The Strained Linear Combination of Bulk Band (SLCBB) method

The SLCBB method is designed to calculate the single particle eigenfunctions and eigenvalues of large nanostructures like self-assembled quantum dots. It is used in conjunction with empirical pseudopotentials like the ESCAN method but uses a different basis. While the ESCAN method uses the conventional plane-wave basis the SLCBB method uses a basis of Bloch orbitals of the underlying bulk:

$$\psi ({\bf x}) = \sum_n^{N_B} \sum_k^{N_k} \sum_{\sigma}^{\uparrow,\downarrow} C_{k,n,\sigma}  \phi^0_{k,n}({\bf x})$$ (1)

$$\phi^0_{k,n}({\bf x})= \frac{1}{\sqrt{N}} u_{k,n}({\bf x}) e^{i {\bf k}\cdot {\bf x}}$$ (2)

$$u_{k,n}({\bf x})= \frac{1}{\sqrt{V_0}} \sum_G^{N_G} A_{k,n}({\bf G}) e^{i {\bf G} \cdot {\bf x}}$$ (3)

where $n$, ${\bf x}$ and ${\bf k}$ stand for the band index, the real space coordinate and the k-vector index. This basis is highly optimized because it is adapted to the system under investigations. The SLCBB method has been used extensively in the past 5 year to investigate InAs (or InGaAs) quantum dots embedded in GaAs. For this special case the SLCBB basis would consist of the strained InAs Bloch functions (the InAs quantum dot is under compressive strain) and unstrained GaAs Bloch functions (the surrounding GaAs matrix is mostly unstrained). Often, physical intuition is required to choose adequate Bloch functions. The Bloch functions can be centered around Gamma for a strongly direct band gap material or around L, X or other points of the Brillouin zone for other materials. The basis must always be tested for convergence with respect to the number of k-points used, the number of bands and the number of different Bloch function types. Typically the SLCBB method does not scale like plane wave methods with the cube of the number of plane-waves ($O(N^3)$) but it scales in a different way. A million atom supercell with pure InAs, for instance, will require for the computation of the band gap about the same computation time than an 8 atom supercell. The scaling is proportional to the size of the basis needed, which does not only depend on system size but on the type of system (unlike the plane-wave basis sets). Unlike the ESCAN method where the Hamiltonian can be easily calculated on the fly, the SLCBB Hamiltonian is expensive and is fully stored. The eigenvalue problem is presently solved via the Arnoldi Restart ARPACK method around a certain reference energy. This procedure differs drastically from ab-initio method that require the calculations of all the bands up to the Fermi energy. Here only informations about the band edges (few conduction band states near the CBM and few valence band states near the VBM), see our description of the EPM method. The SLCBB method has recently been extended to treat electric and magnetic fields and to include the effects of the piezoelectric field (see our description of Piezo effects here). We are currently parallelizing the code and, in collaboration with the group of Jack Dongarra at Oark Ridge, testing different spectral transformations and eigensolvers.

Applications of the SLCBB method

Calculation of the excitonic fine-structure in self-assembled quantum dots (pdf)

Here, SLCBB is used to predict the electron-hole exchange-induced fine-structure and polarization anisotropy in InGaAs/GaAs quantum dots of various shapes and compositions. The origin of the fine-structure splittings is clarified using a simple model where the effects of atomistic symmetry and spin-orbit interaction are separately evident. Remarkably,  polarization anisotropy and fine-structure splittings are shown to occur, even in a cylindrically-symmetric dot. Furthermore, ``dark excitons'' are predicted to be partially allowed. Trends in splittings among different
shapes and compositions are revealed.

Isosurfaces of the wavefunctions squared of the first 3 electron and first 3 hole states for a
flat (dot F: 25.2nm base and 3.5 nm height) and a tall (dot T: b=25.2nm, h= 5nm) In_{0.6}Ga_{0.4}As/GaAs dot. The two isosurfaces enclose 75% and 40% of the state densities.
For analysis purposes we project the wavefunctions on three valence bands x, y, z and the lowest electron band el. The valence bands are labeled by their axial angular momentum values.
The wavefunctions are further decomposed with respect to their axial angular momentum
components (S, P, D).}

Calculation of the excitonic spectra in charged quantum dots (pdf)

In this work, SLCBB has been used to calculate the excitonic X recombination in charged, self-assembled InGaAs/GaAs dots predict striking trends: (i) whereas in alloy InGaAs dots the exciton shifts to the red upon negative charging, in pure InAs dots the exciton shifts to the blue. The opposite behavior is observed upon positive charging. (ii) The recombination peaks of different charge states show peculiar symmetry and alignments, e.g. X^- with X^2- and X^3- with X^4-.
These trends are explained theoretically revealing an underlying systematic in the peak splittings and shifts.
The sign and magnitude of these shifts and splittings can be understood in terms of wavefunction localization and separation.

These cartoons show the initial configurations for e_0 - h_0 exciton recombination of
charged dots. The final configuration is the one where the e_0-h_0 pair connected by the
vertical line is eliminated. The central panel shows the energy change
due to the recombination in the single-configuration approximation.
The constant single-particle energy gap of the e_0-h_0 pair has been omitted.
The energy changes for the positively charged dots
are given by interchanging e with h and vice versa on all the integrals.
When two possible final states are present (exchange splitting), the
results are given for both in parenthesis.

Calculation of the excitonic entanglement in vertically stacked quantum dots (pdf)

SLCBB  has been used to calculate the properties of an exciton in a pair of vertically
stacked InGaAs/GaAs dots. Competing effects of strain, geometry, and band mixing lead to many unexpected features missing in contemporary models. The first four excitonic states are all optically active at small interdot separation, due to the broken symmetry of the single-particle states. We quantify the degree of entanglement of the exciton wavefunctions and show its sensitivity to interdot separation. We suggest ways to spectroscopically identify and maximize the entanglement of exciton states.

(a) Emission spectra in a pair of vertically stacked
InGaAs/GaAs dots. (b) Dot geometry, including a two monolayer
(0.56 nm) InGaAs wetting layer and graded
composition profile.

Prediction of an excitonic ground state in InAs/InSb quantum dots (pdf)

Using SLCBB followed by configuration-interaction many-body calculations, we predict a metal-nonmetal transition and an excitonic ground state in the InAs/InSb quantum dot (QD) system.
For large dots, the conduction band minimum of the InAs dot lies below the valence band maximum of the InSb matrix. Due to quantum confinement, at a critical size calculated here for various shapes, the single-particle gap E_g becomes very small. Strong electron-hole correlation effects are induced by the spatial proximity of the electron and hole wavefunctions, and by the lack of strong (exciton unbinding)
screening, afforded by the existence of fully discrete 0D confined energy levels. These correlation effects overcome E_g, leading to the formation of a bi-excitonic ground state (two electrons in InAs and two holes in InSb) being energetically more favorable (by ~15 meV) than the state without excitons.
We discuss the excitonic phase transition on QD arrays in the low dot density limit.

Wavefunction of the InAs-confined electron states (e1-e4)
and the first two hole states (h1-h2). The transparent lenses indicate
the positions of InAs dots. The isosurface enclose 50 % of the state
density except for e4 which is only weakly confined. For e4, the isosurface
enclose only about 10 % of the state density. The contour plot are slices of
the density taken from choosen planes.