϯ25 meV in the electronic band structures of InAs/GaAs. 18 Optoelectronic properties of InGaAs zinc-blende quantum dot with varying shape and size based on k ជ · p ជ theory have already been studied by Schliwa et al 19 and Veprek et al 20 However, so far, only a small selection of all possible transitions have been studied. In this work we apply the eightband model based on Burt-Foreman formalism derived in Ref.…”

“…14 In order to overcome this ad hoc procedure, Burt formulated the socalled exact envelope-function method 15,16 and soon after Foreman 17 used this method to derive a six-band model for the valence bands of zinc-blende heterostuctures. Pokatilov et al 18 have provided an eight-band model for the conduction and the valence bands. They studied quantum dots using a spherical approximation and compared their model, based on exact envelope-function theory, against the usual symmetrized approach.…”

“…In this work we apply the eightband model based on Burt-Foreman formalism derived in Ref. 18 to zinc-blende InAs/GaAs conical quantum dots. We choose the conical geometry because ͑i͒ it introduces important symmetry lowering effects not present in the earlier spherical models and ͑ii͒ it mimics closely the structures realized in experimental systems, e.g., in Ref.…”

Photoluminescence of tetrahedral quantum-dot quantum wells

“…In order to take the above features into account, we use the nonsymmetrized 8-band Hamiltonian, which was derived earlier by us for quantum-dot heterostructures [5]. To optimize the numerical calculations within the framework of the method described in Ref.…”

Magnetic field implementation in multibandk⋅pHamiltonians of holes in semiconductor heterostructures

“…29 theory in order to account for the spin Zeeman term, and identify the coefficients that should accompany the magnetic terms in this approximation, which were pending clarification. 28 The remote band influence is considered through the zero-field effective masses, which are known to provide a meaningful description even in strongly confined QDs [30][31][32][33] , and effective g-factors. We run calculations comparing this model with the Luttinger approximation and show that the magnitude of the Zeeman splitting we estimate is closer to experimental values for vertically coupled InGaAs QDs.…”