Abstract

Vertically stacked InAs/InP columnar quantum dots (CQDs) for polarization insensitive semiconductor amplifier in telecommunications applications are studied theoretically. An axial model is used to predict mechanical, electronic, and optical properties of these CQDs. A crossover from a dominant transverse electric (TE) optical ground state absorption to a dominant transverse magnetic (TM) absorption is predicted for a number of layers equal to about 9 in good agreement with the experiment. The weight of the light hole component of the valence band ground state increases as a function of the number of layers. The change of the TE/TM polarization ratio is also associated to a symmetry change of the heavy hole component. A modification of the aspect ratio of the CQD seems to be the most important factor to explain the change of the electronic states configuration as a function of the strain distribution.

Recently, vertically stacked quantum dots (QDs) with a very small or zero spacing (columnar QD, CQD) have been investigated in order to obtain polarization insensitive semiconductor amplifiers (SOAs) [1]. These experimental results indicate that the TE/TM mode photoluminescence intensity is inverted for a number of QD layers (N) beyond 9 in the InAs/GaAs system. A first theoretical study for InAs/GaAs columnar system up to shows that the biaxial strain is modified at the center of the CQD by comparison to standard QD [2]. As a consequence, the light hole (LH) component of the hole ground state increases when the number of layer increases or when the QD spacing decreases. This approach has been extended more recently [3] to the InAs/InP system in order to propose SOA operating at 1.55m. These new SOAs will be able to treat random optical signals for general fiber communication. Experimental results have been obtained in our group in the last years concerning the physics and the optimization of devices on similar InAs/InP-QD-based structures [4–8]. We have also used a complete theoretical description of the electronic properties of such nanostructures [9]. An efficient mechanical and electronic axial approximation of the strained Hamiltonian has been proposed for zinc-blende nanostructures with a cylindrical shape on (100) substrates [10, 11]. It is possible with this method to treat large systems using conventional finite element computation [12]. We have demonstrated that it is able to describe the complex inhomogeneous strain distribution and electronic properties of InAs/InP QD using cylindrical coordinates () [9–11]. In this paper, the strain, electronic, and optical properties of InAs/InP CQD are analyzed with this method.

We examine InAs CQD geometries corresponding to the symmetry (rotational symmetry around the z-[001]-axis) as shown by Figure 1(a) for 7 layers of truncated cones. The chosen dimensions are 1.2 nm for the truncated cone height and 7 nm for the radius. The CQDs are embedded into an In_0.85Ga_0.15As_0.33P_0.67 lattice-matched quaternary alloy. The authors of [3] have used an In_0.66Ga_0.34As_0.44P_0.56 tensile strain (1%) quaternary alloy between the layers for strain compensation. The InAs wetting layer usually present during the growth of such Stranski-Krastanow type QD is not specified, so we have decided to use a quaternary alloy with an average composition In_0.75Ga_0.25As_0.58P_0.42 between the layers. It should be pointed out that the results are almost not affected by the composition of this alloy. The axial QD Hamiltonian is block diagonal in a basis, where is the total angular momentum. The basis is constructed in a product form where the first factor corresponds to the band-edge Bloch functions (the bands are, respectively, related to the conduction band (CB), heavy hole (HH), light hole (LH), and split-off bands (SOs)). The variations of the energy of the electronic conduction band (CB) degenerate ground and excited states are represented as a function of the number of layers on Figure 1(b). The CB ground state of the CQD corresponds to independently of the number of layers as observed in the case of a single QD [11]. The most striking feature appears for the valence bands (VBs) behaviour (Figure 1(c)). A crossover between and for the VB ground state is calculated for a number of layers close to 9. We may notice that this number corresponds to an aspect ratio of the CQD on the order of 1. A modification of the main confinement effect from vertical to radial is important.

Figure 1 

As an example, we show InAs CQD geometry corresponding to layers of truncated cones. (a) Dependence of the electronic conduction bands ( straight lines and , dashed lines) (b) and of the electronic Valance Bands ( straight lines and , dashed lines) (c) versus the number of layers.

In Figure 2, the variations of the HH, LH, and CB confinement potentials along the vertical axis are represented for . As it can be seen, the values of the HH and LH potentials are similar at the center of the CQD. The biaxial strain component is almost equal to zero in the CQD which is completely different from what is observed in usual flat QD [9–11]. This is also a result of the change in the aspect ratio. The LH (left) and HH (right) components of the (a)-(b) and VB states are shown in Figure 3 for . In that case (Figure 1(c)), the state is the VB ground state and the is the VB first excited state which is very close in energy. The weights of these two components are indicated above each picture. The weight of the HH component (79%) is the most important one for the VB ground state. In that case, the overlap with the CB component (94%) of the CB ground state is by far the most important one giving rise to a TE-polarized absorption. The LH component of the VB ground state is weaker (12%), but the most important feature is that it is almost antisymmetric with respect to a plane symmetry perpendicular to the z-axis. The overlap with the symmetric CB component of the CB ground state is therefore almost equal to 0, giving rise to a negligible TM-polarized absorption. On the contrary, the LH component of the VB first excited state is symmetric.

Figure 2 

Evolution of the CB, HH (straight lines), LH (dashed line) confinement potentials represented for .

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)

Figure 3 

The LH (a), (e) and HH (b), (f) components of the and VB states are shown for . Idem for with LH (c), (g) and HH (d), (h) components of the and VB states.

The components of the (c), (d) and (g), (h) VB states are represented on Figure 3 in the same order for a number of layers equal to 10 that is beyond the crossover. In that case (Figure 1(c)), the state is now the VB ground state and the is the VB first excited state. It can be observed now (e) that the weight of the LH component of the VB ground state has increased and remains symmetric. The TM-polarized absorption of the ground state optical transition is associated with this component. In addition, a TE-polarized absorption is predicted at the same energy for the optical transition between the CB ground state and the VB ground state. It is related to the LH component of the VB ground state which corresponds to the symmetric LH component of the VB ground state (e). We may notice that the HH component of the VB state (h) is now antisymmetric and therefore the TE-polarized absorption associated to that state disappears.

We have represented on Figure 4, the variation of the TE- (straight lines) or TM (dashed lines)—polarized absorption spectra for structures with . The energy of the ground state optical transition decreases when N increases. It is a size effect. For , the absorption spectrum is predominantly TE at the ground state optical transition and TM at the first excited state optical transition. For , the absorption spectrum at the ground state optical transition is TE and TM with a predominant TM character and the first excited state transition is dark.

Figure 4 

Dependence of the TE- (straight line) and TM- (dashed line) polarized absorption spectra for structures within the interval .

We have used a well-adapted axial model to study the mechanical, electronic, and optical properties of CQD. So, as a conclusion, we confirmed that the crossover from a dominant TE optical ground state absorption to a dominant TM absorption is predicted for a number of layers equal to about 9 in good agreement with the experiment. We have calculated that the electronic state configuration is strongly affected, particularly in the valence band. Then, we have shown that the weight of LH component of the VB ground state increases as a function of the number of layers but the transition of the TE/TM polarization ratio is also associated to a symmetry change of the HH component. We have also illustrated that the variation of the CQD aspect ratio induces changes in the main confinement effect (from vertical to radial) and in the strain distribution.