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Quantum dot modeling

Quantum dot geometry and composition

Strain field

Three-dimensional 8 band

Resolution

Simple view of the electronic structure

A more sophisticated view

The theoretical knowledge of the electronic structure of III-V self-assembled quantum dots is a crucial step towards the interpretation and the understanding of the infrared experimental data. The knowledge of the electronic structure is naturally a precious basis for the spectroscopy but also brings valuable theoretical information that cannot be directly accessed experimentally. Because of the three-dimensional confinement, the density of states is delta-like with well energy separated energy levels, instead of being composed of continuous bands or sub-bands of energy levels as it is the case in quantum wells or bulk semiconductors. Depending on the size and geometry, numerous levels can be confined in the conduction band and even more in the valence band. Non intuitive selection rules, dipole lengths or polarizations can rule the optical transitions between these levels. In the case of two-dimensional heterostructures, a relevant electronic structure can be obtained at a very low computational cost in rather simple single band parabolic envelope function models. As opposed to quantum wells and for many reasons the calculation of the energies of the confined levels, wave functions, and dipole matrix elements in quantum dots is a challenging issue.

On the one hand an exhaustive calculation of the electronic structure should first take into account the specific geometry of the dots and specifically the confinement along the three directions of space, even if the lateral dimensions are ten times longer than along the growth axis. The lack of spatial symmetry in the geometries of the quantum dots usually implies high cost three-dimensional computations. This can be the case for quantum dots with an elongated geometry that presents no continuous nor discrete symmetries. Several electronic bands must be considered because of the large confinement energies. The confinement energies can be 10 times that of quantum wells because the strong confinement of the carriers along the growth axis and in the layer plane. Non-parabolicity of band diagram of underlying semiconductors used to construct the heterostructure is thus no longer negligible. It results in a correction in the energies of about 30% as compared to what can be expected from a parabolic dispersion and leads to strong band mixing. The electronic structure also depends on the strain field which is present in these lattice-mismatched materials. The strain can be inhomogeneous within the dot, involving not only biaxial components but also non negligible shear components. The surrounding of the dot is also strained and can play a significant role on the bound-to-continuum transitions. The effect of strain on core and barrier material band structures should be considered since it modifies the energy band gaps and lifts up the heavy-light hole degeneracy at the zone center. The piezoelectric field originating from the shear component of the strain should also be taken into account since it changes the level energies. There is also an inhomogeneous broadening of the optical properties of dot ensembles due to the size fluctuation from dots to dots and from one dot plane to another and thus there is the necessity to consider dot of different sizes, shapes and compositions. Coulomb interactions finally can play a non negligible role in multi-charged quantum dots.

On the other hand a difficulty comes from a relative uncertainty on the input parameters. Despite numerous experimental characterizations, the exact geometry, the composition, the effect of the In segregation are not perfectly known experimentally. These parameters will depend on the growth conditions and a slight variation on their values (e.g. composition) can lead to strong deviations on the calculated electronic structures (e.g. interband transition energies).

Several theories have been developed to assess the electronic structure of the quantum dots. The simplest and first developed approach is based on the effective-mass theory. Single-band effective-mass calculations have been performed for InAs/GaAs dots with various geometries: cones, square based pyramids , thick lenses, flat lenses. Note that a simple cubic quantum box model gives very crude predictions on the dipole matrix elements and level energies as compared to those provided by a 3D model which accounts for the realistic geometry of the dots. 3D single band effective mass is often a good compromise if one does not require accurate energy predictions. Otherwise

Quantum dot geometry and composition

The InAs quantum dots are grown by molecular epitaxy on [001] GaAs substrate. From structural measurements the dot density is around 4x10

Figure 1 : Cross section image of an InAs
quantum dot imbedded into a GaAs matrix obtained by Transmission
Electron Microscopy (Courtesy of Gilles Patriarche, Laboratoire de
Photonique et Nanostructures, CNRS, Marcoussis). |

Strain field

Self-assembled quantum dots are strained heterostructures since their Stranski-Krastanow growth mode uses the natural lattice mismatch between the substrate and the deposited material. In the case of the InAs/GaAs couple, this mismatch is around 7% and leads to a strong strain field within and around each quantum dot. The 3D components eij of the strain tensor are key parameters on which depends the electronic structure of the bulk underlying materials (InAs and GaAs for the system presented in this article) and thus the electronic structure of the quantum dot. The band gap of InAs is increased by more than 100 meV due to a 7% in-plane applied strain as compared to unstrained InAs. The heavy-light hole degeneracy at the zone center is also strongly lifted pushing the light hole band around 200 meV beneath the heavy hole band. As compared to quantum wells one a priori expects the presence of non-vanishing 3D shear components because of the 3D nature of the quantum dot geometry. This component is at the origin of a piezoelectric field that can be non negligible, especially in the case of high aspect ratio islands.

The strain field can be calculated by minimizing the strain energy Estrain given by a valence force field theory :

(1)

In this microscopic theory the strain energy is expressed from the variation of the bond length and bond angles for all the atoms within the deformed zinc-blend lattice. In equation (1) is the vector going from atom i to atom j, the vector from atom i to atom k in the unstrained crystal, the indexes j and k designing one of the four nearest neighbour of atom i. Taking into account the strain limit conditions set by the substrate, the minimization gives access directly to the displacement of the atoms at an interatomic length scale, in particular around the hetero-interfaces, respecting the 43m symmetry of the crystal and depending solely on the chosen dot geometry and local dot and barrier compositions. One will note that, for each material InAs or GaAs, there are only two input coefficients (bond stretching alpha

Fig. 2 depicts a representation of the strain field of one flat lens-shaped pure InAs self-assembled quantum dot floating over a 0.5 nm thick InAs wetting layer and inserted into a zinc-blend GaAs matrix. For a height of 2.5 nm and a base diameter of 25 nm the calculation shows that the volume dilation is essentially localized into the InAs volume while the GaAs barriers remain nearly unstrained. A more detailed analysis shows that the strain tensor is close to the one of a thin InAs quantum well grown on GaAs: the relative elongations are nearly piecewise constant (i.e. nearly constant in the InAs volume and nearly constant in the GaAs volume) and the shear component in only moderate near the interfaces. The main reason for this specificity is the small aspect ratio (0.1) of the dot. One also expects much smaller piezoelectric field in this flat dot than for high aspect ratio geometry. In quantum dots exhibiting a higher aspect ratio, the piezoelectric potential may lead to significant corrections (~ 10 meV) to the level energies. In what follows we will consider that the strain field is biaxial piecewise constant and we will neglect the effect of piezoelectricity on the level energies.

Figure 2 : Compressive and shear components of
the strain field in an InAs/GaAs self-assembled quantum dot calculated
by minimizing the strain energy given by a microscopic valence force
field theory. The
compressive component (blue) is roughly piecewise constant and located
into
the InAs volume and mimics the strain of a thin InAs quantum well. The
non
negligible shear component corresponds (red) to less than 3% of lattice
deformation. |

Three-dimensional 8 band

The theoretical electronic structure of one quantum dot, i.e. the confined level energies and wave functions, is provided by the three-dimensional resolution of the Schrödinger equation (2) written in a 8 band

(2)

where Hkp8, Hstrain, V, and E are the unstrained 8 band

Resolution

The 8 band

Note that from a mathematical and numerical point of view the resolution of the Schrödinger equation is not straightforward. As far as finite difference is concerned, there are well known and systematic methods applied for parabolic or elliptic differential equations. Unfortunately the 8 band

Simple view of the electronic structure

It is instructive to first consider the electronic structure given by a cubic quantum box with infinite barrier heights. In this case, the electronic structure results from the independent combination of the states of infinite wall quantum wells of respective widths Lx, Ly, Lz along the three x, y and z directions. If the carrier lives in a parabolic single band described by the scalar effective mass matrix (mxx, myy, mzz) then the level energies have the simple following form:

(3)

and the envelope wave functions the form:

(4)

where nx, ny, nz are positive non null integers. One interest of this simple model is to show that the states can be sorted by the numbers of the wave function nodes nx, ny, nz along the three directions of space. According to this notation, the ground state is labelled 000, the first excited state resulting from the confinement along the x-axis is 100 and so on. Allowed optical intersublevel transitions between level i and j only exist between states of odd difference quantum numbers along the same axe, e.g. 000->100, 100->200, 000->300 but not 000->200, 100->001 etc. In a non-cubic quantum box, a convention is to label the states from the denomination they would have in the cubic dot obtained by continuously deforming the original dot.

A more sophisticated view

Figure 3 depicts the electronic structure of one InAs/GaAs self-assembled quantum dot as deduced from the 3D resolution of the 8 band

In the valence band, despite the smaller barrier height a much larger number of states are confined because of the heavier effective masses. Only the very first states are represented in Figure 3. The confinement energy of the ground state h000 is also smaller than in the conduction band. From a general point of view, the valence electronic structure is more complex than the conduction one.

Figure 3 : Electronic structure of a InAs/GaAs
self-assembled quantum dot as deduced from the 3D resolution of the
Schrödinger equation written in a 8 band k.p formalism. The
resolution takes in account a realistic elliptical flat lens-shape
geometry (25x28x2.5 nm^{3}) described in the inset. The
representation of the envelope wave functions considers probability
volumes including 2/3 of the presence probability of the electron. WL
denotes
the wetting layer 2D continuum, beneath the bulk barrier 3D continuum. |