Connection Rules versus Differential Equations for Envelope Functions in Abrupt Heterostructures

The propagation of the classical wave in disordered media at the Anderson localization transition is studied. Our results show that the classical waves may follow a different scaling behavior from that for electrons. For electrons, the effect of weak localization due to interference of recurrent scattering paths is limited within a spherical volume because of electron-electron or electron-phonon scattering, while for classical waves, it is the sample geometry that determines the amount of recurrent scattering paths that contribute. It is found that the weak localization effect is weaker in both cubic and slab geometry than in spherical geometry. As a result, the averaged static diffusion constant D͑L͒ scales like ln͑L͒ / L in cubic or slab geometry and the corresponding transmission follows ͗T͑L͒͘ ϰ ln L / L 2 . This is in contrast to the behavior of D͑L͒ ϰ 1/L and ͗T͑L͒͘ ϰ 1/L 2 obtained previously for electrons or spherical samples. For wave dynamics, we solve the Bethe-Salpeter equation in a disordered slab with the recurrent scattering incorporated in a self-consistent manner. All of the static and dynamic transport quantities studied are found to follow the scaling behavior of D͑L͒. We have also considered position-dependent weak localization effects by using a plausible form of position-dependent diffusion constant D͑z͒. The same scaling behavior is found, i.e., ͗T͑L͒͘ ϰ ln L / L 2 .

Section: ͑29͒mentioning

confidence: 99%

Scaling behavior of classical wave transport in mesoscopic media at the localization transition

Cheung

Zhang

2005

“…Thus, the BCs are derived starting from the requirement of continuity of the components of the wave function at the heterointerface.One can always choose the CRs physically equivalent to the BCs. 22 The both approaches (i) and (ii) are usually used when the wave function inside each layer of a heterostructure can be found analytically, for example in planar or spherical heterostructures. In case of an arbitrary shape of the heterointerface, the approach (ii) can still be used because, when the Hamiltonian is known for the entire heterostructure, one can find an overall numerical solution of the Schrödinger equation.A commonly used heuristic method to obtain a multiband effective-mass Hamiltonian for heterostructures uses symmetrization 23-27 of the corresponding k ·p Hamiltonian.…”

mentioning

confidence: 99%

Development of an eight-band theory for quantum dot heterostructures

Pokatilov

Fonoberov

Fomin³

et al. 2001

107

109

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...

“…The boundary conditions for two media with different crystal structure remains a subject of contention 36 and requires further study. We will not deal with this question in this paper.…”

Section: ͑5͒mentioning

confidence: 99%

Confined modes in finite-size photonic crystals

Yang

Nair

et al. 2005

The real band structure of finite-size photonic crystals ͑PCs͒ is different from the band structure of infinite PCs derived using Bloch's theorem. There are isolated instead of a continuum of modes around a band edge. We use the envelope function approximation ͑EFA͒ method to derive simple formulas to describe the frequency and field pattern of the modes in finite-size PCs. The results are compared with those from finitedifference time-domain ͑FDTD͒ method. We observe that the agreement between the results from EFA with a vanishing boundary condition and FDTD is sensitive to the boundary orientation. In addition, although a simple group-velocity-based analysis explains the long lifetime of the confined modes, this explanation is inappropriate for certain structures. In particular, we find boundary orientations, for which the lifetime of the confined modes can be enhanced by orders of magnitude compared with that anticipated from group velocity analysis. We explain this by total internal reflection. Our results demonstrate the utility of the EFA approach in studying the modes in finite-size PCs and reveal the importance of boundary conditions to develop an EFAbased tool for cavity lasers and for studying nonlinear interactions.